Jhep05(2020)154

Home , False vacuum decay, Pole mass

JHEP05(2020)154 Springer May 3, 2020 May 28, 2020 : : March 15, 2020 : January 20, 2020 : Accepted Revised Published Received , can be larger than 100 GeV if the c inf Published for SISSA by https://doi.org/10.1007/JHEP05(2020)154 H [email protected] and Wen Yin , a,b , is lower than the QCD scale, and if the inﬂation lasts suﬃ- inf H . 3 2001.04464 Fuminobu Takahashi The Authors. a Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs c

The abundance of the QCD axion is known to be suppressed if the Hubble , [email protected] [email protected] Sendai, Miyagi 980-8578, Japan Kavli Institute for theKashiwa, Physics Chiba and 277-8583, Mathematics Japan ofDepartment the of Universe Physics, (WPI), KAIST, UniversityYuseong-gu, of Daejeon 34141, Tokyo, Korea E-mail: Department of Physics, Tohoku University, b c a Open Access Article funded by SCOAP Keywords: Physics ArXiv ePrint: relaxed if the eternala old false inflation vacuum is at drivenfalse by large vacuum the is field located standard-model values. above Higgs theafter Specifically, field intermediate the scale. trapped tunneling We in from also the discuss false the slow-roll vacuum inflation to the electroweak vacuum. Abstract: parameter during inflation, ciently long. We show that the tight upper bound on the inflation scale can be significantly Hiroki Matsui, QCD axion window and false vacuum Higgs inflation JHEP05(2020)154 ]. 10 – (1.1) (1.2) 5 . The ]. ) grows 4 , T 3 ( QCD χ Λ . T at 0 χ the gluon field strength, α (100) MeV, the topological µν 3 F 12 ) 'O pl , M 17 µν α QCD ˜ F Λ , 10 ) α µν 2 14 a T F 16 false ( f a v T χ a 9 f 5 2 = GeV) 2 3 π 2 g – 1 – 11 ) 32 ) was studied by using the lattice QCD [ T 15 10 T GeV ( ) = ( a pl ∼ 11 χ m 5 M the strong gauge coupling, 20 axion ∼ false 9 3 L v g false v , is coupled to the standard model (SM) QCD as a 1 ) is vanishingly small, and so, the axion is almost massless. T ( χ ) is the topological susceptibility. At is the decay constant, its dual. The global U(1) PQ symmetry is explicitly broken by non-perturbative T ]. It predicts the QCD axion, a pseudo Nambu-Goldstone boson, which arises as ( a 2 f χ µν , α 1 ˜ F The QCD axion, 4.1 Thin-wall approximation4.2 (10 Non-degenerate vacua4.3 ( Thick-wall regime4.4 ( Open bubble universe and the dynamics 3.1 A false3.2 vacuum in the The Higgs effective3.3 potential QCD scale The QCD axion window opened wider where susceptibility as the temperature decreases,detailed and temperature it dependence approaches a of When constant the value axion mass becomes comparable to the Hubble parameter, the axion starts where and effects of QCD through theacquires above coupling. a mass If there is no other explicit breaking, the axion a consequence of the spontaneous breakdown of the global U(1) PQ symmetry [ 1 Introduction The Peccei-Quinn (PQ) mechanism islem the [ most plausible solution to the strong CP prob- 5 Slow-roll inflation after the6 bubble nucleation Discussion and conclusions 4 False vacuum decay and slow-roll inflation 3 False vacuum Higgs inflation Contents 1 Introduction 2 The QCD axion and the Bunch-Davies distribution JHEP05(2020)154 . – π for 28 a (1.3) f or the 1 ]. See also ] 46 ] using the 17 – ], the Witten 34 . 14 , is determined 23 3 i – − 2 . The oscillation a ∗ h 20 a , ] for realizing the hilltop p 13 48 , ], is below the QCD scale. Then, ]. See also ref. [ 44 [ 1, which relaxes the upper bound of , is of order unity, the observed dark GeV 47 ], dynamical relaxation [ [ a ], the BD distribution is peaked at the π 12 ], and those axions contribute to dark 2 π /f / 33 46 | ∗ , 10 13 ' , ∗ a inf – ∗ 5 sec.) in which case the bound may be weakened. θ 32 | 43 . θ H = 11 & a ] (including two of the present authors, FT and t ∗ f = – 2 – θ GeV, the axion abundance exceeds the observed 43 . , inf 17 ], and the anthropic selection of the small initial T − 42 must be fine-tuned to be of order 10 38 16 GeV – ∗ θ 8 35 10 = 10 The variance of the BD distribution, a 2 f ], etc. ]. 3 41 43 – ]. For instance, if the axion decay constant is of order the GUT , 39 19 42 . Thus, one can realize , is lower than the QCD scale, and if the inflation lasts long enough. inf inf ) during and after inflation [ H H π ]. If there are no light degrees of freedom which changes the strong CP 45 ], resonant conversion of the QCD axion into a lighter axion-like particle [ ] it was pointed out that the accretion disk formed around the proto-neutron star (or black 18 27 ] for related topics. – and the axion mass, and it can be much smaller than the decay constant 53 24 – inf 49 So far we assume that the QCD scale during inflation is the same as in the present Recently, it was proposed in refs. [ There are several ways to relax the upper bound of the classical axion window. The In the PQ mechanism some amount of coherent oscillations of the axion is necessarily In ref. [ If the QCD axion has a mass mixing with a heavyThe axion cosmological moduli whose problem dynamics of induces string a axions can phase also shift be of relaxed in a similar fashion [ H 2 3 1 ], tachyonic production of hidden photons [ , the initial misalignment angle a hole) may explain the late-time neutrino emission ( one can dynamically realize ainitial hilltop condition initial in condition, a supersymmetric set-up. refs. [ sufficiently small the classical axion window. vacuum. However, this may not be the case. For instance, the Higgs field may take large distribution reaches equilibrium after sufficiently longis inflation when balanced the by quantum diffusion thedistribution [ classical motion.phase (modulo The 2 distribution isCP conserving called point the [ Bunch-Daviesby (BD) WY) that the axionduring overproduction inflation, problem can beThe ameliorated reason if the is Hubble asif parameter follows. the Gibbons-Hawking First, temperature, theeven axion though potential the is axion already mass generated is during much smaller inflation than the Hubble parameter, the axion field effect [ 31 stronger QCD duringoscillation inflation amplitude [ [ scale or the string scale,dark i.e., matter abundance byf many orders of magnitude. Therefore, for such largeaxion values abundance of can be suppressed by late-time entropy production [ matter abundance sets the upper bound of the so-called classical axion window, where the lower boundcooling is neutron due to star the [ neutrino burst duration of SN1987A [ amplitude decreases due to theCP subsequent phase cosmic is expansion, dynamically and suppressed. thus the effective strong produced by the misalignmentmatter. mechanism If [ the initial misalignment angle, to oscillate around the potential minimum with an initial amplitude JHEP05(2020)154 . – ]. 57 we 55 QCD , 4 38 – ]). It was ]. It is also 34 48 54 ]. (See also refs. [ ) GeV in the minimal 5 56 ]. However, it is difficult (10 38 O we briefly review how the BD 2 we estimate the effective QCD ] for more details. ] (see, however, ref. [ 3 37 43 , 42 – 3 – for avoiding the axion overproduction. In section The last section is devoted to discussion and conclusions. inf 5 H The rest of this paper is organized as follows. In section The possibility of a larger Higgs VEV was discussed in different contexts and set-ups. In this paper, we revisit the axion with the BD distribution, focusing on a possibility ] for the Higgs inflation and the multiple point criticality.) We will come back to this 2 The QCD axionLet and us the briefly review Bunch-Davies the distribution refer QCD an axion abundance interested and reader properties to of the the original BD references distribution. [ We distribution suppresses the axionscale abundance. when the In Higgs section fieldbound is on trapped the in a inflationstudy false scale vacuum the at bubble large formation fieldinflation values, and is and discussed the derive in the subsequent section evolution of the universe. The slow-roll almost degenerate vacua at the60 electroweak and the Planckpossibility scales [ later in thisextension paper. which In does the not followingpotential we involve at mainly any the consider false additional the vacuum CP drives SM phases, the and old and its inflation. assume simple that the Higgs In this case, thethe axion inflation, field suppressing can the axionto be isocurvature so suppress perturbation heavy the [ thatgenerically axion it contribute abundance is to the because stabilized effective atalso the strong the pointed CP CP minimum phase out phases [ during that of topological the inflation takes SUSY place soft if parameters the SM Higgs potential has two extension of the SM.e-folding Thus, number the can upper be bound greatly on relaxed. the inflation scaleFor as instance, well in as a the supersymmetricthe required extension of Higgs the field, SM, there and are some flat directions of containing which may take a large VEV during inflation [ into the electroweak vacuum throughinflation the follows bubble the nucleation. quantum We tunneling assumesingle so that bubble. that the our slow-roll What observable universe isto is interesting the contained is enhanced in that the QCDwe the scale shall axion and see acquires reaches a later, the relatively the BD heavy distribution effective mass during QCD due the scale old can inflation. be as As high as to the heavier quark masses, and the effective QCD scalethat becomes much the larger effective than Λ QCDlarger scale expectation during value inflation ofthat is the the higher SM Higgs Higgs than isthe field. the initially eternal old present trapped Our inflation value in takes scenario due a place. is to false After as a vacuum exponentially follows. located long inflation, at We the large assume Higgs field tunnels values where other minimum at a large fieldwhether value. the other It depends minimum is onpossible the lower that precise or value the higher of than Higgs the the potentialhas top electroweak is quark a vacuum uplifted mass vacuum [ by expectation some valueverse, new (VEV) the physics renormalization much effects. larger group If evolution than the of the Higgs the weak field strong scale coupling in constant the is early modified uni- due field values in the early universe. It is well known that the SM Higgs potential allows an- JHEP05(2020)154 ) 4 ], ' ). 2.1 . If 44 0 [ (2.3) (2.4) (2.1) (2.2) inf χ inf T π a, ( 2 a / m m , inf . The axion ' H , the classical H = GeV GeV inf inf a, 17 17 2 a, inf ) during and after m ]. Thus, one needs , a T 153 MeV and 10 10 . It is assumed here /m 62 πf a ' × × f 2 inf are equally likely, such QCD QCD 3 3 . As we are interested in H 12 [ inf T T . a, . & 0 inf QCD & . ≡ T m a a is slightly modified from eq. ( < π H . ' T T f f a eq p ∗ 2 f θ h N QCD 2 inf inf 17 54 T . . eV ≤ a, & H 1 1 DM . & inf 2 m π a N 2 H inf − 1, since otherwise the axion abundance 3 π T inf 8 a GeV ∼ GeV GeV 2 a, f ∗ r a a 12 17 17 m n θ . Then, the axion starts to oscillate around f f ], and we adopt 1 2 10 10 10 = 8 – 4 –

× × ' QCD 6 T 3 3 T QCD 08 [ ) is much smaller than 2 inf − . ], T a θ 4 (

10 GeV for inf 43            V ' , a, q 0 × 12 , we can approximate the potential as the quadratic one, a × χ ∗ 7 n m 42 f . θ 2 √ 5 | ∼ ∗          θ GeV. If all values of | , the axion is almost massless, while it acquires a nonzero mass ∗ 001 12 θ ' . ] 0 10 ) 61 QCD T T and small ( a a 35 . f a 0 m f T ' 2 h . At a 4 is naturally much smaller than unity if (1) for Ω | ∗ θ | During inflation one can define the Gibbons-Hawking temperature, The axion mass is temperature-dependent and it is parametrized by 6 MeV) It is possible that the axion mass during inflation for . cannot be larger than about 10 | O 4 ∗ a θ Thus, that the minimum of the axion potential does notdue change to (modulo the 2 gravitational effects. motion and the quantum diffusionthe are potential balanced, minimum. leading The tovariance typical the of value BD the of distribution BD the peaked distribution misalignment at [ angle is then given by the In the following we assumethe relatively that large Then, after a sufficiently large number of e-folds, necessarily the case if the inflation scale is lowerassociated than with the the QCD scale. horizon.than the Let QCD us scalethe suppose so Higgs that that VEV the is the same QCD temperature as axion in is acquires the present close a vacuum, to small one can but or estimate nonzero smaller it mass, by where we assume that thethere axion is starts no to extra oscillate entropy during productionf the afterwards. radiation-dominated One era can and seewould from exceed the the above equation observed that | dark matter abundance, Ω an initial condition requires a fine-tuning. As we shall see below, however, this is not where the exponent is(75 given by as the temperature decreasesthe down minimum when to its massabundance becomes is comparable given by to [ the Hubble parameter JHEP05(2020)154 , . EW as (3.1) v φ ], where 176 GeV 66 , . false v t 65 [ 246 GeV, which M pl ' . M . EW ]. v 81 max Λ . GeV 10 MeV. 8 ∼ ! inf 0 h H

2 1 to suppress the axion abundance is relaxed. As √ – 5 – = inf GeV for φ H 16 10 ∼ a f ] or compact objects without horizon [ ], where the axion decay constant (or the PQ breaking scale) does 80 – 64 ], but it is under discussion whether it is stable enough in the presence , 69 63 68 , GeV is the reduced Planck mass. Above the turning point the potential ]. For instance, in the case that the Higgs VEV is at the weak scale, the 67 18 43 10 × 4 is the Higgs boson. . 2 h ' Broadly speaking, there are two ways to make the electroweak vacuum absolutely When the Higgs is trapped in the false vacuum, the SM quarks acquire heavier masses In the very early universe, the Higgs may be trapped in a false vacuum at An even more interesting possibility is that the QCD scale is larger during inflation. pl meta-stable. The lifetime ofof the the universe electroweak [ vacuum isof much small longer black than holes the [ present age stable. To this end, the vacuum at large field values must be uplifted. This can be realized The quartic coupling of theloops. Higgs field For receives the negativethe contributions pole SM from Higgs mass the potential top ofM reaches quark the a top turning point quarkstarts at in 10 to the decrease range and of may 171 GeV become negative, implying that the electroweak vacuum is scenario in detail.KSVZ-type For axion simplicity, [ we assumenot that change the during and QCD after axion inflation. is a3.1 string axion or A a false vacuum in the Higgs potential Then, our observable universe isan well inflation contained model in later the in single this bubble. paper. We will providethan such in the electroweakinflation vacuum, lasts and sufficiently so, long, the theby effective initial the QCD misalignment BD angle scale can distribution is be as enhanced. naturally we suppressed If have the seen old before. In the following, we discuss the above If its potential energy dominatesvacuum subsequently the decays universe, into it the drives electroweakan vacuum the through open eternal quantum tunneling, bubble old and inflation. universefield is The drives nucleated. false the slow-roll We inflation assume and that, decays into inside the the SM bubble, particles for another successful scalar reheating. where in a false vacuum drives the eternal old inflation. 3 False vacuum Higgs inflation In the present universespontaneously the breaks SM the Higgs electroweak develops symmetry. a Let nonzero us VEV express of the Higgs doublet axion window is open up to In this case thecurrent one, axion and mass the at upperwe bound the shall on very see in beginning the of next the section, universe this was possibility heavier can than be the realized if the Higgs field trapped the inflation [ JHEP05(2020)154 , ], = S , by t 83 (3.6) (3.3) (3.4) (3.5) (3.2) M is the max 2 R h µ 2 S m P λ + 2 , 4 S λ. is its spin, m c i the second term is the 3 2 NP s denotes the new physics , − is stabilized at the origin. R λ ) = 1. At large fields values, , S µ + h + ∆ i 2 R ( NP s 0, 2 i µ ··· 3 2 3 2 m > + − P − 2 ln 6 for λ / S , 2 ] 4 ) 2 h 2 2 R 2 S h h h = 5 h ( µ ) P 2 t 2 4 i c 2 P h y λ , m ( 0, and λ 171 GeV, the potential has two vacua; one m 4 2 R + . We consider the following potential for eff µ > + ' 4 S λ t 2 S S log . For a certain value of the top quark mass, it S + 1) 2, and MS scheme is given by the general formula [ λ ' and the dots represent higher order terms sup- – 6 – i pl S M m / max[ s 171 GeV in the SM framwork, or (ii) introducing 4! ) λ 2 1 0, 4 M S t , h π (2 y . ( + i > 3 t s 16 2 2 . Due to the minus sign of the second term, the top 2 eff i ' = 0 log 2 S π M S S V h i − 1) m h s 2 S ) 64 2 2 − ) ( R m 2 µ parity on h ( i = 2 for P λ 2 X / S λ Z V ' + ) ) = = 3 2 S h h c ( ( m ( eff eff ]. ]. This is the minimal possibility because no new physics is necessary, is so heavy that the SM is reproduced in the low energy limit. The one- λ V 82 54 4 S , and the other at h 2 1 π EW ) is the effective Higgs self-coupling, and it is approximately given by v h 16 ( 4 GeV [ . = represents degrees of freedom coupled to the Higgs field, ' 0 eff i i λ λ h h Here let us focus on the latter case, and show that one can stabilize the electroweak vac- The other possibility is that the Higgs potential is uplifted at scales above Λ If the top quark pole mass is given by 9 NP . ∆ quark contribution tends to pushcontribution can the be quartic coupling estimated as toward negative values. The scalar Here we have neglectedfirst the term SM is contributions the otherdominant quartic radiative than coupling correction the from at top the thecontribution top Yukawa renormalization coupling. quark from loop, scale the The and extra ∆ scalar where where renormalization scale, the effective potential is conventionally expressed by where we have imposedpressed a by a large cutoff scale.We For assume that loop effective potential for the Higgs in the uum by introducing an extra real scalar field although the required top172 quark mass is not favored by the currentsome measurements, new physics. As weby shall introducing see extra below, heavy particles onefield coupled can is to indeed trapped the stabilize in Higgs the field. the electroweak vacuum false In vacuum either at case, large if scales, the the Higgs eternal old inflation occurs. new physics beyond the SM. is at is even possible to makepoint the criticality two [ vacua almost degenerate, which is known as the multiple by either (i) the top quark mass of JHEP05(2020)154 . is 2 t y On . and S the . GeV (3.7) 1 m S 10 P GeV max . 10 λ Λ 16 < m ) GeV, and × . . Thus, one max R 10 10 3 S . Λ µ GeV; if h in the range of . m false (10 & 11 v R P = 6 O µ λ 10 ) is the finite term , the potential can S , the potential can uplifts the potential 2 = h false P t × . Here we have used m v h λ 3.6 √ y S S for both max 2, one can expand the 4 . max for the false vacuum to m Λ 1 ≈ √ λ . We show in figure and 3 P and R P 2 is equivalent to − , h/ & t λ µ EW /λ S √ y v 4 t 10 is larger, the vacuum at large h / y m = 1 S 10 ≈ Thus, one can indeed uplift the for and q P false m × R 5 v GeV, we have found that the false S , λ t ∼ µ 9 ! 6 false y . . m 10 4 v 2 4 S 2 . GeV. m false ≈ = 0 h 2 . If one further varies v S S ) at 2 10 t π = 10 P S is bounded above to avoid the Landau pole false m m λ v The last term in eq. ( , y 10 R 16 m EW P . µ v λ

× ( – 7 – GeV) = 0 7 and ) that the contribution of O . = 0 eff 10 λ Then, 8 S + . 3.6 R m 6 (10 ≈ > V = 1 at h ) max × 2 S /∂µ P Λ correspond to the central values of the measured top and Higgs λ false m eff false 3 P v 2 ) and ( & 42) λ v λ π ( ∂V h − 3.5 the potential cannot be sufficiently uplifted at . In the logarithm we introduce the renormalization scale in cannot be much larger than the location of the false vacuum 48 6, there appears a Landau pole at the scale eff . and , S . V 1 6, and S 6 t + (39 . S . ) at y ] for the derivation.) This is nothing but the Coleman-Weinberg 0 2 4 m & m to be zero in the loop function as it is sub-dominant around the scale ≈ h π 84 . If the term becomes dominant at . P = 0 ) λ 2 S S λ R t P (16 4 m y µ R / λ ( 3 µ λ h 4 t 6, the false vacuum exists for . . ≈ y GeV. One can see that the potential turns around Λ at large field values. For a proper choice of = 0, ] induced by 1 4 4 t ∼ λ 10 83 can be derived by h EW − y . This can be seen as follows. For v ]. 3 S 7 eff 4 . 82 λ /dh = 10 false & ) v . (See e.g. ref. [ 4 We also note that Here let us comment on the allowed range of One can see from eqs. ( R = 0 > m 2 P The adopted values of µ 5 = λ λh max R P ( description using the highercannot dimensional have operators large breaks hierarchy between down at masses [ be up-lifted and ad local minimum is formedThis is at also the condition thatwith the the dimension dimension 8 6 and higher term.below terms the However, can since be GUT neglected scale, compared the inequality is only marginally satisfied. Also, the effective theory The second termby on integrating the out right-hand side is the effective dimension 6 operator induced h effective potential as vacuum exists for λ smaller, the EW vacuum willfield values be will the be unique thechecked true vacuum; that, vacuum if and for the EWthe vacuum will other be hand, meta-stable. for We have also at the false vacuum isHiggs located potential at at high scales so that there exists aexist. false For vacuum at terms at if have a false vacuum with effective potential of the Higgs field for Λ potential [ such a way that theµ one-loop renormalization groupcontribution equation which of guarantees the almost vanishing Higgs mass and cosmological constant where we have taken JHEP05(2020)154 = R , we GeV, (3.8) (3.9) µ (3.10) 11 max 10 V ∼ . GeV at pl GeV. When false v M 11 11 ∼ 10 . 10 a 2 × f & 3 . and GeV. For comparison, the 10 GeV = 6 false false 10 max v S 11 v as follows. Since the Higgs 4 false H 10 v m 10 ) ∼ × 4 false − 7 . inf v ] 8 ) 10 H ) becomes stronger than that from . ) ≈ − 5 5 = 1, and GeV GeV 3.10 2 − pl [ false false P 100) GeV v 10 v ( M 10 , λ (10 3 − - 10 6 eff . ( O / V 10 (10 h = – 8 – = 0 s |× ∼ t as a function of = H 4 max | 2 , y pl This implies that the QCD axion can only contribute Λ inf inf max M 2 6 . V H 3 = 0 4 t π H 2 pl y λ s 16 M ≡ ∼ . a as large as the Planck scale when f max max Since the false vacuum should have a lower energy than V a H . f . 1 max

inf

0 5 4 3 2 1

ﬀ e

V / ( 10 GeV ) eff

] [ × GeV 10 V H

3)Λ 39 4 - 4 39 − GeV, on the other hand, the bound ( (1 11 . The effective potential for the Higgs field in the presence of the coupling with a singlet ∼ (red solid line). We take 10 S . false In the rest of this section we will see that the axion abundance also gives an upper If the Higgs field is trapped in the false vacuum, the eternal old inflation takes place GeV, for which the false vacuum is located at v Note that the current best-fit values of the top mass and the strong coupling lead to 6 10 false dark matter evenv for the axion abundance with for which the QCD axion can be the dominant dark matter with This gives an upper boundfalse on vacuum. the inflationary scale which can be realizedlimit using on the Higgs the inflationary scale. Specifically we will see that the QCD axion can explain for obtain We can place anpotential upper reaches bound the on turningpotential point maximum due is roughly to given the by top quark contributions, the value of the potential in the SM is also shown (gray dashed line). with the Hubble parameter, Figure 1 scalar 10 JHEP05(2020)154 . R µ 2 (cf. GeV (3.13) (3.11) (3.12) 12 false − v 11 10 ) in figure and . false inf v ( , H false v ! eff QCD ) for the parameters of 5 at the Planck scale as . 0 EW false false , v v v ' i ( 11 / pl m 3 4 g − eff QCD . R ) by the renormalization scale pl 2 ) = µ ) false M pl v false M v Θ false ( ( GeV, on the other hand, the two vacua 3 2 v a 3 ( f g 11 i GeV eff QCD ) was obtained by assuming the initial mis- X is given by a function of 10 5 eff QCD ∗ 1.3 – 9 – Λ 10 θ runs over the quark flavor, and Θ is the Heaviside ). This is naturally realized if 11 + i ' ' − ) false inf false ) to evaluate the typical initial misalignment angle.

v v a, 2 ( 2.4 false 3 3 π m We show the numerical result of Λ v g ( 16 eff QCD 7 . Λ ). For π = eff QCD ) < Λ R 3.12 µ inf 3 T dg of order unity. Now log ( d ∗ ) and ( θ GeV, it can be fitted by 6 3.10 10 denotes the SM quark masses, becomes equal to 4 > i 2 3 g m false Let us recall that the axion window ( As we have seen before, the QCD axion acquires a small but nonzero mass if the v Here we neglect contributions of the exotic quarks in the PQ sector, assuming that the PQ quarks are 7 We substitute this mass into eq. ( alignment angle not coupled to the Higgs and they do not affect the Higgs potential through higher order corrections. due to eqs.must ( be almost degenerate incan energy. be Then, the well axion approximated mass by during the eternal old inflation the false vacuum insufficiently the long next so section, and thatdistribution. here the we probability simply distribution assume of that the the inflation QCDinflation scale lasts axion satisfies reaches the BD where all the SMour quarks interest. are already decoupled at the3.3 Λ The QCDHere axion let window us opened estimate how wider muchtakes the place QCD in axion the window is Higgs relaxed false if vacuum. the We eternal will old come inflation back to the issue of the decay rate of where For where step function. First we fixscales the initial within value the of the SM. strong Tothe gauge be coupling boundary at concrete condition. some we high-energy Then adopt scales, we and solve define the renormalization the group effective equation QCD toward scale lower Λ masses. To see thisgauge we coupling, solve the one-loop renormalization group equation for the strong contributes to the Hubbleaxion dark parameter matter. to saturate the upper3.2 limit and The explain effective theThe QCD QCD effective scale QCD scale during the eternal old inflation gets enhanced due to heavier quark to a fraction of the dark matter. Alternatively, one may introduce another sector that JHEP05(2020)154 , is 3 ]). for − inf 93 a inf for a H – f H can be ∼ 91 inf 176 GeV. . H inf H V ∼ H false ) is stronger v (blue) from top GeV (i.e. below 18 3.10 18 EW ] (see also [ v 10 GeV. 90 as a function of = 18 – is significantly relaxed < from ( i 16 10 85 h inf inf h inf false . H H v H a f . 14 the tunneling probability per unit ] V GeV / GeV / 12 12 as a function of the Higgs VEV GeV represents the classical axion window false v [ GeV (orange), and 12 GeV correspond to the minimal scenario with , the inflation is eternal in a sense that in the GeV (i.e. the green line), the false vacuum is 8 10 eff QCD 10 18 – 10 – inf 11 the upper bound on GeV, the bound on H 10 ∼ Log 10 , the upper bound on 3 11 GeV and 10 a . = 10 f ∼ 10 11 H false , v 10 . false eff 8 v false & v false . If Γ v GeV (green), 10 false V v 11 / 6 GeV may also be possible in a more involved extension of the SM. H 8 V . For 10 )), and in particular, it can be much smaller than unity for sufficiently

. We show in ﬁgure

3 2 1 0 5 4 = Γ < max

10 QCD . The eﬀective QCD scale Λ ] / [ Log

GeV f

H H 3.13 eff , 10 GeV for false eff v ∼ GeV (red), 10 and . Thus, the axion window can be expressed by the upper bound on 18 ) and ( inf = 1. The red lines with Figure 2 false 171 GeV, in which case the false vacuum can be almost degenerate with the true ∗ H v 2.4 θ = 10 ' t false The false vacuum ofthrough the the Higgs bubble field nucleation. istime Let per unstable us unit and denote volume. decaysis by into Then, given Γ the the by effective electroweak Γ decay vacuum whole rate universe per there the are Hubble always regions volume, that continue to inflate [ compared to the SMlarger shown than by the blue line (the bottom one).4 Specifically, False vacuum decay and slow-roll inflation comparable to than that from thethe axion bound. abundance, The and orangeThe line one case approximately needs of corresponds another to inflation theOne top sector can quark to see mass saturate that, depending on with M vacuum in the framework ofthe the red SM. line), On oneto the needs other make to hand, a introduce for false somenot vacuum. new necessarily physics degenerate For which with uplifts the the electroweak one, Higgs because potential the upper bound on eqs. ( small given v to bottom. The left vertical line at JHEP05(2020)154 8 ].

96 (4.1) (4.2) (4.3)

–

inf

GeV

( ] [ )

/ 4 1 94 ) becomes , ρ1/4 [GeV] GeV (red),

inf 91 18 3.10 8 12 11 10 9 . 10 10 10 10 10 = 10 ), we can rewrite 2 18 − false 10 3.13 v ) in a stochastic inﬂation ]. 10 a GeV 10 97 f 17 16 10 ) and ( 10 10 ∼ 3.12 N 2 16 , eq t 10 N ∆ GeV GeV inf

GeV GeV GeV, the upper bound from (

EW 8 -folding number that a randomly picked 18 11 3 H v inf ] ). Using ( e 11 (blue), respectively, from top to bottom. The H 10

10 10 10 = 3 15 10 H 2.3 e = ⟩ ,

= = t EW inf h . GeV v 10 eﬀ

⟨ ∆ [ 11 H v H Γ /

false – 11 – ,

false false = a v 4 typical

v v f i eﬀ ≡ Γ h h 14 − pl -folds the universe experiences before the tunneling, e dec e false M 10 v N -folds extremely large (e.g. as a function of the decay constant for e 4 inf H 13 GeV 10 6 as deﬁned below eq. ( − inf 1 10 2 a, 12 GeV (orange), and = 8 * 10 θ /m . Therefore, in this case, the bubble formation is so rare that it cannot 2 Γ inf V 6 5 4 3 2 2 3 1 1 t - - - 10 H

10 10 10 10 10

10 10 10

= inf GeV H ] [ . The upper bound on eq N , is so large that the BD distribution of the QCD axion is reached. Specifically, we Here we show that the typical GeV (green), 10 It is possible to make the typical 8 11 dec model where the potential has a shallow local minimum around the hilltop [ will show where the above condition as observer experiences is not infinite,universe but finite at (though a exponentially large). laterand Essentially, time in the this is sense, simply the eternity dominated of by eternal inflation those relies regions onN the where volume inflation measure [ continues, by a factor of over a time ∆ terminate the inflation as aIt whole; is some important fraction of to the note, entire however, universe that continues to a inflate. top three lines correspond toplains the dark currently allowed matter range on of each thestronger, line. top and quark Note one mass. that, needs for to The introduce QCD another axion inflation ex- sector to explain the QCD axionOne dark can matter. easily see this by noting that the physical volume of the inflating regions increases Figure 3 10 JHEP05(2020)154 ) (4) , the (4.7) (4.5) (4.6) (4.4) O 3.10 is the 3 b R ], ) is the ρ ( 106 , # ] for the details). sol )) h GeV (see eq. ( 105 11 – false v 10 ( 100 ) . eff pl V false M − v false ) , is bounded above by the QCD v , is hampered by e.g. gravitational in a closed form [ = ) sol inf 0 h A h ( ( H B . In this case, the false vacuum must →∞ eff eff false pl ρ when V , V v h , 4 M d B σ dh on dimensional grounds, where 3 + ( max 2 − 4 2 π H 2 = − b A e 27 h R false sol = 0 and d – 12 – v ' GeV dρ = h 0 0 3 ρ Γ V d 11 dρ → B ρ + | for the standard-model Higgs potential was calculated is the difference between the Euclidean actions for the h 1 2 , one needs to calculate the one-loop fluctuations around 2 A GeV 2 " B is given by A d is of order dρ 11 dh/dρ dρ B 3 A ρ 2 π ). In the following we use a thin-wall approximation to estimate 2 ∞ max 0 Z H = 0 B is the radial coordinate in the Euclidean spacetime, and we have used the fact ] (CDL hereafter). We consider a thin-wall approximation for the CDL bubble ]. In the presence of gravity, the calculations of ρ 98 99 In the thin-wall approximation, one can estimate In the absence of gravity, For precise determination of The false vacuum decay including gravitational effects was studied by Coleman and De has mass dimension four, and with the boundary conditions, gravity, that the dominant bouncesymmetric solution bounce possesses solution an satisfying O(4) the symmetry. Euclidean Here equation of motion, where we have added a subscript of 0 to indicate that it does not include the effect of the bounce solution. The valuein of ref. [ fluctuations and renormalization of theHere graviton we loops simply (see assume refs.physical that [ size of the bubble at the nucleation. where higher order correctionsA are suppressed by the Planckbounce constant. solution and Here the the false prefactor vacuum one. Luccia [ nucleation assuming that thethe dominant semiclassical bounce approximation, the solution CDL possessesin tunneling an the rate following per O(4) form, unit symmetry. time In per volume is given be almost degenerate with theHubble true parameter vacuum. of This the is false because,axion vacuum as abundance, Higgs can and inflation, be it seen isfor in much figure the smaller definition than of the vacuum decay rate. Later we will check the validity of the thin-wall approximation. servable universe must have experiencedinitial a sufficiently axion long field inflation value in obeys the the past BD so distribution. that4.1 the Thin-wall approximation (10 First, we consider the case of 10 If this condition is met, irrespective of whether the volume measure is adopted, our ob- JHEP05(2020)154 ]. 98 1 or (4.8) (4.15) (4.16) (4.13) (4.14) (4.10) (4.11) x < is always x is the same for both and makes the bubble 1. While B B , y > , )] 2 2 1 are often called type A and x x , false ) + . + ) (4.9) v x > ( 2 pl EW xy ). eff xy EW M v v V ( ( inf , ) (4.12) − eff 4.11 1 and 2 eff 1 + 2 pl H ) 1 + 2 V 2 V , h , x, y M p ( p ( 2 pl − ' x < 2 r 2 1) ) + inf eff ) − 3 false σ 0 2 V pl H 3 M − B ζv 3 4 false – 13 – xy 3 2 2 [ false v y M = v ( = ' ( = p 4 ( 2 ]. σ 1 + B 2 2 eff c x dh eff s σ σ V V = 109 = = , false ≡ c v EW ) = 2 y σ x v ), and we have used the fact that the cosmological constant are positive, and in particular, 2 ] Z 107 is defined by − y x, y ≡ ( c ]. Under the thin-wall approximation, a general form of the 108 r σ (10 σ and O 107 x can be negative in a more general case. y is the tension of the bubble wall, is a constant of σ ζ is the energy difference between the two vacua, 1. The weak gravity limit is contained in the former case, while the gravitational effect The above bounce solution can be broadly classified into the two regimes, In the presence of gravity, the bounce action receives gravitational corrections [ x > is very important incases. the The latter bounce case. solutionstype corresponding Note B to that bounces, the respectively expression [ of In our scenario both positive, where the critical tension with nucleation more likely. This isafter because the the nucleation. cosmic expansionthe As assists Gibbons-Hawking the the radiation bounce bubble induces solution to upwardformation expand fluctuations deviates more which from likely also the makes [ bounce thin-wall the action approximation, bubble is given by [ where in the present vacuum is vanishingly small in eq.Here ( we are interestedvacuum in dS or the Minkowski. case In of this case, the the tunneling gravity effect from suppresses the false vacuum dS to the true In our case of the Higgs potential, they are given by and where JHEP05(2020)154 c . In (4.20) (4.21) (4.17) (4.19) (4.18) σ > σ , while false 2 b becomes v /R B 2 false ), we obtain v ). . is reduced to the 6 can be as large as to some value (e.g. 3.12 4.11 − B , in this regime where 1 3 inf inf H H false v inf GeV ), and ( 3 inf H . See eq. ( ) is of order H 11 10 that / , the gravitational effect is neg- 4.10 4.5 , on the other hand, 10 GeV . However, the vacuum decay rate is , 4 false c c c ) v 3 v v 6 v 12 ), ( − 1 3 is so large that the vacuum decay ) is trivially satisfied. & . . 5 & − and the decay rate. The bubble size 2 B 4.3 − 4.12 false (10 B 2 , false v false . GeV v O − pl v 2 2 c ζ false 2 σ σ / 12 v 10 = GeV) M 1 max false For scales only as 10 v V 11 Using ( 9 . For 1, the above expression of 1 + . max inf . So, the gravitational effect becomes relevant defined by the height of the potential barrier. If 4 ∼ 10 V – 14 – H 10 4 b → b 0 12 does not change our conclusion. false GeV false false false 2 v ∼ R y R B v v A max v 2 − 13 ζ π = H . Let us fix the inflation scale 10 2 10 false B B 1 as follows, since we assume that the two vacua are not degenerate. × increases even when ∼ v ) 0, i.e., 0 2 7 takes the smallest value at the smallest possible GeV B false → ' max x > B v (10 V 11 ) GeV, one can see from figure c ], v O is saturated, the thin-wall approximation is not applicable, and ) is essentially cut off around the size of the bounce solution, 11 98 EW & = v 10 4.5 inf ( 0 H B , and the increase of ∼ . Thus, eff false B v V ' may grow as false false B v A v increases in proportion to B denotes the critical value of 1), while it is only a minor effect otherwise. In terms of our model parameters, increases. c , we obtain v ) GeV, which is comparable to b 2 = 10 GeV) for simplicity, and vary x > Let us make an order of magnitude estimate of Now we are ready to estimate In the limit of R false This is justified becauseThe the prefactor upper bound on can be roughly estimated as follows. The first term in ( v 9 inf (10 10 ∼ b ρ mainly determined by the second term isBy of balancing order the two terms, we get, Since the integral of ( O the upper bound on a dedicated analysis is needed. R Therefore, in the caserate of is 10 exponentially suppressed, and the condition ( 4.2 Non-degenerate vacua ( In the case of other words, the vacuum decay ratethe is thin-wall the approximation largest is at applicable. the smallest as H ligible, and independent of we can rewrite the condition where original formula of CDL [ In this limit, one can(i.e. clearly see that the gravitational effect becomes relevant when JHEP05(2020)154 . 4 1 ]. In while (4.22) (4.24) (4.23) x 107 RG × 80 is shown in figure , becomes comparable sol 1 h − dw H − ] for the original topological 4 . false 60 v . The potential is given in figure 1) 112 ρ , ) approaches GeV , in other words, we increase 4 111 x ( , 4.13 5 GeV false v 10 10 GeV false × 10 11 v 40 ⋅ 2 + 1) . 2 ρ 10 2 y , we arrive at – 15 – for this solution and obtained ) ( ] (see refs. [ 4 2 ≈ pl − b 4 x B 56 R -function in ( 10 B M r ' − and increase ) ∼ ) . The bounce solution ∼ 10 inf A 1 . x, y H GeV 20 ( 1, the false r Γ V 10 , the Hubble horizon in the wall, v pl 10 ( x M / ) ). Using RG ∼ sol 3.9 h )/ − false sol . We have calculated v h ρ 2 1 7 6 5 4 3 false - v ) is well satisfied. ( ]. Once such a domain is formed, it will start to expand exponentially, and approaches the Planck mass, the potential becomes relatively flat near the max- false 4.3 v 110 ( . Numerical result of the bounce solution by varying false v To see this, let us again fix We have numerically solved the equation of motion and obtained the bounce solution is fixed. In the limit of inflation). y and the thin-wall approximation breaksas down. a combination Such of a thermalparticular, bounce fluctuations when solution and can the be subsequentto interpreted quantum the tunneling [ wall width,solution [ and thethe bounce topological is Higgs approximately inflation given begins by [ the Hawking-Moss (HM) which agrees well the above estimate, na¨ıve justifying our4.3 conclusion. Thick-wall regimeWhen ( imum. Then, the Higgs stays mostly around the potential maximum in the bounce solution, for the potential shownas in a figure function of where we have used ( for which ( Figure 4 JHEP05(2020)154 ∗ B θ is the (4.25) 0. The 3 2 S < limit of Ω 0019 (68% . d K x 0 0007 , where . 3 2 S Ω = 0 -folding number, because d ) e K ξ ]. In the rest of this paper ( 2 Such high-scale inflation will ρ 116 – , + 12 2 pl 2 113 2 M inf dξ 2 H π GeV. = 8 13 2 ] and the negative curvature . = pl ds ) 127 ]. We note that the thin-wall approximation M 4 pl – false M – 16 – 110 v 2 123 ( ]. Naively, such bubble universes would be almost π can also be applied to this case. eff false 24 5 121 V v ]. Infinitely many open bubble universes are created in – ≈ 119 118 B , 117 , the initial condition of the nucleated bubble is determined by pl 1, the corresponding bounce action reads, M ≈ Then, if the Higgs rolls down to the electroweak vacuum, the inflation is the radial coordinate. One can obtain the metric inside the bubble y 11 ξ false ]. We will study the slow-roll inflation in the next section. v and 122 3 S ]. Although the observation does not give a statistically significant preference to a 62 The created bubble universes might have some predictions such as distinct features The topological Higgs inflation typically lasts only for a few Here we do notThe adopt inflation the model volume discussed measure. in section 11 12 CL) [ nonzero negative spatial curvature, a possibilityyet. of the open bubble universe is not excluded empty with a small energycosmology density, [ but it can be combined with the standardof inflationary the primordial densitylatest perturbations Planck [ data combined with the lensing and BAO gives Ω metric of by an analytic continuation, thefor interior observers in of the the bubble bubblethe [ looks eternally like inflating an false infinite vacuum.context open of The universe the open string bubble landscape universes [ are often discussed in the we are going to focus on the case of 4.4 Open bubble universeIn and the the case dynamics of the O(4) symmetric CDL tunneling solution, will be dominated by thethe quantum initial fluctuations, BD distribution. but it Inof this is case, dark still the matter smaller QCD because than axion of cannot unitythe its be thanks too subdominant the to dominant large component component isocurvature of perturbations.of dark However, non-Gaussianity it matter, of may which still the gives be isocurvature rise perturbations to [ a non-negligible amount ends and the cosmic expansionHM-like becomes instanton decelerated. does Since notinhomogeneous the necessarily unless initial possess another configuration inflation of thesecond starts the O(4) immediately. inflation symmetry, Thus, is the thegenerate considered universe energy quantum will to scale fluctuations be of be of the the of QCD order axion on 10 top of the BD distribution. Thus, obtained in the thin-wall approximation reproducesto the the HM instanton, above which picture. lends support the potential curvature atinside the the maximum domain. is not very different from the Hubble parameter which coincides with theactually HM breaks instanton down [ in this limit. Nevertheless it is assuring that the large When we also take JHEP05(2020)154 , – ϕ to λ ϕ (5.2) (5.3) (5.4) (5.5) (5.6) (5.1) 131 , 51 is much smaller L Higgs [ ϕ − λ cannot exceed 50 or the cutoff scale, and ∗ ]. It is known however M N ], a Coleman-Weinberg 62 , 137 2 is initially around the origin, , 004 [ 6 . ϕ M 0 ϕ 6 , pl 2 min + . 2 , M / ϕ 4 3 pl 965 3 . ϕ 0 M ϕ ∗ 2 ∗ , 4 / λ ' 40 ∗ 40 N 3 N is negligibly small in the following, but it 3 s N − A supersymmetric version of the model was n 0 ∗ 2 M . the mass parameter, − 13 m 40 0 ϕ N ϕ − min 1 0 λ 2 0 ]. As we shall see shortly, ϕ ∼ 2 – 17 – may be identified with the B m 10 7 6 m ' p breaking linear term [ ϕ − − s ϕ × 136 − = n 2 , ] 5 10 10 0 Z V ' ' ' , with the following potential, min 131 135 ϕ ϕ ϕ , ,ϕ λ ) = min as [ 47 inf ϕ ϕ, ( H m inf min V ϕ ]. The inflaton 130 ) is very flat around the origin, and so, if – ) is located at ϕ ( ϕ ( 128 inf inf V V is the e-folding number at the horizon exit of the CMB scales. Since the inflation is a positive coupling constant, ∗ ϕ λ N We briefly summarize here the known properties of the above inflation model. The Let us introduce an inflaton, ] or an axion-like particle [ the energy density during inflation at 0 which is too small to explainthat the the observed spectral value index is extremelya sensitive tiny to the deviation shape from of the theFor quartic inflaton instance, potential potential, and can one even make it may consistent introduce with the a observed value. where scale must be lower thanso. that The of spectral the index false is vacuum Higgs given inflation, by The potential the slow-roll inflation takes place.and the The other CMB parameters such normalizationare as fixes expressed the the inflation in quartic scale terms coupling and of the inflaton mass at the minimum the Higgs field, we will introduce a nonzero massminimum to of realize the slow-roll inflation. V studied in refs. [ 134 than unity. For the momentis we straightforward assume to that take its effect on the inflaton dynamics. In fact, when we couple where The slow-roll inflation isstructure. strongly The supported false by vacuum thedensity Higgs perturbations, observations inflation since of considered the so CMB Higgs potential farroll and is cannot inflation large-scale not explain in sufficiently the flat our to observed after scenario. realize the viable Thus, tunneling slow- event. we need another sector that drives slow-roll inflation 5 Slow-roll inflation after the bubble nucleation JHEP05(2020)154 (5.7) (5.8) (5.9) ). In (5.10) . Then, 5.1 breaking after the ϕ 2 ϕ . After the Z false near the origin. v ϕ = 0 is more likely than h ≈ ϕ ]. While the mass term alone . 138 2 is negligibly small in the low-scale . 0 is only the local minimum during , h ε ) . As we shall shortly, the ≈ . 2 s false 0 2 false acquires a large mass, v ϕ 2 min n V v 4 ϕ ϕ . The second term is introduced to cancel ϕh 2 min h − λ ϕ and ) direction for the fixed 2 η ϕh ϕ ' ϕ ( and λ false direction with the potential given by ( – 18 – 2 v 0 is chosen based on the anthropic argument for ) ϕ ϕh ( 4 ϕ λ ≈ eff false ϕ V ϕ ' m max is extremely small. If the mass is large enough, it makes ( V δV ϕh λ ], or a nonzero mass term [ unless 131 has a coupling to the Ricci curvature with a coupling of order unity, ][ 2 inf ϕ 2 H ϕ remains heavy except when the Higgs field really approaches the “true” , and the other slow-roll parameter ϕ η log[ 2 0) is the coupling between ϕ increases. Note that the linear term does not contribute to one of the slow- the global minimum during the false vacuum Higgs inflation, > s ∝ ( n . One can still argue that ϕ ϕh λ min ϕ To ensure the above-mentioned dynamics, a couple of conditions must be met. First, we Now let us introduce the following renormalizable coupling to Higgs field to make the Here, the main question is if we can realize the hilltop initial condition of ≈ The location of the potential barrier and the false vacuum remain almost intact if The potential energy at the false vacuum is still dominated by the Higgs contribution if the slow-roll inflation startsparticular, along since the vacuum, the previous argument on the bubble nucleation isdo considered not to want remain to valid. modify the Higgs potential significantly by introducing the above coupling. which is larger than the origin the globaltunneling, minimum the Higgs along field the value becomes much smaller, and so does the mass of where the contribution on thefollows. Higgs When the boson Higgs mass is at in the the false EW vacuum, vacuum. The basic picture is as ϕ the successful slow-roll inflation. origin of case one cannot make useOn the of other fine-temperature hand, effects if toit set acquires the a value mass of ofand order can the be Hubble stabilized parameter thethe origin. during false the Considering vacuum false that Higgs vacuum Higgs inflation, inflation, however, it is not certain if inflation. The mass term doesit shift the also inflaton gives field a values negative atlinear contribution the term horizon to exit both will similarly, also but be important for avoiding toobubble long nucleation duration from of the theinside slow-roll the false inflation. bubble vacuum. is almost empty In and the dominated by thin-wall the (negative) approximation, curvature the term. universe In this is not capable ofefficient. increasing the This spectral is indexthe sufficiently, because CMB a scale the non-zero towards linear linear smallerand values term term thus of is shifts the quite inflaton theroll field, inflaton parameters, where field the curvature at is the smaller, horizon exit of potential, JHEP05(2020)154 . ϕ × 14 λ min ϕ 16) ) ] (see / (5.13) (5.11) (5.12) 83 eff true λ v ( (˜ eff ≈ would remain /λ , V This is because ϕh ϕ # λ . Therefore, it is ], where the inflaton 13 . p 139 false min . By integrating out ∼ v . Secondly, the origin ϕ min plane. By adding a certain 2 t h ϕh ϕ true y h ) ˜ v λ − ϕ ≈ p true 6 v h ϕh (˜ = 0 and 10 λ . if )). So, we introduce a nonzero , ϕ at eff 0 that integrating the Higgs does 2 V / pl 5.7 /λ 1 ϕ false ∼ . ϕh v 6 /M λ 0 − = is much smaller than unity, either the to the Higgs quartic coupling should be 4 min V = 0. Thus, the mass of p inf 10 h ϕ GeV √ ϕ ϕ ϕ 2 ∼ ϕh H − λ ∼ λ 7 10 . (see eq. ( ) if − ,ϕ true , we obtain satisfies the above inequality for simplicity. | ) are summarized as v 10 – 19 – inf 5.7 true ϕ v λ to ˜ H ϕh 2 . | false 5.11 GeV v λ < ϕh 13 -direction at . EW | ). Unless 2 min λ v ϕ ϕ 10 ϕ EW , 5.3 v δλ , which contains a flat direction in the 0, there is a tree-level contribution to the quartic potential ϕh | ), and ( . λ false ≈ v ϕ 5.10 2 ϕh ) and ( − +const λ 2 ), but this does not modify the previous argument on the bubble ), ( ]). 10 2 3.9 p h 5.9 " false 142 v ) to cancel this contribution so that the effective mass is negative and its – ( If we require , one can have a successful hilltop inflation around ) + 2 . , it is convenient to rewrite the relevant part of the potential as Min 5.1 ϕ eff eff V . 2 min 140 , the CMB normalization condition would require cancellation between ϕh , 4 ϕ λ ϕh /λ − λ 2 2 ϕh 132 min in eq. ( , ϕ λ GeV] ϕ ( should not be confused with 0 eff 12 131 m is the global minimum in the − true ∼ − The slow-roll inflation model is of the hilltop type, and one may wonder that eternal The conditions ( /λ ˜ v For larger 11 ϕ ϕ ϕh 13 14 λ quartic coupling of The inflaton dynamics is similarnaturally to turns the into alchemical the inflation Higgs model after proposed inflation. in ref. This [ implies that the SM sector is naturally reheated. indeed possible to satisfytaken the to constraints be once sufficiently the low.[10 energy The scale energy of density the of slow-roll the slow-roll inflation inflation is can beinflation as may take large place, as which would erase the BD distribution established during the false where we have usedfirst ( or second term in the bracket gives the strongest condition on not lead to anyhave logarithmic only corrections suppressed to couplings the tothe inflaton matter flat potential. in potential a Also of non-supersymmetric therefs. the set-up, inflaton [ inflaton since should otherwise is spoiled by the Coleman-Weinberg corrections [ For larger and the contribution fromtechnical integrating problems, out the we Higgs.Note assume that Although that this the does Higgs not is induce so any heavy around the origin of positive even after themass Higgs tunnels to ˜ absolute magnitude is still smallerthe than Higgs boson at around δλ Lastly, although we havevacuum required along the the locations Higgs ofvacuum is direction the necessarily potential remain shifted maximum almost from the and unchanged, Higgs the acquires the a false location negative mass of from the ( “true” same value of nucleation. In addition, the loopmuch smaller contribution of than that ofof the top quark. This amounts to In general, one needs to slightly shift the parameters to uplift the false vacuum to have the JHEP05(2020)154 - ϕ e ). In . For (5.14) ∗ , if the θ 5.7 a f drives both ]. The linear ϕ that uplifts the 137 [ S inf L Higgs boson, non- during the inflation. The Higgs fields soon V − . | ϕ EW ) during inflation where λ v | ] or leptogenesis via active In this case, ]). As we have seen before, 5.7 into the Higgs field conden- & 15 97 145 ϕ . ϕh ϕ λ follows the BD distribution and which effectively reduces the total ∗ θ ϕ namely 2 ϕ is identified with the B directly decays into right-handed neutrinos. (1) energy of hh m breaking linear term in ϕ O ϕ 2 & → Z – 20 – 1 to avoid the overproduction of the QCD axion. ϕ 2 min ϕ may be identified with the singlet ). ∗ θ ϕh ϕ λ 3.2 can be largely displaced from because the inflaton is light. This is the main difference h ϕ ] are possible. If GeV. Then, thermal leptogenesis [ 147 , 12 − 11 146 , the reheating can be instantaneous, in which case the reheating temperature ]. This happens when ϕh GeV, one usually assumes λ field does not have to be the inflaton, and it can be another scalar (such as curvaton) or fermion 144 , 12 ϕ 10 143 0. Thus, the field value of The abundance of the QCD axion depends on the initial misalignment angle The reheating is considered to proceed through the perturbative decays The & ' 15 a -folding number. In any case, the BD distribution is not modified during the slow-roll On the other hand,therefore it can be was suppressed, recentlylower if pointed than the out the Hubble QCD that parameter scale, and during if inflation the is inflation comparable lasts to sufficientlycondensates. long. or The To only this requirement end, is one to often keep the Higgs at large values during the eternal inflation. Higgs potential. Into this the case, radiative the corrections falsefrom of vacuum the of scenario the discussed Higgs after ( potential will disappearf due the Higgs acquires aϕ negative mass fromConsequently, the the QCD potential axion window likeinflation can ( scale be is similarly lower opened thanenough. to the large Interestingly, enhanced the values of effective inflaton QCD scale and the inflation lasts long So far we have focused onand the drives possibility that the the eternaldrives Higgs field the old is slow-roll inflation. trapped inflation. into the There After be false is the vacuum larger another than bubble possibilityeternal the formation, that present (or another the one extremely Higgs due scalar long) field tothe field inflation value the observed is inflaton primordial and set field density the perturbations subsequent (see slow-roll e.g. inflation ref. that [ explains neutrino oscillation [ thermal leptogenesis may also take place as 6 Discussion and conclusions particular, the tachyonic resonance transfer sate [ form the SM thermal plasmacoupling, and the reheating iscan successful. be Depending as on high the as size 10 of the inflation. or with the parametric resonance and dissipation effects via the quartic coupling ( folding number is finite andfalse is vacuum not Higgs so inflation large, remains andspectral intact. the index BD We is also distribution obtained note established ifterm that during we shifts a the add the better a location fit tiny to ofe the the observed potential maximum of vacuum Higgs inflation. In fact, if one does not adopt the volume measure, the typical JHEP05(2020)154 ] , Phys. , Phys. Rev. , Phys. Rev. , lattice QCD ]. arXiv:1606.03145 [ SPIRE = 2 + 1 IN f [ N (2016) 498 ]. -folding number that the universe e (1978) 223 B 762 ]. ]. 40 SPIRE IN ][ SPIRE -dependence from The topological susceptibility in finite SPIRE θ Lattice QCD input for axion cosmology IN IN ][ [ Phys. Lett. invariance in the presence of instantons – 21 – , T Phys. Rev. Lett. and , P ]. CP conservation in the presence of instantons Constraints imposed by CP conservation in the presence of (1977) 1791 ]. ), which permits any use, distribution and reproduction in arXiv:1505.07455 [ SPIRE D 16 SPIRE arXiv:1512.06746 IN [ IN [ [ Axion phenomenology and CC-BY 4.0 A new light boson? This article is distributed under the terms of the Creative Commons Problem of strong Phys. Rev. (2015) 034507 (2016) 155 , ]. (1978) 279 (1977) 1440 03 40 38 D 92 SPIRE IN JHEP temperature QCD and axion cosmology [ Lett. Rev. 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