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G. Falcioni

Publications and source records attributed to G. Falcioni.

15 recordsLinked to original sources
arXiv2025

Additional results on the four-loop flavour-singlet splitting functions in QCD

We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $\gamma_{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P_{\rm ik}^{\,(3)}(x)$ to $N = 22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\rm ik}^{\,(3)}(x)$ that we obtained before from the moments $N \leq 20 $ and endpoint constraints, confirming their reliability for collider-physics applications. Due to the additional analytical constraints provided by our new $N = 22$ results, we are now closing in on determining the all-$N$ forms of all non-rational ($\zeta$-function) contributions to $\gamma_{\rm ik}^{\,(3)}(N)$: only the $n_f^{0} \zeta_3$ parts of the quark-to-gluon (gq) and the $n_f^{1} \zeta_3$ parts of the gluon-to-gluon (qg) cases still need to be completed. Finally we extend our approximations for all $P_{\rm ik}^{\,(3)}(x)$ to $n_f^{} = 6$ light flavours and update those for $P_{\rm gq}^{\,(3)}(x)$ at lower $n_f^{}$.

hep-ph↗
arXiv2025

Alien operators for PDF evolution

Understanding the scale dependence of parton distribution functions is vital for precision physics at hadron colliders. The well-known DGLAP evolution equation relates this scale dependence to the QCD splitting functions, which can be calculated perturbatively in terms of the anomalous dimensions of leading-twist gauge-invariant operators. The computation of the latter in general requires one to take into account contributions of gauge-variant (or alien) operators. In this talk, we discuss the systematic study of these alien operators at arbitrary spin. Specifically, using generalized BRST symmetry relations, we derive the one-loop couplings and Feynman rules of the aliens necessary to perform the operator renormalization up to four loops in QCD. This provides an important step towards the determination of the four-loop splitting functions which will be of significant phenomenological importance at future colliders.

hep-ph↗
arXiv2024

Four-loop splitting functions in QCD -- The gluon-gluon case --

We have computed the even-N moments N =< 20 of the gluon-gluon splitting function P_{gg} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general compact simple gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via structure functions in deep-inelastic scattering. Using our new moments and all available endpoint constraints, we construct improved approximations for the four-loop P_{gg}(x) that should be sufficient for a wide range of collider-physics applications. The N^3LO contributions to the scale derivative of the gluon distribution, resulting from these and the corresponding quark-to-gluon splitting functions, amount to 1% or less at x >~ 10^{-4} at a standard reference scale with alpha_s = 0.2.

hep-ph↗
arXiv2024

Constraints for twist-two alien operators in QCD

Parton evolution equations in QCD are controlled by the anomalous dimensions of gauge-invariant twist-two spin-$N$ quark and gluon operators. Under renormalization, these mix with gauge-variant operators of the same quantum numbers, referred to as alien operators. Our work addresses the systematic study of these alien operators at arbitrary spin $N$, using generalized BRST symmetry relations to derive their couplings and Feynman rules at all values of $N$. We observe how the all-$N$ structure of the generalized (anti-)BRST constraints relates the couplings of alien operators with $n+1$ gluons to those with $n$ gluons. Realizing a bootstrap, we present all one-loop results necessary for performing the operator renormalization up to four loops in QCD.

hep-ph↗
arXiv2024

Four-loop splitting functions in QCD -- The quark-to-gluon case

We present the even-N moments N =< 20 of the fourth-order (N^3LO) contribution P_{gq}^(3)(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N =< 10 derived before from structure functions in deep-inelastic scattering. Using the new moments and the available endpoint constraints, we construct approximations for P_{gq}^(3)(x) which improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions x > 0.1. The resulting errors of the convolution of P_{gq} at N^3LO with a typical quark distribution are small at x >~ 10^{-3} and exceed 1% only at x ~< 10^{-4} for a strong coupling alpha_s = 0.2. The present results for P_{gq}^(3)(x) should thus be sufficient for most collider-physics applications.

hep-ph↗
arXiv2024

Four-loop splitting functions in QCD -- The gluon-to-quark case

We have computed the even-$N$ moments $N \leq 20$ of the gluon-to-quark splitting function $P_{\rm qg}$ at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via physical cross sections. Using our new moments and all available endpoint constraints, we construct approximations for the four-loop $P_{\rm qg}(x)$ that should be sufficient for a wide range of collider-physics applications. The N$^3$LO corrections resulting from these and the corresponding quark-quark splitting functions lead to a marked improvement of the perturbative accuracy for the scale derivative of the singlet quark distribution, with effects of 1% or less at $x \gtrsim 10^{\,-4}$ at a standard reference scale with $α_s = 0.2$.

hep-ph↗
arXiv2023

The double fermionic contribution to the four-loop quark-to-gluon splitting function

We have computed the first 30 even-N moments for the double fermionic (nf^2) part of the quark-to-gluon splitting function P_{gq} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. From these results we have determined the all-N form, and hence the exact x-space expression, using systems of Diophantine equations for its coefficients. The dominant and subdominant leading small-x nf^2 contributions to P_{gq}^(3)(x) are of the form 1/x ln x and ln^4 x, respectively; the leading large-x term is ln^4 (1-x). The coefficient of the first of these is new, the other two agree with results obtained before and thus provide checks of our results.

hep-ph↗
arXiv2023

Four-loop splitting functions in QCD -- The quark-quark case

We have computed the even-$N$ moments $N\leq 20$ of the pure-singlet quark splitting function $P_{\,\rm ps}$ at the fourth order of perturbative QCD via the anomalous dimensions of off-shell flavour-singlet operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest six even moments obtained via physical cross sections. Using these results and all available endpoint constraints, we construct approximations for $P_{\rm ps}$ at four loops that should be sufficient for most collider-physics applications. Together with the known results for the non-singlet splitting function $P_{\rm ns}^{\,+}$ at this order, this effectively completes the quark-quark contribution for the evolution of parton distribution at N$^{\:\!3}$LO accuracy. Our new results thus provide a major step towards fully consistent N$^{\:\!3}$LO calculations at the LHC and the reduction of the residual uncertainty in the parton evolution to the percent level.

hep-ph↗
arXiv2018

Five-loop renormalisation of QCD in covariant gauges

We present the complete set of vertex, wave function and charge renormalisation constants in QCD in a general simple gauge group and with the complete dependence on the covariant gauge parameter $ξ$ in the minimal subtraction scheme of conventional dimensional regularisation. Our results confirm all already known results, which were obtained in the Feynman gauge, and allow the extraction of other useful gauges such as the Landau gauge. We use these results to extract the Landau gauge five-loop anomalous dimensions of the composite operator $A^2$ as well as the Landau gauge scheme independent gluon, ghost and fermion propagators at five loops.

hep-ph↗
arXiv2018

The method of global R* and its applications

The global R* operation is a powerful method for computing renormalisation group functions. This technique, based on the principle of infrared rearrangement, allows to express all the ultraviolet counterterms in terms of massless propagator integrals. In this talk we present the main features of global R* and its application to the renormalisation of QCD. By combining this approach with the use of the program Forcer for the evaluation of the relevant Feynman integrals, we renormalise for the first time QCD at five loops in covariant gauges.

hep-ph↗
arXiv2017

The Heavy Quark Form Factors at Two Loops

We compute the two-loop QCD corrections to the heavy quark form factors in case of the vector, axial-vector, scalar and pseudo-scalar currents up to second order in the dimensional parameter $ε= (4-D)/2$. These terms are required in the renormalization of the higher order corrections to these form factors.

hep-ph↗
arXiv2017

Heavy quark form factors at two loops in perturbative QCD

We present the results for heavy quark form factors at two-loop order in perturbative QCD for different currents, namely vector, axial-vector, scalar and pseudo-scalar currents, up to second order in the dimensional regularization parameter. We outline the necessary computational details, ultraviolet renormalization and corresponding universal infrared structure.

hep-ph↗
arXiv2016

The Asymptotic 3-Loop Heavy Flavor Corrections to the Charged Current Structure Functions $F_L^{W^+-W^-}(x,Q^2)$ and $F_2^{W^+-W^-}(x,Q^2)$

We calculate the massive Wilson coefficients for the heavy flavor contributions to the non-singlet charged current deep-inelastic scattering structure functions $F_L^{W^+}(x,Q^2)-F_L^{W^-}(x,Q^2)$ and $F_2^{W^+}(x,Q^2)-F_2^{W^-}(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in Quantum Chromodynamics (QCD) at general values of the Mellin variable $N$ and the momentum fraction $x$. Besides the heavy quark pair production, also the single heavy flavor excitation $s \rightarrow c$ contributes. Numerical results are presented for the charm quark contributions and consequences on the unpolarized Bjorken sum rule and Adler sum rule are discussed.

hep-ph↗
arXiv2011

Unitarity alternatives in the reduced-action model for gravitational collapse

Based on the ACV approach to transplanckian energies, the reduced-action model for the gravitational S-matrix predicts a critical impact parameter b_c ~ R = 2 G sqrt{s} such that S-matrix unitarity is satisfied in the perturbative region b > b_c, while it is exponentially suppressed with respect to s in the region b < b_c that we think corresponds to gravitational collapse. Here we definitely confirm this statement by a detailed analysis of both the critical region b ~ b_c and of further possible contributions due to quantum transitions for b < b_c. We point out, however, that the subcritical unitarity suppression is basically due to the boundary condition which insures that the solutions of the model be ultraviolet-safe. As an alternative, relaxing such condition leads to solutions which carry short-distance singularities presumably regularized by the string. We suggest that through such solutions - depending on the detailed dynamics at the string scale - the lost probability may be recovered.

hep-th↗