Unparticle Contribution to the Hydrogen Atom Ground State Energy

Home , Unparticle physics

arXiv:1603.03319v1 [hep-ph] 10 Mar 2016 ihe .Wondrak F. Michael 3 2 1 bleicher@fias.uni-frankfurt.de nicolini@fias.uni-frankfurt.de wondrak@fias.uni-frankfurt.de rudsaeadw opr twt xeietldt.Bud on Bounds data. experimental Λ with scale of it energy shift unparticle compare energy the we Rayleigh and the using state By obtain ground atom. we hydrogen theory, the stat of Schrödinger modified perturbation levels unparticle energy of the effect on the tentials study we work present the In d aaee einweebud nΛ on bounds where region parameter a natce ol etse naoi hsc experiments. physics atomic in tested be could unparticles U uhMuagSr ,648Fakuta an Germany Main, am Frankfurt 60438 1, Ruth-Moufang-Str. a-o-aeSr ,648Fakuta an Germany Main, am Frankfurt 60438 1, Max-von-Laue-Str. n h opigconstant coupling the and oanWlgn oteUiesta rnfr mMain, am Goethe-Universität Frankfurt Wolfgang Johann PACS: yrgnao rudsaeenergy state ground atom hydrogen a rnfr nttt o dade tde (FIAS), Studies Advandced for Institute Frankfurt natcecnrbto othe to contribution Unparticle 11.k 14.80.-j 11.15.Tk, b ntttfu hoeicePhysik, für Theoretische Institut a,b 1 , ir Nicolini Piero ac 1 2016 11, March U λ Abstract safnto ftesaigdimension scaling the of function a as r eie.W hwta hr exists there that show We derived. are U r tign,sgaln that signalling stringent, are a,b 2 n acsBleicher Marcus and cpo- ic the - a,b 3 1 Introduction

Unparticle physics is an extension of the Standard Model, consisting in the possibility of having a non-trivial scale invariant, yet undiscovered sector of particle physics. At first sight, unparticles appear as a generalization of neutrinos because they share the following properties: scale invariance and only very weak interaction with other fields. Neutrinos enjoy the first property to a good approximation, although their oscillations disclose a small non-zero mass. The second property is a general requirement of any hypothetical particle sector because we want it to be, to a certain extent, hidden from current observations. On a closer inspection, however, we find that unparticles differ drastically from neutrinos. Since we do not restrict the unparticle fields to be massless, we can no longer speak in terms of particle number as in the conventional manner. The unparticle field is controlled by a canonical scaling dimension d which is in general a non-integer number. Due to the U unusual character, one refers to the matter described by such a theory as unlike particles, or unparticle stuff. After Georgi’s seminal paper [1] unparticle effects have been explored in many areas spanning collider physics [2–8], gauge and Higgs interactions [9, 10], cosmology and astrophysics [11, 12], AdS/CFT correspondence [13] as well as gravity short scale deviations [14, 15] and black holes [16–20]. Unparticles also play a crucial role in the fractal properties of a quantum spacetime. A new fractality indicator, called un-spectral dimension, has recently been proposed to address the case of a random-walker problem in terms of an unparticle probe [21]. When the manifold topological dimension is 2, the un-spectral dimension turns to be 2d , i.e., it depends only on the U scaling dimension d . This fact explains the complete “fractalizazion of the U event horizon of un-gravity black holes [18, 19], as well as of metallic plates for the Casimir effect in the presence of an un-photon field [22]. Finally, unparticles have been proposed to explain some anomalies in currents flowing in super-conductors [23] and transport phenomena in cuprates [24]. In this paper we want to address one of the basic questions of unparticle physics, i.e., the value of Λ , the typical energy scale of the theory. U To achieve this goal, we consider the modifications of static potentials that emerge from virtual unparticle exchange. Specifically by considering the corrections to the Coulomb potential we calculate the deviations of the ground state energy of the hydrogen atom. We show that competitive bounds on Λ can be derived by a comparison with experimental data. U The paper is organized as follows. After a short review of the basic formalism of unparticle physics (Section 2), we present the calculation of the energy shift by a perturbative solution of the Schrödinger equation in the presence of an unparticle modified electrostatic potential (Section 3). Finally, in Section 4 we draw our conclusions.

2 2 Unparticle physics and static potentials

We recap the basic motivations of unparticle physics along the lines of Georgi [1]. We start by considering that at some very high energy scale, the Standard Model is accompanied by an additional sector of Banks-Zaks fields ( ). The interaction between the two sectors takes place by exchange BZ of mediating particles having a large mass scale M . If our energy scale of interest falls below M , we can apply effective fieldU theory to integrate out U the mediating field [25] and get the final interaction Lagrangian 1 SM (1) M k O OBZ U where denotes a Standard Model field operator of scaling dimension OSM dSM and is a Banks-Zaks field operator of scaling dimension d . The OkBZ BZ factor M − guarantees the dimensional consistency of (1), being k = dSM + d DUand D is the spacetime dimension. BZ − If the energy is further decreased to a scale Λ < M , the Banks-Zaks fields undergo a dimensional transmutation andU exhibitU a scale invariant behavior with a continuous mass distribution. For energies lower than Λ , U the sector becomes unparticle operators . The matching conditions BZ OU onto the Banks-Zaks operators are imposed at the energy scale Λ and U determine the structure of the coupling between the Standard Model and the unparticle fields based on (1):

dBZ dU C Λ − λ U = (2) U k SM dSM+dU D SM M O OU Λ − O OU U U where d is the scaling dimension of the unparticle operator , C denotes U OU U a dimensionless constant and λ is a dimensionless coupling parameter deﬁned by Λ k λ = C U < 1 . (3) U M U The inequality holds if C < 1 and d > D dSM. Any experimental U BZ − bound on the interaction allows for constraints on the unparticle parameter space, i.e., Λ , λ and d . The scale hierarchy is 1 TeV Λ < M MPl, U U ≤ U U ≤ where MPl is the Planck mass [20]. In the rest of the present paper we assume D = 4. We also assume the customary interval 1 < d < 2 [2]. The U case d = 1 corresponds to no fractalizazion or other continuous dimension U eﬀects of unparticle physics. Unparticles have been largely employed in context of static potential emerging from virtual particle exchange [10, 14, 15, 26, 27]. Such results are instrumental to the working hypothesis of the current investigation. In view of the analysis of the hydrogen atom, we consider an additional contribution to the Coulomb potential for the presence of unparticle exchange in the

3 Figure 1: Electron-proton interaction due to a vector field Aµ and an unparticle scalar field φ . U interaction between electron and proton. In general, we assume electrons and protons to carry unparticle charges λe and λp, respectively. At the same time, they also possess electric charges e allowing them to couple to the ± photon field Aµ. Both interactions are independent of each other since we assume the effective couplings between unparticle stuff and photons to be negligible. Therefore the interaction Lagrangian can be written as 1 = J µA + J φ (4) int µ dU 1 L (Λ ) − U U U µ µ where J = j(~x) δ0 and J = j (~x) and U U j(~x)= e δ(~x ~x )+ e δ(~x ~x ) (5) − − e − p j (~x)= λe δ(~x ~xe)+ λp δ(~x ~xp). (6) U − − Alternatively one can consider the vector unparticle interaction Lagrangian1 and derive the static potential much in the same way as in the scalar case. The two results do not differ apart from a global sign (see e.g. [15]). We recall, however, that conformal invariance can be lost for vector fields if d < U 3, although pure scale invariance can be preserved. For scalar unparticles the issue does not arise [25]. The expression for the unparticle interaction energy V between an elec- U tron and a proton in the static case reads [10, 14, 15, 26, 27]:

λe λp V = ξdU (7) U 2dU 2 2dU 1 − Λ − r − ! U √π Γ d 1 where the coeﬃcient is ξ U − 2 and r ~x ~x denotes dU 2dU e p ≡ (2π) Γ(d ) ≡ | − | the distance between the charges. U For the typical size of the hydrogen atom we can estimate the energy correction with the help of Fig. 2. For r = 0.05 nm the energy shift per unit

1The Lagrangian in such a case reads

L µ 1 µ int = J Aµ + −1 JU (AU )µ (ΛU )dU µ µ where JU = jU (~x) δ0 .

4 Figure 2: Interaction energy per unparticle charge unit vs. r for diﬀerent values of d in the case Λ = 10 TeV. U U unparticle charge varies between 3 102 eV and 6 10 19 eV in the range × × − 1 < d < 2 with Λ = 10 TeV. U U 3 Unparticle eﬀects in the hydrogen atom

The hydrogen atom is a well investigated system. In the non-relativistic limit we can apply the Schrödinger formalism to describe electron dynamics in a static Coulomb potential of the proton, by using the particle reduced mass µ. The energy spectrum in Gaussian/natural units (4πǫ0 = ~ = c = 1) reads µ α2 En = (8) − 2n2 for all n N. Here we used the fine-structure constant α e2. The biggest ∈ ≡ corrections to the Schrödinger description are of order α4 and arise from relativistic effects (i.e. kinetic, spin-orbit and Darwin terms) that can be included via fine-structure modifications of the Hamiltonian. There exist higher order corrections as well. The proton possesses a finite size which affects the form of the Coulomb potential for distances shorter than its radius. Since the proton has a spin which interacts with both the electron angular motion and the electron spin, hyperfine corrections arise. Finally, one can include QED corrections such as the Lamb shift. In Tab. 1, the ground state energy in different descriptions is displayed. Note that the uncertainty of the two most accurate values arises from the uncertainty in the Planck constant h = 4.135 667 662 (25) 10 15 eVs [28]. × − We start our analysis by considering the energy level shift due to unparticle effects within the non-relativistic description. The radial Schrödinger

5 Energy Description Value ES Schrödinger, non-relativistic 13.598 286 eV th − ES, rel Schrödinger, 13.598 467 eV th − incl. fine-structure ED Dirac 13.598 467 eV th − EQED currently best theor. value 13.598 434 49 (9) eV th − 3 288 086 857.127 6 (3 1) MHz h [29] · E currently best exp. value 13.598 434 48 (9) eV exp − 3 288 086 856.8 (0.7) MHz h [30] · Table 1: Theoretical and experimental values of the hydrogen ground state energy. equation of the hydrogen atom in the presence of both electrostatic and unparticle potentials reads

2 2 1 d 1 l (l + 1) e λe λp + ξdU unl(r)= Enlm unl(r) 2 2 2dU 2 2dU 1 −2µ dr 2µ r − r − Λ − r − ! U (9) where the complete wavefunction reads u (r) ψ (r, ϑ, ϕ)= nl Y (ϑ, ϕ) . nlm r lm The above equation is a second order, linear, ordinary differential equation with non-polynomial coefficients. To our best knowledge no exact solution is available in the literature but one can still rely on perturbation theory. This is justified since Λ > 1 TeV by hypothesis, i.e., unparticle physics U can lead only to subleading corrections to the Standard Model. As a result the energy shift of the ground state at the first order in perturbation theory reads:

(1) (0) λe λp (0) ∆ = 100 ξdU 100 100 2dU 2 2dU 1 − Λ − r − D U E λe λp ∞ −2r 2dU +3 a = 4 ξdU dr r− e − 2dU 2 3 Λ − a Z0 U 3 1 (d 1) d λe λp = U − 2 − U , (10) − 2dU 2 2 2dU 2 2dU 1 (2π) − sin(2πd ) (Γ(d )) Λ − a − U U U (0) r 2 a 11 where we used ψ = 3 e− with a 1/(αµ) 5.29 10− m. 100 √4πa ≡ ≃ × (1) For 1 < d < 2, the sign of ∆100 is determined by the product of λe and U λp. This means that like unparticle charges lead to a shift to lower energies while unlike charges cause a shift to higher energies.

6 The theoretical energy value Eth can be divided up into the Schr¨odinger S HO contribution Eth and higher order terms Eth . The same applies to the unparticle-modiﬁed quantities. As a result we can approximate the energy shift as

S HO S HO Eth, Eth = Eth, + Eth, Eth + Eth | U − | U U − S S E th, Eth . (11) ≃ U − HO Here Eth, stands for all the higher order contributions in the presence of unparticles,U contributions we deliberately neglected in our ﬁrst order descrip- HO HO tion. This assumption introduces a theoretical error δEth Eth, Eth . ∼ U − Thus the bound on the energy shift can be written as

S S (1) Eth, Eth = ∆100 < δEth + δEexp , (12) U −

where δEexp is the experimental error. From Tab. 1, one can see that the Schrödinger description differs from the QED result by about 1.5 10 4 eV. × − The theoretical relative error can be estimated to be δE / ES 1.1 th th ≃ × 10 5. It dominates over the experimental relative error that is of the order of − 2.2 10 10 [30]. Accordingly we define where δ (δE + δE )/ES × − max ≡ th exp th ≃ 1.1 10 5 to get × − (1) 2dU 3 2dU 2 3 ∆ 2α − µ − (d 1) d λe λp 100 = U − 2 − U | | < δ . (13) S 2dU 2 2 2dU 2 max Eth (2π) − sin(2d π) (Γ(d )) Λ − U U U

From (13) we obtain 1 3 2dU −2 αµ 2 (d 1) d λe λp U 2 U Λ − − 2 | | . (14) U ≥ 2π α sin(2πd )(Γ(d )) δmax ! U U In Fig. 3 the lower bounds for Λ are illustrated for different values of the U unparticle charge λ, where we assumed λ = λ λ. The area above | e| | p| ≡ each curve is the allowed parameter space for the chosen value of λ. If the parameter λ is larger than a threshold value, λ > λth √2πα δmax 4 ≡ ≃ 7.1 10− , the lower bound on Λ is diverging for d 1 and exceeds 1 TeV × U U → in some parameter region. Here λth is obtained by requiring that the base of the exponentiation in (14) equal 1. The limit Λ corresponds to U → ∞ vanishing unparticle effects since the unparticle sector in the Lagrangian (4) µ disappears yielding the standard electrodynamics limit J Aµ. This Lint → explains the divergent lower bound on Λ expected also in other experiments U such as the electron g 2 anomaly [8] or the Casimir effect [22]. In contrast, − for λ λth, the scale Λ is weakly constrained in the limit d 1. This ≤ U U → means that the unparticle contribution lies within δmax irrespective of the value of Λ . U

7 Figure 3: Lower bounds for Λ with respect to d and λ. U U 4 Discussion and conclusions

In the current work, we have derived the corrections to the hydrogen atom ground state energy due to the presence of an unparticle modified static potential. We obtained this result within a perturbative analysis of the non- relativistic theory of the hydrogen atom. Our result lets obtain compelling limits on the unparticle scale Λ . For d . 1.3, bounds on Λ exceed 1 TeV. U U U Contrary to other proposed investigations (e.g. Newton’s law correction [14], proton-proton collisions at the LHC [5, 6]), the presented analysis can capture the key feature of unparticle physics, i.e., the dependence on the continuous scaling dimension d . U The presented results are filling a gap in the literature and are opening the route for further investigations. For instance one may be interested to find an exact solution of the Schrödinger equation (9) emerging from the inclusion of unparticle static potentials. Higher order corrections to the non-relativistic description can also be included in the analysis in order to approach the bounds currently offered by g 2 analyses [8]. − From the current non-relativistic analysis one can draw an important conclusion: Unparticle effects might be tested in atomic physics experiments.

Acknowledgements This work has been supported by the Studienstiftung des deutschen Volkes and by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoﬀensive zur Entwicklung Wissenschaftlich- Okonomischer¨ Exzellenz) launched by the State of Hesse.

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