for the Hadron Spectrum Collaboration

# Determination of crossing-symmetric $\pi\pi$ scattering amplitudes and the
quark mass evolution of the $\sigma$ constrained by lattice QCD

Arkaitz Rodas<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility,
12000 Jefferson Avenue, Newport News, VA 23606, USA  Jozef J. Dudek
<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility, 12000 Jefferson
Avenue, Newport News, VA 23606, USA  Department of Physics, College of William
and Mary, Williamsburg, VA 23187, USA  Robert G. Edwards<EMAIL_ADDRESS>Thomas Jefferson National Accelerator Facility, 12000 Jefferson Avenue,
Newport News, VA 23606, USA

###### Abstract

Lattice QCD spectra can be used to constrain partial-wave scattering
amplitudes that, while satisfying unitarity, do not have to respect crossing
symmetry and analyticity. This becomes a particular problem when extrapolated
far from real energies, e.g. in the case of broad resonances like the
$\sigma$, leading to large systematic uncertainties in the pole position. In
this manuscript, we will show how dispersion relations can implement the
additional constraints, using as input lattice–determined $\pi\pi$ partial-
wave scattering amplitudes with isospin–0,1,2. We will show that only certain
combinations of amplitude parameterizations satisfy all constraints, and that
when we restrict to these, the $\sigma$ pole position is determined with
minimal systematic uncertainty. The evolution of the now well-constrained
$\sigma$ pole with varying light quark mass is presented, showing how it
transitions from a bound-state to a broad resonance.

††preprint: JLAB-THY-23-3791

##  I Introduction

Pion-pion scattering with isospin–0 plays a key role in nuclear and particle
physics, with the $J^{P}=0^{+}$ component proving to be a feature of many low-
energy phenomena of contemporary interest such as spontaneous symmetry
breaking [1], the long-range nucleon-nucleon interaction [2], and hadronic
contributions to the anomalous magnetic moment of the muon [3]. At low
energies, this partial wave is dominated by the existence of the $\sigma$
particle, the lightest resonance in Quantum Chromodynamics (QCD). This
extremely short-lived state which manifests as a slow energy variation of the
scattering amplitude, has only had its complex energy plane pole position
pinned down precisely relatively recently [4, 5, 6]. How this state arises
within QCD, and its relationship to other light-scalar mesons, like the
$\kappa$, the $f_{0}(980)$, and the $a_{0}(980)$, remains unclear.

Our leading tool to study non-perturbative aspects of QCD, including hadron
scattering and resonances, is Lattice QCD. This is a first-principles
numerical approach in which only controlled approximations are made, and which
features the ability to vary the value of quark masses and to explore the
evolution of physical observables with such variation. Access to scattering
amplitudes comes via computation of the discrete spectrum of states in the
finite-volume of the lattice, extracted from the time-dependence of Euclidean
correlation functions. A finite-volume quantization condition often referred
to as the Lüscher equation, relates the spectrum to infinite-volume partial-
wave scattering amplitudes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In
practice, the computed spectra are used to constrain unitarity-satisfying
parameterizations of the isospin–0 $S$–wave partial-wave amplitude, which when
analytically continued in the complex energy plane, yield the $\sigma$ pole
[19, 20, 21].

With light quark masses chosen so that the pion has a mass of 391 MeV, the
$\sigma$ appears as a well-determined stable _bound-state_ pole below the
$\pi\pi$ threshold. On the other hand, in calculations at a pion mass of 239
MeV it was found that a wide range of parameterizations were capable of
describing the finite-volume spectrum, all giving compatible amplitude
determinations across the elastic scattering region, but these various
amplitudes feature resonant $\sigma$ pole positions lying deep in the complex
plane which are scattered well outside the level of statistical uncertainty
[19].

This observation is closely related to the longstanding challenge of
accurately determining the $\sigma$ pole position from fits to experimental
$\pi\pi$ elastic scattering data [22]. In that case, a solution was found
using _dispersion relations_ which implement the additional fundamental
constraint of _crossing symmetry_ , leading to precise contemporary estimates
of the $\sigma$ pole position [4, 5, 6, 23]. Until recently there had only
been a single attempt to apply crossing symmetry to $\pi\pi$ scattering
amplitudes determined using lattice QCD [20]. In that work the cross-channel
effects were treated only perturbatively in a unitarized chiral perturbation
theory framework.

In this manuscript we will apply dispersion relations, which allow us to
implement crossing symmetry non-perturbatively, to lattice QCD data,
considering two unphysical values of the light quark mass lying in the region
where we believe the $\sigma$ is transitioning from being bound into being a
broad resonance, showing that the dominant systematic error on the $\sigma$
pole position associated with parameterization of partial-wave scattering
amplitudes can be almost entirely removed. Combining these results with
calculations at heavier quark masses, we present a robust estimation within
first-principles QCD of the evolution of the $\sigma$ pole with varying quark
mass.

The Hadron Spectrum Collaboration (hadspec) has performed lattice calculations
with two flavors of dynamical light quarks and a heavier dynamical strange
quark ($N_{f}=2+1$) yielding discrete finite-volume spectra in channels with
the quantum numbers of $\pi\pi$ scattering for four values of the light quark
mass, corresponding to pion masses of $391,330,283$ and $239$ MeV [24, 25, 26,
27, 19, 28, 29]. The $\pi\pi$ elastic partial-wave scattering amplitudes
constrained using these spectra feature a $\rho$ in the isovector $P$–wave
that is a resonance at all quark masses, while the isoscalar $S$–wave has an
accurate interpretation only for the heaviest two quark masses, where the
$\sigma$ appears as a well-determined bound-state pole. Other lattice
calculations exist for $I=0$, with $N_{f}=2$ [20, 21], for $I=2$ with
$N_{f}=2$ [30], and $N_{f}=2+1$ [31, 32, 33]. The $\rho$ has been studied
extensively both for $N_{f}=2$ [34, 35, 36, 37, 38, 39, 40, 41], $N_{f}=2+1$
[42, 43, 32, 44, 45, 46, 47] and for $N_{f}=2+1+1$ [48]. For a review on the
topic, we refer the reader to Ref. [49].

In this manuscript, we will focus on the lightest two quark masses calculated
by hadspec. In both these cases, the $I=2$ $S$–wave amplitude ($S2$) is weak,
repulsive, and acceptably described over the elastic region by a scattering
length approximation. The $I=2$ $D$–wave ($D2$) is extremely weak across the
elastic region. The $I=1$ $P$–wave ($P1$) is dominated by the narrow $\rho$
resonance, which is well described by a Breit-Wigner amplitude, while the
$F$–wave is completely negligible. The $I=0$ $S$–wave amplitude ($S0$) at both
quark masses is a slowly varying function of energy, but there is a
significant change in the behavior at threshold between the two quark masses,
indicating a rapid variation in the scattering length with the quark mass. The
generic functional forms used to describe these amplitudes can be found in
Appendix D, and in the original works mentioned above. Additionally, the $I=2$
channel at $m_{\pi}\sim 239$ MeV, which had not previously appeared in the
literature, has been computed for this work and is presented in Appendix E.

The $S0$ ‘lattice data’ (at both quark masses) proves to be describable by a
large variety of unitarity-respecting parameterization forms, and these
various descriptions lead to a spread of $\sigma$ pole estimates, with a
scatter much larger than their statistical uncertainties, as seen in Ref.
[19]. It is clear, then, that the lattice data in $S0$ alone, even when it has
rather small statistical errors, does not provide sufficient constraint to
uniquely determine the $\sigma$ pole position. We will use the additional
constraint of _crossing symmetry_ , in the form of dispersion relations
applied to the coupled system of lattice data in all the partial waves above,
to pin down the location of the $\sigma$ pole at $m_{\pi}\sim 283$ and $239$
MeV.

The manuscript is organized as follows: we briefly summarize the specific
dispersion relations used in this work, applied to partial-wave amplitudes in
Section II. In Section III, we first define the metrics that quantify the
degree to which our dispersion relations are fulfilled. Then, we apply these
metrics to obtain a constrained system of amplitudes that respect all
$S$-matrix principles. Once this set is obtained, in Section IV, we apply the
dispersive outputs to the study of the subthreshold and resonance regions in
Sections IV.1 and IV.2, respectively, including discussions of the quark mass
behavior of Adler zeroes and the $\sigma$ pole trajectory. Finally, we
summarize our work in Section V.

##  II Dispersion relations

Generically, crossing symmetry relates two-particle scattering behavior in the
$s,t$ and $u$–channels in terms of the same amplitude, $T(s,t,u)$, evaluated
in different kinematical regions. For the case of $\pi\pi$ scattering with its
three isospin configurations, given $T(s,t,u)$ we can construct the amplitudes
of the different $s$-channel isospin processes as

$\displaystyle T^{I=0}(s,t,u)=3T(s,t,u)+T(t,s,u)+T(u,t,s),$ $\displaystyle
T^{I=1}(s,t,u)=T(t,s,u)-T(u,t,s),$ $\displaystyle
T^{I=2}(s,t,u)=T(t,s,u)+T(u,t,s)\,.$ (1)

All isospin amplitudes in $\pi\pi$ scattering are then related through simple
algebraic crossing relations. However, crossing symmetry is obscured when
amplitudes are partial-wave projected in one channel,

$t^{I}_{\ell}(s)=\tfrac{1}{64\pi}\int^{1}_{-1}d\\!\cos\theta_{s}\,T^{I}(s,t,u)\,P_{\ell}(\cos\theta_{s}),$
(2)

with the scattering dynamics in the $t$ and $u$–channels appearing as a _left-
hand cut_ , i.e. as an imaginary part of $t^{I}_{\ell}(s)$ for $s<0$. When the
only constraint on the partial-wave amplitude is data for
$s>s_{\mathrm{thr}}$, typically the left-hand cut remains undetermined.

In the case that a partial-wave is dominated by a _narrow_ resonance (like the
$\rho$), the impact of the left-hand cut singularity for real energies above
threshold is typically negligible, since the nearby resonance pole dominates.
However, when there is no resonance, or when there is a broad resonance, deep
in the complex plane (like the physical $\sigma$), the left-hand cut
contribution can be significant, and there is a need to constrain it
accurately.

Figure 1:  Real part of kernel functions in Eq. 3 contributing to dispersion
of $S0$, $\mathrm{Re}\,K^{0I^{\prime}}_{0\ell^{\prime}}$, plotted in units
where $1$ on the $y$-axis represents a value $4m_{\pi}^{2}$. Left column:
functions evaluated on the real $s^{\prime}$–axis for three values of $s$ –
inset indicates the behavior in the high-energy region. Right column: Kernel
functions evaluated for complex values of $s$.

In the case of $\pi\pi$ elastic scattering, the left-hand cuts of the partial-
waves with $I=0,1,2$, $t^{I}_{\ell}(s)$, can be constrained by crossing
symmetry, and _dispersion relations_ provide an approach to implement this
symmetry using the analytic properties of the amplitudes $T^{I}(s,t,u)$. Such
relations are obtained by application of Cauchy’s theorem to $T^{I}(s,t,u)$
[50], introducing subtractions to ensure that the integrals converge at
infinity. Upon partial-wave projection, the dispersion relations take the
form,

$\tilde{t}^{I}_{\ell}(s)=\tau^{I}_{\ell}(s)+\sum_{I^{\prime},\ell^{\prime}}\int_{4m_{\pi}^{2}}^{\infty}\\!\\!\\!ds^{\prime}\,K^{II^{\prime}}_{\ell\ell^{\prime}}\\!(s^{\prime}\\!,s)\;\mathrm{Im}\,t^{I^{\prime}}_{\ell^{\prime}}(s^{\prime})\,,$
(3)

where scattering amplitudes consistent with crossing symmetry and unitarity
will have $\tilde{t}^{I}_{\ell}(s)=t^{I}_{\ell}(s)$. The
$\\{\tau^{I}_{\ell}(s)\\}$ are low-order polynomials in $s$ featuring a number
of free parameters set by the number of subtractions, while the kernels
$K^{II^{\prime}}_{\ell\ell^{\prime}}\\!(s^{\prime}\\!,s)$ are known functions
(also depending upon the number of subtractions, see Refs. [51, 52]) that
encode the crossing relations111The diagonal kernels,
$K^{II}_{\ell\ell}(s^{\prime},s)$, contain a simple pole
$\sim\frac{1}{\pi}\frac{1}{s^{\prime}-s}$ which ensures that
$\mathrm{Im}\,\tilde{t}^{I}_{\ell}(s)$ is always exactly equal to
$\mathrm{Im}\,t^{I}_{\ell}(s)$ for real energies above threshold.. The
structure of the kernels ensures that the dispersed amplitudes have left-hand
cuts set by the dynamics in the crossed-channels.

For _twice-subtracted_ dispersion relations, commonly referred to as “Roy”
equations [53], the functions $\tau^{I}_{\ell}(s)$ take the form

$\displaystyle\tau^{0}_{0}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{3}\big{(}a^{0}_{0}+5a^{2}_{0}\big{)}+\tfrac{1}{3}\big{(}2a^{0}_{0}-5a^{2}_{0}\big{)}\frac{s}{4m_{\pi}^{2}}\,,$
$\displaystyle\tau^{1}_{1}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{18}\big{(}2a_{0}^{0}-5a^{2}_{0}\big{)}\frac{s-4m_{\pi}^{2}}{4m_{\pi}^{2}}\,,$
$\displaystyle\tau^{2}_{0}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{6}\big{(}2a^{0}_{0}+a^{2}_{0}\big{)}-\tfrac{1}{6}\big{(}2a^{0}_{0}-5a^{2}_{0}\big{)}\frac{s}{4m_{\pi}^{2}}\,,$
(4)

where the $S$-wave scattering-lengths,

$a^{I}_{0}\equiv\mathrm{Re}\,t^{I}_{0}(s=4m_{\pi}^{2})/m_{\pi},$ (5)

are parameters that can be fixed using the $S0$, $S2$ amplitude behaviors at
threshold, since the contribution of the integrals in Eq. 3 goes to zero as
$\sqrt{s}\to 2m_{\pi}$.

Figure 1 illustrates the dispersive kernel functions relevant for obtaining
the $S0$ dispersed amplitude,
$K^{0I^{\prime}}_{0\ell^{\prime}}\\!(s^{\prime}\\!,s)$, where we observe that
crossed-channels will have an influence at elastic scattering energies, and
that all contributions from amplitudes at high energies are heavily suppressed
due to the subtraction scheme. The right column shows the kernels evaluated at
_complex_ values of $s$ where we observe the pole singularity at real
$s^{\prime}=s$ being smoothed out. The lowest entry in the right column is
evaluated in the region where the $\sigma$ pole is eventually found for
$m_{\pi}\sim 239\,\mathrm{MeV}$, and here we observe that the influence of the
crossed-channels, $P1$ and $S2$ (which contribute to the _left-hand cut_),
will be significant.

The integrals in Eq. 3 run over all energies above the elastic scattering
threshold, and inevitably our knowledge of $\\{t^{I}_{\ell}(s)\\}$ stops at
some point, so it proves necessary to parameterize the high-energy behavior,
which we may do using known Regge asymptotics. In our implementation, we
divide the integration from threshold to infinity into two parts: an
integration from threshold to $s_{h}$ where lattice constrained amplitudes are
used, and an integration from $s_{h}$ to infinity where Regge
parameterizations are used. For both cases in our work, $m_{\pi}\sim
239,\,283$ MeV, we use $\sqrt{s_{h}}=0.22\,a_{t}^{-1}$, which for $m_{\pi}\sim
239$ MeV corresponds to $\sqrt{s_{h}}=5.6\,m_{\pi}=1.34\,\mathrm{GeV}$.
Details of the high-energy Regge parameterizations which model $t$-channel
exchanges in the amplitudes $T^{I}(s,t,u)$ are presented in Appendix A.

The dispersed amplitudes $\tilde{t}^{I}_{\ell}(s)$ in Eq. 3 are obtained by
evaluating the integrals numerically. The input lattice-constrained partial-
wave amplitudes carry statistical uncertainties associated with being
constrained by energy levels computed on a finite ensemble of lattice
configurations. These are available in the form of (correlated) statistical
uncertainties on the parameters of each amplitude parameterization. In
addition, as discussed in Section A.2, the high-energy Regge parameterized
amplitudes carry an assigned conservative uncertainty. Given a set of fits to
lattice data with parameters $\\{a_{i}\\}$, and Regge contributions with a
fixed fractional uncertainty, the dispersion relation uncertainties
$\Delta\tilde{f}^{I}_{\ell}(s,\\{a_{i}\\})$ are obtained by linearized
(correlated) error propagation at each sampled energy point.

In this work, we will report results using _minimally-subtracted_ relations
(often referred to as “GKPY equations” [52]), and _twice-subtracted_ relations
(“Roy equations” [53]). A choice can be made to implement _more_ than the
minimal number of subtractions needed for convergence, which will have the
practical consequence of reducing the contribution of the partial-wave
amplitudes at high-energies, in particular in the region where they are Regge-
parameterized, in exchange for increased sensitivity to the behavior of the
amplitudes at the subtraction point, which is typically the elastic threshold.
We observe that the _twice-subtracted_ amplitudes are minimally sensitive to
how we parameterize the high-energy behavior, and while the _minimally-
subtracted_ amplitudes do show sensitivity to estimated high-energy behavior,
nevertheless, compatible results for amplitudes at low energies are obtained.
In the main body of the paper we will present only results applying _twice-
subtracted_ dispersion relations, where, as shown in Eq. 4, the
$\tau^{I}_{\ell}(s)$ are linear in $s$ and depend only upon the $S$–wave
scattering lengths, $a_{0}^{0}$, $a_{0}^{2}$. Results using _minimally-
subtracted_ dispersion relations are presented in Appendix C.

##  III Dispersive evaluation of lattice constrained amplitudes

As described above, the discrete energy spectra extracted from lattice QCD
calculations constrain definite isospin partial-wave amplitudes one-by-one in
the elastic scattering region in a way that exactly respects unitarity, but
which has no constraint from crossing symmetry. Typically multiple
parameterizations prove to be capable of describing the lattice data for real
energies above threshold. We will establish which combinations of partial-wave
amplitude parameterizations are compatible with first-principles using
dispersion relations. Upon input of a selected set of lattice amplitudes,
$\big{\\{}t_{\ell}^{I}(s)\big{\\}}$, into the right-hand-side of Eq. 3, a set
of _dispersed amplitudes_ , $\big{\\{}\tilde{t}_{\ell}^{I}(s)\big{\\}}$, are
produced which have the same imaginary parts, but modified real parts. In
order for these amplitudes to be compatible with unitarity, they must be
statistically compatible with the input amplitudes,
$\big{\\{}t_{\ell}^{I}(s)\big{\\}}$, in the elastic scattering region.

We assess the suitability of the dispersed amplitudes using two metrics. The
first, which we call $d^{2}$, compares the real part of the dispersed
amplitude in one partial-wave
($\tilde{f}^{I}_{\ell}(s)\equiv\mathrm{Re}\,\tilde{t}_{\ell}^{I}(s)$) with the
real part of the input lattice amplitude
(${f}^{I}_{\ell}(s)\equiv\mathrm{Re}\,{t}_{\ell}^{I}(s)$),

$\big{[}d^{2}\big{]}^{I}_{\ell}\equiv\sum_{i=1}^{N_{\mathrm{smpl}}}\left(\frac{\tilde{f}^{I}_{\ell}(s_{i})-f^{I}_{\ell}(s_{i})}{\Delta\big{[}\tilde{f}^{I}_{\ell}(s_{i})-f^{I}_{\ell}(s_{i})\big{]}}\right)^{2}\,,$
(6)

where the difference is sampled at a large number of equally spaced energy
values in the elastic scattering region, and where the uncertainty in the
difference is computed by linearly propagating the correlated uncertainties on
the lattice-fit amplitude parameters, and in addition, the conservative
uncertainty placed on the high-energy Regge-like behavior (see Section A.2 for
details).

Figure 2: Real parts of $S0$, $P1$, $S2$ scattering amplitudes for
$m_{\pi}\sim 239$ MeV. Data-points indicate the constraint provided by
discrete lattice QCD spectra. The top row show an example of parameterizations
(red) which describe well the lattice data, but which generate dispersed
amplitudes (blue) that are in poor agreement, leading to large values of
$d^{2}$ and $\tilde{\chi}^{2}$. The second rows show a case where the lattice
amplitudes (green) prove to be in good agreement with the dispersed
amplitudes. The colored numbers with an error estimate show the $S$-wave
scattering lengths in $m^{-1}_{\pi}$ units. The red(green) italic numbers show
the $\tilde{\chi}^{2}/N_{\text{lat}}$ values for the input amplitudes, whereas
the blue numbers on those same panels show the
$\tilde{\chi}^{2}/N_{\text{lat}}$ values for the dispersed amplitude. Beneath
each panel, values of the quantity in the large parentheses in Eq. 6 at
$N_{\mathrm{smpl}}=91$ points are given, and the numbers listed there show the
$d^{2}/N_{\text{smpl}}$ values.  Figure 3: As Figure 2 but for $m_{\pi}\sim
283$ MeV. The lower panels sampled at $N_{\mathrm{smpl}}=89$ points.

The $d^{2}$ metric can yield small values when at least one of $f_{\ell}^{I}$
or $\tilde{f}_{\ell}^{I}$ has large statistical uncertainties, corresponding
to an (undesired) imprecise amplitude description, so we choose to supplement
it with a second $\chi^{2}$–like metric which compares the central value of
the dispersed amplitude with the lattice data,

$\displaystyle\big{[}\tilde{\chi}^{2}\big{]}^{I}_{\ell}$
$\displaystyle\equiv\sum_{i,j=1}^{N_{\mathrm{lat}}}\left(\tfrac{\mathfrak{f}_{i}-\tilde{f}^{I}_{\ell}(s_{i})}{\Delta_{i}}\right)\mathrm{corr}(\mathfrak{f}_{i},\mathfrak{f}_{j})^{-1}\left(\tfrac{\mathfrak{f}_{j}-\tilde{f}^{I}_{\ell}(s_{j})}{\Delta_{j}}\right)\,,$
(7)

where $\mathfrak{f}_{i}$ are the discrete values of
$\mathrm{Re}\,t^{I}_{\ell}$ extracted from solving the Lüscher finite-volume
condition at energy ${E_{i}=\sqrt{s_{i}}}$. The construction of the
corresponding uncertainty, $\Delta_{i}$, is described in Appendix B.

(a)  Figure 4:  For $m_{\pi}\sim 239$ MeV, for each combination of
parameterizations of $S0$ (columns) and $S2$ (rows), each ordered by the
magnitude of their scattering length in $m^{-1}_{\pi}$ units, boxes are
colored according to the average value of $d^{2}/N_{\mathrm{smpl}}$ (upper
triangle) and $\tilde{\chi}^{2}/N_{\mathrm{lat}}$ (lower triangle) over $S0$,
$P1$, $S2$. The region indicated by the black outline shows parameterization
combinations that have all amplitudes having metric values below the cutoffs
presented in the text.

(a)  Figure 5:  As Figure 4 but for $m_{\pi}\sim 283$ MeV.

Figures 2, 3, show examples of $m_{\pi}\sim 239$ MeV and $m_{\pi}\sim 283$ MeV
$S0$, $P1$ and $S2$ lattice amplitudes 222see Refs. [19, 29] and Appendix D
for details of the particular functional forms used., and their dispersively
modified counterparts, illustrating how the metrics defined above select
consistent and reject inconsistent combinations. In practice, we find that
there is little variation in the metric values with change of $P1$
parameterization, so we opt to fix this to one particular successful form for
both analyses 333a $K$-matrix with a single pole plus a constant, and a
dispersed phase-space subtracted so that its real part vanishes at the bare
$K$-matrix pole.. We compute dispersed amplitudes in the remaining space of
choices of $S0$, $S2$ parameterizations, and retain only those combinations
which have $d^{2}/N_{\mathrm{smpl}}<1$ and
$\tilde{\chi}^{2}/N_{\mathrm{lat}}<2$ for all partial-waves, $S0,P1,S2$ 444The
sensitivity to this particular choice is investigated in Appendix B..

The values of the metrics for a large number of $S0$, $S2$ amplitude
parameterizations are shown in Fig. 4 (for $m_{\pi}\sim 239$ MeV) and Fig. 5
(for $m_{\pi}\sim 283$ MeV), where it is clear that applying cuts on the
metrics, and thus enforcing both unitarity and crossing symmetry, leads to a
much reduced set of amplitudes relative to those which acceptably described
the lattice energy levels in a conventional “partial-wave–by–partial-wave”
analysis. The value of the relevant $S$–wave scattering length is provided for
each parameterization, and it is clear that this approach has significantly
reduced the range of acceptable values of $a_{0}^{0}$ and $a_{0}^{2}$.

In the $m_{\pi}\sim 283$ MeV case shown in Fig. 5, $a_{0}^{0}$ is large and
positive, of much larger magnitude than in the $m_{\pi}\sim 239$ MeV case,
while $a_{0}^{2}$ remains small. Equations (3) and (4) are such that
$a_{0}^{0}$ and $t^{0}_{0}(s>4m_{\pi}^{2})$ feature for _all_ partial-waves,
and in order to get lineshapes compatible with data, including weak scattering
in $I=2$, a delicate cancellation between the integral over $t^{0}_{0}(s)$ and
the contribution of $a^{0}_{0}$ is required. This is reflected in the
relatively small number of amplitudes found with small metric values.

##  IV Dispersed amplitudes away from the elastic scattering region

The acceptable dispersed amplitudes, $\tilde{t}^{I}_{\ell}(s)$, found in the
previous section (by virtue of small values of metrics), have several
desirable properties, owing to the fact that they effectively respect
unitarity, analyticity _and_ crossing symmetry.

An illustration of this comes in their behavior for real energies below the
elastic threshold, which for the lattice-constrained input amplitudes was
essentially unconstrained and varied wildly between parameterizations, with
some even featuring unphysical singular behavior. In a “partial-
wave–by–partial-wave” analysis, this subthreshold region is obtained only as
an extrapolation, and should not be expected to be accurate away from the data
above or near the threshold. On the other hand, our dispersed amplitudes,
$\\{\tilde{t}^{I}_{\ell}(s)\\}$, implicitly include information both from the
right- and the left-hand cuts (via crossing), and we are in effect
_interpolating_ between these, with the additional constraint of analyticity
being imposed by the use of dispersion relations which originate in the Cauchy
theorem. Similarly the extrapolation into the complex energy plane, where
resonance poles are expected to be found, should be rendered more stable.

###  IV.1 Subthreshold region

Figure 6:  $S$-wave amplitudes for $m_{\pi}\sim 239$ MeV. Top panel: Central
values of the real parts of $S0$ and $S2$, both above and below the two-pion
threshold, for all amplitude parameterizations. Those plotted in green
correspond to input parameterizations that produce at least one combination
respecting the metric cuts presented above, while all others are plotted in
red. Open circles on axis indicate the locations of Adler zeroes in the
leading-order of $\chi$PT. Bottom panel: Real parts of the corresponding
dispersive amplitudes for all amplitude combinations respecting the metric
cuts presented above. The uncertainty on one example amplitude is shown by the
gray band.

Figure 7:  As Figure 6 but for $m_{\pi}\sim 283$ MeV.

The amplitude parameterizations used to describe lattice QCD spectra above
threshold (and sometimes slightly below threshold) in a particular partial-
wave are built to exactly obey unitarity above threshold, but they are not
typically guaranteed to be free of unphysical behavior far below threshold.
Usually this is excused as irrelevant since the amplitudes are only needed in
a small region around the real energy axis above threshold (for example when
narrow resonances are sought), but if extrapolation further into the complex
energy plane is required, such parameterization artifacts may be problematic.

The use of dispersion relations can remedy this problem. Since the dispersion
relations take as input the amplitudes only above threshold (where they are
well constrained), while the subthreshold behavior is controlled by the kernel
functions (which have correct analytic properties), the behavior of the
dispersed amplitudes below threshold is rendered free of singularities.

This is illustrated in Figures 6, 7, where the $S0$ and $S2$ amplitudes are
shown for $m_{\pi}\sim 239,\,283$ MeV. The upper panels in each case shows the
input lattice amplitudes (where amplitudes that systematically fail the metric
cuts are shown in red) where subthreshold divergences are observed to be
present, as is a significant scatter of behavior such that one can argue that
the lattice data (above threshold, in a single partial-wave) has not
constrained in any reliable way the amplitude behavior far below threshold. On
the other hand, in the lower panels, we observe that all dispersed amplitudes
satisfying the metric cuts show broadly compatible singularity-free behavior
below threshold. The scatter of behaviors of acceptable amplitudes is observed
to be at the level of the uncertainty (shown for one example amplitude by the
gray band).

In Fig. 6, for $m_{\pi}\sim 239$ MeV, both sets of dispersed amplitudes are
observed to feature a zero-crossing below threshold, located near to
$s/m_{\pi}^{2}\approx 0.8$ for $S0$ and $s/m_{\pi}^{2}\approx 1.6$ for $S2$.
The presence of such zeroes, known as “Adler zeroes”, is an expectation of
chiral perturbation theory [54], and we show in the figure the expected
location at leading order ($s/m_{\pi}^{2}=\tfrac{1}{2}$ for $S0$ and
$s/m_{\pi}^{2}=2$ for $S2$). The zeroes in the dispersed amplitudes are
observed to differ from these expectations, with a non-negligible spread.

In Fig. 7, for $m_{\pi}\sim 283$ MeV, the $S2$ amplitude is seen to feature an
Adler zero near $s/m_{\pi}^{2}\approx 2.3$, while the $S0$ amplitude does not
appear to cross zero between the left-hand cut and threshold, in contradiction
to the expectations of leading-order $\chi$PT. As such, we would argue that
analyses of lattice QCD obtained spectra using amplitudes which _enforce_ an
Adler zero fixed at the leading order location are potentially introducing a
systematic bias and this may impact results such as scattering lengths or low-
lying pole positions.

###  IV.2 Resonance poles in the dispersed amplitudes

Figure 8: For $m_{\pi}\sim 239$ MeV, pole locations in $S0$ amplitude for
lattice-fit amplitudes (gray) and dispersed amplitudes satisfying the metric
cuts described in the text (blue), and modulus of the couplings (as defined in
Refs. [5, 29]) extracted from the pole residues.

Figure 9: As Figure 8 but for $m_{\pi}\sim 283$ MeV. Cases in which the pole
locations feature very large uncertainties have been excluded for clarity of
presentation.

The location of the $\rho$ resonance pole in the dispersed $P1$ amplitudes is
found to be compatible with the small spread observed in the input lattice
amplitudes, as expected for a narrow resonance. On the other hand, for the
$\sigma$ pole in $S0$, which at $m_{\pi}\sim 239$ MeV is lying deep in the
complex plane, the acceptable dispersed amplitudes all have a pole that lies
in a much-reduced region, as shown in Fig. 8. As hoped, the imposition of
analyticity and crossing symmetry, constrained by lattice data in all relevant
isospins and low partial-waves, has led to a robust extraction of the $\sigma$
pole position, which is observed to be independent of any significant
parameterization dependence.

Figure 10:  $\sigma$ pole evolution with changing light-quark mass
corresponding to $m_{\pi}\sim 391$ MeV (green, bound-state, non-dispersive,
taken from Ref. [19]), 330 MeV (blue, bound-state, non-dispersive, taken from
Ref. [29]), 283 MeV (red, virtual bound-state, dispersive), 239 MeV (orange,
resonance, dispersive). Gray points correspond to dispersive extractions from
experimental data given in Refs. [4, 5, 6].

In the $m_{\pi}\sim 283$ MeV case (Fig. 9), the $S0$ lattice amplitudes
indicated that the $\sigma$ could be either a virtual bound-state555a pole on
the real energy axis below threshold on the unphysical Riemann sheet. _or_ a
subthreshold resonance, depending upon parameterization choice, with a spread
in pole positions. Those dispersed amplitudes that meet the metric cuts show a
reduced scatter in pole location. As described in Ref. [29], when the complex
conjugate pole pair of a subthreshold resonance meet on the real axis below
threshold the pole location’s dependence on the amplitude parameters develops
an infinite slope. Near this point, the slopes are large, causing small
uncertainties on the parameters to become large uncertainties on the pole
location. We do not plot the dispersive pole locations for these noisy results
but merely comment that they are compatible with the plotted virtual bound-
state cases. In the current analysis, a definitive statement about whether the
state is a virtual bound-state, or a subthreshold resonance at this pion mass
cannot be made. The $\pi\pi$ couplings of these plotted poles are also
presented in the figure, where we see that the dispersive results have larger
statistical uncertainties but reduced systematic spread with respect to the
conventional analyses presented in Refs. [19, 29].

In our $m_{\pi}\sim 283$ MeV analysis, the selected dispersed central values
find no real Adler zero for $S0$. In those cases, a very noisy third pole,
companion to the pair of $\sigma$ poles, appears close to the left-hand cut in
the unphysical Riemann sheet.

In this work, we have considered two pion masses for which the $\sigma$
appears as a pole in the unphysical Riemann sheet, while at higher pion masses
lattice calculations indicate that the $\sigma$ is a bound-state pole on the
physical Riemann sheet. In these latter cases, a dispersive analysis is
typically not required to determine accurately the $\sigma$ pole location, as
it is tightly constrained by finite-volume energy levels lying close to the
pole. Nevertheless, it is possible to construct applicable dispersion
relations by explicitly including the $\sigma$ in $T^{0}(s,t,u)$ as “fixed”
poles in $s$ and $u$. The $s$–channel pole remains a pole when the amplitude
is projected into the $s$–channel $S$–wave, while the $u$–channel pole
generates a cut that is present in all partial-waves. Recently, Ref. [55]
applied dispersion relations in an approach different to that explored in this
paper, in a case where a bound-state $\sigma$ is present using lattice data at
$m_{\pi}\sim 391$ MeV previously published by hadspec, finding a $\sigma$ pole
compatible with the undispersed analysis in Ref. [28].

Finally, taking the now robust $\sigma$ pole results from dispersive analysis
at $m_{\pi}\sim 239,\,283$ MeV and supplementing them with two heavier quark
masses where dispersive analysis is not required, we show in Fig. 10 the
evolution with changing quark mass of the $\sigma$ pole. In distinction to the
narrow $P$-wave $\rho$ resonance, which exhibits a simple quark-mass
dependence corresponding to an approximately constant coupling to $\pi\pi$
[29, 56, 57], the $S$-wave $\sigma$ undergoes a rapid transition from bound to
resonant state, appearing to pass through a virtual bound-state stage in a
narrow region of pion mass.

##  V Summary

We have presented a dispersive approach to analyze elastic hadron-hadron
scattering information provided by lattice QCD, applied here to the case of
$\pi\pi$ scattering at two pion masses. We have observed that the sensitivity
of $S$–wave scattering lengths and broad resonance pole locations to choice of
parameterization form can be largely eliminated. The method also yields
reliably the scattering amplitudes below threshold, and in this region we have
pinned down the location of amplitude zeroes (Adler zeroes) when they appear.
This indicates that reliable results can be extracted from lattice QCD
calculations without the need to _enforce_ amplitude features, such as Adler
zeroes, not directly motivated by the lattice data. The now well-constrained
amplitudes obtained in this study can be used in future lattice QCD
calculations in which the $\pi\pi$ $S$-wave scattering system is coupled to
external currents [58, 59, 60, 61], where well-defined _form-factors_ of
resonances can be extracted from pole residues, providing structural
information that will aid in the determination of the compositeness of the
$\sigma$ as the light quark mass is varied.

###### Acknowledgements.

We thank our colleagues within the Hadron Spectrum Collaboration. AR, JJD, and
RGE acknowledge support from the U.S. Department of Energy contract DE-
AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and
operates Jefferson Lab. AR, JJD also acknowledge support from the U.S.
Department of Energy award contract DE-SC0018416. This work was done as part
of the ExoHad Topical Collaboration. The software codes Chroma [62], QUDA [63,
64], QUDA-MG [65], QPhiX [66], and QOPQDP [67, 68], and Redstar [69] were
used. The authors acknowledge support from the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research and Office
of Nuclear Physics, Scientific Discovery through Advanced Computing (SciDAC)
program. Also acknowledged is support from the U.S. Department of Energy
Exascale Computing Project. The contractions were performed on clusters at
Jefferson Lab under the USQCD Initiative and the LQCD ARRA project. This
research was supported in part under an ASCR Leadership Computing Challenge
(ALCC) award, and used resources of the 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. This research is also part of the Blue Waters sustained-
petascale computing project, which is supported by the National Science
Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois.
Blue Waters is a joint effort of the University of Illinois at Urbana-
Champaign and its National Center for Supercomputing Applications. This
research used resources of the National Energy Research Scientific Computing
Center (NERSC), a DOE Office of Science User Facility supported by the Office
of Science of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. The authors acknowledge the Texas Advanced Computing Center
(TACC) at The University of Texas at Austin for providing HPC resources. Gauge
configurations were generated using resources awarded from the U.S. Department
of Energy INCITE program at Oak Ridge National Lab, and also resources awarded
at NERSC.

##  Appendix A Dispersive inputs

As described in the text, when implementing Eq. 3 the input
$\mathrm{Im}\,t^{I^{\prime}}_{\ell^{\prime}}(s^{\prime})$ is needed over the
entire energy region from threshold to infinity. In practice, we split this
integral into two pieces, below $s_{h}$ where lattice-obtained partial-wave
amplitudes are used, and above $s_{h}$, where Regge-like parameterizations are
used. Due to the polynomial suppression of the integral kernels at high-
energies, the sensitivity of dispersed amplitudes at low energies to the
details of the amplitudes above $s_{h}$ is weak.

###  A.1 Lattice obtained partial-wave input

The partial-wave lattice inputs are described by using parameterizations
presented in Refs. [19, 29], parameterizations fitted to the results presented
in Ref. [27], or parameterizations fitted to the results presented in Appendix
E. For a detailed summary of the functional forms used, see Appendix D. They
are constrained as follows:

$\mathbf{S2,D2}$: these amplitudes, whose spectra are compatible with there
being no inelasticity, are constrained by fitting energy levels up to around
$a_{t}E_{\mathsf{cm}}=0.22$ for both $m_{\pi}\sim 239,\,283$ MeV masses. They
describe weak repulsive non-resonant scattering.

$\mathbf{P1}$: this amplitude is constrained by fitting energy levels up to
around $a_{t}E_{\mathsf{cm}}=0.185(0.19)$ for $m_{\pi}\sim 239(283)$ MeV, very
slightly above the $K\bar{K}$ threshold. As presented in Ref. [27], there is
negligible inelasticity up to $a_{t}E_{\mathsf{cm}}=0.22$, and as such, we use
an extrapolation of the determined elastic amplitude up to $s_{h}$. This
partial wave is dominated by the $\rho$ resonance in the elastic region, which
can be fitted well by functional forms with few free parameters, such as a
Breit-Wigner form. These amplitudes have a smooth featureless extrapolation up
to $s_{h}$. The fact that the phase-shift is very close to $180^{\circ}$ in
the inelastic region means the imaginary part that enters the dispersion
relations is very close to zero.

$\mathbf{S0}$: this amplitude is constrained by fitting energy levels up to
around $a_{t}E_{\mathsf{cm}}=0.1585(0.16)$ for $m_{\pi}\sim 239(283)$ MeV,
slightly below the $K\bar{K}$ threshold. The amplitudes feature slow energy
dependence in the elastic region. The presence of a scalar resonance analogous
to the $f_{0}(980)$ can cause the $K\bar{K}$ amplitude to turn on rapidly, and
we have not yet attempted coupled-channel descriptions in this region, where
$4\pi$ channels may also be relevant. In practice, we use extrapolations of
the elastic amplitude up to $s_{h}$, and for different parameterizations,
these extrapolations can differ significantly. However, as observed in Figures
4, 5, the main variation in the dispersion relation metrics comes from the
behavior of the amplitudes at threshold (characterized by the scattering
length), so it appears that there is little sensitivity to this extrapolation
region, suppressed as it is by the rapidly decreasing dispersion relation
kernels.

Figure 11:  Regge contributions to $\sigma^{I_{s}=0}(s)$ (left),
$\sigma^{I_{s}=1}(s)$ (center) and $\sigma^{I_{s}=2}(s)$ (right) for
$m_{\pi}\sim 239$ MeV. Parameterizations taken from Ref. [52] (blue) and Ref.
[70] (orange), scaled to be applicable to the current pion mass. In our
dispersive analysis, we use the average of the two (gray) assigning a
conservative 30% fractional uncertainty to each $I_{t}$ amplitude.  Figure 12:
Top row: Blue curves show the real part of the $S0$, $P1$, $S2$ partial-wave
amplitudes for $m_{\pi}\sim 239$ MeV (amplitude combination indicated by the
black star in Fig. 14). The gray curves lying close to zero in each case show
the contribution to the amplitude,
$\tilde{f}^{I}_{\ell}(s)\big{|}_{\mathrm{Regge}}$, coming from the high-energy
amplitude parameterization (as defined in Eq. 10).
Bottom row: Ratio, in absolute value, of
$\tilde{f}^{I}_{\ell}(s)\big{|}_{\mathrm{Regge}}$ and the denominator in the
parenthesis of Eq. 6, indicating the high-energy contribution in units of the
relative uncertainty at each energy.

###  A.2 High-energy parameterization

Generally, it is observed that scattering amplitudes at high energy cease to
show features of individual resonances, becoming smooth functions of energy.
In the limit $s\gg t$, the _complex angular momentum_ , or “Regge” approach
proves to be an efficient method to parameterize scattering amplitudes in
terms of a modest number of parameters describing $t$–channel Regge
trajectories and their residues [71, 72, 73]. This approach has been used
extensively to describe experimental scattering data, with good descriptions
of $\pi\pi$ scattering at high energy being obtained by inclusion of Pomeron,
$\rho$ and $f_{2}$ trajectories [52, 70, 74].

For the current analysis at unphysical values of the light quark mass, we lack
any high-energy scattering data with which to constrain a Regge
parameterization, but we can use the relationship between Regge trajectories
and resonance states exchangeable in the $t$–channel to infer the required
quark-mass scaling. In particular, we will use an average of the
parameterizations given in Refs. [52, 70] (specifically we use the “CFD”
results from Ref. [52]), adapted to our pion masses in a simplistic approach.
These formulas provide us with a description for the three different isospins
in the $t$-channel exchanges $I_{t}=0,\,1,\,2$.

The Pomeron trajectory is typically associated with gluonic exchanges, and as
such we will assume that the parameters obtained from fits to physical data
can be used without adjustment for the changed light quark mass. We will also
assume that the $I_{t}=2$ Regge trajectory, which plays a very minor role,
does not depend on the quark mass. On the other hand, the $\rho$ trajectory
includes the $\rho$ resonance, and the residue of this trajectory in $\pi\pi$
scattering can be related to the decay width for $\rho\to\pi\pi$, which does
change with varying light quark mass. We keep the Regge trajectory
$\alpha_{\rho}(s)$ fixed at the physical value, and only adjust the residue
for the changed quark mass. Equation (E.5) from Ref. [51],

$\Gamma_{\rho}=\frac{\lambda}{96\pi
M_{\rho}^{2}}\left(M_{\rho}^{2}-4M_{\pi}^{2}\right)^{\frac{3}{2}},$ (8)

indicates how to scale the residue (which is proportional to $\lambda$) given
the $M_{\rho},\Gamma_{\rho}$ computed in lattice QCD at an unphysical light
quark mass. For $m_{\pi}\sim 239$ MeV we take $M_{\rho}=793.7$ MeV and
$\Gamma_{\rho}=91.2$ MeV, while for $m_{\pi}\sim 283$ MeV we use
$M_{\rho}=798.6$ MeV and $\Gamma_{\rho}=60.5$ MeV. The residue, $\beta$,
present for $\rho$ exchange, given in Refs. [52, 70], is scaled by the ratio
of the lattice $\lambda$ to the physical value, which we take as
$\lambda_{\mathrm{phys}}=72$. The $\rho$–computed ratio is also used (as a
crude first approximation) to scale the $f_{2}$ residue.

The Regge implementations of Refs. [52, 70], with the quark-mass scaled
residues, are used to calculate the $\pi\pi$ cross-sections in the
$s$-channel, according to

$\sigma^{I_{s}}(s)=\frac{\mathrm{Im}\,T^{I_{s}}(s,0)}{\sqrt{s(s-4m_{\pi}^{2})}}.$
(9)

Figure 11 shows these for the $m_{\pi}\sim 239$ MeV case, along with the
average of the two parameterizations, with a conservatively assigned,
uncorrelated $30\%$ error for each $I_{t}$ total Regge amplitude contribution,
which we propagate through our dispersive analysis.

These amplitudes are used in the dispersion relations from $s_{h}$ up to
infinity, but note that their contributions are heavily suppressed by the
kernels, which as shown in Figure 1, fall rapidly with increasing energy. In
Fig. 12, we present the contribution of the high-energy parameterized
amplitude to the dispersed amplitudes in the elastic scattering region. We
define

$\tilde{f}^{I}_{\ell}(s)\big{|}_{\mathrm{Regge}}\equiv\sum_{I^{\prime},\ell^{\prime}}\int_{s_{h}}^{\infty}\\!\\!ds^{\prime}\;\mathrm{Re}\,K^{II^{\prime}}_{\ell\ell^{\prime}}(s^{\prime},s)\;\mathrm{Im}\,t^{I^{\prime}}_{\ell^{\prime}}(s^{\prime})\big{|}_{\mathrm{Regge}}\,,$
(10)

and compare this to the total dispersed amplitude, which is dominated by the
amplitudes constrained by lattice QCD data for $s<s_{h}$. The plotted ratio
between the high-energy contribution and the relative uncertainty,
$\Delta\big{[}\tilde{f}^{I}_{\ell}(s)-f^{I}_{\ell}(s)\big{]}$, indicates that
the results of the analysis presented above are not sensitive to the detailed
modeling of the Regge amplitudes.

##  Appendix B Dispersed amplitude metrics

Figure 13: Geometric illustration of the construction of a linearized
$\chi^{2}$ contribution for a data-point having $x,y$ correlation of $-1$. The
data-point shown will contribute $1$ to the $\chi^{2}$ in agreement with the
apparent $1\sigma$ deviation from the curve along the diagonal direction.

As discussed in the manuscript, we retain only those dispersed amplitudes
which are both compatible with unitarity, determined by having real part in
agreement with the input amplitude (which itself was exactly unitarity
preserving), and which provide a reasonable description of the original
lattice data. These criteria are assessed using two numerical metrics.

A metric which compares the real part of the dispersed amplitude,
$\tilde{f}^{I}_{\ell}(s)\equiv\mathrm{Re}\,\tilde{t}^{I}_{\ell}(s)$, to the
real part of the input amplitude,
$f^{I}_{\ell}(s)\equiv\mathrm{Re}\,t^{I}_{\ell}(s)$, in order to enforce
unitarity, is

$\big{[}d^{2}\big{]}^{I}_{\ell}\equiv\sum_{i=1}^{N_{\mathrm{smpl}}}\left(\frac{\tilde{f}^{I}_{\ell}(s_{i})-f^{I}_{\ell}(s_{i})}{\Delta\big{[}\tilde{f}^{I}_{\ell}(s_{i})-f^{I}_{\ell}(s_{i})\big{]}}\right)^{2}\,.$
(11)

The sampling of points in the sum runs from just above the $\pi\pi$
threshold666In practice, this difference is calculated at 100 evenly spaced
energy values in the range between the two-pion threshold and
$a_{t}\sqrt{s}=0.14$, of which points above $2a_{t}m_{\pi}+0.005$ are used to
obtain the $d^{2}$ value. The energy of the first point used is around
$a_{t}\sqrt{s}=0.0841(0.0995)$ for $m_{\pi}\sim 239(283)$ MeV.. Excluding the
lowest 30 MeV above threshold reduces the sensitivity to high-order
derivatives of the scattering amplitude at threshold, a sensitivity that is
peculiar to this particular difference. In total, we use 91 equally spaced
points for $m_{\pi}\sim 239$ MeV and 89 points for $m_{\pi}\sim 283$ MeV in
the evaluation of this metric. The uncertainty on the difference at each
energy sample (appearing in the denominator) is calculated by linear
propagation of the (correlated) amplitude parameter uncertainties and the
(uncorrelated) high-energy Regge model uncertainty.

Figure 14: For $m_{\pi}\sim 239$ MeV, for each combination of
parameterizations of $S0$ (columns) and $S2$ (rows), each ordered by the
magnitude of their scattering length in $m^{-1}_{\pi}$ units, the box is
colored according to the average value of $d^{2}/N_{\mathrm{smpl}}$ (upper
triangle) and $\tilde{\chi}^{2}/N_{\mathrm{lat}}$ (lower triangle) over $S0$,
$P1$, $S2$. The different regions indicated by the color contours show the
parameterization combinations respecting the cuts described in the text. Three
example parameterization combinations are indicated by black, gray, and white
stars, which are used for illustration in other figures.  Figure 15: For
$m_{\pi}\sim 239$ MeV, real parts of $S0$, $P1$, $S2$ partial-wave amplitudes
for input (red) and dispersed output (blue). Amplitude combinations correspond
to the stars in Fig. 14: white (upper panel), gray (lower panel). Labels as
described in the caption of Figure 2.

In order to compare the dispersed amplitudes directly to the ‘lattice
amplitude data’, we use a $\chi^{2}$-like construction which accounts for both
the correlation between different lattice data points and the correlation
between the amplitude uncertainty and the energy uncertainty of each point,

$\displaystyle\big{[}\tilde{\chi}^{2}\big{]}^{I}_{\ell}$
$\displaystyle\equiv\sum_{i,j=1}^{N_{\mathrm{lat}}}\left(\tfrac{\mathfrak{f}_{i}-\tilde{f}^{I}_{\ell}(s_{i})}{\Delta_{i}}\right)\mathrm{corr}(\mathfrak{f}_{i},\mathfrak{f}_{j})^{-1}\left(\tfrac{\mathfrak{f}_{j}-\tilde{f}^{I}_{\ell}(s_{j})}{\Delta_{j}}\right)\,,$
$\displaystyle\Delta_{i}^{2}$
$\displaystyle=\big{(}\Delta\mathfrak{f}_{i}\,\,\,\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta
E_{i}\big{)}\begin{pmatrix}1&-c_{i}\\\
-c_{i}&1\end{pmatrix}\begin{pmatrix}\Delta\mathfrak{f}_{i}\\\
\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta E_{i}\end{pmatrix},$ (12)

In the second expression, the “$x,y$” correlation, $c_{i}$, for points
obtained from a linearized, Lüscher finite-volume analysis is $\pm 1$, and in
Fig. 13 we present a geometric illustration of the construction of the
effective uncertainty, compatible with the equation above, for the case
$c_{i}=-1$. $\Delta^{2}_{i}$ is an “effective variance weight”; explicit
examples with and without correlations between the $x$ and $y$ values, and a
justification of the formula above are given in Refs. [75, 76]777The methods
described in Refs. [75, 77, 76] present a solution to fitting data with errors
in both $x,\,y$ variables, for which the $x_{i}$ values are iterated, with the
suggested starting point $x_{i}=E_{i}$ for our case.. The values of $c_{i}$
and $\mathrm{corr}(\mathfrak{f}_{i},\mathfrak{f}_{j})$ are obtained from the
original lattice ensemble distributions888A few data points appear very close
to local maxima or minima of $\tilde{f}^{I}_{\ell}(s_{i})$, for which the
linearized statistical uncertainty on $\mathfrak{f}_{i}$ and/or the derivative
$\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}$ are almost zero. In these cases,
the error is underestimated. For these problematic $\mathfrak{f}_{i}$ points,
we took the maximum statistical or systematic error, when varying the energy
level $E_{i}$ within uncertainties. For $\tilde{f}^{I}_{\ell}(s_{i})$, the
second order correction
$\frac{d^{2}\tilde{f}^{I}_{\ell}(s_{i})}{d^{2}E_{i}}(\Delta E_{i})^{2}/2$ is
sizeable for some points (over $30\%$ of
$|\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta E_{i}|$), and in those
cases we substituted $\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta E_{i}$
by either $\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta
E_{i}+\frac{d^{2}\tilde{f}^{I}_{\ell}(s_{i})}{d^{2}E_{i}}(\Delta E_{i})^{2}/2$
or $\frac{d\tilde{f}^{I}_{\ell}(s_{i})}{dE_{i}}\Delta
E_{i}-\frac{d^{2}\tilde{f}^{I}_{\ell}(s_{i})}{d^{2}E_{i}}(\Delta
E_{i})^{2}/2$, selecting the term that produces the larger value for
$\Delta^{2}_{i}$..

Applying these metrics to the amplitudes obtained from the dispersion
relations generates Figures 4, 5. In the manuscript we apply cuts of
$d^{2}/N_{\mathrm{smpl}}<1$ and ${\tilde{\chi}^{2}/N_{\mathrm{lat}}<2}$,
retaining only those amplitudes which satisfy these for all partial-waves,
$S0,P1,S2$. In Fig. 14, we show the effect of relaxing these cuts somewhat.
The four superimposed boundaries correspond to:

  * white: amplitude combinations with all ${d^{2}/N_{\mathrm{smpl}}<3}$ and $\tilde{\chi}^{2}/N_{\mathrm{lat}}<3$

  * gray: amplitude combinations with all ${d^{2}/N_{\mathrm{smpl}}<1}$ and $\tilde{\chi}^{2}/N_{\mathrm{lat}}<3$

  * dark gray: amplitude combinations with all ${d^{2}/N_{\mathrm{smpl}}<2}$ and $\tilde{\chi}^{2}/N_{\mathrm{lat}}<2$

  * black: amplitude combinations with all ${d^{2}/N_{\mathrm{smpl}}<1}$ and $\tilde{\chi}^{2}/N_{\mathrm{lat}}<2$ ,

and the figure indicates that somewhat looser cuts can _slightly_ increase the
range of acceptable scattering length values, but as shown in Fig. 15 (for the
sample points indicated by the white and gray stars in Fig. 14), they do so by
allowing increasing departures from unitarity, or from an acceptable
description of the lattice data.

Nevertheless, if one allows these looser cuts, we note that the $\sigma$ pole
location remains within the region established by the tighter cuts, as shown
in Fig. 16, indicating that our particular choice of metric cuts is not
introducing a significant systematic error.

Figure 16: For $m_{\pi}\sim 239$ MeV, $S0$ pole locations for lattice-fit
amplitudes (gray), dispersed amplitudes satisfying the tighter metric cuts
described in the text (blue), and for the two sample points (white and gray
stars in Fig. 14) satisfying looser cuts (red).

##  Appendix C Minimally subtracted dispersion relations (“GKPY”)

Figure 17:  For $m_{\pi}\sim 239$ MeV, left panel shows average values of
$d^{2}/N_{\mathrm{smpl}}$ (upper triangle) and
$\tilde{\chi}^{2}/N_{\mathrm{lat}}$ (lower triangle) for each pair of $S0$
(columns) and $S2$ (rows) parameterizations, computed using _minimally-
subtracted_ dispersion relations. The black outline indicates those amplitudes
that passed the metric cuts in the _twice-subtracted_ case. Right panel shows
values of $\left(\frac{\psi}{\Delta\psi}\right)^{2}$ as defined in the text,
testing the degree to which the Olsson sum rule is satisfied.

Figure 18: As Figure 17, but for $m_{\pi}\sim 283$ MeV.

_Minimally-subtracted_ dispersion relations, often referred to as GKPY
equations, are constructed making only those subtractions _required_ to get
convergence, namely one subtraction [52]. The smaller number of subtractions
means the high-energy contributions to the dispersion relations are not as
strongly suppressed, with the kernel functions, in this case, falling off as
$s^{-2}$ rather than $s^{-3}$ as was the case for _twice-subtracted_ (“Roy”)
dispersion relations. The subtraction functions in this case are just
constants which, as before, depend only on the $S$-wave scattering lengths,

$\displaystyle\tau^{0}_{0}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{3}\big{(}a^{0}_{0}+5a^{2}_{0}\big{)}\,,$
$\displaystyle\tau^{1}_{1}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{12}\big{(}2a_{0}^{0}-5a^{2}_{0}\big{)}\,,$
$\displaystyle\tau^{2}_{0}(s)/m_{\pi}$
$\displaystyle=\tfrac{1}{6}\big{(}2a^{0}_{0}+a^{2}_{0}\big{)}\,.$ (13)

In contrast to the _twice-subtracted_ equations, the dispersive integrals do
not go to zero as $\sqrt{s}\to 2m_{\pi}$, and their value in this limit must
conspire with the $\tau$ values above to generate the correct threshold
behavior. These _minimally-subtracted_ equations have two main advantages over
the _twice-subtracted_ variant. Firstly, the error on the dispersed amplitudes
grows less quickly with increasing energy above threshold, which in previous
analyses led to lower uncertainties on resonance pole locations and residues
[5]. Secondly, the scattering length values can be determined, rather than
needing to be supplied as input. The primary disadvantage of _minimally-
subtracted_ equations is their sensitivity to the high-energy region of
scattering, on which, in this lattice application, we have relatively little
constraint. We will return to this later in this section.

Figure 19:  See Fig. 12 for comparison. Dispersed amplitudes produced by
_twice-subtracted_ (blue) and _minimally-subtracted_ (orange) dispersion
relations applied to input amplitudes indicated by the black star in Fig. 14.
Contributions to each from the high-energy parameterization is shown by the
gray and purple curves, respectively. Bottom panels: Ratio, in absolute value,
of $\tilde{f}^{I}_{\ell}(s)\big{|}_{\mathrm{Regge}}$ and the denominator in
the parenthesis of Eq. 6, indicating the high-energy contribution in units of
the relative uncertainty at each energy, described by the same colors as
before.

Figure 20: For $m_{\pi}\sim 239$ MeV. Left: $S0$ pole locations for _twice-
subtracted_ dispersed amplitudes satisfying the metric cuts described in the
text (blue), and for the same amplitude combinations used in _minimally-
subtracted_ dispersion relations (orange). Right: Corresponding couplings
extracted from the pole residues.

In the left panel of Figures 17, 18 we show the $d^{2}$ and $\tilde{\chi}^{2}$
metrics for the output of _minimally-subtracted_ dispersion relations using
the same lattice amplitude parameterizations as shown in Figures 4, 5. The
black boundary superimposed on the plot is the one selected according to the
metric cuts applied to the _twice-subtracted_ results, and we see that a large
fraction of acceptable solutions of the _minimally-subtracted_ dispersion
relations also lie in this region.

In the right panel of Fig. 17 we show a metric designed to show how well the
Olsson sum rule [78],

$\displaystyle\operatorname{Re}\tilde{T}^{I_{t}=1}(4m_{\pi}^{2},0)$
$\displaystyle=\frac{4m_{\pi}^{2}}{\pi}\int_{4m_{\pi}^{2}}^{\infty}\\!ds^{\prime}\,\frac{\operatorname{Im}T^{I_{t}=1}(s^{\prime},0)}{s^{\prime}\left(s^{\prime}-4m_{\pi}^{2}\right)},$
(14)

is satisfied. In this expression, the $t$–channel definite isospin amplitude
can be expressed in terms of the $s$–channel definite isospin amplitudes by

$T^{I_{t}=1}(s,t)=\frac{1}{3}T^{0}(s,t)+\frac{1}{2}T^{1}(s,t)-\frac{5}{6}T^{2}(s,t),$
(15)

and these are expressed in terms of a sum over the partial-wave amplitudes.
The $t$–channel amplitude at threshold is equal to a combination of the
$S$–wave scattering lengths,

$\operatorname{Re}T^{I_{t}=1}(4m_{\pi}^{2},0)=\tfrac{32\pi}{6}(2a^{0}_{0}-5a^{2}_{0})\,m_{\pi},$
(16)

so the quantity plotted, $\left(\frac{\psi}{\Delta\psi}\right)^{2}$, where

$\psi=2a_{0}^{0}-5a_{0}^{2}-\frac{6m_{\pi}}{8\pi^{2}}\int_{4m_{\pi}^{2}}^{\infty}\\!ds^{\prime}\,\frac{\operatorname{Im}T^{I_{t}=1}(s^{\prime},0)}{s^{\prime}\left(s^{\prime}-4m_{\pi}^{2}\right)}\,,$
(17)

should vanish if this dispersive sum rule is fulfilled.

Figure 19 shows an example of the dispersed amplitudes coming from _minimally-
subtracted_ dispersion relations compared to those from _twice-subtracted_
dispersion relations for the same set of input lattice amplitude
parameterizations, indicated by the black star in Fig. 14. The amplitudes are
observed to be compatible within errors over the entire energy region plotted,
but if one examines the contribution to the _minimally-subtracted_ amplitudes
of the high-energy part of the input, one sees it is much larger than for the
_twice-subtracted_ variants. Measured as a fraction of the relative
uncertainty $\Delta\big{[}\tilde{f}^{I}_{\ell}(s)-f^{I}_{\ell}(s)\big{]}$,
depicted in the bottom row, it is seen to contribute at a level where one
would worry about the correctness of the crudely-scaled Regge parameterization
used. As such we do not consider these results to be model-independent
consequences of the lattice QCD calculation. Nevertheless, the amplitude
agreement with the _twice-subtracted_ results inspires us to examine the
$\sigma$ pole location, and as shown in Fig. 20, we see that there is
reasonable agreement with the _twice-subtracted_ results with, as anticipated,
smaller uncertainties on the pole location and the pole residue.

##  Appendix D Parameterization functional forms

We summarize in this section the functional forms used to describe the input
partial-wave amplitudes, with parameters constrained by describing lattice QCD
finite-volume energy levels. A similar discussion can be found in Ref. [29]
where their application to describe lattice spectra is presented.

An elastic partial wave can be recast in terms of a real phase-shift as

$t_{\ell}^{I}(s)=\frac{1}{\rho(s)}\,e^{i\delta^{I}_{\ell}(s)}\sin\delta_{\ell}^{I}(s)=\frac{1}{\rho(s)}\frac{1}{\cot\delta_{\ell}^{I}(s)-i},$
(18)

where $\rho(s)=2k/\sqrt{s}$ is the two-pion phase-space and
${k=\tfrac{1}{2}\sqrt{s-4m_{\pi}^{2}}}$ is the scattering momentum.

At low energies, slow variation of $S$–waves can be described by a low-order
expansion in $k^{2}$, called the _effective range expansion_ ,

$k^{2\ell+1}\cot\delta_{\ell}^{I}=F^{I}_{\ell}(s)\left(\tfrac{1}{a^{I}_{\ell}}+\tfrac{1}{2}r^{I}_{\ell}k^{2}+\dots\right),$
(19)

where the conventional choice is $F^{I}_{\ell}(s)=1$, for which $a_{\ell}^{I}$
is interpreted as the _scattering length_ and $r_{\ell}^{I}$ as the _effective
range_. Additional desired features can be included with other choices of
$F^{I}_{\ell}(s)$, an example being an Adler zero, enforced by using
$F^{I}_{\ell}(s)=(4m_{\pi}^{2}-s_{A})/(s-s_{A})$.

Alternatively, we can make use of a _conformal mapping expansion_ by defining

$\Phi^{I}_{\ell}(s)=\tfrac{2}{\sqrt{s}}k^{2\ell+1}\cot\delta^{I}_{\ell}(s),$
(20)

which is real analytic between the elastic and inelastic thresholds. One can
introduce an effective inelastic threshold, $s_{0}$, and the opening of the
_left-hand-cut_ at $s=0$, by using [79, 80],

$\omega(s)=\frac{\sqrt{s}-\alpha\sqrt{s_{0}-s}}{\sqrt{s}+\alpha\sqrt{s_{0}-s}}\,.$
(21)

In this expression, $\alpha$ and $s_{0}$ are fixed parameters that determine
which energy region is mapped into a unit disk of $\omega$. For $S2$ we set
$s_{0}=0.09\,a_{t}^{-2}$ and $\alpha=0.8$. For $S0$ we use $\alpha=0.8,\,1$,
and consider two values, ${s_{0}=0.032\,a_{t}^{-2},\,0.04\,a_{t}^{-2}}$.
Expanding the desired function as a polynomial in this variable,

$\Phi^{I}_{\ell}(s)=F^{I}_{\ell}(s)\sum_{n=0}^{N}B_{n}\,\omega^{n}\,,$ (22)

we expect convergence with a low-order $N$. Again, one may build in additional
properties by using $F^{I}_{\ell}(s)$ other than $1$. For example, when a
narrow resonance is known to be present, it is convenient to set
$F^{I}_{\ell}(s)=(s-m^{2}_{R})/m_{R}^{2}$. Furthermore, as detailed in Ref.
[81], spurious singularities appearing below threshold can be removed by
adding a function $\gamma^{I}_{\ell}(s)$, so that,

$\Phi^{I}_{\ell}(s)=F^{I}_{\ell}(s)\left(\gamma^{I}_{\ell}(s)+\sum_{n=0}^{N}B_{n}\,\omega^{n}\right)\,.$
(23)

Partial-waves dominated by narrow resonances, like $P1$, are typically well-
described in the vicinity of the pole by a Breit-Wigner form,

$t_{\ell=1}(s)=\frac{1}{\rho(s)}\frac{\sqrt{s}\,\Gamma(s)}{m_{\mathrm{BW}}^{2}-s-i\sqrt{s}\,\Gamma(s)}\,,$
(24)

where the energy-dependent width is given by
${\Gamma(s)=\tfrac{g_{\mathrm{BW}}^{2}}{6\pi}\tfrac{k^{3}}{s}}$. Another, more
flexible, parameterization which allows for the presence of resonances, and
which generalizes nicely to the case of _coupled-channel_ amplitudes, is the
$K$-matrix approach [82],

$\left(t^{I}_{\ell}(s)\right)^{-1}=\left(K^{I}_{\ell}(s)\right)^{-1}-i\rho(s)\,,$
(25)

where a common parameterization choice is a sum of poles plus a finite-order
polynomial,

$K^{I}_{\ell}(s)=(2k)^{2\ell}\left[\sum_{r}\frac{g_{r}^{2}}{m_{r}^{2}-s}+\sum_{p}\gamma_{p}\,s^{p}\right]\,.$
(26)

This form can be modified to include an Adler zero by taking
$K^{I}_{\ell}(s)\to(s-s_{A})\,K(s)$. Another choice is to model
$\left(K^{I}_{\ell}(s)\right)^{-1}$ directly.

The simple phase-space, $\rho(s)$, defined above has an unphysical singularity
at $s=0$, but this can be removed by writing a once-subtracted dispersion
relation leading to the Chew-Mandelstam function [83],

$I(s)=I(s_{M})+\frac{s-s_{M}}{\pi}\int_{s_{thr}}^{\infty}ds^{\prime}\frac{-\rho\left(s^{\prime}\right)}{(s^{\prime}-s_{M})(s^{\prime}-s)},$
(27)

which has $\mathrm{Im}\,I(s)=-\rho(s)$ above the two-pion threshold, as
required by unitarity, and which, when subtracted at threshold, takes the
simple form
$I(s)=\frac{\rho(s)}{\pi}\log\left[\frac{\rho(s)+1}{\rho(s)-1}\right]$.

In this work, the input amplitudes constrained by lattice QCD energy levels
span a range of forms presented above. The $P1$ amplitudes are fixed to the
successful ‘pole plus constant’ $K$-matrix with Chew-Mandelstam phase-space,
but many forms are used for the $S0$ and $S2$ amplitudes, leading to a total
of over 700 input amplitude combinations being tested within dispersion
relations.

Figure 21:  $I=2$ partial-wave amplitudes for $m_{\pi}\sim 239$ MeV. For
$S$–wave (upper two panels), four example parameterizations are shown: a two-
term conformal mapping (black), an effective range expansion with two terms
(green), and two choices with an Adler zero fixed at the leading-order
$\chi$PT location, a two-term conformal mapping (red), and an effective range
expansion with two terms (orange). The $D$–wave phase-shift is shown in the
lower panel illustrated with two parameterizations: a scattering length (red),
and a conformal mapping with one term (black). Data points with very large
uncertainties have been removed from the plots.

The $S0$ and $S2$ partial-waves are described by input amplitudes both with
and without explicit Adler zeroes. As shown in Ref. [52], dispersive analyses
applied to physical $\pi\pi$ scattering data suggest that Adler zeroes may not
be located at the tree-level locations. In our input amplitudes, we fix the
Adler zeroes either to their tree-level values, or to the extremes of the
“CFD” dispersive range in Ref. [52], extrapolated to our pion masses with
$s_{A}=s_{A}^{\mathrm{phys}}\left(m_{\pi}/m_{\pi}^{\mathrm{phys}}\right)^{2}$.
In the case of the $S0$ wave for $m_{\pi}\sim 283$ MeV, we also consider a
couple of parameterizations in which the Adler zero location is allowed to
float in the description of the lattice QCD energy levels, and in this case
alone we find that this does not degrade significantly the precision of the
fit. In other cases, allowing the Adler zero location to float leads to an
extremely imprecise estimate of the location.

##  Appendix E $m_{\pi}\sim 239\,\mathrm{MeV}$ $I=2$ amplitudes

Ref. [19] and Ref. [27] reported on extractions of $I=0$ and $I=1$ finite-
volume spectra from a $32^{3}\times 256$ anisotropic Clover lattice with
light-quark masses such that $m_{\pi}\sim 239$ MeV. The corresponding spectra
on this lattice with $I=2$ have not previously appeared in the literature, but
have been computed for the purposes of this manuscript using the same
techniques, range of energies, and similar operators to the ones reported in
Ref. [29]. As observed in the previous work, the $D$-wave is very weak and
repulsive, and can be described with a single parameter. The $S$-wave shows
stronger repulsion, with a phase-shift whose magnitude grows with energy. Fig.
21 shows discrete values of $S$ and $D$–wave phase-shifts determined from
these spectra along with a sample of parameterizations. The parameterizations
used here to describe the data correspond to a subsect of those types
described above and in Section III of [29]. A compilation of the scattering
lengths obtained from these fits was presented in Figure 5 of that work, and
their explicit values, in $m^{-1}_{\pi}$ units, are given as part of Fig. 4.

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