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Probing the Minimal Length Scale by Precision Tests of the Muon G − 2 Physics Letters B 584 (2004) 109–113 www.elsevier.com/locate/physletb Probing the minimal length scale by precision tests of the muon g − 2 U. Harbach, S. Hossenfelder, M. Bleicher, H. Stöcker Institut für Theoretische Physik, J.W. Goethe-Universität, Robert-Mayer-Str. 8-10, 60054 Frankfurt am Main, Germany Received 25 November 2003; received in revised form 20 January 2004; accepted 21 January 2004 Editor: P.V. Landshoff Abstract Modifications of the gyromagnetic moment of electrons and muons due to a minimal length scale combined with a modified fundamental scale Mf are explored. First-order deviations from the theoretical SM value for g − 2 due to these string theory- motivated effects are derived. Constraints for the fundamental scale Mf are given. 2004 Elsevier B.V. All rights reserved. String theory suggests the existence of a minimum • the need for a higher-dimensional space–time and length scale. An exciting quantum mechanical impli- • the existence of a minimal length scale. cation of this feature is a modification of the uncer- tainty principle. In perturbative string theory [1,2], Naturally, this minimum length uncertainty is re- the feature of a fundamental minimal length scale lated to a modification of the standard commutation arises from the fact that strings cannot probe dis- relations between position and momentum [6,7]. Ap- tances smaller than the string scale. If the energy of plication of this is of high interest for quantum fluc- a string reaches the Planck mass mp, excitations of the tuations in the early universe and inflation [8–16]. string can occur and cause a non-zero extension [3]. The incorporation of the modified commutation re- Due to this, uncertainty in position measurement can lations into quantum theory is not fully consistent in never become smaller than lp = h/m¯ p.Forareview, all approaches. We will follow the propositions made see [4,5]. in [17]. Although a full description of quantum gravity is In our approach, the existence of a minimal length not yet available, there are general features that seem scale grows important for collider physics at high en- to go hand in hand with promising candidates for such ergies or for high precision measurements at low ener- a theory: gies due to the lowered value of the fundamental scale Mf . This new scale is incorporated through the central idea of large extra dimensions (LXDs). The model of LXDs which was recently proposed in [18–22] might E-mail address: [email protected] indeed allow to study first effects of unification or (U. Harbach). quantum gravity in near future experiments. Here, the 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.01.051 110 U. Harbach et al. / Physics Letters B 584 (2004) 109–113 hierarchy-problem is solved or at least reformulated arbitrarily small Compton wavelengths λ = 2π/k and in a geometric language by the existence of d com- that arbitrarily small scales could not be resolved pactified LXDs in which only gravitons can propa- anymore. gate. The standard model particles are bound to our To incorporate this behaviour, we assume a rela- 4-dimensional sub-manifold, often called our 3-brane. tion k = k(p) between p and k which is an uneven This results in a lowering of the Planck scale to a new function (because of parity) and which asymptotically fundamental scale Mf and gives rise to the exciting approaches 1/Lf . Furthermore, we demand the func- possibility of TeV scale GUTs [23]. tional relation between the energy E and the frequency In this scenario the following relation between the ω to be the same as that between the wave vector k and four-dimensional Planck mass mp and the higher- the momentum p. A possible choice for the relations is dimensional Planck mass Mf can be derived: γ = 1/γ p 2 = d d+2 Lf k(p) tanh , (2) mp R M . (1) M f f The lowered fundamental scale would lead to a γ = 1/γ E vast number of observable phenomena for quantum Lf ω(E) tanh , (3) Mf gravity at energies in the range Mf . In fact, the non- observation in past collider experiments of these pre- with a real, positive constant γ . dicted features gives first constraints on the parameters In the following we will study an approximation, of the model, the number of extra dimensions d and namely the regime of first effects including the orders (p/M )3. For this purpose, we expand the function in the fundamental scale Mf [24,25]. On the one hand, f this scenario has major consequences for cosmology a Taylor series for small arguments. and astrophysics such as the modification of inflation Because the exact functional dependence is un- in the early universe and enhanced supernova-cooling known, we assume an arbitrary factor α in front of the 3 due to graviton emission [20,26–29]. On the other order (p/Mf ) -term. Therefore the most general re- hand, additional processes are expected in high-energy lations for k(p) and ω(E) which we will use in the collisions [30,31]: production of real and virtual gravi- following should be tons [32–36] and the creation of black holes at energies p p 3 that can be achieved at colliders in the near future [37– Lf k(p) ≈ − α , (4) Mf Mf 43] and in ultra high energetic cosmic rays [44]. One also has to expect the influence of the extra dimen- 3 ≈ E − E sions on high precision measurements. The most ob- Lf ω(E) α , (5) Mf Mf vious being the modification of Newton’s law at small 1 3 distances [45–47]. Of highest interest are also modifi- p(k) ≈ kLf + α(kLf ) , (6) cations of the gyromagnetic moment of Dirac particles Mf which promises new insight into non-standard model 1 3 E(ω) ≈ ωLf + α(ωLf ) , (7) couplings and effects [48–53]. Mf In this Letter we study implications of these exten- with α being of order one, in general negative values sions in the Dirac equation without the aim to derive of α cannot be excluded. them from a fully consistent theory. Instead we will This yields to 3rd order analyse possible observable modifications that may arise by combining the assumptions of both extra di- 1 ∂p p 2 mensions and a minimal length scale. ≈ 1 + 3α . (8) h¯ ∂k M In order to implement the notion of a minimal f length Lf , let us now suppose that one can increase The quantisation of these relations is straightfor- the momentum p arbitrarily, but that the wave number ward. The commutators between kˆ and xˆ remain in k has an upper bound. This effect will show up the standard form: when p approaches a certain scale M . The physical f [ˆ ˆ]= interpretation of this is that particles could not possess x,k iδij . (9) U. Harbach et al. / Physics Letters B 584 (2004) 109–113 111 Inserting the functional relation between the wave the generalised uncertainty principle, we start with the vector and the momentum then yields the modified relation commutator for the momentum. With the commutator ˆ = 0 i ˆ + relation E(ω) γ γ pi(k) m . (19) ˆ ∂A Including the altered momentum wave vector relation x,ˆ A(k) =+i , (10) ˆ ˆ ∂k p(k) from Eq. (16), this yields again Eq. (18) with the modified momentum operator the modified commutator algebra now reads ˆ − = ∂p p(/ k) m ψ 0. (20) [ˆx,pˆ]=+i . (11) ∂k This equation is Lorentz invariant by construction. This results in the generalised uncertainty relation Since it contains—in position representation—3rd order derivatives in space coordinates, it contains 1 ∂p px . (12) 3rd order time-derivatives too. In our approximation, 2 ∂k we can solve the equation for a single order time With the approximations (4)–(7), the results of derivative by using the energy condition E2 = p2 + Ref. [8] are reproduced up to the factor α: m2. This leads effectively to a replacement of time derivatives by space derivatives: pˆ2 [ˆx,pˆ]≈ih¯ 1 + 3α (13) ˆ 3 ˆi ˆ + 2 2 ˆ αE ˆ p pi m Mf h¯ωˆ ≈ E − = E 1 − α . (21) M2 M2 giving the generalised uncertainty relation f f Inserting the modified E(ω)ˆ and p(k)ˆ and keeping 1 ˆp2 px h¯ 1 + 3α . (14) only derivatives up to 3rd order, we obtain the follow- 2 2 Mf ing expression of the Dirac equation: Quantisation proceeds in the usual way from the 2 ˆi ˆ + 2 0 i ˆ m h¯ k ki m commutation relations. Focusing on conservative po- ω|ψ ≈γ γ ki + 1 − α |ψ . h¯ M2 tentials in quantum mechanics we give the operators f (22) in the position representation which is suited best for this purpose: The task is now to derive the modifications of the anomalous gyromagnetic moment due to the existence ˆ xˆ = x, k =−i∂x, of a minimal length. Therefore we assume as usual pˆ =ˆp(k),ˆ (15) the particle is placed inside a homogeneous and static magnetic field B. Regarding the energy levels of an yielding the new momentum operator electron the magnetic field leads to a splitting of the energetic degenerated values which is proportional to p(ˆ k)ˆ ≈−ih¯ 1 − αL2 ∂2 ∂ . (16) f x x the magnetic field B and the gyromagnetic moment g. In ordinary relativistic quantum mechanics the Since the energy of the particle in the field is not Hamiltonian of the Dirac particle is1 modified (see (18)) there is no modification of the splitting as one might have expected from the fact that ˆ = = 0 i + H ih∂¯ 0 γ ihγ¯ ∂i m .
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