aa r X i v : . [ h e p - ph ] O c t Metastable gauged O’Raifeartaigh
Borut Bajc and Alejandra Melfo , J. Stefan Institute, 1000 Ljubljana, Slovenia and Universidad de los Andes, Merida, Venezuela
We study the possibility of obtaining metastable supersymmetry breaking vacua in a perturbativegauge theory without singlet fields, thus allowing for scenarios where a grand unified symmetry andsupersymmetry are broken by the same sector. We show some explicit SU(5) examples. Theminimal renormalizable example requires the use of two adjoints, but it is shown to inevitably leadto unwanted light states. We suggest various alternatives, and show that the viable possibilitiesconsist of allowing for non-renormalizable operators, of employing four adjoints or of adding at leastone field in a different representation.
PACS numbers: 11.15.Ex, 12.60.Jv, 12.10.-g, 11.30.Pb
I. INTRODUCTION
The idea of trying to combine grand unified theorieswith supersymmetry breaking has been used already inthe early days of supersymmetry [1, 2, 3, 4] followingmainly the suggestion of dimensional transmutation [5].The tree order supersymmetry breaking vacuum enforcedby the O’Raifeartaigh type superpotential automaticallyhas a flat direction, which gets however stabilized at oneloop exactly because of supersymmetry breaking correc-tions. All of these models are in a perturbative regimeand make use of gauge singlets. The mediation of su-persymmetry breaking to the MSSM sector is dominatedby gravity, which cannot predict (although it can fit) thestrong suppression of the flavour changing neutral cur-rents. Later models [6, 7, 8, 9, 10] were able to get ridof gauge singlets, using nonperturbative gauge sectors todynamically break supersymmetry. The minima here arenot global, but local and thus metastable, although witha long enough lifetime. A typical model has more sectorsand gauge groups than usually assumed in phenomeno-logical motivated models like MSSM or grand unification.The results are important and promising: the modelsconsidered are mainly realistic and quite natural (with-out fine-tunings), while the mediation is gauge dominated[1, 3, 11, 12, 13, 14, 15, 16], an important result.What we want to explore in this work is the possi-bility to use as much as possible minimal gauge groups,no singlets and perturbative physics only. The best pos-sibility (and the original motivation) is to use a grandunified group G (we will limit ourselves to SU(5)) with-out singlets and break both G and N = 1 supersymmetryspontaneously (an example of models which break N = 2supersymmetry spontaneously without the use of chiralsinglets is given in [18, 19]). At first glance this seems tobe in contradiction with what we know from perturba-tive spontaneous supersymmetry breaking. In fact, oneneeds a linear term in the superpotential, which must bea singlet, thus naively forbidding for it the use of a gaugemultiplet. However, by choosing properly the basis it iseasy to get rid of the linear term and thus have a formof the superpotential that can be directly employed in gauge theories without any need for singlets. This willbe explicitly shown in section II. As it will be clear, sucha construction is possible only because the consideredvacuum is metastable (a recent revival of models withsuch vacua has been triggered by [17]). For such rea-sons we will call these models of the metastable gaugedO’Raifeartaigh type. It is thus tempting to use this ideain realistic models like for example grand unified theories.Writing a superpotential that exhibits perturbativeand spontaneous supersymmetry breaking without lin-ear terms is only the first part of the story. The secondpart is to make these metastable gauged O’Raifeartaighmodels realistic in the context of grand unified theories.The minimal SU(5) model will be explicitly presentedin section III, together with the main virtues and draw-backs. The virtues are the fact that two adjoint fieldssuffice to break both supersymmetry and SU(5) gaugesymmetry spontaneously. We will show that the modelis locally stable in some range of the vevs. One of thevevs is undetermined at tree order, and we will check thatit can exhibit a metastable local minimum at one loop.The renormalizable superpotential has two terms only, aform which is enforced by a global U(1) R symmetry. Thedrawback of this simple example is the presence of lightstates, which makes it unrealistic. Possible corrections ofthis minimal scenario and the role of supergravity will bedescribed in section IV. We will present explicitly threerealistic cases in which these unwanted light states arenot present: 1) the nonrenormalizable model with two , section IV A, eq. (24); 2) the renormalizable modelwith four , section IV B, eq. (35); 3) the renormal-izable model with two and one , section IV C, eq.(37). Finally, some general remarks and a list of openproblems (among which the suggestion to use this typeof models in hybrid inflation without singlets) to be dis-cussed in more detail elsewhere will be given in sectionV. II. FROM SINGLETS TO GAUGE MULTIPLETS
We start with the simplest model which exhibitsmetastable supersymmetry breaking following the gen-eral analysis [20] W = S (cid:16) ξ + λ ˜ φ (cid:17) . (1)It exhibits a tree level local minimum at h ˜ φ i = 0 , S undetermined , (2)providing |h S i| ≥ (cid:12)(cid:12)(cid:12)(cid:12) ξ λ (cid:12)(cid:12)(cid:12)(cid:12) / . (3)Such a superpotential cannot be directly written in termsof gauge multiplets, due to the existence of the linearterm in S . It is however simple to get rid of it by redefin-ing ˜ φ = φ − h φ i , (4)and choosing h φ i such that ξ + λ h φ i = 0 . (5)We end up with W = µφS + λφ S , (6)i.e., no linear terms, and with a local minimum at h φ i = − µ λ , S undetermined , (7)provided it is in the allowed range |h S i| ≥ |h φ i|√ . (8)This shows that one could start with eq. (6), and sincethere are no linear terms in it, no singlet is really needed:both S and φ in (6) can be part of a gauge multiplet ofa gauge group G , which vevs h S i and h φ i break G spon-taneously to a subgroup H . In the next section we willgive an SU(5) example with two adjoints, both breakingto SU(3) × SU(2) × U(1).
III. THE SIMPLEST EXAMPLE: TWO SU(5)ADJOINTS
Using the results in the previous section, we can imme-diately write down a candidate for a metastable gaugedO’Raifeartaigh SU(5) model: W = µT r Σ Σ + λT r Σ Σ . (9)We expand the adjoints Σ i asΣ i = (cid:18) O i + 2 σ i / √ X i ¯ X i T i − σ i / √ (cid:19) , (10)where σ i are the Standard Model (SM) singlets, O i thecolor octets (8 ,
1; 0), T i the weak triplets (1 ,
3; 0), and X i , ¯ X i the color triplet, weak doublets (3 , ± / v = h σ i is obtained from (cid:28) ∂W∂σ (cid:29) = 0 → v = √ µλ , (11)while supersymmetry breaking is signaled by a nonzero F term: F ∗ ≡ (cid:28) ∂W∂σ (cid:29) = λv √ . (12)The other vev, v (= h σ i ), is undetermined at treeorder, i.e. it is a flat direction. It will be stabilizedby nontrivial 1-loop corrections to the K¨ahler potential,which at tree order is K = T r Σ † i Σ i . (13)Since the vevs of the adjoints are diagonal, the D-termsare vanishing.We have to check two things.First, that the above model does not contain tachyons.That the singlet has non-negative mass square at leastfor some choices of the vevs is expected from (8). Whatremains to be checked are the masses of all other SM mul-tiplets. One pair of the bosons in X i , ¯ X i will provide thewould-be Nambu-Goldstone bosons (mainly from Σ ),while the other pair (mainly from Σ ) will acquire a massproportional to v , so we do not need to worry aboutthem.After SU (5) breaking, the singlet in Σ gets a super-symmetric mass M σ = − λ √ v , (14)while the non-singlet mass matrices have in general theform M = λ √ (cid:18) c v c v c v (cid:19) , (15)with ( c , c ) = (6 ,
4) for color octets, and ( c , c ) =( − , −
6) for weak triplets. The supersymmetry break-ing mass terms in the Lagrangian are δL = λF √ (cid:0) − σ + 2 O − T − X ¯ X (cid:1) + h.c. (16)One can now easily find out that there are no tachyonicstates if the SM singlet scalar σ is not tachyonic, whichis true provided the analogue of (8) is satisfied: | v | ≥ | v |√ . (17)The second thing we need to check is whether the flatdirection σ gets stabilized at 1-loop following the linesof [5]. All is needed is to check what happens with thewavefunction of the field that breaks supersymmetry ( σ )[7]. In fact the potential at one loop gets corrected withrespect to the tree order one by exactly the wavefunc-tion renormalization (neglecting small finite corrections)through V ( σ ) ≈ | F | Z ( | σ | ) , (18)where F can be read from (12) and Z is the wavefunc-tion renormalization at one loop. Obviously the mini-mum of the potential comes from the maximum of Z .At this point one can use the usual rules to write downthe renormalization group equations - RGE’s (a usefuland concise set of rules can be found for example in [15]).For the particle spectrum we take on top of the two ad-joints just the minimal set of three generations of matterfields and one pair of 5 H , 5 H (the results can be easilygeneralized for more Higgs and/or messenger fields). Weobtain ( τ ≡ π ln (cid:16) µM GUT (cid:17) ) the following system ddτ g − = − , (19) ddτ ln λ = − g + 21 λ , (20) ddτ ln Z = 10 g − λ . (21)We have assumed that the couplings between the fun-damental and adjoint Higgses are negligible The extremum of Z fixes one parameter of the super-potential at the minimum λ = 5021 g . (22)That the extremum of the potential is indeed a min-imum can be seen from the negativity of the secondderivative at the extremum This assumption is consistent for example in the simplest of allcases, i.e. W = ¯5 H ( y Σ + M )5 H . Z d Z dτ = − g . (23)The minimum (and thus the GUT scale v ) is deter-mined by the equivalence (22).We have thus checked that the Higgs sector (9) canindeed break both SU(5) to the SM gauge group and su-persymmetry. Also, the original parameters of the model( µ , λ ) can be changed for the physical ones ( F , M GUT ).Notice that all this has been achieved without any finetuning of the model parameters. The gauge coupling wascrucial in this game: the limit of gauge singlets wouldconfirm the observation of [20] that metastable super-symmetry breaking vacua exist only when all values ofthe flat directions are allowed at tree order. In fact,for g → v towards the origin, violating the bound (17) and eventu-ally finishing in one of the two supersymmetry preservingvacua v = 0 or v = √ µ/λ (both with v = 0).The superpotential (9) is the most general renormal-izable superpotential for two SU(5) adjoints that satis-fies a global U(1) R symmetry, under which Σ is neutraland Σ has charge 2. This symmetry is spontaneouslybroken by the v vev and has thus at the perturbativelevel an exact Nambu-Goldstone boson ( σ ). The R-symmetry must be eventually explicitly broken by super-gravity corrections that cancel the cosmological constant[21], which will give a nonzero mass also to this pseudo-Nambu-Goldstone boson.To summarize: SU(5) is broken at v , supersymmetryat v . The adjoint Σ could in principle be used as amessenger.The model is simple and predictive, indeed too predic-tive, leading to inescapable problems. The most pressingone is that either the supersymmetry breaking scale iscomparable to the GUT scale or there are light weaktriplets and colour octets mainly from Σ . In fact from(15) we can see that triplets and octets can have order M GUT mass only if v = O ( v ), i.e. when √ F ≈ v ≈ v ≈ M GUT . Since the most obvious candidate for themessengers are the MSSM multiplets in Σ , the typicalsoft mass is only loop (i.e. ≈ − ) suppressed with re-spect to the triplet and octet masses ≈ F/M
GUT . Keep-ing v as a free parameter one is still able to unify thegauge couplings, but at a too high scale slightly above10 GeV, with the sfermion and gaugino masses around10 GeV. Even if one accepted such a high scale, the cal-culation itself would turn out to be inconsistent, because M GUT ∼ > GeV would make supergravity correctionsto the soft masses dominant. Taking this into accountconsistently changes very little, making such a model un-appealing. In the next section, we describe more realisticscenarios.
IV. MORE REALISTIC OPTIONS
We see that the problem arises because the same scalethat determines the light SM multiplets ( v ) specifies alsothe supersymmetry breaking F ∝ v and thus cannot beat the same time large and small. In order to provide fora different scale, one can resort basically to two possibil-ities: adding non-renormalizable interactions while keep-ing the field content minimal, or adding more fields andkeep renormalizability. We will find three different real-istic models, described in sections IV A, IV B and IV Crespectively. All three of them possess a global U(1) R symmetry, broken by the vacuum expectation value ofthe GUT field that gets a nonzero F term. This is inaccordance with the general theorem [22]. A. Adding non-renormalizable operators
The first option is to keep the number of adjoints at aminimum but increase the number of interaction terms,i.e. allow for non-renormalizable operators. Using higherpowers in Σ is still consistent with the U(1) R symmetry.The simplest correction W = T r h Σ (cid:16) µ Σ + λ Σ + α M Σ + α M T r (cid:0) Σ (cid:1) Σ (cid:17)i (24)is already enough: one can have large enough vev v ≈ v but with F arbitrarily low (with a proper fine-tuning ofthe model parameters), as we now show.From the equation of motion for σ , i.e. ∂W/∂σ = 0we get µ = 2 λ √ v − M (cid:18) α + α (cid:19) v , (25)while the second equation F ∗ = ∂W/∂σ = 0 gives F ∗ = v (cid:20) λ √ − M (cid:18) α + α (cid:19) v (cid:21) . (26)This solution has no tachyonic states provided2 (cid:12)(cid:12)(cid:12)(cid:12) v v (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) F ∗ v − (cid:18) α + α (cid:19) v M (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) F ∗ v (cid:12)(cid:12)(cid:12)(cid:12) . (27)For small enough F this is always the case, so we donot need to worry anymore, allowing large values of v .Assuming all parameters real for simplicity we get for thedeterminants of the octet and triplet mass matrices( − det O ) / = v (cid:20) Fv + (cid:18) α + 10 α (cid:19) v M (cid:21) , (28)( − det T ) / = v (cid:20) Fv + (cid:18) α + 10 α (cid:19) v M (cid:21) . (29) They depend very mildly on the supersymmetry break-ing order parameter F . In the limit F → v ≈ v ≈ M GUT the two eigenvalues of the orderof M GUT /M (barring accidental cancellations). So, al-though there are intermediate states, they are much lessharmful than the ones in the previous examples.There are various comments in order.First, notice that in this example there is a fine-tuningneeded to split the scales v (that we want to be large inorder to avoid too light states) and √ F (that we want tobe small enough, possibly even around 100 TeV). This isseen for example from the constraint (26).Another important point is that the cutoff M cannotbe too large for two reasons: first, eq. (26) tells us that λ can be order 1 as required by (22) only for mild hier-archies M GUT /M ; second, one does not want too lightintermediate states of mass M GUT /M .Finally, one could worry that the new operators intro-duced could influence the RGE’s used to get the min-imum of the effective potential. This is not the case,since at one loop the 1 /M suppressed operators do notcontribute to the renormalization of the wave-functions.Let us now check the gauge coupling unification con-straints. The spectrum is the following: at Λ SUSY wehave the MSSM superpartners (and the second Higgs), at M GUT /M we have two colour octets, two weak tripletsand a pair of X , ¯ X , i.e. one colour octet, one weak tripletand a full SU(5) adjoint. This is completely analogousto the case described in [23] with the result that the finalGUT scale is increased with respect to the usual MSSMcase only by a factor 2, if we assume that the cutoff M is 10 times the GUT scale. Due to the increase of M GUT and the appearence of an extra adjoint multiplet at theintermediate scale, the unification gauge coupling α U in-creases by about 10% with respect to the usual MSSMcase. B. Adding more adjoints
If one wishes to stick to renormalizable models, thesimplest idea is to generalize the model (9) to somethinglike W = T r [Σ N +1 ( µ i Σ i + λ ij Σ i Σ j )] , (30)where now i goes from 1 to some integer N . Notice that λ ij in general does not need to be symmetric, so in gen-eral a SU(5) invariant unitary rotation cannot diagonalize λ . For our purpose it is however enough to concentratejust on the diagonal elements of Σ N +1 and Σ i , so thatthese matrices commute and only the symmetric combi-nation λ ij + λ ji enters, which can be diagonalized. So weobtain in complete generality the N replica of (9), i.e. W = T r (cid:2) Σ N +1 (cid:0) µ i Σ i + λ i Σ i (cid:1)(cid:3) . (31)Repeating the exercise in section III, we get v i = √ µ i λ i ; F N +1 = N X i =1 λ i √ v i . (32)In principle it could be possible to have large v i butsmall F N +1 (by appropriate fine-tuning of the terms inthe sum), but this cannot help, as we shall now see. Themass matrices that generalize (15) are now ( N + 1) × ( N + 1) dimensional, and for the triplet and octet havethe form M = 1 √ (cid:18) λ i c ,i δ ij v N +1 λ i c ,i v i λ i c ,i v i (cid:19) , (33)while the determinant is − det ( M ) = N Y k =1 (cid:18) λ k c ,k v N +1 √ (cid:19) N X i =1 λ i c ,i v i √ c ,i v N +1 . (34)Since all the fields are adjoints, the Clebsch-Gordoncoefficients c ,i , c ,i are the same for each SM state, andtherefore the sum above is proportional to F N +1 : in thelimit F N +1 ≪ v N +1 we get N masses of order v N +1 andone of order F N +1 /v N +1 . Adding more adjoints in thisway cannot give mass to the light colour octets and weaktriplets.This result is a consequence of the superpotential cho-sen, but there is at least another possibility. Namely,since any nonrenormalizable Lagrangian can be in prin-ciple obtained from a renormalizable one by integratingout heavy degrees of freedom, one could use directly therenormalizable potential that gives (24). It turns outthat, due to the linearity in Σ , not one but two addi-tional adjoints (Ω i ) are needed. The following ansatz W = − M T r (Ω Ω ) + T r [Ω ( µ Σ + λ Σ Σ )]+ T r (cid:2) Ω (cid:0) µ Σ + λ Σ (cid:1)(cid:3) (35)will do the job. One can show that this model has theright properties also in its renormalizable version (with-out integrating out Ω , ) for all the mass terms and cou-plings of order 1. Notice that there is still a U(1) R sym-metry, under which Σ and Ω have charge 0 and Σ and Ω have charge 2. The model could presumably begeneralized to W = T r [Σ f Σ (Σ , Ω )] + T r [Ω f Ω (Σ , Ω )] . (36)We will not push this model any further. C. Adding different representations
There is a further possibility to maintain renormaliz-ability. The point is that what precludes to have reallydifferent mass matrices of the MSSM adjoints and thesinglet is the absence of enough terms in the superpoten-tial. In other words, there is only one type of trilinearinvariants for the adjoint fields (although for three differ-ent adjoints there are actually two such invariants, theyare equivalent for diagonal elements that commute). Soone can try to use different SU(5) representations, andthe smallest one for this purpose to add to two adjointsis the . One can write the most general renormalizablesuperpotential as W = µT r (Σ Σ ) + λ T r (Σ Σ )+ λ T r (Φ Σ ) + ηT r (ΦΣ Σ ) , (37)where Φ is the . The supersymmetry breaking isachieved for the SM singlet vevs h Φ i = − √ η λ v , (38) h Σ i ≡ v = − r λ µλ λ − η . (39)We get now F ∗ = − √ λ (cid:18) λ λ − η (cid:19) v , (40)which can be fine-tuned to any desired value by fixingthe expression in brackets. All one has to do now is tomake sure that there are no light states, with massesproportional to the supersymmetry breaking parameter F .There are seven different states in all. Three ofthese are only present in , namely the (8 ,
3; 0), the(3 , ± /
3) and the (6 , ± / λ v since they do not mix. The X, ¯ X provide the Nambu-Goldstone bosons as before. For the other two, namelythe color octets and weak triplets, the determinant of thesupersymmetric mass matrices aredet O = − √ v v √ λ (41) × η − √ √ η λ F v + 845 λ F v ! det T = v √ η λ − F v ! (42)As can be seen, there are no light states left. Thus, thiscan be considered the minimal renormalizable version. D. Supergravity corrections
In supergravity it is possible to spontaneously breaksupersymmetry and SU(5) with just one adjoint [24], al-though with considerable fine-tuning. In this paper wewant to take the opposite limit, i.e. to avoid the dom-ination of terms suppressed by the Planck mass. How-ever, supergravity is there, if nothing else, to cancel thecosmological constant. Here we will shortly check whatsupergravity does to our models. We will limit ourselvesto the most delicate aspects of the above scenario, i.e.the stability of the minimum found through the RGE’sand to the R-axion mass.Consider the nonrenormalizable model with two ad-joints. Although the model has a cutoff lower than thePlanck scale, we assume that the UV completion at thiscutoff, valid all the way to M P l , maintains at least ap-proximately the form of the SM singlets’ superpotential W = F ( φ i ) σ + W ( φ i ) , (43)where F ( h φ i i ) ≡ F sets the scale of supersymmetrybreaking and W ( h φ i i ) ≡ W ≈ F M
P l fine-tunes thecosmological constant to zero. Assuming that all vevsare smaller than M P l the typical supergravity contribu-tion to the potential for σ is schematically F ( σ /M P l ) n and so only the lowest n ’s are relevant. The correctionto the mass is ∆ m σ ≈ (cid:18) FM P l (cid:19) , (44)to be compared with the mass found in the global su-persymmetric case. This can be easily read off from (18)and (23) m σ ≈ (cid:16) α U π (cid:17) (cid:18) FM GUT (cid:19) . (45)We see that the mass square from the solution (45)in the global supersymmetry case is numerically (for M GUT ≈ . GeV, M P l ≈ . GeV, α U ≈ / V ≈ m σ ( σ − M GUT ) + F M P l σ + ... (46)In the limit M P l → ∞ we had M GUT = h σ i ≡ M GUT ,but now the true minimum gets shifted as (we omit num-bers of order one) M GUT = M GUT + F m σ M P l . (47)The two contributions are of the same order and thesupergravity one could even dominate. To settle it onewould need to perform a more precise calculation. Onecan however notice that the value M GUT was defined asthe scale, at which the equality (22) is satisfied. But thenit is enough to shift this scale to a different value, so thatthe final M GUT (47) is what we would like it to be.Another issue is the R-axion mass. The constraint of avanishing cosmological constant requires a constant termof order
F M
P l in the superpotential. This term explicitlybreaks the U(1) R-symmetry. The pseudo R-axion getsthus a non-vanishing mass of order [21] m a ≈ F M GUT M P l . (48)Whether the model is cosmologically safe or not de-pends on the value of F . A weak scale R-axion mass isdangerous, for similar reasons as moduli, see for exam-ple [25, 26, 27] for possible solutions in this case. In theopposite case of small F the R-axion mass is harmless.A short comment is due on D-terms. It is known, thatin general N = 1 D = 4 supergravity, the D and F terms are connected [28, 29]. This means, that if D -terms are non-zero, they are related to F -terms. In ourcase the adjoints are diagonal, so their D -terms are stillzero, similar to the global limit. V. CONCLUSIONS
We have shown that it is possible to construct realis-tic superpotentials that break perturbatively both super-symmetry and a gauge symmetry without using singlets.This is possible only because the minima considered weremetastable. For such models there is no reason to intro-duce extra gauge sectors, which dynamically break super-symmetry. One can thus study just simple gauge groups,a particularly appealing situation in case of grand unifiedtheories. The price to pay is that extra states need to beintroduced.We found three different realistic SU(5) examples:1. nonrenormalizable model with two , sectionIV A, eq. (24);2. renormalizable model with four , section IV B,eq. (35);3. renormalizable model with two and one , sec-tion IV C, eq. (37).All three of them have a spontaneously broken U(1) R global symmetry.There are many issues not touched in this paper, to beaddressed in subsequent work. Let us mention some ofthem. The doublet-triplet problem.
In the minimal case ofthe renormalizable model with two adjoints the doubletsand triplets of a single pair of 5 H and ¯5 H cannot be splitenough even with fine tuning. In fact, Σ should not cou-ple to the fundamentals, because its F term destabilizesthe weak scale. On the other side Σ has a too smallvev (of the order of √ F ) to split enough the doubletsand triplets. It is thus reassuring that in the realisticversions this problem disappears, since now v can be ofthe order of the GUT scale.In models with 75 one can use the missing partnermechanism [30]. Such a model has quite some numberof huge representations (two 24 H , one 75 H and one pairof 50 H and 50 H ), but it should be stressed that no fine-tuning is needed, except the obvious one that createsthe hierarchy √ F ≪ M GUT , needed in all known per-turbative supersymmetry breaking models without lightstates.
Mediation of supersymmetry breaking.
The obviousmediators in all these type of models are the heavy gaugebosons and the adjoints. They can dominate over gravityonly for relatively low M GUT , not much higher than theusual in MSSM. The large number of fields can help forthis purpose. Notice that the potential problem of nega-tive soft mass squared is not necessarily there due to thesubsequent running, as shown recently in [31]. Other pos-sible contributions need the introduction of extra (possi-bly intermediate scale) states, like the usual extra pairsof SU(5) fundamental and anti-fundamentals, or a pairof 15 H and 15 H that can be used also for the neutrinomasses [32]. Non-perturbative contributions.
We have assumed thatthe perturbative part of the superpotential dominates.One could ask, how can the non-perturbative contribu-tions influence the picture. Can one calculate them? Themodels considered are realistic and thus necessarily com-plicated enough to make the usual techniques (use ofholomorphicity, symmetries, etc) hard and probably nonconclusive. Notice that none of the models we presentedis ultraviolet free. The best one can do without furtherwork is to make the most sensitive part of our mecha-nism, i.e. the presence of a U(1) R symmetry, indepen-dent on the quantum non-perturbative corrections. Thiscan be guaranteed by making the U(1) R global symme-try non-anomalous. Of course this depends on the modelchosen. For example, in the non-renormalizable modelwith two adjoints, one needs to add to the usual spec-trum (two adjoint Higgses, a pair of fundamental Hig-gses and three generations of 10 F and ¯5 F matter) alsotwo pairs of (5 i +¯5 i ) chiral multiplets with vanishing R -charge (enforced for example by a term λ i ¯5 i Σ i in thesuperpotential). A general treatment of this issue is veryinteresting, but beyond the scope of this paper. Vacuum metastability.
We have assumed throughoutthe paper that the vacuum lifetime is longer than the age of the universe. This can be checked either with an ex-plicit calculation using the full 1-loop effective potential,or estimated as it is done in [7]. Using the constraint (27)and the method in [7] one finds for such a bounce actionan approximate value of S B ≈ π M GUT / | F | , which ismuch larger than the required value of ≈
500 needed forthe lifetime to be longer than the age of the universe.
Different gauge groups.
We have limited ourselves tothe prototype example of a SU(5) grand unified theory.For many aspects the SO(10) GUT is more successful.Unfortunately the minimal renormalizable version [33]cannot break at the same time the gauge group and su-persymmetry, the reason being the absence of a flat di-rection. An extra problem in such nonminimal groupsis the need for breaking rank, which typically needs anextra fine-tuning.A special role can be played here by partial unifiedgroups, like the Pati-Salam or the Left-Right group. Be-ing possible at lower scales without being necessarily wor-ried about proton decay constraints, they can automat-ically give a low enough supersymmetry breaking scalewithout any fine-tuning. The minimal model with twofields can work in both cases, however, again an addi-tional sector (and additional fine-tuning) is needed inorder to break rank. Of course the whole motivationfor supersymmetry is here less pronounced: no hierarchyproblem because of little or no hierarchy, no one-stepunification because of intermediate scales.
Inflation without singlets.
It is interesting that thesetype of models give possible candidates for a non-singlet(although still MSSM singlet) inflaton. Apart from fewexceptions (for example [34] in MSSM and [35] in a GUT)this would be one of the very few examples of such infla-tons on the market. The simplest model (9) is very simi-lar to the prototype model of F-term hybrid inflation [36].If one is not too ambitious and does not pretend that thesame model describes also supersymmetry breaking, thissimple model could in principle work. In fact, in order toget rid of the unwanted light states, one can think thatthe final state after inflation is in the true minimum, inwhich both adjoints become heavy. Preliminary resultsseem to confirm that inflation can indeed take place, ina similar manner as in the case with singlets introducedin [36]. For example, one can calculate the derivatives ofthe 1-loop potential and find out that the usual require-ments for inflation to happen are satisfied. What wouldbe particularly interesting is to see if there are any dif-ferences in predictions with respect to the case with asinglet. This work is in progress and a detailed analysiswill be presented elsewhere.
Acknowledgements
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