Two-loop Diagrams Research Articles

Towards a Loop-Tree Duality at Two Loops and Beyond

We present an extension of the duality theorem, previously defined by S. Catani et al. on the one-loop level, to higher loop orders. The duality theorem provides a relation between loop integrals and tree-level phase-space integrals. Here, the one-loop relation is rederived in a way which is more suitable for its extension to higher loop orders. This is shown in detail by considering the two-loop N-leg master diagram and by a short discussion of the four master diagrams at three loops, in this sketching the general structure of the duality theorem at even higher loop orders.

A comprehensive analysis of electric dipole moment constraints on CP-violating phases in the MSSM

We analyze the constraints placed on individual, flavor diagonal CP-violating phases in the minimal supersymmetric extension of the Standard Model (MSSM) by current experimental bounds on the electric dipole moments (EDMs) of the neutron, Thallium, and Mercury atoms. We identify the four CP-violating phases that are individually highly constrained by current EDM bounds, and we explore how these phases and correlations among them are constrained by current EDM limits. We also analyze the prospective implications of the next generation of EDM experiments. We point out that all other CP-violating phases in the MSSM are not nearly as tightly constrained by limits on the size of EDMs. We emphasize that a rich set of phenomenological consequences is potentially associated with these generically large EDM-allowed phases, ranging from B physics, electroweak baryogenesis, and signals of CP-violation at the CERN Large Hadron Collider and at future linear colliders. Our numerical study takes into account the complete set of contributions from one- and two-loop EDMs of the electron and quarks, one- and two-loop Chromo-EDMs of quarks, the Weinberg 3-gluon operator, and dominant 4-fermion CP-odd operator contributions, including contributions which are both included and not included yet in the CPsuperH2.0 package. We also introduce an open-source numerical package, 2LEDM, which provides the complete set of two-loop electroweak diagrams contributing to the electric dipole moments of leptons and quarks.

Lorentz violation from the Higgs portal

We study bounds and signatures of models where the Higgs doublet has an inhomogeneous mass or vacuum expectation value, being coupled to a hidden sector that breaks Lorentz invariance. This physics is best described by a low-energy effective Lagrangian in which the Higgs speed-of-light is smaller than c; such effect is naturally small because it is suppressed by four powers of the inhomogeneity scale. The Lorentz violation in the Higgs sector is communicated at tree level to fermions (via Yukawa interactions) and to massive gauge bosons, although the most important effect comes from one-loop diagrams for photons and from two-loop diagrams for fermions. We calculate these effects by deriving the renormalization-group equations for the speed-of-light of the Standard Model particles. An interesting feature is that the strong coupling dynamically makes the speed-of-light equal for all colored particles.

Systematics of high temperature perturbation theory: The two-loop electron self-energy in QED

In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy in high temperature QED. The two-loop bubble diagram contains a linear infrared divergence. Even if regulated with a non-zero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading-order (NLO) contributions at one-loop. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge parameter dependence of the fermion damping rate at both one and two-loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard, nested inside a second loop integral which is soft. There are no hard-hard contributions to the two-loop Sigma at leading order at high T. Provided the same factorization pattern holds for arbitrary ell loops, the NLO high temperature contributions to the electron self-energy come from ell-1 hard loops factorized with one soft loop integral. This hard-soft pattern is both a necessary condition for the resummation over ell to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to the self-energy at scales ~eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multi-loop diagrams at high T.

Structure of the two-point green function of the gauge field for N = 1 supersymmetric Yang-Mills theory regularized by higher covariant derivatives

For the most general renormalizable N = 1 supersymmetric Yang-Mills theory, regularized by higher derivatives, we investigate a contribution to the Gell-Mann-Low function from the two-loop diagrams, containing cubic interaction of the matter superfields. We find that all integrals with respect to the loop momentum in the massless case are factorized to the integrals of total derivatives and can be easily calculated. Thus, we explain the origin of the relation between the two-loop β-function and the one-loop anomalous dimension of the matter superfield.

Electric and anomalous magnetic dipole moments of the muon in the MSSM

We study the electric dipole moment (EDM) and the anomalous magnetic dipole moment (MDM) of the muon in the CP-violating Minimal Supersymmetric extension of the Standard Model (MSSM). We take into account the contributions from the chargino- and neutralino-mediated one-loop graphs and the dominant two-loop Higgs-mediated Barr-Zee diagrams. We improve earlier calculations by incorporating CP-violating Higgs-boson mixing effects and the resummed threshold corrections to the Yukawa couplings of the charged leptons as well as that of the bottom quark. The analytic correlation between the muon EDM and MDM is explicitly presented at one- and two-loop levels and, through several numerical examples, we illustrate its dependence on the source of the dominant contributions. We have implemented the analytic expressions for the muon EDM and MDM in an updated version of the public code CPsuperH2.0.

Chiral expansions of theπ0lifetime

The corrections induced by light quark masses to the current algebra result for the $\pi^0$ lifetime are reexamined. We consider NNLO corrections and we compute all the one-loop and the two-loop diagrams which contribute to the decay amplitude at NNLO in the two-flavour chiral expansion. We show that the result is renormalizable, as Weinberg consistency conditions are satisfied. We find that chiral logarithms are present at this order unlike the case at NLO. The result could be used in conjunction with lattice QCD simulations, the feasibility of which was recently demonstrated. We discuss the matching between the two-flavour and the three-flavour chiral expansions in the anomalous sector at order one-loop and derive the relations between the coupling constants. A modified chiral counting is proposed, in which $m_s$ counts as $O(p)$. We have updated the various inputs needed and used this to make a phenomenological prediction.

Two-loop electroweak fermionic corrections to [formula omitted

We present the first calculation of the two-loop electroweak fermionic correction to the flavour-dependent effective weak-mixing angle for bottom quarks, sin 2 θ eff b b ¯ . For the evaluation of the missing two-loop vertex diagrams, two methods are employed, one based on a semi-numerical Bernstein–Tkachov algorithm and the second on asymptotic expansions in the large top-quark mass. A third method based on dispersion relations is used for checking the basic loop integrals. We find that for small Higgs-boson mass values, M H ∝ 100 GeV , the correction is sizable, of order O ( 10 − 4 ) .

TWO-LOOP VACUUM DIAGRAMS IN BACKGROUND FIELD AND THE HEISENBERG–EULER EFFECTIVE ACTION

We show that in arbitrary even dimensions, the two-loop scalar QED Heisenberg–Euler effective action can be reduced to simple one-loop quantities, using just algebraic manipulations, when the constant background field satisfies F2 = -f2𝟙, which in four dimensions coincides with the condition for self-duality, or definite helicity. This result relies on new recursion relations between two-loop and one-loop diagrams, with background field propagators. It also yields an explicit form of the renormalized two-loop effective action in a general constant background field in two dimensions.

Two-loop electroweak next-to-leading logarithms for processes involving heavy quarks

We derive logarithmically enhanced two-loop virtual electroweak corrections for arbitrary fermion-scattering processes at the TeV scale. This extends results previously obtained for massless fermion scattering to processes that involve also bottom and top quarks. The contributions resulting from soft, collinear, and ultraviolet singularities in the complete electroweak Standard Model are explicitly extracted from two-loop diagrams to next-to-leading-logarithmic accuracy including all effects associated with symmetry breaking and Yukawa interactions.

BOKASUN: A fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams

We present the program BOKASUN for fast and precise evaluation of the Master Integrals of the two-loop self-mass sunrise diagram for arbitrary values of the internal masses and the external four-momentum. We use a combination of two methods: a Bernoulli accelerated series expansion and a Runge–Kutta numerical solution of a system of linear differential equations. Program summary Program title: BOKASUN Catalogue identifier: AECG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECG_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 9404 No. of bytes in distributed program, including test data, etc.: 104 123 Distribution format: tar.gz Programming language: FORTRAN77 Computer: Any computer with a Fortran compiler accepting FORTRAN77 standard. Tested on various PC's with LINUX Operating system: LINUX RAM: 120 kbytes Classification: 4.4 Nature of problem: Any integral arising in the evaluation of the two-loop sunrise Feynman diagram can be expressed in terms of a given set of Master Integrals, which should be calculated numerically. The program provides a fast and precise evaluation method of the Master Integrals for arbitrary (but not vanishing) masses and arbitrary value of the external momentum. Solution method: The integrals depend on three internal masses and the external momentum squared p 2 . The method is a combination of an accelerated expansion in 1 / p 2 in its (pretty large!) region of fast convergence and of a Runge–Kutta numerical solution of a system of linear differential equations. Running time: To obtain 4 Master Integrals on PC with 2 GHz processor it takes 3 μs for series expansion with pre-calculated coefficients, 80 μs for series expansion without pre-calculated coefficients, from a few seconds up to a few minutes for Runge–Kutta method (depending on the required accuracy and the values of the physical parameters).

Electric dipole moments in the MSSM reloaded

We present a detailed study of the Thallium, neutron, Mercury and deuteron electric dipole moments (EDMs) in the CP-violating Minimal Supersymmetric extension of the Standard Model (MSSM). We take into account the complete set of one-loop graphs, the dominant Higgs-mediated two-loop diagrams, the complete CP-odd dimension-six Weinberg operator and the Higgs-mediated four-fermion operators. We improve upon earlier calculations by including the resummation effects due to CP-violating Higgs-boson mixing and to threshold corrections to the Yukawa couplings of all up- and down-type quarks and charged leptons. As an application of our study, we analyse the EDM constraints on the CPX, trimixing and Maximally CP- and Minimally Flavour-Violating (MCPMFV) scenarios. Cancellations may occur among the CP-violating contributions to the three measured EDMs arising from the 6 CP-violating phases in the MCPMFV scenario, leaving open the possibility of relatively large contributions to other CP-violating observables. The analytic expressions for the EDMs are implemented in an updated version of the code CPsuperH2.0.

Two-loop supersymmetric corrections to lepton anomalous dipole moments in split supersymmetry scenarios

An analysis of electroweak corrections to the anomalous dipole moments of lepton from some special two-loop diagrams where a closed neutralino/chargino loop is inserted into relevant one-loop diagrams of the standard model is presented in the split supersymmetry scenarios. Considering the translational invariance of the inner loop momenta and the electromagnetic gauge invariance, we get all dimension-6 operators and their coefficients. After applying equations of motion to the external leptons, we obtain the anomalous dipole moments of lepton. The numerical results imply that there is parameter space where the contribution to the muon anomalous MDM from this sector is perhaps significant, and the contribution to the electron electric dipole moment from this sector is sizable enough to be observed in next-generation experiments.

Electroweak and supersymmetric two-loop corrections to lepton anomalous magnetic and electric dipole moments

Using the effective Lagrangian method, we analyze the electroweak corrections to the anomalous dipole moments of lepton from some special two-loop diagrams where a closed neutralino/chargino loop is inserted into relevant two Higgs doublet one-loop diagrams in the minimal supersymmetric extension of the standard model with CP violation. Considering the translational invariance of loop momenta and the electromagnetic gauge invariance, we get all dimension 6 operators and derive their coefficients. After applying equations of motion to the external leptons, we obtain the anomalous dipole moments of lepton. The numerical results imply that there is parameter space where the contributions to the muon anomalous dipole moments from this sector may be significant.

Expansion schemes for gravitational clustering: computing two-point and three-point functions

Aims. We describe various expansion schemes that can be used to study gravitational clustering. Obtained from the equations of motion or their path-integral formulation, they provide several perturbative expansions that are organized in a different fashion or involve different partial resummations. We focus on the two-point and three-point correlation functions, but these methods also apply to all higher-order correlation and response functions.Methods. We present the general formalism, which holds for the gravitational dynamics and for any similar models, such as the Zeldovich dynamics, that obey similar hydrodynamical equations of motion with a quadratic nonlinearity. We give our explicit analytical results up to one-loop order for the simpler Zeldovich dynamics. For the gravitational dynamics, we compare our one-loop numerical results with numerical simulations.Results. We check that the standard perturbation theory is recovered from the path integral by expanding over Feynman's diagrams. However, the latter expansion is organized in a different fashion, and it contains some UV divergences that cancel out as we sum all diagrams of a given order. Resummation schemes modify the scaling of tree and one-loop diagrams, which exhibit the same scaling over the linear power spectrum (contrary to the standard expansion). However, they do not significantly improve over standard perturbation theory for the bispectrum, unless one uses accurate two-point functions (e.g. a fit to the nonlinear power spectrum from simulations). Extending the range of validity to smaller scales, to reach the range described by phenomenological models, seems to require at least two-loop diagrams.

Classical effective field theory and caged black holes

Matched asymptotic expansion is a useful technique in general relativity and other fields whenever interaction takes place between physics at two different length scales. Here matched asymptotic expansion is argued to be equivalent quite generally to classical effective field theory (CLEFT) where one (or more) of the zones is replaced by an effective theory whose terms are organized in order of increasing irrelevancy, as demonstrated by Goldberger and Rothstein in a certain gravitational context. The CLEFT perspective has advantages as the procedure is clearer, it allows a representation via Feynman diagrams, and divergences can be regularized and renormalized in standard field theoretic methods. As a side product we obtain a wide class of classical examples of regularization and renormalization, concepts which are usually associated with quantum field theories. We demonstrate these ideas through the thermodynamics of caged black holes, both simplifying the nonrotating case, and computing the rotating case. In particular we are able to replace the computation of six two-loop diagrams by a single factorizable two-loop diagram, as well as compute certain new three-loop diagrams. The results generalize to arbitrary compactification manifolds. For caged rotating black holes we obtain the leading correction for all thermodynamic quantities. The angular momentum is found to nonrenormalize at leading order.

Anisotropic conductivity of disordered two-dimensional electron gases due to spin-orbit interactions

We show that the disorder-averaged conductivity tensor of a disordered two-dimensional electron gas becomes anisotropic in the presence of both Rashba and Dresselhaus spin-orbit interactions (SOIs). This anisotropy is a mesoscopic effect and vanishes with vanishing charge dephasing time. Using a diagrammatic approach including zero-, one-, and two-loop diagrams, we show that a consistent calculation needs to go beyond a Boltzmann equation approach. In the absence of charge dephasing and for zero frequency, a finite anisotropy ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}{e}^{2}/{p}_{F}lh$ arises even for infinitesimal SOI, where ${p}_{F}$ is the Fermi momentum and $l$ is the mean free path of an electron.

Further results for the two-loopLccvertex in the Landau gauge

In the previous paper hep-th/0604112 we calculated the first of the five planar two-loop diagrams for the Lcc vertex of the general non-Abelian Yang-Mills theory, the vertex which allows us in principle to obtain all other vertices via the Slavnov-Taylor identity. The integrand of this first diagram has a simple Lorentz structure. In this letter we present the result for the second diagram, whose integrand has a complicated Lorentz structure. The calculation is performed in the D-dimensional Euclidean position space. We initially perform one of the two integrations in the position space and then reduce the Lorentz structure to D-dimensional scalar single integrals. Some of the latter are then calculated by the uniqueness method, others by the Gegenbauer polynomial technique. The result is independent of the ultraviolet and the infrared scale. It is expressed in terms of the squares of spacetime intervals between points of the effective fields in the position space -- it includes simple powers of these intervals, as well as logarithms and polylogarithms thereof, with some of the latter appearing within the Davydychev integral J(1,1,1). Concerning the rest of diagrams, we present the result for the contributions correponding to third, fourth and fifth diagrams without giving the details of calculation. The full result for the Lcc correlator of the effective action at the planar two-loop level is written explicitly for maximally supersymmetric Yang-Mills theory.

Calculating massive Two-Loop Two-Point Functions and Operator Matrix Elements with Mellin-Barnes Integrals

We apply Mellin-Barnes integrals and the gluing operation of Feynman diagrams to calculate massive two-loop diagrams with operator insertions, appearing in the context of operator matrix elements in light-cone expansion.

Effective action of β-deformed [formula omitted] SYM theory: Farewell to two-loop BPS diagrams

Within the background field approach, all two-loop sunset vacuum diagrams, which occur in the Coulomb branch of N = 2 superconformal theories (including N = 4 SYM), obey the BPS condition m 3 = m 1 + m 2 , where the masses are generated by the scalars belonging to a background N = 2 vector multiplet. These diagrams can be evaluated exactly, and prove to be homogeneous quadratic functions of the one-loop tadpoles J ( m 1 2 ) , J ( m 2 2 ) and J ( m 3 2 ) , with the coefficients being rational functions of the squared masses. We demonstrate that, if one switches on the β-deformation of the N = 4 SYM theory, the BPS condition no longer holds, and then generic two-loop sunset vacuum diagrams with three non-vanishing masses prove to be characterized by the following property: 2 ( m 1 2 m 2 2 + m 1 2 m 3 2 + m 2 2 m 3 2 ) > m 1 4 + m 2 4 + m 3 4 . In the literature, there exist several techniques to compute such diagrams. For the β-deformed N = 4 SYM theory, we carry out explicit two-loop calculations of the Kähler potential and F 4 term. Our considerations are restricted to the case of β real.