- Email: [email protected]
On the chiral SU(2)× SU(2) dynamics for A1, ϱ and π mesons
[
~ - . A . 2 Nuclear Physics B24 (1970) 612-622. North-Holland IMblishing Company 8.A.5 j
ON THE CHIRAL SU(2)x SU(2) DYNAMICS FOR MESONS A 1, ~ AND V. I. OGIEVETSKY and B. M. Z U P N I K * Laboratory of Theoretical Physics, Joint Institute f o r Nuclear Research, Dubna, USSR Received 31 August 1970
Abstract: It is remarked that the conventional effective Lagrangian with p and A 1 mesons contains t r i l i n e a r and quadrilinear derivative couplings. Thus one should add independently all t e r m s with three and four field derivatives; the theory should then contain a number of a r b i t r a r y parameters. We suggest a new Lagrangian for the Alp?r system containing no more than two field derivatives in each term. This model is determined by only four p a r a m e t e r s gt> gA, toO, MA and gives the smoothest momentum dependence for the contact ver'tices. W e d e r i v e with n e c e s sity the reasonable value 6 = - 1 for the anomalous magnetic moment of A 1 if the KSI.'R relation holds and also apply our model to ~y scattering and A 1 decay.
1. I N T R O D U C T I O N In r e c e n t y e a r s p r o c e s s e s with A1, p a n d a , m e s o n s have b e e n i n t e n s i v e ly i n v e s t i g a t e d u s i n g c u r r e n t a l g e b r a . I m p o r t a n t i n f o r m a t i o n on the AlPTr s y s t e m h a s b e e n d e r i v e d f r o m the W a r d i d e n t i t i e s for the n - p o i n t f u n c t i o n s in the t r e e a p p r o x i m a t i o n with s o m e r e s t r i c t i o n on the m o m e n t u m d e p e n d e n c e of the c o n t a c t v e r t i c e s [1]. N e v e r t h e l e s s , such a r e s t r i c t i o n l e a d s to the s o l u t i o n of the W a r d i d e n t i t i e s which d e p e n d s on s e v e r a l a r b i t r a r y p a r a m e t e r s [1]. T h e c o n s t r u c t i v e way f o r the r e p r o d u c t i o n of the c u r r e n t - a l g e b r a r e s u l t s is the e f f e c t i v e - L a g r a n g i a n m e t h o d b a s e d on the SU(2)× SU(2) g r o u p [2, 3]. In s e c t . 2 of the p r e s e n t p a p e r we show that the c o n v e n t i o n a l L a g r a n g i a n of the A l P ~ s y s t e m [2] c o n t a i n s t r i l i n e a r a n d q u a d r i l i n e a r d e r i v a t i v e t e r m s due to the e l i m i n a t i o n of the n o n - p h y s i c a l b i l i n e a r Air, c o u p l i n g . T h e n one s h o u l d add a l l i n d e p e n d e n t t e r m s with t h r e e a n d four f i e l d d e r i v a t i v e s a n d t h i s l e a d s to the a p p e a r e n c e of the a r b i t r a r y p a r a m e t e r s in the L a g r a n g i a n . B e s i d e s , the v e r y p r e s e n c e of s u c h d e r i v a t i v e t e r m s is u n d e s i r a b l e in the i n t e r a c t i o n L a g r a n g i a n . In s e c t . 3 we s u g g e s t in t h i s c o n n e c t i o n the f o l l o w i n g n a t u r a l r e q u i r e m e n t : The e f f e c l i v e L a g ~ a n g i a n o f the AlPrr s y s t e m rnust c o n t a i n no m o r e lhan two f i e l d d e r i v a t i v e s in e a c h t e r m . T h i s s e v e r e r e s t r i c t i o n d e t e r m i n e s c o m p l e t e l y the i n v a r i a n t L a g r a n g i a n c o n t a i n i n g f o u r i n d e p e n d e n t p a r a m e t e r s * Dniepropctrovsk State University, USSR.
CHLRAL DYNAMICS
613
gp, g A , rap, M A w i t h t h e a i d of t h e n o n - l i n e a r (in p i o n s ) s u b s t i t u t i o n f o r t h e A1 f i e l d . In t h e t r e e a p p r o x i m a t i o n e a c h c o n t a c t v e r t e x i s a p o l y n o m i a l in the f o u r - m o m e n t a , t h e d e g r e e D of w h i c h i s e q u a l to t h e n u m b e r of f i e l d d e r i v a t i v e s in t h e a p p r o p r i a t e t e r m of t h e L a g r a n g i a n . In s e c t . 4 we e s t a b l i s h t h e c o n n e c t i o n of o u r L a g r a n g i a n m o d e l with the p a r t i a l s o l u t i o n of t h e W a r d i d e n t i t i e s u n d e r lhe condition D ~< 2 f o r any contact c e r t e x . In the AlPV s y s t e m it is i m p o s s i b l e to s t r e n g t h e n t h i s r e s t r i c t i o n on the m o m e n t u m d e p e n d e n c e of t h e c o n t a c t v e r t i c e s . Note t h a t in t h e c o n v e n t i o n a l L a g r a n g i a n m o d e l [2] D ~< 4. In s e c t . 5 we d i s c u s s s o m e c o n s e q u e n c e s f o r t h e a m p l i t u d e s p - ' 2~, A1 - ' pr,, A1 - 3~, r , ; r - ' 7r;r, p~ ~ p,-r, etc. We d e r i v e with n e c e s s i t y t h e r e a s o n a b l e v a l u e
(1.1)
5 : ( g A / g p ) 2 {1 - [ ( n , p / ' g p f v ) 2 - 1] -2} - 1 ~ - 1
f o r the a n o m a l o u s m a g n e t i c m o m e n t of t h e A 1 m e s o n . T h e S - w a v e 7rr, s c a t t e r i n g l e n g t h s h a v e b e e n found to b e in a g r e e m e n t with t h o s e o b t a i n e d in c u r r e n t a l g e b r a [6]. We h a v e a l s o e s t i m a t e d t h e t o t a l width of the A 1.
2. P R E L I M I N A R I E S In t h e p r e s e n t p a p e r we p r o p o s e a n e w m o d e l f o r t h e L a g r a n g i a n of t h e AlPV s y s t e m w h i c h c o n t a i n s no m o r e t h a n b i l i n e a r d e r i v a t i v e c o u p l i n g s . W i t h t h i s e n d in v i e w we s h a l l m e n t i o n h e r e s o m e f e a t u r e s of t h e c o n v e n t i o n a l c h i r a l s c h e m e with A1, p a n d v m e s o n s ( s e e , e . g . , r e f s . [2, 3]). T h e n o n - l i n e a r L a g r a n g i a n f o r v e c t o r , a x i a l - v e c t o r a n d p i o n f i e l d s V~, A~ a n d z c a n be w r i t t e n a s
z _ _ v2, _,A 2,
nZ2o iv2 +A2) +
.
,
/2.1)
with t h e c o v a r i a n t c u r l s
V~v = O~z V v - Ov Vl.t - g Vp.× V - g A .
×Av ,
A ~v : O~tAv - ?vAst - g V ~ × A u + g Vv×A~t
(2.2)
a n d c o v a r i a n t d e r i v a t i v e of t h e p i o n f i e l d
vu ~ : Au ~+ [~(1 + ~)1-1 ~ ( 0 A u ~ ) ,
(2.3)
where (~: ( 1 . ~ 2 ) 1 / 2 ,
Ag,~:
aUq)-gV~×d2+gaAg " .
(2.3a)
In t h e s e e q s . ~ = f ~ l : r r ( f n ~ 95 MeV i s t h e p i o n d e c a y c o n s t a n t ) a n d t o o , g , d are some unrenormalized parameters. This Lagrangian is invariant un-
614
v. I. OGIEVETSKY and B. M. ZUPNIK
d e r the SU(2)× SU(2) group. M o r e o v e r , only the m a s s t e r m for v e c t o r and a x i a l - v e c t o r fields b r e a k s the i n v a r i a n c e under the gauge SU(2)× SU(2) t r a n s f o r m a t i o n s with the c o o r d i n a t e - d e p e n d e n t p a r a m e t e r s a ,fl [2, 3]:
~( vu :LA u) : - ( a ~:~ ) × ( V u ~ A u) _ g - 1 a u ( a ±,a) , 54:
- a×4+fl~.
(2.4)
It follows i m m e d i a t e l y that the interacting fields V., and A/s t r a n s f e r spin 1 only [4] (in the limit of z e r o - m a s s pions), i.e. ~he t r a n s v e r s a l i t y conditions ~ V ~ = 0, 0 ~ A g = 0 a r e satisfied off the m a s s shell. However, the t e r m ( V ~ 4 ) 2 in eq. (2.1) contains the n o n - p h y s i c a l bilinear coupling A / s ~ 4 which is usually eliminated by the following substitution of the a x i a l - v e c t o r field: _9,2
A = a~-(gJf/mo).2)~4.
(2.5)
Then it is i m p o s s i b l e to maintain the condition ~ a p = 0 for the new a x i a l v e c t o r field a~. The introduction of 7 ) ~ 4 = ~ 4 - g V ~ × 4 instead of ~ 4 is n e c e s s a r y to e n s u r e t r a n s v e r s a l i t y for the interacting v e c t o r field V~ [4] since it is important in connection with the v e c t o r - d o m i n a n c e idea [5]. The c r u c i a l point is that substitution (2.5) c o n t r i b u t e s to Vgv the t e r m (/)/x 4 × :7)v 4 w h i c h is bilinear in the field derivatives. Hence, the c o n v e n tional effective L a g r a n g i a n contains the t r i l i n e a r and q u a d r i l i n e a r d e r i v a tive t e r m s ~ Vv" :b~ 4 × 2 ) v 4 , ( a N ×.b v 4 ) ' ( : / ) t x 4 × <~)v4) and (:/')g4 ×~b u4)2 g e n e r a t e d by V~v in eq. (2.1). It should be said that t h e r e is u n d e s i r a b l e to have the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e couplings in the interaction L a g r a n g i a n . N e v e r t h e l e s s , if one a d m i t s the p r e s e n c e of such couplings, then all possible t e r m s with t h r e e and four d e r i v a t i v e s should be added and the n u m b e r of p a r a m e t e r s in the theory should inc r e a s e . Besides, the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e couplings lead to n o n - s m o o t h m o m e n t u m dependence of the amplitudes p --' 2,-r, A 1 ~ 37,, ~ r , - ' r,r., etc.
3. LAGRANGIAN OF THE AlPr, SYSTEM WITHOUT TRILINEAR AND QUADRILINEAR DERIVATIVE COUPLINGS To avoid the above mentioned difficulties of the conventional L a g r a n g i a n model for the AlP~ s y s t e m we p r o p o s e the following natural r e q u i r e m e n t :
Trilinear and quadrilinear in field derivatives terms must be absent in the effective Lagrangian of lhe AlP~ system. Now we s t a r t to c o n s t r u c t the g e n e r a l L a g r a n g i a n which s a t i s f i e s this requirement. It is convenient to define for this p u r p o s e the new c o v a r i a n t quantities
RI~u : V ~v+(C2/g) AW.4×AV 4 ,
(3.17
CHIRAL DYNAMICS
615
S u e : A ~ v - (C2/g (~) 0 × (AI~0 × A v 0 ) ,
(3.2)
with s o m e constant C. The RI ~ v and S11 v have the s a m e t r a n s f o r m a t i o n p r o p e r t i e s as c o v a r i a n t c u r l s Vp. v and A ~ v :
5R~v : - a× Rp.v-flXS~v
,
5 S u e = - a X S u v - fl× R ~v .
(3.3)
H e r e eqs. (2.2), (2.3a) and (2.4) have been used. Let us w r i t e down f i r s t the g e n e r a l e x p r e s s i o n for the i n v a r i a n t L a g r a n gian of the AlP~ s y s t e m (containing t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e terms):
1
2
_¼ZAS2v+~(Zp_ZA)((qb×Rg02_(O×S
)2
- 2crRUv× ~" Sgv)+h[Vl~ 0 x V v 0 ] 2+g(Vl~ 0 " V g 0 ) 2 +
~mo(Vi ~ 12 2 + A 2 9 4 ~ d 2 ( V
0)2 ,
(3.4)
w h e r e Zp, ZA, too, d, h a n d K a r e s o m e a r b i t r a r y c o n s t a n t s . Our main p u r p o s e is the e l i m i n a t i o n of the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e t e r m s f r o m the L a g r a n g i a n (3.4) t o g e t h e r with the e l i m i n a t i o n of the b i l i n e a r AlU coupling. Also the kinetic t e r m s m u s t be c o r r e c t l y n o r malized. All t h e s e r e q u i r e m e n t s can be s a t i s f i e d only with h =K = 0 in a unique way by p e r f o r m i n g the following substitutions for the v e c t o r and a x i a l - v e c t o r fields:
-1/2 e g , V = ZO
A
: ZA1/2[ a u + F ( ~ 2) cD 0 ] .
(3.5)
We shall use below the r e n o r m a l i z e d constants:
go = z-pl/2g
'
gA = Z - 2 / 2 g ,
m
p
= Z-1/2m p
o'
_-112, 2 d2g2) l/2
= z~A
(too+
(3.6)
To exclude the b i l i n e a r a ,rO , t*" 0 coupling we fix in eq. (3.5) the value / F ( O ) = - g2of2/g A m 2 then the function F( 02) is uniquely d e t e r m i n e d by the r e q u i r e r h e n t that the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e t e r m s m u s t be absent in the L a g r a n g i a n C (gPfr')2 ] 1 + ~ F(02) = -gA 0+-Cb) = 2- L
gAmp w h e r e ~ = (1 - 0 2 ) 1/2 and
~
02+
(3.7) "'"
'
616
V.I. OGIEVETSKY and B.M. ZUPNIK
C = [(mp/fT, gp) 2 - 1] -1 .
(3.8)
We have used here the relation [2, 3]: f2 =
m(_~p. 2
) [1-
~
('npEA~ 2] (3.9)
\MAgp/ J ,
which follows from the c o r r e c t normalization of the kinetic t e r m ~(~#x~) 2. In fact we have determined the non-linear substitution of a~x and the value of the constant C to exclude the bilinear derivative t e r m s from RIx ~, (eq. (3.1)) and Slx v (eq. (3.2)) (see appendix), the t e r m s with three and four derivatives being eliminated from eq. (3.4). Now we can write the final e x p r e s s i o n for our Lagrangian of the AlP7, s y s t e m which contains non-derivative couplings and the t e r m s with one and two derivatives only: 2
g.. (~b×Oy.u 72+}(1 _ ( 4 × - /~v) 2 g
, 2 :2 - ie2~.v-¼a~u
1[f~(l+C)12[(,-OlaO)2+CrZPgA'~2 )2] +½H(~2) ( e~v~× a~,) + ~L-I +C~- ] .MAgp. (~a~O ~271+C~
+gA.T.~ iT+CO.] [((~- 1)(:2) Ix 0 ,/f~gpMA', 2
-i(---m~--
) (0x ag)
2+
alx)+ g-l( a~xO ) (Oa O) ]
½m2p 2
2 2 n12f:(~
la + MA"bt + 7,
(3.10) '
where the s y m m e t r y - b r e a k i n g t e r m m 2 f 2 ~ leads to PCAC in this model [3]. In eq. (3.10) the following notations have been used: 2
B((~ 2) : g A L l + c -2
2
II((~2)
-1
F2(~b 2) ,
(3.11)
g2
= &~pgA[I+C-I(~(I__~)IF(~)2),gA
(3.127
CHIRAL I)YNAMICS
617
2
O~u
:
gg
Z 1/2 p R~v
= ~
gp -{-
a
v
Cg A gp _ _ _
(1-C2~)
(abtx:3u0
a
+ /)
× a
u
t~× a v ) ,
(3.13)
= ZA/2[S u-gA F Cb2 '
) O × R / ~ v ] = e /~ a v - ? v a /~ - g p g b t × a v
(
+gp~Ou× a~.-EACO× ( ag× au) .
(3.14)
4. C O N N E C T I O N B E T W E E N OUR M O D E L AND T H E " H A R D - P I O N " M E T H O D IN T H E A L G E B R A O F C U R R E N T S
The n-point functions derived from the effective Lagrangian of the A l P n system are known to be a solution for the Ward identities in SU(2)×SU(2) algebra of currents [1]. In this section we reformulate our basic requirement for the Lagrangian in terms of the "hard-pion" method If]. The vector and axial-vector currents ~/z and "~5g are obtained from our Lagrangian (3.10) by using the Gell-Mann-Levy method (see, e.g. ref. [5]). 2
=
~
mP
~
~
2 mpg A ~5U = --2 [ag +F(~2) gp
'
:7)~b] .
(4.1)
Note t h a t t h e s e c u r r e n t s o b e y t h e e q u a l - t i m e c o m m u t a t i o n r e l a t i o n s of t h e a l g e b r a of f i e l d s [5]: a
b
a
[~ o(X,t),~ g(y,t)] = [ ~ 5 o ( X , t ) ,
~5p(y,t)]
c(
ieabc.~ g x,t) 5( x - y )
=
/ ,\rap 2
-i~-~-).
5abgpk Z -k
5(x-y),
[9:(x,t),'~5~(Y,t)]=[~5o(X,t),~b(y,l)]=Zeabc~)5~(x,t)6(x-y). b
(4.2)
°
(4.3)
Here k=1,2,3 ands=0,1,2,3. T h e p r o o f of t h e s e r e l a t i o n s is e q u i v a l e n t to t h a t r e f . [5]. W e r e m a r k h e r e t h a t in a l l S U ( 2 ) × S U ( 2 ) g a u g e L a g r a n g i a n schemes the corresponding Schwinger terms have the same simple cn u m b e r f o r m if c u r r e n t s a r e c a l c u l a t e d by t h e G e l l - M a n n - L e v y m e t h o d . T h e r e a r e , of c o u r s e , s o m e m o d e l - d e p e n d e n t c o m m u t a t i o n r e l a t i o n s . T h e s t r o n g f o r m of P C A C f3] (2 q ~ = m 2 f :~) is p r e s e n t in o u r m o d e l . " t" " .L J b t d a / ~ . 7, m ne t r e e a p p r o x i m a t i o n t h e c o n t a c t v e r t e x f u n c t i o n s c o r r e s p o n d to t h e v a r i o u s c o n t a c t c o u p l i n g s in the L a g r a n g i a n . T h e d i a g r a m of any n - p o i n t
618
V.I. OGIEVETSKY
and B. M. ZUPNIK
f u n c t i o n i n v o l v e s c o n t a c t v e r t i c e s c o n n e c t e d by p r o p a g a t o r s without f o r m ing loops. E a c h c o n t a c t v e r t e x function is a p o l y n o m i a l in the f o u r - m o m e n ta, the d e g r e e of which is equal to the n u m b e r of field d e r i v a t i v e s in the a p p r o p r i a t e c o n t a c t t e r m . T h i s p o l y n o m i a l d e g r e e D c h a r a c t e r i z e d the m o m e n t u m d e p e n d e n c e of a c o n t a c t v e r t e x function. In the t r e e a p p r o x i m a tion D d e p e n d s on the n u m b e r ng of the p - m e s o n a n d / ' o r A 1 m e s o n lines a t t a c h e d to the v e r t e x i r r e s p e c t i v e of the n u m b e r of pion lines. We have D=2 for ng=0,2, D=l for ng=l,3 andD=0 for ng=4. In the c o n v e n t i o n a l c u r r e n t - a l g e b r a i c a p p r o a c h to the A l P n s y s t e m [ 1 , 2 ] t h e r e a r e v e r t i c e s with D = 3 and D = 4 which c o r r e s p o n d to the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e c o u p l i n g s . In o u r a p p r o a c h the m o m e n t u m d e p e n d e n c e of the c o n t a c t v e r t e x functions is m i n i m a l and cannot be r e d u c e d f u r t h e r . At l e a s t two field d e r i v a t i v e s m u s t be p r e s e n t b e c a u s e the t e r m s ( V ~ b ) 2, (t)~u! 2 etc. a r e a l w a y s p r e s e n t . Thus, using Lagrangian (3.10) we derive n-poinl fitnctions which are lhe minimum momentum-dependent solution of Ward identities in SU(2) x SU(2)
algebra of currents with c-number Schu'inger terms. Note that in m e s o n - n u c l e o n c o u p l i n g s we can l i m i t o u r s e l v e s to v e r t i c e s which a r e l i n e a r in m o m e n t a .
5. SOME C O N S E Q U E N C E S O F THE M O D E L V a r i o u s r e s t r i c t i o n s on the m o m e n t u m d e p e n d e n c e of a m p l i t u d e s a r e often u s e d in the a l g e b r a of c u r r e n t s . In o u r L a g r a n g i a n m o d e l t h e r e a r e such s e v e r e r e s t r i c t i c n s f o r p r o c e s s e s with A1, p and 7r m e s o n s . T h e r e q u i r e m e n t of the m i n i m u m m o m e n t u m d e p e n d e n c e l e a d s to s o m e i m p o r t a n t c o n s e q u e n c e s for the s i m p l e s t n - p o i n t functions. F i r s t , we p r e s e n t the r e s u l t s f o r t h r e e - p o i n t functions. T h e A 1 A l P i n t e r a c t i o n in L a g r a n g i a n (3.101 is d e s c r i b e d by 2 ~AAp-~; A(1-C2)
al-t@v" al.t× av+gp(@lj, x a v -
Ov ×
al-t)'~tzav"
(5.1/
T h e c o r r e s p o n d i n g A 1 A l P v e r t e x Ftav~t(p,q) is l i n e a r in m o m e n t a (i.e. D= 11
AP(p )
j A V ( q ) F ~ v x ( P , q ) = - i g p [ g ~ v ( P + q ) x - g v x P u - "g ~ v qv --NNp~(k )
+(2+5)(k~z~v~t-kvguv) ] .
(5.2/
H e r e 5 is the a n o m a l o u s m a g n e t i c m o m e n t of the A 1 m e s o n .
o:
~A-p 2 I
)
~ mp
2
-2
1-1,
is w h e r e eq. (3.7) f o r C h a s b e e n used. If the KSFR relation (2g2f 2 valid, we obtain 5 = - 1. T h i s v a l u e f o r 5 is the m o s t p r e f e r a b l e " ( a s K s c u s s e d in d e t a i l s in ref. [6]). In the c o n v e n t i o n a l a p p r o a c h [ 1 , 2 ] the a n o m a -
CHI1RAI. DYNAMICS
619
l o u s m a g n e t i c m o m e n t of A 1 is not d e t e r m i n e d . T o obtain the v a l u e (5.3) f o r 6 it is s u f f i c i e n t to r e q u i r e D --< 2 f o r any t h r e e - p o i n t v e r t e x . T h e A l P n c o u p l i n g in o u r L a g r a n g i a n (3.10) is d e s c r i b e d by 2 gA
gP (C+ 1 ) ) ~ a u x + 1 --~
z] .
5.4)
gA Using the r e l a t i o n s [6] 2 2
g A = gp '
2
5.5)
2gpf n = m p ,
we obtain f o r the A l P v v e r t e x -q
1
n Uq,)
5.6)
F u u ( P , q ) = f~r [kg(qu- ~Pu) -gtzu k" (q- ~P)] •
T h i s g i v e s the r a t i o Gs/G D=- m 2, w h e r e G S and G D a r e the c o n s t a n t s which d e t e r m i n e S- and D - w a v e d e c a y of A 1 ~ pn. With 6 = - 1 one d e r i v e s F(A 1 - pn) ~ 45 MeV [6]. T h e pvn c o u p l i n g is v e r y s i m p l e in this a p p r o a c h
~pn~ = -gp 0U . ~ x ~ z l
(5.7)
and the pnr, v e r t e x F~ (p, q) is l i n e a r in m o m e n t a
p p~ (K)
......
:
F u ( p , q ) = igp(p-q)u '
(5.8)
q while in r e f s . [1.2] it h a s D = 3 . The eq. (5.8) g i v e s F ( p -
2v) ~ 135 MeV
[6]. Now we shall obtain e x p r e s s i o n s f o r s o m e i m p o r t a n t f o u r - p o i n t f u n c tions. Since the s y m m e t r y is b r o k e n by the t e r m m2f211 -(r, 'fu) 2] 1/2, then in L a g r a n g i a n (3.10) the nn c o u p l i n g is
2 = "~nT,
1 [(~ 2f:
.z)2_~m2 n
4]+g o 2m ~
[ 2(~ ~ ) 2 _ ( ~ g
~)21 /~ '
(5.9)
w h e r e the f i r s t t e r m is the s a m e a s in ref. [3] and the s e c o n d t e r m is due to the i n c l u s i o n of the f i e l d s O~ and s ~. Note that in the c o n v e n t i o n a l L a g r a n g i a n [2] ,-rn c o u p l i n g c o n t a i n s a l s o the a d d i t i o n a l q u a d r i l i n e a r d e r i v a t i v e t e r m (~ ~t.'-rx ~ i t ) 2. F r o m ~Zvn we obtain the e o n t a c t nv --* nzr v e r t e x Tabcd with D = 2
620
V. I. OGIEVETSKY and B.M. ZUPNIK
• ka
kc
:
-2
2s)-]_]
• ,kb
~ ~ mP
•
" led
(t+u-
+(2 c r o s s e d t e r m s )
1
,
(5.10)
w h e r e s = ( k a + k b ) 2 , t = ( k a - k c ) 2 , u = ( k a - k d ) 2. T h e m a t r i x e l e m e n t f o r ~ s c a t t e r i n g is the s u m of the c o n t a c t t e r m and the t e r m s with the o - m e s o n e x c h a n g e . In this s u m the t e r m s c o n t a i n i n g g ~ c a n c e l at the t h r e s h o l d (s = 4rn~, t = u = 0) and we obtain f o r the S - w a v e s c a t t e r i n g l e n g t h s : F,-
a
~ 0 . 2 0 m -1 O
~
a '
~-0.06m 2
-1 7;
(5.11) '
in a c c o r d a n c e with t h o s e o b t a i n e d by W e i n b e r g [6] u s i n g c u r r e n t a l g e b r a , P C A C and the s m o o t h n e s s condition which in effect c o r r e s p o n d s to o u r r e s t r i c t i o n on the n u m b e r of d e r i v a t i v e s in the L a g r a n g i a n . Note that in the c o n v e n t i o n a l L a g r a n g i a n m o d e l [2] the c o n t a c t ~ ~ w v e r t e x h a s D = 4. T h e n the ~v s c a t t e r i n g l e n g t h s cannot be fixed s i n c e it is p o s s i b l e to include in the L a g r a n g i a n the q u a d r i l i n e a r d e r i v a t i v e t e r m K(Vtz~) 4 (3.4) (K is an a r b i t r a r y constant) which would c h a n g e a o and a 2. The d i r e c t A13~ c o u p l i n g in L a g r a n g i a n (3.10) is gA ~:A3~ : 2f,~ [ 2 ( y t ~ y t ) ( a p : ~ ) - ~ 2 ( a ~ t ) ]
.
(5.12)
T h e c o r r e s p o n d i n g A13~ v e r t e x Ta~bcd is l i n e a r in m o m e n t a
Jeb /
A a~
ig A k c Ta~bcd = - f - [ 6ab 6cd (kc + k d - kb) ~ + 6ac 5bd(kb + led - kc)
/
\ \
kd
+6adSbc(kb +k c - k d ) ~ ] ,
(5.13)
while in the c o n v e n t i o n a l a p p r o a c h [1] it h a s D = 3. Eq. (5.13) g i v e s f o r the p a r t i a l width of the d i r e c t A 1 ~ 3,~ d e c a y mode 35 MeV if r e l a t i o n s (5.5) hold. It is p o s s i b l e to e s t i m a t e r o u g h l y the total width of A 1 adding the d i r e c t A 1 ~ 3,~ and A 1 -~ p~ d e c a y r a t e s . T h i s g i v e s a p p r o x i m a t e l y 80 MeV in a g r e e m e n t with e x p e r i m e n t a l data. M o r e d e t a i l e d p r e d i c t i o n s can be o b t a i n e d f r o m the e x p l i c i t A 1 ~ 3~ a m p l i t u d e which c o n t a i n both the c o n t a c t A 1 ~ 3~ t e r m (5.13) and p - p o l e t e r m s . R e l a t i o n s (5.5) l e a d to the s i m p l e e x p r e s s i o n f o r PvO~ coupling: 2 g o ~:OuO~ = ~ 2 P ( 0 ~ x y t ) 2 - [(~ 8m 2 P which g i v e s the c o n t a c t OTrprr v e r t e x
Qv-0
Og)xTt]2
(5.14)
CHIIRAL I)YNAMICS ", k a
.~ •
kc
rA: :
(go)
621
2
pj
/
×[g~ m2+ ½(pUqp. _g~up.q)] . -COb (p)
(5.15)
,- pd t q)
In conclusion we would like to e m p h a s i z e the i m p o r t a n c e of the r e s t r i c tion on the m o m e n t u m dependence of the n-point functions in connection with the idea of a l g e b r a i c r e a l i z a t i o n s of the c h i r a l s y m m e t r y [7]. We shall p r e s e n t the a n a l y s i s of the a s y m p t o t i c behaviour of t r e e g r a p h s for the p r o c e s s e s in the AlPn s y s t e m elsewhere. The a u t h o r s wish to thank I. V. Polubarinov, B. N. Valuev, A. N. Z a s l a v s ky and e s p e c i a l l y M. A. Eliashwili and M. N. Tugulea for helpful d i s c u s s i o n s .
APPENDIX The elimination of the bilinear d e r i v a t i v e t e r m s f r o m the c o v a r i a n t s R ~ u (3.1) and S~u (3.2) leads to the elimination of the t r i l i n e a r derivative couplings f r o m L a g r a n g i a n (3.4) at h =K = 0. Let us introduce a g e n e r a l substitution for the a x i a l - v e c t o r field: ' n Ag =ZAZ-=[
+F(~b 2) D (~ + G( 4
where only the value F(0) = - (gpfr,)2/gA m 2 " the exclusion of the a p ~ ( b coupling.
2) ~b( ~
~b)] ,
is fixed by the r e q u i r e m e n t
(A.1)
o f
The functions F(~b2), G(~b2) a r e d e t e r m i n e d by the r e q u i r e m e n t that the b i l i n e a r d e r i v a t i v e t e r m s be eliminated f r o m R ~ u , S~u which gives the equations:
gAF
2 _ C2(1 +gA(~F)2 = 0 ,
[FA"A-C2o~ 1 +gAaF)] G gA(G - 2F')
: 0 ,
-C2(1 + g A f F ) ( ~ - I + g A F + g A G ~)2) _ 0 .
(A.2)
Eqs. (A.2) a r e c o n s i s t e n t only with
1] -1
(3.8)
and we have a unique solution (3.7) G =0 ,
F
C gA( 1 +Cc r) m .
__
which is used in sect. 3 (eqs. (3.5), (3.7) and (3.8)).
622
V . I . OGIEVETSKY and B. M. ZUPNIK
REFERENCES [1] H . J . S c h m i t z e r and S. W c i n b e r g , P h y s . Rev. 164 (1967) 1828: J . S . G e r s t e i n and l t . J . S c h m i t z e r , P h y s . Rex,. 170 (1968) ]638. [2] J. S c h w i n g e r , P h y s . L e t t e r s 24B (1967) 473: J. W e s s and B. Z y m i n o , P h y s . Rev. 163 (1967) 1727. [3l P . C h a n g and F. Glirscy, P h y s . Rev. 164 (1967) 1752, 169 (1968) 1397E; S. W e i n b c r g , P h y s . Rev. L e t t e r s 18 (1967) 507, P h y s . Rev. 166 (1968) 1568. B . W . Lee and H . T . Nieh, P h y s . Rev. 166 (1968) 1507; [4] V . I . O g i c v e t s k y and I . V . P o l u b a r i n o v , Ann. of P h y s . 25 (1963) 358. [5] T . D . Lee, S . W e i n b e r g a n d B. Z u m i n o , P h y s . H.ev. L e t t e r s 18 (1967) 1029; T . D . Lce and B. Z u m i n o , P h y s . Rcv. 163 (1967) 1667. [6] S. W e i n b e r g , P r o c . of thc 14th Int. Conf. on high e n c r ~ p h y s i c s , Vienna, 1968, p. 253. [7] S. W e i n b e r g , P h y s . Rev. 177 (1969) 2604.
~ - . A . 2 Nuclear Physics B24 (1970) 612-622. North-Holland IMblishing Company 8.A.5 j
ON THE CHIRAL SU(2)x SU(2) DYNAMICS FOR MESONS A 1, ~ AND V. I. OGIEVETSKY and B. M. Z U P N I K * Laboratory of Theoretical Physics, Joint Institute f o r Nuclear Research, Dubna, USSR Received 31 August 1970
Abstract: It is remarked that the conventional effective Lagrangian with p and A 1 mesons contains t r i l i n e a r and quadrilinear derivative couplings. Thus one should add independently all t e r m s with three and four field derivatives; the theory should then contain a number of a r b i t r a r y parameters. We suggest a new Lagrangian for the Alp?r system containing no more than two field derivatives in each term. This model is determined by only four p a r a m e t e r s gt> gA, toO, MA and gives the smoothest momentum dependence for the contact ver'tices. W e d e r i v e with n e c e s sity the reasonable value 6 = - 1 for the anomalous magnetic moment of A 1 if the KSI.'R relation holds and also apply our model to ~y scattering and A 1 decay.
1. I N T R O D U C T I O N In r e c e n t y e a r s p r o c e s s e s with A1, p a n d a , m e s o n s have b e e n i n t e n s i v e ly i n v e s t i g a t e d u s i n g c u r r e n t a l g e b r a . I m p o r t a n t i n f o r m a t i o n on the AlPTr s y s t e m h a s b e e n d e r i v e d f r o m the W a r d i d e n t i t i e s for the n - p o i n t f u n c t i o n s in the t r e e a p p r o x i m a t i o n with s o m e r e s t r i c t i o n on the m o m e n t u m d e p e n d e n c e of the c o n t a c t v e r t i c e s [1]. N e v e r t h e l e s s , such a r e s t r i c t i o n l e a d s to the s o l u t i o n of the W a r d i d e n t i t i e s which d e p e n d s on s e v e r a l a r b i t r a r y p a r a m e t e r s [1]. T h e c o n s t r u c t i v e way f o r the r e p r o d u c t i o n of the c u r r e n t - a l g e b r a r e s u l t s is the e f f e c t i v e - L a g r a n g i a n m e t h o d b a s e d on the SU(2)× SU(2) g r o u p [2, 3]. In s e c t . 2 of the p r e s e n t p a p e r we show that the c o n v e n t i o n a l L a g r a n g i a n of the A l P ~ s y s t e m [2] c o n t a i n s t r i l i n e a r a n d q u a d r i l i n e a r d e r i v a t i v e t e r m s due to the e l i m i n a t i o n of the n o n - p h y s i c a l b i l i n e a r Air, c o u p l i n g . T h e n one s h o u l d add a l l i n d e p e n d e n t t e r m s with t h r e e a n d four f i e l d d e r i v a t i v e s a n d t h i s l e a d s to the a p p e a r e n c e of the a r b i t r a r y p a r a m e t e r s in the L a g r a n g i a n . B e s i d e s , the v e r y p r e s e n c e of s u c h d e r i v a t i v e t e r m s is u n d e s i r a b l e in the i n t e r a c t i o n L a g r a n g i a n . In s e c t . 3 we s u g g e s t in t h i s c o n n e c t i o n the f o l l o w i n g n a t u r a l r e q u i r e m e n t : The e f f e c l i v e L a g ~ a n g i a n o f the AlPrr s y s t e m rnust c o n t a i n no m o r e lhan two f i e l d d e r i v a t i v e s in e a c h t e r m . T h i s s e v e r e r e s t r i c t i o n d e t e r m i n e s c o m p l e t e l y the i n v a r i a n t L a g r a n g i a n c o n t a i n i n g f o u r i n d e p e n d e n t p a r a m e t e r s * Dniepropctrovsk State University, USSR.
CHLRAL DYNAMICS
613
gp, g A , rap, M A w i t h t h e a i d of t h e n o n - l i n e a r (in p i o n s ) s u b s t i t u t i o n f o r t h e A1 f i e l d . In t h e t r e e a p p r o x i m a t i o n e a c h c o n t a c t v e r t e x i s a p o l y n o m i a l in the f o u r - m o m e n t a , t h e d e g r e e D of w h i c h i s e q u a l to t h e n u m b e r of f i e l d d e r i v a t i v e s in t h e a p p r o p r i a t e t e r m of t h e L a g r a n g i a n . In s e c t . 4 we e s t a b l i s h t h e c o n n e c t i o n of o u r L a g r a n g i a n m o d e l with the p a r t i a l s o l u t i o n of t h e W a r d i d e n t i t i e s u n d e r lhe condition D ~< 2 f o r any contact c e r t e x . In the AlPV s y s t e m it is i m p o s s i b l e to s t r e n g t h e n t h i s r e s t r i c t i o n on the m o m e n t u m d e p e n d e n c e of t h e c o n t a c t v e r t i c e s . Note t h a t in t h e c o n v e n t i o n a l L a g r a n g i a n m o d e l [2] D ~< 4. In s e c t . 5 we d i s c u s s s o m e c o n s e q u e n c e s f o r t h e a m p l i t u d e s p - ' 2~, A1 - ' pr,, A1 - 3~, r , ; r - ' 7r;r, p~ ~ p,-r, etc. We d e r i v e with n e c e s s i t y t h e r e a s o n a b l e v a l u e
(1.1)
5 : ( g A / g p ) 2 {1 - [ ( n , p / ' g p f v ) 2 - 1] -2} - 1 ~ - 1
f o r the a n o m a l o u s m a g n e t i c m o m e n t of t h e A 1 m e s o n . T h e S - w a v e 7rr, s c a t t e r i n g l e n g t h s h a v e b e e n found to b e in a g r e e m e n t with t h o s e o b t a i n e d in c u r r e n t a l g e b r a [6]. We h a v e a l s o e s t i m a t e d t h e t o t a l width of the A 1.
2. P R E L I M I N A R I E S In t h e p r e s e n t p a p e r we p r o p o s e a n e w m o d e l f o r t h e L a g r a n g i a n of t h e AlPV s y s t e m w h i c h c o n t a i n s no m o r e t h a n b i l i n e a r d e r i v a t i v e c o u p l i n g s . W i t h t h i s e n d in v i e w we s h a l l m e n t i o n h e r e s o m e f e a t u r e s of t h e c o n v e n t i o n a l c h i r a l s c h e m e with A1, p a n d v m e s o n s ( s e e , e . g . , r e f s . [2, 3]). T h e n o n - l i n e a r L a g r a n g i a n f o r v e c t o r , a x i a l - v e c t o r a n d p i o n f i e l d s V~, A~ a n d z c a n be w r i t t e n a s
z _ _ v2, _,A 2,
nZ2o iv2 +A2) +
.
,
/2.1)
with t h e c o v a r i a n t c u r l s
V~v = O~z V v - Ov Vl.t - g Vp.× V - g A .
×Av ,
A ~v : O~tAv - ?vAst - g V ~ × A u + g Vv×A~t
(2.2)
a n d c o v a r i a n t d e r i v a t i v e of t h e p i o n f i e l d
vu ~ : Au ~+ [~(1 + ~)1-1 ~ ( 0 A u ~ ) ,
(2.3)
where (~: ( 1 . ~ 2 ) 1 / 2 ,
Ag,~:
aUq)-gV~×d2+gaAg " .
(2.3a)
In t h e s e e q s . ~ = f ~ l : r r ( f n ~ 95 MeV i s t h e p i o n d e c a y c o n s t a n t ) a n d t o o , g , d are some unrenormalized parameters. This Lagrangian is invariant un-
614
v. I. OGIEVETSKY and B. M. ZUPNIK
d e r the SU(2)× SU(2) group. M o r e o v e r , only the m a s s t e r m for v e c t o r and a x i a l - v e c t o r fields b r e a k s the i n v a r i a n c e under the gauge SU(2)× SU(2) t r a n s f o r m a t i o n s with the c o o r d i n a t e - d e p e n d e n t p a r a m e t e r s a ,fl [2, 3]:
~( vu :LA u) : - ( a ~:~ ) × ( V u ~ A u) _ g - 1 a u ( a ±,a) , 54:
- a×4+fl~.
(2.4)
It follows i m m e d i a t e l y that the interacting fields V., and A/s t r a n s f e r spin 1 only [4] (in the limit of z e r o - m a s s pions), i.e. ~he t r a n s v e r s a l i t y conditions ~ V ~ = 0, 0 ~ A g = 0 a r e satisfied off the m a s s shell. However, the t e r m ( V ~ 4 ) 2 in eq. (2.1) contains the n o n - p h y s i c a l bilinear coupling A / s ~ 4 which is usually eliminated by the following substitution of the a x i a l - v e c t o r field: _9,2
A = a~-(gJf/mo).2)~4.
(2.5)
Then it is i m p o s s i b l e to maintain the condition ~ a p = 0 for the new a x i a l v e c t o r field a~. The introduction of 7 ) ~ 4 = ~ 4 - g V ~ × 4 instead of ~ 4 is n e c e s s a r y to e n s u r e t r a n s v e r s a l i t y for the interacting v e c t o r field V~ [4] since it is important in connection with the v e c t o r - d o m i n a n c e idea [5]. The c r u c i a l point is that substitution (2.5) c o n t r i b u t e s to Vgv the t e r m (/)/x 4 × :7)v 4 w h i c h is bilinear in the field derivatives. Hence, the c o n v e n tional effective L a g r a n g i a n contains the t r i l i n e a r and q u a d r i l i n e a r d e r i v a tive t e r m s ~ Vv" :b~ 4 × 2 ) v 4 , ( a N ×.b v 4 ) ' ( : / ) t x 4 × <~)v4) and (:/')g4 ×~b u4)2 g e n e r a t e d by V~v in eq. (2.1). It should be said that t h e r e is u n d e s i r a b l e to have the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e couplings in the interaction L a g r a n g i a n . N e v e r t h e l e s s , if one a d m i t s the p r e s e n c e of such couplings, then all possible t e r m s with t h r e e and four d e r i v a t i v e s should be added and the n u m b e r of p a r a m e t e r s in the theory should inc r e a s e . Besides, the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e couplings lead to n o n - s m o o t h m o m e n t u m dependence of the amplitudes p --' 2,-r, A 1 ~ 37,, ~ r , - ' r,r., etc.
3. LAGRANGIAN OF THE AlPr, SYSTEM WITHOUT TRILINEAR AND QUADRILINEAR DERIVATIVE COUPLINGS To avoid the above mentioned difficulties of the conventional L a g r a n g i a n model for the AlP~ s y s t e m we p r o p o s e the following natural r e q u i r e m e n t :
Trilinear and quadrilinear in field derivatives terms must be absent in the effective Lagrangian of lhe AlP~ system. Now we s t a r t to c o n s t r u c t the g e n e r a l L a g r a n g i a n which s a t i s f i e s this requirement. It is convenient to define for this p u r p o s e the new c o v a r i a n t quantities
RI~u : V ~v+(C2/g) AW.4×AV 4 ,
(3.17
CHIRAL DYNAMICS
615
S u e : A ~ v - (C2/g (~) 0 × (AI~0 × A v 0 ) ,
(3.2)
with s o m e constant C. The RI ~ v and S11 v have the s a m e t r a n s f o r m a t i o n p r o p e r t i e s as c o v a r i a n t c u r l s Vp. v and A ~ v :
5R~v : - a× Rp.v-flXS~v
,
5 S u e = - a X S u v - fl× R ~v .
(3.3)
H e r e eqs. (2.2), (2.3a) and (2.4) have been used. Let us w r i t e down f i r s t the g e n e r a l e x p r e s s i o n for the i n v a r i a n t L a g r a n gian of the AlP~ s y s t e m (containing t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e terms):
1
2
_¼ZAS2v+~(Zp_ZA)((qb×Rg02_(O×S
)2
- 2crRUv× ~" Sgv)+h[Vl~ 0 x V v 0 ] 2+g(Vl~ 0 " V g 0 ) 2 +
~mo(Vi ~ 12 2 + A 2 9 4 ~ d 2 ( V
0)2 ,
(3.4)
w h e r e Zp, ZA, too, d, h a n d K a r e s o m e a r b i t r a r y c o n s t a n t s . Our main p u r p o s e is the e l i m i n a t i o n of the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e t e r m s f r o m the L a g r a n g i a n (3.4) t o g e t h e r with the e l i m i n a t i o n of the b i l i n e a r AlU coupling. Also the kinetic t e r m s m u s t be c o r r e c t l y n o r malized. All t h e s e r e q u i r e m e n t s can be s a t i s f i e d only with h =K = 0 in a unique way by p e r f o r m i n g the following substitutions for the v e c t o r and a x i a l - v e c t o r fields:
-1/2 e g , V = ZO
A
: ZA1/2[ a u + F ( ~ 2) cD 0 ] .
(3.5)
We shall use below the r e n o r m a l i z e d constants:
go = z-pl/2g
'
gA = Z - 2 / 2 g ,
m
p
= Z-1/2m p
o'
_-112, 2 d2g2) l/2
= z~A
(too+
(3.6)
To exclude the b i l i n e a r a ,rO , t*" 0 coupling we fix in eq. (3.5) the value / F ( O ) = - g2of2/g A m 2 then the function F( 02) is uniquely d e t e r m i n e d by the r e q u i r e r h e n t that the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e t e r m s m u s t be absent in the L a g r a n g i a n C (gPfr')2 ] 1 + ~ F(02) = -gA 0+-Cb) = 2- L
gAmp w h e r e ~ = (1 - 0 2 ) 1/2 and
~
02+
(3.7) "'"
'
616
V.I. OGIEVETSKY and B.M. ZUPNIK
C = [(mp/fT, gp) 2 - 1] -1 .
(3.8)
We have used here the relation [2, 3]: f2 =
m(_~p. 2
) [1-
~
('npEA~ 2] (3.9)
\MAgp/ J ,
which follows from the c o r r e c t normalization of the kinetic t e r m ~(~#x~) 2. In fact we have determined the non-linear substitution of a~x and the value of the constant C to exclude the bilinear derivative t e r m s from RIx ~, (eq. (3.1)) and Slx v (eq. (3.2)) (see appendix), the t e r m s with three and four derivatives being eliminated from eq. (3.4). Now we can write the final e x p r e s s i o n for our Lagrangian of the AlP7, s y s t e m which contains non-derivative couplings and the t e r m s with one and two derivatives only: 2
g.. (~b×Oy.u 72+}(1 _ ( 4 × - /~v) 2 g
, 2 :2 - ie2~.v-¼a~u
1[f~(l+C)12[(,-OlaO)2+CrZPgA'~2 )2] +½H(~2) ( e~v~× a~,) + ~L-I +C~- ] .MAgp. (~a~O ~271+C~
+gA.T.~ iT+CO.] [((~- 1)(:2) Ix 0 ,/f~gpMA', 2
-i(---m~--
) (0x ag)
2+
alx)+ g-l( a~xO ) (Oa O) ]
½m2p 2
2 2 n12f:(~
la + MA"bt + 7,
(3.10) '
where the s y m m e t r y - b r e a k i n g t e r m m 2 f 2 ~ leads to PCAC in this model [3]. In eq. (3.10) the following notations have been used: 2
B((~ 2) : g A L l + c -2
2
II((~2)
-1
F2(~b 2) ,
(3.11)
g2
= &~pgA[I+C-I(~(I__~)IF(~)2),gA
(3.127
CHIRAL I)YNAMICS
617
2
O~u
:
gg
Z 1/2 p R~v
= ~
gp -{-
a
v
Cg A gp _ _ _
(1-C2~)
(abtx:3u0
a
+ /)
× a
u
t~× a v ) ,
(3.13)
= ZA/2[S u-gA F Cb2 '
) O × R / ~ v ] = e /~ a v - ? v a /~ - g p g b t × a v
(
+gp~Ou× a~.-EACO× ( ag× au) .
(3.14)
4. C O N N E C T I O N B E T W E E N OUR M O D E L AND T H E " H A R D - P I O N " M E T H O D IN T H E A L G E B R A O F C U R R E N T S
The n-point functions derived from the effective Lagrangian of the A l P n system are known to be a solution for the Ward identities in SU(2)×SU(2) algebra of currents [1]. In this section we reformulate our basic requirement for the Lagrangian in terms of the "hard-pion" method If]. The vector and axial-vector currents ~/z and "~5g are obtained from our Lagrangian (3.10) by using the Gell-Mann-Levy method (see, e.g. ref. [5]). 2
=
~
mP
~
~
2 mpg A ~5U = --2 [ag +F(~2) gp
'
:7)~b] .
(4.1)
Note t h a t t h e s e c u r r e n t s o b e y t h e e q u a l - t i m e c o m m u t a t i o n r e l a t i o n s of t h e a l g e b r a of f i e l d s [5]: a
b
a
[~ o(X,t),~ g(y,t)] = [ ~ 5 o ( X , t ) ,
~5p(y,t)]
c(
ieabc.~ g x,t) 5( x - y )
=
/ ,\rap 2
-i~-~-).
5abgpk Z -k
5(x-y),
[9:(x,t),'~5~(Y,t)]=[~5o(X,t),~b(y,l)]=Zeabc~)5~(x,t)6(x-y). b
(4.2)
°
(4.3)
Here k=1,2,3 ands=0,1,2,3. T h e p r o o f of t h e s e r e l a t i o n s is e q u i v a l e n t to t h a t r e f . [5]. W e r e m a r k h e r e t h a t in a l l S U ( 2 ) × S U ( 2 ) g a u g e L a g r a n g i a n schemes the corresponding Schwinger terms have the same simple cn u m b e r f o r m if c u r r e n t s a r e c a l c u l a t e d by t h e G e l l - M a n n - L e v y m e t h o d . T h e r e a r e , of c o u r s e , s o m e m o d e l - d e p e n d e n t c o m m u t a t i o n r e l a t i o n s . T h e s t r o n g f o r m of P C A C f3] (2 q ~ = m 2 f :~) is p r e s e n t in o u r m o d e l . " t" " .L J b t d a / ~ . 7, m ne t r e e a p p r o x i m a t i o n t h e c o n t a c t v e r t e x f u n c t i o n s c o r r e s p o n d to t h e v a r i o u s c o n t a c t c o u p l i n g s in the L a g r a n g i a n . T h e d i a g r a m of any n - p o i n t
618
V.I. OGIEVETSKY
and B. M. ZUPNIK
f u n c t i o n i n v o l v e s c o n t a c t v e r t i c e s c o n n e c t e d by p r o p a g a t o r s without f o r m ing loops. E a c h c o n t a c t v e r t e x function is a p o l y n o m i a l in the f o u r - m o m e n ta, the d e g r e e of which is equal to the n u m b e r of field d e r i v a t i v e s in the a p p r o p r i a t e c o n t a c t t e r m . T h i s p o l y n o m i a l d e g r e e D c h a r a c t e r i z e d the m o m e n t u m d e p e n d e n c e of a c o n t a c t v e r t e x function. In the t r e e a p p r o x i m a tion D d e p e n d s on the n u m b e r ng of the p - m e s o n a n d / ' o r A 1 m e s o n lines a t t a c h e d to the v e r t e x i r r e s p e c t i v e of the n u m b e r of pion lines. We have D=2 for ng=0,2, D=l for ng=l,3 andD=0 for ng=4. In the c o n v e n t i o n a l c u r r e n t - a l g e b r a i c a p p r o a c h to the A l P n s y s t e m [ 1 , 2 ] t h e r e a r e v e r t i c e s with D = 3 and D = 4 which c o r r e s p o n d to the t r i l i n e a r and q u a d r i l i n e a r d e r i v a t i v e c o u p l i n g s . In o u r a p p r o a c h the m o m e n t u m d e p e n d e n c e of the c o n t a c t v e r t e x functions is m i n i m a l and cannot be r e d u c e d f u r t h e r . At l e a s t two field d e r i v a t i v e s m u s t be p r e s e n t b e c a u s e the t e r m s ( V ~ b ) 2, (t)~u! 2 etc. a r e a l w a y s p r e s e n t . Thus, using Lagrangian (3.10) we derive n-poinl fitnctions which are lhe minimum momentum-dependent solution of Ward identities in SU(2) x SU(2)
algebra of currents with c-number Schu'inger terms. Note that in m e s o n - n u c l e o n c o u p l i n g s we can l i m i t o u r s e l v e s to v e r t i c e s which a r e l i n e a r in m o m e n t a .
5. SOME C O N S E Q U E N C E S O F THE M O D E L V a r i o u s r e s t r i c t i o n s on the m o m e n t u m d e p e n d e n c e of a m p l i t u d e s a r e often u s e d in the a l g e b r a of c u r r e n t s . In o u r L a g r a n g i a n m o d e l t h e r e a r e such s e v e r e r e s t r i c t i c n s f o r p r o c e s s e s with A1, p and 7r m e s o n s . T h e r e q u i r e m e n t of the m i n i m u m m o m e n t u m d e p e n d e n c e l e a d s to s o m e i m p o r t a n t c o n s e q u e n c e s for the s i m p l e s t n - p o i n t functions. F i r s t , we p r e s e n t the r e s u l t s f o r t h r e e - p o i n t functions. T h e A 1 A l P i n t e r a c t i o n in L a g r a n g i a n (3.101 is d e s c r i b e d by 2 ~AAp-~; A(1-C2)
al-t@v" al.t× av+gp(@lj, x a v -
Ov ×
al-t)'~tzav"
(5.1/
T h e c o r r e s p o n d i n g A 1 A l P v e r t e x Ftav~t(p,q) is l i n e a r in m o m e n t a (i.e. D= 11
AP(p )
j A V ( q ) F ~ v x ( P , q ) = - i g p [ g ~ v ( P + q ) x - g v x P u - "g ~ v qv --NNp~(k )
+(2+5)(k~z~v~t-kvguv) ] .
(5.2/
H e r e 5 is the a n o m a l o u s m a g n e t i c m o m e n t of the A 1 m e s o n .
o:
~A-p 2 I
)
~ mp
2
-2
1-1,
is w h e r e eq. (3.7) f o r C h a s b e e n used. If the KSFR relation (2g2f 2 valid, we obtain 5 = - 1. T h i s v a l u e f o r 5 is the m o s t p r e f e r a b l e " ( a s K s c u s s e d in d e t a i l s in ref. [6]). In the c o n v e n t i o n a l a p p r o a c h [ 1 , 2 ] the a n o m a -
CHI1RAI. DYNAMICS
619
l o u s m a g n e t i c m o m e n t of A 1 is not d e t e r m i n e d . T o obtain the v a l u e (5.3) f o r 6 it is s u f f i c i e n t to r e q u i r e D --< 2 f o r any t h r e e - p o i n t v e r t e x . T h e A l P n c o u p l i n g in o u r L a g r a n g i a n (3.10) is d e s c r i b e d by 2 gA
gP (C+ 1 ) ) ~ a u x + 1 --~
z] .
5.4)
gA Using the r e l a t i o n s [6] 2 2
g A = gp '
2
5.5)
2gpf n = m p ,
we obtain f o r the A l P v v e r t e x -q
1
n Uq,)
5.6)
F u u ( P , q ) = f~r [kg(qu- ~Pu) -gtzu k" (q- ~P)] •
T h i s g i v e s the r a t i o Gs/G D=- m 2, w h e r e G S and G D a r e the c o n s t a n t s which d e t e r m i n e S- and D - w a v e d e c a y of A 1 ~ pn. With 6 = - 1 one d e r i v e s F(A 1 - pn) ~ 45 MeV [6]. T h e pvn c o u p l i n g is v e r y s i m p l e in this a p p r o a c h
~pn~ = -gp 0U . ~ x ~ z l
(5.7)
and the pnr, v e r t e x F~ (p, q) is l i n e a r in m o m e n t a
p p~ (K)
......
:
F u ( p , q ) = igp(p-q)u '
(5.8)
q while in r e f s . [1.2] it h a s D = 3 . The eq. (5.8) g i v e s F ( p -
2v) ~ 135 MeV
[6]. Now we shall obtain e x p r e s s i o n s f o r s o m e i m p o r t a n t f o u r - p o i n t f u n c tions. Since the s y m m e t r y is b r o k e n by the t e r m m2f211 -(r, 'fu) 2] 1/2, then in L a g r a n g i a n (3.10) the nn c o u p l i n g is
2 = "~nT,
1 [(~ 2f:
.z)2_~m2 n
4]+g o 2m ~
[ 2(~ ~ ) 2 _ ( ~ g
~)21 /~ '
(5.9)
w h e r e the f i r s t t e r m is the s a m e a s in ref. [3] and the s e c o n d t e r m is due to the i n c l u s i o n of the f i e l d s O~ and s ~. Note that in the c o n v e n t i o n a l L a g r a n g i a n [2] ,-rn c o u p l i n g c o n t a i n s a l s o the a d d i t i o n a l q u a d r i l i n e a r d e r i v a t i v e t e r m (~ ~t.'-rx ~ i t ) 2. F r o m ~Zvn we obtain the e o n t a c t nv --* nzr v e r t e x Tabcd with D = 2
620
V. I. OGIEVETSKY and B.M. ZUPNIK
• ka
kc
:
-2
2s)-]_]
• ,kb
~ ~ mP
•
" led
(t+u-
+(2 c r o s s e d t e r m s )
1
,
(5.10)
w h e r e s = ( k a + k b ) 2 , t = ( k a - k c ) 2 , u = ( k a - k d ) 2. T h e m a t r i x e l e m e n t f o r ~ s c a t t e r i n g is the s u m of the c o n t a c t t e r m and the t e r m s with the o - m e s o n e x c h a n g e . In this s u m the t e r m s c o n t a i n i n g g ~ c a n c e l at the t h r e s h o l d (s = 4rn~, t = u = 0) and we obtain f o r the S - w a v e s c a t t e r i n g l e n g t h s : F,-
a
~ 0 . 2 0 m -1 O
~
a '
~-0.06m 2
-1 7;
(5.11) '
in a c c o r d a n c e with t h o s e o b t a i n e d by W e i n b e r g [6] u s i n g c u r r e n t a l g e b r a , P C A C and the s m o o t h n e s s condition which in effect c o r r e s p o n d s to o u r r e s t r i c t i o n on the n u m b e r of d e r i v a t i v e s in the L a g r a n g i a n . Note that in the c o n v e n t i o n a l L a g r a n g i a n m o d e l [2] the c o n t a c t ~ ~ w v e r t e x h a s D = 4. T h e n the ~v s c a t t e r i n g l e n g t h s cannot be fixed s i n c e it is p o s s i b l e to include in the L a g r a n g i a n the q u a d r i l i n e a r d e r i v a t i v e t e r m K(Vtz~) 4 (3.4) (K is an a r b i t r a r y constant) which would c h a n g e a o and a 2. The d i r e c t A13~ c o u p l i n g in L a g r a n g i a n (3.10) is gA ~:A3~ : 2f,~ [ 2 ( y t ~ y t ) ( a p : ~ ) - ~ 2 ( a ~ t ) ]
.
(5.12)
T h e c o r r e s p o n d i n g A13~ v e r t e x Ta~bcd is l i n e a r in m o m e n t a
Jeb /
A a~
ig A k c Ta~bcd = - f - [ 6ab 6cd (kc + k d - kb) ~ + 6ac 5bd(kb + led - kc)
/
\ \
kd
+6adSbc(kb +k c - k d ) ~ ] ,
(5.13)
while in the c o n v e n t i o n a l a p p r o a c h [1] it h a s D = 3. Eq. (5.13) g i v e s f o r the p a r t i a l width of the d i r e c t A 1 ~ 3,~ d e c a y mode 35 MeV if r e l a t i o n s (5.5) hold. It is p o s s i b l e to e s t i m a t e r o u g h l y the total width of A 1 adding the d i r e c t A 1 ~ 3,~ and A 1 -~ p~ d e c a y r a t e s . T h i s g i v e s a p p r o x i m a t e l y 80 MeV in a g r e e m e n t with e x p e r i m e n t a l data. M o r e d e t a i l e d p r e d i c t i o n s can be o b t a i n e d f r o m the e x p l i c i t A 1 ~ 3~ a m p l i t u d e which c o n t a i n both the c o n t a c t A 1 ~ 3~ t e r m (5.13) and p - p o l e t e r m s . R e l a t i o n s (5.5) l e a d to the s i m p l e e x p r e s s i o n f o r PvO~ coupling: 2 g o ~:OuO~ = ~ 2 P ( 0 ~ x y t ) 2 - [(~ 8m 2 P which g i v e s the c o n t a c t OTrprr v e r t e x
Qv-0
Og)xTt]2
(5.14)
CHIIRAL I)YNAMICS ", k a
.~ •
kc
rA: :
(go)
621
2
pj
/
×[g~ m2+ ½(pUqp. _g~up.q)] . -COb (p)
(5.15)
,- pd t q)
In conclusion we would like to e m p h a s i z e the i m p o r t a n c e of the r e s t r i c tion on the m o m e n t u m dependence of the n-point functions in connection with the idea of a l g e b r a i c r e a l i z a t i o n s of the c h i r a l s y m m e t r y [7]. We shall p r e s e n t the a n a l y s i s of the a s y m p t o t i c behaviour of t r e e g r a p h s for the p r o c e s s e s in the AlPn s y s t e m elsewhere. The a u t h o r s wish to thank I. V. Polubarinov, B. N. Valuev, A. N. Z a s l a v s ky and e s p e c i a l l y M. A. Eliashwili and M. N. Tugulea for helpful d i s c u s s i o n s .
APPENDIX The elimination of the bilinear d e r i v a t i v e t e r m s f r o m the c o v a r i a n t s R ~ u (3.1) and S~u (3.2) leads to the elimination of the t r i l i n e a r derivative couplings f r o m L a g r a n g i a n (3.4) at h =K = 0. Let us introduce a g e n e r a l substitution for the a x i a l - v e c t o r field: ' n Ag =ZAZ-=[
+F(~b 2) D (~ + G( 4
where only the value F(0) = - (gpfr,)2/gA m 2 " the exclusion of the a p ~ ( b coupling.
2) ~b( ~
~b)] ,
is fixed by the r e q u i r e m e n t
(A.1)
o f
The functions F(~b2), G(~b2) a r e d e t e r m i n e d by the r e q u i r e m e n t that the b i l i n e a r d e r i v a t i v e t e r m s be eliminated f r o m R ~ u , S~u which gives the equations:
gAF
2 _ C2(1 +gA(~F)2 = 0 ,
[FA"A-C2o~ 1 +gAaF)] G gA(G - 2F')
: 0 ,
-C2(1 + g A f F ) ( ~ - I + g A F + g A G ~)2) _ 0 .
(A.2)
Eqs. (A.2) a r e c o n s i s t e n t only with
1] -1
(3.8)
and we have a unique solution (3.7) G =0 ,
F
C gA( 1 +Cc r) m .
__
which is used in sect. 3 (eqs. (3.5), (3.7) and (3.8)).
622
V . I . OGIEVETSKY and B. M. ZUPNIK
REFERENCES [1] H . J . S c h m i t z e r and S. W c i n b e r g , P h y s . Rev. 164 (1967) 1828: J . S . G e r s t e i n and l t . J . S c h m i t z e r , P h y s . Rex,. 170 (1968) ]638. [2] J. S c h w i n g e r , P h y s . L e t t e r s 24B (1967) 473: J. W e s s and B. Z y m i n o , P h y s . Rev. 163 (1967) 1727. [3l P . C h a n g and F. Glirscy, P h y s . Rev. 164 (1967) 1752, 169 (1968) 1397E; S. W e i n b c r g , P h y s . Rev. L e t t e r s 18 (1967) 507, P h y s . Rev. 166 (1968) 1568. B . W . Lee and H . T . Nieh, P h y s . Rev. 166 (1968) 1507; [4] V . I . O g i c v e t s k y and I . V . P o l u b a r i n o v , Ann. of P h y s . 25 (1963) 358. [5] T . D . Lee, S . W e i n b e r g a n d B. Z u m i n o , P h y s . H.ev. L e t t e r s 18 (1967) 1029; T . D . Lce and B. Z u m i n o , P h y s . Rcv. 163 (1967) 1667. [6] S. W e i n b e r g , P r o c . of thc 14th Int. Conf. on high e n c r ~ p h y s i c s , Vienna, 1968, p. 253. [7] S. W e i n b e r g , P h y s . Rev. 177 (1969) 2604.