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TAUP 2618/00 February 1, 2008 The Components of the γ γ Cross Section ∗ ∗ E. G O T S M A N1), E. L E V I N2), 0 0 0 U. M A O R3) and E. N A F T A L I4) 2 n a School of Physics and Astronomy J Raymond and Beverly Sackler Faculty of Exact Science 1 Tel Aviv University, Tel Aviv, 69978, ISRAEL 1 1 v 0 8 0 1 0 Abstract: We extend our previous treatment of γ∗p cross section based on Gribov’s 0 0 hypothesis tothecaseofphoton-photonscattering. Withtheaidoftwo parameters, determined / h from experimental data, we separate the interactions into two categories corresponding to short p (”soft”) and long (”hard”) distance processes. The photon-photon cross section, thus, receives - p contributions from three sectors, soft-soft, hard-hard and hard-soft. The additive quark model e is used to describe the soft-soft sector, pQCD the hard-hard sector, while the hard-soft sector h : is determined by relating it to the γ∗p system. We calculate and display the behaviour of the v i total photon-photon cross section and it’s various components and polarizations for different X values of energy and virtuality of the two photons, and discuss the significance of our results. r a 1) Email: [email protected] . 2) Email: [email protected] . 3) Email: [email protected] . 4) Email: [email protected] . 1 Introduction Scattering in the high energy (low x) limit has been studied in perturbative QCD (pQCD) over the past few years, mainly through the analysis of deep inelastic (DIS) events of lepton- hadron and hadron-hadron collisions. Such pQCD investigation requires some knowledge of the non perturbative contribution which is introduced, through the initial input to the evolution equations or put in explicitly. In this paper we present a study of virtual photon-photon ∗ scattering. Our investigation is based on our model for γ p cross section [1] , which provides the framework for the present calculation. Our goal is two fold: 1. Inany QCD process, finding the dynamics for intermediate distances is still anopen prob- lem, as it involves a transition between short distance (“hard” - perturbative) and large distance (“soft” - non perturbative) physics. In Ref. [1] we have suggested a procedure, based on Gribov’s general approach [2], of how to accommodate both contributions in DIS calculations. Two photon physics is an obvious reaction where these ideas can be further studied and re-examined. 2. Virtual photon-photon scattering has been proposed [3, 4, 5, 6] as a laboratory to study the BFKL Pomeron [7], as the total cross section of two highly virtual photons provides a probe of BFKL dynamics. Our study enables one to estimate the background to the proposed BFKL process. This background consists of two contributions: (i) We give an explicit estimate of the soft component in γ∗γ∗ scattering. (ii) Our pQCD estimate for the hard component is based on DGLAP [8] and as such can be used to assess when the BFKL dynamics start to dominate. Impressive attempts have been made [9, 10] to describe two photon physics within the framework of Vector Dominance Model (VDM) mainly as a soft interaction. However, one can consider a two photon interaction as an interesting tool for investigating the interplay between soft and hard physics [11]. The photon can appear as an unresolved object or as a perturbative fluctuation into an interacting quark-antiquark system. A careful analysis of the various components of the total cross section will help us understand the interface of the short distance and large distance interaction. In e+e− colliders, the measurement of the γ∗γ∗ is carried out by double tagging the outgoing leptons close to the forward direction, as most of the initial energy is taken by the scattered electrons. The double tagged cross section falls off with the increase of the photons virtualities due to the photon propagator. The experimental statistics are improved for single and no tag events where one of the colliding photons or both are quasi-real [12]. There is, therefore, a the- oretical interest and an experimental need to better understand and estimate the perturbative and non-perturbative contributions with realistic configurations of the two photon virtualities. 1 Our paper deals with photon-photon collisions in the high energy limit, which confines us to low x values. A pQCD investigation of eγ DIS is non trivial [13] due to the dual nature of the photon target (quasi-real or virtual) which can be perceived as either a hadron like partonic system or a point like object. The resulting difficulties in pQCD calculations of Fγ in the 2 small x limit have been extensively discussed in the literature and several strategies have been devised to bypass these problems [13]. For the purpose of our analysis we follow the approach suggested by Glu¨ck and Reya [14] in which the pQCD calculations has no predictive power regarding the normalization of Fγ but it retains, as for a proton target, the α dependence of 2 S the evolution equations. The above philosophy is very appropriate for our program where we distinguish between the hard pQCD mode and the non-perturbative QCD (npQCD) soft mode of the gluon fields by introducing [1, 15], two separation parameters (M2 and M2 ) in which we match the long and 0,T 0,L short distance components of the transverse (T) and longitudinal (L) contributions to the total ∗ ∗ γ γ cross-section. Our ideology is close to the Semiclassical Gluon Field Approach developed in Ref. [16]. This approach allows one to find a relation between scattering amplitudes and the property of the QCD vacuum based on the Model of the Stochastic Vacuum (MSV) [17]. Whereas the MSV is guided by the assumption of a microscopic structure of the QCD vacuum, our model is phenomenologically oriented based on the Additive Quark Model (AQM)[18]. The MSV has been combined [11] with the two Pomeron model [19]. In the two Pomeron model the hard Pomeron is a fixed J-pole whose Q2 dependence is determined by fitting to data. In a pQCD calculation of the hard Pomeron, one has an effective J-pole whose dependence on x and Q2 is determined by xG(x,Q2). A short review of the various approaches to γ∗γ∗ reactions at high energies, stressing the need for a simultaneous determination of both the soft and the hard contributing components, has just appeared [20]. The plan of our paper is as follows: In section 2 we review the generalization of the ideas ∗ ∗ presented in Ref. [1] and outline the expansion of this model for the γ γ cross section. In section 3 we derive the complete set of formulae for the total cross section components. We present the details of our numerical calculations in section 4 and compare our results with the high energy experimental data available to date. Our conclusions are summarized in section 5. 2 Review of the Approach Our approach follows from the ideas presented in Refs. [21]. This was first suggested in Ref. [15] and successfully applied in Ref. [1]. According to Gribov’s general approach [2], the interaction of a virtual photon, in any QCD description, can be interpreted as a two stage process. The first stage is the fluctuation of the photon into a hadronic system, and in the next stage the hadronic system interacts with the “target”, which in our case is another hadronic system from a different parent photon (see 2 (cid:19)(cid:19)(cid:15) (cid:12)(cid:16)(cid:16) (cid:0) 6 6 @ (cid:3) (cid:3) (cid:13) (cid:0) @ (cid:13) ^_^_^(cid:0)@(cid:0)(M12) M12 M102 (cid:0)(M102)@(cid:0)_^_^_ @ (cid:0) @ ? ? (cid:0) 2 02 2 02 (cid:27)(M1;M1 ;M2;M2 ;s) (cid:0) 6 6 @ (cid:3) (cid:3) (cid:13) (cid:0) @ (cid:13) ^_^_^@(cid:0)(cid:0)(M22) M22 M202 (cid:0)(M202)@(cid:0)_^_^_ @ (cid:0) @ ? ? (cid:0) (cid:18)(cid:18)(cid:14) (cid:13)(cid:17)(cid:17) Figure 1: Gribov’s approach. Fig. 1). These two processes are time ordered and can be treated independently. The vertex function Γ(M2) of the photon fluctuation into a qq¯pair of mass M is given by the experimental value of the ratio σ(e+e− hadrons) Γ(M2) = R(M2) = → . (1) σ(e+e− µ+µ−) → The complete process of two virtual photons fluctuating into two quark-antiquark pairs which then interact with each other, can be expressed by the following dispersion relation: α 2 Γ(M2) Γ(M2) σ(γ∗γ∗) = em dM2 dM2 dM′2 dM′2 1 2 3π 1 2 1 2 (Q2 +M2)(Q2 +M2) × (cid:18) (cid:19) Z 1 1 2 2 Γ(M′2) Γ(M′2) σ(M2,M′2;M2,M′2;s) 1 2 ,(2) 1 1 2 2 (Q2 +M′2)(Q2 +M′2) 1 1 2 2 where σ(M2,M′2;M2,M′2;s) is the cross section of the interaction between two hadronic sys- 1 1 2 2 ′ ′ tems with masses M and M before the interaction and M and M after the interaction. 1 2 1 2 We introduce a separation parameter in the mass integrations, which may be different for longitudinal and transverse polarized virtual photon (M and M respectively). For 0,L 0,T masses below this parameter, the process is soft, long range, and hence one cannot describe the produced hadron state as a qq¯pair. For masses above the separation parameter, the distances between the quark and antiquark are short, and σ(M2,M′2;M2,M′2;s), which depends on the 1 1 2 2 gluon structure function, can be calculated in pQCD. The calculation of the two photon total cross section, according to our approach, is derived ∗ following the same concepts of the σ(γ p) calculations. Each of the two photons can be soft or hard, and we shall derive the formulae on this basis. Without loss of generality, we shall consider the case in which one photon (say, the upper one) is “harder” than the other, hence there are three sectors of the calculation: 3 1. “Hard-hard” when both photons are hard. We treat the interaction between the two qq¯ pairs in pQCD, calculating all the diagrams in which the upper qq¯ pair are harder than the gluons in the ladder, and the gluons in the ladder are harder than the lower pair k2 ℓ2 ℓ2 k2 (see Fig.2). The cross section of the interaction in the hard-hard 1 ≫ 1 ≫ 2 ≫ 2 sector can be expressed through xG , the distribution function of a gluon ladder emitted q from a single quark. To find xG we recall that, in the region of small x, the evolution equation has the form: q d2xG (x,Q2) N q = c α (Q2)xG (x,Q2). (3) dlog 1dlogQ2 π S q x ∗ The solution of this equation in the DLLA has been already given in [22], 1 xG (x,Q2) = G I 2 γ(Q2)log , (4) q 0 0  s x   with G = 0.0453 and 0 12N log Q2 γ(Q2) = c log Λ2 . (5) 11Nc 2nf log Q20 − Λ2   2. “Soft-soft” for two soft photons. As stated, this is the case where neither of the hadronic systems can be treated in pQCD. Here we use the AQM [18] in which the interaction cross-section σ(M2,M′2;M2,M′2;s) is diagonal with respect to M and M′(j = 1,2). 1 1 2 2 j j 3. “Hard-soft” for the case that the upper photon is hard and the lower photon is soft. This sector is related, up to factorization, to the hard interaction between a nucleon and a photon, as the lower system is treated non perturbatively, while the upper hadronic system is a qq¯ pair with small transverse separation. Thus, the interaction cross section σ(M2,M′2;M2,M′2;s) is not diagonal and can be expressed through a nucleon gluonic 1 1 2 2 structure function xG, with a factor of 2, to account for the fact that we replace the 3 nucleon qqq state with a qq¯ state. | i | i As we shall see, our integrations require the knowledge of xG(x,ℓ2) where ℓ2 spans also over small values where the published parameterizations for xG are not valid. We follow Refs. [23] and [1] and introduce an additional gluon scale µ2 and assume that the gluon structure function can be approximated linearly by ℓ2xG(x,µ2). Thus, our approach has µ2 two scales, one separating the hard integration from the soft, which is related to the size of the quark-antiquark pair, and the gluon scale which is related to the size of the quark. For more details – see Refs. [1], [23], and [24]. ∗The function xGq has an additional term proportionalto γ(Q2)(logx1)−1K−1 2 γ(Q2)logx1 , however at low x this term contributes less than 1% and can be neglecqted. (cid:16) q (cid:17) 4 0 0 0 k1 k1 = k1 k1 k1 = k1(cid:0)`1 k1 k1 = k1 (cid:11)(cid:0) (cid:10)(cid:1) (cid:11)(cid:0) (cid:10)(cid:1) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:1)(cid:8)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 (cid:0) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 (cid:0) (cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:11)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 + (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:11)(cid:0) (cid:11)(cid:0) (cid:11)(cid:0) (cid:11)(cid:0) (cid:11)(cid:0) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) 0 0 0 k2 k2 = k2 k2 k2 = k2 k2 k2 = k2+`2 0 0 0 k1 k1 = k1+`1 k1 k1 = k1 k1 k1 = k1+`1 (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:1)(cid:8)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 (cid:0) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:11)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 + (cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:11)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:2)(cid:11)(cid:8)(cid:1)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:0)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)``12 + (cid:1)(cid:1)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) (cid:11)(cid:0) (cid:11)(cid:0) (cid:11)(cid:0) (cid:10)(cid:1) (cid:10)(cid:1) (cid:10)(cid:1) 0 0 0 k2 k2 = k2(cid:0)`2 k2 k2 = k2+`2 k2 k2 = k2+`2 Figure 2: Some of the diagrams which contribute to the pQCD calculations. In the next section, we derive explicitly the formulae for the three sectors described above, taking into account both transverse and longitudinal polarized photons in each sector. In our numerical calculations which are presented in section 4, the choice of our parameters are consistent with Ref. [1]. ∗ ∗ 3 Formulae for the Total γ γ Cross Section 3.1 The “Hard-Hard” components The pQCD calculation for the total cross section of two hard photons is illustrated in diagrams of the type shown in Fig.2. We denote the production amplitude of the two systems of qq¯, one ′ from each “hard” photon, by Mλ1λ′1λ2λ′2 where λ1...λ2 are the helicities of the four quarks. We follow Refs.[1, 15, 25] and write λ λ′λ λ′ in the form: M 1 1 2 2 Mλ1λ′1λ2λ′2 = Nc d2k′1d2z1′ d2k′2d2z2′T1,2ψ1ψ2 (6) q Z Z ′ ′ where = (k ,k ,k ,k ) is the transition amplitude of all the 16 possible diagrams of Fig. T1,2 ′T ′ 1 1 2 2 2, ψ = ψ (k ,z ) j = 1,2 are the wave functions of the qq¯inside the photons, and z (1 z ) j j j j j − j 5 is the fraction of the energy of the j photon that is carried by the quark(antiquark). Here and throughout this paper our momentum variables are defined as the two-dimensional transverse components of a four momentum, i.e. k k⊥. ≡ ′ In the leading log(1/x) approximation z = z and therefore the transition amplitude 1,2 ′ T does not dependent on z ,z . In the limit where k ℓ ℓ k we write: j j 1 ≫ 1 ≫ 2 ≫ 2 4πs d2ℓ d2ℓ ℓ2 = i 1 2∆(k ,k′) ∆(k ,k′) α (ℓ2) f(x, 1) (7) T1,2 2N ℓ2 ℓ2 1 1 2 2 S 1 ℓ2 c Z 1 Z 2 2 where, ′ ′ ′ ′ ∆(k ,k ) = 2δ(k k ) δ(k k +ℓ ) δ(k k ℓ ) (8) j j j − j − j − j j − j − j − j and f is related to the gluons distribution function which will be defined below. Substituting T1,2 into Mλ1λ′1λ2λ′2 and using the delta functions, we get several combinations of products of the two wave functions, 4πs d2ℓ ℓ2 Mλ1λ′1λ2λ′2 = i2√N ℓ2j∆ψλjλ′j(kj,zj) αS(ℓ21) f(x, ℓ21) (9) c Z j j 2 Y with ∆ψ(k ,z ) = 2ψ(k ,z ) ψ(k ℓ ,z ) ψ(k +ℓ ,z ). (10) j j j j j j j j j j − − − For the wave function of the qq¯ pair inside a transverse and longitudinally polarized photon, we shall use the results from Ref. [25]: ± 2ǫ± kj ψλλ′ (kj,zj) = −δλ,−λ′Zfe[(1−2zj)λ∓1] Q2 +· k2 (transverse), (11) j j 1 ψλLλ′(kj,zj) = −2δλ,−λ′ZfeQjzj(1−zj)Q2 +k2 (longitudinal). (12) j j In (11) and (12), Z is the charge of the quark with flavour f in units of the electron charge f e, Q2 z(1 z)Q2 and ǫ± = (0,0,1, i)/√2 is the photon polarization vector. − ≡ − ± Carrying out the angular integration of ∆ψ, we define the functions ϕT and ϕL as follows: 2ǫ± k ǫ± (k ℓ) ǫ± (k+ℓ) d2ℓ · · − · "Q2 +k2 − Q2 +(k ℓ)2 − Q2 +(k +ℓ)2# Z − Q2 k2 Q2 k2 +ℓ2 = πǫ± k dℓ2 − + − · Z Q2 +k2 (Q2 +k2 +ℓ2)2 4k2ℓ2 −  q  πǫ± k dℓ2ϕT(k2,ℓ2,Q2) (13) ≡ · Z 6 2 1 1 d2ℓ "Q2 +k2 − Q2 +(k ℓ)2 − Q2 +(k+ℓ)2# Z − 1 1 = 2π dℓ2 Z Q2 +k2 − (Q2 +k2 +ℓ2)2 4k2ℓ2 −  q  2π dℓ2ϕL(k2,ℓ2,Q2) (14) ≡ Z There are four hard-hard components for the two photon cross section, which we denote by h(T) h(L) σ , σ , etc. . h(T) h(T) h(T) Webeginwiththecalculationofσ . Usingthetransitionamplitude(9),thewavefunction h(T) (11) and the angular integration (13), we write: Z4α2 σh(T) = f em dz z2 +(1 z )2 dz z2 +(1 z )2 h(T) π4N 1 1 − 1 2 2 − 2 P c Z h iZ h i dk2 dk2 dℓ2 dℓ2 ℓ2 1 2 1 2 ϕT(k2,ℓ2,Q2)ϕT(k2,ℓ2,Q2)α (ℓ2)f x, 1 .(15) Z Q12 +k12 Z Q22 +k22 Z ℓ21 Z ℓ22 1 1 1 2 2 2 S 1 ℓ22! In order to perform the integration over z and z , we introduce the variables M and M: 1 2 k2 f M2 = j j z (1 z ) j j − ℓ2 M2 = j . (16) j z (1 z ) j j − f Formally, Eq.(15) now has the form: Z4α2 σh(T) = f em h(T) π4N × P c ℓ2 ℓ2 1 4 1 1 4 2 dM2 dM2 dM2 dM2 dℓ2 − M2 dℓ2 − M2 1 2 1 2 1r 1 2r 2 Z Q21 +M12 Z Q22 +M22 Z Mf12 Z Mf22 Z ℓ21 1−2Mℓ21e2 Z ℓ22 1−2Mℓ22e2 × 1 2 f f ℓ2 ϕT(M12,M12,Q21) ϕT(M22,M22,eQ22) αS(ℓ21) f x, ℓe221!. (17) f f We now make some approximations: 1. In the limit ℓ22 ≫ k22 diagrams with k2 6= k2′ are suppressed, therefore we can neglect the integration over M2 . 2 f 7 2. We can also simplify the ℓ2 and ℓ2 integration in the limits ℓ2 M2 and ℓ2 M2. The 1 2 1 ≪ 1 2 ≪ 2 integrals are dominated by the upper integration limits dictated by the Jacobian, and we can safely replace ℓ2 and ℓ2 in α and f by M2/4 and M2/4 respefctively. f 1 2 S 1 2 3. Integrating by parts over ℓ2, we redefine thefgluon laddfer emitted by a quark: 1 α (ℓ2)xG x, ℓ21 = ℓ21α (ℓ2)dℓ2 ℓ22 dℓ22 f x, ℓ21 . (18) S 1 q ℓ22! Z S 1 1Z ℓ22 ℓ22! Performing these simplifications we obtain: 4 Z4α2 dM2 dM2 dM2 σh(T) = f em 1 2 1 h(T) Pπ4Nc Z Q21 +M12 Z Q22 +M22 Z Mf14 × M2 M2 α 1 xGf x, 1 ϕT(M2,M2,Q2). (19) S f4 ! q Mf22! 1 1 1 f As a last step, according to our approach, we set the limits of the “hard” mass integrations, and replace each 2 Z2 with the ratio R(M2). f P α2 dM2 dM2 M2 σh(T) = em 1 R(M2) 1α ( 1)ϕT(M ,M ,Q ) h(T) π4Nc ZM02 Q21 +M12 1 Z4m2π Mf14 S f4 1 1 1 × f M12 dM22 Rf(M2)xG (x, M12). (20) ZMe02 Q22 +M22 2 q Mf22 h(L) The calculation of σ is straightforward. Using the same assumptions, we find: h(T) α2 dM2 dM2 M2 σh(L) = em Q2 1 R(M2) 1α ( 1)ϕL(M ,M ,Q ) h(T) 4π4Nc 1 ZM02 Q21 +M12 1 Z4m2π Mf14 S f4 1 1 1 × M2 dM2 f M2 f 2 R(M2)xG (x, 1). (21) ZMe02 Q22 +M22 2 q Mf22 h(T) h(T) Westartourcalculationofσ , inthesamewayaswedidforthecaseofσ ,bycollecting h(L) h(T) the transition amplitude (9), the wave function (11) and the angular integration (13): 2 Z4α2 σh(T) = f em Q2 dz z2 +(1 z )2 dz [z (1 z )]2 h(L) π4N 2 1 1 − 1 2 2 − 2 P c Z h iZ dk2 dk2 dℓ2 dℓ2 ℓ2 1 2 1 2 ϕT(k2,ℓ2,Q2) ϕL(k2,ℓ2,Q2)α (ℓ2)f x, 1 .(22) Z Q12 +k12 Z Q22 +k22 Z ℓ21 Z ℓ22 1 1 1 2 2 2 S 1 ℓ22! 8 We consider the limit ℓ2 k2 Q2 where ϕL(k2,ℓ2,Q2) 1 . Using the variables defined 2 ≫ 2 ≫ 2 −→ k22 in Eq.(16), we find the lower photon part of (22) to be: Q2 dz dk2 dℓ2 = 2 2{···} 2{···} 2{···} Z Z Z dM2 dM2 dℓ2 Q2ℓ2 2 2 2 2 2 . (23) Z M22 (1f−2Mℓ222) Z Q22 +M22 Z ℓ2 M22M22 2 f f The maximal value of ℓ2 is M2/4 and the ℓ2 inteegral is dominated by that value. The lower 2 2 2 photon part can now be written in the form: f dM2 dM2 Q2 2 2 2 . (24) M4 Q2 +M2 M2 Z f2 Z 2 2 2 Substituting (24) in (22), and switchinfg the integration variables of the upper photon z ,k2 1 1 into M2,M2 we have: 1 1 f ℓ2 1 4 1 σh(T) dM12 dM12 dℓ21 r − M12 α (ℓ2)ϕT(M2,M2,Q2) h(L) ∝ Z Q21 +M12 Z Mf12 Z ℓ21 1−2Mℓ21e2 S 1 1 1 1 × 1 f f dM2 dM2 4ℓ2 Q2 e 2 2 f x, 1 . (25) 2 Z M22(Q22 +M22) Z Mf24 M22! We can now use the definition (18) of xG , and perform an integrationfby parts fover M2: q 2 M12 dM22 f = 1 xG M12 dM22xG . f (26) ZMe22 Mf24 M22 q −ZMe22 Mf24 q Finally we integrate by parts overfℓ2 and obtain, in the limfit M2 4ℓ2: 1 1 ≫ 1 2α2 dM2 dM2 M2f σh(T) = em Q2 1 R(M2) 1α ( 1)ϕT(M ,M ,Q ) h(L) π4Nc 2 ZM02 Q21 +M12 1 Z4m2π Mf14 S f4 1 1 1 × M12 dM22 R(M2) 1 xGf(x, M12) M12 dMf22xG (x, M12) . (27) ZMe02 M22(Q22 +M22) 2 (M22 q Mf22 − ZMe22 Mf24 q Mf22 ) Following the same procedure, we obtain the last term of the “hard-hafrd” sector, f α2 dM2 dM2 M2 σh(L) = em Q2Q2 1 R(M2) 1α ( 1)ϕL(M ,M ,Q ) h(L) 2π4Nc 1 2 ZM02 Q21 +M12 1 Z4m2π Mf14 S f4 1 1 1 × M12 dM22 R(M2) 1 xG (x, Mf12) M12 dM22xGf(x, M12) . (28) ZMe02 M22(Q22 +M22) 2 (M22 q Mf22 − ZMe22 Mf24 q Mf22 ) f f 9