One-loop Triangle Research Articles

The process gg→h0Z∗ in the inverted hierarchy scenario of the 2HDM type-I at the LHC

While searching at the Large Hadron Collider (LHC) for the production and decay of the CP-odd scalar (A0) in the 2-Higgs-Doublet Model (2HDM) with Natural Flavour Conservation (NFC) via the channels gg→A0 (through one-loop triangle diagrams) and A0→h0Z∗ (with mh0=125 GeV or mh0<125 GeV, with Z off-shell), respectively, a factorisation of the two processes is normally performed, with the A0 state being on-shell. While this approach is gauge-invariant, it is not capturing the presence of either of the following two channels: gg→Z∗→h0Z∗ (through one-loop triangle diagrams) or gg→h0Z∗ (through one-loop box diagrams). As the resolution of the A0 mass cannot be infinitely precise, we affirm that all such contributions should be computed simultaneously, whichever the h0(Z∗) decay(splitting) products, thereby including all possible interferences amongst themselves. The cross section of the ensuing complete process is significantly different from that obtained in the factorisation case, being of the order up to ten percent in either direction at the integrated level and larger (including changes in the shape of kinematical observables) at the differential level. We thus suggest that the complete calculation ought to be performed while searching for A0 in this channel. We illustrate this need for the case of a 2HDM of Type-I in the inverted hierarchy scenario with mh0<125 GeV.

Light-front dynamic analysis of the longitudinal charge density using the solvable scalar field model in ( 1+1 ) dimensions

We investigate the electromagnetic form factor $F(q^2)$ of the meson by using the solvable $\phi^{3}$ scalar field model in $(1+1)$ dimensions. As the transverse rotations are absent in $(1+1)$ dimensions, the advantage of the light-front dynamics (LFD) with the light-front time $x^+ = x^0 + x^3$ as the evolution parameter is maximized in contrast to the usual instant form dynamics (IFD) with the ordinary time $x^0$ as the evolution parameter. In LFD, the individual $x^+$-ordered amplitudes contributing to $F(q^2)$ are invariant under the boost, i.e., frame-independent, while the individual $x^0$-ordered amplitudes in IFD are not invariant under the boost but dependent on the reference frame. The LFD allows to get the analytic result for the one-loop triangle diagram which covers not only the spacelike ($q^{2}<0$) but also timelike region ($q^{2}>0$). Using the analytic results, we verify that the real and imaginary parts of the form factor satisfy the dispersion relations in the entire $q^{2}$ space. Comparing with the results in $(3+1)$ dimensions, we discuss the transverse momentum effects on $F(q^2)$ . We also discuss the longitudinal charge density in terms of the boost invariant variable $\tilde z = p^+ x^-$ in LFD.

Three-body effective potential in general relativity at second post-Minkowskian order and resulting post-Newtonian contributions

We study the post-Minkowskian (PM) and post-Newtonian (PN) expansions of the gravitational three-body effective potential. At order 2PM a formal result is given in terms of a differential operator acting on the maximal generalized cut of the one-loop triangle integral. We compute the integral in all kinematic regions and show that the leading terms in the PN expansion are reproduced. We then perform the PN expansion unambiguously at the level of the integrand. Finding agreement with the 2PN three-body potential after integration, we explicitly present new ${G}^{2}{v}^{4}$-contributions at order 3PN and outline the generalization to ${G}^{2}{v}^{2n}$. The integrals that represent the essential input for these results are obtained by applying the recent Yangian bootstrap directly to their $\ensuremath{\epsilon}$-expansion around three dimensions. The coordinate space Yangian generator that we employ to obtain these integrals can be understood as a special conformal symmetry in a dual momentum space.

GKZ-hypergeometric systems for Feynman integrals

Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller's transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.

The hypergeometric system for one-loop triangle integral

We derive the holonomic hypergeometric system for the [Formula: see text] function with two equal virtual masses, and present the expression of [Formula: see text] in hypergeometric series in corresponding convergent region. Combining the Horn’s convergence theory with Gröbner basis of polynomial ideal, one can calculate the convergence region of the corresponding multiple series concretely. Using the system given here, one can analytically continue [Formula: see text] to whole parameter space.

T -odd correlations in polarized top quark decays in the sequential decay t(↑)→Xb+W+(→ℓ++νℓ) and in the quasi-three-body decay t(↑)→Xb+ℓ++νℓ

We identify the $T$-odd structure functions that appear in the description of polarized top quark decays in the sequential decay $t(\ensuremath{\uparrow})\ensuremath{\rightarrow}{X}_{b}+{W}^{+}(\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}+{\ensuremath{\nu}}_{\ensuremath{\ell}})$ (two structure functions) and the quasi-three-body decay $t(\ensuremath{\uparrow})\ensuremath{\rightarrow}{X}_{b}+{\ensuremath{\ell}}^{+}+{\ensuremath{\nu}}_{\ensuremath{\ell}}$ (one structure function). A convenient measure of the magnitude of the $T$-odd structure functions is the contribution of the imaginary part $\mathrm{Im}{g}_{R}$ of the right-chiral tensor coupling ${g}_{R}$ to the $T$-odd structure functions which we work out. Contrary to the case of QCD, the NLO electroweak corrections to polarized top quark decays admit absorptive one-loop vertex contributions. We analytically calculate the imaginary parts of the relevant four electroweak one-loop triangle vertex diagrams and determine their contributions to the $T$-odd helicity structure functions that appear in the description of polarized top quark decays.

Cuts and coproducts of massive triangle diagrams

Relations between multiple unitarity cuts and coproducts of Feynman inte-grals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to co-products through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.

Trace anomaly and massless scalar degrees of freedom in gravity

The trace anomaly of quantum fields in electromagnetic or gravitational backgrounds implies the existence of massless scalar poles in physical amplitudes involving the stress-energy tensor. Considering first the axial anomaly and using QED as an example, we compute the full one-loop triangle amplitude of the fermionic stress tensor with two current vertices, <T^{mn} J^a J^b>, and exhibit the scalar pole in this amplitude associated with the trace anomaly, in the limit of zero electron mass m -> 0. To emphasize the infrared aspect of the anomaly, we use a dispersive approach and show that this amplitude and the existence of the massless scalar pole is determined completely by its ultraviolet finite terms, together with the requirements of Poincare invariance of the vacuum, Bose symmetry under interchange of J^a and J^b, and vector current and stress tensor conservation. We derive a sum rule for the appropriate positive spectral function corresponding to the discontinuity of the triangle amplitude, showing that it becomes proportional to delta function of k^2, and therefore contains a massless scalar intermediate state in the conformal limit of zero electron mass. The effective action corresponding to the trace of the triangle amplitude can be expressed in local form by the introduction of two scalar auxiliary fields which satisfy massless wave equations. These massless scalar degrees of freedom couple to classical sources, contribute to gravitational scattering processes, and can have long range gravitational effects.

Massless and massive one-loop three-point functions in negative dimensional approach

In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive with the same mass m and three equal masses for the virtual particles. Our results are given in terms of hypergeometric and hypergeometric-type functions of the external momenta (and masses for the massive cases) where the propagators in the Feynman integrals are raised to arbitrary exponents and the dimension of the space-time is D. Our approach reproduces the known results; it produces other solutions as yet unknown in the literature as well. These new solutions occur naturally in the context of NDIM revealing a promising technique to solve Feynman integrals in quantum field theories.

The UV and IR origin of non-Abelian chiral gauge anomalies on noncommutative Minkowski spacetime

We discuss both the UV and IR origins of the one-loop triangle gauge anomalies for noncommutative non-Abelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only originate from planar triangle diagrams, the nonplanar triangle contributions giving rise to no breaking of the Ward identities. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free.

R-current correlators in N = 4 super-Yang-Mills theory from anti-de Sitter supergravity

We test the conjectured relationship between N = 4 super-Yang-Mills theory in four dimensions and IIB supergravity compactified on AdS 5 × S 5 by computing the two- and three-point functions of R-symmetry currents. We observe that the integral expressions describing the general three-point correlator on the supergravity side have a structure similar to one-loop triangle diagrams in N = 4 super-Yang-Mills theory. This allows us to compare the expressions on both sides of the AdS/CFT correspondence without the technical complications of the loop integrations. We confirm that the two- and three-point correspondence arises at only one-loop in the N = 4 super-Yang-Mills theory. Higher-point functions as well as further three-point functions may be analyzed similarly.

Coordinate and Kähler σ-model anomalies and their cancellation in string effective field theories

We discuss the complete set of one-loop triangle graphs involving the Yang-Mills gauge connection, the Kähler connection and the σ-model coordinate connection in the effective field theory of (2, 2) symmetric Z N orbifolds. That is, we discuss pure gauge, pure Kähler and pure σ-model coordinate anomalies as well as the mixed anomalies, such as Kähler-gauge, some of which have been discussed elsewhere. We propose a mechanism for restoring both Kähler and σ-model coordinate symmetry based upon the introduction of two types of counterterms. Finally, we enlarge the σ-model generalization of the Green-Schwarz mechanism to allow the removal of the universal parts of a wider class of anomalies than those previously discussed.

Breakdown of dimensional regularization in the Sudakov problem

An explicit example is presented (a one-loop triangle graph) where dimensional regularization fails to regulate the infra-red singularities that emerge at intermediate steps of studying large- Q 2 Sudakov factorization. The mathematical nature of the phenomenon is explained within the framework of the theory of the As-operation.