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Within the framework of dispersion theory, we study the the processes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><mrow><mo>(</mo><mn>2170</mn><mo>)</mo></mrow><mo>→</mo><mi>ϕ</mi><mi>π</mi><mi>π</mi><mrow><mo>(</mo><mi>K</mi><mover accent="true"><mi>K</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The strong pion–pion final-state interactions, especially the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mover accent="true"><mi>K</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> coupled channel in the <i>S</i> wave, are taken into account in a model-independent way using the Omnès function solution. Through fitting the experimental data of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mi>π</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mi>π</mi></mrow></semantics></math></inline-formula> invariant mass distributions of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><mrow><mo>(</mo><mn>2170</mn><mo>)</mo></mrow><mo>→</mo><mi>ϕ</mi><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula> process, the low-energy constants in the chiral Lagrangian are determined. The theoretical prediction for the cross sections’ ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>σ</mi><mo>(</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><mrow><mo>(</mo><mn>2170</mn><mo>)</mo></mrow><mo>→</mo><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup><mo>)</mo></mrow><mo>/</mo><mrow><mi>σ</mi><mo>(</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><mrow><mo>(</mo><mn>2170</mn><mo>)</mo></mrow><mo>→</mo></mrow></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is given, which could be useful for selecting the physical solution, when the fit to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula> cross-section distribution is available in the future. Our results suggest that above the kinematical threshold of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mi>K</mi><mover accent="true"><mi>K</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, the mechanism <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula>, with the kaons rescattering to a pion pair, plays an important role in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>ϕ</mi><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></mrow></semantics></math></inline-formula> transition.