Sum rule for photon target

Sum rule for photon target E. Bartoˇs,1 S. Dubniˇcka,1 A.Z. Dubniˇckov´a,2 and E. A. Kuraev3

1

Inst. of Physics, Slovak Academy of Sciences, D´ ubravsk´ a cesta 9, 845 11 Bratislava, Slovak Republic

2

Dept. of Theor. Physics, Comenius University, 842 48 Bratislava, Slovak Republic

arXiv:0704.3896v2 [hep-ph] 23 Jul 2007

3

Bogol’ubov Laboratory of Theoretical Physics, JINR, Dubna, Russia

The amplitude of zero angle scattering of electron on photon in the 3-rd QED order of fine structure constant with γ ∗ γ intermediate state converting into quark–antiquark is considered. Utilizing analytic properties of elastic photon–photon scattering amplitude an explicit expression for differential cross- section of quark–antiquark pair production at electron-photon collision in peripheral kinematics is derived apparently. Limiting case of small transferred momenta with an application of the Weizs¨ acker-Williams like relation gives the sum rule for photon target, bringing into the relation the sum of ratios of the four power of the quark charges to squared quark masses with integral over the total γγ → 2jets cross–section. PACS numbers: 11.55.Hx, 13.60.Hb, 25.20.Lj

Keywords: sum rule, photoproduction, cross-section

Sum rules connecting the high energy asymptotic cross-sections of peripheral processes of QED type, which can be studied in colliding electron-positron beams with the fermion form factors (in particular with the slope of Dirac form factor) have been found in 1974 [1]. Applications to more complicated two-loop level QED processes have been investigated in the Appendix of the review paper [2]. In a set of the papers [3]-[6] some applications to baryon, deuteron and meson form factors were considered, where connection of these electromagnetic form factors with Q2 - dependent differential cross-section of deep inelastic electron-hadron scattering in the peripheral kinematics was investigated. In this paper we consider the scattering of electron on photon target with creation of 2 jets in the fragmentation region of the photon. Such kind of problems can be searched at photon-electron colliders constructed on the base of linear electron-positron colliders. Let us consider the two photon exchange electronphoton zero angle scattering amplitude of the process e(p, λ) + γ(k, ε) → e(p, λ) + γ(k, ε),

(1)

in two-loop (α3 ) approximation as presented in Fig. 1, with p2 = m2e , k 2 = 0 and assuming that the total energy squared of the process (1) s = 2p.k ≫ m2e . In an explicit calculation of the corresponding amplitude and a derivation of the sum rule under consideration the Sudakov’s decomposition of the virtual photon transferred momentum q = α˜ p + βk + q⊥ ,

2 q⊥ = (0, 0, ~ q), q⊥ = −~q2 , p˜q⊥ = kq⊥ = 0

into light-like vector p˜ = p − kp2 /s, p˜2 = 0 and the real photon fourmomentum k is suitable. The total energy squared variable s1 of the photon-photon scattering subprocess is then ds1 dβ. (3) 2 Averaging over the initial electron and photon spin states (initial and final spin states are supposed to coincide) one can write down the amplitude of the process (1) in the following form s1 = (q + k)2 = αs − ~q2 ,

=s

α 4π 2

Z

d4 q = d2 ~q

Aeγ→eγ (s, t = 0) = X γγ→γγ pµ pν εα ε∗β d2 ~q ds1 , Aµναβ 2 2 (q ) s2 ε

(4)

where the light-cone projection of the light-by-light (LBL) scattering tensor is the amplitude of γγ → γγ process and takes the form

µ ν α ∗β γγ→γγ p p ε ε Aγγ→γγ (s1 , ~q) = Aµναβ = s2 Z hS S2 S3 i 8α2 1 + + = − 2 Nc Q4q d4 q− π D1 D2 D3

(2) with

(5)

2 p

p q

q

q

q = 2×

k

k

k

+

!

+

k

FIG. 1: Feynman diagram of eγ → eγ scattering with LBL mechanism to be realized by quark-loops

S1 (1/4)T rpˆ(ˆ q− + mq )ˆ p(ˆ q− − qˆ + mq )ˆ ε∗ (ˆ q− − qˆ + kˆ + mq )ˆ ε(ˆ q− − qˆ + mq ) = , 2 2 2 2 2 2 D1 (q− − mq )((q− − q) − mq ) ((q− − q + k) − m2q ) S2 (1/4)T rpˆ(ˆ q− + mq )ˆ p(ˆ q− − qˆ + mq )ˆ ε( qˆ− − qˆ − kˆ + mq )ˆ ε∗ (ˆ q− − qˆ + mq ) = , 2 2 2 2 2 2 D2 (q− − mq )((q− − q) − mq ) ((q− − q − k) − m2q ) (1/4)T rpˆ(ˆ q− + mq )ˆ ε(ˆ q− − kˆ + mq )ˆ p(ˆ q− + qˆ − kˆ + mq )ˆ ε∗ (ˆ q− + qˆ + mq ) S3 = . 2 D3 (q− − m2q )((q− + q)2 − m2q )((q− + q + k)2 − m2q )((q− − k)2 − m2q )

where q− means the quark four-momentum in the quark loop of the process γγ → γγ, Nc is the number of colours s1 –plane

−∞

ds1 ∆u

−4m2q −~ q2 Z ∞

C

=

4m2q +~ q2

2

−q

4m2q

+q

FIG. 2: The path C of an integration in (9)

in QCD and Qq is the charge of the quark q in electron charge units. Regularization of LBL tensor is implied to provide the gauge invariance, which consists in removing some constant symmetrical tensor and the latter has no influence on the final results. Now, taking a derivative of the relation (5) according to d2 ~q and investigating the analytic properties of the obtained expression in s1 -plane one gets the configuration as presented in Fig. 2, where also the path C of the integral expression

Z

C

ds1

dAγγ→γγ (s1 , ~ q) d2 ~ q

(8)

dAγγ→γγ (s1 , ~q) |lef t = d2 ~q

ds1 ∆s

(10)

dAγγ→γγ (s1 , ~q) |right , d2 ~q

where the right s-channel discontinuity by means of the equation (4) is related (due to optical theorem in a differential form)

2

∆s

I=

(7)

cut) one comes to the relation Z

−4m2q

(6)

(9)

is drawn. When the integration contour is closed to the right (on s-channel cut) and to the left (on the u-channel

dAeγ→eγ (s, 0) dσ eγ→eqq¯ = 2s , 2 d ~q d2 ~q

(11)

to the Q2 =~q2 =−q 2 dependent differential cross-section of q q¯ pair creation by electron on photon, to be well known in the framework of QED [7] for l+ l− pair creation ~q2 dσ eγ→eqq¯ 4α3 4 f ( , )N Q = c q 3(q 2 )2 m2q d~q2

(12)

~q2 ) = (~q2 − m2q )J + 1, m2q q q 4 J= q ln[ ~q2 /(4m2q ) + 1 + ~q2 /(4m2q )]. ~q2 (~q2 + 4m2q )

f(

But the right hand cut concerns of two real quark production for s1 > 4m2q , which is associated with 2 jets production. The left-hand cut contribution has the same form as in QED case with constituent quark masses and as a result one obtains X 4α3 dσ eγ→e2jets ~q2 Nc . Q4q f ( 2 ) = 2 2 3(~q ) mq d~q2 q

(13)

3 Finally, for the case of small ~q2 and applying the Weizs¨acker-Williams like relation one comes to the sum rule for photon target as follows 1 14 X Q4q = 2 3 q mq πα2

Z∞

ds1 γγ→2jet σ (s1 ). s1 tot

(14)

4m2q

γγ→2jets The quantity σtot (s1 ) is assumed to degrease with increased values of s1 . It corresponds to the events in γγ collisions with creation of two jets, which are not separated by rapidity gaps and for which till the present days there is no experimental information. The latter complicates a verification of the sum rule (14). An evaluation of the left-hand side with the constituent quark masses [8] mu = md = 280M eV and ms = 405M eV (the contributions of heavy quarks c, b and t give negligible contributions and can be disregarded) gives just 5mb. The saturation of the right-hand side of the pho-

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ton sum rule (14) with an utilization of the data on γγ→X σtot (s1 ) given by Review of Particle Physics [9] on the level of 5mb is achieved with the upper bound of the corresponding integral to be 2 − 3GeV 2 . Unfortunately, the used data are charged by rather large uncertainties and in order to achieve more reliable verification of the γγ→2jets sum rule (14) the data on σtot (s1 ) are highly desirable. I.

ACKNOWLEDGEMENTS

The work was partly supported by Slovak Grant Agency for Sciences VEGA, Grant No. 2/7116/2007 (E.B., S.D. and A.Z.D.). E.A. Kuraev would like to thank Institute of Physics SAS for warm hospitality and Slovak purposed project at JINR for financial support. All authors thank Yu.M. Bystritskiy and M. Seˇcansk´ y for their interest and for discussions at the early stage of this work.

[6] S. Dubniˇcka, A. Z. Dubniˇckov´ a, E. A. Kuraev, Phys. Rev. D 75 (2007) 057901. [7] S. R. Kel’ner, Yad. Fiz. 5 (1967) 1092. [8] M. Nagy, M. K. Volkov, V. L. Yudichev, in: A. Z. Dubnickova, S. Dubnicka, P. Strizenec (Eds.), Proc. of Int. Conf. ”Hadron Structure 2000”, Stara Lesna, Slovak Republic, 2.-7.10.2000., Comenius Univ., Bratislava, 2001, p.188. [9] W.-M. Yao et al, J. Phys. G 33 (2006) 1.