

 12 Jul 1994

Top Quark Mass and Supersymmetry Stefan Pokorskia\Lambda y z aMax-Planck Institute for Physics, Werner Heisenberg Institute, Foeringer Ring 6, 80805 Munich, Germany

We summarize the expectations for the top quark mass in supersymmetric models and discuss the potential implications of its value measured by the CDF group in Fermilab.

1. Introduction

Evidence for the top quark has been recently reported by the CDF group at Fermilab, with the mass Mt = (174 \Sigma 17) GeV. It is, therefore, interesting to summarize the expectations for the top quark mass in supersymmetric models and to discuss the potential implications of the measured value.

2. Top quark mass and precision tests of

the minimal supersymmetric standard model (MSSM)

The MSSM can be viewed as the low energy effective theory of a wide class of supersymmetric models with R parity conservation [1]. It is important to test the model independently of its high energy origin as far as possible. Several groups (Altarelli et al., Chankowski et al., Ellis et al., Haber et al., Langacker et al.) proceed with systematic precision tests of the MSSM based on the electroweak data. This programme is not yet fully completed as a) not all process dependent one-loop supersymmetric corrections have been included, b) the parameter space of the MSSM has not been fully explored in the simultaneous analysis of all available electroweak data. The complete one-loop MSSM analysis exists, so far, only for the corrections to the mass of the W boson [2]: MW = f(G_; ffEM; MZ; Mt; Mh; : : :)

\Lambda On leave from Institute for Theoretical Physics, Warsaw University, Hoza 69, 00-681 Warsaw, Poland. ySupported by the Ministry for Science, Research and Culture of Land Brandenburg, contract II.1-3141-2/8(94). zPartially supported by a grant of the Polish Committee

for Scientific Reseach.

which are conveniently parametrized by the quantity \Delta r:

G_ = p2 = ssffEMi

1 \Gamma M

2 WM

2 Z j M

2W [1+

+ \Delta r (ffEM ; MW ; MZ; Mt; Mh; : : :)] (1) The dots stand for the additional parameters present in the MSSM. In the standard model, the measured value of \Delta r = 0:044 \Sigma 0:015 gives an upper bound on the top quark mass as a function of the Higgs boson mass Mh. In the MSSM, in spite of several additional free parameters, the following "theorem" holds [2]: for the same mass Mh of the standard model Higgs boson and of the lighter Higgs boson in the MSSM, the standard model bound on Mt obtained from \Delta r cannot be softened and is reached only for soft slepton masses M~l * 0(100 GeV) and soft squark masses m~q* 0(200 GeV). Remembering that in the MSSM there exists the upper bound Mh ^ 150 GeV (for Mt ! 250 GeV and m~q! 2TeV), one obtains from \Delta r at 1oe level

Mt ^ 190 GEV (2)

One should, however, be aware of the fact that the measured values of the other electroweak observables (in particular \Gamma Z!b~b) seem to constrain the top quark mass in the standard model from above much stronger than \Delta r. There, the MSSM may help to reconcile the heavy top quark with precision data [3] and it is clear that the extention of the type of analysis performed in ref. [2] to other observables is most welcome. It is also clear that, with the measured value of the top quark mass so close to the bound (2), such an

analysis will put strong constraints on the sparticle masses. Since the value of the top quark mass is measured, the precision electroweak data can be used to constrain the parameters of the MSSM.

Another interesting issue is the significance of the measured value of the top quark mass for the minimal supersymmetric model with unification of strong and electroweak forces. This will be discussed in the rest of this lecture, by gradually supplementing the MSSM with additional assumptions.

3. Top quark mass and perturbative

Yukawa couplings up to 0(1015 \Gamma 1019 GeV).

Present unification scenarios rely on perturbative physics. The requirement of the top quark Yukawa coupling to remain perturbative up to the GUT scale puts the upper bound on its value at MZ:

ht(MZ) ! hMAXt (MZ ) j hIRt (3) where hIRt is the so-called quasi-infrared fixed point value [4] of ht. In the MSSM, assuming its validity all the way up to the GUT scale, one obtains then the following upper bound for the top quark running mass [5]:

M MAXt = h

MAXt

g2 MW sin fi, = (190 \Gamma 210) GeV sin fi (4)

where tan fi= v2=v1 is the ratio of the two Higgs vacuum expectation values and the range of the predicted values corresponds to the uncertainty in ffs: ffs(MZ) = 0:11 \Gamma 0:13. The prediction for M MAXt (pole) (after inclusion of the two-loop QCD corrections to the running mass) as a function of tan fi is plotted in Fig. 1.

The comment worth making at this point is that the measured Mt is stunningly close to its quasi-infrared fixed point values, for generic values of tan fi ! Is there any theoretical reason to expect that ?

1 10

100 120 140 160 180 200

Figure 1. The top quark mass as a function of tan fi. The region bounded by solid lines: the quasi-infrared fixed point prediction with ffs = 0:11 \Gamma 0:13. Dashed curve: the lower bound predicted by the b-o/ Yukawa unification (the upper bound coincides with the IR upper bound). Narrow strip: the prediction following from the t-b-o/ Yukawa unification. Horizontal lines: experimental result Mt = (174 \Sigma 17) GeV.

4. Top quark mass and coupling unification

in the MSSM

The gauge coupling unification is almost insensitive to the value of the top quark mass. However, Mt is an important parameter for the bottom-tau Yukawa unification, which is predicted by simple grand unification scenarios (SU(5), SO(10)). This is due to the fact that the top quark Yukawa coupling, if large enough, affects the renormalization of the bottom Yukawa coupling in its running from MZ to the GUT scale according to the equation

4ss ddt ` hbh

o/ ' =

hb ho/ `

16ffs

3 \Gamma 3

h2b 4ss \Gamma

h2t 4ss + 3

h2o/ 4ss '(5)

with t = 2 ln i MZQ j.

It turns out that the strong interaction renormalization effects present in eq.(5) are, with ffs(MZ) = 0:11 \Gamma 0:13, too strong. Starting with the ratio hb=ho/ = 1 at the GUT scale and neglecting the Yukawa couplings in eq.(5) one obtains the bottom quark pole mass well above the experimental range Mb = (4:9 \Sigma 0:3) GeV (a recent analysis of the B decays based on the QCD sum rules suggests even smaller value of Mb; R. Rueckl, private communication). Thus, indeed, a large top quark Yukawa coupling is necessary for the bottom-tau Yukawa unification in the MSSM.

An important implication of the measured value of the top quark mass is that, in the MSSM, in addition to the gauge couplings, also the b and o/ Yukawa couplings unify at least within (20- 30)% accuracy for a broad range of tan fi values.

To make the relation between Mt and b- o/ Yukawa unification more precise it is convenient to reverse the problem: assume b- o/ Yukawa unification and predict Mt as a function of tan fi (using Mo/ and Mb = (4:9 \Sigma 0:3) GeV as input parameters). This prediction is also shown in Fig. 1 and it almost coincides with the perturbative upper bound M MAXt (tan fi) [6]. The b-o/ Yukawa unification implies the top quark mass to be very close to its quasi-infrared fixed point values. Several comments to Fig. 1 are in order. First, the region in Fig. 1 bounded from below by the dotted line is the prediction which follows from the b-o/ Yukawa and gauge coupling unification. For a detailed discussion of the role played by the gauge coupling unification to get the results shown in Fig. 1 we refer the reader to ref.[7]. Another remark is that relaxing the b- o/ Yukawa unification by 10% gives the same predictions for Mt, with Mb(pole) =4.9 GeV, as the exact unification with Mb = 5:2 GeV. For larger deviations from hb=ho/ , the predicted values of Mt fall below the IR values, e.g. for hb=0.8ho/ the plateau is at , 180 GeV. Finally, the IR fixed point values of Mt and the predictions following from the b-o/ Yukawa unification overlap in almost the whole region of tan fi, but with exception of large tan fi region. There, the b-o/ Yukawa unification does not imply such close proximity

to the IR fixed point because hb renormalizes itself (we reach the region hb? ht). In the large tan fi region the full unification ht =hb=ho/ is possible. If we impose it then the predicted value of Mt is again closer to the IR fixed point values (the narrow strip in Fig. 1).

We can conclude as follows: The b-o/ Yukawa coupling unification (and Yt = Yb = Yo/ at very large tan fi values) implies Yt(MZ) to be very close to its quasi-IR fixed point value. The value of the top quark mass is then strongly correlated with tan fi and a) Mt within its present central experimental values ss (170-180) GeV (or below) corresponds to tan fi! 2 or tan fi,= Mt=Mb (see Fig. 1); b) Mt in the upper range of the presently reported values can even be consistent with tan fi in the plateau region of Fig. 1.

The (Mt,tan fi) correlation which follows from the proximity of the Yt to the IR fixed point has important consequences for the prediction for the lightest Higgs boson mass Mh in the MSSM. As is well known, the upper bound on the mass Mh is the function of tan fi and, therefore becomes now much stronger than the absolute upper bound for a given Mt. This is shown in Fig. 2 and is an encouraging message for the Higgs boson search at LEP2. (I am grateful to R. Barbieri for suggesting this plot to me.)

Finally, even as low a value of Mt as 160 GeV is consistent with all values of tan fi if we do not require the b-o/ Yukawa unification to better than (20-30)% accuracy.

5. Top quark mass and radiative electroweak breaking

The top quark mass is an important parameter for the mechanism of radiative electroweak breaking. In the unification scenario the soft supersymmetry breaking parameters in the scalar potential are postulated to be (almost) universal at the GUT scale and radiative electroweak breaking is the necessary requirement for such models. The discussion in the previous section suggests two systematic ways of investigating the impact of the large top quark mass on predictions of the MSSM with radiative electroweak breaking:

150 160 170 180 190 20040 60 80 100 120 140

top pole mass, [GeV] Higgs mass, [GeV]

Figure 2. The upper bounds on the lightest Higgs boson mass as a function of the top quark mass. Solid upper curve: the absolute upper bound in the MSSM with m~q ! 2TeV. Dotted line: the upper bound with the top quark Yukawa coupling close to its IR fixed point. Solid lower curve: the lower bound in the model with radiative elecrtoweak breaking and Yt close to its IR fixed point value.

(a) study the predictions of the model

with ht ss hIRt as a function of Mt (i.e. as a function of tan fi) for, say, 160 GeV! Mt !200 GeV;

(b) study the predictions with, say, Mt =

180 GeV as a function of tan fi, for small and intermediate values of tan fi (i.e. gradually departing from the ht ss hIRt case)

(The large tan fi case, tan fiss Mt=Mb has several characteristic features which have to be discussed separately [8]).

In case (b), the parameter space consistent with radiative breaking is weakly dependent on tan fi.

The two effects: increase of tan fi and a gradual departure of ht(MZ) from its quasi-infrared fixed point value partially cancel each other. Therefore, for the generic predictions of the model (with radiative electroweak breaking and the heavy top quark) for the superpartner mass spectrum it is sufficient to consider the case of ht ss hIRt .

The main characteristic pattern of the parameter space consistent with radiative electroweak breaking for ht ss hIRt is the correlation between the Higgs mixing parameter _ and the universal gaugino and scalar masses at the GUT scale M1=2 and mo, respectively [9]:

_2 + M

2Z

2 = m

2o 1 + 0:5 tan

2 fi

tan2 fi \Gamma 1

+ M 21=2 0:5 + 3:5 tan

2 fi

tan2 fi \Gamma 1 (6) The relation (6) follows from the requirement that the Higgs potential with quantum corrections included by the renormalization group evolution from the GUT scale has the proper minimum at the electroweak scale. This condition reads

tan2 fi = m

2H

1 + _

2 + M 2Z=2

m2H

2 + _

2 + M 2Z=2 (7)

and the running of the Higgs mass parameters gives

m2H

1 (MZ) = m

2o + 0:5M 2

1=2

m2H

2 (MZ) = \Gamma 0:5m

2o \Gamma 3:5M 2

1=2 (8)

The relation (6) is obtained from eqs.(7) and (8). The above discussion is based on the one-loop RGE which give the correct qualitative insight into the numerical results of ref.[9], obtained with two-loop running and the leading threshold effects included [10].

It follows from eq.(6) that

_ ? M1=2 (9) and, therefore, the lightest neutralino is strongly dominated by the gaugino component [11] (the low energy gaugino mass M2 ss 0:8M1=2, so that _ ? M2). In consequence, the neutralino annihilation proceeds mainly through the slepton exchange (annihilation into quarks mediated by

squark exchange is strongly suppressed: m2~q = 0:5m2o + 6M 21=2 AE m2O/):

oeann , m

2O/

(m2~l + m2O/)2 (10)

where mO/ and m~l are the lightest neutralino and slepton masses, respectively. The requirement that the neutralino relic abundance satisfies the condition \Omega ! 1 (\Omega , oe\Gamma 1ann) puts then strong upper bounds on the values of the neutralino and slepton masses. The bound on mO/ is stronger than the absolute bound O(1 TeV) of ref.[?] mainly because only leptonic annihilation channels are now effectively open. It translates itself in an obvious way (mO/ , 0:4M1=2 and M2 , 0:8M1=2) into an upper bound on M2 and in consequence on the chargino mass. The bound on slepton masses can be relaxed only when 2mO/ ss Mh or 2mO/ ss MZ. The weak couplings of a gaugino-like neutralino to h and Z bosons (vanishing in the limit of pure gaugino) are balanced by the resonance and an acceptable relic abundance can be obtained, no matter how heavy the sleptons are (calculation of the annihilation cross section on a resonance needs special care [13]). Those predictions for the chargino and slepton masses are shown in Fig.3 and are generic for the model with radiative electroweak breaking in the presence of the heavy top quark, if acceptable relic abundance of the lightest neutralino is required. The results for larger values of tan fi are qualitatively similar, with the upper bound on the chargino mass slowly increasing with tan fi.

I am grateful to P. Chankowski, M. Olechowski, M. Carena, J. Rosiek, C. Wagner, A. Dabelstein, W. Hollik, P. Gondolo and W. Moesle for fruitful collaboration and to W. Moesle for his patient help in preparing the figures.

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Figure 3. The region of the chargino and slepton masses consistent with \Omega h2 ! 0:7 in the model with radiative electroweak breaking.

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