LPT 13-74 Bs ! Ds`ν` near zero recoil in and beyond the Standard Model Mariam Atoui∗ and Vincent Mor´enasy Laboratoire de Physique Corpusculaire, Universit´eBlaise Pascal, CNRS/IN2P3, 63177 Aubi`ere Cedex, France Damir Beˇcirevi´cz and Francesco Sanfilippox Laboratoire de Physique Th´eorique(B^at210), Universit´eParis Sud and CNRS, Centre d'Orsay, 91405 Orsay-Cedex, France (Dated: April 28, 2014) We compute the normalization of the form factor entering the Bs ! Ds`ν decay amplitude by using numerical simulations of QCD on the lattice. From our study with Nf = 2 dynamical light quarks, and by employing the maximally twisted Wilson quark action, we obtain in the continuum limit G(1) = 1:052(46). We also compute the scalar and tensor form factors in the region near zero 2 2 2 2 2 2 recoil and find f0(q0 )=f+(q0 ) = 0:77(2), fT (q0 ; mb)=f+(q0 ) = 1:08(7), for q0 = 11:5 GeV . These latter results are useful for searching the effects of physics beyond the Standard Model in Bs ! Ds`ν decays. Our results for the similar form factors relevant to the non-strange case indicate that the method employed here can be used to achieve the precision determination of the B ! D`ν decay amplitude as well. PACS numbers: 12.38.Gc., 13.20.He, 12.39.Hg I. INTRODUCTION average, reads 1 −3 jVcbjincl: = 41:90(70) · 10 : (1) The analyses of exclusive decays, instead, are per- formed by fitting the experimental data to the Inclusive and exclusive semileptonic b ! c`ν de- shapes of the form factors parameterized according cays, with ` 2 fe; µg, have been subjects of inten- to the expressions proposed and derived in ref. [5], sive research over the past two decades. Within so that the final results are then reported in the the Standard Model (SM) the main target of that following form, research was, and still is, the accurate determina- −3 tion of the Cabibbo{Kobayashi-Maskawa matrix el- jVcbjF(1) = 35:90(45) · 10 ; arXiv:1310.5238v3 [hep-lat] 25 Apr 2014 ement jVcbj, which is extracted from the comparison of theoretical expressions with experimental mea- −3 surements of the partial or total decay widths. It jVcbjG(1) = 42:6(1:5) · 10 ; (2) turns out, however, that the value for jVcbj obtained ∗ from the exclusive decays agrees only at the 2:1 σ as obtained from B(B ! D µν) and B(B ! level with the one extracted from inclusive decays. Dµν), respectively. F(1) and G(1) are the rele- More specifically, the two independent analyses of vant hadronic form factors at the zero recoil point. inclusive decays [1, 2] (updated in refs. [3]) give Thanks to the heavy quark symmetry, and up to completely consistent results which, after taking an perturbative QCD corrections, both these form fac- tors are equal to one in the limit of mc;b ! 1 [6]. To compute the deviation of these form factors ∗Electronic address: [email protected] yElectronic address: [email protected] zElectronic address: [email protected] 1 The most recent update of the analysis of inclusive decays xElectronic address: [email protected] using the kinetic scheme can be found in ref. [4]. 2 from unity, it is necessary to include all the non- ification of the proposal of ref. [12], presented in n perturbative order 1=mc;b QCD corrections. The ref. [13] and also implemented in the computation only model independent method allowing to com- of the decay constant fB and the b-quark mass, cf. pute F(1) and G(1) from the first theory principles ref. [14]. The remainder of this paper is organized is lattice QCD. Using the most recent estimates of as follows: In sec. II we define the form factors and the above form factors, F(1) = 0:902(17) [7] and express them in a way that is suitable for the strat- G(1) = 1:074(24) [8], one arrives to egy used for their computation which is described in sec. III. Details of our lattice computations and −3 jVcbjexcl: = 38:56(89) · 10 ; (3) the results for G(1) are given in sec. IV, while the results concerning the scalar and tensor form fac- a number that obviously differs from jVcbjincl: given tors in the region close to zero recoil are discussed in eq. (1). In principle the exclusive decay modes and presented in sec. V. We finally conclude in are better suited for the precision determination sec. VI. of jVcbj because less theoretical assumptions are needed to compute the corresponding decay rates. The main obstacle is, however, the necessity for II. DEFINITIONS a reliable, high precision, lattice QCD estimate of F(1) and G(1). Furthermore, for the required The hadronic matrix element describing the percent precision of jV j it is important to have cb B ! D `ν decay in the SM, hD j¯bγ (1 − γ )cjB i a good control over the structure dependent soft s s ` s µ 5 s ≡ hD j¯bγ cjB i, is parameterized in terms of the photon B ! D(∗)`νγ which could otherwise be s µ s soft hadronic form factors f (q2) as misidentified as pure semileptonic decays. This +;0 problem is less acute for the down-type spectator, hD (k)jV jB (p)i = (p + k) f (q2)+ i.e. B¯0 ! D(∗)+`ν¯, than for the up-type specta- s µ s µ + 2 2 − (∗)0 m − m tor quark, B ! D `ν¯ [9]. For that reason it Bs Ds 2 2 qµ f0(q ) − f+(q ) ; (4) is desirable to consider the charged and neutral B- q2 meson decay modes separately. In this paper we ¯ 2 2 will mainly discuss B ! D `ν decay for which where Vµ = bγµc, q = p − k, and q 2 (0; qmax], s s 2 2 the soft photon pollution is smaller. Moreover, this with qmax = (mBs − mDs ) . The extraction of the mode is much more affordable numerically because form factors becomes particularly simple if we use the valence s-quark is easily accessible in numeri- the projectors cal simulations of QCD on the lattice which is not 2 ! the case with the physical u=d-quark. In this pa- 0 qµ + ~q Pµ = 2 ; Pµ = ; ~q ; (5) per we will also comment on the non-strange case qmax mBs − EDs when appropriate. Finally, we prefer to focus on ∗ B(s) ! D(s)`ν, rather than B(s) ! D(s)`ν, be- so that cause the hadronic matrix element involves much m + m less form factors and the decay rate is therefore 0 hD (k)jV jB (p)i = Bs Ds f (q2) ; Pµ s µ s m − m 0 likely to be less prone to systematic uncertainties. Bs Ds 2m Despite the importance of G(1), only a few lat- + 2 Bs 2 Pµ hDs(k)jVµjBs(p)i = ~q f+(q ) : tice QCD studies have been performed so far. The mBs − EDs methodology described in ref. [10] has been imple- (6) mented in unquenched simulations with Nf = 2+1 dynamical staggered light quarks in refs. [8, 11] In our computations we will consider the situations ~ where the propagating heavy quarks on the lattice with j~pj = 0 and qµ = (mBs − EDs ; −k). Another have been interpreted by means of an effective the- frequently used parameterization of this matrix el- ory approach. In ref. [12], an alternative method ement, motivated by the heavy quark expansion, to compute the B ! D semileptonic form factors reads has been proposed and implemented in quenched 1 approximation. That latter method, based on use p hD (k)jV jB (p)i = (v + v0) h (w) m m s µ s µ + of the step scaling function, allows to compute the Bs Ds 0 same form factors without recourse to heavy quark + (v − v )µ h−(w) ; effective theory. The price to pay, however, is (7) that the method of ref. [12] is computationally very 0 costly and to this date it has not been extended where v = p=mBs , v = k=mDs , and the relative to unquenched QCD. In this paper we use a mod- velocity w = v·v0 = (m2 +m2 −q2)=(2m m ). Bs Ds Bs Ds 3 From the comparison of eqs. (4) and (7) one gets III. STRATEGY 2 2 mBs + mDs mBs − mD h−(w) To extract the form factors f0;+(q ) from numer- f+(q ) = p h+(w) 1 − ; 4mBs mDs mBs + mDs h+(w) ical simulations on the lattice, one first computes p the correlation functions 2 mBs mDs f0(q ) = (w + 1) h+(w) 1− mBs + mDs X y −i~q(~x−~y) Cµ(~q; t) = hPbs(~0; 0)Vµ(~x;t)P (~y; tS)e i ; m + m w − 1 h (w) cs Bs Ds − : ~x;~y mBs − mDs w + 1 h+(w) (14) (8) where the interpolating source operators, the pseu- The form factor G(w) used in experimental analy- doscalar densities Pcs and Pbs, are sufficiently sep- ses of the B(s) ! D(s)`ν decay is proportional to arated in the time direction so that for 0 t tS 2 P − f+(q ), and reads one can isolate the lowest lying states with J = 0 that couple to two source operators, and then ex- tract the matrix element of the vector current be- mBs − mDs h−(w) G(w) = h+(w) 1 − tween the two. The simplest choice would be the m + m h (w) Bs Ds + ¯ " # local operator, Phs = hγ5s, but for practical con- m − m 2 venience one often resorts to the smearing tech- ≡ h (w) 1 − Bs Ds H(w) (9); + nique that helps to significantly reduce the cou- mBs + mDs plings to radially excited states.