”Isochronic” dynamical systems and nullification of amplitudes

We construct the set of theories which share the property that the tree-level threshold amplitudes nullify even if both initial and final states contain the same type of particles. The origin of this phenomenom lies in the fact that the reduced classical dynamics describes isochronic systems. supported by the Lódź University grant No. 269 supported by KBN grant No. 5 P03B 060 21 1 The problem of multiparticle production has attracted much attention in the past decade [1]. It appeared that quite a detailed knowledge concerning the amplitudes of such processes is possible for special kinematics, in particular those involving particles produced at rest [2] ÷ [7]. An interesting phenomenon that appeared here is the nullification of certain tree amplitudes at the threshold. For example, for the process 2 → n, with all final particles at rest, all amplitudes vanish except n = 2 and n = 4 in Φ unbroken theory and except n = 2 if Φ → −Φ symmetry is broken spontaneously [8] ÷ [10]. Other theories were also analysed from this point of view and the nullification of tree 2 → n amplitudes at the threshold has been discovered in the bosonic sector of electroweak model [11] and in the linear σ-model [12]. These results in general do not extend to the one-loop level [13]. One should also mention that in more complicated theories the nullification takes place only provided some relations between parameters are satisfied [11]. The origin of these relations (some hidden symmetry?) remains unclear and is obscured by the fact that nullification does not survive, in general, beyond tree approximation. In the very interesting papers Libanov, Rubakov and Troitsky [14], [15] provided another example of threshold amplitudes nullification in the tree approximation. They considered Φ-theory with O(2) symmetry, the symmetry being softly broken by the mass term. It appeared that the tree amplitudes describing the process of the production of n2 particles φ2 by n1 particles φ1, all at rest, vanishes if n1 and n2 are coprime numbers up to one common divisor 2. Libanov et al. showed that the ultimate reason for nullification is that the O(2)-symmetry survives, in some sense, when the symmetry breaking mass term is introduced. Let us sketch briefly their argument. The starting point is the well-known fact that all Green functions in tree approximation are generated by the solution of classical field equations with additional coupling to external sources and Feynman boundary conditions. Such a solution represents tree-graphs contribution to one-point Green function in the presence of external sources. The consecutive derivatives at vanishing sources provide the relevant Green functions in tree approximation. However, we can do even better [4] (see also [16]). One considers the generating functional for the matrix elements of the field between the states containing arbitrary numbers of inand outon-shell particles. This functional can be obtained as follows [4], [16]. Let the relevant Lagrangian be L(Φ, ∂μΦ) = L0(Φ, ∂μΦ) + LJ (Φ), (1) 2 where Φ ≡ (Φi) is the collection of fields, L0 contains all quadratic terms and LJ describes interactions. Consider the system of integral equations Φi(x | Φ0) = Φ0i(x) + ∫ dy∆Fij(x− y) ∂LJ(Φ) ∂Φj(y) ; (2) here ∆Fij is the operator inverse to δL0 δΦiδΦj with Feynman boundary conditions imposed and Φ0i(x) is the combination, with arbitrary coefficients, of freeparticle wave functions with positive (for incoming particles) and negative (for outgoing particles) energies. Succesive derivatives with respect to these arbitrary coefficients give relevant matrix elements. Graphically, these matrix elements are given by sums of tree graphs with all external lines but one amputated and replaced by relevant wave functions. In order to obtain the corresponding S-matrix element one has only to amputate the remaining propagator and go to mass shell with the corresponding fourmomentum. Eq. (2) implies ( δij +m 2 ij)Φj(x | Φ0)− ∂LJ ∂Φi |Φi→Φi(x|Φ0)= 0 (3a) Φi(x | Φ0) |LJ=0= Φ0i(x) (3b) Things simplify considerably if all particles are at rest. All matrix elements become space-independent and only the time dependence remains to be determined. Eq. (3) is transformed to (∂ t δij +m 2 ij)Φj(t | Φ0)− ∂LJ (Φ) ∂Φi |Φi=Φi(t|Φ0)= 0 (4) We arrive at the set of nonlinear coupled oscillators. Tree expansion arises when we solve (4) pertubatively in LJ (Φ). Libanov et al. have shown that nonvanishing amplitudes are produced if, in the course of solving (4) pertubatively, we are faced with the resonances. Then the solution diverges and this very divergence is cancelled when the external line is amputated. Divergent resonant solution means that we are looking for solution with diverging initial conditions. If, instead, we insist on keeping initial conditions finite while aproaching resonance, the preexponential factor linear (in general-polynomical) in time is produced. So, nonvanishing amplitudes are possible only if the expansion of Φi(t | Φ0), eq.(3), in terms of coupling constant(s) contains terms which are polynomial in time [16]. Libanov et 3 al. have shown that, in the O(2) case, where the corresponding mechanical system in integrable, the symmetry related to the additional integral of motion prevents the resonances to appear. Consequently, the corresponding tree amplitudes vanish. Eventually, this nullification is a result of subtle cancellations of contributions coming from separate graphs. They can be shown to result from Ward identities related to the above symmetry [17]. Libanov et al. argued that the nullification described above should be valid in more general situation. Namely, the reduced classical system, which describes tree amplitudes at the threshold, should exhibit a non-trivial symmetry with the property that the infinitesimal transformation for at least one of the fields contains a term linear in this field or its derivative. This conclusion can be supported by more detailed still simple arguments [18]. One can understand the result of Ref. [15] from slightly different perspective [16]. Assume that the reduced dynamical system (4) is integrable (and confining-this last requirement is, however, not crucial). Then one can introduce action-angle variables (Ji, θi) and expand Φi(t | Φ0) in multiple Fourier series Φi(t | Φ0) = ∑ n1, ..., nr Ai, n1, ..., nr(J, λ)e i r ∑ k=1 nkωk(J, λ)t ; (5) here λ stands for the set of coupling constants. As we have explained above, the resonancees are related to the polynomial preexponential time dependence of separate terms in perturbative expansion. If one expands the righthand side of (5) in λ such terms result from λ-dependence of frequencies ωk(J ; λ). In general, ωk(J, λ) do depend on λ. However, with J = 0 (under appropriate normalization of J ’s), ωk(0; λ) become the frequencies of harmonic part (i. e. the masses of particles) and do not depend on λ. Now, the crucial point is that we are considering the amplitudes with different kinds of particles in incoming and outcoming states. Therefore, in the boundary condition (3b) we can put Φ0i = zie ii, εi = ±1 (6) Then the nontrivial solutions with J = 0 are possible (cf. the explicit solutions given in Ref. [14]), i. e. the coefficients Ai; n1, ..., nr are nonvanishing