On the reduction of Feynman integrals to master integrals

A. V. Smirnov; V. A. Smirnov

Home/Archive/arXiv

arXiv · 0707.3993

On the reduction of Feynman integrals to master integrals

A. V. Smirnov·V. A. Smirnov

View original source ↗Read source document ↗Export metadata ↓

Abstract

The reduction of Feynman integrals to master integrals is an algebraic problem that requires algorithmic approaches at the modern level of calculations. Straightforward applications of the classical Buchberger algorithm to construct Groebner bases seem to be inefficient. An essential modification designed especially for this problem has been worked out. It has been already applied in two- and three-loop calculations. We are also suggesting to combine our method with the Laporta's algorithm.

Explore related subjects

hep-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

A. V. Smirnov, V. A. Smirnov. 2007-07-26. On the reduction of Feynman integrals to master integrals. https://arxiv.org/abs/0707.3993

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Another relation among the neutrino mass-squared differences?

Determining the absolute neutrino mass scale remains a compelling challenge in particle physics. Establishing precise mathematical connections among the neutrino masses could significantly facilitate this task and shed additional light on the underlying mass-generation mechanism. We highlight the possible existence of a simple algebraic relation between the mass-squared differences, $$\frac{\sqrt{Δm^2_{31}}+\sqrt{Δm^2_{21}}}{\sqrt{Δm^2_{31}}-\sqrt{Δm^2_{21}}}=\sqrt{2},$$ which agrees remarkably well with the best-fit values of recent global analyses of neutrino oscillation data. We then investigate the implications of adopting this relation and how it decreases the number of independent parameters in the neutrino sector. We demonstrate that there are alternative representations well known in quark and charged-lepton physics. Furthermore, we place a tight constraint on the solar neutrino mass splitting, $Δm^2_{21}=(7.445 \pm 0.062) \times 10^{-5} \text{ eV}^2$, and also obtain $Δm^2_{31}=(2.548 \pm 0.041) \times 10^{-3} \text{ eV}^2$. Next-generation reactor and accelerator neutrino experiments are expected to significantly reduce systematic uncertainties. In particular, the Jiangmen Underground Neutrino Observatory (JUNO) will measure these parameters independently of each other at the sub-percent level, thereby providing a stringent and direct test of our results.

hep-ph↗

arXiv2026

Improving the robustness of the $δ_{CP}$ determination with $ν$SCOPE

The determination of leptonic CP violation is a primary goal of future long-baseline neutrino experiments such as DUNE and T2HK. The extraction of $δ_{\mathrm{CP}}$ relies on the near-to-far extrapolation and on the assumed knowledge of the cross-section ratios $σ_{ν_e}/σ_{ν_μ}$ and $σ_{\barν_e}/σ_{\barν_μ}$, which are typically inferred under theoretical assumptions such as lepton universality and depend on nuclear modeling. In this work, we quantify how much of the sensitivity of DUNE and T2HK arises from these assumptions by performing a model-agnostic, data-driven estimation of systematic uncertainties in $ν_e$ and $\barν_e$ cross sections. We find that adopting such an agnostic approach can substantially degrade the CP-violation sensitivity, reducing it by nearly $3σ$ at maximal CP violation for DUNE, and $4σ$ for T2HK. We then assess the impact of the proposed $ν$SCOPE experiment, which, through a combination of neutrino tagging and the Narrow-Band Off-Axis technique, can provide percent-level measurements of $σ_{ν_μ}$ and $σ_{\barν_μ}$ and constrain the ratio $σ_{ν_e}/σ_{ν_μ}$ at the $\sim 2\%$ level. We show that including prospective $ν$SCOPE measurements largely restores the lost sensitivity, highlighting that precise external cross-section measurements may be essential for a fully robust determination of $δ_{\mathrm{CP}}$ and for breaking its degeneracy with nuclear mis-modeling or possible new physics affecting neutrino detection.

hep-ph↗

arXiv2026

$CP$ violation in neutral kaon mixing in $D^0\rightarrow K_SK_S$

We study CP violation induced by neutral kaon mixing in $a_{CP}(D^0 \rightarrow K_S K_S)$. We show that the contribution from neutral kaon mixing arises only in connection with second-order weak interactions in $D$ decays. We estimate this effect to be at the $10^{-6}$ level, and thus negligible compared to current experimental sensitivity and to the expected contribution from CP violation in the charm sector. Similar reasoning can be applied to other systems.

hep-ph↗