LBNL44305
UCBPTH99/44
hepph/yymmxxx
[0.03in]
Constraints on Hidden Sector Gaugino Condensation
^{*}^{*}*This work was supported in part by the Director, Office of
Energy Research, Office of High Energy and Nuclear Physics, Division
of High Energy Physics of the U.S. Department of Energy under
Contract DEAC0376SF00098 and in part by the National Science
Foundation under grant PHY9514797 and PHY9404057.
[.1in]
Mary K. Gaillard and Brent D. Nelson
[.05in]
Theoretical Physics Group
Ernest Orlando Lawrence Berkeley National Laboratory
University of California, Berkeley, California 94720
and
Department of Physics
University of California, Berkeley, California 94720
[.1in]
We study the phenomenology of a class of models describing modular invariant gaugino condensation in the hidden sector of a lowenergy effective theory derived from the heterotic string. Placing simple demands on the resulting observable sector, such as a supersymmetrybreaking scale of approximately 1 TeV, a vacuum with properly broken electroweak symmetry, superpartner masses above current direct search limits, etc., results in significant restrictions on the possible configurations of the hidden sector.
Disclaimer
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When considering the subject of effective field theories from strings, the notion of “phenomenological viability” has in the past been a very loose standard. Indeed some of the wellknown problems facing such lowenergy theories seemed quite intractable, depressing the prospects of ever being able to refer to a meaningful superstring phenomenology. The problems to which we refer include the need to generate a hierarchy between the supersymmetrybreaking scale and the Planck scale, the cosmological dangers of moduli fields with Plancksuppressed interactions, the desire for a weaklycoupled effective quantum field theory, and most significantly the need to stabilize the dilaton [1].
Recently [2, 3, 4], however, it was shown that by incorporating postulated nonperturbative stringtheoretical effects in a modular invariant lowenergy field theory the above problems can be addressed in a simple manner with tuning required only in the vanishing of the cosmological constant. Having passed these initial tests it now becomes possible to ask for a slightly higher standard in “viability.”
The philosophy behind this study is to probe this class of models in a series of phenomenological arenas to uncover relations between the dynamics of the hidden sector and the nature of our observable world. After a review in Section 1 of the class of models previously developed in [2, 3, 4, 5], we investigate in Section 2 the initial challenge of setting the supersymmetrybreaking scale that all effective field theories from strings must confront. This is largely a reiteration of results discussed in [4]. In Section 3 we turn to the pattern of soft supersymmetrybreaking parameters and look for the implications of current mass bounds arising from searches at LEP and the Tevatron. Finally, Section 4 considers the question of gauge coupling unification in the context of string theory.
1 Model
1.1 The Effective Lagrangian
The following is a condensation of material more fully presented in [3, 5] and aims to bring together the key points necessary for the subsequent discussion of phenomenological consequences. In those references, as here, the Kähler superspace formalism of [6] is used throughout.
Supersymmetry breaking is implemented via condensation of gauginos charged under the hidden sector gauge group , which is taken to be a subgroup of . For each gaugino condensate a vector superfield is introduced and the gaugino condensate superfields are then identified as the (anti)chiral projections of the vector superfields:
(1) 
The dilaton field (in the linear multiplet formalism [7] used here) is the lowest component of the vector superfield : . Note that none of the individual lowest components will appear in the effective theory component Lagrangian.
In the class of orbifold compactifications we will be considering there are three untwisted moduli chiral superfields and matter chiral superfields with Kähler potential
(2) 
where the are the modular weights of the fields . The relevant part of the complete effective Lagrangian is then
(3) 
where
(4) 
is the Lagrangian density for the gravitational sector coupled to the vector multiplet and gives the kinetic energy terms for the dilaton, chiral multiplets, gravity superfields and treelevel YangMills terms. Here the functions and represent nonperturbative corrections to the Kähler potential arising from string effects. The two functions and are related by the requirement that the Einstein term in (4) have canonical normalization:
(5) 
and obey the weakcoupling boundary conditions: . In the presence of these nonperturbative effects the relationship between the dilaton and the effective field theory gauge coupling becomes .
The second term in (3) is a generalization of the original VenezianoYankielowicz superpotential term [8],
(6) 
which involves the gauge condensates as well as possible gaugeinvariant matter condensates described by chiral superfields [9]. Neither the gaugino nor the matter condensate superfields are taken to be propagating [10]. The coeffecients , and are determined by demanding the correct transformation properties of the expression in (6) under chiral and conformal transformations [3, 11] and yield the following relations:
(7) 
The final condition amounts to choosing the value of so that the effective operator has mass dimension three. In (7) the quantities and are the quadratic Casimir operators for the adjoint and matter representations, respectively. Given the above relations it is also convenient to define the combination
(8) 
which is proportional to the oneloop betafunction coefficient for the condensing gauge group .
The third term in (3) is a superpotential term for the matter condensates consistent with the symmetries of the underlying theory
(9) 
We will adopt the same set of simplifying assumptions taken up in [3], namely that for fixed , for only one value of . Then unless for every value of for which . We next assume that there are no unconfined matter fields charged under the hidden sector gauge group and ignore possible dimensiontwo matter condensates involving vectorlike pairs of matter fields. This allows a simple factorization of the superpotential of the form
(10) 
where the functions are given by
(11) 
Here and the Yukawa coefficients , while a priori unknown variables, are taken to be of . The function is the Dedekind function and its presence in (11) ensures the modular invariance of this term in the Lagrangian.
The remaining terms in (3) include the quantum corrections from light field loops to the unconfined YangMills couplings and the GreenSchwarz (GS) counterterm introduced to ensure modular invariance.^{1}^{1}1Not included in this paper are string loop corrections [13] which vanish for orbifold compactifications with no supersymmetry sector [14]. The latter is given by the expression
(12)  
(13) 
where is proportional to the betafunction coefficient for the group and the coefficients are as yet undetermined.
As for the operators in (3), their rather involved form in curved superspace was worked out in [5] and will not be repeated here. Their importance for this work lies in their contributions to the supersymmetrybreaking gaugino masses at the condensation scale arising from the superconformal anomaly – a contribution that was recently emphasized by a number of authors [12]. We will return to these in Section 3.1.
1.2 Condensation and Dilaton Stabilization
The Lagrangian in (3) can be expanded into component form using the standard techniques of the Kähler superspace formalism of supergravity [6]. In reference [3] the bosonic part of the Lagrangian relevant to dilaton stabilization and gaugino condensation was presented and the equations of motion for the nonpropagating fields were solved. In particular, the equations of motion for the auxiliary fields of the condensates give
(14) 
where and .
Upon substituting for the gauge coupling via the relation we recognize the expected oneinstanton form for gaugino condensation. Expression (14) encodes more information, however, than simply the oneloop running of the gauge coupling. In [11] the loop corrections to the gauge coupling constants were computed using a manifestly supersymmetric PauliVillars regularization. The (moduli independent) corrections were identified with the renormalization group invariant [15]
(15) 
Using the above expression it is possible to solve for the scale at which the term becomes negligible relative to the term – effectively looking for the “all loop” Landau pole for the coupling constant. This scale is related to the string scale by the relation
(16) 
Now comparing the effective Lagrangian given in Section 1.1 with the field theory loop calculation given in [11] shows that the two agree provided we identify the wave function renormalization coefficients with the quantity . This is precisely what is needed to produce the final product in the condensate expression given in (14), indicating that the condensation scale represents the scale at which the coupling becomes strong as would be computed using the socalled “exact” betafunction.
Note that this final factor introduces the unknown Yukawa coefficients into the scale of supersymmetry breaking. Such dependence of the gaugino condensate on the parameters of the superpotential is not unexpected, and has in fact been demonstrated in the case of supersymmetric QCD as well as certain models of supersymmetric YangMills theories coupled to chiral matter [16]. This last Yukawarelated factor has the virtue of allowing two different hidden sector configurations which result in the same betafunction to condense at widely different scales.
In order to go further and make quantitative statements about the scale of gaugino condensation (and hence supersymmetry breaking) it is necessary to specify some form for the nonperturbative effects represented by the functions and . The parameterization adopted in [4] was originally motivated by Shenker [17] and was of the form where is the string coupling constant. A consensus seems to be forming [18] around this characterization for string nonperturbative effects and the function in (4) will be taken to be of the form
(17) 
which was shown [4] to allow dilaton stabilization at weak to moderate string coupling with parameters that are all of . The benefits of invoking stringinspired nonperturbative effects of the form of (17) have recently been explored by others in the literature [19].
The scalar potential for the moduli is minimzed at the selfdual points or , where the corresponding Fcomponents of the chiral superfields vanish. At these points the dilaton potential is given by
(18) 
As an example, the potential (18) can be minimized with vanishing cosmological constant and for and in expression (17).
2 Phenomenological Implications
2.1 Scale of Supersymmetry Breaking
With the adoption of (17) the scale of gaugino condensation can be obtained once the following are specified: (1) the condensing subgroup(s) of the original hidden sector gauge group , (2) the representations of the matter fields charged under the condensing subgroup(s), (3) the Yukawa coefficients in the superpotential for the hidden sector matter fields and (4) the value of the string coupling constant at the compactification scale, which in turn determines the coefficients in (17) necessary to minimize the scalar potential (18).
A great deal of simplification in the above parameter space can be obtained by making the ansatz that all of the matter in the hidden sector which transforms under a given subgroup is of the same representation, such as the fundamental representation. Then the sum of the coefficients over the number of condensate fields labeled by , can be replaced by one effective variable
(19) 
In the above equation is proportional to the quadratic Casimir operator for the matter fields in the common representation and the number of condensates, , can range from zero to a maximum value determined by the condition that the gauge group presumed to be condensing must remain asymptotically free. The redefinition in (19) essentially takes the coefficients , which we are free to choose in our effective Lagrangian up to the conditions given in (7), and assigns the same value to each condensate.
The variable can then be eliminated in (14) in favor of provided the simultaneous redefinition is made so as to keep the product in (14) invariant:
(20) 
With the assumption of universal representations for the matter fields, this implies
(21) 
which we assume to be an number, if not slightly smaller.
From a determination of the condensate value using (14) the supersymmetrybreaking scale can be found by solving for the gravitino mass, given by
(22) 
In [3] it was shown that in the case of multiple gaugino condensates the scale of supersymmetry breaking was governed by the condensate with the largest oneloop betafunction coefficient. Hence in the following it is sufficient to consider the case with just one condensate with betafunction coefficient denoted :
(23) 
As an illustration of this point, the gravitino mass for the case of pure supersymmetric YangMills condensation (no hidden sector matter fields) would be GeV. The addition of an additional condensation of pure supersymmetric YangMills gauginos would only add an additional GeV to the mass.
Now for given values of and the condensation scale
(24) 
and gravitino mass (23) can be plotted in the plane. The sharp variation of the condensate value with the parameters of the theory, as anticipated by the functional form in (14), is apparent in the contour plot of Figure 1.
The dependence of the gravitino mass on the group theory parameters is even more profound. Figure 1 gives contours for the gravitino mass between GeV and TeV. Clearly, the region of parameter space for which a phenomenologically preferred value of the supersymmetrybreaking scale occurs is a rather limited slice of the entire space available.
The variation of the gravitino mass as a function of the Yukawa parameters is shown in Figure 2. On the horizontal axis there are no matter condensates () so there is no dependence on the variable . For values of the contours of gravitino mass in the TeV region lie beyond the limiting value of and are thus in a region of parameter space which is inaccessible to a model in which the unified coupling at the string scale is or larger. For very large values of the effective Yukawa parameter the gravitino mass contours approach an asymptotic value very close to the case shown here for . We might therefore consider the shaded region between the two sets of contours as roughly the maximal region of viable parameter space for a given value of the unified coupling at the string scale.
2.2 Implications for the Hidden Sector
Having examined some of the universal constraints placed on any stringderived model proposing to describe low energy physics in Section 2.1 it is natural to ask whether the region of phenomenological viability (roughly the shaded area in Figure 2) can be used to constrain the matter content of the hidden sector.
Upon orbifold compactification the gauge group of the hidden sector is presumed to break to some subgroup(s) of and the set of all such possible breakings has been computed for orbifolds [20]. Under the working assumption that only the subgroup with the largest betafunction coefficient enters into the lowenergy phenomenology, there are then a finite number of possible groups to consider:
(25) 
For each of the above gauge groups equations (7) and (8) define a line in the plane. These lines will all be parallel to one another with horizontal intercepts at the betafunction coefficient for a pure YangMills theory. The vertical intercept will then indicate the amount of matter required to prevent the group from being asymptotically free, thereby eliminating it as a candidate source for the supersymmetry breaking described in Section 2.1.
In Figure 3 we have overlaid these gauge lines on a plot similar to Figure 2. We restrict the Yukawa couplings of the hidden sector to the more reasonable range of and give three different values of the string coupling at the string scale. The choice of string coupling constant is made when specifying the boundary conditions for solving the dilaton scalar potential, as described in Section 1.2. Changing this boundary condition will affect the scale of gaugino condensation through equation (14), altering the supersymmetrybreaking scale for a fixed point in the plane. Demanding larger values of will result in the shifting of the contours of fixed gravitino mass towards the origin, as in Figure 3. Such large values of have recently been invoked as part of a mechanism for stabilizing the dilaton and/or as a consequence of reconciling the apparent scale of gauge unification in the minimal supersymmetric standard model (MSSM) with the scale predicted by string theory [21]. We will return to such issues in Section 4.
A typical matter configuration would be represented in Figure 3 by a point on one of the gauge group lines. As each field adds a discrete amount to and the fields must come in gaugeinvariant multiples, the set of all such possible hidden sector configurations is necessarily a finite one.^{2}^{2}2For example, one cannot obtain values of arbitrarily close to zero in practical model building. The number of possible configurations consistent with a given choice of and supersymmetrybreaking scale is quite restricted. For example, Figure 3 immediately rules out hidden sector gauge groups smaller than SU(6) for weak coupling at the string scale . Furthermore, even moderately larger values of the string coupling at unification become increasingly difficult to obtain as it is necessary to postulate a hidden sector with very small gauge group and particular combinations of matter to force the betafunction coefficient to small values.
3 Constraints from the LowEnergy Spectrum
3.1 Soft SupersymmetryBreaking Terms
Simply requiring that the scale of supersymmetry breaking be in a reasonable range of energy values (i.e. within an order of magnitude of 1 TeV) can put significant constraints on the dynamics of the hidden sector. Requiring further that the pattern of supersymmetry breaking be consistent with observed electroweak symmetry breaking and direct experimental bounds on superpartner masses can restrict the parameter space even more.
The pattern of supersymmetry breaking is determined by the appearance of soft scalar masses, gaugino masses and trilinear couplings at the condensation scale. The gaugino masses in the onecondensate approximation, including the contribution from the quantum effects of light fields arising at one loop from the superconformal anomaly, are given by [5]
(26) 
The incorporation of scalar masses and trilinear terms in the scalar potential for observable sector matter fields depends on the form of the Kähler potential and the nature of the couplings of observable sector matter fields to the GreenSchwarz counterterm. Adopting the Kähler potential assumed in (2) and the counterterm of (13), the scalar masses are given in the onecondensate approximation by
(27) 
and the trilinear “Aterms” in the scalar potential are given by
(28) 
with
(29) 
As noted in [4], the fact that (27) and (29) are independent of the modular weights of the individual observable sector fields is the result of the vanishing of the auxiliary fields in the vacuum. This is a manifestation of the socalled “dilaton dominated” scenario of supersymmetry breaking [22] for which flavorchanging neutral currents might be naturally suppressed. For this to indeed occur, however, it is also necessary to make the assumption that the couplings are the same for the first and second generations of matter.
To analyze the lowenergy particle spectrum it is necessary to choose a value of for each generation of matter fields. If the GreenSchwarz term (13) is independent of the so that , then from (27) . We will call such a generation “light.” On the other hand, it was postulated in [4] that the GreenSchwarz term may well depend only on the combination , where represents untwisted matter fields. Then for these multiplets and the scalar masses for these fields are in general an order of magnitude greater than the gravitino mass. We will call these generations “heavy.”
The scalar masses (27) and Aterms (29) given above do not include the contributions proportional to the matter field wavefunction renormalization coefficients arising from the superconformal anomaly (the analog to the gaugino mass terms studied in [5] and included in (26)). A systematic treatment of these contributions to the softbreaking terms is currently underway [23], but their general size is comparable to the gaugino masses. In the following it has been checked that varying the initial soft terms by arbitrary amounts of this size has a negligible impact on the conclusions we report here.
Before giving the results of a numerical analysis using the renormalization group equations (RGEs) with the boundary conditions determined by equations (26), (27) and (29), it is worthwhile looking at what patterns of symmetry breaking are expected for various choices of the parameter in the context of the MSSM. For any generation with nonnegligible Yukawa couplings a good indicator that the stable minimum of the scalar potential will yield correct electroweak symmetry breaking is the relation
(30) 
When this bound is not satisfied it is typical to develop minima away from the electroweak symmetry breaking point in a direction in which one of the scalar masses carrying electric or color charge becomes negative. For any heavy matter generation with a nonnegligible coupling to a heavy Higgs field () equation (29) yields and so (30) is already nearly saturated at the condensation scale.
Another key factor in preventing dangerous color and chargebreaking minima is the ratio of scalar masses to gaugino masses and the degree of splitting between any light and heavy matter generations. In this model, both of the hierarchies, and , will turn out to be . This pattern of soft supersymmetrybreaking masses has been shown [24] to lie on the boundary of the region in MSSM parameter space for which light squark masses tend to be driven negative by twoloop effects arising from the heavier squarks. All of the above considerations suggest that compactification scenarios in which the observable sector matter fields couple universally to the GreenSchwarz counterterm with may have trouble reproducing the correct pattern of lowenergy symmetry breaking.
3.2 RGE Viability Analysis Within the MSSM
To determine what region of parameter space in the plane is consistent with current experimental data it is necessary to run the soft supersymmetrybreaking parameters of equations (26), (27) and (29) from the condensation scale to the electroweak scale using the renormalization group equations. For this purpose we take the MSSM superpotential and matter content for the observable sector, keeping only the top, bottom and tau Yukawa couplings. In order to capture the significant two loop contributions to gaugino masses and scalar masses these parameters are run at two loops, while the other parameters are evolved using the oneloop RGEs. The equations used are in the scheme and can be found in [25]. The RGE analysis was performed on four different scenarios:

Scenario A: All three generations light.

Scenario B: Third generation light, first and second generations heavy.

Scenario C: All three generations heavy.

Scenario D: All matter heavy except for the two Higgs doublets which remain light ().
To protect against unwanted flavor changing neutral currents we have chosen the GreenSchwarz coefficients to be universal throughout each matter generation. While our scalars will turn out to be heavy enough that small deviations from universality (such as those arising from the superconformal anomaly discussed above) will not be problematic, the large hierarchies controlled by the values of the would be untenable. The Higgs fields will be taken to couple to the GreenSchwarz counterterm identically to the third generation of matter, as we keep only the third generation Yukawa couplings in the MSSM superpotential. In Scenario D we relax this assumption.
In the boundary values of (26), (27) and (29) the values of and appear only indirectly through the determination of the value of the condensate . It is thus convenient to cast all soft supersymmetrybreaking parameters in terms of the values of and using equation (23). While the gravitino mass itself is not strictly independent of , it is clear from Figure 2 that we are guaranteed of finding a reasonable set of values for consistent with the choice of and provided we scan only over values for weak string coupling. This transformation of variables allows the slice of parameter space represented by the contours of Figure 3 to be recast as a twodimensional plane for a given value of and . The condensation scale (the scale at which the RGrunning begins) is also a function of the gravitino mass in this framework, found by inverting equation (23).
Having chosen a set of input parameters for a particular scenario, the model parameters are run from the condensation scale given by (24) to the electroweak scale , decoupling the scalar particles at a scale approximated by . While treating all superpartners with a common scale sacrifices precision for expediency, the results presented below are meant to be a first survey of the phenomenology of this class of models.
At the electroweak scale the oneloop corrected effective potential is computed and the effective muterm is calculated
(31) 
In equation (31) the quantities and are the second derivatives of the radiative corrections with respect to the uptype and downtype Higgs scalar fields, respectively. These corrections include the effects of all thirdgeneration particles. If the right hand side of equation (31) is positive then there exists some initial value of at the condensation scale which results in correct electroweak symmetry breaking with GeV [26].^{3}^{3}3Note that we do not try to specify the origin of this muterm (nor its associated “Bterm”) and merely leave them as free parameters in the theory – ultimately determined by the requirement that the Zboson receive the correct mass.
A set of input parameters will then be considered viable if at the electroweak scale the oneloop corrected muterm is positive, the Higgs potential is bounded from below, all matter fields have positive scalar masssquareds and the spectrum of physical masses for the superpartners and Higgs fields satisfy the selection criteria given in Table 1.^{4}^{4}4Though the inclusive branching ratio for decays was not used as a criterion, an a posteriori check of the region of the parameter space where this class of models wants to live – namely relatively low and gaugino masses with high scalar masses – indicates no reason to fear a conflict with the bounds from CLEO except possibly in the case for Scenario D [27].
Gluino Mass  GeV  

Lightest Neutralino Mass  GeV  
Lightest Chargino Mass  GeV  
Squark Masses  GeV  
Slepton Masses  GeV  
Light Higgs Mass  GeV  
Pseudoscalar Higgs Mass  GeV 
The first condition to be imposed on the scenarios considered here is correct electroweak symmetry breaking, defined by (31), with no additional scalar masses negative. This criterion alone rules out Scenario C, with all three generations coupling universally to the GScounterterm and having large scalar masses. For the opposite case of no coupling to the GScounterterm (Scenario A) the allowed region is displayed in Figure 4. In this scenario electroweak symmetry breaking requires , the lower bound being the value for which the top quark Yukawa coupling develops a Landau pole below the condensation scale. This restricted region of the parameter space is a result of the large hierarchy between gaugino masses and scalar masses in these models and has been observed in more general studies of the MSSM parameter space [29].
Scenario B with its split generations can exist only for , where the hierarchy between the generations is small enough to prevent the twoloop effects of the heavy generations from driving the righthanded top squark to negative masssquared values. Furthermore, proper electroweak symmetry breaking in this model requires the value of to be in the uncomfortably narrow range , making this pattern of GreenSchwarz couplings phenomenologically unattractive.
As for Scenario D, the large third generation masses give an additional downward pressure on the Higgs masssquareds in the running of the RGEs, allowing for a much wider allowed range of . In fact, electroweak symmetry is radiatively broken in the entire range of parameter space. However, as the value of is raised past the critical range , the scalar mass boundary values at the condensation scale start to become light enough that the righthanded stop is again driven to negative masssquared values. This is shown in Figure 4 where the region between the upper and lower curves is excluded. While this region expands rapidly as the betafunction coefficient is increased, the values of the betafunction coefficient consistent with are nearly saturated when this effect arises.
The direct experimental constraints are most binding for the gaugino sector as they are by far the lightest superpartners in this class of models. Typical bounds reported from collider experiments are derived in the context of universal gaugino masses with a relatively large mass difference between the lightest chargino and the lightest neutralino. For most choices of parameters in the models studied here this is a valid assumption, but when the condensing group betafunction coefficient becomes relatively small (i.e. similar in size to the MSSM hypercharge value of ) the pieces of the gaugino mass arising from the superconformal anomaly (26) can become equal in magnitude to the universal term. Here there is a level crossing in the neutral gaugino sector. The lightest supersymmetric particle (LSP) becomes predominately winolike and the mass difference between the lightest chargino and lightest neutralino becomes negligible. This effect is displayed in Figure 5. The experimental constraints as normally quoted from LEP and the Tevatron cannot be applied in the region where the mass difference between the lightest neutralino and chargino falls below about 2 GeV. The phenomenology of such a gaugino sector has been studied recently in [30]. Note that when any scalar fields couple to the GScounterterm (as in Scenario D) there is a large additional, universal contribution to the gaugino masses at the condensation scale in (26). This eliminates any region with a nonstandard gaugino sector in these cases.
Figure 6 gives the binding constraints from Table 1 for Scenario A with and positive (the most restrictive case). The most critical constraints are for the lightest chargino and gluino.^{5}^{5}5The gluino mass determination takes into account the difference between the running mass () and the physical gluino mass [31]. This difference is neglected for the other mass parameters. The effect of varying on these bounds is negligible over the range , as its effect is solely in the variation in the Yukawa couplings appearing at two loops in the gaugino mass evolution. The region for which the anomalyinduced contributions to the gaugino masses make the normal experimental constraints inoperative is represented by the shaded region in the upper left of the figure. In general, the light gaugino masses at the condensation scale require a large gravitino mass (and hence, a large set of soft scalar masses since in this scenario) in order to evade the observational bounds coming from LEP and the Tevatron. While current theoretical prejudice would disfavor such large soft scalar masses, this pattern of soft parameters may not necessarily be a sign of excessive finetuning [32]. Nevertheless, we refrain from making any statements about the “naturalness” of this class of models as we have not specified any mechanism for generating the muterm.
Figure 7 gives the binding constraints from Table 1 for Scenario D with and positive . Note the change of scale in both axes for these plots relative to those of Scenario A. As in Figure 6, varying over the range has a negligible effect on the gaugino constraint contours and only a very small effect on the contours of constant stop mass. Here the gaugino masses start at much larger values so a lower supersymmetrybreaking scale is sufficient to evade the bounds from LEP and the Tevatron. Though the gravitino mass can now be much smaller, recall that the scalars in this scenario have masses at the condensation scale roughly an order of magnitude larger than the gravitino. Thus the typical size of scalar masses at the electroweak scale continues to be about 1 TeV for the first two generations and a few hundred GeV for the third generation scalars. As opposed to the case where all the matter fields of the observable sector decouple from the GScounterterm, here smaller values of the condensing group betafunction coefficient enhance the gaugino masses via the last term in (26).
4 Gauge Coupling Unification
In Section 2.2 the possibility of larger values of the unified coupling constant at the string scale was considered in a very general way. It is well known [33] that the apparent unification of coupling constants at a scale GeV, assuming only the MSSM field content, is at odds with the string prediction that unification must occur at a scale given by
(32) 
where represents the (schemedependent) oneloop correction from heavy string modes. In [3] this factor was computed for the scheme and it is given by
(33) 
For the vacua considered in this work this parameter is typically .
Even after taking into account oneloop string corrections there is still an order of magnitude discrepancy between the scale of unification predicted by string theory and the apparent scale of unification as extrapolated from low energy measurements under the MSSM framework. One possible solution to the problem is the inclusion of additional matter fields in incomplete multiplets of SU(5) at some intermediate scale which will alter the running of the coupling constants, causing them to converge at some value higher than [34]. These solutions tend to involve slightly larger values of the coupling constant at the string scale than that of the MSSM ().
In the model in question here, the intermediate scale () at which this additional matter might appear is not independent of the scale of the superpartner spectrum (), but the two are in fact related by equation (23). Thus if we assume this additional matter has a typical mass of the condensation scale, each point in the plane can be tested for potential compatibility with string unification given a certain set of additional matter fields. We will not specify the origin of these fields (though such incomplete multiplets are not uncommon in string theory compactifications), but merely posit their existence with masses on the order of the condensation scale.
Our procedure for carrying out this investigation is similar to that used in the literature by a number of authors [35]. The standard model coupling constants , and are determined from , and and these values are converted to the scheme. As we will not be concerned with performing a precision survey, these coupling constants are run at one loop from their values at the electroweak scale using only the standard model field content up to the scale . At this scale the entire supersymmetric spectrum is added to the equations until the scale is reached. Here incomplete multiplets of SU(5) are added and the couplings are run to the scale at which the SU(2) and U(1) fine structure constants coincide. This scale will be defined as the string scale.
We now require at this scale and invert equation (32) to find the implied Planck scale. Consistency requires that this value be the reduced Planck mass of GeV and that the QCD gauge coupling, when the renormalization group equations are solved in the reverse direction, give a value for at within two standard deviations of the measured value.^{6}^{6}6It is worth remarking that even the celebrated supersymmetric SU(5) unification of couplings fails to predict the strong coupling at the electroweak scale at the level of two sigma and calls for a rather large value of [35]. This is usually taken as an indication of the size of modeldependent threshold corrections. We therfore demand no more from the models considered here.
The results of the analysis for a typical choice of extra matter fields are shown in Figure 10, where a pair of vector like and two pairs of vectorlike ’s are introduced at the condensation scale with quantum numbers identical to their MSSM counterparts. The two sigma window about the current bestfit value of can indeed accomodate a consistent Planck mass while allowing for perturbative unification of gauge couplings. From this base configuration additional 5s and 10s of SU(5) can be added at will to increase the value of the unified coupling at the string scale, but the contours of constant implied Planck mass shown in Figure 10 will not move significantly. While these combinations of matter fields have been known to allow for gauge coupling unification for some time [34], the relationships (23) and (32) between the various scales involved makes this a nontrivial accomplishment for this class of models.
Conclusion
The preceeding pages should be cause for guarded optimism with regard to string phenomenology. The initial challenge of dilaton stabilization has been met without resorting to strong coupling in the effective field theory nor requiring delicate cancellations. Reasonable values of the supersymmetrybreaking scale can be achieved over a fairly large region of the parameter space, but a given combination of coupling strength at the string scale and hidden sector matter content will single out a tantalizingly small slice of this space. These successful combinations do not destroy the potential solutions to the coupling constant unification problem by the introduction of additional matter at the condensation scale. Tighter restriction on the hidden sector will require more precise knowledge of the size of Yukawa couplings in the corresponding superpotential.
Requiring a vacuum configuration which gives rise to successful electroweak symmetry breaking seems to demand that either the GreenSchwarz counterterm be independent of the matter fields or that all matter fields couple in a universal way but that the Higgs fields are distinct. The pattern of soft supersymmetrybreaking parameters in the former case pushes the theory towards large gravitino masses and very low values of . The low gaugino masses relative to scalar masses favors larger betafunction coefficients for the condensing group of the hidden sector, while smaller values may result in phenomenology in the gaugino sector similar to that of the “anomaly dominated” scenarios.
In the latter case a proper vacuum configuration and weak coupling at the string scale leave the value of free to take its entire range of possible values. Larger betafunction coefficients for the condensing group allow a promising region with relatively light scalar partners of the thirdgeneration matter fields and light gauginos.
A more realistic model may alter these results to some degree and uncertainty remains in the general size and nature of the Yukawa couplings of the hidden sector of these theories. Nevertheless this survey suggests that eventual measurement of the size and pattern of supersymmetry breaking in our observable world may well point towards a very limited choice of hidden sector configurations (and hence string theory compactifications) compatible with low energy phenomena.
Acknowledgements
We than Pierre Binètruy, Hitoshi Murayama and Marjorie Shapiro for discussions. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DEAC0376SF00098 and in part by the National Science Foundation under grant PHY9514797 and PHY9404057.
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