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

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

PHYSICAL REVIEW D 98, 115008 (2018)

Axion-modified , Coulomb potential, and

† ‡ S. Villalba-Chávez,* A. Golub, and C. Müller Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany

(Received 28 June 2018; published 5 December 2018)

A consistent of a quantum theory of -electrodynamics requires terms beyond the minimal coupling of two to a neutral pseudoscalar . 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 is established and the scopes for searching axionlike 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 radius in muonic hydrogen. A certain loophole remains, though, which is linked to the nonrenormaliz- able nature of axion-electrodynamics.

DOI: 10.1103/PhysRevD.98.115008

I. INTRODUCTION experimental endeavors are currently oriented to detect this elusive degree of freedom or, more generally, an That the path integral measure in quantum chromody- associated class of candidates sharing its main namics (QCD) is not invariant under an axial chiral features, i.e., axionlike particles (ALPs). Some of them Uð1Þ -transformation provides a clear evidence that this A being central pieces in models which attempt to explain the classical does not survive the dark abundance in our Universe [4–8],whereas procedure. As a consequence of this anomaly, QCD others are remnant features of string compactifications should not be a charge- (CP)-preserving framework. [9–12]. CP However, with astonishing experimental accuracy, no - The problematic associated with the ALPs detection violation event is known within the theory of the strong demands both to exploit existing high-precision techniques interactions. This so-called “strong CP-problem” finds a and to develop new routes along which imprints of these consistent theoretical solution by postulating a global hypothetical particles can be observed [13,14]. Descriptions Uð1Þ -invariance in the (SM), which PQ of the most popular detection methods can be found in compensates the CP-violating term via its spontaneous Refs. [15–19]. A vast majority of these searches relies on symmetry breaking [1]. While this mechanism seems to be the axion-diphoton coupling encompassed within axion- the most simple and robust among other possible routes electrodynamics [20]. Correspondingly, many photon- of explanation, it is accompanied by a new puzzle linked to related experiments such as those searching for light the nonobservation of the associated Nambu-Goldstone shining through a wall [21–28] and the ones based on , i.e., the QCD axion [2,3]. As a consequence, polarimetry detections [29–33] have turned out to be constraints resulting from this absence indicate a feeble particularly powerful. interplay between this hypothetical particle and the Contrary to that, precision tests of the Coulomb’slaw well-established SM branch, rendering its detection a via atomic spectroscopy and experiments of Cavendish- very challenging problem to overcome. Still, various type have not been used so far in the search for ALPs, although they are known to constitute powerful probes for other well-motivated particle candidates [34–37].In *[email protected] † [email protected] particular, these setups provide the best laboratory bounds ‡ [email protected] on minicharged particles in the sub μeV mass range. Simultaneously, by investigating the role of ALPs in atomic Published by the American Physical Society under the terms of spectra, one might elucidate whether the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of these hypothetical degrees of freedom may be the source the author(s) and the published article’s title, journal citation, for the large discrepancy > 5σ between the proton radius and DOI. Funded by SCOAP3. that follows from the Lamb-shift in muonic hydrogen

2470-0010=2018=98(11)=115008(18) 115008-1 Published by the American Physical Society S. VILLALBA-CHÁVEZ, A. GOLUB, and C. MÜLLER PHYS. REV. D 98, 115008 (2018) versus the established value based on scattering photon fields, this property introduces an unknown uncer- and the Lamb-shift in ordinary hydrogen [38–41]. Various tainty that cannot be determined, unless the ultraviolet theoretical investigations have been put forward seeking for completion of axion-electrodynamics is found. We argue — — a satisfactory explanation for this anomaly [42–46], some that up to this uncertainty axionlike particles are ruled out as plausible candidates for explaining the proton radius of them including hypothetical scalar particles. anomaly in muonic hydrogen (Hμ). Parallelly, spectro- Against the background of these circumstances, it is scopic results linked to a variety of transitions in hydrogen relevant to derive modifications of the Coulomb potential are exploited to probe the sensitivity of this precision due to quantum vacuum fluctuations of axionlike fields technique in the search for ALPs. We show that, as a and to study their potential consequences. The former are consequence of the mentioned feature, high-precision encompassed in the corresponding vacuum polarization spectroscopy lacks of sufficient sensitivities as to improve tensor whose calculation, however, is not a straightforward the existing laboratory constraints on the parameter space task as far as axion (QED )is A of ALPs. concerned. This is because it requires—first of all—a Our treatment is organized as follows. Firstly, in Secs. II meaningful implementation of the corresponding perturba- A and II B, techniques known from effective field theories tive expansion in this nonrenormalizable framework. In are exploited for establishing the vacuum polarization analogy to [47–53], the expansion in terms tensor within an accuracy of the second order in the of the axion-diphoton coupling gives rise to an infinite axion-diphoton coupling. There we show that, despite number of divergencies that cannot be reabsorbed in the the electrically neutral nature of ALPs, the polarization renormalization constants associated with the parameters tensor closely resembles the one obtained in QED. The and fields of the theory. Unless a similar amount of counter- similarity is stressed even further in Secs. III A and III B, terms is added, this feature spoils the predictivity of the where various asymptotes of the polarization tensor are corresponding scattering matrix, preventing the construc- established and the general expression for the axion- tion of a consistent quantum theory of axion-electrodynam- Coulomb potential is determined. The latter outcome is ics. Hence, the perturbative renormalizability of this presented in such a way that a direct comparison with the theory demands unavoidably the incorporation of higher Uehling potential can be carried out. Also in Sec. III B,we dimensional operators. This, in turn, comes along with the derive the corresponding modification to the Lamb-shift and presence of a large number of free parameters which have to emphasize the problematic introduced by both the effective be fixed from experimental data. It is, however, known that scenario and the use of atomic s-states. In Sec. III C we give this formal aspect relaxes because many of these higher- some estimates and discuss the advantages and disadvantages dimensional terms are redundant, in that the ultimate of testing ALPs via excited states in Hμ, whereas in Sec. IV scattering matrix is not sensitive to their coupling constants [54–57]. As a matter of fact, all contributions of this nature our conclusions are exposed. Some details about the particle- can be formally eliminated while the effective Lagrangian ghost content of the theory are provided in Appendix A. Finally, in Appendix B the sensitivity levels associated with acquires counterterms which allow for the cancellation of precision tests of the axion-modified Coulomb law via the loop divergences. Besides, when working at a certain experiments of Cavendish-type are presented. level of accuracy, only a finite number of counterterms is needed and the cancellations of the involved infinities can be carried out pretty much in the same way as in conventional II. THE MODIFIED PHOTON PROPAGATOR IN renormalizable field theories. AXION-ELECTRODYNAMICS In this paper, axion-electrodynamics is regarded as a A. approach Wilsonian effective theory parametrizing the leading order Axion-electrodynamics relies on an effective action contribution of an ultraviolet completion linked to physics characterized by a natural ultraviolet scale Λ at which beyond the SM. Its quantization is used for determining the UV the Uð1Þ -symmetry is broken spontaneously. It combines self-energy operator of the electromagnetic field with an PQ accuracy of second-order in the axion-diphoton coupling. the standard Maxwell Lagrangian, the free Lagrangian ϕ¯ ð Þ This result is utilized then for obtaining the modified density of the pseudoscalar field x and an interaction Coulomb potential of a static pointlike charge. In con- term coupling two photons and an axion. Explicitly, Z nection, the plausible distortion of the Lamb-shift in 1 1 ¼ 4 − μν þ ∂ ϕ¯ ∂μϕ¯ hydrogenlike atoms is established. Particular attention is Sg¯ d x 4 fμνf 2 μ paid to a limitation caused by the nonrenormalizable 1 1 feature of axion electrodynamics which prevents us from 2 ¯ 2 ¯ ˜ μν − m¯ ϕ þ g¯ ϕ fμνf : ð1Þ having a precise and clear picture of the axion physics at 2 4 distances smaller than the natural cutoff imposed by the axion-diphoton coupling. In contrast to previous studies of Here, fμν ¼ ∂μa¯ ν − ∂νa¯ μ stands for the electromagnetic ˜ μν 1 μναβ the Lamb-shift involving minicharged particles and hidden field tensor, whereas its dual reads f ¼ 2 ϵ fαβ with

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ϵ0123 ¼ 1. Hereafter, we use a metric with signature ordinary action containing higher-order derivatives. Within diagðgμνÞ¼ð1; −1; −1; −1Þ, and a unit system in which a quantum effective theory approach, higher-dimensional the speed of light, the Planck constant and the vacuum operators—like those exhibited in Eq. (2)—are suppressed ¯ permitivity are set to unity, c ¼ ℏ ¼ ϵ0 ¼ 1.Astheaxion- by higher powers of g, so that their consequences at low ¯ ¼ Λ−1 energies relative to Λ ¼ g¯−1 are tiny when compared diphoton coupling g UV has an inverse energy dimension, UV this effective theory belongs to the class of perturbatively with the effects resulting directly from Sg¯ [65,66]. To all nonrenormalizable frameworks. In the following we will effects, they must be treated as perturbations, otherwise Λ suppose that UV is a very large parameter in order to extend a violation of the takes place due to the occurrence integrals over the momentum components to an infinite- of Pauli-Villars ghosts. While this problem has been noted volume Fourier space. Whenever no problem of convergence in previous studies (see for instance [67,68] and references arises, the integrals over the spacetime coordinates will also therein), it is instructive to review it once again in the be extended to the whole Minkowski space. present context. To this end, we first note that the We want to use Eq. (1) to derive radiative corrections up appearance of ghosts emerges quite straightforwardly when to second-order in the axion-diphoton coupling ∼g¯2.For investigating the axion Green function that results from this, we shall use well-established effective field theory combining the corresponding kinetic term and the second techniques which have been applied extensively within the line of Eq. (2): context of quantum gravity [47–53] and chiral perturbation 1 1 theory [58–61] (see also Refs. [62–64] for their application ð 2Þ¼ − ð Þ G p 2 − ¯ 2 2 − ¯ 2 : 3 in nonlinear QED). In connection, we will suppose that Sg¯ p m p ms characterizes—at energies substantially lower than Λ — UV ¼ ¯ 2 the leading order contribution of its UV-completion At the pole p m , the residue of this Green function is 2 1 2 2 ¯ −2 ð Þj 2 2 ¼ 1 ¼ ¯ ¼ð¯b Þ which is Lorentz and gauge invariant. Hence, all higher- ResG p p ¼m¯ , whereas at p ms g ϕ it 2 dimensional operators which are consistent with these turns out to be ResGðp Þj 2¼ ¯ 2 ¼ −1. Hence, the vacuum p ms fundamental symmetries should be included in Eq. (1), excitations linked to the former pole have a positive definite implying that an infinite number of counterterms is norm in the Hilbert space, as should correspond to necessary to cancel out the divergences linked to one- asymptotically single-particle states. Conversely, the square particle irreducible Feynman diagrams [65].However,to of the norm associated with the remaining massive exci- extract quantitative predictions from the radiative correc- tations is nonpositive and no physical state can be asso- tions in the second-order approach in the axion-diphoton ciated with them (ghost states), leading parallely to a coupling, it is sufficient to incorporate the next-to-leading violation of unitarity [67]. Noteworthy, as the associated order term of Sg¯ . Combining the described method with a higher-dimensional operator [see Eq. (2)] contains the ¯ dimensional analysis, we find that the renormalization of square of bϕ, the ghost mass is real and its fictitious the self-energy operators in QEDA should be handled by propagation does not involve a speed faster than the speed two local operators of dimension 6: of light. The term written in the first line of Eq. (2) leads Z to a similar scenario,2 but with a different ghost mass 1 4 2 2 μλ ν ¯ 2 ¯ ¯ −2 2 ¼ ¯ b¯ ð∂ Þð∂ Þ m ¼ðgb Þ [for details, see Appendix A]. The described Sg¯ d x 2 g a μf νf λ gh a situation provides evidences that symmetry arguments are 1 2 ¯ 2 ¯ μ ¯ not enough to obtain a well-behaved quantum theory of the þ g¯ b ð∂μϕÞ□ð∂ ϕÞþ : ð2Þ ϕ S ¼ þ 2 2 fields involved in Sg¯ Sg¯ . However, if the scattering matrix linked to its UV-completion is unitary, it is natural Various local operators sharing both the symmetry of the to expect that the one associated with S, covering ≪ Λ theory and the same dimensionality can be found. However, quantum processes at lower energies UV, is unitary it can be easily verified that all of them reduce to those too. This idea justifies the restriction given above Eq. (3). given in Eq. (2) through integrations by parts. The Wilson We remark that these “ghost-providing” contributions ¯ parameters ba;ϕ determine the strength of the contributions are redundant operators which can be dropped from above. They might be determined by a matching procedure the effective Lagrangian without changing observables – provided the UV-completion of Sg¯ is known. Since we [54 56]. Later on, this outcome is used to remove them ignore the precise form of the latter, they will be considered as arbitrary. We emphasize that the appearance of the square 1Although the exposition is made in terms of bare parameters, 2 — of the Wilson parameters in Sg¯ guarantees that at least at the idea extends straightforwardly when renormalized quantities tree level—the theory is causal [i.e., free of ; see are considered instead. 2The origin of the electromagnetic theory that results from also discussion below]. combining the Maxwell theory with the higher-derivative oper- It is worth remarking that the action resulting from the ator written in the first line of Eq. (2) dates back to the work of P. combination of Eqs. (1) and (2) cannot be considered as an Podolsky [69]. For further developments see Refs. [70–77].

115008-3 S. VILLALBA-CHÁVEZ, A. GOLUB, and C. MÜLLER PHYS. REV. D 98, 115008 (2018) conveniently while the effective Lagrangian acquires coun- axion-diphoton coupling g can be expected, i.e., Zg would terterms which allow for the cancellation of the divergences not deviate from its classic tree-level value Zg ¼ 1. Hence, associated with the loops that are calculated here [see from now on, no distinction between the renormalized and below Eq. (7)]. physical coupling is needed. However, we emphasize that its Now, to carry out the renormalization program, the set of bare counterpart g¯ is still subjected to a renormalization due “ ” ð ¯ ¯ ¯ ¯ ¯ ϕ¯ Þ bare quantities m; g; mgh; ms; aμ; should be replaced to the wave functions renormalization constants Z3 and Zϕ. ð by the respective renormalized parameters m; g; mgh;ms; μ At the quantum level, the dynamical information of the ϕ Þ S ¼ þ 2 þ aR; R . In connection, each term in Sg¯ Sg¯ has system described by Eq. (4) is rooted within the Green to be parametrized by a renormalization constant so that the functions. They can be obtained from the generating action to be considered from now on is functional R R 4 1 μ 2 μ S ¼ S þS þ iSþi d x½− ð∂μa Þ þa j þϕ j R ct ; Dϕ Da e 2ζ R R μ R Z Z½j; j¼ R R R ; ð7Þ 1 1 1 Dϕ D iS S ¼ 4 − 2 þ ð∂ϕ Þ2 − 2ϕ2 R aRe R d x 4fR 2 R 2m R where jðxÞ and jμðxÞ denote the external currents associ- 1 1 1 μ þ gϕ f˜ f þ ð∂f Þ2 − ϕ □2ϕ ; ϕ ðxÞ a ðxÞ 4 R R R 2 2 R 2 2 R R ated with the axion R and the gauge field R . The m ms Z gh contribution in the exponent of Eq. (7) which is propor- 1 1 1 4 2 2 tional to the parameter ζ guarantees a covariant quantization S ¼ d x − ðZ3 −1Þf þ ðZϕ −1Þð∂ϕ Þ ct 4 R 2 R μ ð Þ of aR x . At this point it turns out to be convenient to bring S 1 1 the renormalized Lagrangian in R [see Eq. (4)]toa − m2ðZ −1Þϕ2 þ ðZ −1Þð∂f Þ2 2 m R 2m2 gh R canonical form in which the terms linked to the Pauli- gh Villars ghosts are dropped. This can be achieved by 1 1 þ ðZ −1Þgϕ f˜ f − ðZ −1Þϕ □2ϕ : ð Þ performing the following local field redefinitions within 4 g R R R 2 2 s R R 4 ms the path integral [see Eq. (7)]:

μν μ ν ν μ Here, f ≡ f ¼ ∂ a − ∂ a is the renormalized electro- □ μ μ □ μ R R R R ϕ → ϕ − ϕ ;a→ a − a ; ð8Þ μ ð Þ¼Z−1=2 ¯ μð Þ R R 2m2 R R R 2m2 R magnetic tensor with aR x 3 a x . Likewise, the s gh renormalized axion field ϕ ðxÞ and its bare counterpart ϕ¯ ðxÞ R 2 −1=2 ¯ and by keeping the accuracy to the order g . As these are connected via ϕ ðxÞ¼Zϕ ϕðxÞ, where Zϕ is the μ R transformations are linear in ϕ ðxÞ and a ðxÞ, the asso- corresponding wavefunction renormalization constant. R R ciated Jacobian leads to a decoupling between the corre- Any other bare parameter relates to its respective renormal- sponding Fadeev-Popov ghosts and the fundamental fields. ized quantity following multiplicative renormalizations The equivalence theorem [54,56] generalizes this fact by according to dictating that no change is induced on the scattering matrix sffiffiffiffiffiffiffi through shifts of this nature; still they modify the initial Z Z m¯ ¼ m m; g¯ ¼ g pgffiffiffiffiffiffiffi ; parameters of the theory. Indeed, in our problem the Z Z Z ϕ ð Þ ϕ 3 ϕ redefinition of R x leads to a kinetic term of the form sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi 1 ð1 − m2Þð∂ϕ Þ2 2 2 R . The additional factor contained in this Z Z ms ¯ ¼ 3 ¯ ¼ ϕ ð Þ S mgh mgh Z ; ms ms Z : 5 expression can be reabsorbed in ct [see Eq. (4)]by gh s redefining the wavefunction renormalization constant for m2 0 the axion field Zϕ − 2 → Zϕ. Similarly, we redefine It is worth remarking that in Eq. (4) the shorthand notations ms Z −Z → Z0 Z −Z → Z0 Z → Z0 Z → Z0 2 μν ˜ ˜ μν 2 μ gh 3 , s ϕ s, 3 3 and m m f ≡ f μνf , f f ≡ f μνf , ð∂ϕ Þ ≡ ð∂μϕ Þð∂ ϕ Þ gh R R R R R R R R R R S have been used. When inserting the expressions for the to reabsorb terms arising when transforming ct. Therefore, ghost masses [see below Eq. (3)] in those relations given in apart from this unobservable effect, the Pauli-Villars ghosts the second line of Eq. (5), we link the bare and renormalized have no result other than to remove the divergences that Wilsonian parameters: might arise from the respective one-particle irreducible graphs. 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffi b¯ ¼ b Z Z Z b¯ ¼ b 3 Z ð Þ a a Z 3 gh ϕ; ϕ ϕ Z s; 6 B. Renormalized photon propagator in a g g modified minimal subtraction scheme where the connection between g¯ and g has been used. We pursue our investigation by determining the modi- ð0Þ Noteworthy, within our second-order approximation in the fication to the photon propagator Dαβ ðx; x˜Þ due to quantum ϕ ð Þ coupling constant, no modification on the renormalized vacuum fluctuations of a pseudoscalar axion field R x .

115008-4 AXION-MODIFIED PHOTON PROPAGATOR, COULOMB … PHYS. REV. D 98, 115008 (2018) Z đ4qq q ¼ α γ ð Þ Kαγ 2 2 2 ; 13 q ½ðq − p2Þ − m

which diverges quadratically as jqj → ∞. The regulariza- Π ð Þ FIG. 1. depicting the axion-modified vac- tion of μν p1;p2 is then carriedR out by using a standard 1 ¼ 1 ds uum polarization tensor. While the dashed line represents the free Feynman parametrization [ab 0 ½bþða−bÞsÞ2] and by con- axion propagator, the internal wavy line denotes the free photon þ ð0Þ D ¼ 4 − ϵ ϵ → 0 ð ˜Þ tinuing the loop integral to , dimensions propagator Dμν x; x . Here, the external wavy lines represent via the replacement amputated photon legs. Z Z đ4q… → ðCμÞϵ đDq…; ð14Þ The use of Eq. (7) allows us to express the modified photon propagator Dαβðx; x˜Þ as 1ðγ−1Þ 1 2 where C ¼ e2 =ð4πÞ = and γ ¼ 0.5772… is the Euler- 2 1 δ Z½j; j Mascheroni constant. In this context, μ denotes a dimen- Dαβðx; x˜Þ¼ : ð Þ 2 δjαð Þδjβð˜Þ 9 sionful parameter, i.e., the substracting point that follows i x x j;j¼0 when rescaling the renormalized axion-diphoton coupling ϵ=2 2 g → gμ in D-dimensions so that its mass dimension −1 We then expand Eq. (7) up to the order g and insert the is kept. Also, when going from four to D-dimensions, resulting expression into the formula above. As a conse- the Wilsonian parameters [see Eqs. (2) and (6)] rescale quence, the corrected photon propagator reads −ϵ=2 bi → biμ with i ¼ a; ϕ while their dimensionless fea- Z ture is retained. ð0Þ 4 4 ð0Þ Dαβðx; x˜Þ¼Dαβ ðx; x˜Þþ d yd yD˜ αμ ðx; yÞ Now, we integrate over q and Taylor expand the resulting expression in ϵ. As a consequence, Eq. (12) becomes μν ð0Þ 4 × iΠ ðy; y˜ÞDνβ ðy;˜ x˜ÞþOðg Þ: ð10Þ Πμνð Þ¼δ ð μν 2 − μ νÞπð 2Þ p1;p2 p1;p2 g p2 p2p2 p2 ; μν Z    Here Π ðy; y˜Þ encompasses the expression for the unrenor- 2 1 2 Δð Þ πð 2Þ¼ g Δð Þ − s malized polarization tensor [see Fig. 1] as well as counter- p 2 ds s ln 2 16π 0 ϵ μ terms that allow for the cancellation of the divergences 2 associated with this loop. Analytically, it reads − ðZ0 − 1Þþ p ðZ0 − 1ÞðÞ 3 2 gh 15 mgh μν 2 μαϵτ νβσρ y˜ y ð0Þ y y˜ ð0Þ Π ðy; y˜Þ¼ig ϵ ϵ ½∂σ∂ϵΔ ðy;˜ yÞ½∂τ ∂ρDαβ ðy; y˜Þ F Δð Þ¼ 2 − 2 ð1 − Þ □ with s m s p s s . Manifestly, the term asso- þ Z0 − 1 þ ðZ0 − 1Þ ϵ−1 ϵ → 0 3 2 gh ciated with the factor is singular as . In contrast to mgh QED, such a divergence cannot be reabsorbed fully in the × ½□gμν − ∂μ∂νδ4ðy − y˜Þ; ð11Þ wavefunction renormalization constant of the electromag- 0 netic field Z3 by enforcing that the radiative correction R ð0Þ 4 i ipðx−x˜Þ 4 should not alter the residue of the photon propagator at where Δ ðx; x˜Þ¼ đ p 2 2 e , with đ p ≡ 2 F p −m þi0 p ¼ 0 [65,78]. We solve this problem, by choosing the 4 4 d p=ð2πÞ refers to the unperturbed ALP propagator, counterterms in the following form R − ð0Þð ˜Þ¼ đ4 igαβ ipðx−x˜Þ whereas Dαβ x; x p p2þi0 e denotes the pho- 2 2 2 2 ton propagator in Feynman gauge [ζ ¼ 1]. It is worth 0 g m 0 g mgh Z3 − 1 ¼ lim ; Z − 1 ¼ lim : ð16Þ remarking that,R in momentum space the polarization tensor ϵ→0 16π2ϵ gh ϵ→0 48π2ϵ μν 4 4 −ip1x μν ip2x˜ Π ðp1;p2Þ¼ d xd x˜e Π ðx; x˜Þe reads Notice that the ratio of scales mg acts like a dimensionless coupling constant. Thus, the one-loop renormalized polari- Πμνð Þ¼δ 2ϵμταβϵνσγρ p1;p2 p1;p2 ig gστp2βp2ρKαγ zation tensor in a modified minimal subtraction scheme   2 (MS) scheme reads − Z0 − 1 − p2 ðZ0 − 1Þ ð 2 μν − μ νÞ ð Þ 3 2 gh p2g p2p2 ; 12 μν μ mgh Π ð Þ¼δ ½ 2 μν − νπ ð 2Þ p1;p2 p1;p2 p2g p2p2 p2 ; MS Z MS  2 1 Δð Þ δ ≡ ð2πÞ4δ4ð − Þ 2 g s where the shorthand notation p1;p2 p1 p2 π ð Þ¼− Δð Þ ð Þ MS p 2 ds s ln 2 : 17 has been introduced and 16π 0 μ

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Noteworthy, this expression satisfies the transversality μν μν condition p1μΠ ðp1;p2Þ¼Π ðp1;p2Þp2μ ¼ 0. MS MS Some comments are in order. First, when the QED action is extended with those terms belonging to QEDA, the 0 expression for Z3 − 1 found in Eq. (16) will be added to Z 3 the corresponding one-loop QED-expression [ 3ðQEDÞ]. This operation allows us to define the standard renormal- ¼ Z˜ 1=2 ¯ ized U(1)-charge as usual eR 3ð1 loopÞe, with the bare ¯ Z˜ ¼ Z0 þ Z FIG. 2. Diagrammatic representation of the axion self-energy charge e and 3ð1 loopÞ 3 3ðQEDÞ. Finally, taking operator mediated by quantum vacuum fluctuations of the into account Eqs. (16) and (17), the Fourier transform of electromagnetic field. In contrast to Fig. 1, the external dashed Eq. (10) is, up to an unessential longitudinal term, lines represent amputated ALPs legs, whereas the internal wavy ð0Þð ˜Þ μν μν lines represent photon Dμν x; x . D ðp1;p2Þ¼δ D ðp2Þ; MS p1;p2 MS − μν μν ð Þ¼ ig ½1 þ π ð 2Þ ð Þ Next, we Fourier transform Σðy; y˜Þ and regularize its D p 2 p : 18 MS p MS divergent integral via dimensional as made in Sec. II B. However, in contrast to the case treated there, the This formula constitutes the starting point for further associated divergence at ϵ → 0 is fully reabsorbed here in a considerations. In the next section it will be used to renormalization constant establish the axion-modified Coulomb potential. g2m2 Z0 − 1 ¼ lim s ; ð21Þ C. Axion self-energy operator, renormalized s ϵ→0 32π2ϵ mass vs physical mass 0 whereas Zm does not deviate from its classic tree-level Our aim in this section is to determine the axion self- 0 value [Z ¼ 1]. This feature extends beyond the one-loop energy operator. Its associated Feynman diagram is m approximation because the axion-photon vertex prevents depicted in Fig. 2. This object encloses the way in which the proliferation of self-interacting terms for ALPs con- the quantum vacuum fluctuations of the electromagnetic taining no derivatives [66]. Despite this, the bare mass m¯ field correct the axion propagator. To show this analytically, still is subject to a finite renormalization due to the axion we expand the generating functional for the Green function 2 wavefunction renormalization constant [see Eq. (5)], which [see Eq. (7)] up to first order in g . Once this step has been does not deviate from the classical value Z0 ¼ 1. Keeping carried out, the resulting expression is twice differentiated ϕ functionally with respect to the axion source jðxÞ leading to in mind all these details, we find that—in a MS scheme— the renormalized axion self-energy operator is given by 1 δ2 ½ j Z j; 2 Δ ðx; x˜Þ¼ Σ ðp1;p2Þ¼δ Σ ðp Þ; F 2 δ ð Þδ ð˜Þ MS p1;p2 MS 2 i j x j x j;j¼0 Z   Z 2 2 1 2 3g p Δm→0ðsÞ ð0Þ ð0Þ Σ ð Þ¼ Δ ð Þ ð Þ ¼ Δ ð ˜Þ − 4 4 ˜Δ ð Þ MS p 2 ds m→0 s ln 2 ; 22 F x; x d yd y F x; y 32π 0 μ

ð0Þ 4 Δ ð Þ¼− 2 ð1 − Þ × iΣðy; y˜ÞΔ ðy;˜ x˜ÞþOðg Þ: ð19Þ where m→0 s p s s [see below Eq. (15)]. As F we could have anticipated, this expression is independent Here, Σðy; y˜Þ comprises the expression of the unrenormal- of the renormalized axion mass. It is, perhaps, worth ized axion self-energy operator as well as some possible stressing that—in a MS-scheme—the square of the physi- 2 2 counterterms. Explicitly, cal mass mphy is the value of p for which the real part of the two-points irreducible function4: i 2 μντσ αβργ y y˜ ð0Þ Σðy;y˜Þ¼− g ϵ ϵ ½∂σ∂γ Dμα ðy;y˜Þ 2 Γð Þ¼δ ½ 2 − 2 p1;p2 p1;p2 p2 mphy ; ½∂y∂y˜ ð0Þð˜ Þ − □ðZ0 − 1Þ 2 ¼ 2 − Σ ð 2 ÞðÞ × τ ρDβν y;y ϕ mphy m MS mphy 23 □2 þ ðZ0 −1Þþ 2ðZ0 −1Þ δ4ð − ˜Þ ð Þ vanishes. Whenever the subtracting parameter satisfies 2 s m m y y : 20 μ ≫ ½−32π2 ð 2 2 Þ ≈ ms mphy exp = g mphy , mphy m holds. Hence,

R 4This expression can be established from the identity 3 Z đ4 Γð ÞΔ ð Þ¼− δ Δ ð Þ An explicit expression for 3ðQEDÞ, in dimensional regulari- q p1;q F q; p2 i p1;p2 , where F q; p2 stands for zation, can be found in Eq. (19.37) of Ref. [65]. the Fourier transform of Eq. (19).

115008-6 AXION-MODIFIED PHOTON PROPAGATOR, COULOMB … PHYS. REV. D 98, 115008 (2018) the expression for the polarization tensor [see Eq. (17)]asa can be established easily by taking the integrand above, 2 q → function of mphy would not differ from the one given in with eR. After multiplying the resulting expression by terms of the renormalized mass. eR, we find At this point we find it interesting to make a comparison 2 ðpÞ ðpÞ¼− a ðpÞ¼− escr ð Þ between Eq. (22) and the polarization tensor given in U eR 0 2 ; 27 Eq. (17). To this end, it is convenient to reexpress the latter p as follows: 2 ðpÞ¼ 2 ½1 þ π ðp2Þ where the screened charge escr e has   R MS μ ν been defined. μν p p Π ð Þ¼δ ϰ ð 2Þ μν − 2 2 As we still have freedom of performing finite renorm- p1;p2 p1;p2 MS p g 2 ; MS p2 π ðp2Þ alizations, we can demand that MS vanishes as ϰ ð 2Þ¼ 2π ð 2Þ ð Þ jpj → 0 jxj → ∞ MS p p MS p : 24 . Since the corresponding length scale , 2 ðjxj → ∞Þ escr can be identified with the electrostatic ϰ ð 2Þ In this formula, MS p represents the only nontrivial charge that is measured in experiments at low energies. eigenvalue of the polarization tensor [79,80], which—in the π ð0Þ¼0 This natural renormalization condition [ MS ] holds limit under consideration—turns out to be smaller than for the subtracting parameter μ ¼ m exp½−1=4. The renor- Σ ð 2Þ −2 3 MS p by a factor = . malized polarization tensor then reads Let us finally remark that, in addition to the axion- αβ 2 αβ α β 2 diphoton interplay, axion self-coupling [81] as well as Π ðp1;p2Þ¼δ ½p2g − p p π ðp2Þ; R p1;p2  2 2 R effective interactions with electron, proton and g2m2 1 p2 might occur [16,82–84]. In such a case further one-loop π ðp2Þ¼− 1 − R 64π2 3 m2 contributions to the axion self-energy operator might arise. Z   2 1 Δð Þ However, these contributions depend on coupling constants − g Δð Þ s ð Þ 2 ds s ln 2 ; 28 other than the one mediating the interaction between an 16π 0 m axion and two photons. whereas the axion self-energy operator [see Eq. (22)] reduces to III. AXION-COULOMB POTENTIAL 2 A. Screening of the electric charge and finite Σ ðp1;p2Þ¼δ Σ ðp Þ; R p1;p2 R 2   renormalization: Setting the subtracting parameter 2 4 2 g p p 7 ’ Σ ðp2Þ¼− ln − − : ð29Þ Hypothetical distortions of Coulomb s law can always be R 64π2 m2 6 determined through the temporal component of the electro- magnetic four-potential (hereafter, to simplify notation, μ Theresults obtainedso farare summarizedin Fig. 3, which aμðxÞ a ðxÞ π ð 2Þ Σ ð 2Þ has to be understood as R ) displays the behavior of R p [left panel] and R p Z [right panel] as a function of p2=m2. In both panels the α αβ 4 a ðxÞ¼−i D ðx; x˜Þjβðx˜Þd x;˜ respective real and imaginary parts are shown in green and MS Z red, manifesting by themselves the non-Hermitian feature αβ αβ 4 4 ip1x −ip2x˜ of the polarization tensor and the axion self-energy operator. D ðx; x˜Þ¼ đ p1đ p2e D ðp1;p2Þe ð25Þ MS MS To support this numerical evaluation from an analytic viewpoint, we first determine an exact expression for the αβ β 0 where D ðp1;p2Þ is given in Eq. (18). Here j ðx˜Þ¼ imaginary parts. To this end, we restore the i -prescription MS 2 → 2 − 0 qδβδ3ð˜Þ [m m i ] in Eq. (28) and apply the formula 0 x denotes the four-current density of a pointlike lnð−A − i0Þ¼lnðjAjÞ − iπ with A > 0. Explicitly, q x˜ ¼ 0 static charge placed at the origin of our reference   frame. Particularizing the expression above for α ¼ 0,we g2p2 m2 3 Im½π ðp2Þ ¼ − 1 − Θðp2 − m2Þ; end up with R 96π p2 Z 2ð 2Þ2 đ3 ½Σ ð 2Þ ¼ g p Θð 2 − 2Þ ð Þ p 2 −ip·x Im R p p m ; 30 a0ðxÞ¼q ½1 þ π ðp Þe : ð26Þ 64π p2 MS where ΘðxÞ denotes the unit step function. We note that At this point it is worth emphasizing that, while the for an on-shell photon [p2 ¼ 0], the imaginary part π ðp2Þ Πμν — expression for MS [see Eq. (17)] is finite, its depend- of R vanishes, which implies according to the optical ence on the subtracting point μ introduces an arbitrariness. theorem—that the emission of an ALP from a photon To remove it, we consider the expression of the electrostatic accompanied by the radiation of another photon is energy between two in momentum space UðpÞ.It forbidden. This fact agrees with the outcome resulting

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2 2

1 1 2 2 64 64 4

2 0 0 m m 2 2 g g R R

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 πRðp Þ [see Eq. (28)] is shown as a function 2 2 2 of p =m . The right panel depicts the corresponding dependence of the axion self-energy operator ΣRðp Þ [see Eq. (29)]. The respective real and imaginary parts are displayed in green and red.

from an analysis of the corresponding energy-momentum term of Eq. (30) when the condition p2 ≫ m2 is balance. considered. The expression for the imaginary part of the one-loop self-energy operator Im½Σ ðp2Þ [second line in Eq. (30)] R B. Electrostatic potential and modified Lamb-shift coincides with the result found previously in Ref. [85] through a direct application of the cutting rules. Its on-shell The first contribution in Eq. (26) can be integrated evaluation [p2 ¼ m2] should allow us to determine the straightforwardly, leading to the unperturbed Coulomb a ðxÞ¼q ð4πjxjÞ total rate of the decay process ϕ → γγ via the relation potential C = . The second one, on the other 2 π ðp2Þ ½Σ ð Þ ¼ Γ hand, will be computed by using R rather than Im R m m ϕ→γγ, provided the optical theorem is 2 2 π ðp Þ valid. With accuracy to first order in g , this formula is MS . With all these details in mind we write indeed verified because an expression for Γϕ→γγ—relying a ðxÞ¼a ðxÞþδaðxÞ on the corresponding S-matrix amplitude—can be inferred 0 C ; Z directly from the corresponding neutral decay rate [see đ3p δaðxÞ¼q π ðp2Þ −ip·x ð Þ for instance Eq. (19.119) in Ref. [86]]. This fact evidences p2 R e : 33 that the unitarity is preserved, at least within the second order approximation in the axion-diphoton coupling g. For evaluating δaðxÞ explicitly, it is convenient to integrate Further asymptotic expressions of Eq. (28) are eluci- 2 ≫ 2 π ð 2Þ by parts in Eq. (28) and use an equivalent representation of dated. For m p , we find that R p approaches to π ðp2Þ R instead:   g2p2 3 p2  Z  π ðp2 ≪ m2Þ ≈ 1 − : ð31Þ 2 2 3 1 3 R 144π2 8 2 π ð 2Þ¼ g p 1 − 2 dss ð Þ m R p 2 p 2 2 : 34 144π 2 0 m − p ð1 − sÞ Conversely, for jp2j ≫ m2, its asymptotic behavior turns out to be dominated by the following function Observe that, for applying this formula in Eq. (33), p0 must be set to zero. Taking this into account, the integral over the     2 2 j 2j 7 momentum can be carried out with relative ease. After g p p 2 1=2 π jj 2j≫ 2 ≈ ln − iπΘðp Þ − : ð32Þ developing the change of variable u ¼ 1=ð1 − sÞ , the R p m 96π2 m2 6 axion-modified potential turns out to be ½ 2 0 Notably, when p is timelike p > , the expression 2 2 Z 2 2 q g m ∞ du above gets an imaginary contribution Im½π ðp ≫ m Þ ≈ a ðxÞ¼ 1 þ ½ 2 − 13 −mjxju ð Þ R 0 2 5 u e : 35 −g2p2=ð96πÞ, which coincides with the leading order 4πjxj 48π 1 u

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We remark that spurious contributions containing Dirac deltas δ3ðxÞ have been ignored since the theory is predictive only for distances jxj ≫ g. Although the integral involved in this formula can be calculated analytically, we will keep it as it stands. Mainly, because it will allow us to establish compact expressions for the energy shifts that atomic transitions undergo. Asymptotic formulas for the modified potential can be extracted from Eq. (35) without much efforts. For instance, FIG. 4. Pictorial correction to the Coulomb potential due to −1 at distances larger than the Compton wavelength λ ∼ m quantum vacuum fluctuations of axion and electromagnetic of the axion, i.e., for jxj ≫ λ, the region u ∼ 1 dominates in fields. Leaving aside the electron (proton) legs [external lines the integral involved in Eq. (35), and the axion-modified with arrows in the right (left)], the remaining pieces of this Coulomb potential approaches to diagram are described in Fig. 1.   2 2 −mjxj ð1Þ ð1Þ q g m e δε ¼ δε2 − δε2 ; a ðjxj ≫ λÞ ≈ 1 þ ð Þ s1=2 p1=2 0 2 4 : 36 Z 4πjxj π ðmjxjÞ ∞ ¼ 2δ ð Þ½ 2 ð Þ − 2 ð Þ ð Þ drr U r R2s r R2p r ; 39 0 However, at short distances [jxj ≪ λ], the main contribu- ≡ jxj tion to the integral in Eq. (35) results from the region where r and Rnl stands for a radial hydrogen wave 1 ≤ u ≤ ðmjxjÞ−1, and the integrand can be approached by function. In particular, its most slowly decreasing function in u, which is   − jxj 1 1 − r ∼ m u r 2 ue . Consequently, R2 ðrÞ¼pffiffiffi 1 − e aB ; s 3=2 2 2 a aB B 2 2 1 r − r q g m 1 2 a ðjxj ≪ λÞ ≈ 1 þ ð Þ R2 ðrÞ¼ pffiffiffi e aB : ð40Þ 0 2 2 : 37 p 2 6 5=2 4πjxj 48π ðmjxjÞ aB

Here a ¼ðαm Þ−1 is the Bohr radius with α ¼ 1=137 the This expression is independent of the axion mass. B e fine structure constant and m ¼ 0.511 MeV the elec- Observe that the distance jxj must satisfy the condition e pffiffiffi tron mass. jxj ≫ ð4 3πÞ g= , otherwise our perturbative approach Observe that the integration in Eq. (39) covers the region breaks down.p Incidentally,ffiffiffi the corresponding energy scale ½0; ∞Þ. However, since QED does not provide a precise μ ∼ jxj−1 ∼ 4 3π — A p =g coincides up to a numerical factor information about the form of the axion-Coulomb potential — of the orderpffiffiffi of one with the linked to QEDA: forR distancesR smaller thanR g, the integral over r must be split μ ≈ 4 6π ∞ … ¼ g … þ ∞ … L =g [for details see Ref. [66]]. 0 dr 0 dr g dr . In the following we will The distortion of the Coulomb potential due to ALPs assume that the contribution from the outer region ½g; ∞Þ [see Eq. (35)] allows us to infer the induced modifications dominates over the inner region ½0;g, which we ignore.5 As in the spectrum of a nonrelativistic hydrogenlike . we will see very shortly, the yet undiscarded values for g Since so far no large deviations from the standard QED turn out to be much smaller than any characteristic atomic predictions have been observed, we will assume that these scale. With all these details in mind we integrate over r and energy shifts are very small and, consequently, apply arrive at standard time-independent perturbation theory. When con- Z sidering a first order approximation, the energy shift δε αg2 ∞ du ½u2 − 13 δε ≈− 4 −gmu ð Þ 2 m aB 3 4 e : 41 follows by averaging the correction to the electrostatic 96π 1 u ½1 þ ua m δ ðxÞ¼− δaðxÞ B energy U eR [see Eq. (33) and the Feynman diagram depicted in Fig. 4] over the 0th order wave- Let us study the asymptotes of this expression. We first jψ i ≫ λ functions n;l;j . Explicitly, consider the case in which aB . Under this condition,

ð1Þ 5 δε ¼hψ jδUðxÞjψ i: ð38Þ Strictly speaking, in accordance with the treatment applied in n;l;j n;l;j n;l;j Sec. II [read also below Eq. (35)], the splitting of the integral should be carried out at a certain point d fulfilling the condition d ≫ g. However, in order to avoid uncertainties stemming from We wish to exploit this formula to predict a plausible ¼ 2 − 2 this additional parameter, we set d g. Our corresponding axion Lamb-shift for the s1=2 p1=2 transition in atomic results should be considered as order-of-magnitude estimates, hydrogen. For this case, Eq. (38) leads to accordingly.

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½1 þ 4 the term of the integrand uaBm is dominated by Since this is five orders of magnitude smaller than the 4 4 4 u aBm . The integral resulting from this approximation can existing discrepancy between the experimental measure- be computed exactly. After a Taylor expansion in mg ≪ 1, ment and the theoretical prediction δε¼ 0.31 meV we find the compact expression [38,42,45], virtual neutral are excluded as possible   explanation for the proton radius puzzle. 2 αg 11 Now, we wish to investigate whether the Lamb-shift δε ≈ ln ðgmÞþγ þ : ð42Þ 96π2 3 12 induced by quantum vacuum fluctuations of axionlike aB fields might cure this anomaly. To this end, we will evaluate As in Eq. (14), γ ¼ 0.5772… refers to the Euler-Mascheroni Eq. (41) considering a mass region in which reliable results ≪ λ constant. In the opposite case aB , the integrand in can be extracted. Within a pure spectroscopy context, Eq. (41) turns out to be a function that decreases monoton- this occurs for ALP wavelengths smaller or of the order ically with the growing of u. It is then justified to approach of the Bohr radius of Hμ, i.e., λ ≲ aμ with aμ ≈ 285 fm, it through its most slowly decreasing part which is otherwise interactions of other nature must be included. ∼ 3 −mgu ð1 þ Þ4 u e = uaBm . As a result, the corresponding Correspondingly, we can formally explore ALP masses integral can be computed analytically by using (3.353.1) fulfilling the condition 1 MeV ≲ m. However, in the range in Ref. [87]. In the limit of mg ≪ 1, it allows us to approach 1 MeV ≲ m ≲ 100 MeV the axion-diphoton coupling g     has been constrained severely from various results, αg2 g 11 δε ≈ þ γ þ : ð Þ 96π2 3 ln 6 43 aB aB 10 1 LEP Notice that, the equation above is a good approximation 2 whenever the condition a ≫ g holds. Moreover, although 10 inv. B PVLAS Eqs. (41)–(43) apply for ordinary atomic hydrogen, they can 10 3 e e inv. be adapted conveniently for studying the same transition in other hydrogenlike atoms. When hydrogenlike ions with 10 4 Be am atomic number Z>1 are considered, for instance, the dump – 5 OSQAR correction to the Lamb-shift will be given by Eqs. (41) 1 10 → (43), scaled by the factor Z and aB aB=Z. If a muonic 6 hydrogen atom is investigated instead, a replacement of the GeV 10 KS VZ electron mass me by the reduced mass of the system mr ≈ 186 10 7 me would be required. DFSZ

10 8 CAST SUMICO C. Precision spectroscopy in Hμ and the proton SN1987a radius anomaly 10 9 HBstars Before continuing with the physics of virtual ALPs, we 10 Cosmology will estimate the contribution to the Lamb-shift by a 10 10 8 10 6 10 4 10 2 100 102 104 106 108 whose interaction with the electromagnetic field resembles eV the one exhibited by ALPs [see Eq. (1)], i.e., the neutral pion 0 π . We should however emphasize that its effect should be FIG. 5. Summary of exclusion areas for a pseudoscalar ALP understood as a consequence of the quantum vacuum coupled to two photons. Compilation adapted from Refs. [15–19]. fluctuations of its constituent fields. When thinking The picture includes [inclined yellow band] the predictions of the 0 axion models jE=N − 1.95j¼0.07–7 (the notation of this for- of the axion as π , m → mπ¼ 135 MeV and the coupling mula is in accordance with Ref. [90]). Colored in orange and constant g → α=ðπfπÞ turns out to be determined by α and black appear the regions ruled out by particle decay experiments. the pion decay constant fπ ≈ 92 MeV [65]. Observe that the ≈ 4 97 10−3 While the portion discarded by investigating the energy loss in corresponding value of g . × fm is two orders of the horizontal branch (HB) stars are shown in blue, the excluded ≈ 0 876 magnitude smaller than the proton radius rproton . fm area resulting from the solar monitoring of a plausible ALP flux [38]. The corresponding correction to the 2s1=2 − 2p1=2 (CAST þ SUMICO) has been added in green. In purple the Lamb-shift in hydrogen atoms is δε ¼ −1.07 × 10−12 meV. portion discarded by measuring the duration of the signal This value turns out to be five orders of magnitude smaller of the supernova SN1987A is depicted, whereas the dark gray area −7 results from cosmological studies. The excluded area in dark than the experimental uncertainty jδε2σj¼2 × 10 meV, 2σ turquoise has been established from beam dump experiments. established at confidence level [35,37]. If the previous Besides, the light gray zone has been excluded from electron- evaluation is carried out by considering a muonic hydrogen collider (LEP) investigations. Finally, the colored sectors → instead [me mr], we find that the correction to the energy in olive and red show the exclusion regions corresponding to due to the neutral pion field is δε ¼ −6.81 × 10−6 meV. OSCAR and PVLAS collaborations, respectively.

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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 log10ð−δε½meVÞ. including those dealing with electron beam fixed-target discrepancy or not. Inspired by these arguments, we setups [see compilation of bounds in Refs. [88,89]]. consider as an example the 3p1=2 − 3d1=2 transition. In Conversely, the sensitivities in experiments where ALPs this case, the required wave functions are 100 ≲ ≲ 10 masses MeV m GeV are probed turn out to be pffiffiffi    much weaker [white sector in the right hand side of Fig. 5]. 4 2 1 r r r − 3 R3 ðrÞ¼ 1 − e aB ; A recent investigation based on electron-positron colliders p 9 ð3a Þ3=2 a 6a has constrained g to lie below g<10−2 GeV−1 [89]. pffiffiffi B B  B 2 2 2 1 r A numerical assessment of the axion-modified Lamb- r − 3 R3 ðrÞ¼ pffiffiffi e aB : ð44Þ d ð3 Þ3=2 shift has been carried out by considering this yet undiscarded 27 5 aB aB region. The outcome of this evaluation is summarized in the left panel of Fig. 6. Observe that the energy shift has been An adequate replacement of the radial wave functions in plotted in the form of log10ð−δε½meVÞ. The highest Eq. (39) by those above allows us to determine the value achieved for δε ∼ −10−6 meV corresponds to g ∼ corresponding modification of the transition energy: −2 −1 −2 10 GeV and m ∼ 10 GeV. Toward higher axion Z ∼ 10 α 2 2 ∞ ½ 2 − 13 masses m GeV and lower axion-diphoton couplings δε ≈− g m du u ∼ 10−6 −1 108π2 5 ½1 þ 3 4 g GeV the correction to the Lamb-shift tends to aB 1 u 2 uaBm −14 decrease significantly [δε ∼ −10 meV]. Both estimates 2 1 × 1 − þ : ð45Þ coincide with the values resulting from Eq. (42). The 1 þ 3 ½1 þ 3 2 smallness of these outcomes as compared with the afore- 2 uaBm 2 uaBm mentioned discrepancy rules out the corresponding virtual ≫ λ In the limit aB , the expression above is well approached ALPs as candidates to explain the Hμ anomaly. As we have 2 2 2 5 −7 by δε ≈−αg =ð4374π m a Þ. The expressions associated anticipated above Eq. (41), the chosen values for 10 fm ≲ B −4 with Hμ can be read off from the previous one by replacing g ≲ 10 fm are much smaller than r ≈ 0.876 fm. proton a → a =ð186Þ. Taking as a reference the undiscarded Clearly, the previous statements cannot be considered B B region used previously, 100 MeV ≲ m ≲ 10 GeV with conclusive as our estimation undergoes theoretical uncer- 10−2 −1 tainties arising from both the internal limitation of QED g< GeV ,Eq.(45) has been evaluated. The result A of this assessment is shown in the right panel of Fig. 6.Asin at short distances as well as the finite proton size. The 2 − 2 latter being closely related to the fact that the s-states the s1=2 p1=2 transition, the highest energy-shift linked to 3 − 3 penetrate the nucleus deeply even for the chosen g. This the p1=2 d1=2 transition in Hμ arises from the combina- −2 −1 −2 last problem can be relaxed if transitions between excited tion of g ∼ 10 GeV and m ¼ 10 GeV. In such a case −11 states with nonzero angular momentum are considered δε ∼ −10 meV, which is five orders of magnitude smaller instead. Although the problem of their relatively short than the outcome associated with the axion 2s1=2 − 2p1=2 lifetimes constitutes a major issue for their experimental Lamb-shift. It is worth emphasizing that, while the uncer- investigation, their measurements seem to be a priori a tainty introduced by the use of spherically symmetric orbitals reliable way to have a cleaner picture of whether a could be circumvented as described above, the one linked to certain ALP is the cause of the aforementioned the physics at distances shorter than g remains.

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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 −5 −6 −1 2 −1 10 GeV to 10 GeV. As previously, 10 GeV

D. Sensitivity to ALPs in high-precision bounds on the axion parameter space, an enhancement in hydrogen spectroscopy sensitivity of at least five orders of magnitude is required. It In this section we want to investigate whether the current is worth remarking that this sensitivity gap also manifests in sensitivity in atomic hydrogen can improve the existing other high-precision experiments searching for potential ’ bounds on the axion parameter space. To do this we deviations of the Coulomb s law as those of Cavendish- type. For further details we refer the reader to Appendix B. will analyse the respective energy-shift in both 2s1=2 − 2 1 − 2 This lack of sufficient sensitivity in the context of ALPs is p1=2 and s1=2 R s1=2 transitions. An expression for ∞ 2 2 2 significant when taking into account that these setups have the latter δε ¼ drr δUðrÞ½R ðrÞ − R ðrÞ can be 0 1s 2s allowed for constraining severely the parameter spaces of easily determined by taking into account the formula for other weakly interacting sub-eV particles, including para- R2 ðrÞ in Eq. (40) and the radial part of the 1s1 2-state: s = photons and minicharged particles. However, we should ð Þ¼2 ½− 3=2 R1s r exp r=aB =aB . Explicitly, emphazise that—in contrast to our investigation—these Z particle candidates have been treated within renormalizable αg2m2 ∞ du 1 δε ¼ − ½ 2 − 13 −gmu frameworks and, thus, the bounds have been established on 2 5 u e 1 2 48π a 1 u ½1 þ ua m dimensionless coupling constants. Likewise, we have B  2 B  1 1 2 3 already indicated below Eq. (16) that in the axion theory − 1 − þ : the quantity playing the corresponding role combines two 2 ½1 þ 2 1 þ 2½1 þ 2 uaBm uaBm uaBm unknown parameters ∼gm. Hence, the axion mass m ð46Þ suppresses the limits that can be inferred for g.

While the left panel in Fig. 7 shows the energy shift for IV. CONCLUSION 2s1 2 − 2p1 2 [see Eq. (41)], the one in the right depicts = = Within the effective framework of axion quantum the result associated with the 1s1 2 − 2s1 2 transition [see = = electrodynamics, terms beyond the minimal coupling of Eq. (46)]. Both evaluations have been carried out by two photons to a neutral pseudoscalar field have been used considering the region of the coupling 10−6 GeV−1

115008-12 AXION-MODIFIED PHOTON PROPAGATOR, COULOMB … PHYS. REV. D 98, 115008 (2018) to explain the proton radius anomaly in muonic hydrogen. the grounds that, if the photons couple to a conserved current μ μ By contrasting the experimental result with our theoretical j ðxÞ, i.e., pμj ðpÞ¼0, a term of this nature does not prediction, it was found that—up to the uncertainties contribute to the S-matrix elements. Manifestly, the photon caused by the nature of the transition and the internal Green function in Eq. (A1) resembles Eq. (3).However,the limitations of axion-electrodynamics—ALPs can be particle-ghost content linked to this expression is somewhat excluded as plausible candidates for solving the aforemen- blurred owing to the presence of the metric tensor gαβ.To tioned problem. highlight the emergence of the Pauli-Villars ghost—leaving Our investigation has revealed explicitly that neither aside those unphysical states linked to the quantization atomic spectroscopy nor experiments of Cavendish-type procedure that eventually must cancel each other—we will allow us to infer bounds that improve the existing con- follow a method that has been used previously within the straints on the axion parameter space. This fact contrasts context of quantum gravity [68,104,105].6 Rather than with analogous outcomes linked to scenarios containing dealing with the expression above directly, one introduces minicharged particles and hidden photon fields, in which 2 α 2 β the saturated Green function Gðp Þ¼j Gαβðp Þj and both precision techniques have turned out to be particularly 2 ¼0 2 ¼ ¯ 2 – investigates its residues at each pole: p and p mgh. valuable [34 37]. The loss of sensitivity within the axion 2 ≥ 0 j 2 2 0 — context is conceptually rooted in the nonrenormalizable As for any m the relation j p ¼m < holds see proof — — character of QEDA and manifests at the level of the of Lemma 1 in Ref. [68] a physical particle is linked to modified Coulomb potential Eq. (35)—through the dimen- a nonnegative residue of Gðp2Þ, whereas a ghost emerges sionless factor ∼gm. This ratio of scales accomplishing when the contrary occurs. For the case under consideration ð 2Þj 0 somewhat a role similar to the coupling strengths of then follows that ResG p p2¼0 > (photon) and the photon-paraphoton mixing χ and the parameter ϵ in ð 2Þj 0 ResG p p2¼m¯ 2 < (ghost). the minicharged particles scenario. To a certain extent the gh Noteworthy, the decompositions of the axion and photon described problem justifies the existing demand for new Green functions [see Eqs. (3) and (A1)] suggest that laboratory-based routes looking for ALPs [13,14] by using the effects of the higher-dimensional operators can be strong electromagnetic fields [15–18], e.g., those offered by formulated in terms of auxiliary—fictitious—fields. In this high-intensity lasers [91–101]. Let us finally remark that the expression for context, the action of interest reads μν Π ðp1;p2Þ [see Eq. (28)] constitutes an essential piece Z R 1 1 1 for a more general class of polarization tensors which 4 μν ¯ μ ¯ 2 ¯ 2 ¯ S ¼ d x − fμνf þ ∂μϕ∂ ϕ − m¯ ϕ − Φ□ϕ result when external electromagnetic fields polarize the eff 4 2 2 vacuum [80,91]. 1 1 1 1 2 2 2 2 μν ¯ ˜ μν þ m¯ Φ − m¯ þ fμν þ g¯ ϕ fμνf ; 2 s 2 ghA 2 F 4 ACKNOWLEDGMENTS ðA2Þ The authors thank A. B. Voitkiv and A. E. Shabad for useful discussions. S. V.-C. and C. M. gratefully acknowl- ¼ ∂ − ∂ ¯ 2 ¼ð¯b¯ Þ−2 edge funding by the German Research Foundation (DFG) where Fμν μAν νAμ and ms g ϕ . We remark under Grant No. MU 3149/5-1. that the equations of motion for the auxiliary fields are exact 1 1 APPENDIX A: PARTICLE-GHOST CONTENT OF Φ ¼ □ϕ¯ ¼ − ∂ μ ð Þ ¯ 2 ; Aλ ¯ 2 μf λ: A3 THE GAUGE SECTOR AND AN ALTERNATIVE ms mgh FOUR-FIELDS FORMULATION OF AXION-ELECTRODYNAMICS Hence, when integrating out both ΦðxÞ and AλðxÞ classically, S As mentioned in Sec. II A, the photon sector also contains i.e.,byremovingthemfrom eff using their equations of motion, we reproduce the action of axion-electrodynamics Pauli-Villars ghosts. In order to show this, let us consider the 2 ¯ S → S ¼ þ 2 corresponding Green function resulting from a covariant extended by terms proportional to g ,i.e., eff Sg¯ Sg¯ quantization of a¯ μðxÞ via a path integral representation. [see Eqs. (1) and (2)]. 1 2 ¯ 2 μ 2 ¯ → þ When fixing the gauge via L ¼ − ½ð1 þ g¯ b □Þ∂μa¯ Observe that, as a consequence of the shift a a A gauge 2 a ϕ¯ → ϕ − Φ it turns out to be [102,103]: and , the functional action in Eq. (A2) can be   written as 2 1 1 Gαβðp Þ¼ − þ g ; ðA1Þ 2 2 − ¯ 2 αβ 6 p p mgh We precise that higher-derivate operators in combination with nonlocal terms emerge in many other interesting theoretical ¯ 2 ¼ð¯b¯ Þ−2 scenarios, e.g., in construction of effective quantum field theories where mgh g a is the corresponding bare ghost mass. accounting for quantum conformal and chiral anomalies (see e.g., Here, a longitudinal contribution ∼pαpβ has been ignored on [106] and references therein).

115008-13 S. VILLALBA-CHÁVEZ, A. GOLUB, and C. MÜLLER PHYS. REV. D 98, 115008 (2018) Z 4 1 μν 1 μν 1 2 2 modified Coulomb potential a0ðs; q ¼ 1Þ [see Eq. (35)] S ¼ d x − fμνf þ FμνF − m¯ A eff 4 4 2 gh into the expression above. Notice that, similarly to the case 1 1 1 1 analyzed in Sec.R III B, theR integral overR s must be split into μ 2 2 μ 2 2 g þ ∂μϕ∂ ϕ − m¯ ðϕ − ΦÞ − ∂μΦ∂ Φ þ m¯ Φ r … ¼ … þ r … 2 2 2 2 s two parts: 0 ds 0 ds g ds . Ignoring the 1 contribution coming from the region ½0;g we obtain þ ¯ðϕ − ΦÞ½˜ μν þ 2 ˜ μν þ ˜ μν 4 g fμνf Fμνf FμνF : Z 2 ∞ −2bmu − −gmu γ ≈ g m du ½ 2 − 13 e e ab 2 6 u 96π 1 u b Clearly, the first term contained in this formula is the 2 Maxwell Lagrangian, while the combination of the two − 4 2 þ e bmu sinhðamuÞ þ Oðg m Þ : ðB2Þ remaining contributions in the first line somewhat looks a like the Proca Lagrangian, with the exception that the sign of its kinetic term is not the usual one. Likewise, the kinetic The integral that remains in this formula can be calculated portion linked to the ΦðxÞ field manifests an opposite sign analytically by using (3.351.4) in Ref. [87]. Since both b to the corresponding contribution of ϕðxÞ. Owing to the and a are macroscopic quantities, the conditions a; b; b − described feature the associated Hamiltonian is not positive a ≫ g hold and we can approximate the expression defined leading—upon quantization—to the absence of a above by ground state. The particle-ghost content of the theory is Z elucidated in this alternative formulation to the requirement g2m ∞ du γ ≈ ½ 2 − 13 −gmu ð Þ that free fields of particles (ghosts) have positive (negative) ab 2 6 u e : B3 96π b 1 u energy. Notice that Eq. (B3) is independent of the radius a of the APPENDIX B: SENSITIVITY TO ALPS IN inner sphere. When considering the limit mg ≪ 1 we EXPERIMENTS OF CAVENDISH-TYPE obtain Precision tests of Coulomb’s law via Cavendish-type g γ ≈ ; ðB4Þ experiments have severely constricted the parameter space ab 96π2b of hidden photons in the μeV mass regime [34,107]. Today, these setups also provide the best laboratory bounds on which does not depend on the axion mass m either. μ mini-charged particles in the sub- eV range [36]. Here, we With γab to our disposal, we can proceed to estimate the want to estimate the sensitivity of this type of experiments in sensitivity of this setup in the search for ALPs. For such a the context of ALPs. To this end, we consider a setup purpose, we use the benchmark parameters of the experi- containing two concentric spheres: an outer charged con- ment performed by Plimpton and Lawton [a ¼ 38 cm, — — ducting sphere characterized by a radius b and an b ¼ 46 cm] in which a margin for γab exists, provided it uncharged conducting inner sphere with a radius a. Only lies below jγ j ≲ 3 × 10−10 [109]. By making use of −1 ab if the electrostatic potential follows the r law, the potential Eq. (B4) we find difference between the spheres vanishes and the cavity is free of electromagnetic field. However, deviations from the g ≲ 6.7 × 107 GeV−1: ðB5Þ Coulomb potential like those induced by loop corrections [compare Eq. (35)] could lead to a nontrivial relative voltage — difference γ ¼j½uðQ;b;bÞ − uðQ;a;bÞ=uðQ;b;bÞj that We emphasize that as a consequence of the perturbative ab — can be detected. This observable depends on the potential of condition [see below Eq. (B3)] this result applies for ≪ 15 the charged sphere evaluated on its surface uðQ;b;bÞ as well m eV. Besides, it is trustworthy for axion wave- uð Þ lengths smaller than the typical length scale of the spheres as on the surface of the inner sphere Q;a;b . −6 In general, the electrostatic potential of a sphere with ∼0.1 m, i.e., for axion masses m ≫ 10 eV. We note that −6 radius b and charge Q evaluated at a distance r from its the constraint in Eq. (B5) for the mass region 10 eV ≪ center has the form m ≪ 15 eV has already been discarded by combining the experimental outcomes of collaborations such as PVLAS uð Þ¼ Q ½ ð þ Þ − ðj − jÞ and OSCAR [see Fig. 5]. Hence, the sensitivity in this Q;r;b 2 f r b f r b ; Zbr experiment of Cavendish-type is not high enough to r improve the existing bounds on the axion parameter space. fðrÞ¼ sA0ðsÞds; ðB1Þ 0 It is worth remarking that a more accurate version of this kind of experiments has been carried out by using four where A0ðsÞ is an arbitrary potential in which the charge concentric icosahedrons [110]. For obtaining first esti- of the pointlike particle must be set to unity [108].Now,to mates, they may be treated approximately as four concen- γ determine ab resulting from QEDA we insert the axion- tric spheres. In contrast to the setup of Plimpton and

115008-14 AXION-MODIFIED PHOTON PROPAGATOR, COULOMB … PHYS. REV. D 98, 115008 (2018)

Lawton, here a very high voltage is applied between the Consequently, when the condition md ¼ d=λ ≪ 1 is sat- ¼ 127 ¼ 94 7 outer two spheres with radii d cm and c . cm. isfied, the asymptotic expression for γabcd becomes inde- The voltage difference is measured between the two pendent of the axion mass and quadratic in g: ¼ 94 ¼ 60 internal ones, with radii b cm and a cm, which 2 g c are uncharged. This setup allows us to infer bounds for the γ ≈ abcd 96π2 bðc − dÞðd − bÞ ALPs parameters via the ratio between the voltage   differences: ð − Þ ð − Þ ð − Þ 1 − b d b − d d b þ db d b × ð − Þ ð − Þ ð − Þ : a d a c c b ac c a uðQ;c;bÞ − uðQ;d;bÞ− uðQ;c;aÞþuðQ;d;aÞ γ ¼ : ð Þ abcd 2uðQ;c;dÞ − uðQ;d;dÞ− uðQ;c;cÞ B8 ðB6Þ Since this formula applies for axion wavelengths larger than the typical length scale of the experiment, the out- We insert Eq. (35) into (B1) and evaluate the resulting comes resulting from it can be considered reliable as long formula in the various parameters contained in Eq. (B6).As as the interactions between ALPs and plausible fields/ a consequence, we end up with matter existing outside of the external icosahedron are negligible. Next, the aforementioned experiment achieves a   −16 2 Z precision jγ j≲2×10 [36,110]. Combining this value g mc ∞ 1 3 abcd γ ¼ 1 − −mdu abcd 2 du 2 e with Eq. (B8) we constraint g to lie below 48π δ 1 u    ð Þ ð Þ g ≲ 9.8 × 107 GeV−1 for m ≪ 10−7 eV: ðB9Þ d − δ sinh mbu sinh mau × 1 − e m u − ; ðB7Þ c b a Noteworthy, despite the improvement in the experimental accuracy, the resulting upper limit turns out to be where δ ≡ c − d and terms of the order of ∼g4m2 have been comparable to the one found from the results of disregarded. Notice that, in contrast to Eq. (B2), the Plimpton and Lawton [see Eq. (B5)] and so, no improve- integrand above lacks terms involving ∼e−mgu. This could ment is found as compared with the existing constraints. be anticipated because the numerator of γabcd [see Eq. (B6)] The lack of sensitivity is understood here as a direct does not contain a potential evaluated at the surface of consequence of the quadratic dependence of γabcd on g the spheres [compare with γab given above Eq. (B1)]. [see Eq. (B8)].

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