Cosmological Magnetic Fields from Gauge-Mediated Supersymmetry-Breaking Models

We study the generation of primordial magnetic fields, coherent over cosmologically interesting scales, by gravitational creation of charged scalar particles during the reheating period. We show that magnetic fields consistent with those detected by observation may obtained if the particle mean life \tau_s is in the range 10^{-14} sec \leq \tau_s \leq 10{-7} sec. We apply this mechanism to minimal gauge mediated supersymmetry-breaking models, in the case in which the lightest stau \tilde\tau_1 is the next-to-lightest supersymmetric particle. We show that, for a large range of phenomenologically acceptable values of the supersymmetry-breaking scale \sqrt{F}, the generated primordial magnetic field can be strong enough to seed the galactic dynamo.


Abstract
We study the generation of primordial magnetic fields, coherent over cosmologically interesting scales, by gravitational creation of charged scalar particles during the reheating period. We show that magnetic fields consistent with those detected by observation may be obtained if the particle mean life τ s is in the range 10 −14 sec < ∼ τ s < ∼ 10 −7 sec. We apply this mechanism to minimal gaugemediated supersymmetry-breaking models, in the case in which the lightest staũ τ 1 is the next-to-lightest supersymmetric particle. We show that, for a large range of phenomenologically acceptable values of the supersymmetry-breaking scale √ F , the generated primordial magnetic field can be strong enough to seed the galactic dynamo.

CERN-TH/99-261
Quite homogeneous magnetic fields of intensity B ≃ 3 × 10 −6 Gauss are present in all structures of our Universe: galaxies, galaxy clusters and hydrogen clouds [1,2,3,4]. One of the mechanisms for the generation of magnetic fields is the primordial generation of a seed field that is further amplified by gravitational collapse and/or dynamo [1]. For the origin of the seed field, several mechanisms have been proposed recently: it has been suggested that a primordial field might be produced during the inflationary period if conformal invariance is broken [5,6]; in superstring-inspired models, the coupling between the electromagnetic field and the dilaton breaks conformal invariance and might produce the seed field [7,8]; gaugeinvariant couplings between the electromagnetic field and the space-time curvature also break conformal invariance, but produces in general an uninterestingly small seed field [9]; other mechanisms are based on, for example, first order cosmological phase transitions and on the existence of topological defects [10,11,12].
Recently, a new mechanism for cosmological magnetic field generation was proposed [13], based on the presence during inflation of a charged, minimally coupled scalar field in its invariant vacuum state [14]. When the transition to radiation takes place, quantum creation of charged particles occurs because of the release of gravitational energy. The mean electric current is zero, but stochastic fluctuations around that mean give a non-vanishing contribution. The magnetic field induced by this stochastic current was sufficient to seed the galactic dynamo. However, there remained the important issue of finding a suitable scalar particle to generate the electric current source of the magnetic field.
In this letter we address this problem in the context of gauge-mediated supersymmetrybreaking models (GMSB). In the simplest version of these models, supersymmetry-breaking is communicated to the visible sector through a set of massive fields, called messengers, which carry non-trivial quantum numbers under the gauge group [15,16]. The messengers Φ I ,Φ I are assumed to acquire an explicit mass M I by the vacuum expectation value of a singlet field X I = M I , via a superpotential coupling A vacuum expectation value of the auxiliary component F I of the field X I breaks supersymmetry and induces, through the gauge interactions of the messenger fields, the supersymmetry-breaking masses in the observable sector. For the simplest case of N sets of messenger fields belonging to the fundamental representation of SU(5) and a single field X, one gets gaugino masses where i = 1, 2,3 are associated with the gauge groups U(1) Y , SU(2) and SU(3) c , respectively.
The scalar masses not affected by Yukawa couplings are given by where m S are the supersymmetry-breaking masses for gauginos and scalars, respectively, µ is the renormalization group scale with µ = 0 being identified with the messenger mass scale, c i S is the quadratic Casimir of the scalar particle under the i-gauge group, and α i and b i are the corresponding gauge coupling and MSSM beta function coefficients 1 . From the above, it is easy to see that the right-handed sleptons are the lightest scalars in the spectrum and, for N > 1, the lighest stau can easily become lighter than the lightest neutralino. The lightest stau can also become lighter than the lightest neutralino due to mixing effects, for moderate and large values of tan β, for any value of N. For the characteristic values of the supersymmetry-breaking scale F , however, the lightest supersymmetric particle is the gravitino. Indeed, the gravitino mass is given by where M P l is the Planck scale (we are identifying F with the fundamental supersymmetrybreaking scale F 0 ). Hence, the gravitino is the lightest supersymmetric particle for any messenger mass M much lower than the GUT scale.
In general, under the assumption of R-parity conservation [18], the next-to-lightest SUSY particle will decay into a gravitino and a standard particle with an inverse decay rate [19] where m NLSP is the mass of the NLSP particle andk is a projection factor equal to the component in the NLSP of the superpartner of the particle the NLSP is decaying into. For the case of the stau decaying into a tau and a gravitino,k = 1.
Constraints on the value of the supersymmetry-breaking scale may be obtained, for example, by the requirement that the gravitino density does not overclose the Universe. For instance, if the gravitino mass mG > 1 keV, the temperature at the beginning of the radiation-dominated epoch, called the reheat temperature T γ , should be much smaller than the GUT scale in order to avoid overproduction of gravitinos [20]. The exact bound on T γ depends on the gravitino mass. For relatively large values of the gravitino mass, corresponding to √ F ≃ 10 9 GeV, (M ≃ 10 13 GeV), an upper bound on T γ of the order of 10 7 GeV is obtained 2 . The bound becomes even smaller for smaller values of F . On the other hand, for mG < 1 keV and for any value of the reheat temperature larger than the weak scale, the gravitinos will be in thermal equilibrium at early times and, for these range of masses, the gravitinos are sufficiently light to lead to cosmologically acceptable values of the relic density.
The reheating period can be characterized by the temperature T I obtained by thermalization after preheating and T γ , the temperature at the beginning of the radiation dominated epoch [22]. In Ref. [13] an inflationary model with instantaneous reheating was considered.
In the more realistic case in which the reheating is extended in time, the number of particles created at the two main transitions, namely inflation-reheating and reheating-radiation, as well as during the reheating period itself should be calculated.
We will work in conformal time, which is given by dη = dt/a(t). Defining τ = Hη, where H is the Hubble constant during inflation, and assuming that during reheating the Universe is matter-dominated [23], the scale factors for the different epochs of the Universe read T I and T γ are the temperatures of the Universe at the beginning of reheating and at the beginning of radiation, respectively. We have assumed that during radiation the temperature of the Universe scales with a (τ ) as T ∝ a (τ ) −1 while during reheating it goes as T ∝ a (τ ) −b , with 0 < b < 1 [24].
The evolution of a charged scalar field is given by the Klein-Gordon equation. If we expand the real and imaginary parts of the field as (2π) −3/2 d 3 κφ κ (τ ) e i κ. r + h.c. , the field equation reads where k = H −1 κ (κ being the comoving wavenumber) and where ξ is the coupling to the curvature. We will consider the mass as built up from two contributions, the zerotemperature mass m(0) ≡ mτ and the thermal corrections, so that we have m 2 = m 2 (0) + gT 2 (τ ), where g is of the order of the particle gauge coupling constants.
For the inflationary period, we do not need the thermal corrections, as the temperature of that period is too low to be important. However, in supergravity theories, the possible presence of a non-renormalizable coupling of the inflaton field I to the scalar fields in the Kähler potential [21] would naturally lead to a mass contribution δm 2 ≃ C H H 2 . Hence, in general, an effective mass of the order of the Hubble constant will be generated, although the coefficient C H may be small or even zero in the case when the specific effective coupling is forbidden by symmetries of the theory [25].
The positive-frequency solution to Eq. (9) for inflation reads where H (1) ν are the Hankel functions, with where for the charecteristic values of m(0) and H during inflation, m 2 /H 2 ≃ C H . We will assume throughout this article that the scalar field couples minimally to the curvature, ξ = 0, and that the coefficient C H ≪ 1; we will briefly discuss the implications of different values of these quantities at the end of this article. For reheating and radiation dominance, we propose a WKB solution where radiation It is important to note that the frequency changes from imaginary to real values, at a certain time τ c during reheating.
We match the solutions to the field equation in the different epochs at the transition between them, i.e. at the end of inflation and at the end of reheating. At both times we demand continuity of the corresponding modes and their first time derivatives. Care must be taken to match the WKB solutions through τ c , where ω (τ c ) = 0. We obtain where by φ kγ we denote the modes during radiation and with In the above, we are ignoring the effects produced by the change of the effective Hubble constant during the inflationary period, which results in a change of the Bogoliubov coefficients in the far ultraviolet [26]. These effects, however, are small in the range of wavelengths relevant for the analysis of the generation of magnetic fields, k < ∼ k tod , where k tod is the comoving wave number of the relevant astrophysical scale under study (see below).
In order to proceed with our phenomenological analysis, the values of T I , T γ and H in the previous expressions must be specified. They can be related by the age of the Universe, which can be well approximated by the duration of the matter-dominated epoch. This is given by where T tod ≃ 10 −13 GeV is the present temperature of the Universe, T M ≃ 1 eV is its temperature at the beginning of the matter dominated epoch and for the Hubble constant during inflation, we shall assume that 10 11 GeV ≤ H ≤ 10 13 GeV. From Eq. (19) we obtain where the last equality stems from t tod ≃ M pl /T .
Therefore, independently of the value of b and for the values of the cosmological parameters considered above, the relation T I > T γ is fulfilled for any value of T γ < 10 15 GeV.
In order to compute the magnetic field, we consider the Maxwell equation where σ (τ ) is the time-dependent conductivity of the medium and j is the electric current generated by the charged scalar particles. Although j = 0, the two-point correlation function is different from zero and produces a non-vanishing magnetic field. This field can be expressed in terms of the two-point function of the scalar field as (see Ref. [13] for details) where with Ω (τ ) = τ dτ ′ ω (τ ′ ), φ + kγ (φ − kγ ) the positive (negative) frequency modes of the scalar field during radiation dominance and with G ret |k+k ′ | (τ o , τ ) the retarded propagator for the electromagnetic field. W (λ) kk ′ is the window function that filters scales smaller than λ. Eq. (23) then gives the magnetic energy of a field which is homogenous over volumes of order λ 3 , the intensity of the field therefore being estimated as B 2 λ . From now on it will be understood λ = k −1 tod . where, as mentioned above, k tod is the comoving wavenumber of the astrophysical scale we are interested in.
Real particle propagation can be considered as such from the moment when the frequency becomes real, i.e. from τ c . To evaluate Eq. (23) we shall proceed in the same way as in [13], and consider only the main contribution, which originates from the last term between brackets, which is quartic in the Bogoliubov coefficients and, within this term, from the non-oscillatory contributions. We perform the k integration with the same window function used in [13], i.e. a top-hat one. We propagate the magnetic field during reheating and radiation dominance until the moment of detection with the propagator given by the equation is the electric conductivity of the Universe. After the particles decay, the field propagates conformally. We assume that during all these periods the conductivity of the Universe is given by [27] where α = 2b, τ * = −2 and σ 0 = T I /e 2 H for reheating, and α = 1, τ * = −2 + (T I /T γ ) 1/2b and σ 0 = T 1/2b I The Bogoliubov coefficients are the ones given in Eqs. (17) and (18). Now we are ready to evaluate the time integrals in Eq. (23). It can be checked that the contribution from reheating is negligible with respect to the one from the radiation period. Also, the mass term dominates over the thermal correction for a particle lifetime t max > 10 −14 sec. We therefore consider the time integral Now we are ready to evaluate the magnetic field. For this purpose it is convenient to express the comoving wave number in terms of the present one as k tod = κ tod T γ (T I /T γ ) 1/b /HT tod .
Replacing everything in eq. (23) we obtain Equation (29) gives the intensity of the field at the moment when the electric current vanishes. After that, the field propagates as B phys λ (t) = B 2 λ a (t max ) 2 /a (t) 2 (i.e. magnetic flux conservation). Using the relation a(t) ∝ 1/T , the present value of the magnetic field is given by where T max is the temperature of the Universe when the particles decay, given by In the above, we have only considered the effect induced by the scalar particle and not by the charged particles resulting from its decay. This might seem surprising since, due to charge current conservation, the charged particles coming from the scalar particle decay might also contribute in a relevant way to the magnetic field generation. However, the decay of a massive scalar particle, like the stau in the case under study, will lead mostly to charged fermions (tau leptons, in this case) with wavelengths much shorter than the ones of the original scalar field. These fermions might eventually generate magnetic fields, but, due to the wavelengths involved, these fields will not be coherent in the scales of interest for our study.
In order to apply the above formalism to the case of gauge mediated supersymmetrybreaking models, we should recall Eq. (5), which gives the lifetime of the NLSP, ττ ≡ t max as a function of the supersymmetry-breaking scale and the mass of the lightest stau. Replacing Eqs. (20) and (5) This value of the generated magnetic field is sufficient to seed the galactic dynamo, being also consistent with the bounds imposed by the anisotropies in the CMBR and by primordial nucleosynthesis [28,29].
In the above we have given results for specific values of √ F , H and T γ , for minimal coupling and for C H = 0, that is for ν = 3/2. It is interesting to discuss the dependence on √ F , C H , as well as on departures from minimal coupling. In this case it can be checked that for small k tod , the Bogoliubov coefficients are given by α k ∼ β k ∼ O(1)k −ν , with ν given by Eq. (12). Considering a stau lifetime ττ ≡ t max = 10 n GeV −1 and a stau mass mτ ≃ 100 GeV, the value of the physical magnetic field is given by . ( Acceptable values of the magnetic field, consistent with the cosmological bounds [28,29] are in the range 10 −9 Gauss ≥ B phys λ ≥ 10 −21 Gauss.
For the case ν = 3/2, and for the mean values of the parameters taken above, this implies 158/9 > ∼ n > ∼ 86/9, or, equivalently, The variation of the above bounds with the cosmological parameters can be easily obtained from Eq. (34).
On the other hand, for arbitrary values of ν, the following bound is obtained: n.
In gauge-mediated supersymmetry-breaking models, for a stau mass mτ ≃ 100 GeV and values of the gravitino mass mG < ∼ 1 keV, the stau lifetime, Eq. (5), is such that 11 < ∼ n < ∼ 16, or equivalently For a mimimal stau coupling to the curvature and C H ≪ 1, that is for ν = 3/2, Eq. (38) is in remarkable agreement with the values required to generate an acceptable magnetic field, Eq. (36).
The bounds on n in gauge mediated supersymmetry breaking models also imply bounds on ν Comparing this expression with the value of ν for minimal coupling of the scalar field, ν ≃ 3/2 1 − 4C H /9, we obtain that C H < 0.68 in order to generate cosmologically relevant values of the magnetic field. As follows from Eq. (34), only small modifications of the bound on C H may be obtained for different values of the cosmological parameters.
Consider now the departure from minimal coupling. Assuming that C H ≪ 1 we have ν ≃ 3/2 1 − 16ξ/3. The bounds on the lifetime of the stau are satisfied for 0 ≤ ξ < ∼ 0.06; we thus obtained for a non-negligible interval of coupling values, magnetic fields of an intensity sufficient for these to be cosmologically important. The upper bounds on C H and ξ quoted above can only be obtained for values of √ F (or equivalently n) such that the gravitino mass mG is close to 1 keV.
The results given above were obtained for a reheat temperature T γ ≃ 10 7 GeV. As we emphasized above, for the range of gravitino masses we are concentrating on, the most relevant bound on the reheat temperature comes from Eq. (20), which assures the consistency of the whole approach. Larger values of the magnetic fields may be obtained by lowering the value of the reheat temperature. However, the final result for the magnetic field, Eq. (32), depends very weakly on the value of the reheat temperature T γ . No relevant departures from the obtained values would be obtained even if the reheat temperature were as low as T γ ≃ 10 3 GeV.
In summary, we have shown that cosmologically relevant magnetic fields may be generated by a scalar field, minimally coupled to the curvature, so far its lifetime is bounded by Eq. (36). The bounds on the lifetime are in excellent agreement with those obtained in minimal gauge mediated supersymmetry-breaking models with the lightest stau as the next-to-lightest supersymmetric particle, for values of the supersymmetry-breaking scale such that mG < ∼ 1 keV. This conclusion is very weakly dependent on the assumed values of the cosmological parameters. Moreover, contrary to many models for magnetic field generation proposed in the literature, the present one is related to the properties of the low energy effective theory and these properties can be tested in accelerator experiments in the near future [30].