S- and p-wave structure of S = -1 meson- baryon scattering in the resonance region Supported by NSF (PIF No.1415459, CAREER PHY-1452055), DE-AC05-06OR23177 & DE-SC0016582). HPC support by Forschungszentrum Julich Daniel Sadasivan Maxim Mai Michael Doring 1
Motivation 2
What is a Resonance 1.5 2.5 1.0 2.0 Re f � fm � Re f � fm � 0.5 1.5 0.0 1.0 � 0.5 0.5 0.0 1340 1360 1380 1400 1420 1440 1340 1360 1380 1400 1420 1440 W CMS � MeV � W CMS � MeV � • Seen in peak at a certain energy in scattering cross sections. • Described by certain quantum numbers. • Can be studied through analytic continuation • Useful to relate results to other theories like quark models and lattice QCD. 3
Resonances We Are Looking For 4
Applications • Λ (1405) is dominated by KN interaction • A similar mechanism can be responsible for the generation of K − pp bound states. See for example, S. Ajimura et al arXiv:1805.12275 [nucl-ex]. • The equation of state of neutron stars is sensitive to the antikaon condensate and thus to the propagation of antikaons in nuclear medium. In the era of high- precision measurements of neutron star properties with LIGO, this ultimately can lead to new interconnection between QCD and astrophysical observations 5
Method 6
Model T V T V A depiction of the operator form of the Bethe Saltpeter Equation. The bubble chain summation caused by iteration of the Bethe Salpeter Ansatz The chiral expansion of the driving term, V. 7
Possible Meson Baryon Interactions for S=-1 Possible data for S=-1 interactions. The data that exists in the energy region of interest is shown in red. 8
Data the Free Parameters Were Fit to Below: Total cross sections 10 K - p → K 0 n ( 225 MeV ) K - p → K 0 n ( 235 MeV ) K - p → K 0 n ( 245 MeV ) K - p → K 0 n ( 255 MeV ) fitted by the model. The 8 dashed black line shows the d Ω ( mb ) 6 d σ contribution of the s-wave 4 part of the amplitude only. 2 5.0 K - p → K 0 n ( 265 MeV ) K - p → K 0 n ( 275 MeV ) K - p → K 0 n ( 285 MeV ) K - p → K 0 n ( 295 MeV ) 4.5 4.0 d Ω ( mb ) 3.5 Right: Differential cross d σ 3.0 2.5 sections fitted by the model. 2.0 2.0 K - p → K 0 n ( 235 MeV ) K - p → K 0 n ( 245 MeV ) K - p → K 0 n ( 255 MeV ) K - p → K 0 n ( 265 MeV ) 1.5 d Ω ( mb ) 1.0 d σ 0.5 0.0 2.0 K - p → K 0 n ( 275 MeV ) K - p → K 0 n ( 285 MeV ) K - p → K 0 n ( 295 MeV ) 1.5 d Ω ( mb ) 1.0 d σ 0.5 0.0 - 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 Cos θ Cos θ Cos θ Cos θ In addition, we fit to threshold data including data from the SIDDHARTA Experiment which uses Anti-Kaon Nucleon processes to produce kaonic deuterium. This measurement has a real and imaginary part. 9
More Data arbitrary units W tot = 2. GeV W tot = 2.1 GeV W tot = 2.2 GeV 1.30 1.35 1.40 1.45 1.50 � 0 � 0 3. � + � - d � / dM inv [ � b / GeV ] M �� [ GeV ] � - � + 2. 1. W tot = 2.3 GeV W tot = 2.4 GeV W tot = 2.5 GeV Above: Fit of the generic d � / dM inv [ � b / GeV ] couplingsK − p → Σ (1660) π− and Σ (1660) → ( π−Σ +) π 1 +to the invariant mass distribution in arbitrary 0.5 units. W tot = 2.6 GeV W tot = 2.7 GeV W tot = 2.8 GeV Left: Fit ( χ 2pp= 1.07) to the πΣ invariant mass 0.6 d � / dM inv [ � b / GeV ] distribution (Minv) from γ p → K+( πΣ ) reaction 0.4 0.2 1.35 1.4 1.45 1.35 1.4 1.45 1.35 1.4 1.45 M inv [ GeV ] M inv [ GeV ] M inv [ GeV ] 10
Predictions 11
PWA Amplitudes 0 ( 1 / 2 - ) 1. 0 ( 1 / 2 + ) 16. 12. I [ GeV - 1 ] 8. 0. 4. 0. - 1. f J - 4. - 8. - 2. 1 ( 1 / 2 - ) 1 ( 1 / 2 + ) 4. 1. 3. I [ GeV - 1 ] 0. 2. - 1. 1. f J - 2. 0. - 1. - 3. 1.25 1.30 1.35 1.40 1.45 1.25 1.30 1.35 1.40 1.45 W [ GeV ] W [ GeV ] Real (solid, blue) and imaginary (red, dashed) parts of ̄ KN partial wave amplitudes with I(JP). The vertical dashed lines show the positions of the πΛ , π 0 Σ 0, π + Σ− , π−Σ +,K − p, ̄ K 0 n thresholds, and the dots over the plots represent the available data in the respective channel from top to bottom. 12
Lambda(1405) Left: Pole positions (black stars) for the 0(1/2 − ). The error ellipses are from a re-sampling procedure shown explicitly in the corresponding insets. The shaded squares show the prediction from other literature for the narrow (blue) and broad (orange) pole of Λ (1405) Below: A plot of the amplitude in the complex plane that shows the two peaks. 13
An Anomalous Structure Left: A representation of the position of a pole in the best fit of our model in the 1(1/2 + ) Channel. Below: The amplitudes of the couplings for the poles observed in the best fit. 14
Analysis 15
Possible Explanations of the Anomalous Structure When a structure is observed in a good fit to good data and does not have the quantum numbers of any known state, categorically speaking there are three possibilities. 1. It’s present because the data demonstrate that there exists an undiscovered state in nature. 2. It’s present because the data require the model to account for something it isn’t currently accounting for. 3. It’s completely arbitrary; it’s existence does not improve the fit in any way. 16
Lasso Test of Robustness Above: Plot of Lasso (Least Absolute Shrinkage and Selection Operator) Method. This parametric plot shows how the χ 2 components and distance of the pole are related to the penalty term. λ crit is the value for which the amplitude is no longer visible. λ init is when the amplitude is unpenalized . Right: The amplitude that is penalized. 17
Possible Explanations of the Anomalous Structure When a structure is observed in a good fit to the data and does not have the quantum numbers of any known state, categorically speaking there are three possibilities. 1. It’s present because the data demonstrate that there exists an undiscovered state in nature. 2. It’s present because the data require the model to account for something it isn’t currently accounting for. 3. It’s completely arbitrary; it’s existence does not improve the fit in any way. 18
Σ (1385) Explicit Inclusion Test 10 K - p → K - p ( w = 225 MeV ) K - p → K - p ( w = 235 MeV ) K - p → K - p ( w = 245 MeV ) K - p → K - p ( w = 255 MeV ) 8 Formula for explicitly including a resonance in d σ / d Ω [ mb ] 6 a way that preserves unitarity. 4 2 5.0 K - p → K - p ( w = 265 MeV ) K - p → K - p ( w = 275 MeV ) K - p → K - p ( w = 285 MeV ) K - p → K - p ( w = 295 MeV ) 4.5 d σ / d Ω [ mb ] 4.0 3.5 3.0 2.5 2.0 2.0 K - p → K 0 n ( w = 235 MeV ) K - p → K 0 n ( w = 245 MeV ) K - p → K 0 n ( w = 255 MeV ) 1.5 d σ / d Ω [ mb ] 1.0 Best fit parameters and pole positions found when the Σ (1385) is explicitly included. 0.5 0.0 2.0 K - p → K 0 n ( w = 265 MeV ) K - p → K 0 n ( w = 275 MeV ) K - p → K 0 n ( w = 285 MeV ) K - p → K 0 n ( w = 295 MeV ) 1.5 d σ / d Ω [ mb ] 1.0 0.5 Real Formula for the differential cross section 0.0 - 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 1.0 - 0.5 0.0 0.5 1.0 Cos θ Cos θ Cos θ Cos θ The partial waves of the best fit are plotted as shown Reversed Formula for the differential cross earlier but we additionally include a black line to show section the best fit when the partial waves are reversed. 19
Possible Explanations of the Anomalous Structure When a structure is observed in a good fit to the data and does not have the quantum numbers of any known state, categorically speaking there are three possibilities. 1. It’s present because the data demonstrate that there exists an undiscovered state in nature. 2. It’s present because the data require the model to account for something it isn’t currently accounting for. 3. It’s completely arbitrary; it’s existence does not improve the fit in any way. 20
Summary • The Mai-Meissner Model is fit to differential cross section as well as older data. • Both poles of the Lambda(1405) were reproduced • A new anomalous structure was observed that didn’t have the right parity for the Sigma(1385). • This statistically robust state likely exists because the differential cross section data demand a p-wave resonance and a NLO model cannot give it the right J. 21
Thank You 22
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