The $$\\pi f_0(500)$$ decay of the $$a_1(1260)$$

Abstract

<jats:p>We evaluate the <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1(1260) \\rightarrow \\pi \\sigma (f_0(500))$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1260</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>→</mml:mo>\n <mml:mi>π</mml:mi>\n <mml:mi>σ</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msub>\n <mml:mi>f</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>500</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> decay width from the perspective that the <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1(1260)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1260</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> resonance is dynamically generated from the pseudoscalar–vector interaction and the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>σ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula> arises from the pseudoscalar–pseudoscalar interaction. A triangle mechanism with <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1(1260) \\rightarrow \\rho \\pi $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mn>1260</mml:mn>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>→</mml:mo>\n <mml:mi>ρ</mml:mi>\n <mml:mi>π</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> followed by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\rho \\rightarrow \\pi \\pi $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>ρ</mml:mi>\n <mml:mo>→</mml:mo>\n <mml:mi>π</mml:mi>\n <mml:mi>π</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> and a fusion of two pions within the loop to produce the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>σ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula> provides the mechanism for this decay under these assumptions for the nature of the two resonances. We obtain widths of the order of 13–22\xa0MeV. Present experimental results differ substantially from each other, suggesting that extra efforts should be devoted to the precise extraction of this important partial decay width, which should provide valuable information on the nature of the axial vector and scalar meson resonances and help clarify the role of the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pi \\sigma $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>π</mml:mi>\n <mml:mi>σ</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> channel in recent lattice QCD calculations of the <jats:inline-formula><jats:alternatives><jats:tex-math>$$a_1$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:msub>\n <mml:mi>a</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>