- Email: [email protected]
The mass of the Higgs boson in the Standard Model from precision tests
-_i!!z
17 August 1995
A@2
PHYSICS
LETTERS
B
Physics Letters B 356 (1995) 307-312
ELSEVIER
The mass of the Higgs boson in the Standard Model from precision tests * Piotr H. Chankowski a, Stefan Pokorski b*l a Institute of Theoretical Physics, Warsaw University ul. Hoia 69, 00-681 Warsaw, Poland b Max-Planck-Institute fiir Physik, Werner-Heisenberg-Institute, Fijhringer Ring 6, 80805 Munich, Germany
Received 16 May 1995; revised manuscript received 9 June 1995 Editor: P.V. Landshoff
Abstract
The LEP data and the measurement of MW confirm the electroweak symmetry breaking by the Higgs mechanism with (2-31% accuracy. For the Higgs boson mass, fits to the present data (without the SLD result) give Mh = 121:;: GeV and Mh < 800 GeV at 95% CL. We analyze those fits from the point of view of the impact of different measurements on the final result for Mh. All the relevant observables can be grouped into two categories: those which correlate Mh with mr and those which constrain the range of M,. Using this classification we discuss potential improvements in the limits on Mh. anticipating improvements in the precision of the data. Also the sensitivity of the limits on Mh to the value of (Y~(Mz) is discussed in a toy model which reproduces all the results of the SM but the Z" -+ bb width is increased by hand so that Rb = 0.22 and, in consequence, the fitted value of crs( Mz) M 0.11 (qualitatively similar situation occurs in the Minimal Supersymmetric Standard Model with light chargino and right-handed top squark) .
The LEP data and the measurement of Mw confirm the electroweak symmetry breaking by the Higgs mechanism with few per mill accuracy. For some time they are used by several theoretical groups [l-5] and by the experimental groups [6] to constrain the less precisely known parameters of the Standard Model: the top quark mass m,, the strong coupling constant LY,? ( Mz ) and the Higgs boson mass Mh. The loop corrections to the electroweak observables are O( rnf ) in the top quark mass and only O( log Mh) in the Higgs boson mass. Nevertheless, the increasing precision of the data and in particular the discovery of the top quark *Supported in part by the Polish Committee
for Scientific
Re-
search and European Union Under contract CHRX-CT92-0004. 1On leave of absence from Institute of Theoretical Physics, Warsaw University.
Ekevier Science B.V. SSDI 0370-2693(95)00809-8
[ 7,8] improve the prospects for constraining the Higgs boson mass in a way which may be relevant for planning the future experiments for direct Higgs search. In this short note we analyze the fits to the electroweak data from the point of view of their constraints on Mh. We follow the usual strategy: in terms of the best measured observables GF, (YEMand MZ we calculate Mw, all partial widths of Z” and all asymmetries at the Z” pole and determine m,,a,(Mz) and Mh by a fit to the data. The calculation is performed in the on-shell renormalization scheme, with all “oblique” and process dependent one-loop corrections as well as the leading higher order effects included. The experimental input (i.e., experimental values for the electroweak observables, their errors and correlation matrices) used in the fits is summarized in Ref. [6].
308
P.H. Chankowski, S. Pokorski/Physics
Table I Results of the fits
Letters B 356 (1995) 307-312
to the data (61. Ail masses in GeV
fit
tnr
ANI,
Mh
AMh
a(Mz)
Aa*
X2
d.o.f
-SLD
168
12
121
+2U7 -58
0.124
0.005
11.1
1s
+SLD
166
II
63
+97 4
0.123
0.005
16.5
16
For the top quark mass we use the CDF result m, = ( 176 & 13) GeV [ 71. In the most recent data of Refs. [6,10] there are two significant changes compared to the data of the Glasgow conference [ 11,121: the Ob have increased and central value of Mw and of AFB they now read Mw = (80.33 f 0.17) GeV, AK = 0.1015 5 0.0036. In Table 1 we present the results of our global fit without and with the SLD result for the electron leftright asymmetry [ 91 included. In both cases the lower limits on Mh come from the unsuccessful direct Higgs boson search. The differences between those results and the earlier fits [ 3,5,6] are mainly due to the above mentioned changes in data (as well as the overall better precision of the LEP data). Our main aim, as said above, is to analyze in more detail the limits on Mh, with emphasize on the upper one. To be on the conservative side we shall use the fit without the SLD measurement included because its inclusion would significantly lower the inferred upper
‘888 800 700 600 500 > @ c3
400 300
100 2:: 70 60 50
160
165
170
175 m,
180
185
190
195
200
(GeV)
1.Lines of constant values (for fixed aJ- (Mz) = 0.126) of: I ) Tz, 2) A!:, 3) X2 for Mw, 1- Mf+,/M$, A!$, sin’SFg( (QFB)).
Fig. A,,
AT and A:;.
0.0287
(dashed).
hadr = 0.0280 4) RI for AaEM
(solid lines)
and
bound on Mh. It is instructive to begin with plots in the (m,, Mh) plane of the contoursof constant values of several electroweak observables. In Fig. 1 such lines are shown for the central experimental values Tz = 2.497 1 GeV, AK = 0.1015, RI= 20.800 and also we show the contour of minimal x2 for the set Mw, 1 - M$/M'$,A&,
A!&,sin2 OF< (QFB)), A, and A. The latter can be looked at as the contour of constant Mw with the value of MW determined from the best fit to the above seven measurements: to a very good approximation all those variables are in the SM determined, e.g., in terms of Mw (the oblique corrections are dominated by Ap and the specific, process dependent corrections being negligible) . The contours are shown for two fixed values of the hadronic contribution to (YEM(Mz) which has been estimated to be AcY~$ = 0.0280 f 0.0007 [ 131. The value of LY$(Mz) is 0.124 (as obtained from the fit). We see that in the range 160 < m, < 200 GeV all contours are approximately linear in the variables logMh> Mr. First, let us focus on the plots for the central value of A&$. The striking feature of Fig. 1 is that for A&$ = 0.0280 and the central experimental values of the observables the contours are almost parallel, with theexception of RI. Although RIremains constant along drastically different lines, its dependence on the (m,, Mh) is much weaker and with the experimental error RI = 20.800 f 0.035 the overall x2, without inclusion of Rt, and the CDF result for m,, remains almost constant along the direction of the contours 1, 2, 3. This is clearly seen in Fig. 2 where we show the contours of the overall Ax2 = 1 in a fit without Rb and the CDF value for m, included which, for A&$ = 0.0280, consist of two approximately parallel lines. Thus in this case the fitted value for Mh results in a very transparent way from a combination of effects which can be organized into the following two-step description. A fit to the measured Mw = (80.33 i 0.17)
P.H. Chankowski, S. Pokorski/ Physics Letters B 3% (1995) 307-312
9Q, W
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5 =
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J.
Ih,, 170
,I
I,,
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I
190
I,
60 50
I,
,,., :
,:“‘I’
*’
160
200
m (GeV)
I
’
0,
0
5
170
10
*
5,
160
1, 190
‘I
fl
200
mt (GeV)
Fig. 2. Contours AX* = 1 (without Rb and the CDF direct measurement of WI, in the fit) in the plane (MI,,kfh): a) in the SM with the data of Ref. [6] for different A&$ -... and b) for A&$ = 0.0280 in the SM with anticipated reduced experimental errors (see text) and our toy model which gives an improved Rb. -.‘.
GeV [ lo] and to all measured electroweak observables but Rb = IZO_+~b/I’Z~_+hadronsgives x2 values which are almost independent of the value of the top quark mass m, in the broad range (150-200 GeV) and with the best value of log Mh which is almost linearly correlated with m,.We get, e.g., Mh = 138’_‘,L9 GeV for m, = 170 GeV and Mb = 280*_;!& GeV for m, = 180 GeV. The most recent experimental results [ 6,101, in particular the increase in the central value of Mw by 100 MeV and the new measurement of A:;, evolved in the direction of requiring smaller M,, for a given, fixed m, (or larger m, for a given, fixed Mh). With the data reported at the Glasgow conference [ 111 the analogous result is Mh = 283:::; GeV for m, = 170 GeV and Mh = 511 f_329;)9 GeV for m, = 180 GeV. The (m,,Mh) correlation is the most solid result of the fits which does not depend on whether Rb and/or m, measurement of the CDF [ 71 are included into the fits. It points toward relatively light Higgs boson for nz, in the range ( 170-180) GeV. The x2 isflat as a function of mt unless &, and/or the directly measured in Fermilab mass of the top quark are included into the fit. These are the only two measured observables which in the present case introduce visible x2 dependence along the curves in Fig. 2
and can, therefore, put indirectly (by constraining m,) relevant overall limit on the Higgs mass Mh. We see it very clearly in Fig. 3 where we plot the values of x2 as a function of Mh for several values of m,,in the fits with and without Rb and the CDF result for m,. The best values of m, and Mh aregiven in Table 2 (for fits to the data of Ref. [6]) and the la contour plots are shown in Fig. 4. There is some dependence of the limits on Mh on the value of Aa& = 0.0280 & 0.0007 [ 131. An increase of Aa$$ has the same effect as an increase of the Higgs boson mass, i.e, the same values of the observables are now reproduced with smaller value of Mh. This is shown in Fig. 1. The shift towards lower values of Mh is different for different observables and the flat direction in x2 along the “diagonal” in Fig. 1 disappears for AL&$ larger by 1~ than the Table 2 Best values of WI, and Mh for Aa$‘ -Rb, mr Mtl X2
-I&
undetermined undetermined 3.2
+Rb, 152 41 9.1
--mr
= 0.0280 and various X2 fits -Rb. 174 183 3.5
+tnt
+&,, 168 121 11.1
+lnr
P.H. Chanbwski,
310
S. Pokorski/Physics
25 22 5 20 175 15 “X
125 10 75 5 25 0
Letters B 356 (1995) 307-312
about improvements in those limits anticipating still better precision of the data. experimental Smaller errors for Mw,
sin2~$t((QF-d),
-4, .& A&, Tz, uhadr, Ak and Ag will result in smaller error on A4h for any fixed value of m,, whereas more precise measurement of RI, m, and Rb will constrain the acceptable range of m,. In Fig. 5 we show the results of a fit to the hypothetical experimental data which would give the x2 value of the fit similar to the present data but have the anticipated experimental errors for Mw, AK,, sin’ @Fl’( (QFB)), A, & RI, TZ and rhhadr a factor of 1.5-2 smaller. This hypothetical set of data is obtained by starting with the present experimental values of the observables and by shifting them in the direction of the values obtained in the present best fit and by fixing the central values of the hypothetical data according to the formula: XI -X0
-------=
x2
central value of A&$. Cases a) and b) correspond to x2 without and with the CDF direct measurement of ml included into the x2 respectively. Dashed lines show x2 without Rb in the fit.
central value, as seen in Fig. 2 (now the Ax2 = 1 contour is closed even without Rb and mt contributions). This is due to a combined effect of lowering the curve for R1 = 20.800 and the relative shifts of the curves Tz = constant, Mw = constant which are “less parallel”. The resulting limits on Mh with the CDF result for m, and/or Rb included into the x2 are of course stronger (Fig. 4). Obviously, opposite conhadr. However, as seen in elusion holds for smaller Ann,, Fig, 4, for Ac~pd at lc+, the central values of Mh remain within la contour obtained for A&$ = 0.0280. hadrhas a weak effect Therefore, the uncertainty in AaEM on the global lit with this uncertainty propagating in the errors. Scanning over ALY$ we get the results in Table 1. Given the clear separation of the impact of various measurements in obtaining limits on Mh, we can ask
x0 (1)
c2
(+I
Fig. 3. x2 as a function kfh for different values of ml and the
-
where X1 ((~1), X2 (~2) are the present value (error) and the hypothetical value (anticipated error) for the variable X respectively and Xa is its present best fit value. We use then Mw = (80.332 f 0.100) GeV, Ag = 0.01665 f 0.00080, sin2 $i’( (Qm)) = 0.2319&0.0008,d, = 0.1406f0.0050~ = 0.1419* 0.0050, RI = 20.801 f 0.020, I2 = 2.4977 f 0.0020 and ch& = 4 1.4675 & 0.0500. For the anticipated top quark mass value from the Fermilab we take mFyp = ( 174f8) GeV. Thus a fit to the hypothetical data gives the same results for Mh and mt, with the same x2, with however smaller errors. We see that the considered improvement in the experimental precision would reduce the la uncertainty in the Higgs boson mass to “_:‘,” GeV in the global fit and to, e.g., +I4 _s2 for fixed m, = 170 GeV In particular this conclusion almost does not depend on whether Rb (with its present value) is included into the fit or not. The dependence on cys(Mz) is best illustrated by considering a toy model which reproduces all the results of the SM with, however, a correction to the Z” -+ 6b width given by
r Z”-66 +
= ‘;!t+&b
(7.5 x IO-’ x m, - 3.83 x 10T3 GeV)
(2)
Then, &, M 0.22 for all the values of my. The interest in this possibility stems from the fact that a)
P. H. Chunkowski, S. Pokorski / Physics Letters B 356 (I 995) 307-312
1000 900 800
too0 900
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i
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s 0) c3
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Fig. 4. Contours Ax* = 1 with the CDF result for m, and/or Rb included in the fit in the plane (rnt, Mh) for different Aop$. 1000 900 600
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-I 170
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Fig 5 Contours Ax* = 1 with the CDF result for mr and/or &, included in the fit in the plane (ml, Mh) for Aat$ anticipated reduced errors of the main observables (see text) and for our toy model with gives improved Rb.
= 0.0280 for
312
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S. Pokorski/Physics
as seen in Table 1, in the SM the single variable Rh contributes more than 70% of the total x2; b) a qualitatively similar situation occurs in the Minimal Supersymmetric Standard Model with light chargino and light right-handed top squark [ 14 J. It follows that the best fit gives now LY~ (Mz ) = 0.108. The correlation between the value of rZo_bb and the fitted value of cu,( Mz) is well known [ 15,6] and it is due to the contribution of rZo_hb to RI. From the point of view of the limits on Mh the variable &, becomes irrelevant (its contribution to x2 x 0). The results obtained in that case are shown in Figs. 2, 3. We see that smaller CU,~( Mz) gives similar effect to a decrease in A&$,$ but it is clear from Fig. 2 that the dependence on a.(.( Mz) is very weak. The main effect visible by comparing Fig. 5 and Fig. 4 is due to the fact that Rb is now irrelevant in the fit. Our discussion of the effects of reduced experimental errors applies as well to this model. In summary, precision tests of the Standard Model confirm with very high accuracy the electroweak symmetry breaking by the Higgs mechanism. For the Higgs boson mass, fits to the present data give MI, = 121?$’ GeV and M/, < 800 GeV at 95% CL. Of course, for fixed values of m, one gets stronger bounds, e.g., form, = 170 GeV we have Mh = 139112’ GeV. We have analyzed those fits from the point609f view of the impact of different measurements on the final result for Mh. At the accuracy of the data available at the present and planned accelerators, all the relevant observables can be grouped into two categories: those which correlate MI, with m, (Mw, AEb, sin2 6$’ ( (QFB)), A, A, RI, TZ and ch&) and those which constrain the range of m, (Rb, R, and the direct measurement of m,). Using this clasification we have discussed potential improvements in the limits on M,,, anticipating improvements in the precision of the data. Finally we have studied the sensitivity of MA to a,(Mz) in a toy model which reproduces all the results of the SM with, however, the 2’ --+ 66 width increased by hand so that Rt, = 0.22; the fitted a,$( Mz) is then 0.108. PCh. would like to thank the Max Planck Institute fiir Physik for warm hospitality during his stay in Munich where part of this work was done. Note added. We are grateful to J. Ellis for pointing out to us that also the low energy data break positive
Letters B 354 (1995) 307-312
m, - Mh correlation and it was the inclusion of all those data which made possible to give the first discussion of bounding Mh from precision electroweak data [ 11. Now, wj th better precision of the LEP data and directly measured top quark mass, the role of the low energy data is less important.
References I I] J. Ellis, G.L. Fogli and E. Lisi, Phys. L&t. B 292 (1992) 427; B 318 (1993) 375. [2] J. Ellis, G.L. Fogli and E. Lisi, Phys. Lett. B 324 (1994) 173. 13 1 J. Ellis, G.L. Fogli and E. Lisi, Phys. Lett. B 333 (1994) 118: J. Ellis, plenary talk presented at the Int. Conference on Physics Beyond the Standard Model IV, Lake Tahoe, CA, December 1994. I4 I G. Altarelli and R. Barbieri, Phys. l&t. B 253 (1991) 161; G. Altarelli, R. Barbieri and S. Jadach, Nucl. Phys. B 369 (1992) 3. 15 1 P. Langacker and J. Erler, University of Pennsylvania preprint UPR-0632 T ( 1994). (61 The LEP Electroweak Working Group, CERN preprint
CERN/PPE/94-187, LEPEWWG/95-01. [7) F. Abe et al., The CDF Collaboration, Phys. Rev. Lett. 73 ( 1994) 225 and FERMILAB-PUB-95/022-E. 181 S. Abachi et al., DO Collaboration FERMILAB-PUB951028-E. 191 K. Abe et al.. The SLD Collaboration, Phys. Rev. Lett. 70 (1993) 2515; 73 (1994) 25. IO] U. Uwer, talk at XXX Rencontres de Moriond, March 1995; K. Einsweiler. talk at the 1995 APS Meeting, Washington, D.C. 11 1 D. Schaile, talk at XXVII Int. Conf. on High Energy Physics, Glasgow, July 1994. 121 M. Demarteau et al., Combining W Mass Measurements, CDF/PHYS/CDF/PIJBLIC/2215 and DONOTE 211.5. [ 131 H. Burkhard, E Jegerlehner, G. Penso and C. Verzegnassi Z. Phys. C 43 ( 1989) 497; E Jegerlehner, preprint PSI-PR-91-16 ( 1991), in: Progress in Particle and Nuclear Physics, ed. A. F;issler, Pergamon Press, Oxford. U.K.; A.D. Martin and D. Zeppenfeld, University of Wisconsin preprint MAD/PH/855, November 1994; M. Swartz. SLAC preprint SLAC-PUB-6710, November 1994; S. Eidelman and E Jegerlehner, Paul Scherer Institute preprint, PSI-PR-95-I. January 1995; H. Burkhard and B. Pietrzyk, talk at XXX Rencontres do Moriond, March 1995. ( 141 S. Pokorski and PH. Chankowski, plenary talk presented at the Int. Conference on Physics Beyond the Standard Model IV, Lake Tahoe, CA, December 1994, to be published in the Proceedings, Warsaw University preprint IFT-9515. 1151 A. Blonde1 and C. Verzegnassi, Phys. Lett. B 31 I ( 1993) 346.
17 August 1995
A@2
PHYSICS
LETTERS
B
Physics Letters B 356 (1995) 307-312
ELSEVIER
The mass of the Higgs boson in the Standard Model from precision tests * Piotr H. Chankowski a, Stefan Pokorski b*l a Institute of Theoretical Physics, Warsaw University ul. Hoia 69, 00-681 Warsaw, Poland b Max-Planck-Institute fiir Physik, Werner-Heisenberg-Institute, Fijhringer Ring 6, 80805 Munich, Germany
Received 16 May 1995; revised manuscript received 9 June 1995 Editor: P.V. Landshoff
Abstract
The LEP data and the measurement of MW confirm the electroweak symmetry breaking by the Higgs mechanism with (2-31% accuracy. For the Higgs boson mass, fits to the present data (without the SLD result) give Mh = 121:;: GeV and Mh < 800 GeV at 95% CL. We analyze those fits from the point of view of the impact of different measurements on the final result for Mh. All the relevant observables can be grouped into two categories: those which correlate Mh with mr and those which constrain the range of M,. Using this classification we discuss potential improvements in the limits on Mh. anticipating improvements in the precision of the data. Also the sensitivity of the limits on Mh to the value of (Y~(Mz) is discussed in a toy model which reproduces all the results of the SM but the Z" -+ bb width is increased by hand so that Rb = 0.22 and, in consequence, the fitted value of crs( Mz) M 0.11 (qualitatively similar situation occurs in the Minimal Supersymmetric Standard Model with light chargino and right-handed top squark) .
The LEP data and the measurement of Mw confirm the electroweak symmetry breaking by the Higgs mechanism with few per mill accuracy. For some time they are used by several theoretical groups [l-5] and by the experimental groups [6] to constrain the less precisely known parameters of the Standard Model: the top quark mass m,, the strong coupling constant LY,? ( Mz ) and the Higgs boson mass Mh. The loop corrections to the electroweak observables are O( rnf ) in the top quark mass and only O( log Mh) in the Higgs boson mass. Nevertheless, the increasing precision of the data and in particular the discovery of the top quark *Supported in part by the Polish Committee
for Scientific
Re-
search and European Union Under contract CHRX-CT92-0004. 1On leave of absence from Institute of Theoretical Physics, Warsaw University.
Ekevier Science B.V. SSDI 0370-2693(95)00809-8
[ 7,8] improve the prospects for constraining the Higgs boson mass in a way which may be relevant for planning the future experiments for direct Higgs search. In this short note we analyze the fits to the electroweak data from the point of view of their constraints on Mh. We follow the usual strategy: in terms of the best measured observables GF, (YEMand MZ we calculate Mw, all partial widths of Z” and all asymmetries at the Z” pole and determine m,,a,(Mz) and Mh by a fit to the data. The calculation is performed in the on-shell renormalization scheme, with all “oblique” and process dependent one-loop corrections as well as the leading higher order effects included. The experimental input (i.e., experimental values for the electroweak observables, their errors and correlation matrices) used in the fits is summarized in Ref. [6].
308
P.H. Chankowski, S. Pokorski/Physics
Table I Results of the fits
Letters B 356 (1995) 307-312
to the data (61. Ail masses in GeV
fit
tnr
ANI,
Mh
AMh
a(Mz)
Aa*
X2
d.o.f
-SLD
168
12
121
+2U7 -58
0.124
0.005
11.1
1s
+SLD
166
II
63
+97 4
0.123
0.005
16.5
16
For the top quark mass we use the CDF result m, = ( 176 & 13) GeV [ 71. In the most recent data of Refs. [6,10] there are two significant changes compared to the data of the Glasgow conference [ 11,121: the Ob have increased and central value of Mw and of AFB they now read Mw = (80.33 f 0.17) GeV, AK = 0.1015 5 0.0036. In Table 1 we present the results of our global fit without and with the SLD result for the electron leftright asymmetry [ 91 included. In both cases the lower limits on Mh come from the unsuccessful direct Higgs boson search. The differences between those results and the earlier fits [ 3,5,6] are mainly due to the above mentioned changes in data (as well as the overall better precision of the LEP data). Our main aim, as said above, is to analyze in more detail the limits on Mh, with emphasize on the upper one. To be on the conservative side we shall use the fit without the SLD measurement included because its inclusion would significantly lower the inferred upper
‘888 800 700 600 500 > @ c3
400 300
100 2:: 70 60 50
160
165
170
175 m,
180
185
190
195
200
(GeV)
1.Lines of constant values (for fixed aJ- (Mz) = 0.126) of: I ) Tz, 2) A!:, 3) X2 for Mw, 1- Mf+,/M$, A!$, sin’SFg( (QFB)).
Fig. A,,
AT and A:;.
0.0287
(dashed).
hadr = 0.0280 4) RI for AaEM
(solid lines)
and
bound on Mh. It is instructive to begin with plots in the (m,, Mh) plane of the contoursof constant values of several electroweak observables. In Fig. 1 such lines are shown for the central experimental values Tz = 2.497 1 GeV, AK = 0.1015, RI= 20.800 and also we show the contour of minimal x2 for the set Mw, 1 - M$/M'$,A&,
A!&,sin2 OF< (QFB)), A, and A. The latter can be looked at as the contour of constant Mw with the value of MW determined from the best fit to the above seven measurements: to a very good approximation all those variables are in the SM determined, e.g., in terms of Mw (the oblique corrections are dominated by Ap and the specific, process dependent corrections being negligible) . The contours are shown for two fixed values of the hadronic contribution to (YEM(Mz) which has been estimated to be AcY~$ = 0.0280 f 0.0007 [ 131. The value of LY$(Mz) is 0.124 (as obtained from the fit). We see that in the range 160 < m, < 200 GeV all contours are approximately linear in the variables logMh> Mr. First, let us focus on the plots for the central value of A&$. The striking feature of Fig. 1 is that for A&$ = 0.0280 and the central experimental values of the observables the contours are almost parallel, with theexception of RI. Although RIremains constant along drastically different lines, its dependence on the (m,, Mh) is much weaker and with the experimental error RI = 20.800 f 0.035 the overall x2, without inclusion of Rt, and the CDF result for m,, remains almost constant along the direction of the contours 1, 2, 3. This is clearly seen in Fig. 2 where we show the contours of the overall Ax2 = 1 in a fit without Rb and the CDF value for m, included which, for A&$ = 0.0280, consist of two approximately parallel lines. Thus in this case the fitted value for Mh results in a very transparent way from a combination of effects which can be organized into the following two-step description. A fit to the measured Mw = (80.33 i 0.17)
P.H. Chankowski, S. Pokorski/ Physics Letters B 3% (1995) 307-312
9Q, W
1000 900 600
1000 900 800
700
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500
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309
5 =
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,,., :
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*’
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I
’
0,
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5
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*
5,
160
1, 190
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fl
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mt (GeV)
Fig. 2. Contours AX* = 1 (without Rb and the CDF direct measurement of WI, in the fit) in the plane (MI,,kfh): a) in the SM with the data of Ref. [6] for different A&$ -... and b) for A&$ = 0.0280 in the SM with anticipated reduced experimental errors (see text) and our toy model which gives an improved Rb. -.‘.
GeV [ lo] and to all measured electroweak observables but Rb = IZO_+~b/I’Z~_+hadronsgives x2 values which are almost independent of the value of the top quark mass m, in the broad range (150-200 GeV) and with the best value of log Mh which is almost linearly correlated with m,.We get, e.g., Mh = 138’_‘,L9 GeV for m, = 170 GeV and Mb = 280*_;!& GeV for m, = 180 GeV. The most recent experimental results [ 6,101, in particular the increase in the central value of Mw by 100 MeV and the new measurement of A:;, evolved in the direction of requiring smaller M,, for a given, fixed m, (or larger m, for a given, fixed Mh). With the data reported at the Glasgow conference [ 111 the analogous result is Mh = 283:::; GeV for m, = 170 GeV and Mh = 511 f_329;)9 GeV for m, = 180 GeV. The (m,,Mh) correlation is the most solid result of the fits which does not depend on whether Rb and/or m, measurement of the CDF [ 71 are included into the fits. It points toward relatively light Higgs boson for nz, in the range ( 170-180) GeV. The x2 isflat as a function of mt unless &, and/or the directly measured in Fermilab mass of the top quark are included into the fit. These are the only two measured observables which in the present case introduce visible x2 dependence along the curves in Fig. 2
and can, therefore, put indirectly (by constraining m,) relevant overall limit on the Higgs mass Mh. We see it very clearly in Fig. 3 where we plot the values of x2 as a function of Mh for several values of m,,in the fits with and without Rb and the CDF result for m,. The best values of m, and Mh aregiven in Table 2 (for fits to the data of Ref. [6]) and the la contour plots are shown in Fig. 4. There is some dependence of the limits on Mh on the value of Aa& = 0.0280 & 0.0007 [ 131. An increase of Aa$$ has the same effect as an increase of the Higgs boson mass, i.e, the same values of the observables are now reproduced with smaller value of Mh. This is shown in Fig. 1. The shift towards lower values of Mh is different for different observables and the flat direction in x2 along the “diagonal” in Fig. 1 disappears for AL&$ larger by 1~ than the Table 2 Best values of WI, and Mh for Aa$‘ -Rb, mr Mtl X2
-I&
undetermined undetermined 3.2
+Rb, 152 41 9.1
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= 0.0280 and various X2 fits -Rb. 174 183 3.5
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about improvements in those limits anticipating still better precision of the data. experimental Smaller errors for Mw,
sin2~$t((QF-d),
-4, .& A&, Tz, uhadr, Ak and Ag will result in smaller error on A4h for any fixed value of m,, whereas more precise measurement of RI, m, and Rb will constrain the acceptable range of m,. In Fig. 5 we show the results of a fit to the hypothetical experimental data which would give the x2 value of the fit similar to the present data but have the anticipated experimental errors for Mw, AK,, sin’ @Fl’( (QFB)), A, & RI, TZ and rhhadr a factor of 1.5-2 smaller. This hypothetical set of data is obtained by starting with the present experimental values of the observables and by shifting them in the direction of the values obtained in the present best fit and by fixing the central values of the hypothetical data according to the formula: XI -X0
-------=
x2
central value of A&$. Cases a) and b) correspond to x2 without and with the CDF direct measurement of ml included into the x2 respectively. Dashed lines show x2 without Rb in the fit.
central value, as seen in Fig. 2 (now the Ax2 = 1 contour is closed even without Rb and mt contributions). This is due to a combined effect of lowering the curve for R1 = 20.800 and the relative shifts of the curves Tz = constant, Mw = constant which are “less parallel”. The resulting limits on Mh with the CDF result for m, and/or Rb included into the x2 are of course stronger (Fig. 4). Obviously, opposite conhadr. However, as seen in elusion holds for smaller Ann,, Fig, 4, for Ac~pd at lc+, the central values of Mh remain within la contour obtained for A&$ = 0.0280. hadrhas a weak effect Therefore, the uncertainty in AaEM on the global lit with this uncertainty propagating in the errors. Scanning over ALY$ we get the results in Table 1. Given the clear separation of the impact of various measurements in obtaining limits on Mh, we can ask
x0 (1)
c2
(+I
Fig. 3. x2 as a function kfh for different values of ml and the
-
where X1 ((~1), X2 (~2) are the present value (error) and the hypothetical value (anticipated error) for the variable X respectively and Xa is its present best fit value. We use then Mw = (80.332 f 0.100) GeV, Ag = 0.01665 f 0.00080, sin2 $i’( (Qm)) = 0.2319&0.0008,d, = 0.1406f0.0050~ = 0.1419* 0.0050, RI = 20.801 f 0.020, I2 = 2.4977 f 0.0020 and ch& = 4 1.4675 & 0.0500. For the anticipated top quark mass value from the Fermilab we take mFyp = ( 174f8) GeV. Thus a fit to the hypothetical data gives the same results for Mh and mt, with the same x2, with however smaller errors. We see that the considered improvement in the experimental precision would reduce the la uncertainty in the Higgs boson mass to “_:‘,” GeV in the global fit and to, e.g., +I4 _s2 for fixed m, = 170 GeV In particular this conclusion almost does not depend on whether Rb (with its present value) is included into the fit or not. The dependence on cys(Mz) is best illustrated by considering a toy model which reproduces all the results of the SM with, however, a correction to the Z” -+ 6b width given by
r Z”-66 +
= ‘;!t+&b
(7.5 x IO-’ x m, - 3.83 x 10T3 GeV)
(2)
Then, &, M 0.22 for all the values of my. The interest in this possibility stems from the fact that a)
P. H. Chunkowski, S. Pokorski / Physics Letters B 356 (I 995) 307-312
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Fig. 4. Contours Ax* = 1 with the CDF result for m, and/or Rb included in the fit in the plane (rnt, Mh) for different Aop$. 1000 900 600
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Fig 5 Contours Ax* = 1 with the CDF result for mr and/or &, included in the fit in the plane (ml, Mh) for Aat$ anticipated reduced errors of the main observables (see text) and for our toy model with gives improved Rb.
= 0.0280 for
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as seen in Table 1, in the SM the single variable Rh contributes more than 70% of the total x2; b) a qualitatively similar situation occurs in the Minimal Supersymmetric Standard Model with light chargino and light right-handed top squark [ 14 J. It follows that the best fit gives now LY~ (Mz ) = 0.108. The correlation between the value of rZo_bb and the fitted value of cu,( Mz) is well known [ 15,6] and it is due to the contribution of rZo_hb to RI. From the point of view of the limits on Mh the variable &, becomes irrelevant (its contribution to x2 x 0). The results obtained in that case are shown in Figs. 2, 3. We see that smaller CU,~( Mz) gives similar effect to a decrease in A&$,$ but it is clear from Fig. 2 that the dependence on a.(.( Mz) is very weak. The main effect visible by comparing Fig. 5 and Fig. 4 is due to the fact that Rb is now irrelevant in the fit. Our discussion of the effects of reduced experimental errors applies as well to this model. In summary, precision tests of the Standard Model confirm with very high accuracy the electroweak symmetry breaking by the Higgs mechanism. For the Higgs boson mass, fits to the present data give MI, = 121?$’ GeV and M/, < 800 GeV at 95% CL. Of course, for fixed values of m, one gets stronger bounds, e.g., form, = 170 GeV we have Mh = 139112’ GeV. We have analyzed those fits from the point609f view of the impact of different measurements on the final result for Mh. At the accuracy of the data available at the present and planned accelerators, all the relevant observables can be grouped into two categories: those which correlate MI, with m, (Mw, AEb, sin2 6$’ ( (QFB)), A, A, RI, TZ and ch&) and those which constrain the range of m, (Rb, R, and the direct measurement of m,). Using this clasification we have discussed potential improvements in the limits on M,,, anticipating improvements in the precision of the data. Finally we have studied the sensitivity of MA to a,(Mz) in a toy model which reproduces all the results of the SM with, however, the 2’ --+ 66 width increased by hand so that Rt, = 0.22; the fitted a,$( Mz) is then 0.108. PCh. would like to thank the Max Planck Institute fiir Physik for warm hospitality during his stay in Munich where part of this work was done. Note added. We are grateful to J. Ellis for pointing out to us that also the low energy data break positive
Letters B 354 (1995) 307-312
m, - Mh correlation and it was the inclusion of all those data which made possible to give the first discussion of bounding Mh from precision electroweak data [ 11. Now, wj th better precision of the LEP data and directly measured top quark mass, the role of the low energy data is less important.
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