CIPANP2018Leskovec July 19, 2019
A Lattice QCD study of the resonance
Luka Leskovec
in collaboration with:
Constantia Alexandrou, Stefan Meinel, John Negele, Srijit Paul, Marcus Petschlies, Andrew Pochinsky, Gumaro Rendon, and Sergey Syritsyn
Department of Physics
University of Arizona, AZ85721, USA
We present a lattice QCD study of the resonance with clover fermions at a pion mass of approximately MeV and lattice size fm. We consider two processes involving the . The first process is elastic scattering of two pions in Pwave with isospin . Using the Lüscher method we determine the scattering phase shift, from which we obtain the resonance mass and decay width . The second process is the radiative transition , where we follow the BriceñoHansenWalkerLoud approach to determine the transition amplitude in the invariant mass region near the resonance and for both space and timelike photon momentum. This allows us to determine the coupling between the , the pion and the photon, and the resulting radiative decay width.
PRESENTED AT
Thirteenth Conference on the Intersections of Particle and Nuclear Physics
Palm Springs CA, USA, May 29 – June 3, 2018
1 Introduction
The spectrum of hadrons is a jungle of particles arising from the strong interactions between quarks and gluons. We differentiate between stable hadrons, i.e. those that cannot decay via the strong interactions, and unstable hadrons, i.e. those that can. The simplest example of an unstable hadron is the meson, which is an isotriplet with quantum numbers . While the couples to multiple decay modes, for example , , and , we focus only on the latter two in this report. This simplification can be made because we perform our lattice QCD calculation at light quark masses corresponding to MeV, where the and thresholds are above the energy region we are focusing on. The coupling of the to the channel was previously investigated with lattice QCD in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], and the coupling to in Refs. [15, 16, 17].
To determine the coupling of the resonance to two pions and thus establish its strong decay width, we make use of the Lüscher formalism, which relates the finitevolume spectrum in various moving frames and irreducible representations with the infinitevolume scattering matrix [18, 19]. The appropriate formalism to handle radiative transitions of multihadron states, which we employ here, is the generalization of Lellouch and Lüscher’s work in Ref. [20] by Briceño, Hansen and WalkerLoud (BHWL) [21, 22].
2 Lattice Setup
We use a single lattice gaugefield ensemble with light quark masses corresponding to MeV and a strangequark mass consistent with its physical value. The number of lattice points is . The lattice spacing , determined from the splitting calculated with NRQCD, is equal to fm, leading to a physical spatial volume of approximately fm.
3 wave scattering in
We calculated the lattice spectra in several moving frames and irreducible representations [13] and determined the scattering phase shifts using the Lüscher method. To describe the phase shifts we use the BreitWigner formula
(1) 
where is the invariant mass, is the wave scattering phase shift, is the resonance mass and is the decay width. We investigate two different parametrizations:

BW I: wave decay width:
(2) where is the coupling between the resonance and the scattering channel and is the scattering momentum, , and

BW II: wave decay width modified with BlattWeisskopf barrier factors [23]:
(3) where is the scattering momentum at and is the centrifugal barrier radius.
The resulting fits for both parametrizations are shown in Fig. 2, where the blue line corresponds to BW I and the red line to BW II. The numerical results for the parameters , and are listed in Table 2. There are only minor and statistically not very significant differences between the two parametrizations that appear in the highenergy region , where BW II describes the data slightly better [13]. Overall, we find that both parametrizations describe our results for the elastic wave scattering well.
4 transition amplitude
Because the is not a QCD asymptotic state, but rather a resonance in wave scattering with , the observables related to the resonance photoproduction processes are obtained from the more general process . This process is described by the transition amplitude , which is a function of both the photon fourmomentum transfer and the invariant mass . The transition can however be defined by analytical continuation to the pole located at .
The transition amplitude is associated with the infinite volume matrix element by
(4) 
where is the photon fourmomentum transfer. The transition amplitude has a manifest pole at , and can be written with the help of Watson’s theorem as
(5) 
The form factor is thus free of poles in within the energy region of interest, . Following a general Taylorexpansion approach we parametrize using
(6) 
where is the pole in the channel, and the two variables and are defined as [24, 25, 26]:
(7)  
(8) 
We consider three different families of truncations of the series in Eq. 6 leading to several parametrizations of the transition amplitude [17]. A 3D representation of the transition amplitude is shown for the chosen parametrization “BW II F1 K2” in Fig. 3.
In practice, the resonant form factor is determined by evaluating at the pole:
(9) 
It becomes equal to the photocoupling at zero momentum transfer, . The physical observable we consider is the radiative decay width determined by the photocoupling ,
(10) 
The resonant transition form factor is shown in the left panel of Fig. 4, where the inner shaded region represents the statistical and systematical uncertainties determined on the lattice. The outer shaded region indicates the parametrization uncertainty. The photocouplings determined for each of the parametrizations are shown in the right panel of Fig. 4; we find the photocoupling at MeV to be
(11) 
Due to the largerthan physical pion mass in our calculation the thresholds are much closer to the resonances than in nature. By assuming that the photocoupling is a pionmassindependent quantity, we use the physical values of the pion masses and mass to determine the radiative decay width to be . The number in the first bracket is the combined statistical and systematic uncertainty and the number in the second bracket is the parametrization uncertainty.
5 Summary
We have presented results of two of our recent lattice QCD studies, the determination of the strong decay width of the resonance and the calculation of the resonance radiative decay width. While our calculation was performed at a nonphysical value of the quark mass, the couplings and are already close to their physical values. Future studies are needed to perform chiral extrapolations to the physical point [27, 28, 29] where direct comparisons to experiment can be made.
ACKNOWLEDGEMENTS
We are grateful to Kostas Orginos for providing the gauge field ensemble, which was generated using resources provided by XSEDE (supported by National Science Foundation Grant No. ACI1053575). We thank R. A. Briceño, M. Hansen, and C. B. Lang for valuable discussions. SM and GR were supported in part by National Science Foundation Grant No. PHY1520996; SM, GR, and LL were also supported in part by the U.S. Department of Energy Office of High Energy Physics under Grant No. DESC0009913. SM and SS futher acknowledge support by the RHIC Physics Fellow Program of the RIKEN BNL Research Center. JN and AP were supported in part by the U.S. Department of Energy Office of Nuclear Physics under Grant Nos. DESC0011090 and DEFC0206ER41444. We acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 642069. SP is a Marie SklodowskaCurie fellow supported by the HPCLEAP joint doctorate program. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DEAC0205CH11231. The computations were performed using the Qlua software suite [30].
References
 [1] CPPACS Collaboration, S. Aoki et al., “Lattice QCD Calculation of the rho Meson Decay Width,” Phys. Rev. D76 (2007) 094506, arXiv:0708.3705 [heplat].
 [2] X. Feng, K. Jansen, and D. B. Renner, “Resonance Parameters of the rhoMeson from Lattice QCD,” Phys. Rev. D83 (2011) 094505, arXiv:1011.5288 [heplat].
 [3] C. B. Lang, D. Mohler, S. Prelovsek, and M. Vidmar, “Coupled channel analysis of the rho meson decay in lattice QCD,” Phys. Rev. D84 no. 5, (2011) 054503, arXiv:1105.5636 [heplat]. [Erratum: Phys. Rev.D89,no.5,059903(2014)].
 [4] CS Collaboration, S. Aoki et al., “ Meson Decay in 2+1 Flavor Lattice QCD,” Phys. Rev. D84 (2011) 094505, arXiv:1106.5365 [heplat].
 [5] C. Pelissier and A. Alexandru, “Resonance parameters of the rhomeson from asymmetrical lattices,” Phys. Rev. D87 no. 1, (2013) 014503, arXiv:1211.0092 [heplat].
 [6] Hadron Spectrum Collaboration, J. J. Dudek, R. G. Edwards, and C. E. Thomas, “Energy dependence of the resonance in elastic scattering from lattice QCD,” Phys. Rev. D87 no. 3, (2013) 034505, arXiv:1212.0830 [hepph]. [Erratum: Phys. Rev.D90,no.9,099902(2014)].
 [7] D. J. Wilson, R. A. Briceo, J. J. Dudek, R. G. Edwards, and C. E. Thomas, “Coupled scattering in wave and the resonance from lattice QCD,” Phys. Rev. D92 no. 9, (2015) 094502, arXiv:1507.02599 [hepph].
 [8] RQCD Collaboration, G. S. Bali, S. Collins, A. Cox, G. Donald, M. Gckeler, C. B. Lang, and A. Schfer, “ and resonances on the lattice at nearly physical quark masses and ,” Phys. Rev. D93 no. 5, (2016) 054509, arXiv:1512.08678 [heplat].
 [9] J. Bulava, B. Fahy, B. Hrz, K. J. Juge, C. Morningstar, and C. H. Wong, “ and scattering phase shifts from lattice QCD,” Nucl. Phys. B910 (2016) 842–867, arXiv:1604.05593 [heplat].
 [10] B. Hu, R. Molina, M. Dring, and A. Alexandru, “Twoflavor Simulations of the and the Role of the Channel,” Phys. Rev. Lett. 117 no. 12, (2016) 122001, arXiv:1605.04823 [heplat].
 [11] D. Guo, A. Alexandru, R. Molina, and M. Dring, “Rho resonance parameters from lattice QCD,” Phys. Rev. D94 no. 3, (2016) 034501, arXiv:1605.03993 [heplat].
 [12] Z. Fu and L. Wang, “Studying the resonance parameters with staggered fermions,” Phys. Rev. D94 no. 3, (2016) 034505, arXiv:1608.07478 [heplat].
 [13] C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, “wave scattering and the resonance from lattice QCD,” Phys. Rev. D96 no. 3, (2017) 034525, arXiv:1704.05439 [heplat].
 [14] C. Andersen, J. Bulava, B. Hrz, and C. Morningstar, “The pionpion scattering amplitude and timelike pion form factor from lattice QCD,” arXiv:1808.05007 [heplat].
 [15] R. A. Briceo, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, “The resonant amplitude from Quantum Chromodynamics,” Phys. Rev. Lett. 115 (2015) 242001, arXiv:1507.06622 [hepph].
 [16] R. A. Briceo, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, “The amplitude and the resonant transition from lattice QCD,” Phys. Rev. D93 no. 11, (2016) 114508, arXiv:1604.03530 [hepph].
 [17] C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, “The transition and the radiative decay width from lattice QCD,” Phys. Rev. D98 (2018) 074502, arXiv:1807.08357 [heplat].
 [18] M. Lscher, “Two particle states on a torus and their relation to the scattering matrix,” Nucl. Phys. B354 (1991) 531–578.
 [19] R. A. Briceo, J. J. Dudek, and R. D. Young, “Scattering processes and resonances from lattice QCD,” Rev. Mod. Phys. 90 no. 2, (2018) 025001, arXiv:1706.06223 [heplat].
 [20] L. Lellouch and M. Lscher, “Weak transition matrix elements from finite volume correlation functions,” Commun. Math. Phys. 219 (2001) 31–44, arXiv:heplat/0003023 [heplat].
 [21] R. A. Briceo, M. T. Hansen, and A. WalkerLoud, “Multichannel 1 2 transition amplitudes in a finite volume,” Phys. Rev. D91 no. 3, (2015) 034501, arXiv:1406.5965 [heplat].
 [22] R. A. Briceo and M. T. Hansen, “Multichannel 0 2 and 1 2 transition amplitudes for arbitrary spin particles in a finite volume,” Phys. Rev. D92 no. 7, (2015) 074509, arXiv:1502.04314 [heplat].
 [23] F. Von Hippel and C. Quigg, “Centrifugalbarrier effects in resonance partial decay widths, shapes, and production amplitudes,” Phys. Rev. D5 (1972) 624–638.
 [24] C. G. Boyd, B. Grinstein, and R. F. Lebed, “Constraints on formfactors for exclusive semileptonic heavy to light meson decays,” Phys. Rev. Lett. 74 (1995) 4603–4606, arXiv:hepph/9412324 [hepph].
 [25] C. G. Boyd and M. J. Savage, “Analyticity, shapes of semileptonic formfactors, and ,” Phys. Rev. D56 (1997) 303–311, arXiv:hepph/9702300 [hepph].
 [26] C. Bourrely, I. Caprini, and L. Lellouch, “Modelindependent description of decays and a determination of ,” Phys. Rev. D79 (2009) 013008, arXiv:0807.2722 [hepph]. [Erratum: Phys. Rev.D82,099902(2010)].
 [27] D. R. Bolton, R. A. Briceno, and D. J. Wilson, “Connecting physical resonant amplitudes and lattice QCD,” Phys. Lett. B757 (2016) 50–56, arXiv:1507.07928 [hepph].
 [28] B. Hu, R. Molina, M. Dring, M. Mai, and A. Alexandru, “Chiral extrapolations of the meson in lattice QCD simulations,” Phys. Rev. D96 no. 3, (2017) 034520, arXiv:1704.06248 [heplat].
 [29] P. C. Bruns and M. Mai, “Chiral symmetry constraints on resonant amplitudes,” Phys. Lett. B778 (2018) 43–47, arXiv:1707.08983 [heplat].
 [30] “USQCD software Qlua package.” https://usqcd.lns.mit.edu/w/index.php/QLUA.