Patterns of Soft Masses from General Semi-Direct Gauge Mediation
Riccardo Argurio, Matteo Bertolini, Gabriele Ferretti, Alberto Mariotti
aa r X i v : . [ h e p - ph ] F e b SISSA-71/2009/EPULB-TH/09-40
Patterns of Soft Masses fromGeneral Semi-Direct Gauge Mediation
Riccardo Argurio , Matteo Bertolini , Gabriele Ferretti and Alberto Mariotti Physique Th´eorique et Math´ematique and International Solvay InstitutesUniversit´e Libre de Bruxelles, C.P. 231, 1050 Bruxelles, Belgium SISSA and INFN - Sezione di TriesteVia Beirut 2; I 34014 Trieste, Italy Department of Fundamental PhysicsChalmers University of Technology, 412 96 G¨oteborg, Sweden Theoretische Natuurkunde and International Solvay InstitutesVrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
Abstract
We give a general formulation of semi-direct gauge mediation of supersymmetry breakingwhere the messengers interact with the hidden sector only through a weakly gaugedgroup. Using this general formulation, we provide an explicit proof that the MSSMgaugino masses are vanishing to leading order in the gauge couplings. On the other hand,the MSSM sfermion masses have, generically, a non-vanishing leading contribution. Wediscuss how such a mechanism can successfully be combined with other mediation schemeswhich give tachyonic sfermions, such as sequestered anomaly mediation and some directgauge mediation models.
Introduction
Gauge mediation [1] (see [2] for a comprehensive review) is a popular mechanism tomediate supersymmetry breaking from a hidden sector, where the breaking occurs,to the visible sector (the MSSM or extensions thereof). Its main virtue is, besidescalculability, that of naturally implementing a strong suppression of all soft termsleading to flavor changing neutral currents. In this framework, the only degrees offreedom of the MSSM which interact with the hidden sector are those of the gaugesector.In exploring the phenomenological implications of various mechanisms of super-symmetry breaking, it is desirable to isolate those predictions that are independentof the specific details of the model. Recently, a general formulation of gauge me-diation (GGM) was given that accomplishes this in a very explicit way [3]. In thisformalism, gauge mediation is formulated in terms of the currents (both fermionicand bosonic) that couple to the gauge degrees of freedom. It can be shown thatthe spectrum of soft masses is restricted by two sum rules for the sfermions but isotherwise generic. For instance, these sum rules do not specify any pattern betweenthe masses of gauginos and the masses of the sfermions, which is thus an undefinedfeature in a generic gauge mediation model. For a discussion of such patterns andhierarchies, see e.g. [4, 5].In practice, specific models of gauge mediation will prominently feature mes-senger superfields, which are those fields charged under the MSSM gauge groupsthat also interact with the supersymmetry breaking sector. Once a given model isspecified, patterns and hierarchies among the soft terms immediately arise. As an ex-ample, the minimal gauge mediation scenario (MGM) [6] (often used in phenomeno-logical applications) has messenger chiral superfields which couple trilinearly with aspurion that provides them with both a supersymmetric mass and an off-diagonalnon supersymmetric mass. In this model the gaugino and sfermion masses turn outto be of the same order of magnitude. Other examples, which are denoted directgauge mediation models (DGM) [7], have messengers which are typically compositefields directly participating in the dynamical supersymmetry breaking mechanism.Clearly, those would be the most appealing models (solving the hierarchy problem1ith no tuned parameters), but it turns out that often in such models the gauginomasses are highly suppressed or the sfermion masses are tachyonic. In addition,supersymmetry breaking generally requires a large hidden gauge group giving riseto a Landau pole in the visible couplings. Though models which cure some of theseproblems exist (possibly based on metastable hidden sectors, see e.g. [8]), it wouldbe desirable to single out general properties of models which are phenomenologicallyviable.We focus on a class of models where the messengers interact with the hidden sec-tor only through (non-MSSM) gauge interactions with gauge group G h and coupling g h but, unlike DGM, they do not participate to the supersymmetry breaking dy-namics. These models were dubbed semi-direct gauge mediation (SDGM) in [9]. Asubclass of these models – characterized by the further requirement that letting thehidden coupling g h → g h = 0 – as long as one is allowed to treat perturbatively themasses of the hidden gauge multiplets, although a full analysis of the issues involvedin higgsed gauged mediation is beyond the scope of the present paper.SDGM models lie somewhat in between minimal and direct gauge mediation.Like MGM they have an explicit messenger sector. Like DGM, however, no spurion-like coupling is needed and everything is mediated by gauge interactions alone. Theonly superpotential term for the messenger field is a mass term. From a theoreticalpoint of view the interest of such models lies in their simplicity, and also in the ratherstraightforward way in which they can be generated in string theory inspired quivergauge theories (in which also the mass term arises dynamically, without the need tobe introduced as an external fine tuned parameter) [11]. An important advantage ofthese models with respect to DGM is that they ameliorate the Landau pole problemwhich often afflicts DGM models, since G h can be as small as U (1).Our goal is to discuss the generic features of this class of models by implementing2 formalism very similar to the one of GGM [3]. Our approach will be general sincewe parameterize the supersymmetry breaking sector by currents instead of by aspurion. We aim at computing general expressions relating the MSSM gaugino andsfermion masses to the correlators of the supersymmetry breaking currents.The outcome of our analysis is as follows.The gaugino masses are vanishing at the first order where they would be expectedto appear. This agrees with the results obtained both in [12] and in [9] (see also [13]).In all of those papers, an effective approach was used to provide the argument ofvanishing gaugino masses. Here we provide a derivation of this result based on theprecise cancellation between the two diagrams contributing to the gaugino mass.This cancellation takes place for any supersymmetry breaking current correlator.Hence, the cancellation is a result which does not depend on the existence of a hiddensector spurion encoding supersymmetry breaking, nor on the specific supersymmetrybreaking mechanism occurring in the hidden sector. (Note that this cancellation alsoinvalidates the possibility for unsuppressed gaugino mass in one of the string-inspiredquiver gauge theory models discussed in the final section of [11].)The sfermion masses on the other hand do not vanish at the first order in whichthey are expected to appear, namely, forth order in both the hidden and the vis-ible gauge coupling ( g h g v ). We provide an expression for these masses which isvery reminiscent of the one appearing in general gauge mediation, though there isa complicated kernel appearing in the momentum integral over the hidden sectorcorrelators. This kernel has two effects. Firstly it reverses the sign of the supertrace(i.e., if the hidden sector would have given tachyonic sfermion masses in a directgauge mediation scenario, it will give positive squared masses in a semi-direct gaugemediation scenario). Secondly, it has a soft behavior at low momenta, and a mildlogarithmic growth at large momenta, such that the sfermion masses are safely undercontrol and finite, even though we generically have a non-vanishing supertrace in themessenger sector. This is to be confronted with [14] where a messenger supertracewas introduced by hand and led to UV divergent (and hence UV sensitive) sfermionmasses, due to the fact that once some soft scalar masses are introduced, the othersnecessarily undergo renormalization.One might conclude that semi-direct gauge mediation, having almost vanishing3augino masses, is not phenomenologically interesting. On the contrary, we willargue that it can be quite useful when combined with other mechanisms of mediationof supersymmetry breaking, either with a single or multiple hidden sectors. Inparticular, one could think of combining semi-direct gauge mediation with anomalymediation (with a sequestered hidden sector) [15]. As we shall see, the sfermion masscontribution from SDGM can stabilize the otherwise tachyonic sleptons arising inthe simplest models of anomaly mediation. Interestingly, contrary to what one maysuperficially imagine, it turns out that in this scenario no substantial fine-tuningis required to accomplish such a welcome conspiracy between the two competingeffects. We also discuss another scenario where one combines SDGM with modelsof direct gauge mediation. As we will show in detail, at fixed hidden sector, SDGMand direct gauge mediation provide opposite signs for the squared sfermion masses.Hence in this case SDGM can be useful both in combination with models withtachyonic sfermions, as before, or in models with suppressed gauginos to make theMSSM sparticle spectrum all of the same order (or even to invert the hierarchybetween sfermions and gauginos). In both these cases, however, differently fromanomaly mediation, generically one would need a fine tuning. For related work onconformal gauge mediation, see [16].The rest of this paper is organized as follows. In section 2 we present the semi-direct gauge mediation adapted version of the GGM formalism and explain how touse it to compute the visible mass spectrum. In section 3 we compute the gauginomasses and in section 4 the sfermion masses. We end in section 5 discussing thepossible phenomenological relevance of semi-direct gauge mediation suggested byour analysis. The models we are considering are characterized by three building blocks: • A visible sector with gauge group G v (the MSSM or any extension thereof). • A hidden supersymmetry breaking sector containing, besides confining gaugegroups driving dynamical supersymmetry breaking, a global continuous sym-metry group G h , which is then weakly gauged.4 A pair of messenger superfields Φ and ˜Φ in the bi-fundamentals of G v and G h ,having a supersymmetric mass m but no other superpotential interactions.In this section, for simplicity, we assume that both G h and G v are U (1) factors. Allour results easily generalize to arbitrary gauge groups by adding the appropriategroup theory factors. A pictorial representation of the SDGM scenario is reportedin figure 1. interactions,SM gaugeSUSY breakingsector sectorNon−SMinteractions, gauge G h G v MSSM m Messengers
Figure 1:
A schematic picture of semi-direct gauge mediation. The gauge group G h issingled-out within the hidden sector as the subgroup of the hidden gauge group to whichthe messenger superfields couple. Messengers have a supersymmetric mass m . In the limit where the gauge coupling g h of the gauge group G h is sent to zero,the whole system separates into two completely decoupled ones: the supersymmetrybreaking sector and a supersymmetry preserving sector comprising the messengersector and the MSSM fields. This property allows us to use an approach similar to[3], as we can parametrize all of the supersymmetry breaking effects through thecorrelators of the global G h currents. Namely, we can write, in momentum space h J h ( p ) J h ( − p ) i = C h ( p /M ) , h j hα ( p )¯ j h ˙ α ( − p ) i = p µ σ µα ˙ α C h / ( p /M ) , h j hα ( p ) j hβ ( − p ) i = ǫ αβ M B h ( p /M ) , (1) h j hµ ( p ) j hν ( − p ) i = ( p µ p ν − p η µν ) C h ( p /M ) , where J h , j hα and j hµ are respectively the scalar, spinor and vector components ofthe current superfield J h , and M is a typical scale of the dynamical supersymmetry We use a slightly different sign convention with respect to [3], see Appendix A. C hs and B h . Additionally,we could also have a non vanishing one-point function for J h h J h i = D h . (2)This expectation value is identified with a non zero D -term in the gauge group G h ,which is then spontaneously broken. Our approach can accommodate easily sucha situation, provided the group G h is weakly gauged and the corresponding Higgsgenerated masses can be treated perturbatively (as in [17]). However we will actuallyfind that such D h will not contribute to the soft masses at leading order.The complete Lagrangian of the model reads L = L MSSM + Z d θ (cid:0) Φ † e g v V v + g h V h Φ + ˜Φ e − g v V v − g h V h ˜Φ † (cid:1) + Z d θ m Φ ˜Φ + Z d θ tr W h + h.c. + Z d θ g h V h J h , (3)where, with obvious notation, g h is the gauge coupling of the gauge group G h ,and g v that of the visible sector gauge group G v . The last term in (3) representsthe first order coupling to the supersymmetry breaking dynamics encoded in (1).There are, of course, additional non linear terms required by gauge invariance andsupersymmetry.In contrast, in general gauge mediation one writes for the same Lagrangian L = L MSSM + Z d θ g v V v J v . (4)In turn one encodes all the mediation of supersymmetry breaking to the visiblegauge sector in the functions C vs and B v characterizing the correlators of the variouscomponents of J v . We thus see that we have in this class of models a sort of tumbling where the functions C vs and B v are eventually determined in terms of the functions C hs and B h . This will be our approach in computing the soft masses of the MSSM.It is rather straightforward to write the general structure of the Feynman dia-grams allowing us to extract C vs and B v from C hs and B h . What we have to computeare radiative corrections to the two-point functions of the fields in the visible gauge The superscript v in J v refers to the fact that the currents couple to the visible gauge fields. G h .In order to have supersymmetry breaking at all, the propagators of the latter mustinclude the insertion of a hidden current correlator.In figure 2 we have drawn the five topologically distinct Feynman diagramsentering the computation. p p ll l−p a l−kp p d p p ll l−k ll−p l−k−ppp l−k b e p p llk k kk kll c l−p Figure 2:
The five topologically inequivalent structures contributing to the visible GGMfunctions. External lines, which carry momentum p , correspond to G v -fields ( i.e. fieldsbelonging to the visible gauge group). Internal lines carrying momentum k correspond to G h -fields ( i.e. fields belonging to the hidden sector gauge group G h ) and have attacheda blob which encodes the exact hidden sector non-supersymmetric correction to the cor-responding propagators. Finally, all other internal lines correspond to messenger fields. The external lines are those of the G v -fields (the MSSM gauge bosons, gauginosand auxiliary D -fields). Internal lines carrying momentum k are associated to thefields belonging to the gauge group G h degrees of freedom, and hence will also involvea correlator insertion. All other internal lines are messenger lines, either scalar or7ermionic. All these diagrams have two explicit loops, plus an additional loop factorcoming from the hidden correlator insertion. Hence they all scale like g v g h .A first observation is that while the diagrams of topology a , c , d and e can beeffectively taken into account as one loop diagrams with corrected messenger lines(i.e. they involve only messenger two-point functions), the diagram with topology b is intrinsically two loops since it involves a messenger four-point function. Thecontribution of these latter diagrams cannot be encoded in an effective approachsuch as the one of [14]. As we will see, in computing the visible gaugino masses,there is a dramatic cancellation between the two contributions from the diagrams a and b .A second important observation is that the correlator encoded by the function B , which is complex, is on a different footing with respect to the others, which arereal. By carefully keeping track of the chiral structure of each diagram, one can seethat B v is a function of B h only, while the C vs are functions of a linear combinationof the C hs .The knowledge of the functions C hs and B h and, via tumbling, of the functions C vs and B v , completely determines the soft masses in the MSSM Lagrangian (3).In what follows, we will compute, using the above formalism, both gaugino andsfermion masses to leading order. Obviously, being a model of gauge mediation,general SDGM obeys, if considered in isolation from other mechanisms, the samesfermion sum rules of GGM [3]. In this section we compute the gaugino masses. For this computation, the diagramsof figure 2 are really the end of the story since the external lines are nothing butthe gauginos themselves, and we can set the external momentum to be p = 0.Of the five topologically inequivalent structures of figure 2 only two contributeto the gaugino masses. Indeed, diagrams d and e appear only if the external linesare vector bosons. Diagram c appears only if the internal line (the one with theblob attached) is a vector bosons, and diagrams of this type are prevented fromcontributing to gaugino masses by chirality. The only relevant ones are then those8f type a and b . Their precise structure is depicted in figure 3.The first one is the same as the one discussed in [10] and evaluates to m aλ = − g v g h Z d k (2 π ) Z d l (2 π ) m M B h ( k /M ) k ( l + m ) [( l − k ) + m ] . (5)These are Wick rotated expressions and the conventions used for the Euclideanpropagators are summarized in Appendix A. We have factors of 4 coming from thefour Yukawa vertices, 2 coming from the trace on the internal fermion loop (whichalso gives the overall − sign), and 2 from the messenger multiplicity. a λ m m b λ Figure 3:
The diagrams contributing to the gaugino mass. The left one has two (su-persymmetric) mass insertions, each one represented by a cross on the correspondingmessenger fermionic line.
The second diagram, of type b , gives the following contribution m bλ = 8 g v g h Z d k (2 π ) Z d l (2 π ) l · ( l − k ) M B h ( k /M ) k ( l + m ) [( l − k ) + m ] . (6)The factors are as before, except for the missing − m λ = m aλ + m bλ . (7)In the two expressions above, the integral over the l -momentum can be doneanalytically by standard techniques. We can then write m a,bλ = 8 g v g h Mm Z d k (2 π ) k L a,b ( k /m ) B h ( k /M ) , (8) For convenience we drop the gauge theory factor common to both diagrams. Our result isvalid for any gauge group. L a ( k /m ) = − m Z d l (2 π ) l + m ) [( l − k ) + m ] , (9) L b ( k /m ) = m Z d l (2 π ) l · ( l − k )( l + m ) [( l − k ) + m ] . (10)Evaluating the two integrals above we find L a ( x ) = − L b ( x ) = − π (cid:0) x + 1 x + 4 − f ( x ) x ( x + 4) (cid:1) , (11)where f ( x ) = 4 p x ( x + 4) arctanh r xx + 4 (12)and x = k /m . This result means that the total kernel in the expression for thegaugino mass m λ = 8 g v g h Mm Z d k (2 π ) k L ( k /m ) B h ( k /M ) (13)vanishes, since L ( x ) = L a ( x ) + L b ( x ) = 0 . (14)Hence, m λ = 0 for any function B h , at this order. The fact that the gaugino masswas vanishing at leading order in this class of models was first noted in [12] wherean effective argument based on wave function renormalization was given. There itis also shown that the visible gaugino first obtains a mass at order g h g v . This effecthas been dubbed gaugino screening.Here we have re-derived this important result in a different way. We have shownexplicitly how the cancellation arises, which was not obvious from the start (indeed,even a posteriori, the cancellation would seem rather miraculous if we did not havean independent argument in favor of it taking place). Also, and perhaps moreimportantly, the proof that the kernel L ( x ) is zero means that the cancellation doesnot depend on B h , and hence applies to any model of supersymmetry breaking For the record, note that the same function f ( x ) appears when one evaluates the func-tion B v in a minimal gauge mediation scenario W = ( m + θ F )Φ ˜Φ, namely B v ( p ) =(1 / π )( F/m ) f ( p /m ). B h . This is a counterpart to the argumentgiven in [12] which hinges on the capability of effectively encoding the breakingof supersymmetry in a spurion field, and on the assumption that there are hiddenmessengers mediating supersymmetry breaking to the gauge group G h . We now turn to the computation of the general expression for the MSSM sfermionmasses. In this case the diagrams listed in figure 2 are to be inserted into theself-energy radiative corrections to the scalar propagators. Many such contributionsshould be computed in this case since now the external lines of the diagrams wouldnot only have gauginos but also gauge bosons and D-fields of the visible gauge group G v .In this section we promote G v to the MSSM gauge group, and consider the casewhere the messengers form a complete SU (5) multiplet of index ℓ = 1 for each of theMSSM gauge groups . We still consider G h = U (1) since more general cases can beeasily accommodated by inserting the appropriate group theory factors. In this casethe parameter space spanned by the sfermion masses is contained in that of minimalgauge mediation in the sense that all square masses are proportional to one and thesame dimension-full parameter. However, the dependence of this parameter on thedynamics of the hidden sector is quite different from the one arising in minimalgauge mediation and this is the issue we are going to analyze.For ease of notation we define, for each type of sfermion, an effective coupling g v = g c [ U (1)] + g c [ SU (2)] + g c [ SU (3)] (15)in terms of the SU (3) × SU (2) × U (1) coupling constants g i of the Standard Modeland the Casimir invariants for each of the sfermion representations.A general expression for the mass square was given in [3] m sf = − g v Z d p (2 π ) p (cid:0) C v ( p /M ) − C v / ( p /M ) + 3 C v ( p /M ) (cid:1) . (16) For U (1) Y the “index” is customarily defined as ℓ = 6 Y / c = 3 Y / C hs and C vs ′ is logarithmically UV-divergent, butall divergences cancel in the final linear combinations. In what follows, we thus onlyconsider the finite parts. Our first goal is to compute the C vs ′ in terms of the C hs , andthis is exactly what the diagrams in figure 2 do. Obviously there are now two scales( M and m ) on which C vs ′ depends. Each of the three functions C vs ′ is expressed aslinear combination of contributions from the three functions C hs C vs ′ ( p /M , m /M ) = Z d l (2 π ) d k (2 π ) (cid:0) F s ′ , ( l, k, p, m ) C h ( k /M ) − F s ′ , / ( l, k, p, m ) C h / ( k /M ) + 3 F s ′ , ( l, k, p, m ) C h ( k /M ) (cid:1) , (17)where we have introduced a series of functions F s ′ ,s ( l, k, p, m ) that can be computedexplicitly from the diagrams in Figure 2. The four-momentum k always denotes themomentum going through the current-current correlators in the hidden sector. Toobtain the sfermion masses we must then add the three functions C vs ′ according to(16) and integrate over the four-momentum p . By switching the integrals, leavingthe one over k for last, and combining the various functions F s ′ ,s ( l, k, p, m ) we caneasily rewrite the sfermion masses as m sf = g v g h (4 π ) Z d k (2 π ) k (cid:0) K ( k /m ) C h ( k /M ) − K / ( k /m ) C h / ( k /M ) + 3 K ( k /m ) C h ( k /M ) (cid:1) , (18)where g h (4 π ) k K s ( k /m ) = − Z d ld p (2 π ) p (cid:0) F ,s ( l, k, p, m ) − F / ,s ( l, k, p, m ) + 3 F ,s ( l, k, p, m ) (cid:1) . (19)The expressions K s ( k /m ) represent three (a priori independent) scalar “kernels”depending only on the messenger sector and thus exactly computable irrespectivelyof what strong dynamics is ultimately responsible for supersymmetry breaking. Re-calling that in the supersymmetric limit all C hs are equal and m sf = 0, it followsthat the weighted sum of the three kernels must vanish: K − K / + 3 K = 0.We have computed explicitly the first two of them and found them to be the same12 K = K / ≡ K ), thus implying that the full contribution to the sfermion massescan be written as m sf = g v g h (4 π ) Z dk K ( k /m ) (cid:0) C h ( k /M ) − C h / ( k /M ) + 3 C h ( k /M ) (cid:1) . (20)The kernel K ( x ) is given by a sum of integrals over two loop momenta, which areexactly known. We relegate to Appendix B the details of the computation. Sufficeshere to say that in this case the diagrams of type b involving the messenger four pointfunction are necessary for consistency, for instance to obtain a transverse vectorialcurrent two point function (i.e. a well-defined C v ). K ( x ) is a positive function and has the following asymptotic behavior K ( x ) = 518 x − x + 5437176400 x + O ( x ) for x → , (21) K ( x ) ∼ γ log x for x → ∞ , (22)where we have estimated numerically γ ∼ .
4. A plot of the kernel at large x isgiven in figure 4.From the behavior of the kernel it is clear that the m sf will always be finite sincethe weighted sum of the C hs is soft enough at large momenta, as noted in [3]. Notehowever some important facts.First of all, in our set up we have a non vanishing supertrace in the messengersector. According to the argument of [14] this fact, by itself, leads to an enhancementof sfermion squared masses proportional to the logarithm of a UV scale. Here we donot see any such dangerous enhancement. Since the theory is renormalizable andno soft terms appear in the bare Lagrangian, there cannot be counterterms for thesfermion masses and the only two scales of the problem are M , the hidden sectorsupersymmetry breaking scale, and m , the messenger supersymmetric mass.Secondly, due to the fact that K ( x ) is positive, the sfermion squared masses willbe of the opposite sign with respect to the case where the same hidden supersym-metric sector is directly coupled to the MSSM, cfr. eqs. (16) and (20). As we will seein the next section, this is something useful when trying to use SDGM in concretephenomenological models. 13 K(x) 2000 4000 6000 800020406080100120
Figure 4:
Numerical estimate of the kernel K ( x ) for large x . The fit is represented bythe curve 14 . x − .
3. The numerical errors are about 5% – quite sufficient for thesimple estimates done in this paper.
As a last remark, we note that the fact the kernel vanishes for small mo-menta leads to a vanishing contribution to the sfermion masses from D -terms, ifsuch terms are present. One would expect a first contribution from them at or-der ( D h ) , since the contribution at linear order vanishes after summing over bothcharge conjugate messengers. Two D -tadpole insertions can actually be encodedin C h D ( k ) ∼ ( D h ) δ ( k ) (indeed, two D -tadpoles are equivalent to a D -line cut intwo, and hence with no momentum flowing through it). Obviously, this contributionvanishes when multiplied by K ( x ) ∼ x . This fact can be extracted from the generalexpression given in [14] by tuning the parameters so that only D -terms contributeto the messenger mass matrix. This precise situation and its phenomenology wasanalyzed more recently for instance in [18] where next-to-leading order correctionswere discussed.We can give a rough estimate of the scaling of m sf as a function of M and m , in the two opposite hierarchical limits. We will assume that the weighted sumis a simple step function (which can mimic, roughly, the soft behavior previously14iscussed) ( C h − C h / + 3 C h )( k /M ) ∼ π Θ(1 − k /M ) , (23)with a proportionality factor of either sign depending on the specific hidden sector .In the limit of heavy messengers we get m sf ∼ α v α h (4 π ) M m m ≫ M , (24)while in the opposite limit of light messengers m sf ∼ α v α h (4 π ) γM log (cid:0) M m (cid:1) m ≪ M , (25)where we have defined α h,v = g h,v / π . While we see a signal of the log-enhancementof [14] in the limit m ≪ M , we do not see anything like it in the opposite limit.For completeness, we can estimate the contributions to the mass matrix of themessengers in the limit m ≪ M where the radiative corrections can be comparableto the supersymmetric mass. In this limit we can treat m as a small perturbationand write the following expressions m d = − g h Z d k (2 π ) k (cid:0) C h ( k /M ) − C h / ( k /M ) + 3 C h ( k /M ) (cid:1) , (26) m o = g h Z d k (2 π ) k mM B h ( k /M ) k , (27)where m d and m o are respectively the contributions to the diagonal and off-diagonalelements of the messenger mass matrix. The supertrace over the messenger sectoris proportional to m d .Using the estimate (23) and a similar one for B h , we obtain the estimates for m ≪ M m d ∼ α h (4 π ) M , m o ∼ α h (4 π ) mM log (cid:0) M m (cid:1) . (28)The sign of m d (and thus of the supertrace) is the opposite of the sign in the sfermionmasses (25). Again, for the record, we note that in a minimal gauge mediation scenario we would obtain( C − C / + 3 C )( p ) = (1 / π )(2 | F | /m ) f ′ ( p /m ), where f ′ ( x ) is the derivative of thefunction f ( x ) that we encountered previously, and is everywhere negative.
15e conclude that even if the scale of the supersymmetry breaking sector is muchhigher than the supersymmetric mass of the messengers, it is generally possible toavoid tachyonic eigenvalues in their mass matrix due to the suppression in α h of theradiative corrections to the messenger mass. We now comment on the possible phenomenological relevance of the class of modelsconsidered in this paper. We have seen that the sparticle spectrum produced isessentially one of a strong hierarchy between the sfermions, which are heavy, andthe gauginos which are very light (massless at the order considered above). This isclearly not a satisfying spectrum in itself. Additionally, since there is an extra loopfactor α h in the expression for the sfermion masses, the gravitino is going to be afactor of 1 /α h heavier than in an ordinary gauge mediation model yielding the samevalues of m sf .However, since SDGM gives a contribution mainly to sfermion masses, it can beuseful if combined with other supersymmetry breaking mechanisms which providegaugino masses but tachyonic sfermions. We review below two such situations, inwhich the other mechanisms are respectively anomaly mediation (AM) and directgauge mediation (DGM).We first analyze the scenario where SDGM could address the negative squaredsfermion mass problem of anomaly mediation [15]. We consider the simple set upwhere AM and SDGM have the same supersymmetry breaking sector, and hence thesame supersymmetry breaking scale M . Anomaly mediation gives a gaugino massof the order m λ ∼ α v π M M pl . (29)Assuming a sequestered hidden sector, the slepton masses for the first two genera-16ions are of the order m AM ∼ − α v (4 π ) M M pl . (30)The slepton masses in (30) are tachyonic because of the sign of the beta functioncoefficient (the contribution from the Yukawa couplings can be ignored for the firsttwo generations).We can cure the problem of tachyonic sleptons by combining AM with a SDGMmodel which yields positive sfermion squared masses. Let us denote by δ the ra-tio between the SDGM contribution (24) or (25) to the slepton masses and (30).Depending on the messenger scale, we obtain δ ≡ (cid:12)(cid:12)(cid:12)(cid:12) m SD m AM (cid:12)(cid:12)(cid:12)(cid:12) ∼ α h (4 π ) γ M pl M log M m for m ≪ M α h (4 π ) M M pl m for m ≫ M (31)We require 1 < δ <
10 in order for the SDGM to give a contribution to sfermionmasses which is larger than AM but of the same order. This can be achieved inboth regimes while staying at weak coupling in α h .First of all, in order to have gaugino masses (29) at the TeV scale, we need totake M ∼ GeV.In the first case ( m ≪ M ), assuming α h π ∼ − gives a sensible soft spectrumwhere the messenger mass can be anywhere in the range 10 − GeV. Notethat such a small G h coupling constant could be actually related to the mechanismof sequestering itself. In the second case ( m ≫ M ), one could have for instance m ∼ GeV with α h π ∼ − .We conclude that SDGM can be successfully combined with AM, in both regimesand with no substantial fine-tuning, and cure the slepton problem (some earlierinteresting proposals to cure this problem can be found in [20], and more recently in[21]). It is worth mentioning at this point that such a conspiracy could arise naturally Since we are only interested in order of magnitude estimates, we use a notation similar to(15) and lump all dependence on the visible couplings into α v π ∼ · − . One could be moreprecise, if needed. For instance, focusing on U (1) Y , one has, for the Bino and the right selectron: m ˜ B = (33 / α π M M pl and m e R = − (198 / α (4 π ) M M pl respectively (see e.g. [19]).
17n string theory since SDGM is generic in D-branes embeddings of gauge mediation[11], while anomaly mediation must always be included once gravity effects areconsidered on D-branes.We now turn to the study of the combination of SDGM with DGM. We consideragain the simple set up where the two mechanisms share the same supersymmetrybreaking sector. Note that the currents coupling to the visible gauge group G v and the ones coupling to G h must be different since they couple to different gaugegroups. However we work in the approximation where their correlation functionsare essentially the same. This is not unnatural if the supersymmetry breaking sectorhas only one scale or if the two groups arise from the breaking of a larger group. Theimportant point here is that the contribution to the sfermion masses of SDGM hasan opposite sign with respect to the DGM one. This comes, as already noticed insection 4, by comparing eqs. (16) and (20), given the positivity of the kernel K ( x ).Models of strongly coupled DGM can lead to unsuppressed gaugino masses butnegative squared scalar masses. Using the approximation (23) the DGM contributionto the sfermion masses is m D ∼ − α v (4 π ) M . (32)In this case the positive SDGM contribution could render the sfermion masses nontachyonic, in the limit m ≪ M where it can be larger than (32). To understandif the competition between SDGM and DGM can be realized naturally, we need toestimate the ratio of the sfermion mass contributions (25) and (32) δ = (cid:12)(cid:12)(cid:12)(cid:12) m SD m D (cid:12)(cid:12)(cid:12)(cid:12) ∼ γ α h (4 π ) log M m . (33)We would then demand 1 < δ <
10, as before.Note that, besides the requirement on δ , we have to check that the messen-gers in the SDGM sector do not become tachyonic due to radiative corrections. Insection 4 we estimated the diagonal and off-diagonal corrections to the messengersmass matrix (28) in the limit m ≪ M . Note that in the case at hand the diagonalradiative correction m d is negative since the SDGM contribution to sfermion massesis positive.It is possible, in principle, to satisfy these competing constraints. However, someamount of tuning will be needed in this case. On the one hand, the scale M should18e larger than m to avoid a too strong G h coupling while keeping δ >
1. On theother hand, M should not be too large, in order to avoid tachyons in the messengersector. The possibility of satisfying both these constraints is not generic and it canonly be answered on a case by case basis. However, the possibility of a windowwhere such a mechanism can work is not ruled out. An alternative scenario concerns models of DGM which present an MSSM spar-ticle spectrum where the gaugino mass is suppressed or of the same order of the(positive) sfermion masses. Here the SDGM can provide a negative contribution tothe sfermion masses in order to invert this hierarchy. This scenario can be realizedsimilarly as above, however with a fine tuning of δ , i.e. of the coupling constant α h .The fact that a fine tuning is needed in order to obtain gauginos more massive thansfermions seems a common feature of gauge mediated models (see for instance [4]).In conclusion, our preliminary analysis indicates that models of AM+SDGMcan naturally lead to a sensible MSSM soft mass spectrum and thus seem promisingfor phenomenological applications. On the other hand, the SDGM+DGM scenariosthat we discussed above can possibly lead to sensible phenomenology only in a smallregion of the parameter space, if at all. Acknowledgments
We thank Ken Intriligator for discussions at the beginning of this project and AndreaRomanino for useful comments on a preliminary version of the draft. The researchof R.A. is supported in part by IISN-Belgium (conventions 4.4511.06, 4.4505.86 and4.4514.08). R.A. is a Research Associate of the Fonds de la Recherche Scientifique–F.N.R.S. (Belgium). The research of G.F. is supported in part by the SwedishResearch Council (Vetenskapsr˚adet) contract 621-2006-3337. Contribution fromthe L¨angmanska Kulturfonden and the Wilhelm och Martina Lundgrens Veteskaps-fond are also gratefully acknowledged. A.M. is a Postdoctoral Researcher of FWO-Vlaanderen. A.M. is also supported in part by FWO-Vlaanderen through projectG.0428.06. R.A. and A.M. are supported in part by the Belgian Federal Science Possibly, combining SDGM and DGM with different supersymmetry breaking sectors can ame-liorate these problems, at the price of introducing new scales in the model.
A Conventions
We use the following propagators for the messengers h φ ( p ) φ ∗ ( − p ) i = 1 p + m (34) h ψ α ( p ) ¯ ψ ˙ α ( − p ) i = p µ σ µα ˙ α p + m (35) h ψ α ( p ) ψ β ( − p ) i = mδ βα p + m , (36)while for the hidden gauge sector we have the following propagators, at first orderin the insertions of the (supersymmetry breaking) currents h D h ( p ) D h ( − p ) i = C h ( p /M ) (37) h λ hα ( p )¯ λ h ˙ α ( − p ) i = − p µ σ µα ˙ α p C h / ( p /M ) (38) h λ hα ( p ) λ hβ ( − p ) i = M δ βα p B h ( p /M ) ⋆ (39) h A hµ ( p ) A hν ( − p ) i = p µ p ν − p η µν p C h ( p /M ) . (40)In these conventions that use exclusively Weyl spinors, each φ ∗ ψλ Yukawa vertexcomes with a g √ σ µα ˙ α ¯ σ ν ˙ αβ + σ να ˙ α ¯ σ µ ˙ αβ = − η µν δ βα , (41)and similarly for ¯ σσ . We use the ( − + ++) signature, hence all the above relationsare unchanged after Wick rotation. B Computation of the kernel
In this appendix we collect some more details concerning the computation of thekernels K s ( x ), proving, eventually, that K ( x ) = K / ( x ) = K ( x ). Recall thatupon use of the tumbling equations (17), the sfermion masses (16) depend on the20ernels K s as given in (18). To compute the kernels K s one should write down allthe graphs contributing to the visible C vs ′ as in the tumbling equations (17). Thenone can extract the functions F s ′ ,s and integrate them in order to obtain the kernels(19).Let us focus on the contributions to K , first. Since this is defined as the functionmultiplying C h we need to consider all the diagrams giving the dependence of C vs on C h , namely those diagrams depicted in figure 5. λ D aa AD a DD b DD b AD AD d Figure 5:
The diagrams contributing to K . The external lines represent the auxiliary D -field, the gaugino and the gauge boson of the visible sector gauge group G v . The scalarand spinor messengers (dashed and continuum thin lines, respectively) circulate in theouter loop and the bubble represents the insertion of C h . Each diagram must be countedwith the appropriate coefficient representing the different ways messengers can be inserted. The incoming lines represent the visible gauge particles ( D -field, gaugino andgauge boson), the particles going around the external loop are the bosonic andfermionic messengers and, finally, the internal line represents the insertion of thehidden two-point function h J h ( k ) J h ( − k ) i on a hidden D -line.To completely specify a diagram one would also need to specify the orientationof the internal lines (including the chirality type for the fermions) and the type21f messengers (Φ or ˜Φ). We choose not to write this explicitly in order to keepthe notation simple. Each of the six diagrams in figure 5 thus represents a set ofdiagrams and this is reflected into the numerical coefficients for each contribution.Other numerical coefficients arise from the normalization of the interaction verticesand from the Dirac algebra.The total contribution for each class of diagrams is given by DD a = 4 Z dk dl (2 π ) C h ( k /M )( l + m ) [( l − k ) + m ] [( l − p ) + m ] DD b = 2 Z dk dl (2 π ) C h ( k /M )( l + m ) [( l − k ) + m ] [( l − k − p ) + m ] [( l − p ) + m ] λD a = − Z dk dl (2 π ) ( l − p ) µ σ µα ˙ α C h ( k /M )( l + m ) [( l − k ) + m ] [( l − p ) + m ] AD a = 4 Z dk dl (2 π ) (2 l − p ) µ (2 l − p ) ν C h ( k /M )( l + m ) [( l − k ) + m ] [( l − p ) + m ] (42) AD b = 2 Z dk dl (2 π ) (2 l − p ) µ (2 l − k − p ) ν C h ( k /M )( l + m ) [( l − k ) + m ] [( l − k − p ) + m ] [( l − p ) + m ] AD d = − Z dk dl (2 π ) η µν C h ( k /M )( l + m ) [( l − k ) + m ] . The notation of the above integrals should be self-evident, for instance AD denotesthe contribution of the hidden D -field to the visible gauge boson A . The subscriptrefers to the five topologies introduced in figure 2 (only diagrams of topology a , b and d enter the computation of K ). There are several consistency checks for theabove expressions. For instance, one can check that the sum AD a + AD b + AD d obeys the Ward identities.To obtain K we need to extract the expression for the contribution to C vs fromthe diagrams above by comparing (42) with the definition (1). One then inserts thecontribution thus obtained into (16). The resulting two-loop integral (in p and l )defines the kernel K ( k /m ). It turns out to be convenient to write everything witha common denominator and express the integral over l in terms of three Feynmanparameters. 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