$\Omega_c \gamma \rightarrow\Omega_c^\ast$ transition in lattice QCD

We study the electromagnetic $\Omega_c \gamma \rightarrow\Omega_c^\ast$ transition in 2+1 flavor lattice QCD, which gives access to the dominant decay mode of $\Omega_c^\ast$ baryon. The magnetic dipole and the electric quadrupole transition form factors are computed. The magnetic dipole form factor is found to be mainly determined by the strange quark and the electric quadrupole form factor to be negligibly small, in consistency with the quark model. We also evaluate the helicity amplitudes and the decay rate.

moments, implying that the shapes of N and ∆ deviate from spherical symmetry [21].Quark models predict a nonzero value for E2 and C2 [22], which has also been confirmed experimentally [23,24].However, the results from various theoretical approaches are not in complete agreement with experiment and this issue is still unsettled.
The experimental results for Ω c γ → Ω * c , on the other hand, are not yet precise enough to allow a determination of the transition strengths.In this work, we will mainly focus on the M 1 and E2 transition form factors. Unlike in the case of N γ → ∆, the mass splitting between Ω c and Ω * c can be reproduced on the lattice and an accurate contact can be made to phenomenological observables via these two form factors.We employ near physical 2+1-flavor lattices that correspond to a pion mass of approximately 156 MeV.The data for electromagnetic transition form factors are often noisier than those for elastic form factors, particularly for C2 form factor. Considering also the limited number of gauge configurations we have at the smallest quark mass, we study the M 1 and E2 form factors for the lowest allowed lattice momentum transfer.We, however, make contact with the transition moments at zero-momentum transfer by assuming a simple scaling at low momentum transfer values [12].

II. LATTICE FORMULATION
Electromagnetic transition form factors for Ω c γ → Ω * c can be calculated by considering the baryon matrix elements of the electromagnetic vector current J µ = q 2 3 c(x)γ µ c(x) − 1  3 s(x)γ µ s(x).The matrix element can be written in the following form with the operator O τ µ given in terms of Sachs form factors as [25] O where Here p and p denote the incoming and the outgoing momenta, respectively, q = p − p is the transferred fourmomentum, P = (p + p)/2 and We use the shorthand notation τ µ (P q) = τ µαν P α q ν .The spins are denoted by s, s and the masses of Ω * c and Ω c by m * and m, respectively.u(p, s) is the Dirac spinor and u τ (p, s) is the Rarita-Schwinger spin vector.For real photons, G C2 (0) does not play any role as it is proportional to the longitudinal helicity amplitude.
The Rarita-Schwinger spin sum for the spin-3/2 field in Euclidean space is given by and the Dirac spinor spin sum by We refer the form factors G M 1 , G E2 and G C2 as the magnetic dipole, the electric quadrupole and the electric charge quadrupole transition form factors, respectively.
To extract the form factors we consider the following matrix elements, G with the spin projection matrices defined as Here, α, α are the Dirac indices, σ and τ are the Lorentz indices of the spin-3/2 interpolating field and σ i are the Pauli spin matrices.An initial Ω c state is created at time zero and interacts with the external electromagnetic field at time t 1 .At time t 2 the final Ω * c state is annihilated.The baryon interpolating fields are chosen, similarly to those of ∆ and N as where i, j, k denote the color indices and C = γ 4 γ 2 .It has been shown in Ref. [9] that the interpolating field in Eq. ( 13) has minimal overlap with spin-1/2 states and therefore does not need any spin-3/2 projection.
To extract the form factors, we calculate the following ratio of the two-and three-point functions: In the large Euclidean time limit, t 2 − t 1 and t 1 a, the unknown normalization factors cancel and the time dependence of the correlators can be eliminated.Then the ratio in Eq. ( 15) reduces to the desired form where we can look for a plateau to extract the form factors.We choose the ratio in Eq. ( 15) among several other alternatives used in the literature [12][13][14][15] due to the good plateau region and the quality of the signal it yields.We single out the Sachs form factors by choosing appropriate combinations of Lorentz direction µ and projection matrices Γ.When Ω c is produced at rest and momentum is inserted in one spatial direction, we have [13] where and k and l are two distinct indices running from 1 to 3. When Ω * c is produced at rest, m * = E * in Eqs.(17)(18)(19) and We have run our simulations on gauge configurations generated by PACS-CS collaboration [26] with the nonperturbatively O(a)-improved Wilson quark action and the Iwasaki gauge action.The details of the gauge configurations are given in Table I.The simulations are carried out with near physical u,d sea quarks of hopping parameter κ u,d = 0.13781.This corresponds to a pion mass of approximately 156 MeV [26].The hopping parameter for the sea s quark is fixed to κ s sea = 0.13640 and the hopping parameter for the valence s-quark is taken to be the same.
TABLE I.The details of the gauge configurations used in this work [26].We list the number of flavors (N f ), the lattice spacing (a), the lattice size (L), inverse gauge coupling (β), clover coefficient (cSW ), number of gauge configurations employed and the corresponding pion mass (mπ).For the charm quarks, we apply the Fermilab method [27] in the form employed by the Fermilab Lattice and MILC Collaborations [28,29].A similar approach has been recently used to study charmonium, heavy-light meson resonances and their scattering with pion and kaon [30][31][32].In this simplest form of the Fermilab method, the Clover coefficients c E and c B in the action are set to the tadpole-improved value 1/u 3 0 , where u 0 is the average link.Following the approach in Ref. [30], we estimate u 0 to be the fourth root of the average plaquette.We determine the charm-quark hopping parameter κ c nonperturbatively.To this end, we measure the spin-averaged static masses of charmonium and heavy-light mesons and tune their values accordingly to the experimental results, which yields κ c = 0.1246 [11].
We make our simulations with the lowest allowed lattice momentum transfer q = 2π/N s a, where N s is the spatial dimension of the lattice and a is the lattice spacing.This corresponds to three-momentum squared value of q 2 = 0.183 GeV 2 .In order to access the values of the form factors at Q 2 = −q 2 = 0, we will apply the procedure in Ref. [12] and assume that the momentum-transfer dependence of the transition form factors is the same as the momentum dependence of the Ω * c baryon charge form factor.For instance, the scaling of G M 1 is given by where we consider the scaling of s and c quark sectors separately.The form factors are extracted in two kinematically different cases.In the first case, the Ω * c is produced at rest and the Ω c has momentum −q and in the second case, the Ω c is at rest and Ω * c carries momentum q.In order to increase statistics, we insert positive and negative momentum in one of the spatial directions and make a simultaneous fit over all available data.We also consider current along all spatial directions.The source-sink time separation is fixed to 1.09 fm (t 2 = 12a), which has been shown to be sufficient to avoid excited state contaminations for electromagnetic form factors [11].Using translational symmetry, we have employed multiple source-sink pairs by shifting them 12 lattice units in the temporal direction.All statistical errors are estimated by the single-elimination jackknife analysis.We consider point-split lattice vector current which is conserved by Wilson fermions.
A wall-source/sink method [33] has been employed, which provides a simultaneous extraction of all spin, momentum and projection components of the correlators.The gauge non-invariant wall source/sink requires fixing the gauge.We fix the gauge to Coulomb, which gives a somewhat better coupling to the ground state.The delta function operator is smeared over the three spatial dimensions of the time slice where the source is located, in a gauge-invariant manner using a Gaussian form.In the case of s quark, we choose the smearing parameters so as to give a root-mean-square radius of r l ∼ 0.5 fm.As for the charm quark, we adjust the smearing parameters to obtain r c = r l /3.
FIG. 1.The correlation function ratios Π1 and Π2 in Eq. ( 24) as functions of the current insertion time (t1) for s-and c-quark sectors.We also display G s,c M obtained using Eq. ( 25).The squares (triangles) denote the kinematical case when Ω * c (Ωc) at rest.

III. NUMERICAL RESULTS AND DISCUSSION
We extract the Ω c and Ω * c masses using the two-point correlators in Eqs. ( 9) and (10).Our results for the Ω c and Ω * c masses are given in Table II, together with the experimental values and those obtained by other lattice collaborations.
While we see a few percent discrepancy between our results obtained at a pion mass of m π = 156 MeV and those of PACS-CS obtained at the physical point, the mass splitting m * − m is nicely produced in agreement with experiment.We define the sum of all correlation-function ratios as so that Eq. ( 18) and Eq. ( 19) becomes, Fig. 1 illustrates the Π 1 and Π 2 as functions of the current insertion time, t 1 , for s-and c-quark sectors separately.
The two ratios have opposite sign and they add constructively when they are subtracted.We extract the form factors by fitting the correlation-function ratios by a horizontal line where a plateau develops.We illustrate both kinematical cases giving consistent results within their error bars.A clear plateau can be realized in both kinematical cases, being more flat when Ω c is produced at rest.We fit the correlation function ratios in the range t 1 = [3,6].The statistical errors, on the other hand, are smaller when Ω * c is at rest.The values of the form factors from the two kinematical cases are consistent with each other.
It is straightforward to extract G E2 once we construct the correlation function ratios for G M 1 .The correlation functions have opposite signs and are of similar magnitudes, which result in a vanishing value for G E2 when they are added.We determine G E2 by fitting Π 1 and Π 2 separately and combining the results.This procedure gives consistent results with fitting the sum of the correlation ratios.
Our numerical results are reported in Table III.We give the values of G M 1 and G E2 form factors at the lowest allowed momentum transfer and at zero momentum transfer for the two kinematical cases as explained above.The quark sector contributions to each form factor are given separately.The form factors can be inferred from individual quark contributions by and similarly for G E2 (Q 2 ).The values of the form factors at Q 2 = 0 are extracted using the scaling assumption in Eq. (22).
Similarly to what has been observed in the case of elastic form factors [11], M 1 form factor is dominantly determined by the contribution of the s-quark sector, which is approximately one order of magnitude larger than that of the cquark sector.This pattern is consistent with hyperon transition form factors [12]: The heavier quark contribution is TABLE III.The results for GM1 and GE2 form factors at the lowest allowed four-momentum transfer and at zero momentum transfer for the two kinematical cases.The quark sector contributions to each form factor are given separately.Note that the statistical uncertainty is large in GE2 and results are consistent with zero.systematically smaller than that of the light quarks.From a quark-model point of view, the coupling of the photon to the light quarks prevails in the heavy-quark limit and the heavy quark acts as a spectator.The transition proceeds dominantly through the spin flip of the light degrees of freedom.Our results show that the two quark sectors contribute with opposite signs and yield a value with a statistical error of approximately 5% when combined via Eq.( 27).The values from the two kinematical cases are consistent with each other within their error bars.
In contrast, the extracted values of G E2 at finite and zero momentum transfer are small and consistent with zero within their error bars.A comparison of G M 1 and G E2 reveals that the transition is entirely determined by M 1 transition.In quark model, the quadrupole transition moments arise from the tensor-induced D-state admixtures of the single-quark wavefunctions [22] and the two-quark exchange currents [34,35].In the first, the spins of the quarks remain the same but an S-state quark is changed into a D-state.The latter can be interpreted as the spin flip of a diquark inside the baryon.Given the dependence of the tensor force on the inverse quark mass, one would expect to obtain a smaller G E2 value for heavy baryons as compared to that in the light-baryon sector, in consistency with what we have found.
The Sachs form factors calculated above can be related to phenomenological observables such as the helicity amplitudes and the decay width.The relation between the Sachs form factors extracted in this work and the standard definitions of electromagnetic transition amplitudes f M 1 and f E2 in the rest frame of Ω * c are given by [36,37] where α = 1/137 is the fine structure constant.The helicity amplitudes A 1/2 and A 3/2 can be deduced from the transition amplitudes as follows: Then the decay width is given by [ where we have used the constraint q = (m 2 * − m 2 )/2m * at q 2 = 0.The decay width can also be obtained from the Sachs form factors: Since the above formulas are continuum relations, we use the experimental values of Ω c and Ω * c masses in calculating the helicity amplitudes and the decay width.Our numerical results for the helicity amplitudes in the rest frame of Ω * c and the decay width, at finite and zero momentum transfer, are reported in Table IV.A comparison to the N γ → ∆ transition [1] reveals that, the helicity amplitudes are suppressed roughly by five orders of magnitude due  to diminishing contribution of the heavy quark, the overall reduction in the transition form factors and the larger baryon masses.Since no strong decay channel is kinematically allowed, the total decay rate of Ω * c is almost entirely in terms of the photon decay mode.Eventually a significantly suppressed value of the Ω * c -baryon decay width is yielded, making Ω * c one of the longest living spin-3/2 charmed hadrons.The suppression in the decay width can be mainly attributed to the small Ω * c -Ω c mass splitting.The decay width in Table IV is translated into a lifetime of τ = 1/Γ = 8.901(913) × 10 −18 sec.
The electromagnetic transitions of charmed baryons have also been studied within heavy hadron chiral perturbation theory [16,17] and quark models [18][19][20].It has been found that the charmed baryon electromagnetic decays are suppressed, in qualitative agreement with our result.Of special interest is the Σ * ,+ c → Σ + c γ decay having a similarly small width in the quark model [20].An enhanced width is foreseen in the Σ * + c → Λ + c γ decay, which would be interesting to study on the lattice.The literature on Ω c γ → Ω * c transition is limited.Non-relativistic quark model prediction for Ω * c decay width [18] is one order of magnitude larger than what we have calculated in this work.Note that given the small Ω * c -Ω c mass splitting, such a large width would require a G M 1 value as large as that of N γ → ∆ transition.This cannot be justified as we have found that the heavy-quark contribution diminishes and there is no indication that the light quark contribution is enhanced.
In conclusion, we have computed the Ω c γ → Ω * c transition in lattice QCD.The dominant contribution is due to the magnetic dipole form factor, which we have calculated with a statistical precision of about 5%.The electric quadrupole transition has been found to be negligibly small in consistency with the quark model.We have extracted the helicity amplitudes and the decay width, which have been found to be suppressed.

TABLE IV .
The results for the helicity amplitudes and the decay width in the rest frame of Ω * c .The helicity amplitudes are given at finite and zero momentum transfer.