Dispersive estimate of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>980</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> contribution to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>μ</mml:mi> </mml:msub> </mml:math>

Abstract

A dispersive implementation of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>a</a:mi> <a:mn>0</a:mn> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mn>980</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> resonance to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mo stretchy="false">(</e:mo> <e:mi>g</e:mi> <e:mo>−</e:mo> <e:mn>2</e:mn> <e:msub> <e:mo stretchy="false">)</e:mo> <e:mi>μ</e:mi> </e:msub> </e:math> requires the knowledge of the double-virtual <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>S</i:mi> </i:math> -wave <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>γ</k:mi> <k:mo>*</k:mo> </k:msup> <k:msup> <k:mi>γ</k:mi> <k:mo>*</k:mo> </k:msup> <k:mo stretchy="false">→</k:mo> <k:mi>π</k:mi> <k:mi>η</k:mi> <k:mo>/</k:mo> <k:mi>K</k:mi> <k:msub> <k:mover accent="true"> <k:mi>K</k:mi> <k:mo stretchy="false">¯</k:mo> </k:mover> <k:mrow> <k:mi>I</k:mi> <k:mo>=</k:mo> <k:mn>1</k:mn> </k:mrow> </k:msub> </k:math> amplitudes. To obtain these amplitudes, we used a modified coupled-channel Muskhelishvili–Omnès formalism, with input from the left-hand cuts and the hadronic Omnès matrix. The latter was derived using a data-driven <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>N</p:mi> <p:mo>/</p:mo> <p:mi>D</p:mi> </p:math> method, where the hadronic left-hand cuts were approximated via a conformal expansion. Due to the absence of direct hadronic data in the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>π</r:mi> <r:mi>η</r:mi> </r:math> channel, the expansion coefficients were fitted to various experimental data sets on two-photon fusion processes with <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>π</t:mi> <t:mi>η</t:mi> </t:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>K</v:mi> <v:mover accent="true"> <v:mi>K</v:mi> <v:mo stretchy="false">¯</v:mo> </v:mover> </v:math> final states. The resulting dispersive estimate for the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msub> <z:mi>a</z:mi> <z:mn>0</z:mn> </z:msub> <z:mo stretchy="false">(</z:mo> <z:mn>980</z:mn> <z:mo stretchy="false">)</z:mo> </z:math> contribution to <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mo stretchy="false">(</db:mo> <db:mi>g</db:mi> <db:mo>−</db:mo> <db:mn>2</db:mn> <db:msub> <db:mo stretchy="false">)</db:mo> <db:mi>μ</db:mi> </db:msub> </db:math> is <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:msubsup> <hb:mi>a</hb:mi> <hb:mi>μ</hb:mi> <hb:mtext>HLbL</hb:mtext> </hb:msubsup> <hb:mo stretchy="false">[</hb:mo> <hb:msub> <hb:mi>a</hb:mi> <hb:mn>0</hb:mn> </hb:msub> <hb:mo stretchy="false">(</hb:mo> <hb:mn>980</hb:mn> <hb:mo stretchy="false">)</hb:mo> <hb:msub> <hb:mo stretchy="false">]</hb:mo> <hb:mrow> <hb:mi>resc</hb:mi> </hb:mrow> </hb:msub> <hb:mo>=</hb:mo> <hb:mo>−</hb:mo> <hb:mn>0.44</hb:mn> <hb:mo stretchy="false">(</hb:mo> <hb:mn>5</hb:mn> <hb:mo stretchy="false">)</hb:mo> <hb:mo>×</hb:mo> <hb:msup> <hb:mn>10</hb:mn> <hb:mrow> <hb:mo>−</hb:mo> <hb:mn>11</hb:mn> </hb:mrow> </hb:msup> </hb:math> , which presents an order of magnitude improvement in precision over the narrow resonance approximation.

Topics & Concepts