Gluon Regge trajectory at two loops from Lipatov's high energy effective action
G. Chachamis, M. Hentschinski, J. D. Madrigal, A. Sabio Vera
GGluon Regge trajectory at two loops from Lipatov’shigh energy effective action
G. Chachamis , M. Hentschinski , J. D. Madrigal Mart´ınez , A. Sabio Vera Instituto de F´ısica Corpuscular UVEG/CSIC, E-46980 Paterna (Valencia), Spain. Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA. Instituto de F´ısica Te´orica UAM/CSIC, Nicol´as Cabrera 15 &Universidad Aut´onoma de Madrid, C.U. Cantoblanco, E-28049 Madrid, Spain.
Abstract
We present the derivation of the two-loop gluon Regge trajectory using Lipatov’s highenergy effective action and a direct evaluation of Feynman diagrams. Using a gauge invari-ant regularization of high energy divergences by deforming the light-cone vectors of theeffective action, we determine the two-loop self-energy of the reggeized gluon, after com-puting the master integrals involved using the Mellin-Barnes representations technique.The self-energy is further matched to QCD through a recently proposed subtraction pre-scription. The Regge trajectory of the gluon is then defined through renormalization ofthe reggeized gluon propagator with respect to high energy divergences. Our result is inagreement with previous computations in the literature, providing a non-trivial test ofthe effective action and the proposed subtraction and renormalization framework.
I Introduction
Current applications of high energy factorization to QCD phenomenology range from theanalysis of perturbative observables, such as dijets widely separated in rapidity [1], overtransverse momentum dependent parton distribution functions in the low x region [2], up tothe study of phenomena in heavy ion collisions [3]. Their common base is the factorization ofQCD scattering amplitudes in the limit of asymptotically large center of mass energy, togetherwith the resummation of large logarithmic contributions using the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [4, 5]. Recent phenomenological use of the BFKL resummationcan be found in the analysis of the combined HERA data on the structure function F and F L [6, 7], the study of di-hadron spectra in high multiplicity distributions at the Large HadronCollider [8] or the production of high p T dijets [9, 10, 11] , widely separated in rapidity.In the present work we discuss Lipatov’s high energy effective action [12] and show thatit can serve as a useful tool to reformulate the high energy limit of QCD as an effective fieldtheory of reggeized gluons. While the determination of the high energy limit of tree-level1 a r X i v : . [ h e p - ph ] J u l mplitudes has been well understood for quite some time within this framework [13], it wasonly until recently that progress in the calculation of loop corrections has been achieved.Starting with [14] and extended in [15], a scheme has been developed that comprises the reg-ularization, subtraction and renormalization of high energy divergences. This scheme thenallowed to successfully derive forward jet vertices for both quark and gluon initiated jets atNLO accuracy from Lipatov’s high energy effective action.Here we extend this program to the calculation of the 2-loop gluon Regge trajectory. Thelatter provides an essential ingredient in the formulation of high energy factorization andreggeization of QCD amplitudes at NLO. It has been originally derived in [16, 17] using s -channel unitarity relations. The result was then subsequently confirmed in [18], clarifying anambiguity in the non-infrared divergent contributions of [19]. The original result was furtherverified by explicitly evaluating the high energy limit of 2-loop partonic scattering amplitudes[20]. While the explicit result for the 2-loop gluon Regge trajectory is by now firmly estab-lished, our calculation provides an important confirmation of its universality: unlike previouscalculations, the effective action defines the Regge trajectory of the gluon without makingany reference to a particular QCD scattering process.For the development of a consistent formulation of the effective action, the calculationof the 2-loop gluon trajectory provides an essential and non-trivial test of our scheme. Thelatter has been set up in [21], where partial results, addressing the flavor dependent parts ofthe gluon Regge trajectory have been already presented. The current paper addresses thegluon corrections, which are considerable more complicated than their fermionic counterparts.The outline of this paper is as follows: Sec. II provides a short introduction to Lipatov’seffective action and a list of necessary Feynman rules, together with a discussion of ourregularization and the employed pole prescription. Sec. III recalls the scheme we follow inthe derivation of the gluon Regge trajectory, which has been originally introduced in [21].Sec. IV provides details about our calculation of the 2-loop reggeized gluon self-energy fromthe effective action, together with our result for the 2-loop gluon Regge trajectory. Sec. Vcontains our conclusions and an outlook on future projects. Several technical details of ourcalculations are summarized in the appendix. II Lipatov’s high energy effective action
The effective action [12] describes interactions which are local in rapidity, i.e. which arerestricted to an interval of narrow width ( η ) in rapidity space. The entire dynamics whichextends over rapidity separations larger than η , is on the other hand integrated out andtaken into account through universal eikonal factors. To reconstruct from this setup QCDamplitudes in the limit of large center of mass energies, a new degree of freedom —thereggeized gluon— is introduced on top of the usual QCD fields. The high energy effectiveaction then describes the interaction of this new field with the QCD field content throughadding an induced term S ind. to the QCD action S QCD , S eff = S QCD + S ind. , (1)2here the induced term S ind. describes the coupling of the gluon field v µ = − it a v aµ ( x ) to thereggeized gluon field A ± ( x ) = − it a A a ± ( x ). Due to this particular construction, it is immedi-ately clear that a specific calculational scheme is needed to avoid overcounting and to ensurethe abovementioned locality in rapidity. These requirements can be achieved using the follow-ing two-step procedure: a) calculation of vertices of reggeized gluon fields and QCD degreesof freedom and b) a procedure which matches the resulting field theory of reggeized gluonswith QCD. a) is achieved through Lipatov’s high energy effective action in Eq. (1), whichprovides the gauge invariant couplings of the new reggeized gluon field to the gluon field. Forb), a certain subtraction scheme has been proposed in [14], originally in the context of quark-quark scattering at 1-loop, and later on also verified for the case of gluon-gluon scattering [15].To set the notation it is useful to have a partonic scattering process p a + p b → p + p + . . . in mind with light-like momenta p a = p b = 0 and squared center of mass energy s = 2 p a · p b .Dimensionless light-like four vectors n ± normalized to n + · n − = 2 are then defined througha re-scaling n ± = 2 p a,b / √ s , while a general four-vector k has the decomposition k = k + n − k − n + k , k ± = n ± · k. (2)High energy factorized amplitudes reveal strong ordering in plus and minus components ofmomenta which is reflected in the following kinematic constraint obeyed by the reggeizedgluon field ∂ + A − ( x ) = 0 = ∂ + A + ( x ) . (3)Even though the reggeized gluon field is charged under the QCD gauge group SU( N c ), itis invariant under local gauge transformations: δA ± = 0. Its kinetic term and the gaugeinvariant coupling to the QCD gluon field are contained in the induced term, S ind. = Z d x tr h ( W − [ v ( x )] − A − ( x )) ∂ ⊥ A + ( x ) i + tr h ( W + [ v ( x )] − A + ( x )) ∂ ⊥ A − ( x ) i , (4)with W ± [ v ( x )] = v ± ( x ) 1 D ± ∂ ± , D ± = ∂ ± + gv ± ( x ) . (5)For a more in depth discussion of the effective action we refer the reader to [12] and therecent review [22]. II.1 Feynman rules and regularization
Apart from the usual QCD Feynman rules, the Feynman rules of the effective action comprisethe propagator of the reggeized gluon and an infinite number of so-called induced vertices,which result from the non-local functional Eq. (5). Vertices and propagators needed for thecurrent study are collected in Fig. 1 and Fig. 2.Loop diagrams of the effective action lead to a new type of longitudinal divergenceswhich are not present in conventional quantum corrections to QCD amplitudes, and can3 ,a, ± k,c,ν = − i q δ ac ( n ± ) ν ,k ± = 0 . + a − b q = δ ab i/ q q,a, ± k ,c ,ν k ,c ,ν = gf c c a q k ± ( n ± ) ν ( n ± ) ν ,k ± + k ± = 0 . (a) (b) (c) q, a, ± k , c , ν k , c , ν k , c , ν = ig q f a a e f a ea k ± k ± + f a a e f a ea k ± k ± ! ( n ± ) ν ( n ± ) ν ( n ± ) ν ,k ± + k ± + k ± = 0 . (d) Figure 1:
Feynman rules for the lowest-order effective vertices of the effective action. Wavy linesdenote reggeized fields and curly lines gluons. be regularized introducing an external parameter ρ , evaluated in the limit ρ → ∞ , whichdeforms the light-like vectors n ± into n − → n a = e − ρ n + + n − ,n + → n b = n + + e − ρ n − , (6)without violating the gauge invariance properties of the induced term Eq. (4). While it ispossible to identify ρ with a logarithm in s or the rapidity interval spanned by a certain highenergy process, we refrain from such an interpretation and consider in the following ρ as anexternal parameter, similar to the parameter (cid:15) in dimensional regularization in d = 4 + 2 (cid:15) dimensions. q, a, ± k , c , ν k , c , ν k , c , ν k , c , ν = g q (cid:20) f c c e k ± (cid:18) f e c e f c e a ( k ± + k ± ) k ± + f e c e f c e a ( k ± + k ± ) k ± (cid:19) + f c c e k ± (cid:18) f e a e f c e a ( k ± + k ± ) k ± + f e a e f c e a ( k ± + k ± ) k ± (cid:19) ++ f c c e k ± (cid:18) f e c e f c e a ( k ± + k ± ) k ± + f e c e f c e a ( k ± + k ± ) k ± (cid:19)(cid:21) ( n ± ) ν ( n ± ) ν ( n ± ) ν ( n ± ) ν ,k ± + k ± + k ± + k ± = 0 . Figure 2:
The order g induced vertex. I.2 Pole prescription
The evaluation of loop diagrams requires a prescription to circumvent the light-cone singu-larities in the induced vertices shown in Figs. 1,2. The seemingly natural choice which is tosimply replace the operator D ± in Eq. (5) by e.g. D ± − (cid:15) does not work in this context as itspoils hermiticity of the effective action. At the level of Feynman diagrams this is reflected byterms that violate high energy factorization. Both effects can be traced back to the existanceof new symmetric color tensors, not present in the vertices of Figs. 1, 2. For a more in depthdiscussion we refer to [23]. This problem can be solved by systematically projecting out thesesymmetric color structures, order by order in perturbation theory, sticking in this way tothe color tensors present in the original vertices Fig. 1, 2. The resulting pole prescriptionrespects then Bose symmetry of the induced vertices and high energy factorization [23]. The O ( g ) vertex is taken as a Cauchy principal value: q, a, ± k , c , ν k , c , ν = gf c c a q [ k ± ] ( n ± ) ν ( n ± ) ν , k ± ] ≡ k ± + i(cid:15) + 1 k ± − i(cid:15) ! . (7)For the O ( g ) and O ( g ) vertices the light-cone denominators are to be replaced by certainfunctions g and g : q, a, ± k , c , ν k , c , ν k , c , ν = − ig q (cid:20) f c c e f c ea g ± (3 , , f c c e f c ea g ± (3 , , (cid:21) n ± ν n ± ν n ± ν , (8) q, a, ± k , c , ν k , c , ν k , c , ν k , c , ν = − g q n ± ν n ± ν n ± ν n ± ν · (cid:20) f a a d f d a d f d a c g ± (4 , , ,
2) + f a a d f d a d f d a c g ± (4 , , , f a a d f d a d f d a c g ± (4 , , ,
3) + f a a d f d a d f d a c g ± (4 , , , f a a d f d a d f d a c g ± (4 , , ,
2) + f a a d f d a d f d a c g ± (4 , , , (cid:21) . (9)They are obtained as g ± ( i, j, m ) = (cid:20) − k ± i ][ k ± m ] − π δ ( k ± i ) δ ( k ± m ) (cid:21) . (10) We corrected a typing error present in Eq. (16) of [23] in the expression below. g ± ( i, j, m, n ) = (cid:18) − k ± i ][ k ± n + k ± m ][ k ± n ] − π δ ( k ± n ) δ ( k ± m ) − k ± i ] − π δ ( k ± n ) δ ( k ± i ) 1[ k ± m ] − π δ ( k ± n + k ± m ) δ ( k ± i ) 1[ k ± n ] (cid:19) . (11) III The gluon Regge trajectory from the effective action
A key ingredient in the resummation of high energy logarithms of QCD scattering amplitudesis provided by a universal function associated with the exchange of a single reggeized gluon,known as the Regge trajectory of the gluon. For the real part of QCD scattering amplitudes,where the high energy description is given in terms of single reggeized gluon exchange, thisfunction is known to govern the entire energy dependence at leading logarithmic (LL) andnext-to-leading logarithmic (NLL) accuracy.Multiple reggeized gluon exchanges appear on the other hand for the high energy descrip-tion of the imaginary part of scattering amplitudes and in general for amplitudes beyondNLL accuracy. While this requires new elements, which describe in a nutshell the interactionbetween reggeized gluons, the gluon Regge trajectory remains an essential building block inthe formulation of high energy resummation also in this more general case.To be more precise, for the elastic process p a + p b → p + p with s = ( p a + p b ) and t = q with q = p a − p one finds for amplitudes with gluon quantum numbers in the t -channel atLL and NLL accuracy the following factorized form M ( A ) ( s, t ) M (0) ( s, t ) = Γ a ( t ) "(cid:18) − s − t (cid:19) ω ( t ) + (cid:18) s − t (cid:19) ω ( t ) Γ b ( t ) , (12)where M (0)( A ) is the tree-level amplitude and the subscript ‘ A ’ denotes that the allowed t -channel exchange is restricted to the anti-symmetric color octet channel. The functions Γ ij ( t )are known as impact factors, describing the coupling of the reggeized gluons to scatteringparticles. For the case of gluon and quarks they have been determined within the effectiveaction in [14, 15]. The function ω ( t ) which governs the s -dependence of the scattering ampli-tude is on the other hand the Regge trajectory of the gluon. It is currently known to leading[4] and next-leading order [16] for QCD and to all orders in N = 4 super Yang-Mills theory[25]. The procedure which allows the derivation of the gluon trajectory from the effectiveaction has been originally discussed in [21]. It consists of two steps • determination of the propagator of the reggeized gluon to the desired order in α s ; • renormalization of the rapidity divergences of the reggeized gluon propagator; the gluonRegge trajectory is then identified as the coefficient of the ρ dependent term in therenormalization factor.To obtain the reggeized gluon propagator to order α s it is needed to determine the one- andtwo-loop self-energies of the reggeized gluon. Following the subtraction procedure proposedin [14] these self-energies can be obtained through For a pedagogical review see [24]. determination of the self-energy of the reggeized gluon from the effective action, withthe reggeized gluon treated as a background field; • subtraction of all disconnected contributions which contain internal reggeized gluonlines.Using a symmetric pole prescription as given in Sec. II.2, all diagrams with internal reggeizedgluon lines that would possibly contribute to the one loop self energy can be shown to vanishand no subtraction is necessary. The contributing diagrams are shown in Fig. 3. = + + + + + Figure 3:
Diagrams contributing to the one-loop reggeized gluon self-energy.
Keeping the O ( ρ, ρ ), for ρ → ∞ , terms and using the notation¯ g = g N c Γ(1 − (cid:15) )(4 π ) (cid:15) , (13)we have the following result in d = 4 + 2 (cid:15) dimensions : = Σ (1) ρ ; (cid:15), q µ ! = ( − i q )¯ g Γ (1 + (cid:15) )Γ(1 + 2 (cid:15) ) q µ ! (cid:15) (cid:26) iπ − ρ(cid:15) − (cid:15) ) (cid:15) (cid:20) (cid:15) (cid:15) − n f N c (cid:18) (cid:15) (cid:15) (cid:19) (cid:21)(cid:27) . (14)To determine the 2-loop self energy it is on the other hand needed to subtract disconnecteddiagrams, whereas diagrams with multiple internal reggeized gluons can be shown to yield azero result, if the symmetric pole prescription of Sec. II.2 is used. Schematically one hasΣ (2) ρ ; (cid:15), q µ ! = = − , (15)where the black blob denotes the unsubtracted 2-loop reggeized gluon self-energy, which isobtained through the direct application of the Feynman rules of the effective action, with In the original result presented in [14] and reproduced in [21, 22] a finite result for the second and thirddiagram has been erroneously included. This has been corrected in the result presented here. G (cid:16) ρ ; (cid:15), q , µ (cid:17) = i/ q i/ q Σ ρ ; (cid:15), q µ ! + " i/ q Σ ρ ; (cid:15), q µ ! + . . . , (16)with Σ ρ ; (cid:15), q µ ! = Σ (1) ρ ; (cid:15), q µ ! + Σ (2) ρ ; (cid:15), q µ ! + . . . (17)where the dots indicate higher order terms. As discussed in Sec. II.1 and as directly apparentfrom Eq. (14), the reggeized gluon self-energies are divergent in the limit ρ → ∞ . In [14, 15]it has been demonstrated by explicit calculations that these divergences cancel at one-looplevel, for both quark-quark and gluon-gluon scattering amplitudes, against divergences in thecouplings of the reggeized gluon to external particles. The entire one-loop amplitude is thenfound to be free of any high energy singularity in ρ . High energy factorization then suggeststhat such a cancellation holds also beyond one loop. Starting from this assumption, it ispossible to define a renormalized reggeized gluon propagator through G R ( M + , M − ; (cid:15), q , µ ) = G ( ρ ; (cid:15), q , µ ) Z + (cid:18) M + √ q , ρ ; (cid:15), q µ (cid:19) Z − (cid:18) M − √ q , ρ ; (cid:15), q µ (cid:19) , (18)where the renormalization factors need to cancel against corresponding renormalization fac-tors associated with the vertex to which the reggeized gluon couples with ‘plus’ ( Z + ) and‘minus’ ( Z − ) polarization. For explicit examples we refer the reader to [15, 21]. In their mostgeneral form these renormalization factors are parametrized as Z ± M ± p q , ρ ; (cid:15), q µ ! = exp " ρ − ln M ± p q ! ω (cid:15), q µ ! + f ± (cid:15), q µ ! , (19)with the coefficient of the ρ -divergent term given by the gluon Regge trajectory ω ( (cid:15), q ). Itis assumed to have the following perturbative expansion ω (cid:15), q µ ! = ω (1) (cid:15), q µ ! + ω (2) (cid:15), q µ ! + . . . , (20)and is to be determined by the requirement that the renormalized reggeized gluon propagatormust, at each loop order, be free of ρ divergences. At one loop we get from Eq. (14) ω (1) (cid:15), q µ ! = − g Γ (1 + (cid:15) )Γ(1 + 2 (cid:15) ) (cid:15) q µ ! (cid:15) . (21)The function f ± ( (cid:15), q ) parametrizes finite contributions and is, in principle, arbitrary. Whilesymmetry of the scattering amplitude requires f + = f − = f , Regge theory suggests fixingit in such a way that terms which are not enhanced in ρ are entirely transferred from the8eggeized gluon propagators to the vertices, to which the reggeized gluon couples. With theperturbative expansion f (cid:15), q µ ! = f (1) (cid:15), q µ ! + f (2) (cid:15), q µ ! . . . (22)we obtain from Eq. (14) f (1) (cid:15), q µ ! = ¯ g Γ (1 + (cid:15) )Γ(1 + 2 (cid:15) ) q µ ! (cid:15) ( − (cid:15) )2 (cid:15) (cid:20) (cid:15) (cid:15) − n f N c (cid:18) (cid:15) (cid:15) (cid:19) (cid:21) . (23)The renormalized reggeized gluon propagator is then to one loop accuracy given by G R ( M + , M − ; (cid:15), q , µ ) = 1 + ω (1) (cid:15), q µ ! log M + M − q − iπ ! + . . . (24)The scales M + and M − are arbitrary; their role is analogous to the renormalization scale inUV renormalization and the factorization scale in collinear factorization. They are naturallychosen to coincide with the corresponding light-cone momenta of scattering particles to whichthe reggeized gluon couples. To determine the gluon Regge trajectory at two loops we needin addition the ρ -enhanced terms of the two-loop reggeized gluon self-energy. From Eq. (24)we obtain the following relation ω (2) (cid:15), q µ ! = lim ρ →∞ ρ (cid:20) Σ (2) ( − i q ) + Σ (1) ( − i q ) ! − (cid:16) ρω (1) + 2 f (1) (cid:17) Σ (1) ( − i q )+ ρ (cid:16) ω (1) (cid:17) + 2 ρf (1) ω (1) (cid:21) = lim ρ →∞ ρ (cid:20) Σ (2) ( − i q ) + ρ (cid:16) ω (1) (cid:17) + 2 ρf (1) ω (1) (cid:21) , (25)where we omitted at the right hand side the dependencies on (cid:15) and q /µ ; in the last linewe further expanded Σ (1) in terms of the functions ω (1) and f (1) . We stress that this is anon-trivial definition and that it is not clear a priori whether the right hand side even existsdue to the presence of the second term, linear in ρ . Confirmation of this relation providestherefore an important non-trivial check on the validity of our formalism. IV Computation of the 2-Loop reggeized gluon self-energy
The necessary diagrams for the computation of the unsubtracted reggeized gluon self-energyare shown in Fig. 4. The diagrams (a )-(d ), containing internal quark loops, generating anoverall factor n f , have been computed in [21] and lead to the following result, = − ρ ( − i q )¯ g n f (cid:15)N c Γ (2 + (cid:15) )Γ(4 + 2 (cid:15) ) · − (cid:15) )Γ(1 + (cid:15) )Γ(1 + 2 (cid:15) )Γ (1 − (cid:15) )Γ(1 + 3 (cid:15) ) (cid:15) + O ( ρ ) . (26)9 a) (b) (c ) (c )(d ) (d ) (d ) (e)(f) (g ) (g ) (h )(h ) (h ) (i ) (i )(i ) (j ) (j ) (j )(k ) (k ) (k ) (l )(l ) (l ) (m )(m ) (m ) (m ) (n ) Figure 4:
Diagrams for the two-loop trajectory in the effective action formalism. Tadpole-like con-tributions are zero in dimensional regularization and are omitted. V.1 The scaling argument
For the computation of the remaining diagrams we observe at first that the number of dia-grams, which can be potentially enhanced by a factor ρ k , k ≥
1, is largely reduced by scalingarguments: only those diagrams where both reggeized gluons couple to the internal gluonlines through induced reggeized gluon– n -gluon vertices with n ≥ ρ . This is immediately clear for diagrams where bothreggeized gluons couple through the reggeized gluon–1-gluon vertex Fig. 1 (a) to the internalQCD lines. Those diagrams are a projection of the 2-loop QCD polarization tensor onto thekinematics of reggeized gluons and no ρ enhancement can be expected.To address the case where only one of the reggeized gluons couples through an inducedreggeized gluon– n -gluon ( n ≥
2) vertex to the internal QCD particles, we consider the generaldiagram in Fig. 5. The dependence on the light-cone vectors of the reggeized gluon– n -gluon . . .µ µ µ n ν Figure 5:
General non-enhanced diagram. vertex in Fig. 5 is, up to permutations, of the form n µ a n µ a ··· n µda n a · k n a · k ··· n a · k n − . The denominators n a · k i , i = 1 , . . . n − M µ µ ··· µ n ν . In ageneral diagram such as Fig. 5, the only vectors that are not integrated over in the amplitudeare q , the momentum transfer, and n a , which enters through the denominators of the inducedvertex. The vector n b only contracts with the four-vector index ν . The whole diagram canbe therefore written as n µ a n µ a · · · n µ n a M µ µ ··· µ n ν ( n a , q ) n νb . (27)As a consequence, the tensor structure of M µ µ ··· µ n ν ( n a , q ) can only consist of combinationsof the four vector n µa and the metric tensor g µν , since the external reggeized gluons imply q · n a = q · n b = 0. The only scalar combinations that can appear are therefore q and n a .These factors must give the dimensions required by scale transformations. If s is the numberof metric tensors in the numerator for a given term and l the number of n µa numerators, then n + 1 = 2 s + l and the associated scalar function must scale as1 n n − la = 1( n a ) d − s . (28)Next, we consider the contractions with the vertex currents. If n ρb is contracted through ametric tensor then we obtain( n a ) l n a · n b ( n a ) s − = ( n a ) n − s n a · n b ; (29)11f on the other hand n ρb is directly contracted with one of the n a ’s, we obtain a factor n a · n b ( n a ) s ( n a ) l − = ( n a ) n − s n a · n b . (30)In both cases the factors of n a cancel against corresponding factors in the denominators andno enhancement can occur. Thus, in our case only the diagrams (h ), (i ), (j ), (k ), (l ),(m ) and (n ) are potentially enhanced by (powers of) ρ .(a) (b) Figure 6: (a) typical tadpole contribution to the 2-loop self energy (b) disconnected diagrams withinternal reggeized gluon loops which would contribute to possible subtraction terms. Both contributionscan be shown to vanish.
A further class of diagrams that can be omitted are tadpole diagrams and diagrams withinternal reggeized gluon loops. Tadpole diagrams, such as in Fig. 6 (a), have been verified tovanish in dimensional regularization. Possible loop diagrams with internal reggeized gluonlines, such as Fig. 6 (b) vanish identically due to the symmetry properties obeyed by the poleprescription of the induced vertices.
IV.2 Calculation of the enhanced diagrams
Direct computation reveals that diagram (l ) is identically zero. We use the notation ξ = n a = n b = 4 e − ρ , δ = n a · n b ∼
2, and the following shorthand notation for the master integral[ α , α , · · · , α ] = ( µ ) − d ZZ d d k (2 π ) d d d l (2 π ) d − k − i α [ − ( k − q ) − i α ( − l − i α × − ( l − q ) − i α [ − ( k − l ) − i α · − n a · k ) α ( − n b · k ) α ( − n a · l ) α ( − n b · l ) α , (31)with n a · q = n b · q = 0 and the eikonal factors taken with the pole prescription defined inSec. II.2. More accurately, for master integrals with single poles 1 /n a,b · k ( α = 1 , α = 0, α = 0 , α = 1 and/or α = 1 , α = 0, α = 0 , α = 1) the function g a,b is used, while forterms with two poles 1 /n a,b · k/n a,b · l ( α = 1 , α = 1, and/or α = 1 , α = 1) the function g a,b is employed. Dropping all pieces that cannot generate terms enhanced as ρ → ∞ , we12ave the following contributions from each diagram:[ i M h ] enh = − ig (cid:15) ) S h δ q N c [1 , , , , , , , , S h = 1 . [ i M i ] enh = ig S i ( q ) N c (cid:20) δ (cid:26) q [1 , , , , , , , ,
1] + [1 , , , , , , , , − , , , , , , , , (cid:27) + 8 ξ [1 , , , , , , − , , (cid:21) ; S i = 2 . [ i M j ] enh = − ig S j q N c δ
19 + 12 (cid:15) (cid:15) [1 , , , , , , , , S j = 2 . [ i M k ] enh = 3 ig S k ( q ) N c δ [1 , , , , , , , , S k = 6 . [ i M m ] enh = ig S m ( q ) N c δ (cid:20) − δ q [1 , , , , , , , ,
1] + 2[1 , , , , , , , , ξ [1 , , , , , , , − , (cid:21) ; S m = 1 . [ i M n ] enh = 0 . (32)In some cases, we have used the Mathematica package FIRE [26] that implements the Laportaalgorithm [27] to reduce the number and complexity of master integrals through integration-by-parts identities [28]. Discarding all contributions which are finite or suppressed in thelimit ρ → ∞ , we can express the entire unsubtracted two-loop self-energy in terms of 7master integrals A − G with a certain coefficient associated with each master integral, seeTab. 1. The master integral A can be shown to vanish by symmetry due to the symmetricpole prescription of the eikonal poles of the induced vertices. The ρ -enhanced pieces of theremaining master integrals are computed up to terms of order O ( (cid:15) ) using the Mellin-Barnesrepresentations technique, for a review see e.g. [29].To this end, we first derive multi-contour integral representations for the master integrals,referring the reader for details to Appendix A.1. Having as working environment the code MB.m [30] , we use the Mathematica package MBasymptotics.m [32] to perform an asymptoticexpansion in e − ρ . We remove any terms proportional to e − kρ(cid:15) , k ∈ Z , capturing this waythe leading behavior in ρ . As a final step, we resolve the singularities structure in (cid:15) by usingthe Mathematica packages MB.m and
MBresolve.m . Eventually, some of the final integralsare further simplified by using the Barnes’ lemmas implemented in the Mathematica code barnesroutines.m [33]. The package
MBresolve.m [31] was also used.
A ≡ (cid:2) , , , , , , , , (cid:3) c A = − q B ≡ (cid:2) , , , , , , , , (cid:3) c B = 66 + 42 (cid:15) (cid:15) C ≡ (cid:2) , , , , , , , , (cid:3) c C = − ( q ) D ≡ [1 , , , , , , , , (cid:3) c D = − q E ≡ (cid:2) , , , , , , , − , (cid:3) c E = − ξ q F ≡ (cid:2) , , , , , , − , , (cid:3) c F = − ξ q G ≡ (cid:2) , , , , , , , , (cid:3) c G = 0 Table 1:
Coefficients of the master integrals. Each coefficient is in addition to be multiplied with thecommon overall factor ( − i q ) g N c . Following this procedure we obtain for the master integrals the following results: c B · B = 1(4 π ) " (cid:15) − (cid:15) + 400 + 12Ξ + 396Ξ − π ρ,c C · C = 1(4 π ) (cid:18)(cid:20) − (cid:15) − − Ξ) (cid:15) − π + 8(1 − Ξ) (cid:15) − π (1 − Ξ) − − Ξ) − ζ (3) (cid:21) ρ + (cid:20) (cid:15) + 4(1 − Ξ) (cid:15) + 13 (12(1 − Ξ) − π ) (cid:21) n ρ − iπρ o (cid:19) ,c D · D = 1(4 π ) (cid:20) (cid:15) + 8(1 − Ξ) (cid:15) + 4( π + 6(1 − Ξ) )3 (cid:15) + 8 π (1 − Ξ)3+ 16(1 − Ξ) ζ (3)3 (cid:21) ,c E · E = 0 ,c F · F = 1(4 π ) " − (cid:15) + 8Ξ (cid:15) + 2 π − ) , (33)where we introduced the notation Ξ = 1 − γ E − ln q πµ . (34)Using these results, the (unsubtracted) contribution to the reggeized gluon self-energy (with For details on the computation of imaginary parts, see Appendix A.2. f = 0) reads: gluon cont. = ( − i q ) g N c (4 π ) ( (cid:15) + 4(1 − Ξ) (cid:15) + 4(1 − Ξ) − π ) ρ + (cid:26) (cid:15) − (cid:15) − − π (cid:15) − π − ) + 29 − π − ζ (3) − iπ (cid:20) (cid:15) + 4 1 − Ξ (cid:15) + 13 (12(1 − Ξ) − π ) (cid:21) (cid:27) ρ ! . (35)Expanding in (cid:15) the expression in Eq. (15), one eventually finds for the subtracted reggeizedgluon self-energy for n f = 0:Σ (2) n f =0 ρ, q µ ! = = − = ( − i q ) g N c (4 π ) (cid:26) − (cid:20) (cid:15) + 4(1 − Ξ) (cid:15) + 4(1 − Ξ) − π (cid:21) ρ + (cid:20) (cid:15) + 19 (cid:15) + π (cid:15) − (cid:15) + π (11 − −
29 Ξ + 23 Ξ − ζ (3) (cid:19)(cid:21) ρ (cid:27) + O ( (cid:15) ) + O ( ρ ) . (36)Now we can compare our result for the 2-loop self-energy with the definition of the 2-loopgluon Regge trajectory, Eq. (25). At first we note that all divergent terms ∼ ρ cancel againsteach other since the terms quadratic in ρ in Eq. (36) cancel precisely the term [ ρω (1) ] / i.e. ( ω (1) ) ρ (2) ρ ( − i q ) = 0 , (37)if the first term is expanded up to O ( (cid:15) ). Taking the function f (1) in the limit n f = 0, theremaining terms then yield the 2-loop Regge gluon trajectory for zero flavors, ω (2) ( q ) | n f =0 = ( ω (1) ( q )) "
113 + π − ! (cid:15) + (cid:18) − ζ (3) (cid:19) (cid:15) , (38)which is in complete agreement with the results in the literature [16]. The terms proportionalto n f have been calculated in [21]. With the the flavor-dependent ρ -enhanced terms, thesubtracted 2-loop self-energy is given byΣ (2) n f ρ ; (cid:15), q µ ! = ρ ( − i q )¯ g n f (cid:15)N c Γ (2 + (cid:15) )Γ(4 + 2 (cid:15) ) q µ ! (cid:15) (cid:18) Γ (1 + (cid:15) )Γ(1 + 2 (cid:15) ) 4 (cid:15) − − (cid:15) )Γ(1 + (cid:15) )Γ(1 + 2 (cid:15) )Γ (1 − (cid:15) )Γ(1 + 3 (cid:15) ) (cid:15) (cid:19) , (39)and one obtains for the 2-loop Regge gluon trajectory with n f flavors ω (2) ( q ) = ( ω (1) ( q )) " − n f N c + π − ! (cid:15) + (cid:18) − ζ (3) (cid:19) (cid:15) . (40)15 Conclusions and Outlook
In this paper we have presented a derivation of the two-loop gluon Regge trajectory usingLipatov’s effective action and a recently developed computational scheme, which includes aregularization, subtraction and renormalization procedure. Our result is in precise agreementwith earlier results present in the literature and thus provides a highly non-trivial check ofthe effective action and our proposed computational framework.From a technical point of view, the main result of the paper is the computation of the2-loop reggeized gluon self-energy. Regularizing high energy divergences by slightly movingthe light-like vectors of the effective action away from the light-cone, we first demonstratedthe suppression of a large class of diagrams through a scaling argument. The remainingdiagrams were then expressed in terms of seven master integrals, which have been evaluatedusing multiple Mellin-Barnes representations. Our scheme introduces a consistent generalstrategy to deal with more complex computations, with the hope to easy the path to performfurther calculations with Lipatov’s high-energy effective action.
Acknowledgements
We thank J. Bartels, V. Fadin and L. Lipatov for constant support for many years. We ac-knowledge partial support by the Research Executive Agency (REA) of the European Unionunder the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet), the Comu-nidad de Madrid through Proyecto HEPHACOS ESP-1473, by MICINN (FPA2010-17747), bythe Spanish Government and EU ERDF funds (grants FPA2007-60323, FPA2011-23778 andCSD2007- 00042 Consolider Project CPAN) and by GV (PROMETEUII/2013/007). G.C. ac-knowledges support from Marie Curie Actions (PIEF-GA-2011-298582). M.H. acknowledgessupport from the U.S. Department of Energy under contract number DE-AC02-98CH10886and a “BNL Laboratory Directed Research and Development” grant (LDRD 12-034).
A Appendix
In this appendix we present some details of the derivation of Mellin-Barnes representationsfor the general two-loop master integral considered in this work with propagators to arbitrarypowers. The principal tool in this analysis is the formula1( X + · · · + X n ) λ = 1Γ( λ ) 1(2 πi ) n − Z · · · Z + i ∞− i ∞ dz · · · dz n n Y i =2 X z i i X − λ − z −···− z n × Γ( λ + z + · · · + z n ) n Y i =2 Γ( − z i ) , (41)where the contours of integration are such that poles with a Γ( · · · + z i ) dependence are to theleft of the z i contour and poles with a Γ( · · ·− z i ) dependencies lie to the right of the z i contour.16 .1 Mellin-Barnes Representation for Master Integrals without phases We consider the integral S = Z d d k (2 π ) d − k − i C [ − ( k − q ) − i D [ − ( k − l ) − i E × − n a · k − i µ ( − n b · k − i µ , (42)where the relation n a · q = n b · q = 0 is implied. Unlike the general master integral defined inEq. (31), the contour of integration is in the following always defined to lie above the singu-larities introduced by the light cone denominators. The treatment of alternating descriptions,contained in the functions g and g is summarized in Appendix A.2.Using Schwinger parameters, we can write S = i C + D + E + µ + µ Γ( C )Γ( D )Γ( E )Γ( µ )Γ( µ ) Z ∞ · · · Z ∞ dαdβdγd ˜ δd ˜ σα C − β D − γ E − ˜ δ µ − ˜ σ µ − Z d d k (2 π ) d e i D , D = αk + β ( k − q ) + γ ( k − l ) + ˜ δn a · k + ˜ σn b · k = ( α + β + γ ) k + βq + γl − k · (cid:18) βq + γl − (cid:20) ˜ δ n a σ n b (cid:21)(cid:19) . (43)With a shift in the momentum integral and introducing parameters λ = α + β + γ, ξ = βα + β , η = γα + β + γ ; ˜ δ = 2 λδ, ˜ σ = 2 λσ, and x = 2( δ + σ ) , y = δδ + σ , we arrive at S = i C + D + E + µ + µ Γ( C )Γ( D )Γ( E )Γ( µ )Γ( µ ) Z ∞ dλ λ C + D + E + µ + µ − Z ∞ dx x µ + µ − Z dξ ξ D − (1 − ξ ) C − Z dη η E − (1 − η ) C + D − Z dyy µ − (1 − y ) µ − Z d d k (2 π ) d exp h iλ (cid:0) k − (1 − η ) ξ (1 − ξ ) q − η (1 − η )(1 − ξ )( − l ) − η (1 − η ) ξ [ − ( l − q ) ] − ηx [ y ( − n a · l ) + (1 − y )( − n b · l ) − x (Ψ y (1 − y ) + e − ρ ) (cid:1)i , (44)where Ψ ≡ (1 − e − ρ ) . Performing the integration over momentum and the parameter λ weobtain with Eq. (41) S = i (4 π ) d/ Γ( C )Γ( D )Γ( E )Γ( µ )Γ( µ ) Z dξ ξ D − (1 − ξ ) C − Z dη η E − (1 − η ) C + D − Z ∞ dx x µ + µ − Z dy y µ − (1 − y ) µ − Z · · · Z + i ∞− i ∞ dz πi · · · dz πi Γ( − z ) · · · Γ( − z )Γ( z + z + z + z + z + z + C + D + E + µ + µ − d/ − η ) ξ (1 − ξ ) q ] z [ η (1 − η )(1 − ξ )( − l )] z [ η (1 − η ) ξ ( − ( l − q ) )] z [ ηxy ( − a · l )] z [ ηx (1 − y )( − b · l )] z [ x y (1 − y )] z + z + z + z + z + z + C + D + E + µ + µ − d/ [ x ( e − ρ )] − z , (45)17hich allows to perform the integrations over the parameters ξ, η, x and y . In some casesintegrals of the form Z dy y α − (1 − y ) − α − = Z ∞ dt t − α − = 2 πiδ ( α ) (46)appear which allow for the reduction of contour integrals. Eventually, we arrive at S = i (4 π ) d/ Z · · · Z dz πi dz πi dz πi dz πi dz πi Γ( − z )Γ( − z )Γ( − z )Γ( − z )Γ( − z )Γ( − z )Γ(2 z + z + 2 C + 2 D + 2 E + µ + µ − d )Γ( C )Γ( D )Γ( E )Γ( µ )Γ( µ ) Γ (cid:18) − z − C − D − E + d (cid:19) Γ (cid:18) z − z + C + D + E + µ − d (cid:19) Γ (cid:18) − z − C − D − E − µ + d (cid:19) Γ( − z − z − C − D − E − µ − µ + d )Γ( z + C )Γ( z + D )Γ( − C − D − E − µ − µ + d )Ψ z − z + C + D + E − d/ (cid:16) q (cid:17) z (cid:16) − l (cid:17) z (cid:16) − ( l − q ) (cid:17) z ( − n a · l ) z ( − n b · l ) − z − z − C − D − E − µ − µ + d ( e − ρ ) z , (47)where z ijk... = z i + z j + z k + . . . In an analogous way, we can derive the following Mellin Barnes representation, S = Z d d k (2 π ) d − k − i A [ − ( k − q ) − i B ( − n a · k − i λ ( − n b · k − i λ = i Γ (cid:16) A + B + λ + λ − d (cid:17) π ) d/ Γ( A )Γ( B )Γ( λ )Γ( λ ) Γ (cid:16) d − A − λ + λ (cid:17) Γ (cid:16) d − B − λ + λ (cid:17) Γ( d − A − B − λ − λ )( q ) A + B + λ λ − d × Z dz πi Γ( − z )Γ( − z ) Γ (cid:18) z + λ + λ (cid:19) Γ (cid:18) − z + λ − λ (cid:19) Γ (cid:18) − z − λ − λ (cid:19) (cid:0) e − ρ (cid:1) z . (48)where again n a · q = n b · q = 0 is implied. Iterating the results Eq. (47) and Eq. (48), weobtain the Mellin Barnes representation of the general two-loop master integral S = ZZ d d k (2 π ) d d d l (2 π ) d − k − i α [ − ( k − q ) − i α [ − l − i α [ − ( l − q ) − i α × − ( k − l ) − i α ( − n a · k − i α ( − n b · k − i α ( n a · l − i α ( − n b · l − i α = − (cid:0) q (cid:1) d − α − α π ) d Y i =1 Z dz i πi Γ( − z i ) Γ (cid:16) z + α + α − d (cid:17) Γ( − z + α ) Q j =1 Γ( α j )Γ( − z )Γ( − z )Γ (cid:16) − z − α + α + d (cid:17) Γ (cid:16) − z + z + α − α − d (cid:17) Γ( − z + α )Γ( − z − z − α − α + 2 d )Γ (cid:0) − z − α − α + d (cid:1) Γ (cid:0) − z − α − α + d (cid:1) Γ( z − z + α − d )Γ(2 z + z + 2 α + α − d )Γ(2 z + z + 2 α + α − d )Γ( − z + z − α + d )Γ (cid:0) z + α + α − d (cid:1) Γ( − z + α )Γ( − z − α + d )Γ( − z − z − α − α + d )Γ( z + α )Γ( z + α )Γ( − α + d ) e − z ρ (49)18here z ijk... = z i + z j + z k + . . . and α ijk... = α i + α j + α k + . . . . At this stage one then turnsto explicit values for the parameters α i , i = 1 , . . . , ρ → ∞ and (cid:15) → A.2 Computation of ρ -enhanced imaginary parts Among all integrals, only the masters C and D are, for their ρ -enhanced terms, sensitive tothe details of the pole prescription. For diagram ( k ), which is directly proportional to D andconstitutes the only diagram containing this master, explicit QCD calculations allow to arguethat no enhanced imaginary parts can result from such a diagram. This is immediately clearif one identifies this diagram with the high energy expansion of the quark-quark scatteringamplitude with three gluon exchange (see for instance [34]), which allows to argue that the ρ -enhanced imaginary part of this diagram needs to vanish. We verified that this is indeedthe case and we were able to confirm that the entire ρ -enhanced contribution of this masterintegral coincides with the equivalent integral using the pole prescription of Sec. A.1.The master C possesses on the other hand a ρ -enhanced imaginary part. To this end weconsider the integral C ( ± , ± ) = ( µ ) − (cid:15) Z Z d d k (2 π ) d d d l (2 π ) d − k − i − ( k − q ) − i − l − i − ( l − q ) − i − ( k − l ) − i · − n a · k ± i − n b · k ± i , (50)where the integral C ( −− ) is assumed to be known using the techniques of Sec. A.1 while C (+ , +) = C ( − , − ) holds. Introducing rescaled vectors a, b = e ρ/ n a,b with a = 1 = b and a · b = cosh ρ we find C ( ± , ± ) = 4 e − ρ · ˜ C ( ± , ± ) , ˜ C ( ± , ± ) = ( µ ) − (cid:15) Z Z d d k (2 π ) d d d l (2 π ) d − k − i − ( k − q ) − i − l − i − ( l − q ) − i − ( k − l ) − i · − a · k ± i − b · k ± i e ρ C ( ± , ± ) . (51)As a = 1 = b , the new integral is an analytic function of a · b and q only, ˜ C = ˜ C ( a · b, q ).With 1 − a · k + i − − ( e − iπ a ) · k − i , (52)we have ˜ C ± , ∓ ( a · b, q ) = − ˜ C (+ , +) ( e − iπ a · b, q ) . 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