Hyperons

212 N. C H R I S T AND T . D. L E E 4 -

served amplitude for K; - 2y; therefore, the resulting Phys. Rev. 188, 2033 (1969). fractional change in the lower bound (1) i s expected to be 'B. D. Hyams et al., Phys. Letters B, 521 (1969). of a similarly small magnitude. art from a misprint that gives the factor $ as 2, the

6 ~ e e the review talk by J. Steinberger in Topical Con- inequality (27) i s the s ame a s the inequality (4.1) in the ference on Weak Interactions, CERN, Geneva, Switzer- paper by Martin et al. (Ref. 5). Martin et al. also dis- land, 1969 (CERN, Geneva, 1969), p. 291. cussed modifications to this inequality due to other on-

?M. Banner, J. W. Cronin, J. K. Liu, and J. E. Pilcher, mass-shell intermediate states.

P H Y S I C A L R E V I E W D V O L U M E 4 , N U M B E R I 1 J U L Y 1 9 7 1

Weak Radiative Decay of the 2' and A Hyperons* Glennys R. F a r r a r

Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540 (Received 23 March 1971)

Unitarity i s found to give a reliable, model-independent lower bound on the branching ratio [rate(A-ny)/rate(A - all)] > 8.5X This value i s nearly a s la rge a s the experimentally determined branching ratio for the decay 2 ' 4 p y . Dispersion-theoretic techniques supple- mented with current algebra, PCAC, and pole models a r e then used to determine the r ea l as well as imaginary parts of the amplitudes for the radiative decays of the A and 2' hyperons. Input consists of the experimental nonleptonic decay amplitudes and pion-nucleon phase shifts. The theoretical predictions a r e very sensitive to the hyperon magnetic moments, and until these a r e better known, these results for the radiative decays can only be qualitatively com- pared with experiment.

I. INTRODUCTION peron decays, see the review art icle by ~ a n a k a . ' The principal new result to be presented here i s

The radiative hyperon decays, C+ - py, C0 - ny, in fact virtually model-independent. Unitarity can - - A-ny, Z -C y, z" - Coy, and Z" - A y, a r e inter - be used particularly effectively because the only esting because they can provide insight into the na- purely-hadronic intermediate state that i s energet- ture of the nonleptonic weak interactions. They a r e ically accessible is the N T ~ state. Knowing experi- presumed to result from the combined effect of mentally the photoproduction and A - N T ~ ampli- electromagnetic and weak interactions, so that tudes enables us to give the unitarity lower limitza: working to lowest order in the electromagnetic interaction one has in them a probe of the nonlep-

branching ratio (A - ny)/(n - al l ) 2 8.5 X .

tonic weak interaction. Given the scarci ty of ex- This i s quite a stunning result, being only a factor perimentally accessible nonleptonic processes, of 2 smaller than the experimental branching rat io each source of information on them is particularly for C+ - py given above. The corresponding uni- precious. At present only C+ - py has been seen, tarity lower limit for C+ - py turns out to be "un- with a branching ratio' naturally" small, a s we shall see, leading to

(C' - p y ) / ( ~ + - all) = (1.43 0.26) x lom3 . branching ratio (C+ - py)/(C+ - all) 2 6.9 x lo-' .

The experiment of Gershwin et al.' to determine the asymmetry parameter of the decay C+ - py has added impetus to theoretical efforts to account for these processes. The asymmetry parameter cr is determined from the correlation between the final proton momentum and the polarization of the initial C+ . I t is a measure of the relative magnitudes of the s- and p-wave amplitudes. Gershwin et a l . found a! to be -1.03$:;. Since most theoretical models predict that cr i s approximately zero, this measurement is very challenging to theorists . For a summary of the predictions which various techniques have given when applied to the radiative hy -

With the incentive of a possibly large ra te for 11 - ny, we proceed to make a model-dependent estimate of the rea l part of the amplitudes. For this we exploit our knowledge of the imaginary par t s by assuming the amplitudes A - ny and C - py obey unsubtracted dispersion relations in the m a s s squared of the initial particle. In the dispersion integral, however, we need the absorptive part of the amplitude a s a function of the initial hyperon mass . Approximating the full absorptive part a t al l energies by the contribution of the nucleon- pion intermediate state alone, even at masses for which other hadronic intermediate states a r e ener-

4 - W E A K R A D I A T I V E D E C A Y O F T H E 2' A N D A H Y P E R O N S 213

getically allowed, enables u s to write the imaginary part a s a product of photoproduction and nonleptonic decay amplitudes. The lat ter is only measurable a t the physical value of the hyperon mass, but we assume i t obeys a once-subtracted dispersion re la- tion, thereby defining i t s off-mass-shell behavior.

Both the radiative and nonleptonic hyperon decay amplitudes have poles at the nucleon mass squared, whose residues depend on matrix elements of the weak Hamiltonian. These a r e determined by using current algebra and experimental values of the nonleptonic amplitudes.

Even in the absence of inaccuracies in the procedure outlined above, the prediction of the theory for the rea l par t s is very uncertain due to the sensitivity of the poles of the radiative amplitudes to the hyperon magnetic moments. This is reflected in the large uncertainties quoted in Eq. (22). Never- theless i t can be seen that the theory gives order of magnitude agreement with the C - py branching rat io (tending to underestimate it) and i s capable of pro- ducing large, positive or negative, values of the asymmetry parameter . The A - ny predictions a r e l e s s sensitive to the magnetic moments and indicate a branching rat io of about 2 x lom3 and a positive asymmetry parameter .

The unitarity calculation i s in Sec. If; the dispersion relations a r e in Sec. 111; the determination of the matrix elements of the weak Hamiltonian i s in Sec. IV; the poles of the radiative amplitudes a r e in Sec. V; and the numerical evaluation and con- clusions a r e in Sec. VI. Throughout we use the Bjorken-Drell metric ( a . b= a,&, -8 - 5 ) and y- matrix conventions.

11. UNITARITY CALCULATION

Unitarity determines the imaginary par t of the decay amplitude in t e rms of a sum over physical, experimentally measurable, amplitudes for inter - mediate states. In the case a t hand one may r e - place the sum over intermediate states by the contribution of the Nn intermediate state only. This should be an excellent approximation since al l other s tates energetically allowed involved leptons o r photons with their much smal ler amplitudes. Thus we have the imaginary part of the unknown amplitude in t e rms of known photoproduction and hyperon nonleptonic decay amplitudes, enabling u s to obtain the following expression:

Here n r e f e r s to the N n state, p ' i s the momentum

of the intermediate nucleon, q is the momentum of the intermediate pion, and P i s the initial hyperon (Y) momentum. The most general gauge-invariant expression for the radiative decay amplitude for the process in which a hyperon of momentum P decays into a nucleon of momentum p and a photon of momentum k [Y(P)- N(p) + y(k)] i s

where c i s the photon polarization: c2 = - 1, c . k = 0. It will prove convenient to define the Pauli-space version of Eq. (2):

Throughout, M with no subscripts i s the mass of the initial hyperon, m is the nucleon mass, p i s the pion mass, and A o r C r e f e r s either to the pa r - ticular hyperon o r i t s mass . In t e r m s of these amplitudes, the ra te for radiative decay i s

wx- 1 M 2 - m 2 mkO ( I A ~ I Z + 14 12) = 7 (T) ( 1all2 +la21z), Mll

(4) and the asymmetry parameter , defined by

where 2 i s the polarization of the hyperon and 6 i s the direction of the final nucleon, is

For the amplitudes A,, andAd, we take the stan- dard definitions,

= i X ; ( ~ , + B> .Q)x,,

A,,,,, = X,$[i~l(6 . Z ) + ~ , ( ? r - i j ) Z * (&x 2) (5)

Equation (1) i s then

I ~ A , = -14 I ( M : ~ * B ; / ~ + &f3l2* I- B 2 ' ) '!2

where MI- and Eo+ a r e the photoproduction multipoles given in Berends, Donnachie, and Weaver3 (BDW), the superscripts re fer to the Nn isospin state, and 141 i s the intermediate pion momentum: 141 = 186.7 M ~ V / C for C decay and 141 = 102.9 M ~ V / C

for A decay. The values of these multipoles a t a center-of-mass energy equal to the C or A mass a r e given in Table I . The necessary nonleptonic decay amplitudes a r e given in Table 11.

Numerically evaluating Eq . (6) we find

214 G L E N N Y S R . F A R R A R

TABLE I. Photoproduction multipole amplitudes taken from Berends, Donnachie, and Weaver (Ref. 3) . The numbers a r e to be multiplied by and have the units M~v- ' . Their signs a r e determined by the convention selected in cal- culating the photoproduction Born terms.

Er Na final state (MeV) Multipole 1=1 /2 I =3/2 n.+ p a o

194 EJ? +17.057 not needed 1.273 -19.990 194 MV +2.465 not needed -1.183 -3.685

.-

ImA,(C) = -32.1 ( M ~ V set)-'/',

ImA,(C) = -17.7 ( ~ e ~ s e c ) - " ~ , (7)

ImA,(A) = +14.8 (MeV set)-lh,

ImA,(A) = +279.8 (MeV sec)-l/' . The contributions of these imaginary par t s to the branching rat ios give the lower limits quoted in Sec. I.5

This unitarity lower limit on the A - n y branching rat io i s itself substantial, in contrast to the C+ - py case. Examining Tables I and Il shows that the C - py imaginary part would be much la rger if i t were not for the fact that the large photoproduction amplitude is multiplied by a very small nonleptonic decay amplitude. Thus while we cannot make similar unitarity calculations for the imaginary par t s of the other radiative hyperon decays, we might expect the A - n y imaginary part t o be typical of their order of magnitude.

As s t ressed earl ier , this determination of the A - n y lower limit, which i s virtually model-independent, should be reliable. The fact that the low- e r limit i s s o large is encouraging. It leads us, in the following sections, to use models to estimate the magnitude of the rea l par t s of the amplitudes.

variant amplitude a, o r a,, we need to assume i t is an analytic function of the hyperon mass squared. However, a function has a unique analytic continuation only when i t i s defined on a dense se t of points and the amplitude for a decay is only specified by nature a t one point: the physical mass . Taken seriously, this analysis prevents one from using dispersion techniques for a decay amplitude a t al l . What we do, ra ther than give up the method entirely, is make an ansatz regarding the "best" choice of analytic continuation, guided by experi- ence in scattering problems. Thus our choice of analytic continuation, for both the radiative and nonleptonic amplitudes, is defined by the dispersion relation we write for them. The validi,ty of the particular continuation is justifiable only by i t s "reasonableness" and the success of the final predictions.

One expects the invariant amplitudes a, and a, to be analytic functions of the hyperon m a s s squared with a pole a t the nucleon m a s s squared and a cut beginning a t the Nn threshold. Assuming that it obeys an unsubtracted dispersion relation gives

Ill. DISPERSION RELATION In this expression, R,,, is the residue of a, or a, In order to write a dispersion relation for the in- a t the pole and the threshold has been taken to be

TABLE 11. Nonleptonic decay amplitudes taken from a review by Filthuth (Ref. 4). All the numbers a r e to be multiplied by l o 3 and the units a r e (MeV set)-lf2. Our definition of B2 has the opposite sign from the corresponding amplitude used by Filthuth, and our convention on the phases of the C and a isotriplets lead to 2; amplitudes of the opposite sign. The A I = 8 rule was used to determine A! from AL.

b l b2 Bi B2

4 - W E A K R A D I A T I V E D E C A Y O F T H E C + AND A H Y P E R O N S 215

a t the Nil intermediate state because we have a l - ready agreed to neglect nonhadronic intermediate states. Neglecting al l but the Nn state in unitarity even a t values of s' where other hadronic states a r e accessible, we see that Rea,,,(M2) depends on the nonleptonic decay amplitude at nonphysical values of the hyperon m a s s squared, s'.

In order to determine B: for any sf, we use dispersion techniques on b{ , the corresponding invariant amplitude. To the extent that the nN intermediate state saturates the unitarity relation for C,A - NTI, b{ (s) has the same phase a s the co r r e - sponding ilN partial-wave amplitude (Fermi-Wat- son theorem). This knowledge of the phase of b{(s ) for s greater than (m + y)', along with i t s ana- lyticity properties and i t s value a t s = M', is suffi- cient for determining b:(s) up to a polynomial in s . We can either guess the solution and verify the r e - quired propert ies o r construct i t by the N/D o r

I -

Omnes-Muskhelishvili method. Here we do the former .

For both the Z+ - NT and the A - Nil amplitudes the only pole in s (the variable hyperon mass squared) is a t the nucleon mass squared. Both nonleptonic decay amplitudes have a right-hand cut only, beginning a t (m + P ) ~ . Now consider an amplitude for a final s tate of definite orbital angular momentum and isospin, e.g., b;h, the amplitude for an s-wave, I = $ final state. Along the cut below the inelastic threshold i t has the phase of s-wave, I =;, nN scattering: 6,,,. (The subscript S r e f e r s to s wave, the 3 to isospin 2, and the 1 to total angular momentum $.) We will assume in fact that even above the inelastic threshold it continues to have the phase of elast ic nN scattering. Then the nonleptonic decay amplitude b:h a s a function of the variable hyperon mass squared, s, must be

with r:/' the residue of the nucleon pole in b:/'(s). P ( s ) i s a polynomial in s which i s zero a t s = M'.

It is simple to verify that Eq. (9) has the r e - quired behavior. In order not to introduce addi- tional parameters we make the assumption that P ( s ) i s zero. This is equivalent to the assumption that ~ ( s ) needs only one subtraction and the inte- gra ls over the phase shifts converge.

In order to evaluate Eq. (9) one must know the residues of the poles in the nonleptonic decay amplitude, shown in Fig. 1. These depend on the ma- t r i x elements of the weak Hamiltonian between hyperon and nucleon states:

(n I H, I A' ) = -un(i, + X2y,)uA . and (10)

Treating the pole t e r m s a s Feynman diagrams gives a definite result for the diagrams of Fig. 1. However, the Feynman diagrams make sense only for physical external part icles. In particular, when continuing in the external m a s s squared, it is unclear how the residue of the pole should be ex- tracted from, e .g., the expression

This is in contrast to the usual case in a scat ter- ing process where the external part icles can have

their physical m a s s and yet the total momentum squared can be varied.

In this case, the residue of the pole i s

evaluated a t P2 = m2. However, we a r e unable to evaluate i t at P2= m 2 since P2dA(P) is only defined when PZ= M'. If one knew that this Feynman diagram behaved purely a s a pole a s a function of P2, the residue could be evaluated anywhere (in particular a t P2= M), giving for the residue the constant

Unfortunately, i t is impossible to verify that the Feynman diagram does behave a s a pole only, since that would require knowing the meaning of PU,(P) off the mass shell. Put differently, if one regards igh,(M - m) as the value a t P2= M2 of the function which a t P2 = m 2 is the residue, how does one know whether M is a constant or @, the lat- t e r leading to a vanishing residue.

FIG. 1. Pole contributions to the nonleptonic decay amplitudes.

216 G L E N N Y S R . F A R R A R

The prescription we give, corresponding to tak- Y ~ / ~ ( A ~ ) = - ~ ~ A ~ ( A + m ) , ing the Feynman diagram to be purely a pole in P2, i s to t rea t the P2 in the denominator a s the Y : / ~ ( C + ) = ~ ~ ~ , ( C - m ) , variable and take ii,(p1)pu,(P) t o be fixed a t i t s r :/YE+) = - G g a l ( ~ + m ) , value for physical external particles. With this prescription we obtain the residues Y y 2 ( ~ + ) = 0 ,

r : ' " A 0 ) = 6 g ~ , ( ~ - m ) , ( l l a ) r ;/Y(C+) = o .

IV. MATRIX ELEMENTS OF THE WEAK HAMILTONIAN

Both the pole t e rms in the nonleptonic amplitudes (Fig. 1) and those in the radiative amplitudes (Fig. 2) depend on the matr ix elements of the weak Hamiltonian. These a r e parametrized by quantities called h and a in Eq. (10). One possibility for determining the h's and a's i s to fit the experimental nonleptonic amplitudes by the pole t e rms only. The corresponding choice for the off-mass-shell behavior would be, instead of Eq. (9),

However, one can obtain a better fit to the experimental nonleptonic amplitudes by using a current-algebra approach f i r s t employed by Suzuki and Sugawara and then extended by others to cover the p-wave ampli- t u d e ~ . ~ I describe here the procedure for the second approach. Consider the amplitude of the nonleptonic hyperon decay: Hyperon (Y) of momentum P decays into a nucleon (N) of momentum p' and a pion with third component of isospin i (a" and momentum q. The LSZ (~ehmann-Symanzik-Zimmermann) reduction technique, upon integrating by par t s and dropping the surface term, gives

[~hroughout , normalization factors of ( 2 ~ r ) - ~ / ~ , 1/20, etc . a r e suppressed, since they will be common to - all t e rms in the final result.] Using the partial conservation of axial-vector current (PCAC) choice for the pion interpolating field and again dropping an integration-by-parts surface term, this becomes, in the limit q - 0,

The superscript B means the Born part of the amplitude, Q: i s the charge of the ith component of the axial-vector current, f, i s the pion decay constant (96 MeV), and ~:+,tfi i s the pole contribution to the amplitude for a hyperon to go to a nucleon and an axial-vector current whose SU(3) index i s i . Each te rm in the last bracket i s undefined a s q- 0; however, their sum i s well defined in this limit. In arr iving a t Eq. (13) i t i s assumed that the variation of the amplitude between q = O and q physical i s entirely due to the pole. This i s the PCAC assumption.

Assuming that H, t ransforms like a KO under isospin and that the baryon-axial-vector-current-baryon coupling is SU(3)-invariant, the Born t e rms contributing to Eq. (13) can be evaluated. This gives, e.g., for the C+,

with f / ( f+ d) determined from f i t s to hyperon 0 decay7 to be +0.36. In order to determine the commutation relations of the weak Hamiltonian with the axial-vector charges,

we make use of the SU(3)BSU(3) current algebra proposed by Gell-Mann and a model of H,. Two popular models of H , a r e

W E A K R A D I A T I V E D E C A Y O F T H E C t A N D A H Y P E R O N S 217

f j k . H, = J X J ~ and Hw = d,,,JxJ

J, is the sum of vector and axial-vector hadronic currents , which a r e the Cabibbo currents in the f i r s t case , and the SU(3)-octet current with index i or j in the second case. It is easily seen that either of the above Hamiltonians gives

( n l [ & ~ - ' ~ ~ ~ ] ~ ~ ' ) = o , ( nl[ &g,HW1 1, ) = $iln(h1 + h 2 y 5 ) u ~ ,

( PI[Q:,HWI I C+ ) =%u,(g, + a 2 ~ 5 ) u ~ . These results may be substituted into Eq. (14) to obtain the final equations:

Experimentally the b's have been determined. They a r e tabulated in Table II. Observe that f + d C - m b,(C:) i s virtually zero with small e r ro r s , allow- ing u s to require

n C + m 0 2 -- - - A , & A + m ' ( 1 8 ~ )

Analytically determining the best fit to the five r e - b L f + d [ C - ] A - m ' ( lad) maining equations gives

This fit gives back values for the b's listed in Table 111, and the over-all goodness of the fit is not very sensitive to the choice of X, and a2.

As an indication of the sensitivity of the param- e t e r s h and a to the model used for the nonleptonic decays, one can compare these values with the r e - sul ts of computing them by a pole-only model of the nonleptonic decay amplitudes. Such a model gives, instead of Eq. (16),

FIG. 2. Pole contributions to the radiative decay amplitudes [(a) and (c)l and contributions to the subtraction constant i n the dispersion relations [(b) and (d)].

2 18 G L E N N Y S R . F A R R A R

It can immediately be seen that these equations a r e rather different, expecially in that there i s no coupling between the parity -conserving (XI and a,) and parity -violating (A, and a,) quantities. Using the same input nonleptonic decay amplitudes (from Table 11), we obtain the alternate solutions for ma- t r ix elements of the weak Hamiltonian which we distinguish by primes:

As with the other method of determining these parameters, we find they a r e al l of the same o r - de r of magnitude. This i s contrary to the perfect S U ( 3 ) result that A, and a, a r e identically zero, shaking our confidence in the applicability of SU(3) to H, in any guise. However, with this purely pole model we see that h; has the opposite sign a s A,. This difference leads to a substantial difference in the results for the pole contributions to the radiative decay amplitudes.

The values of the parameters given in Eq. (19) can be substituted into Eqs. (18) in order to see how well o r poorly this pole model fits the hyperon decay data. The results a r e shown in the lower half of Table III.

V. POLE TERMS

The variable in the dispersion relation i s the mass squared of the initial particle so that in the processes A - n y and C + - py pole contributions to Eq. (8) a r i s e from the diagrams shown in Figs. 2(a) and 2(c). The Feynman rules give, e.g., for

TABLE 111. The upper lines contain values of the nonleptonic decay amplitudes corresponding to our best fit for the weak-vertex parameters, given in Eq. (17), and the lower lines, the nonleptonic amplitudes resulting from the alternative pole-model fit given in Eq. (19). All numbers a r e to be multiplied by lo3 . The units a r e (MeV ~ e c ) - ~ / ~ .

~8 r : x i

Fig. 2(a),

where e i s the charge on the proton (e2/4n= &, e > 0), and y,, y ,, y,, and y, a r e the anomalous magnetic moments of proton, neutron, A, and C+ . The presence of the y, t e rms in Eq. (20) keeps

M:,,~ from being gauge invariant; however, the dispersion relation i s gauge invariant so the non- invariant part introduced here must be canceled by a subtraction.

Drawing a cue from Feynman theory, we see that an obvious graph contributing to the subtraction i s the one shown in Fig. 2(b). Because in quantum electrodynamics gauge invariance holds order-by-order, the sum of Figs. 2(a) and 2(b) is gauge-invariant a t the physical mass of the C . If one were interested in the amplitude C+ - p y for unphysical C mass, one would have to use more ingenuity to make i t gauge-invariant for al l values of the external mass squared. Other sources of contributions to the subtraction constant a r e shown in Fig. 3 . These a r e not included for lack of information, and hopefully they a r e unimportant.

In the case of the A pole term, Fig. 2(c), the problem of gauge invariance does not a r i s e . While the contribution of Fig. 2(d) i s not needed for gauge invariance, i t i s still included in the spir i t of approximating the subtraction by "nearby" sin- gularities in other variables.

The sum of the diagrams in Fig. 2, evaluated for physical external mass, a r e the "pole" contributions to radiative hyperon decays, with some of the more important contributions t o the subtraction constant included:

FIG. 3. Contributions to the subtraction constants for the radiative decay amplitudes which a r e not included.

4 - W E A K R A D I A T I V E D E C A Y O F T H E 2' A N D A H Y P E R O N S 219

One can immediately see from Eq. (21) how sensitive these Born contributions a r e to the values of the anomalous magnetic moments of the hyperons. Since p, and p, a r e only poorly known, and pAr (the "transition moment", i.e., the coupling of A-y- c* ) is not measured a t all, the prediction of this model for the radiative amplitudes will be very uncertain until better information i s available for the magnetic moments.

VI. NUMERICAL RESULTS AND SUMM4RY

Using the values of the matrix elements of the weak Hamiltonian given in Eq. (17), the dispersion integral of Eq. (8) may be numerically evaluated using a s well Eqs. (6) and (9). Various approxima- tions a r e necessary in order to use Eq. (9). F i r s t of all, we do not know the elastic phase shifts for s' above 4.7 GeV2, so we cannot integrate to -.* Even if we knew them, the neglect of inelasticity would be a poor approximation a t high values of s ' . Explicitly varying the cutoff shows that the integrals in (9) a r e insensitive to i t . Also, analytically integrating some function which equals the phase shift a t s ' = 4.7 Gev2 and goes smoothly to ze ro indicates that if the integrals converge then they a r e fairly well approximated by the contribution of the region from st to s f = 4.7 GeVZ.

The value of the dispersion contribution to the radiative decay amplitudes thus obtained a r e given in Table IV, column 1. Since the procedure for determining the residues (r:) of the nucleon poles in Y- Nn i s ambiguous (see Sec. III), we give two a l - ternative determinations of bi(s) and their cor re- sponding dispersion contribution to Rea,,, for com- parison.

One of the alternatives considered i s to se t the r : to zero in Eq. (9). This i s essentially the method of Papaioannou5 and Contogouris and Wong,' who never even consider the pole t e rms . One could argue that in the absence of knowledge of the residues, this i s the best procedure, especially if the poles a r e somewhat distant from the cut, since

TABLE IV. Dispersion integral contributions t o the rea l part of the radiative decay amplitudes, in units of ( M ~ V see)-'/'. Various results correspond to the s dependence of the nonleptonic decay amplitude determined using calculated residues and a subtraction [column (I)] , ignoring the pole [column (2)], and assuming no subtraction [column (3)].

(1) (2) (3)

R~A,(AO) +124.5 -98.9 -54.5 ~e A, (A0) +382.8 +181.6 +88.2 ReA,(Bf) +21.7 +17.8 +6.7 Re A2(BC) +175.8 -4.8 +4 .O

i t corresponds to assuming constant off-mass- shell behavior modified by the final-state interaction. When this procedure for obtaining b , ( s ) i s used, the values of the dispersion contribution to ReA,,, a r e those given in Table IV, column 2.

The other possibility in the absence of knowledge of the residue i s t o assume that no subtraction is needed [i.e., b { ( m ) = 01 and to use the physical value of the amplitude to determine the residue at the pole, i.e., use Eq. (12). The resultant values of ReA,,, a r e shown in Table IV, column 3. Since we cut off the integral over Ima, a t a fairly low value of s ( s = 1.82 GeV2), the difference in Rea, due to the difference between choices (2) and (3) for the hi's i s not great . It i s unfortunate that choice (1) gives such a different result from (2) o r (3). Since method (2) and ( 3 ) a r e somehow more "con- servative," their average will be used in the final answer.

As pointed out in Sec. V, the Born contributions to the radiative decay amplitudes a r e very sensitive to the values of the hyperon anomalous magnetic moments. Rather than use the S ~ ~ ( 3 ) predictions for these moments, experimental values, when available, a r e used and corresponding e r - r o r s a r e quoted. SU(3) predictions a r e ambiguous since mass-splitting effects a r e important, e.g., in the SU(3) limit these a r e equivalent:

P A E = - $ F ~ ,

whereas in the r ea l world they a r e quite different. Since no experimental data i s available on pi\=, the average of the above two procedures i s used and the e r r o r i s assigned on the basis of the qual- ity of the corresponding SU(3) predictions for p, and p ,. Using the following values for the magnetic moments:

-- epp - + 1 . 7 9 ~ ( 1 . 6 1 ~ MeV-'), 2m

=(1.57* 0 . 5 2 ) ~ (1.61 x MeV-'), 2 z

one obtains for the Born contributions to the r a - diative amplitudes

220 G L E N N Y S R . F A R R A R - 4

A:@+) = -247* 583 (MeV set)-lh,

At@+) = -131 i 20 ( M ~ V ~ e c ) - ' ' ~ . The uncertainties reflect the uncertainties given above for the magnetic moments.

Using for the dispersion contribution the average of Table IV columns 2 and 3, and the imaginary par t s given in Sec. 11, one finally finds

branching rat io (Z -py) - ( 3 . 4 i 12.5) x + 0.2

a(Z-py)=+0.8-1.1 , branching rat io (A - ny) = (1.9 i 0.8) X lo-=,

a (A - ny) = +0.5 i 0.4.

The uncertainties shown should not be understood in the same sense a s statistical e r ro r s , but a r e meant to give an indication of the sensitivity of the physical quantities to the magnetic moments. The measurements1 give

branching rat io (C -py) = (1 .43i 0.26) X

a ( Z -&)= -1.03::::;.

Thus this result is a factor of 3 too small on the Z rate, a problem common to most models (un- l e s s the magnetic moments a r e chosen most fa- vorably), and the asymmetry parameter i s so sensitive to the magnetic moments that there i s no

*Research sponsored by the Air Force Office of Scien- tific Research under Contract No. A F 49 (638)-1545 and an NSF predoctoral fellowship.

'L. K. Gershwin, M. Alston-Garnjost, R. 0. Banger- t e r , A. Barbaro-Galtieri, T. S. Mast, F. T. Solmitz, and R. D. Tripp, Phys. Rev. 3, 2077 (1969).

'K. Tanaka, Ohio State University report , 1970 (un- published). In particular, see R. E. Behrends, Phys. Rev. 111, 1691 (1958), who f i rs t discussed the structure of the radiative decay amplitudes, and G. Calcucci and G. Furlan, Nuovo Cimento 2, 679 (1961), J. C. Pat i , Phys. Rev. 130, 2097 (1963), and R. H. Graham and S. Palwasa, i hi d. 140, B1144 (1965), who did early pole-model calculations like those of Sec. V. 2 a ~ o t e added in proof: It has been brought to my atten- tion that V. I. Zakharov and A. B. Kaidalov, Yad. Fiz. 5, 369 (1967) [Sov. J. Nucl. Phys. 3 259 (1967)l , using - similar methods, estimated the unitarity lower limit on the A radiative decay rate.

3 ~ . A. Berends, A. Donnachie, and D. L. Weaver, Nucl. Phys. B4, 1 (1967). There a r e three consecutive art icles by these authors dealing with various aspects of the subject.

4 ~ . Filthuth, in Proceedings of the Topical Conference on Weak Interactions, CERN, 1969 (CERN, Geneva, 1969).

5 ~ . M. Papaioannou, Phys. Rev. 178, 2169 (1969), does for Z -py the unitarity calculation described here , also intending to c a r r y out the dispersion calculation. He claims to use the same photoproduction and nonlep-

definite prediction. Nonetheless, a large negative result for a, i s certainly within range of this theory . This i s in contrast t o previous work,'' which has a, = 0 by virtue of approximately incorporating SU(3) which gives 0, and X2=0. The parameters in A - ny decay a r e l e s s sensitive to uncertainties in the magnetic moments.

The importance of this work is l e s s i t s detailed predictions than the following qualitative results:

(i) The unitarity contribution, and thereby the dispersion contribution, can be substantial and must not be casually neglected.

(ii) A theory not incorporating S11(3), instead obtaining matrix elements of the weak Hamiltonian from hyperon nonleptonic decay data, can give large values of a, for C - py decay.

(iii) There i s a firm lower bound on the branching rat io for A - ny:

branching rat io (A - n y ) / ( ~ - all) 2 8.5 X lo-*

and in fact the branching rat io may be several t imes this large. The asymmetry parameter of this decay i s probably nonzero and positive.

ACKNOWLEDGMENT

Professor S. B. Treiman interested me in this subject. I am indebted to him for that and for many helpful discussions.

tonic decay amplitudes a s used here ; however, his tables of amplitudes do not bear this out, and he gets a much larger unitarity contribution to Z -py than we do. %I. Suzuki, Phys. Rev. Letters l5, 986 (1965); H. Su-

gawara, i bi d . 15, 997 (1965) ; L. S. Brown and C. M. Sommerfield, i b i d. 16, 751 (1966); Y. Hara, Y. Nambu, and J. Schechter, i b i d . 16, 380 (1966); S. Obuko, Ann. Phys. (N.Y.) 41, 361 (1968); C. Itzykson and M. Jacob, Nuovo Cimento 9, 655 (196'7); A. Kumar and J. Pat i , Phys. Rev. Letters 5, 1230 (1967).

IN. Brene, R. Veje, M. Roos, and C. Cronstr'dm, Phys. Rev. 149, 1288 (1966).

'phase shifts a r e taken from the compilation of nN partial-wave amplitudes of the Particle Data Group at Berkeley [LRL Report No. UCRL-20030 nN (unyub- lished)] . We use the "CERN theoretical" fit to the Sll, S,,, Pi,, andP,, phase shifts f rom threshold (1.158 G~v ' ) to 4.7 G ~ v ' . This fit i s due to A. Donnachie, R. G. Kirsopp, and C. Lovelace, Phys. Letters K B , 161 (1968), who smooth the experimental f i t by requiring it to have the energy dependence specified by a dispersion relation.

9 ~ . P . Contogouris and How-sen Wong, Nucl. Phys. 40, 34 (1963). T o ~ h e work of M. A. Ahmed, Nuovo Cimento 5&, 728 (1968), giving a(Z-py) =-0.6 has been shown to be in- correct by L. R. Ram Mohan, Phys. Rev. D 3, 785 (1971); L. Heiko and J. Pestieau, Lett. Nuovo Cimento 1, 347 (1971), and B. Holstein, Nuovo Cimento (to be published).

Documents

Hyperons