Finite volume mass gap and free energy of the SU(N)xSU(N) chiral sigma model

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N)xSU(N) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization. We also compute the mass gap in a finite box with periodic boundary conditions.


Introduction
Chiral perturbation theory (χPT) [1,2] is the effective theory describing the low energy dynamics of the lowest lying pseudoscalar mesons. The parameters of the theory are couplings appearing in the effective chiral Lagrangian, the pion decay constant F π (= F in the chiral limit) and other low energy constants (LEC's). These parameters can be determined by phenomenology, or by lattice simulations of QCD. For a detailed summary of various determinations of the LEC's the reader is referred to the FLAG review [3]. For N f = 2 the relevant χPT has SU(2) × SU(2) O(4) symmetry. As a consequence in the past many theoretical χPT computations, in particular those pertaining to finite volume, have been performed for the slightly simpler model with O(n) symmetry. One special environment is the so called δ-regime first discussed by Leutwyler [4] where the system is in a periodic spatial box of sides L s and m π L s is small (i.e. small or zero quark mass) whereas F π L s is large. In 2009 Hasenfratz [5] computed the mass gap in the delta-regime to third order χPT with the hope that a comparison with a precise lattice measurement of the low-lying stable masses in this regime may be used to determine some combination of the LEC's.
In a previous paper [6] we computed the change in the free energy due to a chemical potential coupled to a conserved charge in the non-linear O(n) sigma model with two regularizations, lattice regularization (with standard action) and DR in a general d-dimensional asymmetric volume with periodic boundary conditions (pbc) in all directions. This freedom allowed us for d = 4 to establish two independent relations among the 4-derivative couplings appearing in the effective Langrangians and in turn this allows conversion of results for physical quantities computed by the lattice regularization to those involving scales introduced in DR.
In particular we could convert the computation of the mass gap in a periodic box, by Niedermayer and Weiermann [7] using lattice regularization to a result involving parameters of the dimensionally regularized effective theory, and we verified this result by a direct computation [6] (which disagrees slightly with the previous computation [5]).
Although N f = 2 is the phenomenologically most relevant case due to the low mass of the physical pions, χPT with N f > 2 can also have useful applications [3]. With this in mind in this paper we extend the computations to the case of SU(N ) × SU(N ). After recollecting the structure of the effective Lagrangian in the next section we compute the free energy is in sect. 3 and the mass gap in a finite periodic box in sect. 4.
In this paper we do not analyze explicit chiral symmetry breaking. In QCD the effect of including a small quark mass on the finite volume spectrum has been computed for N f = 2 to leading order in [4], and to next-to-leading order by Weingart [8,9]. Furthermore Matzelle and Tiburzi [10] have studied the effect of small symmetry breaking in the quantum mechanical (QM) rotator picture (N f = 2), and extended the results to small non-zero temperatures. In a related recent paper [11] we have computed the isospin susceptibility in the effective O(n) scalar field theory, to third order χPT in the delta-regime using the QM rotator picture including an explicit symmetry breaking term, and showed consistency with standard χPT computations.

The effective Lagrangian
The dynamical fields are matrices U (x) ∈ SU(N ). In the chiral limit the action is invariant under global SU(N ) L × SU(N ) R transformations of the fields (2.1) In this limit the leading order effective Lagrangian is given by [1]: For N ≥ 4 there are four linearly independent 1 four-derivative terms in the effective Lagrangian [1] The 4-derivative couplings in (2.3) are related to the standard ones [1] as G (i) 4 = −4L i . Since we work here in Euclidean space-time, our couplings differ in sign. 2 Note also the absence of the 4-derivative term tr( U † U ) in the above list; As explained in [12], this term can be eliminated by redefinition of the field U . The argument is reproduced for completeness in Appendix C.
For N < 4 these four operators are not all independent. One has [12] L (0) A proof of (2.9) is given in Appendix B. Accordingly, in (2.3) one can restrict the summation to i = 1, 2 for N = 2 and to i = 1, 2, 3 for N = 3. From these relations it follows that the results obtained for general N should at N = 2 be invariant under the transformation The SU(N ) × SU(N ) model for N = 2 flavors is equivalent to the O(4) non-linear sigma model [13] (with fields S i , i = 0, . . . , 3 and S 2 = 1) where and where σ a are the Pauli matrices, one obtains 4 . (2.18) These and the relations (2.10), (2.11) can serve as checks on the final results.

Perturbative expansion
Here we work in a continuum volume V = L t × L ds s , d s = d − 1. In this section we impose periodic boundary conditions (pbc) on the dynamical variables in all directions. We dimensionally regularize by adding q extra compact dimensions of size L (also with pbc) and analytically continue the resulting loop formulae to q = −2 . We define D = d + q , V D = V L q , and the aspect ratios = L t /L s ,ˆ ≡ L/L s 3 .
For the perturbative expansion we parameterize U with scalar fields ξ a (x) 4 where u is a constant matrix and where the hermitian λ-matrices are defined and some of their properties noted in Appendix A. Further A 2,eff has a perturbative expansion

24)
and where the terms A 2,2 come from the zero mode action derived in Appendix D. The total effective action has a perturbative expansion of the form and The free 2-point function is given by where the sum is over momenta p µ = 2πn µ /L µ , n µ ∈ Z and the prime on the sum means that p = 0 is omitted.

The chemical potential
The chemical potential h is introduced by the substitution: This gives an additional h-dependent part A h to the total action of the form Further writing we have giving up to the order h 2 : Note for an observable X: 14) Averaging over the zero modes, denoting where we used (E.3). So with W given by For the averages we have where [W ] is obtained from W by replacing λ 3 ij λ 3 kl by a λ a ij λ a kl . Using completeness in the form (A.12) we get and have a perturbative expansion 5 Note that for pbc Using (E.3) we get simply This has a perturbative expansion and Expanding (3.13) in a perturbative series we have at leading order (3.36) Here Z 1 is a group factor defined in (A. 16) in Appendix A where also other such factors Z i , i = 2, . . . , 8 appearing below are defined and evaluated. Further the dimensionally regularized sums I nm are formally defined by So we have at leading order This 2-loop function, the "massless sunset diagram", is calculated in detail in [14]. Secondly (3.47) (3.48)

Contribution from the 4-derivative terms
For the averages we have is obtained in Appendix F from C (i) 4 by replacing λ 3 ij λ 3 kl by a λ a ij λ a kl and averaging over the constant modes. From these expressions we obtain (3.50) One can check that F 3,1 = 0 for N = 3 when one sets G

Summary
Collecting the results together, the expansion of the susceptibility with DR is given by with and (3.57) For N = 2, 3 the relations (2.10), (2.11) are satisfied.

Renormalization of the free energy in d = 4
We first recall some results obtained in [14] for the behavior of the functions as q → 0:

59)
(3.60) where the shape functions β 1 ( ), γ i ( ) and W( ) are given in [6], and for the 2-loop function with the non-singular shape function [6]: The shape function W 1 ( ,ˆ ) occurring in (3.59) and (3.62) is not needed here (see below). Below we switch to the conventional couplings L i = −G (i) 4 /4 and express the bare couplings through the renormalized ones by By convention [1] the renormalized couplings are taken at the scale µ = M π , where M π is the mass of the charged pion. Requiring cancellation of the ∝ 1/(D − 4) terms in R 2 one obtains two relations, Finally one has (3.69) For N = 3 one should here omit the term proportional to L r 0 . Similarly, for N = 2 one should omit L r 0 and L r 3 . In addition, to use the conventional notation (stemming from the O(4) formulation), one should make the replacement L r 1 → l r 1 /4, L r 2 → l r 2 /4. This result is also invariant under the transformations corresponding to (2.10) and (2.11).
For the O(n) case one has [6]

Computation of mass gap on a periodic strip
In this section we will compute the mass gap of the 4d chiral SU(N ) × SU(N ) model on a periodic strip. We will follow the method first used in [15] and later in [7]. In the latter references the computation was done using lattice regularization. Here we will employ dimensional regularization as we did in [6]. The dynamical fields U (x) are now defined in a volume 1) with periodic boundary conditions in the D − 1 "spatial" directions, and free boundary conditions in the time direction.
Here we will only give a brief description of the computation since it follows closely that for the O(n) model [6]. We first compute the 2-point function

(4.2)
It follows that the mass gap Since C 0 (x) has a perturbative expansion of the form

equation (4.3) yields the power series
(4.5) If for x 0 → ∞: 1 , (4.7) (2) = −c 1 +c It thus suffices to compute the coefficients c (r) i with i = 0, 1 8 . The fields U (x) are parameterized as in (2.19) but now the ξ(x)-field satisfies Neumann boundary conditions [15] ∂ 0 ξ(x) = 0 for x 0 = ±T , (4.10) and periodic boundary conditions in the spatial directions. The corresponding free 2-point function is given by where where the sum goes over p µ = 2πνµ Lµ , µ = 1, . . . , D − 1 with ν µ ∈ Z, and ω p = |p| . (4.14) with ω 1 = θ 1 0 , (4.17) where . . . c denote connected parts. The interaction terms in the total action have the same form as in the previous section apart from the integration range which is now Λ, and the volume factors V D in the expressions for A  The computation now proceeds as in [6], and here we only give the final results. In lowest orderc where R(0) is dimensionally regularized [6]. So for the energy shift, first computed by Leutwyler [4], In the next orderc Finally at third order we obtain The 3rd order energy shift is given by: Defining the moment of inertia Θ through 9 For O(n) we had [6] for d = 4: and We can check (for d = 4) using (2.18) (and recalling l 1 = −g (2) 4 /4 , l 2 = −g 4 /4) and setting F 2 = 1/g 2 0 that

Renormalization of the mass gap in d = 4
The mass gap does not lead to a new renormalization condition beyond (3.66) required by the free energy considered in this paper. As discussed in [11], the reason is that they are closely related: knowing Θ determines the mass spectrum of Hamiltonian states and these determine the free energy.
In [14] we find  After introducing the renormalized couplings (3.64) one obtains 1 , (4.47) Note that the combination of the renormalized couplings is the same as one of the combinations appearing in (3.57). Again for N = 3 one should omit L r 0 , while for N = 2 the couplings L r 0 and L r 3 should be omitted, and L r 1 , L r 2 should be replaced by 4l r 1 , and 4l r 2 respectively (see Appendix F). Comparing Θ 2 with R 2 in (3.69), using the large-behavior of the shape coefficients from [11] one finds that For the susceptibility calculated in χPT for the long cylinder geometry this gives a remarkably simple result, (4.50) In the O(n) model for → ∞ one obtains L 4 s R 2 = const(n − 2)(n − 4) 2 + O (1), in contrast to the SU(N ) × SU(N ) model. It is interesting to observe that in the cases of n = 2 and n = 4 the manifold S n−1 on which the system is moving is a group manifold, U(1) and SU(2) with symmetries U(1)×U(1) and SU(2), correspondingly. While for general O(n) the expansion parameter for large is /(F 2 L 2 s ), in these special cases the expansion parameter seems to be 1/(F 2 L 2 s ) (see eq. (3.6) of ref. [11]). Eq. (4.50) is obtained assuming L s L t F 2 L 3 s . This is a high-temperature expansion for the spatially constant modes and at the same time a low-temperature expansion for the p = 0 modes. The leading term, Θ/2L s is the classical result. The second one is the leading quantum correction; it appears both for O(n) and for SU(N ) × SU(N ), and does not depend on the dynamics. Note that L 2 s χ ∝ T 2 3 = C 2 /(N 2 − 1) where C 2 is the quadratic Casimir invariant, hence in the more natural choice (N 2 − 1)L 2 s χ the curvature of the SU(N ) manifold, N (N 2 − 1)/12 appears.
A specific feature of the SU(N ) × SU(N ) case is that in the χPT result (4.50) the ∝ L s /Θ ∼ 1/(F 2 L 2 s ) term is absent, i.e. the LEC's to NNL order are hidden in the first term alone. Related to this observation, there is a strong evidence that in the SU(N ) × SU(N ) rotator approximation (describing the contribution of the spatially constant modes) there are no power-like corrections to the first two terms in (4.50) for general N (cf. [18]). For the SU(2) × SU(2) O(4) case this can be shown analytically; writing from (A.38) of [11] it follows that the correction term φ(u) decreases faster than any power of u. In fact it is extremely small already at u = 0.1; one has φ(0.1) = −5.4 × 10 −41 . For N = 3, 4 and 5 this was shown numerically [18]. A derivation of the susceptibility from a SU(N ) × SU(N ) rotator (for general N ) will be presented in a separate paper [18]. Suffice it here to say that in this scenario we have numerically shown absence of power-like corrections for N = 3, 4 and 5.
The absence of a ∼ 1/Θ 2 term in the SU(3) × SU(3) rotator approximation does however not necessarily mean that a term O F −4 L −4 s cannot be present in (4.50), since the simple rotator model requires modifications in order to match χPT at higher orders.
In (4.48) the limit → ∞ is reached exponentially fast, while in (4.49) apart from the exponentially small corrections there are ∝ 1/ corrections as well. This gives for the deviation of the susceptibility χ rot calculated for the standard rotator 11 from the χPT result χ in the NNL order  The omitted terms at this order are vanishing exponentially as → ∞. The 1/ term given above should come from the distortion of the rotator spectrum in the region of energies E 1/L s , much below the threshold for the p = 0 modes. In other words, the true rotator Hamiltonian differs from that of the standard rotator in higher order. A similar situation was found in [6] for the case of the O(n) model. The corresponding correction for the SU(N ) × SU(N ) case is discussed in [18].
Finally we make a few remarks concerning the sensitivity of the observables on the 4-derivative couplings L r i . The sensitivity of the isospin susceptibility at = 1 (hypercubic lattice) is obtained from (3.69) (observing that γ (4) (1) = 0) For a long cubic tube, 1, the sensitivity of the susceptibility and of the mass gap 11 with the standard Hamiltonian proportional to the quadratic Casimir invariant C2. .

(4.54)
Note that all coefficients change sign as varies from 1 to ∞; this feature can be used to select optimal values of for certain purposes e.g. to reduce the influence on the uncertainty of the L r i 's on determination of F .

A SU(N ) Gell-Mann matrices
The N × N Gell-Mann hermitian λ−matrices satisfy tr λ a = 0 , where f abc is totally anti-symmetric and d abc is totally symmetric and Note the identities and f abc f dbc = N δ ad , (A.8) For N = 2 , λ a = σ a , the Pauli matrices. Also for an SU(2) matrix (A.14) Note for N = 3 we have the extra identity [16] d abc d cde + d dbc d ace + d ebc d adc = 1 3 (δ ab δ de + δ ad δ be + δ ae δ bd ) .

(A.15)
A.1 Group factors appearing in the perturbative computation (A.20) B Proof of eq. (2.9) We start from the trivial identity where F is a traceless anti-hermitian matrix. By choosing For the SU(2) case this corresponds to the change of variables which is the transformation used to show the redundancy of the operator tr( S S).
We have still to show that F is indeed traceless. One has Further we can write One has where A µ and B µν are traceless hermitian matrices. From this it follows that Therefore we can conclude that the operator tr( U † U ) can be transformed away by a field redefinition.
A similar discussion to that presented above has been given by Leutwyler in eq. (11.6) and the following paragraph of his article [12].
The group volume is an irrelevant factor. Also for DR we set M [ξ(x)] = 1 , ∀x.
Now we only need Φ[ξ] near the surface x ξ(x) = 0 and we can consider an infinitesimal transformation This induces a change with t(x) obtained by solving (the argument x understood) (1 + iα a λ a + O α 2 )e ig 0 ξ b λ b = e ig 0 ξaλ a +iαat a + O α 2 , G Some relations for the O(4) couplings Some relations between different conventions for the O(4) couplings to connect with those used in ref. [6] l i = l r i +