Perturbative unitarity bounds for fermions composite models
Simone Biondini
备注：
10 pages, 3 figures, Contribution to Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019). Drawn from arXiv:1903.12285
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 experimental exclusion curves for Run 2, the High-Luminosity and High-Energy configurations of the LHC. The results also apply to the searches where a generic single excited state is produced via dimension-6 contact interactions. The unitarity bound, so far overlooked in these effective models, is quite compelling and can serve as a guide for exploring the parameter space ($M,Λ$) in addition to the standard request $M \le Λ$.
The strong nature of Composite Higgs models manifests at high energies through the growing behavior of the scattering amplitudes of longitudinally polarized weak bosons that leads to the formation of composite resonances as well as non-resonant strong effects. In this work the unitarity of these scattering amplitudes is used as a tool to assess the profile of the composite spectrum of the theory, including non-resonant enhancements, vector resonances and the CP-even scalar excitation. These three signatures are then studied in realistic scattering processes at hadron colliders, aiming to estim…
In this paper we present the partial wave unitarity bound in the parameter space of dimension-5 and dimension-6 effective operators that arise in a compositeness scenario. These are routinely used in experimental searches at the LHC to constraint contact and gauge interactions between ordinary Standard Model fermions and excited (composite) states of mass $M$. After deducing the unitarity bound for the production process of a composite neutrino, we implement such bound and compare it with the recent experimental exclusion curves for Run 2, the High-Luminosity and High-Energy configurations of …
We examine a simple Composite Higgs Model (CHM) with vector resonances in addition to the Standard Model (SM) fields in perturbation theory by using the $K$-matrix method to implement unitarity constraints. We find that the $W_LW_L$ scattering amplitude has an additional scalar pole (analogous to the $σ$ meson of QCD) as in generic strongly interacting extensions of the SM. The mass and width of this dynamically generated scalar resonance are large and the mass behaves contrary to the vector one, so that when the vector resonance is lighter, the scalar one is heavier, and vice versa. We also a…
If the di-photon excess at 750 GeV hinted by the 2015 data at the LHC is explained in terms of a scalar resonance participating in the breaking of the electro-weak symmetry, this resonance must be accompanied by other scalar states for perturbative unitarity in vector boson scattering to be preserved. The simplest set-up consistent with perturbative unitarity and with the data of the di-photon excess is the Georgi-Machacek model.
We present a pedagogical case study how to combine micro-causality and unitarity based on a perturbative approach. The method we advocate constructs an analytic extrapolation of partial-wave scattering amplitudes that is constrained by the unitarity condition. Suitably constructed conformal mappings help to arrive at a systematic approximation of the scattering amplitude. The technique is illustrated at hand of a Yukawa interaction. The typical case of a superposition of strong short-range and weak long-range forces is investigated.
We place perturbative unitarity constraints on both the dimensionful and dimensionless couplings in the Next-to-Minimal Supersymmetric Standard Model Higgs Sector. These constraints, plus the requirement that the singlino and/or Higgsino constitutes at least part of the observed dark matter relic abundance, generate upper bounds on the Higgs and neutralino/chargino mass spectrum. We obtain an upper bound of 12 TeV for the charginos and neutralinos and 20 TeV for the heavy Higgses outside defined fine-tuned regions. By using the NMSSM as a template, we describe a method which replaces naturalne…
Unitarity relates the total cross section for neutrino-nucleon scattering to the neutrino-nucleon forward scattering amplitude. Assuming the validity of the perturbative expansion of the forward amplitude in the {\em weak} coupling constant, we derive a unitarity bound on the inelastic cross section. The inelastic cross section saturates this bound at a typical neutrino energy $E_ν\simeq 10^8 {\rm GeV}$. This implies that calculations of the inelastic cross section that use current parton distribution functions and lowest order weak perturbation theory are unreliable above this energy.
Low lying scalar resonances emerge as a necessary part to adjust chiral perturbation theory to experimental data once unitarity constraint is taken into consideration. I review recent progress made in this direction in a model independent approach. Also I briefly review studies on the odd physical properties of these low lying scalar resonances, including in the $γγ\toπ^+π^-, π^0π^0$ processes.
We determine the constraints on the isospin and hypercharge of a scalar electroweak multiplet from partial-wave unitarity of tree-level scattering diagrams. The constraint from SU(2)_L interactions yields T <= 7/2 (i.e., n <= 8) for a complex scalar multiplet and T <= 4 (i.e., n <= 9) for a real scalar multiplet, where n = 2T+1 is the number of isospin states in the multiplet.
There are nowadays strong experimental constraints on supersymmetric theories from the Higgs measurements as well as from the null results in Sparticle searches. However, even the parameter spaces which are in agreement with experimental data can be further constrained by using theoretical considerations. Here, we discuss for the MSSM and NMSSM the impact of perturbative unitarity as well as of the stability of the one-loop effective potential. We find in the case of the MSSM, that vacuum stability is always the stronger constraint. On the other side, the situation is more diverse in the NMSSM…
We propose a Vector-like quark (VLQ) model constrained by the requirement of perturbative unitarity. In this scenario, the neutral and charged couplings of the new heavy VLQs to SM quarks and bosons are strictly related. We derive the corresponding sum rules and display the definite structure of the couplings. We analyse, qualitatively, the phenomenological consequences of this model in terms of expected width, production mechanism and decay channels. We show that the upcoming LHC run 3 could have sensitivity to such particles but the search strategy should be adapted to cover wide resonances …
We study the constraints implied by partial wave unitarity on new physics in the form of spin-zero di-boson resonances at LHC. We derive the scale where the effective description in terms of the SM supplemented by a single resonance is expected to break down depending on the resonance mass and signal cross-section. Likewise, we use unitarity arguments in order to set perturbativity bounds on renormalizable UV completions of the effective description. We finally discuss under which conditions scalar di-boson resonance signals can be accommodated within weakly-coupled models.
We place perturbative unitarity constraints on both the dimensionful and dimensionless parameters in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) Higgs Sector. These constraints, plus the requirement that the singlino and/or Higgsino constitutes at least part of the observed dark matter relic abundance, generate upper bounds on the Higgs, neutralino and chargino mass spectrum. Requiring higher order corrections to be no more than 41% of the tree-level value, we obtain an upper bound of 20 TeV for the heavy Higgses and 12 TeV for the charginos and neutralinos outside defined fine-t…
We discuss the lineshape of a heavy Higgs boson, and the behaviour well above resonance. Previous studies concluded that the energy-dependent Higgs width should be used in the resonance region, but must not be used well away from it. We derive the full result and show that it smoothly extrapolates these limits. It is extremely simple, and would be straightforward to implement in existing calculations.
Dark pion is a promising candidate for the strongly interacting massive particle dark matter. A large pion self-coupling $m_π/f_π$ tends to be required for correct relic abundance, and hence the partial-wave amplitudes can violate the perturbative unitarity even for the coupling within naïve perturbative regime. We improve the partial-wave amplitudes in order to satisfy the optical theorem. We demonstrate that the improvement is relevant only for semi-relativistic pions, and thus this does not affect the self-scattering cross section at the cosmic structures. We also discuss the impact of the …
We study the perturbative unitarity bound given by dimension six derivative interactions consisting of Higgs doublets. These operators emerge from kinetic terms of composite Higgs models or integrating out heavy particles that interact with Higgs doublets. They lead to new phenomena beyond the Standard Model. One of characteristic contributions by derivative interactions appear in vector boson scattering processes. Longitudinal modes of massive vector bosons can be regarded as Nambu Goldstone bosons eaten by each vector field with the equivalence theorem. Since their effects become larger and …
For MSSM phenomenology, soft SUSY breaking dimension-three operators are important, in particular the couplings between Higgs bosons and squarks. In scattering processes, perturbative unitarity is violated at modest center-of-mass energy if these couplings are much larger than the masses of the scalar particles involved. Assuming perturbative unitarity, constraints on the trilinear couplings can be determined using a computer program that we have developed.
We consider the relationship between renormalizability and unitarity at a Lifshitz point in d dimensions. We test tree unitarity for theories containing only scalars and fermions, and for pure gauge theory. In both cases, we find the requirement of weighted power-counting renormalizability is equivalent to that of tree unitarity.
We study bounds from perturbative unitarity in a Composite 2-Higgs Doublet Model (C2HDM) based on the spontaneous breakdown of a global symmetry $SO(6)\to SO(4)\times SO(2)$ at the compositeness scale $f$. The eight pseudo Nambu-Goldstone Bosons (pNGBs) emerging from such a dynamics are identified as two isospin doublet Higgs fields. We calculate the $S$-wave amplitude for all possible 2-to-2-body elastic (pseudo)scalar boson scatterings at energy scales $\sqrt{s}$ reachable at the Large Hadron Collider (LHC) and beyond it, including the longitudinal components of weak gauge boson states as th…
We investigate unitarity within the Complex-Mass Scheme, a convenient universal scheme for perturbative calculations involving unstable particles in Quantum Field Theory which guarantees exact gauge invariance. Since this scheme requires to introduce complex masses and complex couplings, the Cutkosky cutting rules, which express perturbative unitarity in theories of stable particles, are no longer valid. We derive corresponding rules for scalar theories with unstable particles based on Veltman's Largest-Time Equation and prove unitarity in this framework.
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.
The scattering of longitudinally polarized W bosons in extensions of the Standard Model with anomalous Higgs couplings to the gauge sector and higher order $O(p^4)$ operators is considered. The modified couplings should be thought as the low energy remnants of some new dynamics involving the electroweak symmetry breaking sector. By imposing unitarity and causality constraints on $W_LW_L$ scattering amplitudes we relate the possible values of the effective couplings to the presence of new resonances above 300 GeV. We investigate the properties of these new resonances and their experimental dete…
We compute all of the one-loop corrections that are enhanced, $O(λ_i λ_j / 16 π^2)$, in the limit $s \gg |λ_i | v^2 \gg M_{W}^2$, $s \gg m_{12}^2$ to all the $2 \to 2$ longitudinal vector boson and Higgs boson scattering amplitudes in the $CP$-conserving two-Higgs doublet model with a softly broken $\mathbb{Z}_2$ symmetry. In the two simplified scenarios we study, the typical bound we find is $|λ_i(s)| \lessapprox 4$.
We investigate the longitudinal $WW$ scattering in models of dynamical electroweak symmetry breaking featuring a spin one axial and vector state and a composite Higgs. We also investigate the effects of a composite spin two state which has the same properties of a massive graviton. Any model of dynamical electroweak symmetry breaking will feature, depending on the dynamics, some or all these basic resonances as part of the low energy spectrum. We suggest how to take limits in the effective Lagrangian parameter space to reproduce the dynamics of different types of underlying gauge theories, fro…
The coupling of a composite Higgs to the standard model fields can deviate substantially from the standard model values. In this case perturbative unitarity might break down before the scale of compositeness is reached, which would suggest that additional composites should lie well below this scale. In this paper we account for the presence of an additional spin 1 custodial triplet of rhos. We examine the implications of requiring perturbative unitarity up to the compositeness scale and find that one has to be close to saturating certain unitarity sum rules involving the Higgs and the rho coup…
Unitarity cannot be perserved order by order in ordinary perturbation theory because the constraint $UU^\dagger=\1$ is nonlinear. However, the corresponding constraint for $K=\ln U$, being $K=-K^\dagger$, is linear so it can be maintained in every order in a perturbative expansion of $K$. The perturbative expansion of $K$ may be considered as a non-abelian generalization of the linked-cluster expansion in probability theory and in statistical mechanics, and possesses similar advantages resulting from separating the short-range correlations from long-range effects. This point is illustrated in …
In this talk, we report unitarity constraints and phenomenological studies at the Large Hadron Collider for the extra Higgs bosons of a Composite 2-Higgs Doublet Model.
We calculate an $s$-wave amplitude matrix for all the possible 2--to--2 body scalar boson elastic scatterings in models with three scalar doublets, including contributions from the longitudinal component of weak gauge bosons via the Equivalence Theorem Approximation. Specifically, we concentrate on the two cases with two[one] active plus one[two] inert doublet fields, referred to as I(1+2)HDM[I(2+1)HDM], under CP conservation. We obtain three analytically irreducible sub-matrices with the $3\times 3$ form and eighteen eigenvalues for the amplitude matrix as an independent set, where the same f…
We study the scattering of longitudinally polarized W bosons in extensions of the Standard Model where anomalous Higgs couplings to gauge sector and higher order O(p^4) operators are considered. These new couplings with respect to the Standard Model should be thought as the low energy remnants of some new dynamics involving the electroweak symmetry breaking sector. By imposing unitarity and causality constraints on the WW scattering amplitudes we find relevant restrictions on the possible values of the new couplings and the presence of new resonances above 300 GeV. We investigate the propertie…
It has long been known that perturbative calculations in scalar multi-particle production could break down since fast growing amplitudes appear. A recent calculation in the regime $ λn \gg 1$, where $ n $ is the multiplicity and $ λ$ is the self-coupling, gives an amplitude which grows exponentially with the energy, resulting in a divergent propagator and leading to the violation of perturbative unitarity. In this paper the transition rate is calculated from the solution of the self-consistent Schwinger-Dyson equation in spectral representation. We get an amplitude growing quadratically with t…
Perturbative probability conservation provides a strong constraint on the presence of new interactions of the Higgs boson. In this work we consider CP violating Higgs interactions in conjunction with unitarity constraints in the gauge-Higgs and fermion-Higgs sectors. Injecting signal strength measurements of the recently discovered Higgs boson allows us to make concrete and correlated predictions of how CP-violation in the Higgs sector can be directly constrained through collider searches for either characteristic new states or tell-tale enhancements in multi-Higgs processes.
In a series of papers, we examine the physical unitarity in a massive Yang-Mills theory without the Higgs field in which the color gauge symmetry is not spontaneously broken and kept intact. For this purpose, we use a new framework proposed in the previous paper Kondo [arXiv:1208.3521] based on a nonperturbative construction of a non-Abelian field describing a massive spin-one vector boson field, which enables us to perform the perturbative and nonperturbative studies on the physical unitarity. In this paper, we present a new perturbative treatment for the physical unitarity after giving the g…
The Littlest Higgs Model is constrained using the tree-level perturbative arguments, similar to those employed to obtain the Higgs mass bound. It is found that the perturbative unitaity violations set in at energy scales $\sim 1.5χf$, where $f$ is the analog of pion decay constant and sets the scale for the masses in the theory and also the strong interaction scale and $χ$ is a dimensionless parameter which together with $f$ sets the mass scale of heavy scalars in the theory.
We study the quantum information storage capacity of solitons and baryons in renormalizable quantum field theories that do not include gravity. We observe that a 't Hooft-Polyakov magnetic monopole saturates the Bekenstein bound on information when the theory saturates the bound on perturbative unitarity. In this very limit the monopole entropy assumes the form of an area-law, strikingly similar to a black hole entropy in gravity. The phenomenon appears universal and takes place for other solitons and non-perturbative objects. We observe the same behaviour of entropy of a baryon in QCD with la…
Renormalization of Hamiltonian field theory is usually a rather painful algebraic or numerical exercise. By combining a method based on the coupled cluster method, analysed in detail by Suzuki and Okamoto, with a Wilsonian approach to renormalization, we show that a powerful and elegant method exist to solve such problems. The method is in principle non-perturbative, and is not necessarily unitary.
We study perturbative unitarity constraints on generic Yukawa interactions where the involved fields have arbitrary quantum numbers under an $\prod_i SU(N_i) \otimes U(1)$ group. We derive compact expressions for the bounds on the Yukawa couplings for the cases where the fields transform under the trivial, fundamental or adjoint representation of the various $SU(N)$ factors. We apply our results to specific models formulated to explain the anomalous measurements of $(g-2)_μ$ and of the charged- and neutral-current decays of the $B$ mesons. We show that, while these models can generally still e…
The scalar field associated with fluctuations in the positions of the two branes, the ``radion'', plays an important role determining the cosmology and collider phenomenology of the Randall-Sundrum solution to the hierarchy problem. It is now well known that the radion mass is of order the weak scale, and that its couplings to standard model fields are order 1/TeV to the trace of the energy momentum tensor. We calculate longitudinal vector boson scattering amplitudes to explore the constraints on the radion mass and its coupling from perturbative unitarity. The scattering cross section can ind…
Effective Field Theory (EFT) extensions of the Standard Model are tools to compute observables $\big(e.g.$ cross sections with partonic center-of-mass energy $\sqrt{\hat{s}}\,\big)$ as a systematically improvable expansion suppressed by a new physics scale $M$. If one is interested in EFT predictions in the parameter space where $M<\sqrt{\hat{s}}$, concerns of self-consistency emerge, which can manifest as a violation of perturbative partial-wave unitarity. However, when we search for the effects of an EFT at a hadron collider with center-of-mass energy $\sqrt{s}$ using an inclusive strategy, …
We argue that the tree-level graviton-scalar scattering in the Regge limit is unitarized by non-perturbative effects within General Relativity. At Planckian energy the back reaction of the incoming graviton on the background geometry produces a non-perturbative plane wave which, in turn, softens the UV-behavior. Our amplitude then interpolates between the perturbative graviton-scalar scattering at low energy and scattering on a classical plane wave in the Regge limit.
We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cut…
We study perturbative unitarity in the scalar sector of the Myers-Pospelov model. The model introduces a preferred four-vector $n$ which breaks Lorentz symmetry and couples to a five-dimension operator. When the preferred four-vector is chosen in the pure timelike or lightlike direction, the model becomes a higher time derivative theory, leading to a cubic dispersion relation. Two of the poles are shown to be perturbatively connected to the standard ones, while a third pole, which we call the Lee-Wick-like pole, is associated to a negative metric, in Hilbert space, threatening the preservation…
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 investigate perturbative unitarity constraints in a model for a singlet scalar dark matter candidate. Considering elastic two particle scattering processes of the Higgs particle and the dark matter candidate, a real Klein-Gordon scalar field, perturbative unitarity constrains the self-couplings of the scalar fields.
Dark matter that was once in thermal equilibrium with the Standard Model is generally prohibited from obtaining all of its mass from the electroweak phase transition. This implies a new scale of physics and mediator particles to facilitate dark matter annihilation. In this work, we focus on dark matter that annihilates through a generic gauge boson portal. We show how partial wave unitarity places upper bounds on the dark gauge boson, dark Higgs and dark matter masses. Outside of well-defined fine-tuned regions, we find an upper bound of 9 TeV for the dark matter mass when the dark Higgs and d…
A method of deriving bounds on the weak meson form factors, based on perturbative QCD, analyticity and unitarity, is generalized in order to fully exploit heavy quark spin symmetry in the ground state $(L=0)$ doublet of pseudoscalar $(B)$ and vector $(B^*)$ mesons. All the relevant form factors of these mesons are taken into account in the unitarity sum. They are treated as independent functions along the timelike axis, being related by spin symmetry only near the zero recoil point. Heavy quark vacuum polarisation up to three loops in perturbative QCD and the experimental cross sections $σ(e^+…
Composite Higgs models are an attractive scenario, where the discovered Higgs boson is regarded as a Nambu-Goldstone boson associated with spontaneous breakdown of a global symmetry of more fundamental theory. This class of models predicts violation of perturbative unitarity at high energies, and new resonances are expected to appear around TeV scale to maintain the unitarity, while a sizable phase shift is predicted in certain scattering amplitude. We investigate the new resonance scale from the phase shift by drawing analogies with pion physics in QCD. The detectability of the phase shift at…
We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The coefficients of the one-loop divergent terms in the ob…
We evaluate the coordinate space dependence of the matrix elements of the commutator of the electromagnetic and gluon currents in the vicinity of the light-cone but at large distances within the parton model, DGLAP, the resummation approaches to the small x behaviour of DIS processes, and for the Unitarity Bound. We find that an increase of the commutator with relative distance $py$ as $\propto (py)f(py,y^2=t^2-r^2)$ is the generic property of QCD at small but fixed space-time interval $y^2=t^2-r^2$ in perturbative and nonperturbative QCD regimes. We explain that the factor $py$ follows within…
We describe the automation of the calculation of perturbative unitarity constraints including scalars that have colour charges, and its release in SARAH 4.14.4. We apply this, along with vacuum stability constraints, to a simple dark matter model with colourful mediators and interesting decays, and show how it leads to a bound on a thermal relic dark matter mass well below the classic Griest-Kamionkowski limit.
We consider the scenario where the composite Higgs arising as a pNGB in a two-site model with a non-local term included. Constraints from pion scattering and electroweak precision test are considered. We discuss the effects of composite resonances, in particular the one from composite vector-like fermions, on the oblique parameters. It is noticed that the gluon fusion production of Higgs boson is suppressed with respect to the Standard Model for about $6\%$ after imposing the unitarity and electroweak bounds.
Explorations of the violation of null energy condition (NEC) in cosmology could enrich our understanding of the very early universe and the related gravity theories. Although a fully stable NEC violation can be realized in the ``beyond Horndeski'' theory, it remains an open question whether a violation of the NEC is allowed by some fundamental properties of UV-complete theories or the consistency requirements of effective field theory (EFT). We investigate the tree-level perturbative unitarity for stable NEC violations in the contexts of both Galileon and ``beyond Horndeski'' genesis cosmology…
We study bounds on Higgs boson masses from perturbative unitarity in the Georgi-Machacek model, whose Higgs sector is composed of a scalar isospin doublet, a real and a complex isospin triplet fields. This model can be compatible with the electroweak precision data without fine tuning because of the imposed global SU(2)_R symmetry in the Higgs potential, by which the electroweak rho parameter is unity at the tree level. All possible two-body elastic-scattering channels are taken into account to evaluate the S-wave amplitude matrix, and then the condition of perturbative unitarity is imposed on…
We compute a unitarity bound for higgs mass using one--loop corrected s--wave partial amplitude for $Z_L Z_L \rightarrow Z_L Z_L$ scattering. We use the equivalence theorem and show that the higgs mass has to be less than $\approx 600$ GeV in order to save the (perturbative) unitarity in the higgs' resonance region. We also discuss about the validity of perturbation expansion in the symmetry breaking sector.
