By S. L. Dashen, R. F. Adler
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Extra resources for Current Algebras And Applications To Particle Physics
E q . 87) i s q u i t e specific t o t h eq u a r k m o d e l ; f o re x a m p l e , i t d o e s n o t h o l d i n t h e o c t e t a - m o d e l , a s t h er e a d e r c a ne a s i l y v e r i f y . C o n sequences ofthe''quark current hypothesis" a r e discussed b e l o w b yB i e t t i [ P a p e r 16] a n d b y B j o r k e n [ P a p e r 2 2 ] . 4. CURRENT DIVERGENCES M o s t o ft h eapplications o f c u r r e n t a l g e b r a s d i s c u s s e d i n this book also require knowledge o ft h edivergences o f t h e v e c t o r a n d t h ea x i a l - v e c t o r octets.
T h e t i m e c o m p o n e n t s o f t h e v e c t o r a n d a x ial-vector c u r r e n t s i n this m o d e l satisfy t h ea l g e b r a SUg x SU2, and thedivergence of theaxial-vector current is proportional to t h epion field. ^ 0 ] , a n o c t e t o f s c a l a r m e s o n s or = ( a ^ , . . , Q g ) , a u n i t a r y s i n g l e t p s e u d o s c a l a r m e s o n tt a n d a n o c t e t o f p s e u d o s c a l a r m e s o n s tt. T h e L a g r a n g i a n density i s * £ = - q [-iy^a^, etc - g o ( ^ (J' + ^ i ^ ^ s + <^'>' + i7J->^r5)]q * I n t h e e q u a t i o n s t h a t f o l l o w , (7 • X i s s h o r t h a n d f o r 2 ?
77) P V ^ Hk _ P V _ r 5X ~ ^ k H ~ ^^kX^£ 5 X - , ^ k x ^ i J s (PC and P Vdenote, respectively, parity-conserving and p a r i t y - v i o l a t i n g ) , a n d u s e E q s . 14d) t o w r i t e in t h e f o r m eff ni ^ n£PC _ n£PV , . with + 2 s i n e ^ + T 2 V )] cose^(T^'' (1-79) n i P V The parity-violating piece £ i s g i v e n b ya n e i d e n t i c a l t o E q . 79), b u t w i t h P C r e p l a c e d e v e r y w PV. The extended current algebra determines the tion relations o f F^^a n d F ^ with t h ev a r i o u s t e n s o r [F,(xO), T f ^ ( x ) ] = if,,,T^;^(x) .