GLUEBALL SPECTRUM FROM AN ANISOTROPIC . . . PHYSICAL REVIEW D 60 034509

TABLE VII. Final continuum-limit glueball mass estimates TABLE VIII. Glueball mass ratios. mG . When a unique J interpretation for a state cannot be made, the other possibilities are indicated in the second column. States whose m(2ϩϩ)/m(0ϩϩ) 1.39(4) interpretation requires further study are indicated by a dagger. In m(0Ϫϩ)/m(0ϩϩ) 1.50(4) column 3, the first error is the statistical uncertainty from the m(0*ϩϩ)/m(0ϩϩ) 1.54(11) continuum-limit extrapolation and the second is the estimated un- m(1ϩϪ)/m(0ϩϩ) 1.70(5) certainty from the anisotropy. In the final column, the first error m(2Ϫϩ)/m(0ϩϩ) 1.79(5) comes from the combined uncertainties in r m , the second from 0 G m(3ϩϪ)/m(0ϩϩ) 2.06(6) the uncertainty in rϪ1ϭ410(20) MeV. 0 m(0*Ϫϩ)/m(0ϩϩ) 2.11(6) Ϫϩ ϩϩ PC m(0 )/m(2 ) 1.081(12) J Other Jr0mG mG (MeV) 0ϩϩ 4.21 ͑11͒͑4͒ 1730 ͑50͒͑80͒ ϩϩ 2 5.85 ͑2͒͑6͒ 2400 ͑25͒͑120͒ in Table VIII are calculated using the empirical fact that Ϫϩ 0 6.33 ͑7͒͑6͒ 2590 ͑40͒͑130͒ correlations between different symmetry channels were ϩϩ † 0* 6.50 (44)(7) 2670 ͑180͒͑130͒ found to be negligible. Note that the pseudoscalar glueball is ϩϪ 1 7.18 ͑4͒͑7͒ 2940 ͑30͒͑140͒ clearly resolved ͑at the 7␴ level͒ to be heavier than the ten- Ϫϩ 2 7.55 ͑3͒͑8͒ 3100 ͑30͒͑150͒ sor. ϩϪ 3 8.66 ͑4͒͑9͒ 3550 ͑40͒͑170͒ All of the glueball states shown in Fig. 8 are stable against Ϫϩ 0* 8.88 ͑11͒͑9͒ 3640 ͑60͒͑180͒ decay to lighter glueballs. In the PCϭϩϩ sector, the ϩϩ 3 6,7,9,... 8.99 ͑4͒͑9͒ 3690 ͑40͒͑180͒ threshold for decay into two identical 0ϩϩ glueballs having ϪϪ2020 Fall, GPPU Progress report 1 3,5,7,... 9.40 ͑6͒͑9͒ 3850 ͑50͒͑190͒ zero total momentum is twice the mass of the scalar glueball. Ϫϩ † 2* 4,5,8,... 9.50 (4)(9) 3890 ͑40͒͑190͒ Although this lies below the mass of the 3ϩϩ glueball, Bose ϪϪ 2 3,5,7,... 9.59 ͑4͒͑10͒ 3930 ͑40͒͑190͒ symmetrization prohibits odd L partial waves, where L is the ϪϪ Keita Sakai, Nuclear theory group 3 Progress6,7,9,... 10.06 ͑21͒͑10 ͒ in4130 my͑90͒͑200͒ glueballrelative orbital angular momentum,study so that the 3ϩϩ glueball ϩϪ 2 5,7,11,... 10.10 ͑7͒͑10͒ 4140 ͑50͒͑200͒ cannot decay into two identical scalar glueballs. In the PC 0ϩϪ 4,6,8,... 11.57 ͑12͒͑12͒ 4740 ͑70͒͑230͒ ϭϪϩ sector, the lowest-lying two-glueball state consists of 0ϩϩ and 2ϩϩ glueballs in a relative P wave; all of our What I want to know glueballs… in thisOrigin sector have masses of below glueball the sum of the mass ͓2,3,17,18͔ to the continuum limit. Several glueball mass ra- scalar and tensor glueball masses. States of total zero mo- tios are presented in Table VIII. We can determine these mentum and comprised of the 0ϩϩ and 1ϩϪ glueballs with ratios very accurately since, as noted earlier, they are not relative orbital angular momentum L are the lowest-lying contaminatedGlueball by anisotropy errors. The… uncertainties hadron given two-glueball consisted states in the PC ϭϩϪof sectorgluons when L is even only and in the PCϭϪϪ sector when L is odd. Only the 0ϩϪ glueball has sufficient mass to decay into two such glueballs; however, this decay is forbidden because Lϭ1 is required to make a state of zero total angular momentum. To convert our glueball masses into physical units, the Latticevalue of the hadronic QCD scale r0predictsmust be specified. We its used mass Ϫ1 r0 ϭ410(20) MeV from Ref. ͓1͔ to obtain the scale shown on the right-hand vertical axis of Fig. 8. This estimate was obtained by combining Wilson action calculations of a/r0 with values of the lattice spacing a determined using Whatquenched simulation makes results of various this physical quantities, mass? such as the masses of the ␳ and ␾ mesons, the decay con- stant f ␲ , and the 1P-1S splittings in charmonium and bot- tomonium. Note that the errors shown in Fig. 8 do not in- Ϫ1 clude the uncertainty in r0 . For the lowest-lying glueballs, we obtain m(0ϩϩ)ϭ1730(50)(80) MeV and m(2ϩϩ) ϭ2400(25)(120) MeV, where the first error comes from the C. Morningstaruncertainty in r0mG andand the secondM. Peardon, error comes from Phys. the Rev. D60, Ϫ1 uncertainty in r0 . A great deal of care should be taken in FIG. 8. The mass spectrum of glueballs in the pure SU͑3͒ gauge making direct comparisons with experiment since these val- Known mass generationues neglect mechanism the effects of light quarks and mixings with theory. The masses are given in terms of the hadronic scale r0 along the left vertical axis and in terms of GeV along the right vertical nearby conventional mesons. It is this mixing which has axis ͑assuming rϪ1ϭ410 MeV). The mass uncertainties indicated made the search for an incontrovertible experimental signal •Higgs0 Mechanism PC by the vertical extents of the boxes do not include the uncertainty in so difficult. A glueball having exotic J will not mix with not the case setting r0. The locations of states whose interpretation requires fur- conventional hadrons and would be ideal for establishing the ther study•Quark are indicated by the condensate dashed open boxes. (Nambuexistence of glueballs. Mechanism) Unfortunately, our results indicate

034509-11

New mechanism Better understanding of hadron masses Mass measurement in lattice QCD

•From asymptotic behavior of 2pt function Normal way •As a expectation value of EMT operator Need to do

To do the 2nd way: “Good” glueball 2pt func.

† −M τ G(τ + 1) Effective mass ⟨G(τ)G (0)⟩ ≃ e G , Meff(τ) = − log With large t G(τ)

Contribution of excited states “Good” means…

Meff(τ) • Less Plateau • Long m τ To get good 2pt func.

We proposed a new technique: “Spatial flow method” Result Effective mass with spatial flow

Ours (500 flow) By using spatial flow, 0.6 Bali et al. About 2x 0.5 better precision

Effective mass 0.4 Long plateau like behavior 0.3 0 1 2 3 4 5 6 7 Separation time

Plans of research in next few years

1. This year and next year •Study about spatial flow method almost finishing •Glueball mass calc. from EMT

2. After that

Full QCD study about glueball mass Seminar points

GSP:0 Not good… GASP:2

Overseas studies

Symposium etc.

• The 38th International Symposium on lattice field theory (July 25-31, 2021) • The 14th Quark Confinement and the Hadron Spectrum (Aug. 2-7, 2021)

Joint research

(As candidates,) •The University of Stavanger (Norway) •The University of Adelaide (Australia) •The University of Kentucky (U.S.)

(I will decide about this more specifically after submitting my paper…)