Electromagnetic form factors of spin 1/2 doubly charmed baryons

We study the electromagnetic form factors of the doubly charmed baryons, using covariant chiral perturbation theory within the extended on-mass-shell (EOMS) scheme. Vector-meson contributions are also taken into account. We present results for the baryon magnetic moments, charge and magnetic radii. While some of the chiral Lagrangian parameters could be set to values determined in previous works, the available lattice results for $\Xi_{cc}^+$ and $\Omega_{cc}^+$ only allow for robust constraints on the low-energy constant (LEC) combination, $c_{89}(=-\frac{1}{3}c_8+4c_9)$. The couplings of the doubly charmed baryons to the vector mesons have been estimated assuming the Okubo--Zweig--Iizuka (OZI) rule. We also give the expressions for the form factors of the double beauty baryons considering the masses predicted in the framework of quark models. A comparison of our results with those obtained in heavy baryon chiral perturbation theory (HBChPT) at the same chiral order is made.


I. INTRODUCTION
The recent announcement of the observation of the Ξ ++ cc particle by the LHCb collaboration [1] has revived the interest on the physics of doubly heavy baryons. Up to now, the experimental evidence for baryons with two heavy quarks was marginal. Only one cc baryon, Ξ + cc (3520), had been included in the Review of Particle Physics by the PDG [2] and was labeled with one star.
The Ξ + cc (3520) baryon was first observed by the SELEX collaboration in the Λ + c K − π + channel [3] and later corroborated in the pD + K − one [4]. However, neither BABAR [5], nor BELLE [6], nor menta. The corresponding power counting involves some difficulties when baryon loops are included in the calculation and different methods have been developed to overcome this issue such as HBChPT [37], heavy hadron (HH)ChPT, using similar techniques [38,39], and the covariant approaches: infrared [40] and EOMS [41]. All these schemes have been widely and successfully used to investigate the electromagnetic structure of light baryons [36,. There are also some calculations of the electromagnetic properties of baryons with a single heavy quark in HHChPT [39,[72][73][74][75]. Recently, the magnetic moments of doubly heavy baryons with spin 1 2 and 3 2 have been studied in HBChPT [28,76]. Here, we use the covariant EOMS framework instead and we also calculate the electric and magnetic radii of the spin 1 2 triplet. The manifestly Lorentz invariant EOMS scheme has been found to deliver a better chiral convergence than the other schemes for most observables [77] and in particular for the magnetic moment of the light baryons [63,78]. Although the HB techniques are expected to work better the larger the baryon mass is, the differences with the covariant calculation are not negligible. This point is also explored both for di-charm and di-bottom baryons.
Our work is organized as follows. In Section II, the effective Lagrangian describing the interaction of doubly-heavy baryons and Goldstone bosons is given. The form factors of doubly heavy baryons are introduced in Section III and the results are shown in Section IV. Finally, summary and conclusions are given in Section V.

A. Interaction with light pseudoscalar mesons
The effective Lagrangian describing the interaction of double-charm baryons and the Goldstone bosons up to second order was constructed in Refs. [79,80]. It can be written as 1 The relevant pieces of the Lagrangian of order three can be obtained by considering chiral, parity and charge conjugation symmetry. There are two terms contributing to the electromagnetic form factors, The Lagrangians for the double beauty baryons are analogous, only modifying m, their mass in the chiral limit, and the coupling constants. In these equations, U = u 2 , which incorporates the pseudoscalar meson field, is defined as where φ(x) is expressed as The doubly-heavy baryon field ψ with spin 1 2 is a column vector in the flavor space, i.e.
where the subscript Q denotes the charm or beauty quark. In Eqs. (1) and (2), χ, χ ± , f µν , f ± µν , u µ , Γ µ , D µ have the following definitions with A µ the photon field. For the double-charm baryons Q = diag(2, 1, 1), while for the double- The interaction Lagrangian describing the Goldstone-boson interaction with a photon can be extracted from the leading-order meson Lagrangian as follows In Eq. (14),

B. Interaction with vector mesons
It is well known, in the case of light baryons, that the consideration of a pseudoscalar meson cloud plus contact terms, even up to order O(q 4 ), is not sufficient to provide a precise description of the electromagnetic form factors in ChPT [52,53,71]. This is especially true for the Q 2 dependence and thus the charge and magnetic radii. The reason is the importance of the contribution of vector meson mechanisms, see Fig. 1. We expect a similar situation for the case of heavy baryons.
Therefore, in order to model the behavior of the form factors at moderate momentum transfers, the vector-meson contributions are also included. In the case of ideal mixing of the vector-meson singlet and octet, the Lagrangian of the coupling of doubly-heavy baryons to the vector mesons has the following structure According to the OZI rule, Ξ QQ /Ω QQ only couples to (ρ, ω)/φ. Furthermore, given the large breaking of S U(3) symmetry, we take different values for the couplings of Ξ QQ and Ω QQ .
The Lagrangian of the vector-meson coupling to the photon, needed to calculate the contributions of vector mesons to the form factors, is given by [81] Here, In Eq. (17), M V is the mass of the vector meson. F V can be obtained by calculating the decay with α = 1 137 , and C V = 1, 1 3 , − √ 2 3 for ρ, ω and φ, respectively.

A. Definitions
Considering the baryon matrix elements of the electromagnetic vector current, the electromagnetic form factors are defined as where J µ (x) = q e qq (x)γ µ q(x) with q running over the quarks, and B denotes the baryon Ξ ++ cc , The physical mass of the baryon B is given by m B , e q is the charge of the quark q, and F B 1 (q 2 ) and F B 2 (q 2 ) are the Dirac and Pauli form factors. The Dirac spinor of a baryon with four-momentum p µ and mass m is denoted as u(p). The transferred four-momentum q µ = p µ f − p µ i obeys q 2 ≤ 0. The electric and magnetic form factors are defined as Then, the magnetic moment is defined as while the charge and magnetic radii of the baryons can be obtained from the slope of the electric and magnetic form factors For neutral baryons, an exception is made for the electric radius, which reads

B. Calculation of the form factors
In Fig. 2, we show the diagrams derived from the Lagrangians of Eqs. (1), (2), (3) and (15) which contribute to the electromagnetic current matrix element up to order O(q 3 ) in the chiral expansion. We use the standard ChPT definition for the order of a given diagram [33]. The resulting lengthy expressions of the unrenormalized contributions to the Dirac and Pauli form factors F B 1 and F B 2 are given in the Appendix. We renormalize them following the EOMS scheme. As customary, we perform a modified minimal subtraction ( MS) 2 . Later, the terms that still break the nominal power counting, which come from the baryonic loops, are also subtracted. In fact, the only subtraction terms required for the Pauli form factors read where C 4 and C 8 are shown in the Appendix. For the case of the Dirac form factor up to O(q 3 ), the subtraction vanishes exactly due to cancellations between diagrams.
To obtain the final expression, one needs to take into account the wave-function renormalization (WFR) given by Here, the subscript λ denotes π ±,0 , K ±,0 ,K 0 or η, and M λ is the mass of the pseudoscalar meson λ. The C λ,B are shown in Tab. I, and a sum over λ is inferred. The WFR constant only multiplies the O(q) diagrams, since it provides a correction of O(q 2 ). Its effect on other diagrams would only start at O(q 4 ), beyond the order of our calculation. Note that a proper inclusion of the WFR is required to ensure that the total baryon charge F 1 (0) = G E (0) is conserved.
Value of the coefficients C λ,B in Eq. (27).

C. Vector mesons
The contributions to the form factors originating from the coupling to the photon through vector mesons, Fig. 1, are In these equations, B denotes the doubly charmed (beauty) baryons Ξ cc (Ξ bb ) and Ω cc (Ω bb ). The   (28).

IV. RESULTS
For the numerical results presented in this section, we take m Ξ cc = 3.621 GeV [1], m Ω cc = 3.652 GeV [80], F π = 92 MeV, F K = 112 MeV and F η = 110 MeV. Furthermore, we set the renormalization scale in the loop diagrams to 1 GeV, and the chiral limit mass m to the physical baryon mass m B . The coupling of the pseudoscalar mesons to the doubly charmed baryons is fixed at g A = −0.2 [80]. For M π , M K , M η , the nucleon mass m N , the vector meson masses and their widths, we use the averaged PDG values [2].
In order to estimate the relevant low-energy constants c 8 , c 9 , d 1 and d 2 , we use the lattice results from Refs. [30][31][32]. There, the magnetic moments and electromagnetic form factors of Ω + cc and Ξ + cc at different values of Q 2 were obtained for different lattice configurations, and therefore different meson and baryon masses. Since the scale of ChPT is approximately Λ ∼ 1 GeV, we take into account for the fit the lattice results up to Q 2 < 0.4 GeV 2 and M 2 π < 0.4 GeV 2 , meaning a total of 34 data points. For the fit, we set the pion and baryon masses in the ChPT calculations to those given by the lattice collaboration. For the kaon and the η meson masses, not explicit in Refs. [30][31][32], we use the Gell-Mann, Oakes, and Renner relations [33,82] taking into account that the strange quark mass is fixed to its physical value.

A. Magnetic Moments
In Table III, we show our results for the double-charm baryon magnetic moments µ B . The vector mesons do not contribute to this observable. The tree diagram contributions are the same for Ξ + cc and Ω + cc . For the loop terms, we show the analytic expression of the leading-order heavy-baryon expansion, which reproduces the findings of Ref. [28] 3 , and compare the numerical results in HB with the covariant EOMS scheme. We find appreciable differences between the two schemes, especially for Ξ ++ cc .  (24) Up to O(q 3 ), the magnetic moments depend only on c 8 and c 9 and several known parameters. Furthermore, the available lattice data for magnetic moments correspond to the Ω + cc and Ξ + cc baryons. For these two particles, c 8 and c 9 appear just in the combination c 89 = − 1 3 c 8 + 4c 9 . Thus, making a fit to the 13 lattice data for magnetic moments, the only free parameter is c 89 , and we can obtain an estimate for this constant: c 89 = 0.32 (2). 4 The agreement of our µ Ξ + cc and µ Ω + cc results with the simple extrapolations of the lattice data to the physical point done in Refs. [30][31][32] is good considering the uncertainties.
As can be seen, the loop corrections obtained from the relativistic EOMS renormalization are larger than in the HB approach for Ξ + cc and Ω + cc . The main reason for this is that most of the loop diagrams in Fig. 2  In these results, the LEC uncertainties are purely statistical. However, the chiral error estimates, δµ, of the magnetic moments are performed as in Refs. [87,88] and try to account for the systematic error due to the truncation of the chiral series. For our particular case we have where µ (i) are the magnetic moments obtained with our best-fit parameters, up to the order O(p i ).
We show the analogous results for the double-beauty magnetic moments in Table IV. Again, due to the symmetry, the tree-level expressions are the same for the two baryons with the same charge, Ξ − bb and Ω − bb . The only difference between the HB expansions of the charm triplet and the beauty triplet are the baryon masses. We take m Ξ bb = 10.314 GeV and m Ω bb = 10.476 GeV [83] for the numerical calculations. Since the double-beauty baryons are substantially heavier than the double-charm ones, a HB approach is expected to give a better approximation of the full relativistic result. Indeed, for Ξ 0 bb the differences between HB and EOMS results become smaller. However, for the other two baryons the differences are still large, as was the case in the double-charm sector.
In the double-beauty sector, all the magnetic moments are systematically of a smaller magnitude when calculated in EOMS than when determined in a HB approach. 5

B. Electric and magnetic radii
The electric and magnetic radii, r B E,M , measure the derivative with respect to q 2 of the G B E,M form factors. The tree-level results for F 1 and F 2 are shown in Tab. V. Since the LECs c 8 and c 9 appear as the combination c 89 , and d 1 and d 2 as d 12 ≡ − d 1 3 + 4d 2 for both the Ω + cc and the Ξ + cc , if we analyze data for only these latter particles, the number of degrees of freedom from the chiral Lagrangian is reduced to just two.
As it happens for light baryons, we expect the vector mesons to be relevant for these observables. As simplifying conjectures, we assume the OZI rule and ideal mixing, which implies that the vector-meson contributions to the Ξ cc form factors come from ρ and ω and φ is the only vectormeson contributing in the Ω cc case. Still, this amounts to four unknown parameters: g Ξ cc v , g Ξ cc t , g Ω cc v and g Ω cc t . In the magnetic form factors, g Ξ cc v and g Ξ cc t appear as the combination g Ξ cc v − g Ξ cc t ≡ g Ξ cc vt . Therefore, in order to separate them, one needs additionally information on the electric form factors. The same is true for g Ω cc v and g Ω cc t . Since we have lattice results on G calculation at a higher chiral order.
In this case, the error estimates arise mostly due to the large uncertainties of the fitted parameters.
The second number in parentheses corresponds to the uncertainty coming from the chiral truncation calculated as in Eq. (29). These values for the radii support the expectations of the double heavy baryons being substantially smaller than the single heavy ones or the light baryons.
Within their large uncertainties, these radii are compatible with the lattice results from Refs. [30][31][32]. It is worth mentioning, though, that the extrapolation to the physical point with a linear or quadratic fit, as done in Refs. [30][31][32], is not expected to give the correct results at the physical point, since the non-trivial behavior due to chiral loops cannot be taken into account. Specifically, the pion cloud effects, very relevant at low Q 2 values, lead to an unavoidable logarithmic dependence on the pion mass and, therefore, to a rapid curvature of the radii when approaching the physical mass, absent in the extrapolations of Refs. [30][31][32]. radii. We have also compared our results with those extracted in HBChPT. We have found that the differences between the loop term contributions to µ B in HBChPT and EOMS approaches are around 10% ∼ 70% for the double charm sector and 10% ∼ 40% for the double beauty sector.
In order to obtain first estimates for the LECs, we have fitted our model to the available lattice results. We have found that the vector meson contribution is necessary for a good description.