Causality, Unitarity and Symmetry in Effective Field Theory
Timothy Trott
备注：
~50 pages v2: Improved discussion and use of discrete symmetries, some corrections and clarifications, some other minor improvements. v3: Corrected sign error in isospin projectors and resulting bounds (no qualitative changes to the extent investigated), minor improvements in exposition and typing errors corrected
Sum rules in effective field theories, predicated upon causality, place restrictions on scattering amplitudes mediated by effective contact interactions. Through unitarity of the $S$-matrix, these imply that the size of higher dimensional corrections to transition amplitudes between different states is bounded by the strength of their contributions to elastic forward scattering processes. This places fundamental limits on the extent to which hypothetical symmetries can be broken by effective interactions. All analysis is for dimension $8$ operators in the forward limit. Included is a thorough derivation of all positivity bounds for a chiral fermion in $SU(2)$ and $SU(3)$ global symmetry representations resembling those of the Standard Model, general bounds on flavour violation, new bounds for interactions between particles of different spin, inclusion of loops of dimension $6$ operators and illustration of the resulting strengthening of positivity bounds over tree-level expectations, a catalogue of supersymmetric effective interactions up to mass dimension $8$ and $4$ legs and the demonstration that supersymmetry unifies the positivity theorems as well as the new bounds.
The tree-loop duality relation is used as a starting point to derive the constraints of causality and unitarity. Specifically, the Bogoliubov causality condition is ab initio derived at the individual graph level. It leads to a representation of a graph in terms of lower order cut graphs. Extracting the absorptive part gives then the general unitarity relation (Cutkosky rule). The derivation, being carried out directly in momentum space, holds for any local (polynomial) hermitian interaction vertices. This is in contrast to the technical difficulties arising from contact terms in the spacetime…
We canonically quantize $(2+1)$-dimensional electrodynamics including a higher-derivative Chern-Simons term. The effective theory describes a standard photon and an additional degree of freedom associated with a massive ghost. We find the Hamiltonian and the algebra satisfied by the field operators. The theory is characterized by an indefinite metric in the Hilbert space that brings up questions on causality and unitarity. We study both of the latter fundamental properties and show that microcausality as well as perturbative unitarity up to one-loop order are conserved when the Lee-Wick prescr…
We present the first comprehensive analysis of the unitarity thresholds and anomalous thresholds of scattering amplitudes at two loops and beyond based on the loop-tree duality, and show how non-causal unphysical thresholds are locally cancelled in an efficient way when the forest of all the dual on-shell cuts is considered as one. We also prove that soft and collinear singularities at two loops and beyond are restricted to a compact region of the loop three-momenta, which is a necessary condition for implementing a local cancellation of loop infrared singularities with the ones appearing in r…
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.
Proton Compton scattering is analyzed with the chiral Lagrangian. Partial-wave amplitudes are obtained by an analytic extrapolation of subthreshold reaction amplitudes computed in chiral perturbation theory, where the constraints set by electromagnetic-gauge invariance, causality and unitarity are used to stabilize the extrapolation. We present and discuss predictions for various spin observables and polarizabilties of the proton. While for the transition polarizabilities $γ_{E1M2}, γ_{M1E2}$ we recover the results of strict chiral perturbation theory, for the diagonal $γ_{E1E1}, γ_{M1M1}$ ele…
We revisit the previously unsolved problems of ensuring Lorentz invariance and non-perturbative unitarity in Lee-Wick theories. We base our discussion on an ultraviolet completion of QED by Lee-Wick ghost fields, which is argued to be asymptotically safe. We argue that as long as the state space is based upon a suitable choice of distributions of a type invented by Gel'fand and Shilov, the Lee-Wick ghosts can be eliminated while preserving Lorentz invariance to produce a unitary theory. The method for eliminating ghosts is in principle non-perturbatively well-defined, in contrast with some pre…
The hypothesis that the causal properties of space-time, as well as other properties of physical systems like unitarity, charge conservation, etc., might be decided by the higher dimensional structure (in particular, higher-dimensional physical laws), rather than by the four-dimensional one, is explored in order to evade the most awkward problems of higher-dimensional theories with compact extra time-like dimensions: violation of causality and of unitarity.
The weak gravity conjecture states that quantum gravity theories have to contain a charged state with a charge-to-mass ratio bigger than unity. By studying unitarity and causality constraints on higher derivative corrections to the charge-to-mass ratio of extremal back holes, we demonstrate that heavy extremal black holes can play the role of the required charged state under several assumptions. In particular, our argument is applicable when the higher spin states Reggeizing graviton exchange are subdominant in the photon scattering. It covers (1) theories with light neutral bosons such as dil…
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…
We study a class of generalized Abelian gauge field theories where CPT symmetry is violated by a Chern-Simons-like term which selects a preferred direction in spacetime. Such Chern-Simons-like terms may either emerge as part of the low-energy effective action of a more fundamental theory or be produced by chiral anomalies over a nonsimply connected spacetime manifold. Specifically, we investigate the issues of unitarity and causality. We find that the behaviour of these gauge field theories depends on whether the preferred direction is spacelike or timelike. For a purely spacelike preferred di…
We consider the shift of charge-to-mass ratio for extremal black holes in the context of effective field theory, motivated by the Weak Gravity Conjecture. We constrain extremality corrections in different regimes subject to unitarity and causality constraints. In the asymptotic IR, we demonstrate that for any supersymmetric theory in flat space, and for all minimally coupled theories, logarithmic running at one loop pushes the Wilson coefficient of certain four-derivative operators to be larger at lower energies, guaranteeing the existence of sufficiently large black holes with $Q>M$. We ident…
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 argue that symmetry and unification can emerge as byproducts of certain physical constraints on dynamical scattering. To accomplish this we parameterize a general Lorentz invariant, four-dimensional theory of massless and massive scalar fields coupled via arbitrary local interactions. Assuming perturbative unitarity and an Adler zero condition, we prove that any finite spectrum of massless and massive modes will necessarily unify at high energies into multiplets of a linearized symmetry. Certain generators of the symmetry algebra can be derived explicitly in terms of the spectrum and three-…
Stringent limits on the Myers-Pospelov timelike parameter for photons $ξ<10^{-15}$ coming from astrophysical tests suggest exploring more general preferred backgrounds, such as spacelike and lightlike. We take some steps in this direction. We allow the external four vector $n$ characterizing the Lorentz symmetry breaking to have arbitrary directions in spacetime. We study the consistency for the effective field theories in each privileged frame by demanding causality, stability and analyticity. When specializing to a timelike background we found runaway solutions leading to causality and unita…
We study the unitarity bounds of the scattering amplitudes in the extra dimensional gauge theory where the gauge symmetry is broken by the boundary condition. The estimation of the amplitude of the diagram including four massive gauge bosons in the external lines shows that the asymptotic power behavior of the amplitude is canceled. The calculation will be done in the 5 dimensional standard model and the SU(5) grand unified theory, whose 5th dimensional coordinate is compactified on $S^1/Z_2$. The broken gauge theories through the orbifolding preserve the unitarity at high energies similarly t…
We discuss the use of BRST symmetry and the resulting Ward identities for orbifold gauge theories as consistency checks in an arbitrary number of dimensions. We verify that both the usual orbifold symmetry breaking and the recently proposed Higgsless symmetry breaking are consistent with the nilpotency of the BRST transformation. Imposing the Ward identities resulting from the BRST symmetry on the 4-point functions of theory, we obtain relations on the coupling constants that are shown to be equivalent to the conditions for tree level unitarity. We present the complete set of these sum rules a…
We study the general non-minimally coupled charged massive spin 3/2 model both for its low energy phenomenological properties and for its unitarity, causality and degrees of freedom behaviour. When the model is viewed as an effective theory, its parameters (after ensuring the correct excitation count) are related to physical characteristics, such as the magnetic moment g factor, by means of low energy theorems. We also provide the corresponding higher spin generalisation. Separately, we consider both low and high energy unitarity, as well as the causality aspects of our models. None (including…
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 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…
In five dimensional models of Higgsless electroweak symmetry breaking, tree level unitarity in gauge boson scattering is restored by the exchange of gauge boson Kaluza-Klein modes instead of a Higgs boson. Unitarity of scattering amplitudes involving top quarks requires also the Kaluza-Klein modes of the third family quarks. It is shown that the relevant unitarity cancellations are consistent with gauge symmetry breaking by boundary conditions. These results are used to constrain the couplings of the top quark to Kaluza-Klein modes and the implications for collider phenomenology are discussed.
We analyze the unitarity of a modified QED with higher-order terms that violate Lorentz symmetry. We make an explicit calculation to verify unitarity at the one-loop level. As expected we find negative norm states that could in principle lead to a violation of unitarity. However, we show that these states become massive in Euclidean space and do not contribute to the discontinuity.
In this paper, we consider an electrodynamics of higher derivatives coupled to a Lorentz-violating background tensor. Specifically, we are interested in a dimension-five term of the CPT-odd sector of the nonminimal Standard-Model Extension. By a particular choice of the operator $\hat{k}_{AF}$, we obtain a higher-derivative version of the Carroll-Field-Jackiw (CFJ) term, $\frac{1}{2}ε^{κλμν}A_λD_κ\square F_{μν}$, with a Lorentz-violating background vector $D_κ$. This modification is subject to being investigated. We calculate the propagator of the theory and from its poles, we analyze the disp…
Using the analytic properties of two-point functions in QCD, as well as unitarity, bounds on the B meson form factor F(q^2) can be derived. Heavy quark spin symmetry, correctly taken into account, is shown to improve these bounds significantly.
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.
Models with three Higgs doublets (3HDM) are the source of much recent activity, for they are key components of many solutions to the problems of the Standard Model; from extra sources of CP violation to Dark Matter candidates. We compute explicitly the theoretical bounds for all symmetry-constrained 3HDM arising from the perturbative unitarity of two-to-two scattering amplitudes. In addition, we propose a method based on principal minors that foregoes diagonalization and which is preferable in all models (not only 3HDM) dealing with large scattering matrices.
We examine the unitarity properties of spontaneously broken non-commutative gauge theories. We find that the symmetry breaking mechanism in the non-commutative Standard Model of Chaichian et al. leads to an unavoidable violation of tree-level unitarity in gauge boson scattering at high energies. We then study a variety of simplified spontaneously broken non-commutative theories and isolate the source of this unitarity violation. Given the group theoretic restrictions endemic to non-commutative model building, we conclude that it is difficult to build a non-commutative Standard Model under the …
These notes are a fuller version of four lectures given at the 2015 International Summer Workshop in Reaction Theory held at Indiana University, Bloomington. The aim is to provide a simple introduction to how the tools of "the S-matrix era" - i.e. the constraints of unitarity, analyticity and crossing symmetry - can be incorporated into analyses of final state interactions in two- and three-hadron systems. The main focus is on corrections to the isobar model in three-hadron final states, which may be relevant once more as much larger data sets become available.
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^+…
Within a specific texture of the quark mass matrix the notion of a maximal violation of the $CP$ symmetry can be defined. The experimental constraints from weak decays imply that in reality one is close to the case of maximal $CP$ violation, which vanishes as the mass of the $u$--quark approaches zero. The unitarity triangle of the Cabibbo--Kobayashi--Maskawa matrix elements is determined. It is related to the triangle relating the Cabibbo angle to the quark mass ratios in the complex plane. The angle $α$ describing the $CP$ violation in the decay $B_d \rightarrow π^+ π^-$ is close to 90$^{\ci…
We give a detailed analysis of the particle spectrum and the perturbative unitarity of the recently introduced Weyl-invariant version of the new massive gravity in 2+1 dimensions. By computing the action up to second order in the fluctuations of the metric, the gauge and the scalar fields around the anti-de Sitter (AdS) and flat vacua, we find that the theory describes unitary (tachyon and ghost-free) massive spin-2, massive (or massless) spin-1 and massless spin-0 excitations for certain ranges of the dimensionless parameters. The theory is not unitary in de Sitter space. Scale invariance is …
We derive new effective field theory (EFT) positivity bounds on the elastic $2\to2$ scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the $t$ derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the $s$- and $u$-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the $t$-derivative to the forward-limit EFT ampli…
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…
The new results by OPERA collaboration claim the discovery of superluminal neutrinos. Superluminal particles have to break Lorentzian symmetry or causality principle. The method discussed gives us the possibility to reintroduce Lorentzian symmetry without breaking of causality.
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 use conditions for unitarity cancellations to constrain the couplings of the top and bottom quarks to Kaluza-Klein modes in Higgsless models of electroweak symmetry breaking. An example for the mass spectrum of quark resonances in a theory space model is given and the implications for the collider phenomenology in the top sector are discussed, comparing to signatures of Little Higgs and strong electroweak symmetry breaking models.
It is found that in the SM the Ward-Takahashi(WT) identities of the axial-vector currents and the charged vector currents of fermions are invalid after spontaneous symmetry breaking. The spin-0 components of Z and W fields are revealed from the invalidity of these WT identities. The masses of these spin-0 components are at $10^{14}$GeV. They are ghosts. Therefore, unitarity of the SM after spontaneous symmetry breaking is broken at $10^{14}$GeV.
We have considered the structure of the fundamental symmetry violating part of the photon refractive index in vacuum in the presence of constant electric and magnetic fields. This part of the refractive index can, in principle, contain CPT symmetry breaking terms. Some of the terms violate Lorentz invariance, whereas the others violate locality and causality. Estimates of these effects, using laser experiments are considered.
We update and summarize the present status of our understanding of the reparametrization symmetry with $i \leftrightarrow j$ state exchange in neutrino oscillation in matter. We introduce a systematic method called ``Symmetry Finder'' (SF) to uncover such symmetries, demonstrate its efficient hunting capability, and examine their characteristic features. Apparently they have a local nature: The 1-2 and 1-3 state exchange symmetries exist at around the solar- and atmospheric-resonances, respectively, with the level-crossing states exchanged. However, this view is not supported, to date, in the …
We discuss how saturation of unitarity would change phase structure of hadronic matter at very high temperatures emphasizing the role of the vacuum state with spontaneously broken chiral symmetry.
This letter constructs, making use of the on-shell spinor-helicity formalism, a possible ultraviolet completion of gravity following a "bottom-up" approach. The assumptions of locality, unitarity and causality i) require an infinite tower of resonances with increasing spin and quantized mass, ii) introduce a duality relation among crossed scattering channels, and iii) dress all gravitational amplitudes in the Standard Model with a form factor that closely resembles either the Veneziano or the Virasoro-Shapiro amplitude in string theory. As a consequence of unitarity, the theory predicts leadin…
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…
There are six leptonic unitarity triangles (LUTs) defined by six orthogonality conditions of the three-family lepton flavor mixing matrix in the complex plane. In the framework of the standard model or the minimal supersymmetric standard model, the evolutions of sides and inner angles of the six LUTs from a superhigh energy scale $Λ_{\rm H}^{}$ to the electroweak scale $Λ_{\rm EW}^{}$ due to the renormalization-group equation (RGE) running are derived in the integral form for both Dirac and Majorana neutrinos. Furthermore, the LUTs as an intuitively geometrical language are applied to the desc…
We study the temperature evolution of the rho and sigma mass and width, using a unitary chiral approach. The one-loop pion-pion scattering amplitude in Chiral Perturbation Theory at finite temperature is unitarized via the Inverse Amplitude Method. Our results predict a clear increase with T of both the rho and sigma widths. The masses decrease slightly for high T, while the rho-pion-pion coupling increases. The rho behavior seems to be favored by experimental results. In the sigma case, it signals chiral symmetry restoration.
Tree-level unitarity constraints on the masses of the Higgs bosons in the general Two Higgs Doublet Model (THDM) are studied. We first consider the case where the Higgs potential is invariant under a discrete symmetry transformation, and derive strong constraints on the mass of the lightest CP-even Higgs boson ($M_h$) as a function of $\tanβ$. We then show that the inclusion of the discrete symmetry breaking term weakens the mass bounds considerably. It is suggested that a measurement of $M_h$ and $\tanβ$ may enable discrimination between the two Higgs potentials.
Unitarity constraints on the scalar parameters both for the Standard Model and the general Two Higgs Doublet Model (THDM) are examined. In the case of the THDM with an exact discrete symmetry transformation, we show that the mass of the lightest CP-even Higgs boson ($M_h$) and $\tanβ$ are strongly correlated and consequently a strong lower bound can be put on $M_h$ for large $\tanβ$. It is also shown that the inclusion of the discrete symmetry breaking term relaxes the aforementioned correlation.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P-\frac{2}{3}{\cal E})/P$, where $P$ is the pressure and ${\cal E}$ is the energy density. In the high temperature limit this ratio scales as $\frac{zλ}{a}$, where $z$ is the fugacity, $λ$ is the thermal wave length, and $a$ is the scattering length. We show that the bulk viscosity $ζ$ scales as the second power of this parameter, $ζ\sim (\frac{zλ}{a})^2 …
By means of a coupled channel non-perturbative unitary approach, it is possible to extend the strong constrains of Chiral Perturbation Theory to higher energies. In particular, it is possible to reproduce the lowest lying resonances in meson-meson scattering up to 1.2 GeV using the parameters of the O(p^2) and O(p^4) Chiral Lagrangian. We report on an update of these results examining their possible relevance for meson spectroscopy.
We propose new classes of models which predict both tri-bimaximal lepton mixing and a right-angled Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle, alpha approximately 90 degrees. The ingredients of the models include a supersymmetric (SUSY) unified gauge group such as SU(5), a discrete family symmetry such as A4 or S4, a shaping symmetry including products of Z2 and Z4 groups as well as spontaneous CP violation. We show how the vacuum alignment in such models allows a simple explanation of alpha approximately 90 degrees by a combination of purely real or purely imaginary vacuum expectation…
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.
We derive non-linear recursion equations for the leading infrared logarithms in massless non-renormalizable effective field theories. The derivation is based solely on the requirements of the unitarity, analyticity and crossing symmetry of the amplitudes. That emphasizes the general nature of the corresponding equations. The derived equations allow one to compute leading infrared logarithms to essentially unlimited loop order without performing a loop calculation. For the implementation of the recursion equation one needs to calculate tree diagrams only. The application of the equation is demo…
We replace the standard model scalar doublet by a doublet of vector fields and generate masses by dynamical symmetry breaking. Oblique radiative corrections are small if the new vector bosons ($ B^+$, $ B^0 $) are heavy. In this note it is shown that the model has a low momentum scale and above $ Λ\simeq 2 $ TeV it does not respect the perturbative unitarity. From tree-graph unitarity the allowed region of $B ^+ (B^0) $ mass is estimated as $m_{B^+} \geq 369 \hbox{GeV} \quad (m_{B^0} \geq 410 $ GeV) at $Λ= 1 $ TeV.
High-energy unitarity is argued to select a special version of QCD as the strong interaction. Electroweak symmetry breaking has to be due to a new sextet quark sector - that will produce large cross-section effects at the LHC. The sextet sector embeds, uniquely, in a massless SU(5) theory that potentially generates the states and interactions of the Standard Model within a bound-state S-Matrix. Infra-red chirality transitions of the massless Dirac sea play an essential dynamical role.
Pion-pion scattering amplitude obtained from one-loop Chiral Perturbation Theory (ChPT) is crossing symmetric, however the corresponding partial-wave amplitudes do not respect exact unitarity relation. There are different approaches to get unitarized results from ChPT. Here we consider the inverse amplitude method (IAM) and, using the Roskies relations, we measure the amount of crossing symmetry violation when IAM is used in order to fit pion-pion phase-shifts to experimental data in the resonance region. We also show the unitarity violation of the crossing symmetric ChPT amplitude with its pa…
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…
