A lattice QCD calculation of the transverse decay constant of the b1(1235) meson

We review various B meson decays that require knowledge of the transverse decay constant of the b1(1235) meson. We report on an exploratory lattice QCD calculation of the transverse decay constant of the b1 meson. The lattice QCD calculations used unquenched gauge configurations, at two lattice spacings, generated with two flavours of sea quarks. The twisted mass formalism is used.


Introduction and motivation
The transverse decay constant of the b 1 (1235) meson (f T b 1 ) is theoretical input to a number of decays of the B meson. For example f T b 1 is an important QCD input to the following decays: using the light cone formalism [1,2]. The f T b 1 constant is also input to the decay B → b 1 γ [3,4] that use light cone sum rules. Diehl and Hiller [5] discuss studying decays of the B meson with final states that include the b 1 meson.
There are alternative theoretical formalisms [6,7,8,9] to the light cone sum rules such as factorization, that describe the non-leptonic decays of the B meson to final states that include the b 1 meson. So it is important to have a cross-check of the input parameters used in the light cone formalism.
BaBar has experimentally measured the B decays: b 1 π and b 1 K [10,11]. The charmless decays of the B meson, that include those with a b 1 meson in the final state, have been reviewed by Cheng and Smith [12].
The transverse decay constant of the b 1 meson is not accessible to experiment, but can be calculated in models [13] and sum rules. Calculations of the b 1 meson have also been used to tune sum rules [14,15,16]. In particular, the same sum rules are used to simultaneously extract the transverse decay constants of the b 1 and ρ mesons [14,15].
In principle lattice QCD should be able to produce an accurate result for f T b 1 , particularly as modern lattice QCD calculations usually have multiple lattice spacings and volumes, with pion masses below 300 MeV [17]. To the best of our knowledge, there has never been a lattice QCD calculation of f T b 1 before this one.
The b 1 meson is a good test case for lattice techniques that deal with resonances, because it is thought to be a basic quark-antiquark meson that decays via S-wave. The experimental width of the b 1 (1235) is 142(9) MeV, and the bulk of the decays are to ωπ. Hence further motivation for this study is to compute as much information about the b 1 meson from our lattice QCD calculations as possible.
Light cone sum rules and factorization methods, also use the decay constants of the a 0 , π(1300), a 1 mesons to study the decays of the B meson, but there have been previous lattice QCD calculations of those quantities [18,19,20].

The lattice QCD calculation
The transverse decay constant (f T b 1 (µ)) of the b 1 meson is defined [21] by where σ µν = i/2[γ µ , γ ν ], and α (λ) is the polarisation vector of the meson. It is convenient to introduce the tensor current T νµ = ψσ µν ψ. We do not include any momentum in this lattice calculation. For completeness we note that the b 1 meson has J P C = 1 +− .
In the isospin limit the leptonic decay constant of the b 1 meson is zero, because the T ij operator is orthogonal to the vector and axial currents. Although for some quantities, such as the matrix element for ρ − ω mixing [22] or the decay constant of the flavour non-singlet 0 ++ meson [19] an estimate of  isospin violating quantities can be made from a lattice calculation with two degenerate flavours of sea quarks, we don't see how to estimate the leptonic decay constant of the b 1 meson without using non-degenerate light quarks. A formalism to do this has recently been developed for twisted mass QCD [23].
Our lattice calculation uses the twisted mass QCD formalism [24]. Once a single parameter has been tuned, twisted mass QCD has non-perturbative O(a) improvement [25,26]. The twisted mass formalism has been reviewed by Shindler [27].
We have recently reported on the some basic measurements of the ρ, b 1 and a 0 mesons [29]. In this paper we extend that study to the transverse decay constant of the b 1 meson. All the necessary details are in the previous paper [29] and here we provide a brief summary. There is a additional information about the lattice techniques, such as the smearing and variational analysis in the "methods paper for the ETM collaboration [30]." We used the twisted mass Wilson action and the tree level improved Symanzik action. The ensembles used in this analysis are in table 2. Details of the analysis of light pseudo-scalar mesons are in [30,31].
The correlators used to extract the f T b 1 decay constant are in equation 2. We use a smearing matrix of order 2, using basis functions of local and fuzzed interpolating operators, that includes the correlators in equation 2. We fit the smearing matrix to a factorizing fit form [30] with two states. The charged interpolating operator for the b 1 meson in the twisted basis was used [30].
We used ensembles at two different β values. At β = 3.9 we included two volumes. We used the pion decay constant of the π meson to determine the lattice spacing. At β = 4.05 (β = 3.9) the lattice spacing is: a = 0.0667(5) fm (a = 0.0855(5) fm). The lattice spacing from f π was consistent with the value from the nucleon mass [32]. In figures 1 and 2 we show effective mass plots for the b 1 correlators.
The decay constant f T b 1 depends on the value of the renormalisation scale. We used a renormalisation factor obtained from the Rome-Southampton non-   perturbative method [33,34,35]. As traditional in lattice QCD calculations we quote the result at the scale of 2 GeV. The renormalisation group equations can be used to evolve the decay constant to another scale. The results for the decay constant from this calculation are in table 2. In figure 3 the transverse decay constant of the b 1 meson is plotted in physical units as a function of the square of the pion mass. Although the error bars are large, the results for f T b 1 are consistent between the two lattice spacings and volumes.
A common technique to check whether a state is a scattering state or a resonance is look at the volume dependence of the amplitude [36]. The use of the volume dependence of the amplitude and the connection to Lüscher's method has recently been discussed by Meng et al. [37]. The "rule of thumb" is that if our b 1 correlator couples to a scattering state of ωπ, the volume dependence of the f T b 1 decay constant extracted from equation 2 is where V is the spatial volume. For a resonance f T b 1 should be independent of the volume (apart from small corrections if the box size is too small to fit the resonance state).
In our previous paper [29] we showed that the decay of the b 1 meson to ωπ was open in our calculation. However the mass of the lightest state in the 1 −+ channel didn't track the sum of the masses of the ω and π mesons particularly well. In Lüscher's formalism for the study of resonances on the lattice the opening of strong decays is described by an avoided level crossing [38]. The detailed calculations of Bernard et al. [39] for the ∆ baryon suggested that the avoided level crossing is "washed out" by the dynamics, so comparing the mass from the resonant interpolating operator to the sum of the masses of the decay products is probably too simplistic. A similar situation happened with string breaking, where the linearly rising potential was seen to increase beyond the energy that allowed the string to break to two static B mesons [40].
The above considerations suggest that although we don't have full control over the resonant nature of the b 1 meson, our interpolating operators are coupling to the b 1 meson in the range of quark masses in our calculations. We extrapolate f T b 1 linearly in the square of the pion mass to get f T b 1 (2 GeV) = 236(23) MeV at the physical pion mass at β = 3.9.
In table 2 we collect together other estimates for f T b 1 . We have evolved our lattice result to the scale of 1 GeV to compare with the results from sum rules. The formalism to evolve the decay constant with scale is described [29], that uses input from perturbative calculations by Gracey and others [45,46,47,48]. The perturbative factor is 1.095 to evolve f T b 1 from 2 GeV to 1 GeV. In the non-relativistic quark model the b 1 meson is a P-wave meson with a node in the wave-function at the origin. This would suggest the transverse decay constant is small. The partial inclusion of relativistic effects in the quark model [49] increases the decay constant. The results in table 2 show [13] extended NJL quark model 175(9) Ball and Braun [41] sum rule 180(20) Bakulev and Mikhailov [42,43] sum rule 184(5) Bakulev and Mikhailov [42,43] sum rule 181(5) Yang [44,21] sum rule 180(8) This calculation lattice QCD 258(25) that f T b 1 is of the same order of magnitude as the pion decay constant (132 MeV), so this is evidence that local interpolating operators will couple to the b 1 meson. Pragmatically using derivative sources [50,51] with smearing techniques, such as Jacobi, may be useful to get a good signal.

Conclusion
We have presented the first calculation of the transverse decay constant of the b 1 meson from lattice QCD. We obtain f T b 1 (2 GeV) = 236(23) MeV at the physical pion mass. The result is higher than the sum rule results by 3 σ. Future lattice QCD calculations need to reduce the statistical errors on the correlators, and to directly take into account the resonant nature of the b 1 meson.