the recent discovery of a scalar sector in the standard model has profound implications for particle physics , yet it does not settle the issue of what dynamics is behind electroweak symmetry breaking and how the hierarchy problem is resolved . in that respect , the discovery of new particles in the sub - tev region , either from weakly coupled or strongly coupled dynamical extensions , should provide valuable hints . the absence ( so far ) of new - physics states , together with the higgs - like character of the 126  gev scalar , indicates that deviations from the standard model paradigm have to be small . this , together with flavor constraints , puts under strain , if it does not already exclude , most of the natural realizations of supersymmetry and composite models . in this paper composite models were initially introduced as higgsless alternatives to the standard model , borrowing heavily from patterns and characteristics of qcd . while those models are nowadays ruled out , they have been superseded by more elaborate strongly coupled scenarios that accommodate a light scalar through the vacuum misalignment mechanism discussed in  [ 1 , 2 ] . in those dynamical scenarios the higgs - like particle is a pseudo - goldstone mode of some broken global symmetry which develops a potential through quantum corrections , thereby avoiding the hierarchy problem . one of the virtues of the vacuum misalignment is that it is flexible enough to smoothly interpolate between non - decoupling and decoupling new - physics scenarios . recently , such ideas have been formulated in an effective field theory language  [ 39 ] , which allows for a systematic scrutiny of deviations from the standard model paradigm . while an effective field theory is a general description of the physics at low energies , specific models are needed to identify potential signatures and guide searches at colliders . however , without imposing rather ad hoc conditions , the presently allowed size of new - physics effects seems hard to accommodate within models of light ( sub - tev ) states . for instance , constraints on oblique parameters typically push the mass range of vector and axial resonances at a few tev  [ 10 , 11 ] , or else one is lead to a certain degree of fine - tuning by requiring unnaturally sizable one - loop corrections  [ 12 , 13 ] . in this paper i will assume that no large one - loop effects are present , which otherwise would jeopardize the effective field theory expansion , and therefore accept that composite models seem to prefer rather heavy vector and axial states . the question i will then address is whether other lighter spin-1 states can be present . such states should leave a rather subtle phenomenological imprint in order not to upset electroweak precision measurements but might leave at the same time rather clean signals for direct detection . in the following i will work with a minimal dynamical setting , assuming that the new dynamics is invariant under su\documentclass[12pt]{minimal } 
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				 \begin{document}$$(2)_l\times \mathrm{su}(2)_r$$\end{document}(2)lsu(2)r broken down to the custodial \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_v$$\end{document}su(2)v . this coset structure guarantees that the number of pseudo - goldstone bosons does not exceed the experimentally observed ones . in order to be fully general i will further assume that this new dynamics is parity - preserving and generates its lightest resonance state roughly around 700900  gev . this state will be described by an antisymmetric rank - two tensor field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b with the quantum numbers of a pseudovector . this new dynamics will be assumed to be dual to a more fundamental theory of constituents , which sets in at sufficiently high energies . i will show that consistency with this dual picture implies that a pseudovector would leave no trace on effective operators with gauge bosons . at energies much lower than its mass its impact on low - energy effective operators is thus rather elusive , but still it provides distinct signatures for direct detection at colliders , mainly through gluon fusion ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$gg\rightarrow { \mathcal { b}}\rightarrow w^+w^-$$\end{document}ggbw+w- ) and associated production ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$pp\rightarrow { \mathcal { b } } z$$\end{document}ppbz ) . , i discuss some technical aspects of rank - two antisymmetric tensors , such as its decomposition in longitudinal and transverse modes and their connection through a duality transformation . in sect . its indirect traces at low energies are examined in sect .  4 while comments on the most promising signatures for direct detection are addressed in sect .  5 . it is well known that massive spin-1 states can be described by one - form proca fields as well as two - form fields . the proca description successfully accounts for the standard model fundamental gauge spin-1 fields ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$w^{\pm } , z$$\end{document}w,z ) , whose masses are generated after gauge symmetry is spontaneously broken via the higgs mechanism . in contrast , the two - form representation seems to be better suited for composite spin-1 states . in qcd , for instance , the coefficients of the nlo low - energy expansion ( the gasser  leutwyler coefficients ) are expected to be \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(f^2/\lambda ^2)$$\end{document}o(f2/2 ) , with \documentclass[12pt]{minimal } 
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				 \begin{document}$$f$$\end{document}f the pion decay constant and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\lambda $ $ \end{document} the scale of hadronic physics . this is the natural size one would expect from integrated - out hadronic degrees of freedom lying at \documentclass[12pt]{minimal } 
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				 \begin{document}$$m\sim { \mathcal { o}}(\lambda ) $ $ \end{document}mo( ) and agrees rather well with experimental data . a two - form representation of axial and vector mesons naturally accounts for this pattern of vector meson exchange   besides reproducing other key aspects of low - energy chiral dynamics  [ 15 , 16 ] . if mesons are represented by proca fields , however , the contribution to the nlo coefficients from tree level resonance exchange vanishes altogether . following the example of qcd , i will henceforth use the tensorial representation . in this section i will lay out some formal aspects of massive rank - two antisymmetric tensors that will be useful for the sections to come . i will pay special attention to the interplay of the longitudinal and transverse components of the two - forms and a duality transformation connecting them . the most general quadratic form for a second rank antisymmetric tensor with well - defined parity is given by1\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_h=\frac{a}{2 } \partial _ { \sigma } h_{\mu \nu } \partial ^{\sigma } h^{\mu \nu } + b\ , \partial ^{\mu } h_{\mu \nu } \partial _ { \lambda } h^{\lambda \nu } + \frac{c}{2 } h_{\mu \nu } h^{\mu \nu } .\nonumber \\ \end{aligned}$$\end{document}lh = a2hh+bhh+c2hh.generically , the structure of the previous lagrangian does not furnish a representation of the lorentz group . the generic propagator contains two potential poles:2\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta _ { \mu \nu ; \lambda \rho } = \frac{p^{t}_{\mu \nu ; \lambda \rho } } { aq^2+c}+\frac{p^{l}_{\mu \nu ; \lambda \rho } } { ( a+b)q^2+c } \end{aligned}$$\end{document}=ptaq2+c+pl(a+b)q2+cwhere3\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } p^{\mu \nu ; \lambda \rho } _ l&=g^{\mu \lambda } \frac{q^{\nu } q^{\rho } } { 2q^2}-g^{\mu \rho } \frac{q^{\nu } q^{\lambda } } { 2q^2}-g^{\nu \lambda } \frac{q^{\mu } q^{\rho } } { 2q^2}+g^{\nu \rho } \frac{q^{\mu } q^{\lambda } } { 2q^2}\nonumber \\ p^{\mu \nu ; \lambda \rho } _ t&=-\epsilon ^{\mu \nu \alpha \beta } \epsilon ^{\lambda \rho \eta \sigma } g_{\alpha \eta } \frac{q_{\beta } q_{\sigma } } { 2q^2 } \end{aligned}$$\end{document}pl=gqq2q2-gqq2q2-gqq2q2+gqq2q2pt=-gqq2q2are the transverse and longitudinal projectors for rank - two tensors . as such , \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{t}^{\mu \nu ; \lambda \rho } + p_{l}^{\mu \nu ; \lambda \rho } = i^{\mu \nu ; \lambda \rho } $ $ \end{document}pt+pl=i , with \documentclass[12pt]{minimal } 
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				 \begin{document}$$i^{\mu \nu ; \lambda \rho } = \frac{1}{2}(g^{\mu \lambda } g^{\nu \rho } -g^{\mu \rho } g^{\nu \lambda } ) $ $ \end{document}i=12(gg-gg ) , as can easily be checked from  ( 3 ) . the longitudinal mode is isolated by picking \documentclass[12pt]{minimal } 
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				 \begin{document}$$a=0$$\end{document}a=0 , \documentclass[12pt]{minimal } 
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				 \begin{document}$$b=-\frac{1}{2}$$\end{document}b=-12 , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$c=\frac{m_l^2}{2}$$\end{document}c = ml22 , and it leads to4\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta _ { \mu \nu ; \lambda \rho } ^l&=\frac{2}{q^2-m_l^2}\left [ \frac{q^2-m_l^2}{m_l^2}i_{\mu \nu ; \lambda \rho } -\frac{q^2}{m_l^2}p^l_{\mu \nu ; \lambda \rho } \right ] , \end{aligned}$$\end{document}l=2q2-ml2q2-ml2ml2i-q2ml2pl , which corresponds to the lagrangian5\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_l&={\mathrm { tr}}\left [ -\partial ^{\mu } { \mathcal { v}}_{\mu \nu } \partial _ { \rho } { \mathcal { v}}^{\rho \nu } + \frac{m_{\mathcal { v}}^2}{2}{\mathcal { v}}_{\mu \nu } { \mathcal { v}}^{\mu \nu } \right ] \end{aligned}$$\end{document}ll = tr-vv+mv22vvwhere the trace is over the internal symmetry group . the longitudinal field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v is the one currently used to represent spin-1 mesons in qcd  [ 14 , 15 ] . the transverse mode corresponds instead to the parameter choice \documentclass[12pt]{minimal } 
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				 \begin{document}$$a =- b=\frac{1}{2}$$\end{document}a =- b=12 and \documentclass[12pt]{minimal } 
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				 \begin{document}$$c=-\frac{m_t^2}{2}$$\end{document}c =- mt22 , giving6\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \delta _ { \mu \nu ; \lambda \rho } ^t&=- \frac{2}{q^2-m_t^2}\left [ \frac{q^2-m_t^2}{m_t^2}i_{\mu \nu ; \lambda \rho } -\frac{q^2}{m_t^2}p^{t}_{\mu \nu ; \lambda \rho } \right ] \nonumber \\&=\frac{2}{q^2-m_t^2}\left [ i_{\mu \nu ; \lambda \rho } -\frac{q^2}{m_t^2}p^l_{\mu \nu ; \lambda \rho } \right ] , \end{aligned}$$\end{document}t=-2q2-mt2q2-mt2mt2i-q2mt2pt=2q2-mt2i-q2mt2pl , which results from the lagrangian7\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_t&\!=\!{\mathrm { tr}}\left [ \frac{1}{2}\partial _ { \lambda } { \mathcal { b}}^{\mu \nu } \partial ^{\lambda } { \mathcal { b}}_{\mu \nu } -\partial ^{\mu } { \mathcal { b}}_{\mu \nu } \partial _ { \rho } { \mathcal { b}}^{\rho \nu } -\frac{m_{\mathcal { b}}^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } \right ] .\nonumber \\ \end{aligned}$$\end{document}lt = tr12bb-bb-mb22bb.the transverse mode is the natural extension of the gauge two - form field when a mass term is added . a gauged two - form is defined by the free action8\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&=\frac{1}{6}{\mathrm { tr}}\left [ h_{\mu \nu \rho } h^{\mu \nu \rho } \right ] \end{aligned}$$\end{document}l=16trhhwhere \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_{\mu \nu \lambda } \equiv \partial _ { \mu } h_{\nu \lambda } + \partial _ { \lambda } h_{\mu \nu } $ $ \end{document}hh+h+h is the curvature tensor and the metric signature is chosen mostly negative . the previous lagrangian is invariant under the gauge symmetry \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta h_{\mu \nu } = \partial _ { \mu } \lambda _ { \nu } -\partial _ { \nu } \lambda _ { \mu } $ $ \end{document}h=- , which eventually leaves only one independent degree of freedom . it therefore describes a massless spin-0 mode , as can be seen by direct investigation of its helicity structure   or through duality  . in the context of string theory , the field \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_{\mu \nu } $ $ \end{document}h ( both massless and massive ) was found to be the natural dynamical object mediating interstring interactions  [ 18 , 19 ] and is commonly referred to as the kalb ramond field \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_{\mu \nu } $ $ \end{document}b. there , the gauged two - form gets its mass through mixing with a gauge one - form field , absorbs its two degrees of freedom and eventually describes a massive spin-1 mode . the opposite also holds true , namely a gauge field can be made massive by absorbing the scalar degree of freedom hidden inside the gauged two - form  . equation  ( 7 ) can easily be cast in the form9\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_t&={\mathrm { tr}}\left [ \frac{1}{6}{\mathcal { b}}_{\mu \nu \lambda } { \mathcal { b}}^{\mu \nu \lambda } -\frac{m_{\mathcal { b}}^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } \right ] , \end{aligned}$$\end{document}lt = tr16bb-mb22bb,which shows that \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b is a massive kalb therefore , in full generality , a second rank antisymmetric tensor \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_{\mu \nu } $ $ \end{document}h has 6 degrees of freedom , which can be decomposed as two massive spin-1 fields , the transverse ( kalb  ramond ) \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b and the longitudinal \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v. the existence of two tensorial representations for massive spin-1 fields can also be understood by the fact that there are two independent tensor structures for 1-particle creation matrix elements , namely10\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \langle 0| { \mathcal { v}}^x_{\mu \nu } |x\rangle&\propto \frac{i}{m_x}(p_{\mu } \epsilon _ { \nu } -p_{\nu } \epsilon _ { \mu } ) \end{aligned}$$\end{document}0|vx|ximx(p-p)and11\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \langle 0| { \mathcal { b}}^x_{\mu \nu } |x\rangle&\propto \frac{i}{m_x}\varepsilon _ { \mu \nu \lambda \rho } \epsilon ^{\lambda } p^{\rho } , \end{aligned}$$\end{document}0|bx|ximxp,which are the normalizations leading to the propagators discussed above . the previous equations suggest that there is a duality transformation between longitudinal and transverse fields given by12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { v}}_{\mu \nu } \rightarrow \frac{1}{2}\varepsilon _ { \mu \nu \lambda \rho } { \mathcal { b}}^{\lambda \rho } . \end{aligned}$$\end{document}v12b.to be more precise , one can show that a theory generically given by13\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathrm { tr}}\left [ -\partial ^{\mu } { \mathcal { v}}_{\mu \nu } \partial _ { \rho } { \mathcal { v}}^{\rho \nu } + \frac{m^2}{2}{\mathcal { v}}_{\mu \nu } { \mathcal { v}}^{\mu \nu } + { \mathcal { v}}_{\mu \nu } j_{\mathcal { v}}^{\mu \nu } \right ] , \end{aligned}$$\end{document}l = tr-vv+m22vv+vjv,where the interactions are built to describe a particle species \documentclass[12pt]{minimal } 
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				 \begin{document}$$x$$\end{document}x , is dual to another theory14\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathrm { tr}}\left [ \frac{1}{6}{\mathcal { b}}_{\mu \nu \lambda } { \mathcal { b}}^{\mu \nu \lambda } -\frac{m^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } + { \mathcal { b}}_{\mu \nu } j_{\mathcal { b}}^{\mu \nu } \right ] , \end{aligned}$$\end{document}l = tr16bb-m22bb+bjb,which describes the same particles \documentclass[12pt]{minimal } 
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				 \begin{document}$$x$$\end{document}x provided that \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{\mathcal { b}}^{\mu \nu } = \frac{1}{2}\varepsilon ^{\mu \nu \lambda \rho } j_{{\mathcal { v}}\lambda \rho } $ $ \end{document}jb=12jv. as a corollary , in the absence of masses , the original gauge transformation of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \mathcal { b}}_{\mu \nu } = \partial _ { \mu } \lambda _ { \nu } -\partial _ { \nu } \lambda _ { \mu } $ $ \end{document}b=- is cast in terms of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v as \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \mathcal { v}}_{\mu \nu } = \epsilon _ { \mu \nu \lambda \rho } \partial ^{\lambda } \lambda ^{\rho } $ $ \end{document}v=. the duality above means that every spin-1 particle can be equivalently expressed in terms of either a longitudinal or a transverse two - form field . in theories with parity conservation , particles with the same charge conjugation but opposite parity can be described with \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b fields , respectively , while leaving the form of the interaction terms untouched . thus , if an interacting theory for a regular \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{--}$$\end{document}1 - - vector is described by a longitudinal field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v , the corresponding theory for a pseudovector \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{+-}$$\end{document}1 + - can be constructed simply by replacing \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } \rightarrow { \mathcal { b}}_{\mu \nu } $ $ \end{document}vb in the interaction terms . the existence of two tensorial representations for massive spin-1 fields can also be understood by the fact that there are two independent tensor structures for 1-particle creation matrix elements , namely10\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \langle 0| { \mathcal { v}}^x_{\mu \nu } |x\rangle&\propto \frac{i}{m_x}(p_{\mu } \epsilon _ { \nu } -p_{\nu } \epsilon _ { \mu } ) \end{aligned}$$\end{document}0|vx|ximx(p-p)and11\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \langle 0| { \mathcal { b}}^x_{\mu \nu } |x\rangle&\propto \frac{i}{m_x}\varepsilon _ { \mu \nu \lambda \rho } \epsilon ^{\lambda } p^{\rho } , \end{aligned}$$\end{document}0|bx|ximxp,which are the normalizations leading to the propagators discussed above . the previous equations suggest that there is a duality transformation between longitudinal and transverse fields given by12\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { v}}_{\mu \nu } \rightarrow \frac{1}{2}\varepsilon _ { \mu \nu \lambda \rho } { \mathcal { b}}^{\lambda \rho } . be more precise , one can show that a theory generically given by13\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathrm { tr}}\left [ -\partial ^{\mu } { \mathcal { v}}_{\mu \nu } \partial _ { \rho } { \mathcal { v}}^{\rho \nu } + \frac{m^2}{2}{\mathcal { v}}_{\mu \nu } { \mathcal { v}}^{\mu \nu } + { \mathcal { v}}_{\mu \nu } j_{\mathcal { v}}^{\mu \nu } \right ] , \end{aligned}$$\end{document}l = tr-vv+m22vv+vjv,where the interactions are built to describe a particle species \documentclass[12pt]{minimal } 
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				 \begin{document}$$x$$\end{document}x , is dual to another theory14\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathrm { tr}}\left [ \frac{1}{6}{\mathcal { b}}_{\mu \nu \lambda } { \mathcal { b}}^{\mu \nu \lambda } -\frac{m^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } + { \mathcal { b}}_{\mu \nu } j_{\mathcal { b}}^{\mu \nu } \right ] , \end{aligned}$$\end{document}l = tr16bb-m22bb+bjb,which describes the same particles \documentclass[12pt]{minimal } 
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				 \begin{document}$$x$$\end{document}x provided that \documentclass[12pt]{minimal } 
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				 \begin{document}$$j_{\mathcal { b}}^{\mu \nu } = \frac{1}{2}\varepsilon ^{\mu \nu \lambda \rho } j_{{\mathcal { v}}\lambda \rho } $ $ \end{document}jb=12jv. as a corollary , in the absence of masses , the original gauge transformation of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \mathcal { b}}_{\mu \nu } = \partial _ { \mu } \lambda _ { \nu } -\partial _ { \nu } \lambda _ { \mu } $ $ \end{document}b=- is cast in terms of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v as \documentclass[12pt]{minimal } 
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				 \begin{document}$$\delta { \mathcal { v}}_{\mu \nu } = \epsilon _ { \mu \nu \lambda \rho } \partial ^{\lambda } \lambda ^{\rho } $ $ \end{document}v=. the duality above means that every spin-1 particle can be equivalently expressed in terms of either a longitudinal or a transverse two - form field . in theories with parity conservation , particles with the same charge conjugation but opposite parity can be described with \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b fields , respectively , while leaving the form of the interaction terms untouched . thus , if an interacting theory for a regular \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{--}$$\end{document}1 - - vector is described by a longitudinal field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } $ $ \end{document}v , the corresponding theory for a pseudovector \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{+-}$$\end{document}1 + - can be constructed simply by replacing \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { v}}_{\mu \nu } \rightarrow { \mathcal { b}}_{\mu \nu } $ $ \end{document}vb in the interaction terms . if new strongly coupled dynamics trigger electroweak symmetry breaking , a mass gap will typically be generated between their associated goldstone bosons ( at the electroweak scale \documentclass[12pt]{minimal } 
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				 \begin{document}$$v$$\end{document}v ) and bound states ( typically at the tev scale , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\lambda \sim 4\pi v$$\end{document}4v or slightly below ) . in order to work with a minimal model , the following extra assumptions will be made : ( i ) the new strong sector is parity - conserving and invariant under \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_l\times \mathrm{su}(2)_r$$\end{document}su(2)lsu(2)r , spontaneously broken down to the diagonal subgroup \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_v$$\end{document}su(2)v . this is the minimal coset structure required to give gauge bosons a mass ; ( ii ) the higgs - like scalar originates from spontaneous breaking of an extended space - time or internal global symmetry , which occurs at a higher energy scale . i will not model such breaking and instead i will generically introduce the higgs as a singlet under the standard model gauge group ; ( iii ) the first state in the resonance spectrum is a light \documentclass[12pt]{minimal } 
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				 \begin{document}$$j^\mathrm{pc}=1^{+-}$$\end{document}jpc=1 + - pseudovector mode in the mass range 6001,000  gev . vector \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{--}$$\end{document}1 - - and axial \documentclass[12pt]{minimal } 
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				 \begin{document}$$1^{++}$$\end{document}1++ excitations are above the tev , as suggested by electroweak precision constraints . their lowest - lying candidate is the \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_1$$\end{document}b1(1,235 ) meson , which is slightly lighter than the first axial excitation , \documentclass[12pt]{minimal } 
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				 \begin{document}$$a_1$$\end{document}a1(1,260 ) . their properties and phenomenology have been explored in a number of papers , from dispersive analysis involving spectral sum rules  [ 2123 ] to holographic models for spin-1 states  [ 2427 ] , including its low - energy impact on chiral couplings  . here i will be assuming that , unlike in qcd , the pseudovector resonance is the lowest - lying state of the composite spectrum . it will be described by a kalb  ramond field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b , transforming as a triplet under the unbroken custodial group:15\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { b}}_{\mu \nu } \rightarrow \xi _ v{\mathcal { b}}_{\mu \nu } \xi _ v^{\dagger } \end{aligned}$$\end{document}bvbvwhere \documentclass[12pt]{minimal } 
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				 \begin{document}$$\xi _ v\in \mathrm{su}(2)_v$$\end{document}vsu(2)v . the goldstone bosons coming from spontaneous \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_l\times \mathrm{su}(2)_r\rightarrow \mathrm{su}(2)_v$$\end{document}su(2)lsu(2)rsu(2)v breaking will be collected in a nonlinear matrix \documentclass[12pt]{minimal } 
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				 \begin{document}$$u$$\end{document}u , which transforms as a bifundamental:16\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } u\rightarrow \xi _ l u \xi _ r^{\dagger } \end{aligned}$$\end{document}ulurwhere \documentclass[12pt]{minimal } 
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				 \begin{document}$$\xi _ { l , r}\in \mathrm{su}(2)_{l , r}$$\end{document}l , rsu(2)l , r . a convenient parametrization is17\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } u={\mathrm { exp}}(2i\phi /v),\quad \phi = \phi ^at^a=\frac{1}{\sqrt{2}}\left ( \begin{array}{cc } \frac{\phi ^0}{\sqrt{2 } } & { } \phi ^+ \\ \end{aligned}$$\end{document}u = exp(2i/v),=ata=1202+--02.since only \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_l\times u(1)_y$$\end{document}su(2)lu(1)y is gauged , its covariant derivative is given by18\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } d_{\mu } u=\big [ \partial _ { \mu } + igw_{\mu } -ig^{\prime } b_{\mu } \tau _ l\big ] u \end{aligned}$$\end{document}du=[+igw-igbl]uwhere \documentclass[12pt]{minimal } 
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				 \begin{document}$$\tau _ l = ut_3u^{\dagger } $ $ \end{document}l = ut3u. in practice it will be useful to also define \documentclass[12pt]{minimal } 
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				 \begin{document}$$l_{\mu } = iud_{\mu } u^{\dagger } $ $ \end{document}l=iudu. in order to couple the pseudovector to the standard model fields in a gauge - invariant way , it is convenient to introduce quantities that transform only under \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_v$$\end{document}su(2)v . this can be done with the aid of the more fundamental field \documentclass[12pt]{minimal } 
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				 \begin{document}$$u$$\end{document}u19\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } u\rightarrow \xi _ = \xi _ vu\xi _ r^{\dagger } , \quad u^2=u .\end{aligned}$$\end{document}uluv=vur,u2=u.the building blocks covariant under \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_v$$\end{document}su(2)v are then20\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } u^{\dagger } \psi _ l&\rightarrow \xi _ v u^{\dagger } \psi _ l\nonumber \\ u\psi _ r&\rightarrow \xi _ v u\psi _ r\nonumber \\ u_{\mu } & \rightarrow \xi _ v u_{\mu } \xi _ v^{\dagger } \nonumber \\ f_{+}^{\mu \nu } & \rightarrow \xi _ v f_{+}^{\mu \nu } \xi _ v^{\dagger } \end{aligned}$$\end{document}ulvulurvuruvuvf+vf+vwhere \documentclass[12pt]{minimal } 
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				 \begin{document}$$u_{\mu } \!=-u^{\dagger } l_{\mu } u$$\end{document}u=-ulu and \documentclass[12pt]{minimal } 
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				 \begin{document}$$f_{+}^{\mu \nu } \!=u^{\dagger } gw^{\mu \nu } u+ug^{\prime } b^{\mu \nu } t_3u^{\dagger } $ $ \end{document}f+=ugwu+ugbt3u. further derivatives on the fields can be shown to be redundant . the covariant derivative on \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b is defined as21\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \nabla _ { \lambda } { \mathcal { b}}_{\mu \nu } & = \partial _ { \lambda } { \mathcal { b}}_{\mu \nu } + [ \gamma _ { \lambda } , { \mathcal { b}}_{\mu \nu } ] \end{aligned}$$\end{document}b=b+[,b]with22\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \gamma _ { \lambda } & = \frac{1}{2}\bigg [ u(\partial _ { \lambda } -ig^{\prime } b_{\lambda } t_3)u^{\dagger } + u^{\dagger } ( \partial _ { \lambda } -igw_{\lambda } ) u\bigg ] .\end{aligned}$$\end{document}=12[u(-igbt3)u+u(-igw)u].the model of electroweak interactions at energies above \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\mathcal { b}}$$\end{document}mb will be written as23\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathcal { l}}_k+{\mathcal { l}}_m+{\mathcal { l}}_{\mathcal { b } } \end{aligned}$$\end{document}l = lk+lm+lbwhere24\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_k&=-\sum _ { x}\frac{1}{4}x_{\mu \nu } ^ax^{\mu \nu \,a}+i\sum _ j{\bar{\psi } } _ j\gamma _ { \mu } d^{\mu } \psi _ j+\frac{1}{2}\partial _ { \mu } h\partial ^{\mu } h\nonumber \\ { \mathcal { l}}_m&\!=\!\frac{v^2}{4}{\mathrm { tr}}[l_{\mu } l^{\mu } ] \zeta ( h)\!-\!v\big ( { \bar{\psi } } _ iy_{ij}(h)up_{\pm } \psi _ j\!+{\mathrm { h.c.}}\big ) \!-v(h ) \end{aligned}$$\end{document}lk=-x14xaxa+ijjdj+12hhlm = v24tr[ll](h)-v(iyij(h)upj+h.c.)-v(h)is the leading order electroweak lagrangian in the notation of  , where \documentclass[12pt]{minimal } 
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				 \setlength{\oddsidemargin}{-69pt } \begin{document}$$x$$\end{document}x is a generic gauge boson and \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h the higgs - like scalar . \documentclass[12pt]{minimal } 
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				 \begin{document}$$p_{\pm } $ $ \end{document}p simply project out the first and second element of the doublet \documentclass[12pt]{minimal } 
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				 \begin{document}$$\psi _ r$$\end{document}r , respectively . since \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h is introduced as a singlet , \documentclass[12pt]{minimal } 
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				 \begin{document}$$y_{ij}(h)$$\end{document}yij(h ) are generic functions of \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h , whose particular coefficients are model - dependent . the pseudovector sector is chosen as25\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_{\mathcal { b}}&={\mathrm { tr}}\left [ \frac{1}{2}\nabla _ { \lambda } { \mathcal { b}}^{\mu \nu } \nabla ^{\lambda } { \mathcal { b}}_{\mu \nu } -\nabla ^{\mu } { \mathcal { b}}_{\mu \nu } \nabla _ { \rho } { \mathcal { b}}^{\rho \nu } -\frac{m_{\mathcal { b}}^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } \right ] \nonumber \\&\quad + h_t({\bar{\psi } } _ r)\nonumber \\&\quad + { \mathrm { h.c.}}+{\mathrm { tr}}\bigg [ f_1{\mathcal { b}}_{\mu \nu } f_+^{\mu \nu } + if_2{\mathcal { b}}_{\mu \nu } [ u^{\mu } , u^{\nu } ] \bigg ] , \end{aligned}$$\end{document}lb = tr12bb-bb-mb22bb+ht(lubup+r)+hb(lubup-r)+h.c.+tr[f1bf++if2b[u,u]],which contains the most general interaction terms linear in \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}$$\end{document}b . interactions with \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h can be incorporated by appending arbitrary functions of \documentclass[12pt]{minimal } 
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				 \begin{document}$$h$$\end{document}h to those operators . here based on naive dimensional counting , one expects the interaction coefficients to gauge bosons to be \documentclass[12pt]{minimal } 
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				 \begin{document}$$f_i\sim { \mathcal { o}}(v)$$\end{document}fio(v ) . regarding the fermion couplings , chirality suggests that \documentclass[12pt]{minimal } 
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				 \begin{document}$$h_j$$\end{document}hj are proportional to the yukawa couplings . in order to comply with flavor bounds , we will implicitly assume that the model falls into the framework of minimal flavor violation  . accordingly , the pseudovector couples mostly to the third family of quarks . in the following , \documentclass[12pt]{minimal } \usepackage{amsmath } 
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				 \begin{document}$$\psi $ $ \end{document} will stand for a \documentclass[12pt]{minimal } 
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				 \begin{document}$$(t , b)$$\end{document}(t , b ) su\documentclass[12pt]{minimal } 
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				 \begin{document}$$(2)$$\end{document}(2 ) doublet . the presence of a pseudovector in the sub - tev region leaves potential imprints on the low - energy theory through contributions to anomalous couplings . the size of these contributions will show up as \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(v^2/m_{\mathcal { b}}^2)$$\end{document}o(v2/mb2 ) nlo corrections and can be captured efficiently by an effective field theory ( eft ) analysis provided that \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\mathcal { b}}\sim { \mathcal { o}}(\lambda ) $ $ \end{document}mbo( ) . this is the scaling expected from a bound state of strongly coupled dynamics , and guarantees that the nlo new - physics counterterms match the size of the lo loop divergences , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(v^2/\lambda ^2)$$\end{document}o(v2/2 ) . this assumption is the starting point to build a power counting in strongly coupled theories , e.g. in qcd . to be more quantitative , i will choose \documentclass[12pt]{minimal } 
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				 \begin{document}$$m_{\mathcal { b } } > \frac{\lambda } { 5}\sim 600$$\end{document}mb>5600 gev as a lower bound in order not to upset this naive power counting . indirect pseudovector effects can be seen by integrating it out from the model of the previous section . the resulting effective lagrangian will match onto a subset of the nlo terms of the eft worked out in  . writing the model as26\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}&={\mathrm { tr}}\left [ \frac{1}{6}{\mathcal { b}}_{\mu \nu \lambda } { \mathcal { b}}^{\mu \nu \lambda } -\frac{m_{\mathcal { b}}^2}{2}{\mathcal { b}}_{\mu \nu } { \mathcal { b}}^{\mu \nu } + { \mathcal { b}}_{\mu \nu } j^{\mu \nu } \right ] \end{aligned}$$\end{document}l = tr16bb-mb22bb+bjand integrating out the pseudovector at tree level one ends up with the effective lagrangian127\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l } } _ { { { eff}}}&\!=\!\frac{1}{m_{\mathcal { b}}^2}j_{\mu \nu } ^ij_i^{\mu \nu } \!\!=\sum _ j\!\left [ c_j^{(2)}{\mathcal { o}}_j^{(2)}\!+c_j^{(3)}{\mathcal { o}}_j^{(3)}\!+c_j^{(4)}{\mathcal { o}}_j^{(4)}\right ] \end{aligned}$$\end{document}leff=1mb2jiji=jcj(2)oj(2)+cj(3)oj(3)+cj(4)oj(4)where28\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } j_{\mu \nu } ^{j}&\equiv \langle t^jj_{\mu \nu } \rangle = f_1~{\mathrm { tr}}\!\left ( t^jf_{+\mu \nu } \right ) + if_2~{\mathrm { tr}}\left ( t^j[u_{\mu } , u_{\nu } ] \right ) \nonumber \\&\quad + h_t({\bar{\psi } } _ l\sigma _ { \mu \nu } ut^jup_+\psi _ r)+h_b({\bar{\psi } } _ l\sigma _ { \mu \nu } ut^jup_-\psi _ r)\nonumber \\&\quad + { \mathrm { h.c . } } \end{aligned}$$\end{document}jjtjj=f1trtjf++if2trtj[u,u]+ht(lutjup+r)+hb(lutjup-r)+h.c.and use has to be made of the su\documentclass[12pt]{minimal } 
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				 \begin{document}$$(2)$$\end{document}(2 ) relation29\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } t_{ij}^at^a_{kl}&=\frac{1}{2}\delta _ { il}\delta _ { jk}-\frac{1}{4}\delta _ { ij}\delta _ { kl } .\end{aligned}$$\end{document}tijatkla=12iljk-14ijkl . the effective operators \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}_j^{(i)}$$\end{document}oj(i ) describe the anomalous oblique , triple , and quartic vertices depicted in fig . they are listed , together with their corresponding coefficients , in table  1 . from it one would naively conclude that two- , three- , and four - point vertices get modified . however , if one computes explicitly the corresponding diagrams with pseudovector exchange , one realizes that only for the four - point vertices there is actual resonance propagation . pseudovector propagation in two and three - point vertices is forbidden by \documentclass[12pt]{minimal } 
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				 \begin{document}$$p$$\end{document}p and \documentclass[12pt]{minimal } 
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				 \begin{document}$$c$$\end{document}c conservation , and the contribution reported in table  1 corresponds to a contact term interaction . they correspond to anomalous standard model vertices , except the last topology , which is not present in the standard model different low - energy topologies affected by pseudovector exchange . they correspond to anomalous standard model vertices , except the last topology , which is not present in the standard model such contact interactions are licit and physically relevant provided that they do not spoil the consistency of the theory at high energies . even though i have not committed to a uv completion of the present model , any strongly coupled scenario implicitly requires duality , above the deconfinement scale , to a theory of more fundamental constituents . the paradigmatic example is qcd , where hadrons are made of quarks and gluons . in order to link the confined and deconfined phases if the contact terms do not conform to the expected high - energy behavior of the theory , then the model has to be extended with the addition of local operators such that the mismatch is avoided . a similar strategy was adopted in qcd to assess the impact of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$b_1$$\end{document}b1(1,235 ) meson in chiral dynamics  .table 1low - energy effective operators and coefficients resulting from pseudovector integration . , there is only net contribution from the four - fermion operatorsoperators \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}_j^{(i)}$$\end{document}oj(i ) 
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				 \begin{document}$$c_j^{(i)}$$\end{document}cj(i ) 

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				 \begin{document}$${\mathcal { o}}_1^{(2)}=gg^{\prime } \langle w_{\mu \nu } \tau _ l\rangle b^{\mu \nu } $ $ \end{document}o1(2)=ggwlb 

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				 \begin{document}$$f_1 ^ 2/m_{\mathcal { b}}^2$$\end{document}f12/mb2 

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				 \begin{document}$${\mathcal { o}}_1^{(3)}=g\langle w_{\mu \nu } [ l^{\mu } , l^{\nu } ] \rangle $ $ \end{document}o1(3)=gw[l,l] 

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				 \begin{document}$$if_1f_2/m_{\mathcal { b}}^2$$\end{document}if1f2/mb2 

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				 \begin{document}$${\mathcal { o}}_2^{(3)}=g^{\prime } b_{\mu \nu } \langle \tau _ l[l^{\mu } , l^{\nu } ] \rangle $ $ \end{document}o2(3)=gbl[l,l] 

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				 \begin{document}$$if_1f_2/m_{\mathcal { b}}^2$$\end{document}if1f2/mb2 

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				 \begin{document}$${\mathcal { o}}_3^{(3)}=g{\bar{\psi } } _ l\sigma _ { \mu \nu } w^{\mu \nu } up_+\psi _ r$$\end{document}o3(3)=glwup+r 

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				 \begin{document}$$f_1h_t / m_{\mathcal { b}}^2$$\end{document}f1ht / mb2 

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				 \begin{document}$${\mathcal { o}}_4^{(3)}=g{\bar{\psi } } _ l\sigma _ { \mu \nu } w^{\mu \nu } up_-\psi _ r$$\end{document}o4(3)=glwup-r 

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				 \begin{document}$$f_1h_b / m_{\mathcal { b}}^2$$\end{document}f1hb / mb2 

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				 \begin{document}$${\mathcal { o}}_5^{(3)}=g^{\prime } { \bar{\psi } } _ l\sigma _ { \mu \nu } b^{\mu \nu } ut_3p_+\psi _ r$$\end{document}o5(3)=glbut3p+r 

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				 \begin{document}$$f_1h_t / m_{\mathcal { b}}^2$$\end{document}f1ht / mb2 

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				 \begin{document}$${\mathcal { o}}_6^{(3)}=g^{\prime } { \bar{\psi } } _ l\sigma _ { \mu \nu } b^{\mu \nu } ut_3p_-\psi _ r$$\end{document}o6(3)=glbut3p-r 

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				 \begin{document}$$f_1h_b / m_{\mathcal { b}}^2$$\end{document}f1hb / mb2 

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				 \begin{document}$${\mathcal { o}}_1^{(4)}=\langle l_{\mu } l_{\nu } \rangle \langle l^{\mu } l^{\nu } \rangle $ $ \end{document}o1(4)=llll 

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				 \begin{document}$$-f_2 ^ 2/m_{\mathcal { b}}^2$$\end{document}-f22/mb2 

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				 \begin{document}$${\mathcal { o}}_2^{(4)}=\langle l_{\mu } l^{\mu } \rangle \langle l_{\nu } l^{\nu } \rangle $ $ \end{document}o2(4)=llll 

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				 \begin{document}$$f_2 ^ 2/m_{\mathcal { b}}^2$$\end{document}f22/mb2 

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				 \begin{document}$${\mathcal { o}}_3^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_+\psi _ r)({\bar{\psi } } _ l\sigma _ { \mu \nu } up_+\psi _ r)$$\end{document}o3(4)=(lup+r)(lup+r ) 

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				 \begin{document}$$h_t^2/(4m_{\mathcal { b}}^2)$$\end{document}ht2/(4mb2 ) 

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				 \begin{document}$${\mathcal { o}}_4^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_-\psi _ r)({\bar{\psi } } _ l\sigma _ { \mu \nu } up_-\psi _ r)$$\end{document}o4(4)=(lup-r)(lup-r ) 

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				 \begin{document}$$h_b^2/(4m_{\mathcal { b}}^2)$$\end{document}hb2/(4mb2 ) 

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				 \begin{document}$${\mathcal { o}}_5^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_-\psi _ r)({\bar{\psi } } _ l\sigma _ { \mu \nu } up_+\psi _ r)$$\end{document}o5(4)=(lup-r)(lup+r ) 

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				 \begin{document}$$h_th_b/(2m_{\mathcal { b}}^2)$$\end{document}hthb/(2mb2 ) 

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				 \begin{document}$${\mathcal { o}}_6^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_-\psi _ r)({\bar{\psi } } _ rp_+\sigma _ { \mu \nu } u^{\dagger } \psi _ l)$$\end{document}o6(4)=(lup-r)(rp+ul ) 

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				 \begin{document}$$-h_bh_t^*/(2m_{\mathcal { b}}^2)$$\end{document}-hbht/(2mb2 ) 

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				 \begin{document}$${\mathcal { o}}_7^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_-\psi _ r)({\bar{\psi } } _ rp_-\sigma _ { \mu \nu } u^{\dagger } \psi _ l)$$\end{document}o7(4)=(lup-r)(rp-ul ) 

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				 \begin{document}$$-h_bh_b^*/(2m_{\mathcal { b}}^2)$$\end{document}-hbhb/(2mb2 ) 

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				 \begin{document}$${\mathcal { o}}_8^{(4)}=({\bar{\psi } } _ l\sigma _ { \mu \nu } up_+\psi _ r)({\bar{\psi } } _ rp_+\sigma _ { \mu \nu } u^{\dagger } \psi _ l)$$\end{document}o8(4)=(lup+r)(rp+ul ) 

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				 \begin{document}$$-h_th_t^*/(2m_{\mathcal { b}}^2)$$\end{document}-htht/(2mb2 ) 

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				 \begin{document}$${\mathcal { o}}_9^{(4)}={\bar{\psi } } _ l\sigma _ { \mu \nu } [ l^{\mu } , l^{\nu } ] up_+\psi _ r$$\end{document}o9(4)=l[l,l]up+r 

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				 \begin{document}$$if_2h_t / m_{\mathcal { b}}^2$$\end{document}if2ht / mb2 

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				 \begin{document}$${\mathcal { o}}_{10}^{(4)}={\bar{\psi } } _ l\sigma _ { \mu \nu } [ l^{\mu } , l^{\nu } ] up_-\psi _ r$$\end{document}o10(4)=l[l,l]up-r 

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				 \begin{document}$$if_2h_b / m_{\mathcal { b}}^2$$\end{document}if2hb / mb2 
  low - energy effective operators and coefficients resulting from pseudovector integration . , there is only net contribution from the four - fermion operators consider the two - point function \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi _ { wb}$$\end{document}wb , defined as30\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \pi _ { wb}^{\mu \nu } ( q)&=\frac{\delta ^2}{\delta w_{\mu } ^3\delta b_{\nu } } { \mathcal { l}}=\left ( \frac{q^{\mu } q^{\nu } } { q^2}-g^{\mu \nu } \right ) \pi _ { wb}(q^2 ) .\end{aligned}$$\end{document}wb(q)=2w3bl = qqq2-gwb(q2).this correlator is an order parameter of custodial symmetry breaking . at high energies , where the global group su\documentclass[12pt]{minimal } 
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				 \begin{document}$$(2)_l\times \mathrm{su}(2)_r$$\end{document}(2)lsu(2)r is unbroken , one expects perturbative contributions in terms of the fundamental constituents to cancel to all orders . the correlator is therefore power - suppressed at high energies and expected to satisfy an unsubtracted dispersion relation31\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \pi _ { wb}(q^2)&=\int _ 0^{\infty } \frac{\mathrm{d}t}{t - q^2}\frac{1}{\pi } { \mathrm { im}}\pi _ { wb}(t ) .\end{aligned}$$\end{document}wb(q2)=0dtt - q21imwb(t).the contribution from the pseudovector to the absorptive part above is exactly zero , because it is not a propagating mode . actually , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi _ { wb}^{({\mathcal { b}})}(q^2){\,=}-gg^{\prime } \frac{f_1 ^ 2}{m_{\mathcal { b}}^2}q^2$$\end{document}wb(b)(q2)=-ggf12mb2q2 thus effectively generating a subtraction , which would indicate breaking of custodial symmetry in the deep uv and is therefore ruled out by general principles . therefore , the pseudovector model is consistent only when there is no net effect on \documentclass[12pt]{minimal } 
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				 \begin{document}$$\pi _ { wb}$$\end{document}wb . three - point vertices can likewise be examined through a dispersive approach . for qcd - inspired uv completions , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathrm { im}}\pi _ { xx}$$\end{document}imxx scales like a constant in the deep uv . under this assumption 1 must obey once - subtracted dispersion relations of the form32\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } f_{x\rightarrow { \bar{f}}f}(q^2)&=1+q^2\int _ 0^{\infty } \frac{\mathrm{d}t}{t(t - q^2)}\frac{1}{\pi } { \mathrm { im}}f_{x\rightarrow { \bar{f}}f}(t ) \end{aligned}$$\end{document}fxff(q2)=1+q20dtt(t - q2)1imfxff(t)and likewise for \documentclass[12pt]{minimal } 
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				 \begin{document}$$x^i\rightarrow x^jx^k$$\end{document}xixjxk . again , the fact that the pseudovector does not contribute to the absorptive part means that counterterms are needed to avoid the appearance of an unphysical subtraction . there pseudovector exchange has a nonzero absorptive part ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$c$$\end{document}c and \documentclass[12pt]{minimal } 
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				 \begin{document}$$p$$\end{document}p no longer prevent resonance propagation ) and its contribution resembles the one of regular vector resonances . consider the longitudinal gauge - boson scattering \documentclass[12pt]{minimal } 
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				 \begin{document}$$w_l^aw_l^b\rightarrow w_l^cw_l^d$$\end{document}wlawlbwlcwld . by the equivalence theorem , at high energies this corresponds to the scattering of goldstones \documentclass[12pt]{minimal } 
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				 \begin{document}$$\phi ^a\phi ^b\rightarrow \phi ^c\phi ^d$$\end{document}abcd , up to corrections of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(m_w/\sqrt{s})$$\end{document}o(mw / s ) . goldstone scattering is determined by a single function \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { a}}(x , y , z)$$\end{document}a(x , y , z ) , where \documentclass[12pt]{minimal } 
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				 \begin{document}$$x , y , z$$\end{document}x , y , z are the mandelstam variables . since \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { a}}$$\end{document}a is symmetric in the last two arguments , only the first argument need to be kept above and one can define \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { a}}(x)\equiv { \mathcal { a}}(x , y , z)={\mathcal { a}}(x , z , y)$$\end{document}a(x)a(x , y , z)=a(x , z , y ) . the total amplitude reads33\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { m}}(\phi ^a\phi ^b\rightarrow \phi ^c\phi ^d)&=\delta ^{ab}\delta ^{cd}\!{\mathcal { a}}(s)+\delta ^{ac}\delta ^{bd}\!{\mathcal { a}}(t)\nonumber \\&\quad + \delta ^{ad}\delta ^{bc}\!{\mathcal { a}}(u ) .\end{aligned}$$\end{document}m(abcd)=abcda(s)+acbda(t)+adbca(u).in elastic channels with \documentclass[12pt]{minimal } 
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				 \begin{document}$$s\leftrightarrow u$$\end{document}su symmetry , the froissart bound   leads to the following forward ( once - subtracted ) dispersion relation:34\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } a(\nu , t=0)&=c+\nu ^2\!\int _ 0^{\infty } \!\!\frac{\mathrm{d}\eta } { \eta ( \eta ^2-\nu ^2)}\frac{1}{\pi } { \mathrm { im}}a(\eta , 0 ) \end{aligned}$$\end{document}a(,t=0)=c+20d(2-2)1ima(,0)where \documentclass[12pt]{minimal } 
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				 \begin{document}$$\nu = \frac{1}{2}(s - u)$$\end{document}=12(s - u ) . if the new strong dynamics obeys the froissart bound at asymptotically high energies , then at high \documentclass[12pt]{minimal } 
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				 \begin{document}$$\nu ^2$$\end{document}2 the previous amplitudes should go like a constant . explicit evaluation of pseudovector exchange shows that pseudovector exchange is non - propagating in goldstone scattering . this generates a quadratic growth in  ( 34 ) , which again calls for the addition of counterterms . the non - propagating character of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}$$\end{document}b in longitudinal high - energy scattering also suggests that the four - point vertices \documentclass[12pt]{minimal } 
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				 \begin{document}$$x^ax^b\rightarrow { \bar{f}}f$$\end{document}xaxbff will likewise need counterterms . actually , explicit calculation shows that pseudovector contributions to \documentclass[12pt]{minimal } 
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				 \begin{document}$$\phi ^a\phi ^b\rightarrow { \bar{f}}f$$\end{document}abff only generate a contact term . a consistent pseudovector model thus requires to be enlarged with the following counterterms:35\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } { \mathcal { l}}_c&=-\frac{f_1 ^ 2}{m_{\mathcal { b}}^2}{\mathcal { o}}_{1}^{(2)}-i\frac{f_1f_2}{m_{\mathcal { b}}^2}\left ( { \mathcal { o}}_1^{(3)}+{\mathcal { o}}_2^{(3)}\right ) \nonumber \\&\quad -\frac{f_1h_t}{m_{\mathcal { b}}^2}\left ( { \mathcal { o}}_3^{(3)}+{\mathcal { o}}_5^{(3)}\right ) -\frac{f_1h_b}{m_{\mathcal { b}}^2}\left ( { \mathcal { o}}_4^{(3)}+{\mathcal { o}}_6^{(3)}\right ) + { \mathrm { h.c.}}\nonumber \\&\quad + \frac{f_2 ^ 2}{m_{\mathcal { b}}^2}\left ( { \mathcal { o}}_1^{(4)}-{\mathcal { o}}_2^{(4)}\right ) -i\frac{f_2h_t}{m_{\mathcal { b}}^2}{\mathcal { o}}_9^{(4 ) } -i\frac{f_2h_b}{m_{\mathcal { b}}^2}{\mathcal { o}}_{10}^{(4)}+{\mathrm { h.c . } } , \end{aligned}$$\end{document}lc =- f12mb2o1(2)-if1f2mb2o1(3)+o2(3)-f1htmb2o3(3)+o5(3)-f1hbmb2o4(3)+o6(3)+h.c.+f22mb2o1(4)-o2(4)-if2htmb2o9(4)-if2hbmb2o10(4)+h.c.,which precisely balance out the naive pseudovector low - energy contributions . the net result is that , with the exception of four - fermion topologies , low - energy traces of the pseudovector are obliterated . since the conclusions above are based on generic high - energy properties , they are independent of particular resonance models above the tev scale . however , in order to understand the physical entity of the local terms introduced above , consider enlarging the model we have presented here to a model of large-\documentclass[12pt]{minimal } 
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				 \begin{document}$$n_c$$\end{document}nc by inserting a full tower of pseudovector excitations all the way up to infinity . in this case , counterterms would be absent and consistency with dispersion relations would instead be satisfied through the existence of a set of sum rules , where the added contributions of the states in the tower would be required to non - trivially vanish . unlike more conventional sum rules , those would , however , not be spectral sum rules . the counterterms introduced above can therefore be understood as the integrated - out contribution of a full tower of pseudovectors , with the exception of its lowest - lying mode , which at the energies we are interested in is light enough to be dynamical . the results of the previous section show how elusive the light pseudovector can be for indirect detection . its low - energy traces reduce to anomalous four - fermion interactions , which turn out to be mass - suppressed . therefore , only top quark four - point vertices are nonnegligible with \documentclass[12pt]{minimal } 
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				 \begin{document}$$c_{4t}\sim { \mathcal { o}}(1/\lambda ^2)$$\end{document}c4to(1/2 ) , yet extremely challenging to probe . unlike most of the existing new - physics candidates , a pseudovector can in practice only be detected via direct searches . in this section i will discuss the most promising modes for direct detection of pseudovectors in a rather qualitative way . this section is intended to provide a broad - brush picture of the predominant channels for discovery . a more detailed and quantitative study of the specific phenomenology and prospects for detection is postponed to a future publication . ( 25 ) shows that the neutral pseudovector component decays predominantly into a gauge - boson pair and a top pair , \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^{0}\rightarrow w^+w^-$$\end{document}b0w+w- and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^0\rightarrow { \bar{t}}t$$\end{document}b0tt . the charged component instead is dominated by \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^{\pm } \rightarrow w^{\pm } z$$\end{document}bwz and \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^{\pm } \rightarrow { \bar{t}}b$$\end{document}btb.2 explicit computation gives36\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \gamma _ { \mathcal { b}}^{(w^+w^-)}&=\frac{g^4}{48\pi } \left ( \frac{f_2 ^ 2}{m_{\mathcal { b}}}\right ) \frac{(1 + 2x_w^2)\sqrt{1 - 4x_w^2}}{x_w^2 } \end{aligned}$$\end{document}b(w+w-)=g448f22mb(1 + 2xw2)1 - 4xw2xw2where \documentclass[12pt]{minimal } 
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				 \begin{document}$$x_w=\displaystyle \frac{m_w}{m_{\mathcal { b}}}$$\end{document}xw = mwmb . incidentally , if one compares  ( 36 ) with the same decay for a regular vector  , one finds that the pseudovector has a relative \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { o}}(v^2/m_{\mathcal { b}}^2)$$\end{document}o(v2/mb2 ) suppression . assuming cp - conserving interactions , the decay into a top pair is given by37\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \gamma _ { \mathcal { b}}^{({\bar{t}}t)}&=\frac{h_t^2}{24\pi } m_{\mathcal { b}}(1 - 4x_t^2)^{3/2 } \end{aligned}$$\end{document}b(tt)=ht224mb(1 - 4xt2)3/2where \documentclass[12pt]{minimal } 
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				 \begin{document}$$x_t=\displaystyle \frac{m_t}{m_{\mathcal { b}}}$$\end{document}xt = mtmb . notice that since \documentclass[12pt]{minimal } 
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				 \begin{document}$$\gamma _ { \mathcal { b}}^{(ww)}\!\!\sim { \mathcal { o}}(g^2m_{\mathcal { b}})$$\end{document}b(ww)o(g2 mb ) and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\gamma _ { \mathcal { b}}^{({\bar{t}}t)}\!\!\sim { \mathcal { o}}(h_t^2m_{\mathcal { b}})$$\end{document}b(tt)o(ht2 mb ) , both decay modes are of comparable size:38\documentclass[12pt]{minimal } 
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				 \begin{document}$$\begin{aligned } \frac{\gamma _ { \mathcal { b}}^{(w^+w^-)}}{\gamma _ { \mathcal { b}}^{({\bar{f}}f)}}\sim \frac{g^2}{y_t^2}\sim { \mathcal { o}}(1 ) .\end{aligned}$$\end{document}b(w+w-)b(ff)g2yt2o(1).numerically , for the mass range of interest , \documentclass[12pt]{minimal } 
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				 \begin{document}$$\gamma _ { \mathcal { b}}\sim 2\,\%\ , m_{\mathcal { b}}$$\end{document}b2%mb . regarding its detection at the lhc , the pseudovector is most favorably produced through gluon fusion and subsequent decay into \documentclass[12pt]{minimal } 
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				 \begin{document}$${\bar{t}}t$$\end{document}tt jets and \documentclass[12pt]{minimal } 
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				 \begin{document}$$w^+w^-$$\end{document}w+w- ( see upper panel in fig .  2 ) . the latter decay mode offers the cleaner signal for detection , and direct searches should not differ much from the ones already performed for the higgs decay \documentclass[12pt]{minimal } 
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				 \begin{document}$$w$$\end{document}w decay leptonically , and in searches for heavier scalar states  . however , due to the high pseudovector mass , it might be more efficient to let one of the w decay into a quark pair . the main background \documentclass[12pt]{minimal } 
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				 \begin{document}$$w+\,$$\end{document}w+jets can then be efficiently rejected by applying a cut on the neutrino transverse momentum  . unlike the higgs , a specific feature of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}$$\end{document}b is that \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^0\rightarrow zz$$\end{document}b0zz is also loop - induced . therefore , there should be an excess of \documentclass[12pt]{minimal } 
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				 \begin{document}$$ww$$\end{document}ww production over \documentclass[12pt]{minimal } 
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				 \begin{document}$$zz$$\end{document}zz . 
 fig . see main text for a detailed discussion notice also that , contrary to the vector scenario studied in  , drell  yan production does not occur in this case , since a ( propagating ) pseudovector can not couple to a single gauge boson . an interesting alternative is to consider the pseudovector in associated production with a gauge boson , as depicted in the lower panel of fig .  2 . the possible decay modes are \documentclass[12pt]{minimal } 
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				 \begin{document}$$w{\bar{t}}t$$\end{document}wtt , \documentclass[12pt]{minimal } 
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				 \begin{document}$$z{\bar{t}}b$$\end{document}ztb , \documentclass[12pt]{minimal } 
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				 \begin{document}$$www$$\end{document}www , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$wzz$$\end{document}wzz . among them , the cleanest ones are \documentclass[12pt]{minimal } 
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				 \begin{document}$$z{\bar{t}}b$$\end{document}ztb with leptonic decay of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$z$$\end{document}z decaying hadronically.3 the pseudovector should then show up by scanning the invariant mass distribution of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$wz$$\end{document}wz pairs , respectively . at the ilc , detection through two - body decay therefore , the only clean signature at the ilc is the associated production of \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}^{\pm } $ $ \end{document}b with \documentclass[12pt]{minimal } 
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				 \begin{document}$$w^{\mp } $ $ \end{document}w. the cleanest signatures are \documentclass[12pt]{minimal } 
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				 \begin{document}$$w$$\end{document}w decaying into leptons , and \documentclass[12pt]{minimal } 
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				 \begin{document}$$w$$\end{document}w decaying into quarks . the pseudovector can be detected in the invariant mass distribution of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$wz$$\end{document}wz pairs , respectively . the most pressing issue in particle physics is to pin down , or at least shed some light on the dynamical mechanism that triggers electroweak symmetry breaking . the existence of a light scalar with properties close to the standard model higgs does not resolve the issue but poses additional requirements that this underlying mechanism must fulfill , namely ( i ) it must provide a mechanism to stabilize the light scalar mass ; and ( ii ) it should manifest itself around the tev scale yet complying with the strongly constrained deviations from the standard model paradigm . dynamical symmetry breaking is a long - standing candidate for such a mechanism . in such scenarios a light scalar can easily be accommodated as a pseudo - goldstone boson of an underlying broken global symmetry . however , light states ( below the tev scale ) are hard to reconcile with the constraints of electroweak precision data . the smallness of standard model deviations suggests that new - physics states should leave a rather subtle imprint . in this paper i have examined the viability of sub - tev states when confronted with experimental constraints . in particular , i have considered a scenario of new dynamics invariant under parity and \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_l\times \mathrm{su}(2)_r$$\end{document}su(2)lsu(2)r , spontaneously broken down to the custodial \documentclass[12pt]{minimal } 
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				 \begin{document}$$\mathrm{su}(2)_v$$\end{document}su(2)v . the lowest - lying resonance in the spectrum is a pseudovector , described as a kalb  ramond antisymmetric tensor field \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b with couplings to the standard model fields . i have shown that such a state is rather elusive in the low - energy theory , not because of additional ad hoc suppressions of its couplings , but by the requirement that the strong dynamics possess a consistent asymptotically free uv completion . applying this criterion one can show that the pseudovector evades the constraints coming from oblique parameters , electric and magnetic dipole moments , as well as triple and quartic gauge - boson vertices . indirect traces thereof would only affect third - family four - quark operators , which are presently poorly constrained . consequently , the existence of a \documentclass[12pt]{minimal } 
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				 \begin{document}$${\mathcal { b}}_{\mu \nu } $ $ \end{document}b could only be tested by direct detection . at a hadronic collider like the lhc , it is mostly produced through gluon fusion and subsequent decay into a \documentclass[12pt]{minimal } 
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				 \begin{document}$$w^+w^-$$\end{document}w+w- pair or a top dijet . in contrast , at the ilc it could only be detected in its associated production with gauge bosons . its signal could be isolated in \documentclass[12pt]{minimal } 
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				 \begin{document}$$wwz$$\end{document}wwz by scanning through the invariant mass distribution of the \documentclass[12pt]{minimal } 
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				 \begin{document}$$wz$$\end{document}wz pairs , respectively . one could also entertain the possibility of pair production with subsequent decay into heavy quarks . although not a dominant process , for rather light pseudovectors this might well be within the lhc reach . a more detailed quantitative analysis of the collider phenomenology would require a dedicated paper and is left for future work .
if electroweak symmetry breaking is driven by a new strongly coupled dynamical sector , one expects the bound states to appear at the tev scale or slightly below . 
 however , electroweak precision data imposes severe constraints on most of the existing models , putting them under strain . 
 conventional models require the new composite states to come in pairs of rather heavy , close to degenerate spin-1 resonances . in this paper 
 i argue that spin-1 states can actually be lighter without clashing with experimental bounds . as an example 
 , i consider a composite model with a light pseudovector resonance that couples to the standard model gauge boson , fermion , and scalar fields . 
 i show how such a resonance leaves basically no imprint on the nlo corrections to the standard model . 
 this happens not through parameter tuning , but rather as a consequence of generic properties of realistic uv completions . 
 this pseudovector is mostly unconstrained by existing data and could be as light as 600  gev . in the last part of the paper 
 i briefly discuss its most characteristic signatures for direct detection at colliders .