Dark energy from approximate U(1)_{de} symmetry

The PLANCK observation strengthens the argument that the observed acceleration of the Universe is dominated by the invisible component of dark energy. We address how this extremely small DE density can be obtained in an ultraviolet completed theory. From two mass scales, the grand unification scale M_G and the Higgs boson mass, we parametrize this dark energy(DE). To naturally generate an extremely small DE term, we introduce an almost flat DE potential of a pseudo-Goldstone boson of an approximate global symmetry U(1)_{de} from some discrete symmetries allowed in an ultraviolet completed theory. For the DE potential to be extremely shallow, the pseudo-Goldstone boson is required not to couple to the QCD anomaly. This fixes uniquely the nonrenormalizable term generating the potential suppressed by M_G^7 in supergravity models.


I. INTRODUCTION
Recent observations [1] continue to support the existence of dark energy(DE), which is invisible to any method of detection except in Einstein's evolution equation [2] of the Universe. The evidence for DE has been taken into account since the 1998 observation of the accelerated expansion of the Universe [3]. The current DE under this hypothesis is extremely small compared to the fundamental energy density of gravity M 4 P , where M P ≃ 2.44 × 10 18 GeV. Even though there exist several attempts to interpret the accelerated expansion of the Universe, so far none of their parameters is explained from first principles. Therefore, any good idea shedding light on this extremely small magnitude of DE is welcome. In this paper, we suggest such a magnitude from a relation of the recently discovered Higgs boson mass [4] compared to the Planck mass M P . This relation leading to the small DE is drawn from the potential energy of an extremely light pseudo-Goldstone boson originating from a discrete symmetry principle in an ultraviolet complete theory [5][6][7][8][9][10][11]. Such ultraviolet completions with abundant discrete symmetries can be found in the framework of explicit (heterotic) string theory constructions in the sprit of [12][13][14][15][16][17][18]. They provide the quantum-gravity safe discrete symmetries as basis of an approximate global symmetry U(1) de for generating the DE potential.
At present, DE is the dominant form of cosmic energy, constituting roughly 68 %, compared to 27 % CDM density [1]. The DE component drives the accelerated expansion in the ΛCDM cosmology. But, at earlier times the DE was negligible compared to the CDM or radiation energies. When CDM was the dominant component, the cosmological scale factor a(t) grew as the 2/3 power law of the cosmic time, ∝ t 2/3 , and the current acceleration was driven by comparing the data with the 2/3 power law [3]. It is the 'coincidence puzzle' in particle cosmol-ogy why the DE density has overcome the CDM energy density quite recently in the cosmic time scale. In this paper, we will not attempt to attack this problem of coincidence puzzle, but the DE solution along our line of reasoning will be related to the axion CDM density.
In Fig. 1 (a), we present a picture for the potential energy densities responsible for the time evolution of the Universe. The height of the green potential energy represents DE. The simplest form of DE is the cosmological constant (CC) in Einstein's equation [2]. The theoretical CC problem has been to understand why the CC is zero at the vacuum where all equations of motion are satisfied. This vacuum is indicated by the thick lavender arrow in Fig. 1(a). Even though we do not yet have a good theory for understanding the true vacuum with the vanishing CC, it is still meaningful to assume that the CC is zero at the true vacuum [19]. Then, the observed DE is evanescent, eventually converting into φ de oscillations as depicted by the green curves in Fig. 1. The inflaton field Φ inflaton is responsible for the inflationary period in the very early Universe and the quintessential pseudoscalar field φ de [20][21][22] is responsible for the recent accelerated expansion of the Universe by making the decay constant very large M P . Thus our discussion is based on the idea of a quintessential axion in the spririt of refs. [20,21]. We assume in this paper that the magnitude of DE is given by the height of the green shaded area of the quintaxion potential in Fig. 1 (a). This gives us an interesting connection to the idea of the mechanism of an inflationary expenasion of the early universe. We may identify Φ inflaton of Fig. 1(a) as Φ of Fig. 1(b) by closing the green "line" of (a) to a circle. In this case, a scenario of the type of natural-inflation [23] will appear, which could lead to hilltop inflation (at the hilltop of a Mexican hat potential). Natural inflation is based on a cosine potential, and ours uses the quadratic potential as a leading term. They are very similar be- The red φ de potential with the height somewhat larger than 10 −47 GeV 4 is shown for NDW = 2, by closing the green of (a) as a circle. In this case, the 'inflaton' can be Φ, breaking U(1) de , realizing a type of natural inflation [23] in our scheme.
cause both of them give the common features for ∂V /∂v and ∂ 2 V /∂v 2 near v ≈ 0 where v is the inflaton direction, i.e. in our case v = |Φ|. They are very similar because natural inflation uses the shift symmetry of (a nonlinearly realized field) a inflaton → a inflaton + constant and the hilltop inflation uses the linear realization of U(1) de . If natural inflation is based on a confining force at a GUT scale, a inflaton does not lead to small-field inflation since the origin is the minimum as in the familiar QCD axion case. On the other hand, hilltop inflation may be a small field inflation always due to the high temperature effects before the spontaneous symmetry breaking of U(1) de , as depicted as the bullet in Fig. 1(b). Another difference is that the inflaton in natural inflation is probably a pseudoscalar field while the inflaton in hilltop inflation is a scalar field. Apart from theses applications for QCD-axions and inflation, the focus of this paper is on the dynamical origin of dark energy. Its main ingredient is an ultra light quintessential axion with an axion decay constant of order of the string or GUT scale. The height of the very shallow potential is the source of dark energy [20][21][22].
The scale of the potential is protected by an approximate U(1) de , that is derived from discrete symmetries in consistent string theory constructions [12][13][14][15][16][17][18]. These symmetries are of (discrete) gauge symmetry origin and will not be violated by gravitational quantum effects.

II. MASS PARAMETERS FROM FUNDAMENTAL SCALARS
The fundamental mass parameter at the Planck time is the reduced Planck mass M P ≃ 2.44 × 10 18 GeV which is related to Newton's gravitational constant G = 1/8πM 2 P . in addition we will introduce the GUT mass M G ≈ 0.01 M P for the parameter of suppression for higher dimensional operators. This is because our hypothetical fields are arising from some GUT multiplets, and if both the gravity contribution and the GUT contribution to the interactions are present then the GUT contribution dominates. The difficulty in any attempt to understand the magnitude of DE is its smallness compared to the energy scale at the Planck time, i.e. ∼ M 4 P . As a result of discovering the Higgs boson at M h ≃ 1 2 v ew [4], a second fundamental mass parameter, v ew originating from bosons, is now known to exist. v ew is the vacuum expectation value(VEV) of the Higgs scalar at about 246 GeV. In terms of v ew , all the known masses of the quarks and leptons are explained with suitable Yukawa couplings in the standard model(SM) of particle physics. Namely, v ew provides all the masses of the SM, including W ± and Z 0 . Therefore, if DE can be calculated at all in terms of scalar VEVs, its simplest form is expressible in terms of two mass scales, The intermediate scale M int ≃ √ v ew M G is parametrically dependent on M P and v ew , and later the axion scale (about 100 times M int [24]) will be used for it to include all the QCD anomaly couplings. If W/Z had obtained mass by the technicolor idea, the simple mass parameter to use at the TeV scale would have been the technicolor scale Λ 3 , which will be very difficult to be implemented in our parameter fitting.
To compare our suggestion with some well-known suggestions for the acceleration of the Universe, we briefly comment how ours will be different from others in that the quantum-gravity safe discrete symmetries are employed or not. It is summarized in Table I. We start with the so-called modified Newtonian dynamics(MOND) because it is most dramatically contrasted to our DE solution of the accelerating Universe.
One obvious attempt to explain the recent acceleration is the MOND. In MOND, Newton's law is changed by introducing an acceleration parameter a 0 ≃ 1.2 × 10 −8 cm s −2 at the cosmic scale where the measured acceleration was reported. With this, the rotation curves of most galaxies can be explained without the need for CDM [25]. But, MOND fails to explain DM at the cluster scale of galaxies and more importantly the primordial production of light elements, 2 H, 3 He, and 7 Li. Then, MOND also needs the CDM component for nucleosynthesis. Comparing MOND and ΛCDM cosmology in deriving their input parameters from scalar masses in an ultraviolet completed theory, it may be more difficult to obtain a 0 in a MOND [33] than to obtain a reasonable DE scale in a ΛCDM cosmology as shown in this paper.
Since the DE amount is only about 2.5 times that of CDM, the coincidence, "Why is the amount of DE comparable to the amount of matter today?", is intriguing and attempted to be understood by changing the equation of state, most probably via the potential energy of a scalar field [28][29][30][31]. In this regard, the (nonlinearly realized) dilaton was suggested and the dilatonic symmetry was assumed with the spontaneous symmetry breaking scale at ∼ M P [27,32]. The explicit breaking scale of the dilatonic symmetry is via the dimensional transmutation of asymptotically free theories, but the dimensional transmutation scale is not a mass parameter of scalar fields. v ew determined from mass parameters of scalar fields cannot serve as an explicit symmetry breaking of the dilatonic symmetry because it is another VEV breaking the dilatonic symmetry. Another feature of dilaton toward a DE solution is changing Newton's constant, which is unsatisfactory at the moment. In addition, we do not find any discussion on a basic discrete symmetry which we adopt here.
There has been a tremendous effort to understand the DE scale in the ΛCDM cosmology, not changing Newtonian dynamics. The simplest account of DE in this direction is the CC itself, but there is a theoretical difficulty to consider an extremely small CC as commented above that there is not yet a self-tuning solution towards a vanishing CC [34][35][36]. So, if the CC itself is considered as the observed DE, the anthropic bound ρ DE < 550 ρ CDM ≃ (5 × 10 −3 eV) 4 [26] is the most plausible argument.
In Table I, naturalness in the second column is judged from the possibility of obtaining it from a symmetry prin-

Models
References Naturalness Top-down scale MOND [25] No  ciple, in particular from a discrete symmetry principle. We shall explore the possibility to obtain all relevant mass parameters from two mass scales, M G ≈ 0.01 M P and v ew .

III. U(1)DE AND GOLDSTONE BOSON
Thus, we attempt to introduce a flat potential first and then raise it by a tiny amount. For this, it is necessary to introduce a massless particle in the first step. The most plausible theory obtaining an extremely light particle is to trigger spontaneous symmetry breaking of a global U(1) symmetry, leading to a massless Goldstone boson with parity −1 [37]. It appears in the imaginary exponent of the VEV generating complex scalar field, (v + ρ)e iθ . The VEV, f DE of Fig. 1(b), of this complex scalar is taken at the Planck scale and the explicit breaking term of the global U(1) symmetry, making it a pseudo-Goldstone boson, is at the observed DE scale, somewhat bigger than 10 −47 GeV 4 [20]. Even though the height of Fig. 1 (b) is of order M 4 G , the decay constant f DE can be trans-Planckian via a small quartic coupling λ 2 ( 10 −4 ) of the U(1) de breaking field Φ, Here, we do not have the difficulty encountered for the case of B MN in raising their decay constants to M P . String theory has the gravity multiplet in 10-dimensions (M, N = 0, 1, 2, · · · , 9): the symmetric tensor field g MN which contains graviton, the dilaton, and the antisymmetric tensor field B MN . When six of ten dimensions are compactified, B MN leads to pseudoscalar fields, the model-independent(MI) pseudoscalar for {M, N } = {0, 1, 2, 3} and model-dependent(MD) pseudoscalar for {M, N } = {4, · · · , 9}. These were considered before for the DE called 'quintessential axion' [21]. It is known that the quintessential axion has a difficulty for making its potential extremely flat unless massless quarks are introduced [38]. Probably, the MD axions obtain nonnegligible superpotential terms [39], which is the reason we usually neglect these at low energy. For the MI axion, it can become the phase of a U(1) PQ transformation below the string scale if the compactification leads to the anomalous U(1). The very light QCD axion from string theory is usually based on this scheme [40]. Because the decay constants of B MN turn out to be somewhat reduced from M P [41], there is a difficulty obtaining a trans-Planckian decay constants for B MN , and hence 2-flation [42] and N-flation [43] have been considered. Because φ de in our case is not coming from B MN but rather from matter fields of the E 8 ×E ′ 8 representations of the heterotic string in the models considered here [12][13][14][15][16][17][18], we can easily obtain a trans-Planckian f DE by an O(10 −2 ) coupling. Thus, the spontaneous symmetry breaking scale of U(1) de is can be of trans-Planckian as depicted in Fig. 1(b). In Eq. (3), M int is assumed to break also the U(1) de symmetry. Since the height of the GUT scale breaking M 4 G of U(1) de is smaller than M 4 P by a factor of ∼ 10 −8 , there is not much gravity interference of the f DE determination. A generic global symmetry is generically spoiled by gravitational effects [9,44,45], but here we use (discrete) gauge symmetries from string theory that do not suffer from this problem.
In this paper we introduce a 'Dark Energy peudoscalar boson'(DEPS) which does not couple to the QCD (and hidden-sector non-Abelian anomalies, if present). The corresponding approximate global symmetry is called U(1) de , and the DEPS φ de is the corresponding pseudo-Goldstone boson. The first step toward obtaining the DE scale of the Universe is to have an exactly massless Goldstone boson φ de (def) , with superscript (def) meaning the massless Goldstone boson [37] from the U(1) de -defining terms.
To relate the DEPS to v ew , it is necessary to couple it to the Higgs fields H u H d , which implies that U(1) de has a QCD anomaly because H u and H d couple to quarks. Therefore, to have a QCD-anomaly free U(1) de , it is necessary to introduce two U(1) global symmetries, U(1) de and U(1) PQ in our case so that one anomalyfree combination results. The anomaly-free combination is our DEPS. The remaining one is the QCD axion. So, in our discussion the appearance of the QCD axion is inescapable. In the next step we break the anomaly free symmetry slightly to obtain a pseudo-Goldstone boson φ de that feebly contributes to the vacuum energy via the terms in the red part of Fig. 1(b).

IV. EXACT DISCRETE SYMMETRY AND PSEUDO-GOLDSTONE BOSON
We propose to use suitable discrete symmetries toward obtaining our desired approximate global symmetry U(1) de . Even before the 1998 discovery of the accelerating Universe, discrete symmetries were considered for obtaining some approximate global symmetry [46], but being before the 1998 discovery, only a general setup could be given. Ours is the first serious one using a discrete symmetry towards obtaining a (transient) non-vanishing CC, and along the way we have already commented on a new idea of hilltop inflation. Of course, the hypothetical discrete symmetry must satisfy the discrete gauge symmetry rule [7], as it happens in the string model constructions considered here.
Gauge symmetries are not spoiled by gravitational interactions. If a discrete symmetry results from a subgroup of gauge symmetries of string compactifications, gravity does not spoil the discrete symmetry [9]. One can consider a series of interaction terms allowed by the discrete symmetry. This infinite tower of terms, not spoiled by gravity, is shown as the vertical red bar in Fig. 2. If one considers a few lowest order terms of the red column, we can find an accidental global symmetry. Using this global symmetry, one can consider an infinite series of terms as marked in the horizontal green bar in Fig. 2. The terms shown in the lavender part of the vertical column, containing the U(1) de defining terms, satisfy both the discrete and global symmetry transformations. But, the horizontal green bar terms outside the lavender are spoiled by gravity and hence we will not consider them. The vertical red bar terms outside the lavender are not spoiled by gravity, but break the global symmetry. This red part is the source for 10 −47 GeV 4 , making the Goldstone boson the pseudo-Goldstone boson and generating the DE scale.
We identify the U(1) PQ as the anomalous U(1) of string theory for the QCD axion [40] which is spontaneously broken at the intermediate scale. Since we will require the U(1) de not carrying the anomaly including the anomalous U(1), the spontaneous symmetry breaking scale is generically around the Planck scale, M P , as commented above. This is the picture by which we introduce the height of the φ de potential.
where we used M G ≈ 2.5 × 10 16 GeV. The DEPS φ de originates from the complex scalars χ (0) and χ (0) whose VEVs are comparable to the axion decay constant of order 10 11−12 GeV related to the QCD axion. So, χ 0 /M int is of order 10 2 .
Including the soft supersymmetry breaking A-term containing one factor of m 3/2 [47,48], we need an odd number of M G suppression factor so that the resulting potential is split into two groups with the equal number of scalar fields. So, 10 −47 GeV 4 is parametrically expressible in terms of v ew and M G as where m 3/2 is the TeV scale gravitino mass. For m 3/2 v 3 ew ≈ 10 8 GeV 4 and v ew /M G ≃ 10 −14 , this height is roughly 10 −44 GeV 4 . There is some unknown factor in m 3/2 v 3 ew and hence the M −5 G suppression is considered as an adequate one. With M −3 G suppression, the potential is too large compared to 10 −47 GeV 4 , and we would have gone through DE domination much earlier, which would not correspond to the current universe. With M −7 G suppression, the potential is too shallow to have any effect on the recent history of the Universe. In any case, there are three relevant suppression factors, M −1,−3,−5 G . Out of these, M −1 G is responsible for the µ term and the axion [8,49].
In fact, within a scheme based on supergravity, we can consider effective superpotential terms ordered in powers of 1/M G , Here, W (4) defines the PQ symmetry which is explicitly broken by the QCD anomaly. Our definition of the PQ symmetry isà la Ref. [49], H u H d XX/M G with singlet scalar fields X and X. W (4) also contains the Weinberg operator H 2 u ℓℓ/M G where ℓ is the lepton doublet in the SM. The U(1) de symmetry is given by W (6) . To forbid U(1) de symmetric terms in W (4,5) , we typically need dicrete symmetries of large order N as e.g. Z N R . A realization of the Weinberg operator in terms of renormalizable terms is the seesaw model [50]. A realization of the Kim-Nilles operator in terms of renormalizable terms is the "invisible" axion model [51]. For the roles of W (4,5,6) toward the cosmology of pseudo-Goldstone bosons, W (4) defines the QCD axion. W (5) , if present, would not have led to our Universe because of too much CDM without DE at present. W (7) , even if present, would not have affected our Universe so far. Therefore, only the cases W (5) and W (6) are relevant for our discussion of DE. Anthropic arguments might be used to select from the landscape of discrete symmetries of the underlying models, resulting in pseudo-Goldstone bosons from W (5) and W (6) . Being discrete, it is quite possible that an O(0.1) fraction of the allowed models forbids W (5) and chooses W (6) . W (10) contains the term suppressed by M 7 G for Fig. 1 (b).
In the following we will sketch the qualitative picture of our mechanism. We need discrete symmetries of large order. The models involve a large number of fields and a complete definition of the model is beyond the scope of this paper. To see the details of the model, please consult ref. [52], where all symmetries are displayed. The basic picture considers the coupling to H u H d , as in the generation of the axion scale in [49] through H u H d XX/M G , but here we have to consider larger symmetries and higher powers of the superpotential. For this purpose we introduce new fields χ 0 and χ 0 (not to be confused with the X and X discussed in [49]). If in the superpotential j factors of H u H d (i.e. v ew 2 ) are replaced by 2j factors of χ 0 χ 0 , with χ 0 /M int ∼ 10 2 , we obtain an enhancement factor of 10 4j . Thus, two powers of v ew (i.e. j = 2) in Eq.
Anyway, relating χ (0) χ (0) to v ew is needed to make DEPS not couple to the QCD anomaly. This height represents the breaking of U(1) de and is generated by the red part terms of Fig. 2. Let this be composed of 2n external lines (2 fermion lines and (2n − 2) boson lines) in SUSY with dimension (2n + 1) since there are two external fermion lines. So, it has the mass suppression factor (1/M 2n−3 G ), determining n = 5, i.e. 2 fermion lines and 8 boson lines. Fig. 3 shows a typical A-term realization of the potential in supergravity with the heavy internal line with external lines of three H (0) u , three H (0) d , two χ (0) , and two χ (0) . Fig. 3 breaks the U(1) de symmetry if this diagram is obtained by connecting two U(1) de defining diagrams with one common fermion line connecting them. For example, the U(1) de defining diagram consists of two fermion lines and four boson lines as shown in Fig. 4. The quantum numbers are those of U(1) 10R gauge symmetry which become the Z 10R charges mudulo 10. This is obtained from the left/right part of Fig. 3. So, the U(1) de defining X (0) 4 X (0) 4