Nonlinear Sigma model amplitudes to all loop orders are contained in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Tr</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> theory

Abstract

Scattering amplitudes for the simplest theory of colored scalar particles—the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Tr</a:mi> <a:mo stretchy="false">(</a:mo> <a:msup> <a:mi mathvariant="normal">Φ</a:mi> <a:mn>3</a:mn> </a:msup> <a:mo stretchy="false">)</a:mo> </a:math> theory—have recently been the subject of active investigations. In this work we describe an unanticipated wider implication of this work: the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>Tr</f:mi> <f:mo stretchy="false">(</f:mo> <f:msup> <f:mi mathvariant="normal">Φ</f:mi> <f:mn>3</f:mn> </f:msup> <f:mo stretchy="false">)</f:mo> </f:math> theory secretly contains nonlinear sigma model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>Tr</k:mi> <k:mo stretchy="false">(</k:mo> <k:msup> <k:mi mathvariant="normal">Φ</k:mi> <k:mn>3</k:mn> </k:msup> <k:mo stretchy="false">)</k:mo> </k:math> amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in the potential, with extrema spontaneously breaking <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>U</p:mi> <p:mo stretchy="false">(</p:mo> <p:mi>N</p:mi> <p:mo stretchy="false">)</p:mo> <p:mo stretchy="false">→</p:mo> <p:mi>U</p:mi> <p:mo stretchy="false">(</p:mo> <p:mi>N</p:mi> <p:mo>−</p:mo> <p:mi>k</p:mi> <p:mo stretchy="false">)</p:mo> <p:mo>×</p:mo> <p:mi>U</p:mi> <p:mo stretchy="false">(</p:mo> <p:mi>k</p:mi> <p:mo stretchy="false">)</p:mo> </p:math> . The Goldstone amplitudes for this theory coincide with those of pions in the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>U</y:mi> <y:mo stretchy="false">(</y:mo> <y:mi>N</y:mi> <y:mo stretchy="false">)</y:mo> <y:mo>×</y:mo> <y:mi>U</y:mi> <y:mo stretchy="false">(</y:mo> <y:mi>N</y:mi> <y:mo stretchy="false">)</y:mo> <y:mo stretchy="false">→</y:mo> <y:mi>U</y:mi> <y:mo stretchy="false">(</y:mo> <y:mi>N</y:mi> <y:mo stretchy="false">)</y:mo> </y:math> chiral Lagrangian to all orders in the planar limit. We also give a purely on-shell understanding of this correspondence, showing that integrands defined by the kinematic shifts have the correct residues on poles and appropriately produce the Adler zero. Finally, we discuss how similar kinematic shifts produce certain infinite classes of mixed amplitudes of pions and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>Tr</hb:mi> <hb:mo stretchy="false">(</hb:mo> <hb:msup> <hb:mi mathvariant="normal">Φ</hb:mi> <hb:mn>3</hb:mn> </hb:msup> <hb:mo stretchy="false">)</hb:mo> </hb:math> scalars, most of which are not interpretable from the Lagrangian description. Published by the American Physical Society 2024

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