On the renormalization of the beta-decay axial constant

Nuclear Physics 42 (1963) 129-- 133; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission trom the publisher

ON THE RENORMALIZATION OF THE BETA-DECAY

AXIAL CONSTANT NGUYEN-VAN-HIEU

Joint Institute for Nuclear Research, Laboratory o f Theoretical Physics, Dubna Received 20 August 1962 Abstract: The renormalization o f the/]-decay axial constant is considered in the Chew-Low theory with fixed nucleon. The renormalization constant 2 = --GA[G v is expressed in terms of the renormalized constant o f pion-nucleon strong interaction and total cross sections o f pion-nucleon scattering.

1. Introduction

The calculation of the renormalization constant 2 = - G ^ / G v of the fl-decay axial constant is very interesting for confirming the complete agreement of the FeynmanGelt-Mann 1) and Sudarshan-Marshak 2) universal V-A theory of weak interaction with experiment. Because of the well-known difficulties in the strong interaction theory the exact calculation of this renormalization constant is impossible for the time being. However, this question was also considered repeatedly in different models of strong interaction. Bernstein, Gell-Mann and Michel 3) have shown that in the two models suggested by Gell-Mann and Levy 4) it is possible to express ;t in the form of a matrix element of the strong interaction. In the paper of Balachandran 5) this question was considered with the help of the model in which the strong interaction is vs-invariant. In this paper the renormalization constant of the fl-decay axial constant is considered in the Chew-Low theory with fixed nucleon ~). We show that in this theory it is possible to express ;t in terms of the renormalized constant of pion-nucleon strong interaction and total cross sections of scattering of n-mesons on protons. 2. Matrix Element of /Y-Decay in the Non-Relativistic Approximation

The universal V-A theory 1,2) of weak interaction has successfully explained the experimental data on//-decay and decays of/a- and n-mesons. In this theory the weak interaction Lagrangian is of the form . ~ = 2-½GJ.J~

(1)

where Ju is the sum of the lepton currents i~,u(l + ~n)e + i ~ ( 1 + ~5)/~, the strangenessconserving current of strongly interacting particles j~ = j v + j ~ and the strangeness 129

130

NGUYEN

VAN-HIEU

non-conserving current of strongly interacting particles S~ = s V + s ~ . In the first order of weak interaction the last current gives no contribution to the process under consideration. Moreover, in our case we m a y neglect the strange particle effects which are connected with high energies and consider only the terms with nucleons and n-mesons in the current j~. The vector part j v of the strangeness conserving current of strongly interacting particles is an isobaric component of a conserved isovector whose third component is proportional to the isovector electromagnetic current of these particles: jv = i2-½V~+), (2)

(3) N=

,

H=

no

.

I f the strange particle effects are neglected the axial current j~ is also of the form j~ = i2- ~rAt~+),

(4)

The part of Lagrangian responsible for the E-decay is .Lap = 2 - ' G ~ . ( 1 +y,)v(V~+)+A~ +))

(6)

and the matrix element of this decay is

M = 2-~Gv~.~,(t +~s)vvfip~(l+2rs)u..

(7)

2 = -- GA/G v. The conservation of vector current requires that Gv = G.

(8)

N o w we consider the non-relativistic approximation. In this case the matrix element (7) becomes M = 2-aGv[~,~,4(1 +?,)vvX*~*x,+i;tfi.?(1 +rn)v, • * +

(9)

where XN is the isobaric and space spinor which describes the state of the nucleon N. This matrix element m a y be also written in the following form: M = 2-'a[~,r,(l+r,)v,.

< p l V ~ + ] n > + ~ , ? ( l + r s ) v , • ,

(x0)

where IN> denotes the state vector of the real nucleon N. We denote by Z the constant of nucleon wave function renormalization and by n,(k) and n + (k) the annihilation and creation operators for the n-meson with m o m e n t u m k and isobaric index a.

RENORMALIZATION OF fl-DECAY AXIAL CONSTANT

131

in the non-relativistic approximation eqs. (3) and (4) give V,¢'' = z ' Z + 2 E n*~(k)T~,n,(k)

(11)

k

and

A (~) = i~oZ.

(12)

It follows from eqs. (8)-(12) that * "¢"ZNI : ZN1

* "ZNI Z~N2"I~

+2E(N,ln:(k)~,n,(k)lN,)

(13)

k

and 2XN * ~T" OXN

= Z
Ie

IN,>.

(14)

The application of the Chew-Low theory enables us to calculate the matrix elements on the right-hand sides o f eqs. (13) and (14) and therefore to determine 2. 3. C a l c u l a t i o n o f R e n o r m a l i z a t i o n Constant ~.

In the Chew-Low theory the Hamiltonian of the pion-nucleon system is

H = E ['~*(k)'~(k~k + F,(k),~at(k)+ F,*(k)n,*(k)-],

(15)

t-,at

k 2 > K 2, where e, = 0, _+ I; f0 is the unrenormalized constant o f pion-nucleon interaction; m k = (kZ+p2) t, l~ is the meson mass. Now we consider the relation (13) for some a, e.g. a = 0. In this case the terms on both sides of this equation are different flora zero only if the nucleons in the initial and final states have the same charge. For the sake of definiteness let us consider the case in which these nucleons are protons. We have = . It follows from the Hamiltonian (15) and the commutation relations that n,(k)lp) =

1 H+C0k

(p[n.(k)= _
f.lp>,

1 . H+tOk

According to the definition (16)

F*~(k) = -F_=(k). Therefore, (pln*(k)n~(k)[p) -- ~ (Plf*at(k)lv)(vlF-'(k)lP>

,

(e,+o

ky

(17)

i 32

NGUYEN VAN-HIEU

With the help of the Chew-Low method it is possible to express the contribution of intermediate one-nucleon states on the right-hand side of (17) in terms of the renormalized constant f , of pion-nucleon interaction and the contribution of the other intermediate states in terms of the total cross-sections of scattering of rt_,-mesons on protons. We obtain the following result:

l f:Pdk [21ffa,.p(l)-a,_p(l)dl_

z = 1+ ~

co~-

./-2]

co,(,,,, + co,Y

The constant Z was obtained from the relation (13) with a = 0 and N~ but its value does not depend on the choice of these parameters. In a similar way it follows from the relation (14) that

= z r l + d f",',*o(k)+,~,-o(k) k 2-~?Jo co,

(18)

dco~J " =

dk] -t

N 2 =

p,

(19)

j

In eqs. (18) and (19) the renormalization constant 2 is expressed in terms of the renormalized constant f , of pion-nucleon interaction and total cross-sections of scattering of ~±-mesons on protons. The value of 2 depends on the cut-off parameter K. However, from the experimental data s) on a,±v(k ) and the value.f2/4n = 0.08, we obtain 2 < 1 for any K. This result does not agree with experiment. The Chew-Low theory gives results which are in good agreement with experiment at low energies, but is not applicable at high energies. Therefore we must not use experimental data on pion-nucleon scattering, but theoretical values obtained from this theory. Layson 9) has shown that if we use in (17) instead of a cut-off the form factor

v(k) = (1 +k2/K2) -~, K = /~/0.27, F~(k) = i(2cok)-½foa " kz'v(k),

(20) (16')

the (33)-phase obtained in the Chew-Low theory tg ~33

4fr2" t'~k/mo) Iv(k)] 2, = 3/12 COk(1

(21)

where coo is the resonance energy, is in good agreement with experiment. In this case we have in place of (18) and (19) Z = 1 + ~I f o° [v(k)] 2 k4dk -

co,

~21fo" ax÷p(l)-a,-p(l) dl- ./.2 ] [v(l)]2co,(co,+ coy

1,2co~],

(18')

and

2=Z l-l+ #2 ~.ooa..,(k)+a.-p(k)dk']-' L

2,¢).1o

[o(k)]'co,

d

(193

We take into account only the dominant contribution of the elastic scattering cross-

RENORMALlZXTIONOF /~-DECAV^XiAL CONSTANT

133

section in the state (33) a n d neglect the c o n t r i b u t i o n s of the rest. In this case a , - p ( k ) = a ( ~ - p ~ l t - p ) + a ( l t - p ~ g ° n ) = ~o33(k), a,.p(k) and

= a33(k )

a33(k) is o b t a i n e d f r o m (21). We also o b t a i n ;t < 1.

The a u t h o r expresses his gratitude to professor M. A. M a r k o v for the interest in this work, a n d to B. N. Valuev a n d V. A. Meshcheryakov for the discussions.

References I) 2) 3) 4) 5) 6) 7) 8) 9)

R. Fcynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193 E. Sudarshan and R. Marshak, Proc. Intern. Conference at Padova-Venezia (1957) J. Bernstein, GelI-Mann and L. Michel, Nuovo Cim. 16 (1960) 560 M. Gell-Mann and M. Levy, Nuovo Cim. 16 (1960) 705 A. Balachandran, Nuovo Cim. 23 (1962) 428 G. Chew and F. Low, Phys. Rev. 101 (1956) 1570 S. S. Gerstein and Ia. B. Seldovish, JETP 29 (1955) 618 V. S. Barashenkov and V. M. Maltsev, Fortschr. der Phys. 9 (1962) 549 W. Layson, CERN, preprint