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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
<latexit sha1_base64="a+6pgr8uQliY+5Lo6a5V83S0To4=">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</latexit>
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
4 + ↵<latexit sha1_base64="r1elrMZkQ1LRRf0/15BdYMkX/3Q=">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</latexit>
⇒ Gaussian spread:
(in lattice unit)
SU(4),β=11.1 37 2.3
0
2
4
6
8
10
0 2 4 6 8 10
Effe
ctiv
e m
ass
(uni
t: Λ
)
t/a
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
J (0)<latexit sha1_base64="vYNUXuWBVn4Okwz+7suptP29vbw=">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</latexit>
: source op.
C��(t,x� y) ⌘ 1
V
X
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h0 |T [�(x+ r, t)�(y + r, t) · J (0)]| 0i<latexit sha1_base64="A6w4hv2DNBWka4zFYCe34AiBHo8=">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</latexit>
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
0
2x10-8
0.2 0.4 0.6 0.8 1 1.2
BS w
ave
func
tion
(latti
ce u
nit)
r (unit: Λ-1)
BS(1-body src, t=1)-2x10-11
-1x10-11
0
1x10-11
2x10-11
3x10-11
4x10-11
5x10-11
0.2 0.4 0.6 0.8 1 1.2
BS w
ave
func
tion
(latti
ce u
nit)
r (unit: Λ-1)
BS(2-body src, t=1)
-2x10-14
-1.5x10-14
-1x10-14
-5x10-15
0
5x10-15
1x10-14
1.5x10-14
2x10-14
0.2 0.4 0.6 0.8 1 1.2
BS w
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
0
5
10
15
20
0 2 4 6 8 10
Effe
ctiv
e m
ass
of B
S (u
nit: Λ
)
t/a
BS (p=0,2400Conf)Lucini(124,2010) (SU(4),
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.
"1
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R(t, r) ⌘ C��(t, r)
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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
)
r (unit: Λ-1)
1-body src2-body src
(also appeared in the SU(3) case, maybe related w/ the failure of Luescher’s method)
-60
-40
-20
0
20
40
60
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Pote
ntia
l (un
it: Λ
)
r (unit: Λ-1)
1-body src2-body src
SU(3) result
(β = 5.7, 158641 confs)
-40
-20
0
20
40
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Pote
ntia
l (un
it: Λ
)
r (unit: Λ-1)
1-body src2-body src
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
ntia
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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