Phenomenological interaction currents in nuclear systems

1.A

I

Nuclear Physics A220 (1974) 103--113; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout writtenpermissionfrom the publisher

PHENOMENOLOGICAL

INTERACTION CURRENTS

IN NUCLEAR SYSTEMS J. A. LOCK and L. L. FOLDY Case Western Reserve University, Cleveland, Ohio 44106 F. R. BUSKIRK Naval Postgraduate School, Monterey, Calif. 93940 Received 19 October 1973 Abstract: Beginning with the differential conservation law for electric charge, the electromagnetic current density operator for a nuclear system is decomposed into one-body and two-body terms. The two-body or interaction current density is further separated into a longitudinal part which is related to the charge exchange part of the nuclear force and a transverse part, depending on the nucleons' dynamical variables, and containing a large number of arbitrary parameters. In determining all the linearly independent irreducible vectors formed from the relative position vectors Rt and R~ and the nucleon spin vectors tr~ and tTk, which may be present in the transverse current density, the method of group characters is used. Along with these spin-space vectors, all linearly independent irreducible spin-space scalars, pseudoscalars, and pseudovectors are also calculated. These linearly independent vectors are then used together with the various symmetry properties of the current density to determine the most general form the transverse current density can assume. In a similar manner, the most general form of the Fourier transform of this current density operator is determined. Lastly, the form of the axial vector weak current density arising from mesonic exchange effects in fl-decaying nuclei is derived again by using the spin-space pseudovectors and the various symmetry properties of the weak current density.

1. Introduction I n studying electromagnetic effects i n nuclear systems, the general forms of the p h e n o m e n o l o g i c a l exchange a n d i n t e r a c t i o n c u r r e n t densities have been k n o w n for a n u m b e r of years 1 - 5). However, they have n o t b e e n used in calculations because of the u n m a n a g e a b l y large n u m b e r of arbitrary f u n c t i o n s present in the expressions. As a n alternative to this approach, the practice has been to use various m o d e l m e s o n theories to calculate mesonic exchange effects a n d t h e n to use these results i n app r o x i m a t i n g the i n t e r a c t i o n magnetic m o m e n t c o n t r i b u t i o n s for light nuclei 6-13). The present t r e a t m e n t attempts to generate the p h e n o m e n o l o g i c a l expression for the i n t e r a c t i o n c u r r e n t density using g r o u p theory a n d to determine what restrictions s y m m e t r y c o n d i t i o n s impose o n the a r b i t r a r y f u n c t i o n s a p p e a r i n g in the current density expression. F o l l o w i n g previous a u t h o r s 4), we begin with the differential c o n s e r v a t i o n law for electric charge a n d the u s u a l a s s u m p t i o n s that: (1) The i n t e r a c t i o n currents associated with slow m o v i n g nucleons m a y be expressed as functions of the d y n a m i c a l 103

J.A. LOCK et aL

104

variables of the nucleons: their positions ri, spins ai, and isospins ~. This is the socalled static approximation. (2) Only velocity-independent two-body forces give rise to the interaction currents. And (3) there is no interaction charge density and the charge density is localized at the positions of the nucleons. The relaxation of this last assumption would only affect the longitudinal part of the current density and not the transverse part with which we are here primarily concerned. In the absence of external electromagnetic fields and with the neglect of three or more body terms, the total current density for nuclear systems will have the following decomposition:

j(x) = E

E

i

E

i

(1)

i
where j~(x) is the convection current density of the ith nucleon, j~(x) is the spin current density of the ith nucleon, and jlk(X) is the two-body interaction current density between the ith and kth nucleons. The interaction current density may be further decomposed into a longitudinal partjL(x) and a transverse part j r ( x ) with

ilk(x) = jL(x) + j~k(X).

(2)

The longitudinal part takes the form

j.L,k(X) = Vx q~ik(x),

(3)

with ~)ik(X)=

e

Vi~X(r,xlrk)3 (

1

1 i)

(4)

8~hc [xS-ril Ix--rk " In obtaining the form of q~ik(X), it has been assumed that the nucleon-nucleon force consists of an ordinary two-body force and a charge exchange force v = E

+ X

i
i
•

(5)

where the two potentials depend on the spins and relative separation of the nucleons. With this separation, the longitudinal current density is that portion ofj~k(X) which actually represents the exchange of charge from one nucleon to the other. The above decomposition o f j ~ ( x ) into an exchange portion and a transverse portion is not unique. Another possible choice is to have the exchange portion take the form of a straight line filament flow along the line joining the two nucleons 1). The present decomposition has the disadvantage that the magnetic moment operator ½Sx xj(x)dax for the longitudinal current density and for the transverse current density are each conditionally convergent although the total current density operator is unambiguous. The decomposition of ref. 1) on the other hand has the disadvantage that the straight line filament flow current density has no counterpart in the actual physical situation. With any decomposition of the current density, a transverse portion may be written as

j~(x) = Vx x ~,,,

(6)

where the invariance of the interaction Hamiltonian under translations, rotations,

INTERACTION CURRENTS

105

and space inversion requires that ~ik be a pseudovector formed from the relative coordinates R~ = x - r i and R k = x--rk, and the nucleon spins as and trk with the isospin dependence being a function of ~i and ~rk. The vector ~k may be thought of as an interaction magnetization between the ith and kth nucleons and the vectorj~ may be thought of as its magnetization current density. 2. Construction of the transverse interaction current density Considering first the spin-space dependence of ~ik, the most general pseudovector formed from R~, Rk, as and a k will be a linear combination of all the linearly independent pseudovectors which may be formed from these four quantities. This basis set of pseudovectors may be constructed by using the method of group characters. Before continuing with the phenomenological current density, it is worth while to look at the general problem of determining all the linearly independent tensors of various rank formed from the two spin vectors a and X and the two polar "space" vectors a and A. Here a and A are arbitrary but in various applications, they may be taken to be nucleon momenta, positions, or relative position vectors. The characters of unreduced "space" tensors of various ranks were determined by their transformation properties under rotations. Using the orthogonality properties of the group characters, the irreducible representations of the rotation group contained in these unreduced tensors were found t. Since it is assumed that a and • act in a different space than the polar vectors a and A, the irreducible spin tensors and "space" tensors may be computed separately. In what follows, the notation vj will be taken to mean the r a n k j irreducible symmetric tensor formed from the components of the vector v where v = a, A, a, or X. Only rank zero, one, and two irreducible tensors may be formed from the spin operators since tensors quadratic in a or $ may be reduced by the algebra of the Pauli matrices to tensors linear in a and ~. In determining first the irreducible linearly independent scalars and pseudoscalars, the Clebsch-Gordan expansion of the outer product of the spin tensors and the "space" tensors guarantees that no rank higher than two need be considered for the space tensors. This is because the resulting spin-space tensors are to be of rank zero. When the rank zero spin-space tensors are extracted from the outer products, one finds that there are eight linearly independent scalars and eight linearly independent pseudoscalars 15). These are listed in table 1. In this table and in all subsequent listings of the spin-space tensors, arbitrary functions of a • a, a • A and A • A have been factored out of the tensors and are understood to multiply each tensor. In a similar manner, the irreducible linearly-independent spin-space vectors and pseudovectors may be constructed by reducing the outer products of the irreducible spin tensors with the irreducible "space" tensors of up to rank three. Again the Clebsch-Gordan expansion guarantees that no space tensors of rank higher than three need be considered. By this method one finds that there are 24 linearly indeAn alternative method is given in ref. a4).

106

1. A. LOCK et al.

TABI.~ 1 Irreducible linearly-independent spin-space tensors of rank zero and one composed of two spirt vectors or a n d ~ and two polar space vectors a and A Outer product 1

Scalars

Pseudovectors

1

a*A t ffi

axA or

(or" a)n

(71a 2

ala*A * alA z Z~ Zla 2 ZlatA1 ~1A2

or. ( a x A )

a',~'la z o:Z'~aLa. ~

(or.a)(,~.a)

(or- a)(2:'X a), (,~"" a)(or X a)

(or-a)(~;'.A), (or.A)(~;'-a)

(u.$')(nxA),(Or.a)(,Y,×A), (or.A)(2;'xa)

a'2~IA ~

(Or.A)(~.A)

~'. ( a × A )

al2:laSA* at2J*aZA a a*~*a*A z Pseudoscalars az A~ o';a I (r~A ; ala2A ~ (rlalA z ~Vlal ~'lA1 ,~la2At ,,~'lalAZ aW, la ~ a*Z*A*

or • a o-A ~, . a ,g" A

a. (or ×2:) A . (or X~")

a*T-,*a 3

cH~*a2A i aIZ~alAZ a*.S* A 3

(or'a)l~" (a×A)]+(~.a)[or. (a×A)] (or.A)[~.(axA)]q-(,g.A)[or. (a×A)]

(or. a)A, (or.A)a (or. A ) A (~" a)a (~.a)A, (~.A)a (,~. A ) A

(,Z'. a)(or x A ) , (~'.A)(o,x a) (or •A)(-.x'X A), (,~"A)(Or×A) (or. a)(,,~, a)(a × A ) (or. a)(.~. A)(a x A ) + (o". A)(.~. a)(a x A ) (or. A)(~'" A)(a x A ) Vectors a A orXa or×A (17 • a)(a x A ) ((Jr. A)(a ×A) ,~, × a ~'×A ( ~ . a)(a × A ) (,~" A)(a × A ) (or.2;)a, (~r, a)2:, (2;. a)or (or',~r)A, (or.A),~, (X.A)Or (Or. a)(~" a)a ( o r . a ) ( ~ . A ) a , (or.A)(,~.a)a, (or.a)(27.a)A ( o r . A ) ( ~ . a ) A , (or.a)(,~.a)A, (or.a)(,~.A)a (or. A )(~'. A )A

The outer products from which these irreducible tensors are obtained by reduction are given in the first column with the notation ~ standing for the irreducible symmetric tensor of r a n k j formed from the components of the vector v. Irreducible tensors which are space scalar multiples of earlier tensors in the list are omitted. p e n d e n t v e c t o r s a n d 2 4 l i n e a r l y i n d e p e n d e n t p s e u d o v e c t o r s 15),. T h e s e a r e l i s t e d i n t a b l e 1. I n t a b l e 1 it s h o u l d b e n o t e d t h a t f o r e x a m p l e p s e u d o v e c t o r s a r i s i n g f r o m t h e t ReL 2) incorrectly gives the number of pseudovectors as 27. Thus three of their pseudovectors are dependent on the other 24. For example, in line with the results of our table 1, one may consider their numbers (32n)--(32o) and ( 3 2 p ~ ) as the dependent ones. However many other choices are possible.

INTERACTION

CURRENTS

107

outer product alY_,lalAt may be thought to arise from the outer product a t Z t a 2 A 2 also if one remembers that in the present treatment, factors of (a" A) are being deleted. One should also note that none of the vectors or pseudovectors listed in table 1 contain triple products. This is because identities such as [A • (B x C)]D = (,4 • D ) (BxC)+(C'D)(AxB)-(B'D)(AxC) and [A.(BxC)](DxE) = (B.D) (C " E ) A - ( B . E ) ( C . D ) A - ( A . D ) ( C . E ) B + ( A . E ) ( C . D ) B (A " E ) ( B " D ) C + (A • D ) ( B • E ) C reduce any such terms to linear combinations of those which appear in table 1. Also, one should note that the linearly independent rank zero and one tensors formed from a and ? and a single polar vector a are obtained from the results of table 1 by setting* A = 0. U p to now, the polar vectors a and A have been arbitrary. For the case of the phenomenological interaction current density, one identification of interest is to have the relative separation of the ith and kth nucleons be a = p = r k - r i = R i - R k TABLE 2 T h e indexed r a n k one tensors f o r m e d f r o m the spins o f the ith a n d k t h nucleons, ~r~ a n d or~ a n d f r o m t h e p o l a r vectors ~ = r~--r~ a n d g = x - - ½ ( r ~ + r ~ ) where r~ a n d r~ are the positions o f t h e two nucleons

1

@

1

o i-o~

2 3 4

(Oq" O'k)~ (a~. ~)(trk - p ) p (O'l" g)(~ra • g ) ~

2 3 4

Ot XO'~ ~ x (O',. ork)(O × g)

5

(a,. o)(a~, g ) o - (a,. g)(a~ • o)o

5

(a,. o)(a~ • o)(o x g)

6 7

( a , . o)(cr~ • g ) g + ( o , • g ) ( a k • p ) g [(al-t-~ra) " @I(Q x g) [(a,-a~). 0 1 ( o x g) (~tWCrk) ×@ (at-~) x g (try. O)o'~+ (o'~ • p ) o h (or,. g)~ra--((rk • ~)ot

6 7 8 9 10 11 12 13

(~h" g)(Uk • g ) ( p × g) [(or~--crk) • ~1~ [ ( ~ , + ~ ) • g]O [(orL+a~) • ~ ] g [(cr~-aD. g]~ (or,. O)(17kX O)--(O'k " O)(O'l XO) (~q- ~)(ork × 0)q-(ork" g)(crt X ~ ) (O'l" 9)(O'k X 5 ) + (~rk " ~)(O'i X ~)

14

(0"," ~)(0"~x g)--(O'l,• g)(o',× 5)

8 9

10 11 12

V~ (s) 13 14

pV~(ks )

g (tTl" trt)~ (tr,- O)(tr~ • ~ ) 5 (trt. ~)(o'~, • 5 ) 5 ( ~ , . e)(tr.~ • 5 ) 5 - - ( o ' , . ~)(O'k " @)~ (0",. fl)(O'.~ • g)@+(tr, • 5)(Or~ • fl)O [(~,--Or~) • 5 ] ( ~ × 5)

15 16 17 18 19

2o

[(~r,+a~)'o](Ox5)

20

[(a,+a~) • g15

21 22 23 24

(~r~--crg) × ~ (O',+O'k) X g (@,- e ) ~ r ~ - (tr~ • O)tr, (tr,. 5 ) t r a + ( o ' , • g)o',

21 22 23 24

(~rt. 0)(or~ X~)q-(or~ (tr,. ~)(o'~, x # ) - - ( o ' ~ : (try- O)(tr~ x S ) - ( a ~ (o"t • 5)(or, × @)q-(o'a

15 16 17 18

19

tr~+~r~

(tr,.O)(a~.~)(~xg)+(~,.g)(t~.e)(ex 5) [(0",+0'.~) • o l o [Co't--O's) • 5 ] 0 [(~r,--~r~) • ~ ] 5 • p)((r~ × ~) • g)(o', x o )

• e)(~r, xS) • 5)(o', X S )

T h e index ~ labels t h e vector, V, or pseudovector, PV, while (s) or (a) designates s y m m e t r y or a n t i s y m m e t r y u n d e r exchange o f t h e ith a n d k t h nucleons. t F o r the case o f the pseudovectors, o n e obtains the results o f ref. 2).

J.A. LOCK et al.

108

and to have the relative position o f the field point with respect to the center o f mass o f the ith and kth nucleons be A -- ~ = x - R T M = x - ½ ( R ~ + R ~ ) assuming all nucleons t o be o f equal mass. F o r convenience in determining the restrictions symmetries impose on the interaction current density, in table 2, the irreducible linearly independent vectors, V~,, and pseudovectors, PVi~ are written so as to be either antisymmetric under particle interchange [~,~) and PV~,¢~)] or symmetric under particle interchange [I~/~,~) and PV~,~)]. The arbitrary functions multiplying the vectors a n d pseudovectors are o f the f o r m to, = o~,(p 2, ~2, p . ~) and 2~ = 2~(p 2, ¢z, p . ¢) respectively. Considering n o w the isospin dependence o f the interaction current density, the isospin operators must contain ~ and ~rk to no higher power than the first, again because o f the algebra o f the Pauli matrices. In general m a n y such isospin operators m a y be formed but only six c o m m u t e with the total charge operator Q = e ~ [½(1 + ~ ) ] .

(7)

O f these, two are isoscalars, I~k, three are third components o f isovectors, K~k, and one is the 33 c o m p o n e n t o f an isotensor t, L[k. These are listed in table 3. TABLE 3 Isospin operators commuting with the total charge operator

Isoscalar

Third component of an isovector

ll~ 1 = 1

K~k1 -----(rtq-T~) 3

Itk 2 = Ti " T k

Klk 2 = (T/--'/'k) 3

33 component of an isotensor LjkI

=

Ti3"rk 3

K~ 3 ----(r~ ×r~) 3

3. Restrictions on the form ofj~(x) due to symmetry requirements Since the transverse interaction current density is associated with a H a m i l t o n i a n which is gauge invariant and is invariant under translations, rotations, a n d space inversion, the hermitian current density operator must take the f o r m o f the real polar vector Vx x ~k, where, as before, ¢ik is a pseudovector formed f r o m Ri, R k (or equivalently p and ~), ai, ak, ~i, ~ and which c o m m u t e s with the total charge operator 16). It is f o u n d that if ~ik is a linear c o m b i n a t i o n o f the 24 independent spin-space pseudovectors, then its curl is a linear c o m b i n a t i o n o f the 24 independent spin-space vectors. A t the present time, there is some question as to whether the electromagnetic current density operator for any system has an isospin character other than that o f the sum o f an isoscalar and the third c o m p o n e n t o f an isovector *t. * R e f s . to, 11) i n c o r r e c t l y list Lt~ l a s an. i s o s c a l a r .

t t In refs. t o. 1~), the existence of a rlav~3 isospin term was assumed but using a conventional meson model, no contributions from this isospin term were found in a number of processes contributing to the interaction magnetic moment. A discussion of the possibility of an isotensor contribution to the electromagnetic current operator is given in refs. 17. is).

INTERACTION

109

CURRENTS

TABLE 4 S y m m e t r y p r o p e r t i e s o f the irreducible spin-space r a n k o n e t e n s o r s

PsPaP~ (Ps, P~, Pt) (+, +, +) (--, --, +)

( I x, 12) • (c~ = 20, 22) ( I 1, 12) • (tx = 19, 21)

(11, 12) • (0~ = 13, 14, 15, 16, 18, 24) (11, •2). (~x = 17,23)

--

(+, --, +) (--, +, +)

(IX, P ) • ( ~ = 7 , 9 ) (11, 12) • (cx = 8, 10)

(11,12) • ( ~ = 1 , 2 , 3 , 4 , 6 , (11, 1 2 ) . (¢z = 5, 12)

(+, +, +) (+, --, --)

K x • (co = 20, 22) K 2 " (x = 7 , 9 ) K 3"(~= 1,2,3,4,6,11) K 2" (~ = 8, 10) K 3"(~=5,12) K x- (ct = 19, 21)

K x • (m = 13, 14, 15, 16, 18, 24) K 2 " (g = 1,2, 3, 4, 6, 11) K 3 " ( ~ = 7, 9) K 2- (cx = 5, 12) K 3"(~=8,10) K x- (c~ = 17, 23)

K x • (c~ = KX" ( x = K 2 • (~ = K a " (~ = K 2 • (~ = K a • (~t =

K x • (~ = K1 • (~ = K 2 • (ct = K a • (c~ = K 2 • (~ = K a • (ct =

(--, +, --) (--, --, +) (--, +, +) (+, --, +) (+, +, --) ( - - , --, - - )

+ _

+ O "t:;I

8, 10) 7,9) 20, 22) 13, 14, 15, 16, 18, 24) 19, 21) 17,23)

11)

5, 12) 1, 2, 3 , 4 , 6, 11) 13, 14, 15, 16, 18, 24) 20, 22) 17, 23) 19, 21)

(+, +, +) (--, --, +)

(11, •2). (~ = 15, 17, 20) (11, 12) • (ct = 18, 19)

(11, •2). (~ = 16, 21, 24) ( D , 12) • (x = 22, 23)

(+, --, +) (--, +, +)

(I t, 12) • (¢x = 8, 9) ( I x, 12) • (~x = 1, 7, 10)

(I 1, I2) • (~ = 3, 4, 5, 6, 12, 13) ( I l, 12) • (~ = 2, 11, 14)

(+, +, +) (+, --, --)

K x • (oc = K 2- (~z = K 3- (~ = K 2" (~ = K 3.(0~= K x • (~z =

15, 17, 20) 8, 9) 3, 4, 5, 6, 12, 13) 1, 7, 10) 2,11,14) 18,19)

K x • (¢x = g 2. (~x = K 3 • (~x = K 2. (~ = K 3.(~= K x • (¢z =

16, 21, 24) 3, 4, 5, 6, 12, 13) 8, 9) 2, 11, 14) 1,7,10) 22, 23)

K 1.(c¢= K 1 • (c~ = K 2 • (~ = K z • (~ = K 2 • (~ = K 3 • (ct =

1,7,10) 8, 9) 15, 17, 20) 16, 21, 24) 18, 19) 22, 23)

K 1. ( m = K 1 • (0~ = K 2 • (~ = K s • (~ = K 2 • (~ = K 3 • (~t =

2,11,14) 3, 4, 5, 6, 12, 13) 16, 21, 24) 15, 17, 20) 22, 23) 18, 19)

(--, +, --) (--, --, +)

O

I n c o r r e c t t i m e reversal b e h a v i o r

+

+

o

C o r r e c t t i m e reversal b e h a v i o r

(--,+,+) (+, --, +) (+, +, --) (--, --, --)

T h e first t w o c o l u m n s classify the v e c t o r / p s e u d o v e c t o r a n d isoscalar/isovector c h a r a c t e r o f e a c h tensor. T h e t h i r d c o l u m n labels t h e s y m m e t r y o f t h e t e n s o r u n d e r s i m u l t a n e o u s e x c h a n g e o f spin, space, a n d isospin c o o r d i n a t e s o f t h e particles, (PsPRPD. T h e f o u r t h c o l u m n labels the s y m m e t r i e s u n d e r e a c h o f these e x c h a n g e s separately. T h e fifth a n d sixth c o l u m n s list t h e t e n s o r s w i t h correct or i n c o r r e c t b e h a v i o r u n d e r t i m e reversal a s s u m i n g they are to be u s e d to c o n s t r u c t c u r r e n t s o r m a g n e t i z a t i o n s . E a c h e n t r y consists o f the p r o d u c t o f a n i s o s p i n factor f r o m table 3 w i t h a s p i n - s p a c e v e c t o r o r p s e u d o v e c t o r f r o m table 2 d e n o t e d by its index ct. E a c h e n t r y m a y be m u l t i p l i e d by a n a r b i t r a r y scalar f u n c t i o n o f the n u c l e o n c o o r d i n a t e s h a v i n g the s y m m e t r y u n d e r particle exchange given by the s y m m e t r y or a n t i s y m m e t r y listed in the third c o l u m n .

J.A. LOCK et aL

110

In conventional theories of spin 0 and spin 1 isoscalar or isovector mesons, no such exotic terms arise. If we impose these restrictions on the phenomenological current density, then jT(X) assumes the form 24

jT(x) = e Z ~=1

2

24

3

Z I i' k w , , z V i"k + e Z

Z K~t ~o,,, l E~.

/=1

1=1

a=l

(8)

Since j T ( x ) is hermitian, the functions co,, ~ and w',, ~ are real and form a set of 120 arbitrary functions characterizing the transverse interaction current density between the ith and kth nucleons t. The condition that the current density be symmetric under nucleon interchange (ri ~ rk, a~ ~ ak, and ~i ~ zk) will not decrease the number of terms appearing in eq. (8) but will however require that the functions ¢o,, l and ¢o~,,, possess certain symmetry properties with respect to the interchange p ~ - p , ¢ ,--, ~. These symmetry properties are listed in the third column of table 4. The antisymmetry of iT(x) under time reversal places further restrictions on the form for the transverse interaction current density 19). In particular, it requires that a number of the o9~,, and w'~, vanish and thus decreases the number of non-zero terms in eq. (8) to 48. These also are listed in the fifth column of table 4. This would seem to exhaust the list of general symmetries which one could impose on the interaction curient density. However other weaker conditions on o)~,z and oY, z may be imposed. In accordance with eq. (6), Vx "J~k .r = 0. Also, the arbitrary real functions o9,, z and ~o',, t should have ranges in p comparable to the interactionradius of the strong nuclear force. Any further reduction in the number of terms in j ~ ( x ) from the 48 time reversal invariant ones of table 4 must come from special restrictions imposed by the specific nuclear system under consideration and the explicit forms for the arbitrary functions in the theory must come from model meson calculations of specific processes for that nuclear system under consideration tt. The phenomenological form of the interaction magnetization ~ik is also of interest. In analogy with eq. (8), we have 24

(,k(X) = e E ~=1

2

24-

E I[kA,,tPVi] +e Z l=1

~=1

3

E K[k2",tPVi],

(9)

1=1

where 2,,z = 2,,~(p 2, ¢2, p . ~) and 2',,~ = 2",l(p 2, ~2, p . ~) again form a set of 120 new real functions characterizing the transverse interaction current density between the ith and kth nucleons. The restrictions on ffik(x) due to the requirements of symmetry under particle interchange and antisymmetry under time reversal are given in table 4. With these restrictions, there are 54 non-zero contributions to ~a, 20 of them being isoscalars and 34 being isovectors. The utility of using the phenomenological form of ~ik rather than o f j ~ is that when the curl of ~ik is carried out, the resulting * Consideration is given to time reversal invariant but non-hermitian phenomenological magnetic moment operators in xa). tt There are several experimental data which suggest that the isoscalar part of the interaction current is substantially (perhaps an order of magnitude) smaller than the isovector part.

INTERACTION

CURRENTS

111

current density is automatically divergenceless while when beginning with the form of j ~ f r o m table 4, the zero divergence condition must as mentioned above be imposed separately. The Fourier transform of eq. (8),

j~k(k, q) = 1 f ~dapdaCe,k.oelq.~j~(x),

(10)

(2r0°dd is also useful when comparing the results of model meson calculations with the phenomenological theory. Since this current density is also a polar vector formed from k, q, ~ri and %, it must be a linear combination of the same spin-space vectors of table 1 with the identification a = k and A = q. Carrying over the forms of the vectors from table 2 with p ~ k and ~ ~ q and the isospin dependence of table 3, one obtains 24

j~(k,q) = e E ~=1

2

24

EI~ko~.,~i+eY~ 1=1

g=l

3

E r~ko'~,,~i.

(11)

l=1

' _ CO' The real functions co=,t = co~,l(k 2, q2, k" q) and co~,~ =,t~t'k2, q2 k ' q) have in general different dependences on their arguments than do the arbitrary functions of the coordinate space representation of eq. (8). As before, proper behavior under time reversal requires that a number of the functions co~,l and co', z vanish. In particular, since k ---> - k and q ~ - q under time reversal, the same irreducible spin-space vectors which lead to coordinate space currents with the proper time reversal behavior also lead to the m o m e n t u m space currents with the proper time reversal behavior. Thus the fifth and sixth columns of table 4 may be taken over to the m o m e n t u m space current expressions with the substitution p ~ k and ~ ~ q. Then there are 48 Fourier transform terms which give non-zero contributions to the phenomenological transverse interaction current density. For certain other comparisons that may be made with meson theory calculations, the Fourier transforms of the current density with respect to one position vector

j~(k,

x) -- . 1 f d3pea,. (2~) 0

Pj~k(X),

j~(q,

x) = . l.}f.] d3~ e i ' ' gj~(x), (2=)

(12)

(13)

may be useful also. Here one would make the correspondence a = k, A = { or a = p,A = qintablel. 4. Mesonic exchange and interaction processes in nuclear fl-decay

With regards to weak interactions, it is known that mesonic exchange and interaction effects in nuclei play a small role in the fl-decay of nuclei and again, a phenome-

112

J.A. LOCK

etal.

nological description of the weak mesonic interaction current may be made ~1.20-24). The fl-decay Hamiltonian density for a physical nucleus is

G [j(u+)(x)ju(x)+d(u_)(x)jtu(x)],

(14)

where G is the fl-decay coupling constant, Ju = --i~eTu(l+Ts)$v is the charge lowering leptonic four-current density and J~+) is the charge raising nuclear fourcurrent density. Since the Hamiltonian density is hermitian, the relation between the nuclear charge raising and charge lowering weak current densities* is [J~+)(x)]* = Y~-)(x). This nuclear current density has a decomposition into vector and axial vector parts J~+)(x) = YV(+)(x)+JA(+)(X) where the isospin dependence of each term is that of the + component of an isovector. According to the Feynman-Gell-Mann theory of weak interactions, the vector part of the weak current may be obtained from the isovector part of the electromagnetic current for the same nucleus by means of a rotation in isospin space. In particular, if JV(+)(x) is written as a function of the nucleon variables alone, rather than of both the nucleon and meson variables for a given nucleus, then in the velocity-independent static approximation, the portion of jv(+)(x) describing mesonic effects is exactly the isovector portion of eq. (8) and of table 4 with the third component of the isovectors being replaced by the + component of the same isovectors 2 5). For the axial vector weak current density, no such correspondence is possible. Rather again in the velocity-independent static approximation and under the assumptions of the invariance of v/, jA(+) under translations, rotations, and antisymmetry under space inversion, the portion of JUA(+)(X) describing mesonic effects may be written as jA(+) ik . . . . .

24+. ~--- Z [('~i+'gk)+2,, 1 PV/~+('lri--'gk)+2~, 2 P V i ~ + ( ' g i x "gk) A,.3 P V / k ] g=l 3

24 2 ~Try/( " " k + ) ,~,,,PV/k. /=1 x = l

-- X

(15)

Here again J~A+)(X) is written as a function of the nucleon variables alone rather than of the nucleon and meson variables. The PVi~, are the spin-space pseudovectors of table 2 and again since the weak Hamiltonian is hermitian, the functions 2,, z = 2 ~, z(P2, ~2, p . ~) are real and form a set of 72 arbitrary functions parameterizing the portion of the axial weak current which describes the mesonic exchange effects between the ith and kth nucleons. The symmetry of *~kTA(+)me~onunder particle interchange imposes restrictions on the forms of the 2,, t under the interchange p ~ - p and ~ ~ ~. These are listed in table 2. IA(+) IRA(-) Time reversal invariance requires that ,~k . . . . . --* -o~k . . . . . • This condition requires that a number of the 2,. z vanish and decreases the number of non-zero terms in eq. t Using the convention o f 2s), d / , = d/** for/* = 1, 2, 3 and d#* =

--I1.* for t* = 4.

INTERACTION CURRENTS

113

(15) to 34. The pseudovector results of table 4 apply to this situation also if the third components of the isovectors of table 3 are replaced by their _+ components. Again any further decrease in the number of terms in eq. (15) or any explicit forms of the arbitrary functions 2,, ~ must come from model meson theory calculations for specific nuclei. One of the authors (J.A.L.) wishes to thank Dr. P. B. Kantor for several stimulating conversations. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

R. G. Sachs, Phys. Rev. 74 (1948) 433 R. K. Osbom and L. L. Foldy, Phys. Rev. 79 (1950) 795 R. G. Sachs and N. Austern, Phys. Rev, 81 (1951) 705, 710 J. M. Berger and L. L. Foldy, Nuclear Physics Laboratory of Case Institute of Technology, Tech. report no. 18 J. M. Borger, Phys. Rev. 94 (1954) 1698 F. Villars, Phys. Rex,. 72 (1947) 256 A. Thellung and F. Villars, Phys. Rev. 73 (1948) 924 M. Sugawara, Progr. Theor. Phys. 14 (1955) 535 S. Hatano and T. Kaneno, Progr. Theor. Phys. 15 (1956) 63 M. Chemtob, Nucl. Phys. A123 (1969) 449 M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1 D. O. Riska and G. E. Brown, Phys. Lett. 38B (1972) 193 M. Gari and A. H. Huffman, Phys. Rev. C7 (1973) 994 I. M. Gel'Fand, R. A. Minlos and Z. Y. Shapiro, Representations of the rotation and Lorentz groups and their applications (Pergamon Press, New York, 1963) subsect. 5.2 F. R. Buskirk, thesis, Case Institute of Technology (1955) L. L. Foldy, Phys. Rev. 92 (1953) 178 V. (3. Grishin et al., Sov. J. Nucl. Phys. 4 (1967) 90; N. Dombey and P. K. Kabir, Phys. Rex,. Lett. 17 (1966) 730 J. H. Field, Nucl. Phys. 1345 (1972) 88; T, Fujii et aL, Phys. Rev. Lett. 28 (1972) 1672 G. I. Kynch, Phys. Rev. 81 (1951) 1060 R. I. Blin-Stoyle, Phys. Rev. Lett. 13 (1964) 55 R. I. Blin-Stoyle and S. Papageorgiou, Nucl. Phys. 64 (1965) 1 J. S. Bell and R. J. Blin-Stoyle, Nucl. Phys. 6 (1957) 87 R. I. Blin-Stoyle, V. Gupta and H. Primakoff, Nucl. Phys. 11 (1959) 444 R. J. Blin-Stoyle and M. Tint, Phys. Rev. 160 (1967) 803 J. Bernstein, Elementary particles and their currents (Freedman, London, 1968) ch. 9, 10