Axion-Modified Photon Propagator, Coulomb Potential and Lamb-Shift

Axion-modiﬁed photon propagator, Coulomb potential and Lamb-shift

S. Villalba-Chávez,1, ∗ A. Golub,1, † and C. Müller1, ‡ 1Institut fürTheoretische Physik I, Heinrich-Heine-UniversitätDüsseldorf, Universitätsstraße 1, 40225 Düsseldorf,Germany. (Dated: December 7, 2018) A consistent renormalization of a quantum theory of axion-electrodynamics requires terms beyond the minimal coupling of two photons to a neutral pseudoscalar field. This procedure is used to determine the self-energy operators of the electromagnetic and the axion fields with an accuracy of second-order in the axion-diphoton coupling. The resulting polarization tensor is utilized for establishing the axion-modified Coulomb potential of a static pointlike charge. In connection, the plausible distortion of the Lamb-shift in hydrogenlike atoms is established and the scopes for searching axionlike particles in high-precision atomic spectroscopy and in experiments of Cavendish-type are investigated. Particularly, we show that these hypothetical degrees of freedom are ruled out as plausible candidates for explaining the proton radius anomaly in muonic hydrogen. A certain loophole remains, though, which is linked to the nonrenormalizable nature of axion-electrodynamics.

PACS numbers: 11.10.Gh, 11.10.Jj, 12.20.-m, 14.40.n, 14.70.Bh, 14.80-j, 14.80.Ms

I. INTRODUCTION niques and to develop new routes along which imprints of these hypothetical particles can be observed [13, 14]. Descriptions of the most popular detection methods can That the path integral measure in quantum chromo- be found in Refs. [15–19]. A vast majority of these dynamics (QCD) is not invariant under an axial chi- searches relies on the axion-diphoton coupling encom- ral U(1) −transformation provides a clear evidence that A passed within axion-electrodynamics [20]. Correspond- this classical symmetry does not survive the quantiza- ingly, many photon-related experiments such as those tion procedure. As a consequence of this anomaly, QCD searching for light shining through a wall [21–28] and should not be a charge-parity (CP )-preserving frame- the ones based on polarimetry detections [29–33] have work. However, with astonishing experimental accuracy, turned out to be particularly powerful. no CP -violation event is known within the theory of the strong interactions. This so-called “strong CP -problem” Contrary to that, precision tests of the Coulomb’s law finds a consistent theoretical solution by postulating a via atomic spectroscopy and experiments of Cavendish- type have not been used so far in the search for ALPs, al- global U(1)PQ-invariance in the standard model (SM), which compensates the CP -violating term via its spon- though they are known to constitute powerful probes for taneous symmetry breaking [1]. While this mechanism other well-motivated particle candidates [34–37]. In par- seems to be the most simple and robust among other ticular, these setups provide the best laboratory bounds possible routes of explanation, it is accompanied by a on minicharged particles in the sub µeV mass range. Si- new puzzle linked to the nonobservation of the associ- multaneously, by investigating the role of ALPs in atomic ated Nambu-Goldstone boson, i.e. the QCD axion [2, 3]. spectra, one might elucidate whether the quantum vac- As a consequence, constraints resulting from this absence uum of these hypothetical degrees of freedom may be the indicate a feeble interplay between this hypothetical par- source for the large discrepancy > 5σ between the proton ticle and the well-established SM branch, rendering its radius that follows from the Lamb-shift in muonic hydro- detection a very challenging problem to overcome. Still, gen versus the established value based on electron scat- various experimental endeavours are currently oriented tering and the Lamb-shift in ordinary hydrogen [38–41]. arXiv:1806.10940v2 [hep-ph] 6 Dec 2018 to detect this elusive degree of freedom or, more gener- Various theoretical investigations have been put forward ally, an associated class of particle candidates sharing its seeking for a satisfactory explanation for this anomaly main features, i.e. axionlike particles (ALPs). Some of [42–46], some of them including hypothetical scalar par- them being central pieces in models which attempt to ex- ticles. plain the dark matter abundance in our Universe [4–8], Against the background of these circumstances, it is whereas others are remnant features of string compacti- relevant to derive modifications of the Coulomb potential fications [9–12]. due to quantum vacuum fluctuations of axionlike fields The problematic associated with the ALPs detection and to study their potential consequences. The former demands both to exploit existing high-precision tech- are encompassed in the corresponding vacuum polarization tensor whose calculation, however, is not a straight- forward task as far as axion quantum electrodynamics (QEDA) is concerned. This is because it requires–first ∗Electronic address: [email protected] of all–a meaningful implementation of the corresponding †Electronic address: [email protected] perturbative expansion in this nonrenormalizable frame- ‡Electronic address: [email protected] work. In analogy to quantum gravity [47–53], the expan- 2 sion in terms of the axion-diphoton coupling gives rise to spite the electrically neutral nature of ALPs, the polar- an infinite number of divergencies that cannot be reab- ization tensor closely resembles the one obtained in QED. sorbed in the renormalization constants associated with The similarity is stressed even further in Secs. III A and the parameters and fields of the theory. Unless a simi- III B, where various asymptotes of the polarization tensor lar amount of counterterms is added, this feature spoils are established and the general expression for the axion- the predictivity of the corresponding scattering matrix, Coulomb potential is determined. The latter outcome is preventing the construction of a consistent quantum the- presented in such a way that a direct comparison with the ory of axion-electrodynamics. Hence, the perturbative Uehling potential can be carried out. Also in Sec. III B, renormalizability of this theory demands unavoidably the we derive the corresponding modification to the Lamb- incorporation of higher dimensional operators. This, in shift and emphasize the problematic introduced by both turn, comes along with the presence of a large number of the effective scenario and the use of atomic s−states. In free parameters which have to be fixed from experimen- Sec. III C we give some estimates and discuss the ad- tal data. It is, however, known that this formal aspect vantages and disadvantages of testing ALPs via excited relaxes because many of these higher-dimensional terms states in Hµ, whereas in Sec. IV our conclusions are ex- are redundant, in that the ultimate scattering matrix is posed. Some details about the particle-ghost content of not sensitive to their coupling constants [54–57]. As a the theory are provided in Appendix A. Finally, in Ap- matter of fact, all contributions of this nature can be for- pendix B the sensitivity levels associated with precision mally eliminated while the effective Lagrangian acquires tests of the axion-modified Coulomb law via experiments counterterms which allow for the cancellation of the loop of Cavendish-type are presented. divergences. Besides, when working at a certain level of accuracy, only a finite number of counterterms is needed and the cancellations of the involved infinities can be car- II. THE MODIFIED PHOTON PROPAGATOR ried out pretty much in the same way as in conventional IN AXION-ELECTRODYNAMICS renormalizable field theories. In this paper, axion-electrodynamics is regarded as A. Effective field theory approach a Wilsonian effective theory parametrizing the leading order contribution of an ultraviolet completion linked Axion-electrodynamics relies on an effective action to physics beyond the SM. Its quantization is used for characterized by a natural ultraviolet scale Λ at which determining the self-energy operator of the electromag- UV the U(1) −symmetry is broken spontaneously. It com- netic field with an accuracy of second-order in the axion- PQ bines the standard Maxwell Lagrangian, the free La- diphoton coupling. This result is utilized then for ob- grangian density of the pseudoscalar field φ¯(x) and an taining the modified Coulomb potential of a static point- interaction term coupling two photons and an axion. Ex- like charge. In connection, the plausible distortion of the plicitly, Lamb-shift in hydrogenlike atoms is established. Partic- ular attention is paid to a limitation caused by the non- Z 1 1 renormalizable feature of axion electrodynamics which S = d4x − f f µν + ∂ φ∂¯ µφ¯ g¯ 4 µν 2 µ prevents us from having a precise and clear picture of (1) 1 1 the axion physics at distances smaller than the natu- − m¯ 2φ¯2 + g¯φ¯f˜ f µν . ral cutoff imposed by the axion-diphoton coupling. In 2 4 µν contrast to previous studies of the Lamb-shift involving minicharged particles and hidden photon fields, this Here, fµν = ∂µa¯ν − ∂ν a¯µ stands for the electromagnetic property introduces an unknown uncertainty that can- ˜µν 1 µναβ field tensor, whereas its dual reads f = 2 fαβ not be determined, unless the ultraviolet completion of with 0123 = 1. Hereafter, we use a metric with sig- axion-electrodynamics is found. We argue that–up to this nature diag(gµν ) = (1, −1, −1, −1), and a unit system in uncertainty–axionlike particles are ruled out as plausible which the speed of light, the Planck constant and the candidates for explaining the proton radius anomaly in vacuum permitivity are set to unity, c = ~ = 0 = 1. As muonic hydrogen (Hµ). Parallelly, spectroscopic results −1 the axion-diphoton couplingg ¯ = ΛUV has an inverse en- linked to a variety of transitions in hydrogen are exploited ergy dimension, this effective theory belongs to the class to probe the sensitivity of this precision technique in the of perturbatively nonrenormalizable frameworks. In the search for ALPs. We show that, as a consequence of the following we will suppose that ΛUV is a very large pa- mentioned feature, high-precision spectroscopy lacks of rameter in order to extend integrals over the momentum sufficient sensitivities as to improve the existing labora- components to an infinite-volume Fourier space. When- tory constraints on the parameter space of ALPs. ever no problem of convergence arises, the integrals over Our treatment is organized as follows. Firstly, in the spacetime coordinates will also be extended to the Secs. II A and II B, techniques known from effective field whole Minkowski space. theories are exploited for establishing the vacuum polar- We want to use Eq. (1) to derive radiative correc- ization tensor within an accuracy of the second order in tions up to second-order in the axion-diphoton coupling the axion-diphoton coupling. There we show that, de- ∼ g¯2. For this, we shall use well-established effective 3

field theory techniques which have been applied exten- At the pole p =m ¯ 2, the residue of this Green function is 2 sively within the context of quantum gravity [47–53] and Res G(p )|p2=m ¯ 2 = 1, Although the exposition is made chiral perturbation theory [58–61] (see also Refs. [62–64] in terms of bare parameters, the idea extends straightfor- for their application in nonlinear QED). In connection, wardly when renormalized quantities are considered in- 2 2 −2 we will suppose that Sg¯ characterizes–at energies sub- stead. whereas at p =m ¯ s = (¯gb¯φ) it turns out to be 2 stantially lower than Λ –the leading order contribution Res G(p )| 2 2 = −1. Hence, the vacuum excitations UV p =m ¯ s of its UV-completion which is Lorentz and gauge invari- linked to the former pole have a positive definite norm ant. Hence, all higher-dimensional operators which are in the Hilbert space, as should correspond to asymptoti- consistent with these fundamental symmetries should be cally single-particle states. Conversely, the square of the included in Eq. (1), implying that an infinite number of norm associated with the remaining massive excitations counterterms is necessary to cancel out the divergences is nonpositive and no physical state can be associated linked to one-particle irreducible Feynman diagrams [65]. with them (ghost states), leading parallely to a violation However, to extract quantitative predictions from the ra- of unitarity [67]. Noteworthy, as the associated higher- diative corrections in the second-order approach in the dimensional operator [see Eq. (2)] contains the square ¯ axion-diphoton coupling, it is sufficient to incorporate of bφ, the ghost mass is real and its fictitious propaga- the next-to-leading order term of Sg¯. Combining the de- tion does not involve a speed faster than the speed of scribed method with a dimensional analysis, we find that light. The term written in the first line of Eq. (2) leads 1 the renormalization of the self-energy operators in QEDA to a similar scenario, but with a different ghost mass 2 ¯ −2 should be handled by two local operators of dimension 6: m¯ gh = (¯gba) [for details, see Appendix A]. The described situation provides evidences that symmetry argu- Z 4 1 2 2 µλ ν ments are not enough to obtain a well-behaved quantum Sg¯2 = d x g¯ b¯a ∂µf (∂ν f λ) 2 theory of the fields involved in S = Sg¯ + Sg¯2 . However, (2) if the scattering matrix linked to its UV-completion is 1 2 2 ¯ µ ¯ + g¯ b¯ (∂µφ) (∂ φ) + ... . unitary, it is natural to expect that the one associated 2 φ with S, covering quantum processes at lower energies Various local operators sharing both the symmetry of ΛUV, is unitary too. This idea justifies the restric- the theory and the same dimensionality can be found. tion given above Eq. (3). We remark that these “ghost- However, it can be easily verified that all of them reduce providing” contributions are redundant operators which to those given in Eq. (2) through integrations by parts. can be dropped from the effective Lagrangian without changing observables [54–56]. Later on, this outcome is The Wilson parameters ¯ determine the strength of ba,φ used to remove them conveniently while the effective La- the contributions above. They might be determined by grangian acquires counterterms which allow for the can- a matching procedure provided the UV-completion of S g¯ cellation of the divergences associated with the loops that is known. Since we ignore the precise form of the latter, are calculated here [see below Eq. (7)]. they will be considered as arbitrary. We emphasize that the appearance of the square of the Wilson parameters Now, to carry out the renormalization program, the ¯ in Sg¯2 guarantees that–at least at tree level–the theory set of “bare” quantities (m, ¯ g,¯ m¯ gh, m¯ s, a¯µ, φ) should is causal [i.e. free of tachyons; see also discussion below]. be replaced by the respective renormalized parameters µ It is worth remarking that the action resulting from the (m, g, mgh, ms, aR, φR). In connection, each term in combination of Eqs. (1) and (2) cannot be considered as S = Sg¯ + Sg¯2 + ... has to be parametrized by a renor- an ordinary action containing higher-order derivatives. malization constant so that the action to be considered Within a quantum effective theory approach, higher- from now on is dimensional operators–like those exhibited in Eq. (2)– are suppressed by higher powers ofg ¯, so that their con- −1 S = SR + Sct + ..., sequences at low energies relative to ΛUV =g ¯ are Z tiny when compared with the effects resulting directly 4 1 2 1 2 1 2 2 SR = d x − fR + (∂φR) − m φR from Sg¯ [65, 66]. To all effects, they must be treated 4 2 2 as perturbations, otherwise a violation of the unitarity ) (4) takes place due to the occurrence of Pauli-Villars ghosts. 1 ˜ 1 2 1 2 + gφRfRfR + 2 (∂fR) − 2 φR φR , While this problem has been noted in previous studies 4 2mgh 2ms [see for instance [67, 68] and references therein], it is in- Z 4 1 2 1 2 structive to review it once again in the present context. Sct = d x − (Z3 − 1)fR + (Zφ − 1)(∂φR) To this end, we first note that the appearance of ghosts 4 2 emerges quite straightforwardly when investigating the axion Green function that results from combining the corresponding kinetic term and the second line of Eq. (2): 1 The origin of the electromagnetic theory that results from combining the Maxwell theory with the higher-derivative operator 1 1 G(p2) = − . (3) written in the first line of Eq. (2) dates back to the work of 2 2 2 2 P. Podolsky [69]. For further developments see Refs. [70–77]. p − m¯ p − m¯ s 4

1 2 2 1 2 − m (Zm − 1)φR + 2 (Zgh − 1) (∂fR) tions within the path integral [see Eq. (7)]: 2 2mgh µ µ µ 1 1 2 + ( − 1)gφ f˜ f − ( − 1)φ φ . φR → φR − φR, aR → aR − 2 aR, (8) Zg R R R 2 Zs R R 2m2 2m 4 2ms s gh

Here, f ≡ f µν = ∂µaν − ∂ν aµ is the renormalized elec- and by keeping the accuracy to the order g2. As these R R R R µ µ −1/2 transformations are linear in φ (x) and a (x), the asso- tromagnetic tensor with a (x) = a¯µ(x). Likewise, R R R Z3 ciated Jacobian leads to a decoupling between the cor- the renormalized axion field φ (x) and its bare counter- R responding Fadeev-Popov ghosts and the fundamental ¯ −1/2 ¯ part φ(x) are connected via φR(x) = Zφ φ(x), where fields. The equivalence theorem [54, 56] generalizes this Zφ is the corresponding wavefunction renormalization fact by dictating that no change is induced on the scatter- constant. Any other bare parameter relates to its re- ing matrix through shifts of this nature; still they modify spective renormalized quantity following multiplicative the initial parameters of the theory. Indeed, in our prob- renormalizations according to lem the redefinition of φR(x) leads to a kinetic term of 1 m2 2 s the form (1 − 2 )(∂φR) . The additional factor con- 2 ms Zm Zg m¯ = m , g¯ = g p , tained in this expression can be reabsorbed in Sct [see Zφ Z3 Zφ (5) Eq. (4)] by redefining the wavefunction renormalization s r m2 0 Z3 Zφ constant for the axion field Zφ − 2 → Z . Similarly, we ms φ m¯ gh = mgh , m¯ s = ms . 0 0 0 Zgh Zs redefine Zgh − Z3 → Zgh, Zs − Zφ → Zs, Z3 → Z3 and 0 Zm → Zm to reabsorb terms arising when transforming It is worth remarking that in Eq. (4) the shorthand Sct. Therefore, apart from this unobservable effect, the 2 µν ˜ ˜ µν 2 Pauli-Villars ghosts have no result other than to remove notations fR ≡ fRµν fR , fRfR ≡ fRµν fR ,(∂φR) ≡ µ (∂µφR)(∂ φR) have been used. When inserting the ex- the divergences that might arise from the respective one- pressions for the ghost masses [see below Eq. (3)] in those particle irreducible graphs. relations given in the second line of Eq. (5), we link the bare and renormalized Wilsonian parameters: B. Renormalized photon propagator in a modified minimal subtraction scheme ¯ 1 p ¯ Z3 p ba = ba Z3ZghZφ, bφ = bφ Zs, (6) Zg Zg We pursue our investigation by determining the modifi- where the connection betweeng ¯ and g has been used. (0) cation to the photon propagator Dαβ (x, x˜) due to quan- Noteworthy, within our second-order approximation in tum vacuum fluctuations of a pseudoscalar axion field the coupling constant, no modification on the renormal- φR(x). The use of Eq. (7) allows us to express the mod- ized axion-diphoton coupling g can be expected, i.e. Zg ified photon propagator Dαβ(x, x˜) as would not deviate from its classic tree-level value Zg = 1. 2 Hence, from now on, no distinction between the renor- 1 δ Z[j, j] Dαβ(x, x˜) = . (9) malized and physical coupling is needed. However, we i2 δj α(x)δj β(˜x) emphasize that its bare counterpartg ¯ is still subjected j,j=0 to a renormalization due to the wave functions renormal- We then expand Eq. (7) up to the order g2 and insert ization constants and . Z3 Zφ the resulting expression into the formula above. As a At the quantum level, the dynamical information of the consequence, the corrected photon propagator reads system described by Eq. (4) is rooted within the Green Z functions. They can be obtained from the generating (0) 4 4 (0) functional Dαβ(x, x˜) = Dαβ (x, x˜) + d yd y˜ Dαµ (x, y) (10) R 4 1 µ 2 µ µν (0) 4 R iS+i d x[− (∂µa ) +a jµ+φRj] × iΠ (y, y˜)D (˜y, x˜) + O(g ). DφRDaRe 2ζ R R νβ Z[j, j] = R iS , DφRDaRe Here Πµν (y, y˜) encompasses the expression for the un- (7) µ renormalized polarization tensor [see Fig. 1] as well as where j(x) and j (x) denote the external currents associ- counterterms that allow for the cancellation of the diver- ated with the axion φ (x) and the gauge field aµ (x). The R R gences associated with this loop. Analytically, it reads contribution in the exponent of Eq. (7) which is propor- 1 h i tional to the parameter ζ guarantees a covariant quan- µν 2 µατ νβσρ y˜ y (0) µ Π (y, y˜) = ig ∂σ∂ ∆F (˜y, y) tization of aR(x). At this point it turns out to be convenient to bring the renormalized Lagrangian in SR [see h y y˜ (0) i n 0 × ∂τ ∂ρ Dαβ (y, y˜) + Z3 − 1 + 2 (11) Eq. (4)] to a canonical form in which the terms linked mgh to the Pauli-Villars ghosts are dropped. This can be 0 o µν µ ν 4 achieved by performing the following local field redefini- × (Zgh − 1) g − ∂ ∂ δ (y − y˜), 5

with ∆(s) = m2s−p2s(1−s). Manifestly, the term associated with the factor −1 is singular as → 0. In contrast to QED, such a divergence cannot be reabsorbed fully in the wavefunction renormalization constant of the electro- 0 magnetic field Z3 by enforcing that the radiative correction should not alter the residue of the photon propagator FIG. 1: Feynman diagram depicting the axion-modified vac- at p2 = 0 [65, 83]. We solve this problem, by choosing uum polarization tensor. While the dashed line represents the the counterterms in the following form free axion propagator, the internal wavy line denotes the free (0) photon propagator Dµν (x, x˜). Here, the external wavy lines 2 2 g2m2 0 g m 0 gh (16) represent amputated photon legs. Z3 − 1 = lim , Zgh − 1 = lim . →0 16π2 →0 48π2 Notice that the ratio of scales mg acts like a dimension- where ∆(0)(x, x˜) = R d¯4p i eip(x−x˜), withd ¯4p ≡ F p2−m2+i0 less coupling constant. Thus, the one-loop renormalized d4p/(2π)4 refers to the unperturbed ALP propagator, polarization tensor in a modified minimal subtraction (0) R 4 −igαβ ip(x−x˜) whereas Dαβ (x, x˜) = d¯ p p2+i0 e denotes the pho- scheme (MS) scheme reads ton propagator in Feynman gauge [ζ = 1]. It is worth re- Πµν (p , p ) = ¯δ p2 µν − pµpν π (p2), marking that, in momentum space the polarization ten- MS 1 2 p1,p2 2g 2 2 MS 2 µν R 4 4 −ip1x µν ip2x˜ sor Π (p1, p2) = d xd x˜e Π (x, x˜)e reads 2 Z 1 (17) 2 g ∆(s) πMS(p ) = − 2 ds ∆(s) ln 2 . µν n 2 µταβ νσγρ 16π 0 µ Π (p1, p2) = ¯δp1,p2 ig gστ p2βp2ρK αγ " # Noteworthy, this expression satisfies the transversality p2 o (12) µν µν 0 2 0 2 µν µ ν condition p1µΠ (p1, p2) = Π (p1, p2)p2µ = 0. − Z3 − 1 − 2 (Zgh − 1) (p2 g − p2 p2 ) , MS MS mgh Some comments are in order. First, when the QED action is extended with those terms belonging to QEDA, where the shorthand notation ¯δ ≡ (2π)4δ4(p − p ) 0 p1,p2 1 2 the expression for Z3−1 found in Eq. (16) will be added to 2 has been introduced and the corresponding one-loop QED-expression [Z3 (QED)] . Z 4 This operation allows us to define the standard renormal- d¯ q qαqγ 1/2 K αγ = , (13) ized U(1)−charge as usual eR = Z˜ e¯, with the bare q2 [(q − p )2 − m2] 3 (1loop) 2 0 chargee ¯ and Z˜ 3 (1loop) = Z3 + Z3 (QED). Finally, taking which diverges quadratically as |q| → ∞. The regular- into account Eqs. (16) and (17), the Fourier transform of ization of Πµν (p1, p2) is then carried out by using a stan- Eq. (10) is, up to an unessential longitudinal term, dard Feynman parametrization [ 1 = R 1 ds ] and ab 0 [b+(a−b)s)]2 µν µν D (p1, p2) = ¯δp ,p D (p2), by continuing the loop integral to D = 4 − , → 0+ di- MS 1 2 MS −ig µν (18) mensions via the replacement Dµν (p) = 1 + π (p2) . MS p2 MS Z Z 4 D d¯ q . . . → (Cµ) d¯ q . . . , (14) This formula constitutes the starting point for further considerations. In the next section it will be used to 1 (γ−1) 1/2 where C = e 2 /(4π) and γ = 0.5772... is the Euler- establish the axion-modified Coulomb potential. Mascheroni constant. In this context, µ denotes a dimen- sionful parameter, i.e. the substracting point that follows when rescaling the renormalized axion-diphoton coupling C. Axion self-energy operator, renormalized mass g → gµ/2 in D−dimensions so that its mass dimension vs physical mass −1 is kept. Also, when going from four to D−dimensions, the Wilsonian parameters [see Eqs. (2) and (6)] rescale Our aim in this section is to determine the axion self- −/2 bi → biµ with i = a, φ while their dimensionless energy operator. Its associated Feynman diagram is de- feature is retained. picted in Fig. 2. This object encloses the way in which the Now, we integrate over q and Taylor expand the result- quantum vacuum fluctuations of the electromagnetic field ing expression in . As a consequence, Eq. (12) becomes correct the axion propagator. To show this analytically,

µν µν 2 µ ν 2 we expand the generating functional for the Green func- Π (p , p ) = ¯δ ( p − p p )π(p ), 2 1 2 p1,p2 g 2 2 2 2 tion [see Eq. (7)] up to ﬁrst order in g . Once this step 2 Z 1 2 g 2 ∆(s) π(p ) = 2 ds∆(s) − ln 2 16π 0 µ (15) 2 0 p 0 2 An explicit expression for Z3 (QED), in dimensional regulariza- − (Z3 − 1) + 2 (Zgh − 1) mgh tion, can be found in Eq. (19.37) of Ref. [65] 6

tor is given by

2 ΣMS(p1, p2) = ¯δp1,p2 ΣMS(p2),

2 2 Z 1 2 3g p ∆m→0(s) ΣMS(p ) = 2 ds ∆m→0(s) ln 2 , 32π 0 µ (22) 2 where ∆m→0(s) = −p s(1 − s) [see below Eq. (15)]. As we could have anticipated, this expression is independent of the renormalized axion mass. It is, perhaps, worth FIG. 2: Diagrammatic representation of the axion self- stressing that–in a MS−scheme–the square of the physi- energy operator mediated by quantum vacuum fluctuations cal mass m2 is the value of p2 for which the real part of the electromagnetic field. In contrast to Fig. 1, the exter- phy 3 nal dashed lines represent amputated ALPs legs, whereas the of the two-points irreducible function: (0) internal wavy lines represent photon propagators Dµν (x, x˜). 2 2 Γ(p1, p2) = ¯δp1,p2 p2 − mphy , (23) m2 = m2 − Σ (m2 ) has been carried out, the resulting expression is twice dif- phy MS phy ferentiated functionally with respect to the axion source vanishes. Whenever the subtracting parameter satis- 2 2 2 j(x) leading to fies µ mphy exp[−32π /(g mphy)], mphy ≈ m holds. Hence, the expression for the polarization tensor [see 2 2 1 δ Z[j, j] Eq. (17)] as a function of mphy would not differ from ∆F(x, x˜) = i2 δj(x)δj(˜x) the one given in terms of the renormalized mass. j,j=0 At this point we find it interesting to make a compar- Z (19) = ∆(0)(x, x˜) − d4yd4y˜ ∆(0)(x, y) ison between Eq. (22) and the polarization tensor given F F in Eq. (17). To this end, it is convenient to reexpress the (0) 4 latter as follows: × iΣ(y, y˜)∆F (˜y, x˜) + O(g ). pµpν Πµν (p , p ) = ¯δ (p2) µν − 2 2 , Here, Σ(y, y˜) comprises the expression of the unrenormal- MS 1 2 p1,p2 κMS g 2 p2 (24) ized axion self-energy operator as well as some possible counterterms. Explicitly, 2 2 2 κMS(p ) = p πMS(p ). i h i In this formula, κ (p2) represents the only nontrivial Σ(y, y˜) = − g2µντσαβργ ∂y∂y˜D(0)(y, y˜) MS 2 σ γ µα eigenvalue of the polarization tensor [84, 85], which–in h y y˜ (0) i n 0 the limit under consideration–turns out to be smaller × ∂τ ∂ρ Dβν (˜y, y) − Zφ − 1 (20) 2 than ΣMS(p ) by a factor −2/3. 2 Let us finally remark that, in addition to the axion- 0 2 0 o 4 + (Z − 1) + m (Z − 1) δ (y − y˜). diphoton interplay, axion self-coupling [86] as well as m2 s m s effective interactions with electron, proton and neutron might occur [16, 87–89]. In such a case further one-loop Next, we Fourier transform Σ(y, y˜) and regularize its contributions to the axion self-energy operator might divergent integral via dimensional regularization as made arise. However, these contributions depend on coupling in Sec. II B. However, in contrast to the case treated constants other than the one mediating the interaction there, the associated divergence at → 0 is fully reab- between an axion and two photons. sorbed here in a renormalization constant

2 2 0 g ms III. AXION-COULOMB POTENTIAL Zs − 1 = lim , (21) →0 32π2

0 A. Screening of the electric charge and ﬁnite whereas Zm does not deviate from its classic tree-level 0 renormalization: Setting the subtracting parameter value [Zm = 1]. This feature extends beyond the one- loop approximation because the axion-photon vertex prevents the proliferation of self-interacting terms for ALPs Hypothetical distortions of Coulomb’s law can always containing no derivatives [66]. Despite this, the bare be determined through the temporal component of the massm ¯ still is subject to a ﬁnite renormalization due to the axion wavefunction renormalization constant [see

Eq. (5)], which does not deviate from the classical value 3 0 This expression can be established from the identity Zφ = 1. Keeping in mind all these details, we ﬁnd that–in R 4 d¯ q Γ(p1, q)∆F(q, p2) = −i¯δp1,p2 , where ∆F(q, p2) stands for a MS scheme–the renormalized axion self-energy opera- the Fourier transform of Eq. (19). 7

2 2

1 1 LD LD 2 2 Π Π 64 64 H H 4 2 0 0 m m 2 2 g g @ @ R R S Π -1 -1

-2 -2 -4 -2 0 2 4 6 -4 -2 0 2 4 6 p 2 m 2 p 2 m 2

2 FIG. 3: In the left panel, the behavior of the form factor of the vacuum polarization tensor πR(p ) [see Eq. (28)] is shown 2 2 2 as a function of p /m . The right panel depicts the corresponding dependence of the axion self-energy operator ΣR(p ) [see Eq. (29)]. The respective real and imaginary parts are displayed in green and red. electromagnetic four-potential [hereafter, to simpify no- |p| → 0. Since the corresponding length scale |x| → ∞, µ µ 2 tation, a (x) must be understood as aR(x)] escr(|x| → ∞) can be identified with the electrostatic charge that is measured in experiments at low energies. Z α αβ 4 This natural renormalization condition [π (000) = 0] a (x) = −i D (x, x˜)jβ(˜x)d x,˜ MS MS holds for the subtracting parameter µ = m exp[−1/4]. (25) Z The renormalized polarization tensor then reads Dαβ (x, x˜) = d¯4p d¯4p eip1xDαβ (p , p )e−ip2x˜ MS 1 2 MS 1 2 αβ h 2 αβ α βi 2 ΠR (p1, p2) = ¯δp1,p2 p2g − p2 p2 πR(p2), where Dαβ (p , p ) is given in Eq. (18). Here jβ(˜x) = MS 1 2 g2m2 1 p2 β 3 2 δ δ (˜x) denotes the four-current density of a pointlike πR(p ) = − 1 − q 0 64π2 3 m2 (28) static charge q placed at the origin x˜ = 0 of our reference 2 1 frame. Particularizing the expression above for α = 0, g Z ∆(s) − 2 ds ∆(s) ln 2 , we end up with 16π 0 m Z 3 d¯ p 2 −ipp·x whereas the axion self-energy operator [see Eq. (22)] re- a0(x) = q 1 + π (p ) e . (26) p2 MS duces to 2 At this point it is worth emphasizing that, while the ex- ΣR(p1, p2) = ¯δp1,p2 ΣR(p2), 2 pression for πMS(p ) [see Eq. (17)] is finite, its depen- (29) g2p4 p2 7 dence on the subtracting point µ introduces an arbitrari- Σ (p2) = − ln − − . ness. To remove it, we consider the expression of the R 64π2 m2 6 electrostatic energy between two electrons in momentum space (p). It can be established easily by taking the The results obtained so far are summarized in Fig. 3, U 2 integrand above, with → e . After multiplying the which displays the behavior of πR(p ) [left panel] and q R 2 2 2 ΣR(p ) [right panel] as a function of p /m . In both resulting expression by eR, we find panels the respective real and imaginary parts are shown 2 escr(p) in green and red, manifesting by themselves the non- U(p) = −eRa0(p) = − , (27) p2 Hermitian feature of the polarization tensor and the axion self-energy operator. To support this numerical eval- 2 2 2 where the screened charge escr(p) = eR[1 + πMS(p )] has uation from an analytic viewpoint, we first determine an been defined. exact expression for the imaginary parts. To this end, we As we still have freedom of performing finite renor- restore the i0−prescription [m2 → m2 − i0] in Eq. (28) 2 malizations, we can demand that πMS(p ) vanishes as and apply the formula ln(−A − i0) = ln(|A|) − iπ with 8

A > 0. Explicitly, For evaluating δa(x) explicitly, it is convenient to inte- 2 2 2 3 grate by parts in Eq. (28) and use an equivalent repre- g p m 2 Im[π (p2)] = − 1 − Θ(p2 − m2), sentation of πR(p ) instead: R 96π p2 (30) 2 2 Z 1 3 2 g p 3 2 ds s 2 2 2 πR(p ) = 1 − p . (34) 2 g (p ) 2 2 2 2 2 Im[Σ (p )] = Θ(p − m ), 144π 2 0 m − p (1 − s) R 64π where Θ(x) denotes the unit step function. We note Observe that, for applying this formula in Eq. (33), p0 that for an on-shell photon [p2 = 0], the imaginary part must be set to zero. Taking this into account, the integral of Πµν vanishes, which implies–according to the optical over the momentum can be carried out with relative ease. R 1 theorem–that the emission of an ALP from a photon ac- After developing the change of variable u = 1/(1 − s) /2, companied by the radiation of another photon is forbid- the axion-modified potential turns out to be den. This fact agrees with the outcome resulting from an q n g2m2 Z ∞ du analysis of the corresponding energy-momentum balance. x a0(x) = 1 + 2 5 The expression for the imaginary part of the one-loop 4π|x| 48π 1 u (35) 2 3 o self-energy operator Im[ΣR(p )] [second line in Eq. (30)] × u2 − 1 e−m|x|u . coincides with the result found previously in Ref. [90] through a direct application of the cutting rules. Its on- We remark that spurious contributions containing Dirac shell evaluation [p2 = m2] should allow us to determine deltas δ3(x) have been ignored since the theory is pre- the total rate of the decay process φ → γγ via the relation dictive only for distances |x| g. Although the integral Im[Σ (m2)] = mΓ , provided the optical theorem is R φ→γγ involved in this formula can be calculated analytically, valid. With accuracy to first order in g2, this formula is we will keep it as it stands. Mainly, because it will allow indeed verified because an expression for Γ –relying φ→γγ us to establish compact expressions for the energy shifts on the corresponding S−matrix amplitude–can be in- that atomic transitions undergo. ferred directly from the corresponding neutral pion decay Asymptotic formulas for the modified potential can be rate [see for instance Eq. (19.119) in Ref. [91]]. This fact extracted from Eq. (35) without much efforts. For in- evidences that the unitarity is preserved, at least within stance, at distances larger than the Compton wavelength the second order approximation in the axion-diphoton λ ∼ m−1 of the axion, i.e. for |x| λ, the region u ∼ 1 coupling g. dominates in the integral involved in Eq. (35), and the Further asymptotic expressions of Eq. (28) are eluci- 2 2 2 axion-modified Coulomb potential approaches to dated. For m p , we find that πR(p ) approaches to q g2m2 e−m|x| g2p2 3 p2 a0(|x| λ) ≈ 1 + . (36) π (p2 m2) ≈ 1 − . (31) 4π|x| π2 (m|x|)4 R 144π2 8 m2 However, at short distances [|x| λ], the main contri- Conversely, for |p2| m2, its asymptotic behavior turns bution to the integral in Eq. (35) results from the re- out to be dominated by the following function −1 gion 1 6 u 6 (m|x|) , and the integrand can be ap- 2 2 2 g p |p | 2 7 proached by its most slowly decreasing function in u, π | 2 2 ≈ ln − iπΘ(p ) − . (32) R |p |m 96π2 m2 6 which is ∼ ue−m|x|u. Consequently, Notably, when p is timelike [p2 > 0], the expression above q n g2m2 1 o 2 2 a0(|x| λ) ≈ 1 + . (37) gets an imaginary contribution Im πR(p m ) ≈ 4π|x| 48π2 (m|x|)2 −g2p2/(96π), which coincides with the leading order term of Eq. (30) when the condition p2 m2 is con- This expression is independent of the axion mass. Ob- sidered. serve that the√ distance |x| must satisfy the condition |x| g/(4 3π), otherwise our perturbative approach breaks down. Incidentally,√ the corresponding energy B. Electrostatic potential and modified Lamb-shift −1 scale µp ∼ |x| ∼ 4 3π/g coincides–up to a numerical factor of the order√ of one–with the Landau pole linked to The first contribution in Eq. (26) can be integrated QEDA: µL ≈ 4 6π/g [for details see Ref. [66]]. straightforwardly, leading to the unperturbed Coulomb The distortion of the Coulomb potential due to ALPs potential aC (x) = q/(4π|x|). The second one, on the [see Eq. (35)] allows us to infer the induced modifica- 2 other hand, will be computed by using πR(p ) rather than tions in the spectrum of a nonrelativistic hydrogenlike 2 πMS(p ). With all these details in mind we write atom. Since so far no large deviations from the standard QED predictions have been observed, we will as- (x) = (x) + δ (x), a0 aC a sume that these energy shifts are very small and, conse- Z 3 (33) d¯ p 2 −ipp·x quently, apply standard time-independent perturbation δa(x) = q πR(p )e . p2 theory. When considering a first order approximation, 9

10-1 LEP

-2 10 Γ®inv. PVLAS + - 10-3 e e ®Γ+inv.

10-4 Beam dump D -5 1 10 OSQAR -

-6

GeV 10 KSVZ

FIG. 4: Pictorial correction to the Coulomb potential due to @ quantum vacuum ﬂuctuations of axion and electromagnetic g 10-7 ﬁelds. Leaving aside the electron (proton) legs [external lines DFSZ with arrows in the right (left)], the remaining pieces of this 10-8 diagram are described in Fig. 1. C A S T + S U M I C O S N 1 9 8 7 a

10-9 H B s t a r s the energy shift δε follows by averaging the correction to -10 Cosmology the electrostatic energy δ (x) = −e δ (x) [see Eq. (33) 10 U R a -8 -6 -4 -2 0 2 4 6 8 and the Feynman diagram depicted in Fig. 4] over the 10 10 10 10 10 10 10 10 10 eV 0th order wavefunctions |ψn,`,ji. Explicitly, m (1) FIG. 5: Summary of exclusion areas for a pseudoscalar ALP δεn,`,j = hψn,`,j | δU(x) | ψn,`,ji. (38) coupled to two photons. Compilation@ D adapted from Refs. [15– We wish to exploit this formula to predict a plausible 19]. The picture includes [inclined yellow band] the predictions of the axion models |E/N −1.95| = 0.07−7 (the notation axion Lamb-shift for the 2s1/2−2p1/2 transition in atomic hydrogen. For this case, Eq. (38) leads to of this formula is in accordance with Ref. [92]). Colored in or- ange and black appear the regions ruled out by particle decay (1) (1) experiments. While the portion discarded by investigating δε = δε2s − δε2p , 1/2 1/2 the energy loss in the horizontal branch (HB) stars are shown Z ∞ (39) in blue, the excluded area resulting from the solar monitoring 2 2 2 = dr r δU(r) R2s(r) − R2p(r) , of a plausible ALP flux (CAST+SUMICO) has been added 0 in green. In purple the portion discarded by measuring the where r ≡ |x| and Rn` stands for a radial hydrogen wave duration of the neutrino signal of the supernova SN1987A is function. In particular, depicted, whereas the dark gray area results from cosmolog- ical studies. The excluded area in dark turquoise has been 1 1 r − r established from beam dump experiments. Besides, the light R (r) = √ 1 − e 2aB , 2s 3/2 gray zone has been excluded from electron-positron collider 2 a 2aB B (40) (LEP) investigations. Finally, the colored sectors in olive and 1 r r − 2a R2p(r) = √ e B . red show the exclusion regions corresponding to OSCAR and 2 6 5/2 aB PVLAS collaborations, respectively. −1 Here aB = (αme) is the Bohr radius with α = 1/137 the fine structure constant and me = 0.511 MeV the undiscarded values for g turn out to be much smaller than electron mass. any characteristic atomic scale. With all these details in Observe that the integration in Eq. (39) covers the mind we integrate over r and arrive at region [0, ∞). However, since QEDA does not provide a precise information about the form of the axion-Coulomb αg2 Z ∞ du [u2 − 1]3 4 −gmu (41) potential for distances smaller than g, the integral over r δε ≈ − 2 m aB 3 4 e . R ∞ R g R ∞ 96π 1 u [1 + uaBm] must be splitted 0 dr . . . = 0 dr . . .+ g dr . . .. In the following we will assume that the contribution from the Let us study the asymptotes of this expression. We first outer region [g, ∞) dominates over the inner region [0, g], consider the case in which aB λ. Under this condition, 4 4 which we ignore. As we will see very shortly, the yet the term of the integrand [1 + uaBm] is dominated by 4 4 4 u aBm . The integral resulting from this approximation can be computed exactly. After a Taylor expansion in mg 1, we find the compact expression 4 Strictly speaking, in accordance with the treatment applied in Sec. II [read also below Eq. (35)], the splitting of the integral αg2 11 δε ≈ ln (gm) + γ + . (42) should be carried out at a certain point d fulfilling the condi- 96π2a3 12 tion d g. However, in order to avoid uncertainties stemming B from this additional parameter, we set d = g. Our corresponding results should be considered as order-of-magnitude estimates, As in Eq. (14), γ = 0.5772 ... refers to the Euler- accordingly. Mascheroni constant. In the opposite case aB λ, the 10

FIG. 6: Energy-shift induced by hypothetical quantum vacuum fluctuations of axionlike fields on the 2s1/2 − 2p1/2(left panel) and the 3p1/2 − 3d1/2(right panel) atomic transitions in Hµ. The ALPs parameters used in these numerical evaluations are still undiscarded. While the results exhibited in the left panel rely on Eq. (41), the outcomes depicted in the right panel follow from Eq. (45). Observe that the energy shift is plotted in the form of log10(−δε[meV]). integrand in Eq. (41) turns out to be a function that de- out to be determined by α and the pion decay constant creases monotonically with the growing of u. It is then fπ ≈ 92 MeV [65]. Observe that the corresponding value justified to approach it through its most slowly decreas- of g ≈ 4.97 × 10−3 fm is two orders of magnitude smaller 3 −mgu 4 ing part which is ∼ u e /(1 + uaBm) . As a result, than the proton radius rproton ≈ 0.876 fm [38]. The cor- the corresponding integral can be computed analytically responding correction to the 2s1/2 − 2p1/2 Lamb-shift in by using (3.353.1) in Ref. [95]. In the limit of mg 1, hydrogen atoms is δε = −1.07 × 10−12 meV. This value it allows us to approach turns out to be five orders of magnitude smaller than −7 the experimental uncertainty |δε2σ| = 2 × 10 meV, 2 αg g 11 established at 2σ confidence level [35, 37]. If the previ- δε ≈ 2 3 ln + γ + . (43) 96π aB aB 6 ous evaluation is carried out by considering a muonic hydrogen instead [me → mr], we find that the cor- Notice that, the equation above is a good approxima- rection to the energy due to the neutral pion field is tion whenever the condition aB g holds. Moreover, δε = −6.81 × 10−6 meV. Since this is five orders of although Eqs. (41)-(43) apply for ordinary atomic hy- magnitude smaller than the existing discrepancy between drogen, they can be adapted conveniently for studying the experimental measurement and the theoretical pre- the same transition in other hydrogenlike atoms. When diction δε = 0.31 meV [38, 42, 45], virtual neutral pions hydrogenlike ions with atomic number Z > 1 are consid- are excluded as possible explanation for the proton radius ered, for instance, the correction to the Lamb-shift will puzzle. be given by Eqs. (41)-(43), scaled by the factor Z and aB → aB/Z. If a muonic hydrogen atom is investigated Now, we wish to investigate whether the Lamb-shift in- instead, a replacement of the electron mass me by the duced by quantum vacuum fluctuations of axionlike fields reduced mass of the system mr ≈ 186 me would be re- might cure this anomaly. To this end, we will evalu- quired. ate Eq. (41) considering a mass region in which reliable results can be extracted. Within a pure spectroscopy context, this occurs for ALP wavelengths smaller or of C. Precision spectroscopy in H and the proton µ the order of the Bohr radius of Hµ, i.e. λ . aµ with radius anomaly aµ ≈ 285 fm, otherwise interactions of other nature must be included. Correspondingly, we can formally ex- Before continuing with the physics of virtual ALPs, plore ALP masses fulfilling the condition 1 MeV . m. we will estimate the contribution to the Lamb-shift by a However, in the range 1 MeV . m . 100 MeV the meson whose interaction with the electromagnetic field axion-diphoton coupling g has been constrained severely resembles the one exhibited by ALPs [see Eq. (1)], i.e. from various results, including those dealing with elec- the neutral pion π0. We should however emphasize that tron beam fixed-target setups [see compilation of bounds its effect should be understood as a consequence of the in Refs. [93, 94]]. Conversely, the sensitivities in experi- quantum vacuum fluctuations of its constituent quark ments where ALPs masses 100 MeV . m . 10 GeV are 0 fields. When thinking of the axion as π , m → mπ = probed turn out to be much weaker [white sector in the 135 MeV and the coupling constant g → α/(πfπ) turns right hand side of Fig. 5]. A recent investigation based on 11

electron-positron colliders has constrained g to lie below highest energy-shift linked to the 3p1/2 − 3d1/2 transition −2 −1 −2 −1 g < 10 GeV [94]. in Hµ arises from the combination of g ∼ 10 GeV A numerical assessment of the axion-modified Lamb- and m = 10−2 GeV. In such a case δε ∼ −10−11 meV, shift has been carried out by considering this yet undis- which is five orders of magnitude smaller than the out- carded region. The outcome of this evaluation is summa- come associated with the axion 2s1/2 −2p1/2 Lamb-shift. rized in the left panel of Fig. 6. Observe that the energy It is worth emphasizing that, while the uncertainty intro- shift has been plotted in the form of log10(−δε[meV]). duced by the use of spherically symmetric orbitals could The highest value achieved for δε ∼ −10−6 meV cor- be circumvented as described above, the one linked to responds to g ∼ 10−2 GeV−1 and m ∼ 10−2 GeV. the physics at distances shorter than g remains. Toward higher axion masses m ∼ 10 GeV and lower axion-diphoton couplings g ∼ 10−6 GeV−1 the correction to the Lamb-shift tends to decrease significantly [δε ∼ D. Sensitivity to ALPs in high-precision hydrogen −10−14 meV]. Both estimates coincide with the values spectroscopy resulting from Eq. (42). The smallness of these outcomes as compared with the aforementioned discrepancy rules In this section we want to investigate whether the out the corresponding virtual ALPs as candidates to ex- current sensitivity in atomic hydrogen can improve the plain the Hµ anomaly. As we have anticipated above existing bounds on the axion parameter space. To do −7 −4 Eq. (41), the chosen values for 10 fm . g . 10 fm this we will analyse the respective energy-shift in both are much smaller than rproton ≈ 0.876 fm. 2s1/2 − 2p1/2 and 1s1/2 − 2s1/2 transitions. An expres- Clearly, the previous statements cannot be considered R ∞ 2 2 2 sion for the latter δε = 0 dr r δU(r) R1s(r) − R2s(r) conclusive as our estimation undergoes theoretical un- can be easily determined by taking into account the for- certainties arising from both the internal limitation of mula for R2s(r) in Eq. (40) and the radial part of the QEDA at short distances as well as the finite proton size. 3/2 1s1 -state: R (r) = 2 exp[−r/a ]/a . Explicitly, The latter being closely related to the fact that the s- /2 1s B B states penetrate the nucleus deeply even for the chosen αg2m2 Z ∞ du n 1 δε = − [u2 − 1]3e−gmu g. This last problem can be relaxed if transitions between 2 5 1 2 48π aB 1 u [1 + 2 uaBm] excited states with nonzero angular momentum are considered instead. Although the problem of their relatively 1 1 2 3 o − 2 1 − + 2 . short lifetimes constitutes a major issue for their exper- 2 [1 + uaBm] 1 + uaBm 2[1 + uaBm] imental investigation, their measurements seem to be a (46) priori a reliable way to have a cleaner picture of whether While the left panel in Fig. 7 shows the energy shift for a certain ALP is the cause of the aforementioned discrep- 2s − 2p [see Eq. (41)], the one in the right depicts ancy or not. Inspired by these arguments, we consider as 1/2 1/2 the result associated with the 1s1/2 − 2s1/2 transition an example the 3p1 − 3d1 transition. In this case, the /2 /2 [see Eq. (46)]. Both evaluations have been carried out by required wave functions are considering the region of the coupling 10−6 GeV−1 < g < √ −2 −1 10 GeV . In contrast to Hµ, reliable predictions from 4 2 1 r r − r R (r) = 1 − e 3aB , high-precision spectroscopy in ordinary hydrogen require 3p 3/2 9 (3√aB) aB 6aB to deal with ALPs masses m 10−5 GeV, corresponding 2 (44) & 2 2 1 r − r R (r) = √ e 3aB . to wavelengths λ . aB. The highest mass shown in both 3d 3/2 27 5 (3aB) aB panels [m = 10 GeV] has been set in order to preserve the perturbative condition mg 1. An adequate replacement of the radial wave functions When comparing the energy shifts resulting from each in Eq. (39) by those above allows us to determine the panel in Fig. 7 with the corresponding experimental un- corresponding modification of the transition energy: −7 certanities [|δε2σ| = 2 × 10 meV for 2s1/2 − 2p1/2 and −6 αg2m2 Z ∞ du [u2 − 1]3 |δε1σ| = 1 × 10 meV for 1s1/2 − 2s1/2 transition [37]] δε ≈ − we conclude that, in order to improve the current bounds 2 5 3 4 108π aB 1 u [1 + uaBm] 2 (45) on the axion parameter space, an enhancement in sensi- 2 1 tivity of at least five orders of magnitude is required. It × 1 − + . 3 3 2 is worth remarking that this sensitivity gap also man- 1 + 2 uaBm [1 + 2 uaBm] ifests in other high-precision experiments searching for In the limit aB λ, the expression above is well ap- potential deviations of the Coulomb’s law as those of 2 2 2 5 proached by δε ≈ −αg /(4374π m aB). The expressions Cavendish-type. For further details we refer the reader associated with Hµ can be read off from the previous one to Appendix B. This lack of sufficient sensitivity in the by replacing aB → aB/(186). Taking as a reference the context of ALPs is significant when taking into account undiscarded region used previously, 100 MeV . m . that these setups have allowed for constraining severely 10 GeV with g < 10−2 GeV−1, Eq. (45) has been evalu- the parameter spaces of other weakly interacting sub-eV ated. The result of this assessment is shown in the right particles, including paraphotons and minicharged par- panel of Fig. 6. As in the 2s1/2 − 2p1/2 transition, the ticles. However, we should emphazise that–in contrast 12

FIG. 7: Projected sensitivities for the 2s1/2 − 2p1/2 [see Eq. (41)] and 1s1/2 − 2s1/2 [see Eq. (46)] atomic transitions in hydrogen are depicted in the left and the right panel, respectively. In contrast to Fig. 6, the region for the axion mass m is wider, covering from 10−5 GeV to 10 GeV. As previously, 10−6 GeV−1 < g < 102 GeV−1 and the energy-shift is given in the form of log10(−δε[meV]). to our investigation–these particle candidates have been straints on the axion parameter space. This fact con- treated within renormalizable frameworks and, thus, the trasts with analogous outcomes linked to scenarios con- bounds have been established on dimensionless coupling taining minicharged particles and hidden photon fields, constants. Likewise, we have already indicated below in which both precision techniques have turned out to Eq. (16) that in the axion theory the quantity playing be particularly valuable [34–37]. The loss of sensitivity the corresponding role combines two unknown parame- within the axion context is conceptually rooted in the ters ∼ gm. Hence, the axion mass m suppresses the nonrenormalizable character of QEDA and manifests–at limits that can be inferred for g. the level of the modified Coulomb potential Eq. (35)– through the dimensionless factor ∼ gm. This ratio of scales accomplishing somewhat a role similar to the cou- IV. CONCLUSION pling strengths of the photon-paraphoton mixing χ and the parameter in the minicharged particles scenario. To Within the effective framework of axion quantum elec- a certain extent the described problem justifies the exist- trodynamics, terms beyond the minimal coupling of two ing demand for new laboratory-based routes looking for photons to a neutral pseudoscalar field have been used ALPs [13, 14] by using strong electromagnetic fields [15– to renormalize the polarization tensor and the axion self- 18], e.g., those offered by high-intensity lasers [96–106]. energy operator. The former outcome was used to es- Let us finally remark that the expression for µν tablish the photon propagator distorted by the quantum ΠR (p1, p2) [see Eq. (28)] constitutes an essential piece for vacuum fluctuations of axionlike fields, a piece essential a more general class of polarization tensors which result for determining the modification of the Coulomb poten- when external electromagnetic fields polarize the vacuum tial induced by both virtual photons and ALPs. This [85, 96]. result allowed us to evaluate the way in which atomic spectra could change. Particular attention has been paid Acknowledgments to the 2s1/2 − 2p1/2 transition in hydrogenlike atoms as it might constitute the most natural way of verifying our predictions experimentally. Likewise, this sort of axion- The authors thank A. B. Voitkiv and A. E. Shabad modified Lamb-shift has been considered in attempting for useful discussions. S. Villalba-Chávez and C. Müller to explain the proton radius anomaly in muonic hydro- gratefully acknowledge funding by the German Research gen. By contrasting the experimental result with our Foundation (DFG) under Grant No. MU 3149/5-1. theoretical prediction, it was found that–up to the uncertainties caused by the nature of the transition and the internal limitations of axion-electrodynamics–ALPs can Appendix A: Particle-ghost content of the gauge be excluded as plausible candidates for solving the afore- sector and an alternative four-fields formulation of mentioned problem. axion-electrodynamics Our investigation has revealed explicitly that neither atomic spectroscopy nor experiments of Cavendish-type As mentioned in Sec. II A, the photon sector also con- allow us to infer bounds that improve the existing contains Pauli-Villars ghosts. In order to show this, let 13 us consider the corresponding Green function resulting Hence, when integrating out both Φ(x) and Aλ(x) clas- µ from a covariant quantization ofa ¯ (x) via a path inte- sically, i.e. by removing them from Seff using their gral representation. When fixing the gauge via Lgauge = equations of motion, we reproduce the action of axion- 1 2¯2 µ 2 electrodynamics extended by terms proportional tog ¯2, − 2 [(1 +g ¯ ba )∂µa¯ ] it turns out to be [78, 79]: i.e. Seff → S = Sg¯ + Sg¯2 [see Eqs. (1) and (2)]. " # Observe that, as a consequence of the shifta ¯ → a + 2 1 1 A Gαβ(p ) = − + gαβ, (A1) ¯ p2 p2 − m¯ 2 and φ → φ − Φ, the functional action in Eq. (A2) can be gh written as 2 ¯ −2 wherem ¯ = (¯gba) is the corresponding “bare” ghost Z gh 4 1 µν 1 µν 1 2 2 mass. Here, a longitudinal contribution ∼ p p has been Seff = d x − fµν f + Fµν F − m¯ A α β 4 4 2 gh ignored on the grounds that, if the photons couple to µ µ 1 µ 1 2 2 1 µ a conserved current j (x), i.e. pµj (p) = 0, a term of + ∂µφ∂ φ − m¯ (φ − Φ) − ∂µΦ∂ Φ this nature does not contribute to the S-matrix elements. 2 2 2 Manifestly, the photon Green function in Eq. (A1) resem- 1 2 2 1 h ˜ µν ˜ µν + m¯ Φ + g¯(φ − Φ) fµν f + 2Fµν f bles Eq. (3). However, the particle-ghost content linked 2 s 4 to this expression is somewhat blurred owing to the pres- ˜ µν io +Fµν F . ence of the metric tensor gαβ. To highlight the emergence of the Pauli-Villars ghost–leaving aside those unphysical Clearly, the first term contained in this formula is the states linked to the quantization procedure that even- Maxwell Lagrangian, while the combination of the two tually must cancel each other–we will follow a method remaining contributions in the first line somewhat looks that has been used previously within the context of quan- like the Proca Lagrangian, with the exception that the tum gravity [68, 80, 81].5 Rather than dealing with the sign of its kinetic term is not the usual one. Likewise, expression above directly, one introduces the saturated the kinetic portion linked to the Φ(x) field manifests an Green function (p2) = jαG (p2)jβ and investigates G αβ opposite sign to the corresponding contribution of φ(x). its residues at each pole: p2 = 0 and p2 =m ¯ 2 . As for gh Owing to the described feature the associated Hamilto- 2 any m > 0 the relation j |p2=m2 < 0 holds–see proof of nian is not positive defined leading–upon quantization–to Lemma 1 in Ref. [68]–a physical particle is linked to a the absence of a ground state. The particle-ghost content 2 nonnegative residue of G (p ), whereas a ghost emerges of the theory is elucidated in this alternative formulation when the contrary occurs. For the case under considera- to the requirement that free fields of particles (ghosts) tion then follows that Res (p2) > 0 (photon) and G p2=0 have positive (negative) energy. 2 Res G (p ) 2 2 < 0 (ghost). p =m ¯ gh Noteworthy, the decompositions of the axion and photon Green functions [see Eqs. (3) and (A1)] suggest that Appendix B: Sensitivity to ALPs in experiments of the effects of the higher-dimensional operators can be Cavendish-type formulated in terms of auxiliary–fictitious–fields. In this context, the action of interest reads Precision tests of Coulomb’s law via Cavendish-type Z experiments have severely constricted the parameter 4 1 µν 1 ¯ µ ¯ 1 2 ¯2 Seff = d x − fµν f + ∂µφ∂ φ − m¯ φ space of hidden photons in the µeV mass regime [34, 107]. 4 2 2 Today, these setups also provide the best laboratory ¯ 1 2 2 1 2 2 1 µν bounds on mini-charged particles in the sub−µeV range − Φ φ + m¯ Φ − m¯ A + fµν F (A2) 2 s 2 gh 2 [36]. Here, we want to estimate the sensitivity of this 1 type of experiments in the context of ALPs. To this end, + g¯φ¯f˜ f µν , 4 µν we consider a setup containing two concentric spheres: an outer charged conducting sphere–characterized by 2 ¯ −2 a radius b–and an uncharged conducting inner sphere where Fµν = ∂µAν −∂ν Aµ andm ¯ s = (¯gbφ) . We remark that the equations of motion for the auxiliary fields are with a radius a. Only if the electrostatic potential fol- exact lows the r−1 law, the potential difference between the spheres vanishes and the cavity is free of electromag- 1 ¯ 1 µ netic field. However, deviations from the Coulomb po- Φ = 2 φ, Aλ = − 2 ∂µf λ. (A3) m¯ s m¯ gh tential like those induced by loop corrections [compare Eq. (35)] could lead to a nontrivial relative voltage difference γab = |[u(Q , b, b) − u(Q , a, b)]/u(Q , b, b)| that can be detected. This observable depends on the potential of 5 We precise that higher-derivate operators in combination with the charged sphere evaluated on its surface u(Q , b, b) as nonlocal terms emerge in many other interesting theoretical sce- well as on the surface of the inner sphere ( , a, b). narios, e.g. in construction of effective quantum field theories u Q accounting for quantum conformal and chiral anomalies [see e.g. In general, the electrostatic potential of a sphere with [82] and references therein]. radius b and charge Q evaluated at a distance r from its 14 center has the form four concentric icosahedrons [110]. For obtaining first es- Q timates, they may be treated approximately as four con- u(Q , r, b) = [f(r + b) − f(|r − b|)], centric spheres. In contrast to the setup of Plimpton and 2br (B1) Lawton, here a very high voltage is applied between the R r outer two spheres with radii d = 127 cm and c = 94.7 cm. f(r) = sA0(s)ds, 0 The voltage difference is measured between the two in- where A0(s) is an arbitrary potential in which the charge ternal ones, with radii b = 94 cm and a = 60 cm, which of the pointlike particle must be set to unity [108]. are uncharged. This setup allows us to infer bounds for Now, to determine γab resulting from QEDA we insert the ALPs parameters via the ratio between the voltage the axion-modified Coulomb potential a0(s, q = 1) [see differences: Eq. (35)] into the expression above. Notice that, similarly u(Q , c, b) − u(Q , d, b) − u(Q , c, a) + u(Q , d, a) to the case analyzed in Sec. III B, the integral over s must γ = . R r R g R r abcd 2 ( , c, d) − ( , d, d) − ( , c, c) be split into two parts: 0 ds . . . = 0 ds . . . + g ds . . .. u Q u Q u Q Ignoring the contribution coming from the region [0, g] (B6) we obtain We insert Eq. (35) into (B1) and evaluate the resulting formula in the various parameters contained in Eq. (B6). g2m Z ∞ du e−2bmu − e−gmu γ ≈ [u2 − 1]3 As a consequence, we end up with ab 2 6 96π 1 u b (B2) 2 Z ∞ 3 g mc 1 −mdu 2 −bmu 4 2 γ = du 1 − e + e sinh(amu) + O(g m ) . abcd 48π2δ u2 a 1 (B7) d sinh(mbu) sinh(mau) The integral that remains in this formula can be calcu- × 1 − e−mδu − , c b a lated analytically by using (3.351.4) in Ref. [95]. Since both b and a are macroscopic quantities, the conditions where δ ≡ c − d and terms of the order of ∼ g4m2 have a, b, b − a g hold and we can approximate the expres- been disregarded. Notice that, in contrast to Eq. (B2), sion above by the integrand above lacks terms involving ∼ e−mgu. This ∞ g2m Z du could be anticipated because the numerator of γabcd [see γ ≈ [u2 − 1]3e−gmu . (B3) ab 2 6 Eq. (B6)] does not contain a potential evaluated at the 96π b 1 u surface of the spheres [compare with γab given above Notice that Eq. (B3) is independent of the radius a of Eq. (B1)]. Consequently, when the condition md = the inner sphere. When considering the limit mg 1 we d/λ 1 is satisfied, the asymptotic expression for γabcd obtain becomes independent of the axion mass and quadratic in g g: γab ≈ 2 , (B4) 96π b 2 g c which does not depend on the axion mass m either. γabcd ≈ 96π2 b(c − d)(d − b) With γ to our disposal, we can proceed to esti- ab (B8) mate the sensitivity of this setup in the search for ALPs. b(d − b) d(d − b) db(d − b) × 1 − − + . For such a purpose, we use the benchmark parameters a(d − a) c(c − b) ac(c − a) of the experiment performed by Plimpton and Lawton Since this formula applies for axion wavelengths larger [a = 38 cm, b = 46 cm] in which a margin for γab exists, −10 than the typical length scale of the experiment, the out- provided it lies below |γab| . 3 × 10 [109]. By making use of Eq. (B4) we find comes resulting from it can be considered reliable as long as the interactions between ALPs and plausible 7 −1 g . 6.7 × 10 GeV . (B5) fields/matter existing outside of the external icosahe- dron are negligible. Next, the aforementioned experi- We emphasize that–as a consequence of the perturba- ment achieves a precision |γ | 2 × 10−16 [36, 110]. tive condition [see below Eq. (B3)]–this result applies abcd . Combining this value with Eq. (B8) we constraint g to for m 15 eV. Besides, it is trustworthy for axion lie below wavelengths smaller than the typical length scale of the spheres ∼ 0.1 m, i.e., for axion masses m 10−6 eV. We 7 −1 −7 g . 9.8 × 10 GeV for m 10 eV. (B9) note that the constraint in Eq. (B5) for the mass region 10−6 eV m 15 eV has already been discarded by Noteworthy, despite the improvement in the experimen- combining the experimental outcomes of collaborations tal accuracy, the resulting upper limit turns out to be such as PVLAS and OSCAR [see Fig. 5]. Hence, the comparable to the one found from the results of Plimp- sensitivity in this experiment of Cavendish-type is not ton and Lawton [see Eq. (B5)] and so, no improvement high enough to improve the existing bounds on the axion is found as compared with the existing constraints. The parameter space. lack of sensitivity is understood here as a direct conse- It is worth remarking that a more accurate version of quence of the quadratic dependence of γabcd on g [see this kind of experiments has been carried out by using Eq. (B8)]. 15

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