Semi-leptonic $B\to S$ decays in the standard model and in the universal extra dimension model

In this article, we assume the two nonets of scalar mesons below and above 1 GeV are all $\bar{q}q$ states, and study the semi-leptonic decays $B\to S\ell^-\bar{\nu}_{\ell}$, $B\to S\ell^+\ell^-$ and $B\to S\bar{\nu}\nu$ both in the standard model and in the universal extra dimension model using the $B-S$ form-factors calculated by the light-cone QCD sum rules in our previous work. We obtain the partial decay widths and decay widths, which can be confronted with the experimental data in the future to examine the natures of the scalar mesons and constrain the basic parameter in the universal extra dimension model, the compactification scale $1/R$.


Introduction
The natures of the scalar mesons are not well established theoretically, and their underlying structures are under hot debating [1]. Irrespective of the two-quark state, tetraquark state and glueball assignments, the underlying structures determine their productions and decays. In previous work, we assume that the scalar mesons are allqq states, in case I, the scalar mesons {f 0 (600), a 0 (980), κ(800), f 0 (980)} below 1 GeV are the ground states, in case II, the scalar mesons {f 0 (1370), a 0 (1450), K * 0 (1430), f 0 (1500)} above 1 GeV are the ground states; and study the B − S transition form-factors with the light-cone QCD sum rules [2]. The transition form-factors in the semileptonic decays are highly nonperturbative quantities. They not only depend on the dynamics of strong interactions among the quarks in the initial and final mesons, but also depend on the under structures of the involved mesons. In this article, we take the B − S form-factors as basic input parameters, and study the semi-leptonic decays B → Sℓ −ν ℓ , B → Sℓ + ℓ − and B → Sνν both in the standard model and in the universal extra dimension model to examine the nature of the scalar mesons and search for new physic beyond the standard model.
The semi-leptonic B-decays are excellent subjects in studying the CKM matrix elements and CP violations in the standard model. They also serve as a powerful probe of new physics beyond the standard model in a complementary way to the direct searches, the indirect probe plays an important role in identifying the new physics and its properties [3]. At the quark level, the semileptonic B → S decays take place through the transitions b → u(c)ℓ −ν ℓ , b → s(d)ℓ + ℓ − and b → s(d)ν ℓ ν ℓ . In the standard model, the decays b → u(c)ℓ −ν ℓ take place through the exchange of the intermediate W boson at the tree-level, while the decays b → s(d)ℓ + ℓ − and b → s(d)ν ℓ ν ℓ take place through the penguin diagrams and other diagrams at the one-loop level. Those processes induced by the flavor-changing neutral currents b → s(d) provide the most sensitive and stringent test of the standard model at the one-loop level. The branching fractions of the semi-leptonic decaysB 0 (bd) → S(ud)ℓ −ν ℓ , B − (bū) → S(uū)ℓ −ν ℓ ,B 0 s (bs) → S(us)ℓ −ν ℓ are expected to be large, which favors examining the theoretical predictions in the standard model. The branching fractions of the semi-leptonic decaysB 0 (bd) s (bs) → S(ss)ν ℓ ν ℓ are expected to be small, which favors searching for new physics beyond the standard model. New physics effects manifest themselves in the rare B-decays in two different ways, either through new contributions to the Wilson coefficients or through the new operators in the effective Hamiltonians, which are absent in the standard model.

The decay widths in the standard model and in the universal extra dimension model
In the following, we write down the effective Hamiltonian H ef f responsible for the transitions b → uℓ −ν ℓ , b → sℓ + ℓ − and b → sνν in the standard model and in the UED model [6,7,15,16,17], where we have neglected the terms proportional to V ub V * us according to the value |V ub V * us /V tb V * ts | ∼ 10 −2 . No new operators are induced in the ACD model, the effects of the KK contributions are implemented by modifying the Wilson coefficients which also depend on the additional parameter, the compactification radius R. In the present case, we only need to specify the revelent Wilson coefficients C ef f 7 , C ef f 9 , C ef f 10 and X(x t ) [6,7]. In this article, we neglect the long-distance contributions come from the four-quark operators near the cc resonances, such as the J/ψ, ψ ′ , · · · , which can be experimentally removed by applying appropriate kinematical cuts in the neighborhood of the resonances [18]. Now, we write down the Wilson coefficients C ef f 7 , C ef f 9 and C ef f 10 , explicitly, , where η = αs(µW ) αs(µ) , and a i = ( We denote the total squared momentum of the lepton pairs as q 2 and the introduce the variableŝ s and z withŝ = q 2 We take the leading logarithmic approximation for the Wilson coefficients C i with i = 1 − 10 in the standard model [16], where the N DR is the abbreviation for naive dimensional regularization, with the parameters The Wilson coefficients F x t , 1 R generalize the corresponding standard model Wilson coefficients F 0 (x t ) according to the formula, [6,7], and The summation of the coefficients C n (x t , x n ), D ′ n (x t , x n ) and E ′ n (x t , x n ) over n leads to the formula [6,7], where The masses of the KK states increase monotonously with increase of the value of 1/R, in the limit 1/R → ∞. the KK states decouple from the low-energy processes and the standard model phenomenology are recovered. Now we study the semi-leptonic decays B → Sℓ − ν ℓ , B → Sℓ + ℓ − , B → Sν ℓ ν ℓ with the effective Hamiltonian H ef f and write down the transition amplitudes, then we take into account the definitions for the transition form-factors, where P = p ′ + p, p ′ = p + q, q = k 1 + k 2 and Finally we obtain the partial decay widths, where , the E min denotes the missing energy in the decays B → Sνν and λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2bc − 2ca. In calculating the decay widths, it is more convenient to use the form-factors F 1 (q 2 ) and F 0 (q 2 ).
In previous work [2], we calculate the B−S form-factors by taking into account the perturbative O(α s ) corrections to the twist-2 terms using the light-cone QCD sum rules, and fit the numerical values of the form-factors into the single-pole forms, where m B = 5.28 GeV, i = +, −, T , the values of the F i (0) and a i are shown explicitly in Tables  1-3. In calculations, we observe that the uncertainties induced by the uncertainties δa i are greatly  Table 3: The parameters of the transition form-factors F T (q 2 ). 8 discontinuities in the decays to the final states Se + e − and Sµ + µ − , which originate from the discontinuities in the h(z,ŝ) and h(1,ŝ) functions, the discontinuities disappear in the decays to the final states Sτ + τ − , as the value q 2 ≥ 4m 2 τ is large enough to warrant that the variations of the q 2 do not pass the discontinuities in the h(z,ŝ) and h(1,ŝ) functions. From the Figs.1-2, we can see that the branching fractions of the decays to the final states Sℓ −ν ℓ and Sℓ + ℓ − with ℓ = e, µ are much larger than the ones of the corresponding final states Sτ −ν τ and Sτ + τ − due to the much larger available phase-space.
The numerical values of the total branching fractions are shown in Table 4. From the table, we can see that the branching fractions of the decays induced by the transitions b → uℓ −ν ℓ , b → sℓ + ℓ − and b → sνν are of the orders 10 −4 , 10 −7 and 10 −6 , respectively. The magnitudes are compatible with the ones from other works based on the light-cone QCD sum rules [20,21] and perturbative QCD [22] and light-front quark model [23]. The transitions b → uℓ −ν ℓ take place at the tree-level through the intermediate W -boson, while the transitions b → sℓ + ℓ − and b → sνν take place at the loop-level, so the decays B → Sℓ −ν ℓ have the largest branching fractions. Compared to the decays B → Sνν, the decays B → Sℓ + ℓ − have even smaller branching fractions due to the smaller phase-space. The semi-leptonic decays B → Sℓ −ν ℓ are optimal in testing the standard model predictions, we can examine the nature of the scalar mesons by confronting the predictions to the experimental data in the future, while the semi-leptonic B → Sℓ + ℓ − are optimal in searching for new physics beyond standard model.
From Table 4, we can also see that the branching fractions of the decays to the light scalar mesons a 0 (980), κ(800), f 0 (980) below 1 GeV are much larger than that of the corresponding decays to the heavy scalar mesons a 0 (1450), K * 0 (1430), f 0 (1500) above 1 GeV due to the much larger energy released in the decays.
In Figs.4-5, we plot the branching fractions of the semi-leptonic decays B → Sℓ + ℓ − and B → Sνν with variations of the compactification scale 1/R, respectively. From the figures, we can see that the branching fractions decrease monotonously with increase of the values 1/R, at the region 1/R ≥ 800 GeV, the branching fractions almost reach constants, i.e. the KK states almost decouple from the low energy observables, while at the region 1/R ≤ 600 GeV, the impact of the KK states on the decays B → Sℓ + ℓ − are significant, at the region 1/R ≤ 400 GeV, the impact of the KK states on the decays B → Sνν are significant. If the constraint 1/R ≥ 715 GeV obtained from the LHC searches for dilepton resonances is robust [14], the semi-leptonic decays B → Sℓ + ℓ − are not the optimal processes in studying the UED model. In the limit 1/R → ∞ or R → 0, the summation of the coefficients C n (x t , x n ), D ′ n (x t , x n ) and E ′ n (x t , x n ) over n does not vanish, but approach some constants which are independent on the R. The constants modify the Wilson coefficients slightly, and lead to slightly larger branching fractions, it is difficult to distinguish the new physics effects from the standard model contributions.

Conclusion
In previous work, we assume the two scalar nonet mesons below and above 1 GeV are allqq states, in case I, the scalar mesons below 1 GeV are the ground states, in case II, the scalar mesons above 1 GeV are the ground states, and calculate the B − S form-factors by taking into account the perturbative O(α s ) corrections to the twist-2 terms using the light-cone QCD sum rules. In this article, we take those form-factor as basic input parameters, and study the semi-leptonic decays the B → Sℓ −ν ℓ , B → Sℓ + ℓ − and B → Sνν both in the standard model and in the UED model. We obtain the partial decay widths and decay widths, which can be confronted with the experimental data in the future to examine the nature of the scalar mesons and constrain the basic parameters in the UED model, the compactification scale 1/R.