On the role of unitarity in statistical theories of nuclear reactions

It is argued that the approximate statistical S-matrix with energy independent background S-matrix, residues, poles, and with fixed dimensions need not satisfy analytic unitarity but should in general obey the average unitarity condition. The freedom obtained by relaxing analytic unitarity allows a representation where level-level correlations are not present. Different approaches to statistical theories of nuclear reactions employing the pole decomposition of the S-matrix are compared. It is seen that any such approach is characterized by the assumed form of two (matrix) parameters. A model is developed which gives the expected results for the compound cross sections in the limits of strong absorption and weak absorption with statistically equivalent channels, and interpolates between the two extremes. The model depends, however, on the parameter $\text{πΓ}\text{D}$. The possibility of extracting the value of this parameter from experimental data for the variance of cross sections is also investigated.     

Theory of the Trojan-Horse method

The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysically relevant two-body reactions. This is accomplished by studying closely related three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is developed starting from a post-form distorted wave Born approximation of the T-matrix element. In the surface approximation the cross-section of the three-body reaction can be related to the S-matrix elements of the two-body reaction. The essential feature of the Trojan-Horse method is the effective suppression of the Coulomb barrier at low energies for the astrophysical reaction leading to finite cross-sections at the threshold of the two-body reaction. In a modified plane wave approximation the relation between the two- and three-body cross-sections becomes very transparent. The appearing Trojan-Horse integrals are studied in detail.     

Extraction of Astrophysical Cross Sectionsin the Trojan-Horse Method

 The Trojan-horse method has been proposed to extract S-matrix elements of a two-body reaction at astrophysical energies from a related reaction with three particles in the final state. This should be useful in cases where the direct measurement of the two-body reaction at the necessary low energies is experimentally difficult. The formalism of the Trojan-horse method for nuclear reactions is developed in detail from basic scattering theory including spin degrees of freedom of the nuclei and we specify the necessary approximations. The energy dependence of the three-body reaction is determined by characteristic functions that represent the theoretical ingredients for the method. In a plane-wave Born approximation of the T-matrix the differential cross section assumes a simple structure. Received August 31, 1999; revised June 14, 2000; accepted for publication June 30, 2000     

Analyticity and ergodicity in the maximum-entropy approach to statistical nuclear reactions

In the maximum-entropy approach to statistical nuclear reactions one imposes naturally the constraints of unitarity and symmetry of theS-matrix, and of a fixed expectation value ofS. We show that the analytical structure of theS-matrix and the requirement that the problem be ergodic (so that energy averages can be replaced by ensemble averages) impose certain restrictions on the distribution of statisticalS-matrices. Some of these additional constraints are then imposed numerically in a two-channel calculation, and are shown to improve the results for the fluctuation cross sections, the elastic enhancement factor, etc.     

Three-body scattering in Poincaré-invariant quantum mechanics

The relativistic three-nucleon problem is formulated by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. Two-body interactions are included that preserve the Poincaré symmetry, lead to the same invariant two-body S-matrix as the corresponding non-relativistic problem, and result in a three-body S-matrix satisfying cluster properties. The resulting Faddeev equations are solved by direct integration, without partial waves for both elastic and breakup reactions at laboratory energies up to 2?GeV.     

$\mathcal{PT}$-Symmetric Quantum Electrodynamics—$\mathcal{PT}$QED

The construction of PT\mathcal{PT}-symmetric quantum electrodynamics is reviewed. In particular, the massless version of the theory in 1+1 dimensions (the Schwinger model) is solved. Difficulties with unitarity of the S-matrix are discussed.     

Modification of Hauser-Feshbach calculations by direct-reaction channel coupling

If open channels are strongly coupled by direct reactions, the traditional Hauser-Feshbach method of calculating fluctuation cross sections is invalid, because of non-statistical correlations which the direct channel-coupling induces between resonance partial widths in different channels. The fluctuation cross sections can still be computed from the optical S-matrix elements, however, and the formulas necessary for doing so are obtained here with the aid of an “optical background” representation of the full S-matrix. The resulting compound-elastic cross section is increased over the Hauser-Feshbach expression by a factor of 2(Γ ? D) or 3(Γ ? D) in the large-N limit, and compound-reaction cross sections are increased by roughly a factor of $\text{(N + 1)}\text{N}$, where N is the number of directly-coupled open channels.     

Comments on massive and massless Yang-Mills lagrangians with a quartic coupling of Faddeev-Popov ghosts

The complicated Lagrangians for massless as well as massive Yang-Mills fields and their BRS and anti-BRS transformations proposed by Curci and Ferrari are simplified by the help of an auxiliary field. In the massive non-Abelian case, the mechanism of the breakdown of physicalS-matrix unitarity is clarified and contrasted to the situation in the Abelian case.     

Imprints of compound-nucleus and direct-interaction processes in energy-averaged cross sections and durations of nuclear reactions

New analytical expressions for the energy-averaged S-matrix and nuclear-reaction cross sections are obtained on the basis of the unitary S-matrix parametrization. Some approximations are analyzed when compound-nucleus cross sections are described with and without the Hauser-Feshbach formula with the partial-width-fluctuation correction. The connection of the S-matrix parameters in the Simonius representation with the dynamics of compound-nucleus processes is clarified. An explicit expression for the averaged duration of the nuclear reaction is obtained for the range of closely situated resonances. Possible methods of obtaining information on some compound-nucleus and direct-interaction characteristics from the averaged cross sections and durations are examined.     

Separable energy-dependent approximation to local potentials

An extension of the single-pole separable approximation of a two-body t-matrix in which the effects of several poles are included is made. The simple form for the t-matrix derived from a single separable potential is retained. However, the separable potential is constructed using an energy-dependent superposition of the states corresponding to the various poles. The energy dependence is chosen so as to obtain the correct residue of both the on-shell and off-shell t-matrices at each of these poles, while preserving unitarity. The formalism is specialized to the case of s-wave scattering from an attractive square well. Comparison to the exact s-wave cross section gives good results.     

The energy dependence of Delbrück scattering investigated at Z = 73, 82 and 92

Using neutron capture γ-rays from a ¹⁴⁰CeO₂ source installed in the Grenoble high-flux reactor, differential cross sections for the elastic scattering of photons by Ta, Pb and U through θ = 120° have been measured for E = 4.291 and 4.767 MeV. These data have been supplemented by measuring elastic differential cross sections for U, θ = 120° and energies ranging from 0.279 to 1.332 MeV, using radioactive sources. The experimental differential cross sections below 1 MeV confirm the predicted Rayleigh amplitudes based on the second-order S-matrix within 3%. An excellent agreement between experiment and lowest-order Delbrück theory is observed between 1.0 and 1.4 MeV, showing that Coulomb corrections are small close to the threshold for pair production. At 4.291 and 4.767 MeV experiment and lowest-order Delbrück theory agree within ~12%     

Physical unitarity in the Lagrangian Sp(2)-symmetric formalism

The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the Sp(2)-symmetric quantization method. The physical S-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models.     

Model-independent analysis of Airy structures in the differential cross sections for the elastic scattering of 4He and 16O nuclei within a single S-matrix systematics

On the basis of a model-independent S-matrix approach, the differential cross sections for elastic ⁴He-⁹⁰Zr, ¹⁶O-¹²C, and ¹⁶O-¹⁶O scattering at energies of projectile nuclei in the range 13–30 MeV per nucleon are analyzed by using an evolution algorithm. In each of the cases considered in this analysis, the resulting scattering matrix is associated with the same S-matrix systematics and is determined by an absolute value and a nuclear phase, which are smooth and monotonic functions of the orbital angular momentum.     

Statistical theory of nuclear reactions and the Gaussian orthogonal ensemble

Using methods developed in field theory and statistical mechanics, especially in the context of the Anderson model as generalised by Wegner, a novel approach to the statistical theory of nuclear reactions is developed. A finite set of N bound states, coupled to each other by an ensemble of Gaussian orthogonal matrices, is considered and coupled to a set of channels via fixed coupling matrix elements. The ensemble average and the variance of the elements of the nuclear scattering matrix are evaluated, using the method of a generating function combined with the replica trick, followed by the Hubbard-Stratonovitch transformation and a modified loop expansion. In the limit N → ∞, it is shown quite generally that, aside from a trivial dependence on average S-matrix elements, the variance depends only on the transmission coefficients, and that the correlation width of a pair of S-matrix elements is given by a universal function of the transmission coefficients. A modified loop expansion yields an asymptotic series valid for strong absorption. The terms in this series are partly novel, and partly coincide with results obtained earlier in the framework of a model which did not take account of the GOE eigenvalue fluctuations. This suggests that average cross sections are mainly sensitive to the stiffness of the GOE spectrum. Fluctuation properties are also derived, and the link to Ericson fluctuation theory is established.     

Statistical model cross sections for overlapping resonances

The cross section for a reaction dominated by numerous incoherent resonances is studied, as a function of the average resonance width $\text{Λ}$ divided by the spacing D. Both the Bohr and Wheeler formulae for the average cross sections, and the standard Ericson analysis of fluctuations, are found to be valid only for weakly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$, where n is the number of open scattering channels. Two different models are presented which respect unitarity whatever the resonance widths, namely a K-matrix or Weisskopf-Wigner model, and an eikonal type model. The cross sections are found to saturate either the elastic or the inelastic unitarity bound for strongly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$; and the fluctuations in the cross sections are found to have a coherence length of order nD, rather than $\text{Λ}$.     

Gauge quantization for spin-32 fields

The free massless Rarita-Schwinger equation and a recently constructed interacting field theory known as supergravity are invariant under fermionic gauge transformations. Gauge field quantization techniques are applied in both cases. For the free field the Faddeev-Popov ansatz for the generating functional is justified by showing that it is equivalent to canonical quantization in a particular gauge. Propagators are obtained in several gauges and are shown to be ghost-free and causal. For supergravity the Faddeev-Popov ansatz is presented and the gauge fixing and determinant terms are discussed in detail in a Lorentz covariant gauge. The Slavnov-Taylor identity is obtained. It is argued that supergravity theory is free from the difficulty of acausal wave propagation of the type found by Velo and Zwanziger and that pole residues in tree approximation S-matrix elements are positive as required by unitarity.     

Quantization of relativistic systems with constraints

The general dynamical system with constraints is quantized, and the S-matrix is constructed in the most general class of gauges including relativistic ones. In the case when constraints do not form a group a new type of additional diagrams arises securing unitarity of the theory: the four-fermion interaction of ghost fields.     

Strong absorption and the distribution of zeros of the S-matrix

The only distributions normally included in a discussion of the statistical theory of nuclear resonance reactions are the distributions of the widths (Γ) and spacings (D) of the levels of the compound nucleus. However, as the usual Hauser-Feshbach theory makes clear, $\text{Γ}$ and $\text{D}$ alone are not sufficient to determine the ratio $\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>dir</mtext>}}\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>fl</mtext>}}$. In an attempt to determine what further statistical information is sufficient to determine this ratio, in the special limit that it tends to zero for all cc′, c ≠ c′ (the “strong-absorption” limit), we study several “picket fence” S-matrix models, as well as a random-residue model exhibiting Ericson fluctuations. These models indicate that the strong-absorption limit $\text{(}\text{S}{}_{\mathrm{cc\prime }}\text{= 0,}\text{all}\text{cc′)}$ is directly related to the distribution of the zeros of S_cc′(E) in the upper half of the complex E-plane, and that strong absorption is reached only if these zeros are distributed with a high density in the region E → + i ∞. As a by-product, we obtain a generalization of the theorem of Moldauer and Simonius $\text{[|}\text{det}\text{S}\text{| =}\text{exp}\text{(?π}\text{Γ}\text{/}\text{D}\text{)]}$. Our generalization applies to individual optical S-matrix elements (and so to direct-reaction cross sections) rather than just to their determinant.