An ultra-weak sector, the strong CP problem and the pseudo-Goldstone dilaton
aa r X i v : . [ h e p - ph ] S e p An ultra-weak sector, the strong CP problem and the pseudo-Goldstone dilaton
Kyle Allison, ∗ Christopher T. Hill, † andGraham G. Ross ‡ Department of Theoretical PhysicsUniversity of Oxford, 1 Keble RoadOxford OX1 3NP Fermi National Accelerator LaboratoryP.O. Box 500, Batavia, Illinois 60510, USA (Dated: September 20, 2018)In the context of a Coleman-Weinberg mechanism for the Higgs boson mass, we address the strongCP problem. We show that a DFSZ-like invisible axion model with a gauge-singlet complex scalarfield S , whose couplings to the Standard Model are naturally ultra-weak, can solve the strong CPproblem and simultaneously generate acceptable electroweak symmetry breaking. The ultra-weakcouplings of the singlet S are associated with underlying approximate shift symmetries that actas custodial symmetries and maintain technical naturalness. The model also contains a very lightpseudo-Goldstone dilaton that is consistent with cosmological Polonyi bounds, and the axion canbe the dark matter of the universe. We further outline how a SUSY version of this model, whichmay be required in the context of Grand Unification, can avoid introducing a hierarchy problem. PACS numbers: 14.80.Bn,14.80.-j,14.80.-j,14.80.Da
I. INTRODUCTION
In a recent paper [1] we discussed the possibility thatnew gauge singlet fields can have natural ultra-weak cou-plings amongst themselves and to Standard Model (SM)fields. The ultra-weak couplings, ζ i , are protected byshift symmetries that are exact in the limit ζ i → This “classical scale invariance” approach to the Higgsmass is essentially empirical, following from the ex-perimental observation of the low mass Higgs boson.Scale invariance can be viewed as a symmetry of a pure ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] In a field theoretic context, the radiative corrections to the Higgsmass that are quadratically dependent on the loop integral cut-off scale are not physically meaningful as only the renormalized m , the sum of the radiative corrections and the mass counterterm, is measurable. SU (3) × SU (2) × U (1) SM [4], but it would be expected tobe broken in the real world when including GUT, gravi-tational, or any new threshold effects below the scale atwhich the SM couplings are defined ( e.g. , [5, 6]). How-ever, the existence of the fundamental spin-0 Higgs bosonmakes it interesting to examine the possibility that thelower dimension operators, i.e. , the d = 2 renormalizedboson mass terms (and d = 0 cosmological constant), areabsent in the Lagrangian — perhaps as the result of adeeper classical scale invariance of the underlying the-ory. The physical Higgs mass can then be generated byan infrared instability involving new physics.Whether or not the CW mechanism applies to theHiggs boson is a phenomenological question that hasbeen explored in a large number of recent papers [7, 8].However, none of the CW-Higgs models to date have ad-dressed the strong CP problem. We consider this to bean important issue. The usual “invisible” axion solutioninvolves a new SM singlet scalar field S that carries aglobal charge under the Peccei Quinn (PQ) symmetry[9] and develops a very large VEV. Clearly it is impor-tant that the coupling of this field to the Higgs bosondoes not generate an unacceptably large contribution tothe Higgs mass. In this paper, we show that the spon-taneous breaking of the PQ symmetry in an ultra-weaksector via the CW mechanism can lead to an acceptableHiggs boson mass while solving the strong CP problem. The d = 2 and d = 0 terms are special in the sense that if setto zero at a high scale they remain zero in the absence of spon-taneous symmetry breaking, raising the possibility that classicalscale invariance is an emergent symmetry at a high scale. There have been two main suggestions for the natureof the PQ symmetry and the origin of the axion. TheDFSZ axion [10] extends the Higgs sector to include asecond Higgs doublet as well as the complex SM singletscalar field S . The Higgs doublets and the singlet fieldare charged under the PQ symmetry. The KSVZ axion[11] postulates that the Standard Model fields are sin-glets under the PQ symmetry and requires the additionof a “heavy” quark that carries non-zero PQ charge andcouples to S . In both cases the axion is identified withthe phase of S while its modulus is identified with a lightpseudo-dilaton. The origin of a light pseudo-dilaton state can be tracedto the ultra-weak couplings of the S field, which areneeded to avoid generating an unacceptably large massfor the Higgs and to enable CW breaking to generate theEW scale. Such small couplings are natural due to theunderlying shift symmetry of S in the limit its couplingsare zero. As a result, these couplings are multiplicativelyrenormalized in the absence of gravity and there is no un-derlying expectation for their magnitude. Gravitationaleffects will generate S couplings, but these may also besmall due to the shift symmetry. Phenomenologically,the axion acquires its mass via the usual QCD effects m a ∼ Λ QCD /f a , where f a ≡ v s /N DW is the axion de-cay constant for a domain wall number N DW , while thedilaton acquires a mass through mixing with the Higgsof order m s ∼ m h /v s . Indeed, the observation of thepseudo-dilaton together with the axion would provide asmoking gun for this kind of ultra-weak mechanism. II. ELECTROWEAK BREAKING VIA THECOLEMAN WEINBERG MECHANISMA. The DFSZ model
We consider the DFSZ model, which has two Higgsdoublets, H , , whose neutral components couple to theup and down quarks respectively and generate theirmasses. We also include the complex singlet, S , whichcarries only the global PQ charge. The most general clas-sically scale invariant potential for H , and S , consistentwith the PQ symmetry, has the form: V ( H , H , S ) = λ | H | + λ | H | + λ | H | | H | + λ | H † H | + ζ | S | | H | + ζ | S | | H | + ζ S H † H + h.c. + ζ | S | , (1) The large S VEV provides the dominant source of scale breaking,hence the identification of the modulus of S with the pseudo-dilaton. where the fields H , and S are parametrized as H = φ +1 φ √ e iθ /v ! , H = φ +2 φ √ e iθ /v ! ,S = φ s √ e iθ s /v s , (2)with real moduli, φ , φ , and φ s , where h φ i ≡ v , h φ i ≡ v and h φ s i ≡ v s . We simplify the model by tak-ing λ = 0; λ will be generated by gauge interactions,but it remains negligibly small [8]. We also consider theparameter range for which v is small and can be treatedas a perturbation, thereby allowing for an analytic solu-tion to the minimisation conditions. A more completestudy will require a numerical analysis [12].There are two ways in which CW breaking can proceed.In the first, the dominant CW potential term is propor-tional to λ and the interaction with the second Higgsfield drives the quartic coefficient of the first Higgs fieldnegative at some scale. This limit is equivalent to thatstudied in [8]. This requires such a large λ that thereis a Landau pole in the ∼ λ is negligibleand EW breaking is triggered by the VEV of φ s . The H mass squared is then ζ v s and is driven negative byassuming ζ < ζ i in eq(1). The VEV v s gives the ax-ion decay constant f a = v s / N DW = 6 in this model)and hence 2 × GeV . v s . GeV. The singletcouplings ζ , , must therefore be very small: ζ , , ≤ O ( m h /v s ), where m h is the observed Higgs mass. ForCW breaking to proceed, it is necessary for ζ to be evensmaller: ζ ≤ O ( ζ , , ). As mentioned above, this regionof parameter space is natural since the couplings ζ i areforbidden in the shift symmetry limit, S → S + δ [1],and thus are multiplicatively renormalised. The strongerconstraint on ζ is consistent with radiative corrections,as can be seen by noting the couplings are also forbiddenby a partial scale symmetry S → λS , where ζ , , scaleas λ while ζ scales as λ . If the symmetry is broken(perhaps by gravity) by a term scaling as λ , the relativeordering of ζ results.Even though the S couplings are all extremely small,CW breaking in the S sector is still possible. It is conve-nient to consider the phenomenologically relevant limitin which the term proportional to ζ provides the dom-inant CW term. It is in this limit that the additionalHiggs states coming from the second Higgs doublet areheavy enough to have escaped detection to date [13]. Inthis limit the potential, including the dominant one-loop We treat the effect of the ζ term perturbatively as it drivesthe H VEV, which we have assumed to be in the p erturbativeregime. correction, can be written as: V ( φ , φ s ) ≈ λ φ + αφ s ) + 164 π (cid:0) ζ φ s (cid:1) (cid:20) ln (cid:18) ζ φ s M (cid:19) − (cid:21) , (3)where α = ζ /λ , M = 2 M e − π ( ζ − λ α / /ζ , and M is the scale at which the couplings are defined. Thishas a minimum at (minimization of a similar single Higgspotential is discussed in [1]): v s = eM ζ , v = − αv s . (4)Finally, v is driven by the term proportional to ζ ineq(1), giving: v ≈ − ζ ζ v . (5)In the region of parameter space considered here, mix-ing between states is small and the observed Higgs h isapproximately φ . Similarly, the other neutral Higgs H and the pseudo-dilaton are approximately φ and φ s , re-spectively. Then a straightforward calculation gives: m h ≈ − ζ v s ≈ λ v ,m H ≈ ζ v s / ,m s ≈ ζ v s / π . (6)Determining the charged Higgs masses is more subtleas they only acquire mass via the term proportional to ζ in eq(1). This happens because all the other terms arefunctions of | H | and | H | , so: ∂ V /∂φ +1 , ∂φ − , ∝ ∂V /∂φ , = 0 . (7)As a result, we find the “uneaten” charged Higgs statehas a mass: m H ± ≈ − ( v /v )( ζ / v s = ζ v s / . (8)Finally we turn to the phases of the fields. One combi-nation, θ Z ∝ θ v + θ v , (9)provides the longitudinal component of the Z boson. Anorthogonal combination given by: θ A = ( − θ /v + θ /v + 2 θ s /v s ) /N, (10)where: N = 1 /v + 1 /v + 4 /v s , (11)gets mass from the ζ term. Its mass is given by: m A = − ( ζ / v s v v N ≈ − ( ζ / v s v /v = ζ v s / . (12) The orthogonal state to θ Z and θ A is the axion. Theaxion only gets its mass from QCD effects, as usual.To summarise, we have ( ζ < m H = m H ± = m A = − ζ ζ m h ,m s = ζ π m H = − ζ π ζ m h . (13) B. The KSVZ model
In the KSVZ model, the SM states are PQ singlets.However, the SM singlet field S interacts with some newheavy quark X L,R , which is vector-like with respect tothe SM gauge group but carry PQ charge, via the Yukawainteraction L KSV Z = − f X L SX R − f ∗ X R S † X L . (14)Imposing classical scale invariance, the scalar potentialhas the relatively simple form V ( H, S ) = λ | H | + η | S | | H | + η | S | , (15)where H is the SM Higgs doublet.Following from the non-observation of additionalcoloured states up to the TeV range and the need to keepthe Higgs light, one sees from eqs.(14) and (15) that thelargest coupling to the S field is f and the associateddominant loop correction to the S potential involves thenew heavy quark. As a result, the loop correction con-tributes to the potential with a relative minus sign com-pared to that of eq(3) in the DFSZ case. This does not give rise to one-loop EW breaking because, if it triggersEW breaking, it drives the Higgs VEV to an unaccept-ably large scale. Avoiding this problem requires an addi-tional CW radiative correction with the opposite sign todominate. Such a term could arise if there are additionalSM singlet fields. It could also possibly be engineered atthe two-loop level by fermion loops, similar to a modeldiscussed in [8]. We do not explore these possibilitiesfurther here. III. PHENOMENOLOGICAL IMPLICATIONS
The DFSZ model requires the extension of the SMspectrum to include a second doublet of Higgs fields anda complex singlet S which contains the axion a and thepseudo-dilaton φ s . The ultra-weak couplings ζ i ensurethat for collider experiments the phenomenology of themodel is just that of the Type II two Higgs doubletmodel (2HDM) with the common mass scale of the addi-tional Higgs states { H, A, H ± } determined by the ratio R ≡ m H /m h ≃ p ζ / | ζ | . In the 2HDM, additionalHiggs states with masses of roughly 350 GeV or above,which corresponds to R &
3, are allowed in significantregions of parameter space [13]. At the same time, an approximate upper bound R . O (600 GeV).In the usual implementation of the DFSZ model, thepseudo-dilaton φ s is very heavy with a mass of O ( v s ).The novel feature of the model discussed here is that φ s is very light. From eqs(4) and (6), we have: ζ = − m h v s ≈ − . × − (cid:18) GeV v s (cid:19) , (16)and hence from eqs.(4) and (13): m s = − ζ π (cid:18) m H m h (cid:19) m h ≃ (cid:18) GeV v s (cid:19) (cid:18) R (cid:19) eV . (17)Since the pseudo-dilaton is light and couples to quarksthrough its mixing with the SM Higgs, one way to detectit is through fifth force experiments. However, using theestimate for the coupling strength of the pseudo-dilatonto protons: α ∼ π (cid:20)(cid:18) m u + m d v s (cid:19)(cid:21) (18)and ∼ /m s for the effective range of the dilaton ex-change force, it can be seen that the pseudo-dilaton liesoutside the region excluded by Casimir-force and neutronscattering experiments [14].The axion couples to electromagnetic fieldsthrough the axial anomaly in the usual way, ∼ c ( a/v s )( α/ π ) F µν e F µν . Likewise, the dilaton couplesas ∼ c ′ ( φ s /v s )( α/ π ) F µν F µν , with c, c ′ ∼ O (1). Adetailed analysis of the detectibility and limits from theelectromagnetic coupling for the dilaton goes beyond thescope of this paper. It is possible that future terrestrial“5th-force”, nuclear and RF-cavity experiments canbe devised to look for the pseudo-dilaton directly, butthis remains unexplored. It is possible that futureterrestrial experiments can be devised to look for thepseudo-dilaton directly, but this remains unexplored. Atpresent, the only way to constrain the pseudo-dilaton isthrough its cosmological influences, which we turn to adiscussion of now. Note that the convention in studying the Type II 2HDM is tohave H couple to the up-type quarks rather than H . Thereforewhen applying 2HDM limits to this model, one should use thedefinition tan β ≡ v /v rather than the usual tan β ≡ v /v . IV. COSMOLOGY OF THE PSEUDO-DILATON
If the S field acquires its VEV before inflation, theenergy density of the pseudo-dilaton will be diluted away.This is the case if the dilaton mass is larger than theHubble parameter during inflation, which requires a lowscale of inflation: V / . (cid:18) GeV v s (cid:19) / R GeV , (19)and if the reheat temperature is sufficiently low such thatthe PQ symmetry is not restored after inflation. On theother hand, if the PQ symmetry breaking occurs afterinflation, there will be energy stored in the dilaton po-tential that will be released after inflation (the Polonyiproblem [15]) in the form of dilaton oscillations. We shallconsider both cases in turn, starting with the latter case. A. High scale inflation
The energy stored in the dilaton potential depends onthe initial value (VEV) of the dilaton. For the case thatthe Hubble parameter during inflation is much largerthan the dilaton mass, the dilaton will perform a ran-dom walk of step length H inf / π in each Hubble time.The maximum dilaton energy corresponds to the largestinitial value of h φ s i , which in turn corresponds to thecase of the maximum Hubble parameter during inflation, H inf ∼ GeV, consistent with the BICEP2 result [16].To be conservative, let us consider this extreme case sinceall others will have a smaller amount of energy stored inthe dilaton and will be more weakly constrained. For 70e-folds of inflation, one may expect the initial value ofthe dilaton field to be given by h φ s i i ∼ GeV.After inflation and reheat, h φ s i begins to oscillate whenits effective mass becomes larger than the Hubble param-eter. In the presence of a thermal bath, φ s obtains a largethermal mass [17] m s ,th ≃ ζ T , (20)where we have neglected all but the largest coupling of φ s to thermalized particles. For a sufficiently high re-heat temperature, the thermal mass eq.(20) dominatesthe dilaton potential and the dilaton oscillates about azero VEV when it begins to roll. The roll begins when m s, th ∼ H , corresponding to the temperature T roll ≈ × R (cid:18) GeV v s (cid:19) GeV . (21)As the universe expands, the energy density in the dila-ton at the beginning of the roll, ρ s, roll ≃ ζ T h φ s i i , (22)redshifts as radiation, i.e. ∝ T [17]. This is faster thanthe matter redshift that one might expect because thetemperature-dependent thermal mass also redshifts. Asthe temperature drops down to T ∼ R GeV, the ther-mal mass of the dilaton becomes comparable in size tothe 1-loop term in the potential eq.(3) and the minimumat h φ s i = v s appears. However, the tunnelling rate to thetrue vacuum is low and the dilaton continues to oscillateabout a zero VEV. The dilaton oscillates about a zero VEV until the EWsymmetry is ultimately broken at the temperature whenQCD becomes non-perturbative and drives the quarkcondensate, which in turn gives masses to the W andZ bosons as well as the Higgs. Once the temperaturedrops below the masses of these bosons, the stabilizingthermal mass term for the dilaton rapidly vanishes dueto the Boltzmann suppression [17] and the dilaton rollsto its true minimum at v s .After the temperature has dropped below about T ∼ R GeV and until the EW symmetry is broken, theenergy density of the universe is dominated by the po-tential energy in the Higgs and dilaton fields. This willgive rise to a period of thermal inflation with roughlyln(10 R GeV /
200 MeV) ∼ O (1), the re-heating is efficient and gives a reheat temperature of T reh ∼ R GeV. Meanwhile, the potential energy in thedilaton, ρ s ≃ ζ v s π ≃ R m h π , (23)is released as a coherent oscillation of the field that red-shifts as matter, i.e. ∝ T .This energy density is large enough that it will quicklydominate the energy density of the universe, thereby ru-ining late-time cosmology, unless it is somehow dissi-pated. This indeed happens because of a resonant en-hancement of the scattering rate of the coherent state ofzero momentum oscillating dilatons on the thermal back-ground.To illustrate this consider the process s + c → c → SM states involving the scattering of the dilaton off thedistribution of charm quarks. Since the dilaton massis so small, the intermediate c is nearly on-shell andits propagator is dominated by its thermal width Γ c ≃ G F m c / (192 π ). Since this width is small, there is an en-hancement of the scattering rate that leads to a thermal The amplitude of the oscillations scales ∝ T from its initial value h φ s i i at T roll . At T ∼ R GeV, the amplitude of the oscillationsis also too small to reach the minimum at v s . Here we neglect the finite temperature corrections and use the dissipation rate of the dilaton given by [18]Γ s ≃ √ m c π / v s Γ c (cid:18) Tm c (cid:19) / e − m c /T . (24)This rate exceeds the Hubble expansion rate over somerange of temperatures T ∗ < T . m c for v s . × GeV. Thus the dilaton oscillations are dissipatedfor all v s of interest.Note that the inverse dilaton production processes,such as g + q → q → q + s where g is a gluon and q is a thermalized quark, do not have a resonant en-hancement because none of the reactants are zero mo-mentum coherent states. Due to the low reheat temper-ature T reh ∼ R GeV, the number density of the topquark is exponentially suppressed and it is the bottomquark scattering that gives the largest rate of dilatonproduction. An estimate of this rate for T & m b is:Γ prod s ≃ ζ (3) π (cid:18) m b v s (cid:19) α s T, (25)which produces a dilaton population: n s n eq s ∼ Γ prod s H (cid:12)(cid:12)(cid:12)(cid:12) T = m b ∼ . (cid:18) × GeV v s (cid:19) . (26)If the dilaton is sufficiently long lived, it is non-relativistictoday with an abundance:Ω s ∼ . (cid:18) R (cid:19) (cid:18) × GeV v s (cid:19) . (27)To constitute dark matter, however, the dilaton must bestable on cosmological timescales. The dominant directdecay mode of the dilaton is to two axions with the decayrate: Γ s → aa = 132 π m s v s ≃ R m h √ π v s , (28)giving a lifetime: τ s ≃ . × (cid:18) R (cid:19) (cid:18) v s × GeV (cid:19) sec . (29)Constraints on decaying dark matter require the lifetimeto be on the order of 100 Gyr (3 × sec) or longer [19].Thus for v s ∼ × GeV for which a significant dilatonpopulation is produced, the dilaton is unstable on cos-mological time scales and cannot be dark matter. Con-versely, for v s & × GeV for which the dilaton issufficiently stable, dilaton production is negligible.The axion, however, provides a very plausible cold darkmatter candidate. The energy density in the coherent zero temperature width. This is valid for temperatures T . m c ,at which the dissipation rate is sufficiently large anyway. oscillations (zero mode) of the axion through vacuum re-alignment is [20]:Ω a h ≃ . θ i f ( θ i ) (cid:18) v s /N DW GeV (cid:19) / , (30)where N DW = 6 for this model, θ i is the ini-tial misalignment angle, and the function f ( θ i ) = (cid:2) ln( e/ (1 − θ i /π )) (cid:3) / encodes the anharmonic effect.Meanwhile, the higher momentum axion modes and theaxions produced in the decay of strings and domain wallscontribute a comparable amount to the energy density asvacuum realignment [21]. For v s ∼ GeV, the axioncan therefore provide all of the dark matter.There remains the question of how an unacceptableenergy density from the domain walls produced after thePQ breaking transition can be avoided. Since N DW = 6,the energy density in stable domain walls is many ordersgreater than the critical energy density for closing theuniverse and completely unacceptable. However, smallPQ breaking can cause the walls to decay and henceavoid the problem while preserving the axion solutionto the strong CP problem. To see how this can happen,we note that the most general potential V ( H , H , S ) in-cludes the terms λ ( H † H ) + ζ S H † H + ζ S + h.c that break the PQ symmetry and splits the degeneracy ofthe Z ( N DW ) discrete symmetry that leads to the domainwall problem. Note that these couplings multiplicativelyrenormalise and so, following the discussion above, weconclude they can naturally be arbitrarily small. Indeed,with PQ breaking of a similar magnitude to scale break-ing ( λ ∼ ζ , , ) these terms are in the range needed tosolve the domain wall problem without disturbing theaxion solution of the strong CP problem [22].In summary, the large thermal mass of the dilaton pro-duces a period of thermal inflation with approximately 5e-folds after the usual slow roll inflation. For all v s ofinterest, the interactions of the light dilaton with thethermal bath dissipate the energy in its coherent os-cillations. A significant relativistic population of dila-tons is produced in the region of parameter space with v s . × GeV, but the dilaton is too short-lived inthis region to be dark matter; the dilaton can thereforeonly have a negligible contribution to dark matter. Theaxion, however, can comprise all of the dark matter for v s ∼ GeV.
B. Low scale inflation
In the case that eq.(19) is satisfied and the reheat tem-perature is sufficiently low ( T reh .
100 GeV) that thedilaton does not obtain a thermal mass that forces it toroll to v s = 0, the PQ symmetry remains broken duringand after slow roll inflation. As a result, the energy den-sity in the dilaton oscillations are driven exponentiallysmall and the axion field gets homogenized by the expan-sion of the universe during the inflationary phase, thereby preventing the formation of domain walls [21]. The usualresult eq.(30) for the axion contribution to the energydensity from vacuum realignment still holds and the ax-ion can provide all of the dark matter for v s ∼ GeVand θ i ∼ The dilaton, however, never plays a signifi-cant role in cosmology.
V. AN ULTRA-WEAK DFSZ SUSY MODEL
Classical scale invariance of the low-energy theory doesnot apply if there are heavy states coupled to the Higgs,such as Grand Unified states, with mass below the scaleat which the SM couplings are defined. These introduceradiative contributions to the Higgs mass that are pro-portional to the mass of the heavy GUT states. Unlikethe radiative corrections to mass simply proportional tothe cut-off scale, these GUT corrections involve a loga-rithmic dependence on the scale at which they are mea-sured and thus are physical. To avoid the hierarchy prob-lem, the model discussed above must therefore not havea stage of Grand Unification.It is possible to include a stage of Grand Unificationin a scale invariant theory by super-symmetrizing themodel so that the contribution to the Higgs mass comingfrom interactions with the heavy GUT states, althoughpresent, are acceptably small. As we sketch below, CWbreaking in the ultra-weak sector associated with the ax-ion can readily be extended to a supersymmetric theory.The states of the DFSZ model neatly correspond tothe non-SUSY states of the ( N =1) NMSSM, so a super-symmetric version of the model can be constructed easily.After imposing the PQ symmetry, the allowed couplingsare more restricted than those of the NMSSM and corre-spond to those recently discussed in [23]. The only termin the superpotential W involving the S field is: W = ζ ˆ S ˆ H ˆ H , (31)where the scalar components of the super fields ˆ S andˆ H , are the S field and the Higgs doublets.Due to the constraints of supersymmetry, the model isclassically scale invariant in the absence of SUSY break-ing. We are interested in the case that ζ is ultra-weak,which is natural due to the underlying shift symme-try when ζ is zero. Allowing for SUSY breaking, theonly other terms involving just these fields are the soft The contribution from the quantum fluctuations of the axionfield during inflation, which are included by making the replace-ment θ i → θ i + σ θ in eq.(30) where σ θ ≃ ( H I / πf a ) [20], arenegligible for H I satisfying eq.(19). As discussed in [20], with moderate fine-tuning to give θ i ≃ π ,the axion can provide all of the dark matter for smaller values v s due to the anharmonic effect. terms: V ( H , , S ) = m s | S | + m | H | + m | H | + T cl SH H . (32)The quartic scalar terms coming from eq.(31) are pos-itive semi definite, so the only possibility for dynamicalSUSY breaking is through the soft terms. Including ra-diative corrections, m s can be driven negative by radia-tive corrections proportional to ζ m , . This triggers v s at a scale close to the point at which the mass is zero,which can be very large, as required. However, this re-quires that the starting value of m s should be ultra-smallrelative to m , . An ultra-small mass is natural if there isan underlying shift symmetry, which can readily happenif, for example, SUSY is broken in a hidden sector andSUSY breaking is communicated to the S field by grav-itational effects while the SM states receive their SUSYbreaking masses via gauge mediation. In this case, thesoft S mass and the graviton will be much lighter thanthe SUSY breaking masses m , in the visible sector. Thedimensional transmutation mechanism in the UW sectorprovides an economical and elegant origin for the axiondecay constant that does not require the inclusion of anO’Raifeartaigh term involving an explicit mass scale [23].The SUSY phenomenology of the model is essentiallythat of the minimal supersymmetric SM, theMSSM (withgauge mediation) because the additional couplings of theHiggs to the singlet sector are ultra-weak and hence in-significant, apart from providing the origin of the µ termof the MSSM, µ = ζ v s , c.f. eq(31). In this case, EWbreaking proceeds in the usual way through radiative cor-rections that, due to the top Yukawa coupling, drive thesoft Higgs mass squared negative [24–26].The LSP is the axino, the fermion component of S ,with a mass: m ˆ S = µv v v s ∼ − (cid:18) GeV v s (cid:19) eV (33)generated by the see-saw mechanism through its couplingto the Higgsinos. The decay of the lightest MSSM SUSYstate to the gravitino or axino is so slow that it does notoccur within the detector and does not change the MSSMphenomenology. The dark matter component that endsup in the axino depends on the MSSM parameter choiceand has been discussed extensively elsewhere.Due to the quartic couplings associated with the super-potential term in eq(31), the Higgs obtain S dependentmasses as in the non-supersymmetric DFSZ model. As aresult, Higgs oscillations are driven by the dilaton oscil-lations in the manner discussed above. The energy in thedilaton fields is converted to energy in the SM sector ata time before nucleosynthesis and does not significantlychange the usual MSSM cosmology. Soft terms can be generated in a classically scale invariant theorythrough spontaneous breaking, for example via gaugino conden-sation.
VI. SUMMARY AND CONCLUSIONS
The discovery of a Higgs scalar with properties veryclose to that predicted by the SM, together with the ab-sence of any indication for physics beyond the SM, has ledto a re-evaluation of the need for such physics to solve thehierarchy problem. Formally, as a pure field theory, theSM has no hierarchy problem because the radiative cor-rections to the Higgs mass squared that are quadraticallydependent on the cut-off are not physical; only the renor-malised mass is measurable, so any value of m is possibleand only the empirical choice m = 0, which correspondsto classical scale invariance of the theory, is special.With this motivation, we discussed how the SM couldresult from a classically scale invariant theory that alsoaddresses the major questions left unanswered by the SM.While there has been extensive discussion of the possibleorigin of baryogenesis, dark matter and inflation, very lit-tle attention has been paid to the the strong CP problemin such theories. In this paper, we showed how a scaleinvariant version of the DFSZ model can spontaneouslygenerate the large PQ scale through an ultra-weakly cou-pled sector involving a complex SM singlet scalar field S .As discussed in [1], such an ultra-weak sector involvinggauge single fields is technically natural due to an under-lying approximate shift or scale symmetry.Due to the ultra-weak couplings, the DFSZ extensionof the SM contains an anomalously light pseudo-dilatonas well as the usual axion, which come from the complexscalar S . Despite the ultra-weak couplings, there is noPolonyi problem associated with the S field due to a res-onant enhancement of the scattering of the coherent S state off the thermal background after the PQ and EWphase transition are triggered. Unusually, the PQ phasetransition occurs at the EW scale in this model. Mean-while, the dilaton production cross section does not havethe resonant enhancement and its abundance is typicallynegligible. Dark matter can be in the form of axions pro-duced via a combination of vacuum alignment and decayof axion domain walls. Such decay is possible due to ad-ditional PQ breaking terms, which can be consistent withthe axion solution to the strong CP problem as long asthey are also ultra weak and have a strength comparableto the scale breaking terms.Due again to the ultra-weak couplings of the sin-glet fields, the phenomenology of the model is that ofthe usual two-Higgs doublet extension of the SM. Themost significant constraint on the additional heavy Higgsstates comes from the requirement that the little hierar-chy problem is not re-introduced. It may be possible tosearch for the ultra-light dilaton along the lines suggestedin [27], but this remains to be studied.Finally, we outlined the construction of a scale-invariant SUSY version of the model that can accommo-date a stage of Grand Unification without re-introducingthe hierarchy problem. It provides a simple origin forthe µ term and the LSP is the axino, the fermion compo-nent of the super field that contains the DFSZ complexscalar field S . However, since the decay of the lightestMSSM state to the LSP is extremely slow, the colliderphenomenology of the model is just that of the MSSMwith gauge mediated SUSY breaking. Acknowledgements
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