The electric polarizability of the neutron in the static meson theory

Nuclear Physics 24 (1961) 524--526 ;

O North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE ELECTRIC POLARIZABILITY OF THE NEUTRON IN THE STATIC MESON THEORY AKIRA KANAZAWA t Department of Physics, Purdue University, Lafayette, Indiana

tt

Received 19 December 1960 Abstract : The electrh; polarizability of the neutron is recalculated in the framework of the static meson theory. The recalculated values of the electric polarizability are a = 1.1 x 10-42 cma for a Gaussian cut-off with cut-off momentum kc == 5.6 meson mass, and x = 1.0 x 10-42 cms for an inverse square cut-off with the same cut-off momentum . These values are almost half of those which were obtained by Barashenkov and Barbashov.

In recent years many attempts 1) ijave been made to observe or predict the electric polarizability of the neutron, among which is the calculation of Barashenkov and. Barbashov 2) by means of Chew's cut-off theory . However, their calculation is unsatisfactory in several respects. The main objections are that they failed to take into account the contribution from the catastrophic interaction between pions and the external electric field, fig. lb, and that they have not included all the time-ordered diagrams corresponding to fig. la. The polarizability has been recalculated in this paper taking into account the above terms . The fully relativistic calculation of the problem has been carried out by S. Tani and the author and will be published elsewhere 3) . We start from the following interaction Hamiltonian :

HI ~ HI+H2+H3, Hl =

f

*(t)P(x)~aQO~a(x~ t)y t)dx--dm * t

t ,

H2 == e f ~aT«,q ~,q A0 (x)dx,

H3 ` --e2 f (~12+Y'22)A0Apdx .

It should be noted that H3 is the sum of the catastrophic interaction 1 2 (O12+022 )A 16A 0 and the surface dependent interaction ze2 (012+~22) (nP A 0)2 . The function O(x, t) represents the meson field with mass ,u, and ~ is its time derivative. We denote by A 0(x) the scalar potential, which describes the extert tt

On leave of absence from the Department of Physics, Hokkaido University, Sapporo, Japan. Work supported by the National Science Foundation . 524

THE ELECTRIC POLARIZABILITY OF THE NEUTRON

525

nal constant electric field and by V(t) the nucleon field with mass in, which satisfies ( - iat+nt)v(t) = 0.

(2)

The density function p(x) represents the extension of the interaction between nucleon and pion, and f is the rationalized pseudovector coupling constant. By Ta,q we denote the elements of the matrix

The S-matrix element corresponding to any Feynman diagram can be calculated quite analogously to the case of relativistic field theory 4), if we simply note that e-'9o("') d p o

if

The two lowest order diagrams which contribute to the electric polarizability are given in fig. 1. nucleon pion ----electric field

(b)

(a)

Fig. 1

The corresponding matrix elements are (f 2 e2 Ma = -2A(0) (2 1A)

:7)4

(4a)

v(k)v(k")kô (u , k") (a . k)a(q)a(q') - 2 ~Ib, 2-- Wrr-, i,-) 2 k 2 28)2~ ] w r- ja) 2] tkp2k --ZE k [o )Co ( (o 2 v(k)v(kr ') (Q , k") (v , k)a(q ; a (q') 2 (Co__ie)2 Mb = 2 nO (0) -~ -J e (2n) 4 dgdq'd4k - t) rr - je) - ie ) « 2] Cko2[ko2 dgaq,dgk

y

(ko

'

(4b)

where w, w', and w" are the meson energies corresponding to the momenta k, are the Fourier k' = k+q, and k" -= k+q+q' respectively; v (k) and a (q) transform of p(x) and A,(x) ; i.e. e-iq l dx (q), a . P(x) 1 a A ee` ~~ dx ' o(x) * = --iE b (q) = (2a) y(k) _.. îq a f where E is the constant external electric field.

(5)

526

AKIRA KANAZAWA

It should be noted here that the second term in (4a) comes from the ôfunction term of the propagator: o =

(6)

—~xI)—iô~(x—y).

It is well known that the contribution from this a-function term must cancel with that of the surface dependent interaction, which is just one-half of the interaction H3 in the present case. If we carry out the integration with respect to k0 in (4), the first term in M~ vanishes and we have Ma+Mb = ~Mb =

~

2~iâ(O)(i)2e2

dk v(k)v(k”)(a .k”)(o k)cz(q)a(q’) .

(7)

The electric polarizability ~ can be uniquely obtained if we compare (7) with the effective S-matrix element, i.e. Meff

i$ dtH~,(t)= —27?iö(0) (_~ocE2),

where Heff (t)

The result is

(~)(;~)~ e2

/2

=

(8)

2)tp(t).

=

~*

(9)

(t) (—~ocE

(~~)

k4 k6 dv (v2(k) (10~—8 +

2

k4

1

._-~)(;;3).

Numerical values t for two choices of the cut-off function v are ~= 1.1x1042cm3 for v=exp{—(k/5.6)2}, ~= 1.0x10—42cm3 for v= l/{1+(k15.6)2}.

(10) (11)

These values are consistent with current experiments 1). The author wishes to thank Professor M. Sugawara for the hospitality extended to him at Purdue University. t Barashenkov et al. have obtained the numerical values o~ 1.8 for the Gaussian cut and =

=

1.6 for the inverse square cut.

References 1) V. S. Barashenkov and I. P. Stakhanov, Soviet Physics JETP 5 (1957) 144; In. A. Aleksandrov and T. I. Bondarenko, Soviet Physics JETP 4 (1957) 612; A. S. Baldin, Proc. Padua-Venice Conf. on Fundamental Particles (1967); A. Langsdorf, R. 0. Lane and 3. E. Monahan, Phys. Rev. 107 (1957) 1077; R. M. Thaler, Phys. Rev. 114 (1959) 827; V. I. Goldansky, 0. A. Karpukhin, A. V. Kutsenko and V. V. Pavlovskays, Nuclear Physics 18 (1960) 473; G. Breit and M. L. Rustgi, Phys. Rev. 114 (1959) 880; L. L. Foldy, Phys. Rev. Letters 3 (1959) 105 2) V. S. Barashenkov and B. M. Barbashov, Nuclear Physics 9 (1958) 427 3) A. Kanazawa and S. Tani, Progr. Theor. Phys. 25 (1961); see also T. Ueda and S. Sawamura, Progr. Theor. Phys. 24 (1960) 519 At

M

1.1

1ri~.,i

Phi-a

Pg~, O’7 11Q~F~t119~