Electroweak Precision Data, Light Sleptons and Stability of the SUSY Scalar Potential

The light slepton-sneutrino scenario with non-universal scalar masses at the GUT scale is preferred by the electroweak precision data. Though a universal soft breaking mass at or below the Plank scale can produce the required non-universality at the GUT scale through running, such models are in conflict with the stability of the electroweak symmetry breaking vacuum. If the supergravity motivated idea of a common scalar mass at some high scale along with light sleptons is supported by future experiments that may indicate that we are living in a false vacuum. In contrast SO(10) D-terms, which may arise if this GUT group breaks down directly to the Standard Model, can lead to this spectrum with many striking phenomenological predictions, without jeopardizing vacuum stability.

The electroweak precision ( EWP ) tests by the experiments at LEP and SLC[1] are on the whole in excellent agreement with the Glashow -Weinberg -Salam standard model (SM) . However, if some judiciously chosen sub -set of the data is examined, a few unsatisfactory features of the SM fit are revealed [1,2] • The measured values of the parameter sin 2 θ ef f from the observables A LR and A b F B differ at 3.5 σ level.
• Moreover, the value of this parameter as given by the hadronic asymmetries and the leptonic asymmetries also exhibit a considerable discrepancy (at the 3.6 σ level).
• When a global fit is performed a χ 2 /d.o.f = 26/15 corresponding to a C.L. = 0.04 is obtained, which is hardly satisfactory. These observations tempt one to conclude that the hadronic data may be plagued by some hitherto unidentified experimental problem and, hence, the leptonic data should be taken more seriously [2]. This conclusion is challenged by the direct lower bound on the Higgs mass m H > 113 GeV [3] and its indirect determination from EWP data considering the leptonic asymmetries only [4,2]. Using sin 2 θ ef f measured from both hadronic and leptonic asymmetries, the central value of the fitted Higgs mass and the 95 % C.L. upper limit on it happens to be 98 GeV and 212 GeV respectively [1]. These values consistent with the direct search limit, have been confirmed by [2]. However, if sin 2 θ ef f from leptonic data only is employed, the corresponding numbers become 42 GeV and 109 GeV, a situation which is hardly acceptable vis -a -vis the direct limit.
It must be admitted that there are uncertainties in the fitted value of m H [2]. The result has some sensitivity on the value of α QED (m Z ) which is scheme dependent although most of the existing schemes lead to upper bounds on m H in conflict with the direct search limit. Uncalculated higher order effects may have a modest impact on the fitted value of m H [2]. Finally, if the current 1 σ upperlimit of the top mass (m t = 174. 3 + 5.1 GeV ) rather than its central value is used in the fit, then the compatibility of the fitted value of m H with the direct search limit improves.
Although these uncertainties may conspire to produce an agreement between the leptonic EWP data and the direct limit on m H within the framework of the SM, the situation is sufficiently provoking to reanalyse the data in extensions of the SM.
One interesting possibility is to extend the discussion within the framework of supersymmetry [5]. Altarelli et al [2] have found the MSSM parameter space ( PS ) where the SUSY corrections to the elctroweak observables are sufficiently large and act in the direction of impoving the quality of fit. The most significant loop contributions come from the sneutrino (ν), in particular if sneutrino mass is in the in the range 55 -80 GeV, and a perfect agreement with the data is obtained with m H = 113 GeV. The charged left -slepton (l L ) mass is related to mν by the SU(2) breaking D -term: m 2 in a model independent way. Since it must be heavier than 96 GeV according to the LEP direct search limits on charged sleptons [6], the parameter tanβ must be moderately large which is not a severe restriction.
This spectrum, however, is incompatible with the popular mSUGRA [7] scenario with a common scalar mass m 0 at the GUT scale (M G ). Within the framework of mSUGRA such light sneutrinos automatically demand even lighter right-sleptons, which are already ruled out by the LEP mass limits on charged sleptons. Thus one has to look for alternatives with nonuniversal scalar masses at M G . In this paper we shall look for such alternatives and scrutinize them in the light of vacuum stability.
We shall consider only those class of models where the sfermions of the first two generations are nearly degenerate with mass m 0 at M G , as is required by the absence of flavour changing neutral currents. Moreover we shall assume a universal gaugino mass m 1/2 at M G as this assumption is likely to be valid in all GUT models irrespective of the specific choice of the GUT group. Given these parameters the left-slepton and sneutrino masses of the first two generation at the weak scale can be computed by using the standard one loop renormalisation group (RG) equations. Other SUSY paramters may influence the running at the two loop level. Using ISAJET -ISASUSY we have convinced ourselves that these higher order corrections are indeed negligible. We constrain m 0 and m 1/2 by requiring 55GeV < mν < 80 GeV at the weak scale (Fig. 1). The only other relevant SUSY parameter that enters the analysis through the SU(2) breaking D -term is tanβ, although the dependence on it is rather weak. Almost identical allowed PS is obtained for all tanβ > ∼ 5. As long as tanβ is not too large (say, tanβ < ∼ 20),τ L will be degenerate with the sleptons of the first two generations (to a very good approximation). For larger tanβ, it may be somewhat lighter. Since the experimental bound on theτ L mass is considerably weaker (mτ > 68 GeV) than that for the selectron and smuon, higher values of tanβ can also be considered in principle, although we shall not pursue this case further.
The range of m 0 and m 1/2 shown in Fig. 1 may be moderately altered if one considers a large hierarchy among the scalar masses at M G . This happens due to the presence of a particular term in the RG equation which is usually neglected in the mSUGRA approximation (See eq. 4 and the discussions following it). We shall consider below a specific model with this feature.
So far no assumption about the other soft breaking parameters was necessary. However in order to take into account the chargino mass bound mχ± > 100 GeV [6] and to test the stability of the scalar potential [8,9] one has to specify more SUSY parameters. In general mχ± depends on the Higgsino mass parameter (µ) and tanβ in addition to m 1/2 .
The entire range of m 1/2 in Fig 1 is such that µ can be chosen so as to make mχ± consistent with the LEP bound. Of course mχ± is not a very sensative function of µ unless it is very small (µ < ∼ 100 GeV). We next turn our attention on mẽ R and the stability of the scalar potential [8,9] Before looking into specific models it is worthwhile to focus on some generic features of models with light sleptons. In several recent works [9,10,11] on the stability of the standard electroweak symmetry breaking ( EWSB ) vacuum, it has been found that low mass sleptons (to be more specific, sleptons significantly lighter than the electroweak gauginos) are somewhat disfavoured. In view of the fact that there is already a strong lower bound on the chargino mass it is important to check the compatibility of the light sneutrino scenario favoured by the EWP data and vacuum stability.
The unbounded from below 3 ( UFB3 ) direction [9] of the scalar potential, its evaluation procedure and the choice of the generation indices (i, j) which leads to the strongest constraint are elaborately discussed in [9,10]. To clarify why light sleptons are strongly disfavored, eqn. 93 of [9] has to be examined. The required equation is (1) Here λ e j is a leptonic Yukawa coupling and g 1 and g 2 are the U(1) Y and SU (2) gauge couplings respectively. The UFB3 constraint arises from the requirement that V U F B3 must be shallower than the EWSB minima (V 0min ) (See eqn 92 of [9]). To get the strongest constraints i = 1 and j = 3 is considered. Over a large region of the PS corresponding to light sleptons, the first term of eqn. 1 dominates when λ τ is substituted in the second term. The parameters are evaluated at a judiciously chosen renormalisation scaleQ where higher order loop corrections to the scalar potential are small and may be neglected [12,9]. At this scale, the mass parameter m 2 Hu (H u refers to the Higgs bosons coupling to the up-type quarks) gets a large negative value which is required by radiative electroweak symmetry breaking (REWSB). Thus the fisrt term tends to violate the UFB3 constraint for small values of ml L 2 i . Infact it has been shown in reference [11] that the anomaly mediated supersymmetry breaking ( AMSB ) model with light sleptons violate the UFB3 constraint.
We now wish to scrutinize the PS favoured by EWP data ( Fig. 1 ) in the light of the stability of the vacuum. At this stage we have to be more specific about the model since Hu , ml R 2 j and |µ| are also needed to check this point. We first consider a SU(5) SUSY GUT with a common scalar mass m 0 at the Plank scale (M P ≈ 2 × 10 18 GeV ) [13] instead of M G . An attractive feature of this model is that for the first two generations the mass ofl R (denoted by m 10 at M G ) belonging to the 10 plet of SU(5) happens to be larger than that of left slepton belonging to the5 representation ( denoted by m 5 at M G ) due to the running between M P and M G . Thus the conflict between the low mass sneutrino and the LEP limit onl R mass seems to be resolved, at least qualitatively.
For the 3rd generation, m 10 may be somewhat smaller if the relevant Yukawa couplings happen to be large at M G and contribute to the running (all relevant RG eqns are given in ref [13]). This however, may not be a serious problem since the limit on mτ R is considerably weaker as dicussed above.
When we look into the numerical details the situation, however, is far from simple.
According to Polansky et al the GUT scale values m 10 and m 5 for the first two generations are approximately [13] assuming that SUGRA generates the common scalar mass m 0 exactly at M P . Since m 1/2 has to be greater than 130 GeV (approximately; see Fig. 1) we find that m 5 is too large to give mν in the required range at the weak scale even if m 0 ≈ 0. We note that if the common soft breaking mass is generated well below the Plank scale this difficulty may be avoided. Moreover GUT theshold corrections, which cannot be computed precisely without specifying other GUT parameters like masses of heavy multiplets, may affect both m 10 and m 5 to some extent. In view of these uncertainties one can not discard this model on this ground alone. We shall henceforth treat m 10 and m 5 as phenomenological parameters at M G with m 10 > m 5 . Their actual values are to be chosen such that all charged slepton masses at the weak scale satisfy the LEP bound.
The Achilles' heel of the model however, happens to be the running of m 2 Hu between M P and M G . This running is controlled by not only the Yukawa couplings h t and h b but also by λ the coupling of the scalars belonging to the 5,5 and 24 plet of SU (5). In course of running m 2 Hu is usually reduced as one goes below M P , whereas m 5 driven by the gauge coupling alone increases. After considering various scenarios with different magnitudes of these couplings ref [13] has concluded that m 2 Hu < ∼ m 5 in general, while the equality holds if all the Yukawa couplings and λ are negligibly small . We have checked that in such scenarios the UFB3 constraint is always violated for the PS in Fig. 1  If one considers an SO(10) SUSY GUT instead, the matter fields of the first two generations belonging to the 16 plet remain degenerate at M G even if running below M P is considered. This will inevitably lead to a lightl R at the weak scale if the sneutrino mass is required to be in the range preferred by EWP data.
Thus running above the GUT scale alone in a SUGRA type scenario with a common scalar mass generated between M P and M G , is not likely to yield the spectrum preferred by EWP data if the stability of the vacuum is taken into account.
If one gives up the UFB3 constraint by assuming that the standard vacuum is only a false vacuum [14], while the global minimum of the scalar potential is indeed charge color breaking then the above constraints do not apply. If the tunnelling time for transition between the false vacuum and the true vacuum happens to be much larger than the age of the universe, such a model can not be rejected outright, although it seems to be against our intuitive notion of stability. Moreover the tunnelling time, which can be routinely calculated in models with a single scalar, can not be computed reliably in models with multiple scalars. Yet the conclusions derived in the preceeding paragraphs do not loose their significance. If future experimental data confirms light sleptons along with a mass spectrum stemming from a SUGRA motivated common scalar mass at some high scale < ∼ M P , then that would indicate that we may be living in a false vacuum, no matter how counter intuitive it may appear to be at the first sight.
The remaining of this paper shall deal with a type of non-universality which arises when a GUT group breaks down to a group of lower rank leading to non-universal Dterms at M G [15]. This type of models can produce the spectrum preferred by EWP data without violating the UFB3 constraint. As a specific example we consider an SO (10) SUSY GUT breaking down to the SM in a single step. The relevant mass formulae at M G are: where m D is the D-term with unknown magnitude, the common mass of all the members of the 16-plet of SO (10)  The methodology of finding the spectra is same as in [10]. µ and B are determined by the REWSB condition at a scale M S = √ mt L mt R . Then we put the experimental constraints. For a given m 16 and m 1/2 , we consider the smallest m D such that mν < 80 GeV. Larger values of m D may also be considered provided mν is in the range 55 GeV < mν < 80 GeV. However, larger values of m D tends to yield stronger UFB3 constraints.
We first discuss the APS without requiring Yukawa unification, in the m 16 − m 1/2 plane for m 16 = m 10 , A 0 = 0, tanβ = 15 and µ > 0 as shown in Fig. 2. The upper bound on m 1/2 for a given m 16 corresponds to the situation when no m D can give mν e,µ ≤ 80 GeV and the lower bound by experimental lower limit onχ ± . The D-term can control ml L and, hence mν, over a large range of m 16 , which, therefore, is found to be large. If we increase m 16 further, the contribution from τ Yukawa coupling decreases mτ L even for tanβ = 15 thanks to a large ml R . As a result mν τ falls below the experimental bound (43.6 GeV), even though mν e,µ are in the viccinity of 80 GeV. The upper and lower limits on m 16 significantly depends on A 0 and tanβ.
The fact that the allowed range of m 1/2 increases with m 16 is rather puzzling. The origin of this lies in a term in the RG eqn which is ususally neglected in mSUGRA.
The last term on the right hand side is zero at M G in the mSUGRA model. Moreover are not allowed if m h > ∼ 113 GeV is required. However, if m 16 is increased, the higgs mass increases appreciably through radiative corrections. Moreover the running of mτ L and hence of mτ ν , are also modest for low tanβ. Due to these reasons higher values of m 16 are allowed. We find 300(60) GeV < ∼ m 16 < ∼ 700(460) GeV for tanβ=7 (15), while the other parameters are the same as in Fig. 2.
Increasing the absolute value of A 0 makes large difference between mν e,µ and mν τ . As a result m 16 gets a stringent upper bound. It also lowers m H very rapidly giving a strong lower bound on m 16 . For example, 60(120) < ∼ m 16 < ∼ 460(420)GeV for A 0 = 0(m 16 ), the other parameters being the same as in Fig. 2.
There are also appreciable changes in the APS with change in the sign of µ. The masses mχ± and mτ L increase significantly as one change µ < 0 to µ > 0. One need high value of m 1/2 to keep mχ± above experimental bound and high value of m 16 for mτ L above experimental bound for µ < 0. For example 60(140) < ∼ m 16 < ∼ 460(440)GeV for µ > 0(< 0), while the other parameters are as in Fig.2.
We next examine the UFB3 constraint for the APS in Fig. 2. One of the important conclusions of this paper is that the UFB-3 constraint rules out the entire APS for the universal model (throughout this paper we shall use a * (+) for a UFB3 disallwed (allowed) points in the PS).
Next we will consider the effect of nonuniversality (compare Fig. 2 and Fig. 3). The decreases. However, a region is still UFB3 allowed for A 0 > ∼ 0, since m 2 Hu is somewhst larger at M G to begin with.
Next we consider the possibility of Yukawa unification in this model [16]. It has already been shown in [10,17] that full t − b − τ Yukawa unification does not permit low slepton masses even in the presence of D-terms. We shall, therefore, restrict ourselves to partial b − τ unification with an accuracy ≤ 5%. We fix tanβ to its lowest value required by unification. The APS in the universal model (Fig 4) is qualitatively the same as in the fixed tanβ case (compare Fig. 2 & Fig. 4) but its size somewhat smaller. It has been found that for higher values of m D unification requires relatively low values of tanβ ∼ 20 . The phenomenological significance of a light sneutrino has already been discussed at length in the literature [18][19][20][21][22][23][24] If the sneutrino mass happens to be in the range preferred by EWP data then it decays into the invisible channelν → νχ 1 0 with 100% BR and becomes an additional carrier of missing energy. If the lighter chargino mass happens to be near the current lower bound, a situation also preferred by EW precision data, then it would decay into the channelχ ± → ℓν with almost 100% BR (the decay into sleptons are phase space suppressed), while in the coventional mSUGRA scenario it dominantly decay into jets. Finally the second lightest neutralinoχ 0 2 which happens to be nearly degenerate withχ ± in models with gaugino mass unification, also decays dominantly into the invisible channelχ 0 2 →χ 0 1ν and becomes another source of missing energy.
The additional carriers of missing energy which play roles similar to that of the lightest supersymmetric particle ( LSP ), may be termed virtual LSP(VLSP) in the context of collider experiments [18] In the VLSP scenario the collider signatures of squark -gluon production are quite different from the ones in conventional mSUGRA model due to the additional carriers of missing energy. Moreover thanks to the enhanced leptonic decay of the chargino the lepton + jets + E T signal may increase at the cost of jets + E T signature [18,22] The hadronically quiet tri-lepton signature [18] signalling theχ ±χ0 2 production at the hadron colliders may disappear due to the invissible decay ofχ 0 2 . On the other hand the hadronically quiet dilepton + E T signal fromχ ±χ∓ may be boosted at the upgraded Tevatron as well at the e + e − colliders due to the enhanced leptonic decays of charginos [19,21]. Another dramatic signal of the VLSP model could be increase in the e + e − → γ + missing energy events [20].
In the conventional mSUGRA model the SUSY contributions comes only from the channel e + e − → νχ 0 1χ 0 1 which has a modest cross section and is often swamped by the e + e − → γνν background. In the VLSP scenario, however, e + e − → γνν * , γχ 0 1χ 0 2 , γχ 0 2χ 0 2 contributes to the signal in addition to the above conventional mSUGRA process. Implementing some special cuts devised in [20] one can easily suppress the SM background. In particular a suitable cut on the photon energy may kill a large number of e + e − → γνν events arising due to the radiative return to the Z peak at LEP energies above the Z pole without affecting the signal. A reanalysis of the LEP data using such cuts may reveal the VLSP scenario or severly restrict the sneutrino mass range preferred by EWP data.
If m t 1 < mχ±, then the preferred decay mode of the lighter stop (t 1 ) could bet → bℓν rather the loop induced decayt → cχ 0 1 [22]. This would enhance the leptonic signal from the stop at the cost of jets + E T events.
While light sleptons may arise in many scenarios including the ones not based on supergravity (e.g., in the AMSB model), the simultaneous presence of relatively right down squarks and light sleptons would vindicate the SO(10) D-term model. Enhancement of the jets + missing energy signal at the expenses of leptons + jets + E T signal from squark gluino production would be the hall-mark of this scenario [23,25,26]. The effect becomes particularly striking if mg > md R , while all other squarks are much heavier than the gluinos [25,26]. This mass hierarchy is infact obtained over the bulk of the parameter space probed in this paper.  The lower limit on m 1/2 is due to the chargino mass bound from LEP.