We study the dynamics of linear gravitational perturbations on cosmological backgrounds of massive fermionic fields. We observe that, when gravitational and matter action are expanded to quadratic order in gravitational perturbations on cosmological backgrounds, the graviton appears to have an off-shell mass. We derive a consistent set of two equations for the evolution of linear classical and quantum gravitational perturbations on general cosmological backgrounds, and demonstrate that the graviton mass disappears at the level of equations of motion (on-shell). In this work we consider a Universe whose expansion is driven by the one-loop quantum backreaction of fermions, and dynamical gravitons that evolve on that same background. These equations govern the evolution of linear gravitational perturbations on general cosmological matter backgrounds. A concrete one-loop calculation is performed for the simple case of massive Dirac fermions when the temperature of the cosmological fluid changes adiabatically when compared with the expansion rate of the Universe.

China

Leihua Liu

Netherlands

Tomislav Prokopec

Appearances are deceptive: can graviton have a mass?

The fact that graviton propagator contains not only one but two tensorial components excludes a unique definition of the running behavior of the gravitational constant, while at low energies gravitation is characterized solely by Newton’s constant. How these two facts are reconciled when massive quantum fields are present remains unanswered. In this work, by non-minimally coupling gravity to a one-loop massive scalar, we show that this potential conflict is resolved by the non-trivial equivalence between the residues of the two propagator components. Such equivalence is based on a hidden connection between the spin-2 and spin-0 sectors of the propagator. It is verified that this connection also makes the two quantum-corrected gravitational potentials be characterized by the same gravitational constant at large distances. In addition, we find that the potentials in our case as well as the quantum-corrected Coulomb potential can be expressed concisely in a unified formulation. By comparing these results with experiments, we establish a new upper bound on the magnitude of the non-minimal coupling parameter $\textit{ξ}$.

Duojie Jimu

Uniqueness of gravitational constant at low energies from the connection between spin-2 and spin-0 sectors

We derive the quantum effective action up to second order in gradients and up to two-loop order for an interacting scalar field theory. This expansion of the effective action is useful to study problems in cosmological settings where spatial or time gradients are important, such as bubble nucleation in first-order phase transitions. Assuming spacetime dependent background fields, we work in Wigner space and perform a midpoint gradient expansion, which is consistent with the equations of motion satisfied by the propagator. In particular, we consider the fact that the propagator is non-trivially constrained by an additional equation of motion, obtained from symmetry requirements. At one-loop order, we show the calculations for the case of a single scalar field and then generalise the result to the multi-field case. While we find a vanishing result in the single field case, the one-loop second-order gradient corrections can be significant when considering multiple fields. As an example, we apply our result to a simple toy model of two scalar fields with canonical kinetic terms and mass mixing at tree-level. Finally, we calculate the two-loop one-particle irreducible (1PI) effective action in the single scalar field case, and obtain a nonrenormalisable result. The theory is rendered renormalisable by adding two-particle irreducible (2PI) counterterms, making the 2PI formalism the right framework for renormalization when resummed 1PI two-point functions are used in perturbation theory.

Sofia Canevarolo

Gradient corrections to the quantum effective action

Photon propagator for power-law inflation is considered in the general covariant gauges within the canonical quantization formalism. Photon mode functions in covariant gauges are considerably more complicated than their scalar counterparts, except for the special choice of the gauge-fixing parameter we call the simple covariant gauge. We explicitly construct the position space photon propagator in the simple covariant gauge, and find the result considerably more complicated than its scalar counterpart. This is because of the need for explicitly inverting the Laplace operator acting on the scalar propagator, which results in Appell’s fourth function. Our propagator correctly reproduces the de Sitter and flat space limits. We use this propagator to compute two simple observables: the off-coincident field strength-field strength correlator and the energy-momentum tensor, both of which yield consistent results. As a spinoff of our computation we also give the exact expression for the Coulomb gauge propagator in power-law inflation in arbitrary dimensions.

Croatia

Silvije Domazet

Czechia

Dražen Glavan

Photon propagator for inflation in the general covariant gauge

We consider the classical attractor regime of the spectator Abelian Higgs model in power-law inflation, and compute the one-loop corrections to its evolution. For computations we utilize dimensional regularization and the propagators in the unitary gauge. The corrections to both the scalar condensate and the energy-momentum tensor exhibit secular ultraviolet contributions, that tend to slow down the rolling of the scalar down its potential, and drive it away from the classical attractor. These corrections need not be suppressed if the U(1) charge is much larger than the scalar self-coupling, which is seen already in flat space. In addition, at late times the secular corrections necessarily invalidate the perturbative loop expansion. We find the late time secular corrections to be captured by the renormalization group, which opens up the possibility to resum them past the breakdown of perturbativity.

When tadpoles matter: one-loop corrections for spectator Higgs in inflation

The propagator for the massless vector field in de Sitter space cannot maintain de Sitter invariance in the general covariant gauge, except in the exactly transverse limit. This is due to a previously overlooked Ward-Takahashi identity that the propagator must satisfy. We construct the propagator that satisfies all the necessary conditions, and that preserves cosmological symmetries and dilation, but breaks spatial special conformal transformations. Our propagator vanishes in the infrared, contrary to previously reported de Sitter invariant results, which is relevant for loop computations.

Even the photon propagator must break de Sitter symmetry

We consider a free photon field in D-dimensional de Sitter space, and construct its propagator in the general covariant gauge. Canonical quantization is employed to define the system starting from the classical theory. This guarantees that the propagator satisfies both the equation of motion and subsidiary conditions descending from gauge invariance and gauge fixing. We first construct the propagator as a sum-over-modes in momentum space, carefully accounting for symmetry properties of the state. We then derive the position space propagator in a covariant representation, that is our main result. Our conclusions disagree with previous results as we find that the position space photon propagator necessarily breaks de Sitter symmetry, except in the exact transverse gauge limit.

Photon propagator in de Sitter space in the general covariant gauge

We study the dynamics of Majorana fermions in an expanding de Sitter space and to that aim, by making use of the method of mode sums, we construct the vacuum Feynman propagator for Majorana fermions in de Sitter space. The Majorana condition implies nontrivial identities for the mode functions, which we carefully implement. We note that, under charge conjugation, the propagator transforms to minus itself, but do not discuss in detail the topological implications of this observation. We construct the propagator for a general complex mass and find that it is identical to the corresponding Dirac propagator, meaning that the coupling of fermions to the classical de Sitter gravitational background does not violate charge symmetry, i.e. it respects the Majorana condition. The complex mass propagator differs from its real mass counterpart in that the usual positive and negative energy projectors acquire a chiral rotation which depends on the phase of the mass term. We use our propagator to calculate the one-loop effective action for Majorana fermions, and find that it differs from that of Dirac fermions by a factor of 1/2, which accounts for the reduced number of degrees of freedom of the Majorana particle. Finally, we derive the Majorana and Dirac propagators for mixing n fermionic flavors and show that the number of CP-violating phases for Dirac fermions exceeds that of the Majorana fermions by  $${n\left( n-1\right) }/{2}$$  <math> <mrow> <mrow> <mi>n</mi> <mfenced> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mfenced> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </math>  . This is vital for understanding how the dynamics of Dirac and Majorana fermions generate different CP-violating effects in the multi-flavor case.

Vishnu Unnithan

Majorana propagator on de Sitter space

Local measurements of the Hubble parameter obtained from the distance ladder at low redshift are in tension with global values inferred from cosmological standard rulers. A key role in the tension is played by the assumptions on the cosmological history, in particular on the origin of dark energy. Here we consider a scenario where dark energy originates from the amplification of quantum fluctuations of a light field in inflation. We show that spatial correlations inherited from inflationary quantum fluctuations can reduce the Hubble tension down to one standard deviation, thus relieving the problem with respect to the standard cosmological model. Upcoming missions, like Euclid, will be able to test the predictions of models in this class.

Enis Belgacem

Quantum origin of dark energy and the Hubble tension

We derive a general quantum field theoretic formula for the force acting on expanding bubbles of a first order phase transition in the early Universe setting. In the thermodynamic limit the force is proportional to the entropy increase across the bubble of active species that exert a force on the bubble interface. When local thermal equilibrium is attained, we find a strong friction force which grows as the Lorentz factor squared, such that the bubbles quickly reach stationary state and cannot run away. We also study an opposite case when scatterings are negligible across the wall (ballistic limit), finding that the force saturates for moderate Lorentz factors thus allowing for a runaway behavior. We apply our formalism to a massive real scalar field, the standard model and its simple portal extension. For completeness, we also present a derivation of the renormalized, one-loop, thermal energy-momentum tensor for the standard model and demonstrate its gauge independence.