Bc meson and its scalar cousin with QCD sum rules

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Zhi-Gang Wang. The Bc meson and its scalar cousin with the QCD sum rules[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad5a71
Zhi-Gang Wang. The Bc meson and its scalar cousin with the QCD sum rules[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad5a71 shu
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Bc meson and its scalar cousin with QCD sum rules

    Corresponding author: Zhi-Gang Wang, zgwang@aliyun.com
  • Department of Physics, North China Electric Power University, Baoding 071003, China

Abstract: In this study, we use the optical theorem to calculate the next-to-leading order corrections to the QCD spectral densities directly in the QCD sum rules for the pseudoscalar and scalar Bc mesons. We use experimental data for guidance to perform an updated analysis. We obtain the masses and, in particular, decay constants, which are the fundamental input parameters in high energy physics. Ultimately, we obtain the pure leptonic decay widths, which can be compared with experimental data in the future.

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    I.   INTRODUCTION
    • In 1998, at the Fermilab Tevatron, the CDF collaboration observed pseudoscalarBc mesons through the semi-leptonic decay modes B±cJ/ψ±X and B±cJ/ψ±ˉν in the pˉp collisions at an energy of s=1.8TeV. The measured mass was 6.40±0.39±0.13GeV [1, 2]. This was the first time the bottom-charm meson was measured experimentally.

      In 2007, the CDF collaboration confirmed Bc mesons through the non-leptonic decay modes B±cJ/ψπ± with a measured mass of 6275.6±2.9±2.5MeV [3]. In 2008, the D0 collaboration reconstructed non-leptonic decays B±cJ/ψπ± and confirmed Bc mesons with a measured mass of 6300±14±5MeV [4]. The Bc meson is now well established, and the average value listed in the Review of Particle Physics is 6274.47±0.27±0.17MeV [5].

      In 2014, the ATLAS collaboration reported the observation of a structure in the B±cπ+π invariant mass spectrum with a significance of 5.2 standard deviations, which is consistent with the predicted Bc meson with a mass of 6842±4±5MeV [6].

      In 2019, the CMS collaboration observed two excited ˉbc states in the B+cπ+π invariant mass spectrum with a significance exceeding five standard deviations, which are consistent with B+c and B+c, respectively [7]. The two states are separated in mass by 29.1±1.5±0.7MeV, and the mass of B+c was measured to be 6871.0±1.2±0.8±0.8MeV. Additionally, in 2019, the LHCb collaboration observed excited B+c (with a global (local) statistical significance of 2.2σ (3.2σ)) and B+c (with a global (local) statistical significance of 6.3σ (6.8σ)) mesons in the B+cπ+π invariant mass spectrum. The B+c meson has a mass of 6841.2±0.6±0.1±0.8MeV, which is reconstructed without the low-energy photon emitted in B+cB+cγ decay through the process B+cB+cπ+π, whereas the B+c meson has a mass of 6872.1±1.3±0.1±0.8MeV [8].

      The Bc meson emerging heavier than the Bc meson is an odd phenomenon as it conflicts with all the theoretical estimations. This may be owing to impossibility of reconstructing the low-energy photon in B+cB+cγ decay [9]. More precise experimental data are required. Only the Bc and Bc mesons are listed in Review of Particle Physics [5], which is in contrast to the well-established spectroscopy of the charmonium and bottomonium states. Despite the significant developments in heavy quark physics in recent years, the bottom-charm spectroscopy remains poorly known. Thus, further investigations are required.

      Beauty-charm mesons provide an optimal platform for exploring both the perturbative and nonperturbative dynamics of heavy quarks, owing to the absence of contamination from the light quark, and for exploring the strong and electro-weak interactions. This is because they are composed of two different heavy flavor quarks and cannot annihilate into gluons or photons. The excited cˉb states, which lie below the BD threshold, would decay into the Bc meson through radiative or hadronic decays [10, 11]. In contrast, the ground state Bc can only decay weakly through emitting a virtual W-boson; thus, it cannot decay through strong or electromagnetic interactions.

      Several theoretical studies have been conducted on the mass spectroscopy of bottom-charm mesons, such as the relativized (or relativistic) quark model with a special potential [1014], nonrelativistic quark model with a special potential [1523], semi-relativistic quark model using the shifted large-N expansion [24, 25], perturbative QCD [26], nonrelativistic renormalization group [27], lattice QCD [2831], Bethe-Salpeter equation [3234], full QCD sum rules [3541], and potential model combined with the QCD sum rules [15, 16].

      With the continuous developments in experimental techniques, we expect that more cˉb states would be observed by collaborations and facilities such as ATLAS, CMS, and LHCb in the future. The decay constant, which parameterizes the coupling between a current and meson, is essential in exploring the exclusive processes. This is because the decay constants are not only fundamental parameters describing the pure leptonic decays but are also universal input parameters related to the distribution amplitudes, form-factors, partial decay widths, and branching fractions in many processes. By precisely measuring the branching fractions, we can utilize the decay constants to extract the CKM matrix element in the standard model and search for new physics beyond the standard model [42].

      Decay constants of the bottom-charm mesons have been investigated using several theoretical approaches, such as the full QCD sum rules [3541, 43, 44], potential model combined with the QCD sum rules [15, 16], QCD sum rule combined with the heavy quark effective theory [4552], covariant light-front quark model [53, 54], lattice non-relativistic QCD [31], shifted large-N expansion method [25], and field correlator method [55]. However, as the values from different theoretical approaches vary significantly, we are motivated to extend our previous works on vector and axialvector Bc mesons [40] to investigate pseudoscalar and scalar Bc mesons with the full QCD sum rules by including next-to-leading order radiative corrections and using the updated input parameters. Thus, our investigations are performed consistently and systematically. We use experimental data [58] as guides in selecting suitable Borel parameters and continuum threshold parameters and examine the masses and decay constants of pseudoscalar and scalar Bc. mesons with the full QCD sum rules. Therefore, we calculate the pure leptonic decay widths to be compared with experimental data in the future.

      The remainder of this article is arranged as follows: we calculate the next-to-leading order contributions to the spectral densities and obtain the QCD sum rules in Sec. II. In Sec. III, we present the numerical results and discussions. Sec. IV provides our conclusions.

    II.   EXPLICIT CALCULATIONS OF QCD SPECTRAL DENSITIES AT THE NEXT-TO- LEADING ORDER
    • First, we express the two-point correlation functions:

      ΠP/S(p2)=id4xeipx0|T{J(x)J(0)}|0,

      (1)

      where J(x)=JP(x) and JS(x) are

      JP(x)=ˉc(x)iγ5b(x),JS(x)=ˉc(x)b(x),

      (2)

      the subscripts P and S represent the pseudoscalar and scalar mesons, respectively. The correlation functions can be expressed in the form

      ΠP/S(p2)=1π(mb+mc)2dsImΠP/S(s)sp2,

      (3)

      according to the dispersion relation, where

      ImΠP/S(s)π=ρP/S(s)=ρ0P/S(s)+ρ1P/S(s)+ρ2P/S(s)+,

      (4)

      the QCD spectral densities ρP/S(s) are expanded in terms of the strong fine structure constant αs=g2s4π. ρ0P/S(s), ρ1P/S(s), ρ2P/S(s), are the spectral densities of the leading order, next-to-leading order, and next-to-next-to-leading order, . At the leading order,

      ρ0P/S(s)=38π2λ(s,m2b,m2c)s[s(mbmc)2],

      (5)

      where the standard phase space factor is

      λ(s,m2b,m2c)=s2+m4b+m4c2sm2b2sm2c2m2bm2c.

      (6)

      At the next-to-leading order, three standard Feynman diagrams exist, which correspond to the self-energy and vertex corrections, respectively, and contribute to the correlation functions (Fig. 1). We calculate the imaginary parts of these Feynman diagrams using the Cutkosky's rule or optical theorem. The two methods result in the same analytical expressions. Subsequently, we use the dispersion relation to acquire the correlation functions at the quark-gluon level [39, 56]. Ten possible cuts exist, six of which correspond to virtual gluon emissions and four to real gluon emissions.

      Figure 1.  Mext-to-leading order contributions to the correlation functions.

      The six cuts, which are shown in Fig. 2, correspond to virtual gluon emissions and can be classified as self-energy and vertex corrections. We calculate the Feynman diagrams straightforwardly by adopting the dimensional regularization to regularize both the ultraviolet and infrared divergences. We utilize the on-shell renormalization scheme to absorb the ultraviolet divergences by accomplishing the wave-function and quark-mass renormalizations. Subsequently, we observe all contributions, which are shown in Fig. 2, by simply replacing the vertexes in all the currents:

      Figure 2.  Six possible cuts corresponding to virtual gluon emissions.

      ˉu(p1)iγ5u(p2)ˉu(p1)iγ5u(p2)+ˉu(p1)i˜Γ5u(p2)=Z1Z2ˉu(p1)iγ5u(p2)+ˉu(p1)iΓ5u(p2)=ˉu(p1)iγ5u(p2)(1+12δZ1+12δZ2)+ˉu(p1)iΓ5u(p2),

      (7)

      ˉu(p1)u(p2)ˉu(p1)u(p2)+ˉu(p1)˜Γ0u(p2)=Z1Z2ˉu(p1)u(p2)+ˉu(p1)Γ0u(p2)=ˉu(p1)u(p2)(1+12δZ1+12δZ2)+ˉu(p1)Γ0u(p2),

      (8)

      where

      Zi=1+δZi=1+43αsπ(14εUV+12εIR+34logm2i4πμ2+34γ1),

      (9)

      is the wave-function renormalization constant of the i quark, which originates from the self-energy diagram (Fig. 3), and

      Figure 3.  Quark self-energy correction.

      Γ5/0=γ543g2s10dx1x0dydDkE(2π)D×Γ(3)[k2E+(xp1+yp2)2]3{4k2E(112εUV)+2(1xy+2xy)(sm2bm2c)±2(x+y)mbmc+2x(12x)m2b+2y(12y)m2c},

      (10)

      for the vertex diagrams after accomplishing the Wick's rotation (Fig. 4), where γ is the Euler constant, μ is the energy scale of renormalization, and kE=(k1,k2,k3,k4) is the Euclidean four-momentum. We set the dimension D=42εUV=4+2εIR to regularize the ultraviolet and infrared divergences, respectively, where εUV and εIR are positive dimension-less quantities. Moreover, we would include the energy scale factors μ2εUV or μ2εIR if necessary.

      Figure 4.  Vertex correction.

      We accomplish all the integrals over all the variables and observe that the ultraviolet divergences 1εUV in Γ5/0, δZ1, and δZ2 cancel each other out completely, and the offsets are warranted by the Ward identity. Therefore, the total contributions do not have ultraviolet divergences:

      ˜Γ5=43αs4πγ5fP(s),˜Γ0=43αs4πfS(s),

      (11)

      where

      fP/S(s)=¯fP/S(s)+2εIR+3logmbmc4πμ2+4log4πμ2sγ+42(sm2bm2c)λ(s,m2b,m2c)log(1+ω1ω)×(1εIR+logs4πμ2+γ),

      ¯fP/S(s)=4¯V(s)+2(sm2bm2c)[¯V00(s)V10(s)V01(s)+2V11(s)]±2mbmc[V10(s)+V01(s)]+2m2b[V10(s)2V20(s)]+2m2c[V01(s)2V02(s)],ω=s(mb+mc)2s(mbmc)2,

      (12)

      and s=p2. The definitions and explicit expressions of the notations ¯V(s), ¯V00(s), and Vij(s) for i,j=0,1,2 are given in the appendix.

      The contributions of all the virtual gluon emissions to the imaginary parts of the Feynman diagrams in Fig. 1 are

      ImΠVP/S(s)π=43αs4π6πdD1p1(2π)D12Ep1dD1p2(2π)D12Ep2(2π)DδD×(pp1p2)f(s)[s(mbmc)2],

      (13)

      the superscript V denotes the virtual gluon emissions. We determine all the integrals straightforwardly in the dimension D=4+2εIR as ultraviolet divergence does not exist, and we obtain the analytical expression

      ImΠVP/S(s)π=43αsπρ0P/S(s){1εIR2log4π+12γ+12logλ2(s,m2b,m2c)m3bm3cμ8s3+12¯fP/S(s)sm2bm2cλ(s,m2b,m2c)log(1+ω1ω)[1εIR2log4π+2γ2+logλ(s,m2b,m2c)μ4]}.

      (14)

      The four cuts in the Feynman diagrams shown in Fig. 5 contribute only to the real gluon emissions. The corresponding scattering amplitudes are shown explicitly in Fig. 6. From the two diagrams in Fig. 6, we express the scattering amplitudes Ta5,α(p) and Ta0,α(p) as

      Figure 5.  Four possible cuts corresponding to real gluon emissions.

      Figure 6.  Amplitudes of the real gluon emissions.

      Ta5,α(p)=ˉu(p1){igsλa2γαip1+kmbiγ5+iγ5ip2kmcigsλa2γα}v(p2),Ta0,α(p)=ˉu(p1){igsλa2γαip1+kmb+ip2kmcigsλa2γα}v(p2),

      (15)

      where λa is the Gell-Mann matrix. Thus, we obtain the contributions to the imaginary parts of the Feynman diagrams with the optical theorem as follows:

      ImΠRP/S(s)π=12πdD1k(2π)D12EkdD1p1(2π)D12Ep1dD1p2(2π)D12Ep2(2π)DδD(pkp1p2)Tr{Ta5/0,α(p)Ta5/0,β(p)}gαβ

      =2g2sπdD1k(2π)D12EkdD1p1(2π)D12Ep1dD1p2(2π)D12Ep2(2π)DδD(pkp1p2)×{2[s(mbmc)2][m2b(kp1)2+m2c(kp2)2sm2bm2ckp1kp2+sK2kp1kp2](sK2)2kp1kp2},

      (16)

      where we have used the formulas u(p1)ˉu(p1)=p1+mb and v(p2)ˉv(p2)=p2mc for the quark and antiquark, respectively. Additionally, we introduce the symbol K2=(p1+p2)2 for simplicity and the superscript R to denote the real gluon emissions. We determine the integrals in the dimension D=4+2εIR because only infrared divergences exist (no ultraviolet divergences) and obtain the contributions as

      ImΠRP/S(s)π=43αsπρ0P/S(s){1εIR+2log4π2γ+2logλ3(s,m2b,m2c)m2bm2cs2μ4+(sm2bm2c)¯R12(s)¯R11(s)¯R22(s)R112(s)+R21221s(mbmc)2+sm2bm2cλ(s,m2b,m2c)log(1+ω1ω)[1εIR2log4π+2γ2+logλ3(s,m2b,m2c)m2bm2cs2μ4]},

      (17)

      the definitions and explicit expressions of the ¯R11(s), ¯R22(s), ¯R12(s), R112(s), and R212(s) are given in the appendix.

      Now, we obtain the total QCD spectral densities at the next-to-leading order:

      ρ1P/S(s)=43αsπρ0P/S(s){12¯fP/S(s)¯R11(s)¯R22(s)R112(s)+(sm2bm2c)¯R12(s)+R21221s(mbmc)232γ+2+12logm7bm7csλ4(s,m2b,m2c)+sm2bm2cλ(s,m2b,m2c)log(1+ω1ω)logλ2(s,m2b,m2c)m2bm2cs2}.

      (18)

      The infrared divergences of the forms 1εIR and log(1+ω1ω)1εIRfrom the virtual and real gluon emissions cancel each other out completely, and the offsets are guaranteed by the Lee-Nauenberg theorem [57]. The analytical expressions are applicable in many phenomenological analysis in addition to the QCD sum rules.

      Subsequently, we calculate the contributions of the gluon condensate directly. The calculations are easy and do not require extensive explanations. Finally, we obtain the analytical expressions of the QCD spectral densities, obtain the quark-hadron duality below the continuum thresholds s0P/S, and perform the Borel transforms with respect to the variable P2=p2 to acquire the QCD sum rules.

      f2P/SM4P/S(mb±mc)2exp(M2P/ST2)=s0P/S(mb+mc)2ds[ρ0P/S(s)+ρ1P/S(s)+ρconP/S(s)]exp(sT2),

      (19)

      where

      ρconP/S(s)=mbmc24T4αsGGπ10dx[m2cx3+m2b(1x)3]δ(s˜m2Q)±mbmc8T2αsGGπ10dx[1x2+1(1x)2]δ(s˜m2Q)s24T4αsGGπ10dx[(1x)m2cx2+xm2b(1x)2]δ(s˜m2Q),

      (20)

      ˜m2Q=m2b1x+m2cx, T2 is the Borel parameter, and the decay constants are defined by

      0|JP(0)|P(p)=fPM2Pmb+mc,0|JS(0)|S(p)=fSM2Smbmc,

      (21)

      in other words,

      0|JαA(0)|P(p)=ifPpα,0|JαV(0)|S(p)=ifSpα,

      (22)

      the subscripts A and V denote the axial-vector and vector currents, respectively.

      We eliminate the decay constants fP/S and obtain the QCD sum rules for the masses of the pseudoscalar and scalar Bc mesons:

      M2P/S=s0P/S(mb+mc)2dsdd(1/T2)[ρ0P/S(s)+ρ1P/S(s)+ρconP/S(s)]exp(sT2)s0P/S(mb+mc)2ds[ρ0P/S(s)+ρ1P/S(s)+ρconP/S(s)]exp(sT2).

      (23)

    III.   NUMERICAL RESULTS AND DISCUSSIONS
    • The value of the gluon condensate αsGGπ is updated often and changes significantly. We adopt the updated valueαsGGπ=0.022±0.004GeV4 [58]. We use the ¯MS masses of the heavy quarks mc(mc)=1.275±0.025GeV and mb(mb)=4.18±0.03GeV from the Particle Data Group [5]. In addition, we consider the energy-scale dependence of the ¯MS masses,

      mQ(μ)=mQ(mQ)[αs(μ)αs(mQ)]12332nf,αs(μ)=1b0t[1b1b20logtt+b21(log2tlogt1)+b0b2b40t2],

      (24)

      where t=logμ2Λ2, b0=332nf12π, b1=15319nf24π2, b2=285750339nf+32527n2f128π3, and Λ=213MeV, 296MeV, and 339MeV for quark flavor numbers nf=5, 4, and 3, respectively [5]. We set nf=4 and 5 for the c and b quarks, respectively, and then evolve all the heavy quark masses to the typical energy scale μ=2GeV.

      The lower threshold (mb+mc)2 in the QCD sum rules in Eq. (19) decreases rapidly with increasing energy scale. The energy scale should be larger than 1.7GeV, which corresponds to the squared mass of the Bc meson, 39.4GeV2. If we use the typical energy scale μ=2GeV, which corresponds to the lower threshold (mb+mc)236.0GeV2<M2P, selecting such a particular energy scale is reasonable and feasible.

      The experimental masses of Bc and Bc mesons are 6274.47±0.27±0.17MeV and 6871.2±1.0MeV, respectively, from the Particle Data Group [5]. Generally, the scalar Bc meson still escapes the experimental detection. The theoretical mass is 6712±18±7MeV from lattice QCD [30] or 6714MeV from the nonrelativistic quark model [23]. We can tentatively use the continuum threshold parameters as s0P=(3947)GeV2 and s0S=(4555)GeV2 and search for the ideal values by assuming that the energy gap between the ground state and first radial excited states is approximately 0.6GeV. Owing to lack of experimental data, we always resort to such an assumption in the QCD sum rules.

      After trial and error, we obtain the ideal Borel windows and continuum threshold parameters and the corresponding pole contributions (about 70%−85%). The pole dominance is well satisfied. In contrast, the gluon condensate plays a minimal role, and the operator product expansion is adequately convergent. It can be used to reliably extract the masses and pole residues, which are shown in Table 1 and Figs. 78.

      T2/GeV2 s0/GeV2 pole M/GeV f/GeV
      Bc(0) 3.04.0 44±1 (68% − 89%) 6.274±0.054 0.371±0.037
      Bc(0+) 5.46.4 54±1 (69% − 83%) 6.702±0.060 0.236±0.017
      ˆBc(0) 2.43.4 44±1 (75% − 94%) 6.275±0.045 0.208±0.015
      ˆBc(0+) 3.54.5 54±1 (85% − 96%) 6.704±0.055 0.119±0.006

      Table 1.  Borel windows, continuum threshold parameters, pole contributions, masses, and decay constants of the pseudoscalar and scalar Bc mesons, where ^ indicates that the radiative O(αs) corrections have been neglected.

      Figure 7.  (color online) Masses of the pseudoscalar (P) and scalar (S) Bc mesons with variations in the Borel parameters T2.

      Figure 8.  (color online) Decay constants of the pseudoscalar (P) and scalar (S) Bc mesons with variations in the Borel parameters T2.

      The predicted mass MP=6.274±0.054GeV closely agrees with the experimental data 6274.47±0.27±0.17 MeV from the Particle Data Group [5], whereas the predicted mass MS=6.702±0.060GeV is consistent with other theoretical calculations [1016, 1823, 2830, 3234].

      Combined with our previous research [40], we can observe that the relations s0VMVs0PMP0.4 GeV and s0AMAs0SMS0.6GeV exist. We expect that the energy gaps between the ground states and first radial excitations are about 0.6GeV. In practical calculations, we can set the continuum threshold parameter s0 to be any value between the ground state and first radial excitation, i.e., M1S+Γ1S2<s0<M2SΓ2S2, if suitable QCD sum rules can be obtained, where 1S and 2S denote the ground state and first radial excitation, respectively. The energy gaps of 0.4GeV and 0.6GeV are reasonable.

      For the decay constants, even for the pseudoscalar Bc meson, the theoretical values vary in a large range. For example, the values from the full QCD sum rules (QCDSR) [16, 36, 37, 41, 43, 44], relativistic quark model (RQM) [11, 12], non-relativistic quark model (NRQM) [18, 19], light-front quark model (LFQM) [54], lattice non-relativistic QCD (LNQCD) [31], shifted N-expansion method (SNEM) [25], field correlator method (FCM) [55], and Bethe-Salpeter equation (BSE) [33], differ; we present these values in Table 2. Currently, it is difficult to say which value is superior.

      fP/MeV References
      QCDSR 460±60 [16]
      QCDSR 300±65 [36]
      QCDSR 360±60 [37]
      QCDSR 270±30 [41]
      QCDSR 371±17 [43]
      QCDSR 528±19 [44]
      QCDSR 371±37 This work
      RQM 410±40 [11]
      RQM 433 [12]
      NRQM 498 [18]
      NRQM 440 [19]
      LFQM 523±62 [54]
      LNQCD 420±13 [31]
      LNEM 315+2650 [25]
      FCM 438±10 [55]
      BSE 322±42 [33]

      Table 2.  Decay constant of the pseudoscalar Bc meson from different theoretical studies.

      The present prediction fP=371±37MeV closely agrees with the value 371±17MeV from the full QCD sum rules [43]. In our previous study, we obtained the values fV=384±32MeV and fA=373±25MeV for the vector and axial-vector Bc mesons, respectively [40]. Our calculations indicated that fPfVfA>fS. In contrast, in the QCD sum rule combined with the heavy quark effective theory up to the order α3s, the decay constants have the relations ˜fP=fP>fV>fS>˜fS>fA [47], where the decay constants ˜fP and ˜fS are defined by

      0|JP(0)|P(p)=˜fPMP,0|JS(0)|S(p)=˜fSMS.

      (25)

      From Eqs. (21) and (25), we can obtain the relations

      ˜fP=fPMPmb+mc,˜fS=fSMSmbmc,

      (26)

      showing that ˜fP>fP and ˜fS<fS, which are in contrast to the relations obtained in Ref. [47]. Therefore, no definite conclusion can be obtained. Naively, we expect that the vector mesons have larger decay constants than the corresponding pseudoscalar mesons [59].

      If we neglect the radiative O(αs) corrections (in other words, the next-to-leading order contributions), the same input parameters would result in excessively large hadron masses. We must select the energy scales μ=2.1GeV and 2.2GeV for the pseudoscalar and scalar Bc mesons, respectively. Subsequently, we refit the Borel parameters; the corresponding pole contributions, masses and decay constants are given explicitly in Table 1. The table shows that the predicted masses change slightly, whereas the predicted decay constants change significantly, and the decay constants without the radiative O(αs)corrections only account for approximately 56% of the corresponding ones with the radiative O(αs) corrections. As the radiative O(αs) corrections play an essential role, we should consider them.

      The pure leptonic decay widths Γˉν of the pseudoscalar and scalar Bc mesons can be expressed as

      Γˉν=G2F8π|Vbc|2f2P/SMP/SM2(1M2M2P/S)2,

      (27)

      where the leptons =e,μ,τ, the Fermi constant GF=1.16637×105GeV2, the CKM matrix element Vcb=4.08×102, the masses of the leptons me=0.511×103GeV, mμ=1.05658×101GeV, mτ=1.77686GeV, and the lifetime of the Bc meson τBc=0.510×1012s, as reported by the Particle Data Group [5]. We take the masses and decay constants of the pseudoscalar and scalar Bc mesons from the QCD sum rules to obtain the partial decay widths:

      ΓPeˉνe=2.03×1012eV,ΓPμˉνμ=8.68×108eV,ΓPτˉντ=2.08×105eV,ΓSeˉνe=8.78×1013eV,ΓSμˉνμ=3.75×108eV,ΓSτˉντ=9.18×106eV,

      (28)

      and the branching fractions

      BrPeˉνe=1.57×109,BrPμˉνμ=6.73×105,BrPτˉντ=1.61×102.

      (29)

      The largest branching fractions of the Bc(0)ˉν are of the order 102, and the tiny branching fractions may escape experimental detections. By precisely measuring the branching fractions, we can examine the theoretical calculations strictly, although it is a difficult task.

    IV.   CONCLUSION
    • In this study, we extend our previous research on vector and axialvector Bc mesons to investigate pseudoscalar and scalar Bc mesons using the full QCD sum rules by including next-to-leading order corrections and selecting the updated input parameters. In calculating the next-to-leading order corrections, we use the optical theorem (or Cutkosky's rule) to obtain the QCD spectral densities straightforwardly. We utilize dimensional regularization to regularize both the ultraviolet and infrared divergences, which cancel each other out, and the total QCD spectral densities have neither ultraviolet divergences nor infrared divergences. Subsequently, we calculate the gluon condensate contributions and reach the QCD sum rules. We use experimental data as guides in selecting suitable Borel and continuum threshold parameters. We make reasonable predictions for the masses, decay constants, and ultimately, pure leptonic decay widths. These values can be compared with experimental data in the future to examine the theoretical calculations or extract the decay constants, which are fundamental input parameters in high energy physics.

    APPENDIX
    • First, we express all the elementary integrals involving the vertex corrections,

      Vab(s)=16π210dx1x0dydDkE(2π)DxaybΓ(3)[k2E+(xp1+yp2)2]3,V(s)=16π2(112εUV)10dx1x0dydDkE(2π)Dk2EΓ(3)[k2E+(xp1+yp2)2]3,

      (A1)

      and determine all the integrals to acquire the analytical expressions:

      V00(s)=1λ(s,m2b,m2c){log(1+ω1ω)(1εIR+logs4πμ2+γ)+log2(1ω21)4log2(1+ω1)+log2(1ω22)4log2(1+ω2)+2log(ω1+ω2)log(1+ω1ω)logω1log(1+ω21ω2)logω2log(1+ω11ω1)Li2(2ω11+ω1)Li2(2ω21+ω2)+π2},=¯V00(s)1λ(s,m2b,m2c)log(1+ω1ω)(1εIR+logs4πμ2+γ),V10(s)=1s{12log(1ω211ω22)1ω2log(1+ω1ω)+logω2ω1},V01(s)=V10(s)|ω1ω2,V20(s)=12s{ω1ω2ω1+ω2log(1+ω1ω)ω1ω2(ω1+ω2)log(1+ω1ω)+ω1ω1+ω2log(1ω211ω22)+2ω1ω1+ω2logω2ω1+1},V02(s)=V20(s)|ω1ω2,V11(s)=12s{ω1ω2ω1+ω2log(1+ω1ω)ω1ω22(ω1+ω2)log(1ω211ω22)1ω1+ω2log(1+ω1ω)+ω1ω1+ω2logω1ω2+ω2ω1+ω2logω2ω11},V(s)=1εUV+log4πμ2sγ+22ω1ω2ω1+ω2log(1+ω1ω)ω2ω1+ω2log(1ω21)ω1ω1+ω2log(1ω22)2ω1logω1+ω2logω2ω1+ω2+2log(ω1+ω2),=¯V(s)+1εUV+log4πμ2sγ+2,

      (A2)

      where

      ω1=λ(s,m2b,m2c)s+m2bm2c,ω2=λ(s,m2b,m2c)s+m2cm2b,M=mb+mcmbmc,Li2(x)=x0dtlog(1t)t.

      (A3)

      Subsequently, we introduce the notation

      dps=dD1k2EkdD1p12Ep1dD1p22Ep2δD(pkp1p2),

      for simplicity and obtain the elementary three-body phase-space integrals:

      R11(s)=sm2bπ2λ(s,m2b,m2c)(2π)4εIRμ2εIRdps1(kp1)2=12εIRlog4π+γ1+logλ(s,m2b,m2c)3mbmcsμ2s+m2bm2c2λ(s,m2b,m2c)log(1+ω11ω1)m2bm2cλ(s,m2b,m2c)log(1+ω11ω1)sm2b+m2cλ(s,m2b,m2c)log(1+ω1ω)=¯R11(s)+12εIRlog4π+γ1+logλ(s,m2b,m2c)3mbmcsμ2,R22(s)=R11(s)|mbmc,R12(s)=sπ2λ(s,m2b,m2c)(2π)4εIRμ2εIRdps1kp1kp2=1λ(s,m2b,m2c){log(1+ω1ω)[1εIR2log4π+2γ2+2logλ(s,m2b,m2c)3mbmcsμ2]2logmbmclog(M+ωMω)log2(1+ω1ω)+2logsˉslog(1+ω1ω)4Li2(2ω1+ω)+2Li2(ω1ωM)+2Li2(ω1ω+M)2Li2(ω+1ωM)2Li2(ω+1ω+M)12Li2(1+ω12)12Li2(1+ω22)Li2(ω1)Li2(ω2)+log2log[(1+ω1)(1+ω2)]2log222+π212},=¯R12(s)+1λ(s,m2b,m2c)log(1+ω1ω)[1εIR2log4π+2γ2+2logλ(s,m2b,m2c)3mbmcsμ2],R112(s)=sπ2λ(s,m2b,m2c)dpssK2kp1kp2=sλ(s,m2b,m2c){log2(1ω)log2(1+ω)+2log2sˉslog(1+ω1ω)+2Li2(1ω2)2Li2(1+ω2)+2Li2(1+ω1+M)+2Li2(1+ω1M)2Li2(1ω1M)2Li2(1ω1+M)},R212(s)=sπ2λ(s,m2b,m2c)dps(sK2)2kp1kp2=s2λ(s,m2b,m2c){log2(1ω)log2(1+ω)+2log4sˉslog(1+ω1ω)+2Li2(1ω2)2Li2(1+ω2)+2Li2(1+ω1+M)+2Li2(1+ω1M)2Li2(1ω1M)2Li2(1ω1+M)+2ωˉssˉss(1+ω2)log(1+ω1ω)},

      (A4)

      where ˉs=s(mbmc)2.

Reference (59)

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