Muonium-antimuonium oscillations

Alexey A. Petrov Wayne State University

Table of Contents:

• Introduction: modern notations - flavor oscillation parameters: x and y - time-dependent and integrated probabilities • EFT computations of oscillation parameters - mass difference - width difference • Experimental methods and difficulties • Conclusions and things to take home

Based on R. Conlin and AAP, arXiv: 2005.10276 [hep-ph]

Alexey A Petrov (WSU) RF05: CLFV - Muon Decays and Transitions Muonium: just like hydrogen, but simpler!

• Muonium: a bound state of μ+ and e− Spin-0 (singlet) paramuonium – (μ+μ−) bound state is true muonium (anti-symm)

Muonium lifeme • τMμ = 2.2 μs + − – main decay mode: Mμ → e e ν¯μνe

– annihilaon: Mμ → ν¯μνe Spin-1 (triplet) orthomuonium • Muonium’s been around since 1960's (symm) – used in chemistry – QED bound state physics, etc. – New Physics searches (oscillaons)

Hughes (1960) The masses of singlet and triplet are almost the same!

Alexey A Petrov (WSU) 14 RF05: CLFV - Muon Decays and Transitions Muonium oscillaons: just like B0B¯0 mixing, but simpler!

★ Lepton-flavor violang interacons can change Mμ → Mμ Pontecorvo (1957) Feinberg, Weinberg (1961) • Such transion amplitudes are ny in the Standard Model – … but there are plenty of New Physics models where it can happen

∼ (μ¯Γe) (μ¯Γe)

effective operator

– theory: compute transion amplitudes for ALL New Physics models!

– experiment: produce Mμ but see for decay products of Mμ

Alexey A Petrov (WSU) 13 RF05: CLFV - Muon Decays and Transitions Combined evoluon = flavor oscillaons

• If there is an interacon that couples Mμ and Mμ (both SM or NP) – combined me evoluon: non-diagonal Hamiltonian!

– diagonalizaon: new mass eigenstates:

– new mass eigenstates: mass and lifeme differences

} (small)

These mass and width difference are observable quantities

Alexey A Petrov (WSU) 12 RF05: CLFV - Muon Decays and Transitions Combined evoluon = flavor oscillaons

• Study oscillaons via decays: amplitudes for Mμ → f and Mμ → f

– possibility of flavor oscillaons (Mμ → Mμ → f )

with

– me-dependent width:

– oscillaon probability:

Alexey A Petrov (WSU) 11 RF05: CLFV - Muon Decays and Transitions Oscillaon parameters: introducon

• Mixing parameters are related to off-diagonal matrix elements – heavy and light intermediate degrees of freedom

Bi-local at scale : both and Local at scale : only μ = Mμ Δm ΔΓ μ = Mμ Δm lepton number changes: 2 lepton number change (ΔLμ = 1) ΔLμ = 2 or (ΔLμ = 0)(ΔLμ = 2)

– each term has contribuons from different effecve Lagrangians

– … all of which have a form with Λ ∼ �(TeV )

Mass difference = real (dispersive) part; width difference: imaginary (absorptive) part

Alexey A Petrov (WSU) 10 RF05: CLFV - Muon Decays and Transitions Mass difference

• Mass difference comes from the dispersive part

– consider only ΔLμ = 2 Lagrangian contribuons (largest?)

– leading order: all heavy New Physics models are encoded in (the Wilson coefficients of) the five dimension-6 operators

– need to compute matrix elements for both singlet and triplet states

Alexey A Petrov (WSU) 9 RF05: CLFV - Muon Decays and Transitions Effecve Lagrangians and parcular models

• Effecve Lagrangian approach encompasses all models – lets look at an example of a model with a doubly charged Higgs Δ−− – this is common for the le-right models, etc.

– integrate out Δ−− to get

– match to ΔL=2 to see that and ℒeff MΔ = Λ

Alexey A Petrov (WSU) 8 RF05: CLFV - Muon Decays and Transitions Mass difference: matrix elements • QED bound state: know leading order wave funcon! – spacial part is the same as in Hydrogen atom

– can unambiguously compute decay constants and mixing MEs (QED)

– in the non-relavisc limit all decay constants fP = fV = fT = fM

(QED version of Van Royen-Weisskopf)

– NR matrix elements: “vacuum inseron” = direct computaon

Alexey A Petrov (WSU) 7 RF05: CLFV - Muon Decays and Transitions Mass difference: results • Spin-singlet muonium state: – matrix elements:

• Spin-triplet muonium state: – matrix elements

Experimental constraints on x result on experimental constraints on Wilson coefficients ΔL=2 Ck that encode all information about possible New Physics contributions R. Conlin and AAP, arXiv: 2005.10276

Alexey A Petrov (WSU) 6 RF05: CLFV - Muon Decays and Transitions Width difference

• Width difference comes from the absorpve part – light SM intermediate states (e+e−, γγ, ν¯ν, etc . ) – ν¯ν state gives parametrically largest contribuon

New Physics contribution Standard Model contribution ΔLμ = 2 ΔLμ = 0

Alexey A Petrov (WSU) 5 RF05: CLFV - Muon Decays and Transitions Width difference: results

• Spin-singlet muonium state:

• Spin-triplet muonium state:

• Note: y has the same 1/Λ2 suppression as the mass difference!

R. Conlin and AAP, arXiv: 2005.10276

Alexey A Petrov (WSU) 4 RF05: CLFV - Muon Decays and Transitions Experimental setup and constraints

• Similar experimental set ups for different experiments – example: MACS at PSI + – idea: form Mμ by scaering muon (μ ) beam on SiO2 target • A couple of “lile inconveniences”: ➡ how to tell f apart from f ? + − – Mμ → f decay: Mμ → e e ν¯μνe + − – Mμ → f decay: Mμ → e e ν¯eνμ − + – f: fast e (~53 MeV), slow e (13.5 eV) Muonium-Antimuonium ➡ oscillaons happen in magnec field Conversion Spectrometer (MACS) L. Willmann, et al. PRL 82 (1999) 49 – … which selects Mμ vs. Mμ The most recent experimental data comes from 1999! Time is ripe for an update!

Alexey A Petrov (WSU) 3 RF05: CLFV - Muon Decays and Transitions Experimental results

• MACS: observed 5.7 × 1010 muonium atoms aer 4 months of running – magnec field is taken into account (suppression factor)

L. Willmann, et al. PRL 82 (1999) 49

– no oscillaons have been observed

Alexey A Petrov (WSU) 2 RF05: CLFV - Muon Decays and Transitions Experimental constraints

• We can now put constraints on the Wilson coefficients of effecve operators from experimental data (assume single operator dominance) – presence of the magnec field

– no separaon of spin states: average

– set Wilson coefficients to one, set constraints on the scale probed

R. Conlin and AAP, arXiv: 2005.10276 Alexey A Petrov (WSU) 1 RF05: CLFV - Muon Decays and Transitions Conclusions and things to take home

• Muonium is the simplest atom (a bound state of μ+ and e−) – a heavy-light state that can exhibit flavor oscillaons (like K, B, and D mesons) – oscillaons probe New Physics without complicaons of QCD

• We discussed modern approach to phenomenology of muonium mixing – mass differences Δm (heavy NP intermediate states)

– lifeme differences ΔΓ (SM intermediate state, NP in ΔLμ operators)

• We used EFT to compute oscillaon parameters – results can be matched to parcular models of New Physics – found that both Δm and ΔΓ parametrically scale as �(Λ−2)

• Last experimental data is from 1999! Need new data! – we already probe LHC energy domain!

Alexey A Petrov (WSU) 0 RF05: CLFV - Muon Decays and Transitions Alexey A Petrov (WSU) 0 RF05: CLFV - Muon Decays and Transitions Effecve Lagrangians and lifeme difference

• Effecve Lagrangians for ΔLμ = 0, ΔLμ = 1, and ΔLμ = 2

−4 • ΔΓ: naively �(Λ ) from double ΔLμ = 1 inseron! But not always… Alexey A Petrov (WSU) RF05: CLFV - Muon Decays and Transitions