Boundary TBA, trees and loops

We derive a graph expansion for the thermal partition function of solvable two-dimensional models with boundaries. This expansion of the integration measure over the virtual particles winding around the time cycle is obtained with the help of the matrix-tree theorem. The free energy is a sum over all connected graphs, which can be either trees or trees with one loop. The generating function for the connected trees satisfies a non-linear integral equation, which is equivalent to the TBA equation. The sum over connected graphs gives the bulk free energy as well as the exact g-functions for the two boundaries. We reproduced the integral formula conjectured by Dorey, Fioravanti, Rim and Tateo, and proved subsequently by Pozsgay. The method is easily extended to the case of non-diagonal bulk scattering and diagonal reflection matrices. As an example we consider a system with two types of particles solved by nested Bethe Ansatz.


Introduction
The notion of integrability has been extended for systems with boundary by Ghoshal and Zamolodchikov [1]. With the Yang-Baxter equation, unitarity, analyticity and crossing symmetry for both bulk scattering matrix and boundary reflection matrix, a model with integrable boundary is expected to be exactly solved. One of the simplest observable in such a model is its free energy in large volume and finite temperature. Unlike in periodic systems, this free energy contains a volume-independent correction, also known as g-function [2].
The first attempt to compute g-function was carried out by LeClair, Mussardo, Saleur and Skorik [3], using the Thermodynamics Bethe Ansatz (TBA) saddle point approximation. They obtained an expression similar to the bulk TBA free energy where the integration is done with respect to the rapidity variable u. In this equation is the pseudoenergy of the periodic system and Θ contains bulk scattering and boundary reflection matrices. Later it was shown by Woynarovich that another volume-independent contribution is produced by the fluctuation around the saddle point [4]. The result was written as a Fredholm determinant log(g a g b ) fluc = − log det(1 −K + ), (1.2) where the kernelK + involves the pseudo-energy and bulk scattering matrix but not the reflection matrices. In other words, the fluctuation around the saddle point is boundary-independent. A major problem of Woynarovich's approach is that it also predicts a similar fluctuation term for periodic systems, while it is known that there is no such term. Dorey, Fioravanti, Rim and Tateo [5] took a different approach towards the g-function. They started with the definition of the partition function as a thermal sum over a complete set of states labelled by mode numbers. In the infinite volume limit, the sum is replaced by integrals over rapidities. The integrands have been explicitly worked out for small number of particles. Based on these first terms and the structure of TBA saddle point result (1.1), the authors advanced a conjecture about the boundary-independent part of g-function. Their proposal has the structure of a Leclair-Mussardo type series log(g a g b ) = log(g a g b ) saddle where K is the logarithmic derivative of the bulk scattering matrix. Pozsgay [6] argued that the same expression for g-function could be obtained from a refined version of TBA saddle point approximation. He noticed that the mismatch between (1.2) and the series in (1.3) is resolved if one uses a non-flat measure for the functional integration. This non-trivial measure comes from the Jacobian of the change from mode number to rapidities, and represents the continuum limit of the Gaudin determinant.
The computation of the fluctuations around saddle point involves only the diagonal elements of the Gaudin matrix, resulting in the inverse power of the Fredholm determinant det(1 −K + ), with the integration measure the same as in cluster expansion. On the other hand, the functional integration measure contains the off-diagonal elements as well, which constitute another Fredholm determinant det(1 −K − ). Pozsgay rewrote the result (1.3) in terms of these two Fredholm determinants log(g a g b ) = log(g a g b ) saddle + log det The two kernelsK ± can be read off from the asymptotic Bethe equations. For a periodic system, they happen to be the same and the effects from the fluctuation and the measure cancel each other. It is important to distinguish the Jacobians in [5] from the one in [6]. The former appear in each term of the cluster expansion while the latter is obtained from the thermodynamics state that minimizes the TBA functional. Put it simply, the Jacobian in [6] is the thermal average of all the Jacobians in [5].
In this paper, we will derive this known result for the g-function, following the strategy of [5]: writing the partition function as a sum over mode numbers and replacing it by an integral over rapidities in the infinite volume limit. In contrast to [5], we are able to exactly carry out the cluster expansion, by virtue of the matrix-tree theorem [7]. This theorem allows us to write the Jacobian for a finite number of particles as a sum over graphs. Consequently, the g-function is expressed as a sum over graphs with no loops (trees) and graphs with one loop. These combinatorial objects possess simple structure and their sum can be written in the form (1.3) or (1.4). Compared to [6], the gaussian fluctuations and the measure can be respectively interpreted as the sum over two types of loops. Our final result coincides with the one of [6], but our method allows to treat exactly each term in the canonical partition sum, before the thermodynamical limit. This makes our method potentially useful in computing more subtle objects as correlation functions.
The generalization to a theory with non-diagonal bulk scattering (for instance with su(n + 1) symmetry) and diagonal reflection matrices is straightforward. The graphs involve n types of vertices and the Fredholm kernels are n × n matrices, where n is the rank of the symmetry algebra. The same method of graph expansion has been applied for the free energy of a periodic system [8], [9], [10], as well as for one point function [11].
The paper is structured as follows. In section 2 we recall the definition of g-function for a massive theory with diagonal scattering and integrable boundary and spell out the Fredholm determinant formula (1.4). In section 3 we develop the combinatorics needed to sum up the cluster expansion and express the partition function on a cylinder as a sum over (multi)wrapping virtual particles. In section 4 we expand, with the help of the matrix-tree theorem, the canonical partition function on a cylinder as a sum over certain set of Feynman graphs. In section 5 we perform the sum and recover the expression for the g-function. In section 6 we generalise our method to the non-diagonal scattering where we obtain the nested TBA equations and the nested g-function. Section 7 contains the conclusions, and the two appendices present two different proofs of the matrix-tree theorem in the form used in this paper.

Bulk and boundary free energy of a massive integrable field theory
The g-function, also known as boundary entropy or ground-state degeneracy, was first introduced by Affleck and Ludwig [2] and since then has been given many physical interpretations. In this paper we shall look at this multifaceted object as the non-extensive contribution to free energy of a system with boundaries.
Let us consider a 1+1 dimensional field theory with a single massive excitation above the vacuum, defined in an open interval of length L, whose boundaries will be denoted by a and b . The momentum and energy of a particle are parameterized by its rapidity The theory is integrable with a two-particle bulk scattering matrix S(u, v) and reflection matrces R a (u), R b (u) at the boundaries a and b. These two matrices satisfy a set of conditions [1], among which the unitarity condition The bulk scattering matrix does not necessarily depend on the difference between rapidities. We assume a milder condition as well as S(u, u) = −1.
The partition function at inverse temperature R is given by the thermal trace where H ab (L) is the Hamiltonian for the theory defined on a segment of length L with boundary conditions a and b. To extract the boundary free energy, we divide by the thermal partition function for a theory defined on a circle of length L, where H(L) is the Hamiltonian for periodic boundary conditions. The boundary free energy is thus given by the difference The g-function is defined as the contribution of a single boundary to the free energy. To compute it, we pull the two boundaries far away from each other to avoid interference, Compared to the usual definition of g-function given in perturbed CFTs literature, our definition seems to be over-simplifying. This is due to our specific choice of normalization of partition functions. More precisely, we have fixed the ground state energy of both Hamiltonians H(L) and H ab (L) to zero by discarding the bulk energy density as well as its non-extensive boundary contributions. In a relativistic theory there is a mirror transformation exchanging the roles of space and timẽ where u γ means analytical continuation in the rapidity variable which assures that the mirror particle has positive energyẼ and real momentump. The inverse is true only if the mirror theory coincides with the original one. In this case the natural parametrisation is p = m sinh u, E = m cosh u and u γ = u + iπ/2. The product of two mirror transformations, u → −u + iπ, gives a crossing transformation.  Although the partition function is the same, the physics is rather different in the two channels. In the mirror theory, the g-function provides information about overlapping of the boundary states and the ground state at finite volume. To see this, we write (2.9) as a sum over eigenstates |ψ of the periodic HamiltonianH(R) In the large L limit, this sum is dominated by a single term corresponding to the ground state |ψ 0 . The g-function is then given by the overlap between this state and the boundary state An expression for g-function was conjectured in [5] and proven in [6]. Here we write down this result for the case where the bulk scattering matrix is not of difference form. Let us denote respectively by K, K a and K b the logarithmic derivatives respectively of the bulk scattering matrix and the boundary reflection matrices associated with the boundaries a and b It follows from (2.1) and (2.2) that (2.11) Let us denote Then the expression for the g-function (2.6) found in [6] reads where (u) is the pseudo-energy of the same theory on a torus. It can be obtained from the TBA equation at inverse temperature R (2.14) The kernelsK ± have support on the positive real axis and their action is given bŷ In the next sections we will derive the expression (2.13) of the g-function by evaluating the partition function in the R-channel, namely equation (2.3), in the limit when L is asymptotically large. In order to do that, we will insert a decomposition of the identity in a complete basis of eigenstates of the Hamiltonian H ab (L) and perform the thermal trace.
3 Partition function on a cylinder as a sum over wrapping particles

Asymptotic Bethe equations in presence of boundaries
The g-function (2.6) is extracted by taking the limit of large volume L. In this limit, we can diagonalize the Hamiltonian H ab (L) using the technique of Bethe ansatz. Consider an N -particle eigenstate |u = |u 1 , u 2 , ..., u N . To obtain the Bethe Ansatz equations in presence of two boundaries, we follow a particle of rapidity u j as it propagates to a boundary and is reflected to the opposite direction. It continues to the other boundary, being reflected for a second time and finally comes back to its initial position, finishing a trajectory of length 2L. During its propagation, it scatters with the rest of the particles twice, once from the left and once from the right. This process translates into the quantisation condition of the state |u We can write these equations in logarithmic form by introducing a new set of variables: the total scattering phases φ 1 , φ 2 , ..., φ n defined by In terms of these variables, the quantization of state |u reads φ j (u) = 2πn j ∀j = 1, ..., N with n j ∈ Z.
The next step is to impose particle statistics and indistinguishability. In periodic systems, simply taking the mode numbers n j to be all different automatically satisfies both principles. However in the presence of boundary two particles having the same mode numbers but of opposite signs are indistinguishable. To avoid overcounting of states, we should put a positivity constraint on mode numbers. A basis in the N -particle sector of the Hilbert space is then labeled by all sets of strictly increasing positive integers 0 < n 1 < · · · < n N . The corresponding eigenvector of the Hamiltonian H ab (L) is characterised by a set of rapidities 0 < u 1 < · · · < u N , obtained by solving the Bethe equations (3.2) and (3.3).
Inserting a complete set of eigenstates we write the partition function on a cylinder as In this equation, the energy E is a function of mode numbers n 1 , ..., n N . To find its explicit form, one needs to solve the Bethe equations for the corresponding rapidities u 1 , ..., u N . As a function of the rapidities, the energy is equal to the sum of the energies of the individual particles In order to write the sum (3.4) as an integral over rapidities, we first have to remove the constraint between the mode numbers. We do this by inserting Kronecker symbols to get rid of unwanted configurations (1 − δ n j ,0 )e −RE(n 1 ,...,n N ) . (3.5) The first Kronecker symbol introduces the condition that the mode numbers are all different, and the second one eliminates the mode numbers equal to zero. Let us expand in monomials the first factor containing Kronecker symbols, which imposes the exclusion principle. The partition function (3.5) can be written as a sum over all sequences (n r 1 1 , ..., n rm m ) of non-negative, but otherwise unrestricted mode numbers n i with multiplicities r i . Each sequence (n r 1 1 , ..., n rm m ) in the sum corresponds to a state with r j particles of the same mode number n j , for j = 1, 2, ..., m. The total number of particles in such a sequence is N = r 1 + · · · + r m .
For instance, there are four sequences all correspond to unphysical state with three particles of the same mode number n: (n 3 ), (n 2 , n 1 ), (n 1 , n 2 ) and (n 1 , n 1 , n 1 ). They come with the coefficients of 1/3, −1/4, −1/4 and 1/6 respectively. These coefficients sum up to zero, removing this unphysical state from the partition function. Only when n 1 , .., n m are pairwise different and when r 1 , ..., r m are equal to one we have a physicial state.
The coefficients in the expansion are purely combinatorial and have been worked out in [8] (3.6) The rapidities u 1 , . . . , u m of a generalised Bethe states (n r 1 1 , ..., n rm m ) satisfy the Bethe equations where the scattering phases φ j = φ j (u 1 , . . . , u m ) are defined by

From mode numbers to rapidities
In the large L limit, we can replace a discrete sum over mode numbers n by a continuum integral over variables φ We can then use equation (3.8) to pass from (φ 1 , ..., φ m ) to rapidity variables (u 1 , ..., u m ). The only subtle point compared with the periodic case is the factor excluding the mode numbers n j = 0 from the sum (3.6) We would like to incorporate this factor into the Jacobian matrix ∂ u φ. We can do this by first expanding the product as a sum over subsets α ⊂ {1, 2, ..., m}, Here [∂φ/∂u] α,α denotes the diagonal minor of the Jacobian matrix obtained by deleting its α-rows and α-columns. The sum over subsets is the the expansion of the determinant of a sum of two matrices. Hence the unphysical state at u = 0 can be eliminated by adding a term −2πδ(u) to the diagonal elements of the Jacobian matrix when we change variables from φ to u, with Θ a , Θ b defined in (2.12). In order to apply the matrix-tree theorem, we consider the following matrixĜ where we have used the notation In terms of this matrix, the partition function is written as 4 Partition function as a sum over graphs

Matrix-tree theorem
The matrix-tree theorem for signed graphs [7] allows us to write the determinant of the matrix (3.11) as a sum over graphs. This theorem as stated in [7] is quite technical and we provide a brief formulation in the following together with two proofs, one combinatorial and one field-theoretical in the appendices. First, let us define Then the Gaudin-like matrix (3.11) takes the form (j, k = 1, 2, ..., m) The determinant of this matrix can be written as a sum over all (not necessarily connected) graphs F having exactly m vertices labeled by v j with j = 1, ..., m and two types of edges, positive and negative, which we denote by ± jk ≡ v j → v k ± . The connected component of each graph is either: • A rooted directed tree with only positive edges + kl = v k → v l + oriented so that the edge points to the vertex which is farther from the root, as shown in fig. 2. The weight of such a tree is a product of a factor r j D ab (u j ) associated with the root v j and factors r l r k K + lk associated with its edges + kl . The weight of a loop with outgrowing trees is the product of the weights of its edges, with the weight of an edge ± kl given by r l r k K ± lk . In addition, a negative loop carries an extra minus sign. This is why we will call the positive loops bosonic and the negative loops fermionic.
Summarising, we write the determinant of the matrix (4.2) as 3) allows us to express the Jacobian for the integration measure as a sum over graphs whose weights depend only on the "coordinates" {u j , r j } of its vertices. For a periodic system K + = K − and the two families of loops cancel each other, leaving only trees in the expansion of the Gaudin matrix [8].

Graph expansion of the partition function
Applying the matrix-tree theorem for each term in the series (3.13), we obtain a graph expansion for the partition function where the last sum runs over all graphs F with m vertices as constructed above. The next step is to invert the order of the sum over graphs and the integral/sum over the coordinates {u j , r j } assigned to the vertices. As a result we obtain a sum over the ensemble of abstract oriented tree/loop graphs, with their symmetry factors, embedded in the space R + × N where the coordinates u, r of the vertices take values. The embedding is free, in the sense that the sum over the positions of the vertices is taken without restriction. As a result, the sum over the embedded graphs is the exponential of the sum over connected ones. One can think of these graphs as Feynman diagrams obtained by applying the Feynman rules in Fig. 4.
The Feynman rules comprise there kinds of vertices: "root" vertices with only outgoing bosons, "bosonic" vertices with one incoming boson and an arbitrary number of outgoing bosons, "fermionic" vertices with one incoming and one outgoing fermion, together with an arbitrary number of outgoing bosons. The connected diagrams built from these vertices are either trees (figure 2) or bosonic loops ( fig. 3a) or fermionic loops (fig. 3b).
The free energy is a sum over these graphs, In this expression, Y r (u) denotes the sum of over all trees rooted at the point (u, r) and C ± n is the sum over the Feynman graphs having a bosonic/fermionic loop of length n. We have defined Y r (u) in such a way that the all vertices with r outgoing lines, including the root, have the same weight.  5 Summing up the connected graphs: the exact g-function

The tree contribution
In this section, we analyze the part of free energy (4.5) that comes from the tree-diagrams Being the generating function for directed trees rooted at (u, r), Y r (u) obeys a simple equation This equation can be understood diagramatically as in figure 5.
In particular, we have for r = 1 By replacing (5.4) into (5.3), we can express Y r in terms of Y 1 for arbitrary r ≥ 1 This allows us to rewrite (5.4) as a closed equation for This integral can be extended to the real axis by using the parity of the kernel This is nothing but the TBA equation for a periodic system at inverse temperature R. In particular, the periodic partition function can be written in terms of Y 1 Similarly, we can also extend the domain of integration in (5.2) to the real axis, using the parity of D ab (u, r) and Y 1 . By subtracting the periodic free energy (5.6) from the tree part of the free energy (5.2), we obtain the tree contribution to g-function

Loop contribution
Now we turn to the sum over loops and show that they fill the missing part [5] of the g-function (2.13). Let us define For each n ≥ 1, C ± n is the partition sum of K ± loops of length n with the trees growing out of these loops which can be summed separately Y r 1 (u 1 )....Y rn (u n )r 2 r 1 K ± (u 2 , u 1 )....r 1 r n K ± (u 1 , u n ). Figure 6: A K − loop connecting n points (u j , r j ). Trees growing out of a vertex (u j , r j ) sum up to the corresponding Y r (u) function. The factor r j−1 r j of the propagator from (u j−1 , r j−1 ) to (u j , r j ) can be pulled into the adjacent trees. Taking the sum over all r j results in the Fermi-Dirac factor Y 1 (u j )/(1 + Y 1 (u j )) at each vertex j.
In this expression, the sign comes from fermion loop and 1/n is the usual loop symmetry factor. We can use the relation (5.5) to carry out the sum over r r≥1 It follows that where the kernelsK ± are defined in (2.15). The loop contribution is therefore given by The g-function is obtained by combining (5.7) and (5.10) and set a = b log(g a ) = 1 2 F trees aa + F loops

Excited state g-function
In this section, we derive the excited state g-function. This quantity can be regarded as the normalized overlap between the boundary state and an excited bulk eigenstate By setting |ψ to the ground state |ψ 0 , we recover the definition (2.10) of the g-function. We restrict our computation to the case where |ψ is of the form | ± w 1 , ±w 2 , ..., ±w N . We also assume for simplicity that the scattering matrix is a function of the difference of rapidities (relativistic invariance).
First, let us briefly summarize the excited state TBA equations for a periodic system, following [8]. We consider a torus with one large dimension L (physical volume) and a finite dimension R (mirror volume). A mirror state |ϑ = |v 1 , ..., v N propagates along the L direction. Note that the mirrorphysics convention is in reverse order compared to [8]. The Boltzmann weight of a physical particle is dressed by the interaction with these mirror particles The partition function, or equivalently the energy of the state |ϑ is given by where Y ϑ solves for the excited state TBA equation The on-shell condition for the state |ϑ is obtained by transforming a mirror particle of rapidity v j to a physical particle of rapidity v j − iπ/2. The relative factor between the two ways of computing the partition function is −Y (v j − iπ/2). This leads to the finite volume Bethe equations Now let us return to the excited state g-function (5.12). We repeat the same excircise for a long cylinder of length L and radius R with two boundaries a and b together with a state |ψ = | ± w 1 , ±w 2 , ..., ±w N propagating in the L direction. We denote the partition function in this case by Z ab (R, L, ψ).
The idea is, if we can identify the excited energy (5.14) with the extensive part of the partition function Z ab (R, L, ψ) when |ψ ≡ |ϑ , then the rest (intensive part) gives us the excited g-function corresponding to |ψ To compute Z ab (R, L, ψ) we perform the sum over eigenstates of the physical Hamiltonian with boundary H ab . The procedure is similar to that of ground-state g-function: we obtain a sum over trees and loops. The only difference is the Feynman rule for the vertices In particular, the extensive part of the partition function Z ab (R, L, ψ) is given by whereỸ ψ (u) being the sum of trees rooted at vertex u now satisfies the equatioñ As a consequence of the crossing symmetry S(u) = S(iπ − u) we have the identity which means that the functionỸ • ψ (u) is an even function of u. Therefore we can extendỸ ψ to the real axis and identifyỸ ψ with Y ϑ when |ψ = |ϑ . Again we haveỸ ψ (w j ) =Ỹ ψ (−w j ) = −1. We conclude that where the Fermi-Dirac factor in the kernelK ± ψ is now given byỸ ψ /(1 +Ỹ ψ ), c.f. eq. (2.15).

Nested Bethe Ansatz
Our method can be extended to the case of nested Bethe ansatz with diagonal reflection matrices. As an illustration, we consider a theory with two types of particle. Particle of type 1 carries energy E(u) and momentum p(u) parameterized by its rapidity u while particle of type 2 has vanishing energy and momentum. The bulk scattering matrices and the reflection matrices are denoted by S pq and R pa , R pb for p, q ∈ {1, 2}. They are assumed to satisfy the following properties S pp (u, u) = −1, The last property is only needed for system with boundaries.

Periodic system and nested TBA equation
An (N 1 + N 2 )-particle state is characterized by a set of rapidities |u 11 , ..., u 1N 1 , u 21 , ..., u 2N 2 . Particles of the same type must have different rapidities: u 1j = u 1k , u 2j = u 2k . The Bethe equations for such state read The partition function can be written as a sum runs over two sets of mode numbers n 1 = n 11 , ..., n 1m 1 and n 2 = n 21 , ..., n 2m 2 along with two sets of multiplicities (wrapping numbers) r 1 = r 11 , ..., r 1m 1 and r 2 = r 21 , ..., r 2m 2 (−1) m 1 +m 2 m 1 !m 2 ! 0≤n 1 ,n 2 1≤r 1 ,r 2 2 p=1 mp j=1 (−1) r pj r pj e −RE((n 1 ,r 1 ),(n 2 ,r 2 )) . 3) The mode numbers ((n 1 , r 1 ), (n 2 , r 2 )) are related to the rapidities ((u 1 , r 1 ), (u 2 , r 2 )) through Bethe equations with multiplicities The Gaudin matrix has a 2 × 2 block structurê The explicit expressions of each block arê The partition function can be written in terms of the determinant of this matrix Note that E 2 = 0. We apply the matrix-tree theorem for the matrixĜ and obtain a tree expansion of the free energy. Each vertex now carries an index p ∈ {1, 2} to indicate what type of particle it stands for. A branch going from vertex of type p to vertex of type q has a weight of r q r p K qp . All roots are of type 1 and carry a weight of r 1 Lp , where the momentum derivative p should not be mistaken with the vertex label. We denote vertex of type 1 by a disk and vertex of type 2 by a circle.
(u 1 , r 1 ) (u 2 , r 2 ) (u 1 , r 1 ) Let us denote by Y pr (u) the sum over all the trees rooted at (u, r) of type p. The free energy depends only of Y 1r However the TBA equation determining In particular, we still have (u , r) (u , r) For simplicity, let us denote Y 11 and Y 21 simply by Y 1 and Y 2 . We then have

The nested g-function
With diagonal reflection matrices, the Bethe equations for the state |u 11 , ..., u 1N 1 , u 21 , ..., u 2N 2 read (6.10) The rapidities and the mode numbers are taken to be positive. Similar to (3.6), we have (−1) r pj r pj (1 − δ n pj ,0 )e −RE((n 1 ,r 1 ),(n 2 ,r 2 )) . (6.11) The conversion between mode numbers and rapidities under the presence of multiplicities where we have used the notation S pq (u, v) = S pq (u, v)S pq (u, −v). The Gaudin matrix now has a 2 × 2 block structurê The explicit expressions of each block arê where the notations are If we set Θ to zero and K + and K − to equal then we would recover the Gaudin matrix for the periodic system (6.4). The partition function is written in terms of the determinant of this matrix The tree contribution to g-function is obtained in a similar way as before where Y p for p = 1, 2 are solutions of the TBA equations (6.9). Now comes the loop contribution where C ± n denotes the sum over bosonic/fermionic loops of length n. Each of these n vertices can be either of type 1 or 2. The trees growing out of each vertex can be summed to the Fermi-Dirac factor of each type, by virtue of the relation (6.8) The loop contribution can then be written as a sum over cyclic sets p of C({1, 2} n ) where S(p) is the symmetry factor of p. This sum is nothing but the trace of 2 × 2 matricesK ± with elements We obtain two Fredholm determinants with 2 × 2 matrix kernels as a generalization of (5.10) log(g a g b ) loops = log det 1 −K − 1 −K + . (6.18)

Conclusion and outlook
We propose a graph theory-based method to compute the g-function of a theory with diagonal bulk scattering and diagonal reflection matrices. The idea is to apply the matrix-tree theorem to write the Jacobians in the cluster expansion of the partition function by a sum over graphs. The g-function is then written as a sum over trees and loops. The sum over trees gives TBA saddle point result while the sum over loops constitute the two Fredholm determinants. The method was generalized to a theory with non-diagonal bulk scattering and diagonal reflection matrices. We would like to point out the relationship between the expression of the g-function and the overlap between an initial state and the ground-state (2.10). The normalized overlaps play an important role in the study of out of equilibrium dynamics [12], [13], [14], [15], [16] and one point function in AdS/CFT [17], [18], [19], [20]. A direct comparison of the two types of results on the overlaps is not straightforward since they imply different regimes of parameters, but it is an interesting open problem to understand the link between the two.
Several other directions can be investigated in near future. First, one can extend the analysis of the excited state g-function to theories with non diagonal scatterings and/or without relativistic invariance.
In particular one has to verify the parity property (5.21) when the scattering matrix no longer depends explicitly on the difference of rapidities. Second, one can consider the case of non-diagonal reflection matrices. It would be ideal to have a candidate theory which is sufficiently simple to be the working example. Last but not least, our method could also be applied in the hexagon proposal for three point functions in N = 4 super Yang-Mills [21], [22]. This non-perturbative approach is plagued with divergence when one glues two hexagon form factors together [23]. The divergence takes the form of a free energy of particles in the mirror channel. The regularization prescription that leads to this free energy also predicts a finite contribution which bears some similarities to the g-function.
When α 1 = {1, . . . , l} one should repeat again the procedure of splitting the sum over indices, The terms from the second product in the second line above will add a new layer of branches from the branches already grown from the loops of type K − (s 1 s 2 ... sm) , if k ∈ α 3 , or will grow branches from the roots D i , if k ∈ α 2 . The new branches have tips in the ensemble α 12 . The terms in the first product will be treated as in the previous stage. The procedure will be repeated until all the indices are exhausted.
We conclude that after repeating the procedure we are left with an ensemble of disconnected (generalised) trees each growing out from In a similar way, we will introduce the piece k K + jk in M j through the expectation value with respect to n pairs of bosonic variables ϕ i ,φ i (i = 1, ..., n)