Fine corrections in the effective string describing
$\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory in three dimensions

F. Caristo,1 M. Caselle,1,2 N. Magnoli,3 A. Nada,1 M. Panero,1,2 and A.
Smecca1

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1 Department of Physics, University of Turin & INFN, Turin

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Via Pietro Giuria 1, I-10125 Turin, Italy

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2 Arnold-Regge Center, University of Turin

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Via Pietro Giuria 1, I-10125 Turin, Italy

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3 Department of Physics, University of Genoa & INFN, Genoa

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Via Dodecaneso 33, I-16146 Genoa, Italy

We present a study of the effective string that describes the infrared
dynamics of $\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory in three
dimensions. By combining high-precision lattice simulation results for
Polyakov-loop correlators at finite temperatures close to (and less than) the
deconfinement one with the analytical constraints from renormalization-group
arguments, from the exact integrability of the two-dimensional Ising model
that describes the universality class of the critical point of the theory,
from conformal perturbation theory, and from Lorentz invariance, we derive
tight quantitative bounds on the corrections to the effective string action
beyond the Nambu-Gotō approximation. We show that these corrections are
compatible with the predictions derived from a bootstrap analysis of the
effective string theory, but are inconsistent with the axionic string
_Ansatz_.

##  1 Introduction

One of the most promising approaches to understand and model the non-
perturbative behavior of confining Yang-Mills theories is the “effective
string theory” (EST) description, in which the flux tube joining together a
quark-antiquark pair is modeled as a thin vibrating string [1, 2, 3, 4, 5].
Recently, there has been a lot of progress in this context. In particular, it
has been realized that the EST enjoys “low-energy universality” [6, 7, 8, 9,
10, 11, 12]: due to the peculiar features of the string action and to the
symmetry constraints imposed by Poincaré invariance in the target space, the
first few terms of the large-distance expansion of the string action are fixed
and hence universal. This implies that the EST is much more predictive than
typical effective theories, and in fact during the past few years its
predictions have been be confirmed by many simulations in lattice gauge
theories (for recent reviews, see for instance refs. [12, 13, 14]).

At the same time, it was also realized that the simplest Lorentz-invariant
EST, which is the well-known Nambu-Gotō model [1, 2], is an exactly
integrable, irrelevant, perturbation of the two-dimensional free Gaußian model
[15], driven by the $T\overline{T}$ operator of the $D-2$ free bosons111We
will denote in the following with $D$ the number of spacetime dimensions of
the target lattice theory and with $d\equiv D-1$ the number of spacelike
directions. that represent the transverse degrees of freedom of the string
[16]. This observation stimulated much work and led to interesting novel
results, whose relevance extends even beyond the original application to Yang-
Mills theory [17, 18, 19, 20, 21, 22, 23, 24]. In particular, these findings
are at the basis of an S-matrix bootstrap approach that can be used to
constrain the EST action beyond the Nambu-Gotō approximation [25, 26].

Indeed, it is by now clear that the Nambu-Gotō action should be considered
only as a leading-order approximation of the actual EST describing the
infrared dynamics of confining gauge theories. Going beyond this approximation
is one of the most interesting open problems in this context. The terms beyond
the Nambu-Gotō action encode important physical information and their study
could be of great importance to understand the mechanism underlying
confinement or the physical degrees of freedoms from which the confining
string arises.

A natural way to study these corrections would be to perform high-precision
simulations of the interquark potential in different lattice gauge theories.
This approach, however, is hampered by the existence of boundary terms [27,
28, 29, 30] related to the finiteness of the physical flux tubes studied in
lattice simulations. These boundary terms provide a dominant contribution to
the corrections beyond the Nambu-Gotō action and make the detection of other
terms challenging (if possible at all: see below for a detailed discussion of
this issue).

However, it can be shown that these corrections become subleading and can be
neglected if one looks at the interquark potential at finite temperature, in
the neighborhood of the deconfinement transition, but still in the confining
phase. With this motivation, in this work we address a study of the simplest
non-trivial non-Abelian lattice gauge theory: the
$\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory in $(2+1)$ dimensions, in
the range of temperatures $0.8T_{\mbox{\tiny{c}}}\leq T\leq
T_{\mbox{\tiny{c}}}$, where $T_{\mbox{\tiny{c}}}$ denotes the deconfinement
temperature.

This model has been the subject of several lattice studies in the past,
including e.g. refs. [31, 32, 33, 34, 35, 27, 28, 29, 30, 36], because it is a
particularly simple lattice gauge theory based on a non-Abelian Lie group and
allows one to study the non-perturbative features of Yang-Mills theories to
much better numerical precision than one could obtain in lattice simulations
of quantum chromodynamics.

There is another important reason why the choice of the
$\operatorname{\mathrm{SU}}(2)$ lattice gauge theory in $(2+1)$ dimensions is
helpful. Since the deconfinement transition for this model is of the second
order [37, 38], renormalization group arguments show that in the neighborhood
of the deconfinement transition the model is in the same universality class of
the bidimensional Ising model: this is the celebrated Svetitsky-Yaffe
conjecture [39]. According to this correspondence, the Polyakov loop
correlator is mapped to the spin-spin correlator of the two-dimensional Ising
model which, thanks to the exact integrability of the model, is exactly known.
As will be shown in this work, the quantitative accuracy of this mapping is
confirmed by our new sets of high-precision non-perturbative results, obtained
by Monte Carlo simulations; moreover, the knowledge of the exact form of the
spin-spin correlator in the spin model provides us with a tool to extract the
temperature-dependence of the ground-state energy of the theory with high
accuracy, and to compare these results with effective-string predictions. In
particular, this will allow us to precisely quantify the corrections with
respect to the Nambu-Gotō action, which is one of the main goals of this work.

This article is organized as follows. The next four sections contain
introductory material: section 2 is devoted to a description of the lattice
setup, in section 3 we summarize the Svetitsky-Yaffe conjecture, section 4
reviews known results on the spin-spin correlator of the two-dimensional Ising
model, while section 5 presents a brief introduction to the effective string
theory. Our results and a description of lattice simulations are presented in
section 6, while in the last section 7 we summarize our findings and list some
concluding remarks.

##  2 Definitions and lattice setup

As mentioned above, in this work we are focus on the three-dimensional
$\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory at finite temperature. We
regularize the theory on a finite cubic lattice of spacing $a$ and sizes
$aN_{t}$ in the $\hat{0}$ (“Euclidean-time”) direction and $aN_{s}$ in the two
other (“spatial”) directions, labelled as $\hat{1}$ and $\hat{2}$. To simplify
notations, in the following we will set $a=1$. Periodic boundary conditions
are assumed in all directions and we always take $N_{s}\gg N_{t}$. We use the
standard Wilson action [40]

$S_{\mbox{\tiny{W}}}=-\frac{2}{g^{2}}\sum_{x}\sum_{0\leq\mu<\nu\leq
2}\operatorname{\mathrm{Tr}}U_{\mu\nu}(x)$ (1)

where the plaquette
$U_{\mu\nu}(x)=U_{\mu}(x)U_{\nu}\left(x+\hat{\mu}\right)U_{\mu}^{\dagger}\left(x+\hat{\nu}\right)U_{\nu}^{\dagger}(x)$
is defined as the path-ordered product of link variables $U_{\mu}(x)$ (taking
values in the fundamental representation of the
$\operatorname{\mathrm{SU}}(2)$ group) along the elementary square having the
site $x$ as a corner and lying in the oriented $(\mu,\nu)$ plane. $g^{2}$ is
the squared bare coupling, which has energy dimension one. In the following,
we will often use the parameter $\beta$, defined as $\beta=4/g^{2}$.

This lattice model has been studied in various works in the past. These
include, in particular, ref. [32], in which the following scale setting was
presented:

$\sqrt{\sigma_{0}(\beta)}=\frac{1.324(12)}{\beta}+\frac{1.20(11)}{\beta^{2}}+\mathcal{O}(\beta^{-3}),$
(2)

where $\sigma_{0}$ denotes the zero-temperature string tension.222Note that
eq. (2) expresses the square root of the zero-temperature string tension as a
dimensionless quantity, in lattice units. Reinstating the lattice spacing $a$,
the left-hand side of eq. (2) would be replaced by $a\sqrt{\sigma_{0}}$ In the
following, we will also discuss the relationship between $\sigma_{0}$ and the
finite-temperature string tension $\sigma(T)$. For some of our simulations at
$\beta=9.0$, we used the high-precision scale setting that was recently
reported in ref. [36].

The temperature $T$ is related to the extent of the shortest compact size of
the lattice as $N_{t}=1/T$: as a consequence, $T$ can be varied by changing
$N_{t}$, or the lattice spacing (which can be varied continuously by tuning
$\beta$), or both. We will study the system in the confining phase, just below
the deconfinement temperature $T_{\mbox{\tiny{c}}}$, in the temperature range
$0.8\leq T/T_{\mbox{\tiny{c}}}\leq 1$. Very accurate estimates of
$T_{\mbox{\tiny{c}}}$ for various values of $N_{t}$, that we will use in the
following, can be found in ref. [41].

The Polyakov loop through a point of spatial coordinates $\vec{x}$ is defined
as the normalized trace of the closed Wilson line in the $\hat{0}$ direction:

$P\left(\vec{x}\right)=\frac{1}{2}\operatorname{\mathrm{Tr}}\prod_{t=0}^{N_{t}}U_{0}\left(t,\vec{x}\right).$
(3)

The two-point correlation function of Polyakov loops is then defined as

$G(R)=\left\langle\sum_{\vec{x}}\
P\left(\vec{x}\right)P\left(\vec{x}+R\hat{k}\right)\right\rangle,$ (4)

where $\hat{k}$ denotes one of the two spatial directions, the sum is over all
spatial coordinates $\vec{x}$, while the $\langle\dots\rangle$ average is
taken over all values of all of the $U_{\mu}(x)$ variables, with a measure
that is proportional to the product of the Haar measures of all $U_{\mu}(x)$
matrices and to $\exp(-S_{\mbox{\tiny{W}}})$, and normalized in such a way
that the expectation value of the identity operator is $1$.

###  2.1 Finite-temperature interquark potential

In a finite-temperature setting, one can define the “interquark potential”
(or, more precisely, the potential energy associated with a pair of static
fundamental color sources) $V$ from the free energy associated with the
Polyakov-loop two-point correlation function:

$G(R)\equiv\exp\left[-\frac{V(R,N_{t})}{T}\right]=\exp\left[-N_{t}V(R,N_{t})\right].$
(5)

For sufficiently large values of the spatial separation $R$ between the color
sources, in the confining phase one expects $V(R,N_{t})$ to tend to a linearly
rising function of $R$:

$G(R)\simeq\exp\left[-\sigma(T)N_{t}R\right],$ (6)

where $\sigma(T)$ denotes a temperature-dependent string tension. As we will
see below, $\sigma(T)$ is a decreasing function of $T$ and vanishes exactly at
the deconfinement point [42, 43]. From $V(R,N_{t})$ and $\sigma(T)$ it is also
possible to derive the zero-temperature potential $V(R)$ as the $T\to 0$ limit
of $V(R,N_{t})$, and, accordingly, the zero-temperature string tension
$\sigma_{0}$ as the $T\to 0$ limit of $\sigma(T)$.

It is interesting to note that the correlator defined in eq. (5) has an
analogy with the expectation value of an ordinary Wilson loop, except for the
boundary conditions, which in this case are fixed in the spatial directions
and periodic in the compact-time direction. Accordingly, the resulting
geometry for the world-sheet associated with the fluctuating string between
the color sources is that of a cylinder, which is topologically different from
the rectangular geometry associated with a Wilson loop.

The existence of periodic boundary conditions for the gauge fields along the
compactified Euclidean-time direction is the interpretation of the thermal
deconfinement transition in terms of dynamical breaking of a global symmetry
described by a group that is the center $C$ of the gauge group (i.e.
$\mathbb{Z}_{N}$ if the gauge group is $\operatorname{\mathrm{SU}}(N)$) [44,
45, 46]. This symmetry, which can be thought of as the action of multiplying
all timelike links at a given Euclidean time by the same element of $C$, has
the Polyakov loop as an order parameter. In the low-temperature phase
($T<T_{\mbox{\tiny{c}}}$), the center symmetry is realized, and the
expectation value of a Polyakov loop vanishes: this means that the free energy
associated with a static, isolated color source is finite, i.e. color
confinement. Conversely, in the high-temperature phase
($T>T_{\mbox{\tiny{c}}}$), center symmetry is spontaneously broken and the
Polyakov loop has a non-zero expectation value: this signals that the free
energy associated with a static color source is finite, i.e. that the theory
is in its deconfined phase.

##  3 Svetitsky-Yaffe mapping

The peculiar role played by the Polyakov loops in the above discussion
suggests to study the behavior of the theory in the vicinity of the
deconfinement transition using an effective action that can be constructed by
integrating out the spacelike link variables and projecting each Polyakov loop
to the closest element of the center of the gauge group. Starting from a
$(d+1)$-dimensional lattice gauge theory, we end up in this way with an
effective action for the Polyakov loops, which will be a $d$-dimensional spin
model, having the center of the original gauge group as a global symmetry.

This integration cannot be performed exactly, and one usually resorts to some
kind of strong-coupling expansion. Notwithstanding this, some general insight
on the behavior of the model can be deduced by simple renormalization group
arguments [39]. Indeed, if the phase transition is continuous, in the vicinity
of the critical point the fine details of the Hamiltonian describing the
effective spin model can be neglected, and the latter can be shown to belong
to the same universality class of the simplest spin model, with only nearest-
neighbor interactions, sharing the same symmetry-breaking pattern. This means,
in our case, that the deconfinement transition of the
$\operatorname{\mathrm{SU}}(2)$ lattice gauge theory in three dimensions,
which is continuous, belongs to the same universality class of the symmetry-
breaking phase transition of the two-dimensional Ising model. As is well
known, this model is exactly integrable [47] and in particular, as we will see
in the next section, an exact expression for the spin-spin correlator is
known. This fact will play an important role in the following.

Let us list a few important features of gauge-spin mapping suggested by the
Svetitsky-Yaffe conjecture.

  * •

The ordered (low-temperature) phase of the spin model corresponds to the
deconfined (high-temperature) phase of the original gauge theory. These are
the phases in which the order parameters of the two theories (namely, the
Polyakov loop for the gauge theory and the spin for the spin model) have non-
zero expectation values.

  * •

The Polyakov loop is mapped to the spin operator, while the plaquette is
mapped to the energy operator of the effective spin model. Accordingly, the
Polyakov-loop correlator in the confining phase, in which we are interested in
this work, is mapped to the spin-spin correlator in the disordered, high-
temperature phase of the spin model.

  * •

Thermal perturbations from the critical point in the original gauge theory,
which are driven by the plaquette operator, are mapped to thermal
perturbations of the effective spin model, driven by the energy operator.
Notice however the change in sign: an _increase_ in temperature of the
original gauge theory corresponds to a _decrease_ of the temperature of the
effective spin model.

An important consequence of this mapping is that, in the vicinity of the
deconfinement point, we can use the spin-spin correlator of the Ising model to
model the behavior of the Polyakov loop correlator of the lattice theory: this
poses very tight constraints on the EST that describes this correlator. In
this respect, the exact integrability of the two-dimensional Ising model gives
us a unique opportunity to study the dynamics of the
$\operatorname{\mathrm{SU}}(2)$ lattice gauge theory in $(2+1)$ dimensions.

##  4 The spin-spin correlator of the two-dimensional Ising model

The spin-spin correlator of the two-dimensional Ising model can be written as
a finite determinant with a size that depends on the separation of the spins
[48]. In the scaling limit, these determinants can be rewritten as suitable
solutions of an equation of the Painlevé type [49]. These solutions have a
very different form depending on the phase of the model. In particular,
denoting the correlation length by $\xi$ and the $R/\xi$ ratio as $t$, in the
disordered phase which is the focus of our interest in this work they can be
expanded in the short-distance ($R\ll\xi$) and in the large-distance
($R\gg\xi$) limits as follows.

  * •

In the $R\ll\xi$ regime, the two-point spin correlator can be expanded as

$\left\langle{\sigma(0)\sigma(R)}\right\rangle=\frac{k_{s}}{R^{\frac{1}{4}}}\bigg{[}1+\frac{t}{2}\ln\bigg{(}\frac{e^{\gamma}_{\mbox{\tiny{E}}}t}{8}\bigg{)}+\frac{1}{16}t^{2}+\frac{1}{32}t^{3}\ln\bigg{(}\frac{e^{\gamma}_{\mbox{\tiny{E}}}t}{8}\bigg{)}+O(t^{4}\ln^{2}t)\bigg{]},$
(7)

where $\gamma_{\mbox{\tiny{E}}}=0.57721\dots$ denotes the Euler-Mascheroni
constant, while $k_{s}$ is a non-universal constant, which can be evaluated
exactly in the case of the two-dimensional Ising model on a square lattice
[49] (a results which, due to its non-universal nature, is not of relevance
for our present purposes).

  * •

Conversely, in the $R\gg\xi$ regime, the two-point spin correlator admits the
expansion

$\left\langle{\sigma(0)\sigma(R)}\right\rangle={k_{l}}K_{0}(t)$ (8)

where, again, $k_{l}$ is a non-universal constant which can be evaluated
exactly in the case of the two-dimensional Ising model on a square lattice
(but which is not relevant for our discussion), while $K_{0}$ is the modified
Bessel function of order zero, whose large-distance expansion is

$K_{0}(t)\simeq\sqrt{\frac{\pi}{2t}}e^{-t}\left[1+O\left(\frac{1}{t}\right)\right].$
(9)

We conclude this section with some important observations.

  1. 1.

The short-distance expansion of eq. (7) can also be obtained using conformal
perturbation theory (CPT): an approach proposed in 1987 by Zamolodchikov [50],
which for the past thirty years has proved to be a powerful analytical tool to
describe statistical-mechanics models and quantum field theories in the
vicinity of a critical point. As a matter of fact, the full agreement of the
CPT result in the Ising case with the exact expansion of eq. (7) represents
one of the most stringent and successful tests of CPT [51, 52, 53]. While
original applications of CPT were limited to two-dimensional models (see, for
example, refs. [54, 53, 55, 56, 57, 58]), recently it has also been extended
to three-dimensional models [59, 60]. The reason why the CPT approach is
important for our present discussion is that, thanks to it, the results that
we discuss in this article do not necessarily require the exact integrability
of the underlying spin model but can be extended to any pair of lattice gauge
theory and spin model with a second-order deconfinement/symmetry breaking
transition. Indeed this approach was already followed in ref. [61] for the
mapping between the $(3+1)$-dimensional $\operatorname{\mathrm{SU}}(2)$ Yang-
Mills theory and the Ising model in three dimensions.

  2. 2.

The $K_{0}$ function describing the large-distance behavior is the typical
expression that one obtains for the correlator of a two-dimensional Euclidean
quantum field theory with an isolated, massive excitation. In the Ising case,
this is the Majorana fermion which describes the model in the continuum limit.
As for the short-distance expansion, also this result can be extended also to
models that are not exactly integrable. Indeed the symmetric phase of a
generic spin model is always described, in the continuum limit, by an
appropriate set of (possibly interacting) massive particles. In the large-
distance limit, the spin-spin correlator for any of these theories will be
dominated by the lowest mass in the spectrum, whose correlation function in
$d$ dimensions is given by a $K_{(d-2)/2}$ modified Bessel function.

  3. 3.

It is interesting to observe the shift in the exponent of the $1/R$ term in
eq. (7) and in eq. (8). At short distance (where we expect deviations with
respect to the Nambu-Gotō action) the power is fixed by the universality class
of the model (in our case $1/4$), and, in general, will depend on the gauge
group of the underlying gauge theory. At large distances, instead, the
asymptotic behavior of the Bessel function in eq. (9) implies that the power
is always $1/2$: this is in agreement with the Nambu-Gotō result, as we will
discuss in section 5.

##  5 Effective string theory predictions

There is by now a rich literature on the many properties and features of the
effective string theory description of the confining flux tube. Here we will
limit ourselves to a brief discussion of the features that are most relevant
for our problem. For a general introduction to EST, we refer the reader to the
recent reviews [12, 13, 14].

The main idea behind EST is that confinement of color charges can be
associated with the formation of a thin string-like flux tube [1, 2, 3, 4, 5],
which leads, for large separations between the color sources, a linearly
confining potential.

In a finite-temperature setting it is possible to show, with very mild
assumptions, that the EST description implies the following form for the
Polyakov-loop correlator [6, 7]:

$\left\langle{P(0)P^{\dagger}(R)}\right\rangle=\sum_{n}|v_{n}(N_{t})|^{2}2R\bigg{(}\frac{E_{n}}{2\pi
R}\bigg{)}^{\frac{D-1}{2}}K_{(D-3)/2}(E_{n}R),$ (10)

where $D$ denotes the number of spacetime dimensions (in our case $D=3$),
while $E_{n}$ are the energy levels of the string and $v_{n}(N_{t})$ their
amplitudes, which in general depend on the inverse temperature $N_{t}$. The
physical meaning of eq. (10) is that the Polyakov-loop correlation function
can be modelled in terms of an infinite series of modified Bessel functions of
the second kind: this is expected to hold independently from the type of
string that is considered, as long as the energy spectrum is characterized by
the existence of isolated states, and these states are stable against decay by
glueball radiation.

At large distances, the right-hand side of eq. (10) is dominated by the lowest
energy level $E_{0}$ and, setting $D=3$, we end up with exactly the same
expression that we found for the large-distance behavior of the spin spin
correlator in the two-dimensional Ising model eq. (8). It is interesting to
note that this equality is not simply a consequence of the
$\operatorname{\mathrm{SU}}(2)$/Ising correspondence that we are studying
here: instead, it is much more general and holds also for $D>3$ [14]. As we
mentioned above, any spin model with an isolated ground state in the spectrum
is described by a modified Bessel function of the same type that appears in
the EST description. This is an important consistency check of both the EST
picture333In particular, as we will see below, for the Nambu-Gotō action, both
$v(N_{t})$ and $E_{n}$ can be evaluated exactly thanks to the exact
integrability of the model. and of the Svetitsky-Yaffe mapping in this high
temperature limit.

The Nambu-Gotō string model [1, 2] is the simplest Poincaré invariant EST. It
has a simple geometric interpretation, since it associates each possible
configuration that the string can span in the target space with a quantum
weight proportional to the area of the world-sheet surface. As such, the
Nambu-Gotō string action can be thought of as a straightforward generalization
of the relativistic action for a pointlike particle to a bosonic string. The
Nambu-Gotō action can be written as follows:

$\displaystyle S_{\mbox{\tiny{NG}}}=\sigma_{0}\int_{\Sigma}d^{2}\xi\sqrt{g},$
(11)

where $g\equiv\det g_{\alpha\beta}$ and
$g_{\alpha\beta}=\partial_{\alpha}X_{\mu}~{}\partial_{\beta}X^{\mu}$ is the
induced metric on the reference world-sheet surface $\Sigma$, where we denote
the world-sheet coordinates as $\xi\equiv(\xi^{0},\xi^{1})$. This action has
only one free parameter: the string tension $\sigma_{0}$, which has dimension
two.

In order to perform calculations with the Nambu-Gotō action one has first to
fix its invariance under reparametrizations of the string world-sheet
coordinates. A common choice is the so-called “physical gauge”, in which the
two world-sheet coordinates are identified with the longitudinal degrees of
freedom of the string: $\xi^{0}=X^{0}$ and $\xi^{1}=X^{1}$, so that the string
action can be expressed as a function only of the $(D-2)$ degrees of freedom
describing transverse displacements, $X^{i}$, with $i=2,\dots,(D-1)$, which
are assumed to be single-valued functions of the world-sheet coordinates. In
the physical gauge, the determinant of the metric has the form

$g=1+\partial_{0}X_{i}\partial_{0}X^{i}+\partial_{1}X_{i}\partial_{1}X^{i}+\partial_{0}X_{i}\partial_{0}X^{i}\partial_{1}X_{j}\partial_{1}X^{j}-(\partial_{0}X_{i}\partial_{1}X^{i})^{2}$
(12)

and the Nambu-Gotō action can then be written as a low-energy expansion in the
number of derivatives of the transverse degrees of freedom of the string
which, by a suitable redefinition of the fields, can be rephrased as an
expansion around the limit of an infinitely long string. The first few terms
in this expansion are

$S=\sigma_{0}RN_{t}+\frac{\sigma_{0}}{2}\int
d^{2}\xi\left[\partial_{\alpha}X_{i}\cdot\partial^{\alpha}X^{i}+\frac{1}{8}(\partial_{\alpha}X_{i}\cdot\partial^{\alpha}X^{i})^{2}-\frac{1}{4}(\partial_{\alpha}X_{i}\cdot\partial_{\beta}X^{i})^{2}+\dots\right].$
(13)

Despite its apparent complexity, it can be shown that all the additional terms
in the expansion beyond the Gaußian one conspire to yield an exactly
integrable, irrelevant perturbation of the Gaußian term [15], driven by the
$T\overline{T}$ operator of the $D-2$ free bosons [16].

Thanks to this exact integrability, the partition function of the model can be
written explicitly444The explicit expression for the partition function was
actually found even before this $T\overline{T}$ study, first by using the
constraints imposed by the open-closed string duality in ref. [7] and then
using a Dp-brane formalism in ref. [62]., as shown in refs. [63, 64]. For the
two-point Polyakov-loop correlation function that we are considering here, the
expression in $D$ spacetime dimensions is

$G(R)=\sum_{n=0}^{\infty}w_{n}\frac{2R\sigma_{0}N_{t}}{{E}_{n}}\left(\frac{\pi}{\sigma_{0}}\right)^{\frac{D-2}{2}}\left(\frac{{E}_{n}}{2\pi
R}\right)^{\frac{D-1}{2}}K_{(D-3)/2}({E}_{n}R),$ (14)

where the energy levels $E_{n}$ are given by

${E}_{n}=\sigma_{0}N_{t}\sqrt{1+\frac{8\pi}{\sigma_{0}N_{t}^{2}}\left(n-\frac{D-2}{24}\right)}.$
(15)

and the weights $w_{n}$ can be obtained from the expansion in series of $q$ of
the infinite products contained in the Dedekind functions which describes the
large-$R$ limit of eq. (14) (see ref. [62] for a detailed derivation):

$\left(\prod_{r=1}^{\infty}\frac{1}{1-q^{r}}\right)^{D-2}=\sum_{k=0}^{\infty}w_{k}q^{k}.$
(16)

For $D=3$ we have simply $w_{k}=p_{k}$, the number of partitions of the
integer $k$. Similar expressions can be obtained also for the other
geometries, e.g. for the rectangle (relevant for the description of the Wilson
loop) [65] and for the torus (which can model an interface) [66].

From the discussion above we see that, as anticipated, in the Nambu-Gotō case
we have and exact expression both for the amplitudes $v(N_{t})$ and for the
energy levels $E_{n}$. In particular, for the $D=3$ case the lowest state is

$E_{0}=\sigma_{0}N_{t}\sqrt{1-\frac{\pi}{3\sigma_{0}N_{t}^{2}}}=\sigma(T)N_{t},$
(17)

where we defined the “temperature-dependent string tension” as

$\sigma(T)\equiv\sigma_{0}\sqrt{1-\frac{\pi}{3\sigma_{0}N_{t}^{2}}}.$ (18)

As we have seen in the previous section, $E_{0}$ is the inverse of the
correlation length, thus the Nambu-Gotō EST predicts a critical temperature
[67, 68]

$\frac{T_{c,NG}}{\sqrt{\sigma_{0}}}=\sqrt{\frac{3}{\pi(D-2)}}$ (19)

and a critical index $\nu=1/2$. This prediction, however, is inconsistent with
the Svetitsky-Yaffe mapping, which for our model predicts the two-dimensional
Ising value $\nu=1$. Moreover, the prediction for the critical temperature is
quantitatively wrong (albeit close to the correct one). These observation
suggest that, in order to obtain the correct EST describing the gauge theory,
one should necessary go beyond the Nambu-Gotō approximation.

###  5.1 Effective string action beyond the Nambu-Gotō approximation

The discussion above shows that the pure Nambu-Gotō action cannot be the
actual effective string action. Discovering the correct (subleading) terms of
the effective string action beyond the Nambu-Gotō approximation is one of the
major open challenges in present studies of the EST, and is, in fact, the main
goal of this article.

There are essentially two classes of terms which one should address: “bulk
terms” and “boundary terms”. Let us look at them in more detail.

####  5.1.1 Beyond the Nambu-Gotō approximation: bulk terms and low-energy
universality

From an effective-action point of view, there is no reason to constrain the
coefficients of the higher-order terms in eq. (13) to the values they take in
the derivative expansion of the Nambu-Gotō action. One should instead assume
the most general form for such an effective action:

$S=S_{\mbox{\tiny{cl}}}+\frac{\sigma_{0}}{2}\int
d^{2}\xi\left[\partial_{\alpha}X_{i}\cdot\partial^{\alpha}X^{i}+c_{2}(\partial_{\alpha}X_{i}\cdot\partial^{\alpha}X^{i})^{2}+c_{3}(\partial_{\alpha}X_{i}\cdot\partial_{\beta}X^{i})^{2}+\dots\right],$
(20)

and then fix the coefficients (which, in this context, would play the role of
low-energy constants of the effective theory) order by order, either using
results from Monte Carlo simulations or (in the case of quantum
chromodynamics) from experiments.

However, one of the most interesting results of the last few years, known as
“low-energy universality of the EST” is that the $c_{i}$ coefficients are not
arbitrary, but must satisfy a set of constraints to enforce the Poincaré
invariance of the gauge theory in the target space [6, 7, 8, 9, 10, 11]. This
same result can also be obtained in an independent way, using a bootstrap type
of analysis: this was done in refs. [25, 26] in the framework of the S-matrix
approach pioneered in ref. [15]. These constraints are particularly
restrictive for a three-dimensional theory: the first few terms of the
expansion exactly coincide with those that are obtained from the expansion of
the Nambu-Gotō action, while the first correction with respect to the Nambu-
Gotō action, in the high-temperature regime which we are studying here,
appears only at order $1/N_{t}^{7}$. Moreover, even the coefficient of this
correction can be constrained: using the notations of refs. [25, 26], this
additional term can be written as

$-\frac{32\pi^{6}}{225}\frac{\gamma_{3}}{\sigma^{3}N_{t}^{7}}$ (21)

where $\gamma_{3}$ is a new parameter which together with $\sigma_{0}$ defines
the EST. By using a bootstrap analysis it is possible to show that
$\gamma_{3}$ is constrained to be $\gamma_{3}>-\frac{1}{768}$.

The $\gamma_{3}$ parameter encodes some important information on the effective
string theory. For example, it can be shown that the axionic string _Ansatz_
discussed in refs. [18, 19] requires $\gamma_{3}\geq 0$.

####  5.1.2 Beyond the Nambu-Gotō approximation: boundary corrections

Boundary corrections to the EST encode the effect of possible interactions of
the flux tube with the static color sources at its ends. As we will see below,
at zero and very low temperature the boundary correction behaves as $1/R^{4}$,
and hence, is the dominant contribution beyond the Nambu-Gotō approximation.
Its presence makes it almost impossible to detect the much weaker (and more
interesting, being related to the nature of the confining flux tubes) effects
due to bulk correction terms discussed above. However, as will be shown below,
in the high-temperature regime which we studied in this work, the boundary
term actually becomes _subleading_ , making it possible to access the bulk
corrections using Monte Carlo simulations.

Like the bulk terms, also the boundary terms are strongly constrained by
Lorentz invariance: the first boundary correction compatible with the
spacetime symmetries of the underlying gauge theory is [69]

$b_{2}\int
d\xi_{0}\left[\frac{\partial_{0}\partial_{1}X\cdot\partial_{0}\partial_{1}X}{1+\partial_{1}X\cdot\partial_{1}X}-\frac{\left(\partial_{0}\partial_{1}X\cdot\partial_{1}X\right)^{2}}{\left(1+\partial_{1}X\cdot\partial_{1}X\right)^{2}}\right],$
(22)

with an arbitrary, non-universal coefficient $b_{2}$. This coefficient has
been estimated in some recent lattice studies [27, 28, 29, 30, 69]: in
particular, for the $\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory in three
dimensions it was found to be $b_{2}\simeq-0.025/(\sqrt{\sigma})^{3}$. The
lowest-order term of the expansion of eq. (22) is

$S_{b,2}^{(1)}=b_{2}\int d\xi_{0}(\partial_{0}\partial_{1}X)^{2}.$ (23)

The contribution of this term to the interquark potential was evaluated in
ref. [70] using a $\zeta$-function regularization:

$\langle
S^{(1)}_{b,2}\rangle=-b_{2}\frac{\pi^{3}N_{t}}{60R^{4}}E_{4}(e^{-\frac{\pi
N_{t}}{R}})$ (24)

where $E_{4}$ denotes the fourth-order Eisenstein series:

$E_{4}(q)\equiv
1+\frac{2}{\zeta(-3)}\sum_{n=1}^{\infty}\frac{n^{3}q^{n}}{1-q^{n}}~{}$ (25)

and $\zeta(s)$ is the Riemann $\zeta$ function.

In the low-temperature ($N_{t}\gg R$) regime (which is the one that is most
often studied in lattice calculations) eq. (24) amounts to a $1/R^{4}$
contribution proportional to the interquark potential. As it scales with a
larger power of $R$, this term obfuscates the evidence of bulk corrections in
numerical results. However, using the modular properties of the Eisenstein
function,

$E_{4}\left(e^{-\frac{\pi
N_{t}}{R}}\right)=\left(\frac{2R}{N_{t}}\right)^{4}E_{4}\left(e^{-\frac{4\pi
R}{N_{t}}}\right),$ (26)

it is easy to see that in the $R\gg N_{t}$ (“high-temperature”) regime, the
boundary correction becomes

$\langle
S^{(1)}_{b,2}\rangle=-b_{2}\frac{4\pi^{3}}{15N_{t}^{3}}E_{4}\left(e^{-\frac{4\pi
R}{N_{t}}}\right),$ (27)

which does not contain terms linear in $R$ and thus it does not contribute to
the temperature-dependent string tension. We will make use of this property in
the analysis of our numerical results.

##  6 Simulation setting and results

In this section we present the results of a new set of Monte Carlo
simulations. The calculations were run with the parallel C++ code developed
for the studies presented in refs. [71, 72]. The elements of the
$\operatorname{\mathrm{SU}}(2)$ group are stored as four complex numbers in
double precision, and are updated using a combination of local heat-bath [73,
74] and overrelaxation [75] steps.

We performed two sets of simulations. In the first, which was mainly devoted
to testing the Svetitsky-Yaffe mapping and to a general study of the
deviations with respect to the Nambu-Gotō EST predictions, we fixed a few
values of $N_{t}$ and varied the temperature by changing $\beta$. In the
second set of simulations, which was aimed to a high-precision study of the
corrections to the Nambu-Gotō action, we chose the opposite strategy and fixed
three values of $\beta$ and varied the temperature by changing $N_{t}$. Let us
discuss these simulations in detail.

###  6.1 Test of the Svetitsky-Yaffe mapping

We chose four values of $N_{t}$, namely $N_{t}=6$, $7$, $8$, and $9$, for
which a very precise determination of $\beta_{\mbox{\tiny{c}}}$ is known from
ref. [41], and we performed a large set of simulations for different values of
$\beta$, corresponding to temperatures in the range $0.8\leq
T/T_{\mbox{\tiny{c}}}\leq 1$, see tab. 1.

| $N_{t}\times N_{s}^{2}$ | $\beta$ | $T/T_{\mbox{\tiny{c}}}$ | $n_{\mbox{\tiny{conf}}}$
---|---|---|---
$9\times 96^{2}$ | $11.3048$ | $0.80$ | $2.5\times 10^{5}$
$11.72873$ | $0.83$ | $2.5\times 10^{5}$
$12.15266$ | $0.86$ | $2.5\times 10^{5}$
$12.57659$ | $0.89$ | $2.5\times 10^{5}$
$13.00052$ | $0.92$ | $2.5\times 10^{5}$
$13.42445$ | $0.95$ | $2.5\times 10^{5}$
$13.84838$ | $0.98$ | $2.5\times 10^{5}$
$14.131$ | $1.00$ | $1.0\times 10^{5}$
$7\times 96^{2}$ | $9.228023$ | $0.83$ | $2.5\times 10^{5}$
$9.561566$ | $0.86$ | $2.5\times 10^{5}$
$9.895109$ | $0.89$ | $2.5\times 10^{5}$
$10.562195$ | $0.95$ | $2.5\times 10^{5}$
$10.895738$ | $0.98$ | $2.5\times 10^{5}$
$11.1181$ | $1.00$ | $1.0\times 10^{5}$
| $N_{t}\times N_{s}^{2}$ | $\beta$ | $T/T_{\mbox{\tiny{c}}}$ | $n_{\mbox{\tiny{conf}}}$
---|---|---|---
$8\times 96^{2}$ | $10.10736$ | $0.80$ | $2.5\times 10^{5}$
$10.486386$ | $0.83$ | $2.5\times 10^{5}$
$10.865412$ | $0.86$ | $2.5\times 10^{5}$
$11.244438$ | $0.89$ | $2.5\times 10^{5}$
$11.623462$ | $0.92$ | $2.5\times 10^{5}$
$12.00249$ | $0.95$ | $2.5\times 10^{5}$
$12.381516$ | $0.98$ | $2.5\times 10^{5}$
$12.6342$ | $1.00$ | $1.0\times 10^{5}$
$6\times 96^{2}$ | $8.258494$ | $0.86$ | $2.5\times 10^{5}$
$8.546581$ | $0.89$ | $2.5\times 10^{5}$
$8.834668$ | $0.92$ | $2.5\times 10^{5}$
$9.122755$ | $0.95$ | $2.5\times 10^{5}$
$9.410842$ | $0.98$ | $2.5\times 10^{5}$
$9.6029$ | $1.00$ | $1.0\times 10^{5}$
Table 1: Information on the first set of lattice simulations.

In addition, in fig. 1 we present a data sample corresponding to
$T=0.62T_{\mbox{\tiny{c}}}$ (the lowest temperature that we considered in this
work), again from simulations with $N_{t}=6$, $7$, $8$, and $9$: the figure,
showing the Polyakov loop correlator as a function of the distance (in units
of the inverse temperature) reveals a clear collapse of data. The decay of the
correlator can be described very well in terms of a single exponential (which,
in this semilogarithmic plot, manifests itself in the approximately linear
behavior of the data) over a wide range of distances. Leaving aside the points
at values of $R$ of the order of a few lattice spacings, which are affected by
non-negligible discretization effects, the slight bending of the data from
short to intermediate distances, before the onset of the purely exponential
decay, is a signature of the effective string corrections that will be
discussed in detail below.

Figure 1: Collapse of data in the Polyakov-loop two-point correlation function
$G(R)$ obtained for different values of $N_{t}$, from $6$ to $9$, at the same
temperature, $T=0.62T_{\mbox{\tiny{c}}}$. The data, shown using a logarithmic
scale for the vertical axis, are plotted as a function of the distance between
the loops, $R$, in units of the inverse temperature.

In the following, we will discuss in detail only one of our simulations, the
one at $N_{t}=9$ and $\beta=12.15266$, which corresponds to
$T/T_{\mbox{\tiny{c}}}=0.86$. We found analogous results for all the other
cases.

We first tested the Svetitsky-Yaffe mapping, fitting our data to the Ising
expressions for the spin-spin correlator reported in eq. (7) and in eq. (8).
To account for the first periodic copy of the lattice, the large-distance fits
were performed with the following function

$G(R)={k_{l}}\left[K_{0}\left(\frac{R}{\xi}\right)+K_{0}\left(\frac{96-R}{\xi}\right)\right]$
(28)

which, like eq. (8), has only two free parameters: $k_{l}$ and $\xi$. The
results of the fits are listed in tab. 2 and shown in fig. 2 and in fig. 3.
For the sake of completeness, in tab. 3 we also report the results for all the
other values of $\beta$ and $N_{t}$ that we studied.

Figure 2: Fit of the Monte Carlo results obtained at $N_{t}=9$,
$\beta=12.15266$, $T/T_{\mbox{\tiny{c}}}=0.86$ according to the short-distance
approximation of the spin-spin Ising correlator of eq. (7).

From tab. 2 it is clear that both at short and at large distances we could fit
a wide range of data with a good reduced $\chi^{2}$. We observe that the range
of Polyakov-loop separations used for the large-distance fits ($R\in[15-47]$)
has a large overlap with the one for the short-distance fit ($R\in[5-20]$),
and the two estimates of the correlation length agree with each other within
their uncertainties. This means that the two-point Polyakov-loop correlator is
described well by the Ising spin-spin correlation function in the whole range
of distances that we studied (down to $R>4$). We also note that, from the EST
point of view, the distance at which the string description is expected to
break down (due to the tachyonic singularity [12], which at $\beta=12.15266$
is $R_{c}=\sqrt{\frac{\pi}{12\sigma}}\simeq 4.4$), the smallest value of $R$
that we can explore for this $\beta$ is $R=5$.

| $R_{min}$ | $R_{max}$ | amplitude | $\xi$ | $\chi^{2}_{\tiny\mbox{red}}$
---|---|---|---|---|---
eq. (7) | $5$ | $20$ | $k_{s}=0.15226(11)$ | $22.97(3)$ | $1.58$
eq. (28) | $15$ | $47$ | $k_{l}=0.039(3)$ | $23.3(8)$ | $1.14$
Table 2: Results of the fits to the short- and long-distance behaviors, according to the Svetitsky-Yaffe mapping, of the Polyakov loop correlator for $N_{t}=9$, $\beta=12.15266$ which corresponds to $T/T_{\mbox{\tiny{c}}}=0.86$. $N_{t}$ |  | $T/T_{\mbox{\tiny{c}}}$ | $R_{min}$ | $R_{max}$ | amplitude | $\xi$ | $\chi^{2}_{\tiny\mbox{red}}$
---|---|---|---|---|---|---|---
$N_{t}=6$ | eq. (7) | $0.86$ | $5$ | $13$ | $k_{s}=0.16251(13)$ | $14.21(2)$ | $1.22$
eq. (28) | $0.86$ | $6$ | $47$ | $k_{l}=0.0494(12)$ | $14.1(2)$ | $0.62$
eq. (7) | $0.89$ | $5$ | $20$ | $k_{s}=0.16559(10)$ | $18.77(2)$ | $1.38$
eq. (28) | $0.89$ | $8$ | $47$ | $k_{l}=0.049(2)$ | $18.0(3)$ | $1.71$
eq. (7) | $0.92$ | $5$ | $31$ | $k_{s}=0.1663(5)$ | $39.7(2)$ | $1.33$
eq. (28) | $0.92$ | $8$ | $47$ | $k_{l}=0.042(8)$ | $28(3)$ | $1.38$
$N_{t}=7$ | eq. (7) | $0.83$ | $5$ | $13$ | $k_{s}=0.1558(2)$ | $13.43(2)$ | $2.71$
eq. (28) | $0.83$ | $11$ | $47$ | $k_{l}=0.0447(15)$ | $14.08(10)$ | $1.87$
eq. (7) | $0.86$ | $5$ | $18$ | $k_{s}=0.15831(9)$ | $17.10(2)$ | $1.04$
eq. (28) | $0.86$ | $12$ | $47$ | $k_{l}=0.0480(12)$ | $16.53(11)$ | $1.52$
eq. (7) | $0.89$ | $5$ | $21$ | $k_{s}=0.16140(12)$ | $23.25(8)$ | $1.78$
eq. (28) | $0.89$ | $16$ | $47$ | $k_{l}=0.041(3)$ | $23.6(8)$ | $1.29$
$N_{t}=8$ | eq. (7) | $0.80$ | $5$ | $12$ | $k_{s}=0.1491(2)$ | $12.83(2)$ | $0.79$
eq. (28) | $0.80$ | $8$ | $47$ | $k_{l}=0.045(2)$ | $13.14(17)$ | $1.17$
eq. (7) | $0.83$ | $5$ | $16$ | $k_{s}=0.1521(1)$ | $15.74(3)$ | $1.62$
eq. (28) | $0.83$ | $6$ | $47$ | $k_{l}=0.0449(18)$ | $15.75(17)$ | $0.66$
eq. (7) | $0.86$ | $6$ | $19$ | $k_{s}=0.1530(2)$ | $20.97(4)$ | $1.58$
eq. (28) | $0.86$ | $13$ | $47$ | $k_{l}=0.042(3)$ | $20.0(6)$ | $1.39$
eq. (7) | $0.89$ | $5$ | $27$ | $k_{s}=0.1562(5)$ | $29.2(2)$ | $1.25$
eq. (28) | $0.89$ | $13$ | $47$ | $k_{l}=0.039(3)$ | $28.2(1.7)$ | $1.53$
$N_{t}=9$ | eq. (7) | $0.80$ | $5$ | $14$ | $k_{s}=0.1458(2)$ | $14.54(3)$ | $0.91$
eq. (28) | $0.80$ | $11$ | $47$ | $k_{l}=0.042(3)$ | $14.9(2)$ | $1.86$
eq. (7) | $0.83$ | $5$ | $19$ | $k_{s}=0.14877(10)$ | $17.91(2)$ | $1.47$
eq. (28) | $0.83$ | $16$ | $47$ | $k_{l}=0.041(2)$ | $18.3(4)$ | $1.50$
eq. (7) | $0.89$ | $6$ | $24$ | $k_{s}=0.1562(2)$ | $31.96(9)$ | $1.49$
eq. (28) | $0.89$ | $21$ | $47$ | $k_{l}=0.036(5)$ | $33(2)$ | $1.39$
Table 3: Results of the fits to the short- and large-distance behaviors,
according to the Svetitsky-Yaffe mapping, of the Polyakov loop correlator for
different values of $\beta$ and $N_{t}$

In order to appreciate the agreement, in fig. 3 we show the results of both
fits.

As the large-distance regime is the same for the Ising spin-spin correlator
and for the EST description, we can argue that the deviations from the $K_{0}$
Bessel function behavior observed for $R<15$ (which are shown in the inset of
fig. 3) encode the higher-order corrections to the EST that are the main
interest of this work. These deviations are made more clearly visible in fig.
4, where we subtracted the $K_{0}$ Bessel function (i.e. the EST ground state)
from the numerical data. One can see that the deviations are much larger than
the statistical uncertainties. According to the EST picture, these deviations
should be accounted for by higher-order terms of the EST spectrum in eq. (10).
If one assumes the Nambu-Gotō spectrum of eq. (15), then it is easy to see
that the states are not enough to fill the gap between the ground state and
the Monte Carlo data: this is clearly visible in fig. 5, where we plotted the
contribution of the first few states of the spectrum and of the boundary term
discussed in section 5. The energy levels of the Nambu-Gotō spectrum are too
sparse to fill the gap.

However, this gap, by itself, does not allow one to guess the higher-order
corrections to the EST. To obtain information on these corrections, it is more
convenient to use a different strategy, that will be discussed in the next
subsection.

Figure 3: Fit of the data at $N_{t}=9$, $\beta=12.15266$,
$T/T_{\mbox{\tiny{c}}}=0.86$ combining both the short- and large-distance
approximations of the spin-spin Ising correlator, which are respectively given
by eq. (7) and by eq. (28). The zoom in the inset shows the deviations from
the large-distance approximation for $6<R<10$. Figure 4: Same data as in fig.
3 but with the large-distance behavior subtracted. Figure 5: Same as fig. 4,
but taking into account the first few states of the Nambu-Gotō spectrum and
the boundary term.

###  6.2 Identification of EST corrections beyond the Nambu-Gotō approximation

The most efficient way to identify EST corrections beyond the Nambu-Gotō
approximation is to study the behavior of the ground-state energy $E_{0}$ as a
function of the temperature, as the deconfinement transition is approached
from below, at a fixed value of $\beta$ (i.e. at fixed lattice spacing $a$)
and varying the temperature by changing the value of $N_{t}$. For this
purpose, we performed additional sets of simulations, whose details are
reported in tab. 4, in tab. 5, and in tab. 6. For each of the configuration
ensembles produced in these runs, we extracted the value of $E_{0}$ (defined
as the inverse of the correlation length $\xi$) from a fit to eq. (28) at
sufficiently large distances. The results of these fits are reported in tab.
7, in tab. 8, and in tab. 9, respectively.

Studying the EST corrections through the analysis of the ground-state energy
has two main advantages. Firstly, in this setting the Nambu-Gotō expectation
for $E_{0}$ is exactly known and is given by eq. 17, which we recall here

$E_{0}=N_{t}\sigma(N_{t})=N_{t}\sigma_{0}\sqrt{1-\frac{\pi}{3N_{t}^{2}\sigma_{0}}}=N_{t}\sigma_{0}\sqrt{1-\frac{T^{2}}{T_{c,NG}^{2}}}.$
(29)

Note that eq. (29) predicts a mean-field critical index $\nu=1/2$ for the
correlation length $\xi=1/E_{0}$: this is obviously incompatible with the
prediction from the Svetitsky-Yaffe mapping, from which one would expect the
two-dimensional Ising critical index $\nu=1$, i.e. a linear scaling as $T\to
T_{\mbox{\tiny{c}}}$ from below:

$\xi\sim\bigg{(}1-\frac{T}{T_{\mbox{\tiny{c}}}}\bigg{)}^{-1},$ (30)

or, equivalently:

$\xi\sim\bigg{(}1-\frac{N_{t,c}}{N_{t}}\bigg{)}^{-1},$ (31)

which translates into the following _Ansatz_ for the form of the ground-state
energy in the vicinity of the deconfinement point:

$E_{0}\sim 1-\frac{N_{t,c}}{N_{t}}.$ (32)

Thus, the dependence of $E_{0}$ on the temperature is an ideal probe for
corrections with respect to the Nambu-Gotō approximation.

Moreover, this approach follows very closely the one used in the bootstrap
analysis of ref. [25, 26]. From those works, using the low-energy universality
we can search for numerical evidence of the first correction to the Nambu-Gotō
approximation, which is expected to appear at the order $1/N_{t}^{7}$ and can
be parametrized as

$E_{0}(N_{t})=N_{t}\sigma_{0}\sqrt{1-\frac{\pi}{3N_{t}^{2}\sigma_{0}}}-\frac{32\pi^{6}\gamma_{3}}{225\sigma_{0}^{3}N_{t}^{7}}.$
(33)

$\beta$ | $N_{t}$ | $T/T_{\mbox{\tiny{c}}}$ | $n_{\mbox{\tiny{conf}}}$
---|---|---|---
$\beta=9$ | $6$ | $0.935$ | $2.0\times 10^{5}$
$7$ | $0.801$ | $2.0\times 10^{5}$
$8$ | $0.701$ | $2.0\times 10^{5}$
$9$ | $0.623$ | $2.0\times 10^{5}$
$10$ | $0.561$ | $2.0\times 10^{5}$
$11$ | $0.510$ | $2.0\times 10^{5}$
$12$ | $0.468$ | $2.0\times 10^{5}$
Table 4: Information on the simulations at $\beta=9$. $\beta$ | $N_{t}$ | $T/T_{\mbox{\tiny{c}}}$ | $n_{\mbox{\tiny{conf}}}$
---|---|---|---
$\beta=12.15266$ | $8$ | $0.960$ | $2.0\times 10^{5}$
$9$ | $0.853$ | $2.0\times 10^{5}$
$10$ | $0.768$ | $2.0\times 10^{5}$
$11$ | $0.698$ | $2.0\times 10^{5}$
$12$ | $0.640$ | $2.0\times 10^{5}$
$13$ | $0.591$ | $2.0\times 10^{5}$
$14$ | $0.549$ | $2.0\times 10^{5}$
Table 5: Information on the simulations at $\beta=12.15266$. $\beta$ | $N_{t}$ | $T/T_{\mbox{\tiny{c}}}$ | $n_{\mbox{\tiny{conf}}}$
---|---|---|---
$\beta=13.42445$ | $9$ | $0.947$ | $2.0\times 10^{5}$
$10$ | $0.852$ | $2.0\times 10^{5}$
$11$ | $0.775$ | $2.0\times 10^{5}$
$12$ | $0.710$ | $2.0\times 10^{5}$
$13$ | $0.655$ | $2.0\times 10^{5}$
$14$ | $0.609$ | $2.0\times 10^{5}$
$15$ | $0.568$ | $2.0\times 10^{5}$
Table 6: Information on the simulations at $\beta=13.42445$. $N_{t}$ | $R_{min}$ | $R_{max}$ | $k_{l}$ | $E_{0}$ | $\chi^{2}_{\tiny\mbox{red}}$
---|---|---|---|---|---
$6$ | $10$ | $47$ | $0.037(5)$ | $0.025(3)$ | $1.19$
$7$ | $8$ | $47$ | $0.0507(16)$ | $0.088(3)$ | $1.27$
$8$ | $8$ | $47$ | $0.047(3)$ | $0.123(3)$ | $1.56$
$9$ | $6$ | $47$ | $0.051(5)$ | $0.170(4)$ | $1.30$
$10$ | $7$ | $47$ | $0.0439(17)$ | $0.195(3)$ | $2.34$
$11$ | $6$ | $47$ | $0.0452(11)$ | $0.235(2)$ | $0.78$
$12$ | $6$ | $47$ | $0.0413(13)$ | $0.2623(16)$ | $0.33$
Table 7: Best-fit estimates for $E_{0}$ for different values of $N_{t}$ at $\beta=9$. $N_{t}$ | $R_{min}$ | $R_{max}$ | $k_{l}$ | $E_{0}$ | $\chi^{2}_{\tiny\mbox{red}}$
---|---|---|---|---|---
$8$ | $13$ | $47$ | $0.0320(10)$ | $0.0128(3)$ | $1.87$
$9$ | $14$ | $47$ | $0.039(3)$ | $0.0429(14)$ | $1.34$
$10$ | $8$ | $47$ | $0.040(4)$ | $0.068(3)$ | $1.72$
$11$ | $6$ | $47$ | $0.0427(18)$ | $0.093(3)$ | $1.01$
$12$ | $6$ | $47$ | $0.042(3)$ | $0.114(2)$ | $0.38$
$13$ | $6$ | $47$ | $0.0404(19)$ | $0.133(3)$ | $0.28$
$14$ | $6$ | $47$ | $0.0377(11)$ | $0.148(1)$ | $0.60$
Table 8: Best-fit estimates for $E_{0}$ for different values of $N_{t}$ at $\beta=12.15266$. $N_{t}$ | $R_{min}$ | $R_{max}$ | $k_{l}$ | $E_{0}$ | $\chi^{2}_{\tiny\mbox{red}}$
---|---|---|---|---|---
$9$ | $10$ | $47$ | $0.031(3)$ | $0.0134(10)$ | $1.70$
$10$ | $10$ | $47$ | $0.036(10)$ | $0.037(4)$ | $1.56$
$11$ | $6$ | $47$ | $0.040(5)$ | $0.0605(25)$ | $1.05$
$12$ | $6$ | $47$ | $0.0395(15)$ | $0.0785(30)$ | $1.32$
$13$ | $6$ | $47$ | $0.0396(15)$ | $0.097(4)$ | $0.69$
$14$ | $6$ | $47$ | $0.0385(12)$ | $0.1127(35)$ | $0.45$
$15$ | $6$ | $47$ | $0.0362(18)$ | $0.1257(26)$ | $0.92$
Table 9: Best-fit estimates for $E_{0}$ for different values of $N_{t}$ at
$\beta=13.42445$.

Following these observations, we first tried to fit the data using a
functional form motivated by the Nambu-Gotō model ($\nu=1/2$) and one based on
the Ising model ($\nu=1$). Unsurprisingly, we found that neither choice
describes the data accurately: this is clearly visible in fig. 6, in fig. 7,
and in fig. 8. For all the three $\beta$ values the Nambu-Gotō curve fits the
data well at large $N_{t}$ (i.e. at low temperature), but it misses the
approach to the critical point, when the deconfinement transition is
approached. On the contrary, the linear fit agrees with the data near the
critical point, as expected from the Svetitsky-Yaffe correspondence, but this
agreement holds only for the first few values of $N_{t}$. For larger values, a
linear fit is not consistent with the Monte Carlo data.

This failure is indeed in agreement with the low-energy universality: the
latter suggests that the correct behavior at short distances should not be
modelled by assuming an expression like the one in eq. (32), but rather by
adding a suitable $1/N_{t}^{7}$ correction to the Nambu-Gotō approximation.

Moreover, since low-energy universality suggests that, starting from
$O(N_{t}^{-7})$, there may also be any possible higher-order corrections, we
truncate for consistency the Nambu-Gotō expression to this order. Based on
this reasoning, we assume the following form for the $N_{t}$ dependence of the
ground state energy:

$E_{0}(N_{t})=\mbox{Taylor}_{4}(E_{0})+\frac{k_{4}}{(\sigma_{0})^{3}N_{t}^{7}},$
(34)

with

$\mbox{Taylor}_{4}(E_{0})\equiv\sigma_{0}N_{t}-\frac{\pi}{6N_{t}}-\frac{\pi^{2}}{72(\sigma_{0})N_{t}^{3}}-\frac{\pi^{3}}{432(\sigma_{0})^{2}N_{t}^{5}}-\frac{5\pi^{4}}{10368(\sigma_{0})^{3}N_{t}^{7}},$
(35)

where the Taylor expansion is completely known, and the only free parameters
of the fit are the zero-temperature string tension $\sigma_{0}$ and the
$k_{4}$ coefficient.

$\beta$ | $N_{t,min}$ | $N_{t,max}$ | $k_{4}$ | $\sigma_{0}$ | $\chi^{2}_{\tiny\mbox{red}}$ | literature
---|---|---|---|---|---|---
$9$ | $6$ | $12$ | $0.031(13)$ | $0.02587(9)$ | $2.21$ | $0.02583(3)$
$12.15266$ | $8$ | $14$ | $0.054(4)$ | $0.01358(5)$ | $0.57$ | $0.01371(29)$
$13.42445$ | $9$ | $15$ | $0.043(11)$ | $0.01104(11)$ | $0.13$ | $0.01108(23)$
Table 10: Results of the fits of our numerical data to eq.(34). In the last
column we report the values of $\sigma_{0}$ quoted in ref. [36] for $\beta=9$,
and in ref. [32] for $\beta=12.15266$ and for $\beta=13.42445$. Figure 6: Fits
of our data for the ground-state energy $E_{0}$ at $\beta=9$ according to the
prediction from the Nambu-Gotō model, eq. (29) (red line) and according to the
_Ansatz_ of a linear dependence of $E_{0}$ on the temperature, according to
eq. (32) (blue curve). Note that the quantity on the horizontal axis of this
plot is the inverse of the temperature, in units of the lattice spacing.
Figure 7: Same as in fig. 6, but for the data at $\beta=12.15266$. Figure 8:
Same as in fig. 6, but for $\beta=13.42445$. Figure 9: Fit of our numerical
results for the ground-state energy at $\beta=9$ to eq. (34). Figure 10: Same
as in fig. 9, but for $\beta=12.15266$. Figure 11: Same as in fig. 9, but for
$\beta=13.42445$.

Remarkably, these fits yield very good reduced $\chi^{2}$ values for the data
at all the three $\beta$ values. The detailed results are reported in tab. 10,
and shown in fig. 9, in fig. 10 and in fig. 11. In particular, the quality of
the fits improves as one goes toward the continuum limit. Moreover, the best-
fit values obtained for $\sigma_{0}$ are fully consistent with those that were
independently obtained in ref. [36] (for $\beta=9$) and in ref. [32] (for
$\beta=12.15266$ and for $\beta=13.42445$). In fact, the precision of our
results for $\sigma_{0}$ is even better than the one from the latter
reference, so that, in principle, the approach that we used for the
determination of $\sigma_{0}$ could even be used for scale setting (although
much more precise scale-setting methods exist today [76, 77]).

Another non-trivial consistency check of our analysis is that the three values
of $k_{4}$ that we found should agree within their uncertainties, since the
scale dependence of this coefficient is already accounted for by the
$1/\sigma_{0}^{3}$ normalization in eq. (34). As tab. 10 shows, this
expectation is indeed confirmed in our fit results, and the three values are
nearly compatible within their errors. As our final result for $k_{4}$, we
quote the weighted mean of three values: $k_{4}=0.046(8)$ from which we can
extract, using eq. (21),

$\gamma_{3}=-\frac{225}{32\pi^{6}}k_{4}=-0.00034(6),$ (36)

which is well inside the bound $\gamma_{3}\geq-\frac{1}{768}\simeq-0.0013$
obtained in refs. [25, 26].

The fact that we found a negative value for $\gamma_{3}$ is indeed rather non-
trivial: in particular, it rules out the axionic string _Ansatz_ for the EST
describing this gauge theory.

It is interesting to compare this result for $\gamma_{3}$ with the one
obtained in refs. [19, 22, 78] for the $\operatorname{\mathrm{SU}}(6)$ Yang-
Mills theory in three dimensions, using the lattice data from ref. [79], which
was of similar magnitude but opposite in sign. This shows explicitly that, at
this level of resolution, the EST is not universal anymore: instead it
encodes, as it should, the specific properties of the underlying Yang-Mills
theory.

##  7 Concluding remarks

In this work we discussed the results of a set of high-precision simulations
of the Polyakov-loop correlator in the $\operatorname{\mathrm{SU}}(2)$ lattice
gauge theory in three dimensions. All our simulations were run at finite
temperature, in the vicinity of the deconfinement transition, in the range
$0.8T_{\mbox{\tiny{c}}}\leq T\leq T_{\mbox{\tiny{c}}}$, where it is easier to
compare the simulations with the EST predictions. Moreover, invoking the
Svetitsky-Yaffe conjecture, in this regime one can compare the analytical
solution of the two-dimensional Ising model with the gauge-theory data. The
results of this comparison revealed remarkable agreement between our numerical
results for the Polyakov-loop correlator and the exact expression of spin-spin
correlator of the two-dimensional Ising model. We could extract very precise
values for the ground-state energy $E_{0}$ of the effective string describing
this gauge theory, and quantify the deviations from the predictions that can
be derived approximating the effective string action with the Nambu-Gotō
action.

We conclude with some comments on these results.

  1. 1.

The type of comparison that we carried out is not limited to theories having a
critical point in the universality class of the two-dimensional Ising model
(or to another exactly integrable model). Indeed, in principle conformal
perturbation theory allows one to work out the form of the spin-spin
correlator for any model characterized by a continuous phase transition. As a
consequence, the Svetitsky-Yaffe mapping can be used also for spin models that
are not exactly integrable. An example of this approach was discussed in ref.
[61], where results for the $\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory
in four dimensions were compared with the Ising model in three dimensions.

  2. 2.

As we discussed in section 5, an interesting consequence of the effective
string description is that the large-distance behavior of the correlator of a
Yang-Mills theory in $(d+1)$ dimensions is dominated by a $K_{(d-2)/2}(E_{0})$
Bessel function, in exact agreement with the large-distance behavior predicted
by the Svetitsky-Yaffe conjecture for the spin-spin correlator of the
underlying $d$ dimensional spin model. This holds for any EST (under very mild
conditions) and for any spin model, and can be regarded as a non-trivial check
of mutual consistency for the effective string and the Svetitsky-Yaffe
conjecture.

  3. 3.

The value that we found for the $\gamma_{3}$ coefficient for the EST
describing the $\operatorname{\mathrm{SU}}(2)$ Yang-Mills theory in three
dimensions is incompatible with the axionic string _Ansatz_. It would be
interesting to extend our analysis to the $\operatorname{\mathrm{SU}}(4)$
gauge theory, for which results compatible with the axionic string _Ansatz_
were recently obtained in refs. [80, 81]. Similarly, it would also be
interesting to explore the same type of contribution to the EST describing
gauge theories in four dimensions, as the axionic string is expected to play
an important role in the description of the low-energy dynamics of quantum
chromodynamics [21].

  4. 4.

Historically, one of the problems of the EST description of Yang-Mills
theories was its universality, i.e. the fact that it predicted essentially the
same behavior (with only a mild dependence on the number of spacetime
dimensions), for models as different as the three-dimensional $\mathbb{Z}_{2}$
gauge model as the four-dimensional $\operatorname{\mathrm{SU}}(3)$ Yang-Mills
theory. This feature is now understood as a universality that manifests itself
only at low-energy (or, equivalently, a side-effect of the high accuracy of
the Nambu-Gotō approximation of EST), while the details related to the gauge
group (and, possibly, to the confining mechanism into play) may be encoded in
the higher-order EST corrections, which are not expected to be universal. In
view of the fact that our results indicate that the three-dimensional
$\operatorname{\mathrm{SU}}(2)$ gauge theory cannot be described by an axionic
effective string, while the opposite conclusion was recently obtained for the
$\operatorname{\mathrm{SU}}(6)$ Yang-Mills theory [19, 22, 78], quantifying
these corrections and understanding how they depend on the gauge group would
be of great importance. The numerical precision of current Monte Carlo studies
of lattice gauge theory is sufficient to probe the fine details of the
effective string theory at very high orders, and to test the accurate
theoretical predictions that have been formulated during the past few years
[13]. This is possible thanks not only to the increase in computing power, but
also to the deployment of increasingly sophisticated simulation algorithms,
among which we would like to mention non-equilibrium Monte Carlo calculations
[82, 83] based on Jarzynski’s theorem [84, 85]: as an example of their use for
the problems discussed in the present paper, we refer the readers to the
recent study of terms $O(L^{-7})$ in the effective-string description of a
fluctuating interface of linear size $L$ [86].

Acknowledgements

The numerical simulations were run on machines of the Consorzio
Interuniversitario per il Calcolo Automatico dell’Italia Nord Orientale
(CINECA). We acknowledge support from the SFT Scientific Initiative of INFN.

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