Isospin susceptibility in the O($n$) sigma-model in the delta-regime

TL;DR

This paper calculates the isospin susceptibility in the O(n) sigma-model within the delta-regime using chiral perturbation theory, including effects of explicit symmetry breaking and finite-volume corrections, up to third order.

Contribution

It extends previous computations by including a small explicit symmetry breaking term and analyzing finite-volume effects with higher-order chiral perturbation theory in the delta-regime.

Findings

Consistency with previous mass gap and susceptibility results in the chiral limit.

Derived the susceptibility's behavior with explicit symmetry breaking to third order.

Identified a correction to the rotator spectrum proportional to the quadratic Casimir invariant.

Abstract

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 $\ell=L_t/L_s\to\infty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.

Equations

$$H_0(h)=\frac{{\hat{L}}^2}{2\Theta }-h{\hat{L}}_{12},$$

$$Z(h;\Theta )$$

$$=z_0(u)+\frac{1}{2}{h}^2{L}_t^2{z}_1(u)+\mathrm{O}(h^4),$$

$${\mathcal{C}}_{n;l}=l(l+n-2),$$

$$\chi$$

$$=\frac{2\Theta }{n{L}_s^3}\left[1-\frac{L_t}{6\Theta }(n-2)+\frac{L_t^2}{180{\Theta }^2}(n-2)(n-4)+\dots \right].$$

$$H(h)=H_0(h)+\eta {\hat{\mathbf{S}}}_n=\frac{{\hat{L}}^2}{2\Theta }+\eta {\hat{\mathbf{S}}}_n-h{\hat{L}}_{12}.$$

$$Z(h;\Theta ,\eta )$$

$$=z_0(u)-{\eta }^2\Theta L_t{z}_2(u)+\frac{1}{2}{h}^2{L}_t^2\left[z_1(u)-{\eta }^2\Theta L_t{z}_3(u)\right]+\dots$$

$$\chi$$

$$\displaystyle\qquad\qquad-\frac{\eta^{2}L_{t}^{2}}{n(n+2)}\left[1-\frac{1}{6}(2n-5)u+\frac{1}{60}(n^{2}-12n+17)u^{2}\right]+\ldots\Bigg{\}}\,.$$

$$\Theta =F^2{L}_s^3\left[1+\frac{{\tilde{\Theta }}_1}{F^2{L}_s^2}+\frac{{\tilde{\Theta }}_2}{F^4{L}_s^4}+\dots \right],$$

$${\tilde{\Theta }}_1$$

$${\tilde{\Theta }}_2$$

$$-\frac{5(n-2)\rho }{12{\pi }^2}\log (\overline{c}{L}_s{M}_{\pi })-4\rho \left(2l_1^{\mathrm{r}}+n{l}_2^{\mathrm{r}}\right).$$

$$\ln \overline{c}=-\frac{1}{2}\left[\ln (4\pi )-{\gamma }_E+1\right]=-1.476904292,$$

$$c_w=0.0986829798,$$

$$\begin{array}{rl}\chi & =\frac{2}{n}{F}^2[1+\frac{1}{F^2{L}_s^2}\left({\tilde{\Theta }}_1-\frac{\ell }{6}(n-2)\right)\\ & +\frac{1}{F^4{L}_s^4}\left({\tilde{\Theta }}_2+\frac{{\ell }^2}{180}(n-2)(n-4)\right)+\dots ].\end{array}$$

$$\chi =\frac{2}{n}{F}^2\left[1+\frac{{\tilde{R}}_1}{F^2{L}_s^2}+\frac{{\tilde{R}}_2}{F^4{L}_s^4}+\dots \right],$$

$${\tilde{R}}_1=-\frac{(n-2)}{4\pi }\left({\gamma }_2-1\right),$$

$$\begin{array}{rl}{\tilde{R}}_2& =-\frac{(n-2)}{16{\pi }^2}\left[{\left({\gamma }_2-1\right)}^2+8\pi \left({\gamma }_2-1\right){\beta }_1+2\overline{\mathcal{W}}-\frac{(n-2)}{\ell }{\gamma }_3\right]\\ & +\frac{(n-2)}{24{\pi }^2}\left[\frac{3n-7}{\ell }-5\left({\gamma }_1-\frac{1}{2}\right)\right]\log \left(\overline{c}L{M}_{\pi }\right)\\ & -2\left(2l_1^{\mathrm{r}}+n{l}_2^{\mathrm{r}}\right)\left({\gamma }_1-\frac{1}{2}\right)+4\left[(n-1)l_1^{\mathrm{r}}+l_2^{\mathrm{r}}\right]\frac{1}{\ell }.\end{array}$$

$${\tilde{R}}_1+\frac{\ell }{6}(n-2)\simeq {\tilde{\Theta }}_1,$$

$${\tilde{R}}_2-\frac{{\ell }^2}{180}(n-2)(n-4)\simeq {\tilde{\Theta }}_2.$$

$$A=A_0+A_M,$$

$$A_0$$

$$A_M$$

$$A_2$$

$$A_{M;2}$$

$$A_4=\sum _{i=2,3}\frac{g_4^{(i)}}{4}{A}_4^{(i)},$$

$$A_{M;4}=\sum _{i=1,2}{L}_i{A}_{M;4}^{(i)},$$

$$A_{M;4}^{(1)}$$

$$A_{M;4}^{(2)}$$

$$\mathbf{S}\to \mathbf{S}+ϵ\delta (\mathbf{S}),$$

$$\delta (S_n)$$

$$\delta (S_i)$$

$$\delta \left(A_2+A_{M;2}\right)=\frac{1}{M^2{g}_0^2}\left[-A_{M;4}^{(1)}+2A_{M;4}^{(2)}\right].$$

Full text

1 Introduction

In a previous paper Niedermayer:2016yll 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 dimensional regularization (DR) in a general $d$–dimensional asymmetric $L_{s}^{d-1}\times L_{t}$ volume with periodic boundary conditions in all directions. This allowed us, for $d=4$, to establish two independent relations among the 4–derivative couplings appearing in the effective Langrangians and in turn this enables 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 Niedermayer:2010mx using lattice regularization to a result involving parameters of the dimensionally regularized effective theory, and we verified this result by a direct computation Niedermayer:2016yll . The proposal to measure the low-lying stable masses in the delta–regime to constrain some of the low energy constants in the effective chiral Langrangian describing low energy pion dynamics of QCD, was first made by Hasenfratz Hasenfratz:2009mp . For two flavors of massless quarks, the relevant $\chi$PT has SU(2)$\times$SU(2)$\simeq$O(4) symmetry.

The so called $\delta$–regime referred to above, is where the system is in a periodic spatial box of sides $L_{s}$ with $L_{t}\gg L_{s}$ and $m_{\pi}L_{s}$ is small (i.e. small or zero quark mass) whereas $F_{\pi}L_{s}$, ($F_{\pi}$ the pion decay constant) is large. This regime was first analyzed in a pioneering paper by Leutwyler Leutwyler:1987ak . He showed that the low lying dynamics in this regime was described by a quantum rotator for the spatially constant modes.

In that paper Leutwyler also obtained the leading order effects of the quark mass on the low-lying spectrum for O(4). The extension to next-to-leading (NLO) order was presented by Weingart Weingart:2010yv ; Weingart:2010zz . There followed a period of low activity in this field, but recently Matzelle and Tiburzi have, in an interesting paper Matzelle:2015lqk , studied the effect of small symmetry breaking in the QM rotator picture ($N_{\mathrm{f}}=2$), and extended the results to small non-zero temperatures.

The rotator Hamiltonian given in (5) has two parameters, the moment of inertia $\Theta$ and a parameter $\eta$ describing the explicit symmetry breaking O($n$) to O($n-1$). These parameters are themselves functions of $F,L_{s},M$ where $F$ the pion decay constant in the limit of vanishing quark mass, and $M$ is the mass acquired by the Goldstone bosons in leading order $\chi$PT due to the explicit symmetry breaking.

Within this theoretical framework, in this paper we communicate computations which provide, in our opinion, a non-trivial check on our previous rather technical computations Niedermayer:2016yll , and also gives the NNLO (next-to next-to leading order) $\chi$PT expansion for the parameter $\eta$.

In sect. 2 we obtain the expression (7) for the susceptibility for $u\equiv L_{t}/(2\Theta)$ small, under the assumption that in the $\delta$–regime the low lying spectrum is that of the quantum rotator. These computations are themselves rather involved and so we defer many details to Appendix A. There we also review the leading order effects of a small symmetry breaking parameter on the low lying spectrum for general $n$, reproducing Leutwyler’s result Leutwyler:1987ak for $n=4$.

In sub-section 3.1 we first consider the symmetric case $M=0=\eta$. By inserting our expression for $\Theta$ obtained in Niedermayer:2016yll into (4), we obtain the susceptibility for small $u$ as a function of $F,L_{s},\ell\equiv L_{t}/L_{s}$. This can then be compared to the direct $\chi$PT computation of the susceptibility Niedermayer:2016yll in the $\epsilon$–regime for $\ell\gg 1$. The comparison requires knowledge of the large $\ell$–behavior of shape functions and the sunset integral appearing in the latter; these are discussed in Appendix B. The agreement of the two computations provides the non-trivial check referred to above.

In sub-section 3.2 we compute the susceptibility for small quark masses in the framework of $\chi$PT. Subsequently a comparison to the result (7) from the rotator computation determines the term in $\eta$ proportional to $M^{2}$ as a function of $FL_{s}$ to NNLO. Various technical aspects are deferred to Appendices C and D.

In section 4 we investigate how the isospin susceptibility calculated in the $\delta$-regime of $\chi$PT approaches for increasing $\ell$ the result of the rotator approximation. This turns out to be $\sim(FL_{s})^{-4}\ell^{-1}$, and we determine the distortion of the standard rotator spectrum needed to describe this deviation.

2 The free energy of the O($n$)

rotator with isospin chemical potential

We first consider the case with unbroken symmetry and subsequently compute the leading order effect due to the presence of a small O($n$) symmetry breaking mass term.

2.1 Symmetric case

The Hamiltonian of the O($n$) quantum rotator with a chemical potential coupled to the generator $L_{12}$ of rotations in the $12$–plane is

[TABLE]

where $\Theta$ is the moment of inertia; to lowest order $\chi$PT one has $\Theta\simeq F^{2}L_{s}^{3}$. The energy levels and the corresponding multiplicities $g^{(n)}_{lm}$ are found in App. A.

The corresponding partition function for Euclidean time extent $L_{t}$ and its expansion to $\mathrm{O}\left(h^{2}\right)$ is given by 111The notation $l$ is used below because of the analogy with the rotator, here it stands for the isospin.

[TABLE]

where $u=L_{t}/(2\Theta)$ and

[TABLE]

is the eigenvalue of the quadratic Casimir for isospin $l$.

The coefficients $z_{0}(u)$, $z_{1}(u)$ and their expansions for small $u$ are given in App. A. Using these results, the isospin susceptibility of the O($n$) rotator for the unbroken case is given by

[TABLE]

2.2 Rotator in external field

Consider the effective Lagrangian with a term, breaking the O($n$) symmetry down to O($n-1$). In the delta-regime this corresponds to an effective O($n$) rotator in an external “magnetic field”, given by the Hamiltonian (including the isospin chemical potential)

[TABLE]

Here $\hat{\mathbf{S}}_{n}$ is the $n$-th component of the operator defined as $\hat{\mathbf{S}}\psi(\mathbf{S})=\mathbf{S}\psi(\mathbf{S})$, where $\mathbf{S}^{2}=1$. Its matrix elements between energy eigenstates of the Hamiltonian with $\eta=0$ are given in eq. (68).

In the effective theory to leading order $\eta\sim M^{2}\Theta\sim M^{2}F^{2}L_{s}^{3}$. The spectrum and the multiplicities are described in A.1.2

Expanding the partition function in $h$ and $\eta$ to NL order one obtains (cf. A.3.2)

[TABLE]

The expansion of $z_{i}(u)$ for small $u$ is given in Appendix A. The final result for the susceptibility of the rotator in an external magnetic field is

[TABLE]

3 Perturbative computation of the free energy

The expression (7) for the susceptibility was obtained under the assumption that the low lying spectrum in the $\delta$–regime is that of the quantum rotator. In the next sub-section we first consider the symmetric case and show the consistency of (4) with our previous computation of the susceptibility in Niedermayer:2016yll in the $\epsilon$–expansion.

In the $\epsilon$–expansion one considers a 4d box of fixed shape, in the case of an $L_{s}^{3}\times L_{t}$ volume a fixed aspect ratio $\ell=L_{t}/L_{s}$. After separating the constant mode (the direction of the average magnetization over the 4d volume), integration over the $p\neq 0$ modes result in an expansion in powers of $(F^{2}L^{2})^{-1}$, where by convention $L=(L_{s}^{3}L_{t})^{1/4}$. (For simplicity, here we discuss only the case without explicit symmetry breaking.) The coefficients of the expansion contain the shape coefficients depending on $\ell$. Usually one assumes $\ell\sim 1$. For a long tube, $\ell\gg 1$, the spatially constant modes, $p_{0}\neq 0$, $\mathbf{p}=0$, become soft which means that one needs a larger box size $L$ to have a reasonable $\epsilon$–expansion. This is manifested by the fact that the shape coefficients are increasing functions of the aspect ratio $\ell$.

In the $\delta$–expansion one separates the dynamics of the spatially constant modes described by the O($n$) rotator, with the energies $\mathrm{O}\left(F^{-2}L_{s}^{-3}\right)$. For $L_{t}\gg L_{s}/(2\pi)$ the $\mathbf{p}\neq 0$ modes are exponentially suppressed in the partition function, hence the regions of validity of the $\epsilon$– and $\delta$–expansions overlap in this case. The $\mathbf{p}\neq 0$ states are responsible for exponentially small corrections $\sim\exp(-c\ell)$. However, a possible distortion of the higher lying rotator spectrum can still produce a correction of type $\sim\ell^{-\nu}$ to the result obtained by using the “standard” rotator spectrum $\propto\mathcal{C}_{n;l}$. This is discussed in section 4.

The transition to the $\delta$-regime involves relations between the 4d shape coefficients for $\ell\gg 1$ and the 3d shape coefficients in a cubic box, as given e.g. in (132) and (136)-(138).

In subsection 3.2 we compute the susceptibility for small quark masses (i.e. small $M$) in the framework of $\chi$PT and subsequently use the comparison to the result (7) to determine the coefficient of $M^{2}$ in $\eta$ as function of $FL_{s}$ to NNLO.

3.1 The symmetric case

From the mass gap calculation in Niedermayer:2016yll we obtained the moment of inertia:

[TABLE]

with

[TABLE]

Here $\beta_{1}^{(3)}\equiv\beta_{1}^{(3)}(1)$ is a shape function corresponding to a cubic 3d volume, $\rho=8\pi^{2}\beta_{2}^{(3)}(1)$ (136); for notations undefined here we refer the reader to Hasenfratz:1989pk and Niedermayer:2016yll . Further $l_{i}^{\mathrm{r}}$ are renormalized low energy constants (LEC) of Gasser and Leutwyler Gas84 ;

[TABLE]

and the constant $c_{w}$ was determined in Niedermayer:2016ilf as

[TABLE]

differing slightly from the original computation of Hasenfratz Hasenfratz:2009mp .

Inserting (8) into (4) one obtains

[TABLE]

On the other hand the susceptibility calculated by $\chi$PT is given by Niedermayer:2016yll

[TABLE]

with

[TABLE]

and

[TABLE]

The next-to-leading (NLO) and NNLO terms in (13) and (14) agree for large $\ell$ provided that

[TABLE]

These involve relations between the 4d shape functions in the long cylinder ($\ell\gg 1$) to the shape functions for a 3d cube. Derivations of the relations required are supplied in App. B.

Firstly at NLO, using the large $\ell$ behavior of $\gamma_{2}$ given in (137), one indeed verifies (17).

Next, the three relations (132), (136), (138) for the 1-loop shape functions together with a fourth relation (153) referring to the 2-loop sunset diagram, ensure the validity of the NNLO consistency condition (18). This agreement provides, in our opinion, a highly non-trivial check on the various rather technical computations.

3.2 Perturbative computation of the isospin

susceptibility for O($n$) with a mass term using dimensional regularization

Here we restrict attention to a $L_{s}^{d-1}\times L_{t}\,,d=4$ volume with periodic boundary conditions in all directions. For perturbative computations we employ dimensional regularization and add further $D-4$ dimensions of size $\widehat{L}=L_{s}$. In ref. Niedermayer:2016yll we did not fix the value of $\hat{\ell}\equiv\widehat{L}/L_{s}$; confirming independence of physical quantities on $\hat{\ell}$ serves as an additional useful check of the computations.

3.2.1 The effective action with mass terms

The effective action is given by

[TABLE]

where $A_{M}$ vanishes in the chiral limit $M=0$, with

[TABLE]

The lowest order actions are ($\mathbf{S}(x)^{2}=1$)

[TABLE]

There are two independent $4$–derivative terms in the massless case:

[TABLE]

with the $A_{4}^{(i)}$ given in App. C.1.

Terms involving the mass parameter in the next order are

[TABLE]

with

[TABLE]

Actually one of the terms above is redundant for physical quantities. To see this consider the infinitesimal change of variables (preserving $\mathbf{S}^{2}=1$):

[TABLE]

with

[TABLE]

Under this change the leading $\chi$PT action changes according to

[TABLE]

Rearranging $A_{M;4}$

[TABLE]

we see that physical quantities should just depend on the combination $2L_{1}+L_{2}$; this will serve as a check on the computed contributions coming from the $A_{M;4}^{(i)}$.

3.2.2 The chemical potential with mass terms

The chemical potential $h$ is introduced by

[TABLE]

where $(QS)_{1}=iS_{2}$, $(QS)_{2}=-iS_{1}$, $(QS)_{3}=\ldots=0$. This gives the $h$-dependent part of the action:

[TABLE]

where the massless part has contributions

[TABLE]

Here we only consider terms up to and including order $h^{2}$:

[TABLE]

The expressions for $B_{2}\,,C_{2}\,,B_{4}^{(i)}\,,C_{4}^{(i)}$ are given in App. C.1.

To lowest order for the mass term

[TABLE]

with

[TABLE]

where $j_{\mu}(x)$ is defined in (156).

The $h$–dependent part of the free energy $f_{h}$ is given by ($V_{D}=L_{t}L_{s}^{D-1}$):

[TABLE]

where, $\langle...\rangle$ denotes correlation functions with the massless action, and we have only kept terms up to order $M^{4}$ and $h^{2}$. So for the free energy we have

[TABLE]

Noting that

[TABLE]

and that correlation functions with an odd number of spins $S$ vanish:

[TABLE]

many contributions drop out so that to the order we are considering we have

[TABLE]

where “$\mathrm{ho}$” stands for higher order terms in the chiral expansion. In particular we drop terms $\mathrm{O}(M^{4}g_{0}^{0})$.

The terms appearing in the averages above are not O($n$) invariant, and so before starting the perturbative computations we set $\mathbf{S}(x)=\Omega\mathbf{R}(x)$ and average over the rotations $\Omega$. The resulting expressions are given in App. C.2.

The free energy has a small $h$ expansion

[TABLE]

Using the results in App. C.2 the uniform susceptibility (at $h=0$) is given by

[TABLE]

where $\chi_{0}$ is the susceptibility for $M=0$ which we computed previously Niedermayer:2016yll :

[TABLE]

where $W$ is defined in (164) and $\langle C_{4}^{(i)}\rangle$ in (165) and (166).

Next, collecting the terms in App. C.2,

[TABLE]

where $P_{1},P_{2},P_{3},P_{4}$ are given in (170),(174),(C.2), (183) respectively.

Now we can begin with the perturbative computations proceeding as usual by first separating the zero mode and then changing to $\vec{\pi}$ variables according to $\mathbf{R}=(g_{0}\vec{\pi},\sqrt{1-g_{0}^{2}\vec{\pi}^{2}})$. Details of the perturbative computations to NNLO are given in App. C.3.

Summing the terms (for $d=4$) one obtains for $\chi_{0}$ the result (14), (note $F^{2}=1/g_{0}^{2}\,$).

For $\chi_{1}$ it is shown that it has a perturbative expansion of the form

[TABLE]

Here the NLO coefficient is

[TABLE]

Using the behavior of the shape functions $\beta_{1}$ and $\gamma_{2}$ given in (132) and (136) we have for large $\ell$:

[TABLE]

At the NNLO we first show that in order to cancel the $1/(D-4)$ pole terms we must have

[TABLE]

This agrees with the result of Gasser and Leutwyler Gas84 for $n=4$ if we set $L_{1}=-l_{3}\,,L_{2}=0$. Then

[TABLE]

Using the expansions (128) (for $s=2\,,d=4$) and (138) we have for large $\ell$:

[TABLE]

Agreement of the rotator result (7) with the perturbative result for large $\ell$ above requires that the parameter $\eta$ in the rotator Hamiltonian (5) has a $\chi$PT expansion of the form:

[TABLE]

with

[TABLE]

4 Distortion of the rotator spectrum

It is expected that at some higher order the standard rotator spectrum $E_{l}\propto\mathcal{C}_{n;l}$ will be modified. It is interesting that by comparing the NNLO results for the isospin susceptibility obtained from $\chi$PT, eq. (14), and $\chi_{\text{rot}}$ given by (4), one can answer this question under reasonable assumptions.

The two results in NNLO differ by $\propto 1/\ell$ terms for $\ell\gg 1$,

[TABLE]

where

[TABLE]

The omitted terms in (57) at the given order are vanishing exponentially fast for $\ell\to\infty$.222Note that the $\mathbf{p}\neq 0$ modes yield such contributions.

The deviation (57) means that the spectrum of the spatially constant modes, the true rotator spectrum, is not given exactly by the standard rotator spectrum $E_{l}=\mathcal{C}_{n;l}/(2\Theta)$, it is distorted already at energies $E_{l}\ll 1/L_{s}$.

Assuming that the distortion has the form

[TABLE]

one can calculate the leading correction to the susceptibility $\chi_{\text{rot}}$ of the standard rotator.

One has333Note that $w_{1}(l)=w_{0}(l)\mathcal{C}_{n;l}/d_{n}$ where $d_{n}=n(n-1)/2=\mathrm{dim}(\mathrm{O}(n))$.

[TABLE]

From (82) the leading $u\to 0$ behavior of the partition function is $z_{0}(u)\propto u^{-a}$ with $a=(n-1)/2$. This gives for the leading contribution

[TABLE]

(The omitted terms are higher order in $1/(F^{2}L_{s}^{2})$.) The observed deviation (57) requires $\kappa=2$ and $\alpha\propto 1/(FL_{s})^{8}$. Finally, for the coefficient of the distortion one obtains

[TABLE]

The distortion becomes comparable with the leading term at the isospin quantum number $l\sim(FL_{s})^{3}\gg 1$.

Note that the NNLO calculation for the mass gap means determining $L_{s}m_{1}$ up to (and including) the $(FL_{s})^{-6}$ terms. On the other side from eqs. (59), (62) the obtained correction is $\sim(FL_{s})^{-8}$, i.e. a NNNLO term. The reason is that the typical values of the Casimir invariant in the partition function are $\mathcal{C}_{n;l}\sim F^{2}L_{s}^{2}/\ell$. As a consequence the distortion (59) changes the susceptibility by $(FL_{s})^{-4}/\ell$, i.e. in NNL order.

Appendix A The effective O($n$) rotator

A.1 Spectrum and multiplicities

A.1.1 Symmetric case

The energy levels of the Hamiltonian (1) are

[TABLE]

Here $m$ is the quantum number associated with rotation in the 12-plane, with values $m=-l,\ldots,l$.

The subspace of states corresponding to a given energy level can be decomposed according to their transformation properties under the O($n-1$) transformations affecting the first $n-1$ components.

The multiplicity of the energy level (63), $g_{lm}^{(n)}$ satisfies the recursion relation

[TABLE]

For $n=3,4$ one has (assuming $|m|\leq l$)

[TABLE]

The solution of (64), (65) for general $n$ is given by

[TABLE]

The total multiplicity of states with a given $l$ is

[TABLE]

A.1.2 Rotator in external field

Consider now the O($n$) rotator in a small external magnetic field, with Hamiltonian given in (5). For simplicity we take here $h=0$ since its effect is simply adding $-hm$ to the energy levels, as in (63).

The external field splits the energy levels $E^{(n)}(l)=\mathcal{C}_{n;l}/(2\Theta)$ corresponding to a given O($n$) isospin $l$ into levels characterized by the O($n-1$) isospin $k$, where $k=0,\ldots,l$.

To determine the expansion of the energy levels for small breaking parameter $\eta$ one needs the transition matrix elements between the corresponding eigenstates 444unit normalized $\langle l,k,\alpha|l,k,\alpha\rangle=1$. These are

[TABLE]

Here $\alpha$ denotes the remaining quantum numbers besides $k$ characterizing an O($n-1$) eigenvector. (For $n=3$ one has $k=|m|$ and $\alpha$ is the sign of $m$.) The leading order PT gives

[TABLE]

where

[TABLE]

and

[TABLE]

Accordingly the energy of the ground state becomes

[TABLE]

while the $n$–plet of O($n$) is split into a singlet and $(n-1)$–plet under O($n-1$):

[TABLE]

The lower one is the $(n-1)$–plet. For $n=4$ one obtains for the mass gap $E(1,1)-E(0,0)=3/(2\Theta)[1+\eta^{2}\Theta^{2}/15+\ldots]$, in agreement with Leutwyler:1987ak ; Weingart:2010yv .

A useful check for the transition matrix element and the multiplicity is to calculate the trace of $\hat{\mathbf{S}}_{n}^{2}$ in the O($n$) invariant subspace with given $l$ which should give $g^{(n)}_{l}/n$. Written out explicitly one has indeed

[TABLE]

A.2 The transition matrix elements

The squared angular momentum operator in $n$ dimensions, $\hat{L}^{2}$ is given (up to the sign) by the angular Laplacian $\Delta_{S^{n-1}}$ on the unit sphere $S^{n-1}$. The eigenfunctions of $\hat{L}^{2}$ corresponding to the eigenvalue $\mathcal{C}_{n;l}$ can be written as

[TABLE]

where $g_{k}(\phi_{1},\ldots,\phi_{n-2})$ is an eigenfunction of the $n-1$ dimensional squared angular momentum operator $\sum_{i=1}^{n-1}\hat{L}_{i}^{2}$ with eigenvalue $k(k+n-3)$ and $f_{lk}(\theta)$ satisfies the differential equation

[TABLE]

where $x=\cos\theta$ and $f_{lk}(\theta)=y(\cos\theta)\,.$

For $k=0$ the solution is given by the O($n$) Legendre polynomials

[TABLE]

The coefficient is chosen to satisfy the normalization condition $P_{nl0}(1)=1$.

For general $k$ the solution of the differential equation (76) is given by the associated O($n$) Legendre functions:

[TABLE]

Two of these functions corresponding to different values of $l$ (and same $n,k$) are orthogonal with the weight

[TABLE]

For convenience we introduce the notation

[TABLE]

The transition matrix element (68) is then given by

[TABLE]

A.3 Partition function of the O($n$) rotator

A.3.1 Symmetric case

Here we consider the partition function (2) of the O($n$) rotator without an external magnetic field. Expanding it in the chemical potential, the expansion coefficients are given by

[TABLE]

and

[TABLE]

Here the weights $w_{i}(l)$ are (cf. (67))

[TABLE]

The coefficients appearing in (82), (83) can be determined numerically. The sum converges fast enough to separate the different powers of $u$ and then obtain the corresponding polynomials in $n$, “beyond a reasonable doubt”. An exact calculation of the expansions is given in Appendix A.4.

A.3.2 Rotator in external field

For calculating $z_{i}(u)$ to order $\eta^{2}$ and $h^{2}$ one also needs the weights

[TABLE]

and

[TABLE]

With these one obtains

[TABLE]

and

[TABLE]

For the partition function these give

[TABLE]

where $r_{i}(u)=z_{i}(u)/z_{0}(u)$.

Finally, one obtains for the corresponding susceptibility

[TABLE]

from which (7) follows.

A.4 Expansion of $z_{i}(u)$ for small $u$

Consider the sum

[TABLE]

A useful representation to obtain the behavior as $u\to 0$ is

[TABLE]

where $\sigma>(\nu+1)/2$ and $\zeta(s)$ is the Riemann zeta-function

[TABLE]

$\zeta(s)$ has a pole at $s=1$ with residue 1. Also we will need

[TABLE]

where $B_{n}$ are the Bernoulli numbers

[TABLE]

We will use the convention $B_{1}=1/2$ to have $\zeta(0)=-B_{1}$.

Also we note the Gamma function $\Gamma(t)$ has poles at $t=-n\,,\quad n=0,1,2,...$ with residue

[TABLE]

Shifting the integration contour to the left we pick up the residues of the poles, the first one at $t=(\nu+1)/2$ from the $\zeta$ function and then poles at $t=0,-1,...$ from the Gamma function. In this way we get (with the convention $B_{1}=1/2$) for small $u$

[TABLE]

Explicitly

[TABLE]

For even $\nu$ the function $f_{\nu}(u)$ is related to the Jacobi theta-function and its derivatives. In particular

[TABLE]

In this case there are no power corrections from the sum in (99), except the $-1/2$ term for $k=\nu=0$.

A.4.1 Even $n$ values

Consider (84) first for general integer $n$ introducing the variable $q=l+n/2-1$ and

[TABLE]

It is even/odd according to the parity of $n$, $\tilde{w}_{0}(-q)=(-1)^{n}\tilde{w}_{0}(q)$. Expanding in powers of $q$ one has

[TABLE]

where $\left\lfloor{x}\right\rfloor=\operatorname{floor}(x)$, with the leading coefficients

[TABLE]

For the case when $n$ is even one has

[TABLE]

Here we extended the summation range from $k=n/2-1,\ldots,\infty$ to $k=1,\ldots,\infty$ observing that $\tilde{w}_{0}(k)=0$ for $k=1,2,\ldots,n/2-2$. Finally, inserting (99) and (104) we obtain the expansion (82). The calculation goes the same way for $z_{i}(u)$, $i=1,2,3$ yielding the results stated in (83), (88) and (89).

It is interesting to note that the properties listed above for $\tilde{w}_{0}(q)$ are essential for having a proper power series for $z_{i}(u)$, otherwise one would obtain in the small–$u$ expansion a mixture of integer and half-integer powers.

A.4.2 Odd $n$ values

Define

[TABLE]

One has

[TABLE]

In this case

[TABLE]

This yields the same analytic form (82) as obtained for even $n$ values.

Appendix B Shape coefficients for long 4d tube

In this appendix we calculate the behavior of the $1$– and $2$–loop shape functions (appearing in our perturbative computations) in $d=4$ dimensions for a long “tube”, $L_{s}^{3}\times L_{t}$ with $\ell\equiv L_{t}/L_{s}\gg 1\,$. They are related to the shape functions defined for a $3$–dimensional cube.

We first recall some useful relations involving the Jacobi theta–function and some properties of the free massless propagator in an asymmetric periodic volume.

B.1 Some properties Jacobi theta-function

The Jacobi theta-function is defined by

[TABLE]

The first sum above converges quickly for $u\geq 1$ while the second for $0<u\leq 1$. For small and large $u$ it is given by

[TABLE]

Defining

[TABLE]

one has

[TABLE]

[TABLE]

and

[TABLE]

where $\ddot{S}(u,z)=\partial_{z}^{2}S(u,z)$ and $S^{\prime}(u,z)=\partial_{u}S(u,z)$.

Some relations which are also used in the following:

[TABLE]

since

[TABLE]

In particular

[TABLE]

For $D\sim 4$, $a=3/2$ one obtains a pole term

[TABLE]

B.2 Some properties of the free massless propagator

In this paper we employ dimensional regularization and add $q=D-4$ extra dimensions of length $\widehat{L}=L_{s}\,.$ We introduce $L_{0}=L_{t}$ and $L_{\mu}=L_{s}\,,\,\mu\geq 1$ and the volume $V_{D}=\overline{V}_{D}L_{t}$ , $\overline{V}_{D}=L_{s}^{D-1}\,.$

The massless propagator with periodic boundary conditions in all directions is given by

[TABLE]

where the sum is over momenta $p=2\pi(n_{0}/L_{0},\dots,n_{D-1}/L_{D-1})\,,\,\,n_{k}\in{\mathbb{Z}}\,,$ and the prime on the sum means that the zero momentum is omitted: $\sum^{\prime}_{p}=\sum_{p\neq 0}\,$.

For $G(x)$ and $\ddot{G}(x)$ we have the representations Niedermayer:2016ilf (3.55)

[TABLE]

and

[TABLE]

To study the large $\ell$ behavior it is convenient to separate the 1d propagator with periodic b.c. from $G(x)$:

[TABLE]

where

[TABLE]

with the 1d propagator with pbc on the interval $z\in[-1/2,1/2]$,

[TABLE]

Next

[TABLE]

with (see eq. (5.8) in Niedermayer:2010mx )

[TABLE]

The function $R(x)$ is defined in (126) for all $x\in{\mathbb{R}}^{D}\backslash 0\,$; in particular for $|x_{0}|\to\infty$ the function $R(x)$ falls exponentially. The singularity at $x=0$ is regularized dimensionally through an alternative representation in terms of the Jacobi theta function $S(u,z)$:

[TABLE]

B.3 1-loop shape functions

We start with the $1$–loop functions555Note $\beta_{s}$ is only defined for integer values of $s$. $\alpha_{s}^{(d)}(\ell)\,,\beta_{s}(\ell)\,,\gamma_{s}(\ell)$; for notations undefined here we again refer the reader to Hasenfratz:1989pk and Niedermayer:2016yll .

Neglecting terms decreasing exponentially fast with $\ell$ one has

[TABLE]

The pole contributions at $s=0$ and $s=1/2$ cancel and one obtains in the corresponding limits

[TABLE]

[TABLE]

Note that $\alpha_{s-1/2}^{(d-1)}(1)=\alpha_{d/2-s}^{(d-1)}(1)$.

Now

[TABLE]

From this and (128) we get for $d=4$, $k=1$:

[TABLE]

This relation holds numerically extremely well already at $\ell=4$ where it gives $-0.10754837390\simeq-0.10754837389$, illustrating the fact that the approach to the $\ell=\infty$ limit is exponentially fast. The same comment will apply to similar relations obtained in the following.

For $\gamma_{s}(\ell)$ we obtain

[TABLE]

Note that $\alpha_{s-3/2}^{(d-1)}(1)=\alpha_{d/2-s+1}^{(d-1)}(1)$. For $s=1$ this gives

[TABLE]

and for $s>3/2$

[TABLE]

Note since $\zeta(0)=-1/2$ this also reproduces (134) for $s=1$.

For $d=4$ the above relations imply for $s=1$:

[TABLE]

for $s=2$:

[TABLE]

and for $s=3$:

[TABLE]

B.4 The sunset diagram for large $\ell$

The sunset diagram for the susceptibility is (cf. Niedermayer:2010mx (4.1) and (4.36))

[TABLE]

here we have reinstated $\hat{\ell}$, but since we are working with $\hat{\ell}=1$ we have $\mathcal{V}_{D}=\ell\hat{\ell}^{D-4}=\ell$.

For the analogous diagram in the infinite strip one had (cf. Niedermayer:2010mx (5.11) and (5.61))

[TABLE]

Further, for $\ell\gg 1$ one has up to exponentially small corrections (cf. Niedermayer:2010mx (5.20))

[TABLE]

Since $\mathcal{R}(x)\simeq R(x)$ for $\ell\gg 1$ we have

[TABLE]

Writing $G(x)=G_{1}(x)+\mathcal{R}(x)$ we have (noting $\int\mathrm{d}\mathbf{x}\,R(x)=0$)

[TABLE]

The first term here is

[TABLE]

B.4.1 Calculating $\Delta\Psi_{1}$

Since $\Delta\Psi_{1a}$ is regular at $D=4$ we can calculate it in 4-dimensions. Using (114) one has

[TABLE]

From Niedermayer:2010mx (3.74),(3.30) one has

[TABLE]

Combining the results we get

[TABLE]

B.4.2 Calculating $\Delta\Psi_{2}$

[TABLE]

We have

[TABLE]

The second integral cancels with $\Delta\Psi_{2b}$ in (148).

For $\ell^{2}(1/u+1/v)\gg 1$ one has

[TABLE]

Integrating over $z_{0}$ one obtains 666 Note $\int_{0}^{\infty}\mathrm{d}u\mathrm{d}v\,f(u+v)g(u,v)=\frac{1}{2}\int_{0}^{\infty}\mathrm{d}t\,f(t)\int_{-t}^{t}\mathrm{d}\eta\,g((t+\eta)/2,(t-\eta)/2)\,.$ The integral over $\eta$ in our case gives a power of $t$.

[TABLE]

Finally using (131) one gets

[TABLE]

Comparing this with (142) one obtains the desired relation

[TABLE]

The lhs converges exponentially; e.g. for $\ell=4$ it is evaluated as $0.0986829793$ which agrees to 8 significant figures with (12). In fact already at $\ell=2$ the lhs is very close to $c_{w}$ as one can check using the result eq.(4.45) in Niedermayer:2010mx .

Appendix C Details of the perturbative perturbative

computation in subsect. 3.1

C.1 Contributions to the action

The four derivative terms in (24) are given by ($\int_{x}\dots=\int_{V_{D}}\mathrm{d}x\dots$):

[TABLE]

Terms appearing in (35) are given by the following:

[TABLE]

For the operator 2:

[TABLE]

and for the operator 3:

[TABLE]

C.2 Averages over rotations

In this section we set $\mathbf{S}(x)=\Omega\mathbf{R}(x)$ in the correlation functions in (3.2.2) and average over the rotations.

For the mass independent terms we have

[TABLE]

and (using (264)):

[TABLE]

where

[TABLE]

For the 4-derivative terms:

[TABLE]

Turning now to the mass terms we get

[TABLE]

and

[TABLE]

where

[TABLE]

Using (263):

[TABLE]

where

[TABLE]

Next, using (266):

[TABLE]

where

[TABLE]

By partial integration this simplifies to

[TABLE]

where $W$ is given in (164) and

[TABLE]

Next

[TABLE]

where

[TABLE]

and

[TABLE]

where

[TABLE]

Proceeding to the terms $\langle C_{4}^{(i)}A_{M;2}^{2}\rangle\,,$ first

[TABLE]

with

[TABLE]

and

[TABLE]

where

[TABLE]

So

[TABLE]

Similarly

[TABLE]

with

[TABLE]

So

[TABLE]

Together we have simply

[TABLE]

The next two averages in (3.2.2) to be taken into account are

[TABLE]

and

[TABLE]

With similar considerations we can show that the last 4 terms in (3.2.2) can be neglected to the order of interest:

[TABLE]

C.3 Perturbative computations

After separating the zero mode and changing to $\vec{\pi}$ variables according to $\mathbf{R}=(g_{0}\vec{\pi},\sqrt{1-g_{0}^{2}\vec{\pi}^{2}})$ the effective action is given by

[TABLE]

where with DR we have dropped the measure term, and

[TABLE]

The effective action has a perturbative expansion

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

For the details of the perturbative computation of $\chi_{0}$ we refer the reader to Niedermayer:2016yll .

$P_{j}$ have perturbative expansions (to the order we need)

[TABLE]

The expansion of $\chi_{1}$ is then

[TABLE]

with

[TABLE]

Here $R_{1}\,,R_{2}^{(a)},R_{2}^{(b)}$ appear in the perturbative expansions of $\langle W\rangle$ and $\langle C_{4}\rangle$ Niedermayer:2016yll :

[TABLE]

In the following we will need the expansions

[TABLE]

C.3.1 NL order

From (225) we have

[TABLE]

Here $\overline{I}_{10}$ is a particular case of the dimensionally regularized sums

[TABLE]

which are discussed in Niedermayer:2010mx .

There are no contributions from $P_{2},\overline{P}_{3}$ to the NL order:

[TABLE]

So

[TABLE]

Recalling for $D=4$:

[TABLE]

we obtain (49).

C.3.2 NNL order

First we have

[TABLE]

with

[TABLE]

So

[TABLE]

Next

[TABLE]

Finally

[TABLE]

where we have used

[TABLE]

So we obtain

[TABLE]

Note the result depends only on the combination $2L_{1}+L_{2}$ as anticipated from our discussion in subsect. 3.2.

Recalling for $D\sim 4$

[TABLE]

we see that to cancel the pole terms the combination $L_{1}+\frac{1}{2}L_{2}$ must be of the form given in (51). Finally combining the results we obtain (52).

Appendix D Integrals over O($n$) matrices

Here we consider integrals of the form

[TABLE]

with $\Omega$ a real orthogonal $n\times n$ matrix:

[TABLE]

and $\mathrm{d}\Omega$ the O($n$) invariant Haar measure with normalization

[TABLE]

For $r\geq 1$ the integrals are evaluated as sums over $r$ terms:

[TABLE]

where the $T^{(r)(s)}$ are $n$-dependent coefficients times sums of products of $\delta$-functions involving only the $j_{a}$ which are invariant under interchanges $j_{a}\to j_{\sigma(a)}\,,\,\,a=1,..,2r$, where $\sigma$ is any permutation such that

[TABLE]

The sum over permutations in (254) gives the sum over the $N_{r}=(2r)!/(2^{r}r!)$ distinct pairings of the labels $i_{1},\dots,i_{2r}$.

In our computation we only need the integrals for $r=1,2,3$. First for $r=1$:

[TABLE]

For $r=2$ and $n>1$:

[TABLE]

Finally for $r=3$ and $n>2$:

[TABLE]

It follows for averages over O$(n)$ vectors $a,b,c,d,e,f$:

[TABLE]