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Register a new userThe rotator spectrum in the delta-regime of the O(n) effective field theory in 3 and 4 dimensions
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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.
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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.
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.
We solve exactly the Dyson-Schwinger equations for Yang-Mills theory in 3 and 4 dimensions. This permits us to obtain the exact correlation functions till order 2. In this way, the spectrum of the theory is straightforwardly obtained and comparison with lattice data can be accomplished. The results are in exceedingly good agreement with an error well below 1%. This extends both to 3 and 4 dimensions and varying the degree of the gauge group. These results provide a strong support to the value of the lattice computations and show once again how precise can be theoretical computations in quantum field theory.
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 OPE coefficients of the bosonic tensor model of cite{Benedetti:2019eyl} for three point functions with two fields and a bilinear with zero and non-zero spin. We find that all the OPE coefficients are real in the case of an imaginary tetrahedral coupling constant, while one of them is not real in the case of a real coupling. We also discuss the operator spectrum of the free theory based on the character decomposition of the partition function.
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