Introduction Recently, a long awaited calculation of the next-to-leading-logarithmic (NLL) corrections to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equ

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1 EDINBURGH 98/ MSUHEP-8098 Sept 8, 998 hep-ph/9805 Virtual Next-to-Leading Corrections to the Lipatov Vertex Vittorio Del Duca Particle Physics Theory Group, Dept. of Physics and Astronomy University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK and Carl R. Schmidt Department of Physics and Astronomy Michigan State University East Lansing, MI 4884, USA Abstract We compute the virtual next-to-leading corrections to the Lipatov vertex in the helicity-amplitude formalism. These agree with previous results by Fadin and collaborators, in the conventional dimensional-regularization scheme. We discuss the choice of reggeization scale in order to minimize its impact on the next-toleading-logarithmic corrections to the BFKL equation. On leave of absence from I.N.F.N., Sezione di Torino, Italy.

2 Introduction Recently, a long awaited calculation of the next-to-leading-logarithmic (NLL) corrections to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation []-[3] has been completed [4]. The BFKL program is designed to resum the large logarithms of type ln(^s=j^tj) in semihard strong-interaction processes, which are characterized by two large and disparate kinematic scales: ^s, the squared parton center-of-mass energy and j^tj, of the order of the squared momentum transfer. At leading logarithmic (LL) accuracy there can be large dependence on the exact choice of the transverse momentum scale in this resummation, which should be reduced in the NLL calculation. However, the NLL corrections turn out to be large and negative [4], and when the NLL result is applied to Deep Inelastic Scattering (DIS) at small x the phenomenological predictions are unreliable [5]. It has been suggested that the bad behavior of the NLL BFKL resummation is due to the presence of double logarithms of the ratio of two transverse scales, which are ignored in the resummation [6]. On the other hand, an independent check of the NLL resummation [4] is not available yet. A partial check has been performed [7]; however, it relies on the same QCD amplitudes [8, 9, 0,, ] that constitute the building blocks of the original NLL calculation, and it assumes them to have been computed correctly. We have undertaken an independent calculation of the QCD amplitudes relevant to the NLL resummation using the helicity-amplitude formalism [3, 4]. The solution to the LL BFKL equation is an o-shell gluon Green's function, which represents the gluon propagator exchanged in the crossed channel of parton-parton scattering. The building blocks of this equation are the Lipatov vertex [], which summarizes the emission of a gluon along the propagator, and the LL reggeization term [], which summarizes the virtual corrections to the propagator. In NLL approximation, one needs the reggeization term to the corresponding accuracy [], plus the real and virtual corrections to the Lipatov vertex. The real corrections to this vertex are given by the emission of two gluons or of a qq pair at comparable rapidities [8, 3], while the virtual corrections are given by the Lipatov vertex at one-loop accuracy [9, 0, ]. In this paper we compute the Lipatov vertex at one-loop accuracy using the helicity-amplitude formalism. The outline of this paper is as follows. In section we set up the formalism needed for extracting the one-loop Lipatov vertex from the ve-gluon one-loop helicity amplitudes. In order to obtain the one-loop Lipatov vertex, one also needs the helicity-conserving vertices at one-loop accuracy [9, 4, 5, 6], even though these do not enter directly the calculation of the NLL corrections to the BFKL kernel.

3 In section 3 we present these helicity amplitudes, as obtained from the work of Bern, Dixon, and Kosower [7]. In section 4 we consider these amplitudes in the multi-regge kinematics in order to extract the Lipatov vertex to O( 0 ) in the expansion of the spacetime dimension used for regularization of singularities. In section 5 we consider the amplitudes in the limit that the central emitted gluon becomes soft, so as to extract the vertex to O() in this region. This is necessary to get the correct contribution from the infrared singularity when integrating over the momentum of this gluon in the squared amplitude. Finally, in section 6 we discuss the choice of reggeization scale, and we present our conclusions. The ve-gluon amplitude at high-energy We are interested in the ve-gluon amplitude in the multi-regge kinematics, which presumes that the produced gluons are strongly ordered in rapidity and have comparable transverse momenta: y a 0 y y b 0; jp a0?j ' jk? j ' jp b0?j : () In this kinematics the tree-level amplitude for g a g b g a 0 g g b 0 scattering may be written [] M tree ggggg = s h ig f aa0 c C gg(0)? a a 0(?p a ; p a 0) i h ig f cdc0 C g(0) (q a ; q b ) i t t () h i ig f bb 0 c 0 C gg(0)? b b 0(?p b ; p b 0) ; where all external gluons are taken to be outgoing, q i are the momenta transferred in the t-channel, i.e. q a =?p a? p a 0 and q b = p b p b 0, and t i '?jq i?j. The vertices g g g, with g an o-shell gluon, are given by [, 8] C gg(0)? (?p a ; p a 0) =? C gg(0)? (?p b ; p b 0) =? p b 0? ; (3) p b0? and the Lipatov vertex g g g [8, 9] is C g(0) (q a ; q b ) = p q a?q b? k? ; (4) with p? = p x ip y the complex transverse momentum. The C-vertices transform into their complex conjugates under helicity reversal, Cfg (fkg) = C f?g(fkg). The helicity- ip vertex C (0) is subleading in the high-energy limit. For gluon-quark scattering, g a q b

4 g a 0 g q b 0, or quark-quark scattering, q a q b q a 0 g q b 0, we only need to change the relevant vertices C gg(0) to C qq(0) and exchange the corresponding structure constants with color matrices in the fundamental representation [0]. In eq. () the mass-shell condition for the intermediate gluon in the multi-regge kinematics has been used, with s s ab ; s s a 0 ; s s b 0. s = s s jk? j ; (5) The virtual radiative corrections to eq. () in the LL approximation are obtained, to all orders in s, by replacing [, ] t i si (ti ) ; (6) t i in eq. (), with (t) related to the loop transverse-momentum integration (t) g () (t) = s N c t Z d k? () k?(q? k)? t = q '?q? ; (7) and s = g =4. The reggeization scale is much smaller than any of the s-type invariants, s; s ; s, and it is of the order of the t-type invariants, ' t ; t. The precise denition of is immaterial to LL accuracy. The infrared divergence in eq. (7) can be regularized in 4 dimensions with an infrared-cuto mass. Alternatively, using dimensional regularization in d = 4? dimensions, the integral in eq. (7) is performed in? dimensions, yielding with (t) = g () (t) = g N c c? = c? ; (8)?t?( )? (? ) : (9) (4)??(? ) In order to go beyond the LL approximation and to compute the one-loop corrections to the Lipatov vertex, we need a prescription that allows us to disentangle the virtual corrections to the Lipatov vertex (4) from the corrections to the vertices (3) and the corrections that reggeize the gluon (6). Such a prescription is supplied by the general form of the high-energy scattering amplitude, arising from a reggeized gluon in the adjoint representation of SU(N c ) passed in the t - and t -channels [9]. Since only the dispersive part of the one-loop amplitude contributes to the NLL BFKL kernel, we can use the 3

5 modied prescription below. In the helicity basis of eq. () this is given by where h Disp M aa0 dbb 0 a a 0 b 0 b = s ig f aa0 c Disp C gg h ig f cdc0 Disp C g (q a ; q b ) i t s t? a a 0(?p a ; p a 0) i s (t ) (t ) h ig f bb 0 c 0 Disp C gg? b b 0 (?p b ; p b 0) i ; (0) (t) = g () (t) g 4 () (t) O(g 6 ) C g = C g(0) g C g() O(g 4 ) () C gg = C gg(0) g C gg() O(g 4 ) ; are the loop expansions for the reggeized gluon, the Lipatov vertex, and the helicityconserving vertex, respectively. In the NLL approximation to the BFKL kernel it is necessary to compute () (t), C g(), and C gg() ; however, to one loop only C g() and C gg() appear. Expanding eq. (0) to O(g 5 ) and using eq. (), we obtain Disp M aa0 dbb 0 a a 0 b 0 b = M tree 5 ( g () (t ) ln s () (t ) ln s Disp C gg()? a a 0(?p a ; p a 0) C gg(0)? a a 0(?p a ; p a 0) Disp C gg()? b b 0 (?p b ; p b 0) C gg(0)? b b 0 (?p b ; p b 0) g() Disp C (q a ; q b ) C g(0) (q a ; q b ) ) : () Thus, the NLL corrections to C g() can be extracted from the one-loop g g g g g amplitude, by subtracting the one-loop reggeization (8) and the one-loop corrections to the helicity-conserving vertex. The latter has been computed in the HV and CDR schemes [9, 4] and in the dimensional-reduction scheme [4] and is given by Disp C gg()? (?p; p 0 ) C gg(0)? (?p; p 0 ) = c? (?t N c 5 9 N f? 0? ln?t? 3 9? R 6? 0 with 0 = (N c? N f )=3 and the regularization scheme (RS) parameter ) ; (3) R = ( HV or CDR scheme; 0 dimensional-reduction scheme : (4) The general form of the amplitude given by the exchange of reggeized gluons does not hold for the absorptive part of the one-loop amplitudes, because other color structures occur in the high-energy limit. This has been shown for the four-gluon one-loop amplitude in ref. [4] and it is shown for the ve-gluon one-loop amplitude in appendix C. 4

6 The last term in eq. (3) is the modied minimal subtraction scheme MS ultraviolet counterterm. Note that eq. (3) diers from the result in Ref. [4] by the logarithm term, because in that paper the reggeization scale had been taken to be =?t. 3 The one-loop ve-gluon amplitude The color decomposition of a tree-level multigluon amplitude in a helicity basis is [] X M tree n = n= g n? tr( d() d(n) ) m n (p () ; () ; :::; p (n) ; (n) ) ; (5) S n=z n where d ; :::; d n, and ; :::; n are respectively the colors and the polarizations of the gluons, the 's are the color matrices 3 in the fundamental representation of SU(N c ) and the sum is over the noncyclic permutations S n =Z n of the set [; :::; n]. We take all the momenta as outgoing, and consider the maximally helicity-violating congurations (?;?; ; :::; ) for which the gauge-invariant subamplitudes, m n (p ; ; :::; p n ; n ), assume the form [], m n (?;?; ; :::; ) = hp i p j i 4 hp p i hp n?p n ihp n p i ; (6) where i and j are the gluons of negative helicity. The congurations (; ;?; :::;?) are then obtained by replacing the hpki products with [kp] products. We give the formulae for these spinor products in appendix A. Using the high-energy limit of the spinor products (49), the tree-level amplitude for g g g g g scattering may be cast in the form (). The color decomposition of one-loop multigluon amplitudes is also known [3]. For ve gluons it is, M?loop 5 = (7) 5= g 5 4 X X S5=ZZ3 S5=Z5 tr( d () d () d (3) d (4) d (5) ) m 5: ((); (); (3); (4); (5)) tr( d () d () )tr( d (3) d (4) d (5) ) m 5:3 ((); (); (3); (4); (5)) 5 ; where (i) is a shorthand for p (i) ; (i) in the subamplitudes. The sums are over the permutations of the ve color indices, up to cyclic permutations within each trace. The 3 Note that eq.(5) diers by the n= factor from the expression given in ref.[], because we use the standard normalization of the matrices, tr( a b ) = ab =. 5 3

7 string-inspired decomposition of the m 5: subamplitudes [7] is m 5: = N c A g 5 (4N c? N f ) A f 5 (N c? N f ) A s 5 ; (8) where A g 5, A f 5, and A s 5 get contributions from an N = 4 supersymmetric multiplet, an N = chiral multiplet, and a complex scalar, respectively. Also, we have A x 5 = c? m 5 (V x G x ) x = g; f; s : (9) For the NLL BFKL vertex we need the ve-gluon one-loop subamplitudes only in the helicity congurations which are nonzero at tree level. We write the functions for the (; ; 3; 4; 5) color order for the two relevant helicity congurations below. The functions obtained from the N = 4 multiplet, V g and G g, are the same for both helicity congurations [7], V g =? 5X G g = 0 : j=?s j;j 5X j= ln?s j;j ln?s j;j The other functions depend on the helicity conguration. We dene?s j;j? 5?s j?;j? 6? R 3 ; (0) I ijkl = [ij] hjki [kl] hlii : () For the (? ;? ; 3 ; 4 ; 5 ) helicity conguration we have [7], ln V f =? 5? ln?s 3 G f = I 34 I 45?s3 L 0 s s 5?s 5 G s =? Gf 3 I 34 I 45 (I 34 I 45) 3 s 3 s 3 5??s 5 I 35? s 35 I 34 I 45 ; 3 s s 3 s 5 s 6 s s 3 s 5 L?s3?s 5 () while for the (? ; ; 3? ; 4 ; 5 ) helicity conguration, we have ln V f =? 5? ln?s 34 G f =? I 35 I 34?s34 L 0 s 3 s 5?s 5??s 5 I 34 I 34?s3 Ls s 3 s ;?s 34 5?s 5?s 5 6

8 I 35 I 35?s Ls s 3 s 34 G s =? I 34 I 34 s 4 3 s 4 s 5 ;?s 5?s 34?s 34 Ls?s3 ;?s 34?s 5?s 5? I 35 I 35 s 4 3 s 5 s 34 I 3 34 I 34?s3 L 3 s 4 3 s 4 s 3 I 3 35 I 35 5?s 5 3 s 4 3 s 5 s s 3 5 L?s3?s 5 L?s34?s 5?s Ls ;?s 5?s?s5 L L?s 34?s 34?s 34?s 34?s L?s 34?s34 L? I 35 I 34 (I 35 I 34 ) I 3 34 I 34?s 5 s 3 3 I 35 I 34?s34 L 0 6 s 3 s 5?s 5 I 35 I 34 3 s 4 3 s 3 s 5 s 34 s I 34 I 34 I 35 I 35? I 34 I 35 ; 3 s 4 3 s 3 s 4 s 34 s 5 3 s 4 3 s 5 s s 34 s 5 6 s 3s 34 s 5 s 4 3 s 4 I 3 35 I 35 s 4 3 s 5 with the functions L 0 ; L ; L ; Ls dened in Appendix B. For both the helicity congurations above, the functions V s and V f are related by, V s =? V f (3) 3 9 : (4) In the expansion in, eq. (0-4) are valid to O( 0 ). The amplitude (7) dened in terms of eq. (8-4) is MS regulated. Using eq. (9, 0, 4), we can write the m 5: subamplitude (8) as the sum of a universal piece, which is the same for both helicity congurations, and a non-universal piece, which depends on the helicity conguration, with In addition, m 5: = m u 5: m nu 5: ; (5) m u 5: = c? m 5 N c V g ; (6) m nu 5: = c? m 5 0 V f (4N c? N f ) G f (N c? N f ) G s : 9 m 5:3 (4; 5; ; ; 3) = N c X COP (;;3) 4 m 5: ((); (); (3); (4); 5) ; (7) where only the N f -independent, unrenormalized contributions to m 5: are included [7] and COP (;;3) 4 denotes the subset of permutations of S 4 that leave the ordering of (,,3) unchanged up to a cyclic permutation [3]. 7

9 4 The one-loop corrections to the Lipatov vertex To obtain the next-to-leading logarithmic corrections to the Lipatov g g g vertex, we need the amplitude M?loop 5 (B? ; A? ; A 0 ; k ; B 0 ) in the high-energy limit. We must consider each of the color orderings in eq. (7) and expand the subamplitudes (8) in powers of t=s, retaining only the leading power, which yields the leading and next-toleading terms in ln(s=t). In fact, at NLL we only need to keep the dispersive parts of the subamplitudes m 5: and m 5:3. By direct inspection of eq. (9-4) it is straightforward to show that if a given color ordering of m 5 is suppressed by a power of ~t=~s at tree-level, where ~t = t ; t ; jk? j and ~s = s; s ; s, then the corresponding color ordering of m 5: will also be suppressed at one-loop. For the M tree n amplitude in the multi-regge kinematics, the leading color orderings are obtained by untwisting the color ow, in a such a way to obtain a double-sided color- ow diagram, and by retaining only the color-ow diagrams which exhibit strong rapidity orderings of the gluons on both sides of the diagram, without regard for the relative rapidity ordering between the two sides [8, 4]. Easy combinatorics then show that there are n? such color orderings, and because of the reection and cyclic symmetries of the subamplitudes only n?3 need to be determined, e.g. all the ones which have at least (n? )= gluons on one side of the color-ow diagram. For the M tree 5 amplitude, and thus for m 5:, we have eight leading color orderings, out of which four need to be determined, and we can choose them to be (A? ; A 0 ; k ; B 0 ; B? ), (A? ; A 0 ; k ; B? ; B 0 ), (A? ; A 0 ; B 0 ; B? ; k ) and (A? ; k ; B 0 ; B? ; A 0 ). The other four leading subamplitudes are then obtained by taking the ones above in reverse order, which yields an overall minus sign. For positive values of the invariants s ij we use the prescription ln(?s ij ) = ln(s ij )? i. Thus, in the multi-regge kinematics and at NLL the dispersive part of the universal piece (0) becomes, using the spinor products (49), Disp V g =??t jk? j y?t y? ln t t? R ; (8) where we have written the leading logarithms in terms of the physical rapidity intervals y = ln(s = p jk? j) and y = ln(s = p?t jk? j). The non-universal piece (6) becomes, after rewriting all the phases in terms of q a? qb? and some algebraic manipulation, m nu 5: (B? ); (A? ); (A 0 ); (k ); (B 0 ) 8

10 = m 5 (B? ); (A? ); (A 0 ); (k ); (B 0 ) c? (? 3 0? 0?t? 64 9 N c 0 9 N f? 0 (t t q a?q b?) L 0(t =t ) t (9) N c? N f 3 jk? j?[t t (t t jk? j )q a?qb?] L (t =t )? q ) a?qb? ; t 3 t t where the rst term is the MS ultraviolet counterterm, and where the permutations span the eight leading color orderings. Combining eq. (8) and (9), we see that the dispersive parts of the leading m 5: subamplitudes are all proportional to the corresponding treelevel subamplitudes. Therefore, by the tree-level U() decoupling equations [3, 5] the m 5:3 subamplitudes vanish, Disp m 5:3 (B? ); (A? ); (A 0 ); (k ); (B 0 ) = 0 O(t=s) : (30) Thus, we conclude that the dispersive part of the one-loop ve-gluon amplitude is simply proportional to the tree amplitude to leading power in t=s. Combining eq. (8) and (9), it is given to O( 0 ) by Disp M?loop 5 (A? ; A 0 ; k ; B 0 ; B? ) = M tree 5 (A? ; A 0 ; k ; B 0 ; B? ) g c? (N c??t jk? j y y? t?t ln? R t 3 4 3? 3 0? 0?t? 64 9 N c 0 9 N f (3)? 0 (t t q a?q b?) L 0(t =t ) t N c? N f 3 jk? j?[t t (t t jk? j )q a?qb?] L (t =t )? q ) a?qb? : t 3 t t Note that only the real part of this amplitude contributes to the NLL corrections to the BFKL equation. It can easily be obtained using Re(q a?q b?) =?(t t jk? j )=. Using eq. (8,, 3) and eq. (3), we can extract the NLL corrections to the Lipatov vertex to O( 0 ) Disp C g() (q a ; q b ) C g(0) (q a ; q b ) = c? ( N c ln jk? j 9?t ln jk? j

11 ?? t jk? j ln? 0 t 3? 0 (t t q a?qb?) L 0(t =t ) (3) t N c? N f 3 jk? j?[t t (t t jk? j )q a?qb?] L (t =t )? q ) a?qb? O() : t 3 t t Note that the dependence on the RS parameter R has disappeared. If we take =, eq. (3) agrees with the NLL corrections to the Lipatov vertex computed in ref. [9, 0] in the CDR scheme, to O( 0 ). 5 The one-loop Lipatov vertex in the soft limit In the NLL corrections to the BFKL kernel, the one-loop Lipatov vertex is multiplied by the corresponding tree-level vertex with the intermediate gluon k integrated over its phase space. In the soft limit for the intermediate gluon, k 0, an infrared divergence arises 4, which in dimensional regularization manifests itself as a pole of O(=). Thus, in order to generate correctly all the nite terms in the squared amplitude, we need the one-loop Lipatov vertex to O() in the limit that the gluon k is soft. This can be obtained by computing the ve-gluon one-loop amplitude to O() in the soft limit and then matching on to our previous O( 0 ) result, eq. (3). We nd the soft limit of the ve-gluon one-loop amplitude by exploiting the factorization of the soft singularities in the color-ordered subamplitudes: m?loop n (:::; A; k ; B; :::)j k0 = (33) Soft tree (A; k ; B) m?loop n? (:::; A; B; :::) Soft?loop (A; k ; B) m tree n?(:::; A; B; :::) : This requires the use of one-loop soft functions and four-gluon one-loop amplitudes, which have been evaluated to all orders in [6]. Using these results, the ve-gluon one-loop amplitude, in the soft limit for the intermediate gluon, is [6] Disp M?loop 5 (A? ; A 0 ; k ; B 0 ; B? )j k0 = M tree 5 (A? ; A 0 ; k ; B 0 ; B? )j k0 g c (? N c? 4?t ( )? (? ) () ln s (34)?t? R (? ) 3?? 4 (? ) N f (? )(3? )?N c jk? j [ (? )? ( )]? 3 4 No collinear divergence occurs here due to the strong ordering in rapidity. 0 ) : 0

12 Eq. (34) is valid to all orders in for R = 0 and. Expanding it to O(), yields Disp M?loop 5 (A? ; A 0 ; k ; B 0 ; B? )j k0 = M tree 5 (A? ; A 0 ; k ; B 0 ; B? )j k0 g c (? N c? 4?t ln s?t? 64 9? R 380 (3)? 3 7? 8 9 R? 0 N f ? N c jk? j? 3? 3 0 O( ) Matching eq. (34) or eq. (35) to the ve-gluon one-loop amplitude to O( 0 ), eq. (3), one can obtain the ve-gluon one-loop amplitude with soft corrections to all orders in or to O(), respectively. However, for our purposes it suces to perform the matching directly on the one-loop Lipatov vertex. In order to do that, we need the one-loop correction to the helicity-conserving impact factor, which was given to O( 0 ) in eq. (3), to all orders in. It is [6], Disp C gg()? (?p; p 0 ) C gg(0)? (?p; p 0 ) = c? ( (? )?t N c? ln?t? R (3? )? (? ) N f (? )(3? ) ) : (35) ( )? (? ) ()? 0 ) : (36) Eq. (36) is valid to all orders in for R = 0 and. Using eq. (34) and eq. (8,, 36) with t = t = t, and the mass-shell condition (5), we extract the NLL corrections to the Lipatov vertex in the soft limit of the emitted gluon, Disp C g() (q a ; q b ) C g(0) (q a ; q b ) k0 = c? ( N c?t ln (37) jk? j? jk? j [ (? )? ( )]? 0 Eq. (37) is valid to all orders in, and it does not depend on R or N f (except through the MS ultraviolet counterterm). Expanded to O() it reads, Disp C g() (q a ; q b ) ( C g(0) (q a ; q b ) = c? N c ln?t jk k0? j? jk? j?? ) 0 3 O( ) ; (38) which to O( 0 ) agrees with the soft limit of eq. (3). Matching eq. (37) or eq. (38) to the correction to the Lipatov vertex (3), we obtain the one-loop Lipatov vertex with soft ) :

13 corrections to all orders in or to O(), respectively. The matching yields ( Disp C g() (q a ; q b ) = c C g(0)? N c (q a ; q b )? jk? j N c? N f 3? 3 ln jk? j?t ln jk? j? ln t t? 0? 0 (t t q a?q b?) L 0(t =t ) t (39) jk? j?[t t (t t jk? j )q a?qb?] L (t =t )? q ) a?qb? ; t 3 t t where this equation is exact to O() in the k 0 limit and to O( 0 ) elsewhere. Eq. (39) agrees with the NLL corrections to the Lipatov vertex computed in ref. [, 6] in the CDR scheme 5, to O(). 6 Discussion and Conclusions In this paper we have computed the one-loop corrections to the Lipatov vertex, eq.(3), to O( 0 ) in both the CDR and the dimensional reduction schemes. This was done by expanding the complete one-loop ve-gluon helicity amplitudes in the multi-regge kinematics. The result (3) was then augmented by the one-loop corrections to O() in the soft limit for the intermediate gluon, which is necessary due to the infrared divergence which arises in the squared amplitude in this region of phase space. In eq. (37) we computed the soft corrections to all orders in, and in eq. (39) we performed the matching to O() with the one-loop Lipatov vertex (3). Our nal results are independent of the RS parameter R (4), and agree with the one-loop Lipatov vertex computed in the CDR scheme to O( 0 ) in ref. [9, 0], and to O() in ref. []. The one-loop Lipatov vertex (39), however, depends on the arbitrary transverse scale, introduced by the reggeization, eq. (6). This dependence vanishes exactly if we expand the high-energy n-gluon amplitudes to one-loop 6, using the reggeized gluon (6), the 5 In ref. [, 6] the MS subtraction is dened with a factor of?( )=(4)? as opposed to the full c? factor, eq. (9), chosen by us. This yields an accountable dierence at O() between the NLL corrections to the Lipatov vertex of ref. [, 6] and eq. (39). However, it should not make any dierence in the complete NLL corrections to the BFKL resummation as long as the same denition of MS is used in both the virtual and real contributions. This is because both the real and virtual IR singularities will be multiplied by the same overall factor, so that after their cancellation the nite remainder will be the same regardless of what the O() piece of the UV counterterm is. This argument holds as long as one doesn't introduce the two-loop counterterm, which appears at NNLL. 6 We have veried the logarithmic terms in this expansion for six gluons by comparing with the

14 NLL impact factors (36), and the NLL Lipatov vertex (39) as we did for the ve-gluon amplitude (0). However, in the higher-loop contributions at NLL accuracy there will be residual dependence on the scale. In particular, the terms proportional to ln(=jk? j ) in eq. (39), which cancel in the expansion of the amplitudes to one loop, resurface again at two loops as double logarithms. Although formally next-to-next-to-leading-logarithmic (NNLL), these terms would ruin the subtle infrared cancellation between real and virtual gluon integrations when k? becomes soft, thus invalidating the entire BFKL program. The solution to this problem is suggested by the form of the high-energy one-loop ve-gluon amplitude (3). In this equation we see that the high-energy logarithms occur naturally in terms of the physical rapidity intervals. Therefore, it is reasonable to implement a modied prescription for reggeizing the gluons. This prescription is to replace t i t i s i?;i jk i??jjk i?j (ti ) = t i e (t i)y i?;i (40) for each of the t-channel gluons, where y i?;i is the physical rapidity interval between the emitted gluons i? and i. Eectively, this means that the scale \runs as we travel down the reggeon ladder, so that i = jk i??jjk i?j. The terms proportional to ln(=jk? j ) and ln(=? t) in eqs. (39) and (36), respectively, vanish in this prescription so that these potentially troublesome logarithms have been removed at NLL accuracy 7. Although not stated explicitly in this language, the above prescription is in eect used by Fadin and collaborators when they change the scale of the one-loop Lipatov vertex and the one-loop impact factors to remove the dependence on to NLL [6]. A Spinor Algebra in the Multi-Regge kinematics We consider the scattering of two gluons of momenta p a and p b into n gluons of momenta p i, where i = a 0 ; b 0 ; : : : n. Using light-cone coordinates p = p 0 p z, and complex transverse coordinates p? = p x ip y, with scalar product p q = p q? p? q? p? q?? p?q?, the gluon 4-momenta are, p a = p a ; 0; 0; 0 ; maximal-helicity violating N = 4 supersymmetric six-gluon amplitude at one-loop [7]. 7 An alternative to this prescription is to let i = jk i?? jjk i? j, where is some arbitrary constant. Then, the logarithms ln(=jk? j ) and ln(=? t) each get replaced by ln. The troublesome dependence on the k? of the emitted gluon is still removed, but now one can test the sensitivity of the NLL resummation to the arbitrary constant. To be consistent, this prescription also requires the introduction of a corresponding -dependence in the real-gluon contributions. 3

15 p b = 0; p? b ; 0; 0 ; (4) p i = jp i?je y i ; jp i?je?y i ; jp i?j cos i ; jp i?j sin i ; where to the left of the semicolon we have the and - components, and to the right the transverse components. y is the gluon rapidity and is the azimuthal angle between the vector p? and an arbitrary vector in the transverse plane. Momentum conservation gives 0 = X p i? ; p a = X p i ; (4) p? b = X p? i : For each massless momentum p there is a positive and negative helicity spinor, jpi and jp?i, so we can consider two types of spinor products hpqi = hp? jqi [pq] = hp jq?i : (43) Phases are chosen so that hpqi =?hqpi and [pq] =?[qp]. For the momentum under consideration the spinor products are vu u t p j vu u t p i hp i p j i = p i?? p p j? ; i p j s p a hp a p i i =? p i? ; (44) hp i p b i =? hp a p b i =? q q p i p i p? b ; p a p? b =? p s ab ; where we have used the mass-shell condition jp i?j = p i p? i. The other type of spinor product can be obtained from [pq] = hqpi ; (45) where the is used if p and q are both ingoing or both outgoing, and the? is used if one is ingoing and the other outgoing. In the multi-regge kinematics, the gluons are strongly ordered in rapidity and have comparable transverse momentum: y a 0 y : : : y n y b 0; jp i?j ' jp? j : (46) 4

16 Then the momentum conservation (4) in the directions reduces to p a ' p ; a 0 p? b ' p? ; b 0 (47) and the Mandelstam invariants become s ab = p a p b ' p a 0p? b 0 s ai =?p a p i '?p a 0p? i (48) s bi =?p b p i '?p i p? b 0 s ij = p i p j ' jp i?jjp j?je y i?y j = p i p? j (y i y j ) ; where i; j = a 0 ; b 0 ; : : : n. In this limit the spinor products (44) become hp a p b i ' hp a 0p b i '? hp a p b 0i ' hp a 0p b 0i =? hp a p a 0i '?p a0? vu u t p a 0 vu u p b 0 jp b0?j t p a 0 p b 0 p b0? hp b 0p b i '?jp b0?j (49) hp a p i i ' hp a 0p i i =? hp i p b i '? hp i p b 0i '? hp i p j i '? vu u t p i vu u t p i vu u t p i p j p b 0 jp b0?j p b 0 p b0? vu u t p a 0 p i p i? p j? (y i y j ) : B Logarithmic Functions L 0 (x) = ln(x)? x 5

17 L (x) = ln(x)? x (? x) L (x) = ln(x)? x (? x) 3 x Ls (x; y) = (? x? y) [Li (? x) Li (? y) ln(x) ln(y) (? x? y)[l 0 (x) L 0 (y)]? 6 ; (50) where Li is the dilogarithm. C The Absorptive Parts In the subamplitudes of type m 5: the absorptive parts arise entirely from the universal piece (0) Absorp m 5: (A? ; A 0 ; k ; B 0 ; B? ) =?m 5 (A? ; A 0 ; k ; B 0 ; B? ) c? N c Absorp m 5: (A? ; A 0 ; k ; B? ; B 0 ) =?m 5 (A? ; A 0 ; k ; B? ; B 0 ) c? N c Absorp m 5: (A? ; A 0 ; B 0 ; B? ; k ) =?m 5 (A? ; A 0 ; B 0 ; B? ; k ) c? N c Absorp m 5: (A? ; k ; B 0 ; B? ; A 0 ) =?m 5 (A? ; k ; B 0 ; B? ; A 0 ) c? N c jk? j jk? j?? jk? j? jk? j?t?t?t :?t (5) The subamplitudes of type m 5:3 also contribute to the absorptive part. Because of the reection and cyclic symmetries of these subamplitudes [3], m 5:3 (; ; 3; 5; 4) = m 5:3 (; ; 5; 4; 3) =?m 5:3 (; ; 3; 4; 5) ; (5) we may rewrite their contribution to the one-loop ve-gluon amplitude (7) as X S5=ZZ3 tr( d () d () )tr( d (3) d (4) d (5) ) m 5:3 ((); (); (3); (4); (5)) 6

18 = i 4 X S5=ZP3 d () d () f d (3) d (4) d (5) m 5:3 ((); (); (3); (4); (5)) ; (53) where the sums are over the permutations of the ve color indices, up to cyclic permutations within each trace on the left-hand side and up to any permutation within each trace on the right-hand side. This singles out ten dierent subamplitudes of type m 5:3 to be computed. Using the equation for the decomposition of m 5:3 subamplitudes in terms of m 5: subamplitudes, eq. (7), and taking into account the eight leading color orderings of the subamplitudes of type m 5: elucidated in sect. 3, we nd that at NLL there are only three dierent values for the subamplitudes of type m 5:3. They satisfy the following relations m 5:3 (A? ; B? ; A 0 ; k ; B 0 ) = m 5:3 (A 0 ; B? ; A? ; k ; B 0 ) = m 5:3 (A? ; B 0 ; A 0 ; k ; B? ) = m 5:3 (A 0 ; B 0 ; A? ; k ; B? ) ; m 5:3 (A? ; A 0 ; k ; B 0 ; B? ) (54) = m 5:3 (A? ; k ; A 0 ; B 0 ; B? ) = m 5:3 (A 0 ; k ; A? ; B 0 ; B? ) ; m 5:3 (B? ; B 0 ; A? ; A 0 ; k ) = m 5:3 (B? ; k ; A? ; A 0 ; B 0 ) = m 5:3 (B 0 ; k ; A? ; A 0 ; B? ) : Using eq. (7,8, 9) one can easily check that the dispersive parts of the m 5:3 subamplitudes cancel, as expected. The absorptive parts are, using eq.(5), Absorp m 5:3 (A? ; B? ; A 0 ; k ; B 0 ) =?m 5 (A? ; A 0 ; k ; B 0 ; B? ) c?? jk? j?t Absorp m 5:3 (A? ; A 0 ; k ; B 0 ; B? ) (55) =?m 5 (A? ; A 0 ; k ; B 0 ; B? ) c?? jk? j?t Absorp m 5:3 (B? ; B 0 ; A? ; A 0 ; k ) =?m 5 (A? ; A 0 ; k ; B 0 ; B? ) c? Finally, we can combine these to nd? jk? j Absorp M?loop 5 (A? ; A 0 ; k ; B 0 ; B? ) = g 5 c? N c s C gg(0)? a a 0(?p a ; p a 0) C g(0) (q a ; q b ) C gg(0)? t t b b 0(?p b ; p b 0) ( :?t 7

19 jk? j? jk? j?t aa 0 if c if cdc0 if c0 b 0 b i 0 c?t daa f cdc0 d c0 b 0 b i ab f a 0 db 0 a0 b f adb0 ab0 f a0 db a0 b 0 f adb N c? jk? j i N c aa? jk? j i 0 c?t daa d cdc0 f c0 b 0 b 0f db0 b ad f a0 b 0 b a0 d f ab0 b i?t f aa 0 c d cdc0 d c0 b 0 b i bb 0f aa0 d bd f aa0 b 0 b0 d f aa0 b N c : (56) Acknowledgements This work was partly supported by the NATO Collaborative Research Grant CRG-95076, and by the EU Fourth Framework Programme `Training and Mobility of Researchers', Network `Quantum Chromodynamics and the Deep Structure of Elementary Particles', contract FMRX-CT (DG - MIHT). References [] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Zh. Eksp. Teor. Fiz. 7, 840 (976) [Sov. Phys. JETP 44, 443 (976)]. [] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Zh. Eksp. Teor. Fiz. 7, 377 (977) [ Sov. Phys. JETP 45, 99 (977)]. [3] Ya.Ya. Balitsky and L.N. Lipatov, Yad. Fiz (978) [Sov. J. Nucl. Phys. 8, 8 (978)]. [4] V.S. Fadin and L.N. Lipatov, Phys. Lett. B 49, 7 (998) [5] R.D. Ball and S. Forte, preprint hep-ph/980535; J. Blumlein et al, preprint hep-ph/ [6] G.P. Salam, J.High Energy Phys. 9807:09,998. 8

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