Post-Newtonian effects of Dirac particle in curved spacetime - III : the muon g-2 in the Earth’s gravity.

# Post-Newtonian effects of Dirac particle in curved spacetime - III : the muon g-2 in the Earth’s gravity.

Takahiro Morishima    Toshifumi Futamase    Hirohiko M. Shimizu
###### Abstract

The general relativistic effects to the anomalous magnetic moment of muons moving in the Earth’s gravitational field have been examined. The Dirac equation generalized to include the general relativity suggests the magnetic moment of fermions measured on the ground level is influenced by the Earth’s gravitational field as , where is the magnetic moment in the flat spacetime and is the Earth’s gravitational potential. It implies that the muon anomalous magnetic moment measured on the Earth contains the gravitational correction of in addition to the quantum radiative corrections. The gravitationally induced anomaly may affect the comparison between the theoretical and experimental values recently reported: . In this paper, the comparison between the theory and the experiment is examined by considering the influence of the spacetime curvature to the measurement on the muon experiment using the storage ring on the basis of the general relativity up to the post-Newtonian order of .

## 1 Introduction

The g-factor of leptons exactly equals to 2 according to the Dirac theory; (=2, for ), which is one of important consequences of the Dirac equation for free fermions with the electric charge of and the spin of minimally coupling to electromagnetic field222 In this paper we use unit ==. . However, the real value of deviates from 2 with the fraction of about as a result of the quantum radiative corrections. The deviation is referred to as the anomalous magnetic moment defined as

 al≡gl2−1=μlmμlB−1(=0.001...), (1)

where is the magnetic moment and is the Bohr magneton. The Bohr magneton is defined as = for the lepton mass and the spin . For electrons, the comparison between the theoretical value, which was obtained with higher orders of quantum radiative corrections based on the standard model of elementary particles (SM) [1, 4], and the experimental value, which was obtained in the precision measurement [3, 5, 4], shows the extremely precise agreement up to the 12th digit

 ae(EXP)−ae(SM)=−0.91(0.82)×10−12 (2)

as shown in Table 1, which clearly validates the accuracy of the quantum electrodynamics (QED) employed in the standard model of elementary particles [1, 5, 4].

For muons weighting 200 times more than electrons, the comparison is less accurate since it requires the calculation of the electroweak and hadronic contributions in addition to the QED calculation (Table 2). The most recent report [9, 5, 4] claims a disagreement of

 aμ(EXP)−aμ(SM)=28.8(8.0)×10−10(3.6σ). (3)

The significance of the disagreement is being intensively discussed since it may not only be signaling new physics beyond the standard model but also the breaking of equivalence principle [10, 11, 12, 2, 13].

Here we point out that the comparison may not be complete since the experimental value was obtained on the Earth’s surface in the Earth’s gravitational field while the theoretical value was calculated in the flat spacetime. We notice that we consider the influence of the curved spacetime in the general relativity, which is not the effect of the free fall in the Newtonian gravity or the quantum gravity effect at the Planck’s scale. The effective value of the magnetic moment of leptons moving in the curved spacetime in the Earth’s gravitational field is given as

 μeffm≃(1+3ϕ/c2)μm, (4)

where the is the magnetic moment in the flat spacetime and is the gravitational potential on the Earth’s gravity [21].

The gravitational dependence implies that the gravity induces an additional anomaly of according to the definition of the anomalous magnetic moment Eq. (1)  333 For simplicity, we assumed that the Earth is an uniform sphere with the radius of and the gravitational acceleration on the Earth’s surface is . The general relativistic effect can be measured with the scale of . .

Especially for the case of electrons, the additional anomaly introduced by the gravity seems inconsistent with the fact that the experimental value obtained in the gravitational field agrees with the theoretical value calculated in the flat spacetime with the extremely precise accuracy .

In Ref. [22], we showed that the paradoxical situation can be understood, for electrons, within the general relativity as the apparent cancellation in the ratio of experimental values among themselves. The experimental value of the electron anomalous magnetic moment using the Penning trap method is defined as

 ae(EXP)=ΩsΩc−1, (5)

where and are the spin precession frequency and the cyclotron frequency of electrons in the flat spacetime [3]. We define their effective values and in the curved spacetime in the Earth’s gravitational field and should re-define the anomalous magnetic moment measured in the curved spacetime as

 aeffe(EXP)=ΩeffsΩeffc−1. (6)

The effective values of spin precession frequency and the cyclotron frequency are given as ,  , respectively, from the equation of motion of a free particle moving in the curved spacetime with the Schwarzschild metric. The gravitational effect is canceled in the ratio

 aeffe(EXP) = ΩeffsΩeffc−1 (7) = (1+3ϕ/c2)Ωs(1+3ϕ/c2)Ωc−1 = ΩsΩc−1=ae(EXP).

and, consequently, the anomalous magnetic moment measured in the gravitational field accords with that calculated in the flat spacetime [22]. In other words, the Penning trap experiment is the excellent method to cancel the general relativistic effects in the measurement of the anomalous magnetic moment of electrons.

The anomalous magnetic moment of muons has been measured in the spin precession in storage rings at CERN [6] and BNL E821 [7]. We examine the comparison of the experimental values obtained in the storage ring method and the theoretical values calculated in the flat spacetime.

## 2 Interpretation of the Experimental Result of the Muon gμ−2

In the storage ring experiments for such as CERN [6] and BNL E821 [7], the muon anomalous magnetic moment was determined from the relation

 aμ(EXP) ≡ Rλ−R, (8)

where is the ratio of the anomalous spin precession frequency of muons to the proton spin precession frequency measured using the nuclear magnetic resonance and is the ratio of the magnetic moment of muons to that of protons. This can be understood as that the is determined from the relation

 aμ(EXP) = ΩaΩL−Ωa, (9)

where the Larmor precession frequency of muons. This interpretation can be obtained using the classical kinematical framework and it is the basis of the design of the experiment to measure the muon  [7, 6, 5, 4]. We reconsider this interpretation including the spacetime curvature in the Earth’s gravitational field on the basis of general relativity.

### 2.1 Post-Newtonian Representation of Earth’s Gravity

Below we assume that the Earth is an uniform sphere with the mass and the radius . The gravitational potential at the distance of from the center of the Earth is (for ). The gravitational field can be described using the Schwarzschild metric if we ignore the Earth’s rotation. The post-Newtonian representation with the expansion of small quantity of is useful as long as the gravitational field is as weak as . Hereafter, we employ the Schwarzschild metric up to the post-Newtonian order given as

 ds2 = gμνdxμdxν (10) = ϵ−2(1+ϵ22ϕ+ϵ42ϕ2)dt2−(1−ϵ22ϕ)(dx2+dy2+dz2) +O(ϵ4),

to describe the general relativistic effect (See Appendix A).

### 2.2 Cyclotron Motion in Curved Spacetime

We consider the translational motion of a free muon traveling in the Earth’s gravitational field. The general relativity requires that the translational motion of a particle with the electric charge of is described by the covariant equation of motion

 Duμdτ ≡ duμdτ+Γμνλuνuλ=emFμνuν, (11)

where is the proper time, is the four velocity vector, is the electromagnetic tensor and is the Christoffel symbol. Substituting Eq. (10) into Eq. (11), the general relativistic equation of motion of a charged particle in the Earth’s gravitational field up to the post-Newtonian order is obtained as

 dβdt = (1+(2γ2+1)ϵ2ϕ)eγm(E+β×B−(1−4γ2ϵ2ϕ)(β⋅E)β) (12) −ϵ(1+β2)∇ϕ+ϵ(4β⋅∇ϕ)β+O(ϵ4),

where , and

 u0 = dtdτ=1/√(1+2ϵ2ϕ+2ϵ4ϕ2)−(1−2ϵ2ϕ)β2 (13) = 1γ(1+(2γ2−1)ϵ2ϕ)+O(ϵ4).

Substituting into Eq. (12) for evaluating the case of flat spacetime (Minkowski metric), the special relativistic equation of motion of a charged particle is obtained as

 dβdt = eγm(E+β×B−(β⋅E)β). (14)

This equation shows that the translational motion draws, in general, a spiral trajectory which is the combination of the acceleration and the rotational motion. The cyclotron frequency corresponds to the frequency of the rotational motion of the constant-norm vector parallel to the velocity. Here we define the unit vector parallel to the three-dimensional vector as

 β≡β^β,^β⋅^β=β⋅ββ2=1. (15)

We assume that the contribution of the gradient of the gravitational potential can be ignored in the storage ring experiment as explained in the Appendix C. Under this assumption, the rotational component of the muon translational motion is given as

 d^βdt = −(1+(2γ2+1)ϵ2ϕ)em(Bγ−γγ2−1(β×E))×^β (16) = Ωceff×^β.

Using the cyclotron frequency in the flat spacetime the effective cyclotron frequency can be written as

 Ωeffc = (17)

which represents the cyclotron frequency of muon in the Earth’s gravitational field based on the general relativity. In the case of , the effective value of the cyclotron frequency is obtained as , which corresponds to the cyclotron frequency of electrons moving in the Earth’s gravitational field based on the general relativity for the case of the Penning trap experiment [22].

### 2.3 BMT Equations

In this section, we consider the equation of motion of the muon spin in the Earth’s gravitational field. The non-relativistic equation of motion of muon spin in the flat spacetime described in the co-moving coordinate system (rest frame) is given as

 dsdt = μm×B=Ωs×s, (18)

where is the three-dimensional spin vector, is the magnetic moment and is the spin precession frequency. Replacing the three-dimensional spin vector with the four vector , the non-relativistic equation of motion of Eq. (18) can be extended in the Lorentz invariant form as

 dSμdτ = gμ2em(FμνSν+1c2uμ(SλFλνuν))−1c2uμ(Sλduλdτ), (19)

which is known as the BMT equation [8]. We implement the general relativity by replacing the derivative with the covariant derivative in Eq. (19) in the same manner applied in the extension of equation of translational motion Eq. (11) and obtain

 DSμdτ = gμ2em(FμνSν+1c2uμ(SλFλνuν))−1c2uμ(SλDuλdτ), (20)

which is the general relativistic BMT equation: the covariant equation of motion of spin based on the general relativity.

### 2.4 Spin Motion in Flat Spacetime

Here we overview the procedure to evaluate the experimental value of the anomalous magnetic moment using the equation of motion of spin in the flat spacetime (e.g. See Chap.11 in Ref. [18]). The BMT equation Eq. (19) described in the laboratory frame (inertial frame) can be decomposed into a set of equations for time and spatial components as

 dS0dτ = F0+γ2(S⋅dβdτ) dSdτ = F+γ2β(S⋅dβdτ). (21)

The three-dimensional spin vector in the inertial frame and the three-dimensional spin vector are related through the Lorentz transformation

 s = S−γγ+1(β⋅S)β (22) = S−γγ+1S0β.

The equation of motion of the spin in the rest frame can be derived as

 dsdτ = (F−γγ+1F0β)+γ2γ+1s×(β×dβdτ) (23) = F′+γ2γ+1s×(β×dβdτ).

Rewriting the proper time with the time in the inertial frame , we can reproduce the familiar equation

 dsdt = F′γ+γ2γ+1s×(β×dβdt) (24) = gμ2eγms×B′+γ2γ+1s×(β×dβdt).

We note that the last term in the equation of motion of spin in the rest frame (Eq. (24)) represents the relativistic term known as the Thomas precession which does not appear in the non-relativistic equation of motion of spin (Eq. (18)). The electric and magnetic fields in the inertial frame and are related with those in the rest frame and through the Lorentz transformation as

 E′ = γ(E+β×B)−γ2γ+1(β⋅E)β B′ = γ(B−β×E)−γ2γ+1(β⋅B)β. (25)

The equation of motion of spin in the rest frame can be rewritten with the quantities in the inertial frame by substituting Eq. (14) and Eq. (25), which results in

 dsdt = g2ems×(B−β×E−γγ+1(β⋅B)β) (26) = s×em((g2−1+1γ)B−(g2−γγ+1)β×E−(g2−1)γγ+1(β⋅B)β) = Ωs×s.

By using the spin precession frequency , the anomalous spin precession frequency can be written as

 Ωa = Ωs−Ωc (27) = −em(aμB−(aμ−1γ2−1)β×E−aμγγ+1(β⋅B)β),

with . We assume that the motion is confined in the horizontal plane and the magnetic field is applied vertically. In addition, we assume that the electric term (the second term) can be canceled as by selecting the muon momentum of 29.3, which is referred to as the magic momentum. The anomalous magnetic moment of muon in the flat spacetime, which is defined as , can be written as

 aμ(EXP) ≡ −Ωa(eBm)=ΩaΩL−Ωa=Ωa/ΩpΩL/Ωp−Ωa/Ωp (28) = Rλ−R,

by rewriting the magnetic field using the Larmor precession frequency . The Eq. (28) represents the experimental value in the conventional interpretation. The experimental value of the anomalous magnetic moment using Eq. (27) and Eq. (28) are evaluated within the classical kinematical framework. The relations were applied to achieve the high precision by employing the magic momentum to suppress the contribution of electric fields.

### 2.5 Spin Motion in Curved Spacetime

In this section, we apply the procedure of the evaluation of the experimental value of the anomalous magnetic moment in the flat spacetime to that in the curved spacetime.

The BMT equation in the curved spacetime in the inertial frame (Eq. (20)) can be decomposed into the time and spatial components as

 dS0dτ+ϵ2(S0u+u0S)⋅∇ϕ =F0+(1−4γ2ϵ2ϕ)γ2(S⋅dβdτ)+O(ϵ4) dSdτ+ϵu0S0∇ϕ+ϵ2((u⋅S)∇ϕ−(S⋅∇ϕ)u−(u⋅∇ϕ)S) =F+(1−4γ2ϵ2ϕ)γ2β(S⋅dβdτ)+O(ϵ4). (29)

Considering the contribution arising from the gradient of the gravitational potential is sufficiently small to be ignored in the post-Newtonian order (See Appendix C), Eq. (29) can be simplified as

 dS0dτ = F0+(1−4γ2ϵ2ϕ)γ2(S⋅dβdτ)+O(ϵ4) dSdτ = F+(1−4γ2ϵ2ϕ)γ2β(S⋅dβdτ)+O(ϵ4). (30)

Following the procedure in the flat spacetime, we derive the relation between the spins in the inertial frame and the rest frame using the Lorentz transformation. The Lorentz transformation can be naturally defined in the flat spacetime since the Lorentz invariance is satisfied at any point. However, the Lorentz transformation cannot be defined in the general coordinate system (curved spacetime), which disturbs the applicability of the same procedure. We avoid the difficulty by employing the local inertial frame to satisfy the local Lorentz invariance (See Appendix B). The four vector in the general coordinate system can be related with the four vector in the local inertial frame , using the tetrad , as

 x(a) ≡ e(a)μxμ=(~x0,~x). (31)

Using the explicit expression of the tetrad up to the post-Newtonian order in the Schwarzschild metric (shown in Eq.(A7)), the transformation from the general coordinate system into the local inertial frame can be simplified as

 ~S0 = (1+ϵ2ϕ)S0 ~S = (1−ϵ2ϕ)S ~β = (1−2ϵ2ϕ)β ~t = (1+ϵ2ϕ)t ~γ = (1−2ϵ2ϕ(γ2−1))γ ~E = E ~B = (1+2ϵ2ϕ)B (32) ...etc

We rewrite the BMT equation in the curved spacetime (Eq. (30)) using the local inertial frame and obtain

 d~S0dτ=~F0+~γ2(~S⋅d~βdτ)+O(ϵ4) d~Sdτ=~F+~γ2~β(~S⋅d~βdτ)+O(ϵ4), (33)

which is an analogue to the relation in the flat spacetime. The local Lorentz transformation is obtained as

 ~s = ~S−~γ~γ+1(~β⋅~S)~β (34) = ~S−~γ~γ+1~S0~β.

In the same manner as the flat spacetime, using the BMT equation in the curved spacetime (Eq. (33)) and the local Lorentz transformation (Eq. (34)), the equation of motion of spin in the rest frame can be written as

 d~sdτ = (~F−~γ~γ+1~F0~β)+~γ2~γ+1~s×(~β×d~βdτ) (35) = ~F′+~γ2~γ+1~s×(~β×d~βdτ),

which is similar to the case of the flat spacetime. This relation gives the relation similar to the case of the flat spacetime

 d~sd~t = ~F′~γ+~γ2~γ+1~s×(~β×d~βd~t) (36) = gμ2e~γm~s×~B′+~γ2~γ+1~s×(~β×d~βd~t),

by replacing the proper time with the time in the inertial frame . Here we notice that the electric and magnetic fields in the rest frame and is given through a local Lorentz transformation, using the electric and the magnetic fields in the inertial frame and , as

 ~E′ = ~γ(~E+~β×~B)−~γ2~γ+1(~β⋅~E)~β ~B′ = ~γ(~B−~β×~E)−~γ2~γ+1(~β⋅~B)~β. (37)

Thus, the equation of motion of spin in the rest frame can be written using the electric and the magnetic fields in the inertial frame and as

 d~sd~t = gμ2em~s×(~B−~β×~E−~γ~γ+1(~β⋅~B)~β) (38) +~γ2~γ+1~s×(~β×em~γ(~E+~β×~B−(~β⋅~E)~β)) = ~s×em((gμ2−1+1~γ)~B−(gμ2−~γ~γ+1)~β×~E−(gμ2−1)~γ~γ+1(~β⋅~B)~β).

The equation of motion of spin in the rest frame represented in the general coordinate system is obtained as

 dsdt = s×em((gμ2−1+1~γ)(1+3ϵ2ϕ)B (39) −(gμ2−~γ~γ+1)(1−ϵ2ϕ)β×E −(gμ2−1)~γ~γ+1(1−ϵ2ϕ)(β⋅B)β) = s×em((gμ2−1+1γ[1+2ϵ2ϕ(γ2−1)])(1+3ϵ2ϕ)B −(gμ2−γγ+1[1−2ϵ2ϕ(γ−1)])(1−ϵ2ϕ)β×E −(gμ2−1)γγ+1[1−ϵ2ϕ(2γ−1)](β⋅B)β) = Ωeffs×s

Putting and using the effective value of the spin precession frequency in the curved spacetime (Eq. (39)) and the effective value of the cyclotron frequency in the curved spacetime (Eq. (17)), the effective value of the anomalous spin precession frequency is obtained as

 Ωeffa = Ωeffs−Ωeffc (40) = −em((1+3ϵ2ϕ)aμB −aμγγ+1(1−ϵ2ϕ(2γ−1))(β⋅B)β).

When the motion is confined on the horizontal plane, magnetic field is applied vertically and the momentum is chosen to cancel the contribution of electric field, the effective value of the anomalous magnetic moment of muon is interpreted as

 aeffμ(EXP) = ΩeffaΩeffL−Ωeffa. (41)

The Larmor precession frequency in the curved spacetime can be interpreted as the special solution of the BMT equation with = and =, which corresponds to the null translational motion, as

 ΩeffL=−(1+3ϵ2ϕ)gμ2emB=(1+3ϵ2ϕ)ΩL. (42)

Therefore, we obtain

 aeffμ(EXP) = ΩeffaΩeffL−Ωeffa (43) = (1+3ϵ2ϕ)Ωa(1+3ϵ2ϕ)ΩL−(1+3ϵ2ϕ)Ωa = ΩaΩL−Ωa = aμ(EXP),

which show that the effective value of the anomalous magnetic moment in the curved spacetime equals to that in the flat spacetime up to the post-Newtonian order of the general relativity. Thus the effective values of cyclotron frequency, spin precession frequency and Larmor precession frequency have been derived based on the primitive consideration based on kinematical framework. However those effective values in curved spacetime are respectively different from those of values in the flat spacetime, the gravitational contribution is canceled in the ratio Eq. (43) and the anomalous magnetic moment in the curved spacetime coincides with the case in the flat spacetime.

## 3 Discussion

The general relativistic effect of the spacetime curvature of the Earth’s gravitational field introduces additional anomaly in the magnetic moment of muon of  [21]. Its influence is canceled in the ratio based on the kinematical consideration up to the post-Newtonian order using the Schwarzschild metric. Consequently the comparison between the experimental and theoretical values remains valid as before.

However, the conventional choice of the muon momentum to suppress the contribution of the electric field will no longer valid since the anomalous magnetic moment given in Eq. (40) modifies the magic momentum according to the additional terms due to the general relativistic effect. If the was experimentally tuned to minimize the electric field contribution, the quantity, defined as

 amodμ≡aμ−ϵ2ϕ(4+aμ+3γ2−1), (44)

might have been measured. In this case, the general relativistically induced term must be removed from the measured value in the comparison with the theoretical value, which implies that the true experimental value would be modified as

 aμ = amodμ+ϵ2ϕ(4+aμ+3γ2−1) (45) ≃ amodμ−2.8×10−9.

Thus the comparison of theory and experiment would be modified as shown in Table 3. The agreement of theoretical and experimental values of could be understood as one of the evidence of the validity of the general relativity.

## 4 Conclusion

The general relativistic effect in the anomalous magnetic moment of muons moving in the Earth’s gravitational field has been examined. The effective value of the muon magnetic moment is given as based on the Dirac equation generalized to include the general relativity, which implies that the additional anomaly of exists in addition to the quantum radiative corrections of in the muon magnetic moment.

The influence of the additional anomaly has been derived, specifically in the storage ring apparatus, within the kinematical consideration of the spacetime curvature up to the post-Newtonian order in the Earth’s gravitational field. It was found to be canceled out in the anomalous magnetic moment and the comparison between the experimental and the theoretical values remains valid as before.

However, the contribution of the electric field is modified by the general relativistic effect, which suggests a possible change of the magic Lorentz factor in the storage ring experiment. The corresponding correction to the experimental value amounts , which might result in the agreement of theoretical and experimental values of the muon anomalous magnetic moment as shown in Table 3. The possible agreement can be understood as the precise test of the validity of the general relativity. Improved accuracy of planned in experimental projects such as FNAL E989 [19] and J-PARC E34 [20] would clarify the influence of the general relativity.

## Acknowledgment

This work is supported in part by a Grant-in-Aid for Science Research from JSPS (No.17K05453 to T.F).

## A Geometrical values in Schwarzschild metric

The Schwarzschild metric can be written as

 ds2 = (1−ϵ22GM