Dark energy, QCD axion, BICEP2, and trans-Planckian decay constant Dark energy, QCD axion, BICEP2, and trans-Planckian decay constant � Jihn E. Kim Department of Physics, Kyung Hee University, Seoul 130-701, Korea Abstract Discrete symmetries allowed in string compactification are the mother of all global symmetries which are broken at some level. We discuss the resulting pseudo-Goldstone bosons, in particular the QCD axion and a temporary cosmological constant, and inflatons. We also comment on some implications of the recent BICEP2 data. Keywords: Discrete symmetry, QCD axion, Dark energy, Inflation 1. Discrete symmetries The cosmic energy pie is composed of 68% dark energy (DE), 27% cold dark matter (CDM), and 5% atoms [1]. Among these, some of DE and CDM can be bosonic coherent motions (BCMs) [2]. The ongoing search of the QCD axion is based on the BCM. Being a pseudo-Goldstone boson, the QCD axion can be a com- posite one [3], but after the discovery of the fundamen- tal Brout-Englert-Higgs (BEH) boson, the possibility of the QCD axion being fundamental gained much more weight. The ongoing axion search experiment is based on the resonance enhancement of the oscillating E-field following the axion vacuum oscillation as depicted in Fig. 1. It may be possible to detect the CDM axion even its contribution to CDM is only 10% [4]. The BEH boson is fundamental. The QCD axion may be fundamental. The inflaton may be fundamen- tal. These bosons with canonical dimension 1 can af- fect more importantly to low energy physics compared to those of spin- 12 fermions of the canonical dimension 3 2 . This leads to a BEH portal to the high energy scale to the axion scale or even to the standard model (SM) singlets at the grand unification (GUT) scale. Can these �This work is supported by the NRF grant funded by the Korean Government (MEST) (No. 2005-0093841). Email address: jihnekim@gmail.com (Jihn E. Kim) singlets explain both DE and CDM in the Universe? Be- cause the axion decay constant fa can be in the interme- diate scale, axions can live up to now (ma < 24 eV) and constitute DM of the Universe. In this year of a GUT scale VEV, can these also explain the inflation finish? For pseudo-Goldstone bosons like axion, we intro- duce global symmetries. But global symmetries are known to be broken by the quantum gravity effects, es- pecially via the Planck scale wormholes. To resolve this dilemma, we can think of two possibilities of dis- crete symmetries below MP [5]: (i) The discrete sym- metry arises as a part of a gauge symmetry, and (ii) The string selection rules directly give the discrete sym- • • • • • • • • B 〈a〉� Ea Figure 1: The resonant detection idea of the QCD axion. The E-field follows the axion vacuum oscillation. Available online at www.sciencedirect.com Nuclear and Particle Physics Proceedings 273–275 (2016) 389–394 2405-6014/© 2015 Elsevier B.V. All rights reserved. www.elsevier.com/locate/nppp http://dx.doi.org/10.1016/j.nuclphysbps.2015.09.056 http://www.elsevier.com/locate/nppp http://dx.doi.org/10.1016/j.nuclphysbps.2015.09.056 http://dx.doi.org/10.1016/j.nuclphysbps.2015.09.056 http://www.sciencedirect.com Global symmetry D is c r e te sy m m e tr y Gl D Figure 2: Terms respecting discrete and global symmetries. metry. So, we will consider discrete gauge symme- tries allowed in string compactification. Even though the Goldstone boson directions in spontaneously bro- ken gauge symmetries are flat, the Goldstone boson di- rections of spontaneously broken global symmetries are not flat, i.e. global symmetries are always approximate. The question is what is the degree of the approximate- ness. In Fig. 2, we present a cartoon separating effective terms according to string-allowed discrete symmetries. The terms in the vertical column represent exact sym- mmetries such as gauge symmetries and string allowed discrete symmetries. If we consider a few terms in the lavender part, we can consider a global symmetry. With the global symmetry, we can consider the global sym- metric terms which are in the lavender and green parts of Fig. 2. The global symmetry is broken by the terms in the red part. The most studied global symmetry is the Peccei- Quinn (PQ) symmetry U(1)PQ [6]. For U(1)PQ, the dominant breaking term is the QCD anomaly term (θ/32π2)GμνG̃μν where Gμν is the gluon field strength. Since this θ gives a neutron EDM (nEDM) of order 10−16θecm, the experimental upper bound on nEDM re- stricts |θ| < 10−11. “Why is θ so small?” is the strong CP problem. There have been a few solutions, but the remaining plausible solution is the very light axion so- lution [7]. In field theory, it is usually talked about in terms of the KSVZ axion [8] and the DFSZ axion [9], and there are several possibilities even for these one heavy quark or one pair of BEH doublets [10]. For axion detection through the idea of Fig. 1, the axion- photon-photon coupling caγγ is the key parameter. In our search of an ultra-violet completed theory, the cus- tomary numbers of [10] are ad hoc. From string theory, so far there is only one calculation on caγγ [11]. To cal- culate caγγ, the model must lead to acceptable SM phe- nomenology, otherwise the calculation does not lead to a useful global fit to all experimental data. 2. Dark energy and QCD axion It is interesting to note that the QCD axion must arise if one tries to introduce the DE scale via the idea of Fig. 2 [12, 13]. The DE and QCD axions are the BCM ex- amples. Dark energy is classified as CCtmp and QCD axion is classified as BCM1 in [2]. Note that the global symmetry violating terms belong to the red part in Fig. 2. For the QCD axion, the domi- nant breaking is by the QCD anomaly term, which leads to the QCD axion mass in the range of milli- to nano- eV for fa � 109−15 GeV. In the BEH portal scenario, the DE pseudoscalar must couple to the color anomaly since it couples to the BEH doublet and the BEH scalar couples to the quarks. On the other hand, a CCtmp psudoscalar mass is in the range 10−33 ∼ 10−32 eV [14]. Therefore, the QCD anomaly term is too large to ac- count for the DE scale of 10−46 GeV4, and we must find out a QCD-anomaly free global symmetry. It is possible by introducing two global U(1) symmetries [12, 13]. In addition, the breaking scale of U(1)de is trans- Planckian [14]. Including the anharmonic term care- fully with the new data on light quark masses, a recent calculation of the cosmic axion density gives the axion window [15], 109 GeV < fa < 10 12 GeV. (1) It is known that string axions from BMN have GUT scale decay constants [16]; hence the QCD axion from string theory is better to arise from matter fields [17]. For the QCDaxion, the height of the potential is ≈ Λ4QCD. For the DE pseudo-Goldstone boson, the height of the po- tential is ≈ M4GUT, according to the BEH portal idea, Im (Φ) Re (Φ) V (Φ) fDE 10−46 GeV4 � O(M4G) • Figure 3: The DE potential in the red angle direction in the valley of radial field of height ≈ M4GUT. J.E. Kim / Nuclear and Particle Physics Proceedings 273–275 (2016) 389–394390 as shown in Fig. 3. With U(1)PQ and U(1)de, one can construct a DE model from string compactification [13]. Using the SUSY language, the discrete and global sym- metries below MP are the consequence of the full super- potential W. So, the exact symmetries related to string compactification are respected by the full W, i.e. the ver- tical column of Fig. 2. Considering only the d = 3 superpotential W3, we can consider an approximate PQ symmetry. For the MSSM interactions supplied by R- parity, one needs to know all the SM singlet spectrum. We need Z2 for a WIMP candidate. Introducing two global symmetries, we can remove the U(1)de-G-G where G is QCD and the U(1)de charge is a linear combination of two global symmetry charges. The decay constant corresponding to U(1)de is fDE. In- troduction of two global symmetries is inevitable to in- terpret the DE scale and hence in this scenario the ap- pearance of U(1)PQ is a natural consequence. The height of DE potential is so small, 10−46 GeV4, that the needed discrete symmetry breaking term of Fig. 2 must be small, implying the discrete symmetry is of high order. Now, We have a scheme to explain both 68% of DE and 27% of CDM via approximate global symmetries. With SUSY, axino may contribute to CDM also [18]. A typical example for the discrete symmetry is Z10 R as shown in [13]. The Z10 R charges descend from a gauge U(1) charges of the string compactification [19]. Then, the height of the potential is highly suppressed and we can obtain 10−47 GeV4, without the gravity spoil of the global symmetry. In this scheme with BEH por- tal, we introduced three scales for vacuum expectation values (VEVs), TeV scale for Hu Hd , the GUT scale MGUT for singlet VEVs, and the intermediate scale for the QCD axion. The other fundamental scale is MP. The trans-Planckian decay constant fDE can be a derived scale [20]. Spontaneous breaking of U(1)de is via a Mexican hat potential with the height of M4GUT. A byproduct of this Mexican hat potential is the hilltop inflation with the height of O(M4GUT), as shown in Fig. 3. It is a small field inflation, consistent with the 2013 Planck data. 3. Gravity waves from U(1)de potential However, with the surprising report from the BICEP2 group on a large tensor-to-scalar ratio r [21], we must reconsider the above hilltop inflation whether it leads to appropriate numbers on ns,r and the e-fold number e, or not. With two U(1)’s, the large trans-Planckian fDE is not spoiled by the intermediate PQ scale fa because the PQ scale just adds to the fDE decay constant only by a tiny amount, viz. fDE → √ f 2DE + O(1) × f 2a ≈ fDE for | fa/ fDE � 10−7|. Inflaton potentials with almost flat one near the ori- gin, such as the Coleman-Weinberg type new inflation, were the early attempts for inflation. But any models can lead to inflation if the potential is flat enough as in the chaotic inflation with small parameters [22]. A single field chaotic inflation survived until now is the m2φ2 scenario chaotic inflation with m = O(1013 GeV). To shrink the field energy much lower than M4P, a nat- ural inflation (mimicking the axion-type − cos poten- tial) has been introduced [23]. If a large r is observed, Lyth noted that the field value 〈φ〉 must be larger than 15 MP, which is known as the Lyth bound [24]. To obtain this trans-Planckian field value, the Kim-Nilles- Peloso (KNP) 2-flation has been introduced with two axions [20]. It is known recently that the natural infla- tion is more than 2σ away from the central value of BI- CEP2, (r,ns) = (0.2,0.96). In general, the hilltop infla- tion gives almost zero r. This is because ns � 1− 38 r+2η which gives ns = 0.925 for (r,η) = (0.2,0). To raise ns from 0.925 to 0.96, we need a positive η, but the hilltop point gives a negative η. Therefore, for the U(1)de hilltop inflation to give a suitable ns with a large r, one must introduce another field which is called chaoton because it provides the be- havior of m2φ2 term at the BICEP2 point [25]. With this hilltop potential, the height is of order M4GUT and the decay constant is required to be > 15 MP. Cer- tainly, the potential energy is smaller than order M4P for φ = [0, fDE]. Since this hilltop potential is obtained from the mother discrete symmetry, such as Z10 R, the flat valley up to the trans-Planckian fDE is possible, for which the necessary condition is given in terms of quan- tum numbers of Z10 R [25]. 4. The KNP model and U(1)de hilltop inflation A large VEV of a scalar field is possible if a very small coupling constant λ is assumed in V = 14λ(|φ|2 − f 2)2 with a small mass parameter m2 = λ f 2. With a GUT scale m, f can be trans-Planckian of order 10MP for λ < 10−6. But this potential is a single field hilltop type and it is not favored by the above argument with the BICEP2 data [25]. This has led to the recent surge of studies on concave potentials near the origin of the single field. The concave potentials give positive η’s. To cut off the potential exceeding the GUT scale M4GUT, the natural inflation with a GUT scale confin- ing force has been introduced [23]. With two confin- ing forces, it was possible to raise a decay constant of J.E. Kim / Nuclear and Particle Physics Proceedings 273–275 (2016) 389–394 391 the GUT scale axions above MP, which is known as the Kim-Nilles-Peloso (KNP) 2-flation model [20]. In terms of two axions a1 and a2 and two GUT scale (Λ1 and Λ2) confining forces, the minus-cosine potentials can be written as V = Λ41 ( 1 − cos [ α a1 f1 +β a2 f2 ) ]) + Λ 4 2 ( 1 − cos [ γ a1 f1 +δ a2 f2 ) ]) , (2) where α,β,γ, and δ are determined by two U(1) quan- tum numbers. If there is only one confining force, we can set Λ2 = 0 in Eq. (2), which is depicted in Fig. 4 (a). The flat red valley cannot support the inflation energy. The situation with two confining forces is shown in Fig. 4 (b). The inflation path is shown as the arrowed blue curve on top of the red valley on the yellow roof. In this case, we consider a 2 × 2 mass matrix, M2 = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 f 21 ( α2Λ41 +γ 2Λ42 ) , 1f1 f2 (αβΛ41 +γδΛ 4 2) 1 f1 f2 (αβΛ41 +γδΛ 4 2), 1 f 22 (β2Λ41 +δ 2Λ42) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ • • −π −π 2 0 ah/fah aI/faI π 2 π 0 π 2π (a) • −π −π 2 0 ah/fah aI/faI π 2π (b) Figure 4: Two-flation. (a) The flat valley with one confining force, and (b) the KNP model with two confining forces. whose eigenvalues are [26], m2h = 1 2 (A + B) and m2I = 1 2 (A − B) . (3) with A = ⎛⎜⎜⎜⎜⎝α2Λ41 +γ2Λ42 f 21 + β2Λ41 +δ 2Λ42 f 22 ⎞⎟⎟⎟⎟⎠ , B = √ A2 − 4(αδ−βγ)2Λ 4 1Λ 4 2 f 21 f 2 2 . (4) From Eq. (4), we note that a large fI is possible for αδ= βγ+Δ with Δ ≈ 0. Then, the inflaton mass is m2I � Δ2Λ41Λ 4 2 f 22 (α 2Λ41 +γ 2Λ42) + f 2 1 (β 2Λ41 +δ 2Λ42) . For Λ1 =Λ2 and f1 = f2 ≡ f , it becomes m2I � Λ4 (α2 +β2 +γ2 +δ2) f 2/Δ2 . (5) The PQ quantum numbers α,β,γ, and δ are not random priors, but given definitely in a specific model. The per- spectives of 2- and N-flations are given in [26]. Even though a large trans-Planckian decay constant is in principle possible with large PQ quantum numbers in the KNP model, string compactification may not al- low that possibility. The N-flation with a large N has a more severe problem in string compactification [26]. This invites to look for another possibility of generating trans-Planckian decay constants. Since the KNP model already introduced two axions, we look for a possibility of introducing another field (called chaoton before) in the hilltop potential. In effect, the chaoton is designed to provide a positive η. The hilltop potential of Fig. 3 is a Mexican hat poten- tial of U(1)de, i.e. obtained from some discrete symme- try, allowed in string compactification [12]. The discrete symmetry may provide a small DE scale. The trans- Planckian decay constant, satisfying the Lyth bound, is obtained by a small coupling λ in the hilltop potential V . The requirement for the vacuum energy being much smaller than M4P is achieved by restricting the inflaton path in the hilltop region, 〈φ〉 � fDE, as shown in Fig. 5. In Fig. 5, the inflation path affected by chaoton is depicted as the green path. We can compare this hilltop inflation assisted by chaoton with the m2φ2 chaotic inflation. The hilltop in- flation is basically a consequence of discrete symme- tries [5, 12, 13] , allowed in string compactification. If some conditions are satisfied between the discrete quantum numbers of the GUT scale fields and trans- Planckian scale fields, the hilltop potential of Fig. 5 can J.E. Kim / Nuclear and Particle Physics Proceedings 273–275 (2016) 389–394392 φ MP O(M4P) V • 〈φ〉 fDE O(M4GUT) Figure 5: The trans-Planckian decay costant in the hilltop inflation. result [25]. On the other hand, the m2φ2 chaotic infla- tion does not have such symmetry argument, and lacks a rationale forbidding higher order φn terms. This ar- gument was used to forbid many interesting theories by considering the observed slow-roll parameter η from in- flation assumption [27]. But the situation is much worse here than Lyth’s case. For example, for an n = 104 term for the trans-Planckian field Φ and the GUT scale field φ, one must fine-tune the coupling 1 out of 10127 for the trans-Planckian singlet VEV of order 〈Φ〉 ≈ 31MP [25]. 5. PQ symmetry breaking below HI Cosmology of axion models was started in 1982– 1983 [28] with the micro-eV axions [8, 9]. The needed axion scale given in Eq. (1), far below the GUT scale, is understood in models with the anomalous U(1) in string compactification [29]. In addition to the scale problem, there exists the cosmic-string and domain wall (DW) problem [31, 32]. Here, I want to stress that the ax- ion DW problem has to be resolved without the dilution effect by inflation. The BICEP2 finding of “high scale inflation at the GUT scale” implies the reheating temperature after in- flation � 1012 GeV. Then, studies on the isocurvature constraint with the BICEP2 data pin down the axion mass in the upper allowed region [33]. But this axion mass is based on the numerical study of Ref. [34] which has not included the effects of axion string-DW annihi- lation by the Vilenkin-Everett mechanism [31]. In Fig. 6, we present the case for NDW = 2. Topological defects are small balls ((a) and (b)), whose walls separarte θ= 0 and θ= π vacua, and a horizon scale string-wall system. Collisions of small balls on the horizon scale walls do not punch a hole, and the horizon size string-DW system is not erased ((c) and (d)). Therefore, for NDW ≥ 2 axion models, there exists the cosmic energy crisis problem of (a) (b) (c) (d) Figure 6: Small DW balls ((a) and (b), with punches dshowing the inside blue-vacuum) and the horizon scale string-wall system ((c) and (d)) for NDW = 2. Yellow walls are θ = 0 walls, and yellow-green walls are θ= π walls. Yellow-green walls of type (b) are also present. the string-DW system. In Fig. 7, we present the case with NDW = 1. Topological defects are small disks and a horizon scale string-DW system ((a)). Collisions of small balls on the horizon scale walls punch holes ((b)), and the holes expand with light velocity. In this way, the string-wall system is erased ((c)) and the cosmic energy crisis problem is not present in NDW = 1 axion models [37], for example with one heavy quark in the KSVZ model. If the horizon-scale string-DW system is absent, there is no severe axion DW problem. So, with the BICEP2 report, it became of utmost im- portance to obtain NDW = 1 axion models. The first try along this line was the so-called Lazarides-Shafi mech- anism, using the center (discrete group) of GUT gauge groups [35]. A more useful discrete group is a discrete subgroup of continuous U(1)’s, i.e. the discrete points of the longitudinal Goldstone boson directions of gauged U(1)’s [36]. In string theory, the anomalous gauged U(1) is useful for this purpose [29]. This solution has (a) (b) (c) Figure 7: The horizon scale string-wall system with NDW = 1. Any point is connected to another point, not passing through the wall. . J.E. Kim / Nuclear and Particle Physics Proceedings 273–275 (2016) 389–394 393 |g a γ |[G eV − 1 ] maxion[eV] 10−12 10−11 10−10 10−9 10−8 10−7 10−4 10−3 10−2 10−1 1 10 Tokyo 08 CAST Phase I CAST II 4He 3He Tokyo helioscope Lazarus et al. HB stars DAMA SOLAX, COSME H D M 10−15 10−16 IAXO plan 10−310−410−510−6 BICEP2 A xi o n m o d el s< [K S V Z ( e( Q )= 1) ] K S V Z ( e( Q )= 0) D F S Z ( d ,e ) u n if . H K K f lip -S U (5 ) Figure 8: The gaγ(= 1.57 × 10−10 caγγ) vs. ma plot [40]. been recently obtained in Z12−I orbifold compactifica- tion [38]. The QCD-axion string-DW problem may not appear at all if the hidden-sector confining gauge theory con- spire to erase the hidden-sector string-DW system [39]. Here, we introduce just one axion, namely through the anomalous U(1) gauge group, surviving down to the ax- ion window as a global U(1)PQ. Here, we introduce two kinds of heavy quarks, one the SU(Nh) heavy quark Qh and the other SU(3)QCD heavy quark q. Then, the type of Fig. 7 is present with two kinds of walls: one of Λh wall and the other of ΛQCD wall. But, at T ≈ Λh only Λh wall is attached. At somewaht lower tempera- ture Ter (<Λh) the string-DW system is erased à la Fig. 7. The height of the Λh wall is proportional to mQhΛ 3 h with mQh = f 〈X〉. The VEV 〈X〉 is temperature depen- dent, and it is possible that 〈X〉 = 0 below some critical temperature Tc (< Ter). Then, the Λh wall is erased be- low Tc, and at the QCD phase transition only the QCD wall is present. 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