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

We derive the response function for a comoving, pointlike Unruh-DeWitt particle detector coupled to a complex scalar field $\phi$, in the $(3+1)$-dimensional cosmological de Sitter spacetime. The field-detector coupling is taken to be proportional to $\phi^{\dagger} \phi$. We address both conformally invariant and massless minimally coupled scalar field theories, respectively in the conformal and the Bunch-Davies vacuum. The response function integral for the massless minimal complex scalar, not surprisingly, shows divergences and accordingly we use suitable regularisation scheme to find out well behaved results. The regularised result also contains a logarithm, growing with the cosmological time. Possibility of extension of these results with the so called de Sitter $\alpha$-vacua is discussed. In this case even a real conformal scalar is shown to have non-thermal response function. On the other hand, a complex scalar field in this vacua contains some possible ambiguities in the detector response, which are pointed out. We briefly mention the case of a minimal and nearly massless scalar field theory. The variation of response functions with respect to relevant parameters are depicted numerically.


Introduction
An Unruh-DeWitt detector is conventionally a point particle (like an atom) that can couple to a quantum field. The detector has internal discrete energy levels which, along with the field may be excited/de-excited to higher/lower levels. Such (de-)excitation depends upon the trajectory of the detector, the field-detector coupling and also the particular initial and final states we are looking into. One particularly interesting quantity is the response function of the detector, representing the rate of quantum transitions occurring per unit proper time along detector's trajectory. The associated quanta are not necessarily actual created particles which may give rise to flow of energy and momentum, but instead they may be an outcome of application of the external energy required to maintain detector's particular trajectory. We refer our reader to [1] and references therein for a discussion. The response functions for an Unruh-DeWitt detector have been investigated in various contexts in the flat spacetime. This includes various non-inertial trajectories, e.g. [2,3,4,5,6,7,8,9] (also references therein). See also [10,11,12] and references therein for dynamics of entangled detectors interacting with quantum fields.
The de Sitter spacetime is physically very well motivated in the context of the early inflationary as well as the current universe. It has gained considerable attention in the context of the Unruh-DeWitt detector model. The earliest of such discussion can be seen in [1] and references therein, where it was shown that the response function for a comoving detector in the cosmological de Sitter spacetime for a conformal scalar in a conformal vacuum is thermal, although there is no actual particle creation in this scenario. This analysis was later extended in various directions, including scalar fields without conformal symmetry, in the static de Sitter coordinate and also quite extensively in the context of quantum entanglement and decoherence, e.g. [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] and references therein.
In this work we compute at the leading order of the perturbation theory, the response functions of a comoving Unruh-DeWitt detector for complex scalar fields, for both conformally symmetric and massless minimal cases. It is well known that a massless minimal scalar can be a very good candidate for the inflaton. We also wish to discuss the possibility of extending these results in the context of the de Sitter α vacua [33,34,35,36,37,38,39,40,41,42]. Since the field-detector coupling must be hermitian, for a complex scalar it must be at the simplest non-trivial form proportional to φ † φ, 1 unlike the case of a real scalar [1]. Apart from considering this as a theoretical model of handling Dirac fermionic fields, such quadratic couplings can also be expected emerging in low energy effective theories of some interacting theories where a scalar describes composite particles at low energies [43,45]. The same is true for any complex field like a Dirac fermion. Due to such coupling, one obtains product of two Wightman functions in the integral of the response function and thus perhaps not unexpectedly, one obtains divergence needing suitable regularisation. We refer our reader for discussions on the Unruh-DeWitt detector models for complex fields in the Rindler space including entanglement dynamics to [43,45,44,46,47,48,49,50,51]. Expectedly, in curved spacetime, the issue of the non-linear interaction between the detector and the field as well as the corresponding divergences may become much more relevant receiving curvature contributions. In order to analyse the curvature effects, we, therefore, consider the response of the Unruh-DeWitt detector in a maximally symmetric spacetime with constant curvature. This study can be considered as a theoretical approach of handling the divergences appearing in more realistic non-linear couplings in various kind of fields in de Sitter spacetime. We refer our reader to [52] and references therein for discussions on loop effects with a massless minimal complex scalar in the context of scalar quantum electrodynamics in de Sitter spacetime.
A quantum field in the de Sitter spacetime may inherit many inequivalent vacua, depending upon its syemmetry structure. The Bunch-Davies vacuum, for example, is suited for describing early time inflationary modes [59] according to co-moving observers. The de Sitter α-vacua on the other hand, are interesting in the context of the trans-Planckian physics, even though it might not yield a well defined perturbation theory, originating from its inherent non-local characteristics, e.g. [34,35,36,37,38,39,40,41,42]. Since an αvacuum can be expressed as a squeezed state over all Bunch-Davies states, we may expect interesting feature in the response function. Another point of curiosity comes from the fact that unlike the Bunch-Davies case, the timelike parameter that defines the positive frequency α-mode will not be the cosmological time, which is the proper time along a comoving detector's trajectory.
The paper is summarised as follows. In the next section we briefly review the basic set up for the Unruh-DeWitt detector model. We also review the de Sitter α-vacua here and define the detector response function in this context. Using this set up, we compute in Section 3 the the response functions for a real conformal and massless minimal scalar. In Section 4, we discuss complex conformal and massless minimal scalars respectively in the conformal and the Bunch-Davies vacua. For the latter in particular, suitable regularisation scheme is employed in order to find out well behaved result. This regularisation involves in particular, adding a fictitious conformal scalar field with divergent detector-field interaction, in order to cancel a divergence appearing in the response function. This result also possesses a de Sitter symmetry breaking logarithm growing with time analogous to the infrared secular growth, reported earlier in e.g. [53,54,55,56,57]. However, such term is absent for the case of a real scalar [13]. In Section 5, we discuss generalisation of complex fields' results to the α-vacua and we point out some possible ambiguities. The case of the nearly massless minimal scalar is briefly mentioned in Section 6. Finally we conclude in Section 7. Even though we stick to the first order perturbation theory throughout, we argue in Section 7 that the response function for real scalars in the α-vacua can be obtained at any arbitrary order of the perturbation theory.

The set up
The de Sitter metric in the spatially flat cosmological coordinates in (3 + 1)-dimensions reads where H = Λ/3 is the Hubble constant. Defining the conformal time, η = −e −Ht /H, the metric takes a conformally flat form, The generic free action for a real scalar field reads, whereas for a complex scalar field it reads, We are chiefly interested in two cases here : a) a conformal scalar (m 2 + ξR = R/6) and b) a massless minimal scalar (m 2 + ξR = 0). The case of a nearly massless and minimal scalar will be briefly discussed in Section 6. We shall set below = 1 = c.
We shall use the formalism of particle detectors in curved spacetime discussed in e.g. [1] and references therein. Let us first discuss a real scalar field theory. The simplest coupling of this field with a pointlike detector (e.g. an atom) is taken as, where g is a coupling constant and µ is the monopole moment operator of the detector. In the Heisenberg picture, µ(τ ) = e iH0τ µ e −iH0τ , where H 0 is the free Hamiltonian of the detector, and τ is the proper time along detector's trajectory. We shall specialise to a comoving trajectory and hence will take detector's spatial points to be fixed. Thus the first order matrix element for the field-detector combined system to make a transition from an initial state |i to a final state |f is given by where we have taken |i = |E 0 ⊗ |φ i and |f = |E ⊗ |φ f , and E 0 and E are respectively the energy eigencvalues of the detector in these states. The transition probability is given by However, it is more interesting to sum over all possible final states |E and |φ f by using the completeness relation. Thus if we take the initial state |φ i of the field to be its vacuum, the quantity becomes the Wightman function, iG + (x 2 (τ 2 ) − x 1 (τ 1 )). It is then convenient to define two new temporal variables, Assuming further adiabatic turn on and off of the detector-field coupling, taking τ i = 0 and τ f → ∞, the response function of the detector per unit τ + is defined as (in units of g 2 | E|µ|E 0 | 2 which we will stick to in the remaining of the paper) where we have written ∆E = E − E 0 . ∆E > 0 (∆E < 0) denotes excitation (de-excitation) of the detector interacting with the quantum field. The integral Eq. (4) was computed in [13] in the Bunch-Davies vacuum for a conformal, a massless minimally coupled as well as for a minimally coupled and nearly massless scalar field (see also also [1]).
Let us first generalise Eq. (4) to the α-vacua. It is well known that if u(x) is a mode that becomes positive frequency in the asymptotic past corresponding to the Bunch-Davies vacuum, the α-vacua correspond to the Bogoliubov rotated modes, where α is a spacetime independent real parameter. Then the de Sitter invariant Wightman function in these vacua reads [33], where a bar over the spacetime points denotes the antipodal position, which refers to η → −η in Eq. (2). All the G + 's on the right hand side of the above equation stand for the Bunch-Davies vacuum. For a scalar field of mass m and non-minimal coupling ξ, G + (x, x ) reads [33], where , and the de Sitter invariant interval y written in terms of the conformal time reads, where = 0 + . Rewriting things now in the cosmological time t and setting x = x for a comoving detector, we have Likewise we have for the antipodal transformations, Since we have set the comoving spatial separation to zero, the cosmological time t becomes the proper time along detector's trajectory, τ = t. Putting these all in together, the response function for the de Sitter α vacua is then obtained once we replace the Wightman function appearing in Eq. (4) by Eq. (6), and further use Eq. (7), Eq. (8) and Eq. (9) into it.
A few comments pertaining Eq. (10) are in order here. First, we note that we have not changed the e −i∆E∆t term according to the antipodal transformation at all. This is because this term originates purely from the detector, Eq. (3), and not from the field. Since the detector is a pointlike and localised object, the antipodal transformation simply should not act on this term. We also note that the nonlocal characteristic of the α-vacua is manifest from Eq. (6), where we have added Wightman functions corresponding to antipodal points. Thus there seems to be an apparent interpretational problem of coupling a pointlike particle detector to the field in this case. However, we may get rid of this issue by recalling that the Bogoliubov rotation made from the Bunch-Davies modes, Eq. (5), is purely local. Since on the other hand the Wightman function does not represent propagation of the field, the same appearing in Eq. (10) can be interpreted as just an expectation value of the operator φ(x) φ(x ) with respect to the vacuum corresponding to the α modes. With this interpretation, we shall see below that at least for a real scalar field we can compute the response function in the α-vacua without any apparent ambiguity, up to any arbitrary order of the perturbation theory.
Finally, as discussed before, for a complex scalar field the simplest non-trivial detector-field coupling is quadratic in the field, Alike the fermions, e.g. [49], the quadratic coupling is necessary in order to make the field-detector interaction Hamiltonian hermitian. Accordingly, the integral for the response function will contain a product of two Wightman functions. We address this issue in detail in Section 4.

The conformal scalar
We start with the simplest case of a conformally invariant scalar field theory. Even though a conformally invariant field theory does not create particles in a conformally flat spacetime such as the de Sitter in the conformal vacuum, it is well known that the Unruh-DeWitt detector records a thermal response function in the conformal or the Bunch-Davies vacuum [1,13].
In the conformal vacuum (ν = 1/2) Eq. (7) becomes and hence Eq. (10) gives, when written in terms of a dimensionless temporal coordinate, u = H∆t/2, The above integrals can easily be evaluated by using a semicircular contour closing in the lower half plane. We introduce a dimensionless energy difference for convenience, in terms of which the response function is found to be, Setting α = 0 above recovers the thermal spectra associated with the usual conformal vacuum [1]. We have plotted Eq. (14) scaled by the α = 0 result in Fig. 1.
Note that in this case we have an interpretation of pure excitation of the detector. On the other hand, the coefficient of sinh 2 α in Eq. (14) (pHe πp /[4π(e πp − 1)), can be interpreted as a pure de-excitation, corresponds to doing p → −p in the pure excitation term. Thus it is clear that Eq. (14) in general cannot be interpreted as either excitation or de-excitation of the detector. The reason behind this is as follows. First, the time evolution of detector's monopole moment operator m(t) = e iH0t m e −iH0t , is defined with respect to the cosmological time, which is natural for a comoving object. The cosmological time is certainly not the timelike parameter that defines the positive frequency modes corresponding to the α-vacua, Eq. (5). Due to this, an α-vacuum state can be expressed as a squeezed state over all conformal or Bunch-Davies states and thus the response of the detector in an α-vacuum will naturally consist of excitations as well as de-excitations. In different de Sitter vacua the non-thermal response have been obtained from field content analysis as well [16], though an Unruh-DeWitt detector does not always measure the field content.

The minimally coupled massless scalar
As is evident, one cannot simply set ν = 3/2 corresponding to the massless minimal coupling in Eq. (7), owing to the fact that there exists no de Sitter invariant Wightman function for a massless minimal scalar.  14) when scaled by the conformal vacuum (α = 0) result, with respect to the dimensionless energy diference p for different α-values. Note that for each α-value, there is a p-value at which the detector will not respond at all.
One thus needs to find it independently [33], Compared to Eq. (12), the above thus contains additional terms including one that breaks the de Sitter symmetry. Eq. (10) in this case reads, where p = 2∆E/H as earlier and 'c.c.' in the second line denotes complex conjugation. Using Eq. (14), we rewrite the above equation as We note that as → 0, where sgn(u) stands for the 'sign' function. Also, since cosh u is always positive, we can take as → 0, Following [13], then the regularised form of the logarithmic integrals can be found by introducing an infinitesimal positive imaginary part in p and then by integrating them by parts. Some calculations after using Eq. (14) yields, where the suffix "MM" stands for massless and minimal scalar field. Note that putting α = 0 recovers the result of [13]. The term containing the delta function is not interesting, for it cannot represent any excitation or de-excitation (p = 0) of the detector and hence we may just throw it away. We shall argue in Section 7 that the above results for a real conformal or massless minimal scalar goes through arbitrary order of the perturbation theory, without encountering any difficulty.
We have plotted in Fig. 2 the characteristics of the response function (p = 0), by scaling it with the Bunch-Davies result (α = 0).

Complex scalar in the Bunch-Davies vacuum 4.1 Conformal complex scalar in conformal vacuum
Using Eq. (11), the first order response function for a complex scalar field is given by, The second integrand on the right hand side is divergent. For this term, we replace the argument of iG + (0) by an infinitesimal cut-off and obtain a term proportional to δ(p). Hence one can safely ignore it. Such terms are always expected whenever we deal with coupling beyond the linear order [43,44,45]. Also, it was pointed out in [45] that the iG + (0) term can be avoided by just normal ordering the interaction Hamiltonian. Nevertheless, even though we get rid of that term anyway, we shall encounter additional divergences for a massless minimally coupled complex scalar from the first integral, which needs suitable regularisation. However, the first integral does not show any divergence for a conformal scalar in the conformal vacuum. Using Eq. (12), Eq. (21) becomes, We can perform the above integration just like the real scalar field by choosing the integration contour to be a semicircle in the lower half plane and taking the poles lying on the negative imaginary axis. We find Note that the thermal factor remains unchanged compared to the linear coupling of a real scalar field.

The minimally coupled massless complex scalar
Massless complex scalar fields with interactions have previously been studied in context of cosmology [52] and also in BEC systems mimicking gravity systems [58], where study of detector response may be more feasible. Also, as we discussed previously, the quadratic coupling with complex scalars will give us good theoretical exposure of handling the resulting divergences in more physical, e.g. fermionic systems. Therefore, in order to study curvature effects and non-linear couplings in physically realizable systems, we consider complex scalar field interactions as our first step. Further, since we are interested mainly in studying the curvature effects in the detector response, we consider the massless limit first before going to a more realistic small mass limit in Section 6. The case of the minimally coupled massless complex scalar, however, as we shall see will not be as simple as that of the conformal one. In fact there will be finite as well as divergent terms in the expression for the rate of the response function, originating at the coincidence limit of the Wightman functions. We shall not be able to compute the response function in a closed form and will eventually resort to numerical analysis. However, before we do so, we first need to regularise the integral and also need to cast it into a form which can be handled numerically without any ambiguity.
We have the rate of the response function, which can be rewritten as (after excluding the irrelevant δ-function term), Let us evaluate the first integral of Eq. (25) first, which is most non-trivial. It reads, The first integral in Eq. (26) is the same as that of Eq. (22), Let us now evaluate the second integral of Eq. (26), which is problematic due to the branch cut of the logarithm. To tackle this, after using Eq. (18) and expanding the logarithm in powers of e −2u , we rewrite it as where "c.c." denotes complex conjugation and we have used We shall evaluate the above integral first. We write, The poles of the integrand are located at We use a quarter-circular contour in the fourth quadrant to evaluate this integral, as shown in the first of Fig. 3 and let the radius of the quarter-circle go to infinity. The poles are avoided using infinitesimal semicircular deformations, The arc of the quarter-circle does not contribute to the integration. Computing the effect of the deformations and then performing the derivative with respect to , Eq. (30), we have where u I = −Im(u) along the negative imaginary axis. Note that the poles of the integral of Eq. (31) are located at nπ − (−1) n , n = 1, 2, · · · , which are excluded via the first contour of Fig. 3. As a check of consistency, we have where the complex conjugate of the first integral within parenthesis can either be found by using the second contour of Fig. 3 The integral on the right hand side is written as, where = 0 + and hence the integration limits exclude the poles. Note that we have got rid of the term for the integrals in the summation. We could not evaluate any Cauchy principal value for them, for the poles of these integrals are of the second order. Moreover, the first integral diverges as u I → 0 and it is clear that there is no question of defining any principal value for it at all. We tackle this issue by treating all the integrals in an equal footing as follows. We note from Eq. (33) that the coincidence limit of the Wightman function u → 0 on the left hand side corresponds to the points where sin u I = 0 on the right hand side, achieved via the contour of Fig. 3. Thus we shall regularise the integrals of Eq. (34) by using a suitable regulator near each pole, effectively regularising the very short distance divergent correlation as u → 0 and thus giving the response function a physical meaning. This task can be largely simplified if we first set all terms to zero in Eq. (34) and simply rewrite it as Note that the only poles in the above integral are now located at x = 0, π. Performing the sum and further breaking the integration limits, the above can be rewritten as, so that the only pole of the above integration is located at x = 0. Accordingly, we break the above integration into three pieces, (35) The first two integrals in the above expression diverge as x → 0. It is easy to see from Eq. (24) that there is no flat space limit of this divergence as the integrals vanish in the H → 0 limit. Now, this divergence is originating from the short distance correlation and Hadamard states have all similar short distance blow up in the Wightman function. Hence this divergence structure is expected to be present in all Hadamard states. Thus, we will focus only upon the finite terms, by dropping the divergent ones under some suitable regularisation scheme [43,59]. 2 Thus, after getting rid of the infinite correlations in the coincidence limit of the Wightman function, we obtain a regularised form of the 'poles excluded' integral Eq. (33), The first integral of Eq. (28) can also be regularised in a similar manner using the contour of Fig. 3. After computing the effect of deformations around infinitesimal semicircles, we can express it as, The first term on the right hand side corresponds to infinitesimal deformation of the contour in Fig. 3, whereas the second integral corresponds to the 'poles excluded' part as earlier. For the second integral we slightly lift its lower limit, so that we can use the formula 1.441 of [60], to rewrite Eq. (37) as The first term diverges as ζ(1) whereas the integrals diverge as x → . Accordingly, we now separate the above into non-divergent and divergent pieces, The second and third integrals of the above equation can be regularised as earlier by inserting the function, e − /x . However, the first term originated from an infinite summation in Eq. (37), associated with the semicircular deformations in Fig. 3 and hence we cannot regularise it in the same manner.
In order to tackle this issue, we write for the ζ function (e.g. [61]), so that the first term of Eq. (40) now equals − πp e pπ − 1 One way of getting rid of this divergence would be to add in Eq. (11) an interaction term for the same detector with a fictitious real conformal scalar, L = g c µ(τ )φ c (τ ), with Now the detector and scalar fields' state space has three members. We also have Thus while computing the transition probability (following Section 2), the vacuum expectation value of terms like φ † φφ c , φ c φ † φ would vanish. Factoring out the g 2 | E|µ|E 0 | 2 term as earlier, we thus obtain the total response function as the sum of the response functions for this fictitious conformal scalar added with that of our original complex scalar. However, due to the choice made in Eq. (42), it is now evident from Eq. (14) (with α = 0) and Eq. (24) that the formally divergent term of Eq. (41) will cancel. Putting these all in together, the regularised form of Eq. (37) becomes, Combining now the above with Eq. (36) and Eq. (28), we obtain the regularised expression of the second integral of Eq. (26), Finally, we come to the third integral of Eq. (26). After using Eq. (18), it takes the form (after ignoring a term containing δ(p)), After expanding the logarithm, the above integral can be written as, The first two integrals diverge as u → ∞. Such infrared divergence can be regularised by introducing an infinitesimal positive imaginary part in p. Accordingly, we get ∞ 0 du cos pu = 0.
Using this and also integrating by parts, Eq. (46) can be put into a regularised form .
The various summations appearing above, as is evident, are all convergent. Thus Eq. (27) Before we proceed, we summarise the various regularisation procedure we adopted to derive the above expression. First, we used some suitable regularisation scheme in order to get rid of the short distance ultraviolet divergences (such as the first two integrals of Eq. (35)). Second, we introduced a new coupling of the detector with a fictitious real conformal scalar field with a formally divergent coupling constant, Eq. (42), to get rid of the divergent term in Eq. (41). Finally, we also needed to introduce an infinitesimal positive imaginary part in p, in order to regularise the infrared divergence of some of the integrals of Eq. (46).
If we let p → ∞ in Eq. (49), each of the terms of the first two lines as well as the last integral vanish. The remaining three integrals do not vanish individually, but they cancel with each other to yield a vanishing contribution. This is expected, as this limit corresponds to energy level separation of the detector much larger than the spacetime energy scale, H.
Note that there is a logarithm in Eq. (49), increasing monotonically with 2Ht + = H(t + t ), indicating breakdown of the perturbation theory as Ht, Ht 1. This seems to be analogous to the secular growth reported earlier in the context of perturbative quantum field theory in de Sitter space, e.g. [53,54,55,56,57] (also references therein). Such secular growth is absent for a real massless minimal scalar, Eq. (20), for in that case the de Sitter breaking term is accompanied by a δ-function. We also note that in this limit the p-dependence of Eq. (49) becomes qualitatively similar to that of the real scalar, Eq. (20) (with α = 0 and p = 0). Finally, we note that Eq. (49) diverges for small p-values, as of the real scalar, Eq. (20) (see also [13]).
Since all the terms in Eq. (49) are regularised, we can now investigate its behaviour numerically without any trouble, as a function of the dimensionless energy p, Fig. 4.

The ambiguity of complex scalar field with α-vacua
We finally come to the case of complex scalar fields with an α-vacua. However, we argue below that the detector response function is not well defined in this case. The response function in this case is given as, Expanding the square, making some rearrangements of terms and also redefining in some of the integrals, we find Note that in the third term, there is no . These two terms arise due to multiplications, (sinh u−i ) 2 (sinh u+ i ) 2 and (cosh u − i ) 2 (cosh u + i ) 2 , while squaring. All but the integral containing sinh −4 u can be straightforwardly evaluated using semicircular contours in the lower half plane. The integral containing sinh −4 u is problematic because it does not converge on the real line, nor we can attempt to compute any principal value, for it diverges in the presence of poles beyond the first order. We cannot re-insert any i term now, for the answer will depend upon the sign of that term. We cannot insert a regulator like e − /u as the regularisation of Eq. (33), for such regulator is not analytic across u = 0. Also, the contour of Fig. 3 cannot be used here, for we can compute the effect of the infinitesimal semicircular deformations to avoid the poles only if the poles are of first order.
Similar problem with the de Sitter α-vacua in the context of perturbation theory were reported earlier in [34,35,36,37,38,39,40,41,42]. Such ambiguity seems to originate from the inherent non-local characteristic of the α-vacua, coming from the antipodal transformations discussed in Section 2. Thus even though we may give the detector response function for a real scalar in the α-vacua a meaning in the sense of just an expectation value (cf. the discussions at the end of Section 2), it fails for a complex scalar. In [40] (also references therein), it was suggested to modify the Feynman propagator by adding two sources, in order to tackle the non-locality of the α-vacua. However, it is not clear to us how to implement any such analogous modification in the case of a pointlike, localised particle detector.
Nevertheless, we may still try to obtain a regularised version of the problematic integral as follows. However the caveat is, this will not yield as we shall see, a unique result. Let us first rewrite the integral as where is real, no matter positive or negative. Since the pole on the real axis of the integrand on the right hand side is of first order, its principal value is well defined. This allows us to thus define a regularised value of our original integration as, Accepting this definition, the contour for the first integration on the right hand side is taken to be a semicircle with an infinitesimal semicircular deformation of radius centred at u = , in the lower half plane. Thus the poles we pick up are located at u n = inπ + (−1) n (n = −1, −2, −3, . . . ). For the second, we use similar contour in the upper half plane, picking up the poles at u n = inπ + (−1) n (n = 1, 2, 3, . . . ), Fig. 5. Splitting the integral into two parts on the right hand side in the definition of Eq. (52) will ensures its real valuedness. We find Evaluating the rest of the integrals in Eq. (51), we find the regularised detector response function for a conformal complex scalar in the α-vacua, dF α (p) dt + complex conf. = p 3 H 3 384π 3 cosh 4 α + sinh 4 α e πp e πp − 1 + 12 sinh 2α cosh 2 α + sinh 2 α e πp/2 p 2 (e πp/2 − 1) Setting α = 0 recovers the result of Eq. (23). As evident, the above result is not unique, for Eq. (53) would change if we change the integration contour. For example, in the first of Fig. 5, we could have made the infinitesimal semicircular deformation in the upper half plane as well (and the opposite in the second of Fig. 5), which will lead to a change of sign of the second and third terms appearing on the right hand side of Eq. (54). We face similar ambiguity for the case of a massless minimal scalar field as well and hence we shall not pursue it.
where s = 3/2 − ν with |s| 1 is a small parameter and y is given by Eq. (8). Note that the de Sitter symmetry breaking logarithm is absent here compared to Eq. (17). Comparing Eq. (55) and Eq. (17) it is clear that we can compute the detector response for a nearly massless and minimal scalar by just making the replacement, 1 2 ln(a(t)a(t )) + ln 2 − 1 4 → 1 2s + ln 2 − 1 , appearing in any of the expressions for the response function for the massless minimal scalar field (such as Eq. (49)). Thus despite s is large, there will be no term growing with time in this case, as compared to the exactly massless one. On the other hand, if we take a complex scalar field in α-vacua, problems exactly similar to Section 5 will prevail.

Discussions
We have computed, in this work, the response function for the Unruh-DeWitt detector coupled to a complex scalar field at the first order perturbation theory, for both conformal and a massless minimal coupling. The latter requires certain regularisation procedure in order to give the response function a physical meaning.
We have discussed extension of these results to the de Sitter α-vacua and have pointed out some possible ambiguities for a complex scalar.
We have also shown that for a real scalar field theory with a field-detector coupling linear in the field operator, with the interpretation discussed at the end of Section 2, we can indeed compute the response function for the α-vacua. It is easy to argue that such computation extends to any arbitrary order of the perturbation theory. This is because at the n-th order, we have a term like n dτ n φ(τ n ) from the S-matrix extension. Since there are as many integrations as the number of field operators, we shall never have two Wightman functions appearing in a single integral. Accordingly, the cancellation of the i regulators as of Section 5 does not occur in this case. For example, the second order correction in the response function can be evaluated (up to some numerical factors) as which can indeed be computed without any ambiguity. Similarly, the response function can be obtained by accounting for higher order corrections. For the massless and minimal complex scalar in particular (Section 4.2), we needed to introduce a fictitious real conformal scalar field to cancel a divergent term, along with other regularisation procedure. It will thus be interesting to check whether this regularisation scheme can consistently tackle the divergences at higher order of the perturbation theory as well.
Computation of the response function for a fermionic field is more realistic. Since massless fermion is conformally invariant, we may expect in this case the spectrum to be qualitatively similar to that of a complex conformal scalar. However, for fermions we do not expect any de Sitter breaking growing logarithms. It will also be interesting to investigate the massless minimal complex scalar field theory from various perspective of quantum entanglement, e.g. entanglement harvesting. The effect of background primordial electromagnetic fields on a charged scalar will also be interesting, for in this case we expect de Sitter breaking terms indicating instability at late times analogous to that of the growing logarithm as in Eq. (49). We shall come back to these issues in a future work.