About Supersymmetric Hydrogen

Robin Schneider 12

supervised by

Prof. Yuji Tachikawa2

Prof. Guido Festuccia1

August 31, 2017

1Theoretical Physics - Uppsala University2Kavli IPMU - The University of Tokyo

1

Abstract

The energy levels of atomic hydrogen obey an n2 degeneracy at O(α2). It isa consequence of an so(4) symmetry, which is broken by relativistic effects suchas the fine or hyperfine structure, which have an explicit angular momentumand spin dependence at higher order in α.

The energy spectra of hydrogenlike bound states with underlying supersym-metry show some interesting properties. For example, in a theory with N= 1,the hyperfine splitting disappears and the spectrum is described by supermul-tiplets with energies solely determined by the super spin j and main quantumnumber n [1, 2]. Adding more supercharges appears to simplify the spectrumeven more. For a given excitation Vl, the spectrum is then described by a singlemultiplet for which the energy depends only on the angular momentum l andn.

In 2015 Caron-Huot and Henn showed that hydrogenlike bound states inN = 4 super Yang Mills theory preserve the n2 degeneracy of hydrogen forrelativistic corrections up to O(α3) [3]. Their investigations are based on thedual super conformal symmetry of N = 4 super Yang Mills. It is expected thatthis result also holds for higher orders in α.

The goal of this thesis is to classify the different energy spectra of super-symmetric hydrogen, and then reproduce the results found in [3] by means ofconventional quantum field theory. Unfortunately, it turns out that the tech-niques used for hydrogen in (S)QED are not suitable to determine the energycorrections in a model where the photon has a massless scalar superpartner.It can, however, be shown that the so(4) symmetry is preserved, at least inthe heavy proton limit, where mp → ∞ with the help of relativistic quantummechanics.

2

Abstrakt pa svenska

Energinivaerna i vateatomen foljer en n2 degeneration vid O(α2). Detta aren foljd av en so(4)-symmetri vilken ar bruten av relativistiska effekter som fin-och hyperfinstrukturen vilka i sin tur beror explicit pa rorelsemangdsmomentoch spinn vid hogre ordning i α.

Energispektrumen i vateliknande bundna tillstand med underliggande su-persymmetri uppvisar vissa intressanta egenskaper. Till exempel: i en teorimed N= 1 forsvinner hyperfinstrukturen och spektrumet beskrivs av supermul-tipletter vars energier bestams av enbart superspinn j, och huvudkvanttalet n[1, 2]. Tillagg av fler superladdningar verkar forenkla spektrumet ytterligare.For en given excitation Vl beskrivs spektrumet av en enda multiplett dar energinendast beror pa rorelsemangdsmomentet l och n.

Ar 2015 visade Caron-Huot och Henn att vateliknande bundna tillstandi super-Yang-Mills-teori med N = 4 bevarar n2-degenerationen fran vate forrelativistiska korrektioner upp till O(α3) [3]. Deras undersokningar ar baseradepa denna duala superkonforma symmetrin i N = 4 super-Yang-Mills. Manforvantar sig att detta resultat haller aven for hogre ordningar i α.

Malet med denna uppsats ar att klassificera de olika energispektrumen hossupersymmetriskt vate och att darefter reproducera resultaten fran [3] genomatt anvanda konventionell kvantfaltteori. Olyckligtvis visar det sig att teknikernasom anvands for vate i (S)QED inte lampar sig for att bestamma energikorrek-tioner i en modell dar fotonen har en masslos skalar superpartner. Man kan dockvisa att so(4)-symmetrin ar bevarad med hjalp av relativistisk kvantmekanik,atminstone i gransvardet dar protonen ar tung, dar mp →∞.

Contents

1 Introduction 4

2 Theory 62.1 Hydrogen and group theory . . . . . . . . . . . . . . . . . . . . . 62.2 Hydrogen and QED . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Supersymmetric Hydrogen 123.1 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Energy splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Hydrogen in N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 N = 4 Super Yang Mills 204.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Energy splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Discussion 275.1 Relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . 275.2 Cross particle scattering . . . . . . . . . . . . . . . . . . . . . . . 305.3 Radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Outlook 34

A Appendix 37A.1 Conventions and identities . . . . . . . . . . . . . . . . . . . . . . 37A.2 (BS-)Wave function and expectation values . . . . . . . . . . . . 38A.3 Excitation of a non BPS multiplet . . . . . . . . . . . . . . . . . 39

3

Chapter 1

Introduction

The hydrogen atom is one of the best studied systems in theoretical and ex-perimental physics. It has a rich history of new discoveries and progress inboth fields. The beginning of modern atomic hydrogen physics was marked byBohr and Rutherford, who abandoned the classical theories and postulated in1913 that the electron moves on quantized orbits around the proton. From thatassumption, they deduced a n2 energy dependence at O(α2) [4]. The same n2

degeneracy was later derived by Pauli in 1922 when investigating the symmetriesof a central potential, i.e. conservation of angular momentum and Runge-Lenzvector [5]. Solving the time independent Schroedinger equation allows to givemore precise corrections of the energy levels at O(α4) but failed to predict theproper contribution of the fine structure [6]. This was achieved by incorporatingthe spin dependence in terms of the relativistic Dirac equation in 1928 [7]. Oneyear later, Fermi explained the hyperfine splitting by introducing a spin-spincontact interaction [8]. The next big theoretical step was made by Bethe, whocalculated the Lamb shift, which had been found in 1947 [9], in terms of radia-tive corrections [10]. Even though the hydrogen atom has been studied for sucha long time, there is still an active community involved in its research. Thoseactivities include mostly the better measurement of its energy levels, and withit, more accurate theoretical predictions of their splitting [11].

In theoretical physics there has recently been a new interest in investigatinghydrogenlike bound states with different underlying supersymmetries. The con-cept of supersymmetry (SUSY) was introduced in the 70’s as an extension to thePoincare algebra. In a supersymmetric theory every particle has so called su-perpartners. These partners share the same mass and charge; additionally therehas to be an equal number of fermionic and bosonic degrees of freedom. SUSYwas proposed in order to solve several theoretical problems and weaknesses, forexample the hierarchy problem and the unification of electromagnetic weak andstrong force. Furthermore, SUSY offers reasonable candidates for dark matterparticles. The Minimal Supersymmetric Standard Model (MSSM) is a widelyaccepted extension of the standard model, which provides a solution to pre-viously mentioned issues. Since no superpartners have been observed so far,SUSY must be broken at low energies. The search for these particles in particlecolliders such as CERN is ongoing [12].

Supersymmetric bound states were investigated for the first time in 1981 byBuchmueller, Love and Peccei [13]. They analyzed the ground state energy of

4

CHAPTER 1. INTRODUCTION 5

superpositronium in N = 1 and found a lack of hyperfine structure. The differ-ent particle bound state representations can be categorized in a para- and anortho-supermultiplet. This research was extended by Di Vecchia and Schuch-hardt to N = 2, who discovered that the splitting between both supermultipletsdisappears [14] and the ground state is then described by a single large multiplet.

In 2010, Herzog and Klose, [1] and independently Rube and Wacker, [2]investigated the full energy spectrum of supersymmetric hydrogen. They con-firmed the lack of hyperfine splitting found earlier for superpositronium, andderived an easy expression for the energy of every supermultiplet. Their energyonly depends on the superspin j and the main quantum number n. The groundstate contains two supermultiplets with j = 0 and one with j = 1

2 . For a givenangular momentum l > 0 there are four supermultiplets with j = l ± 1

2 , 2 · l.Herzog and Klose motivated their work by the gauge/gravity duality, which

allows us to map problems from field theories to string theory. AdS/CFT corre-spondence was proposed in 1997 by Juan Maldacena [15]. One common exampleof this mapping is the duality between N = 4 super Yang Mills theory and TypeIIb string theory.

It so happens that in 2015 Caron-Huot and Henn published a paper abouthydrogenlike bound states in N = 4 super Yang Mills [3]. They made thefascinating observation that the so(4) symmetry, which is broken for (super-symmetric) hydrogen by relativistic corrections, appears to be preserved in theCoulomb phase of N = 4 super Yang Mills theory. In their paper, the 2 → 2massive vector boson scattering amplitude is analyzed by making use of the dualsuper conformal symmetry properties. The bound state energy is determinedfrom its poles.

The outline of this thesis is as follows: In chapter 2, I will, 1.) review thephysics of atomic hydrogen in terms of its symmetries and then 2.) analyze itsenergy spectrum up to O(α4) using Feynman diagrams. In chapter 3, the energyspectrum of supersymmetric hydrogen is derived in terms of its supermultiplets.The calculations of the relativistic energy corrections are outlined and the mainresults of [1] stated. Using the same field theoretical approach as in the earlierchapters, I will calculate the energy corrections of hydrogen in N = 4 superYang Mills in chapter 4. Unfortunately, it turns out that these calculationscannot confirm the n2 degeneracy found in [3]. This can be explained by severalshortcomings, which make the field theoretical treatment using only tree leveldiagrams incomplete. They are discussed in chapter 5. Making use of relativisticquantum mechanics, I will show that in an equivalent model with the heavyproton limit the so(4) symmetry is preserved up to O(α4). This work forms thebasis of a soon to be published paper [16]. The last chapter will give an outlookto future studies using effective field theories.

Chapter 2

Theory

It is a well known fact that the n2 degeneracy of hydrogen is the result ofa preserved so(4) symmetry. This was shown for the first time by W. Pauliin 1922, [5] who investigated the conserved quantities of angular momentumand Runge-Lenz vector. The so(4) symmetry is broken by relativistic effectsat O(α4). The energy corrections are found from the Dirac equation with thehelp of quantum electro dynamics. In this chapter, I will first derive the n2

degeneracy and then determine all operators which contribute to the energy atO(α4) with the help of Feynman diagrams; this includes the commonly knownfine and hyperfine structure.

Within the framework of this thesis, I will work in natural units ~ = c =ε0 = 1 and neglect the composite particle effects of the proton, hence, treat itas a fermionic point particle like the electron. In the literature, such a systemis usually referred to as muonic hydrogen.

Furthermore, I will follow the notation and conventions of indices, particlesand superfields introduced by Wess and Bagger [17]. Finally, I want to mentionthat I am mainly interested in effects up to O(α4) but will occasionally commenton radiative corrections at higher order in α such as the Lamb shift.

2.1 Hydrogen and group theory

An extensive discussion about hydrogen and group theory is given in [18]. Thissection contains only a short derivation of the degenerate energy levels (a reviewof which can be found in nearly every modern text book of quantum mechanicse.g. [19]) based on W. Pauli’s work. The following Hamiltonian

H =~p2

2µ− α

r(2.1.1)

describes a central potential where µ = (1/me +1/mp) is the reduced mass, andα the fine structure constant. This potential obeys a so(3) symmetry spannedby the angular momentum operators

[Li, Lj ] = iεkijLk . (2.1.2)

6

CHAPTER 2. THEORY 7

The angular momentum is a conserved quantity. In addition, there is anotherconserved quantity which has been known for a long time. The quantum me-chanical hermitian Runge-Lenz vector

~M =1

2µ

(~p× ~L− ~L× ~p

)− α~r

r(2.1.3a)

~M2 =2

µH(~L2 + 1

)+ α2 (2.1.3b)

[Mi,Mj ] = −2i

µεkijHLk (2.1.3c)

is responsible for keeping the rotating electron on a closed disk. Hence, the fullsymmetry of the Hamiltonian is described by

so(3)⊗ so(3) = so(4) . (2.1.4)

This can be easily shown by constructing the combined algebra. Both Runge-Lenz vector and angular momentum commute with the Hamiltonian

[H,M ] = 0 (2.1.5a)

[H, L] = 0 . (2.1.5b)

After appropriate rescaling, M ≡ i√µ/(2E) ~M , one finds the algebra

[Li, Lj ] = iεkijLk (2.1.6a)

[Mi, Lj ] = iεkijMk (2.1.6b)

[Mi, Mj ] = iεkijLk . (2.1.6c)

Combining ~L and M results in two commuting ’angular momentum’ operators,which satisfy

A2 =

[ ~L− M

2

]2

= j1(j1 + 1) (2.1.7a)

B2 =

[ ~L+ M

2

]2

= j2(j2 + 1) . (2.1.7b)

From ~L · M = 0 follows that j1 = j2 = j = 0, 12 , 1, .. . Now acting on a bound

state |ψ〉 with

A2 +B2 |ψ〉 = 2j(j + 1) |ψ〉 =1

2

(~L2 − M

2)|ψ〉

=1

2

(− 1− µ

2Eα2)|ψ〉

(2.1.8)

gives the energy relation

E = −α2µ

2n2(2.1.9)

where n2 = (2j+ 1)2. One finds the well known n2 degenerate energy spectrumfollowing from the preservation of the so(4) symmetry.

CHAPTER 2. THEORY 8

e e

p p

p1 q1

kγ

p2 q2

Figure 2.1: The tree level scattering diagram of an electron and a proton. Theparticles form a bound state, the hydrogen atom, via exchange of a photon.

2.2 Hydrogen and QED

The energy spectrum at O(α4) is determined by including additional effectssuch as the spin-orbit and spin-spin coupling, but also relativistic corrections tothe momentum. They can all be obtained when starting from the Lagrangiandescribing the usual Dirac equation

L = ψe(i /D +me)ψe + ψp(i /D +mp)ψp (2.2.1)

where D is the Abelian covariant derivative and ψ the Dirac spinors of electronand antiproton. The energy levels of the proton-electron bound state are deter-mined from the potential, which is connected to the scattering amplitudeM inthe non relativistic limit via the Born approximation [20]

MNR = −〈q|V |p〉 . (2.2.2)

Hence, in order to find the Hamiltonian, one has to take the Fourier trans-formation of the scattering amplitude described by the Feynman diagram infigure 2.1. The Fourier transformation of MNR is performed with respect to(~p− ~q) = ~k → r while the momentum ~p is kept as a variable.

Here the different normalizations between quantum mechanics 〈q|1 |p〉 =δ3(~p − ~q) and quantum field theory 〈q|1 |p〉 =

√2Eδ3(~p − ~q) have to be taken

into account

MNR =M

4[(m2

e + ~p2)(m2e + ~q2)(m2

p + ~p2)(m2p + ~p2)

] 14

. (2.2.3)

The tree level amplitude from figure 2.1 is determined by using the conven-tional Feynman rules of QED

M = (−ig)2ue(q1)γmue(p1)(−i)[ηmnk2− (1− ξ)kmkn

k4

]vp(p2)γnvp(q2) (2.2.4)

where the gauge dependence is kept explicit. It is most easily calculated in thecenter of mass frame

p1 =

(√~p2 +m2

e , ~p

), q1 =

(√~q2 +m2

e , ~q

), (2.2.5a)

p2 =

(√~p2 +m2

p , −~p), q2 =

(√~q2 +m2

p , −~q). (2.2.5b)

CHAPTER 2. THEORY 9

Figure 2.2: One loop cross photon scattering diagram with leading energy con-tributions to the bound state of O(α4). In Coulomb gauge, however, the leadingcontribution cancels to zero.

Borrowing the notation from [1] the Dirac spinors are

ue(p1) =

(√σ · p1ξe√σ · p1ξe

), ue(q1) =

(−√σ · q1ξ

†e , −

√σ · q1ξ

†e

), (2.2.6a)

vp(q2) =

( √σ · q2ηp

−√σ · q2ηp

), vp(p2) =

(√σ · p2η

†p , −

√σ · p2η

†p

). (2.2.6b)

Using the identities A.1.5 and A.1.6a, A.1.6b, A.1.6c the amplitude becomes [1],

iM' ξ†e oη†p i

4mempg2

(~p− ~q)2

[1 +

1

8(~p+ ~q)2

(1

me+

1

mp

)2

− (1− ξ)4memp

((~p2 − ~q2))2

(~p− ~q)2

− i

2(~p× ~q) · ~σe

(1

2m2e

+1

memp

)+i

2(~p× ~q) · ~σp

(1

2m2p

+1

memp

)+

1

4memp(~p− ~q)2~σe · ~σp −

1

4memp(~p− ~q) · ~σe(~p− ~q) · ~σp

]ξe iηp o ,

(2.2.7)

Here, the non relativistic approximation p0 ≈ m+ ~p2

2m was used, omitting termsof momenta α~p4 which scale with at least O(α5). I will henceforth use Coulombgauge with ξ = 0. In any other gauge one loop diagrams coming from crossphoton scattering as in figure 2.2 have to be considered at O(α4) in bound stateenergy [21]. This is discussed in more detail in section 5.2.

According to 2.2.3, when going from M → MNR the amplitude receivesmany corrections. However, only the ones coming from the normalization of theCoulomb potential, the first term in 2.2.7, are of O(α4)

∆M = −(~p2 + ~q2)

(1

4m2e

+1

4m2p

). (2.2.8)

After transforming the antispinors to spinors with η = iσ2ξ∗, some massaging(use identity A.1.7) and a subsequent Fourier transformation, one finds thepotential

V (~r, ~p) = α

[− 1

r− 1

memp

(~p2

2r+

(~r · ~p)2

2r3+ πδ3(~r)

)+π

2δ3(~r)

(1

me+

1

mp

)2

+~L · ~Ser3

(1

2m2e

+1

memp

)+~L · ~Spr3

(1

2m2p

+1

memp

)+

1

memp

(8π

3~Se · ~Spδ3(~r) +

3~r · ~Se~r · ~SP − ~Se · ~Spr3

)].

(2.2.9)

CHAPTER 2. THEORY 10

Every operator is understood to be normal ordered. Comparing this resultwith the Breit potential [22, 23], one identifies the usual terms such as (i) theCoulomb potential, (ii) the orbit-orbit interaction, (iii) the Darwin term, whichonly contributes to s-waves, (iv) + (v) the spin-orbit interactions and (vi) thespin-spin interactions.

The energy spectrum of atomic hydrogen is then determined with pertur-bation theory. First, however, there is one relativistic correction which is stillmissing; the Hamiltonian of any quantum mechanical system is given by

H = T + V . (2.2.10)

Expanding the kinetic energy

T = E −Mtot =√m2

e + ~p2 +√m2

p + ~p2 − (me +mp)

' me

(1 +

~p2

2m2e

− ~p4

8m4e

)+mp

(1 +

~p2

2m2p

− ~p4

8m4p

)− (me +mp)

=~p2

2

(1

me+

1

mp

)− ~p4

8

(1

m3e

+1

m3p

) (2.2.11)

gives an additional term ∝ ~p4 scaling with O(α4). The full Hamiltonian becomes

⇒ H =~p2

2µ− ~p4

8

(1

m3e

+1

m3p

)+ V (~r, ~p)

=~p2

2µ− α

r+Hint(~r, ~p)

(2.2.12)

where in the last line I decomposed it into an interaction part and the usualCoulomb potential. In first order degenerate perturbation theory, the energyshifts are determined by taking the expectation value (EV)

δEnl = 〈ψnl|Hint |ψnl〉 . (2.2.13)

EVs of the different operators can be found in the appendix A.2. Energy cor-rections scaling with ∼ µα4, such as the electron spin-orbit, the Darwin and the~p4 momentum operator, are often referred to as fine structure

δEfinenj = −µα

4

n4

[n

2j + 1− 3

8

], (2.2.14)

where j = l ± 12 is the total angular momentum. The explicit j dependence

breaks the n2 degeneracy, and with it the so(4) symmetry. Hyperfine struc-ture corrections are all those terms ∝ µα4me/mp. Due to their l and spindependence, they also break the n2 degeneracy. The term hyperfine splitting iscommonly used to describe the splitting of the s-waves into singlet and triplet,breaking the degeneracy of the ground state. This is a consequence of the spin-spin interactions in the last line of equation 2.2.9, which gives different valuesdepending on the alignment of the two spins; i.e. the ground state splits into

Singlet: 1√2

(|↑↓〉 − |↓↑〉

)∆E ∝ −3α4µme

mp

Triplet: 1√2

(|↑↓〉+ |↓↑〉

), |↑↑〉 , |↓↓〉 ∆E ∝ α4µme

mp.

(2.2.15)

CHAPTER 2. THEORY 11

Compared to the fine structure, its magnitude is significantly smaller, as it issuppressed by the proton mass ∼ 1/2000.

In next leading order of α one finds the Lamb shift O(α5 log(α)) comingfrom radiative corrections to the bound state. Even though it comes with ahigher order in α, its contribution is significantly larger than the hyperfinesplitting. It is obtained by calculating the loop diagrams describing correctionsto the vertices, and the vacuum polarization. For a full treatment of (hyper-)finestructure and Lamb shift, Quantum Field Theory by Itzykson and Zuber is avaluable resource [24].

Chapter 3

Supersymmetric Hydrogen

In this chapter I will provide a review of N = 1 supersymmetric hydrogen. Thefocus will be on the structure of the energy spectrum in terms of its supermul-tiplets. The energy shifts between these supermultiplets have to be explicitlycalculated from the Hamiltonian. This was done independently by Herzog andKlose, [1] and Rube and Wacker [2]. I will also briefly comment on an extensionto N = 2. The only explicit N = 2 SQED bound state analysis using Feynmandiagrams was performed by Di Vecchia and Schuchhardt in 1985 for the groundstate of superpositronium [14].

The Lagrangian of supersymmetric hydrogen in terms of N= 1 superfields is

L = Lgauge + Lmatter

=

[ ∫d2θ WW + h.c.

]+

[ ∫d4θ(Φ†e+e2gV Φe+ + Φ†e−e−2gV Φe−

)+

∫d2θme(Φe+Φe− + Φ†e+Φ†e−) + proton terms

].

(3.0.1)

It is build from four massive chiral fields (Φe/p±) and one massless vector field(V ). Each chiral field contains a Weyl spinor and a complex scalar. Twoof these Weyl spinors then combine and form the Dirac spinors of electronand antiproton. In total, the particle spectrum is that of four complex scalars(φe/p±), one electron (ψe) and the antiproton (ψp). The vector field has therole of the gauge field. It is conventional to use Wess-Zumino gauge, such thatits particle spectrum is given by a massless vector boson, the photon γ, and itssuperparticle the gaugino, a massless Majorana fermion λ. W is the Abeliansuperfield strength. The gaugino couples a Weyl spinor to its complex scalar,and is therefore responsible for mixing between different bound states.

3.1 Energy spectrum

A rigorous analysis of the energy and particle spectrum in terms of symmetriescan be found in [2] [25]. The review in the next two sections follows closely [1]and states their main results.

At leading order the bound states are n2 degenerate as a consequence ofthe Coulomb potential coming from the photon exchange. However, at next to

12

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 13

leading order, the bound states receive relativistic and spin dependent energycorrections. The bound state spectrum, in terms of its superparticles, transformsas

S = (Φe+ ⊕ Φe−)⊗ (Φp+ ⊕ Φp−) . (3.1.1)

For the ground state with l = 0 there are in total eight bosonicFermion-Fermion: |↑↑〉 , |↑↓〉 , |↓↑〉 , |↓↓〉Scalar-Scalar: |++〉 , |−+〉 , |+−〉 , |−−〉

(3.1.2)

and eight fermionicFermion-Scalar: |↑ +〉 , |↑ −〉 , |↓ +〉 , |↓ −〉Scalar-Fermion: |+ ↑〉 , |− ↑〉 , |+ ↓〉 , |− ↓〉

(3.1.3)

possible bound state. The total 16 degrees of freedom are matched by twomassive multiplets and one vector multiplet built from a Clifford vacuum Ωj

λ S 2 · Ω0 Ω 12

−1 1 0 1− 1

2 4 1 + 1 20 6 2 + 2 1 + 112 4 1 + 1 21 1 0 1

where λ is the helicity of a given state. Each multiplet Ωj can be decomposedinto 2j + 1 dimensional representations Vj of so(3).

Ωj = Vj− 12⊕ 2 · Vj ⊕ Vj+ 1

2(3.1.4)

The symmetries of the Lagrangian put further constraints on the energy spec-trum. U(1)R-symmetry and parity act on a given superparticle with

PΦ± = Φ∓ R(α)Φ± = e±iαΦ± . (3.1.5)

Hence, parity and a R(α) transformation do not commute; they combine as asemiproduct with U(1)R 1 Z2

∼= O(2)R. The two Ω0 multiplets sit in a doubletof O(2)R and share common eigenenergies while Ω 1

2already forms a doublet

with its CPT conjugate.For excited states, it follows from

Vl ⊗ Vj = V|l−j| ⊕ ...⊕ V|l+j| (3.1.6)

where both and Vl and Vj are representations of so(3) and 3.1.4 that

Vl ⊗ Ωj = Ω|l−j| ⊕ ...⊕ Ω|l+j| . (3.1.7)

Hence, the energy spectrum for a given l decomposes as,

Vl ⊗ (Ω0 ⊕ Ω0 ⊕ Ω 12) = Ωl− 1

2⊕ Ωl ⊕ Ωl ⊕ Ωl+ 1

2. (3.1.8)

In summary, the energy spectrum for arbitrary l is expected to look as presentedin figure 3.4. Surprisingly, there is no energy shift between the multiplets Ωl+ 1

2

and Ωl+1− 12. This accidental degeneracy, however, has to be explicitly calcu-

lated.

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 14

γ γ

Figure 3.1: The three tree level Feynman diagrams describing the eight bosonicbound states of a N = 1 supersymmetric hydrogen model. The first diagramshows the four fermion-fermion bound states, while second and third describethe four different scalar-scalar bound states.

λλλ λ

Figure 3.2: The four tree level Feynman diagrams describing the mixing betweenbosonic i) + ii) and fermionic iii) + iv) bound states of a N = 1 supersymmetrichydrogen model.

3.2 Energy splitting

In order to determine the energies of the supermultiplets, one has to find theHamiltonian. This is a familiar task from section 2.2 where the Hamiltonian foratomic hydrogen was calculated. The relevant Feynman diagrams of the eightbosonic bound states are presented in figure 3.1. The amplitudes of the secondand third diagram representing the scalar-scalar bound states are straightfor-wardly calculated. Their resulting potentials contain the Coulomb attraction,the orbit-orbit interaction and the Darwin correction.

As a consequence of supersymmetry, the different bosonic bound states canmix with each other. They do this by exchanging a gaugino, as shown in figure3.2. From the first two diagrams, one derives the potential

Vmix = − g2

4πξTp

[− i

2√memp

~r · σr3± π

2

mp −me

(memp)32

δ3(~r)± 1

4

mp +me

(memp)32

~L · ~σr3

]ξe .

(3.2.1)

In contrast to other corrections, the first term scales with O(α3). However, itturns out that

〈 ~rr3〉 = 0 (3.2.2)

and Vmix therefore contributes only with second order perturbation theory

δE2 =∑i

| 〈ψi|Hint |ψn〉 |2

En − Ei. (3.2.3)

The energy correction is then proportional to ∝ α4. The integral 〈ψi|Hint |ψn〉is solved with the help of Schwingers solution to the Coulomb Green function[26]. Having determined all energy corrections, one can rewrite the Hamiltonianas a 8x8 matrix spanned by the bound states, given in 3.1.2. From a first guess,this should only be possible for s-waves as one expects states in higher orbitalsto mix for different values of l. However, it turns out that neither second order

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 15

γ γ

Figure 3.3: The two tree level diagrams describing the eight fermionic boundstates of a N = 1 supersymmetric hydrogen model. Each diagram representsthe four different bound states arising from a coupling between one fermion andtwo complex scalar.

perturbation theory nor the spin-spin interactions mix states of different n andl. The analysis of the eight fermionic bound states follows the same reasoning.In figure 3.3 are the two relevant tree level diagrams to determine the fermionicHamiltonian, while the third and fourth diagram in figure 3.2 show the mixingcontributions.

Before calculating the eigenvalues of the Hamiltonians, it is useful to splitthem into two parts

H = Hcommon +Hspecific (3.2.4)

where the common Hamiltonian includes the Coulomb attraction, the orbit-orbit interaction and the relativistic correction, which are the same for all boundstates. Its energy correction is

δEcommon(nl) = −µα2

2n2− µα4

n4

[n

2l + 1− 3

8+

µ

8memp

]. (3.2.5)

Then, Hspecific gives the splitting between the supermultiplets. Since the Darwinterm only contributes to s-waves, while spin-orbit interactions are only nonzerofor l 6= 0, the cases l = 0 and l > 0 are treated separately.

Beginning with the ground state l = 0, one finds two different eigenvalues,each having four eigenstates for bosonic and fermionic bound states

∆E1 = 0

∆E2 = µα4

2n3 .(3.2.6)

The 2 · E1 eigenvalues belong to the 2 · Ω0 super multiplets while the 2 · E2

belong to the Ω 12

super multiplet.

In the case of a fixed l > 0 there are 8 × (2l + 1) bosonic and 8 × (2l + 1)fermionic states. To simplify the calculations, one splits the Hamiltonian intosubspaces described by mj of the total angular momentum. For bosonic boundstates one has, mj = ml = −l, ..., l− 1, l with j = l± 1, l such that the subspacebecomes

|ψ〉 ∈|lml, ↑↑〉 , |lml, ↓↑〉 , |lml, ↑↓〉 , |lml, ↓↓〉 ,|lml,++〉 , |lml,+−〉 , |lml,−+〉 , |lml,−−〉

.

(3.2.7)

While for fermionic bound states, one uses mj + 12 = ml = −l, .., l, l + 1 with

j = l ± 12

|ψ〉 ∈|lml, ↑ +〉 , |lml,− ↑〉 , |lml + 1, ↓ +〉 , |lml + 1,− ↓〉 ,|lml, ↑ −〉 , |lml,+ ↑〉 , |lml + 1, ↓ −〉 , |lml + 1,+ ↓〉

.

(3.2.8)

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 16

l=0 l=1 l=2

n=1

n=2

n=3

L

E

2 · Ω0

Ω 12

2 · Ω0

Ω 12

Ω 12

2 · Ω1

Ω 32

2 · Ω0

Ω 12

Ω 12

2 · Ω1

Ω 32

Ω 32

2 · Ω2

Ω 52

Figure 3.4: Energy spectrum of supersymmetric hydrogen. On the x-axis isthe angular momentum while the y-axis shows the energy level up to n = 3,taken from [2]. The energy value of a given supermultiplet is determined fromequation 3.2.10.

Diagonalization ofHspecific yields once again the same eigenvalues for the fermionicand bosonic sector. In total, one finds three different eigenvalues for the foursupermultiplets

Ωl+ 12

: ∆E1 = µα4

2(l+1)(2l+1)n3

2Ωl : ∆E2 = 0

Ωl− 12

: ∆E3 = − µα4

2l(2l+1)n3

, (3.2.9)

where ∆E1 and ∆E3 each have four eigenstates, and ∆E2 occurs with 2 · 4eigenstates. This specific l dependence makes it possible to express the energy ofa given supermultiplet only in terms of its total angular momentum j (superspin)

E(Ωj) = −µα2

2n2− µα4

n4

(n

2j + 1− 3

8+

µ2

8memp

). (3.2.10)

Therefore multiplets, of different l but with the same j are degenerate in energy.The spectrum is presented in figure 3.4.

Particle representations

The particle representations of Vl±( 12 ,0) which form the multiplets Ωj are

constructed by acting with the supercharges on a given state. In the rest framep = (me +mp, 0, 0, 0) the super algebra is

Qα, Qα = (me +mp)δαα (3.2.11a)

Qα, Qβ = Qα, Qβ = [Qα,H] = 0 . (3.2.11b)

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 17

The particles transform under a supertransformation according to

δξφe/p+ = ±√me/pξαψαe/p δξφ

∗e/p− = ±√me/pξαδ

ααψe/pα (3.2.12a)

δξψe/pα = ±1

2

√me/p(ξαε

ααφe/p+ + ξαφ

∗e/p−) . (3.2.12b)

It is then conventional to define creation and annihilation operators that act onthe Clifford vacuum

a†α =Qα√

me +mp→ a†α |λ〉e/p =

√me/p

me +mp|λ+

1

2〉e/p

(3.2.13)

such that they satisfy

aα, a†α = δαα . (3.2.14)

The particle representations of Vj are found by acting with the creation opera-tors on the state of lowest helicity. For example, given the multiplet Ω0 of theground state with l = 0 and starting from the state |+−〉 one finds

Ω0

V0 = |−+〉

V 12

= a†1 |−+〉=√

me

me+mp|↑ +〉 −

√mp

me+mp|− ↑〉

V 12

= a†2 |−+〉=√

me

me+mp|↓ +〉 −

√mp

me+mp|− ↓〉

V0 = a†1a†2 |−+〉

=mp

me+mp|−−〉+ me

me+mp|++〉+

√mpme

me+mp

(|↑↓〉 − |↓↑〉

).

(3.2.15)

For bound states with fixed l > 0, one must consider that the supercharges carrya spin themselves. Since there is a mixing of mj ’s all 4× 4× (2l+ 1) states fora given l-value have to be considered at the same time. The complete particlerepresentations of excited states can be found in [1, 27].

3.3 Hydrogen in N = 2

A supersymmetric hydrogen model with N = 2 is constructed from two massive(anti-)BPS (Bogomonlyi-Prasad-Sommerfeld) multiplets, built from a Cliffordvacuum Ω0. (Anti-) BPS multiplets are supermultiplets for which half of thecreation and annihilation operators are trivially satisfied. Hence, their particlerepresentation is significantly reduced compared to long multiplets. The N =2 BPS multiplets have the same particle representations as the two N = 1multiplets from the previous section; they contain a Dirac fermion and twocomplex scalars. However, this time the two scalars transform under a su(2)R 'so(3) symmetry as a consequence of N = 2. The massless gauge multipletcontains an additional gaugino (η) and a complex massless scalar (π). It is

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 18

described, in terms of N = 1 superparticles, by one chiral (Π) and one vector(V ) field. Summarizing this, the new Lagrangian reads [28]

L = Lgauge + Lmatter

=

[ ∫d2θ WW + h.c. +

∫d4θΠ†Π

]+

[ ∫d4θ(Φe+e+gV Φ†e+

+ Φe−e−gV Φ†e−)

+

∫d2θ[me(Φe+Φe− + Φ†e+Φ†e−)

+

∫d2θ√

2g[(Φe+Φe−Π + Φ†e+Φ†e−Π†)

]+ proton terms

].

(3.3.1)

The bound states of a BPS and anti-BPS multiplet bind together in leadingorder via exchange of a photon and a massless scalar. The resulting Coulombpotential is twice as strong as in usual QED and preserves the so(4) symmetry. Ifthe second multiplet were another BPS multiplet, contributions from scalar andphoton exchange would exactly cancel, such that there are no existing boundstates. Again, the energy spectrum receives relativistic corrections at O(α4).The states transform

S = (ΦBPSe )⊗ (Φanti-BPS

p ) = Φnon-BPSBS . (3.3.2)

The bound state is described by a massive long (vector) multiplet Φnon-BPS builtfrom a Clifford Vacuum Ω0. A quick check reveals that it contains the samenumber of degrees of freedom

j = 0 →(− 1,4×−1

2,6× 0,4× 1

2, 1

). (3.3.3)

Proposition:A product of a BPS with an anti-BPS multiplet will always transform like anon-BPS (long) multiplet.

Proof:A SUSY algebra with N > 1 in the rest frame p = (mi +mj , 0, 0, 0) is given by,

QAa , QbB = 2mδbaδAB (3.3.4a)

QAa , QBb = εabcAB (3.3.4b)

QaA, QbB = εabcAB (3.3.4c)

where I changed the su(2) indices α, α of the antisymmetric εab tensor to thelittle group so(3) with a, b and m = mi + mj . Rewriting the central chargesas cAB = cJAB where JAB is the antisymmetric tensor of the R-symmetry andc ∈ R, it is conventional to then define the creation and annihilation operatorsof the SUSY algebra as

aAa± ≡1√2

(QAa ± JABεabQbB

)(3.3.5)

which satisfy the reality condition

(aAa±)† = JABεabaBb± (3.3.6)

CHAPTER 3. SUPERSYMMETRIC HYDROGEN 19

πππ π

Figure 3.5: Four additional tree level diagrams of N = 2 supersymmetric hy-drogen corresponding to a massless scalar exchange.

and the anticommutation relations

aIa±, aJb± = (2m± c)JIJεab , aIa±, aJb∓ = 0 . (3.3.7a)

aIa±, (aJb±)† = (2m± c)δIJδba . (3.3.7b)

As a consequence of the positive finiteness of Hilbert space, one finds the BPScondition 2m ≥ |c|. For a BPS (ci = 2mi) multiplet this bound is saturated forhalf the operators such that,

→ aAa− = 0 = (aAa−)† (3.3.8)

are trivially realized. In the case of an anti-BPS (cj = −2mj) multiplet, thesame holds for aAa+ = (aAa+)† = 0. Hence, when taking the product of

ΦBPS ⊗ Φanti-BPS = Φnon-BPS (3.3.9)

all creation and annihilation operators of this model saturate the BPS condi-tion 2mBS > |c|. This follows from the triangle equality, where e.g. in mycase 2mBS > |ci + cj | ⇔ |me + mp| > |me −mp| is always satisfied. The nonBPS multiplet ΦBS is then built from a Clifford vacuum Ω0 and contains 22N−1

bosonic and 22N−1 fermionic states.

Hence, the energy spectrum transforms as a massive vector multiplet. This isin line with results found for positronium with N = 2 [29]. There, the splittingbetween ortho- and para-mutliplet of N = 1 superpositronium disappeared andthe groundstate is described by a single massive super multiplet.

The eigenenergies of the ground state can be determined with the procedureoutlined in the previous section. For that, one has to consider the additionaldiagrams coming from massless scalar exchanges, such as in figure 3.5, and theadditional contributions of the second gaugino, with diagrams similar to thosein 3.2 replacing λ with η.

The non-BPS multiplet transforms trivially under a Vl (so(3)) excitation,leaving a simplified energy spectrum with a single long multiplet for a givenvalue of l and n. This can be shown by working out the combined algebra ofthe annihilation and creation operators, with the so(3) rotations as is done inthe appendix A.3

Vl ⊗ Φnon-BPS = Φnon-BPSl . (3.3.10)

It appears now to be a simple straightforward task to determine the wholeenergy spectrum of the long multiplet. However, it turns out that the masslessscalar exchange gives rise to a number of problems, which I will further discussin chapter 5, since they also arise in the N = 4 model. The corrections foundin [29] for the ground state of N = 2 superpositronium suggest an explicitl-dependence for excited bound states.

Chapter 4

N = 4 Super Yang Mills

I started this thesis by showing that the Coulomb potential obeys a so(3) ⊗so(3) = so(4) symmetry, which implies an n2 degeneracy. Now, in 2015 Caron-Huot and Henn published a paper investigating hydrogenlike bound states in theCoulomb phase of N = 4 super Yang Mills theory. They found that relativisticcorrections up to order O(α3) preserve this n2 degeneracy in the large Nc limit.Their energy corrections are

δE(n) = − λ2m

64π2n2− λ3m

64π2n2

[ n∑k=1

(1

k

)+ log

λ

2πn− 1− 1

2n

]+O(λ4) (4.0.1)

where λ = g2Nc is the ’t Hooft coupling.Their calculations are based on the dual superconformal symmetry, which is

equivalent to so(4, 2) of N = 4 super Yang Mills theory [30]. The 2→ 2 vectorscattering with arbitrary masses is considered where bound states appear asresonances in the process. Introducing masses to the theory appears to leavethe so(4) symmetry unbroken. Their results hold for strong and weak coupling.

Further calculations of massive bound states and their amplitudes have beenperformed for strong coupling in [31, 32, 33, 34].

The goal of this chapter is to confirm their results by calculating the leadingrelativistic correction with the methods introduced in the previous chapters.Starting from the N = 4 Lagrangian, I will construct a hydrogen model con-taining electron and proton generation. Decomposing the bound state multipletin its irreducible representations makes it possible to determine the relativisticO(α4) energy corrections, by simply considering the scalar-scalar bound state.

This chapter is based on my work in the soon to be published paper [16].

4.1 The setup

The N = 4 super Yang Mills Lagrangian was first derived by Gliozzi, Scherkand Olive via Kaluza Klein reduction, from a ten-dimensional supersymmetricLagrangian [35]. A review of various supersymmetric compactification fromhigher dimensions can be found in [36]. Independent of their discovery, Grimm,Sohnius and Wess [37] constructed the same theory from N = 1 superfields infour spacetime dimensions. The authors all use slightly different notations and

20

CHAPTER 4. N = 4 SUPER YANG MILLS 21

conventions. I will begin from the Lagrangian as it was derived in [35]

L = tr

− 1

4FmpF

mp −4∑i=1

iψiσmDmψi

+g

2

6,4,4∑a,i,j

(caijBa[ψj , ψi] + caijBa[ψj , ψi]

)

− 1

2

6∑a=1

DmBaDmBa +

g2

4

6∑a,b=1

[Ba, Bb]2

.

(4.1.1)

The constants caij and caij are related to the Clifford Dirac matrices of the scalarfields so(6) ' su(4) symmetry. Dm is the non-Abelian covariant derivative,defined with the coupling g appearing in the commutator

Dm(·) = ∂m(·) + ig[νm, (·)] (4.1.2)

and Fmp is the non-Abelian field strength of the vector boson field ν

Fmp = ∂mνp − ∂pνm − ig[νm, νp] . (4.1.3)

All fields are in adjoint representation and can be expressed in terms of N ×Nmatrices.

Higgs mechanism

N = 4 super Yang Mills theory is scale invariant. As a consequence, all par-ticles are massless. A realistic hydrogen model, however, contains two fermionswith a mass gap of about 2000 units. Masses in N = 4 are given to the fields byapplying the Higgs mechanism. To keep the model as simple as possible, onlythe first scalar field B1 will acquire a nonzero vacuum expectation value

〈B1〉 = v

m1 0 · · · 00 m2 · · · 0...

.... . .

...0 0 · · · mN

= b1 ⇒ B1 = b1 + B1 (4.1.4a)

〈Ba〉 = 0 ∀ a 6= 1 . (4.1.4b)

The field B1 represents the Goldstone bosons, which arise from the symme-try breaking. By giving B1 a vacuum expectation value, the so(6) scalarsymmetry is broken to so(5). The U(N) representation of B1 is broken toU(N)→ U(1)j×U(N−j) with j being the number of masses for which mj 6= 0.

Mass terms

The Lagrangian in equation 4.1.1 contains a scalar-fermion-fermion interactionterm similar to the Yukawa terms in the standard model. Its B1 interactionthen yields the following fermion mass term

∆L = gcajitr(B1)stψtiuψ

ujv − (B1)stψ

tjuψ

uiv

= gc1ji[(B1)ssψ

siuψ

ujs − (B1)ssψ

sjuψ

uis

]= gc1ji(ms −mu)ψsiuψ

ujs .

(4.1.5)

CHAPTER 4. N = 4 SUPER YANG MILLS 22

There are mass terms (msu = gc1ji(ms −mu)) for all off-diagonal Weyl spinorsof each fermionic field. The factor of two disappears by interchanging i and j.Choosing the values of ms and mu appropriately, it is possible to construct anelectron and proton generation, e.g.

m1 = 0, m2 = 1, m3 = 2000 v = me/g . (4.1.6)

The Weyl spinors with U(1) charge (2, 1) form the electron generation, whilespinors charged under (3, 1) with m31 become the anti-proton generation. Sincethe proton is oppositely charged to the electron, it is useful to consider particlescharged with (1, 3) as the proton generation. The choice of indices is arbitraryas long as both pairs share a common U(1) charge, so that they can interactand form bound states.

Note that the diagonal elements of the fermion fields stay massless; they arethe gauginos that change fermions into scalars. Also due to the non zero valuesof c1ji there are exactly four (eight with the ψ couplings) mass terms in eachgeneration, giving four massive electrons.

The vector mass terms arise from the scalar’s kinetic term

∆L = −1

2trDmBaD

mBa

⇒ g2

2tr

[b1, νm][b1, νm]

= g2

(msmu +mums −mumu −msms

)νm us νsm u

(4.1.7)

which provides mass terms to the N2 −N off-diagonal vector bosons while theN diagonal elements stay massless ’photons’.

Scalar mass terms arise from the four point vertex in the scalar potential

∆L =g2

2tr[b1, Ba]2 = g2

(msmu +mums −mumu −msms

)Bua sB

sa u ,

(4.1.8)

resulting once again in N massless scalars and N2 − N massive ones. TheN2 − N off-diagonal Goldstone bosons B1 are eaten by the massive vectorswhile the N−diagonal elements becomes massless ’Higgs’. They couple to themass of the different particles and are the scalar superpartner to the photons.

Summarizing, one finds the mass

m ≡ |ms −mu| (4.1.9)

for every particle of a given generation (s, u).

Gauge fixing

For a complete picture I will briefly mention gauge fixing. Taking a closer

CHAPTER 4. N = 4 SUPER YANG MILLS 23

look at the coupling between B1 and the gauge bosons

∆L = −1

2trDm(b1 + B1)Dm(b1 + B1)

= −1

2tr∂mB1∂

mB1 − 2gi[b1 + B1, νm]∂mB1 − g2[b1 + B1, ν

m]2

= −1

2tr∂mB1∂

mB1 − 2gi[b1, νm]∂mB1 − 2gi[B1, ν

m]∂mB1

− g2[b1, νm]2 − g2[B1, ν

m]2 − 2g2[b1, νm][B1, ν

m]2

one finds the common terms for propagators, masses of the gauge bosons, inter-action vertices, once coupling to g for the vector boson exchange and the othertime coupling to the mass as in the usual Higgs mechanism. The second term,however, is not physical, since it couples the Goldstone bosons to the vectors.It contributes to the longitudinal mass component of the gauge bosons and canbe canceled by fixing the gauge.

Introducing G = ∂mνm + iξg[b1, B1]:

Lgf = −1

2trξ−1GG

= −1

2trξ−1∂mν

m∂nνn + 2ig[b1, B1]∂mν

m − ξg2[b1, B1]2

= −1

2trξ−1∂mν

n∂nνm + 2ig[b1, ν

m]∂mB1 − ξg2[b1, B1]2 .

(4.1.10)

Here, I partially integrated in the last line so that the second term takes thesame form as in the previous calculation. They have opposite signs and cancelwith each other.

(anti-)BPS multiplets

The Higgs mechanism breaks the so(6) ' su(4) scalar symmetry to so(5) 'usp(4). Each particle generation is described by a (anti-)BPS multiplet. Recall-ing from chapter 3, the creation and annihilation operators are

aIa±, (aJb±)† = −(2m± z)δIJδba . (4.1.11)

Here the indices a, b correspond to the little group so(3), and the indices I, J tothe R-symmetry usp(4). The rest symmetry is then described by so(3)⊗usp(4).The corresponding particle spectrum has been analyzed in the past, e.g. in [38][39]

Φ(anti-)BPS = (1,5)φ ⊕ (2,4)ψ ⊕ (3,1)ν . (4.1.12)

The first index stands for the representation of the little group so(3), while thesecond one is for the representation of the R-symmetry so(5). It confirms theexplicit results found in the previous analysis with five massive scalars, fourmassive fermions and one massive vector boson.

CHAPTER 4. N = 4 SUPER YANG MILLS 24

4.2 Energy spectrum

The previously constructed model allows a total of 256 possible bound states.There are

(ψe ⊗ ψp) = (4particles · 2Spin) · (4particles · 2Spin) = 64

(νe ⊗ νp) = (1particles · 3Spin) · (1particles · 3Spin) = 9

(νe ⊗ φp) = (1particles · 3Spin) · (5particles · 1Spin) = 15

(φe ⊗ νp) = (5particles · 1Spin) · (1particles · 3Spin) = 15

(φe ⊗ φp) = (5particles · 1Spin) · (5particles · 1Spin) = 25

→ 128 bosonic bound states

(4.2.1)

and similarly

(ψe ⊗ νp) = (4particles · 2Spin) · (1particles · 3Spin) = 24

(νe ⊗ ψp) = (1particles · 3Spin) · (4particles · 2Spin) = 24

(ψe ⊗ φp) = (4particles · 2Spin) · (5particles · 1Spin) = 40

(φe ⊗ ψp) = (5particles · 1Spin) · (4particles · 2Spin) = 40

→ 128 fermionic bound states.

(4.2.2)

In analogy to the previous chapter, it is useful to get a better understanding ofthe spectrum by analyzing it in terms of its symmetries. As in the N= 2 case,the two supermultiplets bind together in leading order by exchange of photonand massless ’Higgs’. The Coulomb attraction is twice as strong, but possiblybroken by relativistic effects. The spectrum then transforms as

S = ΦBPSe ⊗ ΦAnti-BPS

p = Φnon BPSBS (4.2.3)

a single multiplet built from the Clifford vacuum Ω0. Here, 2ze = |m1 − m2|and 2zp = |m3 −m1| such that 2mBS > ze + zp. A quick check reveals that thelong non-BPS N = 4 multiplet

1 × λ = ±2

8 × λ = ± 32

28 × λ = ±1

56 × λ = ± 12

70 × λ = 0

(4.2.4)

matches the 128 bosonic and fermionic bound states from 4.2.1 and 4.2.2. Fromappendix A.3 it is known that the long multiplet transforms trivially under so(3)excitations

Vl ⊗ (ΦBPSe ⊗ ΦAnti-BPS

p ) = ΦBSl . (4.2.5)

Since Φnon BPS is determined from the product of BPS and anti-BPS multiplets,it transforms with 4.1.12 as

ΦBPSe ⊗ Φanti-BPS

p

=((1,5)φ ⊕ (2,4)ψ ⊕ (3,1)ν

)e⊗((1,5)φ ⊕ (2, 4)ψ ⊕ (3,1)ν

)p

= 256

(4.2.6)

CHAPTER 4. N = 4 SUPER YANG MILLS 25

ea ea

pb pb

p1 q1

p2 q2

kγ

ea ea

pb pb

p1 q1

p2 q2

kh

ea eb

pc pd

p1 q1

p2 q2

Figure 4.1: The three tree level Feynman diagrams of e p → e p scattering. eand p denote to the scalar superpartners or electron and proton.

under so(5). The 256 can be decomposed into irreducible representation ac-cording to

νm 1 →

(νe, νp) 1⊗ 1 = 1

(νe, ψp) 1⊗ 4 = 4

(νe, Bp) 1⊗ 5 = 5

ψi 4 →

(ψe, νp) 4⊗ 1 = 4

(ψe, ψp) 4⊗ 4 = 16 = 1⊕ 5⊕ 10

(ψe, Bp) 4⊗ 5 = 20 = 4⊕ 16

Ba 5 →

(Be, νp) 5⊗ 1 = 5

(Be, ψp) 5⊗ 4 = 20 = 4⊕ 16

(Be, Bp) 5⊗ 5 = 25 = 1⊕ 10⊕ 14

ψi 4 → · · ·...

. (4.2.7)

Here, the 25 representation of the scalar-scalar bound state decomposes into thetrace 1, the antisymmetric 10 and the traceless symmetric 14 representation.Since the 14 only appears in the scalar-scalar bound states, its eigenvalue getsno contributions from mixing with other bound states. The energy eigenvalueof the long multiplet can then be determined solely from the 14 representation,as all representations have to share the same energy.

4.3 Energy splitting

There are three tree level diagrams presented in figure 4.1 for the scatteringprocess e p → e p. The two particle families interact via exchange of a (1,1)photon and higgs. Additionally, there is a contribution coming from the 4-pointscalar vertex. The first amplitude describing the exchange of a photon is

iM1 '4mempg

2

(~p− ~q)2

[1 + (~p2 + ~q2)

(1

4m2e

+1

4m2p

)− 1

4memp((1− ξ) (~p2 − ~q2)2

(~p− ~q)2− (~p+ ~q)2

)].

(4.3.1)

CHAPTER 4. N = 4 SUPER YANG MILLS 26

The second amplitude coming from a massless scalar exchange is simply

iM2 = 4mempg2

[1

k2

]. (4.3.2)

The first two amplitudes correspond to an overall shift to any scalar-scalarbound state, while the third one

iM3 ∝ ±g2 , (4.3.3)

has a magnitude, which depends on the involved particles. From the followingtable

aa ab ba bbaa 0 0 0 2g2

ab 0 −g2 −g2 0ba 0 −g2 −g2 0bb 2g2 0 0 0

one can determine its contribution to a given bound state. Since only M3 isresponsible for mixing within the scalar-scalar bound state, it solely determinesthe 14-eigenvalue, which is iM14 = −2g2. The normalized amplitude of M =M1 +M2 +M14 then becomes

MNR =g2

(~p− ~q)2

[2 +

1

memp

(~p2 +

(~p− ~q)2

4− (~p · (~p− ~q))2

(~p− ~q)2

)− (~p− ~q)2

4memp

− ~p2 + ~q2

4

(1

m2e

+1

m2p

)− (~p− ~q)2

2memp

],

(4.3.4)

which results in the energy potential

V (~r, ~p) = α

[− 1

memp

(~p2

2r+

(~r · ~p)2

2r3+ πδ3(~r)

)+~p2

r

(1

4m2e

+1

4m2p

)+

3πδ3(~r)

memp

]− ~p4

8

(1

m3e

+1

m3p

),

(4.3.5)

where momentum conservation was used ~p2 + ~q2 ∼ 2~p2. The energy correctionsare then once again determined with first order perturbation theory. Here, onehas to be careful, since the Coulomb potential is twice as strong, all αs comingfrom the EV carry an additional factor of two with them. The results for l 6= 0are then:

δEn,l(α4) =

α4µ

n4

[µ2

memp

(8n

2l + 1− 6

)−(

8n

2l + 1+ 4

)]. (4.3.6)

And the results for l = 0 are:

δEn,0(α4) = δEn,l=0(α4) +µα4

n3

[24µ2

memp

]=α4µ

n4

[µ2

memp

(8n+ 18

)−(

8n+ 4

)].

(4.3.7)

It is obvious due to the explicit l-dependence that the so(4) symmetry is broken.

Chapter 5

Discussion

The results from the previous chapter are in a strong contradiction with theobservation made by Caron-Huot and Henn [3], who found the n2 degeneracyto be preserved for relativistic corrections. In this chapter, I will attempt togive a complete discussion why this contradiction exists. For this, I begin withconsidering the even simpler model of a massive charged scalar coupled to astatic potential via a photon and a massless scalar. This is a simplified versionof my previous model, and also of hydrogen in N = 2 with a heavy proton in themp → ∞ mass limit. The system is evaluated by means of the Klein-Gordonequation and comparison to the time independent Schroedinger equation. Itturns out that there is a discrepancy between the field theoretical approach andrelativistic quantum mechanics.

Furthermore, I will comment on other issues which have been treated incon-sistently or neglected before, such as the choice of Coulomb gauge and othercross particle scattering diagrams, the missing O(α3) corrections, and finallyinteractions with other particles and massive intermediate particle states.

Alongside this I will give a discussion about future studies and directions.

5.1 Relativistic quantum mechanics

Consider a system of two charged scalars, both coupling to a massless vector(photon) and a massless scalar (’Higgs’). In the heavy proton limit, it is equiv-alent to a scalar (selectron) with mass m coupling to a static force field. Thesystem satisfies the slightly modified Klein-Gordon equation[

− (∂t − iVγ)2 + 2− (m− Vϕ)2

]φ = 0 (5.1.1)

where m denotes, from now on, the electron mass. The potentials are given by

Vγ = Vϕ =α

r− ∂t = iEt . (5.1.2)

This is in accordance with the leading order contributions from the field the-oretical approach of photon and scalar exchange. The 4-point vertex has noimpact, since the proton mass is taken to be mp → ∞. Then equation 5.1.1

27

CHAPTER 5. DISCUSSION 28

becomes [(E +m)(E −m+

2α

r) + 2

]φ = 0 . (5.1.3)

The energy is easily found by comparing this to the solution of the time inde-pendent Schroedinger equation, which can be derived via separation of variables[

− 22µ− α

r

]ψ = εψ → ε = −µα

2

2n2

⇔[− 1

2µ

(− (∂r)

2 − 2

r∂r +

l(l + 1)

r2

)− α

r

]ψ = εψ .

(5.1.4)

One finds

E −m = −(E +m

2

)(2α)2

2n2

⇔ E = m

(1− 2α2

n2 + α2

)= m

[1− 2α2

n2+

2α4

n4+O(α6)

].

(5.1.5)

In contradiction to the field theoretical calculations, this result is independentof l and preserves the so(4) symmetry, confirming [3].

It is known that the relativistic QM approach yields the same result as thefield theoretical setup in the heavy proton limit where the proton is approxi-mated as a static Coulomb source [24]. The Klein-Gordon equation for such amodel is given by [(

E +α

r

)2

+ 2−m2

]φ = 0 (5.1.6)

and can be compared to the time independent Schroedinger equation in 5.1.4.The solution is found in many books about relativistic quantum mechanics suchas [40]. The bound state energy is then

E2 −m2

2E= −1

2

α2E

x2x = n− α2

2l + 1+O(α4)

⇔ E = m

[1 +

α2

x2

]− 12

= m

[1− α2

2n2− α4

2n4

(n

l + 1/2− 3

4

)+O(α6)

](5.1.7)

which is equivalent to the result of the field theoretical approach coming fromthe scattering amplitude 4.3.1, and coincides with the fine structure splittingof atomic hydrogen in equation 2.2.14 by replacing j → l due to the lack ofspin-orbit interaction in a scalar-scalar bound state.

Hence, the problem must come from the massless scalar exchange. Thefield theoretical approach only using the ’Higgs’ exchange yields a E(α4) energycoming from the relativistic 2.2.11 and the normalization correction 2.2.8

δE(α4) = − 1

8m3〈φ| ~p4 |φ〉+

1

4m2〈φ| ~p

2

r|φ〉

= − (mα)4

8n4m3

[4n

l + 1/2− 3

]+

(α

m2

)(mα)3

4n4

[2n

l + 1/2− 1

]=mα4

8n4.

(5.1.8)

CHAPTER 5. DISCUSSION 29

The Klein-Gordon equation for a massless scalar exchange coupling to themass is [

E2 + 2−(m− α

r

)2]φ = 0 . (5.1.9)

Comparison with equation 5.1.4 yields for the energy

E2 −m2

2m= −1

2

α2m

x2x = n+

α2

2l + 1+O(α4)

⇔ E = m

[1− α2

x2

] 12

= m

[1− α2

2n2− α4

8n4+

α4

(2l + 1)n3+O(α6)

].

(5.1.10)

The energy corrections at α4 in equations 5.1.8 and 5.1.10 differ. One immedi-ately sees that the difference arises from a ’missing’ factor of two in front of thesecond operator in 5.1.8, coming from the normalization in the field theoreticalapproach.

This can be shown by comparing the operators in both approaches. Theenergy in terms of operators can be naively sketched (ignoring normal ordering)as

E ∼√m2 + ~p2 − 2mV + V 2 ∼ m

[1 +

~p2 − 2mV + V 2

m2

] 12

∼ m+~p2

2m− V +

V2

2m− ~p4

8m3+~p2V

2m2−V2

2m+O(α5)

∼ −mα4

8n4+

mα4

(2l + 1)n3

(5.1.11)

reproducing the energy obtained earlier. While in the field theoretical approach,one finds the operators

H −mp ∼ m+~p2

2m− V − ~p4

8m3+~p2V

4m2+O(α5) ∼ mα4

8n4. (5.1.12)

Here the difference of the previously mentioned factor of 2 becomes even moreclear.

The operator approach for the model with ’Higgs’ and photon exchangedescribed by the Klein-Gordon equation in 5.1.1 yields

E ∼ −V ±[V 2 + ~p2 +m2 − 2mV

] 12

∼ m− 2V +~p2

2m+V2

2m− ~p4

8m3+~p2V

2m2−

V2

2m+O(α5)

∼ 2α4

n4.

(5.1.13)

In the last line one has to be careful when taking the EV that one replacesα0 → 2α, since the Coulomb attraction is twice as strong. As a consequence, thel-dependence from equation 5.1.11 drops out. The energy correction at O(α4) in

CHAPTER 5. DISCUSSION 30

Figure 5.1: One loop cross scattering diagrams describing the exchange of gaug-ino and photon. There are many more possible diagrams involving differentcombinations of on shell scalars and fermions.

equation 5.1.5 is reproduced, showing that this procedure is consistent. Again,the difference between quantum mechanical and the field theoretical approaches

lies in the factor of two in front of the operator ~p2

r . Summarizing, there appearsto be an inconsistency in the field theoretical approach. This might be resolvedby considering one loop diagrams, as discussed in the next section.

5.2 Cross particle scattering

Derivation of all relevant operator terms with energy corrections of O(α4) wasbased on the usage of Coulomb gauge in the case of QED. As mentioned inchapter 2, it has been shown by Lindgren in 1990 [21] that only in this particularchoice of gauge the leading contributions, coming from cross photon scatteringdiagrams, such as shown in figure 2.2 cancel. The gauge dependent term inequation 2.2.7 or 4.3.1 contributes at energy δEcross ∝ α4. This can also naivelybe seen from the cross scattering diagram. Each photon contributes in leadingorder with the Coulomb potential 1

r ∼ α and there are four involved vertices,giving in total another α2.

The study of SQED bound states without an exchanged scalar as in [1] [2]then also made use of the choice ξ = 0. However, the question remains why sev-eral additional ’supersymmetric’ cross scattering diagrams involving gauginosand photons, such as shown in figure 5.1 should not contribute at O(α4). Onecould argue again that the leading operator coming from a single gaugino ex-change contributes with α2 and therefore these new diagrams should contributeat least O(α5). Further studies in the future are needed to indeed show thatcross gaugino scattering diagrams do not contribute at O(α4). The latter twodiagrams in figure 5.1 do not contribute to the 14 scalar-scalar bound staterepresentation of the N= 4 model.

In contrast to the cross gaugino scattering, cross scalar and mixed crossscalar and photon scattering will certainly contribute to the energy spectrumat O(α4). The diagrams presented in figure 5.2 are therefore relevant not onlyfor the N= 4 model, but also for the N= 2 case. This can be argued as Ihave done for the Coulomb gauge and cross photons. It becomes even moreclear when comparing scalar and photon propagators. They both give the sameleading order contribution ∼ µ

r . For a photon in Feynman gauge with ξ = 1, thegauge dependent term in equation 2.2.7 or 4.3.1, which contributes at O(α4)disappears, meaning that the cross scattering diagrams certainly contribute.For future studies it would be interesting if there is a choice of gauge, suchas Coulomb for QED, where all relevant terms contributing at O(α4) can besolely determined from tree level diagrams, in a theory involving a photon with

CHAPTER 5. DISCUSSION 31

ea ea

pb pb

ea ea

pb pb

ea ea

pb pb

Figure 5.2: Additional one loop diagrams which can contribute to the boundstate energy at α4.

a scalar superpartner.

5.3 Radiative corrections

It is known that radiative gaugino and photon correction for supersymmetrichydrogen contribute at O(α5) [2] and are therefore of no relevance for thisthesis. In [3], however, several energy corrections of O(α3) have been found, seeequation 4.0.1. They are attributed to a Lamb shift coming from ultrasoft scalarexchanges. Those are supposed to appear earlier by two orders in α, since theyare, in contrast to the photon, not dipole suppressed. Radiative corrections inQED are best investigated with dimensional regularization, a most symmetrypreserving method, introduced by ’t Hooft and Veltman [41].

The main contribution of the Lamb shift, except for the vacuum polarization,is the correction to the electron vertex. The scalar-scalar-Higgs electron vertexcontributes to a Feynman diagram with

Vssh = 2mg(1 + δVi +O(α2)

)(5.3.1)

where δVi is a correction of order α. Figure 5.3 shows the usual radiative threevertex correction. It is determined by integrating over the loop momentum k

δVi = (2mg)2

∫d4k

(2π)4

1

k2 −m2

1

(k + q)2 −m2

1

(k − p)2. (5.3.2)

The integral is solved by introducing Feynman parameters and Wick rotatingthe time coordinate by 90 degrees counterclockwise in the complex plane toobtain an Euclidean 4-momentum. It can then be analyzed with dimensionalregularization, which has been partly carried out in [42]. The solution suffersfrom infrared (IR) and ultraviolet (UV) divergences. While the UV divergencesshould cancel as a consequence from supersymmetry, and it should be IR pro-tected by the mass, a full treatment of all divergences goes beyond the scope ofthis thesis.

It is important to point out that, in order to get O(α3) energy corrections,one needs to find finite terms for δV1 independent of the momentum operator.This guarantees that they are essentially a correction to the Coulomb poten-tial, which are l independent and therefore automatically preserve the so(4)-symmetry. Moreover, these vertex corrections imply additional l independentcontributions at O(α4) coming e.g. from additional corrections to the protonvertex, which further highlights the incompleteness of the finite mass O(α4)calculation in chapter 4.

CHAPTER 5. DISCUSSION 32

e

h

e

p

k k + qp′

q

p− k

Figure 5.3: Radiative correction to the three scalar-scalar-Higgs vertex due toa Higgs exchange between the two selectrons.

e φ2x

p φx3

e e

p p

e e

p p

Figure 5.4: (i) Decay of the scalar-scalar bound state into two other scalars. (ii)+ (iii) Additional one loop diagrams involving two four point vertices and twovirtual bosonic particles.

5.4 Decay

The hydrogenlike bound state in N= 4 can decay over the four point scalarvertex

B21 aB

13 aB

3x bB

x2 b (5.4.1)

into various possible other particles as shown in the first diagram of figure 5.4The majority of these decays are suppressed by adjusting the masses in

section 4.1, such that the two outgoing particles with U(1) charges (2,x) and(x,3) are very heavy. Due to energy conversation, the two outgoing particleshave to be lighter than the bound state. Hence, all mis from section 4.1 have tobe non-zero. Their values are chosen to be mj >> m3,∀j > 3, so interactionswith particles of other generations can safely be neglected in the non relativisticlimit. The U(N) representation of B1 is therefore broken to U(1)N .

There is one interesting decay left; when the decay involves a massless scalar(φ) of charge (1,1), (2,2) or (3,3) the other decay particle (2,3) or (3,2), fromnow on called χ, has a smaller mass mχ = mp −me < mp +me = mbound thanthe bound state. Hence, there is a significant possibility for the ground state todecay. The decay width is then proportional to [20]

Γ(e p→ χφ) ∝ |M|2|ψ(0)|2 ∼ α5 . (5.4.2)

In section 5.2, contributions from loop diagrams involving the exchange of twobosons have been discussed. This can also happen over two four vertices aspresented in (ii) and (iii) of figure 5.4. These two diagrams, which allow theexchange of any kind of boson, are in fact a correction to the four point vertex.Hence, they contribute at energies of O(α5).

CHAPTER 5. DISCUSSION 33

e e

p p

p1 q1

χ

p2 q2

Figure 5.5: Electron and proton decay into a χ which then decays again into anelectron and proton. This diagram can give contributions of O(α4).

There is also another allowed three vertex decay into the massive vectorboson

e p→ χ→ e p . (5.4.3)

A comparison to positronium shows that the annihilation of electron and positronto a photon and the subsequent decay into electron and positron does give abound state energy contribution at O(α4). This suggests that the diagramshown in figure 5.5 also gives contributions at the same order. However, thereis one major difference to the positronium case χ is a massive vector. It there-fore seems more reasonable to compare its contribution to the exchange of aZ boson in atomic muonium. Here, it was found that it indeed contributes atO(α4), but is once again heavily suppressed by the mass [43]. This also holdsfor the intermediate state of a χ particle. Furthermore, the on-shell χ belongsto a massive BPS multiplet, hence, this diagram will not contribute to the 14representation of the non BPS bound state.

Chapter 6

Outlook

The analysis in the previous chapters of the different energy spectra has shownthat splitting with more supersymmetry is milder and simplified. In real worldhydrogen, the spectrum shows a dependence on the angular momentum l andspin S of the involved fermions; in SQED it only depends on the superspin j ofa given multiplet. My analysis showed that for N= 2 and N= 4 the spectrumcontains only a single massive non-BPS multiplet for any given excitation. Inthe heavy proton limit, I found when using relativistic quantum mechanics thateven the so(4) Coulomb symmetry is preserved for energy corrections at O(α4).

On the other hand, the discussion in the previous chapter showed the limitsof the field theoretical approach used for the majority of this thesis. A fulltreatment of relativistic corrections at O(α4) is a tedious exercise and involvesthe calculation of several different loop diagrams, such as the cross scattering offorce carriers and radiative corrections. It would be interesting to find anothersimpler approach to tackle bound state problems involving a massless scalar asa superpartner of the photon, which can also reproduce the O(α3) correctionsfound in [3].

In the case of bound state problems in QED, there are effective field theoriesthat greatly simplify the calculations. One popular theory is (p)NRQED. Non-Relativistic QED (NRQED) was proposed by Caswell and Lepage in 1986 [44]. Itis based on the fact that there are three different energy scales in non relativisticproblems: the masses of the involved particles (also called hard scale ∼ m), theparticle momentum (soft scale) ∼ mα and the utrasoft scale describing kineticand bound state energy ∼ mα2.

As a consequence of that, usual Feynman diagrams contribute at many dif-ferent orders in the coupling constant α. NRQED integrates out the hard scale∼ m and, after matching the one loop results between QED and NRQED, it isa straightforward task to reproduce the usual Lamb shift by introducing a cutoff and photon mass [45].

pNRQED is then an extension of NRQED. Here, in addition to the hardscale, the soft scale is also integrated out. It is therefore of particular inter-est for the so(4) bound state problem, where the O(α3) corrections found byCaron-Huot and Henn were attributed to ultrasoft scalars. Using the tech-niques of dimensional regularisation pNRQED was used to calculate Lamb shiftcorrections [46].

For future studies we intend to use the pNRQED framework, and extend it

34

CHAPTER 6. OUTLOOK 35

to incorporate the contribution of an additional massless scalar field. At thesame time, the Hamiltonian can be heavily simplified by only considering ascalar-scalar bound state, dropping all spin dependent operators. This wouldthen be similar to the calculations of a static potential in N= 4 super YangMills, which have been carried out in 2007 by Pineda [47].

Acknowledgements

I want to thank my supervisor Professor Yuji Tachikawa for many interesting andproductive discussions and giving me this great opportunity to research one yearat Kavli IPMU. Next, I am grateful to Professor Guido Festuccia for acceptingme as a master student and providing me with valuable feedback from the otherside of the world. I am happy to have enjoyed many interesting discussionsabout physics over the past months with Itamar Yaakov. I would also like tothank Benjamin Thorne and Xiangchong Li for the great company during lunch,tea time or the many hours spend in the office. The collaboration with SakataYusuke and Takemasa Yamaura is greatly acknowledged. Furthermore, I wantto thank Thomas Dawson for proofreading and helping me with the language.Finally, I want to thank my family for all the support no matter where I go.

36

Appendix A

Appendix

A.1 Conventions and identities

Conventions

I use the metric ηnm = diag(−1, 1, 1, 1). The gamma matrices are defined by

γm =

(0 σm

σm 0

)γ5 =

(−i 00 i

)(A.1.1)

where σ = (−1, ~σ), σ = (−1,−~σ). The epsilon tensor used in this thesis is

εαβ =

(0 1−1 0

)= −εαβ , which is needed to raise and lower spinor indices

ψα = εαβψβ . Anti-spinors are defined according to u(k) = u†(k)γ0.

Superfields

The particle representation of a N = 1 chiral field is

Φ = φ+√

2θψ + θθF . (A.1.2)

A vector field in Wess-Zumino gauge contains the particles

V = −θσmθνm + iθθθλ− iθθθλ+1

2θθθθD (A.1.3)

from which follows the Abelian superfield strength

Wα = − iλα +[δβαD −

i

2(σmσn)βα(∂mνn − ∂nνm)

]θβ + θθσmαα∂mλ

α . (A.1.4)

D,F are auxiliary fields that can be eliminated with the equations of motion.

Identities

Making use of the non relativistic approximation p0 ≈ m+ ~p2

2m , one finds

√σ · q ≈ (σ · q +m)√

2(q0 +m)≈

(~σ · ~q + 2m+ ~q2

2m )

2√m(1 + ~q2

8m2 )(A.1.5)

37

APPENDIX A. APPENDIX 38

from which follows

√σ · q√σ · p ≈ m+

1

2~σ(~q + ~p

)+

1

4m

(1

2(~p+ ~q)2 − i(~p× ~q)~σ

), (A.1.6a)(√

σ · q ~σ√σ · p+

√σ · q ~σ √σ · p

)~k ≈ ~σ · (~q + ~p) ~σ · ~k + 2i(~p× ~k) · ~σ ,

(A.1.6b)(√σe · qσe m

√σe · p

)(√σp · pσmp

√σp · q

)≈ −(~q + ~p)2 − (~p− ~q)2~σe · ~σp

− 2i(~p× ~q) · (~σp − ~σe)− (~p− ~q) · ~σe(~p− ~q) · ~σp .

(A.1.6c)

Another useful identity is

(~p+ ~q)2 − (~p2 − ~q2)2

(~p− ~q)2= 4

(~p2 − (~p · (~p− ~q))2

(~p− ~q)2

). (A.1.7)

Propagators

Propagators for gaugino, scalar and photon are given below.

Photon: − i[ηmnk2 + iε

− (1− ξν)kmknk4

]. (A.1.8a)

Gaugino:i/p

p2 + iε. (A.1.8b)

Scalar:i

p2 + iε. (A.1.8c)

so(6) constants

Taken from [35] the so(6) constants of the N= 4 Lagrangian are

c1ij =

(0 σ1

−σ1 0

), c2ij =

(0 −σ3

σ3 0

)c3ij =

(iσ2 00 iσ2

), c4ij =

(0 −σ2

−σ2 0

)c5ij =

(0 i−i 0

), c6ij =

(σ2 00 −σ2

).

A.2 (BS-)Wave function and expectation values

The hydrogen bound state wavefunction is given by,

ψnlm(~r) = (µα)32

2

n2

√(n− l − 1)!

(n+ l)!

(2rµα

n

)lexp

(− rµα

n

)· L2l+1

n−l−1

(2rµα

n

)· Ylm(θ, φ) ,

(A.2.1)

where L and Y are the Laguerre polynomials and spherical functions. Theycan be called in Mathematica under LaguerreL[a,b,x] and SphericalHarmon-icY[l,m,theta,phi].

APPENDIX A. APPENDIX 39

Expectation values

EVs used in this thesis are

〈nlm| 1r|nlm〉 =

µα

n2(A.2.2a)

〈nlm| 1

r2|nlm〉 =

(µα)2

n3(l + 1/2)(A.2.2b)

〈nlm| ~p2

r|nlm〉 =

(µα)3

n4

[2n

l + 1/2− 1

](A.2.2c)

〈nlm| ~p4 |nlm〉 =(µα)4

n4

[4n

l + 1/2− 3

](A.2.2d)

〈nlm|[

(~r · ~p)2

r3+ 2πδ3(~r)

]|nlm〉 =

(µα)3

n4

[n

l + 1/2− 1

](A.2.2e)

〈n00|πδ3(~r) |n00〉 =(µα)3

n3. (A.2.2f)

Additional EVs which are needed for spin-orbit and hyperfine structure correc-tions are found in [1].

A.3 Excitation of a non BPS multiplet

I am interested in how a non-BPS multiplets, the bound state of a BPS with ananti-BPS particle, transforms under Vl ∼ so(3) excitations, i.e.

Vl ⊗ (ΦBPSe ⊗ ΦAnti-BPS

p ) . (A.3.1)

The following analysis is for the case when the R-symmetry is so(5), i.e. N= 4;in the case of N= 2, one has to replace so(5) with so(3) and so(8) with so(6)etc. The analysis for N= 2 and N= 4 is then, in principle, the same. It isuseful to rewrite the creation and annihilation operator (aIa±, (a

Jb±)†) of the

long multiplet with the index pair (I, a) in terms of a single index µ, runningfrom 1 to 8 for usp(4)R⊗su(2)little ' so(5)⊗so(3). After appropriate rescaling,the creation and annihilation operators satisfy the following anti commutationrelations (aµ± → γµ±)

γµ±, γν± = 2δµν , γµ±, γν∓ = 0 . (A.3.2)

It is known that so(3) ⊗ so(5) ∈ so(8), where γµ± is a vector in so(8). Therotations of so(8) Mµν satisfy the algebra

[Mµν ,Mρσ] = δµρMνσ − δµσMνρ − δνρMµσ + δνσMµρ . (A.3.3a)

Then one can define the operators

Sµν± =1

4[γµ±, γ

ν±] (A.3.4)

which satisfy

[Sµν± , γρ±] = δµργν − δνργµ = [Mµν , γρ±] (A.3.5)

APPENDIX A. APPENDIX 40

and

[Sµν± , Sρσ± ] = δµρSνσ − δµσSνρ − δνρSµσ + δµσSνρ

= [Mµν , Sρσ± ] .(A.3.6)

Finally, the operator

Tµν ≡Mµν − Sµν+ − Sµν− (A.3.7)

satisfies the usual so(8) commutation relations, and also commutes with γµ±.The algebra spanned by Tµν , γµ± is equivalent to Mµν , γµ±. Since γµ± is arepresentation of ΦBS, one finds that it transforms trivially under a so(3) Vlexcitation with trivial so(5) transformation.

Bibliography

[1] Christopher P Herzog and Thomas Klose. The perfect atom: boundstates of supersymmetric quantum electrodynamics. Nuclear Physics B,839(1):129–156, 2010.

[2] Tomas Rube and Jay G Wacker. The simplicity of perfect atoms: De-generacies in supersymmetric hydrogen. Journal of Mathematical Physics,52(6):062102, 2011.

[3] Simon Caron-Huot and Johannes M Henn. Solvable relativistic hydrogen-like system in supersymmetric yang-mills theory. Physical review letters,113(16):161601, 2014.

[4] Niels Bohr. On the constitution of atoms and molecules. In Niels Bohr,1913-2013, pages 13–33. Springer, 2016.

[5] Wolfgang Pauli. uber das modell des wasserstoffmolekulions. Annalen derPhysik, 373(11):177–240, 1922.

[6] E. Schrodinger. Quantisierung als Eigenwertproblem. Annalen der Physik,384:361–376, 1926.

[7] P. A. M. Dirac. The Quantum Theory of the Electron. Proceedings of theRoyal Society of London Series A, 117:610–624, February 1928.

[8] E. Fermi. Uber die magnetischen Momente der Atomkerne. Zeitschrift furPhysik, 60:320–333, May 1930.

[9] W. E. Lamb and R. C. Retherford. Fine Structure of the Hydrogen Atomby a Microwave Method. Physical Review, 72:241–243, August 1947.

[10] Hans A Bethe. The electromagnetic shift of energy levels. Physical Review,72(4):339, 1947.

[11] Michael I Eides, Howard Grotch, and Valery A Shelyuto. Theory of lighthydrogenlike atoms. Physics Reports, 342(2):63–261, 2001.

[12] Morad Aaboud et al. Search for new phenomena in a lepton plus high jetmultiplicity final state with the ATLAS experiment using

√s = 13 Tev

proton-proton collision data. 2017.

[13] Wilfried Buchmuller, ST Love, and Roberto D Peccei. Supersymmetricbound states. Nuclear Physics B, 204(3):429–450, 1982.

41

BIBLIOGRAPHY 42

[14] P. Di Vecchia and V. Schuchhardt. N = 1 and N = 2 SupersymmetricPositronium. Phys. Lett., B155:427–431, 1985.

[15] Juan Maldacena. The large-n limit of superconformal field theories andsupergravity. International journal of theoretical physics, 38(4):1113–1133,1999.

[16] Sakata Yusuke, Robin Schneider, Yuji Tachikawa, and Takemasa Yamaura.to be published.

[17] Julius Wess and Jonathan Bagger. Supersymmetry and supergravity.Princeton university press, 1992.

[18] Myron Bander and Claude Itzykson. Group theory and the hydrogen atom(i). Reviews of modern Physics, 38(2):330, 1966.

[19] Jun John Sakurai and Jim Napolitano. Modern quantum mechanics.Addison-Wesley, 2011.

[20] Michael Peskin and Dan Schroeder. An introduction to quantum fieldtheory. 1995.

[21] I Lindgren. Gauge dependence of interelectronic potentials. Journal ofPhysics B: Atomic, Molecular and Optical Physics, 23(7):1085, 1990.

[22] Gregory Breit. The effect of retardation on the interaction of two electrons.Physical Review, 34(4):553, 1929.

[23] WA Barker and FN Glover. Reduction of relativistic two-particle waveequations to approximate forms. iii. Physical Review, 99(1):317, 1955.

[24] Claude Itzykson and Jean-Bernard Zuber. Quantum field theory. CourierCorporation, 2006.

[25] Siavosh R Behbahani, Martin Jankowiak, Tomas Rube, and Jay G Wacker.Nearly supersymmetric dark atoms. Advances in High Energy Physics,2011, 2011.

[26] Julian Schwinger. Coulomb green’s function. Journal of MathematicalPhysics, 5(11):1606–1608, 1964.

[27] Tomas Rube and Jay G. Wacker. The Simplicity of Perfect Atoms: Degen-eracies in Supersymmetric Hydrogen. J. Math. Phys., 52:062102, 2011.

[28] Aleksey Cherman, Saso Grozdanov, and Edward Hardy. Searching for fermisurfaces in super-qed. arXiv preprint arXiv:1308.0335, 2013.

[29] P Di Vecchia and V Schuchhardt. N= 1 and n= 2 supersymmetric positro-nium. Physics Letters B, 155(5):427–431, 1985.

[30] J. M. Drummond, J. Henn, G. P. Korchemsky, and E. Sokatchev. Dualsuperconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory. Nuclear Physics B, 828:317–374, March 2010.

BIBLIOGRAPHY 43

[31] Luis F. Alday, Johannes M. Henn, Jan Plefka, and Theodor Schuster. Scat-tering into the Fifth Dimension of N = 4 Super Yang-Mills. JHEP, 01:077,2010.

[32] Simon Caron-Huot and Donal O’Connell. Spinor Helicity and Dual Con-formal Symmetry in Ten Dimensions. JHEP, 08:014, 2011.

[33] Tristan Dennen and Yu-tin Huang. Dual Conformal Properties of Six-Dimensional Maximal Super Yang-Mills Amplitudes. JHEP, 01:140, 2011.

[34] Jan Plefka, Theodor Schuster, and Valentin Verschinin. From Six to Fourand More: Massless and Massive Maximal Super Yang-Mills Amplitudesin 6D and 4D and Their Hidden Symmetries. JHEP, 01:098, 2015.

[35] Ferdinando Gliozzi, Joel Scherk, and Dl Olive. Supersymmetry, supergrav-ity theories and the dual spinor model. Nuclear Physics B, 122(2):253–290,1977.

[36] Lars Brink, John H Schwarz, and Joel Scherk. Supersymmetric yang-millstheories. Nuclear Physics B, 121(1):77–92, 1977.

[37] R Grimm, M Sohnius, and J Wess. Extended supersymmetry and gaugetheories. Nuclear Physics B, 133(2):275–284, 1978.

[38] J Strathdee. Extended poincare supersymmetry. International Journal ofModern Physics A, 2(01):273–300, 1987.

[39] W. Nahm. Supersymmetries and Their Representations. Nucl. Phys.,B135:149, 1978.

[40] Hans A. Bethe and Roman Jackiw. Intermediate Quantum Mechanics.Addison-Wesley, 3rd edition, 1997.

[41] M Veltman et al. Regularization and renormalization of gauge fields. Nu-clear Physics B, 44(1):189–213, 1972.

[42] M Veltman et al. Scalar one-loop integrals. Nuclear Physics B, 153:365–401,1979.

[43] Michael I Eides. Weak-interaction contributions to hyperfine splitting andlamb shift. Physical Review A, 53(5):2953, 1996.

[44] William E Caswell and G Peter Lepage. Effective lagrangians for boundstate problems in qed, qcd, and other field theories. Physics Letters B,167(4):437–442, 1986.

[45] Patrick Labelle and S. Mohammad Zebarjad. Derivation of the Lamb shiftusing an effective field theory. Can. J. Phys., 77:267–278, 1999.

[46] Antonio Pineda and Joan Soto. The lamb shift in dimensional regularisa-tion. Physics Letters B, 420(3):391–396, 1998.

[47] Antonio Pineda. The Static potential in N = 4 supersymmetric Yang-Millsat weak coupling. Phys. Rev., D77:021701, 2008.