Magnetic Moment of the Mu Meson

Abstract

Assuming spins of 3/2, 5/2, 7/2 for Zr 91 , the expected ratios are 2.3±0.2, 1.00d=0.07, 0.56zb0.04, respectively, according to the formulawhere #\ is the amplitude of the rf field, m the concentration of the nuclei, and y the gyromagnetic ratio. The experimentally determined ratio of the signals is therefore compatible with a spin 5/2 for Zr 91 as measured by Arroe and Mack.Using the ratio of the magnetic moments of H 1 and D 2 as determined by Smaller, 4 At (H 1 )/M(D 2 ) = 3.2571999d=0.0000012, and the value of the proton magnetic moment, 5 M(H 1 ) = 2.79275±0.00003 nm, the diamagnetically uncorrected value of the magnetic moment of Zr 91 can be calculated as M (Zr 91 ) = -1.29802±0.00002 nm, or, diamagnetically corrected, as /x(Zr 91 )= -1.30284±0.00002 nm.When one considers the proton number Z = 40 and the neutron number iV=51 (magic plus one), the spin and sign of the magnetic moment are in agreement with the predictions of the simple single-particle model which places the neutron in a d b / 2 state. The deviation of the magnetic moment of 32% from the Schmidt value ju(Sch)= -1.913 nm, however, is rather large. For this reason one may consider configurations which also include protons, particularly the last two protons outside the closed shell Z=28. In terms of the independent-particle model, the two protons would couple their individual spins j p to a total proton spin J p = 0, 2, or 4, whereas J p would couple with the odd-neutron spin y n =5/2, to the total angular momentum /(Zr 91 ) = 5/2. Possible configurations consistent with the exclusion principle are \jr(pi /2 ) 2 J P =o; vdy. 2~] r>/2, L^igviYjp-o; vd bri~]hl2 , \jr(g9/2) 2 J P -2', vd m~] s/2, and [^(^9/2)^^4; vd h( i~] m , which yield the magnetic moments in nuclear magnetons of -1.913, -1.913, +0.037, and +4.04, respectively. This shows clearly that the ground state of Zr 91 cannot be described by one single configuration of the type mentioned.The observed relaxation time 7\ of Zr 91 in aqueous solution of (ZrF 6 ) is of the order of 10~3 sec and indicates an appreciable electric quadrupole moment.

Precise mass-dependent QED contributions to leptonicg−2at orderα2andα3

“…1, where j is the virtual lepton in the vacuum polarization subgraph. This coefficient was first computed in the late 1950s for the muon g−2 with x = m e /m µ ≪ 1, neglecting terms of O(x) [9]. The exact expression for 0 < x < 1 was reported by Elend in 1966 [10].…”

Theory of light hydrogenlike atoms

“…We also have to consider the electron-loop contribution to the muon anomalous magnetic moment The first two terms in this expression were obtained in [225,226], and an exact analytic result without expansion over m e /m was calculated in [227,228]. Then one readily obtains for the Lamb shift contribution [214] ∆E |l=0 = 1.082 75 Contribution of the mixed polarization graph with one electron-and one muon-loop insertions in the Coulomb photon in Fig.…”

The muon g−2

“…At three loops light-by-light scattering loops show up, etc. As all fermions have different masses, the fermion-loops give rise to mass dependent effects, which were calculated at two loops in [131,132] (see also [133]- [137]), at three loops in [138]- [145], and at four loops in [118]- [120], [126]. For five loops only partial estimates exist [119,120], [146]- [151].…”