Tree Amplitudes Research Articles

Fusing gauge theory tree amplitudes into loop amplitudes

We identify a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities. One-loop amplitudes for massless supersymmetric gauge theories fall into this class; in addition, many non-supersymmetric amplitudes can be rearranged to take advantage of the result. As applications, we construct the one-loop amplitudes for n-gluon scattering in N = 1 supersymmetric theories with the helicity configuration of the Parke-Taylor tree amplitudes, and for six-gluon scattering in N = 4 super-Yang-Mills theory for all helicity configurations.

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.

One-loop n-point gauge theory amplitudes, unitarity and collinear limits

We present a technique which utilizes unitarity and collinear limits to construct ansätze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for n-gluon scattering in N = 4 supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our N = 4 ansatz is correct using general properties of the relevant one-loop n-point integrals. We also give the “splitting amplitudes” which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

Radiation tree amplitudes: Zeroing in on more photons and gluons.

We discuss radiation zeros that are found in gauge tree amplitudes for processes involving multiphoton emission. Previous results are clarified by examples and by further elaboration. The conditions under which such amplitude zeros occur are identical in form to those for the singlephoton zeros, and all radiated photons must travel parallel to each other. Any other neutral particle likewise must be massless (e.g., gluon) and travel in that common direction. The relevance to questions such as gluon jet identification and computational checks is considered. We use examples to show how certain multiphoton amplitudes evade the zeros, and to demonstrate the connection to a more general result, the decoupling of an external electromagnetic plane wave in the "null zone." Brief comments are made about zeros associated with other gauge-boson emission.

Two-dimensional lattice tree exponents and amplitudes: Simulation algorithms versus series

We use a local Monte Carlo algorithm to simulate lattice trees in two dimensions for the site and bond problem. We investigate the properties of radius of gyration, perimeter-to-site ratio, and vertex degree in a tree, adding some new results in the site problem, compare our results on their noncritical properties with those obtained from earlier reversible and slightly nonreversible algorithms, and combine our determinations with new exact series expansion data. On the controversy surrounding the possible lack of universality for the first confluent singularity for the gyration radius, we feel that conclusions must be guarded.

Tree amplitudes at multiparticle threshold in a model with softly broken O(2) symmetry

Tree amplitudes of the production of two kinds of scalar particles at threshold from one virtual particle are calculated in a model of two scalar fields with O(2) symmetric quartic interaction and unequal masses. These amplitudes exhibit interesting factorial and exponential behaviour at large multiplicities. As a by-product we observe that the kinematically allowed decay of one real particle into n real particles of another kind, all at rest, has zero tree amplitude in this model for n>2.

Exact multiparticle amplitudes at threshold in φ4 theories with softly broken O(∞) symmetry

We consider the problem of multiparticle production at threshold in a φ 4-theory with an O( N 1 + N 2) symmetry softly broken down to O( N 1) × O( N 2) by unequal masses. We derive the set of recurrence relations between the multiparticle amplitudes which sums all relevant diagrams with arbitrary number of loops in the large- N limit with fixed number of produced particles. We transform it into a quantum mechanical problem and show how it can be obtained directly from the operator equations of motion by applying the factorization at large N. We find the exact solutions to the problem by using the Gelfand-Dikiĭ representation of the diagonal resolvent of the Schrödinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and masses. The form of the solution is due to the fact that the exact amplitude of the process 2 → n vanishes at n > 2 on mass shell when averaged over the O( N 1,2) indices of incoming particles. We discuss what dynamical symmetry is behind this property. We also give an exact solution in the large- N limit for the model of the O( N) + singlet scalar particle with the spontaneous breaking of a reflection symmetry.

Zeros of tree amplitudes at rest and symmetries of mechanical systems

We consider the tree amplitudes of the production of n 2 scalar particles by n 1 particles of another kind, where both initial and final particles are at rest and on-mass-shell, in a model of two scalar fields with O(2) symmetric interaction and unequal masses. We find that these amplitudes are zero except for the lowest possible n 1 and n 2, and that the cancellation of the corresponding Feynman graphs occurs due to a special symmetry of the classical mechanical counterpart of this theory. This feature is rather general and is inherent in various other scalar field theories.

Trilinear couplings among the electroweak vector bosons and their determination at LEP2

We present a detailed investigation of the potential of LEP2 to measure the (trilinear) couplings of the vector bosons, Z 0W +W − and γW +W − among one another. The restrictions imposed on non-standard trilinear couplings by symmetry requirements, thus excluding violent boson-loop divergences and a violent growth of tree amplitudes for vector-boson scattering, are discussed in detail. We elaborate on how to extract the W ± spin-density matrix and the W +, W − spin correlations from future data. Various theoretically motivated fits to artificial “data” on e +e −→W +W − are carried out in order to evaluate the potential of LEP2 for the measurement of trilinear vector boson couplings. We stress that a direct measurement of the couplings of the vector bosons with one another is indispensable for a verification of the basic SU(2) L×U(1) Y structure of the present electroweak theory. Such measurements at LEP2 improve indirect LEP1 bounds by factors which occasionally reach a full order of magnitude.

Scattering states and symmetries in the matrix model and two-dimensional string theory

We study the correspondence between the linear matrix model and the interacting nonlinear string theory. Starting from the simple matrix harmonic oscillator states, we derive in a direct way scattering amplitudes of two-dimensional strings, exhibiting the nonlinear equation generating arbitrary N-point tree amplitudes. An even closer connection between the matrix model and the conformal string theory is seen in studies of the symmetry algebra of the system.

INTERACTION OF d=2, c=1 DISCRETE STATES FROM STRING FIELD THEORY

Starting from string field theory for 2D gravity coupled to c=1 matter we analyze the off-shell tree amplitudes of discrete states. The amplitudes exhibit the pole structure and we use the off-shell calculus to extract the residues and prove that only the residues are constrained by the Ward identities. The residues generate a simple effective action.

Perturbation theory in two-dimensional open string field theory.

In this paper we develop the covariant string field theory approach to open 2d strings. Upon constructing the vertices, we apply the formalism to calculate the lowest order contributions to the 4- and 5- point tachyon--tachyon tree amplitudes. Our results are shown to match the `bulk' amplitude calculations of Bershadsky and Kutasov. In the present approach the pole structure of the amplitudes becomes manifest and their origin as coming from the higher string modes transparent.

Multiscalar production amplitudes beyond threshold

We present exact tree-order amplitudes for H ∗ → n H , for final states containing one or two particles with non-zero three-momentum, for various interaction potentials. We show that there are potentials leading to tree amplitudes that satisfy unitary, not only at threshold but also in the above kinematical configurations and probably beyond. As a by-product, we also calculated 2 → n tree amplitudes at threshold and show that for the unbroken ifφ 4 theory they vanish for n ⩾ 4, for the standard Model Higgs they vanish for n ⪖ 3 and for a model potential, respecting tree-order unitarity, for n even and n ⩾ 4. Finally, we calculate the imaginary part of the one-loop 1 → n amplitude in both symmetric and spontaneously broken φ 4 theory.

Nonlocal Yang-Mills

We present a very simple and explicit procedure for nonlocalizing the action of any theory which can be formulated perturbatively. When the resulting nonlocal field theory is quantized using the functional formalism — with unit measure factor — its Green functions are finite to all orders. The construction also ensures perturbative unitarity to all orders for scalars with nonderivative interactions, however, decoupling is lost at one loop when vector and tensor quanta are present. Decoupling can be restored (again, to all orders) if a suitable measure factor exists. We compute the required measure factor for pure Yang-Mills at order g 2 and then use it to evaluate the vacuum polarization at one loop. A peculiar feature of our regularization scheme is that the on-shell tree amplitudes are completely unaffected. This implies that the nonlocal field theory can be viewed as a highly noncanonical quantization of the original, local field equations.

OFF-SHELL TREE AMPLITUDES IN HIKKO’S CLOSED STRING FIELD THEORY

Off-shell closed string tree amplitudes are obtained for the HIKKO’s theory, using the method of off-shell conformal field theory.

Arithmetic features of 2-D strings

The on-shell tree amplitudes of 2-D 𝒫-adic strings are constructed. Along with the recently obtained on-shell tree amplitudes of the ordinary (archimedean) 2-D string, they lead to adelic product formulas for all N-point amplitudes. This is in marked difference from 26-D critical strings, where such adelic formulas only obtain for four-point amplitudes. Certain much-discussed operator redefinitions are considered at the 𝒫-adic level where they change radically, the 𝒫-adic gamma function having only one rather than infinitely many real poles. Also considered are q-strings in 2-D.

Continuing between closed and open strings

A family of dual models is defined that interpolates between the tree diagrams of non-orientable bosonic closed-string theory (which has a massless spin-2 state) and the open-string theory with no internal symmetry (in which the lowest-mass spin-2 state is massive). These models are parametrized by the intercept, Δ, of the leading Regge pole. The only models that have an infinite-dimensional conformal invariance and are consequently free of ghosts are the two familiar string theories with Δ = 2 (closed strings) and Δ = 1 (open strings with no internal symmetry). For arbitrary Δ the models are invariant under the finite dimensional conformal group, SO( Δ, 2), which guarantees the crossing symmetry and consistent factorization of tree amplitudes. The spectrum of the level-two states is exhibited explicitly as Δ varies from 2 to 1 in order to illustrate the manner in which the graviton (the lowest-mass spin-2 state) acquires a mass. The scalar ghost generically associated with massive gravity cancels with the “dilaton” precisely at Δ = 1.

The Thermal Virasoro Formula

The thermal stability of non-planar duality is described at any finite temperature through the new-fashioned four-tachyon tree amplitude of closed bosonic thermal strings within the dispersion theoretic approach based upon the thermofield dynamics

Superstring elastic scattering at [formula omitted]: The covariant loop calculus approach

We study quantum gravity by considering the elastic scattering of two on-shell string ground states in the large- s, fixed- t asymptotic limit. The amplitude is computed by means of the covariant loop calculus which turns out to be a powerful method in the study of such a Regge regime. The leading contribution in powers of s is shown to factorize at any g-loop order in a product of g +1 tree amplitudes times the expectation value of a factorized operator. The cumulative effect of these contributions - once it has been resuummed - restores unitarity in the theory and gives a deflection angle in agreement with the linear approximation to general relativity. In the process, it is possible to determine the normalization dictated by unitarity of all terms in the topological expansion. Next, the first sub-leading contribution is calculated at the two-loop order, the so-called H-diagram. A resummation of such terms gives rise to the first non-linear correction in the deflection angle. Our results are in agreement with previous work by Amati, Ciafaloni and Veneziano in the framework of Regge-Gribov techniques. The correspondence between the two approaches is discussed.

Off-shell amplitudes with physical, auxiliary and Faddeev-Popov ghost fields in covariantized light-cone string field theory

We compute the off-shell tree amplitudes involving physical, auxiliary and second-quantized Faddeev-Popov ghost fields for covariantized light-cone string field theory, using off-shell conformal methods. On-shell agreement with the dual-resonance model is shown.