25 February 2002
Physics Letters A 294 (2002) 175–178 www.elsevier.com/locate/pla
Spin–rotation coupling in muon g − 2 experiments G. Papini a,b,∗ , G. Lambiase c,d a Department of Physics, University of Regina, Regina, SK S4S 0A2, Canada b International Institute for Advanced Scientific Studies, 84019 Vietri sul Mare (Sa), Italy c Dipartimento di Fisica “E.R. Caianiello”, Universitá di Salerno, 84081 Baronissi (Sa), Italy d Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Salerno, Italy
Received 5 November 2001; received in revised form 17 January 2002; accepted 17 January 2002 Communicated by P.R. Holland
Abstract Spin–rotation coupling, or Mashhoon effect, is a phenomenon associated with rotating observers. We show that the effect plays a fundamental role in the determination of the anomalous magnetic moment of the muon, is sizable and violates the principle of equivalence. 2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.Pm; 04.20.Cv; 04.80.-y Keywords: Spin–rotation coupling; Inertial effects
Fully covariant wave equations predict the existence of inertial-gravitational effects that can be tested experimentally at the quantum level. Rapid experimental advances also require that inertial effects be identified with great accuracy in precise Earth bound and near space tests of fundamental theories. Experiments already confirm that inertia and Newtonian gravity affect quantum particles in ways that are fully consistent with general relativity down to distances of 10−3 cm for superconducting electrons [1,2] and 10−13 cm for neutrons [3–5].
* Corresponding author.
E-mail addresses: [email protected]
(G. Papini), [email protected]
Spin–inertia and spin–gravity interactions are the subject of numerous theoretical [6–10] and experimental efforts [11–15]. Studies of fully covariant wave equations carried out from different viewpoints [16– 20] identify entirely similar inertial phenomena. Prominent among these is the spin–rotation effect described by Mashhoon  who found that the Hamiltonians H and H of a neutron in an inertial frame F0 and in a frame F rotating with angular velocity ω relative to F0 are related by H = H − (h¯ /2)ω · σ . This effect is conceptually important. It extends our knowledge of rotational inertia to the quantum level and violates the principle of equivalence  that is well-tested experimentally at the classical level. It has, of course, been argued that the principle of equivalence does not hold true in the quantum world. This is certainly supported by the fact that phase shifts
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in particle interferometers [17,22] and particle wave functions depend on the masses of the particles involved [19,23]. In addition, the equivalence principle does not hold true within the context of the causal interpretation of quantum mechanics as shown by Holland . Several models predicting quantum violations of the equivalence principle have also been discussed in the literature [25,26], most recently in connection with neutrino oscillations [27–31]. The Mashhoon term, in particular, yields different potentials for different particles and for different spin states  and cannot, therefore, be considered universal. The relevance of spin–rotation coupling to physical  and astrophysical [19,32] processes has already been pointed out. No direct experimental verification of the Mashhoon effect has so far been reported, though the data given in  can be re-interpreted  as due to the coupling of Earth’s rotation to the nuclear spins in mercury. The effect is also consistent with a small depolarization of electrons in storage rings . The purpose of our work is to show that the spin–rotation effect is sizable and plays an essential role in precise measurements of the g − 2 factor of the muon. The experiment [35,36] involves muons in a storage ring consisting of a vacuum tube, a few meters in diameter, in a uniform vertical magnetic field. Muons on equilibrium orbits within a small fraction of the maximum momentum are almost completely polarized with spin vectors pointing in the direction of motion. As the muons decay, those electrons projected forward in the muon rest frame are detected around the ring. Their angular distribution thence reflects the precession of the muon spin along the cyclotron orbits. Our thesis is best proven starting from the covariant Dirac equation
iγ µ (x) ∂µ + iΓµ (x) − m ψ(x) = 0,
where Γµ (x) represents the spin connection and contains the spin–rotation interaction. The Minkowski metric has signature −2 and units h¯ = c = 1 are used. The calculations are performed in the rotating frame of the muon and do not therefore require a relativistic treatment of inertial spin effects . Then the vierbein formalism yields Γi = 0 and 1 1 Γ0 = − ai σ 0i − ωi σ i , 2 2
where ai and ωi are the three-acceleration and threerotation of the observer, and i i 0 i σ 0 0i σ ≡ γ ,γ = i 0 −σ i 2 in the chiral representation of the usual Dirac matrices. The second term in (1) represents the Mashhoon effect. The first term drops out. The remaining contributions to the Dirac Hamiltonian, to first order in ai and ωi , add up to [16,17] 1 a · x p · α + p · α a · x H ≈ α · p + mβ + 2 σ + . −ω · L (2) 2 For simplicity, all quantities in H are taken to be timeindependent. They are referred to a left-handed tern of axes rotating about the x2 -axis in the clockwise direction of motion of the muons. The x3 -axis is tangent to the orbits and in the direction of the muon momentum. The magnetic field is B2 = −B. Only the Mashhoon term then couples the helicity states of the muon. The remaining terms contribute to the overall energy E of the states, and we indicate by H0 the corresponding part of the Hamiltonian. Before decay the muon states can be represented as ψ(t) = a(t)|ψ+ + b(t)|ψ− , (3) where |ψ+ and |ψ− are the right and left helicity states of the Hamiltonian H0 and satisfy the equation H0 |ψ+,− = E|ψ+,− . The total effective Hamiltonian is Heff = H0 + H , where 1 H = − ω2 σ 2 + µBσ 2 . (4) 2 µ = (1 + (g − 2)/2)µ0 represents the total magnetic moment of the muon and µ0 is the Bohr magneton. Electric fields used to stabilize the orbits and stray radial electric fields can also affect the muon spin. Their effects can, however, be cancelled by choosing an appropriate muon momentum  and will not be considered. The coefficients a(t) and b(t) in (3) evolve in time according to ∂ a(t) a(t) i (5) =M , b(t) ∂t b(t)
G. Papini, G. Lambiase / Physics Letters A 294 (2002) 175–178
where M is the matrix
i ω22 − µB E − i Γ2 , M= −i ω22 − µB E − i Γ2
and Γ represents the width of the muon. The nondiagonal form of M (when B = 0) implies that rotation does not couple universally to matter. M has eigenvalues Γ + 2 Γ h2 = E − i − 2 and eigenstates
h1 = E − i
ω2 − µB, 2 ω2 + µB, 2
1 |ψ1 = √ i|ψ+ + |ψ− , 2 1 |ψ2 = √ −i|ψ+ + |ψ− . 2 The muon states that satisfy (5), and the condition |ψ(0) = |ψ− at t = 0, are −Γ t /2 ˜ ˜ ψ(t) = e e−iEt i e−i ωt − ei ωt |ψ+ 2 ˜ ˜ |ψ− , + ei ωt + e−i ωt
which is precisely the observed modulation frequency of the electron counts  (see also Fig. 19 of Ref. ). This result is independent of the value of the anomalous magnetic moment of the particle. It is therefore the Mashhoon effect that evidences the g − 2 term in Ω by exactly cancelling, in 2µB, the much larger contribution µ0 that pertains to fermions with no anomalous magnetic moment. The cancellation is made possible by the non-diagonal form of M and is therefore a direct consequence of the violation of the equivalence principle. It is perhaps odd that spin–rotation coupling as such has almost gone unnoticed for such a long time. It is, however, significant that its effect is observed in an experiment that has already provided crucial tests of quantum electrodynamics and a test of Einstein’s timedilation formula to better than a 0.1% accuracy. Recent versions of the experiment [39–41] have improved the accuracy of the measurements from 270 to 1.3 ppm. This bodes well for the detection of effects involving spin, inertia and electromagnetic fields or inertial fields to higher order.
where ω2 ω˜ ≡ − µB. 2 The spin–flip probability is therefore 2 Pψ− →ψ+ = ψ+ ψ(t) e−Γ t 1 − cos(2µB − ω2 )t . (8) 2 The Γ -term in (8) accounts for the observed exponential decrease in electron counts due to the loss of muons by radioactive decay . The spin–rotation contribution to Pψ− →ψ+ is represented by ω2 which is the cyclotron angular velocity eB/m . The spin–flip angular frequency is then =
Ω = 2µB − ω2 g − 2 eB eB − = 1+ 2 m m g − 2 eB = , 2 m
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