Decay Constants of Beauty Mesons from QCD Sum Rules

Our recently completed analysis of the decay constants of both pseudoscalar and vector beauty mesons reveals that in the bottom-quark sector two specific features of the sum-rule predictions show up: (i) For the input value of the bottom-quark mass in the $\overline{\rm MS}$ scheme $\overline{m}_b(\overline{m}_b)\approx4.18\;\mbox{GeV},$ the sum-rule result $f_B\approx210$-$220\;\mbox{MeV}$ for the $B$ meson decay constant is substantially larger than the recent lattice-QCD finding $f_B\approx190\;\mbox{MeV}.$ Requiring QCD sum rules to reproduce the lattice-QCD value of $f_B$ yields a significantly larger $b$-quark mass: $\overline{m}_b(\overline{m}_b)=4.247\;\mbox{GeV}.$ (ii) Whereas QCD sum-rule predictions for the charmed-meson decay constants $f_D,$ $f_{D_s},$ $f_{D^*}$ and $f_{D_s^*}$ are practically independent of the choice of renormalization scale, in the beauty sector the results for the decay constants - and especially for the ratio $f_{B^*}/f_B$ - prove to be very sensitive to the specific scale setting.


Correlator, operator product expansion, and heavy-quark mass schemes
The starting point of our QCD sum-rule evaluation of the decay constants [1] of beauty mesons is the time-ordered product of two meson interpolating currents, viz., j 5 (x) = (m b +m)q(x) i γ 5 b(x) for the B meson and j µ (x) =q(x) γ µ b(x) for the B * meson. The correlator of pseudoscalar currents is defined by The Borel transform of this correlation function depends on some Borel parameter τ and takes the form ds e −s τ ρ pert (s, µ) + Π power (τ, µ) .
The B-meson decay constant f B is defined by 0| j 5 (0)|B = f B M 2 B . In order to remove all excited-state contributions, we adopt the standard assumption of quark-hadron duality: the contributions of excited states are compensated by the perturbative contribution above an effective continuum threshold s eff (τ) which differs from the physical continuum threshold. Applying this Ansatz leads, for the B meson, to The right-hand side of the above relation constitutes what we call the dual correlator Π dual (τ, s eff (τ)). The best-known three-loop calculation of the perturbative spectral density ρ pert has been performed as an expansion in terms of the MS strong coupling, a(µ) ≡ α s (µ)/π and the b-quark pole mass M b [2,3]: An tantalizing feature of the pole-mass operator product expansion (OPE) is that each of the known perturbative contributions to the dual correlator is positive. Unfortunately, such a pole-mass OPE does not provide a visible hierarchy of the perturbative contributions and thus poses strong doubts that the O(α 2 s )-truncated pole-mass OPE can provide reliable estimates of the decay constants. An alternative [4] is to reorganize the perturbative expansion in terms of the running MS mass, m b (ν), by substituting in the spectral densities ρ (i) (s, M 2 b ) M b by its perturbative expansion in terms of the running mass m b (ν) The spectral densities in the MS scheme are obtained by expanding the pole-mass spectral densities in powers of a(µ) and omitting terms of order O(a 3 ) and higher; starting with order O(a) they contain two parts: the result of Refs. [2,3] and the impact of lower perturbative orders arising upon expansion of the pole mass in terms of the running mass. In this way, because of the truncation of the perturbative series, one gets an explicit unphysical dependence of both dual correlator and extracted decay constant on the scale µ. In principle, any scale should be equally good. In practice, however, the hierarchy of perturbative contributions to the dual correlator depends on the precise choice of the scale, opening the possibility to choose the scale µ such that the hierarchy of the new perturbative expansion is improved. Let us define the scaleμ by demanding M b = m b (μ). Using the O(a 2 ) relation between running and pole masses, we obtain numericallyμ ≈ 2.23 GeV. The perturbative hierarchy of the MS expansion at this scale is worse than that of the pole-mass expansion: The O(1) spectral densities coincide, whereas the O(a) spectral density in the MS scheme receives a positive contribution compared to the pole-mass scheme. For smaller scales, i.e., µ <μ, the hierarchy of the MS expansion becomes even worse with decreasing µ. For µ >μ, first the hierarchy of the MS expansion improves for increasing µ, see Fig. 1. However, as soon as the scale µ becomes sufficiently larger thanμ, the "induced" contributions, which primarily reflect the poor behaviour of the expansion of the pole mass in terms of the running mass, overtake the "genuine" contributions. This is illustrated by Fig. 1: at µ = 4 GeV, the O(1) contribution to the dual correlator rises steeply with τ, whereas the O(a) contribution becomes negative in order to compensate the rise of the O(1) contribution. Finally, for large values of µ we observe a compensation between the "induced" contributions. One may expect that, in this case, the accuracy of the expansion will deteriorate. This is reflected by a strong scale dependence of the extracted values of f B and f B * : as µ rises, the O(a) term "undercompensates" the rise of the O(1) term for the B meson and f B increases with µ, whereas it overcompensates the rise of the O(1) term for the B * meson and f B * decreases with rising µ; as a consequence, the ratio f B * / f B proves to be particularly sensitive to the precise value of µ. Now, returning to the pole-mass expansion for the b quark, we note that the hierarchy is not too bad and can be easily improved by switching to the running mass and choosing the scale µ slightly aboveμ. Furthermore, we may, for instance, require that the O(α s ) contribution to the dual correlator remains positive in the working range of τ. (For a positive-definite dual correlator, it is not a strictly necessary but, for obvious reasons, a highly welcome feature if each of the perturbative contributions is positive.) In this case, we may expect to arrive at reliable results by setting µ = 2.5-3 GeV for the B * meson and µ = 2.5-3.5 GeV for the B meson. An additional important argument in favour of such "low-µ" choice is that the ratio f B * / f B proves to be definitely less than unity at scales µ ≈ m b , whereas it emerges close to unity for µ = 2.5-3 GeV, in full agreement with heavy-quark expansion and hints from lattice QCD.
The results for the B * -meson decay constant shown in Fig. 1

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To compare the results obtained from, on the one hand, the running-mass OPE and, on the other hand, the pole-mass OPE, we recalculate the pole mass from the O(a 2 ) relation between m b and M b , finding M b = 4.8 GeV. Merely for illustrating the main features of the dual correlators, the sum-rule estimates shown in Fig. 1 are extracted for a τ-independent effective threshold s eff = const. Its value in each case is deduced by requiring maximal stability of the extracted decay constant in the chosen Borel window.

Extraction of observables such as decay constants from QCD sum rules
The well-established procedures of QCD sum rules pave a straightforward path to extract observables: 1. Determine a reasonable Borel window, that is, an interval of the Borel parameter τ defined such that the OPE provides an accurate description of the exact correlator: higher-order radiative and power corrections have to be under control while, at the same time, the ground state contributes "sizeably" to the correlator; our τ window for, e.g., the B meson reads 0.05 τ (GeV −2 ) 0.18.
2. Define and apply an appropriate criterion for fixing the effective continuum threshold s eff (τ). To this end, we employ an earlier developed algorithm [6][7][8][9] that allows for a reliable extraction of the ground-state properties in quantum mechanics and of the charmed-meson decay constants in QCD. We introduce the dual invariant mass M dual and the dual decay constant f dual by defining The dual mass should reproduce the true ground-state mass M B ; its deviation from M B quantifies the contamination of Π dual (τ, s eff (τ)) by excited states. We determine the behaviour of s eff (τ) by starting from a convenient Ansatz for s eff (τ) and minimizing the deviation of the predicted M dual from the observed M B in the τ window by varying s eff (τ). Since we need to know the behaviour of s eff (τ) only in the limited τ window defined before, it suffices to consider merely polynomials in τ (which Ansatz allows, of course, also for the case of s eff (τ) being a τ-independent constant): 3. With the variational result for s eff (τ) at hand, f dual (τ) yields the decay-constant estimate sought.
As all outcomes for hadron observables extracted from QCD sum rules do, also the predictions for decay constants are sensitive to the input values of all the parameters entering in one's OPE-resulting in what we call, for clear reasons, their OPE-related uncertainty-and to the particularities of the route followed to get the effective threshold as a function of τ-contributing to their systematic uncertainty.
OPE-related uncertainty: By assuming a Gaussian distribution for the numerical value of each of the parameters required as OPE input except for the renormalization scales µ and ν, for which we assume uniform distributions in the range 3 GeV ≤ µ, ν ≤ 6 GeV, we may estimate the size of the OPE-related uncertainty by performing a bootstrap analysis. The resulting distribution of decay constants proves to be close to Gaussian shape. So, the quoted OPE-related error should be understood as a Gaussian error.

Systematic uncertainty:
The systematic uncertainty is an immediate consequence of the intrinsically limited accuracy of the QCD sum-rule approach and thus poses, without surprise, a delicate problem. Considering polynomial parameterizations of the effective continuum threshold s eff (τ) for toy models within quantum mechanics, we could demonstrate that the band of results found from linear, quadratic, and cubic Ansätze for s eff (τ) encompasses the true value of the decay constant. Thus, the half-width of this band should be regarded as a realistic estimate for the systematic uncertainty of such an extraction.