Potential of the reaction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> </mml:msup> <mml:mo stretchy="false">→<

The spectrum of light unflavored vector mesons above 2.4 GeV has not yet been firmly established experimentally. In this work, we demonstrate that the process <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>e</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>e</a:mi> <a:mo>−</a:mo> </a:msup> <a:mo stretchy="false">→</a:mo> <a:mi>p</a:mi> <a:mover accent="true"> <a:mi>p</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> <a:mi>π</a:mi> </a:math> serves as a particularly promising channel as a probe for higher isovector <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>ρ</f:mi> </f:math> excitations. In contrast to typical electron-positron annihilation processes into purely light-meson final states, the reaction <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msup> <h:mi>e</h:mi> <h:mo>+</h:mo> </h:msup> <h:msup> <h:mi>e</h:mi> <h:mo>−</h:mo> </h:msup> <h:mo stretchy="false">→</h:mo> <h:mi>p</h:mi> <h:mover accent="true"> <h:mi>p</h:mi> <h:mo stretchy="false">¯</h:mo> </h:mover> <h:mi>π</h:mi> </h:math> benefits from a relatively low continuum background, on the order of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mo stretchy="false">(</m:mo> <m:mn>50</m:mn> <m:mo>∼</m:mo> <m:mn>100</m:mn> <m:mo stretchy="false">)</m:mo> <m:mtext> </m:mtext> <m:mtext> </m:mtext> <m:mi>pb</m:mi> </m:math> around 2.4 GeV, which enhances the visibility of even modest resonance contributions. In addition, the comparatively high production threshold further suppresses potential contamination from lower <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>ρ</q:mi> </q:math> -meson excitations. Our analysis based on effective Lagrangian approach indicates that the observed line shape of the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi>e</s:mi> <s:mo>+</s:mo> </s:msup> <s:msup> <s:mi>e</s:mi> <s:mo>−</s:mo> </s:msup> <s:mo stretchy="false">→</s:mo> <s:mi>p</s:mi> <s:mover accent="true"> <s:mi>p</s:mi> <s:mo stretchy="false">¯</s:mo> </s:mover> <s:mi>π</s:mi> </s:math> cross section measured by the Beijing Spectrometer (BESIII) Experiment can be naturally interpreted by including the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>ρ</x:mi> <x:mo stretchy="false">(</x:mo> <x:mn>5</x:mn> <x:mi>S</x:mi> <x:mo stretchy="false">)</x:mo> </x:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>ρ</bb:mi> <bb:mo stretchy="false">(</bb:mo> <bb:mn>6</bb:mn> <bb:mi>S</bb:mi> <bb:mo stretchy="false">)</bb:mo> </bb:math> contributions, namely the fourth and fifth radial excitations of <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mi>ρ</fb:mi> <fb:mo stretchy="false">(</fb:mo> <fb:mn>770</fb:mn> <fb:mo stretchy="false">)</fb:mo> </fb:math> . In particular, the pronounced rise of the cross section near 2.4 GeV provides strong evidence for the contribution of <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>ρ</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:mn>5</jb:mn> <jb:mi>S</jb:mi> <jb:mo stretchy="false">)</jb:mo> </jb:math> signal. Their branching ratios to the <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mi>p</nb:mi> <nb:mover accent="true"> <nb:mi>p</nb:mi> <nb:mo stretchy="false">¯</nb:mo> </nb:mover> <nb:mi>π</nb:mi> </nb:math> final state are found to be below 4%, which do not contradict with usual expectations. In addition, we evaluate pion-induced production of these two <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:mi>ρ</rb:mi> </rb:math> -meson states as a complementary probe, offering useful guidance for the relevant experiments such as the Common Muon and Proton Apparatus for Structure and Spectroscopy.