Radiative Decay Using

Heavy Quark and Chiral Symmetry

James F. Amundson C. Glenn Boyd, Elizabeth
Jenkins^{†}^{†} On leave from the
University of California at San Diego., Michael Luke,

Aneesh V. Manohar,
Jonathan L. Rosner, Martin J. Savage^{†}^{†} SSC
Fellow and
Mark B. Wise

a) Enrico Fermi Institute and Department of Physics, University of Chicago,

5640 S. Ellis Ave, Chicago, IL 60637

b) CERN TH Division, CH-1211 Geneva 23, Switzerland

c) Department of Physics, University of California at San Diego,

9500 Gilman Drive, La Jolla, CA 92093

d) California Institute of Technology, Pasadena, CA 91125

Abstract

The implications of chiral symmetry and heavy quark symmetry for the radiative decays , , and are discussed. Particular attention is paid to violating contributions of order . Experimental data on these radiative decays provide constraints on the coupling. UCSD/PTH 92-31 CALT-68-1816 EFI-92-45 CERN-TH.6650/92 [email protected]/9209241 September 1992

Recent CLEO data [1] (see Table 1) have brought the and branching ratios into agreement with expectations based on the constituent quark model [2]. In this letter, the rates for decay are described in a model independent framework which incorporates the constraints on the decay amplitudes imposed by the heavy quark and chiral symmetries of QCD.

Table 1: Branching Ratios (%)

Decay Mode — Branching Ratio — \crnorule — —\crnorule — \crnorule — \endtable

At low momentum the strong interactions of the and mesons are described by the chiral Lagrange density [3]

where the ellipsis denotes operators suppressed by factors of and operators with more derivatives or factors of the light quark mass matrix. In Eq. (1), is the four velocity of the heavy meson. The field is written in terms of the octet of pseudo-Nambu-Goldstone bosons

where

At tree level can be set equal to , or . Our normalization convention has . Under chiral transformations,

where and , and is defined implicitly by Eq. (4). is a matrix that contains the and fields:

The index represents light quark flavor, where and . Under heavy quark spin symmetry and chiral symmetry, transforms as

where . The coupling constant is responsible for the decays. At tree level,

The decay width for is a factor of two smaller by isospin symmetry. The experimental upper limit [4] on the width of 131 keV when combined with the and branching ratios in Table 1 leads to the limit .

The axial vector current obtained from the Lagrangian (1) is

In Eq. (8) the ellipsis represents terms containing one or more Goldstone boson fields and is a flavor generator. Treating the quark fields in Eq. (8) as constituent quarks and using the nonrelativistic quark model to estimate the matrix element of the l.h.s. of Eq. (8) gives . (A similar estimate of the pion-nucleon coupling gives .) In the chiral quark model [5] there is a constituent quark-pion coupling. Using the measured pion-nucleon coupling to determine the constituent quark pion coupling gives . Thus various constituent quark model estimates lead to the expectation that is near unity. In this paper, however, we wish to adopt a model independent approach to radiative decay. From the point of view of chiral perturbation theory is a free parameter and its value must be determined from experiment.

The matrix element has the form

where is the transition magnetic moment, is the photon momentum, is the polarization of the photon and is the polarization of the . The resulting decay rate is

The matrix element gets contributions from the photon coupling to the light quark part of the electromagnetic current,

In Eq. (11), is the covariant derivative

where is the strong coupling and is the electromagnetic coupling. The ellipsis denotes terms with more factors of . It is to be understood that the operators and couplings in Eq. (11) are evaluated at a subtraction point , and that corrections of order have been neglected. The last term in Eq. (11) is responsible for a to transition matrix element . By heavy quark symmetry [7],

where is independent of the light quark flavor. Perturbative corrections to the above are computable, while corrections suppressed by a power of are related to those which occur in semileptonic decays [8]. At order , Eq. (13) becomes , where is defined in Ref. [8].

The part of that comes from the photon coupling to the light quark piece of the electromagnetic current, , is not fixed by heavy quark symmetry. The light quark piece of the electromagnetic current transforms as an octet under flavor symmetry. Since there is only one way to combine an 8, 3 and into a singlet, in the limit of symmetry, the are expressible in terms of a single reduced matrix element,

where is an unknown constant and denotes the light quark charges . In the nonrelativistic constituent quark model . Note that Eq. (14) includes effects suppressed by powers of , since it follows from using only symmetry.

The leading -violating contribution to the transition amplitudes has a nonanalytic dependence on of the form which arises from the one-loop Feynman diagrams shown in fig. 1. The strange quark mass, , is not very small, and so the corrections to Eq. (14) from violation may be comparable to , which is suppressed by relative to . Including the leading SU(3) violations, becomes

The difference between using and in Eq. (15) is a higher order effect. We have chosen to use for loops involving kaons and for loops involving pions. For , the one loop contribution to , and is not in the ratio and hence violates . It is easy to understand why the one-loop correction proportional to is different for the and decays. Strong interactions can change a into a virtual pair, while the changes into a virtual pair. In the latter case the virtual kaon is neutral and doesn’t couple to the photon. Thus there is no correction to . The most important correction to Eq. (15) comes from violating terms of order . These terms are analytic in the strange quark mass, and are not determined by the lowest order Lagrangian.

Using

with and given by Eqs. (15) and (13) respectively, determines the rates for , and in terms of and . Combining this with Eq. (7) and using the measured value of gives as a function of the branching ratio for . This in fact gives four different solutions for ; we eliminated three of these by imposing the constraints (as required by Ref. [4]) and 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 .) Note that the favored values for are smaller than what is expected on the basis of the nonrelativistic constituent quark model. Since effects have been included in the radiative decays, the value of extracted in this way is an “effective” value of that includes corrections. From Eq. (7) and our values of we can compute the total width of the as a function of ; this is plotted in fig. 3.

The violation plays an important role in our analysis. Fig. 4 shows the absolute values of the relative contributions to of (dashed-dotted line), (dotted line) and the one-loop nonanalytic contribution to (solid line). The values have all been multiplied by 3/2, so that the dotted line is normalized to . Note that values of near the non-relativistic constituent quark model expectation of favor a small branching ratio, and hence smaller values of . In fig. 5 the value of that follows from neglecting violation (i.e. using Eq. (14) for ) is shown. Larger values of are favored when violation is neglected.

Nonanalytic dependence on similar to what we have found in radiative decay occurs in the mass difference. Including effects up to order [9]

where we have set and is an unknown constant. Experimentally, . The magnitude of the nonanalytic part is about 50% of the mass difference for . This gives us some confidence that the expansion is well behaved for at least some of the range of ’s in fig. 2.

The analysis in this paper allows us to predict the rate as a function of the branching ratio. However, for there is a strong cancellation between and , resulting in a very small width. (Note that is forbidden by isospin.) In this situation, violating terms of order may be very important.

Since heavy quark symmetry ensures that and are the same in the and systems (up to corrections of order ), the results of this paper can be used to predict the widths for radiative decay. Neglecting effects of order and , Eq. (10) becomes

where is given by Eq. (15). An analysis of the radiative decays of charmed baryons using the same methods is possible. Unfortunately, at the present time there is no experimental 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 Science Foundation Presidential Young Investigator award number PHY-8958081. MJS acknowledges the support of a Superconducting Supercollider National Fellowship from the Texas National Research Laboratory Commission under grant FCFY9219.

References

[1][email protected] CLEO Collaboration (F.\text@nobreakspaceButler, et al.), CLNS-92-1143 (July 1992) [2][email protected]\text@nobreakspaceL.\text@nobreakspaceRosner, in Particles and Fields 3, Proceedings of the Banff Summer Institute, Banff Canada 1988, A.\text@nobreakspaceN.\text@nobreakspaceKamal and F.\text@nobreakspaceC.\text@nobreakspaceKhanna, eds., World Scientific, Singapore (1989), p. 395; L.\text@nobreakspaceAngelos and G.\text@nobreakspaceP.\text@nobreakspaceLepage, Phys. Rev. D45 (1992) 3021 [3][email protected]\text@nobreakspaceWise, Phys. Rev. D45 (1992) 2188; G.\text@nobreakspaceBurdman and J.\text@nobreakspaceF.\text@nobreakspaceDonoghue, Phys. Lett. 280B (1992) 287; T.\text@nobreakspaceM.\text@nobreakspaceYan et al., Phys. Rev. D46 (1992) 1148 [4][email protected] ACCMOR Collaboration (S.\text@nobreakspaceBarlag et al.), Phys. Lett. 278B (1992) 480 [5][email protected]\text@nobreakspaceV.\text@nobreakspaceManohar and H.\text@nobreakspaceGeorgi, Nucl. Phys. B234 (1984) 189 [6][email protected]\text@nobreakspaceEichten and B.\text@nobreakspaceHill, Phys. Lett. 234B (1990) 511; H.\text@nobreakspaceGeorgi, Phys. Lett. 240B (1990) 447; A.\text@nobreakspaceF.\text@nobreakspaceFalk, B.\text@nobreakspaceGrinstein and M.\text@nobreakspaceLuke, Nucl. Phys. B357 (1991) 185 [7][email protected]\text@nobreakspaceIsgur and M.\text@nobreakspaceB.\text@nobreakspaceWise, Phys. Lett. 232B (1989) 113; Phys. Lett. 237B (1990) 527 [8][email protected]\text@nobreakspaceLuke, Phys. Lett. 252B (1990) 447 [9][email protected]\text@nobreakspaceL.\text@nobreakspaceGoity, CEBAF-TH-92-16 (1992) [10][email protected]\text@nobreakspaceCho and H.\text@nobreakspaceGeorgi, Harvard Preprint HUTP-02/A043

Figure Captions

[email protected] 1. Diagrams giving the leading non-analytic contributions to . [email protected] 2. The coupling constant as a function of including leading SU(3)-breaking effects. The shaded region indicates the uncertainty due to the variations in and . The arrows indicate the 90\lx@[email protected] confidence level limits on and the width. [email protected] 3. Width of the as a function of including leading SU(3)-breaking effects. The shaded region indicates the uncertainty due to the variations in and . The arrows indicate the 90\lx@[email protected] confidence level limits on and the width. [email protected] 4. Relative contributions to of (dashed-dotted line), (dotted line), and the one-loop nonanalytic term (solid line) to the matrix element for . [email protected] 5. The coupling constant as a function of ignoring violation. The shaded region indicates the uncertainty due to the variations in and . The arrows indicate the 90\lx@[email protected] confidence level limits on and the width.