To the standard model we steer

Volume 255, number 1

PHYSICS LETTERS B

31 January 1991

To the standard m o d e l we steer Giampiero Passarino Dtpartzmento d~ Ftstca Teortca, Untversttgt dt Tormo, and INFN, Seztone dt Tormo, 1-10125 Turm, Italy

Received 2 October 1990

Recent measurements of the Z° mass and partial widths from LEP, as well as of the ratio Mw/Mz from Pl) colhders, are used to constraint the unknown parameters of the standard model of electroweak Interactions through their effects on ra&ative corrections. Within the minimal standard model it followsthat mtop= 151 GeV and mt< 203 (215 ) GeV at 90 (95)% confidence level. Particular emphasis is devoted to a possibledeviation from minimahty. The bound can be modified Ifthep-parameter is allowed to vary but there are no addmonal radmtive corrections coming from the Hlggs sector one obtains mtop=123 GeV In this case the one-sided upper hmit is mtov<298 GeV at 90% confidence level and po= l 0026-t-0.0086-0.0289 also at 90% confidence level. More general posslbilmes, depending on the structure of the Hlggs system and leading to a large range of values for mtop,are also analyzed.

It has become c o m m o n practice to start a paper on radiative corrections by stating that they have entered a new era of high precision. Motivated by the growing set of data made available, in the recent literature, by the four LEP Collaborations on the properties of the Z ° [ 1 ], together with the results from the UA2 a n d C D F Collaborations on the W [ 2 ], we have u n d e r t a k e n the project of comparing experimental data points with the predictions of the standard model of electroweak Interactions. No matter which r e n o r m a l i z a t i o n scheme is used we always start from three data points. O u r scheme has been described in detail in the literature [ 3 ] a n d assumes the precise knowledge of a , GF a n d Mz. There ts no ad hoc definition of the weak mixing angle a n d the p-parameter is not one by construction. After that we are left with two unknowns, namely the top quark mass a n d the Higgs boson mass, not to m e n t i o n as. Therefore we are not really in a position to test the standard model but rather we try to constramt the u n k n o w n masses by their effects on radiative corrections to measurable quantities [4]. Perhaps a precise measure of the partial width Z°--,bl3, being to a large extent i n d e p e n d e n t from mr, will represent a test of the m i n i m a l standard model if the top is not discovered by CDF. We have c o m p u t e d all one-loop radiative correc-

tions and the corresponding formulas have been coded in the F O R T R A N program Q F O R M F [ 5 ]. Next the W mass as well as the Z ° partial widths have been analyzed, including all computable higher order effects, and numerical results are available through the computer program LOOPLIB [ 5 ]. The complete analysis will be presented elsewhere [ 6 ]. Here we only m e n t i o n that the lagrangian of the standard model contains three parameters, the bare coupling constant g, the bare W mass M and the bare weak mixing angle so= sin 0w, not to be confused with the experimental data point s 2 = 1 - M w2/ M z . 2 The bare parameters are then related to a , G F a n d Mz as measured in T h o m s o n scattering, ~t-decay and from the position of the zero of the real part of the inverse Z ° propagator. Every measurable quantity is then expressed in terms o f g ( a , GF, M z ) , etc .... and it is finite by virtue of the renormalizability of the theory. In solving for g, M and so we automatically take into account the proper r e s u m m a t l o n of the large logarithms which appear in the photon self-energy [ 7 ]. Also terms proportional to ( o t m 2 / M 2 ) 2 [7] are included in our formulas. They appear in the expression for the reducible as well as the irreducible twoloop self-energies. In the Z°--,bb partial width there are addittonal contributions proporttonal to a m 2 / M 2 coming from vertex corrections. In this

0370-2693/91/$ 03.50 © 1991 - Elsevier SciencePublishers B.V. ( North-Holland )

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Volume 255, n u m b e r 1

PHYSICS LETTERS B

particular case we have omitted terms quartic in the top mass since the two-loop irreducible vertices are still to be computed. There are already several examples of global fits to the minimal standard model [ 8 ]. In our fit to the data we envisage three different scenarios. F~rst we assume the validity o f the minimal standard model. Here the Higgs comes as a complex isodoublet and the bare p-parameter, often called Po, is just one [ 9 ]. A one-parameter fit to m t was performed to s 2 = 1 - M w2 / M z ,2 1-'z and Rh=Fh/1-'~. The experimental results are for LEP average:

for UA2 + CDF:

in fig. 2. Accordingly we derive the limits at 90% CL shown in tables 1 or 2. For M z = 9 1 . 1 7 2 GeV, m H = 1 0 0 GeV, m t = 1 5 1 GeV and c~s= 0.115 we obtain F z = 2486 M e V , F~= 83.5 M e V , F h = 1737 M e V , Rh=20.80,

4

Mz=91.172+0.031 GeV,

F z = 2 4 9 8 + 2 0 MeV,

31 January 1991

Rh=20.89+0.27,

......

i .........

, .........

i .........

i .........

i .........

i .........

I,,A

.....

35

S~v=0.225 + 0.007.

3

A penalty function was introduced in the fit, X2=X2+ ( m t - 1 0 9 ) 2 / 4 0 0 for m t < 1 0 9 GeV which reflects the lower limit set by CDF, mr> 89 GeV [ 10 ]. The Q C D corrections to the hadronic Z ° width have been mcluded up to terms O (c~3/ n 3 ) according to ref. [ 11 ] and we used c ~ ( M 2) =0.115 _+0.016 [ 12 ]. For a critical discussion we refer to ref. [ 8 ]. The X2 distributions a, functions of m t for different choices o f mH and cts= 0.115 are given in fig. 1, while those for different choices o f as and mH = 100 GeV are shown

a. = 0.131

',

25

,"

2

15

1

05

.......

0 80

i .........

100

i .........

120

i .........

140

i .........

160

, .........

180

i .........

200

, ........

220

240

mtop(GeV) 4

,,, . . . . . . i . . . . . . . . .

35

i .........

"

,

i .........

i .........

, .........

=. . . . . . . . .

__

m H = 1 0 0 GoV

.....

m.-- OOGeV

/

i ....... i

Fig. 2 One-parameter fit to mr. T h e x 2 distributions as functmns of m, for different choices of as and rnH = 100 GeV

J/

:

Table 1 Limits ( m GeV) for mr.

2 5

ot~

0 115

15

m n (GeV) 100

500

1000

151 + 6 4

167+60-73

175+58-76

1

Table 2 Limits ( m GeV) for mr.

0 5

0

........

80

i .........

100

i .........

120

i .........

140

i .........

160

i .........

180

i .........

200

i ........

220

240

mH (GeV)

mtoplGeV) Fig. 1. One-parameter fit to mr. The X2 distributions as funcuons of rnt for different choices of rnH and cq = 0.115.

128

100

Ct, 0.099

0.115

0.131

162+62-70

151 + 6 4

139+66-56

Volume 255, number 1

PHYSICS LETTERS B

Fb6/ Fh =0.218 ,

31 January 1991

450

''1

.........

.

(a) i

S2 =0.22499.

390

One-sided upper limits for mt give mt <203 GeV at 90% CL,

X2=Z2m,, + 1.7,

mt <215 GeV at 95% CL,

X2=g21n +2.7.

330

==

Always within the mlntmal standard model two-parameter fits to m t and mH were performed to s 2 , Fz and Rh. The best fit is for m t = 150 GeV and mH = 61.8 GeV. An addiuonal penalty function was included, X2=X2+ ( m n - 6 2 ) 2 / 4 0 0 , to reflect the constraint mH>41.6 GeV set by the ALEPH Collaboration [ 13 ]. The minimum o f x 2 lives in a very fiat valley, due to the well-known behavior of the one-loop radmtive corrections in the limit of a large mH [ 14 ]. The 68% CL contour (X2=X~,, + 2.30) in the m t - m H plane is shown in fig. 3a, whereas the Xz distributions as functions of mn for different choices of mt are given in fig. 3b. To gave an upper limit for mn is meaningless in this situation where the 1 - a contour extends well beyond the region of applicabdlty of perturbation theory. For Mz=91.172 GeV, m n = 6 1 . 8 GeV, rnt= 150 GeV and o~s= 0.115 we obtain F z = 2497 MeV,

AX2= 2.30 contour

270

~

210

150

90

........

30

i .........

80

i .........

120

i .........

160

i.,

200

280

240

mtop(GeV) )

i

)

i

i

i

i

ii

i

i iJ

I

i

i

i

I

i

i

i

i

ii

i

i

i i

15

~

I

F~= 83.9 MeV, Fh = 1745 MeV, Rh=20.79,

O5

l"b~/Fh=0.216 ,

S2W=0.22475. Finally, the two-parameter fits to mt and as were performed to S~v, Fz and Rh. For ran= 100 GeV the best fit gives mr= 145 GeV and o4=0.122. The minimum o f x 2 for each fixed value of mt is given in fig. 4a. A portion of the 68% CL contour (X 2=Xm,n z +

2.30) in the m t - ~ plane is shown in fig. 4b. There is another scenario beyond the minimal standard model. Actually we have no fundamental reason to assume that po = 1 and the limits on m t can very well be evaded if we change the structure of the Higgs sector [ 15 ]. First we consider Po as just another free parameter in the theory with the assumption that whatever mechanism generates a Po shghtly different from unity it also gives negligible extra ra-

0

10'

i

10= m,,g0,(GeV)

i

103

Fig. 3. (a) Two-parameter fitto mt and m n m the m m l m a l standard model. The 68% C L contour m the m r - m a plane. (b) Twoparameter fitto mt and m . m the m m l m a l standard model. The X2 distributions as funcUons of mH for different choices of mr. The sohd hne shows mr= 120 GeV, the dashed hne shows mr= 140 GeV, the dash-dotted hne shows mt = 160 GeV and the dotted line shows m t = 180 GeV

diative corrections. The two-parameter fits to mt and Po that we have performed to S2w, Fz and Rh give mt= 123 GeV and Po = 1.0026 and consequently F z = 2499 MeV, 129

Volume 255, number 1 1

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, .........

PHYSICS LETTERS B i .........

, .........

, .........

, .........

31 January 1991

The m i n i m u m o f X 2 for each fixed value o f Po is shown in fig. 5a, the 68% CL c o n t o u r ( X 2 = X 2 m

i ........

(a)

+ 2 . 3 0 ) in the mt-Po plane in fig. 5b. By v a r y i n g #o we o b t a i n that larger values f o r mt are allowed. T h e

O8

one-sided u p p e r l i m i t f o r mt is

mt < 298 G e V at 90% CL,

08 _= e4;:,
X2 =X2,.

+ 1.7.

Alternatively the p r e d i c t i o n for Po is O4 . . . . . . . . .

O2

0

i

. . . . . . . . .

i

. . . . . . . . .

i

. . . . . . . . .

i

. . . . . . . .

(a]

OS

........ i ......... i ......... =......... I ......... i ......... i ......... t ......... 100 110 120 130 140 150 180 170 180 mtop(GeV)

OS E ot E 04

0 131

............. (b)! 02 0 123 ,

. . . . . . .

990

AX2

2 3 0 contour ~. . . . . . . . .

1 020

L,,,, 120

.....

i ......... 180

, ....... 200

I ......... 240

~

.........

1 008

1 010

i

. . . . . . . . .

i . . . . . . . . .

I . . . . . .

2

' ' j . . . . . . . . .

Ax = 2 . 3 0 ( b )

1 004 280 1

0

996

i

0 992 0 988 0 984

Fh = 1746 M e V ,

o.o 8'0.......

130

. . . . . . . . .

1 008

F~= 83.9 M e V ,

S2 = 0 . 2 2 5 3 5 .

i

1 002

1 012

Fig. 4. (a) Two-parameter fit to mt and a, m the minimal standard model. The m[mmum ofz 2 for each fixed value of mr. (b) Two-parameterfit to mt and a, m the minimal standard model. Portions of the 68% CL contour m the mr-a, plane.

rb6/Fh=0.217,

. . . . . . . . .

1 016

mtop(GeV)

Rh=20.81 ,

i

0 998

--

oloi . . . . 80

. . . . . . . . .

Po

0 115 i, ,

008g

i

0 994

;;o .......

;;; .......

=;'o .......

;i; ........

,,o

mtop(GeV)

Fig. 5. ( a ) T w o - p a r a m e t e r fit to mt a n d P o m the s t a n d a r d m o d e l w i t h o u t extra Hlggses. The m i n i m u m o f x 2 for each fixed value ofpo. ( b ) T w o - p a r a m e t e r fit to mt a n d Po in the s t a n d a r d m o d e l w~thout extra H~ggses. The 68% C L c o n t o u r m the mt-po plane.

Volume 255, number 1

PHYSICS LETTERS B

Po = 1 . 0 0 2 6 + 0 . 0 0 5 8 - 0 . 0 1 5 7 at 68% CL, 2 2 X =Xmm + 1 ,

Po = 1 . 0 0 2 6 + 0 . 0 0 8 6 - 0 . 0 2 8 9 at 90% CL, 2 2 X =•mln +

2.7.

The possibility of having Po¢ 1 deserves few comments. Introducing a Po in our formulas without specifying the actual mechanism which makes Po ~ 1 can only be a crude approximation. The value of Po is strictly related to the structure o f the Higgs sector and more generally reflects any deviation from minimality. Therefore a fit to mt and Po alone should not be allowed because the same structure which renders Po a free parameter could also introduce new degrees o f freedom and possibly new one-loop effects which m~x thoroughly with those arising from a large m t [ 16 ]. Here we are really waiting for some experimental evidence. For instance if a top quark is not discovered with a mass less than approximately 200 GeV then the minimal standard model is in danger and some deviation should be advocated. To give an illustratmn of this possibility we have parametrized a smooth deviation from minimality by adding to the standard model a Higgs triplet w~th a small vacuum expectation value but large mass splittings among its members [ 16 ]. This structure is not meant to represent a serious belief but only to demonstrate that the primary task o f all experiments should really be to prove the correctness o f the AJ = ½ Higgs rule, or to disprove it on the basis o f a self-consistent extension o f the standard model. We represent the I = 1 Higgs field by a complex symmetrical tensor o f rank 2 and introduce the parameter

~=2x/~G~/2(I=l),

e<
po =

1 1 +~2

"

The H~ggs sector is now given by its minimal version plus two neutral particles V ° and U °, a charge-one particle U -+and a charge-two particle R -+±. All masses in the Higgs sector are free parameters except for mR for which we derive m 2 = 2 m u2- m v o 2 +O(E2). Radiative correctmns are now computed by including these extra contributions, in the limit where we neglect terms of order gEl.2. Large mass splittings give a substantial contribution to Ap, mixing their effects with those o f a Po ~ 1 and o f a large rot. For no better

31 January 1991

reason than illustrating the p h e n o m e n o n we have chosen rnv = 200 GeV while the remaining masses are fixed to 100 GeV. For as = 0.115 the best fit was found for m r = 109 GeV and e=0.034, corresponding to po=0.9988. Portions o f the 68%(90%) CL contour (Z2=Z2mln +2.30(4.61 ) ) in the mt-~ plane are given m fig. 6a. For rnt= 109 GeV, m u = 2 0 0 GeV and Po = 0.9988 we obtain Fz = 2498 M e V , F~ = 8 3 . 9 M e V , Fh = 1746 M e V , Rh=20.83,

Fb6/Fh=0.217,

s2=0.22519.

If instead V o has a large mass compared to the other scalars then the best value for Po requires an imaginary e. Thus in this case the best approximation to the data is given by a triplet with no v a c u u m expectation value and mt will be directly correlated to the scalar masses. In fig. 6b we present the X2 contour plot for mvo --400 GeV. Clearly we need more and more accurate data to disentangle new physics at the tree level from radiative effects, if indeed some incompatiblhty will arise inside the minimal standard model. As mentioned at the beginning there is an important measurement to be considered among the others, namely the decay Z ° - , b b . In Fb6/F h the Z ° propagator with its radiative corrections drops out and moreover the fermions receive a mass only from the I = ½ Higgs scalars and the I = 1 Higgs particles give no contribution to the vertices in Z°~f} ". A high precision measurement o f Fb~/Fh could therefore separate vertex corrections from the so-called obhquecorrections, potentially isolating the mt effect. Even if we obtain a precise constraint on mt it will be very difficult to d~sentangle Po from radiative corrections coming from the I = 1 Hlggs sector. Admittedly this s~tuation may look rather unnatural but the Higgs triplet is meant only to introduce a qualitative example and the main point under discussion is that we cannot safely make any assumption about the Higgs sector w~thout the assistance o f more experimental data. In order to understand this possibility we could use, as an example, the data from the D E L P H I Collaboration [17], namely Fb6/Fh=0.209--+0.030--+ 131

Volume 255, number 1 0 20

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PHYSICS LETTERS B

i .........

i .........

i ...........................

i ..................

In conclusion we add s o m e c o n s i d e r a t i o n s about

) .........

la)

the O(ot~) t e r m in

///////

0 08

0 O0 100

120

140

160

180

220

200

240

280

280

mtop(GoV)

020

.........

i .........

i .........

I .........

i .........

I .........

i .........

I .........

Ib) 016

008

0 04

000

........ '......... '. 150 175 200

.

.

.

225

.

.

250

.

.

.

275

mtop(GeV)

.

.

300

'.........

325

350

Fig. 6. (a) Two-parameter fit to mt and e=2v/2 GV2 (1= 1 ) m the standard model with an extra Hlggs triplet and mu= 200 GeV The dash-dotted (solid) line shows the 68% (90%) CL contour m the mt-~ plane. The dotted hnes show the Z2 contours from 0.1 to l 1. (b) Two-parameter fit to mt and E= 2x/~ G ~j2 ( i = 1) m the standard model with an extra Hlggs tnplet and my0 = 400 GeV. The hnes show the X2 contours from 0.92 to 3.22 m steps of 0.23.

0.021, or those from the L3 Collaboration [18], namely Fb6/Fh= 0.204 + 0.014 + 0.014. However, the precision is not high enough to improve from the previous analysis. 132

is s o m e u n c e r t a i n t y

References

0 04

80

~QCD"T h e r e

for the coefficient and we have repeated the fit to mt in the minimal standard model without it. The best fit goes from 151 GeV to 158 GeV, while now mt< 220 GeV at 95% CL, instead of 215 GeV. From this we can estimate the uncertainty.

016

012

31 January 1991

[ 1 ] E. Fernandez, talk 14th Intern. Conf. on Neutrino physics and astrophysics "Neutrino 90" (CERN, Geneva, 1990), to appear, ALEPH Collab., D Decamp et al., Phys. Lett B 231 (1989) 519, B 234 (1990) 399, B 235 (1990) 399, CERN preprmt CERN-PPE/90-104 ( 1990); DELPHI Collab., P Abreu et al., Phys. Lett B 231 (1989) 539; B 241 (1990) 435, L3 Collab., B. Adeva et al., Phys. Lett. B 231 (1989) 530; B 236 (1990) 109; B 237 (1990) 136; B 238 (1990) 122; B 249 (1990) 341, OPAL Collab., M Z. Akrawy et al., Phys. Lea. B 231 (1989) 530, B 235 (1990) 379, B 240 (1990) 497, B 247 (1990) 458. [ 2 ] D Fraldevaux, talk 14th Intern. Conf. on Neutnno physics and astrophysics "Neutrino 90" (CERN, Geneva, 1990 ) to appear, UA2 Collab., J. Incandela, presentation DPF meeetmg of the American Physical Society (Houston, TX, January 1990), CDF Collab. S. Errede, presentation DPF meeting of the American Physical Society (Houston, TX, January 1990). [ 3 ] G. Passarlno, in: Radiative correctmns for e+e - colhsmns, ed J K. Kuhn (Springer, Berhn, 1989) p 179, G Passanno, m. QED structure functions, ed. G. Bonviom, AlP Conf. Proc No. 201 (ALP, New York, 1990) p. 132, G. Passanno and R. Plttau, Phys Lett. B 228 (1989) 89, G. Passarmo and M Veltman, Phys. Lett. B 237 (1990) 537 [4] U. Amaldl et al., Phys. Rev D 36 (1987) 1385, A. Blondel, in. XXII Rencontres de Monond, ed. J Tran Thahn Van (Edmons Frontl~res, Glf-sur-Yvette, 1987), p 3; CERN prepnnt CERN-EP/87-174; G. Costa et al., Nucl. Phys. B 297 ( 1988 ) 244; J Elhs and G L. Fogh, Phys. Lett B 232 (1989) 139; D. Hmdt, Proc. Europhyslcs Conf. on High energy physics (Madrid, September 1989 ), P. Langacker, UPR preprmt 0400 T (September 1989); A. Blondel, CERN preprmt CERN-EP/90-10 (January 1990). [ 5 ] G Passarlno, QFORMF, a program for the computaUon of one loop form factors, unpublished; G. Passanno, LOOPLIB, a program package for the calculation of electroweak observables, unpublished.

Volume 255, number 1

PHYSICS LETTERS B

[6 ] G. Passarlno, m preparauon. [7] M Consoh and W. Holhk, in Physics at LEPI, eds. G. Altarelh, R. Klelss and C. Verzegnassl CERN report CERN 89-08 (1989) p. 7 [8]A Blondel, CERN prepnnt CERN-EP/90-10 (January 1990), V. Barger, J L Hewett and T.G. R~zzo, Phys. Rev Lett. 65 (1990) 1313, J Elhs and G.L. Fogh, CERN preprmt, CERN-TH. 5817/ 90 [9] M. Veltman, Nucl Phys B 123 (1977) 89; D Ross and M. Veltman, Nucl. Phys. B 95 (1975) 135 [ 10 ] CDF Collab, L. Pondrom, in' Proc. XXV Intern Conf High energy physics (Singapore, August 1990), CDF Collab., K Shwa et al., m: Proc. XXV Rencontres de Monond (Les Arcs, France, March 1990) to appear. [ l l ] Z. Kunszt and P Nason, m: Physics at LEPI, eds G Altarelh, R. Klelss and C Verzegnassl CERN report CERN 89-08 (1989) p 373;

31 January 1991

J H Kiihn and P.M Zerwas, in: Physics at LEP. eds. G. Altarelh, R Klelss and C. Verzegnassl CERN report CERN 89-08 (1989) p. 267. [ 12] J Drees, talk 14th Intern. Conf. on Neutrino physics and astrophysics "Neutrino 90" (CERN, Geneva, 1990), to appear. [ 13] ALEPH Collab., D. Decamp et al., Phys. Lett. B 246 (1990) 306 [14] M Veltman, Nucl. Phys. B 123 (1977) 89. [15] G Passanno, Phys. Lett B231 (1989)458; A Blondel, CERN, preprmt CERN-EP/90-10 (January 1990). [ 16] G. Passarlno, Phys Lett. B 247 (1990) 587. [ 17 ] DELPHI Collab., P Abreu et al., CERN prepnnt CERNPPE/90-118 (August 1990). [ 18] L3 Collab., B Aveda et al., Phys. Lett B 241 (1990) 418.

133