Electroweak Splitting Functions and High Energy Showering

We derive the electroweak (EW) collinear splitting functions for the Standard Model, including the massive fermions, gauge bosons and the Higgs boson. We first present the splitting functions in the limit of unbroken SU(2)xU(1) and discuss their general features in the collinear and soft-collinear regimes. We then systematically incorporate EW symmetry breaking (EWSB), which leads to the emergence of additional "ultra-collinear" splitting phenomena and naive violations of the Goldstone-boson Equivalence Theorem. We suggest a particularly convenient choice of non-covariant gauge (dubbed "Goldstone Equivalence Gauge") that disentangles the effects of Goldstone bosons and gauge fields in the presence of EWSB, and allows trivial book-keeping of leading power corrections in the VEV. We implement a comprehensive, practical EW showering scheme based on these splitting functions using a Sudakov evolution formalism. Novel features in the implementation include a complete accounting of ultra-collinear effects, matching between shower and decay, kinematic back-reaction corrections in multi-stage showers, and mixed-state evolution of neutral bosons (gamma/Z/h) using density-matrices. We employ the EW showering formalism to study a number of important physical processes at O(1-10 TeV) energies. They include (a) electroweak partons in the initial state as the basis for vector-boson-fusion; (b) the emergence of "weak jets" such as those initiated by transverse gauge bosons, with individual splitting probabilities as large as O(30%); (c) EW showers initiated by top quarks, including Higgs bosons in the final state; (d) the occurrence of O(1) interference effects within EW showers involving the neutral bosons; and (e) EW corrections to new physics processes, as illustrated by production of a heavy vector boson (W') and the subsequent showering of its decay products.Comment: 67 pages, 12 figures; v2, published in JHEP, some expanded discussions and other minor revision

Light Dark Matter Showering under Broken Dark U(1)U(1) -- Revisited

It was proposed recently that different chiralities of the dark matter (DM) fermion under a broken dark U(1) gauge group can lead to distinguishable signatures at the LHC through shower patterns, which may reveal the mass origin of the dark sector. We study this subject further by examining the dark shower of two simplified models, the dubbed Chiral Model and the Vector Model. We derive a more complete set of collinear splitting functions with power corrections, specifying the helicities of the initial DM fermion and including the contribution from an extra degree of freedom, the dark Higgs boson. The dark shower is then implemented with these splitting functions, and the new features resulting from its correct modelling are emphasized. It is shown that the DM fermion chirality can be differentiated by measuring dark shower patterns, especially the DM jet energy profile, which is almost independent of the DM energy.Comment: 18 pages, 6 figure

Electroweak Splitting Functions and High Energy Showering

We derive the electroweak (EW) collinear splitting functions for the Standard Model, including the massive fermions, gauge bosons and the Higgs boson. We first present the splitting functions in the limit of unbroken SU(2) × U(1) and discuss their general features in the collinear and soft-collinear regimes. These are the leading contributions at a splitting scale (kT) far above the EW scale (v). We then systematically incorporate EW symmetry breaking (EWSB), which leads to the emergence of additional “ultra-collinear” splitting phenomena and naive violations of the Goldstone-boson Equivalence Theorem. We suggest a particularly convenient choice of non-covariant gauge (dubbed “Goldstone Equivalence Gauge”) that disentangles the effects of Goldstone bosons and gauge fields in the presence of EWSB, and allows trivial book-keeping of leading power corrections in v/kT. We implement a comprehensive, practical EW showering scheme based on these splitting functions using a Sudakov evolution formalism. Novel features in the implementation include a complete accounting of ultra-collinear effects, matching between shower and decay, kinematic back-reaction corrections in multi-stage showers, and mixed-state evolution of neutral bosons (gamma/Z/h) using density-matrices. We employ the EW showering formalism to study a number of important physical processes at O(1-10 TeV) energies. They include (a) electroweak partons in the initial state as the basis for vector-boson-fusion; (b) the emergence of "weak jets" such as those initiated by transverse gauge bosons, with individual splitting probabilities as large as O(35%); (c) EW showers initiated by top quarks, including Higgs bosons in the final state; (d) the occurrence of O(1) interference effects within EW showers involving the neutral bosons; and (e) EW corrections to new physics processes, as illustrated by production of a heavy vector boson (W') and the subsequent showering of its decay products

Helicity amplitudes without gauge cancellation for electroweak processes

We introduce 5-component representation of weak bosons, W and Z bosons of the standard model. The first four components make a Lorentz four vector, representing the transverse and longitudinal polarizations excluding the scalar component of the weak bosons. Its fifth component corresponds to the Goldstone boson. We show that this description can be extended to off-shell weak bosons, with the $5\times5$ component propagators, and prove that exactly the same scattering amplitudes are obtained by making use of the BRST (Becchi-Rouet-Stora-Tyutin) identities among two sub-amplitudes connected by one off-shell weak boson line in the unitary gauge. By replacing all weak boson vertices with those among the 5-component wavefunctions, we arrive at the expression of the electroweak scattering amplitudes, where the magnitude of each Feynman amplitude has the correct on-shell limits for all internal propagators, and hence with no artificial gauge cancellation among diagrams. We implement the 5-component weak boson propagators and their vertices in the numerical helicity amplitude calculation code HELAS (Helicity Amplitude Subroutines), so that an automatic amplitude generation program such as MadGraph can generate the scattering amplitudes without gauge cancellation. We present results for several high-energy scattering processes where subtle gauge-theory cancellation among diagrams takes place in all the other known approaches.Comment: 32 pages, 14 figures, 9 tables; v2: references adde

Helicity amplitudes in light-cone and Feynman-diagram gauges

Recently proposed Feynman-diagram (FD) gauge propagator for massless and massive gauge bosons is obtained from a light-cone (LC) gauge propagator, by choosing the gauge vector along the opposite direction of the gauge boson three-momentum. We implement a general LC gauge propagator for all the gauge bosons of the Standard Model (SM) in the HELicity Amplitude Subroutines (HELAS) codes, such that all the SM helicity amplitudes can be evaluated at the tree level in the LC gauge by using MadGraph. We confirm that our numerical codes produce physical helicity amplitudes which are consistent among all gauge choices. We then study interference patterns among Feynman amplitudes, for a few $2\to3$ scattering processes in QED and QCD, and the process $\gamma\gamma\to W^+W^-$ followed by the $W^\pm$ decays. We find that in a generic LC gauge, where all the gauge boson propagators share a common gauge vector, we cannot remove the off-shell current components which grow with their energy systematically from all the Feynman amplitudes in $2\to3$ processes. On the other hand, the $5\times5$ LC gauge propagator for the weak bosons removes components which grow with energy due to the longitudinal polarization mode of the external bi-fermion currents, and hence can give $2\to2$ weak boson scattering amplitudes which are free from subtle cancellation at high energies. The particular choice of the FD gauge vector has advantages over generic LC gauge, not only because all the terms which grow with energy of off-shell and on-shell currents are removed systematically from all the diagrams, but also because no artificial gauge vector direction dependence of individual amplitudes appears.Comment: 20 pages, 14 figures; references adde