Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit

Nuclear Physics B 568 Ž2000. 211–262 www.elsevier.nlrlocaternpe

Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit Vittorio Del Duca a , Alberto Frizzo b, Fabio Maltoni b a

Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Õia P. Giuria, 1, 10125 Turin, Italy Dipartimento di Fisica Teorica, UniÕersita` di Torino, Õia P. Giuria, 1, 10125 Turin, Italy

b

Received 29 September 1999; accepted 11 October 1999

Abstract In the high-energy limit, we compute the gauge-invariant three-parton forward clusters, which in the BFKL theory constitute the tree parts of the NNLO impact factors. In the triple collinear limit, we obtain the polarized double-splitting functions. For the unpolarized and the spin-correlated double-splitting functions, our results agree with the ones obtained by Campbell–Glover and Catani–Grazzini, respectively. In addition, we compute the four-gluon forward cluster, which in the BFKL theory forms the tree part of the NNNLO gluonic impact factor. In the quadruple collinear limit we obtain the unpolarized triple-splitting functions, while in the limit of a three-parton central cluster we derive the Lipatov vertex for the production of three gluons, relevant for the calculation of a BFKL ladder at NNLL accuracy. Finally, motivated by the reorganization of the color in the high-energy limit, we introduce a color decomposition of the purely gluonic tree amplitudes in terms of the linearly independent subamplitudes only. q 2000 Elsevier Science B.V. All rights reserved. PACS: 12.38.-t; 12.38.Bx; 13.85.Rm Keywords: Perturbative QCD; BFKL; High energy

1. Introduction QCD calculations of multijet rates beyond the leading order ŽLO. in the strong coupling constant a s are generally quite involved. However, in recent years it has become clear how to construct general-purpose algorithms for the calculation of multijet rates at next-to-leading order ŽNLO. accuracy w1–11x. The crucial point is to organize the cancellation of the infrared Ži.e. collinear and soft. singularities in a universal, i.e. process-independent, way. The universal pieces in a NLO calculation are given by the tree-level splitting w12–15x and eikonal w16–18x functions, and by the universal structure of the poles of the one-loop amplitudes w1,4,19x. 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 6 5 7 - 4

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Eventually, the same procedure will permit the construction of general-purpose algorithms at next-to-next-to-leading order ŽNNLO. accuracy. It is mandatory then to fully investigate the infrared structure of the phase space at NNLO. The universal pieces needed to organize the cancellation of the infrared singularities are given by the tree-level double-splitting w20,21x, double-eikonal w18,22x and splitting-eikonal w20,22x functions, by the one-loop splitting w23–27x and eikonal w23–25x functions, and by the universal structure of the poles of the two-loop amplitudes w28x. Another outstanding issue in QCD, at first sight unrelated to the topics discussed above, is the calculation of the higher-order corrections to the BFKL equation w29–31x. In scattering processes characterized by two large and disparate scales, like s, the squared parton center-of-mass energy, and t, a typical momentum transfer, the BFKL equation resums the large logarithms of type lnŽ srt .. The LO term of the resummation requires gluon exchange in the cross channel, which for a given scattering occurs at O Ž a s2 .. The corresponding QCD amplitude factorizes then into a gauge-invariant effective amplitude formed by two scattering centers, the LO impact factors, connected by the gluon exchanged in the cross channel. The LO impact factors are characteristic of the scattering process at hand. The BFKL equation resums then the universal leadinglogarithmic ŽLL. corrections, of O Ž a sn ln n Ž srt .., to the gluon exchange in the cross channel. The building blocks of the BFKL resummation are the Lipatov vertex w32,33x, i.e. the effective gauge-invariant emission of a gluon along the gluon ladder in the cross channel, and the gluon reggeization w29x, i.e. the LL part of the one-loop corrections to the gluon exchange in the cross channel. The accuracy of the BFKL equation is improved by computing the next-to-leading logarithmic ŽNLL. corrections w34–37x, i.e. the corrections of O Ž a sn ln ny1 Ž srt .., to the gluon exchange in the cross channel. In order to do that, the universal building blocks of the BFKL ladder must be computed to NLL accuracy. These are given by the tree corrections to the Lipatov vertex, i.e. the emission of two gluons w38–40x or of a qq pair w40,41x along the gluon ladder, by the one-loop corrections to the Lipatov vertex w42–46x, and finally by the NLL gluon reggeization w47–50x, i.e. the NLL part of the two-loop corrections to the gluon ladder. However, to compute jet production rates at NLL accuracy, the impact factors must be computed at NLO w51–53x. For jet production at large rapidity intervals, they are given by the one-loop corrections w51x to the LO impact factors, and by the tree corrections w38,39,41,54x, i.e. the emission of two partons in the forward-rapidity region. In the collinear or soft limits, the latter reduce to the tree splitting or eikonal functions w55x. To further improve the accuracy of the BFKL ladder one needs to compute the next-to-next-to-leading logarithmic ŽNNLL. corrections, i.e. the corrections of O Ž a sn ln ny2 Ž srt .., to the gluon ladder. At present it is not known whether such corrections can be resummed. If that is the case, the universal building blocks of a BFKL ladder at NNLL would be: the emission of three partons along the gluon ladder, the one-loop corrections to the emission of two partons along the ladder, the two-loop corrections to the Lipatov vertex, and the gluon reggeization at NNLL accuracy. None of them is known at present. In this paper we compute the gluonic NNLO Lipatov vertex, i.e. the emission of three gluons along the ladder. In addition, to compute jet production rates at NNLL accuracy, the BFKL ladder should be supplemented by impact factors at NNLO. They are not known either. In this

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paper we compute their tree components, i.e. the emission of three partons in the forward-rapidity region. By taking then the triple collinear limit of the tree NNLO impact factors, we obtain the polarized double-splitting functions. Summing over the parton polarizations, we obtain the unpolarized and the spin-correlated double-splitting functions, previously computed in Ref. w20x and w21x, respectively, in the conventional dimensional regularization ŽCDR. scheme. Since we sum over two helicity states of the external partons, as it is done in the dimensional reduction ŽDR. scheme w56–58x, our results agree with the ones in the CDR scheme by setting there the dimensional regularization scheme ŽRS. parameter e s 0. For a scattering with production of m partons, we define the n-parton cluster, with m ) n, as the set of n final-state partons where the distance in rapidity between any two partons in the cluster is much smaller than the rapidity distance between a parton inside the cluster and a parton outside. In the BFKL theory, Ž n q 1.-parton forward clusters provide the tree parts of N n LO impact factors, while Ž n q 1.-parton central clusters provide the tree parts of the N n LO Lipatov vertex. n-parton clusters were given also a field-theoretical basis in terms of an effective action describing the interaction between physical partons grouped into gauge-invariant clusters and the gluons exchanged in the cross channel w59x. In addition to computing the three-parton forward clusters and the three-gluon central cluster mentioned above, we compute the four-gluon forward cluster, i.e. the purely gluonic tree part of the NNNLO impact factor. By taking then the quadruple collinear limit, we obtain the polarized triple-splitting functions. They could be used in a gauge-invariant evaluation of the Altarelli–Parisi evolution at three loops w60x. The outline of the paper is: in Section 2 we review the standard color decompositions of the n-parton tree amplitudes, and we present a color decomposition of the gluon amplitudes in terms of the linearly independent subamplitudes only. In Section 3 we review the elastic scattering of two partons in the high-energy limit, which allows for the extraction of the LO impact factors. In Section 4 we review the amplitudes for the production of three partons, with a gauge-invariant two-parton forward cluster; from these, we can extract the tree parts of the NLO impact factors; by taking the collinear limit, we obtain the LO splitting functions. In Section 5 we compute the amplitudes for the production of four partons, with a three-parton forward cluster; then we extract the tree parts of the NNLO impact factors, and by taking the triple collinear limit we obtain the polarized and unpolarized double-splitting functions. In Section 6.1 we compute the amplitude for the production of five gluons, with a four-gluon forward cluster. We extract the tree part of the gluonic NNNLO impact factor, and by taking the quadruple collinear limit we obtain the polarized triple-splitting functions. In addition, by taking the limit in which three gluons are emitted in the central-rapidity region, we obtain the gauge-invariant three-gluon central cluster, i.e. the tree part of the NNLO Lipatov vertex. In Section 7 we draw our conclusions. 2. Tree amplitudes In this section we review the color decomposition of purely gluonic and quark–gluon tree amplitudes. For the purely gluonic tree amplitudes, we introduce a color decomposition in terms of the linearly independent subamplitudes, Eq. Ž2.9..

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2.1. Gluon amplitudes For an amplitude with n gluons the usual color decomposition at tree level reads w61–69x i A Ž g 1 , . . . , g n . s ig ny 2

tr Ž ls1 . . . ls n . A Ž gs 1 , . . . , gs n . ,

Ý

Ž 2.1 .

sgS nrZ n

where SnrZ n are the non-cyclic permutations of n elements. The dependence on the particle helicities and momenta in the subamplitude, and on the gluon colors in the trace, is implicit in labeling each leg with the index i. Helicities and momenta are defined as if all particles were outgoing. The gauge invariant subamplitudes A satisfy the relations w68,69x, proven for arbitrary n in Ref. w70x, A Ž 1,2, . . . ,n y 1,n . s A Ž n,1,2, . . . ,n y 1 .

cyclicity

n

A Ž 1,2, . . . ,n . s Ž y1 . A Ž n, . . . ,2,1 . A Ž 1,2,3, . . . ,n . q A Ž 2,1, . . . ,n . q . . . qA Ž 2,3, . . . ,1,n . s 0

reflection dual Ward identity

Ž 2.2 . The above relations are sufficient to show that, for n ( 6 the number of independent subamplitudes can be reduced from Ž n y 1.! to Ž n y 2.!. For n 0 7 it is still possible to introduce a basis of Ž n y 2.! elements by using Kleiss–Kuijf’s relation w71x A Ž 1, x 1 , . . . , x p ,2, y 1 , . . . , yq . s Ž y1 .

p

A Ž 1,2,  a 4  b 4 . ,

Ý

Ž 2.3 .

s gOP  a 4 b 4

where a i g  a 4 '  x p , x py1 , . . . , x 14 , bi g  b 4 '  y 1 , . . . , yq 4 and OP a 4 b 4 is the set of permutations of the Ž n y 2. objects  x 1 , . . . , x p , y 1 , . . . , yq 4 that preserve the ordering of the a i within  a 4 and of the bi within  b 4 , while allowing for all possible relative orderings of the a i with respect to the bi . The above relation has been checked up to n s 8 in Ref. w70x, and proven for arbitrary n in Ref. w72x. Accordingly, the expression for the summed amplitude squared can be written as Ž ny1 . !

Ý

< A Ž 1, . . . ,n . < 2 s

a1 , . . . , a n

s Cn Ž Nc .

Ý

c i j A i A)j

Ž 2.4 .

i , js1

< A Ž 1, s 2 , . . . , sn . < 2 q O

Ý sgS ny1

1

ž /

Ž 2.5 .

Nc2

Ž ny2 . !

s

Ý c˜i j A i A)j ,

Ž 2.6 .

i , js1

where c i j in Eq. Ž2.4. is ci j s Ž g 2 .

ny2

Ý tr Ž Pi Ž ld , . . . , ld . . tr Ž Pj Ž ld , . . . , ld . . 1

colors

n

1

n

)

,

Ž 2.7 .

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with Pi the ith permutation in SnrZ n . In Eq. Ž2.5., the coefficient CnŽ Nc . is Cn Ž Nc . s

Ž g 2 Nc .

ny 2

2n

Ž Nc2 y 1 . .

Ž 2.8 .

The first term in Eq. Ž2.5. constitutes the Leading Color Approximation ŽLCA.. Up to n s 5, the 1rNc2 corrections in Eq. Ž2.5. vanish and LCA is exact. The reduced color matrix c˜i j in Eq. Ž2.6., has been obtained from c i j applying the linear transformations of Eq. Ž2.3., thus the labels i, j in Eq. Ž2.6. run only on the permutations of the linearly independent subamplitudes. Motivated by the reorganization of the color in the high-energy limit w39,54,73,74x, and using Eqs. Ž2.2. and Ž2.3. we rewrite Eq. Ž2.1. as i A Ž g1 , . . . , gn . si

si

Ž ig .

ny 2

2 g ny 2 2

Ý

f a1 a 2 x 1 f x 1 a 3 x 2 . . . f x ny 3 a ny 1 a n A Ž g 1 , gs 2 , . . . , gs ny 1 , g n .

sgS ny2

Ý Ž Fa

2

. . . F a ny 1 . a1 a n A Ž g 1 , gs 2 , . . . , gs ny 1 , g n . ,

Ž 2.9 .

sgS ny2

where Ž F a . b c ' if b ac. We have checked Eq. Ž2.9. up to n s 7. Eq. Ž2.9. enjoys several remarkable properties. Firstly, it shows explicitly which is the color decomposition that allows us to write the full amplitude i A in terms of the Ž n y 2.! linearly independent subamplitudes only. In the following we shall refer to it as to a color ladder. Hence the color matrix obtained squaring Eq. Ž2.9. yields directly the c˜i j matrix in Eq. Ž2.6.. We have checked it against the explicit results of Ref. w75x, up to n s 5. Moreover, it is quite suggestive to note the formal correspondence with the amplitudes with a quark–antiquark pair and Ž n y 2. gluons, Eq. Ž2.11., where the only difference between the two1 is the appropriate representation for the color matrices, namely the adjoint for the n-gluon amplitude and the fundamental for the one with the qq pair. Finally, the most relevant applications of Eq. Ž2.9. for this work are to the study of the multi-gluon amplitudes in the high-energy limit. As discussed in the following, the color ladder naturally arises w73,74x in the configurations where the gluons are strongly ordered in rapidity, i.e. in the multi-Regge kinematics. Indeed in the strong-rapidity ordering only the subamplitude with the corresponding order in the color coefficient contributes to Eq. Ž2.9.. At NLO, where the strong ordering is relaxed for two adjacent gluons, the leading subamplitudes are the two which differ just by the exchange of the gluon labels in the color ladders w54x. As we shall see this result generalizes at NNLO and beyond. Nonetheless, in the

1

The factor 1r2 in front of Eq. Ž2.9. is due to our choice for the normalization of the fundamental representation matrices, i.e. trŽ l al b . s d a b r2.

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V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

following we have chosen to derive our results starting from Eq. Ž2.1. instead of using directly Eq. Ž2.9.. The former, though more laborious, shows explicitly how the color traces must be recombined to obtain the color ladder and, more importantly, allows us to find the relations necessary to prove the factorization in the multi-collinear limits. For the maximally helicity-Õiolating configurations, Žy,y,q, . . . ,q ., in Eq. Ž2.1. or Eq. Ž2.9., there is only one independent colorrhelicity subamplitude, the Parke–Taylor ŽPT. subamplitude A Ž g 1 , . . . , g n . s 2 n r2

² i j :4 ²1 2: . . . ² Ž n y 1 . n:² n 1:

,

Ž 2.10 .

where the ith and the jth gluons have negative helicity. All other colorrhelicity amplitudes can be obtained by relabelling and by use of reflection symmetry, Eq. Ž2.2., and parity inversion. Parity inversion flips the helicities of all particles, and it is accomplished by the substitution ² i j : w j i x . Subamplitudes of non-PT type, i.e. with three or more gluons of y helicity have a more complicated structure.

l

2.2. Quark–gluon amplitudes For an amplitude with two quarks and Ž n y 2. gluons the color decomposition at tree level is w61–69x, i A Ž q,q ; g 1 , . . . , gŽ ny2. . s ig ny2

Ý Ž ls

1

ı

. . . ls ny 2 . j A Ž q,q ; gs 1 , . . . , gs ny 2 . ,

sgS ny2

Ž 2.11 . where Sny 2 is the permutation group on n y 2 elements. For the maximally helicity-Õiolating configurations, Žy,y,q, . . . ,q ., there is one independent colorrhelicity subamplitude, the Parke–Taylor ŽPT. subamplitude A Ž qq,qy ; g 1 , . . . , gŽ ny2. . s 2 Ž ny2.r2

² qi :² qi :3 ² qq :² q1: . . . ² Ž n y 2 . q :

,

Ž 2.12 .

where gluon g i has negative helicity. Helicity is conserved along the massless-fermion line. All other colorrhelicity amplitudes can be obtained by relabelling and by use of parity inversion, reflection symmetry and charge conjugation. In performing parity inversion, there is a factor of y1 for each pair of quarks participating in the amplitude. Reflection symmetry is like in Eq. Ž2.2., for gluons andror quarks alike. Charge conjugation swaps quarks and antiquarks without inverting helicities. In particular, using reflection symmetry and charge conjugation on Eq. Ž2.12. we obtain A Ž qy,qq ; g 1 , . . . , gŽ ny2. . s 2 Ž ny2.r2 where gluon g i has negative helicity.

² qi :3² qi : ² qq :² q1: . . . ² Ž n y 2 . q :

,

Ž 2.13 .

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For an amplitude with four quarks and Ž n y 4. gluons the color decomposition at tree level is w61x i A Ž q1 ,q1 ;q2 ,q2 ; g 1 , . . . , gŽ ny4. . ny4

s ig ny 2

Ý Ý Ý Ž ls

1

ı

ı

. . . l s k . j 2 1 Ž l r 1 . . . l r l . j1 2

ks0 s gS k r gS l

=A Ž q1 ,q1 ;q2 ,q2 ; gs 1 , . . . , gs k ; g r 1 , . . . , g r l . 1 y Nc

ı

ı

Ž l s 1 . . . l s k . j1 1 Ž l r 1 . . . l r l . j 2 2

=B Ž q1 ,q1 ;q2 ,q2 ; gs 1 , . . . , gs k ; g r 1 , . . . , g r l . ,

Ž 2.14 .

with k q l s n y 4, and where we suppose that the two quark pairs have distinct flavor. The sums are over the partitions of Ž n y 4. gluons between the two quark lines, and over the permutations of the gluons within each partition. For k s 0 or l s 0, the color strings reduce to Kronecker delta’s. For identical quarks, we must subtract from Eq. Ž2.14. the same term with the exchange of the quarks Ž q1 q2 .. For the maximally helicity-violating configurations, Žy,y,q, . . . ,q ., with likehelicity for all of the gluons, the A and B subamplitudes factorize into distinct contributions for the two quark antennae w61–66,68,69x. However, as we shall see in Section 5.4, we need the helicity configurations with two gluons of opposite helicity. For these the above mentioned factorization does not occur.

l

3. The leading impact factors We consider the elastic scattering of two partons of momenta pa and p b into two partons of momenta paX and p bX , in the high-energy limit, s 4 < t <. Firstly, we consider the amplitude for gluon–gluon scattering ŽFig. 1a.. Using Eqs. Ž2.1., Ž2.2., or Eq. Ž2.9., and Eq. Ž2.10., and Appendix B, we obtain w74x A g g ™ g g Ž pan a , panXaX < p bnXbX , p bn b . X

s 2 s ig f a a c C g ; g Ž pan a ; panXaX .

1

X

ig f b b c C g ; g Ž p bn b ; p bnXbX . , Ž 3.1 . t with q s p bX q p b and t , y< q H < 2 . In Eq. Ž3.1. and thereafter we use in the argument of

™

™

Fig. 1. Ža. Amplitude for g g g g scattering and Žb,c. for q g q g scattering. We label the external lines with momentum, color and helicity, and the internal lines with momentum and color.

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™

an amplitude A a vertical bar to separate clusters of particles which are close in rapidity from other clusters. The LO impact factors g ) g g, with g ) an off-shell gluon, are q X C g ; g Ž py a ; pa . s 1

qX C g ; g Ž py b ; pb . s

p b)X H

.

p bX H

Ž 3.2 .

They conserve helicity along the on-shell gluon line and transform under parity into their complex conjugates, C g;g Ž  kn 4 .

)

s C g ; g Ž  ky n 4 . .

Ž 3.3 . 2

In Eq. Ž3.1. four helicity configurations are leading, two for each impact factor . The helicity-flip impact factor C g ; g Ž pq; pXq . is subleading in the high-energy limit. From Eqs. Ž2.11., Ž2.12., we obtain the quark–gluon q g q g scattering amplitude in the high-energy limit w54x,

™

A q g ™ q g Ž pan a , panXaX < p bnXbX , p bn b . n aX s 2 s g lcaX a C q ; q Ž py ; panXaX . a

1 t

X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

Ž 3.4 .

n bX g lcbX b C q ; q Ž py ; p bnXbX . , b

Ž 3.5 .

A g q ™ g q Ž pan a , panXaX < p bnXbX , p bn b . X

s 2 s ig f a a c C g ; g Ž pan a ; panXaX .

1 t

where we have labeled the incoming quarks as outgoing antiquarks with negative momentum, e.g. the antiquark is pa in Eq. Ž3.4. ŽFig. 1b., and p b in Eq. Ž3.5. ŽFig. 1c.. The LO impact factors g ) q q are

™

q X C q ; q Ž py a ; pa . s yi ;

qX C q ; q Ž py b ; pb . s i

p b)X H

ž / p bX H

1r2

.

Ž 3.6 .

Under parity, the functions Ž3.6. transform as C q ;q Ž  k n 4 .

)

s S C q ; q Ž  ky n 4 .

with S s ysign Ž q 0 q 0 . ,

Ž 3.7 .

™

and in general an impact factor acquires a coefficient S for each pair of quarks Žsee Section 2.. Analogously, the antiquark–gluon q g q g amplitude is A q g ™ q g Ž pan a , panXaX < p bnXbX , p bn b . n aX s 2 s g lca aX C q ; q Ž py ; panXaX . a

1 t

X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

Ž 3.8 .

n bX g lcb bX C q ; q Ž py ; p bnXbX . , b

Ž 3.9 .

A g q ™ g q Ž pan a , panXaX < p bnXbX , p bn b . X

s 2 s ig f a a c C g ; g Ž pan a ; panXaX .

2

1 t

All throughout this paper, we shall always write only half of the helicity configurations contributing to an impact factor, the other half being obtained by parity.

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™

where the antiquark is paX in Eq. Ž3.8. and p bX in Eq. Ž3.9., and the LO impact factors g ) q q are C

q ;q

Ž

q X py a ; pa

. si ;

C

q ;q

Ž

qX py b , pb

. s yi

p b)X H

1r2

ž / p bX H

.

Ž 3.10 .

In the amplitudes Ž3.1., Ž3.4., Ž3.5., Ž3.8., Ž3.9., the leading contributions from all the Feynman diagrams have been included. However, the amplitudes have the effective form of a gluon exchange in the t-channel ŽFig. 1., and differ only for the relative color strength in the production vertices w76x. This allows us to replace an incoming gluon with a quark, for instance on the upper line, via the simple substitution X

ig f a a c C g ; g Ž pan a ; panXaX .

™

lg l

cX aa

n aX C q ; q Ž py ; panXaX . , a

Ž 3.11 .

and similar ones for an antiquark andror for the lower line. For example, the quark-quark q q q q scattering amplitude in the high-energy limit is A q q ™ q q Ž pan a , panXaX < p bnXbX , p bn b . n aX s 2 s g lcaX a C q ; q Ž py ; panXaX . a

1 t

n bX g lcbX b C q ; q Ž py ; p bnXbX . . b

Ž 3.12 .

4. The next-to-leading impact factors Let three partons be produced with momenta k 1 , k 2 and p bX in the scattering between two partons of momenta pa and p b , and to be specific, we shall take partons k 1 and k 2 in the forward-rapidity region of parton pa , the analysis for k 1 and k 2 in the forward-rapidity region of p b being similar. Parametrizing the momenta as in Eq. ŽA.1., we have y 1 , y 2 4 y bX ;

< k 1 H < , < k 2 H < , < p bX H < .

4.1. The NLO impact factor gg )

Ž 4.1 .

™ gg

™

We consider the amplitude for the scattering g g g g g ŽFig. 2a.. Only PT subamplitudes contribute, thus using Eqs. Ž2.1., Ž2.2. and Ž2.10., and Appendix C, we obtain w38,39x A g g ™ 3 g Ž pan a ,k 1n 1 ,k 2n 2 < p bnXbX , p bn b . s4 g3

s < qH < 2

C g ; g Ž p bn b ; p bnXbX .

Ý

A g ; g g Ž pan a ;ksns1 1 ,ksns2 2 .

sgS 2 X

X

X

X

=tr Ž l al ds 1 l ds 2 l bl b y l al ds 1 l ds 2 l bl b q l al bl bl ds 2 l ds 1 y l al bl bl ds 2 l ds 1 . X

X

qB g ; g g Ž pan a ;ksns1 1 ,ksns2 2 . tr Ž l al ds 1 l bl bl ds 2 y l al ds 2 l bl bl ds 1 . ,

Ž 4.2 .

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Fig. 2. Amplitudes for the production of three partons, with partons k 1 and k 2 in the forward-rapidity region of parton pa .

with the sum over the permutations of the two gluons 1 and 2, the LO impact factor, C g ; g Ž p bn b ; p bnXbX ., as in Eq. Ž3.2., and A g ; g g Ž pan a ;k 1n 1 ,k 2n 2 . s C g ; g g Ž pan a ;k 1n 1 ,k 2n 2 . An Ž k 1 ,k 2 . , B g ; g g Ž pan a ;k 1n 1 ,k 2n 2 . s C g ; g g Ž pan a ;k 1n 1 ,k 2n 2 . B n Ž k 1 ,k 2 . , with n s signŽ na q n 1 q n 2 ., and

Ž 4.3 .

q q C g ; g g Ž py a ;k 1 ,k 2 . s 1, y q 2 C g ; g g Ž pq a ;k 1 ,k 2 . s x 1 , q y 2 C g ; g g Ž pq a ;k 1 ,k 2 . s x 2 .

Ž 4.4 .

The momentum fractions are defined as kq i xi s q q i s 1,2 Ž x 1 q x 2 s 1 . , k1 q k 2

Ž 4.5 .

and the function Aq as follows: qH x1 1 Aq Ž k 1 ,k 2 . s y'2 , ² k1H x 2 12:

(

Ž 4.6 .

with ²12: a shorthand for ² k 1 k 2 :. Using the dual Ward identity w61x, or UŽ1. decoupling equations w18,67x, the function B n in Eq. Ž4.3., and thus the function B g ; g g , can be written as B n Ž k 1 ,k 2 . s y An Ž k 1 ,k 2 . q An Ž k 2 ,k 1 . .

Ž 4.7 .

™

The function C is subleading to the required accuracy. The function An has a collinear divergence as 2 k 1 P k 2 0, but the divergence cancels out in the function B n where gluons 1 and 2 are not adjacent in color ordering w39x. Using Eq. Ž4.7., and fixing t , y< q H < 2 , the amplitude Ž4.2. may be rewritten as g; g gŽ

q q. pq a ;k 1 ,k 2

A g g ™ 3 g Ž pan a ,k 1n 1 ,k 2n 2 < p bnXbX , p bn b .

½

s 2 s Ž ig . =

1

2

X

Ý

f a ds 1c f c ds 2 cA g ; g g Ž pan a ;ksns1 1 ,ksns2 2 .

sgS 2

5

X X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

t where the NLO impact factor for g ) g six helicity configurations.

Ž 4.8 .

™ g g is enclosed in curly brackets, and includes

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In the multi-Regge limit y 1 4 y 2 , lim A g ; g g Ž pan a ;k 1n 1 ,k 2n 2 . s C g ; g Ž pan a ;k 1n 1 .

y14y 2

1 t1

C g Ž q1 ,k 2n 2 ,q . ,

Ž 4.9 .

with q1 s yŽ pa q k 1 ., and t 1 , y< q1 H < 2 , and with LO Lipatov vertex, g ) g ) w32,33,74x, C g Ž q1 ,kq,q2 . s '2

q1)H q2 H kH

.

™g

Ž 4.10 .

Accordingly, the amplitude Ž4.8. is reduced to an amplitude in multi-Regge kinematics w29,74x, with the effective form of a gluon-ladder exchange in the t channel, A g g ™ 3 g Ž pan a ,k 1n 1 < k 2n 2 < p bnXbX , p bn b . s 2 s ig f a d1 c C g ; g Ž pan a ;k 1n 1 . =

1 t2

1 t1

X

ig f c d 2 c C g Ž q1 ,k 2n 2 ,q2 .

X X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

Ž 4.11 .

with q2 s p b q p bX and t 2 , y< q2 H < 2 . 4.2. The NLO impact factor gg )

™

™ qq

The amplitude g g q q g for the production of a qq pair in the forward-rapidity region of gluon a ŽFig. 2c. is obtained by taking the amplitudes Ž2.11. – Ž2.13. in the kinematics Ž4.1. w41x, n 1 < nXbX A g g ™ q q g Ž pan a ,k 1n 1 ,ky p b , p bn b . 2 X

n1 s 2 s  g 2 Ž lcl a . d 2 d 1 A g ; q q Ž pan a ;k 1n 1 ,ky . 2 X

n1 q Ž l alc . d 2 d 1 A g ; q q Ž pan a ;ky ,k 1n 1 . 2

1

4t

with k 1 the antiquark, the NLO impact factor g ) g

X X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

Ž 4.12 .

™ qq in curly brackets, and with

n1 A g ; q q Ž pan a ;k 1n 1 ,ky . s C g ; q q Ž pan a ;k 1n 1 ,ky2 n 1 . An Ž k 1 ,k 2 . , 2

( . s (x

q y 3 C g q q Ž pq a ;k 1 ,k 2 . s x 1 x 2 , y q C g ; q q Ž pq a ;k 1 ,k 2

™

3 1

x2 ,

Ž 4.13 .

with momentum fractions as in Eq. Ž4.5., An in Eq. Ž4.6. and n s na . The NLO impact factor g ) g qq allows for four helicity configurations. q ) In the multi-Regge limit kq g qq vanishes, since 1 4 k 2 , the NLO impact factor g quark production along the multi-Regge ladder is suppressed.

™

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4.3. The NLO impact factor qg )

™

™ qg

The amplitude q g q g g for the production of a q g pair in the forward-rapidity region of quark a ŽFig. 2b. is obtained by taking the amplitudes Ž2.11. – Ž2.13. in the kinematics Ž4.1. w54x n1 A q g ™ q g g Ž py ,k 1n 1 ,k 2n 2 < p bnXbX , p bn b . a X

n1 s 2 s  g 2 Ž l d 2lc . d1 a A q ; q g Ž py ;k 1n 1 ,k 2n 2 . a 1 X X X n1 q Ž lcl d 2 . d1 a B q ; q g Ž py ;k 1n 1 ,k 2n 2 . 4 ig f b b c C g ; g Ž p bn b ; p bnXbX . , Ž 4.14 . a t with k 1 the final-state quark, and the NLO impact factor q g ) q g in curly brackets. As above, the NLO impact factor includes four helicity configurations, n1 n1 A q ; q g Ž py ;k 1n 1 ,k 2n 2 . s C q ; q g Ž py ;k 1n 1 ,k 2n 2 . An Ž k 1 ,k 2 . , a a n1 n1 B q ; q g Ž py ;k 1n 1 ,k 2n 2 . s C q ; q g Ž py ;k 1n 1 ,k 2n 2 . B n Ž k 1 ,k 2 . , a a

™

q q C q ; q g Ž py a ;k 1 ,k 2 . s yi x 1 ,

(

(

y q 3 C q ; q g Ž pq a ;k 1 ,k 2 . s i x 1 ,

n

Ž 4.15 .

n

3

with A in Eq. Ž4.6., and B given by Eq. Ž4.7., with n s n 2 . As in Section 4.1, the function B q; q g vanishes in the collinear limit. q Ž . Ž . In the multi-Regge limit kq 1 4 k 2 the amplitude 4.14 reduces to Eq. 4.11 , with the substitution Ž3.11. for the upper line, and the LO impact factor C q; q in Eq. Ž3.6.. The treatment of the amplitude q g q g g for the production of a q g pair in the forward-rapidity region of antiquark a is identical to the former, thus the NLO impact factor q g ) q g is the same as in Eq. Ž4.14. up to inverting the color flow on the quark line w54x. The corresponding functions A and B are the same as in Eq. Ž4.15..

™

™

4.4. NLO impact factors in the collinear limit The collinear factorization for a generic amplitude occurs both on the subamplitude and on the full amplitude w61x, since in Eqs. Ž2.1., Ž2.11. and Ž2.14. color orderings where the collinear partons are not adjacent do not have a collinear divergence. Hence in the collinear limit for partons i and j, with k i s zP and k j s Ž1 y z . P, a generic amplitude Ž2.1. can be written as fi fj lim A . . . d i d j . . . Ž . . . ,k in i ,k nj j , . . . . s Ý A . . . c . . . Ž . . . , P n , . . . . Splityf ™ Ž k in i ,k nj j . , n ki<< kj

n

Ž 4.16 . with f denoting the parton species. Accordingly, for k 1 s zP and k 2 s Ž1 y z . P, we can write the amplitudes Ž4.8., Ž4.12. and Ž4.14. as n n X b,p b. lim A f g ™ f 1 f 2 g Ž pan a ,k 1n 1 ,k 2n 2 < py b b k1< < k 2

n n f ™ f1 f 2 X b , p b . P Split s A f g ™ f g Ž pan a , Pyn a < py Ž k 1n 1 ,k 2n 2 . , b b na

with A 3

fg

™ fg

Ž 4.17 .

as in Eq. Ž3.1., Ž3.4. and Ž3.8., respectively, and where we have used

In this context, Eq. Ž4.7. is only a bookkeeping, since the UŽ1. decoupling equation is valid only for the gluino-gluon subamplitudes corresponding to the quark–gluon subamplitudes used in Eq. Ž4.14..

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223

f1 f 2 helicity conservation in the s channel ŽSection 3.. For the collinear factors, Splityf ™ , n we obtain gg gg Splityg ™ Ž k 1n 1 ,k 2n 2 . s ig f c d1 d 2 splityg ™ Ž k 1n 1 ,k 2n 2 . , n n qq qq Splityg ™ Ž k 1n 1 ,k 2n 2 . s g Ž lc . d 2 d 1 splityg ™ Ž k 1n 1 ,k 2n 2 . , n n q™ q g qg Splity Ž k 1n 1 ,k 2n 2 . s g Ž ld 2 . d1 c splityq ™ Ž k 1n 1 ,k 2n 2 . , n n q™ q g qg Splity Ž k 1n 1 ,k 2n 2 . s g Ž ld 2 . c d 1 splityq ™ Ž k 1n 1 ,k 2n 2 . , n n

Ž 4.18 .

with splitting factors w61,72x q ' splityg ™ g g Ž kq 1 ,k 2 . s 2

q ' splitqg ™ g g Ž ky 1 ,k 2 . s 2

™ g g Ž kq1 ,ky2 . s '2

splitqg

y ' splitqg ™ q q Ž kq 1 ,k 2 . s 2 q ' splitqg ™ q q Ž ky 1 ,k 2 . s 2

1

(z Ž 1 y z . ²12: , z2

(z Ž 1 y z . ²12: , 2

Ž1yz . (z Ž 1 y z . ²12: , 1yz ²12: z ²12:

,

,

q™ q g splity Ž kq1 ,kq2 . s splityq ™ q g Ž kq1 ,kq2 . s '2 q™ q g splitq Ž ky1 ,kq2 . s splitqq ™ q g Ž ky1 ,kq2 . s '2

1

'1 y z ²12: , z

'1 y z ²12:

Ž 4.19 .

n 1 yn 2 . f1 f 2 Ž n 1 n 2 . and splitnf ™ f 1 f 2 Ž ky ,k 2 obtained from splityf ™ k 1 ,k 2 by exchanging ² k 1 k 2 : 1 n with w k 2 k 1 x, and multiplying by the coefficient S, Eq. Ž3.7., if the splitting factor includes a quark pair. Summing over the two helicity states of partons 1 and 2, we obtain a two-dimensional matrix, whose entries are the Altarelli–Parisi splitting functions at fixed color and helicity of the parent parton w12–15x

Ý n 1n 2

Splitlf ™ f 1 f 2 Ž k 1n 1 ,k 2n 2 . Split rf ™ f 1 f 2 Ž k 1n 1 ,k 2n 2 .

)

s d cc

X

2 g2 s12

e iŽ fly fr . Plfr™ f 1 f 2 ,

Ž 4.20 .

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224

f ™ f1 f 2 with e iŽ flyf r . a phase, where e iŽ f qyf y. s w21xr²12:, and where by definition Pqq f ™ f1 f 2 f ™ f1 f 2 f ™ f1 f 2 s Pyy , and Pqy s Pyq , and z 1yz g™ gg Pqq s 2CA q qz Ž1yz . , 1yz z g™ gg Pqy s 2CA z Ž 1 y z . , 1 2 2 g ™ qq Pqq s z q Ž1yz . , 2 g ™ qq Pqy sz Ž1yz . ,

1qz ™ ™ , 1yz q™ q g q™ q g Pqy s Pqy s0 . Ž 4.21 . q™ q g For P helicity conservation on the quark line sets the off-diagonal elements equal to zero. P q ™ g q is obtained from P q ™ q g by exchanging Ž z l 1 y z .. Since we sum 2

q qg q qg Pqq s Pqq s CF

over two helicity states of the external partons, Eq. Ž4.21. is valid in the-dimensional reduction ŽDR. scheme w56–58x. Eq. Ž4.21. agrees with the corresponding spin-correlated splitting functions of Ref. w77x in the DR scheme, after contracting the ones of type P g ™ f 1 f 2 with a parent-gluon polarization as in Appendix E. The connection of Eq. Ž4.21. with other regularization schemes ŽRS. is also given in Ref. w77x. Averaging over the trace of P f ™ f 1 f 2 in Eq. Ž4.20., i.e. over color and helicity of the parent parton on the left-hand side of Eq. Ž4.20., we obtain the unpolarized Altarelli– Parisi splitting functions 4 1 2C

Ý

f1 f 2
nn 1n 2

2 g2 s12

² P f ™ f1 f 2 : ,

Ž 4.22 .

with C s Nc2 y 1 for a parent gluon and C s Nc for a parent quark, and where the f ™ f1 f 2 averaged trace of P f ™ f 1 f 2 is ² P f ™ f 1 f 2 : s tr P f ™ f 1 f 2r2 s Pqq . 5. The next-to-next-to-leading impact factors In order to derive the next-to-next-to-leading ŽNNLO. impact factors, we repeat the analysis of Section 4 with one more final-state parton. Let four partons be produced with momenta k 1 , k 2 , k 3 and p bX in the scattering between two partons of momenta pa and p b , with a cluster of three partons, k 1 , k 2 and k 3 , in the forward-rapidity region of parton pa , < k 1 H < , < k 2 H < , < k 3 H < , < p bX H < . y 1 , y 2 , y 3 4 y bX ; Ž 5.1 . 5.1. The NNLO impact factor gg )

™ ggg

™

We begin with the amplitude for the scattering g g g g g g ŽFig. 3a. in the kinematics Ž5.1.. Using Eqs. Ž2.1., Ž2.2. and Ž2.10., and the subamplitudes of non-PT 4

Note that in the DR scheme the unpolarized splitting functions do not coincide with the azimuthally averaged ones. The latter are given in any RS in Ref. w77x.

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225

Fig. 3. Amplitudes for the production of four partons, with partons k 1 , k 2 and k 3 in the forward-rapidity region of parton pa .

type, with three gluons of q helicity and three gluons of y helicity w61x, and Appendix D, we obtain A g g ™ 4 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 < p bnXbX , p bn b . s

s4 g4

< qH < 2

C g ; g Ž p bn b ; p bnXbX .

A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

Ý sgS 3

X

X

X

=tr Ž l al ds 1 l ds 2 l ds 3 l bl b y l al ds 1 l ds 2 l ds 3 l bl b q l bl bl ds 3 l ds 2 l ds 1 l a X

X

yl bl bl ds 3 l ds 2 l ds 1 l a . q B g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 . tr Ž l al ds 1 l ds 2 l bl bl ds 3 X

X

X

yl al ds 1 l ds 2 l bl bl ds 3 q l bl bl ds 2 l ds 1 l al ds 3 y l bl bl ds 2 l ds 1 l al ds 3 . ,

Ž 5.2 .

with the sum over the permutations of the three gluons 1, 2 and 3, and the LO impact factor, C g ; g Ž p bn b ; p bnXbX ., as in Eq. Ž3.2.. From the PT subamplitudes Ž2.10. we obtain the function of Žyqqq . helicities A g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . s C g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . An Ž k 1 ,k 2 ,k 3 . ,

Ž 5.3 .

where n s signŽ na q n 1 q n 2 q n 3 . and Aq Ž k 1 ,k 2 ,k 3 . s y2

qH k1H

(

x1

1

Ž 5.4 .

x 3 ²12:²23:

and xi s

kq i q q kq 1 q k2 q k3

i s 1,2,3

Ž x 1 q x 2 q x 3 s 1. .

Ž 5.5 .

The functions C g ;3 g are a straightforward generalization of the functions C g ; g g defined in Eq. Ž4.4. and read C g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . s

½

1

na s y

x i2

ni s y

i s 1,2,3

with n s q,

Ž 5.6 .

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226

From the non-PT subamplitudes w61x we obtain the function of Žyyqq . helicities q q y A g ;3 g Ž py a ,k 1 ,k 2 ,k 3 .

2 s s12 < k 1 H < q

2

y

s a12 s123

ž

s23

b Ž k 1 ,k 2 ,k 3 . x 1

g Ž k 1 ,k 2 ,k 3 . Ž x 1 x 2 q b Ž k 1 ,k 2 ,k 3 . Ž x 2 q x 3 . .

/

x2 x3 2

y

b Ž k 1 ,k 2 ,k 3 . s a12

2

g Ž k 1 ,k 2 ,k 3 . s123 < k 1H < 2

q

x2

s12 x 12 x 2 < q H < 2

q

s23 x 1 x 2 x 3

s23 Ž x 2 q x 3 .

,

q y q A g ;3 g Ž py a ,k 1 ,k 2 ,k 3 .

2 s s12 < k 1 H < q

2

y

s a12 s123

ž

s23

ya Ž k 1 ,k 3 ,k 2 . x 1

g Ž k 1 ,k 3 ,k 2 . Ž x 1 x 2 y a Ž k 1 ,k 3 ,k 2 . Ž x 2 q x 3 . .

/

x2 x3 2

y

a Ž k 1 ,k 3 ,k 2 . s a12

2

q

g Ž k 1 ,k 3 ,k 2 . s123 < k 1 H < 2

x2

s12 x 12 x 2 < q H < 2

q

s23 x 1 x 2 x 3

y q q A g ;3 g Ž py a ,k 1 ,k 2 ,k 3 . s

2

g Ž k 2 ,k 3 ,k 1 . s123

2 s12

q

s23 Ž x 2 q x 3 .

,

2

y

a Ž k 2 ,k 3 ,k 1 . s a12 x2 < k3H < 2

s23 x 1 x 2 x 3

a Ž k 2 ,k 3 ,k 1 . g Ž k 2 ,k 3 ,k 1 . s a12 s123 Ž x 2 q x 3 . s23 x 2 x 3 < k 3 H < 2

,

Ž 5.7 . with si jk s Ž pi q pj q p k . 2 the three-particle invariant, and

a Ž k 1 ,k 2 ,k 3 . '

b Ž k 1 ,k 2 ,k 3 . '

g Ž k 1 ,k 2 ,k 3 . '

(x

1

k3H

ž (x

1

(

) qH q x 2 w1 2x

s a13

Ž k 1 H qk 2 H . w 1 2 x (x 1 x 2 s a12

(x

1

ž(

/,

,

(

x 2 x 3 w 1 2 x x 1 ²1 3: q x 2 ²2 3: s123

/.

Ž 5.8 .

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227

Using the UŽ1. decoupling equations w18,67x, the function B in Eq. Ž5.2. can be written as B g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . s y A g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . qA g ;3 g Ž pan a ;k 1n 1 ,k 3n 3 ,k 2n 2 . qA g ;3 g Ž pan a ;k 3n 3 ,k 1n 1 ,k 2n 2 . .

Ž 5.9 .

In the triple collinear limit, k 1 < < k 2 < < k 3 , Section 5.6, the function A has a double collinear divergence, while the function B, whose gluon 3 is not color adjacent to gluons 1 and 2, has only a single collinear divergence. Using Eq. Ž5.9., we can rewrite Eq. Ž5.2. as A g g ™ 4 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 < p bnXbX , p bn b .

½

s 2 s Ž ig .

=

1 t

3

X

Ý

X

XX

f a ds 1c f c ds 2 c f c ds 3 c A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

sgS 3

5

X XX

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

where the NNLO impact factor g ) g 14 helicity configurations. 5.2. The NNLO impact factor gg )

Ž 5.10 .

™ g g g is enclosed in curly brackets, and includes

™ gqq

™

We consider the amplitude for the scattering g g g q q g ŽFig. 3b., in the kinematics Ž5.1.. Using Eqs. Ž2.11. – Ž2.13. and the subamplitudes of non-PT type, with two gluons of q helicity and two gluons of y helicity w61x, we obtain n 2 < nXbX A g g ™ g q q g Ž pan a ,k 1n 1 ,k 2n 2 ,ky p b , p bn b . 3 n2 s 2 s I g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3

1

X X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

t

with k 3 the quark, and with NNLO impact factor g g )

™ g q q,

Ž 5.11 .

X

n2 I g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . s g 3 Ž lclald1 . d 3 d 2 A1g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky3 n 2 . 3 X

n2 q Ž l alcl d1 . d 3 d 2 A 2g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3 X

n2 q Ž l d1lcl a . d 3 d 2 A 3g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3 X

n2 q Ž l d1l alc . d 3 d 2 A 4g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3 X

n2 q Ž l al d1lc . d 3 d 2 B1g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3 X

n2 q Ž lcl d1l a . d 3 d 2 B2g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . . 3

Ž 5.12 .

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228

The NNLO impact factor allows for eight helicity configurations. From the PT subamplitudes Ž2.12., Ž2.13. we obtain qH 1 q y q A1g ; g q q Ž pq x 1 x 23 , a ;k 1 ,k 2 ,k 3 . s y2 ² : k1H 1 2 ²2 3:

(

q y q A 2g ; g q q Ž pq a ;k 1 ,k 2 ,k 3 . s 2

qH k3H

q y q A 3g ; g q q Ž pq a ;k 1 ,k 2 ,k 3 . s y2

q y q A 4g ; g q q Ž pq a ;k 1 ,k 2 ,k 3 . s 2

qH k2 H

qH k1H

q y q B1g ; g q q Ž pq a ;k 1 ,k 2 ,k 3 . s y2

q y q B2g ; g q q Ž pq a ;k 1 ,k 2 ,k 3 . s y2

)

x 23 x 32

1

x1

²1 2: ²2 3:

x 22

(x x 1

qH

2 2

qH k2 H

x 22

1

x 1 ²1 3: ²3 2: 1

x3

x2 x3

k3H

x3

(

,

²1 3: ²3 2: 1

k 1 H ²2 3: 1

k 1 H ²2 3:

,

,

,

,

Ž 5.13 .

with momentum fractions as in Eq. Ž5.5.. The impact factors from the non-PT subamplitudes w61x are, q y q A1g ; g q q Ž py a ;k 1 ,k 2 ,k 3 . 2

s2

g Ž k 1 ,k 3 ,k 2 . s123

½

y

w 1 3 x²1 2 :s23 x 1 x 2 x 3

q

(x

1

g Ž k 1 ,k 3 ,k 2 . s123 a Ž k 1 ,k 3 ,k 2 . s3 b bX

(

) s12²2 3 :k 1H x 3 x 3 w1 3x k 2 H

2 Ž k 3 H qq H . a Ž k 1 ,k 3 ,k 2 . s3 b bX

(x

2

x 3 s12 < k 1 H < 2 k 2 H

x 1 s 3 b bX

y

(x

3

s23 < k 1 H < 2 Ž x 2 q x 3 .

(

s12 s23 < k 1 H < x 3

ž

y

s123 x 2 x 3 a Ž k 1 ,k 3 ,k 2 . q

2

= yx 2 q

< q H < 2 x 12 x 23r2

a Ž k 1 ,k 3 ,k 2 . Ž x 2 q x 3 . x1

/5

g Ž k 1 ,k 3 ,k 2 . s123

(x

2

x3

,

Ž 5.14 .

q y q A 2g ; g q q Ž py a ;k 1 ,k 2 ,k 3 .

° ¢

s 2~y

q

2

g Ž k 1 ,k 3 ,k 2 . s123

w 1 3 x²1 2 :s23 x 1 x 2 x 3

(

k 2 H k 3)H y w 2 3 x x 2 x 3

ž

(

s23 k 3)H s1 b bX x 2 x 3

y

(

2

/

q

x2 x3

ž y w2 3x (x

2

( /

q w1 3x x 2

s12 w 2 3 x

x1

(x

1

x3

s23

q

(x

2

¶• ß,

x 3 g Ž k 1 ,k 3 ,k 2 . s123

x 1²1 2 :²2 3 :w 1 3 x

k 3)H

Ž 5.15 .

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229

q y q A 3g ; g q q Ž py a ;k 1 ,k 2 ,k 3 .

s2

½

q

2

(

k 3 H x 2 a Ž k 3 ,k 1 ,k 2 . s1 b bX k 2 H < k 2 H < 2 s23 x 3

2

y

g Ž k 1 ,k 3 ,k 2 . s123 w 1 2 x

(

x 1 x 2 x 3 w 1 3 x s13 s23

a Ž k 3 ,k 1 ,k 2 . s1 b bX(x 2 ž²1 3 :y ²1 2 :(x 2 x 3 / ²1 2 :x 1 q ²1 3 :s23 ²1 3 :< k 2 H < 2 s23(x 3

) w 1 3 x x 2 x 3(x 1 q Ž k 2)H x 1 y q H x 2 x 3 . (x 3 q (x 1 ²1 3 :k 2)H w2 3x

5

,

Ž 5.16 .

q y q A 4g ; g q q Ž py a ;k 1 ,k 2 ,k 3 . 2

) 2 x 1 g Ž k 1 ,k 3 ,k 2 . s123 Ž k 2 H qq H . (x 1 a Ž k 1 ,k 2 ,k 3 . Ž k 2 H qq H . q H s2 y 2 ²1 3 :< k 1 H < k 3 H (x 2 ²2 3 :w 1 3 x < k 1 H < 2 s13(x 1 x 3 x 2

½

q

x 12 < q H < 2(x 2 x 3 w 1 2 x g Ž k 1 ,k 3 ,k 2 . 2 s123 q w 1 3 x s13 s23 x 1 x 2 x 3 < k 1 H < 2 s23 Ž x 2 q x 3 .

y

w 1 2 x x 1 s123 y Ž k 2 H qq H . x 2(x 1 x 3 w 1 3 x q s2 b bX g Ž k 1 ,k 3 ,k 2 . x 2 < k 1 H < 2 s13 s23 w 1 3 x

q

x 1 Ž k 2 H qq H . w 1 2 x g Ž k 1 ,k 3 ,k 2 . s123 Ž x 2 q x 3 . s13 < k 1H < 2 s23 x 1 x 3 x 2

(

5

,

Ž 5.17 .

q y q B1g ; g q q Ž py a ;k 1 ,k 2 ,k 3 .

s2

½

y

y a Ž k 3 ,k 1 ,k 2 . 2 s1 b bX k 3)H s23 k 2 H

'x

q

Ž k 2 H q q H . 2 w 1 3 x x 1 x 3 a Ž k 1 ,k 2 ,k 3 .

'

< k1 H < 2 < k 3 H < 2 k 3 H x 2

'

2 x3

w 1 3 x x 1 x 3 ² 2 3 : x 2 < k 3 H < 2 y Ž k 2 H q q H . s23 Ž x 1 q x 2 . x 3 q k 1 H k 2 H w2 3 x x 3 x 2

'

'

'

'x 'x

y

1

2

< k 1 H < s23 x 2

'

2

2

< k 1 H < < k 3 H < s23

'

) y w1 3x k 1 H x 2 q q H x1 x 3

ž

qH x2 x2 q x3

q y q B2g ; g q q Ž py a ;k 1 ,k 2 ,k 3 .

°yŽ k s 2~ ¢ q

2

1 H qk 2 H

. ž k 1)H q w 1 2 x (x 1 x 2 / (x 2

< k 1 H < 2 k 2)H s3 b bX x 3

(

( ž

(

k 3 H x 2 k 3)H y w 2 3 x x 2 x 3 k 2)H s23 s1 b bX x 33r2

2

/

'

2

q k2 H

/

5

,

Ž 5.18 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

230

x2

y

(

yk 3 H w 2 3 x s3 b bX x 1 x 2 q k 1)H q w 1 2 x x 1 x 2

ž

2

< k 1 H < < k 2 H < 2 s23 x 3

(

= ²2 3 :q H k 3)H y w 2 3 x x 2 x 3 y k 3 H w 2 3 x k 2 H Ž x 2 q x 3 .

ž

y

ž (x

2

< k 1 H < 2 < k 2 H < 2 s23 x 33r2

(

q k 1)H q w 1 2 x x 1 x 2

ž

/

< k 3 H < 2 s 3 b bX x 1 x 2 q < q H < 2 < k 2 H < 2

/Žk

/

/

x 1 x 32 x2 q x3

< <2 X 3 H s1 b b x 2 x 3 q k 2 H k 3 H Ž x 2 q x 3 . .

¶• ß.

Ž 5.19 .

The functions A and B for the remaining helicity configurations are derived using the relations q y g ; g qq na q y q A ig ; g q q Ž pan a ;kq 1 ,k 2 ,k 3 . s yA 5yi Ž pa ;k 1 ,k 3 ,k 2 . ,

Big ; g q q

Ž

q y pan a ;kq 1 ,k 2 ,k 3

.

g ; g qq s yB3yi

y q pan a ;kq 1 ,k 3 ,k 2

Ž

.,

i s 1,2,3,4, i s 1,2.

Ž 5.20 .

™ qgg We consider the amplitude q g ™ q g g g for the production of a quark and two gluons in the forward-rapidity region of quark a ŽFig. 3c. in the kinematics Ž5.1.. Using 5.3. The NNLO impact factor qg )

Eqs. Ž2.11. – Ž2.13. and the subamplitudes of non-PT type, with two gluons of q helicity and two gluons of y helicity w61x, we obtain n1 A q g ™ q 3 g Ž py ,k 1n 1 ,k 2n 2 ,k 3n 3 < p bnXbX , p bn b . a 1 X X n1 s 2 s I q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . ig f b b c C g ; g Ž p bn b ; p bnXbX . , a t with k 1 the final-state quark, and the NNLO impact factor q g ) q g g,

I

q ;q g g

n1 py ;k 1n 1 ,k 2n 2 ,k 3n 3 a

Ž

sg

3

Ý Žl

ds 2

l

ds 3

™

Ž 5.21 .

. X

n1 lc . d1 a A q ; q g g Ž py ;k 1n 1 ,ksns2 2 ,ksns3 3 . a

sgS 2 X

n1 q Ž lcl ds 2 l ds 3 . d1 a B1q ; q g g Ž py ;k 1n 1 ,ksns2 2 ,ksns3 3 . a X

n1 q Ž l ds 2 lcl ds 3 . d1 a B2q ; q g g Ž py ;k 1n 1 ,ksns2 2 ,ksns3 3 . . a

Ž 5.22 .

The NNLO impact factor allows for eight helicity configurations. From the PT subamplitudes Ž2.12., Ž2.13. we obtain qH x1 1 q q q A q ; q g g Ž py , a ;k 1 ,k 2 ,k 3 . s 2 i k 1 H x 3 ²1 2: ²2 3:

(

q q q B1q ; q g g Ž py a ;k 1 ,k 2 ,k 3 . s 2 i

q q q B2q ; q g g Ž py a ;k 1 ,k 2 ,k 3 . s 2 i

qH

x1 x 3

1

x2

k 1 H ²2 3:

k3H

(

qH

x1

k1H

(x

2

1 k 3 H ²1 2:

,

Ž 5.23 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

231

and y q q q ;q g g q q q A q ; q g g Ž pq Ž py a ,k 1 ,k 2 ,k 3 . s yx 1 A a ;k 1 ,k 2 ,k 3 . , y q q q ;q g g q q q Biq ; q g g Ž pq Ž py a ;k 1 ,k 2 ,k 3 . s yx 1 Bi a ;k 1 ,k 2 ,k 3 . , The impact factors from the non-PT subamplitudes w61x are

i s 1,2.

Ž 5.24 .

q y q A q ; q g g Ž py a ;k 1 ,k 2 ,k 3 .

°k ž k s 2 i~ ¢

2

) 2 2 w 1 2 x (x 1 x 2 / g Ž k 1 ,k 2 ,k 3 . g Ž k 1 ,k 3 ,k 2 . s123 q 3r2 < k 1 H < 2 w 1 2 x s3 b bX(x 2 Ž x 1 x 2 x 3 . s12 s23²2 1 :w 1 3 x

2H

) 1H q

w 1 3 x²1 2 :x 1 ž (x 3 ²2 3 :y k 1 H (x 2 / q k 1 H (x 3 g Ž k 1 ,k 3 ,k 2 . s123

q

(

s12 s23 k 1H x 1 x 3

(x

q

1

k 1)H x 3 Ž q H x 2 q k 2 H Ž x 2 q x 3 . . y < q H < 2

2

s23 < k 1 H < x 3

x1 x 2 x 3 x2 q x3

¶

( ž

( / / ß•,

(

qx 1 yq H k 3)H x 2 q k 2 H x 3 k 1)H x 3 y w 2 3 x x 2

ž

Ž 5.25 .

q q y A q ; q g g Ž py a ; k1 , k 2 , k 3 .

s2 i

y

y

½

b Ž k 1 , k 2 , k 3 . 2 s 3 b bX k 2)H w 1 2 x s 12 < k 1 H <

2

2

'

q H g Ž k 1 , k 2 , k 3 . s 132

'

x1 x 3 ² 1 2:

) b Ž k 1 , k 2 , k 3 . s 3 b bX q H

'

x 3 k 2)H 2

w

'

q s 13 2 x 3 x 2

2 g Ž k 1 , k 3 , k 2 . g Ž k 1 , k 2 , k 3 . 2 s 132 )

q x1

s 12 < k 1 H < 2 x 3 x 2 w 3 2 x )

k 2)H x 13r2 s 12 < k 1 H < 2 w 3 2 x

'x

'

b Ž k 1 , k 2 , k 3 . s 3 b bX k 2)H x 1 Ž x 3 q x 2 . q

x 23r2

Ž x1 x 3 x 2 .

3r 2

s 12 s 32² 3 1 : w 1 2 x

x 1 ys 12 < q H < x 22 x 1 x 3 q ² 1 3 : b Ž k 1 , k 2 , k 3 . s 3 b bX k 2)H Ž x 3 q x 2 . 2 q

'

x

s 12 s 32 < k 1 H < 2 x 2 x 3 Ž x 3 q x 2 .

'

q

'x

1

s 3 b bX

w

w 1 2 x² 1 3 : k 2)H

'

x 13 x 2

'

) q b Ž k 1 , k 2 , k 3 . k 2)H s 132 x 1 x 3 y ² 1 3 : k 1)H q H x2

Ž

.x

w 1 2 x s 12 s 32 < k 1 H < 2 x 2 x 3

'

q y q B1q ; q g g Ž py a ;k 1 ,k 2 ,k 3 .

s2i

½

2 Ž k 2 H qq H . w 1 3 x x 1(x 3 2

< k 1 H < k 3 H s 2 b bX

k 2 H w1 3x x 1 x 3

²2 3 :k 1)H < k 3 H < 2(x 2

2

)

y

q

a Ž k 2 ,k 1 ,k 3 . a Ž k 3 ,k 1 ,k 2 . s12b bX < k3H <

2

k 3)H k 2 H s23

(x

1

x2

q

w 1 3 x (x 3 < k 1 H < 2 < k 3 H < 2 s23 x 2

= y< k 3 H < 2 Ž k 2 H qq H . x 1 x 2 q < k 1 H < 2 k 2 H Ž x 1 q x 2 . x 3

5

Ž 5.26 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

232

q

x3 2

k 2 H k 1)H < k 3 H < 2 x 1 q k 2 H < k 3 H < 2 w 1 2 x x 13r2 x 2

(

< k 1 H < < k 3 H < 2 s23 x 2 x 1

) yq H

ž

(

q H < k 3 H < 2 x 12 x 2 x2 q x3

q < k1H < 2 k 2 H Ž x1 q x 2 . x 3

/5

,

Ž 5.27 .

q q y B1q ; q g g Ž py a ;k 1 ,k 2 ,k 3 . 2

)

a Ž k 3 ,k 1 ,k 2 . a Ž k 2 ,k 1 ,k 3 . s12b bX

½

s2i y

y

k 3 H k 2)H < k 3 H < 2 s32 x 1 x 2

(

²1 3 :(x 3 a Ž k 1 ,k 2 ,k 3 . 2 s2 b bX 2

2

q

< k1H < < k 3H < k 3H = w 1 2 x ²1 3 :

(x x (x 1

3

(

q²1 3 :k 3 H x 1 Ž x 3 q x 2 .

(x

3

ž

3

2

< k 1 H < < k 3 H < 2 s23

Ž s2 b bX x 1 y s1 b bX x 2 . q (x 1 ž yk 1 H q H x 33r2

2

) yq H x1

(x

/

< k 3H < 2qH x1

(

x3 q x2

q y q B2q ; q g g Ž py a ;k 1 ,k 2 ,k 3 . s

2i < k1H

y

2

ž

)2 qH y w1 2x y

x1 y x 2

(x

1 x2

(

) w1 2x qH

q ²1 3 :Ž s1 b bX q s2 b bX . x 3

° < ¢

~Žk

2 H qq H

2

,

Ž 5.28 .

. x 1 ž k 1)H q w 1 2 x (x 1 x 2 / k 3 H w 1 2 x (x 2

(

k 2 H k 1)H q w 1 2 x x 1 x 2

ž

/5

/

2

/

w 1 2 x s3 b bX(x 2 y

¶ ß,

2 Ž k 2 H qq H . w 1 3 x x 1(x 3 •

k 3 H s 2 b bX

Ž 5.29 .

q q y B2q ; q g g Ž py a ;k 1 ,k 2 ,k 3 .

2i s

< k1H < 2

½

²1 3 :(x 3 a Ž k 1 ,k 2 ,k 3 . 2 s2 b bX k3H < k3H < 2

k 3)H

(x

²1 2 :

ž

q 1q

q

s12 w 2 1 x x 23r2

²1 2 :x 3 a Ž k 1 ,k 2 ,k 3 . s2 b bX

x1

y

2

b Ž k 1 ,k 2 ,k 3 . s3 b bX k 2)H

) qH x 1 y k 3)H x 2

(x

1 x2

w1 2x

/

1

k3H

b Ž k 1 ,k 2 ,k 3 . s3 b bX

(x

2

5

.

Ž 5.30 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

233

™ qQQ We consider the amplitude q g ™ qQ Q g for the production of three quarks in the forward-rapidity region of quark a ŽFig. 3d. in the kinematics Ž5.1.. Using Eq. Ž2.14. 5.4. The NNLO impact factor qg )

and the subamplitudes of non-PT type, with two gluons of opposite helicities w78x, we obtain n1 n 2 < nXbX A q g ™ q Q Q g Ž py ,k 1n 1 ,k 2n 2 ,ky p b , p bn b . a 3

n1 n2 s 2 s I q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

™q Q Q

with NNLO impact factor q g )

1

X

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

t

Ž 5.31 .

n1 n2 I q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

n1 n2 s g 3 lcd 3 a d d1 d 2 A1q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

1 y Nc

n1 n2 lcd1 a d d 3 d 2 A q2 ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

n1 n2 qlcd1 d 2 d d 3 a B1q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

1 y Nc

n1 n2 lcd 3 d 2 d d1 a B2q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . y dq Q Ž 1 a 3

l 3. .

Ž 5.32 .

The term proportional to d q Q is due to the interference of identical quarks Ži.e. with the same flavor and helicity . in the final state. The NNLO impact factor allows for four helicity configurations. From the non-PT subamplitudes w78x we obtain q y q A1q ; q Q Q Ž py a ;k 1 ,k 2 ,k 3 .

si

½(

x 1 x 3 a Ž k 1 ,k 3 ,k 2 . < k1H <

x2 x1

q

(x

2

x 3 s23 < k 1 H <

2

2

3 1 2

1

< qH < 2

x3

1 y x1

(x

q y q A q2 ; q Q Q Ž py a ;k 1 ,k 2 ,k 3 . s

(x x q

s23 < k 1 H <

a Ž k 3 ,k 1 ,k 2 . s a23 y i

(x

2

x 3 s23

ž

1

q

g Ž k 1 ,k 3 ,k 2 .

(x

1 x2

x 3 s23

g Ž k 1 ,k 3 ,k 2 . s123

g Ž k 1 ,k 3 ,k 2 .

(x

2

(x (

1

5

,

/

q x 1 a Ž k 3 ,k 1 ,k 2 . ,

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

234

q y q B1q ; q Q Q Ž py a ;k 1 ,k 2 ,k 3 . s i

½( (

x 1 x 2 b Ž k 1 ,k 3 ,k 2 . x3

< k1H < 2

x1

a Ž k 3 ,k 1 ,k 2 .

x2 x3

s23

q

(

< k1H <

2

ž(

x2 x3

< qH < 2

x3

s23 < k 1 H < 2

x1

y

(x

(

i x1

3 1 2

1 y x1

= x 1 a Ž k 3 ,k 1 ,k 2 . s a23 y q y q B2q ; q Q Q Ž py a ;k 1 ,k 2 ,k 3 . s

(x x

y

2

x 3 s23 < k 1 H < 2

g Ž k 1 ,k 3 ,k 2 . s123

b Ž k 1 ,k 3 ,k 2 . q

(x

(

x3 x2

1

5

,

a Ž k 1 ,k 3 ,k 2 .

/

Ž 5.33 . with a , b ,g defined in Eq. Ž5.8. and y y q q ;qQQ q y q A qi ; q Q Q Ž pq Ž py a ;k 1 ,k 2 ,k 3 . s A i a ;k 1 ,k 3 ,k 2 . ,

Biq ; q Q Q

Ž

y y q pq a ;k 1 ,k 2 ,k 3

.

s Biq ; q Q Q

Ž

q y q py a ;k 1 ,k 3 ,k 2

i s 1,2,

. , i s 1,2 .

Ž 5.34 .

Note that for each helicity configuration, we have the following relation between the functions A and B: A1q ; q Q Q q B1q ; q Q Q s A q2 ; q Q Q q B2q ; q Q Q ,

Ž 5.35 .

5.5. NNLO impact factors in the high-energy limit The amplitudes Ž5.10., Ž5.11., Ž5.21. and Ž5.31. have been computed in the kinematic limit Ž5.1., in which they factorize into an effective amplitude with a ladder structure, made of a three-parton forward cluster and a LO impact factor connected by a gluon exchanged in the crossed channel ŽFig. 3.. In the limits y 1 , y 2 4 y 3 or y 1 4 y 2 , y 3 , the amplitudes must factorize further into NLO impact factors or into NLO Lipatov vertices for the production of two partons along the ladder. Such limits constitute then necessary consistency checks, and we display them in this section.

Fig. 4. Limits of the amplitude for the production of three gluons in the forward-rapidity region of gluon pa , for y1 4 y 2 , y 3 Ža. and y1 , y 2 4 y 3 Žb..

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

235

™

In the limit, y 1 4 y 2 , y 3 , the NNLO impact factor, g ) g g g g, Eq. Ž5.10., factorizes into a NLO Lipatov vertex for the production of two gluons convoluted with a multi-Regge ladder ŽFig. 4a. lim y14y 2,y 3

½Ž

ig .

3

X

X

XX

½Ž

ig .

f a ds 1c f c ds 2 c f c ds 3 c A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

Ý sgS 3

1

s ig f a d1 c C g ; g Ž pan a ;k 1n 1 .

t1

)

)

X

2

Ý

X

5

XX

5

f c ds 2 c f c ds 3 c A g g Ž q1 ,ksns2 2 ,ksns3 3 ,q2 . ,

sgS 2

Ž 5.36 .

™

with the NLO Lipatov vertex, g g g g, for the production of two gluons k 2 and k 3 given by w38–40x q1)H q2 H x2 1 q A g g Ž q1 ,kq ,k ,q s 2 , . 2 3 2 k2 H x 3 ²23:

(

y A g g Ž q1 ,kq 2 ,k 3 ,q 2 .

s y2

k 2)H k2 H

q

½

k 32H < q1 H < 2

1 y s23

Ž q2 H qk 3 H .

Ž

2

y

y ky 2 q k3

.

kq 3

k 22 H < q2 H < 2

q

Ž

q kq 2 q k3

y q2 H qk 3 H ky 2 q k3

s 3 b bX

ky 2

s23

.

ky 2

k2 H y

q

s 3 b bX k 2 H k 3 H

q kq 2 q k3

kq 3

q ky 2 k3

k 3H

5

Ž 5.37 .

with exchanged momenta in the t channel q1 s yŽ paX q pa ., q2 s p bX q p b , three-parq. ticle invariant s3 b bX s Ž k 3 q q2 . 2 , yŽ< q2 H qk 3 H < 2 q ky 2 k 3 , and with the mass-shell 2 q y conditions k i s < k i H < rk i for i s 2,3. In the collinear limit, k 2 s zP and k 3 s Ž1 y z . P, the NLO Lipatov vertex Ž5.37. reduces to the splitting factor Ž4.19., and amplitude Ž5.10. factorizes into a multi-Regge amplitude Ž4.11. times a collinear factor Ž4.18. lim A g g ™ 4 g Ž pan a ,k 1n 1 < k 2n 2 ,k 3n 3 < p bnXbX , p bn b .

k2<< k3

gg s Ý A g g ™ 3 g Ž pan a ,k 1n 1 < P n < p bnXbX , p bn b . P Splityg ™ Ž k 2n 2 ,k 3n 3 . , n

n

™

In the limit y 1 , y 2 4 y 3 , the NNLO impact factor in Eq. Ž5.10. factorizes into a NLO impact factor, g ) g g g, Eq. Ž4.8., convoluted with a multi-Regge ladder ŽFig. 4b. lim y1,y 24y 3

½

½Ž

s Ž ig . =

2

ig .

X

3

Ý

X

sgS 3 X

Ý

f a ds 1c f c ds 2 cA g ;2 g Ž pan a ;ksns1 1 ,ksns2 2 .

sgS 2

1 t1

XX

f a ds 1c f c ds 2 c f c ds 3 c A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

X

5

5

XX

ig f c d 3 c C g Ž q1 ,k 3n 3 ,q2 . ,

Ž 5.38 .

with q1 s yŽ pa q k 1 q k 2 ., and with LO Lipatov vertex C g Ž q1 ,k 3n 3 ,q2 ., Eq. Ž4.10..

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

236

In the limit y 1 4 y 2 , y 3 , the functions A and B in Eq. Ž5.13. – Ž5.20. fulfill the relations A 2g ; g q q s A 3g ; g q q s 0, B2g ; g q q s yA1g ; g q q, and B1g ; g q q s yA 4g ; g q q, thus the NNLO impact factor, g ) g g q q, Eq. Ž5.12., factorizes into a NLO Lipatov vertex for the production of a qq pair convoluted with a multi-Regge ladder ŽFig. 5a.,

™

lim y14y 2,y 3

n2 I g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3

s ig f a d1 c C g ; g Ž pan a ;k 1n 1 .

1 t1

 g2

X

Ž lclc . d 3 d 2 A q q Ž q1 ,k 2n 2 ,ky3 n 2 ,q2 .

X

n2 q Ž lclc . d 3 d 2 A q q Ž q1 ,ky ,k 2n 2 ,q2 . 3

with the NLO Lipatov vertex, g ) g )

4,

Ž 5.39 .

™ q q, for the production of a qq pair w40,41x

y A q q Ž q1 ,kq 2 ,k 3 ,q 2 .

kq 2

< <2 kq 3 q2 H

kq 3

Ž kq2 q kq3 . s23

( ½

s y2

q

q

< <2 ky 3 k 3 H q1 H y k 2 H Ž ky 2 q k 3 . s 23

q

) kq 3 k 2 H Ž q 2 H qk 3 H .

Ž q2 H qk 3 H . ky2 kq3 y k 2)H k 3 H y Ž q2)H q k 3)H . k 3 H

X kq 2 s3 b b

y

< k3H < 2

k 2 H s23

s23

5

,

Ž 5.40 . with q1 , q2 , and s3 b bX as in Eq. Ž5.37.. In the collinear limit, k 2 s zP and k 3 s Ž1 y z . P, the NLO Lipatov vertex Ž5.40. reduces to the splitting factor Ž4.19., and amplitude Ž5.11. factorizes into a multi-Regge amplitude Ž4.11. times a collinear factor Ž4.18. n 2 < nXbX lim A g g ™ g q q g Ž pan a ,k 1n 1 < k 2n 2 ,ky p b , p bn b . 3

k2<< k3

qq s Ý A g g ™ 3 g Ž pan a ,k 1n 1 < P n < p bnXbX , p bn b . P Splityg ™ Ž k 2n 2 ,ky3 n 2 . . n

n

In the limit y 2 , y 3 4 y 1 , the functions A and B in Eq. Ž5.13. – Ž5.20. fulfill the relations A1g ; g q q s A 4g ; g q q s 0, B2g ; g q q s yA 3g ; g q q, and B1g ; g q q s yA 2g ; g q q thus the

Fig. 5. Same as Fig. 4 for the production of a quark–antiquark pair and a gluon in the forward-rapidity region of gluon pa .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

237

Fig. 6. Same as Fig. 4 for the production of a quark and two gluons in the forward-rapidity region of quark pa .

NNLO impact factor, Eq. Ž5.12., factorizes into a NLO impact factor, g ) g Ž4.12., convoluted with a multi-Regge ladder ŽFig. 5b., lim y 2,y 34y 1

™ q q, Eq.

n2 I g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky . 3

n2 s  g 2 Ž lcl a . d 3 d 2 A g ; q q Ž pan a ;k 2n 2 ,ky . q Ž lalc . d 3 d 2 A g ; q q Ž pan a ;ky3 n 2 ,k 2n 2 . 4 3 1 X = ig f c d1 c C g Ž q1 ,k 1n 1 ,q2 . , Ž 5.41 . t1

with q1 s yŽ pa q k 2 q k 3 .. In the limit y 1 4 y 2 , y 3 , the functions A and B in Ž5.23. – Ž5.30. fulfill the relations n1 n1 B1q ; q g g Ž py ;k 1n 1 ,ksns2 2 ,ksns3 3 . s A q ; q g g Ž py ;k 1n 1 ,ksns3 3 ,ksns2 2 . , a a n1 n1 B2q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . s B2q ; q g g Ž py ;k 1n 1 ,k 3n 3 ,k 2n 2 . a a n1 s y Ž A q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . a n1 qA q ; q g g Ž py ;k 1n 1 ,k 3n 3 ,k 2n 2 . . , a thus the NNLO impact factor, q g q g g, Eq. Ž5.22., factorizes into a NLO Lipatov vertex for the production of two gluons Ž5.37. convoluted with a multi-Regge ladder ŽFig. 6a. )

lim y14y 2,y 3

™

n1 I q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . a

n1 s g lcd1 a C q ; q Ž py ;k 1n 1 . a

=

1 t1

½Ž

ig .

2

X

Ý sgS 2

X

XX

5

f c ds 2 c f c ds 3 c A g g Ž q1 ,ksns2 2 ,ksns3 3 ,q2 . .

Ž 5.42 .

In the limit y 1 , y 2 4 y 3 , the functions A and B in Ž5.23. – Ž5.30. fulfill the relations n1 n1 A q ; q g g Ž py ;k 1n 1 ,k 3n 3 ,k 2n 2 . s B1q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . s 0, a a n1 n1 B2q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . s yA q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . , a a n1 n1 B2q ; q g g Ž py ;k 1n 1 ,k 3n 3 ,k 2n 2 . s B1q ; q g g Ž py ;k 1n 1 ,k 3n 3 ,k 2n 2 . , a a

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238

Fig. 7. Limit of the amplitude for the production of a quark and a quark–antiquark pair in the forward-rapidity region of quark pa , for y1 4 y 2 , y 3 .

™

thus the NNLO impact factor, Eq. Ž5.22., factorizes into a NLO impact factor, q g ) q g, Eq. Ž4.14., convoluted with a multi-Regge ladder ŽFig. 6b. X

lim y1,y 24y 3

n1 n1 I q ; q g g Ž py ;k 1n 1 ,k 2n 2 ,k 3n 3 . s  g 2 Ž l d 2lc . d1 a A q ; q g Ž py ;k 1n 1 ,k 2n 2 . a a X

n1 q Ž lcl d 2 . d1 a B q ; q g Ž py ;k 1n 1 ,k 2n 2 . a

1

=

t1

X

4

XX

ig f c d 3 c C g Ž q1 ,k 3n 3 ,q2 . ,

Ž 5.43 .

with q1 s yŽ pa q k 1 q k 2 .. qQQ In the limit y 1 4 y 2 , y 3 , the function A 2 in Eq. Ž5.33. vanishes, A q; s 0, and 2 ) using Eqs. Ž5.33.-Ž5.35. the NNLO impact factor, q g qQ Q, Eq. Ž5.32., factorizes into a NLO Lipatov vertex for the production of a qq pair Ž5.40. convoluted with a multi-Regge ladder ŽFig. 7.

™

lim y14y 2,y 3

n1 n2 I q ; q Q Q Ž py ;k 1n 1 ,k 2n 2 ,ky . a 3

n1 s g lcd1 a C q ; q Ž py ;k 1n 1 . a

1 t1

 g2

X

Ž lclc . d 3 d 2 A q q Ž q1 ,k 2n 2 ,ky3 n 2 ,q2 .

X

n2 q Ž lclc . d 3 d 2 A q q Ž q1 ,ky ,k 2n 2 ,q2 . 3

4.

Ž 5.44 .

5.6. NNLO impact factors in the triple collinear limit In the triple collinear limit, k i s z i P, with z 1 q z 2 q z 3 s 1 a generic amplitude must factorize as w20,21x lim A . . . d1 d 2 d 3 . . . Ž . . . ,k 1n 1 ,k 2n 2 ,k 3n 3 , . . . .

k1< < k 2 < < k 3

f1 f 2 f 3 s Ý A . . . c . . . Ž . . . , P n , . . . . P Splityf ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 . . n

n

Ž 5.45 .

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239

Accordingly, we must show that taking the triple collinear limit of the NNLO impact factors, we can write the amplitudes Ž5.10., Ž5.11., Ž5.21. and Ž5.31. as n n X b, p b. lim A f g ™ f 1 f 2 f 3 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 < py b b

k1< < k 2 < < k 3

n n f ™ f1 f 2 f 3 X b , p b . P Split s A f g ™ f g Ž pan a , Pyn a < py Ž k 1n 1 ,k 2n 2 ,k 3n 3 . , b b na

Ž 5.46 .

with f denoting the parton species, A f g ™ f g given in Eqs. Ž3.1., Ž3.4. and Ž3.8., and f1 f 2 f 3 the polarized double-splitting functions. with Splityf ™ n In the triple collinear limit, the functions A of Sections 5.1, 5.2, 5.3 and 5.4 yield a quadratic divergence as s123 0 or si j 0 with i, j s 1,2,3. In the same limit, the functions B have a single collinear divergence since only two out of the three partons are color adjacent. However, terms with a single divergence when integrated over the triple collinear region of phase space yield a negligible contribution w20x, thus we ignore them. It is easy to show that a function A g ;3 g , Eqs. Ž5.3. – Ž5.7., differs from its reflection by a term which contains only a single divergence. Using this property and Eq. Ž5.9., we obtain a dual Ward identity and a reflection identity for the functions A g ;3 g , up to singly divergent terms,

™

™

A g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . q A g ;3 g Ž pan a ;k 1n 1 ,k 3n 3 ,k 2n 2 . q A g ;3 g Ž pan a ;k 3n 3 ,k 1n 1 ,k 2n 2 . s 0, A g ;3 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 . s A g ;3 g Ž pan a ;k 3n 3 ,k 2n 2 ,k 1n 1 . .

Ž 5.47 .

Using the identities Ž5.47. in the impact factor in Eq. Ž5.10., we can factorize the color structure on a leg

Ž ig .

3

X

X

XX

f a ds 1c f c ds 2 c f c ds 3 c A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

Ý sgS 3

s

Ž ig .

3

f acc

3 s igf acc

XX

X

Ý

X

f c ds 1c f c ds 2 ds 3 A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

sgS 3 XX

½

g2

Ý Ž Fd

s1

5

F ds 2 . c d 3 A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,k 3n 3 . ,

sgS 2

Ž 5.48 .

where Ž F a . b c ' if b ac. Thus amplitude Ž5.10. can be put in the form of Eq. Ž5.46. with collinear factor 3g Splityg ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 . s g 2 n

Ý Ž Fd

s1

3g F ds 2 . c d 3 splityg ™ Ž ksns1 1 ,ksns2 2 ,k 3n 3 . . n

sgS 2

Ž 5.49 . 3g The splitting factors splityg ™ are the functions A, Eqs. Ž5.3. – Ž5.7., in the triple n collinear limit, up to singly divergent terms, and thus they fulfill the identities, Eq.

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

240

Ž5.47.. The splitting factors of PT type can be soon read off from Eqs. Ž5.3. – Ž5.6., while for the ones of non-PT type we note that the coefficients of Eq. Ž5.8. reduce to z1 z 3 a Ž k 1 ,k 2 ,k 3 . , z1 q z 3

™

b Ž k 1 ,k 2 ,k 3 .

™ y (zP z

g Ž k 1 ,k 2 ,k 3 .

™ (z sz z

1 2 ) H

1 2

3

w1 2x , d Ž 1,2,3 .

Ž 5.50 .

123

with

ž(

(

/

1

1

d Ž 1,2,3 . ' w 1 2 x z 1 ²1 3: q z 2 ²2 3: .

Ž 5.51 .

Thus we obtain q q splityg ™ 3 g Ž kq 1 ,k 2 ,k 3 . s 2

™ 3 g Ž ky1 ,kq2 ,kq3 . s 2

splitqg

y q splitqg ™ 3 g Ž kq 1 ,k 2 ,k 3 . s 2 q y splitqg ™ 3 g Ž kq 1 ,k 2 ,k 3 . s 2

(z

1 z3

²1 2:²2 3:

z 12

(z

1 z3

1 ²1 2:²2 3:

z 22

(z

1 z3

1 ²1 2:²2 3:

z 32

(z

1 z3

1 ²1 2:²2 3:

,

,

,

,

q y splityg ™ 3 g Ž kq 1 ,k 2 ,k 3 .

2 s s12 s23

s12 z 2

Ž 1 y z1 .

q

d Ž 1,2,3 . s123

2

q

(

z2 z1 z 3

Ž 1 y z 3 . d Ž 1,2,3 . ,

q q g™3g splityg ™ 3 g Ž ky Ž kq3 ,kq2 ,ky1 . , 1 ,k 2 ,k 3 . s splity

y q g™3g splityg ™ 3 g Ž kq Ž ky2 ,kq1 ,kq3 . y splityg ™ 3 g Ž kq1 ,kq3 ,ky2 . . 1 ,k 2 ,k 3 . s ysplity Ž 5.52 .

™

In the triple collinear limit of the NNLO impact factor g g ) q q g, the functions A , Eqs. Ž5.13. – Ž5.20., fulfill the relations A 2g ; g q q s yA1g ; g q q and A 4g ; g q q s yA 3g ; g q q, n2. n2 n2. and A 3g ; g q q Ž pan a ;k 1n 1 ,k 2n 2 ,ky s A1g ; g q q Ž pan a ;k 1n 1 ,ky ,k 2 . Thus amplitude Ž5.11. can 3 3 be put in the form of Eq. Ž5.46. with collinear factor g g qq

g qq g qq Splityg ™ Ž k 1n 1 ,k 2n 2 ,ky3 n 2 . s g 2 Ž lcld1 . d 3 d 2 splityg ™ Ž k 1n 1 ,k 2n 2 ,ky3 n 2 . n n

g qq q Ž l d1lc . d 3 d 2 splityg ™ Ž k 1n 1 ,ky3 n 2 ,k 2n 2 . , Ž 5.53 . n

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241

with y q splitqg ™ g q q Ž kq 1 ,k 2 ,k 3 . s 2

)

q y splitqg ™ g q q Ž kq 1 ,k 2 ,k 3 . s 2 z 3

z 23

1

z1

²1 2:²2 3:

(

z2

,

1

z 1 ²1 2:²2 3:

,

y q splityg ™ g q q Ž kq 1 ,k 2 ,k 3 . 2

d Ž 1,3,2 . w 1 2 x

2 sy

w 1 3 x s123

s12 s23

(z

q

2

q

d Ž 1,3,2 .

ž

z3

z 2 Ž z1 y z 3 .

(z

y

1

(

z1 z 2 w2 3x

w1 3x z 3

/

Ž yz 2 s13 q z 3 s23 q z1 z 2 s123 . , (z3 Ž 1 y z1 .

q y splityg ™ g q q Ž kq 1 ,k 2 ,k 3 . 2

d Ž 1,2,3 . w 1 3 x

2 sy s12 s23

q

w 1 2 x s123

d Ž 1,2,3 . Ž 1 y z 3 .

(z

q

(z

2

z 3 s12

Ž 1 y z1 .

1

.

Ž 5.54 .

™

Writing the functions A, Eqs. Ž5.23. – Ž5.30., in the triple collinear limit of the NNLO impact factor q g ) q g g, the amplitude Ž5.21. can be put in the form of Eq. Ž5.46. with collinear factor q™ qgg Splity Ž k 1n ,k 2n 2 ,k 3n 3 . s g 2 n

Ý Ž ld

s2

q™ qgg l ds 3 . d1 c splity Ž k 1n ,ksns2 2 ,ksns3 3 . , n

sgS 2

Ž 5.55 . with q™ qgg splity Ž kq1 ,kq2 ,kq3 . s y q™ qgg splitq Ž ky1 ,kq2 ,kq3 . s

2i

(z

3

1 ²1 2:²2 3:

2 iz 1

(z

3

1 ²1 2:²2 3:

,

,

q™ qgg splity Ž kq1 ,ky2 ,kq3 .

d Ž 1,3,2 .

2i s s12 s23 q

(z

1

z 2 s12

1 y z1

(

2

ž (z

w 1 3 x q (z 2 w 2 3 x / (z2 Ž 1 y z3 . d Ž 1,3,2. q w 1 3 x s123 (z3 1

q z 2 ²2 3 :w 1 3 x ,

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

242

™ q g g Ž kq1 ,kq2 ,ky3 . s

q splity

d Ž 1,2,3 .

2i

2

ž (z

w 1 2 x y (z 3 w 2 3 x / w 1 2 x s123

s12 s23

q

(z

1

Ž 1 y z 3 . d Ž 1,2,3 .

2

(z

q

(z

1

z 2 s23

.

1 y z1

Ž 5.56 .

™

3

In the triple collinear limit of the NNLO impact factor q g ) q Q Q, the functions qQQ Ž . can be put in the A Ž5.33. fulfill the relation A1q; q Q Q s A q; . Thus the amplitude 5.31 2 form of Eq. Ž5.46., with collinear factor q ™ qQQ qQQ Splity Ž k 1n ,k 2n 2 ,k 3n 3 . s 2 g 2 lad1 c lad 3 d 2 P splityq ™ Ž k 1n ,k 2n 2 ,k 3n 3 . n n

½

q ™ qQQ yd q Q l ad 3 c l ad1 d 2 P splity Ž k 3n 3 ,k 2n 2 ,k 1n 1 . , Ž 5.57 . n

5

where the second term occurs for the case of identical quarks, and c is the color index of the parent quark. The splitting factors are q ™ qQQ splity Ž kq1 ,ky2 ,kq3 . s

i s23

(z

ž™

1 z2

z3

q

d Ž 1,3,2 .

1 y z1

s123

/

,

q ™ qQQ splitq Ž 5.58 . Ž ky1 ,ky2 ,kq3 . s splityq q Q Q Ž kq1 ,ky3 ,kq2 . . f ™ f1 f 2 f 3 Ž n 1 n 2 n 3 . f ™ f 1 f 2 f 3 Ž yn 1 yn 2 yn 3 . The factor splitn k 1 ,k 2 ,k 3 can be obtained from splity n k 1 ,k 2 ,k 3 in Eqs. Ž5.52., Ž5.54., Ž5.56. and Ž5.58. by exchanging ² ij : with w ji x, and multiplying by a coefficient S, Eq. Ž3.7., for each quark pair the splitting factor includes. Using Eq. Ž5.49. and Eqs. Ž5.52. – Ž5.58., and summing over the two helicity states of partons 1, 2 and 3, we obtain, as in Section 4.4, the two-dimensional Altarelli–Parisi polarization matrix at fixed color and helicity of the parent parton,

Splitlf ™ f 1 f 2 f 3 Ž k 1n 1 ,k 2n 2 ,k 3n 3 . Split rf ™ f 1 f 2 f 3 Ž k 1n 1 ,k 2n 2 ,k 3n 3 .

Ý

)

n 1n 2 n 3

s d cc

X

4g4 2 s123

Plrf ™ f 1 f 2 f 3 ,

Ž 5.59 .

f ™ f1 f 2 f 3 f ™ f1 f 2 f 3 f ™ f1 f 2 f 3 f ™ f1 f 2 f 3 . ) where Pqq s Pyy , and Pyq s Ž Pqy . For splitting functions of q ™ q f2 f3 q™ q g g q™ q Q Q type P , namely for P , P and P q1 ™ q1 q 2 q 2 , where the last splitting function is for identical quarks, helicity conservation on the quark line sets the off-diagonal elements equal to zero. Averaging over the trace of matrix Ž5.59., i.e. over color and helicity of the parent parton, we obtain the unpolarized Altarelli–Parisi splitting functions w20x

1 2C

Ý

f1 f 2 f 3
nn 1n 2 n 3

4g4 2 s123

² P f ™ f1 f 2 f 3 : ,

Ž 5.60 .

with C defined below Eq. Ž4.22.. For ² P g ™ g 1 g 2 g 3 :, the sum over colors can be immediately done using Eq. Ž2.5., and it yields g1 g 2 g 3
Ý

3g
2

3

sgS 3

Ž 5.61 .

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243

with CnŽ Nc . as in Eq. Ž2.8.. Eq. Ž5.61. shows that for the purely gluonic unpolarized splitting function the color factorizes. f ™ f1 f 2 f 3 Since the averaged trace of P f ™ f 1 f 2 f 3 is ² P f ™ f 1 f 2 f 3 : s tr P f ™ f 1 f 2 f 3r2 s Pqq , f ™ f1 f 2 f 3 we have checked that for the diagonal elements, Pqq , our expressions agree with the unpolarized splitting functions of Ref. w20x by setting there the RS parameter e s 0. Finally, for the off-diagonal elements of the splitting functions of type P g ™ g f 2 f 3 we obtain g ™ g1 g 2 g 3 Pqy s CA2

Ý sgS 3

y

½

ss1s 2

(z

s 2 zs 3

ss1s 3 y

2

s123

y

2 w s 1 s 2 x zs 1 zs 2 ss 1s 2Ž 1 y zs 3 . zs 3

ž

Ds 2 Ds 3 y3 q

3

Ž 1 y zs . zs 3

zs1 zs 2 Ds22Ž 1 y 2 zs 3 .

Ž 1 y zs . zs

2 Ž 1 y zs 2 . zs 2

y

/

3

zs 1 zs 3 Ds23Ž 1 y 2 zs 2 .

Ž 1 y zs . zs

3

2

2

5

,

Ž 5.62 .

with

(

Di s w i j x z j q w i k x z k

(

with i , j,k s 1,2,3 and j,k / i

Ž 5.63 .

and g ™ g1 q2 q3 Pqy s

g ™ g 1 q 2 q 3 Ž nab . Ž C P g ™ g 1 q 2 q 3 Ž ab . q CA Pqy ., 2 F qy where the abelian and non-abelian terms are

1

g ™ g 1 q 2 q 3 Žab. Pqy s

2 s123 s12 s13

g ™ g 1 q 2 q 3 Žnab. Pqy s

Ý sgS 2

2 1

½ z D y 2(z 1

s123 s1 s 2 s1 s 3

½

2

5

z3 D2 D3 ,

1 y z 1 D 12 q zs 2 zs 3 Ds 2 Ds 3 2

(

2

q

2 w s 2 s 3 x zs 2 zs 3 s1 s 2 s1 s 3 ss2 2 s 3Ž 1 y z 1 . z 1

(

Ž 5.64 .

ž

y zs 2 zs 3 Ds 2 Ds 3 1 q

q

s1 s 3 ss 2 s 3

2 zs 2Ž zs 3 y z 1 .

Ž 1 y z1 . z1

yzs 2 Ds22 q

/5

.

2 Ds23 zs2 2 zs 3

Ž 1 y z1 . z1 Ž 5.65 .

We have checked that Eqs. Ž5.62. – Ž5.65. agree with the corresponding spin-correlated splitting functions of Ref. w21x after contracting them with a parent-gluon polarization as in Appendix E, and after setting the RS parameter e s 0.

6. Four-parton forward clusters The procedure of Sections 4 and 5 can be clearly extended to n-parton forward clusters. In a forward cluster there are one incoming and n outgoing partons. Thus, for

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

244

purely gluonic clusters there are 2 nq 1 helicity configurations. However, in the high-energy limit two of these are subleading, thus an n-gluon forward cluster contains 2Ž2 n y 1. helicity configurations. For n-parton forward clusters including q q pairs, all the helicity configurations are leading; then an easy counting yields 2 n helicity configurations for the one including a q q pair, 2 ny 1 for the one including two q q pairs, and so on. For n s 3, we obtain the helicity configurations dealt with in Section 5. 6.1. The NNNLO impact factor gg )

™ gggg

Here we analyse in detail the four-gluon forward cluster. We take the production of five gluons with momenta k 1 , k 2 , k 3 , k 4 and p bX in the scattering between two partons of momenta pa and p b , and we take partons k 1 , k 2 , k 3 and k 4 in the forward-rapidity region of parton pa ŽFig. 8a., < k 1 H < , < k 2 H < , < k 3 H < , < k 4 H < , < p bX H < .

y 1 , y 2 , y 3 , y4 4 y bX ;

Ž 6.1 .

Using Eqs. Ž2.1., Ž2.2. and Ž2.10. and the subamplitudes of non-PT type, with four gluons of q helicity and three gluons of y helicity w75x, we obtain A g g ™ 5 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 < p bnXbX , p bn b . s s s4 g5 C g ; g Ž p bn b ; p bnXbX . Ý A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 . < qH < 2 sgS 4 X

=tr Ž l al ds 1 l ds 2 l ds 3 l ds4 l bl b y l al ds 1 l ds 2 l ds 3 l ds4 l bl b

X

X

X

yl bl bl ds4 l ds 3 l ds 2 l ds 1 l a q l bl bl ds4 l ds 3 l ds 2 l ds 1 l a . X

qB g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 . tr Ž l al ds 1 l ds 2 l ds 3 l bl bl ds4 X

X

yl al ds 1 l ds 2 l ds 3 l bl bl ds4 y l bl bl ds 3 l ds 2 l ds 1 l al ds4 X

ql bl bl ds 3 l ds 2 l ds 1 l al ds4 . q D g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 . X

X

=tr Ž l al ds 1 l ds 2 l bl bl ds 4 l ds 3 y l al ds 3 l ds 4 l bl bl ds 2 l ds 1 . ,

Ž 6.2 .

with the sum over the permutations of the four gluons 1, 2, 3 and 4. From the PT subamplitudes Ž2.10. we obtain the functions of Žyqqqq . helicities A g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . s C g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . An Ž k 1 ,k 2 ,k 3 ,k 4 . , Ž 6.3 . where

n s sign Ž na q n 1 q n 2 q n 3 q n4 . and Aq Ž k 1 ,k 2 ,k 3 ,k 4 . s y2'2

qH k1H

(

x1

1

x 4 ²12:²23:²34:

and xi s

kq i q q q kq 1 q k2 q k3 q k4

,

i s 1,2,3,4

Ž x 1 q x 2 q x 3 q x 4 s 1. .

Ž 6.4 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

245

Fig. 8. Amplitude for the production of five gluons, with gluons k 1 , k 2 , k 3 and k 4 in the forward-rapidity region of gluon pa .

As in Eq. Ž5.6., the functions C g ;4 g are C g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . s

½

1 x i2

na s y n i s y i s 1,2,3,4

with n s q,

Ž 6.5 .

From the non-PT subamplitudes w75x we have obtained the functions of Žyyqqq . helicities. We do not reproduce them here because they are quite lengthy. They are available from the authors upon request. Using the UŽ1. decoupling equations for one and two photons, the functions B and D in Eq. Ž6.2. can be written as B g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . s y A g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . q A g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 4n 4 ,k 3n 3 . qA g ;4 g Ž pan a ;k 1n 1 ,k 4n 4 ,k 2n 2 ,k 3n 3 . q A g ;4 g Ž pan a ;k 4n 4 ,k 1n 1 ,k 2n 2 ,k 3n 3 . ,

Ž 6.6 .

D g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . s A g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . q A g ;4 g Ž pan a ;k 1n 1 ,k 3n 3 ,k 2n 2 ,k 4n 4 . qA g ;4 g Ž pan a ;k 3n 3 ,k 1n 1 ,k 2n 2 ,k 4n 4 . q A g ;4 g Ž pan a ;k 1n 1 ,k 3n 3 ,k 4n 4 ,k 2n 2 . qA g ;4 g Ž pan a ;k 3n 3 ,k 1n 1 ,k 4n 4 ,k 2n 2 . q A g ;4 g Ž pan a ;k 3n 3 ,k 4n 4 ,k 1n 1 ,k 2n 2 . .

Ž 6.7 .

In the quadruple collinear limit, k 1 < < k 2 < < k 3 < < k 4 , Section 6.4, the function A has a triple collinear divergence; the function B, whose gluon 4 is not color adjacent to gluons

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

246

1, 2 and 3, has only a double collinear divergence; the function D, where gluon 1 is adjacent to 2 and gluon 3 is adjacent to 4 but the pairs are not adjacent one to another, has two single collinear divergences. Using Eqs. Ž6.6. and Ž6.7., we can rewrite Eq. Ž6.2. as A g g ™ 5 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 < p bnXbX , p bn b .

½

s 2 s Ž ig .

=

1 t

X

4

X

XX

f a ds 1c f c ds 2 c f c ds 3 c f c

Ý

XX

ds 4 c

XXX

A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 .

sgS 4

5

X XXX

ig f b b c C g ; g Ž p bn b ; p bnXbX . ,

Ž 6.8 .

™

where the NNLO impact factor g ) g g g g g is enclosed in curly brackets, and includes 30 helicity configurations, in agreement with the counting above. 6.2. NNNLO impact factors in the high-energy limit The amplitude Ž6.8. has been computed in the kinematic limit Ž6.1., in which it factorizes into a four-gluon cluster and a LO impact factor connected by a gluon exchanged in the cross channel. In the limits y 1 , y 2 , y 3 4 y4 or y 1 , y 2 4 y 3 , y4 , or y 1 4 y 2 , y 3 , y4 , Eq. Ž6.8. must factorize further into a NNLO impact factor or into a NLO impact factor times a NLO Lipatov vertex, or into a NNLO Lipatov vertex ŽFig. 8., respectively. While the first two limits constitute necessary consistency checks, the last one allows us to derive the so far unknown NNLO Lipatov vertex for the production of three gluons along the ladder. In the limit y 1 , y 2 , y 3 4 y4 , the NNNLO impact factor, g ) g g g g g, in Eq. Ž6.8. factorizes into a NNLO impact factor, g ) g g g g, Eq. Ž5.10., convoluted with a multi-Regge ladder ŽFig. 8a.

™

™

lim y1,y 2,y 34y4

½

= Ž ig .

½

s Ž ig .

=

X

4

X

XX

f a ds 1c f c ds 2 c f c ds 3 c f c

Ý

t1

ds 4 c

XXX

A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 .

sgS 4 3

X

X

XX

f a ds 1c f c ds 2 c f c ds 3 c A g ;3 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 .

Ý sgS 3

1

XX

ig f c

XX

d4 c

XXX

C g Ž q1 ,k 4n 4 ,q2 . ,

5

5 Ž 6.9 .

with q1 s yŽ pa q k 1 q k 2 q k 3 ., q2 s p bX q p b , and with LO Lipatov vertex C g Ž q1 ,k 3n 3 ,q2 ., Eq. Ž4.10..

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

247

™ ™

In the limit y 1 , y 2 4 y 3 , y4 , the NNNLO impact factor in Eq. Ž6.8. factorizes into a NLO impact factor, g ) g g g, Eq. Ž4.8., times a NLO Lipatov vertex for production of two gluons g ) g ) g g Ž5.37., convoluted with a multi-Regge ladder ŽFig. 8b. lim y1,y 24y 3,y4

½

= Ž ig .

X

XX

f a ds 1c f c ds 2 c f c ds 3 c f c

Ý

XX

ds 4 c

XXX

A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 .

sgS 4

½

s Ž ig .

=

X

4

X

2

Ý

f a ds 1c f c ds 2 cA g ; g g Ž pan a ;ksns1 1 ,ksns2 2 .

sgS 2

1 t1

½Ž

ig .

X

2

Ý

XX

f c ds 3 c f c

XX

ds 4 c

5

5

XXX

5

A g g Ž q1 ,ksns3 3 ,ksns4 4 ,q2 . ,

sgS 2

Ž 6.10 .

with q1 s yŽ pa q k 1 q k 2 .. 6.3. The NNLO LipatoÕ Õertex In the limit y 1 4 y 2 , y 3 , y4 , the NNNLO impact factor in Eq. Ž6.8. factorizes into a NNLO Lipatov vertex convoluted with a multi-Regge ladder ŽFig. 8c. lim y14y 2,y 3,y4

½

= Ž ig .

X

4

X

XX

f a ds 1c f c ds 2 c f c ds 3 c f c

Ý

XX

ds 4 c

XXX

A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 .

sgS 4

5

s ig f a d1 c C g ; g Ž pan a ;k 1n 1 . =

1 t1

½Ž

ig .

3

X

Ý

X

XX

f c ds 2 c f c ds 3 c f c

XX

ds 4 c

XXX

A3 g Ž q1 ,ksns2 2 ,ksns3 3 ,ksns4 4 ,q2 .

™

sgS 3

5

Ž 6.11 .

with the NNLO Lipatov vertex, g ) g ) g g g, for the production of three gluons k 2 , k 3 and k 4 enclosed in curly brackets in the right-hand side, with q q ' A3 g Ž q1 ,kq 2 ,k 3 ,k 4 ,q 2 . s y2 2

(

x2

1

x 4 ²2 3:²3 4:

q q A3 g Ž q1 ,ky 2 ,k 3 ,k 4 ,q 2 .

s 2'2

q

½

y

< q1 H < 2 k 2 H Ž k 3)H . 2 Ž q2)H q k 4)H . x 2 s4 b bX k 2)H s23 Ž < k 3 H < 2 x 2 q < k 2 H < 2 x 3 . k2 H

(

s4 b bX ²3 4: k 2)H s23 x 3

Ž q1)H q2)H k 3)H ²3 4:

q1)H q2 H k2 H

,

Ž 6.12 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

248

(

q²2 4: Ž q1)H y k 2)H . Ž q2)H q k 4)H . w 2 3 x . x 3

(

yk 3)H Ž Ž q1)H y k 2)H . s23 q q1)H s34 . x 4 < q1 H < 2 k 2 H Ž k 3)H . 2 x 2 x 3

(

q

²3 4: k 2)H s23

Ž < k3H <

1 q ²3 4: k 2)H s23

( ž yq ž yk

(x

3 x4

2

x2 q < k2 H < 2 x3 . x4

(

ž y< k

2H

< 2 k 3)H

(

q x 2 yq1H Ž q1)H y k 2)H . k 3)H x 2 ) 1H

) 2 H k3H

(x

2

(

(

q ²2 3: k 3)H x 3 q k 2 H w 2 3 x x 3

( /

(

q²2 4: k 3)H x 4 q k 2)H Ž ²2 3: k 3)H q k 2 H w 2 3 x . x 3

ž

( ///

q²2 4: k 3)H x 4

q

q

< q2 H < 2 x 2 ²2 3: x 3 q ²2 4: x 4

( /

( . ž q (x q w 3 4 x (x / ²3 4: k (x < Ž k . k k (x ²2 3:²3 4: s234 x 4

2

q2 H k 2 H Ž q1)H y k 2)H s2 a aX s4 b bX

q

(

ž

< q1 H

) 2H

3

) 2H 2

2

2H

4

4

) ) 3H 4 H

3

²3 4: s23 Ž < q1 H yq2 H < 2 q s234 .Ž < k 3 H < 2 x 2 q < k 2 H < 2 x 3 . x 4

(

q

< q1 H < 2 k 2 H w 3 4 x ²2 4: k 4)H x 3 q ²2 3: k 3)H x 4

(

ž

(x

( /

2

2

x 3 x 4 ²3 4: s23 s234 Ž < q1 H yq2 H < q s234 .

w3 4x q

(

²3 4: s23 s234 x 2 x 3 x 4

( /

ž yq

) 1H k2 H x3

(

x 4 ²2 3: x 3

ž

( ž

q²2 4: x 4 q q2 H x 2 k 4 H Ž q2)H q k 4)H . x 2 x 3 q Ž < k 3 H < 2 x 2 q k 2 H Ž k 2)H y q1)H . x 3 . x 4

y q A 3 g Ž q1 , k q 2 ,k 3 ,k 4 ,q 2 .

'

s2 2

< q2 H <2k3H x 2

' 'x

< q2 H <2 y

3

²2 3 : k 2 H s 34 Ž 1 y x 2 .

'x

2

x 33

²2 3 : s 34 Ž 1 y x 2 .

y

q

< q 2 H < 2 Ž q1 H yk 2 H . k 2)H x 3 s 2 a aX k 2 H s 34 Ž 1 y x 2 .

< q 1 H < 2 Ž k 3 H . 2 k 2)H Ž q 2)H q k 4)H . x 2 s 4 b bX k 2 H s 23 Ž < k 3 H < 2 x 2 q < k 2 H < 2 x 3 .

//

5

,

Ž 6.13 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

q

249

Ž k 3 H . 2 x 2 Ž y Ž q 2)H q k 4)H . w2 3 x x 3 q q 1)H Ž q 2)H x 2 q w2 4 x x 4 . .

'

'

'

'

s4 b bX k 2 H s 23 x 3 q 1)H k 3 H w 2 4 x 2 ²2 3 : Ž q 1 H yk 2 H . x 2 q ²3 4 : k 2 H x 4

' '

Ž

q

' .

k 2 H s 23 s 34 s 234 x 3

'

'

' .

k 3 H k 2)H q 2)H k 2 H x 2 y q 1 H Ž q 2)H q k 4)H . x 2 q k 2 H w 2 4 x x 4

Ž

y

'

s 2 a aX s4 b bX k 2 H x 2

' Ž

'

'

'

k 2)H x 3 yq 1 H q 2 H k 4)H x 2 x 3 y Ž q 1 H yk 2 H . k 3 H w 3 4 x x 2 x 4 q q 2 H k 2 H w 2 4 x x 3 x 4 y

.

'

s 2 a aX k 2 H s 34 x 2 x 4 < q 1 H < 2 Ž k 3 H . 2 k 2)H x 2 x 3 q

'

²3 4 : k 2 H s 23 Ž < k 3 H < 2 x 2 q < k 2 H < 2 x 3 . x 4

' < w 2 4 x x Ž y ² 2 3 : 'x q ² 3 4 : 'x . ² 2 3 : s s 'x w 2 4 x 'x Ž ²3 4 : Ž q q k . 'x y ²2 3 : k 'x . s s s 'x x

< q2 H q

2

3

2

4

34 234

q2 H q

2

4

) 2H

3

) 4H

23 34 234

) 2H

2

2

4

4

< q 1 H < 2 Ž k 3 H . 3 k 2)H k 4)H x 2

y

'x

3 x4

²3 4 : k 2 H s 23 Ž < q 1 H yq 2 H < 2 q s 234 .Ž < k 3 H < 2 x 2 q < k 2 H < 2 x 3 .

< q 1 H < 2 Ž k 3 H . 2 w 2 4 x ² 3 4 : k 4)H x 2 y ² 2 3 : k 2)H x 4

'

Ž

y

' .

x 3 x 4 ²3 4 : k 2 H s 23 s 234 Ž < q 1 H yq 2 H < 2 q s 234 .

'

1 q

'

k 2 H s 23 s 34 x 2 x 4 x 3

Ž x 3 Ž yx 2 Ž k 3 H q 2 H k 3)H w2 4 x q k 3 H w2 3 xw3 4 x .

qq 2 H k 3 H k 2)H w 3 4 x x 2 x 3 y q 2 H < k 2 H < 2 w 2 4 x x 3

'

.

'

qq 1)H k 3 H x 2 q 2 H w 2 4 x x 3 q k 3 H w 3 4 x x 2 x 3 q k 3 H w 2 4 x x 4

Ž

Ž 6.14 .

,

..

q y A3 g Ž q1 ,kq 2 ,k 3 ,k 4 ,q 2 .

s 2'2

k 2)H

' Žq

s2 a aX ²2 3: k 2 H s34 x 2

1 H q2 H

'

²2 3: k 3)H x 2

'

'

y²2 4:w 3 4 x x 2 Ž q1 H y k 2 H .Ž q2 H q k 4 H . q k 2 H x 3 Ž q2 H s23 q Ž q2 H q k 4 H . s34 . . q

q1)H Ž q2 H q k 4 H . 2

Ž

q1 H k 3)H

'x

2

' .

y k 2 H w2 3x x 3

'

s2 a aX s4 b bX ²2 3: k 2 H x 3 q

< q2 H < 2 k 3 H x 2 x 3

'

q

²2 3: k 2 H s34 Ž 1y x 2 . q

< q1 H < 2 Ž k 4 H . 2 Ž ²3 4: k 3)H x 2 q²2 4: k 2)H x 3

'

' .

x 4 x 3 ²2 3:²3 4: k 2 H s234 Ž < q1 H y q2 H < 2 q s234 .

'

< q2 H < 2 Ž q1 H y k 2 H . k 2)H x 3 s2 a aX k 2 H s34 Ž 1y x 2 .

q

' 'x

s4 b bX k 4 H x 2

3

'

' . 'x

q Ž q2 H q k 4 H . Ž Ž ²3 4: k 3)H q k 4 H w 3 4 x . x 2 q²2 4: k 2)H x 3 ²2 3: k 2 H s34 x 4

4

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

250

< q2 H < 2 w 2 3 x Ž ²2 4: x 2 q²3 4: x 3

'

q

' . 'x

4

q

²2 3: s34 s234 w2 3x

ž

q ²2 3: k 2 H s34 s234 x 4

< q2 H < 2 x 2 x 3 x 4

'

²2 3: s34 Ž 1y x 2 .

< q1 H < 2 Ž k 4 H . 2 x 2 q q 2 H

/

y q1)H k 2 H k 4 H Ž k 4 H q q2 H x 4 .

ž

²2 4:< k 2 H < 2

'x

q

²3 4: k 2 H k 3)H

'x

2

3

/(

x 43

Ž 6.15 .

,

where in Eqs. Ž6.13. – Ž6.15. we have used the three-particle invariants, s2 a aX s Ž k 2 y q1 . 2 and s4 b bX s Ž k 4 q q2 . 2 . Eq. Ž6.11. must not diverge more rapidly than 1r< qi H < for < qi H < 0, with i s 1,2, in order for the related cross section not to diverge more than logarithmically. Since Eq. Ž6.11. is proportional to 1r< qi H < 2 , the NNLO Lipatov vertex must be at least linear in < qi H <,

™

lim A3 g Ž q1 ,k 1n 1 ,k 2n 2 ,k 3n 3 ,q2 . s O Ž < qi H < . ,

™0

i s 1,2,

Ž 6.16 .

< qiH<

which is fulfilled by Eqs. Ž6.12. – Ž6.15.. As a consistency check on Eq. Ž6.11., in the further limits y 2 4 y 3 , y4 or y 2 , y 3 4 y4 , the NNLO Lipatov vertex in Eq. Ž6.11. must factorize into a NLO Lipatov vertex convoluted with a multi-Regge ladder, lim y 24y 3,y4

½Ž

ig .

X

3

Ý

X

XX

f c ds 2 c f c ds 3 c f c

XX

ds 4 c

XXX

A3 g Ž q1 ,ksns2 2 ,ksns3 3 ,ksns4 4 ,q2 .

sgS 3

5

X

s ig f c d 2 c C g Ž q1 ,k 2n 2 ,q12 . =

1

½Ž

t 12

ig .

2

X

XX

f c ds 3 c f c

Ý

XX

ds 4 c

XXX

5

A g g Ž q12 ,ksns3 3 ,ksns4 4 ,q2 . ,

sgS 2

Ž 6.17 .

with q12 s q1 y k 2 , and lim y 2,y 34y4

½Ž

½

s Ž ig . =

2

ig .

X

3

Ý

X

XX

f c ds 2 c f c ds 3 c f c

XX

ds 4 c

XXX

A3 g Ž q1 ,ksns2 2 ,ksns3 3 ,ksns4 4 ,q2 .

sgS 3 X

X

XX

f c ds 2 c f c ds 3 c A g g Ž q1 ,ksns2 2 ,ksns3 3 ,q12 .

Ý sgS 2

1 t 12

ig f c

XX

d4 c

XXX

5

5

C g Ž q12 ,k 4n 4 ,q2 . ,

Ž 6.18 .

with q12 s q2 q k 4 . In the triple collinear limit, k 2 s z 2 P, k 3 s z 3 P and k 4 s z 4 P, with z 2 q z 3 q z 4 s 1, the coefficients of the NNLO Lipatov vertex Ž6.12. – Ž6.15. reduce to the splitting functions Ž5.52., and amplitude Ž6.11. factorizes into a multi-Regge amplitude Ž4.11. times a double-collinear factor Ž5.49. lim A g g ™ 5 g Ž pan a ,k 1n 1 < k 2n 2 ,k 3n 3 ,k 4n 4 < p bnXbX , p bn b .

k2 < < k3< < k4

3g s Ý A g g ™ 3 g Ž pan a ,k 1n 1 < P n < p bnXbX , p bn b . P Splityg ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 . . n

n

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

251

6.4. NNNLO impact factors in the quadruple collinear limit In the quadruple collinear limit, k i s z i P, with z 1 q z 2 q z 3 q z 4 s 1 a generic amplitude is expected to factorize as lim k1< < k 2 < < k 3< < k 4

A . . . d1 d 2 d 3 d 4 . . . Ž . . . ,k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 , . . . .

f1 f 2 f 3 f4 s Ý A . . . c . . . Ž . . . , P n , . . . . P Splityf ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . . n

Ž 6.19 .

n

Accordingly, we show that we can write Eq. Ž6.8. as lim k1< < k 2 < < k 3< < k 4

n n X b,p b. A g g ™ 5 g Ž pan a ,k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 < py b b

n n g™4g X b , p b . P Split s A g g ™ g g Ž pan a , Pyn a < py Ž k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . , b b na

Ž 6.20 .

by taking the quadruple collinear limit of the NNNLO impact factor. In the quadruple collinear limit, the functions A g ;4 g of Eq. Ž6.5. yield a cubic divergence as s1234 s Ž k 1 q k 2 q k 3 q k 4 . 2 0 or si jk 0, or si j 0 with i, j,k s 1,2,3,4. Analogously to Section 5.6, a function A g ;4 g differs from its reflection by a term which contains only a quadratic divergence in the vanishing invariants. Using this property and Eqs. Ž6.6. and Ž6.7., we obtain a reflection identity and dual Ward identities, up to quadratically divergent terms,

™

™

™

A g ;4 g Ž pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . s yA g ;4 g Ž pan a ;k 4n 4 ,k 3n 3 ,k 2n 2 ,k 1n 1 . , A

g ;4 g

pan a ;k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4

g ;4 g

Ž 6.21 .

pan a ;k 1n 1 ,k 2n 2 ,k 4n 4 ,k 3n 3

. qA Ž . g ;4 g n n n n n g ;4 g n n n n qA Ž pa ;k 1 ,k 4 ,k 2 ,k 3 . q A Ž pa ;k 4 ,k 1 ,k 2 ,k 3n . s 0, A g ;4 g Ž pan ;k 1n ,k 2n ,k 3n ,k 4n . q A g ;4 g Ž pan ;k 1n ,k 3n ,k 2n ,k 4n . q A g ;4 g Ž pan ;k 3n ,k 1n ,k 2n ,k 4n . q A g ;4 g Ž pan ;k 1n ,k 3n ,k 4n ,k 2n . q A g ;4 g Ž pan ;k 3n ,k 1n ,k 4n ,k 2n . q A g ;4 g Ž pan ;k 3n ,k 4n ,k 1n ,k 2n . s 0 . Ž

a

a

1

1

2

4

2

3

3

a

4

a

1

4

3

1

2

2

3

Ž 6.22 .

4

a

3

1

2

4

a

1

3

4

2

a

3

1

4

2

a

3

4

1

2

Ž 6.23 .

We note, however that the last identity is not independent from the first two. Using the identities Ž6.21. – Ž6.23. in Eq. Ž6.8., we can factorize the color structure on a leg

Ž ig .

4

X

Ý

X

XX

f a ds 1c f c ds 2 c f c ds 3 c f c

XX

ds 4 c

XXX

A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,ksns4 4 .

sgS 4

s igf acc

XXX

½

g3

Ý Ž Fd

s1

F ds 2 F ds 3 . c d 4 A g ;4 g Ž pan a ;ksns1 1 ,ksns2 2 ,ksns3 3 ,k 4n 4 .

sgS 3

5

Ž 6.24 .

thus amplitude Ž6.8. can be put in the form of Eq. Ž6.20. with collinear factor 4g Splityg ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . n

sg3

Ý Ž Fd sgS 3

s1

4g F ds 2 F ds 3 . c d 4 splityg ™ Ž ksns1 1 ,ksns2 2 ,ksns3 3 ,k 4n 4 . . n

Ž 6.25 .

4g The splitting factors splityg ™ are the functions A of Section 6.1 in the quadruple n collinear limit, up to quadratically divergent terms, and thus they fulfill the identities, Eqs. Ž6.21. – Ž6.23.. The splitting factors of PT type can be soon read off from Eqs.

252

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

Ž6.3. – Ž6.5., while the ones of non-PT type can be given in terms of three functions of the collinear momenta, 1 1 q q q ' splityg ™ 4 g Ž kq , 1 ,k 2 ,k 3 ,k 4 . s 2 2 ² :² z 1 z 4 1 2 2 3:²3 4:

(

q q q ' splitqg ™ 4 g Ž ky 1 ,k 2 ,k 3 ,k 4 . s 2 2 y q q ' splitqg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s 2 2 q y q ' splitqg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s 2 2 q q y ' splitqg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s 2 2

z 12

(z

1

1 z4

²1 2:²2 3:²3 4:

z 22

(z

1

1 z4

²1 2:²2 3:²3 4:

z 32

(z

1

1 z4

²1 2:²2 3:²3 4:

z 42

(z

1

1 z4

²1 2:²2 3:²3 4:

,

,

,

,

q q q B1 Ž 4,3,2,1 . , splityg ™ 4 g Ž ky 1 ,k 2 ,k 3 ,k 4 . s yB

y q q splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s B1 Ž 4,3,1,2 . q B1 Ž 4,1,3,2 . q B1 Ž 1,4,3,2 . ,

q y q B1 Ž 1,2,4,3 . y B1 Ž 1,4,2,3 . y B1 Ž 4,1,2,3 . , splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s yB q q y splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s B1 Ž 1,2,3,4 . ,

y q q B2 Ž 4,3,2,1 . , splityg ™ 4 g Ž ky 1 ,k 2 ,k 3 ,k 4 . s yB q y q splityg ™ 4 g Ž ky 1 ,k 2 ,k 3 ,k 4 . s B3 Ž 1,2,3,4 . ,

q q y B3 Ž 1,2,4,3 . q B2 Ž 3,2,4,1 . q B2 Ž 3,2,1,4 . , splityg ™ 4 g Ž ky 1 ,k 2 ,k 3 ,k 4 . s yB y y q B2 Ž 1,4,2,3 . q B3 Ž 3,4,2,1 . y B2 Ž 4,1,2,3 . , splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s yB y q y B3 Ž 4,3,2,1 . , splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s yB q y y splityg ™ 4 g Ž kq 1 ,k 2 ,k 3 ,k 4 . s B2 Ž 1,2,3,4 .

Ž 6.26 .

with B1 Ž 1,2,3,4 . s

2 '2

z 3 d Ž 1,2,3 .

y

(z

²1 2: ²2 3: s34

(

z3

q

z1 z 4

1

Ž z 3 q z 4 . w1 2x

y

(z

2

z 3 z 4 ²1 2:

Ž 1 y z1 . Ž z 3 q z 4 .

Ž 1 y z 4 . e Ž 1,2,3,4 . 2

y

e Ž 1,2,3,4 . Ž ²2 4: w 1 2 x q ²3 4: w 1 3 x . w 3 4 x s1234 s234 2

(

q

q

(z

z 2 e Ž 1,2,3,4 . w 2 3 x z1 4

s234

ž(

²1 2: w 2 3 x z 1 ²1 4: y Ž 2 y z 1 . e Ž 1,2,3,4 .

Ž 1 y z1 . s234

/

,

Ž 6.27 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

B2 Ž 1,2,3,4 . s

(

2'2

z2 z3

q

w 3 4 x s23 Ž 1 y z1 . (z 4

(z

2

253

Ž z1 q z 2 . d Ž 1,2,3 .

(z

1 z4

s12

2

y

d Ž 1,2,3 . e Ž 1,2,3,4 . w 3 4 x

( d Ž 1,2,3 . ž (z

s1234 z 4 s12 s123

q

1

(

/ (

d Ž 3,4,1 . q z 2 d Ž 3,4,2 . q z 1 z 2 z 4 s12 s34

(

s1234 z 4 ²1 2: w 3 4 x q

(

z 1 z 2 d ) Ž 3,4,2 .

y

z 1²3 4: w 1 2 x Ž ²1 3: w 2 3 x q ²1 4: w 2 4 x .

Ž 1 y z1 . s234

s1234 s234

,

Ž 6.28 . B3 Ž 1,2,3,4 .

'

2'2

'z

s12 s23 s34

(

z2

y

y

ž'

z2 z2 z3

s

z1 z 4

1

z1 ²2 3: s34

Ž z1 q z 2 .

y

'z

1y z 4

3

²1 2: s23

z3 q z4

Ž 'z 2 z 3 ²1 2:y z 4²1 3: . d Ž2,4,3. y

/

'

z 2 Ž z 3 e Ž 1,2,4,3 . s23 y²2 3:²3 4:w 2 4 x . Ž z1 q z 2 .

Ž 'z 2 ²1 2:d Ž2,4,3. 2q'z1 ²1 3: Ž z 2 ²1 2:²2 3:q'z 3 'z 4 ²1 3:²3 4:q z 4²1 4:²3 4: . w2 4x 2 .

'z z 2 s23 s34 d ) Ž 1,3,2 .

q

'z

1

'z

s1234 s123

(

y

s12 d Ž 2,4,3 . 2 e Ž 2,3,4,1 .

4

s123

s1234 z1 s234

z3 z1

z 2 ²1 3: s234

²1 3:2 w 2 4 x Ž e Ž 1,2,4,3 . w 2 3 x y e Ž 2,3,4,1 . w 1 2 x .

z 2 ²1 3:2 s34 z1

(

q

'

Ž 1y z 4 . w 2 3 x s123

s34

q

q

1 s1234

Ž 6.29 .

Ž y'z 2 e ) Ž2,3,4,1. q'z1 z 4 w2 4x .

with d Ž1,2,3. as in Eq. Ž5.51., and

(

(

(

e Ž 1,2,3,4 . s z 1 ²1 4: q z 2 ²2 4: q z 3 ²3 4: .

Ž 6.30 .

As in Section 5.6, summing over the helicities of gluons 1, 2, 3 and 4, one can obtain the two-dimensional polarization matrix,

Ý

4g 4g Splityg ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . Splityg ™ Ž k 1n 1 ,k 2n 2 ,k 3n 3 ,k 4n 4 . n n

)

n 1n 2 n 3 n4

s d cc

X

8 g6 3 s1234

Plrg ™ 4 g ,

Ž 6.31 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

254

g™4g g™4g g™4g g ™ 4 g .) where Pqq s Pyy , and Pyq s Ž Pqy . Averaging then over the trace of matrix Ž6.31., i.e. over color and helicity of the parent gluon, one can obtain the unpolarized Altarelli–Parisi gluon triple-splitting function 4g
1 2 Ž Nc2 y 1 .

Ý nn 1n 2 n 3 n4

8 g6 3 s1234

²P g™4g: ,

Ž 6.32 .

g™4g with ² P g ™ 4 g : s Pqq . As in Section 5.6, the sum over colors can be done using Eq. Ž2.5., and we obtain 4g
C5 Ž Nc . s 4C

Ý

4g
2

3

4

Ž 6.33 .

sgS 4

with C5 Ž Nc . as in Eq. Ž2.8.. It is then clear that for the splitting functions P g ™ n g , with n ) 4, the color will not factorize since LCA, Eq. Ž2.5., is not exact any more. We do g™4g g™4g and Pqy , all the information about them being already not compute here Pqq contained in Eqs. Ž6.26. – Ž6.30..

7. Conclusions In this paper, the structure of QCD amplitudes in the high-energy limit and in the collinear limit has been explored beyond NLO. We have computed forward clusters of three partons and four gluons, which in the BFKL theory constitute the tree parts of NNLO and NNNLO impact factors for jet production. In the BFKL theory the NNLO impact factors could be used to compute jet rates at NNLL accuracy. In Sections 5.1, 5.2, 5.3 and 5.4, we have computed the tree parts of the NNLO impact factors for all the parton flavors. On these we have performed in Section 5.5 a set of consistency checks in the high-energy limit, and we have obtained in the triple collinear limit ŽSection 5.6. the polarized, the spin-correlated and the unpolarized double-splitting functions. The last two agree with previous calculations by Catani–Grazzini and Campbell–Glover, respectively. They can be used to set up general algorithms to compute jet rates at NNLO. From the four-gluon forward cluster we have obtained in Section 6.1 the tree part of the purely gluonic NNNLO impact factor. In the quadruple collinear limit, this yields ŽSection 6.4. the purely gluonic unpolarized triple-splitting functions. They could be used to compute the three-loop Altarelli–Parisi evolution, or to compute jet rates at NNNLO. In addition, by separating a central cluster of three gluons out of the four-gluon forward cluster, we have computed the emission of three gluons along the ladder, Eqs. Ž6.11. – Ž6.15., which contributes to the NNLO Lipatov vertex. This constitutes one of the universal building blocks in an eventual construction of a BFKL resummation at NNLL accuracy. Finally, inspired by the color structure in the high-energy limit, we have found a compact color decomposition of the tree multigluon amplitudes in terms of the linearly independent subamplitudes only, Eq. Ž2.9.. It would be interesting to analyse whether this structure generalizes to multigluon amplitudes at one loop, and beyond. The decomposition in rapidity of amplitudes in terms of gauge-invariant parton clusters performed in this work suggests naturally a modular decomposition of a generic

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

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multiparton amplitude, where each module is an n-parton cluster. Such an approximation could be tested against existing approximations of multiparton amplitudes w79,80x. In the high-energy limit, the cluster decomposition seems superior, in that it does not use only PT-type subamplitudes, like the Kunszt–Stirling approximation w79x, and within a cluster it is not limited to collinear kinematics, like the Maxwell approximation w80x.

Acknowledgements We would like to thank Stefano Catani, Walter Giele, David Kosower and Zoltan Trocsanyi for discussions. We are particularly grateful to Lance Dixon for his valuable insight.

Appendix A. Multiparton kinematics We consider the production of n partons of momentum pi , with i s 1, . . . ,n and n 0 2, in the scattering between two partons of momenta pa and p b5 . Using light-cone coordinates p "s p 0 " pz , and complex transverse coordinates ) ) p H s p x q ip y , with scalar product 2 p P q s pqqyq pyqqy p H q H yp H q H , the 4momenta are q q pa s Ž pq a r2,0,0, pa r2 . ' Ž pa ,0;0,0 . , y y p b s Ž py b r2,0,0,y p b r2 . ' Ž 0, p b ;0,0 . , y q y pi s Ž Ž pq i q pi . r2,Re w pi H x ,Im w pi H x , Ž pi y pi . r2 .

' Ž < pi H < e y i , < pi H < eyy i ; < pi H
Ž A.1 .

where y is the rapidity. The first notation in Eq. ŽA.1. is the standard representation p m s Ž p 0 , p x , p y , p z ., while in the second we have the q and - components on the left of the semicolon, and on the right the transverse components. In the following, if not differently stated, pi and pj are always understood for 1 ( i, j ( n. From momentum conservation, n

0s

Ý pi H , is1 n

pq a sy

Ý pqi , is1 n

py b sy

Ý pyi ,

Ž A.2 .

is1

5

By convention we consider the scattering in the unphysical region where all momenta are taken as outgoing, and then we analytically continue to the physical region where pa0 - 0 and p b0 - 0. Thus partons are ingoing or outgoing depending on the sign of their energy. Since the helicity of a positive-energy Žnegative-energy. massless spinor has the same Žopposite. sign as its chirality, the helicities assigned to the partons depend on whether they are incoming or outgoing. Our convention is to label outgoing Žpositive-energy. particles with their helicity; so if they are incoming the actual helicity and charge is reversed.

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

256

the Mandelstam invariants may be written as y y q ) ) si j s 2 pi P pj s pq i p j q pi p j y pi H p j H y pi H p j H so that n

s s 2 pa P p b s

y pq i pj ,

Ý i , js1

n

Ý pyi pqj ,

s ai s 2 pa P pi s y

js1 n

Ý pqi pyj .

s b i s 2 p b P pi s y

Ž A.3 .

js1

Massless Dirac spinors c " Ž p . of fixed helicity are defined by the projection 1 " g5 c "Ž p. s c Ž p. , Ž A.4 . 2 with the shorthand notation c " Ž p . s < p " :, c "Ž p. s² p "< , ² pk : s ² p y < k q : s cy Ž p . cq Ž k . ,

w pk x s ² p q < k y : s cq Ž p . cy Ž k . .

Ž A.5 .

Using the chiral representation of the g-matrices,

g 0s

ž

0 I

I , 0

g is

/

ž

0 si

ys i , 0

/

Ž A.6 .

and the normalization condition ² p " < gm < p " : s 2 pm ,

Ž A.7 . if

and the complex notation p H s < p H < e , the spinors for the momenta ŽA.1. are q i

(p cq Ž pi . s

 0

e if i ,

y i

(p

cy Ž pi . s

0 0

 0  0

cq Ž p b . s yi

(

0 0

6

(

0 0 cy Ž pa . s i 0 y y pq a

,

0 y py b

 0 (

q a

0 0 0

0 0 y yi f i , pi e

y pq i

(y p

cq Ž pa . s i

6

 0  0 ,

(

,

cy Ž p b . s yi

0 0 y py b

(

.

Ž A.8 .

0

The spinors of the incoming partons must be continued to negative energy after the complex conjugation. . For instance, cq Ž pa . s iŽ y pq a ,0,0,0 .

'

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

257

Using the above spinor representation, the spinor products for the momenta ŽA.1. are

( (

pq j

² pi p j : s pi H

² pa pi : s yi

pq i

y pj H

ypq a

)

pq i pq j

,

pi H ,

pq i

q ² pi p b : s i y py b pi ,

(

² pa p b : s y'sˆ ,

Ž A.9 . 2

where we have used the mass-shell condition < pi H < Note that in the present convention the spinors ŽA.8. and the spinor products ŽA.9. differ by phases with respect to the same in Ref. w74x. We consider also the spinor products ² pi q < g P p k < pj q :, which in the spinor representation ŽA.8. take the form y s pq i pi .

² pi q < g P p k < p j q : 1 s

q pq i pj

(

Ž pqi pqj pyk y pqi pj H pk)H y pi)H pqj pk H qpi)H pj H pqk . , ypq a

( (

² pi q < g P p j < p a q : s i

² pi q < g P pj < p b q : s yi

pq i

Ž pqi pyj y pi)H pj H . ,

ypy b pq i

;k ,

; j,

Ž ypqi pj)H q pi)H pqj . ,

; j.

Ž A.10 .

The spinor products fulfill the identities Ž i ' pi , j ' pj . ² ij : s y² ji : ,

w ij x s y w ji x , ² ij :) s sign Ž pi0 pj0 . w ji x , )

Ž ² i q < g m < j q : . s sign Ž pi0 pj0 . ² j q < g m < i q : , ² ij : w ji x s 2 pi P pj s sˆi j ² i q < ku < j q : s w ik x ² kj : , ² i y < ku < j y : s ² ik : w kj x , ² ij :² kl : s ² ik :² jl : q ² il :² kj : ,

w ij xw kl x s w ik xw jl x q w il xw kj x and if

Ý nis1

Ž A.11 .

pi s 0 then

n

Ý w ji x ² ik : s 0 . is1

Ž A.12 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

258

Throughout the paper the following representation for the gluon polarization is used:

em" Ž p,k . s "

² p " < gm < k " :

'2 ² k . < p " : ,

Ž A.13 .

which enjoys the properties

em" ) Ž p,k . s em. Ž p,k . , em" Ž p,k . P p s em" Ž p,k . P k s 0 , emn Ž p,k . ern ) Ž p,k . s ygm r q

Ý

pm kr q pr km

ns"

pPk

,

Ž A.14 .

where k is an arbitrary light-like momentum. The sum in Eq. ŽA.14. is equivalent to use an axial, or physical, gauge.

Appendix B. Multi-Regge kinematics In the multi-Regge kinematics, we require that the gluons are strongly ordered in rapidity and have comparable transverse momentum, < p 1 H < , . . . , < pn H < .

y 1 4 . . . 4 yn ;

Ž B.1 .

Momentum conservation ŽA.2. then becomes n

0s

Ý pi H , is1

q pq a , yp 1 , y py b , ypn .

Ž B.2 .

The Mandelstam invariants ŽA.3. are reduced to, y s s 2 pa P p b , pq 1 pn , y s ai s 2 pa P pi , ypq 1 pi , y sb i s 2 p b P pi , ypq i pn ,

si j s 2 pi P pj , < pi H < < pj H < e < y iyy j <

Ž B.3 .

to leading accuracy. The spinor products ŽA.9. become pq i

) (

² pi p j : , y

pq j

² pa pi : , yi

pj H

pq a pq i

for yi ) yj ,

pi H ,

y ² pi p b : , i pq i pn ,

(

y ² pa p b : , y pq 1 pn .

(

Ž B.4 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

259

Appendix C. NLO multi-Regge kinematics We consider the production of n partons of momenta p 1 , . . . , pn , with partons 1 and 2 in the forward-rapidity region of parton pa , < p 1 H < , < p 2 H < , . . . , < pn H < . y 1 , y 2 4 y 3 4 . . . 4 yn ; Ž C.1 . Momentum conservation ŽA.2. becomes n

0s

Ý pi H , is1

q q pq a , y p1 q p 2 y py b , ypn .

Ž

., Ž C.2 .

The spinor products ŽA.9. become

(Ž p

q q 1 q p2

² pa p b : s y's , y

( (

² pa pn : s yi

² pa p k : s yi

ypq a pq n ypq a pq k

pnH , i

. py n ,

pn H < pn H <

(

p k H , yi

q q y ² p k p b : s i y py b p k , i p k pn ,

² pn

( p : s i(y p

y q b pn

b

( ( (

² p k pn : s p k H

² p1 p 2 : s p1 H

² p k pi : s p k H

(

pq n pq k pq 2 pq 1

pq i pq k

² pa p b : , q pq 1 q p2

pq k

pk H ,

k s 1, . . . ,n y 1,

k s 1, . . . ,n y 1,

, i < pn H < ,

( ( (

y pn H

y p2 H

y pi H

pq k pq n pq 1 pq 2

pq k pq i

, ypn H

(

pq k

,

pq n

k s 1, . . . ,n y 1,

,

, ypi H

(

pq k pq i

,

k s 1,2;i s 3, . . . ,n y 1 .

Ž C.3 . which differ by phases with respect to the same spinor products in Ref. w39x because of the convention for the spinor representation we use in Section A.

Appendix D. NNLO multi-Regge kinematics The extension to the production of n partons of momenta p 1 , . . . , pn , with partons 1, 2 and 3 in the forward-rapidity region of parton pa , < p 1 H < , < p 2 H < , . . . , < pn H < , y 1 , y 2 , y 3 4 y4 4 . . . 4 yn ; Ž D.1 . is straightforward. We mention it here because by taking the further limit y 1 4 y 2 , y 3 , one obtains the kinematics of the NLO Lipatov vertex ŽSection 5.5..

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

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With Eq. ŽD.1., momentum conservation ŽA.2. becomes n

0s

Ý pi H , is1

q q q pq a , y p1 q p 2 q p 3 y py b , ypn .

Ž

., Ž D.2 .

The spinor products ŽA.9. become

(Ž p

q q q 1 q p2 q p3

² pa p b : s y's , y

( (

² pa pn : s yi

² pa p k : s yi

ypq a pq n ypq a pq k

pnH , i

pn H < pn H <

(

p k H , yi

q q y ² p k p b : s i y py b p k , i p k pn ,

² pn

( p : s i(y p

(

y q b pn

b

( (

² p k pn : s p k H

² p k pi : s p k H

pq n pq k pq i pq k

. py n ,

² pa p b : , q q pq 1 q p2 q p3

pq k

pk H ,

k s 1, . . . ,n y 1,

k s 1, . . . ,n y 1,

, i < pn H < ,

( (

y pn H

y pi H

pq k pq n

pq k pq i

( (

, ypn H

, ypi H

pq k

,

pq n

pq k pq i

k s 1, . . . ,n y 1,

,

k s 1,2,3; i s 4, . . . ,n y 1 ,

Ž D.3 .

while the others spinor products remain unchanged. The spinor products ŽD.3. generalize straightforwardly to the kinematics Ž6.1.. Appendix E. The Sudakov parametrization We want to elucidate the relationship between our parametrization of the momenta and the one of Ref. w21x. Recalling the last of Eqs. ŽA.1., we can write, pi s

x i Pq 2

Ž 1,0,0,1 . q Ž 0,Re w pi H x ,Im w pi H x ,0 . q

< pi H < 2 2 x i Pq

Ž 1,0,0,y 1 . ,

Ž E.1 .

where P m is the sum of the three momenta, the x i are the momentum fractions and we y < <2 used the mass-shell condition pq i pi s pi H . This is exactly what is obtained from the general Sudakov parametrization of Ref. w21x, pim s x i

p

m

m qkH iy

2 kH i

nm

xi 2 pPn

through the following choices for the lightlike vectors: Pq pms Ž 1,0,0,1 . and n m s Ž 1,0,0,y 1 . , 2

Ž E.2 .

Ž E.3 .

V. Del Duca et al.r Nuclear Physics B 568 (2000) 211–262

261

and the identification m kH i s Ž 0,Re w pi H x ,Im w pi H x ,0 . .

Ž E.4 .

The spin-correlated splitting functions of Ref. w21x are expressed in terms of the vectors m m k˜ i defined as k˜ im s k H i y z i P H , where, as in our case, the z i variables represent the momentum fractions in the collinear limit. In order to compare Eq. Ž5.59. with the spin-correlated splitting functions of Ref. w21x, we must project the latter onto the helicity basis, namely to contract them with the polarization vector, Eq. ŽA.13., 1 em" Ž P , n . s Ž E.5 . '2 Ž 0,1,. i ,0 . . The contraction of the k˜ im vectors with eq is zi w i j x z j q w i l x zl , k˜ i P eqs 2

(

ž (

( /

Ž E.6 .

with i, j,l s 1,2,3 and j,l / i, with the analogous expressions for ey obtained by complex conjugation. For the off-diagonal terms, P g ™ g f 2 f 3 , we find a relative minus sign between the results of Ref. w21x and ours, which, however, has no physical relevance. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x

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