511 keV Gamma Ray from Moduli Decay in the Galactic Bulge

We show that the $e^++e^-$ decay of a light scalar boson of mass 1-10 MeV may account for the fluxes of 511 keV gamma ray observed by SPI/INTEGRAL. We argue that candidates of such a light scalar boson is one of the string moduli or a scalar partner of the axion in a supersymmetric theory.


Introduction
It has been known in the superstring theory that the compactification with extra-dimensional fluxes stabilizes some moduli in supersymmetric (SUSY) string vacua [1]. It is not yet, however, clear if all of the moduli fields in the sting vacua are stabilized by the flux compactifiction. Therefore, it is a very important task to search a possible evidence for light scalar bosons. In this letter we point out that the 511 keV gamma-ray emission line from the galactic bulge measured by the SPI spectrometer on the space observatory INTEGRAL [2] is explained by the e + + e − decay of a (pseudo)scalar boson of mass O(1) MeV. 1 We also argue that such a light boson can be identified with one of the string moduli or with a scalar partner of the Peccei-Quinn axion field in a SUSY theory. We also stress that the gauge-mediation model with a light gravitino of mass O(1) MeV [4] is very interesting, since the above particles acquires most likely their masses of the order of the gravitino mass m 3/2 if they survive the flux compactification.
We consider a scalar boson φ which has a Yukawa coupling to the electron as If the φ is one of the string moduli, the M * is of the order of the gravitational scale, GeV [5], and if it is a scalar partner (saxion) of the axion, M * ≃ F a where F a represents the breaking scale of the Peccei-Quinn symmetry. The decay width is given by and we get the lifetime as The lifetime of the φ boson should be longer than the age of the universe (≃ 1.3 × 10 10 yr) to give a significant flux of the gamma ray, which leads to a constraint Here we assume that the scalar φ does not have a direct coupling to photons. The decay into neutrinos is negligible because of chirality suppression.

The energy density of the φ boson
It is quite natural to consider that the φ boson has a large classical value φ 0 of the order M * at the end of inflation. The φ starts a coherent oscillation when the Hubble constant of the universe reaches at the mass of the boson, m φ . The energy density of the coherent oscillation easily exceeds the critical density of the universe [6,7,8], which leads to a serious cosmological problem (moduli problem). The thermal inflation [9] is the most promising mechanism to dilute the energy density of the φ oscillation and hence solve the moduli problem. We estimated in [10,11] the energy density, Ω φ , after the thermal inflation.
The Ω φ in the present universe is given by for m φ < ∼ 10 MeV. Here, we have taken h ≃ 0.7 (h: Hubble constant in units of 100km/sec/Mpc) in eq. (20) of Ref. [10] and assumed the initial amplitude of the φ to be φ 0 = κM * with κ being O(1) constant. The lowest density is realized for the reheating temperature after the thermal inflation T R ≃ 10 MeV (see Appendix and [10] for details).

The flux of the 511 keV gamma ray
Now we estimate the 511 keV gamma ray flux from the Galactic center. It was shown in Ref. [12] that e + + e − decay of the dark matter particle of mass m d ∼ O(1 − 100) MeV can produce the 511 keV line emission observed by SPI/INTEGRAL, through e + and (background) e − annihilation. Given a density Ω φ , a mass m φ and a lifetime τ φ of the scalar particle, the 511 keV gamma ray flux Φ 511 is estimated as [12,13] where we have used the present dark matter density Ω dark ∼ 0.3 and the halo density profile ρ halo ∼ 1/r 1.2 [12] (r: distance from the Galactic center). Using eqs. (3) and (5) we It should be remarkable that the prediction of the flux is independent of M * and weakly depends on the mass of the scalar field. Notice that the observed flux is [2] Φ 511 = 9.9 +4.7 −2.1 × 10 −4 cm −2 sec −1 .
We see, from eq. (7), that the positrons emitted by the decay of φ in the Galactic bulge explain naturally (κ ≃ 1) the observed fluxes of the 511 keV line gamma ray if the thermal inflation maximally dilutes the scalar field density.

Conclusions
We have shown that the e + + e − decay of a light scalar particle of mass O(1) MeV diluted by the thermal inflation is capable of producing 511 keV gamma rays observed by the SPI/INTEGRAL.
Since the scalar field does not have a direct coupling to photons in the present model, the two photon decay (φ → 2γ) takes place at most through one-loop corrections and hence its branching ratio is less than ∼ 10 −6 for m φ ∼ 1 MeV, which is well below the present observed gamma ray background [14]. However, the photon flux is only one or two oder of magnitude smaller than the current limit for larger mass ∼ 10 MeV. The detection of line gamma rays with energy m φ /2 by future experiments will confirm the present model, since the line gamma rays emitted by this process is distinctive of the model. On the other hand, Beacom et al. [15] pointed out that "internal bremsstrahlung" emission (φ → e + e − γ) exceeds the observed diffuse gamma ray flux from the Galactic center unless MeV. This is one of the reason why we have assumed m φ ≃ 1 − 10 MeV. We consider that a good candidate for such a light scalar particle is one of the string moduli for M * ≃ M G . However, we also stress that the gamma ray flux is independent of M * as long as it satisfies eq.(4). Thus, the saxion is another candidate if the Peccei-Quinn scale F P Q = M * is ∼ 10 15 GeV.Notice that since the efficient thermal inflation requires the reheating temperature as low as O(1)MeV the axion density is also diluted. However, as shown in [16] the axion with F P Q ∼ 10 15 GeV explains the dark matter density of the universe for the reheating temperature of O(1) MeV. 2 Therefore, the axion may be the dark matter in the present universe.
The above particles most likely acquire masses of the order of the gravitino mass m 3/2 .
The O(1) MeV mass required in the present scenario is expected in the framework of gauge mediation models of SUSY breaking [4].
As for baryon density of the universe, the low reheating temperature makes baryogenesis very hard in general. However, we consider that the late-time Affleck-Dine mechanism [17] may work [18,19]. 3 decay, and the ratio of the final entropy s f to initial entropy s i is estimated as Then, the moduli density is diluted by the entropy production as where ρ cr,0 and s 0 is the present critical and entropy density. Notice that this value is almost the same as the minimum moduli density given by eq. (5) for m φ ≃ 1 MeV.
However, the moduli dilution by the thermal inflation is not so simple because the field value that gives the minimum of the moduli potential is shifted by the Hubble induced mass term during the thermal inflation. This shift produces a new oscillation with amplitude V 0 φ 0 /(m φ M G ) 2 , which leads to Therefore, the total moduli density is determined from contributions of both eqs. (12) and (13). Assuming that the falton decays into two gluons (T R ≃ 10 −3 λ 1/2 m 0 ), we can obtain the minimum moduli density (5) by varing m 0 and λ [10].