Strong isospin violation and mixing in [A S I = 1 weak transition

Transcription

1 Pramina, Vol. 17, No. 5, November 1981, pp ~) Printed in India. Strong isospin violation and mixing in [A S I = 1 weak transition BIJAN K BAGCHI Department of Theoretical Physics, Indian Association for the Cultivation of Science, Calcutta , India MS received 31 July 1981; revised 6 October 1981 Abstract. We consider the effects of,/- ~r mixing on the violation of the I AII = 1/2 rule in I AS I = 1 weak transitions. The processes considered are the K --> 2,r, K --> 3~r, A, ~ and A hyperon decays. Keywords. Isospin violation: ~ -~r mixing; I All = 1/2 rule; kaon decay; hyperon decay. 1. Introduction In the theory of quantum chromodynamics, the strong interaction Lagrangian is written in terms of a small number of parameters namely, a dimensionless colour gauge coupling and a quark mass for all the quark flavours (Gross et ai 1979). By dimensional transmutation the coupling constants may be represented in terms of a mass scale (M), say, the mass of a standard hadron consisting of light quarks. One may as well identify M as the renormalization scale parameter to be determined from experimental observations. The pattern of the quark masses, on the other hand, reflects the character of flavour sy~roetry ofthe strong interactions. If isospin violation is purely accidental then as Weinberg (1978) has stated, there might be special cases where this violation has noticeable effects and which in most cases will be set by the ratio (rnd -- m,)/m. However, the values of the quark mass ratios evaluated from the meson mass spectrum runs counter to such an expectation--the ratio md -- m,/m~ + m u turns out to be ~ 1/3 thereby implying the presence of a fairly substantial isospin breaking in strong interactions whose effects are clearly not observed. This problem was essentially solved by Gross et ai following the observation by 't Hooft that the divergence of the U(1) current contains an anomaly term which ensures that there is no Goldstone boson associated with it even in the chiral limit. Gross et al showed that the existence of the anomaly term reduces the degree ofisospin violation from O (m~--m./m~+m.)to O (md--m./m~) which is phenomcnologically the correct order of magnitude. They also showed that the three cases where isospin violation might produce noticeable effects are the pion mass difference problem, the decay process 7/'--> 3~r and the ~± --> Ae ~ v decay. Recent measurement of the decay rate r(~'--> ~ ) by the Crystal Ball (Oregila et a11980) and the Mark H (Himel et al 1980) groups has given rise to another possibility where also the effects of isospin violation (through isoscalar-isovector pseudoscalar mixing) can be equally 4O5

2 406 Bijan K Bagchi important. In fact the measured experimental values for the branching ratio R given by* R -- r' (w'~ Wzr ) _ (k_z~s 03, (1) P ("F' ~ ~F~) \knl show that the ~7--zt mixing parameter 0 needs to be as large as 0 = (4"5 4-0"6) 10-3 or ( ) 10-2, corresponding to Rex p -----(4 4-1) 10-3 or (6 4-2) 10-3 respectively. Lately, a number of papers have appeared which have attempted to explain this large branching ratio (Lamgacker 1980; Gerard et al 1980; Ioffe and Shifman 1980; Lahiri et al 1980; Lahiri and Bagchi 1981 ; for earlier works see Segre and Weyers 1976; Genz 1978). While some of them (Langacker 1980; Gerard et al 1980) extend the possibility of the mixing phenomenology to include the effects of ~' too, the chief reason being that a combined study of baryon and meson mass splittings, the ~ -~ 3~r decay, and p- oj mixing yield a value for 0 = (Langacker 1980) which is about a factor of 3 too small to explain the decay ratio R; some (Ioffe and Shifman 1980; Lahid et al 1980; Lahiri and Bagchi 1981)hold the view that ~/--~r mixing is sutficient to explain R. In one of these papers the time ordered product of axial currents has been suitably evaluated to estimate the ~/-- ~r mixing angle 0 (Lahiri and Bagchi 1981); in another, Weinberg's spectral function sum rule in the asymptotic limit has been made use of to evaluate 0 (Lahiri et a11980). In both the approaches, 0 has turned out to be very large in comparison with the value just cited above. That the effects of ~7 -- zr mixing may be important and can have an appreciable effect on the violation of the AI = 1/2 rule in AS = 1 Kl 3 decays was pointed out long ago by Oneda et al (1970). It may be mentioned that in an asymptotic symmetry framework they had determined the values of the mixing angles between the *r, ~/ and ~' states. In particular, for the ~7 --,r mixing they had obtained two distinct values for 0, one about ~ and the other approximately They had also pointed out that if the former solution was correct, a sizeable violation of the AI = 1/2 rule was expected while if the AI --- 1/2 rule was well satisfied, the second solution was preferred. More recently, the effects of the ~--,r mixing in the violations of the AI = 1/2 rule in the AS = 1 weak decays in general were studied by Holstein (1979). The result of such a calculation was that the 7/-- ~r mixing did produce a significant effect on the measured size of the intrinsic AI = 3/2 weak amplitudes. However, the value of 0 used to evaluate the ~7- rr mixing effects was the canonical value obtained by Gross et al (1979) in the tadpole approximation viz V'3 md - m. _ (1 4-0'2) ms *If the */-- ~r mixing alone is considered (3)

3 Strong isospin violation 407 As mentioned earlier, this value of 0 is much too small to explain the branching ratio R given by (1). In this paper we inquire into the effects of the ~ --rr mixing once again on the AS = 1 weak transitions in the light of the values of 0 obtained by us (Lahiri et al 1980; Lahiri and Bagchi 1981) and the value estimated by Oneda et al 0970) that differs in sign from the standard values. Our conclusions shall be presented as we go along. 2. Violation of the AI = ½ rule in strangeness changing decays 2.1 K-+ 2,r decays Following Holstein (1979)*, we consider first of all the non-leptonic weak decays K-~ 2rr. The various amplitudes can be expressed in terms of the AI = 1/2 and AI = 3/2 components as A(K -~,:,:) = - A + ~ A, A(K o -,,~+,~-) = 3 ~ + ~ A' A( K+ ~ *r+*rz) =,A fs, (4) where ~ corresponds to the triplet pseudoscalar state that mixes with the octet state to give the physical ~r and ~r = ps + Ops, -- _ Op3 + ps, (5) 0 being the ~ -- 7r mixing parameter. By defining y as y = (6) 2 A(K -->,r+~r-) -- A(K -+ rr% ) ' = 0" "001 (exp.) one can interpret it as a 3 % AI = 1/2 rule violating parameter 1 f (7) *We follow the notations and equations of Holstein (1979).

4 408 Bijan K Bagchi If,7 -- ~r mixing is included, y becomes _ Y vs A 1 fa_t_20. Vg (8) by using (4) and (5) and assuming AJ = 1/2 dominance for A(K->~'~). We now discuss the effects of various values of 0 on (8). For definiteness we consider for 0 the value 0 = ~ obtained by us and which lies within the limits dictated by (1), the value 0 = obtained by Oneda et al and the canonical value given by (3). Case 1 Taking the canonical value of 0 first viz., 0 = 1 x 10 -~, we note that this value of 0 gives (Holstein 1979) (20/~/]),,~ which takes up about 40Yo of the experimental AI= 1/2 violating amplitude so that only 60% need be a result of intrinsic M = 3/2 terms: Case 2 fa ]f~ = 0.02 ~s 0 = 4 x IO-L Such a large value of Ogives (20/~/~) ~ 0"048 which is bigger thanthe experimental value of y and indicates that the intrinsic M = 3/2 terms may be present with the wrong sign 1 fa m Case 3 0 =--6 x 10 -a. This gives 20/~/] ~ Thus the ~7 -- ~r mixing accounts for the experimental AI /2 violating amplitude by about 25 %. The AI /2 effects are then given by 1 fa ~

5 2.2 K~ 3~r decays Strong isospin violation 409 Following the standard parametdsation, the K--> 3rr decays can be written as mk where the mean decay amplitude is giveu by.4 o and I is the slope parameter. The AI rule violating parameter can be introduced as (Holstein 1979) 1 A vl = ~(A--~) --1=0"216-t-0"020, vz = = 0'308 q- 0"051. (I0) In the limit when the ~--~r mixing is neglected, standard calculations yield (Bdcrrtan et al 1978) current algebra vl = 6y ~ 0.19, 27 vz = -~-y m 0.43; (11) however if the ~ -- ~r mixing is invoked, it can be shown that vl = 6y + 2VY O, 27 vz ---- Ty _ 6V30. (12) Case 1 0 = ~. Holstein (1980) has shown that for this value of 0, v 1 and vz become v 1 ~ 0.228, 03 m 0.32, in reasonable agreement with (10). Case 2 0=4 10 -~.

6 410 Bijan K Bagehi In this case, v I and v~ turn out to be v z = = 0.33, v 2 = '416 = Clearly, the present experimental values on v z and v 2 [see (10)] rule oat such a large value of 0. It may be noted that in this case the effects of,7 -- rr mixing on v z and v2 are of the same order as the ones obtained in (11) by neglecting,/-- ~r mixing. Case 3 0= This value of 0 leads to v 1 = 0" = 0.171, vz = = 0.496, which also disagree with their respective experimental values. 2.3 A-hyperon decays Here the AI = 1/2 role violations can be read off from the following ratio R A ---- V'( D--F A ( A -> P 4"-) -k V~ A ( A --> n ~r ) _ P D ~ q- z v5 A - A n:) - (13) which has been obtained by taking into account the effects of A - (Gross et al 1979) 27 mixing as A o = B 8 + P B 3, 2o = _ p B s + B 3, with p,~ In (13), D and F are the octet (B'] Hoj ] B) coupling constants and z = C2 %/(D + 3 F), % being the bonafide AI = 3/2 amplitude contribution. It may be noted that R~ p = (14) Taking the approximation F,,~ -- 2D, we now consider the following cases Case 1 0=1 x 10-2.

7 Strong isospin violation 411 R A turns out to be (Holstein 1979) R A m 0"018 + z which shows that the ~ --,r mixing effects are working opposite in sign to the experimental AI = 1/2 violating effects. Thus the intrinsic AI = 3/2 amplitude contribution is now larger i.e. z ~ than what was expected by neglecting the ~7- = mixing effects viz. z = Case 2 0 = 4 x 10 -z. This gives R A ~ z which implies that the intrinsic AI = 3/2 amplitude contribution ought to be still bigger viz. z ~ than what was obtained in the previous case. Case 3 0= a. This case is quite different from the previous two since here one gets R A = z, indicating that the intrinsic AI = 3/2 amplitude contribution is not much different from naive expectations. 2.4.~. hyperon decay The AI = 1/2 rule violating ratio is given by A(.E- ~ Azr-) + V'2 A(E ~ Art ) RS = V'~ a(.=- ~ A,r-) -- a(=- ~ A,r o) = ~ O- 2PD_ +z', (15) where z' = D -- 3F Experimentally, RF, is R~ xp = -- 0" "011. (16) Case 1 0=1 10 -z. By substituting this value of 0 in (15) one has R.~ = 0"11 + z'.

8 412 Bijan K Bagchi As with the A hyperon decay, here too, the *7 -- ~r mixing effects are present with an opposite sign from the experimental value. The value of z' now is z' = which is larger than what was obtained by neglecting the *7 -- rr mixing: z' = Case 2 0 = 4 x This value leads to R ~ = z', indicating that the bonafide AI = 3/2 contribution needs to be z' = Case 3 0 = In this case the contribution of the.7 -,r mixing effects has a sign similar to the value of/~xp: R ~ = z'. Moreover, the magnitude of the *7-- rr mixing effects is about an order of magnitude smaller than/~xp which is in keeping with the usual expectations, z' in this ease is given by z' = ,~ ~ hyperon decays Here R z is given by RZ =.4 (2:+--> n,~+) - A(Z- --> n~-) + V2 A(Z+ -~ r,e) A(,~- --> mr-) = -- ~/3 sin0 + z u, (17) where tt z" = -- 3%/3 a------l-and R~ p = (18) 4 /g D- F Case 1 0 = 1 x 10-8.

9 Strong isospin violation 413 One obtains here (Holstein 1979) R,~ = z", which, like R A and R~, differs in sign with R~ p. comparison with the naive value z",,~ Here z" needs to be z" ~0.14 in Case 2 0 =4x 10-L R~ turns out to bc R,~ = z", which implies that z" ~ 0.19, slightly larger than what has been obtained in case 1. Case 3 0 = -- 6 x Here one gets R~: = + 0"012 + z", which shows that the magnitude (but differing in sign) of the ~ -- ~r mixing effects is exactly equal to the value of R~ p thus ruling out the possibility of any bonafide AI = 3/2 amplitude contribution. 3. Summary To summarize, we have studied in this paper the effects of the ~ -- 7r mixing on the violation of the A! = 1/2 rule in AS = 1 weak transitions. For the ~/-- 7r mixing angle 0 wc have considered (i) the value 0 = 4 x 10-2 obtained by us in earlier papers and which lies within the limits dictated by the experimental value of F (~F'-~ Tw ) / F (T' -~ ~F~/) ratio, (ii) the value 0 = -- 6 x 10 -s obtained by Oneda et al that differs in sign from the standard values and (iii) the canonical value 0 = 1 x 10-2 obtained by Gross et al. For the weak processes we have considered the K-~ 29, the K-~ 3~r, A, ~ and 27 hyperon decays. We have found that for the values x 10-3 and 0 = 1 x 10-3, the 7/-- ~r mixing effect on the violation of the A[ = 1/2 rule is non-negligibly large although the g-~ 3w decays do seem to rule out the value 0 = 4 x 10-L It may be noted that these decays favour the smaller value of 0 viz L For the negative value of 0 (= -- 6 x l0 -a) we have found that for the A and ~ hyperon decays such a value is favoured since in these processes the intrinsic A! = 3/2 amplitude contributions are not much different from what one expects naively. However, the K-~ 3~r decays disfavour the

10 414 BijanKBagchi value O = -- 6 I0-8. For 2~ hyperon decay, the presence of the ~ -- ~r mixing term rules out the possibility of any bonafide AI = 3/2 amplitude contribution if the negative value of 0 is taken as an input. 3.1 Note After this work was submitted for publication, two papers have appeared (Gusbin 1981; Holstein 1981) which have also dealt with the effects of isoscalar-isovector pseudoscalar mixing in non-leptonie weak decays. In one of these papers, Gusbin (1981) has shown that the 7' mixing effects are negligible in K+ ~ ~r+tr decay and' do not affect the AI = 3/2 transition strength. In the other, Holstein (1981) has argued that the presence of AI = 3/2 term in the K~ ~r~? amplitude seems unlikely. References Bricman C et al 1978 Phys. Lett. B75 1 Genz H 1978 Lett. Nuovo Cimento Gerard J M, Pestieau J, Weyers J 1980 Phys. Lett. B Gross D J, Treiman S B, Wilczek F 1979 Phys. Rev. D Gusbin D 1981 Phys. Rev. D Himel T Met al 1980 Phys. Rev. Lett Holstein B R 1979 Phys. Rev. D Holstein B R 1981 Phys. Rev. D Ioffe B L and Shifman M A 1980 Phys. Lett Lahiri A and Bagchi B 1981 Phys. Lett. B Lahiri A, Bagchi B, Gautam V P 1980 Lett. Niovo Cimento Langacker P 1980 Phys. Lett Bg0 447 Oneda S, Umezawa H, Matsuda S 1970 Phys. Rev. Let Oreglia Met a11980 Phys. Rev. Lett Segre G and Weyers J 1976 Phys. Lett 't Hooft 1976 Phys. Rev. Lett Weinberg S 1978 Trans. N. Y. Acod. Sci. Ser. H38 185