Relaxed Inflation Walter Tangarife,1, ∗ Kohsaku Tobioka,1, 2, † Lorenzo Ubaldi,1, ‡ and Tomer Volansky1, § 1
Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel 2 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
arXiv:1706.00438v1 [hep-ph] 1 Jun 2017
We present an effective model where the inflaton is a relaxion that scans the Higgs mass and sets it at the weak scale. The dynamics consist of a long epoch in which inflation is due to the shallow slope of the potential, followed by a few number of e-folds where slow-roll is maintained thanks to dissipation via non-perturbative gauge-boson production. The same gauge bosons give rise to a strong electric field that triggers the production of electron-positron pairs via the Schwinger mechanism. The subsequent thermalization of these particles provides a novel mechanism of reheating. The relaxation of the Higgs mass occurs after reheating, when the inflaton/relaxion stops on a local minimum of the potential. We argue that this scenario may evade phenomenological and astrophysical bounds while allowing for the cutoff of the effective model to be close to the Planck scale. This framework provides an intriguing connection between inflation and the hierarchy problem.
INTRODUCTION
The mass of the Higgs boson, mh , is sixteen orders of magnitude smaller than the Planck mass. This poses a puzzle, which goes under the name of the naturalness problem. In the Standard Model (SM) of particle physics, we expect large quantum corrections that would raise mh roughly up to the Planck scale. One way to avoid such corrections is to impose additional symmetries to protect mh , and keep it naturally small. Supersymmetry [1] is the most studied extension in this direction and, like most other solutions, predicts the presence of new physics at around the TeV scale that can potentially be accessible at the Large Hadron Collider. Another direction in addressing the problem of naturalness has been put forward in Ref. [2]. The smallness of mh could result from the cosmological evolution of another scalar field, the relaxion, that couples to the Higgs, scans its mass and eventually sets it to the observed value. This solution is based on dynamics rather than symmetry [3], and provides an intriguing connection between naturalness and cosmology. The model is described by an effective Lagrangian valid up to a cutoff scale Λ, and the success in addressing the small Higgs mass is measured by how high Λ is compared to mh , once the constraints from the dynamics are taken into account. In the original proposal, the highest Λ is of order 108 GeV, and can be achieved in a scenario where the relaxation dynamics take place during inflation. Various features of this class of models have been explored in Refs. [4–29]. In this letter, we take the idea of Ref. [2] a step further by promoting the relaxion to an inflaton. The advantages of doing so are that (i) the model is more minimal, as it does not have to rely on an unspecified inflaton sector, and (ii) it evades numerous constraints, allowing the cutoff to lie close to the Planck scale. In the rest of the paper we describe the model, the dynamics of inflation, a novel reheating mech-
anism, and the relaxation of the electroweak (EW) scale, which happens after reheating. The interested reader can find more details in a longer companion paper [30].
THE MODEL
We consider the effective Lagrangian 1 φ 1 L = − ∂µ φ∂ µ φ − Fµν F µν − cγ Fµν F˜ µν 2 4 4f − (gh mφ − Λ2 )H† H − λ(H† H)2 − Vroll (φ) − Vwig (φ) − V0 ,
Vroll (φ) = mΛ2 φ ,
Vwig (φ) = Λ4wig cos
φ , f
(1) (2)
defined in a Friedmann-Robertson-Walker (FRW) metric, ds2 = −dt2 + a2 (t)d~x2 . Here, φ is the relaxion/inflaton, H the Higgs doublet, Fµν the field strength of an Abelian gauge field, F˜µν its dual. f is the scale of spontaneous breaking of a global U (1), of which φ is the Goldstone boson. gh is a dimensionless coupling of order one, cγ is model dependent and can span a large range of values. Λ is the bare Higgs mass and the cutoff of the effective theory. The relaxion potential has three terms: V (φ) = Vroll (φ) + Vwig (φ) + V0 . The first is responsible for the rolling, and is linear in φ (we neglect higher powers, which would come with correspondingly higher powers of the small mass parameter m). The second is responsible for the periodic potential (“wiggles”), which grows proportionally to the Higgs vacuum expectation value (VEV), v, as Λ4wig ∼ (yv)n M 4−n . Here, y is a Yukawa coupling and M is a mass scale smaller than 4πv. Note that for n odd the wiggles are present only when H has a nonzero VEV, while for n even they are present also in the unbroken EW phase [4, 7]. In what follows, we concentrate, for simplicity, on the QCD-like case, n = 1. The third term, V0 , is a constant that we choose to set
Here H is the SM Higgs doublet, is the relaxion/inflaton field, we reheat and the universe becomes radiation dominated. We consider thebare following lagrangian Higgs mass ⇤ is the cuto↵ for our model, we assume m ⌧ In this section we review some aspects of axion inflation that are relevantThe to our framework Before reheating the slow-roll conditions are µ21( is ) =toghgenerate m ⇤2 ,dynamically a small µ2 dimensionless parameter.1 Theµ goal L= @µ @ Fµ⌫ F µ⌫ ↵ Fµ⌫ F˜ µ⌫ V (H, ) , (3 of relaxed inflation. We do not give the full details of the calculations, but refer the 2 the W 4 mass, m f, as representative of the ele 2 mis2Wthe⌧ -dependent ⇤2 , where squared we took mass parameter of W the Higgs potential. 1. 12 ˙ ⌧ V interested reader to Refs. [1–4]. V (H, ) = µ2 ( )H† H + (H† H)2 + Vroll ( ) + Vwig ( ) + V0 , (3 The larger the bare hierarchy mW for andour⇤,model, the more successful The Higgs mass ⇤between is the we assume m ⌧is⇤,thi an 1 cuto↵ 2 f next 2 H 2 section, 2 f and 2 ˙ 2 in the 2 2 2 2 We consider a relaxion with a potential V ( ), that we specify ⇠ 2 ⇠ 2 V ( ) = m⇤ + m + · · · , (3 roll hierarchy the SM dimensionless parameter. The1 problem. goal a small µ , that 2 is to generate dynamically = 2addressing ' ⌧ (4.33) 2 2 , where 2we took m22WThe ⌧ the W ismass, mW , as 2V ↵ V ⇤relaxion 3V↵ potential M= a coupling to SM photons here the same as representative the one used of in the Ref.electrow [5], an ( ) Pl ⇤4 cos . (3 wig
0
f as larger the hierarchy between mW morepossesses successfulthe is this mech V (φ) = 0 at the local minimum where weis obtain the ofThe m is justified by the fact that m and ! 0⇤, thethe model discrete This satisfied We keep onlyfor the linear term in (3.3), expand the around the constant terms 0 and absorb 1 1 addressing SM hierarchy problem. µ µ⌫ µ⌫ Here H+is↵2⇡ the kSMf .Higgs doublet, is themodel relaxion/inflaton field, theoretical issues [ ! As written the poses some ˜ correct EW scale, L @µ @ Finµ⌫VF ↵ Fµ⌫then F readsV ( ) . fThe (2.1)here is the same as the one(4.34) ⌧ relaxion M . potential 0 . The potential used in Ref. [5], and the
2
4
Pl
f
slow roll circumvented [7]. We present the clockwo ⇠ with a clockwork 2axion model 2
µ (m)! = g0hthe m model ⇤ , possesses the discrete shift(3 of1 m is justified by the fact that as 1 1 mW 0 + how to map its parameters to the ones we us 2 we show 4 2 Appendix and there (h, ) = µ 2 ( ) h2 + h4 ++m⇤ coswritten + Vmodel (3.9) 0, ! 2⇡ k + f . ⇤0As the poses some theoretical issues [6], th √ , @hφi ≡ φEW ' h|H|iF≡ v='@ A − mV2W ) .the 1 2. ✏ ⌧(Λ ⇢ , with ˜ 2 4 f 1 Here A , F = ✏ F A photon. is the -dependent squared mass parameter of the Higgs The model µ⌫ µ ⌫λ ⌫ µ µ⌫ ⇢ µ gh m in the3restwith of the paper. axion model [7]. We potential. 2 µ⌫ 3 The model and circumvented a clockwork present the clockwork mo photon-driven The Higgs ✓ bare mass ⇤ is ◆ the cuto↵ for our model, we assume m ⌧ ⇤, and gh i 2( The equation of motion for in a FRW is with µmetric ) = g(3) . We choose ˙V0Appendix such thatand thepoint cosmological constant is zero once h A special in field space is we show there how to map its parameters to the ones2we use in th dynamics hm H dimensionless 1 4The goal is to generate dynamically parameter. a small µ , that is |µ2 | ˙ 2+ We consider the lagrangian and settle to their VEVs:✏ ⌘ the following ' ⇢ Wfollowing (4.35)of the electroweak sca and the of the paper. 2 in⌧ 2rest We consider lagrangian One finds 2 2 m ⇤ , where we took the mass, m , as representative 2 2 W W2M H H 3 ⇤
@V ( )
¨ +23H ˙ +4 Λ4 m2W Λ m V0 = − + + W .@ gh gh 4λ
=
↵ ~ ~ hE · Bi , f
(4)
Pl rspecial radiation in field space m is and ⇤, the TheAlarger thepoint hierarchy between more ,successful is this mechanism W 2 µ22( EW ) ⇠(2.2) 10 ⌘(3.10) 1 dominates gh m 1V 0 problem. hhi = vaddressing = ⇠ 1fthe SMµ8hierarchy 2 µ⌫ 1µ⌫ L↵= F @⇤ ˜ @ µ (4.36) Fµ⌫) ,F µ⌫ ↵ Fµ( µ e L = @ @ F F F V (H, ' 2The2 relaxion f . µ + µ⌫ µ⌫ im g 2 f 2 ⌘ , 0 as the one used in 4 potential here Ref. [5], and the smalln e is the samef 2 4 r 2 ↵ 7 ↵ V M h i= g(3.11) EW where µ, = 0. phase, > 0 , from the broke hm Pl It separates the unbroken EW
2 † † 2 of m is justified by †the fact that m2!)0 = the µ model possesses the(H discrete shift 2 where the dot denotes a derivative with respect to cosmic time t and V†as (H, )H H(+point, + symme Vroll (( V (H, ) = +µ2⇡ ( )H H + (H H)model + happen Vposes ) near + Vtheoretical ) +issues V0itH) , [6],convenien wig roll ( (some As most the dynamics this ! µ2of fseparates . As written the that pha can where = 0. kItinteresting the unbroken EW phase, > 0 , from theis broken with Choosing this V0 corresponds to tuning the cosmoThe first term is small due to (4.34), the second is small for p 2 1 1 2 define This is a regime of warm inflation, with constant photon temperature ⇠ ⇤ . Once 1 e 0 2 2 2 potential around . We m circumvented with a clockwork axion model [7]. We present the clockwork model 2 2 2 0 As of the interesting happen near this point, it is+convenient toinex im 2 2most W dynamics g V ( ) = m⇤ + m · · · , V ( ) V ( ) = m⇤ + m + · · · , (t e roll µ ( ) ' m , ' . (3.12) r roll W EW logical constant. This is crucial, as itHdetermines Appendixaround andEW we show how1 to map its parameters to2the ones we use in this sect ghthere mdefine 2 = p .the (2.3) potential . We 0 ⇠ ⇠ 4 0 paper. = 0 + 4, | (4.37) | ⌧ 0(4.32) . dynamics of the field and ensures the exit of3M inflation V andV in( the fRH V4rest ) 'of ⇢the,= ⇤0 , Pl We have 2 ⇤ m 4point .in field Vwig . 0. ↵ Vwig (m)2WA↵ =special cos ( space is= ( 0 )+ = ⇤ ⇤ , 0 cos | |⌧ 0 before the relaxion settles into the EW vacuum. V0 = + W .f (3.13)f gh radiation We write the4Higgsdominated. doublet as 2 we reheat and the universe becomes ⇤ thatmass is until we exit inflation and We reheat. a a An important ingredient is that the paramwrite the Higgs doublet as 1, the (3 0 ⌘ ih ⌧ Choosing this VHere to SM tuning the cosmological This is crucial for ae 0 corresponds g1h m Here H is Higgs H is the the Higgs doublet, is the SM relaxion/inflaton p H doublet, = hfield, ⌧ a v is ,the relaxion/inflat Beforeporeheating slow-roll conditions areconstant. ih v eter m, which controls the slope of the rolling p H = h e , 2 we describe in the rest the paper. stages. The 2first regime 0 mechanismFIG. 1. Sketch ofof evolution ˙ ⌧ ofV the 3. 3Hsuccess where µ2 = 0. It separates the unbroken EW phase, > 0 , from the broken phase < 1 ˙ 2convenience tential, is tiny. This is technically natural, since inV (black) 2degreevia 2 that For we rewrite the potential where hh is the real freedom eventually gets corresponds toasreal inflation standard 1.future ⌧ ˙where µdegree ( )dynamics =ofgofhfreedom m ⇤ , slow-roll µ2on (gets = ghamvacuum ⇤2 ex , (t is that eventually a vacuum 3H |⇠| As most of the theVinteresting happen near this point, it is)convenient toexpectat expand 2 – 1 – a 2 = f ⌧ 1 , (4.38) ah 1while the limit m → 0 the Lagrangian recovers the disa1 shallow0(VEV) slope. In the second regime (blue), the (dark) equal to v = 246 GeV, while (a = 1, 2, 3) are the three goldstone 2 2 2 2 2 4 ˙ (VEV) equal to v = 246 GeV, h (a = 1, 2, 3) are the three goldstone mode potential around . We define 2 0fMH0 1 ↵ ⇠m 2⇠ f 2 4 W V V V (h, ) = photons gh m h are + responsible h= Vroll (Zgauge ) + Vdissipation. ( ⌧ ) , ⌧1a ⌧are (3.14) a are Pl wig ' (4.33) for the Relaxation eaten W+ and bosons. the Pauli matrices. eaten by+2the W Z bosons. the Pauli matrices. crete shift symmetry φ → φ + 2πf . The scales 2 the isinthe -dependent squared parameter of Higgs potential. 2 4 by 42the is -dependent squared mass parameter of the Higgs 2V ↵mass Vand 3gauge ↵2 M =Pl 0 + , | | ⌧ 0. (3 2 ⇤2 satisfied for m occurs in the last stage, after reheating (red). 1 2 W 1 We We stress that isiscuto↵ still a aclassical field, notnot a⇤ quantum fluctuation. Excuse the and notation the model have the following hierarchic structure The Higgs bare mass ⇤ is the for our model, we assume m ⌧ ⇤, The Higgs bare mass is the cuto↵ for our model stress that still classical field, a quantum fluctuation. Excuse thegno 2 hJ V ( ) = m⇤ + , (3.15) This roll is satisfied for M ↵ 4 as write' the HiggsPl doublet 2 , that is |µ ↵ V +⌧ghWe Vslow2 ⇤ . (4.39) dimensionless parameter. The goal is to generate dynamically a small µ 0 dimensionless parameter. The goal is to generate dynam 1 f ⌧f 2 MPl . (4.34) 0 |⇠| H = p h ei (3.16) , (3 m Λwig < 4πmW Λ < f < MPl , Vwig (5) = ⇤402cos . ⇠mass, 2 W 2 m(2W )⌧ ⇤ , where the mW , we as representative of themelectroweak s 2 the W mass, f we tookm took W , as repres W ⌧ ⇤ , where 2eventually gets a vacuum expectation va where h is the real degree of freedom that 0 ¨ ⌧ V 2.From Λ between The larger the hierarchy between m and ⇤, the more successful is this mechanism W 4. (4.29) we can compute ✏ ⌧ 1 The larger the hierarchy m and ⇤, the mor W H large values of the field, φ > φ0 h≡ –2,4, 3) –where where MPl is the reduced Planck mass. a (a g m 4 Inflationary dynamics (VEV) equal to v = 246 GeV, while =h1, are the three goldstone modes that –problem. 4periodic – addressing hierarchy problem. ✓ bosons. a◆ addressing the SMthere hierarchy noSM ✓ ◆ hasthe VEV, consequently is no eaten and by the W and Z gauge ⌧ are the Pauli matrices. The relaxion is coupled to an Abelian gauge field, ˙ 2 00 H f 1V V ˙ 2 4 ⇠ cosmological In this section we discuss the of fields ,as h0 the Aµ . used For the here is is2the the same2V one in Ref. [5],(4.35) and the Jack. smallu ✏V⌘With 'thatThe +field, ⇢moves ¨The relaxion potential here isright the same the one = potential. 2relaxion ✏f potential +evolution (4.40) φ3and from 2 our still not. a quantum fluctuation. Excuse theas notation 20a classical Aµ . The time-dependent relaxion background evenHstress 2M H 2a We 2conventions, Plbut purpose of our of study we can treat as classical field, we0must treat h possesses and Aµ as the discrete shift symm ↵ V! M ⇡↵ M m is justified by the fact that as m the model Pl Pl of m is justified by the fact that as m ! 0 the model pos to left. In the stage, tually leads to an exponential production longquantumof fields. The equation of motion for first is ⇠ 2 f 2 the8EOM ⇠ V 0 is ! + 2⇡ k f . As written the model poses some theoretical issues [6], that ' 2 + f . (4.36) 00 !↵2 M +22⇡ k7 ↵f . VAs written the model poses somecan t wavelength modes of the gauge bosons. This Vhas⇠ 0. Then In our model ˙axion Pl 0 (φ) (h, ) 3H ↵φ + V =[7]. 0a clockwork ¨ + 3H ˙ +a@Vclockwork ~ ~ circumvented with model We present the(7) clockwork in = h E · Bi . (4.1) circumvented with axion model [7].model We pr two important consequences: (i) it provides a new –4– @ f The first term ¨isand small tofthere (4.34), the to second small for Appendix we⇠due show how its parameters to theto ones weits useparameters in this sec V Appendix f 2map and is we how map source of dissipation for inflation, also maintainto0 a =very and relaxion rolls 2 good ✏ 0 approximation, +2 ⌧ 1show . thethere (4.41) The equation for h is 2 2 and in the rest of the paper. V ↵ V M ⇡↵ M 2 and in rest of the paper. Pl r @V (h, ing slow-roll during the final period of the relax⇠) the0 Pl slowly the shallow linear slope. The speed, (4.37) ¨ + 3Hdue h h˙ + into h+ (4.2) f=V0 . ' ⇢ , point is a2 fieldVspace @h A V 0 (φ) special point in field space is ↵ ation, which necessarily occurs after the end of in- A special ˙ = it3H , slowly increases H decreases going Using the of ✏|φ| above is easy to very that this as condition is again a˙ Heremechanism H ⌘definition an overdotsatisfied a is the Hubble parameter, with a the scale factor. In these equations 2 flation; (ii) it allows for a novel reheating ⇤ that is until we exit inflation and reheat. the potential, but staysof motion small for enough so that given (4.34). denotes a derivative with down respect to cosmic time t. The equation ⇤2 ⌘ , the photon ( that proceeds as follows. The gauge bosons form a ~ · Bi ~ is negligible at this 0stage gh m[see Eq. (11) be- 0 ⌘ g m , h2E is 0 ˙ h 3. 3H ⌧ V d A± (⌧, k) ⇠ ha ⌧ a v
1
strong coherent electric field that produces electron2 + kregime ± 2 k ˙ involves A± (⌧, k) = 0trans-Planckian . (4.3) low]. This field ex3H the |⇠| 2 V EW ⌧ unbroken where µ2 =d⌧0.2 It separates phase, > 0 , from the broken phase < positron pairs via the Schwinger effect. These parti=a very 1number , (4.38) µf V=0 M 0.2 It⌧separates the phase, > cursions, lastsV for large of unbroken e-folds, EW 0 where ↵ Pl As most of the interesting dynamics happen near this point, it is convenient to expand cles quickly thermalize and reheat the universe. We As most ofinto the the interesting happen near this po N > 1030 , and continues brokendynamics EW phase, 1 satisfied for potential around . We define 0 discuss these steps in more detail in the following 2 grows potential define1enough φ < φ0 . Eventually the around speed 0 . We large MPl ↵ V ⌧ Vslow2 ' ⇤40 . (4.39) sections. Note that since the relaxion is very light 2 – 5 – – 9 – |⇠| f , that the gauge-boson=production becomes | | ⌧ 0the . dom( 0+ = that , | |⌧ and weakly coupled, reheating mechanisms via per0+ inant source of friction. To understand how 0 4. ¨ ⌧ V From (4.29) we can compute turbative decays are not effective in this framework. We writehappens, the Higgs we doublet turnasour attention to the EOM of the
a a We write the ✓ Higgs doublet ◆ as
h ⌧ 1 a a massless field. ⇠ V fH2 = pV hV e00i v , 0 ( 1 ¨ = 2gauge i h v⌧ ✏f 2 + 2V . (4.40) p H = h e , 2 2 0 ↵ V modes, the EOM After expanding MPl Aµ⇡↵inMFourier Pl 2 INFLATIONARY DYNAMICS where hfor is the of freedom that [31] eventually gets a vacuum expectation v the real twodegree polarizations reads where h is the real degree of freedom that eventually In our(VEV) modelequal V 00 ⇠ to 0. vThen = 246 GeV, while ha (a = 1, 2, 3) are the three goldstone that equala to v = 246 GeV, while ha (a = 1,modes ~ 2, 3) are t bosons. The equation of motion (EOM) of the infla2 ∂¨2 AZk±gauge (τ⇠)(VEV) ξ eaten by the W and ⌧ are the Pauli matrices. ~ f V2 f k + A ) = 0 bosons. , (8)⌧ a are Z(τgauge the Pauli m =22 eaten ✏ 0 kby2±the +22kW and (4.41) ton/relaxion in the FRW metric is 2 ±⌧ 1 . 0 1 V Mfield, MPl We stress that Vis∂τ still a↵classical aτ quantum fluctuation. Excuse the notation Jack. Pl not⇡↵ 1 We stress that is still a classical field, not a quantum fluctu cγ ~ ~ whereofτ✏ is theitconformal time, dt/a, and we satisfied the definition above is easy to very that dτ this= condition is again · Bi , Using(6) φ¨ + 3H φ˙ + V 0 (φ) = hE f given (4.34). have defined
~ and B ~ are the electric and magnetic fields where E associated with the gauge field. Throughout this paper, an overdot denotes a derivative with respect to cosmic time, t. The energy density of the universe is dominated by the relaxion, that √ drives inflation, V (φ)
so the Hubble parameter is H = √3M . In Eq. (6), Pl we have neglected the term gh mh|H|2 i, because it is always negligible compared to V 0 (φ). The dynamics can be described in three different stages, illustrated in Fig. 1. The rolling starts at
ξ ≡ cγ
φ˙ – 4. – 2f H
(9)
Note that τ and ξ are both negative. Eq. (8) implies that low-momentum (long-wavelength) modes –9– ~ ξ k of A− (τ ), satisfying k−2 τ < 0, experience tachyonic instability and grow exponentially. The solution can be written approximately as 1/4 √ 1 −kτ ~ Ak− (τ ) ' √ eπ |ξ|−2 −2|ξ| k τ , (10) 2 k 2 |ξ|
–4–
3 for |ξ| > 1, and we can use it to compute ~ · Bi ~ ' 2.4 × 10−4 hE ~ 2 i ' 10−4 hE
H 4 2π|ξ| e , |ξ|4
H 4 2π|ξ| e , |ξ|3
~ 2 i ' 10−4 hB
(11) H 4 2π|ξ| e . |ξ|5 (12)
˙ and hence |ξ|, grow large enough, we Once |φ|, smoothly switch from Eq. (7) to the EOM V 0 (φ) =
cγ ~ ~ hE · Bi , f
(13)
where the dissipation is due to gauge-boson production. The solution now is f V 0 (φ) |ξ|4 ˙ = 2|ξ| Hf ' Hf ln |φ| . cγ πcγ 2.4 × 10−4 H 4 cγ (14) In this regime, |ξ| ∼ 20 is roughly constant (only ˙ decreases with the varies logarithmically), and |φ| decreasing H. The energy density of the gauge ~2 + B ~ 2 i, is roughly constant, and bosons, ργ = 21 hE 0 using Eq. (13) we have the relation ργ ' |ξ| cγ f V (φ). One can show that the slow-roll conditions are now satisfied as long as [30] f MPl . < cγ |ξ|
(15)
When the potential V (φ) attains a value smaller than ργ , the energy density is no longer dominated by the inflaton and we exit inflation. The following evolution is still described by Eq. (13), the relaxion keeps slowing down and its kinetic energy remains smaller than Λ4wig . This implies that as the periodic wiggly potential becomes sufficiently large to balance the linear slope, the field stops. Specifically, this condition reads mΛ2 ∼
Λ4wig . f
(16)
This must happen when φ = φEW . By taking m very small, we can achieve a very large Λ, the only bound being Λ < f < MPl . Therefore, with a large f , we can have a cutoff Λ close to the Planck scale. In the original proposal [2], where φ was not the inflaton, Λ was mainly constrained by the requirements that the inflaton dominate the energy density and that the classical motion of the relaxion dominate over its quantum fluctuations. Neither requirement is necessary in our framework, which allows for a significantly larger cutoff scale. Further details of the phenomenological and astrophysical constraints and the corresponding viable parameter space can be found in [30]. The picture of this section seems to describe a successful model of inflation that relaxes the EW
scale. However, there are some subtle complications related to the gauge-boson production that we have to face. They are the subject of the next sections, where we show how they lead to a novel mechanism of reheating. SCHWINGER EFFECT AND REHEATING
The produced gauge bosons have a comoving wavelength comparable to the size of the comoving horizon, (aH)−1 , and an exponentially large occupation number. They form a coherent collection that describes a classical electromagnetic field with the electric field dominating the energy density [see Eq. (12)], and approximately constant within the horizon. If Aµ is the SM photon, the strong electric field allows for electron-positron pair production, via the Schwinger mechanism, with a rate per unit time per unit volume [32, 33] ! ~ 2 (e|E|) −πm2e Γe+ e− = exp . (17) ~ V 4π 3 e|E| ~ ≥ πm2e . The The production is efficient when e|E| + − e and e quickly thermalize the system via annihilations, e+ e− → γγ, and inverse Compton scatterings on the long-wavelength photons, eγ → eγ. The rate of these processes is much faster than the rate at which the electric field is produced by the relaxion. The latter process becomes even less efficient after thermalization, because the photon gets a thermal mass which strongly suppresses the tachyonic instability [21, 26]. As a result the electric field does not grow larger than ∼ πm2e /e and, consequently, the energy density transferred to the e+ e− is of order m4e . This translates into a reheat temperature TRH ∼ me , that is below the Big Bang Nucleosynthesis (BBN) temperature. Moreover, the limited growth of the electric field implies that the c ~ · Bi ~ remains negligible in Eq. (6), and we term fγ hE never enter the regime in which the photon dissipation dominates. The kinetic energy of the inflaton then increases above Λ4wig and we overshoot the minimum at φEW , thus failing to relax the EW scale. Fortunately, there is a simple fix to these problems, as we discuss in the next section. DYNAMICS WITH A DARK PHOTON
Instead of SM photons, let us consider the production of dark photons. We modify the first line of Eq. (1) to 1 1 1 µν L = − ∂µ φ∂ µ φ − Fµν F µν − FD,µν FD 2 4 4 φ κ µν µν − Fµν FD − cγD FD,µν F˜D + eAµ ψ e γ µ ψe . 2 4f (18)
4 The subscript D denotes a dark photon, that kinetically mixes with the SM photon. We assume that φ only couples to AµD and there is no light content in the dark sector other than the dark photon. The field redefinition Aµ → Aµ − κAµD removes the kinetic mixing and introduces a coupling κe between the dark photon and the SM electrons, ψe . The relaxion dynamics proceeds in the same way as described above, the only difference being that now we produce dark electric and magnetic fields. The important point is that the Schwinger rate changes to ! (κe|E~D |)2 −πm2e Γe+ e− . (19) exp = V 4π 3 κe|E~D | ~ D | has to grow larger than This implies that |E πm2e /(κe) for the e+ e− production to occur. The highest value achievable by the electric field, before it saturates the EOM of Eq. (13), is |ξ| |ξ| 4 max 2 ~D | ∼ ρ γD ' |E f V 0 (φ) ∼ Λ . c γD cγD wig
(20)
This imposes a lower bound on κe to allow for the Schwinger pair creation. Meanwhile, to avoid thermal suppression of the tachyonic production of the dark photon, we require that its mean free path through the hot plasma of e+ e− be longer than a Hubble radius. This sets an upper bound on κe and guarantees that AµD does not get a thermal mass. These two bounds restrict κe to the window 1/2 1/2 m2e cγD Λwig . κe . , (21) Λ2wig |ξ| αMPl which implies a lower bound on Λwig : Λwig >
cγ α D MPl m4e |ξ|
1/5
.
(22)
Here α = e2 /(4π). At the beginning of the Schwinger production, the energy density of e+ e− is of order m4e , while that of ~ D | keeps the dark electric field is (κe)−2 larger. As |E growing to its maximum value, it shares its energy with the e+ e− pairs by accelerating them classically. At the end of the process we have ρe+ e− ∼ ργD . This is the energy density available for reheating the visible sector. We can thus achieve a reheat tempera1/4 Λwig , safely above BBN. Due ture TRH ∼ c|ξ| γD to the lack of thermal suppressions, the EOM of the relaxion is still described by Eq. (13) after reheating. Therefore, the continued friction provided by unsuppressed dark photon production crucially slows down the motion of φ and allows it to settle at the EW vacuum. Given the small values of κe under consideration, the dark photons never reach equilibrium with the visible sector and remain cold (they have very low
momentum) throughout the thermal history of the universe. In this way, cosmological bounds on relativistic species are evaded. What we have is a cold dark electric field, whose energy density, ργD , redshifts like radiation and remains comparable to that of the visible sector until the time of matter - radiation equality. After that point the universe enters the matter dominated era, and ργD eventually becomes a negligible component of the energy density budget. There is one more constraint we need to impose on the model. If the gauge-boson production regime lasts too long, we overproduce curvature perturbations, non-Gaussianities and primordial black holes [31, 34–36]. To comply with the corresponding CMB bounds we require that we enter this regime only in the last five e-folds of inflation. This sets a lower bound on f /cγD , and together with the condition of Eq. (15) restricts it to the window 0.2
f MPl MPl . < . |ξ| cγD |ξ|
(23)
The above fixes f to be of order f ' cγ MPl /|ξ|. For values of cγD of order one or larger, f can be close to the Planck scale. This, in turn, allows for a large cutoff Λ.
SUMMARY
We have presented a model where the relaxion, coupled to the Higgs and to a dark photon, drives inflation and relaxes the EW scale after reheating. Inflation proceeds in two stages. In the first, which lasts very long, the relaxion slowly rolls down a shallow slope. In the second, which takes place only in the last five e-folds, the slow-roll is maintained thanks to dark photon production, that provides dissipation. The dark photons, kinetically mixed with the SM photons, form a very large dark electric field which produces SM e+ e− pairs via the Schwinger effect. The e+ e− thermalize the visible sector to a temperature above BBN. After the reheating process, the relaxion keeps rolling and slowing down, due to the continued dark photon dissipation, until it stops on the periodic potential and relaxes the EW scale. The mechanism realizes a low-scale model of inflation (with H ∼ Λ2wig /MPl < m2W /MPl in the final observable e-folds) that fully addresses at the same time the hierarchy problem of the Standard Model. Additional details are presented in a companion paper [30]. The associated CMB signatures deserve further detailed studies, as does the novel reheating mechanism. Both will be presented in a future publication.
5 ACKNOWLEDGMENTS
We would like to thank Tim Cohen, Erik Kuflik, Josh Ruderman, and Yotam Soreq for collaboration at the embryonic stages of this work. We benefited from a multitude of discussions with P. Agrawal, B. Batell, C. Csaki, P. Draper, S. Enomoto, W. Fischler, R. Flauger, P. Fox, R. Harnik, A. Hook, K. Howe, S. Ipek, J. Kearney, H. Kim, G. MarquesTavares, L. McAllister, M. McCullough, S. Nussinov, S. Paban, E. Pajer, M. Peskin, G. Perez, D. Redigolo, A. Romano, R. Sato, L. Sorbo, and M. Takimoto. This work is supported in part by the I-CORE Program of the Planning Budgeting Committee and the Israel Science Foundation (grant No. 1937/12), by the European Research Council (ERC) under the EU Horizon 2020 Programme (ERC- CoG-2015 - Proposal n. 682676 LDMThExp) and by the German-Israeli Foundation (grant No. I-1283- 303.7/2014). The work of LU was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, and was partially supported by a grant from the Simons Foundation.
∗ † ‡ §
[1] [2]
[3] [4] [5] [6] [7]
[8] [9]
[email protected] [email protected] [email protected] [email protected] S. P. Martin, (1997), [Adv. Ser. Direct. High Energy Phys.18,1(1998)], arXiv:hep-ph/9709356 [hep-ph]. P. W. Graham, D. E. Kaplan, and S. Rajendran, Phys. Rev. Lett. 115, 221801 (2015), arXiv:1504.07551 [hep-ph]. This is true modulo the fact that it relies on an argument of technical naturalness, on which we elaborate further in the next section. J. R. Espinosa, C. Grojean, G. Panico, A. Pomarol, O. Pujol` as, and G. Servant, Phys. Rev. Lett. 115, 251803 (2015), arXiv:1506.09217 [hep-ph]. E. Hardy, (2015), arXiv:1507.07525 [hep-ph]. J. Jaeckel, V. M. Mehta, and L. T. Witkowski, Phys. Rev. D93, 063522 (2016), arXiv:1508.03321 [hep-ph]. R. S. Gupta, Z. Komargodski, G. Perez, and L. Ubaldi, JHEP 02, 166 (2016), arXiv:1509.00047 [hep-ph]. B. Batell, G. F. Giudice, and M. McCullough, JHEP 12, 162 (2015), arXiv:1509.00834 [hep-ph]. O. Matsedonskyi, JHEP 01, 063 (2016), arXiv:1509.03583 [hep-ph].
[10] L. Marzola and M. Raidal, Mod. Phys. Lett. A31, 1650215 (2016), arXiv:1510.00710 [hep-ph]. [11] K. Choi and S. H. Im, JHEP 01, 149 (2016), arXiv:1511.00132 [hep-ph]. [12] D. E. Kaplan and R. Rattazzi, (2015), arXiv:1511.01827 [hep-ph]. [13] S. Di Chiara, K. Kannike, L. Marzola, A. Racioppi, M. Raidal, and C. Spethmann, Phys. Rev. D93, 103527 (2016), arXiv:1511.02858 [hep-ph]. [14] L. E. Ibanez, M. Montero, A. Uranga, and I. Valenzuela, JHEP 04, 020 (2016), arXiv:1512.00025 [hepth]. [15] A. Hebecker, F. Rompineve, and A. Westphal, JHEP 04, 157 (2016), arXiv:1512.03768 [hep-th]. [16] N. Fonseca, L. de Lima, C. S. Machado, and R. D. Matheus, Phys. Rev. D94, 015010 (2016), arXiv:1601.07183 [hep-ph]. [17] A. Fowlie, C. Balazs, G. White, L. Marzola, and M. Raidal, JHEP 08, 100 (2016), arXiv:1602.03889 [hep-ph]. [18] J. L. Evans, T. Gherghetta, N. Nagata, and Z. Thomas, JHEP 09, 150 (2016), arXiv:1602.04812 [hep-ph]. [19] F. P. Huang, Y. Cai, H. Li, and X. Zhang, Chin. Phys. C40, 113103 (2016), arXiv:1605.03120 [hepph]. [20] T. Kobayashi, O. Seto, T. Shimomura, and Y. Urakawa, (2016), arXiv:1605.06908 [astroph.CO]. [21] A. Hook and G. Marques-Tavares, JHEP 12, 101 (2016), arXiv:1607.01786 [hep-ph]. [22] T. Higaki, N. Takeda, and Y. Yamada, Phys. Rev. D95, 015009 (2017), arXiv:1607.06551 [hep-ph]. [23] K. Choi and S. H. Im, (2016), arXiv:1610.00680 [hep-ph]. [24] T. Flacke, C. Frugiuele, E. Fuchs, R. S. Gupta, and G. Perez, (2016), arXiv:1610.02025 [hep-ph]. [25] L. McAllister, P. Schwaller, G. Servant, J. Stout, and A. Westphal, (2016), arXiv:1610.05320 [hepth]. [26] K. Choi, H. Kim, and T. Sekiguchi, (2016), arXiv:1611.08569 [hep-ph]. [27] Z. Lalak and A. Markiewicz, (2016), arXiv:1612.09128 [hep-ph]. [28] T. You, (2017), arXiv:1701.09167 [hep-ph]. [29] J. L. Evans, T. Gherghetta, N. Nagata, and M. Peloso, (2017), arXiv:1704.03695 [hep-ph]. [30] W. Tangarife, K. Tobioka, L. Ubaldi, and T. Volansky, to appear. [31] M. M. Anber and L. Sorbo, Phys. Rev. D81, 043534 (2010), arXiv:0908.4089 [hep-th]. [32] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936), arXiv:physics/0605038 [physics]. [33] J. S. Schwinger, Phys. Rev. 82, 664 (1951). [34] N. Barnaby, R. Namba, and M. Peloso, JCAP 1104, 009 (2011), arXiv:1102.4333 [astro-ph.CO]. [35] A. Linde, S. Mooij, and E. Pajer, Phys. Rev. D87, 103506 (2013), arXiv:1212.1693 [hep-th]. [36] J. Garc´ıa-Bellido, M. Peloso, and C. Unal, (2016), arXiv:1610.03763 [astro-ph.CO].
