Short Distance Properties Research Articles

Universality in long-distance geometry and quantum complexity

In physics, two systems that radically differ at short scales can exhibit strikingly similar macroscopic behaviour: they are part of the same long-distance universality class1. Here we apply this viewpoint to geometry and initiate a program of classifying homogeneous metrics on group manifolds2 by their long-distance properties. We show that many metrics on low-dimensional Lie groups have markedly different short-distance properties but nearly identical distance functions at long distances, and provide evidence that this phenomenon is even more robust in high dimensions. An application of these ideas of particular interest to physics and computer science is complexity geometry3–7—the study of quantum computational complexity using Riemannian geometry. We argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered. We conjecture that a new effective metric emerges at larger complexities that describes a broad class of complexity geometries, insensitive to various choices of microscopic penalty factors. We discuss the implications for recent conjectures in quantum gravity.

The Fourier–Bessel expansion: application to experimental probes of gravity and feeble short-range interactions

Two Eöt–Wash torsion balance instruments exploited optimized Fourier–Bessel geometries to test the short-distance properties of gravity and to constrain exotic dipole–dipole and monopole–dipole interactions. We discuss efficient analytic techniques for computing the expected torques in those instruments arising from Newtonian and Yukawa interactions between unpolarized test bodies and dipole–dipole and monopole–dipole torques on polarized test-bodies. We consider systematic effects induced by weak external magnetic fields on the nominally unpolarized test-bodies. We also present a new Fourier–Bessel expansion for inverse-power-law (IPL) potentials and use this to calculate the expected IPL signals in our recent short-distance test of the gravitational inverse-square law. Our results slightly improve limits on inverse-5th-power law potentials.

On some short-distance properties of the fourth-rank hadronic vacuum polarization tensor and the anomalous magnetic moment of the muon

Some short-distance properties of the fourth-rank hadronic vacuum polarization tensor are re-examined. Their consequences are critically discussed in the context of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon.

Properties of the random-singlet phase: From the disordered Heisenberg chain to an amorphous valence-bond solid

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multi-spin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that can not be generated by the SDRG scheme. In the model with multi-spin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

The Operator Product Expansion Converges in Massless $${\varphi_{4}^{4}}$$ φ 4 4 -Theory

It has been shown recently that the mathematical status of the operator product expansion (OPE) is better than was expected before: namely considering massive Euclidean $\varphi^4$-theory in the perturbative loop expansion, the OPE converges at any loop order when considering (as is usually done) composite operator insertions into correlation functions. In the present paper we prove the same result for the massless theory. While the short-distance properties of massive and massless theories may be expected to be similar on physical grounds, the proof in the massless case requires entirely new techniques. In our inductive construction we have to control with sufficient precision the exceptional momentum singularities of the massless correlation functions. In fact the bounds we state are expressed in terms of weight factors associated to certain tree graphs. Our proof is again based on the flow equations of the renormalisation group.

Explaining the proton radius puzzle with disformal scalars

We analyze the consequences of a disformal interaction between a massless scalar and matter particles in the context of atomic physics. We focus on the displacement of the atomic energy levels that it induces, and in particular the change in the Lamb shift between the 2s and 2p states. We find that the correction to the Lamb shift depends on the mass of the fermion orbiting around the nucleus, implying a larger effect for muonic atoms. Taking the cutoff scale describing the effective scalar field theory close to the QCD scale, we find that the disformal interaction can account for the observed difference in the proton radius of muonic versus electronic hydrogen. Explaining the proton radius puzzle is only possible when the scalar field is embedded in nonlinear theories which alleviate constraints from collider and stellar physics. Short distance properties of the Galileon where nonperturbative effects in vacuum are present ensure that unitarity is preserved in high-energy particle collisions. In matter, the chameleon mechanism alleviates the constraints on disformal interactions coming from the burning rates for stellar objects. We show how to combine these two properties in a single model which renders the proposed explanation of the proton radius puzzle viable.

Oblique S and T constraints on electroweak strongly-coupled models with a light Higgs

Using a general effective Lagrangian implementing the chiral symmetry breaking SU(2)L x SU(2)R -> SU(2){L+R}, we present a one-loop calculation of the oblique S and T parameters within electroweak strongly-coupled models with a light scalar. Imposing a proper ultraviolet behaviour, we determine S and T at next-to-leading order in terms of a few resonance parameters. The constraints from the global fit to electroweak precision data force the massive vector and axial-vector states to be heavy, with masses above the TeV scale, and suggest that the W+W- and ZZ couplings of the Higgs-like scalar should be close to the Standard Model value. Our findings are generic, since they only rely on soft requirements on the short-distance properties of the underlying strongly-coupled theory, which are widely satisfied in more specific scenarios.

Short-distance properties of Coulomb systems

We use the operator product expansion to derive exact results for the momentum distribution and the static structure factor at high momentum for a jellium model of electrons in both two and three dimensions. It is shown that independent of the precise state of the Coulomb system and for arbitrary temperatures, the asymptotic behavior is a power law in the momentum, whose strength is determined by the contact value of the pair distribution function $g(0)$. The power-law tails are quantum effects which vanish in the classical limit $\hbar \to 0$. A leading order virial expansion shows that the classical and the high-temperature limit do not agree.

Critical collapse in Einstein-Gauss-Bonnet gravity in five and six dimensions

Einstein-Gauss-Bonnet gravity (EGB) provides a natural higher dimensional and higher order curvature generalization of Einstein gravity. It contains a new, presumably microscopic, length scale that should affect short distance properties of the dynamics, such as Choptuik scaling. We present the results of a numerical analysis in generalized flat slice co-ordinates of self-gravitating massless scalar spherical collapse in five and six dimensional EGB gravity near the threshold of black hole formation. Remarkably, the behaviour is universal (i.e. independent of initial data) but qualitatively different in five and six dimensions. In five dimensions there is a minimum horizon radius, suggestive of a first order transition between black hole and dispersive initial data. In six dimensions no radius gap is evident. Instead, below the GB scale there is a change in the critical exponent and echoing period.

Quasistationary States and the Range of Pair Interactions

"Quasistationary" states are approximately time independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction V(r→∞)~1/r(γ) with γ>0, in d>1 dimensions. We generalize analytic calculations known for gravity in d=3 to determine the scaling parametric dependences of their relaxation rates due to two-body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for γ<d-1, the existence of quasistationary states is ensured by the large distance behavior of the interaction alone, while for γ>d-1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft core in the interaction potential.

Structure and three-body decay ofBe9resonances

The complex-rotated hyperspherical adiabatic method is used to study the decay of low-lying $^9$Be resonances into one neutron and two $\alpha$-particles. We investigate the six resonances above the break-up threshold and below 6 MeV: $1/2^\pm$, $3/2^\pm$ and $5/2^\pm$. The short-distance properties of each resonance are studied, and the different angular momentum and parity configurations of the $^8$Be and $^5$He two-body substructures are determined. We compute the branching ratio for sequential decay via the $^8$Be ground state which qualitatively is consistent with measurements. We extract the momentum distributions after decay directly into the three-body continuum from the large-distance asymptotic structures. The kinematically complete results are presented as Dalitz plots as well as projections on given neutron and $\alpha$-energy. The distributions are discussed and in most cases found to agree with available experimental data.

Weinberg sum rules and the parity doubling of radial Regge trajectories

Assuming that the radial excitations of q¯q meson states exactly follow linear Regge trajectories with constant residues, as prescribed by dual models, and using large Nc arguments and the matching to perturbative QCD in the deep-Minkowski region we obtain that: (a) the region dominated by resonances and the one saturated by perturbation theory approach each other as Nc increases; (b) the scales Λ(V,A) separating the resonance-dominated and the perturbative-saturated region in the V,A channels, respectively, grow as Nc, whereas the difference between (ΛV)2 and (ΛA)2 stays constant or even decreases as Nc increases; (c) the number of visible resonances increases as Nc; (d) in order to satisfy the Weinberg sum rules the slopes of Regge trajectories for mesons of opposite parities must coincide, but the intercepts may differ by a quantity of O(1) at most in the large Nc limit. This suggests that modelizations of QCD where these characteristics are not present and yet the resonances follow linear Regge trajectories are not compatible with the symmetries or short-distance properties of QCD.

Short distance properties of cascading gauge theories

We study the short distance (large momentum) properties of correlation functions of cascading gauge theories by performing a tree-level computation in their dual gravitational background. We prove that these theories are holographically renormalizable; the correlators have only analytic ultraviolet divergences, which may be removed by appropriate local counterterms. We find that n-point correlation functions of properly normalized operators have the expected scaling in the semi-classical gravity (large N) limit: they scale as N_{eff}^{2-n} with N_{eff} proportional to ln(k/Lambda) where k is a typical momentum. Our analysis thus confirms the interpretation of the cascading gauge theories as renormalizable four-dimensional quantum field theories with an effective number of degrees of freedom which logarithmically increases with the energy.

SCALING ALGEBRAS AND SUPERSELECTION SECTORS: STUDY OF A CLASS OF MODELS

We analyze a class of quantum field theory models illustrating some of the possibilities that have emerged in the general study of the short distance properties of superselection sectors, performed in a previous paper (together with R. Verch). In particular, we show that for each pair (G, N), with G a compact Lie group and N a closed normal subgroup, there is a net of observable algebras which has (a subset of) DHR sectors in 1-1 correspondence with classes of irreducible representations of G, and such that only the sectors corresponding to representations of G/N are preserved in the scaling limit. In the way of achieving this result, we derive sufficient conditions under which the scaling limit of a tensor product theory coincides with the product of the scaling limit theories.

A new class of short distance universal amplitude ratios

We propose a new class of universal amplitude ratios which involve the first terms of the short distance expansion of the correlators of a statistical model in the vicinity of a critical point. We will describe the critical system with a conformal field theory (UV fixed point) perturbed by an appropriate relevant operator. In two dimensions the exact knowledge of the UV fixed point allows for accurate predictions of the ratios and in many nontrivial integrable perturbations they can even be evaluated exactly. In three dimensional O(N) scalar systems feasible extensions of some existing results should allow to obtain perturbative expansions for the ratios. By construction these universal ratios are a perfect tool to explore the short distance properties of the underlying quantum field theory even in regimes where the correlation length and one point functions are not accessible in experiments or simulations.

Hadronic light-by-light scattering contribution to the muon g − 2

We briefly review the current status of the hadronic light-by-light scattering correction to the muon g − 2. Then we present our semi-analytical evaluation of the pion-pole contribution, using a description of the pion-photon-photon form factor based on large- N c and short-distance properties of QCD. We also sketch an effective field theory approach to hadronic light-by-light scattering. In view of several still unsolved problems, our conservative estimate for the full hadronic light-by-light scattering contribution is α μ LbyL; had = +8 (4) × 10 −10.

Hadronic light-by-light scattering contribution to gμ − 2

We briefly review the current status of the hadronic light-by-light scattering correction to the muon g − 2. Then we present our semi-analytical evaluation of the pion-pole contribution, using a description of the pion-photon-photon transition form factor based on large- N C and short-distance properties of QCD. We derive a two-dimensional integral representation which allows to separate the generic features from the model dependence, in order to better control the latter. Finally, we sketch an effective field theory approach to hadronic light-by-light scattering which yields the leading logarithmic terms that are enhanced by a factor N C . It also shows that the modeling of hadronic light-by-light scattering by a constituent quark loop is not consistent with QCD.

Heavy quark–antiquark free energy and the renormalized Polyakov loop

We calculate the colour averaged and colour singlet free energies of static quark–antiquark sources placed in a thermal gluonic heat bath. We discuss the renormalization of these free energies using the short distance properties of the zero temperature heavy quark potential. This leads to the definition of the renormalized Polyakov loop as an order parameter for the deconfinement phase transition of the SU(3) gauge theory which is well behaved in the continuum limit.

Hadronic light-by-light corrections to the muong−2:The pion-pole contribution

The correction to the muon anomalous magnetic moment from the pion-pole contribution to the hadronic light-by-light scattering is considered using a description of the ${\ensuremath{\pi}}^{0}\ensuremath{-}{\ensuremath{\gamma}}^{*}\ensuremath{-}{\ensuremath{\gamma}}^{*}$ transition form factor based on the large-${N}_{C}$ and short-distance properties of QCD. The resulting two-loop integrals are treated by first performing the angular integration analytically, using the method of Gegenbauer polynomials, followed by a numerical evaluation of the remaining two-dimensional integration over the moduli of the Euclidean loop momenta. The value obtained, ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};{\ensuremath{\pi}}^{0}}=+5.8(1.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10},$ disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};{\ensuremath{\pi}}^{0}}{|}_{\mathrm{VMD}}=+5.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10},$ and differs only by its overall sign from the value obtained by previous authors. The inclusion of the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ poles gives a total value ${a}_{\ensuremath{\mu}}^{\mathrm{L}\mathrm{b}\mathrm{y}\mathrm{L};\mathrm{P}\mathrm{S}}=+8.3(1.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of ${a}_{\ensuremath{\mu}}$ and its theoretical counterpart in the standard model.

Predictions for high-energy real and virtual photon–photon scattering from color dipole BFKL–Regge factorization

High-energy virtual photon-virtual photon scattering can be viewed as interaction of small size color dipoles from the beam and target photons, which makes $\gamma^{*}\gamma^{*}, \gamma^{*}\gamma$ scattering at high energies (LEP, LEP200 & NLC) an indispensable probe of short distance properties of the QCD pomeron exchange. Based on the color dipole representation, we investigate consequences for the $\gamma^{*}\gamma^{*},\gamma^{*}\gamma$ scattering of the incorporation of asymptotic freedom into the BFKL equation which makes the QCD pomeron a series of isolated poles in the angular momentum plane. The emerging color dipole BFKL-Regge factorization allows us to relate in a model-independent way the contributions of each BFKL pole to $\gamma^{*}\gamma^{*},\gamma^{*} \gamma$ scattering and DIS off protons. Numerical predictions based on our early works on color dipole BFKL phenomenology of DIS on protons are in a good agreement with the experimental data on the photon structure function $F_{2\gamma}$ and most recent data on the $\gamma^*\gamma^*$ cross section $\sigma^{\gamma^*\gamma^*}(Y)$ from OPAL and L3 experiments at LEP200. We discuss the role of non-perturbative dynamics and predict pronounced effect of the Regge-factorization breaking due to large unfactorizable non-perturbative corrections to the perturbative vacuum exchange. We comment on the salient features of the BFKL-Regge expansion for $\gamma^{*}\gamma^{*},\gamma^{*}\gamma$ scattering including the issue of decoupling of subleading BFKL poles and the soft plus rightmost hard BFKL pole dominance .