Spectroscopy, lifetime and decay modes of the T − bb tetraquark
aa r X i v : . [ h e p - ph ] N ov Spectroscopy, lifetime and decay modes of the T − bb tetraquark E. Hern´andez, ∗ J. Vijande, † A. Valcarce, ‡ and Jean-Marc Richard § Departamento de F´ısica Fundamental e IUFFyM,Universidad de Salamanca, 37008 Salamanca, Spain Unidad Mixta de Investigaci´on en Radiof´ısica e Instrumentaci´on Nuclear en Medicina (IRIMED),Instituto de Investigaci´on Sanitaria La Fe (IIS-La Fe)-Universitatde Valencia (UV) and IFIC (UV-CSIC), 46100 Valencia, Spain Universit´e de Lyon, Institut de Physique Nucl´eaire de Lyon, IN2P3-CNRS–UCBL,4 rue Enrico Fermi, 69622 Villeurbanne, France (Dated: November 14, 2019)We present the first full-fledged study of the flavor-exotic isoscalar T − bb ≡ bb ¯ u ¯ d tetraquark withspin and parity J P = 1 + . We report accurate solutions of the four-body problem in a quark model,characterizing the structure of the state as a function of the ratio M Q /m q of the heavy to lightquark masses. For such a standard constituent model, T − bb lies approximately 150 MeV below thestrong decay threshold B − ¯ B ∗ and 105 MeV below the electromagnetic decay threshold B − ¯ B γ .We evaluate the lifetime of T − bb , identifying the promising decay modes where the tetraquark mightbe looked for in future experiments. Its total decay width is Γ ≈ × − GeV and thereforeits lifetime τ ≈ B ∗− D ∗ + ℓ − ¯ ν ℓ and ¯ B ∗ D ∗ ℓ − ¯ ν ℓ among thesemileptonic decays, and B ∗− D ∗ + D ∗ s − , ¯ B ∗ D ∗ D ∗ s − , and B ∗− D ∗ + ρ − among the nonleptonicones. The semileptonic decay to the isoscalar J P = 0 + tetraquark T bc is also relevant but it is notfound to be dominant. There is a broad consensus about the existence of this tetraquark, and its de-tection will validate our understanding of the low-energy realizations of Quantum Chromodynamics(QCD) in the multiquark sector. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] I. INTRODUCTION
Hadronic physics has been much stimulated during the last two decades by the experimental discovery of several newresonances in the hidden-charm sector, resonances that are hardly accommodated in the traditional quark-antiquarkor three-quark picture [1]. These are the so-called
XY Z mesons and LHCb pentaquarks, which belong to the classof ”exotic hadrons”, although they are not flavor exotics. After several years of studies, no definite conclusion hasbeen drawn as to whether such non-flavor exotic states correspond to multiquark structures or to hadron-hadronmolecules. A similar situation was encountered in the light scalar meson sector, where a multiquark picture was firstintroduced [2] as an attempt to explain the inverted mass spectrum (inverted in comparison to the simple quark-antiquark structure favored by the naive quark model) exhibited by the low-lying scalar mesons, some of which werelater on suggested to be meson-meson molecules [3].For years, the sector of flavor-exotic hadrons has been somewhat forgotten, as being less easily accessible thanthe hidden-flavor sector. However, already some decades ago, investigations on flavor-exotic multiquarks concludedthat QQ ¯ q ¯ q four-quark configurations become more and more deeply bound when the mass ratio M Q /m q increases [4].There is nowadays a broad theoretical consensus about the existence of such unconventional tetraquark configurationsfor which all strong decays are energetically forbidden. The most promising candidate is an isoscalar tetraquark withdouble beauty and J P = 1 + quantum numbers, which is stable against strong and electromagnetic decays. The sameconclusion about the stability of this state has been reached in a wide variety of theoretical approaches [4–14]. A novellattice QCD calculation [5] employing a non-relativistic formulation to simulate the bottom quark finds unambiguoussignals for a strong-interaction-stable ( I ) J P = (0)1 + tetraquark, 189(10) MeV below the corresponding two-mesonthreshold, ¯ B ¯ B ∗ . The lattice QCD calculation of Ref. [6] comes to the identical conclusion obtaining a binding energyof 143(34) MeV. With such binding, the tetraquark is stable also with respect to electromagnetic decays. In Ref. [7],the mass of the doubly-charm baryon Ξ ++ cc , discovered by the LHCb Collaboration [15], is used to calibrate the bindingenergy of a QQ diquark. Assuming that the bb diquark binding energy in a T − bb tetraquark is the same as that ofthe cc diquark in the Ξ ++ cc baryon, the mass of the (0)1 + doubly-bottom tetraquark is estimated to be 215 MeVbelow the strong decay threshold ¯ B ¯ B ∗ . In Ref. [8], the heavy-quark-symmetry mass relations linking heavy-lightand doubly-heavy-light mesons and baryons are combined with leading-order corrections for finite heavy-quark mass,corresponding to hyperfine spin-dependent terms and kinetic energy shift that depends only on the light degrees offreedom. This leads to predict that the T − bb state is stable against strong decays. More specifically, using as inputthe masses of the doubly-bottom baryons (not yet experimentally measured) obtained by the model calculations ofRef. [7], Ref. [8] finds an axial-vector tetraquark bound by 121 MeV. In Ref. [9], the Schr¨odinger equation is solvedwith a potential extracted from a lattice QCD calculation for static heavy quarks, in a regime where the pion massis m π ∼
340 MeV, and again, evidence is found for a stable isoscalar doubly-bottom axial-vector tetraquark. Whenextrapolated to physical pion masses it has a binding energy of 90 +43 − MeV. The robustness of these predictionsis reinforced by detailed few-body calculations using phenomenological constituent models based on quark-quarkCornell-type interactions [10, 11], which predict that the isoscalar axial-vector doubly-bottom tetraquark is strong-and electromagnetic-interaction stable with a binding energy ranging between 144–214 MeV for different realisticquark-quark potentials. Recent studies using a simple color-magnetic model have come to similar conclusions [12].The QCD sum rule analysis of Ref. [13] also points to the possibility of a stable doubly-bottom isoscalar axial-vectortetraquark. Finally, the recent phenomenological analysis of Ref. [14] also presents evidence in favor of the existenceof a stable T − bb state.The compelling theoretical evidence for the existence of a T − bb tetraquark has led to preliminary studies of its lifetimeand weak decay modes. In this context, Ref. [16] investigated the amplitudes and decay widths of doubly-heavytetraquarks under the flavor SU(3) symmetry, deriving ratios between decay widths of different channels. Ref. [17]evaluated the semileptonic decay of the T − bb tetraquark to a scalar bc ¯ u ¯ d tetraquark in the framework of QCD sum rules.In spite of the numerous model calculations existing in the literature (see Refs. [11, 12] for a recent compendium) nocomprehensive calculation of the spectroscopy, decay modes, and lifetime of this state has been obtained so far.The purpose of this work is to present the first detailed study of the flavor-exotic T − bb tetraquark with J P = 1 + and isospin I = 0, reporting accurate solutions of the four-body problem, characterizing the structure of the state,evaluating its lifetime and identifying the promising decay modes where the tetraquark might be looked for. A strikingresult deals with the lifetime, which is found significantly longer than for single- b hadrons. With two b quarks, oneexpects either cooperating or conflicting interferences. Also, if one compares a typical meson decay mode B → D x and its tetraquark analog T → B D x , there is a change in the overlap of the final D meson and the c ¯ q system providedby a spectator ¯ q and the c quark coming from one of the b quarks: the color factor and the spatial distribution aremodified. There is also an obvious effect of the phase-space, which is known to be crucial in weak decays, for instancefor β -unstable nuclei. While the Dx invariant mass is 5.3 GeV in B → Dx decay, it is about 4.6 GeV in T → BDx ,depending which sector of the Dalitz plot is reached. Altogether, it looks difficult to attempt a guesstimate of thelifetime before actually performing the calculation.This paper is organized as follows. In Sec. II, we present the masses and wave functions obtained from an accuratefour-body calculation that makes use of a quark model. The calculation of the dominant decay modes and of thelifetime is given is Sec. III. Our conclusions are summarized in Sec. IV.
II. TETRAQUARK MASS AND WAVE FUNCTION
We have studied the spectroscopy of doubly-heavy tetraquarks by two different numerical methods: a hypersphericalharmonic formalism and a generalized Gaussian variational (GGV) approach, both driving to the same results [10].For its later application to the detailed study of the four-quark structure and weak decays, the GGV is more suited.Let us briefly discuss the main characteristics of the method. We shall denote the heavy quark coordinates by r and r , and those of the light antiquarks by r and r . The tetraquark wave function is taken to be a sum over all allowedchannels with well-defined symmetry properties [18, 19]: ψ ( x , y , z ) = X κ =1 χ csfκ R κ ( x , y , z ) , (1)where x = r − r , y = r − r and z = ( r + r − r − r ) / χ csfκ are orthonormalizedcolor-spin-flavor vectors and R κ ( x , y , z ) is the radial part of the wave function of the κ th channel. In order to get theappropriate symmetry properties in configuration space, R κ ( x , y , z ) is expressed as the sum of four components, R κ ( x , y , z ) = X n =1 w nκ R nκ ( x , y , z ) , (2)where w nκ = ±
1. Finally, each R nκ ( x , y , z ) is expanded in terms of N generalized Gaussians R nκ ( x , y , z ) = N X i =1 α iκ exp (cid:2) − a iκ x − b iκ y − c iκ z − d iκ s ( n ) x · y − e iκ s ( n ) x · z − f iκ s ( n ) y · z (cid:3) , (3)where s i ( n ) are equal to ± α iκ , a iκ , · · · , f iκ are thevariational parameters. The latter are determined by minimizing the intrinsic energy of the tetraquark. We followclosely the developments of Refs. [18, 19], where further technical details can be found about the wave function andthe minimization procedure.A four-quark state is stable under the strong interaction if its mass, M T (from now on, T often abbreviates T QQ ),lies below all allowed two-meson decay thresholds. Thus, one can define the difference between the mass of thetetraquark and that of the lowest two-meson threshold, namely:∆ E = M T − ( M + M ) , (4)where M and M are the masses of the mesons constituting the threshold. When ∆ E <
0, all fall-apart decays areforbidden and, therefore, the state is stable under strong interactions. When ∆ E ≥ E <
0. Another quantity of interest is the root-mean-square (r.m.s.) radius of the tetraquark, X T ,given by [10]: X T = " P i =1 m i h ( r i − R ) i P i =1 m i / , (5)where R is the center-of-mass coordinate, and m i are the quark masses M Q or m q .Determining whether stability is reached in this model, i.e., ∆ E <
0, requires a simultaneous and consistentcalculation of the meson masses M and M entering the threshold, and of the tetraquark configurations. For thispurpose, we have adopted the so-called AL1 model by Semay and Silvestre-Brac [22], already used in a number ofexploratory studies of multiquark systems, for instance in our recent investigation of the hidden-charm pentaquarksector ¯ ccqqq [23] or doubly-heavy baryons and tetraquarks [11]. It includes a standard Coulomb-plus-linear centralpotential, supplemented by a smeared version of the chromomagnetic interaction, V ( r ) = −
316 ˜ λ i . ˜ λ j (cid:20) λ r − κr − Λ + V SS ( r ) m i m j σ i · σ j (cid:21) ,V SS ( r ) = 2 π κ ′ π / r exp (cid:18) − r r (cid:19) , r = A (cid:18) m i m j m i + m j (cid:19) − B , (6) TABLE I. Relevant meson masses (in MeV) and r.m.s. radii (in fm) predicted by the AL1 model for the strong, electromagneticand weak decay thresholds of the J P = 1 + T − bb tetraquark.Meson M r.m.s. Meson M r.m.s J = 0 ¯ B K
491 0.283 D π
138 0.298 D s η c J = 1 ¯ B ∗ K ∗
903 0.389 D ∗ ρ
770 0.460 D ∗ s J/ Ψ 3101 0.199 where λ = 0.1653 GeV , Λ = 0.8321 GeV, κ = 0.5069, κ ′ = 1.8609, A = 1.6553 GeV B − , B = 0.2204, m u = m d =0.315 GeV, m s = 0.577 GeV, m c = 1.836 GeV and m b = 5.227 GeV. Here, ˜ λ i . ˜ λ j is a color factor, suitably modified forthe quark-antiquark pairs. We disregard the small three-body term of this model used in [22] to fine-tune the baryonmasses vs. the meson masses. Note that the smearing parameter of the spin-spin term is adapted to the massesinvolved in the quark-quark or quark-antiquark pairs. It is worth to emphasize that the parameters of the AL1potential are constrained in a simultaneous fit of 36 well-established meson states and 53 baryons, with a remarkableagreement with data, as seen in Table 2 of Ref. [22].The meson masses of the threshold in this model are given in Table I, together with the masses of other mesonsthat will be involved in the weak decays discussed in Sec. III. Also shown is the quark-antiquark r.m.s. radius.One can now study the stability of the J P = 1 + T − bb isoscalar state. In the GGV method, if a state is unbound,one observes a slow decrease of its mass toward M + M as N , the number of terms in Eq. (3), increases. It turnsout to be useful to also look at the content of the variational wave function, which comes very close to 100% in acolor singlet-singlet channel in the physical basis [18]. On the other hand, if a variational state converges to a boundstate as N increases, it includes sizable hidden-color components even for low N . We show in Table II the results forthe T QQ tetraquark for different masses of the heavy quark, M Q . In the first line we give the results for the standardmass value of the bottom quark used in the AL1 model, for which we get a binding energy of 151 MeV. We havescrutinized the structure of the T QQ state. For each particular value of M Q we have evaluated the lowest strong-decaythreshold, M + M , the energy of the four-quark state, M T QQ , and the corresponding binding energy B = − ∆ E .We have calculated the probability of the ¯33, P [ | ¯33 i ], and 6¯6, P [ | i ], color components. By using the recouplingtechniques derived in Ref. [18] we have also evaluated the probability of the 11, P [ | i ], and 88, P [ | i ], colorcomponents. Afterwards, we have expanded the wave function in terms of physical states evaluating the probabilityof the pseudoscalar-vector, P MM ∗ , and vector-vector, P M ∗ M ∗ , two-meson physical states. Finally we have calculatedthe average distance between the two heavy quarks, h x i / , between the two light quarks, h y i / , between a heavyand a light quarks, h z i / , and the four-quark r.m.s. radius, X T QQ .As shown in Table II the binding energy of the T QQ tetraquark increases with increasing M Q /m q as predictedin Ref. [4] and recently rediscovered in Refs. [7, 8]. Close to ∆ E = 0 the system behaves like a simple meson-meson molecule, with a large probability in a single meson-meson component, the pseudoscalar-vector channel. The T QQ starts to be bound around the mass of the charm quark used by the AL1 model, m c = 1836 MeV [24]. Suchsmall binding is due to a cooperative effect between the chromoelectric and chromomagnetic pieces in the interactingpotential. Hence, the T cc tetraquark is unbound when the spin-spin interaction is switched off. The ˜ λ i . ˜ λ j contributionof Eq. (6), with a pairwise potential due to color-octet exchange, induces mixing between ¯33 and 6¯6 color states inthe QQ − ¯ q ¯ q basis. The ground state of the QQ ¯ u ¯ d with J P = 1 + has its dominant component with color ¯33, and TABLE II. Properties of the T QQ tetraquark as a function of the mass of the heavy quark M Q for the AL1 model. Energiesand masses are in MeV and distances in fm. For ∆ E >
0, the probabilities and average distances are just an indication thatthe variational calculation will likely not converge toward a bound state. M Q M + M M T QQ ∆ E P [ | ¯33 i ] P [ | i ] P [ | i ] P [ | i ] P MM ∗ P M ∗ M ∗ h x i / h y i / h z i / X T QQ −
151 0.967 0.033 0.344 0.656 0.561 0.439 0.334 0.784 0.544 0.2264549 9290 9163 −
126 0.955 0.045 0.348 0.652 0.597 0.403 0.362 0.791 0.544 0.2423871 7936 7835 −
100 0.930 0.070 0.357 0.643 0.646 0.354 0.411 0.806 0.541 0.2683193 6582 6511 −
71 0.885 0.115 0.372 0.628 0.730 0.270 0.475 0.833 0.536 0.3012515 5230 5189 −
41 0.778 0.222 0.407 0.593 0.795 0.205 0.621 0.919 0.523 0.3691836 3878 3865 −
13 0.579 0.421 0.474 0.526 0.880 0.120 0.966 1.181 0.499 0.5301158 2534 2552 > ≫ ≫ ≫ M Q /m q Charm Bottom
P[|33 〉] P r ob a b ilit y P[|66 〉] (a) 6 11 16 M Q /m q Charm Bottom 〈 y 〉 〈 x 〉 〈 z 〉 〈 r 〉 / (f m ) (b) FIG. 1. (a) Probability of the ¯33 and 6¯6 components of the T QQ color wave function as a function of M Q /m q . (b) Averagedistances h x i / , h y i / and h z i / as a function of M Q /m q . spin { , } in the QQ − ¯ u ¯ d basis. The main admixture consists of 6¯6 with spin { , } and a symmetric orbital wavefunction. Thus, for M Q /m q close to the charm sector, the binding requires both the color mixing of ¯33 with 6¯6, andthe spin-spin interaction [11, 25]. In the most advanced calculations of Ref. [25], it was acknowledged that a pureadditive interaction will not bind cc ¯ q ¯ q , on the sole basis that this tetraquark configuration benefits from the strong cc chromoelectric attraction that is absent in the Q ¯ q + Q ¯ q threshold. In the case where ¯ q ¯ q = ¯ u ¯ d , however, there is inaddition a favorable chromomagnetic interaction in the tetraquark, while the threshold experiences only heavy-lightspin-spin interaction, whose strength is suppressed by a factor m q /M Q .When the ratio M Q /m q increases, the probability of the 6¯6 color component diminishes in such a way that thesystem does not behave any more like a simple meson-meson molecule. The probability of the 6¯6 component ina compact QQ ¯ q ¯ q tetraquark tends to zero for M Q → ∞ . Therefore, heavy-light compact bound states would bealmost a pure ¯33 singlet color state and not a single colorless meson-meson 11 molecule, as shown in Table II. Suchcompact states with two-body colored components can be expanded as the mixture of several physical meson-mesonchannels [10, 26], and thus they can be also studied as an involved coupled-channel problem of physical meson-mesonstates [27].We have shown these results in Fig. 1. In the panel (a), we see how the probability of the 6¯6 color componenttends to zero for M Q → ∞ . On the other hand, we can also see the failure of treating heavy-light tetraquarks as asingle ¯33 color state for charm-light or charm-strange doubly-heavy tetraquarks. In the panel (b) of Fig. 1 we showthe expectation value of the different Jacobi coordinates over the tetraquark wave function, i.e., the average distancebetween the different constituents of the tetraquark [10]. One can see how when the binding increases, i.e. M Q /m q augments, the average distance between the two heavy quarks, h x i / , diminishes rapidly, while that of the twolight quarks, h y i / , although diminishing, remains larger. The heavy-to-light quark distance, h z i / , stays almostconstant for any value of M Q /m q . It is also worth noting how the tetraquark becomes compact in the bottom sector.As can be seen from Tables I and II, for deep binding, X T QQ / r . m . s . ( M + M ) = 0 . / . <
1, the tetraquarkis smaller than the two mesons of the threshold while close to ∆ E = 0, X T QQ / r . m . s . ( M + M ) = 0 . / . > QQ kernel and for the light quarks bound to the stationary color 3 state, to construct a QQ ¯ q ¯ q colorsinglet. As mentioned above, this result is less pronounced for other systems like charm-light ( cc ¯ q ¯ q ) or charm-strange( cs ¯ q ¯ q ) doubly-heavy tetraquarks. On the basis of the results shown in Table II, the schematic evolution of the T QQ state as a function of the ratio M Q /m q , in other words, from deep binding to a close-to-threshold meson-meson state,is shown in Fig. 2 [29]. M Q /m q ≈11 〈 x 〉 〈 y 〉 M Q /m q ≈16 M Q /m q ≈6 〈 z 〉 FIG. 2. From left to right, schematic evolution of a T QQ state as the heavy-quark mass decreases and, thus, the separationbetween the heavy quarks increases. The separation between the light quarks starts to augment close to threshold and theseparation between the heavy and the light quarks remains almost constant. The last scenario, M Q /m q ≈ T QQ molecule or two heavy-light mesons. III. TETRAQUARK LIFETIME AND DECAY MODES.
The double beauty T − bb isoscalar tetraquark with J P = 1 + is stable with respect to strong- and electromagneticinteractions [4, 5, 7–14], and thus it decays weakly. We have studied the semileptonic and nonleptonic decays of T − bb following closely the method developed in Ref. [30]. We present here the results for the most favorable final stateswhere T − bb might be looked for. The remaining channels and a detailed discussion of the technicalities will be presentedelsewhere [31].Among the semileptonic decays one can distinguish between processes with final states with a single meson, see panel(a) of Fig. 3, or those with two mesons, panel (b) of Fig. 3. The first case, T − bb → ¯ B ℓ − ¯ ν ℓ , involves a b ¯ u → W − → ℓ − ¯ ν ℓ transition that at tree level is described by the operator − iV ub G F √ u (0) γ µ (1 − γ )Ψ b (0) Ψ ℓ (0) γ µ (1 − γ )Ψ ν ℓ (0) , (7)where Ψ f is a quark field of a definite flavor f , G F is the Fermi coupling constant and V ub is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element. The decay width is given byΓ = 12 m T Z Z Z d P B (2 π ) E B d p ℓ (2 π ) E ℓ d p ν ℓ (2 π ) E ν ℓ (2 π ) δ (4) ( P T − P B − p ℓ − p ν ℓ ) × | V ub | G F L αβ ( p ℓ , p ν ℓ ) W αβ ( P T , P B ) , (8)where the lepton and hadron tensors are given by, L αβ = 8( p αℓ p βν ℓ + p βℓ p αν ℓ − g αβ p ℓ · p ν ℓ ± iǫ αβρλ p lρ p ν ℓ λ ) , (9) W αβ = 12 J T + 1 X λ,λ ′ h T → Bα ( h T → Bβ ) ∗ (10) h T → Bα = h B, λ ′ P B | Ψ u (0) γ α (1 − γ )Ψ b (0) | T, λ i , (11)where p i is the four-momentum of the particle i , J T stands for the spin of the tetraquark, | M, λ ′ P M i represents the The ± signs correspond respectively to decays into ℓ − ¯ ν ℓ and ℓ + ν ℓ . b bu dT bb ν l l − W − B bdT bb Bb Du l − ν l W − c (a) (b) FIG. 3. Representative diagrams for semileptonic decays of the T bb tetraquark: (a) Final state with a single meson. (b) Finalstate with two mesons. state of an M meson with three momentum P M and spin projection in the meson center of mass λ ′ , and | T, λ i is thestate of the tetraquark at rest. ǫ αβρλ is the fully antisymmetric tensor for which we take the convention ǫ = +1and g αα = (1 , − , − , − | V ub | G F π m T Z Z dE B dE ℓ Θ( m T − E B − E ℓ ) Θ(1 − | cos θ ℓ | ) ˜ L αβ W αβ ( P T , ˜ P B ) , (12)with ˜ P µB = ( E B , , , | P B | ) and cos θ ℓ = ( m T − E B − E ℓ ) − | P B | − | p ℓ | | P B | | p ℓ | . (13)In (12), since all the dependence on ϕ ℓ appears only in the lepton tensor, we have defined˜ L αβ = 116 π Z dϕ ℓ L αβ . (14)The matrix element (11) appearing in the hadron tensor can be expanded as, h T → Bρ = 4 p m T E B Z Z d p x d p z X α ,α α ,α (cid:20) ˆ φ ( B,λ ′ ) α ,α (cid:18) m b m b + m u P B + p x − p z (cid:19)(cid:21) ∗ ˆ φ ( T,λ ) α ,α α ,α ( p x , − p x − P B , p z ) × ( − / − s √ E u E b ¯ v s u (cid:16) − P B − p x − p z (cid:17) γ ρ (1 − γ ) u s b (cid:16) p x + 12 p z (cid:17) δ c c δ f b δ f u , (15)where ˆ φ is the Fourier transform of the radial wave function, obtained in Sec. II using the AL1 constituent model,and α i represents the quantum numbers of spin s i , flavor f i and color c i ( α i ≡ ( s i , f i , c i )) of a quark or an antiquark.For the second class of semileptonic decays represented diagrammatically in panel (b) of Fig. 3, with a b → c transition at the quark level, the operator is given by − iV cb G F √ c (0) γ µ (1 − γ )Ψ b (0) Ψ ℓ (0) γ µ (1 − γ )Ψ ν ℓ (0) , (16)and the decay width can be expressed asΓ = | V cb | G F π m T Z Z Z Z | P B | dE B d cos θ D | P D | dE D | ˜ P B + P D | dE ℓ Θ(1 − | cos θ ℓ | ) × Θ( m T − E B − E D − E ℓ ) ˜ L αβ W αβ ( P T , ˆ P B , ˆ P D ) , (17) du cbT bb Bb Dcud cd D c πη c (a) (b) bT bb Bb d
FIG. 4. Representative diagrams for nonleptonic decays of the T bb tetraquark: (a) Final state with only open flavor mesons.(b) Final state with hidden flavor mesons. The blue box represents a four-quark effective vertex containing the contribution ofthe W boson and radiative corrections as seen in Eq. (21). where ˆ P B = R ′ ˜ P B and ˆ P D = R ′ P D , where R ′ is a rotation that, for a fixed P D , takes ˜ P B + P D → (0 , , | ˜ P B + P D | ).In this case, cos θ ℓ = ( m T − E B − E D − E ℓ ) − | ˜ P B + P D | − | p ℓ | | ˜ P B + P D | | p ℓ | . (18)The matrix element appearing in the hadron tensor h T → M M ρ = h M , λ ′′ P | h M , λ ′ P | Ψ c (0) γ ρ (1 − γ )Ψ b (0) | T, λ i , (19)here written for a T → BD transition (for other cases, the changes are obvious), can be expressed as, h T → BDρ = 4 (2 π ) / p m T E B E D X α ,α α ,α Z Z d p x d p z (cid:20) ˆ φ ( D,λ ′′ ) α ,α (cid:18) − m u m c + m u P D − P B − p x − p z (cid:19)(cid:21) ∗ × (cid:20) ˆ φ ( B,λ ′ ) α ,α (cid:18) m b m b + m u P B + p x − p z (cid:19)(cid:21) ∗ X α ˆ φ ( T,λ ) α ,α α ,α ( p x , − p x − P B , p z ) × √ E c E b ¯ u s c (cid:16) P D + P B + p x + 12 p z (cid:17) γ ρ (1 − γ ) u s b (cid:16) p x + 12 p z (cid:17) δ c c δ f b δ f c . (20)Obviously B could also be a B ∗ and D a D ∗ . If we have a b → u quark transition, one has to change V cb → V ub andthe meson in the final state (apart from B ( B ∗ )) would be a nonstrange meson with u ¯ u or u ¯ d composition.We evaluate now the width of the nonleptonic decays T − bb → B − M M or ¯ B M ′ M ′ represented diagrammaticallyin Fig. 4. These decay modes involve a transition b → c, u at the quark level and they are governed, neglectingpenguin operators, by the effective Hamiltonian [32–34] H eff = G F √ (cid:8) V cb (cid:2) c ( µ ) Q cb + c ( µ ) Q cb (cid:3) + V ub (cid:2) c ( µ ) Q ub + c ( µ ) Q ub (cid:3) + h.c. (cid:9) , (21)where c , c are scale–dependent Wilson coefficients, and Q ib , Q ib , i = u, c , are local four-quark operators of thecurrent-current type given by Q ib = Ψ i (0) γ µ (1 − γ )Ψ b (0) (cid:20) V ∗ ud Ψ d (0) γ µ (1 − γ )Ψ u (0) + V ∗ us Ψ s (0) γ µ (1 − γ )Ψ u (0)+ V ∗ cd Ψ d (0) γ µ (1 − γ )Ψ c (0) + V ∗ cs Ψ s (0) γ µ (1 − γ )Ψ c (0) (cid:21) , (22) Q ib = Ψ d (0) γ µ (1 − γ )Ψ b (0) (cid:20) V ∗ ud Ψ i (0) γ µ (1 − γ )Ψ u (0) + V ∗ cd Ψ i (0) γ µ (1 − γ )Ψ c (0) (cid:21) + Ψ s (0) γ µ (1 − γ )Ψ b (0) (cid:20) V ∗ us Ψ i (0) γ µ (1 − γ )Ψ u (0) + V ∗ cs Ψ i (0) γ µ (1 − γ )Ψ c (0) (cid:21) , (23)where the different V jk are CKM matrix elements.We work in the factorization approximation which amounts to evaluate the hadron matrix elements of the effectiveHamiltonian as a product of two quark-current matrix elements: one is the matrix element for the T bb → BM transition, and the other accounts for the transition from vacuum to the other final meson M , see Fig 4. The lattercoupling is governed by the corresponding meson decay constant. When writing the factorization amplitude, therelevant coefficients of the effective Hamiltonian (21) are the combinations, a ( µ ) = c ( µ ) + 1 N C c ( µ ) a ( µ ) = c ( µ ) + 1 N C c ( µ ) , (24)with N C = 3 the number of colors. The energy scale µ appropriate in this case is µ ≃ m b and the values for a and a that we use are [33]: a = 1 . a = − . . (25)Note that the W -exchange diagrams, that play an important role in the decay of charm, are suppressed in the decayof b since they are proportional to a . The total decay width is given asΓ = 12 m T Z Z Z d P B (2 π ) E B d P (2 π ) E d P (2 π ) E (2 π ) δ (4) ( P T − P B − P − P ) × G F J T + 1 X λ T ,λ B λ ,λ |M λ T λ B λ λ ( P T , P B , P , P ) | . (26)Using invariance arguments as in the semileptonic decay case one finds,Γ = G F π m T Z Z dE B dE Θ(1 − | cos θ | ) Θ( m T − E B − E − M ) × J T + 1 X λ T ,λ B λ ,λ (cid:12)(cid:12)(cid:12) M λ T λ B λ λ ( P T , ˜ P B , ˆ P , ˆ P ) (cid:12)(cid:12)(cid:12) , (27)where cos θ = ( m T − E B − E ) − M − | P B | − | P | | P B | | P | , (28)and M involves the product of a hadron matrix element such as Eq. (19) and meson decay constants that are takenfrom experiment or lattice data. For instance, for a T − bb → B − D + D − decay, one has that M = V cb V ∗ cd a h T → B − D + α if D − P αD − , (29)In particular, for the decays presented in Table VI, we have used the meson decay constants listed in Table III.For the sake of completeness we have also evaluated the decay of the J P = 1 + T − bb isoscalar tetraquark into the J P = 0 + T bc isoscalar tetraquark, decay depicted in Fig. 5. The mass of the J P = 0 + bc ¯ u ¯ d isoscalar state has beenestimated in Ref. [7] where the authors obtain a central value 11 MeV below the ¯ BD threshold, although it is cautionedthat the precision of the calculation is not sufficient to determine whether the tetraquark is actually above or belowthis threshold. A systematic study of exotic QQ ′ ¯ q ¯ q four-quark states containing distinguishable heavy flavors, b and c ,has been recently performed with the AL1 model in Ref. [38]. The J P = 0 + isoscalar state was found to be strong andelectromagnetic-interaction stable with a binding energy of around 23 MeV. Other independent calculations made indifferent frameworks arrive to similar conclusions. Among them, it is important to emphasize the lattice QCD results TABLE III. Meson decay constants, in GeV, used in this work. f π − f π f ρ − ,ρ f D + f D ∗ + f D + s f D ∗ s + b dT bb b u l − ν l W − c T bc FIG. 5. Representative diagram for the semileptonic decay of the T bb J P = 1 + tetraquark to the T bc J P = 0 + tetraquark. of Ref. [39] where it is found evidence for the existence of a strong-interaction-stable ( I ) J P = (0)1 + bc ¯ u ¯ d four-quarkstate with a mass in the range of 15 to 61 MeV below the D ¯ B ∗ threshold. The decay width in this case is givenby (12), changing the final B meson by the T bc tetraquark and V ub by V cb , while the corresponding hadronic matrixelement is h T → T bc ρ = 2 p m T E T bc X α ,α α ,α Z Z Z d p x d p y d p z × (cid:20) ˆ φ ( T bc ,λ ′ ) α ,α α ,α (cid:18) − p x − m b − m c m b + m c ) p z − m b m b + m c P T bc , p y , p z + 2 m u m b + m c + 2 m u P T bc (cid:19)(cid:21) ∗ × X α ˆ φ ( T,λ ) α ,α α ,α ( p x , p y , p z ) 12 √ E c E b ¯ u s c (cid:16) P T bc + p x + 12 p z (cid:17) γ ρ (1 − γ ) u s b (cid:16) p x + 12 p z (cid:17) δ c c δ f b δ f c . (30)Let us now comment on the results. Some aspects could have been anticipated, and are verified. For instance, forthe T → B ( ∗ ) semileptonic decays depicted in Fig. 3(a), and due to the large phase space available in all cases, thedifferences among the widths into the three lepton families are very small. The corresponding results are shown inTable IV. We also note that the overlap in the hadron tensor between the T and the B ( B ∗ ) wave function slightlyfavors the pseudoscalar mesons. Anyhow, decays with a single meson in the final state are suppressed by at leasttwo orders of magnitude as compared to the semileptonic decays with two final mesons, and the leading non-leptonicmodes that are discussed below.For semileptonic decays involving two mesons in the final state, described by panel (b) of Fig. 3, the processesinvolving a b → c vertex are favored compared to those involving a b → u vertex, due to the larger CKM matrixelement | V cb | ∼ .
041 compared to | V ub | ∼ . s − = 0 . × − GeV, for the semileptonic decays with two mesons and a light ℓ = e, µ lepton in the final state. Though much smaller, we also give the widths for the corresponding channels with TABLE IV. Decay widths, in units of 10 − GeV, for processes described by Fig. 3(a).Final state Γ [10 − GeV]¯ B ∗ e − ¯ ν e . ± . B e − ¯ ν e . ± . B ∗ µ − ¯ ν µ . ± . B µ − ¯ ν µ . ± . B ∗ τ − ¯ ν τ . ± . B τ − ¯ ν τ . ± . The errors quoted correspond to the uncertainties of the Monte Carlo numerical integration. TABLE V. Largest decay widths, in units of 10 − GeV, for the processes described by Fig. 3(b). Here ℓ = e, µ .Final state Γ [10 − GeV] Final state Γ [10 − GeV] B ∗− D ∗ + ℓ − ¯ ν ℓ . ± . B ∗− D ∗ + τ − ¯ ν τ . ± . B ∗ D ∗ ℓ − ¯ ν ℓ ¯ B ∗ D ∗ τ − ¯ ν τ B ∗− D + ℓ − ¯ ν ℓ . ± . B ∗− D + τ − ¯ ν τ . ± . B ∗ D ℓ − ¯ ν ℓ ¯ B ∗ D τ − ¯ ν τ B − D ∗ + ℓ − ¯ ν ℓ . ± . B − D ∗ + τ − ¯ ν τ . ± . B D ∗ ℓ − ¯ ν ℓ ¯ B D ∗ τ − ¯ ν τ B − D + l − ¯ ν l . ± . B − D + τ − ¯ ν τ . ± . B D ℓ − ¯ ν ℓ ¯ B D τ − ¯ ν τ a final τ since they could be interesting in the context of studies of lepton-flavor universality violation. Due to spinrecoupling coefficients, the largest decay widths appear for vector mesons in the final state. In short, the largestpreferred semileptonic decay are B ( ∗ ) D ( ∗ ) ℓ ¯ ν ℓ with the various combinations of spins for the mesons, and ℓ = e, µ .Table VI displays now the most important nonleptonic decay modes. All of them contain a b → c vertex and an a factor, and the dominant ones have a D ( ∗ )( s ) meson in the final state. Once again vector mesons are favored inthe final state. As a consequence of the factorization approximation, processes with D s or a light meson final statesarising from vacuum have decay widths comparable to the corresponding semileptonic decay. This is due to the largevalue of the Cabibbo allowed CKM matrix elements | V cs | ∼ | V ud | ∼ .
97 [35] and the fact that the hadronic matrixelements are proportional to a in those cases. Decay channels not shown in Tables V and VI are suppressed by atleast one order of magnitude. For instance, final states with J/ Ψ or η c mesons, are suppressed by more than oneorder of magnitude since their widths are proportional to | V cd | a . According to our study, the promising final statesamong the nonleptonic decays are ¯ B ∗− D ∗ + D ∗ s − , ¯ B ∗ D ∗ D ∗ s − , and ¯ B ∗− D ∗ + ρ − . TABLE VI. Largest decay widths, in units of 10 − GeV, for the processes described by Fig. 4.Final state Γ [10 − GeV] Final state Γ [10 − GeV] B ∗− D ∗ + D − s . ± . B − D ∗ + D ∗ s − . ± . B ∗ D ∗ D − s ¯ B D ∗ D ∗ s − B ∗− D ∗ + D ∗ s − . ± . B − D + D ∗ s − . ± . B ∗ D ∗ D ∗ s − ¯ B D D ∗ s − B ∗− D + D − s . ± . B ∗− D ∗ + ρ − . ± . B ∗ D D − s B ∗− D ∗ + π − . ± . B ∗− D + D ∗ s − . ± . B ∗− D + ρ − . ± . B ∗ D D ∗ s − B ∗− D + π − . ± . B − D ∗ + D − s . ± . B − D ∗ + ρ − . ± . B D ∗ D − s B − D ∗ + π − . ± . Finally, in Table VII we show the results for the semileptonic decay corresponding to Fig. 5 with a J P = 0 + T bc isoscalar tetraquark in the final state. In our calculation, the total semileptonic decay width with a final J P = 0 + T bc isoscalar tetraquark turns out to be 7 . × − GeV, in clear disagreement with the result of Ref. [17] obtained usinga QCD three-point sum rule approach.The total decay width of the T − bb tetraquark, as calculated in this work, is of the order of Γ ≈ × − GeV,which means a lifetime τ ≈ . TABLE VII. Decay widths, in units of 10 − GeV, for the processes described by Fig. 5.Final state Γ [10 − GeV] T bc e − ν e . ± . T bc µ − ν µ . ± . T bc τ − ν τ . ± . IV. SUMMARY AND OUTLOOK
We have presented the first comprehensive study of the flavor-exotic J P = 1 + T − bb isoscalar tetraquark. It includesan accurate solution of the four-body problem within a quark model, which characterizes the structure of the state,and an estimate of the lifetime and of the rates for the leading semileptonic and nonleptonic decay modes which arethe most promising final states where the tetraquark should be looked for. We have shown how pairwise interactionsbased on color-octet exchange induce mixing between the ¯33 and 6¯6 states in the QQ − ¯ q ¯ q basis, enhancing the ¯33components for larger values of M Q due to the attractive chromoelectric interaction of the QQ pair that it is absentin the Q ¯ q threshold. This result is only valid in the bottom sector. In the charm sector, the binding mechanism isdifferent: the ¯33 and 6¯6 components have a similar probability and are mixed by the chromomagnetic interaction.We have shown how the structure of the T QQ state evolves from a molecular-like system to a compact-like structurewhen moving from the charm to the bottom sector.For the first time, the lifetime of the T − bb tetraquark has been calculated in a quark model beyond simple guess-by-analogy estimations. The total decay width of the T − bb found in this work is Γ ≈ × − GeV, corresponding to alifetime τ ≈ . B ∗− D ∗ + l − ¯ ν ℓ and ¯ B ∗ D ∗ ℓ − ¯ ν ℓ among the semileptonic decays,and ¯ B ∗− D ∗ + D ∗ s − , ¯ B ∗ D ∗ D ∗ s − , and B ∗− D ∗ + ρ − among the nonleptonic ones. The T bc ℓ − ν ℓ semileptonic decay isalso relevant but in our calculation is not dominant.Our study complements recent estimates for the production cross sections of T bb tetraquarks based on MonteCarlo event generators pointing towards an excellent discovery potential in ongoing and forthcoming proton-protoncollisions at the LHC [40]. The possible formation of this state in relativistic heavy-ion collisions at the LHC has alsobeen recently discussed in detail within the quark coalescence model using realistic model wave functions with goodprospects [41].The spectroscopy of exotic states with hidden heavy flavor has revealed how interesting the interaction of heavyhadrons is, with presumably a long-range part of Yukawa type, and a short-range part mediated by quark-quark andquark-antiquark forces. A new sector with stable flavor-exotic states, such as the T bb , remains to be investigated. Anexperimental effort towards the detection of this compact tetraquark states is now timely. Its existence is essential tovalidate our understanding of low-energy QCD in the multiquark sector.A long lifetime for the T − bb tetraquark can ease its detection through the method of “displaced vertex” proposed in[42]. V. ACKNOWLEDGMENTS
This work has been funded by Ministerio de Econom´ıa, Industria y Competitividad and EU FEDER under ContractsNo. FPA2016-77177 and FIS2017-84038-C2-1-P, and by the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement no. 824093. [1] H. -X. Chen, W. Chen, X. Liu, and S. -L. Zhu, Phys. Rep. , 1 (2016); R. A. Brice˜no et al. , Chin. Phys. C , 042001(2016); J.-M. Richard, Few-Body Syst. , 1185 (2016); R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl.Phys. , 143 (2017); A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl. Phys. , 123 (2017); A. Esposito, A. Pilloni,and A. D. Polosa, Phys. Rep. , 1 (2017).[2] R. L. Jaffe, Phys. Rev. D , 267 (1977); , 281 (1977).[3] J. D. Weinstein and N. Isgur, Phys. Rev. D , 2236 (1990).[4] J. -P. Ader, J. -M. Richard, and P. Taxil, Phys. Rev. D , 2370 (1982).[5] A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Phys. Rev. Lett. , 142001 (2017).[6] P. Junnarkar, N. Mathur, M. Padmanath, Phys. Rev. D (2019) 034507.[7] M. Karliner and J. L. Rosner, Phys. Rev. Lett. , 202001 (2017).[8] E. J. Eichten and C. Quigg, Phys. Rev. Lett. , 202002 (2017).[9] P. Bicudo, K. Cichy, A. Peters, and M. Wagner, Phys. Rev. D , 034501 (2016).[10] J. Vijande, A. Valcarce, and N. Barnea, Phys. Rev. D , 074010 (2009).[11] J. -M. Richard, A. Valcarce, and J. Vijande, Phys. Rev. C , 035211 (2018).[12] S. -Q. Luo, K. Chen, X. Liu, Y. -R. Liu, and S. -L. Zhu, Eur. Phys. J. C , 709 (2017). We thank A. Ali for calling our attention on this article. [13] M. -L. Du, W. Chen, X. -L. Chen, and S. -L. Zhu, Phys. Rev. D , 014003 (2013).[14] A. Czarnecki, B. Leng, and M. B. Voloshin, Phys. Lett. B , 233 (2018).[15] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , 112001 (2017).[16] Y. Xing and R. Zhu Phys. Rev. D , 053005 (2018).[17] S.S. Agaev, K. Azizi, B. Barsbay, H. Sundu, Phys. Rev. D (2019) 033002.[18] J. Vijande and A. Valcarce, Phys. Rev. C , 035204 (2009).[19] J. Vijande and A. Valcarce, Symmetry , 155 (2009).[20] E. Hiyama, A. Hosaka, M. Oka, and J. M. Richard, Phys. Rev. C , 045208 (2018).[21] M. Oka, S. Maeda, and Y. R. Liu, Int. J. Mod. Phys. Conf. Ser. , 1960004 (2019).[22] C. Semay and B. Silvestre-Brac, Z. Phys. C , 271 (1994).[23] J. -M. Richard, A. Valcarce, and J. Vijande, Phys. Lett. B , 710 (2017).[24] D. Janc and M. Rosina, Few Body Syst. , 175 (2004).[25] J. L. Ballot and J. -M. Richard, Phys. Lett. B , 449 (1983); S. Zouzou, B. Silvestre-Brac, C. Gignoux, and J. -M. Richard,Z. Phys. C , 457 (1986); D. M. Brink and F. Stancu, Phys. Rev. D , 4665 (1994).[26] M. Harvey, Nucl. Phys. , 301 (1981). Phys. Lett. B , 242 (1986).[27] Y. Ikeda, B. Charron, S. Aoki, T. Doi, T. Hatsuda, T. Inoue, N. Ishii, K. Murano, H. Nemura, and K.Sasaki, Phys. Lett.B , 85 (2014).[28] H. J. Lipkin,[29] C. Quigg, in rd Rencontres de Moriond QCD High Energy Interactions Conference , La Thuile, Italy (2018),arXiv:1804.04929 [hep-ph].[30] E. Hern´andez, J. Nieves, and J. M. Verde-Velasco, Phys. Rev. D , 074008 (2006).[31] E. Hern´andez, J. Vijande, and A. Valcarce, to be published.[32] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Rev. D , 094020 (2003).[33] M. A. Ivanov, J. G. K¨orner, and P. Santorelli, Phys. Rev. D , 054024 (2006).[34] P. Colangelo and F. De Fazio, Phys. Rev. D , 034012 (2000).[35] M. Tanabashi et al. , Phys. Rev. D , 030001 (2018).[36] M. Artuso et al. (CLEO Collaboration), Phys. Rev. Lett. , 251801 (2005).[37] D. Becirevic, Ph. Boucaud, J. P. Leroy, V. Lubicz, G. Martinelli, F. Mescia, and F. Rapuano, Phys. Rev. D , 074501(1999).[38] T. F. Caram´es, J. Vijande, and A. Valcarce, Phys. Rev. D , 014006 (2019).[39] A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Phys. Rev. D , 054505 (2019).[40] A. Ali, Q. Qin, and W. Wang, Phys. Lett. B , 605 (2018).[41] J. Hong, S. Cho, T. Song, and S. -H. Lee, Phys. Rev. C , 014913 (2018); S. Cho et al. [ExHIC Collaboration], Phys.Rev. C , 064910 (2011); C. E. Fontoura, G. Krein, J. Vijande, and A. Valcarce, Phys. Rev. D, , 094037 (2019).[42] T. Gershon, A. Poluektov, JHEP1901