Three-body final state interaction in $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$

Three-body final state interaction in $η \to 3 π$

Peng Guo, Igor V. Danilkin, Diane Schott, C. Fernández-Ramírez, V. Mathieu, and Adam P. Szczepaniak (Joint Physics Analysis Center)

Phys. Rev. D 92, 054016 – Published 11 September 2015

Abstract

We present a unitary dispersive model for the $η \to 3 π$ decay process based upon the Khuri-Treiman equations which are solved by means of the Pasquier inversion method. The description of the hadronic final-state interactions for the $η \to 3 π$ decay is essential to reproduce the available data and to understand the existing discrepancies between Dalitz plot parameters from experiment and chiral perturbation theory. Our approach incorporates substraction constants that are fixed by fitting the recent high-statistics WASA-at-COSY data for $η \to π^{+} π^{-} π^{0}$ . Based on the parameters obtained, we predict the slope parameter for the neutral channel to be $α = - 0.022 \pm 0.004$ . Through matching to next-to-leading-order chiral perturbation theory, we estimate the quark mass double ratio to be $Q = 21.4 \pm 0.4$ .

Received 26 May 2015

DOI:https://doi.org/10.1103/PhysRevD.92.054016

Authors & Affiliations

Peng Guo^1,2,3,*, Igor V. Danilkin³, Diane Schott^3,4, C. Fernández-Ramírez³, V. Mathieu^1,2, and Adam P. Szczepaniak^1,2,3 (Joint Physics Analysis Center)

¹Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47403, USA
²Physics Department, Indiana University, Bloomington, Indiana 47405, USA
³Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
⁴Department of Physics, George Washington University, Washington, DC 20052, USA

^*pguo@jlab.org

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Vol. 92, Iss. 5 — 1 September 2015

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Figure 1
A diagrammatic representation of discontinuity relations in Eq. (10). (a) The contribution from the direct s-channel, (b) the contribution from t and u-exchanges.
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Figure 2
Integration contour in the complex $t$ plane. The arrows indicate the direction of increasing $s$ in the interval from $4 m_{π}^{2}$ to $\infty$ . The points labeled $a$ through $i$ correspond to specific values of $s$ , with (a) $t_{-} (\infty) = 0$ , (b) $t_{-} ((m_{η} + m_{π})^{2}) = m_{π} (m_{π} - m_{η})$ , (c) $t_{-} ((m_{η} - m_{π})^{2}) = m_{π} (m_{η} + m_{π})$ , (d) $t_{-} (\frac{m_{η}^{2} - m_{π}^{2}}{2}) = 4 m_{π}^{2}$ , (e) $t_{\pm} (4 m_{π}^{2}) = \frac{m_{η}^{2} - m_{π}^{2}}{2}$ , (f) $t_{+} (m_{π} (m_{η} + m_{π})) = (m_{η} - m_{π})^{2}$ , (g) $t_{+} ((m_{η} - m_{π})^{2}) = m_{π} (m_{η} + m_{π})$ , (h) $t_{+} ((m_{η} + m_{π})^{2}) = m_{π} (m_{π} - m_{η})$ , and (i) $t_{+} (\infty) = - \infty$ , respectively.
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Figure 3
Upper and middle panels are the $x$ - and $y$ -projection plots. Black circles are the data. Red squares and blue squares represent results of the two-body and three-body fits, respectively. The fits are performed on the Dalitz distribution [12] sown in the bottom left panel using a single, $(I, L) = (0, 0)$ wave (upper panels) and two waves, $(I, L) = (0, 0), (1, 1)$ (central panels). For better visualization fit results are shifted horizontally (three-body to right and two-body to left) from the experimental points. The bottom right panel is the Dalitz distribution from the three-body fit with $(I, L) = (0, 0), (1, 1)$ waves.
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Figure 4
Comparison of $R (z)$ plot from [48] (black points) with our predictions from Table 3 that correspond to set 1 (blue band) and set 2 (red band).
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Figure 5
Upper panels: $x$ and $y$ projections of the Dalitz plots. Black circles represent the data. The red squares and blue squares are model results using amplitudes with only two-body and including three-body correlations, respectively. The amplitudes were computed using three partial wave components with $(I, L) = (0, 0), (2, 0), (1, 1)$ . For better visualization fit results are shifted horizontally (three-body to right and two-body to left) from the experimental points. Bottom panels: The comparison of the NLO $χ PT$ amplitudes $M_{I}$ ’s (black curves), with the two-body (red curves) and three-body (blue curves) dispersive amplitudes. Real parts are shown with solid lines and imaginary with dashed lines. In all figures the unknown couplings were fixed by matching to NLO $χ PT$ [see Eq. (38)].
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Figure 6
Integration contour $C^{'}$ in the complex $s$ plane after Pasquier inversion. The black wiggle lines represent cuts attached to two branch points: $(m_{η} \pm m_{π})^{2}$ in the $s$ plane. The points labeled by $a - i$ correspond to (a) $s_{-} (0) = - \infty$ , (b) $s_{-} (4 m_{π}^{2}) = \frac{m_{η}^{2} - m_{π}^{2}}{2}$ , (c) $s_{-} (\frac{m_{η}^{2} - m_{π}^{2}}{2}) = 4 m_{π}^{2}$ , (d) $s_{\pm} ((m_{η} - m_{π})^{2}) = m_{π} (m_{π} + m_{η})$ , (e) $s_{+} (m_{π} (m_{η} + m_{π})) = (m_{η} - m_{π})^{2}$ , (f) $s_{+} (4 m_{π}^{2}) = \frac{m_{η}^{2} - m_{π}^{2}}{2}$ , (g) $s_{+} (0) = \infty$ , (h) $s_{+} (m_{π} (m_{π} - m_{η})) = (m_{η} + m_{π})^{2}$ , and (i) $s_{+} (- \infty) = \infty$ , respectively.
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Physical Review D

covering particles, fields, gravitation, and cosmology

Three-body final state interaction in $η \to 3 π$

Peng Guo, Igor V. Danilkin, Diane Schott, C. Fernández-Ramírez, V. Mathieu, and Adam P. Szczepaniak (Joint Physics Analysis Center)

Phys. Rev. D 92, 054016 – Published 11 September 2015

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Physical Review D

covering particles, fields, gravitation, and cosmology

Three-body final state interaction in η→3π

Peng Guo, Igor V. Danilkin, Diane Schott, C. Fernández-Ramírez, V. Mathieu, and Adam P. Szczepaniak (Joint Physics Analysis Center)

Phys. Rev. D 92, 054016 – Published 11 September 2015

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Three-body final state interaction in $η \to 3 π$