v1 23 May 1995

HEAVY MESON HYPERFINE SPLITTING: A COMPLETE 1/mQ CALCULATION. 1 N. Di Bartolomeo arXiv:hep-ph/9505363v1 23 May 1995 D´epartement de Physique Th´eori...
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HEAVY MESON HYPERFINE SPLITTING: A COMPLETE 1/mQ CALCULATION. 1 N. Di Bartolomeo

arXiv:hep-ph/9505363v1 23 May 1995

D´epartement de Physique Th´eorique, Univ. de Gen`eve

ABSTRACT We compute the chiral corrections to the hyperfine splittings ∆D = (mDs∗ − mDs ) − (mD∗+ − mD+ ) and ∆B = (mBs∗ − mBs ) − (mB∗0 − mB0 ) arising from one-loop chiral corrections, working in a framework of an effective chiral lagrangian incorporating chiral, heavy flavour and spin symmetric terms and first order breaking terms. Among these terms, those responsible for the spin-breaking difference between the couplings gP ∗ P ∗ π and gP ∗ P π are evaluated in the QCD sum rules approach. Their contribution to ∆D and to ∆B appears to cancel previously estimated large chiral effects, giving an estimate in agreement with the experimental data.

to appear in the Proceedings of the XXXth Rencontres de Moriond “QCD and High Energy Hadronic Interactions” Les Arcs, France, March 1995

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Partially supported by the Swiss National Foundation.

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1

Introduction

The spectroscopy of heavy mesons is among the simplest framework where the ideas and the methods of heavy quark expansion can be quantitatively tested. Recently, attention has been focused on the combinations [1, 2, 3, 4]: ∆D = (mDs∗ − mDs ) − (mD∗+ − mD+ )

(1.1)

∆B = (mBs∗ − mBs ) − (mB∗0 − mB0 )

(1.2)

which are measured to be [5]: ∆D ≃ 1.0 ± 1.8 MeV

(1.3)

∆B ≃ 1.0 ± 2.7 MeV

(1.4)

The above hyperfine splitting is free from electromagnetic corrections and it vanishes separately in the SU(3) chiral limit and in the heavy quark limit. In the combined chiral and heavy quark expansion, the leading contribution is of order ms /mQ and one would expect the relation [1]: ∆B =

mc ∆D mb

(1.5)

In the so called heavy meson effective theory [6], which combines the heavy quark expansion and the chiral symmetry, there is only one lowest order operator contributing to ∆D,B . By naive dimensional analysis, its contribution to the hyperfine splitting is of the order (2)

∆D ≃ 20 MeV (2)

∆B ≃ 6 MeV

(1.6) (1.7)

Given the present experimental accuracy, the above estimate is barely acceptable, as an order of magnitude, for ∆B , while it clearly fails to reproduce the data for ∆D . In chiral perturbation theory, an independent contribution arises from one-loop corrections to the heavy meson self energies [3], evaluated from an initial lagrangian containing, at the lowest order, both chiral breaking and spin breaking terms. The loop corrections in turn depend on an arbitrary renormalization point µ2 (e.g. the t’Hooft mass of dimensional regularization). This dependence is cancelled by the µ2 dependence of a counterterm. A commonly accepted point of view is that the overall effect of adding the counterterm consists in replacing µ2 in the loop corrections with the physical scale relevant to the problem at hand, Λ2CSB . Possible finite terms in the counterterm are supposed to be small compared to the large chiral logarithms. With this philosophy in mind, two classes of such corrections has been estimated in ref. [2]: keeping the chiral logarithms and non-analytic contributions of the order m3/2 s , they found a quite large correction to the hyperfine splitting, ∆0D ≃ +95 MeV,

(1.8)

∆0B ≃ +32 MeV,

(1.9)

This provides a rather uncomfortable situation since, to account for the observed data, one should require an accurate and innatural cancellation. Hower, as pointed out in [4], there is another term induced by the difference between the P ∗ P ∗ π and the P ∗ P π couplings (P = D, B). They coincide in the limit MP → ∞ gP ∗ P π = gP ∗ P ∗ π = g 1

(1.10)

because their splitting is a spin breaking effect. To the order 1/mQ we parametrize them in the following way: ! ! a b gP ∗ P π = g 1 + (1.11) gP ∗ P ∗ π = g 1 + mQ mQ The chiral and spin symmetry breaking parameters relevant to the hyperfine splitting are the light pseudoscalar masses mπ , mK and mη , ∆s = MPs − MP , ∆ = MP ∗ − MP and ∆g = gP ∗ P ∗ π − gP ∗ P π . In terms of these quantities, one finds [3, 2, 4]: ∆P =

g 2∆ h 2 Λ2CSB Λ2CSB i Λ2CSB 2 2 ) + 2m ln( ) − 6m ln( ) 4m ln( η π K 16π 2 f 2 m2k m2η m2π

g 2∆ [24πmK ∆s ] 16π 2 f 2 1 3 3 3 g 2 ∆g 3 (m + m − m ) − K 6πf 2 g 2 η 2 π +

(1.12)

The dependence upon the heavy flavour P = D, B is contained in the parameters ∆ and ∆g . In [7] we have provided an estimate of ∆g = gP ∗ P ∗ π − gP ∗ P π based on a QCD sum rule, and, by including this additional spin breaking effect, we have completed the evaluation of ∆D,B in (1.12).

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QCD Sum Rules for gP ∗P π and gP ∗ P ∗π

The coupling gP ∗ P π has been calculated in [8], gP ∗ P ∗ π in [7], by means of QCD sum rules [9]. Without entering into the details of the calculation, we sketch the strategy followed in computing gP ∗ P ∗ π [7]. One starts from the correlator: Aµν (q1 , q) = i

Z

dx < π(q)|T (Vµ(x)Vν† (0)|0 > e−iq1 x = A(q12 , q22, q 2 )ǫµναβ q α q1β + . . .

(2.1)

where Vµ = uγµ Q is the interpolating vector current for the P ∗ meson, computing the scalar function A in the soft pion limit q → 0 (this implies q1 = q2 forcing to use a single Borel transformation). The correlator in (2.1) can be calculated by an Operator Product Expansion: in [7] all the operators with dimension up to five, arising from the expansion of the current Vµ (x) at the third order in powers of x and the heavy quark propagator to the second order, are kept. Proceeding in a standard way, one computes the hadronic side of the sum rule, equating it to the QCD side. The Borel transform enhances the ground state contribution to the sum rule, and has been applied in [7]. Expanding the sum rule in the parameter 1/mQ , keeping the leading term and the first order corrections, allows to derive sum rules for g and for the coefficients a and b in (1.11). In the formula (1.12) for the hyperfine splitting, only the difference ∆g ≡ gP ∗ P ∗ π − gP ∗ P π = g

a−b mQ

(2.2)

enters. The sum rule for the difference a − b turns out to be quite stable, giving [7]: a − b ≃ 0.6 GeV 2

(2.3)

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Discussion and conclusions

From (2.3) and from the formula (1.12) of the hyperfine mass splitting we obtain: ∆B ≈ g 2(27.3 + 61.4 − 75.8) MeV = 12.9g 2 MeV

(3.1)

Notice that we have used in eq. (1.12) f = fπ = 132 MeV for all the light pseudoscalar mesons of the octet. This is suggested by the sum rule for g which shows that g/f is flavour independent. In (3.1) we have taken ΛCSB = 1 GeV . It is evident that there is a large cancellation among the last term and the other ones. In order to be more quantitative we have to fix the value of g. In Ref. [8] the range of values g ≃ 0.2 − 0.4 was found; therefore, putting g 2 = 0.1, we would obtain ∆B ≃ 1 MeV (3.2) The application of our results to the charm case is more doubtful, in view of the large values of the 1/mc correction (a − b)/mc . By scaling the result (3.2) to the charm case, one obtains ∆D =

mb ∆B ≃ 4 MeV mc

(3.3)

In conclusion, our estimate of gP ∗ P ∗ π − gP ∗ P π allows to include a previously neglected term in the loop induced contribution to the hyperfine splitting, providing a substantial cancellation and reconciliating the chiral calculation with the experimental data.

Acknowledgements I would like to thank F. Feruglio, R. Gatto and G. Nardulli, with whom the work presented here was done, for the pleasant collaboration and useful discussions.

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[8] V.L. Eletski and Y.I. Kogan, Z. Phys. C 28 (1985) 155; A.G. Grozin and O.I. Yakovlev, preprint BUDKERINP-94-3 (hep-ph/9401267); P. Colangelo, A. Deandrea, N. Di Bartolomeo, F. Feruglio, R. Gatto and G. Nardulli, Phys. Lett. B 339 (1994) 151; V. M. Belyaev, V. M. Braun, A. Khodjamirian, R. R¨ uckl, preprint MPI-PhT/94-62, CEBAFTH-94-22, LMU 15/94 (hep-ph/9410280). [9] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385, 448.

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