Scattering of massive W bosons into gravitinos and tree unitarity in broken supergravity

The gravitino G is the gauge field of local supersymmetry, namely of Supergravity [1]. In the contest of Cosmology, such a particle plays a relevant role in several scenarios. After the spontaneous breaking of local supersymmetry, through the so-called super Higgs mechanism the gravitino obtains a mass m e G that is proportional to the breaking scale. Accordingly, m e G depends on the particular model considered, and in principle it may range from the eV scale up to the TeV scale and beyond [2]. In general, gauge mediation predicts the gravitino to be the lightest supersymmetric particle, or LSP [3]. Then, if R-parity is conserved, it can be a very attractive candidate for Dark Matter. In other theories, for instance in gravity mediation, the gravitino is unstable and it has a lifetime τ e G which is usually longer than 100 sec. This means that it decays after the beginning of the Big Bang Nucleosynthesis (BBN), affecting the abundances of the primordial light elements and eventually spoiling the success of the BBN [4]. Such problems have been extensively studied in the literature, setting different bounds on the masses of both stable and unstable gravitinos [5].

Inflation may solve these problems, since it dilutes enormously the gravitino abundance and provides with a more natural range for m e G , between O(1 MeV) and O(100 GeV) [6]. The former constraints can then be relaxed and the cosmologically relevant gravitinos were produced during reheating, right after the end of inflation, mostly through hard 2 → 2 scattering processes of particles in the primordial thermal bath [7]. In Ref. [8], the authors calculated the gravitino production rate in supersymmetric QCD at high temperature, to the leading order in the gauge coupling. Ten hard 2 → 2 scatterings with a gravitino in the final state were considered. The total contribution, with appropriate modifications due to the finite temperature of the thermal bath [9,10], provides the collision term for the Boltzmann equation, and therefore an estimate of the gravitino number density. The same approach was then applied to the case of the electroweak interaction in the high-energy limit. By considering massless W bosons, relevant contributions to the total gravitino number density were obtained [11]. By taking into account such results, the BBN constraints from gravitino production have been recently updated in [12], and an analytical procedure that is alternative to the numerical method was proposed in [13].

In this paper, we study for the first time what happens when the gravitinos are produced at a centre of mass energy that is comparable to the EW scale, by assuming a non vanishing mass of the W bosons. In contrast to previous investigations [8,11], and in some analogy with [14], we find that if the gauge bosons are massive, the squared amplitude of the WW scattering contains new terms which violate the unitarity above a certain scale, Eq. (2.15). Such quantities do not factorize any mass splitting which would vanish in the SUSY limit, contrary to what is expected [15]. In fact, in the annihilation diagram the longitudinal degrees of freedom in the W boson propagator disappear from the amplitude by effect of the supergravity vertex. After a comparison with the massless case, we show that the longitudinal modes of the W bosons become strongly interacting at high energies and the divergences hold at any centre of mass energy, as reported in Eq. (2.36).

We now recall that broken supergravity is the effective limit of a more fundamental theory, since it is valid only below the SUSY breaking scale. Our calculations show that unitarity is broken at the same order, e.g. at √ s ≈ O(10 14 GeV) for m e G ≈ O(1 MeV). The result of this paper is thus consistent with the entire energy range allowed by SUGRA, therefore our study is phenomenologically motivated.

In fact, the WW scattering can be observed at the LHC as a secondary process, for instance in gluon fusion [16]. From a more cosmological viewpoint, the fact that the result holds at any energy would be interesting in scenarios with both high and low reheating temperature T R . While the former actually constitutes a rather standard background, T R < ∼ O(10 6 GeV) is favoured in recent investigations on baryogenesis and Dark Matter [17]. Nevertheless, it would be interesting to study our result from a more formal point of view, as it seems to be a general feature of broken supergravity. This provides with an interesting theoretical perspective.

The present article is organized as follows. In Section 2 we calculate the squared scattering amplitude of the process

by using the basis of gauge eigenstates, in the case of massive spin 3/2 gravitino. We find some anomalies leading to violation of unitarity Eq.(2.15), whose origin is discussed in Section 2.1, where an argument concerning the high energy limit provides Eq.(2.36). Finally, at Section 2.2 we calculate the effective limit by considering a gravitino mass

…(Full text truncated)…

This content is AI-processed based on ArXiv data.