Physics Letters B 668 (2008) 312–314 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Anomalous anomalous scaling? ∗ Alessandro Strumia a, ,NikolaosTetradisb a Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy b Department of Physics, University of Athens, Zographou GR-15784, Athens, Greece article info abstract Article history: Motivated by speculations about infrared deviations from the standard behavior of local quantum field Received 15 May 2008 theories, we explore the possibility that such effects might show up as an anomalous running of coupling Received in revised form 9 August 2008 constants. The most sensitive probes are presently given by the anomalous magnetic moments of the Accepted 28 August 2008 electron and the muon, that suggest that α runs 1.00047 ± 0.00018 times faster than predicted by the Available online 3 September 2008 em Standard Model. The running of and up to the weak scale is confirmed with a precision at the % Editor: G.F. Giudice αem αs level. © 2008 Elsevier B.V. All rights reserved. 1. Introduction 2. Speculations The above ideas about a non-local connection between the IR and UV cutoffs do not have a very precise meaning, and one can The range of validity of Quantum Field Theory (QFT) may be debate whether they would lead to any of the effects mentioned limited not only in the UltraViolet (UV), E Λ ,butalsointhe UV above. Rather than arguing in any one direction, we present the InfraRed (IR), E Λ ≡ 1/L, with a non-trivial connection be- IR uncertain issues. tween Λ and Λ . This possibility has attracted interest due to UV IR Firstly, when do these non-local phenomena appear in particle the following reasons. physics? The weakest possibility is only when strong gravity effects, On the theoretical side, requiring that the entropy associ- such as those arising from black holes, are directly relevant. This ated with the QFT degrees of freedom ∼ (Λ /Λ )3 saturates UV IR would practically mean never, as black hole phenomena (Hawk- the Bekenstein entropy [1] ∼ L2 M2 of a black hole with size Pl ing radiation, etc.) are quantitatively irrelevant in all processes we L leads to ∼ 3 M2 . Alternatively, it has been suggested ΛIR ΛUV/ Pl can realistically observe (possibly unless the true quantum gravity that one should require that systems whose size L exceeds their scale is much below MPl). The strongest possibility is that states Schwarzschild radius ∼ m/M2 do not appear in QFT. For m ∼ Pl with energy E ΛUV that propagate for more than L do not exist 3 3 ∼ 2 ΛUV L , this requirement leads to ΛIR ΛUV/MPl [2]. and must be dropped from QFT. However, this possibility seems to On the phenomenological side, Refs. [2,3] discussed possible contradict experience. We see TeV γ rays from the galactic center, connections of these ideas with the cosmological constant and the particles with energies up to 1020 eV from extragalactic sources, supersymmetry breaking puzzles. Indeed, in standard QFT the val- etc. ues of the vacuum energy, scalar masses squared and dimension- We envisage a scenario in which the bulk of the contributions less couplings are given by their bare Planck-scale values plus a from (real of virtual) modes to quantities such as the energy den- 4 2 quantum correction proportional to ΛUV, ΛUV and ln ΛUV,respec- sity or the renormalized couplings is delimited by a scale that tively. Such non-local effects could change this power-counting, depends on the IR cutoff. Individual very energetic modes may still solving or modifying the hierarchy problems associated with mas- exist, but their effect on the above observables is negligible. For ex- sive parameters [2,3]. ample, one may limit the temperature T of a system of size L by We observe here that dimensionless couplings may be similarly demanding that this system does not collapse into a black hole: affected, leading to an anomalous Renormalization Group (RG) run- 2 T MPl/L. States with energy much larger than this limit may ning, and we study how accurately present data test the standard exist, but their contribution to the energy density is negligible. Our QFT prediction. basic, and possibly contentious, assumption is that gravity enforces a similar constraint to the contributions from virtual fluctuations. This leads to limits on the energy density, the quantum-corrected vacuum energy density (i.e. the minimum of the potential) [2],the * Corresponding author. Higgs mass (i.e. the curvature of the potential at its minimum) [3], E-mail address: [email protected] (A. Strumia). and possibly all couplings (i.e. the derivatives of the potential). The 0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.08.061 A. Strumia, N. Tetradis / Physics Letters B 668 (2008) 312–314 313 smallness of the cosmological constant is consistent with such an By using the β function of Eq. (3) within the dimensional reg- assumption. ularization formalism, we get for the one-loop RG running of a Secondly, what is the precise meaning of ΛIR? There are various gauge coupling α possibilities, and we mention two: (i) The IR cutoff can be defined − 1 1 μ1 δ by imposing boundary conditions such that QFT lives “in a box” − β ln +···. (4) 1−δ with size 1/ΛIR. (ii) ΛIR is the minimal energy scale that appears α(μ ) α(μ) μ in loop integrations. In practical cases these choices can lead to This is equivalent to the standard expression, with the Minimal − different answers. For example, only in the first case the IR cutoff Subtraction mass scale μ replaced by μ1 δ. In theories with sev- for the cosmological constant would√ be the Hubble distance 1/H eral particle masses and sizable RG corrections, the factor δ gener- ∼ ∼ (possibly leading to a small ΛUV MPl H eV [2]). On the other ically can become some unknown function of the energy. In order hand, H does not appear in the one-loop correction to the vacuum to compute it, we would need to know the physics around ΛUV. energy, equal to the value of the potential V at its physical local In the next sections, we explore the most sensitive experimen- minimum tal probes of anomalous RG running, assuming for definiteness that ΛUV all Standard Model formula get modified as in Eq. (2), with a con- 4 1 d k 2 stant δ to be extracted from data. V V bare + Str ln k + V , (1) 2 (2π)4 bare Λ IR 3. Running of αem from me to mμ unless ΛIR = H is imposed. The measurements of the anomalous magnetic moments of the We finally arrive to the third issue: what is the precise meaning of electron [4] ΛUV if it is viewed as a function of ΛIR? As we explained above, we as- sume that Λ sets the upper limit for the bulk of the fluctuations UV g /2 = 1.00115965218085(76) (5) that renormalize a quantum field theory. Modes with characteristic e momenta above ΛUV may exist, but they do not contribute signif- and of the muon [5] icantly to the renormalization of the couplings. According to this logic ΛUV (ΛIR) may be used as the upper bound of the momen- gμ/2 = 1.00116592080(63), (6) tum integration is expressions such as (1). together with the assumption of the validity of the Standard If the quantum correction to the minimum of V is naturally Model, allow us to infer the electromagnetic coupling α (μ) at small thanks to an anomalous dependence of Λ on Λ [2],one em UV IR the scales μ = m and m ,inviewofthetheoreticalprediction expects a similar anomalous running of the whole potential, and in e μ particular of its coupling constants. In standard QFT, the one-loop g = 2 + α (m )/π +···, (7) corrections to any dimensionless coupling (e.g. the gauge boson i em i vertex g) has the form where ··· denotes higher-order effects. We recall that ge gives the most precise determination of αem, that is consistent with lower- 3 2 gbare ΛUV energy probes from atomic physics [4]. Assuming the anomalous g(p) = gbare − β ln + finite , (2) 8π p2 running where p is some combination of the external momenta that sets 1 1 1 − δ mμ − = ln +··· (8) the IR cutoff in the loop integration. In standard QFT the physical αem(me) αem(mμ) 3π me coupling g(p) ‘runs’ with the energy p of the process, and the RG one gets the presently most precise determination of δ: coefficient β is a number that depends on the particle content of the theory above p.(InEq.(2) we assumed that all the masses are δ =−(0.047 ± 0.018)%. (9) negligibly small.) If, instead, non-QFT effects produce some physical UV cutoff The central value of δ is about 3σ below zero, because, for δ = 0, ∼ − SM = ± ΛUV that depends on ΛIR p, one generically obtains an anoma- gμ is about 3σ above the SM prediction, (gμ gμ )/2 (23 − lous RG running of g(p). For example, in the one-loop approxima- 9) × 10 10 , with the precise number depending on how one deals tion the running is proportional to with the theoretical uncertainties on higher-order QCD corrections − + to gμ:relyingone e data and/or on τ -decay data [5]. ∂ ln ΛUV β → β 1 − ≡ β(1 − δ). (3) The usual new-physics interpretation of the gμ − 2 anomaly is ∂ ln ΛIR that new particles with heavy mass M, like supersymmetric parti- ∼ 2 2 The correction appears because fewer UV modes contribute to cles, affect gμ giving an extra contribution gμ α2mμ/M .They quantum corrections when the IR cutoff is lowered.