Froissart bound for/from CFT Mellin amplitudes
Parthiv Haldar, Aninda Sinha
Abstract
We derive bounds analogous to the Froissart bound for the absorptive part of CFT _d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mi>d</mml:mi> </mml:msub> </mml:math> Mellin amplitudes. Invoking the AdS/CFT correspondence, these amplitudes correspond to scattering in AdS _{d+1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> . We can take a flat space limit of the corresponding bound. We find the standard Froissart-Martin bound, including the coefficient in front for d+1=4 being \pi/\mu^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>/</mml:mi> <mml:msup> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , \mu <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>μ</mml:mi> </mml:math> being the mass of the lightest exchange. For d>4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , the form is different. We show that while for CFT_{d\leq 6} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , the number of subtractions needed to write a dispersion relation for the Mellin amplitude is equal to 2, for CFT_{d>6} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> the number of subtractions needed is greater than 2 and goes to infinity as d goes to infinity.