Universal features of 2→N scattering in QCD and gravity from shockwave collisions

A remarkable double-copy relation of Einstein gravity to QCD in Regge asymptotics is <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi mathvariant="normal">Γ</a:mi><a:mrow><a:mi>μ</a:mi><a:mi>ν</a:mi></a:mrow></a:msup><a:mo>=</a:mo><a:mfrac><a:mn>1</a:mn><a:mn>2</a:mn></a:mfrac><a:msup><a:mi>C</a:mi><a:mi>μ</a:mi></a:msup><a:msup><a:mi>C</a:mi><a:mi>ν</a:mi></a:msup><a:mo>−</a:mo><a:mfrac><a:mn>1</a:mn><a:mn>2</a:mn></a:mfrac><a:msup><a:mi>N</a:mi><a:mi>μ</a:mi></a:msup><a:msup><a:mi>N</a:mi><a:mi>ν</a:mi></a:msup></a:math>, where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi mathvariant="normal">Γ</d:mi><d:mrow><d:mi>μ</d:mi><d:mi>ν</d:mi></d:mrow></d:msup></d:math> is the gravitational Lipatov vertex in the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>2</g:mn><g:mo stretchy="false">→</g:mo><g:mn>3</g:mn></g:math> graviton scattering amplitude, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi>C</j:mi><j:mi>μ</j:mi></j:msup></j:math> its Yang-Mills counterpart, and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>N</l:mi><l:mi>μ</l:mi></l:msup></l:math> the QED bremsstrahlung vertex. In QCD, the Lipatov vertex is a fundamental building block of the BFKL equation describing the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mn>2</n:mn><n:mo stretchy="false">→</n:mo><n:mi>N</n:mi></n:math> scattering of gluons at high energies. Likewise, the gravitational Lipatov vertex is a key ingredient in a 2D effective field theory framework describing trans-Planckian <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mn>2</q:mn><q:mo stretchy="false">→</q:mo><q:mi>N</q:mi></q:math> graviton scattering. We construct a quantitative correspondence between a semiclassical Yang-Mills framework for radiation in gluon shockwave collisions and its counterpart in general relativity. In particular, we demonstrate the Lipatov double copy in a dilute-dilute approximation corresponding to <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msub><t:mi>R</t:mi><t:mrow><t:mi>S</t:mi><t:mo>,</t:mo><t:mi>L</t:mi></t:mrow></t:msub></t:math>, <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi>R</v:mi><v:mrow><v:mi>S</v:mi><v:mo>,</v:mo><v:mi>H</v:mi></v:mrow></v:msub><v:mo>≪</v:mo><v:mi>b</v:mi></v:math>, where <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mi>R</x:mi><x:mrow><x:mi>S</x:mi><x:mo>,</x:mo><x:mi>L</x:mi></x:mrow></x:msub></x:math>, <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi>R</z:mi><z:mrow><z:mi>S</z:mi><z:mo>,</z:mo><z:mi>H</z:mi></z:mrow></z:msub></z:math> are the respective emergent Schwarzchild radii generated in shockwave collisions and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>b</bb:mi></bb:math> is the impact parameter. We outline extensions of the correspondence developed here to the dilute-dense computation of gravitational wave radiation in the close vicinity of one of the black holes, the construction of graviton propagators in the shockwave background, and a renormalization group approach to compute <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mn>2</db:mn><db:mo stretchy="false">→</db:mo><db:mi>N</db:mi></db:math> amplitudes that incorporates graviton Reggeization and coherent graviton multiple scattering. Published by the American Physical Society 2024