Physics of Muonium and Muonium Oscillations

Physics of muonium and muonium oscillations

Alexey A. Petrov1 1Department of Physics and Astronomy Wayne State University, Detroit, MI 48201, USA Precision studies of a muonium, the bound state of a muon and an electron, provide access to physics beyond the Standard Model. We propose that extensive theoretical and experimental studies of atomic physics of a muonium, its decays and muonium-antimuonium oscillations could provide an impact on indirect searches for new physics.

An especially clean system to study BSM effects in in the Standard Model at tree level, New Physics degrees lepton sector is muonium Mµ, a QED bound state of a of freedom can effectively compete with the SM parti- positively-charged muon and a negatively-charged elec- cles running in the loop graphs, making their discovery + − tron, |Mµi ≡ |µ e i. The main decay channel of the possible. This is, of course, only true provided beyond state is driven by the weak decay of the muon. The av- the Standard Model (BSM) constructions include flavor- erage lifetime of a muonium state τMµ is thus expected violating interactions. In order to probe those we con- to be the same as that of the muon, τµ = (2.1969811 ± sider muonium decays and oscillations. −6 0.0000022) × 10 s [1], apart from the tiny effect due to Denoting the “muon quantum number” by Lµ, FCNC 2 2 2 −10 time dilation, (τMµ −τµ)/τµ = α me/(2mµ) = 6×10 . decays of a muonium would probe ∆Lµ = 1 interactions. Such a lifetime, in principle, is rather long to allow for The effective Lagrangian, L∆Lµ=1, can then be divided precision measurements of muonium’s atomic and parti- eff into the dipole part, LD, and a part that involves four- cle physics properties [2]. In this Letter of Interest (LOI) fermion interactions. we will argue that a robust theoretical and experimental research program in muonium physics will help us better Leff = LD + L`q + .... (1) understand both precision Standard Model (SM) physics and place competitive constraints on New Physics (NP) Here the ellipses denote effective operators that are not interactions. relevant for the following discussion. The dipole part in Eq. (1) is usually written as m h I. MUONIUM: ATOMIC PHYSICS L = − `1 C`1`2 ` σµν P ` D Λ2 DR 2 L 1 i `1`2 µν Muonium’s structure is similar to that of a hydrogen + CDL `2σ PR`1 Fµν + h.c. , (2) without the added complexity of a proton, both muon and electron are point-like, which helps in interpretation where PR,L = (1 ± γ5)/2 is the right (left) chiral projec- of atomic structure measurements. Just like a positron- tion operator. The Wilson coefficients would, in general, ium or a Hydrogen atom, muonium could be produced be different for different leptons ì. in two spin configurations, a spin-one triplet state called The four-fermion dimension-six lepton-quark La- ortho-muonium, and a spin-zero singlet state called para- grangian takes the form [5] muonium. Atomic calculations of muonium properties are impor- ∆L =1 1 X h L µ = − Cf µ γαe + Cf µ γαe jV tant and could be related to other systems. Measure- eff Λ2 VR R R VL L L α ments of its hyperfine splitting ∆ν , its Zeeman ef- f HFS fect, the 1s−2s frequency difference ∆ν could be use- f α q α A 1s2s + CAR µRγ eR + CAL µLγ eL jα ful in connection with similar computations in Hydrogen f f S without the complications of strong interactions and for + memf GF CSR µReL + CSL µLeR j (3) searches for New Physics. For example, the light-by-light f f P (LBL) contribution to the hyperfine splitting in muonium + memf GF CPR µReL + CPL µLeR j is similar to the LBL contribution to (g−2). A three-loop f αβ f αβ T i correction has recently been computed [3] with hadronic + memf GF CTR µRσ eL + CTL µLσ eR jαβ , LBL scattering contribution computed earlier [4]. V A S P where jα = fγαf, jα = fγαγ5f, j = ff, j = fγ5f, T and jαβ = fσαβf are the fermion currents with f repre- II. MUONIUM: DECAYS senting fermions that are not integrated out at the the muonium scale. The subscripts on the Wilson coefficients Flavor-changing neutral current (FCNC) interactions are for the type of Lorentz structure: vector, axial-vector, serve as a powerful probe of physics beyond the standard scalar, pseudo-scalar, and tensor. The Wilson coefficients model (BSM). Since no local operators generate FCNCs would in general be different for different fermions f. 2

The effective Lagrangian of Eq. (1) could be probed ity improved by 2-3 orders of magnitude could exploit in muon decays such as µ → eγ or µ → 3e, etc. These the time evolution of the Mµ − M µ conversion process. decays receive contributions from many possible opera- Integrating over time and normalizing to Γ(Mµ → f) tors in Eqs. (2) and (3), making it difficult to identify we get the probability of Mµ decaying as M µ at some the Lorentz structure of possible NP interactions. This time t > 0 [8], problem could be alleviated by studying decays with con- strained kinematics [5, 6]. Studying decays of para- and ortho-muonia M V → e+e−, M V → 3γ, and M P → γγ Γ(Mµ → f) µ µ µ P (Mµ → M µ) = = RM (x, y). (6) will only select particular combinations of the operators Γ(Mµ → f) in Eq. (3), making possible identification of NP interac- Using the data from [9] to place constraints on the oscil- tions. lation parameters, one must take into to account the fact that the set-up described in [9] had muonia propagating in a magnetic field B0. This magnetic field suppresses os- III. MUONIUM-ANTIMUONIUM cillations by removing degeneracy between Mµ and M µ. OSCILLATIONS It also has a different effect on different spin configurations of the muonium state and the Lorentz structure of In the presence of flavor-violating interactions muo- the operators that generate mixing [10–12]. Experimen- + − tally these effects could accounted for by introducing a nium |Mµi ≡ |µ e i could oscillate into an antimuo- − + factor SB(B0). The oscillation probability is then [9], nium |M µi ≡ |µ e i state. While such processes are strongly suppressed by neutrino masses in the Standard −11 Model, plenty of BSM scenarios boast much larger tran- P (Mµ → M µ) ≤ 8.3 × 10 /SB(B0). (7) sition rates [7]. A mere presence of ∆Lµ = 2 interactions leads to the fact that muonium flavor eigenstates A more thorough analysis of the effects of magnetic fields are no longer its mass eigenstates, leading to the time- would be desirable for better constraints on the oscillat- dependent oscillations of flavor states, and generation of ing parameters. the mass and lifetime splittings of its mass eigenstates, One can use Ref. [8] to relate x and y to the BSM scale which we denote ∆m and ∆Γ, respectively. Λ (or the Wilson coefficients Ci of BSM operators) and Denoting an amplitude for the muonium decay into a constrain their values from the experimental data. Since final state f as Af = hf|H|Mµi and an amplitude for both spin-0 and spin-1 muonium states were produced its decay into a CP-conjugated final state f as Af¯ = in the experiment [9], one should average the oscillation hf|H|Mµi, we can write the time-dependent decay rate probability over the number of polarization degrees of of Mµ into the f [8], freedom,

1 2 2 Γ(M → f)(t) = N |A | e−Γt (Γt) R (x, y), (4) X 1 i i µ 2 f f M P (Mµ → M µ)exp = P (Mµ → M µ ), (8) 2Si + 1 i=P,V where Nf is a phase-space factor and RM (x, y) is the oscillation rate, where P (Mµ → M µ)exp is the experimental oscillation 1 probability from Eq. (7).The values of SB(B0) are given R (x, y) = x2 + y2 , (5) M 2 in Table II of [9]. As can be seen [8], even 20 year old experimental data provide constraints on the New Physics where x = ∆M/ΓMµ and y = ∆Γ/2ΓMµ . Since ∆M scale that are comparable to those probed by the LHC and ∆Γ are dominated by New Physics interactions [8], program of the order Λ ∼ 5 TeV. An improvement of sen- while ΓMµ is given by the SM interactions, x, y 1. sitivity by two orders of magnitude will allow constraint Thus, oscillating functions can be expanded in x and y specific models of New Physics better then the combina- resulting in a power-law dependence displayed in Eq. (5). tions of other measurements. We urge our experimental The most recent experiment studying Mµ −M µ oscilla- colleagues to further study muonium-antimuonium oscil- tions was done in the 1990s [9], and was statistics limited lations. For this, a pulsed µ+ source could significantly by the available µ+ flux. It could not study the time- reduce the ratio of the potential M¯ signal to µ+ and M dependence of Eq. (4). A new experiment with sensitiv- decay related background [2].

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