A warped relaxion

Journal of High Energy Physics, Jul 2018

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 field and show how hierarchically different decay constants for this field can be achieved by different localizations of anomalous terms in the warped space. This framework may also find applications for other axion-like fields. The cutoff of the relaxion model is identified as the scale of the IR brane where the Higgs lives, which can be as high as 106 GeV, while above this scale warping takes over in protecting the Higgs mass.

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A warped relaxion

HJE warped relaxion Nayara Fonseca 0 1 2 Benedict von Harling 0 1 2 Leonardo de Lima 0 1 2 Camila S. Machado 0 1 2 3 Johannes Gutenberg-Universitat Mainz, 0 55099 Mainz , Germany 1 Av. Edmundo Gaievski 1000 , 85770-000 Realeza , Brazil 2 Notkestrasse 85 , 22607 Hamburg , Germany 3 PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics We construct a UV completion of the relaxion in a warped extra dimension. We identify the relaxion with the zero mode of the fth component of a bulk gauge eld and show how hierarchically di erent decay constants for this eld can be achieved by di erent localizations of anomalous terms in the warped space. This framework may also nd applications for other axion-like elds. The cuto of the relaxion model is identi ed as the scale of the IR brane where the Higgs lives, which can be as high as 106 GeV, while above this scale warping takes over in protecting the Higgs mass. Cosmology of Theories beyond the SM; Field Theories in Higher Dimensions - A 1 Introduction 2 Hierarchical decay constants from warped space 2.1 Anomalous couplings from the bulk 2.2 Anomalous couplings from the UV brane 3 Generating the relaxion potential 3.1 General setup 3.2 A warped model 4 Conditions for successful relaxation 4.1 General conditions 4.2 Conditions on the warped model 5 Warping the double-scanner mechanism 5.1 A UV completion 5.2 Constraints 6 Conclusions A An anomalous coupling on the UV brane from two throats B Chern-Simons terms from bulk fermions C Pion-like elds in the relaxion potential 1 4 the cosmological relaxation of the electroweak scale [1] (see also [2{12]). It relies on the scanning of the Higgs mass parameter by a new eld, the relaxion, and a back-reaction { 1 { mechanism that is triggered when the vacuum expectation value (VEV) of the Higgs has reached the electroweak scale, making the relaxion evolution stop.1 This is a radical change of paradigm as it implies that the naturalness problem of the Standard Model ceases to be a reason to expect new physics close to the TeV scale. In what follows we review the relaxation mechanism for which an axion-like scalar is introduced which couples to the Higgs doublet H via the potential which sets the Higgs mass parameter, f the decay constant of the HJEP07(218)3 Here relaxion, the Higgs quartic coupling, g and g0 are small dimensionless couplings, and f (H) is a scale which depends on the Higgs VEV. Assuming a classical time evolution with slow-roll conditions, the second-last term in eq. (1.1) causes the relaxion to move downwards following its potential. The e ective Higgs mass parameter in the background, the rst term in parenthesis in eq. (1.1), then varies accordingly. The relaxion is assumed to start with a VEV such that this mass parameter is initially positive. Due to the evolution of the relaxion, the mass parameter then eventually turns tachyonic, triggering electroweak symmetry breaking. In the presence of a Higgs VEV, the oscillatory barrier from the last term grows, until its slope matches the slope of the linear term. For technically natural parameters in the potential, this causes the relaxion to stop once the Higgs VEV has reached the electroweak scale. There must be some mechanism to dissipate the kinetic energy of the relaxion during its evolution such that the eld does not overshoot the barriers. If the dynamics happens during a period of in ation, Hubble friction can provide the dissipation necessary to slow down the eld [1]. As an alternative to in ation, one can also consider friction due to particle production as proposed in ref. [14] or nite temperature e ects in the early universe as in ref. [15]. Note that the linear terms in are in con ict with the assumption that the relaxion is 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 a period much larger than f . This is realized if the potential takes the form [16{18]:2 V ( ; H) 2H2 + H4 + 4F (H) cos + f4 (H) cos F f ; (1.2) where F f is another decay constant and F (H) another scale that depends on the Higgs in such a way as to reproduce the second and fourth term in eq. (1.1) after expanding in =F . An interesting possibility to obtain this type of potential is the clockwork construction which was rst realized for axion-like elds in refs. [16, 17] and generalized for applications other than the relaxion in ref. [ 26 ]. Further developments regarding the 5D 1See also N naturalness [13], where instead of multiple vacua, many copies of the Standard Model are 2See also refs. [19{24] for similar earlier ideas in in ation model building. For the viability of the relaxation mechanism in string theory in the context of axion monodromy, see ref. [25]. { 2 { continuum limit of the clockwork can be found in refs. [27{30]. Besides the clockwork, one can also generate a potential of the form in eq. (1.2) in realizations inspired by dimensional deconstruction [31, 32], as in ref. [18]. In this work, we show how the required potential for the relaxation mechanism to work can be naturally obtained by embedding the relaxion and Higgs into a warped extra dimension. We consider a slice of AdS5 space which is bounded by two branes, as in the RandallSundrum model [33]. However, in our setup the IR scale or warped-down AdS scale is not of order TeV but can be much larger. We introduce a U(1) gauge eld in the bulk of the extra dimension and break the gauge symmetry on the two branes. The 5th component A5 of the gauge eld then gives rise to one massless scalar mode in 4D which we identify with the laxion as in eq. (1.2) with periods given by the decay constants.3 Due to the warping, these periods can thus naturally be hierarchically di erent as required. We embed the Higgs at or near the IR brane. Its mass parameter is then naturally of order the IR scale which we identify with the cuto of the relaxion theory. The required Higgs-relaxion couplings can be obtained by introducing fermions on the IR brane with higher-dimensional or Yukawa couplings to the Higgs. To summarize, the warping does two things: rstly, 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 g0 in eq. (1.1). Secondly, it provides a UV completion4 for the relaxion. The relaxation mechanism protects the Higgs up to the IR scale above which warping takes over.5 We illustrate this in gure 1. Alternatively, one can think of the relaxation mechanism in our construction as a solution to the little hierachy problem of RandallSundrum models.6 As is well-known, various experimental 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 these models is of order 10 TeV or above. This means that a residual tuning in the permille range is necessary to generate the electroweak scale. In our construction with warping and the relaxion, on the other hand, no such tuning is required. 3A potential for A5 can be generated perturbatively if the underlying gauge eld is coupled to charged bulk states. In the non-abelian case (see e.g. [34]), this includes the gauge elds themselves due to the nonlinear interactions, while the abelian case requires charged scalars or fermions in the bulk (see e.g. [37]). Here we consider a U(1) gauge eld and do not add charged bulk states as we are interested in generating a non-perturbative potential for A5. 4As a caveat, we should stress that the Randall-Sundrum model itself requires a UV completion. In particular, near the IR brane gravity becomes strongly coupled at energies not far above the IR scale. Near that brane, the UV completion therefore needs to kick in at correspondingly low scales. There are known UV completions to the Randall-Sundrum model in string theory [38, 39]. 5See [6, 10, 12] for how the relaxation mechanism can protect the Higgs up to some high supersymmetrybreaking scale instead. 6See [40] for an alternative solution where an accidental form of supersymmetry protects a little hierarchy between the electroweak scale and the IR scale of the Randall-Sundrum model. { 3 { the Higgs into a warped extra dimension. The hierarchy problem is then solved in two steps: the relaxation mechanism protects the Higgs mass up to the IR scale (which can be much larger than the electroweak scale) and from there warping provides protection till the Planck scale. We nd that for an e ective anomalous coupling localized on the UV brane, the decay constant is of order MP2L= IR with MPL and IR being the Planck and IR scale. For an anomalous coupling in the bulk, we instead nd a decay constant of order IR. We then identify F = MP2L= IR and f = IR. Generating a suitable barrier f4 (H) cos( =f ) for the relaxion requires some additional structure. The reason is that this term generically contains a contribution which is independent of the Higgs and which could stop the relaxion before the Higgs VEV has reached the electroweak scale. To avoid this problem, we consider two di erent options. One employs a construction from ref. [1] for which new fermions are introduced which couple to the Higgs. If the masses of these fermions are near the electroweak scale, the Higgs-independent barrier can be su ciently small. The drawback of this construction is a coincidence problem as it requires to introduce the fermions at a scale which is dynamically generated by the relaxation mechanism and thus a priori determined by completely di erent parameters. An interesting alternative is the so-called double-scanner mechanism of ref. [2] (see also [10]). To this end, one introduces another axion-like scalar which dynamically cancels o the Higgs-independent barrier. We identify this axion-like scalar with the 5th component of another U(1) gauge eld in the bulk of the extra dimension. We then show how the potential which is required for the double-scanner mechanism can be obtained. This construction is largely independent of the embedding into warped space and can therefore also be useful for other UV completions of the relaxion. For both options to generate the barrier, we discuss the relevant theoretical and phenomenological constraints for successful relaxation. The highest cuto and IR scale consistent with these constraints in our warped implementation of the relaxation mechanism is = IR . 106 GeV. The plan of this work is as follows. In section 2, we discuss the properties of the A5 and show how hierarchical decay constants can be obtained. In section 3, we generate the desired potential for the relaxation mechanism. We analyse the relevant constraints to guarantee a successful relaxation of the electroweak scale in section 4. In section 5, we present our implementation of the double-scanner mechanism and we conclude in section 6. Additional details are given in three appendices. 2 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 slice { 4 { HJEP07(218)3 where FMN is the U(1) eld strength, g5 the 5D gauge coupling and pg = a5(z). In order to eliminate the mixing between A and A5, we add the gauge xing term (see e.g. [34, 43]) S5D Z d4x dz pg 1 2g52 The bulk equations of motion for the 4D component A and the 5th component A5 then read S5D Z d4x dz pg 1 4g52 FMN F MN ; = 0 = 0 : (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) HJEP07(218)3 of AdS5 space with metric in conformal coordinates given by where a(z) = (kz) 1 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 zUV = 1=k and the IR brane at zIR = ekL=k. The length L of the extra dimension can be stabilized for example by means of the Goldberger-Wise mechanism [42]. The e ective 4D Planck scale for this space is given by MP2L ' M 3=k, where M is the 5D Planck scale. We will assume that the Planck scale and the AdS scale are of the same order of magnitude (and will later often equate them). For later convenience, let us also de ne the IR scale IR k e kL. Let us consider a U(1) gauge boson in the bulk. Its action is given by A5(x; z) = h(z) (x) ; { 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 boundary conditions on A . For consistency, this then requires to impose Neumann boundary conditions for A5. Together the boundary conditions read A jUV;IR = 0 ; Alternatively we could break the gauge symmetry with Higgs elds on the two branes (see e.g. [44, 45]). The above boundary conditions are then obtained in the limit of their VEVs going to in nity. In unitary gauge, ! 1, the bulk equation of motion for A5 gives = 0 : Notice that this equation is consistent with the boundary conditions and there is thus one massless mode from A5. Its other Kaluza-Klein modes are all eaten by A . In particular, there is no massless mode from A , consistent with the fact that the gauge symmetry is broken. As usual, the A5 massless mode can be parameterized as with the 5th component of a U(1) gauge eld in the bulk. Its wavefunction is then localized towards the IR brane. The Higgs is localized on (or near) the IR brane. The UV brane corresponds to the Planck scale. We draw the IR brane with a dashed contour as a reminder that the IR scale in our model can be much larger than the usual TeV scale of the Randall-Sundrum model. where h(z) is its pro le along the extra dimension. From eqs. (2.6) and (2.7), we then see that h(z) = N a(z) 1. Demanding canonically normalized kinetic terms for (x), the normalization constant N of the wavefunction is determined by (2.9) (2.10) (2.11) (2.12) N g 2 5 2 Z zIR dz zUV a(z) = 1 : h(z) ' g4p2kL e kLkz : = B z2 + C ! + 2B N k : At this point, the relaxion is thus an exact Nambu-Goldstone boson which non-linearly realizes a remnant global U(1). By virtue of the 5D gauge invariance, no 5D local, higherdimensional operators can break this shift symmetry (see [46] for a detailed discussion). A { 6 { For kL g4 p 1, this gives N ' g4p2kL e kL, where we de ne the dimensionless coupling g5= L. Altogether, the wavefunction of the massless mode then reads The wavefunction is thus peaked towards the IR brane (see gure 2 for a sketch of the wavefunction pro le in the extra dimension). Furthermore, the fact that N ! 0 for zIR ! 1 shows that the A5 massless mode is indeed localized in the IR. Performing a 5D gauge transformation, AM (x; z) ! AM (x; z) + @M (x; z), we see that the boundary conditions in eq. (2.6) and the bulk equation of motion in eq. (2.7) remain invariant only for the subset of transformations with B and C being independent of x and z. The remaining symmetry in 4D is thus global, again consistent with the fact that the gauge symmetry is broken. Under this remnant symmetry, the massless mode transforms as potential for the relaxion could be generated by non-local e ects in the presence of bulk states which are charged under the U(1) but we assume such states to be absent from the theory.7 Instead we introduce anomalous couplings of the relaxion to con ning non-abelian gauge groups. A potential then arises from instantons, similar to what happens for the axion 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 naturally explain the required hierarchy between the decay constants in the relaxion potential. Anomalous couplings from the bulk Let us add a non-abelian gauge group in the bulk, whose eld strength and coupling we denote respectively as GNP and g5c. We choose boundary conditions for the gauge eld such that the 4D gauge symmetry remains unbroken on the branes. Its tower of Kaluza-Klein modes then contains one massless mode which is the 4D gauge boson. We next introduce a Chern-Simons coupling of the U(1) gauge eld to this gauge group. Including the kinetic term, the action reads where cB is a dimensionless constant and the normalization is chosen for later convenience.8 Under a U(1) gauge transformation AM (x; z) ! AM (x; z) + @M (x; z), the action trans(2.13) forms as S5D ! S5D Z d4x dz (x; z) c B are thus no gauge anomalies. In the 4D e ective theory, this gives rise to an anomalous coupling for . Let us restrict ourselves to the massless mode of the non-abelian gauge eld, whose eld strength we denote as G . Integrating over the extra dimension, eq. (2.13) then in particular gives S4D Z d4x 1 2(g4c)2 Tr [G G ] + (x) 16 2fB where g4c = g5c=pL is the gauge coupling of the massless mode. The decay constant is given 7Alternatively, for example for bulk fermions charged under the U(1) it is su cient if their masses are somewhat larger than the AdS scale in which case any perturbative contribution to the potential is highly 8Note that a factor of 2 arises from the normalization Tr[T aT b] = 12 a;b of the generators of the non{ 7 { HJEP07(218)3 which is of order the IR scale IR and thus warped-down. From eqs. (2.12), (2.14) and (2.15), we see that reproduces the anomaly under a transformation = Bz2. In appendix B, we brie y 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 potential for A5 can be su ciently suppressed. Nevertheless, in the remainder of this paper we will never assume any charged bulk states and will instead include the Chern-Simons terms directly into our e ective 5D theory. Note that eq. (2.13) also yields couplings of to the higher Kaluza-Klein modes of the non-abelian gauge eld. As eq. (2.15) for the massless mode, these couplings are total derivatives (see e.g. ref. [48]) and therefore do not contribute perturbatively to the potential for . We will later assume that the non-abelian gauge group con nes in order to generate a non-perturbative potential for . But we will choose the con nement scale below the IR scale and thus below the Kaluza-Klein masses. The Kaluza-Klein modes of the non-abelian gauge group therefore do not contribute non-perturbatively to the potential either. 2.2 Anomalous couplings from the UV brane We now discuss how a decay constant which is much larger than IR can be obtained. To this end, we consider an e ective anomalous coupling of A5 which is localized on the UV brane [43], S5D Z d4x dz (z zUV) 16 2 k cUV A5 where cUV is a dimensionless constant and GMN is the eld strength of a non-abelian gauge eld in the bulk. As we outline in appendix A, this interaction can for example arise as an e ective 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 UV brane and with @5 (x; z) instead of (x; z). Let us again restrict ourselves to the massless mode of the gauge eld, whose eld strength we denote as G . Using the wavefunction of the massless mode of A5 from eqs. (2.8) and (2.10), this gives with decay constant given by [43] S4D = Z d4x 1 (x) 16 2 fUV p2kL (2.18) (2.19) or fUV MP2L= IR. We see that a warped-up decay constant, much larger than the cuto , appears naturally in this case. This large decay constant can be intuitively understood as being of order the natural scale MPL on the UV brane times an inverse suppression factor from the wavefunction overlap of A5 with the UV brane. Note that super-Planckian decay constants may be constrained by the weak gravity conjecture in theories of quantum gravity [49] (see also [50{53]). Given that the relaxion is an axion-like eld, the conjecture necessarily restricts its eld excursion ( =g0) to be sub-Planckian, setting a lower bound on the coupling g0 in the potential (1.1). The { 8 { weak gravity conjecture is then at odds with any relaxion model with trans-Planckian eld excursions, including our proposal. On the other hand, there are known loopholes to the conjecture [54{58]. For instance, the application of the conjecture to e ective eld theories may result in a much weaker bound on the coupling g0 [57]. Furthermore, in [58], a better understanding of the conclusions of [57] is achieved by considering a string theory embedding. There it is shown that if a clockwork model is successfully embedded in string theory, one may in principle obtain a large cuto , avoiding the naive bound from the weak gravity conjecture, as long as the number of sites in the construction is large. We conclude that two hierarchically di erent decay constants can be obtained, depending on the localization of the anomalous interactions in the warped space. For the relaxion, portional to the warp factor, the potential in eq. (1.2) does not respect a discrete shift symmetry since, in general, F=f is a non-integer number. This is a consequence of the non-local nature of the residual symmetry transformation = Bz2 + C in eq. (2.11) which explicitly depends on the localization. In the following, we build an explicit model that makes use of this toolkit to generate a phenomenologically viable potential in the form of eq. (1.2). 3 3.1 Generating the relaxion potential General setup Let us next discuss the relaxion parameters in more detail and how they can be understood in terms of our UV model. Provided that electroweak symmetry remains unbroken in the con nement phase transition which generates the periodic potentials in eq. (1.2), F;f (H) both depend quadratically on the Higgs (plus generically higher even powers of the Higgs which are, however, not important in the following).9 We can then parametrize 4F;f (H) = 4 F;f 1 + H2 ! MF2;f ; g = 4 F ; F 3 g0 = 4 F F MF2 9As proposed in [1], one can also use the QCD axion as the relaxion. The last term in eq. (1.1) is then the usual QCD axion potential which depends linearly on the Higgs (see e.g. [59]). However, barring additional model building, this spoils the axion solution to the strong CP problem. See also [60{62] for solutions to the strong CP problem in the context of the relaxion. { 9 { where F;f and MF;f can be understood as the scales where the periodic terms and higherdimensional couplings to the Higgs are generated, respectively. The potential in eq. (1.2) then reads V ( ; H) = 2H2 + H4 + 4 F 1 + H2 MF2 cos F + 4 f 1 + f cos : (3.2) H2 ! Mf2 For simplicity, we have dropped terms which may be generated at higher loop-order. We will discuss these terms later in section 4. Assuming that is in the linear regime of the F=2 mod 2 , we can expand it for F=2 . F . After the , this gives the linear part of the relaxion potential in eq. (1.1) low-frequency cosine, rede nition F=2 ! with the identi cations (3.1) (3.3) The last term in eq. (3.2) stops the relaxion once the Higgs VEV has reached the electroweak scale. For this to work, we need to ensure that Mf . vEW, otherwise the Higgs-independent barrier proportional to cos( =f ) would stop the relaxion already before the Higgs VEV has obtained the right value. Note also that the Higgs-independent barrier receives corrections from closing the Higgs loop in the Higgs-dependent one and will thus generically be present. We discuss radiative corrections to the potential in more detail in section 4. But to get a sense of the scales involved, we already note here that radiative stability of the potential demands that f2 . 4 vEWMf and F . 4 MF . To obtain Mf . vEW requires that the higher-dimensional coupling of the Higgs to the periodic potential is generated near the electroweak scale. In the next section, we make 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 new particles at a scale which is dynamically generated by the relaxation mechanism and is thus determined by a priori completely unrelated parameters. One way around this problem is the double-scanner mechanism of ref. [2]. To this end, one introduces another axion-like eld which dynamically cancels o the Higgs-independent barrier in eq. (3.2). This allows HJEP07(218)3 vEW.10 We discuss a UV completion of the relaxation mechanism to work even for Mf this scenario in section 5. 3.2 A warped model We now build a simple explicit model that successfully generates the needed terms in the Higgs-relaxion potential at a phenomenologically viable scale, making use of the results of section 2. We assume that the Higgs is localized on or near the IR brane, so that its mass is warped down to the IR scale (see gure 2). We note that it may also be possible to implement the relaxation mechanism in a model where the Higgs is instead localized on the UV brane. As usual, the relaxion can only protect the Higgs up to some cuto signi cantly below the Planck scale. Such a model would therefore require a UV completion above this cuto on the UV brane. We leave a study of this possibility to future work. As we nd later, the highest IR scale that we can achieve in our implementation of the relaxation mechanism (while still solving the hierarchy problem) is below the GUT scale. If the remaining Standard Model elds are then also localized on the IR brane, higher-dimensional operators violating baryon number lead to too fast proton decay [63]. In order to suppress these 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 and the right-handed top live nearer to the IR brane. This has the added advantage that the hierarchy of Yukawa couplings can then be generated from the warping too. The IR scale in our model can be high enough, on the other hand, to ensure that oblique corrections and avour- and CP -violating processes are su ciently suppressed without imposing custodial or avour symmetries. 10Another proposal for the relaxion that does not require new physics close to the electroweak scale is the particle-production mechanism of ref. [14]. We identify the relaxion with the 5th component of a U(1) gauge eld in the bulk. In order to generate a potential for this eld, we add two non-abelian gauge groups Gf and GF which also live in the bulk. We assume that these gauge groups con ne at the scales GF , respectively. In order to ensure that con nement can be discussed using only the zero-modes of the bulk gauge elds, we take GF to be below the IR scale. This can always be arranged by choosing the 5D gauge couplings and ranks of the gauge groups appropriately. We assume anomalous couplings of the relaxion to the eld strengths G f and GF of the massless 4D gauge elds corresponding to Gf and GF , respectively: S4D Z d4x (x) As we have discussed in section 2, these can arise from a Chern-Simons coupling in the bulk and an e ective anomalous coupling of A5 on the UV brane. But for now, we only assume that F f and postpone a concrete choice for the decay constants to section 4. On the IR brane, we add a pair of chiral fermions and c in the fundamental and antifundamental representation of GF , respectively. These fermions transform under a chiral symmetry which we assume to be broken only by a Dirac mass m . This allows for the terms in the action S5D Z where gIR is the induced metric determinant on the IR brane. We have included a higherdimensional coupling to the Higgs which is generically present and which we expect to be suppressed by a scale near the Planck scale. Note that we will use the symbol H for both the SU(2)-doublet Higgs eld, writing the singlet combination jHj2 as H2 for simplicity, and its VEV. It will be clear from context which one is meant. For simplicity, we also ignore any numerical prefactors for now and set k = MPL. Similarly, we assume that all parameters are real. We will reinstate prefactors and phases later on. Performing the integral over the extra dimension and canonically normalizing the elds gives S4D d4x m 1 + c + h.c. ; Z . Z H2 2 IR H2 2 IR ! ei =F ; where we have rede ned e kLm c. Note in particular that m rede nition ! m , e kLH ! H, e 3kL=2 ! and similarly for IR after the rede nition. Let us next perform the eld while c is left invariant. Due to the non-trivial transformation of the path integral measure, this chiral rotation removes the coupling of in eq. (3.4) and transforms eq. (3.6) to { { { If m is below the con nement scale of GF (which in turn is below to the Higgs-relaxion potential after con nement. Parametrizing11 h IR), this term contributes ci = 3GF , this gives HJEP07(218)3 cos F : This has the same form as the potential with period F in eq. (3.2), including the coupling to the Higgs. We can then make the identi cations 4F = m 3GF ; MF2 = I2R : Next we need to generate the potentials with smaller period f . To this end, we use a construction from ref. [1] and add fermions L and N on the IR brane with the same Standard 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 Gf . We also include fermions Lc and N c in the conjugate representations. Together they allow for the terms in the action Notice that we have not included a higher-dimensional coupling to the Higgs. It could be present but will be subdominant as we will see momentarily. Performing the integral over the extra dimension and canonically normalizing the elds gives Z d4x mL LLc + mN N N c + y HLN c + y~ HyLcN + h.c. ; (3.12) where we have rede ned e kLmL ! mL, e kLH ! H, e 3kL=2L ! L and similarly for mN , N and the conjugated elds. Note in particular that mL; mN . IR after the rede nition. Assuming that mN satis es yy~H2 mL and restricting to a region in eld space where the Higgs VEV m2L, we can integrate out L and Lc. This gives S4D Z d4x mN 11This is thus our de nition of the scale GF . (3.14) (3.15) (3.16) (3.17) We can then perform the chiral rotation N ! ei =f N ; while N c is left invariant. This removes the coupling of transforms eq. (3.13) to to Tr hGf G f i in eq. (3.4) and relaxion potential after con nement. Parametrizing hN N ci = 3Gf , this gives Provided that mN is below the con nement scale of Gf , this term contributes to the Higgscos f This has the form of the potential with period f in eq. (3.2), including the coupling to the Higgs. We can then make the identi cations f4 = mN 3Gf ; Mf2 = mN mL yy~ : For su ciently small mN and mL, this allows for Mf . vEW as required in a technically natural way. Notice that if we had instead relied on the higher-dimensional operator in eq. (3.5) to generate the barrier, we would have obtained Mf IR vEW. We discuss constraints on the parameters of this construction in more detail in section 4. A summary of the matter content on the IR brane is given in table 1. We next reinstate the numerical prefactors and the phases of the parameters which we have ignored so far. Let us denote the prefactor of the Higgs coupling in eq. (3.5) as c H . We absorb possible phases in the fermionic condensates h (relaxion-independent) -terms for GF and Gf into the mass parameters m and mN ; mL, respectively. Redoing the derivation above then gives ci and hN N ci and any cos F + b + jc H j 2 cos + b H H2 IR F jyy~j H2 jmN mLj 3 + 2jmN j Gf cos f + bN f cos + bNH ; (3.18) where the complex phases are given by b = arg(m ), b H = arg(m c H ), bN = arg(mN ) and bNH = arg(yy~=mL). Note that this does generically not match the form of the potential in eq. (3.2). Nevertheless the relaxation mechanism can still work. Indeed expanding the rst two terms in the linear part of the cosines again gives the sliding term for the relaxion and its linear coupling to the Higgs. In order to ensure that these terms have the same sign as required, we need to demand that b b H . As before, the Higgs-independent barrier in the third term should be too small to stop the relaxion by itself. It is then negligible for the dynamics and the phase bN has no consequences. The phase bNH in the Higgs-dependent barrier in the fourth term, on the other hand, slightly shifts the minimum where the relaxion eventually stops but has no other consequences either. To ensure that our calculation of the potentials is applicable, the masses of the fermion pairs ; c and N; N c need to be below their respective condensation scales. This means that the chiral symmetries under which these fermion pairs transform are only weakly broken at the con nement scales. We then expect corresponding pseudo-Nambu-Goldstone bosons in the spectrum of composite states. As we discuss in appendix C, their contribution to the potential factorizes from the remaining potential and they can be trivially integrated out if the spectrum of fermions is doubled. 4 Conditions for successful relaxation We now discuss various conditions that need to be ful lled for the relaxation mechanism to be viable. In section 4.1, we derive general conditions on the parameters in the relaxion potential in eq. (3.2). In section 4.2, we then discuss additional conditions that arise in our warped model with a barrier at the electroweak scale. 4.1 General conditions We begin our discussion of the evolution of the Higgs and relaxion with the Higgs masssquared being positive and of order 2. In order to allow the relaxion to subsequently turn the Higgs mass tachyonic, its average VEV ~ during this stage of the evolution needs to satisfy Since the left-hand side is bounded by 1, this in particular implies the condition The relaxion stops rolling down its potential when the derivatives of the periodic terms balance each other. We will nd below that MF vEW and the term proportional to cos( =F ) is thus dominated by the Higgs-independent part. On the other hand, the term proportional to cos( =f ) needs to be dominated by the Higgs-dependent part as discussed in section 3. The relaxion then stops once the Higgs VEV becomes where we have set sin( ~=F ) 1. This is a good approximation as long as cos( ~=F ) is not very close to its extrema. The parameters need to be chosen such that the combination on the right-hand side gives the electroweak scale vEW. In the following, we will use this relation to trade f for vEW. Notice that the Higgs-dependent barrier H2 cos( =f ) in the potential contributes to the Higgs mass. Imposing that this contribution be less than the electroweak scale (see ~ F cos & 2 MF2 : 4 F 2F & MF : H2 Mf2 Ff 4 F4 ; f (4.1) (4.2) (4.3) e.g. ref. [64])12 gives the constraint f2 . Mf vEW which using eq. (4.3) leads to F . vEW F f 1=4 Together with eq. (4.2), this gives a constraint on the cuto in our model as we discuss in section 4.2. In order to ensure that the Higgs mass is scanned with su cient precision, we need to demand that the change of the Higgs-dependent term proportional to cos( =F ) over one period of the barrier, f , is less than the electroweak scale. This gives the constraint F . (MF vEW)1=2(F=f )1=4 which is weaker than eq. (4.4). Furthermore, there are several requirements on the in ation sector for the relaxation mechanism to be viable. If the relaxion is not the in aton, its energy density should be subdominant compared to the in aton. The energy density in the minimum where the relaxion eventually settles needs to be (close to) zero. This requires an additional constant contribution that is added to the potential and chosen such that the energy density at the minimum (nearly) vanishes. The tuning that is necessary to achieve this is just a manifestation of the cosmological constant problem. The contribution of the relaxion to the energy density relevant for in ation is then determined by how much it changes during its evolution. Using eq. (4.1) in the potential of eq. (3.2) gives the condition where HI is the Hubble rate during in ation. In addition, to ensure that our classical analysis of the eld evolution is applicable, quantum uctuations of the relaxion while it roles down the potential should be su ciently small. Over one Hubble time, the relaxion changes classically by ( )class: are ( )quant: HI . This leads to the condition HI 2dV =d . Its quantum uctuations, on the other hand, Combining the last two inequalities, we get condition Ne( )class: & using eq. (4.1) this gives Finally, the number of e-folds of in ation must be su ciently large to ensure that the relaxion scans the required eld range. Denoting the latter by , this leads to the . Provided that the relaxion is in the linear part of cos( =F ), 12This 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 potential are small. This gives a similar condition as eq. (4.4) but with an additional factor p 4 on the f2 . 4 Mf vEW to ensure that loop corrections to the right-hand side. HI & MF MPL ; HI . 2F & p F 4=3 F F1=3 : MF MPL Ne & HI F MF 4 F 3=2 : 2 : (4.5) (4.6) (4.7) (4.8) The resulting required number of e-folds can be very large. We will not specify the ination sector but will simply assume that it can be arranged to ful ll the conditions in eqs. (4.5), (4.6) and (4.8). Possible complications in achieving this are discussed e.g. in ref. [9]. Note also that the above conditions are somewhat alleviated if the e ect of the time evolution of the Hubble rate during in ation is taken into accout [3]. We also need to ensure that the potential is radiatively stable. The potential is an e ective theory with a cuto determined by the con nement scales gauge groups that give rise to the periodic terms (assuming they are smaller than the cuto s associated with generating the H2-terms in the potential). In the region of the potential where the Higgs mass parameter13 m2H ( ) 4 MFF2 cos F is smaller than these cuto s, the Higgs can give important corrections to the potential. From the one-loop e ective potential, we nd f f 2 Gf # log m2H ( ) ! ; (4.10) GF where we have neglected some subdominant terms. In the opposite region m2H ( ) or 2 , on the other hand, the corrections are strongly suppressed.14 This ensures that 2 Gf the term proportional to m2H ( ) cos( =f ) gives only a small contribution to the Higgsindependent barrier. In order to guarantee that the other term proportional to cos( =f ) is suppressed too, we require that Provided that ; GF ; F . 4 MF the rst two terms in eq. (4.10) give small corrections to the sliding term for the relaxion and do not a ect the dynamics. Finally if f2 . 4 Mf vEW, the cos2( =f )-term is negligible compared to the Higgs-dependent barrier when the Higgs reaches the electroweak scale. Using eq. (4.3), this translates to the constraint Gf . 4 Mf : p 4 vEW F f 1=4 : (4.9) This is less stringent than eq. (4.4). Conditions on the warped model The Higgs is localized on or near the IR brane in our warped model. Its mass parameter is then naturally of order I2R. We therefore identify the cuto of our relaxion model with 13Note that the Higgs mass parameter has an additional contribution from the cos( =f )-term. Since it is subdominant except in a small region of , we de ne eq. (4.9) without this contribution. 14See the one-loop e ective potential e.g. in eq. (2.64) of ref. [66] in the limit U 00 2 . the IR scale: IR : (4.13) As we have discussed in section 2, we can obtain the decay constants fB IR from a Chern-Simons term in the bulk and fUV on the UV brane. Since F MP2L= IR from an e ective anomalous coupling f is required, we identify F = MP2L= IR and f = From the conditions in eqs. (4.2) and (4.4) and using that MF IR, we obtain an upper bound on the IR scale in our warped model: IR . vE2WMPL 1=3 4 104 TeV : (4.14) HJEP07(218)3 Note that this is slightly lower than the maximal cuto found in ref. [1]. The reason is that there the bound on the cuto is partly determined by the requirement of a nite viable window for the Hubble rate. In our warped model, the corresponding contraint in eq. (4.7) is always trivially satis ed as we discuss below. The dominant bound on the cuto instead involves the constraint in eq. (4.2) that the H2 cos( =F )-term in the potential can compensate for a Higgs mass near the cuto . This di erence arises because g is a free parameter in the e ective description of ref. [1], whereas in our warped model g / 1=F is determined in terms of other parameters. We need to ensure that collider and avour bounds on the KK modes in our warped model are ful lled. We have assumed that the Standard Model elds live in the bulk. The dominant constraints then arise from CP -violation in K K-mixing and the electric dipole moment of the neutron. This requires [67, 68]: IR & 10 TeV : (4.15) This also satis es constraints from electroweak precision tests without imposing a custodial symmetry [69, 70] and on the radion (for a typical stabilization mechanism). The potential leads to mixing between the Higgs and the relaxion. This further constrains the IR scale. We use results from ref. [64], where bounds on the parameter 2br = f2 vEW=Mf controlling the mixing have been derived from several experiments ( fth force, astrophysical and cosmological probes, beam dump, avor, and collider searches). Using eq. (4.3), this translates to limits on F and thereby on IR. For our case F = MP2L= IR and f = IR, the most stringent bound comes from the distortion of the di use extragalactic background light spectrum due to relaxion late decays. This gives the constraint which is more stringent than eq. (4.14). We have discussed the con nement of Gf and GF in terms of only the massless modes of the gauge elds in our extra-dimensional model. This is a good approximation provided that the con nement scales are smaller than the KK mass scale:15 15It may be possible to alleviate this condition by including some of the KK modes in the e ective theory. (4.16) (4.17) IR . 4 103 TeV Gf ; GF . IR : IR IR IR IR f IR f 3=2 IR MP1L=2 vEW 10 TeV . IR . 4 103 TeV barrier. The range for the IR scale is allowed by all phenomenological constraints considered in this section. Since F F . GF and MF IR according to eq. (3.10), it then follows from eq. (4.2) that IR is required for successful relaxation. This in turn means that m ; GF Since the fermions ; c are localized on the IR brane, the former condition can be naturally ful lled. In order to discuss the latter condition, let us focus on GF = SU(N ) for de niteness. If we estimate the con nement scale as the scale where the 4D gauge coupling diverges, we nd (see e.g. ref. [71])16 GF MPL IR MPL 24 2 11N(g5c)2k ; (4.18) GF is close to the IR scale if 24 2=(11N (g5c)2k) where g5c is the 5D gauge coupling of GF . From this we see that the con nement scale of 1. This can be achieved for a wide range of values for g5c and N but clearly requires a coincidence between two parameters which are a priori not related. It may be possible to instead trigger the con nement of GF by adding states on the IR brane and thereby achieve We leave a detailed study of this question to future work. GF IR without such a coincidence. We next consider constraints related to the fermions N; N c and L; Lc on the IR brane. The last two terms in eq. (3.12) break the chiral symmetry of N; N c, in addition to their Dirac mass. Loop corrections then contribute to the Dirac mass (see gure 3), leading to the constraint The Higgs-dependent barrier can only stop the relaxion if Mf . vEW. Using eq. (3.17), the loop contribution to mN then implies that mN & y1y~6m2L log( IR=mL) : mL . 4 vEW plog( IR=mL) : (4.19) (4.20) The electroweak doublets L; Lc 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 lighter either. This limits their mass to a region near the electroweak scale. The question why their mass should be near the scale that is dynamically generated via the relaxation mechanism is the coincidence problem that we have mentioned in section 3. This problem does not appear in the double-scanner scenario that we discuss in section 5. 16Brane-localized kinetic terms for the gauge eld would give another factor multiplying one side of this relation. This would change the required relation between g5c and N accordingly. Let us brie y pause to count parameters. The potential in eq. (3.2) has 7 dimensionful parameters. Of these, , MF and F are of order IR, whereas Mf is of order vEW. Furthermore, F and f are given in terms of IR and MPL, while f is xed as a function of the other parameters via eq. (4.3). We can then express all parameters (up to O(1) factors) uniquely in terms of IR (plus MPL and vEW). In table 2, we summarize the corresponding relations and the phenomenologically viable range for the IR scale in our warped model. Additional loop corrections arise in the e ective eld theory at energies below and Gf as discussed in section 4.1. In particular, eq. (4.11) is an upper bound on the con nement scale of Gf . An additional constraint arises from the requirement that the mass of the lightest fermion after diagonalizing eq. (3.12) is smaller than the con nement GF scale (cf. the comment above eq. (3.16)). Together this gives HJEP07(218)3 mN yy~ vE2W 2mL . Gf . 4 vEW ; where we have used Mf vEW and that the largest Higgs VEV of interest is the electroweak scale (as the relaxion stops before the Higgs VEV can grow even further). Using eq. (4.3) and that f . Gf , this upper bound on Gf gives an upper bound on F which is less stringent than eq. (4.4). On the other hand, Gf can be very low provided that y; y~ and mN are su ciently small. In order to ensure that Gf does not contribute to dark radiation during big bang nucleosynthesis, its con nement scale should be larger than a few MeV: Gf & O(few) MeV : From eq. (4.3) and since f . Gf , it follows that such low Gf is only possible for the IR scale near its lower bound in eq. (4.15). Furthermore, we need to ensure that the decay of composite states does not destroy heavy elements during big bang nucleosynthesis. The resulting limits have been worked out in ref. [72]. For Gf = 10 MeV, mL = 500 GeV and y = 2y~, it is found that y; y~ & 0:15 is required. This limit quickly becomes weaker for larger Gf or smaller mL. On the other hand, the Yukawa couplings must not be too large in order to satisfy bounds on the invisible decay width of the Higgs. The corresponding limit is y; y~ . 0:1 for y = y~ and mL = 200 GeV which becomes slightly less stringent for larger mL. Given that the fermions , c, L, Lc, N and N c are all localized on the IR brane, we expect higher-dimensional terms in the action. These include S4D Z d4x c m2 4 ( IR c)2 + cNN IR m42N (N N c)2 + c N m mN 4 IR c N N c + h.c. : (4.21) (4.22) ! (4.23) The coe cients c , cNN and c N could be estimated using naive dimensional analysis. For simplicity, we assume them to be real. After con nement, this gives the additional terms in the Higgs-relaxion potential. Note that higher-dimensional couplings involving LLc either do not directly contribute to the potential as the pair LLc does not condense or the contribution is very suppressed.17 The rst term in eq. (4.24) contributes to the sliding term for the relaxion. But for c . 1 as expected from naive dimensional analysis, this is suppressed compared to the sliding term in eq. (3.2) and can thus be neglected. The second and third term, on the other hand, give additional contributions to the Higgs-independent barrier for the relaxion. Again these are suppressed compared to the barrier in eq. (3.2) and can be neglected. Adding higher-dimensional couplings to the Higgs in eq. (4.23) gives terms which can similarly be neglected. Finally, we check constraints related to in ation. Due to the temperature and quantum uctuations in de-Sitter space, we need to demand that the con nement scales of Gf and GF are larger than the Hubble rate during in ation: HI . Gf ; GF : (4.25) For both GF IR and Gf & f given by eq. (4.3), this is less stringent than eq. (4.6) from requiring that quantum uctuations of the relaxion are negligible for the dynamics. For F = MP2L= IR and since F MF IR, the condition for having a nite viable window for the in ation scale in eq. (4.7) is trivially ful lled. Furthermore, the upper limit on the in ation scale in eq. (4.6) is signi cantly smaller than the IR scale. We will assume that the in ationary sector, which we do not specify further, is located on the UV brane. Then HI IR guarantees that the e ect of in ation on the geometry of the extra dimension is negligible [73, 74]. Similarly, for a typical stabilization mechanism it ensures that the extra dimension is safe from destabilization during in ation. In order to ensure that the barrier for the relaxion is not removed during reheating after in ation, we demand that the reheating temperature be below Gf . This may require a relatively low reheating temperature. As follows from eq. (4.22), it can still be su ciently high to allow for big bang nucleosynthesis though. Under certain conditions, the reheating temperature may also be higher than Gf [1] (see also [75]). To summarize, after imposing all the constraints the usual parameters of the relaxion potential (1.1) in the model discussed in section 3.2 can be written just in terms of IR, vEW and MPL as can be seen from table 2 and using eq. (3.3). The dimensionless couplings of the relaxion potential and the relaxion mass are now determined as g = g0 = m 2 IR ; MP2L 2 IR : MPL (4.26) 17A higher-dimensional coupling ( c)yN N c would give a term proportional to cos( =F =f ) in the potential. These couplings can thus be very small, provided that there is a large hierarchy between the IR scale and the Planck scale. This in turn can be naturally achieved (i.e. without the input of very small numbers) e.g. by means of the Goldberger-Wise mechanism to stabilize the extra dimension [42]. In addition to IR and MPL, the input parameters of the model discussed in section 3.2 include the con nement scales couplings y and y~. Of these, GF and GF and m Gf , the fermion masses m , mN and mL and the are both required to be of order the IR scale. Since the corresponding fermions are localized on the IR brane, the former condition can be naturally ful lled, while the latter condition may require a coincidence of parameters as discussed around eq. (4.18). After imposing this, the electroweak scale is determined Gf , y, y~, mN and mL (plus IR and MPL) as follows from eqs. (3.17) and (4.3). Using eq. (4.19) and the requirement that Mf . vEW as well as imposing that mL & vEW to satisfy electroweak precision tests [72], we see that vEW . mL . 4 vEW plog( IR=vEW) yy~ mL log( IR=mL) 16 2 2 . mN . yy~ vEW : mL Using the range for mL in the range for mN , we then nd yy~ vEW log( IR=vEW) 16 2 . mN . yy~ vEW : The fact that the electroweak doublets need to be close to the electroweak scale is the coincidence problem discussed after eq. (4.20). Note that the condition for the mass of the singlets can be naturally ful lled if it dominantly arises from the loop process in gure 3 (cf. eq. (4.19)). Demanding that the right electroweak scale is obtained, we then see from eq. (4.3) that 3 Gf mL yy~ vE2W MIP2RL ; 6 where y and y~ need to be chosen such that eqs. (4.21) and (4.22) for limits discussed below eq. (4.22) are ful lled. Gf as well as the In the in ationary sector, the allowed window of Hubble scales and the minimum number of e-folds are given by 2 MPL IR . HI . 5=3 MIP2RL=3 ; Ne & M2P2L : IR In table 3, we give numerical values for two benchmark points. For the rst one, we set the cuto to its maximal allowed value in our model, IR = 4 103 TeV, and choose y = 2y~ = 0:2 and mL = 700 GeV. For the second one, we choose the intermediate cuto IR = 500 TeV as well as y = 2y~ = 0:04 and mL = 450 GeV. For both benchmark points, we assume that mN is dominantly generated by the loop process in gure 3 in which case the lower bound in eq. (4.28) is saturated (while our choices for mL satisfy the bound in eq. (4.27)). This in particular leads to Mf vEW as used for table 2. Both benchmark (4.27) (4.28) (4.29) (4.30) (4.31) g; g0 GeV [7 10 6 ; 0:06] min IR = 4 103 TeV, y = 2y~ = 0:2 and mL = 700 GeV, while for the second line, IR = 500 TeV, y = 2y~ = 0:04 and mL = 450 GeV. points satisfy the constraints in eqs. (4.21) and (4.22) in addition to the relevant constraints from colliders and big bang nucleosynthesis as can be seen from gure 10 in ref. [72]. Note that for cuto s IR . 500 TeV, constraints from big bang nucleosynthesis can become problematic. Indeed from eqs. (4.29) and (4.30) and the requirement that mN . see that lower cuto s necessitate smaller values for yy~. If y y~, this leads to longer lifetimes for the lightest N N c bound states which arise from the con nement of Gf (see [72]). For too long lifetimes, these decay during big bang nucleosynthesis. One way out is to choose y 1 y~. The large coupling y then allows for relatively fast decays via an o -shell Z [72]. For example for IR = 10 TeV, y = 1; y~ = 10 9; mL = 800 GeV and assuming that Gf , we the mass of the lightest N N c bound state is 3 Gf , we nd that its lifetime is of order 1000 s while it can kinematically only decay into electron pairs or lighter states. This then satis es the corresponding limit on the lifetime of order 104 s [76]. Alternatively, one could add new decay channels for the bound states which can allow them to decay faster and su ciently long before big bang nucleosynthesis. We leave a further investigation of this possibility for future work. Warping the double-scanner mechanism A UV completion 5 5.1 potential As discussed in section 3.1, the Higgs-dependent barrier in the relaxion potential needs to dominate over the Higgs-independent one once the Higgs VEV has reached the electroweak scale. This requires that Mf . vEW which in turn necessitates to introduce new particles coupled to the Higgs near the electroweak scale. We now discuss an interesting alternative presented in ref. [2]. The idea is to have another axion-like scalar with couplings in the g 3 + 4 f 1 g~ + g~ + H2 ! Mf2 cos f (5.1) and arrange its evolution such that it cancels o the Higgs-independent barrier. Note that we have also included a term cos( =f ) in the potential which will be important. The remaining terms involving the relaxion are as in eq. (1.1). Similar to the relaxion, the shift-symmetry breaking couplings g and g~ of the eld are taken to be very small. Let us assume that begins its evolution at some initial value & ( + g~ )=g~ so that the Higgs-independent term in brackets in eq. (5.1) is unsuppressed. Provided that f4 =f , the barrier term for the relaxion then dominates over its sliding term and the to slide and it eventually reaches the value + g~ )=g~ . This removes the barrier for the relaxion which can subsequently also slide down the potential. Both and then roll down if they track each other according to the relation + g~ )=g~ . The resulting growth of after a while causes the Higgs mass parameter to turn tachyonic and H begins to grow too. Shortly afterwards, the Higgs-dependent barrier in eq. (5.1) then becomes so ' ( big that the relaxion stops again. Provided that can no longer cancel this barrier, the relaxion remains stuck. This mechanism works for certain ranges of parameters which we review below. It then allows the backreaction from the Higgs to stop the relaxion once its VEV has reached the electroweak scale even if Mf vEW. We rst present a construction to generate the required terms in the potential (see also [10, 12]). This construction is, in fact, largely independent of the embedding into warped space and can thus be used in other UV completions of the relaxion as well. It is meant to serve as a proof of principle, and does not preclude the existence of simpler or more complete models. Let us introduce an additional U(1) gauge symmetry in the bulk. We identify the eld with the 5th component of the gauge eld after imposing appropriate boundary conditions. In order to generate the sliding term in eq. (5.1), we add an anomalous coupling of to a non-abelian gauge group GF on the UV brane using the construction in section 2.2. We also introduce two chiral fermions and c on the UV brane, with a Dirac mass m and in respectively the fundamental and anti-fundamental representation of GF . These fermions have no explicit coupling to . Such a coupling is then generated if we perform a chiral rotation of to GF . If the gauge group con nes at some scale GF or c to remove the anomalous coupling of > m , this gives rise to the potential f is the decay constant resulting from the anomalous coupling and b = arg(m ) is the phase of the mass term. As we see later, we again have = IR. Expanding in around the linear part of the trigonometric potential gives the sliding term in eq. (5.1) with g 3 = jm j GF F 3 IR up to factors of order one. Generating the coupling of to the periodic potential for is somewhat more involved. Notice that in eq. (5.1), the periodic potential for appears with the same phase in the last four terms (which for de niteness we have chosen as cos( =f )). Having the same phase to a high precision in these a priori independent terms is in fact necessary for the doublescanner mechanism to work. Let us assume that instead couples to sin( =f ). Keeping the phases for the other periodic terms xed, the barrier in eq. (5.1) then reads (5.2) (5.3) 4 f g~ tan + g~ + f H2 ! Mf2 cos f : (5.4) Even if can then initially cancel o the Higgs-independent terms (which depending on the initial value for may require =g~ ), this cancellation is generically irreversibly spoiled once starts rolling. The same holds for a phase di erence less than =2, if the other periodic terms have di erent phases or if the decay constants in the periodic terms di er from each other (in all cases down to values which are determined by the small couplings in the potential). In order to ensure the required phase and period structure, we extend the gauge symmetry Gf in the bulk from section 3.2 to the product group Gf1 Gf2 Gf3 Gf4 . In addition, we impose discrete symmetries Z2 and Z02 that interchange the groups as follows: HJEP07(218)3 Gf1 x Z02?y Gf3 Z2 ! ! Z2 Gf2 ?yxZ02 Gf4 : This in particular imposes that the underlying groups (e.g. SU(N )) are the same for Gf1 ; Gf2 ; Gf3 and Gf4 . We couple the 5D gauge eld AM that gives rise to to the gauge eld strengths of these four groups via Chern-Simons terms as in section 2.1. We impose that in the resulting anomalous couplings, transforms as under Z2, while it is even under Z02 (by choosing the coe cients cB in eq. (2.13) to transform accordingly). This gives S4D Z Tr hGf1 Gf1 i Tr hGf2 Gf2 i + Tr hGf3 Gf3 i Tr hGf4 Gf4 i ; where the decay constant f IR is equal for all gauge groups by virtue of the symmetries. We also add anomalous couplings of construction in section 2.2. We choose to Gf3 and Gf4 on the UV brane, using the to be even under Z2. This gives S4D Z d4x 1 16 2 F~ Tr hGf3 Gf3 i + Tr hGf4 Gf4 i ; where the decay constant F~ Z02 on the UV brane. We do not add corresponding couplings to Gf1 and Gf2 though. This explicitly breaks the On the IR brane, we next introduce four pairs of chiral fermions 1; 1c, 2; 2c, 3; 3c and 4; 4c in the fundamental and anti-fundamental representation of Gf1 , Gf2 , Gf3 and Gf4 , respectively. The fermion pairs interchange under Z2 consistent with eq. (5.5) but we choose Z02 to be explicitly broken on the IR brane too. Including Dirac masses for the pairs of chiral fermions and higher-dimensional couplings to the Higgs, this gives f is equal for the two gauge groups by virtue of the Z2. (5.5) (5.6) (5.7) S4D Z d4x m 1 [ 1 1c + 2 2] 1 + c 1 2 c + m 3 [ 3 3c + 4 4] 1 + c 3 2 c H2 + h:c: ; (5.8) where the elds are already canonically normalized and m 1 ; m 3 . IR. The coe cients c 1 and c 3 are a priori di erent from each other and could be of order one or be suppressed by a loop factor. We can now perform the chiral rotations i 1 ! e f 1 3 ! ei f +i F~ 3 2 ! e i f 2 4 ! e i f +i F~ while leaving 1c , 2c , 3c and 4c invariant. This moves and from eqs. (5.6) and (5.7) into eq. (5.8). We assume that the gauge groups con ne at energies below the IR scale. By virtue of the Z2 which is unbroken everywhere, the con nement scales of Gf1 and Gf2 are identical, as are those of Gf3 and Gf4 . The condensates then are pairwise equal, 3 Gf1 and h 3 3ci = h 4 4ci = 3Gf3 . The resulting potential at low energies IR (5.9) H2 2 IR (5.10) ; h 1 1ci = h 2 2ci = reads f cos(b 1 ) + jc 1 j cos(d 1 ) 2 cos ~ F + b 3 + jc 3 j cos ~ F + d 3 where b 1 = arg(m 1 ), d 1 = arg(m 1 c 1 ), b 3 = arg(m 3 ) and d 3 = arg(m 3 c 3 ) are given by the complex phases of the parameters. We have kept track of the phases in order to show that all terms are proportional to cos( =f ) without relative phase shifts as required. This is guaranteed by the Z2 under which ! and the potential is invariant. However, note that we have tacitly assumed that the fermionic condensates are real. As we have discussed at the end of section 3.2 and in appendix C, these phases are pion-like elds and thus dynamical. Doubling the spectrum in order to ensure that the potential for these pions factorizes from the remaining potential then xes their phases to the same value for all 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( =f ) between the rst and second line of eq. (5.10) are the same due to the Z02 in the bulk. However, note that this symmetry is broken on the UV brane by the couplings for in eq. (5.7). Nevertheless we expect that this does not a ect the decay constants for in eq. (5.10) by virtue of the non-renomalization properties of anomalous couplings (see e.g. ref. [77]). Also any such e ect would be strongly suppressed since F~ f . We leave a detailed study of this for future work. Furthermore, we have allowed for the masses m 1 and m 3 being di erent which breaks the Z02 also on the IR brane. This generically leads to a di erent running of the gauge couplings of Gf1 and Gf2 compared to those of Gf3 and Gf4 and accordingly di erent con nement scales Gf1 and Gf3 . However, it does not a ect the decay constants for in eq. (5.10) either as these are de ned not involving the gauge couplings of the underlying gauge groups (cf. eqs. (2.15) and (2.16)). As follows from eqs. (3.7) to (3.9), it is precisely the decay constants de ned in this way which determine the period of the periodic potentials. These periods are thus not a ected by the di ering running of the gauge couplings. Note also that the resulting di erence between the con nement scales can be made arbitrarily small for example by increasing the number of colours of the gauge groups. We can match with the potential in eq. (5.1) after expanding both eqs. (5.2) and (5.10) in around regions where the corresponding trigonometric potentials are linear. Both trigonometric potentials can be in the linear part simultaneously for example for F and b b 3 . This also ensures that the right signs in the potential are obtained. In addition to eq. (5.3), we can then identify f4 = jm 1 j 3Gf1 ; Mf = IR pjc 1 j ; g~ 3 jm 3 j Gf3 IR jm 1 j 3Gf1 ~ F up to factors of order one. Notice that eq. (5.10) contains a term cos( =f ) cos( =F~ )H2 which is not included in eq. (5.1). However, provided that for example 3 jm 3 j Gf3 3 jm 1 j Gf1 and jc 3 j is somewhat suppressed compared to jc 1 j, this only gives a small correction to the Higgs-dependent barrier and therefore does not a ect the dynamics. Note that this would not be possible if the Z02 would be unbroken on the IR brane. As in section 3.2, we next introduce fermions and c in the fundamental and antifundamental representation of a non-abelian gauge symmetry GF to generate the sliding term for the relaxion and its coupling to the Higgs. These fermions also allow us to generate the term cos( =f ) in eq. (5.1). To this end, we consider the higher-dimensional operator S4D Z d4x c 1 m m 1 4 which we expect to be present since the relevant fermions live on the IR brane. The elds are already canonically normalized and m ; m 1 . IR. The coe cient c 1 is again of order one or suppressed by a loop factor. Performing the chiral rotations in eqs. (3.7) and (5.9), we nd below the con nement scales GF Gf1 Gf2 Gf3 Gf4 F (5.11) (5.12) (5.13) (5.14) HJEP07(218)3 S4D Z d4x 4jc 1 j 3 3 jm j GF jm 1 j Gf1 cos 4 IR F + b 1 cos f where b 1 = arg(c 1 m m 1 ). Expanding the trigonometric function of =F around its linear part, we can identify up to factors of order one. Note that the coupling in eq. (5.12) with 1 1c; 2 2c replaced by 3 3c; 4 4c gives an additional term cos( =F + =F~ ) cos( =f ) in the potential. We expect g~ = jc 1 j 3 jm j GF I3RF 3 that for example for jm 3 j Gf3 somewhat suppressed compared to c 1 , this does not signi cantly a ect the dynamics. and the corresponding coe cient c 3 being 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 term and coupling to the Higgs of the relaxion. In order to see if the potential parameters in eqs. (5.3), (5.11) and (5.14) (plus eqs. (3.3) and (3.10) for g and g0) can take on values which allow the double-scanner mechanism to work, we next discuss various constraints. We again need to ensure that the conditions discussed in section 4.1 are ful lled. In particular, the Higgs VEV once the relaxion stops is as before given by eq. (4.3). One di erence between the potential parameters for the electroweak-scale barrier and the double scanner is that Mf vEW in the former and Mf IR in the latter. But in both scenarios, by construction the Higgs-independent barrier plays no role and therefore only the combination f2 =Mf is relevant for the dynamics of the relaxion and Higgs. Using eq. (4.3) to x the Higgs VEV, we can express this combination in terms of the decay constants and F . Constraints on these parameters therefore apply for both the electroweak-scale barrier and the double scanner. We can therefore conclude that the allowed range for the IR scale is again given by table 2. Note that f and Mf are di erent from those given in the table but the combination f2 =Mf and the other parameters in the table agree for both scenarios. In particular, we again nd that On the other hand, the constraint on IR and that F GF m IR is required. Gf in eq. (4.17) can always be ful lled as follows from eq. (4.4). Similarly, we see using eqs. (4.3), (4.6) and (4.15) that the constraints in eqs. (4.22) and eq. (4.25) are automatically ful lled. There are new conditions that are speci c to the double-scanner mechanism: the elds and track each other according to the relation + g~ )=g~ once the barrier is ' ( su ciently small provided that [2] where g is given by eqs. (3.3) and (3.10). On the other hand, can no longer cancel the barrier that the Higgs generates once it obtains a VEV if [2] g g~ & g g~ ; g 2 g g~ . g g~ (5.15) (5.16) with being the Higgs quartic coupling. We have F F F~ since these decay constants all arise from anomalous couplings on the UV brane. Comparing eqs. (3.3) and (5.14), we also see that g~ j c 1 jg. On the other hand, the couplings g and g~ can a priori be quite di erent. The gauge group GF that gives rise to the sliding term for can in principle be localized on the UV brane. Nevertheless we should still demand that its con nement scale is below the IR scale to ensure that the e ective description for is valid at the energy scale where the potential is generated. In addition, we need to require that jm j . GF . In order 3 to study one concrete example, let us assume that jm 1 j Gf1 to Z02 being only weakly broken on the IR brane). This gives g~ 3 jm 3 j Gf3 (corresponding g and g & g . The conditions in eqs. (5.15) and (5.16) then simplify to 1 2 g jc 1 j & g ; g j c 1 j . g : (5.17) This can be ful lled for a wide range of g if jc 1 j . 1=(2 ). This example shows that the conditions for the double-scanner mechanism to work can be easily satis ed. Finally, let us consider loop corrections to the potential. The double-scanner mechanism cannot remove barriers from terms like cos2( =f ) [2]. Therefore these must be smaller than the Higgs-dependent barrier when the Higgs reaches the electroweak scale. For loop corrections from the Higgs, this translates to the condition f2 . 4 Mf vEW and in turn to eq. (4.12) which is less stringent than the already imposed eq. (4.4). This also means that eq. (4.11) can always be ful lled. Furthermore, in addition to eq. (5.12) we expect higher-dimensional operators like S4D Z d4x c 4 ( IR c)2 + c 1 1 4 and similar terms involving 3 ; 3c; 4; 4c since the relevant fermions are all localized on the IR brane. The coe cients are again of order one or suppressed by a loop factor and are partly determined by the Z2. Assuming all parameters to be real for simplicity, below the con nement scales this gives The rst term gives a correction to the sliding term for the relaxion which is negligible for c . 1. The second term, on the other hand, gives another type of barrier that cannot be cancelled by the double-scanner mechanism. It is su ciently suppressed compared to the Higgs-dependent barrier provided that condition which for example for c 1 1 c 1 ful lled for the entire range of IR scales in table 2. f2 . vEW I2R=(Mf pc 1 1 ). This in turn leads to a 1 is the same as eq. (4.4) and which is then (5.18) (5.19) 6 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 for a technically natural hierarchy of decay constants. Warped extra dimensions are then a natural choice for its UV completion as they can generate a large hierarchy of scales purely from geometry. In our construction, the relaxion is identi ed with the scalar component of an abelian gauge eld in the bulk, whose pro le automatically has a small overlap with the UV brane. The warping generates the hierarchy from the Planck scale down to the scale of the IR brane, which is then identi ed with the cuto of the relaxion potential. From there onwards, the Higgs mass is relaxed down to its physical value. In section 2, we have presented a model-building toolkit for generating anomalous couplings of the relaxion to new, strong sectors. Depending on the localization of the anomalous terms in the warped interval, hierarchically di erent decay constants for these couplings may be obtained, including decay constants which are super-Planckian. A benchmark model coupling the relaxion to the Higgs was constructed in section 3. The sliding term and its coupling to the Higgs is generated through the condensation of a Dirac pair of SM singlet fermions that live on the IR brane. The barrier term, on the other hand, is generated close to the electroweak scale by the condensation of vector-like fermions with the same quantum numbers as one generation of SM leptons. These are also localized at the IR brane, and have masses near or below the weak scale, but are a priori unrelated to it, leading to the well-known coincidence problem. In order to avoid this and achieve a larger scale for the barrier term, a more elaborate construction is required. In section 5, we have presented a warped UV completion for one such scenario, the double-scanner mechanism of ref. [2]. The constraints for the model, both in general and those speci c to the construction of section 3, were discussed thoroughly in section 4, as well as the stability of the potential under radiative corrections. The requirement of obtaining the correct Higgs VEV may be used to x 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 to be of order the IR scale. Since the SM elds live in the bulk, standard avor constraints of Randall-Sundrum models push the minimum value of the IR scale to & 10 TeV. The maximum cuto that we can achieve while ensuring that all theoretical and phenomenological constraints are ful lled is 4 106 GeV. In this work, we have focused on in ation to provide a friction term for the slowroll of the relaxion, but interesting alternatives such as the particle-production mechanism of ref. [14] exist. It would be interesting to explore how such constructions may be implemented in warped space. The framework that we have described naturally allows for hierarchical decay constants for axion-like elds to be generated. As such it presents many further opportunities for model building, not limited to relaxion models, such as applications to in ation or dark matter. Another interesting possibility for generating this hierarchy is to consider a more general geometry with more than one AdS5 throat [ 78 ]. Acknowledgments LdL thanks DESY for hospitality during his stay, where part of this work was completed and acknowledges support by the S~ao Paulo Research Foundation (FAPESP) under grants 2012/21436-9 and 2015/25393-0. BvH thanks Fermilab for hospitality while part of this work was completed. This visit has received funding/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575. BvH also thanks the Fine Theoretical Physics Institute at the University of Minnesota for hospitality and partial support. The work of CSM was supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Kovalevskaja Award 2016, endowed by the German Federal Ministry of Education and Research. The authors would like to thank Aqeel Ahmed, Enrico Bertuzzo, Zackaria Chacko, Giovanni Grilli di Cortona, Adam Falkowski, Gero von Gersdor , Tony Gherghetta, Christophe Grojean, Roni Harnik, Arthur Hebecker, Ricardo D. Matheus, Enrico Morgante, Eduardo Ponton, Pedro Schwaller, Marco Serone, Geraldine Servant, Alexander Westphal and Alexei Yung for 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 e ective coupling that can for example arise from a Chern-Simons term in a second throat as we now brie y discuss. More details will be presented in [ 78 ]. To this end, we consider a setup with two warped spaces which are glued together at a common UV brane but each slice still has its own IR brane. For simplicity, we assume that both slices have the same AdS scale k. Let us denote the coordinates along the extra dimension in the two throats as z1 and z2, with metric in each throat again given by eq. (2.1). The coordinates match at the common UV brane at zUV1 = zUV2 = 1=k, while the IR branes are at zIR1 = ekL1=k and zIR2 = ekL2=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 conditions in eq. (2.6) but leave it unbroken on the UV brane. This allows for one massless mode from A5 which lives in both throats with wavefunction A5 = N a(zi) 1 in a given throat (the wavefunction is continuous at the UV brane). We will be interested in the case where one throat is signi cantly longer than the other. The normalization constant N is then dominated by the longer throat. Choosing L1 > L2 without loss of generality, we have zIR2, which gives N ' g4p2kL1e kL1 with g4 de ned as before. Let us next introduce a Chern-Simons coupling of AM to a non-abelian gauge group, where we choose the coupling to be localized in the second throat: S5D Z d x 4 Z zIR2 cb2 MNP QRAM Tr [GNP GP Q] : Notice that the coupling to A5 from this resembles eq. (2.17) with the -function replaced by the integral over A5 in the second throat. In the limit of a very short second throat with zIR2 O(few) zUV, we can think of this integral as a smeared-out -function. Correspondingly we expect the decay constant of in this limit to agree with eq. (2.19). Let us again restrict ourselves to the zero-mode of the non-abelian gauge eld. Integrating over the extra dimension, we in particular nd with decay constant given by S4D Z d4x 1 Tr [G fB2 ' cb2 g4p2kL1 k ekL1 2kL2 (A.1) (A.2) (A.3) or fB2 I2R2= IR1. For a very short second throat with L1 2L2, this indeed agrees with eq. (2.19). On the other hand, the two-throat construction allows for more general choices for the decay constant, with a continuum between MP2L= IR1 and IR1 (as IR1 < IR2 by assumption). The resulting phenomenology and the details of the construction will be presented in [ 78 ]. In this appendix, we brie y review how charged bulk fermions can give rise to ChernSimons terms. We consider a bulk fermion which couples to both the non-abelian gauge group and the U(1) from section 2.1. The action reads S5D Z d4xdz pg iD= + m ; iAM with GM being the non-abelian gauge eld (and AM the U(1) gauge eld). In order to see that this gives the same anomaly as a Chern-Simons term, we can perform a eld rede nition [ 79, 80 ] (B.1) (B.2) zIR) : (B.3) (B.4) Z z z0 ! exp i dz~A5(x; z~) ; where the constant z0 can be chosen according to convenience. However, the eld rede nition is anomalous on the branes18 and transforms the action into (see [81{84]) S5D ! S5D + Z d4xdz Z z z0 dz~A5(x; z~) 48 2 Tr [G G ] UV (z zUV ) + IR (z The coe cients UV and IR depend on the boundary conditions on the two branes for the left-handed component L of the bulk fermion (which in turn xes the boundary conditions of the right-handed component R). If L is even (odd) on a given brane, = 1( 1). Let us rst assume UV = IR in which case does not have a massless mode. From eq. (B.3), we then get the anomalous coupling of in eq. (2.15) with cB = IR : 4 Notice that this is independent of z0. In the opposite case UV = IR, on the other hand, cB depends on z0. But then has a massless mode which contributes to the anomaly and which cancels the dependence on z0. If the Chern-Simons term arises from such a bulk fermion, any perturbative contribution to the potential for A5 can be su ciently suppressed by making the bulk mass of the fermion somewhat larger than the AdS scale (see e.g. [34, 47]). C Pion-like elds in the relaxion potential In this appendix, we include the pion-like elds which arise from the condensing fermions on the IR brane and which contribute to the potential. Let us focus on ; c for de niteness. As usual, we can parametrize the pseudo-Nambu-Goldstone boson corresponding to the breaking of the chiral symmetry of ; c by the -model eld U = exp(i =f ) with a 18We note that, e.g. for SU(N), there is an additional SU(N)3 anomaly. It can be canceled by adding another bulk fermion, uncharged under U(1), with opposite boundary conditions from . decay constant of order f this gives GF . After con nement then h 3 U . From eq. (3.8), m where for simplicity we again ignore phases and prefactors. Since F f , generically settles into its minimum min = f f rst after which the potential becomes independent of . This problem is remedied for example by introducing another pair of chiral fermions ~ ~c with the same quantum numbers. Instead of eq. (3.6) we then have ; S4D Z 1 + H2 2 IR [m c + m ~ ~ ~c] + h.c. : (C.2) Similar to the up and down quark in the Standard Model, the fermions transform under an approximate SU(2)L SU(2)R symmetry which is spontaneously broken to a diagonal SU(2)V by the condensates and explicitly but weakly broken by their masses. The corresponding pseudo-Nambu-Goldstone bosons are parametrized as U = ei =f with 0 p 2 +! 0 : We next perform the chiral rotation ! ei 2F ; ~ ! ei 2F ~ with c and ~c left invariant to remove the coupling of to Tr GF GF in eq. (3.4). For this choice of chiral rotation, no kinetic mixing between the relaxion and the pions is induced (see ref. [85]). Choosing m = m ~ for simplicity, from eq. (C.2) we get below the con nement scale V ( ; H) m 3 GF 1 + H2 2 IR cos 2F cos f ; where q ( 0 )2 + 2 + . The potential for the pions and relaxion thus factorizes and no longer vanishes once the pions settle into their minimum. 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Nayara Fonseca, Benedict von Harling, Leonardo de Lima, Camila S. Machado. A warped relaxion, Journal of High Energy Physics, 2018, 33, DOI: 10.1007/JHEP07(2018)033