Heavy Quark and Chiral Symmetry

arXiv:hep-ph/9209241v1 15 Sep 1992Radiative D∗Decay UsingHeavy Quark and Chiral SymmetryJames F. Amundsona, C. Glenn Boyda, Elizabeth Jenkinsb∗, Michael Lukec,Aneesh V. Manoharb ∗, Jonathan L. Rosnera, Martin J. Savagec† and Mark B. Wiseda) Enrico Fermi Institute and Department of Physics, University of Chicago,5640 S. Ellis Ave, Chicago, IL 60637b) CERN TH Division, CH-1211 Geneva 23, Switzerlandc) Department of Physics, University of California at San Diego,9500 Gilman Drive, La Jolla, CA 92093d) California Institute of Technology, Pasadena, CA 91125AbstractThe implications of chiral SU(3)L×SU(3)R symmetry and heavy quark symmetry forthe radiative decays D∗0 →D0γ, D∗+ →D+γ, and D∗s →Dsγ are discussed. Particularattention is paid to SU(3) violating contributions of order m1/2q.

Experimental data onthese radiative decays provide constraints on the D∗Dπ coupling.UCSD/PTH 92-31CALT-68-1816EFI-92-45CERN-TH.6650/92hep-ph@xxx/9209241September 1992∗On leave from the University of California at San Diego.† SSC Fellow1

Recent CLEO data [1] (see Table 1) have brought the D∗0 and D∗+ branching ratiosinto agreement with expectations based on the constituent quark model [2]. In this letter,the rates for D∗decay are described in a model independent framework which incorporatesthe constraints on the decay amplitudes imposed by the heavy quark and chiral SU(3)L ×SU(3)R symmetries of QCD.Table 1: D∗Branching Ratios (%)Decay ModeBranching RatioD∗0 →D0π063.6 ± 2.3 ± 3.3D∗0 →D0γ36.4 ± 2.3 ± 3.3D∗+ →D0π+68.1 ± 1.0 ± 1.3D∗+ →D+π030.8 ± 0.4 ± 0.8D∗+ →D+γ1.1 ± 1.4 ± 1.6At low momentum the strong interactions of the D and D∗mesons are described bythe chiral Lagrange density [3]L = −i Tr Havµ∂µHa + i2 Tr HaHbvµξ†∂µξ + ξ∂µξ†ba+ i2g Tr HaHb γµγ5ξ†∂µξ −ξ∂µξ†ba + · · ·(1)where the ellipsis denotes operators suppressed by factors of 1/mQ and operators with morederivatives or factors of the light quark mass matrix.

In Eq. (1), vµ is the four velocity ofthe heavy meson.

The field ξ is written in terms of the octet of pseudo-Nambu-Goldstonebosonsξ = exp (iM/f) ,(2)whereM =1√2π0 +1√6ηπ+K+π−−1√2π0 +1√6ηK0K−K0−q23η. (3)At tree level f can be set equal to fπ, fK or fη.Our normalization convention hasfπ ≃132 MeV.

Under chiral SU(3)L × SU(3)R transformations,ξ →LξU † = UξR†,(4)2

where L ∈SU(3)L and R ∈SU(3)R, and U is defined implicitly by Eq. (4).

Ha is a 4 × 4matrix that contains the D and D∗fields:Ha = 12 (1 + v/) [D∗µa γµ −Daγ5] ,Ha = γ0H†aγ0 . (5)The index a represents light quark flavor, where (D1, D2, D3) = (D0, D+, Ds) and(D∗1, D∗2, D∗3) = (D∗0, D∗+, D∗s).

Under SU(2)v heavy quark spin symmetry and chiralSU(3)L × SU(3)R symmetry, Ha transforms asHa →S(HU †)a ,(6)where S ∈SU(2)v. The D∗Dπ coupling constant g is responsible for the D∗→Dπ decays.At tree level,Γ(D∗+ →D0π+) =g26πf 2π| ⃗pπ|3 . (7)The decay width for D∗+ →D+π0 is a factor of two smaller by isospin symmetry.

Theexperimental upper limit [4] on the D∗+ width of 131 keV when combined with the D∗+ →D+π0 and D∗+ →D0π+ branching ratios in Table 1 leads to the limit g2 <∼0.5.The axial vector current obtained from the Lagrangian (1) isqa T Aab γνγ5 qb = −g Tr HaHb γνγ5 T Aba + · · · . (8)In Eq.

(8) the ellipsis represents terms containing one or more Goldstone boson fieldsand T A is a flavor SU(3) generator. Treating the quark fields in Eq.

(8) as constituentquarks and using the nonrelativistic quark model to estimate the D∗matrix element ofthe l.h.s. of Eq.

(8) gives g = 1. (A similar estimate of the pion-nucleon coupling givesgA = 5/3.) In the chiral quark model [5] there is a constituent quark-pion coupling.

Usingthe measured pion-nucleon coupling to determine the constituent quark pion coupling givesg ≃0.8. Thus various constituent quark model estimates lead to the expectation that gis near unity.

In this paper, however, we wish to adopt a model independent approachto radiative D∗decay. From the point of view of chiral perturbation theory g is a freeparameter and its value must be determined from experiment.The D∗a →Daγ matrix element has the formM(D∗a →Daγ) = eµa ǫµαβλ ǫ∗µ(γ) vα kβ ǫλ(D∗),(9)3

where eµa/2 is the transition magnetic moment, k is the photon momentum, ǫ(γ) is thepolarization of the photon and ǫ(D∗) is the polarization of the D∗. The resulting decayrate isΓ(D∗a →Daγ) = α3 |µa|2 |⃗k|3 .

(10)The D∗a →Daγ matrix element gets contributions from the photon coupling to the lightquark part of the electromagnetic current,23 uγµu −13 dγµd −13 sγµs, and the photoncoupling to the heavy charm quark part of the electromagnetic current,23 cγµc.Thepart of µa that comes from the charm quark piece of the electromagnetic current, µ(h),is determined by heavy quark symmetry.In the effective heavy quark theory [6], theLagrange density for strong and electromagnetic interactions of the charm quark isL =h(c)v (iv · D) h(c)v+12mch(c)v (iD)2h(c)v−gs2mch(c)v σµνT ah(c)v Gaµν −e3mch(c)v σµνh(c)v Fµν + · · · . (11)In Eq.

(11), Dµ is the covariant derivativeDµ = ∂µ + igsAaµT a + 23ieAµ,(12)where gs is the strong coupling and e is the electromagnetic coupling. The ellipsis denotesterms with more factors of 1/mc.

It is to be understood that the operators and couplings inEq. (11) are evaluated at a subtraction point µ = mc, and that corrections of order αs(mc)have been neglected.

The last term in Eq. (11) is responsible for a D∗to D transitionmatrix element µ(h).

By heavy quark symmetry [7],µ(h) =23mc,(13)where µ(h) is independent of the light quark flavor. Perturbative αs(mc) corrections tothe above are computable, while corrections suppressed by a power of 1/mc are relatedto those which occur in semileptonic B →D∗eνe decays [8].

At order 1/m2c, Eq. (13)becomes µ(h) = (2/3mc) [1 −4ξ+(1)/mc], where ξ+ is defined in Ref.

[8].The part of µa that comes from the photon coupling to the light quark piece of theelectromagnetic current, µ(ℓ)a , is not fixed by heavy quark symmetry.The light quarkpiece of the electromagnetic current transforms as an octet under SU(3) flavor symmetry.4

Since there is only one way to combine an 8, 3 and 3 into a singlet, in the limit of SU(3)symmetry, the µ(ℓ)aare expressible in terms of a single reduced matrix element,µ(ℓ)a= Qaβ ,(14)where β is an unknown constant and Qa denotes the light quark charges Q1 = 2/3, Q2 =−1/3, Q3 = −1/3. In the nonrelativistic constituent quark model β ≃3 GeV−1.

Note thatEq. (14) includes effects suppressed by powers of 1/mc, since it follows from using onlySU(3) symmetry.The leading SU(3)-violating contribution to the transition amplitudes has a nonana-lytic dependence on mq of the form m1/2qwhich arises from the one-loop Feynman diagramsshown in fig.

1. The strange quark mass, ms, is not very small, and so the corrections toEq.

(14) from SU(3) violation may be comparable to µ(h), which is suppressed by 1/mcrelative to µ(ℓ). Including the leading SU(3) violations, µ(ℓ)abecomesµ(ℓ)1= 23β −g2mK4πf 2K−g2mπ4πf 2π,µ(ℓ)2= −13β + g2mπ4πf 2π,µ(ℓ)3= −13β + g2mK4πf 2K.

(15)The difference between using f = fπ and f = fK in Eq. (15) is a higher order effect.We have chosen to use f = fK ≃1.22 fπ for loops involving kaons and f = fπ for loopsinvolving pions.

For mK ̸= mπ, the one loop contribution to µ(ℓ)1 , µ(ℓ)2and µ(ℓ)3is not inthe ratio 2 : −1 : −1 and hence violates SU(3). It is easy to understand why the one-loopcorrection proportional to mK is different for the D∗0 →D0γ and D∗+ →D+γ decays.Strong interactions can change a D∗0 into a virtual K−D∗s pair, while the D∗+ changesinto a virtual K0D∗s pair.

In the latter case the virtual kaon is neutral and doesn’t coupleto the photon. Thus there is no m1/2scorrection to µ(ℓ)2 .

The most important correctionto Eq. (15) comes from SU(3) violating terms of order ms.

These terms are analytic inthe strange quark mass, and are not determined by the lowest order Lagrangian.Usingµa = µ(ℓ)a + µ(h),(16)with µ(ℓ)aand µ(h) given by Eqs. (15) and (13) respectively, determines the rates for D∗0 →D0γ, D∗+ →D+γ and D∗s →Dsγ in terms of β and g. Combining this with Eq.

(7) and5

using the measured value of BR(D∗0 →D0γ)/BR(D∗0 →D0π0) gives g as a function ofthe branching ratio for D∗+ →D+γ. This in fact gives four different solutions for g2; weeliminated three of these by imposing the constraints g < 1 (as required by Ref.

[4]) andµ(ℓ)a> µ(h) i.e., the light quark transition moment is greater than that of the heavy quark.The result is shown in fig. 2.

(We have taken mc = 1.7 GeV.) Note that the favored valuesfor g are smaller than what is expected on the basis of the nonrelativistic constituent quarkmodel.

Since 1/mc effects have been included in the radiative D∗decays, the value of gextracted in this way is an “effective” value of g that includes 1/mc corrections. FromEq.

(7) and our values of g we can compute the total width of the D∗+ as a function ofBR(D∗+ →D+γ); this is plotted in fig. 3.The SU(3) violation plays an important role in our analysis.

Fig. 4 shows the absolutevalues of the relative contributions to µ1 of µ(h) (dashed-dotted line), β (dotted line)and the one-loop nonanalytic contribution to µ(ℓ)1(solid line).

The values have all beenmultiplied by 3/2, so that the dotted line is normalized to β.Note that values of βnear the non-relativistic constituent quark model expectation of ≈3 GeV−1 favor a smallD∗+ →D+γ branching ratio, and hence smaller values of g. In fig. 5 the value of g thatfollows from neglecting SU(3) violation (i.e.

using Eq. (14) for µ(ℓ)a ) is shown.Largervalues of g are favored when SU(3) violation is neglected.Nonanalytic dependence on ms similar to what we have found in radiative D∗decayoccurs in the Ds −D+ mass difference.

Including effects up to order m3/2s[9]mDs −mD+ = Cms −3g264πf 2K2m3K + m3η,(17)where we have set mu = md = 0 and C is an unknown constant. Experimentally, mDs −mD+ ≃100 MeV.The magnitude of the nonanalytic part is about 50% of the massdifference for g = 0.5.

This gives us some confidence that the expansion is well behavedfor at least some of the range of g’s in fig. 2.The analysis in this paper allows us to predict the D∗s →Dsγ rate as a function ofthe D∗+ →D+γ branching ratio.

However, for D∗s →Dsγ there is a strong cancellationbetween µ(ℓ)3and µ(h), resulting in a very small D∗s width. (Note that D∗s →Dsπ isforbidden by isospin.) In this situation, SU(3) violating terms of order ms may be veryimportant.6

Since heavy quark symmetry ensures that g and β are the same in the b and c systems(up to corrections of order 1/mc), the results of this paper can be used to predict the widthsfor radiative B∗decay. Neglecting effects of order 1/mb and 1/mc, Eq.

(10) becomesΓ (B∗a →Baγ) = α3 |µ(ℓ)a |2|⃗k|3(18)where µ(ℓ)ais given by Eq. (15).

An analysis of the radiative decays of charmed baryonsusing the same methods is possible. Unfortunately, at the present time there is no experi-mental information on radiative charmed baryon decays.Work similar to that presented in this paper has also been done by Cho and Georgi[10].

We are grateful to them for communicating their results to us prior to publication.This work was supported in part by the Department of Energy under grant number DOE-FG03-90ER40546 and contract number DEAC-03-81ER40050, and by a National ScienceFoundation Presidential Young Investigator award number PHY-8958081. MJS acknowl-edges the support of a Superconducting Supercollider National Fellowship from the TexasNational Research Laboratory Commission under grant FCFY9219.7

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Figure CaptionsFig. 1.Diagrams giving the leading non-analytic contributions to µ(ℓ)a .Fig.

2.The coupling constant g as a function of BR(D∗+ →D+γ) including leadingSU(3)-breaking effects. The shaded region indicates the uncertainty due to the1σ variations in BR(D∗0 →D0π0) and BR(D∗0 →D0γ).

The arrows indicatethe 90% confidence level limits on BR(D∗+ →D+γ) and the D∗+ width.Fig. 3.Width of the D∗+ as a function of BR(D∗+ →D+γ) including leading SU(3)-breaking effects.The shaded region indicates the uncertainty due to the 1σvariations in BR(D∗0 →D0π0) and BR(D∗0 →D0γ).

The arrows indicate the90% confidence level limits on BR(D∗+ →D+γ) and the D∗+ width.Fig. 4.Relative contributions to µ1 of µ(h) (dashed-dotted line), β (dotted line), and theone-loop nonanalytic m1/2qterm (solid line) to the matrix element for D∗0 →D0γ.Fig.

5.The coupling constant g as a function of BR(D∗+ →D+γ) ignoring SU(3) vio-lation. The shaded region indicates the uncertainty due to the 1σ variations inBR(D∗0 →D0π0) and BR(D∗0 →D0γ).

The arrows indicate the 90% confidencelevel limits on BR(D∗+ →D+γ) and the D∗+ width.9

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