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Submitted to Physical Review Letters

arXiv:hep-ph/9207235v1 11 Jul 1992Submitted to Physical Review LettersEFI 92-31July 1992MESON DECAY CONSTANTS FROM ISOSPIN MASS SPLITTINGSIN THE QUARK MODELJames F. Amundson and Jonathan L. RosnerEnrico Fermi Institute and Department of PhysicsUniversity of Chicago, Chicago, IL 60637Michael A. KellyDepartment of Physics, University of KansasLawrence, Kansas 66045-2151Nahmin Horwitz and Sheldon L. StoneDepartment of Physics, Syracuse UniversitySyracuse, NY 13244-1130Decay constants of D and B mesons are estimated within the frameworkof a heavy-quark approach using measured isospin mass splittings in theD, D∗, and B states to isolate the electromagnetic hyperfine interactionbetween quarks. The values fD = (262 ± 29) MeV and fB = (160 ± 17)MeV are obtained.

Only experimental errors are given; possible theoreti-cal ambiguities, and suggestions for reducing them, are noted.The decay constants fD and fB of mesons containing a single heavy quark are offundamental importance for the understanding of the strong interactions, since theydescribe the behavior of a single light quark bound to a nearly static source of color.The constant fB is crucial for interpreting data on particle-antiparticle mixing in theneutral B meson system, and both constants are essential if one is to anticipate andinterpret new signatures for CP violation. In this Letter we describe a method fordetermination of these constants which relies on the isospin splittings of the D, D∗,and B mesons, and indicate what additional information will be necessary to reducesystematic errors to an acceptable level.The decay constant fM for a meson M is specified by the matrix element of theaxial current between the one-particle state and the vacuum: ⟨0|Aµ|M(q)⟩= iqµfM.In a non-relativistic quark model it is related to |Ψ(0)|, the wave function at theorigin, by [1]fM = (12/MM)1/2|Ψ(0)|.

(1)While we recognize the limitations of the nonrelativistic model and the relation (1),especially for D mesons [2-4], we seek independent information on |Ψ(0)|. We find1

it by comparing isospin splittings in pseudoscalar and vector meson multiplets [5,6].These have recently been measured very precisely, with the results [7]δm(D) ≡M(D+) −M(D0) = (4.80 ± 0.10 ± 0.06) MeV;(2)δm(D∗) ≡M(D∗+) −M(D∗0) = (3.32 ± 0.08 ± 0.05) MeV;(3)δm(D) −δm(D∗) = (1.48 ± 0.09 ± 0.05) MeV. (4)There are previously existing measurements of the mass differenceδm(B) ≡M( ¯B0) −M(B−) =(2.0 ± 1.1 ± 0.3) MeVCLEO 85 [8](−0.4 ± 0.6 ± 0.5) MeVCLEO 87 [9](−0.9 ± 1.2 ± 0.5) MeVARGUS [10](0.12 ± 0.58) MeV(Average).

(5)Within certain assumptions, the values (2) – (5) allow one to determine |Ψ(0)| forthe D and B meson systems, and hence to calculate fD and fB.Three sources of isospin mass splittings in hadrons can be identified [11]. (a) The d-u mass difference leads to splittings both directly and through the QCDhyperfine interaction∆EQCD hfs = const.

⟨σi · σj⟩mimj. (6)Here ⟨σi · σj⟩= (−3, 1) for a total spin Sij = (0, 1) of the ij quark pair.Theconstant depends on the color representation of the quark pair and thus is differentfor mesons and baryons.

One can estimate its value by comparing masses of stateswith different spins, such as nucleon and ∆, or D and D∗. Fits to hadron massesbased on constituent quark masses, with account taken of the term (6), are remarkablysuccessful [12,13].

In a nonrelativistic model with single-gluon exchange, the constantin (6) would be proportional to αs|Ψ(0)|2, where αs is the strong coupling constant. (b) Coulomb interactions among quarks lead to an energy shift ∆ECoul = QiQj⟨1/r⟩ijfor each interaction between quarks of charges Qi and Qj.

(c) The electromagnetic hyperfine interaction leads to an energy shift associatedwith the interaction of quarks i and j:∆Ee.m. hfs = −2πα|Ψ(0)|2QiQj⟨σi · σj⟩3mimj.

(7)In contrast to the strong hyperfine interaction, where the magnitude of the constantin (6) is uncertain a priori (though it can be measured experimentally), the term (7)holds the promise of providing information on |Ψ(0)|2.One can describe the isospin mass splittings in a meson in terms of three unknownparameters: md−mu ≡x, the expectation value of 1/r, and the value of Ψ(0). Quarkmasses are assumed known from fits to meson mass spectra based on the constituentquark model [13]: mav ≡(mu + md)/2 = 310 MeV, mc = 1662 MeV, mb = 5 GeV.2

The isospin mass splittings for charmed mesons are predicted to be [14]:δm(D) = x1 + 3∆MD4mav+ 23α⟨1r⟩D +4πα3mavmc|Ψ(0)|2D;(8)δm(D∗) = x1 −∆MD4mav+ 23α⟨1r⟩D∗−4πα9mavmc|Ψ(0)|2D∗,(9)where ∆MD = 141.4 MeV is the D∗−D mass difference (we use the average for thecharged and neutral states). Here we have expanded the effect of md ̸= mu in theQCD hyperfine interaction term (6) to first order in x ≡md −mu.

For B mesons thecorresponding expressions areδm(B) = x1 + 3∆MB4mav−13α⟨1r⟩B −2πα3mavmb|Ψ(0)|2B;(10)δm(B∗) = x1 −∆MB4mav−13α⟨1r⟩B∗+2πα9mavmb|Ψ(0)|2B∗,(11)where the B −B∗mass splitting term ∆MB is about 46 MeV [15].In the spirit of heavy-quark symmetry [16], we shall assume that the wave functionof a light quark bound to a heavy quark is independent of both the identity and thespin of the heavy quark. With this ansatz, Eqs.

(8) – (10) express the three measuredquantities (2), (3), and (5) in terms of the parameters x = md −mu,⟨1/r⟩≡⟨1/r⟩D = ⟨1/r⟩D∗= ⟨1/r⟩B = ⟨1/r⟩B∗,(12)and|Ψ(0)| ≡|Ψ(0)|D = |Ψ(0)|D∗= |Ψ(0)|B = |Ψ(0)|B∗. (13)Solving, we find [17]x = (1.29 ± 0.35) MeV;α⟨1/r⟩= (3.60 ± 0.53) MeV;|Ψ(0)|2 = (11.3 ± 2.4) × 10−3 GeV3.

(14)Taking MM = [M(pseudoscalar)+3M(vector)]/4 in Eq. (1), we obtain the values (cf.fπ = 132 MeV, fK = 161 MeV)fD = (262 ± 28) MeV;fB = (160 ± 17) MeV.

(15)The value of fD is compatible with the experimental upper limit [18] fD ≤290 MeV(90% c.l.). It is in the range of recent theoretical estimates (see, e.g., [3,5]).

The valueof fB is compatible with analyses of B−¯B mixing in the context of information aboutelements of the Cabibbo-Kobayashi-Maskawa matrix [19], but lies below predictionsof lattice QCD [3]. These, as well as the work of Ref.

[4], suggest that there areimportant corrections of order 1/mQ, where Q is a heavy quark, to the relation (1).The parameters (14) imply thatδm(B∗) = (0.08 ± 0.38) MeV. (16)3

A measurement of δm(B∗) with an experimental error comparable to that in (16)would be desirable as a check of our assumptions.The errors in Eq. (15) are purely experimental.

There are both QCD and O(1/mQ)corrections to the nonrelativistic relations (1) and (7). An example of the first is thatthe ratio fB/fD =qMD/MB implied by the use of identical B and D wave functionsin (1) should be multiplied by a QCD correction of 1.11 [20], while 1/mQ correctionsto (1) have been discussed in Ref.

[4]. However, until a corresponding discussion forthe electromagnetic hyperfine term (7) has been given [21], it does not make sense toincorporate only partial information on such corrections.

Corrections to the ratio ofthe two terms are what we need, and probably make more sense.The errors on light quark masses used in extracting |Ψ(0)|2 from the electromag-netic hyperfine interaction energy are probably about 20%, based on the spread invalues obtained in various constituent-quark fits to mesons and baryons [13]. Theseerrors lead to uncertainties in fD and fB comparable to those due to uncertainties in|Ψ(0)|2 in Eq.

(14).Another way of finding x = md −mu is to use fits to the isospin mass splittings inthe nucleon, Σ, and Ξ states, based on the effects (a) – (c) mentioned above [14]. Anindependent estimate of x allows us to obtain fD by comparing (4) with the differencebetween (8) and (9) (the α⟨1/r⟩contributions cancel in the heavy-quark limit) withoutassuming that wave functions are identical in D and B systems.

However, we shallsee that a much larger value of x arises than one finds in Eq. (14).

This providesus with an idea of systematic errors inherent in the use of the nonrelativistic quarkmodel, and emphasizes the importance of knowing the full set of corrections to theratio of (1) and (7).The QCD hyperfine interaction in baryons may be obtained from the mass dif-ference ∆baryon = M(∆) −M(N) = (1238 −938) = 300 MeV. Neglecting flavor-symmetry-breaking effects in wave functions but keeping them in quark masses, wedefiney ≡α⟨1r⟩baryon;z ≡2πα3m20|Ψ(0)|2baryon;r ≡m0ms,(17)where m0 = 363 MeV is the average u and d quark mass found in a fit to baryonmasses, while ms = 538 MeV in the same fit [13].

The expectation values of the σi ·σjterms in (6) and (7) may be evaluated using quark model wave functions [13], whilepairwise Coulomb interactions of quarks may be summed to obtain the coefficients ofthe term y. To first order in isospin-splitting terms, one can then writeM(n) −M(p) =1 −∆baryon3m0x −13y + 13z = 1.293 MeV;(18)M(Σ−) −M(Σ+) =2 −∆baryon3m0+ 2∆baryon3msx + 13y +43r + 13z == (8.07 ± 0.09) MeV;(19)M(Σ+) + M(Σ−) −2M(Σ0) = y −z = (1.70 ± 0.12) MeV;(20)4

M(Ξ−) −M(Ξ0) =1 + 2∆baryon3msx + 23y + 43rz = (6.4 ± 0.6) MeV. (21)We have used baryon masses from Ref.

[22]. Combining (18) and (20), we find (1 −[∆baryon/3m0])x = (1.86 ± 0.04) MeV, or x = (2.57 ± 0.06) MeV.

The correspondingcentral values of y and z are 3.06 and 1.36 MeV, respectively. Comparing (4) withthe difference between (8) and (9), we then find |Ψ(0)|2D(∗) = (3.9 ± 1.3) × 10−3 GeV3and fD = (154 ± 26) MeV for mav = 310 MeV in (8) and (9).

The comparison ofthis result (based on x from baryon masses) with our previous ones (14) and (15)(based on assumptions motivated by heavy-quark symmetry) shows that these twoindependent ways of deriving values for x give very different answers for fD.It is possible that one should not use the same values of x in mesons and baryons.However, unless one is prepared to make the assumptions leading to (14) and (15),there is not enough independent information available on x From mesons alone. TheK and K∗systems involve wave functions sufficiently different from the D and D∗that their isospin splittings do not provide a reliable estimate of x.We have shown that the values fD = (262 ± 29) MeV and fB = (160 ± 17) MeVcan be obtained from isospin mass splittings in the D, D∗, and B meson systems, ifthe nonrelativistic formulae (1) and (7) relating the wave function at the origin to thedecay constant and to the hyperfine interaction are valid, and if the wave functionsare identical in the three systems.Our use of heavy quark symmetry in the present context can be checked not onlyby measurement of fD (via detection of the decay D →µ+ν), but also by verificationof the very small predicted isospin splitting in the B∗system.One might expect the relation (1) to be better satisfied for B mesons than for Dmesons.

A reliable estimate of x and a measurement of the difference between isospinsplittings in the pseudoscalar and vector B mesons would permit an independentdetermination of fB using the present method.Considerable systematic error comes from uncertainty in constituent-quark masses.Just the uncertainty in whether to use light-quark masses from fits to mesons or tobaryons introduces an additional error of about 10% in the decay constants. A muchmore serious problem is that finding x in another model-dependent manner using fitsto the isospin mass splittings in the baryons gives a value of x about twice as largeas that required to explain the isospin splittings in the D, D∗, and B systems.

Thispotential inconsistency might be resolved once the full set of QCD and O(1/mQ)corrections to the ratio of (1) and (7) have been calculated.This work was supported in part by the U. S. Department of Energy under GrantNo. DE FG02 90ER 40560, and by the National Science Foundation.5

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