Three-particle scattering amplitudes from a finite volume formalism*

Transcription

1 INT-13-53W March 2013 Three-particle scattering amplitudes from a finite volume formalism* Zohreh Davoudi University of Washington *Raul Briceno, ZD, arxiv:

2 Why a finite volume formalism? Lattice QCD: From spectrum to physical specific issue that is a bit frightening at the moment is the density of scattering observables? s in multi-hadron systems 50 L24, p0 L32, p0 L48, p0 L12 fm, p0 0 E MeV He 0 p p n n d p n nn p p nn pp d d 3 He n Beane, et. al. (2012). s far below thresholds are presumably OK, but how do we learn about d d ering? Resonance spectroscopycoupled multi-particle to Maiani-Testa No-go Theorem channels n 2012 Light Nuclei and Hypernuclei from Quantum C the Limit of SU(3) Flavor Symm S.R. Beane, 1 E. Chang, 2 S.D. Cohen, 3 W. Detmold T.C. Luu, 6 K. Orginos, 4, 5 A. Parreño, 2 M.J. Savage, 3 and (NPLQCD Collaboration) 1 Department of Physics, University of New Hampshire, Durha 2 Dept. d Estructura i Constituents de la Matèria. Institut de C Universitat de Barcelona, Martí i Franquès 1, E

3 Dimer formalism and Luscher formula A NR EFT approach L = φ i m φ d i m d g 2 2 (dφ2 +h.c.)+... Eliminate in favor of physical observables: a, r D = = + id (E,q) = imr/2 q cot δ d iq + i D V = = + V id V (E,q) = imr/2 q cot δ d 4π c q 00 (q2 +i)+i The spectrum in FV can be written in a modelindependent way c q 00 (x) = 1 L 3 P Luscher (1986, 1991). Rummukainen, Gottlieb, (1995). Kim, Sachrajda, Sharpe (2005). Beane, et al (2005). k d 3 k (2π) 3 k l 4πY00 (ˆk ) k 2 x k = k q/2

4 Three-body correlation functions with the dimer field (E,P) K 3 g 3 = + g 2 g 2 Kinematic region below fourparticle threshold Expand the correlation function in powers of kernel C V 3 A 3 A 3 = A 3 V A 3 + K 3 V V A 3 K 3 V V K 3 V A L 6 A 3 (q 1 ) id V (E q2 1 2m, P q 1 )ik 3 (q 1, q 2 ; P,E)iD V (E q2 2 2m, P q 2 )A 3 (q 2 ) q 1,q 2 d 3 q 1 (2π) 3 d 3 q 2 (2π) 3? ik 3 (q 1, q 2 ; P,E) ig 3 ig 2 2 E q2 1 2m q2 2 2m (P q 1 q 2 )2 2m + i The poles of three-body kernel cancel with zero s of full FV dimer propagator!

5 Identifying the on-shell states In CM frame (I) Only Luscher poles matter q κ cot δ d =4π c ( 2P 3 q κ ) 00 (q 2 κ) q q 2 κ = me 3 4 q 2 κ COUPLED-CHANNELS q triboson diboson-boson three bosons Three-particle states {q κ,q κ} = { q 0, { } Power-law corrections (me q 2 0 ), q 2 1, 3 (me q 1),..., q N E, 3 (me q 2 N E ) } Off-shell states me < 3 4 q 2 κ Exponential corrections

6 Three-particle quantization conditions (II) Det(1 + M V δ G V ) det oc [det pw (1 + M V δ G V )] = 0 Determinant over open kinematic channels Determinant over partial-wave channels of boson-dimer state M V (p, k; P,E)= M (p, k; P,E) d 3 q (2π) 3 M (p, q; P,E)δD V (E q2 2m, P q ) M V (q, k; P,E) Diagonal in angular momentum Mixes the three particle states M V = + M V + K 3 K 3 K 3 M V V Briceno, ZD (2012). M = + M + K 3 K 3 K 3 M

7 S-wave boson-diboson elastic scattering amplitude M Bd = 3π m 1 q 0 cot δ Bd iq 0 Bound-state particle scatteringrecovering Luscher M V vs. M M Bd? deuteron Key: Diboson is a compact object in sufficiently large volumes 0 q = iγ d + O(e γdl /L) nucleon q 0 cot δ Bd =4π c P 00(q 0)+η e γ dl L A coefficient that needs to be fit to data q 0 = 4 3 (me q 2 0 ) V L 1 γ d Diboson infinite volume binding momentum

8 Bound-state particle scatteringrecovering Luscher Other sources of systematics to the Luscher approximation NLO correction due to size of diboson First off-shell state ignored O O e 2γ d L /L e 4 3 (q 2 1 me )L L Partial-wave mixing, S-wave dimer? (J d,j Bd )={(0, 0), (2, 0), (4, 0), (0, 4), (2, 4), (2, 6),...} (J d,j Bd )={(0, 0), (0, 1), (2, 0), (2, 1), (0, 2), (2, 2),...} See Raul s talk! Triton binding energy? γ Bd + q Bd cot δ Bd q 2 Bd = γ2 Bd = O(e γ BdL )

9 How big must the volumes be? A few percent determination of phase-shifts for dn scatting? Physical pion mass L Presumably there exists a trick to eliminate LO corrections! L L 17 fm L 12 fm Two-body evidence: Do the calculation with different boosts Form suitable linear combinations E MeV [3(0, 0, 1) (0, 0, 0)] ZD, Savage (2011). Κ A Κ B Κ C Κ D Κ E L Three-body problem? Requires more extensive numerical work! 1 4 [(0, 0, 0) + (0, 0, 1) + (0, 1, 1) + (1, 1, 1)] L fm

10 Recombination and breakup processes? Just above the threshold 1 2 deuteron A coupled-channels problem nucleon Relate to physical scattering amplitudes through an integral equation NO ALGEBRAIC APPROXIMATION ABOVE BREAKUP Hansen, Sharpe (2012); Briceno, ZD (2012). Briceno, ZD (2012).

11 Implementation (I) Determine two-body scattering parameters well Obtain the Luscher poles as a function of two-particle boost momentum and energy (II) Calculate three-body spectrum well (III) Numerically solve the quantization condition for three-body kernel.

12 Summary and conclusion The spectrum of three bosons in FV is related to three-particle scattering amplitudes through an integral equation. In theories with two-body bound states a simple Luscher formula exists. The phase shifts should be extrapolated to infinite volume limit. The problem is in general a coupled-channel problem - more than one kinematic channels contributiong Sources of systematics include the presence of nearby off-shell states + FV partial-wave mixing Implementation is an extensive numerical effort. Future work and open questions Nuclear sector? Generalized dimer formalism? See Raul s Talk. Study volume dependence of Triton and dn scattering numerically Three-body problem without dimer? See Max s talk. To determine scattering amplitudes without any reference to the EFT? Breakup recombination processes? Coupled-channel analysis

13 THANKS!

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