Exotic meson $\pi_1(1600)$ with $J^{PC} = 1^{-+}$ and its decay into $\rho(770)\pi$

We study the spin-exotic <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>-</mo><mo>+</mo></mrow></msup></math> amplitude in single-diffractive dissociation of <math display="inline"><mrow><mn>190</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>/</mo><mi>c</mi></mrow></math> pions into <math display="inline"><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> using a hydrogen target and confirm the <math display="inline"><msub><mi>π</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1600</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo><mi>π</mi></math> amplitude, which interferes with a nonresonant <math display="inline"><msup><mn>1</mn><mrow><mo>-</mo><mo>+</mo></mrow></msup></math> amplitude. We demonstrate that conflicting conclusions from previous studies on these amplitudes can be attributed to different analysis models and different treatment of the dependence of the amplitudes on the squared four-momentum transfer and we thus reconcile these experimental findings. We study the nonresonant contributions to the <math display="inline"><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> final state using pseudodata generated on the basis of a Deck model. Subjecting pseudodata and real data to the same partial-wave analysis, we find good agreement concerning the spectral shape and its dependence on the squared four-momentum transfer for the <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>-</mo><mo>+</mo></mrow></msup></math> amplitude and also for amplitudes with other <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup></math> quantum numbers. We investigate for the first time the amplitude of the <math display="inline"><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> subsystem with <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>-</mo><mo>-</mo></mrow></msup></math> in the <math display="inline"><mn>3</mn><mi>π</mi></math> amplitude with <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>-</mo><mo>+</mo></mrow></msup></math> employing the novel freed-isobar analysis scheme. We reveal this <math display="inline"><msup><mi>π</mi><mo>-</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> amplitude to be dominated by the <math display="inline"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> for both the <math display="inline"><msub><mi>π</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1600</mn><mo stretchy="false">)</mo></math> and the nonresonant contribution. These findings largely confirm the underlying assumptions for the isobar model used in all previous partial-wave analyses addressing the <math display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>-</mo><mo>+</mo></mrow></msup></math> amplitude.