Explicit Cancellation Research Articles

Renormalization of twist-two operators in covariant gauge to three loops in QCD

The leading short-distance contributions to hadronic hard-scattering cross sections in the operator product expansion are described by twist-two quark and gluon operators. The anomalous dimensions of these operators determine the splitting functions that govern the scale evolution of parton distribution functions. In massless QCD, these anomalous dimensions can be determined through the calculation of off-shell operator matrix elements, typically performed in a covariant gauge, where the physical operators mix with gauge-variant operators of the same quantum numbers. We derive a new method to systematically extract the counterterm Feynman rules resulting from these gauge-variant operators. As a first application of the new method, we rederive the unpolarized three-loop singlet anomalous dimensions, independently confirming previous results obtained with other methods. Employing a general covariant gauge, we observe the explicit cancellation of the gauge parameter dependence in these results.

The mean values of cubic L-functions over function fields

We obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over F_q[t]. We solve this problem in the non-Kummer setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the theory of metaplectic Eisenstein series. In the non-Kummer setting we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the L-functions.

A Hopf algebra of subword complexes (Extended abstract)

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

Rehabilitating Austin, Reassessing Grice: The Case of Cancellability

Abstract This paper assesses Grice’s work on conversational implicature in the light of one of its early targets: Austin’s claim that we cannot isolate the meaning of an expression from the context in which it is used. Grice argues that we can separate the literal meaning of many utterances from their pragmatic implicatures through the mechanism of explicit cancellation. However, Grice’s conception of cancellation does not account for the fact that an explicit cancellation must be uttered, and that its utterance involves further implicatures that undermine the attempted cancellation. What Grice calls explicit cancellations are better understood as utterances that resolve ambiguities, and hence apply only in cases where there exists an ambiguity that needs resolving. If Grice’s work does not undermine Austin, we are in a position to reassess an Austinian form of philosophical criticism that emphasizes the ordinary usage of expressions deployed in philosophical arguments.

Ironic implicature strength and the test of explicit cancellability

Abstract In this paper, the Gricean notion of explicit cancellability Grice’s original spelling is “cancelability”. is used as a testable characteristic, able to indicate different degrees of strength for different types of (ironic) implicatures. According to the definition adopted for this analysis, implicature strength is determined by the likelihood of retrieval of an implicature in a specific context and, essentially, by the degree of certainty that the hearer maintains about the correctness of the inferred interpretation. Ironic implicature strength is considered the product of various factors (“factors of implicature strength”), some of which are always present (such as the type and strength of assumptions on which a derivation is based), while others are optional and appear in tandem with specific irony strategies. Irony strategies are categorized into two general types (meaning reversal and meaning replacement), which are expected to show different degrees of implicature strength, being influenced by different factors. For the experimental testing of the hypotheses, subjects were presented with the task of judging the acceptability of the explicit cancellation of various implicated (ironic, as well as non-ironic) meanings. Findings show significant differences between irony types in terms of cancellability (measured as acceptability of cancellation – AC), under the influence of (i) type of syllogism and associated assumptions, (ii) co-textual cues, and (iii) humorous framing.

A Hopf algebra of subword complexes

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

A new perspective on the Faddeev equations and the system from chiral dynamics and unitarity in coupled channels

We review recent work concerning the K ¯ N interaction and Faddeev equations with chiral dynamics which allow us to look at the K ¯ N N from a different perspective and pay attention to problems that have been posed in previous studies on the subject. We then show results which provide extra experimental evidence on the existence of two Λ ( 1405 ) states. Then show the findings of a recent approach to Faddeev equations using chiral unitary dynamics, where an explicit cancellation of the two-body off-shell amplitude with three-body forces stemming from the same chiral Lagrangians takes place. This removal of the unphysical off-shell part of the amplitudes is most welcome and renders the approach unambiguous, showing that only on-shell two-body amplitudes need to be used. With this information in mind we use an approximation to the Faddeev equations within the fixed center approximation to study the K ¯ N N system, providing answers within this approximation to questions that have been brought before and evaluating binding energies and widths of this three-body system. As a novelty with respect to recent work on the topic we find a bound state of the system with spin S = 1 , like a bound state of K ¯ -deuteron, less bound that the one of S = 0 , where all recent efforts have been devoted. The width is relatively large in this case, suggesting problems in a possible experimental observation.

Two, three, many body systems involving mesons

In this talk we show recent developments on few body systems involving mesons. We report on an approach to Faddeev equations using chiral unitary dynamics, where an explicit cancellation of the two body off shell amplitude with three body forces stemming from the same chiral Lagrangians takes place. This removal of the unphysical off shell part of the amplitudes is most welcome and renders the approach unambiguous, showing that only on shell two body amplitudes need to be used. Within this approach, systems of two mesons and one baryon are studied, reproducing properties of the low lying 1/2+ states. On the other hand we also report on multirho and K∗ multirho states which can be associated to known meson resonances of high spin.

Construction of infrared finite observables inN=4super Yang-Mills theory

In this paper we give all the details of the calculation that we presented in our previous paper ArXiv:0908.0387 where the infrared structure of the MHV gluon amplitudes in the planar limit for ${\cal N}=4$ super Yang-Mills theory was considered in the next-to-leading order of perturbation theory. Explicit cancellation of the infrared divergencies in properly defined inclusive cross-sections is demonstrated first in a toy model example of "conformal QED" and then in the real ${\cal N}=4$ SYM theory. We give the full-length details both for the calculation of the real emission and for the diagrams with splitting in initial and final states. The finite parts for some inclusive differential cross-sections are presented in an analytical form. In general, contrary to the virtual corrections, they do not reveal any simple structure. An example of the finite part containing just the log functions is presented. The dependence of inclusive cross-section on the external scale related to the definition of asymptotic states is discussed.

Infrared safe observables in [formula omitted] super Yang–Mills theory

The infrared structure of MHV gluon amplitudes in planar limit for N=4 super Yang–Mills theory is considered in the next-to-leading order of PT. Explicit cancellation of the infrared divergencies in properly defined cross-sections is demonstrated. The remaining finite parts for some inclusive differential cross-sections in planar limit are calculated analytically. In general, contrary to the virtual corrections, they do not reveal any simple structure.

Explicit cancellation of triangles in one-loop gravity amplitudes

We analyse one-loop graviton amplitudes in the field theory limit of a genus-one string theory computation. The considered amplitudes can be dimensionally reduced to lower dimensions preserving maximal supersymmetry. The particular case of the one-loop five-graviton amplitude is worked out in detail and explicitly features no triangle contributions. Based on a recursive form of the one-loop amplitude we investigate the contributions that will occur at n-point order in relation to the ``no-triangle'' hypothesis of N=8 supergravity. We argue that the origin of unexpected cancellations observed in gravity scattering amplitudes is linked to general coordinate invariance of the gravitational action and the summation over all orderings of external legs. Such cancellations are instrumental in the extraordinary good ultra-violet behaviour of N=8 supergravity amplitudes and will play a central role in improving the high-energy behaviour of gravity amplitudes at more than one loop.

MECHANISMS FOR ERROR PROPAGATION AND CANCELLATION IN GLIMM'S SCHEME WITHOUT RAREFACTIONS

We derive an a posteriori error bound for Glimm's approximate solutions to convex scalar conservation laws containing only shock waves. Using Liu's wave-tracing method, we show that the L1 norm of the error is bounded by a sum of residuals containing independent contributions from each wave in the approximate solution. We introduce a framework, similar to the method of characteristics, for the analysis of the local errors generated by wave interactions. The analysis allows for explicit cancellation among the errors created by a single wave and for error propagation along discontinuities.

Numerical evaluation of phase space integrals by sector decomposition

In a series of papers we have developed the method of iterated sector decomposition for the calculation of infrared divergent multi-loop integrals. Here we apply it to phase space integrals to calculate a contribution to the double real emission part of the e + e −→2 jets cross section at NNLO. The explicit cancellation of infrared poles upon summation over all possible cuts of a given topology is worked out in detail for a specific example.

Infrared structure of e+e−→2 jets at NNLO

The production of two jets is the simplest exclusive quantum chromodynamics process in electron–positron annihilation. Using this process, we examine the structure of next-to-next-to-leading order (NNLO) corrections to jet production observables. We derive a subtraction formalism including double real radiation at tree level and single real radiation at one loop. For two-jet production, these subtraction terms coincide with the full matrix elements, thus highlighting the phase space structure of the subtraction procedure. We then analytically compute the infrared singularities arising from each partonic channel. For the purely virtual (two-parton) NNLO corrections, these take the well known form predicted by Catani's infrared factorization formula. We demonstrate that individual terms in the infrared factorization formula can be identified with infrared singular terms from three- and four-parton final states, leaving only single poles and a contribution from the one-loop soft gluon current, which subsequently cancels between the three- and four-parton final states. Summing over all different final states, we observe an explicit cancellation of all infrared poles and recover the known two-loop correction to the hadronic R-ratio.

Infrared structure of e+e$minus;$rarr;2Bjets at NNLO

The production of two jets is the simplest exclusive quantum chromodynamics process in electron–positron annihilation. Using this process, we examine the structure of next-to-next-to-leading order (NNLO) corrections to jet production observables. We derive a subtraction formalism including double real radiation at tree level and single real radiation at one loop. For two-jet production, these subtraction terms coincide with the full matrix elements, thus highlighting the phase space structure of the subtraction procedure. We then analytically compute the infrared singularities arising from each partonic channel. For the purely virtual (two-parton) NNLO corrections, these take the well known form predicted by Catani's infrared factorization formula. We demonstrate that individual terms in the infrared factorization formula can be identified with infrared singular terms from three- and four-parton final states, leaving only single poles and a contribution from the one-loop soft gluon current, which subsequently cancels between the three- and four-parton final states. Summing over all different final states, we observe an explicit cancellation of all infrared poles and recover the known two-loop correction to the hadronic R -ratio.

Conformal expansions and renormalons

The large-order behaviour of QCD is dominated by renormalons. On the other hand renormalons do not occur in conformal theories, such as the one describing the infrared fixed-point of QCD at small beta_0 (the Banks--Zaks limit). Since the fixed-point has a perturbative realization, all-order perturbative relations exist between the conformal coefficients, which are renormalon-free, and the standard perturbative coefficients, which contain renormalons. Therefore, an explicit cancellation of renormalons should occur in these relations. The absence of renormalons in the conformal limit can thus be seen as a constraint on the structure of the QCD perturbative expansion. We show that the conformal constraint is non-trivial: a generic model for the large-order behaviour violates it. We also analyse a specific example, based on a renormalon-type integral over the two-loop running-coupling, where the required cancellation does occur.

CP-violating asymmetries in penguin-inducedB-meson decays beyond the leading logarithmic approximation

We investigate CP-violating phenomena both in semi-inclusive and exclusive penguin-inducedB-meson decays by using low energy effective Hamiltonians for |ΔB|=1, ΔC=ΔU=0 transitions including leading and next-to-leading order OCD corrections. In particular, we discuss the renormalization scheme dependences arising beyond the leading order and their explicit cancellation for the physical transition amplitudes. In order to estimate the hadronic matrix elements needed for the exclusive penguin modes, we apply both the Bauer, Stech, Wirbel model and the meson model of Brodsky and Lepage. Numerical results are presented for semi-inclusive\(b \to q'\bar qq(q',q \in \{ d,s\} )\) modes and the exclusive decaysB−→K−K0,\(B^ - \to \pi ^ - \bar K^0 \), and\(B_d^0 \to K^0 \bar K^0 \).

The spin content of the proton in the light-cone quark model without SU(3) flavour symmetry

The authors argue that it is inappropriate to use SU(3) flavour symmetry in analysing the recent EMC measurement of the proton spin structure function and then present a light-cone, quark-proton model analysis without using SU(3). The nucleon is described as a bound state of three relativistic quarks on the light-cone. The constituent quarks have their own internal structure in terms of valence quarks, a quark-antiquark sea and gluons. The EMC result, taken in conjunction with the Bjorken sum rule, gives, separately, the valence quark and an effective sea contribution to the proton spin, demonstrating explicit cancellation of the valence and sea contributions.

Collective-field method for a U(N)-invariant model in the large-N limit

The authors use the collective-field method to discuss the weak- and strong-coupling phases of the one-plaquette U(N)-invariant model. In both phases they obtain a unified description in terms of collective fields and their collections. The ground-state energy including the next-to-leading-order term is finite owing to explicit cancellation of divergences except at the critical point of the phase transition.

Chiral perturbation theory for anomalous processes

A general expression containing all the divergences in Chiral Perturbation Theory at one loop for anomalous processes has been calculated. The nonrenormalization of some amplitudes (Adler-Bardeen theorem) is observed. The different types of corrections are identified. The contributions from vector meson exchange are determined for all anomalous processes. Explicit cancellation of divergences is shown for π0, η and η′ decays to γγ*. The dependence on the γ* mass squared is predicted to be essentially the same for all the three processes. It proceeds mainly through the VMD estimate of the finite parts of the next order Lagrangian and compares reasonably well with the available experimental data.