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Muon Anomalous Magnetic Moment (g-2)_mu and the Randall-Sundrum Model

We study the effects of the Kaluza-Klein gravitons in the Randall-Sundrum model on the recent BNL measurements of the muon (g-2) deviation from the standard model prediction. By examining the J-partial wave unitarity bounds of the elastic process gamma+gamma->gamma+gamma, the cut-off on the number of massive KK gravitons, n_c, has been introduced. We found that the recently measured (Delta a_mu) can be accommodated in the RS model, within the natural parameter space allowed by the perturbative unitarity. For example, dozens (hundreds) of the n_c for Lambda_pi=1-2 TeV (3 TeV) can explain the reported Delta a_mu.

1. Introduction. As one of the most accurately measured quantities in particle physics, the anomalous magnetic moment of the muon, (g − 2) µ , has played an important role in providing constraints on various models for physics beyond the standard model (SM). The recent Brookhaven E821 results for the a µ ≡ (g − 2) µ /2 by a 2.6 standard-deviation from its SM prediction [1], ∆a µ = a µ (exp) − a µ (SM) = (43 ± 16) × 10 −10 , therefore, is very likely to indicate that the SM should be extended just above the electroweak scale. A conservative view still exists such that the uncertainties in the SM calculations of the hadronic vacuum polarization may explain the deviation [2]. The implications of the ∆a µ on new physics have been extensively studied. Supersymmetric models are shown to accommodate this ∆a µ , as being consistent with all other relevant constraints [3]. Interesting results are that the reported ∆a µ value favors large tan β case and imposes observable upper bounds on the masses of supersymmetric particles. The effects from other new physics such as technicolor models [4], lepton flavor violation [5], composite models [6], leptoquark models [7], and others [8] have been also studied.
Based on a theoretical motivation to explain the gauge hierarchy problem, a novel approach has been proposed by introducing extra dimensions: Arkani-Hamed, Dimopoulos and Dvali (ADD) proposed that the large volume of the extra dimensions with factorizable geometry can explain the observed largeness of Planck scale M Pl , whereas the fundamental string scale M S being maintained around TeV scale [9]; based on two branes and a single extra dimension with non-factorizable geometry, Randall and Sundrum (RS) have proposed another high dimensional scenario, where the gauge hierarchy is attributed to a geometrical exponential factor [10]. The phenomenologies of both extra-dimensional models have been extensively studied [11]. In this paper, we examine if the RS model can explain the observed ∆a µ deviation.
In the four-dimensional effective theory, both extra dimensional scenarios yield the to the muon (g − 2) has shown some interesting results such that a given KK graviton generates finite contribution to the ∆a µ [12]; the universal coupling of each KK graviton leads to cancellation among logarithmic divergences of each Feynman diagram; the contribution of the KK graviton much heavier than the muon shows non-decoupling. The finiteness of the contributions of a graviton has been well known in supergravity models [13]. In the We introduce n c , the unitarity cut-off on the number of the KK graviton states. Then, we express the ∆a µ in terms of n c and Λ π . In fact, the necessity of introducing a cut-off in the RS model has been already known in its phenomenological study: Unitarity violation happens at high energies [14]. By examining the J-partial wave amplitudes of the elastic process γγ → γγ, we shall derive the perturbative unitarity constraints on n c and √ s/Λ π .
It is to be shown that the recently measured ∆a µ can be accommodated in the RS model, with the natural values of n c and Λ π allowed by the unitarity constraint.
2. Randall-Sundrum Model and ∆a µ . For the hierarchy problem, Randall and Sundrum have proposed a five-dimensional non-factorizable geometry with the extra dimension compactified on a S 1 /Z 2 orbifold of radius r c [10]. Reconciled with four-dimensional Poincaré invariance, the RS configuration has the following solution to Einstein's equations: where 0 ≤ |ϕ| ≤ π, and k is AdS 5 curvature. Two orbifold fixed points accommodate two three-branes, the hidden brane at ϕ = 0 and our visible brane at |ϕ| = π. The arrangement of our brane at |ϕ| = π renders a fundamental scale m 0 to appear as the four-dimensional physical mass m = e −krcπ m 0 . The hierarchy problem can be answered if kr c ≈ 12. From the four-dimensional effective action, the relation between the four-dimensional Planck scale M Pl and the fundamental string scale M 5 is obtained by M 2 Pl = M 3 5 (1 − e −2krcπ )/k. The compactification of the fifth dimension leads to the following four-dimensional effective Lagrangian [14], where Λ π ≡ e −krcπ M Pl . The coupling of the zero mode KK graviton is suppressed by usual Planck scale M Pl , while those of the massive KK gravitons by the electroweak scale Λ π . The masses of the KK gravitons are also at electroweak scale, where x n is the n-th root of the first order Bessel function. And the total decay width of the n-th KK graviton is Γ n = ρ m n x 2 n (k/M Pl ) 2 , where the ρ, fixed to be one, is a modeldependent parameter which is different according to the decay pattern of the KK graviton.
The condition k < M Pl is to be imposed to maintain the reliability of the RS solution in Eq. (2) [15].
The Lagrangian in Eq. (3) then makes new contributions to the ∆a µ . The relevant Feynman diagrams mediated by the KK gravitons are presented in Fig. 1. We refer readers to Ref. [16] for the corresponding Feynman rules of the graviton, photon and fermion. Note that the gravitational Ward identity renders almost negligible, to leading order, the contributions of the terms containing two or more k α 's in the massive graviton propagator. Here the k α denotes the momentum of the graviton. A given n-th KK graviton state generates the ∆a µ , which is parameterized by [12] where the f runs over five Feynman diagrams and the second equality holds for the case of m n ≫ m µ . As discussed in the previous section, naive summation over all the KK states would yield infinite contribution, i.e., ∆a µ ≡ ∞ n ∆a (n) µ → ∞. A cut-off is needed. This requirement of a cut-off is expected since the RS model described by Eq. (3) is an effective theory. According to the analysis of the process e + e − → µ + µ − with the KK graviton effects, in the large k/M Pl case the partial wave unitarity seems to be violated even at TeV scale as the resonant peaks of the KK gravitons become very wide and the cross section is greatly enhanced [14]. Since in the small k/M Pl < ∼ 0.3 case the unitarity appears preserved at several TeV scales, possible unitarity violation of the RS model at high energy colliders has not been considered seriously yet. However, when we consider the loop effects by using an effective theory (the RS model) with an infinitely large number of states, we should determine the cut-off scale below which the perturbative calculations are reliable, and add only the contributions up to the cut-off scale. We introduce the cut-off on the effective number of KK graviton states, n c . Then, the total RS contribution to the ∆a µ becomes The scale of n c will be estimated by examining the partial wave unitary bound in the process of γγ → γγ. Note that the parameters (Λ π , k/M Pl ) are constrained from the phenomenological studies of virtual exchange of the KK gravitons at LEP II and Tevatron Run I data; Λ π > ∼ 1.1 (0.65) TeV for the k/M Pl = 0.1 (0.3).
In Fig. 2, we plot the ∆a RS µ as a function of n c for Λ π =1, 2, 3 TeV. It can be seen that the RS model can explain the recently reported ∆a µ from the SM prediction with n c ≃ 10, 50, and 150, for Λ π = 1, 2, and 3 TeV, respectively. As shall be shown in the next section, the scale of n c ≃ 10 − 100 is natural in the sense that the partial wave unitarity is well preserved in this range. The scale of the heaviest KK graviton mass, m nc , then becomes of order 10 TeV, which can be regarded as a naive cut-off scale of the four-dimensional effective Lagrangian in the RS scenario. In the next section, we will show that this is consistent with the cut-off scale for the perturbative unitarity.
Some discussions on the other RS effects rather than the KK gravitons are in order here. First in the RS scenario where the modulus field (called the radion) is inherent, its stabilization is needed for the consistency in a cosmological context [17]. According to the Goldberger and Wise stabilization mechanism, this radion can be an order of magnitude lighter than the Λ π [17][18][19]. The radion contributions to the ∆a µ , however, have been too small to explain alone the ∆a µ of ∼ 4 × 10 −9 [20]. The effects of an extended RS model, where the SM gauge and fermion fields, are both in the bulk have been also examined [21].
The dominant contributions from bulk gauge bosons, bulk fermions, and Higgs bosons, however, generate negative contribution to the ∆a µ , which is difficult to be compatible with the recently reported positive deviation in a µ . Moreover, the graviton loop contributions are not well-defined since the cancellation among logarithmic divergences does not occur due to non-universal couplings between the KK gravitons and the bulk SM fields.
3. Unitary Bound from γγ → γγ Scattering. In order to see whether the n c ≃ 10−100 can naturally explain the ∆a µ in the effective theory interpretation, we consider another process to estimate the scale of n c . The elastic process γγ → γγ is to be examined, focused on the unitarity bounds of the RS model: It has some merits such that the RS effects mediated by KK gravitons are dominant due to the absence of the SM contributions at tree level and thus the unitarity bounds of the RS model can be more sensitively probed compared to other processes. The requirement of partial wave unitarity shall constrain √ s/Λ π and the number of the KK graviton states, n c . The J-partial wave amplitude is defined [22] by where the M λ 1 λ 2 λ 3 λ 4 is the helicity amplitudes, µ = λ 1 − λ 2 , µ ′ = λ 3 − λ 4 , and the d J µµ ′ is the Wigner functions [23]. Unitarity implies that the largest eigenvalue (χ) of a J µµ ′ is to be |χ| ≤ 1. The reliability of perturbative calculations is approximately guaranteed by the conditions |χ| = 1 and |Re(χ)| = 1/2. The helicity amplitudes, of which the dominant contribution at high energies comes from the KK gravitons, are where D n (s) = 1/(s − m 2 n + im n Γ n ) with m n being the mass of the n-th KK graviton state. Note that two parameters ( √ s/Λ π , n c ) with a given k/M Pl determine the helicity amplitudes, and thus χ. The odd J-partial wave amplitudes vanish due to Bose-Einstein statistics in the elastic γγ scattering. And we have a 2 22 = a 2 −2−2 and a 2 −22 = a 2 2−2 from the parity arguments. The non-vanishing eigenvalues χ i are a 2 00 and 2 a 2 22 . In two limiting cases where √ s/Λ π ≪ (k/M Pl )x n and √ s/Λ π ≫ (k/M Pl )x n , the two non-vanishing eigenvalues show converging behaviors. When a J(n) µµ ′ denotes the contribution of the n-th KK state to a J µµ ′ , the a 2(n) 00 is approximated as There exist finite unitarity bounds n c < ∼ 1000 at small √ s/Λ π , depending on the value of the parameter k/M Pl . Similarly, we calculate the approximate formula for a 2(n) 22 in two limiting cases: The unitary bounds n c are around n c ≃ 10 ∼ 100. The eigenvalue 2a 2 22 generates stronger constraints compared to that of a 2 00 .

Figures 3 and 4 exhibit the exact numerical calculations of the unitarity bounds on
( √ s/Λ π , n c ) plane from the χ = a 2 00 and χ = 2 a 2 22 constraints, respectively. We consider three values for k/M Pl (= 0.1, 0.3, 0.7). Note that the upper bounds on √ s/Λ π constrained from the perturbative unitarity are almost the same in the range of n c ≃ 10 − 100 (such that √ s < ∼ 4 Λ π , 6.5 Λ π , and 11 Λ π for k/M Pl = 0.1, 0.3, and 0.7 cases, respectively). When we regard this upper bounds on √ s as the cut-off scale of the effective RS model, we can see that this cut-off scale is compatible with the scale of m nc . Thus, the cut-off on the number of KK graviton states (n c ) which explains the reported ∆a µ is within the allowed region by the perturbative unitarity bounds.

Summary.
To summarize, we have studied the effects of the Kaluza-Klein gravitons in the Randall-Sundrum model on the recent BNL measurements of the muon (g − 2) deviation from the SM prediction. It is known that the individual contribution of a heavy KK graviton to ∆a µ is finite. Since the four-dimensional effective theory in the RS model contains an infinitely large number of massive KK gravitons with the TeV suppressed couplings, a naive summation would yield infinite ∆a µ . A cut-off on the number of the KK gravitons has been introduced. Then the reported ∆a µ can be attributed to the RS effects with dozens (hundreds) of the n c for Λ π = 1 ∼ 2 TeV (3 TeV). By examining the J-partial wave unitarity bounds in the elastic process γγ → γγ, we have shown that this range of n c is natural in the effective theory interpretation and the scale of the heaviest graviton mass (m nc ) is compatible with the cut-off scale for the perturbative unitarity. Therefore, we conclude that the recently measured ∆a µ can be accommodated in the RS model.