Chiral logarithms beyond two loops

Bissegger, Moritz; Fuhrer, Andreas

doi:10.1016/j.physletb.2007.01.025

Cited by 25 publications

(43 citation statements)

References 13 publications

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“…Our results for LLs for N ¼ 3 agree with two-loop calculations of amplitude [5] and with five-loop results for the correlator of the scalar currents [6]. Additional check of our method is provided by the case of N ¼ 1.…”

supporting

confidence: 80%

“…The calculation of LLs is a Herculean task-it requires the computation of n-loop diagrams in the nonrenormalizable field theory (1). Presently, the LLs are computed to the two-loop accuracy for the -scattering amplitude [5], to the five-loop accuracy for the correlator of scalar currents [6], and to the three-loop accuracy for the generalized parton distributions (GPDs) [7]. We note that for the case of the chiral corrections to GPDs the summation of LLs is indispensable [7,8], because the smallness of the chiral expansion parameter is compensated by 1=x n Bj .…”

mentioning

confidence: 99%

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Chiral Logarithms in the Massless Limit Tamed

2008

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We derive nonlinear recursion relations for the leading chiral logarithms (LLs) in massless theories. These relations not only provide a very efficient method of computation of LLs (e.g., the 33-loop contribution is calculated in a dozen of seconds on a PC) but also equip us with a powerful tool for the summation of the LLs. Our method is not limited to chiral perturbation theory only; it is pertinent to any nonrenormalizable effective field theory such as, for instance, the theory of critical phenomena, low-energy quantum gravity, etc.

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confidence: 80%

mentioning

confidence: 99%

Chiral Logarithms in the Massless Limit Tamed

2008

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“…In Sec. 4, we consider the O(N) models and apply our relations for the two-point function of the scalar current to present results for general O(N) to a chosen high order up to seven loops, and show that these are consistent with the results of [7] obtained in the O(4)/O(3) model. In Sec.…”

Section: Introductionsupporting

confidence: 68%

Leading logarithms of the two-point function in massless O(N) and SU(N) models to any order from analyticity and unitarity

et al. 2018

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Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints haven been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the O(4)/O(3) model. Later, the all order solutions for the partial waves in any O(N+1)/O(N) model were found. Also, results up to four-loop order exist for massive theories. Here we extend the implications of analyticity and unitarity constraints on the leading logarithms to arbitrary loop order in massless theories. We explicitly obtain the scalar and vector form factors as well as to the scalar two-point function in any O(N) and SU(N) type models. We present relations between the expansion coefficients of these quantities and those of of the relevant partial waves. Our work offers a consistency check on the published results in the O(N) models for form factors, and new results for the scalar two-point function. For the SU(N) type models, we use the known expansion coefficients for partial waves to obtain those for scalar and vector form factors as well as for the scalar two-point function. Our results for the form factor offer a check for the known and future results for massive O(N) and SU(N) type models when the massless limit is taken. Mathematica notebooks which can be used to calculate the expansion coefficients are provided as ancillary files.Large logarithms are a trademark of radiative loop calculations and often signal substantial modifications of tree level results. It is well known that in renormalizable theories, the so called leading logs (lowest power of the coupling constant) can be conveniently summed up by renormalization group techniques and the result of a simple one-loop calculation. Indeed, this often leads to large corrections and a substantial reduction of the (artificial) scale dependence. The next-to-leading logarithms can be treated based on two-loop calculations; and similarly for higher loop orders.It was pointed out by Weinberg [1] that the coefficients of the leading logarithms in effective (nonrenormalizable) field theories at two-loop order could also be obtained from a one-loop calculation. An early application was in demonstrating how to derive these leading logs coefficients from renormalization group (RG) equations for effective field theories by Colangelo [2]. Even earlier, Kazakov [3] argued that renormalization group techniques could be used to relate higher order leading logs to a one-loop calculation in a general setting for non-renormalizable Lagrangians, following notions put forward in ref.[4] in a different context. Soon afterwards, the double chiral logs of the chiral Lagranigan were evaluated along these lines by Bijnens, Colangelo and Ecker [5].Encouraged by these...

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“…This relation for the ππ scattering is well-known, it is a consequence of the Roy equation [10], it was used in Ref. [4] to calculate three-loop LLs in ChPT. Substituting Eq.…”

Section: General Methodsmentioning

confidence: 99%

“…In Ref. [4] the authors, using dispersive methods, calculated the three-loop LLs to ππ scattering in massless Chiral Perturbation Theory (ChPT).…”

Section: Introductionmentioning

confidence: 99%

Leading infrared logarithms from unitarity, analyticity, and crossing

2010

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We derive nonlinear recursion equations for the leading infrared logarithms in massless nonrenormalizable effective field theories. The derivation is based solely on the requirements of the unitarity, analyticity and crossing symmetry of the amplitudes. That emphasizes the general nature of the corresponding equations. The derived equations allow one to compute leading infrared logarithms to essentially unlimited loop order without performing a loop calculation. For the implementation of the recursion equation one needs to calculate tree diagrams only. The application of the equation is demonstrated on several examples of effective field theories in four and higher space-time dimensions.

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Chiral logarithms beyond two loops

Cited by 25 publications

References 13 publications

Chiral Logarithms in the Massless Limit Tamed

Chiral Logarithms in the Massless Limit Tamed

Leading logarithms of the two-point function in massless O(N) and SU(N) models to any order from analyticity and unitarity

Leading infrared logarithms from unitarity, analyticity, and crossing

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