PPrepared for submission to JHEP
DESY 17-228 MITP 17-103
A warped relaxion
Nayara Fonseca, a Benedict von Harling, a Leonardo de Lima b and Camila S. Machado c a DESY, Notkestrasse 85, 22607 Hamburg, Germany b Universidade Federal da Fronteira Sul, Av. Edmundo Gaievski 1000, 85770-000 Realeza, Brazil c PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz, Germany
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We construct a UV completion of the relaxion in a warped extra dimension.We identify the relaxion with the zero mode of the fifth component of a bulk gauge fieldand show how hierarchically different decay constants for this field can be achieved bydifferent localizations of anomalous terms in the warped space. This framework may alsofind applications for other axion-like fields. The cutoff of the relaxion model is identifiedas the scale of the IR brane where the Higgs lives, which can be as high as 10 GeV, whileabove this scale warping takes over in protecting the Higgs mass.
Keywords:
Relaxion, Warped Space, Hierarchy Problem a r X i v : . [ h e p - ph ] S e p ontents The traditional paradigms to approach the hierarchy problem of the Standard Model requirenew physics close to the electroweak scale, attributing the smallness of the Higgs massto a symmetry protection (e.g. supersymmetry) or to the lowering of the cutoff of thetheory (e.g. technicolor). This class of solutions has been a guide to model building ofphysics beyond the Standard Model for many years and one of the leading motivationsof searches for new physics at the LHC. An alternative possibility does not predict newphysics at the TeV scale, but instead requires multiple vacua with a large range of possiblevalues of the Higgs mass and a selection mechanism such that we end up in the vacuumwhere the Higgs is light. Recently, a new dynamical selection mechanism was proposed,the cosmological relaxation of the electroweak scale [1] (see also [2–12]). It relies on thescanning of the Higgs mass parameter by a new field, the relaxion, and a back-reaction– 1 –echanism that is triggered when the vacuum expectation value (VEV) of the Higgs hasreached the electroweak scale, making the relaxion evolution stop. This is a radical changeof paradigm as it implies that the naturalness problem of the Standard Model ceases to bea reason to expect new physics close to the TeV scale.In what follows we review the relaxation mechanism for which an axion-like scalar φ is introduced which couples to the Higgs doublet H via the potential V ( φ, H ) ⊃ − (cid:0) Λ − g (cid:48) Λ φ (cid:1) H + λ H + g Λ φ + Λ f ( H ) cos (cid:18) φf (cid:19) . (1.1)Here Λ is the cutoff which sets the Higgs mass parameter, f the decay constant of therelaxion, λ the Higgs quartic coupling, g and g (cid:48) are small dimensionless couplings, andΛ f ( H ) is a scale which depends on the Higgs VEV. Assuming a classical time evolutionwith slow-roll conditions, the second-last term in Eq. (1.1) causes the relaxion to movedownwards following its potential. The effective Higgs mass parameter in the φ background,the first term in parenthesis in Eq. (1.1), then varies accordingly. The relaxion is assumedto start with a VEV such that this mass parameter is initially positive. Due to the evolutionof the relaxion, the mass parameter then eventually turns tachyonic, triggering electroweaksymmetry breaking. In the presence of a Higgs VEV, the oscillatory barrier from the lastterm grows, until its slope matches the slope of the linear term. For technically naturalparameters in the potential, this causes the relaxion to stop once the Higgs VEV has reachedthe electroweak scale. There must be some mechanism to dissipate the kinetic energy ofthe relaxion during its evolution such that the field does not overshoot the barriers. If thedynamics happens during a period of inflation, Hubble friction can provide the dissipationnecessary to slow down the field [1]. As an alternative to inflation, one can also considerfriction due to particle production as proposed in Ref. [14] or finite temperature effects inthe early universe as in Ref. [15].Note that the linear terms in φ are in conflict with the assumption that the relaxionis a pseudo-Nambu-Goldstone boson as they explicitly break the axion shift symmetry [5].This may be reconciled if the linear terms arise from a second oscillatory potential with aperiod much larger than f . This is realized if the potential takes the form [16–18]: V ( φ, H ) ⊃ − Λ H + λ H + Λ F ( H ) cos (cid:18) φF (cid:19) + Λ f ( H ) cos (cid:18) φf (cid:19) , (1.2)where F (cid:29) f is another decay constant and Λ F ( H ) another scale that depends on the Higgsin such a way as to reproduce the second and fourth term in Eq. (1.1) after expanding in φ/F . An interesting possibility to obtain this type of potential is the clockwork constructionwhich was first realized for axion-like fields in Refs. [16, 17] and generalized for applicationsother than the relaxion in Ref. [26]. Further developments regarding the 5D continuum See also N naturalness [13], where instead of multiple vacua, many copies of the Standard Model areconsidered to explain the smallness of the electroweak scale. The way reheating behaves is such that onlythe copy with the smallest Higgs mass is efficiently reheated. See also Refs. [19–24] for similar earlier ideas in inflation model building. For the viability of therelaxation mechanism in string theory in the context of axion monodromy, see Ref. [25]. – 2 –imit of the clockwork can be found in Refs. [27–30]. Besides the clockwork, one canalso generate a potential of the form in Eq. (1.2) in realizations inspired by dimensionaldeconstruction [31, 32], as in Ref. [18].In this work, we show how the required potential for the relaxation mechanism towork can be naturally obtained by embedding the relaxion and Higgs into a warped extradimension. We consider a slice of AdS space which is bounded by two branes, as in theRandall-Sundrum model [33]. However, in our setup the IR scale or warped-down AdSscale is not of order TeV but can be much larger. We introduce a U (1) gauge field in thebulk of the extra dimension and break the gauge symmetry on the two branes. The 5thcomponent A of the gauge field then gives rise to one massless scalar mode in 4D whichwe identify with the relaxion. In order to generate a potential, we introduce anomalouscouplings of A to two non-abelian gauge groups. The wavefunction of the massless modefrom A is exponentially peaked towards the IR brane (see e.g. [34–36]). Depending onwhere the anomalous terms are localized, this can yield a large hierarchy between thedecay constants for the couplings of the relaxion to the gauge groups. We assume thatthe gauge groups confine at energies below the compactification scale. Instantons thengenerate periodic potentials for the relaxion as in Eq. (1.2) with periods given by the decayconstants. Due to the warping, these periods can thus naturally be hierarchically differentas required. We embed the Higgs at or near the IR brane. Its mass parameter is thennaturally of order the IR scale which we identify with the cutoff of the relaxion theory.The required Higgs-relaxion couplings can be obtained by introducing fermions on theIR brane with higher-dimensional or Yukawa couplings to the Higgs. To summarize, thewarping does two things: Firstly, it generates the hierarchy between the decay constants F and f in Eq. (1.2) and thereby explains the smallness of the couplings g and g (cid:48) in Eq. (1.1).Secondly, it provides a UV completion for the relaxion. The relaxation mechanism protectsthe Higgs up to the IR scale above which warping takes over. We illustrate this in Fig. 1.Alternatively, one can think of the relaxation mechanism in our construction as a solutionto the little hierachy problem of Randall-Sundrum models. As is well-known, variousexperimental constraints (the most stringent ones coming from CP violation in K − ¯ K -mixing and the electirc dipole moment of the neutron) require that the IR scale in thesemodels is of order 10 TeV or above. This means that a residual tuning in the permillerange is necessary to generate the electroweak scale. In our construction with warping and A potential for A can be generated perturbatively if the underlying gauge field is coupled to chargedbulk states. In the non-abelian case (see e.g. [34]), this includes the gauge fields themselves due to the non-linear interactions, while the abelian case requires charged scalars or fermions in the bulk (see e.g. [37]).Here we consider a U (1) gauge field and do not add charged bulk states as we are interested in generatinga non-perturbative potential for A . As a caveat, we should stress that the Randall-Sundrum model itself requires a UV completion. Inparticular, near the IR brane gravity becomes strongly coupled at energies not far above the IR scale. Nearthat brane, the UV completion therefore needs to kick in at correspondingly low scales. There are knownUV completions to the Randall-Sundrum model in string theory [38, 39]. See [6, 10, 12] for how the relaxation mechanism can protect the Higgs up to some high supersymmetry-breaking scale instead. See [40] for an alternative solution where an accidental form of supersymmetry protects a little hierarchybetween the electroweak scale and the IR scale of the Randall-Sundrum model. – 3 – igure 1 . A UV completion for the relaxion model is obtained by embedding the relaxion andthe Higgs into a warped extra dimension. The hierarchy problem is then solved in two steps: therelaxation mechanism protects the Higgs mass up to the IR scale (which can be much larger thanthe electroweak scale) and from there warping provides protection till the Planck scale. the relaxion, on the other hand, no such tuning is required.We find that for an effective anomalous coupling localized on the UV brane, the decayconstant is of order M PL / Λ IR with M PL and Λ IR being the Planck and IR scale. For ananomalous coupling in the bulk, we instead find a decay constant of order Λ IR . We thenidentify F = M PL / Λ IR and f = Λ IR . Generating a suitable barrier Λ f ( H ) cos( φ/f ) forthe relaxion requires some additional structure. The reason is that this term genericallycontains a contribution which is independent of the Higgs and which could stop the re-laxion before the Higgs VEV has reached the electroweak scale. To avoid this problem,we consider two different options. One employs a construction from Ref. [1] for whichnew fermions are introduced which couple to the Higgs. If the masses of these fermionsare near the electroweak scale, the Higgs-independent barrier can be sufficiently small.The drawback of this construction is a coincidence problem as it requires to introduce thefermions at a scale which is dynamically generated by the relaxation mechanism and thusa priori determined by completely different parameters. An interesting alternative is theso-called double-scanner mechanism of Ref. [2] (see also [10]). To this end, one introducesanother axion-like scalar which dynamically cancels off the Higgs-independent barrier. Weidentify this axion-like scalar with the 5th component of another U (1) gauge field in thebulk of the extra dimension. We then show how the potential which is required for thedouble-scanner mechanism can be obtained. This construction is largely independent of theembedding into warped space and can therefore also be useful for other UV completionsof the relaxion. For both options to generate the barrier, we discuss the relevant theo-retical and phenomenological constraints for successful relaxation. The highest cutoff andIR scale consistent with these constraints in our warped implementation of the relaxationmechanism is Λ = Λ IR (cid:46) GeV.The plan of this work is as follows. In Sec. 2, we discuss the properties of the A and show how hierarchical decay constants can be obtained. In Sec. 3, we generate thedesired potential for the relaxation mechanism. We analyse the relevant constraints toguarantee a successful relaxation of the electroweak scale in Sec. 4. In Sec. 5, we presentour implementation of the double-scanner mechanism and we conclude in Sec. 6. Additionaldetails are given in three appendices. – 4 – Hierarchical decay constants from warped space
We will now show how hierarchical decay constants can be obtained from warped space.These will be used in later sections to generate the relaxion potential. We consider a sliceof AdS space with metric in conformal coordinates given by ds = a ( z ) ( η µν dx µ dx ν − dz ) , (2.1)where a ( z ) = ( kz ) − is the warp factor with k being the AdS curvature scale (see e.g. [41]for a review). The slice is bounded by the UV brane at z UV = 1 /k and the IR brane at z IR = e kL /k . The length L of the extra dimension can be stabilized for example by meansof the Goldberger-Wise mechanism [42]. The effective 4D Planck scale for this space isgiven by M PL (cid:39) M ∗ /k , where M ∗ is the 5D Planck scale. We will assume that the Planckscale and the AdS scale are of the same order of magnitude (and will later often equatethem). For later convenience, let us also define the IR scale Λ IR ≡ k e − kL .Let us consider a U (1) gauge boson in the bulk. Its action is given by S ⊃ (cid:90) d x dz √ g (cid:18) − g F MN F MN (cid:19) , (2.2)where F MN is the U (1) field strength, g the 5D gauge coupling and √ g = a ( z ). In orderto eliminate the mixing between A µ and A , we add the gauge fixing term (see e.g. [34, 43]) S ⊃ − (cid:90) d x dz √ g g ξ (cid:20) g µν ∂ µ A ν − g ξa ( z ) ∂ ( A a ( z )) (cid:21) . (2.3)The bulk equations of motion for the 4D component A µ and the 5th component A thenread η µσ η λν (cid:18) ∂ σ F µλ + 1 ξ ∂ λ ∂ µ A σ (cid:19) + a ( z ) − ∂ (cid:16) a ( z ) η µν ∂ A µ (cid:17) = 0 (2.4) η µν ∂ µ ∂ ν A + ξ∂ (cid:16) a ( z ) − ∂ (cid:0) a ( z ) A (cid:1)(cid:17) = 0 . (2.5)We are interested in obtaining a massless scalar mode from the bulk gauge boson.To this end, we break the gauge symmetry on both branes by imposing Dirichlet bound-ary conditions on A µ . For consistency, this then requires to impose Neumann boundaryconditions for A . Together the boundary conditions read A µ | UV , IR = 0 , ∂ (cid:0) a ( z ) A (cid:1)(cid:12)(cid:12) UV , IR = 0 . (2.6)Alternatively we could break the gauge symmetry with Higgs fields on the two branes (seee.g. [44, 45]). The above boundary conditions are then obtained in the limit of their VEVsgoing to infinity. In unitary gauge, ξ → ∞ , the bulk equation of motion for A gives ∂ (cid:16) a ( z ) − ∂ (cid:0) a ( z ) A (cid:1)(cid:17) = 0 . (2.7)Notice that this equation is consistent with the boundary conditions and there is thus onemassless mode from A . Its other Kaluza-Klein modes are all eaten by A µ . In particular,– 5 – igure 2 . Sketch of a slice of AdS space which is bounded by two branes. We identify the relaxionwith the 5th component of a U(1) gauge field in the bulk. Its wavefunction is then localized towardsthe IR brane. The Higgs is localized on (or near) the IR brane. The UV brane corresponds to thePlanck scale. We draw the IR brane with a dashed contour as a reminder that the IR scale in ourmodel can be much larger than the usual TeV scale of the Randall-Sundrum model. there is no massless mode from A µ , consistent with the fact that the gauge symmetry isbroken. As usual, the A massless mode can be parameterized as A ( x, z ) = h ( z ) φ ( x ) , (2.8)where h ( z ) is its profile along the extra dimension. From Eqs. (2.6) and (2.7), we thensee that h ( z ) = N a ( z ) − . Demanding canonically normalized kinetic terms for φ ( x ), thenormalization constant N of the wavefunction is determined by N g (cid:90) z IR z UV dza ( z ) = 1 . (2.9)For kL (cid:29)
1, this gives
N (cid:39) g √ kL e − kL , where we define the dimensionless coupling g ≡ g / √ L . Altogether, the wavefunction of the massless mode then reads h ( z ) (cid:39) g √ kL e − kL kz . (2.10)The wavefunction is thus peaked towards the IR brane (see Fig. 2 for a sketch of thewavefunction profile in the extra dimension). Furthermore, the fact that N → z IR →∞ shows that the A massless mode is indeed localized in the IR.Performing a 5D gauge transformation, A M ( x, z ) → A M ( x, z ) + ∂ M α ( x, z ), we see thatthe boundary conditions in Eq. (2.6) and the bulk equation of motion in Eq. (2.7) remaininvariant only for the subset of transformations α = B z + C (2.11)with B and C being independent of x and z . The remaining symmetry in 4D is thusglobal, again consistent with the fact that the gauge symmetry is broken. Under thisremnant symmetry, the massless mode transforms as φ → φ + 2 B N k . (2.12)– 6 –t this point, the relaxion is thus an exact Nambu-Goldstone boson which non-linearlyrealizes a remnant global U (1). By virtue of the 5D gauge invariance, no 5D local, higher-dimensional operators can break this shift symmetry (see [46] for a detailed discussion). Apotential for the relaxion could be generated by non-local effects in the presence of bulkstates which are charged under the U (1) but we assume such states to be absent from thetheory. Instead we introduce anomalous couplings of the relaxion to confining non-abeliangauge groups. A potential then arises from instantons, similar to what happens for theaxion in QCD. We localize these anomalous couplings in the bulk or on the UV brane.In what follows, we show that these possibilities, thanks to the warp factor, can naturallyexplain the required hierarchy between the decay constants in the relaxion potential.
Let us add a non-abelian gauge group in the bulk, whose field strength and coupling wedenote respectively as G NP and g c . We choose boundary conditions for the gauge field suchthat the 4D gauge symmetry remains unbroken on the branes. Its tower of Kaluza-Kleinmodes then contains one massless mode which is the 4D gauge boson. We next introducea Chern-Simons coupling of the U (1) gauge field to this gauge group. Including the kineticterm, the action reads S ⊃ (cid:90) d x dz (cid:18) √ g − g c ) Tr (cid:2) G MN G MN (cid:3) + c B π (cid:15) MNP QR A M Tr [ G NP G QR ] (cid:19) , (2.13)where c B is a dimensionless constant and the normalization is chosen for later convenience. Under a U (1) gauge transformation A M ( x, z ) → A M ( x, z ) + ∂ M α ( x, z ), the action trans-forms as S → S − (cid:90) d x dz α ( x, z ) c B π (cid:15) µνρσ Tr [ G µν G ρσ ] (cid:0) δ ( z − z UV ) − δ ( z − z IR ) (cid:1) . (2.14)The Chern-Simons term thus induces an anomaly for the U (1) symmetry on the branes.This is not a problem, however, since the symmetry is only global on the branes and thereare thus no gauge anomalies.In the 4D effective theory, this gives rise to an anomalous coupling for φ . Let usrestrict ourselves to the massless mode of the non-abelian gauge field, whose field strengthwe denote as G µν . Integrating over the extra dimension, Eq. (2.13) then in particular gives S ⊃ (cid:90) d x (cid:18) − g c ) Tr [ G µν G µν ] + φ ( x )16 π f B (cid:15) µνρσ Tr [ G µν G ρσ ] (cid:19) , (2.15) Alternatively, for example for bulk fermions charged under the U (1) it is sufficient if their masses aresomewhat larger than the AdS scale in which case any perturbative contribution to the potential is highlysuppressed (see e.g. [34, 47]). Note that a factor of 2 arises from the normalization Tr[ T a T b ] = δ a,b of the generators of the non-abelian gauge group. – 7 –here g c = g c / √ L is the gauge coupling of the massless mode. The decay constant is givenby [43, 46] f B ≡ (cid:20) c B (cid:90) z IR z UV dz h ( z ) (cid:21) − = N c B g (cid:39) k e − kL c B g √ kL (2.16)which is of order the IR scale Λ IR and thus warped-down. From Eqs. (2.12), (2.14) and(2.15), we see that φ reproduces the anomaly under a transformation α = Bz . In Ap-pendix B, we briefly review how Chern-Simons terms can arise from charged bulk fermions.As we also discuss there, any perturbative contribution from such a fermion to the potentialfor A can be sufficiently suppressed. Nevertheless, in the remainder of this paper we willnever assume any charged bulk states and will instead include the Chern-Simons termsdirectly into our effective 5D theory.Note that Eq. (2.13) also yields couplings of φ to the higher Kaluza-Klein modes ofthe non-abelian gauge field. As Eq. (2.15) for the massless mode, these couplings aretotal derivatives (see e.g. Ref. [48]) and therefore do not contribute perturbatively to thepotential for φ . We will later assume that the non-abelian gauge group confines in orderto generate a non-perturbative potential for φ . But we will choose the confinement scalebelow the IR scale and thus below the Kaluza-Klein masses. The Kaluza-Klein modes ofthe non-abelian gauge group therefore do not contribute non-perturbatively to the potentialeither. We now discuss how a decay constant which is much larger than Λ IR can be obtained. Tothis end, we consider an effective anomalous coupling of A which is localized on the UVbrane [43], S ⊃ (cid:90) d x dz δ ( z − z UV ) c UV π A k (cid:15) µνρσ Tr [ G µν G ρσ ] , (2.17)where c UV is a dimensionless constant and G MN is the field strength of a non-abelian gaugefield in the bulk. As we outline in Appendix A, this interaction can for example arise asan effective coupling from a Chern-Simons term in a two-throat geometry. Under a U (1)gauge transformation, the action transforms similar to Eq. (2.14) but restricted to the UVbrane and with ∂ α ( x, z ) instead of α ( x, z ). Let us again restrict ourselves to the masslessmode of the gauge field, whose field strength we denote as G µν . Using the wavefunction ofthe massless mode of A from Eqs. (2.8) and (2.10), this gives S = (cid:90) d x π φ ( x ) f UV (cid:15) µνρσ Tr [ G µν G ρσ ] (2.18)with decay constant given by [43] f UV ≡ kc UV h ( z UV ) (cid:39) k e kL c UV g √ kL (2.19)or f UV ∼ M PL / Λ IR . We see that a warped-up decay constant, much larger than the cutoff,appears naturally in this case. This large decay constant can be intuitively understood as– 8 –eing of order the natural scale M PL on the UV brane times an inverse suppression factorfrom the wavefunction overlap of A with the UV brane.Note that super-Planckian decay constants may be constrained by the weak gravityconjecture in theories of quantum gravity [49] (see also [50–53]). Given that the relaxionis an axion-like field, the conjecture necessarily restricts its field excursion (∆ φ ∼ Λ /g (cid:48) )to be sub-Planckian, setting a lower bound on the coupling g (cid:48) in the potential (1.1). Theweak gravity conjecture is then at odds with any relaxion model with trans-Planckianfield excursions, including our proposal. On the other hand, there are known loopholesto the conjecture [54–58]. For instance, the application of the conjecture to effective fieldtheories may result in a much weaker bound on the coupling g (cid:48) [57]. Furthermore, in [58],a better understanding of the conclusions of [57] is achieved by considering a string theoryembedding. There it is shown that if a clockwork model is successfully embedded in stringtheory, one may in principle obtain a large cutoff, avoiding the naive bound from the weakgravity conjecture, as long as the number of sites in the construction is large.We conclude that two hierarchically different decay constants can be obtained, depend-ing on the localization of the anomalous interactions in the warped space. For the relaxion,we then identify F = f UV ≈ M PL / Λ IR and f = f B ≈ Λ IR . Note that as the ratio F/f isproportional to the warp factor, the potential in Eq. (1.2) does not respect a discrete shiftsymmetry since, in general,
F/f is a non-integer number. This is a consequence of thenon-local nature of the residual symmetry transformation α = Bz + C in Eq. (2.11) whichexplicitly depends on the localization. In the following, we build an explicit model thatmakes use of this toolkit to generate a phenomenologically viable potential in the form ofEq. (1.2). Let us next discuss the relaxion parameters in more detail and how they can be understoodin terms of our UV model. Provided that electroweak symmetry remains unbroken in theconfinement phase transition which generates the periodic potentials in Eq. (1.2), Λ
F,f ( H )both depend quadratically on the Higgs (plus generically higher even powers of the Higgswhich are, however, not important in the following). We can then parametrizeΛ F,f ( H ) = Λ F,f (cid:32) H M F,f (cid:33) , (3.1)where Λ F,f and M F,f can be understood as the scales where the periodic terms and higher-dimensional couplings to the Higgs are generated, respectively. The potential in Eq. (1.2) As proposed in [1], one can also use the QCD axion as the relaxion. The last term in Eq. (1.1) isthen the usual QCD axion potential which depends linearly on the Higgs (see e.g. [59]). However, barringadditional model building, this spoils the axion solution to the strong CP problem. See also [60–62] forsolutions to the strong CP problem in the context of the relaxion. – 9 –hen reads V ( φ, H ) = − Λ H + λ H + Λ F (cid:18) H M F (cid:19) cos (cid:18) φF (cid:19) + Λ f (cid:32) H M f (cid:33) cos (cid:18) φf (cid:19) . (3.2)For simplicity, we have dropped terms which may be generated at higher loop-order. Wewill discuss these terms later in Sec. 4. Assuming that φ is in the linear regime of thelow-frequency cosine, φ ∼ πF/ π , we can expand it for φ − πF/ (cid:46) F . After theredefinition φ − πF/ → φ , this gives the linear part of the relaxion potential in Eq. (1.1)with the identifications g = Λ F F Λ , g (cid:48) = Λ F F M F Λ (3.3)up to factors of order one.The last term in Eq. (3.2) stops the relaxion once the Higgs VEV has reached theelectroweak scale. For this to work, we need to ensure that M f (cid:46) v EW , otherwise theHiggs-independent barrier proportional to cos( φ/f ) would stop the relaxion already beforethe Higgs VEV has obtained the right value. Note also that the Higgs-independent barrierreceives corrections from closing the Higgs loop in the Higgs-dependent one and will thusgenerically be present. We discuss radiative corrections to the potential in more detailin Sec. 4. But to get a sense of the scales involved, we already note here that radiativestability of the potential demands that Λ f (cid:46) π v EW M f and Λ F (cid:46) πM F .To obtain M f (cid:46) v EW requires that the higher-dimensional coupling of the Higgs tothe periodic potential is generated near the electroweak scale. In the next section, wemake use of a construction from Ref. [1] which introduces light fermions for this purpose.The drawback of this scenario is of course a coincidence problem: one has to assume newparticles at a scale which is dynamically generated by the relaxation mechanism and is thusdetermined by a priori completely unrelated parameters. One way around this problem isthe double-scanner mechanism of Ref. [2]. To this end, one introduces another axion-likefield which dynamically cancels off the Higgs-independent barrier in Eq. (3.2). This allowsthe relaxation mechanism to work even for M f (cid:29) v EW . We discuss a UV completion ofthis scenario in Sec. 5.
We now build a simple explicit model that successfully generates the needed terms in theHiggs-relaxion potential at a phenomenologically viable scale, making use of the resultsof Sec. 2. We assume that the Higgs is localized on or near the IR brane, so that itsmass is warped down to the IR scale (see Fig. 2). We note that it may also be possible toimplement the relaxation mechanism in a model where the Higgs is instead localized on theUV brane. As usual, the relaxion can only protect the Higgs up to some cutoff significantlybelow the Planck scale. Such a model would therefore require a UV completion above thiscutoff on the UV brane. We leave a study of this possibility to future work. As we find Another proposal for the relaxion that does not require new physics close to the electroweak scale isthe particle-production mechanism of Ref. [14]. – 10 –ater, the highest IR scale that we can achieve in our implementation of the relaxationmechanism (while still solving the hierarchy problem) is below the GUT scale. If theremaining Standard Model fields are then also localized on the IR brane, higher-dimensionaloperators violating baryon number lead to too fast proton decay [63]. In order to suppressthese operators, we assume that the Standard Model instead lives in the bulk. As usual,the light quarks are localized towards the UV brane, while the top-bottom doublet andthe right-handed top live nearer to the IR brane. This has the added advantage that thehierarchy of Yukawa couplings can then be generated from the warping too. The IR scalein our model can be high enough, on the other hand, to ensure that oblique corrections andflavour- and CP -violating processes are sufficiently suppressed without imposing custodialor flavour symmetries.We identify the relaxion with the 5th component of a U (1) gauge field in the bulk. Inorder to generate a potential for this field, we add two non-abelian gauge groups G f and G F which also live in the bulk. We assume that these gauge groups confine at the scales Λ G f and Λ G F , respectively. In order to ensure that confinement can be discussed using only thezero-modes of the bulk gauge fields, we take Λ G f and Λ G F to be below the IR scale. Thiscan always be arranged by choosing the 5D gauge couplings and ranks of the gauge groupsappropriately.We assume anomalous couplings of the relaxion φ to the field strengths G fµν and G Fµν of the massless 4D gauge fields corresponding to G f and G F , respectively: S ⊃ (cid:90) d x φ ( x )16 π (cid:15) µνρσ (cid:18) F Tr (cid:2) G Fµν G Fρσ (cid:3) + 1 f Tr (cid:104) G fµν G fρσ (cid:105)(cid:19) . (3.4)As we have discussed in Sec. 2, these can arise from a Chern-Simons coupling in the bulkand an effective anomalous coupling of A on the UV brane. But for now, we only assumethat F (cid:29) f and postpone a concrete choice for the decay constants to Sec. 4.On the IR brane, we add a pair of chiral fermions χ and χ c in the fundamental andantifundamental representation of G F , respectively. These fermions transform under achiral symmetry which we assume to be broken only by a Dirac mass m χ . This allows forthe terms in the action S ⊃ (cid:90) d x dz √− g IR δ ( z − z IR ) m χ (cid:18) H M PL (cid:19) χ χ c + h.c. , (3.5)where g IR is the induced metric determinant on the IR brane. We have included a higher-dimensional coupling to the Higgs which is generically present and which we expect to besuppressed by a scale near the Planck scale. Note that we will use the symbol H for boththe SU (2)-doublet Higgs field, writing the singlet combination | H | as H for simplicity,and its VEV. It will be clear from context which one is meant. For simplicity, we alsoignore any numerical prefactors for now and set k = M PL . Similarly, we assume that allparameters are real. We will reinstate prefactors and phases later on. Performing theintegral over the extra dimension and canonically normalizing the fields gives S ⊃ (cid:90) d x m χ (cid:18) H Λ IR (cid:19) χ χ c + h.c. , (3.6)– 11 – χ c N N c L L c G F (cid:3) ¯ (cid:3) – – – – G f – – (cid:3) ¯ (cid:3) (cid:3) ¯ (cid:3) SU (2) L – – – – (cid:3) (cid:3) U (1) Y – – – – − + Table 1 . Matter content on the IR brane with gauge representations for the model with a barrierat the electroweak scale. where we have redefined e − kL m χ → m χ , e − kL H → H , e − kL/ χ → χ and similarly for χ c . Note in particular that m χ (cid:46) Λ IR after the redefinition. Let us next perform the fieldredefinition χ → e iφ/F χ , (3.7)while χ c is left invariant. Due to the non-trivial transformation of the path integral measure,this chiral rotation removes the coupling of φ to Tr (cid:2) G Fµν G Fρσ (cid:3) in Eq. (3.4) and transformsEq. (3.6) to S → S ⊃ (cid:90) d x m χ (cid:18) H Λ IR (cid:19) e iφ/F χχ c + h.c. . (3.8)If m χ is below the confinement scale of G F (which in turn is below Λ IR ), this term contributesto the Higgs-relaxion potential after confinement. Parametrizing (cid:104) χχ c (cid:105) = Λ G F , this gives V ( φ, H ) ⊃ m χ Λ G F (cid:18) H Λ IR (cid:19) cos (cid:18) φF (cid:19) . (3.9)This has the same form as the potential with period F in Eq. (3.2), including the couplingto the Higgs. We can then make the identificationsΛ F = m χ Λ G F , M F = Λ IR . (3.10)Next we need to generate the potentials with smaller period f . To this end, we usea construction from Ref. [1] and add fermions L and N on the IR brane with the sameStandard Model charges as the lepton doublet and the right-handed neutrino, respectively.In addition, these fermions are in the fundamental representation of the gauge group G f .We also include fermions L c and N c in the conjugate representations. Together they allowfor the terms in the action S ⊃ (cid:90) d x dz √− g IR δ ( z − z IR ) (cid:16) m L LL c + m N N N c + y HLN c + ˜ y H † L c N (cid:17) + h.c. . (3.11)Notice that we have not included a higher-dimensional coupling to the Higgs. It could bepresent but will be subdominant as we will see momentarily. Performing the integral overthe extra dimension and canonically normalizing the fields gives S ⊃ (cid:90) d x (cid:16) m L LL c + m N N N c + y HLN c + ˜ y H † L c N (cid:17) + h.c. , (3.12) This is thus our definition of the scale Λ G F . – 12 –here we have redefined e − kL m L → m L , e − kL H → H , e − kL/ L → L and similarly for m N , N and the conjugated fields. Note in particular that m L , m N (cid:46) Λ IR after the redefinition.Assuming that m N (cid:28) m L and restricting to a region in field space where the Higgs VEVsatisfies y ˜ yH (cid:28) m L , we can integrate out L and L c . This gives S ⊃ (cid:90) d x (cid:18) m N − y ˜ y H m L (cid:19) N N c + h.c. . (3.13)We can then perform the chiral rotation N → e iφ/f N , (3.14)while N c is left invariant. This removes the coupling of φ to Tr (cid:104) G fµν G fρσ (cid:105) in Eq. (3.4) andtransforms Eq. (3.13) to S → S ⊃ (cid:90) d x (cid:18) m N − y ˜ y H m L (cid:19) e iφ/f N N c + h.c. . (3.15)Provided that m N is below the confinement scale of G f , this term contributes to the Higgs-relaxion potential after confinement. Parametrizing (cid:104) N N c (cid:105) = Λ G f , this gives V ( φ, H ) ⊃ m N Λ G f (cid:18) − y ˜ y H m N m L (cid:19) cos (cid:18) φf (cid:19) . (3.16)This has the form of the potential with period f in Eq. (3.2), including the coupling to theHiggs. We can then make the identificationsΛ f = m N Λ G f , M f = m N m L y ˜ y . (3.17)For sufficiently small m N and m L , this allows for M f (cid:46) v EW as required in a technicallynatural way. Notice that if we had instead relied on the higher-dimensional operator inEq. (3.5) to generate the barrier, we would have obtained M f ∼ Λ IR (cid:29) v EW . We discussconstraints on the parameters of this construction in more detail in Sec. 4. A summary ofthe matter content on the IR brane is given in Table 1.We next reinstate the numerical prefactors and the phases of the parameters whichwe have ignored so far. Let us denote the prefactor of the Higgs coupling in Eq. (3.5) as c χH . We absorb possible phases in the fermionic condensates (cid:104) χχ c (cid:105) and (cid:104) N N c (cid:105) and any(relaxion-independent) Θ-terms for G F and G f into the mass parameters m χ and m N , m L ,respectively. Redoing the derivation above then gives V ( φ, H ) ⊃ | m χ | Λ G F (cid:20) cos (cid:18) φF + b χ (cid:19) + | c χH | H Λ IR cos (cid:18) φF + b χH (cid:19)(cid:21) + 2 | m N | Λ G f (cid:20) cos (cid:18) φf + b N (cid:19) − | y ˜ y | H | m N m L | cos (cid:18) φf + b NH (cid:19)(cid:21) , (3.18)where the complex phases are given by b χ = arg( m χ ), b χH = arg( m χ c χH ), b N = arg( m N )and b NH = arg( y ˜ y/m L ). Note that this does generically not match the form of the potential– 13 –n Eq. (3.2). Nevertheless the relaxation mechanism can still work. Indeed expanding thefirst two terms in the linear part of the cosines again gives the sliding term for the relaxionand its linear coupling to the Higgs. In order to ensure that these terms have the samesign as required, we need to demand that b χ ∼ b χH . As before, the Higgs-independentbarrier in the third term should be too small to stop the relaxion by itself. It is thennegligible for the dynamics and the phase b N has no consequences. The phase b NH in theHiggs-dependent barrier in the fourth term, on the other hand, slightly shifts the minimumwhere the relaxion eventually stops but has no other consequences either.To ensure that our calculation of the potentials is applicable, the masses of the fermionpairs χ, χ c and N, N c need to be below their respective condensation scales. This meansthat the chiral symmetries under which these fermion pairs transform are only weaklybroken at the confinement scales. We then expect corresponding pseudo-Nambu-Goldstonebosons in the spectrum of composite states. As we discuss in Appendix C, their contributionto the potential factorizes from the remaining potential and they can be trivially integratedout if the spectrum of fermions is doubled. We now discuss various conditions that need to be fulfilled for the relaxation mechanismto be viable. In Sec. 4.1, we derive general conditions on the parameters in the relaxionpotential in Eq. (3.2). In Sec. 4.2, we then discuss additional conditions that arise in ourwarped model with a barrier at the electroweak scale.
We begin our discussion of the evolution of the Higgs and relaxion with the Higgs mass-squared being positive and of order Λ . In order to allow the relaxion to subsequently turnthe Higgs mass tachyonic, its average VEV ˜ φ during this stage of the evolution needs tosatisfy cos (cid:16) ˜ φF (cid:17) (cid:38) Λ M F Λ F . (4.1)Since the left-hand side is bounded by 1, this in particular implies the conditionΛ F (cid:38) Λ M F . (4.2)The relaxion stops rolling down its potential when the derivatives of the periodicterms balance each other. We will find below that M F (cid:29) v EW and the term proportional tocos( φ/F ) is thus dominated by the Higgs-independent part. On the other hand, the termproportional to cos( φ/f ) needs to be dominated by the Higgs-dependent part as discussedin Sec. 3. The relaxion then stops once the Higgs VEV becomes H ≈ M f fF Λ F Λ f , (4.3)where we have set sin( ˜ φ/F ) ∼
1. This is a good approximation as long as cos( ˜ φ/F ) is notvery close to its extrema. The parameters need to be chosen such that the combination– 14 –n the right-hand side gives the electroweak scale v EW . In the following, we will use thisrelation to trade Λ f for v EW .Notice that the Higgs-dependent barrier H cos( φ/f ) in the potential contributes tothe Higgs mass. Imposing that this contribution be less than the electroweak scale (seee.g. Ref. [64]) gives the constraint Λ f (cid:46) M f v EW which using Eq. (4.3) leads toΛ F (cid:46) v EW (cid:18) Ff (cid:19) / . (4.4)Together with Eq. (4.2), this gives a constraint on the cutoff in our model as we discussin Sec. 4.2. In order to ensure that the Higgs mass is scanned with sufficient precision, weneed to demand that the change of the Higgs-dependent term proportional to cos( φ/F )over one period of the barrier, δφ ∼ f , is less than the electroweak scale. This gives theconstraint Λ F (cid:46) ( M F v EW ) / ( F/f ) / which is weaker than Eq. (4.4).Furthermore, there are several requirements on the inflation sector for the relaxationmechanism to be viable. If the relaxion is not the inflaton, its energy density should besubdominant compared to the inflaton. The energy density in the minimum where therelaxion eventually settles needs to be (close to) zero. This requires an additional constantcontribution that is added to the potential and chosen such that the energy density atthe minimum (nearly) vanishes. The tuning that is necessary to achieve this is just amanifestation of the cosmological constant problem. The contribution of the relaxion tothe energy density relevant for inflation is then determined by how much it changes duringits evolution. Using Eq. (4.1) in the potential of Eq. (3.2) gives the condition H I (cid:38) M F Λ M PL , (4.5)where H I is the Hubble rate during inflation. In addition, to ensure that our classicalanalysis of the field evolution is applicable, quantum fluctuations of the relaxion while itroles down the potential should be sufficiently small. Over one Hubble time, the relaxionchanges classically by ( δφ ) class . ∼ H − I dV /dφ . Its quantum fluctuations, on the other hand,are ( δφ ) quant . ∼ H I . This leads to the condition H I (cid:46) Λ / F F / . (4.6)Combining the last two inequalities, we getΛ F (cid:38) √ F (cid:18) M F Λ M PL (cid:19) / . (4.7)Finally, the number of e-folds of inflation must be sufficiently large to ensure that therelaxion scans the required field range. Denoting the latter by ∆ φ , this leads to the condi-tion N e ( δφ ) class . (cid:38) ∆ φ . Provided that the relaxion is in the linear part of cos( φ/F ), using This constraint can be slightly relaxed if one includes the barrier term in the scanning of the Higgs mass[65]. One then still needs to impose that Λ f (cid:46) πM f v EW to ensure that loop corrections to the potentialare small. This gives a similar condition as Eq. (4.4) but with an additional factor √ π on the right-handside. – 15 –q. (4.1) this gives N e (cid:38) (cid:18) H I F M F ΛΛ F (cid:19) . (4.8)The resulting required number of e-folds can be very large. We will not specify the inflationsector but will simply assume that it can be arranged to fulfill the conditions in Eqs. (4.5),(4.6) and (4.8). Possible complications in achieving this are discussed e.g. in Ref. [9]. Notealso that the above conditions are somewhat alleviated if the effect of the time evolutionof the Hubble rate during inflation is taken into accout [3].We also need to ensure that the potential is radiatively stable. The potential is aneffective theory with a cutoff determined by the confinement scales Λ G f and Λ G F of thegauge groups that give rise to the periodic terms (assuming they are smaller than thecutoffs associated with generating the H -terms in the potential). In the region of thepotential where the Higgs mass parameter m H ( φ ) ≡ Λ F M F cos (cid:18) φF (cid:19) − Λ (4.9)is smaller than these cutoffs, the Higgs can give important corrections to the potential.From the one-loop effective potential, we find V ( φ, H ) ⊃ Λ G F m H ( φ )16 π + m H ( φ )16 π log (cid:32) m H ( φ )Λ G F (cid:33) + Λ f Λ G f π M f cos (cid:18) φf (cid:19) + (cid:34) Λ f π M f cos (cid:18) φf (cid:19) + Λ f m H ( φ )8 π M f cos (cid:18) φf (cid:19)(cid:35) log (cid:32) m H ( φ )Λ G f (cid:33) , (4.10)where we have neglected some subdominant terms. In the opposite region m H ( φ ) (cid:29) Λ G f or Λ G F , on the other hand, the corrections are strongly suppressed. This ensures thatthe term proportional to m H ( φ ) cos( φ/f ) gives only a small contribution to the Higgs-independent barrier. In order to guarantee that the other term proportional to cos( φ/f ) issuppressed too, we require that Λ G f (cid:46) πM f . (4.11)Provided that Λ , Λ G F , Λ F (cid:46) πM F the first two terms in Eq. (4.10) give small corrections tothe sliding term for the relaxion and do not affect the dynamics. Finally if Λ f (cid:46) πM f v EW ,the cos ( φ/f )-term is negligible compared to the Higgs-dependent barrier when the Higgsreaches the electroweak scale. Using Eq. (4.3), this translates to the constraintΛ F (cid:46) √ π v EW (cid:18) Ff (cid:19) / . (4.12)This is less stringent than Eq. (4.4). Note that the Higgs mass parameter has an additional contribution from the cos( φ/f )-term. Since itis subdominant except in a small region of φ , we define Eq. (4.9) without this contribution. See the one-loop effective potential e.g. in Eq. (2.64) of Ref. [66] in the limit U (cid:48)(cid:48) (cid:29) Λ . – 16 – .2 Conditions on the warped model The Higgs is localized on or near the IR brane in our warped model. Its mass parameteris then naturally of order Λ IR . We therefore identify the cutoff of our relaxion model withthe IR scale: Λ ∼ Λ IR . (4.13)As we have discussed in Sec. 2, we can obtain the decay constants f B ≈ Λ IR from a Chern-Simons term in the bulk and f UV ≈ M PL / Λ IR from an effective anomalous coupling on theUV brane. Since F (cid:29) f is required, we identify F = M PL / Λ IR and f = Λ IR .From the conditions in Eqs. (4.2) and (4.4) and using that M F ≈ Λ IR , we obtain anupper bound on the IR scale in our warped model:Λ IR (cid:46) (cid:0) v EW M PL (cid:1) / ≈ · TeV . (4.14)Note that this is slightly lower than the maximal cutoff found in Ref. [1]. The reasonis that there the bound on the cutoff is partly determined by the requirement of a finiteviable window for the Hubble rate. In our warped model, the corresponding contraint inEq. (4.7) is always trivially satisfied as we discuss below. The dominant bound on the cutoffinstead involves the constraint in Eq. (4.2) that the H cos( φ/F )-term in the potential cancompensate for a Higgs mass near the cutoff. This difference arises because g is a freeparameter in the effective description of Ref. [1], whereas in our warped model g ∝ /F isdetermined in terms of other parameters.We need to ensure that collider and flavour bounds on the KK modes in our warpedmodel are fulfilled. We have assumed that the Standard Model fields live in the bulk. Thedominant constraints then arise from CP -violation in K − ¯ K -mixing and the electric dipolemoment of the neutron. This requires [67, 68]:Λ IR (cid:38)
10 TeV . (4.15)This also satisfies constraints from electroweak precision tests without imposing a custodialsymmetry [69, 70] and on the radion (for a typical stabilization mechanism).The potential leads to mixing between the Higgs and the relaxion. This furtherconstrains the IR scale. We use results from Ref. [64], where bounds on the parameterΛ = Λ f v EW /M f controlling the mixing have been derived from several experiments (fifthforce, astrophysical and cosmological probes, beam dump, flavor, and collider searches). Us-ing Eq. (4.3), this translates to limits on Λ F and thereby on Λ IR . For our case F = M PL / Λ IR and f = Λ IR , the most stringent bound comes from the distortion of the diffuse extra-galactic background light spectrum due to relaxion late decays. This gives the constraintΛ IR (cid:46) · TeV (4.16)which is more stringent than Eq. (4.14).We have discussed the confinement of G f and G F in terms of only the massless modesof the gauge fields in our extra-dimensional model. This is a good approximation provided– 17 – F Λ F M F f Λ f M f Λ IR M PL Λ IR Λ IR Λ IR Λ IR Λ / IR M / PL v EW
10 TeV (cid:46) Λ IR (cid:46) · TeV
Table 2 . Parameters in the potential in Eq. (3.2) in our warped model with an electroweak-scalebarrier. The range for the IR scale is allowed by all phenomenological constraints considered in thissection. that the confinement scales are smaller than the KK mass scale: Λ G f , Λ G F (cid:46) Λ IR . (4.17)Since Λ F (cid:46) Λ G F and M F ∼ Λ IR according to Eq. (3.10), it then follows from Eq. (4.2) thatΛ F ∼ Λ IR is required for successful relaxation. This in turn means that m χ , Λ G F ∼ Λ IR .Since the fermions χ, χ c are localized on the IR brane, the former condition can be nat-urally fulfilled. In order to discuss the latter condition, let us focus on G F = SU ( N ) fordefiniteness. If we estimate the confinement scale as the scale where the 4D gauge couplingdiverges, we find (see e.g. Ref. [71]) Λ G F M PL ≈ (cid:18) Λ IR M PL (cid:19) π N ( g c ) k , (4.18)where g c is the 5D gauge coupling of G F . From this we see that the confinement scale of G F is close to the IR scale if 24 π / (11 N ( g c ) k ) ≈
1. This can be achieved for a wide rangeof values for g c and N but clearly requires a coincidence between two parameters whichare a priori not related. It may be possible to instead trigger the confinement of G F byadding states on the IR brane and thereby achieve Λ G F ∼ Λ IR without such a coincidence.We leave a detailed study of this question to future work.We next consider constraints related to the fermions N, N c and L, L c on the IR brane.The last two terms in Eq. (3.12) break the chiral symmetry of N, N c , in addition to theirDirac mass. Loop corrections then contribute to the Dirac mass (see Fig. 3), leading to theconstraint m N (cid:38) y ˜ y m L π log(Λ IR /m L ) . (4.19)The Higgs-dependent barrier can only stop the relaxion if M f (cid:46) v EW . Using Eq. (3.17), theloop contribution to m N then implies that m L (cid:46) π v EW (cid:112) log(Λ IR /m L ) . (4.20)The electroweak doublets L, L c can thus not be much heavier than the electroweak scale.On the other hand, due to collider constraints on such particles, they cannot be much It may be possible to alleviate this condition by including some of the KK modes in the effective theory. Brane-localized kinetic terms for the gauge field would give another factor multiplying one side of thisrelation. This would change the required relation between g c and N accordingly. – 18 – igure 3 . Loop correction to m N . lighter either. This limits their mass to a region near the electroweak scale. The questionwhy their mass should be near the scale that is dynamically generated via the relaxationmechanism is the coincidence problem that we have mentioned in Sec. 3. This problemdoes not appear in the double-scanner scenario that we discuss in Sec. 5.Let us briefly pause to count parameters. The potential in Eq. (3.2) has 7 dimensionfulparameters. Of these, Λ, M F and Λ F are of order Λ IR , whereas M f is of order v EW .Furthermore, F and f are given in terms of Λ IR and M PL , while Λ f is fixed as a function ofthe other parameters via Eq. (4.3). We can then express all parameters (up to O (1) factors)uniquely in terms of Λ IR (plus M PL and v EW ). In Table 2, we summarize the correspondingrelations and the phenomenologically viable range for the IR scale in our warped model.Additional loop corrections arise in the effective field theory at energies below Λ G F and Λ G f as discussed in Sec. 4.1. In particular, Eq. (4.11) is an upper bound on theconfinement scale of G f . An additional constraint arises from the requirement that themass of the lightest fermion after diagonalizing Eq. (3.12) is smaller than the confinementscale (cf. the comment above Eq. (3.16)). Together this gives (cid:12)(cid:12)(cid:12)(cid:12) m N − y ˜ y v EW m L (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Λ G f (cid:46) π v EW , (4.21)where we have used M f ≈ v EW and that the largest Higgs VEV of interest is the electroweakscale (as the relaxion stops before the Higgs VEV can grow even further). Using Eq. (4.3)and that Λ f (cid:46) Λ G f , this upper bound on Λ G f gives an upper bound on Λ F which is lessstringent than Eq. (4.4). On the other hand, Λ G f can be very low provided that y, ˜ y and m N are sufficiently small. In order to ensure that G f does not contribute to dark radiationduring big bang nucleosynthesis, its confinement scale should be larger than a few MeV:Λ G f (cid:38) O (few) · MeV . (4.22)From Eq. (4.3) and since Λ f (cid:46) Λ G f , it follows that such low Λ G f is only possible for the IRscale near its lower bound in Eq. (4.15). Furthermore, we need to ensure that the decayof composite states does not destroy heavy elements during big bang nucleosynthesis. Theresulting limits have been worked out in Ref. [72]. For Λ G f = 10 MeV, m L = 500 GeV and y = 2˜ y , it is found that y, ˜ y (cid:38) .
15 is required. This limit quickly becomes weaker forlarger Λ G f or smaller m L . On the other hand, the Yukawa couplings must not be too largein order to satisfy bounds on the invisible decay width of the Higgs. The correspondinglimit is y, ˜ y (cid:46) . y = ˜ y and m L = 200 GeV which becomes slightly less stringent forlarger m L . – 19 –iven that the fermions χ , χ c , L , L c , N and N c are all localized on the IR brane, weexpect higher-dimensional terms in the action. These include S ⊃ (cid:90) d x (cid:32) c χχ m χ Λ IR ( χχ c ) + c NN m N Λ IR ( N N c ) + c χN m χ m N Λ IR χχ c N N c + h.c. (cid:33) . (4.23)The coefficients c χχ , c NN and c χN could be estimated using naive dimensional analysis.For simplicity, we assume them to be real. After confinement, this gives the additionalterms V ( φ, H ) ⊃ c χχ Λ F Λ IR cos (cid:18) φF (cid:19) + c NN Λ f Λ IR cos (cid:18) φf (cid:19) + c χN Λ F Λ f Λ IR cos (cid:18) φF + φf (cid:19) (4.24)in the Higgs-relaxion potential. Note that higher-dimensional couplings involving LL c either do not directly contribute to the potential as the pair LL c does not condense or thecontribution is very suppressed. The first term in Eq. (4.24) contributes to the slidingterm for the relaxion. But for c χχ (cid:46) G f and G F are larger than the Hubble rate during inflation: H I (cid:46) Λ G f , Λ G F . (4.25)For both Λ G F ∼ Λ IR and Λ G f (cid:38) Λ f given by Eq. (4.3), this is less stringent than Eq. (4.6)from requiring that quantum fluctuations of the relaxion are negligible for the dynamics.For F = M PL / Λ IR and since Λ ∼ Λ F ∼ M F ∼ Λ IR , the condition for having a finite viablewindow for the inflation scale in Eq. (4.7) is trivially fulfilled. Furthermore, the upperlimit on the inflation scale in Eq. (4.6) is significantly smaller than the IR scale. We willassume that the inflationary sector, which we do not specify further, is located on the UVbrane. Then H I (cid:28) Λ IR guarantees that the effect of inflation on the geometry of the extradimension is negligible [73, 74]. Similarly, for a typical stabilization mechanism it ensuresthat the extra dimension is safe from destabilization during inflation. In order to ensurethat the barrier for the relaxion is not removed during reheating after inflation, we demandthat the reheating temperature be below Λ G f . This may require a relatively low reheatingtemperature. As follows from Eq. (4.22), it can still be sufficiently high to allow for bigbang nucleosynthesis though. Under certain conditions, the reheating temperature mayalso be higher than Λ G f [1] (see also [75]). A higher-dimensional coupling ( χχ c ) † NN c would give a term proportional to cos( φ/F − φ/f ) in thepotential. – 20 –o summarize, after imposing all the constraints the usual parameters of the relaxionpotential (1.1) in the model discussed in Sec. 3.2 can be written just in terms of Λ IR , v EW and M PL as can be seen from Table 2 and using Eq. (3.3). The dimensionless couplings ofthe relaxion potential and the relaxion mass are now determined as g = g (cid:48) = Λ IR M PL , m φ ∼ Λ IR M PL . (4.26)These couplings can thus be very small, provided that there is a large hierarchy betweenthe IR scale and the Planck scale. This in turn can be naturally achieved (i.e. without theinput of very small numbers) e.g. by means of the Goldberger-Wise mechanism to stabilizethe extra dimension [42].In addition to Λ IR and M PL , the input parameters of the model discussed in Sec. 3.2include the confinement scales Λ G F and Λ G f , the fermion masses m χ , m N and m L and thecouplings y and ˜ y . Of these, Λ G F and m χ are both required to be of order the IR scale.Since the corresponding fermions are localized on the IR brane, the former condition canbe naturally fulfilled, while the latter condition may require a coincidence of parametersas discussed around Eq. (4.18). After imposing this, the electroweak scale is determinedby Λ G f , y , ˜ y , m N and m L (plus Λ IR and M PL ) as follows from Eqs. (3.17) and (4.3). UsingEq. (4.19) and the requirement that M f (cid:46) v EW as well as imposing that m L (cid:38) v EW tosatisfy electroweak precision tests [72], we see that v EW (cid:46) m L (cid:46) π v EW (cid:112) log(Λ IR /v EW ) (4.27) y ˜ y m L log(Λ IR /m L )16 π (cid:46) m N (cid:46) y ˜ y v EW m L . (4.28)Using the range for m L in the range for m N , we then find y ˜ y v EW log(Λ IR /v EW )16 π (cid:46) m N (cid:46) y ˜ y v EW . (4.29)The fact that the electroweak doublets need to be close to the electroweak scale is thecoincidence problem discussed after Eq. (4.20). Note that the condition for the mass ofthe singlets can be naturally fulfilled if it dominantly arises from the loop process in Fig. 3(cf. Eq. (4.19)). Demanding that the right electroweak scale is obtained, we then see fromEq. (4.3) that Λ G f ≈ m L y ˜ y v EW Λ IR M PL , (4.30)where y and ˜ y need to be chosen such that Eqs. (4.21) and (4.22) for Λ G f as well as thelimits discussed below Eq. (4.22) are fulfilled.In the inflationary sector, the allowed window of Hubble scales and the minimumnumber of e-folds are given byΛ IR M PL (cid:46) H I (cid:46) Λ / IR M / PL , N e (cid:38) M PL Λ IR . (4.31)– 21 – , g (cid:48) m φ GeV Λ G f GeV m N GeV H I GeV N min e · − · − . · − , .
06] 4 · · − − . · − [10 − , · − ] 2 · Table 3 . Numerical values of the parameters for two benchmark points. For the first line, we set Λ IR = 4 · TeV, y = 2˜ y = 0 . and m L = 700 GeV, while for the second line, Λ IR = 500 TeV, y = 2˜ y = 0 . and m L = 450 GeV.
In Table 3, we give numerical values for two benchmark points. For the first one, weset the cutoff to its maximal allowed value in our model, Λ IR = 4 · TeV, and choose y = 2˜ y = 0 . m L = 700 GeV. For the second one, we choose the intermediate cutoffΛ IR = 500 TeV as well as y = 2˜ y = 0 .
04 and m L = 450 GeV. For both benchmark points,we assume that m N is dominantly generated by the loop process in Fig. 3 in which casethe lower bound in Eq. (4.28) is saturated (while our choices for m L satisfy the bound inEq. (4.27)). This in particular leads to M f ∼ v EW as used for Table 2. Both benchmarkpoints satisfy the constraints in Eqs. (4.21) and (4.22) in addition to the relevant constraintsfrom colliders and big bang nucleosynthesis as can be seen from Fig. 10 in Ref. [72]. Notethat for cutoffs Λ IR (cid:46)
500 TeV, constraints from big bang nucleosynthesis can becomeproblematic. Indeed from Eqs. (4.29) and (4.30) and the requirement that m N (cid:46) Λ G f ,we see that lower cutoffs necessitate smaller values for y ˜ y . If y ∼ ˜ y , this leads to longerlifetimes for the lightest N N c bound states which arise from the confinement of G f (see[72]). For too long lifetimes, these decay during big bang nucleosynthesis. One way out isto choose y ∼ (cid:29) ˜ y . The large coupling y then allows for relatively fast decays via an off-shell Z [72]. For example for Λ IR = 10 TeV, y = 1 , ˜ y = 10 − , m L = 800 GeV and assumingthat the mass of the lightest N N c bound state is ∼ G f , we find that its lifetime is oforder 1000 s while it can kinematically only decay into electron pairs or lighter states. Thisthen satisfies the corresponding limit on the lifetime of order 10 s [76]. Alternatively, onecould add new decay channels for the bound states which can allow them to decay fasterand sufficiently long before big bang nucleosynthesis. We leave a further investigation ofthis possibility for future work. As discussed in Sec. 3.1, the Higgs-dependent barrier in the relaxion potential needs todominate over the Higgs-independent one once the Higgs VEV has reached the electroweakscale. This requires that M f (cid:46) v EW which in turn necessitates to introduce new particlescoupled to the Higgs near the electroweak scale. We now discuss an interesting alternativepresented in Ref. [2]. The idea is to have another axion-like scalar σ with couplings in the– 22 –otential V ( φ, σ, H ) ⊃ g σ Λ σ + Λ f (cid:32) − ˜ g σ σ Λ + ˜ g φ
Λ + H M f (cid:33) cos (cid:18) φf (cid:19) (5.1)and arrange its evolution such that it cancels off the Higgs-independent barrier. Note thatwe have also included a term φ cos( φ/f ) in the potential which will be important. Theremaining terms involving the relaxion are as in Eq. (1.1). Similar to the relaxion, theshift-symmetry breaking couplings g σ and ˜ g σ of the field σ are taken to be very small.Let us assume that σ begins its evolution at some initial value σ (cid:38) (Λ + ˜ gφ ) / ˜ g σ sothat the Higgs-independent term in brackets in Eq. (5.1) is unsuppressed. Provided that g Λ (cid:46) Λ f /f , the barrier term for the relaxion then dominates over its sliding term and therelaxion is initially stuck in a local minimum. Meanwhile, the first term in Eq. (5.1) causes σ to slide and it eventually reaches the value σ (cid:39) (Λ + ˜ gφ ) / ˜ g σ . This removes the barrierfor the relaxion which can subsequently also slide down the potential. Both σ and φ thenroll down if they track each other according to the relation σ (cid:39) (Λ + ˜ gφ ) / ˜ g σ . The resultinggrowth of φ after a while causes the Higgs mass parameter to turn tachyonic and H beginsto grow too. Shortly afterwards, the Higgs-dependent barrier in Eq. (5.1) then becomes sobig that the relaxion stops again. Provided that σ can no longer cancel this barrier, therelaxion remains stuck. This mechanism works for certain ranges of parameters which wereview below. It then allows the backreaction from the Higgs to stop the relaxion once itsVEV has reached the electroweak scale even if M f (cid:29) v EW .We first present a construction to generate the required terms in the potential (seealso [10, 12]). This construction is, in fact, largely independent of the embedding intowarped space and can thus be used in other UV completions of the relaxion as well. Itis meant to serve as a proof of principle, and does not preclude the existence of simpleror more complete models. Let us introduce an additional U (1) gauge symmetry in thebulk. We identify the field σ with the 5th component of the gauge field after imposingappropriate boundary conditions. In order to generate the sliding term in Eq. (5.1), weadd an anomalous coupling of σ to a non-abelian gauge group G F σ on the UV brane usingthe construction in Sec. 2.2. We also introduce two chiral fermions ρ and ρ c on the UVbrane, with a Dirac mass m ρ and in respectively the fundamental and anti-fundamentalrepresentation of G F σ . These fermions have no explicit coupling to σ . Such a coupling isthen generated if we perform a chiral rotation of ρ or ρ c to remove the anomalous couplingof σ to G F σ . If the gauge group confines at some scale Λ G Fσ > m ρ , this gives rise to thepotential V ( φ, σ, H ) ⊃ | m ρ | Λ G Fσ cos (cid:18) σF σ + b ρ (cid:19) . (5.2)Here F σ (cid:29) f is the decay constant resulting from the anomalous coupling and b ρ = arg( m ρ )is the phase of the mass term. As we see later, we again have Λ = Λ IR . Expanding in σ around the linear part of the trigonometric potential gives the sliding term in Eq. (5.1)with g σ = | m ρ | Λ G Fσ F σ Λ IR (5.3)– 23 –p to factors of order one.Generating the coupling of σ to the periodic potential for φ is somewhat more involved.Notice that in Eq. (5.1), the periodic potential for φ appears with the same phase in thelast four terms (which for definiteness we have chosen as cos( φ/f )). Having the same phaseto a high precision in these a priori independent terms is in fact necessary for the double-scanner mechanism to work. Let us assume that σ instead couples to sin( φ/f ). Keepingthe phases for the other periodic terms fixed, the barrier in Eq. (5.1) then reads V ( φ, σ, H ) ⊃ Λ f (cid:32) − ˜ g σ σ Λ tan (cid:18) φf (cid:19) + ˜ g φ Λ + H M f (cid:33) cos (cid:18) φf (cid:19) . (5.4)Even if σ can then initially cancel off the Higgs-independent terms (which depending onthe initial value for φ may require σ (cid:29) Λ / ˜ g σ ), this cancellation is generically irreversiblyspoiled once φ starts rolling. The same holds for a phase difference less than π/
2, if theother periodic terms have different phases or if the decay constants in the periodic termsdiffer from each other (in all cases down to values which are determined by the smallcouplings in the potential).In order to ensure the required phase and period structure, we extend the gauge sym-metry G f in the bulk from Sec. 3.2 to the product group G f × G f × G f × G f . In addition,we impose discrete symmetries Z and Z (cid:48) that interchange the groups as follows: G f Z ←→ G f Z (cid:48) (cid:120)(cid:121) (cid:120)(cid:121) Z (cid:48) G f ←→ Z G f . (5.5)This in particular imposes that the underlying groups (e.g. SU ( N )) are the same for G f , G f , G f and G f . We couple the 5D gauge field A M that gives rise to φ to the gaugefield strengths of these four groups via Chern-Simons terms as in Sec. 2.1. We impose thatin the resulting anomalous couplings, φ transforms as φ ↔ − φ under Z , while it is evenunder Z (cid:48) (by choosing the coefficients c B in Eq. (2.13) to transform accordingly). Thisgives S ⊃ (cid:90) d x π φf (cid:15) µνρσ (cid:16) Tr (cid:104) G f µν G f ρσ (cid:105) − Tr (cid:104) G f µν G f ρσ (cid:105) + Tr (cid:104) G f µν G f ρσ (cid:105) − Tr (cid:104) G f µν G f ρσ (cid:105)(cid:17) , (5.6)where the decay constant f ∼ Λ IR is equal for all gauge groups by virtue of the symme-tries. We also add anomalous couplings of σ to G f and G f on the UV brane, using theconstruction in Sec. 2.2. We choose σ to be even under Z . This gives S ⊃ (cid:90) d x π σ ˜ F σ (cid:15) µνρσ (cid:16) Tr (cid:104) G f µν G f ρσ (cid:105) + Tr (cid:104) G f µν G f ρσ (cid:105)(cid:17) , (5.7)where the decay constant ˜ F σ (cid:29) f is equal for the two gauge groups by virtue of the Z .We do not add corresponding couplings to G f and G f though. This explicitly breaks the Z (cid:48) on the UV brane. – 24 –n the IR brane, we next introduce four pairs of chiral fermions η , η c , η , η c , η , η c and η , η c in the fundamental and anti-fundamental representation of G f , G f , G f and G f , respectively. The fermion pairs interchange under Z consistent with Eq. (5.5) but wechoose Z (cid:48) to be explicitly broken on the IR brane too. Including Dirac masses for the pairsof chiral fermions and higher-dimensional couplings to the Higgs, this gives S ⊃ (cid:90) d x (cid:18) m η [ η η c + η η c ] (cid:18) c η H Λ IR (cid:19) + m η [ η η c + η η c ] (cid:18) c η H Λ IR (cid:19) + h . c . (cid:19) , (5.8)where the fields are already canonically normalized and m η , m η (cid:46) Λ IR . The coefficients c η and c η are a priori different from each other and could be of order one or be suppressedby a loop factor. We can now perform the chiral rotations η → e i φf η η → e − i φf η η → e i φf + i σ ˜ Fσ η η → e − i φf + i σ ˜ Fσ η (5.9)while leaving η c , η c , η c and η c invariant. This moves φ and σ from Eqs. (5.6) and (5.7)into Eq. (5.8). We assume that the gauge groups confine at energies below the IR scale.By virtue of the Z which is unbroken everywhere, the confinement scales of G f and G f are identical, as are those of G f and G f . The condensates then are pairwise equal, (cid:104) η η c (cid:105) = (cid:104) η η c (cid:105) = Λ G f and (cid:104) η η c (cid:105) = (cid:104) η η c (cid:105) = Λ G f . The resulting potential at low energiesreads V ( φ, σ, H ) ⊃ | m η | Λ G f cos (cid:18) φf (cid:19) (cid:20) cos( b η ) + | c η | cos( d η ) H Λ IR (cid:21) + 4 | m η | Λ G f cos (cid:18) φf (cid:19) (cid:20) cos (cid:18) σ ˜ F σ + b η (cid:19) + | c η | cos (cid:18) σ ˜ F σ + d η (cid:19) H Λ IR (cid:21) , (5.10)where b η = arg( m η ), d η = arg( m η c η ), b η = arg( m η ) and d η = arg( m η c η ) are givenby the complex phases of the parameters. We have kept track of the phases in order toshow that all terms are proportional to cos( φ/f ) without relative phase shifts as required.This is guaranteed by the Z under which φ → − φ and the potential is invariant. However,note that we have tacitly assumed that the fermionic condensates are real. As we havediscussed at the end of Sec. 3.2 and in Appendix C, these phases are pion-like fields andthus dynamical. Doubling the spectrum in order to ensure that the potential for thesepions factorizes from the remaining potential then fixes their phases to the same value forall four condensates and leads to an additional overall minus sign in Eq. (5.10).On the other hand, the decay constants that appear in cos( φ/f ) between the first andsecond line of Eq. (5.10) are the same due to the Z (cid:48) in the bulk. However, note that thissymmetry is broken on the UV brane by the couplings for σ in Eq. (5.7). Nevertheless weexpect that this does not affect the decay constants for φ in Eq. (5.10) by virtue of thenon-renomalization properties of anomalous couplings (see e.g. Ref. [77]). Also any sucheffect would be strongly suppressed since ˜ F σ (cid:29) f . We leave a detailed study of this for– 25 – χ c η η c η η c η η c η η c G F (cid:3) ¯ (cid:3) – – – – – – – – G f – – (cid:3) ¯ (cid:3) – – – – – – G f – – – – (cid:3) ¯ (cid:3) – – – – G f – – – – – – (cid:3) ¯ (cid:3) – – G f – – – – – – – – (cid:3) ¯ (cid:3) Table 4 . Matter content on the IR brane with gauge representations for the double-scanner model. future work. Furthermore, we have allowed for the masses m η and m η being differentwhich breaks the Z (cid:48) also on the IR brane. This generically leads to a different runningof the gauge couplings of G f and G f compared to those of G f and G f and accordinglydifferent confinement scales Λ G f and Λ G f . However, it does not affect the decay constantsfor φ in Eq. (5.10) either as these are defined not involving the gauge couplings of theunderlying gauge groups (cf. Eqs. (2.15) and (2.16)). As follows from Eqs. (3.7) to (3.9),it is precisely the decay constants defined in this way which determine the period of theperiodic potentials. These periods are thus not affected by the differing running of thegauge couplings. Note also that the resulting difference between the confinement scalescan be made arbitrarily small for example by increasing the number of colours of the gaugegroups.We can match with the potential in Eq. (5.1) after expanding both Eqs. (5.2) and(5.10) in σ around regions where the corresponding trigonometric potentials are linear.Both trigonometric potentials can be in the linear part simultaneously for example for F σ ∼ ˜ F σ and b ρ − b η ∼ π . This also ensures that the right signs in the potential areobtained. In addition to Eq. (5.3), we can then identifyΛ f = | m η | Λ G f , M f = Λ IR (cid:112) | c η | , ˜ g σ = | m η | Λ G f | m η | Λ G f Λ IR ˜ F σ (5.11)up to factors of order one. Notice that Eq. (5.10) contains a term cos( φ/f ) cos( σ/ ˜ F σ ) H which is not included in Eq. (5.1). However, provided that for example | m η | Λ G f ≈ | m η | Λ G f and | c η | is somewhat suppressed compared to | c η | , this only gives a small correction tothe Higgs-dependent barrier and therefore does not affect the dynamics. Note that thiswould not be possible if the Z (cid:48) would be unbroken on the IR brane.As in Sec. 3.2, we next introduce fermions χ and χ c in the fundamental and anti-fundamental representation of a non-abelian gauge symmetry G F to generate the slidingterm for the relaxion and its coupling to the Higgs. These fermions also allow us to generatethe term φ cos( φ/f ) in Eq. (5.1). To this end, we consider the higher-dimensional operator S ⊃ (cid:90) d x (cid:18) c χη m χ m η Λ IR χχ c (cid:0) η η c + η η c (cid:1) + h.c. (cid:19) (5.12)– 26 –hich we expect to be present since the relevant fermions live on the IR brane. The fieldsare already canonically normalized and m χ , m η (cid:46) Λ IR . The coefficient c χη is again oforder one or suppressed by a loop factor. Performing the chiral rotations in Eqs. (3.7) and(5.9), we find below the confinement scales S ⊃ (cid:90) d x | c χη | | m χ | Λ G F | m η | Λ G f Λ IR cos (cid:18) φF + b χη (cid:19) cos (cid:18) φf (cid:19) , (5.13)where b χη = arg( c χη m χ m η ). Expanding the trigonometric function of φ/F around itslinear part, we can identify ˜ g = | c χη | | m χ | Λ G F Λ IR F (5.14)up to factors of order one. Note that the coupling in Eq. (5.12) with η η c , η η c replaced by η η c , η η c gives an additional term cos( φ/F + σ/ ˜ F σ ) cos( φ/f ) in the potential. We expectthat for example for | m η | Λ G f ≈ | m η | Λ G f and the corresponding coefficient c χη beingsomewhat suppressed compared to c χη , this does not significantly affect the dynamics.A summary of the matter content on the IR brane is given in Table 4. We have now generated all terms in the potential of Eq. (5.1) as well as the sliding termand coupling to the Higgs of the relaxion. In order to see if the potential parameters inEqs. (5.3), (5.11) and (5.14) (plus Eqs. (3.3) and (3.10) for g and g (cid:48) ) can take on valueswhich allow the double-scanner mechanism to work, we next discuss various constraints. Weagain need to ensure that the conditions discussed in Sec. 4.1 are fulfilled. In particular, theHiggs VEV once the relaxion stops is as before given by Eq. (4.3). One difference betweenthe potential parameters for the electroweak-scale barrier and the double scanner is that M f ∼ v EW in the former and M f ∼ Λ IR in the latter. But in both scenarios, by constructionthe Higgs-independent barrier plays no role and therefore only the combination Λ f /M f isrelevant for the dynamics of the relaxion and Higgs. Using Eq. (4.3) to fix the Higgs VEV,we can express this combination in terms of the decay constants and Λ F . Constraintson these parameters therefore apply for both the electroweak-scale barrier and the doublescanner. We can therefore conclude that the allowed range for the IR scale is again givenby Table 2. Note that Λ f and M f are different from those given in the table but thecombination Λ f /M f and the other parameters in the table agree for both scenarios. Inparticular, we again find that Λ ∼ Λ IR and that Λ F ∼ Λ G F ∼ m χ ∼ Λ IR is required.On the other hand, the constraint on Λ G f in Eq. (4.17) can always be fulfilled as followsfrom Eq. (4.4). Similarly, we see using Eqs. (4.3), (4.6) and (4.15) that the constraints inEqs. (4.22) and Eq. (4.25) are automatically fulfilled.There are new conditions that are specific to the double-scanner mechanism: The fields φ and σ track each other according to the relation σ (cid:39) (Λ + ˜ gφ ) / ˜ g σ once the barrier issufficiently small provided that [2] g ˜ g (cid:38) g σ ˜ g σ , (5.15)– 27 –here g is given by Eqs. (3.3) and (3.10). On the other hand, σ can no longer cancel thebarrier that the Higgs generates once it obtains a VEV if [2] g (cid:16) ˜ g − g λ (cid:17) (cid:46) g σ ˜ g σ (5.16)with λ being the Higgs quartic coupling. We have F ≈ F σ ≈ ˜ F σ since these decay constantsall arise from anomalous couplings on the UV brane. Comparing Eqs. (3.3) and (5.14), wealso see that ˜ g ∼ | c χη | g . On the other hand, the couplings g σ and ˜ g σ can a priori be quitedifferent. The gauge group G F σ that gives rise to the sliding term for σ can in principle belocalized on the UV brane. Nevertheless we should still demand that its confinement scaleis below the IR scale to ensure that the effective description for σ is valid at the energy scalewhere the potential is generated. In addition, we need to require that | m ρ | (cid:46) Λ G Fσ . In orderto study one concrete example, let us assume that | m η | Λ G f ≈ | m η | Λ G f (correspondingto Z (cid:48) being only weakly broken on the IR brane). This gives ˜ g σ ≈ g and g (cid:38) g σ . Theconditions in Eqs. (5.15) and (5.16) then simplify to g | c χη | (cid:38) g σ , g (cid:18) | c χη | − λ (cid:19) (cid:46) g σ . (5.17)This can be fulfilled for a wide range of g σ if | c χη | (cid:46) / (2 λ ). This example shows that theconditions for the double-scanner mechanism to work can be easily satisfied.Finally, let us consider loop corrections to the potential. The double-scanner mecha-nism cannot remove barriers from terms like cos ( φ/f ) [2]. Therefore these must be smallerthan the Higgs-dependent barrier when the Higgs reaches the electroweak scale. For loopcorrections from the Higgs, this translates to the condition Λ f (cid:46) πM f v EW and in turnto Eq. (4.12) which is less stringent than the already imposed Eq. (4.4). This also meansthat Eq. (4.11) can always be fulfilled. Furthermore, in addition to Eq. (5.12) we expecthigher-dimensional operators like S ⊃ (cid:90) d x (cid:32) c χχ m χ Λ IR ( χχ c ) + c η η m η Λ IR (cid:104) ( η η c ) + ( η η c ) (cid:105) + c η η m η Λ IR η η c η η c + h.c. (cid:33) (5.18)and similar terms involving η , η c , η , η c since the relevant fermions are all localized on theIR brane. The coefficients are again of order one or suppressed by a loop factor and arepartly determined by the Z . Assuming all parameters to be real for simplicity, below theconfinement scales this gives V ( φ, H ) ⊃ c χχ Λ F Λ IR cos (cid:18) φF (cid:19) + 4 c η η Λ f Λ IR cos (cid:18) φf (cid:19) . (5.19)The first term gives a correction to the sliding term for the relaxion which is negligible for c χχ (cid:46)
1. The second term, on the other hand, gives another type of barrier that cannot becancelled by the double-scanner mechanism. It is sufficiently suppressed compared to theHiggs-dependent barrier provided that Λ f (cid:46) v EW Λ IR / ( M f √ c η η ). This in turn leads to acondition which for example for c η η ∼ c η ∼ Conclusions
We have implemented the cosmological relaxation mechanism in a warped extra dimension.The relaxion potential trades the hierarchy between the Planck and electroweak scale fora technically natural hierarchy of decay constants. Warped extra dimensions are then anatural choice for its UV completion as they can generate a large hierarchy of scales purelyfrom geometry. In our construction, the relaxion is identified with the scalar component ofan abelian gauge field in the bulk, whose profile automatically has a small overlap with theUV brane. The warping generates the hierarchy from the Planck scale down to the scaleof the IR brane, which is then identified with the cutoff Λ of the relaxion potential. Fromthere onwards, the Higgs mass is relaxed down to its physical value.In Sec. 2, we have presented a model-building toolkit for generating anomalous cou-plings of the relaxion to new, strong sectors. Depending on the localization of the anomalousterms in the warped interval, hierarchically different decay constants for these couplingsmay be obtained, including decay constants which are super-Planckian. A benchmarkmodel coupling the relaxion to the Higgs was constructed in Sec. 3. The sliding term andits coupling to the Higgs is generated through the condensation of a Dirac pair of SM singletfermions that live on the IR brane. The barrier term, on the other hand, is generated closeto the electroweak scale by the condensation of vector-like fermions with the same quantumnumbers as one generation of SM leptons. These are also localized at the IR brane, andhave masses near or below the weak scale, but are a priori unrelated to it, leading to thewell-known coincidence problem. In order to avoid this and achieve a larger scale for thebarrier term, a more elaborate construction is required. In Sec. 5, we have presented awarped UV completion for one such scenario, the double-scanner mechanism of Ref. [2].The constraints for the model, both in general and those specific to the constructionof Sec. 3, were discussed thoroughly in Sec. 4, as well as the stability of the potential underradiative corrections. The requirement of obtaining the correct Higgs VEV may be used tofix the scale where the barrier term is generated in terms of the other parameters. Then,we have found that the scale where the sliding and scanning terms are generated needs tobe of order the IR scale. Since the SM fields live in the bulk, standard flavor constraintsof Randall-Sundrum models push the minimum value of the IR scale to Λ (cid:38)
10 TeV. Themaximum cutoff that we can achieve while ensuring that all theoretical and phenomeno-logical constraints are fulfilled is Λ ≈ · GeV.In this work, we have focused on inflation to provide a friction term for the slow-rollof the relaxion, but interesting alternatives such as the particle-production mechanism ofRef. [14] exist. It would be interesting to explore how such constructions may be im-plemented in warped space. The framework that we have described naturally allows forhierarchical decay constants for axion-like fields to be generated. As such it presents manyfurther opportunities for model building, not limited to relaxion models, such as appli-cations to inflation or dark matter. Another interesting possibility for generating thishierarchy is to consider a more general geometry with more than one AdS throat [78].– 29 – cknowledgments LdL thanks DESY for hospitality during his stay, where part of this work was completedand acknowledges support by the S˜ao Paulo Research Foundation (FAPESP) under grants2012/21436-9 and 2015/25393-0. BvH thanks Fermilab for hospitality while part of thiswork was completed. This visit has received funding/support from the European Union’sHorizon 2020 research and innovation programme under the Marie Sk(cid:32)lodowska-Curie grantagreement No 690575. BvH also thanks the Fine Theoretical Physics Institute at the Uni-versity of Minnesota for hospitality and partial support. The work of CSM was supportedby the Alexander von Humboldt Foundation, in the framework of the Sofja KovalevskajaAward 2016, endowed by the German Federal Ministry of Education and Research. The au-thors would like to thank Aqeel Ahmed, Enrico Bertuzzo, Zackaria Chacko, Giovanni Grillidi Cortona, Adam Falkowski, Gero von Gersdorff, Tony Gherghetta, Christophe Grojean,Roni Harnik, Arthur Hebecker, Ricardo D. Matheus, Enrico Morgante, Eduardo Pont´on,Pedro Schwaller, Marco Serone, G´eraldine Servant, Alexander Westphal and Alexei Yungfor useful discussions and comments.
A An anomalous coupling on the UV brane from two throats
The interaction in Eq. (2.17) should be understood as an effective coupling that can forexample arise from a Chern-Simons term in a second throat as we now briefly discuss.More details will be presented in [78]. To this end, we consider a setup with two warpedspaces which are glued together at a common UV brane but each slice still has its ownIR brane. For simplicity, we assume that both slices have the same AdS scale k . Let usdenote the coordinates along the extra dimension in the two throats as z and z , withmetric in each throat again given by Eq. (2.1). The coordinates match at the common UVbrane at z UV = z UV = 1 /k , while the IR branes are at z IR = e kL /k and z IR = e kL /k .We then introduce an abelian gauge boson which propagates in both throats (see e.g. [44]).We break the gauge symmetry on the two IR branes by imposing the boundary conditionsin Eq. (2.6) but leave it unbroken on the UV brane. This allows for one massless modefrom A which lives in both throats with wavefunction A = N a ( z i ) − φ in a given throat(the wavefunction is continuous at the UV brane). We will be interested in the case whereone throat is significantly longer than the other. The normalization constant N is thendominated by the longer throat. Choosing L > L without loss of generality, we have z IR (cid:29) z IR , which gives N (cid:39) g √ kL e − kL with g defined as before. Let us nextintroduce a Chern-Simons coupling of A M to a non-abelian gauge group, where we choosethe coupling to be localized in the second throat: S ⊃ (cid:90) d x (cid:90) z IR z UV dz c b π (cid:15) MNP QR A M Tr [ G NP G P Q ] . (A.1)Notice that the coupling to A from this resembles Eq. (2.17) with the δ -function replacedby the integral over A in the second throat. In the limit of a very short second throatwith z IR ∼ O (few) · z UV , we can think of this integral as a smeared-out δ -function. Corre-spondingly we expect the decay constant of φ in this limit to agree with Eq. (2.19). Let us– 30 –gain restrict ourselves to the zero-mode of the non-abelian gauge field. Integrating overthe extra dimension, we in particular find S ⊃ (cid:90) d x π φ ( x ) f B (cid:15) µνρσ Tr [ G µν G ρσ ] (A.2)with decay constant given by f B (cid:39) k e kL − kL c b g √ kL (A.3)or f B ≈ Λ IR / Λ IR . For a very short second throat with L (cid:29) L , this indeed agrees withEq. (2.19). On the other hand, the two-throat construction allows for more general choicesfor the decay constant, with a continuum between M PL / Λ IR and Λ IR (as Λ IR < Λ IR byassumption). The resulting phenomenology and the details of the construction will bepresented in [78]. B Chern-Simons terms from bulk fermions
In this appendix, we briefly review how charged bulk fermions can give rise to Chern-Simons terms. We consider a bulk fermion Ψ which couples to both the non-abelian gaugegroup and the U (1) from Sec. 2.1. The action reads S ⊃ (cid:90) d xdz √ g (cid:0) ¯Ψ i /D Ψ + m Ψ ¯ΨΨ (cid:1) , (B.1)where the covariant derivative is D M = ∂ M − i G M − iA M with G M being the non-abeliangauge field (and A M the U (1) gauge field). In order to see that this gives the same anomalyas a Chern-Simons term, we can perform a field redefinition [79, 80]Ψ → exp (cid:20) i (cid:90) zz d ˜ zA ( x, ˜ z ) (cid:21) Ψ , (B.2)where the constant z can be chosen according to convenience. However, the field redefini-tion is anomalous on the branes and transforms the action into (see [81–84]) S → S + (cid:90) d xdz (cid:18)(cid:90) zz d ˜ zA ( x, ˜ z ) (cid:19) (cid:15) µνρσ π Tr [ G µν G ρσ ] (cid:0) α UV δ ( z − z UV ) + α IR δ ( z − z IR ) (cid:1) . (B.3)The coefficients α UV and α IR depend on the boundary conditions on the two branes for theleft-handed component Ψ L of the bulk fermion (which in turn fixes the boundary conditionsof the right-handed component Ψ R ). If Ψ L is even (odd) on a given brane, α = 1( − α UV = − α IR in which case Ψ does not have a massless mode. FromEq. (B.3), we then get the anomalous coupling of φ in Eq. (2.15) with c B = α IR . (B.4) We note that, e.g. for SU ( N ), there is an additional SU ( N ) anomaly. It can be canceled by addinganother bulk fermion, uncharged under U (1), with opposite boundary conditions from Ψ. – 31 –otice that this is independent of z . In the opposite case α UV = α IR , on the other hand, c B depends on z . But then Ψ has a massless mode which contributes to the anomalyand which cancels the dependence on z . If the Chern-Simons term arises from such abulk fermion, any perturbative contribution to the potential for A can be sufficientlysuppressed by making the bulk mass of the fermion somewhat larger than the AdS scale(see e.g. [34, 47]). C Pion-like fields in the relaxion potential
In this appendix, we include the pion-like fields which arise from the condensing fermions onthe IR brane and which contribute to the potential. Let us focus on χ, χ c for definiteness.As usual, we can parametrize the pseudo-Nambu-Goldstone boson corresponding to thebreaking of the chiral symmetry of χ, χ c by the σ -model field U = exp( iπ χ /f χ ) with adecay constant of order f χ ∼ Λ G F . After confinement then (cid:104) χχ c (cid:105) = Λ G F U . From Eq. (3.8),this gives V ( φ, H ) ⊃ m χ Λ G F (cid:18) H Λ IR (cid:19) cos (cid:18) φF + π χ f χ (cid:19) , (C.1)where for simplicity we again ignore phases and prefactors. Since F (cid:29) f χ , generically π χ settles into its minimum π min χ = f χ π − f χ φ/F first after which the potential becomesindependent of φ . This problem is remedied for example by introducing another pair ofchiral fermions ˜ χ ˜ χ c with the same quantum numbers. Instead of Eq. (3.6) we then have S ⊃ (cid:90) d x (cid:18) H Λ IR (cid:19) [ m χ χχ c + m ˜ χ ˜ χ ˜ χ c ] + h.c. . (C.2)Similar to the up and down quark in the Standard Model, the fermions transform underan approximate SU (2) L × SU (2) R symmetry which is spontaneously broken to a diago-nal SU (2) V by the condensates and explicitly but weakly broken by their masses. Thecorresponding pseudo-Nambu-Goldstone bosons are parametrized as U = e i Π χ /f χ with Π χ = (cid:32) π χ √ π + χ √ π − χ − π χ (cid:33) . (C.3)We next perform the chiral rotation χ → e i φ F χ , ˜ χ → e i φ F ˜ χ (C.4)with χ c and ˜ χ c left invariant to remove the coupling of φ to Tr (cid:2) G Fµν G Fρσ (cid:3) in Eq. (3.4).For this choice of chiral rotation, no kinetic mixing between the relaxion and the pions isinduced (see Ref. [85]). Choosing m χ = m ˜ χ for simplicity, from Eq. (C.2) we get below theconfinement scale V ( φ, H ) ⊃ m χ Λ G F (cid:18) H Λ IR (cid:19) cos (cid:18) φ F (cid:19) cos (cid:18) π χ f χ (cid:19) , (C.5)where π χ ≡ (cid:113) ( π χ ) + 2 π + χ π − χ . The potential for the pions and relaxion thus factorizes andno longer vanishes once the pions settle into their minimum. This is similar to what happens– 32 –or the axion and the pion of the Standard Model, see Ref. [59]. For the generalization ofthe potential to the case m χ (cid:54) = m ˜ χ , see also Ref. [59]. The potential after minimizationwith respect to the pion then still leads to a nonvanishing potential for the relaxion butthe latter is no longer a simple cosine. References [1] P. W. Graham, D. E. Kaplan, and S. Rajendran,
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