Low-energy constants of heavy meson effective theory in lattice QCD

We consider effective theory treatment for the lowest-lying $S$- and $P$-wave states of charmed mesons. In our analysis, quantum corrections and contributions from leading chiral and heavy quark symmetry breakings are taken into account. The heavy meson mass expressions have abundance parameters, low-energy constants, in comparison to the measured charmed mesons masses. The experimental and lattice QCD data on charmed meson spectroscopy are used to extract, for the first time, the numerical values of the full set of low-energy constants of the effective chiral Lagrangian. Our results on these parameters can be used for applications on other properties of heavy-light meson systems.

The properties of heavy-light meson systems can be well described using heavy meson chiral perturbation theory. This approach, which is formulated by combining χPT and HQET, can be used in a systematic way to calculate the corrections from chiral and heavy quark symmetry breakings (see e.g., [1][2][3][4][5]). Thomas Mehen and Roxanne Springer in Ref. [6] used this theory to study the masses of the lowest-lying odd-and even-parity charmed mesons. In their analysis, the contributions due to finite masses of light and heavy quarks and one-loop chiral corrections are taken into account. The theory at this, third, order has a large number of unknown LECs in comparison to charmed meson spectrum, and hence a unique fit for them using nonlinear fitting is impossible as concluded in Refs. [6,7].
The work of Mehen and Springer is reconsidered in our paper [8]. There, we employed a different approach to get a unique fit for these unknown LECs. It is based on reducing their number in fit, which is simply done by grouping them into certain linear combinations that equal to the number of charm meson masses, and evaluating the one-loop corrections using physical masses, which unlikes previous approaches ensures that the imaginary parts of loop functions are consistent with the experimental widths of the charmed mesons. By using physical masses in loops, the fit becomes linear and LECs of the effective Lagrangian, which appear in linear combinations, are uniquely determined using the lowest odd-and even-parity charmed spectrum. The fitted parameters from charmed mesons are then used in [8] to predict the spectrum of analog bottom mesons.
It is pointed out in our previous work that to separate the combinations of the LECs into pieces that respect and break chiral symmetry, lattice QCD information on charm mesons ground and excited states with different quark masses are required. The recent lattice calculations on the charm meson spectroscopy undertaken by Cichy, Kalinowski and Wagner in Ref. [9] provide enough information to perform further separations of LECs. Our purpose here is to use the experimental and this lattice data on charmed meson masses to extract the unique values of LECs of the effective Lagrangian used in Refs. [6][7][8]. From extracted LECs, we can examine the consistency of the chiral expansion in HMχPT. We will use the theory to calculate the masses of the lowest odd-and even-parity heavy-light mesons in SU (3) limit, which are useful in extracting the relevant matrix operators in HQET.
Before proceeding, let us first review the mass formula for odd-and even-parity charmed mesons that are presented in [6,8]. In a compact form, up to one-loop corrections, the residual charm meson mass [10] is where A = H, S denotes the odd-and even-parity charm meson states, respectively. In the heavy quark limit, the odd-parity states, i.e., pseudoscalar mesons J P = 0 − (D 0 , D + , D + s ) and vector mesons J P = 1 − (D * 0 , D * + , D * + s ), form members of the 1 2 − -ground-state doublet, and the even-parity states, i.e., scalar meson J P = 0 + (D * 0 0 , D * + 0 , D * 0s ) and axial vector mesons J P = 1 + (D 0 1 , D 1 1 , D 0 1s ), form members of the The quantity Σ A ( * ) q in Eq. (1) refers to the one-loop corrections, which can be obtained by adding all one-loop graphs that are allowed by spin-parity quantum numbers. Their explicit expressions can be found in Appendices of Refs. [6,8]. There are three coupling constants g, g , h entering the one-loop contributions. The coupling g (g ) measures the strength of transitions within states that belong to 1 2 − ( 1 2 + ) doublet which are represented by the chiral function which is defined in the MS-scheme [8]. The renormalization scale is given by µ. The arguments m i and ω are the mass of the Goldstone boson and mass difference between external and internal heavy meson states.
The function F (ω, m i ) is given by [11] F (ω, The transitions between states that belong to different doublets are measured by the coupling strength h and represented by in the MS-scheme [8]. In the SU (3) limit (chiral limit m i → 0), above chiral functions become According to the power counting rules employed in [6,8], terms in above mass expression scale as δ ∼ ∆ ∼ Q, m a ∼ m ∼ Q 2 , and Σ ∼ Q 3 where Q generically denotes the low-energy scales in the theory, i.e., masses and momenta of the Goldstone bosons and splittings between the four lowest states of the charmed mesons introduced above.
There are twelve unknown LECs in Eq. (1) describing eight charm meson masses taking in the isospin limit. It is, thus, hard fixing them using available data alone. To overcome this, LECs can be grouped into [8], so, charmed meson mass given in Eq. (1) can be written as where α q and β ( * ) q are α n = −1, α s = 2, β n = 1/2, β s = −1, β * n = −1/6, and β * s = 1/3. Now, the number of unknown coefficients in Eq. (7) is eight which equals the number of the observed charmed mesons shown in Fig. 1. By using physical values in evaluating chiral loop functions in Eq. (7), as done in [8], one can extract the unique values for the parameters given in Eqs.(5)- (6).
In our fit, the empirical values we use are two masses of the ground-state nonstrange mesons in the isospin limit, two masses of the excited neutral charmed mesons, which are chosen due to their relatively small errors in comparison with the excited charged counterpart, and four masses of strange mesons from both sectors, see  [13]. The coupling constant g is experimentally unknown and the computed lattice QCD value g = −0.122(8)(6) [16] is used in this work. In our previous work [8], the normalization scale was set to the average of pion and kaon masses, µ = 317 MeV. It is worth mentioning that in our approach the extracted parameters and quantities derived from them, e.g., mass splittings, are smoothly varying with µ-scale and their numerical values are in agreement within the associated uncertainties. Therefore, performing calculations at any other values of µ-scale will not make much difference. Here we will use µ = 1000 MeV.  [13] excluding the mass of D 1 , which is reported by the BELLE collaboration [14]. We only take the isospin average of D 0 and D ± (D * 0 and D * ± ) to obtain the mass of nonstrange ground state D (D * ); for details please refer to the text.
To fit parameters in Eq. (7) to the experiment, we need to define the experimental residual masses. For this, we choose m D , the mass of pseudoscalar nonstrange charmed meson, as the reference mass, which yields the following central values for charmed meson residual masses, from fitting the residual mass expression in Eq. (7) to the corresponding experimental masses in Eq. (8).
The associated uncertainties with the fitted parameters, which include the experimental errors of charm meson masses and coupling constants and the error on the coupling g from lattice QCD, are dominated by the uncertainty in the 0 + and 1 + nonstrange masses. Therefore, improved experiments on these mesons are needed to reduce the errors. It is clear from Eq.(6) that the available experimental information is enough to fix the LECs a H , ∆ Nature, however, cannot help us to disentangle chirally symmetric coefficients δ A , ∆ A in Eq. (5) from chiral breaking terms, more precisely σ A and ∆

(σ)
A as a A and ∆

(a)
A already fixed by experiment, see Eq. (13). To make further separations of the LECs in Eq. (5) in the odd-and even-parity sectors, lattice calculations on charm mesons ground and excited states with different quark masses are required. For this purpose, we will use lattice calculations on charmed meson spectroscopy undertaken in Ref. [9].
The computations in [9] were performed using three different lattice spacing and several light quark masses. In this paper, we use the values extracted in ensemble D defined in [9], which has the smallest lattice spacing a = 0.0619 fm, in our fit of LECs. In Table I, we present the continuum masses of odd-and even-parity charmed mesons computed at nonphysical pion masses. The shown values are obtained by performing a continuum extrapolation at the relevant nonphysical pion masses using Strategy 3 illustrated there [12]. For the ground-state charmed meson, they used its mass as an input to fix the charm quark mass for each ensemble, so in our fit we will use the experimental value shown in Fig. 1. In their work, strange valence quark mass was chosen to be close to its physical value. This was achieved by reproducing the physical value of 2m 2 K − m 2 π using measured pion and kaon masses in each ensemble. In leading order chiral perturbation theory, this quantity represents the strange light quark mass and is insensitive to mass of light quark flavor. Consequently, one can use the computed values of pion mass in ensemble D to extract the corresponding masses of kaon and eta particles. This is simply done by using the mass relations ((2m 2 K −m 2 π ) phys +m 2 π,L )/2 and (2(2m 2 K − m 2 π ) phys + m 2 π,L )/3 to get m 2 K and m 2 η , respectively, where m π,L is the lattice measured pion mass, see Table I. The uncertainties associated with the lattice determination of these masses are negligible at our level of precision. The nonstrange ground-state charmed meson mass, mD, was used in [9] to tune the charm quark mass in their lattice computations. In our caculation, we use the experimental value shown in Fig. 1 for this non-measured lattice mass.
Using lattice data from Table I, extracted values of parameters given in Eq. (5) are shown in Figs. 2(a)-2(d) together with that obtained using experimental values, see Eqs. (9) and (11). To fit these parameters, a constrained fitting procedure [17] is employed with priors on the LECs constructing them. For LECs a A and ∆ where associated uncertainties include the experimental errors of charm meson masses and coupling constants and errors from lattice data on charm meson masses. To shrink the uncertainties on the above determined LECs, accurate experimental and lattice results on charmed meson masses are needed.
The extracted values given in Eqs.
which are close to the experimental values given in Eq. (8). By fitting LECs of the effective Lagrangian, we increase the usefulness of HMχPT to other applications of heavy-light meson systems. As a direct application, the theory will be used to extract the masses of charmed and bottom mesons in the SU (3) limit. Here, the form of the residual mass of charm mesons in Eq. (1) becomes m SU(3) The analog expression for bottom mesons can be simply obtained by rescaling the hyperfine operator ∆ A in Eq. (16) that breaks heavy quark symmetry. This is achieved by multiplying it by the mass ratio of charm and bottom quarks. As in our previous work [8], we use the value, m c /m b = 0.305 ± 0.05, for the rescaling factor. To extract the masses, we set pseudoscalar meson mass as a reference mass and define three independent mass splittings in each meson system. As one-loop corrections contain nonlinear functions of meson mass differences, these mass splittings form nonlinear equations and can be solved using an iterative method starting from the tree-level masses [8]. Our results are where the reference masses m are set to the corresponding values given in Ref. [18] to obtain the absolute masses for vector, scalar, and axial vector charmed and bottom mesons. The uncertainties associated with the reference pseudoscalar charmed and bottom masses are the statistical errors [18]. The errors on the other masses include the statistical error associated with pseudoscalar masses, errors from LECs, and the error from rescaling factor for the bottom meson masses only. Above results can be used to