Study of electron-positron annihilation into four pions within chiral effective field theory in the low energy region

In this paper, we employ chiral effective field theory to study the process of electron-positron annihilation into four pions in the low energy region within <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>E</a:mi> </a:mrow> <a:mrow> <a:mi mathvariant="normal">c</a:mi> <a:mo>.</a:mo> <a:mi mathvariant="normal">m</a:mi> <a:mo>.</a:mo> </a:mrow> </a:msub> <a:mo>≤</a:mo> <a:mn>0.6</a:mn> <a:mtext> </a:mtext> <a:mtext> </a:mtext> <a:mi>GeV</a:mi> </a:mrow> </a:math> . The prediction of the cross section is obtained through <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>S</e:mi> <e:mi>U</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>3</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> chiral perturbation theory up to the next-to-leading order, which is smaller than the experimental data in the energy region [0.6–0.65] GeV, though the data has only a few points and poor statistics. Then, the resonance chiral theory is applied to include the resonance contribution, with the lightest scalars and vectors written in the effective Lagrangians. A series of relevant decay widths and the masses of the vectors are studied to fix the unknown couplings. The resonance contribution should be one order larger than that of the chiral perturbation theory but still one to two orders smaller than the data. The significant discrepancy urged the new experimental measurements to give more guidance. We also compute the leading order hadronic vacuum polarization contribution from the four pion channels to the anomalous magnetic moment of the muon, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mo stretchy="false">(</i:mo> <i:mi>g</i:mi> <i:mo>−</i:mo> <i:mn>2</i:mn> <i:msub> <i:mo stretchy="false">)</i:mo> <i:mi>μ</i:mi> </i:msub> </i:math> . In the energy range from threshold up to 0.6 GeV within resonance chiral theory, the contributions are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>a</m:mi> <m:mi>μ</m:mi> </m:msub> <m:mo>=</m:mo> <m:mo stretchy="false">(</m:mo> <m:mn>0.680</m:mn> <m:mo>±</m:mo> <m:mn>0.062</m:mn> <m:mo stretchy="false">)</m:mo> <m:mo>×</m:mo> <m:msup> <m:mn>10</m:mn> <m:mrow> <m:mo>−</m:mo> <m:mn>16</m:mn> </m:mrow> </m:msup> </m:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>a</q:mi> <q:mi>μ</q:mi> </q:msub> <q:mo>=</q:mo> <q:mo stretchy="false">(</q:mo> <q:mn>0.597</q:mn> <q:mo>±</q:mo> <q:mn>0.058</q:mn> <q:mo stretchy="false">)</q:mo> <q:mo>×</q:mo> <q:msup> <q:mn>10</q:mn> <q:mrow> <q:mo>−</q:mo> <q:mn>16</q:mn> </q:mrow> </q:msup> </q:math> for the processes of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mi>e</u:mi> <u:mo>+</u:mo> </u:msup> <u:msup> <u:mi>e</u:mi> <u:mo>−</u:mo> </u:msup> <u:mo stretchy="false">→</u:mo> <u:msup> <u:mi>π</u:mi> <u:mo>+</u:mo> </u:msup> <u:msup> <u:mi>π</u:mi> <u:mo>+</u:mo> </u:msup> <u:msup> <u:mi>π</u:mi> <u:mo>−</u:mo> </u:msup> <u:msup> <u:mi>π</u:mi> <u:mo>−</u:mo> </u:msup> </u:math> , <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:msup> <x:mi>π</x:mi> <x:mn>0</x:mn> </x:msup> <x:msup> <x:mi>π</x:mi> <x:mn>0</x:mn> </x:msup> <x:msup> <x:mi>π</x:mi> <x:mo>+</x:mo> </x:msup> <x:msup> <x:mi>π</x:mi> <x:mo>−</x:mo> </x:msup> </x:math> , respectively.