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Calibration processes for photon–photon colliders
Nuclear Physics B 676 (2004) 481–490 www.elsevier.com/locate/npe
Calibration processes for photon–photon colliders E. Bartoš a,b , A.-Z. Dubniˇcková b , M.V. Galynskii c , E.A. Kuraev a a Joint Institute for Nuclear Research Dubna, 141980 Dubna, Russia b Department of Theoretical Physics, Comenius University, 84248 Bratislava, Slovakia c Stepanov Institute of Physics BAS, Skorina ave. 70, 220072 Minsk, Belarus
Received 6 August 2003; received in revised form 3 October 2003; accepted 17 October 2003
Abstract Processes with creation of a pair charged particles with emission of hard photon and two pairs of charged particles are considered for colliding partially polarized photon–photon beams. The effects of circular and linear polarization of the initial photons are discussed in more details. 2003 Elsevier B.V. All rights reserved. PACS: 12.20.Ds; 13.88.+e Keywords: Polarization; Photon–photon collisions
The planned γ γ colliding beams based on laser backward Compton scattering lepton high energy colliders [1] will provide a new laboratory for investigation of hadron properties. In [2] the general theory of polarization phenomena in colliding photon beams was developed. So a lot of attention was paid to the details of conversion of laser photons in the process of backward Compton scattering and to the effects of density distribution in the photon beams. However, for the purposes of calibration and a measurement of the degree of polarization of photon beams the following QED processes with creation of one and two different pairs of leptons γ (k1 ) + γ (k2 ) → µ+ (q+ ) + µ− (q− ),
(1a)
γ (k1 ) + γ (k2 ) → µ+ (q+ ) + µ− (q− ) + γ (k),
(1b)
¯ + ) + b(q− ) γ (k1 ) + γ (k2 ) → a(p ¯ + ) + a(p− ) + b(q
(1c)
E-mail address: [email protected] (E.A. Kuraev). 0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.10.026
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Fig. 1. Feynman diagram for the process γ (k1 ) + γ (k2) → µ+ (q+ ) + µ−(q− ) (a), the process γ (k1 ) + γ (k2) → µ+ (q+ ) + µ− (q− ) + γ (k) (b) and the process γ (k1 ) + γ (k2 ) → e+ (p+ ) + e− (p− ) + µ+ (q+ ) + µ− (q− ) (c).
can be utilized. Besides the latter, these processes provide an essential background for study of the hadron creation processes as well as ones with heavy vector bosons. It has to be stressed, that the polarization phenomena turns out to be essential in this analysis. A lot of attention [3] was paid to the processes (1a), (1b) as well as to the process (1c) [4,5]. In the one last especially the kinematics of the main contribution to the total cross section was discussed more carefully, namely when the final particles √ move in the narrow cones along the direction of the photon colliding beams θi ∼ (Mi / √s ), s = 2k1 k2 . Its total cross section do not decrease as a function of the total cms energy s. The single pair creation processes (1a), (1b) are considered in the √ kinematical region, when the hard final particles move in the large angles in cms (m/ s θi 1), so the scalar products of all different 4-vectors are large compared with the square masses of particles. The process (1c) is investigated in the quasi peripherical √ kinematical region, where the invariant masses of produced pairs are much smaller than s and large compared with masses of particles. In our opinion this kinematics is useful in the experiments because of absence of background processes. The rough estimation of differential cross sections of processes (a)–(c) in Fig. 1 is α2 , s
α3 α4 , , s smax with smax = max{s1 , s2 } and s1, 2 are the invariant masses squared of electron and muon pairs, respectively. So in a kinematical situation, when smax s, the process (1c) is dominant. Contribution of the rest of Feynman diagrams as well as their interference with the diagrams Fig. 1(c) is suppressed by factor max{s1 , |q 2 |}/s. This fact is the consequence of nondecreasing contribution of Feynman diagrams with t-channel photon exchange in the limit s → ∞ [6]. The analysis of polarization phenomena in these all kinematical regions (which is absent to our knowledge up to now) is the motivation of our paper. Our paper is organized as follows: in part 1 we consider the processes (1a), (1b), in part 2 the process (1c). The polarization effects are studied by using definite expressions for the chiral amplitudes. We consider only the kinematics (mentioned above) the invariant mass of each pair is larger than masses of particles and smaller than cms total energy, moving preferably along the directions of initial photons.
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Process γ γ → µ+ µ− and γ γ → µ+ µ− γ The matrix elements of processes (1a), (1b) are (we put the masses of leptons to be equal to zero) qˆ− − kˆ1 −qˆ+ + kˆ1 Mλδ1 λ2 = −i4πα u¯ δ (q− ) εˆ λ1 (2) εˆ λ2 + εˆ λ2 εˆ λ1 vδ (q+ ) −χ1− −χ1+ and Mλλδ1 λ2 = i(4πα)3/2 u¯ δ (q− ) qˆ− + kˆ −qˆ+ + kˆ2 qˆ− − kˆ1 ∗ −qˆ+ + kˆ2 εˆ λ1 εˆ λ2 + εˆ λ1 εˆ εˆ λ2 × εˆ λ∗ χ− −χ2+ −χ1− λ −χ2+ qˆ− − kˆ1 −qˆ+ − kˆ ∗ qˆ− + kˆ −qˆ+ + kˆ1 + εˆ λ1 εˆ λ2 εˆ λ + εˆ λ∗ εˆ λ2 εˆ λ1 −χ1− χ+ χ− −χ1+ qˆ− − kˆ2 ∗ −qˆ+ + kˆ1 qˆ− − kˆ2 −qˆ+ − kˆ ∗ + εˆ λ2 εˆ εˆ λ1 + εˆ λ2 εˆ λ εˆ λ vδ (q+ ), −χ2− λ −χ1+ −χ2− 1 χ+ (3) with χ1± = 2k1 . q± , and
χ2± = 2k2 . q± ,
χ± = 2k . q±
λ, λ1 , λ2 , δ = ±1,
(4)
µ εˆ λ ≡ ελ γµ ,
u¯ δ , vδ describe the definite chiral states of photons ˆ −λ − kˆ qˆ− qˆ+ ωλ , N = [s1 χ− χ+ /2]−1/2, εˆ λ (k) = N qˆ− qˆ+ kω εˆ λ1 (k1 ) = N1 qˆ− qˆ+ kˆ1 ω−λ1 − kˆ1 qˆ− qˆ+ ωλ1 , N1 = [s1 χ1− χ1+ /2]−1/2 , εˆ λ2 (k2 ) = N2 qˆ− qˆ+ kˆ2 ω−λ2 − kˆ2 qˆ− qˆ+ ωλ2 , N2 = [s1 χ2− χ2+ /2]−1/2 , s1 = 2q− q+ ,
ω±λ = (1 ± λγ5 )/2
(5)
and fermions [7,8] uδ = ωδ u,
vδ = ω−δ v,
u¯ δ = uω ¯ −δ ,
For the process (1a) the chiral amplitudes Mλδ1 λ2 n = 2, 3) ± = ∓N1 N2 sχ2∓ u(q ¯ − )kˆ1 ω∓ v(q+ ), M+− ± M−+ = ±N1 N2 sχ2± u(q ¯ − )kˆ1 ω∓ v(q+ ),
v¯δ = vω ¯ δ.
(6) √ are (further we omit i( 4πα )n factor,
(7)
chiral amplitudes with equal chiralities of photons are equal zero. For a calculating of the cross sections for processes (1a)–(1c) in the case of partially polarized photon beams with momentum ki (i = 1, 2) we will use polarization matrices of density ρi = ρi (ki ) in the helically representation determined by Stokes parameters ξ (i) in the following way [9] (i) (i) (i) iξ1 − ξ3 1 + ξ2 1 ρi = ρi (ki ) = (8) , Tr(ρi ) = 1. (i) 2 −iξ (i) − ξ (i) 1 − ξ2 1 3
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Let us introduce 2 × 2 matrix building from the amplitudes (7) δ δ M++ M+− δ M1 = . δ δ M−+ M−−
(9)
Than the probability of the process (1a) will be reduce to calculation of trace from the product of the next matrices [2] δ 2 T δ δ † M (10) λ1 λ2 → Tr ρ1 M1 ρ2 M1 , where ρ1T is the matrix transposed to ρ1 and M†1 is Hermitian conjugated matrix to M1 . The cross section in general case has a form γ γ →µµ¯ ξ1 ξ2 δ
dσ
dΩµ−
=
(2) (1) (2) α 2 (1) (2) (1) (2) (1) 1 − ξ2 ξ2 R+ − 2 ξ1 ξ1 + ξ3 ξ3 + δ ξ2 − ξ2 R− , 4s (11)
where R± =
2 ± χ2 χ1+ 1−
χ1− χ1+
,
χ1± = χ2∓ .
(12)
For the case of completely circularly polarized photons (right (R): ξ2(1,2) = +1, left (L): (1,2) ξ2 = −1) we will have γ γ →µµ¯
dσLL dΩµ−
γ γ →µµ¯
=
dσRR dΩµ−
γ γ →µµ¯
= 0,
dσLR dΩµ−
γ γ →µµ¯
=
dσRL dΩµ−
=
α2 R+ . s
Further when the colliding beams have equal or reverse complete linear polarization ξ1(1) = ±ξ1(2) = ±1 (ξ3(1) = ±ξ3(2) = ±1) from (11) we have γ γ →µµ¯
dσ±± dΩµ−
=
α2 (R+ ∓ 2). 2s
Finally in the case of unpolarized particles we have the result which coincides with [9] in massless limit dσ γ γ →µµ¯ α2 = R+ . dΩµ− 2s For the process (1b) chiral amplitudes (3) could be written in the form −+ ˆ + v, = NN1 N2 s12 χ+ u¯ kω M++ ++ 2 M+− = NN1 N2 s χ2+ u¯ kˆ2 ω+ v, ++ M−+ −+ M−+ −+ M+− ++ M−−
1 = NN1 N2 s12 χ1+ u¯ kˆ1 ω+ v, = −NN1 N2 s12 χ2− u¯ kˆ2 ω+ v, = −NN1 N2 s12 χ1− u¯ kˆ1 ω+ v, ˆ + v, = −NN1 N2 s12 χ− u¯ kω
+− ˆ − v, M−− = NN1 N2 s12 χ+ u¯ kω −− 2 M−+ = NN1 N2 s χ2+ u¯ kˆ2 ω− v, −− M+− +− M+− +− M−+ −− M++
1 = NN1 N2 s12 χ1+ u¯ kˆ1 ω− v, = −NN1 N2 s12 χ2− u¯ kˆ2 ω− v, = −NN1 N2 s12 χ1− u¯ kˆ1 ω− v, ˆ − v. = −NN1 N2 s12 χ− u¯ kω
(13)
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The matrix element squared for the process (1b) in the case of partially polarized initial beams and summed over the polarizations of final particles is calculated analogously to the process (1a) (see (10)). As a result for differential cross section with taken into account only polarization of photon beams we have α 3 s1 Tin dΓ, D = χ− χ+ χ1− χ1+ χ2− χ2+ , 2π 2 s D d3 q+ d3 q− d3 k 4 dΓ = δ (k1 + k2 − q+ − q− − k), q+ 0 q− 0 k 0 γ γ →µµγ ¯ ξ1 ξ2
dσ
=
(14)
Tin = ξ1(1) ξ1(2) + ξ3(1) ξ3(2) (χ1+ χ2+ + χ1− χ2− )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ) + 4 ξ1(1) ξ3(2) − ξ3(1) ξ1(2) (χ1+ χ2+ − χ1− χ2− )Eq − 4ξ1(1)(χ+ χ2− − χ− χ2+ )Eq + 4ξ1(2) (χ+ χ1− − χ− χ1+ )Eq (1)
+ ξ3 (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ) (2)
+ ξ3 (χ+ χ1− + χ− χ1+ )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ) 2 2 2 2 + χ1− + χ2− − χ2+ χ2− χ2+ + ξ2(1) ξ2(2) χ+ χ− χ+2 + χ−2 − χ1+ χ1− χ1+ 2 2 2 2 + χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1+ + χ2− + χ2+ χ2− χ2+ , µ
ρ
σ = s [q q ] . where Eq = )µνρσ k1 k2ν q+ q− 2 + − z The corresponding cross sections for definite chiral states of initial photons are γ γ →µ− µ+ γ
dσRR(LL)
=
dΓ γ γ →µ− µ+ γ dσRL(LR)
α 3 s1 χ− χ+ (χ−2 + χ+2 ) , D π 2s
2 + χ 2 ) + χ χ (χ 2 + χ 2 ) α 3 s1 χ1− χ1+ (χ1− 2− 2+ 2− 1+ 2+ . (15) dΓ π 2s D For the case when only one photon beam polarized we gained the cross section similar (1) to (14) with the replacement Tin → Tin , where 2 (1) 2 Tin = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1+ 2 (1) 2 + χ2+ χ2− χ2+ − 4ξ1 (χ+ χ2− − χ− χ2+ )Eq + χ2−
=
(1)
+ ξ3 (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ).
(16)
In the case when all particles are unpolarized the differential cross section is γ γ →µ− µ+ γ
dσ0
dΓ
=
2 α 3 s1 2 + χ1+ χ− χ+ χ−2 + χ+2 + χ1− χ1+ χ1− 2 2π s 2 2 /D. + χ2− χ2+ χ2− + χ2+
(17)
The probability of the process (1b) in the case when initial and emitted photons with momenta k1 and k are partially polarized with Stokes parameters ξ (1) and ξ is calculated analogously to (10).
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Summing over the polarizations of final µ+ and µ− particles one can get the differential cross section which takes into account polarization of photons with momentum k1 and k α 3 s1 Tfin dΓ, 4π 2 s D Tfin = ξ1(1) ξ1 + ξ3(1) ξ3 (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ) + 4 ξ1(1) ξ3 − ξ3(1) ξ1 (χ+ χ1+ − χ− χ1− )Eq γ γ →µµγ ¯ ξ1 ξ
dσ
=
(18)
(1)
− 4ξ1 (χ+ χ2− − χ− χ2+ )Eq − 4ξ1 (χ1+ χ2− − χ1− χ2+ )Eq + ξ3(1) (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ) + ξ3 (χ1+ χ2− + χ1− χ2+ )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ) 2 2 (1) 2 2 + ξ2 ξ2 χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ − χ2+ χ2− χ2+ + χ1− + χ2− 2 2 2 2 + χ2+ χ2− χ2+ . + χ1− + χ2− + χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ (19) Obtained expressions (18), (19) allow us to determine Stokes parameters of emitted photon versus polarization of initial photon ξ (1) with momentum k1 f
ξ1 =
f10 + ξ1(1) f11 + ξ3(1) f13 f00 + ξ1(1) f01 + ξ3(1) f03
,
f
ξ3 =
f30 + ξ1(1) f31 + ξ3(1) f33 f00 + ξ1(1) f01 + ξ3(1) f03
,
(1)
ξ2 f22
f
ξ2 =
(1)
(1)
f00 + ξ1 f01 + ξ3 f03
,
(20)
where
2 2 2 2 f00 = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1− + χ2− + χ2+ χ2− χ2+ ,
f01 = −4(χ+ χ2− − χ− χ2+ )Eq , f03 = (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ), f10 = −4(χ1+ χ2− − χ1− χ2+ )Eq , f11 = (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ), f13 = −4(χ+ χ1+ − χ− χ1− )Eq , 2 2 2 2 − χ2+ χ2− χ2+ , + χ1− + χ2− f22 = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ f30 = (χ1+ χ2− + χ1− χ2+ )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ), f31 = −f13 = 4(χ+ χ1+ − χ− χ1− )Eq , f33 = f11 = (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ).
(21)
¯ +) ¯ + )b(q− )b(q Process γ γ → a(p− )a(p The kinematical variables of two pairs production process in quasi peripherical photon collisions are defined as (see (1c)) s = 2k1 . k2 ,
s1 = (p+ + p− )2 ,
q = k1 − p+ − p− = q+ + q− − k2 ,
s2 = (q+ + q− )2 , χ± = 2k1 . p± ,
χ± = 2k2 . q± .
(22)
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Further we will consider the case when the a a¯ pair moves close to the direction of initial photon with the momentum k1 and the b b¯ pair moves close to the direction of second photon. Besides the latter we will restrict ourselves by the case when the invariant masses a a¯ and bb¯ are large√ in comparison with the mass of muon and much less than the center of mass total energy s m2µ s1 , s2 s,
χ± ∼ s1 ,
χ± ∼ s2 .
(23)
A contribution of this region will not depend on s at large s, and will be dominant. For this kinematics the components of the created pairs move inside the cones with polar angle of order θ1,2 ∼ s1,2 /s along the beam axes. A few Feynman amplitudes are relevant in this kinematics, which can be drawn in the form of two block (see Fig. 1). The corresponding matrix element has a factorized form 2(4πα)2 λ1 λe λ2 λµ λ λ + mλ1 2 λe m2 1 µ , m1 m2 M γ γ → e+ e− µ+ µ− = is q2 1 µ 1 λ λ ρ mλ1 1 λe = k2 ε1ν (k1 )M1µν , m2 2 µ = k1σ ε2 (k2 )M2σρ . (24) s s In this two back-to-back kinematical regions the Sudakov parametrization for 4-momenta is convenient [10]. Assuming that the pair a(p− )a(p ¯ + ) belongs to the jet moving along direction of photon k1 we have ⊥ p± = α± k2 + x± k1 + p± ,
α± ≈
p2 χ1± = ±, s sx±
⊥ 2 (p± ) = −p2± ,
s1 = (p+ + p− )2 =
⊥ ⊥ p± . k1 = p± . k2 = 0,
Q2a , x+ x−
Qa = x + p − − x − p + ,
(25) where p± are 2-dimensional Euclidean vectors (p− + p + + q = 0), x± = 2k2p± /s are energy fractions of pair components (x+ + x− = 1) and s1 is squared invariant mass of the pair. ¯ + ) from the opposite jet moving in The similar relations are valid for the pair b(q− )b(q the direction of the photon k2 ⊥ q ± = y ± k 2 + β± k 1 + q ± ,
β± ≈
χ2±
s
=
q 2± , sy±
⊥ q± = −q 2± ,
s2 = (q+ + q− )2 =
⊥ ⊥ q± . k1 = q± . k2 = 0,
Q2b , y+ y−
Qb = y+ q − − y− q + , (26)
where energy fractions y± = 2k1 q± /s, (y+ + y− = 1) and q − + q + − q = 0. We define the chiral amplitudes as a matrix elements calculated with definite chiral states of fermions and photons. We choice the polarization vectors of photons in the form [7] εµj (k) = εµ + iλj εµ⊥ , α β γ εµ = N2 (q− k)q+µ − (q+ k)q−µ , q+ k , εµ⊥ = N2 )µαβγ q− εˆ λ1 = N1 pˆ − pˆ + kˆ1 ω−λ1 − kˆ1 pˆ − pˆ+ ωλ1 , N1 = [s1 χ+ χ− /2]−1/2, εˆ λ2 = N2 qˆ− qˆ+ kˆ2 ω−λ2 − kˆ2 qˆ− qˆ+ ωλ2 , N2 = [s2 χ+ χ− /2]−1/2. λ
(27)
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Chiral states of fermions were defined above. In derivation of cross section we consider only first term in (24), the second one can be obtained by correspondent replacement. Their interference term vanish in limit s → ∞ and is disregarded. M λ1 λe λ2 λµ = is
2(4πα)2 λ1 λe λ2 λµ m1 m2 , q2
λj = ±1, j = 1, 2, e, µ.
(28)
The quantities mλ1 1 λe for lepton pair have a form λ λe
m1 1
=
N1 u(p ¯ − ) δλe λ1 pˆ+ qˆ kˆ2 + δλe (−λ1 ) kˆ2 qˆ pˆ− ωλ1 v(p+ ). s
(29)
For the case of creation of charged pion pair π + π − one gets mλ1 = −N2 {Qa . q − iλ[Qa , q]z }.
(30)
λ λ
Quantities m2 2 µ are constructed analogically. Let us introduce 2 × 2 matrices building from the amplitudes (29) m+ m∗+ m+ m∗− Mλ1 e = , m± = mλ1 1 λe (λ1 = ±1), m− m∗+ m− m∗− ∗ m+ m+ m+ m− ∗ λ λ λ M2 µ = , m± = m2 2 µ (λ2 = ±1), m− m+ ∗ m− m− ∗
(31a) (31b)
then the absolute values of matrix elements squared of the process (1c) will be reduce to calculation of a trace from the product of the matrices as follows λ 1 λ e 2 λ 2 λ µ 2 m m = Tr ρ1 Mλe , = Tr ρ2 Mλµ . (32) 1 1 2 2 A simple calculation leads to the result exp{±2iϕa } x± /x∓ ± 2 2 2 M1(e) = N1 Qa q , exp{∓2iϕa } x∓ /x± 1 exp{2iϕa } 2 2 2
M1(π) = N1 Qa q , ϕa = Q a q. 1 exp{−2iϕa }
(33a) (33b)
Let us now transform the final state phase space volume dΦ4 =
d3 p+ d3 p− d3 q+ d3 q− 4 δ (k1 + k2 − q+ − q− − p+ − p− ) 2p+ 0 2p− 0 2q+ 0 2q− 0
= d4 q d4 p+ d4 p− d4 q+ d4 q− δ 4 (k1 + q − p+ − p− ) 2 2 2 2 δ p− δ q+ δ q− . × δ 4 (k2 − q − q+ − q− )δ p+ Applying Sudakov representation following form dΦ4 =
(d4 q
dx− dy− d2 p− d2 q− d2 q . 8sx+ x− y+ y−
=
s 2
dα dβ d2 q)
(34)
one can rewrite (34) into the
(35)
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For the cross section of two different pairs production (i.e., a = b) one obtains 4 2 2 2 ¯ = α x+ x− y+ y− dx− dy− d q− d p− d q dσ (γ γ → a ab ¯ b) π p 2− q 2− S1(a) S2(b) S1(b) S2(a) , × + (q + q − )2 (q − p− )2 (q − q − )2 (q + p − )2
(36)
and for the case a = b dσ (γ γ → a aa ¯ a) ¯ =
dx− dy− d2 q− d2 p− d2 q S1(a) S2(a) α4 x+ x− y+ y− , 2 2 2 2 π (q + q p− q − − ) (q − p − ) (37)
where S1(a) =
Tr(ρ1 M1(a)) q 2 Q2 N12
,
S2(a) =
Tr(ρ2 M2(a)) q 2 Q2 N22
.
(38)
Explicit form of (38) is S1(π) = 1 − ξ3(1) cos 2ϕa + ξ1(1) sin 2ϕa , (2)
(2)
S2(π) = 1 − ξ3 cos 2ϕb − ξ1 sin 2ϕb , 2 + x2 x− x− − x+ + − λµ ξ2(1) − ξ3(1) cos 2ϕa + ξ1(1) sin 2ϕa , 2x+ x− 2x+ x− 2 y 2 + y+ (2) y− − y+ (2) (2) S2(µ) = − (39) − λµ ξ2 − ξ3 cos 2ϕb + ξ1 sin 2ϕb , 2y+ y− 2y+ y−
with angles ϕj = Q j q. In conclusion we note that the cross section of two pair production in kinematical region, when all hard particles move in large angles even in unpolarized case, has a very complicated form (see, for example, the paper [11], where it was obtained for a cross channel). The ratio of magnitudes of cross sections in peripherical kinematics to the last one is s/smax 1, which underline the importance of quasi peripherical kinematics, considered in this paper.
S1(µ) =
Acknowledgements Two of us (E.B., E.K.) are grateful to Institute of Physics SAS (Bratislava), where part of this work was done. We are also grateful to grant RFBR No. 03-02-17077 and grant INTAS No. 00-0366. The work was in part supported by Slovak Grant Agency for Sciences, Grant No. 2/1111/23 (A.-Z.D.).
References [1] B. Badelek, et al., V. Telnov (Ed.), TESLA technical design report, Part VI, Chapter 1, DESY 2001-011, 2001-209, TESLA report 2001-23, March 2001, hep-ex/0108012;
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I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, V.I. Telnov, Nucl. Instrum. Methods A 219 (1984) 5. [2] I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Yad. Fiz. 38 (1983) 1021. [3] U. Fano, Phys. Rev. 93 (1954) 121; J.M. Jauch, F. Rohrlich, The Theory of Photons and Electrons, 2nd Edition, Springer-Verlag, 1980. [4] E.A. Kuraev, A. Schiller, V.G. Serbo, Nucl. Phys. B 256 (1985) 189. [5] E.A. Kuraev, A. Schiller, V.G. Serbo, D.V. Serebryakova, Eur. Phys. J. C 4 (1998) 631, hep-ph/9710420. [6] V.N. Baier, et al., Phys. Rep. 78 (1981) 293. [7] F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, W. Troost, T.T. Wu, Nucl. Phys. B 206 (1982) 61. [8] M. Galynskii, S. Sikach, Fiz. Elem. Chastits At. Yadra 29 (1998) 1133, hep-ph/9910284. [9] V.B. Berestetsky, E.M. Lifshitz, L.P. Pitaevsky, Relativistic Quantum Theory, Nauka, Moscow, 1968, Vol. IV, Part 1, p. 45. [10] V.V. Sudakov, Sov. Phys. JETP 3 (1956) 65. [11] E.A. Kuraev, A.N. Peryshkin, Yad. Fiz. 42 (1985) 1195.
Calibration processes for photon–photon colliders E. Bartoš a,b , A.-Z. Dubniˇcková b , M.V. Galynskii c , E.A. Kuraev a a Joint Institute for Nuclear Research Dubna, 141980 Dubna, Russia b Department of Theoretical Physics, Comenius University, 84248 Bratislava, Slovakia c Stepanov Institute of Physics BAS, Skorina ave. 70, 220072 Minsk, Belarus
Received 6 August 2003; received in revised form 3 October 2003; accepted 17 October 2003
Abstract Processes with creation of a pair charged particles with emission of hard photon and two pairs of charged particles are considered for colliding partially polarized photon–photon beams. The effects of circular and linear polarization of the initial photons are discussed in more details. 2003 Elsevier B.V. All rights reserved. PACS: 12.20.Ds; 13.88.+e Keywords: Polarization; Photon–photon collisions
The planned γ γ colliding beams based on laser backward Compton scattering lepton high energy colliders [1] will provide a new laboratory for investigation of hadron properties. In [2] the general theory of polarization phenomena in colliding photon beams was developed. So a lot of attention was paid to the details of conversion of laser photons in the process of backward Compton scattering and to the effects of density distribution in the photon beams. However, for the purposes of calibration and a measurement of the degree of polarization of photon beams the following QED processes with creation of one and two different pairs of leptons γ (k1 ) + γ (k2 ) → µ+ (q+ ) + µ− (q− ),
(1a)
γ (k1 ) + γ (k2 ) → µ+ (q+ ) + µ− (q− ) + γ (k),
(1b)
¯ + ) + b(q− ) γ (k1 ) + γ (k2 ) → a(p ¯ + ) + a(p− ) + b(q
(1c)
E-mail address: [email protected] (E.A. Kuraev). 0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.10.026
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Fig. 1. Feynman diagram for the process γ (k1 ) + γ (k2) → µ+ (q+ ) + µ−(q− ) (a), the process γ (k1 ) + γ (k2) → µ+ (q+ ) + µ− (q− ) + γ (k) (b) and the process γ (k1 ) + γ (k2 ) → e+ (p+ ) + e− (p− ) + µ+ (q+ ) + µ− (q− ) (c).
can be utilized. Besides the latter, these processes provide an essential background for study of the hadron creation processes as well as ones with heavy vector bosons. It has to be stressed, that the polarization phenomena turns out to be essential in this analysis. A lot of attention [3] was paid to the processes (1a), (1b) as well as to the process (1c) [4,5]. In the one last especially the kinematics of the main contribution to the total cross section was discussed more carefully, namely when the final particles √ move in the narrow cones along the direction of the photon colliding beams θi ∼ (Mi / √s ), s = 2k1 k2 . Its total cross section do not decrease as a function of the total cms energy s. The single pair creation processes (1a), (1b) are considered in the √ kinematical region, when the hard final particles move in the large angles in cms (m/ s θi 1), so the scalar products of all different 4-vectors are large compared with the square masses of particles. The process (1c) is investigated in the quasi peripherical √ kinematical region, where the invariant masses of produced pairs are much smaller than s and large compared with masses of particles. In our opinion this kinematics is useful in the experiments because of absence of background processes. The rough estimation of differential cross sections of processes (a)–(c) in Fig. 1 is α2 , s
α3 α4 , , s smax with smax = max{s1 , s2 } and s1, 2 are the invariant masses squared of electron and muon pairs, respectively. So in a kinematical situation, when smax s, the process (1c) is dominant. Contribution of the rest of Feynman diagrams as well as their interference with the diagrams Fig. 1(c) is suppressed by factor max{s1 , |q 2 |}/s. This fact is the consequence of nondecreasing contribution of Feynman diagrams with t-channel photon exchange in the limit s → ∞ [6]. The analysis of polarization phenomena in these all kinematical regions (which is absent to our knowledge up to now) is the motivation of our paper. Our paper is organized as follows: in part 1 we consider the processes (1a), (1b), in part 2 the process (1c). The polarization effects are studied by using definite expressions for the chiral amplitudes. We consider only the kinematics (mentioned above) the invariant mass of each pair is larger than masses of particles and smaller than cms total energy, moving preferably along the directions of initial photons.
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Process γ γ → µ+ µ− and γ γ → µ+ µ− γ The matrix elements of processes (1a), (1b) are (we put the masses of leptons to be equal to zero) qˆ− − kˆ1 −qˆ+ + kˆ1 Mλδ1 λ2 = −i4πα u¯ δ (q− ) εˆ λ1 (2) εˆ λ2 + εˆ λ2 εˆ λ1 vδ (q+ ) −χ1− −χ1+ and Mλλδ1 λ2 = i(4πα)3/2 u¯ δ (q− ) qˆ− + kˆ −qˆ+ + kˆ2 qˆ− − kˆ1 ∗ −qˆ+ + kˆ2 εˆ λ1 εˆ λ2 + εˆ λ1 εˆ εˆ λ2 × εˆ λ∗ χ− −χ2+ −χ1− λ −χ2+ qˆ− − kˆ1 −qˆ+ − kˆ ∗ qˆ− + kˆ −qˆ+ + kˆ1 + εˆ λ1 εˆ λ2 εˆ λ + εˆ λ∗ εˆ λ2 εˆ λ1 −χ1− χ+ χ− −χ1+ qˆ− − kˆ2 ∗ −qˆ+ + kˆ1 qˆ− − kˆ2 −qˆ+ − kˆ ∗ + εˆ λ2 εˆ εˆ λ1 + εˆ λ2 εˆ λ εˆ λ vδ (q+ ), −χ2− λ −χ1+ −χ2− 1 χ+ (3) with χ1± = 2k1 . q± , and
χ2± = 2k2 . q± ,
χ± = 2k . q±
λ, λ1 , λ2 , δ = ±1,
(4)
µ εˆ λ ≡ ελ γµ ,
u¯ δ , vδ describe the definite chiral states of photons ˆ −λ − kˆ qˆ− qˆ+ ωλ , N = [s1 χ− χ+ /2]−1/2, εˆ λ (k) = N qˆ− qˆ+ kω εˆ λ1 (k1 ) = N1 qˆ− qˆ+ kˆ1 ω−λ1 − kˆ1 qˆ− qˆ+ ωλ1 , N1 = [s1 χ1− χ1+ /2]−1/2 , εˆ λ2 (k2 ) = N2 qˆ− qˆ+ kˆ2 ω−λ2 − kˆ2 qˆ− qˆ+ ωλ2 , N2 = [s1 χ2− χ2+ /2]−1/2 , s1 = 2q− q+ ,
ω±λ = (1 ± λγ5 )/2
(5)
and fermions [7,8] uδ = ωδ u,
vδ = ω−δ v,
u¯ δ = uω ¯ −δ ,
For the process (1a) the chiral amplitudes Mλδ1 λ2 n = 2, 3) ± = ∓N1 N2 sχ2∓ u(q ¯ − )kˆ1 ω∓ v(q+ ), M+− ± M−+ = ±N1 N2 sχ2± u(q ¯ − )kˆ1 ω∓ v(q+ ),
v¯δ = vω ¯ δ.
(6) √ are (further we omit i( 4πα )n factor,
(7)
chiral amplitudes with equal chiralities of photons are equal zero. For a calculating of the cross sections for processes (1a)–(1c) in the case of partially polarized photon beams with momentum ki (i = 1, 2) we will use polarization matrices of density ρi = ρi (ki ) in the helically representation determined by Stokes parameters ξ (i) in the following way [9] (i) (i) (i) iξ1 − ξ3 1 + ξ2 1 ρi = ρi (ki ) = (8) , Tr(ρi ) = 1. (i) 2 −iξ (i) − ξ (i) 1 − ξ2 1 3
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Let us introduce 2 × 2 matrix building from the amplitudes (7) δ δ M++ M+− δ M1 = . δ δ M−+ M−−
(9)
Than the probability of the process (1a) will be reduce to calculation of trace from the product of the next matrices [2] δ 2 T δ δ † M (10) λ1 λ2 → Tr ρ1 M1 ρ2 M1 , where ρ1T is the matrix transposed to ρ1 and M†1 is Hermitian conjugated matrix to M1 . The cross section in general case has a form γ γ →µµ¯ ξ1 ξ2 δ
dσ
dΩµ−
=
(2) (1) (2) α 2 (1) (2) (1) (2) (1) 1 − ξ2 ξ2 R+ − 2 ξ1 ξ1 + ξ3 ξ3 + δ ξ2 − ξ2 R− , 4s (11)
where R± =
2 ± χ2 χ1+ 1−
χ1− χ1+
,
χ1± = χ2∓ .
(12)
For the case of completely circularly polarized photons (right (R): ξ2(1,2) = +1, left (L): (1,2) ξ2 = −1) we will have γ γ →µµ¯
dσLL dΩµ−
γ γ →µµ¯
=
dσRR dΩµ−
γ γ →µµ¯
= 0,
dσLR dΩµ−
γ γ →µµ¯
=
dσRL dΩµ−
=
α2 R+ . s
Further when the colliding beams have equal or reverse complete linear polarization ξ1(1) = ±ξ1(2) = ±1 (ξ3(1) = ±ξ3(2) = ±1) from (11) we have γ γ →µµ¯
dσ±± dΩµ−
=
α2 (R+ ∓ 2). 2s
Finally in the case of unpolarized particles we have the result which coincides with [9] in massless limit dσ γ γ →µµ¯ α2 = R+ . dΩµ− 2s For the process (1b) chiral amplitudes (3) could be written in the form −+ ˆ + v, = NN1 N2 s12 χ+ u¯ kω M++ ++ 2 M+− = NN1 N2 s χ2+ u¯ kˆ2 ω+ v, ++ M−+ −+ M−+ −+ M+− ++ M−−
1 = NN1 N2 s12 χ1+ u¯ kˆ1 ω+ v, = −NN1 N2 s12 χ2− u¯ kˆ2 ω+ v, = −NN1 N2 s12 χ1− u¯ kˆ1 ω+ v, ˆ + v, = −NN1 N2 s12 χ− u¯ kω
+− ˆ − v, M−− = NN1 N2 s12 χ+ u¯ kω −− 2 M−+ = NN1 N2 s χ2+ u¯ kˆ2 ω− v, −− M+− +− M+− +− M−+ −− M++
1 = NN1 N2 s12 χ1+ u¯ kˆ1 ω− v, = −NN1 N2 s12 χ2− u¯ kˆ2 ω− v, = −NN1 N2 s12 χ1− u¯ kˆ1 ω− v, ˆ − v. = −NN1 N2 s12 χ− u¯ kω
(13)
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The matrix element squared for the process (1b) in the case of partially polarized initial beams and summed over the polarizations of final particles is calculated analogously to the process (1a) (see (10)). As a result for differential cross section with taken into account only polarization of photon beams we have α 3 s1 Tin dΓ, D = χ− χ+ χ1− χ1+ χ2− χ2+ , 2π 2 s D d3 q+ d3 q− d3 k 4 dΓ = δ (k1 + k2 − q+ − q− − k), q+ 0 q− 0 k 0 γ γ →µµγ ¯ ξ1 ξ2
dσ
=
(14)
Tin = ξ1(1) ξ1(2) + ξ3(1) ξ3(2) (χ1+ χ2+ + χ1− χ2− )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ) + 4 ξ1(1) ξ3(2) − ξ3(1) ξ1(2) (χ1+ χ2+ − χ1− χ2− )Eq − 4ξ1(1)(χ+ χ2− − χ− χ2+ )Eq + 4ξ1(2) (χ+ χ1− − χ− χ1+ )Eq (1)
+ ξ3 (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ) (2)
+ ξ3 (χ+ χ1− + χ− χ1+ )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ) 2 2 2 2 + χ1− + χ2− − χ2+ χ2− χ2+ + ξ2(1) ξ2(2) χ+ χ− χ+2 + χ−2 − χ1+ χ1− χ1+ 2 2 2 2 + χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1+ + χ2− + χ2+ χ2− χ2+ , µ
ρ
σ = s [q q ] . where Eq = )µνρσ k1 k2ν q+ q− 2 + − z The corresponding cross sections for definite chiral states of initial photons are γ γ →µ− µ+ γ
dσRR(LL)
=
dΓ γ γ →µ− µ+ γ dσRL(LR)
α 3 s1 χ− χ+ (χ−2 + χ+2 ) , D π 2s
2 + χ 2 ) + χ χ (χ 2 + χ 2 ) α 3 s1 χ1− χ1+ (χ1− 2− 2+ 2− 1+ 2+ . (15) dΓ π 2s D For the case when only one photon beam polarized we gained the cross section similar (1) to (14) with the replacement Tin → Tin , where 2 (1) 2 Tin = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1+ 2 (1) 2 + χ2+ χ2− χ2+ − 4ξ1 (χ+ χ2− − χ− χ2+ )Eq + χ2−
=
(1)
+ ξ3 (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ).
(16)
In the case when all particles are unpolarized the differential cross section is γ γ →µ− µ+ γ
dσ0
dΓ
=
2 α 3 s1 2 + χ1+ χ− χ+ χ−2 + χ+2 + χ1− χ1+ χ1− 2 2π s 2 2 /D. + χ2− χ2+ χ2− + χ2+
(17)
The probability of the process (1b) in the case when initial and emitted photons with momenta k1 and k are partially polarized with Stokes parameters ξ (1) and ξ is calculated analogously to (10).
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Summing over the polarizations of final µ+ and µ− particles one can get the differential cross section which takes into account polarization of photons with momentum k1 and k α 3 s1 Tfin dΓ, 4π 2 s D Tfin = ξ1(1) ξ1 + ξ3(1) ξ3 (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ) + 4 ξ1(1) ξ3 − ξ3(1) ξ1 (χ+ χ1+ − χ− χ1− )Eq γ γ →µµγ ¯ ξ1 ξ
dσ
=
(18)
(1)
− 4ξ1 (χ+ χ2− − χ− χ2+ )Eq − 4ξ1 (χ1+ χ2− − χ1− χ2+ )Eq + ξ3(1) (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ) + ξ3 (χ1+ χ2− + χ1− χ2+ )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ) 2 2 (1) 2 2 + ξ2 ξ2 χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ − χ2+ χ2− χ2+ + χ1− + χ2− 2 2 2 2 + χ2+ χ2− χ2+ . + χ1− + χ2− + χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ (19) Obtained expressions (18), (19) allow us to determine Stokes parameters of emitted photon versus polarization of initial photon ξ (1) with momentum k1 f
ξ1 =
f10 + ξ1(1) f11 + ξ3(1) f13 f00 + ξ1(1) f01 + ξ3(1) f03
,
f
ξ3 =
f30 + ξ1(1) f31 + ξ3(1) f33 f00 + ξ1(1) f01 + ξ3(1) f03
,
(1)
ξ2 f22
f
ξ2 =
(1)
(1)
f00 + ξ1 f01 + ξ3 f03
,
(20)
where
2 2 2 2 f00 = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ + χ1− + χ2− + χ2+ χ2− χ2+ ,
f01 = −4(χ+ χ2− − χ− χ2+ )Eq , f03 = (χ+ χ2− + χ− χ2+ )(χ+ χ− − χ1+ χ1− + χ2+ χ2− ), f10 = −4(χ1+ χ2− − χ1− χ2+ )Eq , f11 = (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ), f13 = −4(χ+ χ1+ − χ− χ1− )Eq , 2 2 2 2 − χ2+ χ2− χ2+ , + χ1− + χ2− f22 = χ+ χ− χ+2 + χ−2 + χ1+ χ1− χ1+ f30 = (χ1+ χ2− + χ1− χ2+ )(χ+ χ− − χ1+ χ1− − χ2+ χ2− ), f31 = −f13 = 4(χ+ χ1+ − χ− χ1− )Eq , f33 = f11 = (χ+ χ1+ + χ− χ1− )(χ+ χ− + χ1+ χ1− − χ2+ χ2− ).
(21)
¯ +) ¯ + )b(q− )b(q Process γ γ → a(p− )a(p The kinematical variables of two pairs production process in quasi peripherical photon collisions are defined as (see (1c)) s = 2k1 . k2 ,
s1 = (p+ + p− )2 ,
q = k1 − p+ − p− = q+ + q− − k2 ,
s2 = (q+ + q− )2 , χ± = 2k1 . p± ,
χ± = 2k2 . q± .
(22)
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Further we will consider the case when the a a¯ pair moves close to the direction of initial photon with the momentum k1 and the b b¯ pair moves close to the direction of second photon. Besides the latter we will restrict ourselves by the case when the invariant masses a a¯ and bb¯ are large√ in comparison with the mass of muon and much less than the center of mass total energy s m2µ s1 , s2 s,
χ± ∼ s1 ,
χ± ∼ s2 .
(23)
A contribution of this region will not depend on s at large s, and will be dominant. For this kinematics the components of the created pairs move inside the cones with polar angle of order θ1,2 ∼ s1,2 /s along the beam axes. A few Feynman amplitudes are relevant in this kinematics, which can be drawn in the form of two block (see Fig. 1). The corresponding matrix element has a factorized form 2(4πα)2 λ1 λe λ2 λµ λ λ + mλ1 2 λe m2 1 µ , m1 m2 M γ γ → e+ e− µ+ µ− = is q2 1 µ 1 λ λ ρ mλ1 1 λe = k2 ε1ν (k1 )M1µν , m2 2 µ = k1σ ε2 (k2 )M2σρ . (24) s s In this two back-to-back kinematical regions the Sudakov parametrization for 4-momenta is convenient [10]. Assuming that the pair a(p− )a(p ¯ + ) belongs to the jet moving along direction of photon k1 we have ⊥ p± = α± k2 + x± k1 + p± ,
α± ≈
p2 χ1± = ±, s sx±
⊥ 2 (p± ) = −p2± ,
s1 = (p+ + p− )2 =
⊥ ⊥ p± . k1 = p± . k2 = 0,
Q2a , x+ x−
Qa = x + p − − x − p + ,
(25) where p± are 2-dimensional Euclidean vectors (p− + p + + q = 0), x± = 2k2p± /s are energy fractions of pair components (x+ + x− = 1) and s1 is squared invariant mass of the pair. ¯ + ) from the opposite jet moving in The similar relations are valid for the pair b(q− )b(q the direction of the photon k2 ⊥ q ± = y ± k 2 + β± k 1 + q ± ,
β± ≈
χ2±
s
=
q 2± , sy±
⊥ q± = −q 2± ,
s2 = (q+ + q− )2 =
⊥ ⊥ q± . k1 = q± . k2 = 0,
Q2b , y+ y−
Qb = y+ q − − y− q + , (26)
where energy fractions y± = 2k1 q± /s, (y+ + y− = 1) and q − + q + − q = 0. We define the chiral amplitudes as a matrix elements calculated with definite chiral states of fermions and photons. We choice the polarization vectors of photons in the form [7] εµj (k) = εµ + iλj εµ⊥ , α β γ εµ = N2 (q− k)q+µ − (q+ k)q−µ , q+ k , εµ⊥ = N2 )µαβγ q− εˆ λ1 = N1 pˆ − pˆ + kˆ1 ω−λ1 − kˆ1 pˆ − pˆ+ ωλ1 , N1 = [s1 χ+ χ− /2]−1/2, εˆ λ2 = N2 qˆ− qˆ+ kˆ2 ω−λ2 − kˆ2 qˆ− qˆ+ ωλ2 , N2 = [s2 χ+ χ− /2]−1/2. λ
(27)
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Chiral states of fermions were defined above. In derivation of cross section we consider only first term in (24), the second one can be obtained by correspondent replacement. Their interference term vanish in limit s → ∞ and is disregarded. M λ1 λe λ2 λµ = is
2(4πα)2 λ1 λe λ2 λµ m1 m2 , q2
λj = ±1, j = 1, 2, e, µ.
(28)
The quantities mλ1 1 λe for lepton pair have a form λ λe
m1 1
=
N1 u(p ¯ − ) δλe λ1 pˆ+ qˆ kˆ2 + δλe (−λ1 ) kˆ2 qˆ pˆ− ωλ1 v(p+ ). s
(29)
For the case of creation of charged pion pair π + π − one gets mλ1 = −N2 {Qa . q − iλ[Qa , q]z }.
(30)
λ λ
Quantities m2 2 µ are constructed analogically. Let us introduce 2 × 2 matrices building from the amplitudes (29) m+ m∗+ m+ m∗− Mλ1 e = , m± = mλ1 1 λe (λ1 = ±1), m− m∗+ m− m∗− ∗ m+ m+ m+ m− ∗ λ λ λ M2 µ = , m± = m2 2 µ (λ2 = ±1), m− m+ ∗ m− m− ∗
(31a) (31b)
then the absolute values of matrix elements squared of the process (1c) will be reduce to calculation of a trace from the product of the matrices as follows λ 1 λ e 2 λ 2 λ µ 2 m m = Tr ρ1 Mλe , = Tr ρ2 Mλµ . (32) 1 1 2 2 A simple calculation leads to the result exp{±2iϕa } x± /x∓ ± 2 2 2 M1(e) = N1 Qa q , exp{∓2iϕa } x∓ /x± 1 exp{2iϕa } 2 2 2
M1(π) = N1 Qa q , ϕa = Q a q. 1 exp{−2iϕa }
(33a) (33b)
Let us now transform the final state phase space volume dΦ4 =
d3 p+ d3 p− d3 q+ d3 q− 4 δ (k1 + k2 − q+ − q− − p+ − p− ) 2p+ 0 2p− 0 2q+ 0 2q− 0
= d4 q d4 p+ d4 p− d4 q+ d4 q− δ 4 (k1 + q − p+ − p− ) 2 2 2 2 δ p− δ q+ δ q− . × δ 4 (k2 − q − q+ − q− )δ p+ Applying Sudakov representation following form dΦ4 =
(d4 q
dx− dy− d2 p− d2 q− d2 q . 8sx+ x− y+ y−
=
s 2
dα dβ d2 q)
(34)
one can rewrite (34) into the
(35)
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For the cross section of two different pairs production (i.e., a = b) one obtains 4 2 2 2 ¯ = α x+ x− y+ y− dx− dy− d q− d p− d q dσ (γ γ → a ab ¯ b) π p 2− q 2− S1(a) S2(b) S1(b) S2(a) , × + (q + q − )2 (q − p− )2 (q − q − )2 (q + p − )2
(36)
and for the case a = b dσ (γ γ → a aa ¯ a) ¯ =
dx− dy− d2 q− d2 p− d2 q S1(a) S2(a) α4 x+ x− y+ y− , 2 2 2 2 π (q + q p− q − − ) (q − p − ) (37)
where S1(a) =
Tr(ρ1 M1(a)) q 2 Q2 N12
,
S2(a) =
Tr(ρ2 M2(a)) q 2 Q2 N22
.
(38)
Explicit form of (38) is S1(π) = 1 − ξ3(1) cos 2ϕa + ξ1(1) sin 2ϕa , (2)
(2)
S2(π) = 1 − ξ3 cos 2ϕb − ξ1 sin 2ϕb , 2 + x2 x− x− − x+ + − λµ ξ2(1) − ξ3(1) cos 2ϕa + ξ1(1) sin 2ϕa , 2x+ x− 2x+ x− 2 y 2 + y+ (2) y− − y+ (2) (2) S2(µ) = − (39) − λµ ξ2 − ξ3 cos 2ϕb + ξ1 sin 2ϕb , 2y+ y− 2y+ y−
with angles ϕj = Q j q. In conclusion we note that the cross section of two pair production in kinematical region, when all hard particles move in large angles even in unpolarized case, has a very complicated form (see, for example, the paper [11], where it was obtained for a cross channel). The ratio of magnitudes of cross sections in peripherical kinematics to the last one is s/smax 1, which underline the importance of quasi peripherical kinematics, considered in this paper.
S1(µ) =
Acknowledgements Two of us (E.B., E.K.) are grateful to Institute of Physics SAS (Bratislava), where part of this work was done. We are also grateful to grant RFBR No. 03-02-17077 and grant INTAS No. 00-0366. The work was in part supported by Slovak Grant Agency for Sciences, Grant No. 2/1111/23 (A.-Z.D.).
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