Lattice Strong Dynamics (LSD) Collaboration

# Goldstone Boson Scattering with a Light Composite Scalar

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  A. Gasbarro
<EMAIL_ADDRESS>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
Physics Department, University of Michigan, Ann Arbor, MI 48109, USA
Theoretical Quantum Physics Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan  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

The appearance of a light composite $0^{+}$ scalar resonance in nearly
conformal gauge–fermion theories motivates further study of the low energy
structure of these theories. To this end, we present a nonperturbative lattice
calculation of s-wave scattering of Goldstone bosons in the maximal-isospin
channel in $\mathrm{SU}(3)$ gauge theory with $N_{f}=8$ light, degenerate
flavors. The scattering phase shift is measured both for different values of
the underlying fermion mass and for different values of the scattering
momentum. We examine the effect of a light flavor-singlet scalar (reported in
earlier studies) on Goldstone boson scattering, employing a dilaton effective
field theory (EFT) at the tree level. The EFT gives a good description of the
scattering data, insofar as the magnitude of deviations between EFT and
lattice data are no larger than the expected size of next-to-leading order
corrections in the EFT.

††preprint: RIKEN-iTHEMS-Report-21††preprint: LLNL-JRNL-823329††preprint: SI-
HEP-2021-18

##  I Introduction

The existence of a light composite $0^{+}$ scalar resonance has been strongly
indicated by lattice calculations in a variety of four-dimensional
gauge–fermion theories near the lower boundary of the conformal window Aoki
_et al._ (2014a, 2013, b, c, 2017); Athenodorou  _et al._ (2015); Fodor  _et
al._ (2015); Rinaldi (2017); Brower  _et al._ (2016); Fodor  _et al._ (2016);
Hasenfratz  _et al._ (2017); Del Debbio  _et al._ (2016); Appelquist  _et al._
(2016, 2019); Gasbarro and Fleming (2017); Gasbarro (2019); Athenodorou  _et
al._ (2017); Fodor  _et al._ (2018); Athenodorou  _et al._ (2021); Brower  _et
al._ (2019). Its presence has sparked renewed interest in theories of a
composite Higgs boson, e.g. Buarque Franzosi  _et al._ (2020); Appelquist  _et
al._ (2021a); Witzel (2019); Brower  _et al._ (2019); Appelquist  _et al._
(2021b); Witzel  _et al._ (2021).

Despite considerable numerical effort to date, little is known from direct
lattice calculation about the light scalar resonance beyond its mass. For
these studies, work is ongoing to control lattice artifacts Fodor  _et al._
(2016) and finite-volume effects Fodor  _et al._ (2020) and to push closer to
the chiral limit Appelquist  _et al._ (2019). Lattice computation near the
conformal window is generally impeded by the long correlation lengths of the
nearly conformal systems, and the disconnected diagrams intrinsic to the
evaluation of correlation functions of flavor singlet operators present
additional challenges. Beyond the singlet scalar two point function, form
factor and scattering amplitudes involving the scalar singlet as an external
state are also desirable to study but present similar or greater numerical
challenges in practice.

In this work, we instead probe the physics of the light singlet scalar through
the lattice computation of pseudo-Nambu–Goldstone boson (PNGB) scattering at
low energies. Chiral symmetry completely constrains PNGB scattering to leading
order when the PNGB momenta and mass $M_{\pi}$ are small compared to other
scales Weinberg (1966). However, the presence of a light $0^{+}$ scalar of
mass $M_{\sigma}$ leads to additional pole terms in the leading-order
expressions for the scattering amplitude whose form cannot be fixed by chiral
symmetry Lai (1969). For the lattice ensembles we consider, $M_{\pi}\approx
M_{\sigma}$, indicating that the scalar pole terms may contribute
significantly if the coupling between the PNGBs and $0^{+}$ scalar is not too
small.

Volume | $am$ | $\tau$ | MDTU | Sep. | $N_{\text{meas}}$ | $L_{\text{Bin}}$ | $N_{\text{Bin}}$
---|---|---|---|---|---|---|---
$24^{3}\times 48$ | 0.00889 | 1.0 | 24480 | 40 | 613 | 4 | 153
$32^{3}\times 64$ | 0.00889 | 1.0 | 5960 | 40 | 150 | 8 | 18
|  |  | 5960 | 40 | 150 | 8 | 18
|  |  | 5960 | 40 | 150 | 8 | 18
|  |  | 5960 | 40 | 150 | 8 | 18
$24^{3}\times 48$ | 0.0075 | 1.0 | 9680 | 40 | 243 | 4 | 60
$32^{3}\times 64$ | 0.0075 | 1.0 | 24600 | 40 | 616 | 4 | 154
$48^{3}\times 96$ | 0.0075 | 1.0 | 19690 | 10 | 1970 | 8 | 246
|  |  | 19580 | 10 | 1959 | 8 | 244
$32^{3}\times 64$ | 0.005 | 2.0 | 2520 | 40 | 64 | 1 | 64
|  |  | 2520 | 40 | 64 | 1 | 64
|  |  | 3160 | 40 | 80 | 1 | 80
$48^{3}\times 96$ | 0.005 | 1.0 | 3950 | 10 | 396 | 8 | 49
|  |  | 3140 | 10 | 315 | 8 | 39
$48^{3}\times 96$ | 0.00222 | 1.0 | 2048 | 16 | 129 | 3 | 43
|  |  | 1168 | 16 | 74 | 3 | 24
$64^{3}\times 128$ | 0.00125 | 0.5 | 1296 | 36 | 37 | 4 | 9
|  |  | 1296 | 36 | 37 | 4 | 9
|  |  | 1296 | 36 | 37 | 4 | 9
|  |  | 1920 | 40 | 49 | 4 | 12
|  |  | 3132 | 36 | 88 | 4 | 22
|  |  | 1080 | 40 | 28 | 4 | 7
|  |  | 3132 | 36 | 88 | 4 | 22
|  |  | 1520 | 40 | 39 | 4 | 9
|  |  | 3132 | 36 | 88 | 4 | 22
|  |  | 920 | 40 | 24 | 4 | 6
Table 1: Summary of statistics for each stream, specifying the lattice volume,
the fermion mass $am$, and the trajectory length $\tau$ (in molecular dynamics
time units, MDTU) used in the configuration generation via the hybrid Monte
Carlo algorithm. All streams use bare coupling $\beta_{F}=4.8$. For each
stream, we perform measurements each “Sep.” MDTU, which provides
$N_{\text{meas}}$ measurements from the stated number of thermalized MDTU. We
reduce autocorrelations by combining $L_{\text{Bin}}$ measurements into
$N_{\text{Bin}}$ ‘binned’ samples, choosing $L_{\text{Bin}}$ by analyzing
autocorrelations of the PNGB and $I=2$ correlation functions.

In this work we investigate the role of the light scalar in s-wave
$\pi\,\pi\to\pi\,\pi$ PNGB scattering at low energies. We employ the Lüscher
finite volume method Lüscher (1991) to extract the scattering phase shift
nonperturbatively on the lattice. The near-conformality of the gauge theory
and the observed spectrum reported in previous publications Appelquist  _et
al._ (2016, 2019) lead us to interpret the lattice results in terms of an
effective field theory (EFT) including a light singlet scalar interpreted as
an approximate dilaton, along with the PNGBs Golterman and Shamir (2016);
Appelquist  _et al._ (2017, 2018a, 2020). Alternative EFT descriptions for the
light scalar have been suggested in Refs. Appelquist  _et al._ (2018b); Hansen
_et al._ (2017).

We focus on a scattering channel, designated as “maximal isospin”, which
contains no fermion-line-disconnected diagrams, allowing us to extract the
scattering length to a high statistical precision. The scalar can then be
exchanged in the $t$ and $u$ channels, contributing to the scattering length
and effective range. In general, the coupling of the $0^{+}$ scalar to the
PNGBs can depend on the details of explicit chiral symmetry breaking, and may
also have dependence on the momenta. In the context of the EFT we employ, it
will be determined by only a small number of EFT parameters.

In Section II, we detail our numerical lattice calculation of low-energy
$\pi\pi$ scattering. In Section III, we apply the dilaton EFT to the
scattering length data, and assess the consistency of the EFT and the ability
of the scattering data to constrain the model. We summarize our results in
Section IV, and discuss open questions.

##  II Lattice Computation of Scattering Phase Shift

Volume | $am_{q}$ | $aM_{\pi}$ | $aF_{\pi}$ | $aM_{\sigma}$ | $a^{2}k^{2}$ | $ak\cot\delta$ | $M_{\pi}^{2}/F_{\pi}^{2}$ | $k^{2}/M_{\pi}^{2}$ | $M_{\pi}/k\cot\delta$
---|---|---|---|---|---|---|---|---|---
$24^{3}\times 48$ | 0.00889 | 0.22754(21) | 0.051862(77) | 0.279(30) | 0.00193(15) | -0.585(38) | 19.250(62) | 0.0374(29) | -0.391(24)
$32^{3}\times 64$ | 0.00889 | 0.225501(85) | 0.052449(41) | – | 0.000745(43) | -0.602(35) | 18.485(24) | 0.01466(85) | -0.376(19)
$24^{3}\times 48$ | 0.00750 | 0.20949(28) | 0.047428(94) | – | 0.00240(17) | -0.491(31) | 19.51(10) | 0.0547(40) | -0.428(25)
$32^{3}\times 64$ | 0.00750 | 0.20616(11) | 0.048171(51) | 0.2573(66) | 0.000830(45) | -0.548(25) | 18.316(36) | 0.0195(11) | -0.377(17)
$48^{3}\times 96$ | 0.00750 | 0.205686(20) | 0.048220(15) | – | 0.0002410(29) | -0.5292(56) | 18.196(11) | 0.005696(68) | -0.3887(41)
$32^{3}\times 64$ | 0.00500 | 0.16725(21) | 0.039196(81) | 0.182(24) | 0.001117(75) | -0.429(25) | 18.207(78) | 0.0399(27) | -0.391(21)
$48^{3}\times 96$ | 0.00500 | 0.165841(52) | 0.039823(28) | – | 0.000264(16) | -0.489(26) | 17.343(24) | 0.00961(57) | -0.340(18)
$48^{3}\times 96$ | 0.00222 | 0.10972(10) | 0.027104(44) | 0.131(20) | 0.000416(25) | -0.330(17) | 16.388(60) | 0.0346(22) | -0.333(17)
$64^{3}\times 128$ | 0.00125 | 0.081971(67) | 0.021419(27) | 0.089(32) | 0.000216(10) | -0.265(11) | 14.646(48) | 0.0321(15) | -0.309(12)
Table 2: Physical observables extracted from PNGB and $I=2$ correlation
functions on each gauge ensemble, with the addition of the $0^{++}$ mass
$M_{\sigma}$ results computed in Ref. Appelquist  _et al._ (2019). Here, $a$
indicates the lattice spacing. Error bars represent statistical plus fit
systematics determined through the AIC procedure.

We carry out our lattice study of maximal isospin $\pi\pi$ scattering on gauge
ensembles for $\mathrm{SU}(3)$ gauge theory with two staggered flavors which
become $N_{f}=8$ dynamical, degenerate light flavors in the continuum limit.
The unrooted staggered fermion action is improved with a single nHYP smearing
step Hasenfratz and Knechtli (2001); Hasenfratz  _et al._ (2007); Cheng  _et
al._ (2012). The gauge action includes both fundamental and adjoint plaquette
terms. The phase structure of this lattice action as well as the low lying
hadronic spectrum have been presented in Refs. Appelquist  _et al._ (2016,
2019); Gasbarro and Fleming (2017); Schaich  _et al._ (2015).

In the continuum limit, the four tastes of each staggered flavor become
degenerate, and the full
$\mathrm{SU}_{L}(8)\times\mathrm{SU}_{R}(8)\rightarrow\mathrm{SU}_{V}(8)$
chiral symmetry breaking pattern is recovered with 63 PNGBs. Any two of the
$N_{f}=8$ fermion flavors can be chosen to transform under an $\mathrm{SU}(2)$
subgroup of this $\mathrm{SU}(8)$, with the remaining 6 transforming as
singlets. For $2\rightarrow 2$ PNGB scattering in the maximal $I=2$ isospin
channel of this $\mathrm{SU}(2)$ subgroup, there is therefore no mixing with
two particle states involving the other 6 flavors. Equivalently, two PNGBs in
an $I=2$ state also belong to an irreducible representation of the full
$\mathrm{SU}(8)$. This property makes a complete coupled channel analysis
unnecessary.

To better understand how lattice artifacts affect the global symmetries,
consider the subgroup
$\mathrm{SU}_{V}(2)\times\mathrm{SU}_{V}(4)\subset\mathrm{SU}_{V}(8)$. This
$\mathrm{SU}_{V}(2)$ rotates between the two staggered flavors and is exact
even at finite lattice spacing, and $\mathrm{SU}_{V}(4)$ is the continuum
symmetry for a single staggered flavor. The four tastes of each staggered
flavor lead to 16 tastes of pseudoscalar bilinears, an $\mathrm{SU}(4)$
adjoint $\mathbf{15}$ and singlet 1. Each of these 16 pseudoscalar tastes may
transform in the adjoint $\mathbf{3}$ or singlet $\mathbf{1}$ representation
of $\mathrm{SU}_{V}(2)$, for a total of 64 pseudoscalars corresponding to the
adjoint $\mathbf{63}$ and singlet $\mathbf{1}$ of $\mathrm{SU}_{V}(8)$. The 16
tastes are designated by the spin-taste structure
$\gamma_{5}\bigotimes\xi_{F}$ Daniel and Sheard (1988).

At finite lattice spacing, the $\mathrm{SU}_{V}(4)$ taste symmetry is
explicitly broken at $\mathcal{O}(a^{2})$ to $\mathrm{SO}(4)$ and at
$\mathcal{O}(a^{2}p^{2})$ to $\text{SW}_{4}$, the discrete lattice spin-taste
group Kilcup and Sharpe (1987); Sharpe and Van de Water (2005). The
$\gamma_{5}\bigotimes\xi_{5}$ non-singlet pseudoscalars become the only three
exact PNGBs in the chiral limit, with the remaining sixty having masses
proportional to the lattice spacing. The spectral study has revealed that the
taste splitting of the 63-plet masses are on the order of 20–30% Appelquist
_et al._ (2019). Lattice artifacts generate irrelevant four-fermion operators
which mix the $\xi_{F}$ tastes, so that maximal isospin $2\rightarrow 2$
scatting can involve coupling between two particle states of other tastes at
finite lattice spacing. However, such mixing should be highly suppressed,
particularly for the scattering of pseudoscalars in an $\mathrm{SO}(4)$ irrep
at low momentum. Therefore, we will assume it sufficient to study a single
(diagonal) taste channel.

We consider the $2\rightarrow 2$ s-wave scattering of pseudoscalars with spin-
taste structure $\gamma_{5}\bigotimes\xi_{5}$ in the maximal isospin $I=2$
channel of $\mathrm{SU}_{V}(2)$. Denoting the staggered flavors as
$(\chi_{1},\chi_{2})$, this spin-taste structure corresponds to the distance-
zero interpolating operator
$\pi^{+}(x)=\bar{\chi}_{2}(x)\epsilon(x)\chi_{1}(x)$, with staggered phase
$\epsilon(x)=(-1)^{x+y+z+t}$. We probe the $I=2$ s-wave scattering channel
with the interpolating operator constructed from two PNGBs at rest.

$\mathcal{O}_{I=2}(t)=\pi^{+}(t)\pi^{+}(t+1)$ (1)

where the individual PNGB operators are projected onto zero spatial momentum

$\pi^{+}(t)=\sum_{\vec{x}}\bar{\chi}_{2}(x)\epsilon(x)\chi_{1}(x).$ (2)

We separate the operators by one time slice to avoid the projection onto
unwanted states from Fiertz rearrangement identities Kuramashi  _et al._
(1993); Fukugita  _et al._ (1995); Fu (2013). The $I=2$ two point correlation
function is then given by

$\displaystyle
C_{I=2}(t,t_{0})=\left\langle\mathcal{O}_{I=2}(t)\mathcal{O}_{I=2}(t_{0})^{\dagger}\right\rangle$
(3)
$\displaystyle=\sum_{\vec{x}_{1},\cdots,\vec{x}_{4}}\left\langle\pi^{+}(t_{4},\vec{x}_{4})\pi^{+}(t_{3},\vec{x}_{3})\pi^{+}(t_{2},\vec{x}_{2})^{\dagger}\pi^{+}(t_{1},\vec{x}_{1})^{\dagger}\right\rangle$

with $t_{1}=t_{0}$, $t_{2}=t_{0}+1$, $t_{3}=t$, and $t_{4}=t+1$. Two types of
Wick contractions contribute to the correlation function, the so called
“direct” (D) and “crossed” (C) channels.

$\displaystyle C_{I=2}(t,t_{0})$ $\displaystyle=C_{D}(13;24)+C_{D}(14;23)$ (4)
$\displaystyle\quad-C_{C}(1324)-C_{C}(1423),$ (5) $\displaystyle C_{D}(ik;jl)$
$\displaystyle=\sum_{\begin{subarray}{c}\vec{x}_{i},\vec{x}_{j},\\\
\vec{x}_{k},\vec{x}_{l}\end{subarray}}\text{Tr}\Big{[}G_{x_{k}x_{i}}^{\dagger}G_{x_{k}x_{i}}\Big{]}\mbox{Tr}\left[G_{x_{l}x_{j}}^{\dagger}G_{x_{l}x_{j}}\right],\;\;\;\;$
(6) $\displaystyle C_{C}(ijkl)$
$\displaystyle=\sum_{\begin{subarray}{c}\vec{x}_{i},\vec{x}_{j},\\\
\vec{x}_{k},\vec{x}_{l}\end{subarray}}\mbox{Tr}\left[G_{x_{k}x_{i}}G_{x_{k}x_{j}}^{\dagger}G_{x_{l}x_{j}}G_{x_{l}x_{i}}^{\dagger}\right],$
(7)

where $\mbox{Tr}\left[\dots\right]$ denotes the color trace and $G_{xy}$ is a
point propagator from $y$ to $x$. The $C_{D}$ and $C_{C}$ correlators may
instead be constructed from Green’s functions computed on wall sources
Kuramashi  _et al._ (1993); Fukugita  _et al._ (1995) satisfying

$\sum_{x^{\prime}}D_{x,x^{\prime}}G_{x^{\prime},t_{0}}^{W}=\sum_{\vec{y}}\delta_{x,(\vec{y},t_{0})}.$
(8)

By inverting on a wall source at $t_{i}=t_{0}$ and $t_{j}=t_{0}+1$, we
construct the Wick diagrams for the $I=2$ correlation function with only
$2N_{c}$ inversions per configuration:

$\displaystyle C_{D}^{W}(ik;jl)$
$\displaystyle=\sum_{\vec{x}_{k},\vec{x}_{l}}\mbox{Tr}\left[G_{x_{k},t_{i}}^{W\;\dagger}G^{W}_{x_{k},t_{i}}\right]\mbox{Tr}\left[G_{x_{l},t_{j}}^{W\;\dagger}G^{W}_{x_{l},t_{j}}\right],$
(9) $\displaystyle C_{C}^{W}(ijkl)$
$\displaystyle=\sum_{\vec{x}_{k},\vec{x}_{l}}\mbox{Tr}\left[G^{W}_{x_{k},t_{i}}G_{x_{k},t_{j}}^{W\;\dagger}G^{W}_{x_{l},t_{j}}G_{x_{l},t_{i}}^{W\;\dagger}\right],$
(10)

which are equivalent to Eqs. (6) and (7) up to terms with open quark lines
which produce gauge-variant noise. We choose not to gauge fix, allowing the
gauge variant terms to average out in the ensemble average rather than being
removed by explicit gauge fixing. This is sometimes referred to as the “moving
wall” method Fu (2013). From the same wall sources, we also construct the two
point correlation function of $\pi^{+}$ at rest and extract the PNGB mass and
decay constant.

In this work we restrict our attention to low momentum scattering, where the
scattering phase shift may be expanded in powers of $k^{2}/M_{\pi}^{2}$.

$k\cot\delta^{I}(k)=\frac{1}{a^{I}}+\frac{1}{2}r^{I}M_{\pi}^{2}\left(\frac{k^{2}}{M_{\pi}^{2}}\right)+\mathcal{O}\left(\frac{k^{4}}{M_{\pi}^{4}}\right),$
(11)

where $a^{I}$ and $r^{I}$ are the s-wave scattering length and effective range
respectively in the isospin $I$ channel. We have suppressed angular momentum
indices on these quantities.

The gauge ensembles used in this study are detailed in Table 1. A
multiexponential fitting procedure is applied to the $I=2$ correlation
function as well as the $\pi^{+}$ two point function. On each gauge ensemble
and for each correlation function, we apply a consistent fitting procedure and
statistical analysis. Energy levels are extracted with a three state $\cosh$
fit using the full covariance matrix between time slices. In the case of the
$I=2$ correlation function, an extra constant is added to the fit function to
account for a wrap around effect where each of the PNGBs propagate in opposite
directions. The second excited state, being the highest fit state in the
correlation function, is assumed to be excited state contaminated and is
discarded. In principle the first $I=2$ excited state can be used to probe
larger scattering momenta at a fixed quark mass. However, we have found that
the first excited state energies on our ensembles correspond to scattering
momenta outside of the radius of convergence of the effective range expansion
Eq. (11). Therefore in this study we focus our attention entirely on the
ground state $I=2$ energy level on each ensemble.

To completely control the fit range systematics, we apply an extension of the
Akaike information criterion (AIC) Akaike (1974) studied recently by Jay and
Neil to average over fits to correlation functions in lattice QCD Jay and Neil
(2021). Each fit is assigned a probability $\text{Pr}(M|D)$ as

$\displaystyle-2\ln\left(\text{Pr}(M|D)\right)=$
$\displaystyle\;\;\;\;-2\ln\left(\text{Pr}(M)\right)+\chi^{2}_{\text{min}}+2k+2N_{\text{cut}},$
(12)

where $\text{Pr}(M)$ is the prior given to each fit, which we take to be
uniform, $\chi^{2}_{\text{min}}$ is the minimum $\chi^{2}$ which is
$-2\times\text{(max log likelihood)}$ for each fit, $k$ is the number of model
parameters used in the fit, and $N_{\text{cut}}$ is the number of data points
cut out by the fit window. The extended AIC probability is then used to
average over the best fit values of the parameters in each fit, as well as to
average the bootstrap error bars. In our fits, we have a good signal to noise
ratio all the way across the lattice, allowing us to always take
$t_{\text{max}}=T/2$. Therefore, we need only control the $t_{\text{min}}$
systematic through this procedure. In all cases, we find that the AIC averaged
result is in good agreement with the value extracted from picking a single fit
where the excited state contamination has plateaued with $t_{\text{min}}$,
though the AIC average generally assigns a slightly larger overall error bar
than choosing a single fit.

The scattering phase shift is computed via the Lüscher procedure from the
ground state energy of the $I=2$ correlation function. For two identical
scalars in an s-wave, the Lüscher finite volume quantization condition Lüscher
(1991) is given by

$\displaystyle
k\cot\delta(k)=\frac{2\pi}{L}\pi^{-3/2}Z_{00}\left(1,\frac{k^{2}L^{2}}{4\pi^{2}}\right),$
(13) $\displaystyle
k^{2}=\frac{1}{4}E_{\pi\pi}^{2}-M_{\pi}^{2}\;,\;Z_{00}(1,q^{2})=\frac{1}{\sqrt{4\pi}}\sum\limits_{\vec{n}\in\mathbb{Z}^{3}}\frac{1}{\vec{n}^{2}-q^{2}}.$
(14)

Each two-PNGB energy level extracted from the $I=2$ correlation function
yields a corresponding determination of the scattering phase shift at
scattering momentum $k^{2}$. The scattering observables $k^{2}$ and
$k\cot\delta$ depend systematically on the fit range choice for both the $I=2$
and PNGB correlation functions. This systematic is controlled by multiplying
the AIC distributions to form a joint distribution in
$(t_{\text{min},\pi},t_{\text{min},I=2})$ and averaging against this
distribution.

The results of the multiexponential fitting and scattering analysis on each
ensemble are shown in Table 2, where errors reflect statistical plus fit
systematic errors processed through the AIC procedure. The values of the PNGB
mass and decay constant reported here are from the new analysis with wall
sources as described above. We use the normalization of the PNGB decay
constant corresponding to the QCD value $F_{\pi}=92.2(1)$ MeV as in Ref.
Appelquist  _et al._ (2019). The results are consistent within statistical
errors compared to Ref. Appelquist  _et al._ (2019), but errors have shrunk
considerably due a combination of the fit range systematic control through the
AIC procedure, higher availability of gauge statistics for certain ensembles,
and perhaps a statistics gain through the use of wall sources. We have
restated the values of the singlet scalar mass computed in Appelquist  _et
al._ (2019) for convenience because we will make use of it in Section. III. In
addition to the basic quantities in lattice units, we have included values for
certain ratios that will be useful in Section III.

The scattering length and effective range are in principle independent
observables, and are sensitive to a different set of low energy parameters in
an EFT. We assess the significance of the effective range contribution to the
scattering phase shift in Fig. 1 by plotting $M_{\pi}/k\cot\delta$ against the
effective range expansion parameter for the gauge ensembles for which we have
two volumes with the same $am$. For comparison we show the leading order
dilaton EFT prediction with the dotted and dashed lines. These curves have
been determined by fixing the dilaton EFT parameters through a global fitting
procedure which we explain in detail below. While the data points for the same
$am$ tend to lie within error bars, data shows a clear downward trend in
correspondence with the dilaton EFT prediction, indicating a non-negligible
sensitivity to the effective range term. The leftmost point with blue
triangular marker corresponds to the ensemble with $am=0.0075$ and volume
$48^{3}\times 96$ where we have very high statistics. This point is in clear
tension with the tree level dilaton EFT prediction. The overall validity of
the tree level dilaton EFT description of the scattering data is discussed
further in Section III.

Figure 1: Plot showing the variation of the scattering phase shift with
scattering momentum for the three largest fermion masses. The points indicate
the lattice data, given in Table 2, whereas the lines represent tree level
dilaton EFT predictions, as explained in Section III.

###  II.1 Accounting for Finite–Volume Effects

To compare the lattice data of Table 2 with the predictions of a continuum
EFT, we must first ensure that finite-volume effects in all lattice determined
quantities are small relative to their quoted uncertainties. We have data
taken for many quantities at multiple lattice volumes using the same fermion
mass (overlapping volumes), enabling estimation of these effects.

It can be seen in Table 2 that central values for $aM_{\pi}$ and $aF_{\pi}$
taken using the same fermion mass change by an amount greater than their
statistical uncertainties between lattice volumes. We therefore perform a
dedicated infinite volume extrapolation of the PNGB masses and decay constants
in this section, and use this extrapolated data in the fit to dilaton EFT in
section III.

We do not have data for $aM_{\sigma}$ at multiple lattice volumes, so the
volume dependence cannot be inferred directly from data, and we will not
perform a dedicated infinite–volume extrapolation for this quantity. We will
attribute the volume dependence of the scattering phase shift in Table 2
entirely to the scattering momentum $k^{2}$ dependence coming from the
effective range, as discussed above and illustrated in Fig 1.

To extrapolate $aM_{\pi}$ and $aF_{\pi}$ to the infinite volume limit, we use
the following model

$\displaystyle M_{\pi}(L)$
$\displaystyle=M_{\pi}\left(1+\alpha\frac{M^{2}_{\pi}}{F^{2}_{\pi}}\frac{e^{-M_{\pi}L}}{(M_{\pi}L)^{3/2}}\right),$
(15) $\displaystyle F_{\pi}(L)$
$\displaystyle=F_{\pi}\left(1-\beta\frac{M^{2}_{\pi}}{F^{2}_{\pi}}\frac{e^{-M_{\pi}L}}{(M_{\pi}L)^{3/2}}\right).$
(16)

In these formulae, $M_{\pi}$ and $F_{\pi}$ represent the extrapolated mass and
decay constant. The parameters $\alpha$ and $\beta$ are extracted from a fit
to lattice data, and crucially, we assume that they have no dependence on the
fermion mass. Expressions similar to Eqs. (15) and (16) have been derived
earlier in the literature Gasser and Leutwyler (1987); Leutwyler (1987);
Gasser and Leutwyler (1988); Lüscher (1986); Colangelo and Durr (2004);
Colangelo and Haefeli (2004), assuming that the PNGBs are well described by
chiral perturbation theory.

We perform a least-squares method fit of Eqs. (15) and (16) to all the data we
have in Table 2 for the PNGB masses and decay constants on overlapping volumes
(14 points total). Correlations between the lattice measurements are accounted
for in this fit. We find that $\chi^{2}_{\text{min}}/N_{\text{dof}}=2.05$,
indicating slight tension between the data and this crude model. The central
values and standard errors for $\alpha$ and $\beta$ indicated by the fit are

$\alpha=1.530(78),\qquad\beta=1.62(12).$ (17)

In Figs 2 and 3, lattice data at $am=0.0075$ are plotted on top of the curves
given by Eqs. (15) and (16), using parameter values taken from the fit. The
plots confirm that the variation in the PNGB mass and decay constant with
lattice size is comfortably larger than the statistical uncertainties, and
that the model describes this variation reasonably well.

We use this model to estimate the infinite volume extrapolated values of
$aM_{\pi}$ and $aF_{\pi}$ at all of the remaining fermion masses, for which
data at only one lattice volume is available. The results of all the
extrapolations are presented in Table 3. This extrapolated data will be
compared with EFT predictions in the following section.

$am$ | $aM_{\pi}$ | $aF_{\pi}$
---|---|---
0.00889 | 0.22525(27) | 0.052491(59)
0.0075 | 0.205680(20) | 0.048233(20)
0.005 | 0.16575(10) | 0.03982(28)
0.00222 | 0.1085(12) | 0.02742(32)
0.00125 | 0.08115(82) | 0.02165(23)
Table 3: Result of the extrapolation $aM_{\pi}$ and $aF_{\pi}$ to the infinite
volume limit. The central values and uncertainties for these quantities will
be used in the fit to Dilaton EFT in section III. Details on how these
extrapolations were performed are provided in the text.

We have estimated the total uncertainty in the extrapolated quantities by
adding in quadrature their statistical uncertainties (shown in Table 2) to the
shift in central value due to the infinite volume extrapolation. This shift is
given by the difference between that quantity at the largest available lattice
volume in Table 2 and the corresponding central value of that quantity in
Table 3. We assigned uncertainties in this conservative way, because the model
we are using for these extrapolations is only approximate. These total
uncertainties are presented in Table 3.

As the uncertainties in Table 3 arise from systematic finite volume effects to
a great extent, which we only estimate using an approximate model, we should
not assume that correlations between these extrapolated quantities can be
properly obtained from the statistical correlations between the lattice
measurements in Table 2. We therefore regard all quantities in Table 3 as
completely uncorrelated.

Figure 2: Dependence of the PNGB mass $aM_{\pi}$ on the inverse of the spatial
extent of the lattice $L$ for the fermion mass $am=0.0075$. The gray line
shows the $L$–dependence predicted by the model in Eq. (15), using the best
fit values for the parameters $\alpha$ and $\beta$ indicated in Eq. (17). For
the range of lattice volumes for which we have data, the finite–volume
correction to $aM_{\pi}$ is clearly significant.  Figure 3: Dependence of the
PNGB mass $aF_{\pi}$ on the inverse of the spatial extent of the lattice $L$
for the fermion mass $am=0.0075$. The gray line shows the $L$–dependence
predicted by the model in Eq. (16).

##  III Dilaton EFT

The EFT which we employ describes the interaction of a single scalar dilaton
with the $N_{f}^{2}-1$ PNGBs associated with the breaking of the approximate
global symmetry group $G=\mathrm{SU}(N_{f})_{L}\times\mathrm{SU}(N_{f})_{R}$
to the diagonal subgroup $\mathrm{SU}(N_{f})_{V}$. It is employed to compute
the low energy properties of an underlying gauge theory capable of producing a
light composite scalar along with the composite PNGBs. In terms of the PNGB
fields $\pi$ and an additional real scalar field $\chi$, the Lagrangian of the
EFT is given below. See Appelquist  _et al._ (2020) and references therein.

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi\,+\mathcal{L}_{\pi}\,+\,\mathcal{L}_{M}\,-\,V_{\Delta}(\chi)\,,$
(18)

where

$\mathcal{L}_{\pi}=\frac{f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{d}}\right)^{2}\,\mbox{Tr}\left[\partial_{\mu}\Sigma(\partial^{\mu}\Sigma)^{\dagger}\right]\,.$
(19)

The matrix field $\Sigma=\exp\left[2i\pi^{a}T^{a}/f_{\pi}\right]$ transforms
as $\Sigma\to\mathrm{U}_{L}\Sigma\mathrm{U}_{R}^{\dagger}$ under the action of
unitary transformations $\mathrm{U}_{L,R}\in\mathrm{SU}(N_{f})_{L,R}$. It also
satisfies the non-linear constraints
$\Sigma\Sigma^{\dagger}=\mathbb{1}_{N_{f}}$. The $T^{a}$ are the generators of
$\mathrm{SU}(8)$, normalized so that
$\mbox{Tr}\left[T^{a}T^{b}\right]=\frac{1}{2}\delta^{ab}$.

The explicit breaking of the global internal symmetry, necessary for the
implementation of lattice computations, is described by the term

$\mathcal{L}_{M}=\frac{m_{\pi}^{2}f_{\pi}^{2}}{4}\left(\frac{\chi}{f_{d}}\right)^{y}\,\mbox{Tr}\left[\Sigma+\Sigma^{\dagger}\right]\,,$
(20)

where $m_{\pi}^{2}\equiv 2B_{\pi}m$ vanishes when the fermion mass $m$ of the
underlying theory is set to zero. The parameter $y$ can be interpreted as the
scaling dimension of the chiral condensate of the underlying gauge theory at
strong coupling Leung  _et al._ (1989).

The dilaton potential $V_{\Delta}(\chi)$, employed in Ref. Appelquist  _et
al._ (2020) and discussed earlier in Refs. Chacko and Mishra (2013); Catà  _et
al._ (2019), is given by

$V_{\Delta}(\chi)=\frac{m_{d}^{2}\chi^{4}}{4(4-\Delta)f_{d}^{2}}\left[1-\frac{4}{\Delta}\left(\frac{\chi}{f_{d}}\right)^{\Delta-4}\right]\,.$
(21)

It contains two contributions. One is a scale-invariant term
($\propto\chi^{4}$) representing the corresponding operators in the underlying
gauge theory. The other ($\propto\chi^{\Delta}$) captures the leading-order
effect of the scale deformation in the underlying theory. The potential has a
minimum at $\chi=f_{d}>0$, with mass $m_{d}$ for the dilaton.

In Ref. Appelquist  _et al._ (2020), this EFT was used to fit lattice data for
the $\mathrm{SU}(3)$ gauge theory with $N_{f}=8$ Dirac fundamental fermions. A
key feature of the EFT is that the dilaton mass (the explicit breaking of
scale symmetry) can be tuned as small as necessary with $f_{d}$ held fixed. It
has been suggested in Refs. Appelquist and Bai (2010); Golterman and Shamir
(2016); Golterman  _et al._ (2020); Golterman and Shamir (2020) that the
explicit breaking in the underlying theory, and also in the EFT as a
consequence, can be made arbitrarily small by tuning the number of flavors
$N_{f}$ arbitrarily close to the critical value $N_{f}^{c}$ at which
confinement gives way to IR conformality. The form of $V_{\Delta}(\chi)$
interpolates among several specific forms found in the literature. The choice
$\Delta=2$ gives the Higgs potential of the standard model, while the choice
$\Delta\rightarrow 4$ corresponds to a nearly marginal deformation of scale
symmetry Goldberger  _et al._ (2008).

The mass deformation in Eq. (20) contributes, in the vacuum
$\left\langle\Sigma\right\rangle=\mathbb{1}$, an additive term to
$V_{\Delta}$. The entire potential is

$W(\chi)=V_{\Delta}(\chi)\,-\,\frac{N_{f}m_{\pi}^{2}f_{\pi}^{2}}{2}\left(\frac{\chi}{f_{d}}\right)^{y}\,,$
(22)

leading to a new minimum for $\chi$ which determines its vacuum value
$\left\langle\chi\right\rangle=F_{d}>f_{d}$. Minimizing the potential leads to
the transcendental equation for $F_{d}$ shown below

$\frac{F_{d}^{4-y}}{(4-\Delta)f^{4-y}_{d}}\left[1-\left(\frac{f_{d}}{F_{d}}\right)^{4-\Delta}\right]=\frac{yN_{f}f^{2}_{\pi}m^{2}_{\pi}}{2f^{2}_{d}m^{2}_{d}},$
(23)

which can be solved numerically to obtain $F_{d}$ for a given choice of the
fermion mass and EFT parameters.

By employing the value $\left\langle\chi\right\rangle=F_{d}$ in Eqs. (19) and
(20), and properly normalising the PNGB kinetic term, one has simple scaling
relations for the PNGB decay constant $F_{\pi}$ and PNGB mass $M_{\pi}$:

$\displaystyle\frac{F_{\pi}^{2}}{f_{\pi}^{2}}$
$\displaystyle=\frac{F_{d}^{2}}{f_{d}^{2}}\,,$ (24)
$\displaystyle\frac{M_{\pi}^{2}}{m_{\pi}^{2}}$
$\displaystyle=\left(\frac{F_{d}^{2}}{f_{d}^{2}}\right)^{\frac{y}{2}-1}\,.$
(25)

There is also a new curvature at the potential minimum, determining the
dilaton mass $M_{d}$. In terms of $F_{d}/f_{d}$:

$\frac{M^{2}_{d}}{F^{2}_{d}}=\frac{m^{2}_{d}}{(4-\Delta)f^{2}_{d}}\left(4-y+(y-\Delta)\left(\frac{f_{d}}{F_{d}}\right)^{4-\Delta}\right).$
(26)

###  III.1 PNGB Scattering

The Feynman diagrams that contribute to the scattering process
$\pi^{a}\,\pi^{b}\to\pi^{c}\,\pi^{d}$ at tree level are shown in Fig. 4. Here
$(abcd)\equiv\mbox{Tr}\left[T^{a}T^{b}T^{c}T^{d}\right]$, all four momenta are
directed inward, and $s\equiv(p_{a}+p_{b})^{2}$, $t\equiv(p_{a}+p_{c})^{2}$
and $u\equiv(p_{a}+p_{d})^{2}$ are the Mandelstam variables.

The dimensionless scattering amplitude is then given at tree level by

Figure 4: Feynman diagrams in dilaton EFT which appear in the $\pi$–$\pi$
scattering amplitude at tree level. The dashed lines represent PNGBs and the
solid lines represent the dilaton.

$\displaystyle i\mathcal{M}^{ab,cd}(s,t,u)=$
$\displaystyle\frac{2i}{3F_{\pi}^{2}}\left\\{[(abcd)+(adcb)](6M_{\pi}^{2}-3t)+[(acdb)+(abdc)](6M_{\pi}^{2}-3u)+[(adbc)+(acbd)](6M_{\pi}^{2}-3s)\right\\}$
$\displaystyle-i\frac{\delta^{ab}\delta^{cd}}{F_{d}^{2}}\frac{(s+(y-2)M_{\pi}^{2})^{2}}{s-M_{d}^{2}}-i\frac{\delta^{ac}\delta^{bd}}{F_{d}^{2}}\frac{(t+(y-2)M_{\pi}^{2})^{2}}{t-M_{d}^{2}}-i\frac{\delta^{ad}\delta^{bc}}{F_{d}^{2}}\frac{(u+(y-2)M_{\pi}^{2})^{2}}{u-M_{d}^{2}}.$
(27)

We examine a scattering process in which the PNGBs belong to the same
$\mathrm{SU}(2)$ triplet, that is, they are each bound states of a single
$\mathrm{SU}(2)$ fermion doublet. When $N_{f}>2$, for example $N_{f}=8$ in our
case, one could imagine that the chosen fermion doublet carries electroweak
quantum numbers and the others are electroweak singlets. The triplet of
$\mathrm{SU}(2)$ PNGBs can scatter in three independent isospin channels
$I=0$, $1$ and $2$. We focus here on the $I=2$ channel since, as noted in
Section II, the lattice computation in the underlying gauge theory then
contains no fermion-line-disconnected diagrams. The scattering amplitude in
this channel takes the form

$\displaystyle\mathcal{M}^{I=2}(s,t,u)=$
$\displaystyle\frac{2M_{\pi}^{2}-s}{F_{\pi}^{2}}-\frac{1}{F_{d}^{2}}\frac{(t+(y-2)M_{\pi}^{2})^{2}}{t-M_{d}^{2}}$
$\displaystyle-\frac{1}{F_{d}^{2}}\frac{(u+(y-2)M_{\pi}^{2})^{2}}{u-M_{d}^{2}}.$
(28)

The first term in this expression comes from the chiral-Lagrangian four-point
interaction while the next two (pole) terms arise from exchange of the dilaton
scalar in the $t$ and $u$ channels.

For our purposes, we develop $\mathcal{M}^{I=2}(s,t,u)$ in a partial wave
expansion:

$\displaystyle\mathcal{M}^{I}(s,t,u)=32\pi\sum_{\ell=0}^{\infty}(2\ell+1)P_{\ell}(\cos\theta)t^{I}_{\ell}(s),$
(29) $\displaystyle
t^{I}_{\ell}(s)=\frac{1}{64\pi}\int_{-1}^{+1}d(\cos\theta)P_{\ell}(\cos\theta)\mathcal{M}^{I}(s,t,u).$
(30)

For the $\ell=0$ partial wave component, this gives

$\displaystyle t^{2}_{0}(s)=$ $\displaystyle\frac{2M_{\pi}^{2}-s}{32\pi
F_{\pi}^{2}}+\frac{2k^{2}-M_{d}^{2}-2M_{\pi}^{2}(y-2)}{16\pi F_{d}^{2}}$
$\displaystyle+\frac{\left[M_{d}^{2}+(y-2)M_{\pi}^{2}\right]^{2}}{64\pi
F_{d}^{2}k^{2}}\log\left(\frac{4k^{2}+M_{d}^{2}}{M_{d}^{2}}\right).$ (31)

with $s=4(k^{2}+M_{\pi}^{2})$ and $k=|\vec{k}|$ is the PNGB spatial momentum
in the center of mass frame.

For any partial wave and isospin channel, the tree-level amplitude
$t_{\ell}^{I}(s)$ can be expressed in terms of the scattering phase shift
$\delta_{\ell}^{I}$ as

$\displaystyle
t^{I}_{\ell}(s)=\frac{\sqrt{M^{2}_{\pi}+k^{2}}}{k\cot\delta^{I}_{\ell}-ik},$
(32)

where $k^{2l+1}\cot\delta_{\ell}$ has a power-series expansion in $k^{2}$
valid for $k^{2}\ll M_{\pi}^{2}$. For the case $\ell=0$, this expansion takes
the form shown in Eq. (11).

For the dilaton EFT, we then have

$\displaystyle M_{\pi}a^{I=2}=-\frac{M^{2}_{\pi}}{16\pi
F^{2}_{\pi}}\left(1-(y-2)^{2}\frac{f^{2}_{\pi}}{f^{2}_{d}}\frac{M^{2}_{\pi}}{M^{2}_{d}}\right),$
(33)

and

$\displaystyle M_{\pi}r^{I=2}=$ $\displaystyle-\frac{16\pi
F^{2}_{\pi}}{M^{2}_{\pi}}\,\cdot\,\frac{1}{1-\frac{(y-2)^{2}f^{2}_{\pi}M^{2}_{\pi}}{f^{2}_{d}M^{2}_{d}}}$
$\displaystyle+\frac{64\pi
F^{2}_{\pi}}{M^{2}_{\pi}}\,\cdot\,\frac{1+\frac{(y-2)f^{2}_{\pi}M^{2}_{\pi}}{f^{2}_{d}M^{2}_{d}}+\frac{(y-2)^{2}f^{2}_{\pi}M^{4}_{\pi}}{2f^{2}_{d}M^{4}_{d}}}{\left(1-\frac{(y-2)^{2}f^{2}_{\pi}M^{2}_{\pi}}{f^{2}_{d}M^{2}_{d}}\right)^{2}}.$
(34)

These expressions give the scattering length and effective range in terms of
quantities directly measured on the lattice ($F_{\pi}^{2}$, $M_{\pi}^{2}$ and
$M_{d}^{2}$) and two dimensionless fit parameters of the dilaton EFT ($y$ and
$f_{\pi}^{2}/f_{d}^{2}$). We note that the expressions take on simpler forms,
derivable from a chiral lagrangian with no dilaton present, when either
$f_{\pi}^{2}/f_{d}^{2}\rightarrow 0$ or $(y-2)\rightarrow 0$. The first is a
limit in which the dilaton decouples from the PNGBs. The second leaves in
place a dilaton contribution to the full scattering amplitude (second line of
Eq. (27)), but eliminates it for the $I=2$ channel (second and third terms of
Eq. (28)) at threshold where $s=4M^{2}_{\pi}$.

###  III.2 Comparison with Lattice Data

In this section, we first assess qualitatively whether the dilaton EFT is
compatible with the lattice data for scattering. We then perform a global fit
of all the lattice data to the leading order dilaton EFT, allowing
quantitative assessment of the goodness of fit and extraction of the best fit
values for the EFT parameters. Finally, we discuss the interpretation of the
global fit.

Figure 5: Points representing the scattering lengths determined from lattice
data are plotted against the finite–volume extrapolated quantity
$M^{2}_{\pi}/F^{2}_{\pi}$. They are compared with the tree level dilaton EFT
prediction given in Eq. (33), and shown here as the straight black dashed
line. The central values from the global fit are inputted to obtain this
dependence. The gray band plotted around the dashed line represents a rough
estimate for the size of higher order corrections to the EFT prediction, and
is given by Eq. (35).

In Fig. 5, we plot the s–wave $I=2$ scattering length against
$M^{2}_{\pi}/F^{2}_{\pi}$. The black dashed line represents the leading order
EFT prediction obtained from Eq. (33). The gray band represents a rough
estimate of the range of values which could be accommodated by higher order
corrections to the scattering length in dilaton EFT. The points represent
lattice data taken for all the fermion masses and lattice volumes appearing in
Table 2 (9 points total), after extrapolation to the infinite volume and zero
scattering momentum limits, needed for direct comparison with the EFT.

To place the points in Fig. 5, we use the infinite volume extrapolated values
for $M^{2}_{\pi}/F^{2}_{\pi}$, shown in Table 3. We also further process the
lattice data for $M_{\pi}/k\cot\delta$ to determine the scattering lengths.
This can be viewed as an extrapolation of lattice data taken at nonzero
scattering momentum to the zero momentum point. To extract the scattering
lengths, we use Eq. (11), along with lattice data for the scattering momenta
and the dilaton EFT expression for the effective range shown in Eq. (34). Use
of the dilaton EFT expressions requires knowledge of parameters, such as $y$
and $f^{2}_{\pi}/f^{2}_{d}$. We take these values from a global fit of the
dilaton EFT to lattice data, which we show in Table 4.

The position of the gray band is given by

$\displaystyle a^{\text{band}}=a^{I=2}\left[1\pm\frac{M^{2}_{\pi}}{(4\pi
F_{\pi})^{2}}\right],$ (35)

where $a^{I=2}$ represents the tree level dilaton EFT prediction for the
scattering length from Eq. (33). The correction term is chosen to scale the
same way as a generic next to leading order (NLO) correction with size given
by naive dimensional analysis.

In Fig. 5, most but not all of the points overlap with the black dashed line
within uncertainties, indicating that the tree level dilaton EFT approximates
the data well. However, with the majority of points lying below the dashed
line, there is mild evidence of a systematic discrepancy between the tree
level theory and lattice data. All of the points lie comfortably within the
gray band though, which suggests that small higher order corrections in the
dilaton EFT could plausibly improve the agreement.

To quantify the level of agreement between the tree level EFT and the
available data, we perform a global fit incorporating the two extrapolated
quantities $aM_{\pi}$ and $aF_{\pi}$ from Table 3, and the raw lattice data
for $aM_{d}$ and $M_{\pi}/k\cot\delta$ shown in Table 2, where $a$ denotes the
lattice spacing. The uncertainties in $aM_{\pi}$ and $aF_{\pi}$ are mainly
systematic rather than statistical, arising from the use of an approximate
model to estimate the size of finite–volume effects, so we treat $aM_{\pi}$
and $aF_{\pi}$ as uncorrelated with all other quantities entering the fit.
Similarly $aM_{d}$ measurements are each taken on their own lattice ensembles,
and so are uncorrelated with the other fitted quantities. This yields 24
quantities to fit. We also use lattice data for $a^{2}k^{2}$, to account for
the momentum dependence expected for the scattering phase shift.

We then fit tree-level dilaton EFT expressions to this data set. First, the
EFT determination of $aF_{\pi}$ can be found from solving the transcendental
equation in Eq. (23) and using the scaling relation in Eq. (24). Then the PNGB
and dilaton masses can be determined from Eqs. (25) and (26) using this
result. The EFT prediction for the scattering phase shift
$M_{\pi}/k\cot\delta$ is made using Eqs. (11), (33) and (34) along with data
for the scattering momentum. There are then 6 independent free parameters from
the EFT that appear in these expressions. We take them to be
$\\{y,\,aB_{\pi},\,f^{2}_{\pi}/f^{2}_{d},\,\Delta,\,a^{2}f^{2}_{\pi},\,m^{2}_{d}/f^{2}_{d}\\}$.

To do the fit, we construct a chi-square function bilinear in the differences
between the four quantities for which we have lattice data and their EFT
predictions. Then, we minimize the chi-square function with respect to the 6
EFT parameters. We find that $\chi^{2}_{\text{min}}/N_{\text{dof}}=3.03$ for
$N_{\text{dof}}=24-6=18$, indicating that the tree level EFT is not a perfect
description.

The parameters’ values at this chi-squared minimum are listed in Table 4. We
use these values to make the dilaton EFT predictions shown in Figs. 1, 5 and
6. The values we get lie close to the allowed ranges found in earlier dilaton
studies of the $N_{f}=8$ theory Appelquist  _et al._ (2020); Golterman  _et
al._ (2020) which reported good agreement between the tree level EFT and the
available lattice data, but did not consider scattering observables. The
estimated uncertainties of the fit parameters are obtained by calculating the
inverse Hessian of the chi-squared function at its minimum, to extract the
standard errors. These parameter uncertainties are somewhat smaller than those
reported in the earlier studies, reflecting the smaller uncertainties in this
more recent lattice data set. However these errors, shown for completeness,
should be interpreted with caution. As they were extracted from a tree level
fit, the best fit values for the EFT parameters may move outside the ranges
indicated once higher order effects within the dilaton EFT are accounted for.

Parameter | Value
---|---
$y$ | 2.1321(61)
$aB_{\pi}$ | 2.210(60)
$f^{2}_{\pi}/f^{2}_{d}$ | 0.0865(42)
$\Delta$ | 3.11(20)
$a^{2}f^{2}_{\pi}$ |  $5.8$(2.1) $\times 10^{-5}$
$m^{2}_{d}/f^{2}_{d}$ | 1.28(26)
Table 4: Best fit values and standard errors for the six dilaton EFT
parameters taken from the global fit. These uncertainty ranges do not take
into account the effects of higher order contributions arising in the EFT.

Even a fundamentally accurate low energy EFT may fail to fit lattice data if
the precision of the data exceeds the theoretical error introduced by
truncating the EFT at a fixed order in its low energy expansion. This
consideration, together with the good qualitative agreement between the tree
level EFT and the data suggest that dilaton EFT is a good low energy
description of this gauge theory, despite the chi-squared per degree of
freedom of the global fit being somewhat larger than one. However, more
precise lattice measurements and higher order calculations within the dilaton
EFT are now well motivated to confirm this picture.

Our global fit favors $(y-2)^{2}f^{2}_{\pi}/f^{2}_{d}\sim 10^{-3}$, ensuring
that Eq. (33) for the scattering length describes a straight line with
gradient $-1/16\pi$ to a high level of approximation. It can be seen in Fig. 5
that this relationship is also roughly satisfied by the lattice data. This is
also the same prediction for the scattering length as in chiral perturbation
theory (without any light scalar state). However we note that the $O(100\%)$
variation of $F_{\pi}$ with fermion mass evident in the lattice data could not
be explained using chiral perturbation theory, and so it would provide a poor
global fit to this dataset considered as a whole. The issues encountered when
comparing chiral perturbation theory with lattice data for scattering
alongside other observables are discussed for the $N_{f}=6$ theory in
Appelquist  _et al._ (2012) and for the $N_{f}=8$ theory in Gasbarro and
Fleming (2017); Gasbarro (2019).

###  III.3 Other Isospin Channels

In future, scattering lengths in other isospin channels could be measured for
the $N_{f}=8$ theory on the lattice, allowing for new independent tests of
this dilaton EFT. Given that the typical size of scattering momenta obtained
on our lattices is much less than the mass splitting between PNGBs in
different taste multiplets, it makes sense to consider scattering processes
involving only the lightest triplet of PNGBs. These PNGBs have insufficient
kinetic energy to scatter into PNGBs with different tastes for the region of
parameter space that we can study on the lattice.

For this lightest triplet of PNGBs, the different possible scattering channels
can be fully specified by their $\mathrm{SU}(2)$ isospin quantum numbers. The
dilaton EFT scattering amplitude in the maximal isospin ($I=2$) channels has
already been calculated in Eq. (28), and its $\ell=0$ partial wave component
shown in Eq. (31). The $I=1$ amplitude will have a vanishing $\ell=0$
component due to Bose symmetry, and we shall not consider it further. Finally,
the $I=0$ scattering amplitude is a promising target for future lattice study,
and we shall calculate the $\ell=0$ scattering length using the dilaton EFT
here.

The $I=0$ amplitude has a pole coming from dilaton exchange in the s-channel.
It can therefore become divergent for certain physical momentum values, if
$M_{d}>2M_{\pi}$. However, the EFT still remains weakly coupled in this
regime. Also, for scattering near threshold, this s-channel contribution to
the amplitude is sensitive to the three-point interaction between a dilaton
and PNGB pairs for dilaton momenta $k^{2}\sim 4M_{\pi}^{2}$. This three-point
interaction is unsuppressed when $y$ is close to 2, ensuring that the dilaton
makes a large contribution to the scattering amplitude near threshold in this
channel.

Using Eq. (30) and the $I=0$ equivalents of Eqs. (32) and (11), we can extract
the scattering length in the $I=0$ channel

$M_{\pi}a^{I=0}=\frac{7M^{2}_{\pi}}{32\pi F^{2}_{\pi}}\\\
+\frac{f^{2}_{\pi}}{f^{2}_{d}}\frac{M^{4}_{\pi}\left(M^{2}_{d}(5y^{2}+4y+20)-8M^{2}_{\pi}(y-2)^{2}\right)}{32\pi
F^{2}_{\pi}M^{2}_{d}\left(M^{2}_{d}-4M^{2}_{\pi}\right)}.$ (36)

The first term in this expression is derivable from the chiral lagrangian, and
includes no dilaton contribution. The second term arising from exchanged
dilatons vanishes in the limit $f^{2}_{\pi}/f^{2}_{d}\rightarrow 0$ when the
dilaton decouples from the PNGBs. However, unlike in the $I=2$ case, this term
remains large when $y-2\ll 1$.

In Fig. 6, the $I=0$ scattering length predicted from dilaton EFT is plotted.
On the same set of axes, the tree-level chiral lagrangian prediction for the
scattering length, corresponding to the first term in Eq. (36) is also plotted
for reference. It can be seen that the dilaton EFT prediction for this
quantity differs quite significantly from the chiral lagrangian prediction
except near the chiral limit, when $M^{2}_{\pi}/F^{2}_{\pi}$ becomes small,
and the PNGBs become significantly lighter than the dilaton. The spike
appearing in the dilaton EFT prediction arises from the s-channel pole in the
amplitude. It occurs when $M^{2}_{d}=4M^{2}_{\pi}$, which is expected nearer
to the chiral limit than current lattice data.

Figure 6: The prediction for the scattering length in the $I=0$ channel,
obtained using tree level dilaton EFT from Eq. (36), and the global fit
central values is shown here as the black solid line. The gray dashed line
representing the prediction from tree level chiral perturbation theory is
shown for comparison.

To make the dilaton EFT prediction, central values from the global fit have
been used to predict the dependence of $M^{2}_{d}/M^{2}_{\pi}$ (which appears
in Eq. (36)) as the chiral limit is approached. Due to the uncertainties in
many of these parameters, there is uncertainty in this prediction away from
the regime where lattice data is available $13<M^{2}_{\pi}/F^{2}_{\pi}<20$. In
particular, the location of the spike depends on the value chosen for
$\Delta$. However, Fig. 6 is sufficient to capture the qualitative behavior of
the scattering length.

The clear difference between the dilaton EFT and the chiral lagrangian
predictions for the $I=0$ scattering length (even at distances from the chiral
limit currently accessible on the lattice) make measurement of this quantity a
worthwhile goal for future lattice studies. Such a measurement provides a
complimentary probe of the interaction strength between the light scalar and
the PNGBs, and would further test the dilaton EFT hypothesis.

##  IV Conclusion

In this work, we have considered the maximal isospin s-wave scattering of
PNGBs in a nearly conformal gauge theory known to possess a light scalar
state. We have investigated the possibility that the light scalar, being
nearly degenerate with the PNGBs on the gauge ensembles considered, may play a
significant role in $\pi\pi$ scattering. While the light scalar pole is not
resonant in the $I=2$ channel, the state is exchanged in the T- and U-channels
and contributes significantly to the scattering length and effective range if
the coupling of the light scalar to the PNGBs in the $I=2$ channel is not too
small.

The nonperturbative lattice determination of the scattering phase shift
through the Lüscher procedure is the first such analysis carried out in a
theory with a confirmed light composite scalar. By utilizing moving wall
sources and the high gauge statistics available to us, we succeeded in
determining the scattering phase shift to high statistical precision. The
calculation was limited by the fact that $(ak)^{2}$ was small on the available
gauge ensembles, leading to a small signal to noise ratio in the difference
$E_{\pi\pi}-2M_{\pi}$ compared to $E_{\pi\pi}$ and $M_{\pi}$ individually.
Nonetheless, our results still showed a significant effective range
contribution as demonstrated in Fig. 1. A more detailed study of the effective
range contribution would be desirable in future studies by utilizing moving
frames and further overlapping volumes.

We compared our data to a leading order dilaton EFT in order to further
understand the lattice results and to assess whether the leading order dilaton
EFT predictions can provide a good global fit of the data. We have presented
for the first time the leading order expressions for $\pi\pi$ scattering in
the dilaton EFT for the $I=2$ and $I=0$ channels. The complete global fit to
the data has revealed slight differences between the predictions of leading
order dilaton EFT and the lattice data.

Our result for the scattering length in the $I=0$ channel expressed in Eq.
(36) and illustrated in Fig. 6 points to a clear difference between the
dilaton EFT and the chiral lagrangian predictions, even at values of
$M_{\pi}^{2}/F_{\pi}^{2}$ currently accessible on the lattice. While the $I=0$
scattering provides additional numerical challenges on the lattice, these new
results from the EFT motivate strongly a dedicated lattice study of this
channel.

With these results, we have demonstrated that PNGB scattering provides
significant information about the dynamics of nearly conformal gauge theories
which exhibit light scalar states. The scattering phase shifts provide
independent tests of effective descriptions such as the dilaton EFT. In future
work, improved precision in the extraction of the scattering phase shifts as
well as the exploration of other scattering momenta will provide stronger
tests of the dilatonic effective description. If discrepancies between the
tree level dilaton EFT and lattice data persist, this is good motivation for
considering NLO terms in the dilatonic effective description.

###### 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. 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. 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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