On-shell Form Factor Research Articles

Gluonic evanescent operators: negative-norm states and complex anomalous dimensions

In this paper, we build on our previous work to further investigate the role of evanescent operators in gauge theories, with a particular focus on their contribution to violations of unitarity. We develop an efficient method for calculating the norms of gauge-invariant operators in Yang-Mills (YM) theory by employing on-shell form factors. Our analysis, applicable to general spacetime dimensions, reveals the existence of negative-norm states among evanescent operators. We also explore the one-loop anomalous dimensions of these operators and find complex anomalous dimensions. We broaden our analysis by considering YM theory coupled with scalar fields and we observe similar patterns of non-unitarity. The presence of negative-norm states and complex anomalous dimensions across these analyses provides compelling evidence that general gauge theories are non-unitary in non-integer spacetime dimensions.

The four loop QCD rapidity anomalous dimension

The rapidity anomalous dimension controls the scaling of transverse momentum dependent observables in the Sudakov region. In a conformal theory it is equivalent to the soft anomalous dimension, but in QCD this relation is broken by anomalous terms proportional to the β-function. In this paper we first give a simple proof of this relation using two different representations of the energy-energy correlator observable. We then calculate the anomalous terms to three loops by computing the three-loop fully differential soft function to \U0001d4aa(ϵ). Combined with recent perturbative data from the study of on-shell form factors and splitting functions, this allows us to derive the four loop rapidity anomalous dimension in QCD.

Two-loop anomalous dimensions of QCD operators up to dimension-sixteen and Higgs EFT amplitudes

We consider two-loop renormalization of high-dimensional Lorentz scalar operators in the gluonic sector of QCD. These operators appear also in the Higgs effective theory obtained by integrating out the top quark loop in the gluon fusion process. We first discuss the classification of operators and how to construct a good set of basis using both off-shell field theory method and on-shell form factor formalism. To study loop corrections, we apply efficient unitarity-IBP strategy and compute the two-loop minimal form factors of length-3 operators up to dimension sixteen. From the UV divergences of form factor results, we extract the renormalization matrices and analyze the operator mixing behavior in detail. The form factors we compute are also equivalent to Higgs plus three-gluon amplitudes that capture high-order top mass corrections in Higgs EFT. We obtain the analytic finite remainder functions which exhibit several universal transcendentality structures.

Konishi form factor at three loops in N=4 supersymmetric Yang-Mills theory

We present the first results on the third order corrections to on-shell form factor (FF) of the Konishi operator in ${\cal N}=4$ supersymmetric Yang-Mills theory using Feynman diagrammatic approach in modified dimensional reduction ($\overline {DR}$) scheme. We show that it satisfies KG equation in $\overline {DR}$ scheme while the result obtained in four dimensional helicity (FDH) scheme needs to be suitably modified not only to satisfy the KG equation but also to get the correct ultraviolet (UV) anomalous dimensions. We find that the cusp, soft and collinear anomalous dimensions obtained to third order are same as those of the FF of the half-BPS operator confirming the universality of the infrared (IR) structures of on-shell form factors. In addition, the highest transcendental terms of the FF of Konishi operator are identical to those of half-BPS operator indicating the probable existence of deeper structure of the on-shell FF. We also confirm the UV anomalous dimensions of Konishi operator up to third order providing a consistency check on the both UV and universal IR structures in ${\cal N}=4$.

Towards a data-driven analysis of hadronic light-by-light scattering

The hadronic light-by-light contribution to the anomalous magnetic moment of the muon was recently analyzed in the framework of dispersion theory, providing a systematic formalism where all input quantities are expressed in terms of on-shell form factors and scattering amplitudes that are in principle accessible in experiment. We briefly review the main ideas behind this framework and discuss the various experimental ingredients needed for the evaluation of one- and two-pion intermediate states. In particular, we identify processes that in the absence of data for doubly-virtual pion–photon interactions can help constrain parameters in the dispersive reconstruction of the relevant input quantities, the pion transition form factor and the helicity partial waves for γ⁎γ⁎→ππ.

Dispersive approach to hadronic light-by-light scattering

Based on dispersion theory, we present a formalism for a model-independent evaluation of the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. In particular, we comment on the definition of the pion pole in this framework and provide a master formula that relates the effect from pi pi intermediate states to the partial waves for the process gamma^* gamma^* --> pi pi. All contributions are expressed in terms of on-shell form factors and scattering amplitudes, and as such amenable to an experimental determination.

One loop standard model corrections to flavor diagonal fermion-graviton vertices

We extend a previous analysis of flavor-changing fermion-graviton vertices, by adding the one-loop SM corrections to the flavor diagonal fermion-graviton interactions. Explicit analytical expressions taking into account fermion masses for the on-shell form factors are computed and presented. The infrared safety of the fermion-graviton vertices against radiative corrections of soft photons and gluons is proved, by extending the ordinary infrared cancellation mechanism between real and virtual emissions to the gravity case. These results can be easily generalized to fermion couplings with massive gravitons, graviscalar, and dilaton fields, with potential phenomenological implications to new physics scenarios with low gravity scale.

Two-photon exchange in a dispersion approach

We calculate the two-photon exchange amplitude for the elastic electron-proton scattering in the framework of dispersion relations. The imaginary part of the amplitude is determined by unitarity. Since in the unitarity relation intermediate states are on shell, off-shell form factors are not needed for the calculation. The real part is then evaluated using analytical properties of the amplitude. The expression for the elastic contribution to the amplitude, obtained in our approach, differs from the results of traditional calculations with on-shell form factors. Nevertheless, numerically the difference is minor for ${Q}^{2}$ up to 6 GeV${}^{2}$.

Sudakov resummations at higher orders

We summarize our recent results on the resummation of hard-scattering coefficient functions and on-shell form factors in massless perturbative QCD. The threshold resummation has been extended to the fourth logarithmic order for deep-inelastic scattering, Drell-Yan lepton pair production and Higgs production via gluon-gluon fusion. The leading six infrared pole terms have been derived to all orders in the strong coupling constant for the photon-quark-quark and the (heavy-top) Higgs-gluon-gluon form factors. These results have many implications, most notably they lead to a new best estimate for the Higgs production cross section at the LHC.

The quark form factor at higher orders

We study the electromagnetic on-shell form factor of quarks in massless perturbative QCD. We derive the complete pole part in dimensional regularization at three loops, and extend the resummation of the form factor to the next-to-next-to-leading contributions. These results are employed to evaluate the infrared finite absolute ratio of the time-like and space-like form factors up to the fourth order in the strong coupling constant. Besides for the pole structure of higher-loop QCD amplitudes, our new contributions to the form factor are also relevant for the high-energy limit of massive gauge theories like QED. The highest-transcendentality component of our results confirms a result recently obtained in N=4 Super-Yang-Mills theory.

Contrasting real-time dynamics with screening phenomena at finite temperature.

We discuss the interpretation of Euclidean correlation functions at finite temperature (T) and their relationship with the corresponding real-time Green's functions. The soluble (2+1)-dimensional Gross-Neveu model in the large-N limit is used throughout as a working example. First, the real-time bound state, identified as an elementary excitation at finite T, is solved. The bound state mass, the dispersion relation at low momenta, the coupling constant, and decay constant are calculated. To characterize the structure of the bound state the on-shell form factor is carefully introduced and calculated. Then we examine the corresponding screening state and contrast the screening mass, coupling constant, decay constant, and the screening Bethe-Salpeter amplitude with the real-time quantities. We find that, although they can be used as qualitative indicators in the low-T regime, the screening states at finite T in general do not reflect the properties of the corresponding real-time bound states. In addition, other relevant issues, such as the subtlety of the real-time manifestation of conservation laws due to some internal symmetries at T\ensuremath{\ne}0, the temperature dependence of the pseudoscalar spectral function and its sum rule, and the high-T limit of the screening state and its implications to the dimensional reduction, are also discussed in detail. \textcopyright{} 1996 The American Physical Society.

Lattice calculation of the branching ratio for some of the exclusive modes of b-->s gamma.

We calculate the leading-order matrix element for exclusive decays of $b \to s\gamma$ in the quenched approximation of lattice QCD on a $24^3\times48$ lattice at $\beta=6.2$, using an O(a)-improved fermion action. The matrix element is used to extract the on-shell form factor $T_1(q^2=0)$ for $B \to K^*\gamma$ and $B_s \to \phi\gamma$, using two different assumptions about the $q^2$ dependence of the form factors for these decays. For $B \to K^*\gamma$, $T_1(q^2=0)$ is determined to be $0.159^{+34}_{-33}\pm 0.067$ or $0.124^{+20}_{-18} \pm 0.022$ in the two cases. We find the results to be consistent (in the Standard Model) with the CLEO experimental branching ratio of $BR(B\to K^*\gamma) = (4.5 \pm 1.5 \pm 0.9) \times 10^{-5}$.

Two-loop calculations with vertex corrections in the Walecka model.

Two-loop corrections with scalar and vector form factors are calculated for nuclear matter in the Walecka model. The on-shell form factors are derived from vertex corrections within the framework of the model and are highly damped at large spacelike momenta. The two-loop corrections are evaluated first by using the one-loop parameters and mean fields and then by refitting the total energy/baryon to empirical nuclear matter saturation properties. The modified two-loop corrections are significantly smaller than those computed with bare vertices. Contributions from the anomalous isoscalar form factor of the nucleon are included for the first time. The effects of the implicit density dependence of the form factors, which arise from the shift in the baryon mass, are also considered. Finally, necessary extensions of these calculations are discussed.

Microscopic model of the timelike electromagnetic form factor of the nucleon.

A microscopic model of the electromagnetic form factor of the nucleon is developed in a hadronic framework, including pions, nucleons and the \ensuremath{\Delta}-resonance explicitly. The spacelike on-shell form factors are reproduced and predictions for the half off-shell dependence are made. The impact of this off-shell dependence in the time like sector (${\mathit{q}}^{2}$1 ${\mathrm{GeV}}^{2}$, thus including the region of vector meson dominance) is of main interest in this investigation.

Lattice calculation of the penguin diagram decay B-->K* gamma.

We calculate the leading-order matrix element for the decay $B \to K^* \gamma$ in the quenched approximation of lattice QCD on a $24^3 \times 48$ lattice at $\beta=6.2$, using an O(a)-improved fermion action. Extrapolating the quark masses to their physical values we obtain an on-shell form factor of $T_1(q^2=0)=0.15+12-14$, where the errors quoted are purely statistical. We find $T_1$ is approximately independent of the spectator quark mass and extract $T_1(q^2=0)=0.15+5-4$ if this independence is assumed. We compare this with the same form factor derived (in the Standard Model) from the CLEO experimental branching ratio of $BR(B \to K^* \gamma) = (4.5 \pm 1.5 \pm 0.9) \times 10^{-5}$ and find the results to be consistent within statistical errors.

Dimensional regularization of mass and infrared singularities in two-loop on-shell vertex functions

We show how massless two-loop on-shell vertex functions can be calculated in a very elegant way if mass and infrared singularities are regularized by n-dimensional regularization. Using dispersion methods one is able to express the Feynman integrals in a product of gamma functions. As an application of our techniques we have calculated a two-loop on-shell form factor. Its infrared behaviour will be compared with the predictions of some resummation formulae which have been derived in the literature.

Nuclear structure effects in high momentum transfer reactions

It is shown that, at large momentum transfers, one can extract nuclear structure information from the response (function) of the nucleus to an external scalar probe by factorizing the on-shell form factor associated to the struck nucleon. This result, derived both for elastic and inelastic scattering, arises from compensations between off-shell effects and exchange current effects generated by the (local) nucleon-nucleon interaction. The corrections to this on-shell factorization are in general found to be small at large momentum transfers. The role of the final state interaction in the nuclear transition form factor is then investigated and it is shown how one can, for instance, correct the orthogonality defect introduced by using a plane wave approximation for the struck nucleon.

Evidence for almost point-like behavior of off-shell particles in field theory

Evidence is given in favor of the possibility that, in contrast to the rapid decrease of on-shell form factors, the off-shell form factors for both elementary and composite hadrons decrease as ( q 2) −1 when the invariant momentum transfer q 2 becomes asymptotically large. Experimental implications are discussed.