Finding B_c(3S) States via Their Strong Decays
Rui Ding, Bing-Dong Wan, Zi-Qiang Chen, Guo-Li Wang, Cong-Feng Qiao
aa r X i v : . [ h e p - ph ] J a n Finding B c (3 S ) States via Their Strong Decays
Rui Ding , Bing-Dong Wan , Zi-Qiang Chen , Guo-Li Wang ∗ ,Cong-Feng Qiao , † School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China CAS Key Laboratory of Vacuum Physics, Beijing 100049, China Department of Physics, Hebei University, Baoding 071002, China
Abstract
The experimentally known B c states are all below open bottom-charm threshold, whichexperience three main decay modes, and all induced by weak interaction. In this work, weinvestigate the mass spectrum and strong decays of the B c (3 S ) states, which just above thethreshold, in the Bethe-Salpeter formalism of and P model. The numerical estimation gives M B c (3 S ) = 7222 MeV, M B ∗ c (3 S ) = 7283 MeV, Γ (cid:0) B c (3 S ) → B ∗ D (cid:1) = 6 . +0 . − . MeV,Γ (cid:0) B ∗ c (3 S ) → BD (cid:1) = 0 . +0 . − . MeV and Γ (cid:0) B ∗ c (3 S ) → B ∗ D (cid:1) = 8 . +0 . − . MeV. Com-pared with previous studies in non-relativistic approximation, our results indicate that therelativistic effects are notable in B c (3 S ) exclusive strong decays. According to the results, wesuggest to find the B c (3 S ) states in their hadronic decays to B and D mesons in experiment,like the LHCb. ∗ gl [email protected] † [email protected] B c meson family is unique in quark model as its states are composed of heavyquarks with different flavors. The B c mesons lie intermediate between ( c ¯ c ) and ( b ¯ b )states both in mass and size, while the different quark masses leads to much richerdynamics. On the other hand, the B c mesons cannot annihilate into gluons or photonsand thus they are very stable. The B c mesons provide an unique window to revealinformation about heavy-quark dynamics and can deepen our understanding of bothstrong and weak interactions.Although there have been many investigations in the literature [1–15] about theproperties of B c mesons, the excited B c states, especially above threshold, are rarelyexplored. The ground state B c meson was first observed by the CDF Collaboration atFermilab [16] in 1998, while there was no reported evidence of the excited B c state until2014, the ATLAS Collaboration reported a structure with mass of 6842 ± B c (2 S ). Recently, the excited B c (2 S )and B ∗ c (2 S ) states have been observed in the B + c π + π − invariant mass spectrum bythe CMS and LHCb Collaboration, with their masses determined to be 6872 . ± . . ± . B ∗ c → B c γ was not reconstructed, the mass of B ∗ c (2 S ) mesonappears lower than that of B c (2 S ).The successful observation of B c (2 S ) states stimulates the interest in searching for B c (3 S ) states. Motivated by this, in this work, we calculate the mass spectrum of B c ( nS ) states up to n = 4 in the framework of Bethe-Salpeter (BS) equation [20]. Along-ranged linear confining potential and a short-ranged one gluon exchange potentialare used in our calculation. Our results indicate that the B c (3 S ) states lie above thethreshold for decay into a BD meson pair. By combining P model with the calculatedrelativistic BS wave functions, we investigate the strong decay properties of B c (3 S )mesons. We also estimate the corresponding numbers of events at the Large HadronCollider (LHC) experimental condition. Although similar topic has been studied inRefs. [3, 8, 11, 13, 14], the relativistic treatment of B c (3 S ) exclusive decays is evidentlymore close to the reality. 2n quantum field theory, the BS equation provides a basic description for boundstates. The BS wave function of a quark-antiquark bound state is defined as χ ( x , x ) = h | T ψ ( x ) ¯ ψ ( x ) | P i , (1)where x and x are the coordinates of the quark and antiquark respectively, P isthe momentum of the bound state, T denotes the time ordering operator. The wavefunction in momentum space is χ P ( q ) = e − iP · X Z d xe − iq · x χ ( x , x ) , (2)where q is the relative momentum between the quark and antiquark. The “center-of-mass coordinate” X and the “relative coordinate” x are defined as: X = m m + m x + m m + m x , x = x − x , (3)with m and m are the masses of the quark and antiquark respectively. Then thebound state BS equation in momentum space reads S − ( p ) χ P ( q ) S − ( − p ) = i Z d k (2 π ) V ( P ; q, k ) χ P ( k ) . (4)Here S i ( ± p i ) = i ± /p i − m i denotes the fermion propagator; V ( P ; q, k ) is the interactionkernel; p and p are the momenta of the quark and anti-quark respectively, which canbe expressed as p i = m i m + m P + J q , (5)where, J = 1 for the quark ( i = 1) and J = − i = 2). With thedefinitions p i P ≡ P · p i M and p µi ⊥ ≡ p µi − P · p i M P µ , the propagator S i ( J p i ) can be decomposedas − iJ S i ( J p i ) = Λ + i ( q ⊥ ) p i P − ω i + iǫ + Λ − i ( q ⊥ ) p i P + ω i − iǫ , (6)where Λ ± i ( q ⊥ ) ≡ ω i (cid:20) /PM ω i ± ( /p i ⊥ + J m i ) (cid:21) ,ω i ≡ q m i − p i ⊥ . (7)3nder the instantaneous approximation, the interaction kernel in the center of massframe takes the form V ( P ; q, k ) | ~P =0 ≈ V ( q ⊥ , k ⊥ ). Then the BS equation can be reducedto χ P ( q ) = S ( p ) η P ( q ⊥ ) S ( − p ) , (8)with η P ( q ⊥ ) = Z d k ⊥ (2 π ) V ( q ⊥ , k ⊥ ) ϕ P ( k ⊥ ) , (9)where ϕ P ( q µ ⊥ ) ≡ i R dq P π χ P ( q ) is the 3-dimensional BS wave function. By introducingthe notation ϕ ±± P ( q ⊥ ) as: ϕ ±± P ( q ⊥ ) ≡ Λ ± ( q ⊥ ) /PM ϕ P ( q ⊥ ) /PM Λ ± ( q ⊥ ) , (10)The wave function can be decomposed as ϕ P ( q ⊥ ) = ϕ ++ P ( q ⊥ ) + ϕ + − P ( q ⊥ ) + ϕ − + P ( q ⊥ ) + ϕ −− P ( q ⊥ ) . (11)And the BS equation (8) can be decomposed into four equations( M − ω − ω ) ϕ ++ P ( q ⊥ ) = Λ +1 ( q ⊥ ) η P ( q ⊥ )Λ +2 ( q ⊥ ) , (12)( M + ω + ω ) ϕ −− p ( q ⊥ ) = − Λ − ( q ⊥ ) η P ( q ⊥ )Λ − ( q ⊥ ) , (13) ϕ + − P ( q ⊥ ) = ϕ − + P ( q ⊥ ) = 0 . (14)To solve the BS equation, one must have a good command of the potential betweentwo quarks. According to lattice QCD calculations, the potential for a heavy quark-antiquark pair in the static limit is well described by a long-ranged linear confiningpotential (Lorentz scalar V S ) and a short-ranged one gluon exchange potential (Lorentzvector V V ) [21, 22]: V ( r ) = V S ( r ) + γ ⊗ γ V V ( r ) ,V S ( r ) = λr (1 − e − αr ) αr + V ,V V ( r ) = − α s ( r ) r e − αr . (15)4ere, the factor e − αr is introduced not only to avoid the infrared divergence but alsoto incorporate the color screening effects of the dynamical light quark pairs on the“quenched” potential [23]. The potentials in momentum space are V ( ~p ) = V S ( ~p ) + γ ⊗ γ V V ( ~p ) ,V S ( ~p ) = − (cid:16) λα + V (cid:17) δ ( ~p ) + λπ ~p + α ) ,V V ( ~p ) = − π α s ( ~p ) ~p + α . (16)Here, α s ( ~p ) is the running coupling which defined as α s ( ~p ) = 12 π
27 1ln (cid:0) a + ~p Λ (cid:1) . (17)The constants λ , α , a , and Λ QCD are the parameters that characterize the potential. Inthe following, we will employ this potential to both the B c system and the heavy-lightquark system as an assumption.In the numerical calculation, following well-fitted parameters [24, 25] are used: α = 0 .
06 GeV , λ = 0 .
21 GeV , Λ QCD = 0 .
27 GeV , m b = 4 .
96 GeV , m c = 1 .
62 GeV ,and V = 0 . S states, V = 0 . S states. Based on theformalism and parameters above, we calculate the masses of B c ( nS ) states up to n = 4.The numerical results are shown in Table I. For comparison, results obtained from otherapproaches are also listed.Our results indicate that the B c (3 S ) states lie above the threshold for decay into a BD meson pair. The corresponding OZI-allowed two body decay can be depicted by P model, where the additional light quark-antiquark pair is assumed to be createdfrom vacuum, as shown in Fig. 1. The usual P model is a non-relativistic model witha transition operator √ g R d x ¯ ψ ( ~x ) ψ ( ~x ), and it can be extended to a relativistic form i √ g R d x ¯ ψ ( x ) ψ ( x ) [25, 26]. The coupling constant g can be parameterized as 2 m q γ ,where m q is the constitute quark mass and γ is a dimensionless parameter which can5 ABLE I: Masses (MeV) of B c ( nS ) mesons.State This work EQ [2] GI [5] Lattice [12] Exp [19] B c (1 S ) 6275(input) 6264 6271 6276 6271 B ∗ c (1 S ) 6339(input) 6337 6338 6331 · · · B c (2 S ) 6853 6856 6855 · · · B ∗ c (2 S ) 6920 6899 6887 · · · B c (3 S ) 7222 7244 7250 · · · · · · B ∗ c (3 S ) 7283 7280 7272 · · · · · · B c (4 S ) 7484 7562 · · · · · · · · · B ∗ c (4 S ) 7543 7594 · · · · · · · · · p p p p p p A BCP, q P , q P , q g FIG. 1: The Feynman diagram of OZI-allowed two-body decay process with a P vertex. be extracted from experimental data. Here we take m u = 0 .
305 GeV, m d = 0 .
311 GeV,and γ = 0 . ± .
010 [27].The transition amplitude for the OZI-allowed two body decay process (with the6omenta assigned as in Fig. 1) can be written as h P P | S | P i = (2 π ) δ ( P − P − P ) M = − ig Z d q (2 π ) d q (2 π ) d q (2 π ) Tr[ χ P ( q ) S − ( p )(2 π ) δ ( p − p ) ¯ χ P ( q )(2 π ) δ ( p − p ) × ¯ χ P ( q ) S − ( p )(2 π ) δ ( p − p )]= − ig (2 π ) δ ( P − P − P ) Z d q (2 π ) Tr[ χ P ( q ) S − ( − p ) ¯ χ P ( q ) ¯ χ P ( q ) S − ( p )] , (18)where q i = q + ( − i +1 (cid:0) m i m + m P − m ii m i + m i P i (cid:1) . The Feynman amplitude takes theform [25]: M = − ig Z d q (2 π ) Tr[ χ P ( q ) S − ( − p ) ¯ χ P ( q ) ¯ χ P ( q ) S − ( p )] ≈ g Z d q ⊥ (2 π ) Tr[ /PM ϕ ++ P ( q ⊥ ) /PM ϕ ++ P ( q ⊥ ) ϕ ++ P ( q ⊥ )] . (19)Note, to get the last line of Eq. (19), we have used the fact that the wave function isstrongly suppressed when | q ⊥ | is large. With the method developed in Refs. [28, 29],the positive energy wave function can be determined by numerically solving Eq. (12).The strong decay widths of B c (3 S ) mesons are shown in Table II. We obtainΓ ( B c (3 S ) → B ∗ D ) = 6 . +0 . − . MeV, Γ ( B ∗ c (3 S ) → BD ) = 0 . +0 . − . MeV andΓ ( B ∗ c (3 S ) → B ∗ D ) = 8 . +0 . − . MeV. The total decay widths here are about5% ∼
10% the widths of Refs. [13, 14], where non-relativistic P model were used.It’s quite a discrepancy but it’s also understandable for the following reasons:1. In Refs. [13, 14], the coupling constant g is set to be about 0 .
264 GeV, whilehere it’s about 0 .
155 GeV according to Eq. (10) of Ref. [27]. This lead to a 3times difference in decay width.2. In the P model with H int = i √ g ¯ ψ ( x ) ψ ( x ), the transition operator contains afactor g ω q , where ω q is the energy of the created light quark. This factor reduce to g m q in the non-relativistic limit. Since the parameter g is extracted from the fitof the non-relativistic P model to the experimental data, our widths are in factsuppressed by a factor m q ω q ∼ .
270 [26]. By multiplying a compensation factor of7
ABLE II: Partial widths (Γ) and branching ratios (Br) of B c (3 S ) mesons. The number ofevents (NE) are estimated under the LHC experimental condition.Meson Decay mode Γ (MeV) Br (%) NE (10 ) B ± c (3 S ) B ∗ D ± . +0 . − . .
58 2.32 B ± c (3 S ) B ∗± D . +0 . − . .
42 2.56 B ± c (3 S ) B D ± . +0 . − . .
09 0.30 B ± c (3 S ) B ± D . +0 . − . .
48 0.79 B ± c (3 S ) B ∗ D ± . +0 . − . .
09 5.77 B ± c (3 S ) B ∗± D . +0 . − . .
34 7.54 about 3 .
7, the strong decay widths here are comparable with those predicted inRef. [3].3. The relativistic effect of the wave functions are non-negligible, as discussed inRef. [30].According to non-relativistic quantum chromodynamics factorization formalism[31], the production rates of B c (3 S ) mesons can be estimated through σ ( B c (3 S )) = σ ( B c (1 S )) | Ψ B c (3 S ) (0) | | Ψ B c (1 S ) (0) | , (20)where Ψ H (0) is the wave function at the origin for meson H . With the σ ( B c (1 S ))predicted in Ref. [32], and the wave functions calculated in Ref. [2], the cross sec-tions for B c (3 S ) mesons at the LHC can be estimated as σ ( B c (3 S )) = 4 .
88 nb and σ ( B ∗ c (3 S )) = 14 . − , the numbers of B c (3 S ) and B ∗ c (3 S ) events are 4 . × and 1 . × , respectively. The numberof events for different decay channels are also presented in Table II.In summary, since the relativistic effects are evidently important in B c (3 S ) ex-clusive decays to B and D measons, we have investigated in this work the massspectrum and strong decay properties of B c (3 S ) states in the framework of BS8quation and relativistic P model. The numerical estimation gives M B c (3 S ) =7222 MeV, M B ∗ c (3 S ) = 7283 MeV, Γ ( B c (3 S ) → B ∗ D ) = 6 . +0 . − . MeV,Γ ( B ∗ c (3 S ) → BD ) = 0 . +0 . − . MeV and Γ ( B ∗ c (3 S ) → B ∗ D ) = 8 . +0 . − . MeV.Note, the relativistic correction diminishes the decay widths in previous studies byabout one order of magnitude. We also estimate the number of events for different de-cay channels in the LHC experimental condition. Since a large number of events maybe produced in experiment, we suggest to find the the B c (3 S ) states in their exclusivestrong decays. Acknowledgments
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