compat=1.0.0

Lattice Strong Dynamics (LSD) Collaboration

# Hidden Conformal Symmetry from the Lattice

T. Appelquist  Department of Physics, Sloane Laboratory, Yale University, New
Haven, Connecticut 06520, USA  R. C. Brower  Department of Physics and Center
for Computational Science, Boston University, Boston, Massachusetts 02215, USA
K. K. Cushman  Department of Physics, Sloane Laboratory, Yale University, New
Haven, Connecticut 06520, USA  G. T. Fleming  Department of Physics, Sloane
Laboratory, Yale University, New Haven, Connecticut 06520, USA  Theoretical
Physics Division, Fermilab, Batavia, IL 60510, USA  A. Gasbarro  AEC Institute
for Theoretical Physics, University of Bern, 3012 Bern, CH  A. Hasenfratz
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
J. Ingoldby<EMAIL_ADDRESS>Abdus Salam International Centre for Theoretical
Physics, Strada Costiera 11, 34151, Trieste, Italy  X. Y. Jin  Computational
Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
E. T. Neil  Department of Physics, University of Colorado, Boulder, Colorado
80309, USA  J. C. Osborn  Computational Science Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA  C. Rebbi  Department of Physics and
Center for Computational Science, Boston University, Boston, Massachusetts
02215, USA  E. Rinaldi  Interdisciplinary Theoretical and Mathematical
Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
D. Schaich  Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 7ZL, UK  P. Vranas  Physical and Life Sciences Division,
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  E. Weinberg  Department of Physics and Center for
Computational Science, Boston University, Boston, Massachusetts 02215, USA
NVIDIA Corporation, Santa Clara, California 95050, USA  O. Witzel  Center for
Particle Physics Siegen (CPPS), Theoretische Physik 1, Naturwissenschaftlich-
Technische Fakultät, Universität Siegen, 57068 Siegen, Germany

###### Abstract

We analyze newly expanded and refined data from lattice studies of an $SU(3)$
gauge theory with eight Dirac fermions in the fundamental representation. We
focus on the light composite states emerging from these studies, consisting of
a set of pseudoscalars and a single light scalar. We first consider the view
that this theory is just outside the conformal window. In this case, the
pseudoscalars arise from spontaneous breaking of chiral symmetry. Identifying
the scalar in this case as an approximate dilaton, we fit the lattice data to
a dilaton effective field theory, finding that it yields a good fit even at
lowest order. For comparison, we then consider the possibility that the theory
is inside the conformal window. The fermion mass provides a deformation,
triggering confinement. We employ simple scaling laws to fit the lattice data,
and find that it is of lesser quality.

††preprint: RIKEN-iTHEMS-Report-23

##  I Introduction

The study of the Higgs boson at the LHC shows it to be well described by the
standard model [1, 2, 3, 4]. This description could, however, evolve into
something more fundamental at higher energy scales, one possibility being
compositeness emerging from underlying strong interactions. The relative
lightness of the Higgs boson could then be accounted for if it is a pseudo-
Nambu-Goldstone boson (pNGB) generated by the spontaneous breaking of an
approximate internal global symmetry [5, 6, 7, 8] or a conformal symmetry, the
latter leading to a description of the Higgs boson as an approximate dilaton
[9]. In recent studies [10, 11], a composite Higgs boson is interpreted as an
admixture of pNGB and dilaton.

Important elucidation of these ideas has come from the study of a class of
lattice $SU(N)$ gauge field theories [12, 13, 14, 15]. With the number $N_{f}$
of light Dirac fermions taken to be relatively large compared to the number of
colors, these theories can be near or within the “conformal window” where
confinement gives way to infrared conformality. For the gauge group $SU(3)$,
the smallest such value is thought to be around $9$, but the exact value is
not known [16, 17, 18, 19].

Lattice studies of the $N_{f}=8$ theory [20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30] have suggested that it is close to the conformal window boundary. If
it is outside and confining, the light pseudoscalars in the spectrum are pNGBs
of a spontaneously broken internal symmetry. The fermion masses employed in
the lattice calculations give them a small mass. Remarkably, the spectrum also
includes a comparably light scalar particle, possibly attributable to the
theory’s nearness to the conformal window. It could be an approximate dilaton
- the pseudo-Goldstone boson that emerges when the breaking of scale
invariance is dominantly spontaneous. A dilaton-effective-field-theory (dEFT)
description of these states [31, 32, 33, 34, 35, 36, 37, 38, 39] has been
found to fit earlier $N_{f}=8$ lattice datasets well [40, 41, 42, 43, 44].

An alternate possibility is that the $N_{f}=8$ theory is inside the conformal
window. The fermion mass would then provide a deformation of the infrared
conformal symmetry, with confinement triggered at a comparable scale. Simple
scaling laws describe the masses and decay constants of all the composite
states [45], which extrapolate to zero in the zero-fermion-mass limit. There
is no approximate conformal or internal global symmetry to be spontaneously
broken. There would be no evident EFT, the scaling laws being described
directly in terms of the underlying gauge theory.

In this letter, we reexamine the light spectrum of the $SU(3)$ gauge theory
with $N_{f}=8$, drawing on our latest lattice data. We first consider the view
that it is outside the conformal window. The dEFT provides a good fit to the
lattice data, now incorporating data for a scalar decay constant $F_{S}$. We
also consider the possibility that the theory is inside the conformal window,
described in the infrared by mass-deformed conformal symmetry. We find that
this description is less consistent with the lattice data than dEFT.

##  II Summary of Lattice Results

We draw on lattice data presented in a companion paper [46] for the
pseudoscalar mass $M_{\pi}$, the pseudoscalar decay constant $F_{\pi}$, the
mass of the light scalar $M_{d}$ (identified with the dilaton in dEFT), and a
scalar decay constant $F_{S}$, in units of the lattice spacing. A common mass
is given to all eight flavors, and we compute data for five different values
of this fermion mass. The bare gauge coupling is the same for all data points.

The expanded and refined dataset in Ref. [46] draws on several new ingredients
relative to previous LSD-collaboration studies of the $N_{f}=8$ theory [26,
27]. These include infinite volume extrapolations and improved estimates of
other systematic errors. We find that the statistical covariances between
different observables are small and we neglect them. The data are summarized
in Table IX of Ref. [46].

We also include data for $2\rightarrow 2$ pNGB scattering presented in Ref.
[44]. These are for $k\cot\delta(k)$ in the s–wave, $I=2$ isospin channel. On
the largest lattice volumes shown there, the scattering momentum $k$ is very
small and can be neglected. We therefore identify these data points with the
inverse scattering length $1/a^{(2)}_{0}$, expressed in units of the lattice
spacing.

Since we consider two interpretations of the lattice data, we employ the
Akaike Information Criterion (AIC) [47] to help assess which provides the
better description. It is given by

$\text{AIC}\equiv\chi_{\text{min}}^{2}+2k\,,$ (1)

where $k$ is the number of free parameters in the model. The term
$\chi^{2}_{\text{min}}$ is the standard chi-squared function for the fit to
data, minimized with respect to the model parameters. Models having a smaller
AIC value may be interpreted as more probable [48] in a Bayesian framework.

##  III Dilaton EFT

To describe the spontaneous breaking of approximate dilatation symmetry, we
introduce a scalar dilaton field $\chi$, which acquires a nonzero vacuum value
$\langle\chi\rangle=f_{d}$. There is also an approximate global symmetry
$SU(N_{f})\times SU(N_{f})$ that is spontaneously broken to a subgroup
$SU(N_{f})$, so we introduce a multiplet of pNGB fields $\Sigma$ satisfying
the constraint $\Sigma^{\dagger}\Sigma=\mathbf{1}$.

The EFT Lagrangian, reviewed in Ref. [39], is given by

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\,+\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{d}}\right)^{2}\,\mbox{Tr}\left[\partial_{\mu}\Sigma(\partial^{\mu}\Sigma)^{\dagger}\right]\\\
+\frac{m_{\pi}^{2}f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{d}}\right)^{y}\,\mbox{Tr}\left[\Sigma+\Sigma^{\dagger}\right]\,-\,V_{\Delta}(\chi)\,,$
(2)

where we include only the leading order (LO) terms in a low energy expansion.
We identify $m^{2}_{\pi}=2B_{\pi}m$, where $m$ is the fermion mass in the
underlying gauge theory and $B_{\pi}$ is a constant. The exponent $y$ is taken
as a free parameter in fits to lattice data.

We take the scalar potential to be [9]

$V_{\Delta}(\chi)=\frac{m_{d}^{2}\chi^{4}}{4(4-\Delta)f_{d}^{2}}\left[1-\frac{4}{\Delta}\left(\frac{\chi}{f_{d}}\right)^{\Delta-4}\right]\,,$
(3)

where $\Delta$ is a free parameter determined from the lattice data. In the
limit $\Delta\rightarrow 4$, this potential takes a logarithmic form.

With $m^{2}_{\pi}$ nonzero, $\chi$ develops a new vacuum value
$\langle\chi\rangle=F_{d}$. By minimizing the potential terms for the dilaton
in Eqs. (2) and (3), we find

$\frac{F_{d}^{4-y}}{(4-\Delta)f^{4-y}_{d}}\left[1-\left(\frac{f_{d}}{F_{d}}\right)^{4-\Delta}\right]=\frac{yN_{f}f^{2}_{\pi}m^{2}_{\pi}}{2f^{2}_{d}m^{2}_{d}}.$
(4)

By normalizing the kinetic term for the pNGB field in the new vacuum, we
arrive at the scaling relations

$\displaystyle\frac{F_{\pi}^{2}}{f_{\pi}^{2}}=\frac{F_{d}^{2}}{f_{d}^{2}}\,,\qquad\qquad\frac{M_{\pi}^{2}}{m_{\pi}^{2}}=\left(\frac{F_{d}}{f_{d}}\right)^{y-2}\,.$
(5)

The physical dilaton mass is given by

$\frac{M^{2}_{d}}{F^{2}_{d}}=\frac{m^{2}_{d}}{(4-\Delta)f^{2}_{d}}\left(4-y+(y-\Delta)\left(\frac{f_{d}}{F_{d}}\right)^{4-\Delta}\right).$
(6)

The dEFT prediction [44] for the s-wave scattering length for two pNGBs in the
isospin-2 channel is given by

$\displaystyle M_{\pi}a^{(2)}_{0}=-\frac{M^{2}_{\pi}}{16\pi
F^{2}_{\pi}}\left(1-(y-2)^{2}\frac{f^{2}_{\pi}}{f^{2}_{d}}\frac{M^{2}_{\pi}}{M^{2}_{d}}\right)\,.$
(7)

The first term in the large parentheses would be present if there was no light
dilaton in the spectrum [49]. The second term comes from the exchange of a
virtual dilaton.

We include an additional quantity that we have recently measured on the
lattice: the scalar decay constant $F_{S}$ defined in terms of the matrix
element

$\left<0\vphantom{J_{S}(x)\chi(p)}\right|J_{S}(x)\left|\chi(p)\vphantom{0J_{S}(x)}\right>\equiv
F_{S}M_{d}^{2}e^{-p\cdot x}\,,$ (8)

where $\left|\chi(p)\right>$ refers to the dilaton state and
$J_{S}(x)=m\bar{\psi}\psi$ is the scheme independent scalar current [50]. The
quantity $F_{S}$ may be extracted from the dilaton contribution to the two
point correlator $\langle J_{S}(x)J_{S}(0)\rangle$. The technique is
demonstrated for the chiral Lagrangian EFT in Refs. [50, 51]. We find

$|F_{S}|=\frac{yN_{f}M^{2}_{\pi}F_{\pi}}{2M^{2}_{d}}\frac{f_{\pi}}{f_{d}}.$
(9)

A related formula for $F_{S}$ was derived using current algebra and compared
with lattice data in Ref. [52].

The quantity $F_{S}$ would control the decay rate of the dilaton if the gauge
theory were modified by including a heavy scalar mediator with a weak Yukawa
coupling to the fermion bilinear and to other light states. The mediator would
then enable the decay of the dilaton to these states. Both $F_{S}$ and this
decay rate would vanish in the chiral limit since the scalar current
$m\bar{\psi}\psi$ defining $F_{S}$ and the Yukawa interaction that enables the
decay both break chiral symmetry.

We perform a global fit at leading order to the lattice data for $M_{\pi}$,
$F_{\pi}$, $M_{d}$, $F_{S}$ and $M_{\pi}\,a^{(2)}_{0}$. This is the first time
that lattice data for $F_{S}$ has been used in an EFT fit. The fit
incorporates $N=25$ lattice data points and $k=6$ free model parameters, which
we take to be
$\\{y,\,B_{\pi},\,\Delta,\,f^{2}_{\pi},\,f^{2}_{\pi}/f^{2}_{d},\,m^{2}_{d}/f^{2}_{d}\\}$.
We obtain dEFT predictions for the quantities measured on the lattice by first
solving Eq. (4) to obtain $F_{d}/f_{d}$. We determine $M_{\pi}$ and $F_{\pi}$
from the scaling relations shown in Eq. (5), and we use Eqs. (6), (7) and (9)
to determine the remaining quantities.

Parameter | LO | NLO
---|---|---
$y$ | 2.091(32) | 2.069(32)
$B_{\pi}$ | 2.45(13) | 2.46(13)
$\Delta$ | 3.06(41) | 2.88(49)
$f^{2}_{\pi}$ | $6.1(3.2)\times 10^{-5}$ | $5.8(3.4)\times 10^{-5}$
$f_{\pi}^{2}/f_{d}^{2}$ | 0.1023(35) | 0.1089(41)
$m_{d}^{2}/f_{d}^{2}$ | 1.94(65) | 2.24(80)
$l_{a}$ | — | 0.78(27)
$\chi^{2}/\text{dof}$ | 21.3/19 | 10.3/18
AIC | 33.3 | 24.3
Table 1: Central values and uncertainties for LO (six-parameter) and NLO
(seven-parameter) fits to the dEFT. Lattice data for $M_{\pi}$, $M_{d}$,
$F_{\pi}$, $F_{S}$ and the $I=2$ scattering length has been incorporated into
this fit, for 5 different vales of the underlying fermion mass $m$,
corresponding to 25 data points. All dimensionful quantities are presented in
units of the lattice spacing.

We construct a chi-square function and minimize it with respect to the six
model parameters to find $\chi^{2}_{\text{min}}/N_{\text{dof}}=1.12$,
indicating that lowest-order dEFT already provides a remarkably good fit to
this expanded dataset.

At the chi-square minimum, the model parameters take the best-fit values shown
in the LO column of Table 1. The fit parameters are consistent with those from
earlier dEFT studies of the $N_{f}=8$ theory [40, 41, 42], which do not use
$F_{S}$ or scattering data. There are also differences with respect to the fit
results of Ref. [44], which we attribute to an improved analysis of systematic
errors. Table 1 also reports the AIC value (Eq. (1)).

Figure 1: Lattice data for $M_{\pi}$, $M_{d}$, $F_{\pi}$, $F_{S}$ plotted
against the underlying fermion mass $m$. The lines represent the dependence of
these quantities on fermion mass predicted by LO dEFT, with the six EFT
parameters set to their central values indicated in Table 1.

In Fig. 1, we plot lattice data for the masses and decay constants for several
values of the fermion mass. The plots confirm that dEFT accurately describes
this lattice data, and show predictions for these quantities at smaller values
of the quark mass, which can be checked in the future. In the $m\rightarrow 0$
limit, we expect $F_{\pi}$ and $M_{d}$ to extrapolate to positive nonzero
values.

Since all the quantities appearing on the right side of Eq. (9) were well
determined in earlier dEFT fits [40, 41, 42, 43, 44], Eq. (9) can be viewed as
a dEFT prediction for $F_{S}$. The new lattice measurements of $F_{S}$ align
nicely with this prediction, providing new evidence for the dilaton
interpretation of the light scalar.

Figure 2: The points represent lattice determinations of the pNGB scattering
length in the $I=2$ channel plotted against $M^{2}_{\pi}/F^{2}_{\pi}$. The
gray dashed line represents the LO dEFT prediction.

In Fig. 2, we plot the s–wave pNGB scattering length in the $I=2$ isospin
channel against $M_{\pi}^{2}/F_{\pi}^{2}$. We assume here that the scattering
length is well approximated by $\left(k\cot\delta\right)^{-1}$ measured on the
largest available lattice volume. The dEFT prediction, represented by the gray
dashed line in Fig. 2, is determined by inputting the LO central values for
the $6$ dEFT parameters from Table 1 into Eqs. (4)–(7). Since $y-2\ll 1$ and
$f_{\pi}^{2}/f_{d}^{2}\ll 1$, the second term in Eq. (7) is negligible
compared with the first, and the gray dashed line is approximately straight
with a gradient of $-1/16\pi$. The dashed line passes close to all the points
in Fig. 2, indicating agreement between leading–order dEFT and the scattering
data, similarly to what was shown in Ref. [44].

Finally we note that including data for $F_{S}$ in the global fit reduced the
uncertainty in $\Delta$ considerably compared to previous fits, such that the
value $\Delta\rightarrow 4$ falls outside the $1\sigma$ range, which runs from
2.5 to 3.5.

##  IV Mass–Deformed Conformal Theory

While dEFT provides a good fit to the lattice data, indicating that the
$N_{f}=8$ theory is outside the conformal window, it is important to test the
alternate possibility that the theory is inside the conformal window,
described in the infrared by mass-deformed conformal symmetry. The analysis is
simplest assuming that the gauge coupling can be approximated by its value at
the infrared fixed point. This approach was employed in Ref. [45] to describe
the $SU(3)$ gauge theory with $N_{f}=12$ Dirac fermions.

We proceed in the same manner, with the caveat that in the present case, the
gauge coupling in the interval between the UV cutoff $\Lambda$ and the induced
confinement scale might not have evolved to be sufficiently close to its fixed
point value [53]. With the assumption that it has, at scales $\mu\ll\Lambda$,
the running fermion mass can be approximated by
$m(\mu)=m(\Lambda/\mu)^{\gamma^{*}}$. Here $\gamma^{*}$ is the mass anomalous
dimension at the fixed point, and $m=m(\Lambda)$ is the input fermion mass
defined at the UV cutoff, which we identify with the inverse lattice spacing.

A simple scaling argument then leads to the result that for each physical
state $X$, the mass $M_{X}$ is given to leading order (LO) by

$\displaystyle M_{X}=C_{X}m^{[1/(1+\gamma^{*})]}\,,$ (10)

With all masses in units of $\Lambda$, the dimensionless coefficients $C_{X}$
are expected to be of order unity. An analogous expression applies to
$F_{\pi}$ and $F_{S}$.

We also expect other quantities with dimensions of mass to scale with $m$ in a
similar way. For example, the inverse scattering length should scale as

$\displaystyle 1/a^{(2)}_{0}=C_{a}\,m^{[1/(1+\gamma^{*})]}\,.$ (11)

We use these LO expressions to fit the lattice data for the masses and decay
constants of the pseudoscalars $\pi$ and scalar $S$, as well as the inverse
s–wave pseudoscalar scattering length in the $I=2$ isospin channel,
$1/a_{0}^{2}$.

Our leading-order fit uses six parameters, the same number as with the
leading-order dEFT fit. The data set is also the same, employing five fermion
mass values. The result is shown in Table 2. A comparison with the leading-
order dEFT fit of Table 1 shows that the CFT fit has an AIC of $60.1$ compared
to the dEFT AIC of $33.3$. The dEFT fit is thus favored over the CFT fit.

Parameter | LO | NLO 1 | NLO 2
---|---|---|---
$C_{M_{\pi}}$ | 2.121(78) | 1.56(11) | 1.57(12)
$C_{F_{\pi}}$ | 0.522(19) | 0.445(21) | 0.448(23)
$C_{M_{d}}$ | 2.97(12) | 2.53(12) | 2.55(13)
$C_{F_{S}}$ | 0.706(33) | 0.599(33) | 0.459(63)
$C_{a}$ | -5.88(22) | -5.05(24) | -5.86(53)
$\gamma^{*}$ | 1.073(28) | 1.207(41) | 1.200(44)
$D_{M_{\pi}}$ | — | 4.80(87) | 4.71(90)
$D_{F_{S}}$ | — | — | 2.77(98)
$D_{a}$ | — | — | 12.9(5.8)
$\chi^{2}/\text{dof}$ | 48.1/19 | 20.9/18 | 6.90/16
AIC | 60.1 | 34.9 | 24.9
Table 2: Central values and uncertainties for the six parameter leading-order
mass-deformed conformal fit are presented in the column labeled LO. Next-to-
leading order fits are shown in the next two columns. All dimensionful
quantities are expressed in units of the lattice spacing.

##  V Next-to-Leading Order

The infrared properties of the $N_{f}=8$ theory can be tested by the inclusion
of next-to-leading order (NLO) terms in each fit.

dEFT - Since the specific forms of the NLO corrections to the dEFT fit
equations are not all known, we adopt a more limited approach to account for
their effects. It is evident from Fig. 2 that the greatest tension in the LO
dEFT fit arises from the lattice data for the $I=2$ scattering length. There
are NLO corrections to this quantity involving higher powers of
$M_{\pi}^{2}/(4\pi F_{\pi})^{2}$ together with chiral logarithms. As a first
NLO step, we modify only the LO expression, Eq. (7), by the addition of a
single term of relative order $M_{\pi}^{2}/(4\pi F_{\pi})^{2}$, disregarding
the possible presence of a chiral logarithm. Thus we employ the expression

$\displaystyle M_{\pi}a^{(2)}_{0}=\frac{-M^{2}_{\pi}}{16\pi
F^{2}_{\pi}}\left(1-(y-2)^{2}\frac{f^{2}_{\pi}}{f^{2}_{d}}\frac{M^{2}_{\pi}}{M^{2}_{d}}+\frac{l_{a}M^{2}_{\pi}}{(4\pi
F_{\pi})^{2}}\right)\,,$ (12)

where $l_{a}$ is a new, seventh fit parameter. The result is shown in the NLO
column of Table 1. The six parameters of the LO fit are changed only slightly,
the parameter $l_{a}$ is of order unity, the $\chi^{2}$/dof falls by roughly a
factor of two, and the AIC improves from $33.3$ to $24.3$. The NLO correction
to the scattering length is less than 10% throughout the range of lattice
data. We also performed fits that included similar NLO corrections to the
other measured quantities. However, the AIC for these fits was always higher
than for the NLO fit shown in Table 1.

CFT - Maintaining the assumption that the gauge coupling takes its infrared
fixed point value, higher order corrections to the fit equations (10)–(11)
enter additively as polynomials in $m$. The leading (NLO) correction in each
case is linear in $m$, arising from physics near the UV cutoff. Each can be
expected to be small compared to the LO term provided only that
$\gamma^{*}>0$. For example, we can modify Eq. (10) to read

$\displaystyle M_{X}=C_{X}m^{[1/(1+\gamma^{*})]}+D_{X}\,m\,,$ (13)

where $D_{X}$ is a new fit parameter.

If we modify only a single fit equation in this way, the one that leads to the
smallest AIC is the equation for $M_{\pi}$. The result is shown in the column
labeled NLO 1 in Table 2. The six parameters of the LO fit are changed by
small amounts. While both the $\chi^{2}$/dof and AIC improve, they remain poor
relative to those of the NLO fit for the dEFT.

Further reduction in the $\chi^{2}$/dof and the AIC for the mass-deformed CFT
fit can be achieved by including the analogous NLO modification of other fit
equations. The smallest AIC comes from adding NLO modifications to only the
equations for $F_{S}$ and the $I=2$ scattering length. The result is shown in
the column of Table 2 labeled NLO 2, where the AIC is reduced to $24.9$, which
is comparable to the AIC of the NLO dEFT fit.

But the column labeled NLO 2 also reveals that some of the NLO corrections
required to fit the data are not much smaller than the leading order terms in
this fit. In particular, the NLO correction to $F_{S}$ grows to 46% of the LO
term size at the top of the fermion mass range, indicating that the expansion
in powers of $m$ is not under good theoretical control.

##  VI Discussion

We analyzed the latest data for an $SU(3)$ gauge theory with $N_{f}=8$ Dirac
fermions in the fundamental representation, focusing on the masses and decay
constants of the lightest composite states. These comprise a set of
pseudoscalars and one additional scalar.

We first adopted the view that the gauge theory is just outside the conformal
window, leading to the interpretation of the pseudoscalars as approximate
Nambu-Goldstone bosons. We interpreted the scalar as an approximate dilaton,
associated with the spontaneous breaking of conformal symmetry. We employed a
dilaton effective field theory (dEFT) to fit the lattice data, newly expanded
to include results for a scalar decay constant and the scattering length in
the $I=2$ channel. At leading order, the result of the dEFT fit is summarized
in Table 1. The goodness of the fit is notable.

We then considered the possibility that the gauge theory is inside the
conformal window. In this case, the fermion mass $m$ provides a small
deformation of infrared conformal symmetry. Simple scaling laws were employed
to describe the masses and decay constants of the light states as a function
of $m$. The result of a leading-order (LO) fit, employing six parameters as in
the case of the LO dEFT fit, is summarized in Table 2. The $\chi^{2}$/dof and
AIC are both larger than those of the dEFT, indicating a lower goodness-of-
fit.

We also examined the quality of fits with the inclusion of a next-to-leading-
order (NLO) term. For both the dEFT and the mass-deformed CFT, the goodness-
of-fit was improved but the dEFT fit remained superior. The addition of extra
NLO parameters in the dEFT fit led to no further improvement, while it did
improve the mass-deformed-CFT fit. But to bring this goodness-of-fit to the
level of the dEFT fit led to uncomfortably large NLO corrections to the LO
term.

Finally we note again that both the dEFT and mass-deformed-CFT fits involved
the assumption that the underlying gauge theory is nearly conformal, with the
gauge coupling approximated by its fixed-point value for energies above the
induced confinement scale. The lesser quality of the mass-deformed-CFT fit
could be a consequence of this assumption not being satisfied [53]. Lattice
simulations at stronger couplings might then be required. For the dEFT fit,
the assumption of near-conformality was critical for the identification of the
scalar as an approximate dilaton. It will be interesting to test whether dEFT
describes other systems which could be approximately conformal [54]. The
success of the dEFT fit presented here, incorporating newly expanded and
refined lattice data, is quite striking.

###### Acknowledgements.

R.C.B. and C.R. acknowledge United States Department of Energy (DOE) Award No.
DE-SC0015845. K.C. acknowledges support from the DOE through the Computational
Sciences Graduate Fellowship (DOE CSGF) through grant No. DE-SC0019323, and
also from the P.E.O Scholar award. G.T.F. acknowledges support from DOE Award
No. DE-SC0019061. A.D.G. is supported by SNSF grant No. 200021_17576. A.H. and
E.T.N. acknowledge support by DOE Award No. DE-SC0010005. J.I. acknowledges
support from ERC grant No. 101039756. D.S. was supported by UK Research and
Innovation Future Leader Fellowship No. MR/S015418/1 and STFC grant
ST/T000988/1. P.V. acknowledges the support of the DOE under contract No. DE-
AC52-07NA27344 (Lawrence Livermore National Laboratory, LLNL). We thank the
LLNL Multiprogrammatic and Institutional Computing program for Grand Challenge
supercomputing allocations. We also thank Argonne Leadership Computing
Facility (ALCF) for allocations through the INCITE program. ALCF is supported
by DOE contract No. DE-AC02-06CH11357. Computations for this work were carried
out in part on facilities of the USQCD Collaboration, which are funded by the
Office of Science of the DOE, and on Boston University computers at the
MGHPCC, in part funded by the National Science Foundation (award No.
OCI-1229059). This research utilized the NVIDIA GPU accelerated Summit
supercomputer at Oak Ridge Leadership Computing Facility at the Oak Ridge
National Laboratory, which is supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE- AC05-00OR22725.

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