Pion couplings to the scalar B meson

We present two-flavor lattice QCD estimates of the hadronic couplings $g_{B^*_0 B \pi}$ and $g_{B_1^* B_0^* \pi}$ that parametrise the non leptonic decays $B^*_0 \to B \pi$ and $B^*_1 \to B_0^* \pi$. We use CLS two-flavour gauge ensembles. Our framework is the Heavy Quark Effective Theory (HQET) in the static limit and solving a Generalized Eigenvalue Problem (GEVP) reveals crucial to disentangle the $B^*_0$($B^*_1$) state from the $B \pi$($B^*\pi$) state. This work brings us some experience on how to treat the possible contribution from multihadronic states to correlation functions calculated on the lattice, especially when $S$-wave states are involved.

1 B * 0 π that parametrise the non leptonic decays B * 0 → Bπ and B * 1 → B * 0 π. Our framework is the Heavy Quark Effective Theory (HQET) in the static limit and solving a Generalized Eigenvalue Problem (GEVP) reveals crucial to disentangle the B * 0 (B * 1 ) state from the Bπ(B * π) state. This work brings us some experience on how to treat the possible contribution from multihadronic states to correlation functions calculated on the lattice, especially when S-wave states are involved.

I. INTRODUCTION
Heavy Meson Chiral Perturbation Theory (HMχPT) [1,2] is commonly used to extrapolate lattice data in the heavy-light sector to the physical point. Relying on Heavy Quark Symmetry and the (spontaneously broken) chiral symmetry, an effective Lagrangian is derived where heavy-light mesons fields [3] couple to a Goldstone field via derivative operators. In the static limit, the total angular momentum of the light degrees of freedom, j l = s l + L, is conserved independently of the total angular momentum J = j l ± 1/2. The pseudoscalar (B) and the vector (B * ) mesons belong to the doublet j P l = (1/2) − corresponding to L = 0 whereas the scalar (B * 0 ) and the axial (B * 1 ) mesons belong to the positive parity doublet j P l = (1/2) + corresponding to L = 1 (see Table I). Equivalently to the low energy constants that parametrize the well known chiral Lagrangian, hadronic couplings enter the effective theory under discussion, that is particularly suitable to describe processes with emission of soft pions, i.e. H 1 (J 1 ) → H 2 (J 2 )π where H i is a heavy-light meson, and p π Λ χ ∼ 1 GeV. The associated pionic couplings are g H1(J1)H2(J2)π and they cannot be computed in perturbation theory. When the negative j P = (1/2) − and positive j P = (1/2) + parity states are taken into account, the effective Lagrangian is parametrized by three couplingsĝ,g and h. The first FIG. 1: Three-point correlation function used by [11] to compute A+(∆ 2 = q 2 ).
coupling,ĝ, relates transitions between mesons belonging to the same doublet J P = (1/2) − and has been precisely measured on the lattice [4] - [8]. On the contrary, the last two couplings are less precisely known. The residue at the poles of form factors in heavy to light semileptonic decays [9] is also expressed in terms of those couplings. In that respect, the channel B * 0 → Bπ is very interesting: The HMχPT Lagrangian tells us that the transition reads also [10] Γ that is appropriate in the heavy quark limit. In the static limit, the couplingg is similar toĝ, but for hadronic transition between positive parity states. Those transitions are energetically not allowed for the B system but are useful for chiral extrapolations in Lattice QCD. So far there is only one computation of h andg [11], using ratio of three-point correlation functions and the techniques of measuring the Fourier transform of the radial distribution to obtain the form factor A + (q 2 π ) in the limit q 2 π → 0, to extract h: Figure 1 and q π = (0, 0, q z ), choosing q z = δ. In Ref. [12] the transition B * 0 → Bπ was directly studied on the lattice, computing two-point correlation functions: the authors claimed that, close to the threshold m Bπ ∼ m B * 0 , the ratio is related to Γ(B * 0 → B + π − ). We follow here this last approach and perform the computation on a set of N f = 2 configurations made available by the Lattice Coordinated Simulations effort. It gives a further check that the extraction of the scalar B meson decay constant on those ensembles, that we report in a forthcoming paper, is under control at ∼ 10% of precision we hope. The plan of the letter is the following: in section II and III we describe the approach we have employed, in section IV we present our lattice set-up and our results are given in section V, that we discuss in section VI.
The transition amplitude under interest is parametrised by with q π = p − p and f π = 130 MeV, the pion decay constant. The Fermi golden rule teaches us that where the density of states ρ reads, for a given energy E π of the pion living on the lattice of spatial volume In lattice units (a being the lattice spacing), we obtain Considering the two-point correlation function C and O Bπ are interpolating fields with vanishing momentum of the B * 0 and the Bπ states respectively, we have We have assumed small overlaps 0|O B * 0 |Bπ and 0|O Bπ |B * 0 and the normalization of states is n|m = 1. Finally, close to the threshold m B * 0 ≈ m Bπ , we get Therefore, one can extract x from the ratio [13] - [15] where C Bπ Bπ are, respectively, two-point correlation functions of a scalar B meson and a Bπ multihadronic state: Further away from the threshold, eq.(1) has to be modified. The most interesting correction for our analysis is the one to the linear term in x. The time dependence of the ratio R is then in where ∆ = m B * 0 − m Bπ . To suppress the contamination by excited states, it is welcome to solve a Generalized Eigenvalue Problem (GEVP) [16] - [20]: Bπ where C Bπ Bπ are from now matrices of two-point correlators and v X are the generalized eigenvectors associated to the ground state in the corresponding channel and (a, Cb) = i a i C ij b j is the scalar product.

III. EXTRACTION OFg
Similarly to the couplingĝ which sets the magnitude of the transition between the pseudoscalar and the vector B mesons by exchanging a single soft pion [10], the couplingg parametrizes the amplitude where B * 0 and B * 1 are respectively the scalar and the axial B mesons at rest and k is the polarization vector of the axial B meson. This matrix element can be extracted using the same technique as discussed in [19] but applied to the first excited heavy-light mesons doublet. Therefore following the method of [19,20], we consider the ratio of three to two-point correlation functions where K ij (t) is the summed three-point correlation function and is the renormalized axial current. The renormalisation constant Z A was determined non-perturbatively by the ALPHA Collaboration [21]. Here, C (3) (t, t 1 ) is again a matrix of correlators and the eigenvectors v B * 1 (t, t 0 ) are defined similarly to eq. (4). Thanks to heavy quark symmetry, the two-point correlation functions C are proportional and only one GEVP needs to be solved. Finally, one can show that, in the static limit of HQET [19], where ∆ mn = E m − E n is the energy difference between the m th and n th excited states of the GEVP and N × N is the size of the matrix of correlators defining the GEVP.

IV. LATTICE SETUP
In our study we have performed measurements on a subset of four N f = 2 CLS lattice simulations, defined with the plaquette gauge action and non perturbatively O(a) improved Wilson-Clover fermions; we collect the main parameters in Table II and we remind the reader that the criterion of our choice is to be very close to the threshold m B * 0 ≈ m Bπ . We have computed static-light correlators with HYP2 static quarks [22] and stochastic all-to-all propagators with full time dilution for the light quarks [23]. A single stochastic source has been used to compute the propagator. Interpolating fields of a static-light meson are defined as [24] where ψ h is the static heavy quark field and ψ l is the relativistic quark field (l = u/d). The Gaussian smearing parameters are κ G = 0.1, r n ≡ 2a √ κ G R n ≤ 0.6 fm and ∆ is a covariant Laplancian made of three times APE-blocked links [25]. Moreover, O Γ,n can be "local" (Γ = γ 0 , γ 5 ) or contain a derivative operator (Γ = γ 0 We have also implemented the isosymmetric interpolating fields of the form which couple to the multihadronic state Using the notation ψ l (y) = G mn l (x, y), ψ h (x)ψ h (y) = G h (x, y) for the smeared light quark propagator and the static quark propagator respectively, the two-point correlation functions constructed from these interpolating fields are Γ = γ 0 Γ † γ 0 , whose the diagram is sketched in Figure 2, whose the direct (12), box (13) and cross (14) diagrams are sketched in Figure 3, Tr G m0 l (y 2 , y 1 )γ 5 G 0n l (y 1 , x)ΓG h (x, y 2 )γ 5 , whose the diagrams are sketched in Figure 4. We have computed the triangle correlators C B * 0 Bπ and C Bπ B * 0 by two methods, either using the one-end-trick and a single inversion to obtain the two light propagators [26,27], or getting the second light propagator by solving the Dirac equation with the first light propagator taken as a generalised source. The second approach is more noisy, as shown on Figure 5.
The box (13) and cross (14) diagrams depicted in Figure 3 require at least one more inversion of the Dirac operator for each time slice and are therefore expensive to compute. They have been computed only in the case of the ensemble E5. Their contributions are small compared to the direct one given by (12), 0.1% and 1%, respectively. Neglecting them, we obtain ax = 0.0241(10) whereas we obtain ax = 0.0228(10) when they are taken into account. The two results are compatible within our errors and the computation of these diagrams does not seem necessary at our level of precision. Since we don't expect the light quark mass dependence to play a major role on that specific point, we neglect these diagrams in our calculation on other ensembles.
Finally, we have also computed the three point correlation functions (9) needed for the extraction of the couplingg using the same basis of interpolating operators: x, y, z tx

V. RESULTS
We show in Figure 6 the ratio R GEVP (t) and its derivative with respect to time x eff (t) = dR GEVP (t)/dt, that corresponds to the quantity ax we are measuring. We observe a nice plateau for every ensemble under study. The very flat behavior of x eff (t) in the plateau region lets us conclude that quadratic and higher terms in t in the formula eq. (2), coming from ∆ = 0, are almost absent. This was expected since in our range of fitting, Concerning the three-point correlation functions, we have checked using either local interpolating operators or interpolating operators built from the insertion of a covariant derivative give compatible results. However in the last case the signal is less noisy as shown in Figure 7. Therefore only these fields are used in the following and some typical plateaus are depicted in Figure 8.
With ax and m B * 0 − m B = 385(17) stat (28) syst MeV [28], we can finally extract Γ/| q π |, h and g B * 0 Bπ ; we collect the values in Table III. In the table, the first error on h comes from the uncertainty on m B * 0 in the continuum limit and the second error comes from the error on ax. The light-quark mass and lattice spacing dependence is so small on our data that it is legitimate to try a fit with a constant: we obtain h = 0.84(3) andg = −0.120 (3). Performing a linear fit in m 2 π , we get compatible results h = 0.86(4) and We used t0/a = 5 for t > t0 and t0 = t − a elsewhere. g = −0.122 (8). A third possibility is to use the NLO formulae of HMχPT [29] h = h 0 1 − 3 4 whereĝ 0 = 0.5(1) [4,8] is the pionic couplings associated to H * → Hπ. We get h = 0.84(3) and g = −0.116 (7). The previous formulae forg take into account corrections from tadpole diagrams to the where the first error is statistical and the second error corresponds to the uncertainty that we evaluate from the discrepancy between the constant and linear fits. We show in Figure 9 the chiral extrapolations of h andg. Rigorously, in the NLO chiral fits, we have neglected the contribution from the heavy-light states of opposite parity, as computed in [30]; they have been studied in [11]. Neglecting them is equivalent to assume m π δ = m B * 0 − m B . Since, for our lattice ensembles, the pion mass lies in the range [280 -440] MeV and the mass difference between the scalar B meson and the ground state B meson is of the order of δ ∼ 400 MeV, the contribution is not negligible. Therefore, we also tried the other fit formulae where the couplingĝ 0 is the same as before and the mass difference δ is given in Table III. The results are h = 0.85(3) andg = −0.116 (7) and is also perfectly compatible with our previous findings. In Refs. [13,14], an alternative method to evaluate such a coupling like h was proposed. Indeed, one can show that the connected contribution to the correlation function C Bπ Bπ (t), which includes box (13) and cross (14) diagrams, has the following behavior: where C conected (t) = − 3 2 C box (t)+ 1 2 C cross (t). As explained before, these diagrams have been computed only for the CLS ensemble E5 and the functionR(t) is plotted in Figure 10. The results are quite precise and the linear dependence in (22) cannot be neglected. Taking this into account, the result reads |ax| = 0.0237 (8), in perfect agreement with the one obtained by the previous method (see Table III). The fit range has been varied from t/a ∈ [9 − 18] to t/a ∈ [13 − 18] where the result is stable to estimate the error.

VI. DISCUSSION AND CONCLUSION
The couplings h andg were explicitly computed on the lattice in Ref [11]. For h, two results are reported for the two different actions used there: h = 0.69(2)( +11 −7 ) and h = 0.58(2)( +6 −2 ). They are lower than what we get but this difference might be explained by the larger quark masses simulated at that time: indeed the chiral extrapolation tends to lower the extrapolated value. Our result is also a bit larger than the QCD sum rules estimates: in Ref [31] the computation of g B * 0 Bπ gives h = 0.56 (28), while in Ref [32] it gives h = 0.74 (23). We can compare our finding with experimental data in the D sector, although the static approximation of HQET is expected to give only a rough estimate due to quite large 1/m c corrections. For example, in the case of the D meson decay constant, a heavy quark spin breaking effects larger than 20% between f D and f D * has been measured [33]. With m D * 0 = 2318(29) MeV and Γ D * 0 = 267(40) MeV [34], we obtain Γ(D * 0 → Dπ)/| q π | = 0.68 (11) and h = 0.74 (8), assuming that the branching ratio B(D * 0 → Dπ) is ∼ 100%. This result is smaller than the one obtained in this work but it is compatible within error bars. In [35] the phase shift of the Dπ scattering state was computed on the lattice: relating the coupling g D * 0 Dπ quoted in that paper to h, one finds that h is around 1. Referring to the Adler-Weissberger sum rule [36] in the Bπ system, in the m Q → ∞ and soft pion limits, δ |X Bδ | 2 = 1, where Γ(I → Fπ) = 1 2πf 2 π | q| 3 2j I +1 |X I→F | 2 [37], we have the boundĝ 2 + h 2 < 1. With the lattice averageĝ = 0.5(1) made with the results [4,8], we obtain that the sum rule would be saturated at 95% by the B * pole and the first orbital excitation. We also confirm the finding of Ref [11] where a small value ofg was obtained. In particular this coupling for positive parity states is smaller than in the case of negative parity statesg g. In conclusion, we have extracted from lattice simulations with N f = 2 dynamical quarks the couplings h andg that parametrise the emission of a soft pion by a scalar B meson. We have observed a very mild quark mass and cut-off dependence of our numbers and we quote h = 0.84(3)(2),g = −0.122(8)(6) as our estimate. Ifg is small, the large value of h compared toĝ ∼ 0.5 outlines the fact that some care is necessary to apply HMχPT for pion masses close to mass splitting m B * 0 −m B ∼ 400 MeV: B meson orbital excitation degrees of freedom cannot be neglected in chiral loops.