Comparison of theoretical and experimental nonleptonic hyperon decay amplitudes

Physics Letters B 266 ( 1991 ) 131-134 North-Holland

PHYSICS LETTERS B

Comparison of theoretical and experimental nonleptonic hyperon decay amplitudes N.N. T r o f i m e n k o f f Heritage College, Hull Canada J8X 3 Y8 Received 15 June 1991

It is shown that fitting Pauli amplitudes instead of Dirac amplitudes reduces substantially the mismatch in fits to experiment of theoretical parity-violating and parity-conserving nonleptonic hyperon decay amplitudes for massless pions obtained from current algebra and PCAC, effective chiral lagrangians, or lowest order chiral perturbation theory.

The weak transition amplitude T for the nonleptonic hyperon decay a ~ f l + n with corresponding momenta

Ps--,pp+p~ and masses rn,~rna+ m~ can be written as [ 1,2 ] T=ap(pp) (A + B75 )Us(Ps) ,

( 1)

where A (B) is the parity-violating (parity-conserving) decay amplitude, and where the Dirac spinors are covariantly normalized such that as (Ps)us ( P . ) = 2ms. This amplitude T can be rewritten in terms of Pauli spinors Z as

T=z*p(a+ btr.,O)X,~ ,

(2)

a=[(ms+ma)2_m~]l/2A,

b=[(ms_mp)Z - m ~:] l/zB ,

(3)

where eqs. (2) and (3) are valid in the rest frame of either a, fl or 7t, and where/~ is a unit vector in the direction of the three-momentump of one of the particles not at rest. For brevity, we will call A and B the Dirac amplitudes and a and b the Pauli amplitudes. Traditionally, theoretical Dirac amplitudes A and B for massless pions obtained by using the simplest results of current algebra and PCAC, effective chiral lagrangians, or lowest order chiral perturbation theory have been compared with experimental Dirac amplitudes Aexp and Bexp (or, equivalently, B and Bexp have been multiplied by a common factor); however, there is a long-standing mismatch, recapitulated briefly below, in this comparison of theory with experiment. In this article, we show that this mismatch can be reduced substantially if, instead, theoretical Pauli amplitudes a and b for massless pions are compared with experimental Pauli amplitudes aexo and bexp. We note that an exception to this tradition of comparing Dirac amplitudes is the old work of ref. [ 3 ], to which our work has some resemblance. The original application of current algebra and PCAC to nonleptonic hyperon decays, supplemented with SU ( 3 ) symmetry and octet dominance for the weak two-body transition amplitudes ( fll Hw Ic~) and with SU (3) symmetry for axial-vector coupling constants (fllAultx), gave the following Dirac decay amplitudes which we will interpret as amplitudes for massless pions:

A(A°_)=(--D-3F)/~/c6f~,

A(Z++)=0,

A(Z=)=(D-F)/f~,

gA(mA+mN)((--D--3F)/w/-6+ (--D+F)qv/2/~/3~

B(A ° ) =

~

\

mA-----'~mN

mz--mN

A(E-)=(-D+3F)/,v/6f~,

(4) (5)

7'

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

131

Volume 266, number 1,2

PHYSICS LETTERS B

gA(mz+mN)( D - F

B(Z+ ) -

f.

(-D+F)(1-q) mz - m N

22 August 1991

(D+3F)#I/3~ - m- A- - -- --

\l/IN

I

'

B(E_)=gA(mz +raN) ((D-F)(I_-q) # (D-I-3F)q/3~ f= \ mz--mN mA--mN i ' gA(mz+mA) ( (D+F)"xfl2/x/3

B(E_-) -

f.

k

m-~=- m ~

(-D+3F)(1-2,)Ixf6) #

m:_ - m a

(5cont'd) l"

The unlisted amplitudes can be obtained from the A I = -~ rules A ° = -A°_/x/r2, Z + = (Z + -Z_- )/x/~, and -o--° _ -.E_-/x/~. Our normalizations are such that f , = 132 MeV, = [(-D-3F)/x/6]GuA. The results of eqs. (4) and (5) are more than twenty years old [ 1 ], and a recent review is contained in, for example, ref. [4]; more recently [5,6], similar results have been obtained in lowest order chiral perturbation theory. The mismatch mentioned above is the following. If D and F in A of eq. (4) are chosen to fit Aexp with D~ F ~ - 0.37, then the fit of B to Boxp is poor for any f/d; notably, B(Z + ) is very small whereas B(Z + )exo is the largest experimental amplitude. On the hand, if D and F in B of eq. (5) are chosen to give the best fit to Bexp, then the fit of A to Aexp is poor; notably, A ( Z - ) is more than twice as large as A (Z-)exp. (The best fit of B to Bexp is obtained with D/F~ - O.88 by first changing (4) axial-vector coupling constants gA (C~fl) to pseudoscalar coupling constants g' (c~fl) by using the generalized Goldberger-Treiman relation gA (C~,8) = g '

(o~fl)f~l(m, +m#) ,

(6)

and then using SU (3) symmetric pseudoscalar coupling constants g' with f ' / d ' ..~0.56. ) Moreover, there is little compromise possible because the fit o f B ( Z + ) to B(Z~_ )exp deteriorates very rapidly as D/F is changed. This mismatch of D/F ratios required to fit best Aexo or Bcxp has been known for more than twenty years; numerous proposals, reviewed in, for example, ref. [4] and continued to the present [7], have been made to alleviate this mismatch. Typically, such proposals introduce unknown or poorly-known parameters or degrade [ 8 ] the major prediction A (Z ++) = 0; our proposal to fit Pauli amplitudes instead of Dirac amplitudes does not suffer from either drawback. In addition to the above mismatch, reasons for considering fits to Pauli amplitudes include the following: (i) The current algebra-PCAC formalism with p2-+0 or p ~ 0 and with p,~=p#+p,~ involves extrapolating the momenta in the amplitude T; the Dirac spinors carry substantial momentum dependence which is not contained in A and B but is explicitly included in a and b. Moreover, quantities such as decay rates and asymmetry parameters [ 1,2 ] involve a and b, not A and B, in a parallel fashion; for example, the decay rate F~: Ia 12+ I b l 2. We do not know of any overwhelming reason for disregarding this momentum dependence or for treating the momentum dependence of the parity-violating and parity-conserving amplitudes in a non-parallel fashion. (it) As the mass differences in the denominators ofeq. (5) show, A ofeq. (4) and B ofeq. (5) are of different order in SU(3) symmetry breaking or, in the language of chiral perturbation theory [5,6], of different order in the strange quark mass rn, which is used as an expansion parameter with mo = md= 0; as eq. ( 3 ) shows, a and b are of the same order so that a comparison of Pauli amplitudes with experiment is a comparison to equal order. Since these reasons alone may be neither compelling nor convincing, we will let our following numerical results speak for themselves. We henceforth deal with theoretical amplitudes for massless pions with mu = rnd = 0 and with momentum conservation Pc<=Pe+P,~maintained so that eq. (3) reduces to

a=(m,~+m#)A,

b=(mc`-m#)B,

(7)

for theoretical amplitudes. First we obtain Pauli amplitudes in the SU (3) and chiral symmetry limits by taking the soft pion limit p~-~0 (that is, m~-*0) so that m,---,ma--,m. Directly from eqs. (4), (5) and (7), we get the following Pauli decay amplitudes in this limit: 132

Volume 266, number 1,2

PHYSICS LETTERS B

a(A°_)=2m(-D-3F)/x/~f~,

a(X*)=0,

a(Z-)

22 August 1991

=2m(D-F)/f~,

a ( -= - ) = 2 m (

-D+3F)/x/~f~, (8)

b(AO)=

b(Z+)= b(X-)=~

gA2mf~ k, ((-D-3F)/x/~+ (-D+F)q(x/~/x/~)mz-mN(m,~-my)) , ga2m( ~-~

gAZm(

(D-F)+(-D+F)(1-rl)+ (D-F)(1-q)+

(D+3F)

(D+3F)

0//3)(m~--mN)~ . . . . , mA --mN ]

ga2m ( (D+F)q(x/~/x/~) (ms--mA)

b(E:) = -~--\

m--z=~--mmz

(q/g)(mz--mN).)

mA--mN

+ (-D+3F)

(1

_2q)/x/~ )

(9)

.

To obtain the ratios of mass differences in eq. (9), we first parametrized the baryon masses in terms of ms with m u = m o = 0 in the form [6]

mN=m--2b2ms, mx=m, mA=m-4(b,+b2)ms, m=_=m-2blms,

(10)

where b, and b2 are two parameters. Then in the limit m s ~ 0 , a ratio of mass differences such as (mA--mN ) / (rnz--raN) remains finite and equal to this ratio with rns # 0, and we therefore use the physical masses in these mass ratios. We note that the parametrization of masses of the kind shown in eq. (10) may not be unique because of possible redefinitions of these parameters [ 9] so that m may be undetermined; however, the ratios of mass differences are unaffected by such redefinitions, and the overall factor m in eqs. (8) and (9) can be absorbed into D and F. Also, as expected from eq. (7), eqs. (8) and (9) can be obtained simply by multiplying the lowest order chiral perturbation theory parity-violating amplitudes listed in ref. [6] by 2rn and the parityconserving amplitudes by m,~- rnp. Fit 1 of table 1 is a typical fit of a and b of eqs. (8) and (9) to experimental amplitudes aexp and besp obtained from ref. [2] and eq. (3). Fit 1 is obtained withf/d=0.56 which is the value suggested in ref. [ 10] and with 2mD/f==2800 MeV and 2mF/f==-3500 MeV so that D/F= - 0 . 8 0 which is near the valence quark (or Wexchange) model of D/F= - 1. Fit 1 shows that the gross mismatch of fits to parity-violating and parity-conserving amplitudes present when Dirac amplitudes are fitted has been substantially reduced by fitting Pauli amplitudes: the fits for A ° and --z decays are to within 10% of experiment and the mismatches for a(£-_ ) and b ( Z $ ) have been reduced to about 50%. In view of the above i m p r o v e m e n t and in the hope of further improving the fit for X decays, we now consider the case p2 = 0 but p ~ 0 with S U ( 3 ) and chiral symmetries broken by baryon masses with rno,#m,o.As in traditional fits, we get best results for b ( Z ++) by using the G o l d b e r g e r - T r e i m a n relation of eq. (6) to change Table 1 Fits of theoretical Pauli amplitudes a and b to experimental Pauli amplitudes aexpand bexpin units of 103 MeV; fit 1 is for eqs. (8) and (9), and fit 2 is for eqs. (11) and (12). Decay

A°_ E+ Z..E2

aex p

3.01 + 0.02 0.13+0.02 4.12_+0.02 -4.96_+0.02

a

bex p

(fit 1)

(fit 2)

3.14 0 6.30 -5.43

2.80 0 6.33 -5.28

1.09 + 0.03 3.95+0.02 -0.14_+0.02 1.13_+0.04

b

(fit 1)

(fit 2)

1.07 2.11 -0.13 1.18

1.21 3.52 -0.14 1.24

133

Volume 266, number 1,2

PHYSICS LETTERS B

22 August 1991

from axial-vector to pseudoscalar coupling constants. Moreover, there is some m i n o r i m p r o v e m e n t if we use an S U ( 3 ) p a r a m e t r i z a t i o n for the Pauli form o f the two-body weak transition a m p l i t u d e s ( i l l H w l a ) . F o r this case, we obtain the following Pauli amplitudes:

a(YT_)=(8-(~)/f~, a ( E _ - ) = ( - 8 + 3 O ) / , , / 6 f ~ , gA((_8_30)/v/~+ (-8+O)q'(xf2/v/~) (mZA--m2)ZmN~

a(A°)=(-8-3O)/x/6f~,

a(X+)=0,

b(A°- ) = f~ k, gA (6-0)+ b(x++)=T

b(E-)=

( ~ 2 ~ m 2 N ~ (~nA +--rex--) (--8+0)

gA ( ( 8 - - 0 ) ( l - - q ' ) m N ~mx

(1--q')mN

(8+3¢) (r/'/3)

mz

(mZ-m 2)

~ (8+30) (q'/3)

(m2A--m~)

gA ((8+~)q'(,,f2/,,/~) (mZ--m2A)2mN (mZ-mZz) (mx+mA) f

b(E_-)= ~-

(11)

j'

(m~--m~)2mN~ (mx + m A )

,/

(m2--m2)2mN~ (mx+mA)

]'

(--8+3~)(l~2q')mN/V/6)

.

(12)

Our n o r m a l i z a t i o n s are such that the pseudoscalar coupling constants are g~NN=d'+f' with ~/'= 1 / ( 1 + f '/d' ), and, for example, ( n l H w l A ) = [ ( - 8 - 30)/x/~]Z*ZA. Also, we have used the G o l d b e r g e r - T r e i m a n relation g~NN=gAxf2mN/f~ to convert g~NN to gA and f~ for easy c o m p a r i s o n o f eqs. ( 1 1 ) and ( 1 2 ) with eqs. ( 8 ) and ( 9 ) or eqs. ( 4 ) a n d (5). Fit 2 o f table 1 is a c o m p a r i s o n o f eqs. ( 11 ) and (12) with experiment o b t a i n e d w i t h f '/d' = 0 . 5 2 5 , 8/f~= 3030 MeV and O/f~= - 3 3 0 0 MeV so that 8 / 0 = - 0 . 9 1 8 which is very close to the valence quark model value o f - 1. Except for a(X_- ) which is about 50% too large, fit 2 is within about 10% o f e x p e r i m e n t for all the other amplitudes; that is, fit 2 reduces substantially the 50% difference between b(X + ) and b(X++ )exp o f fit 1. We have checked for both fits 1 and 2 that it is possible to introduce enough 27 c o m p o n e n t in ( i l l Hw I a ) to fit a ( Z + ) to a ( X ++ )exp and still obtain fits o f the same quality as shown in table 1 for all other amplitudes. We also note that table 1 emphasizes overall fits to the amplitudes; just as in traditional fits o f A to mexp, it is possible to fit all the a to a~xp to within about 10% with D/Fofeq. ( 8 ) or 8 / ~ o f e q . ( 11 ) ~ - 0 . 4 2 , but when the fit o f b to bcxp is poor. We thank the g o v e r n m e n t o f Qu6bec for financial support under the program F o n d s pour la F o r m a t i o n de Chercheurs de l'Aide h la Recherche.

References [ 1] R.E. Marshak, Riazuddin and C.P. Ryan, Theory of weak interactions in particle physics (Wiley-lnterscience,New York, 1969). [2] Particle Data Group, J.J. Hernfindez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1; Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Lett. B 111 (1982) 1. [ 3 ] Y. Hara, Y. Nambu and J. Schechter, Phys. Rev. Lett. 16 (1966) 380. [4] J.F. Donoghue, E. Golowich and B. Holstein, Phys. Rep. 131 (1986) 319. [5] A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189; H. Georgi, Weak interactions and modern particle theory (Benjamin-Cummings, Menlo Park, CA, 1984). [6] J. Bijnens, H. Sonoda and M.B. Wise, Nucl. Phys. B 261 (1985) 185. [7] R.E. Karlsen, W.H. Ryan and M.D. Scadron, Phys. Rev. D 43 ( 1991 ) 157. [8] J.H. Reid and N.N. Trofimenkoff, Nucl. Phys. B 40 (1972) 255. [9] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77; J.F. Donoghue, in: Annual review of nuclear and particle science, eds. J.D. Jackson et al. (Annual Reviews Inc., Palo Alto, CA, 1989) p. 1. [10] F.E. Close, Phys. Rev. Lett. 64 (1990) 361. 134