Also at ]European Centre for Theoretical Studies in Nuclear Physics and
Related Areas (ECT*) & Fondazione Bruno Kessler, Strada delle Tabarelle 286,
38123 Villazzano (TN), Italy

FASTSUM Collaboration

# Non-zero temperature study of spin $1/2$ charmed baryons using lattice gauge
theory

Gert Aarts  [  Chris Allton, M. Naeem Anwar, Ryan Bignell, Timothy J. Burns
Department of Physics, Swansea University,
Swansea, SA2 8PP, United Kingdom  Benjamin Jäger  Quantum Field Theory Center
& Danish IAS, Department of Mathematics and Computer Science
University of Southern Denmark, 5230, Odense M, Denmark  Jon-Ivar Skullerud
Department of Theoretical Physics and Hamilton Institute, National University
of Ireland Maynooth, County Kildare, Ireland  School of Mathematics, Trinity
College, Dublin, Ireland
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the behaviour of spin $1/2$ charmed baryons as the temperature
increases. We make use of anisotropic lattice QCD simulations with $N_{f}=2+1$
dynamical flavours. After determining the positive and negative parity ground
state masses at the lowest temperature, we investigate the effect of rising
temperature using ratios of thermal lattice correlators with both so-called
reconstructed correlators and with simple model correlators. This avoids
difficulties associated with non-zero temperature fitting or spectral
reconstruction. We find that temperature effects are prominent throughout the
hadronic phase for all negative parity channels considered and for some
positive parity channels. Subsequently and where possible, we determine the
masses of the ground states as a function of temperature. Finally we consider
the effect of chiral symmetry restoration and extract an estimate of the
pseudocritical temperature from singly charmed baryonic correlators.

##  I Introduction

This work focuses on the fate of spin $1/2$ charmed baryons as the temperature
increases and the confining hadronic medium at low temperature smoothly
transitions to a quark-gluon plasma (QGP), as predicted by Quantum
Chromodynamics (QCD) [1]. While the light degrees of freedom deconfine at the
crossover [2, 3], there is evidence that heavier hadrons comprised of charm
and bottom quarks can survive as bound states in the QGP. Since the seminal
work by Matsui and Satz [4], this has been studied extensively for charmonium
($c\bar{c}$) and bottomonium ($b\bar{b}$) states, as it provides important
insights into the length scales in the QGP [5, 6, 7]. For results obtained
with lattice QCD, see e.g. the comprehensive reviews [8, 9] and references
therein.

Here we consider baryons containing charm quarks, building upon previous work
studying the behaviour of light and strange baryons at non-zero temperature
[10, 11, 12]. The focus in this earlier work was parity doubling, the
emergence of a degeneracy between positive- and negative-parity correlators
and ground state masses, which is a signal for chiral symmetry restoration.
Due to the heavier mass of the charm quark, one expects a reduction in the
level of parity doubling observed when compared to the light and strange
baryons. Instead, baryons containing multiple charm quarks may be more similar
to heavy-quark mesons (quarkonia) and therefore not immediately dissolve in
the QGP. Hence, it will be interesting to analyse the structure of charmed
baryons across a range of temperatures.

Heavy-light charmed $D$ and $D_{s}$ mesons have been studied using lattice QCD
in Refs. [13, 14]. In particular, Ref. [14] considered the response of
$D_{\left(s\right)}$ mesons to an increasing temperature without the explicit
need for non-zero temperature fits and spectral reconstruction. The outcome of
this analysis was subsequently used to inform the applicability of standard
fits to the correlation function used to determine the ground state mass. The
analysis studied ratios of lattice correlators to single-state model
correlators to determine when the spectral content of the correlator was
different from that of the zero-temperature correlator. Here we extend this
treatment to the baryonic sector for the first time.

Motivated and directed by the outcome of the analysis of correlator ratios
across temperatures, we fit correlators with standard exponential fits when
ratios suggest that thermal effects are small. Hence these fits assume that
the hadron is a (narrow) bound state. Extensive use is made of model averaging
methods [15, 16] in order to produce a reliable extraction of the mass. It is
noted that the behaviour of charmed baryon masses at non-zero temperature may
be of phenomenological interest [17].

To make use of the full range of temperatures available, we also compare
positive and negative parity channels using the correlation functions directly
[11, 12, 18], to reveal the effect of chiral symmetry restoration in the QGP.
Parity doubling is not expected due to the large charm quark mass;
nevertheless, for singly charmed baryons a clear change of behaviour is
observed.

The paper is organised as follows. In Section II we summarise the FASTSUM
Generation 2L ensembles, correlator methods, and the fitting techniques used.
We consider ratios of the lattice correlator with the “reconstructed”
correlator and with a single-state model correlator in Section III. In Section
IV we determine masses across a range of temperatures provided the analysis of
the previous section suggests this is appropriate. The fit-independent parity
doubling ratio is presented in Section V. Conclusions and future work are
discussed at the end.

##  II Lattice Correlators

In the following subsections we discuss the gauge field ensembles, operators,
and fitting procedure used to determine ground state masses at the lowest
temperature.

###  II.1 Ensembles

Table 1: FASTSUM Generation 2L ensembles used in this work. The lattice size is $32^{3}\times N_{\tau}$, with temperature $T=1/\left(a_{\tau}N_{\tau}\right)$. The spatial lattice spacing is $a_{s}=0.11208\left(31\right)$ fm, renormalised anisotropy $\xi=a_{s}/a_{\tau}=3.453(6)$ and the pion mass $m_{\pi}=239(1)$ MeV [19]. We use $\sim 1000$ configurations and eight (random) sources for a total of $\sim 8000$ measurements at each temperature. The estimate for $T_{\rm pc}$ comes from an analysis of the renormalised chiral condensate and equals $T_{\rm pc}=167(2)(1)$ MeV [18, 14]. Full details of these ensembles may be found in Refs. [18, 14].  $N_{\tau}$ | 128 | 64 | 56 | 48 | 40 | 36 | 32 | 28 | 24 | 20 | 16
---|---|---|---|---|---|---|---|---|---|---|---
$T\,\left(\text{MeV}\right)$ | 47 | 95 | 109 | 127 | 152 | 169 | 190 | 217 | 253 | 304 | 380

We make use of the thermal ensembles of the FASTSUM collaboration [18], with
$2+1$ flavours of Wilson fermions on anisotropic lattices. The renormalised
anisotropy is $\xi\equiv a_{s}/a_{\tau}=3.453(6)$ [20, 18]. The lattice action
follows that of the Hadron Spectrum Collaboration [21] and is a Symanzik-
improved [22, 23] anisotropic gauge action with tree-level mean-field
coefficients and a mean-field-improved Wilson-clover [24, 25] fermion action
with stout-smeared links [26]. Full details of the action and parameter values
can be found in Ref. [18]. We use the “Generation 2L” ensembles which have a
pion mass of $m_{\pi}=239(1)$ MeV (in the previous “Generation 2” ensembles
the pion mass was $m_{\pi}=384(4)$ MeV [27, 28, 29]). While this is still
heavier than physical, it represents an important step towards the physical
regime. The strange quark has been tuned to its physical value via the tuning
of the light and strange pseudoscalar masses [27, 30, 31]. The ensembles are
generated using a fixed-scale approach, such that the temperature is varied by
changing $N_{\tau}$, as $T=1/\left(a_{\tau}N_{\tau}\right)$. A summary of the
ensembles is given in Table 1. There are five ensembles below the
pseudocritical temperature $T_{\rm pc}=167(2)(1)$, one close to $T_{\rm pc}$
and five above $T_{\rm pc}$. The estimate for $T_{\rm pc}$ comes from an
analysis of the renormalised chiral condensate [18]. Note that here we have
used the updated lattice spacing of Ref. [19], which has been implemented in
our analysis in Ref. [14].

###  II.2 Operators

The $J=1/2$ singly and doubly charmed baryons, with quark content, are grouped
according to the underlying flavour symmetry of QCD, namely

$\displaystyle SU(3)~{}\mathbf{\bar{3}}:$
$\displaystyle\quad\Lambda_{c}\left(udc\right),~{}\Xi_{c}\left(usc\right)$
$\displaystyle SU(3)~{}\mathbf{6}:$
$\displaystyle\quad\Sigma_{c}\left(udc\right),~{}\Xi^{\prime}_{c}\left(usc\right),~{}\Omega_{c}\left(ssc\right)$
(1)

and for doubly charmed baryons,

$SU(3)\times
U(1)_{\text{charm}}~{}\mathbf{20}_{M}:\quad\Xi_{cc}\left(ccu\right),~{}\Omega_{cc}\left(ccs\right),$
(2)

following the decomposition of $SU(4)$ as in the PDG 2020 [32].

The baryon operators we consider are of the form

$\displaystyle
G^{\alpha\alpha^{\prime}}(x)=\Braket{\mathcal{O}^{\alpha}(x)\overline{\mathcal{O}}^{\alpha^{\prime}}(0)}$
(3)

where $\overline{\mathcal{O}}=\mathcal{O}^{\dagger}\gamma_{4}$ and $\alpha$,
$\alpha^{\prime}$ are Dirac indices. The operators used follow Refs. [33, 34].
Denoting the three quarks as $q_{1}$, $q_{2}$ and $q_{3}$ (from left to
right), the operator for the case $q_{1}=q_{2}$ ($\Xi_{cc}$, $\Omega_{c}$ and
$\Omega_{cc}$) is given by

$\mathcal{O}^{\alpha}_{{\rm
2fl},1/2}\left(q_{1},q_{1},q_{3}\right)=\epsilon_{abc}\,q_{1,\,\alpha}^{a}\left(q_{1,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{3,\,\gamma}^{c}\right).$
(4)

For the non-degenerate case, $\Sigma_{c}$ and $\Xi^{\prime}_{c}$ belong to the
$SU(3)\,\,\mathbf{6}$ flavour multiplet, with operators

$\displaystyle\mathcal{O}^{\alpha}_{\mathbf{6},1/2}\left(q_{1},q_{2},q_{3}\right)=$
$\displaystyle\frac{1}{\sqrt{2}}\epsilon_{abc}\left(q_{1,\,\alpha}^{a}\left(q_{3,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{2,\,\gamma}^{c}\right)\right.$
$\displaystyle+\left.q_{2,\,\alpha}^{a}\left(q_{3,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{1,\,\gamma}^{c}\right)\right),$
(5)

where $q_{1}=u$, $q_{3}=c$ and $q_{2}=d/s$ respectively. $\Lambda_{c}$ and
$\Xi_{c}$ enjoy $SU(3)\,\,\mathbf{\overline{3}}$ flavour symmetry, with
operators

$\displaystyle\mathcal{O}^{\alpha}_{\mathbf{\bar{3}},1/2}\left(q_{1},q_{2},q_{3}\right)=$
$\displaystyle\frac{1}{\sqrt{6}}\epsilon_{abc}\Big{(}2q_{3,\,\alpha}^{a}\left(q_{1,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{2,\,\gamma}^{c}\right)$
$\displaystyle+q_{2,\,\alpha}^{a}\left(q_{1,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{3,\,\gamma}^{c}\right)$
$\displaystyle-
q_{1,\,\alpha}^{a}\left(q_{2,\,\beta}^{b}\left[C\gamma_{5}\right]_{\beta\gamma}q_{3,\,\gamma}^{c}\right)\Big{)},$
(6)

where $q_{1}=u$, $q_{3}=c$ and $q_{2}=d/s$ respectively. The Euclidean gamma
matrices satisfy $\gamma_{\mu}^{\dagger}=\gamma_{\mu}=\gamma_{\mu}^{-1}$ for
$\mu=1,\dots 4$,
$\gamma_{5}=\gamma_{5}^{\dagger}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}$ and
$C$ is the charge conjugation matrix [35].

Figure 1: Mass of the positive parity $\Xi_{cc}$ ground state at the lowest
temperature, from forward constant fits starting at $\tau_{\rm min}$ to the
effective mass. For each $\tau_{\rm min}$, $\tau_{\rm max}$ increases from
left to right in each pane, i.e. $\tau_{\rm min}<\tau_{\rm max}\leq
N_{\tau}/2-0.05\,N_{\tau}$; hence the “middle” $10$% of points are excluded.
X-axis labels are not shown where they would overlap, i.e. on the right side
of the plot.

The contractions of these operators produce terms proportional to

$\displaystyle
S_{1}\mbox{tr}\,\left[C\Gamma_{i}S_{2}C\Gamma_{j}S_{3}\right]\quad\mbox{and}\quad
S_{1}C\Gamma_{i}S_{2}C\Gamma_{j}S_{3},$ (7)

where $S_{1,2,3}$ is a quark propagator of flavour $q_{1,2,3}$ and
$\Gamma_{i/j}$ is the gamma matrix in the source/sink operator respectively.
These are the principal building blocks of the baryon correlation functions.

The positive and negative parity projectors are [35, 33]

$\displaystyle P_{\pm}=\frac{1}{2}\left(1\pm\gamma_{4}\right),$ (8)

and we denote the projected correlation functions at vanishing spatial
momentum as [10, 11, 12]

$G_{\pm}(\tau)=\mbox{tr}\,P_{\pm}G(\tau).$ (9)

These are related as [11]

$G_{\pm}(\tau)=-G_{\mp}(1/T-\tau),$ (10)

implying that the forward- (backward-) propagating states of $G_{+}(\tau)$ are
states with positive (negative) parity.

Gaussian smearing is applied to the source and sink using [36]

$\displaystyle\eta^{\prime}=C_{\rm norm}\left(1+\kappa H\right)^{n}\eta,$ (11)

where $\eta$ is the bare (delta function) source, $\kappa$ and $n$ determine
the amount of smearing, $H$ is the spatial hopping part of the Dirac operator
and $C_{\rm norm}$ is an appropriate normalisation. Here we used $\kappa=5.5$,
$n=100$ at all temperatures. The root-mean-square radius of this profile is
$\sim 6.8$ lattice sites. These parameters were chosen such that the positive
parity nucleon ground state at the lowest temperature displays good ground
state isolation, as in Refs. [11, 12].

###  II.3 Mass extraction

A systematic extraction of hadron masses from lattice QCD is an area of active
development, with methods becoming increasingly sophisticated [37, 38, 39, 40,
41, 16, 42, 15, 43]. Broadly these can be divided into methods which improve
the underlying correlation function [37, 38, 41] and those which aim to more
reliably extract the mass from a given correlation function. As mentioned
above, we use gauge invariant Gaussian source and sink smearing in order to
improve the overlap of the operator with the (zero-temperature) ground state.
We also adopt the model averaging methods of both Refs. [16, 43] and Ref.
[15]. By comparing these two distinct methods we ensure a robust determination
of the mass.

We start at the lowest temperature ($T=47$ MeV, $N_{\tau}=128$) and consider
fit functions of the form

$\displaystyle\overline{G}\left(\tau\right)=\sum_{n=1}^{N}\,A_{n}\text{e}^{-a_{\tau}M_{n}^{+}\tau/a_{\tau}}+B_{n}\text{e}^{-a_{\tau}M_{n}^{-}\left(N_{\tau}-\tau/a_{\tau}\right)},$
(12)

where $\overline{G}(\tau)=G_{+}\left(\tau\right)/G_{+}\left(0\right)$ and
$M_{n}^{\pm}$ is the $n^{\rm th}$ positive (negative) parity state. The number
of exponentials $N$ is allowed to vary from one to three. The final number of
exponentials is set by examining the Gaussian Bayes factor and stepping back
one exponential when it ceases to change as more exponentials are added [44,
45, 39, 46]. This is equivalent to stopping when the data can no longer
support additional exponential terms. We illustrate some of our findings below
using the positive and negative parity $\Xi_{cc}$ ground states as examples.

Figure 2: Mass of the positive parity $\Xi_{cc}$ ground state at the lowest
temperature, extracted from multi-exponential fits as in Eq.
$\left(\ref{eqn:expFit}\right)$. For each $\tau_{\rm min}$, $\tau_{\rm max}$
increases from left to right in each pane. The top $30\%$ highest weighted
fits from Eq. $\left(\ref{eqn:AIC}\right)$ are shown, as are the corresponding
model averaged values from Eqs. $\left(\ref{eqn:pVal}\right)$ and
$\left(\ref{eqn:AIC}\right)$. X-axis tick labels are not shown where they
would overlap.

Typically the correlator $\overline{G}\left(\tau\right)$ is considered from a
starting time $\tau_{\rm min}$ up until an end time $\tau_{\rm max}$. This is
due to excited state contamination at early times and the onset of noise of
late times (on the positive parity side and trivially reversed for the
negative parity side). The choice of “fit window” can have a substantial
effect on the extracted mass; this is particularly evident for the effective
mass [35] fits shown in Figure 1 for the positive parity $\Xi_{cc}$ ground
state at the lowest temperature. Here we have fit the effective mass,

$\displaystyle M_{\rm
eff}=\frac{1}{\delta_{\tau}}\,\operatorname{arccosh}{\frac{\overline{G}\left(\tau+\delta_{\tau}\right)+\overline{G}\left(\tau-\delta_{\tau}\right)}{2\,\overline{G}\left(\tau\right)}},$
(13)

with $\delta_{\tau}=2$ across an interval $\tau_{\rm min}\leq\tau\leq\tau_{\rm
max}$, with varying $\tau_{\rm min}$. Specifically, we used $\tau_{\rm
min}\geq 0.1\,N_{\tau}=13$ and $\tau_{\rm max}\leq
N_{\tau}/2-0.05\,N_{\tau}=57$ (note that in this section we use “temporal
lattice units” for both Euclidean time and quoted mass values). This range was
chosen to remove early time slices dominated by excited state effects and
“noisy” time slices near the middle of the time extent. At small values of
$\tau_{\rm max}$ clear excited state contamination is manifest in the
increased value from the fit. As $\tau_{\rm min}$ and $\tau_{\rm max}$ are
increased, excited state contamination is decreased as the ground state
becomes more dominant, but the statistical uncertainty also increases. While
it may be possible to extract a robust ground state mass from the effective
mass for $N_{\tau}=128$, in general this will not be possible at higher
temperatures, due to a reduction in the number of temporal points.

Exponential fits electing a single fit window ignore the information from
other fit windows. It is therefore beneficial to use model averaging methods
to determine weighted fit parameters. The first method, introduced in Ref.
[16] uses a weight

$\displaystyle\tilde{w}_{f}^{(1)}=\frac{p_{f}/\left(\delta
M_{f}\right)^{2}}{\sum_{f^{\prime}=1}^{N}p_{f^{\prime}}/\left(\delta
M_{f^{\prime}}\right)^{2}},$ (14)

where $p_{f}$ is the p-value of the fit $f$, $\delta M_{f}$ is the uncertainty
in the fit parameter of fit $f$ and there are $N$ fits to be averaged. This
method penalises both poor fits and unconstraining fits [42].

This is in contrast to the second method [15] which uses an exponential of the
modified Akaike information criterion [47, 15]

$\displaystyle\tilde{w}_{f}^{(2)}=\exp\left(-\frac{1}{2}\chi^{2}_{\rm
aug}\left(M_{f}\right)-k-N_{\text{cut}}\right),$ (15)

where $N_{\text{cut}}$ is the number of data points not fit to, $k$ is the
number of fit parameters and $\chi^{2}_{\rm aug}$ is the augmented chi-squared
[44]. Agreement between the two methods provides confidence in the extracted
fit parameter.

A large number of roughly equivalent fit windows is shown in Figure 2. The
exponential fits have range $\tau_{\rm min}=1\leq\tau\leq\tau_{\rm max}\leq
N_{\tau}/2-0.05\,N_{\tau}$. In particular, in Figure 2 only the top $30\%$
highest weighted fits using the weight of Eq. $\left(\ref{eqn:AIC}\right)$ are
shown.

Figure 3: Comparison of the effective mass for the positive and negative
parity $\Xi_{cc}$ ground state at the lowest temperature with the results of
the two model averaging methods of Eqs. $\left(\ref{eqn:pVal}\right)$ and
$\left(\ref{eqn:AIC}\right)$.  Figure 4: Highest weighted fit results for the
positive and negative parity $\Xi_{cc}$ ground state at the lowest temperature
as a function of fit window. Both sides of the correlator are fit
symmetrically and so e.g. “20-57” fits points $\left[20,\,57\right]$ and
$\left[71,\,108\right]$ simultaneously. The weights $\tilde{w}_{f}^{(2)}$ of
Eq. $\left(\ref{eqn:AIC}\right)$ are also shown, as is the resulting model
averaged fit value.

To exclude outlying fits, a further cut is made. The top $30\%$ of weighted
fits using the modified Akaike information criterion weights of Eq.
$\left(\ref{eqn:AIC}\right)$ are separated and re-averaged using both methods.
This is demonstrated in Figure 3, in which the effective masses for both the
positive and negative parity $\Xi_{cc}$ ground state are shown and compared to
the model averaged results from the multi-exponential fits after this cut has
been applied. The resulting fits are in good agreement for both the sectors
giving us assurance that the fitted masses are robustly determined.

Table 2: Positive and negative parity ground state masses $M^{\pm}$ at the lowest temperature, in units of $a_{\tau}$ and in GeV. The systematic error due to the choice of model averaging method has been added in quadrature to the error from the selected model averaging procedure. The final two columns show the Particle Data Group values [48], when known, averaged over charge partners with systematic uncertainties added in quadrature. | Lattice | PDG
---|---|---
Hadron | $a_{\tau}M^{+}$ | $a_{\tau}M^{-}$ |  $M^{+}$ [GeV] |  $M^{-}$ [GeV] |  $M^{+}$ [GeV] |  $M^{-}$ [GeV]
$\Lambda_{c}\left(udc\right)$ | 0.3853(36) | 0.4576(39) | 2.342(22) | 2.782(24) | 2.28646(14) | 2.59225(28)
$\Sigma_{c}\left(udc\right)$ | 0.4118(11) | 0.4777(50) | 2.5034(86) | 2.904(31) | 2.45379(30) | $-$
$\Xi_{c}\left(usc\right)$ | 0.40787(89) | 0.4779(49) | 2.4796(77) | 2.905(30) | 2.46907(36) | 2.79290(71)
$\Xi^{\prime}_{c}\left(usc\right)$ | 0.4271(15) | 0.487(11) | 2.596(11) | 2.961(67) | 2.57845(71) | $-$
$\Omega_{c}\left(ssc\right)$ | 0.4433(21) | 0.5072(87) | 2.695(14) | 3.083(53) | 2.6952(17) | 3.00041(22)
$\Xi_{cc}\left(ccu\right)$ | 0.59642(87) | 0.6575(34) | 3.6258(95) | 3.997(22) | 3.62160(40) | $-$
$\Omega_{cc}\left(ccs\right)$ | 0.6114(17) | 0.6757(26) | 3.717(13) | 4.107(18) | $-$ | $-$

In practice, to produce a single number for a mass, a single averaged value is
chosen and then the difference to the other values is added as a systematic
uncertainty. The averaged value is chosen by examination of the weights, such
that the averaged value clearly represents the fits. This can be seen in
Figure 4 where the averaged value encompasses the majority of the fit weights
(which sum to 1). An unrepresentative case would have the weight dominated by
a single, outlying fit.

###  II.4 Spectrum

We are now in a position to present the spectrum of $J=1/2$ charmed baryons
with positive and negative parity at the lowest temperature, in Figure 5 and
Table 2. While exact agreement with the experimental results is not expected
due to our heavier-than-physical light quarks and lack of continuum and
infinite volume extrapolation, there is encouraging similarity in the observed
masses in both the positive and negative parity channels. Better agreement is
observed for hadrons with more strange and charm quarks; this can be explained
by the tuning of these heavier quarks to their physical masses [27, 30, 31].
We remind the reader that we do not attempt precision zero-temperature
spectroscopy, see e.g. Refs. [30, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62] instead, but that our interest is in the response to heating up
the system, to which we turn in the next section.

Figure 5: Positive (diamonds $\lozenge$) and negative (stars $\star$) parity
ground state masses $M^{\pm}$ at the lowest temperature. Inner error bars
represent the statistical uncertainty and the outer bars incorporate a
systematic error from the choice of averaging method. The grey bars represent
experimental results from the Particle Data Group, when known [48].

##  III Temperature Effects

To investigate the effect of increasing the temperature of the hadronic
medium, we first attempt to assess this using only information contained in
the correlators, i.e., without performing fits at finite temperature. We use
two complementary approaches; first we perform an appropriate resummation of
the correlator at the lowest temperature to account for the difference in
temporal extent at finite temperature. This leads to the so-called
“reconstructed” correlators [63], extended to the fermionic case here.
Subsequently, we consider a minimal model of the thermal correlator using
information only from the well-defined ground state fits at the lowest
temperature and take appropriate ratios, following Ref. [14]. In both these
approaches, the key idea is that we attempt to isolate the effect of changing
the temperature on the spectrum, while removing the effect of reducing the
temporal extent as a trivial kinematic construction on the Euclidean lattice.

###  III.1 Reconstructed Correlator

The spectral relation for baryons reads, see e.g. Eqs. (2.16, 2.17, 2.27) of
Ref. [11],

$\displaystyle
G\left(\tau;T\right)=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,K_{F}\left(\tau,\omega;T\right)\rho\left(\omega\right),$
(16)

where $K_{F}$ is the fermionic kernel

$\displaystyle
K_{F}\left(\tau,\omega;T\right)=\frac{e^{-\omega\tau}}{1+e^{-\omega/T}}.$ (17)

Here $G\left(\tau;T\right)$ and $\rho\left(\omega\right)$ can have Dirac
indices, or be projected with parity operators, see Ref. [11].

This kernel can be compared with the building block for bosonic (mesonic)
correlators

$\displaystyle
K_{B}\left(\tau,\omega;T\right)=\frac{e^{-\omega\tau}}{1-e^{-\omega/T}}.$ (18)

Often this is multiplied with $\exp(\omega/2T)$ in the numerator and the
denominator, to arrive at the usual $\sinh(\omega/2T)$ in the denominator, but
this is not needed here.

To relate a correlator at a higher temperature $T$ to one at a lower
temperature $T_{0}$ by resummation, we use a simple identity which follows
from factorisation. We switch to lattice units such that $T=1/N_{\tau}$,
$T_{0}=1/N_{0}$, and $N_{0}/N_{\tau}=m$ is an (odd) integer. In the bosonic
case

$\displaystyle 1-\text{e}^{-\omega mN_{\tau}}=\left(1-\text{e}^{-\omega
N_{\tau}}\right)\,\left(\sum_{n=0}^{m-1}\text{e}^{-n\omega N_{\tau}}\right),$
(19)

where $m$ is any integer. The fermion case is

$\displaystyle 1+\text{e}^{-\omega mN_{\tau}}=\left(1+\text{e}^{-\omega
N_{\tau}}\right)\,\left(\sum_{n=0}^{m-1}\left(-1\right)^{n}\,\text{e}^{-n\omega
N_{\tau}}\right)$ (20)

for $m$ an odd integer. The bosonic kernel can hence be written as

$\displaystyle K_{B}(\tau,$
$\displaystyle\,\omega;1/N_{\tau})=\frac{\text{e}^{-\omega\tau}}{1-\text{e}^{-\omega
N_{\tau}}}=\sum_{n=0}^{m-1}\frac{\text{e}^{-\omega\left(\tau+nN_{\tau}\right)}}{1-\text{e}^{-\omega
mN_{\tau}}}$
$\displaystyle=\sum_{n=0}^{m-1}K_{B}\left(\tau+nN_{\tau},\omega;1/(mN_{\tau})\right),$
(21)

for integer $m$, and the fermionic kernel as

$\displaystyle K_{F}(\tau,$
$\displaystyle\,\omega;1/N_{\tau})=\frac{\text{e}^{-\omega\tau}}{1+\text{e}^{-\omega
N_{\tau}}}=\sum_{n=0}^{m-1}\frac{\left(-1\right)^{n}\,\text{e}^{-\omega\left(\tau+nN_{\tau}\right)}}{1+\text{e}^{-\omega
mN_{\tau}}}\quad$
$\displaystyle=\sum_{n=0}^{m-1}\left(-1\right)^{n}\,K_{F}\left(\tau+nN_{\tau},\omega;1/(mN_{\tau})\right),$
(22)

for integer odd $m$.

Inserting Eq. (III.1) into the spectral relation (16) for a fermionic
correlator at temperature $T=1/N_{\tau}$ relates this correlator to a
resummation of one at a lower temperature $T_{0}=1/N_{0}=1/(mN_{\tau})$,
assuming that the spectral content content is unchanged. Thus yields the
reconstructed correlator for fermions,

$\displaystyle
G_{\text{rec}}\left(\tau;1/N_{\tau},1/N_{0}\right)=\sum_{n=0}^{m-1}\left(-1\right)^{n}\,G\left(\tau+nN_{\tau};1/N_{0}\right).$
(23)

Switching back to denoting the temperatures with $T$ and $T_{0}$, it allows us
to compare the actual correlator at temperature $T$, $G(\tau;T)$, with the
expected correlator if the spectral content is unchanged from a lower
reference temperature $T_{0}$, namely
$G_{\text{rec}}\left(\tau;T,T_{0}\right)$, via the ratio

$\displaystyle r_{\rm rec}(\tau;T,T_{0})=G\left(\tau;T\right)/G_{\rm
rec}\left(\tau;T,T_{0}\right).$ (24)

For bosons, this technique was used in e.g. Refs. [63, 13]. For fermions, we
are not aware of a previous application.

As $m=N_{0}/N_{\tau}=T/T_{0}$ must be an odd integer for fermions, the lattice
sizes which can be used are limited in principle. This can be resolved by
“padding” the correlator with the minimum value at the minimum. No qualitative
difference is found by padding with zero instead. We consider only adding
points to extend the correlator as this can be done at all temperatures. In
principle one could also consider removing points from the reference
correlator at the lower temperature $T_{0}=1/N_{0}$. No substantial difference
is observed in our tests.

###  III.2 Model Correlator

Alternatively, we may construct simple model correlators at a higher
temperature $T$ by employing the ground state masses determined at the lowest
temperature $T_{0}$ in the previous section, built on the assumption that
spectral content has not changed and using the baryon spectral relation (16)
for the zero-momentum projected positive-parity correlator. We write for the
spectral function

$\displaystyle\rho_{+}\left(\omega\right)=2\pi
A_{+}\delta\left(\omega-M^{+}_{0}\right)+2\pi
A_{-}\delta\left(\omega+M^{-}_{0}\right),$ (25)

where $M^{\pm}_{0}$ is the positive (negative) parity mass at temperature
$T_{0}$ and $A_{\pm}$ are the corresponding amplitudes. The model correlator
at temperature $T$ is then

$\displaystyle G_{\rm model}\left(\tau;T,T_{0}\right)$
$\displaystyle=A_{+}K_{F}\left(\tau,M^{+}_{0}\right)+A_{-}K_{F}\left(\tau,-M^{-}_{0}\right)$
$\displaystyle=\frac{A_{+}\text{e}^{-M^{+}_{0}\tau}}{1+\text{e}^{-M^{+}_{0}/T}}+\frac{A_{-}\text{e}^{M^{-}_{0}\tau}}{1+\text{e}^{M^{-}_{0}/T}}.$
(26)

The main assumption here is that the width of the state extracted is
negligible. We can now take ratios of the actual and the model correlator,

$\displaystyle r\left(\tau;T,T_{0}\right)=G\left(\tau;T\right)/G_{\rm
model}\left(\tau;T,T_{0}\right),$ (27)

as well as double ratios,

$\displaystyle R\left(\tau;T,T_{0}\right)$
$\displaystyle=\frac{r\left(\tau;T,T_{0}\right)}{r\left(\tau;T_{0},T_{0}\right)}$
$\displaystyle=\left.\frac{G\left(\tau;T\right)}{G_{\rm
model}\left(\tau;T,T_{0}\right)}\right/\frac{G\left(\tau;T_{0}\right)}{G_{\rm
model}\left(\tau;T_{0},T_{0}\right)},$ (28)

to potentially eliminate the effects of excited states and other features at
early Euclidean times. This approach has been studied in some detail in Ref.
[14] for the case of $D_{(s)}$ mesons. Note that in all ratios we may consider
separately the positive parity correlator $G_{+}(\tau)$ and the negative
parity correlator $G_{-}(\tau)=-G_{+}(1/T-\tau)$. Moreover, we always divide
the correlators by their value at $\tau=0$, to set a consistent scale.

Figure 6: Ratio of correlator $G\left(\tau;T\right)$ and reconstructed
correlator $G_{\text{rec}}\left(\tau;T,T_{0}\right)$ as described in Eq. 24 in
the $\Sigma_{c}\left(udc\right)$ channel, using as input the correlator at the
lowest temperature $T_{0}=47$ MeV ($N_{0}=128$), for positive parity (left)
and negative parity (right). The top row shows the four lowest temperatures,
$T=47,95,109,127$ MeV; the bottom row the higher ones, $T=127,152,169,190$
MeV. Points past the minimum of the lattice correlator $G\left(\tau;T\right)$
are faded and have open symbols.

Figure 7: As in Fig. 7, for the ratio of the correlator $G\left(\tau;T\right)$
and the model correlator $G_{\rm model}\left(\tau;T,T_{0}\right)$, see Eq.
$\left(\ref{eqn:SRatio}\right)$.

###  III.3 Results

We now present some results for ratios of correlators, using both the
reconstructed correlator and the model correlator in the denominator. We focus
here on the $\Sigma_{c}\left(udc\right)$ channel; Appendix A contains
additional results in the $\Xi_{cc}\left(ccu\right)$ and
$\Omega_{cc}\left(ccs\right)$ channels. These are representative; we have of
course analysed all channels.

We start with the ratio using the reconstructed correlators, see Eq. (24). In
Figure 7 we show the result in the $\Sigma_{c}\left(udc\right)$ channel, for
positive parity (left) and negative parity (right), at low temperatures (top)
and at higher temperatures (bottom). Note that $T=127$ MeV is shown in both
sets of figures. In the positive parity case, we note that the ratio is close
to 1 at the lower temperatures; the bending upwards around
$\tau/a_{\tau}=N_{\tau}/2$ is expected due to the appearance of states with
opposite parity. At the highest temperatures shown, the deviation becomes more
significant around $T\sim T_{\rm pc}$ and is of the order of 10% at $T=190$
MeV. Note the different vertical scales. We conclude that in the positive
parity sector thermal effects are at the percent level up to $T\sim T_{\rm
pc}$. In the negative parity case on the other hand, there are strong
temperature effects at all temperatures, even deep in the hadronic phase.

The reconstructed correlator gives insight into possible changes into the
underlying spectral functions as the temperature increases. A closer look at
the fate of the ground state can be obtained by taking the ratio with the
model correlator, see Eq. (27). This analysis relies on the assumption that at
the lowest temperature the width of the state extracted is negligible and that
it is possible to distinguish the ground state from the excited states. As the
temperature increases the temporal extent of the lattice decreases and so the
ground state may not be distinguishable from the excited states, even if the
spectral information is unchanged. If the ground states are precisely
determined and indeed independent of temperature, the ratio Eq. (27) will be
independent of $\tau$ for large enough Euclidean time, understood as the
region where the ground state dominates, but where the “bending” of the
correlator near $N_{\tau}/2$ is not yet encountered. As the model contains
only the ground state and the correlator $G\left(\tau;T\right)$ has excited
states present, this ratio is not expected to be completely $\tau$
independent.

This is demonstrated in Figure 7, again in the $\Sigma_{c}\left(udc\right)$
channel. On the positive parity side, we do indeed observe a horizontal
plateau for $T\lesssim 127$ MeV, indicating that the ground state is well
described by a single exponential. The negative parity sector is less clear
due to the skewness of the correlation function causing fewer time slices to
be available before the mixing of forward and backward states becomes evident.
This could be improved by using methods to reduce the amount of excited state
contamination in the correlator. The single ratio is similar to an effective
mass in that it shows when excited state effects cannot be ignored. The
presence of plateaus in the single ratio is indicative that the narrow peak
ansatz for the spectral function is suitable for exponential fits as in Eq.
$\left(\ref{eqn:expFit}\right)$. At our lowest temperature, $T=47$ MeV,
excellent plateau behaviour is seen for all channels investigated, providing
further support for the results for the ground states masses presented in
Table 2.

Figure 8: As in Fig. 7, for the double ratio of $G\left(\tau;T\right)/G_{\rm
model}\left(\tau;T,T_{0}\right)$ and $G\left(\tau;T_{0}\right)/G_{\rm
model}\left(\tau;T_{0},T_{0}\right)$, see Eq. $\left(\ref{eqn:DRatio}\right)$.

From the plots with single ratios, it is clear that also the structure at
earlier Euclidean times is approximately temperature independent, in
particular at lower temperatures and in the positive parity sector, see Figure
7 (top left). We can factor out the common behaviour by considering the double
ratio of Eq. $\left(\ref{eqn:DRatio}\right)$, involving temperatures $T$ and
$T_{0}$. The results are shown in Fig. 8. We note that the magnitude of
deviation from 1 is comparable between the double ratio and the ratio with the
reconstructed correlator. Quantitatively, they show different behaviour: as
mentioned, the double ratio only uses the information on the ground state at
$T=T_{0}$, assuming that it is well described by a narrow peak, whereas the
reconstructed correlator uses the full spectral function at $T_{0}$.

To determine where exponential fits can be reasonably performed to extract the
ground state mass in the thermal case, we will require plateau-like behaviour
with the double ratio not exceeding a difference of $\sim 10\%$ from one. This
is illustrated in Fig. 8, where evidence can be seen of a change in behaviour
around the pseudocritical temperature in the positive parity channel. The
double ratio is close to one until $T=169\text{ MeV}\sim T_{\rm pc}$. The
negative parity sector is much noisier — note the different scale — and
deviates from one at much lower temperatures, removing the justification of
using exponential fits for the negative parity state above $T=127$ MeV in this
channel. When the negative parity sector has clear evidence of a change in
spectral content, we do not report a negative parity mass obtained by fitting
to an exponential Ansatz, as it is not justified to do so. As mentioned,
Appendix A contains additional results in the $\Xi_{cc}\left(ccu\right)$ and
$\Omega_{cc}\left(ccs\right)$ channels.

##  IV Masses at non-zero temperature

Figure 9: Ground state masses of singly charmed spin $1/2$ baryons, normalised
with the positive parity ground state mass at the lowest temperature, as a
function of temperature. Filled (open) symbols are used for positive
(negative) parity states. The inner error bar represents the statistical
uncertainty and the outer incorporates the systematic from the choice of
averaging method. Horizontal dashed lines show the result from the lowest
temperature. The uncertainty of the lowest temperature reflects the relative
uncertainty of the lowest temperature mass. Masses are shown only when the
analysis of Section III suggests the mass can be extracted using an
exponential Ansatz.

We now extend the analysis detailed in Section II.3 to non-zero temperature.
We use the analysis and methods of the previous sections as a guide to when
the spectral content has changed such that the exponential Ansatz of Eq.
$\left(\ref{eqn:expFit}\right)$ is no longer appropriate. Consequently we show
mass results for only a subset of the temperatures available to us.

The singly charmed baryon spectrum is presented in Figure 9. All masses have
been normalised with the positive parity ground state mass at the lowest
temperature, $m_{+}(T_{0})$, in the corresponding channel. While the
uncertainties increase as temperature does — reflective of fewer temporal
points and a potential shift away from a zero-width state — the pattern
clearly suggests that in the positive parity sector the masses increase with
temperature and that this starts relatively deep in the confining phase. On
the negative parity side, acceptable results are obtained, even though the
negative parity signal is shorter lived than the positive parity, as
illustrated in e.g. Figure 3. Here the increase in uncertainties is such that
it is difficult to determine if masses of negative parity states are changing
in a systematic way. We note that the ratio analysis of Section III did indeed
suggest that the negative parity sector is more strongly affected by
temperature effects.

Masses of the doubly charmed baryons $\Xi_{cc}\left(ccu\right)$ and
$\Omega_{cc}\left(ccs\right)$ are shown in Figure 10. Temperature effects are
much less evident here, in both parity sectors, with the positive parity
$\Xi_{cc}\left(ccu\right)$ mass remaining approximately constant well into the
deconfining phase, up to $T\sim 190$ MeV. To provide support for this finding
we show the single and double ratios for this baryon in Figures 14 and 16 in
Appendix A. The positive parity double ratio in Figure 16 is particularly
striking, with a value close to one at all temperatures shown here, indicating
the apparent absence of thermal effects in this channel. It would be
interesting to further investigate the difference in thermal effects for
doubly charmed baryons compared to singly charmed baryons using effective
potential models, similar to the well-studied meson (quarkonium) cases;
preliminary work in this direction has been carried out [64].

##  V Parity doubling

The restoration of chiral symmetry is a signature for the formation of a QGP
[2, 3, 65, 66]. We investigate chiral symmetry restoration via the parity
doubling phenomenon [11], in which positive and negative parity correlation
functions become degenerate when chiral symmetry is unbroken. While
correlators for light and strange baryons exhibit a clear parity doubling
signal [11, 12], this is not expected here as the charm quark mass strongly
and explicitly breaks chiral symmetry. Nevertheless, the light quarks
contained within the charmed baryons are susceptible to chiral symmetry,
resulting in a restoration signal as we now demonstrate.

We define the baryon $R$ parameter [67, 10, 11, 12, 18] from the summed ratio
of positive and negative correlators,

$\displaystyle\mathcal{R}\left(\tau\right)$
$\displaystyle=\frac{G_{+}\left(\tau\right)-G_{+}\left(1/T-\tau\right)}{G_{+}\left(\tau\right)+G_{+}\left(1/T-\tau\right)},$
(29) $\displaystyle R\left(n_{0}\right)$
$\displaystyle=\frac{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,\mathcal{R}\left(\tau_{n}\right)/\sigma_{\mathcal{R}}^{2}\left(\tau_{n}\right)}{\sum_{n=n_{0}}^{\frac{1}{2}N_{\tau}-1}\,1/\sigma_{\mathcal{R}}^{2}\left(\tau_{n}\right)}.$
(30)

Here $G_{+}\left(\tau\right)$ is the positive parity correlator,
$G_{+}\left(1/T-\tau\right)=-G_{-}\left(\tau\right)$ is the negative parity
correlator — see Eq. (10), $\sigma_{\mathcal{R}}\left(\tau_{n}\right)$ denotes
the statistical error for $\mathcal{R}\left(\tau_{n}\right)$ and we consider a
sum over time slices $\tau_{n}/a_{\tau}\in\left[n_{0},N_{\tau}/2-1\right]$ at
all temperatures. We choose $n_{0}=4$ to suppress lattice and excited state
artefacts at small $\tau_{n}$ [10]. Shifting $n_{0}$ slightly does not have a
qualitative effect on the results and Gaussian smearing at the source and sink
further suppresses the effect of excited states. Lattice artefacts are due to
e.g. the chiral symmetry violating Wilson term at larger energy scales.

Figure 10: As in Fig. 9, for doubly charmed spin $1/2$ baryons.

As suggested by Eq. $\left(\ref{eqn:mcRratio}\right)$, the $R$ parameter
describes the difference between the positive and negative parity correlators.
When chiral symmetry is unbroken, the correlators are degenerate and $R=0$.
Broken chiral symmetry produces $R\neq 0$. In the limit where the correlators
are dominated by their respective ground states and the positive parity mass
is much smaller than the negative parity mass, $R\approx 1$. We hence expect
that $R$ is close to one in the hadronic phase. Since the charm mass breaks
chiral symmetry at all temperatures considered, we do not expect $R$ to go to
zero.

Figure 11: Crossover behaviour of the $R$ parameter of Eq. (30) for
$J=\frac{1}{2}$ baryons. The lines connecting the data points are cubic
splines; the vertical lines indicate the inflection points for singly charmed
baryons. No inflection point is found for the doubly-charmed baryons.

Figure 11 shows the $R$ parameter for the singly and doubly charmed
$J=\frac{1}{2}$ baryons we considered. We observe that the values are indeed
close to one at the lowest temperature, as expected. As the temperature
increases, $R$ decreases for singly charmed baryons, indicating the
sensitivity of the constituent light quarks to chiral symmetry restoration.
Above the crossover temperature, the $R$ values are distinct from zero but
continue to decrease. It is expected that at very high temperature the effect
of the charm quark mass will vanish eventually, as $m_{c}/T\rightarrow 0$. For
doubly charmed baryons, the $R$ parameter is much less sensitive to
temperature and a reduction only sets in at higher temperatures. It is
interesting to note the ordering of the $R$ values in the QGP: $\Sigma_{c}$,
$\Xi_{c}^{\prime}$, and $\Omega_{c}$ belong to the SU(3) $=\mathbf{6}$ flavour
multiplet, while $\Lambda_{c}$ and $\Xi_{c}$ belong to the
SU(3)$=\mathbf{\bar{3}}$ flavour multiplet. The doubly charmed baryons are
significantly heavier.

The lines connecting the data points in Figure 11 are cubic splines fitted to
the points. For singly charmed baryons, these splines have an inflection point
indicated by the vertical lines and the inflection point temperatures are
listed in Table 3. We note that within a few MeV the inflection point
temperatures agree with the pseudocritical temperature obtained from the
renormalised chiral condensate. This approximate agreement was observed
earlier for the case of light and strange baryons, see Refs. [12, 18]. It is
somewhat remarkable to note that even for singly charmed baryons the
constituent light and strange quarks are sufficiently sensitive to chiral
symmetry restoration to induce inflection points at these temperature values.
On the other hand, the $R$ parameters for doubly charmed baryons do not show
an inflection point in this temperature range.

##  VI Conclusions

In this work we presented a detailed study of the spin $1/2$ charmed baryons
throughout the hadronic and into the deconfining phase. After determining the
spectrum at the lowest temperature, the temperature dependence in each channel
was investigated using ratios of thermal correlators with model or
reconstructed correlators built on spectral content from the lowest
temperature. This permits us to identify where thermal effects become
important. It was found that the negative parity sector is sensitive to
thermal effects at a much lower temperature than the positive parity sector.
For doubly charmed, positive parity baryons we observed an approximate
temperature independence of correlator ratios up to $T\sim 190$ MeV, the
highest temperature we studied. The analysis of these ratios is robust as it
does not rely on a particular Ansatz for spectral content at finite
temperature.

Based upon the analysis of correlator ratios, we justify the use of multi-
exponential fits to determine baryon masses in the cases where the spectral
content appears to be largely unchanged. This allows us to extract positive
parity masses at temperatures up to $T\sim 190$ MeV, albeit with a large
uncertainty for singly charmed baryons, while the negative parity masses can
only be determined up to $T\sim 127$ MeV. The masses of the positive parity
ground states of doubly charmed baryons are approximately independent of
temperature up to $T\sim 190$ MeV.

To study the effect of chiral symmetry restoration for charmed baryons, we
investigated the $R$ parameter which encodes the difference between positive
and negative parity correlators as a function of temperature. While parity
doubling is not manifest — as expected due to the large charm quark mass — a
crossover effect is observed for singly charmed baryons. Interestingly, the
temperatures of the inflection points of this quantity are close to the
pseudocritical temperature obtained from the renormalised chiral condensate,
indicating that the constituent light and strange quarks are sufficiently
sensitive to chiral symmetry restoration to induce this effect also in charmed
correlators.

Future studies could examine the spectrum using either a more sophisticated
operator basis or a lattice with more temporal points. Both these methods are
under investigation. Alternatively one could use spectral function
reconstruction methods to directly examine the mass and width of the ground
states as a function of temperature to determine when thermal effects become
significant.

Table 3: Temperature of the inflection points (in MeV) of the $R$ parameter (30) for singly charmed baryons presented in Figure 11, categorised by strangeness (S) and charm (C). Uncertainties are statistical only. For reference, the pseudocritical temperature from the renormalised chiral condensate is $T_{\rm pc}=167(2)(1)$ MeV. S | C | Inflection point (MeV)
---|---|---
$0$ |  $1$ |  $\Sigma_{c}\left(udc\right)$ $166.1(1.0)$ |  $\Lambda_{c}\left(udc\right)$ $162.6(5)$
$-1$ |  $1$ |  $\Xi^{\prime}_{c}\left(usc\right)$ $164.2(6)$ |  $\Xi_{c}\left(usc\right)$ $162.3(4)$
$-2$ |  $1$ |  $\Omega_{c}\left(ssc\right)$ $167.2(1.3)$ |

Figure 12: Ratio with the reconstructed correlator as in Figure 7, in the
$\Omega_{cc}\left(ccs\right)$ channel.

## Software and Data

Correlation functions were generated using openQCD-hadspec [68], an extension
to openQCD-FASTSUM [69] for correlation functions. openQCD-FASTSUM was used
for ensemble generation [18, 14].

The analysis in this work makes extensive use of the python packages gvar [70]
and lsqfit [46]. Additional data analysis tools included matplotlib [71, 72]
and numpy [73]. The dataset and scripts used for this paper can be found at
[74].

## Authors’ Contributions

  * •

Bignell: Data analysis, plot generation and primary manuscript production.

  * •

Jäger: Data production and preliminary data analysis.

  * •

Aarts: Initial proposal [75], development of the reconstructed correlator for
baryons and draft of the manuscript.

  * •

Allton, Anwar, Burns, Skullerud: Contributions to analysis and physics
interpretations of results, and draft of the manuscript.

###### Acknowledgements.

G.A., C.A., R.B. and T.J.B. are grateful for support via STFC grant
ST/T000813/1. M.N.A. acknowledges support from The Royal Society Newton
International Fellowship. This work used the DiRAC Extreme Scaling service at
the University of Edinburgh, operated by the Edinburgh Parallel Computing
Centre and the DiRAC Data Intensive service operated by the University of
Leicester IT Services on behalf of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). This equipment was funded by BEIS capital funding via STFC
capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC
Operations grants ST/R001006/1 and ST/R001014/1. DiRAC is part of the UK
National e-Infrastructure. We acknowledge the support of the Swansea Academy
for Advanced Computing, the Supercomputing Wales project, which is part-funded
by the European Regional Development Fund (ERDF) via Welsh Government, and the
University of Southern Denmark and ICHEC, Ireland for use of computing
facilities. This work was performed using PRACE resources at Cineca (Italy),
CEA (France) and Stuttgart (Germany) via grants 2015133079, 2018194714,
2019214714 and 2020214714.

##  Appendix A Ratios in other channels

For comparison, we present the ratios of thermal correlators with
reconstructed and model correlators in the $\Xi_{cc}\left(ccu\right)$ and the
$\Omega_{cc}\left(ccs\right)$ channels in Figs. 12-16. Details are as in
Section III.

Figure 13: Single ratio with the model correlator as in Figure 7, in the
$\Xi_{cc}\left(ccu\right)$ channel.

Figure 14: Single ratio with the model correlator as in Figure 7, in the
$\Omega_{cc}\left(ccs\right)$ channel.

Figure 15: Double ratio as in Figure 8, in the $\Xi_{cc}\left(ccu\right)$
channel.

Figure 16: Double ratio as in Figure 8 in the $\Omega_{cc}\left(ccs\right)$
channel.

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