Double-heavy axial-vector tetraquark $T_{bc;\bar{u}\bar{d}}^{0}$

The mass and coupling of the axial-vector tetraquark $T_{bc;\bar{u}\bar{d} }^{0}$ (in a short form $T_{bc}^{0}$) are calculated by means of the QCD two-point sum rule method. In computations we take into account contributions arising from various quark, gluon and mixed vacuum condensates up to dimension 10. The central value of the mass $m=(7105 \pm 155)~\mathrm{ MeV}$ lies below the thresholds for the strong and electromagnetic decays of $T_{bc}^{0}$ state, and hence it transforms to conventional mesons only through the weak decays. In the case of $m=7260~\mathrm{MeV}$ the tetraquark $T_{bc}^{0}$ becomes the strong- and electromagnetic-interaction unstable particle. In the first case, we find the full width and mean lifetime of $ T_{bc}^{0}$ using its dominant semileptonic decays $T_{bc}^{0}\to T_{cc;\bar{ u}\bar{d}}^{+}l\overline{\nu }_{l}$ ($l=e,\ \mu, \tau$), where the final-state tetraquark is a scalar state. We compute also partial widths of the nonleptonic weak decays $T_{bc}^{0}\to T_{cc;\bar{u}\bar{d} }^{+}\pi^{-}(K^{-}, D^{-}, D_{s}^{-})$, and take into account their effects on the full width of $T_{bc}^{0}$. In the context of the second scenario we calculate partial widths of $S$-wave strong decays $T_{bc}^{0}\to B^{\ast -}D^{+}$ and $T_{bc}^{0}\to \overline{B}^{\ast 0}D^{0}$, and using these channels evaluate the full width of $T_{bc}^{0}$. Predictions for $\Gamma_{ \mathrm{full}} =(3.98\pm 0.51)\times 10^{-10}~\mathrm{MeV}$ and mean lifetime $\tau=1.65_{-0.18}^{+0.25}~\mathrm{ps}$ of $T_{bc}^{0}$ obtained in the context of the first option, as well as the full width $\Gamma_{\mathrm{ full}}=(63.5\pm 8.9)~\mathrm{MeV}$ extracted in the second scenario may be useful for experimental and theoretical exploration of double-heavy exotic mesons.

The spectroscopic parameters of the axial-vector tetraquark T 0 bc;ūd are calculated by means of the QCD two-point sum rule method. In computations we take into account contributions arising from various quark, gluon and mixed vacuum condensates up to dimension ten. The mass m = (7105 ± 155) MeV of this particle lies below the thresholds for the strong and electromagnetic decays of T 0 bc;ūd state, and hence it transforms to conventional mesons through the weak decays. The full width and mean lifetime of T 0 bc;ūd are estimated by using its dominant semileptonic decays T 0 bc;ūd → T + cc;ūd lν l (l = e, µ, τ ), where the final-state tetraquark is a scalar state. We calculate partial widths of these processes in terms of the form factors Gi(q 2 ), i = (0, 1, 2, 3) which determine long-distance effects of the relevant weak transitions. Obtained predictions for the full width (3.3 ± 0.5) × 10 −10 MeV and mean lifetime τ = 1.99 +0. 36 −0.26 ps may be useful for experimental searches for the exotic axial-vector meson T 0 bc;ūd .

I. INTRODUCTION
During last two decades double-heavy tetraquarks as real candidates for stable four-quark states became subjects of intensive studies. In the pioneering papers [1][2][3] it was demonstrated that a heavy Q and light q quarks may form the stable exotic mesons QQqq provided the ratio m Q /m q is large enough. These results were obtained in the context of a potential model with the additive pairwise interaction, but even models with relaxed restrictions on the confining potential led to the similar predictions. Indeed, in accordance with Ref. [4] the isoscalar axial-vector tetraquark T − bb;ud turns to be strong-interaction stable state that lies below the BB * threshold. It is worth noting that an only constraint imposed in Ref. [4] on the potential was its finiteness at close distances of two particles. Therefore, T − bb;ud decays to conventional mesons only through weak processes and has a long lifetime, which is important for its experimental exploration. A situation with the tetraquarks T bc;qq ′ and T cc;qq ′ was not clear, because bc and cc diquarks might constitute both stable and unstable states.
In years followed after this progress, various models of high energy physics were used to investigate the double-heavy tetraquarks T QQ [5][6][7][8][9][10][11][12]. Recent interest to these problems was inspired by results of the LHCb Collaboration on properties of the doubly charmed baryon Ξ ++ cc = ccu [13]. Parameters of this baryon were used in Ref. [14] to evaluate the mass and analyze possible decay channels of T − bb;ud . Predictions obtained there confirmed the stability of T − bb;ud against the strong and electromagnetic decays to B − B * 0 and B − B 0 γ, respectively. The strong-interaction stable nature of the tetraquarks T − bb;ud , T − bb;us , and T 0 bb;ds was demonstrated in Ref. [15] by invoking heavy-quark symmetry relations. The mass and coupling of T − bb;ud was evaluated in our work [16] as well, in which we estimated also its full width and mean lifetime using the semileptonic decay channel T − bb;ud → Z 0 bc;ud lν l . Another class of four-quark mesons, namely one that contains the heavy diquarks bc is on agenda of physicists as well. The scalar and axial-vector tetraquarks bcud are particles of special interest, because they may form strong-interaction stable compounds. But calculations performed in the context of different approaches lead controversial results. Thus, the Bethe-Salpeter method predicts the mass of the scalar tetraquark Z 0 bc;ud (in what follows Z 0 bc ) at around 6.93 GeV, which is below the threshold 7145 MeV for S-wave strong decays to heavy mesons B − D + and B 0 D 0 [17]. Recent analysis demonstrated that Z 0 bc lies 11 MeV below this threshold [14], whereas the authors of Ref. [15] found the masses of the scalar and axial-vector tetraquarks bcud equal to 7229 MeV and 7272 MeV, respectively. These predictions make kinematically allowed their strong decays to ordinary B − D + /B 0 D 0 and B * D mesons.
It is interesting that lattice calculations prove the strong-interaction stabile nature of the axial-vector tetraquark udbc, because its mass is below the DB * threshold [18]. However, the authors could not decide would this exotic meson decay weakly or might transform also to the final state DBγ. The stability of J P = 0 + and 1 + isoscalar tetraquarks bcud was confirmed in Ref. [19], in which it was found that J P = 0 + state is a strongand electromagnetic-interaction stable particle, whereas J P = 1 + may also transform through the electromagnetic interaction.
In the context of the QCD sum rule approach the spectroscopic parameters of the scalar tetraquark Z 0 bc were calculated also in our work [16]. For the mass of Z 0 bc computations predicted m Z = (6660 ± 150) MeV, which is considerably below the threshold 7145 MeV. The electromagnetic decay modes Z 0 bc → B 0 D 0 1 γ and B * D * 0 γ are among forbidden processes as well, because relevant thresholds exceed 7600 MeV and are higher than the mass of Z 0 bc . In other words, in accordance with our results the scalar tetraquark Z 0 bc is a strongand electromagnetic-interaction stable particle. The Z 0 bc transforms due to semileptonic decays, which allowed us to find in Ref. [20] its full width and mean lifetime.
In the present article we study the axial-vector tetraquark T 0 bc;ud (hereafter T 0 bc ) by computing its spectroscopic parameters and width. The mass m and coupling f of T 0 bc are obtained in the framework of the QCD two-point sum rule method by taking into account vacuum expectation values of the local quark, gluon and mixed operators up to dimension ten. The central value of m = (7105 ± 155) MeV extracted here is lower than the thresholds 7190 MeV and 7286 MeV for strong S-wave decays to final states B * − D + /B * D 0 and B − D * + /B 0 D * 0 , respectively. This mass is also lower than the threshold 7145 MeV for the electromagnetic decays D + B − γ/D 0 B 0 γ. Therefore, the full width and lifetime of the exotic meson T 0 bc should be determined from its semileptonic decays.
The processes T 0 bc → T + cc;ud lν l , l = e, µ and τ are the dominant weak decay channels of T 0 bc . We treat the final-state tetraquark T + cc;ub as a scalar particle. The differential rates of these semileptonic decays are determined by the weak form factors G i (q 2 ) (i = 0, 1, 2, 3), which are evaluated by employing the QCD three-point sum rule approach. Then, partial width of the decays T 0 bc → T + cc;ud lν l can be found by integrating the relevant differential rates over the momentum transfer q 2 . But the sum rule method does not encompass all kinematically allowed values of q 2 , therefore we introduce fit functions that coincide with sum rule predictions, and can be extrapolated to cover a whole integration region.
This article is organized in the following manner: In Section II, from analysis of the two-point correlation function with an appropriate interpolating current, we derive sum rules to evaluate the spectroscopic parameters of the tetraquark T 0 bc . In the next Section, using the parameters of T 0 bc and ones of the final-state tetraquarks, we calculate the partial width of the weak decays. To this end, we derive the sum rules for the weak form factors and by means of fit functions extrapolate them to the whole region, where an integration over q 2 should be carried out. Section IV is reserved for discussion and concluding notes.

II. MASS AND COUPLING OF THE AXIAL-VECTOR TETRAQUARK T 0 bc;ud
In this section we extract the spectroscopic parameters of the axial-vector tetraquark T 0 bc from the QCD sum rules. To this end, we start from analysis of the correlation function Π µν (p), which is given by the formula Here J µ (x) is the interpolating current to the axial-vector tetraquark T 0 bc . We suggest that T 0 bc is built of the scalar diquark and axial-vector antidiquark, and hence its current has the form Here a and b are the color indices and C is the charge conjugation operator. The current (2) has the antisymmetric color structure [3 c ] bc ⊗ [3 c ] ud and describes a most stable four-quark state with the quantum numbers 1 + , where b T Cγ 5 c and uγ µ Cd T are the scalar diquark and axial-vector antidiquark, respectively. To derive required sum rules we find, in accordance with prescriptions of the method, the correlation function Π µν (p) using the tetraquark's mass m and coupling f . We consider it as a ground-state particle, and isolate the first term in Π Phys Equation (3) is obtained by saturating the correlation function with a complete set of J P = 1 + states and carrying out the integration over x. Contributions of higher resonances and continuum states to Π Phys µν (p) are denoted by the dots.
To simplify further the correlator Π Phys µν (p) it is useful to define the matrix element with ǫ µ being the polarization vector of the T 0 bc state. Then in terms of m and f the correlation function Π Phys µν (p) takes the form The QCD side of the sum rule is determined by the correlation function Π µν (p), but calculated now by employing the quark propagators where S ab q (x) is the heavy (b, c)-or light (u, d)-quark propagators. Their explicit expressions can be found in Ref. [21]. In Eq. (6) we use the shorthand notation The correlation function Π µν (p) contains the different Lorentz structures one of which should be chosen to get the sum rules. The invariant amplitudes Π Phys (p 2 ) and Π OPE (p 2 ) corresponding to the terms ∼ g µν are convenient for our aim, because they do not receive contributions from the scalar particles.
After picking up and equating corresponding invariant amplitudes, we apply the Borel transformation to both sides of the obtained expression. This is necessary to suppress contributions of the higher resonances and continuum states. Afterwards, one has to subtract continuum contributions, which is achieved by invoking suggestion on the quark-hadron duality. The obtained equality acquires a dependence on auxiliary parameters of the sum rules M 2 and s 0 : first of them is the Borel parameter appeared due to corresponding transformation, the second one s 0 is the continuum subtraction parameter that separates the ground-state and higher resonances from each another.
The final sum rule for the mass of the state T 0 bc reads: where For the coupling f one obtains the expression Here ρ OPE (s) is the two-point spectral density, which is determined as an imaginary part of the term in Π OPE µν (p) proportional to g µν , and calculated by taking into account the quark, gluon and mixed vacuum condensates up to dimension ten.
In addition to M 2 and s 0 , numerical values of which depend on the considering problem, the sum rules (8) and (9)  The parameters M 2 and s 0 should satisfy constraints that are standard for the the sum rule computations. Thus, at maximum of the Borel parameter the pole contribution (PC) should be larger than some fixed value, whereas the main criterium to fix the minimum of a Borel window is convergence of the operator product expansion (OPE). Additionally, at minimum M 2 the perturbative contribution has to exceed the nonperturbative terms considerably. Because quantities extracted from the sum rules demonstrate dependence on the auxiliary parameters, the regions for M 2 and s 0 should minimize these side effects, as well. Our analysis proves that the working regions satisfy all aforementioned restrictions. Thus, within the region M 2 ∈ [5.5, 7] GeV 2 the pole contribution decreases approximately from 58% till 34%. A detailed picture for PC is presented in Fig. 1, where we plot the pole contribution as a function of M 2 and s 0 . The minimum M 2 min is found from analysis of the ratio where Π(M 2 , s 0 ) is the Borel transformed and subtracted function Π OPE (p 2 ). In the present work as a measure of the convergence we use the sum of last three terms in OPE DimN = Dim(8 + 9 + 10) and impose the constraint on R(M 2 ): the restriction R(M 2 min ) ≤ 0.01 is fulfilled at 5.5 GeV 2 . The perturbative contribution at M 2 = 5.5 GeV 2 amounts to 68% of the full result and overshoots contribution of the nonperturbative terms. In Fig. 2 we demonstrate the dependence of the mass m on M 2 and s 0 , where weak residual effects of these parameters are seen.
Our results for m and f read: Theoretical errors of the mass is milder than ones of the coupling, nevertheless all these ambiguities do not exceed standard limits of sum rule computations reaching 2.2% and ±20% of the corresponding central values, respectively.The spectroscopic parameters of the axial-vector tetraquark T 0 bc evaluated in this section form a basis for our further investigations. The same as in Fig. 1, but for the mass of the tetraquark T 0 bc .
III. SEMILEPTONIC DECAYS T 0 bc → T + cc;ud lν l As it has been emphasized above T 0 bc is stable against the strong and electromagnetic interactions. To make this conclusion we refer to central value of the mass (13), then m resides 85/190 MeV and 45 MeV below the strong and electromagnetic thresholds, respectively. The semileptonic decays T 0 bc → T + cc;ud lν l of the tetraquark T 0 bc are caused by weak transition b → W − c → clν of the heavy b-quark. It is not difficult to see, that due to large mass difference between the tetraquarks T 0 bc and T + cc;ud , all of the transitions T 0 bc → T + cc;ud lν l with l = e, µ and τ are kinematically allowed. We restrict ourselves by considering only the dominant process b → W − c, because the decay b → W − u due to smallness of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V bu | 2 /|V bc | 2 ≃ 0.01 is suppressed relative to the first one. The W − boson can also create a quark-antiquark pair W − → qq ′ , which afterwards forms a pair of conventional mesons. But creating of two mesons from qq ′ requires additional quarks q ′′ q ′′ appearing due to a gluon g emitted from one of q or q ′ quarks. As a result, multi-meson processes T 0 bc;ud → T + cc;ud M 1 (qq ′′ )M 2 (q ′ q ′′ ) are suppressed relative to the semileptonic decays by the additional factor α 2 s |V qq ′ | 2 , which makes them subdominant among decays of T 0 bc;ud . At the tree-level, the transition b → W − c is described by means of the effective Hamiltonian Here G F is the Fermi coupling constant, and V bc is the element of the CKM matrix. After substituting H eff between the initial and final tetraquark fields and factoring out the leptonic piece we get the matrix element of the current which has to be calculated in terms of the weak form factors G i (q 2 ): they parameterize the long-distance dynamics of the transition In Eq. (16) p and ǫ are the momentum and polarization vector of the T 0 bc , p ′ is the momentum of the scalar tetraquark T + cc;ud . Here we also use the shorthand notations m = m + m T and P µ = p ′ µ + p µ with m T being the mass of the final-state tetraquark. The q µ = p µ −p ′ µ is the momentum transferred to the leptons changing within the limits m 2 l ≤ q 2 ≤ (m − m T ) 2 , where m l is the mass of the lepton l.
The form factors G i (q 2 ) are key quantities to be extracted from the sum rules. To this end, we consider the following three-point correlation function: where J ν (x) and J T (y) are the interpolating currents corresponding to the states T 0 bc and T + cc;ūd , respectively. The current J ν (x) has been introduced by Eq. (2). The interpolating current for the state T + cc;ūd is given by the expression: where ǫ ǫ = ǫ abc ǫ ade .
Here, ǫ abc [c T b Cγ α c c ] and ǫ ade [u d γ α Cd T e ] are the axial-vector diquark and antidiquark, respectively. Then the scalar designation of the final tetraquark T 0 bc;ud stems naturally from the internal structure of the initial four-quark state T 0 bc;ud , which is the axial-vector particle composed of the scalar diquark b T Cγ 5 c and axial-vector antidiquark uγ µ Cd T . The semileptonic decay T 0 bc;ud → T + cc;ud + W − runs through b → W − c, which transforms the scalar diquark bc to the final axial-vector cc, leaving, at the same time, unchanged the initial light antidiquark; the light axialvector antidiquark ud appears both in the initial and final states. The designation of T + cc;ud as an axial-vector requires ud to be a scalar, which implies additional spin-rearrangement in the initial axial-vector ud diquark, which evidently suppresses the corresponding process.
Our strategy to derive sum rules for the form factors G i (q 2 ) is the same as in all of this kind studies. In fact, to determine the phenomenological side of the sum rule Π Phys µν (p, p ′ ) we express the correlation function Π µν (p, p ′ ) in terms of the spectroscopic parameters of particles involving into the decay process. Afterwards we find the QCD side (or OPE) side of the sum rules Π OPE µν (p, p ′ ) by computing the same correlation function in terms of quark propagators. By matching the obtained results and utilizing the quark-hadron duality assumption we extract sum rules and evaluate the physical quantities of interest. Because the quark propagators contain quark, gluon and mixed vacuum condensates, the sum rules express the physical quantities as functions of nonperturbative parameters.
In the context of this approach the function Π Phys µν (p, p ′ ) can be recast into the form where m T is the mass of T + cc;ūd . In the expression above we take into account contribution appearing due to only the ground-state particles, denoting contributions of the higher resonances and continuum states by the dots.
Transformation of the ground-state term in Π Phys µν (p, p ′ ) can be completed by detailing the matrix elements in its expression. The matrix element of T 0 bc and the matrix element for the transition T cc (p ′ )|J tr µ |T (p, ǫ) are given by Eqs. (4) and (16), respectively. The remaining quantity has a simple form and depends only on the mass and coupling f T of the tetraquark T + cc;ūd . Benefiting from these explicit formulas, for Π Phys µν (p, p ′ , q 2 ) we obtain The function Π OPE µν (p, p ′ ) forms the second side of the sum rules: The required sum rules for the form factors G i (q 2 ) can be obtained by equating invariant amplitudes corresponding to the same Lorentz structures both in  Π Phys µν (p, p ′ , q 2 ) and Π OPE µν (p, p ′ ). Because in the threepoint sum rules the invariant amplitudes are functions of p ′2 and p 2 , to suppress contributions of higher resonances and continuum states we have to apply the double Borel transformation over these variables. As a result, the final expressions depend on a set of Borel parameters M 2 = (M 2 1 , M 2 2 ). The continuum subtraction is performed in two channels using two continuum parameters s 0 = (s 0 , s ′ 0 ). The form factor G 0 (q 2 ) is obtained by using the structure g µν and reads: The form factors G i (q 2 ) (i = 1, 2, 3) are derived employing other Lorentz structures in the correlation functions: The sum rules (23) and (24) are written down in terms of the spectral densities ρ i (s, s ′ , q 2 ) which are proportional to the imaginary parts of the corresponding terms in Π OPE µν (p, p ′ ). They contain the perturbative and nonperturbative contributions, and are calculated with dimension-five accuracy.
To compute the weak form factors G i (M 2 , s 0 , q 2 ) we need numerical values of parameters which enter to the sum rules. The vacuum condensates are given in Eq. (10), whereas the spectroscopic parameters of the tetraquark T + cc;ūd is borrowed from our work [22]. The mass and coupling of the initial particle T 0 bc have been calculated in the previous section; these and other parameters are collected in Table I. In computations, we impose on the auxiliary parameters M 2 and s 0 the same constraints as in the mass calculations: the set (M 2 1 , s 0 ) for the initial particle channel is determined by Eq. (11)  Results of sum rule calculations in the case of G 0 (q 2 ), as an example, are shown in Fig. 3. The similar predic- tions have been obtained for the remaining form factors as well. The sum rule results for the functions G i (q 2 ) are necessary, but not enough to calculate the partial width of the process T 0 bc → T + cc;ūd lν l . The reason is that these form factors determine its differential decay rate dΓ/dq 2 (see, Appendix in Ref. [16]). The partial width Γ should be found by integrating dΓ/dq 2 over q 2 within limits allowed by the kinematical constraints m 2 l ≤ q 2 ≤ (m − m T ) 2 . But sum rules do not cover all this region, and give reliable results within the limits m 2 l ≤ q 2 ≤ 8 GeV 2 . Therefore, one has to introduce the model functions F i (q 2 ), which at accessible for the sum rule computations q 2 coincide with G i (q 2 ), but can be extrapolated to the whole integration region.
There are different analytical expressions for the fit functions. The functions are convenient for our purposes. Here F i 0 , c i 1 , c i 2 and m 2 fit are the fit parameters numerical values of which are collected in Table II which are main results of the present work.

IV. DISCUSSION AND CONCLUDING NOTES
In the present work we have explored, in a rather detailed form, the axial-vector tetraquark T 0 bc . As we have emphasized in Section I, there are different predictions for the mass and stability properties of the axial-vector tetraquark T 0 bc in the literature. We have calculated the mass m and coupling f of this tetraquark by means of the QCD sum rule method. The central value of our prediction for the mass 7105 MeV is below both the strong and electromagnetic thresholds, and therefore T 0 bc should transform through the weak transitions.
We have calculated the partial width of the semileptonic decays T 0 bc → T + cc;ūd lν l (l = e, µ and τ ) and evaluated the full width Γ = (3.3±0.5)×10 −10 MeV and mean lifetime τ ≈ 2 ps of the T 0 bc . In our previous work [20] we computed the same parameters of the scalar tetraquark Z 0 bc . It is instructive to compare parameters of the scalar and axial-vector bcud states with each other. The scalar compound Z 0 bc with the mass 6660 MeV has a more stable nature and lives τ ≈ 28 ps which is considerably longer than τ ≈ 2 ps of the T 0 bc . Because the scalar tetraquark T + cc;ūd decays strongly to a pair of conventional D + D 0 mesons [22], it is not difficult to estimate branching ratios of three decay channels of T 0 bc : Obtained in the present work, predictions for the spectroscopic parameters, full width and mean lifetime of the axial-vector tetraquark T 0 bc , as well as information on branching ratios (29) are useful to observe these exotic mesons in existing facilities.