Neutral current couplings and SU(2) ⊗ U(1) gauge models

Volume 82B, number 1

PHYSICS LETTERS

12 March 1979

NEUTRAL CURRENT COUPLINGS AND SU(2)® U(1) GAUGE MODELS Matts ROOS Department of High Energy Physics, University of Ilelsinki, SF-O01 70 ttelsinki 1 7, Finland and Ilkka LIEDE Research Institute for Theoretical Physics, University of Helsinki, ttelsinki, Finland Received 15 December 1978

Combining neutral current (NC) data in a statistically correct way we determine: (i) the weak NC couplings in neutrinonucleon and -electron scattering independently of gauge models; (ii) the weak isospin T3 of the right-handed fermions UR, d R and e~, the Z mass, and sin20w in SU(2) ® U(1) models with standard left-handed fermion multiplet assignments and two Higgs doublets. Both determinations support the Weinberg-Salam model, although neither considerable right-handed fermion singlet-doublet mixing involving new heavy fermions can be excluded, nor the existence of a second Higgs doublet.

1. Introduction. The experimental information on neutral currents can be used to determine the NC coupling constants independently of gauge model assumptions, or to restrict the parameter space within a given gauge model. The first approach has been followed by several groups [ 1 - 5 ] who generally conclude that the data allow one unique set of coupling constants, denoted solution A by Sakurai [1]. This solution includes the standard WS-model (where WS stands for Weinberg, Salam, Ward, Glashow, Iliopoulos, Maiani, Bouchiat, Meyer, Higgs, Kibble, and many others). The second approach has been followed by groups who have tested models of the SU(2) ® U(1) gauge group [1,6] or models o f larger groups such as SU(2)L SU(2)R ® U ( I ) [7,8]. Also here the WS-model is clearly favored. The purpose o f this study is to follow both approaches: to determine the NC couplings independently of gauge models, and to confront a general set o f SU(2) ® U(1) gauge models (with conventional lefthand assignments) with NC data. Although it may seem that we are then repeating earlier analyses, there are two good reasons for doing just that. Firstly the data have changed in important respects during 1978. Secondly, all the earlier analysis [ 1 - 7 ] make incorrect

statistical inferences, their quantitative results being at best qualitatively correct. 2. Statistics. Let us digress here briefly into the theory o f statistics, to describe the four common errors that plague physicists. Firstly, the contour containing probability 3 (at/3 "confidence level") in the space of n free parameters can be approximated (true for normally distributed parameters) in a least-squares fit by the contour X2 = X2min + A(IG, n), where X2in is the least square, and A(3, n) is a constant [8,9]. Physicists customarily use A = 1 for l o contours (/3 = 0.683) and A = 2.7 for 90% contours (3 = 0.90) regardless of n, although this is correct only for n = I. In fact A(0.90,2) = 4.61, &(0.90, 3) = 6.25, A (0.90,4) = 7.78, and A (0.90,5) = 9.24. Secondly, the overlap region of two or more confidence bands or surfaces, each o f known confidence 3, is not a 3 confidence region. In some cases the overlap region o f two regions has confidence 32 , in other cases its confidence is larger than 3- To avoid this problem one should not consider overlap regions at all, but rather use the least-squares method. Thirdly, all probability distributions must be nor-

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Table 1 Experimental constraints Quantity (see refs. [2,8])

u ~ - d~ u2R + u2R u~ - u~ u ~ - u~ GE(0) GM(0) GA(0) Rinc Rinc rinc

Values from fits Value +-1o error

0.304+-0.016 0.029 -+0.009 -0.088 +-0.079 0.030-0.030 a) 0.5 +0.3/-0.5 a) 0.9 +-0.2 0.6 +0.1/-0.2 b) 0,48 +0.17 0.42 +0.13 1,31 -+0.38

Coupling constant fits

Fits in SU (2) ® U (1) parameter space

a,13, y, 6

c~,13,Y, 6 gv, gA

General model

Eqs. (11 ) and (12)

WSmodel

0.305 0.032 -0.091 0.023 -0.05 .0.98 0.62 0.41 0.30 1.18

0.306 0.030 -0.080 0.030 -0.10 1.05 0.62 0.42 0.30 1.15

0.304 0.030 -0.064 0.027 -0.03 1.05 0.63 0.42 0.30 1.12

0.306 0.034 -0.064 0.021 0.02 1.03 0.64 0.43 0.31 1.12

0.296 0.030 -0.060 0.018 0.03 1.05 0.63 0.42 0.29 1.11

o @~e-)

1.35 +-0.48 c)

1.61

1.64

1.54

1.51

o (7,,e-)

1.54 ±0.67 c) 7.6 _+2.2d) 1.86 +-0.48d)

1.78 8.1 1.65

1.77 8.0 1.64

1.44 6.9 1.33

1.33 6.4 1.22

O(Fee-) low en. O(7ee-) high en. Aed(O.21)/Q 2

(-9.5-+1.6) × 10 -s

-8.8 -212

Qw(Bi) Total x2/(x 2) sin20 w o

2.8/6

4.3/9

-8.5 -168 4.3/10 0.242 1.022

-7.7 -127 5.8/13 0.242 1.026

-8.0 -121 6.5/14 0.234 1

a) Asymmetric error, only downwards error is used. b) Asymmetric error, only upwards error is used. c) In units of 10-42Ev (cm2/GeV). d) 111units of 10 -46 (cm2). malized to unity within the physical region of the parameters. Contours extending into unphysical regions must be renormalized to meet this condition. Fourthly, the parameters under study are often strongly correlated, in which case it is very misleading to quote errors alone. Either one should denote the covariance matrix, or plot contours, or transform to less correlated parameters.

3. Coupling constant fits. The NC coupling constants in Sakurai's notation [1] are a, t3, 7, 6 , g v , g A, h v v , hAA, hvA, ~, ~, ~,, 8", and g. We determine first, a,/3, 7, and 6 using 10 experimental constraints. Next we determine g v and gA using 4 experimental constraints. Finally we determine a,/3, 7, 6, g v , and gA simultaneously using 15 experimental constraints and the model-dependent factorization relations [ 1,5 ] 90

= 2gAa/K, ~,=2gAT/g , K=I .

~= 2gv/3/~ ,

(la)

6=2gv8/~, (lb)

Assumptions (la) are true in all single-Z models, and assumption ( l b ) is true for instance in SU(2) ® U(1) models with the simplest Higgs structure. We use the same NC data as in our previous study [8] (including the "Note added in print"). The relations between experimental quantities and coupling constants are well known (see e.g. refs. [1,2,8]). The experimental values used are tabulated in table 1. (a) The a,/3, 7, 6-fit uses inclusive neutrino scattering on isoscalar targets, neutrino pion semi-inclusive scattering on isoscalar targets, elastic neutrino scattering on protons, and inclusive neutrino scattering on protons

Volume 82B, number 1

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12 March 1979 gA 0.2

t~

-q2

b Q2 i

,

(~

i

0.6 p

360 -0.2

i

\\\

24C

1//

Fig. 2. Solid lines: 90% contours from the fit described under 3(b). Broken line: 90% contour in the (gv, gA)-Subspace of the fit described under 3 (c).

12C 120

I

I

140

I

eL

I

160

Fig. 1. (a) 90% contour in the (~,/3, % 8)-space projected onto (0L, 0R)-space from the fit described under 3(a). (b) 71% contour in a subspace defined by (a+ (3)2 + (~¢+ g)2 = 2.25, as obtained from Abbott and Barnett [3]. The WS-model is indicated by an error bar.

Using the minimizing program MINUIT [10], the simultaneous fit yields the best values and 90% C.L. errors n ¢A+0.25 c ~ = u . a ~ 0.12,

1 n o +0.12 ~= ..vv_0.28,

O 99 +0.29 6 = --0.02-+0.34 3' - - - v ' " ~ - - 0 . 3 4 '

(2)

-

and the correlation matrix

~ a

/3 --0.79 -0.65 0.71

fl 0.75 -0.57

,,/

(3) -0.44

The values o f the experimental quantities at the parameter values (2) are tabulated in c o l u m n 3 of table 1. In fig. 1 we plot the 90% confidence region (contour a) in the two-dimensional space o f the parameters 0 L = arctan (UL/dL),

OR = arctan (UR/dR) ,

(4)

where the definition of u L, d L, UR, and d R in terms of a,/3, 7, and 8 is standard [2]. F o r comparison we also plot a c o n t o u r (b) calculated from n e u t r i n o exclusive pion p r o d u c t i o n * i. Obviously the exclusive pion data, which depend quite strongly on p a r t o n model assumptions, would n o t constrain the values of 0 L and OR very m u c h , if they were used. This picture thus justifies neglecting the exclusive pion data in the fits. (b) The g v , g A ' f i t uses vu, ~ , and Ue scattering on electrons, altogether four constraints. As is well k n o w n , there are four ambiguous solutions in the (gv, gA)-P lane Contrary to what the m a n y analyses based o n overlapping confidence bands maintain, n o n e of these solutions can be unambiguously selected (at the 90% confidence level). In fig. 2 we show the 90% confidence contours which form three separate regions. (c) The factorization-dependent a,/3, T, 8, g v , gA -fit uses all the data already m e n t i o n e d , and in addition

4:1 Abbott and Barnett [3] give three contours of the (0L, OR)distribution, conditional on fixed (u~. + ULi ~2 d/2 and (u~ + d}/) 1/2, with A = (1.6) 2 ; thus the confidence is ¢~= 79%. The three contours correspond to (u~ + d~) 1/2 being taken at its central value and at its -+1.6 cr values, thus this confidence is t3 = 90%. Combining the three contours we get the contour b in fag. 1. It represents a section through a 3-dimensional probability distribution, conditional on 2 , j2xl/2 uL 1- UL) having the fixed value 0.53, and containing a total probability of/] = 0.79 X 0.90 = 71%. 91

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0.3(

0.75 0.2~

'~

0.50 0.25

0.26 sinsew 024

tGev]

o

-02

0.2,1

-O,~

o.2o I

-0,7~

Fig. 3.90% contour in tire two-parameter SU(2) U(1) model defined by eqs. (11) and (13). (sin20W, H)-relation in the WS-model is indicated by a straight line.

Fig. 4. Muon forward--backward asymmetry A FB as a function of cms energy for the general 5-parameter SU(2) U(1) model and the WS-model (the shaded region). The curves correspond to maximum and minimum asymmetry over the 90% confidence regions in the parameter space.

polarized electron scattering on deuterium [11]. This experilnent selects among the g v , gA "s°luti°ns, allowing only the bottonr contour in fig. 2. We obtain the parameters and 90% C.L, errors

[T3(eR)[ lie in the continuous range [0,1/2]. Note that non-half-integral effective T 3 values are obtained when mixing occurs, as for example with the assignment

0!70

0 I7 5

Q80

--

H

N AN+0.23 c~= v.vv_0.17,

n Oq +0.18 ~ = v.-~_0.28,

(

N 97 +0-41 6 - O f~A+0"31 7 = -- v ' ~ ' - 0 . 3 3 ' - .... -0.49'

(5)

E° ) , e - sin ~p+ E - cos ~p R

By "effective" T3(e ~ ) we then mean 1

-no~

+0"20

=

gv-~'~"-0.22'

gA

The correlation matrix is given by:

7 6

gv gA

-0.96 -0.92 0.86 0.37 0.49

7

0.93 -0.83 -0.34 --0.45

-0.73 -0.29 -0.31

~

gv (6)

0.34 0.45

0.22

The values of the experimental quantities at the parameter values (5) are tabulated in colmnn 4 of table 1.

4. Tests of SU(2) ® U(1). Let us consider models in which the left-handed fermions are assigned to the conventional doublets. The right-handed fermions UR, dR, e R,/IR, ... can then be assigned to singlets, doublets, etc. We shall restrict ourselves to assignments where the effective values of I T3(UR)I ,IT3(dR)I , 92

,

9

T3(eR) = -- ~ sxn-tp .

~q+0.17 --~'~-0.17" n

~3

(e-c°s~°-E-sin~)R.(7)

(8)

The price o f continuous T 3 would here be the introduction of heavy, as yet unseen fermions with non-diagonal neutral currents [12]. Note that if mixing angles of the order of magnitude of the Cabibbo angle are natural, T3(e R) values as small as 0.025 are physically interesting. The NC data may be used to test for deviations of T3(e~) , T3(UR) , and T3(dR) from zero (the WSmodel). We can also use the NC data to test for the presence of a second complex Higgs doublet, regardless of constraints imposed by the smallness of the ~t -+ e 7 branchin~ ratio [13]. A suitable parameter is then M2

P =-M2

_(

37.286GeV

12

\ M z sin OWCOSOwl .

(9)

In single-Higgs models p = 1, in all SU(2) ® U(1) models with T3(UL) = 1/2, one has P = ~ [1]. Altogether we then have 5 free parameters: sin20w, T3(UR) , T3(dR) , T3(eR) , and p o r M z . The unique solution in this parameter space when fitted simulta-

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neously to all the 15 NC data in table 1, yields the following parameter values and 90% C.L. errors: sin20 w = 0.24+0.15,

T3(UR) = 0.00+0.18,

n c~a+0.19 T3(dR) = - v . v _ _ 0 . 2 1 ,

t) o8 +0.23 T3(eR) = v.v__0.20,

MZ = ~st~ n~+56 18 GeV,

, n-~ +0.25 p = *.,oz 0.21.

(lO)

The values of the experimental quantities at the parameter values (10) are tabulated in column 5 of table 1. Two conclusions can be drawn from this fit. Firstly, the data strongly favor the standard assignment T3(UR) = T3(dR) = T3(eR) = 0 .

(11)

Secondly, the correlation between sin20w and M z (or/9) is - 9 9 % (+ 99%). Although we can see no theoretical reason for this, the data strongly favor a relation of the form M 2 = (1/H)M 2 .

(12)

With this relation, H and sin20 W turn out to be completely uncorrelated. The value of H in the WS-limit obviously is H = cos20w = 0.766 [8]. A two-parameter fit of sin20 W and H with the constraints (11) and (12), yields the results and 90% C.L. errors +0"03 ' sin20w = n. . .~A . -0.04

H = 0.78+0.04

(13)

the correlation being 0.12. In fig. 3 we plot the 90% confidence contour of this fit, with the WS-model indicated. The values of the experimental quantities at the parameters (13) are tabulated in column 6 of table 1. Finally we fit the WS-model to the 15 data, with the result already published [8] sin20 w = 0 . 2 3 4 + 0 . 0 1 3 .

(14)

In (14) the error is exceptionally a l o error (the 90% error being 1.64 times as large). The values of the experimental quantities for this fit are tabulated in column 7 of table 1. Fig. 4 shows predictions for the muon f o r w a r d backward asymmetry in the reaction e÷e - -+ ~+gtwith unpolarized electron beams. The two bands shown are the 90% confidence bands in the general 5-parameter SU(2) ® U(1) model with the parameter values (10), and in the WS-model with 0 w as in eq. (14). Note that the general model allows for a practically zeroasymmetry solution.

12 March 1979

5. Conclusions. We have determined the neutral current coupling constants c~,/3, 7, 6 independently of gauge models, from neutrino neutral current scattering on nucleon targets. The solution is unique and corresponds to Sakurai's solution A [1]. We have also determined the non-unique 90% confidence regions allowed to the coupling constants g v and gA by neutrino-electron scattering. Under the factorization assumption (1) we determine c~, 13, 7,/~, g v , and gA simultaneously from all neutrino scattering data and from polarized electron scattering on deuterium. This solution is unique, and agrees well with the Weinberg-Salam model. We have given the ranges of the parameters of a general SU(2) ® U(1) model, consistent with the neutral current data. We conclude that although the data favor the assignment of right-handed fermions (u R, d R, e~.) to singlet representations, and rule out doublet assignments by more than 99% confidence, the right-handed singlet-doublet mixing involving heavier fermions is still possible with large mixing angles. We also conclude that although the simplest Higgs structure (as in the WS-model) is favored, neutral current data also allow a second complex Higgs doublet within the 90% confidence limits. In the case of two Higgs doublets, the Z-boson mass turns out to be proportional to the W-boson mass, the proportionality factor being independent of sin20 w. The total set of data yield

sin20 w = 0.234+0.013

(68% C.L.),

when fitted to the WS-model. We thank M. Chaichian for having drawn our attention to ref. [13]. We are indebted to J. Maalampi for technical assistance.

References

[1] P.Q. Hung and J.J. Sakurai, Phys. Lett. 63B (1976) 295; 69B (1977) 323; 72B (1977) 208; J.J. Sakurai, preprints UCLA/76/TEP/21 (1976), UCLA/ 77/TEP/15 (1977), UCLA/78/TEP/9 (1978), UCLA/78/ TEP/18 (1978). [2] L.M. Sehgal, Phys. Lett. 71B (1977) 99, and Aachen preprint PITHA 102 (1978). [3] L.F. Abbott and R.M. Barnett, Phys. Rev. Lett. 40 (1978) 1303, and preprint SLAC-PUB-2136. 93

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[4] G. Ecker, Phys. Lett. 72B (1978) 450; P. Langacker and D.P. Sidhu, Phys. Lett. 74B (1978) 233, and preprint BNL-24393 (1978); E.A. Paschos, preprint BNL-24619 (1978). [5 ] M. Konuma and T. Oka, Prog. Theor. Phys. 60 (December 1978); R.M. Barnett, preprint SLAC-PUB-2183 (1978). [6] J. Bernab6u and C. Jarlskog, Phys. Lett. 69B (1977) 71 ; V.S. Mathur and T. Rizzo, Phys. Rev. D17 (1978) 2449; Univ, of Rochester preprint C00-3065-204, UR-662 (1978). [71 A. De Rfljula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 3589; R.N. Mohapatra and D.P. Sidhu, Phys. Rev. D16 (1977) 2843; J.C. Pati, S. Rajpoot and A. Salam, Phys. Rev. D17 (1978) 131.

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[8] I. Liede, J. Maalampi and M. Roos, Nucl. Phys. B (1978). [9] W.T. Eadie, D. Drijard, F.E. James, M. Roos and B. Sadoulet, Statistical methods in experimental physics (North-Holland, Amsterdam, 1971); F. James, Interpretation of the errors on parameters as given by MINUIT, CERN computer centre program library (1977), unpublished. [10] F.E. James and M. Roos, Comput. Phys. Commun. 10 (1975) 343. [11] C.Y. Prescott et al., Phys. Lett. 77B (1978) 347. [12] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [13] J.D. Bjorken and S. Weinberg, Phys. Rev. Lett. 38 (1977) 622; S. Weinberg, Proc. XIX Conf. on High energy physics, (Tokyo, 1978).