Gluonic observables and the scalar spectrum of twelve-ﬂavour QCD · 2013. 8. 5. · Gluonic observables and the scalar spectrum of twelve-ﬂavour QCD Enrico Rinaldi on behalf of

Gluonic observables and the scalar spectrum of twelve-flavour QCD

Enrico Rinaldi

on behalf of the LatKMI collaboration

Higgs Centre for Theoretical Physics, School of Physics and Astronomy University of Edinburgh

Lattice 2013, Mainz

Kobayashi-Maskawa Institutefor the Origin of Particles and the Universe素粒子宇宙起源研究機構

素粒子宇宙起源研究機構

Kobayashi-Maskawa Institute for the Origin of Particles and the Universe




LatKMI collaboration

Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Miura, K.-i. Nagai, H. Ohki,

E. Rinaldi, K. Yamawaki, T. Yamazaki

A. Shibata







Motivations

• a Higgs boson has been found at LHC

• a “composite” Higgs-like scalar can also reproduce the LHC data

• this is realised in Walking Technicolor models

• interest in finding the lower edge of the Conformal Window

• a light scalar appears as pseudo-NG boson

ATLAS, PLB716 (2012) 1-29

CMS, PLB716 (2012) 30-61

Motivations

• a Higgs boson has been found at LHC

• a “composite” Higgs-like scalar can also reproduce the LHC data

• this is realised in Walking Technicolor models

• interest in finding the lower edge of the Conformal Window

• a light scalar appears as pseudo-NG boson

_(µ)

µ

_(µ)

µ

_(µ)

µ

Nf

QCD-like

“walking”

conformal

Lattice simulations of twelve-flavour QCD

• SU(3) gauge field + 12 dynamical fermions

• gauge action → tree level Symanzik

• fermion action → HISQ (Highly Improved Staggered Quarks)

• modified HMC based on MILC code (v7)

• fixed gauge coupling → β = 4.0

• four volumes → V=183, 243, 303, 363

• six bare fermion masses → amf = 0.05 - 0.16

• long trajectories → >15000 MD units

L T amf confs.18 24 0.06 309018 24 0.08 525018 24 0.10 513018 24 0.12 520018 24 0.16 420024 32 0.05 944024 32 0.06 1400024 32 0.08 1534024 32 0.10 948030 40 0.05 750030 40 0.06 1400030 40 0.08 1145030 40 0.10 500036 48 0.05 510036 48 0.06 6000

Glueball spectroscopy

• the glueball spectrum is well studied in SU(N) Yang-Mills theories

• the lightest glueballs are starting to be investigated in full LQCD

• this talk focus on many-flavour QCD

• typically heavy mass, determined by ΛQCD

• typically noisy and require large statistic

21

TABLE XXI: The final glueball spectrum in physical units.In column 2, the first error is the statistical uncertainty com-ing from the continuum extrapolation, the second one is the1% uncertainty resulting from the approximate anisotropy.In column 3, the first error comes from the combined uncer-tainty of r0MG, the second from the uncertainty of r!1

0 =410(20) MeV

JPC r0MG MG (MeV)0++ 4.16(11)(4) 1710(50)(80)2++ 5.83(5)(6) 2390(30)(120)0!+ 6.25(6)(6) 2560(35)(120)1+! 7.27(4)(7) 2980(30)(140)2!+ 7.42(7)(7) 3040(40)(150)3+! 8.79(3)(9) 3600(40)(170)3++ 8.94(6)(9) 3670(50)(180)1!! 9.34(4)(9) 3830(40)(190)2!! 9.77(4)(10) 4010(45)(200)3!! 10.25(4))(10) 4200(45)(200)2+! 10.32(7)(10) 4230(50)(200)0+! 11.66(7)(12) 4780(60)(230)

In the tensor channel, the glueball matrix element isextrapolated to 1.0±0.2 GeV3 in the continuum, which isthe average of results of E and T2 channels. In the calcu-lation, it is found that in the lattice spacing range we use,the glueball mass and matrix elements are approximatelyindependent of the lattice spacing, this implies that thelattice artifacts might be neglected here. If the renor-malization constant ZT ! 0.52(15) of the tensor operatordoes not change much in the range of lattice spacing andapplies to all the ! values in this work, the renormalizedmatrix element of tensor operator is 0.52 ± 0.19 GeV3,which is in agreement with the prediction 0.35 GeV3 fromthe tensor dominance model [17] and QCD sum rule [18]for the tensor mass around 2.2 GeV.

VII. Conclusion

The glueball mass spectrum and glueball-to-vacuummatrix elements are calculated on anisotropic lattices inthis work. The calculations are carried out at five latticespacings as’s which range from 0.22 fm to 0.10 fm. Dueto the implementation of the improved gauge action andimproved gluonic local operators, the lattice artifacts arehighly reduced. The finite volume e!ects are carefullystudied with the result that they can be neglected on thelattices we used in this work.

As to the glueball spectrum, we have carried out cal-culations similar to the previous work [4] on much largerand finer lattices, so that the liability of the continuumlimit extrapolation is reinforced. Our results of the glue-

ball spectrum is summarized in Tab. XXI and Fig. 16.After the non-perturbative renormalization of the local

gluonic operators, we finally get the matrix elements ofscalar(s), pseudoscalar(p), and tensor operator (t) with

0

2

4

6

8

10

12

--+--+++ 0

1

2

3

4

5

r 0 M

G

MG

(GeV

)

0++

2++

3++

0-+

2-+

0+-

1+-

2+-

3+- 1--2--3--

FIG. 16: The mass spectrum of glueballs in the pure SU(3)gauge theory. The masses are given both in terms of r0

(r!10 = 410 MeV) and in GeV. The height of each colored

box indicates the statistical uncertainty of the mass.

the results

s = 15.6 ± 3.2 (GeV)3

p = 8.6 ± 1.3 (GeV)3

t = 0.52 ± 0.19 (GeV)3, (62)

where the errors of s and t come mainly from the errorsof the renormalization constants ZS and ZT . The moreprecise calculation of ZS and ZT will be carried out inlater work.

Acknowledgments

This work is supported in part by U.S. Departmentof Energy under grants DE-FG05-84ER40154 and DE-FG02-95ER40907. The computing resources at NERSC(operated by DOE under DE-AC03-76SF00098) and SC-CAS (Deepcomp 6800) are also acknowledged. Y. Chenis partly supported by NSFC (No.10075051, 10235040)and CAS (KJCX2-SW-N02). C. Morningstar is also sup-ported by NSF under grant PHY-0354982.

[1] B. Berg and A. Billoire, Nucl. Phys. B221, 109 (1983). [2] G. Bali, et al. (UKQCD Collaboration), Phys. Lett. B

Chen et al., PRD73(2006)

0

0.5

1

1.5

2

2.5

3

am

Ground StatesExcitations

A1++ A1

-+ A2+- E++ E-+ T1

+-T1++ T1

-+ T2++ T2

--T2-+

Lucini, Rago, ER, JHEP08(2010)







Gregory, Irving, Lucini, McNeile, Rago, Richards, ER, JHEP10(2012)







Gregory, Irving, Lucini, McNeile, Rago, Richards, ER, JHEP10(2012)

Glueball spectroscopy: operators

• eigenstates of the Hamiltonian are classified according to the irreducible representations of the cubic group

• suitable gauge-invariant operators must be constructed that respect the symmetries

• vacuum contributions must be subtracted in the scalar case

• improved operators are obtained by blocking and smearing algorithms Lucini, Rago, ER

JHEP08(2010)

JHEP08(2010)119

J A1 A2 E T1 T2

0 1 0 0 0 0

1 0 0 0 1 0

2 0 0 1 0 1

3 0 1 0 1 1

4 1 0 1 1 1

Table 1. Subduced representations J ! GO of the octahedral group up to J = 4. This tableillustrates the spin content of the irreducible representations of GO in terms of the continuum J .

3 Extracting glueball masses

In this section we present the construction of our operators and we review the general

methodology for extracting glueball masses. While the standard variational procedure and

the construction of operators in irreducible representations of the cubic lattice group is well

known, this is, up to our knowledge, the first systematic attempt of inserting scattering

and torelon operators into the variational set, in order to rule out from the spectrum

contributions of these spurious states.

Symmetries of the lattice spectrum. At finite lattice spacing a the continuum rota-

tion group is not an exact symmetry of the system. The full continuum rotational symmetry

is dynamically restored only when a " 0. On the lattice, eigenstates of the Hamiltonian

have to fall into the irreducible representations of the octahedral point group GO, the sym-

metry group of the cube. The octahedral point group has 5 irreducible representations A1,

A2, E, T1 and T2 respectively with dimensions 1, 1, 2, 3, 3.

Since we are interested in the glueball spectrum of the gauge theory in the continuum,

we need to consider GO as a subgroup of the complete rotation group SO(3): irreducible

representations of SO(3) are decomposed in terms of those of GO. Irreducible representa-

tions of integer spin J in SO(3) restricted to GO are referred to as subduced representations

J ! GO. When considered as a representation of GO, the (2J +1) degeneracy of the contin-

uum spin J state is split onto di!erent irreducible representations of GO. A simple example

of this kind of pattern is the spin 2 (tensor) glueball, whose 5 polarisations are seen on

the lattice as di!erent states, 2 in the E and 3 in the T2 representation of GO. Due to

the breaking of continuum rotational symmetry on the lattice, the aforementioned pattern

of degeneracies is exact in the limit a " 0, but it is only approximate at finite a. Com-

paring the measured glueball spectrum with the expected pattern of degeneracy can give

information on the relevance of lattice artifacts.

Near the continuum limit, it is possible to identify the masses of spin J glueballs by

matching the patterns of degeneracies of the subduced representations J ! GO from the

degeneracy coe"cients. We report these coe"cients up to J = 4 in table 1. For any given

operator O on the lattice, we define a rotation transformation as Ri(O) where the index i

labels all the elements of the group GO.

Since a generic representation of the group will not be irreducible, in order to create

states that transform only in a given symmetry channel, we will need to create an appro-

– 4 –

{A1(1), A2(1), E(2), T1(3), T2(3)}

OG

(t) =1

L3

X

x2L

3

Tr

0

@Y

l2W(x)

Ul

1

A O(R)G (t) =

24X

↵=1

a(R)↵ R↵ [OG(t)]

O(A1)(t)� h0|O(A1)|0i

• basis of operators ➔

• matrix of correlators ➔

• generalised eigenvalue problem ➔

• ground state correlator fit (α=0) ➔

Glueball spectroscopy: variational analysis

Cij(t) =X

⌧

h0|O†i (⌧ + t)Oj(⌧)|0i

{O1(t), . . . ,On(t)}

Cij(t)v↵j = �↵v↵i

h�†↵(t)�↵(0)i = |c↵|2

⇣e�m↵t + e�m↵(T�t)

⌘

�↵(t) =nX

i=1

v↵i Oi(t)

• the effective mass plateaux is used to determine the fitting window on the correlator

Example: scalar glueball effective mass

The fitted mass is shown on top of the corresponding effective mass

enabling us to establish the goodness of the estimate

0 1 2 3 4 5t/a

0

0.2

0.4

0.6

0.8

1

am

eff

L=18 #5130 cfgs. 171 bins

L=24 #9480 cfgs. 237 bins

L=30 #5000 cfgs. 100 bins

Scalar glueball effective mass !=4.0 amf=0.10

Finite-size effects appear to be negligible at larger fermion masses

Results: scalar glueball spectrum

• The scalar glueball mass decreases with the bare fermion mass

• it is expected to become independent of mf in the quenched limit

• at light quark mass and on small volumes it is consistent with the mass of a fermionic bound state [talk by T.Yamazaki 3F]

• finite volume effects and mixing with a light scalar fermionic state need to be better investigated 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

amf

0

0.2

0.4

0.6

0.8

1

aM

G

L=18L=24L=30L=36! (L=30)" (L=30)

Pseudoscalar and vector data are taken from LatKMI, PRD86(2012)054506

Results: scalar glueball spectrum

• The scalar glueball mass decreases with the bare fermion mass

• it is expected to become independent of mf in the quenched limit

• at light quark mass and on small volumes it is consistent with the mass of a fermionic bound state [talk by T.Yamazaki 3F]

• finite volume effects and mixing with a light scalar fermionic state need to be better investigated

Pseudoscalar and vector data are taken from LatKMI, PRD86(2012)054506Scalar data are taken from LatKMI, arxiv:1305.6006

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

0

0.2

0.4

0.6

0.8

1

aM

G

L=18L=24L=30L=36! (L=30)" (L=30)# (L=30)

Example: spatial torelon effective mass (amf=0.06)

0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

1

1.2

am

eff

L=18L=24L=30L=36am=0.1142(79) [7:11]

am=0.273(31) [7:12]

am=0.533(37) [4:7]

am=0.657(52) [3:5]

amtor

(L) = a2�L� ⇡

3L� ⇡2

18L3

1

a2�• The torelon ground state mass

is obtained from a variational analysis of smeared spatial Polyakov loop correlators

• The string tension is extracted from the L dependence of the torelon ground state mass

• A stable effective mass plateaux is found for each volume and the fitted mass is shown on top of the effective mass data

• The formula applied for the largest volumes implies a common string tension value

Example: spatial torelon effective mass (L=24)

0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

1

1.2

am

eff

amf=0.05

amf=0.06

amf=0.08

amf=0.10

am=0.158(11) [7:12]

am=0.273(31) [7:12]

am=0.615(29) [3:6]

am=0.820(82) [3:5]

• Torelon effective masses on a fixed volume show the bare fermion mass dependence

• When the bare quark mass increases we obtain a larger torelon mass

amtor

(L) = a2�L� ⇡

3L� ⇡2

18L3

1

a2�

Results: string tension

• The string tension is obtained for the full range of simulated masses and volumes with 1% to 4% statistical errors

• The finite-size effects seem under control as date on two volumes with compatible string tension exist for all light fermion masses

• There is a clear dependence of the string tension on the bare fermion mass

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

0

0.1

0.2

a!

1/2

L=18

L=24L=30

L=36

Scaling of the string tension

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

0

0.1

0.2

a!

1/2

L=18L=24L=30L=36(L=30): 0.05(1)+1.5(2)*(am

f)

(L=30): 0.9(2)*(amf)^(0.66(9))

ap� = c am

11+�

fap� = c0 + c1 amf


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

0

0.1

0.2

a!

1/2

L=18L=24L=30L=36(L=30): 0.05(1)+1.5(2)*(am

f)

(L=30): 0.9(2)*(amf)^(0.66(9))

ap� = c am

11+�


γ∼0.5 is compatible with the mesonic spectrum

c0 = 0.05 implies confinement in the chiral limit


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

0

0.1

0.2

a!

1/2

L=18L=24L=30L=36(L=30): 0.05(1)+1.5(2)*(am

f)

(L=30): 0.9(2)*(amf)^(0.66(9))

ap� = c am

11+�


Relevant region to study the chiral behaviour

Region with higher order corrections

Ratios of gluonic and mesonic scales

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

1

1.5

2

2.5

3

3.5

4

m!/"

1/2

L=18L=24L=30L=36

Fermionic pseudoscalar state and string tension


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18am

f

1

1.5

2

2.5

3

3.5

4

m!/"

1/2

L=18L=24L=30L=36

Fermionic pseudoscalar state and string tension


0 0.02 0.04 0.06 0.08 0.1 0.12am

f

0

0.5

1

1.5

2

2.5

3

3.5

4

m!/!

1/2

L=24L=30L=36SU(3) pure gauge

Flavour-singlet fermionic scalar state and string tension


0 0.02 0.04 0.06 0.08 0.1 0.12am

f

0

0.5

1

1.5

2

2.5

3

3.5

4

m!/!

1/2

L=24L=30L=36SU(3) pure gauge

Flavour-singlet fermionic scalar state and string tension

Conclusions

• we study the scalar spectrum in many-flavour QCD, focused on Nf=12

• scalar glueball and scalar meson interpolating operators are used

• both glueball operators and mesonic operators couple to this state

• important to understand the mixing of such contributions

• we measure the string tension in Nf=12 QCD and its fermion mass dependence

• it is difficult to extrapolate the results to the chiral limit

• ratios of fermionic pseudoscalar and scalar state with the string tension are compatible with a constant value, down to the chiral limit

Outlook

• larger volumes at light quark masses are necessary to carefully estimate finite size errors

• possible discretisation effects need to be studied by simulating at different lattice spacing

• lighter quark masses are needed to better identify scaling properties of the string tension

• a variational analysis including both gluonic and mesonic operators in the scalar channel is needed: it will give informations about the mixing between the two kind of operators

• a similar study is in progress on Nf=8 QCD configurations [talks by K.Nagai and H.Ohki 1F]

Thank you

Appendix

additional slides with results

Comparison with gluonic observables Preliminary

0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

am

eff

gluonic operators only

fermionic operators only

Scalar spectrum L=18 T=24 !=4 amf=0.06

0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

am

eff




L = 18 and amf = 0.06 L = 24 and amf = 0.06


L = 18 and amf = 0.08 L = 24 and amf = 0.08

0 2 4 6t/a

0.2

0.4

0.6

0.8

1

am

eff




0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

1

am

eff





L = 30 and amf = 0.06 L = 30 and amf = 0.08

0 2 4 6 8 10t/a

0

0.2

0.4

0.6

0.8

1

am

eff



0 2 4 6 8 10 12t/a

0

0.2

0.4

0.6

0.8

1

am

eff



Documents

Gluonic observables and the scalar spectrum of twelve-ﬂavour QCD · 2013. 8. 5. · Gluonic observables and the scalar spectrum of twelve-ﬂavour QCD Enrico Rinaldi on behalf of