Standard model parameters from a global fit to LEP data

Physics Letters B 303 ( 1993 ) 170-176

North-Holland

PHYSICS LETTERS B

Standard model parameters from a global fit to LEP data Guido Montagna

a,b, O r e s t e

Nicrosini b and Giampiero Passarino c

a Dipartimento di Fisica Nucleare e Teorica. Universit~ di Pavia, 1-27100 Pavia, Italy b INFN, Sezione di Pavia, 1-27100 Pavia, Italy c Dipartimento di Fisica Teorica, Universith di Torino, and INFN, Sezione di Torino, Turin, Italy

Received 20 July 1992

A set of formulae in the framework of the minimal standard model and a corresponding FORTRAN code, TOPAZ0, for the analysis of cross sections and asymmetries around the Z ° resonance are used to analyze the 1989 and 1990 data collected by the four LEP collaborations. The lagrangian parameters of the minimal standard theory of the electroweak and strong interactions are studied. A set of fits with Mz, mr, mn and as as free parameters to the data of each LEP experiment as well as to the weighted average of the LEP data is performed exploiting the sensitivity of the observables to the radiative corrections.

1. Introduction

During 1989 and 1990 each o f the four LEP collaborations ( A L E P H , D E L P H I , L3, O P A L ) has perf o r m e d very precise m e a s u r e m e n t s o f the cross sections and o f the f o r w a r d - b a c k w a r d a s y m m e t r i e s in the leptonic and h a d r o n i c decay channels o f the Z ° resonance. The collected d a t a have been used by each o f the four LEP experiments to extract the p a r a m e ters o f the s t a n d a r d m o d e l for the electroweak and strong interactions by fitting all the experimental line shapes as well as the leptonic a s y m m e t r i e s [ 1,2 ]. The procedure for analyzing the d a t a and the fitting strategy are described in detail in the literature [ 3,4 ]. The determination o f the Z ° line shape parameters at LEP can be s u m m a r i z e d as follows [ 3,4 ]. The lowest-order cross section a o ( e + e - - - , f f ) is first p a r a m e t r i z e d as a relativistic B r e i t - W i g n e r line shape in terms o f some quantities such as the Z ° mass M z , the total and partial decay widths o f the Z ° resonance (Fz, Ff, f'e), the hadronic peak cross section a ° and then, by virtue o f this p a r a m e t r i z a t i o n , the Z ° mass and partial decay widths are d e t e r m i n e d by means o f simultaneous fits to the hadronic and leptonic cross section data. An analogous fitting procedure applied to the data including the leptonic f o r w a r d - b a c k w a r d asymmetries gives the neutral current coupling constants g~v and g~, p r o v i d e d that the correct p a r a m e t r i z a t i o n 170

o f the a s y m m e t r y at the Z ° peak is supplied. The reason why the LEP collaborations a d o p t this strategy is very appealing: it gives a d e t e r m i n a t i o n o f the Z ° parameters which is, in principle, i n d e p e n d e n t from the details o f the minimal standard model ( M S M ) [ 5,6 ]. Several comparisons between all the available experimental data and the standard model expectations have been carried out during the last years [ 7 ]. Usually this is done by using the results quoted by the four LEP collaborations and combining them with the data coming from p/~ colliders ( U A 2 / C D F ) , v N scattering experiments ( C H A R M / C D H S ) and vu(O u ) e scattering ( C H A R M - I I ) . These analyses have been p e r f o r m e d with the final goal o f looking for indirect bounds on the the top quark mass m t and on the Higgs mass rnH which are still unknown p a r a m e t e r s in the electroweak theory. As expected, the conclusions o f these works are in a reasonable agreement since all the investigations, although different in some details, use very similar analysis techniques on the same sample o f experimental results. H o w e v e r an alternative but not antithetic procedure can be pursued in order to derive constraints on the unknown masses, namely one could perform a direct fit to the measured cross sections and to the asymmetries by simply using the predictive power o f the M S M and by exploiting the effects induced by m , and m/~ on the radiative corrections to the measurable quantities. In

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this scheme, the theoretical formulae for the physical cross sections are derived within the MSM framework and the Z ° partial decay widths, the hadronic peak cross section and so on are simply derived quantities and not free parameters. So far, combined fits to the cross sections and to the asymmetries data within the MSM have been performed, to the best of our knowledge, only by the LEP collaborations and their results can be found in refs. [ 1,2 ]. The aim of this work is twofold: the present our formulation of Z ° physics and to provide an independent determination of the MSM parameters through a global fit to most of the LEP data officially published so far. Our analysis is applied to the data quoted by the LEP collaborations for the cross sections extrapolated to the full solid angle, the perfect data, and to the leptonic forward-backward asymmetries obtained via the likelihood method. Our set of data does not include the e + e - ~ e + e - channel because the LEP data available for this process correspond to a different realistic experimental set-up and hence they require a different, more involved, treatment of the radiative corrections. The inclusion of Bhabha scattering in our program is currently under study and will be presented elsewhere. We also would like to stress here that our fitting procedure relies on a set of electroweak radiative corrections which is independent from the electroweak libraries commonly used by the LEP collaborations and is the result o f a general program devoted to the calculation of the radiative corrections to e + e - annihilations into fermionic final states for an inclusive [ 8-10 ] as well as for an exclusive set-up [ 11,12 ]. In general very good agreement is observed between our electroweak library and those used so far. The collection of our analytical results [ 13 ] has been implemented in the F O R T R A N code TOPAZ0 ~j which will be described in detail elsewhere [ 14 ], with a particular emphasis on the comparison with other existing codes.

2. Theory In an inclusive set up initial state Q E D corrections #~ Our FORTRAN code is available upon request to one of the authors (see ref. [ 14] ).

8 April 1993

to a line shape can be implemented by means of a convolution of the form xma~ /w

trc(s)=

dxH(x,s)tro(g),

j

(1)

0

where H(x, s) is the so called radiator [15,9] describing multi-photon emission from initial state and ao(g) is the proper kernel cross section taken at the reduced CM energy g = (1 - x ) s . Xmax is a cut on the invariant mass for the process considered. As far as the forward-backward asymmetry is concerned, we adopt the following formula [ 10 ]:

f ~ dx[~(~)- ~r~(~)]H(x, s) AFB(S) = f ~ x d x [ a ~ ( 2 ) + a ~ ( 2 ) ] H ( x , s ) ,

(2)

where aoFCB) is the forward (backward)cross section. It is worth noting that eq. (2) is an approximate description of the QED corrected forward-backward asymmetry and moreover some of the experiments correct for detector resolution but not for the cuts on acollinearity and momenta. However we checked explicitly that the disagreement with more accurate results [ 11,16 ] is contained within 1 × 1 0 - 3 absolute deviation in the peak region and within a few 10 -3 far from it. Here a comment on the invariant mass cut Xmax is in order. Since the LEP collaborations quote the cross section data after all the corrections for acceptance and efficiency, we choose Xmax = 1 - 4m2r/s for leptonic channels, where my is the final state fermion mass, while for hadronic decay channels we use Xmax = 1 - 4 m ~ / s , where mh is the mass of the lightest hadron produced in the specific q# channel. Actually, as far as we know [ 17 ], OPAL and A L E P H follow a somehow different strategy, putting more severe invariant mass cuts for both leptonic and hadronic channels. This could be in principle a hard problem, since leptonic and hadronic cross sections depend critically on Xmax with a variation of 0.4% for, say, the muons at the peak when we lower Xm~x from 1 4mZ/s to 0.99. This fact could obscure the relevant weak effects in the radiative corrections but we checked that this dependence affects the fit in a statistically irrelevant way. The kernel cross section ao takes into account all the radiative corrections not contained in the radiator H(x, s) but relevant for high precision physics, 171

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namely complete one loop weak corrections together with all the leading higher orders terms [ 8 ], final state QED corrections in the form 1 + ] (a/n)Q~ (when no cut is applied), the contribution coming from leptonic and hadronic pair production [ 18 ] at the Z ° peak, QCD final state corrections to the hadronic shape up to O ( a 3) [19], mixed corrections of O(ototsm2 ) [20] and mass corrections of 2 2 0 (ors2 m b/Mz ) to the axial Z boson decay rate [ 19 ]. From now on by as we imply the next-next-to-leading expression of as [ 19 ]. In order to check to accuracy of our formulae we have performed a comparison of our results with those produced by the code ZFITTER 4.5 [ 16] for an inclusive experimental set up and with a compatible choice of the input flags. From the comparison it emerges that our leptonic, hadronic and total widths differ from those produced in ZFITTER by a 7 × 10-4 relative deviation at worst. Our leptonic and hadronic line shapes agree with the corresponding predictions of ZFITTER within 1.5 per mill (usually much better) for both weakly corrected and QED convoluted observables and for various values of the top quark mass. Also the forward-backward asymmetries show a very good agreement: 1 × 10 -4 absolute deviation for weakly corrected asymmetries and 1 × 10 -3 for QED convoluted ones. Looking at these comparisons, we feel confident that our set of radiative corrections is well under control.

3. Data and errors

Our fits have been performed on the following set of experimental data:/t, z and hadronic line shapes,/z and z forward-backward asymmetries. The data used in our analysis are those collected at LEP during 1989 and 1990 and published in ref. [ 1 ]. For the forwardbackward asymmetries we use the data presented according to the likelihood method. All the observables have been taken in the seven CM energy points corresponding to the highest luminosity values. Since all the LEP collaborations but OPAL correct their line shapes data for the effect of beam energy spread, we corrected the hadronic peak cross section quoted by OPAL for the same effect. Also OPAL data for the asymmetry are not extrapolated to the full solid angle (Icos01 <0.95 for muons and 0.90 for taus) but we 172

8 April 1993

have corrected them for angular acceptance. The four LEP collaborations quote the data with their statistical errors only, but they give detailed indications about the magnitude of the systematic errors and the luminosity uncertainties, when not included in the previous ones. In order to obtain the overall errors on the observables, statistical and systematic errors have to be carefully treated experiment by experiment: in particular, following the procedure described in ref. [ 21 ] the overall uncertainties on the observables measured by each experiment have been obtained by adding statistical and systematic uncertainties in quadrature. Since the data quoted by each collaboration refer to compatible experimental set up, we followed again the strategy described in ref. [21 ] and performed on the LEP data a standard weighted least-squares procedure with the addition of a scale factor applied to the errors. Typical z2/d.o.f, obtained for the line shapes are Z2(/z) = 1.9, Z2(z) =0.5 and y 2(hadrons) = 1.19 and for the asymmetries Za(It)=0.2 and Z z ( T ) = 2 . 2 . Two comments are in order here: the generally good Z 2 values obtained confirm the compatibility of the data considered. Contrary to the other line shapes, a relatively large Z 2 value for the muon line shape implies that the corresponding combined error is of the same order or magnitude as the single experiment error. Bounds on the standard model parameters mr, m/t, as coming from LEP and CDF experiments have also been introduced by means of the proper penalty functions [22 ]. A more complete statistical analysis will be presented elsewhere [ 13 ]. We are well aware of the fact that our analysis has to rely, for the moment, on the 1989+ 1990 data while the Z ° parameters have already been updated with the 1991 data. However the new data will be easily taken into account as soon as the detailed publication of the cross sections and asymmetries will be available.

4. Fits and interpretation

As already stated the complete analysis and its interpretation will be presented elsewhere [ 13 ]. Here we report on the main results. First of all we have performed a simplified one parameter fit (rnt) to Fz, Fh, El, fh/Fl and a ° as quoted by the experiments

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(with no use o f the correlation m~/trix) in o r d e r to test our procedure. Here and in the following when a numerical value is quoted for ors we i m p l y ors(91.2 G e V ) which can be c o m p a r e d with the results o f recent analysis [23 ]. Using several values for M z , we find the results shown in table 1, where we have also reported the corresponding value for as. Going back to our original p r o g r a m we have first o f all p e r f o r m e d a two p a r a m e t e r fit to M z and mr with m~/=300 G e V a n d c q = 0 . 1 2 5 held fixed. The results for each experiment are shown in the upper part o f table 2 where also the Z ° widths corresponding to the best fit are given. The errors presented in the u p p e r part o f table 2 are estimated with the elements o f the covariance matrix o f the regression coefficients. The c o m m o n systematic errors due to LEP absolute energy scale a n d point-to-point energy uncertainty, which affect M z a n d F z respectively, are not included. The results are fully satisfactory, a part

Table 1. Collab.

M z (GeV)

ors

mt (GeV)

z2/d.o.f.

ALEPH DELPHI L3 OPAL AVERAGE AVERAGE

91.182 91.177 91.181 91.161 91.175 91.175

0.121 0.110 0.115 0.118 0.125 0.118

118+47 102-+53 178-+27 130-+49 118+17 129_+26

1.9/(5-1) 1.4/(5-1) 1.2/(5-1) 2.4/(5-1) 0.6/(5-1) 1.8/(5-1)

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from the extremely low value for m t resulting in our analysis o f the O P A L data, a c c o m p a n i e d by an error (0.5 G e V ) that must be considered as an artifact and for this reason is not quoted in the upper part o f table 2. Only for the first three experiments the elements o f the covariance matrix provide a good estimate o f the errors. Actually in the last case the Z 2 is far from being quadratic in its arguments, a fact that we have verified explicitly by assigning to each value o f mr the values o f M z a n d czs that m i n i m i z e the X2 with m n = 300 GeV and assigning confidence levels according to this Z2in. In so doing we find mr=47..+. 24 GeV, where from now on the dots mean that the lower limit falls below open top production, with a corresponding shape o f the X2in which fully accounts for the small error found with a high precision analysis and agrees with the m t and the corresponding error reported for O P A L in the upper part o f table 2 (see fig. 1a) as o b t a i n e d by allowing for a lower accuracy. In the lower part o f table 2 we have shown the results o f a fit to the weighted average o f the LEP data and a weighted average o f the results corresponding to the experiments, with and without O P A L as far as m, is concerned. W h e n our results from O P A L data are included, we assign to the top mass the error quoted in the upper part o f table 2. In order to u n d e r s t a n d better the role played by as we have also p e r f o r m e d a three p a r a m e t e r fit to Mz, mt and ors to the weighted average o f the LEP data. We find

Table 2 (Upper part) Two parameter fit to Mz and mt to the line shapes and the leptonic forward-backward asymmetries for each of the four LEP experiments, mH= 300 GeV and a l = 0.125 are held fixed. For mt we report the 68% error and the 95% CL upper limit. (Lower part) The same as above for the average of the fits [A(F) ] and the fit to the average LEP data [F(A ) ]. For mt we have shown the average with and without our results for OPAL. Fitted quantities

Derived quantities

Mz (GeV)

rnt (GeV)

z2/d.o.f.

Ft (MeV)

Fh (MeV)

Fz (MeV)

Fh/Fr

a°(nb)

ALEPH DELPHI L3 OPAL

91.177+0.013 91.179_+0.013 91.179+0.011 91.165_+0.010

108+84<248 193_+46<269 104+75<230 45_+23<83

1.45 0.88 0.86 0.68

83.33+0.55 84.08_+0.53 83.31+0.48 82.89+0.39

1735+ 9 1747_+ 9 1735+ 8 1726+10

2484+ 13 2502_+13 2483_+12 2471+13

20.82+0.03 20.78+0.03 20.82_+0.03 20.82+0.02

41.29+0.08 41.37_+0.05 41.28_+0.07 41.16+0.08

A(F)

91.174+0.006 78+33(158+36) 91.176+_0.005 101+35<160

0.74

83.31+0.25 83.28+0.22

1736+ 4 1734+ 4

2485+ 6 2482+ 5

20.81+0.01 20.82+0.01

41.30+0.04 41.28+0.04

F(A)

173

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' ' i . . i . . i; . . j

. t .

'

LEP d a t a where to each value o f m, we assign the values o f M z and a s that m i n i m i z e the Z 2. The results are given in figs. I b-1 d, where we also consider the possibility o f introducing penalty functions o f the form

i

b)

1 ~)

0.8 c~

'

///,//

0.6 0.4

i/

0.2 0 30 6

t 60

90

120

r

. . i .

~ c)

, t , 50

i%,

/

6IT°, /

'

r

81

200

,

Z2-~Z2+

4

4 F

108)2 f o r i n t < 108 G e V ,

Z2"--~Z2+ ( m r -

150

100

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(mn-65)2 16.81

formn<65GeV

The results can be s u m m a r i z e d as follows:

3 2

r n n = 6 5 GeV,

~

~L

21 i 0

0

100

go

150

200

mtoo(GeV)

"" i, ," '

0

I 50

100

mi4=300GeV, 150

200

mn=1000GeV,

m,=84.+4°GeV,

m r < 145 GeV,

~ ,• mr= lt~2+35 ,., _32tJev, mt=121+_32GeV,

mt<156GeV, mt<170GeV,

mtop(GeV) where the u p p e r b o u n d refers to 95% CL. In short,

Fig. 1. We assign to each value of mr the values of Mz and as that minimize the Z2with ran= 1000 GeV (65 GeV, 300 GeV ) shown in (b) (c), (d)) and assign confidence levels according to this Z~in. Solid lines include the CDF bound in the form of a penalty function. For OPAL data (a) we do not include the penalty function. rnH = 65 G e V ,

mt =

1/30+35+19 xu.~ 32_18

("-~W ,,3~

.

Finally the correlations between m , - mn, m , - M z and m r - a s are given in figs. 2 a - 2 c where we have shown the 68% and 90% CL contours with the corresponding m o d i f i c a t i o n o f the 68% CL curve when the p r o p e r penalty functions are included.

M z = 9 1 . 1 7 6 _ + 0 . 0 0 5 GeV, mt = 84 + 42 GeV, a s =0.123_+0.014, m n = 300 G e V , M z = 9 1 . 1 7 6 + 0 . 0 0 5 GeV, m t = 101 + 3 7 GeV, cts =0.125_+0.014, m n = 1000 G e V , M z = 9 1 . 1 7 6 _ + 0.005 GeV, m, = 121 _+32 GeV, c%=0.127_+0.014. The 95% CL u p p e r limit for mt is 163 G e V (155 GeV, 174 G e V ) for m , ~ = 3 0 0 G e V (65 GeV, 1000 G e V ) . The agreement between the values found here and those reported in the lower part o f table 2 is highly satisfactory. The total Z ° width is found to be 2484_+ 8 MeV, while R t = 20.824 _+0.098. Next we have considered again a fit to the average

174

5. Summary and conclusions In conclusion Confidence Levels for the p a r a m e ters o f the M S M have been derived by c o m p a r i n g our results directly with the extrapolated observables o f the LEP experiments without the need o f relying on the results o f the fits published by the four collaborations. We find consistency with previously published analyses, typically we obtain m t = 102335+19 G e V to be c o m p a r e d with mt=/-r..,-24OA+53+23G e V as rep o r t e d by the four collaborations [2 ]. Even the small difference between the central values can be explained. I f we neglect pair p r o d u c t i o n in our fit then mt moves from 101 + 3 7 G e V to 9 3 + 4 0 GeV. Actually the most quoted value for mt is . 47-23 GeV as mr= 124 +4°+21 -56-21 GeV [2] (or m~ . . . 12a+37+22 reported by Schaile [ 7] ), but in this case as is constrained to the value as = 0.118 _+0.008. Finally we would like to stress that our prediction for as, namely o q = 0.125 _+0.014 _+0.002 (Higgs), is in excellent agreement with the event shape result [23].

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References

1000

>=

802

904 405

E

208 10 50 200

100 150 mtop(GeV) .

.

.

.

~

.

.

.

200 .

b)

~ ' ~

/ ,o

, i 91.155

91.176

91.195

mz(GeV)

0.14 ..,. : . - , . . : , . . . , - . : . . , . . . , . i , \

~

o.12

~

O.ll

8 April 1993

0.10

/ 50

100 150 mtop(GeV)

200

Fig. 2. (a) The dashed curves correspond to X2ia
Acknowledgement We are grateful to G. Goggi, E. Menichetti and A. Rotondi for useful discussions. We would like also to thank A. Rossi, A.D. Schaile and G. Zumerle for helpful correspondence. Stimulating discussions with M. Cacciari, F. Piccinni and R. Pittau are acknowledged. Finally one of us (G.P.) would like to thank G. Pollarolo for the continuous assistance in numerical and graphical recipes.

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[ 11 ] M. Cacciari, G. Montagna, O. Nicrosini and G. Passarino, Phys. Lett. B 279 (1992) 384; M. Cacciari, A. Deandrea, G. Montagna, O. Nicrosini and L. Trentadue, Phys. Lett. B 268 ( 1991 ) 441; B 271 ( 1991 ) 431. [12] G. Passarino, QFORMFS a computer program for the calculation of one loop scalar form factors, unpublished; M. Cacciari, G. Montagna, O. Nicrosini, G. Passarino and R. Pittau, Phys. Lett. B 286 (1992) 387; G. Montagna, O. Nicrosini and G. Passarino, Analytic final state corrections to e + e - -~ffwith Realistic Cuts, submitted to Phys. Lett. B; F. Piccinini and R. Pittau, Phys. Lett. B 293 (1992) 237. [ 13 ] •. Montagna, O. Nicrosini, G. Passarino, F. Piccinini and R. Pittau, On a semi-analytical and realistic approach to e+e - annihilation into fermion pairs and to Bhabha scattering within the minimal standard model at LEP energies, submitted to Ntfcl. Phys. B. [14] G. Montagna, O. Nicrosini, G. Passarino, F. Piccinini and R. Pittau, TOPAZ0, version 1.0 (July 1992 ), a program for computing observables around the Z ° peak and for fitting cross-sections and forward-backward asymmetries, paper in preparation. (The code is available upon request to: GIAMPIERO @ TORINO.INFN.IT, NICROSIN1 @ PAVIA.INFN.IT. ) [15] E.A. Kuraev and V.S. Fadin, Soy. J. Nucl. Phys. 41 (1985) 466; G. Altarelli and G. Martinelli, Physics at LEP, CERN report 86-02, eds.J. Ellis and R. Peccei (Geneva, 1986); see also F.A. Berends et al., Z ° physics at LEP1, eds. G. Altarelli, R. Kleiss and C. Verzegnassi~ CERN preprint 8908, Vol. 1 (1989) p. 89, and references.therein.

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