Leptophilic U(1) Massive Vector Bosons from Large Extra Dimensions

We demonstrate that the discrepancy between the anomalous magnetic moment measured at BNL and Fermilab and the Standard Model prediction could be explained within the context of low-scale gravity and large extra-dimensions. 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. We show that the KK contribution to $(g-2)_\mu$ is universal with the string scale entering as an effective cutoff. The KK tower provides a unequivocal distinctive signal which will be within reach of the future muon smasher.

Low scale gravity and large extra dimensions offer a genuine solution to the gauge hierarchy problem [1,2]. Within these models one has to address the problem of baryon B and lepton L number violation by higher dimensional operators suppressed only by the low string scale M s . Intersecting D-brane models offer a way out by gauging these symmetries [3][4][5][6][7]. Since the B and L gauge bosons are anomalous they gain masses through a generalization of the Green-Schwarz (GS) anomaly cancellation [8][9][10][11] giving rise to perturbative global symmetries broken only by non-perturbative effects that are suppressed exponentially by the string/gauge coupling. The resulting gauge bosons form in general linear combinations of the various abelian gauge factors orthogonal to the hypercharge combination, that couple to both quark and leptons. However, the Kaluza-Klein (KK) excitations do not mix (to lowest order) and thus those of L couple only to leptons. Such modes can be quite light because LHC constraints are weak but can provide a sizable contribution to the anomalous magnetic moment of the muon a µ = (g − 2) µ /2.
TeV-scale D-brane string compactifications could then provide an innovative framework to explain the extant tension between the Standard Model (SM) prediction of a µ and experiment. Very recently, the Muon g − 2 Experiment at Fermilab reported a measurement reading a FNAL µ = 116592040(54) × 10 −11 [12], which is larger than the SM prediction a SM µ = 116591810(43) × 10 −11 in which contributions from QED, QCD, and electroweak interactions are taken into account with highest precision [13]. This leads to a FNAL µ − a SM µ = (230 ± 69) × 10 −11 , which cor-responds to a 3.2σ discrepancy. Because the Fermilab observation is compatible with the long-standing discrepancy from the E821 experiment at BNL [14], the overall deviation from the SM central value, strengthens the significance to 4.2σ [12]. 1 Even though the discrepancy is not statistical significant yet, it is interesting to entertain the possibility that it corresponds to a real signal of new physics. In this Letter we calculate the massive vector boson contribution to g − 2 from KK excitations of L and we show that it is universal and can accommodate the ∆a exp µ discrepancy of (1). At the leading order in the U(1) L coupling constant g L , the contribution of a massive vector boson to lepton's g − 2 originates from the vertex correction shown in Fig. 1. Note that KK momentum is not conserved in lepton gauge boson vertices since leptons are localized in brane intersections. Figure 1 shows the same diagram that yields the famous α/π in QED, but with the virtual photon replaced by a massive vector boson. The fastest way to compute it is to use the massive propagator in 1 We note in passing that the SM prediction estimated by the latest lattice QCD calculations, a SM,lattice and follow a textbook, for example Ref. [16]. It is easy to see that the longitudinal part of the propagator (second term in Eq. (2)) does not contribute to the magnetic moment. The only difference between the first term and the photon propagator in Feynman gauge is M 2 in the denominator which leads to a slight modification of the integral over Feynman parameters. In the limit of M m, one obtains where we neglected terms of order (m/M) 4 and m is the muon mass. Note that this is a positive correction that brings a µ closer to experimental data. The masses of KK gauge bosons are labelled by integer vectors n, with M 2 ( n) = | n| 2 M 2 , where M is the compactification scale. For M M s the couplings α L ( n) depend very mildly on n when | n| is small [17]. They are approximately 1 until | n| ≈ M s /M and then exponentially suppressed when | n| M s /M. They are given by a Gaussian form α L ( n) = δ − n 2 M 2 /M 2 s with δ < 1 a model dependent constant. In the case of one extra dimension In the case of two extra dimensions, the exponential suppression of α L ( n) at large | n| is crucial for regulating the logarithmic divergence of the sum: Here, α L is the coupling of the lightest KK excitation.
To develop some sense for the orders of magnitude involved, we recall that direct production at LEP provides the best bound on KK couplings and masses. The agreement between the LEP-II measurements and the SM predictions implies that either g L 10 −2 , or else M > 209 GeV, the maximum energy of LEP-II [18]. In Fig. 2 we show contours of constant ∆a exp µ in the g L − M plane for different values of the string scale. We see that there is a large range of masses and couplings that can accommodate the Fermilab result. A point worth noting at this juncture is that the KK contribution to (g − 2) µ is universal, with M s entering as an effective cutoff.
There are two different classes of D-brane constructs that can realize the tower of KK modes. On the one hand, we can envision that L is part of the hypercharge (thus its gauge coupling α L cannot be very small). One can then try to use one of the orthogonal to the hypercharge combinations for explaining the (g − 2) µ discrepancy and make it leptophilic to avoid the LHC bounds. It turns out that this cannot be done because the corresponding U(1) gauge coupling becomes strong. Indeed, the 4 stack model thoroughly analyzed in [19], with gauge group U(3) a × Sp(1) b × U(1) c × U(1) d , typifies this class. Contact with gauge structures at TeV energies is achieved by a field rotation to couple diagonally to hypercharge Y. Two of the Euler angles (ψ, θ, φ) are determined by this rotation. The gauge couplings are related to g Y by and the relation for U(N) unification, g N = g N / √ 2N, holds only at M s because the U(1) couplings (g a , g c , g d ) run differently from the non-abelian SU(3) (g a ) and SU(2) ≡ Sp(1) (g b ) [20]. The zero-mode of the anomalous U(1), hereafter Z , gains a mass via the GS mechanism by absorbing an axionic field from the R-R (Ramond) closed string sector. To get as much contribution to a µ as possible without violating the LHC bounds [21,22], it is natural to consider a leptophilic (in our case meaning large g L ≡ g c ) Z [23]. Next, we compare with the LHC data considering the resonant production cross section of σ(pp → Z → ). Under the narrow width approximation, the cross section can be written in the form of c u w u + c d w d , where w u , w d are given by modelindependent parton distribution functions [24]. The coupling of Z with up and down quarks (assuming same coupling to three families) are encoded in c u , c d . More precisely, for a generic coupling between Z and fermion f the coefficients c u and c d take the following form We compute the branching faction Br( + − ) by including only the decay channels to leptons and quarks. The total decay rate is given by Due to the constraint (6), there are two free parameters (for a given string scale M s ): φ and g d (M s ). Setting the mass of Z to 2 TeV, we then search over the parameter space to get the smallest possible values of c u , c d . For simplicity, the combination of c 2 u + c 2 d is considered. We find that the optimal value of φ generally suppresses the couplings to left-handed quarks and the remaining couplings to the right-handed quarks are controlled by g d . In the best case scenario, g c (M s ) is set to 2π at M s (with 10 M s /TeV 10 3 ), the corresponding cross section (or rather c 2 u + c 2 d ∼ 8.4 × 10 −5 ) is roughly 2 percent of that given by the sequential standard model boson Z SSM [25], saturating the LHC limit [22]. We note that the branching fraction to leptons is close to 1 due to the small coupling to quarks. The signal can be further reduced by including other decay channels. Moreover, the largest possible g c (M s ) also gives the most contribution to a µ . Such a Z boson gives a µ = 9.9 × 10 −11 [19], which is much smaller than the pre-LHC estimate of Ref. [26] and it is not enough to explain the observed discrepancy. The second anomalous U(1) should be much heavier to avoid the LHC bound and its contribution to a µ is negligible. To accommodate the Fermilab data one can advocate the violation of lepton flavor universality [23]. Alternatively, as we have shown in Fig. 2, the Fermilab/BNL data can be interpreted as evidence for massive vector boson contributions to g − 2 from KK excitations of the U(1) c . Note that in contrary to the gauge boson 0-mode which acquires a mass from the anomaly, the masses of KK modes originate from the internal component(s) of the higher dimensional gauge field.
On the other hand, we can envision that L is not part of the hypercharge. If this were the case, the KK tower and even its (anomalous) zero-mode would be completely unconstrained. The generic features of the D-brane constructs (with more than 4 stacks of D-branes) that can realize this class of models can be summarized as follows: • the lepton doublet should lie on the intersection of the weak U(2) w and U(1) L , so that the Abelian charge Q L participates in the hypercharge Y but not L; • the lepton l c should lie on an intersection of a U(1) that participates in Y and U(1) L so that has opposite lepton charge from l; • the quarks should not see U(1) L ; • U(1) L can even be in the bulk (or part of it) with no important accelerator constraints. In summary, we have shown that the exchange of KK excitations of the L (lepton number) gauge boson can provide a dominant contribution to (g − 2) µ and explain the ∆a ex µ discrepancy reported by BNL and Fermilab. In the case of two extra dimensions, the summation of KK modes gives an additional factor of O(10) change in the prediction for ∆a µ compared to that of a single Z -gauge boson, and this is pivotal to avoid the violation of lepton flavor universality in accommodating the data. The KK tower, which will be within reach of the future muon smasher [27], may become the smoking gun of low-scale gravity models and large extra dimensions.