Muonium-Antimuonium Oscillations in Effective Field Theory

PHYSICAL REVIEW D 102, 095001 (2020)

Muonium-antimuonium oscillations in effective field theory

Renae Conlin 1 and Alexey A. Petrov 1,2 1Department of Physics and Astronomy Wayne State University, Detroit, Michigan 48201, USA 2Leinweber Center for Theoretical Physics University of Michigan, Ann Arbor, Michigan 48196, USA

(Received 3 June 2020; accepted 9 October 2020; published 3 November 2020)

Flavor violating processes in the lepton sector have highly suppressed branching ratios in the standard model, mainly due to the tiny neutrino mass. This means that observing a lepton flavor violation (LFV) in the next round of experiments would constitute a clear indication of physics beyond the standard model (BSM). We revisit a discussion of one possible way to search for LFV, muonium-antimuonium oscillations. This process violates the muon lepton number by two units and could be sensitive to the types of BSM physics that are not probed by other types of LFV processes. Using techniques of effective field theory, we calculate the mass and width differences of the mass eigenstates of muonium. We argue that its invisible decays give the parametrically leading contribution to the lifetime difference and put constraints on the scales of new physics probed by effective operators in muonium oscillations.

DOI: 10.1103/PhysRevD.102.095001

I. INTRODUCTION So far, we have not yet observed FCNC in the charged Flavor-changing neutral current (FCNC) interactions lepton sector. This is because in the standard model with serve as a powerful probe of physics beyond the standard massive neutrinos, the charged lepton flavor violating (CLFV) transitions are suppressed by the powers of model (BSM). Since no local operators generate FCNCs in 2 2 the standard model (SM) at tree level, new physics (NP) mν=mW, which renders the predictions for their transition B μ → γ ∼ 10−54 degrees of freedom can effectively compete with the SM rates vanishingly small, e.g., ð e ÞνSM [3,4]. particles running in the loop graphs, making their discovery Yet, experimental analyses constantly push the bounds on possible. This is, of course, only true provided the BSM the CLFV transitions. It might be that in some models of models include flavor-violating interactions. NP, such as a model with the doubly charged Higgs An especially clean system to study BSM effects in the particles [5–8], the effective ΔL ¼ 2 transitions could lepton sector is muonium Mμ, a QED bound state of a occur at a rate that is not far below the sensitivity of positively charged muon and a negatively charged electron, currently operating experiments. Alternatively, it might be þ − jMμi ≡ jμ e i. The main decay channel for both states is that no term that changes the lepton flavor by two units is driven by the weak decay of the muon. The average lifetime present in a BSM Lagrangian. But even in this case, a Δ 1 of a muonium state τ is expected to be the same as that of subsequent application of two L ¼ interactions would Mμ effective ΔL 2 τ 2 1969811 0 0000022 10−6 also generate an ¼ interaction. the muon, μ ¼ð . . Þ× s [1], apart Such a ΔL ¼ 2 interaction would then change the τ − τ τ from the tiny effect due to time dilation, ð Mμ μÞ= μ ¼ muonium state into the antimuonium one, leading to the 2 2 2 −10 α me=ð2mμÞ¼6 × 10 [2]. Just like a positronium or a possibility of muonium-anti-muonium oscillations. As a hydrogen atom, muonium could be produced in two spin variety of well-established new physics models contain configurations, a spin-one triplet state called orthomuo- ΔL ¼ 2 interaction terms [3], the observation of a muonium, and a spin-zero singlet state called paramuonium.We nium converting into an antimuonium could then provide P shall denote the paramuonium state as jMμ i and the especially clean probes of new physics in the leptonic V orthomuonium state as jMμ i. If the spin of the state does sector [4,9]. Theoretical analyses of conversion probability for such transitions have been actively studied, mainly not matter, we shall employ the notation jMμi. using the framework of particular models [10–15]. It would be useful to perform a model-independent computation of the oscillation parameters using techniques of effective theory that includes all possible BSM models encoded in a Published by the American Physical Society under the terms of few Wilson coefficients of effective operators. We do so in the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to this paper, computing all relevant QED matrix elements. the author(s) and the published article’s title, journal citation, Finally, employing similar effective field theory (EFT) and DOI. Funded by SCOAP3. techniques for the computation of the contributions that are

2470-0010=2020=102(9)=095001(9) 095001-1 Published by the American Physical Society RENAE CONLIN and ALEXEY A. PETROV PHYS. REV. D 102, 095001 (2020) nonlocal at the muon mass scale, we present them in terms where we neglected CP violation and employed a con- ∓ of the series of local operators expanded in inverse powers vention where CPjMμi¼ jMμi. The mass and the of mμ [16,17]. width differences of the mass eigenstates are In this paper, we discuss the most general analysis of ¯ Mμ − Mμ oscillations in the framework of effective field Δm ≡ M1 − M2; ΔΓ ≡ Γ2 − Γ1; ð5Þ theory. We review phenomenology of muonium oscillations in Sec. II, taking into account both mass and lifetime where Mi (Γi) are the masses (widths) of the mass Δ ΔΓ differences in the muonium system. We compute the mass eigenstates jMμ1;2i. We defined m and to be either and width differences in Sec. III. In Sec. IV, we constrain positive or negative, which is to be determined by experi- the BSM scale Λ using experimental muonium-anti- ment. It is often convenient to introduce dimensionless muonium oscillation parameters. We conclude in Sec. V. quantities, Appendix contains some details of calculations. Δm ΔΓ x ¼ ;y¼ ; ð6Þ II. PHENOMENOLOGY OF Γ 2Γ MUONIUM OSCILLATIONS where the average lifetime Γ ¼ðΓ1 þ Γ2Þ=2. It is important ¯ The phenomenology of Mμ − Mμ oscillations is very to note that while Γ is defined by the standard model decay similar to phenomenology of meson-antimeson oscillations rate of the muon, x and y are driven by the lepton-flavor [18,19]. There are, however, several important differences violating interactions. It is then expected that both x, y ≪ 1. that we will emphasize below. One major difference is The time evolution of flavor eigenstates follows from related to the fact that both ortho and paramuonium can, in Eq. (2) [18,19], principle, oscillate. While most studies only considered ¯ muonium oscillations due to the BSM heavy states, below jMðtÞi ¼ gþðtÞjMμiþg−ðtÞjMμi; we also discuss the possibility of oscillations via the light ¯ ¯ jMðtÞi ¼ g−ðtÞjMμiþg ðtÞjMμi; ð7Þ states. Since such states can go on a mass shell, these þ contributions would lead to the possibility of a lifetime ¯ where the coefficients gðtÞ are defined as difference in the Mμ − Mμ system. If the new physics Lagrangian includes lepton-flavor 1 −Γ1t=2 −iM1t 1 ΔΓt=2 iΔmt violating interactions, the time development of a muonium gðtÞ¼2 e e ½ e e : ð8Þ and antimuonium states would be coupled, so it would be appropriate to consider their combined evolution, As x, y ≪ 1, we can expand Eq. (8) to get aðtÞ 1 ψ ¯ −Γ 2 − 2 2 j ðtÞi ¼ ¼ aðtÞjMμiþbðtÞjMμi: ð1Þ g t e 1t= e iM1t 1 y − ix Γt ; bðtÞ þð Þ¼ þ 8 ð Þ ð Þ 1 The time evolution of jψðtÞi evolution is governed by a −Γ1t=2 −iM1t − Γ g−ðtÞ¼2 e e ðy ixÞð tÞ: ð9Þ Schrödinger equation, Denoting an amplitude for the muonium decay into a final d jMμðtÞi Γ jMμðtÞi − state f as A ¼hfjHjMμi and an amplitude for its decay i ¯ ¼ m i ¯ : ð2Þ f dt jMμðtÞi 2 jMμðtÞi ¯ ¯ H into a CP-conjugated final state f as Af¯ ¼hfj jMμi,we ¯ can write the time-dependent decay rate of Mμ into the f, CPT invariance dictates that the masses and widths of muonium and antimuonium are the same, m11 ¼ m22, 1 ¯ 2 −Γt 2 Γ11 Γ22 CP ΔLμ 2 Γ → Γ ¼ , while invariance of the ¼ interaction, ðMμ fÞðtÞ¼2 NfjAfj e ð tÞ RMðx; yÞ; ð10Þ which we assume for simplicity, dictates that where N is a phase-space factor and R ðx; yÞ is the Γ Γ f M m12 ¼ m21; 12 ¼ 21: ð3Þ oscillation rate, The presence of off diagonal pieces in the mass matrix 1 2 2 signals that it needs to be diagonalized. The mass eigen- RMðx; yÞ¼2 ðx þ y Þ: ð11Þ states jMμ1;2i can be defined as Integrating over time and normalizing to ΓðMμ → fÞ,we 1 ¯ ffiffiffi ∓ ¯ get the probability of Mμ decaying as Mμ at some time jMμ1;2i¼p ½jMμi jMμi; ð4Þ 2 t>0,

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Γ → ¯ L ¯ ðMμ fÞ Lagrangian, eff, whose Wilson coefficients would be PðMμ → MμÞ¼ ¼ RMðx; yÞ: ð12Þ ΓðMμ → fÞ determined by the UV physics that becomes active at some scale Λ [22,23], This equation generalizes oscillation probability computed X in the classic papers [11,13] by accounting for the lifetime 1 L ¼ − c ðμÞQ ; ð14Þ difference in the muonium system, making it dependent on eff Λ2 i i i both the normalized mass x and the lifetime y differences. We will compute those in the next section. where the c ’s are the short distance Wilson coefficients. We shall use the data from the most recent experiment i They encode all model-specific information. Qi’s are the [9] in order to place constraints on the oscillation param- effective operators that reflect degrees of freedom relevant eters. To do so, we have to account for the fact that the setup at the scale at which a given process takes place. If we described in [9] had muonia propagating in a magnetic field assume that no new light particles (such as “dark photons” B0. This magnetic field suppresses oscillations by remov- ¯ or axions) exist in the low energy spectrum, those operators ing degeneracy between Mμ and Mμ. It also has a different would be written entirely in terms of the SM degrees of effect on different spin configurations of the muonium freedom. In the case at hand, all SM particles with masses state and the Lorentz structure of the operators that larger than that of the muon should also be integrated out, generate mixing [20,21]. Experimentally, these effects were leaving only muon, electron, photon, and neutrino degrees accounted for by introducing a factor SBðB0Þ. The oscil- of freedom. lation probability is then [9] It would be convenient for us to classify effective ¯ −11 operators in Eq. (14) by their lepton quantum numbers. P Mμ → Mμ ≤ 8 3 10 =S B0 : ð Þ . × Bð Þ ð13Þ In particular, we can write the effective Lagrangian as

We shall use different values of SBðB0Þ, presented in L LΔLμ¼0 LΔLμ¼1 LΔLμ¼2 Table II of [9], when placing constraints on the Wilson eff ¼ eff þ eff þ eff : ð15Þ coefficients of effective operators in the next section. The first term in this expansion contains both the standard III. EFFECTIVE THEORY OF OSCILLATIONS model and the new physics contributions. It then follows LΔLμ¼0 Muonium-anti-muonium oscillations could be effective that the leading term in eff is suppressed by powers of Λ probes of flavor-violating new physics in leptons. One of MW, not the new physics scale . We should emphasize the issues is that, at this point, we do not know which that only the operators that are local at the scale of the particular model of new physics will provide the correct muonium mass are retained in Eq. (15). Δ 1 ultraviolet (UV) extension for the standard model. The second term contains Lμ ¼ operators. As we However, since the muonium mass is most likely much integrated out all heavy degrees of freedom, the operators of smaller than the new particle masses, it is not necessary to the lowest possible dimension that governs muonium oscil- know it. Any new physics scenario which involves lepton lations must be of dimension six. The most general dimen- LΔLμ¼1 flavor violating interactions can be matched to an effective sion six effective Lagrangian, eff , has the form [24,25],

1 X ΔLμ¼1 f α f α ¯ f α q α ¯ L − C μ¯ γ e C μ¯ γ e fγαf C μ¯ γ e C μ¯ γ e fγαγ5f eff ¼ Λ2 ½ð VR R R þ VL L LÞ þð AR R R þ AL L LÞ f f μ¯ f μ¯ ¯ f μ¯ f μ¯ ¯γ þ memfGFðCSR ReL þ CSL LeRÞff þ memfGFðCPR ReL þ CPL LeRÞf 5f f μ¯ σαβ f μ¯ σαβ ¯σ þ memfGFðCTR R eL þ CTL L eRÞf αβf þ H:c:; ð16Þ

∼ −2 μ where GF MW is the Fermi constant, and e are the dimension six in the standard model effective field theory μ μ 1 1 γ5 fermion fields, ð ;eÞL;R ¼ PL;Rð ;eÞ. PR;L ¼ 2 ð Þ (SMEFT) but could be generated by higher order operators. are the projection operators, and f represents other fer- This is taken into account by introducing mass and GF mions that are not integrated out at the muonium scale. factors emulating such suppression [24,25]. LΔLμ¼2 The subscripts on the Wilson coefficients are for the type of The last term in Eq. (15), eff , represents the effective Lorentz structure: vector, axial vector, scalar, pseudoscalar, operators changing the lepton quantum number by two and tensor. The Wilson coefficients would in general be units. The leading contribution to muonium oscillations is different for different fermions f. Note that the Lagrangian given by the dimension six operators. The most general Eq. (16) also contains terms that do not follow from the effective Lagrangian,

095001-3 RENAE CONLIN and ALEXEY A. PETROV PHYS. REV. D 102, 095001 (2020) X Δ 2 1 ΔLμ¼2 Lμ¼ ΔL 2 ¯ H ¯ H L − C ¼ μ Q μ ; 17 hMμj effjMμi¼hMμj eff jMμi; ð22Þ eff ¼ Λ2 i ð Þ ið Þ ð Þ i provided that only Q1 − Q5 operators are taken into can be written with the operators written entirely in terms of account. It is easy to see that the relevant contributions the muon and electron degrees of freedom, are only suppressed by Λ2. Other contributions, including ΔLμ 1 the nonlocal double insertions of L ¼ , represented by μ¯ γ μ¯ γα μ¯ γ μ¯ γα eff Q1 ¼ð L αeLÞð L eLÞ;Q2 ¼ð R αeRÞð R eRÞ; the second term in Eq. (21), do contribute to the mass α difference, but are naively suppressed by Λ4. Thus, we shall Q3 ¼ðμ¯ LγαeLÞðμ¯ Rγ eRÞ;Q4 ¼ðμ¯ LeRÞðμ¯ LeRÞ; not consider them in this paper. μ¯ μ¯ Q5 ¼ð ReLÞð ReLÞ: ð18Þ In order to evaluate the mass difference contribution, we need to take the matrix elements. As explained in the We did not include operators that could be related to Introduction, we expect that both spin-0 singlet and spin-1 the presented ones via Fierz relations. It is important to triplet muonium states would undergo oscillations. The note that some of the operators in Eq. (18) are not in- oscillation parameters would in general be different, as the 2 1 variant under the SM gauge group SUð ÞL × Uð Þ. This matrix elements would differ for those two cases. means that they receive additional suppression, as they may Using factorization approach familiar from the meson be generated from the higher-dimensional operators in flavor oscillation, the matrix elements can be easily written SMEFT [23]. in terms of the muonium decay constant fM [27,28], Other ΔLμ ¼ 2 local operators that will be important α 5 P α α V α later in this paper can be written as h0jμγ¯ γ ejMμ i¼ifPp ; h0jμγ¯ ejMμ i¼fVMMϵ ðpÞ; 0 μσ¯ αβ V ϵα β − ϵβ α μ¯ γ ν γαν h j ejMμ i¼ifTð p p Þ; ð23Þ Q6 ¼ð L αeLÞð μL eLÞ; μ¯ γ ν γαν α ’ ϵα Q7 ¼ð R αeRÞð μL eLÞ; ð19Þ where p is paramuonium s four momentum, and ðpÞ is the orthomuonium’s polarization vector. Note that fP ¼ where we only included SMEFT operators that contain left- fV ¼ fT ¼ fM in the nonrelativistic limit. The decay handed neutrinos [23,26]. In order to see how these constant can be written in terms of the bound-state wave operators (and thus new physics) contribute to the mixing function, parameters, it is instructive to consider off diagonal terms in φ 0 2 the mass matrix [18], 2 4 j ð Þj fM ¼ ; ð24Þ MM 1 − i Γ ¯ H m ¼ hMμj effjMμi which is the QED’s version of Van Royen-Weisskopf 2 12 2M M formula. For a Coulombic bound state, the wave function 1 X ¯ H H hMμj effjnihnj effjMμi of the ground state is þ2 − ϵ ; ð20Þ MM n MM En þi 1 − r φðrÞ¼qffiffiffiffiffiffiffiffiffiffi e aMμ ; ð25Þ where the first term does not contain imaginary part, so it π 3 aMμ contributes to m12, i.e., the mass difference. The second term contains bilocal contributions connected by physical where a ¼ðαm Þ−1 is the muonium Bohr radius, α is intermediate states. This term has both real and imaginary Mμ red the fine structure constant, and m ¼ m mμ=ðm þ mμÞ is parts and thus contributes to both m12 and Γ12. red e e the reduced mass. Then,

A. Mass difference: ΔL = 2 operators 3 μ ðm αÞ 1 φ 0 2 red m α 3: 26 We can rewrite Eq. (20) to extract the physical mixing j ð Þj ¼ π ¼ π ð red Þ ð Þ parameters x and y of Eq. (6). For the mass difference, In the nonrelativistic limit, factorization gives the exact 1 result for the QED matrix elements of the six-fermion 2 ¯ H x ¼ Re½ hMμj effjMμi operators. Nevertheless, we explicitly verified that this is 2MMΓ Z indeed the case (see Appendix). ¯ 4 H H 0 þhMμji d xT½ effðxÞ effð ÞjMμi: ð21Þ 1. Paramuonium Assuming the LFV NP is present, the dominant local The matrix elements of the spin-singlet states can be contribution to x comes from the last term in Eq. (15), obtained from Eq. (18) using the definitions of Eq. (23),

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¯ P P 2 2 ¯ P P 2 2 lc l¯ T hMμ jQ1jMμ i¼fMMM; hMμ jQ2jMμ i¼fMMM; where ¼ C is the charge-conjugated lepton state. −− 3 1 Integrating out the Δ field, this Lagrangian leads to the ¯ P P − 2 2 ¯ P P − 2 2 hMμ jQ3jMμ i¼ 2fMMM; hMμ jQ4jMμ i¼ 4fMMM; following effective Hamiltonian [5,8]: 1 g gμμ ¯ P P − 2 2 H ee μ¯ γ μ¯ γα hMμ jQ5jMμ i¼ fMMM: ð27Þ Δ ¼ 2 ð R αeRÞð R eRÞþH:c:; ð32Þ 4 2MΔ Combining the contributions from the different operators below the scales associated with the doubly charged Higgs and using the definitions from Eqs. (24) and (26), we obtain field’s mass MΔ. Examining Eq. (32), we see that this an expression for xP for the paramuonium state, Hamiltonian matches onto our operator Q2 [see Eq. (18)] with the scale Λ ¼ MΔ and the corresponding Wilson 4 m α 3 3 ΔL¼2 2 ð red Þ ΔL¼2 ΔL¼2 − ΔL¼2 coefficient C2 ¼ geegμμ= . xP ¼ πΛ2Γ C1 þ C2 2 C3 1 B. Width difference: ΔLμ = 2 and ΔLμ = 1 operators − ΔL¼2 ΔL¼2 4 ðC4 þ C5 Þ : ð28Þ The lifetime difference in the muonium system can be obtained from Eq. (20) [30]. It comes from the physical This result is universal and holds true for any new physics intermediate states, which is signified by the imaginary part model that can be matched into a set of local ΔL ¼ 2 in Eq. (20) and reads interactions. 1 X ρ ¯ H H y ¼ Γ nhMμj effjnihnj effjMμi; ð33Þ 2. Orthomuonium n ρ Using the same procedure, but computing the relevant where n is a phase space function that corresponds to the ¯ matrix elements for the vector orthomuonium state, we intermediate state that is common for Mμ and Mμ. There are 1 obtain the matrix elements, only two possible intermediate states that can contribute to y, eþe−, and νν¯. The eþe− intermediate state corresponds ¯ V V −3 2 2 ¯ V V −3 2 2 Δ 1 → þ − H hMμ jQ1jMμ i¼ fMMM; hMμ jQ2jMμ i¼ fMMM; to a Lμ ¼ decay Mμ e e , which implies that eff ¼ 3 3 ΔLμ¼1 ¯ V V 2 2 ¯ V V 2 2 H in Eq. (33). According to Eq. (16), it appears that, hMμ jQ3jMμ i¼− f M ; hMμ jQ4jMμ i¼− f M ; eff 2 M M 4 M M quite generally, this contribution is suppressed by Λ4, i.e., 3 will be much smaller than x, irrespective of the values of the ¯ V V − 2 2 hMμ jQ5jMμ i¼ 4fMMM: ð29Þ corresponding Wilson coefficients. Another contribution comes from the νν¯ intermediate Again, combining the contributions from the different state. This common intermediate state can be reached by operators, we obtain an expression for xV for the ortho- the standard model tree level decay Mμ → νμνe interfering ¯ muonium state, with the ΔLμ ¼ 2 decay Mμ → νμνe. Such a contribution is only suppressed by Λ2M2 and represents the parametri- 12 m α 3 1 W − ð red Þ ΔL¼2 ΔL¼2 ΔL¼2 cally leading contribution to y. We shall compute this xV ¼ πΛ2Γ C1 þ C2 þ 2 C3 contribution below. 1 Writing y similarly to x in Eq. (21), i.e., in terms of the ΔL¼2 ΔL¼2 þ 4 ðC4 þ C5 Þ : ð30Þ correlation function, we obtain Z 1 ¯ 4 Again, this result is universal and holds true for any new y ¼ Im hMμji d xT½H ðxÞH ð0ÞjMμi 2M Γ eff eff physics model that can be matched into a set of local M Z Δ 2 1 Δ 2 Δ 0 L ¼ interactions. ¯ 4 H Lμ¼ H Lμ¼ 0 ¼ Im hMμji d xT½ eff ðxÞ eff ð ÞjMμi ; It might be instructive to present an example of a BSM MMΓ model that can be matched into the effective Lagrangian of ð34Þ Eq. (17) and can be constrained from Eqs. (27) and (29).

Let us consider a model that contains a doubly charged ΔLμ¼0 ΔLμ¼0 where the H ¼ −L is given by the ordinary Higgs boson [5,6,29]. Such states often appear in the eff eff standard model Lagrangian, context of left-right models [7,8]. A coupling of the doubly charged Higgs field Δ−− to the lepton fields can be written as 1A possible γγ intermediate state is generated by higher- dimensional operators and therefore, is further suppressed by Λ α L l¯ lcΔ either powers of or the QED coupling than the contributions R ¼ gll R þ H:c:; ð31Þ considered here.

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We can now compute the lifetime difference y by using Eq. (34) and take the relevant matrix elements for the spin singlet and the spin triplet states of the muonium.

1. Paramuonium The matrix elements of the spin-singlet state have been FIG. 1. A contribution to y described in Eq. (34). A white computed above and presented in Eq. (27). Computing the square represents a vertex given by Eq. (19), while a black dot is matrix elements in Eq. (34) using their definitions from given by the SM contribution of Eq. (35). A dotted line represents Eqs. (24) and (26), we obtain an expression for the lifetime the imaginary part. difference yP for the paramuonium state,

G M2 Δ 0 4 ffiffiffiF M α 3 ΔL¼2 − ΔL¼2 Lμ¼ GF α yP ¼ p 2 ðmred Þ ðC6 C7 Þ: ð39Þ L ¼ − pffiffiffi ðμ¯ γαe Þðν γ νμ Þ; ð35Þ 2Λ2 π Γ eff 2 L L eL L ΔL¼2 ΔL¼2 It is interesting to note that if C6 ¼ C7 current conservation assures that no lifetime difference is generated HΔLμ¼2 and eff only contributes through the operators Q6 at this order in 1=Λ for the paramuonium. and Q7. Since the decaying muon injects a large momentum into 2. Orthomuonium the two-neutrino intermediate state, the integral in Eq. (34) Similarly, using the matrix elements for the spin-triplet is dominated by small distance contributions, compared to state computed in Eq. (29), the expression fo Eq. (38) leads the scale set by 1=mμ. We can the compute the correlation to the lifetime difference, function in Eq. (34) by employing a short distance operator product expansion, systematically expanding it in powers 2 GF MM 3 ΔL 2 ΔL 2 of 1=mμ, (for a corresponding diagram, see Fig. 1) y ¼ − pffiffiffi ðm αÞ ð5C ¼ þ C ¼ Þ: ð40Þ V 2Λ2 π2Γ red 6 7

Z We emphasize that Eqs. (39) and (40) represent parametri- 4 HΔLμ¼2 HΔLμ¼0 0 cally leading contributions to muonium lifetime difference, T ¼i d xT½ eff ðxÞ eff ð Þ Z as they are only suppressed by two powers of Λ. 4 μ¯Γ ν¯ γαν μγ¯ ν¯ γβν 0 ¼i d xT½ð αeÞð μL eLÞðxÞð βPLeÞð eL μLÞð Þ: IV. EXPERIMENTAL CONSTRAINTS ð36Þ We can now use the derived expressions for x and y to place constraints on the BSM scale Λ (or the Wilson coefficients Ci) from the experimental constraints on The leading term is obtained by contracting the neutrino muonium-anti-muoium oscillation parameters. Since both fields in Eq. (36) into propagators, spin-0 and spin-1 muonium states were produced in the experiment [9], we should average the oscillation probability over the number of polarization degrees of freedom, ⊓ ν¯μðxÞνμ ð0Þ¼iS ð−xÞ; X 1 F ¯ i ¯ i PðMμ → MμÞ ¼ PðMμ → Mμ Þ; ð41Þ ν ν 0 exp 2 1 eðxÞ e ð Þ¼iSFðxÞ; ð37Þ i¼P;V Si þ ⊔ → ¯ where PðMμ MμÞexp is the experimental oscillation probability from Eq. (13). We shall use the values of where SFðxÞ represents the propagator in coordinate S B0 for B0 2.8 μT from the Table II of [9], as it will representation. In what follows, we will consider neutrinos Bð Þ ¼ provide us the best experimental constraints on the BSM to be Dirac fields for simplicity. scale Λ. We report those constraints in Table I. As one can Using Cutkoski rules to compute the discontinuity see from Eqs. (28), (30), (39), and (40), each observable (imaginary part) of T and calculating the phase space depends on the combination of the operators. We shall integrals, we get assume that only one operator at a time gives a dominant contribution. This ansatz is usually referred to as the single operator dominance hypothesis. It is not necessarily real- G M2 1 ffiffiffiF M ΔL¼2 ΔL¼2 ized in many particular UV completions of the LFV EFTs, DiscT ¼p 2 C6 ðQ1 þQ5Þþ C7 Q3 : ð38Þ 2Λ 3π 2 as cancellations among contributions of different operators

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TABLE I. Constraints on the energy scales probed by different obtained over two decades ago probe the same energy ΔL ¼ 2 operators of Eqs. (18) and (19). We set the corresponding scales as current LHC experiments. 1 Wilson coefficient Ci ¼ . We urge our experimental colleagues to further study muonium-antimuonium oscillations. It would be interesting SBðB0Þ Constraints on the Operator Interaction type (from [9]) scale Λ,TeV to see how far the proposed MACE experiment [31] or similar facility at FNAL could push the constraints on the − − Q1 ðV AÞ × ðV AÞ 0.75 5.4 muonium oscillation parameters. Q2 ðV þ AÞ × ðV þ AÞ 0.75 5.4 − Q3 ðV AÞ × ðV þ AÞ 0.95 5.4 V. CONCLUSIONS Q4 ðS þ PÞ × ðS þ PÞ 0.75 2.7 − − Q5 ðS PÞ × ðS PÞ 0.75 2.7 Lepton flavor violating transitions provide a powerful − − −3 Q6 ðV AÞ × ðV AÞ 0.75 0.58 × 10 engine for new physics searches. In this work, we revisited − −3 Q7 ðV þ AÞ × ðV AÞ 0.95 0.38 × 10 phenomenology of muonium-antimuonium oscillations. We argued that in generic models of new physics both mass and lifetime differences in the muonium system are possible. It is however a useful tool in constraining would contribute to the oscillation probability. We com- L parameters of eff. puted the normalized mass difference x in the muonium Λ2 Since it is the combination Ci= that enters the system with the most general set of effective operators for theoretical predictions for x and y, one cannot separately both spin-singlet and the spin-triplet muonium states. We Λ Λ measure Ci and . We choose to constrain the scale that set up a formalism for computing the lifetime difference is probed by the corresponding operator and set the and computed the parametrically leading contribution to y. corresponding value of the Wilson coefficient Ci to one. Using the derived expressions for x and y, we then put Such anapproach, as any calculation based on effective constraints on the BSM scale Λ. From this, we found that field-theoretic techniques, has its advantages and disadvan- for operators Q1 − Q5 the experimental data provided tages. The advantage of such approach is in the fact that it constraints on scales relevant to the LHC program. allows us to constrain all possible models of new physics ¯ that can generate Mμ − Mμ mixing. The models are ACKNOWLEDGMENTS encoded in the analytic expressions for the Wilson coefficients of a few effective operators in Eq. (18). The This work was supported in part by the U.S. Department disadvantage is reflected in the fact that possible comple- of Energy under Contract No. de-sc0007983. A. A. P. thanks the Institute for Nuclear Theory at the University mentary studies of new physics contributions to ΔL ¼ 1 of Washington for its kind hospitality and stimulating and ΔL ¼ 2 processes are not straightforward. Those can research environment. This research was also supported be done by considering particular BSM scenarios, which is 2 in part by the INT’s U.S. Department of Energy Grant beyond the scope of this paper. The EFT techniques are No. DE-FG02- 00ER41132. then used to simplify calculations of radiative corrections. The results are reported in Table I. As can be seen, the APPENDIX experimental data provide constraints on the scales com- parable to those probed by the LHC program, except In this Appendix, we show that the vacuum insertion for Q6 and Q7. The results indicate that existing bounds approximation leads to the same answer as a direct ¯ on Mμ − Mμ oscillation parameters probe NP scales of the computation of a four-fermion matrix element relevant order of several TeV. The constraints on the lepton-flavor for the muonium-anti-muonium oscillations. We shall show violating neutrino operators Q6 and Q7 are understandably that by computing a matrix element of the Q1 operator as an weaker, as the lifetime difference is suppressed by a factor example. The matrix elements is defined as 2 GF=Λ , while the mass difference is only suppressed by a 2 ¯ μγ¯ μγ¯ α factor of 1=Λ . We would like to emphasize that constraints hQ1i¼hMμjð αPLeÞð PLeÞjMμiðA1Þ on the oscillation parameters come from the data that are over 20 years old [9]. We find it amazing that the data for both pseudoscalar and vector muonium states. In order to compute the matrix element in Eq. (A1), we need to build the muonium states. We can employ the standard Bethe- Salpeter formalism. Since the muonium state is essentially 2An example of such analysis concentrating on models a a nonrelativistic Coulomb bound state of a μþ and an e−, containing doubly charged Higgs states is [29], where it was ¯ we can conventionally define it [32], concluded that 1999 data on Mμ − Mμ oscillations [9] give constraints that are weaker than (but complimentary to) those sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 3 obtained from a combination of constraints on μ → 3e and other 2M d p M φ˜ p p p0 experiments. Other examples include models where the mixing is jMμi¼ 3 ð Þj ; i: ðA2Þ 2mμ2m ð2πÞ generated by loops with neutral particles, such as heavy neutrinos. e

095001-7 RENAE CONLIN and ALEXEY A. PETROV PHYS. REV. D 102, 095001 (2020) 0 This state is normalized as hMμðPÞjMμðP Þi ¼ 1 η ¯γ ¯γα ξ† ξ† γ0γα 1−γ5 3 3 0 ðu αPLvÞðv PLuÞMμ ¼ mμmeð ; Þ ð Þ 2Epð2πÞ δ ðP − P Þ. The muonium state in Eq. (A2) is 4 −η projected from ap two-particleffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffi state of a muon and an ξ e † μ † † † 0 5 0 2 2 ð Þ ð Þ 0 ×ðη ; −η Þγ γαð1−γ Þ ; electron jp; p i¼ Ep Ep0 ap bp0 j i with the help ξ of the Fourier transform of the spatial wave equation describing the bound state φ˜ ðpÞ, ðA9Þ Z or writing out the gamma matrices and spinors from φ˜ ðpÞ¼ d3rφðrÞeipr: ðA3Þ Eqs. (A7) and (A8) and making rearrangements, we find ¯γ ¯γα ðu αPLvÞðv PLuÞMμ We expand each electron and muon field in the operator of 01 0 σα 10 η Eq. (A1) as ξ† ξ† ¼ mμmeð ; Þ α Z 10 σ¯ 0 00 −η 3 X d p 1 − † ψ pffiffiffiffiffiffiffiffi s s ipx s s ipx 01 0 σα 10 ξ ðxÞ¼ 3 ðapu ðpÞe þ bp v ðpÞe Þ: † † ð2πÞ 2Ep × ð η ; −η Þ s 10 σ¯ α 0 00 ξ ðA4Þ ξ†σ¯ αη η†σ¯ ξ ¼ mμmeð Þð α ÞMμ ηξ†σ¯ α ξη†σ¯ We will work in nonrelativistic approximation and neglect ¼ mμmeTr½ Tr½ αMμ : ðA10Þ the momentum dependence of the spinors, which are defined as Projection onto the singlet (spin-0) or the triplet (spin-1) states can be achieved through the substitutions [32], ξ η pffiffiffiffiffiffi pffiffiffiffiffiffi 1 u ¼ m ;v¼ m ; † e ξ e −η ξη ¼ pffiffiffi 12 2 ðA11Þ 2 × pffiffiffiffiffiffi † † 0 pffiffiffiffiffiffi † † 0 u¯ ¼ mμð ξ ; ξ Þγ ; v¯ ¼ mμð η ; −η Þγ : ðA5Þ for the spin-0 state and Here, ξ and η are the two-component spinors [32]. There 1 ξη† ¼ pffiffiffi ϵ⃗ · σ⃗ ðA12Þ are four ways to Wick contract the fields in the operator 2 with those generating the state. Using the anticommutation † 3 3 0 relation fap;a 0 g¼ð2πÞ δ ðp − p Þ results in for the spin-1 state with three possible polarization states, p 1 1 ϵ⃗ 0 0 1 ϵ⃗ pffiffi 1 0 ϵ⃗ pffiffi 1 − 0 1 ¼ð ; ; Þ, 2 ¼ 2 ð ;i; Þ, and 3 ¼ 2 ð ; i; Þ.Itis α α hQ1i¼½ðu¯γαPLvÞðv¯γ PLuÞþðv¯γαPLuÞðu¯γ PLvÞ convenient to introduce polarization four vectors [28], ⃗ ν ν − ¯γ ¯γα − ¯γ ¯γα ϵν ¼ð0; ϵ Þ, σ ¼ð1; σ⃗Þ, and σ¯ ¼ð1; −σ⃗Þ. ðv αPLvÞðu PLuÞ ðu αPLuÞðv PLvÞMμ Z Computing the traces for the singlet spin state, Eq. (A10) 3 2 d p becomes × φ˜ ðpÞ ; ðA6Þ ð2πÞ3 † α † 1 α ηξ σ¯ P ξη σ¯ P σ¯ σ¯ mμmeTr½ Mμ Tr½ αMμ ¼ Tr½ Tr½ α where we indicated that the spinors still need to be 2 projected onto the spin-triplet or the spin-singlet states. ¼ 2mμme: ðA13Þ This projection can be illustrated explicitly by considering α Notice that this expression is zero unless α ¼ 0. Similarly, the first term in Eq. (A6), ðu¯γαPLvÞðv¯γ PLuÞ, the rest can be computed in a complete analogy to that. Employing the for the spin-1 state, Eq. (A10) becomes Weyl basis for the gamma matrices, † α † ηξ σ¯ V ξη σ¯ V mμmeTr½ Mμ Tr½ αMμ 01 0 σα −10 1 0 α 5 ϵ ϵ σ¯ ασμ σ¯ σν γ ¼ ; γ ¼ ; γ ¼ ; ðA7Þ ¼ mμme μ νTr½ Tr½ α 10 σ¯ α 0 01 2 μ ¼ 2mμmeϵμϵ ¼ −6mμme; ðA14Þ where σα and σ¯ α are defined as μ as the sum over polarizations is ϵμϵ ¼ −3. Following the σα ¼ð1; σ⃗Þ; σ¯ α ¼ð1; −σ⃗Þ: ðA8Þ same procedure for the rest of the terms in Eq. (A6) and using σ⃗ Z Note that is a vector comprised of the Pauli matrices, and 3 2 1 2 2 d p φ˜ φ 0 2 is the × identity matrix. Now, expanding the matrix 3 ðpÞ ¼j ð Þj ; ðA15Þ elements, ð2πÞ

095001-8 MUONIUM-ANTIMUONIUM OSCILLATIONS IN EFFECTIVE … PHYS. REV. D 102, 095001 (2020) we get hQ1i for spin-0 and spin-1, which is identical to the definitions in Eqs. (27) and (29), ¯ P P 2 provided that the Van Royen-Weisskopf formula of hMμ jQ1jMμ i¼4M jφð0Þj ; M Eq. (24) is used. The proof for the rest of the operators ¯ V V 2 hMμ jQ1jMμ i¼−12MMjφð0Þj ; ðA16Þ follows the same steps.

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