B_c^+- decays into tetraquarks

The recent observation by the D0 collaboration of a narrow structure X(5568) consisting of four different quark flavors bdus, has not been confirmed by LHCb. More data and dedicated analyses are needed to cover a larger mass range. In the tightly bound diquark model, we estimate the lightest bdus, 0^+ tetraquark at a mass of about 5770 MeV, approximately 200 MeV above the reported X(5568), and just 7 MeV below the B Kbar threshold. The charged tetraquark is accompanied by I=1 and I=0 neutral partners almost degenerate in mass. A bdus, S-wave, 1^+ quartet at 5820 MeV is implied as well. In the charm sector, cdus, 0^+ and 1^+ tetraquarks are predicted at 2365 MeV and 2501 MeV, about 40-50 MeV heavier than D_{s0}(2317) and D_{s1}(2460). bdus tetraquarks can be searched in the hadronic debris of a jet initiated by a b. However, some of them may also be produced in B_c decays. The proposed discovery modes of S-wave tetraquarks are B_c -->X_{b0} + pi with the subsequent decays X_{b0} -->B_s + pi, giving rise to final states such as B_s pi^+ pi^0. We also emphasize the importance of B_c decays as a source of bound hidden charm tetraquarks, such as B_c -->X(3872) + pi.


I. INTRODUCTION
Recently, the D0 experiment reported the observation of a new narrow structure in the B 0 s π + invariant mass 1 [1], which promptly attracted considerable attention, see [2] (but skepticism has been raised in [3]). Based on 10.4 fb −1 of pp collision data at √ s = 1.96 TeV, this candidate resonance, dubbed X(5568), has a mass and width given by M = 5568 MeV and Γ = 22 MeV, respectively.
A state such as X(5568) would be distinct in that a charged light quark pair cannot be created from the vacuum, leading to the unambiguous composition in terms of four valence quarks with different flavors -bdsu (tetraquarks with flavored quantum numbers have also been discussed in [4]).
Exciting a discovery as it would have been, X(5568) has not been confirmed by the LHCb experiment. Their analysis has been reported recently, based on 3 fb −1 of pp collision data at √ s = 7 and 8 TeV, yielding a data sample of B 0 s mesons 20 times higher than that of the D0 collaboration. Adding then a charged pion, the B 0 s π + invariant mass shows no structure from the B 0 s π + threshold up to M B 0 s π + ≤ 5700 MeV and an upper limit on the ratio ρ(X(5568)/B 0 s ) < 0.016(0.018) @ 90 (95) % C.L. is set for p T (B 0 s ) > 10 GeV [5]. The valence quark composition of X(5568) fits into a diquarkonium interpretation [6][7][8][9][10]. In this framework, the constituents are arranged in a tightly bound diquarkantidiquark pair, [bd] 3c [su]3 c , both of them transforming non-trivially under color SU (3). The possible manifestation of these compact tetraquarks follows essentially from symmetry considerations as in the original constituent quark model and their spectrum is rich. However, as outlined below, our computation of the tetraquark mass spectrum with the quark flavorsbdsu yields significantly higher values. The lightest in this sector is the S-state, X + b0 , whose mass is estimated by us to be about 5770 MeV -approximately 200 MeV heavier than the X(5568), and below the B +K 0 threshold by about 7 MeV.
The tetraquark mass spectrum is calculable up to a theoretical error which we estimate to be of the order of ± 30 MeV, judging from the discrepancies of constituent quark masses obtained from baryons and mesons (see e.g. Table I in ref. [7]). Thus, X + b0 and X 0 b0 may lie somewhat above the B +K 0 threshold, in which case X + b0 will decay, perhaps mostly, in the B +K 0 mode, and the B 0 s π + resonance signal would be reduced 2 . An analysis of the B + K − final state has been published by LHCb, based on a limited sample of 1 fb −1 [11].
However, it is also within the margin of errors that the actual masses of these tetraquark S-states are couple of tens of MeV below our estimates, in which case, the B +K 0 mode is not available, and it is logical to anticipate X + b0 and X 0 b0 as resonant B s π states. We pursue this possibility here. An alternative description is found in [12].
With this hindsight, we point out that there are, in principle, two generic different mechanisms for producing X b0 (5770) in high energy pp and pp collisions. These states can be produced as a fragmentation product of a jet initiated by a b-quark, but, subject to phase space, they can also be produced in the decays of the B ± c mesons, B ± c → X b0 (5770) I=1 + π and B ± c → X b0 (5770) I=0 + π ± as a result of weak (c → sud) decays, qq excitation, and quark rearrangement (see Fig. 1). With the anticipated decays X ± b0 → B 0 s π ± and X 0 b0 → B 0 s π 0 , the decay chains will lead to B ± c → B 0 s π ± π 0 etc. A resonating structure in the B s π mode can then be fished out by Dalitz analysis. This mechanism is similar to the production mechanism of many multiquark states, seen in B 0 and B ± decays, such as B → X(3872)(K, Kπ), but also for the pentaquarks, such as P c (4450) + and P c (4380) + , in the decays Λ 0 b → (P c (4380) + , P c (4450) + )K − . We recall that the dominant two-body decay mode B ± c → B 0 s π ± has been measured by LHCb, with a branching ratio of about 10% [13], and we anticipate that some of the B ± c -decays to tetraquarks will be large enough to be measured.
In what follows, we present our estimates of the mass spectrum of the lowest S and P -states with the flavor quantum numbers of the state B 0 s π + = (bs)(du), having the angular momentum quantum numbers J P = 0 + , 1 + together with their counterparts in the charm sector. This is followed by the discussion of the B ± c -decays leading to some of these tetraquark states as well as the bound cc tetraquark states X(3872) in the decays B ± c → X(3872) + π ± .

II. SPECTRUM
Within the constituent quark model the color-spin Hamiltonian describing the interaction between the different constituents of a hadron takes the form where m i are the diquark constituent masses, S i the quark spins and κ ij some effective, representationdependent chromomagnetic couplings. The spin-spin interaction is here understood to be a contact one.
In the most recent and most successful type-II tetraquark model [8,10], the dominant interactions are assumed to be the spin-spin interactions between quarks (antiquarks) inside the same tightly bound diquark (antidiquark). With the composition: [bq] 3c [sq ]3 c with q = q = d, u, this means retaining only κ bq and κ sq and the lightest states will correspond to the heavy-light diquark spins: S [bq] = 0, 1 and S [sq] = 0. The latter case corresponds to the so called 'good diquark' [14], and the two resulting states have J P = 0 + or 1 + , the lightest being the 0 + one. To indicate these particles, we use the In the above approximation, the resulting mass formula for S-wave, [bq][sq ] states is additive in diquark energies, We may compare (3) with the mass formulae of the related tetraquarks a 0 (980) [6], Z b (10610), Z b (10650) [9], obtained with the substitutions: bs → ss and bs → bb (5) and (6) and the known masses [21], we derive In the approximation where tetrquark masses are additive in diquark energies, one finds about 200 MeV more than the X(5568) mass and just 7 MeV below the B +K 0 . To be seen as resonant B s π states, their masses should lie below the BK threshold. A good part of the B s π invariant mass spectrum is excluded by the LHCb, but still there is a window of opportunity left unexplored so far.
As a side remark, we note that in Ref. [7] the value m [sq] = 590 MeV was obtained using the value κ sq 64 MeV obtained from a fit to the baryon masses, which however may be different from the spin-spin coupling inside a diquark. On the other hand, κ ij are expected to scale inversely to the constituent quark masses and this relation is approximately verified by κ bq and κ cq [10] estimated from Z b,c and Z b,c masses, eq. (7b) and eq. (12b) below. If we scale κ sq from κ cq using the strange and charm constituent quark masses, we obtain κ sq 200 MeV (9) leading to The diquark mass thus obtained is close to the sum of constituent light and strange quark masses, 330 and 520 MeV, respectively. The J P = 1 + exotic states lies close by. From Eq. (3) we find The X b1 state is expected to decay into B * 0 s π + followed by B * 0 s → B 0 s γ, with a photon energy of 48 MeV in the B * s rest frame. Such a low energy photon escapes detection at hadron colliders, as pointed out in [1]. As a consequence of this, the observed peak of the X b1 would be shifted towards lower invariant masses and essentially coincide with the X b0 peak.
In the type-II model [8], we estimate the parameters m [cq] and κ cq , from the masses of Z c (3900), Z c (4020) [21], obtaining One might use the previous results to estimate the mass of the analogous X ± cS expected in the charm sector and decaying into D s π The estimates in Eq. (13 -14) set the exotic candidates X ± c0 just above the D K and D * K thresholds (2363 and 2504 MeV, respectively), so that it could be useful to search also in these decay channels.
If the light diquark is in the S = 0 configuration, i.e. it is antisymmetric in spin and color, it must also be antisymmetric in SU(3) F (F for flavor), therefore the tetraquarks [Qq][q q ], with Q = b, c and q, q , q = u, d, s belong to the SU(3) F representation:3 ⊗3 = 3 ⊕6.
In the charm sector, one doubly charged state is present, belonging to the6, e.g. with the flavor content [cū][sd] → D − s π − . In the beauty sector, doubly charged states lie in the symmetric 15 representation of SU (3) F (see, He and Ko in [2]), originating from the product: 3 ⊗ 6 = 3 ⊕ 15. This requires a light diquark with S = 1, the so-called "bad diquarks", which may be argued to have little binding [14].
At present, upper limits on the production at lepton colliders of charmed-strange doubly charged resonances have been given [15] in the D + s π + channel, for masses between 2.25 and 2.61 GeV.
We close this Section by considering the flavour multiplicity of the states X b0 = [bq][sq ], with q, q = u, d, and their decay modes. These states are obviously organised in a isospin triplet and singlet, similar in structure to the scalar light tetraquarks a 0 (980) and f 0 (980). The neutral X b0 states are similarly expected to be nearly degenerate in mass.
The isoscalar state should decay as X (I=0) b0 → B s + η which is most likely phase space forbidden, leaving the possibility of the strong decay X (I=0) b0 → B +K, a situation very similar to the decay f 0 → KK. Should also the latter mode be forbidden by phase space, X (I=0) b0 has to decay by isospin violating interactions: X (I=0) b0 → B s + π 0 , which may occur due to isospin violating mixing with X (I=1) b0 or via η − π 0 mixing, similarly to η decay.
Similar considerations apply to the I = 0 X ± cS states which estimates in Eqs. (13 -14) place only 40 − 50 MeV above the well known D s0 (2317) and D s1 (2460). The mass difference is quite close to the theoretical error so as to suggest X cS to be identified with the latter resonances, the decays into D + s π 0 arising also from isospin breaking interactions, either due to the mixing with the I = 1, I 3 = 0 component or via η − π 0 mixing.

III. TETRAQUARK PRODUCTION IN WEAK DECAYS OF B ± c MESONS
Motivated by the observation of a large number of exotic XY Z mesons in the decays of the B ± and B 0 -mesons, as well as the pentaquark states P c (4450) + and P c (4380) + in the decays of the Λ bbaryons, Λ 0 b → (P c (4380) + , P c (4450) + )K − , with the subsequent decays (P c (4380) + , P c (4450) + ) → J/ψ p, we anticipate production of the charged X b0 (5570) ± and neutral X b0 (5570) 0 tetraquark states in weak decays of the B ± c mesons. We also emphasize that B ± c -decays are a copious, as yet unexplored, source of hidden cc tetraquark states, via decay modes such as B ± c → X(3872)π ± . Other candidate tetraquark states in the same family but having different J P C quantum numbers are, likewise, anticipated in B ± c decays. For the weak decays of B ± c → B 0 s π ± , B ± c → X 0 b0 π ± , and B ± c → X ± b0 π 0 , the active decay at the quark level is c → sud, with theb decay treated as a spectator. This accounts for approximately 50% of the B ± c decays [16]. The effective Hamiltonian for such non-leptonic decays is where G F is the Fermi coupling constant, V ij are the CKM matrix elements, α and β are the color indices, P L = 1 2 (1 − γ 5 ) and C (±) = (C 1 ± C 2 )/2, C 1,2 (µ) being the Wilson coefficients at scale µ = m c , m b . We have dropped QCD penguin contributions and C (±) are QCD renormalization factors [17] computed at a momentum scale equal to the b-quark mass, with [18] 2 C (−) 1. 4 2 C (+) 0.85 (16) The amplitude for B + c → B s π + can be written in the factorized form, see Fig. 1 Here, f ± (q 2 ) are the vector current form factors, evaluated at q 2 = m 2 π , which have been studied in a number of models (see, for example [19] for a comparative evaluation), f π is the pion decay constant, f π = 140 MeV [21], C 1 is the (QCD renormalized) effective Wilson coefficient, estimated to be C 1 ≈ 1.1, and the second term above can be neglected, as it is multiplied by m 2 π . With this, the decay width can be evaluated straightforwardly where |p π | is the π ± 3-momentum in the rest frame of B ± c -meson. The branching ratio for B ± c → B 0 s π ± has been measured by LHCb Here, P (b → B s ) and P (b → B + c ) are the fragmentation probabilities. The ratio of the two probabilities, i.e., the ratio of the production rates of B + c mesons and B s mesons in a b-quark jet is estimated to be about 0.02, yielding a 10% branching ratio for B + c → B s π + . This is the largest branching ratio of any B-meson observed in a single channel.
The decay B ± c → X I=0 b0 (5770)π ± is expected to have a large branching ratio, as this decay amplitude, like B ± c → B 0 s π ± , is factorizable (see Fig. 1(b)). The relevant matrix element can be written down in an analogous way to that of B ± c → B 0 s π ± . One now needs to know the hadronic matrix element X I=0 b0 |s γ µ P L c|B + c . Recalling that X I=0 b0 has J P = 0 + , the transition goes via the axial-vector part of the charged current, yielding an expression similar to the one for M(B ± c → B 0 s π ± ) obtained above. However, in this case, the corresponding hadronic quantity, which we denote by f + (m 2 π ) BcX b0 , is unknown. This can be calculated using QCD sum rules or lattice QCD, as it involves the axial-current matrix element of a single hadron → single hadron transition. In the diquark model at hand, it is expected to be not too different from f + (m 2 π ) BcB 0 s , as the heavy flavor content of the X 0 b0 and B 0 s is the same, namelybs. Denoting the ratio of the two form factors as F (X b0 /B s ) ≡ f + (m 2 π ) BcX b0 /f + (m 2 π ) BcB 0 s , the relative branching ratios can be expressed as With the known masses, and using our estimate m(X I=0 b0 ) = 5.770 GeV, we get a branching ratio of 1(2)% for the decay B ± c → X I=0 b0 π ± for an assumed value of F (X b0 /B s ) 2 = 0.5(1). Given the large sample of B ± c already available and in forthcoming LHC runs, this branching ratio is measurable in the decay mode B ± c → (B 0 s π 0 )π ± , assuming a good π 0 detection efficiency.
We expect the corresponding branching ratio for the decay B ± c → X ± b0 π 0 → (B 0 s π ± )π 0 to be multiplied by a factor C (−)2 /(C (−) + C (+) ) 2 0.62. In fact, O (−) and O (+) contribute equally to B c decay into X I=0 b0 , Fig. 1(b), while only O (−) contributes to the decay into X I=1 b0 , due to color antisymmetry of the final us pair , Fig. 1(c) (this is similar to the Pati and Woo argument [20] to derive the ∆I = 1/2 rule, i.e. flavor antisymmetry, in non-leptonic baryon decays). This pattern could be modified by nonperturbative effects, as also seen in a number of similar B ± and B 0 decays [21].
We now discuss the B + c decays leading to the bound cc tetraquarks. This requires the quark decaȳ b →cud, with the c-quark in B + c acting as an spectator quark. The benchmark decay for this class is B ± c → J/ψ π ± . Requiring now the excitation of a qq pair, followed by quark recombination, leads to decays such as B ± c → X(3872) I=0 π ± and B ± c → X(3872) I=1±,0 π 0,± . These diagrams allow access to both the I = 0 (isosinglet) and the I = 1 (isotriplet) partners of the X(3872), decaying, respectively, to J/ψ ω and J/ψ ρ 0 , as well as the decay of the charged partner X(3872) ± → J/ψ ρ ± , in addition, possibly, toD * D decays. There would be enough phase space to observe the corresponding Pstates as well.
Again, we expect the decay B ± c → X(3872) (I=0) π ± to have a large branching ratio, which is similar to B ± c → J/ψ π ± , as both are factorizable processes and are proportional to C (−) + C (+) . The decays of B ± c to the [cq][cq ]-tetraquarks have the potential to map out a large number of anticipated states in this sector.

IV. CONCLUDING REMARKS
The observation of the X(5568) by D0, with X(5568) → B 0 s π ± , having M = 5568 MeV and Γ = 22 MeV, has not been confirmed by LHCb. It remains to be seen if a state with the quark flavors bsud exists in nature, with a different mass, decay pattern, and width. In this paper we have used the diquark-antidiquark pic-ture to give predictions about the mass spectrum of the lowest S-state, X b0 and its J P = 1 + partners, both in charm and bottom sectors. Our estimates set the mass of the lowest such state in the b-quark sector at around 5770 MeV, somewhat below the BK threshold. Within the errors of our approach, X + b0 could lie just above this threshold, and one has to look for it in the decay X + b0 → B +K 0 . However, X + b0 may as well reveal itself as a resonating B s π state, or not manifest at all, if below threshold, as discussed in [12].
Here we propose to search tetraquark states in the decays of the B ± c mesons, B ± c → X 0 b0 π ± and B ± c → X ± b0 π 0 and have argued that some of these decay modes may have a large branching ratio. This requires a good π 0detection efficiency, which we advocate to improve in hadron collider experiments, such as the LHCb. The two detached vertices (of the B ± c and B 0 s ) may help in reducing the background.
So far, only a handful of B ± c decays have been observed [21], and it is worthwhile to put in a dedicated effort to increase this database. Apart from the possibility of observing the tetraquark states of the B s π variety, we anticipate several bound cc tetraquark states, which emerge from the decay B + c → (cc)ud, followed by a qq excitation from the vacuum. These would lead to decays such as B ± c → X(3872) 0 π ± and B ± c → X(3872) ± π 0 , as well as to other related tetraquark states. They should be searched for at the LHC, and also at Belle-II, if the e + e − center of mass energies could reach the B + c B − c threshold.