LJ~TT~,I~E AL NUOVO CIM~NTO

VOL. 22, N. 1

6 Maggio 1978

JVJV Weak Potential in a Simple Gauge Model and the Polarization of the v-Quanta in n~-p--> d ~ y . V. M. DUBOVIK J o i n t I n s t i t u t e ]or N u c l e a r R e s e a r c h - D u b n a ,

U.S.S.R.

V. S. ZAMIRALOV (*) I s t i t u t o di F i s ~ a

dell' Universit~ - B o l o g n a , I t a l i a

( r i c e v u t o il 13 F e b b r a i o 1978)

T h e unified t h e o r y of w e a k a n d 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 p r o p o s e d b y SAT A~ (i) a n d W~.I~B~.RG (~) a p p r o a c h e s m o r e a n d m o r e t h e s t a t u s of q u a n t u m e l e c t r o d y n a m i e s . T r e m e n d o u s efforts w e r e m a d e n o t o n l y t o p r o v e t h e p r i n c i p a l a s p e c t s of r e n o r m a l i z a b i l i t y (3), b u t also t o o v e r c o m e t h e p r o b l e m of p r a c t i c a l r e n o r m a l i z a t i o n (4). I n t h e l e p t o n s e c t o r w e a k c o r r e c t i o n s to t h e m u o n m a g n e t i c m o m e n t s (~), c o n t r i b u t i o n s of t h e h y p o t h e t i c a l n e u t r a l h e a v y l e p t o n s (6.v) a n d w e a k c o r r e c t i o n s to t h e m a g n e t i c m o m e n t of t h e n e w l y d i s c o v e r e d h e a v y c h a r g e d l e p t o n (s), etc. were c a l c u l a t e d . I t w a s n o t i c e d t h a t t h e c o n t r i b u t i o n s of t h e p a r t i c u l a r d i a g r a m s a r e e n h a n c e d . W e t h i n k t h a t i t is of s o m e i n t e r e s t t o a p p l y a n a l o g o u s d i a g r a m t e c h n i q u e s t o t h e q u a r k s e c t o r a n d , p a r t i c u l a r l y , t o t h e w e a k v%'JC-interaetion. O u r a i m is t o c o n s t r u c t a p a r i t y - v i o l a t i n g 3 ~ - p o t e n t i a l Vpv i n t h e f r a m e w o r k of a s i m p l e g a u g e m o d e l s i m i l a r t o t h a t of W e i n b e r g a n d S a l a m a n d t r y t o e s t i m a t e t h e v a l u e of v - q u a n t u m p o l a r i z a t i o n i n t h e r e a c t i o n n ~ - p - ~ d ~ - y . We shall base our r e s e a r c h o n t h e 4 - q u a r k m o d e l of G I M (') a n d t r e a t all t h e v e c t o r m e s o n s , w h e t h e r i n t e r m e d i a t e w e a k W ~, Z or h a d r o n i c p~.0, co, as g a u g e fields. ( k p o s s i b l e m o d e l of t h i s k i n d w a s c o n s i d e r e d , e.g. i n (10).) H o w e v e r , t h e p's a n d (o's c a n b e t r e a t e d p h e n o m e n o l o g i c a l l y .

(*)

O n l e a v e of a b s e n c e f r o m t h e I n s t i t u t e

of N u c l e a r P h y s i c s of t h e M o s c o w

State University,

Moscow, USSR. (D A. SAL~U~: Elem~n!ary Particle Theory, edited by N. SV.a.RTHOL.'*f (Stockholm, 1968), p. 367. (') S. WEINB~.m~: Phys. Rev. Ldt., 19, 1264 (1967). (8) G. T'HooFT: NucL Phys., 33 B, 173 (1971); 35 B, 167 (1971). (~) P. S~,LOMONSO.Wand Y. U~.DA: Phys. Rev. D, 11, 2606 (1975). (s) K. FUJIKAWA, S. W. LEE and A. T. SAND),: Phys. Rev. D, 6, 2923 (1972). (6) S. ~ . BILEBrKY, S. T. PETCOVand B. l~I. PONTECORVO: J I N R preprint E2-10374, Dubna, USSR (1977); S.T. PETCOV: J I N R preprints E2-10176, E2-10487, Dubna, USSR (1977). (') S. B. TR~.L'~A~r F. Wn~ZE~ and A. ZEE: Phys. R~v. D, 16, 152 (1977). (') S. I~L~RISOh':preprint 77/P.957, Centre de Physique Th~orique, CNRS, ~Iarseille, France. (a) S. GLASHOW, J. ILIOPOULOS and L. MAIA.~I: Phys. P~v. D, 2, 1285 (1970). (,0) H. GALI~, B. GUBEI~I~'Aand D. TADIC: Zeif,s. Phys., 276 A, 65 (1976); Phys. P~v. D, 14, 2327 (1977). 21

22

V. M. I ) U B O V I K

and

v.

s.

ZAMIRALOV

I n t h i s case t h e d i v e r g e n t d i a g r a m s s h o u l d be cut-off a t t h e v a l u e of t h e i r c h a r a c t e r i s t i c m a s s e s ~ 1 GcV. ~r d e s c r i b e t h e d e u t e r o n w i t h t h e J~%N'-potential arisen f r o m t h e e x c h a n g e of a n d co m e s o n s (fig. l a ) ) . Usually, ~ a n d co e x c h a n g e s are i n t r o d u c e d a t t h e nucleon level. T h e c o r r e s p o n d i n g L a g r a n g i a n is g i v e n e.g. in (z~). A t t h e s a m e t i m e w e a k inter-

l=,

Fi~. 1.

a c t i o n s s e e m t o o c c u r a t t h e q u a r k level. So it is r e a s o n a b l e t o i n t r o d u c e t h e ~ J ~ i n t e r a c t i o n v i a p a n d co a t t h e s a m e level (fig. lb)). W e h o p e t h a t it is n o t so d r a s t i c b e c a u s e w e are i n t e r e s t e d m o r e i n t h e r a t i o of t h e w e a k a n d s t r o n g c o n t r i b u t i o n s t h a n in t h e i r a b s o l u t e values. T h e r e l e v a n t p a r t of t h e L a g r a n g i a n can b e w r i t t e n as

(]) w h e r e T's are t h e P a u l i ' s m a t r i c e s a n d q is t h e d o u b l e t of q u a r k s p a n d n. I n a c c o r d a n c e w i t h t h e u s u a l a p p r o x i m a t i o n s (12) w e s t a t e t h a t t h e c o u p l i n g cons t a n t s gpq~, g~a~ can b e e x p r e s s e d t h r o u g h t h e c o u p l i n g c o n s t a n t s , g p ~ , g o ~ , where J~ m e a n s t h e n u c l e o n d o u b l e t , w i t h t h e h e l p of t h e o v e r l a p p i n g f u n c t i o n

(2)

J -~-fd3[r

~3

~ rl -~-r=)d3(r=_r2)l

2

VJ(rl, r 2, r3)]~(rl--r2)

w h e r e ~ ( r l , r=, r 3) is t h e q u a r k w a v e f u n c t i o n .

We have

(3)

_~ gc0Q~J" .

gpo~o~ _~ gpq~.~",

gr

T h e w e a k J ~ J ~ - p o t e n t i a i can be c o n s t r u c t e d in t h e usual w a y (13.z4), b a s i n g it on t h e d i a g r a m s s i m i l a r to t h o s e in fig. 1 b u t w h e r e t h e u p p e r v e r t e x is m a d e w e a k (fig. 2).

e~w Fig. 2.

(11) ~i. KENYA, K. OHYA, T. OKA and Y. Y~'~AMOTO: Prog. Theor. Phys., Suppl., 60, 203 (1976); K. O~YA, T. OKA and Y. YA~IA.~OTO: Prog. Theor. Phys., 56, 875 (1976). Here the reader can find a very full list of references on the subject of P-violation in the ~N~176 (l=) B. DESPL.~..N'QUESand J. MICHELZ: preprint IPNO/TH 77-16, Universit~ de Paris-Sud (1977). (~s) H. J. PIRNER and .~I. L. RUSTGI: Nucl. Phys., 239 A, 427 (1975). (1,) j. F. DONOGHUE: .Phys. Rev. D, 13, 2064 (1976).

,)~, ~r, W E A K POTENTIAL IN A SIMPLE GAUGE MO1)EL ETC.

2~

We use a conventional GIM 4-quark model ('), b u t it is sufficient to confine ourselves to one doublet of quarks p and n. The reason lies in the very small coupling value of the 7 and ~ mesons with nucleons, which permit us to disregard the 7- and %b-exchange diagrams. With these remarks we write the weak-current yielding parity violation in the upper vertex of fig. 2.

J+

=

Wr= ~ P "

cos 0 o ,

(4)

The relevant part of the weak Lagrangian is

(5)

ffw = gj~+w; + H.C. + - - g - ] ~ z ~ ,

g2

SM~

cos 0w

_

G;

v~

Three types of diagrams contribute tothe p-violating vertex of fig. 2. Firstly, there are quark self-energy diagrams whose contributions can be absorbed into the renormal-

W~Z~S,X

q' ~

q'

q~

qlr

q'

.>

..~W ", Z

I

qJ~) q;

Fig. 3.

F i g . 4.

ization of the quark wave functions (fig. 3), secondly there are self-energy graphs of the pc-Z, e-Z, p• • transitions (fig. 4), and finally there are vertex diagrams (fig. 5). The contributions of the diagrams in fig. 4 are suppressed in comparison with those in fig. 5 (for the lepton sector see e.g. (15)). Therefore we confine ourselves to the contributions of the latter, to estimate the effect of the y-polarization in the reaction Z ,X

W+(k), 5l•

e~

F i g . 5.

o.)

b)

(*5) j. NOLLE, J. H. I~EID and L. SCHULKE: Phys. Left.,56 B, 57 (1975).

~ (q)

24

V. M. D U B O V I K a n d

V. S. ZAMIRALOV

n + p ~d-{-V. I n more detail we consider the diagram in fig. 5b) which has the W-exchange. Calculations are made in the t'Hooft gauge and the F e y n m a n rules similar to those in (5) are used. Retaining the p-violating part of the diagram we have

(6)

/#~o~=9= cos= ~

Oo/" ~. ~(p=)[(-- l~ + q) r~(k + 4) - - m(~ - - 4) r~ + m),~,{~ + O)]r, u(p~) o~, ju

~

(~= - - ~ ) [ ( P

- - ~)~ - - m ~3 C(p - - k - - q)= - - , ~ q

'

where ~ is the spinor of the quark p. Upon using the standard program of subtraction of the ultraviolet divergencies we finally obtain the terms proportional to (7)

g' cos= Oo ~(p=)7a~.au(pl)Qc~

--

gpa~.

In M ~ 2i- "9 .M~] ?~bp

64z~

= g ~ a cos 20 o ~ ( p , ) 7 a r 6 u ( p x ) ~ ,

where a : 0.44.10 -4 at M w ----60 GeV. The diagrams in fig. 5a), with the exchange of Z-boson, are given by the same value of a (under the symbol In we p u t M w - - Mz), b u t differ from those in fig. 5b) by the factors of the type cos =0o. The contributions of the diagrams in fig. 5 with the exchange of the unphysical scalar bosons S~, X are suppressed by the factor M~/M~,z in the coupling constants in the gauge chosen, so we safely neglect them. The result can be conveniently written with the help of the effective Hamiltonian describing the p-violation in the 2~f'~N'-interaetions (11). (8)

Hw

=

1, ~

1

--

W'Y~Y~(~+~ +

B--

~-

~+)~ + ~

z=

o

1

--

~'~Y~~ ~ ~ + ~ ~ a y ~ o ~ j

]

,

where f, = V~a(1 - - 2 s i n = Ow)gp V..~/2 c o s 2 0 c = 0 . 6 8 . 1 0 -a , (9)

- - B = 4(cos= 0 o - - 89+ sin20w)/(1-- 2 sin= 0w) ----8 , D ---- 12 V/6go.,~o~(cos~0o + ~ - - sin =Ow)/gp,,v~v(l- - 2 sin 20w) _~ 132,

and ~-l = 6V~, and we have used g~,~r/4~ = 2.4, g~.v~/4~ = 5.4, sin=0w---- 0.3. The large value of the polarization~of the T-quanta in the reaction n + p ~ d + T P~ = (--1.30 • 0.45). 10-6 found in the experiment of LOBASHOVQ*) forced theoreticians on the one hand to analyse once more the validity of their views on the deuteron potential and on the other to try almost every possible models of the weak interactions from Cabibbo's charged currents to the most refined ones based on the Weinberg-Salam model and the notions of asymptotically free giuon fields. However, in all these models the product of the weak nonleptonic currents is studied. We have preferred to use the diagram technique within the G I ~ quark model ('), retaining only the doublet of quarks p and n. The deuteron potentials have been taken from (1~). For the Hamada-Johnston potential we have

(lO)

P~v - J =

(-- 14 + 5.7B--

0 . 6 3 ~ ) " 1 0 - 3 f , - --

1.05.10-v

,

(~') V. ~[. LOBASHEV, D. ~. KA~flNKER, G. I. KHARKEVICH, V. A. KNIAZKOV, N. A. LOZ0VOY, V. A. NAZARENKO, L. F. SAYENKO, IJ. 1~. SMOTRITSKY a n d k . I. YEGOROV: N u c l . Phy@., 1 9 7 A, 241 (1972).

~ '

25

W E A K POT:ENTIAL IN A SIMPLE GAUGE MODEL ETC.

while for the Kishi-Sawada-Watari potential (ll)

P v ~ ( - - 6 . 1 ~- 8 . 2 B - - 0 . 8 2 ~ ) . 10-s~, ---- 1 . 3 8 . 1 0 -7

.

Thus for the realistic potentials of the type quoted in 11 we have obtained a strong divergency with the result of the experiment (1~), although our value of P y is higher than in other works on the subject, if attempts with the nonlocal potentials are not taken into account (1~). We would also m e n t i o n that in (10.1s,1,) the authors have tried to achieve the enhancement b y taking the gluon corrections into account. These attempts were stimulated by (2o), where the enhancement was obtained for the A I = } transitions in the nouleptonic decays with the help of asymptotically free gluon fields. The gluon corrections calculated in (10,1s.19) do not coincide with each other. Moreover, the value of Pv in (~0) is suppressed by a factor of 4 - - 5 in comparison with the usual results. So the problem of gluon corrections is still open. We hope to consider it elsewhere. TABLE I. -- Values o] IPvl. lOs in the Salam-Weinberg model ]or several realistic potentieds. Potential

Usual t r e a t m e n t (11)

Present work

Hamada-Johnston"

2.2

10.5

Reid (soft core)

2.1

Kishi-Sawada-Watari

0.22

13.8

Tamagaki (super soft core)

2.4

7.5

6.2

We state our results in table I and compare them with those of other works based on the construction of the effective nonleptonic Hamiltonian. The enhancement is obtained mainly thanks to the characteristic factor In (mp,,~/Mw,z) ~ of the vertex diagrams. This mechanism proves to be essentially similar to that of gluon corrections.

The authors arc grateful to D. BAI1DIIV, V. PERVUSHIN, profs. G. S. DANILOV and D. V. SHIRKOV for useful discussions. One of us (V.Z.) is grateful to profs. G. GIACOMELLIand G. VE-~'TURIfor the hospitality extended to him at the I n s t i t u t e of Physics of Bologna University where part of this work has been done.

(17) (la) (1,) (10)

K. G. j. ~.

R. LASSEY a n d B. hi. J. ALTARELLI, R. K . ELLIS, F. DO.'COGHUE: Phys. Rev. K . GALLLARD a n d B. W.

~IcKFT.T.A~: .'luSt. Jour~. Phys., 27, 637 (1974). L..~VIAIANI a n d R. PETRONZIO: Nucl. Phys., 88 B, 215 (1975). D, 15, 184 (1975). LEE: Phys. Rev. I~tt., 33, 108 (1974).