Heavy-quark expansion for D and B mesons in nuclear matter

The planned experiments at FAIR enable the study of medium modifications of $D$ and $B$ mesons in (dense) nuclear matter. Evaluating QCD sum rules as a theoretical prerequisite for such investigations encounters heavy-light four-quark condensates. We utilize an extended heavy-quark expansion to cope with the condensation of heavy quarks.


Introduction
The forthcoming experimental perspectives for in-medium heavy-light quark (i. e. D and B) meson spectroscopy, in particular at FAIR, are accompanied by the need for sophisticated theoretical analyses, e. g. [1][2][3][4][5][6]. When utilizing QCD sum rules [7][8][9], this requires a thorough discussion of heavyquark condensates in general, and, in particular, in the nuclear medium [10]. Therefore, the heavyquark expansion (HQE), originally developed for the heavy two-quark condensate Q Q in vacuum, is extended here to four-quark condensates and to the in-medium case, thus going beyond previous approaches, e. g. [11]. Specific formulas are derived and presented which provide important pieces for the complete QCD sum rule analysis of D and B mesons in nuclear matter.

Recollection: HQE in vacuum
In [12], a general method is introduced for vacuum condensates involving heavy quarks Q with mass m Q . The heavy-quark condensate is considered as the one-point function expressed by the heavy-quark propagator S Q in a weak classical gluonic background field in Fock- , incorporating the free heavy-quark propagator S (0) Q (p) = (γ μ p μ + m Q )/(p 2 − m 2 Q ) and the derivative operatorÃ emerging from a Fourier transform defined asÃ μ = ∞ m=0Ã [13,14]. In this way, the heavy-quark propagator interacts with the complex QCD ground state via soft gluons generating a series expansion in the inverse heavy-quark mass. The compact notation (1) differs from [12], but provides a comprehensive scheme easily extendable to in-medium condensates. The first HQE terms of the heavy two-quark condensate (1) reproduce [12]: with the notation The graphic interpretation of the terms in (2) is depicted too: the solid lines denote the free heavyquark propagators and the curly lines are for soft gluons whose condensation is symbolized by the crosses, whereas the heavy quark-condensate is symbolized by the crossed circles [15]. An analogous expression for the mixed heavy-quark gluon condensate can be obtained along those lines which contains, however, a term proportional to m Q . The leading-order term in (2) was employed already in [7] in evaluating the sum rule for charmonia. The vacuum HQE method was rendered free of UV divergent results for higher mass-dimension heavy-quark condensates by requiring at least one condensing gluon per condensed heavy-quark [15,16], which prevents unphysical results, where the condensation probability of heavy-quark condensates rises for an increasing heavy-quark mass.

Application of HQE to in-medium heavy-light four-quark condensates
The above method can be extended to in-medium situations. Our approach contains two new aspects: (i) formulas analogous to equation (1) are to be derived for heavy-quark condensates, e. g. Q / vQ , Q / vσGQ , q/ vt A qQ/ vt A Q , which additionally contribute to the in-medium operator product expansion (OPE) and (ii) medium-specific gluonic condensates, e. g.
enter the HQE of heavy-quark condensates for both, vacuum and additional medium condensates, where . . . denotes Gibbs averaging.
We are especially interested in heavy-light four-quark condensates entering the OPE of D and B mesons, inter alia, in terms corresponding to the next-to-leading-order perturbative diagrams with one light-quark (q) and one heavy-quark (Q) line cut. There are 24 two-flavour four-quark condensates in the nuclear medium [17] represented here in a compact notation by qΓT A qQΓ T A Q , where Γ and Γ denote Dirac structures and T A with A = 0, . . . , 8 are the generators of SU(3) supplemented by the unit element (A = 0). We obtain the analogous formula to (1) for heavy-light four-quark condensates: EPJ Web of C onferences 05007-p.2 The leading-order terms of this HQE are obtained for the heavy-quark propagators S (1) Q containing A (1) μ and S (2) Q with leading-order background fieldsÃ (0) μ : Evaluation of the first term of the expansion (8) for the complete list of two-flavour four-quark condensates in [17] gives three non-zero results: where logarithmic singularities are calculated in the MS scheme, μ is the renormalization scale, and t A = T A for A = 1, . . . , 8. The non-zero contributions for the second term of (8) read where f ABC is the anti-symmetric structure constant of the color group and the corresponding symmetric object d ABC is defined by the anti-commutator {t A , t B } = δ AB /4 + d ABC t C .

Summary and conclusions
The extension of the OPE for QCD sum rules ofqQ andQq mesons by four-quark condensates to mass dimension 6 yields heavy-light condensate contributions requiring HQE in a nuclear medium. The necessary steps to generalize the vacuum HQE [12] to cover in-medium situations are described and a general formula for the HQE of in-medium heavy-light four-quark condensates is presented. The two leading-order terms of this expansion for the complete list of two-flavour four-quark condensates [17] have been evaluated. In leading-order the results contain known condensate structures, thus, reducing the number of condensates entering the sum rule evaluation of mesons composed of a heavy and a light quark. It can be seen that the series does not exhibit a simple expansion in 1/m Q , not even in vacuum. Therefore, the lowest order terms are not suppressed by inverse powers of m Q as for Q Q , challenging the omission of heavy-light four-quark condensates, as often done in previous sum rule analyses.