Physics Letters B 772 (2017) 675–680 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Unparticle Casimir effect ∗ Antonia M. Frassino a,b, Piero Nicolini a,b, , Orlando Panella c a Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, D-60438 Frankfurt am Main, Germany b Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany c Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy a r t i c l e i n f o a b s t r a c t Article history: In this paper we present the un-Casimir effect, namely the study of the Casimir energy in the presence of Received 16 June 2016 an unparticle component in addition to the electromagnetic field contribution. The distinctive feature of Received in revised form 1 June 2017 the un-Casimir effect is a fractalization of metallic plates. This result emerges through a new dependence Accepted 14 July 2017 of the Casimir energy on the plate separation that scales with a continuous power controlled by the Available online 20 July 2017 unparticle dimension. As long as the perfect conductor approximation is valid, we find bounds on the Editor: A. Ringwald unparticle scale that are independent of the effective coupling constant between the scale invariant sector Keywords: and ordinary matter. We find regions of the parameter space such that for plate distances around 5μm Unparticles and larger the un-Casimir bound wins over the other bounds. Scale invariance © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 Casimir effect (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . Plates fractalization 1. Introduction where the unparticle operator OU has dimension dU and kU = dU + dSM − 4. We note that the resulting interaction term de- Recently, a massive extension to the Standard Model was pro- pends on a dimensionless coupling constant λ = (U /MU )k < 1. posed in which scale invariance is preserved, provided these new Unitarity constraints from conformal field theory (CFT) necessitate particles are weakly interacting and appear in non-integer num- a lower bound on the unparticle dimension dU ≥ 1 [3]. Normally bers [1]. The topic has intersected a huge variety of fields, spanning only operators with dU ≤ 2are considered because for dU ≥ 2the astrophysics neutrino physics, AdS/CFT duality and quantum grav- calculations become sensitive to the ultraviolet sector and there- ity. fore less predictive. For a discussion about a more general form of Scale invariance for massive fields can be described through these operators see for instance the HEIDI models [4–7]. Banks–Zaks (BZ) fields [2]. At some very high energy scale MU , The phenomenology of unparticles and the associated signals at the Standard Model fields interact with a sector exhibiting a non- high energy colliders (LEP, LHC) have received attention in recent BZ L = −kO O trivial infrared fixed point, int (MU ) SM BZ , where the literature [8–13]. = + field operators must have dimensions dSM, dBZ and k dSM On the theoretical side, bounds on the parameter space have dBZ − 4. Since the Banks–Zaks fields are not observed in na- been derived by computing the unparticle contribution to the ture, their suppression requires that the scale MU is somewhere muon anomaly [14]. Unparticles also provide a relevant short scale between current experimentally-accessible scales and the Planck modification to gravitational interactions. Black hole solutions have scale. been derived for the case of scalar [15–17] and vector [18] unpar- BZ At a second energy scale U < MU , the sector develops ticle exchange. A significant characteristic of these solutions is the scale-invariant properties and the particle number is controlled by fractalization of the event horizon, whose dimension is a function = a continuous parameter dU dBZ . This is equivalent to saying that of the unparticle parameter dU . This feature has been confirmed BZ fields undergo a dimensional transmutation to become unpar- by subsequent studies of the spectral dimension, as an indicator ticles via of the short-scale spacetime dimension perceived by an unparti- 1 1 cle probe [19]. In addition, such unparticle modifications to gravity O O −−−→ O O SM BZ λ SM U (1) offer compelling effects that can be observed through short-scale (MU )k ∼U (U )kU deviations to Newton’s law [20] and on the energy levels of the hydrogen atom [21]. * Corresponding author. In this paper, we will analyze the Casimir effect in the presence E-mail address: nicolini@fias.uni-frankfurt.de (P. Nicolini). of a weakly-coupled unparticle sector, which we will refer to as the http://dx.doi.org/10.1016/j.physletb.2017.07.029 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 676 A.M. Frassino et al. / Physics Letters B 772 (2017) 675–680 ∞ un-Casimir effect. The Casimir effect has been discussed within var- AdU 2 2 d −2 2 DU (x, x ) = dm (m ) U D(x, x ;m ) ious scenarios beyond the Standard Model including compactified 2 dU −1 2π(U ) extra dimensions [22] and minimal length theories [23,24]. Using 0 the un-Casimir effect, we present a privileged testbed for setting 5/2 + [ ] = 16 π (dU 1/2) bounds on relevant regions of the parameter space U , dU gov- AdU (5) 2dU U − U erning unparticle physics. The un-Casimir effect also offers an in- (2π) (d 1) (2d ) triguing phenomenon of plate fractalization in agreement with the i.e. it is a linear continuous superposition of Feynman propagators above discussion. of fixed mass m. When the conformal dimension tends to unity (dU → 1) the unparticle propagator reduces to that of an ordinary 2 2 2. Unparticle contribution to the Casimir effect massless field DU (p ) → 1/p [25]. The above propagator can be expressed in terms of the unparticle generating functional ZU [ J] In line with (1) we assume the existence of an unparticle vector [33,34]. The net result is U field A of scaling dimension d which couples with the standard μ U = ˆ −1 2 μ ¯ μ DU x, x FU ( ) D x, x (6) model electron Dirac current J = ψγ ψ via the interaction: where D x, x is the Green’s function for massless scalars and λ L = AU ¯ μ ˆ −1 2 int − μ ψγ ψ. (2) FU ( ) is a non-local operator defined as [33,34] dU 1 U − −2 1 dU ˆ 2sin( π dU ) Interactions of the form given in (2) have been extensively used FU (2) ≡ . (7) 2 in the literature in order to study the phenomenology of unparti- AdU U cles [25–27]. In the presence of perfectly conducting parallel plates Eq. (6) shows that the unparticle propagator actually solves the at a distance a, the interaction in (2) will be responsible for a ˆ U massless Green function equation, since [ 2, FU (2) ] = 0. This fact Casimir effect for the field A in much the same way the interac- μ allows us to assume L = A ¯ μ ⎡ ⎛ ⎞ ⎤ tion int e μ ψγ ψ implies the Casimir effect in QED. For ease of presentation, in the following section we discuss the Casimir 3 | U | ≡⎣− i ⎝ ⎠ ⎦ effect mediated by a scalar unparticle field using the scalar field 0 T00 0 ∂μ∂μ DU (x, x ) . (8) 1 2 =0 analogy. This is routinely done in QED, where the actual result μ x=x is just twice that of a scalar field due to the two physical photon The above assumption turns out to be in complete agreement with polarisations. For the case of un-particles such a working hypothe- previous results for the non-local quantum stress tensor in a vari- sis remains valid, since the energy of longitudinal modes related to ˆ ety of contexts (e.g. for a generic operator FU (2)). See for instance superposition of massive fields (see (5) below) do not exceed 1% of [17] for the scalar un-particle mediated gravity and [35–37] for that of the transverse polarizations. Afull treatment of the Casimir other non-local deformations where it is found: effect mediated by a vector unparticle field will be presented in | U | = ˆ −1 2 | | [28]. 0 T00 0 FU ( ) 0 T00 0 , (9) The Casimir energy [29] is often described by the shift in the with 0|T00|0 referring to a massless scalar field. Indeed by using sum of the zero point energies of the normal modes of the electro- (6) in (9) one obtains (8). magnetic field induced by geometrical boundary conditions. Such By inserting (5) in (8) one has that the unparticle Casimir en- a Casimir energy can be written by means of the density of states C ergy EU reads: dN/dω which in quantum field theory (QFT) is related to the imag- ∞ inary part of the trace (over both space and spinor degrees of C AdU 2dU −3 C freedom of the field under consideration) of the Feynman prop- EU = dm m E (m). (10) 2 dU −1 agator [30,31]. It has been shown in [32, p. 47] that for a generic π(U ) 0 massive, scalar field the vacuum energy can be written as Here we used the fact that the derivatives ∂ ∂ and the integra- μ μ E = 3 2 ; 2 tion on m commute. To derive the above result for the Casimir vac(m) i d x ∂0 D(x, x m ) .