Amplitude analysis of resonant production in three pions A. Jackura with M. Mikhasenko & A. Szczepaniak Indiana University, Joint Physics Analysis Center June 2 nd , 2016 14th International Workshop on Meson Production, Properties and Interaction Krak´ ow, Poland June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 1 / 19
Joint Physics Analysis Center (JPAC) The Joint Physics Analysis Center (JPAC) formed in October 2013 We support physics analysis of experimental data for accelerator facilities (JLab12, COMPASS, . . . ) http://www.indiana.edu/ ∼ jpac/ JPAC Talks Vladiszlav Pauk (Today 17:55 in Parallel B) Adam Szczepaniak (Friday 9:00 Plenary) Emilie Passemar (Friday 15:25 in Parallel A) Alessandro Pilloni (Monday 17:15 in Parallel B) Vincent Mathieu (Poster Session) June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 2 / 19
Introduction 3 π at COMPASS Study peripheral resonance production of 3 π systems at COMPASS. High statistics, high purity data allows for detailed analysis JPAC affiliated with COMPASS to perform analysis on data Construct analytic amplitudes to extract resonance information Amplitude satisfy S-matrix principles Emphasize production process and unitarization of amplitude π − π π + R π − p target p recoil [ C. Adolph et al. [COMPASS Collaboration], arXiv:1509.00992] June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 3 / 19
Introduction 3 π Production Mechanisms Peripheral production is advantageous - Effective 2 → 2, 2 → 3, etc. meson scattering By effective we mean particle-reggeon scattering Production mechanisms dictate physics Expect exchange mechanism dominated by pomeron at high-energies Effective 2 → 2, 2 → 3, etc. meson scattering production by particle exchange ρ/ f 2 π − π − π beam π + π + π − ⇒ P π − P π − p target p recoil June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 4 / 19
Introduction PWA of 3 π final state Develop method of analysis satisfying S-matrix principles, study J PC resonances in 3 π In this presentation, we focus on 2 − + , long standing puzzle about π 2 (1670)– π 2 (1880) interplay, 17 waves out of 88 have J PC = 2 − + , S J PC L [C. Adolph et al. [COMPASS Collaboration], arXiv:1509.00992] June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 5 / 19
Formalism The Model Partial wave analysis of 3 π system in π − p → π − π − π + p Use isobar model, with first approximation of stable isobars in ( π − π + ) Pomeron phenomenologically approximated by vector particle, α P ≈ 1 Factorize N → P N vertex from rest of amplitude For J PC = 2 − + , focus on high t event intensities e.g. ρπ F -wave, f 2 (1270) π s F i S - and D -waves, . . . s tot Coupled channel analysis for partial wave amplitudes F i ( s ), s 1 with channel index i = { ρπ ( F ) , f 2 π ( S ) , f 2 π ( D ) , . . . } t P June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 6 / 19
Formalism Unitarity and Analyticity Partial wave unitarity of π − P → ( π − π + ) π − amplitude � t ∗ Disc F i ( s ) = 2 i ij ( s ) ρ j ( s ) F j ( s ) j Rescattering amplitude satisfies its own unitarity equation � t ∗ Im t ij ( s ) = ik ( s ) ρ k ( s ) t kj ( s ) k One can separate F i into LHC and RHC terms, and write dispersive integral equation for F i , with solution given by Omnes � ∞ t ij ( s ) c j + 1 ds ′ ρ j ( s ′ ) b j ( s ′ ) � � F i ( s ) = b i ( s ) + t ij ( s ) s ′ − s π s j j j June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 7 / 19
Formalism K-Matrix Parameterization To preserve unitarity, rescattering amplitude t ij ( s ) is parameterized by K -matrix [ t − 1 ] ij ( s ) = [ K − 1 ] ij ( s ) − I i ( s ) δ ij where I i ( s ) is Chew-Mandelstam phase space factor, with Im I i ( s ) = ρ i ( s ) The real K -matrix is parameterized by resonant and non-resonant contributions g r i g r � � j γ n ij s n K ij ( s ) = r − s + m 2 r n Fit K -matrix parameters to data and extract resonance information June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 8 / 19
Formalism Production Amplitude For the production amplitude b i ( s ), we t model with Deck amplitude Consider π exchange ρ 0 π − Closest LHC to physics region = ⇒ Expected to be significant contribution s π − Ignoring subtleties of π -exchange (May need absorption corrections) π − P Model: g ρππ g P ππ ǫ λ · p 2 ǫ σ ∗ A Deck ( s , Ω) = λ ′ · { p a } t ( s , θ ) − m 2 π b i ( s ) is partial wave projection of A Deck in definite J , M , and L states June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 9 / 19
Current Results Fit Attempts As first attempt, we consider a more simplified model, where the production amplitude is conformal expansion � F i ( s ) = t ij ( s ) α j ( s ) j α i contains no RHCs and has free parameters Also, consider only f 2 π in S - and D -wave π ρ/ f 2 � α j t ij F i = π j P June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 10 / 19
Current Results Simple Production Model Fit June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 11 / 19
Current Results Unitarized Deck Fits The fits for a general production term α i seem too flexible in the current approach Now use unitarized Deck amplitude developed for this analysis � ∞ t ij ( s ) c j + 1 ds ′ ρ j ( s ′ ) b j ( s ′ ) � � F i ( s ) = b i ( s ) + t ij ( s ) s ′ − s π s j j j π π π I I I t ( s ) t ( s ) F i ( s ) = + + π π π P P P � �� � � �� � � �� � Deck projection b 0 Short range production t c Unitarised Deck t /π � ... d s ′ Fit Intensities and phase differences of three channel case June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 12 / 19
Current Results Unitarized Deck Fits Data: three main waves at low | t ′ | (0 . 1 GeV 2 -0 . 113GeV 2 ): 2 − + 0 + ( ππ ) s π D . 2 − + 0 + f 2 π S , 2 − + 0 + f 2 π D , 5000 Data 12000 Data Model curve 40000 Data Model curve Only first pole 10000 Model curve 4000 Only first pole Only second pole Only first pole Only second pole 30000 8000 Only second pole 3000 6000 20000 2000 4000 10000 1000 2000 M 3 π M 3 π M 3 π 1.6 1.8 2.0 2.2 2.4 1.6 1.8 2.0 2.2 2.4 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Figure: Fit model: 3 channel K-matrix with two poles and unitarized ”Deck”. K-matrix assumes elasticity, so simultaneous fit of all decay channels are needed (all 3 π waves), data for 11 | t ′ | intervals are available. | t ′ | -dependence of non-resonance component is fixed by “Deck” model. June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 13 / 19
Outlook Future Developments for COMPASS Analysis Develop Framework to analyze 3 π resonances satisfying S-matrix principles Will investigate Finite Energy Sum Rules to constrain amplitudes We are fitting data based on COMPASS model. Will extend to 4-vectors and for GlueX at JLab [C. Adolph et al. [COMPASS Collaboration], Phys. Rev. Lett. 115, 082001 (2015)] Want to describe entire 3 π spectrum, but some interesting cases along the way (2 − + and 1 ++ ) Will continue the work on COMPASS in 2 − + sector Perform analysis on 1 ++ sector, a 1 (1420) puzzle June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 14 / 19
Outlook Summary We have developed the analysis formalism to analyze 3 π systems for peripheral reactions Formalism satisfies S-matrix principles Applying formalism to COMPASS and extracting resonances Focus on J PC = 2 − + first, then apply to all 3 π J PC Extend formalism for photon beams (JLab12 physics) June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 15 / 19
Backup Backup June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 16 / 19
ρ Backup Phase Space Factors � In stable isobar limit, phase space factor is 2-body: ρ i ∼ ( s − s i ) / s Decaying isobar introduces π + π − scattering amplitude f ( s ) Phase space factors change to quasi-two body phase space factors � √ s − m π ds ′ ρ Isobar − π ( s ′ )Im f ( s ′ ) ρ Quasi ( s ) ∼ 4 m 2 π Affects how we continue to unphysical sheets, new (Woolly) cut introduced 0.035 0.030 0.025 0.020 0.015 quasi - two - body 0.010 two - body 0.005 M 3 π 1.0 1.5 2.0 2.5 June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 17 / 19
Backup Resonance Extraction Analytically continue amplitudes to unphysical sheets to search for poles Stable isobars involve only two-body phase space factors (simple square-roots) For decaying isobars, Woolly cut may hide pole onto a deeper sheet June 2 nd , 2016 Jackura, Mikhasenko (IU, JPAC) Amplitude analysis of 3 π 18 / 19
Recommend
More recommend
