Strong Vertices of Doubly Heavy Spin-3/2 Baryons with Light Pseudoscalar Mesons
aa r X i v : . [ h e p - ph ] F e b Strong Vertices of Doubly Heavy Spin-3/2 Baryonswith Light Pseudoscalar Mesons
A. R. Olamaei , , K. Azizi ,, , , S. Rostami Department of Physics, Jahrom University, Jahrom, P. O. Box 74137-66171, Jahrom, Iran Department of Physics, University of Tehran, North Karegar Ave. Tehran 14395-547, Iran Department of Physics, Doˇgu¸s University, Acibadem-Kadik¨oy, 34722 Istanbul, Turkey School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM),P. O. Box 19395-5531, Tehran, Iran
Abstract
The strong coupling constants are basic quantities that carry informationof the strong interactions among the baryon and meson multiplets as well asinformation on the natures and internal structures of the involved hadrons. Theseparameters enter to the transition matrix elements of various decays as maininputs and they play key roles in analyses of the experimental data includingvarious hadrons. We determine the strong coupling constants among the doublyheavy spin-3 / ∗ QQ ′ and Ω ∗ QQ ′ , and light pseudoscalar mesons, π , K and η , using the light-cone QCD. The values obtained for the strong couplingconstants under study may be used in construction of the strong potentials amongthe doubly heavy spin-3/2 baryons and light pseudoscalar mesons. Introduction
Doubly heavy baryons composed of two heavy quarks and a single light quark are inter-esting objects, investigation of which can help us better understand the nonperturbativenature of QCD. Their investigation provides a good framework for understanding andpredicting the spectrum of heavy baryons. Theoretical studies on different aspectsof these baryons may shed light on the experimental searches of these states. Thesebaryons have been in the focus of various theoretical studies [1–32]. Despite their pre-dictions decades ago via quark model, we have very limited experimental knowledgeon these baryons. Nevertheless, the year 2017 was special in this regard, because itwas marked by the discovery of the doubly charmed Ξ ++ cc baryon by the LHCb Collab-oration [33]. This state was then confirmed in the decay channel Ξ ++ cc → Ξ + c π + [34].Although, the charmed-bottom state Ξ bc has been searched in the Ξ bc → D pK − decayby the LHCb collaboration, no evidence for a signal is found [35]. However, the detectionof Ξ ++ cc has raised hopes for the discovery of other members of doubly heavy baryons.It is wonderful that the LHC opens new horizons in the discovery of heavy baryons andprovides the possibility to investigate their electromagnetic, weak and strong decays.Beyond the quark models [36–38], many effective models like lattice QCD [39],Quark Spin Symmetry [40, 41], QCD Sum Rules [22–26] and Light-Cone Sum Rules[19,20,42–49] have been proposed to describe masses, residues, lifetimes, strong couplingconstants, and other properties of doubly heavy baryons. Given that the expectationsto discover more doubly heavy baryons are growing, we are witnessing the rise of furthertheoretical investigations in this respect.Concerning the strong decays of doubly heavy baryons, the strong coupling constantsare building blocks. They are entered to the amplitudes of the strong decays and thewidths of the related decays are calculated in terms of these constants. They can alsobe used to construct the strong potential energy among the hadronic multiplets. Thesecouplings appear in the low-energy (long-distance) region of QCD, where the runningcoupling gets larger and perturbation theory breaks down. Therefore, to calculatesuch coupling constants, a non-perturbative approach should be used. One of thecomprehensive and reliable methods for evaluating the non-perturbative effects is thelight-cone QCD sum rule (LCSR) which have given many successful descriptions ofthe hadronic properties so far. This method is a developed version of the standardtechnique of SVZ sum rule [50], using the conventional distribution amplitudes (DAs)of the on-sell states. The main difference between SVZ sum rule and LCSR is that thelatter employs the operator product expansion (OPE) near the light-cone x ≈ x ≈
0, and the corresponding matrix elements are parametrized interms of hadronic DAs which are classified according to their twists [51–53].In this study, we calculate the strong coupling constants among the doubly heavyspin-3/2 baryons, Ξ ∗ QQ ′ and Ω ∗ QQ ′ , and light pseudoscalar mesons π , K and η . Here Q and Q ′ both can be b or c quark. The paper is organized as follows: In the next section,we derive the sum rules for the strong coupling constants using the LCSR. Numericalanalysis and results are presented in section 3. The final section 4, is devoted to thesummary and conclusion. 1aryon q Q Q ′ Ξ ∗ QQ ′ u or d b or c b or c Ω ∗ QQ ′ s b or c b or c Table 1: The quark content of the doubly heavy spin-3/2 baryons. B ∗ B ∗ P Strong Cou-plings
In this section we aim to derive the sum rules for the strong coupling constants amongthe doubly heavy spin-3/2 baryons (Ξ ∗ QQ ′ and Ω ∗ QQ ′ ) and light pseudoscalar mesons ( π , K and η ) using the method of LCSR. The quark content of the doubly heavy spin-3/2baryons are presented in Table 1.The first step to calculate the strong couplings is to write the proper correlationfunction (CF) in terms of doubly heavy baryons’ interpolating current, η µ . That isΠ µν ( p, q ) = i Z d xe ipx hP ( q ) |T { η µ ( x )¯ η ν (0) } | i , (1)where P ( q ) represents the pseudoscalar meson carrying the four-momentum q , and p represents the outgoing doubly heavy baryon four-momentum. The above CF can becalculated in two ways: • One by inserting the complete set of hadronic states with the same quantumnumbers of the corresponding doubly heavy baryons, which is called the physicalor phenomenological side of the CF. It is calculated in the timelike region andcontains observables like strong coupling constants. • In the second way, it is calculated in the deep Euclidean spacelike region by thehelp of OPE and in terms of DAs of the on-shell mesons and other QCD degreesof freedom. It is called the theoretical or QCD side of the CF.These two sides are matched via a dispersion integral which leads to the sum rulesfor the corresponding coupling constants. To suppress the contributions of the higherstates and continuum, Borel transformation and continuum subtraction procedures areapplied. In the following, we explain each of these steps in detail.
Constructing Interpolating Current
The procedure to construct the doubly heavy spin-3/2 interpolating current is to make adiquark structure having spin one and then attach the remaining spin-1/2 quark to thatto build a spin-3/2 structure which has to contain the quantum numbers of the relatedbaryon. Following the line of [23] and [45] one can find the corresponding interpolatingcurrent as follows. The diquark structure has the form η diquark = q T C Γ q , (2)2here T and C represent the transposition and charge conjugation operators respec-tively and Γ = I, γ , γ µ , γ µ γ , σ µν . The dependence of the spinors on the spacetime x is omitted for now. Attaching the third quark, the general form of the correspond-ing structure is [ q T C Γ q ]Γ ′ q . Considering the color indices ( a , b and c ), the possi-ble structures have the following forms: ε abc ( Q aT C Γ Q ′ b )Γ ′ q c , ε abc ( q aT C Γ Q b )Γ ′ Q ′ c and ε abc ( q aT C Γ Q ′ b )Γ ′ Q c , where Q ( ′ ) and q are the heavy and light quark spinors respectivelyand the antisymmetric Levi-Civita tensor, ε abc , makes the interpolating current colorsinglet.The Γ matrices can be determined by investigating the diquark part of the inter-polating current. In the first form, as the diquark structure has spin 1, it must besymmetric under the exchange of the heavy quarks Q ↔ Q ′ . Transposing yields to[ ǫ abc Q aT C Γ Q ′ b ] T = − ǫ abc Q ′ bT Γ T C − Q a = ǫ abc Q ′ bT C ( C Γ T C − ) Q a , (3)where we have used the identities C T = C − and C = −
1, and consider the anticommu-tation of the spinor components since they are Grassmann numbers. Since C = iγ γ ,the quantity C Γ T C − would be C Γ T C − = ( Γ for Γ = 1 , γ , γ µ γ , − Γ for Γ = γ µ , σ µν . (4)Therefore, after switching the dummy indices in the LHS of Eq. (3) one gets[ ǫ abc Q aT C Γ Q ′ b ] T = ± ǫ abc Q ′ aT C Γ Q b , (5)where the + and − signs are for Γ = γ µ , σ µν and Γ = 1 , γ , γ γ µ respectively. On theother hand, since the RHS of the above equation has to be symmetric with respect tothe exchange of heavy quarks, we have[ ǫ abc Q aT C Γ Q ′ b ] T = ± ǫ abc Q aT C Γ Q ′ b , (6)where the + and − signs are for Γ = γ µ , σ µν and Γ = 1 , γ , γ γ µ respectively. Moreover,we know that the structure ǫ abc Q aT C Γ Q ′ b is a number (1 × γ µ or σ µν .The above mentioned symmetry of exchanging heavy quarks should be respected bythe two remaining structures, ε abc ( q aT C Γ Q b )Γ ′ Q ′ c and ε abc ( q aT C Γ Q ′ b )Γ ′ Q c , properlywhich leads to the following form: ε abc (cid:2) ( q aT C Γ Q b )Γ ′ Q ′ c + ( q aT C Γ Q ′ b )Γ ′ Q c (cid:3) , (7)where Γ = γ µ or σ µν . Consequently, the interpolating current can be written in twopossible forms as ε abc n(cid:0) Q aT Cγ µ Q ′ b (cid:1) Γ ′ q c + (cid:0) q aT Cγ µ Q b (cid:1) Γ ′ Q ′ c + (cid:0) q aT Cγ µ Q ′ b (cid:1) Γ ′ Q c o , (8)and ε abc n(cid:0) Q aT Cσ µν Q ′ b (cid:1) Γ ′ q c + (cid:0) q aT Cσ µν Q b (cid:1) Γ ′ Q ′ c + (cid:0) q aT Cσ µν Q ′ b (cid:1) Γ ′ Q c o . (9)To determine Γ ′ and Γ ′ , one should consider Lorentz and parity symmetries. As Eqs.(8) and (9) must have the Lorentz vector structure, Γ ′ = 1 or γ and Γ ′ = γ ν or γ ν γ .3ut parity considerations exclude the γ matrix and therefore Γ ′ = 1 and Γ ′ = γ ν .Moreover, Eq. (9) won’t survive if one consider all three quarks the same, and thereforethe only possible choice comes from Eq. (8) as η µ ( x ) = 1 √ ε abc n(cid:2) Q aT ( x ) Cγ µ Q ′ b ( x ) (cid:3) q c ( x ) + (cid:2) q aT ( x ) Cγ µ Q b ( x ) (cid:3) Q ′ c ( x )+ (cid:2) q aT ( x ) Cγ µ Q ′ b ( x ) (cid:3) Q c ( x ) o . (10) Physical Side
At hadronic (low energy) level, first we insert the complete set of hadronic states withthe same quantum numbers of the corresponding initial and final doubly heavy spin-3/2baryons and then perform the Fourier transformation by integrating over four- x . Byisolating the ground state one getsΠ Phys. µν ( p, q ) = h | η µ | B ∗ ( p, r ) ih B ∗ ( p, r ) P ( q ) | B ∗ ( p + q, s ) ih B ∗ ( p + q, s ) | ¯ η µ | i ( p − m )[( p + q ) − m ] + · · · , (11)where B ∗ ( p + q, s ) and B ∗ ( p, r ) are the incoming and outgoing doubly heavy spin-3/2 baryons with the masses m and m respectively. s and r are their spins anddots represent the higher states and continuum. The matrix element h | η µ | B ∗ i ( p, s ) i isdefined as h | η µ | B ∗ i ( p, s ) i = λ B ∗ i u µ ( p, s ) , (12)where u µ ( p, s ) is the Rarita–Schwinger spinor and λ B ∗ i is the residue for the baryon B ∗ i .The matrix element h B ∗ ( p, r ) P ( q ) | B ∗ ( p + q, s ) i can be determined using the Lorentzand parity considerations as h B ∗ ( p, r ) P ( q ) | B ∗ ( p + q, s ) i = g B ∗ B ∗ P ¯ u α ( p, r ) γ u α ( p + q, s ) , (13)where g B ∗ B ∗ P is the strong coupling constant of the doubly heavy spin-3/2 baryons B ∗ and B ∗ with the light pseudoscalar meson P . After substituting the above matrixelements, (12) and (13), in (11), since the initial and final baryons are unpolarized, oneneeds to sum over their spins using the following completeness relation: X s u µ ( p, s )¯ u ν ( p, s ) = − (/ p + m ) g µν − γ µ γ ν − p µ p ν m + p µ γ ν − p ν γ µ m ! , (14)which may lead to the corresponding physical side of the CF. But here we face two majorproblems: First, not all emerging Lorentz structures are independent; and second, sincethe interpolating current η µ also couples to the spin-1/2 doubly heavy baryon states,there are some unwanted contributions from them which must be removed properly.Imposing the condition γ µ η µ = 0, these contributions can be written as h | η µ | B ( p, s = 1 / i = A (cid:16) γ µ − m p µ (cid:17) u ( p, s = 1 / . (15)To fix the above-mentioned problems, we re-order the Dirac matrices in a way thathelp us eliminate the spin-1/2 states’ contributions easily. The ordering we choose is4 µ / p / qγ ν γ which leads us to the final form of the physical sideΠ Phys. µν ( p, q ) = λ B ∗ λ B ∗ [( p + q ) − m )]( p − m ) n g B ∗ B ∗ P m q µ q ν / p / qγ + structuresbeginning with γ µ and ending with γ ν γ , or terms that areproportional to p µ or ( p + q ) ν + other structures o + Z ds ds e − ( s + s ) / M ρ Phys. ( s , s ) , (16)where ρ Phys. ( s , s ) represents the spectral density for the higher states and continuum.One should note that there are many structures like g µν / p / qγ that emerge in the QCDside of the CF which are not shown in the Eq. (16) expansion and can be consideredto calculate the strong coupling constant g B ∗ B ∗ P . We select the structure with largenumber of momenta, q µ q ν / p / qγ , which leads to more stable and reliable results and it isfree of the unwanted doubly heavy spin-1/2 contributions.Now, to suppress the contributions of the higher states and continuum, we performthe double Borel transformation with respect to the squared momenta p = ( p + q ) and p = p , which leads to B p ( M ) B p ( M )Π Phys. µν ( p, q ) ≡ Π Phys. µν ( M ) (17)= 2 g B ∗ B ∗ P m λ B λ B e − m /M e − m /M q µ q ν / p / qγ + · · · , where dots represen tthe suppressed higher states and continuum contributions, M and M are the Borel parameters and M = M M / ( M + M ). The Borel parameters arechosen to be equal since the mass of the initial and final baryons are the same, therefore M = M = 2 M . After matching the physical and QCD sides, we will perform thecontinuum subtraction supplied by quark-hadron duality assumption. QCD Side
After evaluating the physical side of the CF, the next stage is to calculate the QCDside in the deep Euclidean spacelike region, where − ( p + q ) → ∞ and − p → ∞ .The CF in this way is calculated in terms of QCD degrees of freedom as well as non-local matrix elements of pseudoscalar mesons expressed in terms of the DAs of differenttwists [54–56].To proceed, according to Eq. (16), we choose the relevant structure q µ q ν / p / qγ fromthe QCD side and express the CF asΠ QCD µν ( p, q ) = Π( p, q ) q µ q ν / p / qγ , (18)where Π( p, q ) is an invariant function of ( p + q ) and p . To calculate Π( p, q ), we insertthe interpolating current (10) into the CF (1), and using the Wick theorem we find the5CD side of the CF as: (cid:16) Π QCD µν (cid:17) ρσ ( p, q ) = i ǫ abc ǫ a ′ b ′ c ′ Z d xe iq.x hP ( q ) | ¯ q c ′ α (0) q cβ ( x ) | i× ( δ ρα δ βσ Tr h ˜ S aa ′ Q ( x ) γ µ S bb ′ Q ′ ( x ) γ ν i + δ αρ (cid:16) γ ν ˜ S aa ′ Q ( x ) γ µ S bb ′ Q ′ ( x ) (cid:17) βσ + δ αρ (cid:16) γ ν ˜ S bb ′ Q ′ ( x ) γ µ S aa ′ Q ( x ) (cid:17) βσ + δ βσ (cid:16) S bb ′ Q ′ ( x ) γ ν ˜ S aa ′ Q ( x ) γ µ (cid:17) ρα − δ βσ (cid:16) S aa ′ Q ( x ) γ ν ˜ S bb ′ Q ′ ( x ) γ µ (cid:17) ρα + (cid:16) γ ν ˜ S aa ′ Q ( x ) γ µ (cid:17) βα (cid:16) S bb ′ Q ′ ( x ) (cid:17) ρσ − (cid:16) Cγ µ S aa ′ Q ( x ) (cid:17) ασ (cid:16) S bb ′ Q ′ ( x ) γ ν C (cid:17) ρβ − (cid:16) Cγ µ S bb ′ Q ′ ( x ) (cid:17) ασ (cid:16) S aa ′ Q ( x ) γ ν C (cid:17) ρβ + (cid:16) γ ν ˜ S bb ′ Q ′ ( x ) γ µ (cid:17) βα (cid:16) S aa ′ Q ( x ) (cid:17) ρσ ) , (19)where S aa ′ Q ( x ) is the propagator of the heavy quark Q with color indices a and a ′ ,and ˜ S = CS T C . µ and ν are Minkowski and ρ and σ are Dirac indices respectively.The propagators in the above equation contain both the perturbative and the non-perturbative contributions. As we previousely mentioned, the non-local matrix elements hP ( q ) | ¯ q c ′ α (0) q cβ ( x ) | i , are written in terms of DAs of the corresponding light pseudoscalarmeson P ( q ).The next stage is to insert the explicit expression of the heavy quark propagator inEq. (19) as [60]: S aa ′ Q ( x ) = m Q π K ( m Q √− x ) √− x δ aa ′ − i m Q / x π x K ( m Q √− x ) δ aa ′ − ig s Z d k (2 π ) e − ikx Z du " / k + m Q m Q − k ) σ λτ G aa ′ λτ ( ux )+ um Q − k x λ γ τ G aa ′ λτ ( ux ) + · · · , (20)where m Q is the heavy quark mass, K and K are the modified Bessel functions of thesecond kind and G aa ′ λτ is the gluon field strength tensor. It is defined as G aa ′ λτ ≡ G Aλτ t aa ′ A , (21)with λ and τ being the Minkowski indices. t aa ′ A = λ aa ′ A / λ A are the Gell-Mannmatrices with A = 1 , · · · , a, a ′ the color indices. The free propagator contributionis determined by the first two terms and the rest which ∼ G aa ′ λτ are due to the interactionwith the gluon field.Inserting Eq. (20) into (19) leads to several contributions. The first one is due toreplacing both the heavy quark propagators with their free part: S (pert.) Q ( x ) = m Q π K ( m Q √− x ) √− x − i m Q / x π x K ( m Q √− x ) . (22)The DAS in this case are two-particle DAs. Replacing one heavy quark propagator (say S aa ′ Q ) with its gluonic part S aa ′ (non-p.) Q ( x ) = − ig s Z d k (2 π ) e − ikx Z duG aa ′ λτ ( ux )∆ λτQ ( x ) , (23)6here ∆ λτQ ( x ) = 12( m Q − k ) h (/ k + m Q ) σ λτ + 2 u ( m Q − k ) x λ γ τ i , (24)and the other with its free part, leads to the one gluon exchange between the heavyquark Q and the light pseudoscalar meson P . The non-local matrix elements of thiscontribution can be calculated in terms of three-particle DAs of meson P . Replacingboth heavy quark propagators with their gluonic parts involves P meson four-particleDAs which are not yet determined and we ignore them in the present work. Instead,we consider the two-gluon condensate contributions in this study.The non-local matrix elements can be expanded using proper Fierz identities like¯ q c ′ α q cβ → −
112 (Γ J ) βα δ cc ′ ¯ q Γ J q, (25)where Γ J = , γ , γ µ , iγ γ µ , σ µν / √
2. It puts them in a form to be determined interms of the corresponding DAs of different twists which can be found in Refs. [54–56].Putting all things together, one can calculate different contributions to the corre-sponding strong coupling. The leading order contribution corresponding to no gluonexchange which can be obtained by replacing both heavy quark propagators with theirfree part is as follows: (cid:16) Π QCD(0) µν (cid:17) ρσ ( p, q ) = i Z d xe iq.x hP ( q ) | ¯ q (0)Γ J q ( x ) | i ( Tr h ˜ S (pert.) Q ( x ) γ µ S (pert.) Q ′ ( x ) γ ν i(cid:16) Γ J (cid:17) ρσ + Tr h Γ J γ ν ˜ S (pert.) Q ( x ) γ µ i(cid:16) S (pert.) Q ′ ( x ) (cid:17) ρσ + Tr h Γ J γ ν ˜ S (pert.) Q ′ ( x ) γ µ i(cid:16) S (pert.) Q ( x ) (cid:17) ρσ + (cid:16) Γ J γ ν ˜ S (pert.) Q ( x ) γ µ S (pert.) Q ′ ( x ) (cid:17) ρσ + (cid:16) Γ J γ ν ˜ S (pert.) Q ′ ( x ) γ µ S (pert.) Q ( x ) (cid:17) ρσ + (cid:16) S (pert.) Q ′ ( x ) γ ν ˜ S (pert.) Q ( x ) γ µ Γ J (cid:17) ρσ − (cid:16) S (pert.) Q ′ ( x ) γ ν ˜Γ J γ µ S (pert.) Q ( x ) (cid:17) ρσ − (cid:16) S (pert.) Q ( x ) γ ν ˜ S (pert.) Q ′ ( x ) γ µ Γ J (cid:17) ρσ − (cid:16) S (pert.) Q ( x ) γ ν ˜Γ J γ µ S (pert.) Q ′ ( x ) (cid:17) ρσ ) , (26)where the superscript (0) indicates no gluon exchange. The contribution of the exchangeof one gluon between the heavy quark Q and the light pseudoscalar meson P is obtainedas (cid:16) Π QCD(1) µν (cid:17) ρσ ( p, q ) = − ig s Z d k (2 π ) e − ik.x Z du hP ( q ) | ¯ q (0)Γ J G λτ q ( x ) | i× ( Tr h ˜∆ λτQ ( x ) γ µ S (pert.) Q ′ ( x ) γ ν i(cid:16) Γ J (cid:17) ρσ + Tr h γ ν ˜∆ λτQ ( x ) γ µ Γ J i(cid:16) S (pert.) Q ′ ( x ) (cid:17) ρσ + Tr h Γ J γ ν ˜ S (pert.) Q ′ ( x ) γ µ i(cid:16) ∆ λτQ ( x ) (cid:17) ρσ + (cid:16) Γ J γ ν ˜∆ λτQ ( x ) γ µ S (pert.) Q ′ ( x ) (cid:17) ρσ + (cid:16) Γ J γ ν ˜ S (pert.) Q ′ ( x ) γ µ ∆ λτQ ( x ) (cid:17) ρσ + (cid:16) S (pert.) Q ′ ( x ) γ ν ˜∆ λτQ ( x ) γ µ Γ J (cid:17) ρσ − (cid:16) S (pert.) Q ′ ( x ) γ ν ˜Γ J γ µ ∆ λτQ ( x ) (cid:17) ρσ − (cid:16) ∆ λτQ ( x ) γ ν ˜ S (pert.) Q ′ ( x ) γ µ Γ J (cid:17) ρσ − (cid:16) ∆ λτQ ( x ) γ ν ˜Γ J γ µ S (pert.) Q ′ ( x ) (cid:17) ρσ ) , (27)7here the superscript (1) indicates one gluon exchange. By exchanging Q and Q ′ inthe above equation one can simply find the contribution of one gluon exchange betweenthe heavy quark Q ′ and the light pseudoscalar meson P .The general configurations appear in the calculation of the QCD side of the CF (26)and (27) have the form T [ ,µ,µν,... ] ( p, q ) = i Z d x Z dv Z D αe ip.x (cid:0) x (cid:1) n [ e i ( α q + vα g ) q.x G ( α i ) , e iq.x f ( u )] × [1 , x µ , x µ x ν , ... ] K n ( m √− x ) K n ( m √− x ) . (28)The expressions in the brackets on the RHS correspond to different configurations whichmight arise in the calculation and Z D α = Z dα q Z dα ¯ q Z dα g δ (1 − α q − α ¯ q − α g ) . (29)On the LHS, the blank subscript indicates no x µ in the corresponding configuration.There are several representations for the modified Bessel function of the second kindand we use the cosine representation as K n ( m Q √− x ) = Γ( n + 1 /
2) 2 n √ πm nQ Z ∞ dt cos( m Q t ) ( √− x ) n ( t − x ) n +1 / . (30)It is shown that choosing this representation increases the radius of convergence of theBorel transformed CF [42]. To perform the Fourier integral over x , we write the x configurations in the exponential representation as( x ) n = ( − n ∂ n ∂β n (cid:0) e − βx (cid:1) | β =0 ,x µ e iP.x = ( − i ) ∂∂P µ e iP.x . (31) Borel Transformation and Continuum Subtraction
Performing the Fourier transformation, gives us the version of CF that has to be Boreltransformed to supress the divergences arise due to the dispersion integral. Havingtwo independent momenta ( p + q ) and p , we use the double Borel transformation withrespect to the square of these momenta as B p ( M ) B p ( M ) e b ( p + uq ) = M δ ( b + 1 M ) δ ( u − u ) e − q M
21 + M , (32)where u = M / ( M + M ). To be specific, we select the following configuration: T µν ( p, q ) = i Z d x Z dv Z D αe i [ p +( α q + vα g ) q ] .x G ( α i ) (cid:0) x (cid:1) n × x µ x ν K n ( m Q √− x ) K n ( m Q √− x ) , (33)where after Fourier and Borel transformation leads to T µν ( M ) = iπ − n − n e − q M
21 + M M m n Q m n Q Z D α Z dv Z dz ∂ n ∂β n e − m Q z + m Q zz ¯ z ( M − β ) z n − ¯ z n − × ( M − β ) n + n − δ [ u − ( α q + vα g )] h p µ p ν + ( vα g + α q )( p µ q ν + q µ p ν )+( vα g + α q ) q µ q ν + M g µν i . (34)8n Ref. [42] one can find the details of calculation for hadrons containing differentnumbers of heavy quarks (zero to five).The dispersion integral that matches the physical and QCD sides of the CF con-tains contributions from both the ground state and also the excited and continuum oneswhich is calculated in Eq. (16) as a double dispersion integral over ρ Phys. ( s , s ). Tosuppress the latter, one has to perform a proper version of subtraction which enhancesthe contribution of the ground state as well. Using quark-hadron duality one can ap-proximate ρ Phys. ( s , s ), the physical spectral density, with its theoretical counterpart, ρ QCD ( s , s ). One can calculate ρ QCD ( s , s ) directly from the Borel transfomed ver-sion of the CF [58–61]. In this prescription, for the generic factor ( M ) N e − m /M , thefollowing replacement [62] (cid:0) M (cid:1) N e − m /M → N ) Z s m dse − s/M (cid:0) s − m (cid:1) N − , (35)is done for N >
N <
0, where the energy threshold forhigher states and continuum is considered as √ s . Then we restrict the boundaries of z integral in a way to more suppress the unwanted excited states and continuum. Solvingthe equation e − m Q z + m Q zM z ¯ z = e − s /M , (36)for z gives us the proper boundaries as follows: z max(min) = 12 s h ( s + m Q − m Q ) + ( − ) q ( s m Q − m Q ) − m Q s i . (37)Replacing them in the z integral as Z dz → Z z max z min dz, (38)enhances the ground state and suppresses the higher states and continuum as much aspossible.Finally, selecting the coefficient of the corresponding structure q µ q ν / p / qγ , gives us theinvariant function to be used to find the corresponding strong coupling constants. TheBorel transformed subtracted version of the invariant function for the vertex B ∗ B ∗ P can be written asΠ B ∗ B ∗ P ( M , s ) = T B ∗ B ∗ P ( M , s ) + T GGB ∗ B ∗ P ( M , s ) , (39)where the above functions for the vertex Ξ ∗ bb Ξ ∗ bb π , as an example, are given as T Ξ ∗ bb Ξ ∗ bb π ( M , s ) = e − q M
21 + M √ π ( u f π m b m π i ( V k ( α i ) , h ζ − , ( m b ) − ζ , ( m b ) i + 6 u f π m b m π i ( V ⊥ ( α i ) , h ζ − , ( m b ) − ζ , ( m b ) i + µ π − h i ( α g T ( α i ) , v ) − i ( α g T ( α i ) , v ) + i ( α q T ( α i ) , − i ( α q T ( α i ) , v ) ih ˜ ζ (1)0 , ( m b ; s , M ) − ˜ ζ (1)1 , ( m b ; s , M ) i (40)+ ( − ρ π ) u φ σ ( u ) h − ˜ ζ (1)1 , ( m b ; s , M ) + ˜ ζ (1)2 , ( m b ; s , M ) i!) , T GG Ξ ∗ bb Ξ ∗ bb π ( M , s ) = g s h GG i e − q M
21 + M √ M π ( u f π m b m π i ( V k ( α i ) , h ζ − , ( m b ) + ζ − , − ( m b ) i + 6 u f π m b m π i ( V ⊥ ( α i ) , h ζ − , ( m b ) + ζ − , − ( m b ) i + µ π − h i ( α g T ( α i ) , v ) − i ( α g T ( α i ) , v ) + i ( α q T ( α i ) , − i ( α q T ( α i ) , v ) in m b h ˜ ζ (1) − , ( m b ; s , M ) + ˜ ζ (1)0 , − ( m b ; s , M ) i + ˜ ζ (2)0 , ( m b ; s , M ) o − − ρ π ) u φ σ ( u ) h m b ˜ ζ (1) − , ( m b ; s , M )+ m b ˜ ζ (1)0 , − ( m b ; s , M ) + 2 ˜ ζ (2)0 , ( m b ; s , M ) − m b ˜ ζ (1)0 , ( m b ; s , M ) − m b ˜ ζ (1)1 , − ( m b ; s , M ) − ζ (2)1 , ( m b ; s , M ) i!) . (41)Here the function ζ m,n ( m , m ) is defined as ζ m,n ( m , m ) = Z z max z min dzz m ¯ z n e − m M z − m M z , (42)where m and n are integers and ζ m,n ( m , m ) = ζ m,n ( m ). Also ˜ ζ ( N ) m,n ( m , m ; s , M ) isdefined as follows:˜ ζ ( N ) m,n ( m , m ; s , M ) = 1Γ( N ) Z z max z min dz Z s m z + m z dse − s/M z m ¯ z n (cid:16) s − m z − m ¯ z (cid:17) N − , (43)where ˜ ζ ( N ) m,n ( m , m ; s , M ) = ˜ ζ ( N ) m,n ( m ; s , M ). The functions i and i are also definedas i ( φ ( α i ) , f ( v )) = Z D α i Z dvφ ( α q , α ¯ q , α g ) f ( v ) θ ( k − u ) , (44) i ( φ ( α i ) , f ( v )) = Z D α i Z dvφ ( α q , α ¯ q , α g ) f ( v ) δ ( k − u ) , (45)where k = α q − vα g .Finally, matching both the physical and QCD sides of the CF leads to the sum rulesfor the strong coupling constant as g B ∗ B ∗ P = 3 m e m /M e m /M λ B ∗ λ B ∗ Π B ∗ B ∗ P ( M , s ) . (46) In this section, we present the numerical results of the sum rules for the strong couplingconstants of light pseudoscalar meson with doubly heavy spin–3 / m s +11 − MeV m c . +0 . − . GeV m b . +0 . − . GeV m π . ± . m π ± . ± . m η . ± .
017 MeV m K . ± .
013 MeV m K ± . ± .
016 MeV f π
131 MeV f η
130 MeV f K
160 MeVTable 2: The meson masses and leptonic decay constants along with the quark masses[63]. meson a η w η w π .
015 -3 10 0.2K 0.16 0 .
015 -3 0.6 0.2 η µ = 1GeV [54, 55].corresponds to light pseudoscalar mesons that are their masses and decay constantsas well as their nonperturbative parameters appearing in their DAs of different twists,which are listed in Tables (2) and (3) respectively. The other is the masses and residuesof the doubly heavy spin-3/2 baryons and are listed in Table (4).The first step in the numerical analysis is to fix the working intervals of the auxiliaryparameters. These are continuum threshold s and Borel parameter M . To thisend, we impose the conditions of: OPE convergence, pole dominance and relativelyweak dependence of the results on auxiliary parameters. The threshold s , depends onthe energy of the first excited state at each channel. We have not any experimentalinformation on the excited states of the doubly heavy baryons, however, our analysesshow that in the interval m B ∗ + 0 . ≤ √ s ≤ m B ∗ + 0 . M we follow the steps: for itsupper limit we impose the condition of pole dominance and for determination of itslower limit we demand the OPE convergence condition. These requirements lead to theworking regions 14 GeV ≤ M ≤
18 GeV , 8 GeV ≤ M ≤
11 GeV and 4 GeV ≤ M ≤ for bb , bc and cc channels, respectively.In Fig. (1), as an example, we show the dependence of the coupling constant g onthe continuum threshold at some fixed values of the Borel parameter for the vertices11aryon Mass [GeV] Residue [GeV ]Ξ ∗ cc . ± .
16 0 . ± . ∗ bc . ± .
20 0 . ± . ∗ bb . ± . . ± . ∗ cc . ± .
16 0 . ± . ∗ bc . ± . . ± . ∗ bb . ± . . ± . / ∗ bb Ξ ∗ bb π ± , Ω ∗ bb Ω ∗ bb η , Ω ∗ bb Ξ ∗ bb K ± , and Ω ∗ bb Ξ ∗ bb K . As is seen, the strong coupling constantsdepend very weakly on s . As another example, we show the dependence of g on
110 112 114 116 s (GeV ) g Ξ ∗ bb Ξ ∗ bb π ± M = 18 (GeV ) M = 16 (GeV ) M = 14 (GeV )
116 118 120 122 124 s (GeV ) g Ω ∗ bb Ω ∗ bb η M = 18 (GeV ) M = 16 (GeV ) M = 14 (GeV )
116 118 120 122 124 s (GeV ) g Ω ∗ bb Ξ ∗ bb K ± M = 18 (GeV ) M = 16 (GeV ) M = 14 (GeV )
116 118 120 122 124 s (GeV ) g Ω ∗ bb Ξ ∗ bb K M = 18 (GeV ) M = 16 (GeV ) M = 14 (GeV ) Figure 1: Dependence of the strong coupling constant g on the continuum threshold atdifferent values of the Borel parameter M = 14 ,
16 and 18 GeV .the Borel parameter M for the vertices Ξ ∗ bb Ξ ∗ bb π ± , Ω ∗ bb Ω ∗ bb η , Ω ∗ bb Ξ ∗ bb K ± , and Ω ∗ bb Ξ ∗ bb K atfixed values of s in Fig (2). From these plots, we see that the strong coupling constantsshow some dependence on the Borel parameter, which constitute the main parts of theuncertainties. These uncertainties, however, remain inside the limits allowed by themethod.Taking into account all the input values as well as the working intervals for the12 M (GeV ) g Ξ ∗ bb Ξ ∗ bb π ± s = 117 (GeV ) s = 113 (GeV ) s = 109 (GeV )
14 15 16 17 18 M (GeV ) g Ω ∗ bb Ω ∗ bb η s = 125 (GeV ) s = 121 (GeV ) s = 116 (GeV )
14 15 16 17 18 M (GeV ) g Ω ∗ bb Ξ ∗ bb K ± s = 125 (GeV ) s = 121 (GeV ) s = 116 (GeV )
14 15 16 17 18 M (GeV ) g Ω ∗ bb Ξ ∗ bb K s = 125 (GeV ) s = 121 (GeV ) s = 116 (GeV ) Figure 2: Dependence of the strong coupling constant g on the Borel parameter atdifferent values of the continuum threshhold s . The lines correspond to s = 109,113 and 117 GeV for Ξ ∗ bb Ξ ∗ bb π ± channel and s = 116, 121 and 125 GeV for Ω ∗ bb Ω ∗ bb η ,Ω ∗ bb Ξ ∗ bb K ± , and Ω ∗ bb Ξ ∗ bb K ones.auxiliary parameters, we extract the numerical values for the strong coupling constantsfrom the light-cone sum rules (46). The numerical results of g for all the consideredvertices in cc , bc and bb channels are presented in Table (5). There are three sources oferrors in the results. The first is from the input parameters we have used in the calcu-lations. The second, the intrinsic error, is due to the employment of the quark-hadronduality assumption and finally, the third one, is the error because of the variationswith respect to the auxiliary parameters in their working intervals. The results containerrors overall up to 20%. 13ertex M (GeV ) s (GeV ) strong coupling constantDecays to π Ξ ∗ bb Ξ ∗ bb π ≤ M ≤
18 109 ≤ s ≤
117 50 . . . Ξ ∗ bb Ξ ∗ bb π ± ≤ M ≤
18 109 ≤ s ≤
117 70 . . . Ξ ∗ bc Ξ ∗ bc π ≤ M ≤
10 57 ≤ s ≤
63 13 . . . Ξ ∗ bc Ξ ∗ bc π ± ≤ M ≤
10 57 ≤ s ≤
63 18 . . . Ξ ∗ cc Ξ ∗ cc π ≤ M ≤ ≤ s ≤
19 3 . . . Ξ ∗ cc Ξ ∗ cc π ± ≤ M ≤ ≤ s ≤
19 5 . . . Decays to η Ω ∗ bb Ω ∗ bb η ≤ M ≤
18 116 ≤ s ≤
125 125 . . . Ω ∗ bc Ω ∗ bc η ≤ M ≤
10 57 ≤ s ≤
64 18 . . . Ω ∗ cc Ω ∗ cc η ≤ M ≤ ≤ s ≤
20 5 . . . Decays to K Ω ∗ bb Ξ ∗ bb K ≤ M ≤
18 116 ≤ s ≤
125 142 . . . Ω ∗ bb Ξ ∗ bb K ± ≤ M ≤
18 116 ≤ s ≤
125 142 . . . Ω ∗ bc Ξ ∗ bc K ≤ M ≤
10 57 ≤ s ≤
64 27 . . . Ω ∗ bc Ξ ∗ bc K ± ≤ M ≤
10 57 ≤ s ≤
64 27 . . . Ω ∗ cc Ξ ∗ cc K ≤ M ≤ ≤ s ≤
20 8 . . . Ω ∗ cc Ξ ∗ cc K ± ≤ M ≤ ≤ s ≤
20 8 . . . Table 5: Working regions of the Borel parameter M and continuum threshold s aswell as the numerical values for different strong coupling constants extracted from theanalyses. In the present study, we investigated the strong coupling constanst of the doubly heavyspin-3 / ∗ QQ ′ and Ξ ∗ QQ ′ , with the light pseudoscalar mesons, π , K and η , byapplying the LCSR formalism. We used the interpolating currents of the doubly heavybaryons and DAs of the mesons to cunstruct the sum rules for the strong couplingconstants. By fixing the auxiliary parameters, entered to the calculations through theBorel transformation as well as continuum subtraction, we extracted the numericalresults for different vertices possible considering the quark contents of the participatingparticles and other considerations.The obtained results indicate that the heavier the doubly heavy baryons, the largertheir strong couplings with the same pseudoscalar mesons. Also comparing with theresults presented in Ref. [45], one finds out that the values of the strong couplingconstants among the doubly heavy spin-3/2 baryons and light pseudoscalar mesonsare mostly larger than that of the spin-1/2 baryons with the same mesons taken intoaccount in Ref. [45].Our results may help experimental groups in analyses of the data including thehadrons taken into account in the present study. The results may also be used in theconstruction of the strong potentials among the doubly heavy baryons and pseudoscalarmesons. 14 CKNOWLEDGEMENTS
K. Azizi and S. Rostami are thankful to Iran Science Elites Federation (Saramadan)for the financial support provided under the grant number ISEF/M/99171.
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