Anomaly Mediation, Fayet-Iliopoulos D-terms and the Renormalisation Group
aa r X i v : . [ h e p - ph ] S e p LTH 757
Anomaly Mediation, Fayet-Iliopoulos D -terms and the RenormalisationGroup R. Hodgson, I. Jack, D.R.T. Jones
Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK
Abstract
We address renormalisation group evolution issues that arise in the Anomaly MediatedSupersymmetry Breaking scenario when the tachyonic slepton problem is resolved by Fayet-Iliopoulos term contributions. We present typical sparticle spectra both for the originalformulation of this idea and an alternative using Fayet-Iliopoulos terms for a U compatiblewith a straightforward GUT embedding. Introduction
Anomaly mediation ( AM ) [1]- [24] as the main source of supersymmetry breaking is an attractiveidea. In AM , the soft supersymmetry-breaking φ ∗ φ masses, φ couplings and gaugino massesare all determined by the appropriate power of the gravitino mass multiplied by perturbativelycalculable functions of the dimensionless couplings of the underlying supersymmetric theory.Moreover these functions are RG invariant; that is, their renormalisation scale dependence iscorrectly given by the renormalisation scale dependence of the dimensionless couplings. To putit another way, the AM predictions are UV-insensitive [18].In recent papers we have explored a specific version of AM , where the tachyonic sleptonproblem characteristic of a minimal implementation of AM is solved by means of an additional U gauge symmetry, U ′ , that is broken at high energies. The scale of this breaking may beset by a Fayet-Iliopoulos (FI) D -term [23] or via dimensional transmutation [24]. In the formercase we showed how it is quite natural for the effects of U ′ to decouple at low energies apartfrom contributions to the scalar masses, of the form of U ′ FI terms, which are automatically ofthe same order as the AM ones. In the latter we argued that it was possible to dispense withan explicit FI term, generating the U ′ breaking scale via dimensional transmutation, exploitinga flat D -term direction. In our explicit model, the low energy theory consisted of the usual MSSM fields with an additional gauge singlet chiral supermultiplet which is weakly coupled tothe
MSSM fields ; the possible cosmological and phenomenological implications (in particularthe possibility that its fermionic component might be the LSP) remain to be discussed. Inthis paper we will confine ourselves to the first possibility, where the low energy theory simplyconsists of the MSSM fields.In both the above scenarios, there is, however, a subtlety with regard to the afore-mentioned RG invariance, concerning the Fayet-Iliopoulos term associated with the SM (or MSSM ) U , U SM . Suppose for simplicity the U ′ breaking scale coincides with the gauge unification scale M X , and that the U SM FI term is zero there. It turns out that the presence of the U ′ FI termsin the effective field theory means that even though it is zero at M X , the U SM FI term canbecome significant in the evolution to low energies. Thus there will be contributions of FI formfor both the U ′ and the U SM to the scalar masses. Now as emphasised in Ref. [20], these twocontributions can be reparametrised into a contribution of the form of a single U ′′ FI-term. Itshould now be clear, however, that the resulting form of this contribution will be a function ofscale since the size of the U SM FI term generated is a function of scale.The upshot is that if we choose a U ′ with charges for the lepton doublets and singlets chosenso as to solve the tachyonic slepton problem, and also zero FI term for U SM at M X , the resultingspectrum will correspond to a nonzero FI term for U SM at M Z , or a zero FI term for U SM at M Z with a different pair of U ′ leptonic charges.In this paper we shall firstly explain this issue in some detail and then repeat some of theprecision calculations of Ref. [23] but now imposing boundary conditions at M X , and taking theopportunity to update input values and correct some minor bugs in our previous analysis.In the second part of the paper we consider a variation of the same idea where we augment The existence of this light field is in fact a consequence of general arguments concerning AM decoupling givenby Pomarol and Rattazzi [3]. U ′ charge assignments compatible with a GUTembedding; specifically SU , SO or E . Even with the assumption that the low energy theorybelow M X consists only of the MSSM fields, the resulting allowed region for the leptonic chargesand the sparticle spectrum is quite different from the previous case.We also derive some mass sum rules independent of the U ′ charges for this case, similar tothe sum rules given in Refs. [17, 23]. First of all, for completeness and to establish notation, let us recapitulate some standard results.We take an N = 1 supersymmetric gauge theory with gauge group Π α G α and with superpotential W (Φ) = Y ijk Φ i Φ j Φ k + µ ij Φ i Φ j . (1)We also include the standard soft supersymmetry-breaking terms L SB = − ( m ) ji φ i φ j − (cid:16) h ijk φ i φ j φ k + b ij φ i φ j + M λλ + h . c . (cid:17) (2)where φ i = ( φ i ) ∗ .For the moment let us assume that the gauge group has one abelian factor, which we shalltake to be G . We shall denote the hypercharge matrix for G by Y ij = Y j δ ij and its gaugecoupling by g .At one loop we have16 π β (1) g α = g α Q α = g α [ T ( R α ) − C ( G α )] , (3a)16 π γ (1) ij = P ij = Y ikl Y jkl − X α g α [ C ( R α )] ij . (3b)Here β g α are the gauge β -functions and γ is the chiral supermultiplet anomalous dimension, R α is the group representation for G α acting on the chiral fields, C ( R α ) the correspondingquadratic Casimir and T ( R α ) = ( r α ) − Tr[ C ( R α )] , r α being the dimension of G α . For theadjoint representation, C ( R α ) = C ( G α ) I α , where I α is the r α × r α unit matrix. Obviously T ( R ) = Tr[ Y ], [ C ( R )] i j = ( Y ) ij and C ( G ) = 0. At two loops we have(16 π ) β (2) g α = 2 g α C ( G α ) Q α − g α r − α Tr [
P C ( R α )] , (4)(16 π ) γ (2) ij = 2 X α g α C ( R α ) ij Q α − " Y jmn Y mpi + 2 X α g α C ( R α ) pj δ in P np . (5)The one-loop β -functions for the soft-breaking couplings are given by16 π β (1) ijkh = U ijk + U kij + U jki , (6a)16 π β (1) ijb = V ij + V ji , (6b)16 π [ β (1) m ] ij = W ij , (6c)16 π β (1) M α = 2 g α Q α M α , (6d)2here U ijk = h ijl P kl + Y ijl X kl ,V ij = b il P jl + Y ijl Y lmn b mn + µ il X j l ,W ji = Y ipq Y pqn ( m ) j n + Y jpq Y pqn ( m ) ni + 2 Y ipq Y jpr ( m ) qr + h ipq h jpq − X α g α M α M ∗ α C ( R α ) j i , (7)with X ij = h ikl Y jkl + 4 X α g α M α C ( R α ) ij . (8)We have excluded from Eq. (6c) a D -tadpole contribution which arises if we calculate with theauxiliary field D eliminated. If we work in the D -eliminated form of the theory then we haveinstead of Eq. (6c): 16 π [ β (1) m ] ij → W ij + 2 g Y ij Tr[ Y m ] . (9)This extra contribution is only nonvanishing in a theory whose gauge group has an abelianfactor. It can be equivalently viewed as a renormalisation of the Fayet-Iliopoulos parameter, aswe shall now describe.In N = 1 supersymmetric gauge theories whose gauge group has an abelian factor, thereexists a possible invariant that is not otherwise allowed: the Fayet-Iliopoulos D -term, L = ξ Z V ( x, θ, ¯ θ ) d θ = ξD ( x ) . (10)The significance of the ξ term is of course well known. The part of the scalar potentialdependent on the U D -field is V D = − D − D (cid:0) ξ + g φ i Y ij φ j (cid:1) , (11)which upon elimination of the auxiliary field D becomes V D = ( ξ + g φ i Y ij φ j ) , (12)so that to obtain a supersymmetric ground state we require at least one field φ i to have a chargewith the opposite sign to ξ , and to develop a vacuum expectation value. Thus for supersymmetryto be unbroken on the scale set by ξ it is necessarily the case that the corresponding U isspontaneously broken. In Ref. [23] we showed that in the presence of anomaly mediation softsupersymmetry-breaking terms it is quite natural for the U symmetry to be broken at a largescale characterised by ξ while all scalars receive, from the U D -term, (mass) contributionscharacterised by the gravitino (or anomaly mediation) mass.In previous papers [25]– [27] we have discussed the renormalisation of ξ in the presence ofthe soft terms. The result for β ξ is as follows: β ξ = β g g ξ + ˆ β ξ (13)3here ˆ β ξ is determined by V -tadpole (or in components D -tadpole) graphs, and is independentof ξ .We found that 16 π ˆ β (1) ξ = 2 g Tr (cid:2) Y m (cid:3) , (14)16 π ˆ β (2) ξ = − g Tr h Y m γ (1) i . (15)The three-loop contribution was computed in Ref. [26] for an abelian theory and for the MSSM in Ref. [27].
Remarkably the following results are RG invariant [8]: M α = m β g α /g α , (16a) h ijk = − m β ijkY , (16b)( m ) ij = 12 m µ ddµ γ ij , (16c) b ij = κm µ ij − m β ijµ . (16d)Here β Y is the Yukawa β -function, given by β ijkY = γ il Y ljk + γ j l Y ilk + γ kl Y ijl , (17)with a similar expression for β ijµ . It must be emphasised that the RG invariance of Eq. (16c)holds in the D -uneliminated theory. That is to say, given Eq. (16a)-(16d) it follows that β m = 12 m µ ddµ (cid:18) µ ddµ γ (cid:19) (18)where in Eq. (18), β m does not include D -tadpole contributions (that is, at one loop it is givenby Eq. (6c)); the renormalisation of these is dealt with separately by β ξ , as described in the lastsection.Note the arbitrary parameter κ in Eq. (16d); its presence means that we can, in the MSSM ,follow the usual procedure whereby the Higgs B -parameter is determined (along with the µ -term) by the electroweak minimisation. How natural is this procedure is an obvious question,to which we will return later.The approach to the AM tachyonic slepton problem that we will follow is based on the factthat RG invariance is preserved if we replace ( m ) ij in Eq. (16c) by( m ) ij = 12 m µ ddµ γ ij + k ′ ( Y ′ ) ij , (19)where k ′ is a constant and Y ′ is a matrix satisfying( Y ′ ) il Y ljk + ( Y ′ ) j l Y ilk + ( Y ′ ) kl Y ijl = 0 (20)4 u c d c H H ν c − L − e − L e + L − e − L e + L − L − e Table 1: Anomaly free U charges for arbitrary lepton doublet and singlet charges L and e respectively. U SM corresponds to L = − / e = 1.and Tr (cid:2) Y ′ C ( R α ) (cid:3) = 0 , (21)in other words Y ′ is a hypercharge matrix corresponding to a U symmetry (which we shalldenote U ′ ) with no mixed anomalies with the SM gauge group. This U ′ may in general begauged, or a global symmetry.The MSSM (including right-handed neutrinos) admits two independent generation-blindanomaly-free U symmetries. The possible charge assignments are shown in Table 1.Of course the k ′ Y ′ term in Eq. (19) corresponds in form to a FI D -term; we shall assumethat in fact the associated U ′ gauge symmetry is broken at high energy and that the abovecontributions to the scalar masses are the only relic of this breaking that survive in the lowenergy effective field theory. That this is a perfectly natural scenario was demonstrated inRef. [23].Now let us consider a possible FI term ξD associated with the SM (or MSSM ) U , U SM . Here ξ is an independent parameter respecting all the symmetries of the MSSM ; in the vast majority ofanalyses using, for example,
CMSSM boundary conditions at gauge unification, it is assumed tobe zero there. (For an exception, in which ξ is treated as an extra independent parameter at lowenergy, see Ref. [ ? ]). Working in the D -eliminated formalism, the effect of radiative generationof an FI term as we run down to low scales is then automatically taken care of by the termadded in Eq. (9) (and corresponding terms at higher loops). If, on the other hand we work withthe D -uneliminated formalism then obviously if we assume ξ is zero at gauge unification then itis calculable at low energies using β ξ from Eqs. (14,15). The resulting additional contributionsto the masses from Eq. (12) will of course lead to precisely the same results for the masses asobtained directly from the running of the masses using the D -eliminated formalism.How large the radiatively generated ξ is depends on the boundary conditions we assume forthe scalar masses at gauge unification. Let us consider first the standard CMSSM (or
MSUGRA )picture. In that case it is clear that with the assumption of a common scalar mass at gaugeunification, β (1) ξ vanishes there because U SM is free of gravitational anomalies:Tr[ Y ] = 0 . (22)Moreover, and less obviously, β (1) ξ is in fact RG invariant; that is, using Eq. (6c) in Eq. (23) wefind that Tr h Y β (1) m i = 0 (23)where we denote the SM hypercharge by Y . This follows because Y naturally satisfies Eq. (20),5with Y ′ replaced by Y ): Y il Y ljk + Y j l Y ilk + Y kl Y ijl = 0 (24)(similarly for h ijk ) and anomaly cancellation,Tr [ Y C ( R α )] = 0 . (25)So for CMSSM boundary conditions, or indeed any boundary conditions such that Tr (cid:2) Y m (cid:3) =0 at gauge unification, then, in the one-loop approximation, ξ is zero at low energy if it is zero atgauge unification. (If we go beyond one loop then a non-zero but quite small ξ will be generated.)We turn now to the AM scenario. Substituting Eq. (19) in Eq. (14) and Eq. (15) we findthat up to two loops we can write16 π ˆ β ξ = g | m | (cid:18) µ ddµ Tr[ Y ( γ − γ )] + 2 k ′ Tr[ YY ′ (1 − γ )] (cid:19) , (26)and since gauge invariance and anomaly cancellation combined with Eqs. (3b) and (5) yield [25]Tr[ Y γ (1) ] = Tr[ Y ( γ (2) − ( γ (1) ) )] = 0 , (27)this reduces to 16 π ˆ β ξ = 2 k ′ g | m | Tr[ YY ′ (1 − γ )] . (28)Thus in the absence of the Y ′ term (i.e. with the unmodified mass solution of Eq. (16c)) anappreciable U SM FI term will not be generated by the running, and Eq. (16c) will therefore be RG invariant. This was the conclusion of Ref. [27].Using Eq. (19) however, we obtain Eq. (28), which is non-vanishing even at leading orderunless we choose the charges Y ′ so that Tr[ YY ′ ] = 0 . (29)This was in fact the choice made in Ref. [17], the motive there being to suppress kinetic mixingbetween the U SM and the U ′ gauge bosons (in that paper we considered a U ′ broken at ratherlower energies). With such a U ′ , the Y ′ charges L and e satisfy3 L + 7 e = 0 (30)so they are opposite in sign. Consequently Eq. (19) alone would not suffice to escape thetachyonic slepton problem (if it held at low energy). In Ref. [17] it was shown, however, thatreplacing Eq. (19) by ( m ) ij = 12 m µ ddµ γ ij + k ( Y SM ) ij + k ′ ( Y ′ ) ij , (31)(with Y ′ charges satisfying Eq. (30)) could do so. Now since we have shown above that aneffective U SM FI-term is in any event generated by RG running, it is not a priori obvious thathaving simply Eq. (19) at gauge unification even with a U ′ with opposite L, e charges won’t work;however we may expect that the U ′ choice of Ref. [17] clearly will not do, precisely because ofEq. (29); the generated ξ for U SM will be too small. We shall see that this is indeed the case.6ne might hope that it would be possible to choose, for example, U ′ ≡ U B − L ; we shall see,however, that, with Eq. (19), although the region of ( e, L ) parameter space corresponding toan acceptable supersymmetric spectrum does indeed include the possibility of L <
0, it permitsneither Eq. (29) nor L + e = 0, which would have corresponded to U B − L .Let us now follow Ref. [23] by considering a theory with FI-type contributions associatedwith U ′ , and compare the consequences of imposing Eq. (19) (and vanishing FI term for U SM )at (i) gauge unification (ii) a common SUSY scale, M SUSY . It should be clear from the abovediscussion that using the same values of ( e, L ) in the two cases will not give rise to the samespectrum, because imposing it at gauge unification (say) will give rise to a non-vanishing U SM FI term at M SUSY , and corresponding contributions to the sparticle masses.It is easy to see, however, that precisely the same spectrum consequent on a particular choiceof ( e, L ) at at M X can be obtained by by using a different ( e, L ) pair at M SUSY (with in eachcase no U SM FI term). This is simply because we can write m L = m L − k + k ′ L = m L + k ′′ L ′′ m e c = m e c + k + k ′ e = m e c + k ′′ e ′′ m Q = m Q + k + k ′ Q = m Q + k ′′ Q ′′ etc. , (32)where k ′′ Q ′′ = − k ′′ L ′′ , etc.Thus we can absorb the U SM FI term generated by the running into a redefinition of thecharges ( e, L ).Note that the above remarks strictly apply only if we evaluate the spectrum at a commonmass scale, M SUSY . Since in Ref. [23] we systematically evaluated each sparticle pole mass at arenormalisation scale equal to the pole mass itself, small discrepancies were introduced. Fromnow on we will always calculate spectra by running down from M X , inputting ( e, L ) (and zerofor the U SM FI term) there.
The
MSSM is defined by the superpotential: W = H QY t t c + H QY b b c + H LY τ τ c + µH H (33)with soft breaking terms: L SOFT = X φ m φ φ ∗ φ + " m H H + X i =1 M i λ i λ i + h . c . + [ H Qh t t c + H Qh b b c + H Lh τ τ c + h . c . ] (34)where in general Y t,b,τ and h t,b,τ are 3 × π γ H = 3 λ b + λ τ − g − g , π γ H = 3 λ t − g − g , π γ L = λ τ − g − g , π γ Q = λ b + λ t − g − g − g , π γ t c = 2 λ t − g − g , π γ b c = 2 λ b − g − g , π γ τ c = 2 λ τ − g , (35)where λ t,b,τ are the third generation Yukawa couplings. For the first two generations we use thesame expressions but without the Yukawa contributions. The two and three loop results for theanomalous dimensions and the gauge β -functions may be found in Ref. [29].The soft scalar masses are given by m Q = m Q − Lk ′ , m t c = m t c − ( L + e ) k ′ ,m b c = m b c + ( L + e ) k ′ , m L = m L + Lk ′ ,m τ c = m τ c + ek ′ , m H , = m H , ∓ ( e + L ) k ′ , (36)(with similar expressions for the first two generations) where m Q etc are the pure anomaly-mediation contributions, for example: m Q = m µ ddµ γ Q = m β i ∂∂λ i γ Q (37)(here λ i includes all gauge and Yukawa couplings) and k ′ is the effective FI parameter.The 3rd generation A -parameters are given by A t = − m ( γ Q + γ t c + γ H ) ,A b = − m ( γ Q + γ b c + γ H ) ,A τ = − m ( γ L + γ τ c + γ H ) (38)and we set the corresponding first and second generation quantities to zero. The gaugino massesare given by M α = m (cid:18) β g α g α (cid:19) , for α = 1 , , . (39)The manner in which the scale of the effective FI parameter contributions k ′ L etc. to thesparticle masses can naturally be of the same order as the anomaly mediation contributionswhen a U ′ is broken at high energies is explained in Ref. [23] and Ref. [24].Clearly these FI contributions depend on two parameters, Lk ′ and ek ′ . For notationalsimplicity we will set k ′ = 1(TeV) from now on.We begin by choosing input values for m , tan β , L , e and sign µ at M X and then we calculatethe appropriate dimensionless coupling input values at the scale M Z by an iterative procedure8nvolving the sparticle spectrum, and the loop corrections to α ··· , m t , m b and m τ , as describedin Ref. [30]. We define gauge unification by the meeting point of α and α . For the top quarkpole mass we use m t = 170 . β -functions in the calculations. Note that when doing the three-loop calculation,we use in Eq. (37), for example, the three loop approximation for both β i and γ Q , thus includingsome higher order effects. L Figure 1: The region of ( e, L ) space corresponding to an acceptable electroweak vacuum, for m = 40TeV and tan β = 10.The allowed region in ( e, L ) space for µ > m = 40TeV corresponding to an acceptablevacuum is shown in Fig. 1. To define the allowed region, we have imposed m ˜ τ > m ˜ ν τ > m A > m A becoming too light (and quickly imaginary just beyondthe boundary, with breakdown of the electroweak vacuum) and the other two sides to one of thesleptons (usually a stau) becoming too light.Note that as we remarked earlier, the allowed region includes parts with L <
0. To un-derstand this, consider, for example, the point ( e, L ) = (0 . , − . g
925 900 897˜ t
766 757 746˜ t
502 500 487˜ u L
834 819 808˜ u R
774 766 753˜ b
724 712 702˜ b
956 946 936˜ d L
838 822 812˜ d R
965 955 946˜ τ
267 266 266˜ τ
212 199 199˜ e L
262 261 262˜ e R
225 212 212˜ ν e
250 249 249˜ ν τ
248 247 247 χ
106 131 131 χ
354 362 362 χ
569 593 585 χ
580 604 596 χ ±
107 131 131 χ ±
577 601 594 h
114 114 114 H
333 373 361 A
333 373 361 H ±
342 381 370 χ ± − χ (MeV) 226 235 237Table 2: Mass spectrum for m t = 170 . m = 40TeV, tan β = 10, L = 0, e = 1 / YY ′ ] = 4 . . (40)This is positive so from Eq. (28) we see that β ξ for the U SM FI term is positive at M X . Sincewe are running down from M X it follows that a negative ξ SM is generated, and hence a positive contribution to m L , since the U SM charge of the lepton doublet is negative. Evidently the samereasoning means that we cannot have e < M X , as we indeed see to be the case.Although L < L + e = 0 (corresponding to U B − L ) or 3 L + 7 e = 0 (corresponding to Tr[ YY ′ ] = 0).As an example of an acceptable spectrum, we give in Table 2 the results for m = 40TeV , tan β =10 , L = 0 , e = 1 / , sign µ = + as derived using the one, two and three loop approximations forthe anomalous dimensions and β -functions.This point in ( e, L ) space is near the centre of the allowed region (see Fig. 1). As explained10n the previous section, the same spectrum would be obtained to a good approximation byinputting parameters and calculating pole masses at M Z with a different pair of ( e, L ) values,in this case ( e, L ) ≈ (0 . , . U SM term to be zero at M Z , but to a set of ( e, L ) close to but each differingslightly from (0 . , . µ ∼ B ∼ (140GeV) , leading to κ ∼ . µ >> M Z , we have the problem of accounting for the small value of κ , and a degree of fine tuning between the two terms in Eq. (16d). As in Ref. [23] we find thatto obtain a sufficiently high light CP-even Higgs mass, m h and an electroweak vacuum we needto have 25 > ∼ tan β > ∼ AM characteristic phenomenology the reader is referred to Refs. [1]- [24],and in particular Ref. [4]. U and GUTs In the previous sections we have been assuming that our theory has gauge group G SM ⊗ U ′ , brokento G SM at high energies. Let us now ask what modifications ensue if we ask for compatibilitywith a simple GUT embedding; for definiteness let us take SU , and imagine that our matterfields form a set of n f (5 + 10) multiplets as usual, and promote our Higgs multiplets to n h setsof (5 + 5). Then for compatibility with an SU ⊗ U ′ embedding we at once have the relations Q = u c = ed c = L (41)and for U ′ invariance of the Yukawa terms h = − L − eh = − eν c = 2 e − L. (42)Then the SU ⊗ U ′ , SU ⊗ U ′ and ( U SM ) ⊗ U ′ anomalies are all proportional to the quantity A = ( n f − n h )( L + 3 e ) (43)while the ( U ′ ) ⊗ U SM anomaly vanishes. The ( U ′ ) anomaly is proportional to A = ( L + 3 e ) (cid:2) n f − n h )( L + 3 e ) − n f ( L + 3 e ) (cid:3) (44)while the U ′ − gravitational anomaly is proportional to A G = ( L + 3 e )(4 n f − n h ) . (45)Thus if L + 3 e = 0 then the G SM ⊗ U ′ theory is anomaly-free for arbitrary n f , n h . This specialcase corresponds in fact to compatibility with the embedding SO ⊃ SU ⊗ U ′ with each set110 5 ν c H H N e L e − L − e − e − L L + 3 e Table 3: Anomaly free U symmetry for arbitrary lepton doublet and singlet chargesof matter fields forming a 16 and each set of Higgs fields a 10 under SO . (Although SO hascomplex representations they are all anomaly-free). Note the opposite sign charges for L and e ;we argued in Section 3 that this does not preclude starting from M X with an FI term for sucha U ′ , but we shall see that the line L + 3 e = 0 does not cross the allowed ( e, L ) region for ourclass of models. The other way to produce an anomaly-free theory is to first set n h = n f . Then A = 0 while for A and A G we have A = − n f ( L + 3 e ) A G = − n f ( L + 3 e ) (46)so that we can obtain an anomaly-free theory by adding a further set of n f G SM -singlet fields N , with charges L + 3 e . The resulting charge assignments are shown in Table 3.This structure is compatible with SU ⊗ U ′ , and can be embedded in E , when Table 3forms a 27. (Recall that E also has only anomaly-free representations). If L = e we couldhave E ⊃ SO ⊗ U ′ , (with Table 3 forming a 16 ⊕ ⊕ SO ), or, as explained above,for L = − e we could have SO ⊃ SU ⊗ U ′ . Another possibility is to have L = 2 e , in orderthat ν c have zero U ′ charge [33]; evidently this would have model-building advantages if onewants to have a large mass for ν c while breaking U ′ at lower energy. Of course one sees easilythat the cases L = − e and L = 2 e are equivalent from a group theoretic point of view underthe exchanges N ↔ ν c and 5 ↔ H ; obviously in the latter case we could have an anomaly-freetheory with n f sets of (10 , H , N ) and n h sets of ( H, MSSM effective field theory, with three generations and a single pair ofHiggs doublets (of course an explicit construction may lead to a more exotic low energy theory,but here we will confine ourselves to this possibility). We also assume FI contributions to thesparticle masses corresponding to our new U ′ , thus instead of Eq. (36) we have: m Q = m Q + ek ′ , m t c = m t c + ek ′ , m τ c = m τ c + ek ′ ,m b c = m b c + Lk ′ , m L = m L + Lk ′ ,m H = m H − ( e + L ) k ′ , m H = m H − ek ′ , (47)where m Q etc are again the pure anomaly-mediation contributions, and once again we set k ′ = 1.We can then compare the predicted sparticle spectrum with that obtained in the last section.We may expect there to be differences, since evidently if we have both ( e, L ) > contributions. We calculate the12pectrum as described in the previous section, running down from M X ; of course RG invarianceof the AM masses no longer holds because the effective field theory is no longer anomaly-freewith respect to the U ′ . L Figure 2: The region of ( e, L ) space corresponding to an acceptable electroweak vacuum, for m = 40TeV and tan β = 15.The allowed ( e, L ) region with our new charge assignments is shown in Fig. 2. Comparingwith Fig. 1, we see that the most dramatic difference is that increasing ( e, L ) does not leadto loss of the electroweak vacuum as long as L < ∼ e + 0 .
4. Of course increasing ( e, L ) scales upthe squark and slepton masses, | m H , | and hence the (Higgs) µ -parameter, thus increasing thefine-tuning known as the little hierarchy problem. Other scenarios explored recently have alsohad this feature, for example split supersymmetry [34], and the G based model of Ref. [35].For a recent discussion of the little hierarchy problem see (for example) Ref. [36].Another distinctive feature of the new charge assignment is that acceptable spectra areobtained with larger values of tan β than in section 4; here we find an upper limit of tan β = 43.In Table 4 we give results for the sparticle spectrum for a representative point in the allowedregion. Of course L = 1 / e = 1 / AM , so it is not surprising that these massesare quite large for this point. Correspondingly the value of µ determined from electroweakminimisation is quite high at around 1TeV.Both L = e (corresponding to a potential SO ⊗ U ′ embedding) and L = 2 e (corresponding13ass (GeV) 1loop 2loops 3loops˜ g
966 940 938˜ t t
936 923 917˜ u L u R b b
993 1021 966˜ d L d R τ
550 544 544˜ τ
698 697 697˜ e L
556 551 551˜ e R
698 697 697˜ ν e
550 545 545˜ ν τ
548 542 543 χ
111 135 135 χ
362 369 369 χ χ χ ±
111 135 136 χ ± h
115 115 115 H
737 743 737 A
737 743 737 H ±
742 748 742 χ ± − χ (MeV) 185 192 192Table 4: Mass spectrum for m t = 170 . m = 40TeV, tan β = 15, L = 1 / e = 1 / g
934 910 907˜ t
858 847 838˜ t
688 680 672˜ u L
908 891 881˜ u R
911 899 889˜ b
803 789 780˜ b
894 882 872˜ d L
911 894 885˜ d R
916 904 894˜ τ
236 231 231˜ τ
311 308 308˜ e L
282 275 275˜ e R
282 281 281˜ ν e
270 263 263˜ ν τ
266 259 259 χ
109 133 134 χ
358 365 365 χ
820 833 828 χ
826 839 834 χ ±
109 134 134 χ ±
826 839 834 h
115 115 115 H
623 635 629 A
624 636 629 H ±
629 641 634 χ ± − χ (MeV) 192 199 200Table 5: Mass spectrum for m t = 170 . m = 40TeV, tan β = 15, L = e = 0 . U ′ charge for ν c ) are allowed; in the latter case we would need to have e < ∼ .
4. In Table 5we give results for the sparticle spectrum for L = e = 0 .
1, while in Table 6 we give results forthe sparticle spectrum for L = 2 e = 0 . By taking appropriate linear combinations of squark and slepton (masses) so that the ( e, L )contributions cancel it is straightforward to derive a pair of interesting sum rules similar to those15ass (GeV) 1loop 2loops 3loops˜ g
930 906 903˜ t
828 818 809˜ t
650 642 633˜ u L
880 864 854˜ u R
884 873 863˜ b
771 759 749˜ b
893 882 872˜ d L
883 868 857˜ d R
916 905 895˜ τ
290 285 285˜ τ
131 126 127˜ e L
284 278 278˜ e R
165 162 162˜ ν e
272 266 266˜ ν τ
268 262 262 χ
109 133 133 χ
358 365 365 χ
759 774 768 χ
766 780 775 χ ±
109 133 134 χ ±
765 780 775 h
115 115 115 H
585 599 591 A
585 599 592 H ±
591 604 597 χ ± − χ (MeV) 195 203 203Table 6: Mass spectrum for m t = 170 . m = 40TeV, tan β = 15, L = 2 e = 0 . m u L + m d L − m u R − m e R ≈ . m ˜ g ) ,m A + sec 2 β (cid:0) m e R − m e L (cid:1) − M W + M Z ≈ . m ˜ g ) ,m b + m b − m τ − m τ ≈ . m ˜ g ) ,m b + m b − m e L − m e R ≈ . m ˜ g ) ,m d L + m d R − m e L − m e R ≈ . m ˜ g ) . (48)Although these sum rules are derived using the tree results for the various masses they holdreasonably well for the physical masses. The numerical coefficients on the RHS of Eq. (48) arein fact slowly varying functions of tan β ; the above results correspond to tan β = 15. The AM scenario is an attractive alternative to (and distinguishable from) the CMSSM . With AM it is possible to imagine a theory where the only explicit scale in the effective field theoryis the gravitino mass. An explicit realisation of this idea was given in Ref. [24], where the scalecorresponding to the spontaneous breaking of an additional U ′ symmetry (needed to solve thetachyonic slepton problem) was generated by dimensional transmutation. (This theory had theadditional feature of a weakly coupled chiral matter multiplet whose fermionic component isa dark matter candidate). There is no obstacle in principle to extending this idea to a GrandUnified Theory, with the unification scale similarly generated by dimensional transmutation; thisidea led us to consider the alternative charge assignments of section 5. One possibility would bea variation of the inverted hierarchy model of Witten [37], defined by the superpotential W = λ Tr( A Y ) + λ X (Tr A − m ) (49)where A, Y are SU adjoints and X is a singlet. In its original form, supersymmetry is brokenspontaneously in the O’Raifertaigh manner; moreover SU is broken to SU ⊗ SU ⊗ U , with thescale at which this occurs being unrelated to m , and generated by dimensional transmutation.Our variation would be to have m = 0 in Eq. (49), with the SU breaking generated in similarfashion but the supersymmetry breaking provided instead by anomaly mediation. We willexplore this model in more detail elsewhere.We have shown that while a U ′ gauge symmetry broken at high energies can lead in a naturalway to the FI-solution to the AM tachyonic slepton problem, care must be taken with regard tothe FI term associated with U SM . We have also shown how an extension of the minimal modelpermits a gauged U ′ compatible with grand unification, with, in this case, sparticle spectracharacterised by both heavy squarks and heavy sleptons. A discussion of the m → cknowledgements RH was supported by the STFC. DRTJ thanks Luminita Mihaila, Graham Ross and StuartRaby for conversations. We also particularly thank Ting Wang for helpful correspondence.
References [1] L. Randall and R. Sundrum,
Nucl. Phys.
B 557 (1999) 79[2] G.F. Giudice, M.A. Luty, H. Murayama and R. Rattazzi,
JHEP (1998) 27[3] A. Pomarol and R. Rattazzi,
JHEP (1999) 013[4] T. Gherghetta, G.F. Giudice and J.D. Wells,
Nucl. Phys.
B 559 (1999) 27[5] M.A. Luty and R. Rattazzi,
JHEP (1999) 001[6] Z. Chacko, M.A. Luty, I. Maksymyk and E. Ponton,
JHEP (2000) 001[7] E. Katz, Y. Shadmi and Y. Shirman,
JHEP
Phys. Lett.
B 465 (1999) 148[9] J.L. Feng and T. Moroi,
Phys. Rev.
D 61 (2000) 095004[10] G.D. Kribs,
Phys. Rev.
D 62 (2000) 015008[11] S. Su,
Nucl. Phys.
B 573 (2000) 87[12] J.A. Bagger, T. Moroi and E. Poppitz,
JHEP
Nucl. Phys.
B 576 (2000) 3[14] F.E. Paige and J. Wells, hep-ph/0001249[15] N. Okada,
Phys. Rev.
D 65 (2002) 115009[16] M. Luty and R. Sundrum,
Phys. Rev.
D 67 (2003) 045007[17] I. Jack and D.R.T. Jones,
Phys. Lett.
B 482 (2000) 167[18] N. Arkani-Hamed, D.E. Kaplan, H. Murayama and Y. Nomura,
JHEP (2001) 041[19] R. Harnik, H. Murayama and A. Pierce,
JHEP (2002) 034[20] B. Murakami and J.D. Wells,
Phys. Rev.
D 68 (2003) 035006[21] R. Kitano, G. D. Kribs and H. Murayama,
Phys. Rev.
D 70 (2004) 035001[22] M. Ibe, R. Kitano and H. Murayama,
Phys. Rev.
D 71 (2005) 075003[23] R. Hodgson, I. Jack, D.R.T. Jones and G.G. Ross,
Nucl. Phys.
B 728 (2005) 1921824] D.R.T. Jones and G.G. Ross,
Phys. Lett.
B 642 (2006) 540[25] I. Jack and D.R.T. Jones,
Phys. Lett.
B 473 (2000) 102[26] I. Jack, D.R.T. Jones and S. Parsons,
Phys. Rev.
D 62 (2000) 125022[27] I. Jack and D.R.T. Jones,
Phys. Rev.
D 63 (2001) 075010[28] A. de Gouvea, A. Friedland and H. Murayama,
Phys. Rev.
D 59 (1999) 095008[29] P.M. Ferreira, I. Jack and D.R.T. Jones,
Phys. Lett.
B 387 (1996) 80[30] D.M. Pierce, J.A. Bagger, K.T. Matchev and R.J. Zhang,
Nucl. Phys.
B 491 (1997) 3[31] A. Bednyakov et al,
Eur. Phys. J.
C 29 (2003) 87[32] I. Jack, D.R.T. Jones, A.F. Kord,
Ann. Phys.
316 (2005) 213[33] S.F. King, S. Moretti and R.Nevzorov,
Phys. Lett.
B 650 (2007) 57[34] N. Arkani-Hamed and S. Dimopoulos, JHEP (2005) 073;G.F. Giudice and A. Romanino,
Nucl. Phys. B (2004) 65 [Erratum-ibid. B (2005)65].[35] B.S. Acharya, K. Bobkov, G.L. Kane, P. Kumar and J. Shao, arXiv:hep-th/0701034.[36] L.E. Ibanez and G.G. Ross, arXiv:hep-ph/0702046.[37] E. Witten, Phys. Lett.
B 105 (1981) 267[38] M.B. Einhorn and D.R.T. Jones,
Nucl. Phys.