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following the discovery in 1961 of @xmath8 meson in @xmath0 reactions , the measurements of forward - backward asymmetry in @xmath9 suggested the existence of a rho - like resonance in the @xmath2-wave amplitudes , later referred to as @xmath10 scalar meson  @xcite . the cern measurements of @xmath11 and @xmath12 on polarized targets in 1970 s cofirmed the existence of @xmath10 scalar meson . evidence for a narrow @xmath10 was found in amplitude analyses of @xmath11 at 17.2 gev / c  @xcite and in @xmath12 at 5.98 and 11.85  @xcite . additional evidence for @xmath10 came from the amplitude analysis of the itep data on @xmath7 on polarized target at 1.78 gev / c  @xcite . the @xmath2-wave amplitudes and intensities from all these analyses are surveyed in ref . these findings were controversial because the measurements of @xmath13 at cern in 1972 found no evidence for a rho - like meson in the @xmath2-wave amplitudes  @xcite . in 2001 , e852 collaboration at bnl reported high statistics measurements @xmath13 at 18.3 gev / c  @xcite . once again , there was no evidence for a rho - like resonance in the @xmath14 @xmath2-wave amplitudes . the resolution of the puzzle of @xmath10 resonance came from our recent high resolution amplitude analysis of the cern data on @xmath7 for dipion masses 580 - 1080 mev at low momentum transfers  @xcite where @xmath2- and @xmath3-wave amplitudes dominate . the analysis shows that the rho - like resonance in the @xmath2-wave transversity amplitudes arises entirely from the contribution of the @xmath15 resonance to the data component in the expression for the moduli of the @xmath2-wave amplitudes . the @xmath2-wave transversity amplitudes are nearly either in phase or 180@xmath16 out of phase with the resonating @xmath3-wave transversity amplitudes . these facts allow us to identify @xmath10 with @xmath15 . there is a pronounced dip at 980 mev in the moduli of @xmath3-wave amplitudes @xmath17 that is identified with @xmath18 resonance leading to @xmath6 mixing in both @xmath2- and @xmath3-wave amplitudes . a model independent determination of helicity amplitudes from the transversity amplitudes shows a @xmath15 peak in the @xmath2-wave single flip helicity amplitude @xmath19  @xcite . the phase of the amplitude @xmath20 is close to the phase of @xmath15 breit - wigner amplitude  @xcite .    identifying @xmath10 with @xmath15 explains why there is no rho - like resonance observed in @xmath13 since there is no @xmath3-wave amplitude in this process . however , the @xmath6 mixing appears to violate rotational and lorentz symmetry . if this symmetry were violated in strong interactions we should observe a dependence of @xmath15 width @xmath21 on its helicity @xmath22 . in our analysis @xcite we demonstrate the independence of @xmath21 on the helicity confirming the rotational and lorentz symmetry of strong interactions . this suggests that the observed @xmath6 mixing arises from a new kind of interaction independently involved in the pion creation process . the observed @xmath6 mixing can be naturally explained when we assume that the produced partial waves are not isolated spin states but interact with a common environment . we can see how this might work using the following mechanical analogy . consider a pendulum of mass @xmath23 and length @xmath24 oscillating with a natural frequency @xmath25 , and an ensemble of several other penduli with various masses @xmath26 and lengths @xmath27 . when the other penduli are isolated from the oscillating pendulum they do not oscillate . however , when all the penduli are attached to a common rod all penduli begin to oscillate with the same resonant frequency although with different amplitudes . the interaction of the penduli with a common environment - the rod - allows a resonance from one pendulum to `` leak '' into the other penduli which have different natural frequencies . similarly we can imagine that the partial waves produced in @xmath7 process interact with a quantum environment which allows resonances from one partial wave amplitude to `` leak '' into other partial wave amplitudes . then the pion creation process is no longer an isolated system but behaves as an open quantum system described by a non - unitary dynamics . a non - unitary evolution law must evolve the produced final state @xmath1 into an observed final state @xmath28 with new spin mixing partial wave amplitudes . but how do we know that such quantum environment actually exists ? and why its interaction with the @xmath29 process should lead to the @xmath6 mixing ? the basic assumption of the @xmath2-matrix theory is that the minkowski spacetime is empty . it contains no environment with which particle scattering processes could interact . in part i. of this work  @xcite we put forward a hypothesis that the physical space is not empty but contains a universal quantum environment that can interact with some particle scattering processes . the new interaction can not change the identity and the four - momenta of the final state particles produced by the @xmath2-matrix dynamics in order to leave the particle scattering dynamics open to the experimental observation . this means the new interaction must be a pure dephasing interaction that modifies phases of the amplitudes and nothing else . this is a new kind of interaction outside of the standard model .    to test our hypothesis we used unitarity of the @xmath2-matrix to obtain information on the relative phases of the measured partial wave transversity amplitudes @xmath30 and @xmath31 . unitary evolution law evolves pure initial states into pure final states . this fact implies that the relative phases are constrained to @xmath32 in a disagreement with the measured phases in all analyses of @xmath0 processes . an amplitude analysis of the @xmath2- and @xmath3-wave subsystem in @xmath7 assuming these unitary phases is in disagreement with the cern data on polarized target at least at @xmath33 level . we conclude that the cern measurements of @xmath7 and @xmath12 on polarized targets support the hypothesis of the existence of the quantum environment and its dephasing interaction with some particle scattering processes . if the quantum environment is to be an integral part of the nature then its dephasing interactions with particle scattering processes can not contradict standard model . the two aspects of the nature must be mutually consistent . what are the cosequences of this consistency ? the dephasing interaction evolves the produced @xmath2-matrix final state @xmath1 into an observed final state @xmath28 . the non - unitary evolution law is described by kraus representation . in this paper we show that the consistency of the pure dephasing interaction with the standard model in @xmath0 processes requires that it be a dipion spin mixing interaction . in what we call spin mixing mechanism the observed partial wave amplitudes are a unitary transform of the corresponding @xmath2-matrix partial wave amplitudes . the elements of the spin mixing matrix are forward scattering amplitudes of the partial waves with initial and final dipion spins @xmath4 and @xmath5 , respectively . helicities are conserved in this scattering . these amplitudes are matrix elements of kraus operators describing the non - unitary interaction with the environment . the theory predicts @xmath6 mixing in the @xmath2-and @xmath3-wave amplitudes in @xmath7 . the predicted moduli and relative phases of the mixed amplitudes are in an excellent qualitative agreement with the experimental results  @xcite . the theory correctly predicts the absence of @xmath34 mixing . the spin mixing of @xmath2-matrix partial wave amplitudes to form new observable partial wave amplitudes is a new phenomenon outside of standard model and represents a genuine new physics . we discuss kraus representation and kraus operators in section ii . the non - unitary evolution of the produced final state @xmath1 into an observed final state @xmath28 is introduced in section iii . in section iv . we use the purity of the dephasing interaction to derive the form of spin mixing mechanism and define kraus partial wave helicity amplitudes which are mixtures of the @xmath2-matrix partial wave amplitudes . in section v. the requirements of lorentz symmetry , conservation of total angular momentum , @xmath3-parity and total isospin impose consistency constraints on the spin mixing mechanism and kraus helicity amplitudes . in section vi . self - consistency requirements constrain the spin mixing mechanism to a simple unitary transform of the @xmath2-matrix amplitudes . in section vii . we relate the observed effective amplitudes to the kraus amplitudes and demonstrate the conservation of probability by the dephasing intaraction .    in section viii . we discuss resonance mixing in @xmath7 below 1400 mev . we use spin mixing mechanism for the @xmath2- and @xmath3-waves to present qualitative predictions for the moduli and relative phases of the spin mixing @xmath2- and @xmath3-wave amplitudes below 1080 mev which we compare with two analyses of experimental data . section ix . reviews the determination of the dimension @xmath35 of the hilbert space of the environment . in section x. we identify the decoherence free and decohering amplitudes below 1400 mev in @xmath7 and briefly discuss our new analysis using spin mixing mechanism  @xcite . in section xi . we use coleman - mandula no - go theorem to show that the dimension @xmath36 and that the amplitudes in the decoherence free subspace must mix spins while the decohering amplitudes can not mix spins to protect the lorentz symmetry of the @xmath2-matrix . the spontaneous violation of the lorentz symmetry observed in the measured amplitudes is thus consistent with lorentz symmetry of the standard model . we conclude in section xii . with a summary and a brief discussion of the physical nature of the quantum environment . the interaction of an open quantum system @xmath37 with a quantum environment @xmath38 is assumed to be described by a unitary operator @xmath39 . the initial state @xmath40 of the combined system is prepared in a separable state @xmath41 . the interaction evolves @xmath40 into a final state  @xcite @xmath42 unitary evolutions are entangling quantum operations and the joint state @xmath43 is not a separable but an entangled state of the final system @xmath44 and the environment @xmath38 . the observer can perform measurements on the systems @xmath37 and @xmath44 but can not perform direct measurements on the environment @xmath38 . after the transformation @xmath39 , the system @xmath44 no longer interacts with the environment @xmath38 . the quantum state of system @xmath44 is then fully described by reduced density matrix @xmath45 in a sense that we can calculate average values @xmath46 of any observable @xmath47 . the trace in ( 2.2 ) is over the interacting degrees of freedom of the environment @xmath48 . in general , the reduced state @xmath49 is no longer related to the initial state @xmath50 by a unitary transformation @xmath51 . instead it is given by kraus representation  @xcite @xmath52 where we assumed that @xmath53 is a pure state . the evolution operators @xmath54 are called kraus operators . they are acting on the hilbert spaces @xmath55 and @xmath56 of the systems @xmath37 and @xmath44 , respectively , and can be unitary or non - unitary . for trace preserving maps the kraus operators must satisfy completness relation @xmath57 to be physically meaningful , a linear map @xmath58 must be completely positive in order to preserve the positivity of all probabilities . a linear map @xmath59 is completely positive if and only if it has the form of the kraus representation  @xcite . the operator @xmath59 is also called superscattering operator .    the quantum states @xmath60 of the environment form an orthonormal basis in hilbert space @xmath61 with a finite dimension  @xcite @xmath62 when the initial state of the environment is a mixed state @xmath63 the diagonal elements @xmath64 and @xmath65 the reduced density matrix ( 2.2 ) then takes the form @xmath66 where @xmath67 . in order for the reduced density matrix ( 2.8 ) to be completely positive it must acquire the form of the kraus representation ( 2.3 ) . for arbitrary states @xmath68 this will happen if and only if the unitary evolution operator @xmath39 conserves the quantum numbers of the quantum states @xmath60 @xmath69 where @xmath70 . then ( 2.8 ) reads @xmath71 with the replacement @xmath72 we recover the form ( 2.3 ) . we shall refer to @xmath73 as kraus density matrices . the final state @xmath49 is a mixed state even when the initial state @xmath50 is a pure state . for trace preserving maps the kraus operators @xmath74 satisfy a completness relation similar to ( 2.4 ) @xmath75 a trace preserving kraus representation is called bistochastic if it leaves invariant the maximally mixed state of the system @xmath76 the kraus representation ( 2.10 ) is a bistochastic map when the kraus operators @xmath74 are unitary . in that case the map ( 2.10 ) is referred to as a random external field  @xcite . when @xmath77 the effect of the non - unitary interaction with the environment described by kraus representation is a modification ( dephasing ) of the phases of the quantum state @xmath78 . in dissipative dephasing interactions the system @xmath2 and the environment @xmath38 exchange energy - momentum . there is no exchange of energy - momentum or angular momentum in pure dephasing interactions which effect only a change of phases . a very important concept of decoherence free subspace was introduced in ref . it describes a decoupling of a subsystem @xmath79 of the system @xmath2 from the environment . in this case all kraus operators are equal to a unitary operator @xmath80 and the kraus representation is reduced to a unitary evolution law for that subsystem @xmath81 . unitary @xmath2-matrix gives rise to a unitary evolution law @xmath82 the initial state has a general form @xmath83 are the initial four - momenta , helicities and quantum numbers of the state . in the following we suppress the labels @xmath84 and @xmath85 . the completness relation reads @xmath86 the first sum in ( 3.3 ) is over all allowed final states @xmath87 , the second sum is over final state helicities @xmath88 and the integration is over the entire phase space of final state momenta @xmath89 . the symbol @xmath90 labels the quantum numbers of the state @xmath87 . using the completness relation the density matrix @xmath91 has an explicit form @xmath92 where the diagonal terms @xmath93 are the produced @xmath2-matrix states with final particle states @xmath94 and @xmath95 are the off - diagonal terms of @xmath91 . we assume that particle scattering processes are not isolated events in the universe as implied by the unitary evolution law but behave as open quantum systems interacting with a universal quantum environment of quantum states @xmath96 . if the quantum environment and its interaction with particle scattering processes are to be an integral part of the nature then they must be fully consistent with the standard model . this is possible if this new kind of particle interaction does not mix with the fundamental interactions of the standard model and thus does not participate in the scattering or decay process itself . then the interaction with the quantum states @xmath96 can only evolve the state @xmath91 produced by the @xmath2-matrix dynamics into a new observed state @xmath97 . the observed state @xmath97 is described by the non - unitary kraus representation @xmath98 where @xmath99 it is the observed states @xmath28 with final particle states @xmath94 which are the object of experimental measurements . the states @xmath96 and thus the probabilities @xmath100 are poincare invariant . the kraus operators @xmath101 are unitary to preserve in ( 3.8 ) the completness relations ( 3.3 ) . the observed states @xmath28 must communicate the information about the scattering dynamics encoded in the @xmath2-matrix state @xmath1 . the kraus operators therefore must preserve the identities of all final state particles of the produced state @xmath1 including the four - momenta of individual particles . this means that we need to concern ourselves only with the projective measurements of the states @xmath28 as the projections of @xmath102 vanish . there is no exchange of four - momentum and angular momentum between the environment and the produced state @xmath1 . the total four - momentum and total angular momentum are conserved by the kraus operators . the interaction of the produced state @xmath1 with the quantum environment is a non - dissipative pure dephasing interaction . the consistency of this interaction with the symmetries and conservation laws of the standard model imposes constraints on the kraus operators and their matrix elements . how do we observe the changes of the @xmath2-matrix amplitudes in the observed amplitudes from the measurements of the density matrix @xmath28 ? to answer this question consider three complex functions @xmath103 . their relative phases @xmath104 and @xmath105 satisfy the phase condition @xmath106 and the equivalent cosine condition @xmath107 @xmath2-matrix amplitudes @xmath108 are complex valued functions that satisfy the cosine condition ( 3.11 ) . another fundamental property of the @xmath2-matrix amplitudes is that they do not mix resonances of different spins @xmath109 and @xmath110 in the bilinear terms @xmath111 and @xmath112 . to be physically meaningfull the observed density matrix @xmath28 and the kraus densities @xmath113 must have the same bilinear structure as the @xmath2-matrix density @xmath1 . then the measured bilinears @xmath114 and @xmath115 carry the same quantum numbers as the corresponding @xmath2-matrix bilinear terms . from the measured bilinear terms we can determine three cosines @xmath116 . since the observed amplitudes are not @xmath2-matrix amplitudes , these cosines will , in general , violate the cosine condition . for the same reason in some processes certain bilinear terms will exhibit spin mixing of resonances .    only in the case of a decoherence free subspace involving at least three amplitudes will the observed cosines satisfy the cosine condition ( 3.11 ) . if there is no spin mixing observed in these amplitudes then they can be identified with @xmath2-matrix amplitudes . this is the case of all two - body scattering or decay processes . these processes as well as free single particles thus do not interact with the quantum environment . the pion creation process @xmath117 is the very simplest particle scattering process where we can observe non - unitary evolution and the spin mixing interaction . with four - momenta @xmath118 the initial and final particle states of the process are described by state vectors @xmath119 and @xmath120 where @xmath121 and @xmath122 and @xmath123 are the helicities of the target and recoil nucleon , respectively . the direction of @xmath124 in dipion center - of - mass system is descibed by @xmath125 . target nucleon spin density matrix is @xmath126 where @xmath127 is target polarization . the produced final state is given by the @xmath2-matrix unitary evolution law @xmath128 where @xmath129 and where we have used the phase space relation  @xcite @xmath130 in the completness relation . the angular @xmath2-matrix amplitudes are matrix elements @xmath131 prior to the interaction of the produced final state with the quantum environment there is no measurement and thus no measurement projection of this state to @xmath132 ) . the angular state @xmath133 can be expanded in terms of spherical harmonics  @xcite @xmath134 where @xmath135 and @xmath136 are the dipion spin and helicity , respectively . integrating over @xmath137 and @xmath138 and using the relation @xmath139 we find @xmath140 the partial wave @xmath2-matrix amplitudes are matrix elements @xmath141 in general , the final state density matrix ( 4.7 ) is a mixed state of entangled dipion and recoil nucleon helicity helicity states @xmath142 and @xmath143 with different dipion spins @xmath135 . equation ( 4.7 ) can be written in a physically important form @xmath144 where @xmath145 and @xmath146 are superpositions of partial waves @xmath147 given by @xmath148      we now use the kraus representation ( 3.7 ) to determine the nature and the form of the pure dephasing interaction with the quantum environment in @xmath0 processes . the unitary kraus operators @xmath101 act on the superposions ( 4.11 ) of the partial waves @xmath149 by rotating the dipion spin states @xmath150 . applying the completness relation ( 3.3 ) from left to @xmath151 we find @xmath152 since the kraus operators preserve the four - momenta and the identity of the final state particles . using the identity @xmath153 the equation ( 4.12 ) then acquires the form @xmath154 this relation shows the interaction of the produced state @xmath1 with the environment is a dipion spin mixing interaction . its effect is to produce a new superposition of partial waves @xmath155 @xmath156 are kraus partial wave amplitudes @xmath157 which explicitely show the mixing of @xmath2-matrix partial wave amplitudes with different dipion spins @xmath135 and helicities @xmath136 . we refer to the relation ( 4.16 ) as spin mixing mechanism . applying the relation ( 4.13 ) to the combination @xmath158 into an angular form @xmath159 where the kraus angular amplitudes @xmath160 similarly we find for @xmath161 @xmath162 @xmath163 where the kraus amplitudes @xmath164 @xmath165 the kraus representation ( 3.7 ) then reads @xmath166 where @xmath167    the equations ( 4.23 ) have a formal resemblance to the equation ( 4.1 ) for the @xmath2-matrix final state @xmath1 . it is evident from the definitions of kraus partial wave and angular amplitudes ( 4.16 ) , ( 4.19 ) and ( 4.21 ) that these amplitudes can not be written as matrix elements of a unitary operator in contrast to the @xmath2-matrix partial wave amplitudes if not all spin states @xmath168 are allowed and ( 4.13 ) does not apply . in that case the kraus partial wave amplitudes shall violate the unitarity conditions on relative phases of the @xmath2-matrix partial wave amplitudes  @xcite . as a result they are complex valued functions with non - unitary phases . spins @xmath135 are resticted by the conditions ( 5.6 ) .      following a projective measurement of the final state @xmath28 into a state with definite four - momenta @xmath169 and the direction @xmath170 of the pion @xmath124 , the kraus representation ( 4.22 ) takes the form @xmath171 suppressing the four - momenta in the initial and final state vectors @xmath172 and @xmath173 , the kraus density matrices read @xmath174 with @xmath175 we have from ( 4.13 ) @xmath176 where @xmath177 are the new kraus angular amplitudes . following formally the steps employed in the case of the @xmath2-matrix spin formalism @xcite we can redefine the kraus density matrices ( 4.25 ) in terms of new kraus helicity amplitudes @xmath178 the normalization factor @xmath179 where @xmath180 is the dipion mass , @xmath181 is the center - of - mass energy squared , @xmath182 is pion momentum in dipion center - of - mass , @xmath183 and @xmath184 are the energy dependent part of the phase space and flux , respectively  @xcite . the redefined matrices ( 4.25 ) then read @xmath185 angular expansion of kraus helicity amplitudes @xmath186 defines kraus partial wave helicity amplitudes with a definite dipion spin and helicity @xmath187 so far we have suppressed isospin labels of the produced pions . in strong interactions pions are considered as identical particles . the ensuing bose - einstein statistics for two - pion states requires that @xmath188 where @xmath189 is the total dipion isospin  @xcite . only @xmath190 is allowed for identical pions . for different pions @xmath2-matrix partial waves with @xmath191 vanish while for identical pions amplitudes with @xmath192 vanish . labeling the isospin states @xmath193 , the spin mixing mechanism for kraus partial wave helicity amplitudes reads @xmath194 where @xmath195 are the redefined @xmath2-matrix partial wave helicity amplitudes  @xcite . the isospin of the kraus partial wave helicity amplitudes is defined to be @xmath193 although for different pions these amplitudes mix isospin as the result of spin mixing . kraus partial wave amplitudes with @xmath192 still vanish for identical pions due to the bose - einstein statistics for pion spin . we refer to the matrix elements @xmath196 as dephasing amplitudes . they describe a forward two - body scattering of dipion states and recoil nucleons @xmath197 with the kraus operators @xmath101 taking on the role of scattering operators . as in the case of @xmath2-matrix scattering , there is a priori no restriction on the final spin @xmath4 and helicity @xmath22 .    to be consistent with the standard model , the dephasing interaction must be lorentz invariant . that means that all kraus operators must be lorentz invariant . the rescattering process of the dipions with the recoil nucleon must thus conserve the total four - momentum and total angular momentum . the dephasing amplitudes obviously conserve the total four - momentum . they must depend on the lorentz invariants of the four - momenta @xmath198 and @xmath199 , i.e. the total energy @xmath181 and dipion mass @xmath200 . to examine their conservation of the total angular momentum we go into the center - of - mass system in the initial and final states of the process ( 4.33 ) where the dipion has 3-momentum q and direction @xmath201 . then  @xcite @xmath202 where @xmath203 and @xmath204 . the expansion of the dephasing amplitudes defines partial wave amplitudes in @xmath205 which conserve the total angular momentum @xmath206 , @xmath207 and which are independent of @xmath26 due to rotational invariance . then @xmath208 where @xmath209 we see that the dephasing amplitudes are non - zero only for @xmath210 we can write the conservation of total angular momentum @xmath211 in the form @xmath212 where @xmath27 and @xmath213 are the initial and final orbital momenta while @xmath37 and @xmath44 are the initial and final total spins . since the four - momenta of the particles in the initial and final states are equal we have @xmath214 . for total spins we have @xmath215 and @xmath216 . then @xmath217 implies that the only possible values of the spin @xmath135 are @xmath218      we require that the kraus helicity amplitudes ( 4.32 ) be consistent with the standard model , in particular with the conservation of @xmath3-parity , total four - momentum and total angular momentum , total isospin , and with the conservation of probability imposed by the @xmath2-matrix . these requirements will impose constraints on the matrix elements of the kraus operators that ensure the dipion spin mixing interaction is a pure dephasing interaction .    to ensure that all observers can communicate the same information about the production process and its interaction with the quantum environment the kraus partial wave helicity amplitudes ( 4.32 ) must transform as two - body helicity @xmath2-matrix amplitudes  @xcite under lorentz transformations . this in turn requires that the spin mixing mechanism is lorentz invariant . let @xmath219 be a homogeneous lorentz transformation from a frame @xmath220 to a frame @xmath221 . the transformation law of the helicity amplitudes is obtained from @xmath222 in ( 5.7 ) @xmath223 and @xmath224 is the matrix representing wigner rotation @xmath225 . here @xmath226 where @xmath227 is a boost lorentz transformation along the @xmath228-axis followed by a rotation through euler angles @xmath229 .    with lorentz symmetry of the @xmath2-matrix and kraus operators , the transformation laws for the @xmath2-matrix and dephasing amplitudes read  @xcite @xmath230 @xmath231 @xmath232 @xmath233 where we suppressed the isospin labels for the sake of brevity . combining these results in the expression ( 4.32 ) for the kraus helicity amplitudes and using @xmath234 we obtain @xmath235 @xmath236 this is the required transformation law for the kraus helicity amplitudes which also shows that the spin mixing mechanism is lorentz invariant . the requirement of @xmath3-parity conservation implies that the kraus helicity amplitudes satisfy the same parity relations as the corresponding @xmath2-matrix helisity amplitudes , namely  @xcite @xmath237 the relation ( 4.32 ) then requires that @xmath238 @xmath239    since the dephasing interaction preserves the four - momenta of the all final state particles , the total four - momentum is obviously conserved by kraus helicity amplitudes . to show they conserve the total angular momentum we consider the @xmath2-matrix and kraus amplitudes as functions of the c.m.s . scattering angle @xmath240 . then the partial wave expansion of the final two - body state in @xmath2-matrix and kraus amplitudes reads @xmath241 where @xmath242 and @xmath243 are the total angular mometum and its @xmath228-axis component , and @xmath244 , @xmath245 . assuming @xmath246 , or @xmath247 , we can write @xmath248 which ensures the conservation of the total angular momentum by the kraus helicity amplitudes . note that the equality @xmath246 follows from the equation ( 5.5 ) , i.e. from the lorentz symmetry of the kraus operators . the conservation of the total isospin in kraus helicity amplitudes follows from the fact that the isospin identities of the final state particles are preserved by the dephasing interaction . consider @xmath7 process . for @xmath249 isospin state @xmath250 while for @xmath251 the isospin state @xmath252 is a combination of states @xmath253 and @xmath254 . since the total initial isospin is conserved by the @xmath2-matrix amplitudes in both cases , then it must be conserved by the dephasing amplitudes with the states @xmath255 and @xmath256 as well . this propagation of total isospin in the spin mechanism ensures isospin conservation by kraus helicity amplitudes . unitarity of the kraus operators requires @xmath257 the conservation of the total angular momentum constraint ( 5.5 ) requires that @xmath258 for arbitrary @xmath259 . for @xmath260 the unitarity relation is a null identity . the measurements of @xmath0 processes on polarized targets are best analyzed in terms of transversity amplitudes with definite @xmath261-channel naturality . to define kraus transversity amplitudes we first must define kraus helicity amplitudes with definite @xmath261-channel naturality  @xcite . the unnatural exchange amplitudes are defined by @xmath262 while the natural exchange amplitudes are given by @xmath263 where @xmath264 and @xmath265 are nucleon helicities . both amplitudes satisfy the same parity relations ( 5.12 ) as the helicity amplitudes . however to express @xmath266 and @xmath267 in terms of the @xmath2-matrix amplitudes with definite naturality we need another constraint @xmath268 which from the @xmath3-parity relations ( 5.13 ) implies @xmath269 with the resulting expressions we calculate kraus unnatural and natural exchange transversity amplitudes defined by  @xcite @xmath270 for unnatural exchange amplitudes and with a similar definition @xmath271 for the natural exchange amplitudes . in ( 6.5 ) and ( 6.6 ) @xmath272 is the target nucleon transversity . the kraus transversity amplitudes satisfy parity relations @xmath273 for @xmath274 note that @xmath275 so that @xmath276 . starting with the spin mixing mechanism for kraus helicity amplitudes ( 4.32 ) and using the parity relations ( 5.12 ) , ( 6.3 ) and ( 6.4 ) , the spin mixing mechanism for the kraus unnatural and natural transversity amplitudes reads @xmath277 @xmath278 where @xmath279 and @xmath280 are the @xmath2-matrix unnatural and natural exchange transversity amplitudes , respectively , and where @xmath281 we have omitted isospin labels in ( 6.10 ) for the sake of brevity . from the unitarity of the kraus operators @xmath101 follows the unitarity of matrices @xmath282 in ( 6.8 ) @xmath283 the expressions ( 6.8 ) and ( 6.9 ) satisfy the @xmath3-parity relations ( 6.7 ) . we now impose the conditions @xmath284 and @xmath5 that follow from the lorentz symmetry of the kraus operators . the expressions ( 6.8 ) and ( 6.9 ) then read @xmath285 @xmath286 with a change @xmath287 the expressions ( 6.12 ) and ( 6.13 ) read @xmath288 @xmath289 these expressions must satisfy the @xmath3-parity relations ( 6.7 ) for any @xmath22 such that @xmath290 . equating the second sum in the equations ( 6.12 ) and ( 6.13 ) with the second sum in ( 6.14 ) and ( 6.15 ) , respectively , we obtain relations that impose spin mixing on the @xmath2-matrix amplitudes . these constraints are therefore not physical and we must require that @xmath291 then the spin mechanism for the kraus transversity amplitudes takes a simple form for any @xmath290 @xmath292 @xmath293 where we have restored the isospin labels and used the constraint ( 5.6 ) from the lorentz symmetry of kraus operators . the lorentz conditions ( 5.5 ) also require @xmath294 with the only non - zero dephasing amplitudes given by the spin mixing mechanism ( 6.17 ) and ( 6.18 ) , the spin mixing mechanism for the kraus helicity amplitudes ( 4.32 ) reads @xmath295 where @xmath296 we have arrived at a self - consistent form of the spin mixing mechanism for kraus amplitudes that arises from their consistency with the standard model . the consistency of the kraus transversity amplitudes with lorentz symmetry of the kraus operators results in the breaking of the unitarity of the kraus operators into a block unitarity with each block characterized by the helicity @xmath22 . the self - consistency constraint requires that the dephasing interaction with the environment conserves both the dipion helicity and the recoil nucleon helicity and transversity . assuming unitary @xmath2-matrix we used the evolution equation @xmath297 to develop in ref  @xcite a spin formalism to express the measured final state @xmath298 in terms of spin density matrix elements @xmath299 which are bilinear combinations of helicity amplitudes @xmath300 or transversity amplitudes @xmath301 and @xmath302 . on the other hand , the data analyses using this formalism lead to the conclusion that @xmath303 is given by a non - unitary kraus representation ( 4.24 ) . self - consistency requires that the kraus representation preserves the experimental form of the angular intensities describing the final state density matrix and the form of the expressions for density matrix elements in terms of transversity amplitudes . in this section we show that such invariance holds provided that the kraus partial wave amplitudes satisfy the same parity relations as the @xmath2-matrix amplitudes due to the conservation of @xmath3-parity . the form ( 4.25 ) of each term @xmath304 in the kraus representation ( 4,24 ) is the same as the form of the @xmath2-matrix final state in terms of @xmath2-matrix amplitudes @xmath305 we can thus apply the spin formalism develped in ref . @xcite to each such term separately . then each term @xmath304 in ( 4.29 ) has the following form in terms of angular intensities @xmath306 with a decomposition of the intensities @xmath307 in terms of component intensities @xmath308 the components of intensities @xmath309 have angular expansion @xmath310 where the matrix elements @xmath311 are expressed in terms of kraus partial wave helicity amplitudes @xmath312 as the result of linearity of the kraus representation , the observed final state density matrix has the same form as in the case of the @xmath2-matrix @xmath313 where @xmath314 to bring @xmath315 to the experimental @xmath2-matrix form we need to bring to this form the intensities @xmath309 . to this end the density matrix elements @xmath311 must satisfy the same parity relations as the corresponding matrix elements in the @xmath2-matrix theory . this in turn requires that the kraus partial wave amplitudes satisfy the same parity relations as the corresponding @xmath2-matrix partial wave amplitudes . with the parity relation ( 5.12 ) the components @xmath309 then have the desired form  @xcite @xmath316 for @xmath317 and the form @xmath318 for @xmath319 . from ( 7.7 ) it follows that the measured intensities @xmath315 will retain the same form of angular expansions provided that the measured density matrix elements @xmath320 are redefined as @xmath321 the measured density matrix elements are thus environment - averaged elements of the kraus density matrices . l|r @xmath322 & @xmath323 $ ] + @xmath324 & @xmath325 $ ] + @xmath326 & @xmath327 $ ] + @xmath328 & @xmath329 $ ] +    @xmath330 & @xmath331 $ ] + @xmath332 & @xmath333 $ ] + @xmath334 & @xmath335 $ ] + @xmath336 & @xmath337 $ ] +    @xmath338 & @xmath339 $ ] + @xmath340 & @xmath341 $ ] + @xmath342 & @xmath343 $ ] + @xmath344 & @xmath345 $ ] +    @xmath346 & @xmath347 $ ] + @xmath348 & @xmath349 $ ] + @xmath350 & @xmath351 $ ] + @xmath352 & @xmath353 $ ] +    [ table i. ]    the kraus density matrix elements @xmath354 are linear combinations of bilinear terms of kraus transversity amplitudes which are the same as those for the @xmath2-matrix elements and which are given in the table i. the measured density matrix elements @xmath320 are given by ( 5.10 ) in terms of elements of kraus density matrices . because this relation is linear the measured elements @xmath320 will retain the form given in table i. with moduli and bilinear terms of transversity amplitudes now replaced by environment - averaged moduli and bilinear terms of kraus amplitudes , respectively . the expressions for the measured bilinear terms for any two transversity amplitudes @xmath355 and @xmath356 read @xmath357 where we have omitted the spin labels of the amplitudes for the sake of clarity . note that the symbol @xmath358 does not stand for a product of two complex functions @xmath355 and @xmath359 but communicates which amplitudes @xmath360 are being averaged . the averaged bilinear terms can be written in terms of effective transversity amplitudes @xmath361 where @xmath362 and @xmath363 are the moduli of the effective amplitudes and @xmath364 and @xmath365 are their correlations . with this form of measured bilinear terms we can no longer associate complex functions @xmath355 and @xmath356 with the moduli @xmath362 and @xmath363 , respectively , and identify the phases @xmath366 and @xmath367 as relative phases of these complex functions . in general the measured correlations violate the necessary trigonometric identity with @xmath368 . since the phases @xmath366 are not relative phases of complex functions , they violate the phase condition @xmath106 and the equivalent cosine condition @xmath107 which hold for any three complex valued functions @xmath103 .    to better understand the meaning of the measured moduli and correlations we define kraus vectors in the hilbert space @xmath61 @xmath369 and a scalar product @xmath370 comparing ( 5.16 ) with ( 5.11 ) and ( 5.12 ) we find latexmath:[\[\begin{aligned }    \nonumber    \nonumber    provide a new physical interpretation of the measured ( averaged ) bilinear terms : instead of associating the measured effective amplitudes with complex functions we must associate them with complex kraus vectors ( 5.15 ) . the measured bilinear terms of effective amplitudes are real or imaginary parts of scalar products of their kraus vectors . it should be noted that the bilinear terms of kraus amplitudes @xmath360 carry the quantum numbers of the state vectors @xmath60 which define the interacting degrees of freedom of the quantum environment . the bilinear terms of effective amplitudes are mixtures of these quantum numbers . the unitarity of the dephasing amplitudes and the spin mixing mechanism ( 6.17),(6.18 ) imply that for each @xmath372 and @xmath272 @xmath373 depends on the dipion mass @xmath23 . the observed moduli for transversity amplitudes are given by latexmath:[\[\begin{aligned }     \nonumber     ( 7.19 ) that the observed differential cross - section is equal to that given by the @xmath2-matrix amplitudes latexmath:[\[\frac{d^2 \sigma}{dmdt } = \sum \limits_{j \lambda,\tau } |u^j_{\lambda \tau}|^2 + |n^j_{\lambda \tau}|^2 = \sum \limits_{j \lambda,\tau }     embodies the conservation of probability of a particle reaction in dephasing interaction with the environment . the spin mixing mechanism ( 6.20 ) for the helicity amplitudes implies the same conservation of the differential cross - section . we note that the the observed polarized target assymetry is similarly given by the @xmath2-matrix amplitudes @xmath376 recall that unitary evolution law of the @xmath2-matrix imposes relative phases @xmath377 on such transversity amplitudes where @xmath378 are integers  @xcite . then the equations ( 7.23 ) imply @xmath379      the cern - cracow - munich ( ccm ) measurements of @xmath7 at 17.2 gev / c  @xcite measured moments @xmath380 ( unpolarized target ) , @xmath381 ( polarization component perpendicular to the scattering plane ) and @xmath382 ( polarization component in the scattering plane ) . these moments are linear combinations of spin density matrix elements @xmath383 and @xmath384 , respectively , corresponding to different spins @xmath385 . the itep measurements of @xmath7 at 1.78 gev / c  @xcite and the cern - saclay measurements of @xmath386 at 5.98 and 11.85 gev / c  @xcite measured directly the spin density matrix elements @xmath383 and @xmath384 for the @xmath2- and @xmath3-wave subsystem .    to determine the observed production amplitudes ccm used a @xmath387 minimization method  @xcite . the moments are expressed in terms of the transversity amplitudes and these theoretical formulas are fitted to the measured momements . all @xmath387 fits found two solutions @xmath388 for the @xmath2- and @xmath3-wave amplitudes @xmath389 below 980 mev ( @xmath390 threshold ) and a single solution for @xmath2-,@xmath391 and @xmath392-waves above 980 mev . the cosines of the relative phases are treated as free parameters and are not constrained by the cosine conditions ( 3.11 ) in the minimization process . the combinations @xmath393 of the density matrix elements of the @xmath2- and @xmath3-wave subsystem form two separate systems of equations for the amplitudes with transversity `` up '' ( sign + ) and `` down '' ( sign - ) . each system is analytically solvable provided the cosines of the relative phases in each system satisfy the cosine conditions ( 3.11 ) which leads to a cubic equation for each @xmath394 . the two real physical solutions of each cubic equation allow to determine two solutions @xmath395 for the @xmath2- and @xmath3-wave amplitudes @xmath389 below 980 mev . the analytical method has been used to determine the initial values in the @xmath387 minimization analyses  @xcite . it has been fully employed in the itep analysis  @xcite and in the monte carlo analyses of the ccm data on @xmath7 at 17.2 gev / c and the cern - saclay data on @xmath386 at 5.98 and 11.85 gev / c  @xcite .    in monte carlo analyses the amplitudes are calculated for each sampling of the error volume . unphysical solutions are rejected and the physical solutions are used to determine the average values of the amplitudes and their errors . the resulting amplitudes are genuine complex valued functions . kraus amplitudes are complex valued functions . this suggests that the @xmath2- and @xmath3-wave analytical amplitudes may be related to the kraus amplitudes . since the monte carlo amplitude analysis selects data that satisfy the cosine conditions , this data must describe a decoherence free subspace . in this case there is only one set of @xmath2- and @xmath3-wave kraus amplitudes equal to one of the two analytical solutions . in our new analysis of the cern data using spin mixing mechanism  @xcite we show that spin mixing mechanism selects the solution 2 of the analytical as well as the @xmath387 methods . there are no exact analytical solutions of the @xmath2- , @xmath3- and @xmath392-wave system . we thus expect a single effective solution for the @xmath2-,@xmath3-and @xmath392-wave system with some cosines of relative phases violating the cosine conditions . in the section ix . we present evidence from the analysis of rybicki and sakrejda  @xcite that the @xmath392-waves indeed violate the cosine condition . below 1400 mev the process @xmath7 is described by @xmath2- , @xmath3- and @xmath392-wave transversity amplitudes . in the following we shall use a new notation for these amplitudes . the amplitudes @xmath396 and @xmath397 are the @xmath2-wave and @xmath3-wave unnatural exchange amplitudes with helicity @xmath274 . the amplitudes @xmath398 and @xmath399 are the @xmath3-wave unnatural and natural exchange amplitudes with helicity @xmath400 . there are three unnatural exchamge @xmath392-wave amplitudes @xmath401 corresponding to the helicities @xmath402 and two natural exchange @xmath392-wave amplitudes @xmath403 corresponding to helicities @xmath404 .    with the notation @xmath405 in the spin mixing mechanism ( 6.17 ) and ( 6.18 ) the expressions for the @xmath2-,@xmath3- and @xmath392-wave kraus amplitudes in terms of @xmath2-matrix amplitudes read as follows . the three helicity zero amplitudes @xmath406 form a dephasing triplet @xmath407 the amplitudes @xmath408 and @xmath409 form two dephasing doublets . @xmath410 @xmath411 the amplitudes @xmath412 and @xmath413 form two dephasing singlets @xmath414 the matrices in ( 8.1)-(8.4 ) are unitary matrices @xmath415 and @xmath416 . lorentz symmetry of kraus operators imposes a constraint @xmath5 . this means that @xmath417 in ( 8.1 ) . the unitarity of this matrix then implies @xmath418 and ( 8.1 ) takes the form @xmath419 @xmath420 is thus a dephasing singlet and the amplitudes @xmath421 and @xmath422 form a dephasing doublet . the amplitude @xmath420 decouples from the @xmath2- and @xmath3- subsystem with the result that there can be no spin mixing of @xmath423 and @xmath424 amplitudes . the matrix in ( 8.6 ) is a @xmath425 matrix .    in the mass interval below 980 mev at low @xmath261 the @xmath2- and @xmath3-waves dominate and @xmath392-waves are neglected . there is no evidence for the presence of @xmath15 in the measured @xmath426 at high @xmath261 as predicted by the equations ( 8.5 ) . the observed amplitudes @xmath398 and @xmath399 resonate at @xmath15 mass and at high @xmath261 @xmath399 are the largest amplitudes . there is no evidence for the presence of @xmath15 in the amplitudes @xmath427 and @xmath428 at high @xmath261 . this implies @xmath429 . then ( 8.2 ) reads @xmath430 in this mass range there is a consistent body of evidence for @xmath6 mixing in the amplitudes @xmath431 and @xmath432 ( for a review see ref . @xcite ) .      to study the observable effects of the spin mixing mechanism we need to introduce the phases of the @xmath2-matrix transversity amplitudes . in this subsection we omit the label @xmath2 of the @xmath2-matrix transversity amplitudes .    in the part i. of this work  @xcite we have shown that the unitary evolution law imposes a constraint on the relative phases of the @xmath2-matrix amplitudes @xmath433 for all @xmath385 and corresponding @xmath434 . it follows from the unitary conditions ( 8.7 ) that all unnatural exchange amplitudes @xmath30 as well as all natural exchange amplitudes @xmath31 share the same absolute phase up to an integer multiple of @xmath435 @xmath436 where @xmath437 and @xmath438 are integers . the phases @xmath439 and @xmath440 depend on @xmath441 .    the unitary phases ( 8.8 ) imply that if there is a resonance at the mass @xmath442 in the amplitudes @xmath443 ( @xmath444 ) with @xmath445 , then all contributing amplitudes @xmath446 ( @xmath447 ) with @xmath448 will have the same resonant phase near @xmath442 . this does not imply that any of the amplitudes with @xmath448 must resonate and show a resonant peak or a dip at @xmath442 since their non - resonant moduli are mathematically allowed to have resonant phases . although the unitary phases ( 8.8 ) appear to allow such resonance mixing , it is excluded by the the symmetries of the @xmath2-matrix and standard model . indeed , the production as well as the decay of a resonance is described entirely by the resonant @xmath2-matrix amplitudes @xmath449 while the mixing of resonances in the kraus amplitudes @xmath450 arises entirely from the dephasing amplitudes . with ( 8.8 ) the @xmath2-matrix transversity amplitudes then have the form latexmath:[\[\begin{aligned } u^j_{\lambda \tau } & = & e^{i\phi(s_\tau ) } e^{i\pi n(u^j_{\lambda \tau } ) }     \nonumber n^j_{\lambda \tau } & = & e^{i\phi(n_\tau ) } e^{i\pi n(n^j_{\lambda \tau } ) }    phases is presented in the table ii . in the part i. of this work  @xcite . for the @xmath2- and @xmath3-wave subsystem below 980 mev the relative phases read    @xmath452    for masses @xmath453 mev there is a change of sign in the measured interference terms of the amplitudes prompting a change of relative phases for @xmath454  @xcite    @xmath455    in ( 8.10 ) and ( 8.11 ) we have introduced the superscript @xmath456 to label the @xmath2- and @xmath3-wave @xmath2-matrix amplitudes to be used in the next subsection . it follows that for @xmath457 mev    @xmath458    while for the masses above 980 mev the signs of @xmath459 and @xmath460 are reversed . the spin mixing matrix @xmath461 in ( 8.6 ) is a unitary @xmath425 matrix . the most general @xmath425 matrix has the form  @xcite @xmath462 with @xmath463 the spin mixing mechanism for @xmath2- and @xmath3-wave kraus amplitudes @xmath464 reads    @xmath465    in the following we omit the label @xmath466 . first we consider the @xmath15 mass region and @xmath457 mev . with the phases ( 8.12 ) the equations ( 8.14 ) read    @xmath467 the moduli have a form @xmath468 . to fix the phases @xmath469 we require that the cosine @xmath470 of the relative phase @xmath471 is near its unitary value @xmath472 given by ( 8.10a ) @xmath473 from ( 8.15 ) and ( 8.16 ) we find @xmath474 where we have assumed @xmath475 . the consistency with the unitary value requires that @xmath476 . the unitarity of the resulting @xmath461 requires @xmath477 . similarly we require that @xmath478 is near the unitary value @xmath479 . this consistency implies that @xmath480 . the final form of the spin mixing mechanism below 980 mev then reads @xmath481 with @xmath482 the equations ( 8.16 ) take the form latexmath:[\[\begin{aligned }    \nonumber    \nonumber    of @xmath15 resonance in the amplitudes @xmath484 . we now consider the @xmath18 mass region @xmath453 mev . we shall focus on amplitudes with transversity @xmath454 . since there is no change in the amplitudes with @xmath485 we assume we still have @xmath486 and @xmath487 . with the reversed signs of @xmath432 and @xmath488 in ( 8.12 ) the final form of the spin mixing mechanism ( 8.14 ) for @xmath454 reads @xmath489 the moduli for @xmath454 have the form latexmath:[\[\begin{aligned }    \nonumber    \nonumber    moduli retain the form ( 8.21 ) . the equations ( 8.21 ) and ( 8.23 ) both predict the presence of @xmath18 resonance in the amplitudes @xmath394 . expressions for the relative phases can be solved exactly to read @xmath491 where @xmath492 . from the ratio @xmath493 we obtain a quadratic equation for the cotangent @xmath494 which has two solutions @xmath495 only @xmath496 is a physical solution for the assumed @xmath497 .    the spin mixing parameters @xmath498 and @xmath499 depend on the c.m.s . energy @xmath181 and the invariant dipion mass @xmath23 but do not depend on the transversity @xmath500 and momentum transfer @xmath261 . for the expression on the r.h.s . of equation ( 8.28 ) to be independent of @xmath500 and @xmath261 for all @xmath501 we need @xmath502 and @xmath503 to be independent of @xmath500 and @xmath261 . this independence for @xmath503 requires proportionality relations for the @xmath2-matrix amplitudes @xmath504 and @xmath505 with the same @xmath506 . these relations ensure the required independence of @xmath502 . the relations ( 8.29 ) and ( 8.30 ) do not apply to the helicity @xmath400 amplitudes @xmath507 and @xmath508 . the independence of @xmath503 on @xmath500 and @xmath261 means that the relative phases do not depend on the transversity and , at a fixed @xmath181 and @xmath23 , on @xmath261 . the spin mixing mechanism thus predicts that the solutions for the cosines of relative phases in amplitude analyses should be the same for both transversities and independent of @xmath261 at the same energy . there are @xmath509 kraus amplitudes @xmath422 and @xmath421 . the spin mixing parameters @xmath510 and @xmath511 are functions of energy @xmath181 and dipion mass @xmath23 but do not depend on the momentum transfer @xmath261 and the transversity @xmath500 . we can identify the complex valued functions @xmath422 and @xmath421 with the complex valued analytical amplitudes obtained from the experimental data . there are two solutions for the moduli of the @xmath2- and @xmath3-wave amplitudes and the cosines of their relative phases which implies @xmath512 . it is also possible to cross combine the solution 1 ( 2 ) for the @xmath485 amplitudes with the solution 2 ( 1 ) for the @xmath454 amplitudes . such four combinations of the solutions @xmath513 , @xmath514 , @xmath515 and @xmath516 imply @xmath517 . since the cosines do not determine the signs of the relative phases , there is a four - fold sign ambiguity @xmath518 in the relative signs of the relative phases in each solution in both @xmath512 and @xmath517 cases . the amplitudes of the solutions @xmath519 and @xmath520 are simply complex conjugate amplitudes of the solutions @xmath521 and @xmath522 , respectively . the cross solutions @xmath514 and @xmath515 are excluded by the spin mixing mechanism which requires @xmath523 and @xmath524 for each solution @xmath525 . in the following we shall focus on the @xmath512 case with solutions 1 and 2 corresponding to combinations @xmath513 and @xmath516 .    at this point we do not know the @xmath2-matrix amplitudes @xmath526 and @xmath527 , i.e. the ratio @xmath503 , and the spin mixing parameters @xmath528 and @xmath529 so we can not make quantitative predictions for the moduli ( 8.23 ) and the cosines ( 8.24)-(8.26 ) . however we can still make qualitative comparisons with the data and draw from them important conclusions . figure 1 shows the @xmath2-wave moduli from the recent monte carlo analysis of the cern - cracow - munich data ( ccm ) at low momentum transfers @xmath530 ( gev / c)@xmath531  @xcite . these results are nearly identical to the published results of the @xmath387 fit analysis of the same data by the ccm group  @xcite shown in the figure 2 . both analyses clearly indicate the presence of a @xmath15 peak in both solutions for the amplitude @xmath532 . the solution 2 for @xmath532 exhibits a pronounced structure at  930 mev absent in the solution 1 . the results of the recent monte carlo analysis  @xcite are similar to the previously published monte carlo amplitude analyses of the ccm data at smaller monte carlo resolution  @xcite . the recent work includes analysis on unpolarized target and the determination of helicity amplitudes which both confirm the presence of the @xmath15 in the @xmath2-wave . the 1997 @xmath387 fit analysis  @xcite is in agreement with the 1979 @xmath387 analysis using ccm data from a different run  @xcite . independent evidence for @xmath15 in the @xmath2-wave comes from the analysis of itep data on @xmath7 at 1.78 gev / c at low @xmath261  @xcite , and from two analyses at high @xmath261 of ccm data  @xcite and cern - saclay data on @xmath12 at 5.98 and 11.85 gev / c  @xcite . for a full review with figures of all these results for the moduli and intensities see ref . this survey shows a remarkable consistency of the evidence for a rho - like state in the @xmath2-wave in @xmath533 production channel . spin mixing mechanism explains the appearance of the @xmath15 peak in the measured amplitudes @xmath532 as due to the presence of a large resonating @xmath2-matrix amplitude @xmath534 in the expression for the @xmath2-wave modulus ( 8.23 ) for sufficiently large spin mixing parameter @xmath535 . in ccm data this spin mixing appears suppressed in the amplitudes @xmath536 compared to @xmath532 . the monte carlo analysis of ccm data  @xcite shows a dip in the amplitude @xmath17 near the @xmath18 mass in both solutions while there is a sudden drop in @xmath532 at this mass . these effects are due to the mixing of the decreasing @xmath2-matrix amplitude @xmath537 and increasing @xmath2-matrix amplitude @xmath538 above the @xmath390 threshold at 980 mev , indicating @xmath6 mixing in the @xmath3-wave amplitude . [ htp ] -wave amplitudes @xmath536 and @xmath532 from the monte carlo analysis  @xcite.,title="fig:",width=453,height=396 ]    [ hp ] -wave amplitudes @xmath536 and @xmath532 from the @xmath387 fit analysis  @xcite.,title="fig:",width=453,height=396 ]    [ htp ]   from the monte carlo amplitude analysis  @xcite.,title="fig:",width=453,height=396 ]    [ hp ]   from the @xmath387 fit amplitude analysis  @xcite.,title="fig:",width=453,height=396 ]    [ htp ]   from the monte carlo amplitude analysis  @xcite.,title="fig:",width=453,height=396 ]    [ hp ]   from the monte carlo amplitude analysis  @xcite.,title="fig:",width=453,height=396 ]      figures 3 and 4 show the cosines for the relative phases @xmath539 for the monte carlo  @xcite and for the @xmath387 fit  @xcite analyses of the same ccm data , respectively . the results are again nearly identical . the most important but subtle difference is in the cosines for @xmath454 in the mass interval 700 - 800 mev . while in the monte carlo analysis the values of this cosine are always less than 1.0 , in the @xmath387 fit analysis this cosine takes on the value of 1.0 at 749 mev bin . this difference means that in the monte carlo analysis there can be no change of sign of the phase @xmath539 for @xmath454 below 1080 mev while this phase can change sign at the 749 mev bin in the @xmath387 analysis . figures 5 and 6 show the cosines for the relative phases @xmath540 and @xmath541 , respectively , for the monte carlo analysis  @xcite . there are no corresponding figures available from the @xmath387 fit analysis  @xcite .    a careful inspection of the figures 3 , 5 and 6 shows that with the exception of solution 2 for @xmath470 the cosines of relative phases in monte carlo amplitude analysis are near the unitary values ( 8.10 ) and ( 8.11 ) as expected from the spin mixing relations ( 8.18 ) and ( 8.19 ) . also with the exception of solution 2 for @xmath470 , the cosines are approximately independent of the transversity @xmath500 . this consistency is approximate because the amplitude analysis did not impose the spin mixing mechanism constraints @xmath542 and @xmath543 . when these constraints are imposed in the amplitude analysis the consistency for @xmath470 becomes exact  @xcite . we now turn to the question of the dimension @xmath35 of the hilbert space @xmath61 . recall that @xmath544 in a non - unitary process @xmath545  @xcite . in the kraus representation ( 3.7 ) @xmath546 and @xmath547 . the relevant hilbert space @xmath548 is the space of spin states for all final state particles forming the states @xmath1 and @xmath28 . the dimension @xmath35 must be common to all scattering processes .    the simplest scattering process that involves interaction with the environment is @xmath0 . at dipion mass @xmath23 the dimension @xmath549 is determined by the maximum dipion spin @xmath550 . its simplest spin state describes the @xmath2-wave process @xmath551 . in the absence of @xmath3-waves in processes like @xmath552 or @xmath553 this is the only process near threshold . the bnl high statistics meaurements of @xmath552 at 18.2 gev / c show that all @xmath392-wave amplitudes are consistent with zero for @xmath554 mev for all momentum transfers @xmath261 @xcite . the evolution of @xmath555 due to its interaction with the environment then limits the dimension @xmath35 to @xmath556 .    experimental evidence for @xmath36 can only come from the violation of the cosine condition ( 3.11 ) . consider three observed amplitudes @xmath362 , @xmath363 and @xmath557 and their correlations @xmath558 , @xmath559 and @xmath560 . in general the phases @xmath561 , @xmath562 and @xmath563 violate the phase condition ( 3.10 ) but satisfy a phase gap condition @xmath564 which leads to a cosine gap condition @xmath565 where @xmath566 and @xmath567 are the phase gap and cosine gap , respectively . they are related by the condition @xmath568 the cosine gap condition ( 9.2 ) is an unambigous signature of the non - unitary evolution law with @xmath36 .    the first indication for the violation of the phase condition ( 9.1 ) and the cosine condition ( 9.2 ) came in 1985 from the amplitude analysis of cern measurement of @xmath7 at 17.2 gev / c at large momentum transfers @xmath569 ( gev / c@xmath570 by rybicki and sakrejda  @xcite . in that analysis they determined relative phases of @xmath3-wave amplitude @xmath571 ( @xmath572 in our notation ) and @xmath392-wave amplitudes @xmath573 and @xmath574 . figure 10 shows the phase gap @xmath575 and the cosine gap @xmath576 @xmath577 these phase gaps and cosine gaps show large deviations from the unitary values 0 and 1 , respectively , as well as large fluctuations . the analysis of rybicki and sakrejda supports the non - unitary evolution law with @xmath36 . their analysis also finds evidence for a rho - like resonance in the @xmath2-wave , suggesting @xmath6 mixing at large @xmath261 ( see also ref . @xcite ) . [ hp ]   and cosine gap @xmath578 for aplitudes @xmath571 , @xmath579 and @xmath574 . data from ref . @xcite.,title="fig:",width=453,height=396 ]    in our new analysis  @xcite the spin mixing mechanism allows to extract @xmath392-wave observables from the cern data . analysis of the full @xmath392-wave subsystem for transversity @xmath485 reveals a violation of cosine conditions ( 3.11 ) by the amplitudes @xmath580 and @xmath581 . detailed analysis of the amplitudes @xmath580 and @xmath581 determines that the number of interacting degrees of freedom of the environment is @xmath517 . the evidence that @xmath582 brings up the issue of decoherence free and decohering amplitudes . this issue is relevant since the decoherence free subspaces are linked to observations of spin mixing . kraus amplitudes forming decoherence free subspace do not depend on the interacting degrees of freedom @xmath466 since all kraus operators are equal @xmath583 , @xmath584 . in contrast the decohering kraus amplitudes do depend on @xmath466 due to different kraus operators . three or more kraus amplitudes are recognized as decoherence free amplitudes if all cosine conditions on the cosines derived from their measured bilinear terms are satisfied . then we refer to the measured amplitudes also as decoherence free amplitudes . if one or more cosine conditions are violated they evidence one or more decohering kraus amplitudes and we refer to the measured amplitudes as decohering amplitudes . in the absence of a theory of dephasing interactions the spin mixing matrices @xmath585 are only constrained by the experimental data from which we can deduce which amplitudes are decoherence free and which are decohering .    in the sequel paper  @xcite we report a new amplitude analysis of the cern data below 1080 mev using spin mixing mechanism ( 8.20 ) and ( 8.22 ) and the constraints ( 8.29 ) to determine anew the spin mixing @xmath2-and @xmath3-wave transversity amplitudes @xmath586 , and to determine the @xmath2-matrix transversity amplitudes @xmath587 and the spin mixing parameters @xmath125 . the spin mixing mechanism also allows to extract full set of @xmath392-wave amplitudes for the transversity @xmath485 in a simultaneous analysis . the spin mixing mechanism admits only a single physical solution for the @xmath2-and @xmath3-wave amplitudes and for the @xmath392-wave amplitudes @xmath588 corresponding to the solution 2 in figures 1 - 6 . interpreting this solution as kraus amplitudes implies that these amplitudes form a decoherence free subspace because @xmath36 . the spin mixing parameters @xmath125 do not depend on @xmath466 . [ hp ]   and cosine gap @xmath578 for aplitudes @xmath571 , @xmath396 and @xmath579 . data from ref . @xcite.,title="fig:",width=453,height=396 ]    there is no evidence for the presence of @xmath15 in the amplitude @xmath426 in our analysis  @xcite and in the high @xmath261 analysis  @xcite , as predicted by the equations ( 8.5 ) . we do not expect @xmath589 to mix with @xmath397 because @xmath579 still decouples from the @xmath2- and @xmath24-wave amplitudes in agreement with the data . figure 11 shows that in the analysis at low @xmath261 and @xmath590 mev  @xcite the phase condition and cosine condition are satisfied by the relative phases @xmath539 , @xmath591 and @xmath592 for both @xmath500 . the analysis of the full @xmath392-wave subsystem reveals @xmath15 mixing in the amplitudes @xmath593 and @xmath594 which is at variance with the analysis at high @xmath261  @xcite . this mixing requires a small mixing of the @xmath2-matrix amplitudes @xmath595 and @xmath596 in the amplitudes @xmath597 and @xmath598 in ( 8.2 ) and ( 8.3 ) , respectively @xmath599 where the mixing matrix @xmath600 has a general form ( 8.13 ) . the effective amplitudes @xmath601 and @xmath602 have the same @xmath2-matrix phases as @xmath603 and @xmath604 . the amplitudes @xmath595 and @xmath596 have the same phases as @xmath603 and @xmath604 , respectively  @xcite . amplitude analysis determines the effective amplitudes . above 980 mev there is no evidence for the presence of @xmath589 in the amplitudes @xmath398 and @xmath399  @xcite indicating @xmath605 at these masses . our analysis of the amplitudes @xmath574 and @xmath606 in ref . @xcite determines four phases @xmath607 for the two decohering amplitudes @xmath608 and @xmath609 in ( 8.4 ) @xmath610 which sets the dimension @xmath517 . the results for the @xmath392-wave amplitudes mean that the amplitudes @xmath579 , @xmath611 and @xmath612 belong to the same decoherence free subspace with the @xmath2-and @xmath3-wave amplitudes . this subspace includes three pairs of spin mixing amplitudes ( @xmath613 ) , ( @xmath408 ) and ( @xmath409 ) . we conclude with the observation that the spin mixing occurs in decoherence free subspaces while there is no spin mixing in decohering subspaces . we shall use the coleman - mandula no - go theorem  @xcite to show that the non - unitary evolution law ( 3.7 ) is consistent with the @xmath2-matrix scattering dynamics and quantum field theory provided the dimension @xmath36 . the theorem requires that in the decohering subspaces there can be no violation of lorentz symmetry while there must be a lorentz symmetry violating spin mixing in the decoherence free subspaces .    a modern proof of the theorem is given by weinberg in ref . assume that the generators @xmath614 and @xmath615 of the poincare group commute with the @xmath2-matrix . assume that internal symmetry generators @xmath616 ( charges ) commute with the @xmath2-matrix and form an algebra given by commutation relations @xmath617=if^{abc}t^c\ ] ] coleman - mandula no - go theorem asserts that there is no physically meaningful @xmath2-matrix theory where the charges @xmath616 do not commute with the generators of the poincare group . that means there is no mixing of space - time transformations and internal symmetry transformations . supersymmetry evades this no - go theorem by replacing the commutation relations ( 11.1 ) with anticommutation relations  @xcite . the key assumption in the proof of the coleman - mandula theorem is the assumption that a two - body scattering always exists in the @xmath2-matrix theory as opposed to a situation where only a forward scattering is possible . theories that allow only forward scattering evade the no - go theorem . the matrix elements of the operators @xmath618 describing the pure dephasing interaction of the particle final states of @xmath1 all describe forward scattering of these states . do they evade the no - go theorem ?    to answer this question let us first assume that there is only one kraus operator @xmath619 . this operator @xmath619 can not be the @xmath2-matrix for the @xmath2-matrix would allow a non - trivial scattering of the produced particle states . thus @xmath619 is a genuine `` no - scattering '' operator . the lorentz symmetry is conserved by @xmath619 and the coleman - mandula theorem is evaded . then conserved internal charges associated with the kraus operator @xmath619 need not commute with the generators of the poincare group and in fact can carry a lorentz index . these charges would include isospin of particles involved in the forward scattering in a complete contradiction with the @xmath2-matrix theory including supersymmetry . to resolve this contradiction we note that the derivation of the no - go theorem assumes that the evolution of the scattering process is unitary . by assuming that there is more than one dephasing `` no - scattering '' operator in ( 3.7 ) , i.e. by assuming a non - unitary evolution with @xmath620 , we can `` evade the evasion '' and avoid any charges with lorentz index to resolve the contradiction . the non - unitary evolution law does allow for unitary decoherence free channels in the hilbert space @xmath621 . in these subspaces all kraus operators @xmath101 are equal to a unitary operator @xmath622 . in these spaces we can not invoke the condition @xmath36 to `` evade the evasion '' of the no - go theorem . neither can we invoke the breaking of the lorentz symmetry by the kraus operators . the only option the dynamical system now has to `` evade the evasion '' in order to preserve its consistency with the @xmath2-matrix dynamics is to break lorentz symmetry by mixing some @xmath2-matrix amplitudes with different spins in the observed amplitudes . this is a classical case of a spontaneously broken symmetry induced by the environment . the spin mixing of the @xmath2-matrix amplitudes is a pure quantum mechanical effect due to the superposition principle . it arises from the action of the kraus operators on the dipion - recoil nucleon states @xmath623 given by the superposition ( 4.14 ) . since it affects only the observed amplitudes it co - exists with the fundamental lorentz symmetry of the particle dynamics and standard model . quantum superposition principle co - exists with lorentz symmetry in another important quantum effect arising from this principle - quantum entanglement of spin states or other quantum states . the apparent violation of lorentz symmetry by two entangled particles is well known since the famous epr paper . in this sense the spin mixing in particle scattering is analogous to quantum entanglement . in both cases there is an apparent violation of the lorentz symmetry . vector - scalar meson mixing is not a new idea . in 1977 chin studied @xmath624 mixing in nucleon - nucleon interactions in high density matter  @xcite . in 2000 - 2002 gale and collaborators examined the effects of @xmath625 mixing on the dilepton production in relativistic heavy ion collisions  @xcite measured at rhic . the @xmath624 mixing was induced by the ground state of the system of the interacting nucleons , while the @xmath625 mixing originated in nucleon - nucleon excitations in the medium of nuclear matter . in both cases there was no violation of fundamental symmetries because the interaction lagrangian conserved those symmetries including the lorentz symmetry . the meson spin mixing and the spontaneous violation of lorentz symmetry was due to the interaction of the nucleon - nucleon scattering process with its environment . it was this work on vector - scalar mixing that inspired the present author in 2003 to seek the solution to the puzzle of the appearance of a rho - like resonance in the @xmath2-wave in @xmath7 and its absence in @xmath13 in the idea of @xmath6 spin mixing . the cern measurements of @xmath7 on polarized target at 17.2 gev / c are so far the only high statistics measurements of its kind , yet they revealed an interesting new physics beyond the standard model . in part i. of this work we have shown that they reveal a nonunitary interaction of the produced final state with a quantum environment . to preserve the identity of the final state particles this new interaction must be a pure dephasing interaction . in this part ii . we required that the dephasing interaction be fully consistent with the standard model . from the complexity of the consistency requirements emerges the simplicity of the spin mixing mechanism ( 6.17 ) and ( 6.18 ) , and its validation by the conservation of probability ( 7.21 ) . from this consistency alone we have deduced that in @xmath0 processes the new interaction with the environment must be a dipion spin mixing interaction governed by the spin mixing mechanism . applied to the @xmath2- and @xmath3-wave subsystem in @xmath7 the theory predicts @xmath6 mixing in agreement with the evidence for such a mixing in all analyses of five experiments on polarized targets surveyed in ref . @xcite . due to the selection rule @xmath5 the theory correctly predicts the absence of @xmath589 mixing in @xmath7 .    since only @xmath190 are allowed in @xmath13 and in @xmath553 there is no spin mixing in these processes , only a change of the phase of the @xmath2-matrix partial wave amplitudes . consider u - channel processes such as @xmath626 and @xmath627 . in these reactions the two - pion systems @xmath628 and @xmath14 are accompanied by a spin 0 particle . the conservation of the total angular momentum by the kraus operators in @xmath629 then requires @xmath630 . therefore there is no spin mixing also in these processes .    in a sequel paper @xcite a new amplitude analysis of the cern data using spin mixing mechanism determines spin mixing parameters @xmath528 and @xmath529 , @xmath2-and @xmath3-wave @xmath2-matrix amplitudes and selects a single solution for the spin mixing amplitudes of a decoherence free @xmath2- and @xmath3-wave subsystem . this analysis allows to extract @xmath392-wave observables from the cern data . analysis of the full @xmath392-wave subsystem for transversity @xmath485 reveals a @xmath15 mixing in the dephasing doublets @xmath408 and @xmath409 . detailed analysis of the amplitudes @xmath580 and @xmath581 determines that the number of interacting degrees of freedom of the environment is @xmath517 . evidence from the analysis at high @xmath261  @xcite limits the dimension @xmath35 of the hilbert space of the environment to @xmath631 . we use coleman - mandula no - go theorem to show that the consistency of the pure dephasing interactions with the standard model requires @xmath36 . as a result decoherence free subsystems must spontaneously break lorentz symmetry while the lorentz symmetry is conserved by the decohering subspaces . the breaking of lorentz symmetry or the violation of cosine conditions in the observed amplitudes are thus two unambigous signatures of dephasing interactions . the obvious question arises - what is the physical nature of the quantum environment ? the consistency of the quantum environment with the standard model suggests that it is a universal environment in the entire universe . it can be viewed as a sea of quantum states @xmath96 across the universe . the quantum states @xmath96 must be stable and can not directly interact with any of the particles of the standard model including the higgs boson . the pure dephasing interaction involves only the exchange of quantum information between the two quantum states @xmath1 and @xmath68 . as such it stands outside of the standard model . while the dephasing interaction can have strong observable effects , such as @xmath6 mixing , its effects are not observable in the usual studies of particle scattering . since these experiments do not involve diparticle partial wave amplitudes in measurements on polarized targets , they can not observe the resonance mixing or the violations of the cosine conditions . the measurements of relative phases alone can not provide evidence for the quantum environment since the needed unitary phases are known only for @xmath632 processes . the dephasing interaction is largely a `` dark '' interaction . the universal presence of the quantum environment and its pure dephasing interaction with the baryonic matter must manifest themselves in astrophysical observations . such observations provide a convincing evidence for the existence of dark matter and dark energy which are both omnipresent environments in the universe with non - standard interactions with baryonic matter . in ref . @xcite we present a physically motivated model of the quantum states @xmath96 which we propose to identify with the particles of a distinct component of the cold dark matter . in this picture pure dephasing interactions are the interactions of baryonic matter with this form of dark matter . these interactions are not rare events but they require high statistics measurements on polarized targets for their detection . dedicated measurements of production processes such as @xmath0 and @xmath633 on polarized targets would provide new and accessible tools to explore the dark sector of the universe .
in part i. of this work we have presented evidence that the measured relative phases of transversity amplitudes in @xmath0 processes differ from those predicted by the unitary evolution law . 
 we ascribed this difference to a non - unitary interaction of the produced final state @xmath1 with a universal quantum environment in the physical universe . 
 this new kind of interaction must be a pure dephasing interaction in order to render the @xmath2-matrix dynamics of particle scattering processes accessible to experimental observation . 
 if the quantum environment is to be an integral part of the nature then its dephasing interactions must be fully consistent with the standard model . 
 in this work we impose on the dephasing interaction the requirements of the conservation of the identities of the final state particles including their four - momenta , lorentz symmetry , @xmath3-parity and the conservation of total angular momentum and isospin . from this consistency alone 
 we find that the dephasing interaction must be a dipion spin mixing interaction . 
 the observed amplitudes are a unitary transform of the corresponding @xmath2-matrix amplitudes . 
 the elements of the spin mixing matrix are forward scattering amplitudes of dipion spin states with recoil nucleon spin states with initial and final dipion spins @xmath4 and @xmath5 , respectively . 
 dipion helicities are conserved in this scattering . 
 these amplitudes are matrix elements of kraus operators describing the non - unitary interaction with the environment . 
 the theory predicts @xmath6 mixing in the @xmath2-and @xmath3-wave amplitudes in @xmath7 . 
 the predicted moduli and relative phases of the mixed amplitudes are in an excellent qualitative agreement with the experimental results . 
 the mixing of @xmath2-matrix partial wave amplitudes with different dipion spins to form the observable partial wave amplitudes is a new phenomenon beyond the standard model . 
 the spontaneous violation of rotational / lorentz symmetry observed in the measured @xmath2- and @xmath3-wave amplitudes is consistent with the conservation of that symmetry by the @xmath2-matrix and by the standard model lagrangian .