Hunting for X b via hidden bottomonium decays X b → ππχ bJ

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I. INTRODUCTION
In the past decades, a large number of so-called XY Z states has been discovered on the experiments, and tremendous effort has been taken to unravel their nature beyond the conventional quark model [1][2][3][4][5][6][7].
The observation of the X(3872) [8] shows that the meson spectroscopy is far more complicated than the naive expectation of the quark model.It is natural to search for the posited bottomonium counterpart of the X(3872) with J P C = 1 ++ (called X b hereafter) [18,19].As the heavy quark flavor symmetry (HQFS) partner of the X(3872), X b should share some universal properties with the X(3872).The search for X b can provide us with important information on the discrimination between the compact multiquark configuration and the loosely bound hadronic molecule configuration for the X(3872) [20].
The existence of the X b is predicted both in the molecular interpretation [21][22][23][24] and the tetraquark model [25], with a mass coincide with the B B * threshold [2, [21][22][23][26][27][28][29][30][31][32], or in the 10 to 11 GeV/c 2 range [18,25,33,34], respectively.Such a heavy mass of X b makes it less likely to be discovered in current electron-positron collision facilities, although the Super KEKB may provide an opportunity for searching it in the radiative decays of the Υ(5S, 6S) [35].The production of the X b at hadron colliders such as LHCb and Tevatron [31,36] have been extensively investigated [37][38][39][40][41][42] and shows sizeable production rate.Therefore, a suitable decay mode from which the X b state can be reconstructed is imperatively called for. Ulike the X(3872), the isospin breaking decay of the X b to π + π − Υ(1S) should be highly suppressed since the X b may be far below the B B * threshold and the mass difference between neutral and charged bottom mesons is very small, which may explain the null result reported by the CMS, ATLAS Collaborations [43,44] for searching the X b in the π + π − Υ(1S) final state.The isospin conserved hidden bottomonium decay X b → ωΥ(1S) has been investigated in Ref. [45] with the intermediate meson-loop (IML) contributions, but no significant signal is observed in the experiments [20,46].The partial widths of the radioactive decays X b → γΥ(nS) was calculated in Ref. [29] and their magnitude is about 1 keV.
In this work, we will focus on the isospin conserved decay of X b → ππχ bJ (J = 0, 1, 2), which could have sizeable branching fractions and be possible channels for the X b reconstruction.
The X b → ππχ bJ decays are studied in the bottom IML mechanism.The impact of the IML on the heavy quarkonium transitions has been investigated in the Υ decay processes Υ [47][48][49], and gave results consistent with the experimental data.In this work, the partial decay widths of X b → π 0 π 0 χ bJ (1P ) and X b → π + π + χ bJ (1P ) (J = 0, 1, 2) are calculated by using the heavy hadron chiral perturbation theory (HHχPT), including the contribution from the box IML.
The rest of this paper is organized as follows.In Sec.II, we introduce the effective Lagrangians and Feynman diagrams for the hadronic decays of the X b to π 0 χ bJ and ππχ bJ .Our numerical results are presented in Sec.III, and a brief summary is given in Sec.IV.

II. EFFECTIVE LAGRANGIANS AND POWER COUNTING
In this section, we will give the effective Lagrangians and Feynman diagrams for the hadronic decays of X b to π 0 χ bJ and ππχ bJ .The X b is assumed to be an S-wave molecular state with J P C = 1 ++ given by the superposition of B 0 B * 0 + c.c and B − B * + + c.c hadronic configurations as [45] where the charge conjugation conventions B * C → B * and B C → B is used, and φ is a phase angle describing the proportion of neutral and charged constituents, which will be settled in Sec.III by the cancellation of the neutral and charged meson loops in the isospin-violating processes X b → π 0 χ bJ .The effective Lagrangian for the X b coupling to the B * B can be written as where g n and g c denote the coupling constant of the X b to the neutral and charged bottom meson pairs, respectively.As an isoscalar B * B molecular state, the X b state appears as a pole on the real axis in the complex energy plane of the B * + B − − B * 0 B0 coupled channel T matrix with C = + , and the effective couplings g n and g c can be derived from the residues of the T matrix elements at the X b pole and read [50,51] where the binding energies of the X b relative to the B 0 B * 0 and B + B * − thresholds, respectively, and ) are the the reduced masses.
The leading-order heavy hadron chiral perturbation theory (HHχPT) Lagrangian for mesons containing heavy quarks or antiquarks at rest is [52] where ⃗ σ denote the Pauli matrices and a is the light flavor index, the bottom mesons are given by the two-component notation [53] as H a = ⃗ V a • ⃗ σ + P a with P = (B − , B0 ) and V = (B * − , B * 0 ) denote the pseudoscalar and vector heavy mesons, respectively, and the field for the antimesons is Ha = − ⃗ Va • ⃗ σ + Pa with P = (B + , B 0 ) and V = (B * + , B * 0 ).The first two terms in Eq. (5) give the interaction between the heavy mesons and pions.The field ⃗ A ab = − ⃗ ▽Φ ab /f π + • • • is the axial current in the chiral perturbation theory (χPT) and couples to the heave mesons with the axial coupling g = 0.54 [50], where f π = 130 MeV is the pion decay constant, and the Φ field contains the Goldstone bosons as components, The last two terms in Eq. ( 5) describe the B ( * ) B ( * ) ππ interaction, where D The Lagrangian couples the χ bJ to heavy mesons reads where the χ bJ field is expressed as [52] with the coupling constant g 1 = 0. B( * ) B( * ) B( * ) B( * ) B( * )    1(a) and 1(b) scales as v 5 p π /(v 2 ) 2 = p π v and v 5 p π /(v 2 ) 3 = p π /v, respectively.For the decay X b → ππχ bJ , the box diagrams Figs.1(c) and 1(d) scales as v 5 p 2 π /(v 2 ) 4 = p 2 π /v 3 , and the triangle diagrams in Figs.1(e) and 1(f) scale as v 5 p 2 π /(v 2 ) 3 = p 2 π /v and v 5 p 2 π /(v 2 ) 3 = p 2 π /v, respectively.One can see the contributions from the two point bubble diagrams to the X b → πχ bJ decays are suppressed by v 2 relative to the triangle diagrams, and the contributions from the triangle diagrams to the X b → ππχ bJ decays are suppressed by v 2 comparing with the box diagrams.Thus, for a rough estimate of the partial decay width, we only consider the contribution of the triangle diagrams to the decay widths of X b → πχ bJ and the box diagrams to the decay widths of X b → ππχ bJ in this work, and a evaluation of the omitted contributions from the bubble diagrams to the decay widths of X b → πχ bJ and the triangle diagrams to the decay widths of X b → ππχ bJ is given in Appendix B.
Based on the Lagrangians given above, the loop transition amplitudes in Fig. 1 can be expressed in a general form as follows, where V n (n = 1, 2, 3, 4) represent the vertex functions for the initial X b , final χ bJ and π, respectively.The where the symmetry factor S is taken to be 2 in the X b → π 0 π 0 χ bJ decays considering the identical π 0 π 0 particles in the final states, and to be 1 in the decays X b → π 0 χ bJ and X b → π + π − χ bJ , j is the total spin of the initial particle, and there is a sum over all the polarizations of the final-state particles.
The dΦ n (n = 2, 3) is the two-body phase space or three-body phase space, which can be obtained from Ref. [54,55].

III. NUMERICAL RESULTS
In this section, we give the partial decay widths of the X b → π 0 χ bJ , π 0 π 0 χ bJ , and π + π − χ bJ .As the binding energy of the X b is uncertain, covering the mass range of the X b predicted by the molecular and tetraquark model in Ref. [21][22][23]25] could be a good approximation for calculating the decay widths and might be applicable.In Ref. [25], the mass of the lowest-lying 1 ++ bqbq tetraquark was predicated to be 10504 MeV.In the molecular interpretation, it was predicted to be 10562 MeV, which is approximately 42 MeV below the B B * threshold [21], and (10580 +9 −8 ) MeV with a binding energy of (24 +8 −9 ) MeV [23].
values of φ except for φ ∈ [0.81, 0.82] where the charged and the neutral loops cancel with each other.
After the careful arrangement of the charged and neutral components in X b , we give the partial widths of the X b → ππχ bJ process with φ = 0.813 ± 0.005, which are isospin conserved and not sensitive to the angle φ in the neighborhood of φ = 0.813.The partial widths of the X b → ππχ bJ with E n X b = (2, 5, 10, 25, 50, 100) MeV are listed in Table II, where the partial decay widths of X b → ππχ bJ (J = 1, 2) are about tens of keVs, while the partial decay widths of X b → ππχ b0 is two orders of magnitude smaller because it is in higher partial waves as analyzed in the follows.The quantum numbers of the identical particle (π 0 π 0 ) system are I G (J P C ) = 0 + (L ++ ) (L = even).Considering the conservation of the angular momentum and the P -parity, in the X b → π 0 π 0 χ b0 and X b → π 0 π 0 χ b2 processes, the two pions in the identical particle system should be at least in D-wave, and this system should also be at least in D-wave  and S-wave with the χ b0 and χ b2 , respectively.While in the X b → π 0 π 0 χ b1 decay, both the two pions and the two-pion system with the χ b1 can be in S-wave, therefore the partial widths of X b → π 0 π 0 χ b0 are supressed by the higher partial waves comparing with those of X b decaying into π 0 π 0 χ b1 and π 0 π 0 χ b2 .
For the decays X b → π + π − χ bJ , the quantum numbers of the (π + π − ) system satisfy L + I = even with I the total isospin of the (π + π − ) system.For I = 0, the discussions are same as the decays to π 0 π 0 χ bJ .
For I = 1, the two charged pions can be in P -wave and the two-pion system and the χ b0 can also be in P -wave, and the partial width of the X b → π + π − χ b0 process is still suppressed as it is isospin-violated.
And one can see that Γ[π + π − χ bJ ]/Γ[π 0 π 0 χ bJ ] ≃ 2, same as the X(3872) case [52].To see the relation between the partial widths of X b → ππχ bJ (J = 0, 1, 2) and the binding energy of the X b more precisely, the partial widths versus the binding energy E n X b ∈ [2, 100] MeV are demonstrated in Fig. 3, where the partial widths first increase and then decrease with the E n X b increases.This is because the binding energy dependence of the partial width can be influenced by the coupling strength of X b in Eqs. ( 3), (4), and the threshold effects.As the binding energy E n X b increases, the coupling strength of X b increases, while the threshold effects decreases.In the small E n X b region, the contribution from the the coupling strength of X b is dominant, while that from the threshold effects plays an important role in the large E n X b region.One can see that unlike the X b → π + π − Υ which is suppressed by the isospin symmetry, the partial decay widths of the isospin conserved process X b → ππχ b1 is at the same order of magnitude with the partial width of X b → Υ(1S)ω calculated in Ref. [45], and it could be better channel for the X b searching as the ω needs to be reconstructed through the πππ or π 0 γ final states.Therefore our results can be helpful for hunting the X b in the experiments.

IV. SUMMARY
In this work, we investigated the isospin breaking decay X b → π 0 χ bJ and the isospin conserved decay X b → ππχ bJ , where X b is taken to be the HQFS counterpart of X(3872) in the bottomonium sector as a meson-meson molecule candidate.Since the mass of this state may be far below the B B * threshold, the isospin-violating decay channel X b → π 0 χ bJ would be highly suppressed and stimulate the importance of the isospin-conserved decay channel X b → ππχ bJ .The isospin-violating decay channel X b → π 0 χ bJ can helps us determine the charged and neutral components of X b to some extent.For the isospin conserved processes, the calculated partial widths of X b → ππχ bJ (J = 0, 1, 2) are about less than 1 keV, tens of keVs, and a few keVs, respectively.The partial width of X b → ππχ b1 is found to be about tens of keVs, 1 ∼ 2 order(s) of magnitude larger than those of X b → ππχ b2 and X b → ππχ b0 .Taking into account the fact that the total width of X b may be smaller than a few MeV like X(3872), the calculated branching ratios X b → ππχ b1 may reach to orders of 10 −2 , which makes it a possible channel for the experimental searching of the X b .These studies may help us investigate the X b deeply.The experimental observation of X b will provide us further insight into the spectroscopy of exotic states and is helpful to probe the structure of the states connected by the heavy quark symmetry.

V. ACKNOWLEDGMENTS
The authors thank Yun-Hua Chen, Qi Wu, and Shi-Dong Liu for useful discussions, and thank the anonymous referee for useful comments and suggestions.This  In this section, we derive the 2-point, 3-point and 4-point loop integral in the rest frame of the decay particle (p = (M, 0)).The 2-point loop integral can be written as [56] where µ ij = m i m j / (m i + m j ) are the reduced masses, b 12 = m 1 + m 2 − M , and c 1 = 2µ 12 b 12 .The cutoff Λ is taken to be 1 GeV.
The scalar 3-point loop integral is ultraviolet (UV) convergent and can be worked out as [56] where The four-point integrals in the box diagrams are given in Ref. [47].For the Fig. 4(a), the initial particle at rest [p = (M, 0)] reads where ), The m 1 denotes the mass of the top-left intermediate-bottom mesons, and the mass of the other intermediate bottom mesons are labeled as m 2 , m 3 , and m 4 , in counterclockwise order. where For the Fig. 4(c), where ) in Table IV.From these evaluations, the contributions of the triangle diagrams to the decay widths are estimated to be 1 ∼ 3 magnitude smaller than the box diagram contributions.Γ  Γ 53 +0.19 −0.13 GeV −1/2 [47].The Feynman diagrams of X b → π 0 χ bJ and X b → ππχ bJ are shown in Fig. 1 with all the possible combination of the intermediate particles in the loops of each diagram listed in Table I.In Table I, the first particle in each square bracket denotes the top (in the two-point bubble), or top left (in the triangle and box) intermediate bottom meson in the corresponding diagram, and the other intermediate bottom mesons in the same diagram are listed in the square bracket in counterclockwise order along the loop.

Fig. 1
Fig.1(a) [B * , B], [ B * , B] Fig.1(b) [B * , B, B], [ B * , B, B], [ B, B * , B * ], [B, B * , B * ], [B * , B, B * ], [ B * , B, B * ] Since the X b is close to the B B * threshold, the velocity of the intermediate heavy meson v B ( * ) = |E X b |/m B ( * ) should be smaller than 1 (in units of the speed of light) and become a natural small quantity for the power counting of the diagrams in Fig. 1.The diagrams can be counted in the powers of v and p π , where p π ≃ (m X b − m χ bJ )/2 is the momentum of the external pion, and v = (v X b + v χ bJ )/2 with v X b ≃ 0.03 and v χ bJ ≃ 0.40 derived from taking the X b binding energy to be E n Xb = 5 MeV.In each diagram, the nonrelativistic energy counts as v 2 , each loop integral is at v 5 , and each nonrelativistic propagator contributes at v −2 .For the vertices, the B ( * ) B ( * ) ππ vertex is proportional to the square of the energy of the pion E 2 π ∼ p 2 π , the P -wave B * Bπχ bJ and B ( * ) B ( * ) π vertices are proportional to p π , while the X b B ( * ) B( * ) and B ( * ) B( * ) χ bJ vertices are in S-wave and count at v 0 p 0 π .The diagrams of X b → π 0 χ bJ shown in Figs.

FIG. 3 .
FIG. 3. The partial decay widths of X b → ππχ bJ as a function of the binding energy E n X b .
work is partly supported by the National Natural Science Foundation of China under Grant Nos.12075133, 11835015, and 12047503, and by the Natural Science Foundation of Shandong Province under Grant Nos.ZR2021MA082, and ZR2022ZD26.It is also supported by Taishan Scholar Project of Shandong Province (Grant No.tsqn202103062), the Higher Educational Youth Innovation Science and Technology Program Shandong Province (Grant No. 2020KJJ004).

FIG. 5 .
scheme, the contributions from bubble diagrams are suppressed by v 2 compared with the triangle diagrams and therefore the bubble diagrams are not considered in the X b → πχ bJ decay in our calculations in Sec.III.The 2-point integral I[m 1 , m 2 ] given in Appendix A is regularization dependent, and the cut-off dependence of the two body decay widths from the bubble diagram contribution is shown in Fig.5with Λ = 0.5 ∼ 1.5 GeV, φ = 0.813, and E n X b = 5 MeV.The coupling constant c 1 is also taken to be g 1 /m B * and g 1 /( √ 10m B * ) to roughly estimate the contributions from the bubble diagrams, and the results are about 3 ∼ 4 orders of magnitude smaller than the contributions of the triangle diagrams.The cutoff dependence of the decay widths is not significant for Λ below 1 GeV.

TABLE II .
Predicted partial widths (in units of keV) of the X b decays.The units of the binding energy E n X b in first column are MeV.

TABLE III .
Predicted partial widths (in units of keV) of the X b decays with c1 = g1/mB * .