Magnetic and Electric Dipole Constraints on Extra Dimensions and Magnetic Fluxes

The propagation of charged particles and gauge fields in a compact extra dimension contributes to $g-2$ of the charged particles. In addition, a magnetic flux threading this extra dimension generates an electric dipole moment for these particles. We present constraints on the compactification size and on the possible magnetic flux imposed by the comparison of data and theory of the magnetic moment of the muon and from limits on the electric dipole moments of the muon, neutron and electron.


I. INTRODUCTION
The possibility of Universal Extra Dimensions (UED) with size of the order of TeV −1 has been extensively studied for the past several years [1]; in UED models, all fields, not just gravity, have support in the extra dimensions. A size of these dimensions of the order of a TeV −1 allows for the possibility of observing their effects on low energy processes. The simplest enlargements of Minkowski space by one extra dimension, going back to Kaluza and to Klein [2], consists of enlarging ordinary Minkowski space, M 4 , to M 4 × S 1 , where S 1 is a circle of radius R. In this work we introduce a further modification by allowing a magnetic flux b to thread the circle S 1 ; this will result in nontrivial periodicity for the phases of charged particle fields. In addition, such phases will make various interactions, specifically electromagnetism, P and T noninvariant allowing for the presence of electric dipole moments.
Constraints on the radius of the fifth dimension, R, are obtained by attributing the difference between theoretical and experimental values of muon magnetic dipole moment, δ (g−2)/2 , to the extra dimension. In turn, the value of the flux b will either be bounded by limits on the neutron electric dipole moments (edm) or place an upper bound, within this model, on the muon edm. It turns out that for certain values of b/R the corrections to the magnetic moment are very small allowing for large values of R and in turn large contributions to the edm's of some of the charged particles.
In restricting five dimensional QED with a compact fifth dimension one encounters the problem of the fifth component of the gauge potential turning into an unwanted massless scalar particle. The standard method that prevents the appearance of such a state is to compactify the fifth dimension on the orbifold S 1 /Z 2 rather than on S 1 . In addition to this orbifold compactification we shall also eliminate the unwanted massless fields by explicitly introducing a large mass for the fifth component. The values of δ (g−2)/2 and of the edm's are significantly different for these two approaches. Both compactification formalisms are presented in Sec. II Generic results for δ (g−2)/2 and for the edm are discussed in Sec. III while the numerical results and application to δ (g−2)/2 of the muon and the edm's of the muon and neutron are discussed in Sec. IV.

II. FIVE DIMENSIONAL QED WITH A MAGNETIC FLUX
A. Compactification on a circle with explicit A 5 mass On the space M 4 × S 1 , with M 4 denoting ordinary Minkowski space-time and S 1 a circle of radius R, the action for a charged four component fermion, Ψ(x A ), and a gauge potential, The upper case super and subscripts are the five dimensional coordinates, with A = 0, 1, 2, 3, 5, the coordinates x A = (x µ , y), with 0 < y ≤ 2πR and The five dimensional Clifford algebra is spanned by Γ A = (γ µ , iγ 5 ). As discussed in the Introduction, we allow for the possibility of giving A 5 a mass by hand. We find it convenient to develop the formalism using a general mass matrix M AB ; in the end four dimensional Lorentz invariance will be reappear when only M 5,5 = 0. The dimensionful e ′ , upon reduction to four dimensions, will be related to the ordinary electric charge e and to the radius R by All neutral fields will be periodic under y → y + 2πR; the presence of a magnetic flux b threading the fifth dimension will change the charged field periodicities to The equations of motion, obtained by varying (1) are, as usual, Applying ∂ B to the first equation in (4) and using current conservation, The case where only M 5,5 = 0 results in massless A µ 's and A 5 independent of y.
We express all fields as a Fourier series in the y coordinate and impose the periodicity conditions of (3), In terms of the Fourier coefficient fields, ψ n (x µ ), A ν n (x µ ), and A 5 (x µ ) the action of (1) is The fermion mass terms maybe expressed as m nψnŪn U n ψ, with cos 2β n = m m n ; sin 2β n = − n + b m n R .
Using U n ψ n as fermion fields yields a conventional mass term m nψn ψ n at the price of complicating the interaction Lagrangian The transformations of all the fields under parity, P, and time reversal, T , are as usual, with the exception that n → −n, For b = 0 the angle β n = −β −n and the action obtained from the above Lagrangian is invariant under both parity and time reversal. For b = 0 the relation between β n and β −n no longer applies and both parity and time reversal are broken leading to the appearance of an electric dipole moment (edm). The product PT is still conserved, precluding an induced anapole,ψγ 5 γ µ ψ∂ ν F µν , [3] coupling. m 0 is the mass of the lowest fermion in the KK tower, namely the one whose magnetic and electric moments we are interested in. By adjusting the input mass m in (8) and the angles β n of (8) satisfy reality of cos 2β n requires |b| ≤ R. Corrections to the gyromagnetic ratio of the fermion, δ (g−2)/2 , and the value of its electric dipole moment, due to the extra dimension and magnetic flux are obtainable from the Feynman diagrams in Fig. 1. (For a different approach to extra dimensional contributions to the anomalous magnetic moment see Ref. [4]. For a massive A 5 we have the n = 0 term in the summation for δ (g−2)/2 is the usual first order correction, α/π, and thus excluded from the R and b dependent corrections. Although analytic expressions for all the terms appearing in (15) have been obtained, these are quite cumbersome. Rather, we shall present results as integrals over one Feynman parameter, which for subsequent analyses we evaluated numerically. For large n the terms in the summations behave as 1/n making it appear divergent; however, this leading contribution cancels between n and −n resulting in a convergent series for both δ (g−2)/2 and F 3 .
For the F 2 (n, b, R; A µ n ) we have , while for F 3 , (17)

B. Orbifold Compactification
When no mass is introduced for A 5 this component of the gauge potential may be elim- We now turn to results for electric dipole moments. These are presented in two ways: as a function of 1/mR for various values of b/mR, Fig. 3 and as a function of b/mR for fixed R, Fig. 4. The log-log plots continue to be linear to larger values of 1/mR; however as the difference between results for the various b/mR's is more manifest if the results are presented for a limited range of 1/mR.

Muon edm
Using the central value in Ref. [7], namely d µ ≤ 3.7 × 10 −19 e cm as an upper bound for the edm of the muon results in F 3 ≤ 4 × 10 −6 ; for any flux this requires an unreasonably large compactification radius, namely 1/R < 200 m µ for a massive A 5 and 1/R < 20 m µ for orbifolding compactification. For 1/R = 3000, the maximum value of F 3 = 3.8 × 10 −9 yields an upper limit of d µ ≤ 3.5 × 10 −22 e cm for a muon edm.

Neutron edm
A similar study of the neutron edm yields the most stringent limits. In the nonrelativistic quark model the magnetic moments of the neutron and proton can be understood by assigning to quarks a Dirac magnetic moment and a mass of roughly one third that of the hadron [8], implying 1/R ≥ 1000 m quark . The limit |d n | < 6.3 × 10 −26 e cm [9] implies for the constituent up quarks |d u | ≤ 6.3 × 10 −26 or F 3,u ≤ 1.5 × 10 −12 which in turn implies b/R ≤ 2 × 10 −5 . For a massive A 5 , a value of b/R ∼ 0.6 is consistent with the neutron data for 1/R ≥ 60 TeV while the limit is 6 TeV for orbifold compactification. For both compactification schemes the allowed regions in the b − R space are presented in Fig. 5.

V. SUMMARY
For b = 0 the residual discrepancy between theory and experiment on (g − 2)/2 of the muon, Fig 2,provides the best limit on a compactification radius, namely 1/R ≥ 360 GeV for a massive A 5 or 1/R ≥ 50 GeV for orbifold compactification. As The magnetic flux increases these limits become weaker. For b/mR not taken to be very small, limits on the neutron edm (relying on the nonrelativistic quark model) provide the strongest constraints on the compactification radius, reaching into the TeV region for 1/R, Fig. 5.