Pole prescription of higher order induced vertices in Lipatov's QCD effective action
aa r X i v : . [ h e p - ph ] D ec Pole prescription of higher order induced vertices inLipatov’s QCD effective action
Martin HentschinskiInstituto de F´ısica Te´orica UAM/CSIC,Universidad Aut´onoma de Madrid,Cantoblanco, E-28049 Madrid, SpainMay 25, 2018
IFT-UAM/CSIC-11-105LPN11-95
Abstract
We investigate Lipatov’s QCD effective action for the QCD high energy limitand propose a pole prescription for higher order induced vertices. The latter canbe used in the evaluation of loop corrections to high energy factorized matrixelements within the effective action approach. The proposed prescription respectsthe symmetry properties of the unregulated vertices. Explicit results are presentedup to third order in the gauge coupling, while an iterative procedure for higherorders is proposed.
The description of the high energy limit of perturbative QCD is generally given interms of high energy factorization and the BFKL evolution. The latter resums largelogarithms in the center of mass energy √ s at leading (LL) [1] and next-to-leadinglogarithmic (NLL) accuracy [2]. Its derivation is closely related to the observationthat QCD scattering amplitudes reveal in the high energy limit an effective t -channeldegree of freedom, the reggeized gluon, which couples to the external scattering particlesthrough effective vertices. Determination of both effective couplings and higher ordercorrections to reggeized gluon exchange is a highly non-trivial task. This is especiallytrue if one attempts to go beyond leading order accuracy – something which is neededfor e.g. a successful phenomenology of the QCD high energy limit.An efficient tool to address these questions is given by Lipatov’s high energy effectiveaction [3]. It is based on the QCD action with the addition of an induced term.The latter is written in terms of gauge-invariant currents which generate a non-trivialcoupling of the gluon to reggeized gluon fields. The effective action allows to addressboth unitarization of BFKL evolution and the systematic determination of higher orderperturbative corrections to high energy QCD amplitudes. While the high energy limit1f QCD tree-level amplitudes can be obtained in a straightforward manner from theeffective action [4], in loop corrections a new type of longitudinal divergences, notpresent in conventional QCD amplitudes, appear. The treatment of these divergenceshas been at first addressed for leading order transition kernels [5, 6] and the study ofthe leading order reggeized gluon - gluon - 2 reggeized gluon (RG2R) production vertexin [7,8]. Recently the effective action has been used for the first time for the calculationof NLO corrections to the forward quark jet impact factor [9] and the quark part ofthe 2-loop gluon trajectory [10], finding precise agreement with previous results in theliterature.In these studies of loop corrections it has been further necessary to give a prescrip-tion for circumventing light-cone singularities on the complex plane. The latter appeardue to a non-local operator in the induced term of the effective action which describesthe coupling of reggeized gluons to conventional gluons. Locality of the effective ac-tion in rapidity space and high energy kinematics prevent in principle the light-conemomenta in these denominators to take non-zero values. A prescription for these polesseems not to be necessary. That this is true is evident in the case of tree-level am-plitudes in the (Quasi-)Multi-Regge-Kinematics. It seems furthermore reasonable toexpect that a similar statement holds for corresponding virtual corrections. On theother hand all regularizations used up till now need a prescription for these denomina-tors. Other regularizations which contain more physical insight are surely possible, butremain to be explored. In this work we therefore give a pole prescription which can beapplied in combination with regularization methods used so far.To leading order (LO) in the gauge coupling g , the order g induced vertex has so farbeen interpreted as a Cauchy principal value [5–11]. Higher order induced vertices seemto require a more elaborate prescription. In particular interpreting n¨aively every singlepole of the higher vertices as a Cauchy principal value destroys parts of the symmetryof the unregulated vertices and can lead to incorrect results, see for instance [5].In the following we provide a pole prescription of higher order induced verticeswhich respects the symmetry properties of the unregulated vertices and which can bemotivated through the high energy expansion of QCD scattering amplitudes. We willpresent explicit results up to the order g induced vertex which can be directly used forcalculations within the effective action such as the determination of the gluon part ofthe 2-loop gluon Regge trajectory, corrections to gluon induced production processesand studies of multiple reggeized gluon exchanges within the effective action.The paper is organized as follows: in sec. 2 we give a short introduction to Lipatov’seffective action, including a summary of unregulated Feynman rules. In sec. 3 wepresent a derivation of the proposed pole prescription and provide explicit expressionsup to third order in the gauge coupling. Finally in sec. 4 conclusions and suggestionsfor future work are presented. The appendix contains a simple QCD example whichillustrates the relation of the chosen pole prescription with underlying QCD amplitudes.2 The effective action of high energy QCD
To set the notation used in the following for the effective action, it is useful to have inmind a partonic elastic scattering process p a + p b → p + p with p a = 0 = p b light-likemomenta and s = ( p a + p b ) = 2 p a · p b the squared center of mass energy. We thendefine light-like four vectors n ± with n + · n − = 2, related to the momenta of scatteringpartons by the re-scaling, n + = 2 p b / √ s and n − = 2 p a / √ s . The Sudakov decompositionof a general four vector k µ reads k = k + n − / k − n + / k , where k ± = n ± · k and k is transverse w.r.t. the initial scattering axis. The effective action is given as the sumof two terms, S eff = S QCD + S ind. , the QCD action and the induced term. The latterdescribes the coupling of the reggeized gluon field A ± ( x ) = − it a A a ± ( x ) to the gluonicfield v µ ( x ) = − it a v aµ ( x ). It reads S ind. [ v µ , A ± ] = Z d x tr (cid:20)(cid:18) W + [ v ( x )] − A + ( x ) (cid:19) ∂ ⊥ A − ( x ) (cid:21) + Z d x tr (cid:20)(cid:18) W − [ v ( x )] − A − ( x ) (cid:19) ∂ ⊥ A + ( x ) (cid:21) . (1)The infinite number of couplings of the gluon field to the reggeized gluon field arecontained in two functionals W ± [ v ], which are defined through the following operatordefinition W ± [ v ] = v ± D ± ∂ ± where D ± = ∂ ± + gv ± . (2)For perturbative calculations, the following expansion in the gauge coupling g holds, W ± [ v ] = v ± − gv ± ∂ ± v ± + g v ± ∂ ± v ± ∂ ± v ± − . . . (3)The determination of a suitable regularization of the above operators 1 /∂ ± at zero isthen the main goal of this work. Note that the reggeized gluon fields are special in thesense that they are invariant under local gauge transformations, while they transformglobally in the adjoint representation of the SU( N c ) gauge group. In addition strongordering of longitudinal momenta in high energy factorized amplitudes leads to thefollowing kinematical constraint of the reggeized gluon fields, ∂ + A − ( x ) = ∂ − A + ( x ) = 0 , (4)which is always implied. Feynman rules of the high energy effective action have beendetermined in [4]. We depict them in the following using curly lines for the conventionalQCD gluon field and wavy (photon-like) lines for the reggeized gluon field. Followingthe convention of [4], the Feynman rules of the effective action are given by the conven-tional QCD Feynman rules and an infinity number of induced vertices, which include3 , 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)Figure 1: The direct transition vertex (a), the reggeized gluon propagator (b) and the unreg-ulated order g induced vertex (c) q, a, ± k , c , ν k , c , ν k , c , ν = ig q (cid:18) f a a e f a ea k ± k ± + f a a e f a ea k ± k ± (cid:19) ( n ± ) ν ( n ± ) ν ( n ± ) ν k ± + k ± + k ± = 0Figure 2: The unregulated order g induced vertex a direct transition term between gluon and reggeized gluon, fig. 1.a. Within this di-rect transition picture, the propagator of the reggeized gluon receives at tree-level acorrection due to the projection of the gluon propagator on the reggeized gluon stateswhich we absorb here into the bare reggeized gluon propagator, fig. 1.b. Higher or-der reggeized gluon - n gluon transition vertices are up to O ( g ) given by figs. 1.c, 2and 3. Note that there exists a general iterative formula for the O ( g n ) vertex whichis contained in [4]. All of these vertices obey Bose-symmetry, i.e. symmetry undersimultaneous exchange of color, polarization and momenta of the external gluons of theorder g n vertex. This can be verified making use of the constraint P n +1 i k ± i = 0, whichis a direct consequence from eq. (4), and the Jacobi identity. In its original formulation the effective action does not specify a prescription for thepoles of induced vertices. A common choice adapted to the induced vertex fig. 1.c is tointerpret the pole as a Cauchy principal value, [5, 6, 8, 9] q, a, ± k , c , ν k , c , ν = gf c c a q [ k ± ] ( n ± ) ν ( n ± ) ν , k ± ] ≡ (cid:18) k ± + iǫ + 1 k ± − iǫ (cid:19) . (5)This choice has the advantage that it maintains the symmetry of the vertex withoutpole prescription; in particular both Bose-symmetry of the unregulated vertex and4 , 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 ± = 0Figure 3: The unregulated order g induced vertex its anti-symmetry under the substitution k ± → − k ± are kept. The latter propertyis of importance as it can be directly related to negative signature of the reggeizedgluon, see [11]. A straightforward extension of the Cauchy principal value prescriptionto higher order vertices interprets every separate pole as a Cauchy principal value.However, at least if n¨aively applied to the vertices in figs. 2 and 3, such a prescriptionviolates Bose-symmetry and can lead to wrong results, see for instance [5]. This ismainly due to the fact that Cauchy principal values do not obey the eikonal identity inan algebraic sense. One has instead an additional term containing the product of twodelta-functions, 1[ k ± ][ k ± + k ± ] + 1[ k ± ][ k ± + k ± ] = 1[ k ± ][ k ± ] + π δ ( k ± ) δ ( k ± ) , (6)see also [12]. In the following section we suggest a prescription which both respectsBose-symmetry and negative signature of the reggeized gluon and therefore maintainsthe symmetry properties of the unregulated vertices. g induced vertex The most straightforward definition is to replace the operator D ± in eq. (1) by itsregulated counter part D ± − ǫ . This is a commonly used choice in other effectivetheories with Wilson line like operators in the Lagrangian, see for instance [13]. It hasthe advantage that it can be related to the prescription which arises for these polesfrom an high energy expansion of QCD diagrams. Up to the order g induced vertexthis is explicitly demonstrated in Appendix A, see also the discussion in [14]. For this5egulation, the regulated order g induced vertex reads, q, a, ± k , c , ν k , c , ν ′ = ig q (cid:18) tr( t c t c t a ) k ± + iǫ + tr( t c t c t a ) k ± + iǫ (cid:19) n ± ν n ± ν = q (cid:20) f a a c (cid:18) k ± + iǫ + 1 k ± − iǫ (cid:19) + d a a c ǫ )2 πδ ( k ± ) (cid:21) n ± ν n ± ν , (7)with the condition k ± = − k ± implied. Eq. (7) coincides with eq. (5) up to the secondterm in the second line proportional to the symmetric structure constant d abc . Thesymmetric color structure, the dependence on the sign of ǫ and its symmetry behaviorunder k − → − k − indicate that this term corresponds to a reggeized gluon with positivesignature. This is in contrast with the original formulation of the effective action whichcontains only negative signatured reggeized gluons. While it seems at first natural touse eq. (7) instead of eq. (5) and to extend the effective action to reggeized gluons withpositive signature, we do not make use of this possibility in the following and discardterms with symmetric color structure. This choice is due to the following reasons:(i) The dependence on the sign of ǫ leads to a potential conflict with factorization ofQCD amplitudes in the high energy limit. It denotes a dependence of the induced vertexon the energy of the high energy factorized scattering parton (for an illustrative QCDexample we refer to Appendix A). From the point of view of the effective Lagrangianthis dependence on the sign of ǫ can be understood as a violation of hermicity due tothe simple replacement D ± → D ± − ǫ .(ii) A further problem is connected with symmetric color tensors such as d abc itself.Unlike color tensors build from anti-symmetric SU( N c ) structure constants, such as thecolor tensors of the unregulated induced vertices figs. 1.c, 2, 3, the symmetric tensorsdepend generally on the SU( N c ) representation of the fields in the Lagrangian . In thecase of QCD amplitudes this can be backtracked to the SU( N c ) representation of scat-tering particles, signaling another potential breakdown of high energy factorization .In the following we therefore take the most conservative choice and define the poleprescription of the induced vertices through the replacement D ± → D ± − ǫ togetherwith a subsequent projection on the ’maximal antisymmetric’ sub-sector of its colortensors, which is given in terms of anti-symmetric SU( N c ) structure constants alone.The precise meaning of this projection onto the maximal-anti-symmetric sector willbe defined in short. Leaving in eq. (7) aside the projector on the color octet tr( · t a ),where the dot represents any product of SU( N c ) generators, we deal in case of the We are particular thankful to L. N. Lipatov for drawing our attention to this point. That there exists indeed a dependence on the representation of scattering particles apart fromnormalization factors has been directly observed in [15], where an additional contribution for scatter-ing particles in the adjoint representation has been found, which is not present for the fundamentalrepresentation. g , with a color tensor with two adjoint indices c and c . Themaximal anti-symmetric sub-sector of order two is then given by the commutator of thetwo generators, while its symmetric counterpart is defined as the anti-commutator .Projection on the maximal anti-symmetric sub-sector corresponds therefore for theorder g induced vertex to dropping the symmetric term proportional to d abc in eq. (7),which leaves us with the commonly used pole prescription eq. (5). g induced vertex To generalize this projection to higher order induced vertices, we need to find atfirst an appropriate basis that generalizes the decomposition in commutator and anti-commutator of the previous subsection. This basis requires two elements with a double-commutator, which then yield the color structure of the induced vertex fig. 2. In thefollowing we use a short-cut notation[1 ,
2] = [ t c , t c ] , S n (1 . . . n ) = 1 n ! X i ,...i n t c i . . . t c in , (8)denoting the commutator of two generators with color indices c and c and symmetriza-tion of n generators respectively, where in the second expression the sum is taken overall permutations of the numbers 1 , . . . , n . In this notation, a possible decomposition ofa color tensor with three adjoint indices is given by the following basis[[3 , , , [[3 , , , S ([1 , S ([1 , S ([2 , S (123) . (9)As a commutator of two SU( N c ) generators can be expressed in terms of a singlegenerator, expressions such as S ([1 , e.g. S ([1 , t a t a t a − t a t a t a + t a t a t a − t a t a t a . (10)The terms in eq. (9) contains two double-anti-symmetric, three mixed-symmetric andone totally symmetric element, while the third double-antisymmetric element [[1 , , , , , ,
1] by means of the Jacobi-identity. The twodouble-antisymmetric elements [[3 , , , ,
1] define a basis of the maximal anti-symmetric subsector of order two. The above decomposition shares some propertieswith the usual Young-tableau decomposition, while the definition of anti-symmetrizationdiffers in the present case. Apparently the elements [[3 , , , ,
1] define a basisof the maximal anti-symmetric sub-sector of a color tensor with three adjoint indices.To obtain the pole prescription of the induced vertex of order g , we start from theeffective action with the following expression g v ± ∂ ± − ǫ v ± ∂ ± − ǫ v ± ∂ σ A ∓ , (11) Note that without specifying the symmetric counterpart a projection on the anti-symmetric part ismeaningless as the remainder is completely arbitrary and therefore also the resulting pole prescription. D ± → D ± − ǫ in eq. (1) and subsequentexpansion in g . On the level of Feynman rules, it results into the following unprojectedinduced vertex of order g − ig q (cid:18) tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ ) + tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ )+ tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ ) + tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ )+ tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ ) + tr( t c t c t c t a )( k ± + k ± + iǫ )( k ± + iǫ ) (cid:19) n ± ν n ± ν n ± ν , (12)with k ± + k ± + k ± = 0 implied. Leaving aside for the moment the factor − ig q n ± ν n ± ν n ± ν and also the projection on the color octet, tr( · t a ), eq.(12) reads in the basis eq.(9) −
16 [[3 , , (cid:18) k ± − iǫ )( k ± + iǫ ) + 2( k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ )+ 1( k ± − iǫ )( k ± + iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) (cid:19) −
16 [[3 , , (cid:18) k ± − iǫ )( k ± + iǫ ) + 2( k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) (cid:19) + 1( k ± − iǫ )( k ± + iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) (cid:19) − S ([1 , (cid:18) k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) − k ± − iǫ )( k ± + iǫ ) (cid:19) − S ([1 , (cid:18) k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) − k ± − iǫ )( k ± + iǫ ) (cid:19) − S ([2 , (cid:18) k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) − k ± − iǫ )( k ± + iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) − k ± + iǫ )( k ± − iǫ ) (cid:19) − S (123) (cid:18) k ± − iǫ )( k ± + iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ )+ 1( k ± + iǫ )( k ± − iǫ ) + 1( k ± + iǫ )( k ± − iǫ ) + 1( k ± − iǫ )( k ± + iǫ ) (cid:19) . (13)8rojection onto the maximal anti-symmetric sub-sector of order three sets then allterms which contain an S or an S symbol to zero, leaving only the color tensorscontained in the unregulated vertex fig. 2. The pole structure can be further simplifiedmaking use of the eikonal identity1 k ± + iǫ k ± + k ± + iǫ + 1 k ± + iǫ k ± + k ± + iǫ = 1 k ± + iǫ k ± + iǫ , (14)which holds for the above pole prescription not only in the algebraic sense but also inthe sense of the theory of distributions [12]. Evaluating commutators and adding thecolor-octet projection together with the common factor of eq.(12), which correspondsto the substitution [[3 , , → − ig q n ± ν n ± ν n ± ν f a a a f a ac , [[3 , , → − ig q n ± ν n ± ν n ± ν f a a a f a ac , (15)we obtain for the pole prescription of the order g induced vertex 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 , (16)where g ± ( i, j, m ) = (cid:20) − / k ± i − iǫ (cid:18) k ± m + iǫ + 1 / k ± m − iǫ (cid:19) + − / k ± i + iǫ (cid:18) k ± m − iǫ + 1 / k ± m + iǫ (cid:19)(cid:21) . (17)Eq.(17) can be further compactified, making use of the identity,1 k ± + iǫ − k ± − iǫ = − πiδ ( k ± ) , (18)leading to g ± ( i, j, m ) = (cid:20) − k ± i ][ k ± m ] − π δ ( k ± i ) δ ( k ± m ) (cid:21) . (19)Using the eikonal identity for Cauchy principal values eq. (6) and the condition k ± + k ± + k ± = 0 we find that the eikonal function g obeys itself the eikonal identity in thepurely algebraic sense i.e. g ± (3 , ,
1) = − g (1 , , − g (3 , , . (20)9ose-symmetry of eq. (19) is then easily verified. Invariance of eq. (16) under ǫ → − ǫ and symmetry under { k ± , k ± , k ± } → {− k ± , − k ± , − k ± } in analogy to the unregulatedvertex fig. 2 are also satisfied by the regulated vertex.A comment is in order concerning the two delta functions appearing in eq. (19).At first sight they seem to be in conflict with the high energy expansion of underlyingQCD amplitudes which requires non-zero k ± i , i = 1 , ,
3. These delta-functions appearhowever only due to the identity eq. (18) which allows to reduce eq. (17) into the morecompact expression eq. (19). Eq. (17) and the corresponding expressions in eq. (13) areon the other hand free of any delta-functions and can be therefore directly related to anexpansion of QCD Feynman diagrams. In addition one should note that any product oftwo Cauchy principal values can be related to a product of two delta-functions throughthe identity eq. (6) and therefore demanding complete absence of such terms is hard toachieve.
To define the pole prescription of the order g and higher vertices we follow the samepattern. The basis of the color tensor with four adjoint indices t c t c t c t c has now 24independent elements. Again it can be decomposed into a six dimensional maximal anti-symmetric sub-sector which has a basis in terms of combined anti-symmetric structureconstants alone and symmetrization of lower dimensional maximal anti-symmetric sub-sectors. A possible decomposition is given by[[[4 , , , S ([1 , , [3 , S ([[1 , , S ([1 , S (1234)[[[4 , , , S ([1 , , [2 , S ([[3 , , S ([1 , , , , S ([1 , , [2 , S ([[3 , , S ([1 , , , , S ([[1 , , S ([2 , , , , S ([[4 , , S ([2 , , , , S ([[1 , , S ([2 , S ([[4 , , S ([3 , S ([[2 , , S ([[4 , , . (21)To determine the pole prescription of the order g induced vertex we start from thefollowing expression at Lagrangian level − g v ± ∂ ± − ǫ v ± ∂ ± − ǫ v ± ∂ ± − ǫ v ± ∂ σ A ∓ , (22)which is obtained from the replacement D ± → D ± − ǫ and subsequent expansion in g . Re-writing the color tensors of the resulting vertex in terms of the above basis andsetting all tensors to zero, apart from the basis elements of the maximal anti-symmetric10ector of order four, contained in the first column of eq. (21), we obtain the followingregulated order g induced vertex 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) , (23)where the function g ± ( i, j, m, n ) is defined as g ± ( i, j, m, n ) = ( − (cid:26) k ± i + iǫ (cid:20) k ± n + iǫ (cid:18) k ± n + k ± m + iǫ + 1 k ± n + k ± m − iǫ (cid:19) +1 k ± n − iǫ (cid:18) k ± n + k ± m − iǫ + 1 k ± n + k ± m + iǫ (cid:19) (cid:21) + 1 k ± i − iǫ (cid:20) k ± n − iǫ (cid:18) k ± n + k ± m − iǫ + 1 k ± n + k ± m + iǫ (cid:19) + 1 k ± n + iǫ (cid:18) k ± n + k ± m + iǫ + 1 k ± n + k ± m − iǫ (cid:19) (cid:21)(cid:27) . (24)Similarly to eq. (17) this function g can be written in terms of Cauchy-principal valuesand delta-functions, 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) . (25)Using eq.(6), eikonal identities such as g ± (4 , , ,
2) = g ± (2 , , ,
4) + g ± (2 , , ,
1) + g ± ( k − , k − , k − , k − ) , (26)can be shown to hold, which allow to prove Bose-symmetry of the induced vertex withthe above pole-prescription. Invariance under ǫ → − ǫ and anti-symmetry under thesubstitution { k ± , k ± .k ± , k ± } → {− k ± , − k ± − k ± , − k ± } in accordance with the behaviorof the unregulated vertex fig. 3 can be shown to hold for the regulated vertex eq. (23).At this stage the general recipe for the construction of the pole prescription of theinduced vertices should be clear. The starting point is the projector P (1)A which actstrivially on the single generator, P (1)A t a = t a , (27)11nd defines in this way the maximal antisymmetric sector of order one. To arrive atthe projector P ( n )A which projects the color tensor with n adjoint indices t a ....t a n on itsmaximal anti-symmetric sub-sector we proceed as follows. We first construct a basisfor all possible color tensors with n adjoint indices which can be obtained throughsymmetrization of maximal anti-symmetric sub-sectors of order n − n adjoint indices which are orthogonal to this (partial) symmetricsub-space, define then the maximal anti-symmetric sub-sector of order n . P ( n )A thenprojects a generic color tensor with n adjoint indices onto this maximal anti-symmetricsub-sector of order n . We have explicitly verified up to n = 5 that this sub-sector has abasis in terms of ( n − N c ) structureconstants, as contained in the unregulated induced vertices. Given this definition ofprojectors, the pole prescription can be given directly at Lagrangian level, if desired.This requires to replace the unregulated operator W ± ( v ) in the induced Lagrangianeq. (1) by W ǫ ± [ v ] = 12 (cid:20) P A (cid:18) v ± D ± − ǫ ∂ ± (cid:19) + P A (cid:18) v ± D ± + ǫ ∂ ± (cid:19)(cid:21) , (28)where the symmetrization in ǫ has no effect on the resulting Feynman rules and merelyserves to ensure hermicity of the regulated Lagrangian. The projector P A acts orderby order in g on the SU( N c ) color structure of the gluonic fields v ± ( x ) = − it a v a ( x ), P A (cid:18) v ± D ± − ǫ ∂ ± (cid:19) ≡ − i (cid:18) P (1)A ( t a ) v a ± − ( − ig ) v a ± ∂ ± − ǫ v a ± P (2)A ( t a t a )+ ( − ig ) v a ± ∂ ± − ǫ v a ± ∂ ± − ǫ v a ± P (3)A ( t a t a t a ) − . . . (cid:19) , (29)where P ( n )A are the projectors of the color tensors with n adjoint indices on the maximalanti-symmetric sub-sector of order n . We derived a pole prescription for the higher order induced vertices of Lipatov’s highenergy effective action which respects the symmetry properties of the unregulated in-duced vertices and leads to identical color structure for regulated and unregulatedvertices. Explicit expressions have been derived up to the order g induced vertex,while a recipe for the determination of the prescription of the order g n induced vertices n ≥ Acknowledgments
I am deeply indebted for intense and numerous discussion to J. Bartels and L. N. Li-patov. I also want to thank Agustin Sabio Vera for useful comments. Financial sup-port from the German Academic Exchange Service (DAAD), the MICINN under grantFPA2010-17747, the Research Executive Agency (REA) of the European Union underthe Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet) and the DFGgraduate school “Zuk¨unftige Entwicklungen in der Teilchenphysik” is gratefully ac-knowledged.
A Pole prescription and high energy limit of QCD
In the following we take a closer look on the relation between the proposed pole pre-scription and the high energy limit of QCD Feynman diagrams. To start with, weconsider the high energy limit of the QCD gq → gq scattering at tree-level. WithinQCD, this process is described by the sum of the following diagrams k , ν , c p b k , ν , c p ′ b = + + . (30)The effective action describes on the other hand the same process (up to correctionssuppressed by powers of s = 2 p a · p b ) as k , ν , c p b k , ν , c p ′ b = + + O (cid:18) | t | s (cid:19) , (31)with t = q = ( p a − p ) . For covariant gauges, to which we restrict here, the firsteffective diagram can be identified with the high energy limit of the first QCD diagram,while the second effective diagram which contains the induced vertex of order g can beidentified with the high energy limit of the second and third QCD diagram. Leaving13side the polarization vectors of the gluons one has in the high energy limit+ == ¯ u ( p ′ b ) (cid:20) igt c γ ν i ( p (cid:30) b + k (cid:30) )( p b + k ) + iǫ igt c γ ν + igt a γ ν i ( p (cid:30) b − k (cid:30) )( p b − k ) + iǫ igt c γ ν (cid:21) u ( p b )= ¯ u ( p ′ b ) ign (cid:30) − i/ q (cid:20)(cid:18) ig q t c t c k +1 + iǫ/p − b − ig q t c t c k +1 − iǫ/p − b (cid:19) ( n + ) ν ( n + ) ν (cid:21) u ( p b ) + O (cid:18) | t | s (cid:19) , (32)where the expression in the squared bracket can – up to a projection on the color octetsector – be identified with the order g induced vertex as obtained in eq. (7). This is thecore of the statement that the replacement D ± → D ± − ǫ leads to a pole prescriptionin accordance with the high energy expansion of QCD Feynman diagrams.While in the case of anti-symmetric color a dependence of the squared bracket onthe sign of p − b is absent, the same is not true for the symmetric color sector. As aconsequence high energy factorization of eq. (32) is only completely realized in theanti-symmetric color sector. The latter changes the sign under s → − s and thereforecorresponds – from the point of view of Regge theory – to a negative signature. Thesub-leading symmetric color sector, which is invariant under s → − s and thereforecarries positive signature, keeps a residual dependence on the light-cone energy of thescattering quark and therefore does not factorize completely. In addition the resultingcolor tensor depends in this case on the representation of the generators t d in eq. (32).Projecting on the color octet, we obtain for the symmetric sector for generators in thefundamental representation t aF the symmetric SU( N c ) structure constanttr F (cid:16) { t aF , t bF } t cF (cid:17) = 12 d abc . (33)Generators in the adjoint representation t aA yield instead a zero resulttr A (cid:16) { t aA , t bA } t cA (cid:17) = 0 . (34)For the anti-symmetric sector, the commutator leads in both cases to the anti-symmetricstructure constant f abc . The above mapping from the prescription D ± → D ± − ǫ tothe prescription obtained from Feynman diagrams holds also for higher order inducedvertices. For the order g induced vertex one starts in this case with the gq → ggq scattering process in Quasi-Multi-Regge-Kinematics where the final state gluons are of We denote in this paragraph generators in the fundamental representation with a label F andgenerators in the adjoint representation with a label A. Apart from this paragraph t a always denotesa generator in the fundamental representation. g induced vertex read p , ν , c p b p , ν , c p ′ b p , ν , c = + ++ + + + . . . , (35)where the dots indicates Feynman diagrams which can be associated with combinationsof lower induced vertices and pure QCD vertices. Taking now all gluon momenta to beincoming, and leaving again aside the polarization vectors of the gluons one finds forthe sum of diagrams depicted diagrams in eq. (35) in the high energy limit¯ u ( p ′ b ) ign (cid:30) − u ( p b ) i/ q (cid:20)(cid:18) − ig q t c t c t c ( k +1 + iǫ/p − b )( − k +3 + iǫ/p − b ) + − ig q t c t c t c ( k +2 + iǫ/p − b )( − k +3 + iǫ/p − b )+ − ig q t c t c t c ( k +1 + iǫ/p − b )( − k +2 + iǫ/p − b ) + − ig q t c t c t c ( k +3 + iǫ/p − b )( − k +2 + iǫ/p − b )+ − ig q t c t c t c ( k +2 + iǫ/p − b )( − k +1 + iǫ/p − b ) + − ig q t c t c t c ( k +3 + iǫ/p − b )( − k +1 + iǫ/p − b ) (cid:19) · ( n + ) ν ( n + ) ν ( n + ) ν (cid:21) + O (cid:18) q s (cid:19) , (36)from where the relation to expressions such as eq. (12) becomes clear. Similar resultshold then in an apparent way for higher order induced vertices. References [1] V. S. Fadin, E. A. Kuraev and L. N. Lipatov, Phys. Lett. B (1975) 50, E. A. Ku-raev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP (1976) 443 [Zh. Eksp.Teor. Fiz. (1976) 840], L. N. Lipatov, Sov. J. Nucl. Phys. (1976) 338 [Yad.Fiz. (1976) 642], E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP (1977) 199 [Zh. Eksp. Teor. Fiz. (1977) 377], I. I. Balitsky and L. N. Lipatov,Sov. J. Nucl. Phys. (1978) 822 [Yad. Fiz. (1978) 1597].[2] V. S. Fadin and L. N. Lipatov, Phys. Lett. B (1998) 127 [hep-ph/9802290].[3] L. N. Lipatov, Nucl. Phys. B (1995) 369 [hep-ph/9502308]. L. N. Lipatov, Phys.Rept. (1997) 131 [hep-ph/9610276].154] E. N. Antonov, L. N. Lipatov, E. A. Kuraev and I. O. Cherednikov, Nucl. Phys. B (2005) 111 [hep-ph/0411185].[5] M. Hentschinski, Acta Phys. Polon. B (2008) 2567 [arXiv:0808.3082 [hep-ph]].[6] M. Hentschinski, J. Bartels and L. N. Lipatov, arXiv:0809.4146 [hep-ph],M. Hentschinski, PhD-thesis, arXiv:0908.2576 [hep-ph], M. Hentschinski, Nucl.Phys. Proc. Suppl. (2010) 108 [arXiv:0910.2981 [hep-ph]].[7] M. A. Braun, M. Y. .Salykin and M. I. Vyazovsky, [arXiv:1109.1340 [hep-ph]],M. A. Braun, L. N. Lipatov, M. Y. .Salykin and M. I. Vyazovsky, Eur. Phys. J. C (2011) 1639 [arXiv:1103.3618 [hep-ph]].[8] M. A. Braun and M. I. Vyazovsky, Eur. Phys. J. C (2007) 103 [hep-ph/0612323].[9] M. Hentschinski and A. Sabio Vera, arXiv:1110.6741 [hep-ph].[10] G. Chachamis, M. Hentschinski, J. D. Madrigal Martinez and A. Sabio Vera, inpreparation.[11] J. Bartels and M. Hentschinski, in preparation.[12] A. Bassetto, G. Nardelli and R. Soldati, Singapore, Singapore: World Scientific(1991) 227 p[13] M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Nucl. Phys. B (2002)431 [hep-ph/0206152].[14] I. Balitsky, Nucl. Phys. B (1996) 99 [hep-ph/9509348].[15] J. Bartels and M. Hentschinski, JHEP (2009) 103 [arXiv:0903.5464 [hep-ph]], J. Bartels, M. Hentschinski and A. -M. Mischler, Phys. Lett. B (2009)460 [arXiv:0906.3640 [hep-ph]], J. Bartels, C. Ewerz, M. Hentschinski and A. -M. Mischler, JHEP1005