Leptophilic U(1) Massive Vector Bosons from Large Extra Dimensions: Reexamination of Constraints from LEP Data

Very recently, we proposed an explanation of the discrepancy between the measured anomalous magnetic moment of the muon and the Standard Model (SM) prediction in which the dominant contribution to $(g-2)_\mu$ originates in Kaluza-Klein (KK) excitations (of the lepton gauge boson) which do not mix with quarks (to lowest order) and therefore can be quite light avoiding LHC constraints. In this addendum we reexamine the bounds on 4-fermion contact interactions from precise electroweak measurements and show that the constraints on KK masses and couplings are more severe than earlier thought. However, we demonstrate that our explanation remains plausible if a few KK modes are lighter than LEP energy, because if this were the case the contribution to the 4-fermion scattering from the internal propagator would be dominated by the energy and not by the mass. To accommodate the $(g-2)_\mu$ discrepancy we assume that the lepton number $L$ does not partake in the hypercharge and propagates in one extra dimension (transverse to the SM branes): for a mass of the lowest KK excitation of 60 GeV (lower than the LEP energy), the string scale is roughly 10 TeV while the $L$ gauge coupling is of order $\sim 10^{-1}$.

Very recently, we proposed an explanation of the discrepancy between the measured anomalous magnetic moment of the muon and the Standard Model (SM) prediction in which the dominant contribution to (g − 2) µ originates in Kaluza-Klein (KK) excitations (of the lepton gauge boson) which do not mix with quarks (to lowest order) and therefore can be quite light avoiding LHC constraints. In this addendum we reexamine the bounds on 4-fermion contact interactions from precise electroweak measurements and show that the constraints on KK masses and couplings are more severe than earlier thought. However, we demonstrate that our explanation remains plausible if a few KK modes are lighter than LEP energy, because if this were the case the contribution to the 4-fermion scattering from the internal propagator would be dominated by the energy and not by the mass. To accommodate the (g − 2) µ discrepancy we assume that the lepton number L does not partake in the hypercharge and propagates in one extra dimension (transverse to the SM branes): for a mass of the lowest KK excitation of 60 GeV (lower than the LEP energy), the string scale is roughly 10 TeV while the L gauge coupling is of order ∼ 10 −1 .
In [1] we argue that the exchange of Kaluza-Klein (KK) excitations of the lepton number (L) gauge boson could provide a dominant contribution to (g − 2) µ and explain the discrepancy between the Standard Model (SM) prediction of a µ = (g − 2) µ /2 and experiment: ∆a exp µ ≡ a FNAL+BNL µ − a SM µ = (251 ± 59) × 10 −11 [2]. On the other hand, the zero mode of the lepton number gauge boson is anomalous and gains a mass O(M s ) through a four-dimensional generalisation of the Green-Schwarz anomaly cancellation mechanism. Its mass being at the string scale, its contribution to (g − 2) µ is negligible, and therefore only the contributions of the KK modes are relevant to explain the discrepancy. In this addendum we reexamine model constraints from LEP data.
At the leading order in the U(1) L coupling constant g L , the contribution of massive vector bosons to (g − 2) µ comes from the muon vertex correction, and is given by When all KK states have masses much bigger than the muon mass, the sum of the integral (1) over all the KK states can be approximated by where M n is the mass of the nth KK excitation [1]. The bound from LEP data on the so-called compositeness scale associated to 4-fermion operators is given by [3]: where s is the square of the center-of-mass energy 1 . For M n √ s, (3) reduces to n α L (n)/M 2 n < B. Thus, the sum of the KK exchange given in (2) is constrained by the compositeness bound, yielding ∆a (1) µ ∼ O(10 −11 ); a result which is independent on the number of extra dimensions. Hence, one needs at least few KK modes lighter than LEP energy in order to provide a significant contribution able to bridge the gap in the muon anomalous magnetic moment.
A crucial point to take into account is that the gauge coupling is suppressed by the volume of the compact space V ⊥ ∼ (RM s ) d , where g s is the string coupling, R is the compactification scale, M s is the string scale, and d stands for the number of extra dimensions in which L propagates. For d = 1, we have M n = n/R and after substituting these figures into (2), ∆a The observed value of ∆a µ then implies where g s < ∼ 4π to remain in the perturbative regime.
As an illustration, if we take M s = 10 TeV then we have M 1 ∼ g s × 5 GeV, so that the highest possible value for the compactification scale M 1 , obtained for g s = 4π, is of order M 1 ∼ 60 GeV, which is consistent with the condition m µ M 1 √ s for all the approximations. The associated gauge coupling is then of order g L ∼ 10 −1 . Taking √ s LEP = 209 GeV, the total KK contribution to the LEP bound is given by  We also note that we have used a bound on g L for masses lighter than the LEP center-of-mass energy (by neglecting the mass compared to the energy in the exchange Z propagator) using the bound on new physics compatible with the bound on the compositeness scale.
Note that to lower the string scale in the region discussed above, one assumes in general additional large extra dimensions transverse to both SM and L stacks of branes that do not play any role in our analysis.

Case 2 : M ∼ m µ
In the case of a massive boson with a mass of order of the muon mass m µ , its contribution (1) to (g − 2) µ is given by If the lightest KK state have a mass M 1 ∼ m µ , the total contribution to the muon anomalous magnetic moment is therefore the sum of ∆a The (g − 2) µ discrepancy can then be accommodated for a string scale at M s ∼ g s ×3×10 2 TeV, yielding a coupling g L ∼ 5 × 10 −4 , now independent of g s . With M 1 = m µ = 105 MeV, we now get so that the bound (3) is also satisfied.

Case 3 : M m µ
We can also consider the situation where some of the lightest KK states have masses much lower than the muon mass, in which case the integral (1) gives a constant contribution α L 2π . Multiplying by m µ M 1 , the number of states with masses below m µ , and assuming again one extra dimension, we get the contribution The total contribution to the muon anomalous magnetic moment is then the sum of ∆a (12) As an example, let us take again satisfying the bound (3).
Let us note that unlike the discrepancy between the experimental value and the SM prediction of the muon anomalous magnetic moment which is positive, ∆a  [4]. The contributions coming from the KK excitations being positive, they will increase the discrepancy of (g−2) e , and we thus have to check that this contribution is lower than or of order of the experimental error on (g−2) e , that is < ∼ 10 −13 . Assuming For the different values obtained above for M 1 and M s , we get in the case 1 ∆a e ∼ 10 −14 , and in the cases 2 and 3 ∆a e ∼ 10 −13 , indeed smaller than or of order of the error on (g − 2) e .
Finally, one may also worry about LHC bounds using the one loop lepton induced mixing between the L KKexcitations and the photon or Z. The latter couples to quarks while the former can couple to a dilepton pair. The corresponding Drell-Yan exchange can then be estimated as: where E is the dilepton energy, N E/M 1 is the number of KK-modes with mass less than E, g 2 L g s M 1 /M s and 10 −2 counts for the loop factor suppression. It follows that the proposed scenario is compatible with LHC bounds [5].
We thank Greg Landsberg for valuable discussion.