Absorptive corrections to the electromagnetic form factor in high-energy elastic proton-proton scattering

Recently, it was noted that absorptive corrections to the electromagnetic form factor in high-energy proton-proton scattering are important for the theoretical interpretation of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>p</a:mi><a:mo stretchy="false">↑</a:mo></a:msup><a:mi>p</a:mi></a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi>p</d:mi><d:mo stretchy="false">↑</d:mo></d:msup><d:mi>A</d:mi></d:math> analyzing power <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msub><g:mrow><g:mi>A</g:mi></g:mrow><g:mrow><g:mi mathvariant="normal">N</g:mi></g:mrow></g:msub><g:mo stretchy="false">(</g:mo><g:mi>t</g:mi><g:mo stretchy="false">)</g:mo></g:mrow></g:math> measurements with the Hydrogen Jet Target polarimeter (HJET) at RHIC. Here, a concise expression for the absorptive correction was derived within the eikonal approach. The resulting analysis reveals a systematic bias, nearly independent of the beam energy, in the experimental determination of the real-to-imaginary ratio <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>ρ</l:mi></l:math> when absorption effects are overlooked in the data analysis. Quantification of this bias, as <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mrow><n:msup><n:mrow><n:mi>ρ</n:mi></n:mrow><n:mrow><n:mi>meas</n:mi></n:mrow></n:msup><n:mo>=</n:mo><n:mi>ρ</n:mi><n:mo>+</n:mo><n:mo stretchy="false">(</n:mo><n:mn>0.036</n:mn><n:mo>±</n:mo><n:mn>0.016</n:mn><n:msub><n:mrow><n:mo stretchy="false">)</n:mo></n:mrow><n:mrow><n:mi>bias</n:mi></n:mrow></n:msub></n:mrow></n:math>, was achieved using a Regge fit applied to available proton-proton measurements of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msup><r:mi>ρ</r:mi><r:mrow><r:mi>meas</r:mi></r:mrow></r:msup><r:mo stretchy="false">(</r:mo><r:mi>s</r:mi><r:mo stretchy="false">)</r:mo></r:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msubsup><v:mi>σ</v:mi><v:mi>tot</v:mi><v:mrow><v:mi>meas</v:mi></v:mrow></v:msubsup><v:mo stretchy="false">(</v:mo><v:mi>s</v:mi><v:mo stretchy="false">)</v:mo></v:math>. Considering the potential impact of such an effect on the experimentally determined <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi>A</z:mi><z:mi mathvariant="normal">N</z:mi></z:msub><z:mo stretchy="false">(</z:mo><z:mi>t</z:mi><z:mo stretchy="false">)</z:mo></z:math>, one may enhance consistency between the HJET and STAR measurements of the hadronic spin-flip amplitude. While the sign of the bias in the value of <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>ρ</eb:mi></eb:math> aligns with the anticipated effective increase in the proton charge radius in <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>p</gb:mi><gb:mi>p</gb:mi></gb:math> scattering due to absorption, it amplifies the observed discrepancy between <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi>σ</ib:mi><ib:mi>tot</ib:mi><ib:mrow><ib:mi>meas</ib:mi></ib:mrow></ib:msubsup></ib:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msup><kb:mi>ρ</kb:mi><kb:mrow><kb:mi>meas</kb:mi></kb:mrow></kb:msup></kb:math> values at <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msqrt><mb:mi>s</mb:mi></mb:msqrt><mb:mo>=</mb:mo><mb:mn>13</mb:mn><mb:mtext> </mb:mtext><mb:mtext> </mb:mtext><mb:mi>TeV</mb:mi></mb:math> as measured in the TOTEM experiment. Evaluation (using published TOTEM data) of the measured proton-proton <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>d</ob:mi><ob:mi>σ</ob:mi><ob:mo>/</ob:mo><ob:mi>d</ob:mi><ob:mi>t</ob:mi></ob:math> dependence on the absorptive corrections indicated that possible soft photon corrections to the hadronic amplitude slope may be essential for such data analysis.