Transcript [PPT]

Anastasios Taliotis: Un. Of Crete, CCTP
Elias Kiritsis and Anastasios Taliotis Arxiv:[1111.1931]
Outline
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Goals: State Problem/Facts from HIC
Tools: Relating AdS/CFT with Multiplicities
Introduction to TS, an example
Review of earlier works
Possible improvement ingredients: IR applied to several
geometries
Digression: pQCD and the Saturation Scale Qs and
weak coupling matching
Quantized, Normalizable Modes
Results, Data and Predictions
Conclusions/Future Work
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Goals: State Problem/Data
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Goal I.
Finish on Time
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Goal II.: State Problem/Data
• Heavy Ion Collisions:
isentropic evolution
from Yellow Blue
[AdS approach:Kiritsis,Taliotis]
• Stages of Collision
initial state
hadronic phase
and freeze-out
QGP and
hydrodynamic expansion
pre-equilibrium
hadronization
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Multiplicities Nch
hadronic phase
and freeze-out
QGP and
hydrodynamic expansion
initial state
l+
pre-equilibrium
hadronization
N ch = å N ch;i =å ò
i
i
l–
d 3 N ch;i
2
dyd
pT =
2
dy d pT
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Nch from Confining and
non-confining matter
• Find
Nch = Nch (s);
s = sNN = 2E / A
I. Conformal matter (AdS5): Nch = Nch (s / Q2 )
II. Confined matter:
Nch = Nch (s / Q , L
2
2
QCD
/Q )
2
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Relating S with Nch
• 1 Charged part. ÷ ½ Neutral part.
=> Ntot = Nch + Nneu = 3/2Nch
• " Nch $ 5
units of S [Heinz]
=> Sprod=5 × 3/2 × Nch =7.5Nch
Nch = Sprod/7.5
• Use Nch, Ntot, Sprod interchangeably (proportional)
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Tools: Relating AdS/CFT with Nch
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AdS/CFT
• Basic Result AdS/CFT:
SST = SGT
• Conclude: Estimating SprodSTNch
• Estimate Sprod using standard thms of GR
[Penrose, Hawking, Ellis]
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Introduction to TS
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• What this method does not: [Ads:,Albacete,Kovcegov,Taliotis;Romatscke,
Chesler,Yaffe,Heller,Janik,Peschanski…, Flat:D’Eath,Payne,Konstantinu,Tomaras,Spirin,Taliotis…]
• What this method can do: Strap≤Sprod . By reducing to unusual BV
problem [Giddings,Eardly,Nastase,Kung,Gubser,Yarom,Pufu,Kovchegov,
Shuryak,Lin,kiritsis,Taliotis,Aref’eva,Bagrov,Joukovskaya,...]
marginally
trapped
surface
[Picture from GYP]
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Example: 4D Flat
x ± < 0, x^ = x12 + x22
Superimpose two
A/S solutions
ds = h dx dx +[mlog(x^ )d (x )(dx ) +(+ «-)]
2
mn
m
n
+
f+
Head On & f+ = f- = f,
+ 2
f-
1
2
Strap = 2 ´
d
x^
ò
4G4 C(m )
C : Ñ^2 (f - y ) = 0, y |C = 0, (Ñ^yÑ^y )C = 8
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C : Ñ^2 (f - y ) = 0, y |C = 0, (Ñ^yÑ^y )C = 8
fC
y = f - C f- = m log(x^ / x^C )
f-
(Ñ^y )C = 8 => x £ m / 8
C
^
1
2
C 2
2
2
Strap = 2 ´
d
x
~
(x
)
~
m
~
E
~s
ò
^
^
4G4 C(m )
[Giddings & Eardley,03’]
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Review Earlier Works
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Shock Metric in AdS
ds2 = b(r)2 [h mn dx m dxn + (dr)2 +1/ b(r)f (q)d (x+ )(dx + )2 ]
x^2 + (r - r ')2
b(r) = L / r, q =
4rr '
2E(r ')4
+
T++ = ¶g++ =
d
(x
);
2
2 3
p (x^ + (r ') )
• AdS Dictionary:
• BC of TS imply
• Then
[Gubser,Yarom,Pufu,Tanaka,Hotta]
òT
++
=E
G5
G5
1/3
qC £ ( 3 Er ') . Note presence 3
L
L
Strap
E®¥
L3
N ch ³
~
7.5 G5
qC
L3 2
L3 1/3 2 1/3
ò qdq = G qC ~ ( G ) (sr ' )
0
5
5
[Gubser,Yarom,Pufu]
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L3
• To check data must choose
 Lattice
G5
• Nch~ s1/3 [GYP,08’]
[GYP]
Data Nch~ s1/4. Indeed:
Ncharged
7000
6000
5000
PHOBOS,
Arxiv:0210015
Landau
AdS
4000
3000
Plot:[GYP,08’]
2000
1000
100
200
300
400
SNN GeV
• Lessons: (i) A brave effort absorb QFT complexities in a BV problem
Figure 2: A plot of t he total number of charged particles vs. energy. The data points were
taken Worth
from table further
I I of the PHOBOS
results [23]. We show in red the region consist ent
(ii)
investigation
with the bound (15) obtained via t he gauge-string duality, using point -sourced shocks and
• Q: What is missing?
estimates described in the t ext, and assuming the bound (6). The blue curve corresponds to
the prediction of t he Landau model [24].
the latter dependence, predicted by t he Landau model [24],2 seems to hold over a strikingly
large range of energies. Put different ly, t he inequality in (15) is consistent with all heavy-ion
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Possible Improvement Ingredients
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IR physics: Confinement
• According data large fraction of particles
produced low pT~2-300 MeV~ΛQCD. [CMS Col.]
• Suggests possibility non-pQCD effects be
important
• Conclude: confinement may improve AdS/CFT
results
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IHQCD
• Dilaton-Gravity Theories
[Gursoy,Kiritsis,Nitti,Mazzanti,Michalogiorgakis,Gubser,Nelore]
• Appropriate scalar V’s and using
results
Where scale factors b(r) can be
(i) Non-confining: b(r) : (r / L) , a £ -1
r-r
b(r) : (
) , a >1/ 3; e
(ii) Confining:
L
a
0 a
-(r/R)
a
, a > 0; e
-(
R a
)
r-r0
,a > 0
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Entropy from Uniform and NonUniform transverse profiles with or
without confinement
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Uniform Transverse Glueballs
• Using BC
& TS volume
• Cases Analyzed:
3a-1
6a
I. Non-Confining
b(r) : (r / L) , a £ -1 S ~ s
II. Confining
r - r0 a
b(r) : (
) , a >1/ 3; S ~ s
L
III. Confining
IV. Confining
a
1
2
a
3a -1
;1/ 3 <
<1/ 2
6a
3a+1
6a
;1 / 2 <
3a +1
<1
6a
1+a
a
b(r) : e-(r/R) , a > 0; S ~ s log (s)
b(r) : e
-(
R a
)
r-r0
1
2
, a > 0; S ~ s log
1-a
a
(s)
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Non-Uniform Transverse Glueballs
☐ϕ=δ(x-x’)
Cases Analyzed:
I. Power-Like
b(r) : (r / L)a
2 / 3< power <1
Confining
2 / 3< power <1
Non-Confining
II. Exponential
b(r) : e-r/R
(Numerically)
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Most S produced from UV
Observation: According to AdS/CFT for classes of
b(r)’s most S produced in UV part of the TS
Argument:
2
2
x
+
(r
r
')
L
• Have shown b(r) = , (Er ')1/3 ~ ^,C C
r
4rC r '
• => as Elarge, then rUV0
x (E, r )
r
r
S
~
• Have
ò r dr
E
• But integrand singular at UV E
E
• => most S comes from UV
r’
rIR (E )
trap
rUV (E )
2
^
C
3
C
UV
C
IR
3
2
1
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• At UV g<<1=> expect Nch~small=> S ~small.
• Maybe we should not used geometry where it
breaks down
• Way out? Incorporate weak coupling physics..
• How?
• Cut surface at rc1(E)>rUV(E) for all E [GYP]
• But where exactly?
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Digression: pQCD and Qs
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Saturation Scale
• Intuitive def: Qs is a trans. scale in nucleus color
charge becomes dense
Boost
x
+
• Free=interaction:
¶n Am = gAn Am Þ A ~ Qs / g
• Strong classical gluon field g<<1,Qs>>ΛQCD
• Aμ strong, then CGC theory applies and Qs
pertubatively; details:[Dumitru,Jalalian-Marian,Kovchegov,,BNL group:
McLerran,Venugopalan,Khrazeev,…]
, l ' [0.1, 0.15]
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rge multiplicity to be produced. Therefore, as we have argued in t he beg
is section, we will impose asymptotic freedom by cutting-off the trappe
some r 0 as in [31]. It is natural to expect that r 0 may be energy-de
e propose as a nat ural cut -off t he sat urat ion scale Qs (see figure 2) by id
∼ 1/ Qs i.e.
Cutting the TS
• Propose cut TS at rs ~1/Qs provided rs>rUV
R3,1
1/ Qs
R
r
Asym. Freedom Ent ropy Product ion
IR
• Effectively treat weak-strong coupling matching
by step-function (see results follow)
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Localized Transverse Distributions &
Quantized, Normalizable Modes
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e radial solution of equation (6.1) reduces to a (finite) p
An Interesting
Geometry:
at small
r and is normalizable.
Normalizability in the
en the corresponding eigenfunction g1(kn r ) satisfies
•
normalized ò T++ = E
b(r ) 3|g1(kn r )|2dr < ∞ .
, n>0
• Quantized Gravitons:
e set ofLinear
the
normalizable eigenfunctions is given by
glueball trajectories: [Kiritsis, Mazzanti,Nitti]
4
r
al solution of equation (6.1) (2)
reduces
2 to
2 a (finite) polynom
n =pnomials
0, 1, 2, ...
• Then
 4 L n (3r / R ),finite
R Normalizability
mall r and is normalizable.
in the radial
(2)
(2)
2
2
L0 =1, L1 =1- r / R , etc
(2)
he
eigenfunction
g1(knLaguerre
r ) satisfies
erecorresponding
L n are the (finite)
associated
polynomial
alitative behavior in any background
that is confining w
3
2
b(r ) |g1(kn r )| dr < ∞ .
• Normalizable:
eballs and a mass gap. [Kiritsis,
The Mazzanti,Michalogiorgakis,Nitti]
values in (6.23) coincide w
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the radial solution of equation (6.1) reduces t o a (finit e) polynomia
r 4 at small r and is normalizable. Normalizability in the radial dir
when the corresponding eigenfunction g1(kn r ) sat isfies
TS for the n=1 mode
b(r ) 3|g1(kn r )|2dr < ∞ .
• Generally
The set of the normalizable eigenfunctions is given by
r 4 (2) 2 2
L (3r / R ),
R4 n
n = 0, 1, 2, ...
(2)
where L n are the (finite) associated Laguerre polynomials of degre
qualitative behavior in any background that is confining with a disc
glueballs and a mass gap. The values in (6.23) coincide
C C with t he m
g1K 0
2+ + glueballs.
1
• Can show only Ck contributes: yk ~ g1K0 x⊥
•
I
I0
C
0
BC:
(see results)
R
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r
nth mode Strap
• Formulas adequate for numerical analysis
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Recap
•
•
•
•
•
•
Nch = Sprod/7.5
Several b’s* (conf. or not)=> several Strap(s)
None described data Nch ~s1/4 or similar
Most S comes from UV
Cut TS at UV (i) E independent (ii) E depended Qs
Seen quantized, normalizable, graviton
(sm)wave-functions. T++ falls-off exponentially
(Ko)
*It is remarked that out of these geometries only AdS5 reduces (trivially) to AdS5 at the UV.
Results, Data & Predictions
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Results .I
• We have constracted exact (point-like J++) shocks.
• Exponential b’s with UVconst cut yield Strap~ log2(s).
• When b=(r/L)a=1 (confining) with UVconst cut yields Strap~ s1/4 : fits data.
• AdS geometry with unif. profiles produces least Strap
• In confining geometries only normalizable modes result a TS
• Motivate a set of non trivial entropy inequalities,
Define:
a)
b)
GYP when b=L/r. T++ falls as power:~ 1/(x2+x20)3
IHQCD when b=L/r exp[-r2/R2].
Neither has UV-cut. Then *:
*It is remarked that both of these geometries reduce (non-trivially generally) to AdS5 at the UV.
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Results .II: Non trivial inequalities
• Numerically or Analytically found:
I.
E / k = fixed
>
> >
>
>
II.
III.
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Results III. Attempt to Describe DataPredictions (2 Geometries)
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Geometry I. b=L/rexp[-r2/R2] no UV cut-off;n=1
PHOBOS,
Arxiv:0210015
AuAu
PbPb
• Predictions PbPb (A=207): Nch≈19100, 27000,
30500 for 2.76, 5.5 and 7 TeV respectively.
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Geometry II. b=L/r with UV cut at c/Qs
PHOBOS,
Arxiv:0210015
AuAu
Strap =
Lattice;[GYP]
PbPb
Nch » 390(A / AAu )17/18 ( s / GeV)0.483
• Predictions pp (A=1): Nch ≈70, 110, 190, 260, for 0.9, 2.36, 7
and 14 TeV respectively.
• Predictions PbPb (A=207): Nch≈18750, 261800, 29400 for 2.76,
5.5 and 7 TeV respectively.
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Alice Preliminary Results: 2.76 TeV
ALICE,
Arxiv:1107.1973
Dashed line: Our theoretical curve as function of A
at fixed s1/2=2.76 TeV. Data Points: Nch(Npart//2).
• As collision gets more central (our case), data follow
our curve better.
• In particular: at A=190, we predict Nch=17300!!!
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Results III. Conclusions
• Both treatments seem to describe data.
• A more refined investigation required:
 More careful matching with gravity parameters
More Data
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Future Work….
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Thank you
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