25 February 2002

Physics Letters A 294 (2002) 175–178

www.elsevier.com/locate/pla

Spin–rotation coupling in muong− 2 experiments

G. Papinia,b,∗, G. Lambiasec,d

a Department of Physics, University of Regina, Regina, SK S4S 0A2, Canadab 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), Italyd 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 effectplays a fundamental role in the determination of the anomalous magnetic moment of the muon, is sizable and violates theprinciple 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 exis-tence of inertial-gravitational effects that can be testedexperimentally at the quantum level.

Rapid experimental advances also require that iner-tial effects be identified with great accuracy in preciseEarth bound and near space tests of fundamental theo-ries.

Experiments already confirm that inertia and New-tonian gravity affect quantum particles in ways thatare fully consistent with general relativity down to dis-tances of 10−3 cm for superconducting electrons [1,2]and 10−13 cm for neutrons [3–5].

* Corresponding author.E-mail addresses: papini@uregina.ca (G. Papini),

lambiase@sa.infn.it (G. Lambiase).

Spin–inertia and spin–gravity interactions are thesubject of numerous theoretical [6–10] and experi-mental efforts [11–15]. Studies of fully covariant waveequations carried out from different viewpoints [16–20] identify entirely similar inertial phenomena. Prom-inent among these is the spin–rotation effect describedby Mashhoon [10] who found that the HamiltoniansH andH ′ of a neutron in an inertial frameF0 and ina frameF ′ rotating with angular velocityω relative toF0 are related byH ′ =H − (h/2)ω · σ .

This effect is conceptually important. It extends ourknowledge of rotational inertia to the quantum leveland violates the principle of equivalence [21] that iswell-tested experimentally at the classical level.

It has, of course, been argued that the principle ofequivalence does not hold true in the quantum world.This is certainly supported by the fact that phase shifts

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176 G. Papini, G. Lambiase / Physics Letters A 294 (2002) 175–178

in particle interferometers [17,22] and particle wavefunctions depend on the masses of the particles in-volved [19,23]. In addition, the equivalence principledoes not hold true within the context of the causal in-terpretation of quantum mechanics as shown by Hol-land [24]. Several models predicting quantum viola-tions of the equivalence principle have also been dis-cussed in the literature [25,26], most recently in con-nection with neutrino oscillations [27–31]. The Mash-hoon term, in particular, yields different potentials fordifferent particles and for different spin states [19] andcannot, therefore, be considered universal.

The relevance of spin–rotation coupling to physical[20] and astrophysical [19,32] processes has alreadybeen pointed out.

No direct experimental verification of the Mash-hoon effect has so far been reported, though the datagiven in [13] can be re-interpreted [33] as due to thecoupling of Earth’s rotation to the nuclear spins inmercury. The effect is also consistent with a small de-polarization of electrons in storage rings [34]. The pur-pose of our work is to show that the spin–rotation ef-fect is sizable and plays an essential role in precisemeasurements of theg − 2 factor of the muon.

The experiment [35,36] involves muons in a storagering consisting of a vacuum tube, a few meters in di-ameter, in a uniform vertical magnetic field. Muons onequilibrium orbits within a small fraction of the maxi-mum momentum are almost completely polarized withspin vectors pointing in the direction of motion. As themuons decay, those electrons projected forward in themuon rest frame are detected around the ring. Theirangular distribution thence reflects the precession ofthe muon spin along the cyclotron orbits.

Our thesis is best proven starting from the covariantDirac equation

[iγ µ(x)

(∂µ + iΓµ(x)

) −m]ψ(x)= 0,

whereΓµ(x) represents the spin connection and con-tains the spin–rotation interaction. The Minkowskimetric has signature−2 and unitsh= c= 1 are used.The calculations are performed in the rotating frameof the muon and do not therefore require a relativistictreatment of inertial spin effects [37]. Then the vier-bein formalism yieldsΓi = 0 and

(1)Γ0 = −1

2aiσ

0i − 1

2ωiσ

i,

whereai andωi are the three-acceleration and three-rotation of the observer, and

σ 0i ≡ i

2

[γ 0, γ i

] = i

(σ i 00 −σ i

)in the chiral representation of the usual Dirac matri-ces. The second term in (1) represents the Mashhooneffect. The first term drops out. The remaining contri-butions to the Dirac Hamiltonian, to first order inaiandωi , add up to [16,17]

H ≈ α · p+mβ + 1

2

[(a · x)( p · α) + ( p · α)(a · x)](2)− ω ·

(L+ σ

2

).

For simplicity, all quantities inH are taken to be time-independent. They are referred to a left-handed tern ofaxes rotating about thex2-axis in the clockwise direc-tion of motion of the muons. Thex3-axis is tangent tothe orbits and in the direction of the muon momentum.The magnetic field isB2 = −B. Only the Mashhoonterm then couples the helicity states of the muon. Theremaining terms contribute to the overall energyE ofthe states, and we indicate byH0 the correspondingpart of the Hamiltonian.

Before decay the muon states can be represented as

(3)∣∣ψ(t)⟩ = a(t)|ψ+〉 + b(t)|ψ−〉,where |ψ+〉 and |ψ−〉 are the right and left helicitystates of the HamiltonianH0 and satisfy the equation

H0|ψ+,−〉 =E|ψ+,−〉.The total effective Hamiltonian isHeff = H0 + H ′,where

(4)H ′ = −1

2ω2σ

2 +µBσ 2.

µ = (1 + (g − 2)/2)µ0 represents the total magneticmoment of the muon andµ0 is the Bohr magneton.Electric fields used to stabilize the orbits and stray ra-dial electric fields can also affect the muon spin. Theireffects can, however, be cancelled by choosing an ap-propriate muon momentum [36] and will not be con-sidered.

The coefficientsa(t) andb(t) in (3) evolve in timeaccording to

(5)i∂

∂t

(a(t)

b(t)

)=M

(a(t)

b(t)

),

G. Papini, G. Lambiase / Physics Letters A 294 (2002) 175–178 177

whereM is the matrix

(6)M =[

E − i Γ2 i(ω22 −µB

)−i(ω2

2 −µB)

E − i Γ2

],

and Γ represents the width of the muon. The non-diagonal form ofM (whenB = 0) implies that rota-tion does not couple universally to matter.M has eigenvalues

h1 =E − iΓ

2+ ω2

2−µB,

h2 =E − iΓ

2− ω2

2+µB,

and eigenstates

|ψ1〉 = 1√2

[i|ψ+〉 + |ψ−〉],

|ψ2〉 = 1√2

[−i|ψ+〉 + |ψ−〉].The muon states that satisfy (5), and the condition|ψ(0)〉 = |ψ−〉 at t = 0, are

∣∣ψ(t)⟩ = e−Γ t/2

2e−iEt

i[e−iωt − eiωt

]|ψ+〉

(7)+ [e−iωt + eiωt

]|ψ−〉,

where

ω ≡ ω2

2−µB.

The spin–flip probability is therefore

Pψ−→ψ+ = ∣∣⟨ψ+∣∣ψ(t)⟩∣∣2

(8)= e−Γ t

2

[1− cos(2µB −ω2)t

].

The Γ -term in (8) accounts for the observed expo-nential decrease in electron counts due to the loss ofmuons by radioactive decay [36].

The spin–rotation contribution toPψ−→ψ+ is rep-resented byω2 which is the cyclotron angular velocityeB/m [36]. The spin–flip angular frequency is then

Ω = 2µB −ω2

=(

1+ g − 2

2

)eB

m− eB

m

(9)= g− 2

2

eB

m,

which is precisely the observed modulation frequencyof the electron counts [38] (see also Fig. 19 of Ref.[36]). This result is independent of the value of theanomalous magnetic moment of the particle. It istherefore the Mashhoon effect that evidences theg−2term inΩ by exactly cancelling, in 2µB, the muchlarger contributionµ0 that pertains to fermions withno anomalous magnetic moment. The cancellation ismade possible by the non-diagonal form ofM and istherefore a direct consequence of the violation of theequivalence principle.

It is perhaps odd that spin–rotation coupling as suchhas almost gone unnoticed for such a long time. It is,however, significant that its effect is observed in anexperiment that has already provided crucial tests ofquantum electrodynamics and a test of Einstein’s time-dilation formula to better than a 0.1% accuracy. Recentversions of the experiment [39–41] have improved theaccuracy of the measurements from 270 to 1.3 ppm.This bodes well for the detection of effects involvingspin, inertia and electromagnetic fields or inertialfields to higher order.

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