HQE parameters from unquenched lattice data on pseudoscalar and vector heavy-light meson masses

We present a new lattice determination of some of the parameters appearing both in the Operator Product Expansion (OPE) analysis of the inclusive semileptonic $B$-meson decays and in the Heavy Quark Expansion (HQE) of the pseudoscalar (PS) and vector (V) heavy-light meson masses. We perform a lattice QCD (LQCD) computation of PS and V heavy-light meson masses for heavy-quark masses $m_h$ in the range from $m_c^{\rm phys}$ to $\simeq 4m_b^{\rm phys}$. We employed the $N_f = 2+1+1$ gauge configurations of the European Twisted Mass Collaboration (ETMC) at three values of the lattice spacing $a \simeq (0.062, 0.082, 0.089)$ fm with pion masses in the range $M_\pi \simeq (210 - 450)$ MeV. The heavy-quark mass is simulated directly on the lattice up to $\simeq 3m_c^{\rm phys}$. The interpolation to the physical $m_b^{\rm phys}$ is performed using the ETMC ratio method and adopting the kinetic mass scheme. We obtain $m_b^{\rm kin}(1~\mbox{GeV}) = 4.61 (20)$ GeV ($\overline{m}_b(\overline{m}_b) = 4.26 (18)$ GeV in the $\overline{\rm MS}$ scheme). The lattice data are analyzed in terms of the HQE and the matrix elements of dimension-4 and dimension-5 operators are extracted with good precision, namely: $\overline{\Lambda} = 0.552 (26)$ GeV, $\mu_\pi^2 = 0.321 (32)$ GeV$^2$ and $\mu_G^2(m_b) = 0.253 (25)$ GeV$^2$. The data also allow for an estimate of the dimension-6 operator matrix elements.


Introduction
The precise determination of the CKM element V cb is crucial in the search of new physics effects in rare decays (FCNC decays like B s → µ + µ − , K + → π + νν , K L → π 0 νν as well as K ) and particularly interesting nowadays in view of the anomalies in B → D ( * ) τν. As is well known, there is a longstanding tension of about 3 standard deviations between the values of V cb obtained from inclusive and exclusive semileptonic B-meson decays. In the first case the OPE is usually adopted to describe the non-perturbative hadronic physics in terms of few parametersμ 2 i ,ρ 3 i extracted from experimental data on inclusive B → X c ν decays together with V cb (see, e.g., Ref. [1,2] and therein) : Speaker, e-mail: aurora.melis@ific.uv.es. Acknowledges financial support from "La Caixa-Severo Ochoa" scholarship.
In the second case the relevant hadronic inputs are the semileptonic form factors describing the B → D * (D) ν decays, computed using LQCD simulations. Our aim is to present the lattice determination of some of the parameters appearing in the OPE analysis of the inclusive B-meson decays, obtained in Ref. [3]. Since the same parameters appear as coefficients of the HQE for the PS and V heavy-light meson masses, we study the heavy-quark mass dependence of two meson mass combinations, the spin averaged and the hyperfine splitting : where m h = m kin h (µ so f t ) is the renormalized heavy-quark mass in the kinetic scheme [4] at a soft cutoff µ so f t = 1 GeV. The HQE of these quantities reads as where µ 2 i , ρ 2 i refer to matrix elements of asymptotically heavy mesons, directly related toμ 2 i ,ρ 2 i in Eq. (1) that concerns matrix elements at the physical B-meson. Generally only the charmed and beauty meson masses, M D ( * ) and M B ( * ) , are used to constrain the HQE parameters (see Ref. [5]). They could be constrained more effectively having meson masses with the heavy-quark mass between the physical charm and b-quark masses, m as shown in Ref. [6]. The main idea of this contribution is to exploit the ETMC ratio method [7] to employ LQCD as a virtual laboratory and compute these fictitious meson masses with good accuracy.
In Ref. [3] the heavy-quark mass m h defined in the kinetic scheme instead of the pole one, m pole h , was adopted. The main reason is that the relation between the pole mass and the bare lattice masses µ h , and therefore also the HQEs, suffers in perturbation theory from infrared renormalon ambiguities of order O(Λ QCD ) [4,[8][9][10][11]. The kinetic mass m h is a short-distance mass that solves the problem by subtracting from the pole mass its infrared sensitive part [4,12]. The same scheme is often used in the analysis of the inclusive semileptonic B-meson decays relevant for the determination V cb [1,2].
The relation between the simulated bare heavy-quark mass aµ h and the kinetic mass m h can be obtained in few steps. Using the values of the lattice spacing and of the renormalization constant (RC) Z P determined in Ref. [13], we first calculate m h (2 GeV) = (aµ h )/(Z P a) in the MS scheme. Then we evolve the mass from µ = 2 GeV to µ = m h using N 3 LO perturbation theory [14] with four quark flavors (n = 4) and Λ N f =4 QCD = 297(8) MeV [15]. Finally, we make use of the relation between the kinetic mass m h and the MS mass m h (m h ), which is known up to two loops [16] where x ≡ µ so f t /m h (m h ), β 0 = (33 − 2n )/12 and ζ 3 1.20206. The conversion from the MS to the kinetic scheme at the charm mass and specifically the inclusion or not of the last two terms in Eq. (5) give rise to what we will call the () conv uncertainty in the error budget.

Extraction of the ground state meson masses and simulation details
The has In order to obtain a faster suppression of the exited states, it is a common procedure to adopt Gaussiansmeared interpolating quark fields [17] both in the source and/or in the sink, namely C LL where L and S denote local and smeared operators, respectively. As illustrated in Fig. 1, the SL correlations allow for an earlier plateau together with an improved signal to noise ratio, and therefore they are used in our analysis to compute the ground-state masses.
The correlation functions used in this work refer to the gauge ensembles generated by ETMC with N f = 2 + 1 + 1 dynamical quarks, which include in the sea, besides two light mass-degenerate quarks, also the strange and the charm quarks. In the ETMC set-up the gluon interactions are described by the Iwasaki action, while the fermions are regularized in the maximally twisted-mass Wilson formulation. The same ensembles are adopted in Refs. [13,18]   We start by applying the ETMC ratio method [7] to the quantity M av ( m h ) in Eq. (2). To this end we construct a sequence of heavy-quark masses { m (n) h } with a common fixed ratio λ: The series starts at the physical charm quark mass m (1) h = m c = 1.219 (41) (40) conv GeV corresponding to m c (2 GeV) = 1.176(36) GeV, obtained in Ref. [13]. For each gauge ensemble the quantity M av ( m c ) can be computed by a smooth interpolation of the results in the charm region and a subsequent extrapolation to the physical pion mass and to the continuum limit using a simple, combined linear fit in both m and a 2 as shown in Fig. 2(a) evaluated by interpolating the results corresponding to the subset of the bare heavy-quark masses. Then, we can construct the following ratios which have the static limit lim m h →∞ y M ( m h , λ) = 1 (see Eq. (3)). Each ratio is extrapolated to the physical pion mass and to the continuum limit using again a combined linear fit in both m and a 2 , obtaining values denoted by y M ( m (n) h , λ). Considering the ratios (7) has the advantage that discretization effects are suppressed, as illustrated in Fig. 2(b). Thus the m h -dependence of y M can be described as a series expansion in terms of 1/ m h , namely In Fig. 2(c) it can be seen that a linear fit with 2 = 0 is sufficient to fit the data, adopting a correlated χ 2 -minimization procedure. Finally, the chain equation  (10) GeV given in Ref. [18] and consistent with other lattice determinations within one standard deviation (see, e.g., the FLAG review [20]). The chain equation (9) can be easily extended beyond the physical b-quark point, n > K + 1, using the fitting function (8) with 2 = 0. In the case of the spin-averaged meson mass one obtains We have evaluated Eqs. (11) for n 20, i.e. for heavy-quark masses up to m h 4 m b . The results, shown in Fig. 3, exhibit uncertainties at the level of (1 − 2)%, vanishing in the static limit. Taking where the systematic uncertainty includes both the spread of the results of different fits ( m h ≥ m c with σ 4 = 0, m h ≥ 2 m c with σ 4 = 0, m h ≥ m c with σ 4 0) and the bootstrap samplings of the inputs parameters (lattice spacing, RC Z P and physical quark masses) corresponding to the eight branches of the analysis of Ref. [13].

Analysis of the hyperfine meson mass splittings
In this Section we apply the ratio method to the hyperfine meson mass splitting ∆M( m h ) in Eq. (2). As in the case of the spin-averaged case, for each gauge ensemble the quantity ∆M( m c ) is computed by interpolating first to the charm region and then extrapolating to the physical pion mass and to the continuum limit using a combined linear fit in both m and a 2 , as illustrated in Fig. 4(a). We get ∆M phys ( m c ) = 140 (11) which have the static limit lim m h →∞ y ∆M ( m h , λ) = 1 (see Eq. (4)). In Eq. (16) c G ( m h , m b ) is the shortdistance Wilson coefficient that multiplies the matrix element of the HQET chromomagnetic operator. It can be factorized in three parts: where the conversion coefficient c G is known up to three loops [21] and R is the evolution factor (see Ref. [3]) The ratios (16) are extrapolated to the physical pion mass and to the continuum limit using a combined linear fit in both m and a 2 (see Fig. 4(b)), obtaining values denoted by y ∆M ( m (n) h , λ). The m h -dependence of y ∆M can be described as a series expansion in terms of 1/ m h analogous to Eq. (8) with → ∆ . Again, a linear fit (∆ 2 = 0) is sufficient to reproduce the data, as shown in Fig. 4(c).
Using a chain equation analogous to Eq. (9) but expressed in terms of the ratios (16) and adopting the values of the parameters λ and K determined in the previous Section to reach the physical b-quark mass (10) [19], based on the direct investigation of the V to PS meson mass ratios. Going beyond the physical b-quark point, we obtain for K + 1 ≤ n 20. Taking into account the correlations between lattice data, we have performed the HQE fit ansatze (4), as shown in Fig. 5. Our final results for the HQE parameters are:   We have presented a precise lattice computation of the pseudoscalar and vector meson masses up to heavy-quark masses m h 4m phys b . They allow for a new, precise unquenched lattice determination of the dimension-5 and dimension-6 HQE parameters relevant for the determination of the CKM entry V cb from the analysis of the inclusive semileptonic B-meson decays. We have adopted the ETMC gauge configurations with N f = 2 + 1 + 1 dynamical quarks at a (0.06 − 0.09)fm and M π (210−450)MeV. The heavy-quark mass has been simulated directly on the lattice up to 3 times the physical charm mass. The interpolation to the physical b-quark mass has been performed using the ETMC ratio method adopting the kinetic mass scheme in order to work with a short-distance mass free from renormalon ambiguities. Our complete set of results (see Eqs.