Phenomenology of relaxion-Higgs mixing

Journal of High Energy Physics, Jun 2017

We show that the relaxion generically stops its rolling at a point that breaks CP leading to relaxion-Higgs mixing. This opens the door to a variety of observational probes since the possible relaxion mass spans a broad range from sub-eV to the GeV scale. We derive constraints from current experiments (fifth force, astrophysical and cosmological probes, beam dump, flavour, LEP and LHC) and present projections from future experiments such as NA62, SHiP and PIXIE. We find that a large region of the parameter space is already under the experimental scrutiny. All the experimental constraints we derive are equally applicable for general Higgs portal models. In addition, we show that simple multiaxion (clockwork) UV completions suffer from a mild fine tuning problem, which increases with the number of sites. These results favour a cut-off scale lower than the existing theoretical bounds.

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Phenomenology of relaxion-Higgs mixing

Revised: May Phenomenology of relaxion-Higgs mixing Thomas Flacke 0 1 2 4 Claudia Frugiuele 0 1 3 Elina Fuchs 0 1 3 Rick S. Gupta 0 1 3 Gilad Perez 0 1 3 Field Theories, Higgs Physics 0 Weizmann Institute of Science , Rehovot 76100 , Israel 1 Daejeon , 34051 , Korea 2 Department of Physics , Korea University 3 Department of Particle Physics and Astrophysics 4 Center for Theoretical Physics of the Universe, Institute for Basic Science , IBS We show that the relaxion generically stops its rolling at a point that breaks CP leading to relaxion-Higgs mixing. This opens the door to a variety of observational probes since the possible relaxion mass spans a broad range from sub-eV to the GeV scale. We derive constraints from current experiments ( fth force, astrophysical and cosmological probes, beam dump, avour, LEP and LHC) and present projections from future experiments such as NA62, SHiP and PIXIE. We nd that a large region of the parameter space is already under the experimental scrutiny. All the experimental constraints we derive are equally applicable for general Higgs portal models. In addition, we show that simple multiaxion (clockwork) UV completions su er from a mild ne tuning problem, which increases with the number of sites. These results favour a cut-o scale lower than the existing theoretical bounds. Beyond Standard Model; Cosmology of Theories beyond the SM; E ective - 1 Introduction 2 Relaxion-Higgs mixing 3 Review of backreaction models and existing bounds on br 4 New bounds on compact relaxions 5 Laboratory probes of relaxion-Higgs mixing 5.1 The sub-MeV mass range 5.2 Relaxion masses between the MeV- and the weak scale 5.2.1 5.2.2 The 1 MeV{5 GeV range The m > 5 GeV mass range 6 Cosmological and astrophysical probes of relaxion-Higgs mixing 6.1 Cosmological probes 6.2 Astrophysical probes 6.1.1 6.1.2 Relaxion abundance Cosmological bounds on late decays 7 Implications for the relaxion theory space 8 Testing for the CP violating nature of the relaxion 9 Conclusions A Relaxion mass and mixing for the case of j = 1 B Relaxion mass and mixing for the case of j = 2 C Pseudoscalar couplings of the relaxion D The h coupling in j = 2 models E Expressions for relaxion partial widths and lifetime { 1 { Introduction Relaxion models o er a new perspective on the hierarchy problem [1]. The weak scale is obtained in a dynamical way as the Higgs mass depends on a time-dependent vacuum expectation value (VEV) of a scalar eld, . This scalar evolves and eventually halts at a value rendering the e ective Higgs mass much smaller than the cuto . This is achieved due to the fact that the potential of consists of a backreaction term that is switched on once the Higgs mass square gets negative and electroweak symmetry breaking (EWSB) is induced. When compared with conventional models of naturalness, this class of models leads to a completely di erent phenomenology as there is no analog of top or gauge partners go to the low energy, high precision frontier. Let us rst present a very brief review of the relaxion mechanism. In relaxion models the value of 2, the mass squared term in the Higgs potential, changes during the course of in ation as it varies with the classical value of , which slowly rolls because of a potential, V (H; ) = 2 ( )HyH + (HyH)2 ; 2 ( ) = 2 + g + : : : ; V ( ) = rg 3 In these equations g is a coupling1 and is the scale where the Higgs quadratic divergence gets cut o . Note that the operator in eq. (1.3) can be radiatively generated by closing the Higgs loop in the term g (HyH) in eq. (1.1) and thus technical naturalness demands r & 1=16 2. In canonical models the eld slowly rolls down (during in ation) from some initial large eld value > =g, such that 2 is positive and the electroweak symmetry unbroken. It stops rolling shortly after the point c ' electroweak symmetry is broken and the Higgs gets a vacuum expectation value, v2( ) = =g, where 2 becomes negative, 2 ( )= . A crucial ingredient of the relaxation proposal is the feedback mechanism that triggers a backreaction potential once the Higgs gets a VEV, where 1 j 4 is an integer and h^ = (v + h)=p2.2 becomes larger, resulting in a monotonically increasing Higgs VEV and thus increasing the backreaction's amplitude. Eventually the barriers become large enough and the relaxion continues rolling, j 2( )j stops rolling at an arbitrary O(1) value of the phase 0=f , = rg 3 + f sin 0 f = 0 ) = 1Note that the coupling g de ned here is dimensionless and is related to gGKR, the one in ref. [1], via gGKR = g . 2For an alternative proposal where the rolling is stopped due to particle production see ref. [2]. Vbr(h; ) = { 2 { Note that for values of rg 3f , eq. (1.5) is satis ed for j sin( 0=f )j but as M~ 4 j vj grows monotonically and the rolling starts at a random phase, relaxion stops well before it reaches this stage. Therefore, generically the phase an O(1) value. It is basically the result of a balance between the two terms controlling the derivative of the potential in eq. (1.5). We shall return to this point when discussing CP-violation. For a small enough value of g, the cut-o can be raised much above the electroweak scale. Such a small value of g can be radiatively stable as in the g ! 0 limit we recover the discrete symmetry, ! + 2 kf ; k 2 Z : Higher dimensional terms such as g2 2 2; g3 3 : : : contribute at the same order for c ' g and thus do not a ect the above analysis. For the same reason, in variants of the above model where only even powers of appear one can proceed along the same lines to obtain essentially the same results. It is important to emphasize that in order to achieve higher values of the cut-o higher values of br are required. Let us review the three main reasons for this. First of all, cosmological considerations during in ation (classical rolling must dominate over quantum uctuations, see ref. [1]) put an upper bound on the cut-o which decreases if n ' =g f ' n 1=4 r br : { 3 { (1.6) 1, =f , the 0=f has (1.7) ' (1.8) (1.9) (1.10) (1.11) . cq = 4 br f 1 6 pMPl ; where from now onward, for simplicity, by v, M~ and br we will refer to the nal values of these quantities at the relaxion minima = 0. An even stronger bound can be derived if we demand that the relaxion does not have transplanckian excursions, g MPl ) . tp = MPl 4 1 rf br ; which once again favours a large br. As the requirement of subplanckian eld excursions depends on quantum gravity assumptions and can be possibly evaded by UV model building, we will not take this as a strict bound and extend our analysis also to the transplanckian region. Finally, as argued in ref. [3], if the relaxion is a compact eld, as any pseudo Nambu-Goldstone boson (PNGB) must be, the ratio of the distance the relaxion rolls to the periodicity of the backreaction, is, for a single axion sector, generically expected to be an O(1) number. On the other hand, this ratio must be large to raise the cut-o substantially above the weak scale, as eq. (1.5) implies, Thus smaller values of br require larger values of n for a given cut-o scale. In this work we derive several new results relevant for relaxion phenomenology. We emphasise the importance of relaxion-Higgs mixing that is expected in a large class of relaxion models and focus on its experimental and observational implications. As the relaxion-Higgs mixing turns out to be proportional to 4br, these observational constraints put an upper bound on the backreaction scale, br. We also derive a theoretical upper bound on the backreaction scale br. We further consider multiaxion (clockwork) models where n eN , N being the number of sites in these models, and show that for a too large value of N , the clockwork construction becomes tuned unless further structure is assumed. By eqs. (1.8){(1.11) we see that together these considerations favour lower values of the cut-o scale. In the following section we derive the expressions for the relaxion-Higgs mixing and in section 3 review existing backreaction models and bounds on the backreaction scale. In section 4 we consider bounds on models with compact relaxions. We nd a rich variety of experimental and observational probes for the relaxion in the mass range 0.1 eV to 50 GeV described in detail in section 5 and section 6. All our bounds are equally applicable to general Higgs portal models. As the relaxion couplings to SM particles via the mixing are like that of a CP-even scalar, in the sub-eV range fth force experiments can constrain large parts of the relaxion parameter space. In the keV-MeV range constraints on the relaxion parameter space arise from astrophysical star cooling bounds and cosmological probes of late decays, including constraints from entropy injection, BBN observables, CMB distortions and distortion of the extragalactic background light (EBL) spectrum. In the MeV-GeV region we nd that the most important bounds arise from cosmological entropy injection and BBN bounds, cooling rate of the SN 1987 supernova, beam dump experiments and from constraints on rare B- and K-meson decays. Finally for GeV scale masses the bounds arise from LEP Higgs-strahlung data and LHC Higgs coupling bounds on the h ! channel. We also discuss how presently unconstrained parts of the relaxion parameter space would be probed by future data from experiments such as the PIXIE detector for CMB distortions, the NA62 experiment and especially the SHiP beam dump experiment. In section 7 we discuss the implications of our bounds on the theoretical parameter space of relaxion models. In section 8 we brie y discuss how the characteristic CP violation of relaxion models can be probed and nally we conclude in section 9. Useful relations are derived in the appendices. 2 Relaxion-Higgs mixing Relaxion models contain two sources of breaking of the shift symmetry, the one that allows the Higgs mass to scan as the relaxion rolls and the backreaction term. In this section we will see how the presence of both terms can lead to spontaneous CP-violation in the backreaction sector and a measurable relaxion-Higgs mixing (see also ref. [ 4 ]). The full relaxion potential is given by combining the terms in eq. (1.2), eq. (1.3) and eq. (1.4), V = 2 + g To obtain the mixing terms we expand around the minima of the elds and H, therefore, vH = pv2 In models with even j, vH = v = 246 GeV. On the other hand, as we will see in section 3, the backreaction sector breaks electroweak symmetry in models with odd j. In these models, v02, where v0 is the electroweak symmetry breaking (EWSB) VEV in the backreaction sector. The minimisation conditions and explicit mass matrices, Mi2j , for the j = 1 and 2 cases can be, respectively, found in appendix A and B. We nd in both cases that the leading contribution to the mass matrix elements can be written entirely in terms of the parameters of the backreaction sector. In particular, for both j = 1 and j = 2. In addition, as discussed below, we expect implies that the relaxion-Higgs mixing angles is naturally small, sin br . vH which 1. We nd that, to leading order the relaxion Higgs-mixing angle and the relaxion mass m are, 3Note that in j = 1 models the Higgs couplings themselves might di er from their SM values because of the reduced Higgs VEV, vH = pv2 are at most 10 % e ects which we would ignore (see also section 3). v02; in the following we will assume that v0 . 100 GeV so that these { 5 { As anticipated, the mixing angle is proportional to the spontaneous CP-violating spurion sin( 0=f ) in the backreacting sector. In more complicated relaxion models there can be mechanisms to suppress this phase. An example is the model with the QCD axion where a small phase is necessary to be compatible with the non-observation of a strong CP-phase. In such models (see also ref. [5] and ref. [6]), relaxion-Higgs mixing is also suppressed. Couplings: as the relaxion mixes with the Higgs boson, it inherits its couplings to SM particles suppressed by a the mixing angle sin as a universal factor | such as in Higgs portal models.3 For g , the coupling to pairs of matter elds , and g V , the coupling to pairs of V = W or Z, the couplings are given by At the loop level, the relaxion couples via quark loops to gluons and quark and W loops to photons, g f; V = sin ghf;hV : L g 4 g g 4 F F G G ; where with g g = g = AF ( ) = AW ( ) = f ( ) = s sin 8> arcsin2 p 1 4 " log 1 + p 1 p 1 1 1)f ( ) 1 1 i #2 1 where x = m2h=4m2x. Let us nally comment on the pseudoscalar couplings of the relaxion to Standard Model particles, as these may have a signi cant impact on the experimental probes discussed in the following. However, these couplings are model-dependent and as the relaxion potential can be controlled by a sequestered sector [1, 3] these couplings could be in principle suppressed relative to the \Higgs-portal" couplings discussed above (which are at the core of the relaxion construction). As we show in appendix C, this is the case in existing backreaction models (see section 3) where we nd that these couplings are in magnitude generally smaller than or equal to the Higgs portal couplings. An exception is the pseudoscalar coupling to photons which in some backreaction models (see appendix C) can be larger than the one induced via Higgs mixing while in other models is of the same size as the Higgs-portal coupling. As the presence of a large pseudoscalar coupling to photons is thus modeldependent, we will comment on its implications only qualitatively. 3 Review of backreaction models and existing bounds on br As both the cuto of the theory as well as the relaxion-Higgs mixing depends polynomially on the back reaction scale, it is important to examine what is its allowed range. In this section we thus describe the di erent backreaction models in the literature and discuss various bounds on the size of the scale M~ or br that appears in the backreaction potential [see eq. (1.6)]. Note, rst of all, that for odd j = 1 or 3, a non-zero M~ in eq. (1.4) must break electroweak symmetry which already suggests M~ . v, but let us analyze this case in more detail. The simplest relaxion model [1] where the backreacting sector is QCD and the relaxion couples to gluons like the axion, f G Non-perturbative e ects generate a potential for the axion, ~ G is an example of a j = 1 model. Vbr ' mu cos hqqi ' f 4 f 3mu cos ; f (3.1) { 6 { where 4br = 4 f 3mu = 4 f 3yuv=p2 is set by the pion decay constant f and the up quark mass. As the relaxion stops at a generic value of the phase, =f , QCD relaxion models are generally in con ict with the non-observation of a large value of the strong CP phase. This problem can be solved in more complicated variants where there is a dynamical mechanism to make the above phase small. An alternative approach would be to give-up the solution to the strong CP problem and to consider an additional strong sector. For instance, a new technicolor-like strong sector would lead to an EWSB condensate of techniquarks, hULUR + DLDRi ' v03, where UL;R and DL;R are quarks with the same electroweak charges as the SM quarks uL;R and dL;R, but charged under the new strong group and not QCD. If the relaxion is coupled to the operator G0 G~0 , involving the strong sector gauge bosons (G0 corresponds to the new strong sector eld strength), a backreaction is generated with j = 1 and mN cos 4 f 30 y1y2h^2 mL cos : f f = { 7 { It is worth pointing out that such models would not be as strongly constrained as typical technicolor models because the condensate does not need to explain the large top mass and because the presence of an elementary Higgs somewhat alleviates the tension with electroweak precision observables [7]. In this work we have assumed v0 . 100 GeV and ignored O(v02=vH2 ) e ects. A less constrained model with j = 2 was presented in ref. [1]. In this model, couples to G0 G~0 , the gauge bosons of an EW symmetry preserving strong sector. The Higgs couples to two vector-like leptons charged under this strong group as follows, L = y1LHN + y2LcHyN c mLLLc mN N N c + h:c: ; where (L; N ) have the same quantum numbers as the SM lepton doublet and right-handed neutrino, respectively, and (Lc; N c) are in the conjugate representations. If we take mN 4 f 0 mL, only the fermion N forms a condensate that is EW preserving. Upon integrating out L; Lc, the Higgs contributes to the mass of N as follows, so that the relaxion potential gets the backreaction mN = y1y2h^2=mL 4 br ' yv03vH ; p 2 br . 100 GeV: where y is the smaller of the U or D Yukawa coupling with the SM Higgs, and vH is the VEV of the Higgs doublet so that v02 + v H2 = v 2 = (246 GeV)2. Such a scenario is constrained by Higgs and electroweak (EW) precision observables as Higgs couplings deviate from SM values by O(v02=vH2 ). Requiring these deviations to be smaller than 20% gives, v0 . 100 GeV. Together with this upper bound and the fact that the quarks must not have too large an explicit mass, i.e. we must have yvH 4 v0 and hence y . 1, we obtain an upper bound on br, (3.2) (3.3) (3.4) (3.5) L = y1ei =f LHN + y2LcHyN mLLLc 2 mN N N + h:c: where L and Lc have the same quantum numbers as before, and N is a SM singlet fermion. The one-loop Coleman-Weinberg potential of the relaxion reads Vbr ' 1 where m~ is the larger of mL, mN . A theoretical challenge that any j = 2 model faces is that at the quantum level the backreaction term [see eq. (1.4)] generates the term M1~62 2c2 cos f upon closing the h^ loop. This term is independent of the Higgs VEV, which implies the presence of an oscillatory potential even before the Higgs condenses [5]. Thus, the relaxion stops rolling prematurely, before EWSB, unless the scale c at which the Higgs loop is cut-o satis es A perturbative j = 2 model was presented in ref. [3]. In this model the relaxion is a familon, the PNGB of a spontaneously broken avour symmetry. Let us consider the In axion-like models this is automatically satis ed because the instanton contribution are highly suppressed at energy scales larger than the con nement scale, 4 f 0 , so that eq. (3.8) implies f 0 . v. In the model of eq. (3.4) there is actually another contribution to the potential that exists even before EWSB, VN ' 4 f 30 mN cos f , where technical naturalness quirement that in this model requires that mN must be larger than y1y2mL log( =mL)=(16 2) . Demanding the above EW preserving contribution to be smaller than the backreaction generated upon EWSB, mN cos f , we obtain mL . 4 v=plog =mL. Together with these bounds, the remN 4 f 0 mL so that N forms a condensate and L does not, implies (3.6) (3.7) where we have assumed mL & v due to experimental bounds for the rst inequality. In the perturbative familon case, of eq. (3.6), a simple extension can ensure that the constraint in eq. (3.8) is satis ed as followed. The Majorana mass mN is actually induced via a mini see-saw mechanism. A new heavier fermion N c is added to the theory, L mDN N c 2 mNc N cN c : After N c is integrated out, the Majorana mass of N is induced, mN = m2D=mNc . One can show in this case that two-loop corrections to the relaxion potential do not get contributions from energies above the scale mNc so that we get c = mNc , and eq. (3.8) is satis ed as long as mNc . 4 v. As mL;N mNc , this implies mL;N 4 v, and thus 4 br 64 2v4 { 8 { where we have assumed y1;2 < 4 . note that eq. (2.5) implies that for a non-tachyonic , Now let us discuss some model-independent bounds on the backreaction scale. First 2br < ( bmrax)2 = mhv pcos ( 0=f ) j sin ( 0=f ) : Finally notice that in the presence of the mixing the Higgs-like eigenvalue would satisfy, m2h > Mh20h0 . For the j = 1 case this leads to a bound on M~ simply arising from the expression for the Higgs mass that is given by (see eq. (A.8)) HJEP06(217)5 m2h M~ 3 cos( 0=f ) p2vH 2 + 2 vH where the inequality becomes an equality in the limit of no relaxion-Higgs mixing. We must have > 0 to ensure that the potential does not have a runaway direction which implies the following bound, 4 New bounds on compact relaxions In this section we consider simple multiaxion (clockwork) models and then show that these su er from stability issues when the number of sites (axions), N , becomes too large. The instability is, in fact, related to the very same issue of highly irrelevant operators that plagues the two-site construction in ref. [3]. First of all note that in realistic relaxion models the coupling g in eq. (1.2) and eq. (1.3) is obtained from a compact term (at least in QFT constructions where the relaxion is a pNGB), but with a larger periodicity F [3], V (h^; ) = 2 2 cos F + ^2 h which allows us to make the following identi cations for One can now directly obtain eq. (1.11) by demanding V 0( ) = 0 using eq. (4.1), As shown in ref. [8] and [9], the Choi-Kim-Yun (CKY) alignment mechanism [10]4 (also known as the clockwork mechanism in the relaxion context) for multiple axions (or PNGBs) can provide a relaxion potential having two periodicities with a large ratio F=f eN , N being the number of axions. Let us rst review these multiaxion models. We describe 4The mechanism was proposed as a generalization of the Kim-Nilles-Peloso alignment mechanism [11] from 2-axion to an N -axion-alignment. g = ; n = F ' n 1=4 r F f : br: { 9 { (3.12) (3.13) (3.14) (4.1) (4.2) (4.3) here the realization of ref. [9]. Consider N + 1 complex scalar elds 1 to the potential V ( ) = N X j=1 m2 iy i + 4 j iy ij 2 + ! 0) so that the masses m2 and quartic couplings The above potential respects a U(1) symmetry under which the elds 1 ; 2 : : : N have charges Q = 1; 13 : : : ; 31N . For simplicity the symmetry preserving cross terms such as iy i jy j have been ignored and an approximate permutation symmetry has been assumed the radial parts of the elds obtain a VEV, i = pf^ ei i=f^ where f^2 = 4m2= , such are equal for all the elds. For that at low energies only the angular degree of freedom remains. N superpositions of the 2 angular elds obtain masses, but the direction is a at direction that describes a Goldstone boson. The Goldstone mode has an O(1) overlap with the rst site and is exponentially suppressed overlap with the last site, introduce some anomalous breaking of the global U(1) at the rst and last sites, where N = qPjN=+11 32(j1 1) is the norm of the vector de ned by eq. (4.5). Let us now 0 = 1 N 1 + 3 where f = N f^ and we have used eqs. (4.6), (4.7) to rewrite the rst line in terms of . Non-peturbative e ects now generate the desired relaxion potential in eq. (4.1) with F = 3N f so that eq. (4.3) now becomes (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) Thus we see that the CKY/clockwork mechanism can give us a cut-o that grows exponentially with the number of axions, N . Note that the above analysis holds only if so that the potential generated by the anomalous breaking of the U(1) symmetry in eq. (4.1) is subdominant compared to the potential generated from eq. (4.4), the linear combination in eq. (4.5) remains a Goldstone mode, and all the heavier modes can be decoupled. We now show that, for a too large N , these models become nely tuned if we relax the approximate permutation symmetry in eq. (4.4) that was assumed only for convenience of calculation. If we relax this assumption some of the mass square terms might be positive. Let us assume, for instance, that k 1 consecutive elds, n1+1 : : : n1+k 1 have positive mass square terms so that there are no corresponding PNGB modes n1+1 : : : n1+k 1 for these scalars. At rst sight, this breaks the link in the axion chain because | instead of one Goldstone mode as in eq. (4.5) | there are now two decoupled Goldstones elds [in the absence of the subdominant boundary terms of eq. (4.8)], k and N1 and N2 are again normalisation constants. None of the above modes can be identi ed with the relaxion as no single mode above is subject to both the backreaction at the rst site and the rolling potential at the last site. However, the link between the two chains is not completely lost, as a process like 3m n1+k generates a higher dimensional operator that weakly couples the two sectors, n1 1 ! 3 n1 ! : : : ! 2 is an exponentially small number due to = pseudo-Goldstone modes that have masses m2 1. More precisely, as the N 1 heavier 32 f^2=2 (see ref. [9]), must be much lighter than the radial modes with m2 f^2=2, one needs m2 With the above term in the potential we once again obtain that Goldstone mode. With the addition of the explicit breaking terms on the rst and last site in eq. (4.8), the terms relevant for the potential of 01 and 02 are, V ( i) = 4br cos 4br cos f ^1 + N1f ^ 01 + " f^4 2 + " f^4 2 cos cos 3 where we have appropriately rede ned 01 and 02 so that the phase appears only in the last term. The two lightest modes now are superpositions of eq. (4.11) and eq. (4.12). The mass matrix of 01 and 02 is given by M = 0 B f 2 N12brf^4 + 3 2(n1 1) " N1 2 k n1+1 " N1N2 3 k n1+1 " N1N2 32k " N2 2 3 2n2 r 4 A N22 f^4 1 C ; (4.14) + 0 ; (4.15) (4.16) where s = sin , c = cos and is the mixing angle. Let us rst show that in the limit that contribution of the term proportional to 4 to the gradient of the m2 potential is subdominant, i.e. 4 we obtain tan = 3 n1 k 3n2+k" f^4, we recover the usual relaxion potential. In this limit N2=N1, and the rst eigenstate in eq. (4.17) becomes identical to the relaxion mode in eq. (4.5). To obtain the Lagrangian for the lightest mode we rst use the condition to stabilise m2 , which in this limit reads N2 2 sin 3kc N2 m2 ^ f = 0 : Substituting the solution h m2i = 0 in eq. (4.15) and using tan = 3 n1 k N2=N1 yields In the opposite limit, i.e. 4 3n2+k" f 4, the gradient of the m2 potential is domithe potential in eq. (4.1). nated by the term proportional to 4 , sin ^ m2 + 0 = 0 : which drives m2 to the global minimum of the rolling potential giving the Higgs an ) mass. Therefore for the relaxion mechanism to work one needs 4 3n2+k" f^4, which results in, up to normalization factors, the two mass eigenstates, for any positive integer n1, so that the relaxion mechanism cannot even address the little hierarchy problem in this case. How long must the relaxion chain be so that a sequence of k 1 = 3 consecutive positive masses becomes highly probable? To compute this probability we need to nd the number, N3(N ), of sequences of N `+' or `-' signs with at least one chain of 3 consecutive positive signs `+++ ' . It can be shown that N3(N ) obeys the following recursion relation, N3(N + 1) = 2:N3(N ) + 2 N 3 N3(N 3) : Here the rst term comes from the fact that if we already have at least one `+++' chain in a sequence of N axions, by adding either a `+' or `-' sign at the N + 1 th position we obtain an arrangement of size N + 1 satisfying our criterion. This does not include arrangements of N axions with no chain of 3 consecutive positive `+' signs but having a `-++' at the end such that we get a required arrangement if at the N + 1 th position we add a `+' sign; this is taken care of by the second term in eq. (4.22). Finally in the last term we subtract the double counting resulting in cases where the sequence captured by the second term already includes a `+++' in the remaining subchain. To obtain a successful relaxion model, however, we are interested in an N -site sequence with no `+++' chain, N30(N ), which is given by It turns out that N30(N ) satis es the following familiar relation, which is nothing but the recurrence relation of the 3-step Fibonacci sequence.5 By inspection, N30(3) = 7 , the 5th element of the 3-step Fibonacci sequence so that we must have N30(N ) = b3(N + 2). Our arguments can be easily generalized to nd the number of arrangements with no chains of k 1 positive masses which turns out to be just the (k 1)-step Fibonacci sequence. Hence the probability to randomly obtain a sequence with at least one chain of k 1 positive masses is P(k 1; N ) = 1 bk 1(N + 2) 2N : We nd that for N 28 the probability of having at least k 1 = 3 consecutive positive masses in a chain of N axions is P(k 1; N ) ' 90%. Thus for N generically we have 3 TeV as already discussed above. 28, from eq. (4.20) For N . 28 axions there is the possibility of raising the cut-o to a value of . 328=4(16 2 4br)1=4 = 1000 TeV s 2 br ; mhv where we have used eq. (4.9) and the numerical value above is for br ' show in sections 5 and 6, experimental probes can constrain pmhv. As we will br to even smaller values as a function of f (or alternatively the relaxion mass) and this in turn would imply an even lower cut-o in accordance with eq. (4.26). 5 Laboratory probes of relaxion-Higgs mixing In this and the next section we discuss in detail the bounds and the future probes for relaxion-Higgs mixing, distinguishing between laboratory experiments, discussed here, and cosmological and astrophysical probes considered in section 6. As we will show below, the relaxion mass can range from far below the eV-scale to almost the weak scale. Therefore a variety of experiments is needed to look for the relaxion. As the couplings to SM particles are proportional to sin , a convenient plane to present the constraints is the sin2 -m 5The n-step Fibonacci sequence bn is a sequence where any number in the sequence is the sum of the previous n numbers. 100.000 f=103 GeV 106 GeV 109 GeV 1012 GeV 1015 GeV 1018 GeV sin m2 ' j p 4 br ' 2f 2 4 br 2vf m2 h ; p 2 Relaxion mass Relaxion lifetime r * b Λ br arising from the requirement of a non-tachyonic (b) The lifetime also depending on m and sin2 values of the sin2 = 1 line using 1= sin2 . for each f . Here in eq. (3.12) for sin( 0=f ) = 1=p2. with thresholds (vertical gray lines) and example (horizontal gray lines). The lifetime for any other sin2 value can be obtained from for de niteness sin( 0=f ) = cos( 0=f ) = 1= 2,6 p plane. Before going into the details of the various constraints to be presented in gures 2{ 5, let us rst identify the regions of the sin2 -m plane that are relevant for relaxion models. For the convenience of the reader we repeat the expressions for the mass and mixing angle of the relaxion in the small-mixing approximation from eq. (2.4) and eq. (2.5) substituting j = 21=4mhv=j is the maximal allowed value of br that follows from Other O(1) choices of 0=f lead to slightly modi ed numerical values for 7It was shown in ref. [12] that in j = 2 models it is possible to have 6Note that we have assumed vH2 ' v2 which amounts to ignoring, at most, O(10%) e ects (see section 3). br & mhv with smaller then O(1) values for sin( 0=f ) . v2= 2br, such that eq. (3.12) is still satis ed. In this work we take O(1) values of sin( 0=f ), and in accordance with eq. (3.12), br . mhv thus not considering this region of the parameter space. As the backreaction scale, br, is in any case constrained to be less than a few times the weak scale (see section 3), this is actually a narrow region of the parameter space where the constraints are expected 2 to be similar to those we obtain for br mhv. m , sin and bmrax. These can be inverted to obtain 2br(m ; sin ) = f (m ; sin ) = 21=4vm2h j q sin m2 + m2h sin2 ; m2hv sin j(m2 + m2h sin2 ) We use the equations above to make contours of constant br and f in the sin2 -m plane in gure 2, 3 and 5. Although we have made the contours for the j = 2 case, using eq. (5.2) one can easily translate to the j = 1 case by substituting f ! 2f; br ! relaxions heavier than 5 GeV, the mixing can be O(1) and eq. (5.1) and eq. (5.2) are no longer valid. Thus in section 5.2.1 and gure 4 where we consider relaxions in this mass range, we exactly diagonalize the mass matrices in appendix A and B to obtain the br p 2 br. For We see from eq. (5.1) and that if br is much smaller than and sin2 increase with br and we get sin2 m4 . This implies that in this regime, bmrax, for a given f , both a light relaxion has typically a suppressed mixing. However, if we take values of br close bmrax, this tendency does not hold anymore. This behaviour can be seen in gure 1(a) where we plot the relaxion mass as a function of br for di erent values of f taking j = 2. We see that, for all f , the relaxion mass is maximum for for larger values of br the relaxion mass drops rapidly with br as the term within the parenthesis in eq. (5.1) becomes smaller. The relaxion mass can, in fact, be made arbitrarily small by choosing a br that is su ciently close to its maximal value changing sin2 . In the sin2 -m plane this can be seen from the shape of the f contours bmrax, while hardly in gure 2 (and subsequent gures) for which two branches can be clearly identi ed. The region br > br corresponds to the top left part of gure 2 where the f contours become br = br = 2 1=4pmhv=j, and hardly changes but the mass can become arbitrarily small. The nearly horizontal as sin2 thick grey line in gure 2 is the contour this line, which we refer to as the \tuned region", corresponds to the narrow region in the theory space 0:99 bmrax < br < bmrax, marked by the thick black line in Therefore, in the following we will mostly discuss the \untuned region" br < gure 1(a). bmrax, which translates to br = 0:99 bmrax. The whole region above the and implies that in most of the theoretical parameter space, if we make the relaxion lighter, it also becomes more weakly coupled. We would like to point out that this is a general feature of Higgs portal models. For instance, consider the potential [13], where 0 and h0 are as de ned in eq. (2.2) while m^2 ; m^ 2h and x parametrise the couplings in a general Higgs-portal model. For small mixing angles we get m^h = mh, the observed (5.3) (5.4) range 0:98 xmax < x < xmax. consider only the range, Higgs mass, and the condition x < xmax = mh=v to ensure that the lighter eigenstate does not become tachyonic. The region above the grey line in this case corresponds to the small The second restriction on the sin2 m parameter space arises from the fact that we ) 6:5 The lower bound on f arises from the fact that in our analysis of relaxion-Higgs mixing we ignored any new states (for instance radial modes) that must exist below the scale to UV-complete the backreacting sector. Thus our analysis holds only if both the Higgs boson and the relaxion have a mass much smaller than the mass scale of these UV states, i.e. for f & mh. We will call the region de ned by eq. (5.3) and eq. (5.5) the `relaxion parameter space', i.e the region in the sin2 m relevant for relaxion models. Let us now discuss the mass range the relaxion can have given these restrictions. In the untuned region the relaxion can be made lighter either by decreasing br or increasing f . In our analysis we do not consider f > MPl = 2 1018 GeV, but as there is no strict lower bound on br, the relaxion can be made as light as we want by taking su ciently small values of br. As discussed in section 1, however, lower values of br are theoretically disfavoured. For instance if we require relaxion eld excursions to be subplanckian this puts a bound sin2 . 10 27 as shown in gure 2. In the untuned region this can be translated to m > 0:001 eV. As the requirement of subplanckian eld excursions depends on quantum gravity assumptions and can be possibly evaded by UV model building, we will not take this as a strict bound and extend our constraints also to the transplanckian region. We now turn to the question of how heavy the relaxion can be. The largest relaxion mass is obtained for the minimal value, f = mh, and weak scale values of br where the small-mixing approximation in eq. (5.1) no longer holds. In section 5.2.1, by exactly diagonalising the mass matrices in appendix A and B, we nd an upper bound m . 60 GeV (see gure 4). For readers interested in general Higgs portal models our analysis provides the complete constraints in the untuned region of their parameter space apart from the area outside the region de ned by eq. (5.5). Whereas for f > MPl, the constraints in the untuned part of the region arise only from fth force experiments and have been discussed elsewhere (see for instance [13, 14]), the region corresponding to f . mh can be potentially constrained only by some cosmological probes that we will mention in the next section but not fully derive. Before going into the details of the di erent experimental probes, a comment is in order. In the following we are going to study the constraints on the relaxion parameter space driven by its mixing with the Higgs. As it is impossible to include the e ects of the pseudoscalar couplings of the relaxion in a model-independent way we do not consider these. In any case, in existing explicit models, these couplings are generally not larger than the Higgs portal couplings as discussed in appendix C. An exception is the pseudoscalar coupling to photons which can in some backreaction models (see appendix C) be larger than the one induced via Higgs mixing. In section 7 we qualitatively comment on how our constraints would change if a large pseudoscalar coupling to photons is present. In the following we describe the constraints on the relaxion in di erent mass ranges as the relaxion mass spans a wide range from sub-eV values to tens of GeV. While relaxions heavier than a MeV can be potentially probed by collider searches, the only laboratory probes for sub-MeV relaxions are fth force experiments. We discuss these two categories separately starting with sub-MeV relaxions. In this mass range the relaxion has a very large decay length making it impossible for collider searches to probe visible decays of the relaxion. This can be seen from gure 1(b) where we plot, using the expressions in appendix E, the relaxion lifetime as a function of its mass for di erent choices of sin . Eq. (5.3) implies for the considered mass range sin . 10 9, and gure 1(b) shows the corresponding enormous rest frame decay length of c & 1014 m. Therefore the only possible laboratory probes are either fth force experiments, or experiments looking for invisible particles. This last class of experiments, at least at the moment, is not sensitive enough to provide constraints on the very small Higgs-relaxion mixing in this mass range [15]. Fifth force experiments denote experiments which can detect the existence of a new degree of freedom by the corresponding new Yukawa-like force induced between two electrically neutral test bodies. A relaxion induces a spin-independent Yukawa force between two test bodies A and B, de ned by the potential where mA, mB are their respective masses and A, B parametrise the couplings of the relaxion to the two bodies. In Higgs portal models, the couplings are given by [13] where ghNN ' 10 3 and mnuc = 1 GeV. The sensitivity of the various fth force experiments depends on the interaction length which is related to the mediator mass m via V = G A Be r m ; A = B = ghNN p 2MPl s mnuc = m 1 = 1 m 0:2 eV m : Let us start discussing probes of new long-range forces going down from macroscopic length scales to the pm scale of MeV particles. We present the bounds arising from these probes in gure 2. For very low masses (below 3 10 15 GeV), the strongest constraint comes from the Eot-Wash experiments [16, 17] that looked for deviations from Einstein's weak equivalence principle (labelled as EqP in gure 2) by precision measurements of the longrange force between a heavy attractor and two di erent test bodies in a torsion balance. Let us notice that this experiment is able to constrain the Higgs portal down to very small couplings, but for masses lighter than 10 16 GeV the probed parameter space belongs to the (5.6) (5.7) (5.8) tuned region (for other potentially relevant discussion of cosmological and/or low energy probes see for instance [18, 19]). Therefore, in gure 2 we do not show relaxion masses of m < 10 16 GeV although the EqP bound extends even further. On shorter length scales, the mass range 3 10 15-10 11 GeV, the strongest bounds arise from constraints on violations of the inverse square law (labelled as InvSqL) that have been obtained by various experimental groups [20{25]. The excluded region shown in gure 2 is an envelope that contains bounds from all these experiments with the strongest one coming from the Irvine experiment in the mass range 3 10 15-5 10 14 GeV [20, 21], from the Eot-Wash 2006 experiments in the mass range 5 10 14- 2 10 12 GeV [25] and from the Stanford experiment [22, 24] in the mass range 2 10 12- 5 10 11 GeV. Finally we also show the constraints from tests of the Casimir force [26, 27], the force induced by the zero point energy of the electromagnetic eld when two conductors are brought very close to each other. While these bounds from the tests of the Casimir e ect are weaker than the bounds of the torsion balance experiments below 10 11 GeV, they are the strongest bounds above this mass as shown in gure 2. The shaded area below the horizontal light gray, dotted line (sin2 10 27) shows the region where the relaxion has transplanckian excursions for any value of the cut-o scale > 2 TeV (see eq. (1.9)). For heavier particles, i.e. shorter-range forces, the sensitivity is even lower. The intermediate region, between 10 eV and 1 MeV, is the most challenging region to probe in laboratories. The most sensitive experiment in this mass region are neutron scattering experiments that test the existence of a new sub-MeV boson based on their in uence on the neutron-nucleus interaction. These experiments set a very weak bound, s . 0:1 [28, 29], and are therefore incapable of probing a relevant region of the parameter space. In a subset of this mass range from a keV to an MeV (shown later in gure 5) the relaxion parameter space can be probed only by astrophysical and cosmololgical observations to be discussed in detail in the next section. The 10 eV-keV mass range, on the other hand, is largely unconstrained as shown in gure 2. Let us conclude this subsection by commenting that fth force experiments are a unique probe of light states like relaxions that couple to electrons and nucleons as CP-even scalars. Axions, for instance, do not give rise to spin-independent long range forces at leading order because of their pseudoscalar nature and are thus only weakly constrained by fth force experiments. Therefore, di erent laboratory probes have been proposed to circumvent this problem. This is the case for light shining through the wall (LSW) experiments [30], which are also sensitive to Higgs portal models [31]. However, their reach is too limited to compete with fth force experiments and therefore these do not appear in our plot. 5.2 Relaxion masses between the MeV- and the weak scale Let us now study the region of parameter space where the relaxion mass is above the electron threshold and thus it can decay into SM fermions. Furthermore, as shown in gure 1(b), in this region the relaxion has a shorter lifetime and can be directly searched for in laboratory facilities. Let us further distinguish two sub-regions based on the di erent relevant probes. The bounds in the MeV-5 GeV mass range are presented in gure 3, HJEP06(217)5 L q S I V e 14 G 0 1 f= .99 0 r= b Λ ax r)m (Λb 5 r= b Λ eV G Λtp<2 TeV 10 16 GeV and 10 7 GeV. Fifth-force experiments (orange) probe the lightest mass range via the equivalence principle (labelled as EqP), the inverse square law (ISqL) and the Casimir effect (Casimir). Contours of constant br (gray) for solid) and br = 5 GeV (gray, dashed). Here the requirement of a non-tachyonic in eq. (3.12) for sin( 0=f ) = 1=p2. Contours of constant f = MPl; 1016 GeV; 1014 GeV (black, solid). The light gray region below the dotted gray line corresponds to trans-Planckian eld excursions > MPl for = 2 TeV. including also astrophysical and cosmological constraints which will be discussed in the next section. Figure 4 presents the bounds in the GeV region. 5.2.1 The 1 MeV{5 GeV range This region of the parameter space is well covered by rare K- and B-meson decays at proton beam dump and avour experiments. Crucial for both kinds of experiments is the possibility of producing a relaxion in rare decays of K- and B-mesons. In avour experiments that probe rare decays, constraints are put on the branching ratios [32] BR(K ! + ) = 0:002 sin2 BR(B ! K+ ) = 0:5 sin2 2jp j mK ; mB 2jp j F K2 (m ) ; (5.9) (5.10) where p is found using two-body kinematics and FK is de ned in [32]. Even in proton beam dump experiments, rare mesons decays are the main the production mode of the relaxion. The smallness of the branching ratio is overcome by the large luminosity. Electron beam dump experiments do not have any sensitivity to Higgs-relaxion mixing due to the suppressed electron Yukawa coupling. generated by misalignment and thermal production and then use this result to study how these bounds apply to our scenario. Misalignment production: during in ation the expectation value of the eld , h i, satis es the classical equation of motion. Quantum uctuations lead to a spreading of the eld around this classical value. The spreading is given by (see for instance ref. [5]) where = that is for h i and h i)2i, HI is the Hubble scale during in ation and Ne is the number of e-folds. We see that the spreading stops when the r.h.s. above vanishes, d dNe 2 = h( 2 = H2 I . 3 ; where V 0( ) = @V =@ , and to obtain the inequality we have used the requirement that the dynamics of the relaxion is dominated by classical rolling and not quantum uctuations, HI < (V 0( ))1=3 (see ref. [1]). This gives us the misalignment of from its classical value just after in ation. After this, the Universe goes through a phase of radiation domination. If the temperature of the Universe is below the temperature T0 with m 3 H(T = T0) = ) T0 = 45 4 3g 1=4 r m 3 MPl ; the relaxion eld oscillates around the minimum. This leads to an energy density, and an e ective non-relativistic number density, n m, given by and thus results in a comoving number density, m = n m = m2 2 ; 2 =m ; Ym = n m . s m 2s 2 max ; where max is that maximal value of given by eq. (6.2), the entropy density, s = 0:44 gS(Ti) Ti3 and gS(Ti) is the e ective number of degrees of freedom in entropy at the temperature Ti. If the reheating temperature is larger than T0, then Ti = T0, otherwise Ti is the reheating temperature. Thermal production: relaxions can be thermally produced by the process HH ! at temperatures above the Higgs mass, by the processes q(g)+g ! q(g)+ at temperatures below the electroweak critical temperature, TEW, by the pion-relaxion conversion process (6.1) HJEP06(217)5 (6.2) (6.3) m, (6.4) (6.5) (6.6) these processes one by one. at temperatures below QCD, and nally by inverse decays. Let us consider At temperatures above the electroweak critical temperature, TEW mhv=mt the Higgs portal mixing in eq. (2.5) is absent and the relaxion interacts only with the Higgs doublet. The main production mode of the relaxion is then the process HH ! via the coupling g2(HyH) 2 : where we have considered only the top loop for computing the gg coupling as the loop contribution of lighter quarks vanishes for temperatures above their masses. For the Compton P = 0:3 s 3s2T 3 2v2 ; Note that any contribution to the process from the backreaction potential is absent, because in both the non-perturbative axion and the peturbative familon model, the backreaction term dissolves at high temperatures. In the axion case, the potential becomes negligible at high temperatures because instanton e ects become very weak as the non-abelian gauge coupling becomes perturbative. In the familon model the Coleman-Weinberg potential gets no contributions from momenta above mNc so that for T & mNc the backreaction potential vanishes also in this case. The comoving number density for resulting from this process has been computed in ref. [5] to be (6.7) (6.8) (6.9) YH2 ' 13:7 g4 0:278 Mpl ; g (TEW) TEW where we have not considered any contribution above the electroweak critical temperature and g (TEW) is the e ective number of relativistic degrees of freedom in energy density at the temperature TEW. As we shall see in the following, this is negligible compared to the production via the relaxion-Higgs mixing in the EW broken phase. Now let us consider relaxion production in the EW broken phase, that is, production ! q + at temperatures much below the critical temperature of the electroweak phase transition, TEW mhv=mt = 180 GeV. In order to ensure that any nite temperature e ects are negligible, we take T < T0 = 20 GeV so that we always have (T =TEW)2 1. At these temperatures t; h; Z; W are not relativistic and their densities are Boltzmann-suppressed. We thus ignore any contribution to thermal production of relaxions from processes involving these states for T . 20 GeV and ignore any contribution at all from the temperature range 20 GeV < T < TEW where nite temperature e ects become important. We also do not consider any possible contribution from the backreaction sector as this would be impossible to compute model-independently. Consequently, our nal result for the relaxion abundance will be a conservative lower bound and the cosmological bounds we derive can possibly be even stronger. For T . 20 GeV, the dominant production processes are the Primako process q(g) + process q + ! q(g) + , involving the gg vertex and the Compton photoproduction which involves the qq vertex. Using the production rate for the Primako process computed in ref. [68], we get, process the thermally averaged rate is given by [69], f C ' ss2T P Clearly, the dominant contribution is from bottom quarks and the contribution from lighter quarks is negligible. The interference between the Primako and Compton processes also scales as mf2 T , but is suppressed by another power of s with respect to fC in eq. (6.10) and thus we ignore this contribution. We also ignore any contribution from the electromagnetic counterpart of the above processes (that is replacing gluons by photons in the respective diagrams) which are expected to be suppressed by powers of ( em= s). Thus, we nally obtain for the total production rate, With the knowledge of we can now compute the abundance of thermally produced relaxions by solving the Boltzmann equation, Y 0 = xHt Y ; where x = 1=T and the Hubble scale Ht = 4 3g (T ) T 2 MPl . Integrating the above, we get = P + C : 0:278 g p 45 P dx mN T 2 3=2 e T mN m2N s2T 2 4 v2m4 : (6.10) (6.11) (6.12) (6.13) (6.14) (6.15) " Yh = Y pr 1 exp Z 1=Tf 1=T0 xHt X f Z 1=mf 1=T0 f C dx xHt ! # where Y pr = 0:278=gpr and gpr ' 86:25 is the number of relativistic degrees of freedom in energy density in the 1-20 GeV temperature range. In the sum over fermion species we include only the c and b quarks as the contribution due to the other quarks is negligible (see eq. (6.10)). We have taken the nal temperature Tf = 1 GeV for Primako production to justify our use of perturbative QCD and T = mf for the Compton process because below this temperature the respective fermions become non-relativistic. For s & 10 6 the relaxions have an equilibrium density given by Y = Yeq = 0:003 whereas for s . 10 6, the relaxions have a much smaller density, Yh = 2:9 109s2 : Once the Universe cools down to a temperature below the quark/hadron transition, i.e. T . 200 MeV, relaxions can be produced via the pion-relaxion conversion process, i.e. N + ! N + , N being a nucleon. Using gN obtain the following parametric estimate for this process, = mN s and nN = v 2 mN T 3=2 e T mN we One can check that Ht T .200 MeV and hence we ignore this contribution. Finally, inverse decays may become signi cant at temperatures just a bit larger than the relaxion mass. The ratio =Ht, , being the relaxion decay width, is maximal for T & m =5 as below this temperature, the relaxions become non-relativistic and the rate is Boltzmann-suppressed while above these temperatures Ht increases. We check numerically that Ht T =m =5 P + Ht C T &1 GeV and thus the contribution from inverse decays can also be safely ignored. We now show that the contribution to relaxion abundance from the q(g)+ processes in eq. (6.13) by far dominates over the contributions in eq. (6.6) and eq. (6.8). First note that we can rewrite eq. (6.8) as YH2 ' 2 106s4 16 2r 4 using eq. (1.5) and eq. (2.5). As we will discuss in detail in the next subsection, cosmological probes are sensitive only if the relaxion decays after 1 s. As one can see from gure 3, in the region of parameter space which lies in the untuned area de ned in eq. (5.3), if the relaxion decay time is greater than 1 s (below the purple curve) we must have sin < 10 4. In this region YH2 is clearly always smaller than Yh in eq. (6.13), even for a cut-o as low as 3 TeV. As far as the contribution from misalignment, Ym, is concerned we have checked numerically that Ym Yh except in a region of the parameter space where none of the cosmological constraints apply as Yh < Ym < 10 20 are both extremely small. Thus we conclude that, under our assumptions, the abundance is well approximated by eq. (6.13). 6.1.2 Cosmological bounds on late decays In this subsection we study the bounds on late decays of the relaxion. The earliest the relaxion has to decay to have any e ect on cosmology is after 1 s, that is at the neutrino decoupling time, which in the relaxion parameter space corresponds to m < 150 MeV as shown in gure 5. On the other hand for relaxion masses m < 0:1 keV, eq. (5.3) implies that sin2 . 10 17 and thus a lifetime, & 1026 s (see gure 1(b)) much greater than the age of the Universe (1017 s). This means that for masses m < 0:1 keV an exponentially small number of relaxions have decayed by the present time and, as we will soon show more rigorously, there are consequently no bounds in this region. To compute the various constraints from late decays it is important rst to know whether the relaxion decays relativistically or non-relativistically at a given point in the parameter space. If the relaxions are relativistic, their temperature can be computed from their number density, (6.17) ! q(g)+ (6.18) n = Y s = ) T = (32) T 3 1=3 T : g S Y gpr Yepqr (6.19) If T ( ), the relaxion temperature at the time of its decay, is smaller than m =5, it can be safely considered to have become non-relativistic before decaying. If it becomes nonrelativistic, it can even dominate the energy density of the Universe before decaying (as the energy density of non-relativistic matter decreases more slowly compared to that of relativistic matter). As we will see in this section, such a scenario is highly constrained. In most of the parameter space where various bounds on late decays are relevant, the relaxion decays non-relativistically and thus its energy density before decaying is = m Y s. Thus the various bounds on late decays generally put an upper bound on m Y as a function of the lifetime . Let us now discuss the various constraints on the relaxion decays. Entropy injection: if the relaxions decay after the neutrinos have fully decoupled, i.e. for & 1 s, they increase the entropy of the SM plasma by S, HJEP06(217)5 Sbefore Safter = 1 + S S and thus decrease both the baryon-to-photon ratio B and the e ective number of neutrino species, Ne . Let us now proceed to compute S=S. For & 1 s, relaxions decay nonrelativistically except in a small region of the parameter space with sin2 & 10 4 and m < 1 MeV which is outside the region of interest de ned in eq. (5.3). In any case for relativistic decays, S S ' T s = 3 4gS T T 4 . 0:3% and, as we will see, entropy injection smaller than a few percent is unconstrained. To obtain the last inequality above we have used eq. (6.19). In the rest of the parameter space where the relaxion decays non-relativistically, we must di erentiate between the scenario where the relaxion energy density as a fraction of the energy density of radiation, i.e., is smaller than unity, . 1, from the scenario, & 1, where the relaxion dominates the energy density. The entropy injection is given by where x = 1:50 [70] for . 1 whereas x = 1:83 [71] for & 1. Having obtained the expression for S=S, let us proceed to derive the constraints from B and Ne measurements. We rst discuss the bound from Ne . Entropy injection anytime after neutrino decoupling and before recombination leads to the reduction in Ne , that is: = rad = 4 gS m Y 3 g T ( ) S S = x g S 1=4 m Y r MPl ; Ne = 3:046 Sbefore Safter 4=3 (6.20) (6.21) (6.22) (6.23) (6.24) with N SM = 3:046. Following ref. [72] we use the bound Ne > 2:6 and show in pink the e region excluded by this constraint in gure 3 and gure 5. We now discuss bounds arising from the decrease in the baryon-to-photon ratio, B, caused by relaxion decays. Since the baryon-to-photon ratio is inversely proportional to S, B is reduced as follows due to entropy injection, after = before Sbefore : A change of B between BBN and CMB epoch is not supported by observation since the measured value of B during the CMB epoch agrees well with the value after the end of BBN. Therefore, entropy release between these two epochs must be suppressed. In particular, CMB and BBN data constrain S=S to be smaller than 2% [73]. In gure 3 and 5 we show the regions of parameter space excluded by this bound in orange. Big-bang nucleosynthesis: Big-Bang Nucleosynthesis (BBN), the formation of light elements in the early Universe, might be altered by late relaxion decays into SM particles. The e ect depends strongly on the relaxion mass, particularly whether or not it is heavy enough to cause electromagnetic or hadronic cascades. In our region of interest (i.e. for f > mh) relaxions above the pion threshold have a lifetime bigger than 1 s (see gure 5), so they do not a ect cosmology. Decays of lighter relaxions give rise to electromagnetic showers as long as their mass is bigger than twice the minimum photo-disintegration energy of light nuclei (m & 5 MeV). In the relaxion parameter space (see the beginning of section 5) m < 150 MeV for > 1s, so we obtain our bounds from relaxion decays into electrons. BBN bounds put constraints on =s = m Y as a function of the lifetime . We consider here the bounds presented in ref. [73] for the decay of a 140 MeV scalar. The region f < mh in gure 5, while not relevant for relaxion models, can be interesting in general Higgs portal models. This region can be constrained, for instance, by BBN bounds on decays to pions, hadronic showers etc which can be easily derived using our expression for the abundance in eq. (6.13). Distortion of the CMB spectrum: the energy spectrum of the cosmic microwave background (CMB) allows also to constrain energy release in the early Universe. Constraints from CMB distortions become e ective for relaxion decays that take place after 106 s as at earlier times the thermalization process is very e cient. There are two types of distortions: -distortions and y-distortions which dominate at di erent times. At DC = 106 s (T cess ( + e ! 750 eV), the photon number changing double Compton scattering pro + e) freezes out. As a result, the photons can no longer be in a Planck distribution (where the number of particles is xed by the total energy). On the other hand, the Compton process is active until C = 109 s, thus the photons can still maintain a Bose-Einstein (BE) distribution, but with a chemical potential , whereas the observed Planck spectrum corresponds to an almost vanishing chemical potential. Therefore, j j is constrained by the COBE/FIRAS data which give a bound of j j < 0:9 10 4 at 95% CL. The chemical potential generated by these late decays can be computed to be [ 74 ], 1 In the above equation the factor involving exponentials accounts for the fact that only decays in the time period between DC and C contribute to -distortions. If the fractional 1, one can use = m Y s, and nd the constraints whereas & 1 is excluded as it will lead to an O(1) value for which is excluded. We = 152 T 4 to nd that a large portion of the parameter space is excluded by this constraint as shown in If the relaxion decays later than C = 109 s (T 25 eV), even the Compton process freezes out and this leads to a deviation of the CMB spectrum from a BE distribution. The degree of thermalization that the photons can still achieve can be parametrized by y [ 74 ], exp(4y) 1 = C = ) exp( RC= )) : (6.27) and = 152 T 4 to compute the bound. In The region with & 1 is directly excluded whereas in the region 1 we use = m Y s, gure 5, we show the region excluded by the bounds from distortions in a darker shade of green than the one denoting y distortions. We also show by dashed lines the projection for the region PIXIE can exclude at 5-sigma level, given by j j < 1 10 8 and jyj < 5 EBL and reionization: after recombination ( RC 1013 s) the nuclei capture almost all the electrons to form neutral atoms so that the Universe becomes nearly transparent to radiation. The photons injected by relaxion decay can be in principle directly detected, unless their wavelength lies in the ultraviolet range (13.6 eV-300 eV) and they are absorbed in the photoionization process of atoms. In this ultraviolet mass range bounds from reionization can be set. Photons emitted from very late decays that do not lie in this range, can be observed today as a distortion of the di use extragalactic background light (EBL). The above constraints can be used to bound the quantity m Y = of as a function of m . Together these bounds cover the wavelength range between 0.1 and 1000 m, that is roughly the mass range between 0.1 eV and 1 keV. We show in gure 5 the excluded region using the bounds derived in ref. [ 74 ] and [76], but appropriately rescaled to the di erent abundance in our case. Dark matter: if the relaxion decays after the present dark matter density. 6.2 Astrophysical probes 1017 s it forms a very small component of SN1987a supernova: in the core of a supernova, a relaxion can be produced via its couplings to nucleons and thereby contribute to its energy loss. The relevant process is bremsstrahlung N + N ! N + N + . Requiring that the energy loss into the new scalar must be smaller than the measured energy loss into neutrinos leads to bounds on the Higgsrelaxion mixing as long as the relaxion is lighter than 20 MeV. In gure 5 and 3 we show (in light blue) the bounds derived in [77], using the results of ref. [78]. This computation is exponentially sensitive to some uncertainties (see ref. [77]) and thus should be interpreted only as an order of magnitude estimate. At a more conceptual level, even the idea of energy loss via neutrinos has been questioned in the literature [79]. New laboratory constraints that are able to explore this region are therefore required. Globular-cluster star bounds: relaxions can be produced in globular-cluster (GC) stars via processes involving the relaxion electron coupling, g e, such as the Compton and bremsstrahlung processes. Requiring that the total cooling rate is not faster than expected [80, 81] gives us the bound g e < 1:3 ) sin2 < 4 10 17 (6.28) (6.29) (6.30) for m . 10 keV. Limits can be also set on the relaxion-photon coupling considering Primako photon-relaxion conversion [ 74, 81 ], g < 0:6 10 10 GeV 1 ) sin2 < 1 10 11 for m . 30 keV. In gure 5 the GC limit on g e is presented in blue and the one on g CAST experiment: the CERN Axion Solar Telescope (CAST) looks via X-rays for axion-like particles coming from the sun. The present limit on the photon-ALP coupling g < 0:8 10 10 GeV 1 ) sin2 < 2 10 11 for m < 0:02 eV. The limit is slightly weaker than the GC limit and well outside the region of interest in eq. (5.3), hence we omit it in gure 5. In contrast, IAXO [84], the new generation experiment, will be able to improve the limit. However, despite the future progress in this technology this class of experiments is not likely to be relevant for our scenario since it probes a region of the parameter space where fth force experiments provide very strong bounds. 7 Implications for the relaxion theory space In this section, we collect all bounds from laboratory experiments, colliders, astrophysics and cosmology that were shown in gures 2, 3, 4 and 5 for di erent mass regions and translate them, using eq. (5.2), into the underlying theory parameters br and f in gure 6.9 As a connection between both parametrisations, br and f were shown as a grid of contours in the previous plots, whereas in the ( br; f ) plane of gure 6 we show contours m . While the values we provide are for the j = 2 case, as mentioned below eq. (5.2) one can obtain the values for the j = 1 by the simple translation p br ! 2 br; f ! 2f . We show how these bounds push the cut-o to smaller values by the upper horizontal axis, where we translate the br scale in the lower axis to cut-o values using eq. (4.3) for n = 3N = 330. As indicated in the gure these values can be easily rescaled for other values of n or N . 9The color coding for the experimental bounds is the same as in the previous gures. Neff BBN HJEP06(217)5 10-9 10-11 10-13 θ10-15 2 10-17 10-19 f 10-21 10-2130-7 GC-γ GC-e V e 10 G 0 1 = P ixie Λ br= 0 .9 9 (Λb r)m ax CMB-μ V e 0 1 b Λ 1 r= CMB-y EBL G e V h m f= 10-4 10-6 10-5 and m from 100 eV to 0:3 GeV: globular cluster via coupling to electrons (blue) or coupling to photons (turquoise), supernova 1987a (light blue), extragalactic background light (EBL, yellow), CMB y-distortion (light green) and distortion (green), entropy injection S=S bounded by the baryon-to-photon ratio B (orange) and by Ne (pink), BBN (red). The green dotted lines represent the projection for the sensitivity of PIXIE to CMB distortions.The light gray band indicates the possible range of sin2 for j = 1, i.e. the QCD case. The gray lines (from top to bottom) are contours of constant solid), and 1 GeV (dashed). Here of a non-tachyonic in eq. (3.12) for sin( 0=f ) = 1=p2. The black lines (from left to right) are max is the upper bound on br br arising from the requirement br = 0:99 bmrax (thick, contours of constant f = 1010 GeV; 106 GeV (thin) and f = mh (thick). The overview presented in gure 6 shows that large areas in the br f plane are already well covered by existing experimental and observational probes, for instance the high-f region up to MPl is probed by the fth force experiments, on the other hand the cosmological, astrophysical, beam dump and collider observables constrain lower values of f . We see that in the above f ranges, the region with electroweak scale br is practically completely ruled out apart from small gaps that still remain. We also show in gure 6 how some of these gaps in parameter space might be covered soon by future experiments such as SHiP, NA62 and PIXIE. However, the region between f 1010 GeV and 1014 GeV which corresponds to relaxion masses between 0:1 eV and 1 keV, is currently barely constrained by data. For any f (or m ) value, all the constraints can be evaded for su ciently small br values (there are no bounds for br . 0:3 GeV). Small br values are however theoretically disfavoured for several reasons. First of all, as we see from the cq contours in gure 6, the constraints derived here push the relaxion to a region with somewhat lower values for the upper bound on the cut-o derived from cosmological considerations during in ation. If one takes seriously the requirement that the relaxion should not have transplanckian excursions, our bounds have a much stronger impact. This is because, as we see from gure 6, our bounds already cover a large part of the parameter space outside the shaded region where the relaxion travels transplanckian distances for any cut-o larger than 2 TeV. Coming to the issue of the very large global charges that arises due to the compact nature of the relaxion, we see from the upper horizontal axis that even in CKY/clockwork models the number of sites required can become uncomfortably large for very small backreaction scales. For N . 30 (see section 4) our bounds can signi cantly constrain the cut-o . For instance for f = 1000 TeV we nd . 100 TeV. As shown in section 4 the simplest clockwork models start getting tuned for N & 30. As far as the proposal to solve the little hierarchy problem using modest n values is concerned [3], we see that such a proposal would be completely ruled out outside the f 1010 GeV- 1014 GeV (m 0:1 eV - 1 keV) region, as contrary to the philosophy of this approach, too large values of n > (v= br)4 would be required. One should be keep in mind while interpreting these bounds within the clockwork framework that in these models one must have f & from eq. (4.10). Thus even from this point of view the unconstrained f 1010 GeV-1014 GeV (m 0:1 eV-1 keV) window is an interesting region as here the cut-o can be high in these models. Finally let us discuss what impact the pseudoscalar couplings of the relaxion might have on the overall bounds. As explained in appendix C, in the electroweak preserving [1, 3] models discussed in section 3, the relaxion does not have pseudoscalar couplings larger than the Higgs-portal ones, hence our experimental bounds would be qualitatively unchanged. Let us brie y comment on the possible change in our bounds if the pseudoscalar coupling to photons is larger than the one induced by Higgs mixing. As already mentioned, among the models discussed in section 3 this holds only for the pseudoscalar diphoton coupling in the non-QCD j = 1 model where the relaxion has a pseudoscalar coupling to photons suppressed only by 1=f and not by the backreaction scale (see eq. (C.3)). In this case the astrophysical and cosmological bounds discussed in section 6 will be a ected. An analysis of how the cosmological bounds change in the presence of a large g~ coupling is beyond the scope of this work. The enhanced coupling to photons will lead to a stronger bound from globular clusters, that is f & 107 GeV, eq. (6.29). However, this is valid only provided that the relaxion mass is lighter than 30 keV, so we immediately see from gure 6 that this is relevant only for br v. Furthermore, the CAST experiment can put a bound on f of similar order on the coupling to photon eq. (6.30) in the sub-eV region. For large Higgsrelaxion mixing fth force experiments are sensitive, hence the CAST bound is irrelevant. However for v, when the sensitivity to fth force experiments ceases, the CAST bound on the pseudo-scalar coupling can be important for sub-eV relaxions. 8 Testing for the CP violating nature of the relaxion In this section we investigate the feasibility of detecting a signal of spontaneous CPviolation together with a Higgs mixing signal. This would represent a smoking gun for our scenario since what we discussed so far about relaxion phenomenology applies to any 1015 1012 ] V e G [ f 109 106 103 mϕ=μeV meV eV keV 104 Λ/3(N-30)/4 [GeV] tP Λ 1 = 4 5 G e V , Λ cq 105 7 Ge V 1 Λ 2 = T e V , Λ cq E B CM L B ηB Neff 101 th fo rc e 8 Ge V 1 0 T e V , Λ cq SN tP Λ 1 0 Λ b r Λ b Ce m)r G a x 9 Ge V 1 0 K B 102 -LEP -LHC MeV 100 B B N f . The upper horizontal axis bounds the cut-o for N = 30 via eq. (4.9). For other br and , the required N is obtained via N = 4 log3 R + 30 where R = =3(N 30)=4 is the value read o the upper axis. Laboratory: fth force experiments (light orange). Cosmology and astrophysics: EBL (yellow), CMB (green), globular cluster via coupling to electrons (blue, transparent), BBN (red), entropy injection constrained by B (orange) and by Ne (pink), supernova 1987a (light blue). The green dotted lines represent the projection for the sensitivity of PIXIE to CMB distortions. Beam dump experiments: CHARM (dark red) and projections for SHiP (dark red, dotted). For the beam dump projections at NA62 and SeaQuest, see gure 3. Flavour: rare K-meson decays at E949/787, NA48/2, KTEV (dark blue) and projection for NA62 (dark blue, dotted), rare B-meson decays at Belle and LHCb (turquoise). Higgs production and decay at colliders: LEP (green), LHC (purple). The vertical gray band indicates exclusion due to br arising from the requirement of a non-tachyonic The dashed, black lines show (from top to bottom) contours of br > in eq. (3.12) for sin( 0=f ) = 1=p2). max (here br = MPl from the transplanckian (\tp") condition in eq. (1.9). The same contours are obtained for eq. (1.8). The thin, black lines indicate m from 10 15 GeV (uppermost) to 1 GeV (lowest) with a ' 107 GeV; 108 GeV; 109 GeV from the cosmological classical-vs.-quantum (\cq") condition in spacing factor of 103. scalar mixed with the Higgs. However, as already discussed in appendix C, the strength of the relaxion pseudo-scalar couplings depend on the details of the back-reaction sector. Couplings to fermions are typically very suppressed (compared to the one from Higgsrelaxion mixing), while the coupling to photon g~ is in many cases only as large as the scalar one, that is g~ 10 5 sin . In the electroweak breaking non-QCD model discussed in section 3, instead, the coupling to photons is in principle larger since it is not suppressed by the backreaction scale. Despite the model dependence, it is still an interesting question whether a CP-violating signal could be detected at the precision frontier. Let us investigate the relaxion contribution to the electric dipole moments (EDM). In our scenario the leading contribution to the electric dipole moment is generated through its couplings to fermions via Higgs mixing and with the pseudoscalar coupling to photons, g~ .We will focus on the electron EDM, following [85], but similar results hold for the neutron EDM. The rst step is to understand in which relaxion mass range this probe can be e ective. To this end let us estimate the strength of g~ g e since the relaxion one-loop contribution to the electron EDM will be proportional to it. The current upper bound on the electron EDM is de=e 8 10 29cm [86], which corresponds to g~ ge 5 10 14 GeV 1 [85], and improvements of one order of magnitude are expected in the coming years [87]. Let us then see how this compares to relaxion models. For the non-QCD electroweak breaking model we get: g~ g e . me sin 4 v f 3 10 16 m 1GeV 2 GeV 1 (8.1) where we used eq. (2.6) and eq. (C.3). The electroweak preserving models [1, 3] have an additional 4br=v4 suppression from the backreaction scale due to the suppression in g~ in eq. (C.2) as compared to eq. (C.3). We see that in both cases, a relaxion with m ' 1 GeV yields a contribution to the de that is below the current (and near future) sensitivity. The parameter space constrained is therefore in the few GeV region. 9 Conclusions We study various phenomenological aspects of relaxion models. We focus on models where the rolling of the relaxion eld stops due to the presence of a Higgs-relaxion backreaction term. We show that the relaxion generically stops its rolling at a point that breaks the CP symmetry, leading to relaxion-Higgs mixing. We investigate then the implications of this mixing, and analyse current and near future probes involving laboratory, cosmological and astrophysical measurements in terms of reach and sensitivity. In most parts of the parameter space, these observational constraints put the most stringent bound on the backreaction scale, br . On the theoretical front, we show that simple multiaxion (clockwork) UV completions su er from a ne tuning problem, which increases with the number of sites. Let us describe in more detail our main results on the observational probes of relaxionHiggs mixing. The constraints/discovery prospects derived by us are summarised in gures 2{5. In the sub-eV mass range the relaxion lifetime is much larger than the age of the Universe and thus cosmological or direct laboratory probes are not e ective. Fifth force experiments, however, are sensitive in large regions of the parameter space in the sub-eV region because of the low mass of the relaxion and the CP-even nature of its couplings to SM particles via Higgs mixing (see gure 2). The eV-MeV region is practically unconstrained by laboratory probes, but a subset of this region (keV-MeV) can be constrained by astrophysical and cosmological probes as shown in gure 5. The cosmological probes are relevant here because this is the region of parameter space where the relaxion lifetime is between 1 s and 1026 s and thus is tested by a variety of cosmological probes, such as entropy injection constraints from Ne and B measurements, BBN observables, CMB spectral distortions and EBL distortions. Turning to the MeV-GeV region we nd that in some parts of this mass range, the relaxion lifetime is just right for beam dump experiments (O(100 m) in the lab frame) such as the CHARM experiment and experiments probing invisible rare meson decays. We also nd that future data from beam dump experiments like SeaQuest and especially SHiP and the currently running ultra-rare kaon decay experiment NA62 can probe new and interesting regions of the relaxion parameter space. In other parts of this MeV-GeV mass region visible rare meson decays also put signi cant bounds. Finally, for relaxion masses above 5 GeV the constraints arise from LEP bounds on the Higgs-strahlung process and LHC Higgs coupling bounds on the new channel, h ! , as shown in gure 4. In gure 6, we translate these bounds to the relaxion theory space and discuss the theoretical implications. We nally comment that, while the relaxion-Higgs mixing requires CP violation, most of the probes discussed above do not form a strong test of the CP nature of the relaxion. The pseudoscalar couplings of the relaxion tend to be more model-dependent. For instance, in the familon model that was constructed in [3] the relaxion does not couple to F (with F being the QED eld strength) at one loop but only to the orthogonal combination of the electroweak eld strengths. We nd that, in existing models, probes of CP violation are sensitive only for GeV scale relaxion masses. Acknowledgments We thank K r Blum, Josef Pradler, Gordan Krnjaic and Aviv Shalit for helpful discussions and suggestions. We thank Babette Dobrich and Gaia Lanfranchi for useful and detailed discussions about NA62. TF would like to thank the Weizmann theory group for hospitality during the initial stages of the project. TF was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the ministry of Education, Science and Technology (No. 2013R1A1A1062597) and by the Korea- ERC researcher visiting program through NRF (No. 2015K2A7A1036922) and by IBS under the project code, IBS-R018-D1. The work of GP is supported by grants from the BSF, ERC, ISF, Minerva, and the Weizmann-U.K. Making Connections Programme. Note added: as this paper was being completed, ref. [12] appeared which also discusses relaxion phenomenology. Relaxion mass and mixing for the case of j = 1 In this appendix we present the mass matrix for the j = 1 case. The potential in this case is, V = 2 + g (A.1) Expanding around their vacuum expectation values (VEVs), and imposing the minimisation conditions, we get (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) 2 vH 2 + g 0 rg 3 + 2rg2 2 0 + g v 2 2 H + p2f M~ 3vH sin h^ = p ; M~ 3 cos = 0 + 0 ; (A.2) f M~ 3 sin = 0 ; = 0 ; 2 + g M~ 3 = 3 vH2 = g vH + p2f = p2f 2 M~ 3vH cos sin 0 = 2 vH2 + p2vH 0 M~ 3 ' p2f cos 0 + 2rg2 2 ' p2f 2 M~ 3vH cos 0 0 where we have used eq. (A.3), eq. (A.4) and 2 For any given M~ (or by diagonalising the above mass matrix after setting by requiring the heavier eigenvalue to be the physical Higgs mass, mh = 125 GeV. Note that we always have, rvH2 to obtain the approximations above. br) and f the exact relaxion mass and mixing can be determined m2h > Mh20h0 = 2 vH2 + p2vH cos 0 ; M~ 3 a fact we use in section 3 . B Relaxion mass and mixing for the case of j = 2 In this appendix we derive the relevant relation for j = 2 models. To obtain this we expand the potential V (h; ), V = 2 + g M~ 2h^2 cos + h^4 + rg 3 + rg2 2 2 : : : : (B.1) =g and 0=f is expected to be an O(1) phase. Now we calculate the 0 h0 f 2 + g M~ 2 cos = g v + M~ 2v sin 0 2 0 M~ 2v + 2rg2 2 0 sin = 2 v2 ; 0 v2 M~ 2 ' 2 f 2 cos 0 : where we have used eq. (B.2), eq. (B.3) and br) and f , the exact relaxion mass and mixing can be determined rv2 to obtain the approximations above. (B.4) (C.1) by diagonalising the above mass matrix after setting by requiring the heavier eigenvalue to be the physical Higgs mass, mh = 125 GeV. C Pseudoscalar couplings of the relaxion In this appendix we discuss the pseudoscalar couplings of the relaxion that can arise from the backreaction sector. As already mentioned in section 2, these couplings are modeldependent so our discussion here would be limited to the speci c models in section 3, namely the j = 1 non-QCD backreaction model and the j = 2 axion-like and familon models. As far as low energy probes are concerned, the important couplings are the ones to light fermions, photons and gluons. In all the above models, the exotic fermions10 can induce a pseudoscalar coupling of the relaxion to light fermions, g~ f , that is proportional to the light fermion mass as well as the shift symmetry breaking spurion ( 4br) that generates the relaxion mass; thus it has the same suppressions as the Higgs portal coupling, g f in section 2. Furthermore, as the above models involve sequestered sectors, one can check by inspection that these couplings are generated at least one loop order higher than the corresponding Higgs-portal coupling so that around the minimum (vH ; 0). In these models vH = v = 246 GeV. The minimisation conditions yields, v 2 2 + g + g2 2 M~ 2 cos rg 3 + g v 2 2 M~ 2v2 2f sin 0 0 h mass matrix results in As far as the coupling to photons is concerned we need to distinguish between the g~ 4 j = 1 and j = 2 models. By inspection we see that in both the j = 2 models the F F~ coupling can be possibly induced but only with the same shift symmetry breaking 10In the non-perturbative j = 1; 2 models it is the analog of the pion and /or 0 that get the loop induced couplings (to both fermions and photons) and the relaxion obtains its coupling via mixing with these states. g~ f g f : 16 2 suppression ( 4br=v4) and at the same order in perturbation theory as the Higgs portal coupling (in the non-perutbative model the coupling can arise via mixing with the analog of the 0 and in the perturbative familon model via a 2-loop level diagram), In the non-QCD j = 1 model, however, it is possible to have g~ because this backreaction sector is just a scaled-up version of QCD. Thus, as is the case for QCD axions, the relaxion will get an anomaly-induced coupling of the same order via mixing with the 0 and pion analogs of the new strong sector. This generates g~ 4 br em v4 4 f g : g~ em ; 4 f which can be larger than the Higgs portal coupling, g , for values of br v. It is important to mention that while the pseudoscalar coupling of the relaxion to photons is smaller than the Higgs-portal one in the existing j = 2 models, an anomaly induced coupling of the size in eq. (C.3) would exist in simple variants where the relaxion couples directly to the electroweak doublet fermions. One can proceed along the same lines to show that the pseudoscalar coupling of the relaxion to gluons is at least one loop suppressed with respect to the Higgs portal induced coupling to gluons because of the sequestering. We see, therefore, that apart from the g~ coupling in the j = 1 model, the pseudoscalar couplings of the relaxion are either suppressed or of the same order as the Higgs portal coupling in the models in section 3. Our results would thus be qualitatively unchanged by the presence of these couplings apart from the one exception above, on which we comment in the text. D The h coupling in j = 2 models In this appendix we present the expression for the h coupling in j = 2 models. To obtain this we expand the potential V (h; ) in eq. (2.1) around the minimum (v; 0) to obtain all cubic terms and then substituting the gauge eigenstates in terms of the mass eigenstates 0 = s h + c ^ ; h0 = c h + s ^ : e ects, we take sin( 0=f ) = cos( 0=f ) = 1=p2 to nally obtain In order to reduce the complexity the full expression while accounting for the leading mixing gh M~ 2 ' p2f v2s c2 4f 2 vc3 2f vs2c s 3 2 s c2 + 3 vs2c ; where all the terms proportional to powers of g such as the leading contribution g s c2 can be shown to be sub-dominant compared to the terms in eq. (D.2), using eq. (B.3) and assuming 2 v2. We use this expression to derive bounds on the decay of h ! ) in section 5.2.1. In a similar manner we can derive the h coupling for the j = 1 case. While we do not perform the full computation here, we note that the leading term in that case would be gh M~ 3c3=f 2. (C.2) (D.1) (D.2) gh Expressions for relaxion partial widths and lifetime In this appendix we provide the expressions for the relaxion partial widths for di erent channels. The dilepton (ll) and diphoton ( ) partial widths are given by As far as colored states are concerned we use the perturbative description above m 1 GeV. The partial width to quarks (qq) and gluons (gg) is given by ! ll) = sin2 ) = sin2 g ml2 m v2 8 2 m3 ! qq) = sin2 3mq2 m v2 8 ! gg) = 2 m3 4ml2 !3=2 m2 For m < 1 GeV the only hadronic state we consider is the decay to pions. Di erent estimates of the partial width to pions vary over nearly two orders of magnitude [32]. Here we use the leading order calculation of ref. [88] which gives ) = sin2 3 32 v2m 4m2 !1=2 m2 2m2 + 11m2 !2 9 : For m > 1 GeV one should use the partial width to kaons, -mesons etc, but as no reliable estimate exists in this regime [32], our perturbative estimate is su cient in this context. For a given mass, the total width, , can now be obtained by summing over all the kinematically relevant decay modes. Analyzing the ratio =M , we nd that the relaxion is very narrow throughout the whole parameter space of our interest. For example, m 2 10 13; 10 5 sin2 for m = f0:1; 5g GeV : (E.4) For lighter masses, this ratio becomes even smaller. Hence, potential width e ects do not arise. The ratio for intermediate masses between the two example values in eq. (E.4) highly depend on the thresholds of those particles that the relaxion can decay into. The relaxion lifetime, = 1= , is crucial in determining the applicability of various observational constraints. We show the lifetime as a function of m for di erent sin2 values in gure 1(b). 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Thomas Flacke, Claudia Frugiuele, Elina Fuchs, Rick S. Gupta, Gilad Perez. Phenomenology of relaxion-Higgs mixing, Journal of High Energy Physics, 2017, 50, DOI: 10.1007/JHEP06(2017)050