On the F-term problem and quintessence supersymmetry breaking
aa r X i v : . [ h e p - t h ] J a n On the F-term problem and quintessence supersymmetry breaking
Fotis Farakos
Dipartimento di Fisica e Astronomia “Galileo Galilei”Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di PadovaVia Marzolo 8, 35131 Padova, Italy
ABSTRACTInspired by the stringy quintessence F-term problem we highlight a generic contributionto the effective moduli masses that arises due to kinetic mixings between the moduli and thequintessence sector. We then proceed to discuss few supergravity toy models that accommodatesuch effect, and point out possible shortcomings. Interestingly, in the standard 2-derivativesupergravity action there is no term to mediate the supersymmetry breaking from the kineticquintessence sector to the gaugini and generate Majorana masses. Therefore we also propose a2-derivative supersymmetric invariant that plays exactly this role.
Introduction
The existence of de Sitter vacua has been often challenged in string theory [1–3] and one needsto contemplate on viable alternatives. In particular one scenario that has regained attentionis the so-called quintessence phase [4–6]. Quintessence is essentially nothing but a low-scaleinflationary phase, and as such, quintessence is still plagued with some of the problems ofinflation in string theory.However, apart from the challenges that one faces when embedding the quintessence sectorin string theory and achieving moduli stabilization, there is an extra issue that arises whichrelates to the mediation of the supersymmetry breaking to the observable sector. This issuewas highlighted recently in [7], where it was dubbed “F-term problem”, and can be summarizedas follows: Even though the net supersymmetry breaking scale can be (at best) of order TeV (inorder to control loop corrections), the supersymmetry mediation scale is large and gives verylight superpartner masses for the standard model sector. In other words, even though the netsupersymmetry breaking can be TeV, one finds the gaugini masses to be of order 10 − TeV.One could introduce an extra hidden sector (say X ) that also breaks supersymmetry, but thiswill not solve the problem. Indeed, let us assume the quintessence dynamics requires a scalarpotential V quint and that the breaking from X is mediated as usual by a new term in the Kählerpotential of the form δK ∼ α i Λ | X | | Φ i | → α i | F X | Λ | A i | , (1.1)where α i are some parameters that are expected to be of O (1) and Λ is some cut-off (orequivalently a mediation scale). As a result, to have a considerable contribution to the non-supersymmetric masses one will need a large value for h F X i ∼ (TeV) . The latter will then feedinto the supergravity scalar potential giving an extra contribution proportional to δV SUGRA ∼ SUSY breaking ∼ | F X | ≫ V quint , (1.2)which in turn leads to an unrealistic late-time cosmological scenario, or at least spoils the originalquintessence phase that was controlled by V quint . One could cancel this new contribution tothe scalar potential by finding another new contribution to the gravitino mass (recall that V SUGRA = − m / + . . . ), but such resolution posses a challenge as discussed in [7] becauseno new contributions to the gravitino mass seem possible from string theory. Or if they exist,they are going to be sub-dominant.It is important to appreciate that the only consistent way to lower the vacuum energyin N=1 supergravity is either by lowering the net supersymmetry breaking or by increasingthe gravitino mass. Other effects will either fall under these two categories or will includeghosts. This is what makes the F-term problem a very serious issue for quintessence models. A similar issue (but not really of the same nature as the F-term problem) also arises in simple supergravityquintessence models (see e.g. [8]). X -relatednilpotency conditions [11]. However, because the supersymmetry breaking scale from F X is toolow, the goldstino self-interactions remain non-unitary even at rather very low energies (energiescomparable to √ F X ) [12]. This means such a theory will have a very low cut-off (below theTeV scale). Alternatively, one could consider that it is the net supersymmetry breaking (whichis of order TeV) that enters the nilpotency conditions and not only √ F X . It is not clear howrealistic it is to obtain such a scenario from string theory.In this work we want to take a closer look into the quintessence phase, and its supergravityembedding, and see if there is some additional contribution to the scalar masses. We first recallthat the mass splitting between the component fields of a supermultiplet is typically controlledby the overall supersymmetry breaking, therefore one should consider the contributions fromall relevant sectors. Because during quintessence we are on a background where there exists ascalar with non-vanishing kinetic energy, say ˙ φ = 0, then we will have an extra contributionto the supesymmetry breaking. The easiest way to convince oneself that this is the case is bynoticing that 4D N=1 supergravity is equipped with the appropriate goldstino-gravitino mixingterm [13, 14] ∂ m φχ φ σ m σ n ψ m → ˙ φχ φ σ m σ ψ m . (1.3)Such coupling with the gravitino is the signature of a supersymmetry breaking sector, and meansthat ˙ φ is generically bound to contribute to mass-splittings within a supermultiplet or give massto scalar moduli. This also means that the goldstino is not only provided by the multiplet withthe non-vanishing auxiliary field VEV (e.g. F X ) but also gets a contribution from the fermionsuperpartner χ φ of the quintessence scalar. Schematically we conclude that [15]SUSY breaking ∼ “ | F X | ” + “ ˙ φ ” . (1.4)Even though we have established that ˙ φ contributes to the supersymmetry breaking, we areagain faced with a new type of F-term problem because if ˙ φ is too large, then it will endangerthe quintessence phase simply by breaking the slow-roll conditions. This can be also understoodby inserting (1.4) into (1.2). In fact during a slow-roll phase˙ φ ∼ ǫH , (1.5)and therefore the true challenge in these models is to see how realistic a strong mediation termwill be. In other words, we want to see if we can have parametrically large masses (comparedto ǫH ), that are generated now by ˙ φ = 0 instead of F X = 0. As we will see the applicability ofsuch mechanism in quintessence is always model dependent, and it may happen that in modelsderived from string theory such effect cannot truly help ameliorate the F-term problem.2he rest of the article is organized as follows. In the next section we work with a non-supersymmetric gravitational theory and in the third section we turn to supergravity. In bothcases we show that the contribution to the mass of the scalars is model dependent, and wealso discuss few examples where it works and examples where it fails. We also see why suchmechanism (which is intrinsic in supergravity embeddings) may instead pose a catastrophicthreat to otherwise healthy quintessence models, depending on the kinetic mixings. This effectis already studied within the context of inflationary cosmology, because it can ruin inflation,and therefore is dubbed “geometrical destabilization” [16–19]. In the fourth section we turn tothe gaugini masses which are of course also influenced by the F-term problem. We find thatthe term that mediates the breaking from the quintessence sector to the gaugini is not presentin standard supergravity. Then we proceed to construct a new term that does exactly that andwe study its properties. It seems that such term finds its natural place within the new-minimalformulation of supergravity, and in models where the quintessence phase is driven by a reallinear multiplet instead of a chiral. In the fifth section we give a few concluding remarks andan outlook for future work. Before turning to supergravity it is very instructive to work with a non-supersymmetric model.To this end let us consider quintessence driven for simplicity by a real scalar τ slow-rolling downa run-away potential, that is we simply have e − L quint = − R − k ( τ ) ∂τ ∂τ − V . (2.1)Here for later convenience we have also included a kinetic function k ( τ ). On an FLRW back-ground we have ds = − dt + a ( t ) d~x , (2.2)where a is the scale factor and the Hubble scale is H = ˙ a/a . The scalar equation of motion andthe Friedmann equation then read k ( τ )¨ τ + 3 Hk ( τ ) ˙ τ + 12 k ′ ( τ ) ˙ τ + V ′ ( τ ) = 0 , k ( τ ) ˙ τ + V = 3 H , (2.3)where the dot refers to time derivative. One then needs to achieve slow-roll, just as in standardinflation, which requires ǫ = 12 V ′ kV ≪ , η = V ′′ kV − k ′ V ′ k V ≪ . (2.4)Then once the slow-roll conditions are met we have˙ τ ≃ − V ′ Hk . (2.5)3et us assume there is now another scalar in the theory (say ρ ) with canonical kinetic termbut with a kinetic coupling to the quintessence sector of the form k ( τ ) in (2.1) → ˜ k ( τ ) = k ( τ ) − γρ . (2.6)Note that the scalar ρ can also enter the scalar potential V and could also have a mass termdue to other effects. However such scalar gets an extra contribution to its effective mass due tothe kinetic coupling + 12 eγ ∂τ ∂τ ρ → − eγ ˙ τ ρ . (2.7)Therefore this new contribution to the mass has the form δm ρ = γ ˙ τ ≃ γ V ′ V k ≃ γ k ǫV . (2.8)As long as slow-roll holds clearly ǫV is a small number, however the 2 γ/ k term can help givea significant positive contribution to the mass, or reduce it depending on the sign of γ . Inaddition, we can assume that during the quintessence phase ρ | quint = 0 = ˙ ρ | quint such that˜ k | quint = k | quint and so the quintessence phase is left intact. Note that quintessence has acrucial difference compared with inflation: During quintessence it is sufficient to have ǫ . , (2.9)and so if we take roughly V ∼ − , kγ ∼ − , (2.10)we get a mass contribution of order (we restore momentarily M P ) δm ρ ∼ − M P . (2.11)At first sight such small values for k/γ seem to require significant tuning and may seem unreal-istic. Therefore as we said earlier the challenge is to see if such values could be achieved in arealistic model.A model independent discussion could only get us this far, so let us now work with a simpleexample. We set k ( τ ) = 1 τ , V ( τ ) = V τ − / , (2.12)and we remind the reader that we are always working with Planck units, i.e. M P = 1, unlessotherwise noted. Then we can recast the scalar τ into a form with canonical kinetic terms,which means we set φ = lnτ , (2.13)4hich brings the scalar potential for the canonical field φ to the form V = V e − φ/ . (2.14)This scalar potential is typical for quintessence and one can find that the time dependence is φ ≃ ln √ V √ t ! , ˙ φ ≃ H ≃ t . (2.15)Now we can directly evaluate the contribution to the effective mass δm ρ = γ ˙ τ ≃ γe φ ˙ φ ≃ γV (18) H − . (2.16)Taking into account that V H − can take a variety of values we see that the mass contributiondue to the kinetic mixing can be arbitrarily large and can strongly stabilize the scalar as longas γ is positive. In addition the quintessence phase is not broken because the net kinetic energyis still sub-dominant, i.e. we have12 k ( τ ) ˙ τ ≃ ǫH ∼ H ≪ H , (2.17)which means our slow-roll approximations are of course valid.It is interesting to get some feedback from the swampland conjectures to see if such hierarchyhas at least a remote chance of being generated in string theory. The distance conjecture foran inflating theory takes roughly the form [20, 21]∆ φ . λ log UV , (2.18)where Λ UV is some high energy cut-off scale associated to quantum gravity and λ is genericallyassumed to be of order one. This restricts the range of φ but still gives a window that allowsthe V H − fraction in (2.16) to take a variety of different values. Let us assumeΛ UV ∼ TeV ∼ − , λ ∼ , (2.19)which give ∆ φ . . (2.20)Note here that we took the TeV scale (the scale of the superpartners) as the UV cut-off and weset for λ to be one order of magnitude smaller than the generic expectation. Then we can have φ ∼ , V ∼ − , (2.21)which give H ∼ − , δm ρ ∼ √ γ × − . (2.22)5e see that if γ is of order one then such a scenario is marginally within the limits set bythe swampland. Clearly, by changing the behavior of γ we can relax the restrictions from thedistance conjecture even more.In the example we discussed it seems possible that the mass due to kinetic mixing will havea significant positive contribution. This may be true for the specific model we studied but it isstrongly a model-dependent result. Indeed, if for example γ is not a constant, and if it is insteadgiven by γ ∼ k ( τ ) then the contribution to the mass of ρ would be proportional to H and soit would be insignificant. Clearly different choices of γ lead to masses of different magnitude.Another situation is to have γ < ǫ ≪ The bosonic sector of the Lagrangian for chiral scalar superfields coupled to N=1 supergravityhas the form e − L = − R − K ij ∂A i ∂A j − V SUGRA , (3.1)where V SUGRA is the standard scalar potential of supergravity, see e.g. [13, 14], and K is theKähler potential. The A i are the complex scalars that belong to the chiral superfields Φ i .Models and discussions for quintessence in supergravity and in string theory can be found forexample in [8,22–32], a general review focused on string theory models can be found in [33] (andin [7]), and some alternative proposals in string theory can be found in [32, 34, 35]. Thereforewe will not commit ourselves here to finding an appropriate superpotential P (Φ i ) that givesto V SUGRA the required quintessence form. Rather, we will investigate what types of Kählerpotentials can give rise to a significant mass to the scalars due to the kinetic mixing, assumingwe are in a quintessence phase. Since the impact of the kinetic mixing on the mass is highlymodel dependent, we will illustrate the various possibilities with few examples.
Kähler moduli inspired example
Let us first give an example where the effect of the quintessence phase has a different behaviorthan the one we saw in the previous section. We focus on only two complex scalars A and T ,the latter being the quintessence field. We set K = − ln ( T + T − AA/ , (3.2)where T can be some Kähler modulus and A a modulus related for example to the position of aD3-brane (see e.g. [7, 36]). Within a complete string theory setup the K = − . . . would rather6e K = − . . . (for example if we considered Kähler moduli quintessence [25]), but since it doesnot play a significant role we keep the − T are controlled by K T T = 2( T + T ) AA T + T ) + . . . ! . (3.3)To study the dynamics let us split the complex scalar field T as T = τ + iζ , (3.4)with the fields during the quintessence phase given by ζ | quint = 0 , τ | quint = slow-rolling scalar , (3.5)and A | quint = 0 = ˙ A | quint . (3.6)Note that the model we have here gives exactly the same form for the kinetic function as wehad in the previous section k ( τ ) = (1 / τ − . It is then convenient to assume that the dynamicsof τ are similar to the ones we found for the working example in the previous section (i.e. ascalar potential similar to (2.12)) such that τ ∼ V H − , ˙ τ ∼ V H − . (3.7)We can now evaluate the contribution to the effective mass of A due to the kinetic term of T on the quintessence background. From (3.3) we have L kin T ∼ | ˙ T | | A | ( T + T ) ∼ ˙ τ τ | A | ∼ V − H | A | . (3.8)To correctly identify the contribution to the mass of A we have to take into account also thenormalization of its kinetic terms L kin A ∼ − τ | ∂A | ∼ − V − H | ∂A | . (3.9)Therefore after canonical normalization we find a very small negative contribution to the effect-ive mass δm A ∼ − ˙ τ τ ∼ − ǫ H . (3.10)If the scalar A is very light this will lead to a destabilization [16–19], but if there is alreadyanother positive supersymmetric mass (or not supersymmetric) then there is no issue. Notealso that the mass is exactly of the same order as the supersymmetry breaking sourced by thequintessence sector, that is ǫH . 7 different potential can change the impact of such effect and may lead to a significantcontribution to the moduli masses. Indeed, let us assume that we have instead of the scalarpotential in (2.12), a scalar potential of the formˆ V ( τ ) = V τ / , (3.11)but we keep of course the same Kähler potential (3.2). Here the crucial difference is that τ goesto smaller and smaller values as the quintessence phase proceeds. Then we can again recast thescalar τ into a form with canonical kinetic terms by setting φ = − lnτ . (3.12)This brings the scalar potential for the canonical field φ to the formˆ V = V e − φ/ . (3.13)Then we have τ ∼ V − H and thus L kin T ∼ ˙ τ τ | A | ∼ V H − | A | , (3.14)which may give a considerable negative contribution to the moduli masses and completely ruinquintessence, as happens with the geometrical destabilization [16–19]. Note however that oncecanonical normalization is taken into account we will have a mass of the form (3.10) whichseems less dangerous. Still, the H − that comes from the kinetic mixing (3.14) is a hugenegative contribution which has to be matched by another contribution to the second derivativeof the supergravity scalar potential in order to stabilize the scalars.Note that a crude analysis of the fibre quintessence scenario [25] shows that if we includedthe D3-brane moduli as in (3.2) and have T as the quintessence scalar then we would get amass contribution of the form δm ∼ − ǫH . This is of course not an immediate threat as longas there is a sector that contributes to the moduli masses at least as H . Such contribution iswithin reach even if the F-term problem persists.Until now we have seen that the contribution to the moduli masses due to the quintessencesector is of order ǫH , and so rather insignificant. Needless to say that one can add modificationsto the Kähler potential (3.2) such that the contribution from quintessence changes and givesinstead a huge impact to the masses. For example a succinct list of the contributions thatcan enter (3.2) can be found in [7]. However, we checked few simple deformations and foundthat the parameters entering such modifications would require a significant amount of tuningto increase considerably the contribution to the effective mass. Therefore one can wonder if therequired modifications, and in particular the amount of tuning required, would really exist instring theory. We will come back later to this point and give a working example. General expectations
As we have seen the quintessence sector seems to give very small contributions to the effectivemasses of the moduli. In search of possible modifications, let us now see what are the general8xpectations we can have in a supergravity theory. Let us first assume we have a Kählerpotential K ( T + δT, T + δT , A + δA, A + δA ) with a form such that it can be expanded asfollows K = α | δT | + β | δA | − γ | δA | | δT | + . . . (3.15)where the dots are simply higher order terms. The coefficients α , β and γ are of course nothingbut the derivatives of the Kähler potential with respect to the chiral superfields, e.g. α = K T T , β = K AA , and γ = − K T T AA , (3.16)and they are in principle field-dependent. Then the consistency of the kinetic terms will require α ( T, T ) > , β ( T, T ) − γ ( T, T ) | δT | > , (3.17)and we also choose to have γ ( T, T ) >
0. As before, for quintessence we have δT | quint = τ , (3.18)which is a real scalar. Now the kinetic mixing will induce a mass term to the scalar A and oncewe also take into account canonical normalization we have δm A = γ ˙ τ β − γτ . (3.19)For this effect to dominate the mass of the complex scalar A one would essentially ask that γ takes parametrically large values. However, then, the denominator due to the canonicalnormalization would become negative signaling that the A scalar would be a ghost. In otherwords, to guarantee that the denominator is positive one has to ask that β > γτ which can besafely satisfied only when γ is not parametrically large.One can be tempted to take β & γτ in (3.19) such that the denominator remains positivebut approaches zero, in which case the mass becomes arbitrarily large. However such large massshould not be attributed to the quintessence supersymmetry breaking per se, rather it is due tothe singular kinetic term of A . Indeed, one important point we would like to discuss is relatedto the fermionic and the scalar moduli kinetic terms. In principle due to the kinetic mixings wewill have terms of the form − K ij ( τ ) ∂A i ∂A j − iK ij ( τ ) χ j σ m D m ( ω ) χ i , (3.20)which after the field redefinitions may in any case alter the masses. However such terms donot require a specialized discussion here for the following reasons. Firstly, such terms lead to arescaling of the fermions and the scalars of the same supermultiplet in exactly the same way.So as far as the F-term problem is concerned such terms are not so important because theydo not generate a mass splitting per se. Secondly, these terms will influence also the moduli9asses that are generated from the standard scalar potential. Therefore, if these terms do havea significant impact on the masses, then they will also affect the mass splitting coming fromthe F-term breaking. So, again, it is not an effect that should be attributed to the quintessencesupersymmetry breaking per se. Finally, if we want to address the scalar moduli stabilization,then such terms should be in any case taken into account when we evaluate the moduli masses,and so our discussion does not have something extra to add to that. However, we always dohave to check in the end the true effective mass of a scalar, as we have been doing.Our general discussion until now seems to imply that we could never make the kinetic mixinggive a huge positive contribution to the masses in a supergravity setup, and therefore much lessin string theory. However, there is a small detail in the Kähler potential that changes completelythe behavior of such term. In particular we simply have to ask that K AA = β ( A, A ) + 12 ( T − T ) γ ( T, T , A, A ) . (3.21)Then the canonical mass of the scalar A during a quintessence phase (that is Re δT = τ andIm δT = 0), will receive instead a contribution of the form δm A = γ ( τ ) ˙ τ β ( τ ) . (3.22)As a result we see that by making γ arbitrarily large, which we are now at least technicallyallowed to do, we can generate a parametrically large effective mass for the scalar A .The form of the Kähler metric (3.21) implies that in that sector the real part of T has atleast some sort of shift symmetry. In fact the requirements for shift symmetry may becomeeven stronger when one tries to build realistic scalar potentials, i.e. once we introduce a super-potential and check slow-roll, stability, etc. Note also that the form of (3.21) guarantees thatthe quintessence scalar will interact with A only when the derivatives are acting on it. Theseproperties of the quintessence scalar already remind the properties of the axions. In fact it isexactly the derivative couplings of the quintessence sector with the other fields that are neededfor this mechanism to work. We leave a detailed study of the impact of this mechanism onrealistic stringy quintessence models that are based on axions for a future work. Instead nowwe turn to a working example following the strategy we just outlined. An ad hoc working example
Let us now support our previous discussion with a specific example, which is inevitably at thispoint rather ad hoc. We set K = − ln ( T + T ) + AA + 1 M ( T − T ) AA . (3.23)Then we have the kinetic terms e − L kin = − T + T ) ∂T ∂T + 2 M ∂T ∂T | A | − ∂A∂A − ( T − T ) M ∂A∂A + . . . , (3.24)10here the dots stand for terms that are not relevant to us now. As before on an FLRWbackground we have + 2 M ∂T ∂T | A | → − | ˙ T | M | A | , (3.25)which gives rise to a contribution to the effective mass δm A = 2 | ˙ T | M . (3.26)The model we have here gives for the kinetic function the form k ( τ ) = (1 / τ − and so if weworked with a scalar potential similar to (2.12) we would get δm A ∼ ˙ τ M ∼ M V H − . (3.27)We see that in this case we do get a significant contribution to the effective mass of the scalar A , as we anticipated. We stress that the mass (3.27) is truly a product of the supersymmetrybreaking that is sourced from the kinetic energy of the quintessence scalar T . One can explicitlycheck this on the full supergravity action by verifying that the fermion superpartners of A donot get a similar mass term as (3.27). Note that the kinetic terms of A get an extra contributiondue to the term ( T − T ) ∂A∂A ∼ ζ ∂A∂A . However, because in our example here ζ = 0 duringthe quintessence phase, such term vanishes. Let us now turn to the gaugino F-term problem. Until now we have discussed the impactof the F-term problem on the masses of scalar moduli assuming that a similar discussion (atleast qualitatively) will hold for sfermions (scalars that are superpartners of the standard modelfermions). However, one can wonder what happens to the gaugini, which are also influenced bythe F-term problem. Indeed, one needs a term of the form1Λ Z d θXW , (4.1)for some cut-off Λ (or equivalently, a mediation scale), in order to generate standard Majoranamasses of the form F X Λ λ . (4.2)As we already discussed, in the stringy models either Λ is too large or F X is forced to besmall and as a result F X / Λ remains very low [7]. Moreover, we have seen that the quintessencesupersymmetry breaking may instead give a significant contribution to the scalar masses and wewould like to see if it is possible to have a similar effect for the gaugini. However the standard2-derivative supergravity [13,14] is not equipped with a term that gives Majorana masses due tothe quintessence phase. In this section we address exactly this issue. First we see that in rigid11upersymmetry one can construct 2-derivative terms that give rise to gaugini Majorana masses.However, these terms cannot be embedded directly in the old-minimal formulation. Thereforewe turn to an effective field theory approach for their supergravity embedding, but we still askthat they give the 2-derivative terms in the rigid limit. In the new-minimal formulation insteadsuch terms do exist and describe 2-derivative interactions.Note that the supergravity theory does instead contain terms that generate fermionic bilin-ears of the form e − L SUGRA ∼ M P i ( K T − K T ) ˙ τ λσ λ . However such terms do not have thestandard Majorana form as they mix the chiralities, they decouple in the rigid limit, and alsoequivalent terms further contribute both to the masses of the fermions that belong to the chiralmultiplets but also to the gravitino. Therefore we avoid invoking such terms, and we also noticethat they do vanish in our working examples. We believe that they deserve a careful study thatwe leave for future work. Here instead we will construct standard Majorana masses. Tensor multiplets and rigid supersymmetry
Let us start by introducing a supersymmetric interaction that contains up to two derivatives, upto Gaussian auxiliary fields, and gives rise to a gaugino mass on a time dependent background.This term is exactly the missing piece that helps complete the mediation of the quintessencesupersymmetry breaking to matter. It is more convenient to describe such superspace couplingin terms of a real linear multiplet, and then dualize it to derive its form in terms of chiralsuperfields.Let us present the superfields we need. A real linear multiplet is defined in superspace as L = L ∗ , D L = 0 = D L . (4.3)Its component fields are L | = a , D α L | = √ χ Lα , −
12 [ D α , D ˙ α ] L | = σ mα ˙ α H m , (4.4)where H m satisfies the constraint ∂ m H m = 0, which means it is the field strength of a realtwo-form H m = ǫ mnkl ∂ n B kl . Since we want to couple to the gaugini, let us also introducethe standard N=1 gauge multiplet. Its component fields reside in the gauge invariant chiralsuperfield W α = − D D α V , (4.5)and have the form W α | = − iλ α , ( D α W β + D β W α ) | = 2 i ( σ nm ǫ ) αβ F mn , D α W α | = − , (4.6)where F mn = ∂ m v n − ∂ n v m . Here we will work with an abelian gauge vector but our couplingcan be used also for non-abelian. Finally the chiral superfield ( D ˙ α Φ = 0) will be useful laterand has the expansion Φ = A + √ θχ Φ + θ F Φ . (4.7) We use conventions from the book of Wess and Bagger [13]. L med = 1 M Z d θL ( W + W ) = − M Z d θ W ( DL ) + c.c. , (4.8)where in the second equality the D ˙ α L is allowed because it is a chiral superfield in globalsupersymmetry. Our first job is to verify that this is a two-derivative term and that there areno higher order auxiliary field terms. To this end we perform the superspace integral and actwith the superspace derivatives to bring the mediation term to the form L med = 18 M D W | ( DL ) | + 14 M D α W | D α ( DL ) | + 18 M W | D ( DL ) | + c.c. (4.9)From this expansion and from the component field definitions we see that there is not a singleterm that will give rise to more than two derivatives or to more than two auxiliary fields.Moreover, we also see that there are always at least two fermions with no derivatives acting onthem, and there are never more than one derivatives acting on any field. This is crucial becauseit tells us that once we add this term to 4D N=1 gauged chiral models the standard 2-derivativekinetic terms and the scalar potential are left intact. We leave the full component field expan-sion, the investigation of the properties of all the terms in (4.9), and possible generalizationsfor a future work. Here instead we focus on the last term in (4.9) which gives in components L med = − M λ ( ∂ a a + iH a ) + . . . (4.10)where the dots contain the complex conjugate and other fermionic terms. From (4.10) we seethat once the real scalar a or the gauge two-form have a time-dependent profile the gaugini willget an effective Majorana mass term.One can wonder how arbitrary is the coupling we have introduced for the mediation termof the gaugini. As we will show now it is rather unique. As we have discussed until now weare considering backgrounds where either the real scalar a or the gauge two-form get a timedependent profile. As a result h ( ∂ a a + iH a ) i 6 = 0 , (4.11)where with the VEV symbol we refer to the dynamical background. We then recall that h D (cid:16) DLDL (cid:17) i = 2 h ( ∂ a a + iH a ) i 6 = 0 . (4.12)As a result DLDL is a chiral superfield with a non-vanishing VEV for its θ component. Theseproperties suggest that we make the identification X = D ˙ α L D ˙ α L . (4.13)This composite superfield satisfies D ˙ α X = 0 , X = 0 , h F X i 6 = 0 . (4.14)13herefore we have successfully identified a nilpotent goldstino chiral superfield. With thisidentification the term (4.8) takes the standard form (4.1) that corresponds to the mediation ofthe supersymmetry breaking to the gaugini. There is of course further arbitrariness in the exactdefinition of X in terms of DL , but now it is clear that it will be related to the arbitrariness indescribing the goldstino itself in superspace. In the end, it is known that all the formulationsare related to each other [37]. Note that the identification (4.13) was also used in [38] for aslightly different setup, whereas in [39] a similar identification is related to the partial breakingof supersymmetry. In [40] a similar identification is also possible as the supersymmetry breakingis sourced by a modified real linear multiplet. Dualizing to chirals
As we have seen the new coupling has a very simple form in terms of tensor multiplets, butto utilize it in a 4D N=1 supergravity we prefer to recast it in terms of a chiral superfield.We follow this path for two reasons: First, if we couple the term (4.8) to old-minimal N=1supergravity a first inspection shows that we are bound to find non-Gaussian auxiliary fieldsand higher derivatives (these two typically go together). This problem persists also when weturn to the chiral multiplet description of course, as we will see later, so we will in any casetreat the new term only as a perturbation in supergravity, i.e. we will evaluate it on the existingbackground. Second, since we have been working until now with chiral multiplets it is moreconvenient to continue in the same framework. However, we do present the embedding of theterm (4.8) in the new-minimal formulation in a later subsection where we argue that it doesnot contain higher derivatives and the auxiliary fields remain at most Gaussian.Now we recast the mediation superspace coupling in terms of chiral superfields. To dothis we perform the standard chiral-linear duality. Notably, the fact that we can perform thisduality is another indication that there is nothing peculiar with the term (4.8). We considerthe Lagrangian L D = − Z d θL + Z d θL (Φ + Φ) + 1 M Z d θL ( W + W ) , (4.15)where in order to perform the duality we have to pick a kinetic term for L . Note that due to theterm R d θL (Φ + Φ) now L is unconstrained. If we integrate out Φ then L becomes a standardreal linear. Now we vary the real superfield L instead, to get L = (Φ + Φ) + 2 M L ( W + W ) , (4.16)which we solve iteratively as L = (Φ + Φ) (cid:20)
1+ 2 M ( W + W )+ 8 M W W (cid:21) . (4.17)Then we re-insert the solution for L into the dualizer Lagrangian and we find the equivalentLagrangian for Φ. The result is L D = 12 Z d θ (Φ + Φ) (cid:20) M ( W + W ) + 8 M W W (cid:21) . (4.18)14e see that the dual term has a form that looks like a higher derivative term, however, ina hidden way it is not. This is guaranteed because we derived it from a 2-derivative theory.Clearly the two-derivative structure can only be seen after a series of field redefinitions. Theterm that mediates the quintessence supersymmetry breaking to the gaugini is manifest and ithas the form 1 M Z d θ (Φ + Φ) W + c.c. , (4.19)which is the form that we will use in the supergravity theory. Interestingly we see that there isagain a hidden shift symmetry in this coupling, reminding the mediation of the breaking to thescalar moduli. Notice also that if h A + A i = 0 , (4.20)then all the higher order terms in (4.18) will in any case not contribute to the kinetic terms thatwould arise from the superspace derivatives acting on W or its complex conjugate. Therefore(4.18) can be also treated as an effective perturbative expansion around the tree-level interac-tions, and this is in fact how we will treat it in supergravity. Coupling to supergravity
We can now introduce the supergravity term that can mediate the quintessence supersymmetrybreaking to the gaugini. To be compatible with our working example in the previous section wewill use the chiral superfield T for the quintessence sector, and we also preserve the same formfor the mild shift symmetry. To this end we will work with the term L med = 1 M Z d θE ( T − T ) W + c.c. , (4.21)and assume we couple it to a standard gauged 4D N=1 supergravity model. We can conceptuallythink of having included also the higher order contribution 1 /M of (4.18) so that we getthe correct mediation term in the rigid limit, but this will not alter our discussion here insupergravity. First let us recast (4.21) in a more familiar form, that is L med = − M (cid:20)Z d Θ 2 E W D ( T − T ) (cid:21) + c.c. + 2 M (cid:20)Z d Θ 2
E R ( T − T ) W (cid:21) + c.c. (4.22)We will now treat this term in an effective field theory approach and therefore we will assumeit only influences the theory in a perturbative way. To this end we will keep only the leadingcontributions that survive once we set the fields to their background values. In particular, asbefore, we will assume that the complex scalar field T will be given by T = τ + iζ , (4.23)15ith the fields during the quintessence phase to be ζ | quint = 0 , τ | quint = slow-rolling scalar . (4.24)Then on such background there are no bosonic terms arising from (4.22), and the only termswith two fermions are given by e − L med = 2 M ∂τ ∂τ λ + 1 M (cid:18) F mn F mn − − i F mn F kl ǫ klmn (cid:19) ( χ T ) + c.c. , (4.25)where χ T are the fermion superpartners of T . The fact that the terms ( χ T ) are multiplied withD means they get an extra mass contribution if the breaking is sourced also from the gaugemultiplet. Then we perform the standard Weyl rescaling g mn → e K/ g mn , e → e K/ e , λ → e − K/ λ , (4.26)we give to the scalars τ their time-dependent profile, and thus we find the effective masses ofthe gaugini (assuming a trivial gauge kinetic function) δm gaugini = − M e − K/ ˙ τ . (4.27)As we have seen in the previous examples such term can give a significant contribution to theeffective mass. Note of course that we can also include an arbitrary holomorphic function M ( T )in (4.21) to give different properties to the gaugino mass during the evolution of the quintessencephase, and give to the mass (4.27) an extra factor M ( τ ). Such modification will become crucialwhen one focuses on model building, but goes beyond the scope of our work here. Restrictions on the quintessence superpartner
Let us also take the opportunity here to explore if the superspace couplings we are discussing canhave some direct signal to the observable sector, and if such effects lead to restrictions. As wewill see the mediation term (4.21) will impose a very strict phenomenological requirement: thefermion superpartner of the quintessence sector has to decouple from the late-time cosmology.Let us assume that we have a quintessence phase that, as in our working example from thesecond section, leads to h ˙ τ i ∼ V H − ∼ − M P . (4.28)Here we used the values from (2.21) and (2.22). Then if we want the gaugino mass (4.27) to beof order TeV ∼ − M P , we have to set M ∼ − M P ∼ GeV . (4.29)We have assumed e − K/ ∼ M ( T ) that can enter (4.21)16o cancel the e − K/ factor in (4.27). Now, among the various couplings of (4.25), after Weylrescaling, we find the derivative interaction L med = 1 M e − K/ F mn F mn ( χ T ) ˙ α ( χ T ) ˙ α + . . . (4.30)between the superpartner of the quintessence scalar and the standard model gauge bosons.This interaction however is not describing the canonical fermion superpartner of the quintessencescalar. Indeed, it is now important to appreciate that for fields that belong to the same multipletwe are bound to have e − L kin = − k ( τ )( ∂τ ) − k ( τ )( ∂ζ ) − ik ( τ ) χ T σ n D n χ T , (4.31)which means to find the canonical χ T interactions we have to rescale (4.30) with k ( τ ). Thenfrom the consistency of the slow-roll phase one expects to have k ( τ ) ∼ − , (4.32)which means we have to redefine the fermion χ T as χ T ∼ χ T | norm . (4.33)Therefore the true suppression of the higher dimensional operator (4.30) in terms of canonicallynormalized fields is very low, and instead of M , it is˜ M ∼ − GeV . (4.34)Such strong interactions of χ T with the standard model gauge bosons are in sharp contradictionwith the standard model phenomenology. However, the coupling (4.25) itself gives us the answerto this apparent phenomenological shortcoming, because it can give to χ T a large mass. From(4.25), once we integrate out the auxiliary field D, we also find a term of the form L med = − M e − K/ D ( χ T ) ˙ α ( χ T ) ˙ α + . . . (4.35)where now D is the Killing potential related to the gauging, and it is a moduli-dependentfunction. Now, it is realistic to assume that due to some shift in the VEVs of the scalar modulithe function D also gets a VEV, that should be of course very small such that the quintessencephase is not threatened. For example we could have h D i ∼ − M P . (4.36)Then the contribution to the effective mass of the canonically normalized χ T will be δm χ T ∼ − M P ∼ GeV . (4.37)Therefore the fermion superpartner of the quintessence scalar will be very heavy and essentiallydecouple from the late-time cosmological phase. In other words the coupling (4.30) is in fact not17art of the low energy effective field theory. This also means that the supersymmetry breakingin the low-energy effective field theory becomes explicit and not spontaneous.We can also check what type of interactions are generated once the heavy fermion is in-tegrated out. For example, if we do not go to the unitary gauge, we have a mixing withthe gravitino of the form k ( τ ) ˙ τ χ T σ m σ ψ m . Then once we integrate out χ T we find χ T ∼ k ( τ ) ˙ τ σ m σ ψ m M / D . Once we insert this back into (4.30) we find an effective interaction ofthe form F mn ψ l /M which does not pose a threat to the low energy theory as it is suppressed bythe high energy cut-off M . A complete study of the low energy effective theory after integratingout χ T is left for future work. New-minimal supergravity?
Until now we have focused on the old-minimal formulation of supergravity. We would like nowto contemplate on what could be different if one uses the new-minimal formulation instead (fornew-minimal supergravity see e.g. [41, 42]). The duality between the two formulations has beenestablished in [43] but we will see here that some interesting simplifications take place if wediscuss the gaugino mediation term directly in the new-minimal setup. Notably, the dualitybetween new-minimal and old-minimal has been proven in [43] only in the presence of chiraland vector multiplets and without higher derivative terms. Instead from (4.18) we see thatthe chiral superfield version of the gaugino mediation term resembles a higher derivative term,which may be the obstruction to performing the full duality. Such obstruction would mean thatthe new-minimal supergravity version of the gaugino mediation term could not be describedwithin old-minimal supergravity.In the new-minimal formulation of supergravity there is a gauged R-symmetry and so onerequires the chiral superfields that enter the chiral integral R d θ E to have overall chiral weightequal to n=1. The chiral weights of the ingredients we will need for our discussion are n ( L ) = 0 , n ( W ) = 1 , n ( M (Φ i )) = − , (4.38)where L is the real linear superfield, W α is the standard chiral superfield associated to the gaugetheory, and M (Φ i ) is a holomorphic function of the chiral superfields Φ i . The latter are allowedto have arbitrary chiral weight. Now, once we act with a new-minimal superspace derivative onthe real linear superfield we get a weight n ( D ˙ α L ) = 12 . (4.39)The advantage of new-minimal supergravity is that D ˙ α L is also a bona fide chiral multiplet,just as in rigid supersymmetry, therefore we have D L = 0 , D ˙ β (cid:16) D ˙ α L D ˙ α L (cid:17) = 0 . (4.40)As a result we are allowed to introduce a term of the form L new = Z d θ E W M (Φ i ) D ˙ α L D ˙ α L + c.c. (4.41)18he first thing we notice is that, in contrast to old-minimal supergravity, the term (4.41) isin fact Kähler invariant. This happens because in new-minimal supergravity Kähler invariancehas the same form as in the rigid theory, that is (for H chiral) we have Z d θE ˜ K → Z d θE ( ˜ K + H + H ) = Z d θE ˜ K , (4.42)up to boundary terms of course. Here ˜ K is a real function that relates to the Kähler potential ofstandard supergravity, that is K , but is not restricted to have det ˜ K ij > K does indeed correspond to the transformation ˜ K → ˜ K + H + H of˜ K . As a result Kähler invariance is not related to super-Weyl transformations here. The secondobservation is that one could identify M (Φ i ) with P (Φ i ) − , where P (Φ i ) is the superpotential,as long as the gravitino mass is non-vanishing on the background. Finally we see that ondimensional grounds we need [ M (Φ i )] = − . (4.43)Now let us turn to the component field analysis. The chiral density in new-minimal supergravityis E = e n iθσ a ψ a − θθ (cid:16) ψ a σ ab ψ b (cid:17)o . (4.44)Then one can see that expanding (4.41) in components gives a result equivalent to (4.9), and sowill not give rise to terms with more than two auxiliary fields or with higher derivatives. Theproperties of the ingredients involved in such expansion can be found for example in [14, 41, 42].In particular, due to the form of (4.44) the two-fermi terms have again the form (4.9) and giverise to a Majorana mass for the gaugino on a quintessence background L new = e M ( A i )2 λ ( ∂ a a + iH a ) + . . . (4.45)Because again of the structure of the new-minimal supergravity we also notice that we can dothe identification X = D ˙ α L D ˙ α L . (4.46)This is a nilpotent chiral superfield with the properties D ˙ α X = 0 , X = 0 , h F X i 6 = 0 . (4.47)As a result, the term (4.41) is in fact the standard term describing the mediation of the super-symmetry breaking to the gaugini also in supergravity, namely Z d θ E M (Φ i ) X W + c.c. (4.48)From here we can get another indication why in the new-minimal formulation the quintessence-gaugini mediation term is not expected to give rise to higher derivatives. We can keep (4.48)with X un-restricted, and impose that X = D ˙ α L D ˙ α L via a term of the form Z d θ E ZX + 2 Z d θE ZL + c.c. , (4.49)19here Z is a chiral Lagrange multiplier of vanishing chiral weight. From (4.49) once we integrateout Z we get (4.46). The important observation now is that (4.48) and (4.49) are terms that inprinciple belong to the standard 2-derivative supergravity, which means they are not expectedto include higher derivatives or higher order auxiliary fields. We leave a careful study of thefull component expansion of (4.41) for a future work, where one should also couple to a simplenew-minimal supergravity model, integrate out the auxiliary fields and study the dynamics. There has been a renewed interest in the study of quintessence models within string theoryand supergravity. This interest has been sparked from the difficulty to identify controlled deSitter vacua in string theory, and from the various swampland conjectures that restrict de Sitterdirectly [2, 3, 44–57] or indirectly [58–61].In this work we investigated the impact of the quintessence phase on the moduli stabilizationand the induced mass splitting within supermultiplets. We established that there is indeed anadditional contribution to the net supersymmetry breaking that arises due to kinetic mixingsand that it my help in addressing the F-term problem [7]. The effect of such mixing cango either way: in some cases it gives a significant positive mass and helps strongly stabilizethe moduli, in other cases it may be innocuous, and in some instances it may lead to hugetachyonic masses for the moduli and spoil quintessence all together. Therefore one would haveto investigate the impact of these terms independently in each string theory quintessence model.Here instead we have only illustrated the various possibilities with some simple examples in 4DN=1 supergravity. In addition, even if the F-term problem is resolved by some other mechanism,the couplings that we discussed here may in any case play a role in the superpartner masses.We have also presented a specific superspace term that can induce Majorana gaugini masseson a quintessence background and studied few of its properties. We have seen that in thenew-minimal formulation such term takes a very simple form and we have argued that it doesnot lead to higher derivatives (or higher order auxiliary field equations). It would be evenmore surprising if such a completely new term also exists in old-minimal supergravity in away that does not give rise to higher derivatives. Instead, our strategy here was to identifythis term in the rigid limit and then we used it as a perturbation in old-minimal supergravity.However, since the new-minimal formulation of supergravity can consistently accommodate suchterm, one important future direction is to either perform the duality from new-minimal to old-minimal, or otherwise study quintessence directly within new-minimal supergravity. Massivevector multiplets may offer an interesting framework to construct such models and mediate thequintessence supersymmetry breaking [62–65].Finally, we studied the couplings of the quintessence superpartner with matter in a setupwhere the quintessence supersymmetry breaking generates TeV gaugini masses. We found thatgenerically it will have significant interactions with the standard model gauge sector, but, itwill also generically receive a large mass that can be pushed up to the cut-off, and so it will20ecouple from the low energy theory. A careful analysis of the resulting low energy theory andits phenomenological implications is left for future work.
Acknowledgments
I would like to thank A. Hebecker and A. Kehagias for very helpful correspondence and N.Cribiori for discussion. This work is supported by the STARS grant SUGRA-MAX.
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