Introduction
The first observation of gravitational waves (GWs) by LIGO [1 ] established interferometers as a tool for testing gravity in the dynamical regime and motivated the study of possible signatures of quantum gravity in detectors. Coupling a detector to quantized metric perturbations induces fluctuations in the separation of its test masses, with correlations that depend on the gravitational field state [2 ] . These fluctuations can therefore carry information about the quantization of the field, although identifying a measurable noise contribution and establishing its quantum origin are separate tasks.
In the weak-field regime, quantization of gravity introduces the graviton, a spin-2 particle that is massless in the standard description. Massive spin-2 fields were first studied by Fierz and Pauli [3 ] and later became central to theories of massive gravity. In the Lorentz-invariant Fierz-Pauli theory, a massive graviton carries five physical degrees of freedom, rather than the two of linearized general relativity. Its linear coupling to sources exhibits the van Dam–Veltman–Zakharov discontinuity: the massless limit does not reproduce the predictions of general relativity [4 , 5 ] . Generic nonlinear extensions can also introduce the Boulware–Deser ghost [6 ] . Nonlinear Vainshtein screening [7 ] , relevant in DGP [8 , 9 ] and dRGT gravity [10 ] , can recover general-relativistic predictions near sources. Avoiding the ghost is a separate consistency requirement of the nonlinear theory.
These features need not accompany every graviton mass once Lorentz symmetry is relaxed. Very Special Relativity (VSR), introduced in 2006, replaces Lorentz symmetry by a subgroup, here S I M ( 2 ) SIM(2) S I M ( 2 ) [11 ] . Developed independently of the graviton-mass problem, VSR also offers a possible origin of neutrino masses without additional right-handed neutrino fields [12 ] . The subgroup preserves a preferred null direction n μ n^\mu n μ n μ / ( n ⋅ ∂ ) n^\mu/(n \cdot\partial) n μ / ( n ⋅ ∂ )
Such operators allow mass terms for gauge fields without breaking gauge invariance. These constructions have been studied for non-Abelian fields [13 ] , electroweak theory [14 ] , photons [15 , 16 ] , Kalb–Ramond fields [17 ] and Maxwell–Chern–Simons theory [18 , 19 ] . Their extension to the graviton gives Very Special Linear Gravity, a theory with two physical graviton polarizations and dispersion relation ω 2 = ∣ k ∣ 2 + m g 2 \omega^2 = |\mathbf{k}|^2 + m_g^2 ω 2 = ∣ k ∣ 2 + m g 2 [20 , 21 , 22 ] . The preferred direction produces an anisotropic tidal response, and the construction recovers linearized general relativity as m g → 0 m_g \to0 m g → 0
A massive graviton with only two polarizations raises the question of which distinct signatures it could produce in GW detector arms and whether these effects could be observed. We address this question through the detector’s response to a quantized gravitational field. The usual in-out formulation computes transition amplitudes between specified asymptotic states. Here, instead, we are interested in the dynamics of the detector for a given initial state of the gravitational field, without selecting a final state. There are different equivalent ways to work in such cases. One, which has received increasing attention in some fields, is the Schwinger–Keldysh approach, which provides a real-time description [23 , 24 ] . For a general treatment we refer the reader to [25 , 26 , 27 ] . A second, closely related approach, which we use in this work, is the Feynman-Vernon influence functional [28 ] . We closely follow the treatment in Vernon’s thesis [29 ] to integrate out the gravitational field and obtain an effective action for the detector. The resulting dynamics of the detector include dissipative and a noise term induced by the gravitational field. Our aim is to characterize the corrections to both due to the graviton mass m g m_g m g
Therefore, the outline of this work is as follows: Section II introduces the graviton and detector-arm actions in VSR. Section III presents the effective action for the gravitational detector using the Feynman-Vernon method. Section IV gives the detector equations of motion and noise correlations for vacuum, thermal and squeezed initial states. Section V summarizes the results and their observational scale with some comments. The longer calculations and intermediate steps are collected in four appendices. We use c = ℏ = 1 c = \hbar= 1 c = ℏ = 1 η μ ν = diag ( − 1 , + 1 , + 1 , + 1 ) \eta_{\mu\nu} = \operatorname{diag}(-1,+1,+1,+1) η μν = diag ( − 1 , + 1 , + 1 , + 1 ) k μ = ( ω , k ) k^\mu= (\omega,\mathbf{k}) k μ = ( ω , k ) k 2 = − m g 2 k^2 = -m_g^2 k 2 = − m g 2 n μ = ( 1 , n ) n^\mu= (1,\mathbf{n}) n μ = ( 1 , n ) ∣ n ∣ = 1 \lvert\mathbf{n}\rvert= 1 ∣ n ∣ = 1
Action Calculation
VSR action for linearised gravity
The most general quadratic S I M ( 2 ) SIM(2) S I M ( 2 ) h μ ν h_{\mu\nu} h μν [20 ] and it reads
S h = 1 64 π G ∫ d 4 x h μ ν ( x ) O μ ν ρ σ h ρ σ ( x ) , (1) S_h = \frac{1}{64\pi G}\int d^4x\,h_{\mu\nu}(x)\,\mathcal{O}^{\mu\nu\rho\sigma}\,h_{\rho\sigma}(x), \tag*{(1)} S h = 64 π G 1 ∫ d 4 x h μν ( x ) O μν ρ σ h ρ σ ( x ) , (1) where the operator O \mathcal{O} O
O μ ν α β = ∂ μ ∂ ν η α β + ∂ α ∂ β η μ ν − 1 2 ( ∂ μ ∂ α η ν β + ∂ μ ∂ β η ν α + ∂ ν ∂ β η μ α + ∂ ν ∂ α η μ β ) − η μ ν η α β ∂ 2 + 1 2 ( η μ α η ν β + η μ β η ν α ) ∂ 2 − m g 2 N μ N ν ∂ α ∂ β + m g 2 2 N μ N α ∂ ν ∂ β + m g 2 2 N μ N β ∂ ν ∂ α + m g 2 2 N ν N α ∂ μ ∂ β + m g 2 2 N ν N β ∂ μ ∂ α − m g 2 N α N β ∂ μ ∂ ν + ( m g 2 N μ N ν η α β − m g 2 2 N μ N α η ν β − m g 2 2 N μ N β η ν α − m g 2 2 N ν N β η μ α − m g 2 2 η μ β N ν N α + m g 2 N α N β η μ ν ) ∂ 2 − m g 2 η μ ν ( N α ∂ β + ∂ α N β ) + m g 2 2 η μ α ( N ν ∂ β + ∂ ν N β ) + m g 2 2 η μ β ( N ν ∂ α + ∂ ν N α ) + m g 2 2 η ν α ( N μ ∂ β + ∂ μ N β ) + m g 2 2 η ν β ( N μ ∂ α + ∂ μ N α ) − m g 2 η α β ( N μ ∂ ν + ∂ μ N ν ) + m g 2 η μ ν η α β − m g 2 2 η μ α η ν β − m g 2 2 η μ β η ν α , (2) \begin{aligned}
\mathcal{O}_{\mu\nu\alpha\beta}={}&\partial_\mu\partial_\nu\eta_{\alpha\beta}+\partial_\alpha\partial_\beta\eta_{\mu\nu}
-\frac{1}{2}\left(\partial_\mu\partial_\alpha\eta_{\nu\beta}+\partial_\mu\partial_\beta\eta_{\nu\alpha}+\partial_\nu\partial_\beta\eta_{\mu\alpha}+\partial_\nu\partial_\alpha\eta_{\mu\beta}\right)-\eta_{\mu\nu}\eta_{\alpha\beta}\partial^2\\
&+\frac{1}{2}\left(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}\right)\partial^2
-m_g^2N_\mu N_\nu\partial_\alpha\partial_\beta+\frac{m_g^2}{2}N_\mu N_\alpha\partial_\nu\partial_\beta+\frac{m_g^2}{2}N_\mu N_\beta\partial_\nu\partial_\alpha\\
&+\frac{m_g^2}{2}N_\nu N_\alpha\partial_\mu\partial_\beta+\frac{m_g^2}{2}N_\nu N_\beta\partial_\mu\partial_\alpha-m_g^2N_\alpha N_\beta\partial_\mu\partial_\nu\\
&+\left(m_g^2N_\mu N_\nu\eta_{\alpha\beta}-\frac{m_g^2}{2}N_\mu N_\alpha\eta_{\nu\beta}-\frac{m_g^2}{2}N_\mu N_\beta\eta_{\nu\alpha}-\frac{m_g^2}{2}N_\nu N_\beta\eta_{\mu\alpha}-\frac{m_g^2}{2}\eta_{\mu\beta}N_\nu N_\alpha+m_g^2N_\alpha N_\beta\eta_{\mu\nu}\right)\partial^2\\
&-m_g^2\eta_{\mu\nu}\left(N_\alpha\partial_\beta+\partial_\alpha N_\beta\right)+\frac{m_g^2}{2}\eta_{\mu\alpha}\left(N_\nu\partial_\beta+\partial_\nu N_\beta\right)+\frac{m_g^2}{2}\eta_{\mu\beta}\left(N_\nu\partial_\alpha+\partial_\nu N_\alpha\right)\\
&+\frac{m_g^2}{2}\eta_{\nu\alpha}\left(N_\mu\partial_\beta+\partial_\mu N_\beta\right)+\frac{m_g^2}{2}\eta_{\nu\beta}\left(N_\mu\partial_\alpha+\partial_\mu N_\alpha\right)-m_g^2\eta_{\alpha\beta}\left(N_\mu\partial_\nu+\partial_\mu N_\nu\right)\\
&+m_g^2\eta_{\mu\nu}\eta_{\alpha\beta}-\frac{m_g^2}{2}\eta_{\mu\alpha}\eta_{\nu\beta}-\frac{m_g^2}{2}\eta_{\mu\beta}\eta_{\nu\alpha}, \tag*{(2)}
\end{aligned} O μν α β = ∂ μ ∂ ν η α β + ∂ α ∂ β η μν − 2 1 ( ∂ μ ∂ α η ν β + ∂ μ ∂ β η ν α + ∂ ν ∂ β η μα + ∂ ν ∂ α η μ β ) − η μν η α β ∂ 2 + 2 1 ( η μα η ν β + η μ β η ν α ) ∂ 2 − m g 2 N μ N ν ∂ α ∂ β + 2 m g 2 N μ N α ∂ ν ∂ β + 2 m g 2 N μ N β ∂ ν ∂ α + 2 m g 2 N ν N α ∂ μ ∂ β + 2 m g 2 N ν N β ∂ μ ∂ α − m g 2 N α N β ∂ μ ∂ ν + ( m g 2 N μ N ν η α β − 2 m g 2 N μ N α η ν β − 2 m g 2 N μ N β η ν α − 2 m g 2 N ν N β η μα − 2 m g 2 η μ β N ν N α + m g 2 N α N β η μν ) ∂ 2 − m g 2 η μν ( N α ∂ β + ∂ α N β ) + 2 m g 2 η μα ( N ν ∂ β + ∂ ν N β ) + 2 m g 2 η μ β ( N ν ∂ α + ∂ ν N α ) + 2 m g 2 η ν α ( N μ ∂ β + ∂ μ N β ) + 2 m g 2 η ν β ( N μ ∂ α + ∂ μ N α ) − m g 2 η α β ( N μ ∂ ν + ∂ μ N ν ) + m g 2 η μν η α β − 2 m g 2 η μα η ν β − 2 m g 2 η μ β η ν α , (2) and here m g m_g m g
N μ ≡ n μ n ⋅ ∂ . (3) N^\mu\equiv\frac{n^\mu}{n\cdot\partial}. \tag*{(3)} N μ ≡ n ⋅ ∂ n μ . (3) Notice that when the VSR parameter m g m_g m g
∂ μ h μ ν = n μ h μ ν = 0 , h = h μ μ = 0 , (4) \partial^\mu h_{\mu\nu}=n^\mu h_{\mu\nu}=0,\qquad h=h^\mu_{\ \mu}=0, \tag*{(4)} ∂ μ h μν = n μ h μν = 0 , h = h μ μ = 0 , (4) With this choice, the action (1) reduces to
S h = 1 64 π G ∫ d 4 x [ h μ ν ∂ 2 h μ ν − m g 2 h μ ν h μ ν ] , (5) S_h=\frac{1}{64\pi G}\int d^4x\left[h_{\mu\nu}\partial^2h^{\mu\nu}-m_g^2h_{\mu\nu}h^{\mu\nu}\right], \tag*{(5)} S h = 64 π G 1 ∫ d 4 x [ h μν ∂ 2 h μν − m g 2 h μν h μν ] , (5) which gives the correct Klein-Gordon equation in the mostly-plus convention, i.e.,
( ∂ 2 − m g 2 ) h μ ν = 0 , with ∂ 2 = − ∂ 0 2 + ∇ ⃗ 2 . (6) (\partial^2-m_g^2)h_{\mu\nu}=0,\qquad\text{with }\partial^2=-\partial_0^2+\vec{\nabla}^{\,2}. \tag*{(6)} ( ∂ 2 − m g 2 ) h μν = 0 , with ∂ 2 = − ∂ 0 2 + ∇ 2 . (6) Here, we emphasize that, although we have chosen a specific gauge to simplify the algebraic manipulation of the action, the physical observables and the underlying theory remain strictly gauge invariant. Using integration by parts, we can write (5) as
S h = 1 64 π G ∫ d 4 x [ h μ ν h ¨ μ ν − ∇ ⃗ h μ ν ⋅ ∇ ⃗ h μ ν − m g 2 h μ ν h μ ν ] , (7) S_h=\frac{1}{64\pi G}\int d^4x\left[h_{\mu\nu}\ddot{h}^{\mu\nu}-\vec{\nabla}h_{\mu\nu}\cdot\vec{\nabla}h^{\mu\nu}-m_g^2h_{\mu\nu}h^{\mu\nu}\right], \tag*{(7)} S h = 64 π G 1 ∫ d 4 x [ h μν h ¨ μν − ∇ h μν ⋅ ∇ h μν − m g 2 h μν h μν ] , (7) which corresponds to the action for the graviton to be used.
Detector action
In addition, we consider the action for a gravitational wave detector. We start from the equation for the geodesic distance ξ ⃗ \vec{\xi} ξ m 0 m_0 m 0
∂ s 2 ξ i = − R i 00 j ξ j . (8) \partial_s^2 \xi^i=-R_{i00j}\xi^j . \tag*{(8)} ∂ s 2 ξ i = − R i 00 j ξ j . (8) This equation can be obtained from the action
S d e t = 1 2 m 0 ∫ d t ( ξ ˙ i ξ ˙ i − ξ i R i 00 j ξ j ) , (9) S_{\mathrm{det}}=\frac{1}{2}m_0\int dt\left(\dot{\xi}^i\dot{\xi}^i-\xi^iR_{i00j}\xi^j\right), \tag*{(9)} S det = 2 1 m 0 ∫ d t ( ξ ˙ i ξ ˙ i − ξ i R i 00 j ξ j ) , (9) where we have directly included the standard kinetic contribution proportional to the mass of the particle. Using the expression for the linearised Riemann tensor
R i 00 j = − 1 2 ( h 00 , i j + h i j , 00 − h 0 i , 0 j − h 0 j , 0 i ) , (10) R_{i00j}=-\frac{1}{2}\left(h_{00,ij}+h_{ij,00}-h_{0i,0j}-h_{0j,0i}\right), \tag*{(10)} R i 00 j = − 2 1 ( h 00 , ij + h ij , 00 − h 0 i , 0 j − h 0 j , 0 i ) , (10) we can write the action for the detector as
S d e t = 1 2 m 0 ∫ d t ( δ i j ξ ˙ i ξ ˙ j + 1 2 ( h 00 , i j + h i j , 00 − h 0 i , 0 j − h 0 j , 0 i ) ξ i ξ j ) (11) S_{\mathrm{det}}=\frac{1}{2}m_0\int dt\left(\delta_{ij}\dot{\xi}^i\dot{\xi}^j+\frac{1}{2}\left(h_{00,ij}+h_{ij,00}-h_{0i,0j}-h_{0j,0i}\right)\xi^i\xi^j\right) \tag*{(11)} S det = 2 1 m 0 ∫ d t ( δ ij ξ ˙ i ξ ˙ j + 2 1 ( h 00 , ij + h ij , 00 − h 0 i , 0 j − h 0 j , 0 i ) ξ i ξ j ) (11) Full action
Let us write h μ ν h_{\mu\nu} h μν
h μ ν = 1 G ∑ s ∫ d 4 k ( 2 π ) 4 c s ( k ) ϵ μ ν ( s ) ( k ) e i k x (12) h_{\mu\nu}=\frac{1}{\sqrt{G}}\sum_s\int\frac{d^4k}{(2\pi)^4}c_s(k)\epsilon_{\mu\nu}^{(s)}(k)e^{ikx} \tag*{(12)} h μν = G 1 s ∑ ∫ ( 2 π ) 4 d 4 k c s ( k ) ϵ μν ( s ) ( k ) e ik x (12) where s s s ϵ μ ν s \epsilon_{\mu\nu}^{s} ϵ μν s
ϵ μ ν s ( k ⃗ ) ϵ s ′ μ ν ( k ⃗ ) = δ s ′ s (13) \epsilon_{\mu\nu}^{s}(\vec{k})\epsilon_{s'}^{\mu\nu}(\vec{k})=\delta_{s'}^{s} \tag*{(13)} ϵ μν s ( k ) ϵ s ′ μν ( k ) = δ s ′ s (13) We can choose a real basis for the polarization tensor to write the action in a compact way. This is just a matter of convenience. In the next steps we will work with gauge invariant quantities as the sum of polarizations, so they can be computed in any basis, and here we choose a convenient one just to express the action in a convenient way. Together with this choice, we will consider that
ϵ μ ν ( s ) ( − k ) = ϵ μ ν ( s ) ( k ) (14) \epsilon_{\mu\nu}^{(s)}(-k)=\epsilon_{\mu\nu}^{(s)}(k) \tag*{(14)} ϵ μν ( s ) ( − k ) = ϵ μν ( s ) ( k ) (14) With this condition, and the reality of h μ ν h_{\mu\nu} h μν
c s ∗ ( − k ) = c s ( k ) (15) c_s^*(-k)=c_s(k) \tag*{(15)} c s ∗ ( − k ) = c s ( k ) (15) Using all these expressions, the action becomes
S = 1 64 π G 2 ∑ s ∫ d 4 k ( 2 π ) 4 ∣ c s ( k ) ∣ 2 ( ( k 0 ) 2 − ω 2 ) + 1 2 m 0 ∫ d t ( δ i j ξ ˙ i ξ ˙ j − 1 2 G ∑ s ∫ d 4 k ( 2 π ) 4 e i k x c s ( k ) Λ i j ( s ) ξ ˙ i ξ ˙ j ) (16) \begin{aligned}
S={}&\frac{1}{64\pi G^2}\sum_s\int\frac{d^4k}{(2\pi)^4}|c_s(k)|^2\left((k^0)^2-\omega^2\right)\\
&+\frac{1}{2}m_0\int dt\left(\delta_{ij}\dot{\xi}^i\dot{\xi}^j-\frac{1}{2\sqrt{G}}\sum_s\int\frac{d^4k}{(2\pi)^4}e^{ikx}c_s(k)\Lambda_{ij}^{(s)}\dot{\xi}^i\dot{\xi}^j\right) \tag*{(16)}
\end{aligned} S = 64 π G 2 1 s ∑ ∫ ( 2 π ) 4 d 4 k ∣ c s ( k ) ∣ 2 ( ( k 0 ) 2 − ω 2 ) + 2 1 m 0 ∫ d t ( δ ij ξ ˙ i ξ ˙ j − 2 G 1 s ∑ ∫ ( 2 π ) 4 d 4 k e ik x c s ( k ) Λ ij ( s ) ξ ˙ i ξ ˙ j ) (16) where we defined:
ω 2 = ∣ k ⃗ ∣ 2 + m g 2 (17) \omega^2=|\vec{k}|^2+m_g^2 \tag*{(17)} ω 2 = ∣ k ∣ 2 + m g 2 (17) and the object
Λ i j ( s ) = ϵ 00 ( s ) k i k j + ϵ i j ( s ) ( k 0 ) 2 + ( ϵ 0 i ( s ) k j + ϵ 0 j ( s ) k i ) k 0 (18) \Lambda_{ij}^{(s)}=\epsilon_{00}^{(s)}k_i k_j+\epsilon_{ij}^{(s)}(k^0)^2+\left(\epsilon_{0i}^{(s)}k_j+\epsilon_{0j}^{(s)}k_i\right)k^0 \tag*{(18)} Λ ij ( s ) = ϵ 00 ( s ) k i k j + ϵ ij ( s ) ( k 0 ) 2 + ( ϵ 0 i ( s ) k j + ϵ 0 j ( s ) k i ) k 0 (18) Notice that the interaction term depends on the polarization only through Λ i j ( s ) \Lambda^{(s)}_{ij} Λ ij ( s )
δ ϵ μ ν ( s ) = k μ Λ ν ( s ) + k ν Λ μ ( s ) (19) \delta\epsilon^{(s)}_{\mu\nu}=k_\mu\Lambda^{(s)}_\nu+k_\nu\Lambda^{(s)}_\mu \tag*{(19)} δ ϵ μν ( s ) = k μ Λ ν ( s ) + k ν Λ μ ( s ) (19) for arbitrary Λ μ ( s ) \Lambda^{(s)}_\mu Λ μ ( s )
We can see that in the interaction, if we split the k 0 k^0 k 0 k 0 > 0 k^0>0 k 0 > 0 k 0 < 0 k^0<0 k 0 < 0 k → − k k\to-k k → − k k 0 < 0 k^0<0 k 0 < 0 (15) , the fact that Λ i j ( s ) \Lambda^{(s)}_{ij} Λ ij ( s ) Λ i j ( s ) ( − k ) = Λ i j ( s ) ( k ) \Lambda^{(s)}_{ij}(-k)=\Lambda^{(s)}_{ij}(k) Λ ij ( s ) ( − k ) = Λ ij ( s ) ( k )
∫ d 4 k ( 2 π ) 4 e i k x c s ( k ) Λ i j ( s ) = 2 Re ∫ d 3 k ( 2 π ) 3 ∫ 0 ∞ d k 0 2 π e i k x c s ( k ) Λ i j ( s ) ( k ) . (20) \int\frac{\mathrm{d}^4 k}{(2\pi)^4}e^{ikx}c_s(k)\Lambda^{(s)}_{ij}=2\operatorname{Re}\int\frac{\mathrm{d}^3 k}{(2\pi)^3}\int_0^\infty\frac{\mathrm{d}k^0}{2\pi}e^{ikx}c_s(k)\Lambda^{(s)}_{ij}(k). \tag*{(20)} ∫ ( 2 π ) 4 d 4 k e ik x c s ( k ) Λ ij ( s ) = 2 Re ∫ ( 2 π ) 3 d 3 k ∫ 0 ∞ 2 π d k 0 e ik x c s ( k ) Λ ij ( s ) ( k ) . (20) Similarly we can do it for the first term in (16) , and then we can define
q k , s ( t ) = ∫ 0 ∞ d k 0 2 π c s ( k 0 , k ) e − i k 0 t (21) q_{k,s}(t)=\int_0^\infty\frac{\mathrm{d}k^0}{\sqrt{2}\pi}c_s(k^0,\mathbf{k})e^{-ik^0t} \tag*{(21)} q k , s ( t ) = ∫ 0 ∞ 2 π d k 0 c s ( k 0 , k ) e − i k 0 t (21) Then, using the Parseval theorem we can write
∫ − ∞ ∞ d k 0 2 π ∣ c s ∣ 2 = ∫ d t ∣ q k , s ∣ 2 , ∫ − ∞ ∞ d k 0 2 π ( k 0 ) 2 ∣ c s ∣ 2 = ∫ d t ∣ q ˙ k , s ∣ 2 (22) \int_{-\infty}^{\infty}\frac{\mathrm{d}k^0}{2\pi}|c_s|^2=\int\mathrm{d}t|q_{k,s}|^2,\qquad
\int_{-\infty}^{\infty}\frac{\mathrm{d}k^0}{2\pi}(k^0)^2|c_s|^2=\int\mathrm{d}t|\dot{q}_{k,s}|^2 \tag*{(22)} ∫ − ∞ ∞ 2 π d k 0 ∣ c s ∣ 2 = ∫ d t ∣ q k , s ∣ 2 , ∫ − ∞ ∞ 2 π d k 0 ( k 0 ) 2 ∣ c s ∣ 2 = ∫ d t ∣ q ˙ k , s ∣ 2 (22) Moreover, to first order, the free action imposes ( k 0 ) 2 = ω 2 (k^0)^2=\omega^2 ( k 0 ) 2 = ω 2 k 0 > 0 k^0>0 k 0 > 0 k 0 = ω k^0=\omega k 0 = ω
Λ i j ( s ) = ϵ 00 ( s ) k i k j + ϵ i j ( s ) ω 2 ( k ) + ( ϵ 0 i ( s ) k j + ϵ 0 j ( s ) k i ) ω ( k ) (23) \Lambda^{(s)}_{ij}=\epsilon^{(s)}_{00}k_i k_j+\epsilon^{(s)}_{ij}\omega^2(k)+(\epsilon^{(s)}_{0i}k_j+\epsilon^{(s)}_{0j}k_i)\omega(k) \tag*{(23)} Λ ij ( s ) = ϵ 00 ( s ) k i k j + ϵ ij ( s ) ω 2 ( k ) + ( ϵ 0 i ( s ) k j + ϵ 0 j ( s ) k i ) ω ( k ) (23) Also, in the following, we will consider the length-scale of the detector to be smaller than the characteristic scale λ \lambda λ h μ ν h_{\mu\nu} h μν e i k ⋅ x ≈ 1 e^{ik\cdot x}\approx1 e ik ⋅ x ≈ 1
S = 1 64 π G 2 ∑ s ∫ d t ∫ d 3 k ( 2 π ) 3 ( ∣ q ˙ k , s ∣ 2 − ( ∣ k ∣ 2 + m g 2 ) ∣ q k , s ∣ 2 ) + 1 2 m 0 ∫ d t ( δ i j ξ ˙ i ξ ˙ j − 1 2 G ∑ s ∫ d 3 k ( 2 π ) 3 Λ i j ( s ) Re ( q k , s ) ξ ˙ i ξ ˙ j ) (24) \begin{aligned}
S={}&\frac{1}{64\pi G^2}\sum_s\int\mathrm{d}t\int\frac{\mathrm{d}^3k}{(2\pi)^3}\left(|\dot{q}_{k,s}|^2-(|\mathbf{k}|^2+m_g^2)|q_{k,s}|^2\right)\\
&+\frac{1}{2}m_0\int\mathrm{d}t\left(\delta_{ij}\dot{\xi}^i\dot{\xi}^j-\frac{1}{\sqrt{2}G}\sum_s\int\frac{\mathrm{d}^3k}{(2\pi)^3}\Lambda^{(s)}_{ij}\operatorname{Re}(q_{k,s})\dot{\xi}^i\dot{\xi}^j\right) \tag*{(24)}
\end{aligned} S = 64 π G 2 1 s ∑ ∫ d t ∫ ( 2 π ) 3 d 3 k ( ∣ q ˙ k , s ∣ 2 − ( ∣ k ∣ 2 + m g 2 ) ∣ q k , s ∣ 2 ) + 2 1 m 0 ∫ d t ( δ ij ξ ˙ i ξ ˙ j − 2 G 1 s ∑ ∫ ( 2 π ) 3 d 3 k Λ ij ( s ) Re ( q k , s ) ξ ˙ i ξ ˙ j ) (24) Since the interaction only couples to the real part of q k , s q_{k,s} q k , s q q q k \mathbf{k} k
S = S ξ + S q (25) S=S_\xi+S_q \tag*{(25)} S = S ξ + S q (25) with
S ξ = 1 2 m 0 ∫ − T T d t ξ ˙ i ξ ˙ i (26) S_\xi=\frac{1}{2}m_0\int_{-T}^{T}\mathrm{d}t\dot{\xi}^i\dot{\xi}^i \tag*{(26)} S ξ = 2 1 m 0 ∫ − T T d t ξ ˙ i ξ ˙ i (26) S q = ∫ − T T d t [ 1 2 m e f f [ q ˙ 2 − ω 2 q 2 ] + J q ] (27) S_q=\int_{-T}^{T}\mathrm{d}t\left[\frac{1}{2}m_{\mathrm{eff}}[\dot{q}^2-\omega^2q^2]+Jq\right] \tag*{(27)} S q = ∫ − T T d t [ 2 1 m eff [ q ˙ 2 − ω 2 q 2 ] + J q ] (27) where we defined
g = 1 2 G (28) g=\frac{1}{\sqrt{2G}} \tag*{(28)} g = 2 G 1 (28) m e f f = 1 16 π G 2 (29) m_{\mathrm{eff}}=\frac{1}{16\pi G^2} \tag*{(29)} m eff = 16 π G 2 1 (29) ω 2 = ∣ k ∣ 2 + m g 2 (30) \omega^2=|\mathbf{k}|^2+m_g^2 \tag*{(30)} ω 2 = ∣ k ∣ 2 + m g 2 (30) J = − 1 2 m 0 g Λ i j ( s ) ( ω ) ξ ˙ i ξ ˙ j (31) J=-\frac{1}{2}m_0g\Lambda^{(s)}_{ij}(\omega)\dot{\xi}^i\dot{\xi}^j \tag*{(31)} J = − 2 1 m 0 g Λ ij ( s ) ( ω ) ξ ˙ i ξ ˙ j (31) and we have set explicitly limits of integration for the time integral in the action.
Effective action for the detector
In order to construct an effective action for the detector, we will proceed with a path-integral approach. In [14 ] was argued that path integral in VSR produces correct expressions of the propagator, hamiltonian and charge operator in terms of creation and annihilation operators within the canonical second quantization. Moreover, this approach is simpler and covariant, in contrast with canonical quantization, where the possible existence of second class constraints make this task complicated. Following the Vernon method [29 ] , we can construct an effective action for ξ \xi ξ
S e f f [ ξ ] = S ξ [ ξ ] − S ξ [ ξ ′ ] + S I F [ ξ , ξ ′ ] (32) S_{eff}[\xi] = S_{\xi}[\xi] - S_{\xi}[\xi'] + S_{IF}[\xi,\xi'] \tag*{(32)} S e f f [ ξ ] = S ξ [ ξ ] − S ξ [ ξ ′ ] + S I F [ ξ , ξ ′ ] (32) where the part
S I F [ ξ , ξ ′ ] = − i ln F [ ξ , ξ ′ ] (33) S_{IF}[\xi,\xi'] = -i \ln F[\xi,\xi'] \tag*{(33)} S I F [ ξ , ξ ′ ] = − i ln F [ ξ , ξ ′ ] (33) is defined through the influence functional F [ ξ , ξ ′ ] F[\xi,\xi'] F [ ξ , ξ ′ ] q q q
F [ ξ , ξ ′ ] = ∫ d q T d q τ d q T ′ d q τ ′ δ ( q T − q T ′ ) D q D q ′ ρ q ( q τ , q τ ′ ) exp [ i ( S q [ q , ξ ] − S q [ q ′ , ξ ′ ] ) ] (34) F[\xi,\xi'] = \int dq_{T}dq_{\tau}dq'_{T}dq'_{\tau}\,\delta(q_{T}-q'_{T})DqDq'\rho_q(q_{\tau},q'_{\tau})\exp\left[i\left(S_q[q,\xi]-S_q[q',\xi']\right)\right] \tag*{(34)} F [ ξ , ξ ′ ] = ∫ d q T d q τ d q T ′ d q τ ′ δ ( q T − q T ′ ) D q D q ′ ρ q ( q τ , q τ ′ ) exp [ i ( S q [ q , ξ ] − S q [ q ′ , ξ ′ ] ) ] (34) and here we will consider a quadratic initial density matrix:
ρ q ( q τ , q τ ′ ) = N exp [ − m e f f ω 2 ( A ( q τ − a ) 2 + A ∗ ( q τ ′ − a ∗ ) 2 − B ( q τ − a ) ( q τ ′ − a ∗ ) ) ] (35) \rho_q(q_{\tau},q'_{\tau}) = \mathcal{N}\exp\left[-\frac{m_{eff}\omega}{2}\left(A(q_{\tau}-a)^2+A^*(q'_{\tau}-a^*)^2-B(q_{\tau}-a)(q'_{\tau}-a^*)\right)\right] \tag*{(35)} ρ q ( q τ , q τ ′ ) = N exp [ − 2 m e f f ω ( A ( q τ − a ) 2 + A ∗ ( q τ ′ − a ∗ ) 2 − B ( q τ − a ) ( q τ ′ − a ∗ ) ) ] (35) where we choose the coefficients to parametrize in one single expression three possible choices of initial conditions: Vacuum, Thermal and Squeezed states. With this in mind, we choose the coefficients to be
N = m e f f ω tanh ( β ω 2 ) π ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) (36) \mathcal{N} = \sqrt{\frac{m_{eff}\omega\tanh\left(\frac{\beta\omega}{2}\right)}{\pi\left(\cosh(2r)-\cos\phi\sinh(2r)\right)}} \tag*{(36)} N = π ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) m e f f ω tanh ( 2 β ω ) (36) A = cosh r + e i ϕ sinh r cosh r − e − i ϕ sinh r + coth ( β ω ) − 1 cosh ( 2 r ) − cos ϕ sinh ( 2 r ) (37) A = \frac{\cosh r+e^{i\phi}\sinh r}{\cosh r-e^{-i\phi}\sinh r}+\frac{\coth(\beta\omega)-1}{\cosh(2r)-\cos\phi\sinh(2r)} \tag*{(37)} A = cosh r − e − i ϕ sinh r cosh r + e i ϕ sinh r + cosh ( 2 r ) − cos ϕ sinh ( 2 r ) coth ( β ω ) − 1 (37) B = 2 csch ( β ω ) cosh ( 2 r ) − cos ϕ sinh ( 2 r ) (38) B = \frac{2\operatorname{csch}(\beta\omega)}{\cosh(2r)-\cos\phi\sinh(2r)} \tag*{(38)} B = cosh ( 2 r ) − cos ϕ sinh ( 2 r ) 2 csch ( β ω ) (38) We observe that in the following limits we recover those states:
Here N = m e f f ω π \mathcal{N} = \sqrt{\frac{m_{eff}\omega}{\pi}} N = π m e f f ω A = 1 A = 1 A = 1 B = 0 B = 0 B = 0
ρ q ( q τ , q τ ′ ) = N exp [ − m e f f ω 2 ( q τ 2 + q τ ′ 2 ) ] (39) \rho_q(q_{\tau},q'_{\tau}) = \mathcal{N}\exp\left[-\frac{m_{eff}\omega}{2}\left(q_{\tau}^{2}+q_{\tau}'^{2}\right)\right] \tag*{(39)} ρ q ( q τ , q τ ′ ) = N exp [ − 2 m e f f ω ( q τ 2 + q τ ′ 2 ) ] (39) Here N = m e f f ω tanh ( β ω 2 ) π \mathcal{N} = \sqrt{\frac{m_{eff}\omega\tanh\left(\frac{\beta\omega}{2}\right)}{\pi}} N = π m e f f ω t a n h ( 2 β ω ) A = coth ( β ω ) A = \coth(\beta\omega) A = coth ( β ω ) B = 2 csch ( β ω ) B = 2\operatorname{csch}(\beta\omega) B = 2 csch ( β ω )
ρ q ( q τ , q τ ′ ) = N exp [ − m e f f ω 2 ( A ( q τ 2 + q τ ′ 2 ) − B q τ q τ ′ ) ] (40) \rho_q(q_{\tau},q'_{\tau}) = \mathcal{N}\exp\left[-\frac{m_{eff}\omega}{2}\left(A\left(q_{\tau}^{2}+q_{\tau}'^{2}\right)-Bq_{\tau}q'_{\tau}\right)\right] \tag*{(40)} ρ q ( q τ , q τ ′ ) = N exp [ − 2 m e f f ω ( A ( q τ 2 + q τ ′ 2 ) − B q τ q τ ′ ) ] (40) Here N = m e f f ω π ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) \mathcal{N} = \sqrt{\frac{m_{eff}\omega}{\pi\left(\cosh(2r)-\cos\phi\sinh(2r)\right)}} N = π ( c o s h ( 2 r ) − c o s ϕ s i n h ( 2 r ) ) m e f f ω A = cosh r + e i ϕ sinh r cosh r − e − i ϕ sinh r A = \frac{\cosh r+e^{i\phi}\sinh r}{\cosh r-e^{-i\phi}\sinh r} A = c o s h r − e − i ϕ s i n h r c o s h r + e i ϕ s i n h r B = 0 B = 0 B = 0
ρ q ( q τ , q τ ′ ) = N exp [ − m e f f ω 2 ( A ( q τ − a ) 2 + A ∗ ( q τ ′ − a ∗ ) 2 ) ] (41) \rho_q(q_{\tau},q'_{\tau}) = \mathcal{N}\exp\left[-\frac{m_{eff}\omega}{2}\left(A(q_{\tau}-a)^2+A^*(q'_{\tau}-a^*)^2\right)\right] \tag*{(41)} ρ q ( q τ , q τ ′ ) = N exp [ − 2 m e f f ω ( A ( q τ − a ) 2 + A ∗ ( q τ ′ − a ∗ ) 2 ) ] (41) with this we can do the integrations, which are reported in appendix § 1.1.1 . Then, using this influence functional we can employ a typical procedure used in treatments using Schwinger-Keldysh, a change of variable (Keldysh rotation)
ξ c l = 1 2 ( ξ + ξ ′ ) , ξ q = 1 2 ( ξ − ξ ′ ) (42) \xi_{cl}=\frac{1}{2}(\xi+\xi'),\qquad\xi_q=\frac{1}{2}(\xi-\xi') \tag*{(42)} ξ c l = 2 1 ( ξ + ξ ′ ) , ξ q = 2 1 ( ξ − ξ ′ ) (42) to write the effective action for ξ \xi ξ τ = 0 \tau=0 τ = 0
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ ˙ q i + m 0 2 g 2 ( 2 π ) 3 m e f f ∫ d Ω ∫ 0 ∞ d ∣ k ∣ ∣ k ∣ 2 ω ∑ s Λ i j ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ c l j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + 2 i m 0 2 g 2 ( 2 π ) 3 m e f f ∫ d Ω ∫ 0 ∞ d ∣ k ∣ ∣ k ∣ 2 ω ∑ s Λ i j ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + i 2 m 0 g ( 2 π ) 3 ∫ d Ω ∫ 0 ∞ d ∣ k ∣ ∣ k ∣ 2 ∑ s Λ i j ( s ) A + A ∗ − B ∫ 0 T d t ξ c l i ( t ) ξ q j ( t ) R ( t ) (43) \begin{aligned}
S_{\mathrm{eff}}[\xi]={}&2m_0\int_0^T dt\,\dot{\xi}_{cl}^{i}\dot{\xi}_{q}^{i}\\
&+\frac{m_0^2g^2}{(2\pi)^3m_{\mathrm{eff}}}\int d\Omega\int_0^\infty d\lvert k\rvert\frac{\lvert k\rvert^2}{\omega}\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}
\int_0^Tdt\int_0^Tdt'\,\xi_{cl}^{i}(t)\xi_{cl}^{j}(t)\theta(t'-t)\sin\bigl(\omega(t'-t)\bigr)\xi_{cl}^{l}(t')\xi_q^{m}(t')\\
&+\frac{2im_0^2g^2}{(2\pi)^3m_{\mathrm{eff}}}\int d\Omega\int_0^\infty d\lvert k\rvert\frac{\lvert k\rvert^2}{\omega}\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}
\int_0^Tdt\int_0^Tdt'\,\xi_{cl}^{i}(t)\xi_q^{j}(t)M(t,t')\xi_{cl}^{l}(t')\xi_q^{m}(t')\\
&+\frac{i\sqrt{2}m_0g}{(2\pi)^3}\int d\Omega\int_0^\infty d\lvert k\rvert\frac{\lvert k\rvert^2\sum_s\Lambda_{ij}^{(s)}}{A+A^*-B}
\int_0^Tdt\,\xi_{cl}^{i}(t)\xi_q^{j}(t)R(t) \tag*{(43)}
\end{aligned} S eff [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ ˙ q i + ( 2 π ) 3 m eff m 0 2 g 2 ∫ d Ω ∫ 0 ∞ d ∣ k ∣ ω ∣ k ∣ 2 s ∑ Λ ij ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ c l j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + ( 2 π ) 3 m eff 2 i m 0 2 g 2 ∫ d Ω ∫ 0 ∞ d ∣ k ∣ ω ∣ k ∣ 2 s ∑ Λ ij ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + ( 2 π ) 3 i 2 m 0 g ∫ d Ω ∫ 0 ∞ d ∣ k ∣ A + A ∗ − B ∣ k ∣ 2 ∑ s Λ ij ( s ) ∫ 0 T d t ξ c l i ( t ) ξ q j ( t ) R ( t ) (43) where we have assumed that the variables ξ q \xi_q ξ q
M ( t , t ′ ) = ∣ A ∣ 2 − 1 4 B 2 + 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t − t ′ ) ) − ∣ A ∣ 2 − 1 4 B 2 − 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t + t ′ ) ) − ℑ ( A ) 2 ℜ ( A ) − B sin ( ω ( t + t ′ ) ) (44) M(t,t')=\frac{\lvert A\rvert^2-\frac{1}{4}B^2+1}{2(2\Re(A)-B)}\cos\bigl(\omega(t-t')\bigr)-\frac{\lvert A\rvert^2-\frac{1}{4}B^2-1}{2(2\Re(A)-B)}\cos\bigl(\omega(t+t')\bigr)-\frac{\Im(A)}{2\Re(A)-B}\sin\bigl(\omega(t+t')\bigr) \tag*{(44)} M ( t , t ′ ) = 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 + 1 cos ( ω ( t − t ′ ) ) − 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 − 1 cos ( ω ( t + t ′ ) ) − 2ℜ ( A ) − B ℑ ( A ) sin ( ω ( t + t ′ ) ) (44) R ( t ) = i 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) 2 cos ( ω t ) − 2 ( ∣ A ∣ 2 − 1 4 B 2 ) ( a ∗ − a ) sin ( ω t ) (45) R(t)=i\frac{2(Aa+A^*a^*)-B(a^*+a)}{\sqrt{2}}\cos(\omega t)-\sqrt{2}\left(\lvert A\rvert^2-\frac{1}{4}B^2\right)(a^*-a)\sin(\omega t) \tag*{(45)} R ( t ) = i 2 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) cos ( ω t ) − 2 ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) sin ( ω t ) (45) From now on, for simplicity we will consider the displacements a a a a ∗ a^* a ∗ R ( t ) = 0 R(t)=0 R ( t ) = 0 ∑ s Λ i j ( s ) Λ l m ( s ) \sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)} ∑ s Λ ij ( s ) Λ l m ( s ) s s s § 1.1.2 , where for simplicity, we approximate this sum up to order m g 2 m_g^2 m g 2
∫ d Ω ∑ s Λ i j ( s ) Λ l m ( s ) = 4 π ω 4 ( T ~ i j l m ( 0 ) + m g 2 ω 2 Λ i j l m + m g 2 ω 2 ln ( 4 ω 2 m g 2 ) Ξ i j l m ) (46) \int d\Omega\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}=4\pi\omega^4\left(\widetilde{T}_{ijlm}^{(0)}+\frac{m_g^2}{\omega^2}\Lambda_{ijlm}+\frac{m_g^2}{\omega^2}\ln\left(\frac{4\omega^2}{m_g^2}\right)\Xi_{ijlm}\right) \tag*{(46)} ∫ d Ω s ∑ Λ ij ( s ) Λ l m ( s ) = 4 π ω 4 ( T ij l m ( 0 ) + ω 2 m g 2 Λ ij l m + ω 2 m g 2 ln ( m g 2 4 ω 2 ) Ξ ij l m ) (46) where we have defined
T ~ i j l m ( 0 ) = 1 5 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) + 1 15 δ i j δ l m , (47) \widetilde{T}_{ijlm}^{(0)}=\frac{1}{5}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})+\frac{1}{15}\delta_{ij}\delta_{lm}, \tag*{(47)} T ij l m ( 0 ) = 5 1 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) + 15 1 δ ij δ l m , (47) Λ i j l m = 91 60 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) + 11 30 δ i j δ l m + 13 12 ( δ i j n l n m + δ l m n i n j ) − 23 12 ( δ i l n j n m + δ i m n j n l + δ j l n i n m + δ j m n i n l ) + 37 12 n i n j n l n m , (48) \begin{aligned}
\Lambda_{ijlm}={}&\frac{91}{60}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})+\frac{11}{30}\delta_{ij}\delta_{lm}\\
&+\frac{13}{12}(\delta_{ij}n_ln_m+\delta_{lm}n_in_j)\\
&-\frac{23}{12}(\delta_{il}n_jn_m+\delta_{im}n_jn_l+\delta_{jl}n_in_m+\delta_{jm}n_in_l)\\
&+\frac{37}{12}n_in_jn_ln_m, \tag*{(48)}
\end{aligned} Λ ij l m = 60 91 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) + 30 11 δ ij δ l m + 12 13 ( δ ij n l n m + δ l m n i n j ) − 12 23 ( δ i l n j n m + δ im n j n l + δ j l n i n m + δ j m n i n l ) + 12 37 n i n j n l n m , (48) Ξ i j l m = − 1 2 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) − 1 2 ( δ i j n l n m + δ l m n i n j ) + 1 2 ( δ i l n j n m + δ i m n j n l + δ j l n i n m + δ j m n i n l ) − 1 2 n i n j n l n m . (49) \begin{aligned}
\Xi_{ijlm}={}&-\frac{1}{2}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})\\
&-\frac{1}{2}(\delta_{ij}n_ln_m+\delta_{lm}n_in_j)\\
&+\frac{1}{2}(\delta_{il}n_jn_m+\delta_{im}n_jn_l+\delta_{jl}n_in_m+\delta_{jm}n_in_l)\\
&-\frac{1}{2}n_in_jn_ln_m. \tag*{(49)}
\end{aligned} Ξ ij l m = − 2 1 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) − 2 1 ( δ ij n l n m + δ l m n i n j ) + 2 1 ( δ i l n j n m + δ im n j n l + δ j l n i n m + δ j m n i n l ) − 2 1 n i n j n l n m . (49) Therefore, using (46) in (43) , restoring m e f f = 1 16 π G 2 m_{\mathrm{eff}}=\frac{1}{16\pi G^{2}} m eff = 16 π G 2 1 g = 1 2 G g=\frac{1}{\sqrt{2G}} g = 2 G 1 ∣ k ∣ \lvert k\rvert ∣ k ∣ ω \omega ω ξ \xi ξ m g 2 m_{g}^{2} m g 2
S e f f [ ξ ] = − 2 m 0 ∫ 0 T d t ∂ 2 ξ c l i ∂ t 2 ξ q i − 2 G m 0 2 ∫ 0 T d t ∂ 5 ( ξ c l ( t ) ξ c l j ( t ) ) ∂ t 5 T ~ i j l m ( 0 ) ξ c l l ( t ) ξ q m ( t ) + 2 G m 0 2 m g 2 ∫ 0 T d t ∂ 3 ( ξ c l ( t ) ξ c l j ( t ) ) ∂ t 3 S ~ i j l m ξ c l l ( t ) ξ q m ( t ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 t d t ′ ∂ 4 ξ c l i ( t ′ ) ξ c l j ( t ′ ) ∂ t ′ 4 ln ( m g ∣ t − t ′ ∣ ) Ξ i j l m ξ c l l ( t ) ξ q m ( t ) + i 8 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ∂ 2 ( ξ c l i ( t ) ξ q j ( t ) ) ∂ t 2 K i j l m ( t , t ′ ) ∂ 2 ( ξ c l l ( t ′ ) ξ q m ( t ′ ) ) ∂ t ′ 2 (50) \begin{aligned}
S_{\mathrm{eff}}[\xi] ={}& -2m_{0}\int_{0}^{T}dt\frac{\partial^{2}\xi_{\mathrm{cl}}^{i}}{\partial t^{2}}\xi_{q}^{i}\\
&-2Gm_{0}^{2}\int_{0}^{T}dt\frac{\partial^{5}(\xi_{\mathrm{cl}}(t)\xi_{\mathrm{cl}}^{j}(t))}{\partial t^{5}}\widetilde{T}_{ijlm}^{(0)}\xi_{\mathrm{cl}}^{l}(t)\xi_{q}^{m}(t)\\
&+2Gm_{0}^{2}m_{g}^{2}\int_{0}^{T}dt\frac{\partial^{3}(\xi_{\mathrm{cl}}(t)\xi_{\mathrm{cl}}^{j}(t))}{\partial t^{3}}\widetilde{S}_{ijlm}\xi_{\mathrm{cl}}^{l}(t)\xi_{q}^{m}(t)\\
&-4Gm_{0}^{2}m_{g}^{2}\int_{0}^{T}dt\int_{0}^{t}dt'\frac{\partial^{4}\xi_{\mathrm{cl}}^{i}(t')\xi_{\mathrm{cl}}^{j}(t')}{\partial t'^{4}}\ln\left(m_{g}\lvert t-t'\rvert\right)\Xi_{ijlm}\xi_{\mathrm{cl}}^{l}(t)\xi_{q}^{m}(t)\\
&+i\frac{8Gm_{0}^{2}}{\pi}\int_{0}^{T}dt\int_{0}^{T}dt'\frac{\partial^{2}(\xi_{\mathrm{cl}}^{i}(t)\xi_{q}^{j}(t))}{\partial t^{2}}K_{ijlm}(t,t')\frac{\partial^{2}(\xi_{\mathrm{cl}}^{l}(t')\xi_{q}^{m}(t'))}{\partial t'^{2}} \tag*{(50)}
\end{aligned} S eff [ ξ ] = − 2 m 0 ∫ 0 T d t ∂ t 2 ∂ 2 ξ cl i ξ q i − 2 G m 0 2 ∫ 0 T d t ∂ t 5 ∂ 5 ( ξ cl ( t ) ξ cl j ( t )) T ij l m ( 0 ) ξ cl l ( t ) ξ q m ( t ) + 2 G m 0 2 m g 2 ∫ 0 T d t ∂ t 3 ∂ 3 ( ξ cl ( t ) ξ cl j ( t )) S ij l m ξ cl l ( t ) ξ q m ( t ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 t d t ′ ∂ t ′ 4 ∂ 4 ξ cl i ( t ′ ) ξ cl j ( t ′ ) ln ( m g ∣ t − t ′ ∣ ) Ξ ij l m ξ cl l ( t ) ξ q m ( t ) + i π 8 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ∂ t 2 ∂ 2 ( ξ cl i ( t ) ξ q j ( t )) K ij l m ( t , t ′ ) ∂ t ′ 2 ∂ 2 ( ξ cl l ( t ′ ) ξ q m ( t ′ )) (50) where we defined
S ~ i j l m = − 1 2 T ~ i j l m ( 0 ) + Λ i j l m + 2 ( ln ( 2 ) − γ ) Ξ i j l m (51) \widetilde{S}_{ijlm}=-\frac{1}{2}\widetilde{T}_{ijlm}^{(0)}+\Lambda_{ijlm}+2(\ln(2)-\gamma)\Xi_{ijlm} \tag*{(51)} S ij l m = − 2 1 T ij l m ( 0 ) + Λ ij l m + 2 ( ln ( 2 ) − γ ) Ξ ij l m (51) with γ \gamma γ K i j l m K_{ijlm} K ij l m
K i j l m ( t , t ′ ) = I 0 ( t , t ′ ) T ~ i j l m ( 0 ) + m g 2 I 1 ( t , t ′ ) S ~ i j l m + 2 m g 2 I 2 ( t , t ′ ) Ξ i j l m (52) K_{ijlm}(t,t')=I_{0}(t,t')\widetilde{T}_{ijlm}^{(0)}+m_{g}^{2}I_{1}(t,t')\widetilde{S}_{ijlm}+2m_{g}^{2}I_{2}(t,t')\Xi_{ijlm} \tag*{(52)} K ij l m ( t , t ′ ) = I 0 ( t , t ′ ) T ij l m ( 0 ) + m g 2 I 1 ( t , t ′ ) S ij l m + 2 m g 2 I 2 ( t , t ′ ) Ξ ij l m (52) where the objects I 0 I_{0} I 0 I 1 I_{1} I 1 I 2 I_{2} I 2
I 0 = ∫ m g ∞ d ω ω M ( t , t ′ ) (53) I_{0}=\int_{m_{g}}^{\infty}d\omega\,\omega M(t,t') \tag*{(53)} I 0 = ∫ m g ∞ d ω ω M ( t , t ′ ) (53) I 1 = ∫ m g ∞ d ω 1 ω M ( t , t ′ ) (54) I_{1}=\int_{m_{g}}^{\infty}d\omega\,\frac{1}{\omega}M(t,t') \tag*{(54)} I 1 = ∫ m g ∞ d ω ω 1 M ( t , t ′ ) (54) I 2 = ∫ m g ∞ d ω 1 ω ( γ + ln ( ω m g ) ) M ( t , t ′ ) (55) I_{2}=\int_{m_{g}}^{\infty}d\omega\,\frac{1}{\omega}\left(\gamma+\ln\left(\frac{\omega}{m_{g}}\right)\right)M(t,t') \tag*{(55)} I 2 = ∫ m g ∞ d ω ω 1 ( γ + ln ( m g ω ) ) M ( t , t ′ ) (55) and M ( t , t ′ ) M(t,t') M ( t , t ′ ) (44) .
We can arrange the last term of (50) using a Hubbard-Stratonovich transformation in order to define a noise term:
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ q j ( − δ i j ∂ 2 ∂ t 2 − ∂ 2 N i j ∂ t 2 − G m 0 ∂ 5 ( ξ c l l ξ c l m ) ∂ t 5 T ~ l m i j ( 0 ) + G m 0 m g 2 ∂ 3 ( ξ c l l ξ c l m ) ∂ t 3 S ~ l m i j − 2 G m 0 m g 2 ∫ 0 t d t ′ ∂ 4 ( ξ c l l ( t ′ ) ξ c l m ( t ′ ) ) ∂ t ′ 4 ln ( m g ∣ t − t ′ ∣ ) Ξ l m i j ) ξ c l i + i π 8 G ∫ 0 T d t ∫ 0 T d t ′ N i j ( t ) K i j − 1 l m ( t , t ′ ) N l m ( t ′ ) (56) \begin{aligned}
S_{\mathrm{eff}}[\xi]={}&2m_{0}\int_{0}^{T}dt\xi_{q}^{j}\left(-\delta_{ij}\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}\mathcal{N}_{ij}}{\partial t^{2}}-Gm_{0}\frac{\partial^{5}(\xi_{\mathrm{cl}}^{l}\xi_{\mathrm{cl}}^{m})}{\partial t^{5}}\widetilde{T}_{lmij}^{(0)}+Gm_{0}m_{g}^{2}\frac{\partial^{3}(\xi_{\mathrm{cl}}^{l}\xi_{\mathrm{cl}}^{m})}{\partial t^{3}}\widetilde{S}_{lmij}\right.\\
&\left.-2Gm_{0}m_{g}^{2}\int_{0}^{t}dt'\frac{\partial^{4}(\xi_{\mathrm{cl}}^{l}(t')\xi_{\mathrm{cl}}^{m}(t'))}{\partial t'^{4}}\ln\left(m_{g}\lvert t-t'\rvert\right)\Xi_{lmij}\right)\xi_{\mathrm{cl}}^{i}\\
&+i\frac{\pi}{8G}\int_{0}^{T}dt\int_{0}^{T}dt'\mathcal{N}_{ij}(t)K_{ij^{-1}lm}(t,t')\mathcal{N}_{lm}(t') \tag*{(56)}
\end{aligned} S eff [ ξ ] = 2 m 0 ∫ 0 T d t ξ q j ( − δ ij ∂ t 2 ∂ 2 − ∂ t 2 ∂ 2 N ij − G m 0 ∂ t 5 ∂ 5 ( ξ cl l ξ cl m ) T l mij ( 0 ) + G m 0 m g 2 ∂ t 3 ∂ 3 ( ξ cl l ξ cl m ) S l mij − 2 G m 0 m g 2 ∫ 0 t d t ′ ∂ t ′ 4 ∂ 4 ( ξ cl l ( t ′ ) ξ cl m ( t ′ )) ln ( m g ∣ t − t ′ ∣ ) Ξ l mij ) ξ cl i + i 8 G π ∫ 0 T d t ∫ 0 T d t ′ N ij ( t ) K i j − 1 l m ( t , t ′ ) N l m ( t ′ ) (56) Equations of motion from the effective action and noise term correlation
We can obtain the equation of motion for ξ c l \xi_{\mathrm{cl}} ξ cl ξ q \xi_{q} ξ q
δ i j ∂ 2 ξ c l i ∂ t 2 − ( ∂ 2 N i j ∂ t 2 − G m 0 T ~ l m i j ( 0 ) ∂ 5 ( ξ c l l ( t ) ξ c l m ( t ) ) ∂ t 5 + G m 0 m g 2 S ~ l m i j ∂ 3 ( ξ c l l ( t ) ξ c l m ( t ) ) ∂ t 3 − 2 G m 0 m g 2 Ξ l m i j ∫ 0 t d t ′ ∂ 4 ξ c l l ( t ′ ) ξ c l m ( t ′ ) ∂ t ′ 4 ln ( m g ( t − t ′ ) ) ) ξ c l i = 0 (57) \delta_{ij}\frac{\partial^{2}\xi_{\mathrm{cl}}^{i}}{\partial t^{2}}-\left(\frac{\partial^{2}\mathcal{N}_{ij}}{\partial t^{2}}-Gm_{0}\widetilde{T}_{lmij}^{(0)}\frac{\partial^{5}(\xi_{\mathrm{cl}}^{l}(t)\xi_{\mathrm{cl}}^{m}(t))}{\partial t^{5}}+Gm_{0}m_{g}^{2}\widetilde{S}_{lmij}\frac{\partial^{3}(\xi_{\mathrm{cl}}^{l}(t)\xi_{\mathrm{cl}}^{m}(t))}{\partial t^{3}}-2Gm_{0}m_{g}^{2}\Xi_{lmij}\int_{0}^{t}dt'\frac{\partial^{4}\xi_{\mathrm{cl}}^{l}(t')\xi_{\mathrm{cl}}^{m}(t')}{\partial t'^{4}}\ln\left(m_{g}(t-t')\right)\right)\xi_{\mathrm{cl}}^{i}=0 \tag*{(57)} δ ij ∂ t 2 ∂ 2 ξ cl i − ( ∂ t 2 ∂ 2 N ij − G m 0 T l mij ( 0 ) ∂ t 5 ∂ 5 ( ξ cl l ( t ) ξ cl m ( t )) + G m 0 m g 2 S l mij ∂ t 3 ∂ 3 ( ξ cl l ( t ) ξ cl m ( t )) − 2 G m 0 m g 2 Ξ l mij ∫ 0 t d t ′ ∂ t ′ 4 ∂ 4 ξ cl l ( t ′ ) ξ cl m ( t ′ ) ln ( m g ( t − t ′ ) ) ) ξ cl i = 0 (57) which defines the dynamics of the detector in the presence of the gravitational wave. We observe that in the limit m g → 0 m_{g}\to0 m g → 0
δ i j ∂ 2 ξ c l i ∂ t 2 − ( ∂ 2 N i j ∂ t 2 − G m 0 T ~ l m i j ( 0 ) ∂ 5 ( ξ c l l ( t ) ξ c l m ( t ) ) ∂ t 5 ) ξ c l i = 0 (58) \delta_{ij}\frac{\partial^{2}\xi_{\mathrm{cl}}^{i}}{\partial t^{2}}-\left(\frac{\partial^{2}\mathcal{N}_{ij}}{\partial t^{2}}-Gm_{0}\widetilde{T}_{lmij}^{(0)}\frac{\partial^{5}(\xi_{\mathrm{cl}}^{l}(t)\xi_{\mathrm{cl}}^{m}(t))}{\partial t^{5}}\right)\xi_{\mathrm{cl}}^{i}=0 \tag*{(58)} δ ij ∂ t 2 ∂ 2 ξ cl i − ( ∂ t 2 ∂ 2 N ij − G m 0 T l mij ( 0 ) ∂ t 5 ∂ 5 ( ξ cl l ( t ) ξ cl m ( t )) ) ξ cl i = 0 (58) We recover the same structure found in [2 ] , but with differences in the coefficient of the dissipative term. This difference is due to the fact that we have taken here a sum over polarizations instead of only one.
The noise has a big impact on the equation. Depending on the nature of the initial state they can leave different imprints. The correlation of this variable N i j N_{ij} N ij
⟨ N i j ( t ) N l m ( t ′ ) ⟩ = 4 G π K i j l m ( t , t ′ ) (59) \left\langle N_{ij}(t)N_{lm}(t')\right\rangle= \frac{4G}{\pi}K_{ijlm}(t,t') \tag*{(59)} ⟨ N ij ( t ) N l m ( t ′ ) ⟩ = π 4 G K ij l m ( t , t ′ ) (59) Since the integrals I 0 I_0 I 0 I 1 I_1 I 1 I 2 I_2 I 2 K i j l m K_{ijlm} K ij l m N i j N_{ij} N ij [2 ] considering the case when ϕ ≠ 0 \phi\ne0 ϕ = 0
We notice that in the squeezed state, the term I 0 I_0 I 0 sin ϕ \sin\phi sin ϕ t + t ′ t+t' t + t ′ t t t t ′ t' t ′ t , t ′ ≥ 0 t,t' \ge0 t , t ′ ≥ 0 sin ϕ \sin\phi sin ϕ m g m_g m g sin ϕ \sin\phi sin ϕ I 2 I_2 I 2 Ξ i j l m \Xi_{ijlm} Ξ ij l m sin ϕ \sin\phi sin ϕ
Table I. Values of I 0 I_0 I 0 I 1 I_1 I 1 I 2 I_2 I 2 K i j l m K_{ijlm} K ij l m m g 2 m_g^2 m g 2 P F PF P F E i Ei E i
Conclusions
We have studied the response of a model GW detector to a quantized gravitational field in Very Special Linear Gravity. Working in term of quantities that are invariant under gauge transformations (see equation (18) ) is very important for the massive case, since it ensures that the result is free from divergences.
The influence-functional description relates the detector’s causal response and fluctuations to the two physical graviton polarizations. A nonzero mass introduces direction-dependent tensor structures and a logarithmic memory contribution. For the centered Gaussian states considered here, the noise correlations depend on the field state, while the retarded response is the same for vacuum, thermal and squeezed states. The angular corrections vanish smoothly as m g → 0 m_g \to0 m g → 0
The preferred direction suggests looking for orientation-dependent correlations between detector arms, or between detectors sampling the same field. Rotation of the detector relative to this direction could produce a modulation of the correlations.
The scale of the mass correction can be estimated without specifying the state occupation. Restoring units and writing E = h f = ℏ ω E=hf=\hbar\omega E = h f = ℏ ω
R ( f ) = ( m g c 2 h f ) 2 ln [ 4 ( h f ) 2 m g 2 c 4 ] . (60) \mathcal{R}(f)=\left(\frac{m_gc^2}{hf}\right)^2\ln\left[\frac{4(hf)^2}{m_g^2c^4}\right]. \tag*{(60)} R ( f ) = ( h f m g c 2 ) 2 ln [ m g 2 c 4 4 ( h f ) 2 ] . (60) For stationary states, this sets the order of the fractional correction to the noise power, up to geometric coefficients. Taking m g c 2 = 10 − 23 e V m_gc^2=10^{-23}\,\mathrm{eV} m g c 2 = 1 0 − 23 eV R ≃ 3 × 10 − 20 \mathcal{R}\simeq3\times10^{-20} R ≃ 3 × 1 0 − 20 f = 100 H z f=100\,\mathrm{Hz} f = 100 Hz R ≃ 2 × 10 − 10 \mathcal{R}\simeq2\times10^{-10} R ≃ 2 × 1 0 − 10 f = 1 m H z f=1\,\mathrm{mHz} f = 1 mHz 1.92 × 10 − 23 e V / c 2 1.92\times10^{-23}\,\mathrm{eV}/c^2 1.92 × 1 0 − 23 eV / c 2 [30 ] ; we use that result as context, not as a dedicated bound on VSR. While not being a sensitivity forecast, the larger relative correction in the millihertz band motivates considering LISA [31 ] . The estimate assumes m g c 2 ≪ h f m_gc^2\ll hf m g c 2 ≪ h f
The absolute noise level remains essential. In the vacuum and the freely falling, long-wavelength limit, the equivalent strain power spectrum scales as S h v a c ( f ) ∼ t P 2 f S_h^{\mathrm{vac}}(f)\sim t_P^2f S h vac ( f ) ∼ t P 2 f t P = ℏ G / c 5 t_P=\sqrt{\hbar G/c^5} t P = ℏ G / c 5 10 − 42 H z − 1 / 2 10^{-42}\,\mathrm{Hz}^{-1/2} 1 0 − 42 Hz − 1/2 100 H z 100\,\mathrm{Hz} 100 Hz [2 ] . Thermal occupation or squeezing can enhance selected correlations. Fully assessing detectability requires a model of the gravitational state, the interferometer response over the measurement bandwidth, and a comparison with instrumental noise and astrophysical signals. The present results provide kernels for such a calculation; assessing observability requires combining them with a physical source state, the detector transfer function and a finite measurement bandwidth. That analysis however is outside the scope of this paper.
As a final remark, in our work we have kept a general squeezing phase ϕ \phi ϕ [2 ] , where the phase was set to zero for simplicity. For squeezing parameters independent of the frequency, we found that the graviton mass induces a contribution proportional to sin ϕ \sin\phi sin ϕ Ξ i j l m \Xi_{ijlm} Ξ ij l m [32 , 33 , 34 ] . A mechanism to produce squeezed gravitons in a stellar process has been discussed by [35 , 36 ] . There they couple a rotating Black Hole (Kerr) to a massive axion cloud, which is able to radiate squeezed gravitons.
The relevance of the squeezing phase has also been pointed out in [37 ] , where mode-dependent squeezing phases of graviton states such as those of [35 , 36 ] were shown to leave signatures of the gravitational field. Together with our results, this suggests that the squeezing phase may play an important role in identifying different imprints of gravity in general.
Acknowledgements
J.A. acknowledges the partial support of the Institute of Physics PUC.
A.S. acknowledges support from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 10127532 (MALTA4COSMO).
Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency (REA). Neither the European Union nor the granting authority can be held responsible for them.
Alex Soto acknowledges support from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101209941 (Project: BosLong). Also, Alex Soto thanks to Nick Mavromatos for information about graviton squeezed states.
Effective action for ξ \xi ξ
Taking the influence functional F F F (34) with the initial density matrix defined in (35) we have that explicitly can be written as
F [ ξ , ξ ′ ] = ∫ d q T d q T ′ d q T ′ d q T ′ δ ( q T − q T ′ ) D q D q ′ N exp [ i ∫ τ T d t ( 1 2 m e f f ( q ˙ 2 − q ˙ ′ 2 ) − 1 2 m e f f ω 2 ( q 2 − q ′ 2 ) + ( J q − J ′ q ′ ) − m e f f ω 2 ( A ( q T − a ) 2 + A ∗ ( q T ′ − a ∗ ) 2 − B ( q T − a ) ( q T ′ − a ∗ ) ) ) ] . (61) \begin{aligned}
F[\xi,\xi']={}&\int dq_Tdq_T'dq_T'dq_T'\delta(q_T-q_T')\mathcal{D}q\mathcal{D}q'\mathcal{N}\exp\Bigg[i\int_{\tau}^{T}dt\Bigg(\frac{1}{2}m_{\mathrm{eff}}\left(\dot{q}^{\,2}-\dot{q}'^{\,2}\right)-\frac{1}{2}m_{\mathrm{eff}}\omega^2\left(q^2-q'^{\,2}\right)+(Jq-J'q')\\
&\qquad-\frac{m_{\mathrm{eff}}\omega}{2}\left(A(q_T-a)^2+A^*(q_T'-a^*)^2-B(q_T-a)(q_T'-a^*)\right)\Bigg)\Bigg]. \tag*{(61)}
\end{aligned} F [ ξ , ξ ′ ] = ∫ d q T d q T ′ d q T ′ d q T ′ δ ( q T − q T ′ ) D q D q ′ N exp [ i ∫ τ T d t ( 2 1 m eff ( q ˙ 2 − q ˙ ′ 2 ) − 2 1 m eff ω 2 ( q 2 − q ′ 2 ) + ( J q − J ′ q ′ ) − 2 m eff ω ( A ( q T − a ) 2 + A ∗ ( q T ′ − a ∗ ) 2 − B ( q T − a ) ( q T ′ − a ∗ ) ) ) ] . (61) Then, we do the change of variables:
x = 1 2 ( q + q ′ ) , x ′ = 1 2 ( q − q ′ ) , K = 1 2 ( J + J ′ ) , K ′ = 1 2 ( J − J ′ ) (62) \begin{aligned}
x &= \frac{1}{\sqrt{2}}(q+q'), \qquad x'=\frac{1}{\sqrt{2}}(q-q'),\\
K &= \frac{1}{\sqrt{2}}(J+J'), \qquad K'=\frac{1}{\sqrt{2}}(J-J') \tag*{(62)}
\end{aligned} x K = 2 1 ( q + q ′ ) , x ′ = 2 1 ( q − q ′ ) , = 2 1 ( J + J ′ ) , K ′ = 2 1 ( J − J ′ ) (62) to write the functional as
F [ ξ , ξ ′ ] = ∫ d x T d x τ d x T ′ d x τ ′ δ ( 2 x T ′ ) D x D x ′ N exp [ i ∫ τ T d t ( m e f f x ˙ x ˙ ′ − m e f f ω 2 x x ′ + K x ′ + K ′ x ) − m e f f ω 4 ( A ( x τ + x τ ′ − 2 a ) 2 + A ∗ ( x τ − x τ ′ − 2 a ∗ ) 2 − B ( x τ + x τ ′ − 2 a ) ( x τ − x τ ′ − 2 a ∗ ) ) ] (63) \begin{aligned}
F[\xi,\xi']={}&\int dx_Tdx_\tau dx'_Tdx'_\tau\,\delta\left(\sqrt{2}x'_T\right)\mathcal{D}x\mathcal{D}x'\mathcal{N}\exp\left[i\int_\tau^Tdt\left(m_{eff}\dot{x}\dot{x}'-m_{eff}\omega^2xx'+Kx'+K'x\right)\right.\\
&\left.-\frac{m_{eff}\omega}{4}\left(A\left(x_\tau+x'_\tau-\sqrt{2}a\right)^2+A^*\left(x_\tau-x'_\tau-\sqrt{2}a^*\right)^2-B\left(x_\tau+x'_\tau-\sqrt{2}a\right)\left(x_\tau-x'_\tau-\sqrt{2}a^*\right)\right)\right] \tag*{(63)}
\end{aligned} F [ ξ , ξ ′ ] = ∫ d x T d x τ d x T ′ d x τ ′ δ ( 2 x T ′ ) D x D x ′ N exp [ i ∫ τ T d t ( m e f f x ˙ x ˙ ′ − m e f f ω 2 x x ′ + K x ′ + K ′ x ) − 4 m e f f ω ( A ( x τ + x τ ′ − 2 a ) 2 + A ∗ ( x τ − x τ ′ − 2 a ∗ ) 2 − B ( x τ + x τ ′ − 2 a ) ( x τ − x τ ′ − 2 a ∗ ) ) ] (63) Integrating by parts the first term in the exponential and integrating over x ( t ) x(t) x ( t )
F [ ξ , ξ ′ ] = ∫ d x T d x τ d x T ′ d x τ ′ δ ( 2 x T ′ ) D x ′ δ ( m e f f ( x ¨ ′ + ω 2 x ′ ) − K ′ ) N exp [ i ∫ τ T d t K x ′ + i m e f f ( x T x ˙ T ′ − x τ x ˙ τ ′ ) − m e f f ω 4 ( A ( x τ + x τ ′ − 2 a ) 2 + A ∗ ( x τ − x τ ′ − 2 a ∗ ) 2 − B ( x τ + x τ ′ − 2 a ) ( x τ − x τ ′ − 2 a ∗ ) ) ] (64) \begin{aligned}
F[\xi,\xi']={}&\int dx_Tdx_\tau dx'_Tdx'_\tau\,\delta\left(\sqrt{2}x'_T\right)\mathcal{D}x'\delta\left(m_{eff}\left(\ddot{x}'+\omega^2x'\right)-K'\right)\mathcal{N}\exp\left[i\int_\tau^TdtKx'+im_{eff}\left(x_T\dot{x}'_T-x_\tau\dot{x}'_\tau\right)\right.\\
&\left.-\frac{m_{eff}\omega}{4}\left(A\left(x_\tau+x'_\tau-\sqrt{2}a\right)^2+A^*\left(x_\tau-x'_\tau-\sqrt{2}a^*\right)^2-B\left(x_\tau+x'_\tau-\sqrt{2}a\right)\left(x_\tau-x'_\tau-\sqrt{2}a^*\right)\right)\right] \tag*{(64)}
\end{aligned} F [ ξ , ξ ′ ] = ∫ d x T d x τ d x T ′ d x τ ′ δ ( 2 x T ′ ) D x ′ δ ( m e f f ( x ¨ ′ + ω 2 x ′ ) − K ′ ) N exp [ i ∫ τ T d t K x ′ + i m e f f ( x T x ˙ T ′ − x τ x ˙ τ ′ ) − 4 m e f f ω ( A ( x τ + x τ ′ − 2 a ) 2 + A ∗ ( x τ − x τ ′ − 2 a ∗ ) 2 − B ( x τ + x τ ′ − 2 a ) ( x τ − x τ ′ − 2 a ∗ ) ) ] (64) The delta that appeared from this integration defines x ′ ( t ) x'(t) x ′ ( t )
m e f f ( x ¨ ′ + ω 2 x ′ ) − K ′ = 0 (65) m_{eff}\left(\ddot{x}'+\omega^2x'\right)-K'=0 \tag*{(65)} m e f f ( x ¨ ′ + ω 2 x ′ ) − K ′ = 0 (65) Then, we do the integral in x ′ ( t ) x'(t) x ′ ( t ) x ′ x' x ′ x T x_T x T x τ x_\tau x τ
F [ ξ , ξ ′ ] = ∫ d x T ′ d x τ ′ δ ( 2 x T ′ ) δ ( m e f f x ˙ T ′ ) 4 π m e f f ω ( A + A ∗ − B ) exp [ − m e f f ω ( ∣ A ∣ 2 − 1 4 B 2 ) 2 ( A + A ∗ − B ) ( a ∗ − a ) 2 ] × exp [ i ∫ τ T d t K x ′ − m e f f ω ( ∣ A ∣ 2 − 1 4 B 2 ) A + A ∗ − B x τ ′ 2 − i m e f f ω ( A ∗ − A ) A + A ∗ − B x τ ′ x τ ′ − m e f f ω ( A + A ∗ − B ) x ˙ τ ′ 2 − m e f f ω ( ∣ A ∣ 2 − 1 4 B 2 ) A + A ∗ − B 2 ( a ∗ − a ) x τ ′ − i m e f f ( 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) ) 2 ( A + A ∗ − B ) x ˙ τ ′ ] (66) \begin{aligned}
F[\xi,\xi']={}&\int dx'_Tdx'_\tau\,\delta\left(\sqrt{2}x'_T\right)\delta\left(m_{eff}\dot{x}'_T\right)\sqrt{\frac{4\pi}{m_{eff}\omega(A+A^*-B)}}\exp\left[-\frac{m_{eff}\omega\left(|A|^2-\frac{1}{4}B^2\right)}{2(A+A^*-B)}(a^*-a)^2\right]\\
&\times\exp\left[i\int_\tau^TdtKx'-\frac{m_{eff}\omega\left(|A|^2-\frac{1}{4}B^2\right)}{A+A^*-B}x_\tau^{\prime2}-i\frac{m_{eff}\omega(A^*-A)}{A+A^*-B}x'_\tau x'_\tau-\frac{m_{eff}}{\omega(A+A^*-B)}\dot{x}_\tau^{\prime2}\right.\\
&\left.-\frac{m_{eff}\omega\left(|A|^2-\frac{1}{4}B^2\right)}{A+A^*-B}\sqrt{2}(a^*-a)x'_\tau-i\frac{m_{eff}\left(2(Aa+A^*a^*)-B(a^*+a)\right)}{\sqrt{2}(A+A^*-B)}\dot{x}'_\tau\right] \tag*{(66)}
\end{aligned} F [ ξ , ξ ′ ] = ∫ d x T ′ d x τ ′ δ ( 2 x T ′ ) δ ( m e f f x ˙ T ′ ) m e f f ω ( A + A ∗ − B ) 4 π exp [ − 2 ( A + A ∗ − B ) m e f f ω ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) 2 ] × exp [ i ∫ τ T d t K x ′ − A + A ∗ − B m e f f ω ( ∣ A ∣ 2 − 4 1 B 2 ) x τ ′2 − i A + A ∗ − B m e f f ω ( A ∗ − A ) x τ ′ x τ ′ − ω ( A + A ∗ − B ) m e f f x ˙ τ ′2 − A + A ∗ − B m e f f ω ( ∣ A ∣ 2 − 4 1 B 2 ) 2 ( a ∗ − a ) x τ ′ − i 2 ( A + A ∗ − B ) m e f f ( 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) ) x ˙ τ ′ ] (66) With this expression, from the deltas, we observe that the solution for x ′ ( t ) x'(t) x ′ ( t ) T T T x ′ x' x ′ T T T
x ′ ( t ) = 1 m e f f ω ∫ t T sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ (67) x'(t)=\frac{1}{m_{eff}\omega}\int_t^T\sin\left(\omega(t'-t)\right)K'(t')dt' \tag*{(67)} x ′ ( t ) = m e f f ω 1 ∫ t T sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ (67) Inserting this solution in (A6) and rewriting
∫ t T sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ = ∫ τ T θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ (68) \int_t^T\sin\left(\omega(t'-t)\right)K'(t')dt'=\int_\tau^T\theta(t'-t)\sin\left(\omega(t'-t)\right)K'(t')dt' \tag*{(68)} ∫ t T sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ = ∫ τ T θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) d t ′ (68) we have that the functional F F F
F [ ξ , ξ ′ ] = ∫ d x T ′ d x τ ′ δ ( 2 x T ′ ) δ ( m e f f x ˙ T ′ ) 4 π m e f f ω ( A + A ∗ − B ) exp [ − m e f f ω ( ∣ A ∣ 2 − 1 4 B 2 ) 2 ( A + A ∗ − B ) ( a ∗ − a ) 2 ] × exp [ i m e f f ω ∫ τ T d t ∫ τ T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) − 1 m e f f ω ∫ τ T d t ∫ τ T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) + 1 A + A ∗ − B ∫ τ T R ( t ) K ′ ( t ) d t ] (69) \begin{aligned}
F[\xi,\xi']={}&\int dx'_Tdx'_\tau\,\delta\left(\sqrt{2}x'_T\right)\delta\left(m_{eff}\dot{x}'_T\right)\sqrt{\frac{4\pi}{m_{eff}\omega(A+A^*-B)}}\exp\left[-\frac{m_{eff}\omega\left(|A|^2-\frac{1}{4}B^2\right)}{2(A+A^*-B)}(a^*-a)^2\right]\\
&\times\exp\left[\frac{i}{m_{eff}\omega}\int_\tau^Tdt\int_\tau^Tdt'K(t)\theta(t'-t)\sin\left(\omega(t'-t)\right)K'(t')\right.\\
&\left.-\frac{1}{m_{eff}\omega}\int_\tau^Tdt\int_\tau^Tdt'K'(t)M(t,t')K'(t')+\frac{1}{A+A^*-B}\int_\tau^TR(t)K'(t)dt\right] \tag*{(69)}
\end{aligned} F [ ξ , ξ ′ ] = ∫ d x T ′ d x τ ′ δ ( 2 x T ′ ) δ ( m e f f x ˙ T ′ ) m e f f ω ( A + A ∗ − B ) 4 π exp [ − 2 ( A + A ∗ − B ) m e f f ω ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) 2 ] × exp [ m e f f ω i ∫ τ T d t ∫ τ T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) − m e f f ω 1 ∫ τ T d t ∫ τ T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) + A + A ∗ − B 1 ∫ τ T R ( t ) K ′ ( t ) d t ] (69) where we defined
M ( t , t ′ ) = ∣ A ∣ 2 − 1 4 B 2 + 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t − t ′ ) ) − ∣ A ∣ 2 − 1 4 B 2 − 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t + t ′ − 2 τ ) ) − ℑ ( A ) 2 ℜ ( A ) − B sin ( ω ( t − t ′ ) ) − ℑ ( A ) 2 ℜ ( A ) − B sin ( ω ( t + t ′ − 2 τ ) ) (70) \begin{aligned}
M(t,t')={}&\frac{|A|^2-\frac{1}{4}B^2+1}{2(2\Re(A)-B)}\cos(\omega(t-t'))-\frac{|A|^2-\frac{1}{4}B^2-1}{2(2\Re(A)-B)}\cos(\omega(t+t'-2\tau))\\
&-\frac{\Im(A)}{2\Re(A)-B}\sin(\omega(t-t'))-\frac{\Im(A)}{2\Re(A)-B}\sin(\omega(t+t'-2\tau)) \tag*{(70)}
\end{aligned} M ( t , t ′ ) = 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 + 1 cos ( ω ( t − t ′ )) − 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 − 1 cos ( ω ( t + t ′ − 2 τ )) − 2ℜ ( A ) − B ℑ ( A ) sin ( ω ( t − t ′ )) − 2ℜ ( A ) − B ℑ ( A ) sin ( ω ( t + t ′ − 2 τ )) (70) R ( t ) = i 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) 2 cos ( ω ( t − τ ) ) − 2 ( ∣ A ∣ 2 − 1 4 B 2 ) ( a ∗ − a ) sin ( ω ( t − τ ) ) (71) R(t)=i\frac{2(Aa+A^*a^*)-B(a^*+a)}{\sqrt{2}}\cos(\omega(t-\tau))-\sqrt{2}\left(|A|^2-\frac{1}{4}B^2\right)(a^*-a)\sin(\omega(t-\tau)) \tag*{(71)} R ( t ) = i 2 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) cos ( ω ( t − τ )) − 2 ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) sin ( ω ( t − τ )) (71) Considering that the integrals in d x τ ′ d x τ ′ dx'_{\tau}dx'_{\tau} d x τ ′ d x τ ′ δ ( 2 x τ ′ ) δ ( m e f f x ˙ τ ′ ) \delta(\sqrt{2}x'_{\tau})\delta(m_{\mathrm{eff}}\dot{x}'_{\tau}) δ ( 2 x τ ′ ) δ ( m eff x ˙ τ ′ ) x ′ x' x ′ N \mathcal{N} N d x τ ′ dx'_{\tau} d x τ ′ d x ˙ τ ′ d\dot{x}'_{\tau} d x ˙ τ ′ τ → 0 \tau\to0 τ → 0 ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) \int_0^T dt\int_0^T dt'K'(t)M(t,t')K'(t') ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) sin ( ω ( t − t ′ ) ) \sin(\omega(t-t')) sin ( ω ( t − t ′ )) t ↔ t ′ t\leftrightarrow t' t ↔ t ′ t t t t ′ t' t ′
F [ ξ , ξ ′ ] = N 4 π m e f f ω ( A + A ∗ − B ) exp [ − m e f f ω ( ∣ A ∣ 2 − 1 4 B 2 ) 2 ( A + A ∗ − B ) ( a ∗ − a ) 2 ] × exp [ i m e f f ω ∫ 0 T d t ∫ 0 T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) − 1 m e f f ω ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) + 1 A + A ∗ − B ∫ 0 T R ( t ) K ′ ( t ) d t ] (72) \begin{aligned}
F[\xi,\xi']={}&\mathcal{N}\sqrt{\frac{4\pi}{m_{\mathrm{eff}}\omega(A+A^*-B)}}\exp\left[-\frac{m_{\mathrm{eff}}\omega\left(|A|^2-\frac{1}{4}B^2\right)}{2(A+A^*-B)}(a^*-a)^2\right]\\
&\times\exp\left[\frac{i}{m_{\mathrm{eff}}\omega}\int_0^Tdt\int_0^Tdt'K(t)\theta(t'-t)\sin(\omega(t'-t))K'(t')\right.\\
&\left.-\frac{1}{m_{\mathrm{eff}}\omega}\int_0^Tdt\int_0^Tdt'K'(t)M(t,t')K'(t')+\frac{1}{A+A^*-B}\int_0^TR(t)K'(t)dt\right] \tag*{(72)}
\end{aligned} F [ ξ , ξ ′ ] = N m eff ω ( A + A ∗ − B ) 4 π exp [ − 2 ( A + A ∗ − B ) m eff ω ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) 2 ] × exp [ m eff ω i ∫ 0 T d t ∫ 0 T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t )) K ′ ( t ′ ) − m eff ω 1 ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) + A + A ∗ − B 1 ∫ 0 T R ( t ) K ′ ( t ) d t ] (72) with
M ( t , t ′ ) = ∣ A ∣ 2 − 1 4 B 2 + 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t − t ′ ) ) − ∣ A ∣ 2 − 1 4 B 2 − 1 2 ( 2 ℜ ( A ) − B ) cos ( ω ( t + t ′ ) ) − ℑ ( A ) 2 ℜ ( A ) − B sin ( ω ( t + t ′ ) ) (73) M(t,t')=\frac{|A|^2-\frac{1}{4}B^2+1}{2(2\Re(A)-B)}\cos(\omega(t-t'))-\frac{|A|^2-\frac{1}{4}B^2-1}{2(2\Re(A)-B)}\cos(\omega(t+t'))-\frac{\Im(A)}{2\Re(A)-B}\sin(\omega(t+t')) \tag*{(73)} M ( t , t ′ ) = 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 + 1 cos ( ω ( t − t ′ )) − 2 ( 2ℜ ( A ) − B ) ∣ A ∣ 2 − 4 1 B 2 − 1 cos ( ω ( t + t ′ )) − 2ℜ ( A ) − B ℑ ( A ) sin ( ω ( t + t ′ )) (73) R ( t ) = i 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) 2 cos ( ω t ) − 2 ( ∣ A ∣ 2 − 1 4 B 2 ) ( a ∗ − a ) sin ( ω t ) (74) R(t)=i\frac{2(Aa+A^*a^*)-B(a^*+a)}{\sqrt{2}}\cos(\omega t)-\sqrt{2}\left(|A|^2-\frac{1}{4}B^2\right)(a^*-a)\sin(\omega t) \tag*{(74)} R ( t ) = i 2 2 ( A a + A ∗ a ∗ ) − B ( a ∗ + a ) cos ( ω t ) − 2 ( ∣ A ∣ 2 − 4 1 B 2 ) ( a ∗ − a ) sin ( ω t ) (74) Taking the logarithm of this expression for all the modes, and from the equations (32) and (33) we can construct the effective action for ξ \xi ξ
S e f f [ ξ ] = 1 2 m 0 ∫ 0 T d t ( ξ ˙ i ξ ˙ i − ξ ˙ ′ i ξ ˙ ′ i ) + ∑ s ∫ d 3 k ( 2 π ) 3 1 m e f f ω ∫ 0 T d t ∫ 0 T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) K ′ ( t ′ ) + ∑ s ∫ d 3 k ( 2 π ) 3 i m e f f ω ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) − ∑ s ∫ d 3 k ( 2 π ) 3 i 2 ℜ ( A ) − B ∫ 0 T R ( t ) K ′ ( t ) d t (75) \begin{aligned}
S_{\mathrm{eff}}[\xi]={}&\frac{1}{2}m_0\int_0^Tdt\left(\dot{\xi}^i\dot{\xi}^i-\dot{\xi}^{\prime i}\dot{\xi}^{\prime i}\right)+\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{1}{m_{\mathrm{eff}}\omega}\int_0^Tdt\int_0^Tdt'K(t)\theta(t'-t)\sin(\omega(t'-t))K'(t')\\
&+\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{i}{m_{\mathrm{eff}}\omega}\int_0^Tdt\int_0^Tdt'K'(t)M(t,t')K'(t')\\
&-\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{i}{2\Re(A)-B}\int_0^TR(t)K'(t)dt \tag*{(75)}
\end{aligned} S eff [ ξ ] = 2 1 m 0 ∫ 0 T d t ( ξ ˙ i ξ ˙ i − ξ ˙ ′ i ξ ˙ ′ i ) + s ∑ ∫ ( 2 π ) 3 d 3 k m eff ω 1 ∫ 0 T d t ∫ 0 T d t ′ K ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t )) K ′ ( t ′ ) + s ∑ ∫ ( 2 π ) 3 d 3 k m eff ω i ∫ 0 T d t ∫ 0 T d t ′ K ′ ( t ) M ( t , t ′ ) K ′ ( t ′ ) − s ∑ ∫ ( 2 π ) 3 d 3 k 2ℜ ( A ) − B i ∫ 0 T R ( t ) K ′ ( t ) d t (75) where we discarded the first line in (A12) since it is just a constant which is irrelevant to the classical equation of motion of ξ \xi ξ
Now, we return the K K K K ′ K' K ′ J J J J ′ J' J ′ ξ \xi ξ
S e f f [ ξ ] = 1 2 m 0 ∫ 0 T d t ( ξ ˙ i ξ ˙ i − ξ ˙ ′ i ξ ˙ ′ i ) + ∑ s ∫ d 3 k ( 2 π ) 3 m 0 2 g 2 8 m e f f 2 ω ∫ 0 T d t ∫ 0 T d t ′ Λ i j ( s ) ( ω ) Λ l m ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) + ξ ′ i ( t ) ξ ′ j ( t ) ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ( ξ l ( t ′ ) ξ m ( t ′ ) − ξ ′ l ( t ′ ) ξ ′ m ( t ′ ) ) + ∑ s ∫ d 3 k ( 2 π ) 3 i m 0 2 g 2 8 m e f f 2 ω ∫ 0 T d t ∫ 0 T d t ′ Λ i j ( s ) ( ω ) Λ l m ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) − ξ ′ i ( t ) ξ ′ j ( t ) ) M ( t , t ′ ) ( ξ l ( t ′ ) ξ m ( t ′ ) − ξ ′ l ( t ′ ) ξ ′ m ( t ′ ) ) + ∑ s ∫ d 3 k ( 2 π ) 3 i m 0 g 2 2 ( A + A ∗ − B ) ∫ 0 T d t Λ i j ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) − ξ ′ i ( t ) ξ ′ j ( t ) ) R ( t ) (76) S_{\mathrm{eff}}[\xi]=\frac{1}{2}m_0\int_0^T dt\left(\dot{\xi}^i\dot{\xi}^i-\dot{\xi}^{\prime i}\dot{\xi}^{\prime i}\right)
+\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{m_0^2g^2}{8m_{\mathrm{eff}}^2\omega}\int_0^Tdt\int_0^Tdt'\Lambda_{ij}^{(s)}(\omega)\Lambda_{lm}^{(s)}(\omega)\left(\xi^i(t)\xi^j(t)+\xi^{\prime i}(t)\xi^{\prime j}(t)\right)\theta(t'-t)\sin\left(\omega(t'-t)\right)
\left(\xi^l(t')\xi^m(t')-\xi^{\prime l}(t')\xi^{\prime m}(t')\right)
+\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{im_0^2g^2}{8m_{\mathrm{eff}}^2\omega}\int_0^Tdt\int_0^Tdt'\Lambda_{ij}^{(s)}(\omega)\Lambda_{lm}^{(s)}(\omega)\left(\xi^i(t)\xi^j(t)-\xi^{\prime i}(t)\xi^{\prime j}(t)\right)M(t,t')
\left(\xi^l(t')\xi^m(t')-\xi^{\prime l}(t')\xi^{\prime m}(t')\right)
+\sum_s\int\frac{d^3k}{(2\pi)^3}\frac{im_0g}{2\sqrt{2}(A+A^*-B)}\int_0^Tdt\Lambda_{ij}^{(s)}(\omega)\left(\xi^i(t)\xi^j(t)-\xi^{\prime i}(t)\xi^{\prime j}(t)\right)R(t) \tag*{(76)} S eff [ ξ ] = 2 1 m 0 ∫ 0 T d t ( ξ ˙ i ξ ˙ i − ξ ˙ ′ i ξ ˙ ′ i ) + s ∑ ∫ ( 2 π ) 3 d 3 k 8 m eff 2 ω m 0 2 g 2 ∫ 0 T d t ∫ 0 T d t ′ Λ ij ( s ) ( ω ) Λ l m ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) + ξ ′ i ( t ) ξ ′ j ( t ) ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ( ξ l ( t ′ ) ξ m ( t ′ ) − ξ ′ l ( t ′ ) ξ ′ m ( t ′ ) ) + s ∑ ∫ ( 2 π ) 3 d 3 k 8 m eff 2 ω i m 0 2 g 2 ∫ 0 T d t ∫ 0 T d t ′ Λ ij ( s ) ( ω ) Λ l m ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) − ξ ′ i ( t ) ξ ′ j ( t ) ) M ( t , t ′ ) ( ξ l ( t ′ ) ξ m ( t ′ ) − ξ ′ l ( t ′ ) ξ ′ m ( t ′ ) ) + s ∑ ∫ ( 2 π ) 3 d 3 k 2 2 ( A + A ∗ − B ) i m 0 g ∫ 0 T d t Λ ij ( s ) ( ω ) ( ξ i ( t ) ξ j ( t ) − ξ ′ i ( t ) ξ ′ j ( t ) ) R ( t ) (76) Now, we do a change of variables:
ξ c l = 1 2 ( ξ + ξ ′ ) , ξ q = 1 2 ( ξ − ξ ′ ) (77) \xi_{\mathrm{cl}}=\frac{1}{2}(\xi+\xi'),\qquad\xi_q=\frac{1}{2}(\xi-\xi') \tag*{(77)} ξ cl = 2 1 ( ξ + ξ ′ ) , ξ q = 2 1 ( ξ − ξ ′ ) (77) and we discard elements with more than two ξ q \xi_q ξ q Λ i j \Lambda_{ij} Λ ij i ↔ j i\leftrightarrow j i ↔ j
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + m 0 2 g 2 m e f f ∫ d 3 k ( 2 π ) 3 ∑ s Λ i j ( s ) Λ l m ( s ) ω ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + 2 i m 0 2 g 2 m e f f ∫ d 3 k ( 2 π ) 3 ∑ s Λ i j ( s ) Λ l m ( s ) ω ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + i 2 m 0 g ∫ d 3 k ( 2 π ) 3 ∑ s Λ i j ( s ) A + A ∗ − B ∫ 0 T d t ξ c l i ( t ) ξ q j ( t ) R ( t ) (78) S_{\mathrm{eff}}[\xi]=2m_0\int_0^Tdt\dot{\xi}_{\mathrm{cl}}^i\xi_q^i+\frac{m_0^2g^2}{m_{\mathrm{eff}}}\int\frac{d^3k}{(2\pi)^3}\frac{\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}}{\omega}\int_0^Tdt\int_0^Tdt'\xi_{\mathrm{cl}}^i(t)\xi_q^j(t)\theta(t'-t)\sin\left(\omega(t'-t)\right)\xi_{\mathrm{cl}}^l(t')\xi_q^m(t')
+\frac{2im_0^2g^2}{m_{\mathrm{eff}}}\int\frac{d^3k}{(2\pi)^3}\frac{\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}}{\omega}\int_0^Tdt\int_0^Tdt'\xi_{\mathrm{cl}}^i(t)\xi_q^j(t)M(t,t')\xi_{\mathrm{cl}}^l(t')\xi_q^m(t')
+i\sqrt{2}m_0g\int\frac{d^3k}{(2\pi)^3}\frac{\sum_s\Lambda_{ij}^{(s)}}{A+A^*-B}\int_0^Tdt\xi_{\mathrm{cl}}^i(t)\xi_q^j(t)R(t) \tag*{(78)} S eff [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ cl i ξ q i + m eff m 0 2 g 2 ∫ ( 2 π ) 3 d 3 k ω ∑ s Λ ij ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ cl i ( t ) ξ q j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ξ cl l ( t ′ ) ξ q m ( t ′ ) + m eff 2 i m 0 2 g 2 ∫ ( 2 π ) 3 d 3 k ω ∑ s Λ ij ( s ) Λ l m ( s ) ∫ 0 T d t ∫ 0 T d t ′ ξ cl i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ cl l ( t ′ ) ξ q m ( t ′ ) + i 2 m 0 g ∫ ( 2 π ) 3 d 3 k A + A ∗ − B ∑ s Λ ij ( s ) ∫ 0 T d t ξ cl i ( t ) ξ q j ( t ) R ( t ) (78) which is used in the main text writing the momentum integral in spherical coordinates.
Angular integral of ∑ s Λ i j ( s ) Λ l m ( s ) \sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)} ∑ s Λ ij ( s ) Λ l m ( s )
Define the rectangular kernel
k ^ i μ = ω δ i μ + k i δ 0 μ , k ^ i μ k μ = 0 (79) \hat{k}^{\mu}_{\ i}=\omega\delta^\mu_i+k_i\delta^\mu_0,\qquad\hat{k}^{\mu}_{\ i}k_\mu=0 \tag*{(79)} k ^ i μ = ω δ i μ + k i δ 0 μ , k ^ i μ k μ = 0 (79) where the second identity follows from k 0 = − ω k_0=-\omega k 0 = − ω Λ \Lambda Λ
Λ i j ( s ) = k ^ i μ k ^ j ν ε μ ν ( s ) . (80) \Lambda_{ij}^{(s)}=\hat{k}^{\mu}_{\ i}\hat{k}^{\nu}_{\ j}\varepsilon_{\mu\nu}^{(s)}. \tag*{(80)} Λ ij ( s ) = k ^ i μ k ^ j ν ε μν ( s ) . (80) Define the polarization sum
S μ ν ρ σ ≡ ∑ s ε μ ν ( s ) ε ρ σ ( s ) . (81) S_{\mu\nu\rho\sigma}\equiv\sum_s\varepsilon_{\mu\nu}^{(s)}\varepsilon_{\rho\sigma}^{(s)}. \tag*{(81)} S μν ρ σ ≡ s ∑ ε μν ( s ) ε ρ σ ( s ) . (81) The two physical polarizations are normalized by
ε μ ν ( s ) ε ( s ′ ) μ ν = δ s s ′ , S μ ν μ ν = 2. (82) \varepsilon_{\mu\nu}^{(s)}\varepsilon^{(s')\mu\nu}=\delta^{ss'},\qquad S^{\mu\nu}_{\ \ \mu\nu}=2. \tag*{(82)} ε μν ( s ) ε ( s ′ ) μν = δ s s ′ , S μν μν = 2. (82) Its transversality and tracelessness are
k μ S μ ν ρ σ = n μ S μ ν ρ σ = η μ ν S μ ν ρ σ = 0. (83) k^\mu S_{\mu\nu\rho\sigma}=n^\mu S_{\mu\nu\rho\sigma}=\eta^{\mu\nu}S_{\mu\nu\rho\sigma}=0. \tag*{(83)} k μ S μν ρ σ = n μ S μν ρ σ = η μν S μν ρ σ = 0. (83) Thus
∑ s Λ i j ( s ) Λ l m ( s ) = k ^ i k ^ j k ^ l k ^ m S μ ν ρ σ = 1 2 ( P i l P j m + P i m P j l − P i j P l m ) , (84) \sum_s \Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}=\hat{k}_i\hat{k}_j\hat{k}_l\hat{k}_m S_{\mu\nu\rho\sigma}=\frac{1}{2}\left(\mathcal{P}_{il}\mathcal{P}_{jm}+\mathcal{P}_{im}\mathcal{P}_{jl}-\mathcal{P}_{ij}\mathcal{P}_{lm}\right), \tag*{(84)} s ∑ Λ ij ( s ) Λ l m ( s ) = k ^ i k ^ j k ^ l k ^ m S μν ρ σ = 2 1 ( P i l P j m + P im P j l − P ij P l m ) , (84) where
P i j = ω 2 δ i j − ( 1 + m g 2 ( n ⋅ k ) 2 ) k i k j + ω m g 2 ( n ⋅ k ) 2 ( n i k j + n j k i ) − ω 2 m g 2 ( n ⋅ k ) 2 n i n j . (85) \mathcal{P}_{ij}=\omega^2\delta_{ij}-\left(1+\frac{m_g^2}{(\boldsymbol{n}\cdot\boldsymbol{k})^2}\right)k_i k_j+\omega\frac{m_g^2}{(\boldsymbol{n}\cdot\boldsymbol{k})^2}(n_i k_j+n_j k_i)-\omega^2\frac{m_g^2}{(\boldsymbol{n}\cdot\boldsymbol{k})^2}n_i n_j. \tag*{(85)} P ij = ω 2 δ ij − ( 1 + ( n ⋅ k ) 2 m g 2 ) k i k j + ω ( n ⋅ k ) 2 m g 2 ( n i k j + n j k i ) − ω 2 ( n ⋅ k ) 2 m g 2 n i n j . (85) With
ϵ = m g 2 ω 2 , L = ln ( ω + ∣ k ⃗ ∣ ω − ∣ k ⃗ ∣ ) = ln ( 4 ω 2 m g 2 ) − m g 2 2 ω 2 + O ( m g 4 ω 4 ) , (86) \epsilon=\frac{m_g^2}{\omega^2},\qquad L=\ln\left(\frac{\omega+\lvert\vec{k}\rvert}{\omega-\lvert\vec{k}\rvert}\right)=\ln\left(\frac{4\omega^2}{m_g^2}\right)-\frac{m_g^2}{2\omega^2}+O\left(\frac{m_g^4}{\omega^4}\right), \tag*{(86)} ϵ = ω 2 m g 2 , L = ln ( ω − ∣ k ∣ ω + ∣ k ∣ ) = ln ( m g 2 4 ω 2 ) − 2 ω 2 m g 2 + O ( ω 4 m g 4 ) , (86) the angular integral through first order can be written in the form used in Eq. (46) :
∫ d Ω ∑ s Λ i j ( s ) Λ l m ( s ) = 4 π ω 4 [ T ~ i j l m ( 0 ) + m g 2 ω 2 Λ i j l m + m g 2 ω 2 ln ( 4 ω 2 m g 2 ) Ξ i j l m + O ( m g 4 ω 4 L ) ] , (87) \int d\Omega\sum_s\Lambda_{ij}^{(s)}\Lambda_{lm}^{(s)}=4\pi\omega^4\left[\widetilde{T}_{ijlm}^{(0)}+\frac{m_g^2}{\omega^2}\Lambda_{ijlm}+\frac{m_g^2}{\omega^2}\ln\left(\frac{4\omega^2}{m_g^2}\right)\Xi_{ijlm}+O\left(\frac{m_g^4}{\omega^4}L\right)\right], \tag*{(87)} ∫ d Ω s ∑ Λ ij ( s ) Λ l m ( s ) = 4 π ω 4 [ T ij l m ( 0 ) + ω 2 m g 2 Λ ij l m + ω 2 m g 2 ln ( m g 2 4 ω 2 ) Ξ ij l m + O ( ω 4 m g 4 L ) ] , (87) where
T ~ i j l m ( 0 ) = 1 5 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) + 1 15 δ i j δ l m , (88) \widetilde{T}_{ijlm}^{(0)}=\frac{1}{5}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})+\frac{1}{15}\delta_{ij}\delta_{lm}, \tag*{(88)} T ij l m ( 0 ) = 5 1 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) + 15 1 δ ij δ l m , (88) and the nonlogarithmic and logarithmic corrections are, respectively,
Λ i j l m = 91 60 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) + 11 30 δ i j δ l m + 13 12 ( δ i j n l n m + δ l m n i n j ) − 23 12 ( δ i l n j n m + δ i m n j n l + δ j l n i n m + δ j m n i n l ) + 37 12 n i n j n l n m , (89) \begin{aligned}
\Lambda_{ijlm}={}&\frac{91}{60}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})+\frac{11}{30}\delta_{ij}\delta_{lm}\\
&+\frac{13}{12}(\delta_{ij}n_l n_m+\delta_{lm}n_i n_j)\\
&-\frac{23}{12}(\delta_{il}n_j n_m+\delta_{im}n_j n_l+\delta_{jl}n_i n_m+\delta_{jm}n_i n_l)\\
&+\frac{37}{12}n_i n_j n_l n_m, \tag*{(89)}
\end{aligned} Λ ij l m = 60 91 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) + 30 11 δ ij δ l m + 12 13 ( δ ij n l n m + δ l m n i n j ) − 12 23 ( δ i l n j n m + δ im n j n l + δ j l n i n m + δ j m n i n l ) + 12 37 n i n j n l n m , (89) Ξ i j l m = − 1 2 ( δ i l δ j m + δ i m δ j l − δ i j δ l m ) − 1 2 ( δ i j n l n m + δ l m n i n j ) + 1 2 ( δ i l n j n m + δ i m n j n l + δ j l n i n m + δ j m n i n l ) − 1 2 n i n j n l n m . (90) \begin{aligned}
\Xi_{ijlm}={}&-\frac{1}{2}(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}-\delta_{ij}\delta_{lm})\\
&-\frac{1}{2}(\delta_{ij}n_l n_m+\delta_{lm}n_i n_j)\\
&+\frac{1}{2}(\delta_{il}n_j n_m+\delta_{im}n_j n_l+\delta_{jl}n_i n_m+\delta_{jm}n_i n_l)\\
&-\frac{1}{2}n_i n_j n_l n_m. \tag*{(90)}
\end{aligned} Ξ ij l m = − 2 1 ( δ i l δ j m + δ im δ j l − δ ij δ l m ) − 2 1 ( δ ij n l n m + δ l m n i n j ) + 2 1 ( δ i l n j n m + δ im n j n l + δ j l n i n m + δ j m n i n l ) − 2 1 n i n j n l n m . (90) Both tensors depend only on δ i j \delta_{ij} δ ij n i n_i n i L L L ln ( 4 ω 2 / m g 2 ) \ln(4\omega^2/m_g^2) ln ( 4 ω 2 / m g 2 ) ϵ 2 \epsilon^2 ϵ 2 (87) . The mass-dependent correction vanishes as ϵ ln ( 1 / ϵ ) \epsilon\ln(1/\epsilon) ϵ ln ( 1/ ϵ ) ϵ → 0 \epsilon\to0 ϵ → 0 m g m_g m g
Effective action arrangements
From equation (43), with a = a ∗ = 0 a=a^*=0 a = a ∗ = 0 m e f f = 1 16 π G 2 m_{eff}=\frac{1}{16\pi G^2} m e f f = 16 π G 2 1 g = 1 2 G g=\frac{1}{\sqrt{2G}} g = 2 G 1 ∣ k ∣ \lvert\mathbf{k}\rvert ∣ k ∣ ω \omega ω ω = ∣ k ∣ 2 + m g 2 \omega=\sqrt{\lvert\mathbf{k}\rvert^2+m_g^2} ω = ∣ k ∣ 2 + m g 2
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + 4 G m 0 2 π ∫ m g ∞ ω 2 − m g 2 d ω ( ω 4 T ~ i j l m ( 0 ) + m g 2 ω 2 ( Λ i j l m + ln ( 4 ω 2 m g 2 ) Ξ i j l m ) ) × ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t ) ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + i 8 G m 0 2 π ∫ m g ∞ ω 2 − m g 2 d ω ( ω 4 T ~ i j l m ( 0 ) + m g 2 ω 2 ( Λ i j l m + ln ( 4 ω 2 m g 2 ) Ξ i j l m ) ) × ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (91) \begin{aligned}
S_{eff}[\xi] ={}& 2m_0\int_0^T dt\dot{\xi}_{cl}^i\xi_q^i \\
&+\frac{4Gm_0^2}{\pi}\int_{m_g}^{\infty}\sqrt{\omega^2-m_g^2}\,d\omega\left(\omega^4\widetilde{T}_{ijlm}^{(0)}+m_g^2\omega^2\left(\Lambda_{ijlm}+\ln\left(\frac{4\omega^2}{m_g^2}\right)\Xi_{ijlm}\right)\right)\\
&\qquad\times\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\sin(\omega(t'-t))\xi_{cl}^l(t')\xi_q^m(t')\\
&+i\frac{8Gm_0^2}{\pi}\int_{m_g}^{\infty}\sqrt{\omega^2-m_g^2}\,d\omega\left(\omega^4\widetilde{T}_{ijlm}^{(0)}+m_g^2\omega^2\left(\Lambda_{ijlm}+\ln\left(\frac{4\omega^2}{m_g^2}\right)\Xi_{ijlm}\right)\right)\\
&\qquad\times\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)M(t,t')\xi_{cl}^l(t')\xi_q^m(t') \tag*{(91)}
\end{aligned} S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + π 4 G m 0 2 ∫ m g ∞ ω 2 − m g 2 d ω ( ω 4 T ij l m ( 0 ) + m g 2 ω 2 ( Λ ij l m + ln ( m g 2 4 ω 2 ) Ξ ij l m ) ) × ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) sin ( ω ( t ′ − t )) ξ c l l ( t ′ ) ξ q m ( t ′ ) + i π 8 G m 0 2 ∫ m g ∞ ω 2 − m g 2 d ω ( ω 4 T ij l m ( 0 ) + m g 2 ω 2 ( Λ ij l m + ln ( m g 2 4 ω 2 ) Ξ ij l m ) ) × ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (91) Since the sum of polarizations used was approximated up to order m g 2 m_g^2 m g 2
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + 4 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 5 sin ( ω ( t ′ − t ) ) T ~ i j l m ( 0 ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + 4 G m 0 2 m g 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 3 sin ( ω ( t ′ − t ) ) S ~ i j l m ′ ξ c l l ( t ′ ) ξ q m ( t ′ ) + 8 G m 0 2 m g 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 3 ln ( ω m g ) sin ( ω ( t ′ − t ) ) Ξ i j l m ξ c l l ( t ′ ) ξ q m ( t ′ ) + i 8 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) ∫ m g ∞ d ω ( ω 5 T ~ i j l m ( 0 ) + m g 2 ω 3 S ~ i j l m ′ + 2 m g 2 ω 3 ln ( ω m g ) Ξ i j l m ) × M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (92) \begin{aligned}
S_{eff}[\xi] ={}& 2m_0\int_0^Tdt\dot{\xi}_{cl}^i\xi_q^i\\
&+\frac{4Gm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\int_{m_g}^{\infty}d\omega\,\omega^5\sin(\omega(t'-t))\widetilde{T}_{ijlm}^{(0)}\xi_{cl}^l(t')\xi_q^m(t')\\
&+\frac{4Gm_0^2m_g^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\int_{m_g}^{\infty}d\omega\,\omega^3\sin(\omega(t'-t))\widetilde{S}'_{ijlm}\xi_{cl}^l(t')\xi_q^m(t')\\
&+\frac{8Gm_0^2m_g^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\int_{m_g}^{\infty}d\omega\,\omega^3\ln\left(\frac{\omega}{m_g}\right)\sin(\omega(t'-t))\Xi_{ijlm}\xi_{cl}^l(t')\xi_q^m(t')\\
&+i\frac{8Gm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\int_{m_g}^{\infty}d\omega\left(\omega^5\widetilde{T}_{ijlm}^{(0)}+m_g^2\omega^3\widetilde{S}'_{ijlm}+2m_g^2\omega^3\ln\left(\frac{\omega}{m_g}\right)\Xi_{ijlm}\right)\\
&\qquad\times M(t,t')\xi_{cl}^l(t')\xi_q^m(t') \tag*{(92)}
\end{aligned} S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + π 4 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 5 sin ( ω ( t ′ − t )) T ij l m ( 0 ) ξ c l l ( t ′ ) ξ q m ( t ′ ) + π 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 3 sin ( ω ( t ′ − t )) S ij l m ′ ξ c l l ( t ′ ) ξ q m ( t ′ ) + π 8 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∫ m g ∞ d ω ω 3 ln ( m g ω ) sin ( ω ( t ′ − t )) Ξ ij l m ξ c l l ( t ′ ) ξ q m ( t ′ ) + i π 8 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) ∫ m g ∞ d ω ( ω 5 T ij l m ( 0 ) + m g 2 ω 3 S ij l m ′ + 2 m g 2 ω 3 ln ( m g ω ) Ξ ij l m ) × M ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (92) where we defined
S ~ i j l m ′ = − 1 2 T ~ i j l m ( 0 ) + Λ i j l m + 2 Ξ i j l m ln ( 2 ) (93) \widetilde{S}'_{ijlm}=-\frac{1}{2}\widetilde{T}_{ijlm}^{(0)}+\Lambda_{ijlm}+2\Xi_{ijlm}\ln(2) \tag*{(93)} S ij l m ′ = − 2 1 T ij l m ( 0 ) + Λ ij l m + 2 Ξ ij l m ln ( 2 ) (93) The integrals can be rearranged extracting time derivatives as:
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + 4 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ 5 ∂ t 5 ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) T ~ i j l m ( 0 ) ξ c l l ( t ′ ) ξ q m ( t ′ ) − 4 G m 0 2 m g 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ 3 ∂ t 3 ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) S ~ i j l m ′ ξ c l l ( t ′ ) ξ q m ( t ′ ) − 8 G m 0 2 m g 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ 3 ∂ t 3 ∫ m g ∞ d ω ln ( ω m g ) cos ( ω ( t ′ − t ) ) Ξ i j l m ξ c l l ( t ′ ) ξ q m ( t ′ ) + i 8 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) ∂ 2 ∂ t 2 ∂ 2 ∂ t ′ 2 K i j l m ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (94) \begin{aligned}
S_{eff}[\xi] ={}& 2m_0\int_0^Tdt\dot{\xi}_{cl}^i\xi_q^i\\
&+\frac{4Gm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\frac{\partial^5}{\partial t^5}\int_{m_g}^{\infty}d\omega\cos(\omega(t'-t))\widetilde{T}_{ijlm}^{(0)}\xi_{cl}^l(t')\xi_q^m(t')\\
&-\frac{4Gm_0^2m_g^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\frac{\partial^3}{\partial t^3}\int_{m_g}^{\infty}d\omega\cos(\omega(t'-t))\widetilde{S}'_{ijlm}\xi_{cl}^l(t')\xi_q^m(t')\\
&-\frac{8Gm_0^2m_g^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\theta(t'-t)\frac{\partial^3}{\partial t^3}\int_{m_g}^{\infty}d\omega\ln\left(\frac{\omega}{m_g}\right)\cos(\omega(t'-t))\Xi_{ijlm}\xi_{cl}^l(t')\xi_q^m(t')\\
&+i\frac{8Gm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\xi_{cl}^i(t)\xi_q^j(t)\frac{\partial^2}{\partial t^2}\frac{\partial^2}{\partial t'^2}K_{ijlm}(t,t')\xi_{cl}^l(t')\xi_q^m(t') \tag*{(94)}
\end{aligned} S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ q i + π 4 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ t 5 ∂ 5 ∫ m g ∞ d ω cos ( ω ( t ′ − t )) T ij l m ( 0 ) ξ c l l ( t ′ ) ξ q m ( t ′ ) − π 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ t 3 ∂ 3 ∫ m g ∞ d ω cos ( ω ( t ′ − t )) S ij l m ′ ξ c l l ( t ′ ) ξ q m ( t ′ ) − π 8 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) θ ( t ′ − t ) ∂ t 3 ∂ 3 ∫ m g ∞ d ω ln ( m g ω ) cos ( ω ( t ′ − t )) Ξ ij l m ξ c l l ( t ′ ) ξ q m ( t ′ ) + i π 8 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ c l i ( t ) ξ q j ( t ) ∂ t 2 ∂ 2 ∂ t ′2 ∂ 2 K ij l m ( t , t ′ ) ξ c l l ( t ′ ) ξ q m ( t ′ ) (94) with the definition of
K i j l m ( t , t ′ ) = ∫ m g ∞ d ω ( ω T ~ i j l m ( 0 ) + m g 2 1 ω S ~ i j l m ′ + 2 m g 2 1 ω ln ( ω m g ) Ξ i j l m ) M ( t , t ′ ) (95) K_{ijlm}(t,t')=\int_{m_g}^{\infty}d\omega\left(\omega\widetilde{T}_{ijlm}^{(0)}+m_g^2\frac{1}{\omega}\widetilde{S}'_{ijlm}+2m_g^2\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)\Xi_{ijlm}\right)M(t,t') \tag*{(95)} K ij l m ( t , t ′ ) = ∫ m g ∞ d ω ( ω T ij l m ( 0 ) + m g 2 ω 1 S ij l m ′ + 2 m g 2 ω 1 ln ( m g ω ) Ξ ij l m ) M ( t , t ′ ) (95) Let us to see the integrals
∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) , and ∫ m g ∞ d ω ln ( ω m g ) cos ( ω ( t ′ − t ) ) (96) \int_{m_g}^{\infty} d\omega\cos\bigl(\omega(t' - t)\bigr), \quad\text{and} \quad\int_{m_g}^{\infty} d\omega\ln\left(\frac{\omega}{m_g}\right)\cos\bigl(\omega(t' - t)\bigr) \tag*{(96)} ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) , and ∫ m g ∞ d ω ln ( m g ω ) cos ( ω ( t ′ − t ) ) (96) For the first one, splitting in two pieces:
∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) = ∫ 0 ∞ d ω cos ( ω ( t ′ − t ) ) − ∫ 0 m g d ω cos ( ω ( t ′ − t ) ) (97) \int_{m_g}^{\infty} d\omega\cos\bigl(\omega(t' - t)\bigr) = \int_{0}^{\infty} d\omega\cos\bigl(\omega(t' - t)\bigr) - \int_{0}^{m_g} d\omega\cos\bigl(\omega(t' - t)\bigr) \tag*{(97)} ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) = ∫ 0 ∞ d ω cos ( ω ( t ′ − t ) ) − ∫ 0 m g d ω cos ( ω ( t ′ − t ) ) (97) we observe that the second term is finite, and the first one, can be understood in the distributional sense via the relation:
∫ − ∞ ∞ d k k n e i k ( t − t ′ ) = 2 π 1 i n δ ( n ) ( t − t ′ ) (98) \int_{-\infty}^{\infty} dk k^n e^{ik(t-t')} = 2\pi\frac{1}{i^n}\delta^{(n)}(t-t') \tag*{(98)} ∫ − ∞ ∞ d k k n e ik ( t − t ′ ) = 2 π i n 1 δ ( n ) ( t − t ′ ) (98) so we have that the derivative of that term, which is what we have in (C4), becomes
∂ 5 ∂ t 5 ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) = π ∂ 5 ∂ t 5 δ ( t − t ′ ) − ∂ 5 ∂ t 5 sin ( m g ( t ′ − t ) ) t ′ − t (99) \frac{\partial^5}{\partial t^5}\int_{m_g}^{\infty}d\omega\cos\bigl(\omega(t'-t)\bigr)=\pi\frac{\partial^5}{\partial t^5}\delta(t-t')-\frac{\partial^5}{\partial t^5}\frac{\sin\bigl(m_g(t'-t)\bigr)}{t'-t} \tag*{(99)} ∂ t 5 ∂ 5 ∫ m g ∞ d ω cos ( ω ( t ′ − t ) ) = π ∂ t 5 ∂ 5 δ ( t − t ′ ) − ∂ t 5 ∂ 5 t ′ − t sin ( m g ( t ′ − t ) ) (99) If we expand the second term the result is of order higher than m g 2 m_g^2 m g 2
∫ m g ∞ d ω ln ( ω m g ) cos ( ω ( t ′ − t ) ) = m g ∫ 0 ∞ d x ln ( x ) cos ( x m g ( t ′ − t ) ) − m g ∫ 0 1 d x ln ( x ) cos ( x m g ( t ′ − t ) ) (100) \int_{m_g}^{\infty}d\omega\ln\left(\frac{\omega}{m_g}\right)\cos\bigl(\omega(t'-t)\bigr)=m_g\int_{0}^{\infty}dx\ln(x)\cos\bigl(xm_g(t'-t)\bigr)-m_g\int_{0}^{1}dx\ln(x)\cos\bigl(xm_g(t'-t)\bigr) \tag*{(100)} ∫ m g ∞ d ω ln ( m g ω ) cos ( ω ( t ′ − t ) ) = m g ∫ 0 ∞ d x ln ( x ) cos ( x m g ( t ′ − t ) ) − m g ∫ 0 1 d x ln ( x ) cos ( x m g ( t ′ − t ) ) (100) As before, the second term is finite, and the first one can be arranged as:
∫ 0 ∞ d x ln ( x ) cos ( x m g ( t ′ − t ) ) = − ∫ 0 ∞ d x 1 m g ∂ ∂ t [ ln ( x ) sin ( x m g ( t ′ − t ) ) x ] (101) \int_{0}^{\infty}dx\ln(x)\cos\bigl(xm_g(t'-t)\bigr)=-\int_{0}^{\infty}dx\frac{1}{m_g}\frac{\partial}{\partial t}\left[\ln(x)\frac{\sin\bigl(xm_g(t'-t)\bigr)}{x}\right] \tag*{(101)} ∫ 0 ∞ d x ln ( x ) cos ( x m g ( t ′ − t ) ) = − ∫ 0 ∞ d x m g 1 ∂ t ∂ [ ln ( x ) x sin ( x m g ( t ′ − t ) ) ] (101) Interchanging integral and derivative, that integral can be done easily and after deriving we obtain
∫ m g ∞ d ω ln ( ω m g ) cos ( ω ( t ′ − t ) ) = − π γ δ ( t ′ − t ) + π 2 ∂ sgn ( t ′ − t ) ln ∣ m g ( t ′ − t ) ∣ ∂ t + Si ( m g ( t ′ − t ) ) t ′ − t (102) \int_{m_g}^{\infty}d\omega\ln\left(\frac{\omega}{m_g}\right)\cos\bigl(\omega(t'-t)\bigr)=-\pi\gamma\delta(t'-t)+\frac{\pi}{2}\frac{\partial\operatorname{sgn}(t'-t)\ln\lvert m_g(t'-t)\rvert}{\partial t}+\frac{\operatorname{Si}\bigl(m_g(t'-t)\bigr)}{t'-t} \tag*{(102)} ∫ m g ∞ d ω ln ( m g ω ) cos ( ω ( t ′ − t ) ) = − π γ δ ( t ′ − t ) + 2 π ∂ t ∂ sgn ( t ′ − t ) ln ∣ m g ( t ′ − t )∣ + t ′ − t Si ( m g ( t ′ − t ) ) (102) where γ \gamma γ S i Si S i m g 2 m_g^2 m g 2
S e f f [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ c l i ξ ˙ c l i + 4 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) θ ( t ′ − t ) ∂ 5 ∂ t 5 δ ( t − t ′ ) T ~ i j l m ( 0 ) ξ c l l ( t ′ ) ξ m ( t ′ ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) θ ( t ′ − t ) ∂ 3 ∂ t 3 δ ( t − t ′ ) S ~ i j l m ξ c l l ( t ′ ) ξ m ( t ′ ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) θ ( t ′ − t ) ∂ 4 sgn ( t ′ − t ) ln ( m g ∣ t ′ − t ∣ ) ∂ t 4 Ξ i j l m ξ c l l ( t ′ ) ξ m ( t ′ ) + 8 i G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) ∂ 2 ∂ t 2 ∂ 2 ∂ t ′ 2 K i j l m ( t , t ′ ) ξ c l l ( t ′ ) ξ m ( t ′ ) (103) \begin{aligned}
S_{\mathrm{eff}}[\xi] ={}& 2m_0\int_0^Tdt\,\dot{\xi}_{\mathrm{cl}}^i\dot{\xi}_{\mathrm{cl}}^i \\
&+4Gm_0^2\int_0^Tdt\int_0^Tdt'\,\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\theta(t'-t)\frac{\partial^5}{\partial t^5}\delta(t-t')\widetilde{T}_{ijlm}^{(0)}\xi_{\mathrm{cl}}^l(t')\xi^m(t') \\
&-4Gm_0^2m_g^2\int_0^Tdt\int_0^Tdt'\,\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\theta(t'-t)\frac{\partial^3}{\partial t^3}\delta(t-t')\widetilde{S}_{ijlm}\xi_{\mathrm{cl}}^l(t')\xi^m(t') \\
&-4Gm_0^2m_g^2\int_0^Tdt\int_0^Tdt'\,\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\theta(t'-t)\frac{\partial^4\operatorname{sgn}(t'-t)\ln\bigl(m_g\lvert t'-t\rvert\bigr)}{\partial t^4}\Xi_{ijlm}\xi_{\mathrm{cl}}^l(t')\xi^m(t') \\
&+\frac{8iGm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\,\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\frac{\partial^2}{\partial t^2}\frac{\partial^2}{\partial t'^2}K_{ijlm}(t,t')\xi_{\mathrm{cl}}^l(t')\xi^m(t') \tag*{(103)}
\end{aligned} S eff [ ξ ] = 2 m 0 ∫ 0 T d t ξ ˙ cl i ξ ˙ cl i + 4 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) θ ( t ′ − t ) ∂ t 5 ∂ 5 δ ( t − t ′ ) T ij l m ( 0 ) ξ cl l ( t ′ ) ξ m ( t ′ ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) θ ( t ′ − t ) ∂ t 3 ∂ 3 δ ( t − t ′ ) S ij l m ξ cl l ( t ′ ) ξ m ( t ′ ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) θ ( t ′ − t ) ∂ t 4 ∂ 4 sgn ( t ′ − t ) ln ( m g ∣ t ′ − t ∣ ) Ξ ij l m ξ cl l ( t ′ ) ξ m ( t ′ ) + π 8 i G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) ∂ t 2 ∂ 2 ∂ t ′2 ∂ 2 K ij l m ( t , t ′ ) ξ cl l ( t ′ ) ξ m ( t ′ ) (103) where we redefine:
S ~ i j l m = S ~ i j l m ′ − 2 γ Ξ i j l m = − 1 2 T ~ i j l m ( 0 ) + Λ i j l m + 2 ( ln ( 2 ) − γ ) Ξ i j l m (104) \widetilde{S}_{ijlm}=\widetilde{S}'_{ijlm}-2\gamma\Xi_{ijlm}=-\frac{1}{2}\widetilde{T}_{ijlm}^{(0)}+\Lambda_{ijlm}+2\bigl(\ln(2)-\gamma\bigr)\Xi_{ijlm} \tag*{(104)} S ij l m = S ij l m ′ − 2 γ Ξ ij l m = − 2 1 T ij l m ( 0 ) + Λ ij l m + 2 ( ln ( 2 ) − γ ) Ξ ij l m (104) and with it,
K i j l m ( t , t ′ ) = I 0 ( t , t ′ ) T ~ i j l m ( 0 ) + m g 2 I 1 ( t , t ′ ) S ~ i j l m + 2 m g 2 I 2 ( t , t ′ ) Ξ i j l m (105) K_{ijlm}(t,t')=I_0(t,t')\widetilde{T}_{ijlm}^{(0)}+m_g^2 I_1(t,t')\widetilde{S}_{ijlm}+2m_g^2 I_2(t,t')\Xi_{ijlm} \tag*{(105)} K ij l m ( t , t ′ ) = I 0 ( t , t ′ ) T ij l m ( 0 ) + m g 2 I 1 ( t , t ′ ) S ij l m + 2 m g 2 I 2 ( t , t ′ ) Ξ ij l m (105) where we define
I 0 = ∫ m g ∞ d ω ω M ( t , t ′ ) (106) I_0=\int_{m_g}^{\infty}d\omega\,\omega M(t,t') \tag*{(106)} I 0 = ∫ m g ∞ d ω ω M ( t , t ′ ) (106) I 1 = ∫ m g ∞ d ω 1 ω M ( t , t ′ ) (107) I_1=\int_{m_g}^{\infty}d\omega\,\frac{1}{\omega}M(t,t') \tag*{(107)} I 1 = ∫ m g ∞ d ω ω 1 M ( t , t ′ ) (107) I 2 = ∫ m g ∞ d ω 1 ω ( γ + ln ( ω m g ) ) M ( t , t ′ ) (108) I_2=\int_{m_g}^{\infty}d\omega\,\frac{1}{\omega}\left(\gamma+\ln\left(\frac{\omega}{m_g}\right)\right)M(t,t') \tag*{(108)} I 2 = ∫ m g ∞ d ω ω 1 ( γ + ln ( m g ω ) ) M ( t , t ′ ) (108) For the second term in (C13) we use
∂ 5 ∂ t 5 δ ( t − t ′ ) = − ∂ 5 ∂ t ′ 5 δ ( t − t ′ ) (109) \frac{\partial^5}{\partial t^5}\delta(t-t')=-\frac{\partial^5}{\partial t'^5}\delta(t-t') \tag*{(109)} ∂ t 5 ∂ 5 δ ( t − t ′ ) = − ∂ t ′5 ∂ 5 δ ( t − t ′ ) (109) and after integrations by parts, discarding terms containing products of deltas with derivatives of Heaviside Theta functions, which are ill defined, and considering that ξ q \xi_q ξ q
∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) θ ( t ′ − t ) ∂ 5 ∂ t ′ 5 δ ( t − t ′ ) T ~ i j l m ( 0 ) ξ c l l ( t ′ ) ξ q m ( t ′ ) = θ ( 0 ) ∫ 0 T d t ∂ 5 ( ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) ) ∂ t 5 T ~ i j l m ( 0 ) ξ c l l ( t ) ξ q m ( t ) (110) \int_0^T dt\int_0^T dt'\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\theta(t'-t)\frac{\partial^5}{\partial t'^5}\delta(t-t')\widetilde{T}_{ijlm}^{(0)}\xi_{\mathrm{cl}}^l(t')\xi_q^m(t')=\theta(0)\int_0^T dt\frac{\partial^5(\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t))}{\partial t^5}\widetilde{T}_{ijlm}^{(0)}\xi_{\mathrm{cl}}^l(t)\xi_q^m(t) \tag*{(110)} ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) θ ( t ′ − t ) ∂ t ′5 ∂ 5 δ ( t − t ′ ) T ij l m ( 0 ) ξ cl l ( t ′ ) ξ q m ( t ′ ) = θ ( 0 ) ∫ 0 T d t ∂ t 5 ∂ 5 ( ξ ˙ cl i ( t ) ξ ˙ cl j ( t )) T ij l m ( 0 ) ξ cl l ( t ) ξ q m ( t ) (110) We proceed similarly for the third term in (C13) and for the fourth term we integrate by parts, and we will consider that the terms in the boundary cancel, which is equivalent to assume an adiabatic switching. Then, after removing the terms with deltas and its derivatives, which are treated in the same way as for the other cases, we have
∫ 0 T d t ∫ 0 T d t ′ ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) θ ( t ′ − t ) ∂ 4 sgn ( t ′ − t ) ln ( m g ∣ t ′ − t ∣ ) ∂ t ′ 4 Ξ i j l m ξ c l l ( t ′ ) ξ q m ( t ′ ) = ∫ 0 T d t ∫ 0 t d t ′ ∂ 4 ( ξ ˙ c l i ( t ′ ) ξ ˙ c l j ( t ′ ) ) ∂ t ′ 4 ln ( m g ∣ t − t ′ ∣ ) Ξ i j l m ξ c l l ( t ) ξ q m ( t ) (111) \begin{aligned}
&\int_0^T dt\int_0^T dt'\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t)\theta(t'-t)\frac{\partial^4\operatorname{sgn}(t'-t)\ln(m_g|t'-t|)}{\partial t'^4}\Xi_{ijlm}\xi_{\mathrm{cl}}^l(t')\xi_q^m(t')\\
&=\int_0^T dt\int_0^t dt'\frac{\partial^4(\dot{\xi}_{\mathrm{cl}}^i(t')\dot{\xi}_{\mathrm{cl}}^j(t'))}{\partial t'^4}\ln(m_g|t-t'|)\Xi_{ijlm}\xi_{\mathrm{cl}}^l(t)\xi_q^m(t) \tag*{(111)}
\end{aligned} ∫ 0 T d t ∫ 0 T d t ′ ξ ˙ cl i ( t ) ξ ˙ cl j ( t ) θ ( t ′ − t ) ∂ t ′4 ∂ 4 sgn ( t ′ − t ) ln ( m g ∣ t ′ − t ∣ ) Ξ ij l m ξ cl l ( t ′ ) ξ q m ( t ′ ) = ∫ 0 T d t ∫ 0 t d t ′ ∂ t ′4 ∂ 4 ( ξ ˙ cl i ( t ′ ) ξ ˙ cl j ( t ′ )) ln ( m g ∣ t − t ′ ∣ ) Ξ ij l m ξ cl l ( t ) ξ q m ( t ) (111) With all that, after integrating by parts in the first and last terms we have that the effective action now is
S e f f [ ξ ] = − 2 m 0 ∫ 0 T d t ∂ 2 ξ c l i ∂ t 2 ξ q i − 4 G m 0 2 θ ( 0 ) ∫ 0 T d t ∂ 5 ( ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) ) ∂ t 5 T ~ i j l m ( 0 ) ξ c l l ( t ) ξ q m ( t ) + 4 G m 0 2 m g 2 θ ( 0 ) ∫ 0 T d t ∂ 3 ( ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) ) ∂ t 3 S ~ i j l m ξ c l l ( t ) ξ q m ( t ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 t d t ′ ∂ 4 ( ξ ˙ c l i ( t ′ ) ξ ˙ c l j ( t ′ ) ) ∂ t ′ 4 ln ( m g ∣ t − t ′ ∣ ) Ξ i j l m ξ c l l ( t ) ξ q m ( t ) + i 8 G m 0 2 π ∫ 0 T d t ∫ 0 T d t ′ ∂ 2 ( ξ ˙ c l i ( t ) ξ ˙ c l j ( t ) ) ∂ t 2 K i j l m ( t , t ′ ) ∂ 2 ( ξ ˙ c l l ( t ′ ) ξ ˙ c l m ( t ′ ) ) ∂ t ′ 2 (112) \begin{aligned}
S_{\mathrm{eff}}[\xi]={}&-2m_0\int_0^Tdt\frac{\partial^2\xi_{\mathrm{cl}}^i}{\partial t^2}\xi_q^i\\
&-4Gm_0^2\theta(0)\int_0^Tdt\frac{\partial^5(\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t))}{\partial t^5}\widetilde{T}_{ijlm}^{(0)}\xi_{\mathrm{cl}}^l(t)\xi_q^m(t)\\
&+4Gm_0^2m_g^2\theta(0)\int_0^Tdt\frac{\partial^3(\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t))}{\partial t^3}\widetilde{S}_{ijlm}\xi_{\mathrm{cl}}^l(t)\xi_q^m(t)\\
&-4Gm_0^2m_g^2\int_0^Tdt\int_0^t dt'\frac{\partial^4(\dot{\xi}_{\mathrm{cl}}^i(t')\dot{\xi}_{\mathrm{cl}}^j(t'))}{\partial t'^4}\ln(m_g|t-t'|)\Xi_{ijlm}\xi_{\mathrm{cl}}^l(t)\xi_q^m(t)\\
&+i\frac{8Gm_0^2}{\pi}\int_0^Tdt\int_0^Tdt'\frac{\partial^2(\dot{\xi}_{\mathrm{cl}}^i(t)\dot{\xi}_{\mathrm{cl}}^j(t))}{\partial t^2}K_{ijlm}(t,t')\frac{\partial^2(\dot{\xi}_{\mathrm{cl}}^l(t')\dot{\xi}_{\mathrm{cl}}^m(t'))}{\partial t'^2} \tag*{(112)}
\end{aligned} S eff [ ξ ] = − 2 m 0 ∫ 0 T d t ∂ t 2 ∂ 2 ξ cl i ξ q i − 4 G m 0 2 θ ( 0 ) ∫ 0 T d t ∂ t 5 ∂ 5 ( ξ ˙ cl i ( t ) ξ ˙ cl j ( t )) T ij l m ( 0 ) ξ cl l ( t ) ξ q m ( t ) + 4 G m 0 2 m g 2 θ ( 0 ) ∫ 0 T d t ∂ t 3 ∂ 3 ( ξ ˙ cl i ( t ) ξ ˙ cl j ( t )) S ij l m ξ cl l ( t ) ξ q m ( t ) − 4 G m 0 2 m g 2 ∫ 0 T d t ∫ 0 t d t ′ ∂ t ′4 ∂ 4 ( ξ ˙ cl i ( t ′ ) ξ ˙ cl j ( t ′ )) ln ( m g ∣ t − t ′ ∣ ) Ξ ij l m ξ cl l ( t ) ξ q m ( t ) + i π 8 G m 0 2 ∫ 0 T d t ∫ 0 T d t ′ ∂ t 2 ∂ 2 ( ξ ˙ cl i ( t ) ξ ˙ cl j ( t )) K ij l m ( t , t ′ ) ∂ t ′2 ∂ 2 ( ξ ˙ cl l ( t ′ ) ξ ˙ cl m ( t ′ )) (112) Now we use as prescription θ ( 0 ) = 1 / 2 \theta(0)=1/2 θ ( 0 ) = 1/2
Integrals for the noises
In this appendix we evaluate the integrals I 0 I_0 I 0 I 1 I_1 I 1 I 2 I_2 I 2
Vacuum
For the vacuum case, A = 1 A = 1 A = 1 B = 0 B = 0 B = 0
M ( t , t ′ ) = 1 2 cos ( ω ( t − t ′ ) ) (113) M(t,t') = \frac{1}{2}\cos(\omega(t-t')) \tag*{(113)} M ( t , t ′ ) = 2 1 cos ( ω ( t − t ′ )) (113) Then, we first compute I 0 I_0 I 0
I 0 = 1 2 ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ ) ) − 1 2 ∫ 0 m g d ω ω cos ( ω ( t − t ′ ) ) (114) I_0 = \frac{1}{2}\int_0^\infty\mathrm{d}\omega\,\omega\cos(\omega(t-t'))-\frac{1}{2}\int_0^{m_g}\mathrm{d}\omega\,\omega\cos(\omega(t-t')) \tag*{(114)} I 0 = 2 1 ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ )) − 2 1 ∫ 0 m g d ω ω cos ( ω ( t − t ′ )) (114) The second integral is finite and can be computed:
∫ 0 m g d ω ω cos ( ω ( t − t ′ ) ) = − 1 + cos ( m g ( t − t ′ ) ) + m g ( t − t ′ ) sin ( m g ( t − t ′ ) ) ( t − t ′ ) 2 (115) \int_0^{m_g}\mathrm{d}\omega\,\omega\cos(\omega(t-t')) = \frac{-1+\cos(m_g(t-t'))+m_g(t-t')\sin(m_g(t-t'))}{(t-t')^2} \tag*{(115)} ∫ 0 m g d ω ω cos ( ω ( t − t ′ )) = ( t − t ′ ) 2 − 1 + cos ( m g ( t − t ′ )) + m g ( t − t ′ ) sin ( m g ( t − t ′ )) (115) To compute the first integral we need to introduce a regulator ϵ > 0 \epsilon> 0 ϵ > 0
∫ 0 ∞ d ω ω cos ( ω ( t − t ′ ) ) → ∫ 0 ∞ d ω ω e − ϵ ω cos ( ω ( t − t ′ ) ) (116) \int_0^\infty\mathrm{d}\omega\,\omega\cos(\omega(t-t')) \to\int_0^\infty\mathrm{d}\omega\,\omega e^{-\epsilon\omega}\cos(\omega(t-t')) \tag*{(116)} ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ )) → ∫ 0 ∞ d ω ω e − ϵ ω cos ( ω ( t − t ′ )) (116) and after doing all the manipulations we will take the limit ϵ → 0 \epsilon\to0 ϵ → 0
∫ 0 ∞ d ω ω e − ϵ ω cos ( ω ( t − t ′ ) ) = ℜ ∫ 0 ∞ d ω ω e ω ( i ( t − t ′ ) − ϵ ) = − ℜ ( 1 ( ( t − t ′ ) + i ϵ ) 2 ) (117) \int_0^\infty\mathrm{d}\omega\,\omega e^{-\epsilon\omega}\cos(\omega(t-t')) = \Re\int_0^\infty\mathrm{d}\omega\,\omega e^{\omega(i(t-t')-\epsilon)} = -\Re\left(\frac{1}{((t-t')+i\epsilon)^2}\right) \tag*{(117)} ∫ 0 ∞ d ω ω e − ϵ ω cos ( ω ( t − t ′ )) = ℜ ∫ 0 ∞ d ω ω e ω ( i ( t − t ′ ) − ϵ ) = − ℜ ( (( t − t ′ ) + i ϵ ) 2 1 ) (117) and after taking the limit ϵ → 0 \epsilon\to0 ϵ → 0 m g 2 m_g^2 m g 2
I 0 = − 1 2 P F ( 1 ( t − t ′ ) 2 ) − m g 2 4 (118) I_0 = -\frac{1}{2}\mathit{PF}\left(\frac{1}{(t-t')^2}\right)-\frac{m_g^2}{4} \tag*{(118)} I 0 = − 2 1 PF ( ( t − t ′ ) 2 1 ) − 4 m g 2 (118) where P F PF P F
For I 1 I_1 I 1
I 1 = − 1 2 Ci ( m g ( t − t ′ ) ) (119) I_1 = -\frac{1}{2}\operatorname{Ci}(m_g(t-t')) \tag*{(119)} I 1 = − 2 1 Ci ( m g ( t − t ′ )) (119) Where C i Ci C i I 1 I_1 I 1 m g 2 m_g^2 m g 2 K i j l m K_{ijlm} K ij l m m g m_g m g
I 1 = − 1 2 ( γ + ln ( m g ∣ t − t ′ ∣ ) ) (120) I_1 = -\frac{1}{2}\left(\gamma+\ln(m_g|t-t'|)\right) \tag*{(120)} I 1 = − 2 1 ( γ + ln ( m g ∣ t − t ′ ∣ ) ) (120) For I 2 I_2 I 2
I 2 = γ 2 ∫ m g ∞ d ω 1 ω cos ( ω ( t − t ′ ) ) + 1 2 ∫ m g ∞ d ω 1 ω ln ( ω m g ) cos ( ω ( t − t ′ ) ) (121) I_2 = \frac{\gamma}{2}\int_{m_g}^\infty\mathrm{d}\omega\,\frac{1}{\omega}\cos(\omega(t-t'))+\frac{1}{2}\int_{m_g}^\infty\mathrm{d}\omega\,\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)\cos(\omega(t-t')) \tag*{(121)} I 2 = 2 γ ∫ m g ∞ d ω ω 1 cos ( ω ( t − t ′ )) + 2 1 ∫ m g ∞ d ω ω 1 ln ( m g ω ) cos ( ω ( t − t ′ )) (121) The first integral is basically what we got from I 1 I_1 I 1 u = ω ( t − t ′ ) u=\omega(t-t') u = ω ( t − t ′ ) x = m g ( t − t ′ ) x=m_g(t-t') x = m g ( t − t ′ )
∫ m g ∞ d ω 1 ω ln ( ω m g ) cos ( ω ( t − t ′ ) ) → ∫ x ∞ d u 1 u ln ( u ) cos ( u ) − ln ( x ) ∫ x ∞ d u 1 u cos ( u ) (122) \int_{m_g}^\infty\mathrm{d}\omega\,\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)\cos(\omega(t-t')) \to\int_x^\infty\mathrm{d}u\,\frac{1}{u}\ln(u)\cos(u)-\ln(x)\int_x^\infty\mathrm{d}u\,\frac{1}{u}\cos(u) \tag*{(122)} ∫ m g ∞ d ω ω 1 ln ( m g ω ) cos ( ω ( t − t ′ )) → ∫ x ∞ d u u 1 ln ( u ) cos ( u ) − ln ( x ) ∫ x ∞ d u u 1 cos ( u ) (122) Both integrals can be solved in the computational software Mathematica, and since I 2 I_2 I 2 m g 2 m_g^2 m g 2 K i j l m K_{ijlm} K ij l m m g m_g m g
∫ m g ∞ d ω 1 ω ln ( ω m g ) cos ( ω ( t − t ′ ) ) = 1 2 γ 2 − 1 24 π 2 + 1 2 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 + γ ln ( m g ∣ t − t ′ ∣ ) (123) \int_{m_g}^\infty\mathrm{d}\omega\,\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)\cos(\omega(t-t')) = \frac{1}{2}\gamma^2-\frac{1}{24}\pi^2+\frac{1}{2}\left(\ln(m_g|t-t'|)\right)^2+\gamma\ln(m_g|t-t'|) \tag*{(123)} ∫ m g ∞ d ω ω 1 ln ( m g ω ) cos ( ω ( t − t ′ )) = 2 1 γ 2 − 24 1 π 2 + 2 1 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 + γ ln ( m g ∣ t − t ′ ∣ ) (123) Therefore,
I 2 = 1 4 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 1 4 γ 2 − 1 48 π 2 (124) I_{2} = \frac{1}{4}\left(\ln(m_{g}|t-t'|)\right)^{2} - \frac{1}{4}\gamma^{2} - \frac{1}{48}\pi^{2} \tag*{(124)} I 2 = 4 1 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 4 1 γ 2 − 48 1 π 2 (124) Since the noise enters in the equation of motion only through the second derivative of N i j N_{ij} N ij I 0 I_{0} I 0 I 1 I_{1} I 1 I 2 I_{2} I 2
I 0 = − 1 2 PF ( 1 ( t − t ′ ) 2 ) (125) I_{0} = -\frac{1}{2}\operatorname{PF}\left(\frac{1}{(t-t')^{2}}\right) \tag*{(125)} I 0 = − 2 1 PF ( ( t − t ′ ) 2 1 ) (125) I 1 = − 1 2 ln ( m g ∣ t − t ′ ∣ ) (126) I_{1} = -\frac{1}{2}\ln(m_{g}|t-t'|) \tag*{(126)} I 1 = − 2 1 ln ( m g ∣ t − t ′ ∣ ) (126) I 2 = 1 4 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 (127) I_{2} = \frac{1}{4}\left(\ln(m_{g}|t-t'|)\right)^{2} \tag*{(127)} I 2 = 4 1 ( ln ( m g ∣ t − t ′ ∣ ) ) 2 (127) Thermal
In the thermal case, we have that A = coth ( β ω ) A=\coth(\beta\omega) A = coth ( β ω ) B=2\csch(\beta\omega) , so
M ( t , t ′ ) = 1 2 tanh ( β ω 2 ) cos ( ω ( t − t ′ ) ) = 1 2 ( 2 e β ω − 1 + 1 ) cos ( ω ( t − t ′ ) ) (128) M(t,t') = \frac{1}{2\tanh\left(\frac{\beta\omega}{2}\right)}\cos(\omega(t-t')) = \frac{1}{2}\left(\frac{2}{e^{\beta\omega}-1}+1\right)\cos(\omega(t-t')) \tag*{(128)} M ( t , t ′ ) = 2 tanh ( 2 β ω ) 1 cos ( ω ( t − t ′ )) = 2 1 ( e β ω − 1 2 + 1 ) cos ( ω ( t − t ′ )) (128) With this,
I 0 = 1 2 ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ ) ) − 1 2 ∫ 0 m g d ω ω cos ( ω ( t − t ′ ) ) + ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ ) ) e β ω − 1 − ∫ 0 m g d ω ω cos ( ω ( t − t ′ ) ) e β ω − 1 (129) \begin{aligned}
I_{0} ={}& \frac{1}{2}\int_{0}^{\infty}d\omega\,\omega\cos(\omega(t-t'))-\frac{1}{2}\int_{0}^{m_{g}}d\omega\,\omega\cos(\omega(t-t')) \\
&+\int_{0}^{\infty}d\omega\,\frac{\omega\cos(\omega(t-t'))}{e^{\beta\omega}-1}-\int_{0}^{m_{g}}d\omega\,\frac{\omega\cos(\omega(t-t'))}{e^{\beta\omega}-1} \tag*{(129)}
\end{aligned} I 0 = 2 1 ∫ 0 ∞ d ω ω cos ( ω ( t − t ′ )) − 2 1 ∫ 0 m g d ω ω cos ( ω ( t − t ′ )) + ∫ 0 ∞ d ω e β ω − 1 ω cos ( ω ( t − t ′ )) − ∫ 0 m g d ω e β ω − 1 ω cos ( ω ( t − t ′ )) (129) We observe that the first line in (D17) will give the same integrals computed for the vacuum case, so we focus here only in the second line. The last term in the second line is finite, and that full integral computed in the software Mathematica is composed by Hurwitz Zeta functions and Hypergeometric functions, but expanding up to order m g 2 m_{g}^{2} m g 2
∫ 0 m g d ω ω cos ( ω ( t − t ′ ) ) e β ω − 1 ≈ m g β − m g 2 4 (130) \int_{0}^{m_{g}}d\omega\,\frac{\omega\cos(\omega(t-t'))}{e^{\beta\omega}-1}\approx\frac{m_{g}}{\beta}-\frac{m_{g}^{2}}{4} \tag*{(130)} ∫ 0 m g d ω e β ω − 1 ω cos ( ω ( t − t ′ )) ≈ β m g − 4 m g 2 (130) To solve the first integral in the second line of (D17) we start from the following integral:
∫ 0 ∞ d ω sin ( ω ( t − t ′ ) ) e β ω − 1 = ∫ 0 ∞ d ω sin ( ω ( t − t ′ ) ) ∑ n = 1 ∞ e − β n ω (131) \int_{0}^{\infty}d\omega\,\frac{\sin(\omega(t-t'))}{e^{\beta\omega}-1}=\int_{0}^{\infty}d\omega\sin(\omega(t-t'))\sum_{n=1}^{\infty}e^{-\beta n\omega} \tag*{(131)} ∫ 0 ∞ d ω e β ω − 1 sin ( ω ( t − t ′ )) = ∫ 0 ∞ d ω sin ( ω ( t − t ′ )) n = 1 ∑ ∞ e − β nω (131) after interchanging integral and sum, the integration gives
∫ 0 ∞ d ω sin ( ω ( t − t ′ ) ) e β ω − 1 = − 1 2 ( t − t ′ ) + π 2 β coth ( π β ( t − t ′ ) ) (132) \int_{0}^{\infty}d\omega\,\frac{\sin(\omega(t-t'))}{e^{\beta\omega}-1}=-\frac{1}{2(t-t')}+\frac{\pi}{2\beta}\coth\left(\frac{\pi}{\beta}(t-t')\right) \tag*{(132)} ∫ 0 ∞ d ω e β ω − 1 sin ( ω ( t − t ′ )) = − 2 ( t − t ′ ) 1 + 2 β π coth ( β π ( t − t ′ ) ) (132) With this integral, we can derive respect to t − t ′ t-t' t − t ′
∫ 0 ∞ d ω ω cos ( ω ( t − t ′ ) ) e β ω − 1 = 1 2 ( t − t ′ ) 2 − π 2 2 β 2 ( sinh ( π β ( t − t ′ ) ) ) 2 (133) \int_{0}^{\infty}d\omega\,\frac{\omega\cos(\omega(t-t'))}{e^{\beta\omega}-1}=\frac{1}{2(t-t')^{2}}-\frac{\pi^{2}}{2\beta^{2}\left(\sinh\left(\frac{\pi}{\beta}(t-t')\right)\right)^{2}} \tag*{(133)} ∫ 0 ∞ d ω e β ω − 1 ω cos ( ω ( t − t ′ )) = 2 ( t − t ′ ) 2 1 − 2 β 2 ( sinh ( β π ( t − t ′ ) ) ) 2 π 2 (133) We notice that the first term is cancelled by the vacuum term coming from the first term in first line of (D17). Thus, putting all the pieces together
I 0 = − π 2 2 β 2 ( sinh ( π β ( t − t ′ ) ) ) 2 − m g β (134) I_{0}=-\frac{\pi^{2}}{2\beta^{2}\left(\sinh\left(\frac{\pi}{\beta}(t-t')\right)\right)^{2}}-\frac{m_{g}}{\beta} \tag*{(134)} I 0 = − 2 β 2 ( sinh ( β π ( t − t ′ ) ) ) 2 π 2 − β m g (134) For I 1 I_1 I 1
I 1 = ∫ m g ∞ d ω 1 ω e β ω − 1 cos ( ω ( t − t ′ ) ) + 1 2 ∫ m g ∞ d ω 1 ω cos ( ω ( t − t ′ ) ) (135) I_1=\int_{m_g}^{\infty}d\omega\frac{1}{\omega e^{\beta\omega}-1}\cos\left(\omega(t-t')\right)+\frac{1}{2}\int_{m_g}^{\infty}d\omega\frac{1}{\omega}\cos\left(\omega(t-t')\right) \tag*{(135)} I 1 = ∫ m g ∞ d ω ω e β ω − 1 1 cos ( ω ( t − t ′ ) ) + 2 1 ∫ m g ∞ d ω ω 1 cos ( ω ( t − t ′ ) ) (135) This can be rewritten as
I 1 = 1 2 ∫ m g ∞ d ω 1 ω coth ( 1 2 β ω ) cos ( ω ( t − t ′ ) ) (136) I_1=\frac{1}{2}\int_{m_g}^{\infty}d\omega\frac{1}{\omega}\coth\left(\frac{1}{2}\beta\omega\right)\cos\left(\omega(t-t')\right) \tag*{(136)} I 1 = 2 1 ∫ m g ∞ d ω ω 1 coth ( 2 1 β ω ) cos ( ω ( t − t ′ ) ) (136) Using the identity:
coth ( x ) = 1 x + ∑ n = 1 ∞ 2 x x 2 + π 2 n 2 (137) \coth(x)=\frac{1}{x}+\sum_{n=1}^{\infty}\frac{2x}{x^2+\pi^2n^2} \tag*{(137)} coth ( x ) = x 1 + n = 1 ∑ ∞ x 2 + π 2 n 2 2 x (137) it can be arranged as
I 1 = 1 β ∫ m g ∞ d ω 1 ω 2 cos ( ω ( t − t ′ ) ) + 2 β ∑ n = 1 ∞ ∫ 0 ∞ d ω cos ( ω ( t − t ′ ) ) ω 2 + 4 π 2 n 2 β 2 − 2 β ∑ n = 1 ∞ ∫ 0 m g d ω cos ( ω ( t − t ′ ) ) ω 2 + 4 π 2 n 2 β 2 (138) I_1=\frac{1}{\beta}\int_{m_g}^{\infty}d\omega\frac{1}{\omega^2}\cos\left(\omega(t-t')\right)+\frac{2}{\beta}\sum_{n=1}^{\infty}\int_{0}^{\infty}d\omega\frac{\cos\left(\omega(t-t')\right)}{\omega^2+\frac{4\pi^2n^2}{\beta^2}}-\frac{2}{\beta}\sum_{n=1}^{\infty}\int_{0}^{m_g}d\omega\frac{\cos\left(\omega(t-t')\right)}{\omega^2+\frac{4\pi^2n^2}{\beta^2}} \tag*{(138)} I 1 = β 1 ∫ m g ∞ d ω ω 2 1 cos ( ω ( t − t ′ ) ) + β 2 n = 1 ∑ ∞ ∫ 0 ∞ d ω ω 2 + β 2 4 π 2 n 2 cos ( ω ( t − t ′ ) ) − β 2 n = 1 ∑ ∞ ∫ 0 m g d ω ω 2 + β 2 4 π 2 n 2 cos ( ω ( t − t ′ ) ) (138) All those integrals can be done in Mathematica, giving terms that contain Sine Integrals and Exponential Integral functions. In any case, expanding up to order zero in m g m_g m g I 1 I_1 I 1 m g 2 m_g^2 m g 2
I 1 = − π 2 β ∣ t − t ′ ∣ + 1 β m g + ∑ n = 1 ∞ 1 2 n e − 2 π n β ∣ t − t ′ ∣ (139) I_1=-\frac{\pi}{2\beta}|t-t'|+\frac{1}{\beta m_g}+\sum_{n=1}^{\infty}\frac{1}{2n}e^{-\frac{2\pi n}{\beta}|t-t'|} \tag*{(139)} I 1 = − 2 β π ∣ t − t ′ ∣ + β m g 1 + n = 1 ∑ ∞ 2 n 1 e − β 2 π n ∣ t − t ′ ∣ (139) The sum corresponds to a logarithm and after arranging we obtain
I 1 = 1 β m g − 1 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) (140) I_1=\frac{1}{\beta m_g}-\frac{1}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}|t-t'|\right)\right) \tag*{(140)} I 1 = β m g 1 − 2 1 ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) (140) To compute I 2 I_2 I 2
I 2 = γ I 1 + ∫ m g ∞ d ω 1 ω ln ( ω m g ) M ( t , t ′ ) (141) I_2=\gamma I_1+\int_{m_g}^{\infty}d\omega\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)M(t,t') \tag*{(141)} I 2 = γ I 1 + ∫ m g ∞ d ω ω 1 ln ( m g ω ) M ( t , t ′ ) (141) and it can be arranged as
I 2 = γ β m g − γ 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) + 1 2 ∫ m g ∞ d ω 1 ω ln ( ω m g ) coth ( 1 2 β ω ) cos ( ω ( t − t ′ ) ) (142) I_2=\frac{\gamma}{\beta m_g}-\frac{\gamma}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}|t-t'|\right)\right)+\frac{1}{2}\int_{m_g}^{\infty}d\omega\frac{1}{\omega}\ln\left(\frac{\omega}{m_g}\right)\coth\left(\frac{1}{2}\beta\omega\right)\cos\left(\omega(t-t')\right) \tag*{(142)} I 2 = β m g γ − 2 γ ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) + 2 1 ∫ m g ∞ d ω ω 1 ln ( m g ω ) coth ( 2 1 β ω ) cos ( ω ( t − t ′ ) ) (142) Using the identity (D25) we can arrange ths expression as
I 2 = γ β m g − γ 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) + 1 β ∫ m g ∞ d ω 1 ω 2 ln ( ω m g ) cos ( ω ( t − t ′ ) ) + 2 β ∫ 0 ∞ d ω ∑ n = 1 ∞ 1 ω 2 + 4 π 2 n 2 β 2 ln ( ω m g ) cos ( ω ( t − t ′ ) ) − 2 β ∫ 0 m g d ω ∑ n = 1 ∞ 1 ω 2 + 4 π 2 n 2 β 2 ln ( ω m g ) cos ( ω ( t − t ′ ) ) (143) \begin{aligned}
I_2={}&\frac{\gamma}{\beta m_g}-\frac{\gamma}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}|t-t'|\right)\right)+\frac{1}{\beta}\int_{m_g}^{\infty}d\omega\frac{1}{\omega^2}\ln\left(\frac{\omega}{m_g}\right)\cos\left(\omega(t-t')\right)\\
&+\frac{2}{\beta}\int_{0}^{\infty}d\omega\sum_{n=1}^{\infty}\frac{1}{\omega^2+\frac{4\pi^2n^2}{\beta^2}}\ln\left(\frac{\omega}{m_g}\right)\cos\left(\omega(t-t')\right)\\
&-\frac{2}{\beta}\int_{0}^{m_g}d\omega\sum_{n=1}^{\infty}\frac{1}{\omega^2+\frac{4\pi^2n^2}{\beta^2}}\ln\left(\frac{\omega}{m_g}\right)\cos\left(\omega(t-t')\right) \tag*{(143)}
\end{aligned} I 2 = β m g γ − 2 γ ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) + β 1 ∫ m g ∞ d ω ω 2 1 ln ( m g ω ) cos ( ω ( t − t ′ ) ) + β 2 ∫ 0 ∞ d ω n = 1 ∑ ∞ ω 2 + β 2 4 π 2 n 2 1 ln ( m g ω ) cos ( ω ( t − t ′ ) ) − β 2 ∫ 0 m g d ω n = 1 ∑ ∞ ω 2 + β 2 4 π 2 n 2 1 ln ( m g ω ) cos ( ω ( t − t ′ ) ) (143) The last term will give only an object that expanded in series starts at order m g m_g m g I 2 I_2 I 2 m g 2 m_g^2 m g 2 m g m_g m g
I 2 = γ + 1 β m g − γ 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) − π 2 β ∣ t − t ′ ∣ ( 1 − γ − ln ( m g ∣ t − t ′ ∣ ) ) + ∑ n = 1 ∞ 1 2 n ln ( 2 π n β m g ) e − 2 π n β ∣ t − t ′ ∣ + ∑ n = 1 ∞ 1 4 n ( e 2 π n β ∣ t − t ′ ∣ E i ( − 2 π n β ∣ t − t ′ ∣ ) − e − 2 π n β ∣ t − t ′ ∣ E i ( 2 π n β ∣ t − t ′ ∣ ) ) (144) \begin{aligned}
I_2={}&\frac{\gamma+1}{\beta m_g}-\frac{\gamma}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}|t-t'|\right)\right)-\frac{\pi}{2\beta}|t-t'|\left(1-\gamma-\ln\left(m_g|t-t'|\right)\right)\\
&+\sum_{n=1}^{\infty}\frac{1}{2n}\ln\left(\frac{2\pi n}{\beta m_g}\right)e^{-\frac{2\pi n}{\beta}|t-t'|}\\
&+\sum_{n=1}^{\infty}\frac{1}{4n}\left(e^{\frac{2\pi n}{\beta}|t-t'|}Ei\left(-\frac{2\pi n}{\beta}|t-t'|\right)-e^{-\frac{2\pi n}{\beta}|t-t'|}Ei\left(\frac{2\pi n}{\beta}|t-t'|\right)\right) \tag*{(144)}
\end{aligned} I 2 = β m g γ + 1 − 2 γ ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) − 2 β π ∣ t − t ′ ∣ ( 1 − γ − ln ( m g ∣ t − t ′ ∣ ) ) + n = 1 ∑ ∞ 2 n 1 ln ( β m g 2 π n ) e − β 2 π n ∣ t − t ′ ∣ + n = 1 ∑ ∞ 4 n 1 ( e β 2 π n ∣ t − t ′ ∣ E i ( − β 2 π n ∣ t − t ′ ∣ ) − e − β 2 π n ∣ t − t ′ ∣ E i ( β 2 π n ∣ t − t ′ ∣ ) ) (144) where E i Ei E i
As in the vacuum case, we remove the constant terms and we have:
I 0 = − π 2 2 β 2 ( sinh ( π β ( t − t ′ ) ) ) 2 (145) I_0=-\frac{\pi^2}{2\beta^2\left(\sinh\left(\frac{\pi}{\beta}(t-t')\right)\right)^2} \tag*{(145)} I 0 = − 2 β 2 ( sinh ( β π ( t − t ′ ) ) ) 2 π 2 (145) I 1 = − 1 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) (146) I_1=-\frac{1}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}\lvert t-t'\rvert\right)\right) \tag*{(146)} I 1 = − 2 1 ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) (146) I 2 = − γ 2 ln ( 2 sinh ( π β ∣ t − t ′ ∣ ) ) − π 2 β ∣ t − t ′ ∣ ( 1 − γ − ln ( m g ∣ t − t ′ ∣ ) ) + ∑ n = 1 ∞ 1 2 n ln ( 2 π n β m g ) e − 2 π n β ∣ t − t ′ ∣ + ∑ n = 1 ∞ 1 4 n ( e 2 π n β ∣ t − t ′ ∣ E i ( − 2 π n β ∣ t − t ′ ∣ ) − e − 2 π n β ∣ t − t ′ ∣ E i ( 2 π n β ∣ t − t ′ ∣ ) ) (147) \begin{aligned}
I_2={}&-\frac{\gamma}{2}\ln\left(2\sinh\left(\frac{\pi}{\beta}\lvert t-t'\rvert\right)\right)-\frac{\pi}{2\beta}\lvert t-t'\rvert\left(1-\gamma-\ln(m_g\lvert t-t'\rvert)\right)\\
&+\sum_{n=1}^{\infty}\frac{1}{2n}\ln\left(\frac{2\pi n}{\beta m_g}\right)e^{-\frac{2\pi n}{\beta}\lvert t-t'\rvert}\\
&+\sum_{n=1}^{\infty}\frac{1}{4n}\left(e^{\frac{2\pi n}{\beta}\lvert t-t'\rvert}Ei\left(-\frac{2\pi n}{\beta}\lvert t-t'\rvert\right)-e^{-\frac{2\pi n}{\beta}\lvert t-t'\rvert}Ei\left(\frac{2\pi n}{\beta}\lvert t-t'\rvert\right)\right) \tag*{(147)}
\end{aligned} I 2 = − 2 γ ln ( 2 sinh ( β π ∣ t − t ′ ∣ ) ) − 2 β π ∣ t − t ′ ∣ ( 1 − γ − ln ( m g ∣ t − t ′ ∣) ) + n = 1 ∑ ∞ 2 n 1 ln ( β m g 2 π n ) e − β 2 π n ∣ t − t ′ ∣ + n = 1 ∑ ∞ 4 n 1 ( e β 2 π n ∣ t − t ′ ∣ E i ( − β 2 π n ∣ t − t ′ ∣ ) − e − β 2 π n ∣ t − t ′ ∣ E i ( β 2 π n ∣ t − t ′ ∣ ) ) (147) Squeeze
For the squeeze state, A = cosh r + e i ϕ sinh r cosh r − e i ϕ sinh r A=\frac{\cosh r+e^{i\phi}\sinh r}{\cosh r-e^{i\phi}\sinh r} A = c o s h r − e i ϕ s i n h r c o s h r + e i ϕ s i n h r B = 0 B=0 B = 0
ℜ ( A ) = 1 cosh ( 2 r ) − cos ϕ sinh ( 2 r ) (148) \Re(A)=\frac{1}{\cosh(2r)-\cos\phi\sinh(2r)} \tag*{(148)} ℜ ( A ) = cosh ( 2 r ) − cos ϕ sinh ( 2 r ) 1 (148) ℑ ( A ) = sin ϕ sinh ( 2 r ) cosh ( 2 r ) − cos ϕ sinh ( 2 r ) (149) \Im(A)=\frac{\sin\phi\sinh(2r)}{\cosh(2r)-\cos\phi\sinh(2r)} \tag*{(149)} ℑ ( A ) = cosh ( 2 r ) − cos ϕ sinh ( 2 r ) sin ϕ sinh ( 2 r ) (149) ∣ A ∣ 2 = cosh ( 2 r ) + cos ϕ sinh ( 2 r ) cosh ( 2 r ) − cos ϕ sinh ( 2 r ) (150) \lvert A\rvert^2=\frac{\cosh(2r)+\cos\phi\sinh(2r)}{\cosh(2r)-\cos\phi\sinh(2r)} \tag*{(150)} ∣ A ∣ 2 = cosh ( 2 r ) − cos ϕ sinh ( 2 r ) cosh ( 2 r ) + cos ϕ sinh ( 2 r ) (150) and hence,
M ( t , t ′ ) = 1 2 cosh ( 2 r ) cos ( ω ( t − t ′ ) ) − 1 2 cos ϕ sinh ( 2 r ) cos ( ω ( t + t ′ ) ) − 1 2 sin ϕ sinh ( 2 r ) sin ( ω ( t + t ′ ) ) (151) M(t,t')=\frac{1}{2}\cosh(2r)\cos(\omega(t-t'))-\frac{1}{2}\cos\phi\sinh(2r)\cos(\omega(t+t'))-\frac{1}{2}\sin\phi\sinh(2r)\sin(\omega(t+t')) \tag*{(151)} M ( t , t ′ ) = 2 1 cosh ( 2 r ) cos ( ω ( t − t ′ )) − 2 1 cos ϕ sinh ( 2 r ) cos ( ω ( t + t ′ )) − 2 1 sin ϕ sinh ( 2 r ) sin ( ω ( t + t ′ )) (151) We can see that the first term can be treated as in the vacuum case multiplied by cosh ( 2 r ) \cosh(2r) cosh ( 2 r ) t − t ′ t-t' t − t ′ t + t ′ t+t' t + t ′ − cos ϕ sinh ( 2 r ) -\cos\phi\sinh(2r) − cos ϕ sinh ( 2 r ) sinh ( ω ( t + t ′ ) ) \sinh(\omega(t+t')) sinh ( ω ( t + t ′ ))
∫ m g ∞ d ω ω sin ( ω ( t + t ′ ) ) = ∫ 0 ∞ d ω ω sin ( ω ( t + t ′ ) ) − ∫ 0 m g d ω ω sin ( ω ( t + t ′ ) ) (152) \int_{m_g}^{\infty}d\omega\,\omega\sin(\omega(t+t'))=\int_{0}^{\infty}d\omega\,\omega\sin(\omega(t+t'))-\int_{0}^{m_g}d\omega\,\omega\sin(\omega(t+t')) \tag*{(152)} ∫ m g ∞ d ω ω sin ( ω ( t + t ′ )) = ∫ 0 ∞ d ω ω sin ( ω ( t + t ′ )) − ∫ 0 m g d ω ω sin ( ω ( t + t ′ )) (152) The second term is just a finite part that starts at order m g 3 m_g^3 m g 3
∫ m g ∞ d ω ω sin ( ω ( t + t ′ ) ) ≈ − π δ ′ ( t + t ′ ) (153) \int_{m_g}^{\infty}d\omega\,\omega\sin(\omega(t+t'))\approx-\pi\delta'(t+t') \tag*{(153)} ∫ m g ∞ d ω ω sin ( ω ( t + t ′ )) ≈ − π δ ′ ( t + t ′ ) (153) Proceeding similarly for the other integrals with sine, where the split term that goes from 0 0 0 m g m_g m g m g 2 m_g^2 m g 2
∫ m g ∞ d ω 1 ω sin ( ω ( t + t ′ ) ) ≈ π 2 (154) \int_{m_g}^{\infty}d\omega\,\frac{1}{\omega}\sin(\omega(t+t'))\approx\frac{\pi}{2} \tag*{(154)} ∫ m g ∞ d ω ω 1 sin ( ω ( t + t ′ )) ≈ 2 π (154) ∫ m g ∞ d ω 1 ω ( γ + ln ( ω m g ) ) sin ( ω ( t + t ′ ) ) ≈ − π 2 ln ( m g ∣ t + t ′ ∣ ) (155) \int_{m_g}^{\infty}d\omega\,\frac{1}{\omega}\left(\gamma+\ln\left(\frac{\omega}{m_g}\right)\right)\sin(\omega(t+t'))\approx-\frac{\pi}{2}\ln(m_g\lvert t+t'\rvert) \tag*{(155)} ∫ m g ∞ d ω ω 1 ( γ + ln ( m g ω ) ) sin ( ω ( t + t ′ )) ≈ − 2 π ln ( m g ∣ t + t ′ ∣) (155) With all that, the terms I 0 I_0 I 0 I 1 I_1 I 1 I 2 I_2 I 2
I 0 = − 1 2 cosh ( 2 r ) P F ( 1 ( t − t ′ ) 2 ) + 1 2 cos ϕ sinh ( 2 r ) P F ( 1 ( t + t ′ ) 2 ) − m g 2 4 ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) + π 2 sin ϕ sinh ( 2 r ) δ ′ ( t + t ′ ) (156) I_0=-\frac{1}{2}\cosh(2r)PF\left(\frac{1}{(t-t')^2}\right)+\frac{1}{2}\cos\phi\sinh(2r)PF\left(\frac{1}{(t+t')^2}\right)-\frac{m_g^2}{4}\left(\cosh(2r)-\cos\phi\sinh(2r)\right)
+\frac{\pi}{2}\sin\phi\sinh(2r)\delta'(t+t') \tag*{(156)} I 0 = − 2 1 cosh ( 2 r ) P F ( ( t − t ′ ) 2 1 ) + 2 1 cos ϕ sinh ( 2 r ) P F ( ( t + t ′ ) 2 1 ) − 4 m g 2 ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) + 2 π sin ϕ sinh ( 2 r ) δ ′ ( t + t ′ ) (156) I 1 = − 1 2 cosh ( 2 r ) ( γ + ln ( m g ∣ t − t ′ ∣ ) ) + 1 2 cos ϕ sinh ( 2 r ) ( γ + ln ( m g ∣ t + t ′ ∣ ) ) − π 4 sin ϕ sinh ( 2 r ) (157) I_1=-\frac{1}{2}\cosh(2r)\left(\gamma+\ln(m_g|t-t'|)\right)+\frac{1}{2}\cos\phi\sinh(2r)\left(\gamma+\ln(m_g|t+t'|)\right)-\frac{\pi}{4}\sin\phi\sinh(2r) \tag*{(157)} I 1 = − 2 1 cosh ( 2 r ) ( γ + ln ( m g ∣ t − t ′ ∣ ) ) + 2 1 cos ϕ sinh ( 2 r ) ( γ + ln ( m g ∣ t + t ′ ∣ ) ) − 4 π sin ϕ sinh ( 2 r ) (157) I 2 = 1 4 cosh ( 2 r ) ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 1 4 cos ϕ sinh ( 2 r ) ( ln ( m g ∣ t + t ′ ∣ ) ) 2 − 1 4 ( γ 2 + π 2 12 ) ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) + π 4 sin ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (158) I_2=\frac{1}{4}\cosh(2r)\left(\ln(m_g|t-t'|)\right)^2-\frac{1}{4}\cos\phi\sinh(2r)\left(\ln(m_g|t+t'|)\right)^2-\frac{1}{4}\left(\gamma^2+\frac{\pi^2}{12}\right)\left(\cosh(2r)-\cos\phi\sinh(2r)\right)
+\frac{\pi}{4}\sin\phi\sinh(2r)\ln(m_g|t+t'|) \tag*{(158)} I 2 = 4 1 cosh ( 2 r ) ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 4 1 cos ϕ sinh ( 2 r ) ( ln ( m g ∣ t + t ′ ∣ ) ) 2 − 4 1 ( γ 2 + 12 π 2 ) ( cosh ( 2 r ) − cos ϕ sinh ( 2 r ) ) + 4 π sin ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (158) Removing the constant term as in the previous cases:
I 0 = − 1 2 cosh ( 2 r ) P F ( 1 ( t − t ′ ) 2 ) + 1 2 cos ϕ sinh ( 2 r ) P F ( 1 ( t + t ′ ) 2 ) + π 2 sin ϕ sinh ( 2 r ) δ ′ ( t + t ′ ) (159) I_0=-\frac{1}{2}\cosh(2r)PF\left(\frac{1}{(t-t')^2}\right)+\frac{1}{2}\cos\phi\sinh(2r)PF\left(\frac{1}{(t+t')^2}\right)+\frac{\pi}{2}\sin\phi\sinh(2r)\delta'(t+t') \tag*{(159)} I 0 = − 2 1 cosh ( 2 r ) P F ( ( t − t ′ ) 2 1 ) + 2 1 cos ϕ sinh ( 2 r ) P F ( ( t + t ′ ) 2 1 ) + 2 π sin ϕ sinh ( 2 r ) δ ′ ( t + t ′ ) (159) I 1 = − 1 2 cosh ( 2 r ) ln ( m g ∣ t − t ′ ∣ ) + 1 2 cos ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (160) I_1=-\frac{1}{2}\cosh(2r)\ln(m_g|t-t'|)+\frac{1}{2}\cos\phi\sinh(2r)\ln(m_g|t+t'|) \tag*{(160)} I 1 = − 2 1 cosh ( 2 r ) ln ( m g ∣ t − t ′ ∣ ) + 2 1 cos ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (160) I 2 = 1 4 cosh ( 2 r ) ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 1 4 cos ϕ sinh ( 2 r ) ( ln ( m g ∣ t + t ′ ∣ ) ) 2 + π 4 sin ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (161) I_2=\frac{1}{4}\cosh(2r)\left(\ln(m_g|t-t'|)\right)^2-\frac{1}{4}\cos\phi\sinh(2r)\left(\ln(m_g|t+t'|)\right)^2+\frac{\pi}{4}\sin\phi\sinh(2r)\ln(m_g|t+t'|) \tag*{(161)} I 2 = 4 1 cosh ( 2 r ) ( ln ( m g ∣ t − t ′ ∣ ) ) 2 − 4 1 cos ϕ sinh ( 2 r ) ( ln ( m g ∣ t + t ′ ∣ ) ) 2 + 4 π sin ϕ sinh ( 2 r ) ln ( m g ∣ t + t ′ ∣ ) (161) References 