We describe an application of generalised unitarity to the computation of one-loop amplitudes with massive external fermions. We present analytic results for the cut-constructible parts of the leading colour contributions to the all-plus helicity configuration of the $t\bar{t}gg$ amplitude. Using a special choice for the helicity basis of the massive fermions we are able to obtain extremely compact analytic expressions. In particular we describe how one can fix the divergent contributions from tadpole and wave-function renormalisation using universal UV and IR behaviour.
We present a novel set of Feynman rules and generalised unitarity cut-conditions for computing one-loop amplitudes via d-dimensional integrand reduction algorithm. Our algorithm is suited for analytic as well as numerical result, because all ingredients turn out to have a four-dimensional representation. We will apply this formalism to NLO QCD corrections.
In this letter, we exploit generalised unitarity in order to calculate the cut-constructible part of one-loop amplitudes in non-supersymmetric Yang-Mills theory. In particular, we rederive the n-gluon MHV amplitudes for both the adjacent and non-adjacent gluon helicity configurations from 3- and 4-particle cuts alone.
We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions.
We study the upper bound on the scale of fermion mass generation in a two-Higgs-doublet model. If the model is weakly-coupled, the scale of fermion mass generation is much less than the Appelquist-Chanowitz unitarity bound. However, if we allow some dimensionless Higgs self-couplings to become large, the Appelquist-Chanowitz unitarity bound can be saturated. The unitarity bound on the scale of top-quark mass generation is about 3 TeV, which may be within the reach of future colliders.
We analyze the scattering of fermions, Higgs and electroweak gauge bosons in order to obtain the partial-wave unitarity bounds on dimension-six effective operators, including those involving fermions. We also quantify whether, at the LHC energies, the dimension-six operators lead to unitarity violation after taking into account the presently available constraints on their Wilson coefficients. Our results show that for most dimension-six operators relevant for the LHC physics there is no unitarity violation at the LHC energies, and consequently, there is no need for the introduction of form fac…
Two-particle unitarity-cuts of scattering amplitudes can be efficiently computed by applying Stokes' Theorem, in the fashion of the Generalised Cauchy Theorem. Consequently, the Optical Theorem can be related to the Berry Phase, showing how the imaginary part of arbitrary one-loop Feynman amplitudes can be interpreted as the flux of a complex 2-form.
We review the Four-Dimensional-Formulation variant of the Four-Dimensional-Helicity scheme, by showing two applications of this regularisation scheme. The first one is the computation of one-loop helicity amplitudes, for which we present preliminary results for the analytic expressions of the one-loop Higgs plus five- gluon amplitudes. In the second part, we study the Colour-Kinematics duality for off-shell diagrams in gauge theories coupled to matter, showing in a diagrammatic way that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial g…
Problems related to fermions, unitarity and chiral anomaly in high energy electroweak interactions, are investigated. Particular attention is paid to the correct functional integration over fermion fields in the background of instanton-anti\-instanton type configurations. This leads to an expansion of correlation functions in terms of a small parameter, $ρ/R$, when the instanton-antiinstanton separation ($R$) is large compared to their sizes ($ρ$). Applying such a method to widely discussed cases of fermion-number violation in the electroweak theory, we conclude that there are no theoretical b…
We extend the generalized D-dimensional unitarity method for numerical evaluation of one-loop amplitudes by incorporating massive particles. The issues related to extending the spinor algebra to higher dimensions, treatment of external self-energy diagrams and mass renormalization are discussed within the context of the D-dimensional unitarity method. To validate our approach, we calculate in QCD the one-loop scattering amplitudes of a massive quark pair with up to three additional gluons for arbitrary spin states of the external quarks and gluons.
Some years ago Appelquist and Chanowitz considered the scattering of fermion--anti-fermion into a pair of longitudinal gauge bosons. Their calculation established that unitarity implies that the physics giving mass to a quark of mass $m_f$ must be below a mass scale of $16 πv^2 / m_f$ ($v=246$ GeV). This bound is a bit difficult to interpret, because the unitarity of gauge boson scattering requires in any case that there be new physics, such as a Higgs boson, with a mass lighter than this. This paper re-examines the Appelquist-Chanowitz bound in order to clarify its meaning. This work uses toy…
Perturbative unitarity is a powerful tool for inferring the range of validity of a given effective field theory. Here, we study such a bound in the parameter space of dimension-5 and dimension-6 effective operators that arise in a scenario of fermion compositeness. These operators are routinely used in experimental searches at the LHC to constraint contact and gauge interactions between ordinary Standard Model fermions and excited states of mass $M$. We derive the unitarity bound for the production process of an excited neutrino, then we implement such bound and compare it with the recent expe…
If a new massive vector boson with nonzero axial couplings to fermions will be observed at LHC, then an upper limit on the scale of new physics could be derived from unitarity of $\mathcal{S}$-matrix. The new physics will involve either new massive fermions, or scalars, or even a strongly coupled sector. We derive a model independent bound on the scale of new physics. If $M_{G}/ g_{A} < 3$ TeV and the fermion is a top quark, the upper limit is 78 TeV.
We explore the possibility of a beyond the standard model scalar ($ϕ$) as a possible explanation of the diphoton resonance at 750~GeV invariant mass reported by the ATLAS and CMS collaborations at the large hadron collider (LHC). We first present in a model-independent way the scalar-gluon-gluon and scalar-photon-photon effective couplings needed for obtaining the required diphoton cross-section at the LHC for different total widths. We investigate here two new-physics possibilities that can generate these effective couplings, namely, (i) the 2-Higgs-doublet model (2HDM) in the alignment limit…
We study a simple extension of the Standard Model that accounts for neutrino masses and dark matter. The Standard Model is augmented by two Higgs doublets and one Dirac singlet fermion, all charged under a new dark global symmetry. It is a generalised version of the Scotogenic Model with Dirac fermion dark matter. Masses for two neutrinos are generated radiatively at one-loop level. We study the case where the singlet fermion constitutes the dark matter of the Universe. We study in depth the phenomenology of the model, in particular the complementarity between dark matter direct detection and …
We discuss the consistency of fermion mass generation by boundary conditions and brane localized terms in higher dimensional Higgsless models of gauge symmetry breaking. The sum rules imposed by tree-level unitarity and Ward Identities are applied to check the consistency of mass generation by orbifold projections and more general boundary conditions consistent with the variational principle. We find that the sum rules are satisfied for boundary conditions corresponding to brane localized mass and kinetic terms consistent with the reduced gauge symmetry on the brane.
We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in $d$ dimensions. We then show that divergent wavefunction cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in $4-2ε$ dimensions and ultraviolet poles in $6-2ε$ dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-lev…
We extract the shear viscosity to entropy density ratio η/s of cold fermionic atoms in the unitarity limit from experimental data on the damping of collective excitations. We find that near the critical temperature η/s is roughly equal to 1/2 in units of \hbar/k_B. With the possible exception of the quark gluon plasma, this value is closer to the conjectured lower bound 1/(4π) than any other known liquid.
We present an extension of the spinor integration formalism of one loop amplitudes from the double-cut to the single-cut case. This technique can be applied for the computation of the tadpole coefficients. Moreover we describe an off-shell continuation of one loop amplitudes that allows a finite evaluation of the unitarity cuts in the channel of a single massive external fermion.
We report the "state of the art" of the problem of $B+L$ violation in high-energy electroweak scatterings. Results of various analyses point toward (though do not prove rigorously yet) the "half-suppression", i.e., that the $B+L$ violating cross section remains suppressed at least by the negative exponent of the single instanton action, at all energies. Most interesting techniques developed in this field are reviewed. Particular attention is paid to unitarity constraints on the anomalous cross section, and to some conceptual problem involving the use of the optical theorem in the presence of i…
We consider various constraints on Higgsless models of electroweak symmetry breaking based on a bulk SU(2)_L x SU(2)_R x U(1)_{B-L} gauge group in warped space. First we show that the S parameter which is positive if fermions are localized on the Planck brane can be lowered (or made vanishing) by changing the localization of the light fermions. If the wave function of the light fermions is almost flat their coupling to the gauge boson KK modes will be close to vanishing, and therefore contributions to the S parameter will be suppressed. At the same time the experimental bounds on such Z' and W…
By employing the dictum that axiomatic principles are devoid of predictive power, we find that the elastic unitarity constraint, applied to strong W$_L$W$_L$ scattering, does not alter the assumed spectrum of intermediate states. We consider intermediate states involving a heavy Higgs and heavy fermions of a hypothetical fourth generation doublet. In contrast to recent studies, we find no p-wave resonance, and therefore no violation of the S parameter upper bound. We conclude that the elastic unitarity constraint sheds no light on the existence of a heavy fourth generation.
It shows that because of the axial-vector coupling and the coupling with fermions of different masses the untarity of the SM is broken at the scale of 10^{14}GeV. A new theory of EW interactions without Higgs is proposed. We obtain M^2_W=g^2 m^2_t/2, M^2_Z=(g^2+g'^2)m^2_t/2, and G_F=1/(2\sqrt{2}m^2_t}.
We investigate decay modes of spin-1 heavy vector bosons ($V'$) from the viewpoint of perturbative unitarity in a model-independent manner. Perturbative unitarity requires some relations among couplings. The relations are called unitarity sum rules. We derive the unitarity sum rules from processes that contain two fermions and two gauge bosons. We find the relations between $V'$ couplings to the SM fermions $(f)$ and $V'$ couplings to the SM gauge bosons ($V$). Using the coupling relations, we calculate partial decay widths for $V'$ decays into $VV$ and $ff$. We show that Br($W' \to WZ) \lesss…
In this paper, we consider the low-energy scale inverse seesaw mechanism in which the observed neutrino mass and lepton mixing are explained by introducing right handed neutrinos and the gauge-singlet fermions with experimentally testable energy scale. Moreover, the presence of such new fermions leads to unitarity violation in lepton mixing due to significantly large mixing between active neutrinos and the heavy fermions. In addition to this, such large lepton mixing also gives rise to potentially large lepton flavor violation, which allows to constrain the non-unitarity parameters via lepton …
Recently, Jaeger and Willenbrock have shown that the Appelquist and Chanowitz bound on the scale of top-quark mass generation can formally be saturated in a particular limit of a two-Higgs doublet model. In this note I present an alternate derivation of their result. I perform a coupled channel analysis for f fbar to V_L V_L and V_L V_L to V_L V_L scattering and derive the conditions on the parameters required for f fbar to V_L V_L scattering to be relevant to unitarity. I also show that it is not possible to saturate the bound on fermion mass generation in the two-Higgs model while maintainin…
We present compact analytic expressions for all one-loop helicity amplitudes contributing to ttbar production at hadron colliders. Using recently developed generalised unitarity methods and a traditional Feynman based approach we produce a fast and flexible implementation.
The scale of fermion mass generation can, as shown by Appelquist and Chanowitz, be bounded from above by relating it to the scale of unitarity violation in the helicity nonconserving amplitude for fermion-anti-fermion pairs to scatter into pairs of longitudinally polarized electroweak gauge bosons. In this paper, we examine the process t tbar -> W_L W_L in a family of phenomenologically-viable deconstructed Higgsless models and we show that scale of unitarity violation depends on the mass of the additional vector-like fermion states that occur in these theories (the states that are the deconst…
We use unitarity and analyticity of scattering amplitudes to constrain fermionic operators in the standard model effective field theory. For four-fermion operators at mass dimension 8, we scatter flavor superpositions in fixed standard model representations and find the Wilson coefficients to be constrained so that their contraction with any pair of pure density matrices is positive. These constraints imply that flavor-violating couplings are upper-bounded by their flavor-conserving cousins. For instance, LEP data already appears to preclude certain operators in upcoming $μ\to 3e$ measurements…
We combine the six-dimensional helicity formalism of Cheung and O'Connell with D-dimensional generalized unitarity to obtain a new formalism for computing one-loop amplitudes in dimensionally regularized QCD. With this procedure, we simultaneously obtain the pieces that are constructible from four-dimensional unitarity cuts and the rational pieces that are missed by them, while retaining a helicity formalism. We illustrate the procedure using four- and five-point one-loop amplitudes in QCD, including examples with external fermions. We also demonstrate the technique's effectiveness in next-to-…
We present a numerical implementation for virtual corrections to multi-jet production at Next-to-Leading order. Using the algorithm of generalised unitarity we compute primitive amplitudes from tree-level input. These basic ingredients are then used to compute full colour and helicity summed corrections.
A generalized two-brane Randall-Sundrum warped braneworld model admits of solutions of the warp factor for both positive or negative cosmological constant on the visible 3-brane which can resolve the naturalness problem in connection with the fine tuning of Higgs mass in the standard model of elementary particles. To explore the location of the standard model fermions in such a generalized warped model, we, in this work, determine the dependence of the localization profile of a bulk fermion on the brane cosmological constant brane tension and the bulk fermion mass. Our results reveal that for …
The main challenge faced by Higgsless models of electroweak symmetry breaking is to reconcile the experimental constraints imposed by the precision electroweak data and the top quark phenomenology with the unitarity constraints imposed by longitudinal gauge boson scattering amplitudes. In this paper we expand on previous work, giving details of how delocalized fermions can be used to adjust the $S$ parameter to zero, while keeping the $T$ and $U$ parameters naturally suppressed. We also show that it is possible to obtain the top quark mass, without affecting the delay of unitarity violation of…
We study perturbative unitarity constraints on general W' models by considering the high energy behavior of fermion scattering into gauge bosons. In most cases we survey, a Z' boson with a comparable mass must be present for the theory to be consistent, with fixed couplings to the standard model gauge bosons and fermions. Applying these results to a class of W' models which explains the top quark forward-backward asymmetry observed at the Tevatron, we find that a Z' must exist with a mass below 7-8 TeV and sizable coupling to the light quarks. While such a Z' is strongly constrained by existin…
We study a model of N-component complex fermions with a kinetic term that is second order in derivatives. This symplectic fermion model has an Sp(2N) symmetry, which for any N contains an SO(3) subgroup that can be identified with rotational spin of spin-1/2 particles. Since the spin-1/2 representation is not promoted to a representation of the Lorentz group, the model is not fully Lorentz invariant, although it has a relativistic dispersion relation. The hamiltonian is pseudo-hermitian, H^\dagger = C H C, which implies it has a unitary time evolution. Renormalization-group analysis shows the …
We derive cutting rules for loop integrals containing propagators with complex masses. Using a field-theoretical model of a heavy vector boson interacting with a light fermion, we demonstrate that the complex-mass scheme respects unitarity order by order in a perturbative expansion provided that the renormalized coupling constant remains real.
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form $\mathrm{Tr}\, F^n$ where $F$ is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a gen…
``Three-mass triangles'' are a class of integral functions appearing in one-loop gauge theory amplitudes. We discuss how the complex analytic properties and singularity structures of these amplitudes can be combined with generalised unitarity techniques to produce compact expressions for three-mass triangle coefficients. We present formulae for the N=1 contributions to the n-point NMHV amplitude.
We present a study of the effective action approach to incorporate higher-order effects in e^+e^- -> n fermions. In its minimal version, the effective action approach is found to exhibit problems with unitarity and high-energy behaviour. We identify the origin of these problems by investigating the zero-mode solutions of the Ward Identities. A numerical analysis of the importance of the zero-mode solutions is presented for four-fermion production processes.
We compute a complete set of independent leading-color two-loop five-parton amplitudes in QCD. These constitute a fundamental ingredient for the next-to-next-to-leading order QCD corrections to three-jet production at hadron colliders. We show how to consistently consider helicity amplitudes with external fermions in dimensional regularization, allowing the application of a numerical variant of the unitarity approach. Amplitudes are computed by exploiting a decomposition of the integrand into master and surface terms that is independent of the parton type. Master integral coefficients are nume…
We study the loop corrections to the scalar propagator and the fermionic self-energy for the mass dimension one fermionic dark matter with the Yukawa interaction. We find, in the former case, there is a non-vanishing Lorentz-violating term while the later is Lorentz-invariant. Our study of the fermionic loop correction shows that unitarity demands the fermionic mass must be at least half of the bosonic mass and that the Lorentz-violating term makes a non-trivial correction to the bosonic propagator. We discuss what these results mean in the context of the Standard Model and the possibility of …
The fermionic fields constructed from Elko have several unexpected properties. They satisfy the Klein-Gordon but not the Dirac equation and are of mass dimension one instead of three-half. Starting with the Klein-Gordon Lagrangian, we initiate a careful study of the symmetries and interactions of these fermions and their higher-spin generalisations. We find, although the fermions are of mass dimension one, the four-point fermionic self-interaction violates unitarity at high-energy. Therefore, it cannot be a fundamental interaction of the theory. It follows that for the spin-half fermions, the …
We present lattice results for spin-1/2 fermions at unitarity, where the effective range of the interaction is zero and the scattering length is infinite. We measure the spatial coherence of difermion pairs for a system of 6, 10, 14, 18, 22, 26 particles with equal numbers of up and down spins in a periodic cube. Using Euclidean time projection, we analyze ground state properties and transient behavior due to low-energy excitations. At asymptotically large values of t we see long-range order consistent with spontaneously broken U(1) fermion-number symmetry and a superfluid ground state. At int…
The scale of mass generation for fermions (including neutrinos) and the scale for electroweak symmetry breaking (EWSB) can be bounded from above by the unitarity of scattering involving longitudinal weak gauge bosons or their corresponding would-be Goldstone bosons. Including the exact n-body phase space we analyze the 2 --> n ($n \geq 2$) processes for the fermion-(anti)fermion scattering into multiple gauge boson final states. Contrary to naive energy power counting, we demonstrate that as $n$ becomes large, the competition between an increasing energy factor and a phase-space suppression le…
We present a compact analytic expression for the leading colour two-loop five-gluon amplitude in Yang-Mills theory with a single negative helicity and four positive helicities. The analytic result is reconstructed from numerical evaluations over finite fields. The numerical method combines integrand reduction, integration-by-parts identities and Laurent expansion into a basis of pentagon functions to compute the coefficients directly from six-dimensional generalised unitarity cuts.
We study perturbative unitarity in a Lorentz-symmetry-violating QED model with higher-order derivative operators in the light of the results of Lee and Wick to preserve unitarity in indefinite metric theories. Specifically, we consider the fermionic sector of the Myers-Pospelov model, which includes dimension-five operators, coupled to standard photons. We canonically quantize the model, paying attention to its effective character, and show that its Hamiltonian is stable, emphasizing the exact stage at which the indefinite metric appears and decomposes into a positive-metric sector and a negat…
