γ5-invariance and fermion mass

Volume 4, number 3

PHYSICS

75-INVARIANCE

LETTERS

AND

1April 1963

FERMION

MASS

*

W. THIRJtING Institute for Theoretical Physics, University of Vienna, Austria Received 12 March 1963

R e c e n t l y S c h w i n g e r 1) p r e s e n t e d a n e x p l i c i t e x a m p l e of a gauge i n v a r i a n t t h e o r y w h e r e the photon a c q u i r e s a m a s s . T h e m e c h a n i s m which p r o d u c e s t h e photon m a s s i s f o r m a l l y i d e n t i c a l with the one which p r o d u c e s the M e i s s n e r O c h s e n f i e l d effect 2). T h e p r o p e r v a c u u m p o l a r i s a t i o n p a r t i s in both c a s e s p r o p o r t i o n a l to the t r a n s v e r s a l p r o j e c t i o n o p e r a t o r t i m e s a f u n c t i o n which h a s a f i r s t o r d e r p o l e f o r k2 = 0. T h i s i m p l i e s a t o t a l q u e n c h i n g of the e x t e r n a l c u r r e n t a n d h e n c e l e a d s to the P r o c a o r London e q u a t i o n s r e s p e c t i v e l y . In t h i s n o t e we want to p o i n t out that the s a m e s i t u a t i o n , n a m e l y a pole i n the p r o p e r s e l f e n e r g y p a r t l e a d s to a f e r m i o n m a s s i n a ~ ' 5 - i n v a r i a n t t h e o r y . On g e n e r a l grounds ~5-invariants implies either mass zero or a d e g e n e r a c y 3) ** c o r r e s p o n d i n g to f e r m i o n C P d u b l e t s . H o w e v e r , no d y n a m i c a l m e c h a n i s m h a s b e e n g i v e n which could r e a l i s e the s e c o n d p o s s i b i l i t y . Although i t i s d i f f i c u l t to find a n e x p l i c i t and r e a l i s t i c m o d e l which shows t h e s e f e a t u r e s we s h a l l s e e what the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s a r e f o r a m a s s to e m e r g e i n a v 5 - i n v a r i a n t t h e o r y . In a C P - i n v a r i a n t r e l a t i v i s t i c t h e o r y the f e r m i o n p r o p a g a t o r i s of the f o r m 1 s(p)

-

mo

_ P z(p)

'

(1)

w h e r e r. i s the p r o p e r s e l f e n e r g y p a r t . F o r Yuhawa c o u p l i n g s the c a n o n i c a l c o m m u t a t i o n r u l e s i m p l y lira ~ - 0 and we h a v e the s p e c t r a l r e p r e s e n t a t i o n ** *

r,(p) = ? --OO

da p(a) a-~O

(2)

F o r m e t r i c and e n e r g y s p e c t r u m both p o s i t i v e p >/0. In a 7 5 - i n v a r i a n t t h e o r y s(-p) = - S(p) so that m o = 0, p(-a) = p(a) and h e n c e * The research reported in this document has been partly sponsored by the U.S. Government. ** We do not consider a degenerate ground state as Z umino 4). *** Note the lower limit of integration which makes only one spectral function necessary.

a - p2

-oo

"

F r o m t h i s f o r m it i s u s u a l l y c o n c l u d e d that S h a s a pole f o r ig = 0 and h e n c e that t h e r e a r e m a s s l e s s f e r m i o n s . T h i s i s t r u e u n l e s s p i s of the f o r m

p(a) = cS(a) + ½(r(a) ,

(r -= 0

for

ral < M .

(4)

In t h i s e a s e the p o l e s of S o c c u r f o r p2 = Z ff

D(Z) : Z - C + f ~ da__~(_a)Z _ 0 . M

(5)

a2 - Z

Since D(0) < 0 a n d D ' > 0 f o r Z < M 2 we have one z e r o if D(M2) > 0 o t h e r w i s e n o n e . In the f i r s t c a s e t h e r e i s a p a i r of f e r m i o n s with m a s s = 0, in the s e c o n d t h e r e a r e no s t a b l e f e r m i o n s . In p a r t i c u l a r if Z i s p r o p o r t i o n a l to S, o r =0,

c =rn2,

(6)

w e have s(p) = - ~

+

,

(7)

e . g . , a f e r m i o n d o u b l e t with m a s s rn. T h i s kind of s i t u a t i o n i s r e a l i s e d f o r the photon p r o p a g a t o r in two d i m e n s i o n a l e l e c t r o d y n a m i e s w h e r e the s p e c t r a l f u n c t i o n of the p r o p e r s e l f e n e r g y p a r t i s s i m p l y (e2/rr)6(a). It i s not e a s y to find a r e a l i s t i c d y n a m i c a l m e c h a n i s m which g i v e s a of the f o r m (4) f o r f e r m i o n s . H e n c e t h e s e c o n s i d e r a t i o n s m a y be of a c a d e m i c i n t e r e s t only. Howe v e r , they show at l e a s t w h e r e t h e r e i s a hole in the u s u a l a r g u m e n t s .

1) Schwinger, Phys. Rev. 128 (1962) 2425. 2) Schafroth, Phys. Rev. 100 (1958) 463, 3) W. Thirring, Nuclear Phys. 10 (1959) 97. B.Tousehek, Nuovo Cimento 13 (1959) 395. 4) B.Zumino, Heisenberg Festschrift (Vieweg, 1961) p. 234.

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