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  • Open Access

Pole position of the a1(1260) from τ-decay

M. Mikhasenko, A. Pilloni, A. Jackura, M. Albaladejo, C. Fernández-Ramírez, V. Mathieu, J. Nys, A. Rodas, B. Ketzer, and A. P. Szczepaniak (Joint Physics Analysis Center Collaboration)
Phys. Rev. D 98, 096021 – Published 28 November 2018
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  • INTRODUCTION
  • THE REACTION MODEL
  • FIT RESULTS AND RESONANCE PARAMETERS
  • ANALYTIC CONTINUATION THE POLE POSITION
  • SYSTEMATIC UNCERTAINTIES
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We perform an analysis of the three-pion system with quantum numbers JPC=1++ produced in the weak decay of τ leptons. The interaction is known to be dominated by the axial meson a1(1260). We build a model based on approximate three-body unitary and fix the free parameters by fitting it to the ALEPH data on τ−→π−π+π−ντ decay. We then perform the analytic continuation of the amplitude to the complex energy plane. The singularity structures related to the ππ subchannel resonances are carefully addressed. Finally, we extract the a1(1260) pole position mp(a1(1260))−iΓp(a1(1260))/2 with mp(a1(1260))=(1209±4−9+12)  MeV, Γp(a1(1260))=(576±11−20+89)  MeV.

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    • Received 10 October 2018

    DOI:https://doi.org/10.1103/PhysRevD.98.096021

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Hadron-hadron interactions
    1. Physical Systems
    Light mesonsTau leptons
    Particles & FieldsNuclear Physics

    Authors & Affiliations

    M. Mikhasenko1,*, A. Pilloni2,3, A. Jackura4,5, M. Albaladejo2,6, C. Fernández-Ramírez7, V. Mathieu2, J. Nys8, A. Rodas9, B. Ketzer1, and A. P. Szczepaniak4,5,2 (Joint Physics Analysis Center Collaboration)

    • 1Universität Bonn, Helmholtz-Institut für Strahlen- und Kernphysik, 53115 Bonn, Germany
    • 2Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
    • 3European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) and Fondazione Bruno Kessler, I-38123 Villazzano (TN), Italy
    • 4Center for Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47403, USA
    • 5Physics Department, Indiana University, Bloomington, Indiana 47405, USA
    • 6Departamento de Física, Universidad de Murcia, E-30071 Murcia, Spain
    • 7Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de México 04510, Mexico
    • 8Department of Physics and Astronomy, Ghent University, Ghent 9000, Belgium
    • 9Departamento de Física Teórica, Universidad Complutense de Madrid, E-28040 Madrid, Spain

    • *mikhail.mikhasenko@hiskp.uni-bonn.de

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    Issue

    Vol. 98, Iss. 9 — 1 November 2018

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    Images

    • Figure 1
      Figure 1

      Diagram for the decay τ−→π−π+π−ντ. The momenta of the τ lepton and ντ are denoted by pτ and pν. The pions momenta are labeled by pi, i=1, 2, 3. s is the invariant mass of the three pions.

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    • Figure 2
      Figure 2

      The phase space ρ(s) calculated for different models. The black solid line shows the symmetrized ρSYMM from Eq. (12a). The dashed curve represents ρQTB from Eq. (12b), which neglects the interference between the two ρπ decay chains. For reference we draw the two-body ρπ phase space given by (s−(mρ+mπ)2)(s−(mρ−mπ)2)/(8πs) with a solid red line. Due to the chosen normalization in Eq. (10), all functions approach the same asymptotic limit. The dotted line shows the difference in the interference terms calculated in two different ways for s+iε as discussed in Sec. 4b.

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    • Figure 3
      Figure 3

      Fit to ALEPH data with the four models described in the text. The models differ by either including the effect of interference between two ρπ decay channels (SYMM) or not (QTB), and either using the dispersive integral over the phase space (DISP), or not. The lower panels show the normalized residues.

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    • Figure 4
      Figure 4

      The left plot shows the complex plane of the integrand of Eq. (20), for s=0.6−0.35i  GeV2. The red circular markers are the square-root branch points, the crosses indicate positions of the poles. The integration paths from Eq. (21) are shown by the solid lines with arrows. The right plot presents the location of the ρπ cut for the different integration paths.

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    • Figure 5
      Figure 5

      Analytic continuation of the amplitude t(s) in Eq. (14) for different models: QTB-DISP (Left plot), SYMM-DISP (Right plot). Lines indicate the |t(s)| equipotential levels. The poles of the amplitude are the bright spots. The red dots indicate branch points corresponding to the opening of decay channels.

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    • Figure 6
      Figure 6

      Integration paths in the complex σ-plane: while σ1 is moving along the path σth→σlim, the integration endpoints σ3+(s,σ1)(σ3−(s,σ1)) are traveling in the complex plane along the lines shown by black solid (dashed) curve The left plot shows the integration ranges of σ1 and σ3 for a real value of s=1.5  GeV2. The red line is the straight integration path in σ1. The yellow circles indicate the border of the integration domain when the integration endpoints in σ3 coincide. In the right plot, the same lines are shown in the complex σ plane combined for σ1 and σ3 when s=1.5−0.6i  GeV2. The points 4mπ2 and (s−mπ)2 are shown by the small orange dots. While σ1 moves along the contour C(hook) indicated by the red line, the integration limits σ3± follow the dashed and the solid lines analogously to the left plot. The shaded area indicates the additional contribution to the phase-space integral discussed in Eq. (28).

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    • Figure 7
      Figure 7

      The complex sheets of the isobar amplitude fρ(σ) and fρ*(σ). The left plot shows the analytic continuation of the function fρ(σ) above and below the real axis. The function for positive imaginary part is the same as fρ(I)(σ); it is continuously connected to fρ(II)(σ) plotted for the negative imaginary part of σ. The right plot shows the analytic continuation of fρ*(σ), where the sheets are inverted. The lines are |fρ(σ)| equipotential surfaces. The circular spots are the poles (see also red crosses in the left plot of Fig. 4). The markers on the real axis are the branch points of the left-hand cuts: the square marker shows the branch point from the break-up momentum located at σ=4mπ2, the diamond marker the σ=0 branch point, the circular marker indicates the branch point related to the Blatt-Weisskopf factors in the numerator of the fρ(σ) in Eq. (25).

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    • Figure 8
      Figure 8

      The change of the χ2 is plotted against the ρ-meson parameters in Eq. (9): the mass mρ, the width Γρ and the Blatt-Weisskopf size parameter R. The vertical lines indicate the estimated values where the minimum is found.

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    • Figure 9
      Figure 9

      Extracted a1(1260) pole positions in the models QTB-DISP and SYMM-DISP. The ellipses show the 2σ contour of the systematic uncertainties obtained by the bootstrap method. The results of the systematic tests are shown by the open ellipses. The numerical labels correspond to the indices of the studies described in Table 2.

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    • Figure 10
      Figure 10

      Analytic continuation of the amplitude 1/DBW(s) from Eq. (a1). Lines indicate the |DBW| equipotential levels. The poles of the amplitude are the bright spots. The red dots indicate branch points for channel openings.

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    • Figure 11
      Figure 11

      t(s) in the model sQTB-DISP(k). Lines indicate equipotential levels for the |t^sQTB-DISP(k)(s)| function from Eq. (a4). The poles of the amplitude are the yellow spots. The red dots indicate branch points for channel openings: 3π-branch point and ρπ-branch point. The complex plane for the model sQTB-DISP(2) (the models model sQTB-DISP(3)) fitted to the data is shown in the left (right) plot. The quality of the fit is indicated in the legend box on the right.

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    • Figure 12
      Figure 12

      Extracted pole positions in the models QTB-DISP and SYMM-DISP: the resonance poles are on the right, the spurious poles are on the left. The ellipses show the 2σ contours of the statistical uncertainties obtained by the bootstrap method. The results of the systematic tests are shown by the open ellipses. The numerical labels correspond to the indexes of the systematic tests described in Table 2.

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    • Figure 13
      Figure 13

      The left (right) plot presents the real (imaginary) part of the σ3± as a function of σ1 for a fixed value of s+iε. The σ1 is changed linearly between the integration limits. The zoomed plots show how the σ3− passes the real axis first below the branch point σth=4mπ2, then returns above the branch point performing the circling. The red line indicates the closest point on the σ1-path to the (s−1)/2 since it does not go exactly through it.

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    • Figure 14
      Figure 14

      Analytical structure of the integrand in Eq. (c2). The branch points are shown by the red dots with the cuts indicated by the solid red lines. The arbitrary integration path σth→P→σlim is shown by the dashed green line.

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