Pramfi0a, Vol. 8, No. 6, 1977, pp. 518-523. © Printed in India.
The Us(W)-gauge theory III: Atomic physics parity-violation
L K P A N D I T
Tata Institute of Fundamenfal Research, Bombay 400005
MS received 19 February 1977
Abstract. It is shown how a slight natural generalization of the mechanism of the spontaneous gauge symmetry breaking in the U3(W) gauge theory (Pandit 1976) can accommodate the degree of parity-violation in atomic physics suggested by some recent experiments, along with the neutral current processes involving the neutrinos.
Keywords. Ua(W)-gauge theory ; parity-violation; atomic physics; spontaneous symmetry breaking; neutrinos.
1. Introduction
Some time ago we h a d proposed the unified U3~W)-gauge t h e o r y o f weak a n d electromagnetic interactions ( P a n d i t 1976, referred t o h e r e o n as I). In a subsequent publication (Pandit 1977, referred to h e r e o n as II) we h a d examined the impli- cations o f this t h e o r y for some typical p h e n o m e n a resulting f r o m its weak neutral currents. It was f o u n d t h a t the results for the neutral c u r r e n t processes involving the neutrinos fared quite as well as the popular s t a n d a r d W S - G I M theory, whert c o m p a r e d with the available experimental information. It was further noted t h a t i m p o r t a n t differences between the two theories arose in some o t h e r effects o f inte- rest. One o f these is the predicted violation o f p a r i t y in atomic physics. Our result for this effect was a good bit smaller in m a g n i t u d e t h a n that o f the W S -
G I M model; a n d further, what is even more i m p o r t a n t , o f the opposite sign.
In a recent joint letter to Nature, tWo experimental groups have reported their interim results o f independent measurements o f optical r o t a t i o n in atomic bismuth, carried o u t at the University o f Oxford (B~ird et al 1976) a n d at the University o f W a s h i n g t o n ( F o r t s o n et al 1976). For the relevant parameter chaxacterising parity violation t h e y quote the results:
R ( 2 = 648 rim) : - ( + 10 i 8) × 10 -s (Oxford), (I) R ( 2 = 876 n m ) = ( - - 8 ~: 3) × 10 -'s (Washington). (2) T h e errors quoted are two standard deviations a n d the systematic errors (though not well u n d e r s t o o d yet) are believed by the experimenters not to exceed 4- 10
× 10 -s. It is clear t h a t the experiments stand in need o f f u r t h e r refinement, with a firmer understanding of the systematic errors, before a definite sign a n d magni- tudes o f the R values can be considered as established. All the same, even at
518
U3(W)-gauge lheory and atomic physics parity-violation 519 this preliminary stage, the values predicted by the standard W S - G I M model, R (2
= 6 4 8 r i m ) = - - 4 0 × 10-8 and R ( 2 ~ - 8 7 6 n m ) = - - 3 0 × 10-8, seem already in severe difficulty.
The purpose of the present paper is to show that the U3(W)-gauge t h e o r y is perfectly capable of coping with the emerging situation with regard to the ques- tion of parity-violation in atomic physics. In our theory there are two neutral intermediate weak interaction vector bosons, Z 1 and Z2. Only the weak neutral current coupled to Z t involves the neutrinos. On the other hand, both the currents coupled to Z t as well as to Z2 involve the electron, so that both contribute to the parity-violating electron-nucleus weak potential relevant for atomic physics.
In paper I we had adopted a particularly simple and elegant mechanism for the spontaneous breaking of the gauge symmetry leading to specific relations between the masses and the couplings of Zt and Z 2. This gave rise to a parity-violating potential of sign opposite to that of the W S - G I M theory, and of a magnitude around 1.6 times smaller. This result cannot be ruled out by the Oxford result (1), though it would disagree in sign with the Washington result (2).
In the present paper we show that the mechanism of the spontaneous gauge symmetry breaking of paper I may be generalized quite naturally in such a way that the resulting masses and couplings strengths of Z~ and Z 2 conspire to give much smaller strengths for the parity-violating potential. The eventual outcome of the relevant experiments' will then be able to fix more precisely the nature e f the generalized gauge symmetry breaking.
In section 2 we briefly describe the proposed generalization of our spontaneous U3(W)-gauge symmetry breaking and indicate the ensuing changes in the "vector boson masses and coupling stlengths. The question of the resulting parity-viola- tion in atomic physics is then discussed on this basis in section 3. In section 4 we show that the reasonable agreement of our theory with experiments on the neutral currents processes involving the neutrinos is still very well maintained.
2. Mechanism of the spontaneous breaking of the gauge symmetry
To implement the spontaneous breaking of the U3(W)-gauge symmetry, as des- cribed in detail in paper I, we introduce three SU3(W) triplets of Higgs scalar fields
~,~ _~ (~1c,~, ~c,~, ~3t~), i = 1, 2, 3, havingfor the Ul(W) generator G O the values G o = - - ~ f o r 9 tl~ and G o = + ½ f o r ~ and ~ 3 ~ The " v a c u u m ' is then arranged to be such (by a suitable choice of the potential function in the Lagran- gian) that St ~a~, ~2 ~2~ and q~a ~3~ attain non-zero vacuum expectation values: (#l~a))o
= (~b2~2))o = (ffa~a))o ---- ~/:~ 0; all the other scalar fields having zero vacuum expec- tation values. This simple and elegant choice ensures that the stringent require- ment (necessary in any gauge theory seeking to unify weak and electromagnetic interactions), that only the gauge vector boson A~, coupled to the electromagnetic current turns out to be mass-less while all the other eight (weak) vector bosons become massive, is successfully met. This requirement is still satisfied even if we adopt the somewhat more general choice ($1 ~1~)0 =~/' ~ 0; ( ~ 2 ( 2 ) ) 0 = ( ~ 3 ( 3 ) ) 0 = ?] =?& 0 and r / ' # ~/, It is imperative, however, that the vacuum expectation values of ~bz t~
and ~b3 c8~ remain still equal.
In the present paper we shall indicate some phenomenological consequences of adopting the above more general spontaneous gauge symmetry breaking mecha-
520 L K Pandit
nism, taking ~' a n d r/ defined above to be, in general, different. For ~ / ' = q, o f course, we shall get back to the old results of I and II.
The most important modification o f phenomenological consequence is in the values o f the (weak) intermediate gauge vector boson masses. We now find the results [to be compared with eq. (3.31) of I]:
m s ( W ~) = m 9- ( R ~z) = f~rl~ (1 + ½ 6), rn ~ (Zrj ---- m s (H, _T-I) --- f" q~,
m~(Zx) = (1 + ~rs)f2qs(1 q- ~6),
(3)
where we have defined
-
1 + 6, (6 > - 1). (4)
F o r ~/' = 0, we have 6 = 0 a n d the results (3) go over to those o f I. The SU3(W ) gauge coupling c o n s t a n t is denoted by f , the Ua(W)-gauge coupling constant by f ' a n d we use the parameters (introduced in I a n d II)
t a n X ~- a ~ ( 2 f ' / ~ / ~ f ) . (5)
The expressions for the gauge vector fields o f definite mass after symmetry breaking, Wt,+, Rt` +, 1-I u, Ht`*, Z l t ,, Z2t` and At,, in terms o f the hermitian gauge fields, Wu,, a -~ 1, ..., 8, and Bt`, remain the same as before [see eq. (3.30) o f I]
as also the f o r m o f their interactions with the leptons a n d quarks [see eq. (5.4) o f I a n d eqs (6) a n d (7) o f II]. The effect of the changes in the vector boson masses, eq. (3), i f we generalize t o 6 ~: 0, will be seen in the effective four-fermion Lagrangians mediated by these vector bosons. Keeping to the fixed normaliza- tion:
f 2 2e
G = 4 ~/2 mS (W+) ' f = ~ sin X ' (6)
where G is the standard Fermi coupling c o n s t a n t for the W+-mediated weak interaction, we now have to make the following replacements in the effective four- fermion interactions o f I a n d II:
G--* G, in R~Z-mediated int., G ~ G 61, in Z : m e d i a t e d int., G --* G 62, in Z2-mediated int., G -~ G 62, i n (H, H)-mediated int., where we have introduced the abbreviations:
( 7 )
(8) (9) (10)
6
6x----1 6 + 4 6 ' 6 2 ~ 1 + ½ 6 . (11)
I f we take ~/' = ~/, i.e., 6 --- 0, we have 61 = 62 = 1 a n d we get back to the old results o f I and II.
U3(BO-gauge theory and atomic physics parity-violation 3. Parity-violation in atomic physics
521
The effective parity-violating electron-nucleus potential (see section 9 of II), arising from the Z l and the Z 2 exchanges, is proportional to the paranneter
Q (Z, N) given by:
Q ( z , N) ---- (62 -- 61) N + 2 (62 -- 61 sin ~ Z) Z, (12) where N is the neutron number and Z the proton number of the nucleus in question. For 6 --- 0, i.e., for 61 = 62 = 1, we get back the expression, derived in
II,
depending only on Z and sin 2 Z. In the standard W S - G I M theory, where only one neutral weak vector boson occurs, the expression isQ (Z, N) ---- (1-- 4 sin ~ Ow) Z -- N, (WS-GIM theory). (13) As discussed in II, the neutralcurrent processes involving the neutrinos are reason- ably well described by 6 ~ 0 and sin ~ Z -~ J~ in our theory, as well as ill the W S - G I M theory with sin ~ 0a, ~- zs-. The approximations in the theoretical models used (chiefly t h a t of using the valence quark parton model) and the uncertainties in the experimental results, however, still allow us the freedom in our theory to, admit small non-zero values for 6.
To probe the effect of taking small non-zero values of 6 in the atomic physics parity-violation, let us fix sin S X ~ ~, and consider the example of the experi- mentally interesting case of Bi (Z---- 83, N = 126).
We then find
Q (Bi)= {
+ 7 , for 6 = - - k , + 5 8 , for 6 = - - 1 , + 104, for 6 = 0, + 148, for 6 = +¼.
(14)
In the W S - G I M theory, for sin ~ 0w - j}, Q (Bi) ~_ -- 168. Thus the preliminary experimental results quoted in eqs (1) and (2), while in almost certain disagreement with the W S - G I M theory, can yet be very well accommodated in our theory with a suitable choice of 6. More definitive future experiments should be able t o determine the value of 6. Coupling the results (14) with the discussion of the next section on neutrino neutral current processes, suggests t h a t values of 6 ~ - - ¼ up to 0 are rather likely.
4. Neutral current processes of neutrinos
The value of 6 is also important for the neutral current processes involving t h e neutrinos, since the mass of the neutral vector boson ZI, mediating them also n o w depends on it according to eq. (3). According to eq. (8), the Z~-mediated effective Lagrangians, relevant for these processes, given in paper II must then be multi- plied by a factor 61 defined in eq. (11).
As a first example of incorporating this change, let us consider the deep inelastic inclusive scatterings of v~ and ~ off the isospin averaged nucleon (,~') (see section 5 of II). The ratio of the cross-sections for these processes, of course, does n o t
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522 L K Pandit
depend on t$ but only on the value of sin ~ Z- Taking sin2z _ ], we have ( g stand- ing for " a n y t h i n g "):
[ o ( ~ + ,w'--, ~ + X)/o(v,, + gO--, v~, + X)] -~ 0.55, (I5) which agrees quite well with the value 0" 59 + 0.14 of the Gargamelle experiment (Blietschau e t a l 1976). The values for the parameters
Rv -~ [o (vt, + ~ ~ v~, + X ) / o (v~ + ,~r--, # - +
X)], (16)
R~ -- [o(~ t, + ,gV'--, ~, + X ) / o ( ~ , + ~W'--, !~+ + X)], (17) however, must now be multiplied by 61 ~. The values of R~ and R~,, obtained in the approximation of the valence quark parton model, are given in table 1 for various values of 6, along with the values of the Gargamelle experiment (Blietschau et al 1976).
F r o m table 1, we see that the agreement with the experiment is quite reasonable for 6 >~- ¼. In judging the theoretical values we must, of course, remember that the approximation of the valence quark parton model has been used. The same model #yes for the well established charged current interaction process the result: [ o ( ~ + , ~ ' - , It+ -t: X)/o(vu + ~ r _ , I t - + X ) ] _~ 0.33 (independently of sin~x and 6), whereas the value of the Gargamelle experiment (Blietschau et al
1976) is 0- 38 ± 0.02.
The above result coupled with the results of eq. (14), relevant to the parity- violation in atomic physics, leads us to suggest that values of 6 _~ - - ¼ up to 0 would be rather likely.
As another example of the Zl-mediated processes, we consider the v # - e and
~, -- e scattering cross-sections (see section 3 of II). In the commonly employed notation:
a (v~ e~ ~ C (v~ e) × 10 -41 (Ev/GeV) cm 2,
o ( ~ e) ~ C ( ~ e) × 10 -~1 (E,/GeV) cm 2, (18) we obtain, with 6 ~- --¼, sin 2 X -~ ], the values:
C(v~ e) -~ 0.09, C(p~ e) = 0-27. (19
These are quite reasonable in view of the rat~er uncertain experimental situation.
Thus the Gargamelle experiment (Musset 1976) gives C ( P ~ e ) = 0.11 +0.2~ - - 0 - 0 g '
C ( v ~ e ) < 0.26; and the Aachen-Padova experiment (Faissner e t a l 1976) gives C(~tte ) --- 0.54 + 0.17, C(vt, e) = 0-24 + 0-12.
Table 1. (R, and R~- for sin ~ X -- ~.)
3 = - - ½ 8---- ¼ ~ = 0 8 = +¼ Experiment (Gargamelle) Rv 0"42 0"36 0"33 0 " 3 1 0"26-1-0"04 R~, 0" 67 0" 58 0" 53 0" 49 0" 39 4-0" 06
All the Z I a n d Z2 mediated interactions, discussed in p a p e r II, m a y be similarly modified if di:/: 0. T h e changes are obvious and m a y be r e a d o f f using the re- placements indicated in eqs (7) t o (10). Thus f o r the weak i n t e r a c t i o n effects in the proce3s e+e - ~ # + # - , the parameters o f section 8 o f II, must n o w be c h a n g e d to h v v = ½ ~ 2 + ~ ~1 ( ½ - - 3 sin 2X) 2, h , t A = ½ ~ 2 + ~ ~1, h v A = ½ ~ +
~1 ( ½ - 3 sin ~ X). These particular changes, however, will be o f interest in the experiments o f a more d i s t a n t future a n d are given here f o r completeness.
We h o p e t h a t more definitive experimental results o n the a t o m i c physics parity violation, as well as o n neutrino neutral current processes, will be available in t h e near future to enable a full scale testing of o u r t h e o r y .
Acknowledgements
T h e a u t h o r wishes t o r e c o r d his thanks to K V L Sarma a n d P P D i v a k a r a n for useful conversations.
References
Baird P E G etal 1976 Nature 264 528 Blietschau J etal 1976 CERN/EP]PHYS 76-55
Faissner I-I et al 1976 Report at the XVIII lnt. Conf. High Energy Phys, Tbilisi Fortson E N e t al 1976 Nature 264 528
Musset P e t al 1976 Report at the X-VIII Int. Conf. High Energy Phys. Tbilisi Pandit L K 1976 PramS.ha 7 291
Pa~:dit L K 1977 Pram~n.a 8 68
