Eikonal Amplitudes and Nonglobal Logarithms from the BMS Equation


The Banfi–Marchesini–Smye (BMS) equation accounts for resummation of nonglobal logarithms to all orders in perturbation theory in the large-\({{{\text{N}}}_{{\text{c}}}}\) approximation. We show that the squared amplitudes for the emission of soft energy-ordered gluons are correctly embedded in this equation, and explicitly verify that they coincide with those derived in our previous work in the large-\({{{\text{N}}}_{{\text{c}}}}\) limit up to sixth order in the strong coupling. We perform analytical calculations for the nonglobal logarithms up to fourth order for the specific hemisphere mass distribution in \({{e}^{ + }}{{e}^{ - }}\) collisions, thus confirming our previous semi-numerical results. We show that the solution to the BMS equation may be cast into a product of an infinite number of exponentials each of whichresums a class of Feynman diagrams that manifest a symmetry pattern, and explicitly carry out the computation of the first of these exponentials.

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Fig. 1.
Fig. 2.


  1. 1.

    When permuting the irreducible parts, e.g. \(\bar {\mathcal{W}}_{{ijk}}^{{{\text{RRR}}}}\), the indices must always be ordered such that \(i < j < k\).

  2. 2.

    The Feynman diagrams corresponding to these terms look like a ladder (see Fig. 1).

  3. 3.

    Notice that the evolution parameter \(t\) is denoted as \(\delta \) in [3].


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This work is supported by:

• Deanship of Research at the Islamic University of Madinah (research project no. 40/107);

• PRFU: B00L02UN050120190001 (Algeria).

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Correspondence to H. Benslama or Y. Delenda or K. Khelifa-Kerfa or A. M. Ibrahim.

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Benslama, H., Delenda, Y., Khelifa-Kerfa, K. et al. Eikonal Amplitudes and Nonglobal Logarithms from the BMS Equation. Phys. Part. Nuclei Lett. 18, 5–18 (2021). https://doi.org/10.1134/S1547477121010039

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