BUDKERINP/2001-33 DFCAL-TH 01/2 June 2001 PHOTON-REGGEON INTERACTION VERTICES IN THE NLA ∗ V.S. Fadina,b †, D.Yu. Ivanovc,e ‡ and M.I. Kotskya,d †† a Budker Institute for Nuclear Physics, 630090 Novosibirsk, Russia 2 b Novosibirsk State University, 630090 Novosibirsk, Russia 0 c Institute of Mathematics, 630090 Novosibirsk, Russia 0 d Istituto Nazionale di Fisica Nucleare, Gruppo collegato 2 di Cosenza, Arcavacata di Rende, I-87036 Cosenza, Italy n a e Regensburg University, Germany J 5 1 Abstract 2 v 9 Wecalculatetheeffectiveverticesforthequark-antiquarkandthequark-antiquark- 9 gluon production in the virtual photon - Reggeized gluon interaction. The last ver- 0 6 tex is considered at the Born level; for the first one the one-loop corrections are 0 obtained. These vertices have a number of applications; in particular, they are 1 necessary for calculation of the virtual photon impact factor in the next-to-leading 0 / logarithmic approximation. h p - p e h : v i X ∗Work supported in part by INTAS and in part by the Russian Fund of Basic Researches. r a †e-mail address: [email protected] ‡e-mail address: [email protected] ††e-mail address: [email protected] 1 Introduction Investigation of processes with the Pomeron exchange remains to be one of the important problems of high energy physics. A special attention is attracted by the so called semi- hard processes, where large values of typical momentum transfers Q2 give a possibility to use perturbative QCD for their theoretical description. The most common basis for such description is given by the BFKL approach [1]. It became widely known since discovery at HERA of the sharp rise of the proton structure function at decrease of the Bjorken variable x (see, for example, [2]). Recently the total cross section of the interaction of two highly virtual photons was measured at LEP. This process, being the one-scale process, seems to be even more natural for the application of the BFKL approach than the two- scale process of the deep inelastic scattering at small x, since here the evolution in x described by the BFKL equation does not interfere with the evolution in Q2 described by the DGLAP equation. For a consistent comparison with the experimental data the theoretical predictions must be obtained in the next-to-leading approximation (NLA), where together with the leading terms (α ln(s))n the terms α (α ln(s))n are also resumed. The radiative correc- s s s tions to the kernel of the BFKL equation were calculated several years ago [3]-[8] and the explicit form of the kernel of the equation in the NLA is known now [9, 10] for the case of forward scattering . But the problem of calculation in the NLA of the so called impact factors, which describe the coupling of the Pomeron to the scattering particles, remains unsolved. Let us remind (see, for example, Ref. [11] for the details), that in the BFKL approach the relevant to the irreducible representation of the colour group in the t-channel part ( )A′B′ of the scattering amplitude for the Rprocess AB A′B′ at large c.m.s. energy AR AB → √s and fixed momentum transfer q q ( means transverse to the initial particle ⊥ → ∞ ≈ ⊥ momenta plane) is expressed in terms of the Mellin transform of the Green function of the two interacting Reggeized gluons G(R) and of the impact factors of the colliding particles ω (R,ν) (R,ν) Φ and Φ : A′A B′B A′B′ s dD−2q1 dD−2q2 m ( ) = I s AR AB (2π)D−2 ~q 2(~q ~q)2 ~q 2(~q ~q)2 Z 1 1 − Z 2 2 − δ+i∞ dω s ω Φ(R,ν)(~q ,~q,s ) G(R)(~q ,~q ,~q) Φ(R,ν)( ~q , ~q,s ), (1.1) × ν A′A 1 0 Zδ−i∞ 2πi (cid:20)(cid:18)s0(cid:19) ω 1 2 (cid:21) B′B − 2 − 0 X where m means the s-channel imaginary part, the vector sign is used for denotation of s I the transverse components, ν enumerates the states in the representation , D = 4+2ǫ R is the space-time dimension different from 4 to regularize both infrared and ultraviolet divergencies and the parameter s is artificial and introduced for a convenience. While 0 the Green function obeys the generalized BFKL equation [11] dD−2k ωG(R)(~q ,~q ,~q) = ~q 2(~q ~q)2δ(D−2)(~q ~q )+ (R)(~q ,~k,~q)G(R)(~k,~q ,~q) ω 1 2 1 1 − 1 − 2 ~k 2(~k ~q)2K 1 ω 2 Z − (1.2) 1 with the NLA kernel (R) and is completely defined by this equation, the impact factors K should be calculated separately. The definition of the NLA impact factors has been given in Ref. [11]; in the case of definite colours of c and c′ of the Reggeized gluons the impact factor has a form [12] s 12ω(−q~12) s 21ω(−(q~1−q~)2) Φcc′ (~q ,~q,s ) = 0 0 AA′ 1 0 ~q12! (~q1 −~q)2! dκθ(s κ)dρ Γc Γc′ ∗ × 2π Λ − f {f}A {f}A′ X{f}Z (cid:16) (cid:17) 1 dD−2k Φc1c′1(Born)(~k,~q,s ) Born c′1c′(~k,~q ,~q)ln s2Λ , (1.3) − 2 ~k 2(~k ~q)2 AA′ 0 Kr c1c 1 s (~k ~q )2! Z − (cid:16) (cid:17) 0 − 1 where ω(t) is the Reggeized gluon trajectory and the intermediate parameter s should go Λ to infinity. The integration in the first term of the above equality is carried out over the phase space dρ and over the squared invariant mass κ of the system f produced in the f { } fragmentation region of the particle A, Γc are the particle-Reggeon effective vertices {f}A for this production and the sum is taken over all systems f which can be produced in { } the NLA. The second term in Eq. (1.3) is the counterterm for the LLA part of the first one, so that the logarithmic dependence of both terms on the intermediate parameter s disappears in their sum; Born is the part of the leading order BFKL kernel Λ → ∞ Kr related to the real gluon production (see Refs. [12] for more details). It was shown in Ref. [13] that the definition (1.3) guarantees infrared finiteness of the colourless particle impact factors. It is clear from above that for complete NLA description in the BFKL approach one needs to know the impact factors, analogously as in the DGLAP approach one should know not only the parton distributions, but also the coefficient functions. This paper is an extended version of the short note [14], which can be considered as the first step in the calculation of the virtual photon impact factor in the NLA. We calculate here the virtual photon-Reggeon effective vertices which enter the definition (1.3) in the case when the particle A is the virtual photon. In the NLA the states which can be produced in the Reggeon-virtual photon collision are the quark-antiquark and the quark-antiquark-gluon ones. In the next Section we present the effective vertices for production of these states in the Born approximation. This approximation is sufficient to find in the NLA the contribution to the virtual photon impact factor from the quark- antiquark-gluon state. In the case of the quark-antiquark state we need to know the effective production vertex with the one-loop accuracy. Sections 3-5 are devoted to the calculation of the one-loop corrections. In Sections 3 and 4 we consider the two-gluon and the one-gluon exchange diagrams correspondingly; in Section 5 the total one-loop correction is presented. The results obtained are discussed in Section 6. Some details of the calculation are given in Appendix A. In the following the photon-Reggeon effective vertices presented in this paper will be used for the calculation of the photon impact factor. But they could have many other applications, for example, in the diffractive production of quark jets and so on. 2 2 The Born interaction vertices In this section we present the vertices for the qq¯and the qq¯g production in the Reggeon- virtual photon collision in the Born approximation for the case of completely massless QCD. These vertices can be obtained from the high energy amplitudes with the octet colour state and the negative signature in the t-channel for collision of the virtual photon with any particle, if the corresponding system is produced in the virtual photon frag- mentation region. For simplicity we always consider collision of the virtual photon with the momentum p and the quark with the momentum p . We use everywhere below the A B Feynman gauge for the gluon field, the Sudakov decomposition of momenta p2 +p~ 2 p = βp +αp +p , α = , p2 = p2 = 0, 1 2 ⊥ sβ 1 2 s = 2p p , p~ 2 p2, (2.1) 1 2 → ∞ ≡ − ⊥ with the lightcone basis in the longitudinal space defined by Q2 p = p p , p2 = Q2, p = p , p2 = 0 , (2.2) A 1 − s 2 A − B 2 B and the usual trick of retaining only the first term in the decomposition of the metric tensor 2pµpν 2pµpν 2pµpν gµν = 2 1 + 1 2 +gµν 2 1 (2.3) s s ⊥ → s in the numerator of the gluon propagator connecting vertices µ and ν with momenta predominantly along p and p respectively. The virtual photon polarization vector e is 1 2 taken in the gauge ep =0, so that 2 2ep 1 e = e + p . (2.4) ⊥ 2 s Then the polarization vector e˜ in the usual gauge e˜p = 0 is A ep 1 e˜= e+ p , (2.5) A Q2 so that in the case of the longitudinal polarization, when e˜2 = 1, we have e p = Q. L L 1 Let us start with the calculation of the quark-antiquark production vertex. The dia- grams of the production process contributing in the Regge asymptotics are shown in Fig. 1. As was already mentioned, the quark-antiquark pair is produced in the photon frag- mentation region, so thattheinvariant mass √κofthepair isoforder oftypicaltransverse (0) momentaanddoesn’tgrowwiths. TheReggeformoftheproductionamplitude AQγ∗→Qqq¯ is 2s (0) c(0) c(0) = Γ Γ , (2.6) AQγ∗→Qqq¯ γ∗qq¯ t QQ where t = q2 and Γc(0) and Γc(0) are corresponding particle-Reggeon effective vertices in γ∗qq¯ QQ the Born approximation. Let us note, that the amplitude of Fig. 1 has automatically 3 k k 1 1 p p A A k k 2 2 q − q − pB pB′ pB pB′ Figure 1: The lowest order Feynman diagrams for the process γ∗Q (qq¯)Q. → k k 1 1 p p A A k k 2 2 − − q, c q, c c(0) Figure 2: Schematic representation of the vertex Γ . γ∗qq¯ only the octet colour state and the negative signature in the t-channel, so that it is not necessary here to perform any projection. The notations for all momenta are shown in Fig. 1. The quark-Reggeon vertex is known up to the NLA accuracy and its Born part is p ΓcQ(Q0) = gtcB′Bu¯B′6 s1uB, (2.7) where g is the coupling constant, tc are the colour group generators in the fundamental representation and u is a quark spinor wave function. Then, comparing Eqs. (2.6), (2.7) with the explicit form of the amplitude given by the diagrams of Fig. 1, one can easily c(0) obtain for Γ the diagrammatic representation of Fig. 2 (see Ref. [13]), where the γ∗qq¯ zig-zag lines represent the Reggeon with the momentum κ+Q2 +~q 2 q = p +q , t = q2 = q2 = ~q 2, (2.8) − s 2 ⊥ ⊥ − and the colour index c. The lowest order effective vertices for interaction of the Reggeon with quarks and gluons are defined in Fig. 3 (see Ref. [13]). c(0) The vertex Γ can be obtained from the diagrams of Fig. 2 by the usual Feynman γ∗qq¯ rules as the amplitude of the quark-antiquark production in collision of the virtual photon with the Reggeon. This procedure gives us the result Γˆ Γˆ p Γˆ p Γcγ(∗0q)q¯= −eqfgtci1i2u¯1 t11 − t22! 6 s2v2 = −eqfgtci1i2 "u¯1 t116 s2v2#−[1 ↔ 2]!, (2.9) where eq is the electric charge of the produced quark, i and i are the colour indices of f 1 2 the quark and antiquark correspondingly, v is the spinor wave function of the produced 4 k , a, µ k , b, ν 1 2 = igtc6p2. = igTc gµνp2(k2−k1) + pµ2(2k +k )ν s − ab s s 1 2 (cid:20) pν2(2k +k )µ +2(k1+k2)2 pµ2pν2 . c k +k , c − s 2 1 s p2(k1−k2) 1 2 (cid:21) (a) (b) Figure 3: The quark-quark-Reggeon and the gluon-gluon-Reggeon effective vertices. Tc is the colour group generator in the adjoint representation. antiquark, 1 1 Γˆ = (2x (ek ) e k ), Γˆ = (2x (ek ) k e ), 1 x 2 1 − 6 ⊥6 1⊥ 2 x 1 2 −6 2⊥ 6 ⊥ 1 2 ~k 2 +x x Q2 t = (p k )2 = i 1 2 , (2.10) i A i − − x i with the variables x defined by the Sudakov decompositions of the produced quark and i antiquark momenta ~k 2 k = x p + i p +k , k2 = 0 , i = 1,2 . (2.11) i i 1 sx 2 i⊥ i i The substitution 1 2 in Eq. (2.9) means replacement quark antiquark , i.e. x 1 ↔ ↔ ↔ ~ ~ x ,k k togetherwithreplacement ofthepolarizations. Validityofthesecondequality 2 1 2 ↔ in (2.9) can be easily verified using the charge conjugation matrix. We will need later the c(0) Born effective vertex Γ also in the helicity representation for the case of the space-time γ∗qq¯ dimension D equal 4. To obtain it we use the polarization matrix 1 1 p µ p ρˆ (v u¯ ) = x k +x k κ 2 2iξeµνσρk k 2ρ γ (1 ξγ ), (2.12) 2 1 2 1 1 2 2ν 1σ µ 5 ≡ √x x 4 − s − s − 1 2 (cid:20)(cid:18) (cid:19) (cid:21) where e0123 = 1 , γ = iγ0γ1γ2γ3 , (2.13) 5 and ξ = 1 is a double helicity of the produced quark. The polarization matrix satisfies ± the evident relations k ρˆ= ρˆ k = (1 ξγ )ρˆ= ρˆ(1+ξγ ) = 0 . (2.14) 2 1 5 5 6 6 − For the virtual photon polarization vector we also use the helicity representation 1 p 2pµ eµ(λ) = (δ +δ ) qµ +2iλeµνσρq p 2ρ +δ 2tQ2 2 , λ = 0, 1. √−2t " λ,1 λ,−1 (cid:18) ⊥ ν 1σ s (cid:19) λ,0q− s # ± (2.15) Using Eqs. (2.9) - (2.15) we get 2eq gtc x Γc(0) = f i1i2 2 δ √2qQx x γ∗qq¯ −√ 2tx x t λ,0 1 2− − 1 2 (cid:18)(cid:20) 1 n 5 k k k 1 1 1 p p p A A A k k k k k k 2 2 2 − − − q q q k k k 2 2 2 p − p − p − A A A k k k k k k 1 1 1 q q q k k 1 2 p p − A A k k 2 1 − k k q q c(0) Figure 4: Schematic representation of the vertex Γ . γ∗qq¯g ~ (k ~q +iλP)(x δ x δ ) [1 2] = 1 2 λ,−ξ 1 λ,ξ − − − ↔ 2eq gtc oix x (cid:17) = f i1i2 δ √2qQx x 1 2 λ,0 1 2 √ 2tx x t − t − 1 2 (cid:20) (cid:18) 2 1(cid:19) x x 2 ~ 1 ~ +(x δ x δ ) (k ~q +iλP)+ (k ~q iλP) , (2.16) 2 λ,−ξ 1 λ,ξ 1 2 − t t − (cid:18) 1 2 (cid:19)(cid:21) where q = ~q , | | p P = 2eµνσρk k p 2ρ, (2.17) 1µ 2ν 1σ s with the property P2 =~k 2~k 2 (~k ~k )2, and the replacement (1 2) is x x , ~k 1 2 − 1 2 ↔ 1 ↔ 2 1 ↔ ~ k , ξ ξ. 2 ↔ − Next we do is the calculation of the quark-antiquark-gluon production effective vertex c(0) Γ . It can be obtained through the usual Feynman rules with the elementary Reggeon γ∗qq¯g vertices defined at Fig. 3 as the amplitude of the quark-antiquark-gluon production in the virtual photon-Reggeon collision represented by the diagrams of Fig. 4, where the denotations of momenta arepresented. The colour indices ofthe Reggeonand the emitted gluon are c and b respectively. The Reggeon momentum is given by Eq. (2.8), where κ c(0) now is the quark-antiquark-gluon squared invariant mass. The vertex Γ obtained γ∗qq¯g in this way is invariant with respect to the gauge transformations of the emitted gluon polarization vector e and can be simplified by appropriate choice of the gauge. We use g the axial gauge 2(e k ) e p = 0, e = g⊥ ⊥ p +e , (2.18) g 2 g − sβ 2 g⊥ where β is defined by k = βp +~k 2/(sβ)p +k . In this gauge the last nonlocal term in 1 2 ⊥ the expression for the gluon-Reggeon interaction vertex of Fig. 3(b) disappears and we 6 obtain 1 p Γcγ(∗0q)q¯g(eqfg2)−1 = h1|tbtc|2i"u¯1((pA k1)2(k2 +q)2 6 e(6 pA− 6 k1) 6 e∗g(6 k2+ 6 q)6 s2 − 1 p 1 + e∗( k+ k )6 2( p k ) e e∗( k+ k ) e (k +k )2(p k )2 6 g 6 6 1 s 6 A− 6 2 6 − (k +k )2(k +q)2 6 g 6 6 1 6 × 1 A 2 1 2 − µ p γ ( p k ) e e( p k )γ 1 p ( k + q)6 2 + µ 6 A− 6 2 6 6 6 A− 6 1 µ βe∗µ 2(2qe∗) v × 6 2 6 s (pA k2)2 − (pA k1)2 ! (k +q)2 g − s g !) 2# − − + 1 tctb 2 [1 2] = h | | i ↔ 1 p 1 = 1 tbtc 2 u¯ e( p k ) e∗( k + q)6 2 + h | | i" 1((pA k1)2(k2 +q)2 6 6 A− 6 1 6 g 6 2 6 s (k +k1)2(pA k2)2× − − p 1 p e∗( k+ k )6 2( p k ) e e∗( k+ k ) e( k + q)6 2 × 6 g 6 6 1 s 6 A− 6 2 6 − (k +k )2(k +q)2 6 g 6 6 1 6 6 2 6 s 1 2 γ ( p k ) e e( p k )γ 1 pµ + µ 6 A− 6 2 6 6 6 A− 6 1 µ βe∗µ 2(2qe∗) v (pA k2)2 − (pA k1)2 ! (k +q)2 g − s g !) 2# − − 1 p 1 + 1 tctb 2 u¯ 6 2( k + q) e∗( p k ) e+ h | | i" 1((pA k2)2(k1 +q)2 s 6 1 6 6 g 6 A− 6 2 6 (k +k2)2(pA k1)2× − − p 1 p e( p k )6 2( k+ k ) e∗ 6 2( k + q) e( k+ k ) e∗ × 6 6 A− 6 1 s 6 6 2 6 g − (k +k )2(k +q)2 s 6 1 6 6 6 6 2 6 g 2 1 µ e( p k )γ γ ( p k ) e 1 p + 6 6 A− 6 1 µ µ 6 A− 6 2 6 βe∗µ 2(2qe∗) v . (2.19) (pA k1)2 − (pA k2)2 ! (k +q)2 g − s g !) 2# − − 3 The one-loop correction: the two-gluon exchange diagrams In this section we consider the contribution of the two gluon exchange diagrams to Γc . γ∗qq¯ There are six diagrams of such kind for the process we consider; they are shown at Fig. 5. Now we have to perform the projection on the negative signature and the octet colour state in the t-channel. It is done by the following replacement of the colour factor of the lowest line of the diagrams Fig. 5: 1 1 tbta tbta tatb = Tc tc . (3.1) B′B → 2 − B′B 2 ab B′B (cid:16) (cid:17) (cid:16) (cid:17) Then we obtain 1 (2g)(8,−)(1) = Ntc tc [(D +D ) (1 2)] [s s] , (3.2) AQγ∗→Qqq¯ 4 i1i2 B′B{ 1 2 − ↔ − ↔ − } 7 k k k 1 1 1 p p p A A A k k k 2 2 2 − − − pB pB′ pB pB′ pB pB′ (1) (2) (3) k k k 1 1 1 p p p A A A k k k 2 2 2 − − − pB pB′ pB pB′ pB pB′ (4) (5) (6) Figure 5: The two-gluon exchange Feynman diagrams for the process γ∗Q (qq¯)Q. → where N is the number of colours, D is the amplitude represented by the diagram of Fig. 1 5(1) with omitted colour generators in any its vertex, and 2D is such amplitude for the 2 diagram of Fig. 5(2). The calculation of D is quite straightforward. Note here that since our final goal is 1 the virtual photon impact factor in the physical space-time, and since the integration over the quark-antiquark states in Eq. (1.3) is not singular, we need to retain in the vertex Γc and consequently in the amplitude (2g)(8,−)(1) only the terms which do not vanish γ∗qq¯ AQγ∗→Qqq¯ at ǫ 0. Therefore all one-loop results are presented in this paper with such accuracy. → In the convenient for us form we have 4 p Γˆ Γˆ p s (D1 D1(1 2)) (s s) = gu¯B′6 1uB( )eqfgu¯1 1 2 6 2v2 ω(1)(t) − ↔ − ↔ − N s − t1 − t2 ! s t s s p 2s Γ(2 ǫ) 1 × ln t +ln −t +4gu¯B′6 s1uB t eqfg3 (4π)−2+ǫ 2ǫ (cid:18) (cid:18)− (cid:19) (cid:18)− (cid:19)(cid:19) ˆ Γ p 1 u¯ 16 2v 2( t)ǫ +1+2(1+ǫ)lnx +ǫ 5ǫψ′(1) 1 2 2 × t s − ǫ − (cid:26)(cid:20) 1 (cid:18) (cid:18) (cid:19) 1 dy + 2(1+ǫ)(t 2(t t )yǫ) 1 2 , (3.3) Z0 ( (1 y)t yt1)1−ǫ − − 1 (cid:19)(cid:21)−(cid:20) ↔ (cid:21)(cid:27) − − − (8,−) where the first term is responsible for the Reggeization of the amplitude with AQγ∗→Qqq¯ ω(1) being the one-loop Reggeized gluon trajectory, Γ(1 ǫ) ǫ Γ2(ǫ) ω(1)(t) = g2N − ~q 2 , (3.4) − (4π)2+ǫ Γ(2ǫ) (cid:16) (cid:17) and Γ(z) and ψ(z) are the Euler Γ-function and its logarithmic derivative correspond- ingly. The calculation of D is more complicated and we present some details of it in the 2 Appendix. Here we write down only the result p 2s Γ(2 ǫ) 1 (D2 −D2(1 ↔ 2))−(s ↔ −s) = 4gu¯B′6 s1uB t eqfg3 (4π)−2+ǫ 2ǫ 8 1 1 dy dy u¯ 1 2 yǫ−1(1 y )y−ǫ ×(cid:26)(cid:20) 1Z0 Z0 [(1 y2)( (1 y1)t y1t2)+y2( (1 y1)t1 +y1Q2)]2−ǫ(cid:18) 1 − 1 2 − − − − − − xǫx−ǫ 2ǫ2ψ′(1) 2tΓˆ +(1 y )4t(ek ) × 1 2 − 1 − 1 1 (cid:16) (cid:17) p + yǫy−ǫxǫx−ǫ 1 4x t(ep ) 6 2v 1 2 . (3.5) 1 2 1 2 − 2 1 s 2 − ↔ (cid:16) (cid:17) (cid:19) (cid:21) (cid:20) (cid:21)(cid:27) Note that the imaginary parts of D2 (in the (pB′ + k2)2-channel) and D2(1 2) (in ↔ the (pB′ + k1)2-channel) which would destroy the Reggeization cancel in the amplitude (8,−) . Althoughin the NLA BFKL approach there is no requirement ofthe Reggeiza- AQγ∗→Qqq¯ tion of full amplitudes (the Reggeization of their real parts is sufficient), we see that nevertheless the Reggeization holds also without omitting of any imaginary part for the process Qγ∗ Qqq¯. → The amplitude Qγ∗→Qqq¯with the octet colour state and the negative signature in the A t-channel has the following Reggeized form s s ω(t) s ω(t) 2s s A(Q8γ,−∗→) Qqq¯= Γcγ∗qq¯t "(cid:18)−t(cid:19) +(cid:18)−−t(cid:19) #ΓcQQ ≈ Γcγ(∗0q)q¯ t ΓcQ(Q0) +Γcγ(∗0q)q¯tω(1)(t) s s 2s 2s c(0) c(0) c(1) c(1) c(0) × ln t +ln −t ΓQQ +Γγ∗qq¯ t ΓQQ +Γγ∗qq¯ t ΓQQ. (3.6) (cid:20) (cid:18)− (cid:19) (cid:18)− (cid:19)(cid:21) Let us now split the one-loop contributions to this amplitude and both of the effective vertices according to the three sets of the one-loop diagrams for Qγ∗→Qqq¯: the two-gluon A exchange diagrams, the t-channel gluon self-energy diagrams and the one-gluon exchange diagrams 2s 2s (2g)(8,−)(1) (se)(8,−)(1) (1g)(8,−)(1) (2g)c(1) c(0) c(0) (2g)c(1) + + = Γ Γ +Γ Γ AQγ∗→Qqq¯ AQγ∗→Qqq¯ AQγ∗→Qqq¯ γ∗qq¯ t QQ γ∗qq¯ t QQ (cid:26) s s s +Γcγ(∗0q)q¯tω(1)(t) ln t +ln −t ΓcQ(Q0) (cid:20) (cid:18)− (cid:19) (cid:18)− (cid:19)(cid:21) (cid:27) 2s 2s 2s 2s (se)c(1) c(0) c(0) (se)c(1) (1g)c(1) c(0) c(0) (1g)c(1) + Γ Γ +Γ Γ + Γ Γ +Γ Γ , (3.7) γ∗qq¯ t QQ γ∗qq¯ t QQ γ∗qq¯ t QQ γ∗qq¯ t QQ (cid:26) (cid:27) (cid:26) (cid:27) where the self-energy diagrams and one-gluon exchange diagrams have automatically only the octet colour state and negative signature in the t-channel, so that (se)(8,−)(1) (se)(1) (1g)(8,−)(1) (1g)(1) , . (3.8) AQγ∗→Qqq¯ ≡ AQγ∗→Qqq¯ AQγ∗→Qqq¯ ≡ AQγ∗→Qqq¯ We remind that in our case of completely massless quantum field theory the contribution from the renormalization of the external lines is absent in the dimensional regularization. (1g)c(1) Now, from the representations of Eqs. (3.7), (3.8) it is easy to see that Γ is given by γ∗qq¯ the radiative corrections to the amplitude of the quark-antiquark production in collision of the virtual photon with the gluon having momentum q, colour index c and polarization vector pµ/s, whereas Γ(1g)c(1) is defined by the radiative corrections to the vertex of − 2 QQ interaction of this gluon with the quark Q. In both cases the gluon self-energy is not 9