Radiative decays of the doubly charmed baryons in chiral perturbation theory

We have systematically investigated the spin-$\frac{3}{2}$ to spin-$\frac{1}{2}$ doubly charmed baryon transition magnetic moments to the next-to-next-to-leading order in the heavy baryon chiral perturbation theory (HBChPT). Numerical results of transition magnetic moments and decay widths are presented to the next-to-leading order: $\mu_{\Xi_{cc}^{*++}\rightarrow\Xi_{cc}^{++}}=-2.35\mu_{N}$, $\mu_{\Xi_{cc}^{*+}\rightarrow\Xi_{cc}^{+}}=1.55\mu_{N}$, $\mu_{\Omega_{cc}^{*+}\rightarrow\Omega_{cc}^{+}}=1.54\mu_{N}$, $\Gamma_{\Xi_{cc}^{*++}\rightarrow\Xi_{cc}^{++}}=22.0$ keV, $\Gamma_{\Xi_{cc}^{*+}\rightarrow\Xi_{cc}^{+}}=9.57$ keV, $\Gamma_{\Omega_{cc}^{*+}\rightarrow\Omega_{cc}^{+}}=9.45$ keV.

We have systematically investigated the spin- 3 2 to spin-1 2 doubly charmed baryon transition magnetic moments to the next-to-next-to-leading order in the heavy baryon chiral perturbation theory (HBChPT). Numerical results of transition magnetic moments and decay widths are presented to the next-to-leading order: µ Ξ * ++

I. INTRODUCTION
SELEX Collaboration first reported the possible candidates of the doubly charm baryons [1], which were unfortunately not confirmed by other experimental collaborations like FOCUS [2], BABAR [3] and Belle [4]. Recently, LHCb collaboration discovered Ξ ++ cc in the mass spectrum of Λ + c K − π + π + with the mass M Ξ ++ cc = 3621.40 ± 0.72 ± 0.27 ± 0.14MeV [5]. The mass and decay properties of double heavy charm and bottom baryons have been studied extensively in literature .
As the electromagnetic properties characterize fundamental aspects of the inner structure of baryons, it is very important to investigate the baryon electromagnetic form factors, especially the magnetic moments. Lichtenberg first investigated the magnetic moments of spin- 1 2 doubly charmed baryons with nonrelativistic qurak model in Ref. [50]. Since then, more and more approaches were developed to investigate the magnetic moments of spin- 1 2 doubly charmed baryons [51][52][53][54][55][56][57][58][59][60]. However, as the degenerate partner state of the spin- 1 2 doubly charmed baryons in the heavy quark limit, the spin- 3 2 doubly charmed baryons were rarely studied. The spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moment deserves more attention as it probes the inner structure and possible deformation of both the spin-1 2 and spin- 3 2 doubly charmed baryons. The spin- 3 2 to spin-1 2 baryon transition amplitude has been systematically investigated in Refs. [61,62]. The transition amplitude contains the magnetic dipole (M1), electric quadrupole (E2), and Coulumb quadrupole (C2) contributions with the spin-parity selection rule. For the doubly charmed baryons, the radiative transitions have been studied in the MIT bag model [63,64], SU(4) chiral constituent quark model (χCQM) [65] and manifestly Lorentz covariant constituent three quark model [66].
In this paper, we focus on the transition magnetic moment of the doubly charmed baryons in the framework of chiral perturbation theory (ChPT) [67]. ChPT is a very useful framework in the low-energy hadron physics and was first developed to deal with the pseudoscalar meson system. When extended to include the baryons, heavy baryon chiral perturbation theory (HBChPT) was proposed [68][69][70][71].
In this work, we will employ HBChPT to calculate the the one-loop chiral corrections to the spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments order by order. We use quark model to estimate some low energy constants (LEC) in ChPT including the leading-order axial coupling and tree level transition magnetic moments. We explicitly consider the mass splitting δ between spin-3 2 and spin-1 2 doubly charmed baryons. We present our numerical results up to the next-to-leading order while we derive the analytical results to the next-to-next-to-leading order.
This paper is organized as follows. In Section II, we discuss the spin-3 2 to spin-1 2 baryon electromagnetic transition form factors. We introduce the effective chiral Lagrangians of the doubly charmed baryons in Section III. In Section IV, we calculate the transition magnetic moments order by order. We estimate the low-energy constants and present our numerical results in Section V. We give a short summary in Section VI and collect some useful formulae and the coefficients of the loop corrections in the Appendix A.
With the constraint of Lorentz covariance, gauge invariance, parity conservation and time reversal invariance, the matrix element of electromagnetic current between spin-3 2 and spin-1 2 baryon states can be written as [62,74]: with Here we use B to denote the spin-1 2 baryons and T to denote spin-3 2 baryons. p and p ′ are the corresponding momenta of the baryons. In the above equations, P = 1 2 (p ′ + p), q = p ′ − p, M B and M T are the corresponding baryon masses, and u ρ (p) is the Rarita-Schwinger spinor satisfying p ρ u ρ (p) = 0 and γ ρ u ρ (p) = 0 for an on-shell heavy baryon.
In the framework of HBChPT, the baryon field B can be decomposed into the large component N and the small component H.
where v µ = (1, 0) is the velocity of the baryon. For the Rarita-Schwinger field, the large component of spin-3 2 degrees of freedom is denoted as T ρ . Now the spin-3 2 and spin-1 2 matrix elements of the electromagnetic current J µ can be parameterized as The tensor O ρµ can be parameterized in terms of two Lorentz invariant form factors.
In this paper, we will use Eq. (6) to define the electro quadrupole (E2) and magnetic-dipole (M1) multipole transtion form factors. The multipole form factors are where |q| = δ in the rest frame of spin-3 2 baryon. The decay width and transition magnetic moment are expressed as where α = e 2 4π = 1 137 is the electromagnetic fine structure constant.

A. The strong interaction chiral Lagrangians
We follow the basic definitions of the pseudoscalar mesons and the spin-1 2 doubly charmed baryon chiral effective Lagrangians in Refs. [60,71]. The pseudoscalar meson fields are defined as: The chiral connection and axial vector field are defined as [71]: where We use the pseudoscalar meson decay constants f π ≈ 92.4 MeV, f K ≈ 113 MeV and f η ≈ 116 MeV. For the spin-3 2 doubly charmed baryon field, we adopt the Rarita-Schwinger field [72].
The leading order pseudoscalar meson and baryon interaction Lagrangians read where M H is the spin-1 2 doubly charmed baryon mass, M T is the spin-3 2 doubly charmed baryon mass, We also need the second order pseudoscalar meson and doubly charmed baryon interaction Lagrangians. Recall that When the product of u µ and u ν belongs to the 8 1 and 8 2 flavor representation, we can write down two independent interaction terms of the second order pseudoscalar meson and doubly charmed baryon Lagrangians: where M B is the nucleon mass and h 1,2 are the coupling constants.
In the framework of HBChPT, we denote the large component of the spin-3 2 doubly charmed baryon as T µ . The leading order nonrelativistic pseudoscalar meson and baryon Lagrangians read where S µ is the covariant spin-operator, δ = M T −M H is the spin-1 2 and spin-3 2 doubly charmed baryon mass splitting. We adopt δ = 0.1 GeV approximatively [25]. The φHH couplingg A = − 2 5 g A = −0.50 [60]. With the help of quark model, we have estimated the φT T couplingg B = − The above Lagrangians contribute to the spin- Following the definitions in Ref. [60], the lowest order O(p 2 ) Lagrangians contribute to the magnetic moments of the spin- 1 2 doubly charmed baryons at the tree level read where the operatorF + µν = F + µν − 1 3 Tr(F + µν ) is traceless and transforms as the adjoint representation. We can also write the lowest order O(p 2 ) Lagrangians which contribute to the magnetic moments of the spin-3 2 doubly charmed baryons and the spin-3 2 to spin-1 2 transition magnetic moments,

C. The higher order electromagnetic chiral Lagrangians
We also need the O(p 4 ) electromagnetic chiral Lagrangians at the tree level to calculate the magnetic moments to O(p 3 ). Recalling flavor representation in Eqs. (19), (20) and considering that we only need the leading-order terms of the fields F + µν and χ + which are diagonal matrices, only three independent terms contribute to the magnetic moments of the doubly charmed baryons up to O(p 3 ), where χ + =diag(0,0,1) at the leading order and the factor m s has been absorbed in the LECs d 1,2,3 . There are two more terms which also contribute to the doubly charmed baryon magnetic moments.

IV. FORMALISM UP TO ONE-LOOP LEVEL
Considering the standard power counting scheme of HBChPT, the chiral order D χ of a given diagram [73] where N L is the number of loops, I M is the number of internal meson lines, I B is the number of internal baryon lines and N n is the number of the vertices from the nth order Lagrangians. Thus, the chiral order of the transition magnetic moments is (D χ − 1). The tree-level Lagrangians in Eqs. (27), (28) contribute to the transition magnetic moments at O(p 1 ) and O(p 3 ) as shown in Fig. 1. The Clebsch-Gordan coefficients for the various processes are collected in Table I. All tree level transition magnetic moments are given in terms of a 5 , a 6 , d 1 , d 2 and d 3 .
There are sixteen Feynman diagrams contribute to the spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments at one-loop level as shown in Fig. 2. In diagrams (a-b), the photon vertex is from the meson photon interaction term while the meson vertex is from the interaction terms in Eq. (23). In diagram (c), the photonmeson-baryon vertex is from the high-order expansion of the O(p 2 ) tree level transition magnetic moment interaction in Eq. (27). In diagram (d), the meson-baryon vertex is from the second order pseudoscalar meson and baryon Lagrangian in Eq. (24) while the photon vertex is also from the meson photon interaction term. In diagrams (eh), the photon-baryon vertex from the O(p 2 ) tree level magnetic moment interaction in Eqs. (25), (26), (27) while the meson vertex is from the interaction terms in Eq. (23). In diagrams (i-l), the two vertices are from the strong interaction and seagull terms respectively. In diagrams (m-p), the meson vertex is from the strong interaction terms while the photon vertex from the O(p 2 ) tree level spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moment interaction in Eq. (27).
The diagrams (a) and (b) contribute to the transition magnetic moment at O(p 2 ) while the diagrams (c-p) contributes at O(p 3 ). The diagrams (i-l) vanish in the heavy baryon mass limit for the same reason as in Ref. [74]. The diagrams (m-p) are the wave function renormalization corrections.
Summing all the one-loop level contributions in Fig. 2, the loop corrections to the spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments can be expressed as where λ = 4πf π is the renormalization scale. The coefficients β φ a and β φ b arise from the spin-3 2 and spin-1 2 doubly charmed baryon intermediate states respectively. γ φ c ,γ φ d ,γ φ e , γ φ f , γ φ T H and γ φ w arise from the corresponding diagrams in Fig. 2. We collect their explicit expressions in Tables VI, VII, VIII in the Appendix A.
With the low energy counter terms and loop contributions (33,34), we obtain the transition magnetic moments, where µ T H and µ (3,tree) T H are the tree-level magnetic moments from Eqs. (27)- (28). to spin-1 2 doubly charmed baryon transition magnetic moments to the next-to-leading order (in unit of µN ). Process

V. NUMERICAL RESULTS AND DISCUSSIONS
There are not any experimental information on the spin-3 2 to spin-1 2 doubly charmed baryon transitions yet. Thus, we adopt the same strategy as in Ref. [60]. We use the quark model to estimate the leading-order tree level transition magnetic moments. At the leading order O(p 1 ), as the charge matrix Q H is not traceless, there are two unknown LECs a 5,6 in Eq. (27). Notice the second column in Table I, the a 5 parts are proportional to the light quark charge within the doubly charmed baryon and the a 6 parts are the same for all three transition processes.
At the quark level, the flavor and spin wave functions of the spin-1 2 and spin-3 2 doubly charmed baryons Ξ ccq and Ξ * ccq read: where the arrows denote the third-components of the spin s 3 . q can be u,d and s quark. The spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments in the quark model are the matrix elements of the following operator between Eq. (45) and Eq. (46), where µ i is the magnetic moment of the quark: We adopt the m u = m d = 336 MeV, m s = 540 MeV, m c = 1660 MeV as the constituent quark masses and give the results in the second column in Table II. The light quark magnetic moments contributes to the LEC a 5 , which is proportional to the light quark charge. The heavy quark magnetic moments contributes to the LEC a 6 , which are the same for all three transitions. Up to O(p 2 ), we need include both the leading tree-level magnetic moments and the O(p 2 ) loop corrections. We use the quark model to estimate the leading-order tree level transition magnetic moments. Thus, there exist only three LECs:g A ,g B andg C at this order.g A = − 2 5 g A = −0.50 has been estimated in Ref. [60]. After similar calculations, one obtains the φT T couplingg B = − Process 17 52 we obtain the numerical results of the O(p 2 ) spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments in Table II. As the E2 transtion moments are much smaller than M1 transtion moments, using Eq. (9) we can also calculate the decay width of the spin-3 2 to spin-1 2 doubly charmed baryon transitions in Table III. Up to O(p 3 ), there are ten unknown LECs: a 1−6 , h 1 , d 1,2,3 . It is impossible to to fix all these LECs with the present experimental data. We present our analytical results in Eqs. (33)- (34) and Table I. We also calculate the spin-3 2 to spin-1 2 transition magnetic moments for the doubly bottomed baryons and charmed bottomed baryons. The results are given in Table IV. For the charmed bottomed baryons, we use the subscript {bc} and [bc] to label the systems with symmetric and antisymmetry spin wave functions in the heavy quark sector, respectively. It is very interesting to note that the light quark does not contribute to the ({bc}q) * → ([bc]q) transition magnetic moments at the tree-level in the quark model. Meanwhile, for these processes, the O(p 2 ) loop contribution also vanishes for the vanishing axial coupling constants. In the quark model, the pions only couple to the light quark sector of the heavy baryons. However, the heavy quark spin wave functions of the ({bc}q) * and ([bc]q) systems are orthogonal to each other. Thus, the pionic couplings vanish.

VI. CONCLUSIONS
The recently observation of Ξ ++ cc has aroused tremendous attention to the doubly charmed baryons. The chiral dynamics of the doubly charmed baryons is simpler than the nucleon, which allows one to study the chiral dynamics of the light quarks more directly. Moreover, the spin-3 2 to spin-1 2 doubly charmed baryon transition electromagnetic properties probe the inner structure and possible deformation of both the spin-1 2 and spin-3 2 doubly charmed baryons. In this paper, we have systematically calculated the chiral corrections to the spin-3 2 to spin-1 2 doubly charmed baryon transition magnetic moments up to the next-to-next-to-leading order in the framework of the heavy baryon chiral perturbation theory. In principle, the low energy constants at the tree-level should be extracted through fitting  to spin-1 2 doubly charmed baryon transition magnetic moments in literature including nonrelativistic quark model with effective quark mass (SCR) [35], SU(4) chiral constituent quark model (χCQM) [65], MIT bag model [64] and nonrelativistic quark model(NRQM) (in unit of µN ).
Process β to the experimental data. However, the current experimental information is very poor. With the help of quark model, we have estimated the LECs such as the leading-order axial coupling and tree level transition magnetic moments. Because of lack of enough experimental inputs, we only present the numerical results up to next-to-leading order: µ Ξ * ++ cc →Ξ ++ cc = −2.35µ N , µ Ξ * + cc →Ξ + cc = 1.55µ N , µ Ω * + cc →Ω + cc = 1.54µ N . In Table V, we compare our results in HBChPT with those from other approaches such as nonrelativistic quark model with effective quark mass (ENRQM) [35], SU(4) chiral constituent quark model (χCQM) [65], MIT bag model [64] and nonrelativistic quark model(NRQM).
We also calculate the decay width of the doubly charmed baryon transitions: Γ Ξ * ++ cc →Ξ ++ cc = 22.0 keV, Γ Ξ * + cc →Ξ + cc = 9.57 keV, Γ Ω * + cc →Ω + cc = 9.45 keV. As we use the quark model to estimate the leading-order tree level transition magnetic moments, the final transition magnetic moments and decay width may be enhanced to some extent.
We hope our results may be useful for future experimental measurement of the transition magnetic moments. Moreover, the analytical expressions derived in this work may be useful to the possible chiral extrapolation of the lattice simulations of the double charmed baryon transition electromagnetic properties in the coming future.