a phase space integrals for 2-, 3-, and 4-particle...
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Tord Riemann Standardmodell & Beschleunigerphysik Draft 2014-01-30 20:58
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.
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→ 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
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= 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.
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= = 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)
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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
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= (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.
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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
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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.
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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)
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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
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=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.
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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
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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)
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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
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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)
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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)
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= = dΦ4(1,2,3,4)
Figure A.3: A sequential parametrization of the 4-particle phase space.
= = dΦ5(1,2,3,4,5)
Figure A.4: A sequential parametrization of the 5-particle phase space.
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)
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(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.
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