Scattering Amplitudes

Multiparticle production is described by the obvious generalization of Eq. (l), when all invariants are large: We obtain Aa-s-l-NM-2NB at s-~0, for the exclusive cross section integrated over a (CM angular) region where the pi. pj / s are held fixed and NM and NB are the total number of external mesons and baryons in the process. Page 7, first paragraph: Replace I1 Fig. 2 (c)l1 by 1' Fig. 2(b)", and replace the sentence in parenthesis by: The above discussion assumes the elementary subprocess is qq-nn orqn-qn in the case of pion production and 16 qP-4P, 4"-p<G or Gi-pp for proton production. In addition to this, as pointed out by Bjorken, proton production can also occur through the elementary subprocess qq-pq. This process, as well as the indirect processes 17 such as q=-. qn , with final state fragmentation, givesN = 4 for PP-pX and Z-P-pX.

HELAC is a FORTRAN based package that is able to compute efficiently helicity amplitudes for arbitrary scattering processes within the standard electroweak theory. The algorithm exploits the virtues of the Dyson-Schwinger equations as compared to the traditional Feynman graph approach. All electroweak vertices are included in both the unitary and Feynman gauges, and computations including all mass effects are available. A version performing multi-precision computations with arbitrary-user defined-accuracy is also included, allowing access to any phase space point for arbitrary high energies.

We prove by explicit computation that 6-point matrix elements of D 4 R 4 and D 6 R 4 in N = 8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E 7(7) symmetry. The soft limits precisely match automorphism constraints. Together with previous results for R 4 , this provides a direct proof that no E 7(7) -invariant candidate counterterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators D 4 R 6 , R 8 , ϕ 2 R 8 ,. . . with n > 4 fields and prove that they all violate E 7(7) symmetry. This means that the 4-graviton amplitude determines whether or not the theory is finite at 7-loop order. We show that the corresponding candidate counterterm D 8 R 4 has a nonlinear supersymmetrization such that its single-and double-soft scalar limits are compatible with E 7(7) up to and including 6-points. At loop orders 7, 8, 9 we provide an exhaustive account of all independent candidate counterterms with up to 16, 14, 12 fields, respectively, together with their potential single-soft scalar limits.

We discuss lattice simulations of light nuclei at leading order in chiral effective field theory. Using lattice pion fields and auxiliary fields, we include the physics of instantaneous one-pion exchange and the leading-order S-wave contact interactions. We also consider higher-derivative contact interactions which adjust the S-wave scattering amplitude at higher momenta. By construction our lattice path integral is positive definite in the limit of exact Wigner SU(4) symmetry for any even number of nucleons. This SU(4) positivity and the approximate SU(4) symmetry of the low-energy interactions play an important role in suppressing sign and phase oscillations in Monte Carlo simulations. We assess the computational scaling of the lattice algorithm for light nuclei with up to eight nucleons and analyze in detail calculations of the deuteron, triton, and helium-4.

A d−dimensional rational polytope P is a polytope whose vertices are located at the nodes of Z d lattice. Consider the number kP ∩ Z d of points inside the inflated P with coefficient of inflation k (k = 1, 2, 3, ...). The Ehrhart polynomial of P counts the number of such lattice points inside the inflated P and (may be) at its faces (including vertices). In Part I [ JGP 55 (2005) 50] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) physical models reproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g. those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on ππ scattering , etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [PNAS 93 (1996)14238 ].

The AdS/CFT correspondence between string theory in AdS space and conformal field theories in physical space-time leads to an analytic, semi-classical model for strongly-coupled QCD which has scale invariance and dimensional counting at short distances and color confinement at large distances. The AdS/CFT correspondence also provides insights into the inherently nonperturbative aspects of QCD such as the orbital and radial spectra of hadrons and the form of hadronic wavefunctions. In particular, we show that there is an exact correspondence between the fifth-dimensional coordinate of AdS space z and a specific impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron in ordinary space-time. This connection leads to AdS/CFT predictions for the analytic form of the frameindependent light-front wavefunctions (LFWFs) of mesons and baryons, the fundamental entities which encode hadron properties. The LFWFs in turn predict decay constants and spin correlations, as well as dynamical quantities such as form factors, structure functions, generalized parton distributions, and exclusive scattering amplitudes. Relativistic light-front equations in ordinary space-time are found which reproduce the results obtained using the fifth-dimensional theory and have remarkable algebraic structures and integrability properties. As specific examples we describe the behavior of the pion form factor in the space and time-like regions and determine the Dirac nucleon form factors in the space-like region. An extension to nonzero quark mass is used to determine hadronic distribution amplitudes of all mesons, heavy and light. We compare our results with the moments of the distribution amplitudes which have recently been computed from lattice gauge theory.

Coherence factors are a hallmark of superconductivity as a pair-condensation phenomenon. When electrons pair, quasi-particles develop an acute sensitivity to different types of scattering potential, described by the appearance of coherence factors in the scattering amplitudes. While the effects of coherence factors are well established in isotropic superconductors, they are much harder to detect in their anisotropic counterparts, such as high-T c cuprates. Here we demonstrate a new approach which highlights the momentum-dependent coherence factors in Ca 2-x Na x CuO 2 Cl 2. Using Fourier-transform scanning tunnelling spectroscopy to detect quasi-particle interference effects, our experiments reveal a magnetic-field dependence in quasi-particle scattering which is sensitive to the sign of the anisotropic gap. This result can be understood in terms of d-wave coherence factors and it exposes the role of vortices as quasi-particle scattering centers. We also show that a magnetic field gives rise to an enlarged gapless region around the gap nodes.

We discuss conceptual aspects of renormalization in the context of effective field theories for the two-nucleon system. It is shown that, contrary to widespread belief, renormalization scheme dependence of the scattering amplitude can only be eliminated up to the order the calculations are performed. We further consider an effective theory for an exactly solvable quantum mechanical model which possesses a long-and short-range interaction to simulate pionful effective field theory. We discuss the meaning of low-energy theorems in this model and demonstrate their validity in calculations with a finite cutoff Λ as long as it is chosen of the order of the hard scale in the problem. Removing the cutoff by taking the limit Λ → ∞ yields a finite result for the scattering amplitude but violates the low-energy theorems and is, therefore, not compatible with the effective field theory framework.

We compute the next-to-maximally-helicity-violating one-loop n-gluon amplitudes in N=1 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and constitute the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We assemble ingredients from the N=4 NMHV tree super-amplitude into previously unwritten double cuts and use the spinor integration technique to calculate all bubble coefficients. We also derive the missing box coefficients from quadruple cuts. Together with the known formula for three-mass triangles, this completes the set of NMHV one-loop master integral coefficients in N=1 SYM. To facilitate further use of our results, we provide their Mathematica implementation.

The novel massive spinor-helicity formalism of Arkani-Hamed, Huang and Huang provides an elegant way to calculate scattering amplitudes in quantum chromody-namics for arbitrary quark spin projections. In this note we compute two families of tree-level QCD amplitudes with one massive quark pair and n − 2 gluons. The two cases include all gluons with identical helicity and one opposite-helicity gluon being color-adjacent to one of the quarks. Our results naturally incorporate the previously known amplitudes for both quark spins quantized along one of the gluonic momenta. In the all-multiplicity formulae presented here the spin quantization axes can be tuned at will, which includes the case of the definite-helicity quark states.

The BCFW recursion relation allows to find out the tree-level scattering amplitudes for gluons and tensor gauge bosons in generalized Yang-Mills theory. We demonstrate that the corresponding MHV amplitudes for the tensor gauge bosons of spin-s and n gluons are invariant under conformal group transformations. We discuss and compare the treelevel scattering amplitudes for the charged tensor bosons with the corresponding scattering amplitudes for gravitons, stressing their differences and similarities.

Any gravitational scattering amplitude takes a remarkably simple factorized form at tree level in multi-Regge kinematics (MRK), where the produced particles are strongly ordered in rapidity. Very recently, it was shown that the scattering equations also have a very simple structure in MRK. In this paper we study Einstein gravity amplitudes in MRK in the framework of the scattering equations. We present a new derivation of the multi-Regge factorization of tree-level amplitudes with any number of external gravitons and any helicity configuration.

SUa| 3 ehiral algebra and PCAC have been employed to derive pion and kaon scattering amplitudes on nuclei from sum rules of the Fubini-Furlan type. The nuclear states are described as finite systems in a simple, nonrelativistic impulse approximation. The effects of the break down of chiral SUn| SU a symmetry are described in a (3, 3*)-]-(3", 3) model. The 7:~f ~ sigma-term turns out to be about 461V[eV and we find good agreement between the model by Gell-Mann, Oakes and Renner and both pion and kaon-nueleus data.

Within the = 2 superspace perturbative formulation of ABJM theory, we derive the set of scalar Feynman integrals describing the four-point scattering amplitude of chiral matter fields. Using dimensional regularization (D = 3 - 2ϵ), we evaluate the integrals on-shell and we compare the result with that of a Wilson-loop computed in a light-like tetragon contour: we find matching results between both calculations after identifying the kinematical invariants. We discuss further implications and open questions. This work is a contribution to the proceedings of the ‘XVIIth European Workshop on String Theory 2011’ held in Padua, Italy, on September 2011.

In heavy ion collision simulations many hadron states and/or parton degrees of freedom are included in order to obtain the observables. Meson spectroscopy, for example, considers the 0 ++ meson as a mixture of qq and glue. This fact is usually not considered in heavy ion collision physics. In the present work we consider two extreme possibilities for the constitution of the 0 ++ meson, either as a pure glueball or as qqmeson. The scattering amplitude and cross-sections with constituent interchange are determined for the two situations. The comparison showed that the glueball-glueball elastic scattering cross-section for a color singlet state is between one to two orders of magnitude smaller than the corresponding qq state. The 2 ++ glueball-glueball interaction is also evaluated with similar behavior. Thus, glueball-glueball scattering is not very likely to introduce significant changes in heavy ion collision observables.

When two non-relativistic particles scatter in one dimension, they can become entangled. This entanglement process is constrained by the symmetries of the scattering system and the boundary conditions on the incoming state. Applying these constraints, three different mechanisms of entanglement can be identified: the superposition of reflected and transmitted modes, momentum correlations of the reflected mode due to inversion of the relative momentum, and momentum correlations in the transmitted and reflected modes due to dependence of the scattering amplitude on the relative momentum. We consider three standard potentials, the hard core, Dirac delta, and double Dirac delta, and show that the relative importance of these mechanisms depends on the interaction and on the properties of the incoming wave function. We find that even when the momenta distributions of the incoming articles are sharply peaked, entanglement due to the momentum correlations generated by reflection can be quite large for particles with unequal mass.

Gravitational couplings in bulk space-time include those terms which are fixed by scattering amplitude of strings and ambiguous terms that are coming from the field redefinitions. These field redefinitions can be fixed in the bulk by ghost-free condition. In this paper we have revised the effective gravitational couplings on D-branes by including the field redefinitions. We find the gravitational effective action up to α 2-order.

We examine the passage of ultracold two-level atoms through two separated laser fields for the nonresonant case. We show that implications of the atomic quantized motion change dramatically the behavior of the interference fringes compared to the semiclassical description of this optical Ramsey interferometer. Using two-channel recurrence relations we are able to express the doublelaser scattering amplitudes by means of the single-laser ones and to give explicit analytical results. When considering slower and slower atoms, the transmission probability of the system changes considerably from an interference behavior to a regime where scattering resonances prevail. This may be understood in terms of different families of trajectories that dominate the overall transmission probability in the weak field or in the strong field limit.

Elastic and excitation ͑single and double͒ cross sections ͑n ഛ 3͒ of orthopositronium-orthopositronium ͑o-Pso-Ps͒ scattering system are predicted using eigenstate close-coupling approximation at medium energies. The partial wave cross sections have shown a peculiar feature. The static exchange results and other inelastic cross sections for even parity transitions only survive for even values of partial wave and vice versa like Born-Oppenheimer ͑BO͒ cross sections. To the best of our knowledge, this feature has not been noticed for other systems. Our results are nearly convergent with added eigenstates and elastic and excitation cross sections are compared with their corresponding BO predictions.

The AdS/CFT correspondence is a powerful tool to study the properties of conformal QCD at strong coupling in terms of a higher dimensional dual gravity theory. The power-law falloff of scattering amplitudes in the non-perturbative regime and calculable hadron spectra follow from holographic models dual to QCD with conformal behavior at short distances and confinement at large distances. String modes and fluctuations about the AdS background are identified with QCD degrees of freedom and orbital excitations at the AdS boundary limit. A description of form factors in space and time-like regions and the behavior of light-front wave functions can also be understood in terms of a dual gravity description in the interior of AdS.

The AdS/CFT correspondence between string theory in AdS space and conformal field theories in physical spacetime leads to an analytic, semi-classical model for strongly-coupled QCD which has scale invariance and dimensional counting at short distances and color confinement at large distances. Although QCD is not conformally invariant, one can nevertheless use the mathematical representation of the conformal group in five-dimensional anti-de Sitter space to construct a first approximation to the theory. The AdS/CFT correspondence also provides insights into the inherently non-perturbative aspects of QCD, such as the orbital and radial spectra of hadrons and the form of hadronic wavefunctions. In particular, we show that there is an exact correspondence between the fifth-dimensional coordinate of AdS space z and a specific impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron in ordinary space-time. This connection allows one to compute the analytic form of the frame-independent light-front wavefunctions, the fundamental entities which encode hadron properties and allow the computation of decay constants, form factors, and other exclusive scattering amplitudes. New relativistic lightfront equations in ordinary space-time are found which reproduce the results obtained using the 5-dimensional theory. The effective light-front equations possess remarkable algebraic structures and integrability properties. Since they are complete and orthonormal, the AdS/CFT model wavefunctions can also be used as a basis for the diagonalization of the full light-front QCD Hamiltonian, thus systematically improving the AdS/CFT approximation.

Recently, a new boson h has been discovered at the LHC which, so far, is compatible with the properties of the SM Higgs. However, the SM is not the most general low-energy dynamics for the minimal electroweak symmetry breaking sector with three Goldstone bosons and one light scalar. By using non-linear effective Lagrangian for these four particles we study different processes at one-loop precision, identifying the counterterms needed to cancel the divergences. Then we apply the IAM unitarization method on the partial waves, both to make more realistic pre-dictions which could be tested at the LHC and to discuss the limitations of the one-loop computations. The studied processes are the elastic scattering amplitude for both the longitudinal components of the gauge bosons V = W, Z and the hh → hh, as well as the inelastic VV → hh. Keywords:

We apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. After a set of master integrals has been found using the integration-by-parts method, the crucial point of this strategy is to introduce a new basis where all master integrals are pure functions of uniform transcendentality. In this paper, we apply this method to all planar three-loop four-point massless on-shell master integrals. We explicitly find such a basis, and show that the differential equations are of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all orders in the dimensional regularization parameter ǫ, including all boundary constants, in a purely algebraic way. The solution is expressed in terms of harmonic polylogarithms. We explicitly write out the Laurent expansion in ǫ for all master integrals up to weight six.

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are formfactor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, p 2 2 = 0. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with ǫ = (4 − D)/2. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small p 2 2 to our results at p 2 2 = 0 and to identify various terms in these solutions according to expansion by regions. The second family consists of fourpoint non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called K 4 graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in ǫ up to weight six.

Exclusive B decays can be factorized as convolutions of hard scattering amplitudes involving the weak interaction with universal hadron distribution amplitudes, thus providing a new QCD-based phenomenology. In addition, semi-leptonic decay amplitudes can be computed exactly in terms of the diagonal and off-diagonal ∆n = 2 overlap of hadronic light-cone wavefunctions. I review these formalisms and the essential QCD ingredients. A canonical form of the light-cone wavefunctions, valid at low values of the transverse momenta, is presented. The existence of intrinsic charm Fock states in the B meson wavefunction can enhance the production of final states of B-decay with three charmed quarks, such as B → J/ψDπ, as well as lead to the breakdown of the CKM hierarchy.