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Home , Muonium, True muonium, X17 particle

arXiv:2003.07207v2 [hep-ph] 2 Jun 2020 fteosre etms f16 of mass because rest “X17” observed the the as of to referred commonly cle, and results find experimental to of Standard predictions. deviation the quest theoretical a of the on extension based low-energy been hitherto Model, possible has 7]), a [6, of success, signs Refs. with e.g., crowned (see, accuracy. not motivation unprecedented third with a fundamen- [5] Yet, important constants en- of physical number and a tal [2–4], determine to constants the us fundamental on abled limits of ac- stringent variation The to time led [1]. have higher of measurements systems predictions curate atomic theoretical simple in and transitions experiments of accuracy oo naoi ytm.I oegnrltrs eask we terms, general X17 more the In by induced systems. effects atomic the in of calculation a out carry eiv ht ihteavn fcnitn observations consistent in of transitions we nuclear advent However, two the in 15]). with [14, that, Refs. see believe interpretations results, ATOMKI alternative experimen- the conceivable other of inde- and of be searches overview to an tal remain (for and verified experiment pendently other any by firmed to . couples offers as which well second [13]), as the Ref. and (see light 12]), “massive, particle (a [11, pseudoscalar particle Refs. a vector see a ”, involves dark these of Model. Standard first the The to additions be- low-energy both on forward, based conceivableing put Two been have nuclei. emission explanations beryllium theoretical the and in transitions peak excited nuclear in of a pairs on - based of spectrum [8–10], Refs. in tigated eety h osbeeitneo fhfreparti- fifth-force a of existence possible the Recently, the push to tried have physicists atomic decades, For h nig fRf.[–0 aentytbe con- been yet not have [8–10] Refs. of findings The eateto hsc,Msor nvriyo cec n T and Science of University Missouri Physics, of Department n 0 and u oteX7aesont edatclyehne o muonic an ( for are enhanced exchange true X17 drastically muonium, by be do , mediated to effects but shown hyperfine effects, of are hyperfine detection X17 to X1 the leads virtual to exclusively due one nucleus of the exchange the and hypothesis, pseudoscalar the msino h lcrnpsto ar ae ntesymmet the on Based pair. electron-positron the of emission fruhy1 e.Tes-aldX7pril osiue v a constitutes particle X17 so-called The MeV. 17 roughly of electr in of states pattern excited emission from angular the (regarding Sciences bevto fteX7mdae ffcsi on ytm con systems bound potentials, in X17-generated effects the for X17-mediated th of the to range of contribution short observation its the and of annihilation Because virtual of effect The bandi eea omwihi plcbet ohelectr hypothese both X17 to vector applicable is the which as form well i general a hyperfine as in for pseudoscalar obtained X17, the the both by for generated potentials effective the w eeteprmna bevtosa h TMIInstitu ATOMKI the at observations experimental recent Two S .INTRODUCTION I. tts r ie ymdfiain fsotrne(Dirac- short-range of modifications by given are states, − → it oc n yefieSltigi on Systems Bound in Splitting Hyperfine and Force Fifth 0 + ,teX7cudete eavco,o suoclrparticl pseudoscalar a or vector, a be either could X17 the ), 8 eand Be . 8 e,hsbe inves- been has MeV, 7 eand Be 4 e ti utfidto justified is it He, 4 e niaetepsil xsec fapril fars ma rest a of particle a of existence possible the indicate He) µ lihD Jentschura D. Ulrich + µ − on ytm,adpositronium. and system), bound n fncersz ffcs eoeteX7sga ol be could understand- signal X17 accurate the reliably. an before extracted of effects, nuclear-size terms was of in it ing effort required view However, additional considerable be in that would number. [16] X17, charge Ref. the in nuclear range concluded by ion, larger the induced carbon the approaches potential muonic of Yukawa radius a the Bohr on of effective based the [16], been where Ref. has cor- circumvention nuclear-finite-size in possible par- the A discussed X17 energy. from the the shift to of Lamb rection effect the the where on distinguish one, ticle to muonic a hope even could system, one atomic an find to hard yrgnao is hydrogen 5 16 [13]. Ref. in po- envisioned a atomic as from in spectra, particle expected Higgs”-type be pseudoscalar could “light tential effects which question the ue,i edn re,b h Dirac- the by for Con- order, least leading effects. in at duced, (67) hyperfine effects, Eq. excludes in hyperfine explicitly e.g., versely, places, [18], other Ref. many in of used and [17] Ref. swl smoi on ytm 1] vnmr im- particle more X17 Even the 11 of [16]. (about wavelength systems Compton bound the electronic muonic portantly, both as for scale well energy as binding energy the mass than rest larger X17 is observed The experiments. physics h 1 r adt itnus rmncersz effects [16]. nuclear-size of experiments from effects distinguish the atomic to in potential, hard Yukawa are the X17 of the deter- range particle X17 Be- the the mines systems. of wavelength bound muonic Compton the as well cause as electronic both for × h ento fteLm shift Lamb the of definition The oehtufruaey h etms ag of range mass rest the unfortunately, Somewhat ercl htteBh aisaonsto amounts radius Bohr the that recall We . e ae h 1 atcehr odtc natomic in detect to hard particle X17 the makes MeV 7 10 npsto ar rmncertransitions nuclear from pairs on-positron 4 yefiesltigas sconsidered. is also splitting hyperfine e rulsaei h rcs,peeigthe preceding process, the in state irtual m hl h ffcieBh aiso muonic a of radius Bohr effective the while fm, nca ela uncbudsystems. bound muonic as well as onic yo h ula rniin (1 transitions nuclear the of ry cnlg,Rla isui649 USA 65409, Missouri Rolla, echnology, unu ewe h on bound the between quantum 7 trcin nsml tmcsystems, atomic simple in nteractions δ sntaetteLm hf.Effects shift. Lamb the affect not es h otpoiigptwyfrthe for pathway promising most the lzdfrmoi etru,muonic , muonic for alyzed . oetasi oriaesae For space. coordinate in potentials ) m ssalrta h ffcieBh radius Bohr effective the than smaller is fm) 8 en yefieitrcin,which, interactions, hyperfine .Teeetv aitnasare Hamiltonians effective The s. on ytm.Popcsfrthe for Prospects systems. bound eo h ugra cdm of Academy Hungarian the of te a 0 ∼ .Hr,w calculate we Here, e. ~ / ( αm µ c ) senergy ss ∼ + L δ 5 m ti thus is It fm. 256 → sevsgdin envisaged as , S eko h mag- the of peak 0 tts r in- are states, + a 0 ∼ 2 netic dipole field of the at the origin [see literature. We write the interaction Lagrangian LX,V for Eq. (9) of Ref. [19]]. The Fermi contact interaction, which the interaction of an X17 with the gives rise to the leading-order contribution to the hyper- fields f = e,µ (electron and ) and the fine splitting for S states, is proportional to a Dirac–δ in N = p,n ( and ) as follows, coordinate space, commensurate with the fact that the µ µ atomic nucleus has a radius not exceeding the femtometer LX,V = εf e ψ¯f γ ψf Xµ εN e ψ¯N γ ψN Xµ , scale. The effect of short-range potentials is thus less sup- − − Xf XN pressed when we consider the hyperfine splitting, as com- (1a) pared to the . The afici´onados of bound states where we follow the conventions delineated in the re- thus realize that, if we consider hyperfine effects, we have marks following Eq. (1) of Ref. [11] and Eq. (10) of a much better chance of extracting the effect induced by Ref. [12]. Here, εf and εN are the flavor-dependent cou- the X17, which, on the ranking scale of the contribu- pling parameters for the fermions and nucleons, while tions, occupies a much higher place than for the Lamb e = √4πα = 0.091 is the electron charge. The − − shift alone. Despite the large mass mX of the X17 par- fermion and field-operators (the latter, inter- ticle, which leads to a short-range potential proportional preted as field operators for the composite ) and ~ to exp( mX r) (in natural units with = c = ǫ0 = 1, denoted as ψ and ψ , while the X is the X17 field op- − f N µ which will be used throughout the current paper), the erator. For reasons which will become obvious later, we effect of the X17 could thus be visible in the hyperfine use, in Eq. (1), the alternative conventions, splitting in muonic . h′ h′ Here, we shall elaborate on this idea, and derive the f = εf e , N = εN e , (1b) leading corrections to the hyperfine splitting of nS, nP1/2 and nP3/2 states in ordinary as well as muonic hydro- for the coupling parameters to the hypothetical X17 vec- genlike systems, due to the , by matching tor boson. Our conventions imply that for εN > 0, h′ the nuclear- dependent terms in the scattering ampli- the coupling parameter N parameterizes a “negatively tude with the effective Hamiltonian. Anticipating some charged” nucleon under the additional U(1) gauge group results, we can say that the relative correction (expressed of the vector X particle. in terms of the leading Fermi term) is proportional to According to a remark following the text after Eq. (9) mr/mX , where mr is the reduced mass of the two-body of Ref. [12], conservation of X charge implies that the bound system, while mX is the X17 boson mass. The couplings to the proton and neutron currents fulfill the effect is thus enhanced for muonic in comparison to elec- relationships tronic bound systems. This paper is organized as follows. In Sec. II, we sum- εp =2εu + εd , εn = εu +2εd , (2) marize the interaction Lagrangians for both a hypotheti- cal X17 vector exchange [11, 12], as well as a pseudoscalar where the up and down couplings are denoted by exchange [13], with corresponding conventions for the the subscripts u and d. Numerically, one finds [see the coupling parameters. In Sec. III, we derive the effective detailed discussion around Eqs. (38) and (39) of Ref. [12]] hyperfine Hamiltonians for both vector and pseudoscalar that the electron-positron field coupling εe needs to fulfill exchanges. In Sec. IV, we evaluate general expressions the relationship for the corrections to hyperfine energies induced by the 2 10−4 <ε < 1.4 10−3 . (3) X17 particle, for S and P states. In Sec. V, we derive × e × bounds on the coupling parameters for both models in the muon sector, based on the muon g factor. Finally, in Furthermore, in order to explain the experimental obser- Sec. VI, we apply the obtained results to muonic hydro- vations [8, 9], one needs the neutron coupling to fulfill gen, muonic deuterium, muonium, (bound [see Eq. (10) of Ref. [11]] µ+µ− system), and . We also discuss the 3 1 measurability of the X17 effects in the hyperfine struc- ε = ε +2ε ε . (4) | n| | u d|≈ 2 d ≈ 100 ture of the mentioned atomic systems. Conclusions are reserved for Sec. VII. Because the hypothetical X vector particle acts like a “” which is hardly distinguishable from the ordinary photon in the high-energy domain, the proton II. INTERACTION LAGRANGIANS coupling εp is highly constrained. According to Eq. (8) and (9) of Ref. [12], and Eq. (35) of Ref. [12], one needs In the following, we intend to study both the interac- to have tion of X17 vector and pseudoscalar particles with bound −4 ( and ) and nucleons ( and εp = 2εu + εd . 8 10 . (5) deuterons). Vector interactions will be denoted by the | | | | × subscript V , while pseudoscalar interactions will carry This is why the conjectured X17 vector boson is referred the subscript A, as is customary in the to as “protophobic” in Refs. [11, 12]. 3

Following Ref. [13], we write the interaction La- f f f f grangian for the fermions interacting with the pseu- doscalar candidate of the X17 particle as follows,

L = h ψ¯ i γ5 ψ A h ψ¯ i γ5 ψ A, X A X,A − f f f − N N N Xf XN (6) where A is the field operator of the pseudoscalar field. Inspired by an analogy with putative pseudoscalar Higgs N N N N couplings [20], the pseudoscalar couplings have been es- timated in Refs. [13, 20] to be of the functional form (a) (b)

mf mN FIG. 1. The one-quantum exchange scattering ampli- h = ξ , h = ξ , (7) f f v N N v tude for the X17 particle is matched against the effective Hamiltonian, for the vector hypothesis [diagram (a)] and where v = 246 GeV is the vacuum expectation value of the pseudoscalar hypothesis [diagram (b)]. The arrow the Higgs (or Englert–Brout–Higgs, see Refs. [21, 22]) of time is from left to right. field, mf is the fermion mass, and mN is the nucleon’s mass. Furthermore, the parameters ξf and ξN could in principle be assumed to be of order unity. Note that the spin-parity of the has re- use the bispinors in the representation [cf. Eq. (83.7) of cently been determined to be consistent with a scalar, Ref. [28]] not pseudoscalar, particle [23], but it is still intuitively 2 ~pf,N suggested to parameterize the couplings to the novel pu- 1 wf,N tative pseudoscalar X17 in the same way as one would − 8m2 uf,N =  !  , (11) otherwise parameterize the couplings to the Higgs parti-  ~σ ~p  cle.  · wf,N   2m  According to Eq. (2.7) and the remark following   Eq. (3.12) of Ref. [13], the nucleon couplings can roughly where f,N stands for the bound fermion, or the nu- be estimated as cleus, and wf,N are the nonrelativistic spinors. Of course, two two-component (sic!) spinors constitute the four- mp −3 hp = ( 0.40 ξu 1.71 ξd) 2.4 10 , (8a) component bispinor u of the same field. The mas- v − − ≈− × f,N sive photon propagator (for the X17 vector hypothesis) mn −4 hn = ( 0.40 ξu +0.85 ξd) 5.1 10 , (8b) is used in the following normalization (we may ignore the v − ≈ × frequency of the photon in the order of approximation where we have assumed ξu ξd 0.3. For the electron- relevant for the current article), positron field, based on other≈ ≈ constraints detailed in Ref. [13], one has to require that [see Eq. (4.2) of Ref. [13]] 1 D00(~q)= 2 2 , (12a) − ~q + mX ! ! i j h 4 me −6 1 q q ξe > 4 , e > =8.13 10 . (9) D (~q)= δij . (12b) v × ij − ~q 2 + m2 − ~q 2 + m2 X  X  Based on a combination of experimental data [24] and The derivation of the massive vector boson propagator in theoretical considerations [25–27], one can also derive an the Coulomb gauge, which is best adapted to bound-state upper bound, calculations and involves a certain subtlety, is discussed in Appendix A. The pseudoscalar propagator is used in ! ! 500 m − ξ < 500 , h < e = 10 3 , (10) the normalization e e v 1 which will be used in the following. DA(~q)= 2 2 , (13) −~q + mX where we also ignore the frequency. The scattering am- III. MATCHING OF THE SCATTERING plitude for the X17 vector particle reads as AMPLITUDE h h ′ 0 ′ 0 Mfi,V = f N u¯f γ u¯f u¯N γ u¯N D00 In order to match the scattering amplitude (see Fig. 1) + u¯′ γi u¯ u¯′ γj u¯ D , (14) with the effective Hamiltonian, we use the approach out- f  f N N ij lined in Chap. 83 of Ref. [28], but with a slightly al- and   tered normalization for the propagators, better adapted ~ h h ′ 5 ′ 5 to natural unit system ( = c = ǫ0 = 1). Specifically, we Mfi,A = f N u¯f i γ u¯f u¯N i γ u¯N DA , (15)   4 for the pseudoscalar case. Here, we denote the final states cle, ′ ′ ′ of the scattering process by a prime, uf = uf (~pf ), uN = ′ h′ h′ uN (~pN ), while the initial states are uf = uf (~pf ) and f N 8π (3) HHFS,V = δ (~r) ~σf ~σN uN = uN (~pN ), and the bar denotes the Dirac adjoint. 16 πmf mN − 3 · Analogous definitions are used for the w′ and w in  f,N f,N m2 ~σ ~r~σ ~r r2 ~σ ~σ ′ ′ X f N f N −mX r Eq. (11). Furthermore, we have ~pf +~pN = ~p f +~p N . The · ·3 − · e ′ ′ − r  momentum transfer is ~q = ~p f ~pf = ~pN ~p N . 2 − − 3 ~σf ~r~σN ~r r ~σf ~σN − (1 + m r) · · − · e mX r The form (11) is valid for the bispinors if the Dirac − X r5 equation is solved in the Dirac representation of the Dirac mf ~σN L~ − matrices, 2+ (1 + m r) · e mX r . (19) − m X r3  N  # i 0 ½2×2 0 i 0 σ γ = , γ = i , Taking the limit mX 0, and replacing 0 ½2×2 σ 0 →  −   −  ′ ′ gN ( e) gN e 2 5 0 ½2×2 h e , h − = | | , e =4πα , γ = . (16) f → N → 2 2 ½2×2 0   (20) one recovers the Fermi Hamiltonian HF [see Eq. (10) of The scattering amplitudes are matched against the effec- Ref. [19]], tive Hamiltonian by the relation g α π H = N ~σ ~σ δ(3)(~r) F m m 3 f N ′+ ′+ f N · Mfi = (w w ) U(~pf , ~pN , ~q) (wf wN ) , (17) h f N 2 − 3 ~σf ~r~σN ~r r ~σf ~σN ~σN L~ + · · 5− · + ·3 , (21) 8 r 4 r # where U(~pf , ~pN , ~q) is the the effective Hamiltonian. The scattering amplitude Mfi is a matrix element involving where we have ignored the reduced-mass correction pro- four spinors, two of which represent the final and initial portional to m /m in the ~σ L~ term in Eq. (19). For states of the two-particle system. f N N a pseudoscalar exchange, one has· The scattering amplitude, evaluated between four spinors [cf. Eq. (83.8) of Ref. [28]], must now be matched h h 4π H = f N δ(3)(~r) ~σ ~σ against a Hamiltonian which acts on only one wave func- HFS,A 16 πm m 3 f · N f N  tion in the end. We need to remember that the scat- 2 m ~σf ~r~σN ~r − tering amplitude corresponds to a matrix element of the X · · e mX r Hamiltonian. Any matrix element of the Hamiltonian, − r3 2 even in the one-particle setting, is sandwiched between 3 ~σf ~r~σN ~r ~σf ~σN r − +(1+ m r) · · − · e mX r . two wave functions, not one. Then, going into the center- X r5 ′ ′  of-mass frame ~q = ~p f ~pf = ~pN ~p N means that the (22) wave function is written− in terms of− a center-of-mass co- ordinate R~, and a relative coordinate ~r. In the center- Note that the Hamiltonian given in Eq. (22) constitutes of-mass frame, one eliminates the dependence on the the complete Hamiltonian derived from pseudoscalar ex- center-of-mass coordinate R~ and the total momentum change, which, in view of the γ5 matrix in the Lagrangian P~ = ~pf +~pN . Fourier transformation under the condition given in Eq. (6), contributes only to the hyperfine split- ′ ′ ting, but not to the Lamb shift, in leading order [i.e., via ~p f + ~p N = ~pf + ~pN leads to the effective Hamiltonian [cf. Eq. (83.15) of Ref. [28]]. the exchange of one , as given in Fig. 1(a)]. For a deuteron nucleus, the spin matrix ~σ has to be For the record, we note that in the X17 vector case, the N ~ ~ 00 component of the photon propagator gives the leading, replaced by 2 IN , where IN is the spin operator of the spin-independent term in the effective Hamiltonian, deuteron, corresponding to the spin-1 particle. Impor- h′ h tant bounds on the coupling parameters µ and µ can be derived from the muon anomalous magnetic moment h′ h′ H = f N exp( m r) . (18) (see Fig. 2). 0 4πr − X

Under the replacements h′ e and h′ e, in the IV. CORRECTIONS f → N → − massless limit mX 0, one recovers the Coulomb po- 2 e → α A. X17 Boson Exchange tential, H0 4πr = r . One finally extracts the terms responsible→ − for the− hyperfine structure, i.e., those involving the nuclear spin operator ~σN , and obtains the In order to analyze the S state hyperfine splitting, following hyperfine Hamiltonian for a vector X17 parti- we extract from Eqs. (19) and (22) the terms which are 5 nonzero when evaluated on a spherically symmetric wave We have neglected relative corrections of higher than function. This entails the replacements first order in αmf /mX and mf /mN . For the pseu- dovector hypothesis, one finds that EX,A(nS1/2) = 1 2 ~σ ~r~σ ~r r ~σ ~σ , ~σ L~ 0 , (23a) nS1/2 HHFS,A nS1/2 is given as follows, f · N · → 3 f · N N · → | | ′ ′ h h ~σf ~σN 4 4 f N · (Zα) mr HHFS,V EX,A(nS1/2)= hf hN → − 24 πmf mN 3πn3 m m m  f N X m2 5 5 (3) X −mX r 5(Zα) 1 mr 4πδ (~r) e , (23b) 1+ ~σf ~σN . × − r − 6πn3 5n2 m m m2 h · iS1/2,F     f N X  h h ~σ ~σ (28) H f N f · N HFS,A → 48 πm m f N The S state splitting is obtained from the following ex- 2 m − pectation values, 4πδ(3)(~r) X e mX r . (23c) × − r   ~σf ~σN =1 , ~σf ~σN = 3 . h · iS1/2,F =1 h · iS1/2,F =0 − The expectation value of the Fermi Hamiltonian is (29) Expressed as a relative correction to the leading term, 3 3 α (Zα) mr given in Eq. (21), one has the following corrections due EF (nS)= nS1/2 HF nS1/2 =gN 3 ~σf ~σN . | | 3 n mf mN h · i to the X17 particle, (24) h′ h′ EX,V (nS ) 2 Zm Here, Z is the nuclear charge number, and mr = 1/2 f N r , (30a) mf mN /(mf + mN ) is the reduced mass of the system. EF (nS1/2) ≈ − gN π mX We use the nuclear g factor in the normalization EX,A(nS ) h h Zm 1/2 f N r , (30b) e ~σN EF (nS1/2) ≈ gN π mX ~µN =gN | | , (25) 2mN 2 depending on the vector (V ) or pseudoscalar (A) hypoth- which can more easily be extended to more general two- esis. One notices the different sign of the correction, de- body systems than a definition in terms on the nuclear pending on the symmetry group of the new particle. We observe that the relative correction to the Fermi split- magneton. For the proton, one has gp = 5.5856 ... as the proton’s g factor [29, 30], while the definition (25) ting is enhanced for muonic bound systems, by a factor mr/mX mµ/mX as compared to electronic bound sys- implies that gd =1.713 ... for the deuteron [5]. For true ∼ muonium (µ+µ− bound system) and positronium, one tems, because the corresponding factor me/mX is two orders of magnitude smaller. has gN = 2 according to the definition (25). By contrast, in the limit m , one verifies that For nP1/2 states, whose wave function van- X → ∞ ishes at the nucleus in the nonrelativistic ap- 2 proximation, one finds for the first-order correc- m − lim X e mX r =4πδ(3)(~r) , (26) →∞ tions EX,V (nP1/2) = nP1/2 HHFS,V nP1/2 and mX r | |   E (nP )= nP H nP , X,A 1/2 1/2| HFS ,A| 1/2 and the two Hamiltonians given in Eqs. (23b) and (23c) (Zα)5 1 vanish in the limit of an infinitely heavy X17 particle. E (nP )= h′ h′ 1 X,V 1/2 f N πn3 − n2 This implies that the expectation values of S states of   5 the Hamiltonians in Eqs. (23b) and (23c) have to carry mr ~σf ~σN , (31a) at least one power of mX in the denominator, and in 2 nP1/2,F × mf mN mX h · i particular, that the correction to the hyperfine energy 5 4 3 (Zα) 1 will be of order α(Zα) , not α(Zα) , as one would other- E (nP )= h h 1 X,A 1/2 f N 2πn3 − n2 wise expect from the two individual terms in Eqs. (23b)   and (23c). For the vector hypothesis, one finds that 5 mr ~σf ~σN . (31b) EX,V (nS1/2) = nS1/2 HHFS,V nS1/2 , for the leading 2 nP1/2,F × mf mN m h · i and subleading terms in| the expansion| in inverse powers X of mX , can be expressed as In these results, matrix elements of tensor structures proportional to ~σf ~r ~σN ~r in Eqs. (19) and (22) 4 4 h · · i 2(Zα) m have been reduced to simpler structures ~σf ~σN by an- E (nS )= h′ h′ r h · i X,V 1/2 f N − 3πn3 m m m gular algebra reduction formulas, which are familar in  f N X h′ h 5 5 atomic physics [31]. Under the replacement f f 5(Zα) 1 mr h′ h → + 1+ ~σf ~σN . and N N , the correction, for a vector X17 particle, 3πn3 5n2 m m m2 h · iS1/2,F →   f N X  assumes the same form as for the pseudoscalar hypoth- (27) esis, up to an additional overall factor 1/2. For nP1/2 6 states, the expectation values are X A

1 ~σf ~σN = , ~σf ~σN =1 . µ µ µ µ h · iP1/2,F =1 −3 h · iP1/2,F =0 (32) The leading term in the hyperfine splitting for nP1/2 states is well known to be equal to

(a) (b) E (nP )= nP ) H nP F 1/2 1/2 | F | 1/2 3 3 FIG. 2. The X17 particle induces vertex corrections to the α (Zα) mr = gN ~σf ~σN . anomalous magnetic moment of the muon. For the X17 3 nP1/2,F − 3 n mf mN h · i vector hypothesis, one obtains diagram (a), while the pseu- (33) doscalar hypothesis leads to diagram (b). The interaction with the external magnetic field is denoted by a zigzag line. Expressed in terms of the leading term, one obtains the following corrections due to the X17 particle for nP3/2 states, f f f f

h′ h′ EX,V (nP ) 3 Zm 1 Zαm 1/2 f N r r X A 1 2 , EF (nP ) ≈ − gN π mX − n mX 1/2     f f f f (34a)

EX,A(nP ) 3h h Zm 1 Zαm 1/2 f N r 1 r . (a) (b) E (nP ) ≈ − 2g π m − n2 m F 1/2 N X    X  (34b) FIG. 3. For bound systems consisting of a lepton and anti- lepton there is an additional correction to the energy levels Parametrically, these are suppressed with respect to induced by virtual annihilation (the arrow of time is from left the results for S states, by an additional factor to right). We here consider f = e (positronium) and f = µ (true muonium). The resulting effective potential is propor- Zαmr/mX . For the nP3/2 states, one considers the tional to a Dirac-δ and affects S states. Diagram (a) is rel- corrections E (nP ) = nP H nP and X,V 3/2 3/2| HFS,V | 3/2 evant for orthopositronium and ortho true muonion (S = 1, EX,A(nP3/2)= nP3/2 HHFS,A nP3/2 , with the results annihilation into a vector X17 boson) while diagram (b) is | | relevant for para states (with total spin S = 0, which al- lows for an annhilation into a hypothetical pseudoscalar X17 (Zα)5 1 m5 E (nP )= 1 r boson). Both virtual processes contribute to the hyperfine X,V 3/2 − 12πn3 − n2 m2 m2 splitting.   N X h′ h′ f N ~σf ~σN , (35a) × h · inP3/2,V 2(Zα)6 1 m6 E (nP )= 1 r X,A 3/2 45πn3 − n2 m m m3 states,   f N X hf hN ~σf ~σN . (35b) h′ h′ × h · inP3/2,A EX,V (nP3/2) 5 f N Zmr mf EF (nP3/2) ≈ − 4gN π mX mN Here, the expectation values are 1 Zαmr 1 2 , (38a) × − n mX 5     ~σ ~σ =1 , ~σ ~σ = . 2 f N P3 2,F =2 f N P3 2,F =1 EX,A(nP ) 2h h Zm 1 Zαm h · i / h · i / −3 3/2 f N r 1 r . (36) E (nP ) ≈ 3g π m − n2 m F 3/2 N X    X  The leading term in the hyperfine splitting for nP3/2 (38b) states is well known to be equal to

Parametrically, in comparison to nP1/2 states, the correc- E (nP )= nP ) H nP tion for nP3/2 states is suppressed for a vector X17 by an F 3/2 3/2 | F | 3/2 3 3 additional factor mf /mN , while for a pseudoscalar X17, α (Zα) mr =g ~σ ~σ . (37) the suppression factor is Zαmr/mX . For electrons bound N 3 f N nP3 2 15 n mf mN h · i / to protons and other nuclei, both suppression factors are approximately of the same order-of-magnitude, while for Expressed in terms of the leading term, one obtains the muonic hydrogen and deuterium, the vector contribution following corrections due to the X17 particle for nP1/2 dominates over the pseudoscalar one. 7

B. Virtual Annihilation In the calculation, one precisely follows Eqs. (83.18)– (83.24) of Ref. [28] and adjusts for the mass of the X17 For bound systems consisting of a particle and an- in the propagator denominator. The relative correction tiparticle, virtual annihilation processes also need to be to the hyperfine splitting due to annihilation channel, for considered (see Fig. 3). The resulting effective poten- a virtual vector X17 particle (for S states), is tials are local (proportional to a Dirac-δ) and affect S ′ 2 2 E (nS) (h ) m states. We here consider positronium and true muonium ANN,V f f = 2 1 2 . (46) EANN,γ(nS) 4πα m m (f = e,µ), for which one has Z = 1, N = f (antifermion), f − 4 X mr = mf /2, mN = mf . In this case, the Fermi energy, For the virtual annihilation into a pseudoscalar parti- as defined in Eq. (24), assumes the form (it is no longer cle, one can also follow Eqs. (83.18)–(83.24) of Ref. [28], equal to the full leading-order result for the hyperfine but one has to adjust for the different interaction La- splitting, as we will see)” grangian, which now involves the fifth current, and one also needs to adjust for the mass of the X17 in the α4 m f pseudoscalar propagator denominator. The result in EF,ff (nS)= 3 ~σf ~σf . (39) 12 n h · i Eq. (83.22) of Ref. [28] for the Fierz transformation of When one replaces the vector X17 boson in Fig. 3(a) the currents has to be adapted to the pseudoscalar cur- by a photon, one obtains the annihilation potential [see rent, i.e., to the last entry in Eq. (28.17) of Ref. [28]. The Eq. (83.24) of Ref. [28]], result is h2 3 f (3) πα (3) HANN,A = ~σf ~σ 1 δ (~r) . (47) HANN,γ = ~σf ~σ +3 δ (~r) . (40) 2 1 2 f 2 f 8(mf 4 mX ) · − 2mf · −     The expectation value of this effective Hamiltonian is Based on the identity nonvanishing only for para states (S = 0), as had to be expected in view of the pseudoscalar nature of the virtual ~σ ~σ =2S(S + 1) 3 , (41) h f · f i − particle (the intrinsic parity of para states of positron- ium and true muonium is negative, allowing for the vir- where S = 1 for an ortho state, and S = 0 for a para tual transition to the pseudoscalar X17). In contrast to state, one can see that the annihilation process into a Eq. (22), we observe a small shift of the hyperfine cen- virtual vector particle is relevant only for ortho states, troid for the fermion-antifermion system, due to the term consistent with the conservation of total angular momen- that is added to the scalar product of the spin operators. tum in the virtual transition to the photon, which has an The general expression for the expection value in and intrinsic spin of unity. The virtual annihilation contribu- nS state is (we select the term relevant to the hyperfine tion to the hyperfine splitting is splitting) 4 α mf 3h2 (αm )3 EANN,γ (nS)= ~σf ~σ . (42) f f 16 n3 h · f i EANN,A(nS)= ~σf ~σ . (48) 64(m2 1 m2 )n3 h · f i f − 4 X The difference between the expectation values for ortho The relative correction to the hyperfine splitting due to and para states is the well-known result [ ~σ ~σ = hh f · f ii the pseudoscalar annihilation channel, for S states, is 1 ( 3) = 4] − − 2 2 EANN,A(nS) 3(hf ) mf 4 7 α mf = 2 2 . (49) ∆E (nS)= E (nS)+ E (nS) = . EANN,γ(nS) 4πα mf mX HFS hh F ANN,γ ii 12 n3 − (43) For the fermion-antifermion bound system, the correc- The exchange of a virtual photon contributes a fraction tions given in Eq. (30) specialize as follows, of 4/7 to this result, while the virtual annihilation yields h′ 2 the remaining fraction of 3/7. EX,V (nS) ( f ) mf , (50a) The generalization of Eq. (40) to a vector X17 ex- EF (nS) ≈ − 2π mX change is immediate, E (nS) (h )2 m X,A f f , (50b) (h′ )2 EF (nS) ≈ 4π mX H = f ~σ ~σ +3 δ(3)(~r) , (44) ANN,V 2 1 2 f f The relative correction to the total hyperfine splitting, 8(mf 4 mX ) · −   due to the X17 exchange and annihilation channels, is with the expectation value (we select the term relevant 4 EX,V (nS) 3 EANN,V (nS) to the hyperfine splitting) χV (nS)= + , (51a) 7 EF (nS) 7 EANN,γ (nS) h′ 2 3 ( f ) (αmf ) 4 EX,A(nS) 3 EANN,A(nS) EANN,V (nS)= ~σf ~σ . (45) χA(nS)= + , (51b) 64(m2 1 m2 )n3 h · f i 7 EF (nS) 7 EANN,γ(nS) f − 4 X 8 with the individual contributions given in Here, because the correction is negative and decreases Eqs. (30), (46), (49) and (50). the value of aµ, the experimental-theoretical discrepancy given in Eq. (52) can only be enhanced by the pseu- doscalar X17 particle. If we demand that the discrepancy V. X17 PARTICLE AND MUON ANOMALOUS not be increased beyond 6σ, then we obtain the condition MAGNETIC MOMENT −4 that hµ could not exceed a numerical value of 3.8 10 . In the| following,| we take the maximum permissible× value One aim of our investigations is to explore the possi- of bility of a detection of the X17 particle in the hyperfine −4 splitting of muonic bound systems. To this end, it is hµ = (hµ)max =3.8 10 , (57) instructive to derive upper bounds on the coupling pa- × h′ h rameters µ and µ, for the muon. The contribution of in order to estimate the magnitude of corrections to the a massive pseudoscalar loop to the muon anomaly [see hyperfine splitting in muonic bound systems, induced by Fig. 2(b)] has been studied for a long time [32–36], and a hypothetical pseudoscalar X17 particle. recent updates of theoretical contributions [37–39] has confirmed the existence of a (roughly) 3.5 discrepancy of theory and experiment. The contribution of a massive VI. NUMERICAL ESTIMATES AND vector exchange [see Fig. 2(a)] has recently been revisited EXPERIMENTAL VERIFICATION in Ref. [36]. Specifically, the experimental results for the muon anomaly aµ [see Eqs. (1.1) and (3.36) of Ref. [38]] A. Overview are as follows, (exp) The relative corrections to the hyperfine splitting due aµ =0.00116592091(54)(33) , (52) to the X17 particle, expressed in terms of the lead- (thr) aµ =0.001165918204(356) . (53) ing Fermi interaction, for S and P states, are given in

(exp) (thr) −9 Eqs. (30), (34a), (34b), (38a), and (38b). All of the for- The 3.7σ discrepancy aµ aµ 2.7 10 needs − ≈ × mulas involve at least one factor of mr/mX , and so, ex- to be explained. periments appear to be more attractive for muonic rather According to Eq. (41) of Ref. [36], we have the follow- than electronic bound systems. Furthermore, paramet- ing correction to the muon anomaly due to the vector X rically, the corrections are largest for S states, which is vertex correction in Fig. 2(a), understandable because the range of the X17 potential 1 is limited to its Compton wavelength of about 11.8 fm, h′ 2 2 2 ( µ) mµ dx x (2 x) and so, its effects should be more pronounced for states ∆a = 2 2 µ 2 2 m− m 8π mX µ µ whose probability density does not vanish at the ori- Z (1 x) 1 m2 + m2 x 0 − − X X gin, i.e., for S states. This intuitive understanding is −3 h′ 2 h i =8.64 10 ( µ) , (54) confirmed by our calculations. Note that the formulas × for the corrections to the hyperfine splitting, given in where we have used the numerical value m = 16.7 MeV. X Eqs. (27), (28), (31a), (31b), (35a), and (35b), are gen- The following numerical value erally applicable to bound systems with a heavy nucleus, h′ = (h′ ) =5.6 10−4 (55) upon a suitable reinterpretation of the nuclear spin ma- µ µ opt × trix ~σN in terms of a nuclear spin operator. is “optimum” in the sense that it precisely remedies the discrepancy described by Eq. (52) and will be taken as the input datum for all subsequent evaluations of cor- B. Muonic Deuterium rections to the hyperfine splitting in muonic bound sys- tems. Note that, even if the vector X17 particle does not provide for an explanation of the muon anomaly dis- In view of a recent theoretical work [40] which describes crepancy, the order-of-magnitude of the coupling param- a 5σ discrepancy of theory and experiment for muonic eter h′ could not be larger than the value indicated in deuterium, it appears indicated to analyze this system µ first. Indeed, the theory of nuclear-structure effects in Eq. (55), because otherwise, the theoretical value of aµ would increase too much beyond the experimental result. muonic deuterium has been updated a number of times According to Eq. (20) of Ref. [36], the vertex correction in recent years [40–42]. Expressed in terms of the Fermi due to a virtual pseudoscalar X17 leads to the following term, the discrepancy δEHFS(2S) observed in Ref. [40] correction, can be written as 1 ∆E (2S ) 2 2 3 HFS 1/2 (hµ) mµ dx x =0.0094(18) . (58) ∆a = 2 E (2S ) µ 2 2 m F 1/2 − 4π m µ mµ 2 X (1 x) 1 2 + x Z0 m mX − − X In order to evaluate an estimate for the correction due to = 1.19 10−3 (h )2 . h i (56) − × µ the X17 vector particle, we observe that the interaction is 9 protophobic [11, 12]. Hence, we can assume that the cou- for S states, given in Eq. (58). The result is that the pling parameter of the deuteron is approximately equal achievable theoretical uncertainty should be better than to that of the neutron. We will assume the opposite sign 10−7, which would make the effect given in Eq. (62) for the coupling parameter of the deuteron (neutron), measurable. Also, according to Ref. [43], the experi- as compared to the sign of the coupling parameter in mental accuracy should be improved into the range of Eq. (55). This choice is inspired by the opposite charge 10−6 ... 10−7 in the next round of planned experiments. of the muon and nucleus with respect to the U(1) gauge Results for the Sternheim [44] weighted differences 3 group of . In view of Eq. (10) [n EX,V (nS1/2) EX,V (1S1/2)]/EF (1S1/2) and corre- 3 − of Ref. [11] and Eq. (4) here, we thus have the estimate spondingly [n EX,A(nS1/2) EX,A(1S1/2)]/EF (1S1/2) are of the same order-of-magnitude− as for the individual ′ ′ 1 − h h = √4πα = 3.02 10 3 . (59) P states. d ≈ n −100 − × 1/2 Because of the numerical dominance of the proton cou- pling to the pseudoscalar particle over that of the neutron C. Muonic Hydrogen [see Eq. (8)], we estimate the pseudoscalar coupling of the deuteron to be of the order of The considerations are analogous to those for muonic −3 deuterium. However, the coupling parameter for the nu- hd hp = 2.4 10 . (60) ≈ − × cleus, for the protophobic vector model, is constrained In view of Eq. (30), we obtain the estimates by Eq. (5), (µd) E (nS ) ′ −4 −5 X,V 1/2 −6 h 8 10 √4πα = 2.42 10 , (63) ˙ 3.8 10 , (61a) p ≈− × − × EF (nS1/2) ≈ × (µd) which is much smaller than for the deuteron nucleus. The E (nS1/2) − X,A ˙ 1.0 10 6 , (61b) coupling parameter of the proton, for the pseudoscalar EF (nS1/2) ≈ − × model, can be estimated according to Eq. (8a). One ob- tains where the symbol ˙ is used to denote an estimate for the quantity specified≈ on the left, including its sign, based (µp) EX,V (nS1/2) − on the estimates of the coupling parameters of the hy- ˙ 8.8 10 9 , (64a) E (nS ) pothetical vector and pseudoscalar X17 particle, as de- F 1/2 ≈ × scribed in the current work. As already explained, the (µp) EX,A (nS1/2) − modulus of our estimates for the coupling parameters is ˙ 2.9 10 7 , (64b) E (nS ) ≈ − × close to the upper end of the allowed range; the same F 1/2 thus applies to the absolute magnitude of our estimates for the S state effects, and for the X17-mediated corrections to hyperfine energies. Note that the vector X17 contribution slightly decreases (µp) E (nP1/2) 1 the discrepancy noted in Ref. [40], while the hypotheti- X,V ˙ 5.8 10−10 1 , (65a) E (nP ) ≈ × − n2 cal pseudoscalar effect slightly increases the discrepancy, F 1/2   yet, on a numerically almost negligible level. (µp) E (nP1/2) 1 Similar considerations, based on Eqs. (34a) and (34a), X,A ˙ 1.8 10−8 1 , (65b) E (nP ) ≈ × − n2 lead to the following results for P states, F 1/2   (µd) for P1/2 states. Results of the same order-of-magnitude E (nP1/2) − 1 X,V ˙ 2.5 10 7 1 , (62a) as for individual P states are obtained for the Stern- E (nP ) ≈ × − n2 1/2 F 1/2   heim difference of S states. The effects, in muonic hy- (µd) drogen, for the vector model, are seen to be numer- EX,A (nP1/2) − 1 ˙ 6.6 10 8 1 , (62b) ically suppressed. The contributions of the X17 par- E (nP ) ≈ × − n2 F 1/2   ticle need to be compared to the proton structure ef- which might be measurable in future experiments. fects, which have recently been analyzed in Refs. [45– Specifically, there is a nuclear-structure correction to the 49]. According to Ref. [48], the numerical accuracy of P1/2 state hyperfine splitting due to the lower compo- the theoretical prediction for the 2S hyperfine splitting in nent of the Dirac nP1/2 wave function, which has S- muonic hydrogen is currently about 72 ppm [EHFS(2S)= state symmetry. However, the lower component of the 22.8108(16) meV]. wave function is suppressed by a factor α, which implies that the P1/2 state nuclear-structure correction is sup- 2 D. Muonium pressed in relation to EF (nP1/2) by a factor α . An order-of-magnitude of the achievable theoretical uncer- tainty for the P1/2-state result can be given by an ap- Muonium is the bound system consisting of a pos- propriate scaling of the current theoretical uncertainty itively charged antimuon (µ+), and an electron (e−). 10

Its ground-state hyperfine splitting has been studied in F. Positronium Ref. [50] with a result of ∆ν(exp) = 4463302765(53)Hz, HFS − i.e., with an accuracy of 1.2 10 8. The theoretical Quite considerable efforts have recently been invested × uncertainty is about one order-of-magnitude worse and in the calculation of the mα7 corrections to the positro- − amounts to 1.2 10 7 (see Ref. [51]), with the current nium hyperfine splitting, and related effects [58–71]. In × status being summarized in the theoretical prediction positronium, effects of the X17 particle are suppressed in (thr) ∆νHFS = 4463302872(515)Hz. view of the smaller reduced mass of the bound system. Coupling parameters for the muon have been estimated Under these assumptions, the estimates for S states are in Eqs. (55) and (57) for the vector and pseudoscalar as follows, [see Eq. (51)] models, respectively, and we take the antimuon coupling ˙ −9 ˙ −9 estimate as the negative value of the coupling parameters χV (nS) 3.6 10 , χA(nS) 5.1 10 . ≈ − × ≈ − × (70) for the muon. For the coupling parameters, we use the 7 maximum allowed value for the electron in the vector The effects are thus numerically smaller than the mα model [see Eq. (3)], effects currently under study [59–71]. h′ −3 −4 e ˙ 1.4 10 √4πα =4.2 10 , (66) ≈ × × VII. CONCLUSIONS and for the pseudoscalar model [see Eq. (10)],

m − The conceivable existence of the X17 particle [8–10] h ˙ 500 e =1.0 10 3 . (67) e ≈ v × provides atomic physicists with a long-awaited opportu- nity to detect a very serious candidate for a low-energy One obtains the estimates (fifth force) addition to the Standard Model. The energy (µµ) range of about 17MeV provides for a certain challenge E (nS1/2) − X,V ˙ 2.3 10 9 , (68a) from the viewpoint of atomic physics; the range of the EF (nS1/2) ≈ × X17-induced interaction potentials is smaller than the (µµ) effective Bohr radii in electronic and muonic bound sys- EX,A (nS1/2) − ˙ 1.8 10 9 . (68b) tems. Rather than looking at the Lamb shift [16], we E (nS ) ≈ − × F 1/2 here advocate a closer look at the effects induced by the Because the reduced mass of muonium is close to the X17 on the hyperfine splitting, for both S and P states, electron mass, the effect of the X17 boson is parametri- in simple electronic and muonic bound systems. This no- cally suppressed. It will take considerable effort to in- tion is based on two observations: (i) Hyperfine effects crease experimental precision beyond the level attained for S states are induced by a contact interaction (the in Ref. [50]. On the other hand, hadronic vacuum po- Fermi contact term), and thus, naturally confined to a larization effects are suppressed in muonium, and their distance range very close to the atomic nucleus. As far uncertainty [52] is less than the X17-induced effect in the as hyperfine effects are concerned, the virtual exchange hyperfine splitting of muonium. It is thus not completely of an X17 is thus not masked by its small range. (ii) In hopeless to detect X17-induced effects in muonium in the the pseudoscalar model [13], the one-quantum exchange future. of an X17 exclusively leads to hyperfine effects, but leaves the Lamb shift invariant. or a fermion-antifermion sys- tem, this statement should be taken with a small grain − E. True Muonium (µ+µ ) System of salt, namely, it holds up to the numerically tiny shift of the hyperfine centroid, induced by the virtual anni- Taking into account the exchange and annihilation hilation channel [see Eq. (47)]. This finding could be channels, and using the same coupling parameter esti- of interest irrespective of whether the results of the ex- mates as for muonium, one obtains the estimates [see perimental observations reported in Refs. [8–10] can be Eq. (51)] independently confirmed by other groups. We have derived limits on the coupling parameters of −6 −6 χV (nS) ˙ 1.3 10 , χA(nS) ˙ 2.1 10 . (69) the X17 particle in the muonic sector in Sec. V and com- ≈ × ≈ × piled estimates for the X17-induced effects in Sec. VI, for The annihilation channel contribution numerically domi- a number of simple atomic systems. The results can be nates over the exchange channel. For S states, the contri- summarized as follows. bution of hadronic vacuum polarization in the annihila- tion channel has been improved to the level of 2 ppm, as a We show in Sec. V that the pseudoscalar model [13] result of gradual progress achieved over the last decades • enhances the experimental-theoretical discrepancy [see Eq. (41) of Ref. [53, 54] as well as Refs. [53, 55, 56], in the muon anomaly, while the vector model [11, and the recent work [57]]. A very modest progress in 12]) could eliminate it. Stringent bounds on the the theoretical determination of the hadronic vacuum- magnitude of the muon coupling parameters to the polarization contribution would make the effect of the X17 particle can be derived based on the muon X17 visible. anomaly. Note that the order-of-magnitude of 11

the maximum permissible coupling to the pseu- two orders of magnitude lower than current theo- doscalar, given in Eq. (57), also leads to a tension retical predictions [51], we can say that, at least, with the parameterization hf = ξf (mf /v) given the uncertainty in the theoretical treatment of the in Ref. [13] (applied to the case f = µ, i.e., to hadronic vacuum polarization [52] would not im- the muon). Namely, the parameterization could pede an experimental detection of the X17. Still, it be read as suggesting a likely increase of the pseu- would seem that more attractive possibilities exist doscalar coupling parameter with the mass of the in muonic systems. particle. While ξe is bound from below by the con- dition ξe > 4 [see Eq. (9)] the corresponding pa- rameter in the muon sector must fulfill ξµ < 0.9 For true muonium, one expects effects (for the hy- • −6 [see Eq. (57)]. perfine splitting of S states) on the order of 10 for the vector X17 model as well as the pseudoscalar The relative correction to the hyperfine splitting for • model. These have to be compared to the un- both S and P states is enhanced in muonic as com- certainty from the hadronic vacuum polarization, pared to electronic bound systems by two orders which has recently been improved to the level of of magnitude, in view of the scaling of the relative 2 ppm [57]. This implies that a modest progress in corrections with mr/mX , where mr is the reduced the determination of the R ratio, namely, the ratio mass of the two-body bound system. of the cross section of an electron-positron pair go- ing into hadrons, versus an electron-positron pair In muonic deuterium, the correction, for S states, − − going into muons [see Eq. (9) of Ref. [57]], could • is of order 3.8 10 6 (vector X17) and 1.0 10 6 render the effect visible in true muonium. (pseudoscalar× X17) in units of the Fermi− energy,× while the experimental accuracy for the S state hy- perfine splitting is of order 10−3, and there is a For positronium, the effects of the X17 are sup- 5 σ discrepancy of theory and experiment, in view • pressed by the small reduced mass of the system. of a recent calculation of the nuclear polarizabil- − They are bound not to exceed the level of 10 9 for ity effects [40]. One concludes that the experimen- the vector model and pseudoscalar models. Thus, tal accuracy would have to be improved by three even taking into account all the mα7 corrections orders of magnitude before the effects of the X17 currently under study [59–71], the detection of an become visible, and the understanding of the nu- X17-induced signal in the hyperfine splitting ap- clear effects would likewise have to be improved by pears to be extremely challenging in positronium. a similar factor. In muonic deuterium, for the hyperfine splitting of • One concludes that the most promising approach to- P1/2 states, the X17-mediated correction to the hy- ward a conceivable detection of the X17 in high-precision −7 perfine splitting is of order 2.5 10 for the vec- atomic physics experiments would probably concern the tor model, and of order 6.6 ×10−8 for the pseu- × hyperfine splitting of P1/2 states in muonic deuterium, doscalar model. These effects are not suppressed by and the related Sternheim difference, where the effects challenging nuclear structure effects and could be are enhanced because of the large mass ratio of the re- measurable in the next round of experiments [43]. duced mass of the atomic system to the mass of the X17, The same applies to the Sternheim weighted dif- and nuclear structure effects are suppressed because the ference [44] of the hyperfine splitting of S states, P state wave function (as well as the weighted difference where the effect induced by the X17 particle is of of the S states) vanishes at the origin. Furthermore, very the same order-of-magnitude as for the P1/2 split- attractive prospects in true muonium [53–57] should not ting. be overlooked. In muonic hydrogen, because of the protophobic • character of the vector model, effects of the vec- tor X17 are suppressed (order 10−9 for the S state −10 splitting and order 10 for the P state splitting ACKNOWLEDGMENTS and the Sternheim weighted difference). For the pseudoscalar model, the S state splitting is affected at relative order 10−7, and the P state splitting as The author acknowledges support from the National well as the Sternheim difference are affected at or- Science Foundation (Grant PHY–1710856). Also, the au- der 10−8. These effects could be measurable in the thor acknowledges utmost insightful conversations with future. K. Pachucki. Helpful conversations with A. Kraszna- horkay, U. Ellwanger, S. Moretti, and W. Shepherd are For muonium, the situation is not hopeless: While also gratefully acknowledged. The author also thanks the • the effects induced by the X17 shift the hyper- late Professor G. Soff for insightful discussions on general fine splitting only on the level of 10−9, which is aspects of the true muonium bound system. 12

Appendix A: Coulomb–Gauge Propagator for In order to generalize the result to a massive vector Massive Vector particle, it is necessary to observe that, in a more gen- eral sense, the “Coulomb gauge” for the calculation of Even if the Coulomb-gauge propagator for massive vec- bound states is defined to be the gauge in which the pho- tor bosons, given in Eqs. (12a) and (12b), has been used ton propagator component D00 is exactly static, i.e., has in the literature before [see Eq. (6) of Ref. [72] and no dependence on the photon frequency. Otherwise, one Eqs. (16) and (17) of Ref. [73]], separate notes on its would incur additional corrections in the Breit Hamilto- derivation can clarify the role of the “Coulomb gauge” nian, which is generated by the spatial components of the for massive vector bosons. photon propagator. Generally, in order to calculate the vector boson prop- The generalization of Eq. (A1) to the massive vector agator in a specific gauge, one can write the relation of exchange reads as [gµν = diag(1, 1, 1, 1)] the vector potential Aµ to the currents J ν (in the gauge − − − under investigation), and observe that the operator me- g 1 D (k)= µν + (f µ kν + f ν kµ) . diating the relation is the propagator. However, one can µν k2 m2 2(k2 m2) also ask the question which terms could possibly be added − − (A3) to the propagator, initially obtained in a specific gauge, Here, m = mX denotes the vector boson mass, and we without changing the fields. 2 2 start from the Feynman gauge result gµν /(k m ) [see Let us start with the massless case, i.e., the photon. Eqs. (3.147) and (3.149) in Ref. [75], with λ =− 1, in the One should remember [74] that the most general form of notation of Ref. [75]]. Now, choosing a gauge transformation of the photon propagator is [in relativistic notation with kµ = (ω,~k)] k0 ki f 0 = , f i = , (A4) gµν 1 ~k2 + m2 −~k2 + m2 D (k)= + (f µ kν + f ν kµ) . (A1) µν k2 2k2 one (almost) derives the result given in Eqs. (12a) The first term is the Feynman gauge result. The added and (12b) [and previously used in Eq. (6) of Ref. [72] terms do not change the fields, because the term pro- and in Eqs. (16) and (17) of Ref. [73]], with one caveat. portional to f µ kν vanishes in view of current conserva- ν µ Namely, in the spatial components of the propagator tion, while the term proportional to f k amounts to a (denoted by the Latin indices ij), one replaces k2 = gauge-transformation of the four-vector potential Aµ, as ω2 ~k2 ~k2 in the order of approximation of in- a careful inspection shows. The choice terest− here.→ − This is because the frequency dependence k0 ki of the propagator denominator leads to higher-order cor- f 0 = , f i = , (A2) rections, which, for the photon exchange, are summarized ~ 2 −~ 2 k k in the Salpeter recoil correction [76]. lead to the Coulomb gauge. The fact that the fields re- The transition to the massive Coulomb gauge is well main unaffected holds even for a “non-covariant” form of known to be useful in bound-state theory but is perhaps the f µ, i.e., for a form where the f µ do not transform as less familiar in the particle physics community; pertinent components of a four-vector under Lorentz transforma- remarks are thus in order when it comes to the possible tions. detection of a new particle in low-energy experiments.

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