Search for mCPs
2.2 Physical constraints in the mCP mass/charge
The Standard Model of particle physics [40] is the theory describing all known ele-mentary particles and their possible interactions. It is known to be incomplete as it does not explain problems such as the neutrinos masses or matter-antimatter asymmetry, for example. However, being the base to correctly explain a vast number of natural phenom-ena, the addition of new particles within its framework or modications of any kind at the theory is very restricted. Expanding the SM usually have consequences incompatible with the experimental knowledge in well studied physical process. This argument can be directly applied to the mCP, a new fermion inserted into the theory, whose characteristics such as mass and charge should be constrained by the known physics. Some examples of experimental limits imposed to the mCP mass and charge derived from this idea are given below. Most of them are related with physics at early stages of the Universe or stellar physics, systems with sucient energy density to create mCPs. The combination of the latest constraints imposed in the mCP parameters space (not only the ones discussed in details here) is shown in gure 1.1
I. Big Bang Nucleosynthesis
In Cosmology, Big Bang nucleosynthesis (BBN) [46] refers to the production of light chemical elements (atomic masses lighter or equal than7Li) during the early stages of the Universe. One of the greatest success of the big bang theory, it predicts the abundance of these elements as function of the barion-photon ratio η= nnB
γ based on the SM.
The nucleosynthesis happens in radiation dominated era of the Universe when the energy density was given by:
where the index i runs over all possible particles, bosons or fermions with respectively gibosons and gifermions degrees of freedom, contributing for the energy density and T is the
Universe temperature. The nucleosynthesis starts when the temperature falls to MeV scale and the only relativistic species are photons, electrons, positrons, neutrinos and anti-neutrinos, yielding a energy density ρrad = π302 2 + 74 +74Nν
T4. The number of neutrino avors Nν were left as a variable on purpose for further discussions. In the standard BBN calculation,Nν = 3is used. The important weak interactions for BBN are
n+e+ ↔p+ ¯νe n+νe ↔p+e−
n ↔p+e−+ ¯νe
(2.6)
For temperatures T 1 MeV, these reactions are in thermodynamical equilibrium with an equal number of protons and neutrons. Due to the temperature drop with the Universe expansion, at some point the weak interaction rates will no longer be fast enough to sustain the equilibrium, a process known as freeze-out. This condition is reached at T = 0.8 MeV, when the reaction rate Γ ∼ GFT5 falls below the expansion rate H2 ∼ 8πG3Nρ. The proton and neutron number densities after the freeze-out can be calculated the using Maxwell-Boltzmann distribution. So, for T = 0.8MeV, the neutron-proton ratio is nnnp =
mn
mp
3/2
exp ∆m/T ∼ 1/6, where ∆m is the neutron-proton mass dierence. This value goes to1/7if ones consider neutrons decaying before nucleosynthesis begins.
The elements formation chain starts with the Deuterium production p+n ↔ D+ γ. Because of the large number of photons relative to nucleons (η−1 ∼ 1010), with a signicative fraction of them carrying energy above the Deuterium binding energy ED = 2.2 MeV, the formation of Deuterium is delayed until the temperature reaches T = 0.1 MeV. At this point the production rate overcomes the destruction. Then, the chain continues with the Helium formation through D+D → 4He. Using a crude estimation which assumes that all neutrons are converted in Helium, the expected abundance of this
element is given by equation 2.7
YHe ≈ 2n
n+p = 2 (n/p)
1 + (n/p) ≈0.25 (2.7)
The other light elements are produced in less abundance: D and 3He at ∼ 10−5 level and 7Li at ∼ 10−10 level. The BBN predictions are compatible with experimental observations and the addition of the mCP into the theory must not change this fact.
New components in the Universe content with masses below the MeV scale are relativis-tic just before BBN starts, contributing in equation 2.5 and modifying the abundances.
This argument was rst applied to constraint the number of possible neutrino avors. It can be shown that the abundances of light elements are still compatible with observations for Nν < 3.3 Majorana neutrino avors [4749]. In this way, BBN allows for 0.3 extra two-component neutrinos or their equivalent in other relativistic particles at MeV mass scale. For a model without a hidden photon, this argument rules out mCP withMmCP <1 MeV. As this constraint is predicated on the assumption that these particles are in ther-mal equilibrium with the electrons and photons, it is valid for charges withε >10−8 [21].
In order to reach the equilibrium at BBN, their interaction cross section must fall below the expansion rate before temperatures ofT = 5MeV. If one consider models including a massless hidden photonγ0, the exclusion is even more stringent. As a massless boson, the hidden photon would contribute as 8/7 of a neutrino and, for this reason, they are not allowed to be in thermal equilibrium with the ordinary photons. Instead, they are only allowed to exist if 87Tγ40 = 0.3Tγ. The photons must therefore be heated by annihilation after the hidden photon decouple. It can be shown that this implies the photons and hidden photons must be out of equilibrium at QCD phase transition (T ∼200 MeV) [21].
This fact can be used to constraint the mCP masses below7 + 0.4 lnε GeV scale [50].
II. Flatness of the Universe
Another physical constraint that bounds the mCP parameter space is the requirement of a particle density below the critical density ρc. In Cosmology, the expansion of the
Universe is governed by the Friedmann equations [51, 52]. Within the context of general relativity, the Cosmological Principle, the notion of a spatially homogeneous and isotropic Universe at large scales (justied at scales larger than ∼ 100 Mpc), leads to a generic metric in the form:
ds2 =a(t)dr32−c2dt2 (2.8) where, dr32 is a three-dimensional spatial metric and a(t) is a scale factor. This is called the Friedmann-Lemaître-Robertson-Walker metric. Einstein's eld equations relates the evolution of the scale factor to the pressurep and energy densityρ of the matter content in the Universe as:
where G is gravitational constant. The energy density and the pressure term are related by a perfect uid equation of state p = ρc2ω (dierent values of the constant ω for each Universe component). The actual Universe's energy content, a crucial parameter to determine the Universe evolution, is one of the biggest open questions of the century.
A large number of astronomical observations [5355] indicate that the regular baryonic matter represents only a small fractional, 4.9%, of the energetic content of the universe.
About 26.8% is in the form of the so called dark matter (introduced in chapter 1 and described in more details in chapter 3). The remaining 68.3% is called dark energy, a negative pressure term inserted in Friedmann equation via a cosmological constant Λ, is responsible for the accelerate expansion of the universe. The great open question resides in what is the composition or physical phenomena responsible for these last two terms, the major content of the Universe.
In this context, the critical densityρcwas rst introduced as a means to determine the spatial geometry of the universe, representing the density for which the spatial geometry
is at. By settingΛ = 0andk = 0, one nds the expression for the critical density to be:
ρc= 3H02
8πG (2.10)
where is usual to dene the density parameter as Ω ≡ ρρ
c. If Ω is larger than unity, the space sections of the universe are closed and the universe will eventually stop expanding, then collapse. IfΩis less than unity, they are open and the universe expands forever. So, by requiring the mCP particle density below the critical densityρc, one assure that they will not overclose the Universe.
The abundance observed today of any thermally produced particle (including mCPs) in the Universe can be obtained via the Boltzmann equation [56]:
dn
dt + 3Hn=< σav >(n−neq) (2.11) where the second term includes the density dilution due to the Universe expansion and the last is an interaction term taking into account the particle annihilation. The annihilation rate is given by the thermally averaged annihilation cross section< σav >and the particle number density.
It is assumed that the particles were created in thermal equilibrium between the cre-ation and annihilcre-ation. As the Universe expands, it also cools down, stopping the particle creation and reducing exponentially the particle number density due to annihilation. Fi-nally, for a rate of expansion higher than the interaction rate, the particle number density freezes in a certain value. This value is controlled by the annihilation cross-section. This process is shown in gure 2.1 taken from [57]. In the case of mCPs, dierent masses and charge fraction leads to dierent number density that are constrained requiringρmCP < ρc.
III. Stellar physics
The stellar evolution and supernova theory is well described by the standard model.
In a simplied picture, a star is the result of the dynamics between the gravitational collapse and the pressure of the stellar material. In this thermodynamical system, fusion
Figure 2.1: Comoving number density evolution as a function of the ratio mass tempera-ture ratiom/T in the context of the thermal freeze-out. The size of the annihilation cross section determines the relic abundance.
reactions driven by the gravitational collapse and high temperatures liberate energy while creating heavy elements. In parallel, neutrinos carry energy from the star as they are produced in the photo-neutrino process γe− → e−νν¯, the bremsstrahlung process e− + (A, Z) → (A, Z) +e− +νν¯ and the plasmon decay γ → νν¯. Plasmons are excitations of the dense electron-proton plasma in which the photons obtain non-trivial dispersion relation. In other words, they acquire an eective mass while propagating through this medium, allowing them to decay without violating gauge invariance or conservation of energy and momentum. The stellar theory based on the standard model is in good agreement with experimental data. New particles that could carry extra energy from the star, such as mCPs with small couplingε (or other weakly interacting particle), may lead to observational modications on the standard course of its evolution depending on its properties [58]. In general, the hot dense medium inside a star has sucient energy to produce mCPs and stellar physics can also be used to constraint these particles parameter
space.
The dominant mechanism that would produce mCPs in the stellar medium is the plasmon decay γ∗ → ff¯. Other possible mCP production channels are negligible in comparison with the plasmon decay. The small number of positrons suppresses the mCP production via the annihilatione+e− →ff¯. The fusion of two photonsγγ →ff¯is in high order of the small coupling ε. Also, the fusion of two hidden photons γ0γ0 → ff¯would be suppressed by the small number of these particle. Based on this energy loss argument, red giants (RG), horizontal-branch (HB) stars, white dwarf (WD), supernova (SN) 1987A and sun data were used to derive constraints on the mCP parameter space. A red giant is a luminous giant star, of roughly 0.3 to 8 solar masses M, in a late phase of stellar evolution. In this stage, the star is still fusing hydrogen into helium, with a hydrogen shell surrounding an inert helium core. Changes in the electron density and temperature, that occurs in the presence of a new energy loss channel, would reect in the brightness at the tip of the red-giant branch in globular clusters and therefore could be observed [812].
The horizontal branch is a stage of stellar evolution that immediately follows the red giant for stars with masses similar to the Sun's. They are powered by helium fusion in the core and hydrogen fusion in a shell surrounding the core. New energy loss mechanisms means an accelerated consumption of nuclear fuel for these stars. This can be measured by number counts of HB stars in globular clusters [814]. The population of White Dwarfs can also be used to limit the mCPs characteristics. The present theory explaining the cooling for these stars by surface emission of photons and volume emission of neutrinos produced in plasmon decay via Standard Model interactions, seems to agree with the observed luminosity distribution of their population. This observable would be changed with a new decay channel for the plasmon [8, 1518]. For supernovas, the number of neutrinos detected at Earth after the SN 1987A event agrees roughly with theoretical expectations. Other particles contributing to the cooling of the proto neutron star would imply in a reduced neutrino ux and in short duration of the neutrino signal [8,19,20]. In the case of the Sun, the proximity of the star allow the same study using observables such
as neutrino ux and sound speed prole. Solar models with exotic energy loss mechanism, lead to measurable changes on these observables and this is also used to limit the mCP parameter space [6].