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In 1998, at the Fermilab Tevatron, the CDF collaboration observed pseudoscalar
Bc mesons through the semi-leptonic decay modesB±c→J/ψℓ±X andB±c→J/ψℓ±ˉνℓ in thepˉp collisions at an energy of√s=1.8TeV . The measured mass was6.40±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 modesB±c→J/ψπ± with a measured mass of6275.6±2.9±2.5MeV [3]. In 2008, the D0 collaboration reconstructed non-leptonic decaysB±c→J/ψπ± and confirmedBc mesons with a measured mass of6300±14±5MeV [4]. TheBc meson is now well established, and the average value listed in the Review of Particle Physics is6274.47±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 predictedB′c meson with a mass of6842±4±5MeV [6].In 2019, the CMS collaboration observed two excited
ˉbc states in theB+cπ+π− invariant mass spectrum with a significance exceeding five standard deviations, which are consistent withB′+c andB∗′+c , respectively [7]. The two states are separated in mass by29.1±1.5±0.7MeV , and the mass ofB′+c was measured to be6871.0±1.2±0.8±0.8MeV . Additionally, in 2019, the LHCb collaboration observed excitedB′+c (with a global (local) statistical significance of2.2σ (3.2σ )) andB∗′+c (with a global (local) statistical significance of6.3σ (6.8σ )) mesons in theB+cπ+π− invariant mass spectrum. TheB∗′+c meson has a mass of6841.2±0.6±0.1±0.8MeV , which is reconstructed without the low-energy photon emitted inB∗+c→B+cγ decay through the processB′∗+c→B∗+cπ+π− , whereas theB′+c meson has a mass of6872.1±1.3±0.1±0.8MeV [8].The
B′c meson emerging heavier than theB∗′c 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 inB∗+c→B+cγ decay [9]. More precise experimental data are required. Only theBc andB′c 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 theBD threshold, would decay into theBc meson through radiative or hadronic decays [10, 11]. In contrast, the ground stateBc 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 [10−14], nonrelativistic quark model with a special potential [15−23], semi-relativistic quark model using the shifted large-N expansion [24, 25], perturbative QCD [26], nonrelativistic renormalization group [27], lattice QCD [28−31], Bethe-Salpeter equation [32−34], full QCD sum rules [35−41], 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 [35−41, 43, 44], potential model combined with the QCD sum rules [15, 16], QCD sum rule combined with the heavy quark effective theory [45−52], 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 scalarBc 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 [5−8] as guides in selecting suitable Borel parameters and continuum threshold parameters and examine the masses and decay constants of pseudoscalar and scalarBc . 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.
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First, we express the two-point correlation functions:
ΠP/S(p2)=i∫d4xeip⋅x⟨0|T{J(x)J†(0)}|0⟩,
(1) where
J(x)=JP(x) andJS(x) areJP(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)s−p2,
(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 constantαs=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)=38π2√λ(s,m2b,m2c)s[s−(mb∓mc)2],
(5) where the standard phase space factor is
λ(s,m2b,m2c)=s2+m4b+m4c−2sm2b−2sm2c−2m2bm2c.
(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.
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:
ˉu(p1)iγ5u(p2)→ˉu(p1)iγ5u(p2)+ˉu(p1)i˜Γ5u(p2)=√Z1√Z2ˉu(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)=√Z1√Z2ˉu(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+34γ−1),
(9) is the wave-function renormalization constant of the i quark, which originates from the self-energy diagram (Fig. 3), and
Γ5/0=γ543g2s∫10dx∫1−x0dy∫dDkE(2π)D×Γ(3)[k2E+(xp1+yp2)2]3{4k2E(1−12εUV)+2(1−x−y+2xy)(s−m2b−m2c)±2(x+y)mbmc+2x(1−2x)m2b+2y(1−2y)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 dimensionD=4−2εUV=4+2εIR to regularize the ultraviolet and infrared divergences, respectively, whereεUV andεIR are positive dimension-less quantities. Moreover, we would include the energy scale factorsμ2εUV orμ−2εIR if necessary.We accomplish all the integrals over all the variables and observe that the ultraviolet divergences
1εUV inΓ5/0 ,δZ1 , andδZ2 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+4log4πμ2s−γ+4−2(s−m2b−m2c)√λ(s,m2b,m2c)log(1+ω1−ω)×(1εIR+logs4πμ2+γ),
¯fP/S(s)=4¯V(s)+2(s−m2b−m2c)[¯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−(mb−mc)2,
(12) and
s=p2 . The definitions and explicit expressions of the notations¯V(s) ,¯V00(s) , andVij(s) fori,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=43αs4π6π∫dD−1→p1(2π)D−12Ep1dD−1→p2(2π)D−12Ep2(2π)DδD×(p−p1−p2)f(s)[s−(mb∓mc)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 expressionImΠVP/S(s)π=43αsπρ0P/S(s){1εIR−2log4π+12γ+12logλ2(s,m2b,m2c)m3bm3cμ8s3+12¯fP/S(s)−s−m2b−m2c√λ(s,m2b,m2c)log(1+ω1−ω)[1εIR−2log4π+2γ−2+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) andTa0,α(p) asTa5,α(p)=ˉu(p1){igsλa2γαi⧸p1+⧸k−mbiγ5+iγ5i−⧸p2−⧸k−mcigsλa2γα}v(p2),Ta0,α(p)=ˉu(p1){igsλa2γαi⧸p1+⧸k−mb+i−⧸p2−⧸k−mcigsλa2γα}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=−12π∫dD−1→k(2π)D−12EkdD−1→p1(2π)D−12Ep1dD−1→p2(2π)D−12Ep2(2π)DδD(p−k−p1−p2)Tr{Ta5/0,α(p)Ta†5/0,β(p)}gαβ
=−2g2sπ∫dD−1→k(2π)D−12EkdD−1→p1(2π)D−12Ep1dD−1→p2(2π)D−12Ep2(2π)DδD(p−k−p1−p2)×{2[s−(mb∓mc)2][m2b(k⋅p1)2+m2c(k⋅p2)2−s−m2b−m2ck⋅p1k⋅p2+s−K2k⋅p1k⋅p2]−(s−K2)2k⋅p1k⋅p2},
(16) where we have used the formulas
∑u(p1)ˉu(p1)=⧸p1+mb and∑v(p2)ˉv(p2)=⧸p2−mc for the quark and antiquark, respectively. Additionally, we introduce the symbolK2=(p1+p2)2 for simplicity and the superscript R to denote the real gluon emissions. We determine the integrals in the dimensionD=4+2εIR because only infrared divergences exist (no ultraviolet divergences) and obtain the contributions asImΠRP/S(s)π=43αsπρ0P/S(s){−1εIR+2log4π−2γ+2−logλ3(s,m2b,m2c)m2bm2cs2μ4+(s−m2b−m2c)¯R12(s)−¯R11(s)−¯R22(s)−R112(s)+R21221s−(mb∓mc)2+s−m2b−m2c√λ(s,m2b,m2c)log(1+ω1−ω)[1εIR−2log4π+2γ−2+logλ3(s,m2b,m2c)m2bm2cs2μ4]},
(17) the definitions and explicit expressions of the
¯R11(s) ,¯R22(s) ,¯R12(s) ,R112(s) , andR212(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)+(s−m2b−m2c)¯R12(s)+R21221s−(mb∓mc)2−32γ+2+12logm7bm7csλ4(s,m2b,m2c)+s−m2b−m2c√λ(s,m2b,m2c)log(1+ω1−ω)logλ2(s,m2b,m2c)m2bm2cs2}.
(18) The infrared divergences of the forms
1εIR andlog(1+ω1−ω)1εIR from 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 variableP2=−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)=∓mbmc24T4⟨αsGGπ⟩∫10dx[m2cx3+m2b(1−x)3]δ(s−˜m2Q)±mbmc8T2⟨αsGGπ⟩∫10dx[1x2+1(1−x)2]δ(s−˜m2Q)−s24T4⟨αsGGπ⟩∫10dx[(1−x)m2cx2+xm2b(1−x)2]δ(s−˜m2Q),
(20) ˜m2Q=m2b1−x+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)⟩=fSM2Smb−mc,
(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 scalarBc 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) -
The value of the gluon condensate
⟨αsGGπ⟩ is updated often and changes significantly. We adopt the updated value⟨αsGGπ⟩=0.022±0.004GeV4 [58]. We use the¯MS masses of the heavy quarksmc(mc)=1.275±0.025GeV andmb(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)]1233−2nf,αs(μ)=1b0t[1−b1b20logtt+b21(log2t−logt−1)+b0b2b40t2],
(24) where
t=logμ2Λ2 ,b0=33−2nf12π ,b1=153−19nf24π2 ,b2=2857−50339nf+32527n2f128π3 , andΛ=213MeV ,296MeV , and339MeV for quark flavor numbersnf=5 ,4 , and3 , respectively [5]. We setnf=4 and5 for the c and b quarks, respectively, and then evolve all the heavy quark masses to the typical energy scaleμ=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 than1.7GeV , which corresponds to the squared mass of theBc meson,39.4GeV2 . If we use the typical energy scaleμ=2GeV , which corresponds to the lower threshold(mb+mc)2≈36.0GeV2<M2P , selecting such a particular energy scale is reasonable and feasible.The experimental masses of
Bc andB′c mesons are6274.47±0.27±0.17MeV and6871.2±1.0MeV , respectively, from the Particle Data Group [5]. Generally, the scalarBc meson still escapes the experimental detection. The theoretical mass is6712±18±7MeV from lattice QCD [30] or6714MeV from the nonrelativistic quark model [23]. We can tentatively use the continuum threshold parameters ass0P=(39−47)GeV2 ands0S=(45−55)GeV2 and search for the ideal values by assuming that the energy gap between the ground state and first radial excited states is approximately0.6GeV . 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. 7−8.
T2/GeV2 s0/GeV2 pole M/GeV f/GeV Bc(0−) 3.0−4.0 44±1 (68% − 89%) 6.274±0.054 0.371±0.037 Bc(0+) 5.4−6.4 54±1 (69% − 83%) 6.702±0.060 0.236±0.017 ˆBc(0−) 2.4−3.4 44±1 (75% − 94%) 6.275±0.045 0.208±0.015 ˆBc(0+) 3.5−4.5 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 radiativeO(αs) corrections have been neglected.Figure 7. (color online) Masses of the pseudoscalar (P) and scalar (S)
Bc mesons with variations in the Borel parametersT2 .Figure 8. (color online) Decay constants of the pseudoscalar (P) and scalar (S)
Bc mesons with variations in the Borel parametersT2 .The predicted mass
MP=6.274±0.054GeV closely agrees with the experimental data6274.47±0.27±0.17 MeV from the Particle Data Group [5], whereas the predicted massMS=6.702±0.060GeV is consistent with other theoretical calculations [10−16, 18−23, 28−30, 32−34].Combined with our previous research [40], we can observe that the relations
√s0V−MV≈√s0P−MP≈0.4 GeV and√s0A−MA≈√s0S−MS≈0.6GeV exist. We expect that the energy gaps between the ground states and first radial excitations are about0.6GeV . 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<M2S−Γ2S2 , if suitable QCD sum rules can be obtained, where 1S and2S denote the ground state and first radial excitation, respectively. The energy gaps of0.4GeV and0.6GeV 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.Table 2. Decay constant of the pseudoscalar
Bc meson from different theoretical studies.The present prediction
fP=371±37MeV closely agrees with the value371±17MeV from the full QCD sum rules [43]. In our previous study, we obtained the valuesfV=384±32MeV andfA=373±25MeV for the vector and axial-vectorBc mesons, respectively [40]. Our calculations indicated thatfP≈fV≈fA>fS . In contrast, in the QCD sum rule combined with the heavy quark effective theory up to the orderα3s , 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=fSMSmb−mc,
(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 scalesμ=2.1GeV and2.2GeV for the pseudoscalar and scalarBc 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 radiativeO(αs) corrections only account for approximately 56% of the corresponding ones with the radiativeO(αs) corrections. As the radiativeO(αs) corrections play an essential role, we should consider them.The pure leptonic decay widths
Γℓˉνℓ of the pseudoscalar and scalarBc mesons can be expressed asΓℓˉνℓ=G2F8π|Vbc|2f2P/SMP/SM2ℓ(1−M2ℓM2P/S)2,
(27) where the leptons
ℓ=e,μ,τ , the Fermi constantGF=1.16637×10−5GeV−2 , the CKM matrix elementVcb=4.08×10−2 , the masses of the leptonsme=0.511×10−3GeV ,mμ=1.05658×10−1GeV ,mτ=1.77686GeV , and the lifetime of theBc mesonτBc=0.510×10−12s , as reported by the Particle Data Group [5]. We take the masses and decay constants of the pseudoscalar and scalarBc mesons from the QCD sum rules to obtain the partial decay widths:ΓP→eˉνe=2.03×10−12eV,ΓP→μˉνμ=8.68×10−8eV,ΓP→τˉντ=2.08×10−5eV,ΓS→eˉνe=8.78×10−13eV,ΓS→μˉνμ=3.75×10−8eV,ΓS→τˉντ=9.18×10−6eV,
(28) and the branching fractions
BrP→eˉνe=1.57×10−9,BrP→μˉνμ=6.73×10−5,BrP→τˉντ=1.61×10−2.
(29) The largest branching fractions of the
Bc(0−)→ℓˉνℓ are of the order10−2 , 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. -
In this study, we extend our previous research on vector and axialvector
Bc mesons to investigate pseudoscalar and scalarBc 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. -
First, we express all the elementary integrals involving the vertex corrections,
Vab(s)=16π2∫10dx∫1−x0dy∫dDkE(2π)DxaybΓ(3)[k2E+(xp1+yp2)2]3,V(s)=16π2(1−12εUV)∫10dx∫1−x0dy∫dDkE(2π)Dk2EΓ(3)[k2E+(xp1+yp2)2]3, (A1) and determine all the integrals to acquire the analytical expressions:
V00(s)=1√λ(s,m2b,m2c){−log(1+ω1−ω)(1εIR+logs4πμ2+γ)+log2(1−ω21)4−log2(1+ω1)+log2(1−ω22)4−log2(1+ω2)+2log(ω1+ω2)log(1+ω1−ω)−logω1log(1+ω21−ω2)−logω2log(1+ω11−ω1)−Li2(2ω11+ω1)−Li2(2ω21+ω2)+π2},=¯V00(s)−1√λ(s,m2b,m2c)log(1+ω1−ω)(1εIR+logs4πμ2+γ),V10(s)=1s{12log(1−ω211−ω22)−1ω2log(1+ω1−ω)+logω2ω1},V01(s)=V10(s)|ω1↔ω2,V20(s)=12s{−ω1ω2ω1+ω2log(1+ω1−ω)−ω1ω2(ω1+ω2)log(1+ω1−ω)+ω1ω1+ω2log(1−ω211−ω22)+2ω1ω1+ω2logω2ω1+1},V02(s)=V20(s)|ω1↔ω2,V11(s)=12s{ω1ω2ω1+ω2log(1+ω1−ω)−ω1−ω22(ω1+ω2)log(1−ω211−ω22)−1ω1+ω2log(1+ω1−ω)+ω1ω1+ω2logω1ω2+ω2ω1+ω2logω2ω1−1},V(s)=1εUV+log4πμ2s−γ+2−2ω1ω2ω1+ω2log(1+ω1−ω)−ω2ω1+ω2log(1−ω21)−ω1ω1+ω2log(1−ω22)−2ω1logω1+ω2logω2ω1+ω2+2log(ω1+ω2),=¯V(s)+1εUV+log4πμ2s−γ+2,
(A2) where
ω1=√λ(s,m2b,m2c)s+m2b−m2c,ω2=√λ(s,m2b,m2c)s+m2c−m2b,M=mb+mcmb−mc,Li2(x)=−∫x0dtlog(1−t)t.
(A3) Subsequently, we introduce the notation
∫dps=∫dD−1→k2EkdD−1→p12Ep1dD−1→p22Ep2δD(p−k−p1−p2),
for simplicity and obtain the elementary three-body phase-space integrals:
R11(s)=sm2bπ2√λ(s,m2b,m2c)(2π)−4εIRμ−2εIR∫dps1(k⋅p1)2=12εIR−log4π+γ−1+log√λ(s,m2b,m2c)3mbmcsμ2−s+m2b−m2c2√λ(s,m2b,m2c)log(1+ω11−ω1)−m2b−m2c√λ(s,m2b,m2c)log(1+ω11−ω1)−s−m2b+m2c√λ(s,m2b,m2c)log(1+ω1−ω)=¯R11(s)+12εIR−log4π+γ−1+log√λ(s,m2b,m2c)3mbmcsμ2,R22(s)=R11(s)|mb↔mc,R12(s)=sπ2√λ(s,m2b,m2c)(2π)−4εIRμ−2εIR∫dps1k⋅p1k⋅p2=1√λ(s,m2b,m2c){log(1+ω1−ω)[1εIR−2log4π+2γ−2+2log√λ(s,m2b,m2c)3mbmcsμ2]−2logmbmclog(M+ωM−ω)−log2(1+ω1−ω)+2logsˉslog(1+ω1−ω)−4Li2(2ω1+ω)+2Li2(ω−1ω−M)+2Li2(ω−1ω+M)−2Li2(ω+1ω−M)−2Li2(ω+1ω+M)−12Li2(1+ω12)−12Li2(1+ω22)−Li2(ω1)−Li2(ω2)+log2log[(1+ω1)(1+ω2)]2−log222+π212},=¯R12(s)+1√λ(s,m2b,m2c)log(1+ω1−ω)[1εIR−2log4π+2γ−2+2log√λ(s,m2b,m2c)3mbmcsμ2],R112(s)=sπ2√λ(s,m2b,m2c)∫dpss−K2k⋅p1k⋅p2=s√λ(s,m2b,m2c){log2(1−ω)−log2(1+ω)+2log2sˉslog(1+ω1−ω)+2Li2(1−ω2)−2Li2(1+ω2)+2Li2(1+ω1+M)+2Li2(1+ω1−M)−2Li2(1−ω1−M)−2Li2(1−ω1+M)},R212(s)=sπ2√λ(s,m2b,m2c)∫dps(s−K2)2k⋅p1k⋅p2=s2√λ(s,m2b,m2c){log2(1−ω)−log2(1+ω)+2log4sˉslog(1+ω1−ω)+2Li2(1−ω2)−2Li2(1+ω2)+2Li2(1+ω1+M)+2Li2(1+ω1−M)−2Li2(1−ω1−M)−2Li2(1−ω1+M)+2ωˉss−ˉss(1+ω2)log(1+ω1−ω)},
(A4) where
ˉs=s−(mb−mc)2 .
Bc meson and its scalar cousin with QCD sum rules
- Received Date: 2024-04-20
- Available Online: 2024-10-15
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.