Abstract
We show that the Mellin transform of an n-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of (n−2) linear first order partial differential equations corresponding to (n−2) positive helicity gluons. Although these equations closely resemble Knizhnik-Zamoldochikov equations for SU(N) current algebra there is also an additional “correction” term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Similar equations can also be written down for the momentum space tree level MHV scattering amplitudes. We also propose a way to deal with the non closure of subleading current algebra generators under commutation. This is then used to compute some subleading terms in the mixed helicity gluon OPE.
Highlights
It is generally believed that any consistent theory of quantum gravity on a space-time with asymptotic boundary should have a holographic dual description in terms of a theory living on the boundary at infinity
We show that the Mellin transform of an n-point tree level Maximal Helicity Violating (MHV) gluon scattering amplitude, known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of (n−2) linear first order partial differential equations corresponding to (n−2) positive helicity gluons
In the case of asymptotically flat space-time the observables are the S-matrix elements and it has been proposed that the dual theory is a conformal field theory [11,12,13,14,15,16,17,18,19,20, 23,24,25, 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51], dubbed “celestial conformal field theory (CCFT)”, which lives on the celestial sphere
Summary
It is generally believed that any consistent theory of quantum gravity on a space-time with asymptotic boundary should have a holographic dual description in terms of a theory living on the boundary at infinity. The third term in (1.1) is an additional contribution coming from the (local) subleading soft gluon symmetry This has no analog in the usual KZ equation and is most likely related to the fact that there is no Sugawara stress tensor in CCFT. In appendix B we present a detailed calculation of the first subleading correction to the leading celestial OPE of positive helicity gluons using the Mellin transform of the 5-point tree level MHV gluon amplitude in Yang-Mills theory. In appendix C we use the 4-point MHV Mellin amplitude to extract the first set of subleading terms in the OPE between opposite helicity gluon primaries
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