Phase space integrals for 2-, 3-, and 4-particle production. f 2 Decay of a particle into three particles, e.g.

Transcription

1 A Phase space integrals for 2-, 3-, and 4-particle production We introduce the definition of the general final state phase space integral 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 + +ν µ,e + +ν e, H f+ f, Z f+ f, W ± f 1 + f 2 Decay of a particle into three particles, e.g. µ e +ν+ ν, p n+e + +ν Scattering processes with two-particle final states, e.g. e + e f+ f,z+ Z,Z+ H,W + + W, p p t+ t, ep e+ X, νn ν+ N,µ+ N,ν+ X,µ+ X Scattering processes with three-particle final states, e.g. e + e f+ f+γ,q+ q+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) f 1 + f 2 + f 3 + f 4, e + e f+ f+ 2γ, gg q+ q+2g It is common to all these reaction that the squared matrix elements may depend on invariants only. After summing and avaraging over 109

2 spin degrees of freedom, there are left the scalar products p i p j, and we also know that p 2 i= m 2 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 momentum 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 different 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 colliding 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 parameterizations, 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: P a (p a )+ P b (p b ) F 1 (p 1 )+ F n (p n ), (A.1) 110

3 is related to the squared matrix M element by the following relation, with s=(p a + p b ) 2 : dσ(2 n) = (2π) 4 2 λ(s,m 2a,m M 2 dφ n (p a + p b ; p 1,..., p n ), 2b ) (A.2) n dφ n (p ini ; p 1,..., p n ) = P δ 4 p ini p i d 4 p i δ ( p 2 i + ) m2 i θ(p 0 i ),(A.3) i i=1 λ(s,m 2a,m 2b ) = 2 (p a p b ) 2 m 2 am 2 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) = a 2 + b 2 + c 2 2ab 2bc 2ca = (a b c) 2 4bc = a 2 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 ). Further, P is a combinatorial factor arising if k groups of p j identical particles (e.g. several photons) are produced: k 1 P = p j!. (A.6) j=1 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 2 dφ n (p; p 1,..., p n ). (A.7) For the applications in these lectures, final states with two to four particles are of interest. 111 (A.5)

4 A.2 The 3-momentum of one particle in the rest frame of another particle Sometimes we will need the velocityβ(p), or the module of the 3- momentum p of one particle (with 4-momentum p and mass m): β(p) p p 0= p p 2 + m 2, (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 λ[p 2M 2,k 2,(k+ p) 2 ] k=0. (A.9) The last equation may be got as follows. When it is: Thus, k = 0, (A.10) k 2 = M 2, (A.11) p 2 = m 2, (A.12) (p+k) 2 = m 2 + M 2 + 2p 0 M, (A.13) p 0 = 1 [ (p+k) 2 m 2 M 2] k=0 2M, (A.14) and with use of p = (p 0 ) 2 m 2, 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<4m 2, thenβ=0. Theβ= λ(s,m 2 1,m2 2 )) is the threshold function in particle production, as will become clear in section A

5 The special case of production of a massive and a massless particle corresponds to β = λ(s, M 2,0))= 1 M2 s. (A.16) A.3 Two-particle final state Production of two particles, e.g. in the reaction: e + (p a )+e (p b ) f 1 (p 1 )+ f 2 (p 2 ), (A.17) may be described by two 4-momentum dependent functions, leaving aside for a moment spin. This may be easily seen. With account of global 4-momentum conservation and the mass-shell conditions p 2 i = m2 i, only two of the six possible products of 4-momenta p ip j are independent. One may choose e.g.: s = (p a + p b ) 2, t = (p a p 1 ) 2. (A.18) (A.19) Besides s and t, a third invariant u often is introduced: u = (p a p 2 ) 2. (A.20) These three invariants are related by 4-momentum conservation: s+t+ u = m 2 a+ m 2 b + m2 1 + m2 2 (A.21) A more symmetric notation considers all momenta as incoming, where then s i j = (p i + p j ) 2 = s,t,u for i j=ab,a1,a2, with s+t+ u=σm (A.22) 2 i. The most general differential cross-section for two-particle production is thus single-differential: dσ dt. 113 (A.23)

6 = dφ 2 (1,2) Figure A.1: The 2-particle phase space. This, and the total cross-section σ tot = t1 t 0 dσ dt (A.24) are the measurable quantities in this specific case. The integration boundaries are either the extreme values allowed by the process kinematics or limited by the specific experimental set-up. Any additional final state particle n+1 adds an additional 4-momentum vector, with boundary condition p 2 n+1 = m2 n+1, and thus three additional degrees of freedom. Now we derive a parametrization of the final state phase space which is adapted for applications. We will use for the D 4 introduced in (A.3): dxg(x)δ[ f (x) a] = g(x) f (x) from which follows for a particle: f (x)=a, D 4 p d 4 pδ ( p 2 m 2) θ(p 0 )= d3 p 2p 0 and introducing spherical coordinates: (A.25) (A.26) D 4 p = = p 2 2p0d p d cosθdφ. (A.27) 114

7 In e + e annihilation, e + e f f, a convenient kinematical system is the center of mass system. Forµpair production: p a + p b = p 1 + p 2 = 0, (A.28) s = 4Ee= 2 4Eµ, 2 β e = β µ = 1 4m2 e s, 1 4m2 µ s. (A.29) (A.30) (A.31) For Z boson decay, Z f f, one has to replace s MZ 2. If, as in W ± decay, W f 1 f 2, two particles with different masses are produced, this generalizes because then the 3-momenta agree, but no more E 1 E 2. In fact, from (A.9) it follows: 1 p 2 = 2 λ(m 2 2 M, M2 W,(p 2+ p W ) 2 ) W 1 = 2 λ(m 2 2 M, M2 W,m2 1 ) (A.32) W because (p 2 + p W ) 2 = ( p 1 ) 2 = m 2 1. The energies are then easily derived: 1 E 1,2 = 2 ( M 2 W + m 2 1,2 M ) m2 2,1. (A.33) W We now are prepared to study the general two particle phase space: dφ 2 (1,2) = δ 4 (p 12 p 1 p 2 ) D 4 p 1 D 4 p 2 = d3 p 1 δ 4 (p 2p 0 12 p 1 p 2 ) D 4 p 2 1 = d3 p 1 δ 1[ (p 2p 0 12 p 1 ) 2 m2] 2. (A.34) 1 A symbolic figure is shown in Fig. A.1. The further evaluation is convenient in the rest system of p 12. This situation is experimentally realized at e + e colliders when a pair of 115

8 particles (often with equal masses) is produced since the momenta of the incident electron and positron are balanced. Then it is: p 12 = 0 (A.35) and p 1 and p 2 are opposite to each other. It is further understood that p 2 12 = M2 12 s. (A.36) Introducing spherical coordinates, we may go on: p 1 2 d p 1 dφ 2 (1,2) = d cosθ 12 dφ 12 δ [ M 2p M0 12 p0 1 m2 1 + ] m2 2 1 p 1 = d cosθ 12 dφ 12 4M 12 λ 1/2 (M12 2 = d cosθ 12 dφ,m2 1,m2 2 ) 12 (A.37) 8M12 2 We used the relation p 1 d p 1 = p 0 1 dp0 1, see (A.9). The angles are chosen in the center-of-mass system (c.m.s.), i.e.θ 12 is the scattering angleθof one of the produced fermions, e.g. of f. The integration over dφ 12 yields a factor of 2π, because usually the quared matrix element is independent of that angle. 14 Thus, finally we obtain: dφ 2 (1,2) = π λ 1/2 (s,m 2 1,m2 2 ) d cosθ, 4 s (A.38) where, for two equal mass particles, the ratioβ=λ 1/2 (s,m 2,m 2 )/s is the velocity of the produced particles. Theβis 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) 14 Evidently, 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??. 116

9 The invariant differential cross-section with respect to t may be obtained from Eq. (A.19). It is: t = 2E e E 1 (1 β e β 1 cosθ 1 )+m 2 e+ m 2 1, (A.40) whereβare the velocities andθ 1 the angle between e and particle 1. E.g. in the cms, it is 2E e E 1 = (s+m 2 1 m2 2 )/2 andθ 1 the scattering angle. As mentioned, the cosine ofθ 1 varies in the interval (-1,+1). This way, the integration limits t 0,t 1 in Eq. (A.24) may be determined. Further, it is: dt = 1 λ(s,m 2 2 e,m 2 e) λ(s,m 2 1,m2 2 )d cosθ 1. (A.41) The third invariant introduced, u, is analogously: u = 2E e E 2 (1 β e β 2 cosθ 2 )+m 2 e+ m 2 2, (A.42) and for two-particle production in the cms it is cosθ 1 = cosθ 2. Easily, relation (A.21) is recovered. If additionally m 1 = m 2, we also have β 1 =β 2. 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: and the following relations hold: t = (p a p 1 ) 2 = s 2 β2 (1 cosθ), (A.43) u = (p a p 2 ) 2 = s 2 β2 (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) t = T+ 2m 2, (A.48) u = U+ 2m 2, (A.49) s+t+ u = 4m 2, (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 The kinematical limits are reached at cosϑ=±1, where t vanishes exactly. 1 t. 117 (A.52)

10 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): V 2 (s,m 1,m 2 ) = dφ 2 = π λ(1,m /s,m2 2 /s), (A.53) V 2 (s,0,0) = π 2. (A.54) If a function F(s)β(s,m 2 1,m2 2 ) = F(s) λ(s,m 2 1,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 ofλ. This happens e.g. if a photonic propagator, F(s)=1/s, appears. Although for small masses the integrand gets nearly everywhere F(s)β(s,m 2 1,m2 2 ) 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: s min = (m 1 + m 2 ) 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 show 15 : s ds 1 β(s,m 2,m 2 ) 4m 2 s s = ln 1+β(s,m2,m 2 )/s 1 β(s,m 2,m 2 )/s 2β(s,m2,m 2 )/s (A.58) 15 With Mathematica or with the aid of some integrals of Appendix D

11 and s ln s 4m2+ 2ln(2) 2 for β/s 1,(A.59) ds 1 = ln s 4m 2 s 4m 2, (A.60) s ds 1 β(s,m 2,m 2 ) = 1 ( β(s,m 2,m 2 )/s ) 3, (A.61) 4m 2 s 2 s 6m 2 s 4m 2 ds 1 s 2 = 1 4m 2 ( β(s,m 2,m 2 )/s ) 2. (A.62) Clearly, there are finite deviations between the integrals with and without the threshold function even for vanishing mass. These deviations disappear in the limit of vanishing masses for a lower cut on the integration region. The squared matrix element often depends only on final state momenta. This happens typically for processes with one-particle s-channel exchange. The only scalar products are then p 2 1 = m2 1, p2 2 = m2 2, and 2p 1 p 2 = (p 1 + p 2 ) 2 m 2 1 m2 2 = s m2 1 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. p 1,µ p 2,ν. For the first case, the phase space integrations are simple: I = dφ 2 F(p 1 p 2 ) = V 2 F(s m 2 1 m2 2 ), (A.64) with V 2 to be taken from (A.53). The second case may be treated by the so-called tensor integration method with the following ansatz (Q= p 1 + p 2 = p a + p b ): I µν = dφ 2 F(2p 1 p 2 )p 1,µ p 2,ν 119

12 = F(s m 2 1 m2 2 )( I 0 g µν + Q µ Q ν I 2 ). (A.65) The tensor integral is evidently independent of the initial state momenta p a and p b, and so the only momentum which might be used for its representation is p 1 + p 2 = p a + p b. Multiplying the ansatz by g µν or by Q µ Q ν yields two equations for the coefficients I 0, I 2 : 1 2 (s m2 1 m2 2 )V 2 = si 2 + 4I 0, (A.66) 1 4 (s+m2 1 m2 2 )(s m2 1 + m2 2 )V 2 = s 2 I 2 + si 0. (A.67) We used here g µν g µν = 4. It is now easy to determine the tensor coefficients. In the massless case: V 2 = π 2, (A.68) I = V 2 F(s), (A.69) I µν = V 2 12 F(s)[ ] sg µν + 2Q µ Q ν. (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: I αβγ = dφ 2 p 1,α p 2,β p 2,γ (A.71) The ansatz is now: I αβγ = AQ α Q β Q γ + B(Q γ g αβ + Q β g αγ )+CQ α g βγ (A.72) This, again, is easily resolved by multiplying both sides with Q α Q β Q γ, Q γ g αβ, and (Q γ g αβ + Q β g αγ ). References [1] E. Byckling, K. Kajantie, Particle Kinematics (Nauka, Moscow, 1975), in Russian. 120

13 = = dφ 3 (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: with dφ 3 (1,2,3) = d3 p 1 2p 0 1 d 3 p 2 d 3 p 3 δ 4 (p 123 p 12 p 3 ), (A.73) 2p 0 2 2p 0 3 p 123 = p 1 + p 2 + p 3, (A.74) p 12 = p 1 + p 2. (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: and after resorting it is: 1 = d 4 p 12 δ 4 (p 12 p 1 p 2 ), (A.76) dφ 3 (1,2,3) = dφ 2 (1,2) d3 p 3 d 4 p 2p 0 12 δ 4 (p 123 p 12 p 3 ).(A.77) 3 Insert another unity factor: 1 = dm12 2 δ(p2 12 M2 12 ), (A.78) 121

14 and get with the definitions s = M 2 123, s = M 2 12 = (p 1+ p 2 ) 2, (A.79) (A.80) the following expression: dφ 3 (1,2,3) = dφ 2 (1,2) dm12 2 d 3 p 3 2p 0 3 δ 4 (p 123 p 12 p 3 ) d3 p 12 2p 0 12 = dφ 2 (1,2) ds λ 1/2 (s, s,m 2 3 d cosθ 3 dφ ) 3, (A.81) 8M123 2 which is again a compact representation: dφ 3 (1,2,3) = dφ 2 (1,2) ds dφ 2 (12,3). (A.82) The scattering is symmetric with respect to the azimuthal angle (but: spin phenomena), and then it is simply dφ 3 = 2π. Of course, the phase space boundaries crucially depend on the parametrization chosen. Here: 1 cosθ i +1, (A.83) 0 φ i 2π, (A.84) (m 1 + m 2 ) 2 s ( s m 3 ) 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, p 123 = 0, p = s. Then, the definition of s becomes independent of any angle: s = (p 1 + p 2 ) 2 122

15 = (p 123 p 3 ) 2 = s+m se 3, (A.86) and the corresponding integral is replaced: ds = 2 sde 3. (A.87) The integration boundary (??) will transform into: m 3 E [ s+m 2 s 3 (m 1 + m 2 ) 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 explicitely depends onθ γ in the cms, on two anglesθ 12,φ 12 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 se γ, (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 crosssection in, e.g., cosθ 1 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 production 16 The described phase space parametrization has been used for the study of anomalousττγ production at LEP 1 energies in [?]. A lot of technical stuff, includuing a simple cut, needed for explicit calculations is given there. 123

16 is studied originally. Again this invariant mass is related to the cms energy E 3 : s = s+m se γ, (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 : dφ 3 F = π 32 ds λ(s, s,m 2 3 ) s λ(s,m 2 2,m2 1 ) s d cosθ 12 dφ 12 d cosθ 3 F, (A.91) where F is a function of all variables and where we used that the integrand is independent of the angleφ 3. As simplest application, we get the volume V 3 of the three-aprticle phase space: V 3 = = π2 4 dφ 3 ( s m3 ) 2 (m 1 +m 2 ) 2 ds λ(s, s,m 2 3 ) s λ(s,m 2 2,m2 1 ). (A.92) If one of the final state particles is a photon, m 3 = 0, this simplifies: V 3 = π2 4 s (m 1 +m 2 ) 2 ds s s s β, s (A.93) whereβ 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

17 production, the two fermion masses are equal and we get, with help of integrals given in Appendix D.8.2: V 3 = π2 s [β ( s,m 2,m 2)( ) 1+ ) (1 2m2 4m2 m2 log 1+β ] (A.94). 8 s s s 1 β For exactly massless particles 17 : V 3 = π2 s 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 angleθ 1 and to invariant mass s = (p 1 + p 2 ) 2. The former is the variable in which the Born cross-section is differential, the latter one in which soft photon exponentiation may 17 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 combinationβ /s. The integral of this deviates from the integral without this factor by a finite term. See for details in section ZZZ. 125

18 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, , 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 (p 1 ), f (p 2 ), and cosθ is the angle between f and e +. s = (k 1 + k 2 ) 2, s = (p 1 + p 2 ) 2 > 0, (Remark: any scalar product pk is positive.) One proves easily the relation: The final state phase space factorizes into: (A.96) (A.97) v 1 = 2pp 1 > 0, (A.98) v 2 = 2pp 2 > 0. (A.99) s= s + v 1 + v 2. (A.100) dφ 3 ( f, f,γ) = dφ 2 ( f,γ) dm 2 fγ d cosθ f 2π λ1/2 (s, M 2 fγ,m2 ) (A.101). 8s From the definition: M 2 fγ = (p+ p 1) 2 = v 1 + m 2 = m 2 + s s v 2. (A.102) From this relation we may get (if v 2 is fixed already): dm 2 fγ = ds. 126 (A.103)

19 Now we analyze in some detail the recoiling 2-pa<rticle phase space: λ 1/2 (M 2 fγ dφ 2 ( f,γ) = d cosθ fγ dφ,m2,0) fγ. (A.104) 4M fγ The λ,θ fγ,φ fγ are module of 3-momentum and angles in the rest system of the compound, It is (from the calculational rule...): λ(m 2 fγ,m2,0) = M 2 fγ m2 p+ p 1 = 0. (A.105) = v 1 + m 2 = m 2 + s s v 2. (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 boundaries after transformations, derive naturally from the condition in the cms: p+ p 1 + p 2 = 0. (A.107) There are two essentially different angles between these 3 vectors. One is that between p 2 and p; remembering that f (p 2 ) is recoiling from the comppound of the two others, then it is clear that the angleθ( f,γ) may be used as the angleθ fγ. The other one,θ( f, 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: cosθ( f,γ) (cms) p 1 2 = p 2 + p 2 = p p p 2 p cosθ (cms) ( f,γ), (A.108) = λ f λ f λ γ 2 λ f λγ 127

20 = s (s s ) v 2 (s+ s ) (s s ). (A.109) λ f In the last equation we used already the representation of 3-momenta withλfunctions introduced earlier. λ f λ f = λ( s,m 2, (k 1 + k 2 ± p 1 ) 2 )=(s v 2 ) 2 4m 2 s, (A.110) = λ( s,m 2, (k 1 + k 2 ± p 2 ) 2 )=(s v 1 ) 2 4m 2 s = (s + v 2 ) 2 4m 2 s, (A.111) λ γ = λ( s,0, (k 1 + k 2 ± 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 p 2 p sinθ (cms) ( f,γ) = 1 λ f λγ 2 2 s 2 s = 1 16s 1 cos 2 θ (cms) ( f,γ) λ(λ f,λ f,λ γ ). From the sin and cos we get the 2 conditions: (A.113) λ(λ f,λ f,λ γ ) 0, (λ f λ f λ γ ) 2 4λ fλ γ. (A.114) (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( f,γ). The phase space element dφ 2 ( f,γ) is invariant, so we may go into the cms as we did already. 128

21 We choose f moving along the z axis: p 2 = (0,0, p 2,ip 0 2 ), (A.116) k 1 =... (A.117) k 2 =... (A.118) p = p 0 (sinθ( f,γ)cosφ( f,γ),sinθ( f,γ)sinφ( f,γ),cosθ( f,γ),i). (A.119) When we introduced s into the phase space, we had also to introduce a dependence on v 2. Make use of this and insert unity: 1=dv 2 δ(v 2 + 2pp 2 ). in all the variables introduced above in the cms, it is: 2pp 2 = 2p 0 p 0 2 2p0 p 2 cosθ( f,γ), (A.120) (A.121) and p 2 = λ f/2 s. With this insertion it is trivial to get: dφ 2 ( f,γ) = dφ fγdv 2 4. (A.122) λ f Collecting everything, the λ f cancels and we get the relatively simple expression: dφ 3 = constds d cosθ (cms) dv f 2 dφ (cms) ( f,γ) (A.123) The boundaries of s and v 2 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 + (k 1 )+e (k 2 ) q(p 1 )+ q(p 2 )+g(p 3 ), (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

22 final-state emission. So, as shown in section??, the squared matrix element depends only on the scalar products p i p j, and we first show that they all may be expressed by the cms-energies of the quarks and gluons: We define: (p 1 + p 2 + p 3 ) 2 = (k 1 + k 2 ) 2 = s, (A.125) (p 1 + p 2 ) 2 = M 2 12 = s. (A.126) x i 1 s 2p ip 123 It follows immediately = 1 s [2m2 i + 2p ip j + 2p i p k ], withj i k. (A.127) x 1 + x 2 + x 3 = 2 s p2 123 = 2. (A.128) We now express all final state scalar products by the x i, using: p a p b = p a p 123 m 2 a p a p c, witha b c. (A.129) This yields a system of three linear equations: p 1 p 2 + p 1 p 3 = p1p 123 m 2 1 = s 2 x 1 m 2 1, (A.130) p 2 p 1 + p 2 p 3 = p2p 123 m 2 2 = s 2 x 2 m 2 2, (A.131) p 3 p 1 + p 3 p 2 = p3p 123 m 2 3 = s 2 x 3 m 2 3. (A.132) The solution is: 2p 1 p 2 = s 2 [x 1+ x 2 x 3 ] [m m2 2 m2 3 ], (A.133) 2p 1 p 3 = s 2 [x 1 x 2 + x 3 ] [m 2 1 m2 2 + m2 3 ], (A.134) 2p 2 p 3 = s 2 [ x 1+ x 2 + x 3 ] [ m m2 2 + m2 3 ]. (A.135) 130

23 In the cms with p 123 = 0, the x i are just the (properly normalized) energies of the final state particles: x i = 1 s 2p0 i p0 123 = 2 E cms s i, m i x i 1, (A.136) because Ei cms is at most E beam = 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 x i. This may be derived as follows. The three-particle phase space is: dφ 3 (1,2,3) = d3 p 1 2E 1 d 3 p 2 2E 2 d 3 p 3 2E 3 δ 4 (p 123 p 1 p 2 p 3 ), = d3 p 1 2p 0 1 d 3 p 2 2p 0 2 Using properties of theδ-function, we may rewrite: and have now: 1 2E 3 δ 1 ( s E 1 E 2 E 3 ). (A.137) δ 1 ( s E 1 E 2 E 3 )= 1 s/2 δ 1 (2 x 1 x 2 x 3 ), (A.138) dφ 3 (1,2,3) = d3 p 1 d 3 p E 1 2p 0 δ 1 (2 x 1 x 2 x 3 ).(A.139) 2E 2 3 s/2 If the integrand is independent of angles between beams and final particles, which would introduce a dependence on sclar products k i p j, we may rewrite one of the three-momentum integrals by assuming that the angle be that related to a beam in the cms: d 3 p 1 2E 1 = 4π p1 2E 1 de 1. (A.140) This introduces x 1, because x 1 = E 1 /( s/2). The other integral depends, besides on E 2, i.e. on x 2, on an angleθwhich may be chosen as 131

24 the opening angle of p 2 and p 3, measured in the cms: 2p 2 p 3 = 2[E 2 E 3 p 2 p 3 cos(θ 23 )] = 2 s 4 [x 2x 3 x 22 4m22 /s x3 2 4m2 3 /scos(θ 23)] = s 2 [ x 1+ x 2 + x 3 ] [ m m2 2 + m2 3 ] (A.141) In the massless case, this reduces to 2p 2 p 3 = 2 s 4 x 2x 3 [1 cos(θ 23 )]. (A.142) We may derive here the relation of the differentials. But an easier way is the one starting from expressing E 3 in the cms: This allows to derive: E 2 3 = m2 3 + ( p 1+ p 2 p 123 ) 2 = m ( p 1+ p 2 ) 2 = m [ p2 1 + p p 1 p 2 cos(θ 12 )]. (A.143) 2E 3 de 3 = 2 p 1 p 2 d cos(θ 12 ) (A.144) or just: s 4 x 3dx 3 = p 1 p 2 cos(θ 12 ). (A.145) Collecting everything, one arrives at: dφ 3 (1,2,3) = const dx 1 dx 2 dx 3 δ 1 (2 x 1 x 2 x 3 ) const 1 0 dx x 1 dx 2. (A.146) The integration limits are easiest seen as follows. At a given value of x 1 (0,1), consider x 2 : x 2 = (1 x 1 )+(1 x 3 ) (A.147) 132

25 By now varying x 3 (0,1), one derives the boundaries of x 2. Remark: The result is correct, compare to the literature, e.g. [?,?]. I may, however, use an analogue of (A.141): 2p 2 p 1 = 2[E 2 E 1 p 2 p 1 cos(θ 21 )] = 2 s 4 [x 2x 1 x 22 4m22 /s x1 2 4m2 1 /scos(θ 21)] = s 2 [x 1+ x 2 x 3 ] [m m2 2 m2 3 ]. (A.148) In the massless case, this reduces to 2p 2 p 1 = 2 s 4 x 2x 1 [1 cos(θ 21 )]. (A.149) Now deriving a relation between dx 3 and d cos(θ 21 ), I get a linear relation between the two, in contradiction to (A.145). That the relations both are true may be checked using 2= x 1 + x 2 + x 3 (not yet done). I see no explanation, but there is one. TR Checked with Mathematica that really for massless case: x 1 x 2 cos(θ 21 ) = x 1 x 2 x 1 x 2 + x 3 = 1 2 (x2 3 x2 1 x2 2 ) (A.150) if 2= x 1 + x 2 + x 3 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: dφ 4 (1,2,3,4) = dφ 2 (1,2) dφ 2 (3,4) dm34 2 dm2 12 d cosθ λ 1/2 (M dφ, M2 12, M2 34 ) 34 (A.151), 8M

26 = = dφ 4 (1,2,3,4) Figure A.3: A sequential parametrization of the 4-particle phase space. = = dφ 5 (1,2,3,4,5) Figure A.4: A sequential parametrization of the 5-particle phase space. or: dφ 4 (1,2,3,4) = dφ 2 (1,2)) dm 2 12 dφ 2(3,4) dm 2 34 dφ 2(12,34). The limits to the invariant variables are: (A.152) (m 1 + m 2 ) 2 s 12 = M 2 12 ( s m 3 m 4 ) 2, (A.153) (m 3 + m 4 ) 2 s 34 = M 2 34 ( s M 12 ) 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 fiveparticle final state parametrization is needed. From the above, it is evident that the result is: dφ 5 (1,2,3,4,5) = dφ 2 (1,2) dm12 2 dφ 2(3,4) dm34 2 dφ 2 (12,34) dm dφ 2(5,1234). (A.155) The boundaries of the invariants are: s = p , (A.156) 134

27 (m 1 + m 2 + m 3 + m 4 ) 2 s = M ( s m 5 ) 2, (A.157) (m 1 + m 2 ) 2 s 12 = M12 2 ( s m 1 m 2 ) 2, (A.158) (m 3 + m 4 ) 2 s 34 = M34 2 ( s s 12 ) 2. (A.159) Another order of integrations corresponds to: s = p , (A.160) (m 1 + m 2 ) 2 s 12 = M12 2 ( s m 1 m 2 ) 2, (A.161) (m 3 + m 4 ) 2 s 34 = M34 2 ( s s 12 ) 2, (A.162) ( s 12 + s 34 ) 2 s = M ( s m 5 ) 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