This letter investigates the contribution of the <math altimg='si9.svg'><mrow><msqrt><mrow><mo>&#8722;</mo><mi>g</mi></mrow></msqrt><mi>&#958;</mi><mi>R</mi><msup><mi>&#981;</mi><mn>2</mn></msup></mrow></math> non-minimal interaction to the long range gravitational potential for massive scalar fields, found from the non-relativistic limit of the 2-2 scattering amplitude with graviton exchanges. Such coupling is most naturally motivated from the renormalisation of a scalar field theory with quartic self interaction in a curved spacetime. This is qualitatively different from the minimal ones like <math altimg='si10.svg'><mrow><msqrt><mi>G</mi></msqrt><msup><mi>h</mi><mrow><mi>&#956;</mi><mi>&#957;</mi></mrow></msup><msub><mi>T</mi><mrow><mi>&#956;</mi><mi>&#957;</mi></mrow></msub></mrow></math>, as the vertices corresponding to the former does not explicitly contain any scalar momenta, but instead explicitly contains the momentum carried by graviton line. For the minimal vertex, the long range gravitational potential up to one loop <math altimg='si11.svg'><mrow><mo>(</mo><mi mathvariant='script'>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi mathvariant='script'>O</mi><mrow><mo>(</mo><msup><mi>G</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>)</mo></mrow></math> was obtained earlier from the terms non-analytic in the transfer momentum, <math altimg='si12.svg'><mrow><msup><mi>q</mi><mrow><mo>&#8722;</mo><mn>2</mn></mrow></msup><mo>,</mo><mspace width='0.33em' /><msup><mi>q</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace width='0.33em' /><mi>ln</mi><msup><mi>q</mi><mn>2</mn></msup></mrow></math>, yielding potentials respectively like <math altimg='si13.svg'><msup><mi>r</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup></math>, <math altimg='si14.svg'><msup><mi>r</mi><mrow><mo>&#8722;</mo><mn>2</mn></mrow></msup></math>, <math altimg='si15.svg'><msup><mi>r</mi><mrow><mo>&#8722;</mo><mn>3</mn></mrow></msup></math>. However owing to the aforesaid explicit appearance of graviton&#8217;s transfer momentum for the non-minimal vertices, the leading contribution in this case comes at <math altimg='si6.svg'><mrow><mi mathvariant='script'>O</mi><mo>(</mo><mi>&#958;</mi><msup><mi>G</mi><mn>2</mn></msup><mo>)</mo></mrow></math>, and turns out to be subleading compared to even <math altimg='si15.svg'><msup><mi>r</mi><mrow><mo>&#8722;</mo><mn>3</mn></mrow></msup></math>. In order to complement this &#8216;screening&#8217; effect, we consider the three graviton vertex generated by the <math altimg='si16.svg'><mrow><mo>&#8764;</mo><mstyle mathvariant='normal'><mi>&#923;</mi></mstyle><msqrt><mrow><mo>&#8722;</mo><mi>g</mi></mrow></msqrt><mo linebreak='goodbreak'>/</mo><mi>G</mi></mrow></math> term in the action, where &#923; is the cosmological constant. This vertex does not contain any graviton momentum explicitly. With this vertex, and assuming short scale scattering much small compared to the Hubble horizon, we compute the seagull, the vacuum polarisation and the fish diagrams and obtain the 2-2 scattering amplitudes. The leading two body gravitational potential at <math altimg='si17.svg'><mrow><mi mathvariant='script'>O</mi><mo>(</mo><mi>&#958;</mi><mstyle mathvariant='normal'><mi>&#923;</mi></mstyle><msup><mi>G</mi><mn>2</mn></msup><mo>)</mo></mrow></math> behaves like <math altimg='si13.svg'><msup><mi>r</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup></math>, even though it is much subleading compared to Newton&#8217;s potential due to the appearance of &#923;. We also discuss the scenario where this potential dominates the aforesaid <math altimg='si6.svg'><mrow><mi mathvariant='script'>O</mi><mo>(</mo><mi>&#958;</mi><msup><mi>G</mi><mn>2</mn></msup><mo>)</mo></mrow></math> one.

India

Avijit Sen Majumder

Sourav Bhattacharya

<italic>&#958;R&#981;</italic> <sup>2</sup> coupling, cosmological constant and quantum gravitational corrections to Newton&#8217;s potential

We have studied the decoherence mechanism in a fermion and scalar quantum field theory with the Yukawa interaction in the Minkowski spacetime, using the non-equilibrium effective field theory formalism appropriate for open systems. The scalar field is treated as the system whereas the fermions as the environment. As the simplest realistic scenario, we assume that an observer measures only the Gaussian 2-point correlator for the scalar field. The cause of decoherence and the subsequent entropy generation is the ignorance of information stored in higher-order correlators, Gaussian and non-Gaussian, of the system and the surrounding. Using the 2-loop 2-particle irreducible effective action, we construct the renormalised Kadanoff–Baym equation, i.e., the equation of motion satisfied by the 2-point correlators in the Schwinger–Keldysh formalism. These equations contain the non-local self-energy corrections. We then compute the statistical propagator in terms of the 2-point functions. Using the relationship of the statistical propagator with the phase space area, we next compute the von Neumann entropy, as a measure of the decoherence or effective loss of information for the system. We have obtained the variation of the entropy with respect to various relevant parameters. We also discuss the qualitative similarities and differences of our results with the scenario when both the system and the environment are scalar fields.

Nitin Joshi

Shagun Kaushal

Decoherence and entropy generation in an open quantum scalar-fermion system with Yukawa interaction

We study the vacuum polarization effects of the Dirac fermionic field induced by a pointlike global monopole located in the cosmological de Sitter spacetime. First we derive the four orthonormal Dirac modes in this background in a closed form. Quantizing the field using these modes, we then compute the fermionic condensate, <math><mrow><mo>⟨</mo><mn>0</mn><mo>|</mo><mover><mrow><mi>Ψ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>Ψ</mi><mo>|</mo><mn>0</mn><mo>⟩</mo></mrow></math>, as well as the vacuum expectation value of the energy-momentum tensor for a massive Dirac field, regularized in a particular way. We have used the Abel-Plana summation formula in order to extract the global monopole contribution to these quantities and have investigated their variations numerically with respect to relevant parameters.

Md. Sabir Ali

Vacuum polarization of Dirac fermions in the cosmological de Sitter global monopole spacetime

We investigate the degradation of quantum entanglement in the Schwarzschild-de Sitter black hole spacetime, by studying the mutual information and the logarithmic negativity for maximally entangled, bipartite states for massless minimal scalar fields. This spacetime is endowed with a black hole as well as a cosmological event horizon, giving rise to particle creation at two different temperatures. We consider two independent descriptions of thermodynamics and particle creation in this background. The first involves thermal equilibrium of an observer with either of the horizons. We show that as of the asymptotically flat/anti–de Sitter black holes, in this case, the entanglement or correlation degrades with increasing Hawking temperatures. The second treats both the horizons combined in order to define a total entropy and an effective equilibrium temperature. We present a field theoretic derivation of this effective temperature and show that unlike the usual cases, the particle creation does not occur here in causally disconnected spacetime wedges but instead in a single region. Using the associated vacua, we show that in this scenario, the entanglement never degrades but increases with increasing black hole temperature and holds true no matter how hot the black hole becomes or how small the cosmological constant is. We argue that this phenomenon can have no analogue in the asymptotically flat/anti–de Sitter black hole spacetimes.

Entanglement degradation in multi-event horizon spacetimes

The violation of the Bell inequality for Dirac fermions is investigated in the cosmological de Sitter spacetime, in the presence of background electromagnetic fields of constant strengths. The orthonormal Dirac mode functions are obtained and the relevant in-out squeezed state expansion in terms of the Bogoliubov coefficients are found. We focus on two scenarios here: strong electric field and heavy mass limits (with respect to the Hubble constant). Using the squeezed state expansion, we then demonstrate the Bell violations for the vacuum and some maximally entangled initial states. Even though a background magnetic field alone cannot create particles, in the presence of background electric field and or spacetime curvature, it can affect the particle creation rate. Thus, our chief aim here is to investigate the role of the background magnetic field strength in the Bell violation. Qualitative differences in this regard for different maximally entangled initial states are shown. Further extension of these results to the so-called <math><mi>α</mi></math> vacua are also discussed.

Md Sabir Ali

Shankhadeep Chakrabortty

Fermionic Bell violation in the presence of background electromagnetic fields in the cosmological de Sitter spacetime

We derive the response function for a comoving, pointlike Unruh-DeWitt particle detector coupled to a complex scalar field ϕ, in the (3 + 1)-dimensional cosmological de Sitter spacetime. The field-detector coupling is taken to be proportional to ϕ † ϕ. We address both conformally invariant and massless minimally coupled scalar field theories, respectively in the conformal and the Bunch-Davies vacuum. The response function integral for the massless minimal complex scalar, not surprisingly, shows divergences and accordingly we use suitable regularisation scheme to find out well behaved results. The regularised result also contains a de Sitter symmetry breaking logarithm, growing with the cosmological time. Possibility of extension of these results with the so called de Sitter α-vacua is discussed. While we find no apparent problem in computing the response function for a real scalar in these vacua, a complex scalar field is shown to contain some possible ambiguities in the detector response. The case of the minimal and nearly massless scalar field theory is also briefly discussed.

Md Ali

Kinjalk Lochan

Unruh-DeWitt detector responses for complex scalar fields in de Sitter spacetime

In this work we consider two complex scalar fields distinguished by their masses coupled to constant background electric and magnetic fields in the <math><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></math>-dimensional Minkowski spacetime and subsequently investigate a few measures quantifying the quantum correlations between the created particle-antiparticle Schwinger pairs. Since the background magnetic field itself cannot cause the decay of the Minkowski vacuum, our chief motivation here is to investigate the interplay between the effects due to the electric and magnetic fields. We start by computing the entanglement entropy for the vacuum state of a single scalar field. Second, we consider some maximally entangled states for the two-scalar field system and compute the logarithmic negativity and the mutual information. Qualitative differences of these results pertaining to the charge content of the states are emphasised. Based upon these results, we suggest some possible effects of a background magnetic field on the degradation of entanglement between states in an accelerated frame, for charged quantum fields.

Hironori Hoshino

Background magnetic field and quantum correlations in the Schwinger effect

We investigate two measures of quantum correlations and entanglement, namely the violation of the Bell-Mermin-Klyshko (BMK) inequalities and the quantum discord, for Dirac fermions in the cosmological de Sitter background of dimension four. The BMK violation is focused on the vacuum for two- and four-mode squeezed states and the maximum violation is demonstrated. For the quantum discord, we investigate a maximally entangled in-vacuum state. Qualitative similarities as well as differences of our results with that of different coordinations of de Sitter in the context of scalar and fermionic field theory are discussed.

Himanshu Gaur

Some measures for fermionic entanglement in the cosmological de Sitter spacetime

We discuss the field quantization of a free massive Dirac fermion in the two causally disconnected static patches of the de Sitter spacetime, by using mode functions that are normalizable on the cosmological event horizon. Using this, we compute the entanglement entropy of the vacuum state corresponding to these two regions, for a given fermionic mode. Further extensions of this result to more general static spherically symmetric and stationary axisymmetric spacetimes are discussed. For the stationary axisymmetric Kerr-de Sitter spacetime in particular, the variations of the entanglement entropy with respect to various eigenvalues and spacetime parameters are depicted numerically. We also comment on such variations when instead we consider the nonextremal black hole event horizon of the same spacetime.

Shivang Goyal

Dirac fermion, cosmological event horizons, and quantum entanglement

We report a non-trivial feature of the vacuum structure of free massive or massless Dirac fields in the hyperbolic de Sitter spacetime. Here we have two causally disconnected regions, say R and L separated by another region, C. We are interested in the field theory in $$R\cup L$$ <math><mrow><mi>R</mi><mo>∪</mo><mi>L</mi></mrow></math>  to understand the long range quantum correlations between R and L. There are local modes of the Dirac field having supports individually either in R or L, as well as global modes found via analytically continuing the R modes to L and vice versa. However, we show that unlike the case of a scalar field, the analytic continuation does not preserve the orthogonality of the resulting global modes. Accordingly, we need to orthonormalise them following the Gram–Schmidt prescription, prior to the field quantisation in order to preserve the canonical anti-commutation relations. We observe that this prescription naturally incorporates a spacetime independent continuous parameter, $$\theta _{\mathrm{RL}}$$ <math><msub><mi>θ</mi><mi>RL</mi></msub></math> , into the picture. Thus interestingly, we obtain a naturally emerging one-parameter family of $$\alpha $$ <math><mi>α</mi></math> -like de Sitter vacua. The values of $$\theta _{\mathrm{RL}}$$ <math><msub><mi>θ</mi><mi>RL</mi></msub></math>  yielding the usual thermal spectra of massless created particles are pointed out. Next, using these vacua, we investigate both entanglement and Rényi entropies of either of the regions and demonstrate their dependence on $$\theta _{\mathrm{RL}}$$ <math><msub><mi>θ</mi><mi>RL</mi></msub></math> .