Neutral SU(2) and neutral currents

Volume 96B, number 3, 4

PHYSICS LETTERS

3 November 1980

NEUTRAL SU(2) AND N E U T R A L CURRENTS M. CLAUDSON 1, H. GEORGI 1,2 and A. YILDIZ 1,3 L yman Laboratory o f Physics, Harvard University, Cambridge, MA 02138, USA Received 29 July 1980

We examine the neutral current structure of a class of topless gauge theories in which b decay and CP violation are mediated by neutral gauge bosons, N. We find a good fit to aLl data (better than the fit to the standard model) for sin20 = 0.205 and M~I/M~ ~ 10.

While the failure o f e+e - annihilation experiments to produce evidence for the existence of the t-quark in the expected energy range m a y simply indicate a lack of success in predicting rot, a number o f attempts to eliminate the t-quark from the theory of weak interactions have recently been made [1,2]. In several of these proposed models, the phenomenological gauge group is enlarged to contain a second SU(2) factor with neutral gauge bosons N +'. It is interesting to note that these bosons may have significant effects on the structure of neutral current interactions provided that their masses are not substantially greater than those of the conventional weak interaction bosons [2]. In conjunction with this, it is observed that present experimental determinations of the weak mixing parameter yield sin20 w = 0.22 + 0.02, while renormalization group calculations give sin20 w = 0 . 2 0 - 0 . 2 1 at phenomenological energies for a large class of unified theories [3]. Although this disceprancy fails within the range of experimental errors or may even be attributed to a lack of understanding of physics at large mass scales, an alternative explanation is to be desired if current experimental and theoretical trends persist. In this note we analyze the neutral current interactions of a class of models discussed in ref. [2] 1 Research supported in part by the National Science Foundation under Grant No. PHY77-22864, and Department of Energy. 2 Alfred Sloan Foundation. 3 Permanent address: College of Engineering and Physical Sciences, University of New Hampshire, Durham, NH 03824, USA. 340

with SU(2)W X SU(2)N X U(1) weak interactions. Within these models, we find that, while the presently available neutral current data favor sin20 w m 0.23, a fit significantly better than that of the standard model occurs for the theoretically expected value sin20 w 0 . 2 0 - 0 . 2 1 and a range of the new parameters needed to describe the neutral current interactions of the model, Experimental limits on these parameters are also investigated leading to an interesting result concerning the mass of the neutral analogs of the W ± (N and N*). The models which we consider include fermions transforming as singlets, (2, 1)'s, (1,2)'s, and (2, 2)'s of SU(2) w × SU(2)N. Explicitly we make the assignments for quarks UR'

ud ) L ,

bL'

(d,b)R '

(la)

and leptons eR ,

R

,

e L

.

(lb)

Here vertical pairs represent doublets o f SU(2) w ; horizontal pairs represent doublets o f SU(2)N. Also, n r is a massive neutral lepton, while v'r denotes a linear combination of nrL and the presumably massless r neutrino. The orthogonal combination n'r has not been included as its proper assignment is still unclear. We shall return to this question later. A similar set involving, c, s, tl, u, and the unobserved fermions g, o, no, and uo is also present. These assignments are discussed in detail in ref. [2]. While we make no attempt to incorporate

Volume 96B, number 3, 4

PHYSICS LETTERS

this structure into a unified theory, we note that the fermions listed above comprise a (5, 1), a (10, 1), and a (5, 2) under SU(5) X SU(2). Together with an additional (1,2) of neutral leptons, this set constitutes a 27 of E6 [4]. Although the neutral current analysis is largely independent of the details of the Higgs sector, we consider it briefly for motivational purposes. Masses for u and c and for b, ~, r and o may respectively be obtained by the vacuum expectation values (vev's) (

0 ), 2-1/2H w

(0, 2-1/2HN),

(2)

vev's

(ii): ( H0 0 ) .

(3)

Finally, we note that either of these sets of vev's may be obtained in a natural way as the minimum of the most general scalar potential consistent with the above symmetries. A relevant fact in analyzing the neutral current interaction is that either vev of eq. (3) leads to mixing of the vector bosons ZU and N~ through the mass term ~

~(Z

(5c)

with the -+ sign referring to cases (i) and (ii), respectively. Denoting the mass eigenstate vector fields by Z~ and

z~,

(z l: cos Z~/

\-sing cost]

N~ '

(6)

with masses M 1 and M2, we find the mass relations M~ cos2~ + M2 sin2~ = M2wsec20w ,

M1 sin2 +

cos2

=

M 2) s i n z ~ = M 2 ,

(7)

2 22 = MNM(v 2 2 sec 20 W - M 4 M1M Turning to the neutral current interactions, we note that the current coupling to Nu and N *~ contains only terms involving heavy fermions (b, r, etc.) and may be ignored in the present context. In terms of the currents coupling to Z ~*, J~ = CI'7~ [TL3(1 + 3,5)/2 -- Q sin20w] xP ,

,

(Z~) " 2 N3)MzN N3,*

=½(Z*',N')(MM22sec20w ; ~ ) ( N 3 Zu . ) '

(8b)

we may write the currents coupling to Z~ and Z~ as J ~ l = ( c°s~

sin~l(gsecOwJ~

J~/

cos~]\gNJ~q

\-sin~

]"

(9)

The effective low energy, neutral current interaction between fermions ~ and/3 is then (c~ g= 3)

- j~ur3 1,,2 + j2taJ~2ta/M2 . "L~eff1 "lp/~"l

(10)

Upon using the above mass relations and introducing the parameters

2 2 2 = ( H 2 + 21HI2)/(H2w+ 21HI2) m 2 =g2M~q/gNM~v (4)

(8a)

and to N~, J~ = q*3," [TNR3(1-- 3,5)/2 + TL3(1 + 3,5)/21g',

(i): (00 /0/),

~

M 2 = + gggNsec 0w IH[ 2

la(M2

for Higgs fields in the representations (2, 1) and (1,2) of SU(2)W X SU(2)N. In order to save the GIM mechanism which excludes flavor changing neutral currents arising from mixing in the fermion sector, it is necessary to avoid mixings of d and s with b and ~ and of e and /l with r and a. This may be accomplished by imposing a global symmetry corresponding to a conserved additive quantum number (+1 for b L and ~t, --1 for (n r r)TR, and (n o o) T and the (1,2) of Higgs, and zero for all other fields). To give masses to d, s, e and/~, we must also include a (2, 2) of Higgs fields with either of the

3 November 1980

(11)

7n2 = (gM2/gN M2) cos 0 w = + 2 [/4]2/(H2 + 2[HI 2), we have

in which M w is the mass of the usual We bosons, M2 = I g2(H2 + 2 IHI 2) = x/~g2/8GF,

(5a)

and

(12)

- (m2/mZ)(J~uJ~ + J~uJz~' ~") + m-2 J~uJ~N" ]

M N is the mass of the N and N* bosons,

M2N=lg2(" 2 + 21HI2),

~Oeef = (GF/X/~) 8 (1 -- rn4/m2) -1 t"Zu"Ztra ,eu

(5b)

It is evident that ~2 must lie in either the range 0 ~2 ~ 1 [case (i)] or the range - 1 ~ 1 is needed for phenomenological 341

Volume 96B, number 3, 4

PHYSICS LETTERS 100

consistency, while for m 2 >> 1 the SU(2)N bosons have only negligible effects on observed neutral current processes. Note that if the two SU(2)'s are embedded within a unified theory such as E 6 there is a renormalization group prediction g/gN ~ 1, so that m 2 is a good measure of M2 N/M2. The form of the effective neutral current interactions allows predictions for the neutral current couplings measured in several experiments to be written in the general manner f = (1

-

ff/a/m2)-I [a --

(ffl2/m2)b +m-2c],

3 November 1980

80

60 40

2O m2

(13)

in which f i s an observed coupling parameter and a, b, and c are linear functions of sin20 w. These may be obtained from the general forms of J~ and d~ together with the fermion transformation properties. Table 1 lists a, b, and c for the parameters ~ +~.~l~ and 3 + ~ measured in polarized electron-deuteron scattering (SLAC), the parameters g v and gA measured in electron-neutrino scattering, and the parameters t~,/3, 7, and 5 measured in neutrino-nucleon scattering. Also given are the observed values of these parameters and their one standard deviation errors o [5] (see also ref. [3] and references therein). For comparison with the observed values, we compute the quantity chi-squared for the fit as a function of sin20w, m 2, and r~ 2, The best fit is obtained for sin20 w = 0.235, m 2 = 8, and tX 2 = 0.1, while Xmin 2 = 3.1 for five degrees of freedom (DOF) is a measure of the goodness of the fit. In contrast, the best fit to the one parameter standard model occurs for sin20 w = 0.22 with X2min = 6.8 for seven DOF. Fig. 1 shows the best fit along with contours of X2min + 1 (representing roughly l o errors in meaasurement of sin20w, m 2, and

/

/ :

y

sin28w,.22 sin28w,.24 sin20 w ,.26

® Best Fit

sin28w-235 2"

m2 "8 ~2. o.I

X2min• 3,1 for 5 DOF ]

I

-I0 -3

I

I

I

-.6 -.4 -.2

0

t

I

L

I

.2

.4

.6

.8

1.0

~2

F~g. 1. Best fit for arbitrary sin20w, along with contours of ×min + 1 (representing roughly le errors) for various values of sin20W. /~2) in the m 2 - ~ 2 plane for various values of sin20 w and yields roughly sin20 w = 0.235 4--0 . 0 2 5 , m 2=10+4,

(14)

~2=0.1 +0.4.

Table 1 Experiment

f

eD (SLAC)

c~+ ~ q ,

a 1_

+ ~" ev

gv gA

vN

b

--1

- ~ + ~ 20s m.

- 1 + 4 sin20w

- ~2

- 5 + 2 sinaOw 1 - 3

-sin20w 0

1 - 2sin20w # ~"

1

- ] sin20w

0 342

+ ~-sin20w

-

2

0w

~_sin20w

-½ + 2sin20w -½ - 5 + ~sin20w

-5

fobserved

c +~1 1

-0.60 ±0.16 0.31 ±0.51

5 1 3

0.00 ±0.18 -0.56 ±0.14

1

0.597±0.055 0.935±0.050 -0.254±0.082 0.127±0.083

-3 - 31 5

Volume 96B, number 3, 4

PHYSICS LETTERS

The second o f these corresponds to MN/M w ~ 3. The existence of a lower bound on m 2 is hardly surprising. The presence of an upper limit (due largely to the lower bound on ~ and upper bounds on/3 and 3') is both less expected and extremely interesting• The source of our elation with the result m 2 ~ 10 is that kaonic CP violation in this model originates in the relative phase of the W+W- and NN* exchange box diagrams and is therefore proportional to m -4. Thus m 2 10 is entirely consistent with the observed size of CP violation• This success could be subject to a further test in the near future• Provided that the mixing of r and a relative to e and/~ is not too large, interference between W mediated and N mediated r decays should lead to violations of e-/2 universality at the level of m - 2 (the e - branching ratio should be larger than the /a- by ~10%). Another point concerns rh 2. Our bounds on N2 favor small I/4]2 setting the l o limit IHI 2 < ½H2 w and suggesting that ZU-N~ mixing should not be substan-

3 November 1980

tial. We also note that the data do not distinguish between the cases (i) and (ii) for the (2, 2) vev. A more interesting question is that of the assignment of the nrL. If nrL is a singlet under SU(2)W X SU(2)N , the nrL--nrR mass term is generated by H w, while, if nrL is part of a (1,2), it is generated by IHr 2. Thus, if~) denotes the nrL--V~ mixing angle, we expect tan ~ H N / H w in the former case and tanq~ ~ H N / I H [ in the latter. The rate of ~"decay in this model is suppressed by ~cos2~ compared to the prediction for the standard model. Thus tan q~cannot be too large. Although the bound [HI 2 < ~ H 2 does not conclusively distinguish between the two cases, it tends to favor the singlet assignment. As a note of caution to temper our enthusiasm, we present fig. 2 which depicts contours o f X2in +4 (representing roughly 20 errors). At this level of confidence in the data, a wide range of the parameter space is allowed, particularly in the large m 2 region. 100

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- _ _

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,,, ooo

sin2Ow 18 sin&Ow •..20

~

sin2O w

-~-

sin2Ow-24

....

sin2Ow-.26

® Best Fit m2- B

]0- errors ---

~L-O.I

20-

errors

X~min-4,2 for B OOF

I

I-

I

I

I

-.2

.2

.4

.6

.8

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I

1.0 -.B

-.6

I -.4

t -2

L

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I

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.2

.4

.6

.8

].0

1.0

N2 2 Fig. 2. Contours of xmin + 4 relative to the best fit of fig. 1 (representing roughly 2a errors) for various values of sin2ow .

Fig. 3. Best fit for the theoretically expected value sin20w = • 2 +2 '~• 0.205, along wzth contours of ×min I and Xmin + 4 relatlve to this best fit (roughly la and 2a errors respective/y). 343

Volume 96B, number 3, 4

PHYSICS LETTERS

A final point concerns sin20 w. Upon adopting the theoretically expected value sin2'0W = 0.205, the best fit occurs for m 2 = 8 and ~ 2 = - 0 . 1 with X2in = 4.2 for six DOF. Fig. 3 illustrates the contours of X2in + 1 and X2in +4 and gives m 2 = 13 + 7.5 and ~ 2 = - 0 . 2 + 0.5. Although a lower X2 can be obtained for higher values of sin20 w, it is interesting that both the best fit X2 for sin20w = 0.205 and that for arbitrary sin20 w represent a confidence level of roughly 65% for acceptance. In contrast, the best fit to the standard model corresponds to a similar confidence level of only 50%. By way o f conclusion, we note that while it is not surprising that we can fit the data with this model, it is interesting that the fit gives MN/Mw in just the right range to explain the observed CP violation in K decays. Of course, the really crucial test of this model will be the observation of peculair semileptonic decays of the b quark. But we hope that we have convinced the reader that the model gives an attractive description o f neutral current phenomenology.

References [1] P. Langacker, G. Segr~ and H.A. Weldon, Phys. Lett. 73B (1978) 87, and references therein; F. Gtirsey, P. Ramond and P. Sildvie, Phys. Lett. 60B (1976) 177;

344

3 November 1980

F. Giirsey and M. Serdaroglu, Nuovo Cimento Lett. 21 (1978) 28; and to be published; Y. Achiman, Phys. Lett. 70B (1977) 187; J.D. Bjorken and K. Lane, Proc. Intern. Conf. Neutrino 77 (Elbrus, 1977); P. Minkowski, Nucl. Phys. B138 (1978) 527; Y. Achiman and B. Stech, Phys. Lett. 77B (1978) 389; H. Georgi and A. Pais, Phys. Rev. D 19 (1979) 2746; H. Georgi and M. Machacek, Phys. Rev. Lett. 43 (1979) 83; P. Cox, M. Claudson and A. Yildix, Harvard preprint HUTP-80/A013. [21 H. Georgi and S.L. Glashow, Nucl. Phys. B167 (1980) 173. [31 P. Langacker et al., Univ. of Pennsylvania preprint COO3071-243;J.E. Kim et al., to be published. [41 H. Georgi, in: First workshop on grand unification, eds. P.H. Frampton, S.L. Glashow and A. Yildiz (Math. Sci. Press, Brookline, 1980). [51 We use values quoted by J.J. Sakurai, UCLA preprint UCLA/80/TEP/5,to be published in: Proc. 8th Hawaii Topical Conf. on Particle physics (1979).