Interglueball potential in lattice gauge theory

2019/04/24 YITP Kyoto

Nodoka Yamanaka(YITP)

In Collaboration withH. Iida (FEFU), A. Nakamura (FEFU), M. Wakayama (RCNP)

Dark = invisibleSomething invisible around us!

Dark ≠ black

What is it?Nobody knows!

Why do we know it exists?Let us see…

Unveiling it is one of the most important goal of cosmology, astrophysics, particle physics.

What is dark matter ??

Velocity of the disc cannot be explained by visible stars

Suggesting additional something invisible surrounding the Milky way.

(Zwicky, 1930’s)

Galactic rotation curve

DM Halo

20kpc

DM density at the Earth: 0.3GeV/cm3

Our galaxy is surrounded by a halo of dark matter

DM halo : ⇒ Weakly interacting

with star, gas, and each other

⇒ Nonrelativistic

Dark matter halo

Bullet cluster

Magellan telescope Chandra Xray image

Difference between luminous (baryonic) and total mass distributions!

A more powerful proof : Galactic collision

Cosmic "makeup". Credit: ESA/Planck

From the cosmic microwave background analysis (Planck), fraction of dark matter can be derived

⇒ Most of matter in our Universe is dark.

Dark matter : 27% of the energy component of the Universe

Dark matter is required to speed up the formation of galaxies

DM clump

Baryon concentration catalyzed by dark matter clumps during the cooling (Early Universe, high temperature)

Baryons

BaryonsBaryons

Baryons

⇒ Dark matter absolutely required in our existence!

If no dark matters, galaxy formation is much slower.

Formation of galaxy

MACHO : Massive Compact Halo Object

Example : primordial blackholes, brown dwarfsAlmost non luminous astronomical body

Can be probed with gravitational lensing

MACHOs are not favored by observations, even if a window (around MPBH/M◉～10-12) is still left

⇒ Dark matter is likely to be particles?

H. Niikura et al., Nature Astronomy (2019) (arXiv:1701.02151 [astro-ph.CO])

Is the dark matter a MACHO?

WIMP : weakly interacting massive particleWIMP = particle physics

Property of WIMPs:No charge, no colorNot neutrino (ruled out by Bigbang nucleosynthesis)No candidates in standard model of particle physics

Challenge in particle physics:

⇒ Find theory explaining dark matter!

WIMP dark matter

SU(N) Yang-Mills theory

⇒ Theory with very high naturalness

(Suppose a GUT which generates SM and DM, the difference of mass scales between SM and DM is not serious)

Lightest particles are glueballs ! ⇒ SU(N) glueballs are candidate of DM

LYM = �1

4Fµ⌫a Fµ⌫,a

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Important properties:

LYM does not have apparent scale, but scale is dynamically generated (dimensional transmutation)

Dark matter in hidden YM theory:

(summarized in the report of USQCD Collaboration : arXiv:1904.09964 [hep-lat])

Renormalizable theory, running coupling has logarithmic scale variation,

difference of Nc can generate ΛYM’s which differ by orders of magnitude

⇒ The simplest interacting theory(a =1,…,Nc2–1)

⇒ No important fine-tuning problem in the choice of ΛYM !

No scalars and massive fermions ⇒ Free from quadratic divergences

Self-interacting dark matter

There are (were?) several problems in the galactic DM distribution:

Core vs Cusp problem:

Introducing DM self-interaction changes its distribution smaller than Mpc

N-body simulation predicts cuspy DM distribution near the galactic center, whereas observations suggest flat ones.

Too-big-to-fail problem:

Missing satellite problem:

DM-DM self-interaction ↔ DM-DM scattering ↔ DM-DM potential

must be studied

The DM distribution can be predicted in N-body simulation with gravity only

⇒ Successful in describing the large scale structure (scale > Mpc)

More satellite galaxies than those predicted by the N-body simulation are observed (resolved?).

Satellite galaxies are less dense than those predicted by the N-body simulation.

DM density

radius

core

cusp

Object of study

In this work, we study the interglueball interaction on lattice which is the only way to quantify nonperturbative physics of nonabelian gauge theory.

In this work, we study the interglueball interaction of SU(N) Yang-Mills theory on lattice.

Object:

(Please be careful, SU(2), SU(3), and SU(4) may alternate, but the global feature is the same).

Setup

Standard SU(N) plaquette action :

Improvement of glueball operator :

Confs. generated with pseudo-heat-bath method

APE smearing

We consider the SU(2), SU(3), and SU(4) pure Yang-Mills theory

We use all space-time translational and cubic rotational symmetries to effectively increase the statistics (like the all-mode average for meson and baryon observables)

Lattice spacings : β = 2.5 (Nc=2), 5.7 (Nc=3), 10.789 (Nc=4),

Volume : 163x2410.9 (Nc=4)

Scale determination (example of SU(4))

10.789 0.2706(8)

a√σβ

10.9 0.228(7)

11.1 0.197(8)

String tension for several β in SU(4) YM :

We do not know the scale of the YM theory, so we leave it as a free parameter Λ

Nevertheless, all quantities calculated on lattice depends on Λ

⇒ We express all quantities in unit of Λ.

Relation between Λ and string tension:

ΛMS

√σ = 0.503(2)(40) +0.33(3)(3)

N2

C. Allton et al., JHEP 0807 (2008) 021 M. Teper, Acta Phys. Polon. B 40 (2009) 3249

Fitted from the analysis of the running coupling

11.4 0.14277(72)

B. Lucini et al., JHEP 0406 (2004) 012

M. Teper, Phys. Lett. B 397 (1997) 223; hep-th/9812187

B. Lucini et al., JHEP 0406 (2004) 012

M. Teper, Phys. Lett. B 397 (1997) 223; hep-th/9812187

0.524(40)= (for SU(4))

Scale determination (example of SU(4))

10.789 0.2706(8)

a√σβ

10.9 0.228(7)

11.1 0.197(8)

String tension for several β in SU(4) YM :

We do not know the scale of the YM theory, so we leave it as a free parameter Λ

Nevertheless, all quantities calculated on lattice depends on Λ

⇒ We express all quantities in unit of Λ.

Relation between Λ and string tension:

ΛMS

√σ = 0.503(2)(40) +0.33(3)(3)

N2

C. Allton et al., JHEP 0807 (2008) 021 M. Teper, Acta Phys. Polon. B 40 (2009) 3249

Fitted from the analysis of the running coupling

⇒ Lattice spacing is now expressed in unit of Λ

11.4 0.14277(72)

0.524(40)= (for SU(4))

a (in unit of Λ-1)

0.142(11)

0.119(10)

0.103(9)

0.075(6)

Glueball operator and operator improvement

APE smearing :

= +α x

Re Tr [ U(n+1) V(n)†]

V(n)

U(n+1) so as to maximize

where

Optimal parameters:

n α

SU(4),β=10.789 17 2.3

SU(4),β=10.9 21 2.3

Ape Collaboration, PLB 192 (1987) 163 N. Ishii et al., PRD 66, 094506 (2002)

0++ glueball operator:

Φ = —Glueball has expectation value → subtract

Sum over cubic rotational invarianceΣ

2

rn

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⇒ Gaussian spread:

(in lattice unit)

SU(4),β=11.1 37 2.3

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176000Conf 17x smr102000Conf 17x smr

Lucini(124,2010)

(SU(4) 0++ glueball, β=10.789)

Nambu-Bethe-Salpeter amplitude

For the glueball, caution is needed :

Multi-glueball operators also have expectation value!

(often called “VEV”, but it corresponds to the divergence caused by the mixing with the identity operator)

⇒ We then have to subtract the “VEV” of both source and sink

(removing the source “VEV” will automatically remove sink “VEV”:

⇒ Important consequence : fulfills the cluster decomposition!

<(φsrcφsrc-<φsrcφsrc>)(φsnkφsnk-<φsnkφsnk>)>=<(φsrcφsrc-<φsrcφsrc>)φsnkφsnk> )

2-glueball (0++) state mixes with all other multi-glueball states:

⇒ The source may be chosen as 1-body, 2-body, etc, on convenience

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: source op.

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The source is smeared, but the sinks are not

Glueball NBS wave function plot1-body source: 2-body source:

3-body source:

(case of SU(2), β=2.5)

1-body src BS is 0 at large r due to cluster decomposition

2-body src BS is finite at large r ⇒ Two free glueballs

3-body src BS should be finite at large r, but large error

t

r

-1.2x10-7

-1x10-7

-8x10-8

-6x10-8

-4x10-8

-2x10-8

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2x10-8

0.2 0.4 0.6 0.8 1 1.2

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(latti

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r (unit: Λ-1)

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-1x10-11

0

1x10-11

2x10-11

3x10-11

4x10-11

5x10-11

0.2 0.4 0.6 0.8 1 1.2

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(latti

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r (unit: Λ-1)

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0

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1x10-14

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2x10-14

0.2 0.4 0.6 0.8 1 1.2

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ave

func

tion

(latti

ce u

nit)

r (unit: Λ-1)

BS(3-body src, t=1)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

am

G+S+T G+S G+T

mixS ~ 16%mixT ~ 27%

mixS ~ 36% mixT ~ 34%

Luescher’s method

Calculate the scattering phase shift : need the modulation of the energy of NBS wavefunction in momentum

Problem for the interglueball scattering :

⇒ The glueball 2-body state mixes

with 1-body state (at least for 0++)

⇒ GS saturation of 2-body scattering

dominated by 1-glueball state !

What about diagonalization?

⇒ Many glueball states with energy close to 2mGB…?

(remove 1-body state)

Difficult to calculate interglueball scattering with Luescher’s method

⇒ Maybe difficult to distinguish the 2mGB+ΔE level

from other glueball states

(momentum modulation may be visible, but challenging)B. Lucini et al., JHEP 1008 (2010) 119

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β=10.789)

Time-dependent HALQCD method

Crucial advantage : do not need ground state saturation

Extract the potential from the NBS wave function

Inelastic threshold for glueball = 3mφ : high enough to consider t=2,3

N. Ishii et al., PLB 712 (2012) 437.

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Almost mandatory to use time-dependent HAL method for the glueball analysis, since the glueball correlator becomes very noisy before ground state saturation

SU(4) result : potential plot (local central only)

3 regions :

(β = 10.789, 176000 confs)

(β = 10.9, 210000 confs)

Very short range (lattice unit 0 and 1) : artifact due to ?

Short range (r < 0.4 Λ-1) : looks repulsive (determined from 1-body src)

Long range (r > 0.4 Λ-1) : flat (determined from 2-body src)

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Pote

ntia

l (un

it: Λ

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r (unit: Λ-1)

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(also appeared in the SU(3) case, maybe related w/ the failure of Luescher’s method)

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40

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4

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SU(3) result

(β = 5.7, 158641 confs)

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-20

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20

40

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Pote

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l (un

it: Λ

)

r (unit: Λ-1)

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SU(2) result

(β = 2.5, 1045000 confs)

-300

-200

-100

0

100

200

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Pote

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l (un

it: Λ

)

r (unit: Λ-1)

1-body src2-body src

• Glueballs of the SU(N) Yang-Mills theory are good candidates of dark matter : study of self-interaction is important.

• We studied the interglueball potential in the SU(2), SU(3), and SU(4) Yang-Mills theory.

• Luescher’s method has difficulty in the calculation of glueball scattering due to the mixing between 1-body and 2-body states.

• HALQCD method probes the spatial modulation of the correlator: we think it is OK for the glueball potential calculation.

• Time-dependent HALQCD method is important for the interglueball potential because the signal becomes noisy before the ground state saturation.

• Interglueball potential repulsive for r <0.4Λ-1 ? flat at r >0.4Λ-1.

Summary

Homeworks:• Extraction of the scattering cross section. • Reduce statistical error with cluster decomposition principle. • Operator dependence (artifact) at the short distance to be discussed.

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