- Email: [email protected]
Radiative corrections to ϕ → KK decay
-_~
Nuclear Physics B9 (1969) 451-459. North-Holland Publ. Comp., A m s t e r d a m
B
RADIATIVE
CORRECTIONS
TO
~ --.. K K D E C A Y
E. C R E M M E R and M. G O U R D I N
Laboratoire de Physique Th~orique et Hautes Energies, Orsay Received 19 December 1968 Abstract: We compute the electromagnetic c o r r e c t i o n s to the ~P--~ KK decay as the product of p h a s e - s p a c e c o r r e c t i o n s due to the K +, Ko mass difference by r a d i a tive c o r r e c t i o n s in the s e c o n d - o r d e r approximation. The final result is F ((fl -~ K+K-) = 1.60 F (eft ~ KOKo)
1. I N T R O D U C T I O N T h e a i m of t h i s p a p e r is a c o m p a r i s o n of t h e two d e c a y m o d e s of t h e q~ meson cp -~ K + + K - ,
(1)
q~ ~ K ° + K ° ,
(2)
t a k i n g into a c c o u n t t h e e l e c t r o m a g n e t i c c o r r e c t i o n s to t h e i s o b a r i c s p i n i n variance. L e t us f i r s t d e s c r i b e o u r n o t a t i o n f o r t h e d e c a y of a v e c t o r m e s o n V of m a s s M into t w o p s e u d o s c a l a r ( o r s c a l a r ) m e s o n s B an d B of m a s s rn (M > 2m). T h e L o r e n t z - i n v a r i a n t T - m a t r i x e l e m e n t i s g i v e n by
(3)
where (a) P , p+, p . a r e t h e e n e r g y - m o m e n t u m v e c t o r s r e s p e c t i v e l y f o r t h e V, B, B m e s o n s w i t h t h e c o n s e r v a t i o n l a w P = p++p_, (b) e~(P, ~) i s t h e p o l a r i z a t i o n v e c t o r f o r t h e V - m e s o n of e n e r g y m o m e n t u m P and h e l i c i t y ;% r e s t r i c t e d by th e s u p p l e m e n t a r y c o n d i t i o n Pp e~(P, X)= =0. T h e c o m p l e x c o u p l i n g c o n s t a n t g i s d e f i n e d a s t h e v a l u e of t h e V B B v e r t e x f u n c t i o n f o r t h e t h r e e p a r t i c l e s on t h e m a s s s h e l l . T h e d e c a y w i d t h F ( V ~ B + B) i s r e l a t e d to g by t h e w e l l - k n o w n e x p r e s sion [I]
Laboratoire associg au C. N . R . S . Postal address: Laboratoire de Physique Th~orique et Hautes E n e r g i e s , B~ttiment 211, Facult~ des Sciences, 91-Orsay, F r a n c e .
452
E. CREMMER and M. GOURDIN l F(V ~ B +B) = ~
lg[ 2 4n M v3 ,
(4)
w h e r e the v e l o c i t y v is g i v e n in t e r m s of m a s s e s by ,=
( 1 - 4m2~½
.
(5)
T h e r a t i o of the two widths f o r the d e c a y m o d e s (1) and (2) of the (p m e s o n is given by Rq~ -
F(~K+K -)_ F ( q ~ ~ K ° K °)
g + 2 (v+ j3 go \~-o]
.
(6)
L e t us a s s u m e f o r a m o m e n t that the i s o b a r i c spin s y m m e t r y is an e x a c t i n v a r i a n c e . T h e c h a r g e d and n e u t r a l K - m e s o n m a s s e s a r e then equal and v+ = v o. On the o t h e r hand the ~ m e s o n b e i n g an i s o s i n g l e t g+ = go. It f o l l o w s that in the a b s e n c e of e l e c t r o m a g n e t i c i n t e r a c t i o n s we s i m p l y have Rq~ = 1. We now i n t r o d u c e the e l e c t r o m a g n e t i c i n t e r a c t i o n s giving r i s e f i r s t to a m a s s splitting i n s i d e the K+K ° doublet and s e c o n d l y to a d e v i a t i o n f r o m unity of the c o u p l i n g c o n s t a n t r a t i o g+/go. Both t y p e s of e f f e c t s a r e of c o u r s e s t r o n g l y c o r r e l a t e d b e c a u s e of t h e i r c o m m o n origin. N e v e r t h e l e s s , it is u s u a l , as an a p p r o x i m a t e t r e a t m e n t , to split the e l e c t r o m a g n e t i c c o r r e c t i o n s into two p a r t s : (a) the p h a s e - s p a c e c o r r e c t i o n s by i n s e r t i n g the e x p e r i m e n t a l v a l u e s f o r the c h a r g e d and n e u t r a l K - m e s o n m a s s e s in the p h a s e - s p a c e r a t i o (v+/vo) 3 (ref. [2]) _~)3 ~ 1.54 ,
(7)
(b) the e l e c t r o m a g n e t i c c o r r e c t i o n s to the c o u p l i n g c o n s t a n t , e v a l u a t i n g the finite p a r t of the v e r t e x f u n c t i o n in the s e c o n d - o r d e r a p p r o x i m a t i o n . T h e r e s u l t of o u r c a l c u l a t i o n g i v e s g_n+ 2_~ 1.042 . C o m b i n i n g this r e s u l t and t h a t of f o r m u l a (7) we d e d u c e , f o r the r a t i o Rq~, the v a l u e R(p ~ 1.60 , to be c o m p a r e d with p r e c i s e e x p e r i m e n t s . Sect. 2 is d e v o t e d to g e n e r a l c o n s i d e r a t i o n s about the e l e c t r o d y n a m i c s of K m e s o n and (p m e s o n . T h e r a d i a t i v e c o r r e c t i o n s to the VBB v e r t e x due to v i r t u a l p h o t o n s a r e d i s c u s s e d in s e c t . 3; as a c o n s e q u e n c e of the W a r d identity, the u l t r a v i o l e t divergences cancel. In sect. 4, we c o m p u t e the finite p a r t of the v e r t e x f u n c t i o n u s i n g d i s p e r sion r e l a t i o n t e c h n i q u e in the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to electromagnetic interactions.
RADIATIVE CORRECTIONS
453
T h e i n f r a r e d d i v e r g e n c e s a r e e l i m i n a t e d , c o n s i d e r i n g in addition the r a d i a t i v e c o r r e c t i o n s a s s o c i a t e d to a s o f t - p h o t o n e m i s s i o n . T h e b r e m s s t r a h l u n g p r o c e s s is c o m p u t e d in s e c t . 5. F i n a l l y , the r e s u l t s a r e c o l l e c t e d and d i s c u s s e d in s e c t . 6.
2. G E N E R A L C O N S I D E R A T I O N S T h e H a m i l t o n i a n d e n s i t y f o r the i n t e r a c t i o n of a c h a r g e d s p i n l e s s p a r t i cle d e s c r i b e d by a field ~(x) with the e l e c t r o m a g n e t i c field Aiz(x ) h a s the familiar form Hint(X ) = i eAl.L(x)[ ~*(x) 8~)(x) 8x~t
8eP*(x) 8xtz ~(x)] - e 2 ~ * ( x ) ~ ( x ) A g ( x ) A g ( x ) .
T h e l a s t t e r m of eq. (8), g e n e r a l l y c a l l e d a c o n t a c t t e r m , e n s u r e the g a u g e i n v a r i a n c e of the t h e o r y .
(8)
is n e c e s s a r y to
2.1. K-meson self-energy part We c o n s i d e r a v i r t u a l K m e s o n of e n e r g y - m o m e n t u m p. T h e s e l f - e n e r g y f u n c t i o n ~(p2) c o r r e s p o n d s to a c l a s s of d i a g r a m s s y m b o l i c a l l y d r a w n in fig. 1.
Fig. 1. General self-energy diagrams for K mesons. T h e f u n c t i o n ~(p2) can be c o n v e n i e n t l y w r i t t e n in the f o r m ~(p2) = A + (p2 + m2) B + (p2 + m 2) ~f(p2) ,
(9)
w h e r e the infinite (with u l t r a v i o l e t d i v e r g e n c e s ) c o n s t a n t s A and B a r e d e fined by A = ~(-m2) ,
B = ~'(-m2) ,
and the finite p a r t F4(P2) v a n i s h e s f o r a r e a l K m e s o n ~ f ( - m 2) = 0 . F o r i n s t a n c e , in the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to e l e c t r o m a g n e t i c i n t e r a c t i o n s , the f u n c t i o n ~(2)(p2) c o r r e s p o n d s to the s u m of the two d i a g r a m s (figs. 2(a) and 2(b)) w h e r e the s o l i d line r e p r e s e n t s a K m e s o n and the wavy line a photon.
(a) (b) Fig. 2. Meson self-energy diagrams in the second electromagnetic approximation order.
454
E. CREMMER and M. GOURDIN
2.2. V e r t e x p a r t The v e r t e x function f o r an off-shell v e c t o r m e s o n decaying into two p s e u d o s c a l a r m e s o n s B and ~ has the g e n e r a l s t r u c t u r e A~(p+, p_) = g o ( s ) ( p + - p _ ) l ~ [ L + Af(s)] ,
(10)
where p+, p_ a r e the m e s o n e n e r g y - m o m e n t u m v e c t o r s and s = - ( p + + p _ ) 2 . The infinite constant L contains u l t r a v i o l e t d i v e r g e n c e s and is defined so that the finite p a r t Af(s) s a t i s f i e s Af(O) = 0 . In the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to e l e c t r o m a g n e t i c i n t e r a c tions the function A!2)(p+, p_) c o r r e s p o n d s to the sum of the t h r e e d i a g r a m s figs. 3(a), 3(b), 3(c).~Here the double solid line r e p r e s e n t s the v e c t o r meson.
a4
"
( b ) ~
Fig. 3. Second-order electromagnetic corrections to the VBB vertex. 2.3. W a r d i d e n t i t y The coupling of the v e c t o r m e s o n to p s e u d o s c a l a r m e s o n s is analogous to the coupling of a photon to p s e u d o s c a l a r m e s o n s . We then have a Ward identity which r e l a t e s the s e l f - e n e r g y function E(p 2) and the v e r t e x function A~(p+, p_) in the following way apP E(P 2) p2 = -m2 : - ~ 1
A/~(p, -p) .
F r o m eqs. (9) and (10) we simply deduce B+L
3. RADIATIVE
CORRECTIONS
TO
THE
(11)
: 0 .
VBB
VERTEX:
VIRTUAL
PHOTONS
Let us use the following notation as introduced in sect. 1. p++p_
= P,
p+-p-
= ± ,
4. P = 0 ,
p2 = -s.
The T - m a t r i x element for the V -* B +B t r a n s i t i o n has the s t r u c t u r e given by eq. (3) and we denote by go the coupling constant in the a b s e n c e of e l e c tromagnetic interactions T(0) = go ~
e~(p,~) .
(12)
RADIATIVE CORRECTIONS
455
3.1. M a s s and m e s o n p r o p a g a t o r r e n o r m a l i z a t i o n We f i r s t c o n s i d e r the d i a g r a m s of s e l f e n e r g y f o r the m e s o n s (figs. 4(a) and 4(b)). T h e s e l f - e n e r g y f u n c t i o n a s s o c i a t e d to d i a g r a m 4(a) h a s b e e n s t u d i e d in s e c t . 2 (eq. (9)). T h e d i a g r a m 4(b) c o r r e s p o n d s to a c o u n t e r t e r m of m a s s r e n o r m a l i z a t i o n and c a n c e l s the c o n t r i b u t i o n A in ~(p2).
Fig. 4. Meson self-energy diagrams for the VBB vertex. T h e s u m of d i a g r a m s 4(a) and 4(b) is well known and equal to 1 B go Al~ e U ( P , X) -~
T h e m e s o n s e l f - e n e r g y c o n t r i b u t i o n s f o r K + and K- a r e i d e n t i c a l and we obtain: TSE = B go Ap e ~ ( P , X ) .
(13)
3.2. V e r t e x p a r t and o b s e r v a b l e e f f e c t T h e s t r u c t u r e of the v e r t e x f u n c t i o n h a s b e e n given in eq. (10) we h a v e the v e r t e x c o n t r i b u t i o n T V = [L +Af(M2)] g o A~ e ~ ( P , X) .
(14)
S u m m i n g now eqs. (12) - (14) we d e d u c e T = [l+S+L+Af(M2)]go
A~ e~(P,X) .
(15)
While s e p a r a t e l y , TSE and T V a r e not g a u g e i n v a r i a n t , the s u m T = TSE + + TV is g a u g e i n v a r i a n t and a l s o the finite p a r t Af. T a k i n g now into a c c o u n t the W a r d identity (11) we find that the u l t r a v i o l e t d i v e r g e n c e s c a n c e l and the s i m p l e r e s u l t is g : [1 + Af(M2)] go •
4. T H E F U N C T I O N
(16)
Af(s)
4.1. G e n e r a l i t i e s We r e s t r i c t o u r s e l v e s to the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to the e l e c t r o m a g n e t i c i n t e r a c t i o n s .
456
E. CREMMER
and M. GOURDIN
A s t r a i g h t f o r w a r d method to obtain the function Af(s) is to compute the sum of the t h r e e d i a g r a m s drawn in fig. 3 applying the F e y n m a n rules. F o r r e a s o n s of simplicity we have p r e f e r r e d to use the technique of d i s p e r s i o n r e l a t i o n to obtain d i r e c t l y Af(s). The function Af(s) is an analytic function of s in the complex plane of s with a cut on the r e a l positive axis s t a r t i n g f r o m 4m2. We then write a d i s p e r s i o n relation with a subtraction at s = 0. Using the n o r m a l i z a t i o n condition Af(0) = 0, we deduce the s p e c t r a l r e p r e s e n t a tion Af(s)
s r°°
~(t)
-~ I m 2 t(t - s - i ¢ )
dt
(17)
0
The s p e c t r a l function ~(t) is known f r o m the unitarity p r o p e r t y of the Smatrix. 4.2. E x p l i c i t r e s u l t f o r Af(s) In the s e c o n d - o r d e r approximation the s p e c t r a l function ~(2)(t) is simply the imaginary p a r t of the d i a g r a m 3(a). In o r d e r to t r e a t the i n f r a r e d div e r g e n c e s c o r r e c t l y we introduce a small m a s s ~ for the photon. The r e sult is 1
(1
2m 2
4m2) ~
{1 +log U -~
½log
t - 4m2~ O(t - 4m 2) m2 j ,
(18)
t "
w h e r e a is the fine s t r u c t u r e constant. The r e s u l t is the following: ReA~2)(M2)
~/l+v2 = n L 4v
2_(l+log~)(1
l+v 2 1-v +-~log ~ - ~ )
1 +v 2 2v [ £ 3 2 ( v ) - £ 3 2 ( - v ) ] - ~vV2v v2 [ £ 3 2 ( 1 - ~ ) - £32(1_--~)]}, ImA~2)(M2) =
l+V2{l+log~-½1og
- a ~
~
4v2 }
.
(19)
(20)
The velocity v has been introduced in sect. 1 v = (1 - 4m2) ½ M2 ' and the Spence functions £32 a r e defined by [3] x
£32(x) = _ f o
log(I-u), du. u
Taking into account the analytic p r o p e r t i e s of the Spence functions, we can check by analytic continuation the n o r m a l i z a t i o n condition A(~2)(0)" = 0 . I
RADIATIVE CORRECTIONS 5. RADIATIVE
CORRECTIONS
TO
THE
VBB
457
VERTEX:
REAL
PHOTONS
5.1. Generalities In t h e l i m i t g -~ 0 t h e f u n c t i o n A~2)(M 2)'" e x h i b i t s i n f r a r e d d i v e r g e n c e s of t h e f o r m l o g ( g / m ) . In o r d e r to c a n c e l t h e s e d i v e r g e n c e s , we m u s t t a k e i n to a c c o u n t t h e p o s s i b l e e m i s s i o n of r e a l s o f t p h o t o n s . In t h e s e c o n d - o r d e r a p p r o x i m a t i o n t h e c o n s i d e r a t i o n of one p h o t o n i s enough. T h e d e c a y V -~ B + B + 7 i s d e s c r i b e d by t h e g a u g e i n v a r i a n t s u m of t h e t h r e e d i a g r a m s of fig. 5, in t h e l o w e s t - o r d e r a p p r o x i m a t i o n .
(a~ )
(b)~
~ e)% ( %% % ,
Fig. 5. Diagrams for the V ~ B + B +T decay. T h e p h o t o n h a s an e n e r g y - m o n e n t u m v e c t o r k, a p o l a r i z a t i o n v e c t o r Ct~(k) w i t h t h e c o n d i t i o n k . E = 0. T h e T - m a t r i x e l e m e n t f o r t h e t r a n s i t i o n V -~ B + B +~ h a s t h e s t r u c t u r e : : Eg (k)e P(P + k) T up(P, A, k) = e go Tfi • The weak gauge invariance condition is simply kg Tgp = 0 . T h e r e i s no i n t e r f e r e n c e b e t w e e n t h e z e r o and one p h o t o n a m p l i t u d e s . U s i n g eq. (4) w e c a n w r i t e in t h e s e c o n d - o r d e r a p p r o x i m a t i o n
r(v-~
B~r)
~
1
F ( V - * B B) - 2 ~ M 2 v 3
fdpf ~
w h e r e dpf i s t h e t w o - d i m e n s i o n a l p h a s e - s p a c e photon.
pol
ITfil 2 ,
(21)
density associated to the
5.2. E m i s s i o n of soft photons W e now s t u d y t h e e m i s s i o n of s o f t p h o t o n s . W e h a v e a g a i n a n i n f r a r e d d i v e r g e n c e f o r a z e r o p h o t o n e n e r g y and, a s in s e c t . 4, w e i n t r o d u c e a s m a l l m a s s g f o r t h e photon. P u t t i n g ¢o = ~ we r e s t r i c t t h e p h a s e s p a c e i n t e g r a t i o n to t h e p h o t o n e n e r g y r a n g e /~ < w < AE. Eq. (21) i m p l i e s , in t h e a p p r o x i m a t i o n A E / m << 1
458
E. CREMMER and M. GOURDIN F(V --* BBT)soft _ a f F(V BB) 2n
AE
k d~
fdZ
\k.p+
pol
k.p_/
w h e r e Z = c o s 0V and 0y the photon angle with r e s p e c t to a m e s o n in the r e s t s y s t e m of the v e c t o r meson. Neglecting the t e r m s which tend to z e r o with AE we obtain F(V ~ B~Y)~of t : ~ {2 log 2 ~ F(V ~ BB) 1 + v 2 rj?
+ ~
[1 + ~l + v 2 log i1--~v ] _ l l o g
-1 + v
1 + v2
(l~_v)]
[ 2 (-~-)--(22
v
1 +v 2
4v
1 -v
log ~
1 - vv 1+
[ • 2 ( v ) - •2(-v)] 1-v 2 log ~ } + O(AE) .
(22)
6. RESULTS
[2
6.1. Final expression for Ig+/go Combining the s e c o n d - o r d e r r a d i a t i v e c o r r e c t i o n s due to v i r t u a l and r e a l photons we find the i n f r a r e d d i v e r g e n c e s cancel, as expected, and the final r e s u l t t a k e s the f o r m g+ 2 = 1 + - a- ~1 + v2 ~2 - 2 (1 +log ---~-) 2AE. [1 + l~+ v 2 log ~-;~] 1-v go 7r t 2v 2(1+v2)[2~2(v )_22(_v) ] U
l+v2
- - -12
2
- Z2(l_d]
2 ~
-
1
log
1-v "
(23)
The f i r s t t e r m and dominant one, a~r(1 + v2)/(2v), is usually r e c o g n i z e d as the Coulomb t e r m . 6.2. Numerical results The n u m e r i c a l value of [g+/go 12 depends on AE, e.g. on the e x p e r i m e n tal conditions of detection. F o r t u n a t e l y , such a dependence is weak enough to allow a good e s t i m a t e of the r a d i a t i v e c o r r e c t i o n s . F o r a s t a n d a r d AE = = 1 MeV, eq. (23) g i v e s the n u m e r i c a l value g+2 g°° : 1.042 , f r o m which we deduce, using eqs. (6) and (7), the b r a n c h i n g r a t i o R99 ~ 1.60 . 6.3. Concluding remarks B e c a u s e of the s m a l l p h a s e s p a c e a v a i l a b l e f o r the K m e s o n s in the q~ -~ KI~ decay, the e l e c t r o m a g n e t i c c o r r e c t i o n s to the p h a s e - s p a c e r a t i o (eq. (7)) a r e c o n s i d e r a b l y l a r g e r than the e l e c t r o m a g n e t i c c o r r e c t i o n s to the coupling constant r a t i o (eq. (24)). It follows that the splitting of the
(24)
RADIATIVE CORRECTIONS
459
e l e c t r o m a g n e t i c c o r r e c t i o n s a s e x p l a i n e d i n s e c t . 1 i s not c o m p l e t e l y j u s t i fied and m u s t be c o n s i d e r e d as an a p p r o x i m a t i o n . Due to t h e ~ d o m i n a n c e of the v i r t u a l p h o t o n p r o p a g a t o r a r o u n d the q~ m a s s , the v a c u u m p o l a r i z a t i o n e f f e c t s e s s e n t i a l l y c a n c e l in the r a t i o g+/go. N e v e r t h e l e s s , it i s r a t h e r d i f f i c u l t to give a c o m p l e t e a n d c o n s i s t e n t t r e a t m e n t of t h e s e e f f e c t s f o r the r e a s o n e x p l a i n e d in the p r e v i o u s p a r a g r a p h [4]. T h e s t o r a g e r i n g e x p e r i m e n t s e+e - --* K+K - a n d e+e - -~ K°F~° f o r a t o t a l e n e r g y a r o u n d the ~ m a s s , a r e c e r t a i n l y the b e s t way to s t u d y the r a t i o Rq~, b e c a u s e they c o r r e s p o n d to the d e c a y of a q~ m e s o n at r e s t . F o r a d i g e s t of the a c t u a l e x p e r i m e n t a l s i t u a t i o n we s u g g e s t to the r e a d e r to c o n s u l t P a r t i c l e P r o p e r t i e s T a b l e s [5] to f r a m e h i s own o p i n i o n . A c o m p l e t e t r e a t m e n t of the r a d i a t i v e c o r r e c t i o n s f o r t h e s e r e a c t i o n s i s now i n p r o g r e s s a n d the v a l u e AE w i l l b e g i v e n by the e x p e r i m e n t a l a r r a n g e ment. We t h a n k X. A r t r u , A. N e v e u a n d J. S c h e r k f o r u s e f u l d i s c u s s i o n s .
RE F E R E N C E S [1] J . J . Sakurai, Phys. Rev. Letters 9 (1962) 472. [2] M.Gourdin, in: Unitary s y m m e t r i e s (North-Holland Publ. Comp., Amsterdam, 1967). [3] K.Mitchell, Phil. Mag. 40 (1949) 351. [4] Z. Z. Aydin and A. O. Barut, preprint I. C ./68/101 Trieste. [5] Particle Data Group, UCRL 8030 (August 1968).
Nuclear Physics B9 (1969) 451-459. North-Holland Publ. Comp., A m s t e r d a m
B
RADIATIVE
CORRECTIONS
TO
~ --.. K K D E C A Y
E. C R E M M E R and M. G O U R D I N
Laboratoire de Physique Th~orique et Hautes Energies, Orsay Received 19 December 1968 Abstract: We compute the electromagnetic c o r r e c t i o n s to the ~P--~ KK decay as the product of p h a s e - s p a c e c o r r e c t i o n s due to the K +, Ko mass difference by r a d i a tive c o r r e c t i o n s in the s e c o n d - o r d e r approximation. The final result is F ((fl -~ K+K-) = 1.60 F (eft ~ KOKo)
1. I N T R O D U C T I O N T h e a i m of t h i s p a p e r is a c o m p a r i s o n of t h e two d e c a y m o d e s of t h e q~ meson cp -~ K + + K - ,
(1)
q~ ~ K ° + K ° ,
(2)
t a k i n g into a c c o u n t t h e e l e c t r o m a g n e t i c c o r r e c t i o n s to t h e i s o b a r i c s p i n i n variance. L e t us f i r s t d e s c r i b e o u r n o t a t i o n f o r t h e d e c a y of a v e c t o r m e s o n V of m a s s M into t w o p s e u d o s c a l a r ( o r s c a l a r ) m e s o n s B an d B of m a s s rn (M > 2m). T h e L o r e n t z - i n v a r i a n t T - m a t r i x e l e m e n t i s g i v e n by
(3)
where (a) P , p+, p . a r e t h e e n e r g y - m o m e n t u m v e c t o r s r e s p e c t i v e l y f o r t h e V, B, B m e s o n s w i t h t h e c o n s e r v a t i o n l a w P = p++p_, (b) e~(P, ~) i s t h e p o l a r i z a t i o n v e c t o r f o r t h e V - m e s o n of e n e r g y m o m e n t u m P and h e l i c i t y ;% r e s t r i c t e d by th e s u p p l e m e n t a r y c o n d i t i o n Pp e~(P, X)= =0. T h e c o m p l e x c o u p l i n g c o n s t a n t g i s d e f i n e d a s t h e v a l u e of t h e V B B v e r t e x f u n c t i o n f o r t h e t h r e e p a r t i c l e s on t h e m a s s s h e l l . T h e d e c a y w i d t h F ( V ~ B + B) i s r e l a t e d to g by t h e w e l l - k n o w n e x p r e s sion [I]
Laboratoire associg au C. N . R . S . Postal address: Laboratoire de Physique Th~orique et Hautes E n e r g i e s , B~ttiment 211, Facult~ des Sciences, 91-Orsay, F r a n c e .
452
E. CREMMER and M. GOURDIN l F(V ~ B +B) = ~
lg[ 2 4n M v3 ,
(4)
w h e r e the v e l o c i t y v is g i v e n in t e r m s of m a s s e s by ,=
( 1 - 4m2~½
.
(5)
T h e r a t i o of the two widths f o r the d e c a y m o d e s (1) and (2) of the (p m e s o n is given by Rq~ -
F(~K+K -)_ F ( q ~ ~ K ° K °)
g + 2 (v+ j3 go \~-o]
.
(6)
L e t us a s s u m e f o r a m o m e n t that the i s o b a r i c spin s y m m e t r y is an e x a c t i n v a r i a n c e . T h e c h a r g e d and n e u t r a l K - m e s o n m a s s e s a r e then equal and v+ = v o. On the o t h e r hand the ~ m e s o n b e i n g an i s o s i n g l e t g+ = go. It f o l l o w s that in the a b s e n c e of e l e c t r o m a g n e t i c i n t e r a c t i o n s we s i m p l y have Rq~ = 1. We now i n t r o d u c e the e l e c t r o m a g n e t i c i n t e r a c t i o n s giving r i s e f i r s t to a m a s s splitting i n s i d e the K+K ° doublet and s e c o n d l y to a d e v i a t i o n f r o m unity of the c o u p l i n g c o n s t a n t r a t i o g+/go. Both t y p e s of e f f e c t s a r e of c o u r s e s t r o n g l y c o r r e l a t e d b e c a u s e of t h e i r c o m m o n origin. N e v e r t h e l e s s , it is u s u a l , as an a p p r o x i m a t e t r e a t m e n t , to split the e l e c t r o m a g n e t i c c o r r e c t i o n s into two p a r t s : (a) the p h a s e - s p a c e c o r r e c t i o n s by i n s e r t i n g the e x p e r i m e n t a l v a l u e s f o r the c h a r g e d and n e u t r a l K - m e s o n m a s s e s in the p h a s e - s p a c e r a t i o (v+/vo) 3 (ref. [2]) _~)3 ~ 1.54 ,
(7)
(b) the e l e c t r o m a g n e t i c c o r r e c t i o n s to the c o u p l i n g c o n s t a n t , e v a l u a t i n g the finite p a r t of the v e r t e x f u n c t i o n in the s e c o n d - o r d e r a p p r o x i m a t i o n . T h e r e s u l t of o u r c a l c u l a t i o n g i v e s g_n+ 2_~ 1.042 . C o m b i n i n g this r e s u l t and t h a t of f o r m u l a (7) we d e d u c e , f o r the r a t i o Rq~, the v a l u e R(p ~ 1.60 , to be c o m p a r e d with p r e c i s e e x p e r i m e n t s . Sect. 2 is d e v o t e d to g e n e r a l c o n s i d e r a t i o n s about the e l e c t r o d y n a m i c s of K m e s o n and (p m e s o n . T h e r a d i a t i v e c o r r e c t i o n s to the VBB v e r t e x due to v i r t u a l p h o t o n s a r e d i s c u s s e d in s e c t . 3; as a c o n s e q u e n c e of the W a r d identity, the u l t r a v i o l e t divergences cancel. In sect. 4, we c o m p u t e the finite p a r t of the v e r t e x f u n c t i o n u s i n g d i s p e r sion r e l a t i o n t e c h n i q u e in the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to electromagnetic interactions.
RADIATIVE CORRECTIONS
453
T h e i n f r a r e d d i v e r g e n c e s a r e e l i m i n a t e d , c o n s i d e r i n g in addition the r a d i a t i v e c o r r e c t i o n s a s s o c i a t e d to a s o f t - p h o t o n e m i s s i o n . T h e b r e m s s t r a h l u n g p r o c e s s is c o m p u t e d in s e c t . 5. F i n a l l y , the r e s u l t s a r e c o l l e c t e d and d i s c u s s e d in s e c t . 6.
2. G E N E R A L C O N S I D E R A T I O N S T h e H a m i l t o n i a n d e n s i t y f o r the i n t e r a c t i o n of a c h a r g e d s p i n l e s s p a r t i cle d e s c r i b e d by a field ~(x) with the e l e c t r o m a g n e t i c field Aiz(x ) h a s the familiar form Hint(X ) = i eAl.L(x)[ ~*(x) 8~)(x) 8x~t
8eP*(x) 8xtz ~(x)] - e 2 ~ * ( x ) ~ ( x ) A g ( x ) A g ( x ) .
T h e l a s t t e r m of eq. (8), g e n e r a l l y c a l l e d a c o n t a c t t e r m , e n s u r e the g a u g e i n v a r i a n c e of the t h e o r y .
(8)
is n e c e s s a r y to
2.1. K-meson self-energy part We c o n s i d e r a v i r t u a l K m e s o n of e n e r g y - m o m e n t u m p. T h e s e l f - e n e r g y f u n c t i o n ~(p2) c o r r e s p o n d s to a c l a s s of d i a g r a m s s y m b o l i c a l l y d r a w n in fig. 1.
Fig. 1. General self-energy diagrams for K mesons. T h e f u n c t i o n ~(p2) can be c o n v e n i e n t l y w r i t t e n in the f o r m ~(p2) = A + (p2 + m2) B + (p2 + m 2) ~f(p2) ,
(9)
w h e r e the infinite (with u l t r a v i o l e t d i v e r g e n c e s ) c o n s t a n t s A and B a r e d e fined by A = ~(-m2) ,
B = ~'(-m2) ,
and the finite p a r t F4(P2) v a n i s h e s f o r a r e a l K m e s o n ~ f ( - m 2) = 0 . F o r i n s t a n c e , in the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to e l e c t r o m a g n e t i c i n t e r a c t i o n s , the f u n c t i o n ~(2)(p2) c o r r e s p o n d s to the s u m of the two d i a g r a m s (figs. 2(a) and 2(b)) w h e r e the s o l i d line r e p r e s e n t s a K m e s o n and the wavy line a photon.
(a) (b) Fig. 2. Meson self-energy diagrams in the second electromagnetic approximation order.
454
E. CREMMER and M. GOURDIN
2.2. V e r t e x p a r t The v e r t e x function f o r an off-shell v e c t o r m e s o n decaying into two p s e u d o s c a l a r m e s o n s B and ~ has the g e n e r a l s t r u c t u r e A~(p+, p_) = g o ( s ) ( p + - p _ ) l ~ [ L + Af(s)] ,
(10)
where p+, p_ a r e the m e s o n e n e r g y - m o m e n t u m v e c t o r s and s = - ( p + + p _ ) 2 . The infinite constant L contains u l t r a v i o l e t d i v e r g e n c e s and is defined so that the finite p a r t Af(s) s a t i s f i e s Af(O) = 0 . In the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to e l e c t r o m a g n e t i c i n t e r a c tions the function A!2)(p+, p_) c o r r e s p o n d s to the sum of the t h r e e d i a g r a m s figs. 3(a), 3(b), 3(c).~Here the double solid line r e p r e s e n t s the v e c t o r meson.
a4
"
( b ) ~
Fig. 3. Second-order electromagnetic corrections to the VBB vertex. 2.3. W a r d i d e n t i t y The coupling of the v e c t o r m e s o n to p s e u d o s c a l a r m e s o n s is analogous to the coupling of a photon to p s e u d o s c a l a r m e s o n s . We then have a Ward identity which r e l a t e s the s e l f - e n e r g y function E(p 2) and the v e r t e x function A~(p+, p_) in the following way apP E(P 2) p2 = -m2 : - ~ 1
A/~(p, -p) .
F r o m eqs. (9) and (10) we simply deduce B+L
3. RADIATIVE
CORRECTIONS
TO
THE
(11)
: 0 .
VBB
VERTEX:
VIRTUAL
PHOTONS
Let us use the following notation as introduced in sect. 1. p++p_
= P,
p+-p-
= ± ,
4. P = 0 ,
p2 = -s.
The T - m a t r i x element for the V -* B +B t r a n s i t i o n has the s t r u c t u r e given by eq. (3) and we denote by go the coupling constant in the a b s e n c e of e l e c tromagnetic interactions T(0) = go ~
e~(p,~) .
(12)
RADIATIVE CORRECTIONS
455
3.1. M a s s and m e s o n p r o p a g a t o r r e n o r m a l i z a t i o n We f i r s t c o n s i d e r the d i a g r a m s of s e l f e n e r g y f o r the m e s o n s (figs. 4(a) and 4(b)). T h e s e l f - e n e r g y f u n c t i o n a s s o c i a t e d to d i a g r a m 4(a) h a s b e e n s t u d i e d in s e c t . 2 (eq. (9)). T h e d i a g r a m 4(b) c o r r e s p o n d s to a c o u n t e r t e r m of m a s s r e n o r m a l i z a t i o n and c a n c e l s the c o n t r i b u t i o n A in ~(p2).
Fig. 4. Meson self-energy diagrams for the VBB vertex. T h e s u m of d i a g r a m s 4(a) and 4(b) is well known and equal to 1 B go Al~ e U ( P , X) -~
T h e m e s o n s e l f - e n e r g y c o n t r i b u t i o n s f o r K + and K- a r e i d e n t i c a l and we obtain: TSE = B go Ap e ~ ( P , X ) .
(13)
3.2. V e r t e x p a r t and o b s e r v a b l e e f f e c t T h e s t r u c t u r e of the v e r t e x f u n c t i o n h a s b e e n given in eq. (10) we h a v e the v e r t e x c o n t r i b u t i o n T V = [L +Af(M2)] g o A~ e ~ ( P , X) .
(14)
S u m m i n g now eqs. (12) - (14) we d e d u c e T = [l+S+L+Af(M2)]go
A~ e~(P,X) .
(15)
While s e p a r a t e l y , TSE and T V a r e not g a u g e i n v a r i a n t , the s u m T = TSE + + TV is g a u g e i n v a r i a n t and a l s o the finite p a r t Af. T a k i n g now into a c c o u n t the W a r d identity (11) we find that the u l t r a v i o l e t d i v e r g e n c e s c a n c e l and the s i m p l e r e s u l t is g : [1 + Af(M2)] go •
4. T H E F U N C T I O N
(16)
Af(s)
4.1. G e n e r a l i t i e s We r e s t r i c t o u r s e l v e s to the s e c o n d - o r d e r a p p r o x i m a t i o n with r e s p e c t to the e l e c t r o m a g n e t i c i n t e r a c t i o n s .
456
E. CREMMER
and M. GOURDIN
A s t r a i g h t f o r w a r d method to obtain the function Af(s) is to compute the sum of the t h r e e d i a g r a m s drawn in fig. 3 applying the F e y n m a n rules. F o r r e a s o n s of simplicity we have p r e f e r r e d to use the technique of d i s p e r s i o n r e l a t i o n to obtain d i r e c t l y Af(s). The function Af(s) is an analytic function of s in the complex plane of s with a cut on the r e a l positive axis s t a r t i n g f r o m 4m2. We then write a d i s p e r s i o n relation with a subtraction at s = 0. Using the n o r m a l i z a t i o n condition Af(0) = 0, we deduce the s p e c t r a l r e p r e s e n t a tion Af(s)
s r°°
~(t)
-~ I m 2 t(t - s - i ¢ )
dt
(17)
0
The s p e c t r a l function ~(t) is known f r o m the unitarity p r o p e r t y of the Smatrix. 4.2. E x p l i c i t r e s u l t f o r Af(s) In the s e c o n d - o r d e r approximation the s p e c t r a l function ~(2)(t) is simply the imaginary p a r t of the d i a g r a m 3(a). In o r d e r to t r e a t the i n f r a r e d div e r g e n c e s c o r r e c t l y we introduce a small m a s s ~ for the photon. The r e sult is 1
(1
2m 2
4m2) ~
{1 +log U -~
½log
t - 4m2~ O(t - 4m 2) m2 j ,
(18)
t "
w h e r e a is the fine s t r u c t u r e constant. The r e s u l t is the following: ReA~2)(M2)
~/l+v2 = n L 4v
2_(l+log~)(1
l+v 2 1-v +-~log ~ - ~ )
1 +v 2 2v [ £ 3 2 ( v ) - £ 3 2 ( - v ) ] - ~vV2v v2 [ £ 3 2 ( 1 - ~ ) - £32(1_--~)]}, ImA~2)(M2) =
l+V2{l+log~-½1og
- a ~
~
4v2 }
.
(19)
(20)
The velocity v has been introduced in sect. 1 v = (1 - 4m2) ½ M2 ' and the Spence functions £32 a r e defined by [3] x
£32(x) = _ f o
log(I-u), du. u
Taking into account the analytic p r o p e r t i e s of the Spence functions, we can check by analytic continuation the n o r m a l i z a t i o n condition A(~2)(0)" = 0 . I
RADIATIVE CORRECTIONS 5. RADIATIVE
CORRECTIONS
TO
THE
VBB
457
VERTEX:
REAL
PHOTONS
5.1. Generalities In t h e l i m i t g -~ 0 t h e f u n c t i o n A~2)(M 2)'" e x h i b i t s i n f r a r e d d i v e r g e n c e s of t h e f o r m l o g ( g / m ) . In o r d e r to c a n c e l t h e s e d i v e r g e n c e s , we m u s t t a k e i n to a c c o u n t t h e p o s s i b l e e m i s s i o n of r e a l s o f t p h o t o n s . In t h e s e c o n d - o r d e r a p p r o x i m a t i o n t h e c o n s i d e r a t i o n of one p h o t o n i s enough. T h e d e c a y V -~ B + B + 7 i s d e s c r i b e d by t h e g a u g e i n v a r i a n t s u m of t h e t h r e e d i a g r a m s of fig. 5, in t h e l o w e s t - o r d e r a p p r o x i m a t i o n .
(a~ )
(b)~
~ e)% ( %% % ,
Fig. 5. Diagrams for the V ~ B + B +T decay. T h e p h o t o n h a s an e n e r g y - m o n e n t u m v e c t o r k, a p o l a r i z a t i o n v e c t o r Ct~(k) w i t h t h e c o n d i t i o n k . E = 0. T h e T - m a t r i x e l e m e n t f o r t h e t r a n s i t i o n V -~ B + B +~ h a s t h e s t r u c t u r e :
r(v-~
B~r)
~
1
F ( V - * B B) - 2 ~ M 2 v 3
fdpf ~
w h e r e dpf i s t h e t w o - d i m e n s i o n a l p h a s e - s p a c e photon.
pol
ITfil 2 ,
(21)
density associated to the
5.2. E m i s s i o n of soft photons W e now s t u d y t h e e m i s s i o n of s o f t p h o t o n s . W e h a v e a g a i n a n i n f r a r e d d i v e r g e n c e f o r a z e r o p h o t o n e n e r g y and, a s in s e c t . 4, w e i n t r o d u c e a s m a l l m a s s g f o r t h e photon. P u t t i n g ¢o = ~ we r e s t r i c t t h e p h a s e s p a c e i n t e g r a t i o n to t h e p h o t o n e n e r g y r a n g e /~ < w < AE. Eq. (21) i m p l i e s , in t h e a p p r o x i m a t i o n A E / m << 1
458
E. CREMMER and M. GOURDIN F(V --* BBT)soft _ a f F(V BB) 2n
AE
k d~
fdZ
\k.p+
pol
k.p_/
w h e r e Z = c o s 0V and 0y the photon angle with r e s p e c t to a m e s o n in the r e s t s y s t e m of the v e c t o r meson. Neglecting the t e r m s which tend to z e r o with AE we obtain F(V ~ B~Y)~of t : ~ {2 log 2 ~ F(V ~ BB) 1 + v 2 rj?
+ ~
[1 + ~l + v 2 log i1--~v ] _ l l o g
-1 + v
1 + v2
(l~_v)]
[ 2 (-~-)--(22
v
1 +v 2
4v
1 -v
log ~
1 - vv 1+
[ • 2 ( v ) - •2(-v)] 1-v 2 log ~ } + O(AE) .
(22)
6. RESULTS
[2
6.1. Final expression for Ig+/go Combining the s e c o n d - o r d e r r a d i a t i v e c o r r e c t i o n s due to v i r t u a l and r e a l photons we find the i n f r a r e d d i v e r g e n c e s cancel, as expected, and the final r e s u l t t a k e s the f o r m g+ 2 = 1 + - a- ~1 + v2 ~2 - 2 (1 +log ---~-) 2AE. [1 + l~+ v 2 log ~-;~] 1-v go 7r t 2v 2(1+v2)[2~2(v )_22(_v) ] U
l+v2
- - -12
2
- Z2(l_d]
2 ~
-
1
log
1-v "
(23)
The f i r s t t e r m and dominant one, a~r(1 + v2)/(2v), is usually r e c o g n i z e d as the Coulomb t e r m . 6.2. Numerical results The n u m e r i c a l value of [g+/go 12 depends on AE, e.g. on the e x p e r i m e n tal conditions of detection. F o r t u n a t e l y , such a dependence is weak enough to allow a good e s t i m a t e of the r a d i a t i v e c o r r e c t i o n s . F o r a s t a n d a r d AE = = 1 MeV, eq. (23) g i v e s the n u m e r i c a l value g+2 g°° : 1.042 , f r o m which we deduce, using eqs. (6) and (7), the b r a n c h i n g r a t i o R99 ~ 1.60 . 6.3. Concluding remarks B e c a u s e of the s m a l l p h a s e s p a c e a v a i l a b l e f o r the K m e s o n s in the q~ -~ KI~ decay, the e l e c t r o m a g n e t i c c o r r e c t i o n s to the p h a s e - s p a c e r a t i o (eq. (7)) a r e c o n s i d e r a b l y l a r g e r than the e l e c t r o m a g n e t i c c o r r e c t i o n s to the coupling constant r a t i o (eq. (24)). It follows that the splitting of the
(24)
RADIATIVE CORRECTIONS
459
e l e c t r o m a g n e t i c c o r r e c t i o n s a s e x p l a i n e d i n s e c t . 1 i s not c o m p l e t e l y j u s t i fied and m u s t be c o n s i d e r e d as an a p p r o x i m a t i o n . Due to t h e ~ d o m i n a n c e of the v i r t u a l p h o t o n p r o p a g a t o r a r o u n d the q~ m a s s , the v a c u u m p o l a r i z a t i o n e f f e c t s e s s e n t i a l l y c a n c e l in the r a t i o g+/go. N e v e r t h e l e s s , it i s r a t h e r d i f f i c u l t to give a c o m p l e t e a n d c o n s i s t e n t t r e a t m e n t of t h e s e e f f e c t s f o r the r e a s o n e x p l a i n e d in the p r e v i o u s p a r a g r a p h [4]. T h e s t o r a g e r i n g e x p e r i m e n t s e+e - --* K+K - a n d e+e - -~ K°F~° f o r a t o t a l e n e r g y a r o u n d the ~ m a s s , a r e c e r t a i n l y the b e s t way to s t u d y the r a t i o Rq~, b e c a u s e they c o r r e s p o n d to the d e c a y of a q~ m e s o n at r e s t . F o r a d i g e s t of the a c t u a l e x p e r i m e n t a l s i t u a t i o n we s u g g e s t to the r e a d e r to c o n s u l t P a r t i c l e P r o p e r t i e s T a b l e s [5] to f r a m e h i s own o p i n i o n . A c o m p l e t e t r e a t m e n t of the r a d i a t i v e c o r r e c t i o n s f o r t h e s e r e a c t i o n s i s now i n p r o g r e s s a n d the v a l u e AE w i l l b e g i v e n by the e x p e r i m e n t a l a r r a n g e ment. We t h a n k X. A r t r u , A. N e v e u a n d J. S c h e r k f o r u s e f u l d i s c u s s i o n s .
RE F E R E N C E S [1] J . J . Sakurai, Phys. Rev. Letters 9 (1962) 472. [2] M.Gourdin, in: Unitary s y m m e t r i e s (North-Holland Publ. Comp., Amsterdam, 1967). [3] K.Mitchell, Phil. Mag. 40 (1949) 351. [4] Z. Z. Aydin and A. O. Barut, preprint I. C ./68/101 Trieste. [5] Particle Data Group, UCRL 8030 (August 1968).