# Local-duality QCD sum rules for strong isospin breaking in the decay constants of heavy–light mesons

## Abstract

We discuss the leptonic decay constants of heavy–light mesons by means of Borel QCD sum rules in the local-duality (LD) limit of infinitely large Borel mass parameter. In this limit, for an appropriate choice of the invariant structures in the QCD correlation functions, all vacuum-condensate contributions vanish and all nonperturbative effects are contained in only one quantity, the effective threshold. We study properties of the LD effective thresholds in the limits of large heavy-quark mass \(m_Q\) and small light-quark mass \(m_q\). In the heavy-quark limit, we clarify the role played by the radiative corrections in the effective threshold for reproducing the pQCD expansion of the decay constants of pseudoscalar and vector mesons. We show that the dependence of the meson decay constants on \(m_q\) arises predominantly (at the level of 70–80%) from the calculable \(m_q\)-dependence of the perturbative spectral densities. Making use of the lattice QCD results for the decay constants of nonstrange and strange pseudoscalar and vector heavy mesons, we obtain solid predictions for the decay constants of heavy–light mesons as functions of \(m_q\) in the range from a few to 100 MeV and evaluate the corresponding strong isospin-breaking effects: \(f_{D^+} - f_{D^0}=(0.96 \pm 0.09) \ \mathrm{MeV}\), \(f_{D^{*+}} - f_{D^{*0}}= (1.18 \pm 0.35) \ \mathrm{MeV}\), \(f_{B^0} - f_{B^+}=(1.01 \pm 0.10) \ \mathrm{MeV}\), \(f_{B^{*0}} - f_{B^{*+}}=(0.89 \pm 0.30) \ \mathrm{MeV}\).

## 1 Introduction

The method of QCD sum rules [1], based on the exploitation of Wilson’s operator product expansion (OPE) in the study of properties of individual hadrons, has been extensively applied to the decay constants of heavy mesons [2, 3, 4]. An important finding of these analyses was the observation of the strong sensitivity of the decay constants to the precise values of the input OPE parameters and to the algorithm used for fixing the effective threshold [5, 6, 7, 8, 9]. For any given approximation of the hadronic spectral density based on quark–hadron duality, the effective threshold determines to a large extent the numerical prediction for the decay constants inferred from QCD sum rules: even if the parameters of the truncated OPE are known with high precision, the decay constants may be predicted with only a limited accuracy, which represents their systematic uncertainty. In a series of papers [10, 11, 12, 13, 14], we proposed a new algorithm for fixing the effective threshold within the Borel QCD sum rules which allowed us to obtain realistic estimates of the systematic uncertainties. Our procedure opened the possibility to get predictions for the decay constants with a controlled accuracy [15, 16, 17, 18] and thus allowed us to address subtle effects that call for a profound accurate treatment, such as the ratios of the decay constants of heavy vector and pseudoscalar mesons [19, 20] or the strong isospin-breaking (IB) effects in the decay constants of heavy–light mesons [21], generated by the mass difference (\(m_d - m_u\)) between up and down quarks.

Here, we discuss the application of another variant of QCD sum rules to the evaluation of the strong IB effects in the decay constants of heavy–light pseudoscalar and vector mesons. Our analysis takes advantage of the fact that the OPE provides the analytic dependence of the correlation functions on the quark masses; this allows us to study, e.g., the impact of the light-quark mass on heavy-meson decay constants, thus providing access to the strong IB effects. The approach we describe in this work seems quite promising for studying the dependence of a generic hadron observable on quark masses.

### 1.1 QCD sum rule in the local-duality limit

*H*of mass \(M_H\), consisting of a heavy quark

*Q*with mass \(m_Q\) and a light quark

*q*with mass \(m_q\), has the form

*N*is an integer that depends on the Lorentz structure in the correlation function chosen for the sum rule and on the number of subtractions in the corresponding dispersion representation, and \(s_\mathrm{eff}\) is the effective threshold such that \(\sqrt{s_\mathrm{eff}}\) lies between the mass of the ground-state and the first excited state [1], namely \(s_\mathrm{eff}=(M_H + z_\mathrm{eff})^2\) with \(z_\mathrm{eff} \simeq \) 0.4–0.5 GeV.

Nonperturbative effects appear on the r.h.s. of (1) at two places: as power corrections given in terms of vacuum condensates and in the effective threshold \(s^{(N)}_\mathrm{eff}\). Depending on the chosen value of *N*, nonperturbative effects are distributed in a different way between power corrections and the effective threshold. Perturbative effects are encoded in the spectral density \(\rho _\mathrm{pert}\), in the effective threshold \(s^{(N)}_\mathrm{eff}\) and in the power corrections \(\varPi ^{(N)}_\mathrm{power}\).

*s*above some \(s_\mathrm{up}\), but is a “weak” relation and requires an appropriate smearing for

*s*in the mid-energy region above the physical hadron continuum threshold \(s_\mathrm{th}\). An appropriate smearing is reached by performing the Borel transform

^{1}Obviously, at \(\tau =0\) an appropriate smearing in (2) is guaranteed by the integration; on the other hand, the excited states are not suppressed and one can doubt that modelling the hadron continuum as the effective continuum remains a good approximation at small \(\tau \).

First, note that power corrections contain singular terms of the form \(\tau ^{2-N}\log (\tau )\). Therefore, the limit \(\tau \rightarrow 0\) cannot easily be taken in the sum rule (1) for \(N\ge 2\). For \(N=0\) and \(N=1\), the limit \(\tau \rightarrow 0\) in (1) is mathematically well defined. To demonstrate that this limit is also physically meaningful, one needs to show that the corresponding \(s_\mathrm{eff}\) indeed lies in the expected range.

It is clear that for the \(N=1\) case the OPE is under good control in a broad range of \(\tau \) and therefore the lower boundary of the Borel window can be safely extended down to \(\tau =0\). For \(N=0\) only at relatively small \(\tau \) the OPE is under control and the approximation of a \(\tau \)-independent effective threshold may work well. The relevance of the unknown higher-order power corrections is reflected by the sharp rise of the effective threshold visible in Fig. 1a. Obviously, the standard QCD sum-rule analysis based on a stability window with a constant effective threshold may be problematic. In this respect an alternative approach based on a \(\tau \)-dependent effective threshold seems to be more appropriate, but this issue goes well beyond the scope of the present paper. Here, the only important property is that the effective threshold at \(\tau =0\) has the value expected on the basis of the standard considerations [1], i.e., it is around \((M_{B^*} + z_\mathrm{eff})^2\) with \(z_\mathrm{eff} \sim \) 0.4–0.5 GeV. Therefore, for the correlators with \(N<2\), modelling the hadron continuum as an effective continuum remains a valid and equally accurate approximation as \(\tau \rightarrow 0\), which does not represent a point of discontinuity.

In this work we show that the sum rule (1) for \(N=0\) can be of particular interest. Obviously, considering the sum rule at only one point, \(\tau =0\), does not allow for the use of the usual sum-rule stability criteria [1] for determining \(s_\mathrm{eff}\). Consequently, the decay constants cannot be determined entirely within the QCD sum rules; some “external” inputs are needed to determine \(s_\mathrm{eff}\). Nevertheless, in this work we will show that, besides the reduction of the uncertainties related to the absence of the condensates, the LD sum rules represent an efficient tool to investigate the dependence of the pseudoscalar and vector meson decay constants on quark masses and their perturbative behaviour in QCD. Moreover, the LD sum rules turn out to be particularly suitable for the analysis of the strong IB effects in the two-point functions when implemented with only few “external” inputs, e.g., from experiment or lattice QCD.

### 1.2 Strong isospin breaking from a QCD sum rule in the LD limit

We are interested in the dependence of the decay constants of heavy–light mesons on the quark masses, in particular, in the strong IB effects in the decay constants (i.e., the difference between the decay constants of \(\bar{Q}d\) and \(\bar{Q}u\) mesons induced by the small mass difference \(\delta m=m_d-m_u\)). We therefore need to properly take into account all effects depending on the light-quark flavour *q* in the correlation function of the appropriate \(\bar{Q} q\) interpolating currents.

Clearly, the \(m_q\)-dependence on the l.h.s. of (1) is encoded both in the decay constant \(f_H\) and in the meson mass \(M_H\). On the r.h.s, the IB effects come from several sources: the \(m_q\)-dependence of \(\rho _\mathrm{pert}(s,m_Q,m_q,\alpha _s)\), the \(m_q\)-dependence of the effective threshold \(s_\mathrm{eff}\), the \(m_q\)-dependence of the power corrections, and the flavour dependence of the quark condensates, in particular, of \(\langle \bar{q}q\rangle \). In general, all these effects mix together, which renders the goal of isolating the IB effects in \(f_H\) a complicated task. A careful analysis has been carried out recently in [21], following the standard choice \(N=2\) for pseudoscalar and \(N=1\) for vector mesons.

*u*- and

*d*-quark masses in QCD.

We will demonstrate that the main \(m_q\) dependence of the decay constants originates from the calculable \(m_q\) dependence of the perturbative spectral densities. Therefore, the LD limit opens the possibility of a reliable analysis of the \(m_q\) dependence and the strong IB effects in the decay constants of heavy–light mesons (and, in principle, also in other quantities).

This paper is organized as follows: in Sect. 2, we recall the spectral densities of the QCD correlation functions relevant for our LD sum-rule analysis. In Sect. 3, we study the \(m_Q\)- and \(m_q\)-dependences of the effective thresholds by making use of an appropriate mass scheme (pole mass and running mass) for the heavy quarks. In Sect. 4, we perform the numerical analysis of the decay constants of heavy–light pseudoscalar and vector mesons and obtain predictions for strong IB effects in the decay constants. Section 5 gives our conclusions. The Appendix collects some details of treating the IB effects within the OPE, which, in our opinion, deserve to be presented.

## 2 Local-duality sum rules for \(f_P\) and \(f_V\)

*P*) and vector (

*V*) mesons built up of one massive quark

*Q*with mass \(m_Q\) and one light quark

*q*with mass \(m_q\). We consider the axial-vector current

In Eq. (15), we may employ different definitions of the quark masses: The most advanced calculation of the pseudoscalar and vector spectral densities including order-\(O(a^2)\) terms was performed [35, 36], for a massless light quark, in terms of the heavy-quark pole mass. The expansion in terms of the heavy-quark pole mass is appropriate for considering the heavy-quark limit, which we address in Sect. 3.1.

However, the pole-mass expansion leads to a rather slow convergence of the perturbative expansion for the decay constants [15, 16, 17, 18, 37]. The convergence improves considerably when one rearranges the perturbative expansion in terms of the running \(\overline{\mathrm{MS}}\) masses. Therefore, for the practical analysis of the \(m_q\)-dependences of the meson decay constants in Sect. 4, we make use of the perturbative expansion in terms of the running \(\overline{\mathrm{MS}}\) masses of the light and the heavy quarks. The corresponding NLO and NNLO functions \(\rho _{P}^{(i)}\) (\(i=1,2\)) in (15) necessary for such an analysis are found from the spectral densities of the pseudoscalar correlation function given in [37] by multiplying them by \(1/s^2\). Similarly, the transverse spectral densities \(\rho _{V}^{(i)}\) in (15) are found from the spectral densities of [32, 33, 34] by multiplying them by 1/*s*. In our analysis, we make use of the exact LO perturbative spectral density given by (16), at the NLO we keep the terms \(O(a m_q^0)\) and \(O(a m_q^1)\), and in the NNLO we keep only the known terms of order \(O(a^2m_q^0)\).

We would like to emphasize that the perturbative spectral density (15) does not generate terms of order \(m_q\log (m_q)\) in the dual correlator (14). This observation will be crucial for discussing properties of the effective thresholds in the next section.

## 3 Dependence of the effective thresholds on the quark masses

Let us now consider the dependences of the effective threshold on the quark masses \(m_Q\) and \(m_q\).

### 3.1 Heavy-quark limit in the pole-mass scheme

We start with the heavy-quark limit of the decay constants, originally discussed in Refs. [38, 39] within the Heavy Quark Effective Theory (HQET). In what follows, however, we do not consider the static decay constants and we work in full QCD.

Since only the near-threshold behaviour of the spectral densities is relevant for the leading behaviour in the large-\(M_Q\) limit, we may obtain also the \(O(\bar{a}^2)\) terms in the dual correlation functions [i.e., the r.h.s. of (19)] from the analytical expressions for these spectral densities given by Eqs. (30) and (31) of [35, 36].

^{2}

### 3.2 Combined heavy-quark and chiral limits in the pole-mass scheme

### 3.3 Quark-mass dependences of the effective threshold in the running-mass scheme

## 4 Numerical analysis of the sum rules

Parameters of the effective thresholds and resulting IB in the decay constants of heavy pseudoscalar and vector mesons. The parameter \(z_L\) in the effective threshold for the “linear + log” ansatz is fixed by ChHQET in the heavy-quark limit

Meson | Threshold | \(z_0\;(\mathrm{GeV})\) | \(z_1\,[\mathrm{GeV}^{-1}]\quad \) | \(f_{M_d}-f_{M_u} \ (\mathrm{MeV})\) |
---|---|---|---|---|

| Constant | \(1.363\pm 0.213\) | \(1.222\pm 0.219\) | |

Linear | \(1.366\pm 0.203\) | \(- \ 0.365\pm 0.301\) | \(1.050\pm 0.102\) | |

Linear + log | \(1.225\pm 0.194\) | \(- \ 1.422\pm 0.304\) | \(0.929\pm 0.088\) | |

\(D^*\) | Constant | \(1.207\pm 0.147\) | \(1.276\pm 0.217\) | |

Linear | \(1.207\pm 0.138\) | \(0.006\pm 0.464\) | \(1.281\pm 0.389\) | |

Linear + log | \(1.087\pm 0.139\) | \(- \ 0.978\pm 0.524\) | \(1.080\pm 0.381\) | |

| Constant | \(1.501\pm 0.143\) | \(0.792\pm 0.081\) | |

Linear | \(1.499\pm 0.134\) | \(\;\,0.498\pm 0.076\) | \(1.113\pm 0.108\) | |

Linear + log | \(1.365\pm 0.136\) | \(- \ 0.639\pm 0.147\) | \(0.918\pm 0.091\) | |

\(B^*\) | Constant | \(1.534\pm 0.163\) | \(0.839\pm 0.076\) | |

Linear | \(1.533\pm 0.152\) | \(0.227 \pm 0.401\) | \(1.010\pm 0.317\) | |

Linear + log | \(1.395\pm 0.152\) | \(- \ 0.938\pm 0.448\) | \(0.786\pm 0.311\) |

- 1.
“Constant” threshold: the \(\bar{z}_1(\mu )\) term in the effective threshold (32) and the chiral logs \(z_L^\mathrm{HQ}\) are neglected; the only unknown parameter \(\bar{z}_{0}(\mu )\) is fixed from the lattice results for the decay constants of the isospin-symmetric heavy mesons, with \(m_q=m_{ud}\).

- 2.
“Linear” threshold: the chiral logs \(z_L^\mathrm{HQ}\) are neglected and the parameters \(\bar{z}_{0}(\mu )\) and \(\bar{z}_1(\mu )\) are fixed by the lattice QCD results for the decay constants at two \(m_q\) values, for the isospin-symmetric and the strange heavy mesons.

- 3.
“Linear + log” threshold: the known leading chiral logs represented by \(z_L^\mathrm{HQ}\) are included; the parameters \(\bar{z}_{0}(\mu )\) and \(\bar{z}_1(\mu )\) are fixed from the lattice QCD results for the decay constants at two \(m_q\) values, for isospin-symmetric and strange heavy mesons.

Table 1 summarizes the effective thresholds corresponding to our three Ansätze and presents estimates of the strong IB effect. For our final estimates, we perform a bootstrap analysis of the uncertainties assuming that the OPE parameters in (33) have Gaussian distributions with corresponding Gaussian errors, whereas the scale \(\mu \) has a flat distribution in the range \(1< \mu \;(\mathrm{GeV}) <3\) for charmed mesons and \(3< \mu \;(\mathrm{GeV}) < 6\) for beauty mesons.

Notice that the results corresponding to a constant effective threshold [ansatz (1)] are quite close to those obtained including the \(m_q\)-dependence [ansatz (2)] and to the results of Ref. [21], which contain effects in the decay constants at any order in the light-quark mass. So, an important conclusion to be drawn from our results is that effects at order \(\mathcal{{O}}(m_q^2)\) in the effective threshold are not crucial for describing the \(m_q\)-dependence of the decay constants and for estimating the slope of the IB effect at the physical value of the light-quark mass: the latter are both determined to a large extent by the *known* \(m_q\)-dependence of the spectral densities and can thus be reliably controlled in our approach.

## 5 Summary and conclusions

We addressed the local-duality (LD) limit, \(\tau =0\), of the Borel QCD sum rules for the decay constants of heavy–light pseudoscalar and vector mesons. An invaluable feature of the LD limit is that for a proper choice of the correlation function, all vacuum-condensate contributions vanish and the full nonperturbative QCD dynamics is parameterized in terms of merely one quantity – the effective threshold. Our analysis demonstrates that the effective threshold has a nontrivial functional dependence on the masses of the heavy and the light quarks, \(m_Q\) and \(m_q\), respectively. This dependence has been parameterized in the form suggested by the behaviour of the decay constants in the known limits: the chiral limit for \(m_q\) and the heavy-quark limit for \(m_Q\). In the heavy-quark limit, we clarify the role played by the radiative corrections in the effective threshold for reproducing the pQCD expansion of the decay constants of pseudoscalar and vector mesons.

This paper elucidates the dependence of the decay constants on a light-quark mass \(m_q\) in the range \(m_{ud} < m_q<\)\(m_s\). Fixing a few numerical parameters of the effective threshold by using the available accurate inputs from lattice QCD, we have derived the full analytic dependence of the decay constants \(f_H(m_q)\) on the light-quark mass \(m_q\). The resulting dependence of the decay constants \(f_H(m_q)\) on \(m_q\) emerges from two sources: (i) from the \(m_q\)-dependence of the QCD perturbative spectral densities known explicitly as expansion in powers of \(\alpha _s\) and (ii) from the \(m_q\)-dependence of the effective threshold known approximately. An important outcome of our analysis is that the variation of the decay constants with respect to \(m_q\) comes to a great extent (70–80% of the full effect) comes from the rigorously calculable dependence on \(m_q\) of the perturbative spectral densities and is therefore under a good theoretical control.

Noteworthy, the known perturbative expansion of the correlation functions [32, 33, 34, 35, 36, 37], where the sea-quark mass effects are neglected, limits the accuracy of the decay constants of the heavy–light mesons to \(O(m_s \bar{a}^2)\) accuracy, \(\bar{a}\sim 0.1\) at the appropriate renormalisation scales. Therefore the accuracy of the individual decay constants obtained from QCD sum rules does not exceed a few MeV. Nevertheless, we would like to emphasize that the IB difference of the decay constants, \(f_{M_d}-f_{M_u}\), where the sea-quark contributions of order \(O(m_{s,u,d} \bar{a}^2)\) cancel each other, may be predicted with a much higher accuracy, \(O(\delta m\bar{a}^2)\). Therefore, the proposed method can potentially provide a higher accuracy of the IB effects than other approaches.

Very recently [48] a new precise determination of the strong IB effect in the decay constants of D- and B-mesons has been carried out by the FNAL and MILC lattice collaborations.

In the charm sector their result is \(f_{D^+} - f_{D^0} = 1.13 (15)\) MeV, which nicely agrees with our findings (35) and (39). As for the bottom sector, it is shown that the available HPQCD and RBC/UKQCD calculations [49, 50, 51] overestimate significantly the strong IB effect because of an inappropriate use of unitary lattice points (i.e. those having the same mass for valence and sea light-quarks). The FNAL/MILC result is \(f_{B^0} - f_{B^+} = 1.12 (15)\) MeV, which is in excellent agreement with our findings (37) and (41).

Thus, our sum-rule predictions are nicely confirmed quantitatively by lattice QCD both for the central values and the overall uncertainties. This is reassuring: the strong IB effect and its uncertainty in the decay constants of heavy–light mesons can be reliable and accurately estimated within the QCD sum-rule approach.

It should be emphasized that the present approach based on the combination of OPE and a few inputs from lattice QCD potentially has fewer theoretical uncertainties than other formulations of QCD sum rules: first, the condensate contributions, in particular, those of the quark condensate, which produced the main OPE error in the decay constants, vanish in the LD limit; second, the systematic uncertainty of the sum-rule method is now encoded in only one quantity – the effective threshold, which may be fixed to good accuracy due to the use of the few accurate lattice inputs.

Thus, QCD sum rules for the mass dimension-2 Borelized invariant amplitudes at \(\tau =0\) (i.e., an infinitely large Borel mass parameter) provide an efficient tool for the analysis of the dependence of decay constants (and potentially of other hadron observables) on quark masses.

Finally, we want to mention that, besides the strong IB effect due to the up and down quark mass difference, there are other isospin violating effects due to electromagnetism, i.e. to the difference between the up and down quark electric charges. However, the inclusion of such electromagnetic corrections within a sum-rule approach is not a trivial task and it requires the development of new strategies going beyond the traditional QCD sum-rule approaches. In this respect it is worth mentioning a new lattice strategy [52] developed to deal with QCD + QED effects on quantities that require the cancellation of infrared divergences in the intermediate steps of the calculation, like, e.g., the decay rate of charged pseudoscalar mesons [53].

## Footnotes

- 1.
The LD limit in Borel sum rules was introduced and discussed in [22, 23, 24, 25] in connection with the pion and nucleon elastic form factors, and later applied to the analysis of meson transition form factors in [26, 27, 28, 29]. A specific feature of this limit is the vanishing of power corrections in the two- and three-point Borelized correlation functions of axial-vector and vector currents of light quarks.

- 2.

## Notes

### Acknowledgements

The authors are grateful to A. Grozin for interesting comments. S. S. warmly thanks S. R. Sharpe for providing the extension of the calculation of the chiral logs of Ref. (S. R. Sharpe, private communication; see the appendix in [9]) [42] to the case of \(N_f = 2 + 1\) dynamical quarks. D. M. was supported by the Austrian Science Fund (FWF) under project P29028.

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