Lecture 2: two-body William Detmold

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1 Lecture 2: two-body William Detmold Massachusetts Institute of Technology INT Lattice Summer School, Seattle, August 13 th -17 th, 212 Hadron Interactions and Many-body Physics

2 Lecture content Two particles in finite volume: Lüscher method Scattering Resonances Two-particle bound states

3 Theory

4 Maiani-Testa no-go theorem Scattering and decays are real-time processes How can Euclidean space (imaginary time) calculations address generic Minkowski space correlations? Maiani & Testa [91]: Euclidean correlators with initial/final states at kinematic thresholds allow access to physical information (matrix elements, weak decays) In infinite volume away from kinematic thresholds, scattering continuum masks the physically interesting information Example: K π π weak decay K(p=) π(-p) Consider C(t 1,t 2 )=ho K (t 1 )O weak ()O (t 2 )i Take large t1,2 to get single state?! hk O weak (ˆp ) ( ˆp )i +... π(p) [see also Michael 89]

5 Maiani-Testa no-go theorem Scattering and decays are real-time processes How can Euclidean space (imaginary time) calculations address generic Minkowski space correlations? Maiani & Testa [91]: Euclidean correlators with initial/final states at kinematic thresholds allow access to physical information (matrix elements, weak decays) In infinite volume away from kinematic thresholds, scattering continuum masks the physically interesting information Example: K π π weak decay π(-p) Consider K(p=) π(p) mk E Scattering continuum Physical ππ state C(t 1,t 2 )=ho K (t 1 )O weak ()O (t 2 )i 2m π Take large t1,2 to get single state?! hk O weak (ˆp ) ( ˆp )i +... [see also Michael 89]

6 Maiani-Testa no-go theorem Scattering and decays are real-time processes How can Euclidean space (imaginary time) calculations address generic Minkowski space correlations? Maiani & Testa [91]: Euclidean correlators with initial/final states at kinematic thresholds allow access to physical information (matrix elements, weak decays) In infinite volume away from kinematic thresholds, scattering continuum masks the physically interesting information Example: K π π weak decay π(-p) K(p=) E FV scattering states Consider π(p) mk Physical kinematics C(t 1,t 2 )=ho K (t 1 )O weak ()O (t 2 )i 2m π Take large t1,2 to get single state?! hk O weak (ˆp ) ( ˆp )i +... [see also Michael 89]

Two particles in a box Long realised that forcing particles to be in a finite volume shifts their energy in a way that depends on their interactions Uhlenbeck 193 s; Bogoliubov 194 s; Lee, Huang,

7 Two particles in a box Long realised that forcing particles to be in a finite volume shifts their energy in a way that depends on their interactions Uhlenbeck 193 s; Bogoliubov 194 s; Lee, Huang, Yang 195 s,... Lüscher (1986,1991) demonstrated that this is also true in QFT up to inelastic thresholds (see also Hamber, Marinari, Parisi & Rebbi) Energy eigenvalues of discrete scattering states well defined no issue in Minkowski-Euclidean connection Bypasses Maiani-Testa NGT E Bound state 2M Scattering continuum

8 Two particles in a box Long realised that forcing particles to be in a finite volume shifts their energy in a way that depends on their interactions Uhlenbeck 193 s; Bogoliubov 194 s; Lee, Huang, Yang 195 s,... Lüscher (1986,1991) demonstrated that this is also true in QFT up to inelastic thresholds (see also Hamber, Marinari, Parisi & Rebbi) Energy eigenvalues of discrete scattering states well defined no issue in Minkowski-Euclidean connection Bypasses Maiani-Testa NGT E Bound state E Bound state 2M Scattering continuum V 2M Scattering states

9 Two particles in a box p = 2 L n, n i 2 Z E n (L) =2 r m L 2 n 2 E L

10 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 E L

11 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 E L

12 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 (E) E E L

13 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 E L

14 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 (E) E E L

15 Two particles in a box Consider first the non-interacting two particle system with particles of mass m in a box of dimensions L 3 with zero CoM momentum Particles constrained to have momenta p = 2 L n, n i 2 Z hence the spectrum is given by r E n (L) =2 m L 2 n 2 (E) E E L Modified by interactions in a calculable way

16 Scattering at finite volume Consider simple effective field theory of a scalar particle interacting via contact interactions + M 2 + C 4 + C 2 D Scattering amplitude given by bubble sum + + i C2i p 2i In infinite volume A 4 /M p cot (p) ip = Pn C 2np 2n 1 I (p) P n C 2np 2n Making non-relativistic approx (doing relativistically is not much harder) and using power divergent subtraction regularisation scheme [Kaplan, Savage, Wise] Z I PDS (p) =µ 4 d d d 1 k E k 2 /M + i = M (µ + ip) 4

17 Finite volume energies In finite volume, integral restricts to allowed mode sum Pn A(L) = C 2np 2n 1 I L(p) P n C p cot (p) = 4 2np 2n M 1 Pn C 2np 2n + µ =A 1 (L) =p cot (p) Mµ 4 I L (p) Define PDS regulated sum as I L (p) 1 PDS X 1 L 3 E k 2 /M k = 1 k < X L 3 = M 4 " k 1 L 1 E k 2 /M + X n n 2 1 Z d 3 k/(2 ) 3 L 2 EM 4 2 k 2 /M 4 L µ # Z PDS d 3 k/(2 ) 3 k 2 /M

18 Finite volume energies Energies satisfy eigenvalue equation (Lüscher s method) 1 L 2 p cot (p) L S p = X 1 where S(x) = n 2 4 x Result valid for momenta up to inelastic threshold n [3D zeta function] Valid up to exponentially small corrections Eg: lowest energy level (zero rel. mom.) E = 4 a ML 3 apple a a 2 1+c 1 L + c L known Lüscher coefficients Calculation of energy levels on the lattice determines scattering parameters HW: 1. derive the energy shift ΔE from the Lüscher formula above assuming small a/l. 2. Calculate the coefficient c1.

19 Finite volume energies Energies satisfy eigenvalue equation (Lüscher s method) 1 L 2 p cot (p) L S p = X 1 where S(x) = n 2 4 x Result valid for momenta up to inelastic threshold n [3D zeta function] Valid up to exponentially small corrections Eg: lowest energy level (zero rel. mom.) E = 4 a ML 3 apple a a 2 1+c 1 L + c L known Lüscher coefficients Calculation of energy levels on the lattice determines scattering parameters HW: 1. derive the energy shift ΔE from the Lüscher formula above assuming small a/l. 2. Calculate the coefficient c1.

20 Two particle energies S p L 2 p 2 /4 2 Intersections correspond to eigen-energies of states in lattice volume

21 Two particle energies p cot (p) = 1/a S p 2 x x x x x L 2 p 2 /4 2 Intersections correspond to eigen-energies of states in lattice volume

22 Two particle energies 3 S p p cot (p) = 1/a p cot (p) = 1/a + r/2 p 2 1 x x x x x x x x L 2 p 2 /4 2 Intersections correspond to eigen-energies of states in lattice volume

23 Asymptotic expansions Ground state energy shift E = 4 a apple a a 2 ML 3 1+c 1 L + c L First excited state energy shift E 1 = 4 12 tan 1+c ML 2 ML 2 1 tan + c 2 tan where Each new level extracted or new volume used adds information on the phase shift at a different energy Bound states can also be described apple 2 12 E 1 = 1+ M L(1 r 3 ) e L +... Expansions also for L/a << 1 [Beane et al.] = (p E1 )

24 Bound states at finite volume Lüscher eigenvalue equation also includes solutions with p 2 < (bound state?) Two particle scattering amplitude in infinite volume bound state at p 2 = A 2 4 /M p cot (p) when i p cot (i )=i Binding energy EB related to binding momentum as = p 2ME B Scattering amplitude in finite volume (another way of expanding Lüscher eqn) cot (iapple) =i i X ~m6= e ~m applel ~m applel apple L!1! Multiple volumes required to show a negatively shifted state is bound

25 Boosted systems Boost of the two body system CoM relative to boundary conditions (lab) changes the effective shape of the box as seen by the interacting system First studied by Rummukainen & Gottlieb [95]; Further study by Kim, Sachrajda & Sharpe [5]; Kim Christ & Sachrajda [5];!!! " "! Generalised to other frames [Feng, Renner & Jansen]!!!! Allows access to phase shift at different momenta Effects on bound states investigated [Bour et al; Davoudi & Savage] Can be used to cancel leading exponential FV corrections to binding energies E MeV A B C D E L L fm

Asymmetry box of geometry Eigenvalue equation modified S( p 2 )!

hep-lat/435] 1 L 2 L L X n n 2 L 1 L 1 where ñ = ñ 2 p 2 4 n L n1, n 2,n 3 1 2

Asymmetries exist where sub-leading FV effects are suppressed: ci(η 1,η 2 )= apple 1+c

26 Asymmetry box of geometry Eigenvalue equation modified S( p 2 )! S( p 2, 1, 2 )= Asymmetric boxes [ Li & Liu hep-lat/31135; WD & Savage hep-lat/435] 1 L 2 L L X n n 2 L 1 L 1 where ñ = ñ 2 p 2 4 n L n1, n 2,n Asymptotic expansion E = 4 a 1 2 ML 3 Geometric coefficients c 1 ( 1,2 )= Asymmetries exist where sub-leading FV effects are suppressed: ci(η 1,η 2 )= apple 1+c 1 ( 1,2 ) a a 2 L + c 2( 1,2 ) +... L X n ñ6= 1 ñ n A Η c 2 Η 1,Η 2 c 2 Η 1,Η 2 c 1 Η 1,Η Η 1

27 Resonances at finite volume Pion is light so very few stable hadrons in the real world often the lightest particle of a given set of conserved quantum numbers and not much else Ex: ρ(77) decays to ππ and to ππππ; Δ(1232) decays to Nπ Ex: I=1 π π phase shift 15 ρ 1 Γ ρ Extensive experimental efforts to understand excited spectrum of hadrons At finite volume, spectrum is discrete: how do resonances manifest? Spectrum gets large modifications as a function of volume near resonance energy embodies in solutions of Lüscher eigenvalue equation See excellent lectures of Jo Dudek at HUGS 212 for details [

28 Resonances at finite volume Consider simple spin model in 3+1 D [Rummukainen & Gottlieb hep-lat/95988] S = κ φ x;ˆµ φ x φ x+ˆµ κ ρ x;ˆµ ρ x ρ x+ˆµ + g x;ˆµ ρ x φ x φ x+ˆµ Left hand case g= (no resonance); right hand case g=.21 (ρ appears as resonance in ϕϕ channel).9 P = 1.4 P = 1.2 W.7 W L L

29 Inelastic scattering In most physical scattering processes, inelastic contributions are important Ex: I= KK Derivation of Lüscher method breaks down at inelastic thresholds Various attempts to get around this Treat the system purely quantum mechanically [He, Liu et al.] Effective field theory at finite volume [Bernard et al.; Döring et al.; Briceno&Davoudi; Hansen&Sharpe] Introduces some level of systematic uncertainty (high order effects,...) Active area of research f (98)

Bethe-Salpeter wave functions An alternate way of learning about scattering is based on determination of (Nambu-)Bethe-Salpeter wavefunction [Lüscher; Lin et al.; Aoki et al.

for large x-y asymptotic behaviour of partial waves given by can be used to determine phase shift k(x y) =#h T [ (x) (y)] (k) ( k)i in ` sin( k x k(x)!

30 Bethe-Salpeter wave functions An alternate way of learning about scattering is based on determination of (Nambu-)Bethe-Salpeter wavefunction [Lüscher; Lin et al.; Aoki et al.; HALQCD] chosen interpolating operator probes content of state Satisfies Schrödinger equation for non-local BS kernel U(x,y) 1 r 2 + k 2 Z k(x) = d 3 yu(x, y) k (y) (*) 2µ Provided U(x,y)= for large x-y asymptotic behaviour of partial waves given by can be used to determine phase shift k(x y) =#h T [ (x) (y)] (k) ( k)i in ` sin( k x k(x)! A` ` /2+ `(k) k x HALQCD method: invert (*) to determine a potential by approximating U(x, y) =V (x, r) 3 (x y) = 3 (x y) V (x)+o(r 2 )] giving V (x) = 1 2µ r 2 + k 2 k(x) k( bf x) Dropping these terms renders V(x) energy dependent [perhaps weakly]

31 Bethe-Salpeter wave functions BS wavefunctions can be determined from LQCD correlation functions C(r,t)=h T [ (x + r,t) (x,t)]j () i = X n h T [ (x + r,t) (x,t)] (n) ih (n) J () i = X Z n e E nt k n (r) n t!1! Z e E t k (r) Eigen-energies contain scattering information Various extensions considered see recent review [Aoki et al ] Potentials obtained from this method are: Energy dependent (weakly in some cases see [Murano et al, ]) Only guaranteed to reproduce phase shift at E Sink dependent not an issue as observable is phase shift at measured energy Extraction of phase shift from lattice potential introduces model dependence as a functional form must be fit to finite lattice data probably a mild problem

32 Numerical Investigations

33 Two body scattering studies Meson-meson π π (I=2,1,): [CP-PACS; NPLQCD; Feng et al; HadSpec; Fu;... many others] π K (I=1/2, 3/2): [NPLQCD; Z Fu; Nagata et al; PACS-CS; Lang et al.] K K (I=1): [NPLQCD; Z Fu] Meson-baryon Five simple octet baryon octet meson channels studied [NPLQCD] J/ψ nucleon [Liu et al; Kawanai & Sasaki] Baryon-baryon Various octet baryon octet baryon scattering [HALQCD, NPLQCD] Omega (sss) Omega scattering [Buchoff, Luu & Wasem] A rapidly growing field

34 Example: I=2 ππ I=2 π π easy as no disconnected contractions Measure multiple energy levels of two pions in a box for multiple volumes and with multiple PCM bt E n=5 n=4 n=3 n=2 n=1 n= L ê b s = 32, P cm = têb m π = 39 MeV [prd]

35 Example: I=2 ππ I=2 π π easy as no disconnected contractions Measure multiple energy levels of two pions in a box for multiple volumes and with multiple PCM.24 P cm =, n=1 P cm =, n=2 bt E P cm =1, n= P cm =, n=1 P cm =1, n=1 P cm = 2, n= ê P cm =, n=1 Dashed lines are non-interacting energy levels.16 P cm =1, n=.14 P cm =, n= P cm =, n= P cm =, n= P cm =, n=.12 L ê b s = m π = 39 MeV [prd]

36 Example: I=2 ππ Input into Lüscher eigenvalue equation Allows phase shift to be extracted at multiple energies -1 Ï -2 Ï Ï Ú kcotdêmp -3 Ï Ï -4-5 Ú Ï Ú Ï L ê b s = 16 L ê b s = 2 L ê b s = 24 L ê b s = k 2 êm p m π = 39 MeV [prd]

37 Example: I=2 ππ Combine with chiral perturbation theory (low-momentum interactions turn off in the chiral limit) to interpolate to physical pion mass d HdegreesL -1-2 Ù Ï ÙÏ Ï Ù Ï Ï Ï Ï Ù Ù Ï Ù Hoogland et al '77 Cohen et al '73 Durosoy et al '73 Losty et al '74 NPLQCD '11-3 Ï Ï Ù k 2 HGeV 2 L [prd]

38 Meson-baryon scattering a + + SU(2) Also constrained in the chiral limit Study of channels with no annihilation using DWF on MILC lattices a + + (fm) pcot /m versus E/m m1 -.6 pcot /m K + p K + n K + m π =35 MeV a + (fm) m (MeV) a + SU(2) E/m m (MeV) [Torok et al ]

39 Studies by L Liu,; Kawanai &Sasaki J/Ψ-h scattering Interactions are purely gluon/sea quark effects Phenomenological studies suggest J/Ψ might bind in a nucleus through attraction of multiple nucleons Interactions are small Scattering length a [fm] η c -N J/ψ-N (J=1/2) J/ψ-N (J=3/2) M π [GeV ] amphjêy-pl m p ê f p m p ê f p amphjêy-n,spin-3ê2l

40 NN phase shifts Potential extracted for BS wave function method at E~, use to evaluate phase shift NB: energy dependence of potential neglected 8 NN-online ~ expt 1 S T lab (MeV) [N Ishii, Chiral Dynamics 212] 3 S T lab (MeV)

41 Resonance studies: rho Map out phase shift to identify a resonance Having multiple PCM frames is crucial Phase shift can be well determined at heavy quark masses PRELIMINARY 15 ρ 1 5 Γ ρ Experiment [HadSpec Collaboration] Properties of resonance requires modelling eg fit with modified Breit-Wigner Much current work addressing best way to get the most information

42 Resonance studies: Delta(1232) Delta baryon studied by QCDSF-Bonn-Jülich collaboration [Meißner QNP12] a more challenging case [see Schierholz talk tomorrow at INT program] Multiple volumes at light quark masses

colour, flavour singlet H-dibaryon (uuddss) J=I=, s=-2 most stable Bound in a many hadronic models Experimental searches H

43 H-dibaryon First QCD bound state observed in LQCD NPLQCD [PRL 16, 1621 (211)] and HALQCD [PRL 16, 1622 (211)] Jaffe [1977]: chromo-magnetic interaction between quarks hh m i 1 4 N(N 1) S(S + 1) C2 c + C 2 f most attractive for spin, colour, flavour singlet H-dibaryon (uuddss) J=I=, s=-2 most stable Bound in a many hadronic models Experimental searches H = 1 p 8 + p 3 +2 N Emulsion expts, heavy-ion, stopped kaons No conclusive evidence for or against KEK-ps (27) K - 12 C K + ΛΛ X

44 the or ensembl it is possible to interactions. Therefore, o Hbaryon dibaryon in QCD der in the expo ensemble (m L32= 5.7 e-volume limit to (m behaviour L =of two-point 7.71) are us from large Euclidean time Extract energy eigenstatesensemble d state. correlators Writing conclude that the lore regarding fi X fted Cin(t) energy in = h (x, t) () i!in Z eparticular, that m L> 2 C (t) e R(t) =! Ze nce of the bindc (t) X ume C (t) = h (x, t) () i! Z e e ects to beenegligibly smal M t follows directly M the study of multi-baryon system Correlator ratio allows direct access to energy shift Lu scher s method assumes t 3. fm 4. fm energy-momentum relation is sat 2 L +..., (3) calculated energy eigenvalues. In j 5, 2 dof , fit j 3, b M x M t t!1.44 SS.15 PP t! E t M MeV E MeV b M 323 t!1.221 j 4, 2 dof , fit dof , fit b M b E M MeV b E 243 b M E t b E x t b.3 j 3, 2 dof , fit 22 j 3, t b t b b E dof , fit fit 21 j 6, 2 dof ,.2 4 k b E E t b k2.1. E k2 b E E MeV. k2 b E 243 E MeV t b 3 3 t b

Simple extrapolations 1. After volume extrapolation H bound at unphysical quark masses Quark mass extrapolation is uncertain and unconstrained -i cothdl.6.2 3. fm 4. fm B quad H B lin H = +11.5 ± 2.

45 Simple extrapolations 1. After volume extrapolation H bound at unphysical quark masses Quark mass extrapolation is uncertain and unconstrained -i cothdl fm 4. fm B quad H B lin H = ± 2.8 ± 6. MeV =+4.9 ± 4. ± 8.3 MeV Hq êm p L 2 Other extrapolations, see [Shanahan, Thomas & Young PRL. 17 (211) 924, Haidenbauer & Meissner ] Suggests H is weakly bound or just unbound More study required at light masses * 23 MeV point preliminary (one volume)

46 Deuteron and Dineutron Deuteron, di-neutron also investigated NPLQCD PACS-CS More work needed at lighter masses Also s=-4 ΞΞ dibaryon is found to be bound [Yamazaki et al ] ΔE( 3 S 1 ) [GeV] experiment Fukugita et al. [7] NPLQCD mixed [8] Aoki et al. [11] PACS-CS V oo [2] NPLQCD 2+1f V oo [3] NPLQCD 3f V max [4] This work 2+1f V oo m π 2 [GeV 2 ] ΔE( 1 S ) [GeV] Fukugita et al. [7] NPLQCD mixed [8] Aoki et al. [11] PACS-CS V oo [2] NPLQCD 2+1f V oo [3] NPLQCD 3f V max [4] This work 2+1f V oo

47 Deuteron and Dineutron Deuteron, di-neutron also investigated NPLQCD PACS-CS More work needed at lighter masses Also s=-4 ΞΞ dibaryon is found to be bound [Yamazaki et al ] ΔE( 3 S 1 ) [GeV] experiment Fukugita et al. [7] NPLQCD mixed [8] Aoki et al. [11] PACS-CS V oo [2] NPLQCD 2+1f V oo [3] NPLQCD 3f V max [4] This work 2+1f V oo m π 2 [GeV 2 ] ΔE( 1 S ) [GeV] Fukugita et al. [7] NPLQCD mixed [8] Aoki et al. [11] PACS-CS V oo [2] NPLQCD 2+1f V oo [3] NPLQCD 3f V max [4] This work 2+1f V oo

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