Nucl.Phys.B v.576

Nuclear Physics B 576 (2000) 328www.elsevier.nl/locate/npe

Phenomenology of deflected anomaly-mediationRiccardo Rattazzi a, Alessandro Strumia b, James D. Wells c

a INFN and Scuola Normale Superiore, I-56100 Pisa, Italyb Dipartimento di fisica, Universit di Pisa and INFN, I-56126 Pisa, Italyc Physics Department, University of California, Davis, CA, 95616, USA

Received 23 December 1999; revised 8 February 2000; accepted 28 February 2000

Abstract

We explore the phenomenology of a class of models with anomaly-mediated supersymmetrybreaking. These models retain the successful flavor properties of the minimal scenario while avoidingthe tachyons. The mass spectrum is predicted in terms of a few parameters. However variousqualitatively different spectra are possible, often strongly different from the ones usually employedto explore capabilities of new accelerators. One stable feature is the limited spread of the spectrum,so that squarks and gluinos could be conceivably produced at TEVII. The lightest superpartner ofstandard particles is often a charged slepton or a neutral higgsino. It behaves as a stable particlein collider experiments but it decays at or before nucleosynthesis. We identify the experimentalsignatures at hadron colliders that can help distinguish this scenario from the usual ones. 2000Elsevier Science B.V. All rights reserved.

PACS: 12.60.Jv; 11.30.Pb; 14.80.Ly

1. Introduction

The origin of supersymmetry breaking is the central issue in the construction of arealistic supersymmetric extension of the Standard Model (SM). If supersymmetry is tobe of any relevance to the hierarchy problem the sparticle masses should be smaller thanabout a TeV. Then, flavor violating processes mediated by virtual sparticles constraintheir masses to preserve flavor to a high degree. One main goal of model building isto provide flavor symmetric soft terms in a simple and natural way. Gauge mediatedsupersymmetry breaking (GMSB) [13] represents an elegant solution to this problem: softterms are calculable and are dominated by a flavor symmetric contribution due to gaugeinteractions. Supergravity, on the other hand, provides perhaps the simplest way to mediatesupersymmetry breaking [46]. However, in the absence of a more fundamental theory, softterms are not calculable in supergravity, so there is little control on their flavor structure.More technically, one could say that soft terms are dominated by extreme ultraviolet

0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0550-3213(00) 00 13 0- 9

4 R. Rattazzi et al. / Nuclear Physics B 576 (2000) 328

dynamics in supergravity and consequently are sensitive to all possible new sources offlavor violation, not just the low-energy Yukawa couplings. This can be considereda generic problem of soft terms mediated by supergravity. Various solutions have beensuggested, including special string inspired scenarios (dilaton dominance) and horizontalsymmetries.

Recently, important progress has been made in our understanding of a class of calculablequantum effects in supergravity [7,8]. These effects can be characterized as the puresupergravity contribution to soft terms. This is because they are simply determined by thevacuum expectation value of the auxiliary scalar field F in the graviton supermultiplet.The couplings of F to the minimal supersymmetric standard model (MSSM) area purely quantum effect dictated by the conformal anomaly. The resulting anomalymediated contribution to sparticle masses is of order F/4pi m3/2/4pi . In a genericsupergravity scenario, this calculable effect would only represent a negligible correction tothe uncalculable m3/2 tree level terms. However it is consistent to consider a situationwhere Anomaly Mediation (AM) is the leading effect. Indeed, as pointed out by Randalland Sundrum [7], this may happen in an extra-dimensional scenario, for example, when theMSSM lives on a 3-brane, while the hidden sector lives on a brane that is far-away in a bulkwhere only gravity propagates. Recently an explicit realization of this setup has been givenin Ref. [9]. A more conventional situation, where the anomaly mediated contribution to justthe gaugino masses and A-terms dominates, is dynamical hidden sector models withoutsinglets [8]. Various technical aspects of AM have been further discussed in Refs. [1012],the latter of which gives a more formal derivation along with a comparison to previouscomputations of quantum contributions to soft terms [1315].

In pure Anomaly Mediation sfermion masses are dominated by an infrared contribution,so they are only sensitive to the sources of flavor violation that are relevant at low energy,as encoded in the fermion masses and CKM angles of the SM. Therefore AM, like the SM,satisfies natural flavor conservation. Sfermion masses are in practice family independent,since the gauge contributions dominate, like in GMSB. Unfortunately, this is not the fullstory: flavor is fine but the squared slepton masses are predicted to be negative.

Various attempts have been made to save the situation. In principle adding anextra supergravity contribution ruins predictivity. Nevertheless, if one assumes thatsome unspecified flavor universal contribution lifts the sleptons, then the low-energyphenomenology is quite peculiar [16,17]. Other proposals involve extra fields at, or justabove, the weak scale [11,18]. In this paper we will focus on the idea of Ref. [10], whichwe outline below.

The fact that AM provides a special Renormalization Group (RG) trajectory where allunwanted ultraviolet (UV) effects on soft terms decouple is very suggestive. Indeed, inorder to solve the supersymmetric flavor problem, it would be enough to remain on thistrajectory only down to a scale M0 somewhat below the scale of flavor. In Ref. [10] itwas pointed out that a theory can be kicked off the AM trajectory when an intermediatethreshold is governed by the vacuum expectation value (VEV) of a field X that is masslessin the supersymmetric limit. This does not truly violate the UV insensitivity of AM, sincethe low energy theory is not just the MSSM but contains also the modulus X. While this

R. Rattazzi et al. / Nuclear Physics B 576 (2000) 328 5

field is coupled to the MSSM only by 1/X suppressed operators, its presence affects thesoft masses in a relevant way. Ref. [10] used this remark to build a realistic class of models,with flavor universal and positive sfermion masses. The intermediate threshold is given bya messenger sector similar to that of GMSB models. However the sparticle spectrum ofthese models strongly differs from both GMSB and conventional supergravity. Indeed theprediction for gaugino mass ratios is also distinguished from minimal AM. The mostimportant features of the spectrum are a reduced hierarchy between coloured sparticlesand the rest, and the lightest spartner being either a slepton or a higgsino-like neutralino.The lightest supersymmetric particle (LSP) is the fermionic partner of the modulus X,so the lightest sparticle in the MSSM can be charged.

The purpose of the present paper is to study the implications of these novel featuresin collider physics and cosmology. It is organized as follows. In Section 2 we recall thebuilding blocks of the model and the corresponding high-scale boundary conditions for softterms. In Section 3 we study the low-energy spectrum and consider the constraints fromelectroweak symmetry breaking. In Section 4 we focus on the signatures at both TEVII andLHC and draw a comparison to those of GMSB and minimal supergravity (mSUGRA).Supersymmetric corrections to rare processes are studied in Section 5. In Section 6 wediscuss the NLSP decays and the bounds on it placed by big-bang nucleosynthesis.Section 7 contains our conclusions. In Appendix A we write the one-loop RG evolutionfor the soft terms in terms of a minimal number of semi-analytic functions, starting fromthe most general boundary conditions.

2. The model

Anomaly Mediated soft terms can be defined in a very simple operational way. Considerfirst any model in the supersymmetric limit and assign R-charge 2/3 to all its chiralmatter superfields. Notice that in general this is not a true symmetry. For instance, in thesuperpotential only the trilinear couplings are invariant. Consider then the introduction of aspurion (classical external field) with R-charge 2/3 and scaling-dimension 1, and coupleit to the original Lagrangian in order to make it formally both R and scale invariant. Forinstance for a generic superpotentialW(Q) we have

W(Q)=M1Q2 + Q3 + 1M1

Q4 + M1Q2 + Q3 + 1M1

Q4 + = 3W(Q/). (1)

When the choice = 1+ 2F is made, some special soft terms are generated: they areproportional to the dimension of the original superpotential coupling. Notice that theyvanish for a purely cubic W . The same game can be played with the gauge interactionterms. Like for Yukawas, the coupling to is absent because gauge interactions are scaleinvariant and R symmetric at tree level. So in a theory with only gauge and Yukawacouplings no soft term arises at tree level. However, a coupling to arises at the quantumlevel due to anomalous breaking of scale (and R) invariance. Indeed, in order to formally


restore the two symmetries one should also couple the regulator Lagrangian to . Forinstance in supersymmetric QED the PauliVillars mass should be multiplied by a factor ,like in Eq. (1). The quantum dependence on can be effectively accounted for byconsidering superfield matter wave functions and gauge couplings [19,20]

Zi ()Z(

), R() g2

(

), (2)

where Zi() and g2() are the running parameters in the supersymmetric limit. Eq. (2) isderived by noticing that the quantity/

is the only scale andR invariant combination

of and [7,8]. By Eq. (2) the A-terms, scalar and gaugino masses in the AM scenarioare

Aijk()= 12(i()+ j ()+ k()

)F, i = d lnZi

d ln, (3a)

m2i ()= 14 i()|F |2, i =di

d ln, (3b)

M()= (g2())

2g2()F, = dg

2

d ln, (3c)

whereAijk is the dimensionful scalar-Yukawa analogous to the Yukawa coupling ijk . Thepure gauge contribution to scalar masses is proportional to (g2), which is positive forasymptotically free gauge theories and negative otherwise. In the MSSM neither SU(2)Lnor U(1)Y is asymptotically free. So the slepton squared masses, which are dominated bythe SU(2)L U(1)Y contribution, are negative and the model is ruled out.

The models constructed in Ref. [10] eliminate the tachyons while preserving thesuccessful flavor properties of AM. In these models n flavors of messengers i , i inthe 5+ 5 of SU(5) and a singlet X are added to the MSSM fields. These fields interact viathe superpotential

Wmess = Xii (4)so the basic structure is that of GMSB models. However it is assumed that soft termsare generated by AM already in supergravity. It is also assumed that X has no otherinteractions, so that the direction X 6= 0 , = 0 is flat. We are interested in a situationwhere X gets a large VEV so that the messengers are ultra-heavy. If, contrary to ourassumption, X were fixed by supersymmetric dynamics, for example by a superpotentialW(X), then the relation FX/X = F would hold in the presence of the spurion . Themessenger supermultiplets would then be split, and upon integrating them out a gauge-mediated correction to the sparticle masses would arise. By the relation FX/X = F , thiscorrection would precisely adjust the soft terms to the AM trajectory of the low-energytheory, i.e., to the beta functions of the theory without messengers. This is just an exampleof the celebrated decoupling of heavy thresholds in AM.

However in our model,X is a flat direction in the supersymmetric limit only lifted by theeffects of F 6= 0. Below the scale X, the messengers are integrated out and X becomesa free field. By RG-invariance, supersymmetry and R-symmetry the effective action for X


is simply obtained by replacing XX in the running wave function Z() of Eq. (2)[10,1921]

Leff =d4 ZX

(XX/

)XX. (5)

By this equation we obtain the standard result that the potential along X is determined bythe running soft mass m2X(= |X|)

V (X)=m2X(|X|)|X|2 ' F16pi2

2n2 (X)[c2 (X) cig2i (X)]|X|2 , (6)where c, ci > 0, and a sum over the gauge couplings gi of the messengers isunderstood. If the running mass m2X is positive at large X and crosses zero at some pointX =M0, the potential has a stable minimum around this point [22]. There exists a choiceof parameters for which this happens: the positive Yukawa term in Eq. (6) may dominatein the UV while the negative gauge contribution may balance it at a lower scale. For thismechanism to work better one may imagine the presence of a new and strongly UV freemessenger gauge interaction. This is because SU(3) SU(2) U(1) ends up IR free bythe addition of the messengers. Around the minimum, Re(X) gains a mass (/4pi)3Fwhich could be of order a few GeV, while Im(X) is an axion. The crucial result, evidentfrom Eq. (5), is FX/X = X(M0)F/2, a 1-loop quantity instead of the tree level resultFX/X = F we mentioned above. Therefore, when the messengers are integrated out,their gauge-mediated contribution to sparticle masses is O(2F), which represents anegligible correction to the original O(F) anomaly mediated masses. Thus while thegauge beta functions are modified by eliminating the messengers, the soft terms arentadjusted to the beta functions of the low energy theory. Below the scaleM0, the RG flow isdeflected from the AM trajectory. That is why we call this scenario Deflected AnomalyMediation (DAM). Practically the phenomenology of this model is that of the MSSMwith boundary conditions for soft terms at scale M0 given by AM in the MSSM plus nfamilies of messengers. We give these boundary conditions below. Notice that the additionof messengers apparently worsens the situation in that it makes the beta functions morenegative. However the gaugino masses are also changed: it is the gaugino RG contributionfrom M0 to mZ that eliminates all tachyons. An example of this behaviour for a DAMmodel with n= 5 and M0 = 1015 GeV is shown in Fig. 1.

The model is completed by a sector whose dynamics generate and B. We remindthe reader that the generation of these parameters is yet another problem of simple AM.As in GMSB, it is quite easy to obtain the right , but it is hard to avoid B F mweak.These problems are avoided in DAM by considering the addition of one singlet S coupledvia the superpotential

d2[HSHdHu + 13SS3 + 12XS2X

]. (7)

Along X 6= 0, the field S is massive and by integrating it out the following effectiveoperator is generated

d4

{HdHu

HX

XXZ(XX/

)+ h.c.

}, (8)


Fig. 1. Sample RG evolution of soft terms and qualitatively different sparticle spectra possible inDAM models. Notations are explained in the caption of Fig. 4.

where Z() is the running wave function mixing between X and S. Eq. (8) leads to thefollowing expressions for and B at the scale M0

= HX

(XZ+ Z

)F 2, B =

2XZ ZXZ+ Z

F

2, (9)

where the dots represent derivatives with respect to ln. Both parameters are F mweak. Notice that even though the effective operator Eq. (8) resembles those of typicalGMSB models, and B are the right size since FX is a 1-loop quantity.

2.1. Predictions for the soft terms renormalized at M0

The DAM predictions for the soft terms, renormalized at the high scale M0, in units ofm F/(4pi)2, are

Mi =big2i m, (10a)ARRR =

(cRi + cR

i + cR

i

)g2i m (10b)

for any fields RRR except

At =AQUHu +(t 2H

)m. (10c)

The scalar masses of the fields R without significant Yukawa interactions (sleptons,d-squarks and first and second generation of u-squarks) are

m2R =bicRi g4i m2. (10d)The soft masses of Higgses and third generationQ3 andU3 squarks also receive significantYukawa contributions

m2Hd/m2 =bicLi g4i + , (10e)


m2Hu/m2 =bicLi g4i + + 2t

( 3t + 32H ), (10f)m2U3/m

2 =bicUi g4i + 2t( 2t + 22H ), (10g)

m2Q3/m2 =bicQi g4i + 2t

( t + 2H ), (10h)where all running parameters are renormalized at M0, bi = bMSSMi + bmessi = (33/5,1,3)i + n, the quadratic Casimir coefficients cRi are listed in Table 4 and

t =(cQi + cLi + cUi

)g2i 62t , = 2H

(42H + 32t + 2cLi g2i

).

Finally, H and are unknown parameters, related to the unknown parameters in the modelLagrangian as

2 =2

ZZZX(1 ||2), 2X =

2XZ2SZX(1 ||2)3

,

2S =2S

Z3S(1 ||2)3, 2H =

2HZHdZHuZS(1 ||2)

,

= ||2(n2 + 52 2X)+ 22S + 2X (S X + h.c.),where = Z/ZXZS . Notice that ||< 1 is required for the model to be stable (positivekinetic terms). Then is positive definite and is positive. We will see below that theseextra positive contributions to the Higgs mass parameter, together with the requirement ofcorrect electroweak symmetry breaking (EWSB), lead to an upper bound on /m.

As is often the case, the model-dependent couplings introduced to generate the andB terms also affect the Higgs mass parameters. In this concrete model they also affectthe soft parameters of the third generation squarks. Since 4 soft masses depend on onlytwo unknown parameters (H and ) there are testable predictions. On the contrary the and the B terms are determined by more than two additional unknown combinations ofparameters; therefore, we consider them as free parameters and do not give their explicitexpression in terms of model parameters. Even assuming real Yukawa couplings in themessenger sector, the observable sign of the B term is not predicted. However, if forsome reason the kinetic mixing term Z is small, CP phases can be rotated away. The modelthen predicts the sign of B and gives one relation between , B and the soft terms.

We have here neglected the effects of the other Yukawa couplings, including the possiblysignificantly and b ones. If tan is large their effect should be added. They should alsobe taken into account when studying the predictions for fine details of the spectrum (likethe mass splitting between 1 ' R and eR, R and the q/q mixing angles at the gauginovertices induced by the CKM matrix).

The soft terms at the electroweak scale are obtained by renormalizing their values atM0 listed in this section with the usual MSSM RG equations. The standard semi-analyticsolutions cannot be applied in this case since gaugino masses do not obey unificationrelations, Mi i . In Appendix A we write the RG evolution for the soft terms startingfrom the most general boundary conditions in terms of a minimal number of semi-analytic functions. DAM models predict Mi (bMSSMi + n)i . In this particular case thesemi-analytic solutions could be further simplified.


3. The sparticle spectrum

The predictions for the soft terms depend on 7 parameters. The gaugino masses dependonly on n (the messenger contribution to the gauge functions); soft terms of first andsecond generation sfermions depend only on n and M0 (the messenger mass); while softterms of third generation sfermions and higgses depend also on the (imprecisely known)top Yukawa coupling at M0, and on the possible messenger couplings H and . The and B terms can be considered as free parameters, and are fixed in our analysis by theconditions of successful EWSB.

The dependence on t is stronger than in gauge mediation or supergravity models. Theunknown parameters H and can give important corrections when n is not too large(n . 10): in these cases they always increase the value of m2Hu(Q), and thus reduce thevalue of that gives a correct EWSB.

Even if all parameters are important, M0 and n are the ones that control mostof the sparticle spectrum (the gauginos and the sfermions). In Fig. 2 we show thephenomenologically acceptable range of (M0, n) for t (MGUT)= 0.5 and small messengercouplings. Shaded regions are excluded because the gauge couplings run to infinity beforethe unification scale (if n is too large), or because one slepton is tachyonic (if n is toolow). If n < 4 there are tachyonic sleptons, as in pure AM where n= 0. If n= 4 sleptonscan have positive squared masses, but also m2Hu is positive. When n > 4 it is possibleto have negative m2Hu and positive sfermion masses unless M0 is too low. In all theparameter space there exist unphysical deeper minima (since m2` < 0 at high field values,see Fig. 1). There is no reason for excluding the model for this reason. Quantum and

(a) (b)Fig. 2. Allowed values of the main unknown model parameters, n and M0 for tan = 4,t (MGUT) = 0.5, and (a) small (H = 0) or (b) significant (H = 1) Yukawa messengers. Inthe unshaded regions of the (n,M0) plane tachyonic sleptons are avoided without too many lightmessengers. Below the dashed line m2

Huis positive, so that EWSB is possible only with appropriate

correlation between the parameters. Inside (outside) the dotted lines the lightest superpartner is ahiggsino (almost always a slepton).


thermal tunneling rates are negligible [23]. Moreover within standard cosmology thereexist plausible mechanisms [24] that naturally single out the desired physical minimumcloser to the origin. A possible source of cosmological problems is the modulus X, sinceits mass cannot exceed a few GeV. Therefore to avoid large modulus fluctuations we mustassume X to be already around its minimum when the temperature of the universe issomewhat below X. Then, since X is only coupled to the MSSM by non-renormalizableinteractions at low energy, thermal fluctuations will not affect it.

3.1. EWSB and naturalness

In most of the acceptable parameter space only the Higgs fieldHu has a negative squaredmass term, m2Hu < 0, so that EWSB is induced by supersymmetry breaking in the usualway. However,m2Hu is positive for certain values of the parameters: this happens for n= 4(unless t and H are small); it also happens for higher values of n below the dashedlines in Fig. 2 if H 1. With a positive m2Hu it is still possible to break electroweaksymmetry, but only in the narrow region of the parameter space where the and B termsgive appropriate mixings in the Higgs mass matrix. Moreover this situation tends to givevalues of tan close to 1, so that the lightest Higgs mass is below its experimental boundunless the sparticles are very heavy. For these reasons we do not consider this possibilityattractive, and we will restrict our analysis to the more interesting case n> 5.

Strong, non-preliminary constraints on the parameter space are now given by LEPand Tevatron experiments. The bounds m & 90 GeV, mh & 85 GeV and M3 & (180250)GeV are satisfied only in a small portion of the parameter space of conventionalsupersymmetric models (like mSUGRA and GMSB), implying that the EWSB scale isunexpectedly smaller than the unobserved sparticle masses. How unnatural this situation isin any given model depends on two different characteristics of the model:

1. How light is the Z boson mass with respect to the soft terms? Since EWSB is inducedby supersymmetry breaking,M2Z is predicted to be a sum of various squared soft massterms (often dominated by the gluino contribution).

2. How strong are the bounds on model parameters induced by the experimental boundson sparticle masses? The naturalness problem becomes more stringent in the presenceof an indirect bound on M3 stronger than the direct Tevatron bound on M3.

Concerning the second point, in SUGRA and GMSB gaugino masses obey unificationrelations so that the LEP bound on the chargino mass gives an indirect bound on the gluinomass, M3 & 300 GeV, somewhat stronger than the direct Tevatron bound, M3 & 220 GeV(valid if mq M3, as in our model). This undesired feature is not present in the scenariounder study, basically for all appealing values of the parameters. However, as it happens inGMSB, the bound on the selectron mass gives an indirect bound on M3 which is strongerthan the Tevatron bound. In conclusion, for what concerns point 2, DAM is not better thanconventional models.

On the contrary DAM makes a somewhat more favourable prediction regardingpoint 1. It predicts a cancellation in the EWSB conditions for M2Z , because the positiveradiative O(M23 ) contribution to M2Z is partially canceled by negative radiative O(m2q )


(a) (b)Fig. 3. The size of the allowed regions (empty regions) of the parameter space (/m,B/m) indicateshow natural is the model. (a) refers to our reference DAM1 model while in (b) we show forcomparison a mSUGRA model with m0 = m1/2 and A0 = 0. The shaded regions are excludedbecause correct EWSB is not possible, while regions marked with different symbols are nowexperimentally excluded (see text).

contributions (in DAM models all sfermion squared masses are negative, before includingRG corrections).

Putting it all together, DAM models suffer from some naturalness problem. This ismainly because the experimental bounds on sparticle masses are satisfied only in a smallregion of parameter space [25]. This is shown in Fig. 3a, where we display the allowedportion of the parameter space for fixed n = 5, M0 = 1015 GeV and t (MGUT)= 0.5 andassuming that the Yukawa couplings of the messengers are negligible. With this assumptionthe soft terms only depend on 3 parameters:m (the overall scale of anomaly mediated softterms), the -term and B . The EWSB condition allows to compute the overall SUSYscale m and tan in terms of two dimensionless ratios (/m and B/m in Figs. 3, allrenormalized at M0).

In Fig. 3 we have shaded the regions where correct EWSB is not possible, and markedwith different symbols the points of the parameter space where some sparticle is too light.Sampling points marked with a

(@ , , ,

)are experimentally excluded because a

(gluino, chargino, selectron, higgs) is too light. Regions where BR(BXs ) differs fromits SM value by more than 50% are marked with a . We have restricted our plots to signsof and B such that the interference between charged higgs and chargino contributionsto the b s decay amplitude is destructive. With a constructive interference the indirectbounds on sparticle masses from BR(B Xs ) are stronger than the direct acceleratorbounds and restrict the allowed parameter space to a very small region, smaller than ourresolution of Fig. 3.

We see that different portions of the parameter space are excluded by differentcombination of the bounds on gluino, charged higgsino, slepton and higgs masses. Since


DAM models look somewhat disfavoured by naturalness considerations, 1 we also showin Fig. 3b that a typical mSUGRA model (assuming A0 = 0 and m0 = m1/2 in order tomake a plot in the (/m0,B/m0) plane) has similar problems. Moreover, also gaugemediated models have a naturalness problem, mainly because they predict light right-handed sleptons. As for pure AM (n = 0), it predicts tachyonic sleptons, and it also hassome naturalness problem: a chargino heavier than the LEP2 kinematical reach limit,M2 &MZ , would imply that the contribution from m2Hu to M

2Z is 100 times larger than M2Z

itself. Adding a universal contribution to scalar masses [7,16,28] eliminates the tachyonsbut does not improve naturalness. On the other hand, DAM models also do better on theproblem of naturalness.

3.2. The sparticle spectrum

We now continue our analysis studying the spectrum of sparticles in the allowed portionof the parameter space.

Before going on, we must anticipate (see the discussion in Section 6) that the LSP of ourmodels is the fermionic component of the X modulus. This fact is important as it allows acharged NSLP (sometimes a slepton). However, over the parameter space allowed in Fig. 2,the NLSP decays into LSP always outside the detector. Therefore, the NLSP is practicallya stable particle and the LSP plays no role in collider phenomenology.

In Fig. 4 we plot the spectrum as a function of n for M0 = 1015 GeV (Fig. 4a) and asa function of M0 for n = 5 (Fig. 4b). In both cases we have assumed M3 = 500 GeV,tan = 4, t (MGUT)= 0.5 and negligible messenger Yukawa couplings and computed the-term from the condition of correct EWSB. Although no unique pattern emerges overall the parameter space, we try to summarize the main features of the spectrum in thefollowing way:

0. The NLSP is usually a slepton or a neutral higgsino. The mass splitting betweensleptons receives three different computable contributions; all of them (apart from aless important RG effect) tend to make an almost right-handed state the lightestslepton. Although the R is often lighter than the higgsino (see Fig. 2), it isalways possible to force an higgsino NLSP by increasing the value of the unknownmessenger Yukawas which decreases the value of that gives the correct EWSB.When n= 4 the NLSP can be a stop, while for large n& 10 the NLSP can be a bino.

1. When n = 5 the NLSP is most often a neutral higgsino, sleptons are light, and allgauginos have a comparable mass above the squark masses.

2. When n = 6,7,8 the electroweak gauginos are lighter than the squarks, but heavierthan the higgsinos.

3. When n 1 the sfermion and gaugino masses are dominated by the pure anomalymediated contribution to gaugino masses.

1 In DAM models with n 5 the fine tuning (FT [26]) of MZ with respect to the soft terms is typically low. Bychoosing appropriate values of the unknown Yukawa couplings it is even possible to get FT 1. However thisdoes not mean that DAM models are perfectly natural: since the soft terms depend on unknown Yukawa couplingsthe FT with respect to just the soft terms is not an adequate measure of naturalness [27].


(a) (b)Fig. 4. Spectrum of sparticles as function of n for M0 = 1015 GeV (a) and as function of M0 forn = 5 (b) at fixed M3 = 500 GeV, t (M) = 0.5 and negligible Yukawa messengers. Dashed (longdashed, continuous) lines refer to sfermions (higgses, fermions). Thin (thick and red) lines refer touncoloured (coloured) sparticles. Black (blue) lines refer to the neutralinos (charginos and sleptons).

As discussed in the next section, features 1 and 2 listed above give characteristicmanifestations at hadronic colliders. It is more difficult to distinguish DAM models withlarger n from mSUGRA or GMSB at hadron colliders, even if for quite large values of n themass spectrum remains significantly different from the one with unified gaugino masses.For example if n= 20 the ratioM1/M3 (connected in a simple way to the measurable ratiobetween the bino and the gluino masses) is still 50% higher than in the unified gauginocase.

In the following section we perform more detailed studies by selecting three referencepoints in the DAM parameter space that capture the main characteristics of the model:

DAM1: we choose n= 5, M0 = 1015 GeV, H = 0, t (MGUT)= 0.5, M3 = 500 GeV inorder to have a characteristic DAM model with n= 5 and higgsino NLSP.

DAM2: we choose n= 6, M0 = 1015 GeV, H = 0, t (MGUT)= 0.5, M3 = 500 GeV inorder to have a characteristic DAM model with n= 6 and slepton NLSP.

DAM3: we choose n = 6, M0 = 1015 GeV, H = 1, t (MGUT) = 0.5, M3 = 500 GeV.DAM3 is similar to DAM2, except the NLSP is a neutral higgsino.

The spectra corresponding to these three sets of parameters are shown in Fig. 1 and listedin Tables 1 and 2. Using these three examples we will now illustrate the phenomenologyat high-energy colliders.


Table 1Masses of the SUSY particles, in GeV, for the DAMmodel point 1

Sparticle spectrum in DAM model 1

Sparticle Mass Sparticle Mass

g 5001 145

2 481

NLSP= 01 136 02 15203 462

04 483

uL 432 uR 384dL 439 dR 371t1 306 t2 454b1 371 b2 406eL 257 eR 190e 246 2461 190 2 257h0 98 H 0 297A0 293 H 303

Table 2Masses of the SUSY particles, in GeV, for the DAM model point 2 (left columns) and for DAMmodel point 3 (right columns)

Sparticle spectrum in DAM model 2 Sparticle spectrum in DAM model 3

Sparticle Mass Sparticle Mass Sparticle Mass Sparticle Mass

g 500 g 5001 176

2 381

1 151

2 381

01 165 02 187 NLSP= 01 141 02 162

03 337 04 382

03 337

04 382

uL 435 uR 399 uL 435 uR 399dL 441 dR 392 dL 441 dR 392t1 326 t2 465 t1 313 t2 470b1 392 b2 412 b1 392 b2 410eL 218 coNLSP= eR 154 eL 218 eR 154e 205 205 e 205 205

coNLSP= 1 154 2 218 1 154 2 218h0 99 H 0 283 h0 101 H 0 290A0 278 H 289 A0 286 H 296


4. Signals at collider

The experimental manifestation of supersymmetry at hadron colliders like the Tevatronand the LHC depends strongly on how the supersymmetric particles are ordered inmass, and on the nature of the lightest superpartner of ordinary particles (stable/unstable,charged/neutral). The model under study has strong dependencies on the parameters ofthe theory, and therefore does not make unique predictions for these important issuesrelevant to collider physics. Furthermore, measuring the parameters at a high-energyhadron collider is not a straightforward task. Nevertheless, we would like to point outsome expectations for these models at hadron colliders despite the above difficulties.

The most important feature of the model we are presenting here is the relatively smallmass gap between all the gauginos. One immediate consequence of this is a changedinterpretation of gluino mass bounds from LEP2 results. The e+e LEP2 collider doesnot produce gluinos directly, yet it does probe the Winos very effectively. Limits on thecharged Wino mass from the four LEP collaborations are nearly 100 GeV (see, e.g, [29]),the exact value depending on the details of the full supersymmetric spectrum. This canbe interpreted as a limit on the gluino mass of about mg & 300 GeV, provided we assumegaugino mass unification. Therefore, if the Tevatron finds a gluino with mass less than300 GeV, by any of the known discovery channels, that would be one piece of evidence forthe AM models. Current direct limits on the gluino mass are approximatelymg & 185 GeVin R-parity conserving supersymmetric models with mq mg , and mg & 220 GeV whenmq =mg [3032].

To be convinced that the DAM model is correct, much additional evidence mustbe gathered consistent with the model. Useful observables at hadron colliders includetotal rates above background in large lepton/jet multiplicity events with missing energy,invariant mass peaks of decaying heavy particles, kinematic edges to lepton or jet invariantmass spectra, and exotic signatures such as a highly ionizing track associated with a stable,heavy, charged particle track passing through the detector. All of these methods can beused to uncover evidence for supersymmetry and to help determine precisely what modelis being discovered.

DAM models have several gross features that may be keys to distinguishing themfrom other models, such as mSUGRA and minimal GMSB. One such feature that wementioned above is the relatively small mass difference between all the sparticles in thespectrum. Typical parameter choices in models of mSUGRA and especially GMSB havenearly an order of magnitude difference between the lightest supersymmetric partner (notcounting the gravitino) and the heaviest partner. The heaviest of these sparticles are usuallythe strongly interacting squarks and gluinos. Consequently, unless sparticles are muchheavier than the top quark, in DAM models the decays g t2t and t1 tN are usuallykinematically forbidden (t1 is the lighter stop and t2 is the heavier stop). Therefore in DAMmodels it is not unusual to have at most two top quarks per event, while four top quarkscan be present in mSUGRA and GMSB models.


4.1. Total rates with two stable sleptons

Over much of the DAM parameter space the lightest supersymmetric partner to beproduced in the detector is the lR . For example, analyzing DAM model point 2 (seeTable 2), we find that the NLSP is mlR = 155 GeV and M1,M2,M3, are 334,364,500and 176 GeV, respectively. Production of gauginos, squarks and sleptons all end upproducing the lightest state lR , which can be discovered rather easily by the detectors.The total supersymmetry production rate at the Tevatron with

s = 2 TeV is more than

200 fb, and with several fb1 expected at Tevatron run II, this choice of parameters forthe model would be detected, despite superpartners not being kinematically accessible atLEP2. A careful analysis of run I data may even be able to discover or definitively rule outthe parameter choices made for this example.

GMSB is another model that has a large parameter space for (quasi)-stable sleptons. Ifstable, charged tracks are discovered at the Tevatron, the first task will be to find the massof the particle, and then determine the rest of the spectrum that gave rise to this sparticle.Finding the mass is relatively straightforward once there is a significant signal. Timinginformation along with dE/dx measurements as the particle passes through the detectorare useful in this regard. Determining what model these stable tracks come from is muchmore difficult. One beginning step will be to estimate total supersymmetry production ratebased on all (lRlR +X) signatures. This can then be compared between the DAM modelpresented here and, say, minimal GMSB.

If we apply the slepton and chargino mass limits from LEP2 to GMSB, and then analyzeexpectations for the Tevatron, we find that squark and gluino production are not significantin supersymmetry searches at the Tevatron. This is even true when the lR is the NLSPand does not decay in the detector perhaps the most likely possibility [33] in GMSBwithN5+5 > 2. Neglecting potentially important detector efficiency issues, every event thatproduces superpartners will be registered and tagged as a supersymmetry event since stablesleptons yield such an exotic signature in the detector [35,36]. Production of sleptons,gauginos, higgsinos, and squarks all will decay ultimately to two charged sleptons plusstandard model particles. Therefore, we can speak about the discovery of these modelssolely by analyzing the two sleptons and ignoring all other associated particles in theevents, just as we did for the DAM. In this case, there is very little variability in the total ratefor lR lR +X, and the rate depends mostly on the number of 5+ 5 messengers. In Fig. 5we plot the range allowed for total supersymmetry production (event simulations wereperformed with ISAJET 7.42 [34]) in GMSB with moderate to small tan as a function ofmeR . (Distinguishing between low and high tan can be accomplished by careful analysisof the associated particles in X [36].) The upper line corresponds to N5+5 = 2 and thelower line corresponds to N5+5 =. In contrast, recall from the paragraph above that atypical DAM set of parameters yielded a total cross-section of 200 fb for mlR = 155 GeVbecause the squarks and gluinos are much lighter and contribute to the signal. Therefore,a first step in distinguishing between DAM models and GMSB models is to measure meRdirectly from stable, charged particle track analysis, and then compare the total measuredrate of (lRlR +X) to Fig. 5.


Fig. 5. Total cross-section for supersymmetry production at the Tevatron. The upper line is for GMSBwithN5+5 = 2 messengers, and the lower line is forN5+5 =messengers. Minimal GMSB modelsare expected to fall within these two lines. DAM models, by contrast, are expected to be have muchhigher cross-sections since squark and gluinos masses are generally much lighter for the same meR .

4.2. Lepton multiplicity and pT distributions

Other important observables in supersymmetric events are the lepton multiplicity andpT distributions. These are often sensitive to the mass hierarchies in the supersymmetricmodel. For example, a large source of high pT leptons in mSUGRA models is the cascadedecays through 1 l01 . The mass difference between m1 and m01 is large, and the1 state is expected to participate significantly in the cascade decays of the heavier squarksand gluino down to the LSP.

In contrast, the DAM model has relatively fewer sources of high pT leptons becauseof the near degeneracy of the NLSP and the next least massive chargino and neutralino.For example, in DAM model point 1 (see Table 1), we find the quasi-stable NLSP is01 H , M1,M2,M3 = 461,468,500 GeV and 380 GeV.mq . 440 GeV. Production ofgluinos and squarks, while large in this model, more rarely produce sleptons becausemq .M1,M2. Instead, q like to decay directly to a quark and a higgsino with no intermediateleptons in a cascade decay. Leptons can arise however from 1 l01 , but these leptonsare somewhat softer because of the near degeneracy between the mostly higgsino 1 and01 states. In the particular example given here, the mass splitting between the lightestchargino and the lightest neutralino is about 10 GeV. The other significant source of leptonscomes from third family superpartner production and decay. Since the stop and sbottomsquarks are rather light in this example, many leptons do get produced from decays of theW and b particles in t bW decays.

The lepton multiplicity and lepton pT depend on the M1 and M2 masses. We illustratethis dependence by first calculating lepton observables for our example model point 1, and


Table 3From DAM model point 1, the lepton multiplicity in 1000 simulated LHC events with at least200 GeV of total missing energy. Leptons are counted if they have < 3 and pT > 10 GeV. Thenumbers in parenthesis have no pT cut on the leptons. In the third column the spectrum is the sameas DAM model point 1, except the M1 and M2 masses are GUT normalized. In the last column, thelepton multiplicity is given for DAM model point 3, which has significant production of leptons dueto on-shell cascades of q W l

Number of DAM Model 1 DAM Model 1 DAM Model 3leptons with Mi = iM3/s

0 813 (741) 714 (700) 161 (122)1 85 (129) 105 (117) 169 (137)2 24 (48) 12 (13) 233 (248)3 1 (5) 1 (2) 99 (117)4 0 (0) 1 (1) 57 (84)5 0 (0) 0 (0) 9 (17)6 0 (0) 0 (0) 2 (5)

7+ 0 (0) 0 (0) 0 (0)

Table 4Values of the RG coefficients in the MSSM

i bi cQi c

Ui c

Di c

Li c

Ei c

ui c

di c

ei

1 3351

308

152

15310

65

1315

715

95

2 1 32 0 032 0 3 3 3

3 3 83 83 83 0 0 163 163 0

then doing the calculation for the same model but with M1 and M2 redefined to be equalto Mi = iM3/s , consistent with gaugino mass unification, while M3 remains the same.In this case, M1 andM2 are reset to 75 GeV and 145 GeV, respectively, andM3 remains at500 GeV. In Table 4 we list the total multiplicities of leptons in 1000 simulated LHC eventsfor the DAM example model, and the DAM example model with M1 and M2 redefined.The lepton multiplicity is defined to be the number of charged leptons of first and secondgeneration with pseudo-rapidity < 3 and transverse momentum pT > 10 GeV present ineach supersymmetry event. We have also required the missing energy to be greater than200 GeV in these events to reduce standard model background, and we have not countedleptons that originate in a QCD jet (isolation requirement).

The lepton multiplicity tends to higher values for the GUT normalized gaugino spectrumrather than the untampered DAM gaugino spectrum. This is largely because more leptonspass the pT > 10 GeV cut due to the large mass mass gap between the mostly bino NLSPand the next higher mass chargino and neutralino. If we put no cut on the pT of the lepton,the number of leptons from the DAM model would be larger than the number of leptons


generated in the cascade decays of the GUT normalized gaugino version of the spectrum.(The number of leptons produced with arbitrarily low pT values is listed in parenthesis inthe table.) This is indicative of the importance of looking carefully at the pT spectrum ofthe leptons to see the imprint of different mass hierarchies in the spectrum.

We demonstrate the softer lepton pT distributions of the DAM model in Fig. 6. We havesimulated 1000 supersymmetric events at the LHC and plotted the pT distribution of theleading lepton with < 3 and pT > 10 GeV. The effect is present as anticipated, and themagnitude of the effect is rather sizeable. In the first bin there is nearly a 50% differencebetween the models. We expect this observable, along with other observables [3739],such as kinematic endpoint distributions, to play a key role in helping to distinguish DAMmodels from their competitors. In this analysis we have been assuming that the signalwith large missing energy, large lepton multiplicity and large overall rate will render thestandard model background not significant enough to diminish our conclusions, but ofcourse a full investigation of the background, and simulations of real detector effects arenecessary to make definitive statements about parameter determinations in supersymmetricmodels. Nevertheless, we are encouraged that distinctions between closely related modelsof supersymmetry can be made at hadron colliders.

If n > 6 it is still possible to have higgsino NLSP: the change in the scalar masses ofHu and ti with H 1 can alter the conditions for EWSB to allow a higgsino NLSP. Forexample, if we employ the same choices of parameters that we used to generate DAMmodel point 2, except now we set H = 1, the resulting spectrum has a higgsino NLSP.

Fig. 6. The pT distribution of the leading lepton in simulated events of supersymmetry productionat the LHC. The solid line is for DAM model point 1 described in the text. The dashed line is for thesame model except the electroweak gaugino masses are GUT normalized with respect to the gluino(Mi = iM3/3). The fainter dotted line represents DAM model point 3. All lines are normalizedto 1 and not the total cross-section. (The total lepton+X cross-section of DAM3 is a factor of 2.8times that of DAM1.)


This is model point 3 given in the right two columns of Table 2. The phenomenology of thismodel with n = 6 and higgsino NLSP is dramatically different than the phenomenologyof point 1. In contrast to DAM1, DAM3 has a high multiplicity of leptons and high pTdistribution of leptons. Table 3 lists the lepton multiplicities for model point 3, and thefaint dotted curve of Fig. 6 demonstrates the flat distribution of lepton pT , characteristic ofa high pT spectrum of leptons. These results are readily understood by inspecting the masshierarchies of point 3 compared to point 1. In point 3 the strongly interacting sparticles(squarks and gluinos) will almost always cascade decay to a lepton. The most effectivepath is through q W l, where at least one lepton results. The mass hierarchies ofpoint 1 do not allow these high lepton multiplicities. Therefore, close inspection of thelepton observables may provide a handle on the parameter H in addition to measurementsof the various sparticle masses.

5. Signals in rare processes

In DAM models the soft terms could contain no extra flavour or CP violating termsbeyond the ones induced by the CKM matrix. There are however two possible exceptions.

1. The and B terms could be complex: in this case they would typically generate toolarge electron and neutron electric dipoles, unless their phases are so small (less thanabout 0.01) [40] 2 that do not significantly affect collider observables.

2. Extra Yukawa couplings not present in the SM can affect the soft terms in a way thatcrucially depends on how the soft terms are mediated. In supergravity heavy particlesaffect the soft terms, while in pure AM models soft terms are not affected by fieldsabove the supersymmetry breaking scale. Like in GMSB, in DAM models the softterms are not affected by interactions of fields heavier than the messenger mass M0.The effective theory at M0 however might not be the MSSM. For example, some ofthe right-handed neutrinos N often introduced in order to generate the observedneutrino masses could be lighter than M0. If they have order one Yukawa couplingsN NLHu they imprint lepton flavour violation in the soft mass terms of left-handedsleptons L inducing significant rates for processes like e . Unlike in GMSBmodels, in DAM models these effects are not suppressed by a (RG-enhanced) loopfactor. However for a right-handed neutrino mass MN 10911 GeV, optimal forleptogenesis, the Yukawa couplings N must be small, N . 0.005, in order to get aleft-handed neutrino mass smaller than 1 eV.

If none of these exceptions is realized, in DAM models supersymmetric loop effects onlygive new contributions to processes already present in the SM (b s , g 2 of ,K , 1mB , K pi decays) but cannot give rise to new effects (like e decay,electric dipoles, contributions to K,Bd,Bs,D physics with non-CKM and/or non-SMchiral structure). Taking into account the accelerator bounds on sparticle masses, few rareprocesses can receive interesting contributions:

2 For a recent useful analysis see [41].


Supersymmetric corrections can significantly enhance BR(B Xs ) [42] over itsSM value. For example in all the reference points studied in the previous section theb s effective operator (with all fields and couplings renormalized at the relevantscale QmB ) is

Heff =[0.29(SM) 0.08(charged higgs) 0.07(chargino)]VtbV t s

eg22(4pi)2

mb

2M2W

[(sLF

bR)+ h.c.].

Unless the chargino contribution compensates the charged higgs contribution (itssign depends on the relative sign between m, and B), BR(B Xs ) is twotimes larger than in the SM, conflicting with experimental bounds. Even assuming adestructive interference (otherwise the sparticles must have unnaturally heavy masses)a detectable supersymmetric correction to the B Xs branching ratio remainslikely. In these models the gluino/bottom contribution is computable, and turns outto be negligible. Since EW gauginos are heavier than in mSUGRA or GM models, a supersymmetric

correction to the anomalous magnetic moment of the [4345], at a level detectablein forthcoming experiments [46] is rather unlikely (but not impossible). The supersymmetric corrections to K and B mixing [42] can be larger than in

mSUGRA and GMSB models, because coloured sparticles can be lighter. With areasonable sparticle spectrum,1mB can be enhanced by (2025)% with respect toits SM value. Such corrections are comparable to the present theoretical uncertaintieson the relevant QCD matrix elements. Larger corrections are present in small cornersof the parameter space with light stops.

6. NLSP decays and nucleosynthesis

The lightest supersymmetric particle is the fermionic partner of the modulusX. Indeedby studying the effective action in Eq. (5) one finds m =O(/4pi)2F . Therefore, unlesssome coupling in the messenger sector is strong, we expect m to be smaller than a fewGeV, so that is the LSP. The is a welcome fact: the LSP of our model is automaticallyneutral and unwanted charged relics are avoided. On the other hand, the lightest sparticlein the SM sector, the NLSP, can be charged (a right-handed slepton) as it decays into .Now, the effective couplings governing this decay are suppressed by inverse powers of themessenger mass and by loop factors. Indeed plays a role similar to that of the goldstinoin gauge mediated models. In the range of allowed M0, the NLSP lifetime is so long thatit behaves as a stable particle in collider experiments. However, lifetimes in excess of 1 s,can dangerously affect the big-bang predictions of light element abundances. In the rest ofthis section we will discuss the constraints placed on M0 by nucleosynthesis.

Let us first derive the couplings of to the SM particles. For a chiral matter multipletQ the effective Lagrangian, computing loop corrections with superfield techniques, is [10]

Leff =d4 ZQ

(2/,XX/

)QQ (11)


leading to a coupling

Lqq = q FM0

qq + h.c., (12a)where

q = (ln2 + lnXX)lnXX lnZQ. (12b)The above expression is easily obtained by expanding lnZQ in powers of ln and lnX/M0and by noting that the leading contribution to Lqq comes from second order cross terms ln lnX/M0. In the case of right-handed sleptons we have

eR =1

8pi22n(n+ 33/5)

11(21(M0) 21(MZ)

), (13)

where we have taken = mZ in Leff. Notice that the coupling from Eqs. (12b), (13)is qualitatively similar to the goldstino coupling for a gauge mediated model withFX/X = F . However in gauge mediation, unlike here, q = m2q (M20/FX)2 by currentalgebra.

In the case of a higgsino NLSP the relevant term is the one generating

Leff =d4 HuHd

X

XZ(XX/

). (14)

As discussed in Section 2, the effective term is equal to (XZ)|2/M0. By writing X =M0+ X, it is easy to see that, at the leading order in an expansion in 1/M0 and , Eq. (14)leads to a superpotential coupling

Leff =d2

M0HuHd X. (15)

Notice that the coupling of to the Higgs sector is stronger than that to sfermions. It isproportional to the supersymmetric mass (1-loop) rather than to the mass splitting B(2-loop). This is consistent, since is not the goldstino. The most important consequenceof Eq. (15) is that it can mediate the decay N1 h whenever allowed by phase space.For a higgsino-like NLSP, we have mN1 ' with N1 ' H 0 = (H 0d H 0u ), depending onthe sign of . The width of a higgsino-like NLSP is then

N1h =(cos sin)2

64pi3

M20

(1

2

m2h

)2, (16)

corresponding to a lifetime shorter than a second over most of parameter space already forM0 . 1015 GeV. We conclude that nucleosynthesis does not place significant bounds on ahiggsino LSP whenever >mh, which is almost required by experimental bounds.

Let us consider now the bounds on a stau NLSP. The coupling to is smaller than forthe higgsino NLSP (2-loop versus 1-loop). The correspondingly longer lifetime is wellapproximated, as a function of m and M0, by

=(

M01013 GeV

)2(200 GeVm

)3s. (17)


This quantity is larger than 1 s over a significant fraction of parameter space, where the decay can dangerously affect nucleosynthesis. The most stringent bounds come fromdecays processes involving hadronic showers. These showers break up the ambient 4Heinto D and 3He and can lead to an overabundance of the two latter elements. The showerscan also overproduce 6Li and 7Li from hadrosynthesis of 3He, T or 4He. The decay leads to hadronic showers as the further decays hadronically with a largebranching ratio. Using the results in Ref. [47,48], it was concluded in Ref. [49] thatlifetimes larger than 104 s lead to unacceptable overproduction of 7Li. Ref. [49] showsa careful analysis, including a computation of the relic NLSP density at nucleosynthesis,for gauge mediated models with a stau NLSP. A similarly detailed analysis is beyond theaim of the present paper, but we expect that the results of [49] can be carried over to ourcase. This is because the bounds do not depend very strongly on the relic density, whichin our model is not going to differ drastically from that in gauge mediation. Therefore weconclude that overproduction of 7Li gives the bound < 104 s. By Eq. (17) this boundroughly translates into M0 < 1014 GeV.

A stronger bound, forbidding decays between 102 and 104 s can come from the deute-rium abundance XD normalized to hydrogen. However there is, at the moment a contro-versy in the measurement of XD from astrophysical observation. Two values are quoted inthe literature, a high one XD = (1.9 0.5) 104 from Ref. [50] and a low one XD =(3.39 0.25) 105 from Ref. [51]. In Ref. [49] it was concluded that no further boundsare obtained when the high value of XD is assumed. On the other hand, the low XD valuecan give a stronger constraint < 102 s.

We conclude that nucleosynthesis places a significant bound on the messenger masswhen the NLSP is a stau. This bound on M0 can range between 1013 and 1015 GeVdepending upon the model parameters m and n and on the astrophysical input data. Westress that while the bound is not negligible, there remains a large allowed region 1010 mq for DAM1, squark production leads to a cascade with fewer highpT leptons than in standard bino LSP scenarios. The softness of the leptons is due to thesmall mass splitting among the charged and neutral higgsinos produced in the cascade,while in mSUGRA and GMSB the LSP is well split from the next higher mass neutralinoand chargino. Also, now the squarks often decay directly to the lightest higgsino, withoutproducing any lepton.

On the other hand for mW < mq (e.g., DAM3), more high pT leptons are producedthan usual. This is because squarks can decay via qL W H 01,2, H+ and qL W l H 0. Energetic leptons are then produced in W decays and/or the l decays, whileadditional softer leptons are produced when H2 and H+ further decay to H1. The lepton pTdistribution for the above cases is shown in Fig. 6 where it is compared to a standard binoLSP scenario, and lepton multiplicities are given in Table 3. Of course similar signaturesare obtained in any scenario where the higgsinos are somewhat lighter than winos and bino.However, the unique mass hierarchy of the charginos, neutralinos and sleptons in DAM,as illustrated by the spectrum of DAM3, rarely occurs in GMSB or mSUGRA. To furthertell DAM from these other possibilities one can resort to other observables. One additionalconsequence of the compact DAM spectrum is that more than 2 tops in the gluino cascadeare often forbidden by phase space, whereas higher multiplicity of top quarks may exist infinal states of GMSB and mSUGRA.

We conclude that DAM provides an interesting alternative to conventional soft termscenarios from both the theoretical and the phenomenological point of view. Depending onthe model parameters DAM can manifest itself at hadron colliders in a few qualitativelydifferent ways. We have studied the distinguishing signatures that characterize the variouscases. If sparticles exist and if DAM is the source of their masses, then we may not haveto wait until the LHC to discover superpartners: TEVII may have enough luminosity andenergy.


Acknowledgements

We thank K. Matchev, S. Mrenna, M. Nojiri, and F. Paige for useful and stimulatingdiscussions. R.R. and J.D.W. wish to thank the ITP, Santa Barbara for its support duringpart of this work (NSF Grant No. PHY94-07194).

Appendix A. RG evolution of soft terms with non-unified gaugino masses

In this appendix we present semi-analytic solutions for the one-loop RG evolution ofthe soft terms in presence of the large Yukawa coupling of the top. We give the soft termsat an arbitrary energy scale Q, starting from an arbitrary scale M0 with arbitrary gauginomasses Mi0, sfermion masses m2R0, A terms A

fg0, -term 0, and B-term B0. We do not

assume unification of the gauge couplings. Here i = {1,2,3} runs over the three factors ofthe SM gauge group, f= u,d, e, g = 1,2,3 is a generation index and R runs over all thescalar sparticles (Qg,Ug,Dg,Eg,Lg,Hu,Hd). These formul, obtained with superfieldtechniques [19,20,5254], are significantly simpler than equivalent ones already existingin the literature [55] 3 because they never involve double integrals over the renormalizationscale. The running soft terms renormalized at an energy scale Q are

Mi(Q)=Mi0/fi, (18a)(Q)=0 ybu1Eh, (18b)B(Q)=B0 + 2xLi1Mi0 bu1I /bt , (18c)Afg(Q)=Afg0 + xf1i(E)Mi0 bfgI (E)/bt , (18d)m2R(Q)=m2R0 + xRi2M2i0 btRI 35 YRb1 IY , (18e)

where

t (Q) 2(4pi)2

lnM

Q, fi

(t (Q)

) i(M0)i(Q)

,

E(t)i

fc1 /bii (t), x

in

ci

bi(1 fni )

and M is any scale. All b-factors are simple numerical coefficients: the bi are thecoefficients of the one-loop functions, {b1, b2, b3} = {33/5,1,3}. The YR are thehypercharges of the various fields R, normalized as YE =+1. The btR coefficients vanishfor all fields R except the ones involved in the top Yukawa coupling: btHu = 1/2, btQ3 = 1/6and bt

U3= 1/3. The factor IY = (11/f1)Tr[YRm2R0] takes into account a small RG effect

induced by the U(1)Y gauge coupling. The t effects are contained in

3 Compact analytical solutions of the RGE, analogous to those given in this appendix, have been recentlypresented in [56]. These expressions obtained via superfield expansion of the iterative solution of thesupersymmetric RGE [57] can be applied even if the b- and -Yukawa couplings are not negligible.


I = [At0+Mi0Xi ], (19a)I = (m2Q30 +m2U30 +m2Hu0)+ (1 )A2t0 [(1 )At0 Mi0Xi]2 + M2i0Xi2 + Mi0Mj0Xij , (19b)

where At0 =Au30 is the top A-term at M0 and

X(M0,Q)t (Q)

t (M0)

Eu(t) dt, Xin(M0,Q)=Euxuin dtE dt

,

Xij (M0,Q)=Euxu

i1xui1dt

E dt. (20)

All the integrals are done in the same range as the first one. The semi-analytic functionsXin are needed only for n = 1 and 2. In practice one has to compute numerically fewfunctions of two variables, Q and M0. A more efficient computer implementation isobtained rewriting the X(M0,Q) functions in terms of 1+ 3+ 9 functions with only oneargument

F(M0) F(Q)=t (Q)

t (M0)

Eu(t) dt, Fin (M0) Fin(Q)=Eu

f nidt,

Fij (M0) Fij (Q)=

Eu

fifjdt.

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Nuclear Physics B 576 (2000) 2955www.elsevier.nl/locate/npe

Effects of CP-violating phases on Higgs bosonproduction at hadron colliders in the Minimal

Supersymmetric Standard ModelA. Dedes a, S. Moretti a,b

a Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UKb Department of Radiation Sciences, Uppsala University, P.O. Box 535, 75121 Uppsala, Sweden

Received 4 October 1999; accepted 2 March 2000

Abstract

If the soft Supersymmetry (SUSY) breaking masses and couplings are complex, then theassociated CP-violating phases can in principle modify the known phenomenological pattern ofthe Minimal Supersymmetric Standard Model (MSSM). We investigate here their effects on Higgsboson production in the gluongluon fusion mode at the Tevatron and the Large Hadron Collider(LHC), by taking into account all experimental bounds available at present. The by far most stringentones are those derived from the measurements of the Electric Dipole Moments (EDMs) of fermions.However, it has recently been suggested that, over a sizable portion of the MSSM parameter space,cancellations among the SUSY contributions to the EDMs can take place, so that the CP-violatingphases can evade those limits. We find a strong dependence of the production rates of any neutralHiggs state upon the complex masses and couplings over such parts of the MSSM parameter space.We show these effects relatively to the ordinary MSSM rates as well as illustrate them at absolutecross section level at both colliders. 2000 Elsevier Science B.V. All rights reserved.

PACS: 12.60.Jv; 12.60.Fr; 14.80.Ly; 13.85.-tKeywords: Supersymmetry; Higgs bosons; Supersymmetric partners of known particles; Hadron colliders

1. Introduction and plan

The soft SUSY breaking parameters of the MSSM can well be complex. Even in theabsence of flavour non-conservation in the sfermion sector, the higgsino mass term, thegaugino masses, the trilinear couplings and the Higgs soft bilinear mass need not be real.Assuming universality of the soft gaugino masses at the Grand Unification (GUT) (orPlanck) scale, the effects of complex soft masses and couplings in the MSSM Lagrangian(see Appendix A) can be parametrised in terms of only two independent phases [1,2],0550-3213/00/$ see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0550-3213(00) 00 14 4- 9

30 A. Dedes, S. Moretti / Nuclear Physics B 576 (2000) 2955

and A, associated to the (complex) higgsino mass term, , and to the trilinear scalarcoupling A 1 , respectively. In other terms,

ei = || , eiA = A|A| . (1)

Their presence is a potentially dangerous new source of violation of the CP-symmetry inthe MSSM. But their size can in principle strongly be constrained by the measurementsof the fermionic EDMs (mainly, of electron and neutron) and several analyses [35] haveindicated that and A must be small in general. However, recent investigations [614]have shown that, in a restricted but still sizable part of the parameter space of the MSSM,the bounds drawn from the EDM measurements are rather weak, so that such phases caneven be close to pi/2. This is a consequence of cancellations taking place among the SUSYloop contributions to the EDMs. Although, in order to be effective, these require a certainamount of fine-tuning among the soft masses and couplings [14], it has recently beensuggested that such cancellations occur naturally in the context of Superstring models[15,16]. If the SUSY loop contributions to the EDMs do vanish, then, as emphasised bythe authors of Ref. [17], SUSY parameters with large imaginary parts may have a non-negligible impact on the confrontation of the MSSM with experiments. In particular, manyof the SUSY (s)particle production and decay processes develop a dependence on andA, so that, in view of the importance of searches for New Physics at present and futureaccelerators, their phenomenology needs a thorough re-investigation.

Various sparticle processes including the effect of such phases have recently beenconsidered. For example, neutralino [1820] and chargino [21,22] production at LEP, atthe CERN LHC [23] as well as at future electronpositron linear colliders (LCs) [24,25].Direct CP-asymmetries in decays of heavy hadrons, such as BXs+ , BXd+ andB Xsl+l, have been investigated in the context of the Supergravity inspired MSSM(M-SUGRA): in Refs. [26,27], [28] and [29,30], respectively.

In this paper, we are concerned with the Higgs sector of the MSSM. Here, although thetree-level Higgs potential is not affected by the CP-violating phases, since M2H1 , M

2H2

andtan (the mass parameters of the two Higgs fields and the ratio of their vacuum expectationvalues (VEVs), respectively) are real and enters only through ||2, it should be noticedthat this is no longer the case if one includes radiative corrections. In their presence, onefinds [3133] that the three neutral Higgs bosons can mix and that their effective couplingsto fermions can be rather different at one loop. However, for the MSSM parameter spacethat we will consider here, such corrections turn out to be negligible, of the order of just afew percent, as compared to those induced at the lowest order by the CP-violating phasesin the squark sector. There are some reasons for this. First of all, the induced radiativecorrections to the Higgs-quarkquark vertices can be parametrised in terms of the massof the charged Higgs boson, MH (MA0 ). Then, one can verify that they essentiallydepend only upon the input values given to ||, |A| and MSUSY (the typical mass scale ofthe SUSY partners of ordinary matter). Here, we will mainly be concerned with trilinear

1 For simplicity, hereafter (except in the appendices), we assume AAu =Ad at the electroweak (EW) scale,i.e., O(MZ), where u and d refer to all flavours of up- and down-type (s)quarks.

A. Dedes, S. Moretti / Nuclear Physics B 576 (2000) 2955 31

Fig. 1. SM-like contributions from top (t ) and bottom (b) quarks to Higgs boson production viagg0 in the MSSM.

couplings in the range |A|. 700 GeV, higgsino masses || of the order of 600 GeV or so,and MSUSY ' 300 GeV. According to the analytic formulae of Ref. [32,33], in the aboveMSSM regime, one finds negligible corrections to the tree-level h0t t and h0bb couplings.(Similarly, for the case of the lightest Higgs boson mass.) In contrast, the strength of theH 0t t vertex can significantly be modified for not too heavy masses of the charged Higgsboson (say, MH MA0 < 200 GeV) and rather large values of |A| and || (typically,|A| ' || ' 2 TeV), a region of parameter space that we will avoid, whereas that of theH 0bb one is generally small because we shall limit ourselves to the interval 2 . tan. 10. As for papers allowing instead for the presence of non-zero values of and/or Aand studying the Higgs sector, one can list Refs. [3436]. The first publication deals withdecay rates whereas the second one with Higgs production channels probable at a futureLC. In such papers though, no systematic treatment of the limits imposed by the EDMmeasurements was addressed. Such effects ought to be incorporated in realistic analyses ofthe MSSM Higgs dynamics.

Here, we have necessarily done so, since it is our purpose to study the effect of finitevalues of and/or A on Higgs production via the gg 0 channel [3742], where0 = H 0, h0 and A0 represents any of the three neutral Higgs bosons of the MSSM.(A preliminary account in this respect was already given by the authors in Ref. [43].)These processes proceed through quark (mainly top and bottom: i.e., t and b, see Fig. 1)and squark (mainly stop and sbottom: i.e., t1, t2 and b1, b2, see Fig. 2, each in increasingorder of mass) loops, in which the (s)fermions couple directly to 0. Needless to say, asin the MSSM the lightest of the Higgs particles is bound to have a mass not much largerthan that of the Z boson,MZ , much of the experimental effort at both the Tevatron (Run 2)and the LHC will be focused on finding this Higgs state, h0. In this respect, we remindthe reader that direct Higgs production via gluongluon fusion is the dominant mechanismover a large portion of the MSSM parameter space at the LHC and a sizable one at theTevatron [4450].

The plan of the paper is as follows. The next section describes our theoretical frameworkand discusses experimental limits on the parameters of the model. The following onebriefly sketches the way we have performed the calculation. Section 4 presents somenumerical results, whereas Section 5 summarises our analysis and draws the conclusions.Finally, in the two appendices, we introduce our notation and explicitly derive the Feynmanrules and cross section formulae needed for our numerical analysis.


Fig. 2. SUSY-like contributions from top (t1,2) and bottom (b1,2) squarks to Higgs boson productionvia gg0 in the MSSM. (Notice that, if the CP-symmetry is conserved, then 0 6=A0.)

2. The theoretical model and its parameters

We work in the theoretical framework provided by the MSSM, the latter includingexplicitly the CP-violating phases and assuming universality of the soft gaugino massesat the GUT scale and universality of the soft trilinear couplings at the EW scale. We defineits parameters at the EW scale, without making any assumptions about the structure of theSUSY breaking dynamics at the Planck scale, whether driven by Supergravity (SUGRA),gauge mediated (GMSB) or proceeding via other (yet unknown) mechanisms. We treat theMSSM as a low-energy effective theory, and input all parameters needed for our analysisindependently from each other. However, we require these to be consistent with currentexperimental bounds. In fact, given the dramatic impact that the latter can have on theviability at the Tevatron and/or the LHC of the CP-violating effects in the processes we aredealing with, we specifically devote the two following subsections to discuss all availableexperimental constraints. The first focuses on collider data, from LEP and Tevatron; thesecond on the measurements of the fermionic EDMs. (Some bounds can also be derivedfrom the requirement of positive definiteness of the squark masses squared.) Followingthis discussion, we will establish the currently allowed ranges for the Higgs and sparticlemasses and couplings.

Before proceeding in this respect though, we declare the numerical values adopted forthose MSSM parameters that are in common with the Standard Model (SM). For thetop and bottom masses entering the SM-like fermionic loops of our process, we haveused mt = 175 GeV and mb = 4.9 GeV, respectively. As for the gauge couplings, thestrong, electromagnetic (EM) coupling constants and the sine squared of the Weinbergangle, we have adopted the following values: s(MZ)= 0.119, EM(MZ)= 1/127.9 andsin2 W (MZ)= 0.2315, respectively.


2.1. Limits from colliders

The Higgs bosons and sparticles of the MSSM that enter the gg 0 productionprocesses can also be produced via other channels, both as real and virtual objects. Fromtheir search at past and present colliders, several limits on their masses and couplings havebeen drawn. As for the neutral Higgs bosons of the MSSM, the most stringent bounds comefrom LEP. For both Mh0 and MA0 these are set after the 1998 LEP runs at around80 GeV by all Collaborations [51], for tan > 1. 2 The tightest experimental limits onthe squark masses come from direct searches at the Tevatron. Concerning the t1 mass, forthe upper value of tan that we will be using here, i.e., 10, the limit on mt1 can safely bedrawn at 120 GeV or so [53], fairly independently of the SUSY model assumed. As forthe lightest sbottom mass, mb1 , this is excluded for somewhat lower values, see Ref. [54].Besides, D also contradicts all models with mq 6=t1,b1 < 250 GeV for tan . 2, A = 0and < 0 [55] (in scenarios with equal squark and gluino masses the limit goes up tomq 6=t1,b1 < 260 GeV).

2.2. Limits from the EDMs

These are possibly the most stringent experimental constraints available at present on thesize of the CP-violating phases. The name itself owns much to the consequences induced inthe QED sector. In fact, to introduce a complex part into the soft SUSY breaking parametersof the MSSM corresponds to explicitly violating CP-invariance in the matrix elements(MEs) involving the EM current, as the phases lead to non-zero TP form factors, which inturn contribute to the fermionic EDMs. In contrast, within the SM, it is well known thatcontributions to the EDMs arise only from higher-order CP-violating effects in the quarksector, and they are much smaller than the current experimental upper bounds. At 90% CL,those on the electron, de [56,57], and neutron, dn [58], read as:

|de|exp 6 4.3 1027 e cm, |dn|exp 6 6.3 1026 e cm. (2)As mentioned in the introduction, if cancellations take place among the SUSY contribu-tions to the electron and neutron EDMs, so that their value in the MSSM is well belowthe above limits, i.e., |de|MSSM |de|exp and |dn|MSSM |dn|exp, then and A can belarge. To search for those combinations of soft sparticle masses and couplings that guaran-tee vanishing SUSY contributions to the EDMs for each possible choice of the CP-violatingphases, we have scanned over the (,A) plane and made use of the program of Ref. [14].This returns those minimum values of the modulus of the common trilinear coupling, |A|,above which the cancellations work. For instance, in the case of the neutron EDMs, thedominant chargino and gluino contributions appear with opposite sign over a large portionof the MSSM parameter space. Thus, for a given ||, a chargino diagram can cancel agluino one and this occurs for certain values of the gaugino/squark masses and a specific

2 See also [52] for a more recent and somewhat higher limit that we adopt here on Mh0 from ALEPH, of

about 85.2 GeV for tan > 1 at 95% confidence level (CL), using data collected at collider centre-of-mass (CM)energies in the range 192 GeV .see . 196 GeV and a total luminosity of about 100 pb1.


choice of |A| [614]. In general, internal cancellations are more likely among the SUSYcontributions to the neutron EDMs, than they are in the case of the electron. So much sothat, in the former case, it is even possible to remain consistently above the experimentallimits (2) if one only assumes the phase of || to be non-zero [14].

However, not all the surviving combinations of , A and |A| are necessarily allowed.In fact, one should recall that physical parameters of the MSSM depend upon these threeinputs. In particular, the squark masses (entering the triangle loops of the productionprocesses considered here) are strongly related to , A and |A|. Given the assumptionsalready made on the trilinear couplings (i.e., their universality), and further setting (seeAppendix A for the notation)

Mq3 MQ3 =MU3 =MD3 , (3)Mq1,2 MQ1,2 =MU1,2 =MD1,2 , (4)

with Mq1,2 >Mq3 , where Mq1,2,3 are the soft squark masses of the three generations, onegets for the lightest stop and sbottom masses the following relations:

m2t1=M2q3 +m2t + 14M2Z cos 2

{( 56M

2Z 43M2W

)2cos2 2

+ 4m2t[|A|2+ ||2 cot2 + 2|A||| cos( A) cot]}1/2, (5)

m2b1=M2q3 +m2b 14M2Z cos 2

{( 16M

2Z 23M2W

)2cos2 2

+ 4m2b[|A|2 + ||2 tan2 + 2|A||| cos( A) tan]}1/2. (6)

For some choices of , A and |A| and a given value ofMq3 , || and tan , the two abovemasses (squared) can become negative. This leads to a breaking of the SU(3) symmetry,that is, to the appearance of colour and charge breaking minima. In order to avoid this,some points on the plane (,A) will further be excluded in our study.

3. Numerical calculation

We have calculated the Higgs production rates in presence of the CP-violating phasesexactly at the leading order (LO) and compared them to the yield of the ordinary MSSM(that is, phaseless) at the same accuracy. In our simulations, we have included only thet-, b-, t1-, t2-, b1- and b2-loops, indeed the dominant terms, because of the Yukawa typecouplings involved. In order to do so, we had to compute from scratch all the relevantanalytical formulae for A0 production. In fact, one should notice that for such a Higgsstate there exist no tree-level couplings with identical squarks if = A = 0, whereasthey appear at lowest order whenever one of these two parameters is non-zero. Besides,being the quark loop contributions antisymmetric (recall that A0 is a pseudoscalar state)and the squark loop ones symmetric, no interference effects can take place between theSM- and the SUSY-like terms in the ME for gg A0. (That is to say that - and A-induced corrections are always positive if 0 = A0.) The full amplitude is given explicitly


in Appendix B. For completeness, we have also recomputed the well known expressions forscalar Higgs production,0 = h0,H 0, finding perfect agreement with those already givenin literature (again, see Appendix B). Here, CP-violating effects can produce correctionsof both signs.

It is well known that next-to-leading order (NLO) corrections to gg 0 processesfrom ordinary QCD are very large [5964]. However, it has been shown that they affect thequark and squark contributions very similarly [64]. Thus, as a preliminary exercise, onecan look at the LO rates only in order to estimate the effects induced by the CP-violatingphases. In contrast, for more phenomenological analyses, one ought to incorporate theseQCD effects. We have eventually done so by resorting to the analytical expressions for theheavy (s)quark limit given in Ref. [64]. These are expected to be a very good approximationfor Higgs masses below the quarkquark and squarksquark thresholds. Therefore, we willconfine ourselves to combinations of masses which respect such kinematic condition.

As Parton Distribution Functions (PDFs), we have used the fits MRS98-LO(05A) [65]and MRS98-NLO(ET08) [66], in correspondence of our one- and two-loop simulations,respectively. Consistently, we have adopted the one- and two-loop expansion for the strongcoupling constant s(Q), with all relevant (s)particle thresholds onset within the MSSMas described in [67]. The running of the latter, as well as the evolution of the PDFs, wasalways described in terms of the factorisation scale Q QF , which was set equal to theproduced Higgs mass, M0 . In fact, the same value was adopted for the renormalisationscale QQR entering the Higgs production processes, see Eqs. (B.1) and (B.6)(B.7).

Finally, for CM energy of the LHC, we have assumedspp = 14 TeV; whereas for the

Tevatron, we have takenspp = 2 TeV.

4. Results

A sample of , A and |A| values that guarantee the mentioned cancellations can befound in Fig. 3, for the two representative choices ofMq1,2 given in Table 1. Here, both ||and Mq3 are held constant at, e.g., 600 and 300 GeV, respectively. The soft gluino massis given too, in Table 1, as it enters our analysis indirectly, through the EDM constraints(recall the discussion in Section 2.2). The allowed values for the modulus of the (common)trilinear couplings |A| are displayed in the form of a contour plot over the (,A) plane,where both phases are varied from 0 to pi . (Same results are obtained in the interval(pi,2pi), because of the periodic form of the SUSY couplings and mixing angles: see

Table 1Two possible parameters setups of our model. (Apart from thedimensionless tan , all other quantities are given in GeV.)

tan || Mq1,2 Mq3 Mg MA0

2.7 600 2500 300 1000 20010 600 5000 300 2000 200


Fig. 3. Contour plots for the values of the modulus of the common trilinear coupling, |A|, needed inorder to obtain the cancellations of the SUSY contributions to the one-loop EDMs, over the (,A)plane for small (left-hand plot) and large (right-hand plot) tan . The other MSSM parameters areas given in Table 1. Here and in the following, symbols denote points excluded because ofthe negativity of the squark masses squared; symbols denote points excluded by the two-loopZeeBarr type contributions to the EDMs; and symbols denote points excluded from Higgsboson and squark direct searches, respectively.

Fig. 4. Contour plots for the values of the lightest top squark mass, mt1 , corresponding to those of|A| in Fig. 3, over the (,A) plane for small (left-hand plot) and large (right-hand plot) tan . Theother MSSM parameters are as given in Table 1.

Appendix A.) In the same plots, we have superimposed those regions (to be excludedfrom further consideration) over which the observable MSSM parameters assume valuesthat are either forbidden by collider limits (dots for the lightest stop mass and squares forthe lightest Higgs mass: see Figs. 45) or for which the squark masses squared become


Fig. 5. Contour plots for the values of the lightest Higgs boson mass, Mh0 , corresponding to those of|A| in Fig. 3, over the (,A) plane for small (left-hand plot) and large (right-hand plot) tan . Theother MSSM parameters are as given in Table 1.

negative (crosses), for a given combination of the other soft SUSY breaking parameters.Typically, we obtain that for small phases, i.e., . pi/300 0.01, the value of |A|tends to be zero for almost all values of A. In the region where both phases are quite large(,A pi/2), the modulus of the trilinear coupling must be around 700 GeV, for theEDM constraints to be satisfied.

In Table 1, in order to completely define our model for the calculation of the gg0processes, we also have introduced the Higgs sector parameters: the mass of one physicalstates, e.g., MA0 , and the ratio of the VEVs of the two doublet fields, i.e., tan . Wehave fixed the former to be 200 GeV, whereas two possible choices of the latter havebeen a

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