The topological susceptibility in the large

Transcription

New Frontiers in Theoretical Physics
Cortona@GGI, 17-20 May 2016
The topological susceptibility
in the large-N limit of Yang–Mills theory
Marco Cèab
in collaboration with
M. García Veracd , L. Giustief , S. Schaeferc
a) Scuola Normale Superiore
b) INFN, Sezione di Pisa
c) NIC, DESY, Zeuthen
d) Humboldt-Universität zu Berlin
e) Università di Milano-Bicocca
f) INFN, Sezione di Milano-Bicocca
19 May 2016
Chiral symmetry breaking
chiral symmetry: SU(3)
L ×SU(3)R
quark condensate  = ψ̄ψ
Pseudoscalar mesons [PDG 2014]
- spontaneous χSB
Meson
Quark content
Mass [MeV]
SU(3)L × SU(3)R → SU(3)V
+
π
d̄ À
139.57018(35)
p
π0
d d̄ − ̄ 2
134.9766(6)
- 8 massless NG bosons:
π−
d ̄
139.57018(35)
π, K, η pseudoscalar
+
K
s̄
493.677(16)
mesons
K0
ds̄
497.614(24)
quark masses
M = diag(m , md , ms )
K̄ 0
K−
- explicit χSB
- pseudo NG
bosons:
M2π = (  F 2 )mq
Marco Cè – SNS & INFN, Pisa
19 May 2016
sd̄
s̄
497.614(24)
493.677(16)
η
cos θη8 + sin θη0
η′
− sin θη8 + cos θη0
957.78(6)
Àp
η8 = (d d̄ + ̄ − 2ss̄) 6
Àp
η0 = (d d̄ + ̄ + ss̄) 3
θ ≃ −11.4°
547.862(18)
1
Chiral symmetry breaking
flavour singlet sector: U(1)B × U(1)A
Pseudoscalar mesons
U(1)B : baryon number
- conserved
[PDG 2014]
Meson
Quark content
Mass [MeV]
π+
π0
π−
d̄ À
p
d d̄ − ̄ 2
139.57018(35)
134.9766(6)
d ̄
139.57018(35)
K+
K0
K̄ 0
K−
s̄
ds̄
sd̄
s̄
493.677(16)
497.614(24)
497.614(24)
493.677(16)
η
cos θη8 + sin θη0
547.862(18)
η′
− sin θη8 + cos θη0
957.78(6)
Àp
η8 = (d d̄ + ̄ − 2ss̄) 6
Àp
η0 = (d d̄ + ̄ + ss̄) 3
θ ≃ −11.4°
U(1)A
- no parity partner in nature
- broken explicitly by quark
masses
- the η′ is too heavy to be
the 9thppseudo NG boson:
Mη′ < 3Mπ [Weinberg 1975]
η′
The U(1)A problem: Why is the
much heavier than other
pseudo Nambu–Goldstone bosons?
19 May 2016
1
Topological charge
The topological charge
∫
Q=
d4  q() ∈ Z
q() =
1
64π 2


εμνρσ Fμν
()Fρσ
()
- divergence of a gauge-dependent current q() = ∂μ Kμ ()
- U(1)A is broken by an anomaly ∝ 2Nƒ q(),
a purely quantum effect
19 May 2016
2
Topological charge
∫
Q=
d4  q() ∈ Z
q() =
1
64π 2


εμνρσ Fμν
()Fρσ
()
θ-term in YM (or QCD) Lagrangian: θ-vacua
L=
1
4g20


Fμν
()Fμν
() − θq()
- imaginary in Euclidean space: sign problem for θ ̸= 0
- strong CP problem: neutron EDM ⇒ |θ| < 10−10
19 May 2016
2
Topological charge
∫
Q=
d4  q() ∈ Z
q() =
1
64π 2


εμνρσ Fμν
()Fρσ
()
θ-term in YM (or QCD) Lagrangian: θ-vacua
L=
1
4g20


Fμν
()Fμν
() − θq()
- imaginary in Euclidean space: sign problem for θ ̸= 0
- strong CP problem: neutron EDM ⇒ |θ| < 10−10
The topological susceptibility is the two-point function of q()
∫
χ=
d4  〈q()q(0)〉
19 May 2016
2
The Witten–Veneziano mechanism
A mechanism to solve the U(1)A problem based on the
vanishing of the anomaly in the large-N limit
[Witten 1979; Veneziano 1979]
- anomalous chiral Ward identity
∂μ A0μ = 2mP0 () − 2Nƒ q()
- N → ∞: U(1)A is restored, the η′ is a NG boson
- in Yang–Mills theory, χYM ̸= 0 and O(1) in large-N
- dynamical quarks are an O(1/ N) effect
- no θ dependence in QCD with m = 0 quarks ⇒ χ = 0
⇒ The η′ gets a mass M2η′ = O(1/ N)
The WV formula, valid in the chiral and N → ∞ limits
F 2 M2η′
2Nƒ
= χYM
- links Mη′ in QCD with χ measured in YM theory
- is a non-perturbative test of the quantization of QCD
19 May 2016
3
large-N chiral perturbation theory
Simultaneous expansion in mq , 1/ N and p2
[Di Vecchia, Veneziano 1980, Kaiser, Leutwyler 2000]
M2η′ = 2mq

F2
+
2Nƒ τ
F2

+ O m2q , 1 N2 , p2
- M2η′ linear in expansion parameters mq , 1/ F 2 ∝ 1/ N
- τ = limN→∞ χYM is a low-energy constant
- ﬁtting LO χPT with real world [Veneziano 1979]

Fπ2 2
+ O( 1/3 )
Mη + M2η′ − 2M2K = (180 MeV)4 = χYM
6
N→∞
- N → ∞ YM theory is not physical
19 May 2016
4
large-N chiral perturbation theory
Simultaneous expansion in mq , 1/ N and p2
[Di Vecchia, Veneziano 1980, Kaiser, Leutwyler 2000]
M2η′ = 2mq

F2
+
2Nƒ τ
F2

+ O m2q , 1 N2 , p2
- M2η′ linear in expansion parameters mq , 1/ F 2 ∝ 1/ N
- τ = limN→∞ χYM is a low-energy constant
- ﬁtting LO χPT with real world [Veneziano 1979]

Fπ2 2
+ O( 1/3 )
Mη + M2η′ − 2M2K = (180 MeV)4 = χYM
6
N→∞
- N → ∞ YM theory is not physical
⇒ measure χYM
in lattice-discretized large-N YM theory
N→∞
- using lattice SU(3) YM [Cè, Consonni, Engel, Giusti 2015]
χYM
= (180.5(43) MeV)4 = χYM
+ O 1 32
N=3
N→∞
- O 1 N2 corrections are likely small
19 May 2016
4
Topological charge on the lattice (I)
Problem: define the topological charge on the lattice
⇒ naïve discretization of q()
qL () =
1
64π 2


εμνρσ Fμν
()Fρσ
()
 () given in terms of plaquettes (clover definition)
- Fμν


Fμν
() =
1
4

P 


ν
μ



with P (U) projecting over su(3) Lie algebra
19 May 2016
5
Topological charge on the lattice (I)
Problem: define the topological charge on the lattice
⇒ naïve discretization of q()
qL () =
1
64π 2


εμνρσ Fμν
()Fρσ
()
 () given in terms of plaquettes (clover definition)
- Fμν


Fμν
() =
1
4

P 


ν
μ



with P (U) projecting over su(3) Lie algebra
- qL () need multiplicative renormalization [Allés et al. 1995]
- | − y| → 0: singularities in 〈qL ()qL (y)〉
⇒ qL () does not satisfy anomalous chiral Ward identities
19 May 2016
5
Topological charge on the lattice (II)
Definitions of Q satisfying Ward identities:
- fermionic definition, using a Dirac operator satisfying
Ginsparg–Wilson [Ginsparg, Wilson 1982], as Neuberger–Dirac
operator DN [Neuberger 1998]
qN () = −
▶
▶
▶
̄
24
tr γ5 DN (, )
∑
index theorem: QN = 4 qN () = − 2̄ Tr γ5 DN = n+ − n−
∑
no renormalization of 4  〈qN ()qN (0)〉
satisfy anomalous chiral Ward Identities
[Giusti, Rossi, Testa, Veneziano 2002; Giusti, Rossi, Testa 2004; Lüscher 2004]
▶
computationally very expensive
19 May 2016
6
Topological charge on the lattice (II)
Definitions of Q satisfying Ward identities:
- fermionic definition, using a Dirac operator satisfying
Ginsparg–Wilson [Ginsparg, Wilson 1982], as Neuberger–Dirac
operator DN [Neuberger 1998]
qN () = −
▶
▶
▶
̄
24
tr γ5 DN (, )
∑
index theorem: QN = 4 qN () = − 2̄ Tr γ5 DN = n+ − n−
∑
no renormalization of 4  〈qN ()qN (0)〉
satisfy anomalous chiral Ward Identities
[Giusti, Rossi, Testa, Veneziano 2002; Giusti, Rossi, Testa 2004; Lüscher 2004]
▶
computationally very expensive
- spectral projectors [Giusti, Lüscher 2009]
- naïve qL () at positive YM gradient flow times
19 May 2016
[Lüscher 2010]
6
The Yang–Mills gradient flow
The gradient flow is the solution of the initial value problem
[Lüscher 2010]
d
dt
Vμ (t, ) = −g20 ∂μ, SW [V(t)] Vμ (t, )
Vμ (0, ) = Uμ ()
- proportional to the negative gradient of YM action
p
- Vμ (t, ): configurations smoothed within a radius 8t
- universality: (multi)local fields at t > 0 are finite and do
not require renormalization [Lüscher, Weisz 2011]
- numerical simulations: integrate numerically ODE with
structure-preserving Runge–Kutta method
- scale setting with energy density E(t): t0 reference scale
t 2 〈E(t)〉t=t0 = 0.3
t0 = (0.176(4) fm)2 in YM theory
19 May 2016
7
Universality of gradient flow definition
Result in [Cè, Consonni, Engel, Giusti 2015]:
- in the continuum, using small-t expansion
∂t q(t, ) = ∂ρ ρ (t, ), density-chain expression of Q, OP
expansion, t-independence of χ
∫
∫
d4  〈q(t, )q(t, 0)〉 =
d4  〈q()q(0)〉
- on the lattice, with Neuberger definition qN (),
∑
∑
〈qN (t, )qN (t, 0)〉 = lim 4
〈qN ()qN (0)〉
lim 4
→0
→0

19 May 2016

8
∫
∫
d4  〈q(t, )q(t, 0)〉 =
d4  〈q()q(0)〉
∑
∑
〈qN ()qN (0)〉
〈qL (t, )qL (t, 0)〉 = lim 4
lim 4
→0
→0


- Universality at t > 0 ⇒ naïve discretization qL (t, )
19 May 2016
8
∫
∫
d4  〈q(t, )q(t, 0)〉 =
d4  〈q()q(0)〉
∑
∑
〈qN ()qN (0)〉
〈qL (t, )qL (t, 0)〉 = lim 4
lim 4
→0
→0


- Universality at t > 0 ⇒ naïve discretization qL (t, )
Cumulants CL,n (t) of the topological charge given by the
naïve definition at strictly positive flow time have a finite and
unambiguous continuum limit that satisfies anomalous chiral
Ward identities
lim CL,n (t) = lim CN,n = Cn ,
→0
→0
19 May 2016
(t > 0)
8
Autocorrelation
Topological observables are known to be changed slowly by
the Monte Carlo updates
PBC
Q(t0 )
10
- Q is changed through
lattice artifacts
5
- topological charge
freezing in the continuum
0
- and for larger N
[Del Deb-
bio, Panagopoulos, Vicari 2002]
−5
Q
τint ∝ ecN
−10
Monte carlo history of Q at t0 ,
SU(6) lattice at  ≃ 0.1 fm, PBC
−1
- applies also to χ
1 2
χ = TV
Q
19 May 2016
9
Autocorrelation
Topological observables are known to be changed slowly by
the Monte Carlo updates
PBC
OBC
10
- our solution: open
boundary conditions
(OBC) in the time direction
[Lüscher, Schaefer 2011]
Q(t0 )
5
τint ∝ −2 (N?)
Q
0
- Q is no more an integer
−5
- time translation
invariance is lost
−10
Monte carlo history of Q at t0 ,
SU(6) lattice at  ≃ 0.1 fm, OBC
- excited states
- O() discretization effects
- ⇒ modify χ definition
19 May 2016
9
Topological susceptibility with OBC
Definition based on correlators
of topological charge on a
∑
time-slice: q̄(t, 0 ) = ⃗ q(t, )
C̄(t, Δ) =
1
(T−2d−Δ)V
T−1−d−Δ
∑
χ(t) =
〈q̄(t, 0 )q̄(t, 0 + Δ)〉
0 =d
+r
∑
C̄(t, |Δ|)
Δ=−r
- d fixed in physical units to avoid excited states from
open boundaries (and boundaries cut-off effects)
p
d = 7.5 t0 ≃ 1.3 fm
- r fixed in physical units to avoid summing over noise
p
r = 6.9 t0 ≃ 1.2 fm
- systematics checked to be negligible
19 May 2016
10
(preliminary)
Numerical results
[Cè, Garcia Vera, Giusti, Schaefer to appear]
19 May 2016
11
Simulation results
N = 4, 5, 6, three lattice spacings
- gradient flow up to t0
 [fm]
7.4
×10−4
0.05
0.07
0.08
0.09
0.10
2
t e(t, ) SU(4)
SU(5)
7.2
SU(6)
= 0.1125 N N−1
2
t=t0
7.0
- t0 used as a scale: t02 χ(t0 )
t02 χ
6.8
- ∼ 1 % statistical error on
the single lattice result
- no clear 2 or 1/ N2 scaling
6.6
6.4
6.2
6.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
2 / t0
19 May 2016
12
Simulation results
N = 4, 5, 6, three lattice spacings
- gradient flow up to t0
 [fm]
7.4
×10−4
0.05
0.07
0.08
0.09
0.10
2
t e(t, ) SU(4)
SU(5)
7.2
SU(6)
= 0.1125 N N−1
2
t=t0
SU(3)
7.0
- t0 used as a scale: t02 χ(t0 )
t02 χ
6.8
- ∼ 1 % statistical error on
the single lattice result
6.6
6.4
6.2
6.0
0.00
0.05
0.10
0.15
0.20
2 / t0
0.25
0.30
0.35
- including SU(3) results
[Cè, Consonni, Engel, Giusti 2015]
19 May 2016
12
Discretization effects vs. N
t-dependent discretization effects can be studied taking the
ratio χ(t)/ χ(t0 ) at different flow times t
 [fm]
0.05
1.05
0.07
0.08
0.09
0.10
1.00
0.95
χ(t)/ χ(t0 )
0.90
- very small errors, smaller
than marker size
- universality in the
continuum
0.85
lim χ(t)/ χ(t0 ) = 1
0.80
→0
0.75
0.70
0.65
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
2 / t0
fit with
1 + c1 (2 / t0 ) + c2 (2 / t0 )2
- t-dependent discretization
effects nearly
N-independent
- true also for SU(3) data
19 May 2016
13
Continuum and N → ∞ limit
- combined fit with
 [fm]
7.4
×10−4
0.05
0.07
0.08
0.09
0.10
c0,0 + c1,0 2 / t0 + c0,1 1/ N2
SU(4)
SU(5)
7.2
SU(6)
SU(3)
7.0
t02 χ
6.8
6.6
6.4
6.2
6.0
0.00
0.05
0.10
0.15
0.20
2 / t0
0.25
0.30
0.35
- using only two finest
lattices
- 2 / t0 discretization effects
mainly fixed by SU(3) data
- χ2 dof = 0.90
- ∼ 1.5 % statistical error
(on a d = 4 observable!)
Our (preliminary) result:
t02 χN→∞ = 6.98(11) × 10−4
19 May 2016
14
Comparison with other results
Our (preliminary) result:
t02 χN→∞ = 6.98(11) × 10−4 ,
χN→∞ = (182.3(7)(44) MeV)4
5 % correction with respect to
t02 χN=3 = 6.67(7) × 10−4 ,
[Cè, Consonni, Engel, Giusti 2015]
χN=3 = (180.5(5)(43) MeV)4
Status of the art ≳ 10 years ago [Review by: Vicari, Panagopoulos 2009]
- cooling definition of topological charge
- coarser lattice spacings
- statistical errors ≳ 6 %
p
- scale setting with σ = 440 MeV
χN→∞ = 0.0200(43)σ 2 = (165(9) MeV)4
2
χN→∞ = 0.0221(14)σ = (170(3) MeV)
4
χN→∞ = 0.0248(18)σ 2 = (175(3) MeV)4
19 May 2016
[Lucini, Teper 2001]
[Del Debbio, Panagopoulos, Vicari 20
[Lucini, Teper, Wenger 2005]
15
Conclusions & Outlook
Conclusions:
- we measured the topological susceptibility of SU(N) YM
theory in the N → ∞ limit
▶
▶
with an O(1 %) statistical accuracy
and systematics under control
- we obtained this result with open boundary conditions in
the time direction
▶
they work to avoid topological charge freezing with N
19 May 2016
16
Conclusions & Outlook
Conclusions:
- we measured the topological susceptibility of SU(N) YM
theory in the N → ∞ limit
▶
▶
with an O(1 %) statistical accuracy
and systematics under control
- we obtained this result with open boundary conditions in
the time direction
▶
they work to avoid topological charge freezing with N
Outlook:
- 1 % accuracy should be sufficient for the future
- verify the Witten–Veneziano relation on both sides:
compute Mη′ on the lattice for N > 3, with dynamical
fermions
▶
▶
▶
exponential degradation in S/N of correlators
topological charge freezing with dynamical fermions
long term project
19 May 2016
16
Thanks
for your attention!
Backup
Quantum Chromodynamics
QCD is the theory of strong interactions
L=
1
4g20
[Fritzsch, Gell-Mann 1972]


Fμν
()Fμν
()
- SU(3) gauge group, imaginary-time YM Lagrangian
- gluons Aμ : 8 vector bosons, g20 : bare gauge coupling
- asymptotic free theory [Gross, Wilczek 1973; Politzer 1973]
- confined spectrum: mass gap, glueballs
19 May 2016
i
Quantum Chromodynamics
QCD is the theory of strong interactions
L=
1
4g20
[Fritzsch, Gell-Mann 1972]


Fμν
()Fμν
() + ψ̄()(γμ Dμ + M)ψ()
- SU(3) gauge group, imaginary-time QCD Lagrangian
- gluons Aμ : 8 vector bosons, g20 : bare gauge coupling
- asymptotic free theory (if Nƒ < ( 11/2 )N)
- confined spectrum: mass gap, mesons, barions
- quarks ψ: 3Nƒ Dirac fermions, mass matrix M
- Nƒ : number of flavours (Nƒ = 2 or 3 light quarks)
- massless quarks M = 0 ⇒ global symmetry of the action
SU(Nƒ )L × SU(Nƒ )R × U(1)B × U(1)A (at classical level)
19 May 2016
i
The dilute instanton gas
A solution of the U(1)A problem proposed by ’t Hooft using
instantons: dilute instanton gas approximation [’t Hooft 1976]
- Semiclassical approximation
- Free energy dependence on θ
F(θ) = − ln Z(θ) = −VA(cos θ − 1)
- Arbitrary normalization A ⇒ no prediction for χYM
2F 1
d
1
Q2 =
=A
χYM =
2
V
V dθ θ=0
- No IR bound on instantons size
- Expected to be valid at high temperature
19 May 2016
ii
Higher moments
Higher moments of the probability distribution of the
topological charge Q in Yang–Mills theory
- to confront with dilute instanton gas model prediction
- higher cumulants are obtained deriving F(θ)
1 2n con (−1)n+1 d2n
=
Cn ≡
Q
F(θ)
V
V
dθ2n
θ=0
DIG: Cn = A
large-N: Cn ∝ N−2(n−1)
- we define the ratio between the 4th and the 2nd cumulant
con
2
C2
Q4
Q4 − 3 Q2
R≡
= =
C1
Q2
Q2
DIG: R = 1
large-N: R ∝
19 May 2016
1
N2
iii
Lattice regularization (I)
Euclidean YM theory is discretized on a four-dimensional
hypercubic lattice with lattice spacing  [Wilson 1974]
ν

μ
- gauge field defined on links between lattice sites:
Uμ () =
μ

 + μ̂
- plaquettes: Uμν () =
∈ SU(3)
ν
μ

- finite volume V = L4 with periodic boundary conditions
- 18 × 4 × L4 real variables
19 May 2016
iv
Lattice regularization (II)
under gauge transformation g() ∈ SU(3)
Uμ () → g()Uμ ()g−1 (+ μ̂)
Uμν () → g()Uμν ()g−1 ()
Wilson plaquette action [Wilson 1974]
∑ ∑
1
1 − Re tr Uμν ()
SW [U] = β
N
 μ<ν
β=
2N
g20
- gauge invariant at finite lattice spacing
¦
©
- Uμ () → exp Aμ ()T 

∑ 1

SW [U] → 4
F  ()Fμν
() + O 2
2 μν
 4g0
- naïve continuum limit: lim→0 SW = SYM
- O 2 discretization artefacts,  = (g0 )
- continuum limit  → 0 ⇔ critical point g0 → 0 (β → ∞)
19 May 2016
v
Path integral Monte Carlo
YM theory path integral
∫
1
〈O〉 =
D[U]O[U] exp{−SYM [U]}
Z
- V = (10)4 lattice: 320 000 integrations
19 May 2016
vi
Path integral Monte Carlo
YM theory path integral
∫
1
〈O〉 =
D[U]O[U] exp{−SYM [U]}
Z
- V = (10)4 lattice: 320 000 integrations
Markov chain Monte Carlo
- importance sampling: generate n gauge field
configurations {U } with pdf dp = D[U]e−SYM [U]
∑n
n→∞
- Ō = (1/ n) =1 O[U ] −−−→ 〈O〉
Cabibbo–Marinari updates [Cabibbo, Marinari 1982]
- local updates: sweep all the links on the lattice
- heat bath + overrelaxation sweeps
Configurations are correlated
- critical slowing down: τint ∝ −2
- topological charge changed through lattice artefacts:
 → 0 ⇒ topological freezing
19 May 2016
vi
Universality of lim→0 χ(t) (I)
- In the continuum, to all order of perturbation theory,
using ∂t q(t, ) = ∂ρ ρ (t, ) is valid the small-t expansion
∫
t
〈q(t, )O(y)〉 = 〈q()O(y)〉 + ∂ρ
dt ′ ρ (t ′ , )O(y) ,
0
where
ρ (t, ) =
1
ε
8π 2 μνρσ
tr Fμν (t, )Dα Fασ (t, )
is a gauge-invariant, pseudovector dimension-5 field
- No local d < 5 fields with ρ transformation properties ⇒
not singular at t = 0
〈q()O(y)〉 = lim 〈q(t, )O(y)〉,
t→0
( ̸= y)
or
〈q(t, )O(y)〉 = 〈q()O(y)〉 + O(t),
19 May 2016
( ̸= y)
vii
Universality of lim→0 χ(t) (II)
- With
∑ O = q, using the density-chain expression
4  q()
=
∑
5
20
−m 
z 〈P51 (z1 )S12 (z2 )S23 (z3 )S34 (z4 )S45 (z5 )〉F the
leading order divergence in
〈q()P51 (z1 )S12 (z2 )S23 (z3 )S34 (z4 )S45 (z5 )〉
for  → zj , computable in perturbation theory, is
→0
q()Sj (0) −−→ c()Pj (0)
where c() ∝ ||−4 for  → 0
- Applying small-t expansion
q(t, )Sj (z)O(y) =
∫
t
q()Sj (z)O(y) + ∂ρ
dt ′ ρ (t ′ , )Sj (z)O(y) ,
0
c() = ∂μ μ () and do not contribute to the integrated
correlation function
19 May 2016
viii
Universality of lim→0 χ(t) (III)
- No other d ≤ 4 gauge-invariant pseudoscalar operators
implies
∑
〈qN (t, 0)qN ()〉 = finite,
lim Zq 4
(t > 0)
→0

- Replacing qN () with density-chain expression
4
∑
〈qN (t, 0)qN ()〉 =

− m5 20
∑
〈qN (t, 0)P51 (z1 )S12 (z2 ) . . . S45 (z5 )〉
z
the r.h.s. is finite for t > 0
- This implies that Zq = 1 and therefore
lim 〈qN ()qN (0)〉 = 〈q()q(0)〉,
→0
19 May 2016
( ̸= 0)
ix
Universality of lim→0 χ(t) (IV)
- Using small-t expansion again and t-independence of χ,
∑
∑
〈qN ()qN (0)〉 = lim 4
〈qN (t, )qN (t, 0)〉
lim 4
→0
→0


- Since qN () and the naïve discretization qL () have the
same naïve continuum limit, at t > 0 thanks to
universality
∑
∑
〈qN ()qN (0)〉 = lim 4
〈qL (t, )qL (t, 0)〉
lim 4
→0
→0


- By iterating the procedure, it extends to higher
cumulants
lim CN,n = lim CL,n (t),
→0
→0
19 May 2016
(t > 0)
x
Lattices details
N
β
T/ 
L/ 
t0 / 2
[fm]
L[fm]
4
10.92
11.14
11.35
64
80
96
16
20
24
2.9900(7)
4.5207(8)
6.4849(16)
0.098
0.080
0.067
1.58
1.60
1.60
5
17.32
17.67
18.01
64
80
96
16
20
24
3.0636(7)
4.6751(8)
6.8151(17)
0.097
0.079
0.065
1.55
1.57
1.56
6
25.15
25.68
26.15
64
80
96
16
20
24
3.0824(4)
4.8239(9)
6.9463(13)
0.097
0.077
0.065
1.55
1.55
1.55
Physical units using
p
t0 = 0.17 fm
19 May 2016
xi
Reference flow-time
The reference flow-time t0 is defined by
t 2 〈E(t)〉t=t0 = 0.3
- Easy to measure
with great statistical
accuracy
- Our result
p
8t0
= 0.941(7)
r0
- using FK measured
in quenched QCD
for scale setting
t0 = (0.176(4) fm)2
19 May 2016
xii