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Soft pion production in deep inelastic electromagnetic and weak processes
Volume 37B, number 1
SOFT
PHYSICS
PION
PRODUCTION AND
LETTERS
IN DEEP INELASTIC WEAK PROCESSES
1November 1971
ELECTROMAGNETIC
B. L. I O F F E
Institute for Theoretical and Experimental Physics, Moscow, USSR Received 5 July 1971
On the basis of the PCAC hypothesis and current algebra, we calculate the cross section of one and two soft pion production in the deep inelastic scattering of electrons and neutrinos on nue[eons. The experimental measurement of these processes would make it possible to investigate the one-nucleon m a trix elements of the axial and isovector current commutators and contribute to elucidation of the scale invariance mechanism. As is w e l l known, the t o t a l c r o s s s e c t i o n s of the e l e c t r o n and n e u t r i n o - n u c l e o n s c a t t e r i n g a r e e x p r e s s e d t h r o u g h the c u r r e n t c o m m u t a t o r m a t r i x e l e m e n t s . It a l l o w s one in p a r t i c u l a r , u s i n g the e x p e r i m e n t a l d a t a r e l a t e d to the d e e p i n e l a s t i c r e g i o n , to find the b e h a v i o u r of the c u r r e n t c o m m u t a t o r o n e n u c l e o n m a t r i x e l e m e n t s n e a r the light c o n e [1]. H o w e v e r , u s i n g only the d a t a on the n e u t r i n o and e l e c t r o n - n u c l e o n s c a t t e r i n g one c a n n o t s e l e c t s e p e r a t e l y the v e c t o r and a x i a l c u r r e n t c o n t r i b u t i o n in c a s e of the n e u t r i n o s c a t t e r i n g and i s o v e c t o r and i s o s c a l a r c u r r e n t c o n t r i b u t i o n in c a s e of e l e c t r o p r o d u c t i o n . T h i s p r o b l e m m a y be s o l v e d by c o n s i d e r i n g the e l e c t r o p r o d u c t i o n p r o c e s s e s w i t h one o r two soft pion p r o d u c t i o n b a s i n g on the P C A C h y p o t h e s i s . (An a n a l o g o u s p r o b l e m f o r the a n n i h i l a t i o n p r o c e s s e + + e h a d r o n s w a s c o n s i d e r e d by P a i s and T r e i m a n [2].) L e t us f i r s t c o n s i d e r the e l e c t r o n - p r o t o n d e e p i n e l a s t i c s c a t t e r i n g w i t h the soft c h a r g e d ~ m e s o n p r o d u c t i o n . T h e pion m o m e n t u m in lab. s y s t e m w i l l be t a k e n a s s m a l l , I k i / t x ~ < < l . T h e h a d r o n i c p a r t of the m a t r i x e l e m e n t of the p r o c e s s e + p .... e + 7r+ (soft) + h a d r o n i c s t a t e h can be written as +
M p = f d4xexpikx ~J -
p2~r) (h
IT{co(x)l jep1 (0)}i p>
(1)
w h e r e ¢ ( x ) i s the m e s o n i c field, j~l the e l e c t r o m a g n e t i c c u r r e n t of h a d r o n s ,
jel= ½(j(3p)+ flOp)). Sub-
stituting ~(x) into eq. (1) as the axial current divergence
= ca~Ap/p27r;
c = 1/frr = 1 / 0 . 9 3 / x ~
(2)
c o n s i d e r i n g k to be s m a l l and u s i n g the c o n v e n t i o n a l P C A C t e c h n i q u e we get + = M~ c ( h l A ~ ( 0 ) lP> +
ickv fd4xexp(ikx)
.el
3p
(0) l}p)
(3)
T h e s e c o n d t e r m in the r i g h t - h a n d p a r t of eq. (3) is c o n t r i b u t e d only by the p o l e t e r m s c o r r e s p o n d i n g to the soft p i o n e m i s s i o n by i n i t i a l and f i n a l n u c l e o n s as w e l l a s by the p o l e t e r m a p p e a r i n g due to the a b s o r p t i o n of the v i r t u a l photon by the soft pion. It is o b v i o u s , that at l a r g e q2 (q b e i n g the v i r t u a l p h o ton m o m e n t u m ) the p o l e t e r m c o n t r i b u t i o n of the l a t t e r type is s m a l l , < k/Iq 2 L1/2. When c o n s i d e r i n g the c o n t r i b u t i o n of the p o l e t e r m s c o r r e s p o n d i n g to the B r e m s s t r a h l u n g of the soft p i o n s by n u c l e o n s , we s h a l l t a k e the c a s e of l a r g e e n e r g y t r a n s f e r r e d to h a d r o n s but not enough that the n u c l e o n - a n t i n u c l e o n p a i r p r o d u c t i o n w o u l d g i v e a n o t i c e a b l e c o n t r i b u t i o n into the t o t a l c r o s s s e c t i o n of the e l e c t r o p r o d u c t i o n . T h e n the f i n a l s t a t e , b e s i d e s m e s o n s , w o u l d c o n t a i n only one b a r y o n . The p o l e t e r m c o r r e s p o n d i n g to the soft pion e m i s s i o n by i n i t i a l n u c l e o n is p r o p o r t i o n a l to O(/~ -
k
-
m) -1 Y#Y5
u(p)kp
(4)
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Volume 37B. n u m b e r 1
PHYSICS
LETTERS
1 November 1971
w h e r e u(p) i s t h e s p i n o r d e s c r i b i n g t h e i n i t i a l s t a t e of n u c l e o n w i t h t h e m o m e n t u m p, O i s s o m e o p e r a t o r w h i c h a t s m a l l k i s s u p p o s e d to b e i n d e p e n d e n t of k. N e g l e c t i n g t h e t e r m s q u a d r a t i c in k, eq. (4) m a y b e e a s i l y r e d u c e d to t h e f o r m
-o
k "o .(p) 1 / k o
(5)
a n d , c o n s e q u e n t l y , a t ] k ] ]k o << I t h e c o n t r i b u t i o n of t h i s p o l e t u r n s out to b e s m a l l . T h e pole t e r m a p p e a r i n g in t h e s o f t p i o n e m i s s i o n b y t h e f i n a l n u c l e o n is c a l c u l a t e d a n a l o g o u s l y . H e r e t h e f i n a l n u c l e o n m o m e n t p ' m a y b e c o n s i d e r e d a s s m a l l , IP' ] / m << 1. I n d e e d , a c c o r d i n g to t h e pp s c a t t e r i n g e x p e r i m e n t a l d a t a [3,4], t h e s q u a r e d m o m e n t u m t r a n s f e r to h a d r o n s i s of o r d e r -1 = p , 2 (0.3 - 0 . 4 ) m 2. It i s n a t u r a l to e x p e c t t h a t in t h e c a s e of e l e c t r o p r o d u c t i o n a n a n a l o g o u s s i t u a t i o n w i l l t a k e p l a c e a n d t h e t e r m s of t h e o r d e r of p ' 2 / m 2 m a y b e n e g l e c t e d w i t h t h e a c c u r a c y of o r d e r of s e v e r a l d o z e n s p e r c e n t . C a l c u l a t i n g u n d e r t h i s a s s u m p t i o n t h e p o l e t e r m c o r r e s p o n d i n g to t h e e m i s s i o n of t h e s o f t p i o n by t h e f i n a l n u c l e o n , we find t h a t up to t e r m s of h i g h e r o r d e r in IP' !/m a n d I k ! / k o it i s p r o p o r t i o n a l to
,,(Y)r/5(~'+/~-.,)-1 o k,~(p') ( mi.e. a l s o g i v e s a s m a l l c o n t r i b u t i o n . It m a y b e e x p e c t e d t h a t j u s t t h e s a m e the s m a l l c o n t r i b u t i o n i n t o eq. (3) w i l l b e g i v e n b y t h e e m i s s i o n of s o f t p i o n s by h y p e r o n s o r b a r y o n i c r e s o n a n c e s a p p e a r i n g in t h e final state. N e g l e c t i n g t h e c o n t r i b u t i o n of t h e s e c o n d t e r m in t h e r i g h t - h a n d p a r t of (3) we get f o r t h e c r o s s s e c t i o n of t h e p r o c e s s e + p - * e + ~+ (soft) + h a d r o n s de7r+
d~dvdE
_ 47ra 2 E '
Av
1
A P ( v ' q2)}
q4 m E { c°s2½° w2 (v, q2)+2sin 2 ~ o w 1
k
(2~)2/2
U)
Ag w h e r e E, E ' a r e i n i t i a l a n d f i n a l e n e r g i e s of t h e e l e c t r o n , 0 i s i t s s c a t t e r i n g a n g l e , v = pq = mE, w 1 w A~ a r e i n v a r i a n t f o r m f a c t o r s due to t h e a x i a l c u r r e n t ( i n d e x Y i n d i c a t e s t h e i d e n t i t y of t h e s e f o r m f a c t o r s w i t h t h e f o r m f a c t o r s due to t h e a x i a l c u r r e n t in t h e p r o c e s s v + p - [ + h a d r o n s ) . Its i s o b v i o u s t h a t f o r t h e c r o s s s e c t i o n of t h e s o f t ~ - p r o d u c t i o n w i l l h o l d a n a n a l o g o u s e x p r e s s i o n r e p l a c i n g ~ -~ v, w h i l e t h e c r o s s s e c t i o n of t h e s o f t ~o p r o d u c t i o n m u s t v a n i s h in t h e a p p r o x i m a t i o n u n d e r c o n s i d e r a t i o n . L e t u s now d i s c u s s t h e a c c u r a c y of t h e r e s u l t o b t a i n e d . A s it w a s m e n t i o n e d a b o v e t h e p o l e t e r m c o n t r i b u t i o n we h a v e n e g l e c t e d in eq. (3) m a y b e of the o r d e r of s e v e r a l d o z e n s p e r c e n t . (It m a y b e s h o w n t h a t t h e i n t e r f e r e n c e of t h e p o l e a n d n o n - p o l e t e r m s in eq. (3) l e a d s to t e r m s of t h e s a m e o r d e r in t h e t o t a l c r o s s s e c t i o n ) . T h e s a m e i s t h e a c c u r a c y of t h e P C A C at s m a l l e n e r g i e s . In c a s e of h i g h e n e r g i e s of t h e v i r t u a l p h o t o n , t h e P C A C h y p o t h e s i s m a y b e , in p r i n c i p l e , i n a p p l i c a b l e b e c a u s e of t h e v a l u e qk w h i c h in t h e P C A C m e t h o d i s c o n s i d e r e d to b e s m a l l , i s r e a l l y not s m a l l (qk .-~ qo Prr) a s c o m p a r e d to t h e m a s s e s of v i r t u a l p a r t i c l e s . In p a r t i c u l a r , t h i s s i t u a t i o n m a y h o l d f o r t h e p r o c e s s e s of t h e s o f t p i o n p r o d u c t i o n a t t h e e l e c t r o n - n u c l e o n e l a s t i c s c a t t e r i n g a t ]q2 [ >> m2 ( c o n s i d e r e d in r e f s . [5] a n d [6]) o r at h i g h e n e r g y p h o t o p r o d u c t i o n * . T h e d e e p i n e l a s t i c r e g i o n i s m o r e f a v o u r a b l e f r o m v i e w of t h e P C A C a p p l i c a b i l i t y , a s it m a y b e e x p e c t e d t h a t due to t h e s c a l e i n v a r i a n c e w h i c h h o l d s in t h i s r e g i o n , t h e m a s s e s of t h e v i r t u a l s t a t e s a r e i n e s s e n t i a l h e r e a n d qk m a y e n t e r only a s r a t i o s qk/q 2, qk/qp, s o t h a t t h e a c c u r a c y w i l l b e of t h e s a m e o r d e r a s a t s m a l l e n e r g i e s . L e t u s now c o n s i d e r t h e d e e p i n e l a s t i c e l e c t r o p r o d u c t i o n i n v o l v i n g t h e two s o f t p i o n e m i s s i o n . In c a s e of t h e s o f t g+Tr-, t h e h a d r o n i c p a r t of t h e m a t r i x e l e m e n t w i t h p r o d u c t i o n of h a d r o n i c s t a t e h c a l culated using the conventional PCAC technique, is M + - : c 2 ( h I j ! 3) (0) !p) #
(8)
* If -/-quantum emits the soft pion going over iat9 some virtual state A, the propagator of this state D~ 1 equals to D-A1 - q2 _2kq+k 2 - M R and at not v e r y large q2 but large v essentially depends on qk ifM£~ is not large. On the ot~er hand. as it was "shown in ref. [1] in the virtual photon-nucleon s c a t t e r i n g at s m a l l qZ and large v of i m p o r tance are the large longitudinal d i s t a n c e s , i.e. finite (and comparatively small) MA. Thus. it should be expected that the PCAC method would appear to be inapplicable for the electroproduction with the soft pion e m i s s i o n at Iq2 i< m 2 and v >> m 2 and for the photoproduction at v>> m2.
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Volume 37B, number 1
LETTERS
1November 1971
In eq. (8) we n e g l e c t the t e r m c o n t a i n i n g the c o m m u t a t o r [A + (x), }~)~A~(0)]Xo= 0 p r o p o r t i o n a l , a s it is u s u a l l y e x p e c t e d to the s m a l l p a r a m e t e r p2. The c o n t r i b u t i o n in M +- of the pole t e r m s c o r r e s p o n d i n g to the B r e m s s t r a h l u n g of the soft pions by i n i t i a l n u c l e o n (baryon) is s m a l l for the s a m e r e a s o n s as in c a s e of one pion e m i s s i o n . The c o n t r i b u t i o n in M ~ - of the pole t e r m s due to the e m i s s i o n of two soft p i o n s by the fast pion (or n u c l e o n ) is a l s o s m a l l b e c a u s e of that, a s it was shown by W e i n b e r g [7], in this c a s e the soft p i o n s m u s t be e m i t t e d in the p state. C a l c u l a t i n g with the help of eq. (8) the c r o s s s e c t i o n of the e l e c t r o p r o d u c t i o n of the soft 7r+~- m e s o n s with m o m e n t a k l , /(2, we have
dan+Tr-
_ 4nX 2 E' {cos 2
(3 2sin 2 ½0 ( 4 Ikl i ik2 ! ½0 w2)(~' q2) + Wl3)(v' q2)~ (2~)~4-f~ ---
q4 m E
dq2 d v d E 1 dE 2
(9)
w h e r e w (3), w (3) c o r r e s p o n d to the c o n t r i b u t i o n of the i s o v e c t o r p a r t of the e l e c t r o m a g n e t i c c u r r e n t ½j(3). The c r o ~ s s e c t i o n s of the e l e c t r o p r o d u c t i o n with p r o d u c t i o n of the soft ~+Tro o r ~-~o m e s o n s a r e ~ o b t a i n e d f r o m eq. (9) by r e p l a c i n g w(3) by 4 w v5 o r 4 wv~ r e s p e c t i v e l y . As the a m p l i t u d e of the n e u t r i n o ( a n t i n e u t r i n o ) - n u c l e o n s c a t t e r i n g is d e t e r m i n e d by the s u m of the a x i a l and v e c t o r i s o v e c t o r c u r r e n t s then b e c a u s e of the i s o t o p i c i n v a r i a n c e of s t r o n g i n t e r a c t i o n s , the equality 2 4 2 G q .fn
1
dcv7r+
1
dcr~
-
(2n)2f2
ilkTr+i dq2dudE + + Ik~.-I dq2dudETr _÷
a2
da~+~ -
ik~r+ilkTr-I dq2dudEzr+dETr I k _ k + - ~ O
= (10)
d(7 up -
dcr~P +
dcr un
- - - +
dcr ~n - -
dq2
+dq2
takes place. (In the r i g h t - h a n d p a r t of eq. (10), we take the s u m of c r o s s s e c t i o n s of the n e u t r i n o and a n t i n e u t r i n o n e u t r o n and p r o t o n s c a t t e r i n g in o r d e r to e x c l u d e the c o n t r i b u t i o n of the f u n c t i o n w3.) An a n a l o g o u s c o n s i d e r a t i o n m a y be p e r f o r m e d for the soft p i o n e m i s s i o n at the deep i n e l a s t i c s c a t t e r i n g of n e u t r i n o (or a n t i n e u t r i n o ) on n u c l e o n s . T h u s , for i n s t a n c e , for the m a t r i x e l e m e n t s of the p r o cesses ~+p-4~+~(soft) + h a d r o n s , ~ + p - 4 1 + ~ + (soft) + h a d r o n s we get
Mup- orf- Zh = - MUP~
= c(hlj(p3) ip)
(11)
while the m a t r i x e l e m e n t s of the p r o c e s s e s ~+ p -- [ + v+ (soft) + h a d r o n s , u + p -~ l + v - (soft) + h a d r o n s v a n i s h in o u r a p p r o x i m a t i o n . F r o m eq. (11) we have the following r e l a t i o n s for the c r o s s s e c t i o n s
+all) dq2 dudE - ! k _ ~ 0
(2~)2f 2 d c r ( p p ~ [ ~ -
II~rr- ]
_ do ~p -
-
dcr ~n +
dq2du
dq2dv
davP
daun
i ?r (2~r)2f 2 d a ( u p
IkT;+L
(12)
~hr+ +alI) k +
dq2dudETr+
~0
dq2du + dq2d~
It should be noted in conclusion,, that the experimental measurement of the soft pion production cross section in the deep inelastic electromagnetic and weak processes (and, particularly, the check ot relations (I0), (12) would be of considerable interest from the view of determination of the scale invariance mechanism. Namely, as was discussed above, at conventional concepts on the nature of the scale invariance when the effective masses of all the virtual states are large, of the order of l q2 L or u the ralations (10), (12) must hold while they must be, generally speaking violated in the region of large a n d c o m p a r a t i v e l y s m a l l I q21 <~m2. T h a n k s a r e due to A. I. V a i n s t e i n , V.I. Z a k h a r o v , M . V . T e r e n t y e v a n d V. D. K h o v a n s k y for u s e f u l d i s cussions. 103
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LETTERS
References [1] B. L. Ioffe, Phys. L e t t e r s 3OB (1969) 123. [2] A. P a l s , S. B. T r e i m a m , Phys. Rev. L e t t e r s 25 (1970) 975. [3] G. J. M a r m e r et al. P h y s . Rev. L e t t e r s 23 (1969) 1469; M. A. Abolins et a l . , Phys. Rev. L e t t e r s 25 (1970) 126. [4] J. V. Allaby et al., P r e p r i n t CERN 70-12 (1970). [5] A. M. G l e e s o n , M. G. Gundzik, J. C. K u r i y a n , Phys. Rev. 173 (1968) 1708. [6] Y. Nambu, M. Y o s h i m u r a , P h y s . Rev. L e t t e r s 24 (1970) 25. [7] S. W e i n b e r g , Phys. Rev. L e t t e r s 17 (1966) 336.
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1 N o v e m b e r 1971
SOFT
PHYSICS
PION
PRODUCTION AND
LETTERS
IN DEEP INELASTIC WEAK PROCESSES
1November 1971
ELECTROMAGNETIC
B. L. I O F F E
Institute for Theoretical and Experimental Physics, Moscow, USSR Received 5 July 1971
On the basis of the PCAC hypothesis and current algebra, we calculate the cross section of one and two soft pion production in the deep inelastic scattering of electrons and neutrinos on nue[eons. The experimental measurement of these processes would make it possible to investigate the one-nucleon m a trix elements of the axial and isovector current commutators and contribute to elucidation of the scale invariance mechanism. As is w e l l known, the t o t a l c r o s s s e c t i o n s of the e l e c t r o n and n e u t r i n o - n u c l e o n s c a t t e r i n g a r e e x p r e s s e d t h r o u g h the c u r r e n t c o m m u t a t o r m a t r i x e l e m e n t s . It a l l o w s one in p a r t i c u l a r , u s i n g the e x p e r i m e n t a l d a t a r e l a t e d to the d e e p i n e l a s t i c r e g i o n , to find the b e h a v i o u r of the c u r r e n t c o m m u t a t o r o n e n u c l e o n m a t r i x e l e m e n t s n e a r the light c o n e [1]. H o w e v e r , u s i n g only the d a t a on the n e u t r i n o and e l e c t r o n - n u c l e o n s c a t t e r i n g one c a n n o t s e l e c t s e p e r a t e l y the v e c t o r and a x i a l c u r r e n t c o n t r i b u t i o n in c a s e of the n e u t r i n o s c a t t e r i n g and i s o v e c t o r and i s o s c a l a r c u r r e n t c o n t r i b u t i o n in c a s e of e l e c t r o p r o d u c t i o n . T h i s p r o b l e m m a y be s o l v e d by c o n s i d e r i n g the e l e c t r o p r o d u c t i o n p r o c e s s e s w i t h one o r two soft pion p r o d u c t i o n b a s i n g on the P C A C h y p o t h e s i s . (An a n a l o g o u s p r o b l e m f o r the a n n i h i l a t i o n p r o c e s s e + + e h a d r o n s w a s c o n s i d e r e d by P a i s and T r e i m a n [2].) L e t us f i r s t c o n s i d e r the e l e c t r o n - p r o t o n d e e p i n e l a s t i c s c a t t e r i n g w i t h the soft c h a r g e d ~ m e s o n p r o d u c t i o n . T h e pion m o m e n t u m in lab. s y s t e m w i l l be t a k e n a s s m a l l , I k i / t x ~ < < l . T h e h a d r o n i c p a r t of the m a t r i x e l e m e n t of the p r o c e s s e + p .... e + 7r+ (soft) + h a d r o n i c s t a t e h can be written as +
M p = f d4xexpikx ~J -
p2~r) (h
IT{co(x)l jep1 (0)}i p>
(1)
w h e r e ¢ ( x ) i s the m e s o n i c field, j~l the e l e c t r o m a g n e t i c c u r r e n t of h a d r o n s ,
jel= ½(j(3p)+ flOp)). Sub-
stituting ~(x) into eq. (1) as the axial current divergence
= ca~Ap/p27r;
c = 1/frr = 1 / 0 . 9 3 / x ~
(2)
c o n s i d e r i n g k to be s m a l l and u s i n g the c o n v e n t i o n a l P C A C t e c h n i q u e we get + = M~ c ( h l A ~ ( 0 ) lP> +
ickv fd4xexp(ikx)
.el
3p
(0) l}p)
(3)
T h e s e c o n d t e r m in the r i g h t - h a n d p a r t of eq. (3) is c o n t r i b u t e d only by the p o l e t e r m s c o r r e s p o n d i n g to the soft p i o n e m i s s i o n by i n i t i a l and f i n a l n u c l e o n s as w e l l a s by the p o l e t e r m a p p e a r i n g due to the a b s o r p t i o n of the v i r t u a l photon by the soft pion. It is o b v i o u s , that at l a r g e q2 (q b e i n g the v i r t u a l p h o ton m o m e n t u m ) the p o l e t e r m c o n t r i b u t i o n of the l a t t e r type is s m a l l , < k/Iq 2 L1/2. When c o n s i d e r i n g the c o n t r i b u t i o n of the p o l e t e r m s c o r r e s p o n d i n g to the B r e m s s t r a h l u n g of the soft p i o n s by n u c l e o n s , we s h a l l t a k e the c a s e of l a r g e e n e r g y t r a n s f e r r e d to h a d r o n s but not enough that the n u c l e o n - a n t i n u c l e o n p a i r p r o d u c t i o n w o u l d g i v e a n o t i c e a b l e c o n t r i b u t i o n into the t o t a l c r o s s s e c t i o n of the e l e c t r o p r o d u c t i o n . T h e n the f i n a l s t a t e , b e s i d e s m e s o n s , w o u l d c o n t a i n only one b a r y o n . The p o l e t e r m c o r r e s p o n d i n g to the soft pion e m i s s i o n by i n i t i a l n u c l e o n is p r o p o r t i o n a l to O(/~ -
k
-
m) -1 Y#Y5
u(p)kp
(4)
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w h e r e u(p) i s t h e s p i n o r d e s c r i b i n g t h e i n i t i a l s t a t e of n u c l e o n w i t h t h e m o m e n t u m p, O i s s o m e o p e r a t o r w h i c h a t s m a l l k i s s u p p o s e d to b e i n d e p e n d e n t of k. N e g l e c t i n g t h e t e r m s q u a d r a t i c in k, eq. (4) m a y b e e a s i l y r e d u c e d to t h e f o r m
-o
k "o .(p) 1 / k o
(5)
a n d , c o n s e q u e n t l y , a t ] k ] ]k o << I t h e c o n t r i b u t i o n of t h i s p o l e t u r n s out to b e s m a l l . T h e pole t e r m a p p e a r i n g in t h e s o f t p i o n e m i s s i o n b y t h e f i n a l n u c l e o n is c a l c u l a t e d a n a l o g o u s l y . H e r e t h e f i n a l n u c l e o n m o m e n t p ' m a y b e c o n s i d e r e d a s s m a l l , IP' ] / m << 1. I n d e e d , a c c o r d i n g to t h e pp s c a t t e r i n g e x p e r i m e n t a l d a t a [3,4], t h e s q u a r e d m o m e n t u m t r a n s f e r to h a d r o n s i s of o r d e r -1 = p , 2 (0.3 - 0 . 4 ) m 2. It i s n a t u r a l to e x p e c t t h a t in t h e c a s e of e l e c t r o p r o d u c t i o n a n a n a l o g o u s s i t u a t i o n w i l l t a k e p l a c e a n d t h e t e r m s of t h e o r d e r of p ' 2 / m 2 m a y b e n e g l e c t e d w i t h t h e a c c u r a c y of o r d e r of s e v e r a l d o z e n s p e r c e n t . C a l c u l a t i n g u n d e r t h i s a s s u m p t i o n t h e p o l e t e r m c o r r e s p o n d i n g to t h e e m i s s i o n of t h e s o f t p i o n by t h e f i n a l n u c l e o n , we find t h a t up to t e r m s of h i g h e r o r d e r in IP' !/m a n d I k ! / k o it i s p r o p o r t i o n a l to
,,(Y)r/5(~'+/~-.,)-1 o k,~(p') ( mi.e. a l s o g i v e s a s m a l l c o n t r i b u t i o n . It m a y b e e x p e c t e d t h a t j u s t t h e s a m e the s m a l l c o n t r i b u t i o n i n t o eq. (3) w i l l b e g i v e n b y t h e e m i s s i o n of s o f t p i o n s by h y p e r o n s o r b a r y o n i c r e s o n a n c e s a p p e a r i n g in t h e final state. N e g l e c t i n g t h e c o n t r i b u t i o n of t h e s e c o n d t e r m in t h e r i g h t - h a n d p a r t of (3) we get f o r t h e c r o s s s e c t i o n of t h e p r o c e s s e + p - * e + ~+ (soft) + h a d r o n s de7r+
d~dvdE
_ 47ra 2 E '
Av
1
A P ( v ' q2)}
q4 m E { c°s2½° w2 (v, q2)+2sin 2 ~ o w 1
k
(2~)2/2
U)
Ag w h e r e E, E ' a r e i n i t i a l a n d f i n a l e n e r g i e s of t h e e l e c t r o n , 0 i s i t s s c a t t e r i n g a n g l e , v = pq = mE, w 1 w A~ a r e i n v a r i a n t f o r m f a c t o r s due to t h e a x i a l c u r r e n t ( i n d e x Y i n d i c a t e s t h e i d e n t i t y of t h e s e f o r m f a c t o r s w i t h t h e f o r m f a c t o r s due to t h e a x i a l c u r r e n t in t h e p r o c e s s v + p - [ + h a d r o n s ) . Its i s o b v i o u s t h a t f o r t h e c r o s s s e c t i o n of t h e s o f t ~ - p r o d u c t i o n w i l l h o l d a n a n a l o g o u s e x p r e s s i o n r e p l a c i n g ~ -~ v, w h i l e t h e c r o s s s e c t i o n of t h e s o f t ~o p r o d u c t i o n m u s t v a n i s h in t h e a p p r o x i m a t i o n u n d e r c o n s i d e r a t i o n . L e t u s now d i s c u s s t h e a c c u r a c y of t h e r e s u l t o b t a i n e d . A s it w a s m e n t i o n e d a b o v e t h e p o l e t e r m c o n t r i b u t i o n we h a v e n e g l e c t e d in eq. (3) m a y b e of the o r d e r of s e v e r a l d o z e n s p e r c e n t . (It m a y b e s h o w n t h a t t h e i n t e r f e r e n c e of t h e p o l e a n d n o n - p o l e t e r m s in eq. (3) l e a d s to t e r m s of t h e s a m e o r d e r in t h e t o t a l c r o s s s e c t i o n ) . T h e s a m e i s t h e a c c u r a c y of t h e P C A C at s m a l l e n e r g i e s . In c a s e of h i g h e n e r g i e s of t h e v i r t u a l p h o t o n , t h e P C A C h y p o t h e s i s m a y b e , in p r i n c i p l e , i n a p p l i c a b l e b e c a u s e of t h e v a l u e qk w h i c h in t h e P C A C m e t h o d i s c o n s i d e r e d to b e s m a l l , i s r e a l l y not s m a l l (qk .-~ qo Prr) a s c o m p a r e d to t h e m a s s e s of v i r t u a l p a r t i c l e s . In p a r t i c u l a r , t h i s s i t u a t i o n m a y h o l d f o r t h e p r o c e s s e s of t h e s o f t p i o n p r o d u c t i o n a t t h e e l e c t r o n - n u c l e o n e l a s t i c s c a t t e r i n g a t ]q2 [ >> m2 ( c o n s i d e r e d in r e f s . [5] a n d [6]) o r at h i g h e n e r g y p h o t o p r o d u c t i o n * . T h e d e e p i n e l a s t i c r e g i o n i s m o r e f a v o u r a b l e f r o m v i e w of t h e P C A C a p p l i c a b i l i t y , a s it m a y b e e x p e c t e d t h a t due to t h e s c a l e i n v a r i a n c e w h i c h h o l d s in t h i s r e g i o n , t h e m a s s e s of t h e v i r t u a l s t a t e s a r e i n e s s e n t i a l h e r e a n d qk m a y e n t e r only a s r a t i o s qk/q 2, qk/qp, s o t h a t t h e a c c u r a c y w i l l b e of t h e s a m e o r d e r a s a t s m a l l e n e r g i e s . L e t u s now c o n s i d e r t h e d e e p i n e l a s t i c e l e c t r o p r o d u c t i o n i n v o l v i n g t h e two s o f t p i o n e m i s s i o n . In c a s e of t h e s o f t g+Tr-, t h e h a d r o n i c p a r t of t h e m a t r i x e l e m e n t w i t h p r o d u c t i o n of h a d r o n i c s t a t e h c a l culated using the conventional PCAC technique, is M + - : c 2 ( h I j ! 3) (0) !p) #
(8)
* If -/-quantum emits the soft pion going over iat9 some virtual state A, the propagator of this state D~ 1 equals to D-A1 - q2 _2kq+k 2 - M R and at not v e r y large q2 but large v essentially depends on qk ifM£~ is not large. On the ot~er hand. as it was "shown in ref. [1] in the virtual photon-nucleon s c a t t e r i n g at s m a l l qZ and large v of i m p o r tance are the large longitudinal d i s t a n c e s , i.e. finite (and comparatively small) MA. Thus. it should be expected that the PCAC method would appear to be inapplicable for the electroproduction with the soft pion e m i s s i o n at Iq2 i< m 2 and v >> m 2 and for the photoproduction at v>> m2.
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In eq. (8) we n e g l e c t the t e r m c o n t a i n i n g the c o m m u t a t o r [A + (x), }~)~A~(0)]Xo= 0 p r o p o r t i o n a l , a s it is u s u a l l y e x p e c t e d to the s m a l l p a r a m e t e r p2. The c o n t r i b u t i o n in M +- of the pole t e r m s c o r r e s p o n d i n g to the B r e m s s t r a h l u n g of the soft pions by i n i t i a l n u c l e o n (baryon) is s m a l l for the s a m e r e a s o n s as in c a s e of one pion e m i s s i o n . The c o n t r i b u t i o n in M ~ - of the pole t e r m s due to the e m i s s i o n of two soft p i o n s by the fast pion (or n u c l e o n ) is a l s o s m a l l b e c a u s e of that, a s it was shown by W e i n b e r g [7], in this c a s e the soft p i o n s m u s t be e m i t t e d in the p state. C a l c u l a t i n g with the help of eq. (8) the c r o s s s e c t i o n of the e l e c t r o p r o d u c t i o n of the soft 7r+~- m e s o n s with m o m e n t a k l , /(2, we have
dan+Tr-
_ 4nX 2 E' {cos 2
(3 2sin 2 ½0 ( 4 Ikl i ik2 ! ½0 w2)(~' q2) + Wl3)(v' q2)~ (2~)~4-f~ ---
q4 m E
dq2 d v d E 1 dE 2
(9)
w h e r e w (3), w (3) c o r r e s p o n d to the c o n t r i b u t i o n of the i s o v e c t o r p a r t of the e l e c t r o m a g n e t i c c u r r e n t ½j(3). The c r o ~ s s e c t i o n s of the e l e c t r o p r o d u c t i o n with p r o d u c t i o n of the soft ~+Tro o r ~-~o m e s o n s a r e ~ o b t a i n e d f r o m eq. (9) by r e p l a c i n g w(3) by 4 w v5 o r 4 wv~ r e s p e c t i v e l y . As the a m p l i t u d e of the n e u t r i n o ( a n t i n e u t r i n o ) - n u c l e o n s c a t t e r i n g is d e t e r m i n e d by the s u m of the a x i a l and v e c t o r i s o v e c t o r c u r r e n t s then b e c a u s e of the i s o t o p i c i n v a r i a n c e of s t r o n g i n t e r a c t i o n s , the equality 2 4 2 G q .fn
1
dcv7r+
1
dcr~
-
(2n)2f2
ilkTr+i dq2dudE + + Ik~.-I dq2dudETr _÷
a2
da~+~ -
ik~r+ilkTr-I dq2dudEzr+dETr I k _ k + - ~ O
= (10)
d(7 up -
dcr~P +
dcr un
- - - +
dcr ~n - -
dq2
+dq2
takes place. (In the r i g h t - h a n d p a r t of eq. (10), we take the s u m of c r o s s s e c t i o n s of the n e u t r i n o and a n t i n e u t r i n o n e u t r o n and p r o t o n s c a t t e r i n g in o r d e r to e x c l u d e the c o n t r i b u t i o n of the f u n c t i o n w3.) An a n a l o g o u s c o n s i d e r a t i o n m a y be p e r f o r m e d for the soft p i o n e m i s s i o n at the deep i n e l a s t i c s c a t t e r i n g of n e u t r i n o (or a n t i n e u t r i n o ) on n u c l e o n s . T h u s , for i n s t a n c e , for the m a t r i x e l e m e n t s of the p r o cesses ~+p-4~+~(soft) + h a d r o n s , ~ + p - 4 1 + ~ + (soft) + h a d r o n s we get
Mup- orf- Zh = - MUP~
= c(hlj(p3) ip)
(11)
while the m a t r i x e l e m e n t s of the p r o c e s s e s ~+ p -- [ + v+ (soft) + h a d r o n s , u + p -~ l + v - (soft) + h a d r o n s v a n i s h in o u r a p p r o x i m a t i o n . F r o m eq. (11) we have the following r e l a t i o n s for the c r o s s s e c t i o n s
+all) dq2 dudE - ! k _ ~ 0
(2~)2f 2 d c r ( p p ~ [ ~ -
II~rr- ]
_ do ~p -
-
dcr ~n +
dq2du
dq2dv
davP
daun
i ?r (2~r)2f 2 d a ( u p
IkT;+L
(12)
~hr+ +alI) k +
dq2dudETr+
~0
dq2du + dq2d~
It should be noted in conclusion,, that the experimental measurement of the soft pion production cross section in the deep inelastic electromagnetic and weak processes (and, particularly, the check ot relations (I0), (12) would be of considerable interest from the view of determination of the scale invariance mechanism. Namely, as was discussed above, at conventional concepts on the nature of the scale invariance when the effective masses of all the virtual states are large, of the order of l q2 L or u the ralations (10), (12) must hold while they must be, generally speaking violated in the region of large a n d c o m p a r a t i v e l y s m a l l I q21 <~m2. T h a n k s a r e due to A. I. V a i n s t e i n , V.I. Z a k h a r o v , M . V . T e r e n t y e v a n d V. D. K h o v a n s k y for u s e f u l d i s cussions. 103
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References [1] B. L. Ioffe, Phys. L e t t e r s 3OB (1969) 123. [2] A. P a l s , S. B. T r e i m a m , Phys. Rev. L e t t e r s 25 (1970) 975. [3] G. J. M a r m e r et al. P h y s . Rev. L e t t e r s 23 (1969) 1469; M. A. Abolins et a l . , Phys. Rev. L e t t e r s 25 (1970) 126. [4] J. V. Allaby et al., P r e p r i n t CERN 70-12 (1970). [5] A. M. G l e e s o n , M. G. Gundzik, J. C. K u r i y a n , Phys. Rev. 173 (1968) 1708. [6] Y. Nambu, M. Y o s h i m u r a , P h y s . Rev. L e t t e r s 24 (1970) 25. [7] S. W e i n b e r g , Phys. Rev. L e t t e r s 17 (1966) 336.
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