No-particle-production classification for supersymmetric two-dimensional theories with multiple superfields or extended supersymmetry

Two-dimensional integrable quantum field theories scatter purely elastically: amplitudes for two particles to turn into three or more particles vanish exactly. Bercini and Trancanelli (Phys. Rev. D 97, 105013 (2018)) imposed this vanishing at tree level on the generic superpotential of a single real $\mathcal{N}=(1,1)$ scalar superfield and found a unique answer within the $Z_2$-symmetric class, the supersymmetric sinh-Gordon superpotential, while showing that without $Z_2$ the fermionic production processes cannot be cancelled simultaneously with the bosonic ones, so that apparently no supersymmetric Bullough-Dodd extension exists; they posed the extension to theories with more than one superfield and to $\mathcal{N}=(2,2)$ multiplets as an open problem. This problem asks for that classification: determine all superpotentials of two-dimensional theories built from two or more real $\mathcal{N}=(1,1)$ scalar superfields, and of $\mathcal{N}=(2,2)$ scalar multiplets, for which every tree-level $2\to n$ amplitude ($n\geq 3$, arbitrary boson/fermion external mixtures) vanishes, or prove that no nontrivial such superpotential exists. The bosonic analogue is settled - tree-level elasticity forces affine-Toda couplings - making the supersymmetric multi-field classification the missing piece of this rediscovery program.

Background

In many two-dimensional quantum field theories the scattering is integrable: the two-particle S-matrix determines all amplitudes, collisions are purely elastic (momenta are exchanged but never redistributed), and processes in which two incoming particles turn into three or more outgoing particles have exactly zero amplitude. This last property - absence of particle production - is a sharp diagnostic of integrability that can be tested directly on a Lagrangian, order by order in perturbation theory. At tree level (the lowest, classical order in the couplings), one may write down a general class of interactions and ask which choices make every production amplitude $M_{2\to n}$ with $n\geq 3$ vanish identically. For a single real bosonic scalar this singles out the sinh-Gordon and Bullough-Dodd potentials (the integrable single-exponential and two-exponential relativistic potentials); recent work re-derived this from recursion relations in a high-energy (multi-Regge) limit, in which a light-cone momentum parameter is taken to infinity, and extended the analysis to several bosonic scalar fields, showing that vanishing tree-level production forces the couplings to satisfy the area rule of affine Toda field theory (the area rule fixes the three-point couplings in terms of the particle masses in a universal way; affine Toda theories are the integrable bosonic theories whose exponential potentials are built from the root systems of affine Lie algebras).

Supersymmetry ties bosons and fermions together and makes the question richer. A two-dimensional theory with $\mathcal{N}=(1,1)$ supersymmetry (one supercharge of each chirality) can be written in terms of superfields: a real scalar superfield $\Phi$ packages a real boson $\phi(x)$, a two-component Majorana fermion $\psi(x)$ (a fermion that is its own antiparticle), and an auxiliary field $F(x)$ that is eliminated by its algebraic equation of motion. All interactions are then encoded in a single function, the superpotential $W(\phi)$: a mass term plus an arbitrary power series in $\phi$ (a scalar has classical scaling dimension zero in two dimensions, so arbitrary powers are allowed without spoiling power-counting renormalizability). Eliminating $F$ yields the component Lagrangian

$$\mathcal{L} \;=\; \tfrac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi \;+\; \bar\psi\, i\slashed\partial\, \psi \;-\; \tfrac{1}{2}\big(W'(\phi)\big)^2 \;-\; W''(\phi)\,\bar\psi\psi ,$$

where the square of $W'$ gives the bosonic potential while $W''$ sets the fermion bilinear couplings: supersymmetry locks the two sectors together.

Bercini and Trancanelli carried out the no-particle-production program for one such superfield. Restricting to potentials with the $Z_2$ symmetry $\phi\to-\phi$ (even powers only), vanishing of the tree-level $M_{2\to n}$ for all $n$ imposes a recursion relation on the couplings, $\lambda_n=\lambda_4^{(n-2)/2}$ for even $n$ (in the paper's normalization, which sets the mass parameter to one without loss of generality), whose resummation is $W(\phi)=\frac{1}{\lambda_4}\big(\cosh(\sqrt{\lambda_4}\,\phi)-1\big)$, the superpotential of the $\mathcal{N}=1$ supersymmetric sinh-Gordon model (analytic continuation $\lambda_4\to-\lambda_4$ gives the supersymmetric sine-Gordon); they verified that all production processes involving external fermions, such as $\bar\psi\psi\to\phi^4$, then also vanish, recursively in $n$. Relaxing $Z_2$ opens the odd-power branch, on which the bosonic amplitudes are consistent with the Bullough-Dodd potential, but the simplest fermionic process $\bar\psi\psi\to\phi^3$ requires $\lambda_5=18\lambda_3^3$, incompatible with the bosonic Bullough-Dodd value $\lambda_5=15\lambda_3^3$, so particle production is unavoidable: apparently no supersymmetric extension of the Bullough-Dodd model exists within a single real superfield. The supersymmetric sinh-Gordon model is thus the unique $Z_2$-symmetric single-superfield answer.

The paper closes by posing the natural completion, which is the present problem: extend the analysis to theories with more than one superfield, and to $\mathcal{N}=(2,2)$ multiplets (two supercharges of each chirality, with scalar matter in chiral superfields, meaning superfields whose lowest component is a complex scalar, and holomorphic superpotentials). The bosonic analogue suggests that multi-superfield answers, if they exist, should be supersymmetric relatives of the affine-Toda interactions; known integrable $\mathcal{N}=2$ models such as the $\mathcal{N}=2$ sine-Gordon provide candidate endpoints for the extended-supersymmetry sector. But the single-field obstruction already shows that supersymmetry can remove entire bosonic branches, so nonexistence results in some sectors are a genuine possibility.

Problem Statement

For two-dimensional supersymmetric theories of scalar superfields, determine the complete classification of superpotentials with no tree-level particle production, beyond the known single-superfield case. Concretely, two linked sectors jointly complete the classification program posed by the source:

(1) Multi-superfield sector: consider theories of $k\geq 2$ real $\mathcal{N}=(1,1)$ scalar superfields $\Phi_i$ (components $\phi_i$, $\psi_i$, $F_i$, with the component Lagrangian obtained by eliminating the auxiliary fields via $F_i=-\partial W/\partial\phi_i$), with a generic superpotential $W(\phi_1,\dots,\phi_k)$ given by an arbitrary power series in the fields, containing mass terms and interactions, with no discrete symmetry assumed (solutions may or may not respect one). Determine all $W$ for which every tree-level amplitude $M_{2\to n}$ with $n\geq 3$ vanishes identically, for arbitrary choices of the two incoming and $n$ outgoing particles among the bosons and fermions of the multiplets, and for all on-shell momenta.

(2) Extended-supersymmetry sector: determine all superpotentials of $\mathcal{N}=(2,2)$ scalar multiplet contents (for example, sets of chiral superfields, with holomorphic superpotentials) for which the same property holds - vanishing of all tree-level $2\to n$ production amplitudes in every external-particle mixture allowed by the multiplets.

A complete answer consists of, for each sector, either an exhaustive list of nontrivial production-free superpotentials together with (i) a proof that each listed superpotential has vanishing $M_{2\to n}$ for all $n$ and all external mixtures, and (ii) a proof that any other superpotential in the class produces particles at tree level; or a proof that no nontrivial such superpotential exists, so that only the free (quadratic) theory qualifies. The expected shape of the answer in sector (1), suggested by the bosonic multi-field result that tree-level elasticity forces affine-Toda couplings, is a supersymmetric affine-Toda-type family, but this must be derived rather than assumed, and negative results - as occurred for the single-superfield Bullough-Dodd - are possible in either sector. Partial results, such as a complete classification for two superfields only, or for one fixed $\mathcal{N}=(2,2)$ multiplet content, constitute partial progress toward the full classification.

References

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