Scale and quality of Peccei-Quinn symmetry and weak gravity conjectures
SScale and quality of Peccei-Quinn symmetry andweak gravity conjectures
Wen Yin
Department of Physics, Faculty of Science, The University of Tokyo,Bunkyo-ku, Tokyo 113-0033, Japan,Department of Physics, Korea Advanced Institute of Science and Technology,Daejeon 34141, Korea
Abstract
The promising solution to the strong CP problem by a Peccei-Quinn (PQ)symmetry may introduce quality and hierarchy problems, which are both rel-evant to Planck physics. In this paper, we study whether both problems canbe explained by introducing a simple hidden gauge group that satisfies a weakgravity conjecture (WGC) or its variant. As a concrete example, we pointout that a weakly-coupled hidden SU( N ) gauge symmetry, which is brokendown to SO( N ), can do this job in the context of a Tower/sub-Lattice WGC.Cosmology is discussed. a r X i v : . [ h e p - ph ] A ug Introduction
A global Peccei-Quinn (PQ) symmetry, U(1) PQ , is a leading candidate to solve afine-tuning problem, the strong CP problem, of the standard model (SM) [1, 2].Through the spontaneously breaking of U(1) PQ , an axion, which is a pseudo-NambuGoldstone boson (pNGB), arises [3–8]. Since the anomaly of U(1) PQ -SU(3) C is non-vanishing, the axion gets a potential with a CP-conserving minimum due to the non-perturbative effect of the QCD, and thus at the vacuum the strong CP problem issolved. Because of the coherent oscillation in the early universe, the axion condensatecan contribute to the matter density and hence can explain the dark matter [9–11].(See e.g. Refs. [12–19] for reviews.)The solution, however, suffers from hierarchy and quality problems. The firstproblem is due to that the PQ scale, or the decay constant of the QCD axion, f a , isconstrained to be within the so-called classical axion window:10 GeV (cid:46) f a (cid:46) GeV , (1)which is smaller than the reduced Planck scale, M pl = 2 . × GeV. The lowerbound comes from the duration of the neutrino burst in the SN1987a [20] (See alsoRefs. [21, 22]). The upper bound comes from the axion abundance constraint. Onesimple way to address this hierarchy is to open the window. This is possible if theHubble parameter during the inflation, which lasts long enough, is lower than theQCD scale [23, 24]. Another way is to extend the PQ sector to make the scaleof the classical axion window natural, e.g. the axion is composite [38, 39], or withsupersymmetry. Our proposal will belong to the latter.The quality problem, on the other hand, is somewhat related to quantum gravity.It is believed that any global symmetry should be explicitly broken by Planck-scalephysics (See e.g. Refs. [40–42]). Thus the PQ symmetry should be explicitly bro-ken. It was pointed out that a PQ symmetry with good enough quality can beobtained as an accidental symmetry of discrete gauge symmetries [43–47], abeliangauge symmetries [48–50], and non-abelian gauge symmetries [51–54]. A relevantcriterion for quantum gravity is weak-gravity conjecture (WGC) [55], which suggeststhat gravity is the weakest long-range force. From this conjecture, given the charged This low scale inflation can also alleviate the moduli problem at the same time [25]. See alsorelated topics [26–32]. It is also possible to introduce other degrees of freedom to, e.g., dilute ortransfer the axion abundance [10, 33–37]. B − L symmetry which is weakly coupled, the hierarchybetween the electroweak and Planck scales was discussed within the WGC [56]. Itmay be also important to discuss the scale of the PQ symmetry in the context ofthe WGC via the gauge symmetry introduced for the quality problem.In this paper, we find that by introducing a U(1) (cid:48) gauge symmetry in a KSVZaxion model [5, 6], the hierarchy between the scale of axion window and the Planckscale can be explained within the context of the WGC, which restricts an unbrokenabelian gauge symmetry. However it cannot solve the quality problem because if thePQ symmetry appears as the accidental symmetry of U(1) (cid:48) , U(1) (cid:48) must be brokenin order to have the spontaneous PQ symmetry breaking. From this finding, westudy a fundamental axion model with a hidden SU( N ) gauge symmetry, which isincompletely broken down via the PQ symmetry breaking, and solves the qualityproblem. In this case, a Tower/Sub-Lattice WGC (sLWGC) [57–60] can set a cutoffto the energy scales of the field theory [61]. If the cutoff is around the axion window,the PQ scale cannot be higher than that and, as a result, both the quality andhierarchy problems are solved. We also point out that within this low-cutoff theory,one may have a consistent cosmology.This paper is organized as follows. In the next section we review the hierarchyproblem for the PQ scale, and discuss the possible solution by introducing an unbro-ken abelian gauge symmetry with mild-version of WGC. In the Sec. 3, we study thenon-abelian gauge theory with tower/sLWGC and show that both the quality andhierarchy problems can be solved. The cosmology of the scenario is also discussed.The last section is devoted to conclusions and discussion. Let us consider a KSVZ model with the following particle contents q : (1 , r SM ) , ¯ q : (0 , ¯ r SM ) , H PQ : ( − ,
1) (2)Here q and ¯ q are exotic PQ quarks, and H PQ is a PQ Higgs field, under the repre-sentation of (U(1) PQ , G SM ) . r SM denotes the representation of the SM gauge group3 SM = (U(1) Y , SU(2) L , SU(3) C ), and we take r SM = ( Y, L , C ) here and hereafter.Then it is allowed to write down the Yukawa coupling of L ⊃ y q ¯ qH PQ q. (3)The potential of the H PQ is given as V PQ = − m | H PQ | + λ | H PQ | . (4) m ( >
0) and λ are the mass parameter and quartic coupling, respectively. Thusone readily gets that the H PQ obtains a non-vanishing vacuum expectation value(VEV) as v PQ ≡ (cid:104) H PQ (cid:105) = m PQ √ λ . (5)The quarks get mass of M q = y q v PQ . (6)Then the PQ symmetry is spontaneously broken and a (pseudo) NGB, a , appearswhich couples to q and ¯ q . Since the anomaly of U(1) PQ - G is non-vanishing, byintegrating out the heavy quarks one obtains L eff ⊃ π a √ v PQ (cid:16) Y g Y F Y ˜ F Y + g tr[ F C ˜ F C ] (cid:17) . (7)Thus a is the QCD axion and f a = √ v PQ (8)which should satisfy the classical axion window (1).Next, let us introduce a hierarchy problem of this model, which can be regardedas a radiative instability problem. The radiative correction to the mass parameterof the PQ field is δm = O (max [ λ, y q ]) Λ . o . π (9)where Λ c . o . is the cutoff scale of this model. We will take into account the quantumgravity effects, which we have omitted here, later. By assuming the cutoff scaleΛ c . o ∼ M pl , the radiative correction is 10 GeV for O (1) couplings. This implies that m in the conventional axion window (8) requires fine-tuning of m /δm (cid:46) − between the bare mass squared and δm . Notice that by simply taking the couplingsmall the tuning cannot be relaxed. Since v = λ − m , a small λ implies an evensmaller m , i.e. the ratio δm /m remains.4 .2 WGC and PQ scale: case of U(1) (cid:48)
The WGC states that gravity is the weakest long-range force. More precisely, it saysthat in the effective theory of U(1) (cid:48) gauge symmetry consistent with the quantumgravity, there is at least a charged particle with mass m and charge q (cid:48) satisfying [55] m (cid:46) q (cid:48) g (cid:48) M pl (WGC) (10)where g (cid:48) is the coupling of the U(1) (cid:48) at the scale of m . In this section, we take q (cid:48) = 1for simplicity. If the WGC is violated, charged black holes would become stablewhich is unnatural. There is not yet any counterexample found in string theory to(the mild version of) the WGC. Moreover, there are several proof of the WGC undercertain assumptions [62–66].Now let us follow [56] to explain the hierarchy by using Eq. (10). Let us introducean unbroken U(1) (cid:48) gauge symmetry under which q : 1 ¯ q : − . (11)The other fields (including the SM particles) are all supposed to be charge-less forsimplicity. For later convenience, let us rewrite the gauge coupling g (cid:48) ≡ y q ˜ f √ M pl (cid:39) × − y q (cid:32) ˜ f GeV (cid:33) , (12)by the dimensionful parameter ˜ f .From Eq. (10), it turns out that M q ≤ y q ˜ f √ . (13)Consequently, from Eq. (6), one obtains that v PQ (cid:46) ˜ f √ One notices that the anomalies of U(1) PQ -U(1) Y -U(1) (cid:48) and U(1) PQ -U(1) (cid:48) are non-vanishingif Y is non-vanishing. This, and possible kinetic mixing to a photon, may lead to the axioncouplings to the hidden photon field strength, F (cid:48) , as L eff ⊃ Y g Y g (cid:48) π v PQ aF Y ˜ F (cid:48) + g (cid:48) π v PQ aF (cid:48) ˜ F (cid:48) ≡ g aγγ (cid:48) aF Y ˜ F (cid:48) + g aγ (cid:48) γ (cid:48) aF (cid:48) ˜ F (cid:48) which is studied in e.g. Refs. [67–70]. m PQ (cid:39) (cid:114) λ f , (15)i.e. the WGC bound is saturated. As a result, if ˜ f is within or slightly above theaxion window, f a can be naturally within (1).In the explanation of the hierarchy between the v PQ and M pl , we have introduceda small parameter g (cid:48) . Although the small parameter is technical natural, the mildesttuned parameter set is g (cid:48) ∼ − (for ˜ f (cid:39) GeV) , y q = O (1) . (16)One may wonder if U(1) (cid:48) with a proper charge assignment can lead to an ac-cidental PQ symmetry, and solve the quality problem. However, it is difficult. Ifthe PQ symmetry were an accidental symmetry relevant to U(1) (cid:48) , a PQ Higgs field,which breaks the PQ symmetry, should be also charged under U(1) (cid:48) . Therefore U(1) (cid:48) symmetry must be broken, which means that the WGC cannot apply. To sum up, inthe context of the mild version of the WGC, an unbroken U(1) (cid:48) symmetry can solvethe hierarchy problem of the PQ symmetry but cannot solve the quality problem. To have an unbroken gauge symmetry relevant to the quality of the accidental PQsymmetry, we will introduce a non-abelian gauge symmetry instead of U(1) (cid:48) . Thisgauge symmetry is spontaneously broken down to an unbroken gauge symmetryvia the PQ symmetry breaking. In this case, the WGC can apply to the remnantunbroken gauge symmetry. Strictly speaking, the total gauge group including the G SM is a product group, and one shouldapply the convex hull condition [56]. We have checked that this does not change much our conclusionas an order of estimate. One option is to introduce another gauge symmetry to solve the quality problem, namely thequality and hierarchy problems are solved by different gauge symmetries. .1 A hidden SU( N ) gauge model for precise PQ symmetry To be concrete, let us assume that the exotic quarks are charged under a hiddenSU( N ) gauge group. The charge assignments of (SU( N ) , G SM ) are given as q : ( ¯ N , r SM ) , ¯ q : ( ¯ N , ¯ r SM ) , ψ a : ( N,
1) (17)where N is the fundamental representation of SU( N ) , and ψ a are needed to cancelthe gauge anomaly of SU( N ) with a = 1 · · · r SM ] . To give masses to the exotic quarks let us introduce a Higgs field who is a sym-metric tensor of the second rank, H PQ ≡ H { ij } PQ : (cid:18) N + N , (cid:19) . (18)We have explicitly written the symmetric indices, i, j = 1 · · · N of SU( N ). (We con-sider this representation because of simplicity, and because that N required for thePQ quality, as discussed below, is smallest. From the discussion in Ref. [54], it iseasy to consider other representations with good PQ quality. Another simple possi-bility is discussed in Appendix A, in which however, SU( N ) is completely broken.)The renormalizable Yukawa terms are given by L ⊃ y q ¯ qH PQ q + y abψ ψ a H ∗ PQ ψ b (19)where y q , and y ψ are the Yukawa couplings.In fact, it was pointed out in Ref. [54] that a large N SU( N ) gauge theory cangenerically lead to an accidental U(1) B H global symmetry (hidden baryon numbersymmetry) originating from the N -ality due to the group structure. The U(1) B H charge assignment is automatically obtained as q, ¯ q : − , ψ a : 1 , H PQ : 2 , (20)by counting the number of the indices with the sign corresponding to the complexrepresentation. One can check that in the Yukawa term and the following Higgspotential that U(1) B H manifests with good quality. The leading operator that breaks U (1) B H has a dimension N as L ⊃ c N det [ H PQ ] M N − . (21)Here c N is a constant and this term may be generated through a quantum gravityeffect (e.g. Refs. [42, 71, 72]). One notices thatU(1) B H - G ⊃ U(1) B H -SU(3) c (22)7s anomalous. It turns out that the U(1) B H is the PQ symmetryU(1) PQ ≡ U(1) B H . (23)The PQ symmetry can be precise enough against Planck-scale suppressed terms andsolve the strong CP problem if N (cid:38) c N = O (1) , f a = 10 GeV [41]. Consequently, the notorious quality problem ofthe PQ symmetry can be solved.It may be non-trivial whether the U(1) PQ can be spontaneously broken down,although we have implicitly assumed. The potential of the H PQ is obtained as V = λ H † PQ H PQ ] + λ H † PQ H PQ ) ] − m H † PQ H PQ ] . (25)Here λ , λ are quartic couplings. At the minimum of the potential, one obtains thenon-vanishing VEV (cid:104) H PQ (cid:105) = v PQ diag [1 , · · ·
1] if λ > λ ≤ (cid:104) H PQ (cid:105) ∝ diag [1 , · · · v = m N λ + λ . (27)Thus, SU( N ) × U(1) PQ → SO( N ) × Z , (28)where Z is the remnant of the U(1) PQ . Thus, H PQ not only breaks SU( N ) but alsoU(1) PQ incompletely. A QCD axion appears with the coupling of √ N π av PQ (cid:16) Y g Y F Y ˜ F Y + g tr[ F C ˜ F C ] (cid:17) . (29)It turns out that f a = 1 √ N v PQ . (30)8 .2 WGC and PQ scale: case of SO( N ) Since there is an unbroken gauge symmetry SO( N ) , the WGC can apply. Howeverthe mild-version of the WGC for any U(1) in the subgroup of SO( N ) is satisfieddue to the charged massless gauge bosons of SO( N ) and the charged massive gaugebosons eating the massless NGBs in H PQ . On the other hand, it is considered thatthe WGC should be somewhat sharpened. One reason is that the mild-version is notinvariant under the dimensional reduction.The Tower/sub-lattice WGC (sLWGC) [57–60], belonging to the stronger variantsof the WGC motivated by the invariance under dimensional reduction, states: aninfinite tower of particles/resonances of different charges satisfying (10) exists. Thisconjecture also clears various theoretical tests. Since a large number of particlesexist, they come into the loop of gravity and makes the gravity strongly coupled ata scale Λ QG . Λ QG satisfies [73] M pl (cid:38) (cid:112) N states Λ QG , (31), where N states is the number of states below the Λ QG . Λ QG can be seen as the cutoffscale for the quantum field theory. If N is so large that the number of states in thetower increases fast enough [61], Λ QG (cid:46) g N M pl (32)where g N is the coupling of the large N gauge theory at the scale g N M pl . In thesLWGC, log N states ∼ N log (Λ QG /g N M pl ) for large N and one can get (32) from(31) [61].Now let us discuss the hierarchy between the m PQ and M pl . Since there is anunbroken SO( N ) gauge symmetry, following (32) one gets the cutoff of the quantumfield theory of Λ QG (cid:46) GeV (cid:16) g N − (cid:17) . (33)By identifying Λ c . o . = Λ QG , (34)the radiative correction is then given by δm = O (max[ | λ | N , | λ | , | y q | , | y ψ | , g N ]) Λ π . (35)There is no fine-tuning between the radiative correction and the bare mass if O ( Λ QG √ π ) (cid:46) v PQ (cid:46) Λ QG . (36)9hen the lower bound is satisfied, f a ∼ × GeV (cid:114) N (cid:18) Λ QG GeV (cid:19) . (37)Consequently, both the quality and hierarchy problems of the PQ symmetry aresolved in this model in the context of the Tower/sLWGC. The cosmology of models solving the quality problem is usually troublesome. Forinstance, in our model Z symmetry in (28) stabilizes q, ¯ q, and ψ a which could causecosmological problem , although some of the fermion could be the dark matter. Also,the PQ symmetry breaking, if happens during the thermal history, may generatedomain walls. If the PQ symmetry is not restored during inflation and the axion isthe dominant dark matter, there can be an isocurvature problem. These problemsare all solved if the inflation scale is small enough. However, again, a hierarchybetween the inflation and the Planck scales is introduced.In our scenario, the inflation scale must be smaller than Λ QG , i.e. the Hubbleparameter during inflation, H inf , satisfies H inf < × GeV (cid:18) Λ QG GeV (cid:19) . (38)Thus the above problems can be solved with small inflation scales with the hierarchyexplained in the context of the Tower/sLWGC and the marginally small g N . A directprediction of the scenario is the suppressed tensor/scalar ratio, r ≈ . × − (cid:18) H inf GeV (cid:19) . (39)If we maximize the inflation scale, and if the QCD axion is dominant dark matter,the induced isocurvature perturbation is close to the current bound [74]. Thus itmay be searched for in the near future.Interestingly, the cutoff scale Λ QG ∼ GeV is close to the seesaw scale. If theright-handed neutrino masses are around the cutoff scale, and the neutrino Yukawa This problem may be solved if we consider q : ( ¯ N ( ¯ N + 1) / , r SM ) , ¯ q : ( ¯ N ( ¯ N + 1) / , ¯ r SM ) , ψ a :( N ( N + 1) / ,
1) instead. They have charges 2 and − B H , and thus neutral underthe Z symmetry. In this case, dimension 4 or 5 terms involving SM particles are allowed if Y = ± / ± / . QG , c N ∼ (cid:16) M pl Λ QG (cid:17) N − , although it is model dependent (See c.f. Refs. [81, 82]).With v PQ / Λ QG ∼ (cid:112) / π , the quality can be good enough if N (cid:38) . If the higherdimensional terms of the SM particles are generated at the scales of Λ QG , the axiondark matter with f a (cid:39) v PQ / √ N ∼ GeV implies a cutoff of Λ QG ∼ − GeV . In this scenario, the neutrino masses can be generated by the higher dimensionaloperators of c ab Λ QG H SM L a H SM L b (40)correctly for the dimensionless coefficient c ab ∼ O (1) . Here H SM and L a are the SMHiggs and lepton doublet fields with the flavor indices a, b = e, µ, τ. In this case, thebaryon number violating operators may also exist although it again depends on thedetail of the UV model (See e.g. Ref. [81–83]). To avoid the proton decays, onecan introduce a Z symmetry under which SM leptons are odd but the baryons areeven or vise versa. In this case, the baryon asymmetry may be correctly obtainedfrom neutrino oscillation with the higher dimensional term (40) [84, 85] and also thequark oscillation could be important via the baryon-number-violating but baryon-parity-conserving operator [86].Lastly, let us mention the confinement of SO( N ). The unbroken SO( N ) becomesnon-perturbative at low energy scales. However, g N is tiny, which means that the“QCD” scale Λ SO( N ) = e − π /bg N f a with b = 11( N − / , is extremely low. For N < , g N < − , f a ∼ GeV, the size of the instanton 1 / Λ SO( N ) is muchlarger than 1 /H , where H is the current Hubble parameter, and thus we canneglect the non-perturbative effect in phenomenology. An anomalous PQ symmetry solves the strong CP problem, which is a fine-tuningproblem. The solution, however, suffers from other fine-tuning problems, the qualityand hierarchy problems. Moreover, to solves the potentially existing domain wall,stable PQ fermions, and isocurvature problems, a low inflation scale may be needed,11hich may introduce another hierarchy problem.In this paper, we have studied whether the hierarchy and the quality problemsfor the PQ symmetry can be both explained by introducing a simple hidden gaugegroup which satisfies the WGC or its variant. By introducing an unbroken U(1) (cid:48) gauge symmetry under which some PQ fermions are charged, the mild version ofthe WGC can constrain the PQ scale to be below 10 GeV for the coupling g (cid:48) (cid:46) − . However, the quality problem cannot be solved. A non-abelian hidden gaugesymmetry which is partially broken down via the spontaneous PQ breaking can solvethe quality problem. In this case, according to a stronger version of the WGC, theTower/sLWGC, the cutoff scale of the field theory is reduced to be around the QCDaxion window if the coupling satisfies g N (cid:46) − . As a result the small PQ andinflation scales can be simultaneously explained. Interestingly, the seesaw scale isalso around the cutoff scale.A Λ QG < − GeV may not be compatible with the ordinary grand unifiedtheory (cf. Ref. [83]), which explains the quantized charges of the SM. However thecharge quantization can be achieved if supersymmetry restores at the scale aroundΛ QG and a slepton is the SM Higgs field [87], given the three generations.Although we have focused on the QCD axion, in general a light pNGB with asmall decay constant, has both the problems of quality and hierarchy. Our mecha-nism can also apply to a general pNGB or ALP. Then the tiny mass of the ALPcan be generated via the higher dimensional terms or the non-perturbative effect ofSO( N ) dynamics if U(1) B H -SO( N ) is anomalous and g N N is large enough. Thenon-perturbative effect, if sizable, requires an extremely large N for a small Λ QG . Note added:
While preparing this paper, we found Ref. [98] where the model withSU( N ) hidden gauge symmetry broken down to SO( N ) was discussed in the contextof quality problem. In this paper, we mainly focus on the small g N regime, andfind that due to the unbroken SO( N ) the spectra may be restricted by weak gravityconjectures. We showed that the much lower PQ scale than M pl and safe cosmologycan be obtained due to the resulting low cutoff scale. These are not discussed in [98]. Such ALP may be important in various contexts e.g. ALP inflation [88–90], and the explanationof the XENON1T excess [91] [92–97]. For instance, we may change the fermion contents, (17) to ψ α : ( N, , ˜ ψ : ( ¯ N ( ¯ N + 1) ,
1) where α = 1 · · · N + 4 , we find U(1) B H -SO( N ) anomaly is non-vanishing. Here, ˜ ψ has U(1) B H charge 2accidentally. cknowledgments The author thanks Prof. Hye-Sung Lee for discussion and reading the manuscript.This work was supported by JSPS KAKENHI Grant No. 16H06490 and by NationalResearch Foundation Strategic Research Program (NRF-2017R1E1A1A01072736).
A Alternative models with sLWGC
It was discussed that the cutoff Λ QG may be set even with the non-abelian gaugefield completely broken by higgsing [61, 99]. In this case, we can consider instead ofEqs. (17) and (18) q : ( ¯ N , r SM ) , ¯ q l : (1 , ¯ r SM ) , ψ a : ( N, . (41)and H l PQ : ( N, , (42)where l = 1 · · · N and a = 1 · · · dim[ r SM ]. The Yukawa interaction is given by L ⊃ ( y q ) ml ¯ q m H l PQ q. (43)Again we get accidental U(1) PQ where q : − , ψ a : 1 , H i PQ : 1 , ¯ q l : 0 . Supposing thatwe have a potential at the minimum (cid:10) ( H l PQ ) i (cid:11) = v i δ lj (cid:54) = 0 , we obtain SU( N ) × U(1) PQ completely broken, and an axion appears. The axion couples to the gluon with thedecay constant given by f a = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i (cid:32) v i (cid:80) j /v j (cid:33) . (44)The hierarchy are quality problems are similarly solved as in the main part.If a spontaneous broken U(1) (cid:48) symmetry is also restricted by the sLWGC, thecutoff is set by Λ QG (cid:46) ( g (cid:48) ) / M pl [61]. Then both the quality and scale of the PQsymmetry can be explained with g (cid:48) ∼ − and with certain charge assignment [48–50]. References [1] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440 (1977).doi:10.1103/PhysRevLett.38.1440 132] R. D. Peccei and H. R. Quinn, Phys. Rev. D , 1791 (1977).doi:10.1103/PhysRevD.16.1791[3] S. Weinberg, Phys. Rev. Lett. , 223 (1978). doi:10.1103/PhysRevLett.40.223[4] F. Wilczek, Phys. Rev. Lett. , 279 (1978). doi:10.1103/PhysRevLett.40.279[5] J. E. Kim, Phys. Rev. Lett. , 103 (1979). doi:10.1103/PhysRevLett.43.103[6] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B , 493(1980). doi:10.1016/0550-3213(80)90209-6[7] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. , 199 (1981).doi:10.1016/0370-2693(81)90590-6[8] A. R. Zhitnitsky, Sov. J. Nucl. Phys. , 260 (1980) [Yad. Fiz. , 497 (1980)].[9] L. F. Abbott and P. Sikivie, Phys. Lett. , 133 (1983). doi:10.1016/0370-2693(83)90638-X[10] M. Dine and W. Fischler, Phys. Lett. , 137 (1983). doi:10.1016/0370-2693(83)90639-1[11] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. , 127 (1983).doi:10.1016/0370-2693(83)90637-8[12] J. E. Kim and G. Carosi, Rev. Mod. Phys. , 557 (2010) Erratum: [Rev.Mod. Phys. , no. 4, 049902 (2019)] doi:10.1103/RevModPhys.91.049902,10.1103/RevModPhys.82.557 [arXiv:0807.3125 [hep-ph]].[13] O. Wantz and E. P. S. Shellard, Phys. Rev. D , 123508 (2010)doi:10.1103/PhysRevD.82.123508 [arXiv:0910.1066 [astro-ph.CO]].[14] A. Ringwald, Phys. Dark Univ. , 116 (2012) doi:10.1016/j.dark.2012.10.008[arXiv:1210.5081 [hep-ph]].[15] M. Kawasaki and K. Nakayama, Ann. Rev. Nucl. Part. Sci. , 69 (2013)doi:10.1146/annurev-nucl-102212-170536 [arXiv:1301.1123 [hep-ph]].[16] D. J. E. Marsh, Phys. Rept. , 1 (2016) doi:10.1016/j.physrep.2016.06.005[arXiv:1510.07633 [astro-ph.CO]].[17] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner and K. A. vanBibber, Ann. Rev. Nucl. Part. Sci. , 485 (2015) doi:10.1146/annurev-nucl-102014-022120 [arXiv:1602.00039 [hep-ex]].[18] I. G. Irastorza and J. Redondo, Prog. Part. Nucl. Phys. , 89 (2018)doi:10.1016/j.ppnp.2018.05.003 [arXiv:1801.08127 [hep-ph]].1419] L. Di Luzio, M. Giannotti, E. Nardi and L. Visinelli, arXiv:2003.01100 [hep-ph].[20] J. H. Chang, R. Essig and S. D. McDermott, JHEP , 051 (2018)doi:10.1007/JHEP09(2018)051 [arXiv:1803.00993 [hep-ph]].[21] R. Mayle, J. R. Wilson, J. R. Ellis, K. A. Olive, D. N. Schramm andG. Steigman, Phys. Lett. B , 188 (1988). doi:10.1016/0370-2693(88)91595-X[22] G. Raffelt and D. Seckel, Phys. Rev. Lett. , 1793 (1988).doi:10.1103/PhysRevLett.60.1793[23] P. W. Graham and A. Scherlis, Phys. Rev. D , no. 3, 035017 (2018)doi:10.1103/PhysRevD.98.035017 [arXiv:1805.07362 [hep-ph]].[24] F. Takahashi, W. Yin and A. H. Guth, Phys. Rev. D , no. 1, 015042 (2018)doi:10.1103/PhysRevD.98.015042 [arXiv:1805.08763 [hep-ph]].[25] S. Y. Ho, F. Takahashi and W. Yin, JHEP , 149 (2019)doi:10.1007/JHEP04(2019)149 [arXiv:1901.01240 [hep-ph]].[26] T. Tenkanen and L. Visinelli, JCAP , 033 (2019) doi:10.1088/1475-7516/2019/08/033 [arXiv:1906.11837 [astro-ph.CO]].[27] R. T. Co, E. Gonzalez and K. Harigaya, JHEP , 162 (2019)doi:10.1007/JHEP05(2019)162 [arXiv:1812.11186 [hep-ph]].[28] N. Okada, D. Raut and Q. Shafi, arXiv:1910.14586 [hep-ph].[29] G. Alonso- ´Alvarez, T. Hugle and J. Jaeckel, JCAP , 014 (2020)doi:10.1088/1475-7516/2020/02/014 [arXiv:1905.09836 [hep-ph]].[30] D. J. E. Marsh and W. Yin, arXiv:1912.08188 [hep-ph].[31] H. Matsui, F. Takahashi and W. Yin, JHEP , 154 (2020)doi:10.1007/JHEP05(2020)154 [arXiv:2001.04464 [hep-ph]].[32] S. Nakagawa, F. Takahashi and W. Yin, JCAP , 004 (2020)doi:10.1088/1475-7516/2020/05/004 [arXiv:2002.12195 [hep-ph]].[33] P. J. Steinhardt and M. S. Turner, Phys. Lett. , 51 (1983).doi:10.1016/0370-2693(83)90727-X[34] N. Kitajima and F. Takahashi, JCAP , 032 (2015) doi:10.1088/1475-7516/2015/01/032 [arXiv:1411.2011 [hep-ph]].[35] M. Kawasaki, F. Takahashi and M. Yamada, Phys. Lett. B , 677 (2016)doi:10.1016/j.physletb.2015.12.075 [arXiv:1511.05030 [hep-ph]].1536] N. Kitajima, T. Sekiguchi and F. Takahashi, Phys. Lett. B , 684 (2018)doi:10.1016/j.physletb.2018.04.024 [arXiv:1711.06590 [hep-ph]].[37] P. Agrawal, G. Marques-Tavares and W. Xue, JHEP , 049 (2018)doi:10.1007/JHEP03(2018)049 [arXiv:1708.05008 [hep-ph]].[38] J. E. Kim, Phys. Rev. D , 1733 (1985). doi:10.1103/PhysRevD.31.1733[39] K. Choi and J. E. Kim, Phys. Rev. D , 1828 (1985).doi:10.1103/PhysRevD.32.1828[40] C. W. Misner and J. A. Wheeler, Annals Phys. , 525 (1957). doi:10.1016/0003-4916(57)90049-0[41] S. M. Barr and D. Seckel, Phys. Rev. D , 539 (1992).doi:10.1103/PhysRevD.46.539[42] T. Banks and N. Seiberg, Phys. Rev. D , 084019 (2011)doi:10.1103/PhysRevD.83.084019 [arXiv:1011.5120 [hep-th]].[43] E. J. Chun and A. Lukas, Phys. Lett. B , 298 (1992) doi:10.1016/0370-2693(92)91266-C [hep-ph/9209208].[44] M. Bastero-Gil and S. F. King, Phys. Lett. B , 27 (1998) doi:10.1016/S0370-2693(98)00124-5 [hep-ph/9709502].[45] K. S. Babu, I. Gogoladze and K. Wang, Phys. Lett. B , 214 (2003)doi:10.1016/S0370-2693(03)00411-8 [hep-ph/0212339].[46] A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D , 015007 (2004)doi:10.1103/PhysRevD.69.015007 [hep-ph/0210172].[47] K. Harigaya, M. Ibe, K. Schmitz and T. T. Yanagida, Phys. Rev. D , no. 7,075022 (2013) doi:10.1103/PhysRevD.88.075022 [arXiv:1308.1227 [hep-ph]].[48] H. Fukuda, M. Ibe, M. Suzuki and T. T. Yanagida, Phys. Lett. B , 327(2017) doi:10.1016/j.physletb.2017.05.071 [arXiv:1703.01112 [hep-ph]].[49] M. Duerr, K. Schmidt-Hoberg and J. Unwin, Phys. Lett. B , 553 (2018)doi:10.1016/j.physletb.2018.03.054 [arXiv:1712.01841 [hep-ph]].[50] Q. Bonnefoy, E. Dudas and S. Pokorski, Eur. Phys. J. C , no. 1, 31 (2019)doi:10.1140/epjc/s10052-018-6528-z [arXiv:1804.01112 [hep-ph]].[51] L. Randall, Phys. Lett. B , 77 (1992). doi:10.1016/0370-2693(92)91928-3[52] L. Di Luzio, E. Nardi and L. Ubaldi, Phys. Rev. Lett. , no. 1, 011801 (2017)doi:10.1103/PhysRevLett.119.011801 [arXiv:1704.01122 [hep-ph]].1653] B. Lillard and T. M. P. Tait, JHEP , 199 (2018)doi:10.1007/JHEP11(2018)199 [arXiv:1811.03089 [hep-ph]].[54] H. S. Lee and W. Yin, Phys. Rev. D , no. 1, 015041 (2019)doi:10.1103/PhysRevD.99.015041 [arXiv:1811.04039 [hep-ph]].[55] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, JHEP , 060 (2007)doi:10.1088/1126-6708/2007/06/060 [hep-th/0601001].[56] C. Cheung and G. N. Remmen, Phys. Rev. Lett. , 051601 (2014)doi:10.1103/PhysRevLett.113.051601 [arXiv:1402.2287 [hep-ph]].[57] B. Heidenreich, M. Reece and T. Rudelius, JHEP , 140 (2016)doi:10.1007/JHEP02(2016)140 [arXiv:1509.06374 [hep-th]].[58] B. Heidenreich, M. Reece and T. Rudelius, JHEP , 025 (2017)doi:10.1007/JHEP08(2017)025 [arXiv:1606.08437 [hep-th]].[59] M. Montero, G. Shiu and P. Soler, JHEP , 159 (2016)doi:10.1007/JHEP10(2016)159 [arXiv:1606.08438 [hep-th]].[60] S. Andriolo, D. Junghans, T. Noumi and G. Shiu, Fortsch. Phys. , no. 5,1800020 (2018) doi:10.1002/prop.201800020 [arXiv:1802.04287 [hep-th]].[61] B. Heidenreich, M. Reece and T. Rudelius, Eur. Phys. J. C , no. 4, 337 (2018)doi:10.1140/epjc/s10052-018-5811-3 [arXiv:1712.01868 [hep-th]].[62] S. Hod, Int. J. Mod. Phys. D , no. 12, 1742004 (2017)doi:10.1142/S0218271817420044 [arXiv:1705.06287 [gr-qc]].[63] Z. Fisher and C. J. Mogni, arXiv:1706.08257 [hep-th].[64] M. Montero, JHEP , 157 (2019) doi:10.1007/JHEP03(2019)157[arXiv:1812.03978 [hep-th]].[65] C. Cheung, J. Liu and G. N. Remmen, JHEP , 004 (2018)doi:10.1007/JHEP10(2018)004 [arXiv:1801.08546 [hep-th]].[66] Y. Hamada, T. Noumi and G. Shiu, Phys. Rev. Lett. , no. 5, 051601 (2019)doi:10.1103/PhysRevLett.123.051601 [arXiv:1810.03637 [hep-th]].[67] J. Jaeckel, J. Redondo and A. Ringwald, Phys. Rev. D , 103511 (2014)doi:10.1103/PhysRevD.89.103511 [arXiv:1402.7335 [hep-ph]].[68] K. Kaneta, H. S. Lee and S. Yun, Phys. Rev. Lett. , no. 10, 101802 (2017)doi:10.1103/PhysRevLett.118.101802 [arXiv:1611.01466 [hep-ph]].1769] K. Choi, H. Kim and T. Sekiguchi, Phys. Rev. D , no. 7, 075008 (2017)doi:10.1103/PhysRevD.95.075008 [arXiv:1611.08569 [hep-ph]].[70] R. Daido, F. Takahashi and N. Yokozaki, Phys. Lett. B , 538 (2018)doi:10.1016/j.physletb.2018.03.039 [arXiv:1801.10344 [hep-ph]].[71] L. F. Abbott and M. B. Wise, Nucl. Phys. B , 687 (1989). doi:10.1016/0550-3213(89)90503-8[72] S. R. Coleman and K. M. Lee, Nucl. Phys. B , 387 (1990). doi:10.1016/0550-3213(90)90149-8[73] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, hep-th/0501082.[74] Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].[75] T. Yanagida, Conf. Proc. C , 95 (1979).[76] S. L. Glashow, NATO Sci. Ser. B , 687 (1980). doi:10.1007/978-1-4684-7197-7 15[77] M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C , 315 (1979)[arXiv:1306.4669 [hep-th]].[78] P. Minkowski, Phys. Lett. , 421 (1977). doi:10.1016/0370-2693(77)90435-X[79] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. , 912 (1980).doi:10.1103/PhysRevLett.44.912[80] M. Fukugita and T. Yanagida, Phys. Lett. B , 45 (1986). doi:10.1016/0370-2693(86)91126-3[81] N. Arkani-Hamed and S. Dimopoulos, Phys. Rev. D , 052003 (2002)doi:10.1103/PhysRevD.65.052003 [hep-ph/9811353].[82] N. Arkani-Hamed and M. Schmaltz, Phys. Rev. D , 033005 (2000)doi:10.1103/PhysRevD.61.033005 [hep-ph/9903417].[83] K. R. Dienes, E. Dudas and T. Gherghetta, Nucl. Phys. B , 47 (1999)doi:10.1016/S0550-3213(98)00669-5 [hep-ph/9806292].[84] Y. Hamada, R. Kitano and W. Yin, JHEP , 178 (2018)doi:10.1007/JHEP10(2018)178 [arXiv:1807.06582 [hep-ph]].[85] S. Eijima, R. Kitano and W. Yin, JCAP , 048 (2020) doi:10.1088/1475-7516/2020/03/048 [arXiv:1908.11864 [hep-ph]].1886] T. Asaka, H. Ishida and W. Yin, JHEP , 174 (2020)doi:10.1007/JHEP07(2020)174 [arXiv:1912.08797 [hep-ph]].[87] W. Yin, Phys. Lett. B , 585 (2018) doi:10.1016/j.physletb.2018.09.023[arXiv:1808.00440 [hep-ph]].[88] R. Daido, F. Takahashi and W. Yin, JCAP , 044 (2017) doi:10.1088/1475-7516/2017/05/044 [arXiv:1702.03284 [hep-ph]].[89] R. Daido, F. Takahashi and W. Yin, JHEP , 104 (2018)doi:10.1007/JHEP02(2018)104 [arXiv:1710.11107 [hep-ph]].[90] F. Takahashi and W. Yin, JHEP , 095 (2019)doi:10.1007/JHEP07(2019)095 [arXiv:1903.00462 [hep-ph]].[91] E. Aprile et al. [XENON Collaboration], arXiv:2006.09721 [hep-ex].[92] F. Takahashi, M. Yamada and W. Yin, arXiv:2006.10035 [hep-ph].[93] I. M. Bloch, A. Caputo, R. Essig, D. Redigolo, M. Sholapurkar and T. Volansky,arXiv:2006.14521 [hep-ph].[94] T. Li, arXiv:2007.00874 [hep-ph].[95] P. Athron et al. , arXiv:2007.05517 [astro-ph.CO].[96] C. Han, M. L. L´opez-Ib´a˜nez, A. Melis, O. Vives and J. M. Yang,arXiv:2007.08834 [hep-ph].[97] F. Takahashi, M. Yamada and W. Yin, arXiv:2007.10311 [hep-ph].[98] M. Ardu, L. Di Luzio, G. Landini, A. Strumia, D. Teresi and J. W. Wang,[arXiv:2007.12663 [hep-ph]].[99] M. Reece, JHEP1907