Reformulation of the theory of the neutrino

Reformulation of the theory of the neutrino

I ~ Nuclear Physics 7 (1958) 411--420; (~)North-Holland Publishing Co., Amsterdam [ Not to be reproduced by photoprint or microfilm without writte...

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I

~

Nuclear Physics 7 (1958) 411--420; (~)North-Holland Publishing Co., Amsterdam

[

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

R E F O R M U L A T I O N OF T H E T H E O R Y OF T H E N E U T R I N O S. K A M E F U C H I

Universitetets Institut ]or teoretisk Fvsik , Copenhagen Received l i March 1958 A r e f o r m u l a t i o n of the t h e o r y of the n e u t r i n o is a t t e m p t e d b y generalizing t h e free L a g r a n g i a n so as to include f r o m the o u t s e t t e r m s violating p a r i t y and particle n u m b e r conservations. I t is s h o w n t h a t , so far as the free field is concerned, our generalized field is equivalent to t h e usual field a n d t h a t a difference a p p e a r s only w h e n interactions are included. An application is also given to the discussion of coupling c o n s t a n t i n v a r i a n t s of fl-decay.

Abstract:

1. Introduction Since the parity conservation is violated in some weak interactions comprising neutrinos, it would be a tempting idea to attribute the origin of this violation to the intrinsic nature of the neutrino. When we take such a viewpoint, it becomes possible to formulate a field theory of the neutrino in a w a y more general than the usual theories. This is because, in this case, we need not take into account the restriction of parity conservation and can thus start with a free Lagrangian which is not invariant under space reflection. The two-component theory developed by Lee and Yang, and others, is an example of this kind of approach 1). The question of the conservation of the lepton number is not yet fully settled b y experiments and thus there exists, at present, further latitude to extend the theory in this direction, i.e., so as to introduce into the free Lagrangian terms violating this conservation law. In this note, an attempt is made to reformulate the theory of the neutrino b y generalizing the Lagrangian in the above sense. An application of our formalism is also given to the problem of coupling constant invariants, discussed recently b y Pauli and Pursey ~). Our main conclusion is the following: so far as the free field is concerned, our generalized field is equivalent to the usual Dirac, Majorana or twocomponent neutrino field, which m a y have, in general, indefinite parities under the transformations of space reflection and of charge conjugation. Characteristic features of our field become explicit, however, when the interaction comes in. It is also found that our formalism is convenient for discussing coupling constant invariants of the /~-decay interaction; it can easily be extended to the case of non-vanishing mass of the neutrino and, 411

412

S. KAMEFUCHI

moreover, with the use of a perturbation expansion, it enables us to reach somewhat more explicit conclusions than the above mentioned authors. An appendix is devoted to the discussion of some identities which exist between Pauli's covariants of coupling constants.

2 N e u t r i n o Field of V a n i s h i n g M a s s In this section, the case of a neutrino field with vanishing mass will be considered. The most general form of the free Lagrangian, which takes into account both the violation of parity and that of the lepton number, is given b y t

L : --l[~/(1-[-avs)I'~v+~vFX(l+aTsX)~v+~vlr'C~--~*~C-1I'~!,

(1)

where / ' is defined b y ya O/Ox,, ~ and a are real constants, ~ in general a complex constant and C the usual charge conjugation matrix. Now, the Lagrangian (1) shows different features according to the values of the parameters, and therefore we shall consider several cases separately.

2.1.

~2(1--a2)--[~12 # 0,

a :/: ~ 1

The quantization in this case is carried out b y using the standard method and most easily b y using Umezawa-Takahashi's formula a) tt. The anticommutators for field quantities are as follows: {~,(x), ~v(x')} : --1 ~ So(X--x')C, i ~2(1--a2)--l~12

(2)

(W(x), ~(x')} ----T $2(1--a2)--[~12 (l+a~'5)S°(x--x'); So(x ) is defined b y I'D(x), where D(x) is the usual invariant function for massless fields. In this case, we see from the consistency of the theory that there exist further restrictions on the parameters, which can be found in the following way. First, as seen from the second of eqs. (2), the constant a has to satisfy the condition [al < 1, which is required in order that each component of W has the anti-commutators with the right sign. And, if this is the case, we can eliminate the ~,5-terms from (1) b y making a linear transformation ~v : kl~v'+k2ys~V', where k12+k~2 = 1, 2klk 2 = a. * Terms like ~sFC~, w h i c h d o n o t c o n t r i b u t e t o t h e v a r i a t i o n of t h e a c t i o n f u n c t i o n , h a v e b e e n o m i t t e d . W e c a n e a s i l y see t h a t eSf~)psI'CCpdax = 0, for a n y v a r i a t i o n s u c h t h a t

{~(x), ~(x')} = 0. ?t I n o r d e r t o a p p l y t h i s f o r m u l a t o o u r case, w e h a v e o n l y t o r e w r i t e t h e L a g r a n g i a n (1) t h e f o r m L = ½grTA~r*, w h e r e g * = ( ~ ) -a a[~l*F'C-x, grr(l+a75T)~] . . . .T. h e n a ....... i, ~F(1--aTa), *IFG t h e a n t i - c o m m u t a t o r is i m m e d i a t e l y g i v e n b y t h e f o r m u l a {~tVT(x), ~ ( x ' ) } = i R D ( x - - x ' ) , w h e r e R is t h e s o l u t i o n t o A R = ([')~. in

~.FO,HULAT~ON OF THE THEORYOF THE ~EUTR~NO

413

After this transformation the Lagrangian becomes L =--½V'l--a2 X [2~ ~/1 -- a 2(p' Fy' + ~Tgp'FC~'-- ~* ~o' C -1 Fv2'], which can further be rewritten in terms of two real fields $1 and $3, defined b y y' = (1/V'2) (~1+i$2), as follows: @l--a

s

2

where we have used the Majorana representation for the },-matrices and changed the phase of ~0' in such a way that ~/ becomes real. Thus, from the requirement that the commutators for 41 and ~b2 have the right signs, follow the conditions ~ V ' i - - a 2 q - ~ / > 0, which, in general, lead to ~ > 0 and ~2(1--a2)--[~[2> 0. When these conditions are satisfied, we can further eliminate ~/-terms from the Lagrangian b y making a transformation ~o'= k3~o"+k4C~", where k32+k42= 1, 2k3k 4 = --~/~X/1--a 2. Thus, it is shown that the Lagrangian (1) in this case can ultimately be brought into a "canonical" form b y making a linear transformation of the type = ~.,.~ + ~ , },~~ + # ~ c,~ +#.~ },~ c~. (s) Here, by the canonical form of the Lagrangian we mean the familiar form Lo = - - ~ r ~ .

(4)

It is also to be noted that the transformation (3) reduces to Pauli's canonical transformation e) only when some conditions are imposed on the coefficients ~'s and fl's t. For convenience in later discussions we here define the following quantities: K ~ I~.,l~'+l~d~+l#ll"+l#.-,I% L -- ~ ~ * + ~ * ~,~+#~ #3"+#1" #3, z - 2(-~*#1"+~*#~*),

U

~

(5)

I~112- I~d ~ - I#xl~+ I#d~+~l~*--~* ~ - # ~ #3"-+-#1"J~2"

In order to bring (1) into the form (4), the coefficients a's and fl's have only to satisfy the conditions K -IMI~,

L a

K'

I* ~

IMI2.

(6)

In terms of quantities defined b y (5) the anti-commutators (2) can be rewritten in the form t I t can be p r o v e d t h a t o u r t r a n s f o r m a t i o n (3), w h i c h c o n t a i n s 8 real p a r a m e t e r s , re duc e s t o a c a n o n i c a l t r a n s f o r m a t i o n , p r o v i d e d t h a t t h e following four c o n d i t i o n s a re satisfied, viz., K = L = I = I * = 0. (This is also seen from (9.') below.) Thi s r e s u l t m e a n s t h a t t h e m o s t g e n e r a l t y p e of c a n o n i c a l t r a n s f o r m a t i o n s for t h e n e u t r i n o fields c ons i s t s of a 4-param e t e r group, j u s t as g i v e n b y Pauli.

414

S. K A M E F U C H I

1 I* So(x--x')C ,

=

T

=

T

(e')

] (K+L75)So(x_x,),

where we h a v e used the i d e n t i t y K 2 - - L 2 - - [ I ] 2 = [M] 2. Our results (2) or (2') can be regarded as a f u r t h e r generalization of the a n t i - c o m m u t a t o r s discussed b y J a u c h and others 9 ; if we p u t a = 0, the expressions (2) or (2') go into J a u c h ' s a n t i - c o m m u t a t o r s . Thus, the fields ~v corresponding to a = 0, V = 0 and a = .0, 0 < I~l < $ describe the usual Dirac n e u t r i n o and w h a t m a y be called the J a u c h neutrino, respectively. As ]Vl-+ ~:, the a n t i - c o m m u t a t o r s of the J a u c h n e u t r i n o t e n d to those of the Majorana neutrino. 2.2.

~e2(l--a2)--]~/[ 2 -~ 0,

a ¢ ~1

As m e n t i o n e d above, we can r e g a r d V as real a n d positive w i t h o u t loss of generality; la[ < 1 in this case. First, we shall consider the case # > 0. T h e n ~ = ~V/(1--a2), and, b y a t r a n s f o r m a t i o n ~v' ----cI y~+ c2 Y5~vwith real cl, c 2 such t h a t Cl > c2, c~2+ c22 = 1, a = - - 2 Q c 2 , we can bring the L a g r a n g i a n into the following form:

This means t h a t in this case the L a g r a n g i a n describes just the usual M a j o r a n a field. As is evident from the above argument, the case ~ < 0 is unphysical. We also r e m a r k t h a t in general a n y Majorana field can be w r i t t e n in the same form as (3) in t e r m s of the Dirac field, b u t with the additional conditions t h a t ~ x * = f1 and ~ 2 " = --/~2. 2.3.

a=~l

First, let u s c o n s i d e r the case $ @ 0. In this case, the ~-terms in (1) just correspond to the free L a g r a n g i a n of a n e u t r i n o with a definite helicity, whereas the ~-terms describe couplings between this n e u t r i n o and a n o t h e r kind with opposite helicity, the free L a g r a n g i a n of which is missing in (1), however. To exclude such an u n p h y s i c a l case, we m u s t p u t ~ / = 0. Needless to say we m u s t have ~ > 0. Therefore, this case just corresponds to the wellk n o w n t w o - c o m p o n e n t neutrino field, which can again be w r i t t e n in a form like (3) with the conditions ~1 = :J= ~ and fx = -4- f12. Now the case ~ = 0 has to be excluded: for the L a g r a n g i a n in this case contains only the coupling t e r m between neutrinos with different helicities and no t e r m corresponding to free Lagrangians. Or, w h a t is the same thing, the Lagrangian, being written in t e r m s of Majorana fields, in the form L

=

-

1 r

r¢1-

¢5

REFORMULATION OF THE THEORY OF THE NEUTRINO

415

evidently leads to some inconsistency after quantization, i.e. to commutators with wrong signs irrespective of ~?. It is also to be remembered here that the cases 2.2 and 2.3 differ from each other only b y their representations. As shown b y many authors 5), they are completely equivalent so far as the free field is concerned. In fact, we can easily show that b y changing the variable ~vinto (1/~/2){(cl+c~7s)~v + (q--c275)C~}, with c1 and c2 defined above, the Lagrangian of the twocomponent neutrino L = - - ~ ( 1 ± 75)F~v can be transformed into the form --l~{2~/(l+ays)F~v+~(1--a2) ((pFC~--~vC-1Fw)}, which is nothing else than the Lagrangian of the Majorana neutrino discussed above. From the above arguments we conclude that, so far as the free field is concerned, our generalized Lagrangian can always be brought into the canonical form, and so our field is equivalent to the usual field of Dirac, Majorana and the two-component theory, respectively. It is to be noted, however, that in general the latter fields would have indefinite transformation properties under both space reflection and charge conjugation, as is easily seen from (3). The difference between our field and the usual fields appears only when the interaction is introduced. For, if the interaction is present, the transformation (3) induces corresponding changes in the interaction terms, toot.

3. Neutrino Field with Non-Vanishing Mass Our method can be extended to the case of non-vanishing mass. Along similar lines as above we shall consider the general field defined by v,, =

where ~ is the usual Now, it two fields {v,,(x),

r5

(7)

a Dirac field with non-vanishing mass K, which therefore satisfies anti-commutators, and ,q's and fl{s are arbitrary constants. is of special interest to note the commutation relations between ~v~ and ~vj, defined b y (7), with different coefficients cq, fl~ and

v,,(x')}

{V,,(x), ~,(x')}

=

1

1

=TS(x--x'){K,s--L,,ys}+

TxA ( x - - x

, ){--K , , , + L , ,J Ys},

(s)

where A (x) is the usual delta-function for a field with mass K and S(x) = /'zJ (x). The constants I , etc. are defined in the following way: t However, we can also s a y t h a t the Dirac neutrino w i t h t h e m o s t general kind of interaction, whose L a g r a n g i a n is, for instance, L e (9) + L t a t (0tl~0"~-~Xa7~ ~O+ HI C~ + ~2 7 5C~), is entirely equivalent to our n e u t r i n o w i t h a simple form of interaction, whose L a g r a n g i a n is L(~v)+Lmt(~ ). I n this connection, see ref. 4).

416

S. KAMEFUCHI *

*

-- -~1~ ~lj+~, * ~*

*

*

*

~*. + / ~ ,* ~*,

* /~*_~_~* * ~, , V'li ~ 2 i - - P 2 i ~ l J , * *

O~lt t2i--O~2i/"15 *

K. =

K~

L~ =

L *~ ---

r.

~;,

=

~1, ~ + ~

~ . + ~ , ¢ ~ j + ¢ 2 , ~*~,

(9)

--Otli P l i - - 0 ~ 2 i D 2 1 - - P l i O t l i - - f l 2 i 0~i,

J;, =

Jj, ~

t

t,

t

t,

~ + ~ , ~ . + ~ :~,~+ ~2~ ~ *~, ~1, ~ * ~ - ~ ,

O~2i-F~li ~ 1*~ - - ~ 2 i fl2i' * * ,

These definitions are just the same as those given b y Pauli and others 2)t. The anti-commutators (8) include those of (2) or (2') as a special case. In the case of K = 0, they are invariant under Pauli's canonical transformation.

4. Interacting Fields In this section, the case of interacting fields will be considered; as an example, let us take the fl-decay problem. In terms of our field quantities ~v~, the most general interaction of fl-decay is written in the form n ~ = ~. (CppOi~n)(~eOi~oi)-~h.c. ,

(10)

i

where the O,'s (i = 1, 2 . . . . . 5) denote the usual covariant 7-matrices describing S, V, T, A, and P couplings, and the yJ,'s are defined b y (7). Now it is to be recalled that any physical quantity which does not depend upon the neutrino variables explicitly, such as the S-matrix element for a neutrino-less double fl-process or the cross-section of a simple fl-decay summed over neutrino variables, can always be written in terms of vacuum expectation values of chronological products of neutrino field quantities (P(~p,(x), ~o;(x')))0 or (P(~p~(x), ~(x')))o. It can also easily be shown that these vacuum expectation values are given b y the same expressions as the right-hand sides of (8), but with a simple replacement A (x) -+ - - { i A r ( x ) , where A v ( x ) is the usual Feynman functiontt. From this argument we can derive the conclusion that the coupling constants ~ and fit enter into the physical quantities mentioned above only through the combinations given b y (9). Thus, we have generalized Pauli's result to the case of non-vanishing mass. It might be somewhat unexpected that in the case of K = 0 some of Pauli's covariants (viz. M , , Nu~ and Niuj) do not appear in our result, and so the quantities t T h e r e l a t i o n b e t w e e n o u r n o t a t i o n a n d P a u l i ' s is as follows:

~li : gI*i' ~,i : 1~, --ill' =glli*, fl2i : /I'*'* t t T h i s r e s u l t s f r o m t h e f a c t t h a t for o u r fields ~p~ w e still h a v e t h e p r o p e r t y ~p~c+~I0) = 0, w h e r e ( + ) m e a n s t h e p o s i t i v e f r e q u e n c y p a r t a n d I0 ) is t h e v a c u u m s t a t e .

REFORMULATION

OF

THE

THEORY

OF

THE

417

NEUTRINO

under consideration, being written in terms of the invariant combinations only, remain invariant under Pauli's canonical transformation. However, we show in the appendix that there exist many identities between (Mi~, Nii~, Niiij) and (Kit, L~j, I,~, Jij): thus we can say that the former can appear in the physical quantities only through the intermediary of these identities. We also remark that as far as the spin states of neutrinos are not measured, the quantities to be compared directly with experiments are limited to those which can be written in terms of the vacuum values of neutrino field operators. Thus, the properties of coupling constants can be discussed only through the combinations given b y (9). Now, we shall consider the P, C and T transformations. Under these transformations all the field operators in (10) obey the well-known transformation laws, except the neutrino operator, which can, in general, be a mixture of different "parities". Therefore, if we require the physical quantity under consideration to be invariant under these transformations, the vacuum expectation values like


× (~e(xl,

tl)O,F,,(Xl--X2, tl--te)o,T(ve(X2, t2)),

where we have put


id(--xl)dtlld(--x2)dt2(Pp*PnPe*) -2

S = const. X i1

"

x

tl)O, Vn(-x

× (Vye(--xl,

tl)Oiy4F,~(x1--x2, tl--t2)y4To~T~'e(--Xo, t2)).

,

t2))

Therefore, in order that this matrix element be invariant under P-transformation, we have to require that F i ~ ( - - ~ , tl--t2) = (pp* pnpe*)-274 F (x~--x2, tl--t2)74 T. This relation is nothing but the usaal transformation property for the quantity

~v'(--x, t) = (pp*pnpe*)-~y4~(x, t). The other cases can be discussed in the same manner. Now, from the above requirement we can obtain the necessary and sufficient conditions for invariance under P, C, and T transformations. These are as follows:

418

S. K A M E F U C H I

for P: L~j = L',j = O, and

(all i, j) either I o . = I ' ~ j = 0 ,

or

J,j=J',~=

(11)

0;

for C: Ki,. real, L,j pure imaginary.

K' o. real, L',j pure imaginary,

I~ I~* real. j()i j oT0* real, kl

(all i, i, k, l)

I;~ J(k{* pure imaginary; for T: K~j, L,j, K',j, L'~j real, I~ I~* real, j oiJ d7o* real, kl

(an i, i, k, Z)

(13)

I~ J~* real; here I~, for example, means the same quantity with or without prime. The above result is a generalization of that of Kahana and Purser 2),who have given the conditions for the unprimed quantities only. It is to be noted that, if the neutrino has a non-vanishing mass, however small it m a y be, the number of conditions is almost doubled. In this connection, it should also be noted that the conditions for P, C and T invariance as given by (11), (12) and (13) are weaker than those usually imposed on the interaction Hamiltonian itself (or the coupling constants). (For example, in the case of only one type of coupling present and of K = 0, (13) gives no restrictions at all for T invariance). This results from the fact that, in our case, the invariance is required only for the expressions which are quadratic in coupling constants. In the case of K ~- 0, it can also be explained by the fact that the existence of Pauli transformation allows us to generalize the operation of P, C and T transformations, and so for the invariance of the theory one has only to require the invariance under these generalized operations which are defined by P' ~ UI TU2, C' ~ U 1 CU2, and T ' = U 1 TU2, respectively, where P, C and T are the usual operations and U 1 and U z appropriate Pauli transformations. The necessary and sufficient condition for lepton number conservation in our formalism is given by vanishing of the expression
(all i, [).

(14)

Our method can also be applied to the similar problem in/,-meson decay interaction, which was recently discussed by Ferretti and others 6).

REFORMULATION

OF

THE

THEORY

OF

THE

NEUTRINO

4~19

The author likes to thank Professors G. K~ll6n and L. Rosenfeld for their critical comments and valuable discussions. He wishes to express his gratitude to Professor Niels Bohr for the hospitality at the Institute for Theoretical Physics. It is also a pleasure to acknowledge the financial aid received from the Ford Foundation.

Appendix Here we shall discuss the identities which hold between the unprimed quantities listed in (9) and M,~, NI~~ and NH~~ defined by Pauli 2). To this purpose, it is convenient to use the quantities F, and G~ defined by

(Eli ~ Fi

*

*

\F2i]

\fl2i--fllJ

Gii

( li-~0¢2'~

G~ =

(A1)

G2i = \fl2~+¢~1]

Under Pauli's canonical transformation with respect to the neutrino variables, F, and G~, apart from phase factors, transform according to the spinor representation of the three-dimensional rotation group. In terms of F, and Gi the quantities under consideration can be written in the following way:

K.+L.

= G*~G. + G*~G2~, ---- Fli FI~+ F2~F2j,

I . -~-JiJ = Gli F2~--G2i FI~, I. -J.

=

F2~GI~ FI~ G2j, - -

Mi~ = F'~ Glj +

(A2)

F*~G2j ,

NI~ = Fu F2j-- F~ F . , NH. = Gas G~--G2~ Glj. Now, M,~, Nu~ and NlliJ are not invariant quantities under Pauli's transformation, but change by phase factors exp (2i~), exp (--2i~) and exp (2i~), respectively; therefore, all possible invariant quantities built up from M,~, NI~j and Niu ~. only are given by the following expressions: M~j Mk*, NujNik v NimNnk~, NiijNiik~, NIijM~ and NimMk*. Therefore, it is to be expected that all these quantities can be expressed in terms of the invariants K,a, L~j, I,j and J,.~, and, in fact, the following identities hold:

420

S.

M~j Mk* : N~,j N*kz = N,,~jN,I~Z = N,.. N,,~ = N,i~ M~ = NH. M*, =

KAMEFUCHI

(Kzj +L,~)(K~--L~)--(Ijk +Jjk)(I* -J~),*

(Kk,--Lk~) (K~j--L~)--(Kk~--Lkj) (Kz,--L~,),

(Kk~ +L~)(Kzj+L~j )-- ( K k j + L k ~ ) ( K . + L . ), (A3) (I,~ +L~) (1,~ --Yi~)--(Ik~÷ ]~)(L~ - - ] . ), (I~ +L~)(K~,--L~,)-- (L~ +].)(K~--L~), (L~ --A,) (K~ + L~ )-- ([~ -- ]~) (K. + L. ), (~ll i, i, k, l). These are easily checked by substituting the expressions given in (A2).

References 1) T. D. Lee and C. N. Yang, Phys. Rev. 105 (1957) 1671; L. D. Landau, Nuclear Physics 3 (1957) 127; A. Salam, Nuovo Cimento 5 (1957) 299 2) W. Pauli, Nuovo Cimento 6 (1957) 2Ct; C. P. Enz, ibid. 6 (1957) 250; D. L. Pursey, ibid. 6 (1957) 266; S. K a h a n a and D. L. Pursey, ibid. 6 (1957) 1469; G. Liiders, ibid. 7 (1958) 171 3) Y. Takahashi and H. Umezawa, Prog. Theor. Phys. 9 (1953) 14, 501 4) J. M. Jauch, Helv. Phys. Acta 27 (1954) 89; Y. Takahashi, Nuovo Cimento 1 (1955) 414; S. Kamefuchi and S. Tanaka, Prog. Theor. Phys. 14 (1955) 225 5) J. Serpe, Physica 18 (1952) 295; Nuclear Physics 4 (1957) 183; J. A. McLennan, Phys. Rev. 106 (1957) 821; K. M. Case, ibid. 107 (1957) 307; L. A. Radicati and ]3. Touschek, Nuovo Cimento 5 (1957) 1693 6) ]3. Ferretti, Nuovo Cimento 6 (1957) 997; R. Gattu and G. Liiders, ibid. 7 (1958) 806