A Phase space integrals for 2-, 3-, and 4-particle...

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A Phase space integrals for 2-, 3-, and 4-particle production

We introduce the definition of the general final state phase space inte-

gral and work out the details for 2- to 4-particle phase spaces.

Important applications, which are treated in these lectures, are:

• Decay of a particle into two particles, e.g.

π+→ µ++ νµ,e++ νe,H → f + f̄ ,

Z → f + f̄ ,

W±→ f1+ f̄2

• Decay of a particle into three particles, e.g.

µ−→ e−+ ν+ ν̄,

p → n+ e++ ν

• Scattering processes with two-particle final states, e.g.

e+e−→ f + f̄ ,Z+Z,Z+H,W++W−,

pp̄ → t+ t̄,

ep → e+X,

νN → ν+N,µ+N′, ν+X,µ+X′

• Scattering processes with three-particle final states, e.g.

e+e−→ f + f̄ +γ,q+ q̄+g,

ep → eXγ,

νN → ν+N +γ,

µ+N′+γ

• Scattering processes with four-particle final states, e.g.

e+e−→ (W++W−,ZZ,ZH) → f1+ f̄2+ f3+ f̄4,

e+e−→ f + f̄ +2γ,

gg → q+ q̄+2g

It is common to all these reaction that the squared matrix elements

may depend on invariants only. After summing and avaraging over

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spin degrees of freedom, there are left the scalar products pip j, and we

also know that p2i= m2

i. Momentum conservation holds, so there are

only n+ 1 independent momenta. For a 2 → n reaction, this leads to

(n+1)n/2 different products (i , j) which have to be expressed by the

chosen kinematical variables. The true number of degrees of freedom

is smaller than that number. For a 2 → 2 reaction, it is effectively one,

namely besides the initial state invariant mass squared s, there is yet

the scattering angle θ (or, equivalently, t). Any additional final state

particle adds three degrees of freedom due to its three spatial momen-

tum components, so we get, for spinless problems with n final state

particles N(n) = 2+ 3(n− 2) degrees of freedom, and so the reaction

depends on only N different scalar products of invariants, and all the

others may be (and have to be) expressed by them.

The various decays and scattering processes depend on quite differ-

ent sets of observable kinematical quantities, often defined in a specific

reference frame, e.g.:

• the center-of-mass system (cms), where the compound of two col-

liding beam particles, or a decaying particle, is at rest;

• the laboratory system, where the target hit by a beam is at rest.

For this reason, we have to study several final state phase space param-

eterizations, and we will do this in a systematic way.

A systematic presentation of elementary particle kinematics may be

found in [1].

A.1 Kinematics and Phase Space Parameterizations

Cross-sections and decay rates are the basic observables in the study

of elementary particles and their interactions. The cross-section for the

reaction:

Pa(pa)+Pb(pb) → F1(p1)+ · · ·Fn(pn), (A.1)

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is related to the squared matrix M element by the following relation,

with s = (pa+ pb)2:

dσ(2 → n) =(2π)4

2

√

λ(s,m2a,m

2b)

× |M|2× dΦn(pa+ pb; p1, . . . , pn),

(A.2)

dΦn(pini; p1, . . . , pn) = P×δ4

pini−∑

i

pi

n∏

i=1

d4piδ(

p2i +m2

i

)

θ(p0i ),(A.3)

√

λ(s,m2a,m

2b) = 2

√

(papb)2−m2am2

b. (A.4)

For vanishing initial state masses, it is just√λ(s,0,0) = 2s. Here we

use the Kallen function:

λ(a,b,c) = a2+b2+ c2−2ab−2bc−2ca

= (a−b− c)2−4bc

= a2−2a(b+ c)+ (b− c)2

=

[

a−(√

b+√

c)2] [

a−(√

b−√

c)2]

=(√

a−√

b−√

c) (√

a+√

b+√

c) (√

a−√

b+√

c) (√

a+√

b−√

c)

.

(A.5)

Further, P is a combinatorial factor arising if k groups of p j identical

particles (e.g. several photons) are produced:

P =

k∏

j=1

1

p j!. (A.6)

Often one writes this factor into the definition of the matrix element

square.

The definition of the decay width is similar:

dΓ(1 → n) =(2π)4

2M|M|2dΦn(p; p1, . . . , pn). (A.7)

For the applications in these lectures, final states with two to four par-

ticles are of interest.

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A.2 The 3-momentum of one particle in the rest frame of another particle

Sometimes we will need the velocity β(p), or the module of the 3-

momentum |~p| of one particle (with 4-momentum p and mass m):

β(p) ≡ |~p|p0

=|~p|

√

|~p|2+m2, (A.8)

in the rest frame of another particle (with 4-momentum k and mass M).

This relation may be elegantly expressed using the Kallen function

(A.5):

|~p| = 1

2M

√

λ[p2,k2, (k+ p)2]∣

∣

∣~k=0. (A.9)

The last equation may be got as follows. When

~k = 0, (A.10)

k2 = M2, (A.11)

p2 = m2, (A.12)

it is:

(p+ k)2 = m2+M2+2p0M, (A.13)

Thus,

p0 =1

2M

[

(p+ k)2−m2−M2] ∣

∣

∣~k=0, (A.14)

and with use of |~p| =√

(p0)2−m2, eq. (A.9) follows immmediately.

For the production of two particles with equal masses m from a state

at rest with invariant mass M =√

s, one obtains e.g.:

β =

√

1− 4m2

s. (A.15)

If s < 4m2, then β = 0. The β =√

λ(s,m21,m2

2)) is the threshold function

in particle production, as will become clear in section A.3.

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The special case of production of a massive and a massless particle

corresponds to

β =√

λ(s,M2,0)) =

√

1− M2

s. (A.16)

A.3 Two-particle final state

Production of two particles, e.g. in the reaction:

e+(pa)+ e−(pb) → f1(p1)+ f2(p2), (A.17)

may be described by two 4-momentum dependent functions, leav-

ing aside for a moment spin. This may be easily seen. With ac-

count of global 4-momentum conservation and the mass-shell condi-

tions p2i=m2

i, only two of the six possible products of 4-momenta pip j

are independent. One may choose e.g.:

s = (pa+ pb)2, (A.18)

t = (pa− p1)2. (A.19)

Besides s and t, a third invariant u often is introduced:

u = (pa− p2)2. (A.20)

These three invariants are related by 4-momentum conservation:

s+ t+u = m2a+m2

b+m21+m2

2 (A.21)

A more symmetric notation considers all momenta as incoming, where

then

si j = (pi+ p j)2 = s, t,u for i j = ab,a1,a2, with s+ t+u = Σm2

i .(A.22)

The most general differential cross-section for two-particle production

is thus single-differential:

dσ

dt. (A.23)

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= dΦ2(1,2)

Figure A.1: The 2-particle phase space.

This, and the total cross-section

σtot =

∫ t1

t0

dσ

dt(A.24)

are the measurable quantities in this specific case. The integration

boundaries are either the extreme values allowed by the process kine-

matics or limited by the specific experimental set-up.

Any additional final state particle n+1 adds an additional 4-momentum

vector, with boundary condition p2n+1

= −m2n+1

, and thus three addi-

tional degrees of freedom.

Now we derive a parametrization of the final state phase space which

is adapted for applications.

We will use for the D4 introduced in (A.3):∫

dxg(x)δ[ f (x)−a] =g(x)

f ′(x)

∣

∣

∣ f (x)=a , (A.25)

from which follows for a particle:

D4p ≡ d4pδ(

p2−m2)

θ(p0) =d3~p

2p0(A.26)

and introducing spherical coordinates:

D4p = =|~p|22p0

d|~p|d cosθdφ. (A.27)

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In e+e− annihilation, e+e−→ f̄ f , a convenient kinematical system is

the center of mass system. For µ pair production:

~pa+ ~pb = ~p1+ ~p2 = 0, (A.28)

s = 4E2e = 4E2

µ, (A.29)

βe =

√

1−4m2

e

s, (A.30)

βµ =

√

1−4m2µ

s. (A.31)

For Z boson decay, Z → f̄ f , one has to replace s → M2Z.

If, as in W± decay, W → f̄1 f2, two particles with different masses

are produced, this generalizes because then the 3-momenta agree, but

no more E1 , E2. In fact, from (A.9) it follows:

|~p2| =1

2√

MW

√

λ(m22,M2

W, (p2+ pW)2)

=1

2√

MW

√

λ(m22,M2

W,m2

1) (A.32)

because (p2+ pW)2 = (−p1)2 =m21. The energies are then easily derived:

E1,2 =1

2√

MW

(

M2W +m2

1,2−m22,1

)

. (A.33)

We now are prepared to study the general two particle phase space:

dΦ2(1,2) = δ4(p12− p1− p2)×D4p1×D4p2

=d3~p1

2p01

δ4(p12− p1− p2)×D4p2

=d3~p1

2p01

δ1[

(p12− p1)2−m22

]

. (A.34)

A symbolic figure is shown in Fig. A.1.

The further evaluation is convenient in the rest system of ~p12. This

situation is experimentally realized at e+e− colliders when a pair of

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particles (often with equal masses) is produced since the momenta of

the incident electron and positron are balanced. Then it is:

~p12 = 0 (A.35)

and ~p1 and ~p2 are opposite to each other. It is further understood that

p212 = M2

12 ≡ s. (A.36)

Introducing spherical coordinates, we may go on:

dΦ2(1,2) = d cosθ12dφ12

|~p1|2d|~p1|2p0

1

δ[

−M212+2M0

12p01−m2

1+m22

]

= d cosθ12dφ12

|~p1|4M12

= d cosθ12dφ12

λ1/2(M212,m2

1,m2

2)

8M212

(A.37)

We used the relation |~p1|d|~p1| = p01dp0

1, see (A.9). The angles are cho-

sen in the center-of-mass system (c.m.s.), i.e. θ12 is the scattering an-

gle θ of one of the produced fermions, e.g. of f −. The integration over

dφ12 yields a factor of 2π, because usually the quared matrix element

is independent of that angle.14 Thus, finally we obtain:

dΦ2(1,2) =π

4

λ1/2(s,m21,m2

2)

sd cosθ, (A.38)

where, for two equal mass particles, the ratio β = λ1/2(s,m2,m2)/s is

the velocity of the produced particles. The β is the so-called threshold

function because at too small values of s (i.e. below the production

threshold) it vanishes. The integration boundaries are trivially known:

−1 ≤ cosθ ≤ +1. (A.39)14Evidently, it is important to know on which variables the integrand depends and how they are related to the integration

variables. If one wants to determine a forward-backward asymmetry, then it is mandatory to use the corresponding

scattering angle as one of the phase space parameters. For a three-particle phase space, this complicates the situation, see

section xxx ??.

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The invariant differential cross-section with respect to t may be ob-

tained from Eq. (A.19). It is:

t = −2EeE1 (1−βeβ1 cosθ1)+m2e +m2

1, (A.40)

where β are the velocities and θ1 the angle between e− and particle 1.

E.g. in the cms, it is 2EeE1 = (s+m21−m2

2)/2 and θ1 the scattering

angle. As mentioned, the cosine of θ1 varies in the interval (-1,+1).

This way, the integration limits t0, t1 in Eq. (A.24) may be determined.

Further, it is:

dt =1

2

√

λ(s,m2e,m

2e)

√

λ(s,m21,m2

2)d cosθ1. (A.41)

The third invariant introduced, u, is analogously:

u = −2EeE2 (1−βeβ2 cosθ2)+m2e +m2

2, (A.42)

and for two-particle production in the cms it is cosθ1 =−cosθ2. Easily,

relation (A.21) is recovered. If additionally m1 = m2, we also have

β1 = β2.

In cases like Bhabha scattering, the total cross-section without applying cuts (here: angular cuts) doesn’t

even exist. This arises from the photon propagator in the t-channel. It is:

t = (pa− p1)2 = − s

2β2(1− cosθ), (A.43)

u = (pa− p2)2 = − s

2β2(1+ cosθ), (A.44)

β =

√

1− 4m2

s, (A.45)

T =s

2(1−β2 cosθ), (A.46)

U =s

2(1+β2 cosθ), (A.47)

and the following relations hold:

t = −T +2m2, (A.48)

u = −U +2m2, (A.49)

s+ t+u = 4m2, (A.50)

s−T −U = 0. (A.51)

The two matrix elements of Bhabha scattering contain a photon propagator exchange in the s and t channels,

respectively, and the latter behaves like

1

t. (A.52)

The kinematical limits are reached at cosϑ = ±1, where t vanishes exactly.

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A.3.1 Some phase-space integrals

The volume of the two-particle phase space is the integral over (A.38

) in the boundaries (A.39):

V2(s,m1,m2) =

∫

dΦ2

=π

2

√

λ(1,m21/s,m2

2/s), (A.53)

V2(s,0,0) =π

2. (A.54)

If a function

F(s)β(s,m21,m

22) = F(s)

√

λ(s,m21,m2

2)

s(A.55)

undergoes a subsequent integration over s, it may well happen that

even for extremely small masses the integral has to be taken with exact

treatment of λ. This happens e.g. if a photonic propagator, F(s) = 1/s,

appears. Although for small masses the integrand gets nearly every-

where

F(s)β(s,m21,m

22) ≈ 1

s, (A.56)

this is a bad approximation when the singularity at s= 0 is approached.

The 1/s gets largest at the kinematical limit:

smin = (m1+m2)2. (A.57)

There, at the same time, the integrand vanishes identically as may be

seen from eq. (A.5). In fact, it is easy to show15:∫ s

4m2

ds′1

s′β(s′,m2,m2)

s′= ln

1+β(s,m2,m2)/s

1−β(s,m2,m2)/s−2β(s,m2,m2)/s

(A.58)15With Mathematica or with the aid of some integrals of Appendix D.8.2.

118


→ lns

4m2+2ln(2)−2 for β/s → 1,(A.59)

∫ s

4m2

ds′1

s′= ln

s

4m2, (A.60)

and∫ s

4m2

ds′1

s′2β(s′,m2,m2)

s′=

1

6m2

(

β(s,m2,m2)/s)3, (A.61)

∫ s

4m2

ds′1

s′2=

1

4m2

(

β(s,m2,m2)/s)2. (A.62)

Clearly, there are finite deviations between the integrals with and with-

out the threshold function even for vanishing mass. These deviations

disappear in the limit of vanishing masses for a lower cut on the inte-

gration region.

The squared matrix element often depends only on final state mo-

menta. This happens typically for processes with one-particle s-channel

exchange. The only scalar products are then p21= m2

1, p2

2= m2

2, and

2p1p2 = (p1+ p2)2−m21−m2

2 = s−m21−m2

2, (A.63)

i.e. the squared matrix element is in fact independent of the final state

momenta, and thus of the scattering angle. Or, it depends additionally

on some final state tensors like e.g. p1,µp2,ν.

For the first case, the phase space integrations are simple:

I =

∫

dΦ2F(p1p2)

= V2F(s−m21−m2

2), (A.64)

with V2 to be taken from (A.53).

The second case may be treated by the so-called tensor integration

method with the following ansatz (Q = p1+ p2 = pa+ pb):

Iµν =

∫

dΦ2F(2p1p2)p1,µp2,ν

119


= F(s−m21−m2

2)(

I0gµν+QµQνI2

)

. (A.65)

The tensor integral is evidently independent of the initial state mo-

menta pa and pb, and so the only momentum which might be used for

its representation is p1+ p2 = pa+ pb. Multiplying the ansatz by gµν or

by QµQν yields two equations for the coefficients I0, I2:

1

2(s−m2

1−m22)V2 = sI2+4I0, (A.66)

1

4(s+m2

1−m22)(s−m2

1+m22)V2 = s2I2+ sI0. (A.67)

We used here gµνgµν = 4. It is now easy to determine the tensor coeffi-

cients. In the massless case:

V2 =π

2, (A.68)

I = V2F(s), (A.69)

Iµν =V2

12F(s)

[

sgµν+2QµQν]

. (A.70)

Other tensor integrals may be also evaluated with this method. An

interesting application is the case arising in the determination of the

W propagator effect to the muon decay rate. For this, the following

integral has to be performed:

Iαβγ =

∫

dΦ2p1,αp2,βp2,γ (A.71)

The ansatz is now:

Iαβγ = AQαQβQγ+B(Qγgαβ+Qβgαγ)+CQαgβγ (A.72)

This, again, is easily resolved by multiplying both sides with QαQβQγ,Qγgαβ,

and (Qγgαβ+Qβgαγ).

References

[1] E. Byckling, K. Kajantie, “Particle Kinematics” (Nauka, Moscow, 1975), in Russian.

120


= = dΦ3(1,2,3)

Figure A.2: A sequential parametrization of the 3-particle phase space.

A.4 Three-particle phase space

The three-particle phase space is:

dΦ3(1,2,3) =d3~p1

2p01

d3~p2

2p02

d3~p3

2p03

δ4(p123− p12− p3), (A.73)

with

p123 = p1+ p2+ p3, (A.74)

p12 = p1+ p2. (A.75)

A.4.1 Sequential parameterization

Often it is useful to describe the n-particle phase space as a sequence of

2-particle phase spaces. A derivation begins with introducing a unity

factor:

1 = d4p12×δ4(p12− p1− p2), (A.76)

and after resorting it is:

dΦ3(1,2,3) = dΦ2(1,2)× d3~p3

2p03

d4p12δ4(p123− p12− p3).(A.77)

Insert another unity factor:

1 = dM212×δ(p2

12−M212), (A.78)

121


and get with the definitions

s = M2123, (A.79)

s′ = M212 = (p1+ p2)2, (A.80)

the following expression:

dΦ3(1,2,3) = dΦ2(1,2)×dM212

d3~p3

2p03

δ4(p123− p12− p3)d3~p12

2p012

= dΦ2(1,2)×ds′d cosθ3dφ3

λ1/2(s, s′,m23)

8M2123

, (A.81)

which is again a compact representation:

dΦ3(1,2,3) = dΦ2(1,2)×ds′×dΦ2(12,3). (A.82)

The scattering is symmetric with respect to the azimuthal angle (but:

spin phenomena), and then it is simply∫

dφ3 = 2π. Of course, the

phase space boundaries crucially depend on the parametrization cho-

sen. Here:

−1 ≤ cosθi ≤ +1, (A.83)

0 ≤ φi ≤ 2π, (A.84)

(m1+m2)2 ≤ s′ ≤ (√

s−m3)2. (A.85)

The s′ is the effective mass of the compound particle consisting of

particles 1 and 2.

The parameterization derived here is useful when the part of the

matrix element, which is related to particles 1 and 2, is independent of

the rest of the matrix element. This is quite often the case.

One may go one step further and choose a coordinate system, namely

the rest system of the initial state, ~p123 = 0, p0123

=√

s. Then, the defi-

nition of s′ becomes independent of any angle:

s′ = (p1+ p2)2

122


= (p123− p3)2 = s+m23−2

√sE3, (A.86)

and the corresponding integral is replaced:

ds′ = −2√

sdE3. (A.87)

The integration boundary (??) will transform into:

m3 ≤ E3 ≤1

2√

s

[

s+m23− (m1+m2)2

]

. (A.88)

The phase space parameterization derived here will be used e.g. for

the derivation of the muon life time.

Old material to be used somehow

(a)

Particle 3 is chosen to be a photon. Then, the cross-section ex-

plicitely depends on θγ in the cms, on two angles θ12,φ12 being defined

in a boosted (!) system, and on the invariant mass s′ of the pair of

particles 1,2 in the cms. The latter is related to the photon energy in

the cms,

s′ = s−2√

sEγ, (A.89)

so if one is interested in photon variables this choice is convenient But,

this choice of variables does not allow to calculate a differential cross-

section in, e.g., cosθ1 in the cms 16

(b)

Particle 3 is chosen not to be the photon. Then, one may determine

the differential cross-section in the scattering angle of particle 3. This

is along the lines of Born physics, perturbed by the inclusive photon.

But, the interesting variable s′, the invariant mass of the two other

particles (2,γ) is not that of the pair of particles whose production16 The described phase space parametrization has been used for the study of anomalous ττγ production at LEP 1

energies in [?]. A lot of technical stuff, includuing a simple cut, needed for explicit calculations is given there.

123


is studied originally. Again this invariant mass is related to the cms

energy E3:

s′ = s+m23−2

√sEγ, (A.90)

For certain applications, this is useful.

do we have some good reference for details etc.?

See also section A.4.4.

A.4.2 Phase-space volume

The formulae derived so far allow to write down the general phase

space integral in terms of three angles and of the two invariant energies

s, s′:

∫

dΦ3F =π

32

∫

ds′

√

λ(s, s′,m23)

s

√

λ(s′,m22,m2

1)

s′d cosθ12dφ12d cosθ3F,

(A.91)

where F is a function of all variables and where we used that the inte-

grand is independent of the angle φ3. As simplest application, we get

the volume V3 of the three-aprticle phase space:

V3 =

∫

dΦ3

=π2

4

∫ (√

s−m3)2

(m1+m2)2

ds′

√

λ(s, s′,m23)

s

√

λ(s′,m22,m2

1)

s′. (A.92)

If one of the final state particles is a photon, m3 = 0, this simplifies:

V3 =π2

4

∫ s

(m1+m2)2

ds′s− s′

sβ′, (A.93)

where β′ is the threshold function of the pair of the two other particles

with reduced invariant energy squared s′. For the case of fermion-pair

124


production, the two fermion masses are equal and we get, with help of

integrals given in Appendix D.8.2:

V3 =π2s

8

[

β(

s,m2,m2)

(

1+2m2

s

)

− 4m2

s

(

1−m2

s

)

log1+β

1−β

]

.(A.94)

For exactly massless particles 17:

V3 =π2s

8. (A.95)

A.4.3 d dimensions and soft gluon or photon corrections

The evaluation should be done in dimensional regularization, because

the integrals diverge due to soft photon/gluon singularities, and also

due to collinear singularities. The cases of massive and massless fermions

are treated differently, because there is no smooth transition between

them.

Now reproduce the many formulae a la Mathematica files

• bhabha-nf-brems.nb with complete integration of the massive case

• soft-photon-phase-space-integrals.nb with a nice comparison of

the massive and massless cases

A.4.4 Photon angle and invariant lepton pair mass

What one mostly is interested in for e+e− annihilation into two fermiosn

with photonic corrections is:

(c)

Access to both the scattering angle θ1 and to invariant mass s′ =

(p1+ p2)2. The former is the variable in which the Born cross-section

is differential, the latter one in which soft photon exponentiation may17 If the integrand contains a photon propagator at reduced invariant mass s′ due to initial state photon emission, then

the limit of small masses has to be taken with care due to the combination β′/s′. The integral of this deviates from the

integral without this factor by a finite term. See for details in section ZZZ.

125


be performed. Since these higher order corrections are big (at LEP 1

e.g.), an access to s′ is substantial for precise predictions.

now a text piece comes where I have to give up for today,

14 05 1997, with a reasonable presentation.

I scetch only the line of thinking.

begin of piece.

Because the dependence of the squared matrix element on momenta

is much more involved, we may profit from introducing more of the

invariants:

For convenience I use the following traditional abbreviations (and will

change to our book conventions in the final form, when I know what I

want):

γ(p), f (p1), f̄ (p2), and cosθ is the angle between f̄ and e+.

s = −(k1+ k2)2, (A.96)

s′ = −(p1+ p2)2 > 0, (A.97)

v1 = −2pp1 > 0, (A.98)

v2 = −2pp2 > 0. (A.99)

(Remark: any scalar product −pk is positive.)

One proves easily the relation:

s = s′+ v1+ v2. (A.100)

The final state phase space factorizes into:

dΦ3( f , f̄ ,γ) = dΦ2( f ,γ)×dM2fγd cosθ f̄ 2π

λ1/2(s,M2fγ,m

2)

8s.(A.101)

From the definition:

M2fγ = −(p+ p1)2 = v1+m2

= m2+ s− s′− v2. (A.102)

From this relation we may get (if v2 is fixed already):

dM2fγ = −ds′. (A.103)

126


Now we analyze in some detail the recoiling 2-pa<rticle phase space:

dΦ2( f ,γ) = d cosθ fγdφ fγ

λ1/2(M2fγ,m

2,0)

4M fγ

. (A.104)

The√λ,θ fγ,φ fγ are module of 3-momentum and angles in the rest

system of the compound,

~p+ ~p1 = 0. (A.105)

It is (from the calculational rule...):√

λ(M2fγ,m2,0) = M2

fγ−m2

= v1+m2

= m2+ s− s′− v2. (A.106)

next kinematics considerations should be shifted to earlier

piece of 3-particle section, since it is general. The

dependences of the final state variables, especially the kinematic bound-

aries after transformations, derive naturally from the condition in the

cms:

~p+ ~p1+ ~p2 = 0. (A.107)

There are two essentially different angles between these 3 vectors. One

is that between ~p2 and ~p; remembering that f̄ (p2) is recoiling from the

comppound of the two others, then it is clear that the angle θ( f̄ ,γ) may

be used as the angle θ fγ. The other one, θ( f̄ , f ), is also of importance.

It is related to the so-called acollinearity of the final state fermion pair

and will play a crucial role in realistic experimental analyses. Both

angles are accessible as follows:

|~p1|2 = |~p2+ ~p|2

= |~p2|2+ |~p|2+2|~p2||~p|cosθ(cms)( f̄ ,γ), (A.108)

cosθ( f̄ ,γ)(cms) =λ f −λ f̄ −λγ2√

λ f̄

√

λγ

127


=s′(s− s′)− v2(s+ s′)

(s− s′)√

λ f̄

. (A.109)

In the last equation we used already the representation of 3-momenta

with λ functions introduced earlier.

λ f = λ(−s,m2,−(k1+ k2± p1)2) = (s− v2)2−4m2s, (A.110)

λ f̄ = λ(−s,m2,−(k1+ k2± p2)2) = (s− v1)2−4m2s

= (s′+ v2)2−4m2s, (A.111)

λγ = λ(−s,0,−(k1+ k2± p)2) = (s− s′)2. (A.112)

The cos has limits and may be used for derivation of boundaries.

Another variable of interest is the area of the triangle spanned by the

three 3-vectors in the cms. It is positive and reads:

A =1

2|~p2||~p|sinθ(cms)( f̄ ,γ)

=1

2

√

λ f̄

2√

s

√

λγ

2√

s

√

1− cos2 θ(cms)( f̄ ,γ)

=1

16s

√

−λ(λ f ,λ f̄ ,λγ). (A.113)

From the sin and cos we get the 2 conditions:

λ(λ f ,λ f̄ ,λγ) ≤ 0, (A.114)

(λ f −λ f̄ −λγ)2 ≤ 4λ f̄λγ. (A.115)

Similarly, conditions involving the acollinearity may be derived and

used. This will be done later. end of piece.

What remains at this stage is now the dealing with the photonic

angles in the boosted frame. (In the simple case of muon decay, we

could perform the tensor integration and thus forget about details. In

the general case this is impossible.)

We need an expression for cosθ( f̄ ,γ). The phase space element

dΦ2( f ,γ) is invariant, so we may go into the cms as we did already.

128


We choose f̄ moving along the z axis:

p2 = (0,0, |~p2|, ip02), (A.116)

k1 = ... (A.117)

k2 = ... (A.118)

p = p0(sinθ( f̄ ,γ)cosφ( f̄ ,γ),sinθ( f̄ ,γ) sinφ( f̄ ,γ),cosθ( f̄ ,γ), i).

(A.119)

When we introduced s′ into the phase space, we had also to intro-

duce a dependence on v2. Make use of this and insert unity:

1 = dv2δ(v2+2pp2). (A.120)

in all the variables introduced above in the cms, it is:

−2pp2 = 2p0p02−2p0|~p2|cosθ( f̄ ,γ), (A.121)

and |~p2| =√

λ f̄/2√

s. With this insertion it is trivial to get:

dΦ2( f ,γ) =dφ fγdv2

4√

λ f̄

. (A.122)

Collecting everything, the√

λ f̄ cancels and we get the relatively

simple expression:

dΦ3 = constds′d cosθ(cms)

f̄dv2dφ(cms)( f̄ ,γ) (A.123)

The boundaries of s′ and v2 have to be determined yet. The phase

space volume may be checked then against the original expression.

A.4.5 A symmetric parameterization using the particle energies in 4 and in d dimensions

For processes like 3-jet production in e+e− annihilation,

e+(k1)+ e−(k2) → q(p1)+ q̄(p2)+g(p3), (A.124)

a quite different phase space parameterization proves to be useful. It

depends crucially on the fact that here the gluon emission can only be

129


final-state emission. So, as shown in section ??, the squared matrix

element depends only on the scalar products

pip j,

and we first show that they all may be expressed by the cms-energies

of the quarks and gluons:

(p1+ p2+ p3)2 = (k1+ k2)2 = s, (A.125)

(p1+ p2)2 = M212 = s′. (A.126)

We define:

xi ≡1

s2pip123

=1

s[2m2

i +2pip j+2pipk], withj , i , k. (A.127)

It follows immediately

x1+ x2+ x3 =2

sp2

123 = 2. (A.128)

We now express all final state scalar products by the xi, using:

papb = pap123−m2a− papc, witha , b , c. (A.129)

This yields a system of three linear equations:

p1p2+ p1p3 = p1p123−m21 =

s

2x1−m2

1, (A.130)

p2p1+ p2p3 = p2p123m22 =

s

2x2−m2

2, (A.131)

p3p1+ p3p2 = p3p123−m23 =

s

2x3−m2

3. (A.132)

The solution is:

2p1p2 =s

2[x1+ x2− x3]− [m2

1+m22−m2

3], (A.133)

2p1p3 =s

2[x1− x2+ x3]− [m2

1−m22+m2

3], (A.134)

2p2p3 =s

2[−x1+ x2+ x3]− [−m2

1+m22+m2

3]. (A.135)

130


In the cms with ~p123 = 0, the xi are just the (properly normalized) en-

ergies of the final state particles:

xi =1

s2p0

i p0123 =

2√

sEcms

i , mi ≤ xi ≤ 1, (A.136)

because Ecmsi

is at most Ebeam =√

s/2 (this happens when the other

two particles are collinear to each other and move opposite to particle

i).

In this situation, we like to have a phase space parameterization in

terms of the xi. This may be derived as follows. The three-particle

phase space is:

dΦ3(1,2,3) =d3~p1

2E1

d3~p2

2E2

d3~p3

2E3

δ4(p123− p1− p2− p3),

=d3~p1

2p01

d3~p2

2p02

1

2E3

δ1(√

s−E1−E2−E3). (A.137)

Using properties of the δ-function, we may rewrite:

δ1(√

s−E1−E2−E3) =1

√s/2δ1(2− x1− x2− x3), (A.138)

and have now:

dΦ3(1,2,3) =d3~p1

2E1

d3~p2

2p02

1

2E3

1√

s/2δ1(2− x1− x2− x3).(A.139)

If the integrand is independent of angles between beams and final par-

ticles, which would introduce a dependence on sclar products kip j, we

may rewrite one of the three-momentum integrals by assuming that the

angle be that related to a beam in the cms:∫

d3~p1

2E1

= 4π

∫ |~p1|2E1

dE1. (A.140)

This introduces x1, because x1 = E1/(√

s/2). The other integral de-

pends, besides on E2, i.e. on x2, on an angle θ which may be chosen as

131


the opening angle of ~p2 and ~p3, measured in the cms:

2p2p3 = 2[E2E3− |~p2||~p3|cos(θ23)]

= 2s

4[x2x3−

√

x22−4m2

2/s

√

x23−4m2

3/scos(θ23)]

=s

2[−x1+ x2+ x3]− [−m2

1+m22+m2

3] (A.141)

In the massless case, this reduces to

2p2p3 = 2s

4x2x3[1− cos(θ23)]. (A.142)

We may derive here the relation of the differentials. But an easier way

is the one starting from expressing E3 in the cms:

E23 = m2

3+ (~p1+ ~p2− ~p123)2

= m23+ (~p1+ ~p2)2

= m23+ [~p2

1+ ~p22+2|~p1||~p2|cos(θ12)]. (A.143)

This allows to derive:

2E3dE3 = 2|~p1||~p2|d cos(θ12) (A.144)

or just:

s

4x3dx3 = |~p1||~p2|cos(θ12). (A.145)

Collecting everything, one arrives at:

dΦ3(1,2,3) = const

∫

dx1dx2dx3δ1(2− x1− x2− x3)

const

∫ 1

0

dx1

∫ 1

1−x1

dx2. (A.146)

The integration limits are easiest seen as follows. At a given value of

x1 ∈ (0,1), consider x2:

x2 = (1− x1)+ (1− x3) (A.147)

132


By now varying x3 ∈ (0,1), one derives the boundaries of x2.

Remark:

The result is correct, compare to the literature, e.g. [?, ?].

I may, however, use an analogue of (A.141):

2p2p1 = 2[E2E1− |~p2||~p1|cos(θ21)]

= 2s

4[x2x1−

√

x22−4m2

2/s

√

x21−4m2

1/scos(θ21)]

=s

2[x1+ x2− x3]− [m2

1+m22−m2

3]. (A.148)

In the massless case, this reduces to

2p2p1 = 2s

4x2x1[1− cos(θ21)]. (A.149)

Now deriving a relation between dx3 and d cos(θ21), I get a linear rela-

tion between the two, in contradiction to (A.145).

That the relations both are true may be checked using 2 = x1+ x2+ x3

(not yet done).

I see no explanation, but there is one. TR 2009-12 15.

Checked with Mathematica that really for massless case:

x1x2 cos(θ21) = x1x2− x1− x2+ x3

=1

2(x2

3− x21− x2

2) (A.150)

if 2 = x1+ x2+ x3 is valid.

But evidently, the differentials get different. ?????????

A.5 Four- and five particle phase spaces

In a next step, the same way one may derive:

dΦ4(1,2,3,4) = dΦ2(1,2)×dΦ2(3,4)

× dM234dM2

12d cosθ34dφ34

λ1/2(M21234,M2

12,M2

34)

8M21234

,(A.151)

133


= = dΦ4(1,2,3,4)


= = dΦ5(1,2,3,4,5)


or:

dΦ4(1,2,3,4) = dΦ2(1,2))×dM212×dΦ2(3,4)×dM2

34×dΦ2(12,34).

(A.152)

The limits to the invariant variables are:

(m1+m2)2 ≤ s12 = M212 ≤ (

√s−m3−m4)2, (A.153)

(m3+m4)2 ≤ s34 = M234 ≤ (

√s−M12)2. (A.154)

Finally, if one studies photonic corrections to the production of pairs

of vector bosons decaying into four fermion final states, even a five-

particle final state parametrization is needed. From the above, it is

evident that the result is:

dΦ5(1,2,3,4,5) = dΦ2(1,2)×dM212×dΦ2(3,4)×dM2

34

× dΦ2(12,34)×dM21234×dΦ2(5,1234).(A.155)

The boundaries of the invariants are:

s = −p212345, (A.156)

134


(m1+m2+m3+m4)2 ≤ s′ = M21234 ≤ (

√s−m5)2, (A.157)

(m1+m2)2 ≤ s12 = M212 ≤ (

√s′−m1−m2)2, (A.158)

(m3+m4)2 ≤ s34 = M234 ≤ (

√s′− √s12)2. (A.159)

Another order of integrations corresponds to:

s = −p212345, (A.160)

(m1+m2)2 ≤ s12 = M212 ≤ (

√s−m1−m2)2, (A.161)

(m3+m4)2 ≤ s34 = M234 ≤ (

√s− √s12)2, (A.162)

(√

s12+√

s34)2 ≤ s′ = M21234 ≤ (

√s−m5)2. (A.163)

We remember once again: For applications, it is often necessary to

use different angular variables than introduced above. In contrast to

the case of two-particle production, where the angle used here is at

once the scattering angle in the cms, in the other cases this is not the

case. We will come back to this point where needed.

135

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