Particle decays and stability on the de Sitter universe
Abstract
We study particle decay in de Sitter spacetime as given by first order perturbation theory in a Lagrangian interacting quantum field theory. We study in detail the adiabatic limit of the perturbative amplitude and compute the “phase space” coefficient exactly in the case of two equal particles produced in the disintegration. We show that for fields with masses above a critical mass there is no such thing as particle stability, so that decays forbidden in flat spacetime do occur here. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with mass lower than critical have a completely different behavior: the masses of their decay products must obey quantification rules, and their lifetime is zero.
1 Introduction
Some important progress in the astronomical observations of the last ten years [1, 2] have led in a progressively convincing way to the surprising conclusion that the recent universe is dominated by an almost spatially homogeneous exotic form of energy density to which there corresponds an effective negative pressure. Such negative pressure acts repulsively at large scales, opposing itself to the gravitational attraction. It has become customary to characterize such energy density by the term ”dark”.
The simplest and best known candidate for the ”dark energy” is the cosmological constant. As of today, the CDM (Cold Dark Matter) model, which is obtained by adding a cosmological constant to the standard model, is the one which is in better agreement with the cosmological observations, the latter being progressively more precise. Recent data show that dark energy behaves as a cosmological constant within a few percent error. In addition, if the description provided by the CDM model is correct, Friedmann’s equation shows that the remaining energy components must in the future progressively thin out and eventually vanish thus letting the cosmological constant term alone survive.
In the above scenario the de Sitter geometry [3, 4], which is the homogeneous and isotropic solution of the vacuum Einstein equations with cosmological term, appears to take the double role of reference geometry of the universe, namely the geometry of spacetime deprived of its matter and radiation content and of geometry that the universe approaches asymptotically. On the other hand, it seems reasonable to imagine that the presence of a small cosmological constant, while having a huge impact on our understanding of the universe as a whole, would not influence microphysics in its quantum aspects. However this conclusion may have to be reassessed, because in the presence of a cosmological constant, however small, it is the notion of elementary particle itself which has to be reconsidered: indeed, the usual asymptotic theory is based on concepts which refer closely to the global structure of Minkowski spacetime and to its Fourier representation, and do not apply to the de Sitter universe which is not asymptotically flat. Secondly, even if one may think that interactions between elementary particles happen in a ”laboratory” so that ”infinity” is a distance of the order of meters, our present understanding of perturbative quantum field theory is also based on global concepts; in particular, the calculation of perturbative amplitudes involves integrations over the whole spacetime manifold and it should be expected that different topological global structures result in different physical properties in the ”small”.
The literature about de Sitter quantum field theory is very extensive, but there is no comparison with the understanding one has of Minkowskian field theory as regards both general and structural results as a well as its operative and computational possibilities. This second point is particularly doleful: calculations of perturbative amplitudes which in the Minkowskian case would be simple or even trivial become rapidly prohibitive or impossible in the case of de Sitter or anti de Sitter universes: this in spite of the fact that one is dealing with maximally symmetric manifolds which have invariance groups of the same dimension as the Poincaré group. The technical, but also the physical, difference lies precisely in the above mentioned fact that much of the usual quantum field theory is based on concepts which are characteristic of the global structure of Minkowski spacetime and which do not persist in the presence of curvature, already in the presence of a mere cosmological constant, where Minkowskian spacetime is replaced by the de Sitter or by the anti de Sitter one.
In this paper we give a full description of how to solve the problem of calculating the mean lifetime of unstable scalar particles on de Sitter spacetime at first order in perturbation theory. This interesting physical problem provides also an example of a concrete perturbative calculation in presence of the cosmological constant. The task already presents considerable mathematical difficulties.
To our knowledge this calculation was first taken up by O. Nachtmann [5] in 1968. He showed, in a very special case, that while a Minkowskian particle can never decay into heavier products, a dSparticle can, although this effect is exponentially small in the dSradius.
The subject has acquired a greater physical interest with the advent of inflationary cosmology. In particular, the idea that particle decays during the (quasi)de Sitter phase may have important consequences on the physics of the early universe has been suggested recently [6, 7, 8]. The mathematical and physical difficulties related to the lack of timetranslation symmetry of the de Sitter universe, and more generally of nonstatic cosmological backgrounds, have been tackled [6, 7, 8] by using the SchwingerKeldysh formalism, which is suitable for studying certain aspects of the quantum dynamics of systems out of equilibrium. An important ingredient of this approach is the so called Dynamic Renormalization Group [9] which allows a kind of resummation of an infinite series of infrared diverging quantities. That method is however based on the introduction of a practical notion of lifetime of an unstable particle which is quite different from the definition commonly used in quantum physics. Also, the hard technical difficulties of the concrete calculation involved in solving a complicated integrodifferential equation have only been faced in the favorably special conformal and minimally coupled massless cases although in principle the method can be used to deal with particles of generic mass [6, 7, 8].
In this paper we perform a computation which is similar to the one outlined by Nachtmann and follows the conventional quantum field theoretical perturbative approach for computing probability amplitudes. Our work gives significantly wider results w.r.t. [5], e.g. regarding the socalled adiabatic limit, complementaryseriesparticles, and explicit expressions of the relevant KällénLehmann weights. On the other side comparing our result with those of [6, 7, 8] is not easy because of the non standard (but interesting) definition of lifetime chosen in [6, 7, 8].
These findings have been summarized in a recent short communication [10]. The results exhibit significant differences compared to the Minkowski case, and decay processes which are normally forbidden become possible and, viceversa, processes that are normally possible are now forbidden. The maximal symmetry of the de Sitter universe implies the existence of a global squaremass operator, one of the two Casimir operators of the de Sitter group (see e.g. [11]); this quantity is conserved for de Sitter invariant field theories. However, in contrast with the Poincaré group case, the tensor product of two unitary irreducible representations of masses and decomposes into a direct integral of representations whose masses do not satisfy the ‘subadditivity condition’ : all representations of mass larger than a certain critical value (principal series) appear in the decomposition. This fact was shown in [5] for the twodimensional case and will be established here in general. This means that the de Sitter symmetry does not prevent a particle with mass in the principal series from decaying into e.g. pairs of heavier particles. This phenomenon also implies that there can be nothing like a mass gap in that range. This is a major obstruction to attempts at constructing a de Sitter Smatrix; the Minkowskian asymptotic theory makes essential use of an isolated point in the spectrum of the mass operator, and this will generally not occur in the de Sitter case. We will also show that the tensor product of two representations of sufficiently small mass below the critical value (complementary series) contains an additional finite sum of discrete terms in the complementary series itself (at most one term in dimension 4). This implies a form of particle stability, but the new phenomenon is that a particle of this kind cannot disintegrate unless the masses of the decay products have certain quantized values. Stability for the same range of masses has also been recently found [8] in a completely different context. Other remarks about the physical meaning and applicability of our results will be presented in the concluding section.
1.1 Notation
We denote the open upper complex halfplane. Let , . The function is defined as holomorphic on and real on and as . It is entire in . If and , then . If and , then . We define as holomorphic on and asymptotic to at large . It is Herglotz, negative on and positive on .
2 Free fields in Minkowski and de Sitter spacetimes
In this section we give a short summary of the theory of free and generalized free quantum fields on de Sitter spacetime. Since there are infinitely many inequivalent representations of the field algebra, a (mathematical) choice has to be made on physical grounds. Ours is based on the analyticity properties of the vacuum expectation values: see the condition (W2) below. In the Minkowski space, this is equivalent to the positivity of the energy. In the de Sitter case, it admits a thermal interpretation ([12, 13, 14, 15]). The reader can find in [13, 14, 15] a general approach to to de Sitter QFT based on such analytic properties. It includes the so called BunchDavies, also called Euclidean vacuum of de Sitter scalar KleinGordon fields as a basic example.
The real (resp. complex) dimensional Minkowski spacetime (resp. ) is (resp. ) equipped with the Lorentzian inner product
(2.1) 
w.r.t. an arbitrarily chosen Lorentz frame . When no ambiguity arises, . The real (resp. complex) de Sitter spacetime (resp. ) with radius are the hyperboloids
(2.2) 
equipped with the pseudoriemannian metric induced by (2.1). is the connected Lorentz group acting on , and is the connected complex Lorentz group acting on . The connected group of displacements on (resp. ) is (resp. ), sometimes denoted (resp. ). These groups act transitively. Note that our definition of etc. arbitrarily selects a particular orthonormal basis in or in . These particular Lorentz frames will be useful in the sequel. In the future and past open cones and the future and past lightcones are given by
(2.3) 
The future and past tubes in the complex Minkowski spacetime are given by:
(2.4) 
The future and past tuboids in are the intersections of the future and past tubes in with the complex de Sitter manifold :
(2.5) 
We will use the letter to denote either or when the same discussion applies to both, denoting the complexified object. will denote the standard invariant measure on , i.e. using the frame , in the case of , and for .
A (neutral scalar) generalized free field on is entirely specified by its 2point function. This is a tempered distribution on (we denote ), which we require to have the following properties:

(W1) Hermiticity:
(2.6) 
(W2) Analyticity and invariance: there is a function of one complex variable, holomorphic in the cut plane , with tempered behavior at infinity and at the boundaries, such that, in the sense of tempered distributions,
(2.7) hence
(2.8) For complex such that we will denote . Note that this implies
(2.9) and
(2.10) (W1) and (W2) also imply
(2.11) 
(W3) Positivity: For every ,
(2.12)
Conversely, given and having these properties, after having identified the kernel we can construct a Hilbert space by completing equipped with the scalar product , and then exponentiate into a Fock space
(2.13) 
The vacuum is the unit vector . There is a continuous unitary representation of the Poincaré or de Sitter group acting on and preserving the , with . The generalized free field is defined on a dense domain in and .
As a result of the analyticity property (W2), the Wick powers of a generalized free field are welldefined local fields operating in the same Fock space. Their vacuum expectation values are obtained by the standard Wick formulae as sums of products of .
We note that a function on possessing the properties (W1) and (W2) automatically extends (through (2.7)) to a function with the same properties on , so that a generalized free field on has an extension as a generalized free field on . However the extension of need not satisfy (W3) on even if it does on .
A free field of mass on is a generalized free field such that is a solution of the KleinGordon equation with mass in both arguments, and is normalized so as to obey the canonical commutation relations. In that case is uniquely determined by and will be denoted . In the Minkowskian case, the representation is irreducible and equivalent to the representation of the Poincaré group.
As usual, the representation provides a representation of the Lie algebra of the (Poincaré or de Sitter) group and its envelopping algebra by selfadjoint (or selfadjoint) operators on . In particular the squaremass operator is given by in the Minkowskian case, and by in the de Sitter case. In both cases, for every . See e.g. [11].
2.1 Special features of free fields in de Sitter spacetime
In the de Sitter case the mass can be related to a dimensionless parameter as follows
(2.14) 
(2.15) 
In this case, if no ambiguity arises, we shall often denote to mean , and similarly and . Explicitly, if , , and hence does not belong to the real interval ),
(2.16)  
(2.17) 
Since is entire in , , and the rhs of (2.17) is meromorphic in with simple poles at , an integer. In other words extends to a holomorphic function of , and in the domain . However possesses the positivity property (W3) (see (2.12)) only if either

(1) is real, i.e. . In this case is an irreducible unitary representation of the “principal series”.
or

(2) is pure imaginary with , i.e. . In this case is an irreducible unitary representation of the “complementary series”.
We shall need a small part of the harmonic analysis on the de Sitter spacetime as developed in [14]. If and , then , so that is welldefined and holomorphic in in . The role of plane waves on is played by the distributions
(2.18) 
An important formula expressing the de Sitter case twopoint as a Fourier superposition of planewaves is the following (see [14]):
(2.19) 
where , , and
(2.20) 
In (2.19), is a ()cycle in homologous to the sphere . The ()form is given, in the standard coordinates, by
(2.21) 
If a smooth function on is homogeneous of degree , the form is closed, so that the linear functional
(2.22) 
is independent of . This implies that it is Lorentzinvariant. We often denote the measure defined on by the restriction of . In particular the restriction of to the sphere is the standard volume form on that sphere, normalized by . It is possible to take the limit of (2.19), in the sense of distributions, when and tend to the reals:
(2.23) 
Comparing (2.17) with (2.19) and (2.20) gives
(2.24) 
Both sides of this equation are holomorphic in in the domain , hence the equation (2.24) holds in this domain.
Remark 2.1
If is a homogeneous distribution of degree on , it can be restricted to any submanifold of dimension which is transversal to the generators of , in particular to hyperplanar sections such as and . If is of this type and compact, is welldefined and, if , it is independent of .
Remark 2.2
For any complex , is in and holomorphic in on and it is an entire function of . For each it has a limit in the sense of tempered distributions on as tends to the reals, and this has been denoted . It is an entire function in . Furthermore its invariance under implies that, if , is in . Indeed any small displacement of can be effected by a group transformation close to the identity, which can be transferred to and thence to . In the same way, is in (as well as homogeneous) when integrated with a smooth testfunction in . This explains the meaning of formulae such as (2.23). Note that the integral in this formula is entire in . For similar reasons, for any , is in and meromorphic in .
2.2 More features common to Minkowski and de Sitter spacetime
An important formula, which holds in Minkowski as well as in de Sitter spacetime (but in this case only if ), is the projector identity:
(2.25) 
Here
(2.26)  
(2.27) 
The proof of the above identity is trivial in the Minkowskian case. For the de Sitter case it will be provided in Appendix D. Note that tends to as for a fixed .
The KällénLehmann decomposition theorem exists in both and . In the case of , (see [16], p. 360), it asserts that, for every having the properties (W1) and (W2) there is a tempered such that
(2.28) 
If satisfies (W3), then is a tempered positive measure. The same holds in the dS case provided satisfies some decrease property. In this case, the integral runs on masses of the principal series, i.e. . For proofs and details, see [14, 17]. In particular if and, in the dS case, for ,
(2.29) 
Here in the de Sitter case, in the Minkowski case.
3 Particle decays: general formalism
There is at the moment nothing like the HaagRuelle asymptotic theory (HRT) (see [18, 19, 16]) for the de Sitter universe. Indeed all the ingredients of that theory are missing in the de Sitter case. For example, as it will be shown in this paper, even in a free field theory of mass , the mass is not an isolated point in the mass spectrum. Moreover the solutions of the KleinGordon equation do not have the kind of localization at infinity which plays an essential role in the HRT. The concept of a particle is therefore not obvious in de Sitter spacetime, except for localized observations. Here we adopt Wigner’s point of view: a oneparticle vector state is a state belonging to an invariant subspace of the Hilbert space in which the representation of the invariance group reduces to an irreducible representation. In the dS case, we also require that this irreducible representation belong to the principal or complementary series, i.e. it should be equivalent to one of the representations which occur in the of a free field.
We shall study the decay of a particle using firstorder perturbation theory. The initial framework and calculations are the same for the Minkowski and de Sitter cases: its ingredients are the projector identity and the KällénLehmann representation. (It can also be extended to the Minkowskian thermal case ([20]) although there is no KällénLehmann representation there). Let
(3.30) 
be independent free scalar fields with masses , acting in a common Fock space , the tensor product of the individual Fock spaces for the :
(3.31) 
(3.32) 
We denote
(3.33) 
This is the subspace of states in containing particles. denotes the hermitian projector onto this subspace. We now switch on an interaction term
(3.34) 
Here the are nonnegative integers, and we denote . is a small constant. is a smooth, rapidly decreasing function over . In the end, should be made to tend to 1 (adiabatic limit). According to perturbation theory, the transition amplitude between two normalized states and in is given by , where is the formal series in
(3.35) 
In (3.35), denotes the (renormalized) timeordered product of . In the first order in , the transition amplitude between two orthogonal states and is
(3.36) 
We take
(3.37)  
(3.38) 
where and are smooth rapidly decreasing functions. The states of the form (3.37) generate and the states of the form (3.38) generate . The probability of transition from to any state in is:
(3.39)  
(3.40) 
From now on, we suppose, in the dS case, that , , i.e all particles belong to the principal series. We may then replace the central twopoint function in and by its KällénLehmann decomposition:
(3.41) 
Here occurs times as an argument of . This gives
(3.42)  
(3.43) 
The next step would be the socalled adiabatic limit, and should consist in letting the cutoff tend to 1 in this formula. It is however easier to set first only one of the ’s equal to 1, say in (3.43). It then becomes possible to perform the integration over by using the projector identity (2.25) and we find for the transition probability:
(3.44) 
where
(3.45) 
This formula exhibits an interesting factorization: the first factor depends only on the wavepacket , the mass of the incoming particle and the switchingoff factor ; the adiabatic limit still remains to be done there; the second factor contains all the information about the decay products.
If we now attempt to set in (3.45) and to integrate over using again (2.25), the result is proportional to , i.e. the integral diverges. This difficulty was resolved in the 1930’s by aiming at the average transition probability per unit time (see e.g. [21], pp. 6062). We first review the wellknown Minkowski case, in a form which can serve as a model for the de Sitter case. In fact even this famous old case deserves some reexamination on its own right and it is possible, in this case, to allow the two in (3.43) to tend to 1 simultaneously, or even at different rates. This is done in Appendix A. It is found that, if both are taken as in (4.48), the result is the same as found above. But this is not necessarily the case for other . Nevertheless the procedure announced above (i.e. setting the first in (3.43) be equal to 1, then discussing the time average of the limit as the second tends to 1) will be used in the de Sitter case, since it gives good results in the Minkowski case, and since calculations in the dS case would become much more difficult otherwise. Note that in the de Sitter case (3.44) and (3.45) are applicable only when and the range of integration over in (3.43) contains only values (). In the case of the decay into two particles of mass , it will be seen below that this includes the case , but also the case .
4 Minkowski case
4.1 Adiabatic limit: the Fermi golden rule
The simplicity of the Minkowskian case arises from being able to use of the Fourier representations:
(4.46) 
Then the factor in (3.45) becomes
(4.47) 
We now specialize the cutoff to depend only on the time coordinate of the chosen frame , i.e. we think of the interaction as smoothly switched on and then turned off. The Fourier representation is then and eq. (4.47) becomes
(4.48) 
If we choose for the indicator function of a timeslice of thickness , i.e. , , we get
(4.49) 
Therefore, as noted above, removing the cutoff produces infinity. However, according to the Fermi golden rule, what is physically meaningful is not the amplitude but the amplitude per unit time. Therefore, dividing this by and taking the limit as (a particularly trivial operation in this case) we finally get the following expression for the transition probability per unit time:
(4.50) 
The reciprocal of this expression is the lifetime of the 0particle in the state . The dependence on the wavepacket is a crucial feature of the special relativistic Minkowski case as it will be readily recognized. For instance to compute the lifetime of a particle at rest in the chosen frame we may let tend to , e.g. by taking
(4.51) 
with , and letting . Then (4.50) tends to
(4.52) 
We may act with a Lorentz boost on the same particle by replacing in (4.50) the wavepacket by , ; the amplitude is modified as follows