Multiparticle Amplitudes Research Articles

Basis decompositions of genus-one string integrals

One-loop scattering amplitudes in string theories involve configuration-space integrals over genus-one surfaces with coefficients of Kronecker-Eisenstein series in the integrand. A conjectural genus-one basis of integrands under Fay identities and integration by parts was recently constructed out of chains of Kronecker-Eisenstein series. In this work, we decompose a variety of more general genus-one integrands into the conjectural chain basis. The explicit form of the expansion coefficients is worked out for infinite families of cases where the Kronecker-Eisenstein series form cycles. Our results can be used to simplify multiparticle amplitudes in supersymmetric, heterotic and bosonic string theories and to investigate loop-level echoes of the field-theory double-copy structures of string tree-level amplitudes. The multitude of basis reductions in this work strongly validate the recently proposed chain basis and stimulate mathematical follow-up studies of more general configuration-space integrals with additional marked points or at higher genus.

Advanced tools for basis decompositions of genus-one string integrals

In string theories, one-loop scattering amplitudes are characterized by integrals over genus-one surfaces using the Kronecker-Eisenstein series. A recent methodology proposed a genus-one basis formed from products of these series of chain topologies. A prior work further deconstructed cyclic products of the Kronecker-Eisenstein series on this basis. Building on it, our study further employs advanced and comprehensive combinatorial techniques to decompose more general genus-one integrands including a product of an arbitrary number of cyclic products of Kronecker-Eisenstein series, supplemented by Mathematica codes. Our insights enhance the understanding of multiparticle amplitudes across various string theories and illuminate loop-level parallels with string tree-level amplitudes.

The breakdown of resummed perturbation theory at high energies

Calculations of high-energy processes involving the production of a large number of particles in weakly-coupled quantum field theories have previously signaled the need for novel non-perturbative behavior or even new physical phenomena. In some scenarios, already tree-level computations may enter the regime of large-order perturbation theory and therefore require a careful investigation. We demonstrate that in scalar quantum field theories with a unique global minimum, where suitably resummed perturbative expansions are expected to capture all relevant physical effects, perturbation theory may still suffer from severe shortcomings in the high-energy regime. As an example, we consider the computation of multiparticle threshold amplitudes of the form 1 → n in φ6 theory with a positive mass term, and show that they may violate unitarity of the quantum theory for large n, even after the resummation of all leading-n quantum corrections. We further argue that this is a generic feature of scalar field theories with higher-order self-interactions beyond φ4, thereby rendering the latter unique with respect to its high-energy behavior.

Scattering amplitudes as multi-particle higher-spin charges in the correspondence space

Following the proposal of Gelfond and Vasiliev (2016 J. High Energy Phys. JHEP10(2016)067), multi-particle scattering amplitudes are represented as conserved higher-spin (HS) charges. The advantage of such reformulation is that multi-particle amplitudes acquire the form of an integral of a closed form in the correspondence space unifying usual space-time with the twistor-like spinor space. This allows one to identify seemingly different formulae for amplitudes in terms of twistor and space-time integrals. An example of tree MHV amplitudes of Yang–Mills theory is considered in detail. Our results are derived using unfolded dynamics formulation of massless fields. In these terms all information on the amplitude is contained in a single function that, even for lower spin amplitudes, can be interpreted as a HS symmetry parameter. The proposed technique can be useful to relate different approaches to amplitude calculations.

Multi-particle amplitudes from the four-point correlator in planar mathcal{N} = 4 SYM

A non-trivial consequence of the super-correlator/super-amplitude duality is that the integrand of the four-point correlation function of stress-tensor multiplets in planar mathcal{N} = 4 super Yang-Mills contains a certain combination of n-point amplitude integrands for any n. This combination is the sum of products of all helicity super-amplitudes with their corresponding helicity conjugates. The four-point correlator itself is described by a single scalar function whose loop level integrands possess a hidden permutation symmetry facilitating its computation up to ten loops. We discover that assuming Yangian symmetry and an appropriate basis of planar dual conformal integrands it is possible to disentangle the contributions from the individual amplitudes from this combination. We test this up to seven points and up to two loops. This suggests that any scattering amplitude for any n, with any helicity structure and at any loop order may be extractable from the four-point correlator.

Computational complexity of exterior products and multiparticle amplitudes of noninteracting fermions in entangled states

Noninteracting bosons were proposed to be used for a demonstration of quantum-computing supremacy in a boson-sampling setup. A similar demonstration with fermions would require that the fermions are initially prepared in an entangled state. I suggest that pairwise entanglement of fermions would be sufficient for this purpose. Namely, it is shown that computing multi-particle scattering amplitudes for fermions entangled pairwise in groups of four single-particle states is #P hard. In linear algebra, such amplitudes are expressed as exterior products of two-forms of rank two. In particular, a permanent of a NxN matrix may be expressed as an exterior product of N^2 two-forms of rank two in dimension 2N^2, which establishes the #P-hardness of the latter.

Multi-Regge form of gluon-exchange amplitudes in supersymmetric Yang-Mills theories

Within supersymmetric Yang-Mills theories, all effective vertices for the interaction of Reggeized gluons with particles were found both in the leading-logarithm approximation and in the next-to-leading-order approximation. The contributions of scalars to the eigenfunction of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel for the adjoint representation of the gauge group and to impact factors similar to those in QCD were calculated. The impact factors that arise in supersymmetric Yang-Mills theories, but which are not present in QCD, are also found. Fulfillment of all bootstrap conditions, which guarantees the multi-Regge form of multiparticle amplitudes in supersymmetric Yang-Mills theories in the next-to-leading-order approximation, was proven. A method for testing bootstrap relations without calculating explicit expressions for vertices and impact factors was developed. This method was used to prove fulfillment of bootstrap conditions in theories featuring any number of fermions and scalar particles transformed according to any representations of the color group and involved in interactions with one another via Yukawa coupling.

Impact factor for gluon production in multi-Regge kinematics in the next-to-leading order

The one-loop correction to the impact factor for gluon production upon the transition of a one-Reggeon state in the t channel to a two-Reggeon state is found. This impact factor is an element of multiparticle amplitudes in multi-Regge kinematics. The correction in question is necessary for developing the theory of Regge and multi-Regge processes. In particular, it is necessary for proving the multi-Regge form of the amplitude in the next-to-leading-logarithm approximation. This correction also makes it possible to complete the verification of the last of the unproven bootstrap conditions for gluon Reggeization and to prove, in this approximation, the validity of the multi-Regge form of the amplitude. All necessary calculations are presented, and an explicit expression for the impact factor in front of all possible color states in the t channel is given.

Infrared approach to Reggeization

We present a new approach to Reggeization of gauge amplitudes based on the universal properties of their infrared singularities. Using the "dipole formula", a compact ansatz for all infrared singularities of massless amplitudes, we study Reggeization of singular contributions to high-energy amplitudes for arbitrary color representations, and any logarithmic accuracy. We derive leading-logarithmic Reggeization for general cross-channel color exchanges, and we show that Reggeization breaks down for the imaginary part of the amplitude at next-to-leading logarithms and for the real part at next-to-next-to-leading logarithms. Our formalism applies to multiparticle amplitudes in multi-Regge kinematics, and constrains possible corrections to the dipole formula starting at three loops.

The infrared structure of gauge theory amplitudes in the high-energy limit

We develop an approach to the high-energy limit of gauge theories based on the universal properties of their infrared singularities. Our main tool is the dipole formula, a compact ansatz for the all-order infrared singularity structure of scattering amplitudes of massless partons. By taking the high-energy limit, we show that the dipole formula implies Reggeization of infrared-singular contributions to the amplitude, at leading logarithmic accuracy, for the exchange of arbitrary color representations in the cross channel. We observe that the real part of the amplitude Reggeizes also at next-to-leading logarithmic order, and we compute the singular part of the two-loop Regge trajectory, which is universally expressed in terms of the cusp anomalous dimension. Our approach provides tools to study the high-energy limit beyond the boundaries of Regge factorization: thus we show that Reggeization generically breaks down at next-to-next-to-leading logarithmic accuracy, and provide a general expression for the leading Reggeization-breaking operator. Our approach applies to multiparticle amplitudes in multi-Regge kinematics, and it also implies new constraints on possible corrections to the dipole formula, based on the Regge limit.

Gluon Reggeization in QCD

The Reggeization hypothesis can be proved using the “bootstrap” relations, required by the compatibility of the gluon Reggeization with the s-channel unitarity. It is extremely nontrivial, that an infinite number of the bootstrap relations for the multi-particle production amplitudes can be fulfilled if the Reggeon vertices and trajectory satisfy several bootstrap conditions. All these conditions are derived and all but one are proved to be satisfied.

Multiparticle amplitudes at one-loop: an algebraic/numeric approach

We discuss algebraic/numeric methods to compute one-loop corrections for multiparticle/jet production cross sections. By using efficient reduction algorythms a compact expression for the g g g γ γ → 0 amplitude is obtained. Further a numerical approach for 6-point 1-loop diagrams is presented.

Calculation of 1-loop hexagon amplitudes in the Yukawa model

We calculate a class of one-loop six-point amplitudes in the Yukawa model. The construction of multi-particle amplitudes is done in the string inspired formalism and compared to the Feynman diagrammatic approach. We show that there exists a surprisingly efficient way of calculating such amplitudes by using cyclic identities of kinematic coefficients and discuss in detail cancellation mechanisms of spurious terms. A collection of formulas which are useful for the calculation of massless hexagon amplitudes is given.

Renormalization group corrections to threshold amplitudes

The threshold multiparticle amplitudes in Φ 4 theory are analysed from a point of view of renormalization group. The formulae for leading and next-to-leading terms are given.

Infinity cancellation, type I′ compactification and string S-matrix functional

Non-vanishing tadpoles and possible infinities associated in the multiparticle amplitudes are discussed with regard to the disk and RP 2 diagrams of the type I′ compactification. We find that the infinity cancellation of the SO(32) type I theory extends to this case as well despite the presence of tadpoles localized in the D-brane world-volume and the orientifold surfaces. Formalism of the string S-matrix generating functional is presented to find a consistent string background as the c-number source function: we find this only treats the cancellation of tadpoles in the linearized approximation. Our formalism automatically provides a representation of the string amplitudes on this background to all orders in α′.

MULTIPARTICLE THRESHOLD AMPLITUDES EXPONENTIATE IN ARBITRARY SCALAR THEORIES

Threshold amplitudes are considered for n-particle production in arbitrary scalar theory. It is found that, like in Φ4, leading-n corrections to the tree level amplitudes, being summed over all loops, exponentiate. This result provides more evidence in favor of the conjecture on the exponential behavior of the multiparticle amplitudes.

GEOMETRICAL ASPECTS OF THE HELICITY FORMALISM

A bilinear combination of Dirac spinors is expressed in terms of Lorentz vectors. The result is represented in a convenient form, which can lead to using it for analytical calculations of multiparticle amplitudes. A geometrical interpretation of the introduced vectors is given and the correspondence between Dirac spinors and two-forms in complexified Minkowski space is clarified.

Exponentiation of multiparticle amplitudes in scalar theories. II. Universality of the exponent.

It has been shown recently that the amplitude of the creation of $n$ real scalar particles by one virtual boson near $n$--particle threshold exhibits exponential behavior at $n\sim1/\lambda$. We extend this result to the processes of multiparticle production at threshold by two virtual bosons. We find that both the tree--level amplitude and leading--$n$ loop corrections have the same exponential behavior, with the common exponent for $1\to n$ and $2\to n$ processes, for various kinematics. This result strongly indicates that the exponent for multiparticle amplitudes is independent of the initial state and that there may exist a semiclassical approach to the study of multiparticle production.

Exponentiation of multiparticle amplitudes in scalar theories.

It is argued that the amplitudes of the production of [ital n] soft scalar particles by one or a few energetic ones in theories such as [lambda][ital cphi][sup 4] have the exponential form [ital A][sub [ital n]][proportional to] [radical][ital n] exp[1/[lambda][ital F]([lambda][ital n],[epsilon])], in the regime [lambda][r arrow]0, [lambda][ital n]=fixed, [epsilon]=fixed, where [epsilon] is the typical kinetic energy of outgoing particles. Existing results support this conjecture. Several new analytical and numerical results in favor of the exponential behavior of multiparticle amplitudes are presented.

Exact multiparticle amplitudes at threshold in large-N component phi4 theory.

I derive the set of recurrence relations between the amplitudes of multiparticle production at threshold in the standard large-N limit of the O(N)-symmetric phi^4$ theory which sums all relevant diagrams with arbitrary number of loops. I find an exact solution to the recurrence relations using the Gelfand--Dikii representation of the diagonal resolvent of the Schrodinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and mass. The form of the solution is due to the fact that the exact amplitude of the process 2->n at n>2 vanishes on mass shell when averaged over the O(N)-indices of incoming particles for dynamical reasons because of the cancellation between diagrams. I discuss some possible applications of large-N amplitudes, in particular, for the renormalon problem.