Mass of Heavy Mesons from Lattice QCD

Transcription

1 Mass of Heavy Mesons from Lattice QCD David Richards Jefferson Laboratory/Hadron Spectrum Collaboration Temple, March 2016

2 Outline Heavy Mesons Lattice QCD Spectroscopy Recipe Book Results and insight Light-meson spectroscopy and isoscalar The search for Glue Heavy-quark States Summary

3 Heavy Mesons I will adopt a liberal interpretation: States high in the spectrum - gluonic excitations! States with exotic structures: four-quark, hybrid, Mesons with Heavy Constituents (c,b, etc..) q 3 > q 2 q>

4 Meson Spectrum S2 L S1 Simple quark model (for neutral mesons) admits only certain values of J PC P = ( 1) l+1 C = ( 1) l+s Exotic Mesons are those whose values of J PC are in accessible to quark model: 0 +-, 1 -+, 2 +- Multi-quark states: Hybrids with excitations of the flux-tube Study of hybrids: revealing gluonic degrees of freedom of QCD. Glueballs: purely, or predominantly, gluonic states

5 Variational Method Subleading terms Excited states Construct matrix of correlators with judicious choice of operators C ij (t, 0) = 1 V 3 X Z N i ~x,~y ho i (~x, t)o j (~y, 0)i = X N hn O i (0) 0i Delineate contributions using variational method: solve C(t)v (N) (t, t 0 )= N (t, t 0 )C(t 0 )v (N) (t, t 0 ). N (t, t 0 )! e E N (t t 0 ), Zi N Zj N 2E N e E N t Eigenvectors, with metric C(t 0 ), are orthonormal and project onto the respective states v (N 0 ) C(t 0 )v (N) = N,N 0 Z N i = p 2m N e m N t 0 /2 v (N) j C ji (t 0 ).

6 Glueball Spectroscopy - I Morningstar, Peardon 97,99 M 2 M E M T2 Observe emergence of degeneracies a 2

7 Glueball Spectrum - II In QCD with quarks, glueballs mix with two-pi and conventional quark states - not a smoking gun for gluonic excitations!

8 Glueballs in Dynamical LQCD Glueballs high in the spectrum Gregory et al, JHEP10 (2012) 170

9 Spectroscopy with Quarks Anisotropic lattices - to precisely resolve energies Variational method - with sufficient operator basis to delineate states Identification of spin - Many Values of Lattice Spacing? Anisotropic fermion action Edwards, Joo, Lin, PRD78 (2008) 8 < S G [U] = Nc g : X x,s>s 0 S F [U,, ] = X x (x) 1 ũ t ( 1 2 " 1 2 apple 5 3u 4 P ss 0 s g f ũ t ũ 2 s 1 12u 6 s R ss 0 ũ t ˆm 0 + Ŵt + 1 X f X s s Ŵ s + X x,s ts ˆFts + 1 f 1 ũ 3 s apple X 4 3u 2 su 2 t s<s 0 ss 0 ˆFss 0 1 P st 12u 4 su 2 R st t #) (x). Two anisotropy parameters to tune, in gauge and fermion sectors 9 = ; =3.5 g = 0 f = 0 / Dispersion Relation a s ' 0.12 fm a t ' fm

10 Specifying Quark Masses Tuning performed for three-flavor theory Challenge: setting scale and strange-quark mass Lattice coupling fixed Proportional to ms to LO ChPT Omega Express physics in (dimensionless) (l,s) coordinates H-W Lin et al (Hadron Spectrum Collaboration), PRD79, (2009 ) Proportional to ml to LO ChPT

11 Variational Method: Meson Operators Aim: interpolating operators of definite (continuum) JM: O JM Starting point Introduce circular basis: h0 O JM J 0,M 0 i = Z J J,J 0 M,M 0 ( x, t) Di D j... ( x, t)! D m= 1 = i p 2!D x! D m=0 = i! D z! D m=+1 = ( D [1] J=1 )J,M = X i! D y pi!d 2 x + i D! y. Straighforward to project to definite spin - for example J = 0, 1, 2 m 1,m 2 1,m1 ;1,m 2 J, M m1! D m2.

12 Isovector Meson Spectrum - I Dudek et al, PRL 103: (2009) Isovector spectrum with quantum numbers reliably identified Nf = 3 theory - three mass-degenerate strange quarks { Exotic

13 Interpretation of Meson Spectrum Z N i = p 2m N e m N t 0 /2 v (N) j C ji (t 0 ). D [2] J=1 Vanishes for unit gauge field In each Lattice Irrep, state dominated by operators of particular J

14 rd excited state is dominantly hybrid? with some look at the overlaps nd excited state is dominantly with some st excited state is dominantly with some hybrid? Anti-commutator of covariant derivative: vanishes for unit gauge! ground state is dominantly Use lattice QCD to build phenomenology of bound states Dudek, arxiv:

15 Isoscalar Meson Spectrum negative parity positive parity exotics isoscalar isovector YM glueball Diagonalize in 2x2 flavor space C = C +2D 2 D s Dudek et al, arxiv: , arxiv: D s C ss + D ss. J. Dudek et al., PRD73, 11502

16 Baryon Operators Aim: interpolating operators of definite (continuum) JM: O JM Lattice does not respect symmetries of continuum: cubic symmetry for states at rest Starting point Introduce circular basis: h0 O JM J 0,M 0 i = Z J J,J 0 M,M 0! D m= 1 = i p 2!D x! D m=0 = i! D z! D m=+1 = i! D y pi!d 2 x + i D! y. Straighforward to project to definite spin: J = 1/2, 3/2, 5/2

17 Interpolating Operators Examine overlaps onto different NR operators, i.e. containing upper components of spinors: ground state has substantial hybrid component

18 Hybrid Baryon Spectrum hybrid operators of form D [2] l=1,m 3.0 Hybrid: gluons structural 2.5 No exotic baryons R.G.Edwards et al., arxiv: Dudek, Edwards, arxiv:

19 Putting it Together Subtract ρ Subtract N Common mechanism in meson and baryon hybrids: chromomagnetic field with Eg GeV

20 Heavy-Quark States

21 Charmonium Operator construction follows light-quark Liuming Liu et al, arxiv: Ignore annihilation contributions Charm quark mass set from ηc with scale set using Ω D s D s Exotics atm 0.65 DD Volume-dependence small quote results at larger volume

22 Charmonium - II D-wave Ds Ds Appearance of multiplets from n 2s+1 LJ quark potential model M-Mhc HMeVL P-wave DD 500 S-wave Hybrid supermultiplet Z 6 4 1,3 Hyb 8p NR, r NR < D J=1 1,3 Hyb J M-Mhc HMeVL D s D s DD HT 2 L 2 -+ HEL

23 Precision Spectroscopy Durr et al., BMW Collaboration Science 2008 Control over: Quark-mass dependence Continuum extrapolation finite-volume effects (pions, resonances)

24 Twisted-mass Nf = Tuning quark masses Cichy et al, arxiv:

25 Ambiguity: breaking of rotational symmetry

26 Charmonium Predictions

27 Esposito et al, arxiv: Experimental status of spectrum STATES ABOVE THRESHOLD

28 Momentum-dependent I = 2 ππ Phase Shift Include two-body operators O, ( p ) = X Operator basis m Total momentum zero - pion momentum ±p 0.40 S,m, X ˆp Dudek et al., Phys Rev D83, (2011) Y m (ˆp) O (p)o ( p) 0.35 Luescher: energy levels at finite volume phase shift at corresponding k

29 Reinventing the quantum-mechanical wheel Thanks to Raul Briceno L (in 1+1 dimensions) (x) e ipx Periodicity: Lp n =2 n

30 Reinventing the quantum-mechanical wheel Two particles:

31 Reinventing the quantum-mechanical wheel Two particles: p* x

32 Reinventing the quantum-mechanical wheel Two particles: infinite volume scattering phase shift p* x (x) e ip x +i2 (p ) Spectrum: Asymptotic wavefunction

33 Reinventing the quantum-mechanical wheel Two particles: infinite volume scattering phase shift p* x (x) e ip x +i2 (p ) Spectrum: Asymptotic wavefunction Periodicity: Lp n +2 (p n)=2 n

34 Reinventing the quantum-mechanical wheel Lp n +2 (p n)=2 n p [MeV] [degrees] n=3 n=2 n=1 n=0 L[fm] p [MeV]

35 Reinventing the quantum-mechanical wheel Lp n +2 (p n)=2 n p [MeV] [degrees] n=3 n=2 n=1 n=0 L[fm] p [MeV]

36 Reinventing the quantum-mechanical wheel Lp n +2 (p n)=2 n p [MeV] [degrees] n=3 n=2 n=1 n=0 L[fm] p [MeV]

37 Reinventing the quantum-mechanical wheel Lp n +2 (p n)=2 n p [MeV] [degrees] n=3 n=2 n=1 n=0 L[fm] p [MeV]

38 Reinventing the quantum-mechanical wheel Lp n +2 (p n)=2 n p [MeV] [degrees] n=3 n=2 n=1 n=0 L[fm] p [MeV]

39 I=2 and Resonant I = 1 ππ Phase Shift det h e 2i (k) U k L 2 i = Matrix in l lattice irrep -20 Dudek et al., Phys Rev D83, (2011); arxiv: Feng, Renner, Jansen, PRD83, PACS-CS, PRD84, Alexandru et al Lang et al., PRD84, Dudek, Edwards, Thomas, Phys. Rev. D 87, (2013)

40 Inelastic in ππ KK channel Wilson, Briceno, Dudek, Edwards, Thomas, arxiv: Decreasing Pion Mass Inelastic Threshold O, ( p ) = X m S,m, X ˆp Y m (ˆp) O (p)o ( p)

41 det h First - and Successful - inelastic ij JJ 0 + i i t (J) ij (E cm) i P JJ 0 + im ~ JJ (p 0 i L) =0 Parametrized as phase shift + inelasticity t ii = ( e2i i 1) 2i i,t ij = p 1 2 e i( i + j ) 2 p i j Dudek, Edwards, Thomas, Wilson, PRL, PRD

42 Resonances in Charmonium - I Lang et al, arxiv: Click to edit Master text styles Energies from Variational Method C(t)v (N) (t, t 0 )= N (t, t 0 )C(t 0 )v (N) (t, t 0 ).

43 DDbar in p-wave

44 DDbar in p-wave - II

45 Energy levels obtained from variational analysis DDbar in S wave

46 Take-Away Precision calculations of charmonium spectrum, with predictions First calculations of scattering amplitudes for states above threshold Important take-home message for workshop. Gluonic excitations key component of both excited nucleon and excited meson spectrum, with a common mechanism Can we formalize the definition of gluon hybrids? Form Factors?