A BPH approach to weak interactions

Volume 40B, number 2

PHYSICS LETTERS

A BPH

APPROACH

TO WEAK

26 June 1972

INTERACTIONS*

G. PRE PARATA

Istituto di Fisica. Universit~ di Roma, Rome, Italy Received 31 March 1972

A Bogoliubov-Parasiuk-Hepp renormalization program for the usual Fermi Lagrangian is proposed. All second order processes are given in terms of only two subtraction constants associated with "counter terms" exhibiting a high degree of lepton-hadron universality.

In the l a s t few y e a r s the bright p i c t u r e of the weak i n t e r a c t i o n s , c u l m i n a t i n g in the s u c c e s s of the Cabibbo theory [1], has been definitely d e t e r i o r a t i n g under the impact of the d i s c o v e r y of C P - v i o l a t i o n , the r e a l i z a t i o n of the g r a v i t y of the higher o r d e r d i v e r g e n c e s , and finally the p o s sible b r a n d new p r o b l e m of the decay KL ~ / x + p [21. In this l e t t e r we would like to offer a partial solution to the p r o b l e m of n o n - r e n o r m a l i z a b i l i t y of the u s u a l F e r m i weak i n t e r a c t i o n L a g r a n g i a n ZF(X) = ¼G ~ { J

(x), JU+(x)}

(11

or e q u i v a l e n t l y the usual i n t e r m e d i a t e vector boson Lagr angian 2IVB(X) = g(W+p(x) JU(x)+h.c.) .

(2)

A s i m i l a r approach to leptonic weak i n t e r a c tions has r e c e n t l y b e e n independently proposed by Appelquist and Bjorken [3]. The s u c c e s s of lowest o r d e r p e r t u r b a t i v e c a l culations c a r r i e d out with (1) and (2) suggests that a p e r t u r b a t i o n theory approach to weak i n t e r a c t i o n s is indeed adequate, and that higher o r d e r c o r r e c t i o n s , in spite of t h e i r s e v e r e f o r m a l d i v e r g e n c e s , a r e quite s m a l l . In a cut-off app r o a c h * * this m e a n s that the weak i n t e r a c t i o n cut-off A2 is such that G A2 << 1, otherwise one would have higher o r d e r c o n t r i b u t i o n s giving i m p o r t a n t c o r r e c t i o n s to the lowest o r d e r c a l c u l a tions. In the following toward the vexing p r o b l e m of d i v e r g e n c e s in weak i n t e r a c t i o n s we shall take an attitude which is based on the Bogoliubov, * Supported in part by the Air Force Office of Scientific Research, through the European Office of Aerospace Research under contract No. F: 61/052 67 C0084. ** See ref. [1], p. 696 and following.

P a r a s i u k , Hepp [4] (BHP) approach to p e r t u r b a tion theory, when s u p p l e m e n t e d with the a s s u m p tion that the weak i n t e r a c t i o n s cut-off is such that GA2<< 1. In this way one r e a c h e s the conclusion that the r e l e v a n t expansion p a r a m e t e r s of weak i n t e r a c t i o n s a r e of the f o r m , Gs, Gt, . .. thus leading to the f a i l u r e of p e r t u r b a t i o n theory only at e n e r g i e s s ~ G -1 ~ 105 m 2 (the well known u n i t a r i t y l i m i t energy). Very briefly, the BHP approach [4] c o n s i s t s in c o n s t r u c t i n g i t e r a t i v e l y out of a chosen i n t e r action L a g r a n g i a n a S - m a t r i x which obeys L o r e n t z - i n v a r i a n c e , c a u s a l i t y and u n i t a r i t y . F o r n o n - r e n o r m a l i z a b l e L a g r a n g i a n s like (1) and (2), this c o n s t r u c t i o n is shown to be ambiguous in that in e v e r y o r d e r one has to introduce "counter t e r m s " to r e n d e r the G r e e n ' s - f u n c t i o n s of the theory t e m p e r e d d i s t r i b u t i o n s , and these c o u n t e r t e r m s do not have the s a m e o p e r a t o r s t r u c t u r e as the o r i g i n a l L a g r a n g i a n , thus putting into the theory an i n c r e a s i n g n u m b e r of uncalculable subt r a c t i o n constants. However the p r o l i f e r a t i o n of these counter t e r m s is not wholly c a p r i c i o u s . T h e i r o p e r a t o r s t r u c t u r e can be s e v e r e l y l i m i t e d by c a r r y i n g out a detailed a n a l y s i s of the s h o r t d i s t a n c e behavior of the p r o d u c t s of local weak interaction Lagrangians. This is done by use of the Wick theorem for the leptonic part and can be immediately generalized to the whole weak interactions by use of Wilson's expansions in their canonical gluon-quark form [5,6]; in this way a very stringent form of universality emerges also for counterterms. We emphasize that this scheme does not basically contradict the interesting efforts to construct a theory of the weak interactions which is free f r o m d i v e r g e n c e s [7]; the s u c c e s s of these efforts would then amount to computing the s u b t r a c t i o n c o n s t a n t s which m u s t be introduced in this approach. 253

Volume 40B, number 2

PHYSICS LETTERS

We a r e now going to c a r r y out the a n a l y s i s of s h o r t d i s t a n c e s i n g u l a r i t i e s in second o r d e r ; the g e n e r a l i z a t i o n to higher o r d e r s is s t r a i g h t f o r ward although quite tedius and at p r e s e n t u n i n t e r e s t i n g . We s h a l l f i r s t c o n s i d e r the p u r e l y leptonie c a s e , in the F e r m i ease, and then d i s c u s s the extension to the hadronic p a r t by m e a n s of c a n o n i c a l W i l s o n ' s expansion. Our L a g r a n g i a n is then g+ L(x) =½Gv~:l#(x)l (x): (3)

leg(x)

26 June 1972

= : ~ x ) y (1 - y5)[~-+,T-]~(x):

(7 ') lA(x) --:~P-(xSr.(1 - ~55{r +, r-}Cix): , and a l o g a r i t h m i c one with o p e r a t o r

o (x5

s% ssc;x5

+ ½:A c

~(x)A c

^(x) :

+ :AA

Ax)A A ~(x) :

(8)

5{:ao

where where I

~ : ~ ( x ) ~ (1 - ~55 ~(x): u(x) -- l =e,

s~2(~) = ½:~(x)C¢~v¢ +'g~v~5(1 - v55~(x5:

= - ~ x ) T+~]~(1-75) ~(x):

A.~(x) -- ½:~(xS(Vqv¢ - a~v.)(1 -v5)~(x):

10!

and ~-+ is the m a t r i x r+ =e v~

0 0 0

0 0 0

and C(A) r e f e r to the c o m m u t a t o r ( a n t i c o m m u t a tot) of r + and r - . Thus to second o r d e r we can write S(2)(x, Y5 =

.

T '(L(x) L(y))

The S-operator in second order is given by: S (2)

: ½ fdxdy S(2)(x,y);

(4)

where S(2)(x, y) = i2T

(L(x)L(y))

+ i A2(x , y)

(5)

and A2(x,y ) is so c o n s t r u c t e d as to r e m o v e the s h o r t d i s t a n c e s i n g u l a r i t i e s which r e n d e r T (L(x)L(y)) a n o n - t e m p e r e d d i s t r i b u t i o n . The g e n e r a l f o r m of A2(x ,y) is thus given by 64(x-y) and its d e r i v a t i v e s m u l t i p l y i n g "coefficient" local o p e r a t o r s , whose s t r u c t u r e is given by a p p l i c a tion of the Wick t h e o r e m . We have in fact

4

T(L(x)L(y)) = :L(x)L(y): + ~ n=l

(6) (n-contractions).

÷

l~(x)lg(x): +

- ~Tr(~+v-)

where

:

(9) + 6 4(x - y ) i S 1 O(1/)(x) + K20(125(x) ]

where T ' stands for the T - p r o d u c t with its non i n t e g r a b l e s i n g u l a r i t y removed. Eq. (9) is ext r e m e l y i n t e r e s t i n g b e c a u s e it shows that for all leptonic p r o c e s s e s up to second o r d e r we m u s t introduce only 2 s u b t r a c t i o n c o n s t a n t s K 1 and K 2. A c c o r d i n g to the p r e v i o u s d i s c u s s i o n we assume t h a t K 1 ~ G2A2/47r 2 ~ 10 - 5 - 10 -6 G, and K 2 ~ (G2/47r2) In A2 ~ 10 -6 - 10 -7 G, so that for m o d e r ate e n e r g i e s K 1 d o m i n a t e s K2. The g e n e r a l i z a t i o n to the hadronic p a r t is quite easy in t e r m s of W f l s o n ' s e x p a n s i o n s in the canonical q u a r k - g l u o n model, we need only add to (7') their hadronic c o u n t e r p a r t s

jC(x)

:~(x)v~(1 -v 55[x, x+]~(x):

jA(x)

= :~7(x)~ ~z(1-y5){~t, )t+}~(x) :

(10)

The 1 - c o n t r a c t i o n t e r m is i n t e g r a b l e , 3- and 4c o n t r a c t i o n s a r e u n i n t e r e s t i n g , we c o n c e n t r a t e on the 2 - c o n t r a c t i o n s t e r m . C a r r y i n g out some tedious a l g e b r a we find that we have two d i v e r g e n t s t r u c t u r e s when a s u i t a b l e cut-off A2 is sent to 0% a quadratic d i v e r g e n c e with a coefficient operator

o (x) :

~

(7)

where ~(x) is the quark field, and i ~=

cos0 0 0

sin0 0 0

i

Obviously (105 are the n e u t r a l m e m b e r s of the set of 18 c h i r a l U(3) × U(35 c u r r e n t s . As for the l o g a r i t h m i c d i v e r g e n c e , we have to add to (8')

s (h)~(x5 = ~:~(x5 C~ ~ + ~ ) ( 1

-y 55~(x5: (11)

A (h)(x~ -- ½ : ~ ( x ) ( ~ - ~ ' ~ 5 (1 -~ 55~(x): 254

(8')

Volume 40B, number 2

PHYSICS LETTERS

w h e r e A~ = "~ot+2igBot, and Bol is the gluon field. O p e r a t o r s ( I i ) appear in the light cone e x p a n s i o n s of two weak c u r r e n t s , and t h e i r m a t r i x e l e m e n t between b a r y o n s can be computed f r o m e l e c t r o n and n e u t r i n o deep i n e l a s t i c s e a t t e r i n g [6]. Thus we see that all the c o u n t e r t e r m s in s e c ond o r d e r can be e x p r e s s e d in t e r m s of two s u b t r a c t i o n constants m u l t i p l y i n g the o p e r a t o r s Ol(X) and O2(x) obtained from (7) and (8) by substituting C C C C C

26 June 1972

K 1 ~ ( 3 - 6 ) x 10 -11 mp 2 ,

(15)

a quite r e a s o n a b l e r e s u l t . K L - /~+/~-: The decay amplitude is given by

= !~ KlCOSOsin0 <0 ]j6/~[KL >~-(K+)),p(i Y5)u(K-) = ~ KlCOS0 sin0 x / 2 f K 2 m ~ ~-(K+)v5u(K_) .

(16)

By use of (15) one gets the e s t i m a t e

F(KL ~ / i + # - ) F(KL -~ all) ~ (1.6 - 6.5) × 10-9

(17)

S(1)C~-.~(h)Ca n d A(1)Cwith AC =A(1)C+A(h)AC We finally discuss some implications of the theory developed so far (K1 - K 2) mass difference: We have (Am)=-

I

Ren(m2)

mK

well in the range of the e.m. contribution; however a more meaningful appraisal of (17) must await further experiments on this branching ratio. Neutral currents: From (9) we can readily estimate the relative importance of neutral currents compared with lowest order charged currents, as Ampl. for n e u t r a l c u r r e n t s Ampl. for charged c u r r e n t s

where : - i f d
16 K1 ~ 5 G

(1.4- 2.8) × 10 -5

is the Ko - K--o self energy function. According to the preceding discussion we have 2 ~(mK) = (finite part) +K 1
(12)

where we have neglected K2 with respect to K 1. We note that (K t O1(0)IK} = e°s20 sin20 ~
~ mlf 2~sin20

_

cos20

(13)

where this estimate follows through SU~ invarlance from the fact that the AI = ~ decay matrix element for K+ ~ ~+~o is, by use of PCAC, given by the same reduced matrix element [8]. The finite part can be estimated from ~, ~7, P, intermediate states [1], the 2~ contribution has also been estimated [9]. The result has the wrong sign (Am)finite ~ - 0.5 ~.-1 S

(14)

and the reliability of this estimate should not be better than a factor of two. Experimentally (Am) ~ 0.5~'~I, so we estimate

(18)

AS = -AQ c u r r e n t s : It is easy to see that these a m p l i t u d e s a r e not affected by the c o u n t e r t e r m s , and can t h e r e f o r e be e s t i m a t e d by s i m p l e p e r t u r bation theory. A r e a s o n a b l e model makes use of b a r y o n poles d i a g r a m s to e s t i m a t e , for i n s t a n c e

A(Z+-~ ne+ve) ~ I0 -6 A ( Z - ~ ne- re)

(19)

an order of magnitude smaller than (18) due to the absence of subtractions. Leptonic interactions: For purely leptonic interactions we can phase our predictions in the language of ref. [3] . Up to second order we have c~10 = 4,/2-,

a20 = 0,

Cql=

-2K2/G 2

, (20')

K1

~io = ~ -G- ~ ( 4 - 8) x io -5 , f120YlO-

24K2 62 ,

24 K1

Y20 = -

s

G ~

16K2 G2 ,

~11-

12K2 G2 ,

(20")

(1.5 3)x 10-5, -

Yll

16K2 G2

(20'")

* See in particular sect. 3 of this very interesting paper 255

Volume 40B, number 2

PHYSICS LETTERS

We se e that all t h e s e s u b t r a c t i o n t e r m s a r e exp r e s s e d in t e r m s of the single unknown p a r a m e t e r K 2. We conclude by s t r e s s i n g that it is at p r e s e n t p o s s i b l e , without a l t e r i n g d r a m a t i c a l l y the p i c t u r e of weak i n t e r a c t i o n s e m b o d i e d in the L a g r a n g i a n s (1) and (2), to c a r r y out c a l c u l a t i o n s of h i g h e r o r d e r weak i n t e r a c t i o n s in t e r m s of a l i m i t e d n u m b e r of unknown s u b t r a c t i o n constants. F o r c o n s i s t e n c y of the p e r t u r b a t i o n t h e o r y expansion such s u b t r a c t i o n constants n u m e r i c a l l y should follow the o r d e r i n g d ic a t e d by the F e r m i coupling constant G. We have in this p a p e r shown how a s h o r t d i s tance a n a l y s i s of the p e r t u r b a t i v e expansion can l i m i t the c o u n t e r t e r m s ' s t r u c t u r e through use of the W i l s o n ' s expansion in its c a n o n ic a l g l u o n - q u a r k f o r m . In second o r d e r we find that all s u b t r a c tions can be e x p r e s s e d in t e r m s of two c o n s t a n t s , K 1 and K2, m u l t i p l y i n g o p e r a t o r s which exhibit a high d e g r e e of l e p t o n - h a d r o n universality. F i n a l l y an e s t i m a t e of K 1 f r o m K L - KS m a s s d i f f e r e n c e h as a l s o been p o s s i b l e and found c o n s i s t e n t with the b a s i c i d e a s of this paper.

256

26 June 1972

We thank P r o f e s s o r s N. Gabibbo and R. Gatto for discussions.

References [1] For an extensive review of weak interactions, see R. E. Marshak, Riazuddin and C. P. Ryan, Theory of weak interactions in particle physics (WileyInterscience, New York, 1969). [2] A. R. Clark et al., Phys. Rev. Letters 26 (1971) 1667. [3] R. Appelquist and J. D. Bjorken, SLAC-PUB-950 (1971). [4] N. N. Bogoliubov and D. V. Sbirkov, Introduction to the theory of quantized fields (Interscienee, New York 1959); N. N. Bogoliubov and O. S. Parasiuk, Aeta Math. 97 (1957) 227; K. Hepp, Comm. Math. Phys. 2 (1966) 301. [5] K. G. Wilson, Phys. Rev. 179 (1969) 1499. [6] For a discussion of canonical gluon-quark model see R. Brandt and G. Preparata, Lectures at the 1971 Hamburg School, to be published. [7] S. Weinberg, Invited talk at the 1972 Coral Gables Conf., Miami (1972). [8] G. Preparata, Phys. Letters 34]3 (1971) 412. [9] K. Kang and J. Land, Phys. Rev. Letters 18 (1967)

503.