Field theoretical derivation of Lüscher’s formula and calculation of finite volume form factors
We initiate a systematic method to calculate both the finite volume energy levels and form factors from the momentum space finite volume two-point function. By expanding the two point function in the volume we extracted the leading exponential volume correction both to the energy of a moving particle state and to the simplest non-diagonal form factor. The form factor corrections are given in terms of a regularized infinite volume 3-particle form factor and terms related to the Lüsher correction of the momentum quantization. We tested these results against second order Lagrangian and Hamiltonian perturbation theory in the sinh-Gordon theory and we obtained perfect agreement.
Quantum Field Theories play an important role in many branches of physics. On the one hand, they provide the language in which we formulate the fundamental interactions of Nature including the electro-weak and strong interactions. On the other hand, they are frequently used in effective models appearing in particle, solid state or statistical physics. In most of these applications the physical system has a finite size: scattering experiments are performed in a finite accelerator/detector, solid state systems are analyzed in laboratories, even the lattice simulations of gauge theories are performed on finite lattices etc. The understanding of finite size effects are therefore unavoidable and the ultimate goal is to solve QFTs for any finite volume. Fortunately, finite size corrections can be formulated purely in terms of the infinite volume characteristics of the theory, such as the masses and scattering matrices of the constituent particles and the form factors of local operators Luscher (); Luscher:1986pf (); Pozsgay:2007kn (); Pozsgay:2007gx (). For a system in a box of finite sizes the leading volume corrections are polynomial in the inverse of these sizes and are related to the quantization of the momenta of the particles Luscher:1986pf (). In massive theories the subleading corrections are exponentially suppressed and are due to virtual processes in which virtual particles “travel around the world” Luscher ().
The typical observables of an infinite volume QFT (with massive excitations) are the mass spectrum, the scattering matrix, the matrix elements of local operators, i.e. the form factors, and the correlation functions of these operators. The mass spectrum and the scattering matrix is the simplest information, which characterize the QFT on the mass-shell. The form factors are half on-shell half off-shell data, while the correlation functions are completely off-shell information. These can be seen from the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, which connects the scattering matrix and form factors to correlation functions: The scattering matrix is the amputated momentum space correlation function on the mass-shell, while for form factors only the momenta, which correspond to the asymptotic states are put on shell. Clearly, correlation functions are the most general objects as form factors and scattering matrices can be obtained from them by restriction. Alternatively, however, the knowledge of the spectrum and form factors provides a systematic expansion of the correlation functions as well.
The field of two dimensional integrable models is an adequate testing ground for finite size effects. These theories are not only relevant as toy models, but, in many cases, describe highly anisotropic solid state systems and via the AdS/CFT correspondence, solve four dimensional gauge theories Mussardo:2010mgq (); Samaj:2013yva (); Beisert:2010jr (). Additionally, they can be solved exactly and the structure of the solution provides valuable insight for higher dimensional theories. For simplicity we restrict our attention in this paper to a theory with a single massive particle, which does not form any boundstate.
The finite size energy spectrum has been systematically calculated in integrable theories. The leading finite size correction is polynomial in the inverse of the volume and originates from momentum quantization Luscher:1986pf (). The finite volume wave-function of a particle has to be periodic, thus when moving the particle around the volume, , it has to pick up the translational phase. If the theory were free this phase should be , in an interacting theory, however, the particle scatters on all the other particles suffering phase shifts, , which adds to the translational phase and corrects the free quantization condition. These equations are called the Bethe-Yang (BY) equations. The energy of a multiparticle state is simply the sum of infinite volume energies but with the quantized momenta depending on the infinite volume scattering matrix. The exponentially small corrections are related to virtual processes. In the leading process a virtual particle anti-particle pair appears from the vacuum, one of them travels around the world, scatters on the physical particles and annihilates with its pair. Similar process modifies the large volume momentum quantization of the particles BaJa (). The total energy contains not only the particles’ energies, but also the contribution of the sea of virtual particles. The next exponential correction contains two virtual particle pairs and a single pair which wraps twice around the cylinder Bombardelli:2013yka (). For an exact description all of these virtual processes have to be summed up, which is provided by the Thermodynamic Bethe Ansatz (TBA) equations Zamolodchikov:1989cf (). TBA equations can be derived (only for the ground state) by evaluating the Euclidean torus partition function in the limit, when one of the sizes goes to infinity. If this size is interpreted as Euclidean time, then only the lowest energy state, namely the finite volume ground state contributes. If, however, it is interpreted as a very large volume, then the partition function is dominated by the contribution of finite density states. Since the volume eventually goes to infinity the BY equations are almost exact and can be used to derive (nonlinear) TBA integral equations to determine the density of the particles, which minimize the partition function in the saddle point approximation. By careful analytical continuations this exact TBA integral equation can be extended for excited states Dorey:1996re ().
The similar program to determine the finite volume matrix elements of local operators, i.e. form factors, is still in its infancy. Since there is a sharp difference between diagonal and non-diagonal form factors they have to be analyzed separately. For nondiagonal form factors the polynomial finite size corrections, besides the already explained momentum quantization, involve also the renormalization of states, to conform with the finite volume Kronecker delta normalization Pozsgay:2007kn (). The polynomial corrections for diagonal form factors are much more complicated, as they contain disconnected terms. They were conjectured in Saleur:1999hq (); Pozsgay:2007gx () and confirmed in Bajnok:2017bfg (). For exponential corrections the situation is the opposite. Exact expressions for the finite volume one-point function can be obtained in terms of the TBA minimizing particle density and the infinite volume form factors by evaluating the one-point function on an Euclidean torus where one of the sizes is sent to infinity Leclair:1999ys (). The analytical continuation trick used for the spectrum can be generalized and leads to exact expressions for all finite volume diagonal form factors Pozsgay:2013jua (). For non-diagonal form factors, however, not even the leading exponential correction is known for theories without boundstates. In case of boundstates the leading exponential volume correction is in fact the so called term, which originates from a process in which the particle can virtually decay in a finite volume into its constituents. This idea was used to calculate the leading term explicitly for the simplest non-diagonal form factor in Pozsgay:2008bf (). As we would like to calculate the leading volume correction coming from virtual particles travelling around the world, i.e. the F-term, we focus on theories without boundstates. The aim of this paper is to initiate research into the calculation of these corrections.
We develop a framework which provides direct access both to excited states’ energy levels and finite volume form factors. The idea is to calculate the Euclidean torus two-point function in the limit, when one of the sizes is sent to infinity. The exact finite volume two-point function then can be used, similarly to the LSZ formula, to extract the information needed: the momentum space two-point function, when continued analytically to imaginary values, has poles exactly at the finite volume energy levels whose residues are the products of finite volume form factors. Of course, the exact determination of the finite-volume two-point function is hopeless in interacting theories, but developing any systematic expansion leads to a systematic expansion of both the energy levels and the form factors. We analyze two such expansions in this paper: in the first, we expand the two-point function in the volume, which leads to the leading exponential corrections. We perform the calculation for a moving one-particle state. In the second expansion, we calculate the same quantities perturbatively in the coupling in the sinh-Gordon theory. By comparing the two approaches in the overlapping domain we find complete agreement.
The paper is organized as follows: In the next section we give an overview of the method and present our main result for the leading exponential volume correction of the simplest nondiagonal form factor. In section 3 we present the details of the calculation in the mirror channel and derive the correction explicitly. In section 4 we specify the results for the sinh-Gordon theory in preparation for a perturbative check. We use Hamiltonian perturbation theory in section 5 to derive the leading finite size correction in the coupling both to the one-particle energy and form factor. Technical details are relegated to Appendix B. We then expand these results in the volume and confirm the previously derived leading exponential finite size corrections. Finally, we finish the main body of the paper with conclusions in section 6. We have several Appendices. Appendix A contains the perturbative expansion of the exact TBA equations. In Appendix C we make a perturbative expansion of the finite volume two point function in the sinh-Gordon theory and extract the leading correction to the finite volume energy and form factors confirming the results of section 4. Appendix D shows the equivalence of the finite volume regularizations of Pozsgay:2010cr () with our infinite volume regularizations.
2 Overview of the method and summary of the results
In the following we analyze a relativistic integrable QFT in two dimensions with a single particle of mass and scattering matrix , which satisfies unitarity and crossing symmetry and does not have any pole in the physical strip. Such theory is the sinh-Gordon theory and the generalization for more species with diagonal scatterings is straightforward. We put this QFT in a finite volume of size and focus on the finite size energy spectrum and the finite size form factors.
2.1 Finite size energy spectrum
We analyze the energy of an particle state with rapidities , . As explained in the introduction the polynomial corrections come from the quantization of momenta formulated by the Bethe-Yang equations
where, by the superscript , we indicated that only the polynomial volume corrections are kept. Given integers the rapidities can be determined leading to the energy formula
The leading exponential correction was conjectured in BaJa () and has two sources. First one has to take into account how the sea of virtual particles changes the quantization condition
where denotes . We then have to add the direct energy contribution of the virtual particles. By expressing all contributions in terms of the leading rapidities, , we have:
where is the inverse of the matrix with entries .
The exact equations come either from an analytical continuation of the groundstate TBA result Dorey:1996re (); Bajnok:2010ke () or from a continuum limit of a solved integrable lattice regularization Teschner:2007ng (). The quantization condition for the exact rapidities is
where satisfies the coupled non-linear integral equation
and the energy is
In particular, for a moving one-particle state at leading order we obtain
and the corresponding energy is
The leading exponential correction of the quantization condition contains an extra term of the form
We will reproduce this result from the study of the finite volume two-point function.
2.2 Finite size form factors
Form factors are defined as the matrix elements of local operators sandwiched between finite volume energy eigenstates. These states are normalized to Kronecker- functions
opposed to infinite volume states, which are normalized to Dirac- functions: . The finite volume states can be equivalently labeled by the rapidities . The space-time dependence of the form factors can be easily calculated
where and with , while we simply abbreviated by .
The polynomial finite size corrections purely change the normalization of states and give Pozsgay:2007kn ():
where denotes the infinite volume form factor
and all the rapidities can be taken at the leading order values with superscript . Since even the leading exponential correction is not known for these form factors we develop a systematic method based on the two-point function to calculate them.
In particular, for the one-particle form factor the formulae simplify as
and the aim of our paper is to calculate the leading exponential corrections to these formulae.
2.3 Finite volume two-point function
Let us focus on the Euclidean finite volume two-point function, which is defined by the path integral555We restrict our attention to the case when the two operators are the same. The generalization for different operators is straightforward.
where configurations are periodic in with and . The momentum space form is obtained by its Fourier transform
where periodicity in requires that . Taking as Euclidean time the two point function is the vacuum expection value of the time ordered product:
We can insert a complete system of finite volume energy-momentum eigenstates and use the Euclidean version of the space-time dependence (2.13). By performing the integrals we obtain
For a fixed we can determine the energy levels by searching for poles in the analytically continued . For a generic volume and fixed momentum the energy levels are never degenerate. Thus the poles are located at with residue
which is nothing but the square of the finite volume form factor.
In order to obtain the exponential corrections of these form factors we have to expand the two point function on the space-time cylinder in . The Euclidean version of this cylinder can be thought of as the large size limit of the torus. On the torus we can exchange the role of the Euclidean time and space and represent the two point function as
Inserting two complete system of (mirror) states denoted by and and exploiting the symmetry together with we obtain:
where . Note that the expansion in naturally corresponds to expansions in Lüscher orders. In the bulk of the paper we perform a systematic expansion related to a moving one-particle state. Let us summarize the result we got.
For a one-particle state we focus on the one-particle finite volume pole
where is the exact finite volume energy with momentum and is the corresponding exact finite volume form factor. We choose the phase of the one-particle state so that is real and positive. We used the momentum variable to label the state, which is related to the rapidity as , such that the corresponding energy is . We can expand around the large volume Bethe-Yang pole at . At the leading Lüscher order we have first and second order poles
such that the leading exponential corrections of the energy and form factor can be written as
We calculate in the mirror channel (2.24). The leading order result comes from terms, when is the vacuum state and is a one-particle state. The leading Lüscher corrections, and , come from terms when is a one-particle state and is either the vacuum or a two-particle state.
Having performed the calculations we could reproduce the Lüscher correction of the 1-particle energy (2.1). For the form factors we obtained the result
where the density of states at the leading exponential order is
and the regularized form factor is defined to be
In the rest of the paper we derive this result and check it by a second order perturbative calculation in the sinh-Gordon theory.
3 Mirror representation
We perform our calculation starting from the mirror representation (2.24). The denominator has the Hilbert space representation
and we see that its Lüscher expansion,
is divergent. The divergent constant comes from the continuum normalization. As we will see, this divergence cancels with a similar term from the numerator. However, we need some regularization to make intermediate steps well-defined. In the main text we will use continuum regularization, but as shown in Appendix D, this is completely equivalent to finite volume regularization.
The leading (0th order) term in the Lüscher expansion of is
It is easy to see that this is regular in (around the 1-particle pole) unless is a 1-particle state. Indeed, the -particle contribution can be written as
After changing the integration variables to the relative rapidities , and the global rapidity this becomes:
where the matrix element (-particle form factor ) only depends on the relative rapidities666Remember we are considering a scalar operator . and is the -particle invariant mass. Performing the integral with the help of the delta function we get
Since there is a singularity at only for .
Similarly, the first Lüscher correction in (2.24) (i.e. terms where is a one-particle state) is regular unless is the vacuum or a 2-particle state. As we will see, the -particle contribution can be evaluated (after regularization) by shifting the integration contour for the rapidity integration away from the real axis. Performing the integration first, we notice that the matrix element (form factor)
has a pole singularity in the variable at and so the total contribution consists of a residue term plus a shifted integral. As we will see, the shifted integral is not singular in at the 1-particle pole, while the residue term is proportional to
where is the global rapidity of the remaining -particle system, , is the invariant mass of this remaining system and , . After performing the integral, the denominator
leads to pole singularities only for .
The (potentially) singular part of the first Lüscher correction to the 2-point function is of the form
Here we introduced the notations
The first term in (3.11) comes from the combination of the 1-particle contribution to the th order correlation function with the first order term, proportional to , coming from the denominator in (2.24) (see (3.6) and (3.2)). The second term comes from (2.24) when is the vacuum state and finally the third term is the 2-particle contribution when . Note that we restricted the integration to the range . This is possible since for the contribution is subleading to the second Lüscher order, which is O and which we neglect. This restriction is also necessary for some of our later estimates to be valid.
The matrix element can be represented in terms of the S-matrix and the 3-particle form factor as Smirnov:1992vz ()
The integral of its square is divergent and needs to be regularized.
We will use the regularized delta function
in (3.14) and take the limit only at the end of the calculation.
The regularized delta function terms can be nicely combined with those coming from the pole terms in the 3-particle form factor and the regularized matrix element becomes
Here is the finite part of the form factor, defined by
The finite part is obtained by explicitly removing the pole singularities required by the form factor axioms Smirnov:1992vz (). is finite at , . For later use, we now also define the modified form factor :
, defined by (2.30), can be written as
Next we introduce the variables , by
and integrate (3.12) over using the delta function. This means that after this integration stands for the solution of
Next we make use of the analyticity of the form factors and shift the integral from real to , where is small. We have to pay attention to the following.
The right hand side of (3.22) must not cross the cut of the arcsinh function (which runs from to along the imaginary axis).
Avoid points where .
Take into account the poles of the regularized matrix elements at .
Problems A) and B) can be easily avoided if and the parameter is small enough. The form factor poles can be taken into account explicitly, using the residue theorem. (Only two of the poles lie above the real axis.) After a long computation, we find (up to terms vanishing in the limit):
Here the notation
is used and is the shifted integral ():
The (negative) divergent term coming from the denominator is accompanied with a (positive) divergent term coming from the calculation of the numerator. They both multiply the same function. Our main assumption is that the divergences cancel777Note that putting blindly to the definition of the regularized delta function gives . and the remaining finite terms are correct. Indeed, in appendix D we show that our heuristic regularization is completely equivalent to the well-defined finite volume regularization. We will make the substitution
where is a finite renormalization constant, which will be fixed later.
3.2 Analytic continuation
(3.25) is our final result for the Fourier space 2-point function for real . We need to analytically continue this function towards . We will do it in two steps. First we extend it to a small region where is just above the real axis. The explicit terms are analytic, so we have to concentrate on the integral . In this region there is no problem with A) and B), but as we increase the imaginary part of , the integration contour will cross the double pole at coming from the form factor function squared. We can take into account the effects of this pole explicitly, using the residue theorem. After a second long calculation, we find that adding these new contributions to (3.25) many terms cancel and we have
where is the same integral as (3.28), but with the integration contour moved back to the real axis. (We are allowed to do this after is already above the contour.)
In the second step we continue further towards . We can show (for ) that is analytic in in the vicinity of the imaginary axis, except for a cut starting at . The cut appears as the consequence of the definition and the limit in the language of the variable becomes
where . The sign is according to whether we go around the branch point from the right or from the left. Since no pole terms are coming from the integral, we are left with the explicitly evaluated terms in (3.30) and the singular terms of the Lüscher correction can be written
Finally we calculate the residues of the simple and double poles of the Lüscher term:
3.3 Lüscher’s formula
The S-matrix is real analytic and satisfies crossing:
from which we conclude that for real is real and satisfies
Thus is real and independent of the sign.
3.4 Finite volume form factor
is real and independent of the sign, since using the form factor axioms we can show that
If we require that at infinite energy the interaction can be neglected and the form factor is given by its free field value,
then this fixes the integration constant to .
Finally if we notice that in the first Lüscher approximation (2.29) can be written