QCD Axion Window and False Vacuum Higgs Inflation
TTU-1096IPMU20-0004
QCD Axion Window and False Vacuum Higgs Inflation
Hiroki Matsui , Fuminobu Takahashi , , Wen Yin Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), University ofTokyo, Kashiwa 277–8583, Japan Department of Physics, KAIST, Daejeon 34141, Korea
Abstract
The abundance of the QCD axion is known to be suppressed if the Hubble parameterduring inflation, H inf , is lower than the QCD scale, and if the inflation lasts sufficientlylong. We show that the tight upper bound on the inflation scale can be significantlyrelaxed if the eternal old inflation is driven by the standard-model Higgs field trappedin a false vacuum at large field values. Specifically, H inf can be larger than 100 GeV ifthe false vacuum is located above the intermediate scale. We also discuss the slow-rollinflation after the tunneling from the false vacuum to the electroweak vacuum. a r X i v : . [ h e p - ph ] J a n Introduction
The Peccei-Quinn (PQ) mechanism is the most plausible solution to the strong CP prob-lem [1, 2]. It predicts the QCD axion, a pseudo Nambu-Goldstone boson, which arises as aconsequence of the spontaneous breakdown of the global U(1) PQ symmetry [3, 4].The QCD axion, a , is coupled to the standard model (SM) QCD as L axion = g π af a F αµν ˜ F µνα , (1)where f a is the decay constant, g the strong gauge coupling, F αµν the gluon field strength,and ˜ F µνα its dual. The global U(1) PQ symmetry is explicitly broken by non-perturbativeeffects of QCD through the above coupling. If there is no other explicit breaking, the axionacquires a mass m a ( T ) = χ ( T ) f a , (2)where χ ( T ) is the topological susceptibility. At T (cid:29) Λ QCD (cid:39) O (100) MeV, the topologicalsusceptibility χ ( T ) is vanishingly small, and so, the axion is almost massless. χ ( T ) grows asthe temperature decreases, and it approaches a constant value χ at T (cid:46) Λ QCD . The detailedtemperature dependence of χ ( T ) was studied by using the lattice QCD [5–10]. When theaxion mass becomes comparable to the Hubble parameter, the axion starts to oscillate aroundthe potential minimum with an initial amplitude a ∗ . The oscillation amplitude decreases dueto the subsequent cosmic expansion, and thus the effective strong CP phase is dynamicallysuppressed.In the PQ mechanism some amount of coherent oscillations of the axion is necessarilyproduced by the misalignment mechanism [11–13], and those axions contribute to dark mat-ter. If the initial misalignment angle, θ ∗ = a ∗ /f a , is of order unity, the observed dark matterabundance sets the upper bound of the so-called classical axion window,10 GeV (cid:46) f a (cid:46) GeV , (3)where the lower bound is due to the neutrino burst duration of SN1987A [14–17] or thecooling neutron star [19]. For instance, if the axion decay constant is of order the GUT scaleor the string scale, i.e., f a = 10 − GeV, the axion abundance exceeds the observed darkmatter abundance by many orders of magnitude. Therefore, for such large values of f a , theinitial misalignment angle θ ∗ must be fine-tuned to be of order 10 − . In Ref. [18] it was pointed out that the accretion disk formed around the proto-neutron star (or blackhole) may explain the late-time neutrino emission ( t (cid:38) H inf , is lower than the QCD scale, and if the inflation lasts long enough.The reason is as follows. First, the axion potential is already generated during inflationif the Gibbons-Hawking temperature, T inf = H inf / π [44], is below the QCD scale. Then,even though the axion mass is much smaller than the Hubble parameter, the axion fielddistribution reaches equilibrium after sufficiently long inflation when the quantum diffusionis balanced by the classical motion. The distribution is called the Bunch-Davies (BD) dis-tribution [45]. If there are no light degrees of freedom which changes the strong CP phase(modulo 2 π ) during and after inflation, the BD distribution is peaked at the CP conservingpoint [43, 46]. The variance of the BD distribution, (cid:112) (cid:104) a (cid:105) , is determined by H inf and theaxion mass, and it can be much smaller than the decay constant f a for sufficiently small H inf . Thus, one can realize | θ ∗ | (cid:28)
1, which relaxes the upper bound of the classical axionwindow. So far we assume that the QCD scale during inflation is the same as in the presentvacuum. However, this may not be the case. For instance, the Higgs field may take largefield values in the early universe. It is well known that the SM Higgs potential allows anotherminimum at a large field value. It depends on the precise value of the top quark mass whetherthe other minimum is lower or higher than the electroweak vacuum [53]. It is also possiblethat the Higgs potential is uplifted by some new physics effects. If the Higgs field has avacuum expectation value (VEV) much larger than the weak scale in the early universe,the renormalization group evolution of the strong coupling constant is modified due to theheavier quark masses, and the effective QCD scale becomes much larger than Λ
QCD .In this paper, we revisit the axion with the BD distribution, focusing on a possibilitythat the effective QCD scale during inflation is higher than the present value due to a larger If the QCD axion has a mass mixing with a heavy axion whose dynamics induces a phase shift of π .one can dynamically realize a hilltop initial condition, θ ∗ (cid:39) π [47]. See also Ref. [48] for realizing the hilltopinitial condition in a supersymmetric set-up. The cosmological moduli problem of string axions can also be relaxed in a similar fashion [46]. See alsoRefs. [49–52] for related topics. O (10 ) GeV in the minimal extensionof the SM. Thus, the upper bound on the inflation scale as well as the required e-foldingnumber can be greatly relaxed.The possibility of a larger Higgs VEV was discussed in different contexts and set-ups.For instance, in a supersymmetric extension of the SM, there are flat directions containingthe Higgs field, and some of which may take a large VEV during inflation [34–38, 54]. In thiscase, the axion field can be so heavy that it is stabilized at the minimum during the inflation,suppressing the axion isocurvature perturbation [38]. However, it is difficult to suppress theaxion abundance because the CP phases of the SUSY soft parameters generically contributeto the effective strong CP phase [37] (see [48]). It was also pointed out that topologicalinflation takes place if the SM Higgs potential has two almost degenerate vacua at theelectroweak and the Planck scales [55]. (See also Refs. [56–59] for the Higgs inflation andthe multiple point criticality.) We will come back to this possibility later in this paper. Inthe following we mainly consider the SM and its simple extension which does not involveany additional CP phases, and assume that the Higgs potential at the false vacuum drivesthe old inflation.The rest of this paper is organized as follows. In Sec. 2 we briefly review how the BDdistribution suppresses the axion abundance. In Sec. 3 we estimate the effective QCD scalewhen the Higgs field is trapped in a false vacuum at large field values, and derive the boundon the inflation scale H inf for avoiding the axion overproduction. In Sec. 4 we study thebubble formation and the subsequent evolution of the universe. The slow-roll inflation isdiscussed in Sec. 5 The last section is devoted to discussion and conclusions. Let us briefly review the QCD axion abundance and properties of the BD distribution. Werefer an interested reader to the original references [42, 43] for more details.3he axion mass is temperature-dependent and it is parametrized by m a ( T ) (cid:39) √ χ f a (cid:18) T QCD T (cid:19) n T (cid:38) T QCD . × − (cid:18) GeV f a (cid:19) eV T (cid:46) T QCD , (4)where the exponent is given by n (cid:39) .
08 [8], and we adopt T QCD (cid:39)
153 MeV and χ (cid:39) (75 . . At T (cid:29) T QCD , the axion is almost massless, while it acquires a nonzero massas the temperature decreases down to T QCD . Then, the axion starts to oscillate aroundthe minimum when its mass becomes comparable to the Hubble parameter H . The axionabundance is given by [60]Ω a h (cid:39) . (cid:18) θ ∗ . (cid:19) × (cid:18) f a × GeV (cid:19) . f a (cid:46) × GeV (cid:18) f a × GeV (cid:19) . f a (cid:38) × GeV , (5)where we assume that the axion starts to oscillate during the radiation-dominated era andthere is no extra entropy production afterwards. One can see from the above equation that f a cannot be larger than about 10 GeV for θ ∗ ∼
1, since otherwise the axion abundancewould exceed the observed dark matter abundance, Ω DM h (cid:39) .
12 [61]. Thus, one needs | θ ∗ | (cid:28) O (1) for f a (cid:29) GeV. If all values of − π ≤ θ ∗ < π are equally likely, such an initialcondition requires a fine-tuning. As we shall see below, however, this is not necessarily thecase if the inflation scale is lower than the QCD scale.During inflation one can define the Gibbons-Hawking temperature, T inf = H inf / π [44],associated with the horizon. Let us suppose that the temperature is close to or smaller thanthe QCD scale so that the QCD axion acquires a small but nonzero mass, m a, inf . If theHiggs VEV is same as in the present vacuum, one can estimate it by m a, inf (cid:39) m a ( T inf ). Inthe following we assume that m a, inf is much smaller than H inf . As we are interested in therelatively large f a and small θ ∗ , we can approximate the potential as the quadratic one, V ( a ) (cid:39) m a, inf a . (6)Then, after a sufficiently large number of e-folds, N (cid:38) N eq ≡ H /m a, inf , the classicalmotion and the quantum diffusion are balanced, leading to the BD distribution peaked at It is possible that the axion mass during inflation for T inf (cid:38) T QCD is slightly modified from Eq. (4) dueto the gravitational effects. | θ ∗ | ∼ (cid:113) (cid:104) θ (cid:105) = (cid:114) π H m a, inf f a . (7)Thus, | θ ∗ | is naturally much smaller than unity if H inf (cid:28) (cid:112) m a, inf f a . It is assumed here thatthe minimum of the axion potential does not change (modulo 2 πf a ) during and after theinflation [43]. For instance, in the case that the Higgs VEV is at the weak scale, the axionwindow is open up to f a ∼ GeV for H inf ∼
10 MeV.An even more interesting possibility is that the QCD scale is larger during inflation. Inthis case the axion mass at the very beginning of the universe was heavier than the currentone, and the upper bound on H inf to suppress the axion abundance is relaxed. As we shallsee in the next section, this possibility can be realized if the Higgs field trapped in a falsevacuum drives the eternal old inflation. In the present universe the SM Higgs develops a nonzero VEV of v EW (cid:39)
246 GeV, whichspontaneously breaks the electroweak symmetry. Let us express the Higgs doublet φ as φ = 1 √ (cid:32) h (cid:33) (8)where h is the Higgs boson.In the very early universe, the Higgs may be trapped in a false vacuum at v false (cid:29) v EW .If its potential energy dominates the universe, it drives the eternal old inflation. The falsevacuum subsequently decays into the electroweak vacuum through quantum tunneling, andan open bubble universe is nucleated. We assume that, inside the bubble, another scalarfield drives the slow-roll inflation and decays into the SM particles for successful reheating.Then, our observable universe is well contained in the single bubble. We will provide suchan inflation model later in this paper.When the Higgs is trapped in the false vacuum, the SM quarks acquire heavier massesthan in the electroweak vacuum, and so, the effective QCD scale is enhanced. If the oldinflation lasts sufficiently long, the initial misalignment angle can be naturally suppressed bythe BD distribution as we have seen before. In the following, we discuss the above scenarioin detail. For simplicity, we assume that the QCD axion is a string axion or a KSVZ-type5xion [62, 63], where the axion decay constant (or the PQ breaking scale) does not changeduring and after inflation. The quartic coupling of the Higgs field receives negative contributions from the top quarkloops. For the pole mass of the top quark in the range of 171 GeV (cid:46) M t (cid:46)
176 GeV , the SM Higgs potential reaches a turning point at 10 GeV (cid:46) Λ max (cid:46) M pl [64, 65], where M pl (cid:39) . × GeV is the reduced Planck mass. Above the turning point the potentialstarts to decrease and may become negative, implying that the electroweak vacuum is meta-stable. The lifetime of the electroweak vacuum is much longer than the present age of theuniverse [66, 67], but it is under discussion whether it is stable enough in the presence ofsmall black holes [68–79] or compact objects without horizon [80].Broadly speaking, there are two ways to make the electroweak vacuum absolutely stable.If the top quark pole mass is given by M t (cid:39)
171 GeV, the potential has two vacua; one isat (cid:104) h (cid:105) = v EW , and the other at (cid:104) h (cid:105) (cid:39) M pl . For a certain value of the top quark mass, itis even possible to make the two vacua almost degenerate, which is known as the multiplepoint criticality [53]. This is the minimal possibility because no new physics is necessary,although the required top quark mass is not favored by the current measurements, M t =172 . ± . max by some new physics. As we shall see below, one can indeed stabilize theelectroweak vacuum by introducing extra heavy particles coupled to the Higgs field. Ineither case, if the Higgs field is trapped in the false vacuum at large scales, the eternal oldinflation occurs.Here let us see that one can stabilize the electroweak vacuum by introducing an extrareal scalar field S . We consider the following potential for S , V S = m S S + λ S S + λ P h S + · · · , (9)where we have imposed a Z parity on S and the dots represent higher order terms suppressedby a large cutoff scale. For m S > λ S >
0, and λ P > S is stabilized at the origin. Weassume that S is so heavy that the SM is reproduced in the low energy limit. By integratingout S , the effective Higgs potential at large scales can be well approximated by the quarticterm, V eff ( h ) (cid:39) λ eff ( h )4 h , (10)6here λ eff ( h ) is the effective Higgs self-coupling, and at one-loop level it is approximatelygiven by λ eff ( h ) (cid:39) λ ( µ RG ) − y t π (cid:18) log (cid:18) y t h µ (cid:19) − (cid:19) + ∆ NP λ. (11)Here we have neglected the SM contributions other than the top Yukawa coupling. Thefirst term is the quartic coupling at the renormalization scale µ RG , the second term is thedominant radiative correction from the top quark loop, and ∆ NP λ denotes the contributionof the extra scalar S . Due to the minus sign of the second term, the top quark contributiontends to push the quartic coupling toward negative values. The scalar contribution can beestimated as∆ NP λ (cid:39) π h (cid:18) ( m S + λ P h ) (cid:18) log (cid:18) m S + λ P h max[ µ , m S ] (cid:19) − (cid:19) − λ P m S h (cid:19) , (12)where we have taken λ to be zero in the loop function as it is sub-dominant around thescale Λ max . (See e.g. Ref. [82] for the derivation.) In the logarithm we introduce therenormalization scale in such a way that the one-loop renormalization group equation of λ for both µ RG < m S and µ RG > m S can be derived by ∂V eff /∂µ RG = 0 . The last term inEq. (12) is the finite term contribution which guarantees the almost vanishing Higgs massat v EW (cid:28) µ RG (cid:28) m S . One can see from Eqs. (11) and (12) that the contribution of S uplifts the potential if λ P (cid:38) y t at large field values. For a proper choice of m S and λ P , the potential can have afalse vacuum with V eff ( v false ) > V eff ( v EW ) at v false (cid:29) v EW . We show in Fig. 1 the effectivepotential for the Higgs where we have chosen λ = 0 , y t = 0 . , λ S = 0 . , and m S = 10 GeVat µ RG = 10 GeV. In this case the false vacuum is located at v false (cid:39) . × GeV. Thus, one can indeed uplift the Higgs potential at high scales so that there exists a falsevacuum at v false (cid:38) Λ max .If the Higgs field is trapped in the false vacuum, the eternal old inflation takes place withthe Hubble parameter, H inf = (cid:115) V eff ( v false )3 M . (13)We can put an upper bound on H inf as a function of v false as follows. Since the Higgspotential reaches the turning point due to the top quark contributions, the value of the The adopted values of y t and λ correspond to the central values of the measured top and Higgsmasses [81]. h × - [ GeV ] V e ff × - [ G e V ] h /(10 GeV) V e ff / ( G e V ) Figure 1: The effective potential for Higgs in the presence of a coupling with a singlet scalar S (red solid line). We take λ = 0 , y t = 0 . , λ P = 0 .
8, and m S = 10 GeV at µ RG = 10 GeV,and the false vacuum is located at v false (cid:39) . × GeV. For comparison, the potential inthe SM is also shown (gray dashed line).potential maximum is roughly given by V max ∼ y t π Λ = O (10 − − − ) v (14)for v false ∼ (1 − max . Since the false vacuum should have a lower energy than V max , weobtain H inf (cid:46) H max ≡ (cid:115) V max M ∼ (10 − (cid:16) v false GeV (cid:17) . (15)This gives an upper bound on the inflationary scale which can be realized using Higgs falsevacuum.In the rest of this section we will see that the axion abundance also gives an upper limitto the inflationary scale. Specifically we will see that the QCD axion can explain dark mattereven for f a as large as the Planck scale when v false (cid:38) GeV. When v false (cid:46) GeV, onthe other hand, the bound (15) becomes stronger than that from the axion abundance with f a (cid:46) M pl . This implies that the QCD axion can only contribute to a fraction of the darkmatter. In this region the QCD axion window is open to (cid:38) M pl . One needs to introduce8nother sector that contributes to the Hubble parameter to saturate the upper limit andexplain the QCD axion dark matter. The effective QCD scale during the eternal old inflation gets enhanced due to heavier quarkmasses. To see this we solve the one-loop renormalization group equation for the stronggauge coupling, dg d log ( µ RG ) = g π (cid:32) −
11 + (cid:88) i
23 Θ (cid:18) µ RG − m i v false v EW (cid:19)(cid:33) , (16)where m i denotes the SM quark masses, i runs over the quark flavor, and Θ is the Heavisidestep function. First we fix the initial value of the strong gauge coupling at some high-energyscales within the SM. To be concrete we adopt g ( M pl ) = g pl3 (cid:39) . effQCD ( v false ) by the renormalization scale µ RG where g becomes equal to 4 π . We show the numerical result of Λ effQCD ( v false ) in Fig. 2. For v false > GeV, it can be fitted byΛ effQCD ( v false ) (cid:39) GeV (cid:18) v false M pl (cid:19) / , (17)where all the SM quarks are already decoupled at the Λ effQCD ( v false ) for the parameters of ourinterest. Here let us estimate how much the QCD axion window is relaxed if the eternal old infla-tion takes place in the Higgs false vacuum. We will come back to the issue of the decayrate of the false vacuum in the next section, and here we simply assume that the inflationlasts sufficiently long so that the probability distribution of the QCD axion reaches the BDdistribution.As we have seen before, the QCD axion acquires a small but nonzero mass if the inflationscale satisfies T inf < Λ effQCD ( v false ). This is naturally realized if v false (cid:46) − GeV due to Note that the current best-fit values of the top mass and the strong coupling lead to v false ∼ GeV,for which the QCD axion can be the dominant dark matter with H inf ∼ H max and f a ∼ M pl . Here we neglect contributions of the exotic quarks in the PQ sector, assuming that the PQ quarks arenot coupled to the Higgs and they do not affect the Higgs potential through higher order corrections. Log [ v false / GeV ] L og [ (cid:1) Q C D e ff / G e V ] Figure 2: The effective QCD scale Λ effQCD as a function of the Higgs VEV v false . Eqs. (15) and (17). For v false (cid:29) GeV, on the other hand, the two vacua must be almostdegenerate in energy. Then, the axion mass during the eternal old inflation can be wellapproximated by m a, inf (cid:39) Λ effQCD ( v false ) f a . (18)We substitute this mass into Eq. (7) to evaluate the typical initial misalignment angle.Let us recall that the axion window (3) was obtained by assuming the initial misalignmentangle θ ∗ of order unity. Now θ ∗ is given by a function of H inf and v false (cf. Eqs. (7) and (18)),and in particular, it can be much smaller than unity for sufficiently small H inf . Thus, theaxion window can be expressed by the upper bound on H inf for a given v false and f a . We showin Fig. 3 the upper bound on H inf as a function of f a for v false = 10 GeV (red), 10 GeV(green), 10 GeV (orange), and (cid:104) h (cid:105) = v EW (blue) from top to bottom. The left vertical lineat f a ∼ GeV represents the classical axion window with θ ∗ = 1. The red lines with v false = 10 GeV correspond to the minimal scenario with M t (cid:39)
171 GeV, in which case thefalse vacuum can be almost degenerate with the true vacuum in the framework of the SM. Onthe other hand, for v false < GeV (i.e. below the red line), one needs to introduce somenew physics which uplifts the Higgs potential to make a false vacuum. For v false ∼ GeV(i.e. the green line), the false vacuum is not necessarily degenerate with the electroweak10 - - - f a [ GeV ] H i n f [ G e V ] ( (cid:1) i n f ) / [ G e V ] ρ / i n f [ G e V ] v f a l s e = G e V v f a l s e = G e V v f a l s e = G e V ⟨ h ⟩ = v E W θ * = 1 Figure 3: The upper bound on H inf as a function of the decay constant for v false = 10 GeV(red), 10 GeV (green), 10 GeV (orange), and (cid:104) h (cid:105) = v EW (blue), respectively, from top tobottom. The top three lines correspond to the currently allowed range of the top quark mass.The QCD axion explains dark matter on each line. Note that, for v (cid:46) GeV, the upperbound from (15) becomes stronger, and one needs to introduce another inflation sector toexplain the QCD axion dark matter.one, because the upper bound on H inf is comparable to H max . For v false (cid:46) GeV, thebound on H inf from (15) is stronger than that from the axion abundance, and one needsanother inflation sector to saturate the bound. The orange line approximately correspondsto the top quark mass ∼
176 GeV. The case of v false < GeV may also be possiblein a more involved extension of the SM. One can see that, depending on v false , the upperbound on H inf is significantly relaxed compared to the SM shown by the blue line (thebottom one). Specifically, H inf can be larger than ∼
10 GeV for v false (cid:38) GeV and10 GeV (cid:46) f a (cid:46) GeV. 11
False vacuum decay and slow-roll inflation
The false vacuum of the Higgs field is unstable and decays into the electroweak vacuumthrough the bubble nucleation. Let us denote by Γ / V the tunneling probability per unittime per unit volume. Then, the effective decay rate per the Hubble volume, V H ∼ H − ,is given by Γ eff , H = Γ V H / V . If Γ eff , H (cid:46) H inf , the inflation is eternal in a sense that in thewhole universe there are always regions that continue to inflate [83–88] (see also [89–91]).One can easily see this by noting that the physical volume of the inflating regions increasesby a factor of e − Γ eff , H ∆ t e H inf ∆ t (19)over a time ∆ t . Therefore, in this case, the bubble formation is so rare that it cannotterminate the inflation as a whole; some fraction of the entire universe continues to inflate.It is important to note, however, that a typical e -folding number that a randomly pickedobserver experiences is not infinite, but finite (though exponentially large). Essentially, theuniverse at a later time is simply dominated by those regions where inflation continues, andin this sense, the eternity of eternal inflation relies on the volume measure [89, 92–94]. Here we show that the typical e -folds the universe experiences before the tunneling, N dec ,is so large that the BD distribution of the QCD axion is reached. Specifically, we will show N dec ≡ H inf Γ eff , H (cid:29) N eq , (20)where N eq = H /m a, inf as defined below Eq. (6). Using (17) and (18), we can rewrite theabove condition asΓ V (cid:28) − GeV (cid:18) v false M pl (cid:19) / (cid:18) H inf GeV (cid:19) (cid:18) f a GeV (cid:19) − . (21)If this condition is met, irrespective of whether the volume measure is adopted, our observableuniverse must have experienced a sufficiently long inflation in the past so that the initial axionfield value obeys the BD distribution. GeV (cid:28) v false (cid:28) M pl ) First, we consider the case of 10 GeV (cid:28) v false (cid:28) M pl . In this case, the false vacuum mustbe almost degenerate with the true vacuum. This is because, as can be seen in Fig. 3, the It is possible to make the typical e -folds extremely large (e.g. N ∼ ) in a stochastic inflationmodel where the potential has a shallow local minimum around the hilltop [95]. H inf , is bounded above by the QCDaxion abundance, and it is much smaller than H max when v false (cid:29) GeV (see Eq. (15) forthe definition of H max ). In the following we use a thin-wall approximation to estimate thevacuum decay rate. Later we will check the validity of the thin-wall approximation.The false vacuum decay including gravitational effects was studied by Coleman and DeLuccia [96] (CDL hereafter). We consider a thin-wall approximation for the CDL bubblenucleation assuming that the dominant bounce solution possesses an O(4) symmetry. In thesemiclassical approximation, the CDL tunneling rate per unit time per volume is given inthe following form, Γ V (cid:39)
A e − B , (22)where higher order corrections are suppressed by the Planck constant. Here the prefactor A has mass dimension four, and B is the difference between the Euclidean actions for thebounce solution and the false vacuum one.For precise determination of A , one needs to calculate the one-loop fluctuations aroundthe bounce solution. The value of A for the standard-model Higgs potential was calculated inRef. [97]. In the presence of gravity, the calculations of A is hampered by e.g. gravitationalfluctuations and renormalization of the graviton loops (see Refs. [98–103] for the details).Here we simply assume that A is of order R − b on dimensional grounds, where R b is thephysical size of the bubble at the nucleation.In the absence of gravity, B is given by B = (cid:90) ∞ π ρ dρ (cid:34) (cid:18) ddρ h sol (cid:19) + ( V eff ( h sol ) − V eff ( v false )) (cid:35) , (23)where we have added a subscript of 0 to indicate that it does not include the effect of gravity, ρ is the radial coordinate in the Euclidean spacetime, and we have used the fact that thedominant bounce solution possesses an O(4) symmetry. Here h sol ( ρ ) is the O (4) symmetricbounce solution satisfying the Euclidean equation of motion, d dρ h + 3 ρ ddρ h = ddh V eff ( h ) , (24)with the boundary conditions, dh/dρ | ρ → = 0 and h ρ →∞ = v false . In the thin-wall approximation, one can estimate B in a closed form [104], B = 27 π σ (cid:15) , (25)13here σ is the tension of the bubble wall, σ ≡ (cid:90) v false v EW dh (cid:112) V eff ( h ) − V eff ( v false )] , (26)and (cid:15) is the energy difference between the two vacua, (cid:15) ≡ V eff ( v false ) − V eff ( v EW ) (27)In our case of the Higgs potential, they are given by σ = ζv , (28) (cid:15) (cid:39) H M , (29)where ζ is a constant of O (10 − ), and we have used the fact that the cosmological constantin the present vacuum is vanishingly small in Eq. (29).In the presence of gravity, the bounce action receives gravitational corrections [96]. Herewe are interested in the case of the tunneling from the false vacuum dS to the true vacuum dSor Minkowski. In this case, the gravity effect suppresses B and makes the bubble nucleationmore likely. This is because the cosmic expansion assists the bubble to expand after thenucleation. As the bounce solution deviates from the thin-wall approximation, the Gibbons-Hawking radiation induces upward fluctuations which also makes the bubble formation morelikely [105]. Under the thin-wall approximation, a general form of the bounce action is givenby [107] B = B r ( x, y ) (30)with r ( x, y ) = 2 1 + xy − (cid:112) xy + x x ( y − (cid:112) xy + x , (31) x = σ σ c = 3 σ (cid:15)M , (32) y = V eff ( v false ) + V eff ( v EW ) (cid:15) , (33)where the critical tension σ c is defined by σ c = (cid:115) (cid:15)M (cid:39) H inf M . (34)14n our scenario both x and y are positive, and in particular, y >
1. While x is alwayspositive, y can be negative in a more general case.The above bounce solution can be broadly classified into the two regimes, x < x > B is the same for both cases. Thebounce solutions corresponding to x < x > V eff ( v EW ) →
0, i.e., y →
1, the above expression of B is reduced to theoriginal formula of CDL [96], B = B (cid:18) σ σ c (cid:19) − . (35)In this limit, one can clearly see that the gravitational effect becomes relevant when σ > σ c (i.e. x > x > v false (cid:38) v c (cid:39) × GeV (cid:18) ζ − (cid:19) − (cid:18) H inf
10 GeV (cid:19) , (36)where v c denotes the critical value of v false . So, the gravitational effect becomes relevant as v false increases.Now we are ready to estimate B . Let us fix the inflation scale H inf to some value (e.g. H inf = 10 GeV) for simplicity, and vary v false . For v false (cid:46) v c , the gravitational effect isnegligible, and B increases in proportion to v . For v false (cid:38) v c , on the other hand, B becomes independent of v false . Thus, B takes the smallest value at the smallest possible v false . In other words, the vacuum decay rate is the largest at the smallest v false in thisregime where the thin-wall approximation is applicable. Using (30), (28), and (29), weobtain B (cid:39) B = O (10 ) (cid:18) ζ − (cid:19) (cid:16) v false GeV (cid:17) (cid:18) H inf GeV (cid:19) − . (37)Therefore, in the case of 10 GeV (cid:28) v false (cid:28) M pl , B is so large that the vacuum decay rateis exponentially suppressed, and the condition (21) is trivially satisfied. This is justified because the upper bound on H inf scales only as v / . See Eq. (17). The prefactor A may grow as v false increases even when v false (cid:38) v c . However, the vacuum decay rate ismainly determined by B , and the increase of A does not change our conclusion. .2 Non-degenerate vacua ( v false ∼ GeV ) In the case of v false ∼ GeV, one can see from Fig. 3 that H inf can be as large as O (10 ) GeV, which is comparable to H max defined by the height of the potential barrier. Ifthe upper bound on H inf is saturated, the thin-wall approximation is not applicable, and adedicated analysis is needed.Let us make an order of magnitude estimate of B and the decay rate. The bubble size R b can be roughly estimated as follows. The first term in (23) is of order v /R b , whilethe second term is of order V max since we assume that the two vacua are not degenerate. Bybalancing the two terms, we get, R b ∼ v false V / . (38)Since the integral of (23) is essentially cut off around the size of the bounce solution, ρ ∼ R b ,we obtain B ∼ π R b V max = O (10 − ) , (39)where we have used (14). Using A ∼ R − b , we arrive atΓ V (cid:46) − (cid:16) v false GeV (cid:17) GeV (40)for which (21) is well satisfied.We have numerically solved the equation of motion and obtained the bounce solution forthe potential shown in Fig. 1. The bounce solution v false − h sol is shown in Fig. 4 as a functionof ρ . We have calculated B for this solution and obtained B (cid:39) . × , (41)which agrees well the above na¨ıve estimate, justifying our conclusion. v false ∼ M pl ) When v false approaches the Planck mass, the potential becomes relatively flat near the max-imum. Then, the Higgs stays mostly around the potential maximum in the bounce solution,and the thin-wall approximation breaks down. Such a bounce solution can be interpreted as acombination of thermal fluctuations and the subsequent quantum tunneling [105]. In partic-ular, when v false ∼ M pl , the Hubble horizon in the wall, H − , becomes comparable to the wallwidth, and the bounce is approximately given by the Hawking-Moss (HM) solution [108].Once such a domain is formed, it will start to expand exponentially, and the topologicalHiggs inflation begins [55] (see Refs. [109, 110] for the original topological inflation).16 (cid:1) × (cid:2) RG ( v false - h sol )/ (cid:2) RG ( v false − h sol )/(10 GeV) ρ ⋅ 10 GeV
Figure 4: Numerical result of the bounce solution by varying ρ . The potential is given inFig. 1.To see this, let us again fix H inf and increase v false , in other words, we increase x while y is fixed. In the limit of x (cid:29)
1, the r -function in (31) approaches r ( x, y ) (cid:39) x ( y + 1) ( x (cid:29) . (42)When we also take y ≈
1, the corresponding bounce action reads, B ≈ π M V eff ( v false ) = 8 π M H , (43)which coincides with the HM instanton [108]. We note that the thin-wall approximationactually breaks down in this limit. Nevertheless it is assuring that the large x limit of B obtained in the thin-wall approximation reproduces the HM instanton, which lends supportto the above picture.The topological Higgs inflation typically lasts only for a few e -folding number, becausethe potential curvature at the maximum is not very different from the Hubble parameterinside the domain. Then, if the Higgs rolls down to the electroweak vacuum, the infla-tion ends and the cosmic expansion becomes decelerated. Since the initial configuration of Here we do not adopt the volume measure. GeV. Such high-scale inflation willgenerate quantum fluctuations of the QCD axion on top of the BD distribution. Thus, θ ∗ will be dominated by the quantum fluctuations, but it is still smaller than unity thanks tothe initial BD distribution. In this case, the QCD axion cannot be the dominant componentof dark matter because of its too large isocurvature perturbations. However, it may still bethe subdominant component of dark matter, which gives rise to a non-negligible amount ofnon-Gaussianity of the isocurvature perturbations [111–114]. In the rest of this paper we aregoing to focus on the case of v false (cid:28) M pl . In the case of v false (cid:28) M pl , the initial condition of the nucleated bubble is determined by theO(4) symmetric CDL tunneling solution, ds = dξ + ρ ( ξ ) d Ω S , where d Ω S is the metric of S and ξ is the radial coordinate. One can obtain the metric inside the bubble by an analyticcontinuation, the interior of the bubble looks like an infinite open universe for observers inthe bubble [115, 116]. Infinitely many open bubble universes are created in the eternallyinflating false vacuum. The open bubble universes are often discussed in the context of thestring landscape [117–119]. Naively, such bubble universes would be almost empty with asmall energy density, but it can be combined with the standard inflationary cosmology [120].We will study the slow-roll inflation in the next section.The created bubble universes might have some predictions such as distinct features ofthe primordial density perturbations [121–125] and the negative curvature K <
0. The latestPlanck data combined with the lensing and BAO gives Ω K = 0 . ± . The slow-roll inflation is strongly supported by the observations of CMB and large-scalestructure. The false vacuum Higgs inflation considered so far cannot explain the observeddensity perturbations, since the Higgs potential is not sufficiently flat to realize viable slow-roll inflation in our scenario. Thus, we need another sector that drives slow-roll inflation The inflation model discussed in Sec.5 can also be applied to this case. ϕ , with the following potential, V inf ( ϕ ) = V − m ϕ − λ ϕ ϕ + ϕ M , (44)where λ ϕ is a positive coupling constant, m the mass parameter, M the cutoff scale, and V the energy density during inflation at ϕ ∼ . A supersymmetric version of the model wasstudied in Refs. [126–128]. The inflaton ϕ may be identified with the B − L Higgs [50,129–132]or an axion-like particle [47, 133, 134]. As we shall see shortly, λ ϕ is much smaller thanunity. For the moment we assume that m is negligibly small in the following, but it isstraightforward to take its effect on the inflaton dynamics. In fact, when we couple ϕ to theHiggs field, we will introduce a nonzero mass to realize the slow-roll inflation.We briefly summarize here the known properties of the above inflation model. Theminimum of V inf ( ϕ ) is located at ϕ min = (cid:112) λ ϕ M (cid:28) M pl . (45)The potential V inf ( ϕ ) is very flat around the origin, and so, if ϕ is initially around the origin,the slow-roll inflation takes place. The CMB normalization fixes the quartic coupling λ ϕ ,and the other parameters such as the inflation scale and the inflaton mass at the minimumare expressed in terms of ϕ min as [129] λ ϕ (cid:39) × − (cid:18) N ∗ (cid:19) , (46) H inf ,ϕ (cid:39) − (cid:18) N ∗ (cid:19) / (cid:18) ϕ M pl (cid:19) , (47) m ϕ, min (cid:39) − ϕ min (cid:18) N ∗ (cid:19) / , (48)where N ∗ is the e-folding number at the horizon exit of the CMB scales. Since the inflationscale must be lower than that of the false vacuum Higgs inflation, N ∗ cannot exceed 50. Thespectral index is given by n s (cid:39) − N ∗ , (49)which is too small to explain the observed value n s (cid:39) . ± .
004 [61]. It is known howeverthat the spectral index is extremely sensitive to the shape of the inflaton potential, and evena tiny deviation from the quartic potential can make it consistent with the observed value. For instance, one can introduce a nonzero mass term [135], a Z breaking linear term [136] or a Coleman-Weinberg potential, ∝ ϕ log[ ϕ ] [129]. ϕ after the bubblenucleation from the false vacuum Higgs. In the thin-wall approximation, the universe insidethe bubble is almost empty and dominated by the (negative) curvature term. In this case onecannot make use of fine-temperature effects to set the value of ϕ near the origin. On the otherhand, if ϕ has a coupling to the Ricci curvature with a coupling of order unity, it acquiresa mass of order the Hubble parameter during the false vacuum Higgs inflation, and can bestabilized the origin. However, considering that ϕ ≈ ϕ ≈ ϕ ≈ ϕ min . Now let us introduce the following renormalizable coupling to Higgs field to make theorigin of ϕ the global minimum during the false vacuum Higgs inflation, δV = λ ϕh ϕ − ϕ ) h . (50)where λ ϕh ( >
0) is the coupling between ϕ and h . The basic picture is as follows. When theHiggs is in the false vacuum, ϕ acquires a large mass,( m false ϕ ) (cid:39) λ ϕh v , (51)which is larger than H unless λ ϕ is extremely small. In fact, if the mass is large enough, itmakes the origin the global minimum along the ϕ direction for the fixed h = v false . After thetunneling, the Higgs field value becomes much smaller, and so does the mass of ϕ . Then, theslow-roll inflation starts along the ϕ direction with the potential given by (44). In particular,since ϕ remains heavy except when the Higgs field really approaches the “true” vacuum, theprevious argument on the bubble nucleation is considered to remain valid.To ensure the above-mentioned dynamics, a couple of conditions must be met. First, wedo not want to modify the Higgs potential significantly by introducing the above coupling.The location of the potential barrier and the false vacuum remain almost intact if V max (cid:29) λ ϕh ϕ v . (52)The potential energy at the false vacuum is still dominated by the Higgs contribution if V eff ( v false ) (cid:29) V . (53)In general, one needs to slightly shift the parameters to uplift the false vacuum to have thesame value of V eff ( v false ), but this does not modify the previous argument on the bubble One can still argue that ϕ ≈ ϕ to the Higgs quartic coupling should bemuch smaller than that of the top quark. This amounts to λ ϕh (cid:28) y t . Secondly, the originof ϕ is the global minimum in the ϕ -direction at h = v false if λ ϕh ϕ v (cid:29) λ ϕ ϕ ∼ V . (54)Lastly, although we have required the locations of the potential maximum and the falsevacuum along the Higgs direction remain almost unchanged, the location of the “true”vacuum is necessarily shifted to v true ∼ (cid:112) λ ϕ /λ eff ( v true ) ϕ min . This is because the Higgsacquires a negative mass from (50) if ϕ = 0. Thus, the mass of ϕ would remain positiveeven after the Higgs tunnels to v true . (see Eq. (50)). So, we introduce a nonzero mass m in Eq. (44) to cancel this contribution so that the effective mass is negative and its absolutemagnitude is still smaller than H inf ,ϕ ∼ √ V /M pl at h ≈ v true . Then, the slow-roll inflationstarts along the ϕ direction, whose dynamics is well described by the hilltop quartic inflationmodel. Only when ϕ approaches ϕ min after the end of slow-roll inflation, the Higgs fieldapproaches the electroweak vacuum.The conditions (52), (53), and (54) are summarized as ϕ min (cid:28) Min (cid:34) − (cid:112) λ ϕh v false , GeV (cid:18) H inf GeV (cid:19) , (cid:112) λ ϕh v false (cid:35) , (55)where we have used (14) and (46). Unless λ ϕh is much smaller than unity, either the firstor second term in the bracket gives the strongest condition on ϕ min . Therefore, it is indeedpossible to satisfy the constraints once the energy scale of the slow-roll inflation is taken to besufficiently low. The energy density of the slow-roll inflation can be as large as [10 − GeV] . The slow-roll inflation model is of the hilltop type, and one may wonder that eternalinflation may take place, which would erase the BD distribution established during the falsevacuum Higgs inflation. In fact, if one does not adopt the volume measure, the typical e -folding number is finite and is not so large, and the BD distribution established during thefalse vacuum Higgs inflation remains intact. We also note that a better fit to the observedspectral index is obtained if we add a tiny Z breaking linear term in V inf [136]. The linearterm shifts the location of the potential maximum of ϕ which effectively reduces the total e -folding number. In any case, the BD distribution is not modified during the slow-rollinflation.The reheating is considered to proceed through the perturbative decays ϕ → hh (56)21r with the parametric resonance and dissipation effects via the quartic coupling (50). De-pending on the size of the coupling, λ ϕh , the reheating can be instantaneous, in which casethe reheating temperature can be as high as 10 − GeV . Then, thermal leptogenesis [137]is possible. If ϕ is identified with the B − L Higgs boson, non-thermal leptogenesis may alsotake place as ϕ directly decays into right-handed neutrinos. Alternatively, by introducing thedimension five Majorana neutrino mass terms, the leptogenesis via active neutrino oscillationis also possible [138, 139]. So far we have focused on the possibility that the Higgs field is trapped in the false vacuumand drives the eternal old inflation. After the bubble formation, another scalar field ϕ drivesthe slow-roll inflation. There is another possibility that the Higgs field value is set to belarger than the present one due to the inflaton field ϕ . In this case, ϕ drives both eternal(or extremely long) inflation and the subsequent slow-roll inflation that explains the observedprimordial density perturbations ( see e.g. Ref. [95]). As we have seen before, the Higgsacquires a negative mass from the potential like (50) during inflation where ϕ (cid:39)
0. Thus,the field value of h can be largely displaced from v EW during the inflation. Consequently, theQCD axion window can be similarly opened to large values of f a , if the inflation scale is lowerthan the enhanced effective QCD scale and the inflation lasts long enough. Interestingly, theinflaton ϕ may be identified with the singlet S that uplifts the Higgs potential. In this case,the false vacuum of the Higgs potential will disappear due to the radiative corrections of ϕ because the inflaton is light. This is the main difference from the scenario discussed after(9).The abundance of the QCD axion depends on the initial misalignment angle θ ∗ . For f a (cid:38) GeV, one usually assumes θ ∗ (cid:28) θ ∗ follows the BD distribution andtherefore can be suppressed, if the Hubble parameter during inflation is comparable to orlower than the QCD scale, and if the inflation lasts sufficiently long. To this end, oneoften needs to introduce another sector for the very long inflation. In this paper we havepointed out that the false vacuum Higgs inflation can do the job. Furthermore, the effectiveQCD scale is enhanced because of the Higgs field value larger than the present value, whichsignificantly relaxes the upper bound on the inflation scale. We have found that the Hubble The ϕ field does not have to be the inflaton, and it can be another scalar (such as curvaton) or fermioncondensates. The only requirement is to keep the Higgs at large values during the eternal inflation. e -folding number that theuniverse experiences before the bubble nucleation is so large that the BD distribution of theQCD axion is realized. After the tunneling event, another slow-roll inflation must follow togenerate the primordial density perturbation and to make the universe filled with radiationinstead of the negative curvature. In a simple model based on the hilltop quartic inflation,the hilltop initial condition can be naturally realized if the inflaton has a quartic couplingwith the Higgs field. Alternatively, through the coupling, the inflaton may be able to upliftthe Higgs potential to erase the false vacuum, and it keeps the Higgs field at large valuesduring inflation. The latter provides another attractive scenario, which warrants furtherinvestigation. Acknowledgments
W.Y. thanks particle physics and cosmology group at Tohoku University for the kind hos-pitality. This work is supported by JSPS KAKENHI Grant Numbers JP15H05889 (F.T.),JP15K21733 (F.T.), JP17H02875 (F.T.), JP17H02878(F.T.), by World Premier Interna-tional Research Center Initiative (WPI Initiative), MEXT, Japan, and by NRF StrategicResearch Program NRF-2017R1E1A1A01072736 (W.Y.).
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