Quantum effects due to a moving Dirichlet point

We study quantum effects induced by a point-like object that imposes Dirichlet boundary conditions along its world-line, on a real scalar field φ in 1, 2 and 3 spatial dimensions. The boundary conditions result from the strong coupling limit of a term quadratic in the field and localized on the particle’s trajectory. We discuss the renormalization issues that appear and evaluate the effective action. Special attention is paid to the case of 2 spatial dimensions where the coupling constant is adimensional.


Introduction
Quantum Field Theory predicts that an open system is capable of evolving, from the vacuum, to a state characterized by a non-vanishing number of (real) quanta [1]. That is indeed the case, among other related phenomena, of the Dynamical Casimir Effect (DCE), one of the most studied manifestations of quantum dissipation [2]. The DCE consists of the emission of real to dealing with d = 2, and d = 3. The main distinction between d = 2 or d = 3 and d = 1 are, as we shall see, due to the different UV properties induced by the coupling between the particle and the field. Indeed, the usual renormalization which is required to make sense of a δ-like potential in Quantum Mechanics in two and three spatial dimensions, also manifests itself here; moreover, the resulting divergences can be cured by applying a similar procedure.
The d = 2 case will also be relevant for future developments regarding the quantum properties of center vortices, and support the construction of phenomenological ensembles for these magnetic defects. They are topological variables that are believed to capture the infrared behavior in Yang-Mills theories. For a given realization, the calculation of the effective action involves regularity field conditions on world-lines and world-surfaces in three and four Euclidean dimensions, respectively, which are problems of codimension 2. The effective action for a single center vortex without curvature was analyzed in Refs. [11]- [15]. This may involve singular spectral problems with different gyromagnetic ratios and regularity conditions on the YM off-diagonal sector, which depend on how the fluctuations are parametrized.
The structure of this paper is as follows: in Sect. 2 we introduce the kind of system that we study, as well as some general expressions for its effective action in the small departure limit, in the path integral framework. Then, in Sect. 3 we evaluate the effective action for the massless field, by considering a perturbative expansion in powers of the departure of the worldline from the one of a static particle, assuming the mirror moves non-relativistically, for d = 1 and d = 3. Because of its particularities, related to scale invariance of the coupling between field and mirror, the d = 2 case is considered separately in Sec. 4. Finally, in Sec. 5 we present our conclusions.

The system
The system that we shall deal with throughout this paper consists of a real scalar field ϕ in d + 1 dimensions, with d = 1, 2, or 3, coupled to point-like objects which are meant to implement Dirichlet-like conditions.
For an object imposing Dirichlet conditions, the effective action will be denoted by Γ(C), since it is a functional of the world-line C. In a functional integral approach, and using Euclidean conventions 1 , where Z(C) (Z 0 ) denotes the Euclidean vacuum transition amplitude corresponding to the scalar field in the presence (absence) of the particle. Z(C) and Z 0 are given, explicitly, by where S 0 is the action which describes the free propagation of the field, and a functional δ-function has been introduced to account for the Dirichlet conditions; namely, the vanishing of the field at the position of the particle. The former is given by: where we have introduced a shorthand notation for the integration, in this case over all of the spacetime coordinates x = (x 0 , x 1 , . . . , x d ). Namely, in the case above, x ≡ d d+1 x. Greek indices will be assumed to run over the values 0, 1, . . . , d, and space-time is endowed with the Euclidean metric: g µν = δ µν . Regarding δ C (φ), it should select, among the configurations appearing in the functional integration measure, just the ϕ-field configurations which satisfy Dirichlet boundary conditions on C. As already advanced, those conditions will be reached as the limit of a local term, namely: we add to the free action a term, quadratic in ϕ and localized on C, with a strength λ which, when λ → ∞, imposes Dirichlet boundary conditions: where we have assumed that τ → y µ (τ ) (µ = 0, 1, . . . , d) is a parametrization of C, and g(τ ) ≡ẏ µ (τ )ẏ µ (τ ) (we have ignored, as customary in the functional integral context, irrelevant factors which in this case are independent of the curve and the field). The g(τ ) factor has been introduced in order to have reparametrization invariance. It is rather convenient to use an auxiliary field ξ(τ ), in order to have an alternative representation for the functional above, where ϕ may be integrated out in a simpler fashion. Indeed, where Integrating out ϕ, we see that: with a kernel K which may be rendered as follows: where s(τ ) denotes the Euclidean version of the proper time, namely, the arc length, and we have used a bra-ket notation for the kernel of an operator. Then, integration of the auxiliary field yields and finally We proceed in the next Section to perform a perturbative expansion of the effective action in powers of the departure of the particle as measured with respect to a static situation.

Small-departure expansion for the massless field
We consider a worldline C parametrized with the 'Lab' time τ ≡ t ≡ x 0 . Therefore, (y µ ) = (y µ (t)) = (t, η i (t)) (i = 1, . . . , d), and we assume that η i (t), the departure from a static situation: (t, 0), is small. By an adequate choice of the spatial origin, we can always assume that the average position of the particle is 0; thus: t η i (t) = 0. Besides, we deal with non-relativistic motions, so that the √ g factors will be replaced by 1.
We proceed to perform an expansion in powers of η i (t). Using an index to denote, in a given object, the order in η i in that expansion, we shall have: where the explicit form of the first few terms is: It goes without saying that the Γ (0) λ (C), independent of the particle's motion, may be safely discarded, and we shall do so (it only contributes a constant to the static vacuum energy).
To evaluate the remaining terms, we need to consider the kernels K (i) , i = 0, 1, 2. We see that: where: where (we have used ω to denote the k 0 component of the momentum).
It is rather straightforward to see that K (1) λ , we obtain: We see that: It is evident that the first-order term Γ (1) λ vanishes. Let us the calculate Γ (2) λ , the only surviving contribution: whereη i is the Fourier transform of η i , and: Note that the first term in the second line of (22) is a mass renormalization; we shall focus in what follows on the properties of the second one. The treatment of such a term differs depending on the number d of spatial dimensions. Indeed, we see that f depends on I(ω), both explicitly and also through K (0) λ , and I(ω) diverges, except for d = 1. We note that the very same divergence appears when considering the δ-function potential in d > 1. This requires to renormalize the coupling λ, something which we will implement here as well.
One can also see that, since K (1) vanishes, the expression for the fourthorder term simplifies to: The d = 1 case has been previously studied [17]. In this case, no renormalization of λ is required, since the integral I(ω) is convergent. Indeed, .
The ν integral in the expression for f can then be explicitly evaluated, the result being: A large-λ expansion of the previous expression yields where one observes the different nature of the terms; the second one is the well-known Dirichlet result, and the first one amounts to a renormalization of the kinetic energy of the particle.

d = 3
In 3 spatial dimensions, we see that: Inserting this into K (0) λ , we now obtain instead where: Let us evaluate the kernel f (ω) for d = 3, for the case λ r → ∞. We see that: The last integral may be obtained as: where, after a standard calculation, we find: We can also calculate explicitly the subleading terms. We have found that all the terms involving even powers of the frequency ω, and therefore not contributing to the dissipative effects (imaginary part of the analytically continued Γ), are divergent. The terms which are odd in |ω|, including the λ r → ∞ one, are finite: Therefore, performing the rotation back to real time, In d = 2, we note that the integral in I(ω) is logarithmically divergent.
Introducing an UV cutoff Λ, we see that, for large values of Λ, Following the usual treatment of the δ-function potential, we introduce a renormalization scale µ, and rewrite Inserting this into K λ , we see that where we have introduced the renormalized coupling constant: and the scale-dependent function Note that I(ω) also appears in the denominator of (23); using (38), one sees that: were the first two terms amount to a mass and kinetic term renormalizations, while which does not involve divergent objects in its integrand. Let us now evaluate the previous integral, which may be rendered in the following fashion: where we have indicated the range of integration explicitly.
To perform the integral, we first perform a shift in the integration variable, and symmetrize it explicitly with respect to ν. Then we obtain the equivalent expression: The last integral is UV divergent; to cope with those divergences, we subtract from the integrand its Taylor expansion around ω = 0, up to the second order. This procedure does not erase information related to dissipation, as the subtracted terms give rise to a renormalization of the kinetic term and the mass of the particle. This leads (after some algebra) to the subtracted integral f s : The previous integral is UV-convergent. Performing a rescaling in the integration variable, where ψ(y) may be evaluated numerically, and it turns out to be finite and smooth for every y > 0, as can be seen in Figs 1 and 2, which were generated with Mathematica. We have thus succeeded in renormalizing the codimension 2 case in d = 2. Since d = 2 exhibits interesting features, let us now consider, for the sake of completeness, also a massive field with mass m.
In what follows, we shall consider the arc-length (s) of the curve, and introduce a regularization in s-parameter space. Our starting point is Eq.
is singular when s → s . This object can be treated by means of a regularization which enables the identification of the singular part in the sense of distributions. For example, we could use dimensional regularization (d = 2 − ) and, of course, introduce a mass parameter λ → λµ so as to keep λ adimensional: Another possibility is to consider the regularized quantity In order to simplify the (s-independent) finite part, which will be absorved in the renormalized coupling constant, we shall adopt the second procedure. It is easy to see that in the limit m → 0 this coincides with the usual dimensional regularization.

Regularization in s-space
We can initially rewrite where H l is the contribution of a line. In fact, the regulator can be removed in D , as D = H C − H l is regular when s → s . To see this, we can expand y(s ) around s: and use that for the arc-length parameter it is verified (e(s) = y (s)) |e(s)| 2 = 1 , e(s) · e (s) = 0 , e(s) · e (s) = −|e (s)| 2 .
In other words, where the dots represent orders higher than (s − s ) 2 . Therefore, is manifestly regular when s approaches s . In particular Now, let us analyze which is the product of a regular factor times a distribution with singularities. It is well-known that the distribution |x| α has a simple pole at α = −1 whose residue is 2δ(x) [16]. Then, defining the dimensionless variable x ≡ µ(s − s ), subtracting and adding the polar part of I l (s, s ), and then multiplying by the regular factor e −m|s−s | , we get We can check that R (s, s ) is regular when → 0 by acting on a test function Indeed, this is well-defined in the limit → 0. Then, introducing the renormalized coupling constant λ r , the contribution to the effective action is obtained from the → 0 limit of up to an irrelevant constant.

The weak λ r -limit
We may perform an expansion for small λ r : The first and second order contributions read, respectively, Because of Eq. (59), the first term in Eq. (66) vanishes, while the second term, in spite of the regularization, is still an ill-defined divergent quantity. The same happens with the R 2 -contribution in Eq. (67). However, any improved regularization that keeps the natural dependence of R in s − s , which represents the translation symmetry of the contribution along the line, will give a divergence proportional to the length of the curve ds = L → ∞. This is associated to a renormalization of the string tension. In the → 0 limit, the cross term in Eq.
This expression is, by construction, regular. Since D(s, s) = 0, this can be written in the simpler form: Note that D(s, s ) ≥ 0, so that this is a negative contribution to the effective action, as well as that originated from the D 2 -term.

The small curvature limit
An interesting physical situation to analyze is when the acceleration of the particle is small, so that, due to the mass gap, it does not radiate. To obtain the lowest order contribution of acceleration, it will be useful to perform an expansion of Γ λ (C) in powers of D, which tends to zero when C → l. For this objective, we can rewrite Let us analyze the second term, where Q(s − s ) is the kernel of the operator (λ −1 r + R ) Now, if the kernel of the operator R is R (s − s ), then, in terms of the Fourier transforms we have . (74) The explicit form of R (s − s ) is obtained from Eqs. (60), (61) Performing the Fourier transform, using that for > 0 [16] F(e −m|x| |x| −1 ) = F(e −mx x −1 we arrive atR Here, we can safely take the limit → 0 to conclude Finally, to get rid of γ, we redefine µ → µe −γ , which implies Then, to lowest order, we find where Γ λ (l) is the effective action of a straight line, and χ(m, µ) is a constant that depends on the mass and the arbitrary scale µ. For m >> µ e 2π λr , χ is finite and negative. In Fig. 3, we show the dependence of the χ-coefficient with m in this regime.

Conclusions
In this work we have defined and studied quantum dissipation in a moving DCE setting involving δ interactions with codimension larger than one. Specifically, we studied a real scalar field in d + 1 dimensions (d = 2, 3) coupled to an imperfect one dimensional mirror. We found that the singular nature of the problem requires a renormalization of the coupling (λ) between the field and the mirror. In particular, for d = 3, there is a finite scaleindependent imaginary part for the effective action. In the Dirichlet limit, similarly to the well-known d = 1 result, this imaginary part contains a |ω| 3 dependence, which is what one expects on dimensional grounds, assuming no renormalization scale dependence is generated.
The case d = 2, where λ is dimensionless, is special, for the coupling not only gets renormalized but also acquires a dependence on a mass scale µ. This phenomenon parallels that observed when dealing with the δ-potential in quantum mechanics for a planar system [9]. In this case, we have shown how the mass scale µ is generated for the massless field, and how it intervenes in the construction of the renormalized effective action, in the small-departure approximation. We have also found that, for a massive field in d = 2, apart from inducing renormalizations for the kinetic energy and the mass of the mirror, the Euclidean effective action is finite and negative for small λ r . Finally, we considered a massive field coupled to an imperfect mirror with small acceleration, where no imaginary part is expected, and found that the effective action is lower than that of a static mirror.
It would be very interesting to extend this analysis to other cases with codimension 2 that arise when considering quantum fluctuations around vortex-like defects in three and four Euclidean dimensions. In particular, it would be important to study the singular problem associated with curved thin center vortices in Yang-Mills theories, and obtain quantum properties such as stiffness from a fundamental point of view.