Extracting 3-particle scattering amplitudes from the finite-volume spectrum

Transcription

1 Extracting 3-particle scattering amplitudes from the finite-volume spectrum Steve Sharpe University of Washington Based on soon to be submitted work with Max Hansen S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 1 /68

2 The fundamental issue Lattice simulations are done in finite volumes Experiments are not How do we connect these? S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 2 /68

3 The fundamental issue Lattice QCD can calculate energy levels of multiple particle systems in a box How are these related to scattering amplitudes? L L? L S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 3 /68

4 The fundamental issue Lattice QCD can calculate energy levels of multiple particle systems in a box How are these related to scattering amplitudes? E 2 (L) E 1 (L)? im n!m E 0 (L) Discrete energy spectrum Scattering amplitudes S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 4 /68

5 When is spectrum related to scattering amplitudes? L R (interaction range) L L<2R No outside region. Spectrum NOT related to scatt. amps. Depends on finite-density properties L>2R There is an outside region. Spectrum IS related to scatt. amps. up to corrections proportional to e M L [Lüscher] S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 5

6 Problems considered today L>2R L>3R (?) Previously solved; solution used by simulations; will sketch as warm-up problem Will present new solution; practical applicability under investigation S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 6

7 Outline Motivation & background Set-up and main ideas 2-particle quantization condition 3-particle quantization condition Utility of result: truncation Important check: threshold expansion Conclusions and outlook S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 7 /68

8 Some references M. Rubin, R. Sugar & G. Tiktopoulos, Dispersion Relations for Three-Particle Scattering Amplitudes.1, Phys. Rev. 146 (1966) 1130 M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories: 2. Scattering States, Commun. Math. Phys. 105 (1986) 153 M. Lüscher, Signatures of unstable particles in finite volumes, Nucl. Phys. B354 (1991) 531 C.Kim, C. Sachrajda & S.R. Sharpe, Finite-volume effects for two-hadron states in moving frames, Nucl. Phys. B737 (2005) 218, hep-lat/ S.R. Beane, W. Detmold & M.J. Savage, n-boson Energies and Finite Volume and Three-Boson Interactions, Phys. Rev. D76 (2007) , arxiv: S. Tan, Three-boson problem at low energy and implications for dilute Bose- Einstein condensates, Phys. Rev. A78 (2008) , arxiv: K. Polejaeva & A. Rusetsky, Three particles in a finite volume, Eur. Phys. J. A48 (2012) 67, arxiv: M.T. Hansen & S.R. Sharpe, Multi-channel generalization of Lellouch-Lüscher formula, Phys. Rev. D86 (2012), arxiv: R.A. Briceno & Z. Davoudi, Three-particle scattering amplitudes from a finite volume formalism, arxiv: M.T. Hansen & S.R. Sharpe, arxiv:13??.xxxx, in preparation S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 8 /68

9 Motivation & background S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 9 /68

10 Motivation Study of resonances: ρ( ππ), K* ( Kπ), Δ( Nπ),... Resonances are not asymptotic states; show up in behavior of phase-shift S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 10/68

11 Motivation Study of resonances: ρ( ππ), K* ( Kπ), Δ( Nπ),... Many results now obtained using Lüscher s method & generalizations, e.g. [Dudek et al., 2013] ρ phase shift m = 391 MeV S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 11 /68

12 Motivation Study of resonances: ρ( ππ), K* ( Kπ), Δ( Nπ),... Calculating electroweak decay amplitudes, e.g. K ππ Lattice QCD can calculate <K HW ππ>l, but to use this requires determining the composition of the finite volume ππ state (which contains several partial waves with different normalizations) [Lellouch & Lüscher] For I=2 ππ, results obtained with physical kinematics; for I=0, pilot study completed [RBC/UKQCD] In ~3-5 years, we may be able to determine Standard Model prediction for direct CP violation in K ππ, and compare to experimental result (ε /ε) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 12 /68

13 Motivation Study of resonances: ρ( ππ), K* ( Kπ), Δ( Nπ),... Calculating electroweak decay amplitudes, e.g. K ππ Calculating CP violating parts of D ππ, KK Hints (sadly fading) of larger-than-expected CP violation from LHCb & others Existing theoretical methods fail when multiparticle channels open, e.g. D 4π These channels are expected to be significant for the D (though not for the ρ) Finite volume ππ state will be a mixture of ππ, KK, ηη, 4π,... (each in several partial waves) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 13 /68

14 Motivation Study of resonances: ρ( ππ), K* ( Kπ), Δ( Nπ),... Calculating electroweak decay amplitudes, e.g. K ππ Calculating CP violating parts of D ππ, KK Hints (sadly fading) of larger-than-expected CP violation from LHCb & others Existing theoretical methods fail when multiparticle channels open, e.g. D 4π These channels are expected to be significant for the D (though not for the ρ) Finite volume ππ state will be a mixture of ππ, KK, ηη, 4π,... (each in several partial waves) Need to develop tools to deal with n>2 particles in a box S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 13 /68

15 Motivation for n=3 n=4 particles is too hard, so start with n=3! Caveat: n=4 is intrinsically relativistic, while n=3 can be treated non-relativistically Many interesting applications: ω πππ, K πππ, determining NNN 3-body interaction, ND NNN,... We want a generalization of Lüscher s method and also of Lellouch-Lüscher formula Discuss only the former today S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 14/68

16 Theoretical status for 2 particles Underlying idea is simple in 1-d: e 2i (k) = e ikl Generalizations to 3-d in QM [Huang & Yang 57,...] [Lüscher 86 & 91] derived quantization formula for energies below inelastic threshold (and for P=0) by converting QFT problem to one in NRQM [Rummukainen & Gottlieb 85] generalized to general P (using rel.qm) [Lellouch & Lüscher 00] generalized to weak decay amplitudes [Kim, Sachrajda & SS 05] gave alternate derivation directly in QFT allowing generalization of LL formula to general P (see also [Christ, Kim & Yamazaki]) [Various authors] generalized the quantization and LL formulae to the case of any number of two particle channels (e.g. ππ, KK, ηη) First step on the way to D decays S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 15 /68

17 Theoretical status for 3 particles [Polejaeva & Rusetsky 12] Showed in NREFT that spectrum determined by infinite-volume amplitudes, using integral equation [Briceno & Davoudi `12] Used a dimer approach in NREFT to determine relation between spectrum and a finite volume quantity, itself related to infinite-volume amplitudes by an integral equation [Beane, Detmold & Savage `07 and Tan `08] derived threshold expansion for n particles in NRQM, and argued it applied also in QFT Our aim: work in general, relativistic QFT and determine an algebraic relation between spectrum and scattering amplitudes S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 16/68

18 Set-up & main ideas S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 17 /68

19 Set-up Work in continuum (assume that LQCD can control discretization errors) L Cubic box of size L with periodic BC, and infinite (Minkowski) time Spatial loops are sums: P 1 ~ L 3 ~ k k = 2 L ~n Consider identical particles with physical mass m, interacting arbitrarily except for a Z2 (G-parity-like) symmetry Only vertices are 2 2, 2 4, 3 3, 3 1, 3 5, 5 7, etc. Even & odd particle-number sectors decouple L L S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 18/68

20 Methodology Calculate (for some P=2πn P/L) CM energy is E * = (E 2 -P 2 ) Poles in C L occur at energies of finite-volume spectrum For 2 & 3 particle states, σ ~ π2 & π 3, respectively E.g. for 2 particles: Infinite-volume vertices Boxes indicated summation over finite-volume momenta Full propagators Normalized to unit residue at pole S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 19/68

21 3-particle correlator C L (E, ~ P )= Notation here slightly different Boxes still enclose summed momenta Lines without blobs are here full propagators Open circles are (for now) infinite-volume vertices S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 20/68

22 Key step 1 Replace loop sums with integrals where possible Drop exponentially suppressed terms (~e -ML, e -(ML)^2, etc.) while keeping power-law dependence S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 21 /68

23 Key step 1 Replace loop sums with integrals where possible Drop exponentially suppressed terms (~e -ML, e -(ML)^2, etc.) while keeping power-law dependence Exp. suppressed if g(k) is smooth and scale of derivatives of g is ~1/M S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 21 /68

24 Key step 2 Use sum=integral [sum-integral] if integrand has pole, with [KSS] dk0 2 1 X L 3 = Z k Z 1 d 4 k A (2 ) 4 f(k) d q d q 0 f j (ˆq )F jj (q,q 0 )g j (ˆq 0 ) 1 k 2 m 2 j i 1 (P k) 2 m 2 j i g(k) exp. suppressed q * is relative momentum of pair on left in CM Kinematic function f & g evaluated for ON-SHELL momenta Depend only on direction in CM Example f is left-hand part of integrand P-k Focus on this loop g is right-hand part of integrand P =(E, ~ P ) k S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 22/68

25 Key step 2 Use sum=integral [sum-integral] where integrand has pole, with [KSS] dk0 2 1 X L 3 = Z k Z 1 d 4 k A (2 ) 4 f(k) d q d q 0 f j (ˆq )F jj (q,q 0 )g j (ˆq 0 ) Decomposed into spherical harmonics, F becomes 1 k 2 m 2 j i 1 (P k) 2 m 2 j i g(k) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 23/68

26 Key step 2 Use sum=integral [sum-integral] where integrand has pole, with [KSS] dk0 2 1 X L 3 = Z k Z 1 d 4 k A (2 ) 4 f(k) d q d q 0 f j (ˆq )F jj (q,q 0 )g j (ˆq 0 ) 1 k 2 m 2 j i 1 (P k) 2 m 2 j i g(k) Diagrammatically 1 X L 3 ~ k Z ~ k finite-volume residue off-shell on-shell S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 24/68

27 Key step 2 Use sum=integral [sum-integral] where integrand has pole, with [KSS] dk0 2 1 X L 3 = Z k Z 1 d 4 k A (2 ) 4 f(k) d q d q 0 f j (ˆq )F jj (q,q 0 )g j (ˆq 0 ) 1 k 2 m 2 j i 1 (P k) 2 m 2 j i g(k) Can also use in 3-particle case (with some subtleties discussed below) Bottom line first set on-shell off-shell = F on-shell S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 25/68

28 Kinematic functions = x 2 = x 2 P=0 [Luu & Savage, `11] S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 26/68

29 Key step 3 Identify potential singularities: can use time-ordered PT (i.e. do k 0 integrals) Example S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 27/68

30 2 out of 6 time orderings: Key step P 1 E! 1 0! 2 0! 3 0! 4 0! 5 E! E! 1! 2! 3! 4! 5 0 j=1,6! 1! 2! 5 j On-shell energy! j = q ~k 2 j M 2 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 28/68

31 2 out of 6 time orderings: Key step P 1 E! 1 0! 2 0! 3 0! 4 0! 5 E! E! 1! 2! 3! 4! 5 0 j=1,6! 1! 2! 5 j If restrict M < E* < 5M then only 3-particle cuts have singularities, and these occur only when all three particles to go on-shell S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 28/68

32 Combining key steps For each diagram, determine which momenta must be summed, and which can be integrated In our 3-particle example, find: Can integrate Must sum momenta passing through box S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 29/68

33 Combining key steps For each diagram, determine which momenta must be summed, and which can be integrated In our 2-particle example, find: Can replace sum with integral here But not here S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 30/68

34 Combining key steps For each diagram, determine which momenta must be summed, and which can be integrated In our 2-particle example, find: Can replace sum with integral here But not here Then repeatedly use sum=integral sum-integral to simplify S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 30/68

35 2-particle quantization condition Following method of [Kim, Sachrajda & SS 05] S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 31 /68

36 Apply previous analysis to 2-particle correlator (0 < E* < 4M) C L (E, ~ P )= these loops are now integrated S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 32

37 Apply previous analysis to 2-particle correlator (0 < E* < 4M) C L (E, ~ P )= these loops are now integrated Collect terms into infinite-volume Bethe-Salpeter kernels C L (E, ~ P )= ik S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 32

38 Apply previous analysis to 2-particle correlator Collect terms into infinite-volume Bethe-Salpeter kernels C L (E, ~ P )= ik S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 33

39 Apply previous analysis to 2-particle correlator Collect terms into infinite-volume Bethe-Salpeter kernels C L (E, ~ P )= ik Leading to C L (E, ~ P )= ik ik ik S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 33

40 Next use sum identity C L (E, ~ P )= ik F ik ik ik ik ik ik F F F F S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 34

41 Next use sum identity C L (E, ~ P )= ik F ik ik ik ik ik ik F F F F And regroup according to number of F cuts C L (E, ~ P )=C 1 (E, ~ P ) ik ik A zero F F one F A 0 matrix elements: S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 34

42 Next use sum identity C L (E, ~ P )= ik F ik ik ik ik ik ik F F F F And keep regrouping according to number of F cuts C L (E, ~ P )=C 1 (E, ~ P ) two F s A ik ik ik A 0 F im A F A 0 the infinite-volume, on-shell 2 2 scattering amplitude F S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 34

43 Final result: C L (E, ~ P )=C 1 (E, ~ P ) A A 0 A im A 0 A im im C L (E, ~ P )=C 1 (E, ~ P ) F F F F F F 1X n=0 A 0 A 0 if [im 2!2 if ] n A Correlator is expressed in terms of infinite-volume, physical quantities and kinematic functions encoding the finite-volume effects S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria /68 35

44 Final result: C L (E, ~ P )=C 1 (E, ~ P ) A A 0 im A 0 A im im C L (E, ~ P )=C 1 (E, ~ P ) A F F F F F F 1X n=0 C L (E, ~ P )=C 1 (E, ~ P )A 0 if no poles, only cuts A 0 A 0 if [im 2!2 if ] n A 1 1 im 2!2 if A matrices in l,m space no poles, only cuts C L (E, ~ P ) if 1 1 im 2!2 if diverges whenever diverges S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 36/68

45 2-particle quantization condition C L (E, ~ P )=C 1 (E, ~ P )A 0 if 1 1 im 2!2 if A At fixed L & P, the finite-volume spectrum E 1, E2,... is given by solutions to L, ~ P (E) =det (if ) 1 im 2!2 =0 M is diagonal in l,m: im 2!2;`0,m 0 ;`,m / `,`0 m,m 0 F is off-diagonal, since the box violates rotation symmetry To make useful, truncate by assuming that M vanishes above l max For example, if l max=0, obtain im 2!2;00;00 (E n)=[if 00;00 (E n, ~ P,L)] 1 Generalization of s-wave Lüscher equation to moving frame [Rummukainen & Gottlieb] S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 37/68

46 3-particle quantization condition S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 38 /68

47 Skeleton expansion C L (E, ~ P )= Full propagator 3 3 ik 2!2 ik e 3!3 Infinite-volume quantities Contains single-particle cuts Sum momenta passing through boxes If remove endcaps, drop first diagram, and change internal sums to integrals, then have skeleton expansion for M3 3 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 39/68

48 Skeleton expansion C L (E, ~ P )= Summing ik 3 3 is straightforward; difficulties arise from multiple ik2 2 s Switches present the greatest challenge (related to issue raised by Akaki Rusetsky) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 40/68

49 Skeleton expansion C L (E, ~ P )= 0 switches: 1 switch: 2 switches: switch state Summing ik 3 3 is straightforward; difficulties arise from multiple ik2 2 s Switches present the greatest challenge (related to issue raised by Akaki Rusetsky) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 40/68

50 No switch diagrams C (1) L k k k k S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 41 /68

51 No switch diagrams C (1) L k k k k C (1) L Do k 0 integral, keeping only on-shell pole at k0=ωk Other poles give terms in which remaining sums can be replaced by integrals, and thus contribute to C (1) 1 X 1 L 3 2! k ~ k S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 41 /68

52 No switch diagrams C (1) L k k k k C (1) L Do k 0 integral, keeping only on-shell pole at k0=ωk Other poles give terms in which remaining sums can be replaced by integrals, and thus contribute to C (1) 1 X 1 L 3 2! k ~ k Substitute identity = F & proceed as for 2-particle case C (1) L = C 1 (1) 1 X 1 L 3 2! k ~ k F F F F F F F F F F im 2!2 Quantities on either side of F s are on-shell S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 41 /68

53 No switch diagrams C (1) L = C(1) 1 A 2 3 Dimer propagator A = if 1 2!L 3 1 im 2!2 if Present due to mismatch of symmetry factors. Absent if bottom particle non-interacting Matrix notation: indices are expanded compared to 2-particle case [ spectator momentum: k=2πn/l] x [2-particle CM angular momentum: l,m] if k 0,`0,m 0 ;k,`,m = k,k 0 if`0,m 0 ;`,m(e! k, P ~ ~ k) im k 0,`0,m 0 ;k,`,m = k,k 0 im`0,m 0 ;`,m(e! k, P ~ ~ k) 4-momentum of non-spectator pair S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 42/68

54 No switch diagrams C (1) L = C(1) 1 A 2 3 Dimer propagator A = if 1 2!L 3 1 im 2!2 if Present due to mismatch of symmetry factors. Absent if bottom particle non-interacting Matrix notation: indices are expanded compared to 2-particle case k, l, m parametrizes three particles with fixed total (E, P) (E! k, ~ P ~ k) â! `, m (! k, ~ k) BOOST S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 43/68

55 No switch diagrams C (1) L = C(1) 1 A 2 3 Dimer propagator A = if 1 2!L 3 1 im 2!2 if Present due to mismatch of symmetry factors. Absent if bottom particle non-interacting Matrix notation: indices are expanded compared to 2-particle case k, l, m parametrizes three particles with fixed total (E, P) k = 2π n / L remains discrete Essential to obtain correct quantization condition if bottom particle is non-interacting S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 44/68

56 No switch diagrams C (1) L = C(1) 1 A 2 3 A = if 1 2!L 3 1 im 2!2 if Important technical observations: 4-momentum in non-spectator pair can lie below threshold [Polejaeva & Rusetsky] Occurs if (E-ω k) 2 -(P-k) 2 < 4M 2 Projection from sum-integral gives analytic continuation of on-shell quantities below threshold This is standard when using Lüscher formula slightly below threshold, but here we require this to be extended by 2M below threshold Boost to CM fails if (E-ω k) 2 -(P-k) 2 < 0 Definition of on-shell projection breaks down We remove these cases by introducing a smooth regulator function which runs from 1 at and above threshold to 0 at (E-ωk) 2 -(P-k) 2 = 0 Changes F by exponentially small corrections (since excised region has non-singular integrand) S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 45/68

57 One-switch diagrams C (2) L = k 0 k Two spectator momenta, at least one of which must be on-shell for a FV contrib. Only get new feature if both are on shell C (2) L = C(2) 1 A A Terms with C (1) L form, but with modified endcaps Term between A s is our first contribution to the 3 3 on-shell amplitude im (2,unsym) 3!3;k 0,`0,m 0 ;k,`,m `, m ~ k ~ k 0 `0,m 0 Yl,m on left appear in lower pair, due to switch S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 46/68

58 One-switch problem im (2,unsym) 3!3;k 0,`0,m 0 ;k,`,m `, m ~ k ~ k 0 `0,m 0 Amplitude is singular for some choices of k, k in physical regime Propagator goes on shell if top two (and thus bottom two) scatter elastically Not a problem per se, but leads to difficulties when amplitude is symmetrized Occurs when include three-switch contributions! im 3!3;k0,`0,m 0 ;k,`,m `, m ~ k ~ k 0 `0,m 0 Singularity implies that decomposition in Y l,m will converge very slowly Truncation of problem (as in 2-particle case) will fail due to non-locality of full 3 3 amplitude Related to concern discussed by Akaki Rusetsky S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 47/68

59 One-switch solution Define divergence-free amplitude by subtracting singular part Include regulator H so that on-shell projection for M s is well defined Utility of subtraction noted in [Rubin, Sugar & Tiktopoulos, 66] im (2,unsym) df,3!3 im (2,unsym) 3!3 im 2!2 ih( ~ k, ~ k 0 ) 2! b (E! k! k 0! b ) im 2!2 Off-shell except at pole i G(k,k ) Always on-shell `, m ~ k b ~ k 0 `0,m 0 `, m ~ k b ~ k 0 `0,m 0 Key point: M df,3 3 is local and its expansion in harmonics can be truncated Subtracted term must be added back; its form is (in our matrix notation) im 2!2 ig im 2!2 Truncation of expansion in harmonics in M 2 2 G M2 2 enforced by properties of M2 2 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 48/68

60 All orders divergence-free amplitude im df,3!3 = im 3!3 S h im 2!2 ig im 2!2 R i ~q im 2!2 ig im 2!2 ig im 2!2... Include also 3 3 B-S kernel Symmetrizer ~q Also divergent M df,3 3 is free of singularities to all orders & has key properties Depends only on on-shell scattering amplitudes Expected to be local and thus have an expansion in spherical harmonics that can be usefully truncated 3-switch and higher-order terms are not divergent for identical particles, but are if non-degenerate Arises naturally in our derivation S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 49/68

61 Final result Extension to any number of switches is tedious but introduces no new features Symmetrized M 3 3 appears only when work to all orders Inclusion of K 3 3 (3-particle Bethe-Salpeter amplitude) is straightforward Spectrum is determined (for given L, P) by solutions of L,P (E) =det[1 im df,3!3 if 3 ]=0 if 3 1 2!L 3 apple (2/3)iF 1 [if ] 1 [1 imig] 1 im 2 2 amplitude Depends only on physical scattering amplitudes (although must analytically continue 2 2 amplitude below threshold) Finite volume effects enter through quantization of spectator momentum & F Surprisingly simple result, superficially similar to 2-particle form det [1 im 2!2 if ]=0 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 50/68

62 Utility of result: truncation S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 51 /68

63 Truncation in 2 particle case det [1 im 2!2 if ]=0 If M (which is diagonal in l,m) vanishes for l > lmax then can show that need only keep l lmax in F (which is not diagonal) and so have finite matrix condition which can be inverted to find M(E) from energy levels S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 52/68

64 Truncation in 3 particle case L,P (E) =det[1 im df,3!3 if 3 ]=0 if 3 1 2!L 3 apple (2/3)iF 1 [if ] 1 [1 imig] 1 im 2 2 amplitude For fixed E & P, as spectator momentum k increases, remaining two-particle system drops below threshold, so F becomes exponentially suppressed Explicitly implemented by H factor in F (and in G) Thus k sum truncated (with, say, N terms required) l is truncated if both M and M df, 3 3 vanish for l > lmax Yields determinant condition truncated to [N(2l max1)] 2 block S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 53/68

65 Truncation in 3 particle case L,P (E) =det[1 im df,3!3 if 3 ]=0 if 3 1 2!L 3 apple (2/3)iF 1 [if ] 1 [1 imig] 1 im 2 2 amplitude Given prior knowledge of M 2 2, each energy level Ei of the 3 particle system gives information on Mdf,3 3 at the corresponding 3-particle CM energy Ei * Could proceed by parameterizing M df,3 3, in which case one would need at least as many levels as parameters at given energy Given M 2 2 and Mdf,3 3 one can reconstruct M3 3 Clearly very challenging in practice! S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 54/68

66 Isotropic approximation L,P (E) =det[1 im df,3!3 if 3 ]=0 if 3 1 2!L 3 apple (2/3)iF 1 [if ] 1 [1 imig] 1 im 2 2 amplitude Assume M df,3 3 depends only on E * (and thus is indep. of k, l, m) Also assume M 2 2 only non-zero for s-wave ( lmax=0) and known Truncated [N x N] problem simplifies: M df,3 3 has only 1 non-zero eigenvalue, and problem collapses to a single equation: im df,3!3 (E n)=[if 3,iso (E n, ~ P,L)] 1 N if 3,iso X ~k,~p if 3;k,p s-wave only S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 55/68

67 Important check: threshold expansion S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 56 /68

68 Threshold expansion Given complexity of derivation & new features of result, checks are important Can do so for P=0 and near threshold: E=3mΔE, with ΔE~1/L3... In other words, study energy shift of three particles (almost) at rest Dominant effects (L-3, L -4, L -5 ) involve 2-particle interactions, but 3-particle interaction enters at L -6 For large L, particles are non-relativistic (ΔE m) and can use NREFT methods This has been done previously by [Beane, Detmold & Savage, ] and [Tan, ] S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 57/68

69 NR EFT results 2 particles 3 particles [Beane, Detmold & Savage, ] 2-particle result agrees with [Luscher] Scattering length a is in nuclear physics convention r is effective range I, J, K are zeta-functions 3 particle result through L -4 is 3x(2-particle result) from number of pairs Not true at L -5,L -6 where additional finite-volume functions Q, R enter η3(μ) is 3-particle contact potential, which requires renormalization S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 58/68

70 NR EFT results 2 particles 3 particles [Beane, Detmold & Savage, ] 2-particle result agrees with [Luscher] Scattering length a is in nuclear physics convention r is effective range I, J, K are zeta-functions 3 particle result through L -4 is 3x(2-particle result) from number of pairs Not true at L -5,L -6 where additional finite-volume functions Q, R enter η3(μ) is 3-particle contact potential, which requires renormalization Tan has 36 instead of 24, but a different definition of η3 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 58/68

71 NR EFT results [Beane, Detmold & Savage, ] zeta-functions additional finite-volume quantities dim. reg. S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 59/68

72 Expanding our result det [1 im df,3!3 if 3 ]=0 if 3 = h if 1 2!L M 1 F G F i Take M to be purely s-wave and M df,3 3 to be a constant (i.e. lmax=0) F 3, F, G are then truncated to matrices in spectator momentum space Can show that [F 3]0,0 dominates other matrix elements by at least L 2, so quantization condition becomes im df,3!3 =(i[f 3 ] 0,0 ) 1 F is O(L0 ), so to cancel the 1/L 3 in F3 need [M -1 FG] -1 ~L 3 Roughly speaking this requires the cancellation of L0, L -1 & L -2 terms in [M -1 FG], which requires tuning E and determines the L -3, L -4 & L -5 in ΔE The L-6 term in ΔE is then determined by the quantization condition S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 60/68

73 Our threshold expansion S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 61/68

74 Our threshold expansion agrees with [Beane et al.] and [Tan] modulo definitions of Q & R UV convergent! ( ˆQ of [Beane et al.] Log divergent after NR expansion, so requires regulation as in [Beane et al.] Similar situation for R S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 62/68

75 Our threshold expansion G required to get correct factors in these terms S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 63/68

76 Our threshold expansion [Beane et al] have 24, [Tan] has 36, we have 72 Physical, finite quantity, with no μ dependence Directly related to scattering amplitudes In [Beane et al.] this term is [Beane et al.] and [Tan] do not have this term S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 64/68

77 Interpretation of differences vs. [Beane et al.] [Hansen & SRS] We do not know a priori the relation between M df,3 3 and η3 M df,3 3 is physical, while η3 is a short-distance parameter, indirectly related to physical quantities We can view this comparison as providing the relation between M df,3 3 and η3 if we equate the two expressions As far as we can see, there is nothing forbidding this relation to include the finite a 2 and a 3 r terms Indeed, a similar finite difference is required to match [Beane et al.] with [Tan] It would clearly be good to check this purported relation in another context S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 65/68

78 Conclusions & Outlook S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 66 /68

79 Summary Confirmed that 3-particle spectrum determined by infinite-volume scattering amplitudes, although the 2 2 amplitude is needed below (& above) threshold Obtained an algebraic result directly in terms of these amplitudes Derivation leads us to introduce divergence-free 3 3 scattering amplitude, which is a natural quantity to appear Threshold expansion check gives us confidence in the expression & provides an example of how it can be used in practice Warning (in case you want to run off and apply this!): σ couples also to the single pion state (which is why we kept E * > M). In Euclidean space this pole will be the lowest lying state. All 3 pion states will thus be excited states. S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 67/68

80 Plans Extend result to non-degenerate masses Onward to four particles?! Further studies of practical utility using simple forms for amplitudes Detailed comparison with [Polejaeva & Rusetsky] and [Briceno & Davoudi] Derive generalization of Lellouch-Lüscher formula S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 68/68

81 Plans Extend result to non-degenerate masses Onward to four particles?! Further studies of practical utility using simple forms for amplitudes Detailed comparison with [Polejaeva & Rusetsky] and [Briceno & Davoudi] Derive generalization of Lellouch-Lüscher formula Thank you! S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 68/68

82 Backup Slides S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 69 /68

83 Relation to dimer approach Roles of F and G are almost symmetrical Previous form: Dimer form : Sum of subtractions in finite volume 1iMiG (imig) 2... Finite-volume scattering amplitude a.k.a. dimer propagator = im imifim imifimifim... May allow relation to dimer approach of [Briceno & Davoudi, arxiv: ] to be worked out S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 70/68

84 Examples of expansions ) UV finite quantity (though, strictly speaking, need to regulate the sum & integral separately before taking difference---which we do) Real part (Imag. part cancels with M) NR expansion: q is momentum of each of non-spectator pair contributes to R G k,p = 1 2! p L 3 2! pk (E! p! k! pk ) NR expansion: ~ L 0 ~ L -1 ~ L -1 Need spectator-momentum matrix structure of F & G to evaluate [Fthree]0,0 S. Sharpe, 3-particle quantization condition STRONGnet 2013, Graz, Austria 71/68