Non-unitarity in quantum affine Toda theory and perturbed conformal field theory

There has been some debate about the validity of quantum affine Toda field theory at imaginary coupling, owing to the non-unitarity of the action, and consequently of its usefulness as a model of perturbed conformal field theory. Drawing on our recent work, we investigate the two simplest affine Toda theories for which this is an issue –a₂⁽¹⁾ and a₂⁽²⁾. By investigating the S-matrices of these theories before RSOS restriction, we show that quantum Toda theory (with or without RSOS restriction) indeed has some fundamental problems, but that these problems are of two different sorts. For a₂⁽¹⁾, the scattering of solitons and breathers is flawed in both classical and quantum theories, and RSOS restriction cannot solve this problem. For a₂⁽²⁾ however, while there are no problems with breather-soliton scattering there are instead difficulties with soliton-excited soliton scattering in the unrestricted theory. After RSOS restriction, the problems with kink-excited kink may be cured or may remain, depending in part on the choice of gradation, as we found earlier [Nucl. Phys. B 489 [FS] (1997) 557]. We comment on the importance of regradations, and also on the survival of R-matrix unitarity and the S-matrix bootstrap in these circumstances.     

Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-l B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ affine Lie algebras, are baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D_n+1⁽²⁾, A_2n⁽²⁾ and B_n⁽¹⁾R-matrices of Bazhanov and Jimbo. For the D_n+1⁽²⁾ and B_n⁽¹⁾ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.     

Algebraic construction of higher-rank dilute A models

Starting from representations of the Birman-Wenzl-Murakami algebra underlying the critical B_n⁽¹⁾, C_n⁽¹⁾ and D_n⁽¹⁾ RSOS models of Jimbo, Miwa and Okado, we derive four series of solvable, critical RSOS models associated with the twisted affine Lie algebra A_n⁽²⁾. Two of these are the critical limit of the A_2n−1⁽²⁾ and A_2n⁽²⁾ models obtained previously by Kuniba. The other two series, again one of the A_2n−1⁽²⁾ and one of the A_2n⁽²⁾ type, are new, and the latter generalizes the dilute A models to arbitrary rank n. For the two new series we present an elliptic extension which satisfies the Yang-Baxter equation, and show that for certain values of the parameters the higher-rank dilute A models break the ℤ₂ symmetry of the underlying adjacency graphG, whereGis the level-l C_n⁽¹⁾ weight lattice.     

On the relation between Φ(1,2) and Φ(1,5) perturbed minimal models and unitarity

We consider the RSOS S-matrices of the Φ_(1,5) perturbed minimal models which have recently been found in a companion paper. These S-matrices have some interesting properties, in particular, unitarity may be broken in a stronger sense than seen before, while one of the three classes of Φ_(1,5) perturbations (to be described) shares the same thermodynamic Bethe ansatz as a related Φ_(1,2) perturbation. We test these new S-matrices by the standard Truncated Conformal Space method, and further observe that in some cases the BA equations for two particle energy levels may be continued to complex rapidity to describe (a) single particle excitations and (b) complex eigenvalues of the Hamiltonian corresponding to non-unitary S-matrix elements. We make some comments on identities between characters in the two related models following from the fact that the two perturbed theories share the same breather sector.     

Curiosities at c = 1

We consider conformal field theories on a torus with central charge c = 1, and in particular models based upon modding out string propagation on the SU(2) group manifold by its finite subgroups. We find that the partition functions for these models coincide with the continuum limit partition functions of a recently introduced class of RSOS models, defined in terms of the extended Dynkin diagrams of simply-laced Lie algebras, thus giving an alternative interpretation for the primary fields in these latter theories. Three of the models have no massless moduli and thus do not lie on the same line of critical points with the rest. The particular correspondence between simply-laced Lie algebras and finite subgroups of SU(2) that emerges coincides with that which has already appeared in other mathematical contexts.     

Solution of functional equations of restricted An−1(1) fused lattice models

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A_n−1⁽¹⁾ lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra A_n−1⁽¹⁾ with GKO pair A_n−1⁽¹⁾ ⊕ A_n−1⁽¹⁾ ⊃ A_n−1⁽¹⁾.     

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.     

Effect of Wigner energy on the symmetry energy coefficient in nuclei

The nuclear symmetry energy coefficient(including the coefficient a_(sym)~((4)) of the I~4 term) of finite nuclei is extracted by using the differences of available experimental binding energies of isobaric nuclei.It is found that the extracted symmetry energy coefficient a_(sym)~*(A,I) decreases with increasing isospin asymmetry I,which is mainly caused by Wigner correction,since e_(sym)~* is the summation of the traditional symmetry energy e_(sym) and the Wigner energy ew.We obtain the optimal values J = 30.25±0.10 MeV,a_(ss)=56.18±1.25 MeV,a_(sym)~((4)) = 8.33±1.21 MeV and the Wigner parameter x= 2.38 ±0.12 through a polynomial fit to 2240 measured binding energies for nuclei with20 ≤ A ≤ 261 with an rms deviation of 23.42 keV.We also find that the volume symmetry coefficient J■ 30 MeV is insensitive to the value x,whereas the surface symmetry coefficient a_(ss) and the coefficient a_(sym)~((4)) are very sensitive to the value of x in the range 1≤x≤4.The contribution of the a_(sym)~((4)) term increases rapidly with increasing isospin asymmetry I.For very neutron-rich nuclei,the contribution of the a_(sym)~((4)) term will play an important role.     

Quantum corrections to the classical reflection factor in a2(1) Toda field theory

The O(β²) quantum correction to the classical reflection factor is calculated for one of the integrable boundary conditions of a₂⁽¹⁾ affine Toda field theory. This is found to agree with the conjectured exact reflection factor of the quantum theory. We consider the existence of other exact reflection factors consistent with our perturbative answer and examine the question of how duality transformations might relate theories with different boundary conditions.     

Lowest excited states of 2-aminopurine

Absorption and fluorescene spectra and quantum yields of emission over a wide range of pH and solvent polarity were measured for 2-aminopurine and 2-dimethylaminopurine. An increase in dipole moment and a rise of pK values for N₁(N₃) ring atom protonation in the excited state were demonstrated. Thus, the lowest excited singlet state shows partially the (1, a_π¹) configuration. The polarization, lifetimes and the phosphorescence-to-fluorescence ratios indicate that spin-orbit coupling of ¹(n, π¹) and ¹(1, a_π¹) with the lowest triplet state is the main source of phosphorescence intensity. The matrix element of spin-orbit coupling of ¹(n, π¹) to the T₁ state is about 33 times larger than that of ¹(1, a_π¹). The temperature- dependent process of 2-aminopurine fluorescence quenching in either is presumably due to the thermal population of the second excited triplets state of (n, π¹) configuration, which can approximately be located at 370 cm^-1 above the lowest (1, a_π¹) singlet.     

Form factors of exponential operators and exact wave function renormalization constant in the Bullough-Dodd model

We compute the form factors of exponential operators e^kgϕ(x) in the two-dimensional integrable Bullough-Dodd model (a₂⁽²⁾ affine Toda field theory). These form factors are selected among the solutions of general non-derivative scalar operators by their asymptotic cluster property. Through analytical continuation to complex values of the coupling constant these solutions permit to compute the form factors of scaling relevant primary fields in the lightest-breather sector of integrable /gf_1,2 and /gf_1,5 deformations of conformal minimal models. We also obtain the exact wave-function renormalization constant Z(g) of the model and the properly normalized form factors of the operators ϕ(x) and : ϕ²(x) :.     

The methane ν1(a1) vibrational state rotational structure obtained from high-resolution CARS-spectra of the Q-branch

The gaseous methane ν₁(a₁), Q-branch coherent anti-Stokes Raman scattering (CARS) spectra have been investigated at a resolution of 0.002 cm^?1. A complex rotational structure of the resolved Q-branch has been experimentally observed. This structure can be ascribed to strong tetrahedral splitting of the rotational levels of the upper vibrational state, which possibly occurs due to Fermi resonance between the ν₁(a₁) and 2ν₂(a₁) vibrational energy levels which are close to each other. An assignment of the observed spectral lines has been made, yielding the rotational constants B, D, and D_t for the ν₁(a₁) vibrational state of the methane molecule. The absolute Raman frequency ν₁ of the purely vibrational transition has been found.     

Reflection K-matrices for 19-vertex models

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A₁⁽¹⁾ model, Izergin-Korepin or A₂⁽²⁾ model, sl(2|1) model and the osp(2|1) model. We find that there is a general solution for A₁⁽¹⁰⁾ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A₂⁽²⁾ and os(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.     

Photoelectron spectra of pyromellitic dianhydride,dithioanhydride and diimide and of trimellitic anhydride

The photoelectron (He(I)) spectra of the tricyclic tetracarbonyl compounds pyromellitic dianhydride, dithioanhydride and diimide and of the tetracyclic hexacarbonyl compound trimellitic anhydride have been investigated. To aid the interpretation of the main features of the spectra, i.e. the ordering and splitting of the ⁿCO ionisations and the behaviour of the ‘benzenic’ and heteroatom π ionisations, MO calculations based on a ZDO pragmatic model and semiempirical SCF-PP calculations have been carried out. The evolution of the ⁿCO and π_X ionisations upon progressive fusion of anhydride moieties with a benzene nucleus is analysed in detail. The proposed orbital sequences for the n orbitals are: a_g(S⁺) $\text{\$}\text{?}$b_1u(AS⁺) $\text{\$}\text{?}$b_2u(S^?) $\text{\$}\text{?}$b_3g(AS^?) for the tetracarbonyls and a′₁(S⁺) $\text{\$}\text{?}$e′(AS⁺) $\text{\$}\text{?}$a′₂ (AS^?) ≈ e′(S^?) for the hexacarbonyl.     

Local state probabilities for solvable restricted solid-on-solid models:A n,D n,D n (1) , andA n (1)

The local state probabilities (LSPs) are exactly computed for four hierarchies of solvable lattice models. They are restricted solid-on-solid (RSOS) models whose local states and their adjacent conditions are specified by Dinkin diagrams of typesA _n,D _n,D _n ⁽¹⁾ andA _n ⁽¹⁾ . The LSPs are expressed in terms of modular functions characterized by branching identities among the theta functions. Their automorphic properties are used to study the critical behaviors. Some fine structures are found in the spectrum of the critical exponents.     

The :φ2: Field in theP(φ)2 model

Euclidean Field Theory techniques are used to study the Schwinger functions and characteristic function of the :φ²: field in evenP(φ)₂ models. The infinite volume limit is obtained for Half-Dirichlet boundary conditions by means of correlation inequalities. Analytic continuation yields Lorentz invariant Wightman functions. It is shown that, in the infinite volume limit, <:φ(x)²:>≧0 for both the Half and the Full-Dirichlet (λφ⁴)₂ model. This result also holds for a finite volume with periodic boundary conditions.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

Rotational and vibrational temperatures of electronically excited BiN radicals in a chemiluminescent flame

Rotational and vibrational temperatures of electronically excited BiN radicals in a low-pressure Bi_x+N/N₂*/N₂+Ar chemiluminescent flame have been deduced from high-resolution Fourier-transform emission spectra. Bands of three electronic transitions, a³Σ⁺(a₁1)→X¹Σ⁺(X0⁺), b⁵Σ⁺(b₁0⁺)→X¹Σ⁺(X0⁺), and b⁵Σ⁺(b₁0⁺)→a ³Σ⁺(a₁1), were analysed to determine the optical temperatures in the a³Σ⁺(a₁1) and b⁵Σ⁺(b₁0⁺) states. The rotational temperatures characterising the rotational populations in the a₁1, v=0 and 1 states were determined from the a₁→X, 0-2, 0-3, 0-4, 1-1, and 1-2 bands. The b₁→X, 0-8 and 0-11 bands, and the b₁→a₁, 0-0 bands served to determine the rotational temperature of the radicals in the b₁0⁺, v=0 state. The temperatures derived from the various bands and transitions were well consistent and the mean rotational temperature was determined to be 353±18 K, which is close to the translational temperature of the gas.Vibrational temperatures of the radicals in the a₁1 and b₁0⁺ states were derived from band intensities of the a₁→X and from the b₁→X as well as b₁→a₁ systems, respectively. The Franck-Condon factors needed were calculated with RKR potentials deduced from literature values of the rotational and vibrational constants in the three states involved. The a₁1 vibrational temperature (336±21 K) was close to the rotational temperature, while the b₁0⁺ vibrational temperature (438±36 K) differed, likely due to the previously observed perturbation of the b₁0⁺ state.     

Broken N=2 super Yang-Mills theories with a finite vacuum energy

It is shown that an explicit symmetry breaking of the N=2 ultra-violet finite super Yang-Mills theories through the addition of soft mass terms does not affect the finiteness of the vacuum energy above one loop. One-loop vacuum energy is finite provided the mass terms satisfy the sum rules Σ_J(2J+1)(−1)^2Jm_J^2K = 0 (K = 0, 1, 2).