#### 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