This research has been co-financed by the European Union (European - - PowerPoint PPT Presentation
This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF), under the
This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), under the grants schemes ”Funding of proposals that have received a positive evaluation in the 3rd and 4th Call
1
Gauge/Gravity Duality 2013 Munich, 29 July 2013
University of Crete APC, Paris
2-
Based on recent work with: Umut Gursoy, (Utrecht), Ioannis Iatrakis (Crete), Francesco Nitti, (APC), Andy O’Bannon (Cambridge) arXiv:1212.3894 [hep-ph] and based also on past work with:
(APC) arXiv:0903.2859 [hep-th] arXiv:0707.1349 [hep-th]
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 3
YM theory.
vacua, and a new coupling constant (the instanton angle θ that breaks the CP symmetry in YM).
4
are exponentially small in perturbation theory,
λ
IR coupling of the YM Theory that is strong.
instantons, the dynamical contributions of instantons to many YM processes are un-calculable.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 4-
λ
is exponentially suppressed with Nc, instanton effects should be exponentially small in the large-Nc limit.
times false.
stantons cannot be treated like a gas (because large instantons dominate), and instan- ton effects are NOT exponentially suppressed, but only power suppressed, like other dynamical effects.
Witten ’79, Veneziano, ’79
U(1)A in QCD) is Mη′ ∼ Nf Nc ΛQCD
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 5
conservation in QCD Jµ
5
=
Nf
∑
i=1
ψi γµ γ5 ψi ∂µJµ
5 = − Nf
16π2 ϵµνρσ TrFµνFρσ = − Nf 8π2 TrF ∧ F .
distinct vacua of SU(Nc) YM which cannot be connected with small gauge
NCS ≡ 1 8π2
∫
∂M d3x ϵijk Tr
[
Ai∂jAk − 2ig 3 AiAjAk
]
where i, j, k = 1, 2, 3.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 6
and is given by the two-point function of q(xµ), ΓCS ≡ ⟨(∆NCS)2⟩ V t =
∫
d4x ⟨q(xµ)q(0)⟩symmetric
k)= Fourier transform of the retarded Green function of q(xµ) by: ΓCS = − lim
ω→0
2T ω Im GR(ω,⃗ k = 0) ,
sphaleron configurations which produce non-zero ΓCS upon decay.
anomaly of the axial current. These are domains of more left-handed than right-handed quarks or the opposite.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 7
R3
∼ 1018 (1019) at RHIC (LHC). Or eB ∼ 5 − 15 m2
π.
Gauss
8
fermions (blue is spin, brown is momentum).
field, will generate, charge asymmetry on an event-by-event basis.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 8-
are the fluctuations of the chiral asymmetry.
mation and (and B¨
1 log 1
αs ≪ 1)
ΓCS = 0.21 Ncg2T m2
D
(
log mD γ + 3.041
) N2
c − 1
N2
c
(Nc αs)5 T 4 m2
D = 2Nc + Nf
6 g2T 2 , γ = Ncg2T 4π
(
log mD γ + 3.041
)
, as = g2 4π
Giudice+Shaposhnikov, ’93, Moore, ’97, ’00, Moore+Tassler, ’10
Son+Starinets, ’02
ΓCS = λ2 28π3 T 4
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 9
nomenological parameters in V (λ)) S = M3
p N2 c
∫
d5x√−g
[
R − 4 3 (∂λ)2 λ2 + V (λ)
]
where λ = eϕ and (Mpℓ)3 = 45π2.
Gursoy+Kiritsis+Nitti, ’07
V (λ) = 12 ℓ2
1 +
∞
∑
n=1
Vnλn
, λ → 0 V (λ) ∼ λ
4 3
√
log λ + · · · , λ → ∞
as well as the finite temperature thermodynamics.
m2
n ∼ n.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 10
this is the usual IIB axion in ten dimensions.
Sa = M3
p
2
∫
d5x√g Z(λ) (∂a)2 + O
(
(∂a)4 N2
c
)
is exact.
c
) compared with the IHQCD action. UV λ → 0 Z(λ) = Z0
1 +
∞
∑
n=1
cnλn
IR λ → ∞ Z(λ) ∼ c4λ4 + · · · , lim
n→∞
m2
n(0−+)
m2
n(0++) = 1
Gursoy+Kiritsis+Mazzanti+Nitti, ’09 Holography and the Chern-Simons diffusion rate, Elias Kiritsis 11
θ-angle
⟨Tr[F ∧ F]n⟩ are UV finite), and Seiberg-Witten type solutions.
¨ a +
(
3 ˙ A + ˙ Z(λ) Z(λ)
)
˙ a = 0 , ds2 = e2A(r)(dr2 + dxµdxµ)
a(r) = θUV + 2πk + C
∫ r
0 dre−3A
Z(λ) , C = ⟨q(x)⟩ = 1 16π2⟨Tr[F ∧ F]⟩
a(r) ∼ r4 ∼ e− 4
b0λ 12
E(θUV ) = −M3
p
2
∫
d5x√g Z(λ) (∂a)2 = −M3
p
2 Ca(r)
r=0
a(∞) = 0. This determines the solution
Witten, ’79
C = ⟨q(x)⟩ = −θUV + 2πk
∫ ∞
0 dre−3A Z(λ)
E(θUV ) = EIHQCD + M3
p
2 Mink (θUV + 2πk)2
∫ ∞
dr e3AZ(λ)
E(θ) = N2
c E0 + 1
2χ θ2 + O
(
θ4 N2
c
)
, χ = M3
p
∫ ∞
dr e3AZ(λ)
Z(λ) = Z0
(
1 + c4λ4)
0−+ glueball mass (lattice, m0−+
m0++ = 1.50). The predicted next mass agrees well with lattice (m0⋆ −+ m0++ = 2.11).
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 12-
temperature, T ≥ Tc.
ered in the black hole background, ds2 = e2A
(
dr2 f − fdt2 + dxidxi
)
,
[
1 Z(λ)√−g ∂r
(
Z(λ)√−g grr∂r
)
− gµνkµkν
]
δα(r, kµ) = 0 , with in-going wave boundary conditions at the horizon.
Son-shell
α
=
∫
d4k (2π)4 a(−kµ) F(r, kµ) a(kµ)
, F(r, kµ) ≡ −M3
p δα(r, −kµ) Z(λ) √−g grr ∂rδα(r, kµ)/2
ˆ GR(ω,⃗ k) = −2 lim
r→0 F(r, kµ).
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 13
ΓCS sT/N2
c
= Z(λh) 2π , which has implicit dependence on T through Z(λh). λh is the value of the dilaton at the black hole horizon.
c ) is bounded from below by
its value in the T → ∞ limit, lim
T→∞
ΓCS sT/N2
c
= Z0 2π .
1 2 3 4 5 6 TTc 0.99990 0.99995 1.00000 1.00005 1.00010 1.00015 1.00020 CSΚ2 Z02 Π s TNc2
14
(a) The parametrization was crude (b) λh < 1
from linear terms so we reparametrize Z(λ) as Z(λ) = Z0
(
1 + c1λ + c4λ4)
glueball masses fall within one σ of the lattice values for the two first 0−+ glueball masses 0 < c1 < 5 , 0.06 < c4 < 50
ball masses that deviate from lattice results more than 10%.
ΓCS sT/N2
c
≤
Z0 2π is always valid.
14-
IHQCD masses of the 0−+ glueball states with excitation number n, normalized to the 0++ mass for various (c1, c4). From the top (red) dots to the bottom (blue) dots: (c1, c4) = (0, 0.26), (0.5, 0.87), (1, 2.2), (5, 24), (10, 75) The two horizontal blue lines with surrounding blue bands indicate the results and errors, respectively, of the large-N YM lattice masses of the lowest and first excited states, n = 1 and n = 2, (Morningstar-Peardon). 14-
1 2 3 4 5 6 TTc 2 4 6 8
CSΚ2 Z02 Π s TNc2
,
1 2 3 4 5 6 TTc 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
CSΚ2 Z02 Π s TNc2
(a) The numerical result for ΓCS/(sT/N2
c ), normalized to the T → ∞ value κ2Z0/2π, as functions of T/Tc,
for different values of the parameters (c1, c4). From the bottom (red) curve to the top (blue) curve, (c1, c4) = (0, 0.26), (0.5, 0.87), (1, 2.2), (5, 24), (10, 75), (20, 230), (40, 600). (b) Close-up of the curves for (from bottom to top) (c1, c4) = (0, 0.26), (0.5, 0.87), (1, 2.2). 14-
transition is ΓCS(Tc) T 4
c
= 0.31 × Z(λc) 2π > 0.31 × Z0 2π = 1.64 , where we have used the lattice result for the entropy density, s(Tc) = 0.31N2
c T 3 c , (Lucini-Teper-Wenger).
1.64 ≤ ΓCS(Tc) T 4
c
≤ 2.8
standard values, λ ≃ 6π to get a result that is at least 40 times smaller. N = 4 sYM : ΓCS(T) T 4 ≃ 0.045
αs = 0.5, gives the right order of magnitude, ΓMT
CS (Tc)
T 4
c
= 30α5
s
≃ 1 (1)
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 14-
5 10
k Tc
20 40
Ω Tc
0.002 0.001 0.000 0.001
Im GR Tc Mp
3
Our numerical results for ∆Im GR(ω,⃗ k = 0; Tc, 2Tc)/(TcM3
p ) as a function of
ω/Tc and |⃗ k|/Tc. As |⃗ k| increases up to |⃗ k|/Tc ≈ 10, the largest peak shifts from ω/Tc ≈ 22 up to ω/Tc ≈ 30. The width of the peak changes very little.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 15
is bounded below.
tation that has the same order of magnitude mass as the 0−+ glueball.
2-point function of q(xµ) is an important future plan.
as well (V-QCD).
holographically upon addition of the flavor in the pure glue background. In this case, a vacuum state with a magnetic field and net chirality should be constructed.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 16
.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 17
This research has been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), under the grants schemes ”Funding of proposals that have received a positive evaluation in the 3rd and 4th Call
18
∆NCS =
∫
d4x q(xµ) q(xµ) ≡ 1 16π2
∫
M Tr [F ∧ F] =
1 32π2 ϵµνρσTrFµνFρσ .
∆N5 = (NL − NR)|t=∞ − (NL − NR)|t=−∞ = 2Nf ∆NCS .
19
by a Langevin-like process d(∆N5) dt = −ξCS ∆N5 + η(t) , ⟨η(t)η(t′)⟩ ≃ ΓCSδ(t − t′) where ΓCS is the Chern-Simons diffusion rate.
relator while the friction coefficient ξCS is controlled by the retarded corre- lator.
times the change in axial number A ∆N5 = 2
∫
d4x⟨ψ2|q(x)|ψ1⟩ Taking the square of the above equation, summing over all possible |ψ2(t2)⟩ and using the completeness relation I = ∑ |ψ2⟩⟨ψ2| we obtain ⟨(∆N5)2⟩ = 4
∫
d4x d4y ⟨ψ1| q(x) q(y) |ψ1⟩ RETURN
19-
and is given by the two-point function of q(xµ), ΓCS ≡ ⟨(∆NCS)2⟩ V t =
∫
d4x ⟨q(xµ)q(0)⟩symmetric
ΓCS = 2T ξCS = − lim
ω→0
2T ω Im GR(ω,⃗ k = 0) , where GR(ω,⃗ k)= Fourier transform of the retarded Green function of q(xµ).
sphaleron configurations which produce non-zero ΓCS upon decay.
handed than right-handed quarks or the opposite) because of the anomaly of the axial current.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 19-
100 200 300 400 500 600 E MeV 0.0 0.2 0.4 0.6 0.8 1.0 Θ ΘUV
4 (1 + 0.26λ4)
vanishes in the IR θ(U) = θ(1 − U3
0/U3)
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 20
k) for non-zero ω and |⃗ k|. We first set ⃗ k = 0
k = 0) ∝ ω.
k = 0) ∝ ω4 because the theory is conformally invariant in the UV and q(xµ) has dimension four.
21
15 20 25 30 35 40
Ω Tc
0.1 0.2 0.3 0.4 0.5
Im GR Tc Mp
3
Im GR(ω,⃗ k = 0)/(TcM3
p ) as a function of ω/Tc, at Tc. The red dots are our
numerical results that match the solid blue curve, (1.6×10−7)×(ω/Tc)4.051, which is the expected large ω behavior of the spectral function.
21-
k = 0) at large ω comes from the UV (free) part of the two-point function.
k), rendering them practically invisible.
k) at two different temperatures, T1 and T2, and then take the difference, ∆GR(ω,⃗ k; T1, T2) ≡ GR(ω,⃗ k)
k)
21-
10 20 30 40 50 60
Ω Tc
0.0010 0.0005 0.0005
Im GR Tc Mp
3
The difference ∆Im GR(ω,⃗ k = 0; Tc, 2Tc)/(TcM3
p ) as a function of ω/Tc.
The difference goes to zero as ω/Tc → ∞. The prominent minimum at ω/Tc ≈ 10 and maximum at ω/Tc ≈ 22 indicate a shift in spectral weight with increasing T, possibly from the motion of a peak in the spectral function. There is a peak at ω ≃ 20Tc ≃ 1600MeV and width ∼ 10Tc ∼ 1300MeV. Holography and the Chern-Simons diffusion rate, Elias Kiritsis 21-
n 3000 4000 5000 6000 M n 3000 4000 5000 6000 M
(a) (b) Comparison of glueball spectra from our model with b0 = 4.2, l0 = 0.05 (boxes), with the lattice QCD data from Ref. I (crosses) and the AdS/QCD computation (diamonds), for (a) 0++ glueballs; (b) 2++ glueballs. The masses are in MeV, and the scale is normalized to match the lowest 0++ state from Ref. I.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 22
JPC Ref I (MeV) Our model (MeV) Mismatch Nc → ∞
Mismatch
0++ 1475 (4%) 1475 1475 2++ 2150 (5%) 2055 4% 2153 (10%) 5% 0−+ 2250 (4%) 2243 0++∗ 2755 (4%) 2753 2814 (12%) 2% 2++∗ 2880 (5%) 2991 4% 0−+∗ 3370 (4%) 3288 2% 0++∗∗ 3370 (4%) 3561 5% 0++∗∗∗ 3990 (5%) 4253 6% Comparison between the glueball spectra in Ref. I and in our model. The states we use as input in our fit are marked in red. The parenthesis in the lattice data indicate the percent accuracy.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 23
Fit and comparison
HQCD lattice Nc = 3 lattice Nc → ∞ Parameter [p/(N2
c T 4)]T=2Tc
1.2 1.2
Lh/(N2
c T 4 c )
0.31 0.28 (Karsch) 0.31 (Teper+Lucini) V 3 = 170 [p/(N2
c T 4)]T→+∞
π2/45 π2/45 π2/45 Mpℓ = [45π2]−1/3 m0++/√σ 3.37 3.56 (Chen ) 3.37 (Teper+Lucini) ℓs/ℓ = 0.15 m0−+/m0++ 1.49 1.49 (Chen )
χ (191MeV )4 (191MeV )4 (DelDebbio)
Tc/m0++ 0.167
m0∗++/m0++ 1.61 1.56(11) 1.90(17) m2++/m0++ 1.36 1.40(4) 1.46(11) m0∗−+/m0++ 2.10 2.12(10)
gemeier and B. Petersson, “Thermodynamics of SU(3) Lattice Gauge Theory,” Nucl. Phys. B 469, 419 (1996) [arXiv:hep-lat/9602007].
phase transition in SU(N) gauge theories,” JHEP 0502, 033 (2005) [arXiv:hep-lat/0502003]; “SU(N) gauge theories in four dimensions: Exploring the approach to N =∞,” JHEP 0106, 050 (2001) [arXiv:hep-lat/0103027].
lattices,” Phys. Rev. D 73 (2006) 014516 [arXiv:hep-lat/0510074].
SU(3) gauge theory,” Phys. Rev. Lett. 94, 032003 (2005) [arXiv:hep- th/0407052].
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 24-
Marco Panero arXiv: 0907.3719 Holography and the Chern-Simons diffusion rate, Elias Kiritsis 25
Marco Panero arXiv: 0907.3719 Holography and the Chern-Simons diffusion rate, Elias Kiritsis 26
Marco Panero arXiv: 0907.3719 Holography and the Chern-Simons diffusion rate, Elias Kiritsis 27
2
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 28
2
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 29
c2
s ≃ 1
3 − 1 2 V ′2 V 2
1 2 3 4 5 6 0.15 0.2 0.25 0.3 0.35
Gursoy (unpublished) 2009
formula.
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 30
31
Holography and the Chern-Simons diffusion rate, Elias Kiritsis 31-