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Small-x behaviour of the polarized structure function $F_3^{gamma}(x,Q^2)$

NTZ 15/89

arXiv:hep-ph/9806439v2 3 Jul 1998

Small-x behaviour of the polarized γ photon structure function F3 (x, Q2) 1
B. Ermolaev?$ , R. Kirschner? , and L. Szymanowski?#

Naturwissenschaftlich-Theoretisches Zentrum und Institut f¨ ur Theoretische Physik, Universit¨ at Leipzig Augustusplatz 10, D-04109 Leipzig, Germany

Io?e Physico-Technical Institute, St. Petersburg 194021, Russia


Soltan Institut for Nuclear Studies, Ho˙ za 69, 00-681 Warsaw, Poland

Abstract: We study the small-x behaviour of the polarized photon structure γ function F3 , measuring the gluon transversity distribution, in the leading logarithmic approximation of perturbative QCD. There are two contributions, both arising from two-gluon exchange. The leading contribution to small-x is related to the BFKL pomeron and behaves like x?1?ω2 , ω2 = O(αS ). The other contribution includes in particular the ones summed by the DGLAP equation and (+) √ (+) behaves like x1?ω0 , ω0 = O( αS ).

Supported by German Bundesministerium f¨ ur Bildung, Wissenschaft, Forschung und Technologie, grant No. 05 7LP91 P0, and by Volkswagen Stiftung





The deep-inelastic scattering o? a photon attracted much attention over 20 years. The idea of extracting the direct contribution to the structure function, which is calculable from theory without additional assumptions [1], motivated experimental studies more involved compared to the standard lepton-nucleon reactions ([2] and references therein). The extension of the deep-inelastic photon cross section on a target of spin 1 involves structure functions which have no analogon in the usual spin 1 case. 2 The Lorentz structure of photon-photon amplitudes have been studied by many authors, e.g. [3]. The expression for the deep-inelastic cross section on general spin 1 target is given in [4]. We recall the expression for the photon case d2 σ ? L?ν W?ν , dq 2 dx q?q? 1 γ ρ g ?ν ? 2 ?? W ?ν = F1 ? F3 ⊥ ?⊥ρ 2 q pq pq 1 γ ρ p? ? 2 q ? pν ? 2 q ν ?? ? F2 ? F3 ⊥ ?⊥ρ 2 q q 1 γ ?? ν ν ? ?νλα + F3 (?⊥ ?⊥ + ?? qλ Sα . ⊥ ?⊥ ) + i g1 ? 2


Here ?? ⊥ is the transverse (with respect to p and q ) part of the target photon helicity vector. Sα is the spin vector given by Sα = i ?αβγτ ?⊥β ?? ⊥ γ pτ . γ Comparing with the proton target case we see that indeed F3 is the interesting new structure function. It can be measured if the target photon is polarized; the polarization of the virtual photon, i.e. of the lepton scattered with large momentum transfer, is not necessary. This structure function measures the parton polarization in an essentially di?erent way compared to g1 . It is related to the virtual photon - photon scattering amplitude with helicity ?ip [4], [5]. In order to give this amplitude the probability interpretation of a parton distribution one has to change the basis from helicity (circular polarization) to linear polarizations. γ F3 measures the asymmetry of linear polarizations, the transversity of gluons. γ F3 is similar to the structure function h1 , which appears in the Drell-Yan cross-section and measures the transversity of quarks [6]. We show that there is γ a far-reaching analogy of the small-x behaviour of both F3 and h1 . The small-x behaviour of h1 has been analyzed recently [7]. There are recent results on the Q2 evolution of h1 [8]. Related questions have been discussed in the workshop on spin physics at HERA [9]. In the present paper we investigate the polarized photon structure function γ F3 at small-x. In this asymptotics it is appropriate to consider the t-channel exchange of the corresponding forward scattering amplitude and to study the


leading singularities on the complex angular momentum plane of the corresponding t-channel partial wave. In general, P parity and a minimal angular momentum are the relevant tchannel quantum numbers for polarized structure functions. The unpolarized parton distributions are related to positive parity exchange, the helicity distributions to negative parity exchange. The second type of polarized parton distributions, the transverse polarization asymmetries, are related to the exchange of longitudinal angular momentum 1 (quark transversity) or 2 (gluon transversity). It is convenient to study the perturbative Regge asymptotics in the framework of the high-energy e?ective action [10], [11], leading in particular to reggeized gluons and quarks. Two leading reggeized gluons interacting via e?ective vertices result in the BFKL pomeron. Non-leading gluonic reggeons have been studied recently [12]. The latter are relevant in particular for polarized structure functions since they are able to to transfer odd parity or non-vanishing longitudinal angular momentum projection (s-channel helicity) σ . The leading gluonic reggeon 1 corresponds to σ = 0 and the leading reggeized quark to σ = 2 . The multiple exchange of reggeons gives rise to the tree-level energy behaviour sα0 , where α0 = 1 ? σi . (1.2)

This can be considered as the extension of the known Asimov rule [13] to nonleading exchanges. Helicity 2 exchange by two gluons, relevant for our case, can occur in two ways: (1) by non-leading reggeons (σ1 = 0, σ2 = 2), α0 = ?1 (2) by leading reggeons (σ1 = σ2 = 0) with a non-vanishing longitudinal projection of orbital momentum, α0 = +1. We shall encounter contributions of both cases. After recalling in sec. 2 the known results about the photon spin-?ip amplitude [3], [14], [4], [5], about the one-loop coe?cient function and the parton splitting function [16], [15], [17], we sum in sec. 3 the double-logarithmic correc(+) γ tions. This concerns a contribution to F3 behaving like x1?ω0 , corresponding to the case (1) above with α0 = ?1. We calculate the partial wave whose singularity position gives rise to this behaviour and to the related resummed anomalous dimensions near angular momentum j = ?1. The contribution of the BFKL pomeron to the photon spin-?ip amplitude is known [22]. In sec. 4 we recall the impact factors relevant for the photon structure function and the branch point singularity of the conformal spin or helicity γ n = 2 term in the BFKL solution. This contribution to F3 behaves like x?1?ω2 corresponding to the case (2) above with α0 = +1.



Lowest order contribution and Q2 evolution

γ The spin structure function of the photon F3 is given by the imaginary part of the virtual photon-photon forward scattering amplitude with spin ?ip of both scattered photons [5] γ ?ν ;αβ ?ν ;αβ 2 = P(+ ?;?+) + P(?+;+?) F3 (x, Q ) ?=2 ?ν ;αβ P(+ ?;?+) ;αβ + P(?ν ?+;+?) =

W ?ν ;αβ

1 ?α νβ ?β να ?ν αβ g⊥ g⊥ + g⊥ g⊥ ? g⊥ g⊥ 2



W ?ν in (1.1) is related to (2.1) by W ?ν = W ?ν ;αβ ?⊥α ?? ⊥β . The transverse subspace 2 2 is the one orthogonal to the momentum q (q = ?Q ) of the virtual photon and p (p2 = 0) of the photon target. Consider ?rst the lowest order contribution to this amplitude arising from a fermion loop, Fig. 1. This has been calculated by many authors long ago [3], [14]. We use calculations by Zima [14] and pick up the projection Eq. (2.1) from the result e4 2 (0) 2 (2.2) F3 (x, Q ) = 2 x N , 4π where e is the electromagnetic coupling constant and N denotes the number of colours.
q p



Figure 1: Fermion loop contribution to the imaginary part of the γ ? γ forward scattering amplitude.

The lowest order fermion loop does not contribute to the anomalous dimension. Notice that the resulting amplitude (2.1), (2.2) is even in exchanging s and u channels. We see that the positive signature amplitude is relevant for the photon spin structure function.


It is known that contributions to the anomalous dimensions (Q2 evolution) arise from the exchange of two gluons and these contributions are proportional to x, i.e. they dominate (2.2) at small x [15], [16]. We calculate the lowest order contribution with two gluon exchange in the asymptotics of small x (Fig. 2).
q p







Figure 2: Two-gluon exchange contribution to γ ? γ forward scattering. As building blocks we have the imaginary parts of the fermion loop photongluon scattering amplitude, on one side with the virtual photon q and on the other with the real photon p. Both can be read o? from [14], where the same 2 ′ ′ projector (2.1) applies (k? = ?αq? + βp? + κ⊥? , q? = q? ? 2qpq p? ) F γ (β s, κ⊥ ; Q2 ) = 8 e2 g 2

2 κ2 ⊥Q ln (sβ )2 2 (κ2 ⊥) , (sα)2

2 κ2 ⊥Q

, Q2 ? |κ2 ⊥| , (2.3)

2 2 F γ (α s, κ⊥ ; Q2 0 ) = ?2 e g

2 Q2 0 ? |κ⊥ | .

2 F γ for Q2 0 ? |κ⊥ | takes into account the direct photon contribution. A resolved (non-perturbative) contribution is to be added. The results are approximate according to the Regge kinematics appropriate at small x:

We obtain (x =

Q2 ) s

sαβ ? κ2 s ? κ2 ⊥ , ⊥ , 2 2 sα ? κ⊥ , sβ ? κ⊥ + Q2 .


d4 k ? F γ (βs, κ⊥ ; Q2 )F γ (αs, κ⊥ ; Q2 0) 2 2 (k + i?) π2 ? 1 + O(x2 ) = ?e4 g 4 N 2 2πx ln2 x + 8πx 12 5



The resulting contribution to the structure function behaves indeed like x with logarithmic corrections. The Q2 evolution arises from the interaction of the exchanged gluons with strong ordering of the transverse momenta [18] (see Figs. 3a,b)
γ g F3 = F3 ? F3 , (0)

g 2 (Q2 ) d g 2 F ( x, Q ) = d ln Q2 3 8π 2



dz z g P ( ) F3 (z, Q2 ) . z x


g The photon structure function is obtained from the gluon transversity F3 by (0) convolution with the coe?cient function proportional to F3 (2.2).




k q (a) p (b) p

Figure 3: One-loop contribution to the Q2 evolution. The initial-value x-distribution at Q2 0 for eq.(2.6) is the sum of the perturbative contribution of F γ (2.3) and a non-perturbative resolved contribution. The evolution kernel P (z ) is readily obtained, e.g. by calculating the graph in Fig. ′ 3b in the axial gauge, A? q? = 0, P (z ) = 2Nz . 1?z (2.7)

The anomalous dimensions corresponding to this result has been presented ?rst γ in [17] without relation to F3 . Later this splitting function was derived in [15] and in [16]. In (2.7) we do not write the contributions ? δ (z ? 1) which are irrelevant for the small-x asymptotics. We shall see that this leading ln Q2 evolution leads to a small-x behaviour of γ F3 proportional to x up to logarithmic corrections. Correspondingly, the oneloop anomalous dimension has a pole at j = ?1. The logarithmic correction to the small-x asymptotics are not completely accounted for by (2.7) as we shall discuss below. 6


Double log contributions at small x
The gluon ladder graphs

We consider the contribution of a s-channel intermediate state of gluons in multiRegge kinematics to the interaction of the exchanged gluons (see Fig. 4a).

b β1 1 k 1

b β 2 2 k 2

a α 1 1 (a)

a α2 2 (b)

Figure 4: The gluon ladder contribution. From the DGLAP equation (2.6) we see that there is a double log contribution 1 ? g 2(ln ) ln Q2 x from each loop arising from the con?guration of strongly ordered transverse momenta. We show that there are further double logarithms resulting in a contri1 bution ? g 2 ln2 x from each loop which determine the small-x behaviour. As the ?rst step we analyze the one-rung contribution (Fig. 4b) in the appropriate projection (2.1)
⊥ σ β1 β2 Γα2 α1 σ (k2 , k1 )f a2 a1 c gσσ (k1 , k2 )f cb1 b2 ′Γ δa1 b1 δa2 b2 (P(+?;?+) + P(?+;+?) )α1 β1 ,α2 β2 N2 ? 1 2 = 4N (κ2 1⊥ + κ2⊥ ) ,


where Γ is the triple-gluon vertex. Note that only the transverse gluon polarizations contribute to the s-channel intermediate states. In order to have a double log contribution from each loop we have to obtain a logarithmic contribution from the transverse momentum κi⊥ integral. For this a factor of κ2 i⊥ from the numerator as is provided by the result (3.1) is essential. The coe?cient 4N in (3.1) determines the size of the double log contribution. The ladder graphs shown in Fig. 4a are summed by an integral equation the kernel of which is obtained in an obvious way using (3.1). The peculiarity are the limits of integration in the longitudinal momentum fraction β and the 7

transverse momentum κ, which show whether there are double logs beyond the strong ordering region in κ: x ? . . . βi ? βi+1 . . . ? 1 , |κ2 | |κ2 Q2 i| ? ... ? i+1 . . . ? ?2 . x βi βi+1 (3.2)

Here and in the following we denote the transverse vectors by means of the 2 complex number (κ = κ1 ⊥ + iκ⊥ ). Indeed we ?nd that there is strong ordering 2 | , which leads to a double log region larger than the one in the in β ’s and in |κ β DGLAP equation. The discussion above relies on the simple ladder graphs assuming that there are no further double log contributions. This is indeed the true for the negative signature channel, but not obvious. However we ?nally have to calculate the positive signature partial wave, as we have pointed out in Sec. 2. We apply the method of Ref. [19].


The soft t-channel intermediate state

The leading ln s approximation in the considered channel results in a sum of ladder graphs summed by an BFKL-like equation. The ladders are built of e?ective vertices and reggeized gluons and each of them is a gauge invariant sum (in the considered approximation) of Feynman graphs. The leading ln s contributions in the vicinity of j = ?1 and for negative signature, i.e. the ln s corrections to the power s?1 contribution to the amplitude, include in particular the double log contributions in question. Consider the transverse momentum κi integral in one of the ladder loop. The region where κi is smaller than all other transverse momenta in the loops is to be investigated in particular. The question about the double log beyond the DGLAP equation reduces to the question of whether the transverse momentum integral is logarithmic in this region of smallest κ. The ladder loop at κ → 0, Fig. 5a, is related to the product of graphs with scattering gluons Fig. 5b. In this way the exchanged (t-channel) reggeized gluons are related to scattered (s-channel) gluons. Due to the particular projection (2.1) the gluon scattering is accompanied by helicity ?ip. The scattering graph Fig. 5b stands for gauge invariant sum of graphs involving e?ective scattering vertices. In the tree approximation there are no e?ective scattering vertices changing helicity with leading (near j = 1) or subleading (near j = 0) gluonic reggeons [12]. A helicity-?ip e?ective vertex appears only at the next-to-subleading (j = ?1) level. We conclude that the contribution to the considered channel (2.1) arises from the exchange of one leading (j ? 1) and one next-to-subleading (j ? ?1) gluonic reggeon. Now we consider again the relation of the scattering graphs Fig. 5b to the ladder loop Fig. 5a (t-channel unitarity). In order to get Fig. 5a from Fig. 5b one has to turn the scattering gluons into exchanged reggeons and add the 8




Figure 5: Gluon ladder element with the smallest transverse momentum (a) and the related gluon scattering graphs (b).

propagators. The trajectory functions α(κ) can be disregarded in the double log approximation. The relation of a leading gluonic reggeon to a scattering gluon in the small κ region has been analysed in [11]. It is important that there arises a factor of κ at each vertex in turning the scattering gluon into the leading gluonic reggeon. The subleading gluonic reggeon is obtained from the scattering gluon without such additional factor. As a result we have a factor |κ|2 and together with the two propagators we obtain a logarithmic transverse momentum integral. We compare the situation to the other channels: In the case of two leading gluonic reggeons we would have instead a factor |κ|4 and in the case of two subleading reggeons no additional factors of κ. In both cases no double log contribution from the soft κ region appears. The scattering graphs Fig. 5b are directly related to the parton splitting kernel P (z ) in the DGLAP equation (2.6). Each of the scattering graphs can be compared with the graph Fig. 3b, from which we have obtained P (z ) using axial gauge q ′ A = 0. Indeed, the gauge invariant e?ective scattering vertices coincide with the bare vertices in the corresponding axial gauge. The double log contribution of the BFKL ladders can be summed by the equation given graphically in Fig. 6. Introducing the partial waves we can write it in the following form (ω = j + 1) f (?) (ω ) = 1 1 g 2a0 + 2 (f (?) (ω ))2 . ω 8π ω (3.3)

The Born term corresponds in fact to one of the scattering graphs in Fig. 5b;




Figure 6: Equation summing the double log contribution from soft t-channel intermediate states.

a0 j +1

is the leading contribution at j → ?1 of the moment transformation of P (x) a0 = j+1


d x xj ?1 P (x) |j →?1


and we have a0 = 2N . Eq. (3.3) is readily solved f (?) (ω ) = 4π 2 ω ?1 ?

4αS N ? 1? . πω 2



However this is not the ?nal result for the double log contributions to the parton spin-?ip amplitude (2.1) because we have calculated so far only the negative signature contribution.


Soft bremsstrahlung contributions

The leading ln s e?ective ladder equation of BFKL type is obtained under the essential assumption of colour singlet state in t-channel. Only in this case the infrared divergencies cancel. To analyze amplitudes and partial waves corresponding to non-singlet channels we restrict the integrations over the transverse momentum κ by the condition |κ|2 > ?2 . In non-singlet channels there are more double log contributions besides of those contained in the ladder. Loops with a single gluon being soft, i.e. carrying the smallest κ, give rise to double logs. This is di?erent from the two-gluon t-channel intermediate state discussed above. The contribution of the soft gluon loop is easily calculated relying on Gribov’s bremsstrahlung theorem: The gluon with the smallest transverse momentum is effectively emitted and absorbed from the external lines. With the bremsstrahlung contribution the equation in Fig. 6 generalizes to the one shown in Fig.7. The soft bremsstrahlung contributions change the gauge group quantum numbers in the t-channel. Now instead of a single partial wave f (ω ) we have to consider a column vector involving the colour singlet partial wave discussed so far







Figure 7: Equation summing the double log contributions both from soft tchannel intermediate states and from soft bremsstrahlung.

together with the colour octet and further colour channnels
? ? ? ? ? ? ? ?

?= f

f f8 · · ·

? ? ? ? ? ? ? ?



We shall concentrate on these ?rst two entries, disregarding the remaining representations which appear in the symmetric part of the tensor product of two adjoint (gluon) representations.. Let us notice that the octet contribution in question is the one arising by antisymmetrization, i.e. by projection with f abc . In colour space the interaction induced by gluons attached to the external lines is expressed by multiplying the column of partial waves by a matrix. This 11

matrix depends on whether the pair of those external lines belongs to the s- or t- channels. The upper limits of the double logarithmic transverse momentum integrals are respectively s, u or t. Therefore the t-channel type soft gluons are not important in our case. Nevertheless it is reasonable to introduce all three ? t is diagonal in the representation with de?nite quantum matrices. Obviously M numbers in the t-channel. Restricting to the two relevant colour channels we have ?t = M N 0 0
N 2



The sum of the three types of colour interactions taken with the same weight (the interaction in momentum space leads to di?erent weights, however) acts as the identity up to the factor N ?s + M ?u + M ?t = N I ? M (3.8) Working out the relations of SU (N ) representations leads to ?s = M 0
N N 2 ?1

N 4



The complete matrices for all colour representations appearing in the two-gluon exchange are given in [20]. The soft bremsstrahlung contribution is readily calculated in momentum space. The sum of 4 diagrams, two of s-channel and two of u-channel type, results in g2 4π 2


d |κ|2 ? ?s ? u ln( s ) A ?(σ) (s, |κ|2) . Ms ln( 2 ) + M 2 |κ| |κ| |κ|2


The square bracket in (3.10) involves an odd contribution in s, which leads to a ?(σ) (s, |κ|2) is the column vector of (almost on-shell) change in signature σ . Here A gluon forward scattering amplitude where the double log corrections are included with κ as the lower cut-o? in all transverse momentum integrals

?(σ) (s, |κ|2) = A


d ω ?(σ) s f (ω ) 2πi |κ|2


ζ (σ) (ω ) .


Taking into account the signature factor approximated for small ω = j + 1 1 (3.12) ζ (?) (ω ) ≈ 1 , ζ (+) (ω ) ≈ ? iπω , 2 we have in terms of partial waves ?(?) (ω ) f

= =

g2 ? ?(?) (ω ) ?u 1 d f Ms + M 2 4π ω dω (3.13)

?(+) (ω ) f

g2 1 ? g2 ? 1 d (+) ? ?(?) (ω ) . ? ?u f ( ω f ( ω )) ? M + M Ms ? M s u 4π 2 ω 2 dω 4π 2 ω 2 12

The signature changing contribution is relevant in our approximation (small ω ) only if multiplied with the negative signature amplitude and therefore leads to an additional inhomogeneous term in the positive signature equation. We write the resulting partial wave equations in matrix form ?(?) (ω ) = aM ?t g f ω 2 g ?(+) (ω ) = aM ?t f ω 2 g ?s ? M ?u ? 2 M 4π

g2 ? ?(?) (ω ) + 1 1 (f ?(?) 2 (ω )) ?u 1 d f Ms + M 2 4π ω dω 8π 2 ω g2 ? 1 d ?(+) (ω )) ? + 2 M (ω f s + Mu 4π ω 2 dω 1 1 ?(?) ?(+) 2 (ω )) . f (ω ) + 2 (f (3.14) 2 ω 8π ω +

?(±) 2 ) denotes the column vector with the squares of the partial waves of the (f corresponding colour channel. Comparing with a0 in (3.3) for the colour singlet channel we identify a = 2. This is the part of the residue a0 of the one-loop anomalous dimensions at j = ?1 not related to colour factors but merely to the helicity state. Comparing to the gluonic contribution to the double log small-x asymptotics of the helicity asymmetry g1 [21] and to the ?avour non-singlet (quark-antiquark) contributions [7] we see that there we have a = 4 and 1, respectively. ?s + M ? u )11 vanishes the equation for the colour Since the matrix element (M singlet component for both signatures is just an algebraic equation of second order. The negative signature solution has been discussed above. For the positive signature we obtain f0

= 4π 2 ω ?1 ?



2αS N 1 (?) 2 ? 2 f8 (ω ) ? . 2 πω 2π ω



The solution is expressed in terms of the negative signature octet partial wave. The corresponding equation is di?erential of Ricatti type. The solution can be expressed in terms of the logarithmic derivative of the parabolic cylinder function Dp (z ) [19] f8

= 4παS N αS N , 2π

ω2 ω d ln exp( 2 )Dp ( ) dω 4? ω ω ? p=2 . (3.16)

ω ?2 =

γ The small-x dependence of F3 (x, Q2 ) is obtained from the partial wave f0 inverse Mellin transform i∞ γ F3 (x, Q2 )




d ω (+) f (ω )ζ (+) (ω ) x?ω . 2πi 0 13


In the relation we did not include explicitely the convolution of the reggeon Green (+) function f0 with the appropriate impact factors, which is not important neither for the leading small-x behaviour nor for the leading term of the anomalous dimensions. The small-x asymptotics is determined by the right-most singularity (+) (+) ω0 of f0 (ω )
γ F3 (x, Q2 ) double?log (+) (?)

? x1?ω0





is close to ω0 , the branch point of f0 (ω ) ω0



4αS N . π


Assuming αS = 0.2, N = 3 we have ω0 ≈ 0.87. As a rough estimate of the (?) signature changing contribution we replace f8 by its Born term. Unlike the case of two fermion exchange the latter is not suppressed in the large N approximation, (+) therefore this estimate is worse in the present case. We obtain ω0 ≈ 0.61. In the double-log approximation also the resummed anomalous dimensions (+) (+) ν (j ) near j = ?1 are obtained in terms of the partial wave f0 (ω ), ω = j + 1, ν (+) (?1 + ω ) = 1 (+) f (ω ) . 8π 2 0 (3.20)


The perturbative pomeron contribution

We calculate in the leading ln s approximation the contribution to the virtual photon spin-?ip amplitude (2.1) with the asymptotics s1+ω2 leading to a contribution to the structure function behaving like x?1?ω2 . There are well known results for the real and virtual photon impact factors with the exchange of two leading gluonic reggeons, describing the coupling of the scattering photon to the BFKL pomeron [24] via a quark loop [25], [22], [23]. These impact factors involve a helicity-?ip contribution at vanishing momentum transfer. For a virtual photon the helicity-?ip (?λ = ±2) contribution to the impact factor reads
(+,?) ΦA (κ, Q2 )

αe.m. αS = √ ·4 2π



q 2 0 ? 0


x(1 ? x)y (1 ? y ) κ?2 |κ|2 x(1 ? x) + Q2 y (1 ? y ) + m2 q (4.1)



(κ, Q ) = Φ


(κ, Q )


Q2 ? m2 ρ .

For quasi-real photons (Q2 ≤ m2 ρ ) this expression is applicable only for the heavy ?avour contribution. Non-perturbative contributions are essential and can be roughly described by applying the Borel transform with respect to Q2 to (4.1)


and identifying the variable conjugated to Q2 with the ρ-meson mass mρ [22],
(+,?) ΦB (κ, Q2 )

+H.Q. ,

Q2 ≤ m2 ρ .

|κ|2 x(1 ? x) x(1 ? x)κ?2 αe.m. αS exp ? · 4 · Nq · dx dy = √ m2 m2 2π ρ ρ y (1 ? y ) 0 0 (4.2)



Here Nq is the number of light quarks (Nq = 3) and H.Q. denotes the expression (4.1) with the sum now restricted to the heavy quark ?avours. In the Regge asymptotics the partial wave of the photon spin-?ip forward scattering amplitude can be calculated from the impact representation d2 κ1 d2 κ2 (+,?) (?,+) Φ (κ1 , Q2 ) F (ω ; κ1 , κ2 ) ΦB (κ2 , Q2 0) . |κ1 |4 |κ2 |4 A (4.3) In our case the reggeon Green function F (ω ; κ1 , κ2 ) is obtained from the well known solution of the BFKL equation [23], [24]. In the sum over the conformal spin n only the term n = 2 contributes f+,?;?,+ (ω, Q2 ) =
? 2 ?iν 2 ?2 ? 2 +iν |κ2 κ1 κ2 dν |κ2 2| 1| , 2 g N ω ? 8π2 ?(2, ν ) n 1 n 1 ?(n, ν ) = 4ψ (1) ? ψ ( + iν + ) ? ψ ( ? iν + ) 2 2 2 2 1 n 1 n ?ψ ( + iν ? ) ? ψ ( ? iν ? ) . (4.4) 2 2 2 2 The spin-?ip partial wave has a cut with the right branch point at angular momentum j = 1 + ω2 ,

1 F (ω ; κ1, κ2 ) = 2 2π



ω2 =

g2 N g2 N ?(2 , ν ) = (ln 2 ? 1) . 8π 2 π2


g = 0.2 as in the usual estimate for the BFKL pomeron intercept Choosing αS = 4 π and N = 3 we have ω2 = ?0.23. Investigating the Q2 behaviour of (4.4) we ?nd that this perturbative pomeron contribution does not in?uence the anomalous dimension (in the vicinity of moment number j = 1) at the leading ln Q2 level. Its in?uence on the anomalous 1 dimension arises at the next-to-leading level (as a pole term ? j ? ) from the 1 iteration of both considered contributions.



There are two contributions to the small-x asymptotics of the polarized photon γ structure function F3 (x, Q2 )
γ γ F3 (x, Q2 ) = F3 (x, Q2 ) double?log γ + F3 (x, Q2 ) BF KL




The ?rst is closely related to the DGLAP evolution. The small-x asymptotics is obtained in the double logarithmic approximation extending the double-log contribution to the DGLAP equation beyond the region of strong ordering in the transverse momenta. The t-channel partial wave describing this contribution and the related anomalous dimension near angular momentum j = ?1 in all orders of perturbation theory have been calculated. The small-x behaviour of (+) (+) this contribution is found to be x1?ω0 , where the displacement ω0 is of order √ (+) αS and is estimated to be ω0 ≈ 0.6. The second contribution is not directly related to the DGLAP evolution. Its trace appears in the anomalous dimensions as pole terms at j = +1 starting from the two-loop approximation. This contribution arises from the conformal spin n = 2 term of the BFKL pomeron solution [23], [24]. It is a term which is not essential in the usual phenomenological applications of the BFKL pomeron. γ Therefore the experimental study of F3 (x, Q2 ) would allow to test the detailed structure of the BFKL pomeron. This contribution dominates at small x and behaves like x?1?ω2 , where the displacement ω2 is of order αS and is estimated to be ω2 ≈ ?0.23. Both contributions arise from the exchange in t-channel of two (reggeized) gluons interacting by s-channel gluons. s-channel helicity σ = 2 is transferred in both cases. In the ?rst case one of the reggeized gluons is the leading one carrying σ = 0 and the other a twice-subleading one carrying σ = 2. The latter couples to gluons scattering with helicity ?ip, whereas the ?rst does not feel the helicity of scattering partons. The resulting Regge singularity is a branch cut at j = (+) ?1 + ω0 . In the second case both reggeized gluons are the leading ones (σ = 0), which are, however, in a state with the longitudinal projection of the orbital angular momentum equal to 2. The resulting Regge singularity is a branch cut at j = 1 + ω2 . We notice that also in the small-x asymptotics of the structure function h1 (x, Q2 ) measuring the quark transversity we have encountered two contributions in analogy to (5.1). There we have two reggeized quarks in the t-channel which carry s-channel helicity σ = 1. The ?rst contribution (? x plus corrections) related to the DGLAP evolution, arises from the exchange of one leading and one subleading quark reggeon. The second contribution (constant in x) arises from two leading quarks with parallel helicities. We have given the couplings (impact factors) of these exchanges to the scattering photons explicitely. We did not perform here a numerical analysis of the γ expressions obtained for the amplitude related to F3 (x, Q2 ). It is clear, however, that the couplings of the angular momentum 2 state are weaker. Therefore (+) we expect that the ?rst contribution (? x1?ω0 ) will dominate at not too small


values of x and will be overcome by the second (? x?1?ω2 ) only at very small x.

L.Sz. would like to acknowledge the warm hospitality extended to him at University of Leipzig where this work was completed.

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