Missouri University of Science and Technology Scholars' Mine

Physics Faculty Research & Creative Works Physics

01 Jun 2020

Fifth Force and Hyperfine Splitting in Bound Systems

Ulrich D. Jentschura Missouri University of Science and Technology, [email protected]

Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork

Part of the Physics Commons

Recommended Citation U. D. Jentschura, " and Hyperfine Splitting in Bound Systems," Physical Review A, vol. 101, no. 6, American Physical Society, Jun 2020. The definitive version is available at https://doi.org/10.1103/PhysRevA.101.062503

This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. PHYSICAL REVIEW A 101, 062503 (2020)

Fifth force and hyperfine splitting in bound systems

Ulrich D. Jentschura Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA

(Received 10 March 2020; accepted 5 May 2020; published 1 June 2020)

Two recent experimental observations at the ATOMKI Institute of the Hungarian Academy of Sciences (regarding the angular emission pattern of - pairs from nuclear transitions from excited states in 8Be and 4He) indicate the possible existence of a of a rest mass energy of roughly 17 MeV. The so-called X17 particle constitutes a virtual state in the process, preceding the emission of the electron-positron pair. Based on the symmetry of the nuclear transitions (1+ → 0+ and 0− → 0+), the X17 could either be a vector, or a pseudoscalar particle. Here, we calculate the effective potentials generated by the X17, for hyperfine interactions in simple atomic systems, for both the pseudoscalar as well as the vector X17 hypotheses. The effective Hamiltonians are obtained in a general form which is applicable to both electronic as well as muonic bound systems. The effect of virtual annihilation and its contribution to the hyperfine splitting also is considered. Because of the short range of the X17-generated potentials, the most promising pathway for the observation of the X17-mediated effects in bound systems concerns hyperfine interactions, which, for S states, are given by modifications of short-range (Dirac-δ) potentials in coordinate space. For the pseudoscalar hypothesis, the exchange of one virtual X17 quantum between the bound and the nucleus leads to hyperfine effects, but does not affect the Lamb shift. Effects due to the X17 are shown to be drastically enhanced for muonic bound systems. Prospects for the detection of hyperfine effects mediated by X17 exchange are analyzed for muonic deuterium, muonic hydrogen, , true muonium (μ+μ− bound system), and .

DOI: 10.1103/PhysRevA.101.062503

I. INTRODUCTION general terms, we ask the question of which effects could be expected from a potential “light pseudoscalar Higgs”-type For decades, atomic physicists have tried to push the accu- particle in atomic spectra, as envisioned in Ref. [13]. racy of experiments and theoretical predictions of transitions Somewhat unfortunately, the rest mass of 16.7 MeV makes in simple atomic systems higher [1]. The accurate measure- the X17 particle hard to detect in atomic physics experiments. ments have led to stringent limits on the time variation of The observed X17 rest mass energy is larger than the binding fundamental constants [2–4], and enabled us to determine a energy scale for both electronic as well as muonic bound number of important fundamental physical constants [5] with systems [16]. Even more importantly, the Compton wave- unprecedented accuracy. Yet, a third motivation (see, e.g., length of the X17 particle (about 11.8 fm) is smaller than the Refs. [6,7]), hitherto not crowned with success, has been the effective Bohr radius for both electronic as well as muonic quest to find signs of a possible low-energy extension of the bound systems. Because the Compton wavelength of the X17 , based on a deviation of experimental results particle determines the range of the Yukawa potential, the and theoretical predictions. effects of the X17 are hard to distinguish from nuclear-size Recently, the possible existence of a fifth-force particle, effects in atomic spectroscopy experiments [16]. 4 commonly referred to as the “X17” particle because of the We recall that the Bohr radius amounts to a0 ∼ 5 × 10 observed rest mass of 16.7 MeV, has been investigated in fm, while the effective Bohr radius of a muonic hydrogen Refs. [8–10], based on a peak in the emission spectrum of is a0 ∼ h¯/(αmμc) ∼ 256 fm. It is thus hard to find an electron-positron pairs in nuclear transitions of excited atomic system, even a muonic one, where one could hope and beryllium nuclei. Two conceivable theoretical explana- to distinguish the effect of the X17 particle on the Lamb tions have been put forward, both being based on low-energy shift from the nuclear-finite-size correction to the energy. A additions to the Standard Model. The first of these involves a possible circumvention has been discussed in Ref. [16], based vector particle (a “massive, dark ”; see Refs. [11,12]), on a muonic carbon ion, where the effective Bohr radius and the second offers a pseudoscalar particle (see Ref. [13]), approaches the range of the Yukawa potential induced by the which couples to light as well as . X17, in view of the larger nuclear charge number. However, it The findings of Refs. [8–10] have not yet been confirmed was concluded in Ref. [16] that considerable additional effort by any other experiment and remain to be independently would be required in terms of an accurate understanding of verified (for an overview of other experimental searches and nuclear-size effects, before the X17 signal could be extracted conceivable alternative interpretations of the ATOMKI re- reliably. sults, see Refs. [14,15]). However, we believe that, with the The definition of the Lamb shift L, as envisaged in advent of consistent observations in two nuclear transitions in Ref. [17] and used in many other places, e.g., in Eq. (67) of 8Be and 4He, it is justified to carry out a calculation of the Ref. [18], explicitly excludes hyperfine effects. Conversely, effects induced by the X17 in atomic systems. In more hyperfine effects, at least for S states, are induced, in leading

2469-9926/2020/101(6)/062503(13) 062503-1 ©2020 American Physical Society ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020) order, by the Dirac-δ peak of the magnetic dipole field of where we follow the conventions delineated in the remarks the at the origin (see Eq. (9) of Ref. [19]). following Eq. (1) of Ref. [11] and Eq. (10) of Ref. [12]. Here, ε ε The Fermi contact interaction, which gives rise to the leading- f and N are the flavor-dependent coupling√ parameters for order contribution to the hyperfine splitting for S states, is pro- the fermions and , while e =− 4πα =−0.091 is portional to a Dirac δ in coordinate space, commensurate with the electron charge. The and field operators the fact that the atomic nucleus has a radius not exceeding (the latter, interpreted as field operators for the composite the femtometer scale. The effect of short-range potentials is ) and denoted as ψ f and ψN , while the Xμ is the X17 thus less suppressed when we consider the hyperfine splitting, field operator. For reasons which will become obvious later, as compared to the Lamb shift. The aficionados of bound we use, in Eq. (1), the alternative conventions, states thus realize that, if we consider hyperfine effects, we  = ε ,  = ε , have a much better chance of extracting the effect induced h f f e hN N e (1b) by the X17, which, on the ranking scale of the contributions, for the coupling parameters to the hypothetical X17 vector occupies a much higher place than for the Lamb shift alone. boson. Our conventions imply that for εN > 0, the coupling  Despite the large mass mX of the X17 particle, which leads to parameter h parametrizes a “negatively charged” nucleon − N a short-range potential proportional to exp( mX r) (in natural under the additional U (1) gauge group of the vector X = =  = units withh ¯ c 0 1, which are used throughout the particle. current paper), the effect of the X17 could thus be visible in According to a remark following the text after Eq. (9) of the hyperfine splitting in muonic . Ref. [12], conservation of X charge implies that the couplings Here, we elaborate on this idea, and derive the leading cor- to the and currents fulfill the relationships rections to the hyperfine splitting of nS, nP / , and nP / states 1 2 3 2 ε = ε + ε ,ε= ε + ε , in ordinary as well as muonic hydrogenlike systems, due p 2 u d n u 2 d (2) to the X17 particle, by matching the nuclear-spin-dependent where the up and down couplings are denoted by the terms in the scattering amplitude with the effective Hamilto- subscripts u and d. Numerically, one finds (see the detailed nian. Anticipating some results, we can say that the relative discussion around Eqs. (38) and (39) of Ref. [12]) that the correction (expressed in terms of the leading Fermi term) is electron-positron field coupling ε needs to fulfill the relation- / e proportional to mr mX , where mr is the reduced mass of the ship two-body bound system, while m is the X17 boson mass. The X × −4 <ε < . × −3. effect is thus enhanced for muonic in comparison to electronic 2 10 e 1 4 10 (3) bound systems. Furthermore, in order to explain the experimental observa- This paper is organized as follows. In Sec. II,wesum- tions [8,9], one needs the neutron coupling to fulfill (see marize the interaction Lagrangians for both a hypothetical Eq. (10) of Ref. [11]) X17 vector exchange [11,12], as well as a pseudoscalar ex- |ε |=|ε + ε |≈ 3 ε ≈ 1 . change [13], with corresponding conventions for the coupling n u 2 d 2 d 100 (4) parameters. In Sec. III, we derive the effective hyperfine Because the hypothetical X vector particle acts like a “dark Hamiltonians for both vector and pseudoscalar exchanges. In photon” which is hardly distinguishable from the ordinary Sec. IV, we evaluate general expressions for the corrections photon in the high-energy domain, the proton coupling εp is to hyperfine energies induced by the X17 particle, for S highly constrained. According to Eqs. (8) and (9) of Ref. [12], and P states. In Sec. V, we derive bounds on the coupling and Eq. (35) of Ref. [12], one needs to have parameters for both models in the sector, based on the |ε |=| ε + ε |  × −4. muon g factor. Finally, in Sec. VI, we apply the obtained re- p 2 u d 8 10 (5) sults to muonic hydrogen, muonic deuterium, muonium, true This is why the conjectured X17 is referred to as μ+μ− muonium (bound system), and positronium. We also “protophobic” in Refs. [11,12]. discuss the measurability of the X17 effects in the hyperfine Following Ref. [13], we write the interaction Lagrangian structure of the mentioned atomic systems. Conclusions are for the fermions interacting with the pseudoscalar candidate reserved for Sec. VII. of the X17 particle as follows: 5 5 II. INTERACTION LAGRANGIANS LX,A =− h f ψ¯ f i γ ψ f A − hN ψ¯N i γ ψN A, (6) In the following, we intend to study both the interaction f N of X17 vector and pseudoscalar particles with bound where A is the field operator of the pseudoscalar field. Inspired ( and ) and nucleons ( and deuterons). by an analogy with putative pseudoscalar Higgs couplings Vector interactions are denoted by the subscript V , while pseu- [20], the pseudoscalar couplings have been estimated in doscalar interactions carry the subscript A, as is customary Refs. [13,20] to be of the functional form in the literature. We write the interaction m = ξ f , = ξ mN , Lagrangian LX,V for the interaction of an X17 vector boson h f f hN N (7) = ,μ v v with the fermion fields f e (electron and muon) and the = nucleons N = p, n (proton and neutron) as follows: where v 246 GeV is the vacuum expectation value of the Higgs (or Englert-Brout-Higgs [21,22]) field, m f is the μ μ LX,V =− ε f e ψ¯ f γ ψ f Xμ − εN e ψ¯N γ ψN Xμ, fermion mass, and mN is the nucleon’s mass. Furthermore, the f N parameters ξ f and ξN could in principle be assumed to be of (1a) order unity. Note that the spin parity of the Standard Model

062503-2 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020)

f f f f (we may ignore the frequency of the photon in the order of approximation relevant for the current article): 1 D (q) =− , (12a) 00 q2 + m2 X A X 1 qi q j D (q) =− δij − . (12b) ij 2 + 2 2 + 2 q mX q mX NN NN The derivation of the massive vector boson propagator in the Coulomb gauge, which is best adapted to bound-state (a) (b) calculations and involves a certain subtlety is discussed in the Appendix. The pseudoscalar propagator is used in the FIG. 1. The one-quantum exchange scattering amplitude for the normalization X17 particle is matched against the effective Hamiltonian, for (a) the 1 D (q) =− , (13) vector hypothesis and (b) the pseudoscalar hypothesis. The arrow of A q2 + m2 time is from left to right. X where we also ignore the frequency. The scattering amplitude for the X17 vector particle reads as has recently been determined to be consistent = {  γ 0  γ 0 with a scalar, not pseudoscalar, particle [23], but it is still M fi,V h f hN (¯u f u¯ f )(¯uN u¯N ) D00 intuitively suggested to parametrize the couplings to the novel + (¯u γ i u¯ )(¯u γ j u¯ ) D }, (14) putative pseudoscalar X17 in the same way as one would f f N N ij otherwise parametrize the couplings to the Higgs particle. and According to Eq. (2.7) and the remark following Eq. (3.12) =  γ 5  γ 5 , M fi,A h f hN (¯u f i u¯ f )(¯uN i u¯N ) DA (15) of Ref. [13], the nucleon couplings can roughly be estimated as for the pseudoscalar case. Here, we denote the final states of     the scattering process by a prime, u = u f (p ), u = uN (p ), mp −3 f f N N hp = (−0.40 ξu − 1.71 ξd ) ≈−2.4 × 10 , (8a) =  =  v while the initial states are u f u f (p f ) and uN uN (pN ), and m the bar denotes the Dirac adjoint. Analogous definitions are = n − . ξ + . ξ ≈ . × −4,  hn ( 0 40 u 0 85 d ) 5 1 10 (8b) used for the w f ,N and w f ,N in Eq. (11). Furthermore, we have v    p + p = p + p . The momentum transfer is q = p − where we have assumed ξ ≈ ξ ≈ 0.3. For the electron- f N f N f u d  =  −   positron field, based on other constraints detailed in Ref. [13], p f pN p N . one has to require that (see Eq. (4.2) of Ref. [13]) The form (11) is valid for the bispinors if the Dirac equa- tion is solved in the Dirac representation of the Dirac matrices, ! ! 4 me −6 1 × 0 σ i ξe > 4, he > = 8.13 × 10 . (9) γ 0 = 2 2 ,γi = 0 , v −σ i 0 −12×2 0 Based on a combination of experimental data [24] and theo- 0 1 × retical considerations [25–27], one can also derive an upper γ 5 = 2 2 . (16) bound, 12×2 0 ! ! 500 m The scattering amplitudes are matched against the effective ξ < 500, h < e = 10−3, (10) e e v Hamiltonian by the relation =− + +  ,  ,  , which is used in the following. M fi (w f wN )U (p f pN q)(w f wN ) (17)

where U (pf , pN , q) is the effective Hamiltonian. The scatter- III. MATCHING OF THE SCATTERING AMPLITUDE ing amplitude M fi is a matrix element involving four spinors, In order to match the scattering amplitude (see Fig. 1) with two of which represent the final and initial states of the two- the effective Hamiltonian, we use the approach outlined in particle system. Chap. 83 of Ref. [28], but with a slightly altered normalization The scattering amplitude, evaluated between four spinors for the propagators, better adapted to a natural unit system (cf. Eq. (83.8) of Ref. [28]), must now be matched against (¯h = c = 0 = 1). Specifically, we use the bispinors in the a Hamiltonian which acts on only one wave function in representation (cf. Eq. (83.7) of Ref. [28]) the end. We need to remember that the scattering amplitude ⎛  ⎞ 2 corresponds to a matrix element of the Hamiltonian. Any p f ,N 1 − w , matrix element of the Hamiltonian, even in the one-particle = ⎝ 8m2 f N ⎠, u f ,N (11) setting, is sandwiched between two wave functions, not one. σ·p  w f ,N = − = 2m Then, going into the center-of-mass frame q p f pf −  where f , N stands for the bound fermion, or the nucleus, pN p N means that the wave function is written in terms  and w f ,N are the nonrelativistic spinors. Of course, two two- of a center-of-mass coordinate R, and a relative coordinate r. component spinors constitute the four-component bispinor In the center-of-mass frame, one eliminates the dependence  u f ,N of the same field. The massive photon propagator (for the on the center-of-mass coordinate R and the total momen-  X17 vector hypothesis) is used in the following normalization tum P = pf + pN . Fourier transformation under the condition

062503-3 ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020)

 +  = + p f p N pf pN leads to the effective Hamiltonian (cf. IV. HYPERFINE STRUCTURE CORRECTIONS Eq. (83.15) of Ref. [28]). A. X17 boson exchange For the record, we note that in the X17 vector case, the 00 component of the photon propagator gives the leading, spin- In order to analyze the S-state hyperfine splitting, we independent term in the effective Hamiltonian, extract from Eqs. (19) and (22) the terms which are nonzero   when evaluated on a spherically symmetric wave function. h hN H = f exp(−m r). (18) This entails the replacements 0 4πr X 1   2  → →− σf · r σN · r → r σf · σN , σN · L → 0, (23a) Under the replacements h f e and hN e, in the mass- 3 less limit m → 0, one recovers the Coulomb potential, H → X 0   2 2 α h h σf · σN m − e =− f N (3) X −mX r π . One finally extracts the terms responsible for H , →− πδ r − , 4 r r HFS V π 4 ( ) e the hyperfine structure, i.e., those involving the nuclear spin 24 m f mN r operator σN , and obtains the following hyperfine Hamiltonian (23b) for a vector X17 particle: 2 h f hN σf · σN m −   → πδ(3) − X mX r . h h π HHFS,A 4 (r) e f N 8 (3) π H , = − δ (r) σ · σ 48 m f mN r HFS V 16 π m m 3 f N f N (23c) 2 2 m (σf · r σN · r − r σf · σN ) − − X e mX r The expectation value of the Fermi Hamiltonian is r3 α α 3 3 2 (Z ) mr 3 σ · r σ · r − r σ · σ EF (nS) =nS1/2|HF |nS1/2 =gN σf · σN . f N f N −mX r 3 − (1 + m r) e 3 n m f mN X r5 (24)  = / + m f σN · L − Here, Z is the nuclear charge number, and mr m f mN (m f − 2 + (1 + m r) e mX r . (19) m X r3 mN ) is the reduced mass of the system. We use the nuclear g N factor in the normalization Taking the limit mX → 0, and replacing |e| σ μ = N , g (−e) g |e| N gN (25)  → ,  → N = N , 2 = πα, 2mN 2 h f e hN e 4 (20) 2 2 which can more easily be extended to more general two-body one recovers the Fermi Hamiltonian HF (see Eq. (10) of systems than a definition in terms of the nuclear magneton. = . ... Ref. [19]), For the proton, one has gp 5 5856 as the proton’s g fac- = . ... α π tor [29,30], while definition (25) implies that gd 1 713 gN (3) + − HF = σf · σN δ (r) for the deuteron [5]. For true muonium (μ μ bound system) m f mN 3 and positronium, one has g = 2 according to the definition N 2  (25). 3 σf · r σN · r − r σf · σN σN · L + + , (21) By contrast, in the limit mX →∞, one verifies that 8 r5 4 r3 2 m − lim X e mX r = 4πδ(3)(r), (26) →∞ where we have ignored the reduced-mass correction pro- mX r  portional to m f /mN in the σN · L term in Eq. (19). For a and the two Hamiltonians given in Eqs. (23b) and (23c) vanish pseudoscalar exchange, one has π in the limit of an infinitely heavy X17 particle. This implies h f hN 4 (3) H , = δ r σ · σ that the expectation values of S states of the Hamiltonians HFS A π ( ) f N 16 m f mN 3 in Eqs. (23b) and (23c) have to carry at least one power of m2 σ · r σ · r mX in the denominator, and in particular, that the correction X f N −mX r − e + (1 + mX r) to the hyperfine energy will be of order α(Zα)4, not α(Zα)3, r3 as one would otherwise expect from the two individual terms 2 3 σf · r σN · r − σf · σN r − × e mX r . (22) in Eqs. (23b) and (23c). For the vector hypothesis, one finds 5 r that EX,V (nS1/2 ) =nS1/2|HHFS,V |nS1/2 , for the leading and subleading terms in the expansion in inverse powers of m , Note that the Hamiltonian given in Eq. (22) constitutes the X can be expressed as complete Hamiltonian derived from pseudoscalar exchange, γ 5 α 4 4 α 5 which, in view of the matrix in the Lagrangian given in   2(Z ) mr 5(Z ) E , (nS / ) = − + Eq. (6), contributes only to the hyperfine splitting, but not to X V 1 2 h f hN 3 3 3πn m f mN mX 3πn the Lamb shift, in leading order [i.e., via the exchange of one 5 , as given in Fig. 1(a)]. For a deuteron nucleus, 1 mr × 1 + σ · σ , . σ   2 2 f N S1/2 F thespinmatrix N has to be replaced by 2 IN , where IN is 5n m f mN mX the spin operator of the deuteron, corresponding to the spin-  (27) 1 particle. Important bounds on the coupling parameters hμ and hμ can be derived from the muon anomalous magnetic We have neglected relative corrections of higher than first or- moment (see Fig. 2). der in α m f /mX and m f /mN . For the pseudovector hypothesis,

062503-4 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020) one finds that EX,A(nS1/2 ) =nS1/2|HHFS,A|nS1/2 is given as Expressed in terms of the leading term, one obtains the follows: following corrections due to the X17 particle for nP3/2 α 4 4 α 5 (Z ) mr 5(Z ) states: EX,A(nS1/2 ) = h f hN − 3πn3 m m m 6πn3   f N X E , (nP / ) 3h hN Zm 1 Zαm X V 1 2 ≈− f r 1 − r , (34a) 5 π 2 1 mr EF (nP1/2 ) gN mX n mX × 1 + σ · σ , . 2 2 f N S1/2 F 5n m f mN mX E , (nP / ) 3h h Zm 1 Zαm X A 1 2 ≈− f N r 1 − r . (34b) 2 (28) EF (nP1/2 ) 2gN π mX n mX The S-state splitting is obtained from the following expecta- Parametrically, these are suppressed with respect to the re- tion values: sults for S states, by an additional factor Zαmr/mX .For σ · σ = , σ · σ =− . the nP3/2 states, one considers the corrections EX,V (nP3/2 ) = f N S1/2,F=1 1 f N S1/2,F=0 3 (29) nP3/2|HHFS,V |nP3/2 and EX,A(nP3/2 ) =nP3/2|HHFS,A|nP3/2 , Expressed as a relative correction to the leading term, given with the results in Eq. (21), one has the following corrections due to the X17 α 5 5 particle, (Z ) 1 mr EX,V (nP3/2 ) =− 1 − 2h h 12πn3 n2 m2 m2 EX,V (nS1/2 ) ≈− f N Zmr , N X (30a)   / π × σ · σ , EF (nS1 2 ) gN mX h f hN f N nP3/2,V (35a) EX,A(nS1/2 ) h f hN Zmr α 6 6 ≈ , (30b) 2(Z ) 1 mr EX,A(nP3/2 ) = 1 − EF (nS1/2 ) gN π mX π 3 2 3 45 n n m f mN mX depending on the vector (V ) or pseudoscalar (A) hypothesis. × σ · σ . h f hN f N nP3/2,A (35b) One notices the different sign of the correction, depending on the symmetry group of the new particle. We observe that Here, the expectation values are 5 the relative correction to the Fermi splitting is enhanced for σ · σ , = = , σ · σ , = =− . f N P3/2 F 2 1 f N P3/2 F 1 3 (36) muonic bound systems, by a factor mr/mX ∼ mμ/mX as com- pared to electronic bound systems, because the corresponding The leading term in the hyperfine splitting for nP3/2 states is factor me/mX is two orders of magnitude smaller. well known to be equal to For nP1/2 states, whose wave function vanishes at the E (nP / ) =nP / |H |nP / nucleus in the nonrelativistic approximation, one finds for F 3 2 3 2 F 3 2 3 3 the first-order corrections E , (nP / ) =nP / |H , |nP / α (Zα) m X V 1 2 1 2 HFS V 1 2 = r σ · σ . = | | gN f N nP3/2 (37) and EX,A(nP1/2 ) nP1/2 HHFS,A nP1/2 , 15 n3 m m f N α 5   (Z ) 1 Expressed in terms of the leading term, one obtains the E , (nP / ) = 1 − X V 1 2 h f hN 3 2 πn n following corrections due to the X17 particle for nP1/2 5 states: mr × σ · σ , ,   2 f N nP1/2 F (31a) m m m EX,V (nP3/2 ) 5h f hN Zmr m f 1 Zαmr f N X ≈− 1 − , π 2 (Zα)5 1 EF (nP3/2 ) 4gN mX mN n mX E , (nP / ) = h h 1 − X A 1 2 f N πn3 n2 (38a) 2 α 2 m5 EX,A(nP3/2 ) 2h f hN Zmr 1 Z mr × r σ · σ . ≈ 1 − . (38b) f N nP / ,F (31b) 2 2 1 2 EF (nP3/2 ) 3gN π mX n mX m f mN mX In these results, matrix elements of tensor structures pro- Parametrically, in comparison to nP1/2 states, the correction portional to σ · r σ · r in Eqs. (19) and (22) have been for nP3/2 states is suppressed for a vector X17 by an additional f N / reduced to simpler structures σ · σ by angular algebra factor m f mN , while for a pseudoscalar X17, the suppres- f N α / reduction formulas, which are familiar in atomic physics sion factor is Z mr mX . For electrons bound to protons and  →  → other nuclei, both suppression factors are approximately of [31]. Under the replacement h f h f and hN hN ,the correction, for a vector X17 particle, assumes the same form the same order of magnitude, while for muonic hydrogen as for the pseudoscalar hypothesis, up to an additional overall and deuterium, the vector contribution dominates over the factor 1/2. For nP1/2 states, the expectation values are pseudoscalar one. 1 σ · σ , = =− , σ · σ , = = 1. (32) f N P1/2 F 1 3 f N P1/2 F 0 B. Virtual annihilation The leading term in the hyperfine splitting for nP1/2 states is For bound systems consisting of a particle and , well known to be equal to virtual annihilation processes also need to be considered (see Fig. 3). The resulting effective potentials are local (propor- E (nP / ) =nP / |H |nP / F 1 2 1 2 F 1 2 tional to a Dirac δ) and affect S states. We here consider α α 3 3 (Z ) mr positronium and true muonium ( f = e,μ), for which one has =−g σ · σ , . (33) N 3 f N nP1/2 F = = = / = 3 n m f mN Z 1, N f (antifermion), mr m f 2, mN m f .Inthis

062503-5 ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020)

X A The difference between the expectation values for ortho σ · σ = − and para states is the well-known result [ f f 1 (−3) = 4] 4 µ µ µ µ 7 α m f E (nS) =E (nS) + E ,γ (nS) = . (43) HFS F ANN 12 n3 The exchange of a virtual photon contributes a fraction of 4/7 to this result, while the virtual annihilation yields the remaining fraction of 3/7. (a) (b) The generalization of Eq. (40) to a vector X17 exchange is immediate, FIG. 2. The X17 particle induces vertex corrections to the (h )2 f (3) anomalous magnetic moment of the muon: (a) X17 vector hypothesis H , = (σ · σ + 3)δ (r), (44) ANN V 2 − 1 2 f f and (b) pseudoscalar hypothesis. The interaction with the external 8 m f 4 mX magnetic field is denoted by a zigzag line. with the expectation value (we select the term relevant to the hyperfine splitting) case, the Fermi energy, as defined in Eq. (24), assumes the  2 α 3 (h f ) ( m f ) form (it is no longer equal to the full leading-order result for E , (nS) = σ · σ . (45) ANN V 2 − 1 2 3 f f the hyperfine splitting, as we will see) 64 m f 4 mX n α4 m In the calculation, one precisely follows Eqs. (83.18)–(83.24) E (nS) = f σ · σ . (39) F, f f 12 n3 f f of Ref. [28] and adjusts for the mass of the X17 in the prop- agator denominator. The relative correction to the hyperfine When one replaces the vector X17 boson in Fig. 3(a) by a splitting due to annihilation channel, for a virtual vector X17 photon, one obtains the annihilation potential (see Eq. (83.24) particle (for S states), is of Ref. [28]),  2 2 πα EANN,V (nS) (h f ) m f (3) = . (46) HANN,γ = (σf · σ + 3)δ (r). (40) 2 1 2 2 f EANN,γ (nS) 4πα m − m 2m f f 4 X Based on the identity For the virtual annihilation into a pseudoscalar particle, one can also follow Eqs. (83.18)–(83.24) of Ref. [28], but σ · σ = + − , f f 2S(S 1) 3 (41) one has to adjust for the different interaction Lagrangian, which now involves the fifth current, and one also needs to where S = 1 for an ortho state, and S = 0 for a para state, adjust for the mass of the X17 in the pseudoscalar propagator one can see that the annihilation process into a virtual vector denominator. The result in Eq. (83.22) of Ref. [28]forthe particle is relevant only for ortho states, consistent with the Fierz transformation of the currents has to be adapted to the conservation of total angular momentum in the virtual transi- pseudoscalar current, i.e., to the last entry in Eq. (28.17) of tion to the photon, which has an intrinsic spin of unity. The Ref. [28]. The result is virtual annihilation contribution to the hyperfine splitting is 3h2 4 f (3) α m H , = (σ · σ − 1)δ (r). (47) f ANN A 2 1 2 f f EANN,γ (nS) = σf · σ . (42) 8 m − m 16 n3 f f 4 X The expectation value of this effective Hamiltonian is nonva- nishing only for para states (S = 0), as had to be expected f f f f in view of the pseudoscalar nature of the virtual particle (the intrinsic parity of para states of positronium and true X A muonium is negative, allowing for the virtual transition to the pseudoscalar X17). In contrast to Eq. (22), we observe a small f f f f shift of the hyperfine centroid for the fermion-antifermion system, due to the term that is added to the scalar product of (a) (b) the spin operators. The general expression for the expectation value in an nS state is (we select the term relevant to the FIG. 3. For bound systems consisting of a lepton and antilepton hyperfine splitting) there is an additional correction to the energy levels induced by 2 α 3 virtual annihilation (the arrow of time is from left to right). We 3h f ( m f ) E , (nS) = σ · σ . (48) here consider f = e (positronium) and f = μ (true muonium). The ANN A 2 − 1 2 3 f f 64 m mX n resulting effective potential is proportional to a Dirac δ and affects f 4 S states. (a) Diagram is relevant for orthopositronium and ortho The relative correction to the hyperfine splitting due to the true muonium (S = 1, annihilation into a vector X17 boson) and pseudoscalar annihilation channel, for S states, is = (b) diagram is relevant for para states (with total spin S 0, which 2 2 E , (nS) 3(h ) m allows for an annihilation into a hypothetical pseudoscalar X17 ANN A = f f . (49) πα 2 − 2 boson). Both virtual processes contribute to the hyperfine splitting. EANN,γ (nS) 4 m f mX

062503-6 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020)

For the fermion-antifermion bound system, the corrections value of aμ would increase too much beyond the experimental given in Eq. (30) specialize as follows: result.  2 According to Eq. (20) of Ref. [36], the vertex correction E , (nS) (h ) m X V ≈− f f , (50a) due to a virtual pseudoscalar X17 leads to the following E (nS) 2π m F X correction: 2 2 1 3 2 μ mμ dxx EX,A(nS) (h f ) m f =−(h ) ≈ . (50b) aμ 2 2 2 mμ mμ 2 E (nS) 4π m 4π m 0 − − + F X X (1 x) 1 2 x mX mX The relative correction to the total hyperfine splitting, due to =− . × −3 2. the X17 exchange and annihilation channels, is 1 19 10 (hμ) (56) Here, because the correction is negative and decreases the χ = 4 EX,V (nS) + 3 EANN,V (nS), V (nS) (51a) value of aμ, the experimental-theoretical discrepancy given 7 EF (nS) 7 EANN,γ (nS) in Eq. (52) can only be enhanced by the pseudoscalar X17 4 EX,A(nS) 3 EANN,A(nS) particle. If we demand that the discrepancy not be increased χA(nS) = + , (51b) 7 EF (nS) 7 EANN,γ (nS) beyond 6σ , then we obtain the condition that |hμ| could not . × −4 with the individual contributions given in Eqs. (30), (46), (49), exceed a numerical value of 3 8 10 . In the following, we and (50). take the maximum permissible value of −4 hμ = (hμ)max = 3.8 × 10 (57) V. X17 PARTICLE AND MUON ANOMALOUS in order to estimate the magnitude of corrections to the MAGNETIC MOMENT hyperfine splitting in muonic bound systems, induced by a One aim of our investigations is to explore the possibility hypothetical pseudoscalar X17 particle. of a detection of the X17 particle in the hyperfine splitting of muonic bound systems. To this end, it is instructive to derive VI. NUMERICAL ESTIMATES  upper bounds on the coupling parameters hμ and hμ,forthe AND EXPERIMENTAL VERIFICATION muon. The contribution of a massive pseudoscalar loop to the A. Overview muon anomaly [see Fig. 2(b)] has been studied for a long time [32–36], and recent updates of theoretical contributions The relative corrections to the hyperfine splitting due to [37–39] has confirmed the existence of a (roughly) 3.5σ the X17 particle, expressed in terms of the leading Fermi discrepancy of theory and experiment. The contribution of interaction, for S and P states, are given in Eqs. (30), (34a), a massive vector exchange [see Fig. 2(a)] has recently been (34b), (38a), and (38b). All of the formulas involve at least / revisitedinRef.[36]. Specifically, the experimental results for one factor of mr mX , and so experiments appear to be more the muon anomaly aμ (see Eqs. (1.1) and (3.36) of Ref. [38]) attractive for muonic rather than electronic bound systems. are as follows: Furthermore, parametrically, the corrections are largest for (expt) = . , S states, which is understandable because the range of the aμ 0 00116592091(54)(33) (52) X17 potential is limited to its Compton wavelength of about (theor) = . . 11.8 fm, and so its effects should be more pronounced for aμ 0 001165918204(356) (53) states whose probability density does not vanish at the origin, (expt) (theor) −9 i.e., for S states. This intuitive understanding is confirmed by The 3.7σ discrepancy aμ –aμ ≈ 2.7 × 10 needs to be explained. our calculations. Note that the formulas for the corrections to According to Eq. (41) of Ref. [36], we have the following the hyperfine splitting, given in Eqs. (27), (28), (31a), (31b), correction to the muon anomaly due to the vector X vertex (35a), and (35b), are generally applicable to bound systems correction in Fig. 2(a), with a heavy nucleus, upon a suitable reinterpretation of the σ  2 2 1 2 nuclear spin matrix N in terms of a nuclear spin operator. (hμ) mμ dxx (2 − x) aμ = 2 2 m2 m2 8π m 0 − − μ + μ X (1 x) 1 2 2 x B. Muonic deuterium mX mX −3  2 In view of a recent theoretical work [40] which describes = 8.64 × 10 (hμ) , (54) a5σ discrepancy of theory and experiment for muonic where we have used the numerical value mX = 16.7 MeV. The deuterium, it appears indicated to analyze this system first. following numerical value Indeed, the theory of nuclear-structure effects in muonic   −4 hμ = (hμ)opt = 5.6 × 10 (55) deuterium has been updated a number of times in recent years [40–42]. Expressed in terms of the Fermi term, the is “optimum” in the sense that it precisely remedies the dis- discrepancy δEHFS(2S) observed in Ref. [40] can be written crepancy described by Eq. (52) and will be taken as the input as datum for all subsequent evaluations of corrections to the EHFS(2S1/2 ) hyperfine splitting in muonic bound systems. Note that, even = 0.0094(18). (58) E (2S / ) if the vector X17 particle does not provide for an explanation F 1 2 of the muon anomaly discrepancy, the order of magnitude In order to evaluate an estimate for the correction due to  of the coupling parameter hμ could not be larger than the the X17 vector particle, we observe that the interaction is value indicated in Eq. (55), because otherwise, the theoretical protophobic [11,12]. Hence, we can assume that the sign of

062503-7 ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020) the coupling parameter of the deuteron is approximately equal experiments. Results for the Sternheim [44] weighted dif- 3 to that of the neutron. We will assume the opposite sign for the ferences [n EX,V (nS1/2 ) − EX,V (1S1/2 )]/EF (1S1/2 ) and cor- 3 coupling parameter of the deuteron (neutron), as compared to respondingly [n EX,A(nS1/2 ) − EX,A(1S1/2 )]/EF (1S1/2 )areof the sign of the coupling parameter in Eq. (55). This choice is the same order of magnitude as for the individual P1/2 states. inspired by the opposite charge of the muon and nucleus with respect to the U (1) gauge group of quantum electrodynamics. C. Muonic hydrogen In view of Eq. (10) of Ref. [11] and Eq. (4) here, we thus have The considerations are analogous to those for muonic the estimate deuterium. However, the coupling parameter for the nucleus, 1 √ h ≈ h =− 4πα =−3.02 × 10−3. (59) for the protophobic vector model, is constrained by Eq. (5), d n 100 √ h ≈−8 × 10−4 4πα =−2.42 × 10−5, (63) Because of the numerical dominance of the proton coupling to p the pseudoscalar particle over that of the neutron [see Eq. (8)], which is much smaller than for the deuteron nucleus. The we estimate the pseudoscalar coupling of the deuteron to be of coupling parameter of the proton, for the pseudoscalar model, the order of can be estimated according to Eq. (8a). One obtains μ −3 ( p) h ≈ h =−2.4 × 10 . (60) EX,V (nS1/2 ) − d p ≈˙ 8.8 × 10 9, (64a) E (nS / ) In view of Eq. (30), we obtain the estimates F 1 2 (μp) (μd) EX,A (nS1/2 ) − / ≈− . × 7, EX,V (nS1 2 ) −6 ˙ 2 9 10 (64b) ≈˙ 3.8 × 10 , (61a) EF (nS1/2 ) EF (nS1/2 ) S (μd) for the -state effects, and E nS / X,A ( 1 2 ) −6 (μp) ≈−˙ 1.0 × 10 , (61b) / EX,V (nP1 2 ) −10 1 EF (nS1/2 ) ≈˙ 5.8 × 10 1 − , (65a) 2 EF (nP1/2 ) n where the symbol ≈˙ is used to denote an estimate for the (μp) quantity specified on the left, including its sign, based on EX,A (nP1/2 ) − 1 ≈˙ 1.8 × 10 8 1 − , (65b) 2 the estimates of the coupling parameters of the hypothetical EF (nP1/2 ) n vector and pseudoscalar X17 particle, as described in the for P / states. Results of the same order of magnitude as current work. As already explained, the modulus of our es- 1 2 for individual P / states are obtained for the Sternheim timates for the coupling parameters is close to the upper end 1 2 difference of S states. The effects, in muonic hydrogen, for of the allowed range; the same thus applies to the absolute the vector model are seen to be numerically suppressed. The magnitude of our estimates for the X17-mediated corrections contributions of the X17 particle need to be compared to the to hyperfine energies. Note that the vector X17 contribution proton structure effects, which have recently been analyzed in slightly decreases the discrepancy noted in Ref. [40], while Refs. [45–49]. According to Ref. [48], the numerical accuracy the hypothetical pseudoscalar effect slightly increases the of the theoretical prediction for the 2S hyperfine splitting discrepancy, yet on a numerically almost negligible level. in muonic hydrogen is currently about 72 ppm [E (2S) = Similar considerations, based on Eqs. (34a) and (34a), lead HFS 22.8108(16) meV]. to the following results for P states, (μd) E , (nP1/2 ) 1 D. Muonium X V ≈˙ 2.5 × 10−7 1 − , (62a) 2 EF (nP1/2 ) n Muonium is the bound system consisting of a positively + − (μd) charged antimuon (μ ) and an electron (e ). Its ground-state E , (nP1/2 ) 1 X A ≈˙ 6.6 × 10−8 1 − , (62b) hyperfine splitting has been studied in Ref. [50] with a result 2 EF (nP1/2 ) n ν(expt) = of HFS 4 463 302 765(53) Hz, i.e., with an accuracy of −8 which might be measurable in future experiments. Specifi- 1.2 × 10 . The theoretical uncertainty is about one order of magnitude worse and amounts to 1.2 × 10−7 (see Ref. [51]), cally, there is a nuclear-structure correction to the P1/2-state hyperfine splitting due to the lower component of the Dirac with the current status being summarized in the theoretical ν(theor) = nP1/2 wave function, which has S-state symmetry. However, prediction HFS 4 463 302 872(515) Hz. the lower component of the wave function is suppressed by Coupling parameters for the muon have been estimated in a factor α, which implies that the P1/2 state nuclear-structure Eqs. (55) and (57) for the vector and pseudoscalar models, correction is suppressed in relation to EF (nP1/2 ) by a factor respectively, and we take the antimuon coupling estimate as α2. An order of magnitude of the achievable theoretical uncer- the negative value of the coupling parameters for the muon. tainty for the P1/2-state result can be given by an appropriate For the coupling parameters, we use the maximum allowed scaling of the current theoretical uncertainty for S states, value for the electron in the vector model [see Eq. (3)], √ given in Eq. (58). The result is that the achievable theoretical  − − h ≈˙ 1.4 × 10 3 4πα = 4.2 × 10 4, (66) uncertainty should be better than 10−7, which would make e the effect given in Eq. (62) measurable. Also, according to and for the pseudoscalar model [see Eq. (10)], Ref. [43], the experimental accuracy should be improved me −3 −6 −7 h ≈˙ 500 = 1.0 × 10 . (67) into the range 10 –10 in the next round of planned e v

062503-8 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020)

One obtains the estimates in simple electronic and muonic bound systems. This notion (μμ) is based on two observations: E , (nS1/2 ) X V ≈˙ 2.3 × 10−9, (68a) (i) Hyperfine effects for S states are induced by a contact EF (nS1/2 ) interaction (the Fermi contact term) and, thus, naturally con- (μμ) fined to a distance range very close to the atomic nucleus. As E , (nS1/2 ) X A ≈−˙ 1.8 × 10−9. (68b) far as hyperfine effects are concerned, the virtual exchange of EF (nS1/2 ) an X17 is thus not masked by its small range. Because the reduced mass of muonium is close to the elec- (ii) In the pseudoscalar model [13], the one-quantum ex- tron mass, the effect of the X17 boson is parametrically change of an X17 exclusively leads to hyperfine effects, but suppressed. It will take considerable effort to increase ex- leaves the Lamb shift invariant. For a fermion-antifermion perimental precision beyond the level attained in Ref. [50]. system, this statement should be taken with a small grain of On the other hand, hadronic vacuum polarization effects are salt, namely, it holds up to the numerically tiny shift of the suppressed in muonium, and their uncertainty [52]islessthan hyperfine centroid, induced by the virtual annihilation channel the X17-induced effect in the hyperfine splitting of muonium. [see Eq. (47)]. This finding could be of interest irrespective of It is thus not completely hopeless to detect X17-induced whether the results of the experimental observations reported effects in muonium in the future. in Refs. [8–10] can be independently confirmed by other groups.

+ − We have derived limits on the coupling parameters of the E. True muonium (μ μ )system X17 particle in the muonic sector in Sec. V and compiled Taking into account the exchange and annihilation chan- estimates for the X17-induced effects in Sec. VI, for a number nels, and using the same coupling parameter estimates as for of simple atomic systems. The results can be summarized as muonium, one obtains the estimates [see Eq. (51)] follows. (a) We show in Sec. V that the pseudoscalar model χ ≈ . × −6,χ ≈ . × −6. V (nS) ˙ 1 3 10 A(nS) ˙ 2 1 10 (69) [13] enhances the experimental-theoretical discrepancy in The annihilation channel contribution numerically dominates the muon anomaly, while the vector model [11,12] could over the exchange channel. For S states, the contribution of eliminate it. Stringent bounds on the magnitude of the muon hadronic vacuum polarization in the annihilation channel has coupling parameters to the X17 particle can be derived based been improved to the level of 2 ppm, as a result of gradual on the muon anomaly. Note that the order of magnitude of progress achieved over the last decades (see Eq. (41) of the maximum permissible coupling to the pseudoscalar, given Ref. [53,54]aswellasRefs.[53,55,56], and the recent work in Eq. (57), also leads to a tension with the parametrization = ξ / = μ [57]). A very modest progress in the theoretical determination h f f (m f v)giveninRef.[13] (applied to the case f , of the hadronic vacuum-polarization contribution would make i.e., to the muon). Namely, the parametrization could be read the effect of the X17 visible. as suggesting a likely increase of the pseudoscalar coupling parameter with the mass of the particle. While ξe is bound from below by the condition ξe > 4 [see Eq. (9)] the corre- F. Positronium sponding parameter in the muon sector must fulfill ξμ < 0.9 Quite considerable efforts have recently been invested in [see Eq. (57)]. the calculation of the mα7 corrections to the positronium (b) The relative correction to the hyperfine splitting for hyperfine splitting, and related effects [58–71]. In positron- both S and P states is enhanced in muonic as compared to ium, effects of the X17 particle are suppressed in view of electronic bound systems by two orders of magnitude, in view the smaller reduced mass of the bound system. Under these of the scaling of the relative corrections with mr/mX , where assumptions, the estimates for S states are as follows [see mr is the reduced mass of the two-body bound system. Eq. (51)]: (c) In muonic deuterium, the correction, for S states, is of order 3.8 × 10−6 (vector X17) and −1.0 × 10−6 (pseu- χ (nS) ≈−˙ 3.6 × 10−9,χ(nS) ≈−˙ 5.1 × 10−9. (70) V A doscalar X17) in units of the Fermi energy, while the experi- The effects are thus numerically smaller than the mα7 effects mental accuracy for the S-state hyperfine splitting is of order − currently under study [59–71]. 10 3, and there is a 5σ discrepancy of theory and experiment, in view of a recent calculation of the nuclear polarizability effects [40]. One concludes that the experimental accuracy VII. CONCLUSIONS would have to be improved by three orders of magnitude The conceivable existence of the X17 particle [8–10]pro- before the effects of the X17 become visible, and the un- vides atomic physicists with a long-awaited opportunity to derstanding of the nuclear effects would likewise have to be detect a very serious candidate for a low-energy (fifth force) improved by a similar factor. addition to the Standard Model. The energy range of about (d) In muonic deuterium, for the hyperfine splitting of P1/2 17 MeV provides for a certain challenge from the viewpoint states, the X17-mediated correction to the hyperfine splitting of atomic physics; the range of the X17-induced interaction is of order 2.5 × 10−7 for the vector model, and of order potentials is smaller than the effective Bohr radii in electronic 6.6 × 10−8 for the pseudoscalar model. These effects are not and muonic bound systems. Rather than looking at the Lamb suppressed by challenging nuclear structure effects and could shift [16], we here advocate a closer look at the effects induced be measurable in the next round of experiments [43]. The by the X17 on the hyperfine splitting, for both S and P states, same applies to the Sternheim weighted difference [44]ofthe

062503-9 ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020) hyperfine splitting of S states, where the effect induced by the APPENDIX: COULOMB-GAUGE PROPAGATOR FOR X17 particle is of the same order of magnitude as for the P1/2 MASSIVE VECTOR splitting. Even if the Coulomb-gauge propagator for massive vector (e) In muonic hydrogen, because of the protophobic char- bosons, given in Eqs. (12a) and (12b), has been used in the acter of the vector model, effects of the vector X17 are − literature before (see Eq. (6) of Ref. [72] and Eqs. (16) and suppressed (order 10 9 for the S-state splitting and order − (17) of Ref. [73]), separate notes on its derivation can clarify 10 10 for the P-state splitting and the Sternheim weighted the role of the “Coulomb gauge” for massive vector bosons. difference). For the pseudoscalar model, the S-state splitting − Generally, in order to calculate the vector boson propagator is affected at relative order 10 7, and the P-state splitting as − in a specific gauge, one can write the relation of the vector well as the Sternheim difference are affected at order 10 8. potential Aμ to the currents Jν (in the gauge under investiga- These effects could be measurable in the future. tion), and observe that the operator mediating the relation is (f) For muonium, the situation is not hopeless: While the the propagator. However, one can also ask the question which effects induced by the X17 shift the hyperfine splitting only on − terms could possibly be added to the propagator, initially the level of 10 9, which is two orders of magnitude lower than obtained in a specific gauge, without changing the fields. current theoretical predictions [51], we can say that, at least, Let us start with the massless case, i.e., the photon. One the uncertainty in the theoretical treatment of the hadronic should remember [74] that the most general form of a gauge vacuum polarization [52] would not impede an experimental transformation of the photon propagator is [in relativistic detection of the X17. Still, it would seem that more attractive μ = ω,  possibilities exist in muonic systems. notation with k ( k)] (g) For true muonium, one expects effects (for the hy- −6 gμν 1 μ ν ν μ perfine splitting of S states) on the order of 10 for the Dμν (k) = + ( f k + f k ). (A1) 2 2 vector X17 model as well as the pseudoscalar model. These k 2k have to be compared to the uncertainty from the hadronic The first term is the Feynman gauge result. The added terms vacuum polarization, which has recently been improved to do not change the fields, because the term proportional to the level of 2 ppm [57]. This implies that a modest progress f μkν vanishes in view of current conservation, while the term in the determination of the R ratio, namely, the ratio of proportional to f ν kμ amounts to a gauge transformation of the the cross section of an electron-positron pair going into four-vector potential Aμ, as a careful inspection shows. The hadrons versus an electron-positron pair going into muons choice (see Eq. (9) of Ref. [57]), could render the effect visible in true k0 ki f 0 = , f i =− , muonium.   (A2) (h) For positronium, the effects of the X17 are suppressed k2 k2 by the small reduced mass of the system. They are bound leads to the Coulomb gauge. The fact that the fields remain − not to exceed the level of 10 9 for the vector model and unaffected holds even for a “noncovariant” form of the f μ, pseudoscalar models. Thus, even taking into account all the i.e., for a form where the f μ do not transform as components mα7 corrections currently under study [59–71], the detection of a four-vector under Lorentz transformations. of an X17-induced signal in the hyperfine splitting appears to In order to generalize the result to a massive vector particle, be extremely challenging in positronium. it is necessary to observe that, in a more general sense, One concludes that the most promising approach toward the “Coulomb gauge” for the calculation of bound states a conceivable detection of the X17 in high-precision atomic is defined to be the gauge in which the photon propagator physics experiments would probably concern the hyperfine component D00 is exactly static, i.e., has no dependence on splitting of P1/2 states in muonic deuterium, and the related the photon frequency. Otherwise, one would incur additional Sternheim difference, where the effects are enhanced because corrections in the Breit Hamiltonian, which is generated by of the large mass ratio of the reduced mass of the atomic the spatial components of the photon propagator. system to the mass of the X17, and nuclear structure effects The generalization of Eq. (A1) to the massive vector ex- are suppressed because the P-state wave function (as well as change reads as [gμν = diag(1, −1, −1, −1)] the weighted difference of the S states) vanishes at the ori- gin. Furthermore, very attractive prospects in true muonium gμν 1 μ ν ν μ Dμν (k) = + ( f k + f k ). (A3) [53–57] should not be overlooked. k2 − m2 2(k2 − m2 )

Here, m = mX denotes the vector boson mass, and we start 2 2 from the Feynman gauge result gμν /(k − m ) (see Eqs. ACKNOWLEDGMENTS (3.147) and (3.149) in Ref. [75], with λ = 1, in the notation The author acknowledges support from the National Sci- of Ref. [75]). Now, choosing ence Foundation (Grant No. PHY-1710856). Also, the au- k0 ki thor acknowledges utmost insightful conversations with K. f 0 = , f i =− , (A4) 2 + 2 2 + 2 Pachucki. Helpful conversations with A. Krasznahorkay, U. k m k m Ellwanger, S. Moretti, and W. Shepherd are also gratefully one (almost) derives the result given in Eqs. (12a) and (12b) acknowledged. The author also thanks the late Professor G. (and previously used in Eq. (6) of Ref. [72] and in Eqs. Soff for insightful discussions on general aspects of the true (16) and (17) of Ref. [73]), with one caveat. Namely, in muonium bound system. the spatial components of the propagator (denoted by the

062503-10 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020)

Latin indices ij), one replaces k2 = ω2 − k2 →−k2 in the The transition to the massive Coulomb gauge is well order of approximation of interest here. This is because the known to be useful in bound-state theory but is perhaps less frequency dependence of the propagator denominator leads to familiar in the particle physics community; pertinent remarks higher-order corrections, which, for the photon exchange, are are thus in order when it comes to the possible detection of a summarized in the Salpeter recoil correction [76]. new particle in low-energy experiments.

[1] T. W. Hänsch, Nobel Lecture: Passion for precision, Rev. Mod. [13] U. Ellwanger and S. Moretti, Possible explanation of the Phys. 78, 1297 (2006). electron positron anomaly at 17 MeV in 8Be transitions [2] M. Fischer, N. Kolachevsky, M. Zimmermann, R. Holzwarth, T. through a light pseudoscalar, J. High Energy Phys. 11 (2016) Udem, T. W. Hansch, M. Abgrall, J. Grunert, I. Maksimovic, S. 039. Bize, H. Marion, F. P. Dos Santos, P. Lemonde, G. Santarelli, P. [14] D. Banerjee et al. (NA64 Collaboration), Search for a Hypo- Laurent, A. Clairon, C. Salomon, M. Haas, U. D. Jentschura, thetical 16.7 MeV and Dark in the NA64 and C. H. Keitel, New Limits on the Drift of Fundamental Experiment at CERN, Phys.Rev.Lett.120, 231802 (2018). Constants from Laboratory Measurements, Phys.Rev.Lett.92, [15] B. Koch, X17: A new force, or evidence for a hard γ + γ 230802 (2004). process? arXiv:2003.05722. [3] R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, [16] U. D. Jentschura and I. Nándori, Atomic physics constraints on L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, the X boson, Phys.Rev.A97, 042502 (2018). K. Bongs, and P. Gill, Frequency Ratio of Two Optical Clock [17] J. Sapirstein and D. R. Yennie, Theory of hydrogenic bound Transitions in 171Yb+ and Constraints on the Time Varia- states, in Quantum Electrodynamics, Advanced Series on Di- tion of Fundamental Constants, Phys.Rev.Lett.113, 210801 rections in High Energy Physics Vol. 7, edited by T. Kinoshita (2014). (World Scientific, Singapore, 1990), pp. 560–672. [4] N. Huntemann, B. Lipphardt, C. Tamm, V.Gerginov, S. Weyers, [18] U. Jentschura and K. Pachucki, Higher-order binding correc-

and E. Peik, Improved Limit on a Temporal Variation of mp/me tions to the Lamb shift of 2P states, Phys. Rev. A 54, 1853 from Comparisons of Yb+ and Cs Atomic Clocks, Phys. Rev. (1996). Lett. 113, 210802 (2014). [19] U. D. Jentschura and V. A. Yerokhin, Quantum electrodynamic [5] P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATA recom- corrections to the hyperfine structure of excited S states, Phys. mended values of the fundamental physical constants: 2014, Rev. A 73, 062503 (2006). Rev. Mod. Phys. 88, 035009 (2016). [20] H.-Y. Cheng and C.-W. Chiang, Revisiting scalar and pseu- [6] D. Tucker-Smith and I. Yavin, Muonic hydrogen and MeV doscalar couplings with nucleons, J. High Energy Phys. 07 forces, Phys.Rev.D83, 101702(R) (2011). (2012) 009. [7] V. Barger, C.-W. Chiang, W.-Y. Keung, and D. Marfatia, Con- [21] F. Englert and R. Brout, Broken Symmetry and the Mass of straint on Parity-Violating Muonic Forces, Phys. Rev. Lett. 108, Gauge Vector , Phys.Rev.Lett.13, 321 (1964). 081802 (2012). [22] P. W. Higgs, Broken Symmetries and the Masses of Gauge [8] A. J. Krasznahorkay, M. Csatlós, L. Csige, Z. Gácsi, J. Gulyás, Bosons, Phys.Rev.Lett.13, 508 (1964). M. Hunyadi, I. Kuti, B. M. Nyakó, L. Stuhl, J. Timár, T. G. [23] G. Aad et al. (ATLAS Collaboration), Study of the spin and Tornyi, Zs. Vajta, T. J. Ketel, and A. Krasznahorkay, Observa- parity of the Higgs boson in diboson decays with the ATLAS tion of Anomalous Internal Pair Creation in 8Be: A Possible detector, Eur. Phys. J. C 75, 476 (2015). Indication of a Light, Neutral Boson, Phys.Rev.Lett.116, [24] A. Anastasi, D. Babusci, G. Bencivenni, M. Berlowski, 042501 (2016). C. Bloise, F. Bossi, P. Branchini, A. Budano, L. Caldeira [9] A. J. Krasznahorkay, M. Csatlós, L. Csige, J. Gulyás, T. J. Balkestahl, B. Cao, F. Ceradini, P. Ciambrone, F. Curciarello, E. Ketel, A. Krasznahorkay, I. Kuti, A. Nagy, B. M. Nyako, N. Czerwinski, G. D’Agostini, E. Dane, V. De Leo, E. De Lucia, Sas, and J. Timar, On the creation of the 17 MeV X boson in A. De Santis, P. De Simone, A. Di Cicco, A. Di Domenico, the 17.6 MeV M1 transition of 8Be, Eur. Phys. J.: Web Conf. R. Di Salvo, D. Domenici, A. D’Uffizi, A. Fantini, G. Felici, 142, 01019 (2017). S. Fiore, A. Gajos, P. Gauzzi, G. Giardina, S. Giovannella, E. [10] A. J. Krasznahorkay, M. Csatlós, J. Gulyás, M. Koszta, B. Graziani, F. Happacher, Heijkenskjöld, W. Ikegami Andersson, Szihalmi, J. Timár, D. S. Firak, A. Nagy, N. J. Sas, and A. T. Johansson, D. Kaminska, W. Krzemien, A. Kupsc, S. Krasznahorkay, New evidence supporting the existence of the Loffredo, G. Mandaglio, M. Martini, M. Mascolo, R. Messi, S. hypothetic X17 particle, arXiv:1910.10459. Miscetti, G. Morello, D. Moricciani, P. Moskal, A. Palladino, [11] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, M. Papenbrock, A. Passeri, V. Patera, E. Perez del Rio, A. T. M. P. Tait, and P. Tanedo, Protophobic Fifth-Force Interpre- Ranieri, P. Santangelo, I. Sarra, M. Schioppa, M. Silarski, F. tation of the Observed Anomaly in 8Be Nuclear Transitions, Sirghi, L. Tortora, G. Venanzoni, W. Wislicki, and M. Wolke, Phys.Rev.Lett.117, 071803 (2016). Limit on the production of a low-mass vector boson in e+ e− → [12] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, Uγ , U → e+e− with the KLOE experiment, Phys. Lett. B 750, T. M. P. Tait, and P. Tanedo, Particle physics models for the 633 (2015). 17 MeV anomaly in beryllium nuclear decays, Phys. Rev. D 95, [25] Daniele S. M. Alves and Neal Weiner, A viable QCD in 035017 (2017). the MeV mass range, J. High Energy Phys. 07 (2018) 092.

062503-11 ULRICH D. JENTSCHURA PHYSICAL REVIEW A 101, 062503 (2020)

[26] J. Liu, C. E. M. Wagner, and X.-P. Wang, A light complex scalar [47] O. Tomalak, Two-photon exchange correction to the hyper- for the electron and muon anomalous magnetic moment, J. High fine splitting in muonic hydrogen, Eur. Phys. J. C 77, 858 Energy Phys. 03 (2019) 008. (2017). [27] U. Ellwanger (private communication, 2020). [48] O. Tomalak, Hyperfine splitting in ordinary and muonic hydro- [28] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum gen, Eur.Phys.J.A54, 3 (2019). Electrodynamics, Volume 4 of the Course on Theoretical Physics, [49] O. Tomalak, Two-photon exchange correction to the Lamb shift 2nd ed. (Pergamon Press, Oxford, UK, 1982). and hyperfine splitting of S levels, Eur. Phys. J. A 55, 64 [29] J. DiSciacca and G. Gabrielse, Direct Measurement of the (2019). Proton Magnetic Moment, Phys.Rev.Lett.108, 153001 [50] W. Liu, M. G. Boshier, S. Dhawan, O. van Dyck, P. Egan, (2012). X. Fei, M. Grosse Perdekamp, V. W. Hughes, M. Janousch, [30] A. Mooser, S. Ulmer, K. Blaum, K. Franke, H. Kracke, C. K. Jungmann, D. Kawall, F. G. Mariam, C. Pillai, R. Prigl, Leiteritz, W. Quint, C. de Carvalho Rodegheri, C. Smorra, and G. zu Putlitz, I. Reinhard, W. Schwarz, P. A. Thompson, and J. Walz, Direct high-precision measurement of the magnetic K. A. Woodle, High Precision Measurements of the Ground moment of the proton, Nature (London) 509, 596 (2014). State Hyperfine Structure Interval of Muonium and of the Muon [31] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Magnetic Moment, Phys.Rev.Lett.82, 711 (1999). Quantum Theory of Angular Momentum (World Scientific, [51] M. I. Eides, Hyperfine splitting in muonium: Accuracy of the Singapore, 1988). theoretical prediction, Phys. Lett. B 795, 113 (2019). [32] W. A. Bardeen, R. Gastmans, and B. Lautrup, Static quantities [52] S. G. Karshenboim and V. A. Shelyuto, Hadronic vacuum and Weinberg’s model of weak and electromagnetic interac- polarization contribution to the muonium hyperfine splitting, tions, Nucl. Phys. B 46, 319 (1972). Phys. Lett. B 517, 32 (2001). [33] J. R. Primack and H. R. Quinn, Muon g − 2 and other con- [53] D. A. Owen and W. W. Repko, Vacuum-polarization corrections straints on a model of weak and electromagnetic interactions to the hyperfine structure of the μ+μ− bound system, Phys. Rev. without neutral currents, Phys. Rev. D 6, 3171 (1972). A 5, 1570 (1972). [34] J. P. Leveille, The second-order weak correction to (g − 2) of [54] U. D. Jentschura, G. Soff, V. G. Ivanov, and S. G. Karshenboim, the muon in arbitrary gauge models, Nucl. Phys. B 137, 63 Bound μ+μ− system, Phys. Rev. A 56, 4483 (1997). (1978). [55] S. G. Karshenboim, U. D. Jentschura, V. G. Ivanov, and G. Soff, [35] H. E. Haber, G. L. Kane, and T. Sterling, The fermion mass Next-to-leading and higher-order corrections to the decay rate scale and possible effects of Higgs bosons on experimental of dimuonium, Phys. Lett. B 424, 397 (1998). observables, Nucl. Phys. B 161, 493 (1979). [56] S. J. Brodsky and R. F. Lebed, Production of the Smallest QED [36] F. S. Queiroz and W. Shepherd, New physics contributions to Atom: True Muonium (μ+μ−), Phys.Rev.Lett.102, 213401 the muon anomalous magnetic moment: A numerical code, (2009). Phys.Rev.D89, 095024 (2014). [57] H. Lamm, Hadronic vacuum polarization in true muonium, [37] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reeval- Phys. Rev. A 95, 012505 (2017). uation of the hadronic vacuum polarisation contributions to the [58] C. Peset and A. Pineda, Model-independent determination of − α 2 Standard Model predictions of the muon g 2and (MZ )using the two-photon exchange contribution to hyperfine splitting in newest hadronic cross-section data, Eur.Phys.J.C77, 827 muonic hydrogen, J. High Energy Phys. 04 (2017) 060. (2017). [59] M. W. Heiss, G. Wichmann, A. Rubbia, and P. Crivelli, The [38] A. Keshavarzi, D. Nomura, and T. Teubner, Muon g − 2and positronium hyperfine structure: Progress towards a direct mea- α 2 3 → 1 (MZ ): A new data-based analysis, Phys. Rev. D 97, 114025 surement of the 2 S1 2 S0 transition in vacuum, J. Phys.: (2018). Conf. Ser. 1138, 012007 (2018). [39] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new [60] A. Czarnecki, K. Melnikov, and A. Yelkhovsky, Positronium evaluation of the hadronic vacuum polarisation contributions Hyperfine Splitting: Analytical Value at O(mα6 ), Phys. Rev. α 2 to the muon anomalous magnetic moment and to (MZ ), Eur. Lett. 82, 311 (1999). Phys. J. C 80, 241 (2020). [61] A. Czarnecki, K. Melnikov, and A. S. Yelkhovsky, Positronium [40] M. Kalinowski, K. Pachucki, and V. A. Yerokhin, Nuclear- S-state spectrum: Analytic results at O(m α6 ), Phys. Rev. A 59, structure corrections to the hyperfine splitting in muonic deu- 4316 (1999). terium, Phys. Rev. A 98, 062513 (2018). [62] V. Barger, L. Fu, J. G. Learned, D. Marfatia, S. Pakvasa, and [41] K. Pachucki, Nuclear Structure Corrections in Muonic Deu- T. J. Weiler, Glashow resonance as a window into cosmic terium, Phys. Rev. Lett. 106, 193007 (2011). sources, Phys.Rev.D90, 121301(R) (2014). [42] K. Pachucki and A. Wienczek, Nuclear structure in deuterium, [63] G. S. Adkins and R. N. Fell, Positronium hyperfine splitting Phys.Rev.A91, 040503(R) (2015). at order mα7: Light-by-light scattering in the two-photon- [43] R. Pohl (private communication, 2020). exchange channel, Phys.Rev.A89, 052518 (2014). [44] M. M. Sternheim, State-dependent mass corrections to hyper- [64] M. I. Eides and V.A. Shelyuto, Hard nonlogarithmic corrections fine structure in hydrogenic atoms, Phys. Rev. 130, 211 (1963). of order mα7 to hyperfine splitting in positronium, Phys.Rev.D [45] A. P. Martynenko, 2S Hyperfine splitting of muonic hydrogen, 89, 111301(R) (2014). Phys.Rev.A71, 022506 (2005). [65] G. S. Adkins, C. Parsons, M. D. Salinger, R. Wang, and R. N. [46] R. Pohl, R. Gilman, G. A. Miller, and K. Pachucki, Muonic Fell, Positronium energy levels at order mα7: Light-by-light hydrogen and the proton radius puzzle, Annu. Rev. Nucl. Part. scattering in the two-photon-annihilation channel, Phys. Rev. Sci. 63, 175 (2011). A 90, 042502 (2014).

062503-12 FIFTH FORCE AND HYPERFINE SPLITTING IN BOUND … PHYSICAL REVIEW A 101, 062503 (2020)

[66] M. I. Eides and V. A. Shelyuto, Hard three-loop corrections to [71] M. I. Eides and V. A. Shelyuto, One more hard three-loop hyperfine splitting in positronium and muonium, Phys.Rev.D correction to parapositronium energy levels, Phys.Rev.D96, 92, 013010 (2015). 011301(R) (2017). [67] G. S. Adkins, C. Parsons, M. D. Salinger, and R. Wang, [72] A. Veitia and K. Pachucki, Nuclear recoil effects in antiprotonic Positronium energy levels at order mα7: Vacuum polarization and muonic atoms, Phys. Rev. A 69, 042501 (2004). corrections in the two-photon-annihilation channel, Phys. Lett. [73] U. D. Jentschura, Relativistic reduced-mass and recoil cor- B 747, 551 (2015). rections to vacuum polarization in muonic hydrogen, muonic [68] G. S. Adkins, M. Kim, C. Parsons, and R. N. Fell, Three- deuterium and muonic helium ions, Phys. Rev. A 84, 012505 Photon-Annihilation Contributions to Positronium Energies at (2011). Order mα7, Phys.Rev.Lett.115, 233401 (2015). [74] K. Pachucki, Higher-order binding corrections to the Lamb [69] G. S. Adkins, L. M. Tran, and R. Wang, Positronium energy shift, Ann. Phys. (NY) 226, 1 (1993). levels at order mα7: Product contributions in the two-photon- [75] C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw- annihilation channel, Phys. Rev. A 93, 052511 (2016). Hill, New York, 1980). [70] M. I. Eides and V. A. Shelyuto, Hyperfine splitting in muonium [76] E. E. Salpeter, Mass corrections to the fine structure of and positronium, Int. J. Mod. Phys. A 31, 1645034 (2016). hydrogen-like atoms, Phys. Rev. 87, 328 (1952).

062503-13