Strong and Electromagnetic Mass Splittings in Heavy Mesons

The contributions to heavy meson mass diﬀerences by the strong hyperﬁne interaction, the light quark masses and the electromagnetic interaction are obtained from the empirical values of the D , D ∗ , B and B ∗ masses by means of a mass formula based on the heavy quark mass expansion. The three diﬀerent types of contributions are determined with signiﬁcant accuracy to next to leading order in that expansion.


I. INTRODUCTION
Hadron masses reveal key aspects of QCD. The light pseudoscalar meson masses reveal the spontaneous breaking of chiral symmetry as well as its explicit breaking by the light quark masses, and in a rather direct manner they permit to extract the ratios of these quark masses which are fundamental parameters of QCD. They also give access to the electromagnetic contributions to the masses, for instance, in the mass difference between charged and neutral pions. In the framework of chiral perturbation theory, a rather accurate understanding of the effects of light quark masses and electromagnetic corrections on the pseudoscalar octet has been achieved, and to a lesser extent in baryons as well. Heavy mesons represent another kind of hadronic system where one can arrive at a good determination of the various effects that determine their masses. Also, to a first degree of approximation in the heavy quark expansion, the strong hyperfine effects, which involve the heavy quark spin, serve to determine the ratio m c /m b , which is another fundamental input in QCD.
In this work, we show that the current knowledge of the heavy meson masses allows for a quantitative determination of the different effects that contribute to the differences of heavy meson masses. The approach followed here is similar to the one given by Rosner and Wise [1], and the improvement in the results is possible thanks to the better empirical accuracy in the heavy meson masses and also in the better knowledge of the heavy quark masses.
In the limit of infinite heavy quark masses and of light quark SU(3) symmetry, heavy mesons fill multiplets of U(2N F ) × U(3) × SU j (2), where N F = 2 is the number of heavy quarks, SU j (2) is the rotation group associated with the light degrees of freedom of the heavy meson. In particular, the ground state mesons, namely D, D * , B and B * , fill the multiplet (4, 3, 2) of that symmetry. The symmetry is broken by several effects: • The finite masses of the heavy c and b quarks break is the rotation group is associated with the spin of the meson.
• The light quark masses break The analysis in this work has the objective of sorting out these three sources of symmetry breaking from the current knowledge of the heavy meson masses. The analysis only involves meson mass differences. Those in the D mesons are very well known with errors smaller than 2.5% [2]. In the B mesons, not all mass splittings are established; the mass differences (B * − − B − ) and (B * 0 − B 0 ) are not separately known and, in addition, the errors are significantly larger than in the D-system [2]. Nonetheless, the available information is sufficient for the analysis to lead to significant conclusions.
The analysis is based on the mass formuli that result from the expansions in 1/m Q , where m Q is the heavy quark mass (Q = c, b), in m u , m d and m s , and in the fine structure constant α. These mass formuli are similar to the ones given long ago by Rosner and Wise [1], except that the QCD running of some of the parameters are taken into account. Using the notation H = M H , etc., they read as follows: where q and Q denote respectively the light and heavy quark flavors of the heavy meson, the label * is used for the vector mesons, and Q q and Q Q are the respectively the light and heavy quark charges.
The first term in the mass formula is made out of the contributions in the limit m q → 0 and α → 0 plus SU ( For dimensional purposes, we use the ρ-meson mass as the reference mass scale. The explicitly displayed terms O(1/m Q ) and higher represent the mass difference produced by the strong hyperfine interaction. h 1 (m Q ) has a non-trivial dependence on m Q that results from the QCD running of the effective heavy quark operator in the 1/m Q expansion associated with it. The heavy quark effective Lagrangian in the 1/m Q expansion is given in standard notation by [3]: where the third term, which couples to the heavy quark spin, gives rise to h 1 . The coefficient where µ is the renormalization scale, is given by [3]: with the initial condition given by the matching to full QCD [3]: Here C A = 3 , C F = 4/3 , and β 0 = 11 − 2 3 N f is the first coefficient of the β-function with N f the number of flavors lighter than m Q . Because h 1 is proportional to C M , we can express it in the following renormalization group invariant form: whereh 1 is m Q and µ independent. In what follows, all over-lined coefficients are m Q and µ independent. The term proportional to h 2 receives contributions from two terms in the heavy quark Lagrangian at O(1/m 2 Q ) [4]. Taking into account the running of h 2 with m Q is thus impossible in this analysis. Fortunately, this is not important because the O(1/m 2 Q ) terms play a minor role in the B mesons, and thus neglecting the running of the h 2 is a good approximation.
For the light quark mass effects, m q (µ) is defined in MS scheme and one has: Because the spin-independent term in the O(1/m Q ) heavy quark Lagrangian is scale independent, κ 0 and κ 1 are independent of m Q and their dependence on µ is given by the running On the other hand, because κ ′ 1 is proportional to C M , it has an extra running factor similar to that of h 1 : The µ dependence given above is rather immaterial in our analysis, where the same µ is used for D and B mesons.
In the case of electromegnetic effects, we need to discuss seperately the two terms. The coefficients of the self-energy can be expressed in the following most general form: We note that a ′ 1 will run with m Q in a similar form as h 1 . The spin of the heavy quark will affect very little the self-energy term, and therefore a ′ 1 will be small. As shown later, a ′ 1 can be eliminated because of linear dependencies, which means that the effect associated with it cannot be distinguished from other effects on the meson masses.
On the other hand, the electromagnetic terms involving the interaction between the light degrees of freedom and the heavy quark have the form: b 0 andb 1 give the leading and subleading in 1/m Q contributions to the Coulomb interaction, and both are scale independent as it is known from the renormalization of current operators of the heavy quark, such as the electromagnetic current. The electromagnetic hyperfine effect proportional to b ′ 1 can be expressed as follows: 1 .
It receives contributions from two general types of diagrams shown in Fig (1). The first term results from the coupling of the photon to the heavy quark spin and therefore it has no m Q dependence, while the second term corresponds to the coupling of a gluon to the heavy quark spin, and is therefore proportional to C M and thus m Q dependent.

II. ANALYSIS
In this analysis, we consider the five different mass splittings possible in each multiplet, The mass formuli leave one parameter independent mass relation, which reads: where we denote This mass relation is violated by terms . Note that this relation was discovered in Ref.
[1], where the evolution factor χ(m Q ) was not included.
If one disregards the term O(1/m 2 Q ) in the strong hyperfine interaction, i.e. the term proportional to h 2 , one obtains an additional relation: The deviations from this relation are a measure of the importance of the 1/m 2 Q term in the hyperfine interaction.
In the mass formuli, there is a total of twelve parameters that enter in mass differences.
Since there are ten mass differences and one parameter free mass relation, there must be three linearly dependent terms in the mass formuli that we can eliminate. The linear dependencies are such thatā 1 andb 1 can be absorbed intoā 0 andb 0 , and a ′ 1 (m Q ) into h 1 (m Q ) and b ′ 1 (m Q ). Since no dependencies appear if one stays at leading order in 1/m Q , it is natural to eliminate sub-leading terms. We, therefore, eliminate thereforeā 1 , a ′ 1 andb 1 . The linear dependencies imply that, in this analysis, one cannot determine the 1/m Q corrections to the self-energy and to the Coulomb effects independently from other effects. In what follows, we therefore set:ā The quark masses are the key input parameters in the mass formuli. If we would only The first ratio is obtained from the ratio M 2 K /M 2 π , which after next to leading order chiral corrections, gives a value of 24.4 ± 1.5, and the second ratio requires the input of isospin breaking observables, in particular the mass ratio (M K 0 − M K + )/M 2 K , and the rates for η → 3π, with a result m s /(m d − m u ) = 42.5 ± 3.2 [6]. For further reference, this latter ratio corresponds to having the electromagnetic mass difference (K + − K 0 ) EM = 2.0 ± 0.4 MeV.
The final input is Λ QCD required by α s ; we use Λ QCD = 200 MeV.
With our inputs for the heavy quark masses, we obtain for the left hand side of Eqn. (13) a value equal to 0.90 ± 0.08, which gives some evidence for the need of the O(1/m 2 Q ) term in Eqn. (2). For Eqn. (13) to hold, it would be necessary to have m c /m b = 0.40. From Eqn.
(12) we obtain the combination: which is not known experimentally because the mass difference (B * 0 − B 0 ) has not been established separately from the one for charged ones. The improvement over similar prediction given in [1] is primarily due to the improved accuracy of the various inputs, especially the heavy quark masses and the running effect characterized by the factor χ(m Q ).
In Table I, we give the results of our fits, displaying the partial contributions, and in Table II Table II.  Table II, show that they are determined by this analysis with an accuracy that is in general better than 10%. There is, however, one important and still unresolved problem concerning the light quark mass effects, and this has to do with the non-analytic contributions proportional to O(m 3/2 q ) [7], which are expected to be large [7,8], according to the estimated value of the coupling g [9] that gives the amplitudes D * → Dπ. A consistent analysis should include up to the m 2 q effects, which is beyond the current analysis. This problem, therefore, introduces some uncertainty in the determination of the light quark mass effects that is difficult to estimate.
As mentioned earlier, we can separate the electromagnetic effects into self-energy and Coulomb plus hyperfine type terms. The self-energy has only a spin independent piece and it represents an effect of less than 1 MeV, and is determined with about 30% error.
It has the same sign and comparable magnitude to results from calculations based on the Cottingham formula for electromagnetic mass shifts in a VMD approximation [10,11]. The effect of the Coulomb interaction is given by the term proportional tob 0 and its subleading piece proportional tob 1 , as explained earlier, has been absorbed into other terms. The fit determines the Coulomb effect with an error of 11%. The electromagnetic hyperfine effects are associated with the two parametersb 1 , which amounts to ignoring the m Q dependence inb (2) 1 , absorbing the rest of it intob (1) 1 . Our analysis is, therefore, insensitive to the QCD running of the electromagnetic hyperfine effects, which is not surprising. We have moreover checked that our results are almost insensitive to the interpretation of the input (B * − B) as an arbitrary combination of the charged and neutral mass differences. The hyperfine effects are significant in the D-mesons, for instance, in the (D + − D 0 ) case, it is about 60% of the Coulomb effect. On the other hand, in the B-system, the hyperfine effects are much smaller than the experimental uncertainties in the mass differences. Comparison, with the calculations in Ref. [11], shows agreement with the results in the elastic approximation to the Cottingham formula using VMD. Note that the inelastic contributions in the Cottingham formula that correspond to the interaction Coulomb and hyperfine terms are suppressed by 1/m Q and, therefore, to the order we are working here, they can be neglected.
It is instructive to make some comparisons. The electromagnetic shifts for the pseu-  In summary, we have analyzed the different contributions to the mass splittings in heavy ground state mesons. The analysis shows that, with the current empirical accuracy of the heavy meson masses, one can determine these contributions with significant precision even at the sub-leading order in 1/m Q . The results obtained here can be useful for constraining models of heavy mesons, and perhaps also for lattice QCD calculations of heavy mesons masses where it is possible to study the light quark mass dependence.