On the rotator Hamiltonian for the SU$(N)\times\,$SU$(N)$ sigma-model in the delta-regime

We investigate some properties of the standard rotator approximation of the SU$(N)\times\,$SU$(N)$ sigma-model in the delta-regime. In particular we show that the isospin susceptibility calculated in this framework agrees with that computed by chiral perturbation theory up to next-to-next to leading order in the limit $\ell=L_t/L\to\infty\,.$ The difference between the results involves terms vanishing like $1/\ell\,,$ plus terms vanishing exponentially with $\ell\,$. As we have previously shown for the O($n$) model, this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions for $N=3\,.$


Introduction
The low energy dynamics of QCD in the δ−regime is to lowest order in chiral perturbation theory (χPT) described by a quantum rotator for the spatially constant Goldstone modes [1]. We recall that for a system in a periodic spatial box of sides L the δ−regime is where the "time" extent L t L and m π L is small (i.e. small or zero quark mass) whereas F π L, (F π the pion decay constant) is large.
Many other systems described by non-linear sigma models, also in d = 2, 3 dimensions, are similarly approximated by a quantum rotator to leading order in the analogous perturbative domain. Accordingly, the lowest energy momentum zero states in a representation r of the symmetry group have, to leading order perturbation theory, energies of the form where C 2 (r) is the eigenvalue of the quadratic Casimir (of the symmetry group) in the representation r . At 1-loop level it turns out that the Casimir scaling (1.1) still holds, but it is of course expected that at some higher order the standard rotator spectrum will be modified. The standard rotator describes a system where the length of the total magnetization on a timeslice does not change in time. This is obviously not true in the full effective model given by χPT.
In a previous paper [2] we pointed out that by comparing the already obtained NNLO results for the isospin susceptibility from χPT at large ≡ L t /L with that computed from the standard rotator, one can establish, under reasonable assumptions, that at 3-loops there is a correction to the rotator Hamiltonian proportional to the square of the Casimir operator, with a proportionality constant determined by the NNLO low energy constants (LEC's) of χPT.
In ref. [3] we considered the QM rotator for the group O(n). In this paper we extend the analysis of the QM rotator to the group SU(N ) × SU(N ) , which has for N > 2 , to our knowledge, not been frequently considered in the literature. This paper is organized as follows: In section 2 we recall the definition and results for the isospin susceptibility of the standard quantum rotator coupled to a chemical potential for the SU(2)×SU(2) and the SU(3)×SU(3) cases. Results for the general SU(N )×SU(N ) case are given in section 3. The results in this section are new, in particular Eq. (3.4) gives the eigenvalue of the quadratic Casimir invariant for a generic SU(N ) representation. In section 4 we discuss the corrections to the simple rotator formula calculated in chiral perturbation theory. In sect. 5 we consider the case of d = 2 . For SU(3) × SU(3) Kazakov and Leurent [4] have computed the lowest energies of two representations using an alternative to the thermodynamic Bethe ansatz (TBA). Their NLIE (nonlinear integral equation), in contrast to the infinite component TBA, is formulated in terms of finitely many unknown functions and allows for a much better numerical precision than the corresponding TBA calculation. Their data clearly show that Casimir scaling is valid to a very good approximation for M L < 1 , however it was not sufficiently precise to see the expected deviations. Here we present more precise numerical data allowing us to clearly see the deviation from the simple rotator spectrum. Our data are completely consistent with the results of the perturbative calculations. The details of our calculations are given in various appendices, in particular the algorithm of Ref. [4] to use the NLIE equations for the calculation of the finite size spectrum of the model (for N = 3) is reviewed in appendix E.
The contribution of the main author of this paper, Ferenc Niedermayer, was essential in the formulation of the bulk of this paper. His untimely death on 12 August 2018 denied him the completion of the numerical calculations. We devote this paper to the memory of Ferenc.

The isospin susceptibility
Here we consider the Hamiltonian of the SU(N ) × SU(N ) standard quantum rotator with a chemical potential coupled to generators J L3 , J R3 : where J 2 X are the quadratic Casimir operators of the left and right SU(N ) groups:

2)
and Θ is the moment of inertia. In d = 4 dimensions to lowest order χPT one has Θ F 2 L 3 . The isospin susceptibility is defined as the second derivative of the free energy wrt h: where u = 2L t /Θ . The partition function has for small h , the expansion The isospin susceptibility is then given by We wish to compute χ for small u for general N , however the reader may find it instructive to first consider the special cases N = 2, 3 which we treat in the following subsections.

SU(2) × SU(2) case
The quantum mechanics (QM) of a symmetric rotor (rigid body) in 3 dimensions is equivalent to QM of a point particle moving in the SU(2) group manifold, which is the sphere S 3 . It can be considered as a special case of the O(n) rotator (point particle moving on the sphere S n−1 ) for n = 4 . At the same time it is a special case of a particle moving on the SU(N ) group manifold with N = 2 . The coordinates in the two descriptions are: U ∈ SU(2), where U = s 0 + is k σ k ∈ SU(2) , (σ k the Pauli matrices) or equivalently, in the O(4) picture s = (s 0 , s 1 , s 2 , s 3 ) ∈ S 3 , (s 2 = 1) . The wave functions have the form ψ(U ) or ψ(s) . The symmetry group of H 0 for h = 0 is G = SU(2) × SU(2) SO(4) , and the transformation of a wave function under The symmetry generators are J Li for SU(2) L and J Ri for SU(2) R transformations, (i = 1, 2, 3), or alternatively the 6 generators of SO(4) .
In the SU(2) × SU(2) picture the wave functions are constructed using the U variables. The set of four wave functions ψ(U ) ∈ U 11 , U 12 , U 21 , U 22 belong to the representation with j L = j R = 1/2. In general, the Hilbert space of the system splits into irreps of SU(2)×SU (2) for which j L = j R = j 1 .
To label the irreps of SU(2) we adopt the convention which is a special case to be used for SU(N ) with N ≥ 3 below. The representation with given j is denoted by (p) where p = 2j = 0, 1, 2, . . ., with the corresponding dimension p + 1 = 2j + 1 . Accordingly, the eigenstates of the Hamiltonian (2.1) |j, m L × |j, m R , −j ≤ m L , m R ≤ j belong to the representation (p) × (p) with multiplicity (p + 1) 2 2 .
The eigenvalue of the quadratic Casimir invariant in a representation (p) is given by which differs by a factor 4 from the O(4) Casimir invariant l(l + 2) 3 . The kinetic energy is then given by consistent with our conventions in ref. [3].
In the O(4) picture the isospin chemical potential is coupled to generator of rotations in the 12-plane, L 12 . It has eigenvalues m = −l, . . . , l for SO (4). The corresponding multiplicities are g lm = l − |m| + 1 . For l = 1 one has: m = ±1: s 1 ± is 2 , for m = 0: 1 In the classical description a given trajectory U (t) of the particle can be reached in two equivalent ways, by left rotations U (t) = g L (t)U 0 or by right rotations, U (t) = U 0 g † R (t). Obviously, the energy of a given eigenstate or its multiplicity should not depend on the description chosen. 2 Note that in the equivalent O(4) language the eigenstates with given l = 0, 1, 2, . . . have multiplicity (l + 1) 2 (cf. [3]). 3 Note that exp(iJ 3 φ) in SU(2) rotates by an angle φ/2 around the 3rd axis, while for O(4) exp(iJ 3 φ) rotates by angle φ.
The partition function with zero chemical potential is then where S(x) is the Jacobi theta-function For small u it has an expansion For z 1 we have ∂z 0 (u) ∂u . (2.14) The SU(2) × SU(2) rotator susceptibility is then given by

The SU(3) × SU(3) case
Next we consider the QM of a point particle moving on the group manifold of SU (3). For the SU(3) irreducible representations we shall in this subsection use the familiar notation (p, q) where p, q = 0, 1, 2 . . . i.e. the first and second rows of the corresponding Young tableaux have p + q and q boxes respectively.
The corresponding value of the quadratic Casimir invariant is while the dimension of the representation is given by We consider a system described by coordinates U ∈ SU(3), and wave functions ψ(U, U ) which transform under g = g L × g R ∈ SU(3) × SU(3) according to The 9 wave functions ψ(U ) = U ab , where a, b ∈ {1, 2, 3} belong to the representation (1, 0) × (0, 1) of SU(3) × SU(3). The first index, a is for 3 ≡ (1, 0), while b for 3 ≡ (0, 1). At this stage we assume that the irreps appearing in the partition function sum over states are of the type (p, q) × (q, p) ; the motivation for this will be given in subsection 3.2 4 . The corresponding energy is given by (cf. (2.9)) with the multiplicity d(p, q) 2 (cf. (2.17)). We discuss the full dependence of Z(u; h) on h in Appendix C.5 although this information will not be needed in this paper. From eqs. (2.5),(2.6) we obtain 2 ((p, q)) , where s are eigenstates of J 3 with eigenvalues λ(s). One has (see (3.3)) 2 ((p, q)) = 1 8 C 2 ((p, q))d(p, q) .
In z 0 , z 1 we have a double sum over integers, hence analytic expressions are not so simple. However the leading terms for small u can be determined analytically. After separating the constant term in (2.16), the remaining expressions are homogeneous iñ p = p + 1 andq = q + 1 : For small u we can replace the sums by integrals to obtain To investigate the corrections to (2.24) one can first proceed numerically e.g. evaluating the difference to 500 digits at 0.1 ≤ u ≤ 1 one has for the relative deviation (z 0 (u) − z 0A (u))/z 0 (u) at u = 1.0: ∼ 10 −13 , and at u = 0.1: ∼ 10 −164 , i.e. it decreases faster than any power of u. Fitting the difference one obtains the next approximation The correction to the leading first term is exponentially small for small u, and has a structure similar to (2.13). For N = 3 using (2.19) this gives for the susceptibility of the SU(3) × SU(3) rotator We stress again that for u → 0 the omitted terms decrease faster than any power of u. The leading term in (2.15), (2.27) is the classical result for the high temperature expansion of the corresponding rotator (rigid body). The next one, ∝ is the leading quantum correction, which does not depend on Θ, only on the corresponding group. It is interesting to note that for N = 2, 3 the 1/(F 2 L 2 ) term (for d = 4) is absent in the expansion, a property which we will see holds for arbitrary N .

The quadratic Casimir invariant
As proposed by Gelfand and Tsetlin [5], an irrep of SU(N ) can be conveniently described by a non-increasing series of N integers (cf. [6] and references therein) m 1 ≥ m 2 ≥ . . . ≥ m N . Two series differing in a constant, m k = m k + c , ∀k where c ∈ Z describe the same irrep. One can choose m N = 0, however, for some purposes it is convenient to use the redundant form with N integers. If one sets m N = 0 then m k corresponds to the number of boxes in the k'th row of the corresponding Young tableau. The more conventional description of an SU(N ) irrep, like (p, q) for SU (3) , is given by the differences (p 1 , p 2 , . . . , p N −1 ) where p k = m k − m k+1 ≥ 0 .
Following the notation in [6], let J where s are eigenstates of J 3 with eigenvalues λ 1 (s). Q (N ) 0 (r) is the dimension of a given irrep r and is explicitly given by [6] The quadratic Casimir invariant can be calculated using the basis of the su(N ) algebra described in [6]. Alternatively one can use recursion relations for Q where s , (s = 0, 2) contain N − 1 nested summations. For not too large N one can perform these analytically. We have done this for N ≤ 5 , and obtained a very simple result, which is easy to generalize to arbitrary N . We conjecture Note that this expression is invariant under a constant shift m k → m k + c as it should.
Denoting n k = m k + N − k one has for the factor appearing in Q The Casimir invariant of the representation in terms of n's is is proportional to the curvature of the SU(N ) manifold. Note that apart from the constant in C

Wave functions
First we note that for U ∈ SU(N ) the complex conjugate of a matrix element equals the corresponding cofactor of the matrix, As a consequence, a function written in terms of products containing U 's and U 's can be written in terms of the U 's alone. Under a general SU(N ) × SU(N ) transformation one has Under separate left/right transformations i.e. U belongs to the representation (1, 0, . . . , 0) × (0, . . . , 0, 1), according to its 1st and 2nd index, respectively. Similarly for an arbitrary representation (r) . Hence the elements of the matrix D (r) (U ) belong to a representation with complex conjugate pair (r) Strictly one should still show also that each such representation enters only once in the Hilbert space of the SU(N )×SU(N ) rotator; here we accept this as a reasonable hypothesis.

The partition function and susceptibility
The partition function is given by Changing to the variables n k = m k + N − k the condition m k ≥ m k+1 transforms into n k > n k+1 . Also the irreps with n k = n k + c where c ∈ Z are equivalent and should be taken only once in the partition function. Again a convenient choice is to set n N = 0 and one has z (N ) 16) where the second equality follows since the summand is invariant under permutations.
In the conventional "p-notation" the Casimir invariant and Q 0 for the r p = (p, 0, 0, . . . , 0) representation is (3.17) In particular for the ground state p = 0 and for p = 1 For the adjoint representation 5 r A = (1, 0, . . . , 0, 1) one obtains 2 (r 2 ) the mass gap is given by the states in the representation r 1 × (0, . . . , 0, 1), and its conjugate, with a total multiplicity 2N 2 . The formula for the mass gap is then (cf. [2] eq. (4.31)) The contribution of these states together with the ground state gives The behavior of z Thus the susceptibility is obtained from z 0 (u) as The susceptibility is then given by which is in agreement with eq. (4.48) of [2] obtained by χPT to NNL order for general N . 5 Here we assume N ≥ 3

The 1/ term in χPT
As mentioned in [2] the susceptibility calculated in χPT to NNLO for → ∞ approaches the result obtained in the rotator approximation. However, the approach is not exponentially fast, to this order one obtains besides the exponentially vanishing contribution a ∝ 1/ term (but no −k , k ≥ 2 terms!). More precisely for the deviation δχ = χ − χ rot in [2] (cf. eq. (4.52)) we found From here one concludes that the rotator spectrum should be distorted at some higher order in g 2 0 already at small energies, not only the energies ∼ L −1 of the p = 0 modes. Let us assume that the distortion of the spectrum has the form then one obtains Taking z 0 (u) ∝ u −a with a = (N 2 − 1)/2 one gets for the leading term The observed deviation (4.1) requires then κ = 2 and since Θ ∼

Delta regime in d = 2
The susceptibility computed in χPT is for d = 2 given by [2] χ = 1 where g MS (q) is the minimal subtraction (MS) scheme running coupling at momentum scale q , and The large behavior of the shape functions appearing in (5.1) and (5.2) are discussed in [3]. In particular we find for 1: On the other hand the susceptibility computed from the simple rotator is given by: where g FV is the LWW running coupling [7] defined through the finite volume mass gap: Its expansion in terms of the running coupling in the MS scheme of dimensional regularization (DR) is given by The first two coefficients are obtained using the methods of ref. [7]: Combining the results we arrive at On the other hand, from our considerations of the modified rotator in the previous subsection we would expect where Φ 3 is the leading coefficient in the perturbative expansion of Φ(L), assuming the expansion starts at order g 8 MS : Comparing (5.11) with (5.12) determines where The low-lying spectrum to order g 8 MS is given by Hence we conclude, for example, In subsection 5.2 we test this prediction for N = 3 .

Running coupling functions
First, following Balog and Hegedus [8] we introduce a function g 2 J (L) of the box size L through 1 g 2 J (L) where b 0 , b 1 are the universal first perturbative coefficients of the β−function 6 7 : and Λ FV is the Λ−parameter of the LWW finite volume coupling in (5.7). We chose the solution which is small for Λ FV L small, which has the property 8 We consider g 2 J as a function of z = M L where M is the infinite volume mass gap: The ratio M/Λ FV is known using the result in ref.
The LWW coupling has the following expansion in terms of g 2 J : In Table 1 we reproduce the data for the energy gaps E(r p ) calculated from the numerical results given in Table 2 in appendix E. f 3,est appearing in the last column is defined in (5.27). From the fifth column of Table 1, we see that the ratio E(r 2 )/E(r 1 ) is close to the ratio 10/4 of the Casimir eigenvalues. However, our numerical precision is sufficient to establish that the simple effective rotator model requires corrections. To see even more clearly the agreement of our analysis with the data in Table 1 in Fig. 1 we plot estimates for f 3 given by for the case N = 3. In the figure the extrapolation to zero volume is also shown. We do not have measured values close enough to α J = 0 to make linear fits. Our extrapolation is based on a quadratic least squares fit (weighted by the error bars given in the last column in Table 1) to the first 11 data points in the range α J < 0.063 giving f 3 (quadratic, 11) = 0.147. If we use the first 10 data points only, we get f 3 (quadratic, 10) = 0.127. The green line is a constrained fit where the zero volume limit (5.15) is kept fixed at α J = 0. This shows that our measurements are completely consistent with our prediction in (5.15).  In the next subsection we show for the irrep r specified by (m 1 , . . . , m N ) and n k ≡ m k + N − k Q . . .
The summation goes over l 1 , . . . , l N −1 which satisfy the condition To prove the formula for C

A.1 Proof of the recursion relations
To prove the recursion relations (A.1) we generalize the notion of summation to the case of symbolic limits for polynomial summands.
in an analogous way, calculating the rhs for integer values of the limits with a l < b l and using the obtained expression as a definition for symbolic limits a l , b l .
Using the invariance under n k → n k − 1 the recursion relation (A.1) can for k = 0 be rewritten as Q Since the limits for t 1 and t 2 coincide, and the summand Q (N −1) 0 (t 1 , t 2 , . . . , t N −1 ) changes sign for t 1 ↔ t 2 , the rhs vanishes for x 3 = x 1 , and therefore it contains a factor (x 1 − x 3 ). Obviously, it also contains all factors of type (x k − x k+2 ). ] respectively) appear twice, and from the antisymmetry wrt. t 1 ↔ t 2 (respectively t 2 ↔ t 3 ) one concludes that the rhs vanishes also for x 4 = x 1 . In this way one can show that the rhs contains all factors (x k − x k ), 1 ≤ k < k ≤ N appearing in Q (N ) 0 (x 1 , x 2 , x 1 , . . . , x N ). Since the orders of the polynomials on the two sides also coincide, they must be equal, apart from a possible constant factor, which can be shown to be 1.

B The partition function
The function φ(u, α) can be expressed through the Jacobi theta-function The relation is given by The function φ(u, α) satisfies the duality relation For u < 1 it is convenient to use the fast converging expression (B.7) Here the n = 0 terms in (B.7) are suppressed exponentially for u → 0. For u 1 and defining w = 4π 2 /u , one obtains for N = 2, 3, 4 , the expansions

C The partition function Z(u; h)
In this appendix we consider the full dependence of the partition function on the chemical potential h. Below we use the short-hand notationĥ = hL t

C.1 U(1) case
The irreps of the U(1) group are labeled by m ∈ Z, and all have dimension 1, while the Casimir invariant is C = m 2 . The partition function is

C.3 SU(N ) case
Similarly to the steps in (B.2) one obtains where φ(u, α) is as in (B.5) . For small u one can use the expansion (B.7).

C.4 SU(2) case, again
Using m 3 ] are One has (using the symmetry h → −h) D Relation of z 0 (u) to the heat kernel K(U, t) The heat kernel K(U, t) on the group manifold SU(N ) in the U → I limit (where I is the identity) is related to our partition function z 0 (u) at u = t. It is given by (see eqs. (3), (6) from [10]) where r runs over all irreducible unitary representations, χ (r) (U ) is the character of the representation, d (r) = χ (r) (I) its dimension and C For SU(2), eq. (9),[10] where U = S diag[φ, −φ] S † , and the prefactor N 2 does not depend on φ. Note, however, that it depends on the parameter t, a fact which was irrelevant for the discussion of [10]).
In the limit U → I this gives From our result in eq. (2.12) we deduce Note that for t → 0 the square bracket goes to 1 exponentially fast and the important information for the isospin susceptibility in this limit is hidden entirely in the undetermined t-dependence of N (t).

E Hirota dynamics and NLIE for the SU(3) principal model
In this appendix we give formulas necessary to numerically compute the finite volume energy spectrum of the SU(N ) principal chiral model for N = 3 using the NLIE equations constructed in Ref. [4]. [4] discusses the case for general N ≥ 3, but here for simplicity we restrict our attention to N = 3 only. (The N = 2 case was discussed before in [11].) We give all formulas necessary to perform the numerical computation in a "cookbook style" and refer to the original paper [4] for the derivation of the equations and further details.

E.1 SU(3) T-system and Y-system
Based on previous experience with integrable models, where similar equations were constructed by starting from integrable lattice regularizations and/or by bootstrap methods the following double-infinite T-system is proposed as the basis for the description of the finite volume spectrum of the SU(3) principal model: where the T-functions T a,s (θ) are indexed by a = 0, 1, 2, 3 and s = 0, ±1, ±2, . . . and by definition Here for any function f (θ) the notation f ± (θ) stands for For the description of a particular state in the spectrum of the model we have to specify the corresponding solution of the T-system (E.1). Starting from the T-system (Hirota equations) one can go to the corresponding double-infinite Y-system, which is used to construct the TBA integral equations, or, alternatively, to the finite Q-system, which is used in the NLIE approach. The SU(3) Y-system for the Y-functions Y a,s (θ) , a = 1, 2, s = 0, ±1, ±2, . . . is , This Y-system is illustrated by Fig. 2, where the s = 0 nodes are black indicating that they are the massive nodes with asymptotic behavior Here M is the mass of the particles in infinite volume and L is the size of the system. All other (magnonic) nodes behave as The relation between the T-functions and Y-functions is . (E.9) We will consider the U(1) sector only, where all particles are highest weight states in the defining representation of SU (3). For these states one can establish, using the T-Y relations (E.9) that 1 + Y 1,0 (θ) has zeroes at : θ = θ 1,j + 3i 1 + Y 1,0 (θ) has poles at : θ = θ 2,j − i 4 (E.10) and 1 + Y 2,0 (θ) has zeroes at : 1 + Y 2,0 (θ) has poles at : θ = θ 1,j + i 4 . (E.11) The position of singularities are not independent; they are related by the T-Y relations. They are parameterized in terms of two complex quantities θ 1,j and θ 2,j , which are deformations of (and for large L are exponentially close to) the real asymptotic rapidities (Bethe roots) θ j . The index j (j = 1, . . . , N ) labels the particles. The two Y-functions are conjugates of each other: and consequently θ * 1,j = θ 2,j . (E.13) Whether one uses the infinite set of TBA equations, which can be derived from the Y-system (E.4), or the finite set of NLIE equations of Ref. [4], it is always necessary to construct at least the s = 0 Y-functions corresponding to the massive nodes, since these are entering the energy formula 11 . (E.14) This energy formula was conjectured in Ref. [4], but it can also be systematically derived [12] from an integrable lattice regularization of the model.

E.2 Asymptotic solutions
We are not going to use the infinite set of TBA equations in our numerical calculations but we used them to derive some large volume asymptotic formulas to determine the exponentially small shifts of the parameters θ a,j and to calculate the first exponentially small corrections to the energy (E.14). In this subsection we will use the "natural" normalization of the rapidity parameters (to avoid confusion we denote them by T instead of θ). The relation between the two normalizations is To leading order the energy is just the sum of the individual free particle energies given by where the asymptotic rapidities satisfy the Bethe quantization conditions where n j are integer quantum numbers and δ(θ) is the phase shift for the scattering of highest weight triplet particles. In the simplest 1-particle case the quantization condition reduces to sinh The next (leading exponential, also called Lüscher) corrections consist of two parts: where the mu-term E (µ) comes from the deformation of the rapidity parameters in the first line of (E.14) and the F-term E (F ) is the leading exponential approximation of the second line (the integral term). We can write the change of the rapidities as where both x j and y j are exponentially small. The sign of y j determines whether when moving away from the L = ∞ limit the exact rapidities move upwards or downwards in the complex plane. We now write down the Lüscher order asymptotic formulas for N = 1 and for the lowest energy zero total momentum N = 2 state.
First we introduce some notations and definitions.
We note that in (E.22) A(θ) and B(θ) are real for θ real. Further For N = 1 we find The coefficient of the exponential in the mu-term (E.29) (for the standing particle T 1 = 0) is different from that given by Eq. (95) of Ref. [4]. Although our coefficient is larger by a factor π/3 ≈ 1.05 only, we have demonstrated numerically that the difference between the two formulas is clearly visible and that it is indeed (E.29) that agrees asymptotically with the exact result. For the lowest energy parity symmetric N = 2 state with we find y 1 = y 2 = −σA e −σz coshT K(2T ) , (E.33) (E.35)

E.3 Alternative energy formula
In Ref. [4] an alternative energy formula is used: (E.36) The reason for suggesting this alternative formula is, as we will see later, that it is more suitable for the NLIE approach. On the other hand, there is a problem with (E.36) since as can be seen from (E.10) and (E.11) there are zeroes/poles dangerously close to the integration contours. These singularities tend to the integration contours in the L → ∞ limit and asymptotically coincide. For this reason the energy formula (E.36) should be applied with care. First of all the equivalence of it with the established formula (E.14) should be proved. In Ref. [4] the starting point for the proof of this equivalence is the energy formula which is what one obtains from (E.36) by formal partial integration. The strategy of the proof is to shift the integration contour to the real line in order to match the integral with the integral part of (E.14). During this deformation of the contours we encounter singularities at the poles/zeroes given by (E.10) and (E.11), provided they lie inside the strip bordered by the contours. There are two cases. If Im θ 1,j > 0 (case I) , (E.38) then the corresponding zeroes are inside the strip (poles are outside). If then the corresponding zeroes are outside the strip (poles are inside). One can show that during the contour deformation using Cauchy's theorem we pick up residue contributions that make the formulas (E.37) and (E.14) exactly coincide in case I only. In case II (E.37) and (E.14) are definitely different. Using the asymptotic result (E.28) we can see that the 1-particle states belong to case I only for even quantum numbers n 1 . (E.33) shows that the lowest energy N = 2 symmetric state also belongs to case I. Thus luckily the states we are interested in (lowest energy 1 and 2-particle states) are all case I states.
Another problem is that (except for the ground state) (E.36) is not equal to its formally partially integrated version (E.37). During partial integration the boundary terms at infinity of course vanish since Y a,0 (θ) are exponentially small there. However, there are boundary terms coming from certain points on the contour. As we have seen there are (at least for large L) poles and zeroes very close to each other and to the integration contour. This implies that there must be some realθ such that 1 + Y 1,0 (θ − i/4) is real and negative. Using the standard definition of the log function, there are extra boundary contributions coming from the fact that ln[1 + Y 1,0 (θ − i/4)] jumps at θ =θ by ±2iπ. Simultaneously ln[1 + Y 2,0 (θ + i/4)] jumps at the same point by ∓2iπ since it is the complex conjugate. For example for the standing 1-particle state we find that as θ goes along the integration contour from −∞ to ∞ the value of 1 + Y 1,0 (θ − i/4), starting from +1, crosses the negative real axis from above at θ =θ and than goes back to +1 in the lower half plane. The corresponding extra term is Then also giving the contribution We find that by parity symmetryθ = 0 and so To summarize, we can use the energy formula (E.36) for the zero momentum 0,1, and 2-particle states with the standard definition of the log function but the correct energy is given by E = E KL + N M cosh(vB N ) (E.47) with B 1 = 0 and B 2 determined from the requirements We emphasize that for states belonging to case II (for example moving 1-particle states with odd momentum quantum numbers) further modifications are necessary.

E.4.1 The integral equations
We introduce the notation The two NLIE equations are given by Eq. (82) of Ref. [4] with Z = 1. They can be written in the form (E.63) The above formulas are obtained from Eqs. (75), (49) of Ref. [4]. D 1 and D 2 are shorthand for the complicated expressions of the right hand side of Eq. (82) of Ref. [4] with Z = 1 and they will be given explicitly below. We can express f 2 and f 3 from the NLIE equations. Defining D 3 = AB −ĀB (E.64) we have (E.65)

E.5 Numerical results
We have calculated the energies for the SU(3) states r p = (p, 0, 0) for p = 0, 1, 2 (vacuum, standing 1-particle and parity symmetric 2-particle states) to 12 digits numerically for small volumes. These energies are denoted by E p in Table 2. The results agree with those of Ref. [4] up to the numerical precision given there. Studying the volume dependence of the 2-particle β 1 parameter for small volumes we conjecture that it is given by a perturbative expansion in the running coupling α J (see subsection 5.1) with coefficients a 1 , a 2 , a 3 , . . .