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Nuclear Physics B 901 (2015) 115-138

www.elsevier.com/locate/nuclphysb

Discontinuities of BFKL amplitudes and the BDS ansatz *

V.S. Fadin a b, R. Fiorec*

a Budker Institute of Nuclear Physics of SD RAS, 630090 Novosibirsk, Russia b Novosibirsk State University, 630090 Novosibirsk, Russia c Dipartimento di Fisica, Universita della Calabria, and Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza, Arcavacata di Rende, I-87036 Cosenza, Italy

Received 1 August 2015; accepted 12 October 2015

Available online 20 October 2015

Editor: Herman Verlinde

Abstract

We perform an examination of discontinuities of multiple production amplitudes, which are required for further development of the BFKL approach. It turns out that the discontinuities of 2 ^ 2 + n amplitudes obtained in the BFKL approach contradict to the BDS ansatz for amplitudes with maximal helicity violation in N = 4 supersymmetric Yang-Mills theory with large number of colors starting with n = 2. Explicit expressions for the discontinuities of the 2 ^ 3 and 2 ^ 4 amplitudes in the invariant mass of pairs of produced gluons are obtained in the planar N = 4 SYM in the next-to-leading logarithmic approximation. These expressions can be used for checking the conjectured duality between the light-like Wilson loops and the MHV amplitudes.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

The BFKL (Balitsky-Fadin-Kuraev-Lipatov) approach [1-4] is based on the multi-Regge form of scattering amplitudes with gluon quantum numbers in all cross-channels. For the ampli-

* Work supported in part by the Ministry of Education and Science of the Russian Federation, in part by RFBR, grant 13-02-01023, and in part by Ministero Italiano dell'Istruzione, dell'Università e della Ricerca.

* Corresponding author.

E-mail addresses: fadin@inp.nsk.su (V.S. Fadin), roberto.fiore@cs.infn.it (R. Fiore).

http://dx.doi.org/10.1016/j.nuclphysb.2015.10.006

0550-3213/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

tude A2^n+2 of the process A + B ^ A' + G1 +-----+ Gn + B' of production of n gluons with

momenta ki, k2,kn in the multi-Regge kinematics (MRK) this form can be written as

= i if1 ^ ^ ] — ( ^ r^V^1, (1.1)

\f=\ tA \kl-1\\kl\) YRiRi+1) tn+i\\kn\\qn+l\' BB y '

where w(t) is called gluon trajectory (in fact, the trajectory is 1 + w(t)), rR,A and rR,B are

the particle-particle-Reggeon (PPR) vertices, or the scattering vertices, and YaR + are the Reggeon-Reggeon-gluon (RRG) vertices, or the production vertices. Moreover,

s = (PA + pb)2, si = (ki-1 + kt)2, i = 1,...,n+ 1, ko = Pa', kn+1 = Pb', q1 = pa - p'a , qj+1 = qj - kj, j = 1,...,n, qn+1 = pb' - pb , (1.2)

the vector sign means transverse to the pA, pB plane components. In the MRK

1T+1 ss » si >>\t' \ —q2, s - '' = ' S . (1.3)

nn=1 kt2

The Reggeon vertices and the gluon trajectory are known in the next-to-leading order (NLO), that means the one-loop approximation for the vertices and the two-loop approximation for the trajectory, in SYM as well as in QCD. It is just the accuracy which is required for the derivation of the BFKL equation in the next-to-leading logarithmic approximations (NLLA), taking into account all radiative corrections of the type as (as ln s)n. To be precise, note that in this approximation one has to consider not only the amplitudes (1.1), but also amplitudes obtained from them by replacement of one of final particles by a couple of particles with fixed (of order of transverse momenta) invariant mass.

The sign iR in the Eq. (1.1) means the real part. It is important that this simple factorized form is valid only for the real part of the amplitudes. Fortunately, the imaginary parts are not essential for the derivation of the BFKL equation in the NLLA, because they are suppressed by one power of ln s' in comparison with the real ones, and with the NLLA accuracy do not contribute in the unitarity relations. But understanding of properties of the imaginary parts which are associated with the discontinuities in the variables s'j = (k + kj)2 is very important. First, it is necessary for the justification of the BFKL approach, that means a proof of the multi-Regge form of multiple production amplitudes. Second, account of the imaginary parts is indispensable in further development of the BFKL approach. As it was pointed above, they are not essential for derivation of the BFKL equation in the NLLA, but they must be taken into account in the NNLLA.

The idea of the multi-Regge form appeared in Refs. [1,5] from results of fixed order calculations. Later it was proved in the leading logarithmic approximation (LLA) [6] with use of the s-channel unitarity. The proof of the multi-Regge form in the NLLA is based also on the s-channel unitarity [7].

Here it is necessary to recall that as compared with ordinary particles, Reggeons in the Regge-Gribov theory of complex angular momenta possess an additional quantum number, called signature. At large s' the signature means parity with respect to the substitution s' ^ -s'. The signature of the Reggeized gluon is negative, and the real part of the amplitude presented in Eq. (1.1) coincides with the real part of the amplitude A-l2+n with the Reggeized gluons (and,

consequently, with the negative signatures) in all ti channels. Amplitudes with the positive signature in the si -channel are suppressed because of the cancellation of leading powers of log si, so that with the NLLA accuracy ^A2^2+n = ^A-i2+n.

Compatibility of unitarity with the multi-Regge form leads to the bootstrap relations [8] connecting discontinuities of the amplitudes with products of their real parts and gluon trajectories:

\i=j+1

Here ^A2^n+2 is the multi-Regge form (1.1) and the sij -channel discontinuities must be calculated using this form into the unitarity conditions. Note that for multi-particle amplitudes the discontinuities are not pure imaginary, since a discontinuity in one of the channels can have, in turn, a discontinuity in another channel. But these double discontinuities are sub-sub-leading, so that they are neglected in Eq. (1.4) and in the following.

It turns out [7] that the fulfillment of an infinite set of the relations (1.4) guarantees the multi-Regge form of scattering amplitudes and that all bootstrap relations are fulfilled if several conditions imposed on the Reggeon vertices and the trajectory (bootstrap conditions) hold true. The most complicated condition, which includes the impact factors for Reggeon-gluon transition, was proved recently, both in QCD [9-11] and in its supersymmetric generalizations [12].

The proof that the fulfillment of the bootstrap relations (1.4) is ensured by the bootstrap conditions is based on the form of the discontinuities derived from the unitarity in Ref. [7]. Besides of the Reggeon vertices and the trajectory entering in Eq. (1.1), the discontinuities contain as building blocks the impact factors for particle-particle and Reggeon-particle transitions, the kernel of the BFKL equation and the four-Reggeon gluon production vertex. In fact, the bootstrap conditions are conditions on these building blocks. But since the impact factors for particle-particle and Reggeon-particle transitions, the kernel of the BFKL equation and the four-Reggeon gluon production vertex are expressed in terms of the Reggeon vertices and the trajectory, one can say that the bootstrap conditions are imposed on the Reggeon vertices and the trajectory.

The expressions for discontinuities obtained in Ref. [7] are rather formal, since the impact factors, the kernel and the four-Reggeon gluon production vertex are not given explicitly. In this paper we obtain explicit expressions for the discontinuities of multiple production amplitudes in N = 4 SYM with large number of colors (in the planar approximation). Consideration of the discontinuities in this theory is also interesting for two reasons. First, it provides a simple demonstration of imperfection of the BDS (Bern-Dixon-Smirnov) ansatz [13,14] MBDS for multi-particle amplitudes with maximal helicity violation (MHV amplitudes). Second, the discontinuities can be used for the verification of the hypotheses used for the calculation of corrections to this ansatz. It is believed (but not yet proved) that the true amplitudes can be presented as the product of MBDS and the remainder function R, where MBDS contains all infrared divergences and R depends only on the anharmonic ratios of kinematic invariants [15-21]. This property is called dual conformal invariance. Another property is the conjecture (also not yet proved) of correspondence between the MHV amplitudes and expectation values of Wilson loops [19,20,22-25]. All this makes important the direct calculation of the discontinuities.

The paper is organized as follows. In the next Section we introduce the notation, give the general expression for the discontinuities and use it for the calculation of the discontinuity of the amplitude A->2. Discontinuities of the amplitude A-l3 are found in Section 3. Section 4 is devoted to the calculation of discontinuities of the amplitude A-}4. Discontinuities of ampli-

J2discs,j I Ai,Jn+2 = {a(tj+1) - a(tj)) ^A2^n+2 . (1.4) 1=0 I

tudes with a larger number of final particles are considered in Section 5. Conclusions are drawn in Section 6. Appendices A, B and C contain some details of calculations.

2. Definitions, notation and the A-}., discontinuity

Let us first present explicit forms of the gluon trajectory and the Reggeon vertices in N = 4 SYM with the accuracy up to terms vanishing in the limit e ^ 0. In the NLO the vertices, as well as the impact factors, are scheme-dependent. We will use the scheme introduced in Ref. [26] and then developed in Ref. [7], which we call standard one. But since usual dimensional regulariza-tion is incompatible with supersymmetry, we will use the dimensional reduction instead of the dimensional regularization. The NLO trajectory is given by [27-35]

o(t) = -2g 2(- + ln(-O) 4

Z(2)( I + 21n(-t) ) - Z(3)

where Z(n) is the Riemann zeta-function,

g2NcV(1 - e)

6 (4n)2+e ' 2 '

T(x) being the Euler gamma-function, and D is the space-time dimension.

For the gluon polarization vectors in the Reggeon vertices and impact factors we will use the L and R light-cone gauges (eLn2) = 0 and (eRn1) = 0 respectively, with the light-cone vectors n2 and ni such that

(n 1 n2) = 1, (PAPB) — (:PAn2)(PBn\)

eL = ^ -■

(ejk±) kn2

eR = eR -

(eRk±)

Note that the transverse parts of the polarization vectors in the left and right gauges are different. It is easy to see that the polarization vectors are connected by the gauge transformation:

eL = eR - 2 (-^k k2

For transverse components this means

— ^uve

— ^uve

Qßv = = giv 2

kißkiv

VuvVvP = gP.

Using the results of Refs. [28,36,37] and [38] for the one-loop gluon, quark and scalar corrections correspondingly, for the gluon-gluon-Reggeon vertex we have

rR>o = gTRG(e*'e )

1 + g2(q2)M -2 + 5Z(2)

Here q is the Reggeon momentum, e and e *' are the polarization vectors of the initial and final gluons G and G' respectively (they have to be taken in the same gauge), TRR,G is the color group

generator in the adjoint representation. For simplicity, here and in the following we use for color indices the same letters as for particles and Reggeons.

The Reggeon—Reggeon-gluon vertex was obtained in the Born approximation in Ref. [5] and looks as

YrR) = STr\r2 el(k)C'^(q2 ,qi), (2.9)

(qj2 kp2\ I q2 kp\ \

---+ 2- - P2^ ---+ 2-

kp1 P\P2J \kp2 P1P2 J

= - - 2-P-) (ki - 2q2J + - 2q22J • (2.

The vertex is gauge invariant, being C^(q2, q\)k^ = 0. In the light cone gauges (2.4) we get

^(k)C^(q2,qi) = eIf CIL{q2,qi) = eR_*CR(q2,q!) , (2.11)

nLr \ n r \ n2C(q2,q1)1 0/ , qu.

Cj_(q2,q1) = C^(q2,q1)----kf = -2 1 qi± - kf —j-

kn2 \ k|

Cf (q2,q1) = C±(q2,qx) - ^^l^kf = -2 ( q2f + kf ) . (2.12)

kni ^ kf J

It makes sense to note that using the light-cone gauges does not mean loss of generality. One can restore any vertex in a gauge invariant form from its form in one of the gauges (2.4). Let us demonstrate it here for the vertex (2.9), denoting C(q2, q1) there as C for brevity. Note that C can be changed by adding terms proportional to k without changing the vertex (2.9), as well as Cf and CR defined in formulas (2.12), and without loss of the gauge invariance. Let us choose these terms in such a way that C goes to CL subject to the condition (CLn2) = 0. Then, we have

L L n1CL

CL = CL + n-n2 . (2.13)

On the other hand, from kCL = 0 we have L (n1CL)(n2k)

CLkf + —-= 0 , (2.14)

f n1 n2

so that

CL = CL - n2 • (2.15)

As it has been said, using in Eq. (2.9) CL instead of C does not change the vertex leaving it gauge invariant. Thus, we obtain the gauge-invariant form of the vertex from its form in the light-cone gauge. Using the relations (2.12) one can see that CL is equal to C - k, where C is the original form given by Eq. (2.10).

One-loop gluon corrections to the vertex were calculated in Refs. [36,39-41]. In the last paper they were obtained at arbitrary D = 4 + 2e dimension. With the same accuracy, the quark and scalar corrections were obtained in Refs. [42] and [35] respectively. In the N = 4 SYM, with the

accuracy resulting when the terms singular at small k are given at arbitrary D, but the other terms in the limit e ^ 0, we have in the dimensional reduction

YR1R2 (qi,q2) = Y^iquq}(1 - ¿2[^ - y + M$)])

(2.16)

A general representation for the discontinuities was derived in Ref. [7] (it is presented also in Ref. [43]). The discontinuity of A-L+2

in the si,j -channel is represented as i=j

-4i(2n)D-2S(qi± - q{j+x)± - ^ki±) disc,SiJ A2^n+2

= / S1 yi^i A yr-!ri / Si \®(ti)\

x(G'Ritn( k^)"(jjj+1 }

sl \m(tl) YRilRi+1\ ( Sn+1 y^n+^r^B1 (2 17)

=+t\ \kl-l\ \kl \' ti j\ \kn \ \qn+1 \' t(n+1) ' .

Here the bra- and ket-states (GiRi \ and \GjRj+1} denote the impact factors for the Reggeon-gluon transitions, K and Gl are the operators of the BFKL kernel and the gluon production, which acts in the space of states \GiG2} of two t-channel Reggeons with the orthonormality property

G1G2G1G2} = ?12?22S(n - - ?2/)5C1C'1, (2.18)

where ri and r/ are the Reggeon transverse momenta and ci and ci are their color indices. The operators are specified by their matrix elements and the states are defined by their projections on the two-Reggeon states.

If i = 0 we must omit all factors to the left of (G0R0\ and replace (G0R0\ by the impact factors of A ^ A' transition {A'A\ and k0 - q0 by pA> - Pa; in the case j = n + 1 we must omit all factors to the right of \Gn+\Rn+2} and perform the substitutions \GJn+\Rn+2} ^ \B>B}, kn+1 + qn+2 ^ Pb> - Pb .

For the discontinuity discS A2^2 = A2^2(s + i0) - A2^2(s - i0) we have

fit ln 1 s 1

-4i(2n)D-2S(q-qB) discs A2^2 = 2s{A'A\e Vq 2>\B'B} , (2.19)

where q = pa - Pa>, qB = Pb> - Pb .

We have to pay attention here on the fundamental difference between the sense of the representation (2.19) used here and that of the formally quite similar representation of the discontinuities of amplitudes with the Pomeron exchange. The BFKL Pomeron means the positive signature and the color singlet in the t -channel, while the amplitudes considered in this paper are the amplitudes with the negative signature and the adjoint representation of the color group in the t-channel. The gluon Reggeization makes the discontinuities (2.19) much simpler than the discontinuities of amplitudes with the Pomeron exchange. Indeed, the bootstrap conditions of the gluon Reggeization [7] tells us that

(A'A\ = r«Ag(RoAq)\ , \B'B)=g\Rm(qBWl,B , (2.20)

¡C\Ra(q))=a(t)\Ra(q)) , (2.21)

2(2gJ)D-1 Ra($ ')\Ra(q)) = &R-R&(q - q ')a(t) , (2.22)

where rR,A and rR,B are the scattering Reggeon vertices entering in the form (1.1), \Ra(q)) is the process independent eigenstate of the kernel K with eigenvalue a(t) and normalization (2.22). It is transformed according to the adjoint representation of the color group. In the right side of Eq. (2.22) R' and R are the color indices of the eigenstates; in Eq. (2.20) summation over the color indices R is assumed. Note that the bra and ket vectors are related by the left-right substitution, where A ^ B, A' ^ B', n ^ n2, that means, in particular, replacement of the left and right gauges.

Fulfillment of the bootstrap conditions (2.20)-(2.22) was proved in the NLO both in QCD [44,45] and SYM [38]. Using these conditions we have from the representation (2.19)

(2s\ ( s \a(t)

discs^2 = -in\ — )a(t)rR/A[ J2) rR'B- (2.23)

It is easy to see that the result (2.23) with account of the form (1.1) at n = 0 is in agreement with the bootstrap relation (1.4).

Finally, let us present the eigenstate \Ra(q)). From Refs. [44] (see also Ref. [9]) and [12] we obtain with the accuracy up to terms vanishing in the limit e ^ 0

G1 G2\Ra( q))

„ / „ r 1 (r,2 \ / r„2 \ 1 \

, (2.24)

= Hq-h -T2)Tr& 1 +g2

-r(2)-2ln ÏW&2:

where r1 and r2 are the momenta of the Reggeons Q\ and G2 respectively.

It is necessary to note here that the accuracy of Eq. (2.24) does not provide preservation of nonvanishing in the limit e ^ 0 terms of the g2 order in the product

(K(q')\R»(a))= £ f R(a'WG)^^^-¿i -h){GiG2\R»(a))

G1G2 h h

(the summation here is performed over color states of the Reggeons G1 and G2) because of the infrared divergency of the integration measure. To provide the preservation one has to keep in (GiG2\Rm(q)) terms of the order O(g2e).

3. Discontinuities of the 2 ^ 3 amplitude

3.1. Discontinuities in the s1 and s2 channels

For the s1-channel discontinuity we obtain from the general form (2.17)

-4i(2n)D-2S(qi - k- q2) disc^ ^2^3

= 2s(A'A\/KA)\GR2) 1 (-S2-, (3.1)

t2^ \ k1 \ \ q2 \'

where q1 = pA - pa', k and q2 are the momenta of the gluon G and the Reggeon R2, \GR2) is the impact factor for the Reggeon-gluon transition. The bootstrap conditions (2.20) and (2.21) give us

-4i(2n)D 2S(qi - k-q2) disc^ ^2^3

„ / Si X^l) 1 / S2 \m(t2) R

= 2sTR,J--(-rR2»g{Ra,(qi)\GR2) . (3.2)

AAV \ki\\qi\/ t2\\ki\\hV BBg

The impact factors for Reggeon-gluon transitions were calculated in Refs. [10,12] in the special scheme (the so-called bootstrap scheme) which simplifies the proof of the most complicated bootstrap condition

{GRi\-gqi2{Rm(qi)\g = gy^R {Rm(qi — k)\ , (3.3)

where G is the gluon production operator, k is the gluon momentum. The Reggeon R in the condition (3.3) has the momentum qi - k and the same color indices as the eigenstate {Rm(qi — k)\; summation over them is assumed. The eigenfunction {Rm(qi)\GiG2) in the bootstrap scheme also was obtained in Ref. [i2]. We could calculate the matrix element {Rm(qi)\GRi) in Eq. (3.2) just in this scheme. It turns, however, that it is much more convenient, especially in the further calculations, to use the scheme which we call conformal. It is associated with the modified kernel ftm, introduced in Ref. [46], which is obtained from the usual BFKL kernel in the adjoint representation, by subtraction of the gluon trajectory depending on the total t-channel momentum. One of advantages of this kernel is its infrared safety, which permits to consider this kernel at physical transverse dimension D — 2 = 2. But the most important advantage is its behavior under Möbius transformations in the two-dimensional transverse momentum space. It is not difficult to see that in the leading order Km is Möbius invariant. But in the NLO in the standard scheme, in which the kernel was initially calculated [47,48], it is not Möbius invariant. The existence of the scheme where the modified kernel is Möbius invariant (Möbius scheme) was conjectured in Ref. [49] and then proved in Ref. [50], where the transformation from the standard Km to the conformal (Möbius invariant) kernel Kc was found. It reads

K — K - -

Kc — Km 4

in (q i2 q 22), k 1

where KB is the LO kernel. Note that since K and Km differ only for the trajectory depending on the total t-channel momentum, which is a C -number, in all commutators K can be replaced by Km and vice versa. We will use the following representations for the kernel:

GG^\Ki\GxG2} = ^(rx +r2 -ri-rJ>)q 2J2(PrG§ KR(h,h; 1). (3.5)

Here r and r{> are the Reggeon momenta, q = r\ + r2, l = r - r>, PR is the projection operator on the representation R of the color group, and

r 2r 2

KR(h, 12; l) = KR(h,T2; i) + r-qrr (v(-r12)8(n - r() + u(-r22)8(r2 - l)) , (3.6)

where KR is called the real part of the kernel.

In general, the kernel KR(r\, i2; l) depends on R. But at large Nc only the antisymmetric and symmetric adjoint representations do survive in the decomposition (3.5), with

(PAafgfg] - —fiQiSifiG'lG2, ^Asfgfgl — NdiG'Q2diG'lG2 , (3J)

and the same kernel KR(r', r2; I). Therefore in the following we will omit the index of representation R.

In the LO the real part of the kernel is given by

KB(r p.h g2 N (2 + r22r/2 A (38)

whereas the gluon trajectory has the representation

B g2 Nct f d 1 „ 2 ojB(t) = —-C— -— , t = -q2 . (3.9)

() 2(2n)D-1 J q2( q-1)2 ' q ' '

The difference with usual denotation is in the factor q 2 in the representation (3.5). Its extraction is necessary to make the modified kernel

Km(ril) = K(ri, 12; l) - ^S(ri - r;)S(h - l) v(t) (3.10)

q 2 1 2

explicitly invariant at D = 4 with respect to the Möbius transformations

azi + b

Zi ^ —^T , (3.11)

czi + di

where a, b, c and d are complex numbers, zi = xi + iyi, xi and yi are the Cartesian components of the "dual" transverse momenta pi such that

l1 = P1 - P2, 12 = P4 - P1, r[ = P3 - p2, r2,= P4 - P3 . (3.12)

The need for the factor q 2 is clear from another point of view: the kernel could be explicitly Möbius invariant only when the corresponding normalization condition is Möbius invariant. The condition (2.18) is not invariant; to make it invariant one needs to multiply both sides on 1/q 2 and include 1/q 2 in the left-hand side in definition of the two-Reggeon states. This can be seen from the invariance of the corresponding measure,

d?1/d?2/l7|l72 5(11 + ?2 - ?/ - I) = ^ , (3.13)

where q — z' + r2 and z — r'2+r+/(r[+r+) is invariant. Here and in the following we use the chiral components r + — x + iy and r — — x — iy for the two-dimensional vectors r — (x, y). Vice versa, the two conjugate complex numbers z and z* are confronted with the vector Z through the components (z + z*)/2 and (z — z*)/(2i). At the same time, dr — dxdy — dr+dr—/2 , 8(r) — 2S(r+)S(r—) and we define S2(z) in such a way that S2(z) — 5(z+ )S(z—)/2 — 8(z).

The transformation (3.4) gives [50] Kc(r', r2; I) — Kc(z), where z — r+r',+/(r+r'+) and

B,^ [ g2Nc^\, ,(2)„ _Jg2 Nc, 1 [g2 Nc

Kc(z) = K!(z)[1- + S(2\ 1 - 3Z(3) + 8^v 8n2

1 1 + |z I 2, ,2 1-kP , , ,2, I 1 - z |4 ' ln2 I z |2 - —-^ ln I z |2 ln ■

2 |1 - z|2/ 2| 1 - z|2 |z|2

1 - z 1 - z*

(z - z*)

\x - z|2

KB(z) =

z + z*

32n3 \ \1 - z\2

— 5(2)(1 - z) f

dl l +1* ÏF\1 -1\2,

with the properties

Kc(z) = Kc(z*) = Kc(1/z), Kc(0) = 0

(3.14)

(3.15)

(3.16)

The transformation (3.4) has to be accompanied by the corresponding transformation of the impact factors and the eigenstate (Rm\. The eigenstate (Rm\c which corresponds to the kernel Kc (3.4) is

1 „ t

(Rm(q)\c = {Ra(q)\- 4 (Rm(q)\1

(3.17)

where Kf is the real part of the LO kernel (3.8), ii and i2 are the Reggeon momentum operators. Using (see Appendix A for details)

{Rm(q)\B

\ G1G2)

= -2g2S(q -l - Г2)Т*д2 ln ( ^ )ln( T*

(3.18)

we obtain from Eq. (2.24)

{R«(q)\GxG2)c = 8(q-n -Г2)Тд1д2 (1 -g2Z(2)) .

(3.19)

Now turn to the impact factor \GR2}. The impact factor corresponding to the kernel Kc (3.4) is obtained from \ GR2} in the standard scheme by the transformation

\GR2)^\GR2) + 4

\GR2)b

(3.20)

It was found, however, in Ref. [51] that impact factors for Reggeon-gluon transitions acquire the most simple form in the scheme in which not only the kernel, but also the energy evolution parameter is conformal invariant. Transition to this scheme, which is called conformal scheme, means the additional transformation for the impact factor \GR2),

\GR2) ^ \GR2) - 2ln ^q^J KBm\GR2)B Together with the transformation (3.20) it gives

(3.21)

\GR2)^\GR2)c = \GR2)- 4

(l?l22) ■

\GR2)b

|j KB\GR2)B

(3.22)

Note that the transformation (3.21) does not affect the matrix element {Rm(q1)\GR2) because {R0J \ is the eigenstate of Km with the eigenvalue equal to 0.

For amplitudes with the negative signature, the impact factors are antisymmetric with respect to the G1 ^ G2 exchange. In fact, putting

{GRl\ = {GRl\s - {GR1 \u , (3.23)

we have

{GR1\Ç1Ç2)u = {GR1\G2G1)s . (3.24)

As it follows from Ref. [51], in the conformal scheme, the impact factors {GR1 \Q1Q2)s of gluons with the polarization vectors

et = (-x + ixly ) , eL * = -1= (-x - ixly ) (3.25)

for helicities X = ±1 have the form

{GR1\Ç1Ç2)S _

_ ' )2\ tt 2лe

{GR1\G1G2)B

1+g2(h(z) - M |)- V+2z(2)

(3.26)

Here z = -q+r+/(k+r+),

{GR1\Ç1Ç2)B = -g2S(q1 -r1 -2 - k)(TR1 TG\ eL *C L(h,q1), (3.27)

^ ' д 1G2

C L(n,q1) = -2 - (-1 - , (3.28) and I+1(z) = I(z), I-1 (z) = I*(z) = I(z*), where

>m = ^z (J H^! Ï ^ - z\4

\z\2 \z\

- 6Li2(z) + 6Li2(z*) - 3ln\z\2ln-

1 - z*

1 ,2, \1 - z\2 3.

-- ln !1 - zfln^-^- -- ln2 \z\2 , (3.29)

2 \z\2 8

— ln(1 - xz). (3.30)

Note that I(0) = 0, I(1/z) = I(z)/z. In the two-dimensional transverse momentum space, with the polarization vectors (3.25) we have

i+ *C L(i1,i1) = V2 q1 r\ , e- *C l(-1 ,i1) = q1 r\_ . (3.31)

(q1 - r1)+ (q1 - r1)

The set of diagrams for the process A + B ^ A' + G + B' is evidently invariant with respect to rotating around the gluon line and the exchange A ^ B. It means that the impact factor <G1G2\GR2} can be obtained from <GR1\G1G2} by the replacement

ni ^ n2, qi ^ _q2,

h,2 ^-h,2 , (tRi TG) ^ (TR2 TG) ■ (3.32)

V ' G1G2 \ ' G1G2

The replacement n1 ^ n2 means also eR ^ eR. With account of Eqs. (2.6) and (2.7) we have

eR = _ (p) e_LÀ = _ (p) & _ uey) ■ (3.33)

Using the substitutions (3.32) and formulas (3.33) we obtain (Q1Q2\GR2)S = {G1G2GR2)

,2/ r^ 1^2 «ï \ (k2)

1 +g2( n(z) _ 2 M 7*) + 2Z(2)

2\ te 2^e

(3.34)

where z = q+r+ /(k+r+),

(G1G2\GR2)B = g2Kh +r2 _ k_q2)(TR2 TG\ eR *C R(h ,n) (3.35)

Ce R(q2, n) = -2^12 + (r1 _ q2)e_h)2^ ■ (3.36)

Using now Eqs. (3.19) and (3.34)-(3.36) we arrive to {Rm(q1)\GR2)s

g-Nr&(q1 _ k_q2)TGR2 eR * f dr^n-q^ 5(?1 +r2 _ k _ q2) 2q12 1 2 J e12r22

2 \ n„2\e

x c r(h,h)

1 + g2[ i_x(z) _ |ln^q^) _ (eLL + Z(2)

(3.37)

where X = ±1 is the gluon helicity, I+1(z) = I(z), I_1 (z) = I(z*), I(z) is defined in Eq. (3.29), and z = q+r+ /(k+r+). The integral with I_X(z) in Eq. (3.37) is not singular and can be calculated in two-dimensional transverse momentum space. Using the measure (3.13) and formulas

1 _ + _ + _

eR*C R(q2,h) = V2k— q2 r1 =V2 J2 q1 , + (q k+ (r1 _ q2)_ k+(1 _ z*) '

*C R(h, h) = = , (3.38)

k (r1 _ q2)+ k (1 _ z)

we obtain that the contribution of the term with I_X(z) in Eq. (3.37) is equal zero. Indeed, for the positive helicity it is proportional to

I(z*) = 0 ■ (3.39)

\z\2(1 _ z*)

The result (3.39) follows from the fact that in the expansion of the integrand in powers of (z*)n at \z\ < 1 and in powers of (1/z*)n at \z\ > 1 there are only terms with n > 0 (remind that, as pointed out previously, I(z) = 0, I(z) = zI(1 /z)). For the negative helicity the result is obtained

by complex conjugation. It means that the term with I-x(z) in Eq. (3.37) can be omitted. The remaining integral (details of the calculation are given in Appendix B) is

q 2 q 2

dridr2-r2^2 8(h + h - k-q2)\q2 + (h - <?2—^r, h2r22 \ (ri - q2)2 ,

= n 1+T(1 - e)(q2 + kIM i1 + ln( tq~1k I I • (3.40)

k2/ \ q22

For discS1 from Eqs. (3.2) and (1.1) we get

-4f(2^)D-2S(q1 - k- i2)discs1. ^

gt1<Rw(q1)\GR2} 2 gt1<Rw(q1)\GR2}s

= VG =2 yG . (3.41)

YR1R2 YR1R2

Here the last equality comes from antisymmetry of <Rm\G1G2} with respect to G1 ^ G2 exchange. Then, using Eqs. (2.16), (2.9), (3.37) and the equalities

i+R *C R(l2, q1) = iL *C L(q2, q1) = ^ ,

eR *C R(q2,q1) = e-L *C L(q2,q1) = V2^^ , (3.42)

which means

G(B) yrir2

= -gTGiR2V2^, y™ , = -gTRGiR,^-^ , (3.43)

^ = 1 + .n ^ (1 - 2^(2^ .3.44)

A.=+1 0 R!R2' k+

we obtain

' ' ' ' j2^ 2

Here, it is necessary to make the note analogous to that given at the end of Section 2. The accuracy of Eq. (3.19) for (Rm(q)\GiG2)c and Eq. (3.34) for (GiG2\GR2)s does not provide preservation of nonvanishing in the limit e ^ 0 corrections of the g2 order in the integral (3.37) (and therefore in the discontinuity discS1 a2-13 ) because of the infrared divergency of the integration measure in Eq. (3.37). To provide the preservation one has to find (Rm(q)\GiG2)c and (GiG2\GR2)s with higher accuracy. This issue requires special consideration. It applies also to other discontinuities discussed below.

Evidently, the s2-channel discontinuity can be obtained by the replacement (3.32) and is given by the relations

.... D-2*r t ^ discs2 gt2(GR1\RAq2)) -4i(2n) 5(qx - k - q2) ——-=-G- , (3.45)

gt2 (GRl\Rm(q2)) YRi R2

- 8Q1 - k-h)g2n 1+^(1 - e)( 1 - 2g 2r(2^(1 +ln ( ^22")) , (3.46)

^ -"W1 + "(' -(347)

. ----A I + ln I -

3.2. Discontinuity in the s channel

According to the representation (2.17), for the s -channel discontinuity we have

-4i(2n)D-2S(q1 - k - q2) disc, A{2_

- 2s(A'A\eKK w)Ç,e1CKimi)\B'B) , (3.48)

where Ç is the gluon production operator. Using the bootstrap conditions (2.20) and (2.21), we obtain

,D-2S/- f -+,discs A2__3 g2q12q22(R®(q1)\Ç^\R®(q2)) -4i(2n)D 8(q1 - k-q2) Y - 12 - • (3.49)

Then, due to the bootstrap condition (3.3), we have

gqi2(Rm(q1)\^\Rm(q2)) - (GRi\Rm(q2))-gYG1R (R»(fo - k)\R„(q2)) • (3.50)

Both matrix elements here are known: the first comes from the calculation of the s2-channel discontinuity, see Eq. (3.46), and the second from the bootstrap condition (2.22). Thus, we obtain

^t^f = -nig2(1 + ^ ( 1^)) i1 - 2g2Z(2)) - ™(<2)

„ 1 / q,2q2\ „ / / q,2q2 \ \

(3.51)

- nig2

1 + J m + 2g2 U - z(2) ln if

In fact, it was not needed at all to calculate neither the s2-channel, nor the s-channel discontinuities, because they can be expressed in terms of ,-channel discontinuities from the bootstrap relations (1.4). Indeed, for the amplitude a2~Î3 there are three relations:

disc.. A{-K disc. A{-K

- -inw(t\) ,

- -inm(t2) ,

--in(m(h) - v(t2)) • (3.52)

disc,2 A2_^ 3 disc, A2_}3

disc,1 _y 3 discs2 A2_y 3

However, they are not independent: the third of them is the difference of the first two. Therefore, there are two relationships between the discontinuities, so that only one of them is independent. It

is easy to see that the discontinuities calculated above satisfy the relations (3.52). The fulfillment of the third of them, with account of

o,(h) - oAt2) = 2g2ln||^(l - 2g2£(2)) , (3.53)

follows from Eqs. (3.44) and (3.47) and fulfillment of the second follows from Eqs. (3.47) and (3.51).

4. Discontinuities of the 2 ^ 4 amplitude

4.1. Discontinuities in the s1 and s3 channels

From the representation (2.17), for the s1-channel discontinuity we have -4i(2n)D-28(qi - ki -q2) discs1 A^4

t2V jjkljjjk2K Y*2*313^\k2\\q3W B'

therefore, using the bootstrap relations (2.20) and (2.21) and the representation (1.1) of the MRK amplitude, we obtain

D 2 -> -> discs1 ji-l4 g211 {Rm(q1)\G1R2)

-4i(2n)D 8(q1 - k1 - q2) = g A 1 • (4.2)

The ratio in the right-hand side of Eq. (4.2) is the same as in Eq. (3.41) with the replacement G ^ G1, so that using Eq. (3.44) we arrive to

= n,g2(l + (1 - 2g2Z(2)). (4.3)

Obviously, such ratio for the s3-channel discontinuity can be obtained by the replacement k1 ^ k2, q1 ^ -q3, q2 ^ -q2; it reads

= nig2(I + „(f2)) (1 -2g2Z(2)) . (4.4)

4.2. Discontinuity in the s2 channel

For the s2-channel discontinuity, using the modified kernel Km, K = 1Cm + m(t2), we have from the representation (2.17)

-4i(2n)D-28(q-qB) discs2

= 2sr*1 1 I( s3 \

A'AtA iq,iik,i/ h\ \ko\\03\J

AAt1 V ||1||*1K t3\ \k2\jq3\' BB

( s2 )m(t2) KKm lnl ,r % , I

xi^^) {G1R1 \ e Vjklj fe^ G2R3)) . (4.5)

v\ k1 \ \ k2 \ '

Here we meet two new important aspects. First, the energy dependence of the s2 channel discontinuity (4.5) evidently differs from that predicted by the BDS ansatz [14], where this dependence is the same as for the real part of Instead, according to Eq. (4.5), there is an additional

dependence coming from the matrix element with Km. For agreement with the BDS ansatz the impact factors for Reggeon-gluon transitions have to be proportional to the eigenvector of Km with zero eigenvalue, what is obviously not so. Note that the discrepancy is manifested already in the leading logarithmic approximation.

Actually, it is well known that the BDS ansatz for n-gluon amplitudes is incomplete at n > 6. The first indications of the incompleteness were obtained in Ref. [52] in the strong coupling regime using the Maldacena hypothesis [53] about the ADS/CFT duality, and in Ref. [24] using the hypothesis of the scattering amplitude/Wilson loop correspondence. Then the incompleteness was shown by direct two-loop calculations in Ref. [54]. Moreover, disagreement of the BDS ansatz with the BFKL approach is also known [55]. Dignity of the demonstration of the discrepancy presented here is its simplicity.

The second new aspect is seen from the expressions for the impact factors in Eq. (4.5)

G1R1IG1G2) = <GiRi\giG2)s - (G2Rl\G2Gl)s ,

G1G2G2R3) = <GlG2\G2R3>s - <G2Gl\G2R3>s , (4.6)

where <GiRi\GiG2>s is given by Eqs. (3.26)-(3.29) and <GiG2IG2R3) by Eqs.(3.34)-(3.36) with the replacement k ^ k2, q2 ^ q3. The new aspect is the appearance in the discontinuity of the color structure D^^D^^, where Dlc = dabc, in addition to the structure T^RT^R in the real part of the amplitude A2^n+2 presented in Eq. (1.1). Indeed, using

Nc h Nc

^Xfcj = ^ ^^cj = (4.7)

we have at large Nc

(TR1 TG1 )i(TR3 tg2 )j = Nc (TG1 tg2 + dg1 Dg2^

j j 4 V / R1R3 '

(TR1 TG1 )ij(TR'3 TG2)ji = — (-TG1TG2 + DG1 DgA . (4.8)

J J 4 V / R1R3

Writing explicitly all color factors, we obtain using Eqs. (3.23) and (3.24)

K m l^ irS^rr) <G1R1\e V|k1l|k2l/|G2R3)

= I Tg1 Tg2 + Dg1 Dg2 | 2 V / R1R3

+ — (-TG1 TG2 + DG1DG2) <G1R1\setCm \gg>h ' (4.9)

2 R1R3

where the tilde sign in the impact factors means rejection of color factors.

It is very convenient to use the conformal representation for calculation of the matrix elements in the right side of Eq. (4.9), but transition to the two-dimensional transverse momentum space in this representation must be done with caution because of the infrared divergency in the first term in the right side of Eq. (4.9). In the leading logarithmic approximation, this problem was considered in details in Ref. [46]. In principle, nothing has changed at the transition to the next-to-leading approximation.

The divergency emerges because of the singularity of the integration measure (3.13) at zero momenta of intermediate Reggeized gluons. As it follows from Eqs. (3.26), (3.27), (3.34) and (3.35), the s-pieces of the impact factors {G1R1 \G1G2)s and {G1G2\G2R3)s vanish at r1 = 0, but not at r2 = 0. It means that the first matrix element in the right side of Eq. (4.9) is divergent. Fortunately, the divergence exists only in the zero term of the expansion in powers of the BFKL kernel due to its property (3.16). Therefore, writing

_ ftm lnf^i^^^ ( ftm , _

{gR\se Vjklj \k2\/\ g2R3)s = {gR\s |e Vjklj ^J - 1| \g2R3)

+ G1R1 j s j G2 R3)s, (4.10)

we can use for the first term in the right side the conformal representation directly in the two-dimensional space. Using Eqs. (3.26), (3.27), (3.31) and (2.16), (3.43) we have for the positive helicity of the gluon G1(X1 = 1)

{G1R1 j GhG2)s „ - „ „ 1 2

V'^27s = gs(q1 - k1 -h -n)-— [1 + g2(i(z1) - z(m, (4.11)

yr1r2 1 - Z1

where z1 = -q+r+/(k+r+) and the tilde signs means omission of the color factors. Analogously, for the positive helicity of the gluon G2 (X2 = 1), we obtain using Eqs. (3.34), (3.35), (3.38) and (2.16), (3.43)

{G1G2 j g2r3)s „ - „ „ 1 2

2 2 s = -gS(q2 - k2 - h1 - ?2)--2 [1 + g2(I(z*2) - Z(m , (4.12)

Yr2r3 1 - Z2

where z2 = q+r+ /(k+ r+). The corresponding results for negative helicities are obtained by complex conjugation of Eqs. (4.11) and (4.12).

In the conformal representation, the energy evolution parameter in Eq. (4.10) is s2q22/(jh j jh j jk1 j jk2j) (instead of s2/(jk1 j j k2j ), and in the two-dimensional transverse momentum space the kernel takes the form (3.14). It has the representation

{G1G2jKc jGG) = f dv m(v, n) {G1G2jv,n){v,njGG) , (4.13)

n=-cx-w

with the eigenfunctions [50]

1 / + \ f+iv I -\-n+iv

{G1G2jv,n) = 8(r1 +n -h)rTj yr-J ' (4.14)

which form an orthonormal set with the integration measure (3.13), the eigenvalues being [49]

g2Nc (1 j n j jn j jnj \

m(v,n) = ^2 ^ - f(1 +iv - n) + f(1 - iv + n + 2f(1)

g2Nc^ , ig2N

X I1 - Z(2V + v 8n2

X - f"(1 + iv + n) + f"(1 - iv + n)

21VU'(1 - tv + M) - f '(1 + iv + M)) \ 1 I n| (v2 - £)

+ —--2-- + 3Z(3) + --^-3

v2 + n2 ) 4 (v2 + f

(v2+n2)3

(4.15)

Here f(x) = (ln r(x))'. Note that n) has the important property

m(0, 0) = 0 , (4.16)

in accordance with the bootstrap conditions. Using the representation (4.13) and Eqs. (4.11), (4.12), we obtain for positive helicities of both gluons

K m ln( )

Î2G1R11 s le V^ Mj - 1| | g2R3) s

YR1 r2 yr2r3 = sfa - k! - k2 - h)g2(1 - 2g2z(2))

x 1 +Y f dv \ea(v'n) ^I I r^ Ik2l ) - 1 | w'n+ '"(w*)-2+'v

f (' + g2«z'))zn^2 ^

f^T-ZÎ (' +g2'M ^^n(4'17)

X I -r(1 +g2I *(Z?)))(ZÏ) 2-

n IZ21 2 1 - z

where w = k+q+/(k+q+).

The second term in Eq. (4.10) must be calculated at D = 4 + 2e. Using Eqs. (3.26)-(3.28) and (3.25) for GhRx |s, Eqs. (3.34)-(3.36) and (3.33) for | G^R3)s, and the results obtained in Appendix C, we have for positive gluon helicities

t2((hR1ls 'g2r3)s ^ t 7 - . 2

-:-:-= à(qi - kl - k2 -q3)g

YR1R2 yr2r3

x^ 1 + ^^ (1 - 2gg2Z(2)) • (4.18)

Calculation of the second term in the right side of Eq. (4.9) is simplified because the infrared divergency is absent in this term, since (G1G2 'G2R3)u = 0 at r2 = 0 according to Eq. (3.24). Therefore, we have for positive helicities of both gluons

Î2G1R1 |s I eKm ta(- 1 | I G2R3)s YR1R2 KR2R3

:Sq - k1 - k2 -q3)g2(1 - 2g2Z(2))

X 1 +Y / dv \emiv'n) ^| qn | ^H |k2| ) - 1 | wnn +tv{w*)-n+'v

(i + +iv(z')

n | zi | 2 1 - zi \ > 1

where w = k+q+/(k+q+).

4.3. Discontinuities in the s02, s13 and s channels

The discontinuities in the s02, s13 and s channels can be expressed through the ones calculated above with the help of the bootstrap relations (1.4). There are four such relations, but only three of them are independent. In general, for A2^2+n, there are n + 2 bootstrap relations (1.4), for j = 0, 1, ..., n + 1, but their sum is identically zero. Denoting discsij = R{-', we

have for j = 0, 1, 2 in the relations (1.4)

R0-} + R0-} + *0-} = -inm(t1) , r{-} + R{~} - R0-} = -in (m(t2) - mt)) , *2-} - R{-} - *0-} = -in (m(t3) - m(t2)) . (4.20)

This result gives

discs02 = discs3 A-4 - discs2 a2-14 - in (m(t2) - mt)) ^A2^4 ,

discs13 j2-14 = discs1 A-4 - discs2 A-l4 - in (m(t2) - m(t1)) ,

discs <Af_^^ — discs2 A.f_^ 4 discs1 <Af_discs3 <Af_^4

- in(m(t3) + m(t1) - m(t2)) ^2^4 .

(4.21)

The same relations can be obtained from the representation of the discontinuities in terms of matrix elements of the evolution operators and the gluon production operators between the impact factor states and use of the bootstrap conditions (2.20)-(2.22) and (3.3).

Thus, all the discontinuities are expressed through the discontinuities in s1, s3 and s2 channels, and the last one evidently disagree with the BDS ansatz. It is necessary to note that in the total imaginary part of A-}4 in the channel where all sij are positive, which is defined by the sum of all the discontinuities, the contribution of the s2-channel discontinuity cancel, so we get

discsij A-!4 = discS1 a2-14 + discS3 A-|4 - inœ{t2)^A2.^4 . (4.22)

i=0 j=i + 1

5. Discontinuities of amplitudes with larger number of particles

In general, there are (n + 1)(n + 2)/2 sij -channel discontinuities for the amplitude A2^2+n, that means ten discontinuities for A2^5. The bootstrap relations (1.4) give n + 1 connections between them. For A2^5 one can choose as independent discontinuities in the channels s1, s2, s3, s4, s13 and, for example, s04. The ratios discsij A-^5/^A2^5 for the first four channels can

be obtained from the results for A-4 by evident substitutions. But the discs13 A->5 contains the new matrix element

<G1R11 e V|k11 |k21 )<G(k2)e V|k21 |k3l7| G3R4)

where G(k2) is the gluon production operator and k2 is the gluon momentum. To calculate it one needs to know its matrix elements <G1 G'21 G(k2) \ GiG2). They are known in the LO, but in the NLO only matrix elements <Rm(q2)I<G(k2)\GiG2) are known in the "bootstrap scheme" (see, for instance, Refs. [9,11,38]), which was introduced to simplify the proof of validity of the bootstrap conditions. Of course, the matrix elements <G[G2lG(k2)|GiG2) are necessary for calculation of discontinuities of amplitudes with larger number of particles. We intend to discuss this matrix element in subsequent paper.

A few words about the total imaginary part of A->5 in the channel where all stj are positive. With account of the bootstrap conditions it can be greatly simplified, so that its ratio to the real part is expressed through gluon trajectories and the ratios of the type shown in Eqs. (4.3), (4.4).

6. Conclusion

In this paper, using the BFKL approach, we have performed an analysis of the discontinuities of multiple production amplitudes in invariant masses of pairs of produced gluons in the multi-Regge kinematics. We have discovered, in particularly, that the discontinuities of the four gluon production amplitudes contradict the BDS ansatz for MHV amplitudes in planar N = 4 super-symmetric Yang-Mills theory. This contradiction is almost obvious and is already apparent in the leading logarithmic approximation. It appears also in amplitudes with more than four produced gluons.

We have obtained explicit expressions of all discontinuities for production of three and four gluons, as well as of some of discontinuities for production of a greater number of gluons in the next-to-leading logarithmic approximation. It turns out that certain discontinuities have a rather complicated form. In particular, their color structure differs from the color structure of the real part of the corresponding amplitude. In the sum of all discontinuities the complicated pieces cancel due to the bootstrap conditions, so that the sum acquires a relatively simple form and the same color structure as the real part. This result can be important for further development of the BFKL approach.

Appendix A

First, consider Eq. (3.18). Using Eq. (2.24) and Eqs. (3.5), (3.6) and (3.8) we have

g2S(r:1 +r2 -q)^R f dr/dr^r/ + r2' - q)

IG1G2) = ■

- e)n1+e

r(1 - e)n

? 2r / 2 1 2

2i ' 2

(h - r{)

r'2r ' 2 r1 r2

1'n ( W )

Due to the symmetry under the r1 ^ r2, r1/ ^ r2' exchange, it is sufficient to calculate in

f 2 ->

Eq. (A.1) the integral with ln(4y-) and to add in the answer the term with r1 ^ r2. The integral

is not infrared divergent and can be evaluated at e = 0. It can be done using the decomposition

1 /h2^2 + ^1'2 -2* / 1 1 \ 1 1

r( 2h2' 2\ (n -h)2 " J {r-+ + r+ - r1V \r- - r[- q--r'1~/

1 1 1 1 + \-T= +

r1 r1 - r\ \ri - ri q + - r1 ,

and the integral

, ^ ^ +-1--T--T- I'n ^\0(A2 - I2)

n \(a+- 1+) (b-- 1-) (a-- 1-)(b+- 1+y \a2 /

= + 'i^)^) (A.3)

The upper integration limit A is introduced because the separate terms of the decomposition (A.2) give divergent integrals. In the sum the divergencies cancel and that leads to the result

dridr^h(r{ + h - q) I r2^2 + h2?/2 A r; 2h'2

r1'2h2'2 \ (h -h/)2 q Jlni h2h2

=-2n ln( q2)ln( p). (A.4)

Using this result in Eq. (A.1) we come to Eq. (3.18). Appendix B

Let us consider the integral in Eq. (3.40). The piece of this integral with the term q2 is known from the calculation of mB(t1) and gives (with k = q1 - q2)

hj d nd h2 S(?1 +h -h) = 2n 1+T(1 - e)q2^ 1 + ln q2^ . (B.1)

The integral with the second term can be represented as

dhdS(rl +h2 -qlWl -q2)- 2

t12h22 (n - q_2)2

q{2q2f d f d n _ _ 2

q1 q 1 1 lnfa -hf . (B.2)

- 2 J ^

The last integral can be written as sum of two integrals:

dr1 w- -^2

ln(h1 - q2)

hi (h - hl)

mi^ +in(q22k 2)i. (B.3)

1 f (h2 - hHk + h2 \\q2 k2) + ln (h2 k )

Here the second integral is known, whereas in the first one the contributions of the singularities at (q2 - I) = 0 and (k + I) = 0 cancel and the integral can be calculated at e = 0 using the decomposition

1 1/1 1 \ / 1 1 \

+ —— + (B.4)

(q2 - l)2(k + l)2 q12 \q+ - l+ k+ + l+j y q2- - l- k- + l and the integral (A.3). As a result, we have

dr1 , ^ ,2 ■ ln (h - —)

(h -q\)2

= n 1+eT(1 - e)q-2 Un (h-k 2) + lnh2) + ^ ' —

q{ \ v 7 ve / 2 \q2

Substituting this result in Eq. (B.2) and using Eq. (B.1), we obtain

dr:ldr:2т-^-5(?1 +?2 -q1)[q2 + (h -q2)- 2

r12 r22 y (?1 -q2)2J

= n 1+eT(1 - e)L + kS V1 + ln C^^ • (B.5)

\ r v v \ q22 //

Appendix C

Let us calculate now the chiral components of the tensor

/¡J = 1 [ dh (_ (h- r) )!'((h - r) ) (C

n 1+er(1 - e) J r 2(h2-h)2\q2 q-h)2J vq32 q-h)2J • (.)

Writing

/qi _ (q1 - r) y /_ k\_\!' /_ (q1 - r) Y (C 2)

vq12 (q1 -?)2/ = Iq2 k2) + Vk2 (q1 -i)2' ( )

we can split the tensor in the sum of two pieces:

Jij = fj + Jj, (C.3)

ij _ / _ _kiV 1 f dr /_ (q3 - r) \ j

1 = vq2 kv n ^ra -e)J r2q-T)2\q32 q-?)2/ ,

The first tensor can be obtained from Eq. (B.5) by the replacement q1 ^ q2, q2 ^ q3; we get

' '2i2'

/J-(ê -ê№+if)'¿T+-teh2 • (C5)

The tensor Jlj is infrared finite and can be calculated at e = 0. The calculation of its chiral components can be performed easily using the decomposition of the integrand into a sum of terms of the type (a+ - r+)-1(b- - r-)-1 and the integral

d2r A2

0(A2 -r2) = ln -(C.6)

n(a+- r+)(b-- r-) \(a- b)2

It gives

T++- (k _ kl\ + (+ kkl^

J2 = \k 2 k2) + ^2 % \ к22ql2/

j++-=( n - k)+{n+k|) ), (c.7)

2 W k12/ Vq32 k22/ q22 \h 2q2J

where k = k1 + k2 and J2 = (J2++)*, J2 + = (J+)*. Therefore, for the +- component

j% - ko+(n+k|)-U -+,nf ^

V-i2 Vq32 k22/ q22\6 \ h2

q - k - q3 k+ \ e

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