Feynman Parameter Integrals Research Articles

Towards analytic structure of Feynman parameter integrals with rational curves

We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum of discontinuities induced from the integral, together with a geometric method for computing their singularities on the principal sheet. For integrals that yield multiple polylogarithms we expect the data collected in this strategy to be sufficient for the construction of their symbols. We motivate this analysis by the Aomoto polylogarithms, and further check its validity and illustrate technical details using examples with quadric integrand singularities (which the one-loop Feynman integrals belong to). Generalizations to higher-loop integrals are commented at the end.

A Method of Fast Calculaion of Lepton Magnetic Moments in Quantum Electrodynamics

A new method of divergence subtraction in Feynman parametric integrals is presented. The method is suitable for calculating the lepton anomalous magnetic moments (AMM) in quantum electrodynamics (QED). The subtraction procedure eliminates all divergences before integration and leads to a finite Feynman parametric integral for each individual Feynman diagram. It is based on a forest formula with linear operators applied to the Feynman amplitudes of ultraviolet-divergent subdiagrams. The formula is similar to BPHZ; the difference is only in the linear operators used and in the way of combining them. The subtraction is equivalent to the on-shell renormalization from the beginning: for obtaining the final result we should only sum up the contributions of all Feynman diagrams after subtraction. The developed method is an improvement of the method presented by the author in 2016. The modification is specifically designed for calculating the contributions dependent on the relations of particle masses. In comparison with the old version, the new subtraction formula does not contain redundant terms and possesses some flexibility that can be used for improving the precision of calculations. Numerical test results are presented up to four loops.

Chromomagnetic and chromoelectric dipole moments of quarks in the reduced 331 model

The one-loop contributions to the chromomagnetic dipole moment and electric dipole moment of the top quark are calculated within the reduced 331 model (RM331) for non-zero . It is argued that the results are gauge independent and thus represent valid observable quantities. In the RM331, receives new contributions from two heavy gauge bosons, namely and , and one neutral scalar boson , along with a new contribution from the standard model's Higgs boson via flavor changing neutral currents. The latter, which is also mediated by the gauge boson and the scalar boson , can provide a non-vanishing if there is a -violating phase. The analytical results are presented in terms of both Feynman parameter integrals and Passarino-Veltman scalar functions, which are useful to cross-check the numerical results. Both and are numerically evaluated for parameter values still allowed by the constraints from experimental data. It is found that the new one-loop contributions of the RM331 to the real (imaginary) part of are of the order of ( ), which means at least three orders of magnitude smaller than the standard model prediction but larger than the predictions of other models of new physics. In the RM331, the dominant contribution arises from the gauge boson for in the 30-1000 GeV interval and a mass of the order of a few hundreds of GeV. As for , it receives its largest contribution from exchange and can reach values of the order of , i.e., smaller than the contributions predicted by other standard model extensions.

New estimate of the chromomagnetic dipole moment of quarks in the standard model

A new estimate of the one loop contributions of the standard model to the chromomagnetic dipole moment (CMDM) $\hat \mu_q(q^2)$ of quarks is presented with the aim to address a few disagreements arising in previous calculations. We consider the most general case with an off-shell gluon with transfer momentum $q^2$ and obtain analytical results in terms of Feynman parameter integrals and Passarino-Veltman scalar functions, which are then expressed in terms of closed form functions when possible. The calculation is done via a renormalizable linear $R_\xi$ gauge and the background field method, which allows one to corroborate that the resulting $\hat \mu_q(q^2)$ is gauge independent and thus a valid observable quantity. It is found that the QCD contribution from a three-gluon Feynman diagram has an infrared divergence, which agrees with a previous evaluation and stems from the fact that the static CMDM [$\hat\mu(0)$] has no sense in perturbative QCD. For the numerical analysis we consider the region 30 GeV$<\|q\|<$ 1000 GeV and analyze the behavior of $\hat \mu_q(q^2)$ for all the standard model quarks. It is found that the CMDM of light quarks is considerably smaller than that of the top quark as it is directly proportional to the quark mass. In the considered energy interval, both the real and imaginary parts of $\hat\mu_t(q^2)$ are of the order of $10^{-2}-10^{-3}$, with the largest contribution arising from the QCD induced diagrams, though around the threshold $q^2=4m_t^2$ there are also important contributions from diagrams with $Z$ gauge boson and Higgs boson exchange.

Three point energy correlators in the collinear limit: symmetries, dualities and analytic results

Energy Correlators measure the energy deposited in multiple detectors as a function of the angles between the detectors. In this paper, we analytically compute the three particle correlator in the collinear limit in QCD for quark and gluon jets, and also in mathcal{N} = 4 super Yang-Mills theory. We find an intriguing duality between the integrals for the energy correlators and infrared finite Feynman parameter integrals, which maps the angles of the correlators to dual momentum variables. In mathcal{N} = 4, we use this duality to express our result as a rational sum of simple Feynman integrals (triangles and boxes). In QCD our result is expressed as a sum of the same transcendental functions, but with more complicated rational functions of cross ratio variables as coefficients. Our results represent the first analytic calculation of a three-prong jet substructure observable of phenomenological relevance for the LHC, revealing unexplored simplicity in the energy flow of QCD jets. They also provide valuable data for improving the understanding of the light-ray operator product expansion.

Manifestly dual-conformal loop integration

Local, manifestly dual-conformally invariant loop integrands are now known for all finite quantities associated with observables in planar, maximally supersymmetric Yang-Mills theory through three loops. These representations, however, are not infrared-finite term by term and therefore require regularization; and even using a regulator consistent with dual-conformal invariance, ordinary methods of loop integration would naïvely obscure this symmetry. In this work, we show how any planar loop integral through at least two loops can be systematically regulated and evaluated directly in terms of strictly finite, manifestly dual-conformal Feynman-parameter integrals. We apply these methods to the case of the two-loop ratio and remainder functions for six particles, reproducing the known results in terms of individually regulated local loop integrals, and we comment on some of the novelties that arise for this regularization scheme not previously seen at one loop.

Chromomagnetic and chromoelectric dipole moments of the top quark in the fourth-generation THDM

The chromomagnetic dipole moment (CMDM) and chromoelectric dipole moment (CEDM) of the top quark are calculated at the one-loop level in the framework of the two-Higgs doublet model with four fermion generations (4GTHDM), which is still consistent with experimental data and apart from new scalar bosons (${H}^{0}$, ${A}^{0}$, and ${H}^{\ifmmode\pm\else\textpm\fi{}}$) and quarks (${b}^{\ensuremath{'}}$ and ${t}^{\ensuremath{'}}$) predicts new sources of $CP$ violation via the extended $4\ifmmode\times\else\texttimes\fi{}4$ CKM matrix. Analytical expressions for the CMDM and CEDM of a quark are presented both in terms of Feynman parameter integrals, which are explicitly integrated, and Passarino-Veltman scalar functions, with the main contributions arising from loops carrying the scalar bosons accompanied by the third- and fourth-generation quarks. The current bounds on the parameter space of the 4GTHDM are discussed and a region still consistent with the LHC data on the 125 GeV Higgs boson and the oblique parameters is identified. It is found that the top quark CMDM, which is induced by all the scalar bosons, can reach values of the order of ${10}^{\ensuremath{-}2}--{10}^{\ensuremath{-}1}$. As for the top quark CEDM, it only receives contributions from the charged scalar boson and can reach values of the order of ${10}^{\ensuremath{-}20}--{10}^{\ensuremath{-}19}\text{ }\text{ }\mathrm{ecm}$ for relatively light ${m}_{{H}^{\ifmmode\pm\else\textpm\fi{}}}$ and heavy ${m}_{{b}^{\ensuremath{'}}}$, with the dominant contribution arising from the $b$ quark. The CEDM would be the most interesting prediction of this model as it can be larger than the value predicted by the usual THDMs by one order of magnitude.

A quasi-finite basis for multi-loop Feynman integrals

We present a new method for the decomposition of multi-loop Euclidean Feynman integrals into quasi-finite Feynman integrals. These are defined in shifted dimensions with higher powers of the propagators, make explicit both infrared and ultraviolet divergences, and allow for an immediate and trivial expansion in the parameter of dimensional regularization. Our approach avoids the introduction of spurious structures and thereby leaves integrals particularly accessible to direct analytical integration techniques. Alternatively, the resulting convergent Feynman parameter integrals may be evaluated numerically. Our approach is guided by previous work by the second author but overcomes practical limitations of the original procedure by employing integration by parts reduction.

TENTH-ORDER QED CONTRIBUTION TO THE ELECTRON g-2 AND HIGH PRECISION TEST OF QUANTUM ELECTRODYNAMICS

This paper presents the current status of the theory of electron anomalous magnetic moment ae ≡(g-2)/2, including a complete evaluation of 12,672 Feynman diagrams in the tenth-order perturbation theory. To solve this problem, we developed a code-generator which converts Feynman diagrams automatically into fully renormalized Feynman-parametric integrals. They are evaluated numerically by an integration routine VEGAS. The preliminary result obtained thus far is 9.16 (58) (α/π)5, where (58) denotes the uncertainty in the last two digits. This leads to ae( theory ) = 1.159 652 181 78 (77) ×10-3, which is in agreement with the latest measurement ae ( exp :2008) = 1.159 652 180 73 (28) ×10-3. It shows that the Feynman–Dyson method of perturbative QED works up to the precision of 10-12.

Minimal Ward—Takahashi Vertices and Light Cone Pion Distribution Amplitudes from the GND Quark Model

The gauge-invariant, nonlocal, dynamical quark model is shown to generate minimal vertices that satisfy the Ward—Takahashi identities and the flat-like form of the light-cone pion distribution amplitudes. Non-flat form amplitudes can be produced only if we take a finite momentum cutoff and include nonzero pion mass corrections or go beyond the minimal vertices. A by-product of our investigation shows that the variable u appearing in light-cone pion distribution amplitudes is just the standard Feynman parameter in the Feynman parameter integrals.

Scalar vertex operator for bound-state QED in the Coulomb gauge

Adkins's result [Phys. Rev. D 34, 2489 (1986)] for the time component of the renormalized vertex operator in Coulomb-gauge QED is separated according to its tensor structure and some of the Feynman parameter integrals are carried out analytically, yielding a form suited for numerical bound-state QED calculations. This modified form is applied to the evaluation of the self-energy shift to the binding energy in hydrogenic ions of high nuclear charge.

Algorithms for special functions

In this thesis we present our contributions to symbolic summation, extending WZFasenmyer methods to handle definite hypergeometric sums with nonstandard boundary conditions and to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms. We also include concrete applications of these methods to Feynman integral calculus, as well as for proving identities involving definite integrals and special functions. First we give a short introduction to WZ-summation methods, including K. Wegschaider’s approach to this method and its implementation in the package MultiSum. Inspired by work of Sister Celine Fasenmyer, these techniques were introduced by H. Wilf and D. Zeilberger to algorithmically compute recurrences for multiple sums over hypergeometric terms. Their procedure is based on finding a certificate recurrence satisfied by the hypergeometric summand and summing over this difference equation to obtain a recurrence for the nested sum. Our proofs of two nontrivial special function identities involving Gegenbauer polynomials, provide classic applications of the method. As part of the collaboration between RISC and DESY coordinated by C. Schneider, we developed an algorithmic approach to compute Feynman parameter integrals after rewriting them as multisums over hypergeometric terms to fit the input class of classic summation algorithms. Since these definite sums have nonstandard boundary conditions, the WZ-method delivers inhomogeneous recurrence relations. We designed a recursive procedure to determine the inhomogeneous parts of these recurrences and implemented it in the Mathematica package FSums which builds on the already existing packages, MultiSum and C. Schneider’s Sigma. Another approach to evaluate Feynman integrals is by representating them in terms of nested Mellin-Barnes integrals. These complex contour integrals can also be viewed as sums of residues at certain poles of the integrands and they are connected to the inversion formula for the Mellin transform. In the last part, we show how WZ-methods can be used to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms, eliminating the need to search for sum representations. We applied this new algorithmic technique to prove typical entries from the Gradshteyn-Ryzhik table of integrals using the Mellin transform and to find recurrences for a class of Ising integrals.

Muon anomaly from lepton vacuum polarization and the Mellin-Barnes representation

We evaluate, analytically, a specific class of eighth--order and tenth--order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons $l=e,\mu$, and $\tau$. The results are given in the form of analytic expansions in terms of the mass ratios $m_e/m_\mu$ and $m_\mu/m_\tau$. We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin--Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.

SECTOR DECOMPOSITION

Sector decomposition is a constructive method to isolate divergences from parameter integrals occurring in perturbative quantum field theory. We explain the general algorithm in detail and review its application to multiloop Feynman parameter integrals as well as infrared divergent phase-space integrals over real radiation matrix elements.

QCD corrections to triboson production

We present a computation of the next-to-leading order QCD corrections to the production of three $Z$ bosons at the Large Hadron Collider. We calculate these corrections using a completely numerical method that combines sector decomposition to extract infrared singularities with contour deformation of the Feynman parameter integrals to avoid internal loop thresholds. The NLO QCD corrections to $pp\ensuremath{\rightarrow}ZZZ$ are approximately 50% and are badly underestimated by the leading order scale dependence. However, the kinematic dependence of the corrections is minimal in phase space regions accessible at leading order.

Next-to-leading-order corrections to theγ*impact factor: First numerical results for the real corrections toγL*

We analytically perform the transverse momentum integrations in the real corrections to the longitudinal \gamma*_L impact factor. The resulting integrals are Feynman parameter integrals, and we provide a MATHEMATICA file which contains the integrands. The remaining integrals are carried out numerically. We perform a numerical test, and we compute those parts of the impact factor which depend upon the energy scale s_0: they are found to be negative and, with decreasing values of s_0, their absolute value increases.

Numerical determination of Feynman-parametric QED integrals

Some fourth-order Feynman-parametric QED integrals are calculated numerically and their exact values inferred. Newton-Cotes formulas are used to compute the integrals, using an increasing number of intervals. This yields a series of values from which an often quite accurate result is found via repeated extrapolations.

EXTRACTION OF INFRARED DIVERGENCES IN THE DIMENSIONAL REGULARIZED TWO-LOOP LADDER GRAPH

We consider the evaluation of the fundamental scalar integral in the on-shell two-loop ladder graph with different external masses and arbitrary transfer momentum. A method for cleanly extracting the infrared divergences in the Feynman parameter integrals using dimensional regularization is presented, and we analyze one of the finite part contributions to this integral.

Dimensionally-regulated pentagon integrals

We present methods for evaluating the Feynman parameter integrals associated with the pentagon diagram in 4-2∈ dimensions, along with explicit results for the integrals with all masses vanishing or with one non-vanishing external mass. The scalar pentagon integral can be expressed as a linear combination of box integrals, up to O(∈) corrections, a result which is the dimensionally-regulated version of a D = 4 result of Melrose, and of van Neerven and Vermaseren. We obtain and solve differential equations for various dimensionally-regulated box integrals with massless internal lines, which appear in one-loop n-point calculations in QCD. We give a procedure for constructing the tensor pentagon integrals needed in gauge theory, again through O(∈ 0).

Two-loop ladder-diagram contributions to Bhabha scattering (I). Structure of the matrix elements

We discuss contributions to Bhabha scattering from the gauge-invariant set of six t-channel, photon-exchange, two-loop, ladder-like diagrams. The contributions from these diagrams to the unpolarized cross section can be expressed in terms of four scalar functions. These functions are given in terms of Feynman-parameter integrals in the limit s ⪢ | t | ⪢ m 2.