LeClairMussardo series for twopoint functions in Integrable QFT
Accepted: May
LeClairMussardo series for twopoint functions in
B. Pozsgay 0 1 2 3 5
I.M. Szecsenyi 0 1 2 3 4
0 H1111 Budapest , Hungary
1 Budapest University of Technology and Economics
2 1111 Budapest , Budafoki ut 8 , Hungary
3 BME Statistical Field Theory Research Group, Institute of Physics
4 Department of Mathematics , City , University of London
5 Department of Theoretical Physics, Budapest University of Technology and Economics
We develop a wellde ned spectral representation for twopoint functions in relativistic Integrable QFT in nite density situations, valid for spacelike separations. The resulting integral series is based on the in nite volume, zero density form factors of the theory, and certain statistical functions related to the distribution of Bethe roots in the nite density background. Our nal formulas are checked by comparing them to previous partial results obtained in a lowtemperature expansion. It is also show that in the limit of large separations the new integral series factorizes into the product of two LeClairMussardo series for onepoint functions, thereby satisfying the clustering requirement for the twopoint function.
Integrable Field Theories; Bethe Ansatz

Integrable
1 Introduction 2
The LeClairMussardo series
Onepoint functions
3
4
5
6
7
2.1
2.2
2.3
4.1
4.2
4.3
4.4
6.1
6.2
c
c
G0
G1
Conclusions
A Contour transformation
1
Introduction
Earlier results for twopoint functions in iQFT
LeClairMussardo series for twopoint functions
Form factors of the bilocal operators
Form factor properties of the bilocal form factors
Lorentz transformation
Exchange property
Kinematical poles
Other properties and discussion
Compact representations of the LeClairMussardo series
Comparison to our earlier work
One dimensional integrable models are special interacting many body systems, where the
eigenstates and eigenenergies can be computed with exact methods. Various forms of the
method called Bethe Ansatz have been developed that apply to a wide variety of models
including spin chains, continuum models, and Quantum Field Theories [1{3]. Interest in
integrable models has been sparked by recent experimental advances [4]: it has become
possible to measure physical quantities of such systems both at equilibrium and in
farfromequilibrium situations. In order to compare to experimental data it is essential to
calculate the correlation functions.
However, computing exact correlations in Bethe Ansatz solvable models is a notoriously
di cult problem. Depending on the models a number of methods have been developed;
they include the Algebraic Bethe Ansatz [1], methods based purely on symmetry arguments
(for example the vertex operator approach to spin chains [5]), or the socalled form factor
approach. Here we do not attempt to review the vast literature, instead we focus on the
form factor method.
{ 1 {
The main idea of the form factor approach is to evaluate twopoint functions as a
spectral sum over intermediate states.
By de nition, the form factors are the matrix
elements of local operators on in nite volume scattering states, and they are naturally
related to
nite volume onshell matrix elements [6, 7]. Traditionally there are two ways
to obtain the form factors: either by solving a set of functional relations that follow from
factorized scattering and relativistic invariance [1, 7{11], or by explicitly embedding the
local operators into the Algebraic Bethe Ansatz framework [12{15]. In many cases these
methods lead to explicit and compact representations of the form factors. On the other
hand, the summation of the spectral series is typically a very challenging problem, and its
treatment depends on the speci cs of the physical situation.
First of all, the form factor approach was applied in massive Integrable QFT in order
to obtain twopoint functions in the physical vacuum [10]. In these cases only the socalled
elementary form factors (matrix elements between the vacuum and a multiparticle state)
are needed, and the resulting integral series has good convergence properties. Typically
it can not be summed up analytically, but a numerical treatment gives highly accurate
results [16].
A completely di erent situation arises when there is a nite density of excitations in
the system. Examples include the ground states of certain nonrelativistic models such
as the antiferromagnetic spin chains or the LiebLiniger model, or
situations. Quantum quenches also belong to this class of problems: the longtime limit of
local correlation functions can be evaluated on a nite density background given by the
socalled Generalized Gibbs Ensemble [17{19]. In these cases the form factors and correlation
functions display di erent types of singular behaviour, depending on how the nite density
state is constructed. Starting with the in nite volume form factors with a xed number of
particles one encounters the socalled kinematical poles, whose treatment requires special
care.
On the other hand, starting with a
nite volume and increasing the number of
particles proportionally with the volume can lead to a nontrivial scaling behaviour for the
transition matrix elements [20]. In the XXZ spin chain and related models this approach
was used to compute the long distance behaviour of correlations (see [21] and references
therein), by studying the asymptotics of explicit determinant representations of the form
factors, and performing the relevant summations.
In the context of integrable QFT (iQFT) a framework was proposed in [22] to deal
with the kinematical poles of the form factors in
nite temperature situations. Integral
series were derived for onepoint and twopoint functions. The resulting series (today
known as the LeClairMussardo series) are built on the basic form factors with a
nite
number of particles. Additionally, they involve a thermodynamic function that describes
the distribution of Bethe roots, and in the case of the twopoint function a thermodynamic
dressing of the energy and momenta of the intermediate particles. Arguments and
counterexamples against the LM series for twopoint functions were presented in [23{27], whereas
the result for the onepoint function was still believed to be true [23].
In [28] it was shown that the LM series for onepoint functions follows from a nite
volume expansion of mean values, which uses the socalled connected limit of the in nite
volume form factors. The expansion itself was conjectured in full generality in [29] (see
{ 2 {
also [23]), and for the nonrelativistic LiebLiniger model it was proven using Algebraic
Bethe Ansatz in [28]. Finally, the expansion was proven also for iQFT in the recent
work [30], which thus completed the proof of the LM series for onepoint functions.
On the other hand, the problem of the
nite temperature or nite density twopoint
functions has remained unresolved. Multiple works performed a lowtemperature expansion
and obtained explicit formulas for the rst few terms [31{36]. These results were free of
any singularities, and they could be understood as the rst few terms in an expansion of a
hypothetical LMtype series, but it was not clear what the general structure of this integral
series should be.
As an alternative approach it was suggested in [25{27] that nite temperature cor
relations should be computed using form factors that take into account the dressing due
to the
nite density background. This is in contrast to the logic of the LM series, which
uses the zerodensity form factors calculated over the vacuum. Even though there is clear
physical motivation for this approach, the proposed program has only been applied to free
theories. Quite interestingly, a similar picture emerged in the recent work [37], which
considered correlation functions in generic inhomogeneous nonequilibrium situations within
the framework of Generalized Hydrodynamics. An integral series was derived for the large
time and long distance limit of correlations, which would apply as a special case also to
static correlations in an arbitrary
nite density background. The results of [37] only
concern the large time limit, nevertheless it is remarkable that the integral series takes the
same form as suggested by the works [25{27].
It is somewhat overlooked in the literature, that the formalism of the LeClairMussardo
series was already developed much earlier in the context of the in nite volume,
nonrelativistic Quantum Inverse Scattering Method [12, 13, 38{41]. In the particular case
of the 1D Bose gas explicit formulas were obtained for the form factors of the eld
operators [12] and the particle current [39] from the socalled Quantum GelfandLevitan method.
In [40] these were shown to satisfy a set of functional relations that are known today as
\form factor axioms" in iQFT. Furthermore, an integral series for the twopoint function
was also derived in [13, 40], that has the same structure as the LM series for onepoint
functions. In these works the twopoint function is treated as a composite object, and the
resulting series is built on the form factors of the bilocal operator. In this approach there is
no need for an insertion of intermediate states, because the underlying method (the
Quantum GelfandLevitan equation) allows for an explicit representation of the bilocal product
of eld operators in terms of the FaddeevZamolodchikov creation/annihilation operators.
This program remained con ned to the 1D Bose gas, essentially due to the fact that
in iQFT the form factors are determined from the solution of the form factor axioms,
their structure is considerably more complicated (for example the operators don't preserve
particle number), and there is no e cient method to treat the matrix elements of bilocal
operators. Moreover, even in the 1D Bose gas alternative approaches (such as the
nite
volume Algebraic Bethe Ansatz) became dominant, because they lead to intermediate
formulas that are more convenient for subsequent analytic or numerical analysis.
In a completely independent line of research the paper [42] developed an alternative
framework to deal with
nite density states in integrable models. Here the goal was to
{ 3 {
provide a eld theoretical derivation of the TBA equations, by computing the mean value
of the Hamiltonian density using its form factor series. The main idea of this work is to
consider smeared states such that one does not hit the singularities of the form factors
directly. Although this work only considered the Hamiltonian density, its derivations only
rely on the general properties of the form factors, therefore all of the intermediate results
(before specifying the operator through its form factors) are valid for general onepoint
functions. This would imply an independent proof of the LeClairMussardo series for
onepoint functions. We believe that this connection has not yet been noticed in the literature.
An important lesson of the works [12, 13, 39, 40] is that the bilocal operators satisfy the
same kinematical pole equation as the local ones. In the 1D Bose gas this was established
using explicit form factors for the bilocal operators. Therefore, any type of regularization
procedure that treats the kinematical singularities has to work equally well for the onepoint
and twopoint functions. In the present work we build on these ideas: we present arguments
for the validity of the kinematical pole equation for the bilocal operators even in integrable
QFT, and derive a wellde ned LeClairMussardo formula for the twopoint function.
The article is composed as follows. In section 2 we review previous approaches
towards the LM series, both for the onepoint and twopoint functions. In section 2.3 we
also formulate our main result using the form factors of the bilocal operators. These are
computed in section 3 by inserting a complete set of states between the two operators.
The properties of the bilocal form factors are studied in section 4. Two di erent compact
representations for the LM series are derived in section 5, where we explicitly prove the
clustering property of the integral series for the twopoint function. Our results are
compared in section 6 to earlier calculations in a lowtemperature expansion. Finally, section 7
includes our conclusions.
2
The LeClairMussardo series
In this work we consider onepoint and twopoint functions in massive, relativistic,
integrable QFT. We limit ourselves to theories with one particle species with mass m. The
scattering phase shift will be denoted by S( ), where
is the rapidity variable. We put
forward that the generalization of our results to more particles with diagonal scattering is
straightforward, but theories with nondiagonal scattering pose additional technical
challenges, which are not considered here.
Let us denote the incoming and outgoing scattering states as
j 1; : : : ; ni;
h#n; : : : ; #1j;
where the rapidities are real numbers such that j > k and #j > #k for j > k.
We consider local operators O(x; t) in 2 dimensional Minkowski space. Their form
factors are uniquely de ned as the matrix elements
FnO;m(#n; : : : ; #1j 1; : : : ; m) = h#n; : : : ; #1jOj 1; : : : ; mi;
where O
O(0; 0). Originally de ned for sets of rapidities with the ordering given above,
the form factor functions are extended analytically to the whole complex plane. The
{ 4 {
(2.1)
(2.2)
analytic properties of these functions have been investigated in great detail in [10, 43] and
they will be discussed below.
Our goal is to derive integral representations for the mean values of the onepoint and
twopoint functions in
nite density situations. A
nite density state can be characterized
by a density of rapidities r( ) such that in a
nite volume L the number of particles
between
and
+
is
N = 2
( )
. As usually we also de ne the density of holes
h( ), which satis es the integral equation
Z 1 d 0
i dd log S( ) is the scattering kernel and p0( ) = m cosh is the derivative of
the oneparticle momentum. We also de ne the lling fraction
f ( ) =
r( )
h( ) + r( )
:
The physical applications include the standard Gibbs and the Generalized Gibbs (GGE)
ensembles [17]. In the rst case the thermal average is de ned as
where
= 1=T . Similarly, for the GGE
hO1(0; 0)O2(x; t)iT =
Tr e
H
O1(0; 0)O2(x; t)
Tr (e
H )
;
hO1(0; 0)O2(x; t)iGGE =
Tr e
P
j jQj O1(0; 0)O2(x; t)
Tr e
P
j jQj
:
In both cases the ensemble average can be simpli ed to a single mean value on a
representative state j i, whose root density r( ) is determined by the Thermodynamic Bethe
Ansatz (TBA) equations [44]. De ning the pseudoenergy as "( ) = log hr(( )) , the standard
TBA equation reads
Z 1 d 0
charges also enter the source terms.
Our main goal is to develop integral series for the objects
h jO1(0)O2(x; t)j i
using their in nite volume, zerodensity form factors, for arbitrary root distributions. We
allow for two di erent operators O1 and O2, even though in practice they are typically
the same or simply just adjoints of each other. We will restrict ourselves to spacelike
separations x2
t2 > 0, for reasons to be discussed below.
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
{ 5 {
In evaluating the twopoint function there are two main di culties that need to be
solved. First of all, the
nite density state j i is not a well de ned object if one starts
from the in nite volume directly. Instead, di erent regularization schemes need to be
applied that increase the number of particles gradually. Second, one has to deal with the
singularities of the form factors, including the disconnected pieces and the kinematical
poles. Even after the subtraction of the singular parts it is highly nontrivial to nd the
remaining
nite contributions.
In 2.1 we review previous approaches to the onepoint functions that lead to the
corresponding LeClairMussardo series. Later in 2.2 we review previous attempts for the
twopoint function and in 2.3 we formulate the main results. However, before turning to
the LM series we list here the main analytic properties of the form factors of local operators,
and discuss the diagonal limit with a
nite number of particles.
A generic form factor can be expressed with the socalled elementary form factors
(matrix elements between the vacuum and a multiparticle state) by applying the socalled
crossing relation
FmO;n( 10; : : : ; m0j 1; : : : ; n) = FmO 1;n+1( 10; : : : ; m0 1j m0 + i ; 1; : : : ; n)
n
+ X
k=1
2
( m0
the relations [43, 45{48]
I. Lorentz transformation:
FnO( 1 + ; : : : ; n + ) = esO FnO( 1; : : : ; n);
{ 6 {
n
Y S(
k=1
1
hOi
where sO is the Lorentzspin of the operator.
II. Exchange:
FnO( 1; : : : ; k; k+1; : : : ; n) = S( k
k+1)FnO( 1; : : : ; k+1; k; : : : ; n):
III. Cyclic permutation:
IV. Kinematical singularity:
FnO( 1 + 2i ; 2; : : : ; n) = FnO( 2; : : : ; n; 1):
i R=es0 FnO+2( + i ; 0 ; 1; : : : ; n) =
1
!
k) FnO( 1; : : : ; n):
(2.13)
There is a further relation related to the bound state structure of the theory, but it will not
be used in the present work. On the other hand, we will assume that the form factors show
the asymptotic factorization property, when a subset of the rapidities is boosted to in nity:
!1
lim FnO( 1 + ; : : : ; m + ; m+1; : : : ; n) =
FmO( 1; : : : ; m)FnO m( m+1; : : : ; n):
(2.9)
(2.10)
(2.11)
(2.12)
(2.14)
i Res FnO;m(#n; : : : ; #1j 1; : : : ; m) =
=
1
m
Y S( 1
k=2
n
k=2
The diagonal limit is reached by setting n = m and letting #j !
j . In this limit the
form factor has an apparent nfold pole, but a straightforward calculation shows that the
residue is actually zero. It can be shown that around this point the form factor behaves as
HJEP05(218)7
FnO;n(#n; : : : ; #2j 2; : : : ; n)
;
Onepoint functions
point functions:
In the seminal work [22] the following result was proposed for the nite temperature
onehOiT =
n=0
j=1 1 + e"( j) A FnO;c( 1; : : : ; n);
(2.15)
(2.17)
(2.18)
(2.19)
where the coe cients Ai1i2:::in are symmetric in the n indices.
There are two natural ways to de ne a regularized diagonal form factor. The socalled
connected form factor is de ned as
FnO;c( 1; : : : ; n)
F.P. nF2On( 1 + "1; : : : ; n + "nj n; : : : ; 1) ;
o
where F.P. stands for nite part, i.e. the terms which are free of any singularities of the
form "j ="k. According to the expansion (2.16) this coincides with n!A12:::n. The second
possibility is to de ne the symmetric limit as
FnO;s( 1; : : : ; n)
"li!m0 F2On( 1 + "; : : : ; n + "j n; : : : ; 1):
Both diagonal form factors are symmetric in their variables. The linear relations between
them can be found using (2.15); they were studied in detail in [29]. Further analytic
properties of the diagonal form factors were studied in [50].
First observed by Smirnov [43] and later proven in [49], this relation holds for relevant
scaling operators, and it is used to identify solutions to the form factor axioms with
concrete operators.
For future use it is useful to display the kinematical pole relation in the form
where "( ) is the solution of the TBA equations (2.7) and FnO;c are the connected diagonal
form factors de ned in (2.17).
The main idea of [22] was to consider the nitetemperature problem in nite volume
L and in the lowtemperature limit T
m. In this case the volume parameter can be
chosen to satisfy Le m=T
1 such that the partition functions in (2.5) are dominated
by states with few particles. In this case it is possible to perform an expansion in the
{ 7 {
small parameter e m=T such that the disconnected terms in the numerator are canceled by
the Boltzmannsums in the denominator. A formal calculation gives disconnected terms
proportional to (0), which were interpreted in [22] as diverging terms proportional to the
volume. It was argued that each order in the cluster expansion becomes nite after the
cancellation of all Diracdeltas. However, this procedure only kept to most divergent pieces
in L, and subleading singularities can also a ect the remaining
nite answer.
A more rigorous approach was initiated in [29] which aimed to evaluate the
average (2.5) in the small temperature limit by keeping all diverging pieces polynomial in L.
This was achieved by developing a precise description of the nite volume diagonal matrix
elements. In the following we brie y review the results of [29].
Let us denote nite volume states by
j 1; : : : ; N iL:
Here it is understood that the rapidities solve the Bethe equations
eiQj
eipjL Y S( j
k6=j
k) = 1;
j = 1 : : : N;
where pj = p( j ) = m sinh( j ). For transition matrix elements it was found in [6]
10; : : : ; M0 Oj 1; : : : ; N iL =
p
where N and
M are Gaudin determinants:
FNO;M ( 10; : : : ; M0 j 1; : : : ; N )
M ( 10; : : : ; M0 ) N ( 1; : : : ; N )
+ O(e
L);
N ( 1; : : : ; N ) = det J ij ;
J
ij =
They can be interpreted as the density of states in rapidity space, and in nonrelativistic
models as the exact norm of the Bethe Ansatz wave function. The relation (2.20) simply
states that (apart from the physically motivated normalization) the form factors are the
same in
nite and in nite volume. For a coordinate Bethe Ansatz interpretation of this
statement see [7].
For nite volume mean values the following result was found in [29]:
h 1; : : : ; N jOj 1; : : : ; N iL =
F2On;c f
g
N n f +gjf
1
N ( 1; : : : ; N )
X
f +g[f g
N n(f +gjf
g) = det J+;
(2.20)
(2.21)
g ;
(2.22)
(2.23)
where we used the restricted determinant
where J+ is the submatrix of J corresponding to the particles in the set f +g. There
is also an alternative representation built on the symmetric diagonal form factors (2.18),
but it will not be used here. A rigorous proof of (2.22) was given for local operators of
the LiebLiniger model in [28], and for relativistic QFT it was nally proven in [30]. It is
{ 8 {
important that [30] used the kinematical pole relation and the result (2.20) for o diagonal
In [29] the expansion (2.22) was used to perform a rigorous lowtemperature expansion
of the Gibbs average, and the LM series was con rmed up to third order. These calculations
were performed in the regime Le m=T
low particle numbers.
1, such that it was enough to consider states with
An alternative, allorders proof was later given in [28]. Here the idea was to consider
a representative state at some temperature (or with some
xed root distribution) and to
perform the thermodynamic limit directly on the formula (2.22). In this approach the
physical amplitudes F2On;c f
g of (2.22) enter the integrals of the LM series, and the
ratios of the determinants produce the weight functions 1=(1 + e"( )). More generally it
was shown that for an arbitrary state we have
where f ( ) is the lling fraction (2.4). It was also shown that the LM series can be expressed
using the socalled symmetric diagonal form factors de ned in (2.18) as
h jOj i =
n=0
(2.24)
(2.25)
(2.26)
0 n
j=1
Z d 0
2
{ 9 {
1
1
where
!( ) = exp
f ( 0)'(
0) :
The two formulas (2.24) and (2.25) can be considered to be partial resummations of each
other. Results similar to (2.25) (including the weight function !( )) had been obtained for
the LiebLiniger model using Algebraic Bethe Ansatz [1, 51].
It is remarkable, that results of the form (2.24) were obtained much earlier in the
context of the 1D Bose gas in [12, 13, 38{40]. These works considered both onepoint
and twopoint functions, for example the
eld eld correlation
y(x) (0), which in the
x ! 0 limit becomes the particle density operator. An integral series with the same
structure as the LM series was derived for y(x) (0), and in the x ! 0 limit the YangYang
thermodynamics was obtained completely independently from the TBA arguments [38].
The main idea of these works is to introduce a regularization involving a Galilean boost
operator [38], which renders the various singular terms nite. A very important result of
this approach, presented in [40], is that the form factors of the two point function
y(x) (0)
satisfy the same kinematical pole equation (2.13) as those of the local operators, see for
example (3.11) of [40].
An alternative in nite volume regularization scheme for onepoint functions was
developed in [42], which considered smeared states to avoid the singularities of the form factors.
The main goal of [42] was to provide a eld theoretical derivation of the TBA equations,
by calculating the mean value of the energy density operator in a
nite density state.
Although the regularization in [42] is di erent from that of [38], the treatment of the form
factors is the same and the calculation relies only on the kinematical pole property. The
work [42] only considered the energy current operator, but its methods could be adapted
in a straightforward way to arbitrary local operators, and this would give an independent
rigorous proof of the LM series (2.19). We believe that this connection has not been noticed
in the literature before.
Earlier results for twopoint functions in iQFT
In [22] the following series was proposed for the nite temperature twopoint functions:
hO(x; t)O(0; 0)iT
hOiT
1
X
N=1
1
N !
X
i= 1
Z d 1 : : :
2
d N
2
where f j ( j ) = 1=(1 + e j"( j)) and kj = k( j ), where k( ) can be interpreted as the
dressed momentum and it is given by
Z
k( ) = m sinh( ) +
d 0 (
0) 1( 0);
where 1( ) is the solution of the integral equation
2
1( )(1 + e"( )) = m cosh( ) +
d 0'(
0) 1( 0):
Z
The form factors appearing in (2.27) are de ned by
h0jOj 1 : : : N i 1::: N = FNO( 1
i ~1; : : : ; N
i ~N )
~j = (1
j )=2 2 f0; 1g:
It is an important feature that the x and t dependent phase factors involve the dressed
energies and momenta of the particles, where the dressing is due to the
nite density
background. On the other hand, the form factors are the bare quantities.
An explicit counterexample to (2.27) was found in [23]; this counterexample involved
a chemical potential, and it was not clear whether (2.27) could still hold for the
= 0 case
for which it was originally derived. In [24] the temperature dependent twopoint function
of the stressenergy operator T (x) was evaluated in the scaling LeeYang model, in the
massless limit. The results were compared to benchmark calculations from CFT. The
work [24] only considered terms of the LM series up to N = 2, but after an investigation
of the convergence properties of the series it was concluded that (2.27) can not be correct.
In [25{27] it was shown that the LM series is not correct in the Ising model for most
operators; it only works for operators with at most two free eld factors.
Also, there is a central problem with the proposal (2.27), which is independent from
the previous counterexamples: the higher terms with N
3 involve illde ned integrals
and it is not speci ed how to subtract the singular pieces. For N
3 all the terms for
which the
variables are not equal have poles in the variables. These poles are squared
and there is no prescription given for the subtraction of these singularities. As far as we
HJEP05(218)7
(2.28)
(2.29)
know, this problem has not been emphasized in the literature. The calculation of [24] did
not encounter this problem, because they only included terms up to N = 2, for which the
prefactor of the double pole is zero.
Later works attempted to perform a wellde ned lowtemperature expansion of the
Gibbs average for the twopoint function. In [31{34] form factor series were developed for
spin chains and
eld theoretical models. The paper [
34
] includes both
nite and in nite
volume regularization, involving also a constant shift in the rapidity parameters to treat the
singularities of the form factors; this corresponds to adding a Lorentzboost operator into
the de nition of the partition function. Adding a boost is essentially the same technique
that was used in [12, 13, 38{40], where the Galilean boost operator was used due to the
nonrelativistic kinematics of the LiebLiniger model.
The nite volume regularization was applied later in [35, 36], where the main goal was
to derive a regularized form factor expansion irrespective of the details of the model or the
operator. The summation over Bethe states in the Gibbs average was transformed into
contour integrals such that the in nite limit volume could be taken in a straightforward
way. However, a shortcoming of this method was that it required a term by term analysis,
and it was not evident how to express the general higher order terms.
All of these approaches use the bare form factors, that is, the in nite volume zero
density form factors of the theory. In [25{27] it was suggested that
correlations should be described by dressed form factors that already take into account
certain e ects of the background. New form factor axioms were set up for the
nite
temperature case, and they were solved for problems involving Majorana fermions. However,
the approach has not yet been worked out for interacting theories. On the other hand, an
exact result was computed in [37] for the large scale correlations in generic inhomogeneous,
nonequilibrium situations, which also include static backgrounds as special cases. Here we
do not discuss the results of [37] in detail, as they apply to the large time limit; however,
in the Conclusions we give remarks about the possible relations to our work.
2.3
LeClairMussardo series for twopoint functions
In the following we derive a new form factor series for the twopoint function in nite density
situations. The structure of our result is essentially the same as that of the formula (2.19),
therefore it can be called the LMseries for the twopoint function. The central idea is
to treat the product of two local operators as a composite object, and to investigate its
matrix elements.
Let us de ne the form factors of the bilocal operator O1(0; 0)O2(x; t) as
Gxn;;tm(#n; : : : ; #1j 1; : : : ; n) = h#n; : : : ; #1jO1(0; 0)O2(x; t)j 1; : : : ; mi:
(2.30)
In order to shorten the notations, we have omitted the superscript O1O2 for Gxn;;tm and
we assume that we will deal with the same two (unspeci ed) local operators throughout
the work.
It is important that we restrict ourselves to spacelike separations x2 t2 > 0, in which
case the two operators commute with each other. This restriction is essential for some
of our arguments to be presented below. The form factors are always Lorentzinvariant,
therefore we could transform any jtj < jxj to t = 0 with a proper boost; however, the nite
density state j i is typically not Lorentzinvariant, therefore we reserve the possibility of
a nite time displacement.
In the following we present three independent arguments that show that the bilocal
form factors satisfy the same kinematical pole property (2.15) as the usual form factors.
Our rst argument is based on the analytic properties of the Operator Product
Expansion (OPE)1 Writing the bilocal operator as the OPE
O1(0; 0)O2(x; t) =
X cjO1O2 (x; t)Oj (0; 0)
j
(2.31)
each term on the r.h.s. satis es the analytic properties, therefore their sum does too. This
argument relies on the existence and the absolute convergence of the OPE. Whereas this
has not been rigorously proven, it is generically believed to be true in QFT for spacelike
separations [52{56].
Our second argument relies on the coordinate Bethe Ansatz wave function. It was
shown in [7] that in nonrelativistic cases the kinematical pole is easily proven by
investigating the real space integrals in the form factor. For this argument we require t = 0, but
here we deal with only a nite number of rapidity variables (independent of the background
j i), therefore we can always set t to zero by an appropriate Lorentztransformation. We
argue that the bilocal form factor satis es the relation
#1= 1
i Res Gxn;;tm(#n; : : : ; #1j 1; : : : ; m) =
=
1
m
Y S( 1
k=2
n
k=2
k) Y S(#k
#1)
!
(2.32)
Gxn;t 1;m 1(#n; : : : ; #2j 2; : : : ; m);
which has the same form as (2.15). The singularity can be understood easily by real space
calculation of the form factors [7]. The two terms in the pole represent divergent real space
integrals when the particles with rapidities 1 and #1 are before or behind the operator
and all the other particles, and the prefactor re ects the change of the phase of the wave
functions as the particles with 1 and #1 are moved from the rst position to the last.
The kinematical pole arises from in nite x !
1 real space integrations, therefore it is
not sensitive to the precise locality properties of the operators, the only requirement being
the product should have a
nite support in real space. Similarly, relativistic e ects that
modify the Bethe Ansatz wave function on small distances (comparable to the Compton
wavelength of the particles) do not play a role either, because the nonrelativistic derivation
of [7] only uses the long distance behaviour of the wave function, which is described by the
Bethe Ansatz even in QFT. Therefore, this argument also supports the kinematical pole
property even for the bilocal operators.
The third argument is based on explicit representations of the bilocal form factors
in terms of the form factors of the individual operators; such integral formulas will be
1This argument was suggested to us by Gabor Takacs.
physical understanding.
functions:
presented in section 3. The kinematical pole is then evaluated explicitly in section 4.3. This
derivation is mathematically rigorous, but it leaves the physical meaning of the kinematical
residue somewhat obscure. We believe that it is our rst two arguments which provide the
In 2.1 we reviewed three di erent approaches that lead to the LM series for onepoint
The in nite volume regularization of [12, 13, 38{40] that uses a boost operator.
The in nite volume regularization of [42] that uses smeared states.
HJEP05(218)7
The nite volume regularization of [29] (supplied with the proof of [30] for the diagonal
matrix elements), and the presentation of [28] regarding the thermodynamic limit.
All three approaches use only the kinematical pole property, and we have argued above
that this property holds for the bilocal operators as well. It follows that nite density
twopoint functions are given by the LeClairMussardo series
where Gxn;;tc is the connected diagonal form factor of the bilocal operator, which is de ned
as the nite part of
Gxn;;tn( 1 + "1; : : : ; n + "nj n; : : : ; 1);
which is free of any singularities of the form "j ="k. Alternatively, the LM series can be
expressed in the same form as (2.25):
2
0Yn Z d j f ( j )!( j )A Gxn;;ts( 1; : : : ; n);
1
where
"!0
Gxn;;ts( 1; : : : ; n) = lim Gxn;;tn( 1 + "; : : : ; n + "j n; : : : ; 1);
and !( ) is given by (2.26). The equivalence between (2.33) and (2.34) is guaranteed by the
theorems for the connected and symmetric diagonal form factors presented in [29]; these
theorems only use the kinematical pole property, therefore they apply to the twopoint
function as well.
Expressions (2.33){(2.34) are implicit: they do not specify how to compute the form
factors of the bilocal operator, they only describe how to deal with the singularities of
this object. In the next section we also show how to compute the bilocal form factors
using those of the local operators. This is achieved by inserting a complete set of (in nite
volume) states between the two operators, which makes the LM series completely explicit.
It is very important that the composite form factors depend on the position (x; t), and the
diagonal limit has to be taken by considering a full dependence on the rapidities, including
(2.33)
(2.34)
(2.35)
the kinematical factors involving (x; t). Examples in section 6 show that this can lead to
secular terms.
We remark again, that in the nonrelativistic case an LM series of the form (2.33) has
been already established in [40], see for example eq. (3.18) in that work, together with
(3.14), which agrees with our de nition for Gxn;;tc. An important di erence between the two
situations is that in the 1D Bose gas the eld operators change particle number by one,
and the bilocal form factors could be obtained explicitly as sums of algebraic expressions.
On the other hand, in the eld theoretical case the operators can have matrix elements between any N  and M particle states, and the bilocal form factors are obtained as an in nite integral series, to be presented in the next section.
3
Form factors of the bilocal operators
In this section we derive explicit integral representations for the generic nm form factor
Gxn;;tm(#n; : : : ; #1j 1; : : : ; m) = h#n; : : : ; #1jO1(0; 0)O2(x; t)j 1; : : : ; mi
(3.1)
by inserting a complete set of states between the two local observables. The idea and the
methods of this section are essentially the same as in the original works by Smirnov [43].
The analytic properties of the resulting series are investigated in section 4.
A naive insertion of states would lead to an encounter with the kinematical poles: this
happens when intermediate rapidities approach one of the # or
variables. In order to
avoid these singularities we employ a wellde ned expansion of the local operators in terms
of the FaddeevZamolodchikov (FZ) creation/annihilation operators Zy( ), Z( ) [57, 58].
The FZ operators satisfy the commutation relations
Zy( 1)Zy( 2) = S( 1
Z( 1)Z( 2) = S( 1
2)Zy( 2)Zy( 1) ;
2)Z( 2)Z( 1) ;
Z( 1)Zy( 2) = S( 2
1)Zy( 2)Z( 1) + 2
( 1
2)1 :
Local operators are represented in terms of the FZ operators as [43, 59, 60]
1
X
k;l=0
O(x; t) =
Hk;l(x; t);
where
Hk;l(x; t) =
Yl d j fkO;l( 1; : : : ; kj l; : : : 1)Kx;t(f gjf g
)
Zy( 1) : : : Zy( k)Z( l) : : : Z( 1);
(3.2)
(3.3)
(3.4)
where the functions f can be expressed in terms of the form factors
fkO;l( 1; : : : ; kj l; : : : ; 1) = FlO+n( k + i + i0; : : : ; 1 + i + i0; 1
i0; : : : ; l
i0) (3.5)
HJEP05(218)7
and Kx;t is the phase factor due to the displacement of the operator:
Kx;t(f gjf g) = eit(Pi E( i) Pi E( i)) ix(Pi P ( i) Pi P ( i)) ;
(3.6)
where E( ) = m cosh( ) and P ( ) = m sinh( ) are the energy end momentum of the
oneparticle states.
An important ingredient in the expansion above is the presence of the in nitesimal
shifts of
i0. They are irrelevant for the o diagonal form factors of O, but they are
necessary to obtain a wellde ned integral representation for the bilocal form factor (3.1).
The bilocal form factor is obtained by inserting two instances of the expansion (3.3){
(3.4) into (3.1) and computing the contractions of the FZ operators using the algebra (3.2).
It is easy to see that only those terms contribute, where there are at least n FZ creation and
m annihilation operators. Also, there can not be more than m annihilation (or n creation)
operators immediately acting on the ket (or bra) states. These constraints are satis ed by
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
the triple sum
where
where
Gxn;;tm(#n; : : : ; #1j 1; : : : ; m) =
n
X
m
X
1
X
k=0 l=0 p=0
Hkn;;lm;p ;
Hkn;;lm;p = h#n; : : : ; #1jHn k;p+l(0; 0)Hp+k;m l(x; t)j 1; : : : ; mi :
In Hkn;;lm;p the indices k and l show the numbers of the disconnected # and
rapidities.
We introduce the notation for the ordered set of rapidities f j1 ; j2 ; : : : ; jn g = f gJ<
and f#jn ; #jn 1 ; : : : ; #j1 g = f#gJ> , where the elements of J are ordered as ji < ji+1. In;<
denotes the ordered set f1; 2; : : : ; ng and In;> denotes fn; n
1; : : : ; 1g.
For each term in Hkn;;lm;p we need to evaluate a contraction of the form
f#gIn;> Zy( 1) : : : Zy( n k)Z( p+l) : : : Z( 1)Zy( 1) : : : Zy( p+k)Z( m l) : : : Z( 1) f gIm;< ;
The contractions lead to various Diracdeltas and phase factors, which can be treated using
straightforward calculations, leading to
Gxn;;tm(#In;> j Im;< ) = X1 1 Yp Z d i
p=0 p!
i=1
2
X
X
+ i + i0; f gIp;> + i + i0; f gA+<
i0)
S<(
f gA+< jf gA< )S>(f#gB> jf#gB>+ )S$(f#gB> jf gA< ) ;
f#gIm;> = h0jZ(#m) : : : Z(#1):
where the two inner summations run over bipartite partitions of the index sets In and Im,
and we omitted the particle number subscript of the form factors.
f#B+ g
The phase factors S<, S> and S$ above result from the exchanges of the FZ operators
rapidities as
ordered sets f A+< g and f A< g
to O2. They are de ned as follows. For a given partitioning Im = A+
such that the set of particles f A g can be connected to the operator O1, and the set f#B g
by ff A+< g; f A< gg the ordered set of rapidities that is obtained by concatenating the two
. The phase factor S< follows from the rearrangement of
Similarly, S> is de ned as the phase factor coming from the rearrangement
where ai+, i = 1; : : : ; jA+j and aj , j = 1; : : : ; jA j are the indices in the sets A+ and A .
with the explicit expression
Finally, the mutual phase factor S$ is given by
f#gIn;> = S>(f#gB> jf#gB>+ )Df#gB> ; f#gB>+
S>(f#gB> jf#gB>+ ) =
S(#bi+
#bj ):
S$(f#gB> jf gA< ) =
S(#i
j ):
The intuitive interpretation of the formula (3.11) is the following: the incoming and
outgoing particles are connected to one of the local operators, whereas they are disconnected
from the other one; this choice is given by the partitioning of the index sets. On the
other hand, the integration variables j represent additional intermediate particles between
the two operators. The phase factors in (3.11) arise from the interchange of positions of
the particles.
We stress that the i0 shifts for the rapidities are essential to obtain a regular expression for the bilocal form factor. A graphical representation of (3.11) is given in gure 1.
It is useful to derive an alternative representation of the bilocal form factor, where the
integration contour over the intermediate particles is well separated from the incoming and
Y
bi+>bj
Y
Y
i2B j2A
jf gi
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
outgoing rapidities. To this order we shift the integration contour in (3.11) to R + i with
being a small real number. The advantage of this step is that afterwards the kinematical
poles at #j !
k can be studied directly, without paying attention to the integrals. On
the other hand, if some of the
integrals would run between the rapidities #j and k,
then the contour could be pinched, and such cases would complicate the analysis of the
kinematical singularities.
First we assume that
> 0, then the contour shift picks up poles of the form factors
of O1. After a straightforward, but tedious calculation we obtain the formula
Gxn;;tm(#In;> j Im;< ) =
1
X Gxn;;tm;p(#In;> j Im;< );
where
d i
X
X
F O1 (f#gB>+ + i ; f gIp;< ; f gA< )F O2 (f gIp;> + i ; f#gB>
f gA+< jf gA< )S>(f#gB>+ jf#gB> ) ;
where we omitted the
i0 shifts, because they are not relevant anymore. Details of this
calculation are presented in appendix A; here we just note that the changes in the phase
factors (including the reordering of the rapidities in the form factor F O2 ) at a given p result
from adding pole contributions from terms with p0 > p.
We can also shift the contours to the negative imaginary direction, which results in
the alternative formula
Gxn;;tm;p(#In;> j Im;< ) =
A+[A =Im B+[B =In
Kx;t(f gIp ; f#gB jf gA+ )
F O1 (f#gB>+ + i ; f gA< ; f gIp;< )F O2 (f#gB>
+ i ; f gIp;> + i ; f gA+< )
S<(
f gA< jf gA+< )S>(f#gB> jf#gB>+ ) :
The di erences between the integrands in (3.11) and those of (3.17) and (3.18) lie in the
phase factors, including the ordering of the rapidities in the form factors functions.
The convergence of the integrals in the above expressions depends on the separation
between the operators. The dependence for large j j is given solely by the Kfactors,
given that the asymptotic factorization property (2.14) holds. It follows that the
integrations towards
!
1 are convergent in (3.17) if x < 0, and in (3.18) if x > 0. If the two
operators O1;2 are not identical then the twopoint function might not have the x !
symmetry, and only one of the representations can be used, depending on the sign of x. On
x
the other hand, the integrals would never be convergent for timelike separations: they
would diverge either for
! +1 or
!
It is important that the integral series is e ectively a largedistance expansion for the
bilocal form factor: each
integral carries a weight of e ms with s2 = x
2
t2. This is
analogous to the case of the vacuum twopoint function.
We note that essentially the same integral representations as presented in this section
were used by Smirnov to prove the local commutativity theorem [43]. This theorem states
(3.17)
(3.18)
that if there are two operators such that the form factors satisfy the axioms as given above,
then the two operators commute for spacelike separations. We do not replicate the proof
here. Instead, we just point that it also uses a shift of the integration contours in the
complex plane, which is only possible for spacelike displacement between the operators.
4
Form factor properties of the bilocal form factors
In this section we investigate the analytic properties of the bilocal form factors based on
the integral representations (3.17){(3.18). Most importantly we con rm explicitly that the
bilocal form factor satis es the kinematical pole relation (2.32), which is the basis for the
derivation of the LeClairMussardo series. In 4.4 we also discuss the implications of the
form factor axioms for the bilocal operators.
Lorentz transformation
After a Lorentz transformation every rapidity is shifted by . The Smatrix factors are
invariant under the shift, since they only depend on rapidity di erences. Each elementary
form factor satis es the Lorentz transformation axiom (2.10). The phase factor
transforms as
Kx;t(f g +
jf g + ) = Kx0;t0 (f gjf g) ;
where x0 = x cosh( ) t sinh( ) and t0 = t cosh( )
x sinh( ) are exactly the
transformation rules for spacetime coordinates under Lorentz transformation. As a consequence, the
bilocal form factor satis es the following Lorentz transformation rule
Gxn;;tm(f#gIn;> +
jf gIm;< + ) = e(sO1 +sO2 ) Gxn0;;mt0 (f#gIn;> jf gIm;< ) ;
(4.1)
(4.2)
(4.3)
where sOi is the Lorentz spin of the operator Oi.
4.2
Exchange property
appropriate ordering.
4.3
Kinematical poles
It is easy to see that the form factors of the bilocal operators satisfy the exchange property
for rapidities both sides of the operators
Gxn;;tm(: : : ; #i+1; #i; : : : j : : : ) = S(#i+1
Gxn;;tm(: : : j : : : ; i; i+1; : : : ) = S( i
#i)Gxn;;tm(: : : ; #i; #i+1j : : : ) ;
i+1)Gxn;;tm(: : : j : : : ; i+1; i; : : : ) :
For those terms where the rapidities to be exchanged are in the same sets A
or B
the
property follows from the property of the elementary form factors (2.11). In those cases,
where the rapidities are in di erent sets, the extra Smatrix factor exactly cancels out
or introduces the terms in the S<( A+< j A< ) and S>(#B>+ j#B> ) phase factors to get the
The kinematical pole property (2.13) is the essential ingredient for the proof of the
LeClairMussardo formula. Here we prove that it holds for the form factors of the bilocal operators
too. We use the representation (3.17) in which there are no singularities associated with
the intermediate particles. However, the elementary form factors in the summation (3.17)
have direct kinematical poles for #1
1, and these need to be summed up to obtain the
total residue of the bilocal form factor.
The singularities come from terms where #1 and 1 are in the same elementary form
factors, namely if #1 2 f#gB+ and 1 2 f gA
or #1 2 f#gB
and 1 2 f gA+ . Introducing
the notation A~
= A
n f1g and B~
= B
n f1g, the residues in the rst situation are
F O1 (f#gB~>+ + i ; f gIp;< ; f gA~< )
i[S$(f gIp j 1)
S$( 1jf#gB~+ + i )S$( 1jf gA~ )]
F O2 (f gIp;> + i ; f#gB~> + i ; f gA~+< )
S<(
f gA~+< jf gA~< )S>(f#gB~>+ jf#gB~> )S$( 1jf gA~+ )S$(f#gB~ j 1) ;
(4.4)
while in the second situation they take the form
F O1 (f#gB~>+ + i ; f gIp;< ; f gA~< )F O2 (f gIp;> + i ; f#gB~> + i ; f gA~+< )
(4.5)
i[1
S<(
S$( 1jf gIp + i )S$( 1jf#gB~ + i )S$( 1jf gA~+ )]
f gA~+< jf gA~< )S>(f#gB~>+ jf#gB~> ) :
Adding up the two kinds of residues we arrive to the kinematical pole property of the
bilocal form factor
#1= 1
Res Gxn;;tm(f#g(Innf1g)> ; #1j 1; f g(Imnf1g))< ) = i 1 S$( 1jf gImnf1g)S$(f#gInnf1gj 1)
Kx;t(f gIp ; f#gB~ jf gA~+ )
Gxn;;tm(f#g(Innf1g)> jf g(Imnf1g)< ) :
(4.6)
iom (2.13).
representation (3.18).
This is identical to the crossed version (2.32) of the form factor kinematical pole
ax
We remark that an analogous calculation can be performed also for the alternative
4.4
Other properties and discussion
We have checked that the crossing and periodicity properties (2.9) and (2.12) also hold
for the bilocal form factors. However, these relations are not relevant for the LM series,
therefore we refrain from presenting the proof. We also remark that the asymptotic
factorization property (2.14) does not hold in the bilocal case, which can be understood simply
from a physical point of view, or from the integral representations.
So far we have argued that the bilocal form factors satisfy all axioms that were
originally derived for the local ones. Here we discuss the implications of this statement.
First of all, it is an interesting idea to apply the local commutativity theorem to
the composite objects. This theorem by Smirnov states that if the form factors of two
operators satisfy the set of axioms presented in section 2, then they commute at spacelike
separations [43]. On the other hand, our results suggest an extension of this theorem to
bilocal objects. Such a theorem would state that for any three operators with the required
form factor properties we have
[O1(x1; t1); O2(x2; t2)O3(x3; t3)] = 0;
(4.7)
given that all three separations are spacelike. The trivial way of proving (4.7) is to apply
Smirnov's theorem twice to commute the two objects. However, a second proof can be given
by using the form factor properties of the bilocal operator, and then repeating Smirnov's
calculation for the composite object. We refrain from presenting a rigorous proof of this
extended theorem, as it is not relevant to the present work. Nevertheless, we stress that
the local commutativity theorem can not be applied backwards: it does not imply that
only the local operators can satisfy the form factor axioms.
Finally we comment on the possibility of a form factor bootstrap for the bilocal objects.
In the case of the local operators the form factor axioms (together with the asymptotic
factorization property) include enough constraints to x the form factors completely, both
in relativistic and nonrelativistic situations. The reason for this is the following: the form
factors can always be written as a product of a \minimal" part (which is twoparticle
irreducible) and a physical amplitude, which is a symmetric polynomial in the appropriate
variables. Then the kinematical pole axiom is used to
x this polynomial. In the bilocal
case the polynomial would depend also on the displacement between the two operators, and
the same number of constraints can not x it. Therefore, the standard bootstrap procedure
can not be applied to the bilocal form factors.
5
Compact representations of the LeClairMussardo series
Formulas (3.17){(3.18) give an integral representation for Gxn;;tm in terms of the elementary
form factors. The LM series is then given by (2.33) and (2.34), and this gives a wellde ned
way to evaluate the twopoint function.
The nal integral formulas (2.33) and (2.34) require the evaluation of the connected
and symmetric diagonal form factors. It follows from the proof of the kinematical pole
property, that these operations can be performed on the objects Gxn;;tn;p for each n and p
separately. On the other hand, computing the diagonal limit term by term in (3.17){(3.18)
leads to singularities, and it is only the sum over the partitions in (3.17){(3.18) that has
the desired residue structure
Gxn;;tn(#In;> j In;< )
P
;
One way towards singularity free expressions is to develop an integral formula which
automatically produces the diagonal limit of the form factors. This can be achieved by
where Cj are small contours around j . These contours can be opened to encircle the whole
real line. Opening the contours we do not encounter additional nfold poles, therefore we
can write
Gxn;;tc( ) =
1 Yn Z d#j Gxn;;tn(#In;> j In;< )
n! j=1 C 2 i Qn
k=1 sinh(#k
k)
;
where C is a narrow contour around the real line and the factor of 1=n! has been inserted
due to the possible permutations of the set f#g. It is important that C is narrow enough so
that it does not hit the integration contours R + i or R
i for the intermediate particles
in (3.17){(3.18).
It follows that the LeClairMussardo series can be written nally as
An alternative compact formula can be given by using the symmetric diagonal limit
in (2.34). The idea is to exchange the " ! 0 limit with the summations. This is justi ed
(at least in the thermal situation) due to the exponentially decaying factors e mR cosh( )
and e mx cosh( ) associated with the numbers n and p, and the limits on the growth of the
form factors following from the asymptotic factorization property (2.14). We then obtain
introducing auxiliary integration variables. It follows from (5.1) that
Gxn;;tc( ) = Yn Z
j=1 Cj 2 i Qn
k=1 sinh(#k
d#j Gxn;;tn(#In;> j In;< )
k)
;
h jO1(0)O2(x; t)j i =
= "li!m0 4
Substituting for example (3.17) gives
h jO1(0)O2(x; t)j i = lim
X
X
A+[A =In B+[B =In
0Yn Z d j f ( j )!( j )A Gxn;;tn( 1 + "; : : : ; n + "j n; : : : ; 1)5 :
1 3
"!0
n;p=0 n! p! j=1 R 2
" 1
X
1 1 Yn Z d j f ( j )!( j ) Yp Z
d i
#
F O1 (f gB>+ +i"+i ; f gIp;< ; f gA< )F O2 (f gIp;> +i ; f gB>
+i"+i ; f gA+< ) :
(5.6)
We stress that it is important to keep the i" shift in the kinematical factors Kx;t, because
they can combine with the poles the form factors to produce terms proportional to x and
t. Examples for this are given in section 6. On the other hand, the shifts could be omitted
from the phase factors, because these factors only depend on the rapidity di erences within
a given set. It is also important that all " variables have to be identical, because any other
choice with some "j ="k 6= 1 would lead to di erent nite terms.
(5.2)
(5.3)
(5.4)
(5.5)
jf gi
the representative state j i that interact with the twopoint function, whereas f g are intermediate
particles between the two operators. Each physical amplitude is associated with additional phase
factors, that depend on the partitions (ordering of the particles) and the contour for the integrals.
HJEP05(218)7
Representation (5.6) is perhaps the most transparent from a physical point of view:
the rapidities f g stand for particles that are present in the representative state j i and
interact with the twopoint function, whereas the f g are additional intermediate particles
between the two operators. A graphical interpretation is given in gure 2. We also point
out the striking similarity with the original proposal (2.27), however, there are crucial
di erences. In (2.27) the energy and momenta of the intermediate particles is dressed,
and it is not speci ed how to deal with the kinematical poles in the higher order terms.
In contrast, here the energy and momenta are not dressed, and all terms are regular due
to the wellde ned shifts i and ". The only \dressed" quantities in (5.6) are the weight
functions f ( ) and !( ), and the derivation in [28] shows that these are statistical weights
that are determined by the Bethe Ansatz description of the state j i.
To conclude this section we discuss the clustering property of the twopoint function.
In the limit of large separations it is expected that
jxj!1
lim h jO1(0)O2(x; t)j i = h jO1j ih jO2j i:
(5.7)
This identity is motivated by physical requirements about the pure state j i, but its explicit
con rmation is a highly nontrivial test of the various integral formulas [35, 36]. Here we
prove that the LM series satis es the clustering property; the simplest way is to use the
representation (5.6).
The kinematical factors Kx;t include multipliers of the form e ip( )x, which result in
terms of O(e mjxj) after integration in . Therefore, in the large distance limit only those
terms survive where all Kfactors are trivial. This happens when there are no intermediate
and A
= B+. In these cases
the incoming and outgoing j rapidities are always attached to the same operator, and
we obtain
jxj!1
lim h jO1(0)O2(x; t)j i = lim
"!0 n=0
n! j=1 R 2
"X1 1 Yn Z d j f ( j )!( j )
X
A+[A =In
S<(
f gA> jf gA+> )
F O1 (f + i"gA>
+ i ; f gA< )F O2 (f + i"gA+> + i ; f gA+< ) :
#
(5.8)
jf gi
(5.9)
HJEP05(218)7
hf gj
function.
lim
f A+ + i"g
f A + i"g
1 limit only those terms survive in the twopoint function, where f g = ; and the
partitionings of the index sets A and B are complementary, leading to a factorization of the twopoint
The phase factors cancel each other by de nition, and then the integrals completely
factorize according to the partitions. By permutation symmetry we obtain
lim h jO1(0)O2(x; t)j i =
" 1
X
X
1
n=0 k=1 k!(n k)! j=1 R 2
Yk Z d j f ( j )!( j ) F O1 (f +i"gIk;> +i ; f gIk;<)
j=1 R 2
n k
Y Z d~j f (~j )!(~j )F O2 (f~+i"gIn k;> +i ; f~gIn k;<) = h jO1j ih jO2j i:
#
A graphical interpretation of this identity is shown in gure 3.
6
Comparison to our earlier work
Here we compare the formula (5.4) to our previous results for the nite temperature case [35, 36]. These articles approached the evaluation of nite temperature twopoint
functions through nite volume regularization, leading to a double series
X
N;M
h jO1(0; 0)O2(x; t)j i =
DN;M :
(6.1)
Here j i is a representative state of the Gibbs ensemble, and the r.h.s. is the result of a
linked cluster expansion, where the DN;M are well de ned L !
1 limits of nite volume
regularized expressions. For the details of the calculation we refer the reader to the original
articles; here we only describe the main properties of the expansion (6.1) and cite a few
relevant formulas.
The series (6.1) is organized as a low temperature expansion, such that each DN;M
carries a thermal weight e NmR cosh( ) and M refers to the number of intermediate particles
in the nite volume regularization scheme. It is important that N does not correspond to
the index n in (2.33): the LeClairMussardo series involves the weights f ( ) = 1=(1 + e"( )),
which have a lowtemperature expansion themselves. Therefore the terms with a given n
in (2.33) include certain terms from DN;M with N > n. Similarly, although the index M
plays a similar role as the number p of intermediate integrals in (3.17){(3.18), the terms
can not be matched one to one. This is an e ect of the di erences in the regularization
schemes, and it can be seen explicitly in the example presented below.
We compare the LM series to (6.1) up to rst order in e mR cosh( ). Due to the in nite
number of possible intermediate particles between the two operators this is already a strong
independent con rmation of (2.33).
In the rst order approximation the (2.7) gives simply f ( ) = e mR cosh( ) and we
expect to match the zeroth and rst order contributions as
M
M
X D0M = G0x;;ct(;)
These two terms are evaluated in the following two subsections.
0Yp Z
d j
1
A Kx;t(f gj;)F O1 f g< F O2 f g> + i
;
(6.4)
which is nothing more that the spectral series for the zerotemperature twopoint function.
In this case there is no singularity for the variables, and the integration contour could
be pulled back to the real line. The same result was given for P
M D0M in [35].
We calculate the connected part of G1, which is de ned as the " independent part of the
bilocal form factor G1( + i"j ). Twoparticle form factors don't have kinematical poles,
therefore in this case the connected part coincides simply with the " ! 0 limit. From (3.17)
we have the integral representation
G1( + i"j ) =
X1 1
n
0Yp Z
duj
1
A
+ Kx;t(fug; + i"j;)F O1 fug<;
F O2 fug> + i ; + i + i" o
:
(6.5)
The rst and fourth terms are nite in the " ! 0 limit, but the second and third terms
have apparent rst order poles at " = 0. These singularities cancel each other, and care
needs to be taken to obtain the nite leftover pieces.
+i X
+i X
k
k
We introduce an expansion of the form factors to make the singularities apparent
F O1
+i ; fugI;< = FrOeg1
+i ; fugI;<
F O2 fugI;> +i ;
= FrOeg2 fugI;> +i ;
Qj<k S(uj uk) h1 Qj6=k S(uk uj)i F O1 fugInfkg;<
Qj>k S(uj uk) h1 Qj6=k S(
uj)i F O2 fugInfkg;> +i
;
:
With the help of the expansion we can express the " independent part of the integrands in
the second and third lines of (6.9). Following [35, 36] these terms will also be called the
\connected parts". They are given by
+i ; fugI;<;
F:P: FcO1
+i +i"; fugI;<;
FcO1
= FrOeg1
+i ; fugI;<;
+i X F O1 fugInfkg;<;
F~O2 fugI;> +i ; +i j
c
= FrOeg2 fugI;> +i ; +i ;
k
k
+i X F O2 fugInfkg;> +i ; +i
2
+41
Y S(uj
j2I
3 0
X '(uj
j2I
Qj<k S(uj uk) 1
h
Qj6=k S(uk uj )S(uk
)
i
;
F:P: Kx;t(fugI ; +i"j )F O2 fugI;> +i ; +i +i";
Qj>k S(uj uk) h1+Qj6=k S(
uj )
i
In the (6.8) the tilde denotes, that the connected form factor incorporates the contribution
form the energy and momentum phase factor as well.
With the regular pieces introduced above the connected bilocal form factor is
written as
G1( j ) = X1 1
c
0Ya Z
j=1 R+i 2
duj
1
A
n
+ Kx;t(fugj;)FcO1
+ i jfug<;
+ Kx;t(fug; j;)F O1 fug<;
F O2 fug> + i ; + i
o
:
Kx;t(fugI j;)
uk
1
(6.6)
(6.7)
(6.8)
(6.9)
du1 Kx;t (u1j ) F 1 ( +i ; u1) F 2 (u1 +i ; ) ;
)
Z 2d e mR cosh( )
( 1
duj
1
j=1 R+i 2 A Kx;t(fugj )F O1
+i ; fug< F O2 fug> +i ;
1
1
1
0M 1
Y Z
0M 1
Y Z
0M 2
Y Z
duj
duj
1
1
1
j=1 R+i 2 A Kx;t(fugj;)FcO1
+i jfug<; F O2 fug> +i
duj
j=1 R+i 2 A Kx;t(fugj;)F O1 fug<; F~O2 fug> +i ; +i ;
c
j=1 R+i 2 A Kx;t(fug; j;)F O1 fug<; F O2 fug> +i ; +i
R
R
Z
R
:
(6.10)
(6.11)
Comparing the form of D1M to G1c we see, that
as required. Note that D1;M includes terms that have M , M
1 and M
2 intermediate
integrals, whereas in (6.9) the number of integrals for the terms with a given p is always p.
Nevertheless, the summed quantities exactly correspond to each other.
7
In this work we derived a LeClairMussardo formula for the twopoint function at spacelike
separations. The general structure of the series is given by the two implicit
representations (2.33) and (2.34). These formulas involve the form factors of the bilocal operators,
for which the integral series are given by (3.17) or (3.18), depending on the sign of x. More
explicit representations are given by (5.4), and nally (5.6). We believe that the latter form
is the physically most transparent; its graphical interpretation is plotted on
gure 2. The
result can be considered as a large distance expansion, because each intermediate particle
(each integral) carries a weight of
e ms with s2 = x2
t2.
It is an important property of the nal formulas that each term is explicit, with a well
de ned prescription to deal with the kinematical singularities. This was missing in the
previous works: the original proposal of [22] had singularities in the higher order terms,
and the regularization of [35, 36] was only performed on a case by case basis.
Our results are expressed in terms of the bare form factors of the theory, and the
momenta and energy of the intermediate particles between the operators are also the bare
quantities. This follows from our approach to treat the bilocal operator as a composite
object, and to develop the LM series based on the bilocal form factors. The only dressed
quantities are the statistical lling fractions f ( ) and the weight function !( ). These two
are derived from the Bethe Ansatz description of the background j i, and they are the
same for all local and bilocal operators.
The original proposal of [22] used the bare form factors of the theory, but it suggested a
dressing for the intermediate particles. This might seem intuitive, but our derivation shows
that it is inconsistent. We expressed the two point function with the bare quantities, and
if there is a partial resummation of the series, then it should lead to a dressing for both
the form factors and the kinematical factors. Examples for such dressing can be found in
the context of nonrelativistic models [1], and a similar structure was also implied by the
results of [37], at least in the large time limit.
The methods and results of this paper also apply to nonrelativistic models, such as
the 1D Bose gas. In fact, most of these results were already derived for the Bose gas
in [12, 13, 38{40], and these papers served as a motivation for the present work. The main
addition of our work is the realization that the kinematical pole structure of the bilocal
operator is the same even in the relativistic case, and that this property is enough to
establish the LM series. We presented three arguments for the kinematical pole; the most
rigorous is the explicit check presented in 4.3.
The form factors of the bilocal operator were obtained by inserting a complete set of
states between the two operators. This is the same technique that was used by Smirnov [43],
which lead to the proof of the local commutativity theorem. There is one essential step
in both calculations: a certain shift of integration contours, which is possible only for
spacelike separations.
As tests of our results for the LM series we evaluated the rst order corrections in the
lowtemperature expansion in section 6 and compared them to existing results; complete
agreement was found for an arbitrary number of intermediate particles. As an additional
test we performed the large distance limit, and con rmed that the integral series factorizes
into the product of two LeClairMussardo series, thus ful lling the clustering property.
This is a highly nontrivial test of the nal formula (5.6).
The most important open problem is to nd methods for the actual implementation
of the
nal formulas. As shown by the examples in section 6, this can be a cumbersome
task, both numerically and analytically.
It is also important to consider the case of timelike separations. Our rst two
arguments for the validity kinematical pole property (those based on the OPE and the Bethe
wave function) do not apply in this situation, but preliminary calculations show that the
integral series for the bilocal form factor can be modi ed such that the kinematical pole
property can be proven nevertheless. However, the convergence properties of the resulting
integral series can be drastically di erent, and we don't have a decisive answer whether a
LM series can be established for this case too.
Coming back to spacelike separations, it would be interesting to consider the rst
corrections in a large distance expansion. Generally it is expected that
h jO1(0)O2(x; 0)j i = h jO1j ih jO2j i + O(e x= );
(7.1)
where is a correlation length. In the thermal situation Euclidean invariance implies, that
the above object is equal to a twopoint function with timelike separation, evaluated in
the ground state of a nite volume QFT with volume R = 1=T . In this case the correlation
length is 1=
= E1
E0, where E0 and E1 are the exact ground state and
rst excited
state energies in the given volume. It is an interesting question whether these quantities
(or possibly even the exact
nite volume transition matrix elements that determine the
prefactors of the correction terms in (7.1)) could be extracted from the LM series.
Finally, it would be interesting to establish a connection to the results of [37].
Equation (3.38) in that work is reminiscent of our formulas, even though it was derived for
timelike separations in the large time limit. The concrete relations between the two
integral series have yet to be investigated.
We hope to return to these questions in future research.
Note added.
After this work was
nished we became aware of the work [61] which
derives the socalled rst Luscher's corrections to nite volume form factors in iQFT. The
methods and results there are related to our work. In [61] the new results were shown to
be consistent with the previous works [35, 36], thus they are consistent with our present
results too.
Acknowledgments
for useful discussions.
We would like to thank Gabor Takacs, Benjamin Doyon, Zoltan Bajnok, and Lorenzo Piroli
B.P. acknowledges support from the \Premium" Postdoctoral Program of the
Hungarian Academy of Sciences, the K2016 grant no. 119204 and the KH17 grant no. 125567 of
the research agency NKFIH. This work was partially supported also within the Quantum
Technology National Excellence Program (Project No. 20171.2.1NKP201700001).
I.M.SZ. work was partially supported by the EPSRC Standard Grant \Entanglement
Measures, Twist Fields, and Partition Functions in Quantum Field Theory" under reference
number EP/P006108/1, and has received funding from the People Programme (Marie Curie
Actions) of the European Union's Seventh Framework Programme FP7/2007 2013/ under
REA Grant Agreement No 317089 (GATIS).
A
Contour transformation
Here we perform the contour transformation in the variables in the original formula (3.11).
The goal is to shift the contour to R + i , thereby hitting the kinematical singularities
whenever the integrals cross one of the # variables.
Let us focus on a term in Gxn;;tm with given A
and B
subsets, and the residue, when
r
B+j number of s coincide with same number of #s, denoted by the set B~. Due to
the exchange axiom (2.11) and relabeling the integration variables, we only calculate the
residue when #~bi =
factor p!=(p
residue to evaluate is
i and take into account the other permutation by the combinatorial
r)!. Furthermore, we use the notation B^ = B+
n B~ and ~i =
i+r. The
1
X
p=r (p
1
Z
p r
Y d~i
r)! R+i i=1
2
X
Yr Z
d i
B~[B^=B+ i=1 Ci 2
Kx;t(f gIr ; f~gIp r ; f#gB jf gA+ )
F O1 (f#gB>+ + i ; f gIr;< ; f~gIp r;< ; f gA< )
+ i ; f~gIp r;> + i ; f gIr;> + i ; f gA+< )
S<(
f gA+< jf gA< )S>(f#gB> jf#gB>+ )S$(f#gB> jf gA< ) ;
(A.1)
where Ci is contour encircling #~bi with positive orientation. We omitted the
i0 shifts in
the formula, since there are no more poles on the integration contours, the form factor is
o diagonal (#i 6= j ) and the shifts are not necessary anymore. We note, that in the case
A j = 0, p = jB+j and p = r, the residue is missing, since the twoparticle form factor is
regular. With (2.11) and (2.13) the residue evaluates to
S$(f#g~bi jf#gB^< + i )S$(f#g~bi jf~gIp r;< )S$(f#g~bi jf gA< )]
1
X
p=r (p
1
Z
r)! R+i i=1
p r
Y d~i
X
B~[B^=B+
F O1 (f#gB^> + i ; f~gIp r;< ; f gA< )
S>(f#gB^> jf#gB~> ) Y[1
2
r
i=1
F O2 (f#gB>
+ i ; f~gIp r;> + i ; f#gB~> + i ; f gA+< )
S<(
f gA+< jf gA< )S>(f#gB> jf#gB>+ )S$(f#gB> jf gA< ) :
Kx;t(f#gB~ ; f~gIp r ; f#gB jf gA+ )
The next step is to expand the product in the formula. We separate the set B~ into two
disjoint subsets B~I [ B~S = B~. For indices in B~I we pick the 1term in the product, for the
indices in B~S we pick the product of the Smatrices. Using the exchange axiom (2.11) we
simplify the expression to
F O1 (f#gB^> + i ; f gIp;< ; f gA< )Kx;t(f#gB~I ; f#g(B [B~S); f gIp jf gA+ )
F O2 (f#g(B [B~S)> + i ; f gIp;> + i ; f#gB~I;>
S$(f#g(B [B~S)> jf gA< )S>(f#g(B [B~S)> jf#gB^> )S>(f#gB~S;> jf#gB~I;>
S>(f#gB> jf#gB~I;> )S<(
f gA+< jf gA< )S>(f#gB^> jf#gB~I;> ) ;
(A.2)
(A.3)
~
where for the set B
[ BS we imply the rearrangement of the elements to increasing order,
and we relabeled the integration variables.
It is important to note that apart from the sign of the expression, it only depends on
the sets B^, B~I and B
[ B~S and p. Since we have to sum for all disjoint B+ and B
sums up to B, there is a chance for cancellation of terms. For xed B^, B~I and B
and p the sum of the sign prefactors is
jB [B~Sj
X
jB~Sj=0
jB~Sj
B
[ B~Sj ( 1)jB~Sj =
jB [B~Sj;0 :
that
(A.4)
After this cancellation we relabel the sets as B^ ! B+ and B~I ! B
to arrive to the
result (3.17).
We should also investigate the situation when jA j = 0, p = jB+j and p = r. The
general formula for the residues (A.3) is not valid in this case, since the twoparticle form
factor is regular, and we don't get corrections from residues to the jBj = 0, jA j = 0
and p = 0 term. However, these terms already are in the shape of (3.17), hence the nal
^
formula is valid.
Open Access.
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