The initial phase of the inspiral process of a binary system producing gravitational waves can be described by perturbation theory. At the third post-Minkowskian order a two-loop double box graph, known as H-graph contributes. We consider the case where the two objects making up the binary system have equal masses. We express all master integrals related to the equal-mass H-graph up to weight four in terms of multiple polylogarithms. We provide a numerical program which evaluates all master integrals up to weight four in the physical regions with arbitrary precision.
We consider a two-loop sunrise integral with two different internal masses at pseudo-threshold kinematics and we solve it in terms of elliptic polylogarithms to all orders of the dimensional regulator.
We review recent developments in the study of multiple polylogarithms, including the Hopf algebra of the multiple polylogarithms and the symbol map, as well as the construction of single valued multiple polylogarithms and discuss an algorithm for finding fibration bases. We document how these algorithms are implemented in the Mathematica package PolyLogTools and show how it can be used to study the coproduct structure of polylogarithmic expressions and how to compute iterated parametric integrals over polylogarithmic expressions that show up in Feynman integal computations at low loop orders.
We summarize recent computations with a class of elliptic generalizations of polylogarithms, arising from the massive sunrise integral. For the case of arbitrary masses we obtain results in two and four space-time dimensions. The iterated integral structure of our functions allows us to furthermore compute the equal mass case to arbitrary order.
We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral i…
We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain …
We study one and two-loop triangle integrals with massless propagators and all external legs off shell. We show that there is a kinematic region where the results can be expressed in terms of a basis of single-valued polylogarithms in one complex variable. The relevant space of single-valued functions can be determined a priori and the results take strikingly a simple and compact form when written in terms of this basis. We study the properties of the basis functions and illustrate how one can easily analytically continue our results to all kinematic regions where the external masses have the …
We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form $G(a_1,\ldots,a_w;1)$ where the indices $a_i$ are equal to zero or a sixth root of unity, with $a_1\neq 1$. For $w\leq 6$, we present bases of the linear spaces generated by the real and imaginary parts of $G(a_1,\ldots,a_w;1)$ and present a table for expressing them as linear combinations of the elements of the bases.
We present fully analytic results for all master integrals for the three-loop banana graph with four equal and non-zero masses. The results are remarkably simple and all integrals are expressed as linear combinations of iterated integrals of modular forms of uniform weight for the same congruence subgroup as for the two-loop equal-mass sunrise graph. We also show how to write the results in terms of elliptic polylogarithms evaluated at rational points.
We study several multiscale one-loop five-point families of Feynman integrals. More specifically, we employ the Simplified Differential Equations approach to obtain results in terms of Goncharov polylogarithms of up to transcendental weight four for families with two and three massive external legs and massless propagators, as well as with one massive internal line and up to two massive external legs. This is the first time this computational approach is applied to cases involving internal masses.
We present analytic results for one of two types of planar QED two-loop integrals for $μe$ scattering including finite lepton masses. No approximations are made, such that the results are valid not only in the limit of a small electron mass. The results are expressed in terms of multiple polylogarithms with algebraic function arguments in a representation which allows for fast numerical evaluation in the physical phase-space.
We use the method of differential equations to analytically evaluate all planar three-loop Feynman integrals relevant for form factor calculations involving massive particles. Our results for ninety master integrals at general $q^2$ are expressed in terms of multiple polylogarithms, and results for fiftyone master integrals at the threshold $q^2=4m^2$ are expressed in terms of multiple polylogarithms of argument one, with indices equal to zero or to a sixth root of unity.
The initial phase of the inspiral process of a binary black-hole system can be described by perturbation theory. At the third post-Minkowskian order a two-loop double box graph, known as H-graph, contributes. In this talk we report how all master integrals of the H-graph with equal masses can be expressed up to weight four in terms of multiple polylogarithms. We also discuss techniques for the unequal mass case. The essential complication (and the focus of the talk) is the occurrence of several square roots.
We study the analytic continuation of Feynman integrals from the kite family, expressed in terms of elliptic generalisations of (multiple) polylogarithms. Expressed in this way, the Feynman integrals are functions of two periods of an elliptic curve. We show that all what is required is just the analytic continuation of these two periods. We present an explicit formula for the two periods for all values of $t \in {\mathbb R}$. Furthermore, the nome $q$ of the elliptic curve satisfies over the complete range in $t$ the inequality $|q|\le 1$, where $|q|=1$ is attained only at the singular points…
We present a method to compute the Laurent expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation $\varepsilon$. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.
We present a model for the lepton sector, with $A_4$ horizontal-symmetry group, in which two of the Higgs doublets in an $A_4$ triplet of Higgs doublets have equal vacuum expectation values. The model makes well-defined predictions for the effective light-neutrino Majorana mass matrix. We show that those predictions are compatible with the experimental data.
The dynamical scenario is considered for N=1 SQCD, with N_c colors and N_c<N_F<3N_c flavors with small but nonzero current quark masses m_Q\neq 0, in which quarks form the diquark-condensate phase. This means that colorless chiral quark pairs condense coherently in the vacuum, <{\bar Q}Q>\neq 0, while quarks alone don't condense, <Q>=<\bar Q>=0, so that the color is confined. Such condensation of quarks results in formation of dynamical constituent masses μ_C \gg m_Q of quarks and appearance of light "pions" (similarly to QCD). The mass spectrum of SQCD in this phase is described and compariso…
We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter $\varepsilon$. In order to do so, we transform the system of differential equations for the master integrals to an $\varepsilon$-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space ${\mathcal M}_{1,3}$ of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on $\overline{\mathcal M}_{1,3}$. On the hypersu…
We present the two-loop master integrals relevant to the ${\mathcal O}(αα_s)$-corrections to the decay $H \rightarrow b \bar{b}$ through a $H t \bar{t}$-coupling. We keep the full dependence on the heavy particle masses, but neglect the $b$-quark mass. The occurring square roots can be rationalised and the result is expressed in terms of multiple polylogarithms.
We describe the calculation of all planar master integrals that are needed for the computation of NNLO QCD corrections to the production of two off-shell vector bosons in hadron collisions. The most complicated representatives of integrals in this class are the two-loop four-point functions where two external lines are on the light-cone and two other external lines have different invariant masses. We compute these and other relevant integrals analytically using differential equations in external kinematic variables and express our results in terms of Goncharov polylogarithms. The case of two e…
The $O(α_s^3 T_F^2 C_F (C_A))$ contributions to the transition matrix element $A_{gg,Q}$ relevant for the variable flavor number scheme at 3--loop order are calculated. The corresponding graphs contain two massive fermion lines of equal mass leading to terms given by inverse binomially weighted sums beyond the usual harmonic sums. In $x$-space two root-valued letters contribute in the iterated integrals in addition to those forming the harmonic polylogarithms. We outline technical details needed in the calculation of graphs of this type, which are as well of importance in the case of two diffe…
We propose a mechanism that the soft supersymmetry breaking masses can be induced from the dynamical rearrangement of local U(1) symmetry in a five-dimensional model. The U(1) symmetry possesses several extraordinary features. The eigenstates of U(1) do not equal to those of boundary conditions. The U(1) charge of standard model particles does not equal to that of superpartners. A large U(1) charge hierarchy among superpartners and a standard model gauge singlet is necessary in order to obtain the masses of O(1)TeV.
We integrate three-loop sunrise-type vacuum diagrams in $D_0=4$ dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in $D_0=2$ dimensions we make use of the known closed form $p$-space result to express the moment of…
We present the calculation of all non-planar master integrals that are needed to describe production of two off-shell vector bosons in collisions of two massless partons through NNLO in perturbative QCD. The integrals are computed analytically using differential equations in external kinematic variables and expressed in terms of Goncharov polylogarithms. These results provide the last missing ingredient needed for the computation of two-loop amplitudes that describe the production of two gauge bosons with different invariant masses in hadron collisions.
The self-consistent model of classical field interactions formulated as the counterpart of the quantum electroweak model leads to homogeneous boson ground state solutions in presence of non-zero extended fermionic charge density fluctuations. Two different types of electroweak configurations of fields are analyzed. The first one has non-zero electric and weak charge fluctuations. The second one is electrically uncharged but weakly charged. Both types of configurations have two physically interesting solutions which possess masses equal to 126.67 GeV at the value of the scalar fluctuation poten…
We present the two-loop sunrise integral with arbitrary non-zero masses in two space-time dimensions in terms of elliptic dilogarithms. We find that the structure of the result is as simple and elegant as in the equal mass case, only the arguments of the elliptic dilogarithms are modified. These arguments have a nice geometric interpretation.
The neutrino masses and oscillation are closely related to how neutrino propagates. In this paper, we first derive a new form of four-momentum conservation that connects the four-momentum $(E_α,~p_α)$ of the neutrino flavor eigenstate and the four momenta $(E_i,~p_i)$ of its mass eigen-components. Then we use the assumption that the mass eigen-components travel at equal velocity to derive the energy-momentum square difference (EMSD) $E_α^2-p_α^2$ for the flavor eigenstate. It is shown that for the equal velocity assumption, $E_α^2-p_α^2$ will be a fixed constant in events with different $E_α$ …
We consider the on-shell mass and wave function renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $\sqrt{1-τ^2}$ and $\sqrt{1-τ^2}/τ$ which extends the findings from Ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order ${\cal O}(ε^2)$ and ${\cal O}(ε)$ at two- and three-loop order which are crucial ingrediants for a future four-loop calculation. Compact results for…
