# Electromagnetic Structure of Spin-$\frac{1}{2}$ Doubly Charmed Baryons in
Lattice QCD

H. Bahtiyar  Department of Physics, Faculty of Science and Letters, Mimar
Sinan Fine Arts University, Bomonti 34380, Istanbul, Turkey

###### Abstract

We compute the electromagnetic properties of spin-$\frac{1}{2}$ doubly charmed
baryons on 2+1 flavor lattices that have a pion mass of $\sim$ 156 MeV. A
relativistic fermion action is employed for the charm quark in addition to the
standard isotropic Clover action to quantify the $\mathcal{O}(m_{q}a)$
effects. We calculate the electric and magnetic Sachs form factors and extract
the magnetic moments, and electric and magnetic charge radii. We also
investigate the individual quark sector contributions to the charge radii and
the magnetic moments. The results provide vital information to understand the
size and shape of the doubly charmed baryons. We find that the two heavy charm
quarks drive the charge radii and the magnetic moments to smaller values than
that of light baryons. The central values of the observables that are obtained
using the relativistic action for the charm quark are $5$ to $10\%$ larger
than those obtained using the Clover action. Utilizing the available lattice
data, we reexamine the quark mass dependence of the observables.

charmed baryons,magnetic moment, lattice QCD

###### pacs:

14.20.Lq, 12.38.Gc, 13.40.Gp

##  I Introduction

In the last decades, measurements of the charmed baryons have accelerated with
the developments in experimental facilities. Many states are discovered, yet
many states need to be confirmed; therefore charmed baryon sector is
theoretically appealing. All of the singly charmed ground-state baryons have
been experimentally observed [1, 2, 3, 4, 5]. However, observation of doubly
charmed baryons has been long overdue. In 2002, SELEX collaboration reported
the doubly charmed baryon, which contains two $c$ and one $d$ quark, and the
mass of the $\Xi_{cc}^{+}$ reported as $3519\pm 1$ MeV/c2 [6]. However, none
of the FOCUS [7], BABAR [8] , BELLE [9], and LHCb [10] could confirm SELEX’s
result. In 2017, LHCb Collaboration discovered the isospin partner of
$\Xi_{cc}^{+}$, namely $\Xi_{cc}^{++}$ [11], containing two $c$ quarks and one
$u$ quark and the mass of the $\Xi_{cc}^{++}$ reported by LHCb is $3621.40\pm
0.72\pm 0.27\pm 0.14$ MeV/c2, around $100$ MeV larger than the SELEX result.
The mass difference has been studied within various theoretical approaches
[12, 13, 14, 15, 16, 17].

Doubly charmed baryons sit at the top layer of spin-$\frac{1}{2}$ flavor-mixed
symmetric 20-plet of the SU(4) multiplet. This layer consists of three
baryons; the isospin doublets $\Xi_{cc}^{++}($ucc$)$ and
$\Xi_{cc}^{+}($dcc$)$, and the isospin singlet $\Omega_{cc}$ ($scc$). Although
$\Omega_{cc}$ has not been experimentally observed yet [18], several
theoretical studies have been conducted on its mass and form factors [19, 20,
21, 22, 23, 24, 25].

Studying the electromagnetic properties of baryons provide us with valuable
information about their internal structures. One can extract information about
their sizes, shapes, decay widths and compare them with the experimental
results. Baryons containing two charm quarks are especially exciting to
explore since examining the electromagnetic properties of two heavy-quarks
bound to a light-quark helps us understand the internal interaction dynamics
of baryons containing heavy-quarks. In addition, the results may shed light on
our understanding of the fundamental properties of QCD, such as confinement
and flavor effects. Electromagnetic properties of doubly charmed baryons have
been previously studied in quark models [26, 27, 28, 29, 30, 31, 32], MIT bag
model [33, 34, 35], chiral perturbation theory [36, 37, 38, 39], QCD sum rules
[40, 25] and lattice QCD [41, 24].

This work focuses on calculating the electromagnetic form factors and
extracting the charge radii and magnetic moments of doubly charmed baryons
with near physical $u$, $d$ sea quarks in $2+1$ flavor lattice QCD. A
relativistic fermion action is employed for the charm quark in addition to the
standard isotropic Clover action to quantify the $\mathcal{O}(m_{q}a)$
effects. Since the lattice pion mass is close to the physical pion mass, we
also employ the chiral extrapolation of the results calculated with Clover
action using the results of Refs [41, 24]. This paper is organized as follows:
Theoretical formalism, details of the lattice setup, and definitions of the
form factors are given in Section II. We present our results, compare them to
other works and give a discussion in Section III. Section IV summarizes our
results.

##  II Theoretical Formalism and Lattice Setup

Electromagnetic form factors is encoded into baryon matrix elements written in
the following form:

$\langle{\cal B}(p^{\prime})|J_{\mu}|{\cal
B}(p)\rangle=\overline{u}(p^{\prime})\Big{[}\gamma_{\mu}F_{1}(q^{2})+i\frac{\sigma_{\mu\nu}q^{\nu}}{2m_{\cal
B}}F_{2}(q^{2})\Big{]}u(p),$ (1)

where $F_{1}(q^{2}),F_{2}(q^{2})$ are the Dirac and Pauli form factors
respectively, $u(p^{\prime})$ and $u(p)$ are the Dirac spinor of the baryon
with mass of $m_{\cal B}$, and $q^{\mu}=p^{\mu}-p^{\prime\mu}$ is the
transferred four-momentum. The baryon matrix elements of the electromagnetic
vector current are written in the following form:

$J_{\mu}=\frac{2}{3}\overline{c}\gamma_{\mu}c+\frac{2}{3}\overline{c}\gamma_{\mu}c+C_{\ell}\overline{\ell}\gamma_{\mu}\ell,$
(2)

where $\ell$ denotes the flavor of the lighter quarks (u, d, and s) and
$C_{\ell}$ is its charge ($2/3$ or $-1/3$). The electric and magnetic Sachs
form factors are defined in terms of Dirac and Pauli form factors as follows:

$\displaystyle G_{E}(q^{2})$ $\displaystyle=F_{1}(q^{2})+\frac{q^{2}}{4m_{\cal
B}^{2}}F_{2}(q^{2}),$ (3) $\displaystyle G_{M}(q^{2})$
$\displaystyle=F_{1}(q^{2})+F_{2}(q^{2}).$ (4)

The form factors are extracted using the two-point and three-point correlation
functions,

$\displaystyle\langle F^{{\cal B}\,{\cal B}}(t;p;\Gamma_{4})\rangle$
$\displaystyle=\sum\limits\limits_{x}\,e^{i\,p\,x}\Gamma^{\beta\,\alpha}_{4}\,\langle\Omega|T(\chi_{\cal
B}^{\beta}(x)\overline{\chi}_{\cal B}^{\alpha}(0))|\Omega\rangle,$ (5)
$\displaystyle\langle F^{{\cal B}J_{\mu}{\cal
B}}(t_{2},t_{1};p^{\prime},p;\Gamma)\rangle$
$\displaystyle=-i\,\sum\limits\limits_{x_{1},x_{2}}\,e^{-i\,p\,x_{2}}\,e^{i\,q\,x_{1}}\Gamma^{\beta\,\alpha}\langle\Omega|T(\chi^{\beta}_{{\cal
B}}(x_{2})J_{\mu}(x_{1})\overline{\chi}^{\alpha}_{\cal B}(0))|\Omega\rangle,$
(6)

where $t_{1}$ is the time when the electromagnetic current is inserted to
quark, and $t_{2}$ is the time when the baryon is annihilated. Gamma matrices
are defined as,

$\displaystyle\Gamma_{4}=\frac{1}{2}\begin{bmatrix}1&0\\\
0&0\end{bmatrix},\quad\Gamma_{i}=\frac{1}{2}\begin{bmatrix}\sigma_{i}&0\\\
0&0\end{bmatrix}.$ (7)

The baryon interpolating fields are chosen as the nucleon-like form

$\displaystyle\chi_{\cal B}$
$\displaystyle=\varepsilon_{abc}\big{(}c^{T}_{a}(C\gamma_{5})\ell_{b}\big{)}c_{c},$
(8)

where $\ell$ denotes $s$, $u$ and $d$ quarks for $\Omega_{cc}$,
$\Xi_{cc}^{++}$ and $\Xi_{cc}^{+}$ baryons respectively. The $a,b,c$ are color
indices and the charge conjugation matrix is defined as
$C=\gamma_{4}\gamma_{2}$.

In order to eliminate the normalization factors and extract the
electromagnetic form factors, we define the ratio,

$\displaystyle R(t_{2},t_{1},p^{\prime},p,\Gamma,\mu)=$
$\displaystyle\frac{\langle F^{\cal
BJ_{\mu}B}(t_{2},t_{1};p^{\prime},p;\Gamma)\rangle}{\langle F^{\cal
BB}(t_{2};p^{\prime};\Gamma_{4})\rangle}\times$ (9)
$\displaystyle\sqrt{\frac{\langle F^{\cal
BB}(t_{2}-t_{1};p;\Gamma_{4})\rangle\langle F^{\cal
BB}(t_{1};p^{\prime};\Gamma_{4})\rangle\langle F^{\cal
BB}(t_{2};p^{\prime};\Gamma_{4})\rangle}{\langle F^{\cal
BB}(t_{2}-t_{1};p^{\prime};\Gamma_{4})\rangle\langle F^{\cal
BB}(t_{1};p;\Gamma_{4})\rangle\langle F^{\cal
BB}(t_{2};p;\Gamma_{4})\rangle}}.$

In the large Euclidean-time limit, $t_{2}-t_{1}\gg a$ and $t_{1}\gg a$, time
dependences are eliminated and the ratio in Eq.(9) reduces to

$R(t_{2},t_{1};p^{\prime},p;\Gamma;\mu)\xrightarrow[t_{1}\gg a]{t_{2}-t_{1}\gg
a}\Pi(p^{\prime},p;\Gamma;\mu).$ (10)

One can extract the Sachs form factors by choosing the appropriate
combinations of projection matrices $\Gamma$ and the Lorentz index $\mu$:

$\Pi(p^{\prime},p;\Gamma_{4};\mu=4)=\sqrt{\frac{(E_{{\cal B}}+m_{{\cal
B}})}{2E_{\cal B}}}G_{E}(q^{2}),$ (11)
$\Pi(p^{\prime},p;\Gamma_{j};\mu=i)=\sqrt{\frac{1}{2E_{\cal B}(E_{{\cal
B}}+m_{{\cal B}})}}\epsilon_{ijk}q_{k}G_{M}(q^{2}).$ (12)

Here $G_{E}$ gives the electric charge of the baryon at zero transferred
momentum. Similarly, the magnetic moment can be obtained by extrapolating the
$G_{M}$ to zero transferred momentum.

Simulations have been run on $32^{2}\times 64$ lattices with $2+1$ flavors of
dynamical quarks that have been generated by the PACS-CS collaboration [42]
with the non-perturbatively O(a)-improved Wilson quark action and the Iwasaki
gauge action. Simulations are carried out with near physical $u$, $d$ sea
quarks of hopping parameter $\kappa^{u,d}_{\text{sea}}=0.13781$. Hopping
parameter for the sea $s$ quark is fixed to $\kappa^{s}_{\text{sea}}=0.13640$.
Features of the lattice setup are explained in detail in Ref. [43].

We use the Clover action for the $u$, $d$, and $s$ valence quarks. Hopping
parameter of the light-quarks is taken equal to that of the light sea quark
$\kappa^{u,d}_{\text{val}}=\kappa^{u,d}_{\text{sea}}=0.13781$ which
corresponds to a pion mass of approximately $156$ MeV [42]. As for the strange
quark, it is reported that the hopping parameter overestimates the
experimental value by $6\%$ [44]. Therefore hopping parameter of the strange
quark is taken as $\kappa^{s}_{\text{val}}=0.13665$, as suggested in Refs.
[45, 46].

We utilize two different fermion actions for the charm quark. Firstly, we
apply a Clover action in the form used by Fermilab and MILC Collaborations
[47, 48, 49]. We use the value of the hopping parameter
$\kappa^{c}_{\text{Clover}}=0.1246$ as determined in our previous work [24].
The second fermion action we implement is the relativistic heavy-quark action
(Tsukuba) proposed by Aoki et al. [50]. This action is designed to reduce the
leading cutoff effects, which can be removed by tuning the action’s parameters
non-perturbatively. As a result, only $\mathcal{O}((a\Lambda_{QCD})^{2})$
discretization errors remain. The hopping parameter is taken as
$\kappa^{c}_{\text{Tsukuba}}=0.10954007$. The tuning of the action’s
parameters is explained in detail in Ref. [44].

We insert positive and negative momenta in all spatial directions and make a
simultaneous fit over all data to increase statistics. We also calculate
current insertion along all spatial directions. Data are binned at the bin
size of $20$ in order to account for autocorrelations. The source-sink time
separation is fixed to $12$ lattice units to prevent the excited state
contamination [24, 44]. We also use multiple source-sink pairs, shifting them
$12$ lattice units along the temporal direction. The point-split lattice
vector current is employed

$J_{\mu}=\frac{1}{2}[\overline{q}(x+\mu)U_{\mu}^{\dagger}(1+\gamma_{\mu})q(x)-\overline{q}(x)U_{\mu}(1-\gamma_{\mu})q(x+\mu)],$
(13)

which is conserved by Wilson fermions and therefore does not need any
renormalization. Momentum is inserted up to $9$ lattice units and averaged
over equivalent momenta. All statistical errors are estimated by the single-
elimination jackknife analysis. The wall-source/sink method [51] is employed;
therefore, one can simultaneously extract all spin, momentum, and projection
components. The wall source/sink is a gauge-dependent object; thus, the gauge
is fixed to Coulomb.

The finite-size effects should be negligible when $m_{\pi}L\geq 4$. On the
other hand, the ensemble that we used in the analysis yields $m_{\pi}L=2.3$,
which is below this bound. However, it is confirmed that the finite-size
effects on this particular setup are under control for physical quantities
related to strange and charmed baryons [51].

Only connected diagrams are computed in this work. We performed our
computations using a modified version of Chroma software system [52] on CPU
clusters and with QUDA [53, 54] for propagator inversion on GPUs.

##  III Results and Discussion

We begin our work by extracting the masses using the two-point correlation
function given in Eq.(5). If the sink operator is projected to zero momentum,
the two-point correlation functions reduce to

$\langle F^{{\cal B}\,{\cal B}}(t;p;\Gamma_{4})\rangle\simeq Z_{\cal
B}(\mathbf{p})\bar{Z}_{\cal B}(\mathbf{p})e^{-E_{\cal
B}(\mathbf{p})t}(1+\mathcal{O}(e^{-\Delta Et})+\dots),$ (14)

where the mass of a baryon is encoded into the leading exponential and can be
identified for the p = (0, 0, 0) case when the excited states are properly
suppressed. We perform standard effective mass analysis using the forms given

$\displaystyle m_{eff}\bigg{(}t+\frac{1}{2}\bigg{)}$
$\displaystyle=ln\frac{C(t)}{C(t+1)},$ (15) $\displaystyle C(t)$
$\displaystyle=Ze^{-mt}.$ (16)

When the ground state is dominant, the signal obtained from Eq.(15) forms a
plateau. To this end, we seek a plateau to estimate suitable fit ranges for
the one-exponential fit function given in Eq.(16). Then we extract the ground
state masses from the correlation functions. It is possible to take the
contribution of the first excited state as a correction term to Eq.(16) to
pinpoint the ground state precisely; however, we find it to be an excessive
treatment considering the precision and agreement of our results. We calculate
the masses of the baryons created using the Clover and Tsukuba actions
separately. Our results are given in Table 1, together with the experimental
values and those obtained by other lattice collaborations.

Table 1: Extracted $\Xi_{cc}$, and $\Omega_{cc}$ masses as well as those of other lattice collaborations and experimental values. The errors in this work are statistical only, while those quoted by other collaborations correspond to statistical and various systematical errors if given. |  $m_{\Xi_{cc}}$ [GeV] |  $m_{\Omega_{cc}}$ [GeV]
---|---|---
Tsukuba | 3.626 (30) | 3.720 (12)
Clover | 3.627 (32) | 3.726 (12)
Experiment [11] | 3.62140(72)(27)(14) | —
PACS-CS [55] | 3.603(22) | 3.704(17)
ETMC [56] | 3.568(14)(19)(1) | 3.658(11)(16)(50)
Briceno et al. [57] | 3.595(39)(20)(6) | 3.679(40)(17)(5)
Brown et al. [58] | 3.610(23)(22) | 3.738(20)(20)
RQCD [59] | 3.610(21) | 3.713(16)

For the $\Xi_{cc}$ baryon, the mass results using the Clover and Tsukuba
actions agree with those from the LHCb experiment. Therefore, it is seen that
charm quark actions are correctly tuned. The $\Omega_{cc}$ baryon is not
experimentally observed yet; nevertheless, our results agree with the results
obtained by other lattice collaborations. As seen from Table 1, the mass of
the $\Omega_{cc}$ is expected to be between $3.650-3.750$ GeV.

$G_{E}$ gives the electric charge of the baryon at zero transferred momentum,
which can be computed directly with the formulation on the lattice. However,
the formulation restricts to make a direct measurement of the magnetic form
factor at zero momentum $G_{M}(0)$. To this end, we use the following dipole
form to describe the $-q^{2}\equiv Q^{2}$ dependence of the form factors

$G_{E,M}(Q^{2})=\dfrac{G_{E,M}(0)}{\left(1+Q^{2}/\Lambda_{E,M}^{2}\right)^{2}}.$
(17)

We examine the contribution of individual quark sectors to the magnetic
properties to gain a deeper understanding of quark dynamics. The analysis is
performed by coupling the electromagnetic field only to the $u/s$ or the $c$
quark. The baryon form factors are calculated from individual quark
contributions by

$G_{M}(Q^{2})=2\times\frac{2}{3}G_{M}^{c}(Q^{2})+C_{\ell}G_{M}^{\ell}(Q^{2}),$
(18)

where $c_{\ell}=-1/3$ for the $d$, $s$ quark, and $C_{\ell}=2/3$ for the $u$
quark. We combine the individual quark contributions using Eq.(18) for each
momentum transfer $Q^{2}$ and extrapolate the combined form factor values to
$Q^{2}=0$. The light and $c$ quark contributions for the $\Xi_{cc}$ have
opposite signs, for $\Xi_{cc}^{+}$ $d$ and $c$ quark contributions are
multiplied with electric charges of the opposite sign and added
constructively. In contrast, $u$ and $c$ quark contributions cancel each other
and result in noisy data for $\Xi_{cc}^{++}$. Consequently, in this work, we
can only focus on $\Xi_{cc}^{+}$ and $\Omega_{cc}$ magnetic form factors,
$\Xi_{cc}^{++}$ form factor could not be extracted due to the poor signal-to-
noise ratio, and limited number of gauge configurations. Figure 1 displays the
magnetic form factors of $\Xi_{cc}^{+}$ and $\Omega_{cc}$ as functions of
$Q^{2}$. Our results for the magnetic form factors are given in Table 2. As
can be seen from the Figure 1, the dipole form describes the lattice data
successfully. The central values of the magnetic form factors calculated using
the Tsukuba action are approximately $5\%$ greater than those of the Clover
action. A similar pattern has already been observed in our spin-$3/2\to$
spin-$1/2$ radiative transition work [44].

Figure 1: $Q^{2}$ dependence of the magnetic form factors of $\Xi_{cc}^{+}$
and $\Omega_{cc}$.

One can extract the charge radii of the baryons from the slope of the form
factor at $Q^{2}=0$,

$\langle
r_{E,M}^{2}\rangle=-\frac{6}{G_{\text{E,M}}(0)}\frac{d}{dQ^{2}}G_{E,M}(Q^{2})\bigg{|}_{Q^{2}=0}.$
(19)

To evaluate the charge radii with the above formula, we will assume the dipole
form given in Eq.(17) for the form factors leading to the relation,

$\langle r_{E,M}^{2}\rangle=\frac{12}{\Lambda_{E,M}^{2}}.$ (20)

Then the charge radii can be directly calculated using the values of the fit
parameters obtained from the dipole fit to the form factor data. Our results
for the electromagnetic charge radii are given in Table 2. One can compare the
baryons with the same electric charge to figure out the effects of internal
dynamics. It is seen in Table 2 that electric charge radii of $\Xi_{cc}^{+}$
and $\Omega_{cc}$ baryons are very close to each other, and much smaller
compared to proton’s $\langle r_{E}^{2}\rangle^{p}=0.707$ fm2 [60]. When
comparing $\Xi_{cc}^{+}$ and $\Omega_{cc}$, the only difference is that a
light-quark is changed to a strange quark, albeit this change shows that the
strange quark has minimal effect on the electric charge radius.
$\Xi_{cc}^{++}$ has the largest charge radius, as it has two units of electric
charge. When we examine the contribution of individual quarks, it is apparent
that the light-quark contributions are greater than the charm-quarks, and the
main difference comes from the electric charges in the baryon. Overall the
results agree with our previous findings [24, 23], and it can be summarized
from a quark model point of view as the heavy $c$ quark core acts to shift the
center of mass towards itself, reducing the size of the baryon. The results
calculated using different actions differ from each other by $5$ to $10$%, and
the electric charge radii calculated with the Tsukuba action are greater than
those from the Clover action.

Magnetic charge radii of $\Xi_{cc}^{+}$ and $\Omega_{cc}$ have similar
behavior to their electric charge radii and are comparable to each other. The
light-quark contribution is observed to be greater than the charm-quark
contribution. Note that since the magnetic form factor of $\Xi_{cc}^{++}$
could not be extracted due to poor signal-to-noise ratio, we could not compute
its magnetic properties (magnetic charge radius and magnetic moment).

In order to calculate the magnetic moment, first one needs to extract the
$G_{M}$ at zero transferred momentum using the dipole form in Eq.(17), then
magnetic moment can be calculated using the equation below

$\mu_{\cal B}=G_{M}(0)\Big{(}\frac{e}{2m_{\cal
B}}\Big{)}=G_{M}(0)\Big{(}\frac{m_{N}}{m_{\cal B}}\Big{)}\mu_{N},$ (21)

where $m_{N}$ is the physical nucleon mass and $m_{\cal B}$ is the baryon mass
obtained in this work given in Table 1. Our results for the magnetic moments
are given in Table 2.

The Tsukuba and Clover nomenclature in Table 2 applies only to charm quarks.
All light and strange quarks in this work are calculated using the Clover
action. Since all the quarks in the baryon are bound to each other, light
quark results of baryon calculated by the Tsukuba action of the charm quark
propagator differ from the baryon calculated with Clover action.

The magnetic moment of $\Xi_{cc}^{+}$ and $\Omega_{cc}$ are similar. The
light-quark contribution is found to be negative, and the absolute value of
the light-quark contribution is greater than the charm-quark contribution. The
signs of the magnetic moments disclose the interaction of the spins of the
quarks. The opposite signs of the light- and heavy-quark magnetic moments
mainly indicate that their spins are anti-aligned. As given in Table 2, the
light- and heavy-quark magnetic moments are of opposite signs, so by simple
deduction, the charm quarks are paired in a spin-1 state with their spins
aligned, which leads to a significant charm quark contribution to the total
spin and magnetic moment [24].

Table 2: The form factor values extrapolated to $Q^{2}=0$, together with the magnetic moments in units of nuclear magneton. The Tsukuba and Clover nomenclature applies only to charm quarks. All light and strange quarks in this work are calculated using the Clover action. | $G_{M}(0)$ | Magnetic moment [$\mu_{N}$] |  $\langle r^{2}_{E}\rangle$ [fm2] |  $\langle r^{2}_{M}\rangle$ [fm2]
---|---|---|---|---
| Tsukuba | Clover | Tsukuba | Clover | Tsukuba | Clover | Tsukuba | Clover
$\Xi_{cc}^{\ell}$ | -1.395 (181) | -1.331 (189) | -0.384 (47) | -0.336 (45) | 0.443 (37) | 0.468 (51) | 0.441 (41) | 0.324 (38)
$\Xi_{cc}^{c}$ | 0.891 (77) | 0.863 (95) | 0.231 (19) | 0.223 (25) | 0.090 (5) | 0.084 (5) | 0.087 (3) | 0.083 (3)
$\Xi_{cc}^{+}$ | 1.649 (145) | 1.572 (171) | 0.426 (37) | 0.406 (45) | 0.024 (8) | 0.016 (8) | 0.135 (3) | 0.120 (3)
$\Xi_{cc}^{++}$ | — | — | — | — | 0.149 (7) | 0.143 (9) | — | —
$\Omega_{cc}^{s}$ | -1.541 (70) | -1.539 (70) | -0.390 (8) | -0.388 (5) | 0.296 (3) | 0.309 (10) | 0.363 (29) | 0.356 (21)
$\Omega_{cc}^{c}$ | 0.920 (39) | 0.872 (37) | 0.232 (11) | 0.219 (9) | 0.094 (2) | 0.087 (2) | 0.092 (3) | 0.081 (3)
$\Omega_{cc}$ | 1.707 (77) | 1.654 (71) | 0.430 (19) | 0.416 (18) | 0.039 (3) | 0.025 (3) | 0.131 (5) | 0.127 (5)

The ensemble we used in this work has almost physical light-quark masses at
$m_{\pi}\approx 156$ MeV; therefore, extrapolation to the chiral limit is not
strictly necessary. Nevertheless, we combine our findings of Clover action
with our previous results given in Ref. [24] to investigate pion-mass
dependence as we approach the physical point. To obtain the values of the
observables at the chiral point, we perform fits that are linear and quadratic
in $m_{\pi}^{2}$:

$\displaystyle f_{lin}$ $\displaystyle=am_{\pi}^{2}+b$ (22) $\displaystyle
f_{quad}$ $\displaystyle=cm_{\pi}^{4}+dm_{\pi}^{2}+e$ (23)

where a,b,c,d,e are the fit parameters. In order to keep consistency, we only
make chiral extrapolation to observables of $\Xi_{cc}^{+}$ since the hopping
parameter of strange valance quark differs from our previous work [24].

Figure 2: Chiral extrapolations of magnetic moment (left), and magnetic
(middle) and electric (right) charge radii of $\Xi_{cc}^{+}$. Gray data points
are taken from Ref. [24], blue data points are our results calculated with
Clover action, red diamonds and red squares represent the quadratic and linear
fits respectively.

Figure 2 shows the chiral fits of $\Xi_{cc}^{+}$. On the left panel, we show
the magnetic moment as a function of the square of the pion mass in lattice
units. The magnetic moment appears to be more compatible with the linear form
showing almost no quark mass dependence. The figure in the middle shows the
chiral fits of the magnetic charge radius. It is seen that the quark mass
dependence is described by linear form better. In the figure on the right, the
electric radius is given. The errors agree with our previous findings, and
there seems to be mild pion mass dependence.

It is not possible to make a similar extrapolation for $\Omega_{cc}$ as a
different hopping parameter is used compared to Ref [24]. However, a quick
qualitative analysis is applicable; values of the observables increase around
$10-15\%$ due to the strange-quark retuning [44]. If we take this into account
and make a chiral extrapolation, we observe similar quark mass dependence
behavior to Figure 2. These findings suggest that observables of doubly
charmed baryons are mildly dependent on the sea-quark mass.

As a result, one can conclude that the electromagnetic observables we have
calculated near the physical point using the Tsukuba action can be taken as
our final values. These conclusions corroborate the findings of our previous
works [61, 62, 44, 43].

Table 3: Comparison of our results with various other models. All values are given in nuclear magnetons. |  $\mu_{\Omega_{cc}^{+}}~{}$[$\mu_{N}$] |  $\mu_{\Xi_{cc}^{+}}~{}$[$\mu_{N}$]
---|---|---
This Work | 0.430 (19) | 0.426 (37)
Lattice QCD [24] | 0.413 (24) | 0.425 (29)
QCD S.R. [25] | 0.39 (9) | 0.43 (9)
R. Quark Model [27] | 0.72 | 0.86
N. R. Quark Model I [28] | 0.785 | 0.860
N. R. Quark Model II [29] | $0.635^{+0.012}_{-0.015}$ | $0.785^{+0.050}_{-0.030}$
R. T. Quark Model [30] | 0.67 | 0.74
C.C. Quark Model [32] | 0.697 | 0.84
MIT Bag model I [33] | 0.668 | 0.722
MIT Bag model II [34] | 0.645 | 0.719
MIT Bag model III [35] | 0.86 | 0.91
EOMS BHCPT I [36] | 0.40 (3) | 0.37 (2)
EOMS BHCPT II [37] | 0.397 (15) | 0.392 (13)
HB ChPT [39] | 0.41 | 0.62

Electromagnetic properties of doubly charmed baryons have been previously
studied in quark models [26, 27, 28, 29, 30, 31, 32], MIT bag model [33, 34,
35], chiral perturbation theory [36, 37, 38, 39], QCD sum rules [40, 25] and
Lattice QCD [41, 24]. Our final results for the magnetic moments are given in
Table 3 along with a comparison to the literature. The signs of the magnetic
moments are correctly determined. However, there is a discrepancy among the
results. The moments seem to be underestimated with respect to quark models
[27, 28, 29, 30, 32], bag models [33, 34, 35], however, our findings are in
agreement with those obtained from QCD sum rules [25] and extended on-mass-
shell chiral perturbation theory [36, 37]. These findings are also in
agreement with our earlier observations [41, 24].

##  IV Conclusion

The present work has aimed to examine the electromagnetic properties of doubly
charmed baryons from $2+1$-flavor near physical light-quark masses simulations
on $32^{3}\times 64$ lattice. We have extracted the magnetic moments, and the
electric and magnetic charge radii of $\Xi_{cc}$ and $\Omega_{cc}$. We have
also determined individual quark contributions to the observables, which give
valuable insight into the dynamics of the quarks having masses at different
scales. The results of this investigation show that the doubly charmed baryons
are compact in comparison to light baryons. The analysis of quark sector
contributions to the charge radii has shown that light quark distributions are
larger, and the heavy quark decreases the size of the baryon. The light- and
heavy-quark contributions are opposite signs, which indicate that the charm
quarks are paired in a spin-1 state. The magnetic moments seem to be
underestimated compared to quark and bag models, but our findings agree with
those obtained from QCD sum rules and extended on-mass-shell chiral
perturbation theory. Lastly, this study has found that the observables of
doubly charmed baryons are mildly dependent on the sea-quark mass.

## Acknowledgments

The unquenched gauge configurations employed in our analysis were generated by
PACS-CS collaboration [42]. We used a modified version of Chroma software
system [52] along with QUDA [53, 54]. The publicly available configurations
are downloaded via the ILDG/JLDG network [63]. The author thanks G. Erkol, and
K. U. Can from the TRJQCD Collaboration for valuable discussions and their
comments on the manuscript. The author thanks Esra Yüksel for reading early
drafts of the manuscript and comments.

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