Regge meets collinear in strongly-coupled \( \mathcal{N}=4 \) super Yang-Mills

Journal of High Energy Physics, Jan 2017

We revisit the calculation of the six-gluon remainder function in planar \( \mathcal{N}=4 \) super Yang-Mills theory from the strong coupling TBA in the multi-Regge limit and identify an infinite set of kinematically subleading terms. These new terms can be compared to the strong coupling limit of the finite-coupling expressions for the impact factor and the BFKL eigenvalue proposed by Basso et al. in [1], which were obtained from an analytic continuation of the Wilson loop OPE. After comparing the results order by order in those subleading terms, we show that it is possible to precisely map both formalisms onto each other. A similar calculation can be carried out for the seven-gluon amplitude, the result of which shows that the central emission vertex does not become trivial at strong coupling.

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Regge meets collinear in strongly-coupled \( \mathcal{N}=4 \) super Yang-Mills

Received: November meets collinear in strongly-coupled Martin Sprenger 0 Open Access 0 c The Authors. 0 0 Institut fur Theoretische Physik, Eidgenossische Technische Hochschule Zurich We revisit the calculation of the six-gluon remainder function in planar N = 4 super Yang-Mills theory from the strong coupling TBA in the multi-Regge limit and identify an in nite set of kinematically subleading terms. These new terms can be compared to the strong coupling limit of the nite-coupling expressions for the impact factor and the BFKL eigenvalue proposed by Basso et al. in [1], which were obtained from an analytic continuation of the Wilson loop OPE. After comparing the results order by order in those subleading terms, we show that it is possible to precisely map both formalisms onto each other. A similar calculation can be carried out for the seven-gluon amplitude, the result of which shows that the central emission vertex does not become trivial at strong coupling. Scattering Amplitudes; AdS-CFT Correspondence; Supersymmetric gauge 1 Introduction 2 The 3 ! 3{amplitude at strong coupling Six-point amplitude from the TBA Multi-Regge kinematics Calculation of the remainder function R3!3 R3!3 in the Mandelstam region R3!3 in the Mandelstam region and subleading kinematics Subleading kinematics from the OPE Extracting the subleading pieces Mapping TBA $ BFKL OPE Subleading kinematics for the 2 ! 5{amplitude A Derivation of the impact factor at strong coupling A.1 u^; v^ > 1 A.2 u^; v^ < 1 Introduction The high-energy behavior of scattering amplitudes in planar N = 4 super Yang-Mills number of loops to nite coupling still seems di cult. One might therefore hope that for full kinematics. Indeed, in a series of papers it was shown that the six-point remainder function has a 15], before a nite-coupling proposal was put forward in [1]. Expanding the dispersion integral at weak coupling leads to an expansion of the reof this paper. not the case is another point of this paper. Another special kinematic con guration is the collinear limit, which is governed by the Wilson loop OPE [33{36]. This expansion takes the form of a ux-tube spanned expressions for the BFKL eigenvalue and the impact factor from nite-coupling expressions governing the energies and momenta of certain ux-tube excitations. Remarkably, the and the strong coupling result of [14, 15] and pass all tests. In this paper, we identify kinematically subleading terms at strong coupling which allow a more detailed check of the nite-coupling expressions with the result of the TBA potential nite-coupling expression for the central emission vertex. This paper is organized as follows. In section 2 we brie y review the calculation of the 3 ! 3{amplitude at strong coupling from the TBA. We then show how to obtain 2 ! 5{amplitude in section 4 before concluding in section 5. The technical details for the appendix A. The 3 ! 3{amplitude at strong coupling In this section, we brie y review the calculation of the 3 ! 3{amplitude at strong coupling Six-point amplitude from the TBA remainder function,1 which depends only on the three dual-conformal cross ratios u1 = u2 = u3 = only arise in contributions which are subleading in p . 1Note that at strong coupling, there is no distinction between the MHV and the NMHV case. Di erences xi, with xi+N normalization, because the prefactor in eq. (2.1) is subleading in p . The remainder xi and xi;j := xi xj . The amplitude is only xed up to an overall function is given by several terms, The simplest piece of eq. (2.3) is which is directly given in terms of the cross ratios and reads R6 := (Afree + Aper + Afree = jmj Z Aper = u1 = to the remainder function are given by d cosh log 1 + Ye1( ) 1 + Ye3( ) 1 + Ye2( ) The cross ratios (2.2) can be calculated from the Yea-functions through the relations u2 = u3 = Those Yea-functions satisfy a set of TBA-like equations, log Yea( ) = ma cosh where the parameters are given by and the integration kernels read m1 = m3 = jmj; m2 = p C1 = C3 = C; C2 = 0 in terms of the two functions Kaa0 ( ) = B@K2( ) 2K1( ) K2( )C A K1( ) K2( ) K1( ) K1( ) = K2( ) = a dependence on the parameter ' arises. For completeness, let us mention that the Yeafunctions satisfy a recursion relation, Yea( ) = This relation allows us to easily construct the Yea-functions far away from the real axis, tion kernels. All of the above has a nice generalization to the general n-gluon case, but the expressions become more complex. We therefore refer the reader to [23] for details. Multi-Regge kinematics Regge limit of the 3 ! 3{amplitude. As described in [58], this limit is characterized by the following behavior of the cross ratios, reduced cross ratios remain constant. This di ers from the 2 ! 4{amplitude only in that the large cross ratio from the exact relation C = cosh 1 = cosh 1 1 + u~2 + u~3 2 u1u~2u~3 which can be derived from the recursion relation (2.11) as well as the exact relation than minus one, which leads to with Ce being real. For comparison, in the 2 ! 4{case C is purely imaginary. This, however, the analysis of [59] for the 2 ! 4{case, and use the relations between the Yea-functions and = e2C ; C = i + Ce; limit. The result is that the limit jmj ! 1; C const: which behave as " ! 0 and w ! const: in the multi-Regge limit, we nd the following parametrization of the cross ratios in terms of the TBA parameters u1 = 1 + 2 cosh C ; u2 = "w; u3 = with corrections of O("2). This parametrization nicely shows the behavior (2.12) once we take into account eq. (2.16). Calculation of the remainder function R3!3 of the remainder function (2.3). The two contributions and Aper are easily computed. Simply plugging in the parametrization (2.18) and expanding in " we obtain Aper = + O (" log ") ; are exponentially suppressed in jmj. Indeed, a careful analysis shows that and is therefore negligible in the limit " ! 0, see [14] for details. Summing up all contributions, we nd that the remainder function is a constant Afree = O(" log ") R3!3 = This constant, however, comes solely from the Li2-part of and cancels with a similar term there is a kinematic region of the 3 ! 3{amplitude, in which a Regge cut is known to continuation in the cross ratios as = 0 : : : ; to nd a non-trivial remainder function. and refer the reader to those references for details. As explained in the last section, we want to perform an analytic continuation in the cross ratios. This is trivial for the -contribution to the remainder function (2.4), as it is subtle for the following reason. For any given Yea-function there are locations in the complex -plane where Yea( ) = 1. The location of these points, of course, depends on the TBA parameters. Hence these points will move in the complex -plane during the Yea-function they are associated to, i.e. we have Yea(ea;i) = 1 for i = 1; : : : ; na: TBA equations as Kaa0 ( ) =: @ log Saa0 ( ): S1( ) = i S2( ) = log Ye a0( ) = m0a cosh 0) log 1 + Ye a00 ( 0) a0 i=1 X sign(Im(ea;i)) log Saa0 ( The quantities Saa0 ( ) appearing in eq. (2.24) are related to the kernels via For the basic kernels K1( ) and K2( ) they explicitly read contributions to the remainder function using A0per = A0free = jmj0 Z d cosh log 1 + Ye 10( ) 1 + Ye 30( ) 1 + Ye 20( ) a i=1 X sign(Im(ea;i)) sinh ea;i: of the residue terms in both the TBA equations and the A0free-contribution. To obtain an explicit result for the 3 ! 3{remainder function for the continuation (2.22), all we need to do is gure out how many crossing solutions there are for the three Yea-functions and what their locations ea;i at the end of the continuation are. The key di culty in those calculations is to gure out which path the TBA parameters jmj and C have to follow for a given path in terms of the cross ratios. Basically, it the equations Yea( ) = 1 numerically to see whether any of those solutions cross the real proceed with the analytic continuation. the continuation we go to the multi-Regge regime jmj0 ! 1, where we can neglect the the continuation, the Yea-functions can be evaluated at the locations of the crossed solutions which by de nition yields locations ea;i. Therefore, our though it involves numerical intermediate steps. nal result for the remainder function will be exact, even Following this algorithm for the path (2.22) for the cross ratios, we solutions of the equation Ye1( ) = 1 cross the real axis,2 as shown in nd that two gure 1. No solutions of the other Yea-functions cross the real axis. We therefore have two crossing solutions, which we will call in the following. Furthermore, solving the Bethe ansatz 2The fact that we nd crossing solutions for Ye1 is related to our choice of Re (Ce) > 0 in our numerical analysis. Choosing Re (Ce) < 0 would lead to crossing solutions in Ye3, which, however, gives rise to the same result for the remainder function. a0 j=1 −1 −1 −0.5 1 during the analytic continuation (2.22). The w = 1 and C = i We switch colors from blue to red when two of the solutions cross the real axis. The convergence of the endpoint position of the crossing solutions against clearly visible. equations (2.29) we nd that is exactly the same as in the 2 ! 4{case [15]. We have now assembled all necessary ingredients to calculate the remainder function in the Mandelstam region. R3!3 in the Mandelstam region and subleading kinematics considered in those references. We begin from the modi ed TBA equations valid in the Mandelstam region. Since we send jmj0 ! 1 at the end of the continuation, we can neglect all integrals and nd log Ye a0( ) = m0a cosh Ca0 + log From the Ye a0-functions we can then calculate the cross ratios at the endpoint of the continuation through the relations (2.10), from which we nd u02 = cos '0 + sin '0 u03 = 2 + cos '0 + sin '0 ; (2.32) tions of O("0 2). Using our choice of path (2.22) we then demand that u02 = u03 = eters " and w. Using '0 = tan 1 logarithm in eq. (2.35) is positive. For Aper, we use eq. (2.34) and nd4 A0per = log2 w0 = Aper 3Note that for any given numerical value of " there is, of course, an exponent N such that log N " and when indeed all corrections of the form log n " are smaller than the corrections of the form "n. 4Note that due to the quadratic term log2 "0 in Aper we always have to expand the parameters "0, w0 in eq. (2.34) one order higher than the order we want to compute the remainder function to. "0 = w0 = w 2p2. These are the subleading kinematic corrections we are after in this paper. In the previous analysis, only the leading terms (i.e. without any factors of log n ") " ! 0, since both the integrals we neglect in the TBA equations as well as the higher order terms neglected in eq. (2.32) are of the form O("n) and therefore much smaller than -contribution to the remainder function is easily evaluated, since it is a function tion (2.22) to nd 0 = + 2 cosh C llooggw"00 , these equations can be solved order by order Lastly, the A0free-contribution can be calculated via eq. (2.28) once we specify the number and endpoints of the crossing solutions. After neglecting the integrals, we obtain A0free = 2 jmj0 = p 2 log "0 1 + = p Note again that as described in section 2.3 the integrals in A0free are of O(" log ") and all contribution to nd the remainder function 2 ) log " + + 2 cosh C coe cients from this approach. Subleading kinematics from the OPE As mentioned in the introduction, a conjecture for a nite-coupling expression for the m= 1 where the kinematic variables are related to the cross ratios via log u2u3; = log w: variables , for the TBA variables and nd in eq. (3.1) agrees with the phase of the TBA result (2.39). The nite factor (or BFKL measure, both expressions will be used interchangeably) given in [1]. Here, we only spell out their form in the strong coupling limit p derived in [1] and read !( ) SC = ( ) SC = 2 2 cosh 2 2 sinh 1 + i 2 sinh i 2 sinh need the behavior of the impact factor BFKL(u) at strong coupling. This quantity is not related object, the OPE measure of the small fermion.5 Given the technical nature of which reads 5We would like to point out that the result (3.5), while unpublished, was already derived by Benjamin Basso and thank him for sharing the nal expression (3.5) with us. we can evaluate the integral by means log BFKL( ) SC = cosh(2 1) cosh(2 2) cosh( 1 which holds in the region u^ < 1. Extracting the subleading pieces Since all quantities in eqs. (3.4), (3.5) scale like p of a saddle point approximation, i.e. we rst solve the equation 0 = @ log BFKL( ) + i log w @ log " @ !( ) for the saddle point 0 and then obtain the remainder function as elog BFKL( 0)+i log w ( 0) log " !( 0): not know which m-mode is dominant. We therefore write in eq. (3.7). Similar to the TBA-case, we can expand the result in orders of lo1g " . For example, it is easy to see that the leading order result is given by 0 = 0 + O(log 1 ") which leads to 2 log(1+ 2 2 log(1+ 2 ) + log2(1+ We have performed this calculation up to ten orders in log n " and found perfect matching of the BFKL OPE and the TBA results, which strongly supports the conjectured in the next section that this is indeed the case. Mapping TBA $ BFKL OPE remainder function | rst the kinematic aspect of nding the saddle point for the OPE remainder function on these solutions. It is therefore natural to expect that the saddle point equation (3.6) can be mapped to the equation determining '0 (2.33). To see this, we start from the de nition of '0, log w = tan '0 log " + h5('0); '0 = tan 1 and use eqs. (2.32), (2.33), (2.18) to rewrite this in the form i log w g1('0) log " g2('0) + g3('0) = 0; perfectly, once we identify 0 = i'0: Similarly, it should then be possible to map the expressions for the remainder functhe TBA side can be written as in terms of the parameters jmj0 and jmj. Keeping in mind the identi cation (3.12) as well to rewrite the TBA remainder function (3.13) in the form e p2 (i log w h1('0) log " h2('0)+h3('0)): freedom in rearranging terms due to the relation which we do not spell out explicitly. We then obtain e 2 (i log w(h1('0)+h4('0)) log "(h2('0) ih4('0) tan '0)+(h3('0) ih4('0)h5('0))); (3.16) two formalisms are identical at strong coupling. Subleading kinematics for the 2 ! 5{amplitude u11 = 1 u12 = 1 + 2 cosh C2 ; u21 = "2w2; u31 = + 2 cosh C1 ; u22 = "1w1; u32 = in the multi-Regge limit where "i ! 0, the wi are real and constant, and Ci are purely imaginary and constant. Corrections to the cross ratios in eq. (4.1) are of O("2). Furthermore, there is another, dependent cross ratios u~, which behaves as 1 analytic continuation with all other cross ratios held xed. Subtleties in probing this region, usually denoted as , from the TBA are discussed in [27], but do not play a role here. In this region, the all-loop remainder function is expected to be of the form [25] = i !( 2;n2)jz2j2i 2 ( 2; n2) sub + : : : ; by zi = w3 i 1 eC3 i . While the BFKL eigenvalue !( ; n) and the impact factors ( ; n) in investigated in [27] with the result that where again e2 = = 7; Note that the phase ~7; in eq. (4.5) already slightly di ers from the predicted valued in [24, 25] by an additional piece. This di erence, however, could well arise from the not trivial at strong coupling and links the two integrations. parametrize the remainder function as k1= 1 k2= 1 ck1;k2 (w1; w2) log k1 "1 log k2 "2: In this notation, the leading terms of eq. (4.4) correspond to the terms c 1;0, c0; 1 and c0;0. Some of the lowest subleading terms read6 c1;0(w1; w2) = log w1 + const: ; (4.7) c0;1(w1; w2) = c1;0 (1=w2; 1=w1) via target-projectile symmetry and c1;1(w1; w2) = 6 log w1 log w2 + 6 log 1 + p 2 2 log w2 + const: 6We provide the full form of the rst four orders of subleading terms in the le 7pt subleading.m attached to the arXiv submission of this paper. While their explicit form is not particularly simple, it is very interesting that we subleading terms of the form log 1 "1 log 1 "2. These terms couple the two triplets of cross coupling, assuming that the form (4.3) holds. Conclusions can actually be precisely mapped onto each other at strong coupling. in the Mandelstam region P7; and found that there are subleading terms which couple an open question for future investigations. Acknowledgments Science Foundation through the NCCR SwissMAP. Derivation of the impact factor at strong coupling at strong coupling, eq. (3.5). We are interested in the limit g ! 1, while keeping the rescaled rapidity u^ = 2ug xed and the mode number m of O(1). To derive the impact as derived in [1], g2(x[+m]x[ m] x[+m]x[+m] eA+2fB(3F)KL;m(u) 2fB(4F)KL;m(u); The constant A is given by x[ m] = x u i x(u) = (u + pu2 A = 2 J0(2gt)2 and the source terms M = (1 + K) 1; Qij = ij( 1)i+1i (A.4) BFKL = Z 1 dt Jj(2gt) et jeven + ( 1)jet jodd cos(ut)e mt=2 J0(2gt) ; sin(ut)e mt=2; f B(4F)KL;m(u) are de ned as f B(3F)KL;m(u) = 2eBmFKL(u) Q M f B(4F)KL;m(u) = 2 BmFKL(u) Q M BFKL(u) and for the functions f B(3F)KL;m(u), f (4) expanding at strong coupling, we easily see that The strong coupling limit g ! 1 can be readily performed for all parts of eq. (A.1) except BFKL;m(u), which are more involved. Rescaling u ! u^ and nd that the leading term of the prefactor eq. (A.1) is of order g0. Since we drop the prefactor from now on. For the constant A we make the substitution t ! 2tg and expand at strong coupling to obtain = x(u^)jSC ; AjSC = where we used that cusp = 2 f B(3F)KL;m(u), f B(4F)KL;m(u). The de nition of these functions given in eq. (A.6) is well-suited at strong coupling. Let us now turn to the functions to nite size, since Kij entries are of the same order Kij gi+j at weak coupling. At strong coupling, however, all matrix g and only become numerically smaller as i; j grow. string [37]. Indeed, at strong coupling we have BFKL(u^) SC = 2 Jj (t) where SF stands for small fermion, see appendix B of [37]. Similarly we have that SC = SC = 4g ejSF(u^) SC : We can therefore use the strong coupling expansion of the functions fS(3F;4) for the small drop the subscript SC from now on. We start from a slight generalization of the functions f (3;4), namely f B(3F)KL(u^; v^) := 2 BFKL(u^) Q f B(4F)KL(u^; v^) := 2 BFKL(u^) Q BFKL(v^); BFKL(v^); and equivalently for the small fermion functions fS(3F;4)(u^; v^). We will determine these functions and take the limit v^ ! u^ in the end. As in the case of the BFKL eigenvalue !(u^) both examine in the following. of the function fS(3F)(u^; v^),7 fS(3F)(u^; v^) = 7For the calculations in the small fermion case, we use the notation of [60] for all quantities with the to leading order at strong coupling. In eq. (A.14), the function W +( ; u^) is de ned as where P denotes the principal value. To carry out the integrals, we use the relation W +( ; u^) := 2 Z 1 cos( k) d sin(v^ ) cos(k ) = where the function ef ;u^( ) is shown to be given by ef ;u^( ) = W +( ; u^) + u^ + 1 4 using the identity for jpj > 1, see e.g. [60], we obtain the result fS(3F)(u^; v^) = 32g u^ + v^ u^ + 1 4 u^ + 1 4 We then obtain8 f B(4F)KL(u^; v^) = 2 BFKL(u^) Q BFKL(1) = = 16g2 Taking the limit v^ ! u^ then gives the expression f B(4F)KL(u^) needed for the BFKL measure. 8In the second step we assume that summation and integration commute, which is supported by numerSimilarly the function fS(4F)(u^; v^) has an integral representation as fS(4F)(u^; v^) = 1 Z 1 d cos(v^ ) +f;u^( ); where the function +f;u^( ) is given by +f;u^( ) = 4g 4 u^ + 1 W ( ; u^) + W ( ; u^) to leading order at strong coupling. Furthermore, the function W ( ; u^) is de ned as W ( ; u^) := Performing the integrals as before one obtains the result fS(4F)(u^; v^) = 32g u^ + v^ u^ + 1 4 v^ To integrate this expression to the BFKL case we use the boundary values BFKL(1) = BFKL(u^) = ( (u^) 4gu^) ; sion of eq. (3.4) for the region u^ > 1 and is given by ( ) = sinh + i 2u^ + 2 1 is obtained by noting that the measure at strong coupling is symmetric under u^ $ see [1]. We then nd log (u^) = + ( (u^) 4gu^) 4g f B(3F)KL(u^; v^) = 16g2 d 2 fS(4F)( 1; 2) + ( (v^) 4gv^) Putting all contributions together, we obtain the full measure for u^ > 1 as sinh(2 1) sinh(2 2) cosh( 1 slightly di erent form and is given by in the main text, as the saddle point turns out to be close to u0 0. This region is ef ;u^( ) = W +( ; u^) + 1 + u^ 4 now all integrals are principal value integrals. Partitioning principal values as where jpj < 1, we obtain the result fS(3F)(u^; v^) = 64g u^ + v^ and performing the integrals using the identity f B(4F)KL(u^; v^) = 2 = 16g2 To integrate to the BFKL case, we again use relation eq. (A.19) and nd f B(4F)KL(u^) = = 12 tanh 1(u^). cosh(2 1) cosh(2 2) cosh( 1 cosh( 1 + 2) In the same way, we obtain the result for f B(3F)KL(u^). We begin with the modi cations for the small fermion function fS(4F)(u^; v^), where the only change from eq. (A.21) is in +f;u^, which now reads +f;u^( ) = W ( ; u^) + 1 + u^ 4 Going through the same steps as before we obtain the result fS(4F)(u^;v^) = 64g u^ + v^ This can be integrated to the BFKL case using the boundary condition e BFKL(0) = 0 f B(4F)KL(u^; v^) = 2 BFKL(u^) Q M BFKL(v^) = 2 1 Z Z 2 0 0 cosh(2 1) cosh(2 2) cosh( 1 cosh( 1 + 2) leading order at strong coupling is given by 0 0 cosh(2 1) cosh(2 2) cosh( 1 cosh( 1 + 2) cosh(2 1) cosh(2 2) cosh( 1 cosh( 1 + 2) cosh(2 1) cosh(2 2) cosh( 1 eq. (3.5). 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Martin Sprenger. Regge meets collinear in strongly-coupled \( \mathcal{N}=4 \) super Yang-Mills, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP01(2017)035