The Deck Effect in πn πππn with Adam Szczepaniak, Indiana U.
I =1,G= πp πππp mesons have been seen in e.g. π + a 2,a 1,π 2... s Partial wave analysis exposes the details standard PWA based on isobar model p J P M ² I(W, t, s 1, angles) = p isobar L S π + s 2 Pb C b(w, t)a b (s 1, angles) } s 1 A b (s 1, angles) P λ DJ Mλ (Ω 1)Dλ S (Ω 3)hL; Sλ J λif L (p 1 )F S (p 3 ) isobar S (s1) we fix the s1 dependence in isobar model: isobar S (s1) = Breit- 2 W Wigner
πππ πππ partial wave analysis I b (W )= R dt... C b (W, t)a b 2 1 + S + ρπ 2 + D1 + ρπ 2 S + f 2 π data φ(2 + D1 + ρπ 1 + S + ρπ)
diffractive dissociation and the Deck long been known that this is not the only diagram possible, see e.g. R.T.Deck (1964) isobar π + π + p p p p diffractive dissociation π ρπ + near-on-shell πp πp large enhancement near threshold
Stodolsky s Deck s Stodolsky demonstrated kinematic origin of the enhancement and its partial wave structure 1 use a conventional pion propagator t m 2 ρ π p t π t p W s 2 diffractive (Pomeron) πn πn A s 2 t m 2 π is 2 e at in the interesting kinematic region we find s 2 s m2 t π W 2 m 2 π t t min,sàw 2,m p,m π,m ρ A(t t min,sà W 2 ) s W 2 m 2 π Stodolsky pole
This simple model tells us a lot the propagation and scattering of the virtual pion is described by Stodolsky s Deck πρ so this model produces with πρ s W 2 m 2 π πρ M = and the are in a relative S -wave which has no angular dependence Stodolsky s Deck will generate amplitude dominantly in the A 1 the imbroglio Deck or Resonance? S + ²π 1 + S + ρπ 2 S + f 2 π waves a 1 need for an resonance finally established in τ decays
beyond Stodolsky s Deck These properties are modified if the pion propagation is not just for example, with an off-shell pion form-factor at either vertex or with a Reggeised pion t π π ei 2 α(t π ) t m 2 π 1 t m 2 π e bt π t m 2 π the extra dependence puts some Deck amplitude into higher L -waves sophisticated models developed to describe the Deck I ll use the Ascoli et al. model to demonstrate the ingredients
Ascoli Deck model (t R ) π + p(~p B ) p(~p D ) ππ expt al phase-shift Regge pion exchange low s πn - expt al πn phase shift high s πn - Regge param n kinematics are not simple, but note one important dependence: amplitude has s 1 dependence beyond just the isobar factor this will not be captured correctly by the isobar-model PWA
Ascoli cont sort of shape we expect, some amplitude in P wave, but, too much Deck? The Ascoli scheme has some interpretative problems By Reggeising the pion (and by indirectly using the Pomeron) we ve modeled the entire amplitude in the large limit. πn πππn W, s πn,s W But we don t have large - we re in the resonance region Concept of Regge duality comes up this amplitude is approximately dual to resonances in? W multi-regge theory
why go back to this? This was the state of the art circa 198, and little consideration has been given since - the Deck effect has not gone away! good, high statistics data in the 21 st century: 25, events 2,, events publications to come
do we need the Deck effect? a 1 π 2 (167) The a 1 is the classic example but there is an arguably more important case - the, often used as a reference wave to extract other res. 3 3 2 2 1 3 E852(22) 3 2 S + f 2 π 3 2 2 1 BW m =166MeV Γ = 3 MeV 1 1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 45 4 35 3 25 6 4 3 BW m =184MeV Γ =27 MeV 2 15 1 5 2 D + f 2 π 2 1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 /E852(22) data
do we need the Deck effect? a 1 π 2 (167) The a 1 is the classic example but there is an arguably more important case - the, often used as a reference wave to extract other res. 3 3 3 3 2 2 2 2-2 1-4 1-6 1-8 1 3 3 E852(22) E852(22) 3 2 S + f 2 π phase S-D 3 2 2 1 1 BW m =166MeV Γ = 3 MeV -1-12 -14 1.3 1.3 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.7 1.8 1.8 1.9 1.9 2 2.1 2.1-16 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2-18 -2 45 45-22 4 4 23-24 35 35 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 3 3 25 25 2 2 6 4 3 BW m =184MeV Γ =27 MeV 15 15 1 1 5 5 2 D + f 2 π 2 1 1.3 1.3 1.4 1.4 1.5 1.5 1.6 1.6 1.7 1.7 1.8 1.8 1.9 1.9 2 2 2.1 2.1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 /E852(22) data
could Deck explain the Try something simple to test if Deck-style amplitude can help: π 2 ( 166 184)? add direct resonance production to a Deck background of the Ascoli type M = A + B m,γ 2 S + f 2 π Perform a simultaneous fit to the waves 2 D + f 2 π 3 3 2 2 1 1 Deck + BW Deck BW 2 S + f 2 π 2 D + f 2 π 4 m =178MeV 3 Γ 2 =25 MeV 6 Deck + BW Deck BW 1 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
could Deck explain the π 2 ( 166 184)? promising suggests the Deck bump is in the right place to account for the peak shift but, large phase motion is not explained, and we worry about double counting more theoretically justified scheme promoted by Aitchison & Bowler and others Deck as a Born term with subsequent rescattering limits itself to an isobar picture, but implements two-particle unitarity fits in this vein will follow shortly using both simple Stodolsky and more sophisticated Deck models
the new data new data from E852 opens up new possibilities enough events to consider fine-binned t -dependence of partial waves.8 t.1 ;.1 t.12 ;.12 t.14 ;.14 t.16 ;.16 t.18 ;.18 t.23 ;.23 t.28 ;.28 t.33 ;.33 t.38 ;.38 t.43 ;.43 t.48 ;.48 t.53 ;.53 t.58 2 S1 + f 2 π 2 D1 + f 2 π minor waves, such as are statistically significant multiple wave-sets considered to ensure robustness new charge combination available π π
conclusion resonances are not the only features of the S matrix and if we want to properly understand the meson spectrum we need to take into account these other dynamical effects the Deck effect, while not fully understood theoretically, has a simple kinematic origin. It affects low L partial waves near threshold, manifesting itself as an asymmetric bump interpreting PWA phase information has been done with reference to the which may be polluted by an as yet unknown degree of Deck π 2 (167) attempts are underway using past and future data to understand this