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Register a new userCasimir squared correction to the standard rotator Hamiltonian for the O($n$) sigma-model in the delta-regime
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In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1/ell,,$ for $ell=L_t/Ltoinfty$ and further showed that this deviation could 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, by Balog and Hegedus for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.
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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 $ell=L_t/Ltoinfty,.$ 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 $N=3,.$
We compute the isospin susceptibility in an effective O($n$) scalar field theory (in $d=4$ dimensions), to third order in chiral perturbation theory ($chi$PT) in the delta--regime using the quantum mechanical rotator picture. This is done in the presence of an additional coupling, involving a parameter $eta$, describing the effect of a small explicit symmetry breaking term (quark mass). For the chiral limit $eta=0$ we demonstrate consistency with our previous $chi$PT computations of the finite-volume mass gap and isospin susceptibility. For the massive case by computing the leading mass effect in the susceptibility using $chi$PT with dimensional regularization, we determine the $chi$PT expansion for $eta$ to third order. The behavior of the shape coefficients for long tube geometry obtained here might be of broader interest. The susceptibility calculated from the rotator approximation differs from the $chi$PT result in terms vanishing like $1/ell$ for $ell=L_t/L_stoinfty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.
TBA integral equations are proposed for 1-particle states in the sausage- and SS-models and their $sigma$-model limits. Combined with the ground state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear $sigma$-model and the results are compared to 3-loop perturbation theory and Monte Carlo data.
The low lying spectrum of the O(n) effective field theory is calculated in the delta-regime in 3 and 4 space-time dimensions using lattice regularization to NNL order. It allows, in particular, to determine, using numerical simulations in different spatial volumes, the pion decay constant F in QCD with 2 flavours or the spin stiffness rho for an antiferromagnet in d=2+1 dimensions.
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.
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