Some properties of combined PC-transformation and associated selection rules

Nuclear

Not to

SOME

Physics be

4 (1957)

repmduced

by

PROPERTIES AND

664-678;

photoprint

OF

@ or

North-Holland

mimfilm

without

COMBINED

ASSOCIATED

Publishing

written

permission

Co., Amsterdam from

the

publisher

PC-TRANSFORMATION

SELECTION

RULES

P. ROMAN Department

of Theoretical Received

Physics,

University of Manchester

29 April 1957

Abstract: We derive the transformation rules of the creation and annihilation operators of baryon and lepton fields under a combined charge-conjugation and space-reflection operation, and find eigenstates of the corresponding operator. Also properties of the ‘squared’ operation are studied, giving a distinction between baryon and lepton fields from which some selection rules are derived. x- and K-meson fields are also studied and some consequences for K-decay are investigated. Then, assuming a universal Fermi interaction, it is proved that the number of baryons is conserved if we demand the invariance of the interaction under a combined charge-conjugation and space-reflection transformation. Some selection rules for allowed processes are also studied.

1. Introduction As a consequence of some experiments 1$“) showing definitely the violation of parity conservation in weak interactions, recently considerable interest has arisen with respect to the less stringent PC-transformation, i.e. a transformation composed of a charge conjugation and a subsequent space inversion (see e.g. ref. 3*4, 5, “)). In this paper we develop the formal properties of this transformation in more detail, with a view to deriving selection rules for processes governed by weak interactions. In particular, it can be seen that baryon number conservation in Fermi interactions is a consequence of PC invariance. 2. Fermions 2.1. CHARGE

CONJUGATIOX

The field operator

of any fermion y = ;

~.u(p)

can be written

e”P’“+b,*v(p)

e-fp’x

as (1)

where np is the annihilation operator for a particle with momentum p, and bz the creation operator for the corresponding antiparticle. The adjoint field is y = 2 a,*&(p) e-‘*“+b,b(p) eip’x. (2) The charge conjugation

is defined

by

Nuclear Physics 4 (1957) 564---578; (~) North-Holland Publishing Co., Am~teldam Not to be reproduced by photoprint or microfilm without written permission ~rom the publisher

SOME PROPERTIES

OF C O M B I N E D P C - T R A N S F O R M A T I O N

AND ASSOCIATED SELECTION RULES P. R O M A N

Department o/ Theoretical Physics, University o] Manchester Received 29 April 1957 W e derive t h e t r a n s f o r m a t i o n rules of t h e c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s of b a r y o n a n d l e p t o n fields u n d e r a c o m b i n e d c h a r g e - c o n j u g a t i o n a n d space-reflection operation, a n d find e i g e n s t a t e s of t h e c o r r e s p o n d i n g operator. Also p r o p e r t i e s of t h e ' s q u a r e d ' o p e r a t i o n are studied, g i v i n g a d i s t i n c t i o n b e t w e e n b a r y o n a n d l e p t o n fields f r o m w h i c h s o m e selection rules are derived. ~- a n d K - m e s o n fields axe also s t u d i e d a n d s o m e c o n s e q u e n c e s for K - d e c a y are i n v e s t i g a t e d . T h e n , a s s u m i n g a u n i v e r s a l F e r m i i n t e r a c t i o n , it is p r o v e d t h a t t h e n u m b e r of b a r y o n s is c o n s e r v e d if we d e m a n d t h e inv a r i a n c e of t h e i n t e r a c t i o n u n d e r a c o m b i n e d c h a r g e - c o n j u g a t i o n a n d space-reflection t r a n s f o r m a t i o n . S o m e selection rules for allowed processes are also studied.

Abstract:

1. I n t r o d u c t i o n As a consequence of some experiments a, 2) showing definitely the violation of parity conservation in weak interactions, recently considerable interest has arisen with respect to the less stringent PC-transformation, i.e. a transformation composed of a charge conjugation and a subsequent space inversion (see e.g. ref. a, 4, 5, e)). In this paper we develop the formal properties of this transformation in more detail, with a view to deriving selection rules for processes governed by weak interactions. In particular, it can be seen that baryon number conservation in Fermi interactions is a consequence of PC invariance. 2. F e r m i o n s 2.1. C H A R G E C O N J U G A T I O N

The field operator of any fermion can be written as VJ :

• a p u ( p ) e ' P ' X + b * v ( p ) e -'p'x (1) p where ap is the annihilation operator for a particle with momentum p, and b* the creation operator for the corresponding antiparticle. The adjoint field is * a ( p ) e - ' P ' = + b p ~ ( p ) e 'p'x. (2) The charge conjugation is defined by 564

COMBINED

565

PC-TRANSFORMATION

where C-Ip.C = --~,,

C -1 = C*,

O =

-c.

(2b)

Further, it is well known that u c ~vc - -

Ca

=

v

=

u

(3)

and therefore from (2) ~ve -- • a*v(p) e-'P'X+bpu(p) e 'p'x.

(4)

2.2. INVERSION: THE TWO CLASSES OF FERMIONS

The striking fact that in any observed process the number of baryons minus antibaryons remains unchanged has raised considerable interest /or a longtime. The first attempt of an explanationis due to Yangand TiomnoV). They ascribed definite space-reflection parities to the different fermions and demanded invariance under space reflection.Their argumentation was refined by Votruba 8). The strongest objection against this standpoint was raised by Wigner et al. 9) and explained very clearly by Shapiro 10). It consists in the observation that a system composed of a lepton and a baryon (such as the H-atom) would not form any possible representation of the extended Lorentz-group (it would transform under a reflection according to CH = iCH)" Later the ideas of Yang were indeed dropped and completely different, in fact rather bold considerations were put forward to explain somehow the stability of baryons (see e.g. Lee and Yang 1,) and Ferretti 12)). Now, after the beautiful experiments performed b y Wu et al. (see ref. '. 2)), proving definitely the violation of space-parity in weak interactions, the situation has changed decisively and the question has to be thoroughly re-investigated. In the first place, it is impossible to demand invariance under reflection, in other words, the extended Lorentz group (including inversion) is not a symmetry group of Nature. However, as was shown by several authors (see Landau 3), Salam e), Lee and Yang *)) it is still possible to impose the condition of invariance under a combined PC transformation, i.e. to demand invariance under a charge conjugation followed by a spaceinversion. Therefore we re-adopt the view that baryons belong to the 'real' class and leptons to the 'imaginary' class, i.e. baryons are transformed in case of inversion with -k-Y4 whereas leptons are transformed with 4-i74. It is further well known that [or the real class u and u c = v have the opposite space-parity (one trans/ormation with q-y, and the other with --74) and [or the imaginary class they have the same relative parity. Let us first consider the case of a baryon field.

566

P. ROMAN

2.3. BARYONS

P e r f o r m i n g on (4) the space-inversion transformation, i.e. going over from a right s y s t e m to a left s y s t e m ( x - - > - - x ) and t r a n s f o r m i n g the spinors in the p r o p e r w a y as s t a t e d above, we obtain with regard to (3)

(5)

IpPC = ~ - - a * Yi v ( p ) e/p" x + bp ~'4 u ( p ) e -/p" x Now b o t h u and v satisfy the Dirac equation yipi+Y4c+mC

or

(6)

= o.

v(p) Multiplying with )q one finds

(

E ) y4u(p) +mc or --YiPi+Y4 c y4v(P) -

-

=

(7)

O.

Comparing (7) with (6) one sees t h a t y4u(P) = u ( - - p ) y4v(P) = v ( - - p ) "

(8)

Inserting this into (5) ~vPc = ~ - - a * v ( - - p ) e t p ' X + b p u ( - - p ) e -ip'x = ~--a_*pV(p)

e-ip'x+b_pu(p)

e ip'x.

(9)

Comparing this result (9) with (1) we see t h a t the PC t r a n s f o r m a t i o n a m o u n t s in case of a baryon field to the following t r a n s f o r m a t i o n of the creation a n d annihilation operators: ap --~ Qap-(2 -1 = b_p,

b$ --~ Qb~Q -1 = --a*_p.

(10)

In a similar fashion (or directly from (10)) we obtain also ap* -> Da* I2 -1 = b ' p ,

bp -+ Dbp ~2-1 = -- a p .

(11)

L e t us investigate now the corresponding t r a n s f o r m a t i o n of definite physical states. F o r instance, a o n e - b a r y o n state with m o m e n t u m p

~ n(p)

* 40

= ap

(where 4 ° is the v a c u u m state) behaves as follows: (~n(p) ---> ~r~)n(p) :

t h a t is, larly,

~¢~a~ 4 0 = ~¢2a~ ~¢~-1 ~¢~0 :

b*pq~O ~ (~ii(-p),

goes over into a one-antibaryon state with opposite momentum. ~ ( p ) --> ~¢'2(~
Simi-

--~n(--p)"

One-particle states are therefore no eigenstates of the PC-operation ~2. B u t consider e.g.

COMBINED PC-TRANSFORMATION

56~

1 = - ~ (apa_p--bpb_p) 1 fl = ~-~ (apa_p+bpb_p).

(12)

By (10) and (11) one finds, taking into account the anticommutation rules {ap, a p} = {bp, b_p} = O, that O~ "--P Q~z,Q - 1 =

_~ ~ Q - 1

(X

(13)

= _~.

Therefore the symmetric and antisymmetric states composed of a system containing two baryons a n d another system containing two antibaryons, the particles in each having opposite momenta, will be antisymmetric or symmetric eigenstates of g2, respectively: 1 1 V 2 ($.(p)a(_p)+~.~(p)~(_p)) ~ =[z ~

(~bn(p)n(_p)_/_cb.~n(p)~(_p)).

(14)

2.4. LEPTONS

Now let us turn to the properties of lepton fields. Equation (4) will of course be unchanged, but when performing the inversion we have to multiply both terms with +i74, and therefore we obtain with (8) ~vPc = ~ia*_pV(p) e - ' p ' X + i b _ p u ( p )

e 'p'=.

(15)

Comparing this with (1), a~ -+ Q a p ~ - 1 = ib_p,

b* -+ ~b.*Q-~ = ia_*p

(16)

and similarly ap* ~ Q a ~ Q -I :

--ib*, bp -+ Q b p Q -1 = --ia_p.

(17)

One-lepton states are again not eigenstates, e.g.

Here the ~ and ~ combinations (12) behave also in an opposite way: one finds indeed _+ ~ - 1

__ ~

(IS)

and therefore I

1

(19)

568

P. ROMAN

One can also investigate 'mixed' composed states and find e.g. 1 1 V ~~ (¢n(p)l(p) ~- (~(--p)l(--p)) --> : ~ i ~ - ~ (¢n(p)l(p)-~-¢~(--p)i(--p))"

(20)

2.5. SELECTION RULES More interesting is, however, the investigation of a 'squared' PC-transformation, i.e. the properties of the operator Q~. One easily obtains with the aid of eqs. (10, 11) and eqs. (16, 17) respectively, the important fact t h a t

both one-baryon (antibaryon) and one-lepton (antilepton) states are eigenstates o~ ~22, with opposite eigenvalues. In fact ~14,o = -,/,.,

~¢~ = -~

(21)

~ ¢ , = +¢1,

Q2¢~ = +¢~.

(22)

One further finds that also two, t h r e e . . , baryon, antibaryon, lepton, antilepton and mixed one-baryon -- one-lepton etc. states are all eigenstates of D 2 alternatingly with eigenvalues -4-1, and one can write for any state Q 2 = (__ 1)N+N (23) where N, ~ are the occupation number operators for baryons and antibaryons. Now we can postulate that all interactions should be invariant under PC-transformation. This imposes by (23) the selection rule Nt+-Nt = Nt+2Vr+2k

(23a)

where the subscript t refers to the initial and f to the final state, and k means any integer. No restrictions follow for the number of leptons involved. This selection rule prohibits a number of processes violating conservation of baryon number, such as n ~ I+1+i n -+ n + n + i n -+ n + f i + l

n ~ n+n+i but still allows n-~ fi+l+l. For the latter a more detailed discussion is needed (see section 6.1), 3. ~ - M e s o n s

The pion field can be written as ~--- ~ p p e

• X+ q p* e - i p ' x *p"

4" ~-~ Z qp eiP'x+ib~ e-iP'X • -X * (~0 ~--- 2 Sp e 'p --~Sp

e -`p'x

(24)

COMBINED PC-TRANSFORMATION

509

where the operators have the following meaning:

creation annihilation

p* p

q* q

s* s

Charge conjugation means 7~+ ~ ~--, 7~-- ~

~+~ 7l:0 ---> 7~0

and can be obtained e.g. b y

p~q,

q~p,

s-~s.

The charge-conjugate fields of eq. (24) are therefore 6c = X qtp e ' P " + p * e -'p'x ¢*e = • pp e,p'X+q* e-,p'= ¢~ = X Sp e i p ' ' + s * e -'°'x

(25)

The consecutive inversion is obtained simply b y x --> - - x and b y taking into account the pseudoscalar property of z, i.e. b y introducing a negative sign. After changing the summation index from p to --p, one has finally ePC = - - X q-p e'p'x+~b*p e-'p'x ¢.ec = _ • p_p e,p.~+q.p e-,p.x (26) ¢oPc = - - X S_p e'P'X+S*o e -'p'~ Comparing this with eq. (24) we see that for pions the PC-transformation means

pp -~ - q _ p q,, -~ - p _ p

q~ ~ -p*_p p* -~ - q * p

Sp ~

~t Sp ---> - - S * p .

--S_p

(27)

One meson states are of course again no eigenstates, e.g. ¢.+(p) -->" ~¢~+(p) =

--¢~-(_p) .

However, in this case, contrary to the fermions, there are relatively simple one-particle eigenstates. So one finds e.g. immediately that 1

1

~ / 2 (¢nO(p)-~-~(_p}) - + :~: ~

(¢nO(p)-~-¢nO(_p))

(28)

(¢n+(P)-~-~)~-(--P))"

(2o)

and 1

1

V 2 (¢n+(P)2r-¢n-(--P)) - ~ :~ ~

We mention also that here, of course, we have always

~2=

+1.

(30)

570

P. ~OMAN

4. K - M e s o n s 4.1. TRANSFORMATION PROPERTIES

The K-meson field can be described by

= ~ %u e'P'=+b*v e-~p'=+%v e'P'*+d~u e -'p'~ 9 " = ~ ap*U* e - ~ P ' X ÷ b p V * e *p

•X

*

÷ ¢ p o* e - ~ P ' X ÷ d p u * e ~p'x

(31)

where the meaning of the operators is

creation annihilation

K+

K°

K-

a* a

¢* ¢

b* b

~o d* d

m

(K +, K °) and (K-, K °) are understood, as usually, to be distinguishable particles and antiparticles, forming in isobaric space a two-component spinor. The functions u and v in eq. (31) are the corresponding isospinors, depending only on isobaric variables, u referring, say, to isospin +½ and v to isospin --½. Charge conjugation means K + --> K-, K ° --~ K °, which is achieved by a-+b,

c---~ d etc.,

and, therefore, from (31),

~o = 7~ bpu e'P'*+a* v e-'~'*+d~v e'~'*+c* u e-'P'* and similarly for 9 *0. The subsequent inversion yields, if considering the K to be a pseudoscalar or scalar, respectively, 9 l'c ----- T ~ , b_pu e~P'X+a_*pV e-~P'f+d_pv e-O'=+C_*pU e -~p'x

(32)

Comparison with (31) shows that ap -+ :7~b_p,

Cp --~ T d _ p

b~ "-+ =[=a*p,

d ; --> =FC_*p

(33a)

and, similarly, ap* -+ T b*_p,

* --+ =Fd_*p Cp

bp --+ =~a_p,

dp --+ :~Xc p.

(33b)

An important eigenstate is the linear combination of a K ° with momentum p and a K ° with opposite momentum. In fact, 1 p:~x 1 ~ / 2 (~K0(p)'-~-~K°(--P)) ~ S: -~- ~//----2(¢K°(P)-~-~K°(-P))

and, similarly, for the superposition of K+(p) with K - ( - - p ) .

(34)

COMBINED PC-TRANSFORMATION

571

4.2. K-DECAY

The decay 0 ° --> zr++rr -

(35)

was frequently observed. Now, in the centre-of-mass system, the final state is (for a decay at rest)

¢,,+lpl,,-c-pl = Pp* Cp #. With (27) one finds immediately that this is an eigenstate of K2 with the eigenvalue + 1. Therefore, assuming conservation of f2 in any process, the initial state has to be also an eigenstate with f2 ~-- 1. The superposition 1

¢01 ~

~

(¢KO(p)--¢~O(_p))

(36)

has just this property (for pseudoscalar K) as seen from (34). Therefore the 0° state undergoing the decay (35) is just this superposition. One can now understand w h y a three-particle decay o] uncharged Kmesons never was observed. The final state n + + n - + : ~ 0 is (in the restsystem of the :#, say) of the type = g p ~t_p ~0

This is also an eigenstate of f2, but bdonging to the eigenvalue £2 = --1. Therefore state (36) cannot disintegrate in this fashion. Of course the superposition 1

¢o, --

(37)

can very well undergo a three-pion decay. But, because of the smaller available phase-volume, this has a smaller probability. Further, in an actual experiment, a beam of 'pure' K°'s is produced, which of course can be written as a superposition of 01 and 0~: 1 cKo =

(¢0,+¢0,).

But from this wave packet first the ¢0x will rapidly decay and one could only observe the 3z~-decay when the 0~-part is predominant. As one has generally only weak beams, this will hardly be detectable, the less so if 0~ has also an other decay-channel (e.g. into/~+ and/z-, which is, as is easily verified, not possible for 01). 5. Transformation

of Fermi

Interaction

In the following sections we will treat the case of four interacting fermion fields in more detail. For this purpose it is more convenient to work in configuration space. We shall denote any baryon field b y the field operator

572

P. ROMAN

~v~ and any lepton field b y Wt. The charge-conjugate field will be, as in general, identified with the corresponding antiparticle field t and thus ¢

C

= w . , ~ = ~7, 9~ = ~ ,

~ = ,e~

(3s)

where ~1)¢ = C ~ , ~'c = 9 e =

(C--l"~)~o

(38a)

Now determine the behaviour of the fields under the combined PC-transformation. Following the suggestions of section 2.2, we obtain the following transf6rmation rules: W~--~ 74 ~v~ 9n -+ 9~Y4 ~v~ --> - - 7 4 ~'n 9~ --> - - g n Y4 (39) ~vl --> i74 ~vi 91 ~ - - i 9 w 4 y~ --+ iy4 ~ l --> --i91Y4" (Remember that inversion and charge conjugation as well are interchangeable with adjunction.) Forgetting all we said in the preceding, we could also start from (39), taking it as the definition o/ the P C trans]ormation. Now the most general Hermitian Fermi interaction between any four Dirac fields has the form L = 9a P r Wb 9c ]ws '~pd-Ii-~Pd*I ~ : ~o - * ~b* r * 9 " .

(40)

Here a, b, c, d refer either to particles or to antiparticles and we have chosen the matrices /'~ in the following w a y TABLE

Type S P A V T T'

1

r

/', 1 75 iy5 ~u

112i (Tu ~',,-- r, ;.'~,) 1/ 2i (~,V7,-- ~, YJ*)Y5

Of course, in (40) summation over the components of F, and F, is understood (tensor product). The second term in (40) can be rewritten in a more advantageous way:

w* 1 " 9 " w* r * 9* = 9dr, r * r, Wogbr, r * r,w. = o,,o,~9drswo9br,~a where the o~ are defined b y r 4 F * = oJ, F . r4.

With help of the definitions and numerical annotation of table 1 one finds immediately for the possible combinations (see below) ~" We do not adopt the two-component theory of the neutrino.

COMBIS~D PC-TRA~SFORMATIO~

57~

and therefore eq. (40) can be written L = ~ F, y~b~c F, ~Vd+ (-- 1)'+* ~d F, ~Pc~b F, ~a.

(40a)

Let us introduce a simplification in notation. Define X~b ~ ~aF, W etc.;

(41)

' rr L = X ~"b X o*d + ~/- - . jl ~ ' + ' Y"'d¢''b~"

(42)

we then have As we did not choose, as is usually done, r = s, this interaction will not in general be inversion-invariant and thus we allowed for parity violation. But so as to ensure invariance under the restricted Lorentz-group, some restrictions are of course necessary: e.g. one cannot combine a scalar and a vector, etc. Anyway, the eleven possibilities encountered are Group I

SS SP

( r = 1, s---- 1) ( r = 1, s = 2)

PS

(r=2, s=l)

PP

(r=2,

s=2)

Group II

AA AV VA VV

(r---- 3, (r----- 3, (r=4, (r=4,

s = 3) s=4) s:3) s:4)

Group I I I

TT TT' T'T

(r ---- 5, s = 5) (r---- 5, s : 6) (r = 6, s = 5)

The interaction (42) is a little ambiguous in so far as the order of the four fermion fields is not fixed. But this can be remedied by making use of Michel's theorem is) according to which any permutation of the fields is equivalent with a suitably chosen linear combination of all possible interaction types, the order of the fields taken in a fixed (but arbitrary) manner. (Of course, instead of the 5 possible interactions in the older parity conserving theory we have now to sum over the 11 possibilities listed above.) This will be understood throughout. Now we are going to calculate the change of the interaction (42) under the combined PC-transformation (39). Using the definitions (41) and those of table 1, one finds immediately for S, A and T-types the transformations listed in table 2, and for P, V and T' the same changes but with opposite sign. (To save space we have written symbolically simply (ab) instead of Xab, and, similarly for X~b, the transformed quantity). The values listed are symbolical also in the sense that in the A, V, T-case they refer to the spatial part; the time-components carry an opposite sign, but evidently this will cause no confusion in the following.

574

P. ROMAN TABLE 2 Xtab

Xab

Xab

X'ab

(nn)

(~)

(nl)

(nh) (~n) (~)

(~i)

-- (~n)

(ni)

i(~1)

-

(li) (il) (ii)

(r~)

(~1)

(nn) ([l)

(hi) (In)

(il)

(]~)

(li)

(in)

(n)

{i~)

-i(ni) -i(nl) -i(ifi) i (in) -~(m) i (In)

Comprising all possibilities into one formula, X:~ ---- (--1)rYx7 (a, b)X:b

(43)

where y(a, b) is - b l or + i , according to table 2. Now we are p r e p a r e d to calculate in complete generality the trans/ormed interaction. B y m a k i n g r e p e a t e d l y use of (43) one obtains from (42) b y a s t r a i g h t f o r w a r d calculation L ' = ~ [ ( - 1 )r+s X~bX~d+Xa~X~ r_ s_ s r 3 (44) with 0t = y ( a , b) • y(c, d)

(44a)

As we demand invariance under the combined PC-trans/ormation (39), only those interactions are allowed, /or which (44) and (42) are identical. T o c o m p a r e eq. (44) a n d eq. (42), some more or less familiar connections between X:b and X ~ etc. h a v e to be t a k e n into account. F o r convenience we summarise t h e m here. As a consequence of eqs ( 8 8 ) ' a n d (2b) we h a v e

~ = -~,,,~c-~, w~ = c G ~

(45)

and therefore e.g.

¢,, V',, = (--~,, °c-~) (CCb °) = --~',,°'h, ° = --(~b~ W,,~) -. Now we m a y either assume t h a t all fermion fields a n t i c o m m u t e , or t h a t

identical /ermion fields anticommute, but different /ermion fields commute. In view of the conclusions which will be reached in sections 6.1 and 6.2, we shall only develop in detail the consequences of the l a t t e r assumption. We t h e n have

- (G~ G~)" = ~

(w~ G~) = ,1~b G~ w~ = ,1~b ~b ° w~°

where Aab ---- + 1 if a = b, and Aab = --1 if a :/: b. (We have suppressed the (infinite) value of the singular S-function, which amounts, as is well known, to the s u b t r a c t i o n of the v a c u u m value. T h u s L has to be u n d e r s t o o d as the N - p r o d u c t of the operators involved.) So we have obtained (applying the definitions (38))