η’ meson mass from lattice QCD
External Review on CCS, 2007/10/31
T.Yoshie, CCS, Tsukuba

CP-PACS/JLQCD Collaborations

1. Introduction
2. Configurations
3. Method
4. Results
5. Summary and Future WorkPoS(LAT2006)204

Introduction• U(1) problem : one of outstanding issues in light
hadron spectroscopy– η’ (960MeV) is much heavier thanπ(140MeV)– naive expectation (1975,S.Weinberg)
mh ' = 960 MeV < 3mp = 234MeV
mh + mh ' = 1510 MeV ¹ 2mK = 990MeV– QCD: topological vacuum structure, axial U(1) anomaly• Deriving the η’ mass from first principles of QCD– give solution of the problem– demonstrate departure of QCD from models• Lattice QCD calculations– in quenched QCD ~< 2000
• importance of disconnected (2-loop) loop diagram• relation with topological structure• mass calculation, theoretically not consistent– in two-flavor full QCD ~< 2005
• large mass of η’ is demonstrated in e.g.
CP-PACS Collab. “Flavor singlet meson mass in the continuum
limit in two-flavor lattice QCD'', Phys. Rev. D67 (2003) 074503– Nf=2+1 simulation is essential for quantitative studies• ms >> mu = md mixing of singlet and octet PS
leads to physical η and η’• first study in 2+1 flavor full QCD, taking account ofmixingConfigurations• a set of 2+1 flavor full QCD configurations from a joint
project of CP-PACS/JLQCD collaborations• Actions– Iwasaki RG gauge action– Non-perturbatively O(a) improved Wilson quark action• Parameters– β=1.83 a=0.122fm (coarsest of three lattice spacings we have)– 16^3x32 L=2 fm– 10 combinations of (Kud, Ks)
Kud = 0.13655, 0.13710, 0.13760, 0.13800, 0.13825
mPS/mV = 0.78 – 0.61 (for Light-Light)
Ks = 0.13710 and 0.13760
mPS/mV = 0.75, 0.70
physical strange quark mass mPS/mV = 0.73
– 7000 – 8000 traj for each (Kud, Ks), measurement at each 10 traj

Method• numerical challenge– disconnected loop diagram– extract mass from G(t) at small t– η’ as an excited state• a combination of known techniques– Stochastic Noise Estimator technique– Smearing method （Coulomb gauge)– Propagator matrix diagonalization(variational method)• smearing kernel   K (n - m) = A exp(- B | n - m |) n ¹ m

K (0) = 1, A = 0.12, B = 0.10
– point sink
– with doubly smeared source,a good estimate of octet PS meson
masses at relatively small time slices

typical effective mass plot• stochastic noise for 2-loop diagram
prepare U(1) random numbers for allsites exp(iq (n ,t )) and solve quark
propagator q (n, t )   x (t src ) = å exp(-iq (l , t src )) K (l - m) K (m - n )q(n , t src )– l ,m,nz (t snk ) = å exp(-iq (n , t snk ))q (n , t snk )

P P

P P

S SS Sn1
G2 -loop (t ) = 3
áTr (g 5z (t snk ))Tr (g 5x (t src ))ñ noiseå
L T t =t snk -t src– 20 sets of stochastic noise for eachcolor and spin 240 prop. on a config.
– all information of configuration• projection to zero momentum made at bothsource and sink
• average over time slice, time difference of
two loops fixed• statistical error of 2-loop diagram– statistical error consists of1) stochastic noise2) configuration fluctuations
Error of 2-loop diagram for all configurations (800),
bin=(10 conf, i.e. 100 traj.)– as increase of #noise, error reaches plateaus– error are (almost) dominated by configuration
fluctuations• propagator matrix and mass extraction– prepare flavor SU(2)-singlet and strange PS operators

h n = (ug 5u + dg 5 d ) / 2

h s = ( sg 5 s )– calculate 2x2 correlators

æh n (t )h n (0) h n (t )h s (0) ö÷÷G (t ) = ççèh s (t )h n (0) h s (t )h s (0) ø– normalize the correlator matrix at some reference time slice t0 =4,
and diagonalize it to extact masses of the ground stateη and thefisrt excited stateη’:-1-1

C G (t )G (t0 ) C » d ij exp(-mi (t - t0 ))æh ö æ h öC çç n ÷÷ = çç ÷÷èh s ø èh ' ø– assuming that the ground and 1st excited states dominate at td =3,
C is estimated from diagonalization at td

– diagonalize at t=td=3– diagonal parts decay exponentially– off-diagonal parts are negligible for all t– η η’ contribution dominates our propagators already at t~2

diagonalized propagator

effective mass plotKud = 0.13655, 0.13710, 0.13760, 0.13800, 0.13825– determine η mass from fit with t=3-5– estimate η’ mass from fit with t=2-4

Results
Ks = 0.13710

Ks = 0.13760

m h=A+Bud m ud+Bs m s+Dud m 2ud
m ud=(1/Kud-1/Kc)/2m h'=A'+Bud' m ud+Bs'm sm s=(1/Ks-1/Kc)/2

• masses at the physical point

mh'=0.871(46) GeV

exp. 0.960 GeV

mh =0.545(16) GeV

exp. 0.550 GeV(input p r K)

– η mass: consistent with experiment– η’ mass: much heavier than octet masses
smaller than experiment only by 100MeV (2 sigma)• comparison with another work
– similar work by UKQCD Collab. with KS action– η’ ~ 0.99 GeV, η~ 0.60 GeV (an estimate fromvariational propagator at a simulation point)Summary and Future Work• η and η’ masses are estimated in Nf=2+1 lattice QCD,
albeit at one lattice spacing• Mass of η’ larger than octet PS masses,
only 100 MeV below experiment• Importance of U(1) contribution demonstrate• lighter quarks, finer lattices toward a quantitative solution• reduce computational cost
– implement an efficient SET algorithm with e.g. a variancereduction, truncated eigen-mode ex., domain-decomposition ….

• remove systematic errors from excited states– fine-tune smearing function– use more operators