19 August 1999

Ž .Physics Letters B 461 1999 149–154

Measuring WWZ and WWg coupling constants with Z0-pole data

Peter Molnar, Martin Grunewald´ ¨Humboldt UniÕersity Berlin, Institute of Physics, InÕalidenstr. 110, D-10115 Berlin, Germany

Received 19 February 1999Editor: K. Winter

Abstract

Triple gauge boson couplings between Z0, g and the W boson are determined by exploiting their impact on radiativecorrections to fermion-pair production in eqey interactions at centre-of-mass energies near the Z0-pole. Recent values ofobservables in the electroweak part of the Standard Model are used to determine the four parameters e , e , e and e . In a1 2 3 b

second step the results on the four e parameters are used to determine the couplings Dg1 and Dk . For a wide range ofZ g

scales, these indirect coupling measurements are more precise than recent direct measurements at LEP-2 and at theTEVATRON. The Standard Model predictions agree well with these measurements. q 1999 Elsevier Science B.V. All rightsreserved.

1. Introduction

One of the most prominent goals of the LEP-2program performed at the Large Electron Positron

Ž .Collider LEP is the precise measurement of thecouplings between the neutral electroweak bosons

0 " w xZ , g and the charged boson W 1 . Analogousmeasurements were performed at the TEVATRONmeasuring mainly the coupling between the photonand the W ". These two measurements were the firstones which were able to prove the non-Abeliancharacter of the electroweak part of the Standard

w xModel 2 . Even more precise determinations will bepossible at future hadron or electron-positron-col-lider.

However, before the LEP-2 program with centre-of-mass energies above the W-pair productionthreshold of about 161 GeV, LEP was running atenergies around the Z0-pole at 91 GeV allowing toperform very precise measurements of fermion pair

production properties. The experiments at LEP-1 andalso at SLAC measure radiative corrections to theZ0 ff vertex. These radiative corrections involve con-

Ž 0 .tributions with WWV VsZ , g vertices as shownŽ . Ž .in Fig. 1 a and b and WWV-independent contri-

Ž Ž . Ž ..butions Fig. 1 c , d . Therefore precise measure-ments of fermion-pair production allow the determi-nation of the WWV coupling constants. This was

w xnoted already in the beginning of the LEP era 3,4 .The phenomenological effective Lagrangian of

the WWZ and WWg vertices, respecting onlyLorentz-invariance, contains 14 triple gauge coupling

Ž .constants TGCs as free parameters. All of these canbe accommodated in the Standard Model requesting

Ž . Ž .SU 2 =U 1 gauge invariance, if one considersŽ . Ž .higher dimensional SU 2 =U 1 gauge invariant

operators. The neglect of higher dimensional opera-tors leads automatically to relations between TGCs.The model which is discussed in the following ne-glects operators having a higher dimension than six.

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00815-1

( )P. Molnar, M. GrunewaldrPhysics Letters B 461 1999 149–154´ ¨150

Fig. 1. Radiative correction to the decay width of the Z0 into0fermions, Z ™ ff. This process is used to constrain the Higgs-bo-

Ž . Ž .son and top-quark mass. Graphs a and b depend on the WWVŽ . Ž .coupling constants, while graphs c and d depend only on

fermion to boson couplings.

Loop corrections in this model lead to a logarithmicw xdivergence of low energy observables 3 . However

it was shown that three dimension-six operators, thatinduce non-standard TGCs do not have this propertyw x4 . Assuming the existence of a light Higgs boson,created by the Higgs-doublet field F , one can apply

Ž . Ž .a linear realization of the SU 2 =U 1 symmetry.Then one obtains in addition to the SM Lagrangian

w xthe following three terms 4 :

Dk ycos2u Dg1g W Z †X mnD LLs ıg D F B D FŽ .Ž .m n2mW

cos2u Dg1W Z † mnˆq ıg D F tPW D FŽ .Ž .m n2mW

lg m n rˆ ˆ ˆq ıg WB P WB =WB . 1Ž .ž /n r m26mW

In this model the TGC-relations are:

cos2uW 1Dk s Dk yDg , 2Ž .Ž .g Z Z2sin uW

l sl . 3Ž .g Z

The remaining nine coupling constants are zero. TheSM predicts that all 14 parameters are zero. The

TGCs Dk and Dg1 parametrise the difference ofV V

g1 and k to its SM expectation of unity:V V

Dk sk y1, 4Ž .V V

Dg1 sg1 y1. 5Ž .V V

In almost all models the electromagnetic gauge in-variance is taken for granted, such that Dg1 , theg

divergence of the W-charge from the unit charge, isalways zero. The parameter l is also set to zero ing

our analysis, since we are not aware of any computa-tion of the dependence of e , e and e on l .1 2 3 g

2. Analysis and results

The preliminary measurements of electroweak pa-rameters performed at LEP-1, SLAC and TEVA-TRON are listed in Table 1. The SM predictions

w xagree well with these measurements 5 . The analysis

Table 1Preliminary electroweak parameters that are used in the fit to the e

parameters. The correlations among the observables in the b and cquark sector as well as the one between m , G , s , R andZ Z had e

e w xA is taken properly into account. Consult 5 and referencesFB

therein for details. m is only used in the Standard Model calcula-t

tion of the e-parameters

Parameter Central value ErrorsŽ5.Ž .1ra m 128.878 0.090Z

m 91.1867 0.0021Z

G 2.4939 0.0024Z

s 41.491 0.058had

R 20.765 0.026eeA 0.01683 0.00096FB

PP 0.1479 0.0051e

PP 0.1431 0.0045t2 effŽ .sin u Q 0.2321 0.0010w f b2 effŽ .sin u A 0.23109 0.00029w LRŽ .m LEP2 80.37 0.09WŽ .m pp 80.41 0.09W

R 0.21656 0.00074b

R 0.1735 0.0044cbA 0.0990 0.0021FBcA 0.0709 0.0044FB

A 0.867 0.035b

A 0.647 0.040c

m 173.8 5.0t

( )P. Molnar, M. GrunewaldrPhysics Letters B 461 1999 149–154´ ¨ 151

of this data set proceeds via two steps. In the firstw xstep, the e parameters e , e , e and e 6 :1 2 3 b

e sDr , 6Ž .1

sin2u 0 DrW W X2 0 2 0e scos u Drq y2sin u Dk ,2 W W2 0 2 0cos u ysin uW W

7Ž .

e scos2u 0 Drq cos2u 0 ysin2u 0 DkX , 8Ž .Ž .3 W W W

g bA

e s y1 andb lg A

g bV X4 2 0e s y 1y 1qDk sin u , 9Ž . Ž .Ž .b W3lg A

where:

pa m2Ž .Z2 0 2 0sin u cos u s ,W W 2'2 G mF Z

sin2u eff s 1qDkX sin2u 0 , g 2 s 1qDr r4Ž . Ž .w W A

10Ž .

are extracted. These parameters are very sensitive toradiative corrections and thus the influence of physicsbeyond the SM, hence also very sensitive to non-SMTGCs. It is interesting to note that e and e do not,2 b

on the one-loop level, depend on the yet unknownHiggs-mass m . Here Dr stands for radiative cor-H

rections to the Z leptonic width, similar to those onw xthe r-parameter 7 , Dr describes corrections to thew

G -M relation and DkX relates sin2u 0 to sin2u eff,F W W w

the effective electroweak mixing angle, extractedfrom asymmetries to the Z pole. The complete set of

w xrelations is given in 6,8 .

As the fermion coupling constants depend on thee-parameters one can extract these from the Z-pole

Žmeasurements reported in Table 1 except the top-. 2 effquark mass , which all depend on g , g or sin u ;V A w

w xsee 9,10 , for example. A simultaneous fit to all fourparameters and in addition to the electromagnetic

Ž .coupling constant a m , the strong coupling con-em ZŽ .stant a m and m gives the numbers quoted ins Z Z

Table 2. The computation of the SM expectationsshows that these values are in good agreement withthe measured ones, and they are also in good agree-

w xment with other recent computations 8,11 . Onefinds strong correlations between e and a as wellb s

as for e and e . The latter is visible in Fig. 2,1 3

showing the two-dimensional contours of each pairof e-parameters. These contour curves are comparedwith the evolution of the e-parameters as a functionof the TGC coupling constants.

The dependence of the e-parameters on the WWVcouplings is shown in the following equationsw x12,13,4 :

2 212p m LZ27 2y De s y tan u ln1 W2 2 2½a m mW W

2 2 29 m m L 1Z Hq ln q Dkg4 2 52 2m mW H

2 2q tan u ycot uW W½2 29 m L 1H 1y ln q Dg , 11Ž .Z2 2 52 2m mW H

Table 2Ž 2 .The e values in the SM and from a fit to the electroweak data summarised in Table 1 x rNdfs11.6r11, probability 39%

Fit parameter Measured MSM Correlation matrix

1 a m e e e es Z 1 2 3 bŽ5.a

Ž5.Ž .1ra m 128.878"0.090 – 1 .00 0.00 0.00 0.00 y0.07 0.46 0.00ZŽ .a m 0.1244"0.0045 – 0.00 1 .00 0.00 y0.45 y0.22 y0.31 y0.62s Z

m 91.1866"0.0021 – 0.00 0.00 1 .00 y0.06 y0.01 y0.02 0.00Z3e =10 4.2"1.2 4.6"1.1 0.00 y0.45 y0.06 1 .00 0.44 0.80 y0.0113e =10 y8.9"2.0 y7.5"0.3 y0.07 y0.22 0.00 0.44 1 .00 0.26 y0.0123e =10 4.2"1.2 5.8"0.7 0.46 y0.31 y0.02 0.80 0.26 1 .00 0.0033e =10 y4.5"1.9 y5.8"0.5 0.00 y0.62 0.00 y0.01 y0.01 0.00 1 .00b

( )P. Molnar, M. GrunewaldrPhysics Letters B 461 1999 149–154´ ¨152

Fig. 2. The contours of the e parameters. The arrows indicate the change of the SM prediction if the coupling parameters Dg1 and Dk areZ g

varied according to the direct measurements of LEP-2 and TEVATRON.

( )P. Molnar, M. GrunewaldrPhysics Letters B 461 1999 149–154´ ¨ 153

12p m2 L2Z 2De s ln sin u Dk2 W g2 2a m mW W

L22 1qcot u ln Dg , 12Ž .W Z2mW

2 212p m LZ34 2De s cos u y7cos u y ln3 W W 4 2 2½a m mW W

2 23 m L 1Hy ln q Dkg2 2 54 2m mW W

L232 1q 10cos u q ln Dg , 13Ž .� 4W Z2 2mW

2 2 2 2 2 2 2m m L cot u m m LZ t W Z tDe s ln Dk y lnb g2 4 2 2 4 264p m m 64p m mW W H W

2 2 23cot u m LW t 1q ln Dg . 14Ž .Z2 2 232p m mW W

These expressions are based on the constraintsbetween TGCs quoted earlier. All non-standard con-tributions are logarithmically divergent. The cou-pling parameters, that are used here, are defined independence on the new physics scale L and a formfactor f coming from the new physics effect, e.g.

m2Z1Dg s f . 15Ž .Z 2L

In the following the new physics scale L is set to 1TeV; higher values of L imply tighter constraints onTGCs. In addition the Higgs-mass is set to 300 GeVand varied between 90 GeV, the lower limit on the

w xHiggs mass derived from the direct search 14 , and1000 GeV. Since we look for effects beyond the SM,we cannot make use of constraints on the Higgsmass derived from a SM analysis of radiative correc-

w xtions such as 5 .Ž . Ž .A fit using Eqs. 11 and 14 and the difference

of the measured values of the e-parameters and theones expected in the SM as shown in Table 2 is usedto determine the TGC coupling parameters Dg1 andZ

Dk . The errors on the SM predictions of the e-g

parameters are included, neglecting their correla-tions. The x 2 curves of a fit to each of thesecoupling constants, setting the other to its SM value

Fig. 3. The Dx 2 curves for the TGC couplings and the contribu-tions of the different e parameters. The combined curve is the addup of the single curves taking the correlation coefficients properlyinto account. The parameter e has almost no sensitivity to TGCs.2

of zero, is shown in Fig. 3. One finds the followingresults:

Dg1 sy0.017"0.018q0 .018 m 16Ž . Ž .Z y0.003 H

or

Dk sq0.016"0.019q0 .009 m . 17Ž . Ž .g y0.013 H

If both couplings are allowed to vary in the fit,one finds the contour plot in Fig. 4. The correspond-ing numerical values of the TGC-parameters are

Dg1 sy0.013"0.027q0 .023 m ,Ž .Z y0.001 H

Dk sq0.005"0.029q0 .011 m ,, 18Ž . Ž .g y0.001 H

( )P. Molnar, M. GrunewaldrPhysics Letters B 461 1999 149–154´ ¨154

Fig. 4. The contour curves for the two dimensional fit, Dg1Z

versus Dk . The dot shows the SM expectation.g

with a correlation of 75.5 percent. The SM expecta-tion of zero for both TGC parameters agrees wellwith this measurement. For other values of the newphysics scale L, both fitted central values and fittederrors of the TGC parameters scale approximately as1rln L2. Thus the significance of the compatibilityof the TGC with the SM, i.e., valuererror, is approx-imately independent of L. The systematic uncer-tainty arising from the Higgs mass variation is quoted

Ž . Ž .as second error in Eqs. 16 – 18 ; large Higgs massesmove the TGCs closer to their SM expectation ofzero. The error of 5 GeV on m , as quoted in Table 1t

has a negligible impact on the result.The results presented above are more precise than

recent direct measurements of the LEP and TEVA-w x 1 q0.12TRON collaborations 5 : Dg s0.00 and DkZ y0.11 g

s0.28q0 .33. Here the parameters are negatively cor-y0.27

related with y54 percent. The direct measurement ishowever more suitable for a general test of the TGCswhile the indirect measurement tests TGCs only inparticular models.

w xRecent computations 12,13 parametrise also thedependence of e on the coupling constants l andb g

g 5 giving access to a more general view of the TGCZ

couplings. Computations of the dependence of e , e1 2

and e on the TGCs l and g 5 would be most3 g Z

useful to measure also these coupling constants moreprecisely.

Acknowledgements

We are very grateful to S. Riemann for bringingthe possibility of the indirect measurement of TGCsto our attention. We thank F. Caravaglios and G.Altarelli for clarifying discussions on the e parame-ters and T. Hebbeker, W. Lohmann and T. Riemannfor useful comments.

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