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29 February 1996

Physics Letters B 369 (1996) 337-340

PHYSICS LETTERS 6

The radiative charmed baryon decay @$ + gFlr

Ming Lu , Martin J. Savage 2, James Walden 3 Department of Physics. Carnegie Mellon University, Pittsbutgh, PA 15213. USA

Received I 1 November 1995; revised manuscript received I9 December I996 Editor: H. Georgi

Abstract

U-spin symmetry (S +-+ n symmetry) forbids the radiative decay a$ -+ al!,r in the SU(3) limit. The quark mass term breaks U-spin symmetry and the leading nonanalytic contribution to the radiative decay amplitude is computable in heavy baryon chiral perturbation theory. The radiative decay branching ratio is determined by the coupling constant g2 and at leading order in chiral perturbation theory is given by Br( E$ --+ $!,-y) = 1 .O x IO- g2. Measurement of this branching fraction will determine Ig21.

Recently CLEO reported the discovery of E$

(J = ;+, sextet) with a mass m(BzT) = 2643 f 2 MeV [ I]. The dominant decay mode of $J is to

$1 (J = f, anti-triplet) and a pion. Radiative de- cay zo* -c2 --t eg,, is forbidden in the SU(3) limit by U-spin symmetry (a symmetry of strong and electro- magnetic interactions under the interchange of strange and down quarks), an SU(2) subgroup of flavor SU( 3). We point out that the leading contribution to the radiative decay amplitude of ZFT is nonanalytic in

quark masses (N 0( HI:*) ), finite and computable in heavy baryon chiral perturbation theory. This leading order calculation gives a radiative decay branching ratio Br( @z --+ @,r ) = 1.0 x 10m3g$ where g2 is the 6(*)6(*)71. coupling in the heavy baryon chiral Lagrangian. Using the value of g2 in the large N limit of QCD (where N is the number of colors) yields a branching ratio of N f%. (The nonrelativistic quark

Email address: lu@fermi.phys.cmu.edu. ? Email address: savage@thepub.phys.cmu.edu.

Email address: walden@fermi.phys.cmu.edu.

Elsevier Science B.V.

PII SO370-2693(96)00032-9

model gives a branching ratio which is about an order of magnitude smaller than what one gets in chiral perturbation theory.) Although it may be difficult to observe a branching fraction of 1 O-* at CLEO [ 2 1, the E78 1 experiment at Fermilab may be able to reach the lo- level [ 31. This would be sufficient to deter- mine IgzJ to an accuracy of N 30%. Previous work on radiative charmed baryon decays found that the decay amplitude of a:z -+ $,r is small [4]. However, measurement of this branching ratio will be one of the only ways to determine ]g2]. Since g2 enters in many heavy baryon loop calculations, its determina- tion is vital if we wish to go beyond tree level in the heavy baryon sector. Furthermore, it is important to test the large N and quark model predictions for axial coupling constants such as gz.

Physics of hadrons containing a single heavy quark simplifies significantly in the limit where the mass of the heavy quark becomes infinitely greater than the scale of strong interactions [5]. In the heavy quark limit, heavy hadrons can be classified according to the spins of the light degrees of freedom, st. The lowest

338 M. Lu et al./Physics Letters B 369 (1996) 337-340

lying charm baryons contain the $1 = O(J = f)

states which transform as 3 under flavor SU(3)

(A:,@, and E,!j), and the se = l(P = f,;) states which transform as 6 under flavor SU(3) (xt+* x+(*),Cc(*) =;+(*) +(*) and @*) [6].

It is convement ti intzucd 7;: superfillds K(u) (which transforms as 3) and @(u) (which trans- forms as 6), where o (the velocity of charmed baryons) is conserved in the heavy quark limit. The two superfields are

+d = -&, + v,)Y52 Bij + !+y. (1)

Here

B, = a;, , B2 = -Z:,i , BJ = A:, (2)

while

B13 = -+A t B23 = +F2, B33 = Sz;, (3)

with the corresponding B,i* fields for spin-; partner of6.

Interactions of heavy hadrons with soft pions and photons (of energy < A, N 1 GeV) can be de- scribed by using both heavy quark symmetry and chi- ral symmetry. At lowest order in the heavy quark ex- pansion and chiral expansion the heavy baryon chiral Lagrangian is given by [6]

+ pio. 07;: - s;iu . D$ + ATiTi

+ ig2ep,,,&vY ( AA)$,jk

+ g3 (E@( Ap)$; + h. C. ) , (4)

where A is the mass difference between the 6 and 5 states, and D, is the chiral covariant derivative. The vector and axial vector chiral fields V, and A, are formed from the pseudo-Goldstone boson fields

VP =

A, = ; (!t+a/k - &!$+) 9 (5)

and t2 = X = exp (2iM/f), where M is the meson octet

M=

(

$r + 57) lrTT+ K+

a- -&To+&7 @

)

,

K- P -;?j J

(6)

and f N 132 MeV is the pion decay constant. The axial coupling constants g2, g3 are unknown

parameters in the effective theory and must be deter- mined experimentally. It has been shown that in the

large N limit they are given by gs = $ $&I?

g2 =

- $gA [ 7,8], where gA is the nucleon axial coupling. (Experimentally gA = 1.25.) One expects that the deviations from these N = oc relations occur at the 1 /N level, i.e., N 30% [ 81. (The nonrelativistic quark model gives g3 = &, g2 = -2.) The total hadronic decay width of a$ (+ ZFi7r, a: rr-) is given at tree level by

(7)

The CLEO result TO < 5.5 MeV [I] gives an upper bound ]gs] < 1.4. At present there is no experimental information on g2. It is important to determine g2, not only because g2 enters in many heavy baryon loop computations, but also because this will indicate how well the large N and quark model predictions work in the heavy baryon sector.

We now turn to the radiative decays of the Jp = g baryons in the 6 of SU( 3). The leading contribution to 6 -+ 3~ is the magnetic dipole (Ml ) radiation with the electric dipole (E2) component suppressed by a factor of l/m, [ 91. In chiral Lagrangian the Ml transition arises from a dimension 5 operator

(8)

M. Lu et al./Physics Letters B 369 (1996) 337-340 339

no* +(*) EC2 cc , ELF ;;O - cl

Fig I. Feynman diagram contributing to SF: + Ey,y at leading order in heavy baryon chiml perturbation theory.

Q is the charge matrix for the light quarks

(10)

nnd CI is some unknown constant of order 0( 1) by dimensional analysis. This term gives the leading con- tribution to the radiative decays Xz(*) + A$J and -if*) (2 - E.,ir. However, it doesnt contribute to =I)1 * 1 t-2 -f a:, y because of the U-spin symmetry. The U-spin symmetry is broken by quark masses (md + /jr,). and the leading contribution to $!i* -+ a:, y arises from the 1 -loop graph (Fig. 1) by keeping the masses of K and r in the loop. This gives rise to an

amplitude of order 0( m:). An explicit calculation gives the partial width for a$ --+ $y4

1 2

-.I(& Ey)) , (11)

where

./fin. E) = dxJm2 - x~I?? - ie

-I xE X r-2tan

Jnr2 - x2@ - k > (12)

The same calculation gives the width of Eh. (Fionic transitions

from E,z to E, 1 are kinematically forbidden.)

and E, is energy of the photon. (In the limit A - 0, J(m*) -+ rrm.) From Eqs. (11) and (12), we arrive at the branching ratio for 2:; + $!,r

4 -q 1 = --___ 37r l&-J (47rf)2

- (J(m;,.E,)

1 2

-J(m;, E,))

= 1.0 x lo-g;. (13)

Inserting the large N value for g2 yields a radiative branching ratio of N 5 %. This is probably within the reach of E781, and may also be seen at CLEO with some luck.

The leading counter terms which contribute to =o* -c2 + Ez,r have the form

L = b, f$Q/({my[ + ~+m,~+)~,y~yYsS~ Fp,,~,;I x

where my is the light quark mass matrix

and bk, b2 are unknown constants of order O( 1). These contributions are suppressed by a factor of or-

der 0( m,i*) relative to the leading nonanalytic piece. There are also U( l/m,) SU( 3) breaking corrections to the above result. From naive dimensional analy- sis we expect that these higher order contributions are down by a factor of O(mK/m,, mK/Ax) and therefore we expect our result to hold with an approximately 30% uncertainty.

The electric quadrupole (E2) contribution to zo* -0 -c2 -+ oc,y violates both the heavy quark spin sym-

5 The nonrelativistic quark model gives a radiative branching

ratio which is about an order of magnitude smaller. With typical

values for strange and down quark masses in the nonrelativistic

quark model, one gets a partial width for radiative decay of -

2.0 x lop3 MeV, and a branching ratio of - 4 x IO- if the total

width of E;r, is taken to be 5 MeV.

340 M. Lu et al./Physics Letters B 369 (1996) 337-340

metry and flavor SU(3). (The E2 transition to decay Cy -+ h,y was considered in Ref. [9] .) Formally, the leading contribution to the E2 amplitude comes from the same graph (Fig. 1) as the MI amplitude but with the 6* - 6 mass difference explicitly retained. A straightforward calculation gives the ratio of the E2 amplitude to the Ml amplitude to be approximately I %. (The mass splitting between the 6* and 6 states is taken to be N 64MeV [lo].) This is too small to be seen in the near future.

To conclude, we have studied the SU( 3) breaking charm baryon decay Zzl --) $!,r at leading order in chiral perturbation theory and found a branching ra- tio of 1.0 x 10m3gi, where g2 is the 6(*)6(*)7r ax- ial coupling constant in the heavy baryon chiral La- grangian. We estimate that the theoretical uncertainty of our result is approximately 30%. Measurement of this branching fraction may prove to be the best way to determine IgIl.

We thank J. Yelton and J. Russ for useful discus- sions. This work was supported in part by DOE under contract DE-FG02-91ER40682. MJS