Cut Diagrams Research Articles

From tree- to loop-simplicity in affine Toda theories II: higher-order poles and cut decompositions

Recently we showed how, in two-dimensional scalar theories, one-loop threshold diagrams can be cut into the product of one or more tree-level diagrams [1]. Using this method on the ADE series of Toda models, we computed the double- and single-pole coefficients of the Laurent expansion of the S-matrix around a pole of arbitrary even order, finding agreement with the bootstrapped results. Here we generalise the cut method explained in [1] to multiple loops and use it to simplify large networks of singular diagrams. We observe that only a small number of cut diagrams survive and contribute to the expected bootstrapped result, while most of them cancel each other out through a mechanism inherited from the tree-level integrability of these models. The simplification mechanism between cut diagrams inside networks is reminiscent of Gauss’s theorem in the space of Feynman diagrams.

Cutkosky rules and unitarity (violation) in D-instanton amplitudes

In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained after integrating out the open string modes, we prove that the D-instanton amplitudes actually satisfy these cutting rules, provided the effective action is real. The violation of unitarity in the closed string sector of two dimensional string theory can be traced to the failure of this reality condition. In the critical superstring theory, multi-instanton and multi anti-instanton amplitudes satisfy the reality condition. Contribution to the amplitudes from the instanton anti-instanton sector satisfies the reality condition if we make a specific choice of integration cycle over the configuration space of string fields, whereas contribution due to the non-BPS D-instantons will need to either vanish or have an overall real normalization in order for it to give real contribution. We use Picard-Lefschetz theory to argue that these conditions are indeed satisfied in superstring theories.

On the singular behavior of the chirality-odd twist-3 parton distribution e(x)

The first moment of the chirality-odd twist-3 parton distribution e(x) is related to the pion-nucleon σ-term, which is important for phenomenology. However, the possible existence of a singular contribution proportional to δ(x) in the distribution prevents the determination of the σ-term with e(x) extracted from experimental data. There are two approaches to show the existence: the first one is based on an operator identity; the second one is based on a perturbative calculation of a single quark state with finite quark mass. We show that all contributions proportional to δ(x) in the first approach are canceled. For the second approach we find that e(x) of a multiparton state with a massless quark has no contribution with δ(x). Considering that a proton is essentially a multiparton state, the effect of the contribution with δ(x) is expected to be suppressed by light quark masses with arguments from perturbation theory. A detailed discussion of the difference between cut diagrams and uncut diagrams of e(x) is provided.

Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories

We prove the unitarity of the Euclidean nonlocal scalar field theory to all perturbative orders in the loop expansion. The amplitudes in the Euclidean space are calculated assuming that all the particles have purely imaginary energies, and afterwards they are analytically continued to real energies. We show that such amplitudes satisfy the Cutkowsky rules and that only the cut diagrams corresponding to normal thresholds contribute to their imaginary part. This implies that the theory is unitary. This analysis is then exported to nonlocal gauge and gravity theories by means of Becchi-Rouet-Stora-Tyutin or diffeomorphism invariance, and Ward identities.

Unitarity of superstring field theory

We complete the proof of unitarity of (compactified) heterotic and type II string field theories by showing that in the cut diagrams only physical states appear in the sum over intermediate states. This analysis takes into account the effect of mass and wave-function renormalization, and the possibility that the true vacuum may be related to the perturbative vacuum by small shifts in the string fields.

General-mass treatment for deep inelastic scattering at two-loop accuracy

We present a next-to-next-to-leading order (NNLO) realization of a general quark mass scheme (S-ACOT-$\ensuremath{\chi}$) in deep inelastic scattering and explore the impact of NNLO terms on heavy-quark structure functions ${F}_{2,L}^{c}(x,Q)$. An amended QCD factorization theorem for DIS is discussed that validates the S-ACOT-$\ensuremath{\chi}$ scheme to all orders in the QCD coupling strength. As a new feature, kinematical constraints on collinear production of heavy quarks that are crucial near the heavy-quark threshold are included in the amended factorization theorem. An algorithmic procedure is outlined for implementing this scheme at NNLO by using mass-dependent and massless results from literature. At two loops in QCD cut diagrams, the S-ACOT-$\ensuremath{\chi}$ scheme reduces scale dependence of heavy-quark DIS cross sections as compared to the fixed-flavor number scheme.

SCATTERING MECHANISM OF HIGH ENERGY STRONG INTERACTION SOFT PROCESSES

Starting from the viewpoint that the constituent quark has its own structure and incorporating the hypothesis of maximum nonperturbative strong interaction reaction (MNSIR), which should be obeyed in high energy strong soft processes, we modified the field theory for soft Pomeron (ℙ) of Landshoff and Nachtmann and proposed a new structure model for the soft Pomeron ℙ and Reggeon ℝ. In this model a pair of constituent quarks coming from colliding hadron in strong soft processes are first disaggregated into a valance quark with a nonperturbative gluon cloud G wrapped about, or, according to the case, into a nonperturbative gluon wrapped by spinor cloud. Corresponding to such mechanism, the relevant structure of soft Pomeron ℙ or Region ℝ is represented by the sum of a series of cut ladder diagrams. In the multi-Regge region, where the energy of the system is very large and the momentum transfer |t| is very small, and in the approximation where only the leading order of log s is preserved, we calculated the [Formula: see text] scattering amplitude and its total cross-section from the sum of such ladder diagrams. The Regge-behavior factor in the form of an exponential power law of s does arise. We also get compact formulas for the trajectories of soft ℙ and ℝ. As example of application, an alternative mechanism of J/Ψ nonquarkonic-hadron decay is proposed.

A New Effect in the QCD Fusion of Nuclear Partons**The project supported in part by the Major State Basic Research Development Program under Contract No. G20000774 and the National Natural Science Foundation of China under Grant Nos 10075020 and 19875024

The parton fusion in nucleus at the leading order of recombination is investigated based on perturbative QCD. We compute various cut diagrams including the nuclear parton fusion, and find that the parton-fusion effects depend on the nuclear QCD structure.

Unitarized diffractive scattering in QCD and its application to virtual photon total cross sections

The problem of restoring Froissart bound to the BFKL-Pomeron is studied in an extended leading-log approximation of QCD. We consider parton-parton scattering amplitude and show that the sum of all Feynman-diagram contributions can be written in an eikonal form. In this form dynamics is determined by the phase shift, and subleading-logs of all orders needed to restore the Froissart bound are automatically provided. The main technical difficulty is to find a way to extract these subleading contributions without having to compute each Feynman diagram beyond the leading order. We solve that problem by using nonabelian cut diagrams introduced elsewhere. They can be considered as colour filters used to isolate the multi-Reggeon contributions that supply these subleading-log terms. Illustration of the formalism is given for amplitudes and phase shifts up to three loops. For diffractive scattering, only phase shifts governed by one and two Reggeon exchanges are needed. They can be computed from the leading-log-Reggeon and the BFKL-Pomeron amplitudes. In applications, we argue that the dependence of the energy-growth exponent on virtuality $Q^2$ for $\gamma^*P$ total cross section observed at HERA can be interpreted as the first sign of a slowdown of energy growth towards satisfying the Froissart bound. An attempt to understand these exponents with the present formalism is discussed.

A New Derivation of Standard Quantum Chromodynamics Evolution Equation

Standard evolution equation including virtual graphs is derived in time ordered perturbative theory. We propose a new cutting rule in deep inelastic scattering processes, which provides a useful tool for discussing the connections among the real and virtual cut diagrams and better understanding of evolution equation.

A new approach to parton recombination in the QCD evolution equations

Parton recombination is reconsidered in perturbation theory without using the AGK cutting rules in the leading order of the recombination. We use time-ordered perturbation theory to sum the cut diagrams, which are neglected in the GLR evolution equation. We present a set of new evolution equations including parton recombination.

Cutting rules in the real-time formalisms at finite temperature

In this paper, we review the set of rules specific to the calculation of the imaginary part of a Green's function at finite temperature in the real-time formalisms. Emphasis is put on the clarification of a recent controversy concerning these rules in the time-ordered version of the real-time formalism, more precisely on the issue related to the interpretation of these rules in terms of cut diagrams, like at T = 0. Also, new results are presented, enabling one to calculate the imaginary part of thermal Green's functions in other formulations of the real-time formalism, like the “retarded/advanced” formalism in which a lot of simplifications occur.

Multiple Reggeon exchange from summing QCD Feynman diagrams

Multiple Reggeon exchange supplies subleading logarithms that may be used to restore unitarity to the Low-Nussinov Pomeron, provided it can be proven that the sum of Feynman diagrams to all orders gives rise to such multiple Regge exchanges. This question cannot be easily tackled in the usual way except for very low-order diagrams, on account of delicate cancellations present in the sum which necessitate individual Feynman diagrams to be computed to subleading orders. Moreover, it is not clear that sums of high-order Feynman diagrams with complicated crisscrossing of lines can lead to factorization implied by the multi-Regge scenario. Both of these difficulties can be overcome by using the recently developed non-Abelian cut diagrams. We are then able to show that the sum of $s$-channel-ladder diagrams to all orders does lead to such multiple Reggeon exchanges.

Cutting rules in the real-time formalisms at finite temperature

In this paper, we review the set of rules specific to the calculation of the imaginary part of a Green's function at finite temperature in the real-time formalisms. Emphasis is put on the clarification of a recent controversy concerning these rules in the "1/2" formalism, more precisely on the issue related to the interpretation of these rules in terms of cut diagrams, like at T=0. On the second hand, new results are presented, enabling one to calculate the imaginary part of thermal Green's functions in other formulations of the real-time formalism, like the "retarded/advanced" formalism in which a lot of simplifications occur.

Cut diagrams for high energy scatterings.

A new approach is introduced to study QCD amplitudes at high energy and comparatively small momentum transfer. Novel cut diagrams, representing the resummation of Feynman diagrams, are used to simplify the calculation and to avoid delicate cancellations encountered in the usual approach. An explicit calculation to the sixth order is carried out to demonstrate the advantage of cut diagrams over Feynman diagrams.

Eikonal diagrams in multiparton semihard interactions.

We study the set of eikonal diagrams, derived from perturbative QCD, at the lowest order in the coupling constant and with vacuum quantum number exchange, in the three-body interaction of a high-energy parton with two target partons. The contribution to the semihard component of the inelastic cross section is worked out by evaluating the leading behavior of all the dominant cut diagrams. The different cut amplitudes are shown to be proportional to one another, with the same weights of the cutting rules which have been derived in the context of multi-Pomeron exchange. As a consequence of the dominant configuration in the loop integrals, corresponding to the projectile parton on shell between successive interactions, the process is represented by the simplest probabilistic picture, where the three-body interaction is factorized as the product of two-body interaction probabilities.

Cancellation of the Infrared Singularity through the Unitarity Relation

It is shown that the infrared singularities (both soft and collinear divergences) in usual field theories including QCD are cancelled among topologically equal double cut diagrams by appropriate summation over initial and/or final cuts. The basic tools of the analysis are the largest time equation and the power counting method in the space consisting of time plus three momentum components.

Double-Cut Diagram Method for Mass Singularities in QCD

A method to handle hard mass singularities in QCD is given based on the use of double­ cut diagrams. By this method the cancellation of mass singularities in differential cross sections can be easily seen after the summation over degenerate (emission) processes. The procedure is applicable to all orders in perturbation theory unless massless particle;; are present in both initial and final states.

Unitarity of supergravity in a general covariant gauge

Unitarity of supergravity is studied in a general covariant gauge. Except for the Feynman gauge, propagators of gauge fields have double poles. However these poles cannot contribute to the cut diagrams, and 't Hooft's diagramatic prescription is applicable to the theory in a general covariant gauge.

Unitarity of supergravity in a general covariant gauge

Unitarity of supergravity is studied in a general covariant gauge. Except for the Feynman gauge, propagators of gauge fields have double poles. However these poles cannot contribute to the cut diagrams, and 't Hooft's diagramatic prescription is applicable to the theory in a general covariant gauge.