Diagonal form factors and hexagon form factors II. Non-BPS light operator

Journal of High Energy Physics, Jan 2017

We study the asymptotic volume dependence of the heavy-heavy-light three-point functions in the \( \mathcal{N}=4 \) Super-Yang-Mills theory using the hexagon bootstrap approach, where the volume is the length of the heavy operator. We extend the analysis of our previous short letter [1] to the general case where the heavy operators can be in any rank one sector and the light operator being a generic non-BPS operator. We prove the conjecture of Bajnok, Janik and Wereszczynski [2] up to leading finite size corrections.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://link.springer.com/content/pdf/10.1007%2FJHEP01%282017%29021.pdf

Diagonal form factors and hexagon form factors II. Non-BPS light operator

Received: November Diagonal form factors and hexagon form factors II. Non-BPS light operator Open Access 0 1 c The Authors. 0 1 0 nite size corrections 1 Institut fur Theoretische Physik, ETH Zurich We study the asymptotic volume dependence of the heavy-heavy-light threepoint functions in the N = 4 Super-Yang-Mills theory using the hexagon bootstrap approach, where the volume is the length of the heavy operator. We extend the analysis of our previous short letter [1] to the general case where the heavy operators can be in any rank one sector and the light operator being a generic non-BPS operator. We prove the conjecture of Bajnok, Janik and Wereszczynski [2] up to leading AdS-CFT Correspondence; Bethe Ansatz; Integrable Field Theories 1 Introduction The set-up 2 3 A direct check of BJW conjecture Diagonal limit and kinematical poles N = 1 case N = 2 case Generalization to N magnons Factorization property Proof of BJW conjecture The recursion relation Proof of BJW conjecture 6 In nite volume form factors Conclusions and discussions A Explicit expression for F2s(u1; u2) B Singlet state and factorization C The ratio of phase factors C.1 Scalars C.3 Fermions D Computation of n Introduction Form factor bootstrap program [3] is a powerful method to obtain non-perturbative results the large Nc limit is a new kind of integrable system with very rich structure. In recent years, there has been many solid progress in the computation of three-point functions in this problem at nite coupling, it is desirable to relate the three-point functions to form factors and apply the bootstrap methodology. There are at least three proposals in this direction so far. Klose and McLaughlin proposed a set of bootstrap axioms for the worldsheet form factors [33]. This is a direct generalization of the form factor bootstrap program in 2d integrable eld theories. However, due to the non-relativistic nature of the light-cone gauge xed string theory and the complicated spectrum of the theory, it is highly challenging to solve the bootstrap axioms. Also, relating the worldsheet form factors to three-point functions is a non-trivial problem. Inspired by the structure of lightcone string eld theory, which has been used to calculate three-point functions in the BMN regime, Bajnok and Janik proposed a set of axioms for the so-called generalized Neumann coe cient [34]. This object can be de ned for any integrable eld theories and is obtained by taking a special decompacti cation limit of the structure constant. In contrary to the usual form factors in integrable eld theories, the generalized Neumann coe cient corresponds to form factors of non-local operators. This fact modi es the form factor bootstrap axioms by some extra phase factors. Again the set of axioms is quite challenging to solve, but some progress has been made recently in [35]. At weak coupling, similar ideas have led to the proposal of the spin vertex formalism [36{38]. Very recently, Basso, Komatsu and Vieira [39] proposed a di erent method called the hexagon bootstrap program. In this method, one cut the three-point function, which is represented by a pair of pants, into two more fundamental objects called the hexagons or the hexagon form factors. The authors of [39] proposed a set of bootstrap axioms for the hexagon form factor which can be solved explicitly. Gluing back the two hexagons by taking into account the mirror excitations, one obtains the structure constant. This method has been veri ed by many non-trivial checks [39{41]. Apart form these proposals, there is yet another way of relating form factors to a special type of three-point functions called the heavy-heavy-light (HHL) three-point function. Here heavy (light) means the quantum number of the operator is large (small). This type of three-point function is rst investigated in the dual string theory in [42, 43]. It can be seen as a kind of \perturbation" of classical string solutions with light supergravity modes. If we regard the two heavy operators as incoming and outgoing states and the light operator as some operator sandwiched between these states, then the HHL three-point function can be seen as a diagonal form factor or the mean value of the light operator in the state corresponding to the heavy operator. This idea is made more concrete in [2]. The form factor bootstrap method gives us non-perturbative result in in nite volume. In the context of three-point function, the \volume" is the length of the operator and should be nite. It is therefore an important question to take into account the volume corrections. There are in general two type of volume corrections. The rst type is called from imposing the periodic boundary condition which changes the quantization condition of the excitations. The second type is called wrapping corrections or nite volume corrections, which is due to the propagation of virtual particles and takes an exponential form e E L where E is the energy of the virtue particle and L is the length that it propagates. While the asymptotic volume corrections can be taken into account in a systematic manner, it is notoriously hard to take into account the wrapping corrections. Based on previous studies in the 2d integrable eld theories [45], Bajnok, Janik and Wereszczyski proposed a conjecture concerning the asymptotic volume dependence of the HHL structure constant at any coupling. This conjecture was checked at strong coupling by the same authors for several examples and at weak coupling in [44] in the su(2) sector. Using the hexagon form factor approach, the BJW conjecture is also checked at coupling in the su(2) sector for the light operator being the BMN vacuum [1]. In this paper, we generalize the result of [1] and show that the BJW conjecture is valid for nonBPS light operator. We prove the conjecture for all the rank one sectors, namely su(2), sl(2) and su(1j1) sectors. As in [1], due to the fact that it is not yet clear how to take into account all the mirror excitations, we restrict our proof to only the physical excitations. The rest of the paper is organized as follows. In section 2, we describe the set-up of the problem. In section 3, we present a method to check the validity of BJW conjecture directly for few excitations. When the rapidities of two excitations on the two physical edges coincide, they will decouple and the hexagon form factor is proportional to the one without these two excitations. In section 4, we study this decoupling limit in detail. We call the relation of the hexagons before and after decoupling the factorization properties. In section 5, we prove the BJW conjecture up to mirror excitations. In section 6, we give some comments on the in nite volume form factors which appear in the BJW conjecture. We conclude in section 7 and discuss future directions to explore. Some complementary details are presented in the appendices. The set-up In this section, we give the set-up of our problem. For HHL three-point functions, the two heavy operators O1 and O2 are conjugate to each other. They can be chosen in any of the three rank one sectors, namely the su(2), sl(2) and su(1j1) sectors. The excitations in these three sectors are scalars, (covariant) derivatives and fermions, respectively. We denote a generic excitation by and its conjugate by . There are 8 pairs of excitations: The polarizations of the excitations of the two heavy operators are chosen such that O1 : 2 so that by performing 2 transformations of the excitations on O2 to the edge of O1 the two sets of excitations are conjugated to each other. Let us denote the length and the number of excitations of the heavy operators by L and N . The two heavy operators take the following form O1 = Tr ZL N N + O2 = Tr ZL N N + We denote the two sets of rapidities of by u = fu1; ; uN g and v = fv1; respectively. The length of the third operator is denoted by 2l0, where l0 L. In the Z + Z + Y Y . In the current paper, we will consider general non-BPS operators and put excitations on the BMN vacuum. The set of excitations of the light operator is denoted by1 XAA_ (w) = fXA1A_1 (w1); XA2A_2 (w2); ; XAnA_n (wn)g: [[ ==vu [ =w where XAkA_k denotes a generic excitation. According to the hexagon approach [39], the asymptotic structure constant with three non-BPS operators is given by the following sumover-partition formula C123 = ( 1)j j+j j+j j!l31 ( ; )!l12 ( ; )!l23 ( ; ) The arrangement of excitations is depicted in gure 1. l12 = L l0; l23 = l0; l31 = l0: on the ordering of the excitations. However, the normalized structure constant (2.8) does not depend on the ordering. We choose the same ordering as in [1], namely the rapidities u are reverse ordered. We use the Bethe Ansatz Equations (BAE) for the rst splitting factor !l31 ( ; ) and write The second splitting factor !l12 ( ; ) can be written as 1If there are more than one type of excitations, we need to use the nested Bethe ansatz and take proper linear combinations of the excitations in order the third operator to have well-de ned scaling dimension. In that case, what we consider is one of the terms in the linear combination. We show that the BJW conjecture holds for each term and thus holds for the whole operator. ! l( ; ) = !l( ; ) = We do not need the explicit form of the third splitting factor2 and we simply denoted by !l0 ( ; ). The un-normalized HHL structure constant is de ned as the diagonal limit CHHL = vli!mu C123 The quantity we want to study is the following normalized HHL structure constant CHHL = QiN=1 a (ui) a (u) = ( 1)f_ 2 f +1n The normalization constant a (u) is de ned where f and f_ is the fermionic number of the un-dotted and dotted indices of the excitations, (u) is the measure introduced in [39] and n is a simple number which will be de ned in section 4. In (2.8), su(2), sl(2) and su(1j1) sector for ;N (u) is the asymptotic Gaudin determinant of the being scalars, derivatives and fermions, respectively. The asymptotic Gaudin determinant is proportional to the norm of the Bethe state and is ;N (u) = det k = p(uk)L l6=k where S (u; v) is the S-matrix in the corresponding subsectors. When we consider a subset u, we can de ne two quantities related to determinant with respect to the rapidities uj 2 Gaudin determinant N (u) with respect to uj 2 . While rapidities in the set , cN ( ) depends on all the rapidities u. We will prove that s ( ) depends only on the N (u). We de ne s ( ) as the Gaudin and c ( ) as the diagonal minor of the CHHL = CHHL = unambiguously de ned. instead of a scalar function. where F w;s( ) is some well-de ned quantity in in nite volume, which we shall call the in nite volume form factor. A theorem in [45] states that (2.11) has another equivalent expansion in terms of cN ( ) where F w;c( ) is di erent from F w;s( ) in general, but they are related by the relations given in [45]. The fact that we have two expansions reveals the ambiguity of the diagonal form factor in the in nite volume. Nevertheless, the nite volume form factor CHHL is 2If there are more than one type of excitations for the light operator, the splitting factor can be a matrix formation and the right diagram corresponds to a 4 transformation. Finally we comment on the mirror excitations. In order to obtain the complete result of the structure constant, we need to take into account all the mirror excitations on the three mirror edges, as is shown in gure 1. The mirror excitations on the opposite edge to the edge of the light operator corresponds to the physical wrapping corrections, which are of order e E L and can be neglected safely since we are considering the large L limit. The mirror excitations on the edges that are adjacent to the edge of the light operator leads to the so-called bridge wrapping corrections, which is of order e E l0 . Since l0 is nite, we should take into account all the mirror excitations on the adjacent mirror edges. However, in the hexagon approach, when two mirror excitations on the two adjacent edges coincide, there is a double pole in the integrand and so far it is not yet clear how to deal with this divergence. Due to this restriction, we will not consider any mirror excitations in this paper and leave this question for future investigations. We stress here that our proof in this paper is only up to mirror excitations. A direct check of BJW conjecture In this section, we describe a method to check the BJW conjecture (2.11) explicitly for a few magnons. For simplicity, we consider the case where the excitations for the heavy operators are the transverse scalars 11_ ; 22_ and the light operator being the BPS operator method can be readily applied to more general cases. Diagonal limit and kinematical poles In order to calculate the hexagon form factors, one needs to use mirror/crossing transformations to move all the excitations on the same edge. In our current example, we choose to move all the excitations on the edge which corresponds to O1. There are two possible transformations, as is shown in gure 2. The two crossing transformations lead to the same nal result, as is should be. However, the intermediate steps are rather di erent. In the diagonal limit where ui ! vi, there is a kinematical pole in the hexagon form factor. The hexagon form factor can be written as the product of a scalar or dynamical part and a matrix part. If we perform the 2 transformation, the kinematical pole appears in the matrix part while if we perform the 4 transformation, the kinematical pole appears in the dynamical part. Since the dynamical part is a simple product of the scalar functions h(u; v), it is much easier to keep track of the kinematical poles. At the same time, when performing crossing transformations, there will be some phase factors which originate from changing between the string frame and the spin chain frame. In the 4 this phase factor is usually simpler. For the current case, it is simply 1. Therefore, we will proceed our calculation by performing 4 transformations for excitations of O2. The general hexagon form factor in our example takes the form H(ujv) = phase4 H(v4 ; u) where phase4 excitations on di erent edges is denoted as H(ujwjv) while the one with excitations on the same edge is denoted by H(u; w; v). The latter is called the fundamental hexagon and can be written as a product of the dynamical part and matrix part3 H(v4 ; u) = Hdyn(v4 ; u)Hmat(v4 ; u) Hdyn(v4 ; u) = h<(v; v)h>(u; u) Here we have introduced the short-hand notation F <(v; v) = Y F (vi; vj ); F >(u; u) = Y F (ui; uj ); F (u; v) = Y F (ui; vj ) function h(u; v) can be written as h(u; v) = h~(u; v) = 1=x1 x2 ) 1=x1 x2 )(1 1=x1+x2+) 12 u=g and 12 is the square root of BES dressing phase [50]. The scalar function h~(u; u) is related to the measure (u) as It it clear that in the diagonal limit v ! u, there is a kinematical pole in the function 3In general there is also a phase factor ( 1)f taking into account the proper grading. For scalar excitations, this phase factor is simply 1. (u) = h~(u; u) as expected. We can thus write the dynamical part as Hdyn(v4 ; u) = 1 h<(v; v)h>(u; u) where the kinematical poles in the diagonal limit are all in the rst factor of (3.8). The matrix part of the hexagon is given in terms of Beisert's S-matrix elements [49] with the dressing phase setting to 1. Under 4 transformation, the Zhukowsky variables the hexagon form factor are also invariant Hmat(v4 ; u) = Hmat(v; u): N = 1 case sum-over-partition formula Here and later in this section, we omit the upper index of Hmat to simplify the notation. The next step is to expand each term T0 and T1 in terms and keep only the leading term. The diagonal limit of the nite volume form factor is well de ned, so we should have rearranged as ! 0. The sum t1 + t2 can be C123 (u1; v1) = t1 + t2 t1 = t2 = Hmat(u1; v1) h~(v1; u1) Hmat(v1; u1) h~(u1; v1) t1 + t2 = T0 + T1 T0 = h~(v1; u1) H(u1; v1) T1 = h~(v1; u1) H(u1; v1) H(v1; u1) H(v1; u1) h~(u1; v1) T0 = T0;0 + T0;1 + T1 = T1;1 + 2 T1;2 + in the next subsection that for more magnons, the fact that the diagonal form factors are well-de ned is ensured by the factorization properties of the hexagon. The un-normalized structure constant reads CHHL(u1) = T0;0 + T1;1 = (u1) ( 1(u1) + F1s(u1)) ; H(1;0)(u; u) = CHHL(u1) = C123 (u1; u2; v1; v2) = X ti: h(u2; u1)h(v1; v2) H(u2; u1; v1; v2) h(v1; u2)h(v2; u1) h~(v1; u1)h~(v2; u2) H(u1; v1) H(v2; u2) h~(v1; u1) h~(u2; v2) H(u2; v2) H(v1; u1) h~(v2; u2) h~(u1; v1) e i(p(u1) p(v1))lS(u2; u1)S(v1; v2) h(u2; u1)h(v1; v2) H(v1; v2; u2; u1) h(u1; v2)h(u2; v1) h~(u1; v1)h~(u2; v2) 1(u) = L p0(u); F1s(u) = iH(0;1)(u; u) iH(1;0)(u; u) + ih~(0;1)(u; u) ih~(1;0)(u; u) We see indeed that the volume dependence is encoded in the function 1(u). The in nite volume form factor for one magnon is given by F1s(u). Here the upper indices (1; 0) and (0; 1) denote partial derivatives. For example, We con rm that the normalized structure constant for one excitation indeed takes the form N = 2 case For two magnon case, there are 6 terms t1 = t2 = t3 = t4 = t5 = t6 = H(u1; v2) H(v1; u2) h(v2; u1) h(u2; v1) H(u2; v1) H(v2; u1) h(v1; u2) h(u1; v2) X ti = T0 + T1 + 2 T2: Then we perform the un-normalized diagonal structure constant is given by CHHL(u1; u2) = T0;0 + T1;1 + T2;2: As alluded before, the disappearance of Tk;n (n < k) is guaranteed by the factorization property of the hexagon form factors. For example, T2;0 = h~(u1; u1)h~(u2; u2) [H(u1; u2; u2; u1) + H(u2; u1; u1; u2) 2H(u1; u1)H(u2; u2)] In the diagonal limit, we take vk = uk and arrange the sum as which does not vanish automatically. However, notice that there are coinciding rapidities in the matrix part of the hexagon, they can be written in terms of hexagons with less excitations. In fact, we will derive in the next section that automatically, but will vanish if we take into account (3.26) as well as the relations of the following type H(1;0;0;0)(u1; u2; u2; u1) = H(1;0)(u1; u1) + h(0;1)(u2; u1) h(u2; u1) h(u2; u1) S(1;0)(u1; u2) S(u1; u2) Here H(1;0;0;0)(u1; u2; u2; u1) stands for H(1;0;0;0)(u1; u2; u2; u1) H(v; u2; u1; u2) v=u1 The relation (3.27) comes from taking the derivatives with respect to v of the following factorization relation 2(u1; u2) The mechanism works also for more magnons. By using the factorization properties and the corresponding derivatives, the terms Tk;n with n < k will vanish. Taking into account the normalization, we nd that the normalized symmetric structure constant takes the following form CHHL(u1; u2) = [ 2(u1; u2) + s1(u1)F1s(u2) + s1(u2)F1s(u1) + F2s(u1; u2)] where F1s(u) is derived in (3.16) and F2s(u1; u2) a rather complicated function in terms of the momenta p(u), su(2) scattering matrix S(u; v), the scalar factor h(u; v), the matrix part of the hexagon for 2 and 4 excitations H(u1; v1), H(u1; u2; v1; v2) and their derivatives. The explicit form of F2s(u1; u2) can be found in appendix A. Generalization to N magnons ! 0, we can organize the result as Then we expand each Tk in terms of The un-normalized symmetric structure constant is given by Tk = X Tk;n n: CHHL(u) = X Tk;k: In order to check (2.11) for N magnons, we need to know the expression of all the in nite volume form factors Fns(u1; ; un) with n < N in terms of p(u); h(u; v); S(u; v) and H. Then by subtracting the volume dependence from the nite volume form factor of N magnons, we obtain the in nite volume form factor of N magnons FNs (u1; We can check the structure (2.11) for a few excitations. The expression for the in nite volume form factors become complicated very quickly. Although a general proof is very hard to achieve following this method, we can give an argument for (2.11) based on our In our previous calculations for one and two excitations, we do not specify the explicit form of p(u); h(u; v); S(u; v) and H. The calculation is exactly the same whether we take the leading order expressions or the all-loop expressions. The only di erences are the explicit form of s and in nite volume form factors F s. As far as the structure (2.11) is concerned, they are equivalent. If we can nd a \representation" of the quantities p(u), h(u; v), S(u; v) and H such that the structure (2.11) holds, then the BJW conjecture should hold in general. In our case, such a \representation" indeed exists, where we take all the quantities p; h; S; H at the leading order. In [44], we have shown that (2.11) holds for any magnons at the leading order using the solution of the quantum inverse scattering problem and the Slavnov determinant formula. Based on this argument and the explicit calculations of the rst few magnons, we already see that the structure (2.11) should hold at nite coupling.4 The rigorous proof will be given in section 5. 4Again up to mirror excitations. Factorization property In this section, we derive the factorization property of the hexagon form factor. These properties are used in the previous section to insure the diagonal form factors to be wellde ned and will be used in the next section to prove the BJW conjecture. The main f = f2 Hmat (u; u; u) = ( 1)f n Hmat (u) where the polarizations of the excitations are ( (u); XA1A_1 (u1); with XA A_(u) = XA1A_1 (u1) XAN A_N (uN ) being arbitrary. The phase factor ( 1)f takes into account the proper grading and is given in (4.5) and (4.6) and n is a simple number de ne in (4.14) and calculated in appendix D. To prove the factorization properties, we compute the following hexagon form factor = hhj (u)XAA_ (u)ij in the limit v ! u. We compute the hexagon by performing crossing transformations for . We can choose either a 2 transformation or a 4 transformation, as is shown in gure 3. By comparing the expressions for two di erent crossing transformations, we obtain the factorization properties. The two mirror transformations should lead to the same result = phase2 H ;2 = phase 4 H ; 4 where phase2 , phase 4 are the phase factors coming from the crossing transformations and H ;2 , H ; 4 are the corresponding fundamental hexagons. From (4.3), we have H ; 4 = The fundamental hexagons can be written as a product of the dynamical part, the matrix part and the phase factor which takes into account the grading. Let us denote the ratio of the two phase factors by ( 1)f, namely 2 = (f_ + f_ )fA + f_ f ; f 4 = f_ fA + f_ _ f 2 + f_ f 2 : A Here the symbol f and f_ denote the fermionic number for the corresponding excitation of the 8 pairs of excitations ; , we always have f_ + f_ (mod 2). For the l.h.s. of (4.4), i=1 fAi , f_A_ = Pn i=1 f_A_i . Let us notice that for 0 (mod 2). Therefore f2 Hdy;n4 = h(v; u) h(v; u) Hm;at4 = Hm;at4 (u; u; v): In the limit v ! u, there is a kinematical pole in the dynamical part while the matrix part Res Hdy;n4 = i (u); lim Hm;at4 = Hm;at4 (u; u; u): P = For the r.h.s. of (4.4), Hdy;2n = h(v2 ; u)h(u; u); Hm;a2t = Hm;a2t (v2 ; u; u): In the decoupling limit v ! u, the dynamical part is regular while the matrix part has a pole. The residue of the matrix part can be worked out by using the same argument as lim Hdy;2n = h(u2 ; u)h(u; u) Res Hm;a2t = Res Hm;a2t (v2 ; u) Hmat(u) h(u2 ; u)h(u; u) excitations XAA_ (u) The derivation of (4.10) can be found in appendix B. The ratio of the phase factors phase2 and phase 4 is computed in appendix C and reads Combining (4.5), (4.8), (4.10) and (4.12), we have p = p(u): Hm;at4 (u; u; u) = ( 1)f Hm;a2t (u) n n = i Res Hm;a2t (v2 ; u) is a simple number and is computed in appendix D. We list n for the 8 pairs of excitations in the following table For our purpose, we are concerned with the following type of factorization property Hmat(u; u; w; v; u) = ( 1)f n Hmat(u; w; v) where fu; ug and fv; ug are rapidities of the excitations of type The polarizations of w can be arbitrary. If the coinciding rapidities are not on the leftmost 2 , respectively. and rightmost, we can use the following relation to move the excitations Hmat(?; ?; ; ?; ?) = S (ui; uj ) h(ui; uj ) ) = S (ui; uj ) h(ui; uj ) Equation (4.15) and (4.16) together give the factorization property. Proof of BJW conjecture In this section, we prove the BJW conjecture up to mirror excitations. We rst prove a recursion relation for the un-normalized HHL structure constant and then prove the BJW conjecture based on the recursion relation. This is a generalization of the proof presented The recursion relation As we can see from the examples, the explicit L-dependence comes from taking derivatives zi-dependence of the structure constant. Let us rst introduce some notations. We denote the expression in the sum-over-partition formula (2.3) by C2wN (ujv) C123 . The diagonal limit of this quantity is denoted by Note that the sum-over-partition formula gives the structure constant in the large but nite volume, therefore the diagonal limit is unambiguously de ned and does not depend on the one, or equivalently we can take i = i and then take i ! 0 one by ! 0, they give the same result. This is di erent from the diagonal limit in in nite volume where the result is divergent and depends on how one takes the diagonal limit. Another useful quantity in the diagonal limit is given by FNw(u) = lim v!u C2wN (ujv)j (vi)= (ui) In terms of words, we rst put the phase factor eilp(vi) ! eilp(ui) and then take the diagonal limit. As we discussed before, the explicit l-dependence originates from derivatives of the factor (vi). Replacing these factors by (ui) before taking the diagonal limit eliminates the l-dependence. Therefore FNw(u) does not depend on l and is a well de ned quantity in the in nite volume. In both (5.1) and (5.2), after taking the diagonal limit, we impose the BAE to replace the phase factors e ilp(ui) by eil0p(ui) together with products of S-matrices. w The dependence of CHHL(u) on zk is linear and is given by the following relation where the set u n uk means the rapidity uk is deleted from the original set and a (uk) = ( 1)f_ f +1 n The index \mod" stands for the following replacement zi ! zimod = zi + '(ui; uk); '(u; v) = i log S(u; v): We rst prove the recursion relation for zN . The quantity zN comes from taking derivatives of the factor eip(vN )l, therefore we must have vN 2 factor. On the other hand, we also need to have uN 2 in order to have such a because otherwise uN and vN are on di erent hexagons and there is no kinematical pole and hence not necessary to take derivatives. Consider a generic such term in the sum-over-partition formula (2.3) denoted by t(fuN g [ ; [ fvN g; ). The splitting factors satisfy ! l( ; fuN g [ )!l( ; [ fvN g)!l0( ; ) = e ilp(uN )+ilp(vN ) The hexagon form factor that we are interested in takes the following form H( ; vN j juN ; ) = phase4 We want to study the relation between this hexagon form factor and the one without uN H( j j ) = phase04 form factor is the product of a phase factor ( 1)f, the dynamical part and the matrix part. H( 4 ; 2 ; ) by ( 1) f. The dynamical parts of the fundamental hexagons satisfy Let us denote the ratio of the phase factors of the hexagons H( 4 ; vN4 ; 2 ; uN ; ) and h( ; vN ) h(uN ; ) h( 2 ; uN ) h(uN ; ) h( ; vN ) h( 2 ; vN ) h(uN ; vN ) The splitting factor and the dynamical part are universal in the sense that they do not depend on the polarizations of excitations. For the matrix part of the hexagon, we apply the factorization property = ( 1)f n S ( ; uN )S (uN ; ) h(uN ; ) h( ; uN ) h( ; uN ) h(uN ; ) WNw(u) = [ =u WNw(u) = CHwHL(u): where again the index \mod" stands for the replacement rule (5.5). After taking vi ! ui for the rest of the rapidities, we obtain @zN CHwHL(u) = a ;N CHwH;mLod(u n uN ): Finally let us notice that the structure constant is symmetric with respect to the rapidities, hence (5.3) is valid for any k. Proof of BJW conjecture partition u = Now we are ready to prove to the BJW conjecture up to nite size corrections. For a given where n = 1 depending on the polarizations. One can show straightforwardly that ( 1) f+f = ( 1)f_ f : Combining (5.6), (5.9) and (5.10) and summing over the partitions, we obtain We further de ne a quantity As a rst step, we want to show Noticing that @zk sN;l(u) = sN;mo1d;l(u n uk) @zk WNw(u) = a ;k WNw;m1od(u n uk): Assume that (5.16) holds for n From (5.3) and (5.18) we nd that the zi dependence of the two quantities are the same. It remains to show that the terms independent of zi is also the same. Putting zi ! 0, all On the other hand, form the de nition of FNw(u) (5.2), we rst put eilp(vi) to eilp(ui) and then take the diagonal limit, which prevents the appearance of zi and thus This proves (5.16) and we have Taking l1 = L and l2 = l0, we have WNw(u)jzi=0 = FNw(u): CHwHL(u)jzi=0 = FNw(u): CHwHL(u) = Y a (uk) [ =u [ =u sN;l1+l2(u) = CHwHL(u) = Y a (uk) Fjwj;s( ) = [ = Finally we go from length l to length L, this can be done by the following relation Taking into account the normalizations, the normalized structure constant indeed takes the form predicted by BJW conjecture (2.8). In nite volume form factors The normalized structure constant takes the same form as diagonal form factors in volume. For the later case, the coe cients in front of s and c are identi ed with the diagonal form factor in in nite volume. Keeping this analogy in mind, we also call our coe cient F w;s( ) or F w;c( ) as the in nite volume form factor. From the de nition of these coe cients (5.20) and (5.24), we can calculate them in terms of p(u), S(u; v), h(u; v), Hmat and their derivatives. The explicit expression becomes cumbersome very quickly. For the moment, we do not have a good understanding of the structure of the in nite volume form factors. This is an interesting question to explore in the near future. One possible direction is to formulate a set of bootstrap axioms directly for the diagonal form factors and solve these axioms. In the case where the light operator is BMN vacuum and the heavy operators are in the su(2) sector, we can expand F c at weak coupling and compare with the known results in [44] where a perfect match is found. At tree level, the in nite volume form factor F c(0) F c(0)(u) = 1(0)'(102)'(203) N 1;N + permutations (0)(u) = u2 + 1=4 '(0)(u; v) = Interestingly, it is checked in [1] that at one loop the form (6.1) still holds5 with the following corrections (1)(u) = '(1)(u; v) = u2 + 1=4 (u2 + 1=4)3 (u2 + 1=4)(v2 + 1=4)((u It is possible that the structure still holds at higher loop orders6 with proper modi cations of (u) and '(u; v). This may give us some hints about the general structure of the diagonal form factors in the in nite volume and lead to more e cient ways of calculating them. Conclusions and discussions In this paper, we prove the conjecture of Bajok, Janik and Wereszczyski concerning the asymptotic volume dependence of the heavy-heavy-light structure constant at all loops in one sectors with generic non-BPS light operators. In order to complete the proof, we need to take into account the bridge wrapping corrections. Once the double pole problem of the hexagon form factor approach is resolved 5We checked up to 4 excitations. 6This requires to take into account also the bridge wrapping corrections. The explicit results we have obtained so far are restricted to the su(2) case and the light properly, we can try to use the similar method to complete the proof. Most probably, the bridge wrapping corrections will not modify the asymptotic volume dependence but will correct the in nite volume form factors. Another kind of mirror excitations give rise to physical wrapping corrections of the form e E L. For the diagonal form factor, there are conjectures of the nite volume form factor with both asymptotic volume corrections and wrapping corrections taken into account [46{48]. It will be very interesting to incorporate the wrapping corrections for the HHL structure constant within the hexagon approach and compare with the proposals of nite volume diagonal form factors in the literature. operator being BMN vaccum. In order to gain a general understanding of HHL structure constant, it is useful to explore other con gurations. One of the most interesting case is the light operator being the dilaton. In this case, the HHL structure constant is known to be related to the derivative of the scaling dimension of the heavy operator with respect to the coupling constant g2 [43]. This allows us to make contact with the results of the spectral problem. In addition, since the relation is valid for any coupling, it may shed some light on taking into account bridge wrapping corrections. It will also be interesting to perform the strong coupling expansion and compare the results with the string theory calculation in the literature [2, 42, 43]. In this direction, one particularly interesting example is taking the giant magnon solution for the heavy operators and dilaton for the light operator. Finally, the BJW conjecture only concerns the rank one sectors, namely there is only one type of excitation for the heavy operators. This is also the case that has been studied in 2d integrable eld theories. A natural direction of further investigation is to study the HHL structure constant in higher rank sectors and nd out the form of asymptotic volume corrections. For the operators in higher rank sectors, one needs to apply the nested Bethe ansatz and there will be richer structures to explore. Acknowledgments It is my pleasure to thank Andrei Petrovskii and Laszlo Hollo for initial collaborations on the project and collaborations on related works. I'm indebted to Benjamin Basso and Shota Komatsu for many useful discussions and correspondences. I would also like to thank Zoltan Bajnok and Shota Komatsu for helpful comments on the manuscript. The work of Y.J. is partially supported by the Swiss National Science Foundation through the NCCR SwissMap. Explicit expression for F2s(u1; u2) The explicit expression for the in nite volume form factor with 2 excitations is given by F2s(u1; u2) = F1s(u1)F1s(u2) + 2 (u1) (u2) h(u2; u1)h(u1; u2) cos(p(u1) + H(0;1)(u1; u1)H(0;1)(u2; u2) + H(1;0)(u1; u1)H(1;0)(u2; u2) 2h(0;1)(u1; u2)h(1;0)(u1; u2) H(u1; u2) h(u1; u2)2 H(u1; u2) h(u1; u2) S(1;1)(u1; u2) S(u1; u2) H(0;0;1;1)(u2; u1; u1; u2) H(1;1;0;0)(u1; u2; u2; u1): j112i = We need to scatter all the excitations with each other. The scattering can be organized as follows, we rst scatter the rst two excitations in the decoupling limit v ! u. The result is divergent due to the kinematical pole and the residue is proportional to Beisert's singlet state [49] up to a Z vR!eus S12j 11(v2 ) 22(u)i / jZ 112i: Then we scatter the singlet with rest of the excitations, which is trivial up to a scalar factor Ani = Y k=1 h(u2 ; uk)h(u; uk) jZ Finally we scatter the rest of the rapidities, they contribute to Hm;a2t (u). It is clear that Hmat(u) h(u2 ; u)h(u; u) The analysis is similar for other polarizations. The Z makers usually leads to some global phase factors, which needs to be taken with some care. In order to nd these factors, we notice that when forming the singlet state, there is a di erence of Z maker between bosonic and fermionic excitations Singlet state and factorization (v) = 22_ (v) as an example. The matrix part of the hexagon form factor H2 (v2 ; u; u) 11_ (u) and is computed by Let us focus on the part also there is a Z singlet state + marker di erence between the bosonic and fermionic excitations in the being scalars, one scatters the rst two bosonic excitations a and form a singlet marker, then move the singlet to the rightmost, nally contract the singlet with the scalar excitations of the right sector where we need to take into account the Z We need to move the Z + marker to the leftmost in order to pull it out. The Z + marker. markers cancel each other. However, when moving the Z pick up the phase factor eiP by the rule of moving makers. + markers to the leftmost, we being derivatives, one scatters the fermionic excitations and form a singlet. Then move the singlet to the rightmost and nally contract the singlet with fermionic excitations of the right sector. No markers are involved in the process, hence the phase factor is 1. being fermions, there are two types of process. The rst corresponds to scattering the scalar excitations and contract the singlet with fermionic excitations in the right sector. This involves a Z maker on the left and no Z a phase factor e 2i P . The second case corresponds to scattering the fermionic excitations and + maker. Pulling it out, we get contract the singlet with bosonic excitations. This involves a Z markers. Moving the Z + marker to the leftmost picks up a phase eiP , pulling + marker on the rightmost it out gives another phase e 2i P . In total the phase factor is e 2i P . To summarize, the phase factors for the three kind of excitations are given by Combining (B.7) and (B.8), we obtain (4.10) in the main text. The ratio of phase factors In this appendix, we calculate the ratio of phase factors in (4.12) of the main text. form factor For scalar excitations, we can take 22_ and consider the following hexagon = hhj 11_ (u)XAA_ (u)ij 22_(v)ij0i The phase factor contains three parts Phases come from changing from spin chain frame to string frame before crossing; Phases come from crossing transformation of Phases come from changing from string frame to spin chain frame after crossing. The rst part is the same for both 2 and 4 transformations. The second part is for 2 transformation and 1 for 4 transformation. In order to nd the ratio of the two phase factors, it is enough to consider only the third part. Let us remind here that the transformation rules between spin chain frame and string frame for the derivatives, scalars and fermions are given by7 [39] Dstring = Dspin; string = p string = Z1=4 spinZ1=4: transformation: hhj 22_ (v2 ) 11_ (u)XAA_ (u)ij0ij0istring Here FA is the phase factor coming from moving all the Z-markers of XAA_ (u) to the left. Since we allow any kind of excitations, n does not have to be equal to N and not even have to be an integer. We can then move all the Z-markers to the leftmost and then pull them out using the rule Again we only consider the third step of changing back from string frame to spin chain frame since we are considering the ratios. 7We thank S. Komatsu for informing us the transformation rule for the fermions. i = znhhj i; z = e ip=2 where p is the total momentum of the state j i. The result is given by transformation: hhj 11_(u)XAA_ (u) 22_ (v 4 )ij0ij0istring e 2i (n 1)p1 2i (n+1)p2 2i (n+2)P FA: where FA is the same phase factor as in (C.3). By moving and pulling out the Z-markers, From (C.5) and (C.7) and taking into account the relative minus sign from crossing transformation, it is clear that For derivatives, we take = D33_ , HD = hhjD33_ (u)XAA_ (u)ijD44_ (v)ij0i: ij ij istring = h 4 transformation: ij ij istring = h Comparing (C.10) and (C.11), we conclude that for the derivatives = h For fermions, we take and consider the following con guration 2 transformation: 4 transformation: ij ij istring = h p +P )(n+1) ij ij istring = h Comparing the two results, we obtain = e 2 Computation of n In this appendix, we compute n for di erent polarizations. Therefore n = 1. (v ; u) = The result is summarized in the following table iRes Hmat(v2 ; u) = Res E(v2 ; u) = (u) Hmat(v2 ; u) = (D(v2 ; u) iRes Hmat(v2 ; u) = Res E(v2 ; u) = Hmat(v2 ; u) = iRes Hmat(v2 ; u) = Res C(v2 ; u) = (u) e 2i p Hmat(v2 ; u) = (A(v2 ; u) iRes Hmat(v2 ; u) = Res B(v2 ; u) = Hmat(v2 ; u) = (D(v2 ; u) + E(v2 ; u)) Therefore n = therefore n = 1 therefore n = Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. (2016) 120 [arXiv:1511.06199] [INSPIRE]. factors, JHEP 09 (2014) 050 [arXiv:1404.4556] [INSPIRE]. Math. Phys. 14 (1992) 1 [INSPIRE]. JHEP 08 (2004) 055 [hep-th/0404190] [INSPIRE]. Yang-Mills at one loop, JHEP 09 (2005) 070 [hep-th/0502186] [INSPIRE]. JHEP 09 (2004) 032 [hep-th/0407140] [INSPIRE]. integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE]. Classical tunneling, JHEP 07 (2012) 044 [arXiv:1111.2349] [INSPIRE]. [arXiv:1111.4663] [INSPIRE]. [arXiv:1205.4400] [INSPIRE]. 494018 [arXiv:1205.4412] [INSPIRE]. Phys. A 47 (2014) 245401 [arXiv:1403.0358] [INSPIRE]. [INSPIRE]. Theta-morphism, JHEP 04 (2014) 068 [arXiv:1205.5288] [INSPIRE]. JHEP 04 (2014) 019 [arXiv:1401.0384] [INSPIRE]. [arXiv:1311.6404] [INSPIRE]. 173 [arXiv:1404.4128] [INSPIRE]. [INSPIRE]. contribution, JHEP 12 (2011) 095 [arXiv:1109.6262] [INSPIRE]. [INSPIRE]. integrability, JHEP 09 (2012) 022 [arXiv:1205.6060] [INSPIRE]. JHEP 03 (2014) 052 [arXiv:1312.3727] [INSPIRE]. operators, JHEP 03 (2014) 096 [arXiv:1311.7461] [INSPIRE]. Regge spins, JHEP 04 (2015) 134 [arXiv:1410.4746] [INSPIRE]. operators in the Regge and Lorentzian OPE limits, JHEP 04 (2014) 094 [arXiv:1311.4886] [INSPIRE]. [34] Z. Bajnok and R.A. Janik, String eld theory vertex from integrability, JHEP 04 (2015) 042 [35] Z. Bajnok and R.A. Janik, The kinematical AdS5 S5 Neumann coe cient, JHEP 02 B 897 (2015) 374 [arXiv:1410.8860] [INSPIRE]. the gauge/gravity duality, JHEP 11 (2010) 141 [arXiv:1008.1070] [INSPIRE]. [1] Y. Jiang and A. Petrovskii, Diagonal form factors and hexagon form factors, JHEP 07 [4] K. Okuyama and L.-S. Tseng, Three-point functions in N = 4 SYM theory at one-loop, [5] L.F. Alday, J.R. David, E. Gava and K.S. Narain, Structure constants of planar N = 4 [6] R. Roiban and A. Volovich, Yang-Mills correlation functions from integrable spin chains, [7] J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and [8] N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. [9] O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [10] O. Foda and M. Wheeler, Partial domain wall partition functions, JHEP 07 (2012) 186 [11] I. Kostov, Classical limit of the three-point function of N = 4 supersymmetric Yang-Mills theory from integrability, Phys. Rev. Lett. 108 (2012) 261604 [arXiv:1203.6180] [INSPIRE]. [12] I. Kostov, Three-point function of semiclassical states at weak coupling, J. Phys. A 45 (2012) [13] E. Bettelheim and I. Kostov, Semi-classical analysis of the inner product of Bethe states, J. [14] N. Gromov and P. Vieira, Quantum integrability for three-point functions of maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 211601 [arXiv:1202.4103] [15] N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. [16] Y. Jiang, I. Kostov, F. Loebbert and D. Serban, Fixing the quantum three-point function, [17] O. Foda, Y. Jiang, I. Kostov and D. Serban, A tree-level 3-point function in the SU(3)-sector [18] P. Vieira and T. Wang, Tailoring non-compact spin chains, JHEP 10 (2014) 35 [19] J. Caetano and T. Fleury, Three-point functions and su(1j1) spin chains, JHEP 09 (2014) [20] Y. Jiang, S. Komatsu, I. Kostov and D. Serban, The hexagon in the mirror: the three-point function in the SoV representation, J. Phys. A 49 (2016) 174007 [arXiv:1506.09088] [21] E. Sobko, A new representation for two- and three-point correlators of operators from sl(2) sector, JHEP 12 (2014) 101 [arXiv:1311.6957] [INSPIRE]. [22] R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: the AdS [23] Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, JHEP 01 (2012) 110 [Erratum ibid. 06 (2012) 150] [arXiv:1110.3949] [24] Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from [25] Y. Kazama and S. Komatsu, Three-point functions in the SU(2) sector at strong coupling, [26] V. Kazakov and E. Sobko, Three-point correlators of twist-2 operators in N = 4 SYM at Born approximation, JHEP 06 (2013) 061 [arXiv:1212.6563] [INSPIRE]. [27] I. Balitsky, V. Kazakov and E. Sobko, Structure constant of twist-2 light-ray operators in the Regge limit, Phys. Rev. D 93 (2016) 061701 [arXiv:1506.02038] [INSPIRE]. [28] I. Balitsky, V. Kazakov and E. Sobko, Three-point correlator of twist-2 light-ray operators in [29] T. Bargheer, J.A. Minahan and R. Pereira, Computing three-point functions for short [30] J.A. Minahan and R. Pereira, Three-point correlators from string amplitudes: mixing and [31] M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [32] M.S. Costa, J. Drummond, V. Goncalves and J. Penedones, The role of leading twist [33] T. Klose and T. McLoughlin, Worldsheet form factors in AdS/CFT, Phys. Rev. D 87 (2013) [36] Y. Jiang, I. Kostov, A. Petrovskii and D. Serban, String bits and the spin vertex, Nucl. Phys. [37] Y. Jiang and A. Petrovskii, From spin vertex to string vertex, JHEP 06 (2015) 172 [38] Y. Kazama, S. Komatsu and T. Nishimura, Novel construction and the monodromy relation for three-point functions at weak coupling, JHEP 01 (2015) 095 [Erratum ibid. 08 (2015) [39] B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar [40] B. Eden and A. Sfondrini, Three-point functions in N = 4 SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE]. [41] B. Basso, V. Goncalves, S. Komatsu and P. Vieira, Gluing hexagons at three loops, Nucl. Phys . B 907 ( 2016 ) 695 [arXiv:1510.01683] [INSPIRE]. [42] K. Zarembo , Holographic three-point functions of semiclassical states , JHEP 09 ( 2010 ) 030 [43] M.S. Costa , R. Monteiro , J.E. Santos and D. Zoakos , On three-point correlation functions in [44] L. Hollo , Y. Jiang and A. Petrovskii , Diagonal form factors and heavy-heavy-light three-point functions at weak coupling , JHEP 09 ( 2015 ) 125 [arXiv:1504.07133] [INSPIRE]. [45] B. Pozsgay and G. Takacs , Form factors in nite volume. II. Disconnected terms and nite temperature correlators, Nucl . Phys . B 788 ( 2008 ) 209 [arXiv:0706.3605] [INSPIRE]. [46] A. Leclair and G. Mussardo , Finite temperature correlation functions in integrable QFT, Nucl . Phys . B 552 ( 1999 ) 624 [hep-th /9902075] [INSPIRE]. QFT , JHEP 07 ( 2013 ) 157 [arXiv:1305.3373] [INSPIRE]. [47] B. Pozsgay , Form factor approach to diagonal nite volume matrix elements in integrable [48] B. Pozsgay , I.M. Szecsenyi and G. Takacs , Exact nite volume expectation values of local operators in excited states , JHEP 04 ( 2015 ) 023 [arXiv:1412.8436] [INSPIRE]. [49] . Beisert , The analytic Bethe ansatz for a chain with centrally extended su(2j2) symmetry , J. Stat. Mech. 01 ( 2007 ) P01017 [nlin/0610017] . [50] N. Beisert , B. Eden and M. Staudacher , Transcendentality and crossing, J. Stat. Mech. 0701


This is a preview of a remote PDF: http://link.springer.com/content/pdf/10.1007%2FJHEP01%282017%29021.pdf

Yunfeng Jiang. Diagonal form factors and hexagon form factors II. Non-BPS light operator, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP01(2017)021