The B c meson and its scalar cousin with the QCD sum rules

In the present work, we use 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 B c mesons. We take the experimental data as guides to perform updated analysis, and obtain the masses and decay constants, especially the decay constants, which are the fundamental input parameters in the high energy physics, therefore the pure leptonic decay widths, which can be confronted to the experimental data in the future

decays or hadronic decays [10,11], while the ground state B c can only decay weakly through emitting a virtual W -boson, thus it cannot decay through strong or electromagnetic interactions.
With the continuous developments in experimental techniques, we expect that more c b states would be observed by the ATLAS, CMS, LHCb, etc in the future.The decay constant, which parameterizes the coupling between a current and a meson, plays an important role in exploring the exclusive processes, because the decay constants are not only a fundamental parameter describing the pure leptonic decays, but also are an universal input parameter related to the distribution amplitudes, form-factors, partial decay widths and branching fractions in many processes.By precisely measuring the branching fractions, we can resort to 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 in a number of theoretical approaches, such as the full QCD sum rules [35,36,37,38,39,40,41,43,44], the potential model combined with the QCD sum rules [15,16], the QCD sum rule combined with the heavy quark effective theory [45,46,47,48,49,50,51,52], the covariant light-front quark model [53,54], the lattice non-relativistic QCD [31], the shifted large-N expansion method [25], the field correlator method [55], etc.However, the values from different theoretical approaches vary in a large range, it is interesting and necessary to extend our previous works on the vector and axialvector B c mesons [40] to investigate the pseudoscalar and scalar B c mesons with the full QCD sum rules by including next-to-leading order radiative corrections and choose the updated input parameters, thus our investigations are performed in a consistent and systematic way.We take the experimental data [5,6,7,8] as guides to choose the suitable Borel parameters and continuum threshold parameters, examine the masses and decay constants of the pseudoscalar and scalar B c mesons with the full QCD sum rules, therefore we calculate the pure leptonic decay widths to be confronted to the experimental data in the future.
The article is arranged: we calculate the next-to-leading order contributions to the spectral densities and obtain the QCD sum rules in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusions.

Explicit calculations of the QCD spectral densities at the next-to-leading order
We write down the two-point correlation functions firstly, where J(x) = J P (x) and J S (x), the subscripts P and S represent the pseudoscalar and scalar mesons, respectively.The correlation functions can be written in the form, Figure 1: The next-to-leading order contributions to the correlation functions.according to the dispersion relation, where the QCD spectral densities ρ P/S (s) are expanded in terms of the strong fine structure constant , the ρ 0 P/S (s), ρ 1 P/S (s), ρ 2 P/S (s), • • • are the spectral densities of the leading order, nextto-leading order, and next-to-next-to-leading order, • • • .At the leading order, where the standard phase space factor, At the next-to-leading order, there exist three standard Feynman diagrams, which correspond to the self-energy and vertex corrections respectively, and make contributions to the correlation functions, see Fig. 1.We calculate the imaginary parts of those Feynman diagrams resorting to the Cutkosky's rule or optical theorem, the two methods result in the same analytical expressions, then we use dispersion relation to acquire the correlation functions at the quark-gluon level [39,56].There exist ten possible cuts, six cuts make attributions to virtual gluon emissions and four cuts make attributions to real gluon emissions.
The six cuts which are shown in Fig. 2 make attributions to virtual gluon emissions, and could be classified as the self-energy and vertex corrections, respectively.We calculate the Feynman diagrams straightforwardly by adopting the dimensional regularization to regularize both the ultraviolet and infrared divergences, and resort to the on-shell renormalization scheme to absorb the ultraviolet divergences by accomplishing the wave-function and quark-mass renormalizations.Then we take account of all contributions which are shown in Fig. 2 by the simple replacements of  the vertexes in all the currents, where is the i quark's wave-function renormalization constant which originates from the self-energy diagram, see Fig. 3, and for the vertex diagrams after accomplishing the Wick's rotation, see Fig. 4, where the γ is the Euler constant, the µ is the energy scale of renormalization, and the k E = (k 1 , k 2 , k 3 , k 4 ) is Euclidean four-momentum.We set the dimension D = 4 − 2ε UV = 4 + 2ε IR to regularize the ultraviolet and infrared divergences respectively, where the ε UV and ε IR are positive dimension-less quantities, and we would add the energy scale factors µ 2εUV or µ −2εIR if necessary.
We accomplish all the integrals over all the variables, and observe that the ultraviolet divergences 1  εUV in the Γ 5/0 , δZ 1 and δZ 2 are canceled out completely with each other, the offsets are warranted by the Ward identity.So the total contributions do not have ultraviolet divergences, where and s = p 2 , the definitions and explicit expressions of the notations V (s), V 00 (s) and V ij (s) with 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, the superscript V denotes the virtual gluon emissions.We accomplish all the integrals straightforwardly in the dimension D = 4 + 2ε IR as there does not exist ultraviolet divergence, and obtain the analytical expressions, The four cuts in the Feynman diagrams shown in Fig. 5 only make contributions to the real gluon emissions, the corresponding scattering amplitudes are shown explicitly in Fig. 6.From the two diagrams in Fig. 6, we write down the scattering amplitudes T a 5,α (p) and T a 0,α (p) , where the λ a is the Gell-Mann matrix.Then we obtain the contributions to the imaginary parts of the Feynman diagrams with the optical theorem, where we have used the formulas u(p 1 )ū(p 1 ) = p 1 + m b and v(p 2 )v(p 2 ) = p 2 − m c for the quark and antiquark respectively, and we introduce the symbol K 2 = (p 1 + p 2 ) 2 for simplicity, and introduce the superscript R to denote the real gluon emissions.We accomplish the integrals in the dimension D = 4 + 2ε IR because there only exist the infrared divergences (no ultraviolet divergences), and obtain the contributions, the definitions and explicit expressions of the R 11 (s), R 22 (s), R 12 (s), R 1 12 (s) and R 2 12 (s) are given in the appendix.Now we obtain the total QCD spectral densities at the next-to-leading order, The infrared divergences of the forms 1 εIR , log 1+ω 1−ω 1 εIR from the virtual and real gluon emissions are canceled out with each other completely, the offsets are guaranteed by the Lee-Nauenberg Figure 6: The amplitudes for the real gluon emissions.
theorem [57].The analytical expressions are applicable in many phenomenological analysis besides the QCD sum rules.
Then we calculate the contributions of the gluon condensate directly, the calculations are easy and no much to say.Finally, we obtain the analytical expressions of the QCD spectral densities, take the quark-hadron duality below the continuum thresholds s 0 P/S and perform the Borel transforms in regard to the variable P 2 = −p 2 to acquire the QCD sum rules, where x , the T 2 is the Borel parameter, and the decay constants are defined by, in other words, the subscripts A and V denote the axial-vector and vector currents, respectively.We eliminate the decay constants f P/S and obtain the QCD sum rules for the masses of the pseudoscalar and scalar B c mesons, (m b +mc) 2 ds ρ 0 P/S (s) + ρ 1 P/S (s) + 3 Numerical results and discussions The value of the gluon condensate αsGG π has been updated from time to time, and changes greatly, we adopt the updated value αsGG π = 0.022 ± 0.004 GeV 4 [58].We take the M S masses of the heavy quarks m c (m c ) = 1.275 ± 0.025 GeV and m b (m b ) = 4.18 ± 0.03 GeV from the Particle Data Group [5].In addition, we take account of the energy-scale dependence of the M S masses, where , Λ = 213 MeV, 296 MeV and 339 MeV for the quark flavor numbers n f = 5, 4 and 3, respectively [5].We choose n f = 4 and 5 for the c and b quarks, respectively, and then evolve all the heavy quark masses to the typical energy scale µ = 2 GeV.
The lower threshold (m b + m c ) 2 in the QCD sum rules in Eq.( 19) decreases quickly with increase of the energy scale, the energy scale should be larger than 1.7 GeV, which corresponds to the squared mass of the B c meson, 39.4 GeV 2 .If we take the typical energy scale µ = 2 GeV, which corresponds to the lower threshold (m b + m c ) 2 ≈ 36.0GeV 2 < M 2 P , it is reasonable and feasible to choose such a particular energy scale.
The experimental masses of the B c and B ′ c mesons are 6274.47± 0.27 ± 0.17 MeV and 6871.2 ± 1.0 MeV respectively from the Particle Data Group [5].The scalar B c meson still escapes the experimental detection, roughly speaking, the theoretical mass is 6712 ± 18 ± 7 MeV from the lattice QCD [30] or 6714 MeV from the nonrelativistic quark model [23].We can tentatively take the continuum threshold parameters as s 0 P = (39 − 47) GeV 2 and s 0 S = (45 − 55) GeV 2 , and search for the ideal values by assuming the energy gap between the ground state and first radial excited states is about 0.6 GeV, if lacking 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.On the other hand, the gluon condensate plays a tiny important role, the operator product expansion is well convergent.It is reliable to extract the masses and pole residues, which are shown in Table 1 and Figs.7-8.
Combined with our previous work [40], we can observe that there exist the relations GeV. Usually, we expect that the energy gaps between the ground states and first radial excitations are about 0.6 GeV.In practical calculations, we can set the continuum threshold parameter √ s 0 to be any values between the ground state and first radial excitation, i.e.
, if good QCD sum rules can be obtained, where the 1S and 2S stand for the ground state and first radial excitation, respectively.The energy gaps 0.4 GeV and 0.6 GeV are all make sense.
As for the decay constants, even for the pseudoscalar B c 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], the relativistic quark model (RQM) [11,12], the non-relativistic quark model (NRQM) [18,19], the light-front quark model (LFQM) [54], the lattice non-relativistic QCD (LNQCD) [31], the shifted N -expansion method (SNEM) [25], the field correlator method (FCM) [55], the Bethe-Salpeter equation (BSE) [33], etc; and we present those values in Table 2 for clearness.At the present time, it is difficult to say which value is superior to others.The present prediction f P = 371±37 MeV is in very good agreement with the value 371±17 MeV from the full QCD sum rules [43].In our previous work, we obtain the values f V = 384±32 MeV and f A = 373 ± 25 MeV for the vector and axial-vector B c mesons, respectively [40].Our calculations indicate that f P ≈ f V ≈ f A > f S .While in the QCD sum rule combined with the heavy quark effective theory up to the order α 3 s , the decay constants have the relations fP = f P > f V > f S > fS > f A [47], where the decay constants fP and fS are defined by From Eq.( 21) and Eq.( 25), we can obtain the relations, it is obvious that fP > f P and fS < f S , which are in contrary to the relations obtained in Ref. [47], so 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(α s ) corrections (in other words, the next-to-leading order contributions), the same input parameters would lead to too large hadron masses, we have to choose the energy scales µ = 2.1 GeV and 2.2 GeV for the pseudoscalar and scalar B c mesons, respectively.Then we refit the Borel parameters, the corresponding pole contributions, masses and decay constants are given explicitly in Table 1.Form the Table, we can see explicitly that the predicted masses change slightly, while the predicted decay constants change greatly, the decay constants without the radiative O(α s ) corrections only count for about 56% of the corresponding ones with the radiative O(α s ) corrections.The radiative O(α s ) corrections play an important role, we should take it into account.
The pure leptonic decay widths Γ ℓν ℓ of the pseudoscalar and scalar B c mesons can be written as, and the branching fractions, Br P →eνe = 1.57× 10 −9 , Br P →µνµ = 6.73 × 10 −5 , The largest branching fractions of the B c (0 − ) → ℓν ℓ are of the order 10 −2 , the tiny branching fractions maybe escape experimental detections.By precisely measuring the branching fractions, we can examine the theoretical calculations strictly, although it is a hard work.

Conclusion
In this work, we extend our previous works on the vector and axialvector B c mesons to investigate the pseudoscalar and scalar B c mesons with the full QCD sum rules by including next-to-leading order corrections and choose the updated input parameters.In calculating the next-to-leading order corrections, we use optical theorem (or Cutkosky's rule) to obtain the QCD spectral densities straightforwardly, and resort to the dimensional regularization to regularize both the ultraviolet and infrared divergences, which are canceled out with each other separately, the total QCD spectral densities have neither ultraviolet divergences nor infrared divergences.Then we calculate the gluon condensate contributions and reach the QCD sum rules.We take the experimental data as guides to choose the suitable Borel parameters and continuum threshold parameters, and make reasonable predictions for the masses and decay constants, therefore the pure leptonic decay widths, which can be confronted to the experimental data in the future to examine the theoretical calculations or extract the decay constants, which are fundamental input parameters in the high energy physics.

Appendix
At first, we write down all the elementary integrals involving the vertex corrections, and accomplish all the integrals to acquire the analytical expressions, where where s = s − (m b − m c ) 2 .

Figure 2 :
Figure 2: Six possible cuts correspond to virtual gluon emissions.

Figure 5 :
Figure 5: Four possible cuts correspond to real gluon emissions.

Table 1 :
The Borel windows, continuum threshold parameters, pole contributions, masses and decay constants of the pseudoscalar and scalar B c mesons, where the denotes that the radiative O(α s ) corrections have been neglected.The masses of the pseudoscalar (P ) and scalar (S) B c mesons with variations of the Borel parameters T 2 .
Figure 8: The decay constants of the pseudoscalar (P ) and scalar (S) B c mesons with variations of the Borel parameters T 2 .

Table 2 :
The decay constant of the pseudoscalar B c meson from different theoretical works.