THEORETICAL FOUNDATIONS OF ASTROPARTICLE PHYSICSsigl/astroparticle-lectures.pdfTHEORETICAL...
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1 THEORETICAL FOUNDATIONS OF ASTROPARTICLE PHYSICS Prof. G¨ unter Sigl II. Institut f¨ ur Theoretische Physik der Universit¨ at Hamburg Luruper Chaussee 149 D-22761 Hamburg Germany email: [email protected] tel: 040-8998-2224
Transcript of THEORETICAL FOUNDATIONS OF ASTROPARTICLE PHYSICSsigl/astroparticle-lectures.pdfTHEORETICAL...
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THEORETICAL FOUNDATIONS OF ASTROPARTICLE PHYSICS
Prof. Gunter Sigl
II. Institut fur Theoretische Physik der Universitat Hamburg
Luruper Chaussee 149
D-22761 Hamburg
Germany
email: [email protected]
tel: 040-8998-2224
2
Contents
I. Fundamentals of Particle Physics 4A. Neutrinos and Weak Interactions 4B. Fermi Theory of Nuclear β-Decay 4C. Free Neutrinos: Inverse β-decay 5D. Parity Violation in β-Decay 6E. Helicity of the Neutrino 6F. Dirac Fermions and the V–A Interaction 6
1. Dirac Fermions as Representations of Space-Time Symmetries 72. The V –A Coupling 10
G. Pion and Muon Decay 101. Muon Decay and Michel Parameter 102. Branching Ratio of Pion Decay as a Signature of V –A Interactions 11
H. Weak Neutral Currents, the GIM model and Charm 11I. Divergences in the Weak Interactions 12J. Renormalizability 13K. Gauge Symmetries and Interactions 13
1. Symmetries of the Action 132. Gauge Symmetry of Matter Fields 143. Gauge Theory of the Electroweak Interaction 15
L. The Gravitational Interaction 18M. Limitations of the Standard Model 19N. Extensions: Supersymmetry and Extra Dimensions 19
II. Fundamentals of Cosmology 19A. From the Big Bang to Today 20B. Sources powered by nuclear energy: Stars 21C. Sources powered by gravitational energy: Accretion 21D. The Hubble Law 21E. The Cosmological Principle and the Friedmann-Lemaitre Equation 23F. Structure Formation 24G. Equilibrium Thermodynamics and the Cosmic Microwave Background (CMB) 27H. Relics from the Early Universe: Freeze-Out 28I. Big Bang Nucleosynthesis (BBN) 30J. Phase Transitions 32K. Inflation and Reheating 33
III. High Energy Astrophysics 33A. Cosmic Ray Acceleration 34B. Cosmic Ray Propagation 40
1. Galactic Cosmic and γ−rays 422. Extragalactic Cosmic Rays: The GZK effect 43
C. γ−ray astrophysics: The Principal Electromagnetic Processes 431. Synchrotron Radiation 432. Inverse Compton Scattering (ICS) 443. Pair Production (PP) 454. Higher Order Processes 465. Electromagnetic Cascades 47
D. Indirect Dark Matter Detection 48
IV. Neutrino Astrophysics 51A. Neutrino Scattering 51
1. Some High Energy Astrophysical Neutrino Sources 53B. Dirac and Majorana Neutrinos 54C. Neutrino Oscillations 57D. Selected Applications in Astrophysics and Cosmology 58
1. Stellar Burning and Solar Neutrino Oscillations 58
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2. Atmospheric Neutrinos 603. Flavor Composition from High Energy Neutrino Sources 624. Neutrino Hot Dark Matter 625. Leptogenesis and Baryogenesis 64
References 65
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I. FUNDAMENTALS OF PARTICLE PHYSICS
A. Neutrinos and Weak Interactions
Good introductory texts on particle physics are contained in Ref. [1] (more phenomenologically and experimentallyoriented) and in Refs. [2–4]. Here we will only recall the most essential facts.Neutrinos only have weak interactions. Historically, experiments with neutrinos obtained from decaying pions andkaons have shown that charged and neutral leptons appear in three doublets:
TABLE I: The lepton doublets
q Le = 1 Lµ = 1 Lτ = 1
0−1
(
νee−
) (
νµµ−
) (
νττ−
)
Charge q and lepton numbers Le, Lµ, and Lτ are conserved separately, apart from flavor mixing in the neutrinochannel to which we will come in chapter 4. There are corresponding doublets of anti-leptons with opposite chargeand lepton numbers, denoted by νi for the anti-neutrinos and by the respective positively charged anti-leptons.The “neutrino” is thus defined as the neutral particle emitted together with positrons in β+-decay or followingK-capture of electrons. The “anti-neutrino” accompanies negative electrons in β−-decay.Lifetimes for weak decays are long compared to lifetimes associated with electromagnetic (∼ 10−19 s) and strong(∼ 10−23 s) interactions. A weak interaction cross section at ∼ 1GeV interaction energy is typically ∼ 1012 timessmaller than a strong interaction cross section.Weak interactions are classified into leptonic, semi-leptonic, and non-leptonic interactions.We will usually use natural units in which h = c = k = 1, unless these constants are explicitly given.
B. Fermi Theory of Nuclear β-Decay
Let us consider nuclear β-decays involving
n→ p+ e− + νe , (1)
which in the quark picture writes
d→ u+ e− + νe . (2)
Fermi’s golden rule yields for the rate Γ of a reaction from an initial state i to a final state f the expression
Γ =2π
h|Mif |2
dN
dEf, (3)
where Mif ≡ 〈f |Hint|i〉 is the matrix element between initial and final states i and f with Hint the interaction energy,and dN/dEf is the final state number density evaluated at the conserved total energy of the final states.To compute from this the rate Γ of the decays Eq. (1), we use the historical Fermi theory after which such interactionsare described by point-like couplings of four fermions, symbolically Hint = GF
∫
d3xψ4, with Fermi’s coupling constantGF. This yields
Γ =2π
hG2
F|M |2 dNdEf
, (4)
where symbolically M =∫
d3xψ4 which incorporates the detailed structure of the interaction. If we normalize thevolume V to one, M is dimensionless and of order unity, otherwise M scales as V −1. In fact, it is roughly the spinmultiplicity factor, such that |M |2 ≃ V −2 if the total leptonic angular momentum is 0, thus involving no change ofspin in the nuclei (”Fermi transitions”), whereas |M |2 ≃ 3V −2 if the total leptonic angular momentum is 1, thusinvolving a change of spin in the nuclei (”Gamow-Teller transitions”). The final state density of a free particle is
V d3p
(2πh)3, (5)
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so that, taking into account energy-momentum conservation, we get for the phase space factor
dN
dEf=
∫
V d3pe(2πh)3
V d3pν(2πh)3
V d3pn(2πh)3
(2π)3δ3(pe + pν + pn)
Vδ(Ee + Eν + En −Q) , (6)
where pe, pν , pn, Ee, Eν , En, are momenta and kinetic energies of the electron, neutrino, and final state nucleus,respectively, and Q = mf −mi is the mass difference of initial and final state nucleus, the ”Q-value”. Since Q ∼MeV,the recoil energy ≃ p2
n/2mf <∼ 1 keV of the final state nucleus can be neglected. Then integrating out pn against the
momentum-delta function, and using p2νdpν = pνEνdEν , pν = (E2ν−m2
ν)1/2, with mν the neutrino mass and pν ≡ |pν |
etc., and integrating out Eν = Q− Ee against the energy delta-function, the phase space factor simplifies to
dN
dEf≃ V 2
4π4
∫ (Q2−m2
e)1/2
0
dpep2e(Q− Ee)
2
[
1−(
mν
Q− Ee
)2]1/2
, (7)
where it was furthermore assumed that |M |2 does not vary significantly over phase space. Plotting[d2N/dEfdEe/p
2e]
1/2 against Ee is called a ”Curie plot” and holds information on the neutrino mass mν in thatit vanishes at Ee = Q−mν .Neglecting the electron and neutrino masses, we can integrate Eq. (7) and insert into Eq. (4) to obtain for the decayrate
Γ ≃ G2FQ
5
60π3. (8)
Comparing this with experimental lifetimes of nuclei of various Q results in GF ≃ 10−5 GeV−2.
C. Free Neutrinos: Inverse β-decay
Anti-neutrinos produced in reactions Eq. (1) can undergo inverse β-decay
νe + p→ n+ e+ . (9)
In the center of mass (CM) frame the phase space factor for the two body final state becomes
dN
dEf=
∫
V d3pe(2πh)3
V d3pn(2πh)3
(2π)3δ3(pe + pn)
Vδ(Ee + En − E0) , (10)
where E0 is the total initial energy. Integrating out one of the momenta gives pf ≡ pe = pn so that energy conservation
E0 = (p2f +m2e)
1/2 + (p2f +m2n)
1/2 gives the factor dpf/dE0 = (pf/Ee + pf/En)−1 = v−1
f with vf being the relativevelocity of the two final state particles. This yields
dN
dEf=
V
2π2
p2fvf. (11)
We are now interested in the cross section σ of the two-body reaction Eq. (9) defined by
Γ = σnivi , (12)
where ni = V −1 and vi are density and velocity, respectively, of one of the incoming particles in the frame where theother one is at rest. Putting this together with Eqs. (4) and (11) finally yields
σ(νep→ ne+) =G2
F
π|Mif |2
p2fvivf
. (13)
For pf ≃ 1MeV this cross section is ∼ 10−43 cm2. In a target of proton density np this gives a mean free path definedby lνnpσ(νep→ ne+) ∼ 1.The first detections of this reaction was made by Reines and Cowan in 1959. The source were neutron rich fissionproducts undergoing β-decay Eq. (1). A 1000 MW reactor gives a flux of ∼ 1013 cm−2 s−1 νes which they observedwith a target of CdCl2 and water. Observed are fast electrons Compton scattered by annihilation photons from thepositrons within ∼ 10−9 s of the reaction (”prompt pulse”) γ−rays from the neutrons captured by the cadmium about10−6 s after the reaction (”delayed pulse”).
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D. Parity Violation in β-Decay
Wu et al. tested parity conservation in 1957 by studying the pure Gamow-Teller transition
60Co(J = 5) →60 Ni∗(J = 4) + e− + νe . (14)
The 60Co spins were aligned by a magnetic field at 0.02K. The intensity of electrons emitted with an angle θ relativeto the 60Co spin was found to be distributed as
I(θ) ∝ 1 + α
(
σ · peEe
)
= 1 + αve cos θ , (15)
with α = −1, σ a unit vector in the direction of the electron spin and ve = pe/Ee the electron velocity. Here, boththe electron and neutrino spin (J = 1/2) have to be in the direction of the 60Co spin because angular momentum isconserved and orbital angular momentum is zero for a point interaction (s-wave). Thus, the helicity polarization isgiven by
P =I+ − I−I+ + I−
= αve , (16)
where I± are the intensities in the H = ±1 states. Experimentally, α = +1 for e+ and α = −1 for e−.Note that Eq. (15) violates parity since spins are invariant under parity, whereas momenta change sign. Thus, paritytransformation corresponds to α→ −α.
E. Helicity of the Neutrino
The neutrino helicity was established by Goldhaber et al. in 1958. It consisted of the following steps:
• 152Eu(J = 0)−→
K− capture152 Sm∗(J = 1). Again, angular momentum conservation requires that the Samarium
spin is parallel to the electron spin, but opposite to the neutrino spin. Thus, the recoiling 152Sm∗ has the samepolarization as the neutrino.
• The γ−rays produced in the de-excitation 152Sm∗(J = 1) →152Sm(J = 0) + γ take up the Samarium spin.These γ−rays will therefore be polarized in the same or opposite sense as the neutrinos, depending on whetherthey are emitted towards or against the direction of flight of the 152Sm∗ nucleus. ”Forward” γ−rays thus havethe same polarization as the neutrino.
• For the produced γ−rays to undergo resonance scattering,
γ +152 Sm →152 Sm∗ → γ +152 Sm , (17)
since 152Sm∗ recoils, the γ−ray has to be slightly more energetic than the decay photon. Thus only the ”forward”γ−rays can scatter which according to the second step are polarized as the neutrino.
• To measure the ”forward” γ−ray polarization, before reaching the Samarium target, they passed through mag-netized iron. There, electrons are preferentially polarized in the direction opposite to the magnetic field B tominimize interaction energy µe ·B with the electron magnetic moment µe parallel to the electron spin. Angularmomentum conservation then implies that for B along the γ−ray beam, spin flip can only occur for right-handedγ−rays, whereas left-handed γ−rays are not absorbed.
As a result, it was confirmed that neutrinos are left-handed, thus
H = α , (18)
with α = −1 for ν and α = +1 for ν.
F. Dirac Fermions and the V–A Interaction
Given the experimental results we now want to work out the detailed structure of the electroweak interactions. Inorder to do that we first have to introduce the Dirac fermion.
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1. Dirac Fermions as Representations of Space-Time Symmetries
The Poincare group is the symmetry group of special relativity and consists of all transformations leaving invariantthe metric
ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2 , (19)
where x0 is a time coordinate and x1, x2, and x3 are Cartesian space coordinates. These transformations are of theform
x′µ = Λµνxν + aµ , (20)
where aµ defines arbitrary space-time translations, and the constant matrix Λµν satisfies
ηµνΛµρΛ
νσ = ηρσ , (21)
where ηµν = diag(−1, 1, 1, 1). The unitary transformations on fields and physical states ψ induced by Eq. (20) satisfythe composition rule
U(Λ2, a2)U(Λ1, a1) = U(Λ2Λ1,Λ2a1 + a2) . (22)
Important subgroups are defined by all elements with Λ = 1 (the commutative group of translations), and by allelements with aµ = 0 [the homogeneous Lorentz group SO(3, 1) of matrices Λµν satisfying Eq. (21)]. The lattercontains the subgroup SO(3) of all rotations for which Λ0
0 = 1, Λµ0 = Λ0µ = 0 for µ = 1, 2, 3.
The general infinitesimal transformations of this type are characterized by an anti-symmetric tensor ωµν and a vectorǫµ,
Λµν = δµν + ωµν aµ = ǫµ . (23)
Any element U(1+ω, ǫ) of the Poincare group which is infinitesimally close to the unit operator can then be expandedinto the corresponding hermitian generators Jµν and Pµ,
U(1 + ω, ǫ) = 1 +1
2iωµνJ
µν − iǫµPµ . (24)
It can be shown that these generators satisfy the commutation relations
i [Jµν , Jρσ] = ηνρJµσ − ηµρJνσ − ησµJρν + ησνJρµ
i [Pµ, Jρσ] = ηµρP σ − ηµσP ρ (25)
[Pµ, P ν ] = 0 .
The Pµ represent the energy-momentum vector, and since the Hamiltonian H ≡ P 0 commutes with the spatialpseudo-three-vector J ≡ (J23, J31, J12), the latter represents the angular-momentum which generates the group ofrotations SO(3).The homogeneous Lorentz group implies that the dispersion relation of free particles is of the form
E2(p) = p2 +M2 . (26)
for a particle of mass M , momentum p, and energy E. If one now expands a free charged quantum field ψ(x) into itsenergy-momentum eigenfunctions and interprets the coefficients a(p) of the positive energy solutions as annihilatorof a particle in mode p, then the coefficients b†(p) of the negative energy contributions have to be interpreted ascreators of anti-particles of opposite charge,
ψ(x) =∑
p,E(p)>0
a(p)u(p)e−iE(k)t+ip·x +∑
p,E(p)<0
b†(p)v(p)eiE(p)t−ip·x . (27)
Canonical quantization, shows that the creators and annihilators indeed satisfy the relations,[
ai(p), a†i′(p
′)]
±=[
bi(p), b†i′(p
′)]
±= δii′δ(p− p′) , (28)
where i, i′ now denote internal degrees of freedom such as spin, and [., .]± denotes the commutator for bosons, andthe anti-commutator for fermions, respectively.
8
Fields and physical states can thus be characterized by their energy-momentum and spin, which characterize theirtransformation properties under the group of translations and under the rotation group, respectively. Let us firstfocus on fields and states with non-vanishing mass. In this case one can perform a Lorentz boost into the rest framewhere Pµ = (M, 0, 0, 0) with M the mass of the state. Pµ is then invariant under the rotation group SO(3). Theirreducible unitary representations of this group are characterized by a integer- or half-integer valued spin j suchthat the 2j + 1 states are characterized by the eigenvalues of Ji which run over −j,−j + 1, · · · , j − 1, j. Note thatan eigenstate with eigenvalue σ of Ji is multiplied by a phase factor e2πiσ under a rotation around the i−axis by2π, and a half-integer spin state thus changes sign. Given the fact that a rotation by 2π is the identity this mayat first seem surprising. Note, however, that normalized states in quantum mechanics are only defined up to phasefactors and thus a general unitary projective representation of a symmetry group on the Hilbert space of states canin general include phase factors in the composition rules such as Eq. (22). This is indeed the case for the rotationgroup SO(3) which is isomorphic to S3/Z2, the three-dimensional sphere in Euclidean four-dimensional space withopposite points identified, and is thus doubly connected. This means that closed curves winding n times over a closedpath are continuously contractible to a point if n is even, but are not otherwise. Half-integer spins then correspondto representations for which U(Λ1)U(Λ2) = (−)nU(Λ1Λ2), where n is the winding number along the path from 1 toΛ1, to Λ1Λ2 and back to 1, whereas integer spins do not produce a phase factor.With respect to homogeneous Lorentz transformations, there are then two groups of representations. The first one isformed by the tensor representations which transform just as products of vectors,
W ′µ···ν··· = ΛµρΛ
σν · · ·W ρ···
σ··· . (29)
These represent bosonic degrees of freedom with maximal integer spin j given by the number of indices. The simplestcase is a complex spin-zero scalar φ of mass m whose standard free Lagrangian
Lφ = −1
2
(
∂µφ†∂µφ+m2φ†φ
)
, (30)
leads to an equation of motion known as Klein-Gordon equation,
(
∂µ∂µ −m2
)
φ = 0 . (31)
In the static case p0 = 0 this leads to an interaction potential
V (r) = g1g2e−mr
r, (32)
between two “charges” g1 and g2 which correspond to sources on the right hand side of Eq. (31). The potential forthe exchange of bosons of non-zero spin involve some additional factors for the tensor structure. Note that the rangeof the potential is given by ≃ m−1. In the general case p0 6= 0 the Fourier transform of Eq. (31) with a delta-functionsource term on the right hand side is ∝ −i/(p2 +m2). A four-fermion point-like interaction of the form GFψ
4 canthus be interpreted as the low-energy limit p2 ≪ m2 of the exchange of a boson of mass m. Later we will realize thatthe modern theory of electroweak interactions is indeed based on the exchange of heavy charged and neutral ”gaugebosons”. In the absence of sources, Eq. (31) gives the usual dispersion relation E2 = p2 +m2 for a free particle.The second type of representation of the homogeneous Lorentz group can be constructed from any set of Dirac matricesγµ satisfying the anti-commutation relations
γµ, γν = 2ηµν , (33)
also known as Clifford algebra. One can then show that the matrices
Jµν ≡ − i
4[γµ, γν ] (34)
indeed obey the commutation relations in Eq. (25). The objects on which these matrices act are called Dirac spinorsand have spin 1/2. In 3+1 dimensions, the smallest representation has four complex components, and thus the γµ
are 4× 4 matrices. A possible representation of Eq. (33) is
γi =
(
0 −iσiiσi 0
)
, i = 1, 2, 3 , γ0 = i
(
1 00 −1
)
, (35)
where σi are the Pauli matrices.
9
The standard free Lagrangian for a spin-1/2 Dirac spinor ψ of mass m,
Lψ = −ψ(γµ∂µ +m)ψ , (36)
where ψ ≡ ψ†iγ0, leads to an equation of motion known as Dirac equation,
(γµ∂µ +m)ψ = 0 . (37)
Its free solutions also satisfy the Klein-Gordon equation Eq. (31) and are of the form Eq. (27) where, up to anormalization factor N ,
u(p) = N
(
uσ·pE+m u
)
, v(p) = N
(
σ·pE+m vv
)
. (38)
Here, u and v are 4-spinors, whereas u and v are two-spinors.It is easy to see that the matrix
γ5 ≡ −iγ0γ1γ2γ3 = −(
0 11 0
)
(39)
is a pseudo-scalar because the spatial γi change sign under parity transformation, and satisfies
γ25 = 1 γµ, γ5 = 0 [Jµν , γ5] = 0 . (40)
A four-component Dirac spinor ψ can then be split into two inequivalent Weyl representations ψL and ψR which arecalled left-chiral and right-chiral,
ψ = ψL + ψR ≡ 1 + γ52
ψ +1− γ5
2ψ . (41)
Note that according to Eqs. (40) and (41) the mass term in the Lagrangian Eq. (36) flips chirality, whereas the kineticterm conserves chirality.The general irreducible representations of the homogeneous Lorentz group are then given by arbitrary direct productsof spinors and tensors. By transforming into the rest frame, we see that massive states form representations of SO(3)leaving invariant Pµ. In contrast, massless states form representations of the group SO(2) leaving invariant Pµ. Thegroup SO(2) has only one generator which can be identified with helicity, the projection of spin onto three-momentum.For fermions this is the chirality defined by γ5 above.In the presence of mass the relation between chirality and helicity H ≡ σ · p/p is more complicated:
1± γ52
u(p) =N
2
(
1∓ σ · pE +m
)(
u∓u
)
(42)
=N
2
[(
1∓ p
E +m
)
1 +H
2+
(
1± p
E +m
)
1−H
2
](
u∓u
)
1± γ52
v(p) = ∓N2
[(
1∓ p
E +m
)
1 +H
2+
(
1± p
E +m
)
1−H
2
](
v∓v
)
.
From this follows that in a chiral state uL,R the helicity polarization is given by
PL,R =IL,R+ − IL,R−
IL,R+ + IL,R−
= ∓ p
E, (43)
where IL,R± are the intensities in the H = ±1 states for given chirality L or R, i.e. the squares of the amplitudes inEq. (42). Note that due to Eq. (27) the physical momentum of anti-particles described by the v spinor is −p in thisconvention, and therefore the helicity polarization for anti-particles in pure chiral states are opposite from Eq. (43):Left chiral particles are predominantly left-handed and left-chiral anti-particles are predominantly right-handed in therelativistic limit. Furthermore, helicity and chirality commute exactly only in the limit p ≫ m, v → 1. Comparisonof Eq. (43) with the experimental results Eq. (16) and (18) now imply that both electrons and neutrinos and theiranti-particles are fully left-chiral in charged current interactions.
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2. The V –A Coupling
Since Dirac spinors have 4 independent components, there are 16 independent bilinears listed in Tab. II. Using theequality
ㆵ = γ0γµγ0 , (44)
which can easily be derived from Eq. (35), one sees that the phase factors of the bilinears in Tab. II are chosen suchthat their hermitian conjugate is the same with ψ1 ↔ ψ2.
TABLE II: The Dirac bilinears. For ψ1 = ψ2 these are real.
ψ1ψ2 scalar S
iψ1γµψ2 4-vector V
iψ1γµγνψ2 tensor T
iψ1γµγ5ψ2 axial 4-vector A
iψ1γ5ψ2 pseudo-scalar P
Lorentz invariance implies that the matrix element of a general β-interaction is of the form
M = GF
∑
i=S,V,T,A,P
Ci(ψ1Oiψ2)(ψ3Oiψ4) , (45)
such that only the same types of operators Oi from Tab. II couple and common Lorentz indices are contracted over.Eq. (45) is a Lorentz scalar. However, we know that electroweak interactions violate parity and thus we have to addpseudo-scalar quantities to Eq. (45). Equivalently, we can substitute any lepton spinor ψ in Eq. (45) by 1
2 (1 + γ5)ψ.This is correct at least for the interactions with charge exchange, the so called charged current interactions, for whichwe know experimentally that both neutrinos and charged leptons are fully left-chiral. Using Eq. (40), this leads toterms of the form
l−i,LOνi,L = l−i (1− γ5)O(1 + γ5)νi , l = e, µ, τ , (46)
which implies that only the V and A type interactions from Tab. II can contribute. The general form of chargedcurrent interactions involving neutrinos is therefore usually written as
Mνcc =
GF√2
[
ψ1γµ(CV + CAγ5)ψ2
]
[
l−i γµ(1 + γ5)νi
]
, (47)
G. Pion and Muon Decay
1. Muon Decay and Michel Parameter
With
x ≡ EeEe,max
≃ 2Eemµ
(48)
one has for the fractional muon decay rate
1
Γµ
dΓµdx
= 12x2[
1− x+2
3ρ
(
4
3x− 1
)]
. (49)
Here, the Michel parameter is ρ = 3/4 for the V –A interaction, consistent with experiment, and ρ = 1 for the S–Pinteraction. The total muon decay rate is
Γµ =G2
Fm5µ
192π3. (50)
Note that this has the same structure as Eq. (8).
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2. Branching Ratio of Pion Decay as a Signature of V –A Interactions
Pions can decay into several channels, among them eνe and µνµ. Note that due to lepton number conservation, thefinal state always has one particle and one anti-particle. From a generalization of Eq. (46) to arbitrary chiralities wesee that S, P , and T couplings produce particle anti-particle pairs of opposite chirality, whereas V and A couplingsproduce particle anti-particle pairs of equal chirality. Chirality and helicity are related by Eq. (43) for particles andEq. (43) with the opposite sign for anti-particles. For relativistic final states, the S, P , and T couplings thereforetend to produce particle anti-particle pairs of equal helicity, whereas V and A couplings tend to produce particleanti-particle pairs of opposite helicity. Since the pion has spin zero, the two final state particles have to have equalhelicity. Neutrinos and anti-neutrinos are always left-chiral, see Eq. (18), thus it follows that the S, P , and T couplingsare enhanced by a factor 1 + v, whereas V and A couplings are suppressed by a factor 1 − v, where v here is thevelocity of the massive final state product, i.e. of the electron or muon of mass m and momentum p.The rest is kinematics: One has mπ = p+(p2+m2)1/2, thus p = (m2
π−m2)/(2mπ), (p2+m2)1/2 = (m2
π+m2)/(2mπ),
and the phase space factor becomes
dN
dEf= p2
dp
dEf=
(m2π +m2)(m2
π −m2)2
4m4π
, (51)
whereas the charged lepton polarization enhancement and suppression factors yield
1 + v =2m2
π
m2π +m2
1− v =2m2
m2π +m2
From this we obtain the predicted branching ratios
V ,A coupling : R =π → eνeπ → µνµ
=m2e
m2µ
1
(1−m2µ/m
2π)
2≃ 1.275× 10−4
S , P , T coupling : R =π → eνeπ → µνµ
=1
(1−m2µ/m
2π)
2≃ 5.5 . (52)
In the first case, the charged lepton is produced in the “wrong” helicity, which gives a strong suppression for thehighly relativistic e±. Experimental results are completely consistent with this V –A case. The suppression of piondecay into the electron channel can have consequences for the observed flavor of high energy astrophysical neutrinosources, as we will see in Sect. IVD3.
H. Weak Neutral Currents, the GIM model and Charm
Before 1970 only up (u), down (d), and strange (s) quarks were known. The leptonic electroweak doublets Tab. Iwere complemented by one hadronic doublet
(
udC
)
=
(
ud cos θC + s sin θC
)
, (53)
where one accounts for the fact that the “down-type” quark flavor eigenstate dC is rotated by the Cabibbo angle θCwith respect to the mass eigenstates d and s.Besides the charged current interactions connecting the upper and lower entries in the electroweak doublets, e.g.J+ ∝ ud cos θC , there are also neutral currents. These latter do not echange electrical charge and thus are bilinearsof the individual entries themselves. They consequently have the form
[
uu+ dd cos2 θC + ss sin2 θC)]
+(
sd+ sd)
sin θC cos θC , (54)
where the first term conserves strangeness and the second changes it by one unit. Experimentally, such flavor changingprocesses have not been seen. As first suggested by Glashow, Iliopoulos and Maiani, this can be “fixed” by introducinga charm (c) quark and combining if with the orthogonal combination of d and s to a further electroweak doublet:
(
csC
)
=
(
cs cos θC − d sin θC
)
. (55)
12
The new contribution of the lower component in Eq. (55) to the neutral current then exactly cancels the flavorchanging part in Eq. (54). This is called the GIM mechanism.Finally, a third quark doublet involving top and bottom quarks gives rise to a 3×3 mixing matrix connecting flavor andmass eigenstates. This Cabibbo Kobayashi Maskawa (CKM) matrix will be characterized by 3 angles and one phase.The phase can give rise to CP -violation in the quark sector, similar to the neutrino mixing matrix, see Sect. IVBbelow.
I. Divergences in the Weak Interactions
An incoming plane wave ψi ≡ eikz of momentum k in the z-direction can be expanded into incoming and outgoingradial modes e−ikr and eikr, respectively, in the following way
eikz =i
2kr
∑
l
(2l + 1)[
(−1)le−ikr − eikr]
Pl(cos θ) , (56)
where Pl(x) are the Legendre polynomials and cos θ = z/r. Scattering modifies the outgoing modes by multiplyingthem with a phase e2iδl and an amplitude ηl with 0 ≤ ηl ≤ 1. The scattered outgoing wave thus has the form
ψscatt =eikr
kr
∑
l
(2l + 1)ηle
2iδl − 1
2iPl(cos θ) ≡
eikr
rF (θ) , (57)
where F (θ) is called the scattering amplitude.Let us now imagine elastic scattering in the CM frame, where momentum p∗ = k and velocity v are equal beforeand after scattering. The incoming flux is then v|ψi|2 = v and the outgoing flux through a solid angle dΩ isv|ψscatt|2r2dΩ = v|F (θ)|2dΩ. The definition Eq. (12) of the scattering cross section then yields
(
dσ
dΩ
)
el
= |F (θ)|2 . (58)
Using orthogonality of the Legendre polynomials,∫
dΩPl(Ω)Pl′(Ω) = 4πδll′/(2l + 1), in Eq. (57), we obtain for thetotal elastic scattering cross section
σel =4π
p2∗
∑
l
(2l + 1)
∣
∣
∣
∣
ηle2iδl − 1
2i
∣
∣
∣
∣
2
. (59)
For scattering of waves of angular momentum l this results in the upper limit
σel,l ≤4π
p2∗(2l + 1) , (60)
which is called partial wave unitarity.On the other hand, in Fermi theory typical cross sections grow with p∗ as in Eq. (13) and violate Eq. (60) for s-waves(l = 0) for
4G2Fp
2∗
π>∼
4π
p2∗, (61)
where we used∑ |Mif |2 ≃ 4 for the sum over polarizations. This occurs for p∗ >∼ (π/GF)
1/2 ≃ 500GeV. Suchenergies are nowadays routinely achieved at accelerators such as in the Tevatron at Fermilab. As will be seen in thenext section, this is ultimately due to the fact that the coupling constant GF has negative energy dimension andcorresponds to a non-renormalizable interaction. This will be cured by spreading the contact interaction with the
propagator of a gauge boson of mass M ≃ G−1/2F ∼ 300GeV. This corresponds to multiplying the l.h.s. of Eq. (61)
with the square of the propagator, ≃ (1 + p2∗/M2)−2, which thus becomes 4/(πp2∗) for p∗ → ∞. This scaling with p∗
is of course a simple consequence of dimensional analysis. As a result, partial wave unitarity is not violated any moreat high energies, at least within this rough order of magnitude argument.
13
J. Renormalizability
Let us now discuss the divergence of loop diagrams at large energies. We assume that all momenta go to infinity witha common factor κ and we are interested in the power D of the resulting integral
∫∞dκκD−1 which we obtain by
power counting: First, we write the mass dimensionality of a quantum field as 1 + sf , where sf is its effective spin.The Lagrangians Eq. (30) and (36) imply that for massless gauge bosons and scalar fields sf = 0, whereas spin-1/2fermions sf = 1/2.The propagator of a field is the solution of its free wave equation for a delta-function in space-time. In momentumspace, Eq. (31) implies that the boson propagator has mass dimensionality -2, wheres Eq. (37) implies dimensionality-3/2 for Dirac fermion propagators. In general, the dimensionality of the propagator for a field is −2 + 2sf .An interaction i with nif fields of type f and di derivatives has dimensionality di +
∑
f nif (1 + sf ). Since theaction is dimensionless, the Lagrangian has dimensionality +4, and thus the coupling constant of this interaction hasdimensionality ∆i = 4− di −
∑
f nif (1 + sf ).The momentum space amplitude of a connected Feynman graph with Ef external lines of type f is the Fouriertransfom over 4
∑
f Ef coordinates of the vacuum expectation value of a product of fields with total dimensionality∑
f Ef (1 + sf ), and thus has dimensionality∑
f Ef (−3 + sf ). Of this, −4 is due to an overall momentum space
delta-function and∑
f Ef (−2 + 2sf ) is the dimensionality of the propagators of the external lines.Therefore, the momentum space integral without the coupling constants has dimensionality
D = 4−∑
f
Ef (1 + sf )−∑
i
Ni∆i , (62)
where Ni is the number of interaction vertices of type i.This implies that in a theory where ∆i ≥ 0 for all interactions, there is only a finite number of graphs divergingat large energies, i.e. with D ≥ 0. It turns out that these divergences can be absorbed into the finite number ofparameters of the theory which is therefore called renormalizable. These are thus exactly the theories which haveno coupling constants of negative mass dimensionality. Fermi theory is thus non-renormalizable, whereas the gaugetheory of electroweak interactions discussed below is renormalizable.Good examples of non-renormalizable interaction terms are given by
ie
2Mψ[γµ, γν ]ψF
µν ,e
2Mψγ5[γµ, γν ]ψF
µν , (63)
where M is some large mass scale presumably related to grand unification and e is the (positive) electric chargeunit. The gauge invariant field strength tensor Fµν = ∂µAν − ∂νAµ in terms of the gauge potential Eq. (78) belowrepresents the electric field strength Ei = −∂0Ai − ∂iA
0 = F 0i and magnetic field strength Bi = ǫijk∂jAk = ǫijkFjkwhere Latin indices represent spatial indices and ǫijk is totally anti-symmetric with ǫ123 = 1. As a consequence, inthe non-relativistic limit, Eq. (63) reduce to magnetic and electric dipole moments of the ψ field, µ · B and µ · E,respectively, of size 4e/M . Note that these are even and odd, respectively, under parity and time reversal.
K. Gauge Symmetries and Interactions
1. Symmetries of the Action
Lorentz invariance suggests that the action should be the space-time integral of a scalar function of the fields ψi(x, t)and their space-time derivatives ∂µψi(x, t), and thus that the Lagrangian should be the space-integral of a scalarcalled the Lagrangian density L,
S[ψ] =
∫
d4xL[ψi(x), ∂µψi(x)] , (64)
where x ≡ (x, t) from now on. In this case, the equations of motion read
∂µ∂L
∂(∂µψi)=
∂L∂ψi
, (65)
which are called Euler-Lagrange equations and are obviously Lorentz invariant if L is a scalar.
14
Symmetries can be treated in a very transparent way in the Lagrangian formalism. Assume that the action is invariant,δS = 0, independent of whether ψi(x) satisfy the field equations or not, under a global symmetry transformation,
δψi(x) = iǫFi[ψj(x), ∂µψj(x)] , (66)
for which ǫ is independent of x. Here and in the following explicit factors of i denote the imaginary unit, and not anindex. Then, for a space-time dependent ǫ(x), the variation must be of the form
δS = −∫
d4xJµ[x, ψj(x), ∂µψj(x)]∂µǫ(x) . (67)
But if the fields satisfy their equations of motion, δS = 0, and thus
∂µJµ[x, ψj(x), ∂µψj(x)] = 0 , (68)
which implies Noethers theorem, the existence of one conserved current Jµ for each continuous global symmetry. IfEq. (66) leaves the Lagrangian density itself invariant, an explicit formula for Jµ follows immediately,
Jµ = −i ∂L∂(∂µψi)
Fi , (69)
where we drop the field arguments from now on.An important example is invariance of the action S under space-time translations, for which Fi = −i∂µψi(x) inEq. (66). In this case, the currents Jν for each µ are given by
T νµ = δνµL − ∂L∂(∂νψi)
∂µψi , (70)
which represents the energy momentum tensor. Using the equations of motion Eq. (65) one can show that the energymomentum tensor is indeed conserved,
∂νTνµ = 0 . (71)
As opposed to a global symmetry, Eq. (66), which leaves a theory invariant under a transformation that is the sameat all space-time points, a gauge symmetry is more powerful as it leaves invariant a theory, i.e. δL = δS = 0, undertransformations that can be chosen independently at each space-time point. Gauge symmetries are usually also linearin the (fermionic) matter fields which we represent here by one big spinor ψ(x) that in general contains Lorentz spinorindices as well as some internal group indices on which the gauge transformations act. For real infinitesimal ǫα(x) wewrite
δψ(x) = iǫα(x)tαψ(x) , (72)
where α labels the different independent generators tα of the gauge group. A finite gauge transformation would bewritten as ψ(x) → exp(iΛα(x)tα)ψ(x) and reduces to Eq. (72) in the limit Λα(x) = ǫα(x) → 0. The hermitianmatrices tα form a Lie algebra with commutation relations
[tα, tβ ] = iCγαβtγ , (73)
where the real constants Cγαβ are called structure constants of the Lie algebra, and are anti-symmetric in αβ.
2. Gauge Symmetry of Matter Fields
If the Lagrangian contained no field derivatives, but only terms of the form ψ(x)† · · ·ψ(x), there would be no differencebetween global and local gauge invariance. However, dynamical theories contain space-time derivatives ∂µψ whichtransform differently under Eq. (72) than ψ, and thus would spoil local gauge invariance. One can cure this byintroducing new vector gauge fields Aαµ(x) and defining covariant derivatives by
Dµψ(x) ≡ ∂µψ(x)− iAαµ(x)tαψ(x) . (74)
The gauge variation of this from the variation of ψ alone (i.e. assuming Aαµ constant for the moment) reads
δψDµψ(x) = iǫα(x)tαDµψ(x) + i[
∂µǫα(x) + CαβγA
βµ(x)ǫ
γ(x)]
tαψ(x) , (75)
15
The new term proportional to the structure constants Cαβγ results from moving the gauge variation of ψ in Eq. (74) to
the left of the matrix gauge field Aαµ(x)tα, using Eq. (73). It is only present in non-abelian gauge theories for which thetα do not all commute. The variation δψSm of the matter action Sm can then be obtained from Eq. (67), generalizedto several ǫα, and with ∂µǫ
α(x) substituted by the corresponding first factor of the second term in Eq. (75),
δψSm = −∫
d4xJµα(x)[
∂µǫα(x) + CαβγA
βµ(x)ǫ
γ(x)]
, (76)
where the gauge currents Eq. (69) now read [compare Eqs. (66) and (72)]
Jµα = −i ∂Lm
∂(∂µψ)tαψ (77)
in terms of the matter Lagrangian Lm. Realizing now that [∂Sm/∂Aαµ(x)] = [∂Lm/∂(∂µψ)](−itαψ) = Jµα , we see that
δSm = δψSm + [∂Sm/∂Aαµ(x)]δA
αµ(x) vanishes identically if we adopt the gauge transformation
δAαµ(x) = ∂µǫα(x) + CαβγA
βµ(x)ǫ
γ(x) , (78)
for the gauge field Aαµ(x).The standard gauge-invariant term for fermions is then given by the matter Lagrange density
Lm = −ψ(γµDµ +m)ψ = −ψ(γµ∂µ +m)ψ +AαµJµα , (79)
where m is the fermion mass matrix. The second equality shows how the matter Lagrangian splits into the free partquadratic in the fields, Eq. (36), and the fundamental coupling of the gauge field to the gauge current Eq. (77). SinceLm is real and the gauge current is hermitian, Jµα = J†
µα, the gauge fields Aαµ are also real.
3. Gauge Theory of the Electroweak Interaction
In the electroweak Standard Model the elementary fermions are arranged into three families or generations which
here are labeled with the index i. Each family consists of a left-chiral doublet of leptons,
(
νil−i
)
L
, a left-chiral
doublet of quarks,
(
uidi
)
L
, and the corresponding right-chiral singlets l−iR, uiR, and diR. Here, left- and right-chiral
is understood as in Eq. (41), and each quark species comes in three colors corresponding to the three-dimensionalrepresentations of the strong interaction gauge group SU(3) whose index is suppressed here. The three known leptonsare the electron, muon, and tau with their corresponding neutrinos. The three up-type quarks are called up, charm-,and top-quark, and the down-type quarks are the down-, strange-, and bottom-quarks. The fermion masses risesteeply with generation from about 1 MeV for the first generation to up to ≃ 175GeV for the top-quark whose directdiscovery occurred as late as 1995 at Fermilab in the USA.Note that no right-handed neutrino appears and thus neutrino mass terms of the form νLνR+h.c. (h.c. denoteshermitian conjugate here and in the following) are absent in the Standard Model. Implications of recent experimentalevidence for neutrino masses for modifications of the Standard Model will not be discussed here. To simplify the
notation we assemble all fields into lepton and quark doublets, li ≡(
νil−i
)
, and qi ≡(
uidi
)
, including the right-
handed components. We will also use the Pauli matrices
(τ0, τ ) = (τ0, τ1, τ2, τ3) (80)
≡(
1 00 1
)
,
(
0 1/21/2 0
)
,
(
0 −i/2i/2 0
)
,
(
1/2 00 −1/2
)
.
The electroweak gauge group is given by
G = SU(2)L × U(1)Y , (81)
where the first factor only acts on the left-handed doublets. Denoting the dimensionless coupling constants corre-sponding to these two factors with g and g′, we write the four generators in the leptonic and quark sector as
tl = tq ≡ (t1, t2, t3) ≡ g1 + γ5
2τ
16
tY l = g′[
1 + γ52
τ02
+1− γ5
2τ0
]
(82)
tY q = g′[
−1 + γ52
τ06
− 1− γ52
(τ06
+ τ3
)
]
.
These correspond to the generators tα from the previous section, and we denote the corresponding gauge fields by Aµ
and Bµ. It is easy to see that the electric charge operator is then given by the combination
q =e
gt3 −
e
g′tY , (83)
where e is the (positive) electric charge unit.We are here only interested in the part of the Lagrangian involving matter fields. This is then given by Eq. (79) whereψ now represents all lepton and quark multiplets li and qi. Using Eq. (74), where, from comparing Eq. (82) withEq. (73), Cαβγ = gǫαβγ for SU(2)L, and zero for U(1), we can write the matter part of the electroweak Lagrangian as
Lew,m = −3∑
i=1
liγµ (∂µ − iAµ · t− iBµtY l) li (84)
−3∑
i=1
qiγµ (∂µ − iAµ · t− iBµtY q) qi .
It will be more convenient to use charge eigenstates as basis of the electroweak gauge bosons and to identify thephoton Aµ as carrier of the electromagnetic interactions. There is then one other neutral gauge boson Zµ and twogauge bosons W± of charge ±e. They are defined by
A1µ =
1√2
(
W−µ +W+
µ
)
A2µ =
1√2
(
W−µ −W+
µ
)
(85)
A3µ = cos θewZµ − sin θewAµ
Bµ = sin θewZµ + cos θewAµ ,
(86)
where the electroweak angle θew is defined by
g =e
sin θew, g′ =
e
cos θew. (87)
The interaction terms in Eq. (84) can then be written as
Aµ · t+BµtY =g√2
1 + γ52
(
W+µ τ
+ +W−µ τ
−)
(88)
− g
2 cos θewZµ
(
1 + γ52
τ3 − q sin2 θew
)
+Aµq ,
where τ± ≡ τ1 ± iτ2 are the weak isospin raising and lowering operators, respectively.Up to this point all fields are massless. Unbroken gauge symmetry implies strict masslessness of the gauge fields.In addition, fermion mass terms would have to be of the form liRliL+h.c. etc. and are thus also inconsistent withthe particular SU(2)L × U(1)Y gauge symmetry of electroweak physics. We know, however, experimentally that allfields except for the photon have mass. In order to create mass terms we need to break the gauge symmetry downto U(1)em. The theoretically favored mechanism is the Higgs mechanism which, however, is not yet experimentally
confirmed. It introduces a Higgs doublet φ ≡(
φ+
φ0
)
of complex scalar fields whose couplings to the electroweak gauge
group are given by
tφ = g τ
tY φ = −g′
2τ0 , (89)
17
which also confirms via Eq. (83) that φ+ has charge +e and φ0 is uncharged. The most general renormalizable andLorentz- and gauge-invariant term involving the Higgs doublet is given by
LφA = −1
2|(∂µ − iAµ · tφ − iBµtY φ)φ|2 −
µ2
2φ†φ− λ
4
(
φ†φ)2, (90)
where stability of the corresponding Higgs potential V (φ) = (µ2/2)φ†φ+(λ/4)(
φ†φ)2
requires λ > 0. For µ2 < 0 this
potential is minimized for a non-vanishing vacuum expectation value of the Higgs field given by 〈φ〉†〈φ〉 = v2 ≡ |µ2|/λ.We can now apply a suitable electroweak gauge transformation to a unitary gauge in which at each point in space-time
φ+ = 0 , φ0 = v +H ,where 〈H〉 = 0 , H† = H , (91)
where v is real and positive. The Higgs Lagrangian Eq. (90) then yields mass terms for the gauge bosons,
−1
2|(Aµ · tφ +BµtY φ)φ|2 = −v
2g2
4W+µ W
µ− − v2
8
(
g2 + g′2)
ZµZµ . (92)
The photon is therefore left massless, which corresponds to the remaining unbroken electromagnetic gauge symmetrygenerated by q, q〈φ〉 = 0, whereas the W± and Z bosons attain the masses
mW =v|g|2
, mZ =v√
g2 + g′2
2. (93)
The vacuum expectation value for φ therefore has induced a spontaneous breakdown of the electroweak gauge symme-try into the electromagnetic gauge symmetry. Three of the four original real degrees of freedom of the Higgs doubletφ now appear as the longitudinal components of the now massive gauge bosons W± and Z, whereas H = φ0 − v isleft as a neutral, real scalar usually called the Higgs boson which experimentalists are still trying to find. Electroweakprecision measurements currently constrain the Higgs mass to 46GeV ≤ mH ≤ 154GeV to 90% confidence level,and in fact a possible hint at mH ≃ 115GeV has been seen before the Large Electron Positron collider (LEP) at theEuropean Organization for Nuclear Research CERN was shut down.The experimentally measured values mW = 80.403 ± 0.029GeV, and mZ = 91.1876 ± 0.0021GeV then allow todetermine the electroweak angle by comparing Eqs. (87) and (93), as sin2 θew = 1−m2
W /m2Z ≃ 0.23122.
The Higgs doublet also allows the following gauge-invariant terms involving matter fields,
Lφm = −3∑
i,j=1
huij
(
uiLdiL
)(
φ0
−φ−)
ujR −3∑
i,j=1
hdij
(
uiLdiL
)(
φ+
φ0
)
djR
−3∑
i=1
hli
(
νiLl−iL
)(
φ+
φ0
)
l−iR + h.c. , (94)
where φ− ≡ (φ+)† is the conjugate of φ+, and huij , hdij , and h
li are constant c-numbers, so called Yukawa couplings. By
applying suitable unitary transformations qiL,R →∑3j=1 UijL,RqjL,R we can introduce real and diagonal masses mu
i
for the u-type quarks,∑3k,l=1 U
∗kiLvh
uklUljR = δijm
ui (it is, of course, only a convention to diagonalize huij instead of
hdij), and analogously masses mli for the charged leptons. After electroweak symmetry breaking and in unitary gauge
Eq. (91), Eq. (94) reads
Lφm = −3∑
i=1
uiL (mui +H)uiR −
3∑
i,j=1
diL(
mdij +H
)
djR
−3∑
i=1
l−iL(
mli +H
)
l−iR + h.c. , (95)
where mdij =
∑3k,l=1 U
∗kiLvh
dklUljR. When diagonalizing this matrix by applying another pair of unitary matrices, one
of them can be absorbed into the definition of the right-handed d-type quarks, but the other one cannot be elimi-nated and is known as Cabibbo-Kobayashi-Maskawa matrix which connects left-handed down quark mass eigenstateswith flavor eigenstates. A similar matrix appears in the leptonic sector if neutrinos are not massless as now seemsexperimentally verified.
18
Let us now consider processes involving the exchange of a W± or Z boson with energy-momentum transfer qmuch smaller than the gauge boson mass, |q2| ≪ m2
W,Z , such that the boson propagator can be approximated
by −iηµν/m2W,Z . In this case the second order terms in the perturbation series give rise to effective interactions of
the form
1
m2W
JµccJ†µcc +
1
m2Z
JµncJµnc . (96)
Here, the charged current and neutral current are gauge currents given by comparing Eq. (79) with Eq. (84), andusing Eq. (88),
Jµcc = ig√2
3∑
i=1
[
liγµ 1 + γ5
2τ+li + qiγ
µ 1 + γ52
τ+qi
]
(97)
Jµnc = −i g
2 cos θew
3∑
i=1
[
liγµ
(
1 + γ52
τ3 − q sin2 θew
)
li
+qiγµ
(
1 + γ52
τ3 − q sin2 θew
)
qi
]
.
Eqs. (96) and (97) provide an effective description of all low energy weak processes. This is an instructive exampleof how a more fundamental renormalizable description of interactions at high energies, in this case electroweak gaugetheory, can reduce to an effective non-renormalizable description of interactions at low energies which are suppressedby a large mass scale, in this case mW or mZ . In fact, the latter is identical in form with the historical “V–A” theoryEq. (47) which, for example, for the muon decay µ− → e−νeνµ reads
GF√2
[
e−γµ(1 + γ5)νe
]
[
νµγµ(1 + γ5)µ−]
+ h.c. , (98)
where the Fermi constant GF by comparison with Eqs. (96) and (97) is given by
GF =g2
4√2m2
W
= 1.16637(1)× 10−5 GeV−2 . (99)
Radioactive β−decay processes are described by the terms in Eq. (96) containing e−γµ(1 + γ5)νe or its hermitianconjugate for one of the charged currents Jµcc or J†
µcc, and a quark term for the other current. For example, neutron
decay, n→ pe−νe is due to the contribution uγµ(1 + γ5)d to Jµcc, which causes one of the d-quarks in the neutron totransform into a u-quark under emission of a W− boson which in turn decays into e−νe, (udd) → (uud)e−νe.Inverting Eq. (85) to Zµ = cos θewA
3µ + sin θewBµ and writing out the mass term of the neutral gauge boson sector
12m
2ZZµZ
µ implies
mW = mZ cos θew , (100)
because the mass term of A3µ has to be identical to the one for W±
µ . From this it follows immediately that the neutralcurrent part of Eqs. (96), (97) involving neutrinos can be written as
GF√2[νiγ
µ(1 + γ5)νi][
ψγµ (gL(1 + γ5) + gR(1− γ5))ψ]
, (101)
where ψ stands for quarks and leptons and
gL = τ3 − q sin2 θew
gR = −q sin2 θew . (102)
L. The Gravitational Interaction
In analogy to the Fermi constant GF in the original Fermi theory of electroweak interactions, Newton’s constantGN has inverse squared mass dimension. But whereas the Standard Model provides a renormalizable description ofelectroweak interactions without dimensionful coupling constants, no such theory has yet been found for gravity.
19
At energies (or masses) E and distances r for which the Newtonian potential per mass VN = GNE/r ≪ 1, one canchoose coordinate systems in which the deviations of the metric from the Minkowsky metric ηµν can be treated assmall perturbations and Newtonian gravity is recovered. The Newtonian force can thus be interpreted as due to theexchange of gravitons which are represented by the metric gµν and thus have spin two. In contrast, at energies and
distances close to the Planck scale mPl = G−1/2N ≃ 1.22 × 1019 GeV the ratio GNE
2 of one-graviton to zero-gravitonamplitudes approaches unity. It is thus at this energy scale where quantum gravity effects become important. Onthe other hand gravity has the peculiar property that for E ≫ mP it becomes more classical again: The Comptonwavelength of a relativistic particle hc/E becomes much smaller than the Schwarzschild radius 2GNE/c
4. Here wekept the h and c dependence explicit to reveal the quantum and classical nature of these two length scales, respectively.
M. Limitations of the Standard Model
While the Standard Model of particle physics is in excellent agreement with all experimental data, it has considerabletheoretical limitations:
• gauge hierarchy problem: The strong interactions are not unified with the electroweak interactions, neither areany unified with gravity. The gravity scale is 17 orders of magnitude higher than the other scales.
• Whereas vector boson and fermion masses are stabilized by the gauge symmetries, the Standard Model hasa fundamental scalar, the Higgs boson which is not stabilized; radiative corrections would drive it to the UVcut-off scale.
• There is absolutely no explanation of fermion masses and mixing; rather the Yukawa couplings are put in ”byhand”.
• The Standard Model turns out to not explain the observed baryon asymmetry of our Universe: The amount ofCP violation (so far only observed in the quark sector) is too small and the electroweak phase transition is asmooth cross over.
• There is no suitable scalar field that could lead to inflation.
N. Extensions: Supersymmetry and Extra Dimensions
Supersymmetry is a symmetry between fermions and bosons. It stabilizes scalar masses because it links them tothe mass of the corresponding fermionic partner which is protected by gauge symmetry. In many versions of thesupersymmetric Standard Model there is an additional discrete ”R-symmetry” under which supersymmetric partnersof Standard Model particles are odd. The lightest one is thus stable against decay and constitutes a candidate fordark matter.For a fundamental gravity scale meff in n extra dimensions of volume Vn, one has the relation m2
Pl = mn+2eff Vn. If all
dimensions have the same size, we have
rn ≃ m−1eff
(
mPl
meff
)2/n
≃ 2× 10−17
(
TeV
meff
)(
mPl
meff
)2/n
cm . (103)
In such scenarios there may thus be Kaluza-Klein excitations with masses multiples of 1/rn, either excitations ofelectroweak particles, if Standard Model particles also propagate in the extra dimensions, or massive gravitons.
II. FUNDAMENTALS OF COSMOLOGY
There are many textbooks on cosmology. One of the more recent ones is Ref. [5]. A classic but now somewhatoutdated is Ref. [2]. A classic on particle cosmology is Ref. [12].
20
A. From the Big Bang to Today
In this part we will discuss the evolution of our visible Universe from its origin until today. Note that there aretheories, notably string theory, which predict the existence of ”multiverses” with a whole ”landscape” of universesmost of which are not directly observable by us. The structure and evolution of the Universe we live in may then beexplainable by some form of “anthropic principle” which essentially states that our very existence requires a Universewith properties similar to the one we observe. We will not go into such issues here, but rather restrict attention tothe observable Universe. Its rough history is depicted in Fig. 1. We will go back in time, starting from today.
FIG. 1: A short history of the visible part of our Universe.
21
B. Sources powered by nuclear energy: Stars
The basic energy source of stars is fusion of hydrogen into helium. The nuclear binding energy of helium is about 28MeV. Since the effective reaction is 4p→ 4He+2e+ +2νe, and the average energy going into positrons and neutrinosis of order 0.5 MeV, each such reaction turns out to release about 26.7 MeV in thermal energy, or in other words,≃ 6× 1018 ergs g−1. In the last chapter we will see that the neutrinos released can be used as a test beam to measureneutrino properties.On the main sequence, stars are stabilized against gravitational contraction by the thermal pressure. Assuming fora rough estimate a homogeneous shere of mass M , radius R and temperature T we have p ≃ (3M/4πmNR
3)T ≃GNM
2/(4πR4), where mN is the nucleon mass. This leads to
T ≃ GNM
R
mN
3. (104)
The first factor in Eq. (104) is sometimes called the compactness parameter of the star. According to general relativityit is always smaller than unity.Once the thermonuclear fuel of the star is exhausted the only possibilty to stabilize it against gravitational collapseis via fermionic degeneracy pressure. In a star of mass M there are roughly M/mN fermions. The mean distancebetween the fermions is thus ∼ R(mN/M)1/3, which corresponds to a Fermi momentum of pF ∼ (M/mN )1/3/R and,for a fermion mass m, to a Fermi energy EF ≃ (m2 + p2F)
1/2. The total energy per fermion is thus
E ≃(
m2 +(M/mN )2/3
R2
)1/2
−m− GNMmN
R. (105)
The Chandrasekhar limit mass is given by setting E = 0,
MCh ≃ 1
G3/2N m2
N
=m3
Pl
m2N
∼ 1.5M⊙ , (106)
with mPl the Planck mass. For M >∼Mch, the sign of the total energy in Eq. (105) is negative and can be decreasedwithout bound by decreasing R, thus no stable solution is possible. In contrast, for M <∼ MCh, stability can beachieved at a radius
R ∼ 1
m(GNm2N )1/2
=mPl
mmN. (107)
The resulting compactness parameter for neutron stars is ∼ 0.2 and the one for white dwarfs, stabilized by electrondegeneracy pressure, is ∼ 10−4.
C. Sources powered by gravitational energy: Accretion
In the previous section we have derived the compactness parameters for white dwarfs and neutron stars. The compact-ness parameter of black holes is 0.5 because the Schwarzschild radius is 2GNM . In accretion, most of the gravitationalenergy can be released as electromagnetic energy. This results in a release of ∼ 1017 ergs g−1, ∼ 1020 ergs g−1, and∼ 5× 1020 ergs g−1 for white dwarfs, neutron stars, and black holes, respectively.Consider spherical accretion of plasma onto a compact object of mass M at a distance r from its center. The gravita-tional force on a proton of mass mp is Fgrav = GNMmp/r
2. On the other hand, if this object emits electromagneticradiation with luminosity L, its flux at radius r will be j = L/(4πr2) exerting an outward electromagnetic force ofsize Fem = jσT where σT ≃ 0.665 × 10−24 cm2 is the Thomson cross section. The maximal luminosity is thus givenby the Eddington luminosity
LEdd =4πGNMmp
σT≃ 1.3× 1038
(
M
M⊙
)
erg s−1 . (108)
D. The Hubble Law
In the 1920s Edwin Hubble observed that the spectral lines from distant galaxies are shifted to the red by an amountproportional to the distance d on average,
z =∆λ
λ≃ H0d , (109)
22
where H0 is the so-called Hubble constant. If interpreted as a Doppler effect, this corresponds to a recession velocityof v/c = z. In order to measure the distance and thus the Hubble constant, nowadays one uses certain “standardcandles” which are calibrated using a ”distance ladder”. For example, variable stars of the Cepheid type have a well-known relation between their variability time scale and their absolute luminosity, calibrated by statistically averagedparallax measurements which depend only on the (known) peculiar motion of our Sun within the Galaxy and thedistance of the variable star. Using the resulting distance ruler, astronomers have realized that there is also a tightcorrelation between the rise and decay time of the light curves of type Ia supernovae and their absolute luminosity.Such supernovae are thought to be thermonuclear explosions of white dwarfs triggered by accretion from a binarystar. The physical reason for this correlation is not completely understood.The absolute luminosity L is related to the apparent luminosity F via the ”luminosity distance” which by definitionis
dL ≡√
L/4πF . (110)
Astronomers use magnitudes as a logarithmic measure for luminosities: Five magnitudes correspond to a factor 100in luminosity and the absolute magnitude M is defined as equaling the apparent magnitude m at a distance of 10 pc.Thus, one has for the ”distance modulus”
m−M = 5 log10(dL/10 pc) . (111)
A modern example of the resulting Hubble relation is shown in Fig. 2. It leads to H0 = 100h kms−1Mpc−1 withh = 0.73+0.04
−0.03.
FIG. 2: The Hubble relation from the HST key project [6].
23
E. The Cosmological Principle and the Friedmann-Lemaitre Equation
The observable Universe appears to be homogeneous and isotropic on scales larger than 100 Mpc. The cosmolog-ical principle states that on average over such scales it looks the same everywhere and in every direction. In thisapproximation the Universe just undergoes a self-similar expansion and its 4-metric can be written as
ds2 = gµνdxµdxν = dt2 −R(t)2dΩ2
3 = a(t)2(
dη2 −R20dΩ
23
)
, (112)
where ds2 is the Lorentz-invariant distance element, t is cosmic time, R(t) ≡ R0a(t) is the time-dependent scale factorwhich we separate into a dimensionless expansion factor, with a(t0) = 1, and the curvature scale R0 today, and dΩ2
3 isthe ”comoving” volume 3-element of a static homogeneous and isotropic 3-space. In the second equality of Eq. (112),the time dependence appears only in the global factor when conformal time is defined as dη ≡ dt/a(t). In terms of a“coordinate distance” r, in Einstein gravity it turns out that one has
dΩ23 =
dr2
1− kr2+ r2dΩ2
2 , (113)
where dΩ22 ≡ dθ2 + sin θ2dϕ2 is the geometry of a 2-sphere, and k can take the values +1, -1, or 0, for a closed, open
or flat spatial geometry, respectively.Light rays propagating along radial directions then have 0 = ds2 = dt2 − R(t)2dr2/(1 − kr2). The wavelengths oflight rays are stretched by the expansion just as any other physical length, l(t) = a(t)l0 = l0/(1+ z), where the index0 refers to today when by definition we have a ≡ 1 and thus l0 ≡ R0r is the comoving length. Two points at fixedspatial positions in the comoving geometry separated by a comoving distance R0r thus recede from each other witha velocity v = a(t)R0r = (a/a)(t)l(t). Comparing this with Eq. (109) results in
H =a
a=R
R. (114)
We can now actually derive most aspects of the equations of motion of the Universe expansion without having toknow general relativity. Consider a sphere of physical radius l(t) = a(t)R0r. A particle of mass m on the surface ofthe sphere will move with a velocity v = aR0r = H l(t) with respect to the center. It experiences attractive forcesof the surrounding matter of density ρ(t) and a repulsive force proportional to a possible cosmological constant Λ.According to Birkhoff’s theorem, the particle has a conserved total (kinetic plus potential) energy
E =1
2mH2l2(t)− 4πGNmρ(t)l
2(t)
3− Λm
6l2(t) , (115)
where GN is Newton’s constant and we have normalized Λ conveniently. This implies
H2(t) ≡(
a
a
)2
=8πGNρ
3− K
a2+
Λ
3, (116)
where K ≡ −2E/(mR20r
2) =const. A full treatment based on the the equations of general relativity yields K = k/R20
with the above defined dimensionless k =+1,-1, or 0.Due to the dilution of particle number in an expanding universe, the physical energy density of non-relativistic matterscales as ρm ∝ (1 + z)3. Relativistic matter or ”radiation” has an additional redshift factor for the energy and thusρr ∝ (1 + z)4. Finally, a cosmological constant corresponds to a vacuum energy ρv ≡ Λ/(8πGN) =const. Furtherdefining the ”curvature energy density” ρk ≡ −3K/(8πGNa
2) = −3k/[8πGNR2(t)] we can rewrite Eq. (116) as
H2(t) =8πGN(ρm + ρr + ρv + ρk)
3(117)
This implies that the Universe is flat, K = 0, when the density ρtot ≡ ρm + ρr + ρv is critical, ρtot = ρc with
ρc ≡3H2
8πGN, (118)
and with Ωi ≡ ρi/ρc we obtain the ”sum rule”
Ω ≡ Ωm +Ωr +Ωv = 1− Ωk . (119)
24
This can also be written as
Ω− 1 ∝ 1
a(t)2H(t)2(120)
The expansion age of the Universe at redshift z depends on the density parameters as t(z) =∫
dt =∫ z
0dz′/[(1 +
z′)H(z′)]. Within orders of magnitude we thus have t(z) ∼ H(z)−1. In terms of comoving distance, the maximaldistance a particle can travel from the begin of the Universe up to a given cosmic time t or redshift z is given by
conformal time η(t) = R0
∫ r
0dr′/
√
(1 − kr′2) =∫ t
0dt′/a(t′), which sometimes is also called ”proper distance” dp, as
opposed to the coordinate distance r to which it is related by r = arcsin(dp/R0), r = dp/R0, and r = sin(dp/R0) fork = −1, 0, 1, respectively. Using da = −dz/(1 + z)2 we have η =
∫ z
0dz′/H(z′). The corresponding physical length
scale η/(1 + z) is known as the ”causal horizon” because no signal can propagate faster than light. This physicallength scale is thus comparable to the expansion age t(z) ≃ H(z)−1 and is thus also called ”Hubble scale”. At anygiven time, events and properties of the Universe separated by more than the causal horizon must be uncorrelated.We note that both in the matter and radiation dominated epochs, the horizon H(t)−1 grows more quickly with timethan the scale factor R(t). This leads to the cosmological ”horizon problem”: Today’s Hubble scale, when redshiftedinto the early Universe could not have been causally connected and thus should look inhomogeneous in a Universedominated by ”ordinary” matter and radiation. In addition, Eq. (120) implies that since today’s Universe is close toflat, it must have been incredibly fine-tuned to Ω = 1 in the early Universe. This is called the ”flatness problem”.Finally, we saw in the first chapter that if a grand unified symmetry is broken such that an U(1) symmetry survives,the formation of magnetic monopoles is inevitable. According to the Higgs-Kibble mechanism, about one monopoleper Hubble volume at the epoch of symmetry breaking will form. Since these monopoles do not annihilate, theirdensity just redshifts as (1 + z)−3, leading to a value today of
ρ
ρc∼ H3(TGUT)TGUT
ρc
(
T0TGUT
)3
∼(
8πGN
3
)5/2(gπ2
30
)3/2T 4GUTT
30
H20
≃ 8× 1018(
TGUT
1016 GeV
)4
, (121)
where we have assumed that monopoles of mass M ∼ TGUT form at a temperature TGUT in presence of g ∼ 100thermal degrees of freedom, and the radiation temperature today is T0 = 2.3697× 10−4 eV. This would overclose thepresent Universe by many orders of magnitude !We can now also compute the luminosity distance of a source at redshift z as follows: The proper area of the spheresurrounding a source at proper distance dp today is 4πd2p. In addition, photon energies are redshifted and time scales
are dilated by a factor (1 + z) each, therefore, F = L/[4πd2p(1 + z)2], giving for the luminosity distance Eq. (110)
dL = (1 + z)dp = (1 + z)
∫ z
0
dz′
H(z′)=
1 + z
H0
∫ z
0
dz′
E(z′), (122)
where
H(z) = H0E(z) ≡ H0
[
Ωm(1 + z)3 +Ωr(1 + z)4 +Ωk(1 + z)2 +Ωv]1/2
. (123)
Using Eqs. (111) and (122) one can now predict the distance modulus of standard candles as a function of redshiftand compare with data. This is shown in Fig. 3.The proper distance is dp = R0r, dp = R0 arcsin r, and dp = R0 arcsinh r for a flat, closed, and open Universe,respectively. We see that the geometry of a closed Universe is that of a sphere of radius R0 where R0r is thedistance from an axis through the center of the sphere. A physical length scale λ will thus appear under an angleα = λ/(R0r) = λ/(R0 sin dp/R0), where the latter expression is for a closed Universe, k = 1. Since Eq. (122) showsthat dp does not depend sensitively on Ωk, this angle essentially depends on the curvature Ωk ∝ 1/R2
0 and thus, viaEq. (119), on the total density Ω. We will see in the next chapter that the cosmic microwave background providesnatural physical length scales λ which thus allow to measure Ω.
F. Structure Formation
We start with the Newtonian theory of small perturbations. Assume a non-relativistic fluid of density ρ, pressure pand velocity v, subject to Newtonian gravitational forces g. The continuity equation, Euler equation and gravitationalfield equations are, respectively,
∂ρ
∂t+∇ · (ρv) = dρ
dt+ ρ∇ · v = 0 ,
25
FIG. 3: Hubble diagram measured by two major experimental collaborations [7].
26
dv
dt≡ ∂v
∂t+ (v ·∇)v = −1
ρ∇p+ g ,
∇× g = 0 , (124)
∇ · g = −4πGNρ ,
where d/dt ≡ ∂/∂t + v ·∇ is the time derivative along the worldline of the particles. In an expanding Universe wehave the following homogeneous solution of Eq. (124):
ρ0(t) = ρ0
[
R0
R(t)
]3
,
v0(r, t) = rH(t) , (125)
g0(t) = −r4πGNρ0(t)
3.
Introducing small perturbations by writing ρ = ρ0 + ρ1, etc. and expanding into comoving momentum modes withρ1(r, t) = ρ1(t) exp [ik · r/a(t)], etc., to lowest order Eqs. (124) give
ρ1 + 3H(t)ρ1 + iρ0(t)k · v1
a(t)= 0 ,
v1 +H(t)v1 = − ic2sρ0(t)a(t)
kρ1 + g1 , (126)
g1 =4πiGNρ1a(t)k
k2,
where cs ≡ (dp/dρ)1/2 is the speed of sound of the fluid. Eq. (126) shows that the component of v1 perpendicularto k just decays as a−1(t), whereas the relative density perturbation δ(t) ≡ ρ1/ρ0(t) and the longitudinal velocitycomponent
δ(t) ≡ ρ1ρ0(t)
=ρ1ρ0
[
R(t)
R0
]3
,
ε ≡ − ia(t)k · v1
k2, (127)
respectively, obey
δ =k2
a(t)2ε , (128)
ε =
(
−c2s +4πGNρ0(t)
k2
)
δ .
This can be combined to the second order equation
δ + 2H(t)δ +
(
c2sk2
a(t)2− 4πGNρ0(t)
)
δ = 0 . (129)
Eq. (129) implies immediately that one can define the Jeans wavenumber
kJa(t)
≡(
4πGNρ0(t)
c2s
)1/2
, (130)
such that at small length scales k >∼ kJ , one has oscillatory, non-growing solutions. Growing solutions can only resultin the matter dominated regime because otherwise the matter term (note that ρ0 is the average matter density)
−4πGNρmδ is always much smaller than the damping term 2H(t)δ ∼ 2H2(t)δ ∼ 16πGNρtotδ/3. In the matterdominated regime, growing solutions result at large scales k ≪ kJ , corresponding to matter masses M ≫ MJ withthe Jeans mass
MJ ≡ 4πρm3
(
2πa(t)
kJ
)3
=4π5/2c
3/2s
3G3/2N ρ
1/2m
∼ 3.4× 1020 c3sM⊙ . (131)
27
After recombination, c2s ∼ Tgas/mN ≪ 1 where Tgas is the gas temperature.The Newtonian description is only applicable at scales much smaller than the horizon length and if the pressureis negligible. While a detailed description of scales comparable to the Hubble scale requires general relativity, it issufficient for us to know that perturbations at scales larger than the horizon cannot grow and are ”frozen in”. Theycan start to grow only once they cross inside the horizon during the radiation or matter dominated regime.
G. Equilibrium Thermodynamics and the Cosmic Microwave Background (CMB)
In thermal equilibrium at temperature T , in momentum mode k the occupation number of g fermionic or bosonicdegrees of freedom of mass m is
nk =g
exp (Ek/T )± 1, (132)
respectively, where Ek ≡ (k2+m2)1/2 is the energy of a particle in mode k. We here use units in which the Boltzmannconstant kB is unity. The spectral number density is then
dn
dk=gk2
2π2
1
exp (Ek/T )± 1, (133)
and for massless particles the total number and energy densities are accordingly
n =ζ(3)
2π2gNnT
3
ρ =π2
30gNρT
4 , (134)
where ζ(x) is the Riemann zeta function and Nn = 3/2, Nρ = 7/8 and Nn = 2, Nρ = 1 for fermions and bosons,respectively.In thermal equilibrium at temperature T , a system with density ρ, pressure p, and volume V has an entropy S whichsatisfies
dS(V, T ) =d(ρ(T )V ) + p(T )dV
T. (135)
Since S is an extensive quantity, S(V, T ) = s(t)V , we have for the entropy density
s(T ) =ρ(T ) + p(T )
T. (136)
For a relativistic gas we have p(T ) = ρ(T )/3 and thus
s(T ) =4ρ(T )
3T=
4π2
90gNρT
3 . (137)
Note that in absence of non-adiabatic processes, S(T ) ∝ s(T )R(T )3 is conserved since T ∝ R−1.Using the effective number of degrees of freedom
g ≡ gb +7
8gf , (138)
where gb and gf are the number of bosonic and fermionic degrees of freedom, respectively, in the relativistic regimewe can then write
ρ(T ) =π2
30gT 4 ,
s(T ) =4π2
90gT 3 . (139)
For the CMB we have two photon polarizations and thus g = 2. Today the CMB has a temperature T0 = 2.725 ±0.001K. Together with H0 = 100h km s−1 Mpc−1 where h ≃ 0.73, this gives Ωγ = (2.471 ± 0.004) × 10−5h−2.Extrapolating back, we thus get for the redshift of matter-radiation equality
1 + zeq =ΩmΩγ
≃ 5100
(
Ωmh2
0.127
)
. (140)
28
Furthermore, when the CMB photon energies fall below the hydrogen ionization energy ∼ 13.6 eV, at a redshift∼ 13.6 eV/T0 ∼ 5× 104, electrons should recombine with the protons and hydrogen should form. However, since thenumber of photons is larger than the number of baryons by a factor ≃ nγ(T0)/(Ωbρc/mp) ≃ 1.68× 109(0.0224/Ωbh
2),hydrogen remains actually ionized down to considerably lower redshifts. A detailed statistical equilibrium analysisresults in the recombination redshift
1 + zrec ≃ 1100 . (141)
In the presence of sources Φ such as primordial density fluctuations, the wave equation for temperature perturbationsΘ ≡ δT/T in the Fourier space of comoving momenta k reads
Θ + c2sk2Θ = Φ , (142)
where cs ≡ (dp/dρ)1/2 is the speed of sound of the photon fluid of pressure p and density ρ, and the time derivativesare with respect to conformal time η. Note that because ∂η = a(t)∂t, in terms of physical time Eq. (142) reads[
∂2tΘ+H∂t + c2s(k/a)2]
Θ = Φ/a(t)2. Eq. (142) holds as long as the baryon-photon fluid is strongly coupled by thefree electrons. After recombination, the temperature fluctuations become frozen into the CMB and can be observedtoday. They are thus of the form
Θk ∝ cos ks⋆ , (143)
where s⋆ ≡∫ trec0
dtcs(t)(1+z) =∫∞
zrecdzcs(z)/H(z) is the comoving sound horizon. Since cs is roughly constant before
recombination, one can approximate this as s⋆ ≃ csη⋆, with η⋆ the conformal time at recombination.The CMB temperature perturbations projected onto the sky, Θ(n, can be decomposed into spherical harmonics,
Θl,m ≡∫
dnY ⋆l,m(n)Θ(n) , (144)
where Yl,m(n) are the spherical harmonics functions. We can then define
Cl ≡ T 2⟨
|Θl,m|2⟩
(145)
The resulting power spectrum is shown in Fig. 4.At recombination, zrec ∼ 1100, the comoving sound horizon is roughly (1 + zrec)H(zrec)
−1 ≃ (1 + zrec)−1/2H−1
0 ≃120Mpc. Today, this scale appears under an angle α ≃ 120Mpc/(R0r). For an approximately flat Universe onehas R0r ≃ η(t0) =
∫ zrec0
dz′/H(z′) ≃ 2/H0 since H(z) ∝ (1 + z)3/2 in the matter-dominated regime. Thus, α ≃120MpcH0/2 ≃ 0.015 ≃ 0.8 which corresponds to a multipole l ≃ 180/α ≃ 200. Indeed the first peak of thebaryon acoustic oscillations in the CMB shown in Fig. 4 appears at roughly this scale. After recombination thebaryons decouple and form a non-relativistic gas of temperature T ∼ eV. The speed of sound drops precipitously toc2s ∼ T/mN and thus the scale of baryon acoustic oscillations changes little. It is indeed also visible at the same scale∼ 100Mpc in the large scale correlation function of galaxies at various redshifts [9].Fig. 4 also shows that at zrec ∼ 1100, the baryonic density perturbations at length scales below the comoving soundhorizon are of order 10−5. In the previous chapter we saw that since then they could have grown only by a factor∼ (1 + zrec) ∼ 103. On the other hand, we know that matter perturbations have turned non-linear by today at scales<∼ 8Mpc. An ingredient is thus missing. If there was a significant component of non-baryonic dark matter not couplingto baryons, their density perturbations would have grown already before recombination without communicating withthe baryons which were tightly coupled to the photons. Later on, the baryons could have fallen into the potentialwells of the dark matter, thereby turning non-linear. CMB observations indicate thus the necessity of dark matter.The distance modulus versus redshift measurements of type Ia supernovae shown in Fig. 3 together with CMBmeasurements result in the constraints of cosmological density parameters shown in Fig. 5. In fact, one can see fromEq. (122) that the luminosity distance and thus the constraint from type Ia supernovae is relatively insensitive toΩ but rather depends on Ωm − Ωv. In contrast, the position of the acoustic peaks in the CMB depend mostly onΩ ≃ Ωm +Ωv. This is clearly reflected in Fig. 5.
H. Relics from the Early Universe: Freeze-Out
A species interacting with a rate Γ is in thermal equilibrium as long as Γ ≫ H. In general Γ decreases fasterwith the expansion of the Universe than the Hubble rate H. Let us first consider neutrinos which at temperaturesT >∼ 1MeV interact mostly with the electron-positron plasma, with a cross section similar to Eq. (13), i.e. σv ∼
29
FIG. 4: Multipole power spectrum of the CMB temperature fluctuations from WMAP and other CMB anisotropy experi-ments, where the Cl are defined in Eq. (145). The position of the first peak is mostly sensitive to Ω and indicates a Universequite close to being flat. The height of the first peak determines the matter density, and the ratio of the first to secondpeaks determines the baryon density. Combined with SDSS large-scale structure data, the resulting best fit parametersare: Ω = 1.003 ± 0.010, Ωm = 0.24 ± 0.02, Ωb = 0.042 ± 0.002, and Ωv = 0.76 ± 0.02 [8], where Ωb is the dimensionlessbaryon density. This “concordance cosmology” predicts the curve shown, where the band indicates cosmic variance. Fromhttp://lambda.gsfc.nasa.gov/product/map/.
G2FT
2 ∼ 10−43(T/MeV)2 cm2. According to Eq. (134), with 4 degrees of freedom the plasma density is n ∼ T 3 andthus Eq. (12) gives
Γ ∼ G2FT
5 ∼ 0.4
(
T
MeV
)5
s−1 (146)
On the other hand, Eq. (116) yields
H(T ) ∼(
8π3GNgT4
90
)1/2
∼ 0.2 g1/2(
T
MeV
)2
s−1 , (147)
where g is the number of relativistic degrees of freedom. We thus conclude that neutrino interactions freeze out ataround 1 MeV.The Boltzmann equation for a dark matter species X with average annihilation cross section 〈σXv〉 and density nX is
dnXdt
+ 3H(t)nX = −〈σXv〉[
n2X − (neqX )2]
, (148)
where the equilibrium density is given by the integral of Eq. (133). For T ≪ m, neqx ∝ exp(−m/T ) is Boltzmann-suppressed and can thus be neglected on the r.h.s. of Eq. (148). For such temperatures nX ∝ T 3 thus just redshifts,consistent with the fact that in the radiation dominated regime the l.h.s. of Eq. (148) decreases ∝ T 5, whereas the
30
FIG. 5: Constraints on cosmological density parameters [10].
r.h.s. decreases ∝ T 6. One can then show that the dark matter density today can be estimated as
ΩXh2 ∼ 10−37 cm2
〈σXv〉. (149)
It is interesting (and maybe no coincidence !) that a weak-scale annihilation cross section leads to the correct orderof magnitude for the dark matter density.The freeze-out process is demonstrated in Fig. 6.
I. Big Bang Nucleosynthesis (BBN)
For more detailed introductions to the following three topics we refer the reader to standard text books [12, 13].The early universe consisted of a mixture of protons, neutrons, electrons, positrons, photons and neutrinos. Theirrelative abundances were determined by thermodynamic equilibrium until the weak interactions ”froze out” once
31
1 10 100 1000
0.0001
0.001
0.01
FIG. 6: Comoving number density of dark matter particles in the early Universe as a function of x ≡ m/T . The solid line isthe equilibrium abundance, whereas the dashed lines represent the actual abundances for different annihilation cross sections.From Ref. [11].
the temperature of the expanding universe dropped below Tf ∼ 1MeV where their rates became smaller than theexpansion rate. The interaction rates of nucleons νep↔ ne+ and e−p↔ nνe are similar to the neutrino interaction rateEq. (146) at temperatures 100GeV >∼ T >∼ 1MeV where the neutron-proton mass differencemn−mp = 1.293MeV and
the electron mass are negligible and the e± and electron neutrino densities n ∼ T 3. This becomes indeed comparableto the expansion rate Eq. (147) once T approaches Tf ≃ 1MeV. The equilibrium neutron to proton ratio at thattemperature is given by thermodynamics as
nnnp
= exp [−(mn −mp)/Tf ] (150)
At that time, the free neutrons were quickly bound into helium which could not be broken up any more by the coolingthermal radiation. The helium abundance was thus determined by the freeze out of electroweak interactions. Sinceequating Eq. (146) with Eq. (147) yields Tf ∝ g1/6 we also see that the helium abundance should increase with g.Since the number Nν of stable neutrino species with mass below ∼ 1MeV contributes to g, this number is constrainedby the observed helium abundance. Since this abundance depends essentially on the expansion rate of the Universe,helium is a good chronometer. Apart from the expansion rate, in the absence of a significant asymmetry betweenneutrinos and anti-neutrinos, elemental abundances depend only on the effective number of relativistic neutrinos Nνand the baryon to photon ratio
η10 ≡ 1010nBnγ
≃ 274Ωbh2 . (151)
Predictions for standard big bang nucleosynthesis (SBBN) with Nν = 3, the number of active neutrinos consistentwith the Z boson width, are shown in Fig. 7. As is apparent there, the steep and monotonous dependence of itsabundance on η makes it a good baryometer. A detailed comparison of measured and predicted abundances shownin Fig. 7 with η10 and Nν free parameters yields the following: The universal density of baryons η10 inferred from
32
SBBN and the measured light element abundances, 4.7 ≤ η10(SBBN) ≤ 6.5, is in excellent agreement with the baryondensity derived from CMB data [14], η10(CMB) = 6.11 ± 0.2, as also seen in Fig. 7. However, as Fig. 7 also shows,there is a ∼ 2σ tension between the abundances predicted by SBBN with this range of η10 for 4He and deuterium onthe one hand and for 7Li on the other hand. Could this suggest that BBN is non-standard, for example that Nν 6= 3,or that some other as yet unknown physics might play a role ? Inhomogeneous nucleosynthesis can alter abundancesfor a given ηBBN but will overproduce 7Li. Entropy generation by some non-standard process could have decreasedη between the BBN era and CMB decoupling, however the lack of spectral distortions in the CMB rules out anysignificant energy injection upto a redshift z ∼ 107.
FIG. 7: Predicted abundances of primordially synthesized light nuclei as a function of the baryon-to-photon ratio (smoothlines/bands). Predictions are confronted with measured abundances (denoted by subscript p) inferred from measurements(smaller boxes: 2σ statistical errors, larger boxes 2σ statistical plus systematic errors). Y denotes the mass fraction of 4He.The vertical shaded band is the CMB measure of the cosmic baryon density. (taken from the Review of Particle Properties,Particle Data Group).
J. Phase Transitions
In chapter I we saw that in the context of the Standard Model, there are two phase transitions, the electroweaktransition at T ∼ 100GeV and the QCD transition at T ∼ 100MeV.Physics beyond the Standard Model may give rise to other phase transitions, such as the one associated with theGrand Unified Symmetry and the Peccei Quinn symmetry related to axions.These phase transitions are also thought to play a role in the creation of extragalactic magnetic fields.
33
K. Inflation and Reheating
The Lagrangian of a scalar field φ moving in a potential V (φ) in the expanding Universe is
Lφ = R(t)3
[
V (φ)− φ2
2
]
, S[φ] =
∫
dtr2dr√1− kr2
sin θdθdϕL . (152)
For a homogeneous scalar field the equations of motion Eq. (65) thus read
φ+ 3H(t)φ+ V ′(φ) = 0 . (153)
The energy momentum tensor of an ideal fluid has the form Tµν = diag(ρ, p, p, p). From its general form Eq. (70) wethus have
ρ =φ2
2+ V (φ) , p =
φ2
2− V (φ) . (154)
Inflation takes place as long as scalar field evolution is dominated by its potential, i.e. φ2 ≪ V (φ). In this casethe equation of state is w ≡ p/ρ ≃ −1. Inspection of Eq. (116) shows that the scalar field then effectively acts asa cosmological constant and H(t) ∼ const., leading to exponential expansion a(t) ∝ exp(Ht). Typical cases for theinflaton potential are shown in Fig. 8.
φ
V(φ)(a)
φ
V(φ)
(b)
FIG. 8: Two typical models for the inflaton potential.
Fig. 9 shows how inflation solves the horizon and homogeneity problem. According to Eq. (120) it also solves theflatness problem because H(t) ∼const. and a(t) grows exponentially. It finally also solves the monopole problembecause their density is exponentially diluted.It is interesting to note, however, that electroweak symmetry breaking should occur after inflation, in which casedifferent types of gauge bosons would be frozen out in different directions in the sky, as pointed out by R. Penrose [15].Due to the exponential expansion during inflation, practically all scales observable today have been at physical lengthscales much smaller than the Planck length at the begin of inflation. Sometimes this is seen as a problem becausewe have no theory of quantum gravity that would describe such scales. On the other hand, signatures of such”trans-Planckian physics” may show up in the CMB.Note that it is currently unclear if there is any connection between the current phase of exponential expansion due todark energy and early Universe inflation.After inflation, the energy stored in the inflaton field potential V (φ) is released in form of thermal radiation, throughthe inflaton coupling to ”ordinary” particles. This is achieved through a complicated process of parametric resonancesof the inflaton (”preheating”) and subsequent thermalization (”reheating”). The reheating temperature Tr is roughlygiven by conservation of energy, V (φf ) ∼ ρ ∝ gT 4
r , where φf is the value of the inflaton field upon exit from the ”slow
roll” phase in which φ2 ≪ V (φ).
III. HIGH ENERGY ASTROPHYSICS
Recent books on cosmic rays are Ref. [16, 17]. A rather technical recent presentation of theoretical high eneryastrophysics is Ref. [18]. An older detailed book on theoretical cosmic ray physics is Ref. [19].Let us start by presenting summary figures of the diffuse cosmic ray (CR), γ−ray, and neutrino fluxes in Figs. 10, 11,and 12, respectively.
34
FIG. 9: The scale factor R(t) versus cosmic expansion time with and without inflation.
A. Cosmic Ray Acceleration
The force of an electromagnetic field E,B on a particle of charge eZ is given by
F = eZ (E+ v ×B) . (155)
Consider a particle of momentum p and energy E gyrating with an angular frequency ωg in a homogeneous magneticfield B0. It will experience a force ωgp = eZ(p×B0)/E, resulting in
ωg =eZB⊥
E, (156)
where B⊥ = |p × B0|/p is the modulus of the component of B0 perpendicular to the motion of the charge. Thegyro-radius is
rg(p/Z) =v
ωg=p/E
ωg=
p
eZB⊥
. (157)
Note that for ultra-relativistic particles, p ≃ E, v ≃ 1, the equation of motion only depends on rigidity, E/Z.
35
FIG. 10: The all-particle cosmic ray spectrum and experiments relevant for its detection (taken from S. Swordy).
Astrophysical plasmas are often highly conducting, thus when the plasma flow velocity is v, we have E ≃ −v × B,and thus E = 0 in the plasma rest frame.Charged particles with a gyro-radius Eq. (157) much smaller than the size of the magnetized region are diffusing in themagnetic field. In a general diffusion process a particle with a mean scattering length over which it changes direction lwill propagate an average squared distance 〈d2〉 ∼ l t over time t. The in general energy dependent scattering length lis often called the diffusion coefficient D(E). In case of diffusion in magnetic fields, charged particles typically scatteron inhomogeneities of the magnetic field and energy loss due to collisions with ambient gas or low energy photonsis often negligible, except at the highest energies. In this case the diffusion coefficient D(E) will be depend on thedetailed structure of the magnetic field, but will typically be of the order the gyro-radius Eq. (157),
D(E) ≃ 1
3rg(E) . (158)
A shock is basically a solution of the hydrodynamics euqtaions with a discontinuity in the flow of a fluid or plasma.We will here only consider plane shocks. In the ”shock frame” in which the shock is at rest, the plasma is moving
36
FIG. 11: The diffuse ”grand unified photon spectrum”, taken from Ref. [20].
toward the shock with a velocity u1 from the ”upstream” side, and moving away from the shock on the opposite,”downstream” side with a velocity u2 < u1. The situation is depicted in Fig. 13 where we define the positivez−direction as perpendicular to the shock front pointing toward the downstream region. In the Fermi accelerationprocess, cosmic rays cross back and forth across astrophysical shocks by scattering on inhomogeneities of the magneticfields. As we will show in the following, in each shock crossing, on average they gain energy, but also have a finiteprobability for escaping the acceleration region. This results in a power law spectrum extending up to the energywhere the gyro-radius of the particle becomes so large that it escapes the magnetized region of the accelerator withprobability one. For relativistic shocks the maximal energy is thus
Emax ∼ 1018(
B
µG
)(
L
kpc
)
eV , (159)
where B and L are the magnetic field strength and size of the shock, respectively. This maximal energy is shown inFig. 14 for various astrophysical objects.Let us now consider one cycle of shock crossing for a non-relativistic shock. In the plasma rest frame where E = 0,
37
FIG. 12: The diffuse grand unified neutrino spectrum. Solar neutrinos, burst neutrinos from SN1987A, reactor neutrinos,terrestrial neutrinos and atmospheric neutrinos have been already detected. Another guaranteed although not yet detected fluxis that of neutrinos generated in collisions of ultra-energetic protons with the 3K cosmic microwave background (CMB), theso-called GZK (Greisen-Zatsepin-Kuzmin) neutrinos. See for AGN and GZK neutrinos Sect. IVA1. Whereas GZK and AGNneutrinos will likely be detected in the next decade, no practicable idea exists how to detect 1.9 K cosmological neutrinos (theanalogue to the 3K CMB).
the particle momentum is constant. Let us denote the angle of its direction of motion relative to the z−directionwith θi, and µi ≡ cos θi, for i = 1, 2. A relativistic particle of momentum p1 in the upstream rest frame approachingthe shock front has µ1 > −u1 and will have a momentum p2 = Γp1(1 + vµ1) in the downstream rest frame, wherev = (u1 − u2)/(1 − u1u2) with corresponding Lorentz factor Γ = (1 − v2)−1/2. Similarly, a particle with momentump2 in the downstream rest frame approaching the shock has µ2 < −u2 and will have a momentum p1 = Γp2(1− vµ2)in the upstream rest frame. After one cycle we thus have
p′1 = Γ2p1(1− vµ2)(1 + vµ1) . (160)
Assuming an isotropic distribution in both plasma rest frames and averaging the fluxes ∝ µi over the shock front over
38
u1 > Cs1 u2 < Cs2
P1 << P2 P2 >> P1
ρ1 ρ2 = rρ1
⇒ ⇒
Bl1 Bl2 = Bl1
Bt1 Bt2 = rBt1
FIG. 13: For a plane, adiabatic shock the jumps in mass density, velocity, and pressure are given by the Rankine-Hugoniotconditions and determine the compression ratio r ≡ ρ2/ρ1. Upstream and downstream regions are denoted with index 1 and2, respectively.
−u1 ≤ µ1 ≤ 1 and −1 ≤ µ2 ≤ −u2, for non-relativistic shocks, u1, u2, v ≪ 1 leads to an average momentum gain of
〈∆p〉 ≃ 4
3(u1 − u2)p (161)
during one cycle, where p is the initial momentum.We can estimate the downstream escape probability as the ratio of the convective flux far from the shock front tothe flux entering the downstream region from the upstream region. Assuming a constant cosmic ray mass density ndownstream, one has
Pesc ≃nu2
n4π2π
∫ 1
0dµ2µ2
= 4u2 . (162)
We can now estimate the spectrum as follows: Starting with a momentum p0, after n cycles, the cosmic rays thathave not escaped the downstream region yet will on average have momentum pn ≃ p0 (1 + 〈∆p〉/p)n, and a density ofn(> pn) = n(> p0)(1 − Pesc)
n. Writing the integral spectrum of such cosmic rays as n(> pn) = n(> p0)(pn/p0)1−α,
taking the logarithm yields
α = 1− ln [n(> pn)/n(> p0)]
ln (pn/p0)=r + 2
r − 1, (163)
where r ≡ u1/u2 = ρ2/ρ1 > 1 is the shock compression ratio because mass conservation implies ρ1u1 = ρ2u2, whereρ1,2 are the energy densities on the two sides of the shock. The second equality holds for non-relativistic shocks.Let us now compute the compression ratio r in terms of the properties of the plasma. The flow of plasma across theshock front conserves mass, momentum and energy,
ρ1u1 = ρ2u2 ,
ρ1u21 + P1 = ρ2u
22 + P2 , (164)
39
FIG. 14: The famous Hillas plot shows potential cosmic-ray accelerators in a diagram where the horizontal direction signifiesthe size L of the accelerator, and the vertical the magnetic field strength B. The maximal energy E which an accelerator canachieve is proportional to ZLBβ, with β being the shock velocity in units of the speed of light and Z the particle charge;the limiting energy results from the requirement to confine the particle to the acceleration region while it gains energy. Aparticular energy corresponds to a diagonal line in this diagram and can be achieved e.g. with a large, low field accelerationregion or with a compact, high-field region. Assuming a shock velocity β ∼1, neutron stars, AGN, Radio Galaxies or Galacticclusters can accelerate protons to E ∼ 1020 eV. For non-relativistic shocks (β ∼ 1/300), no object class with sufficient size andmagnetic field to produce 1020 eV protons is known. Note that the actual maximal energy reached by a particular acceleratorcan be smaller than implied by this plot which neglects energy loss processes during acceleration. Such losses can be especiallyimportant in relatively compact objects, for example in neutron stars.
ρ1u1
(
1
2u21 + h1
)
= ρ2u2
(
1
2u22 + h2
)
,
where Pi and hi are pressure and specific enthalpy, respectively, of the plasma on the two sides, i = 1, 2. The specificenthalpy is the ”available” energy of a system under constant pressure. It is given by h = (ρkin + P )/ρ, where ρkin isthe kinetic (excluding rest mass) energy density. For an ideal, non-relativistic gas P ∝ ρ and thus the speed of soundc2s = dP/dρ = P/ρ. Furthermore, P/ρkin = γ − 1, where γ is the adiabatic index. For an ideal, non-relativistic gas it
40
is γ = 5/3, so that one has P = 23ρkin = nT = ρ
mT , where m and T are mass and temperature of the particles and
we have used ρkin = 12nm〈v2〉 = 3
2nT with 〈v2〉 the average squared particle velocity at temperature T . Thus, in this
case c2s = T/m. In general, one thus has h = γγ−1c
2s =
γγ−1P/ρ.
Eqs. (164) are often called the Rankine-Hugoniot jump conditions. Dividing the last one by ρ1u1u22 = ρ2u
32 = ρ1u
31/r
2
[these equalities are a consequence of the first condition in Eqs. (164)], we obtain
r2(
1
2+
γ
γ − 1
1
M2
)
=1
2+
γ
γ − 1
P2
ρ2u22=
1
2+
γ
γ − 1rP2
ρ1u21, (165)
where we have introduced the upstream Mach number M ≡ u1/cs,1. Using the second condition in Eqs. (164), we canexpress P2 in terms of upstream quantities, P2 = P1 +
r−1r ρ1u
21. Wit this, the right hand side of Eq. (165) becomes
−(γ + 1)/[2(γ − 1)] + γ(
1 +M−2)
r/(γ − 1). This yields a quadratic equation for r whose largest solution is
r =γ + 1
γ − 1 + 2γ/M2. (166)
We now realize that for non-relativistic shocks with γ = 5/3 we have r ≤ 4 and from Eq. (163) α ≥ 2, where thelimiting values are obtained in the limit of large Mach number, M ≫ 1.We can estimate the time scales associated with acceleration as follows: The escape time is given by equating
the downstream convection and diffusion distances, u2Tesc ≃ [2D2(p)Tesc]1/2
, with D2(p) the downstream diffusioncoefficient, or
Tesc(p) ≃2D2(p)
u22(167)
Furthermore, the cycle time scale is Tcyc ≃ PescTesc and the acceleration time scale is Tacc ≃ (p/〈∆p〉)Tcyc, giving
Tcyc(p) ≃ D2(p)
2u2
Tacc(p) ≃ 3D2(p)
8(u1 − u2)u2, (168)
where we have used Eqs. (162) and (161). In cases where collisional energy losses can not be neglected, one can obtainthe maximal energy by comparing Tacc(p) to the relevant energy loss time scales, such as inverse Compton scattering(ICS), synchrotron radiation, pion production, to be discussed below, as well as with the shock lifetime.Relativistic shocks with u1, v → 1 are more complicated to treat because the particle distributions can not be assumedisotropic anymore. Nevertheless, in the most optimistic scenarios, one has
Tacc(p) ∼ D(p) ∼ rg(p) ∼p
eZB. (169)
It is obvious from Eq. (160) that as long as particles are isotropically distributed in the intervals −u1 ≤ µ1 ≤ 1 and−1 ≤ µ2 ≤ −u2, one has 〈p′1〉 ∼ 2Γ2p1 for relativistic shocks and thus particles gain energy very efficiently in one cycle.However, if particles have not enough time to isotropize after the first cycle, by performing Lorentz transformationsone has µ1 ≡ p1,z/p1 = (p2,z − vp2)/ [p2(1− vµ2)] = (µ2 − v)/(1− vµ2) and the factor (1+ vµ1) in Eq. (160) becomes1/[Γ2(1 − vµ2)] ∼ 1/(2Γ2) since the condition for crossing from downstream to upstream requires µ2 ≃ −1 in anycycle. Again using Eq. (160) then yields 〈p′1〉 ≃ p1 and thus there is little energy gain after the first cycle.
B. Cosmic Ray Propagation
We first set up some general notation and present the general equations governing cosmic ray propagation. Theinteraction length l(E) of a CR of energy E and mass m propagating through a background of particles of mass mb
is given by
l(E)−1 =
∫
dεnb(ε)
∫ +1
−1
dµ1− µββb
2σ(s) , (170)
where nb(ε) is the number density of the background particles per unit energy at energy ε, βb = (1−m2b/ε
2)1/2 and
β = (1−m2/E2)1/2 are the velocities of the background particle and the CR, respectively, µ is the cosine of the angle
41
between the incoming momenta, and σ(s) is the total cross section of the relevant process for the squared center ofmass (CM) energy
s = m2b +m2 + 2εE (1− µββb) . (171)
The most important background particles turn out to be photons with energies in the infrared and optical (IR/O)range or below, so that we will usually have mb = 0, βb = 1. A review of the universal photon background has beengiven in Ref. [20].It proves convenient to also introduce an energy attenuation length lE(E) that is obtained from Eq. (170) by mul-tiplying the integrand with the inelasticity, i.e. the fraction of the energy transferred from the incoming CR to therecoiling final state particle of interest. The inelasticity η(s) is given by
η(s) ≡ 1− 1
σ(s)
∫
dE′E′ dσ
dE′(E′, s) , (172)
where E′ is the energy of the recoiling particle considered in units of the incoming CR energy E. Here by recoilingparticle we usually mean the “leading” particle, i.e. the one which carries most of the energy.In the most general case, propagation is three-dimensional due to the effect of electromagnetic (EM) fields exerting theforce Eq. (155). If the spatial distribution is irrelevant, for example in a homogeneous situation or because deflectionis negligible, formally, the Boltzmann equations for the evolution of a set of species with densities per energy ni(E)are given by
∂tni(E) = −ni(E)
∫
dεnb(ε)
∫ +1
−1
dµ1− µβbβi
2
∑
j
σi→j
[
s = m2b +m2 + 2εE(1− µβbβi)
]
(173)
+
∫
dE′
∫
dεnb(ε)
∫ +1
−1
dµ∑
j
1− µβbβ′j
2nj(E
′)dσj→i
dE
[
s = m2b +m2 + 2εE′(1− µβbβj), E
]
+Φi
for an isotropic background distribution (here assumed to be only one species) with our notation of Eqs. (170)and (171) extended to several species.If one is mostly interested in this leading particle, the detailed transport equations for the local density of particlesper unit energy, n(E), are often approximated by the simple “diffusion equation”
∂tn(E) = −∂E [b(E)n(E)] + Φ(E) (174)
in terms of the energy loss rate b(E) = E/lE(E) and the local injection spectrum Φ(E) per energy E, volume and timet. Eq. (174) applies to a particle which loses energy at a rate dE/dt = b(E), and is often referred to as the continuousenergy loss (CEL) approximation. The CEL approximation is in general good if the non-leading particle is of a differentnature than the leading particle, and if the inelasticity is small, η(s) ≪ 1. For an isotropic source distribution withinjection rate density Φ(E, z) per physical volume in the cosmology described by Eq. (123), integrating Eq. (174) overtime and using dt = dz/[(1 + z)H(z)] yields a differential flux today at energy E, j(E), as
j(E) =1
4πH0
∫ zi,max
0
dzi
(1 + zi)4 [Ωm(1 + z)3 +Ωr(1 + z)4 +Ωk(1 + z)2 +Ωv]1/2
dEi(E, zi)
dEΦ [Ei(E, zi), zi] , (175)
where Ei(E, zi) is the energy at injection redshift zi in the CEL approximation, i.e. the solution of dE/dt = b(E),with b(E) including loss due to redshifting and the initial condition Ei(E, z = 0) = E. The maximum redshift zi,max
corresponds either to an absolute cutoff of the source spectrum at Emax = Ei(E, zi,max) or to the earliest epoch whenthe source became active, whichever is smaller. For a homogeneous production spectrum Φ(E), this simplifies to
j(E) ≃ 1
4πlE(E)Φ(E) , (176)
if lE(E) is much smaller than the horizon size such that redshift and evolution effects can be ignored. Eqs. (175)and (176) are often used in the literature for approximate flux calculations.If the gyro-radius of a charged CR is much smaller than the distance scale over which propagation is considered andif energy loss processes can be treated in the CEL approximation, one can formulate the problem as Eq. (174) withan additional, in general location and energy dependent diffusion coefficient D(r, E):
∂tn(r, E) = −∂E [b(E)n(r, E)] +∇ [D(r, E)∇n(r, E)] + Φ(E) . (177)
If D(r, E) is independent of r, an analytical solution of this “energy loss-diffusion equation” given by Syrovatskii [21]can be employed.
42
1. Galactic Cosmic and γ−rays
CR above ∼ 1GeV have to be of extra-solar origin. The average energy density of CR is thus expected to be uniformat least throughout most of the Galaxy. If CR are universal, their density should be constant throughout the wholeUniverse. As a curiosity we note in this context that the mean energy density of CR, uCR, is comparable to the energydensity of the CMB. It is not clear, however, what physical process could lead to such an “equilibration”, which isthus most likely just a coincidence. A universal origin of the bulk of the CR is now-a-days not regarded as a likelypossibility.If the CR accelerators are Galactic, they must replenish for the escape of CR from the Galaxy in order to sustainthe observed Galactic CR differential intensity j(E). Their total luminosity in CR must therefore satisfy LCR =(4π/c)
∫
dEdV tCR(E)−1Ej(E), where tCR(E) is the mean residence time of CR with energy E in the Galaxy and Vis the volume. tCR(E) can be estimated from the mean column density, X(E), of gas in the interstellar medium thatGalactic CR with energy E have traversed. Interaction of the primary CR particles with the gas in the interstellarmedium leads to production of various secondary species. From the secondary to primary abundance ratios of GalacticCR it was infered that [22]
X(E) = ρgtCR(E) ≃ 6.9
(
E
20Z GeV
)−0.6
g cm−2 , (178)
where ρg is the mean density of interstellar gas and Z is the mean charge number of the CR particles. The mean energydensity of CR and the total mass of gas in the Milky Way that have been inferred from the diffuse Galactic γ−ray,X-ray and radio emissions are uCR = (4π/c)
∫
dEEj(E) ≃ 1 eV cm−3 and Mg ∼ ρgV ∼ 4.8 × 109M⊙, respectively.Hence, simple integration yields
LCR ∼Mg
∫
dEEj(E)
X(E)∼ 1.5× 1041 erg sec−1 . (179)
This is about 10% of the estimated total power output in the form of kinetic energy of the ejected material in Galacticsupernovae which, from the energetics point of view, could therefore account for most of the CR. We note that theenergy release from other Galactic sources, e.g. ordinary stars or isolated neutron stars [19] is expected to be toosmall, even for UHECR. Together with other considerations (see below) this leads to the widely held notion that CRat least up to the knee predominantly originate from first-order Fermi acceleration in SNRs.Another interesting observation is that the energy density in the form of CR is comparable both to the energy densityin the Galactic magnetic field (∼ 10−6 G) as well as that in the turbulent motion of the gas,
uCR ∼ B2
8π∼ 1
2ρgv
2t , (180)
where ρg and vt are the density and turbulent velocity of the gas, respectively. This can be expected from a pressureequilibrium between the (relativistic) CR, the magnetic field, and the gas flow. If Eq. (180) roughly holds not onlyin the Galaxy but also throughout extragalactic space, then we would expect the extragalactic CR energy density tobe considerably smaller than the Galactic one which is another argument in favor of a mostly Galactic origin of theCR observed near Earth (see Sect.3.2). We note, however, that, in order for Eq. (180) to hold, typical CR diffusiontime-scale over the size of the system under consideration must be smaller than its age. This is not the case, forexample, in clusters of galaxies if the bulk of CR are produced in the member galaxies or in cluster accretion shocks.Supernova remnants are now routinely observed in TeV γ−rays, see, for example, Fig. 15. If such supernova remnantsare hadronic accelerators, then the γ−rays are produced by pp→ppπ0 → γγ with cross section
σpp(s) ∼[
35.49 + 0.307 ln2(s/28.94GeV2)]
mb . (181)
This cross section is almost constant so that secondary spectra have roughly the same shape as primary fluxes as longas meson cooling time is much larger than the decay time.It is even likely that diffusion of cosmic rays away from a one-time source close to the galactic center has been observedindirectly via secondary γ−rays, as presented in Fig. 16.Fig. 17 shows that the typical γ−ray spectra from galactic sources are ∝ E−2.2, indicating a similar primary CRspectrum. This is consistent with the spectra expected from non-relativistic shock acceleration, as discussed inSect. IIIA.
43
FIG. 15: The supernova remnant RXJ713-3946 in γ−rays above 800 GeV from Ref. [23].
2. Extragalactic Cosmic Rays: The GZK effect
The threshold for the reaction Nγ → Nπ, for a head-on collision of a nucleon N of energy E with a photon of energyε is given by
E ≥ mπ(mN +mπ/2)
2ε≃ 3.4× 1019
( ε
10−3 eV
)−1
eV , (182)
where ε ∼ 10−3 eV represents the energy of a typical CMB photon.
C. γ−ray astrophysics: The Principal Electromagnetic Processes
1. Synchrotron Radiation
A charged particle gyrating in a magnetic field, as considered in Sect. IIIA, produces a power spectrum of synchrotronphotons given by
dPsyn
dEγ=
√3
2π
eZB⊥
mG(Eγ/Ec) , (183)
where G(x) ≡ x∫
xdξK5/3(ξ) with K5/3 the modified Bessel function of the third kind, m the particle mass and
Ec =3eZB⊥E
2
2m3. (184)
44
FIG. 16: HESS has observed γ−-rays from objects around the galactic center which correlate well with the gas density inmolecular clouds for a cosmic ray diffusion time of T ∼ R2/D ∼ 3 × 103(θ/1)2/ξ years where θ is the apparent angulardistance from the source and D = ξ1030 cm2/s is the diffusion coefficient for protons of a few TeV. From Ref. [24].
The emission Eq. (183) peaks at
Esyn ≃ 0.23Ec ≃ 1.53× 104Z
(
B⊥
µG
)(
E
1015 eV
)2(me
m
)3
eV , (185)
where me is the electron mass. The total power emitted is obtained by integrating Eq. (183),
dE
dt
∣
∣
∣
∣
syn
= −Psyn = −4
3σTuB
(
Zme
m
)4(p
me
)2
= − 2
9πuB
(
Ze
m
)4
p2 , (186)
where σT ≡ e4/(6πm2e) is the Thomson cross section and uB ≡ B2/(8π) is the magnetic energy density. The resulting
synchrotron cooling time is
tsynch =Ee
dEe/dt=
6πm2e
σTEeB2≃ 3.84 kpc
(
Ee1015 eV
)−1(B
µG
)−2
. (187)
2. Inverse Compton Scattering (ICS)
The total cross section for ICS is
σICS =3
8σT
m2e
sβ
[
2
β(1 + β)
(
2 + 2β − β2 − 2β3)
− 1
β2
(
2− 3β2 − β3)
ln1 + β
1− β
]
, (188)
where s is the relativistic squared center of mass (c.m.) energy and β ≡ (s − m2e)/(s + m2
e) is the velocity of theoutgoing electron in the c.m. In the Klein-Nishina limit, s ≫ m2
e, most of the electron energy goes to the ”leading”photon, whereas in the Thomson regime, the average energy lost by the electron per interaction is δEe ∼ 4ǫE2
e/3m2e,
where ǫ is the background photon energy.The total inverse Compton power per electron is
dE
dt
∣
∣
∣
∣
ICS
= −PICS = −4
3σTuγ
(
Zme
m
)4(p
me
)2
= − 2
9πuγ
(
Ze
m
)4
p2 , (189)
where uγ is the energy of the background photon field. This is of exactly the same form as Eq. (186), with uBsubstituted by uγ . It shows that ICS is fundamentally the same process as synchrotron radiation, except that in thelatter case, the interaction takes place with the virtual photons of the magnetic field.Fig. 18 shows the typical double-peaked power spectrum of a typical AGN. In leptonic models, the high energy peak iscaused by ICS of accelerated electrons on the ambient photon field, whereas the low energy peak is due to synchrotronemission of these same electrons. Eqs. (186) and (189) show that Psyn/PICS = uB/uγ and thus if the magnetic fieldis known, the radiation energy density can be inferred, or if the latter is known, one can determine the magnetic fieldstrength.
45
FIG. 17: Spectra of some galactic γ−ray sources.
3. Pair Production (PP)
The total cross section for pair production is
σPP =3
16σT (1− β2)
[
(
3− β4)
ln1 + β
1− β− 2β
(
2− β2)
]
, (190)
where in this case the velocity of the outgoing electron in the c.m. is β = (1− 4m2e/s)
1/2.The γ−ray threshold energy for PP on a background photon of energy ε is
Eth =m2e
ε≃ 2.6× 1011
( ε
eV
)−1
eV , (191)
whereas ICS has no threshold. In the high energy limit, the total cross sections for ICS, Eq. (188) and PP (190) are
σPP ≃ 2σICS ≃ 3
2σT
m2e
sln
s
2m2e
(s≫ m2e) . (192)
46
109
1011
1013
1015
1017
1019
1021
1023
1025
ν [Hz]
1010
1011
1012
1013
1014
νFν
[Jy
Hz]
3C279
P1 (June 1991 flare)P2 (Dec. 92 / Jan. 93)June 2003Jan. 15, 2006
FIG. 18: The spectrum of a typical AGN.
For s ≪ m2e, σICS approaches the Thomson cross section σT , whereas σPP peaks near the threshold Eq. (191).
Therefore, the most efficient targets for electrons and γ−rays of energy E are background photons of energy ε ≃ m2e/E.
For the propagation of photons of energy UHE E >∼ 1018 eV, as they are produced, for example, by the GZK effect,this corresponds to ε <∼ 10−6 eV ≃ 1GHz. Thus, radio background photons play an important role in UHE γ-raypropagation through extragalactic space.The cosmic infrared (IR) background is produced by the combined emission of galaxies. Its photons have energiesaround 1 eV and thus AGN spectra will start to be suppressed above ∼ 100GeV...An overview over the diffuse extragalactic photon background is given in Fig. 11.
4. Higher Order Processes
The lowest order cross sections, Eq. (192), fall off as ln s/s for s ≫ m2e. Therefore, at EHE, higher order processes
with more than two final state particles start to become important because the mass scales of these particles can enterinto the corresponding cross section which typically is asymptotically constant or proportional to powers of ln s.Double pair production (DPP), γγb → e+e−e+e−, is a higher order QED process that affects UHE photons. TheDPP total cross section is a sharply rising function of s near the threshold that is given by Eq. (191) with me → 2me,and quickly approaches its asymptotic value [25]
σDPP ≃ 172α4
36πm2e
≃ 6.45µbarn (s≫ m2e) . (193)
47
For extragalactic propagation DPP begins to dominate over PP above ∼ 1021−1023 eV, where the higher values applyfor stronger radio background.For electrons, the relevant higher order process is triplet pair production (TPP), eγb → ee+e−. This process has beendiscussed in some detail in Refs. [26] and its asymptotic high energy cross section is
σTPP ≃ 3α
8πσT
(
28
9ln
s
m2e
− 218
27
)
(s≫ m2e) , (194)
with an inelasticity of
η ≃ 1.768
(
s
m2e
)−3/4
(s≫ m2e) . (195)
Thus, although the total cross section for TPP on CMB photons becomes comparable to the ICS cross section alreadyaround 1017 eV, the energy attenuation is not important up to ∼ 1022 eV because η <∼ 10−3. The main effect of TPPbetween these energies is to create a considerable number of electrons and channel them to energies below the UHErange. However, TPP is dominated over by synchrotron cooling, and therefore negligible, if the electrons propagatein a magnetic field of r.m.s. strength >∼ 10−12 G.
5. Electromagnetic Cascades
FIG. 19: Effective penetration depth of EM cascades, as defined in the text, for a strong URB estimate (solid lines), and a lowURB estimate (dashed lines), and for an EGMF ≪ 10−11 G (thick lines), and 10−9 G (thin lines), respectively.
In the extreme Klein-Nishina limit, s≫ m2e, either the electron or the positron produced in the process γγb → e+e−
carries most of the energy of the initial UHE photon. This leading electron can then undergo ICS whose inelasticity(relative to the electron) is close to 1 in the Klein-Nishina limit. As a consequence, the upscattered photon which
48
is now the leading particle after this two-step cycle still carries most of the energy of the original γ−ray, and caninitiate a fresh cycle of PP and ICS interactions. This leads to the development of an electromagnetic (EM) cascadewhich plays an important role in the resulting observable γ−ray spectra. An important consequence of the EMcascade development is that the effective penetration depth of the EM cascade, which can be characterized by theenergy attenuation length of the leading particle (photon or electron/positron), is considerably greater than just theinteraction lengths. As a result, EM cascade fluxes can be considerably larger than that calculated by consideringonly the absorption of photons due to PP. For the case of cascades propagating in extragalactic case this is shown inFig. 19.Cascade development accelerates at lower energies due to the increasing cross sections until most of the γ−rays fallbelow the PP threshold on the low energy photon background at which point they pile up with a characteristic E−1.5
spectrum below this threshold [19, 27–29]. The source of these γ−rays are predominantly the ICS photons of averageenergy 〈Eγ〉 ∼ 4εE2
e/3m2e arising from interactions of electrons of energy Ee with the background photons of energy ε
in the Thomson regime. The relevant background for cosmological propagation is constituted by the universal IR/Obackground, corresponding to ε <∼ 1 eV in Eq. (191), or Eth ≃ 1011 eV. Therefore, most of the energy of fully developedEM cascades ends up below ≃ 100GeV where it is constrained by measurements of the diffuse γ−ray flux by EGRETon board the CGRO [30, 31], as well as by effects on nucleosynthesis and the CMB. Flux predictions involving EMcascades are therefore an important source of constraints of UHE energy injection on cosmological scales.The development of EM cascades depends sensitively on the strength of the extragalactic magnetic fields (EGMFs)which is rather uncertain. The EGMF typically inhibits cascade development because of the synchrotron coolingof the e+e− pairs produced in the PP process. For a sufficiently strong EGMF the synchrotron cooling time scaleof the leading electron (positron) may be small compared to the time scale of ICS interaction, in which case, theelectron (positron) synchrotron cools before it can undergo ICS, and thus cascade development stops. In this case,the UHE γ-ray flux is determined mainly by the “direct” γ-rays, i.e., the ones that originate at distances less thanthe absorption length due to PP process. The energy lost through synchrotron cooling does not, however, disappear;rather, it reappears at lower energies and can even initiate fresh EM cascades there depending on the remaining pathlength and the strength of the relevant background photons. Thus, the overall effect of a relatively strong EGMF isto deplete the UHE γ-ray flux above some energy and increase the flux below a corresponding energy in the “low”(typically few tens to hundreds of GeV) energy region.
D. Indirect Dark Matter Detection
Dark matter can not only be detected directly in dedicated experiments searching for nuclear recoils from the scat-tering of dark matter particles, or produced in particle accelerators such as the LHC, but can also reveal its existenceindirectly: Although, apart from dilution from cosmic expansion, the density of dark matter does not change sig-nificantly after having been ”frozen out” in the early Universe, the very self-annihilation playing a central role inthis freeze-out described around Eq. (148) can give rise to significant fluxes of γ−rays, neutrinos, and even someantimatter such as anti-protons and positrons, especially in regions with large dark matter densities. The energies ofthe secondary particles can reach up to the dark matter particle mass which typically would be a few hundred GeV.The positrons can annihilate with the electrons of the interstellar plasma and give rise to a 511 keV line emission.In fact, such a line emission has already been observed by INTEGRAL [32] but it is currently unclear what themain production mechanism of the relevant positrons is. Secondary electrons and positrons can even give rise tosynchrotron radiation in the galactic magnetic field that can be detected in the radio band. Therefore, cosmic andγ−ray detectors, neutrino telescopes, and even radio telescopes can be used for indirect dark matter detection as well.Since dark matter annihilation depends on the square of the dark matter density, indirect detection is even moresensitive to cosmological and astrophysical processes than direct detection. Such processes include the formation ofcusps around the central super-massive black holes in our own as well as other galaxies and of dark matter ”clumps”that could condense, for example around intermediate mass black holes. These processes constitute one of the mainuncertainties in the predictions of indirect signals.Just as for ordinary cosmic rays, the fluxes and spectra of dark matter annihilation and decay products will bemodified during propagation to Earth. Charged particles at GeV energies essentially diffuse in the galactic magneticfield, whereas photons and neutrinos will propagate on straight lines and can therefore reveal the structure of the darkmatter distribution. Eventually, this could lead to an experimental test of theoretical modeling of the Dark Mattersmall scale structure involving cusps and clumps.The observed flux of neutral secondaries can be written as
Φi(ψ,E) = σvdNidE
1
4πm2X
∫
line of sightd sρ2 (r(s, ψ)) (196)
49
where the index i denotes the secondary particle observed (we focus on γ−rays and neutrinos) and the coordinate sruns along the line of sight, in a direction denoted by ψ. σXv is the same annihilation cross section as in Eqs. (148)and (149), dNi/dE is the spectrum of secondary particles per annihilation and mX is again the dark matter particlemass.In order to separate the factors depending on the dark matter profile from those depending only on particle physics,we introduce, following [33], the quantity J(ψ)
J (ψ) =1
8.5 kpc
(
1
0.3GeV/cm3
)2 ∫
line of sightd sρ2 (r(s, ψ)) . (197)
We define J(ψ,∆Ω) as the average of J(ψ′) over a spherical region of solid angle ∆Ω, centered around ψ. Thesevalues depend strongly on the dark matter profile.We can then express the flux from a solid angle ∆Ω as
Φi(∆Ω, E) ≃ 5.6× 10−12 dNidE
(
σvpb
)
(
1TeVM
)2J (∆Ω)
× ∆Ωcm−2s−1 . (198)
If instead of annihilation dark matter dominantly decays with lifetime τX and a spectrum dNi/dE of secondaryparticles per decay, the flux observed at Earth Eq, (196) would instead read
Φi(ψ,E) =dNidE
1
4πτXmX
∫
line of sightd sρ (r(s, ψ)) (199)
Energy (TeV)1 10
)-1
s-2
dN
/dE
(T
eV c
m× 2
E
-1310
-1210
-1110
2004 (H.E.S.S.)2003 (H.E.S.S.)MSSMKK
-τ+τ, 30% b70% b
FIG. 20: Spectral energy density E2× dN/dE of γ−rays from the galactic centre source seen by H.E.S.S. (points). Upper
limits are 95% CL. The shaded area shows the power-law fit dN/dE ∼ E−Γ. The dashed line illustrates typical spectra ofphenomenological minimal supersymmetric standard model annihilation for best fit neutralino masses of 14 TeV. The dottedline shows the distribution predicted for Kaluza-Klein dark matter with a mass of 5 TeV. The solid line gives the spectrum ofa 10 TeV dark matter particle annihilating into τ+τ− (30%) and bb (70%). Adapted from Ref. [34].
Indirect detection of WIMPs requires disentangling the fluxes of Dark Matter annihilation products from conven-tional astrophysical contributions such as interaction of cosmic rays with interstellar and inter-galactic gas. Severalobservations have already been suggested as possible Dark Matter annihilation signatures. These include the H.E.S.S.observations of TeV γ−rays [34] from the galactic centre, see Fig. 20, the INTEGRAL observation of a 511 keV γ−rayline also from the galactic centre [32], see Fig. 21, and a GeV gamma-ray excess emission from our Galaxy observedby the EGRET detector onboard the Compton gamma-ray Observatory, which appears difficult to interpret in termsof conventional cosmic ray interactions [35], see Fig. 22. However, with more recent data, these three signatures seemto be more compatible with normal astrophysical processes: The TeV γ−rays seen by H.E.S.S. extend to > 30TeV
50
0 10-2
1 10-2
2 10-2
3 10-2
4 10-2
490 495 500 505 510 515 520 525 530
observationbackground model: continuumbackground model: linebackground model: total
rate
(c/
s/de
t/keV
)
E (keV)
-1 10-3
0 10-3
1 10-3
2 10-3
3 10-3
4 10-3
-30-25-20-15-10-50
observationmodel
rate
(c/
s/de
t)
longitude (degrees)
FIG. 21: The 511 keV line seen by INTGRAL/SPI (left panel) and the distribution of its flux around the galactic centre (rightpanel), from Ref. [32].
10-5
10-4
10-1
1 10 102
E [GeV]
E2 *
flux
[GeV
cm
-2 s
-1sr
-1]
EGRETbackgroundsignal
mWIMP=50-70 GeV
Dark MatterPion decayInverse ComptonBremsstrahlung
energy, MeV 1 10 210 310 410 510 610
MeV
-1
s-1
sr
-2. i
nte
nsi
ty, c
m2
E
-410
-310
-210
-110 galdef ID 44_500190
0.25<l<29.75 , 330.25<l<359.75 -4.75<b<-0.25 , 0.25<b< 4.75
IC
bremssπο
total
EB
FIG. 22: Left panel: The GeV γ−ray ”excess” from our Galaxy as seen by EGRET. The background is from cosmic rayinteractions for a standard propagation model with constant parameters throughout the Galaxy. In blue is the contributionfrom WIMP annihilation, far a range of WIMP masses. Adapted from Ref. [35]. Right panel: The EGRET excess can be fittedby an ”optimized” cosmic ray propagation model [36].
which would require an unnaturally heavy dark matter primary and the spectrum seems compatible with comingfrom an accelerated primary cosmic ray component [37]. The INTEGRAL 511 keV flux distribution seems not to bespherically symmetric as expected if its origin were in a dark matter halo, but seems to correlate with the Galacticbulge instead [38]. Finally, there are rumors that the FERMI/GLAST satellite which has begun releasing results endof 2008 and is the successor of the old EGRET experiment does not see the excess claimed by this experiment.On the other hand, the PAMELA satellite has recently strengthened the case for a positron fraction in the electronplus positron flux that is increasing between ∼ 10GeV and ∼ 100GeV [39], see Fig. 23 left panel. This is naively notexpected if positrons are produced as secondaries of primary cosmic rays diffusing in the Galaxy, simply because thecolumn depth seen by primary cosmic rays and thus their interaction probability decreases with increasing energy.However, a positron fraction increasing with energy can also be explained by nearby astrophysical sources such aspulsars. Interpretation in terms of annihilating or decaying dark matter needs “leptophilic” models with end products
51
FIG. 23: This figure is from Ref. [42]. Left panel: The GeV positron ”excess” of Galactic positrons as seen by the PAMELA(filled circles) and other experiments. Right panel: The electron plus positron flux as observed by the FERMI LAT experiment(filled circles), H.E.S.S. (downward pointing triangles), ATIC (filled squares) and other experiments. In both panels, thedashed lines show the standard cosmic ray contributions for homogeneous source distributions. The dotted and solid lines showcontributions from dark matter of mass mX = 600Gen and mX = 3000GeV, respectively, decaying into W±µ∓ with a lifetimefir to the data. The best fit for mX = 3000GeV corresponds to τX = 2.1× 1026 s.
dominated by leptons because PAMELA does not seem to be any significant enhancement of the anti-proton fluxbeyond what is expected from interactions of cosmic rays with the gas in our Galaxy [40]. A particular problem ofa dark matter self-annihilation origin of the positron excess is that it requires an annihilation cross section a factor100−−1000 higher than the “natural” cross section Eq. (149) required for dark matter produced by thermal freeze-out. A remedy of this problem requires either an enhancement of the annihilation cross section at the small velocitiesrelevant for Galactic dark matter, relative to the cross section at semi-relativistic velocities relevant at freeze-out, ascan occur in a so-called “Sommerfeld enhancement” due to resonances, or a strong “boost factor” due to small-scaledark matter clumps with large over-densities. This problem does not occur for decaying dark matter since the darkmatter lifetime is not fixed by the freeze-out process.Finally, the LAT experiment onboard the FERMI/GLAST satellite has also observed an excess in the electron pluspositron flux beyond what would be expected from primary cosmic rays with a homogeneous source distributionbetween ∼ 100GeV and ∼ 1000GeV [41], see Fig. 23 right panel. This is consistent with the flux measured by theH.E.S.S. above 340 GeV which starts to steepen above ∼ 1TeV [43].In order to sort out any dark matter induced contributions to cosmic ray, anti-matter, γ−ray and also neutrino fluxes,a truly multi-messenger approach is, therefore, called for that includes a detailed understanding of acceleration andpropagation of cosmic rays, and the production of secondary γ−rays and neutrinos.
IV. NEUTRINO ASTROPHYSICS
Good book presentations of neutrino physics and astrophysics are given by Refs. [13, 44]. The role of neutrinos andother particles in stellar evolution is extensively presented in Ref. [45].
A. Neutrino Scattering
Imagine a neutrino of energy Eν scattering on a parton i carrying a fraction x of the 4-momentum P of a state X ofmass M . Denoting the fractional recoil energy of X by y ≡ E′
X/Eν and the distribution of parton type i by fi(x,Q),in the relativistic limit Eν ≫ mX the contribution to the νX cross section turns out to be
dσνX
dxdy=
2G2FMEνx
π
(
M2W,Z
2MEνxy +M2W,Z
)2∑
i
fi(x,Q)[
g2i,L + g2i,R(1− y)2]
. (200)
Here, gi,L and gi,R are the left- and right-chiral couplings of parton i, respectively, given by Eq. (102). Eq. (200)applies to both charged and neutral currents, as well as to the case where X represents an elementary particle suchas the electron, in which case fi(x,Q) = δ(x− 1).
52
FIG. 24: From Ref. [46]. Cross sections for νℓN interactions at high energies, according to the CTEQ4–DIS parton distributions:dashed line, σ(νℓN → νℓ+anything); thin line, σ(νℓN → ℓ−+anything); thick line, total (charged-current plus neutral-current)cross section.
As usual, if the four-momentum transfer Q becomes comparable to the electroweak scale, Q2 ≫ M2W,Z , the weak
gauge boson propagator effects, represented by the factor M2W,Z/(Q
2 +M2W,Z) in Eq. (200), become important. We
have used that in the limit |Q2| ≫M2 one has 0 ≃ −M2 = (xP +Q)2 ≃ Q2+2P ·Qx with P ·Q ≃ −ME′X = −MEνy
evaluated in the laboratory frame, i.e. the rest frame of X before the interaction. Q2 ≃ 2MEνxy is also called thevirtuality because it is a measure for how far the exchanged gauge boson is form the mass shell Q2 = −M2
W,Z .
We will not derive Eq. (200) in detail, but it is easy to understand its structure: First, the overall normalization isanalogous to Eq. (13), using the fact that for Eν ≫ M the CM momentum p2∗ ≃ MEν/2. Second, if the helicities ofthe parton and the neutrino are equal, the total spin is zero and the scattering is spherically symmetric in the CMframe. In contrast, if the parton is right-handed, the total spin is 1 which introduces an angular dependence: After arotation by the scattering angle θ∗ in the CM frame the particle helicities are unchanged for the outgoing final stateparticle and one has to project back onto the original helicities in order to conserve spin. If a left-handed particleoriginally propagated along the positive z-axis, its left-handed component after scattering by θ∗ in the x− z plane is
1
2
(
1− σ · pp
)(
01
)
=1
2
(
− sin θ∗1 + cos θ∗
)
, (201)
giving a projection [(1+cos θ∗)/2]2. Now, Lorentz transformation from the CM frame to the lab frame gives E′
ν/Eν =(1 + vX cos θ∗)/2 ≃ (1 + cos θ∗)/2 in the relativistic limit and thus the projection factor equals (E′
ν/Eν)2 = (1 −
E′X/Eν)
2 = (1 − y)2, as in Eq. (200) for the right-handed parton contribution. Integrated over 0 ≤ y ≤ 1 this gives1/3, corresponding to the fact that only one of the three projections of the J = 1 state contributes.We now briefly consider neutrino-nucleon interaction. From Eq. (200) it is obvious that at ultra-high energies 2EνM ≫M2W,Z , the dominant contribution comes from partons with
x ∼M2W,Z
2EνM. (202)
53
FIG. 25: Estimated fluxes and evolution of instrumental sensitivity for neutrinos, compared to fluxes of charged cosmic rays andγ−rays. Primary cosmic ray fluxes (data and a model fit) are shown in black, the secondary γ-ray flux expected from protoninteractions with the Cosmic Microwave Background (p+CMB→ γ) in red and the guaranteed neutrino fluxes per neutrinospecies in blue: atmospheric ν, galactic ν resulting from cosmic ray interactions with matter in our Galaxy, and GZK neutrinosresulting from cosmic ray interaction with the Cosmic Microwave Background, p+CMB→ ν. These secondary fluxes dependto some extent on the distribution of the (unknown) primary cosmic ray sources for which active galaxies were assumed above1018 eV. Cosmic ray interactions within these sources can also produce neutrinos whose fluxes depend on models for which oneexample is given (AGN ν). The flux of atmospheric neutrinos has been measured by underground detectors and AMANDA.Also shown are existing upper limits and future sensitivities to diffuse neutrino fluxes from various experiments (dashed anddotted light blue lines, respectively), assuming the Standard Model neutrino-nucleon cross section extrapolated to the relevantenergies. The maximum possible neutrino flux is given by horizontally extrapolating the diffuse γ−-ray background observedby EGRET [30, 31].
Since, very roughly, xfi(x,Q) ∝ x−0.3 for x ≪ 1, it follows that the neutrino-nucleon cross section grows roughly∝ E0.3
ν . This is confirmed by a more detailed evaluation of Eq. (200) shown in Fig. 24.
1. Some High Energy Astrophysical Neutrino Sources
Let us use this to do a very rough estimate of event rates expected for extraterrestrial UHE neutrinos in neutrinotelescopes. Such neutrinos are usually produced via pion production by accelerated UHE protons interacting withintheir source or with the cosmic microwave background (CMB) during propagation to Earth. At GZK energiesEq. (182), the secondary neutrino flux should be very roughly comparable with the primary UHE cosmic ray flux,within large margins. Fig. 25 shows a scenario where neutrinos are produced by the primary cosmic ray interactionswith the CMB. Using that the neutrino-nucleon cross section from Fig. 24 roughly scales as σνN ∝ E0.363
ν for1016 eV <∼ Eν <∼ 1021 eV, and assuming water or ice as detector medium, we obtain the rate
Γν ∼ σνN (Eν)2πEνj(Eν)nNVeff
54
∼ 0.03
(
Eν1019 eV
)−0.637(E2νj(Eν)
102 eVcm−2sr−1s−1
)(
Veff
km3
)
yr−1 , (203)
where nN ≃ 6 × 1023 cm−3 is the nucleon density in water/ice, Veff the effective detection volume, and j(Eν) is thedifferential neutrino flux in units of cm−2eV−1sr−1s−1.Eq. (203) indicates that at Eν >∼ 1018 eV, effective volumes >∼ 100 km3 are necessary. Although impractical for con-ventional neutrino telescopes, big air shower arrays such as the Pierre Auger experiment can achieve this. In contrast,if there are sources such as active galactic nuclei emitting at Eν ∼ 1016 eV at a level E2
νj(Eν) ∼ 102 eVcm−2sr−1s−1,km-scale neutrino telescopes should detect something.If AGNs are hadronic accelerators, one can make a rough estimate of their neutrino flux as follows:
• Size of accelerator R ∼ ΓT , where jet boost factor Γ ∼ 10 and duration of observed bursts T ∼ 1 day.
• Magnetic field strength in jet B2 ∼ ρelectron ∼ 1 erg cm−3 (equipartition).
• ”Hillas condition” on maximal proton energy Emax ∼ eBR and from pγ → Nπ kinematics Emax,ν ∼ 0.1Emax ∼1018 eV.
• Neutrino luminosity related to γ−ray luminosity by Lν ≃ 313Lγ from pγ → Nπ kinematics.
• Assume proton spectrum dNp/dEp ∝ E−2−εp and γ−ray spectrum dNγ/dεγ ∝ ε−2−α
γ . If jet is optically thinagainst pγ then
dNνdEν
∝ dNpdEp
(10Eν)
∫ ∞
εthrγ
dεγdNγdεγ
∝ E−2−εν
(
εthrγ
)−1−α ∝ E−1−ε+αν ,
since the pion production threshold εthrγ ∝ E−1p ∝ E−1
ν .
• Combine with normalization:
dNνdEν
≃ 3
13
LγEmax,ν
1− ε+ α
Eν
(
EνEmax,ν
)−ε+α
.
• Fold with luminosity function of AGNs in GeV γ−rays.
The diffuse fluxes resulting from the GZK effect and from hadronic AGN models, among others, are shown in Figs. 12and 25.
B. Dirac and Majorana Neutrinos
Up to now we have assumed that neutrinos and anti-neutrinos are separate entities. This is true if lepton number isconserved, see Tab. I, and corresponds to pure ”Dirac neutrinos”. However, lepton number may be violated in theneutrino sector and neutrinos may be indistinguishable from anti-neutrinos. In order to elucidate this, let us firststudy some symmetries of the Dirac equation (37). From Eqs. (33), (35) one can easily show that
γ∗µ = γ2γµγ2 . (204)
Complex conjugating the Dirac equation (37) and multiplying it with ξ∗γ2 from the left, it then follows that it isinvariant under the ”charge conjugation transformation”
Cψ(x)C−1 ≡ ψc ≡ ξ∗γ2ψ∗ , (205)
where ξ is an arbitrary complex number with |ξ| = 1. This transformation exchanges particles and anti-particlesand satisfies (ψc)c = ψ. Note that γ2 appears because according to Eq. (204) it is the only real Dirac matrix in thisconvention.A Majorana neutrino satisfies the reality condition
φ(x) = γ2φ∗(x) , or φc = ξ∗φ . (206)
55
These spinors are of the form
φ =
(
−iσ2χ∗
χ
)
, (207)
where χ is a two-spinor. This implies that for any Dirac spinor ψ one can construct a Majorana spinor by
φ ≡ ψ + ξψc . (208)
Note that this spinor is not an eigenstate of lepton number because under a phase transformation ψ → ψeiα one hasψc → ψce−iα. Defining
ψcL,R ≡ (ψc)L,R =1± γ5
2ψc =
(
1∓ γ52
ψ
)c
, (209)
one can define Majorana fields
φ∓ ≡ ψL,R + (ψL,R)c = ψL,R + ψcR,L . (210)
Note that both these fields now contain both left and right-handed fields. What before experimentally was calledneutrino and anti-neutrino now is called left- and right-handed neutrino, respectively. We can now introduce Majoranamass terms of the form
LM = −1
2
(
mLφ−φ− +mRφ+φ+)
(211)
= −1
2mL
(
ψLψcR + ψcRψL
)
− 1
2mR
(
ψRψcL + ψcLψR
)
,
where mL and mR are real. Together with the Dirac mass term this can be written as
LM + LD = −1
2
(
ψL, ψcL)
(
mL mD
mD mR
)(
ψcRψR
)
+ h.c. , (212)
where we have used (ψ1ψ2)† = ψ2ψ1, see around Tab. II and (using γ†2 = γ2 and the fact that ψ anti-commutes)
ψcLψcR = [(ψR)
c]†iγ0
1− γ52
ξ∗γ2ψ∗ = ψTRγ
†2iγ
0 1− γ52
γ2ψ∗
= −iψTR1− γ5
2γ0ψ∗ = i
[
ψTR1− γ5
2γ0ψ∗
]T
= ψLψR
for any Dirac spinors ψ1,2 and ψ. Note that under ψ → ψeiα Dirac terms are invariant, whereas Majorana termspick up the phase e±2iα, according to lepton number conservation and non-conservation, respectively. Furthermore,we see that for mR ≫ mD, mL ≃ 0, the two mass eigenvalues in Eq. (212) are ≃ mR and m2
D/mR. The latterare very small and thus may explain the sub-eV masses involved in left-chiral neutrino oscillations. This is calledthe see-saw mechanism which would imply that the mass eigenstates are Majorana in nature. The existence of oneheavy right-handed Majorana neutrino per lepton generation is motivated by Grand Unification extensions of theelectroweak gauge group to SO(10) which has 16-dimensional representations that could fit 15 Standard Model leptonand quark states plus one new state, see, e.g., Ref. [13].Finally, in the exactly massless case, Dirac and Majorana particles are exactly equivalent, since the two fields Eq. (210)completely decouple, see Eq. (212). We also mention that in supersymmetric extensions of the Standard Model, thefermionic super-partners of the gauge bosons are Majorana fermions. As a consequence, they can self-annihilate whichplays an important role to their being candidates for cold dark matter.For n > 1 neutrino flavors, mass eigenstates |νi〉 of mass mi and interaction eigenstates |να〉 in general are notidentical, but related by a unitary n× n matrix U :
|να〉 =∑
i
Uαi |νi〉 , (213)
where for anti-neutrinos U has to be replaced by U∗. Such a matrix in general has n2 real parameters. Subtracting2n − 1 relative phases of the n neutrinos in the two bases, one ends up with (n − 1)2 physically independent realparameters. Of these, n(n−1)/2 are mixing angles, and the remaining (n−1)(n−2)/2 are ”CP -violating phases”. In
56
order to have CP -violation in the Dirac neutrino sector thus requires n ≥ 3. Once the relative phases of the differentflavors have been fixed, for non-vanishing Majorana masses there will in general be n− 1 Majorana phases that cannot be projected out by ψ → ψeiα in Eq. (211). Thus, the number of independent real parameters is larger, namelyn(n− 1), in this case. Note that the CKM matrix discussed in Sect. I H is pure Dirac because Majorana terms wouldviolate electric charge conservation in the quark sector.The neutrino mixing matrix, also called Maki-Nakagawa-Sakata-Pontecorvo (MNSP) matrix, can be parametrized as
U =
(
1 0 00 c23 s230 −s23 c23
)(
c13 0 s13e−iδD
0 1 0−s13eiδD 0 c13
)(
c12 s12 0−s12 c12 00 0 1
)(
eiα1/2 0 00 eiα2/2 00 0 1
)
=
(
c12c13 s12c13 s13e−iδD
−s12c23 − c12s23s13eiδD c12c23 − s12s23s13e
iδD s23c13s12s23 − c12c23s13e
iδD −c12s23 − s12c23s13eiδD c23c13
)(
eiα1/2 0 00 eiα2/2 00 0 1
)
, (214)
where c12 ≡ cos θ12, s12 ≡ sin θ12, etc., δD is the Dirac CP-violating phase, and α1,2 are the two Majorana phases.If at time t = 0 a flavor eigenstate |να〉 =
∑
i Uαi |νi〉 is produced in an interaction, in vacuum the time developmentwill thus be
|ν(t)〉 =∑
i
Uαie−iEit |νi〉 =
∑
i,β
UαiU∗βie
−iEit |νβ〉 . (215)
Since masses and energies of anti-particles are equal according to the CPT theorem, from this we obtain the followingtransition probabilities
P (να → νβ) =
∣
∣
∣
∣
∣
∑
i
UαiU∗βi exp(−iEit)
∣
∣
∣
∣
∣
2
P (να → νβ) =
∣
∣
∣
∣
∣
∑
i
U∗αiUβi exp(−iEit)
∣
∣
∣
∣
∣
2
. (216)
From this follows immediately
P (να → νβ) = P (νβ → να) , (217)
which is due to the CPT theorem. Furthermore, if the mixing matrix satisfies a reality condition of the form
Uαi = U∗αiηi , (218)
with ηi phases, corresponding to CP -conservation, one also has
P (να → νβ) = P (να → νβ) . (219)
We mention two other important differences between Dirac and Majorana neutrinos:
• Neutrino-less double beta-decay is only possible in the presence of Majorana masses because the final state e−e−
violates lepton number. In this case the rate is proportional to the square of
mee =
∣
∣
∣
∣
∣
∑
i
|Uei|2mieiαi
∣
∣
∣
∣
∣
, (220)
see Eq. (212), where only one of the Majorana phases αi can be projected out. Apart from these phases, thisequation results from Eq. (215) for α = β = e. There is even evidence claimed for this kind of decay, and thusfor an electron neutrino Majorana mass around 0.4 eV, see Ref. [47]. The issue is expected to be settled by nextgeneration experiments such as CUORE.
In contrast, in β−decay with neutrinos, the electron spectra are influenced by the individual eigenstates of realmass mi, and not by any phases. The current best experimental upper limit is given by the Mainz experimentbased on tritium β−decay 3H →3 Hee−νe [48],
mνe =
√
∑
i
|Uei|2m2i<∼ 2.2 eV (221)
at 95% confidence level (CL). The KATRIN experiment aims at a sensitivity down to 0.2 eV within the nextfew years.
57
• Majorana neutrinos cannot have magnetic dipole moments between equal neutrino flavors, as seen from thefollowing identity using Eqs. (206), (204), (44), and the reality of the spinors in Tab. II:
iψ1[γµ, γν ]ψ2 = −ψT1 γ2γ0[γµ, γν ]γ2ψ∗2 = −ψT1
(
γ0[γµ, γν ])∗ψ∗2 =
= −(
ψ†1γ
0[γµ, γν ]ψ2
)∗
=(
ψ†1γ
0[γµ, γν ]ψ2
)†
= −iψ2[γµ, γν ]ψ1 ,
where in the second-last identity we have used that transposition changes the order of the fermionic fields, thuspicking up a minus sign. As a consequence, only transition magnetic moments between different flavors arepossible for Majorana neutrinos.
C. Neutrino Oscillations
Let us now restrict to two-neutrino oscillations, n = 2, between |νe〉 and |νµ〉, say, and write
U =
(
cos θ0 sin θ0− sin θ0 cos θ0
)
(222)
for the mixing matrix in Eq. (213) which is characterized by one real vacuum mixing angle θ0. Since id |νi〉 /dt = Ei |νi〉for i = 1, 2 in the mass basis, and since Ei = (m2
i + p2)1/2 ≃ p +m2i /(2p) ≃ E +m2
i /(2E) in the relativistic limitp≫ mi, using the trigonometric identities cos2 θ0−sin2 θ0 = cos 2θ0, 2 cos θ0 sin θ0 = sin 2θ0, it follows from Eqs. (215),(222) that
id
dt
(
νeνµ
)
E
=
[(
E +m2
1 +m22
4E
)
+∆m2
4E
(
cos 2θ0 − sin 2θ0− sin 2θ0 − cos 2θ0
)](
νeνµ
)
E
, (223)
where we consider a given momentum mode p, and ∆m2 ≡ m21 −m2
2. From now on we will consider the relativisticlimit with p ≃ E. The first term in Eq. (223) is a common phase factor and can be ignored.The integrated version of this is Eq. (215). Then applying Eq. (216), one can show that this has the solution
P (να → νβ) =1
2sin2 2θ0
(
1− cos∆m2 L
2E
)
for α 6= β , (224)
for oscillations over a length L. The oscillation length in vacuum is thus
L0 = 4πE
|∆m2| ≃ 2.48
(
E
MeV
)( |∆m2|eV2
)−1
m . (225)
Neutrino oscillations are modified by forward scattering amplitudes in matter. Since neutral currents are by definitionflavor-neutral, they only contribute to the common phase factor which in the following will be ignored. The chargedcurrent interaction is diagonal in flavor space and, according to Eqs. (96), (97), and (99), the low-energy limit of itsforward scattering part for νe(p) has the form
2√2GF
∑
p′
[
e(p′)†γ0γµ1 + γ5
2νe(p)
] [
νe(p)†γ0γµ
1 + γ52
e(p′)
]
, (226)
where we have used ψ = ψ†iγ0. We need to express this in terms of Dirac bilinears of the form Tab. II for electronsand neutrinos separately. In order to do that we use the fact that every 4× 4 matrix O can be expanded according to
O =∑
i
tr(OOi)
tr(O2i )
Oi , (227)
where Oi are the 16 matrices appearing in Tab. II which satisfy tr(OiOj) = 0 for i 6= j. Using this one can show that
(
γ0γµ1 + γ5
2
)
αβ
(
γ0γµ1 + γ5
2
)
γδ
=
(
1 + γ52
)
γβ
(
1 + γ52
)
αδ
+1
8
(
[
γλ, γκ] 1 + γ5
2
)
γβ
([γλ, γκ])αδ . (228)
58
Applying this to Eq. (226) and noting that the last term in Eq. (228) does not contribute in the rest frame of theelectron plasma, one obtains
−2√2GF
[
νe(p)† 1 + γ5
2νe(p)
]
∑
p′
[
e(p′)†1 + γ5
2e(p′)
]
, (229)
where we have picked up an extra minus from the anti-commutation of fermionic fields. The electron and neutrino fields
have the form Eq. (27). The matter-dependent part of the sum in Eq. (229) thus takes the form∑
p(a†e,L(p)ae,L(p)−
b†e,L(p)be,L(p). When tracing out the charged lepton density matrix, this reduces to the density of left-chiral electrons
minus the density of left-chiral positrons. Thus, for an unpolarized plasma, the contribution to the |νe〉 self-energy
finally is −√2GFNe, where Ne is the electron-number density, i.e. the electron minus the positron density, and
analogously for the other active flavors. The non-trivial part of Eq. (223) is thus modified to
id
dt
(
νeνµ
)
E
=
(
∆m2 cos 2θ04E −
√2GFNe −∆m2 sin 2θ0
4E
−∆m2 sin 2θ04E −∆m2 cos 2θ0
4E −√2GFNµ
)
(
νeνµ
)
E
. (230)
It is illustrative to write this in terms of the hermitian density matrix ρp(t) ≡ |νp(t)〉 〈νp(t) |. If we expand this intoan occupation number np and polarization Pp, ρp = 1
2 (np +Pp · σ), one can easily show that Eq. (230) is equivalentto
Pp = Bp ×Pp , (231)
where the precession vector Bp ≡ B(p)
B1(p) =∆m2
2Esin 2θ0 ,
B3(p) =∆m2
2Ecos 2θ0 −
√2GFN , (232)
where N = Ne −Nµ,τ for νe − νµ,τ mixing and N = Ne,µ,τ −Nn for active-sterile mixing, with Nn a combination ofnucleon densities. Thus, neutrino oscillations are mathematically equivalent to the precession of a magnetic momentin a variable external magnetic field.Eq. (232) shows immediately that at the resonance density
Nr ≡∆m2 cos 2θ0
2√2GFE
(233)
the two diagonal entries in Eq. (230) become equal or, equivalently, B3(p) = 0. More generally, Eqs. (230), (231),(232) are diagonalized by a mixing matrix Eq. (222) where θ0 is replaced by the mixing angle in matter θE given by
tan 2θE =tan 2θ0
1−N/Nr. (234)
Maximummixing, θE = π/4, thus occurs at the so-calledMichaev-Smirnow-Wolfenstein (MSW) resonance atN = Nr.For N ≪ Nr one thus has vacuum mixing, θE ≃ θ0, whereas for N ≫ Nr one has θE ≪ θ0. Eq. (231) now showsthat propagation from N ≫ Nr to N ≪ Nr can lead to an efficient transition from one flavor to another, as longas |Bp| <∼ |Bp|2. Such transitions are called adiabatic. Note that since masses of anti-particles and particles areequal, whereas the lepton numbers Ne,µ,τ change sign under charge conjugation, resonances in matter occur eitherfor neutrinos or for anti-neutrinos.In subsequent sections we will apply this to various observational evidence for neutrino oscillations.
D. Selected Applications in Astrophysics and Cosmology
1. Stellar Burning and Solar Neutrino Oscillations
Weak interactions are crucial in cosmology and stellar physics. In main sequence stars the first stage of hydrogenfusion into helium is the weak interaction
p+ p→ 2H+ e+ + νe . (235)
59
The subsequent reactions 2H+p→3He+γ and 3He+3He→4He+p+ p lead to the net reaction
4p→ 4He + 2e+ + 2νe + 26.73MeV . (236)
For many more details see Ref. [45].When normalized to the solar energy flux arriving at Earth, one can calculate the expected neutrino fluxes within theso-called Standard Solar Model. The resulting fluxes in this “pp” channel as well as in other reaction channels areshown in Fig. 26
FIG. 26: The predicted solar neutrino energy spectrum from Ref. [49]. For continuum sources, the neutrino fluxes are givenin number per cm−2sec−1MeV−1 at the Earth’s surface. For line sources, the units are number per cm−2sec−1. The totaltheoretical uncertainties are shown. The CNO neutrino fluxes are not shown.
However, less than half of the expected solar electron-neutrino flux at a few MeV has been observed. On the otherhand, neutral current experiments with the Sudbury Neutrino Observatory (SNO) have shown that the sum of theelectron, muon- and tau neutrino flux coincides with the expected electron neutrino flux. This can be explained byan MSW transition of νe into νµ and ντ within the Sun with a ∆m2
solar ∼ 10−5 eV2. Note that this corresponds toa vacuum oscillation length Eq. (225) of a few hundred kilometers. Recently this has been confirmed independentlyby the KamLAND experiment which measured the disappearance of the νe neutrinos produced by nuclear reactors afew hundred kilometers from the detector. The best fit parameters for the parameters of mixing of two neutrinos invacuum from all solar and reactor data are [50]
∆m2solar ≃ 8.2+0.3
−0.3(+1.0−0.8)× 10−5 eV2 ; tan2 θsolar ≃ 0.39+0.05
−0.04(+0.19−0.11) , (237)
where 1σ and 3σ errors are given. The relevant contour plots are shown in Fig. 27. It is interesting to note thatmaximal mixing is strongly excluded, tan θsolar ≤ 1.0 at 5.8σ.Finally, since solar and reactor neutrino experiments deal with νe and νe, respectively, comparison of the two corre-sponding oscillations parameters allows to set limits on CPT -violation in the neutrino sector.
60
∆m21
2 (e
V2 )
∆m21
2 (e
V2 )
tan2θ12 tan2θ12
FIG. 27: From Ref. [50]. Allowed oscillation parameters: Solar vs KamLAND. The two left panels show the 90%, 95%, 99%,and 3σ allowed regions for oscillation parameters that are obtained by a global fit of all the available solar data. The two rightpanels show the 90%, 95%, 99%, and 3σ allowed regions for oscillation parameters that are obtained by a global fit of all thereactor data from KamLAND and CHOOZ. The two upper (lower) panels correspond to the analysis of all data available before(after) the Neutrino 2004 conference, June 14-19, 2004 (Paris). The new KamLAND data are sufficiently precise that mattereffects discernibly break the degeneracy between the two mirror vacuum solutions in the lower right panel.
2. Atmospheric Neutrinos
Cosmic rays interact in the atmosphere and produce, among other particles, pions and kaons whose decay productscontain neutrinos. The observed ratio of upcoming to down-going atmospheric muon neutrinos is about 0.5 in the GeVrange. Since upcoming neutrinos travel several thousand kilometers, this can be interpreted as vacuum oscillationsbetween muon and tau-neutrinos. The best fit parameters are
∆m2atmos ≃ 2.1× 10−3 eV2 ; sin2 θatmos ≃ 0.5 , (238)
consistent with maximal mixing. The corresponding contours resulting from atmospheric and long baseline neutrinooscillation data from K2K are shown in Fig. 28. Recently, the L/E dependence, see Eq. (224), characteristic forneutrino oscillations has been confirmed by the Superkamiokande experiment, thereby strongly constraining alternativeexplanations of muon neutrino disappearance such as neutrino decay [51]. In addition, oscillations into sterile neutrinosare strongly disfavored over oscillations into tau-neutrinos via the following discriminating effects: Neutral currentswould be non-diagonal for oscillations into sterile states, thus modifying oscillation amplitudes and total scatteringrates, and charged current interactions of tau-neutrinos imply τ appearance.In 3-neutrino oscillation schemes θsolar and θatmos are usually identified with θ12 and θ23, respectively. The “normalmass hierarchy” in which m1,m2 ≪ m3 seems the most natural for neutrino mass modeling in Grand Unificationscenarios [53]. However, the inverted hierarchy is also possible, see Fig. 29. As Fig. 30 shows, these two hierarchieshave an important influence on the neutrino-less double beta decay rates.According to the discussion around Eq. (213), there is one more mixing angle θ13 and at least one Dirac CP -violatingphase called δD. Note that solar and atmospheric neutrinos only decouple exactly for θ13 = 0 in which case CP would
61
0 0.25 0.5 0.75 1
sin2θ
0
2
4
6
∆m2 [1
0−3 e
V2 ]
0
5
10
15
20
∆χ2
3σ
2σ
0 5 10 15 20
∆χ2
3σ2σ
atmospheric onlyatmospheric + K2K
FIG. 28: From Ref. [52]. Allowed (sin2 θatmos, ∆m2atmos) regions at 90%, 95%, 99%, and 3σ CL for two degrees of freedom. The
regions delimited by the lines correspond to atmospheric data only, while for the colored regions also K2K data are added. Thebest fit point of atmospheric (atmospheric + K2K) data is marked by a triangle (star). Also shown is the ∆χ2 as a functionof sin2 θatmos and ∆m2
atmos, minimized with respect to the undisplayed parameter.
also be conserved. Whereas there are at most weak indications for leptonic CP -violations yet [54], the third mixingangle is constrained at 90% CL
sin2 2θ13 ≤ 0.19 . (239)
The present knowledge on the MNSP matrix can be approximately summarized as follows,
UMNSP =
(
0.8 0.5 ?0.4 0.6 0.70.4 0.6 0.7
)
, (240)
which is in marked contrast to the CKM matrix for quark flavor mixing
UCKM =
(
1 0.2 0.0050.2 1 0.040.005 0.4 1
)
. (241)
We finally stress that neutrino oscillations are sensitive only to differences of squared masses, not to absolute massscales. To probe the latter requires laboratory experiments discussed earlier such as β−decay, the study of cosmologicaleffects such as the influence of neutrino mass on the power spectrum, see Sect. IVD4, or measuring time delays ofastrophysical neutrino bursts from γ−ray bursts and supernovae relative to the speed of light.
62
FIG. 29: The two possible mass arrangements based on oscillation data, left the normal hierarchy, right the inverted one.Colors indicate the contribution from the different weak eigenstates, reflecting the known facts about mixing angles.
3. Flavor Composition from High Energy Neutrino Sources
For sources at cosmological distances the oscillating terms in the transition probability Eq. (216) average out and oneobtains:
P (να → νβ) = P (να → νβ) =∑
i
|Uαi|2 |Uβi|2 . (242)
For flavors α injected with relative weights wα at the source, the flux of flavor β at the observer is then
φβ(E) ∝∑
α
wα P (να → νβ) ≃∑
α,i
wα |Uαi|2 |Uβi|2 . (243)
Eventually, the observed flavor ratios will depend on both the flavor ratios injected at the source, and the mixingmatrix.There are at least three kinds of sources:
• At energies where the neutrino flux is dominated by neutron decay, essentially only νe are present.
• Since pion decay into electrons is helicity suppressed, pion decay produces essentially only muon neutrinos,whereas muon decay produces similar fluxes in electron and muon type neutrinos. Therefore, if both pions andmuons decay before loosing energy, we have we : wµ : wτ ≃ 1
3 : 23 : 0.
• Since tµ ∼ 100tπ, there are energies at which pions decay before loosing energy, but muons loose energy beforedecaying. In this case one has we : wµ : wτ ≃ 0 : 1 : 0.
4. Neutrino Hot Dark Matter
Finally, massive neutrinos in the eV range contribute to the density of non-relativistic matter in today’s universe,
Ωνh2 =
∑
mν
92.5 eV(244)
63
FIG. 30: Allowed effective neutrino mass Eq. (220) vs. mass of the lightest mass state (adopted from S. Petcov). The differentlines and colors for theoretical predictions correspond to various assumptions on CP violating phases.
in terms of the critical (closure) density for mν ≫ 10−4 eV, today’s temperature. Eq. (244) results from the fact thatneutrinos have been relativistic at decoupling at T ∼ 1MeV, thus constituting hot dark matter, and their numberdensity is simply determined by the redshifted number density at freeze-out, in analogy to Sect. II I.Since neutrinos are freely streaming on scales of many Mpc, the matter power spectrum is reduced by a relativeamount ∆Pm/Pm = −8Ων/Ωm, where Ωm is the total matter density. A combination of data on the large scalestructure and the CMB then leads to the limit [55]
∑
mν <∼ 1.0 eV . (245)
There was even a claim for a positive detection with [56]
∑
mν ≃ 0.56 eV , (246)
but newest analyses suggest upper bounds even slightly below this [57].It is intriguing that direct experimental bounds Eq. (221) and cosmological bounds Eq. (245) have reached comparablesensitivities. In addition, both a combination of future CMB data from the Planck satellite with large scale structuresurveys [55] and next generation laboratory experiments such as KATRIN will probe the 0.1 eV regime.Assuming three active neutrino oscillations with the parameters discussed in Sects. IVD1 and IVD2 has an interestingcosmological consequence: Flavor equilibrium is reached before the BBN epoch and the asymmetry parameter ξν =µν/T , where µν is the common neutrino chemical potential, is constrained by [58]
|ξν | <∼ 0.07 . (247)
As a consequence, neutrino degeneracy is unobservable in the large scale structure and the CMB.
64
5. Leptogenesis and Baryogenesis
Neutrino masses may also play a key role in explaining the fact that we live in a universe dominated by matterrather than anti-matter. The heavy right-handed Majorana neutrinos involved in the seesaw mechanism discussedin Sect. IVB could have been produced in the early Universe and their out-of-equilibrium decays could give riseto a non-vanishing net lepton number L. Non-perturbative quantum effects related to the non-abelian character ofthe electroweak interactions can translate this into a net baryon number B while conserving B − L. The amountof baryon number nB created in this scenario is related to the low-energy leptonic CP -violation phase δD [59]. Itscompatibility with the observed value for the baryon per photon number nB/nγ ≃ 6 × 10−10 implies a lower boundmR >∼ few 1010 GeV. Via the see-saw relation mν ≃ m2
D/mR for the light neutrino mass, this corresponds to an
optimal range 10−3 eV <∼ mν <∼ 0.1 eV, in remarkable agreement with the observed atmospheric and solar neutrinomass scales [59]. In general baryogenesis requires violation of baryon number B, charge conjugation C, combinedcharge and parity conjugation CP , and a departure from thermal equilibrium, usually caused by the expansion of theUniverse. These conditions are known as the Sakharov conditions [60].
65
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