Radiative decays of the heavy-quark-spin molecular partner of T + cc

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(Dated: November 30, 2023) With the assumptions that the T + cc discovered at LHCb is a D * D hadronic molecule, using a nonrelativistic effective field theory we calculate the radiative partial widths of T * cc → D * Dγ with T * cc being a D * D * shallow bound state and the heavy-quark-spin partner of T + cc .The I = 0 D * D rescattering effect with the T cc pole is taken into account.The results show that the isoscalar D * D rescattering can increase the tree-level decay width of T * + cc → D * + D 0 γ by about 50% and decrease that of T * + cc → D * 0 D + γ by a similar amount.The two-body partial decay widths of the T * + cc into T + cc γ and T + cc π 0 are also calculated, and the results are about 6 and 3 keV, respectively.Considering that the D * needs to be reconstructed from the Dπ or Dγ final state in an experimental measurement, the four-body partial widths of the T * + cc into DDγγ and DDπγ are explicitly calculated, and we find that the interference effect between different intermediate D * Dγ states is small.The total radiative decay width of the T * cc is predicted to be about 24 keV.Adding the hadronic decay widths of T * cc → D * Dπ, the total width of the T * cc is finally predicted to be (65 ± 2) keV.

I. INTRODUCTION
The LHCb Collaboration has reported a narrow resonance, the double-charm exotic candidate T cc with probable quantum numbers I(J P ) = 0(1 + ), in the D 0 D 0 π + invariant mass distribution [1,2].Its mass and decay width were reported as [1,2] δm BW = m BW − (m D * + + m D 0 ) = −273 ± 61 ± 5 +11 −14 keV, parametrizing the T + cc using a relativistic P -wave two-body Breit-Wigner function with a Blatt-Weisskopf form factor, and using a unitarized Breit-Wigner profile [2].An analysis of the LHCb data with the full DDπ three-body effects taken into account gives [3] δm pole = −356 +39 −38 keV, Γ pole = (56 ± 2) keV. ( By analyzing the line shape of the T cc or the low-energy S-wave DD * scattering parameters [3][4][5][6][7], it has been concluded that the T cc is an excellent candidate of a DD * hadronic molecule [8][9][10][11][12][13].It was predicted to have a heavy-quark-spin symmetry (HQSS) partner T * cc , a D * D * hadronic molecule with the quantum numbers I(J P ) = 0(1 + ) [3,4].In particular, the mass of the T * cc relative to the D * D * threshold is predicted to be B = 2m D * − m T * cc = (503 ± 40) keV in Ref. [3], which is called the binding energy of the T * cc in the following.Precise knowledge of the T * cc decay width is valuable for its searching in experiments, and it can be calculated in a nonrelativistic effective field theory called XEFT.
The XEFT is a nonrelativistic effective field theory which was first constructed to systematically study the long-range properties of the exotic X(3872) [14,15], also known as χ c1 (3872) with a mass coinciding with the D 0 D * 0 threshold.The D, D * , D, D * , and pions are the effective degrees of freedom in XEFT and are all treated nonrelativistically [16].The partial decay widths of the T cc , including T cc → DDπ and T cc → DDγ, are calculated using XEFT in Ref. [17]; the result of the total width of T cc about 58 keV is in good agreement with Eq. (3) extracted in Ref. [3].In Ref. [18], the next-to-leading-order (NLO) contributions to the strong decay width of the T cc are calculated including the contributions from one-pion exchange and final-state interaction (FSI).
In Ref. [19], we have calculated the hadronic partial widths of the spin partner T * cc decaying into D * Dπ(→ DDππ) including contributions of the D * D and D * π FSIs.The total hadronic decay width of the T * cc is predicted to be about 41 keV.The T * cc can also decay radiatively into the D * Dγ (subsequently to DDπγ and DDγγ) final states.In this work, we compute the partial widths of such radiative decays and will give the total width of the T * cc by summing up the hadronic and radiative partial widths.This paper is organized as follows.In Sec.II, we introduce the XEFT effective Lagrangian for the charmed mesons, photons, and pions and the power counting of the Feynman diagrams for where m T * cc , m D , m D * , and m π are the masses of T * cc , D, D * , and π, respectively.Then the momentum of the emitted pion in the rest frame of the T * cc is p π ≲ 110 MeV.Despite that, the pion in triangle loops (involving the D * π rescattering) will still be treated nonrelativistically, while the phase spaces of the decays are treated relativistically.Since such diagrams provide only a small correction (to be calculated later), this simplification presents a good approximation.
The XEFT Lagrangian we use for the decays of T * cc reads [16,20] with the pseudoscalar H = (D 0 , D + ) T , vector heavy mesons H * = (D * 0 , D * + ) T , the magnetic field B k = ϵ ijk ∂ i A j , and the pions Here m H , m H * , and m π are the masses of the H, H * , and π particles, respectively; δ = ∆ − m π ≃ 7 MeV with ∆ = m D * 0 − m D 0 comes from a field redefinition shifting the residual D * mass from the D * kinetic term to the pion kinetic term [16], and it introduces a small momentum scale MeV appearing in the pion propagator [16,21]; the pion decay constant is taken as F π = 92.2MeV, and τ a with a = 1, 2, 3 are the Pauli matrices in the isospin space in which the traces (⟨ ⟩) act.
The first line in Eq. ( 5) includes the kinetic terms for the charmed mesons and pions.The second line contains the contact interactions of the D * + and D * 0 , where the first term mediates the D * D * scattering in the isoscalar channel and the second term mediates the scattering in the isovector channel.In the third line, the first term describes the coupling between the charmed mesons and a pion, with the coupling constant ḡ ≃ 0.27; 1 the second term gives the magnetic couplings for the charmed mesons and a photon [17,22,23] with the matrix of transition magnetic where fm obtained in the analysis in Ref. [3].Here the isospin-breaking effect, which is a higher-order effect [27], is neglected in the isoscalar D * D → D * D rescattering.
We also ignore the isovector DD * FSI, which is much weaker than the isoscalar one, since there is no evidence for an isovector double-charm tetraquark near the D * D threshold.The last two terms with C 1 2 π and C 3 2 π in Eq. ( 5) are the D * π → D * π contact interactions for I = 1/2 and I = 3/2, respectively, and the coefficients C 1 2 π = 25.2GeV −1 and C 3 2 π = −6.8GeV −1 are derived by matching to the D * π scattering lengths, which should be approximately equal to the Dπ ones in Ref. [28] (for detailed derivations, see Appendix A in Ref. [19]) due to HQSS.
The effective Lagrangian for the T * cc coupling to D * D * in S wave can be written as where ε ijk is the three-dimensional antisymmetric Levi-Civita tensor.The effective coupling g s can be derived from the residue of the D * + D * 0 → D * + D * 0 scattering amplitude at the T * + cc pole as [20,29,30] With the above Lagrangians in Eqs. ( 5) and ( 8), the leading-order (LO) amplitude for the In the following, we will briefly introduce the power counting of all these diagrams in Fig. 2 following the analysis for the decays of the X(3872) and T cc in Refs.[16,21,31].The relevant small momenta involved in the decays of the T * cc are {p D , p D * , q γ , γ, µ}, where q γ is momentum of the emitted photon.They are of the same order and denoted generically by Q.Each nonrelativistic propagator is of O(Q −2 ), and, as the nonrelativistic energy is of O(Q 2 ), each nonrelativistic loop integral measure counts as O(Q 5 ).The isoscalar contact interaction C 0D between the D * and D is replaced with T DD * in Eq. ( 7) and, thus, contributes at O(Q −1 ) [16,20].For the diagrams in In this section, all the decay amplitudes of the T * + cc → D * + D 0 γ and T * + cc → D * 0 D + γ processes in Fig. 2 are given, as well as expressions of their partial differential decay rates.The Breit-Wigner form of the D * propagator, G D * (p), is used to include the contribution of the D * self-energy, i.e., where 83.4 keV [15], and Γ D * 0 = 55.3 keV [25].
A. Partial decay rate of First, we consider the three-body decay T * + cc → D * + D 0 γ.The LO amplitude from the tree diagram in Fig. 2(a) reads where ⃗ p D * + is the three-momentum of the external D * + , ⃗ q γ is the three-momentum of the final state γ, and ϵ i (T * + cc ), ϵ j * (D * + ), and ϵ j(i) * (γ) are the polarization vectors of the incoming particle T * + cc and the outcoming particles D * + and γ, respectively.
The LO amplitudes from the D * D rescattering diagrams in Figs.2(b) and 2(c) read where 0D are the contact terms for the D * + D 0 → D * + D 0 and the D * 0 D + → D * + D 0 , respectively, and I b (q γ ) and I c (q γ ) are the three-point scalar loop integrals, whose explicit expressions can be obtained from I(q) given in Appendix A [21,32] as follows: m 1 , m 2 and m 3 in Eq. (A1) are taken to be the masses of D * 0 , D * + and D 0 for I b (q γ ), and the masses of D * + , D * 0 and D + for I c (q γ ), respectively.
The decay rate is given by where the overall factor comes from the normalization of nonrelativistic particles, with M being the mass of the T * cc , E 1 and E 2 being the energies of two nonrelativistic final-state particles in the T * cc rest frame, respectively, j = 1 is the spin of the T * cc , and there is a sum over the polarizations of spin-1 particles.Here the three-body phase space integration is given by where ⃗ p 1 and ⃗ p 2 are the three-momenta for two of the final-state particles in the rest frame of the initial particle.
The LO partial differential rate for the For the decay T * + cc → D * 0 D + γ, the LO amplitude from the tree diagram in Fig. 2(d) reads where ⃗ p D * 0 and ϵ j (D * 0 ) are the three-momentum and polarization vector of the external D * 0 , respectively.The LO amplitudes from the D * D rescattering diagrams in Figs.2(e) and 2(f) are where The LO partial differential rate for the where the coupling constants g 0 = 1.03/ 2m T + cc and g + = −0.99/2m T + cc are taken from the analysis of scheme III in Ref. [3]. 2or the decay T * + cc → T + cc γ, using the effective Lagrangians in Eqs. ( 5) and ( 21), the amplitudes in Figs.3(a) and 3(b) are where ⃗ q γ is the three-momentum of the external photon and the ϵ j * (T + cc ) is the polarization vector of the T + cc .
The differential decay width is given by where the two-body phase space is where |⃗ p 1 | is the magnitude of the three-momentum of particle 1 in the rest frame of the initial state, dΩ 1 = d cos θ 1 dφ 1 is the solid angle of particle 1, and E is the energy of the initial-state particle in the same reference frame.
The partial decay width for For the isospin-breaking decay T * + cc → T + cc π 0 , the amplitudes from the diagrams in Figs.3(c) and 3(d) read where ⃗ p π 0 is the three-momentum of the external π 0 .The partial width of This decay breaks isospin symmetry; thus, if we use the isospin-averaged masses for all the involved mesons, the contributions in Figs.3(c) and 3(d) would vanish.cc with a binding energy B = (503 ± 40) keV.Γ Tree contains the contributions from the tree-level diagrams, and Γ LO is the LO decay width which includes the contributions from the tree-level and D * D rescattering diagrams.The errors come from that of the binding energy B predicted in Ref. [3].
In this section, we present the partial decay widths for the decays T * cc → D * Dγ, T cc γ, and T cc π.In Table I I, and the partial widths with the binding energy varying from 0.01 to 0.80 MeV are shown in Fig. 6.Here, we do not consider the unknown correlations between the binding energies of the T cc and T * cc .
Combining the hadronic and radiative decay widths of the T + cc calculated in Ref. [17] and the In Ref. [19], the obtained hadronic decay width of the T * cc is about (41 ± 2) keV, so the predicted total width of the which is larger than that of the T cc and can be regarded as the main result of our work.decay.The four-body phase space in Eq. ( 32) reads (for details of the derivations, see Refs.[19,33]) where s 12 = (p 1 + p 2 ) 2 , s 34 = (p 3 + p 4 ) 2 , ⃗ q = (|⃗ q|, Ω) is the three-momentum of the (1, 2) particle system in the rest frame of the initial particle is the three-momentum of particle 1 in the c.m. frame of the (1, 2) particle system, and is the three-momentum of particle 3 in the c.m. frame of the (3, 4) particle system.The magnitudes of the three-momenta are given by with λ(x, y, z) ≡ x 2 + y 2 + z 2 − 2(xy + xz + yz) being the Källén triangle function.
The differential decay rate for the T * + cc → DDγγ at LO including the D * D FSI reads where "1," "2," "3," and "4" denote the γ, D 0 , γ, and D + particles, respectively, p 0 2 and p 0 4 are the energies of the D 0 and D + mesons in the T * cc rest frame, respectively, ⃗ p * 1 and ⃗ p ′ 3 are the where "1," "2," "3," and "4" denote the γ, D 0 , π 0 (π + ) and D + (D 0 ) particles, respectively, for the T * + cc → D + D 0 γπ 0 (D 0 D 0 γπ + ) decay, p0  ratios in Fig. 13.One can see that the difference between the decay widths with and without the interference between the intermediate three-body D * Dγ states is marginal for the T * cc binding energy larger than 200 keV, and the binding energy (503 ± 40) keV predicted in Ref. [3] is within this region.

VII. SUMMARY
In this work, we calculate the radiative partial decay widths of T * cc → D * + D 0 γ/D * 0 D + γ taking into account the D * D rescattering contributions where the T * + cc is an isoscalar 1 + D * + D * 0 shallow bound state and the spin partner of the T cc (3875).We found that the I = 0 D * + D 0 /D * 0 D + rescattering, which generates a T + cc pole just below the threshold, contributes at LO and has a sizable constructive contribution to the partial width of the T * + cc → D * + D 0 γ and destructive influence on the T * + cc → D * 0 D + γ.The two-body partial decay widths of the T * + cc → T + cc γ and T + cc π 0 are calculated to be about 6 and 3 keV, respectively.Since the D * further decays into the Dγ and Dπ final states, we also calculate the four-body decay widths of T * + cc → DDγγ and DDγπ, and find that the interference effect between different intermediate D * Dγ and D * Dπ states is small.Thus, the T * cc radiative decay width can be well approximated by summing over the D * Dγ partial widths for the T * cc binding energy larger than 200 keV.Taking the binding energy (503 ± 40) keV predicted in Ref. [3], the obtained T * cc radiative decay width is about 24 keV.Adding the hadronic decay width 41 keV calculated in Ref. [19], the total width of the T * cc is about 65 keV.The results calculated here should be useful for searching the T * + cc state at LHCb and testing the molecular nature of the T cc in the future. where and the two-point function B(c) in the power divergence subtraction (PDS) scheme [34] reads with Λ PDS a scale in the PDS scheme.
The width of the unstable D * can be included by considering the D * self-energy contribution shown in Eq. ( 10) by the following replacement: with s 12 = q 2 , s 34 = k 2 , and s ij = (p i + p j ) 2 , i, j = 1, ..., 4. Here, p j 1 and p k 3 are the three-momenta of the two photons in the final state in the T * cc rest frame, respectively, and q µ = (q 0 , ⃗ q), k µ = (k 0 , ⃗ k) are the four-momenta of the (1, 2) and (3, 4) two-particle systems in the T * cc rest frame, respectively.Considering the crossed-channel effects of the two identical photons in the final state, we also have where s 23 = l 2 , s 14 = t 2 , and l µ = (l 0 , ⃗ l), t µ = (t 0 , ⃗ t) are the four-momenta of the (2, 3) and (1,4) two-particle systems in the T * cc rest frame, respectively.For the decay T * + cc → D + D 0 γπ 0 , the LO amplitude from the tree diagram in Fig. 9(a) reads The LO amplitudes from the D * + D 0 /D * 0 D + rescattering diagrams are ) the T * cc → D * Dγ processes.The amplitudes and partial decay rates of the T * cc → D * Dγ including contributions from the D * D FSIs are derived in Sec.III.The amplitudes and partial decay rates of the T * cc → T cc γ and T * cc → T cc π are derived in Sec.IV, and the numerical results for the partial decay widths of the T * + cc are shown in Sec.V.The four-body decays T * cc → DDγγ and DDγπ including the corrections from the D * D and D * π FSIs are discussed in Sec.VI.Finally, all the results are summarized in Sec.VII.Some expressions are relegated to the Appendixes.II.EFFECTIVE LAGRANGIAN AND POWER COUNTING In this section, the effective Lagrangian and the power counting rules of the diagrams for the decays of the T * + cc are introduced.For the T * + cc being an S-wave isoscalar D * D * shallow bound state with quantum numbers J P = 1 + and a binding energy B = (503 ± 40) keV [3], the typical momentum and velocity of the D * mesons in T * + cc are p D * ≃ γ D * D * ≡ √ 2µ D * D * B ≲ 33 MeV and v D * ≃ B/(2µ D * D * ) ≲ 0.02, respectively, where µ D * D * is the reduced mass of D * + and D * 0 .Therefore, the D * and D mesons can be treated nonrelativistically in the decays of T * cc → D * Dγ, DDγγ, and DDγπ.The maximal energy of the emitted pion in the T * cc → DDγπ decays is

FIG. 1 .
FIG. 1. Resumming the D * D rescattering diagrams.The single thin lines represent the D + (D 0 ), the double lines represent the D * 0 (D * + ), and the wavy lines represent the photon.
where µ D + = −0.15GeV −1 and µ D 0 = 0.55 GeV −1 are obtained by reproducing the partial widths Γ[D * + → D + γ] = 1.33 keV[24] and Γ[D * 0 → D 0 γ] = 19.52 keV[25]; the last term is the isoscalar contact interaction for D * D → D * D. Because of the existence of the T + cc , the resummation effect shown in Fig.1needs to be considered[21] by replacing C 0D with the near-threshold T matrix for the isoscalar D * D → D * D[26] DD * is the reduced mass of D * and D, p = |⃗ p D * − ⃗ p D |/2 is the relative momentum between D * and D in the D * D center-of-mass (c.m.) frame, and the D * D scattering length a is set to be 1 Here, the ḡ parameter is related to the g parameter in Ref. [17] by ḡ = g/2.a = − 6.72 +0.36 −0.45 − i 0.10 +0.03 −0.03 Dγ including the effects of the D * D FSIs is shown in Fig. 2, where Figs.2(a)-2(c) are the diagrams for the decay T * + cc → D * + D 0 γ and Figs.2(d)-2(f) are for the decay T * + cc → D * 0 D + γ.

FIG. 3 .
FIG. 3. Feynman diagrams for calculating the partial decay widths of T * + cc → T + cc γ and T * + cc → T + cc π 0 .The double lines represent the spin-1 mesons, T * + cc , T + cc , D * 0 and D * + ; the single thin lines represent the pseudoscalar charmed mesons, D + and D 0 ; the wavy lines represent the photon; and the dashed lines represent the pion.
, we list the decay widths with the binding energy of the T * cc being B = (503 ± 40) keV[3].The second column denoted by Γ Tree lists the decay widths including only the contributions from the tree-level diagrams, and the third column marked by Γ LO lists the LO decay widths including the tree-level and the D * D rescattering contributions.One sees that the isoscalar D * D rescattering which contains the T cc pole indeed contributes sizably, increases the tree-level results by about 50% for T * + cc → D * + D 0 γ, and decreases the tree-level results by about 58% for T * + cc → D * 0 D + γ.To see the contributions of the D * D rescattering to the decay widths more clearly, the differential decay rates as a function of the D * D invariant mass √ s 12 for T * + cc → D * + D 0 γ and T * + cc → D * 0 D + γ are shown in Fig. 4. One can clearly see the constructive interference and destructive interference effects for these two decays.Since the binding energy of the T * cc is uncertain, we further give the partial width of T * cc → D * Dγ with the binding energy varying from 0.01 to 0.80 MeV in Fig. 5, where the blue dot-dashed lines show the decay widths from the tree-level diagram and the black solid-dashed lines show the LO decay width including both the tree-level and the D * D rescattering contributions.For the decays T * cc → T cc γ and T cc π, the decay widths with the binding energy of the T * cc being B = (503 ± 40) keV are shown in the second column in Table

12 (FIG. 5 . 0 FIG. 6 .
FIG. 4. Differential decay rates for T * cc → D * Dγ with √ s 12 representing the D * D invariant mass.The dashed and solid curves show the result including only the tree-level diagrams and that with the isoscalar DD * rescattering in addition, respectively.
VI. PARTIAL DECAY WIDTHS FOR T *cc → DDγγ AND DDγπIn the three-body decay T * cc → D * Dγ, the resonant D * in the final states can further decay into the Dγ or Dπ, so the T * cc will decay into the stable final states D + D 0 γγ, D + D 0 γπ 0 , and D 0 D 0 γπ + .Since the D * + D 0 γ intermediate state can decay into the same D + D 0 γγ final states as D * 0 D + γ, the same D + D 0 γπ 0 final states as D * + D 0 π 0 and D * 0 D + π 0 , and the same D 0 D 0 γπ + final states as D * 0 D 0 π + , it interferes with these processes.In the following, we will calculate the decay widths of the T * + cc → D + D 0 γγ, D + D 0 γπ 0 , and D 0 D 0 γπ + to show that the interference between the intermediate three-body states is small, and it is a good approximation that we consider only the three-body D * Dγ final states to calculate the T * cc radiative decay width.The diagrams for the four-body decays T * + cc → D + D 0 γγ, T * + cc → D + D 0 γπ 0 , and T * + cc → D 0 D 0 γπ + are shown in Figs. 8, 9, and 10, respectively.The amplitudes for all the diagrams are collected in Appendix B. The four-body decay rate is given by cc *+ →D *+ D 0 γ]/Γ[T cc *+ →D *+ D 0 π 0 ] Γ[T cc *+ →D *0 D + γ]/Γ[T cc *+ →D *0 D + π 0 ] Γ[T cc *+ →D * Dγ]/Γ[T cc *+ →D * Dπ]

FIG. 9 .
FIG. 9. Feynman diagrams for calculating the partial decay width of T * + cc → D + D 0 γπ 0 .The double lines represent the spin-1 mesons, T * + cc , D * 0 , and D * + ; the single thin lines represent the pseudoscalar charmed mesons, D + and D 0 ; the wavy lines represent the photon; and the dashed lines represent the pion.

FIG. 10 .
FIG. 10.Feynman diagrams for calculating the partial decay width of T * + cc → D + D 0 γπ + .The double lines represent the spin-1 mesons, T * + cc , D * 0 , and D * + ; the single thin lines represent the pseudoscalar charmed mesons, D + and D 0 ; the wavy lines represent the photon; and the dashed lines represent the pion.

3 and p0 4
are the energies of π 0 (π + ) and D + (D 0 ) in the T * cc rest frame, respectively, and A NLO is the NLO amplitude including only the D * π rescattering diagrams.The second term in the curly brackets includes the correction of the D * π rescattering, which is the interference term between the LO and NLO amplitudes.Table II shows the radiative decay widths of the T * cc .The second column denoted by Γ Tree includes the contribution from only the tree-level diagram.The third column marked by Γ LO TABLE II.Partial decay widths of the T * cc → DDγγ and DDγπ for T * cc with a binding energy B = (503 ± 40) keV.The second column lists results from the tree-level diagrams, the third column gives the LO decay widths including contributions from both the tree-level and D * D rescattering diagrams, and the last column lists the final results including corrections from the D * π rescattering to Γ LO .There are no D * π scatterings in the decays of T * cc → DDγγ.

)
Appendix B: Four-body decay amplitudesIn this section, we show all the amplitudes for the diagrams in Figs.8-10 of the four-body T * cc → DDγγ and DDγπ decays.

TABLE I .
Partial decay widths of the T * +