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The continuum limit of the static-light meson spectrum

We investigate the continuum limit of the low lying static-light meson spectrum using Wilson twisted mass lattice QCD with N_f = 2 dynamical quark flavours. We consider three values of the lattice spacing a ~ 0.051 fm, 0.064 fm, 0.080 fm and various values of the pion mass in the range 280 MeV<m_PS<640 MeV. We present results in the continuum limit for light cloud angular momentum j = 1/2, 3/2, 5/2 and for parity P = +, -. We extrapolate our results to physical quark masses, make predictions regarding the spectrum of B and B_s mesons and compare with available experimental results.


Introduction
A systematic way to study B and B s mesons from first principles is with lattice QCD. Since am b > 1 at currently available lattice spacings for large volume simulations, one needs to use for the b quark a formalism such as Heavy Quark Effective Theory (HQET) [1,2] or Non-Relativistic QCD [3]. An alternative procedure has recently been proposed [4] which is based on HQET but does not make use of the static point. Here we follow the standard HQET route, which enables all sources of systematic error to be controlled.
In the static limit a heavy-light meson will be the "hydrogen atom" of QCD. Since in this limit there are no interactions involving the heavy quark spin, states are doubly degenerate, i.e. there is no hyperfine splitting. Therefore, it is common to label static-light mesons by parity P and the total angular momentum of the light degrees of freedom j with j = |l ± 1/2|, where l and ±1/2 denote respectively angular momentum and spin. An equivalent notation is given by l ± , which reads S ≡ (1/2) − , P − ≡ (1/2) + , P + ≡ (3/2) + , D − ≡ (3/2) − , D + ≡ (5/2) − , F − ≡ (5/2) + , F + ≡ (7/2) + ... The total angular momentum of a static-light meson is either J = j + 1/2 or J = j − 1/2, where both states are of the same mass. Note that in contrast to parity, charge conjugation is not a good quantum number, since static-light mesons are made from non-identical quarks.
The static-light meson spectrum has been studied comprehensively by lattice methods in the quenched approximation with a rather coarse lattice spacing [5]. Lattice studies with N f = 2 flavours of dynamical sea quarks have also explored this spectrum [6,7,8,9,10,11,12]. Here following our initial study [13,14], we use N f = 2 flavours and are able to reach lighter dynamical quark masses, which are closer to the physical u/d quark mass, so enabling a more reliable extrapolation. Note that in our formalism, maximally twisted mass lattice QCD, mass differences in the static-light spectrum are O(a) improved so that the continuum limit is more readily accessible. We now extend our study to include three different lattice spacings, which gives us confidence that we are indeed extracting the continuum limit.
In this paper, we approach the B meson spectrum by concentrating on the unitary sector, where valence quarks and sea quarks are of the same mass. This is appropriate for static-light mesons with a light quark, which is u or d.
We also estimate masses of B s mesons with s quarks of physical mass, where the s quark is treated as a valence quark in the sea of light u and d quarks (so this is a partially quenched study). We took our s quark mass values from ETMC studies of strange mesons [15,16].
Within the twisted mass formalism, it is feasible to use N f = 2 + 1 + 1 flavours of dynamical sea quarks, which would give a more appropriate focus on the static-strange meson spectrum if strange quark sea effects were significant. This is under study by ETMC.
In HQET the leading order is just the static limit. The next correction will be of order 1/m Q , where m Q is the mass of the heavy quark. This correction is expected be relatively small for b quarks, but larger for c quarks. Lattice methods to evaluate these 1/m Q contributions to the B meson hyperfine splittings have been established and tested in quenched studies [17,18,19,20,21]. We intend to explore these contributions using lattice techniques subsequently. An alternative way to predict the spectrum for B and B s mesons is to interpolate between D and D s states, where the experimental spectrum is rather well known, and the static limit obtained by lattice QCD assuming a dependence as 1/m Q . Thus the splittings among B and B s mesons should be approximately m c /m b ≈ 1/3 of those among the corresponding D and D s mesons.
For excited D s mesons, experiment has shown that some of the states have very narrow decay widths [22]. This comes about, since the hadronic transitions to DK and D s M (where M is a flavour singlet mesonic system, e.g. η ′ , ππ or f 0 ) are not allowed energetically. The isospin violating decay to D s π together with electromagnetic decay to D s γ are then responsible for the narrow width observed. A similar situation may exist for B s decays and we investigate this here using our lattice mass determinations of the excited states. This will enable us to predict whether narrow excited B s mesons should be found.
As well as exploring this issue of great interest to experiment, we determine the excited state spectrum of static-light mesons as fully as possible. This will help the construction of phenomenological models and will shed light on questions such as, whether there is an inversion of the level ordering with l + lighter than l − at larger l or for radial excitations as has been predicted [23,24,25,26,27].
Since we measure the spectrum for a range of values of the bare quark mass parameter µ q for the light quark, we could also compare with chiral effective Lagrangians appropriate to HQET. This comparison would be most appropriate applied to heavy-light decay constants in the continuum limit (see ref [28]). Since that study awaits more precise renormalization constants, we do not discuss it further here.
Since we have discussed the basic methods in a previous paper [14], in this paper we present only briefly the details of our computation of static-light meson mass differences. We give a full discussion of our extrapolation to the continuum and to physical light quark masses. We also discuss the interpolation to the physical b quark mass.

Lattice details
We use N f = 2 flavour gauge configurations produced by the European Twisted Mass Collaboration (ETMC). The gauge action is tree-level Symanzik improved [29], while the fermionic action is Wilson twisted mass at maximal twist (cf. e.g. [30] and references therein). As argued in [14] this ensures automatic O(a) improvement for static-light spectral quantities, e.g. mass differences of static-light mesons, the quantities we are focusing on in this work.
We use three different values of the lattice spacing a ≈ 0.051 fm , 0.064 fm , 0.080 fm and various values of the pion mass in the range 280 MeV < ∼ m PS < ∼ 640 MeV. All lattice volumes are big enough to fulfill m PS L > 3.2. The ensembles we are considering are listed in Table 1. Details regarding the generation of gauge configurations and analysis procedures for standard quantities (e.g. lattice spacing, pion mass) can be found in [31,32].
In Table 1 we also list the number of gauges, on which we have computed static-light correlation functions, and the number and type of inversions performed to estimate light quark propagators stochastically. Note that in contrast to our previous work [13,14] we treat B s mesons in a partially quenched approach, where the mass of the valence quark is approximately the mass of the physical s quark, taken from the study of strange mesons using the same configurations [15,16], • β = 4.05 → µ q,valence = µ q,s = 0.017, • β = 4.20 → µ q,valence = µ q,s = 0.015, while the sea is considerably lighter (cf. the listed µ q values in Table 1).

Static-light mass differences
The determination of static-light mass differences is essentially identical to what we have done in [13,14].
For each of our ensembles characterised by the gauge coupling β and the twisted light quark mass µ q (cf. Table 1) and each of the lattice angular momentum representations A 1 , E and A 2 we compute 6 × 6 static-light correlation matrices. The corresponding meson creation operators differ in their (twisted mass) parity, in their γ matrix structure and in their spatial size. They are precisely the same we have been using before and are explained in detail in [14], section 3, Table 3.
From these correlation matrices we compute effective mass plateaux using variational methods [33,34] (cf. [35] for exemplary plots showing the quality of our plateaus). We extract mass differences by fitting constants to these plateaus at sufficiently large temporal separations T min . . . T max . We determine T min and T max by requiring that the reduced χ 2 is O(1). T min values are listed in Table 2, while T max = 11 for β = 3.90 and β = 4.05 and T max = 17 for β = 4.20 in most cases (for some of the excited states smaller values had to be chosen, because the signal was lost in statistical noise). Note, however, that the choice of T max is essentially irrelevant for the resulting mass (on the "T max side" of the effective mass plateau statistical errors are rather large and, therefore, data points only have a very weak effect on the fit). Since we are only interested in mass differences ∆M (j P ) = M (j P ) − M (S), the jackknife analysis has been applied directly to the mass difference and not to the individual masses. The samples for M (j P ) and M (S) entering for such a mass difference have been obtained with the same value of T min .  Table 2: T min for fitting constants to effective mass plateaus.
The resulting mass differences ∆M (j P )a (in lattice units), where j P ∈ {P − , P + , D ± , D + , F ± , S * }, together with the pion masses m PS a (in lattice units; cf. Table 1 and [32]) and the lattice spacings a (in physical units; cf. Table 1) serve as input for the extrapolation procedure to physical u/d quark masses described in the next section.
We checked the stability of our results by varying T min by ±1 as well as by fitting superpositions of exponentials to the elements of the correlation matrices (as done in [14]) instead of solving generalised eigenvalue problems. We found consistency within statistical errors.
4 Continuum limit and extrapolation to physical u/d quark masses

Numerical results
The mass differences ∆M (j P ) obtained for all our ensembles are plotted against (m PS ) 2 in Figure 1 (unitary, i.e. "B mesons") and Figure 2 (partially quenched, i.e. "B s mesons"). Note that, although we use three different values of the lattice spacing, points corresponding to the same mass difference fall on a single curve. This is reassuring, since we use Wilson twisted mass lattice QCD at maximal twist, where static-light mass differences are O(a) improved [14]. In Table 3 and Table 4 we collect the values of the mass differences in MeV 1 for all simulation points for B and B s mesons respectively.
Because our ground state mass values enter into all of the mass differences we study, we simultaneously fit to all the meson mass differences we have computed. We find that fits which are independent of the lattice spacing and which are linear in the light quark mass (represented by the mass squared of the light-light pseudoscalar meson) are acceptable, i.e. yield χ 2 /dof < ∼ 1.
For the B s mesons, our results depend on the strange quark mass we choose. We have taken these values from studies of strange-light mesons [15,16] as discussed above. The possible systematic error arising from an incorrect value for the strange quark mass is very small: because the mass differences we measure turn out to be very weakly dependent on that mass. This will be seen when we compare our results for the B and B s mesons extrapolated to physical light quark masses.
The details of our fitting procedure are collected in appendix A.
As already mentioned both fits (one for B mesons, the other for B s mesons) are of good quality in a sense that χ 2 /dof < ∼ 1. This shows that at the present level of statistical accuracy the continuum limit has already been reached at our largest value of the lattice spacing a ≈ 0.080 fm. Moreover, these fits enable us to extrapolate to physical u/d quark masses.
Extrapolations of static-light mass differences to physical u/d quark masses are listed in Table 5 in MeV both for B mesons and for B s mesons. Note that both fits give χ 2 /d.o.f. ≈ 1, i.e. are consistent with our assumption that static-light meson mass differences as functions of (m PS ) 2 can be parameterised by straight lines.  To check the stability of these fits, we have varied T min by ±1. Within statistical errors mass differences obtained with T min − 1, with T min and with T min + 1 are in agreement.
The extrapolations are shown in Figure 1 and Figure 2. The red dots represent the maximum likelihood estimates ofz = ((m PS ) 2 , ∆M (j P )) obtained during the fitting procedure. In addition to x-y-error bars we also plot covariance ellipses, which reflect the correlations between (m PS ) 2 and ∆M (j P ) induced by the lattice spacing a, that is they are generated from the inverses of the corresponding 2 × 2 submatrices of the covariance matrix C.

Contamination of static-light meson masses by multi particle states
The radially and orbitally excited static-light mesons P − , P + , D − , D + , F − , F + and S * can decay into multi particle states S + n × π with relative angular momentum such that quantum numbers j P are identical. In particular the P − static-light meson is not protected by angular momentum, i.e. it can decay via an S wave into S + π, whose wave function is not suppressed at the origin. In the following we argue that the effect of S + π states on our P − mass is small compared to its statistical error. To this end we resort to a model presented and to numerical results obtained in [36,37,38].
We consider the P − static-light meson at β = 3.90 and our lightest u/d quark mass at this β value (µ q = 0.0040). In that ensemble the masses of the P − state and of the S + π state are quite similar: m(P − )a ≈ 0.57 and (m(S) + m(π))a ≈ 0.53 (we consider the case, where the pion has zero momentum). Therefore, we expect mixing of P − and S + π with respect to the eigenstates of the Hamiltonian H, mixing which will be different in different spatial volumes. Consequently, we do not focus on the eigenvalues of these states, but rather on m(P − ) = P − |H|P − (|P − is a state with j P = (1/2) + created by single particle operators, e.g. operators of typeQu or Qd, which we have used in the construction of trial states). At very large temporal separation the correlators we are studying will inevitably yield the eigenvalues of the Hamiltonian. At intermediate temporal separations, however, one can expect to read off m(P − ) as we will explain in the following.
In [38] the effective coupling strength of the decay P − → S + π has been estimated by a lattice computation: Γ/k ≈ 0.46. Moreover, some evidence has been obtained that this quantity is fairly independent of the light quark mass. Using this result one can determine the mixing element xa of the energy matrix via eqn. (5) in [38] for our situation (L/a = 24, m π a ≈ 0.14): As detailed in [36,37,38] for large temporal separations the P − correlator is of the form while the corresponding effective mass is At t/a = 12 (the maximum temporal separation we have considered) the estimated systematic error of m P − coming from mixing with S + π is tanh((xa)(t/a))xa ≈ 0.0063, i.e. roughly a 1% effect. This correction is significantly smaller than the statistical error of m effective,P − a in that t region.
For the other temporal separations and/or ensembles we obtain similar estimates. We, therefore, expect that at the present level of statistical accuracy the effect of multi particle states on our static-light meson masses, in particular on P − , is negligible.
Our conclusions are in agreement with those obtained in [9], where a study of the static-light meson spectrum with similar techniques has been performed using two different lattice volumes. No volume dependence of the eigenvectors of static-light meson states has been observed, which is a sign that contributions of multi particle states are strongly suppressed.

Extrapolation to the physical b quark mass
To make contact with experimentally available results on the spectrum of B mesons, we need to correct for the non-infinite mass of the b quark. In Heavy Quark Effective Theory, the leading correction will be of order 1/m H , where m H is the heavy quark mass. It is possible, in principle, to evaluate the coefficients of this correction from first principles on a lattice [19,20]. This we intend to explore in the future, but here we use a more direct method to establish the size of this small correction between static quarks and b quarks of realistic mass. These 1/m H terms will break the degeneracy of mesonic states found in the static limit.
We evaluate for physical b quarks by interpolating between static heavy quarks and the charm quark, where experimental data is available. As a measure of the heavy quark mass, we take the mass of the ground state heavy-light meson (D or B). This measure is equivalent to another (such as using quark masses in some scheme) to the order 1/m H we are using. One test of this interpolation can be made. The hyperfine splitting between D * and D of 141 MeV when interpolated from the static limit (namely zero) gives for B * and B a splitting reduced by m(D)/m(B) = 0.35 to 49 MeV which agrees with the observed splitting [22] of 46 MeV to within 6%.
For the fine splitting, the kinetic term (rather than the chromo-magnetic) is relevant and the experimental results for the spectrum are rather incomplete -indeed this current study is to establish the spectrum from a theoretical input. Lattice studies do confirm [19,20] that a 1/m H behavior is dominant down to masses near the charm quark mass.
We interpolate our lattice results for static-light mass differences of P and S wave states to the physical b quark mass at m(D)/m(B) = 0.35 linearly in m(D)/m H , making use of experimental data on D and D s mesons as input [22]. For details regarding this method of extrapolation cf. [14]. Results are listed and compared to experimental results in Table 6. The corresponding extrapolations are shown in Figure 3.
For D mesons the assignment of the two J P = 1 + states to B * 1 and B 1 is easy, because their widths differ by more than an order of magnitude (we associate the narrow state with B 1 [one of the two degenerate j P = (3/2) + states in the static limit, which can only decay to S + π via a D wave and is, therefore, protected by angular momentum]; the wide state with B * 1 [one of the two degenerate j P = (1/2) + states in the static limit, which can readily decay to S + π via an S wave]). In contrast to that the situation is less clear for D s mesons, where both J P = 1 + states have similar (narrow) widths. Therefore, we show both possibilities in Table 6 Table 6: lattice and experimental results for P wave B and B s states (B * J and B * sJ denote rather vague experimental signals, which can be interpreted as stemming from several broad and narrow resonances possibly including the j = 1/2 P wave states B * 0 , B * 1 , B * s0 and B * s1 ; the two lattice values listed for B * s1 and B s1 correspond to the two possibilities of assigning experimental J P = 1 + D results [cf. text for more details]).
Compared to our previous study [13,14] at a single lattice spacing, the above results are similar for the B (unitary) case. For B s mesons we now employ a partially quenched s quark which allows a more realistic treatment of the light quark sea. So our new results supersede those obtained previously for B s . Indeed we find a significant dependence on the sea quark mass (cf. Figure 2), which is now the physical u/d quark mass, while it previously corresponded to the significantly heavier s quark mass.
In our lattice study we have extracted the continuum limit and have extrapolated to physical light quarks using a linear dependence. We have then interpolated to the physical b quark assuming that a 1/m H behavior is valid down to the charm quark mass. These assumptions induce systematic errors and, in principle, they can be quantified by further lattice studies.
The assumption of a linear extrapolation to physical light quarks is sensitive to possible admixtures of two body states which become more important at lighter quark masses as thresholds for decay open. We have explored this possibility and found no evidence of such effects, so it is difficult to estimate the magnitude of a possible systematic error from this. If there was a significant difference between the light quark behavior for the ground state and an excited state, this would introduce an error on our extrapolation to the physical value which could be as large as 10 MeV.
The test of the 1/m H assumption for the chromo-magnetic term, discussed above, was found to be valid within 6%. This suggests that an estimate of the systematic errors for the B and B s meson mass splittings coming from 1/m effects should also be at least of order 6%. Since the 1/m correction to the P wave states is of order 100 MeV, this implies a systematic error of order 6 MeV.
One further possible source of systematic error is from our neglect of the strange contribution to the sea. This will be addressed in a future study making use of the N f = 2 + 1 + 1 sea which includes dynamical s quarks from ETMC [39,40].
Overall, it seems prudent to assign systematic errors on our mass differences (for P − and P + relative to S) of order 20 MeV from these effects, even though we have little evidence for such effects.
The experimental determination of the spectrum of excited B and B s mesons is quite limited [22]. Assuming that the relatively narrow states seen correspond to our P + state (since a J P = 2 + state must have that assignment), the mass difference we see of over 500 MeV does not agree closely with the experimental results of around 450 MeV. We do get a mass difference of around 450 MeV from our P − states, although such states cannot have J P = 2 + .
In view of this discrepancy with experimental results, it is also interesting to compare with independent existing lattice computations, in particular with the rather recent study reported in [12]. There the light quark extrapolation is only performed in the valence quark mass (from which static-light mass differences essentially seem to be independent, as can be seen by comparing our B and B s results and also from corresponding plots and numbers presented in [12]), while the sea quark mass is kept fixed. More generally, a comparison of the dependence of static-light mass differences on the sea quark mass, which we have computed down to m PS ≈ 280 MeV, with existing lattice studies is not possible: there the number of investigated sea quark masses is rather small and they are quite heavy, around the mass of the s quark. What one can do, however, is to compare meson mass differences for a given value of the sea quark mass. Before comparing results (in physical units) with those quoted in [12] it should be noted that in [12] the scale is set by identifying r 0 with 0.49 fm, while our result for this quantity is r 0 = 0.42 fm [32]. Therefore, to perform a meaningful comparison, one should express all quantities in units of r 0 or equivalently scale all masses in physical units listed in [12] by a factor of around 0.49/0.42 ≈ 1.14. For the lightest sea quark mass considered in [12] corresponding to m PS ≈ 461 MeV it is most appropriate to compare with our results at β = 3.90, µ q = 0.0100 (m PS ≈ 517 MeV). For the P wave mass differences one finds (m(P − ) − m(S)) ETMC (m(P − ) − m(S)) [12] ≈ 474(29) MeV 454(19)(9) MeV ≈ 1.04 (11) (m(P + ) − m(S)) ETMC (m(P + ) − m(S)) [12] ≈ 481(27) MeV 446(17)(9) MeV ≈ 1.08(11), ratios, which are within statistical errors fully consistent with the expected factor 1.14.
It is interesting to note that the ratios between our lattice results and the experimental values (see Table 6) are on the same ballpark of the ratio between two values of r 0 used above, i.e. ≈ 1.14.
While there is no reason to doubt the precise determination of the lattice spacing performed in [32], it would be interesting, although beyond the scope of this paper, to investigate, whether simulations at lighter quark masses and/or with N f = 2 + 1 + 1 dynamical flavours will improve the agreement with experimental results.
One interesting issue is whether the B s states are stable to the strong decay to BK. This decay has a threshold at 408 MeV above the ground state B s meson. Our P − states (the upper two in Table 6) do indeed have masses which are close to (or below) this threshold. That would imply that these two states (B * s0 and B * s1 ) should have a very small decay width. This is consistent with the experimental observation that only two candidate P wave B * states have been seen so far: corresponding to the heavier P + states. All the other states B s we study, including the S * , lie higher than this BK threshold and so would have a strong decay open.
Moreover, our findings clearly indicate that there is no inversion of level ordering for P wave states, neither for B mesons nor for B s mesons. B * 0 and B * 1 (B * s0 and B * s1 ) are considerably lighter than B 1 and B * 2 (B s1 and B * s2 ) as can be read off from Table 6 and Figure 3. This is in contrast to predictions obtained from certain phenomenological models [23,24,25,26,27] and, therefore, might provide valuable input for future model building.

Conclusions
We have determined the continuum limit for static-light mesons on a lattice using N f = 2 flavours of light quarks. The removal of O(a) effects by using maximally-twisted mass fermions for meson mass differences in the static limit is confirmed.
We have investigated the light sea quark mass dependence of B and B s mesons down to m PS ≈ 280 MeV, which is significantly lighter than what has been achieved in previous studies of static-light mesons. We find that our results are compatible with a linear extrapolation in the light quark mass to its physical value. We see no sign of any mixing with two body effects and this is consistent with our estimate that such effects should be too small to see on our lattices.
We have determined masses for a wide variety of excited states in the continuum limit and this will be a valuable resource for model builders.
We have employed the assumption of a 1/m H dependence on the heavy quark mass together with experimental results for charm-light mesons to allow us to estimate the spectrum that one would obtain for physical b quarks.
Our results imply that there will be a J P = 0 + and J P = 1 + B s meson which has a narrow width since its strong decay to BK is suppressed (or zero) due to phase space effects.

A Details of the fitting procedure
Data points (m PS ) 2 and ∆M (j P ), j P ∈ {P − , P + , D ± , D + , F ± , S * } corresponding to the same β are correlated via the lattice spacing a. We take that into account via a covariance matrix, which we estimate by resampling m PS a, ∆M (j P )a and a (100, 000 samples). Consequently, we do not fit straight lines to the Data points ((m PS ) 2 , ∆M (j P )) individually for every static-light state j P , but perform a single correlated fit of six straight lines to the six mass differences of interest. During the fitting we take statistical errors both along the horizontal axis (errors in (m PS ) 2 ) and along the vertical axis (errors in ∆M (j P )) into account.
The method of performing the two-dimensional fits is based on what has been used in [41].
• C denotes the estimated covariance matrix for z (a 56 × 56 matrix for B mesons, a 42 × 42 matrix for B s mesons).
• The linear fits y(j) = a(j)x + b(j) are parameterised by a(j) and b(j) (the quantities, which will finally allow the extrapolation to physical u/d quark masses).
The constraint minimization is equivalent to solving a system of non-linear equations, which we do by means of the scaled-hybrid algorithm of the GSL library [42]. It needs initial parameters, which should preferably be close to the global extremum. Such initial parameters can be obtained by individual standard one-dimensional straight line fits: • λ n (j) = 0, • a(j) and b(j) minimizing n a(j)x n + b(j) − y n (j) 2 C yn(j),yn(j) , •x = x andȳ(j) = y(j).
To judge the quality of the resulting fit, we define a "reduced χ 2 " via The resulting straight lines allow an extrapolation to physical u/d quark masses (corresponding to m PS = 135 MeV). The corresponding statistical errors are obtained by repeating this fitting and extrapolation procedure 100 times with randomly sampled sets z A (we randomly sample the input data and compute z A ≡ ((m PS ) (n) a) 2 /a 2 and z A ≡ ((∆M ) (n) (j)a)/a) and taking the variance.