Testing String Theory with Jets
William Horowitz
The Ohio State University
Columbia University
Frankfurt Institute for Advanced Studies (FIAS)
November 13, 2008
LHC Predictions: Phys. Lett. B666:320, 2008 (arXiv:0706.2336)
RHIC Predictions: J. Phys. G35:044025, 2008 (arXiv:0710.0703)
With many thanks to Miklos Gyulassy
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
1
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A Little History:
QCD as Theory of Strong Force
1935: Yukawa proposes pion as nuclear mediator
1947: Powell, et al., definitively distinguishes p from m
1947-: Particle zoo => 1962: Gell-Mann’s Eightfold Way => 1964: W- found at BNL
1965: Nambu and Hahn propose color to solve Pauli problem
1969-73: Feynman’s partons—weakly-coupled point-like subnuclear particles
1973: Coleman and Gross—Asymptotic freedom unique to nonabelian QFTs
1975: Jets—quarks (’75) and gluons (’79)
1992: SU(Nc = 3)
p
m
Two and three jet
C.events:
M. G. Lattes, H. Muirhead, G. P. S. Occhialini, and C. F. Powell,
R. Brandelik et al.
PROCESSES
(TASSO), INVOLVING CHARGED MESONS, Nature 159, 694
Phys. Lett. B86, (1947).
243 (1979)
V. E. Barnes et al., OBSERVATION OF A HYPERON WITH
STRANGENESS -3, Phys. Rev. Lett. 12, 204 (1964)
mW- = 1686 +/- 12 MeV/c2
D. Decamp et al. (ALEPH), Phys. Lett. B284, 151 (1992)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
2
Traditional Toolbox for QCD
Previously only two methods:
Lattice QCD
Two 10 Tflops QCDOC Computers: RBRC and DOE
11/13/08
Nuclear Seminar, The Ohio State University
pQCD
Diagrams!
William Horowitz
3
Traditional Tools (cont’d)
• Successful
Lattice QCD
pQCD
Davies et al. (HPQCD),
PRL 92, 022001 (2004)
• But limited
• All momenta
• Euclidean correlators
11/13/08
de Florian, Sassot, Stratmann, Phys.Rev.D75:114010,2007
• Any quantity
• Small coupling
(large momenta)
Nuclear Seminar, The Ohio State University
William Horowitz
4
Maldacena Conjecture
Large Nc limit of d-dimensional conformal field theory
dual to string theory on the product of d+1-dimensional
Anti-de Sitter space with a compact manifold
J Maldacena, Adv.Theor.Math.Phys.2:231-252,1998
Bosonic part of IIB low energy
effective action
TPlasma = THawking
Geometry of bosonic part of 10D supergravity, near horizon limit
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
5
Regime of Applicability
– Large Nc, constant ‘t Hooft coupling (
)
Small quantum corrections
– Large ‘t Hooft coupling
Small string vibration corrections
– Only tractable case is both limits at once
Classical supergravity (SUGRA)
Q.M. SSYM
=> C.M. SNG
J Friess, S Gubser, G Michalogiorgakis, S Pufu, Phys Rev D75:106003, 2007
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
6
Strong Coupling Calculation
• The supergravity double conjecture:
QCD  SYM  IIB
– IF super Yang-Mills (SYM) is not too
different from QCD, &
– IF Maldacena conjecture is true
– Then a tool exists to calculate stronglycoupled QCD in SUGRA
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Testing String Theory
Adapted from P Sorensen, WWND ‘08, arXiv:0808.0503
=> 1/4p?
Kallosh and Linde, JCAP 0704:017,2007:
Too small to be detected
11/13/08
Huovinen et al., Phys. Lett. B503 (2001) 58
Nuclear Seminar, The Ohio State University
William Horowitz
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What’s All the Fuss About?
…data [from RHIC] appear to be
more accurately described using
string theory methods than with
more traditional approaches.
Hold yer horses!
Will Horowitz (OSU)
Let’s look at the details
Brian Greene (TV)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
9
QGP Creation
– Robust prediction of QCD phase transition
Walecka:
Hagedorn:
S. C. Frautschi, Phys. Rev. D3, 2821 (1971)
J. D. Walecka, Theoretical Nuclear and Subnuclear Physics, 2nd ed.
Lattice:
Karsh et al., Phys. Rev. D62, 034021 (2000), Nucl. Phys. A698, 199 (2002),
PoS LAT2005, 193 (2006)
11/13/08
M. Cheng et al., Phys. Rev. D77, 014511 (2008)
Nuclear Seminar, The Ohio State University
William Horowitz
10
Probing the QGP
• Low momentum (low-pT) particles
– Collective dynamics of the bulk
• Statistical Models: temperature
• Hydrodynamics: spectra, elliptic flow
• HBT (Hanbury-Brown Twiss): freeze-out surface
• High momentum (high-pT) particles
– Parton jets, vacuum fragmentation
• Learn about medium (jet tomography)
• Learn about energy loss mechanism (pQCD, ST)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
11
Geometry of a HI Collision
M Kaneta, Results from the
Relativistic Heavy Ion Collider (Part II)
• Hydro propagates IC
T Ludlum and L McLerran, Phys. Today 56N10:48 (2003)
– Results depend strongly on initial conditions
• Viscosity reduces eventual momentum
anisotropy
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Perfect Fluidity:
AdS + Hydro’s Most Famous Success
– Hydro h/s small ~ .1
• QGP fluid near-perfect
liquid
– Naïve pQCD => h/s ~ 1
• New estimates ~ .1
Z Xu, C Greiner, and H Stoecker, PRL101:082302 (2008)
– Lowest order AdS result:
h/s = 1/4p
• Universality?
P Kovtun, D Son, and A Starinets, Phys.Rev.Lett.94:111601 (2005)
P Kats and P Petrov, arXiv:0712.0743
M Brigante et al., Phys. Rev. D77:126006 (2008)
11/13/08
D. Teaney, Phys. Rev. C68, 034913 (2003)
Nuclear Seminar, The Ohio State University
William Horowitz
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IC, Viscosity, and Hydro
T Hirano, et al., Phys. Lett. B636:299-304, 2006
• Sharper IC (CGC) => viscosity
• Softer IC (Glauber) => “perfect”
• Test IC with fluctuations?
• Control over hadronization?
P Sorensen, WWND ‘08, arXiv:0808.0503
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Why High-pT Jets?
• IC smaller effect
• Vacuum fragmentation well controlled
• Compare unmodified p+p collisions to
A+A:
pT
pT
2D Transverse directions
Longitudinal
(beam pipe) direction
Figures from http://www.star.bnl.gov/central/focus/highPt/
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
15
Jet Physics Terminology
Naïvely: if medium has no effect, then RAA = 1
Common variables used are transverse
momentum, pT, and angle with respect to the
reaction plane, f
Convenient to Fourier expand RAA:
11/13/08
Nuclear Seminar, The Ohio State University
pT
f
William Horowitz
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pQCD Success at RHIC:
(circa 2005)
Y. Akiba for the PHENIX collaboration,
hep-ex/0510008
– Consistency:
RAA(h)~RAA(p)
– Null Control:
RAA(g)~1
– GLV Prediction: Theory~Data for reasonable
fixed L~5 fm and dNg/dy~dNp/dy
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
17
Trouble for wQGP Picture
Hydro
h/s
too small
e-2Rtoo
too
•v
wQGP
notsmall
ruled
out, but what if we try
AA large
strong coupling?
A.
H. Feng,
and J. Jia,
C71:034909
(2005)
M. Drees,
Djorjevic,
M. Gyulassy,
R.Phys.
Vogt,Rev.
S. Wicks,
Phys. Lett.
(first
byD.E.Teaney,
Shuryak,
Phys.
Rev.
C66:027902
(2002))
Rev.
C68,
034913 (2003)
B632:81-86
(2006) Phys.
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Qualitative AdS/CFT Successes:
-R1~
sMach
=(3/4)
wave-like
s
structures
,
similar
• h/s
e-strong
RAA
~
p,
h
R
;
e
)to Lattice
~
1/4p
<<
weak
AA
AA(fh/s
AdS/CFT
pQCD
Naïve AdS/CFT
S. S. Gubser, S. S. Pufu, and A. Yarom,
PHENIX, Phys.Rev.Lett.101:082301,2008
arXiv:0706.0213
J. P. Blaizot, E. Iancu, U. Kraemmer, A. Rebhan, hep-ph/0611393
T. Hirano and M. Gyulassy, Nucl. Phys. A69:71-94 (2006)
PHENIX, Phys. Rev. Lett. 98, 172301 (2007)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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AdS/CFT Energy Loss Models
• Langevin model
– Collisional energy loss for heavy quarks
– Restricted to low pT
– pQCD vs. AdS/CFT computation of D, the diffusion
coefficient
Moore and Teaney, Phys.Rev.C71:064904,2005
Casalderrey-Solana and Teaney, Phys.Rev.D74:085012,2006; JHEP 0704:039,2007
• ASW model
– Radiative energy loss model for all parton species
– pQCD vs. AdS/CFT computation of
– Debate over its predicted magnitude
BDMPS, Nucl.Phys.B484:265-282,1997
Armesto, Salgado, and Wiedemann, Phys. Rev. D69 (2004) 114003
Liu, Ragagopal, Wiedemann, PRL 97:182301,2006; JHEP 0703:066,2007
• ST drag calculation
– Drag coefficient for a massive quark moving through a
strongly coupled SYM plasma at uniform T
– not yet used to calculate observables: let’s do it!
Gubser, Phys.Rev.D74:126005,2006
Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013,2006
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
20
AdS/CFT Drag
• Model heavy quark jet energy loss by
embedding string in AdS space
dpT/dt = - m pT
m = pl1/2 T2/2Mq
J Friess, S Gubser, G Michalogiorgakis, S Pufu, Phys Rev D75:106003, 2007
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Energy Loss Comparison
D7 Probe Brane
t

– AdS/CFT Drag:
zm = 2pm / l1/2
Q, m
dpT/dt ~ -(T2/Mq) pT
zh = pT
z=0
v
x
3+1D Brane
Boundary
D3 Black Brane
(horizon)
Black Hole
– Similar to Bethe-Heitler
dpT/dt ~ -(T3/Mq2) pT
– Very different from LPM
dpT/dt ~ -LT3 log(pT/Mq)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
22
RAA Approximation
– Above a few GeV, quark production
spectrum is approximately power law:
• dN/dpT ~ 1/pT(n+1), where n(pT) has some
momentum dependence
y=0
RHIC
– We can approximate RAA(pT):
• RAA ~ (1-e(pT))n(pT),
where pf = (1-e)pi (i.e. e = 1-pf/pi)
LHC
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Looking for a Robust, Detectable Signal
– Use LHC’s large pT reach and identification of c
and b to distinguish between pQCD, AdS/CFT
• Asymptotic pQCD momentum loss:
erad ~ as L2 log(pT/Mq)/pT
• String theory drag momentum loss:
eST ~ 1 - Exp(-m L),
m = pl1/2 T2/2Mq
S. Gubser, Phys.Rev.D74:126005 (2006); C. Herzog et al. JHEP 0607:013,2006
– Independent of pT and strongly dependent on Mq!
– T2 dependence in exponent makes for a very sensitive probe
– Expect: epQCD
0 vs. eAdS indep of pT!!
• dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
24
Model Inputs
– AdS/CFT Drag: nontrivial mapping of QCD to SYM
• “Obvious”: as = aSYM = const., TSYM = TQCD
– D 2pT = 3 inspired: as = .05
– pQCD/Hydro inspired: as = .3 (D 2pT ~ 1)
• “Alternative”: l = 5.5, TSYM = TQCD/31/4
• Start loss at thermalization time t0; end loss at Tc
– WHDG convolved radiative and elastic energy loss
• as = .3
– WHDG radiative energy loss (similar to ASW)
•
= 40, 100
– Use realistic, diffuse medium with Bjorken expansion
– PHOBOS (dNg/dy = 1750); KLN model of CGC (dNg/dy = 2900)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
25
LHC c, b RAA pT Dependence
WH, M. Gyulassy, arXiv:0706.2336
– Significant
NaïvePrediction
LHC
Unfortunately,
Large
suppression
expectations
rise large
inZoo:
Rleads
met
suppression
What
(pTin
to
) for
full
flattening
a Mess!
pQCD
numerical
pQCD
Rad+El
similar
calculation:
to AdS/CFT
AA
– Use
Let’sofgorealistic
through
dRAA
geometry
step
(pT)/dp
by step
> 0 Bjorken
=> pQCD;
expansion
dRAA(pTallows
)/dpT <
saturation
0 => ST below .2
Tand
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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An Enhanced Signal
• But what about the interplay between
mass and momentum?
– Take ratio of c to b RAA(pT)
• pQCD: Mass effects die out with increasing pT
RcbpQCD(pT) ~ 1 - as n(pT) L2 log(Mb/Mc) ( /pT)
– Ratio starts below 1, asymptotically approaches 1.
Approach is slower for higher quenching
• ST: drag independent of pT, inversely
proportional to mass. Simple analytic approx.
of uniform medium gives
RcbpQCD(pT) ~ nbMc/ncMb ~ Mc/Mb ~ .27
– Ratio starts below 1; independent of pT
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
27
LHC RcAA(pT)/RbAA(pT) Prediction
• Recall the Zoo:
WH, M. Gyulassy, arXiv:0706.2336 [nucl-th]
– Taking the ratio cancels most normalization differences seen previously
– pQCD ratio asymptotically approaches 1, and more slowly so for increased
quenching (until quenching saturates)
WH, M.times
Gyulassy,
arXiv:0706.2336
– AdS/CFT ratio is flat and many
smaller
than[nucl-th]
pQCD at only moderate pT
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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Not So Fast!
– Speed limit estimate for
applicability of AdS drag
• g < gcrit = (1 + 2Mq/l1/2 T)2
~ 4Mq2/(l T2)
– Limited by Mcharm ~ 1.2 GeV
• Similar to BH
LPM
– gcrit ~ Mq/(lT)
– No Single T for QGP
D7 Probe Brane
Worldsheet boundary
Spacelike if g > gcrit
x5
Trailing
String
“Brachistochrone”
• smallest gcrit for largest T
T = T(t0, x=y=0): “(”
• largest gcrit for smallest T
T = Tc: “]”
11/13/08
Q
Nuclear Seminar, The Ohio State University
D3 Black Brane
“z”
William Horowitz
29
LHC RcAA(pT)/RbAA(pT) Prediction
(with speed limits)
WH, M. Gyulassy, arXiv:0706.2336 [nucl-th]
– T(t0): (, corrections unlikely for smaller momenta
– Tc: ], corrections likely for higher momenta
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
30
Measurement at RHIC
– Future detector upgrades will allow for identified c
and b quark measurements
– RHIC production spectrum significantly
harder than LHC
•
• NOT slowly varying
y=0
RHIC
– No longer expect
pQCD dRAA/dpT > 0
• Large n requires
corrections to naïve
Rcb ~ Mc/Mb
11/13/08
Nuclear Seminar, The Ohio State University
LHC
William Horowitz
31
RHIC c, b RAA pT Dependence
WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]
• Large increase in n(pT) overcomes reduction in
E-loss and makes pQCD dRAA/dpT < 0, as well
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
32
RHIC Rcb Ratio
pQCD
pQCD
AdS/CFT
AdS/CFT
WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]
• Wider distribution of AdS/CFT curves due to large n:
increased sensitivity to input parameters
• Advantage of RHIC: lower T => higher AdS speed limits
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
33
Conclusions
• Previous AdS qualitative successes inconclusive
• AdS/CFT Drag observables calculated
• Generic differences (pQCD vs. AdS/CFT Drag)
seen in RAA
– Masked by extreme pQCD
• Enhancement from ratio of c to b RAA
– Discovery potential in Year 1 LHC Run
• Understanding regions of self-consistency crucial
• RHIC measurement possible
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
34
Backup Slides
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
35
Another AdS Test: Correlations
B Betz, M Gyulassy, J Noronha, and G Torrieri, arXiv:0807.4526
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
36
Geometry of a HI Collision
Medium density and jet production
are wide, smooth distributions
Use of unrealistic geometries strongly
bias results
S. Wicks, WH, M. Djordjevic, M. Gyulassy,
Nucl.Phys.A784:426-442,2007
1D Hubble flow => r(t) ~ 1/t
=> T(t) ~ 1/t1/3
M. Gyulassy and L. McLerran, Nucl.Phys.A750:30-63,2005
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
37
Langevin Model
– Langevin equations (assumes gv ~ 1 to neglect
radiative effects):
– Relate drag coef. to diffusion coef.:
– IIB Calculation:
AdS/CFT here
• Use of Langevin requires relaxation time be large
compared to the inverse temperature:
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
38
But There’s a Catch (II)
• Limited experimental pT reach?
ALICE Physics Performance Report, Vol. II
– ATLAS and CMS do not seem to be limited in this
way (claims of year 1 pT reach of ~100 GeV) but
systematic studies have not yet been performed
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
39
LHC p Predictions
WH, S. Wicks, M. Gyulassy, M. Djordjevic,
in preparation
11/13/08
• Our predictions show a
significant increase in RAA as a
function of pT
• This rise is robust over the
range of predicted dNg/dy for
the LHC that we used
• This should be compared to
the flat in pT curves of AWSbased energy loss (next slide)
• We wish to understand the
origin of this difference
Nuclear Seminar, The Ohio State University
William Horowitz
40
Asymptopia at the LHC
Asymptotic pocket formulae:
DErad/E ~ a3 Log(E/m2L)/E
DEel/E ~ a2 Log((E T)1/2/mg)/E
11/13/08
WH, S. Wicks,
M. The
Gyulassy,
Djordjevic,
in preparation
Nuclear
Seminar,
Ohio M.
State
University
William Horowitz
41
K. J. Eskola, H. Honkanen, C. A. Salgado, and U.
A. Wiedemann, Nucl. Phys. A747:511:529 (2005)
K. J. Eskola, H. Honkanen, C. A. Salgado, and
U. A. Wiedemann, Nucl. Phys. A747:511:529
(2005)
11/13/08
A. Dainese, C. Loizides, G. Paic, Eur. Phys. J. C38:461-474 (2005)
Nuclear Seminar, The Ohio State University
William Horowitz
42
Pion RAA
• Is it a good measurement for tomography?
– Yes: small experimental error
– Maybe not: some models
appear “fragile”
• Claim: we should not be so immediately dismissive of the pion RAA as a tomographic tool
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
43
Fragility:
A Poor Descriptor
• All energy loss models with a formation time
saturate at some RminAA > 0
• The questions asked should be quantitative :
– Where is RdataAA compared to RminAA?
– How much can one change a model’s controlling
parameter so that it still agrees with a measurement
within error?
– Define sensitivity, s = min. param/max. param that
is consistent with data within error
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
44
Different Models have Different
Sensitivities to the Pion RAA
• GLV:
s<2
• Higher Twist:
s<2
• DGLV+El+Geom:
s<2
• AWS:
s~3
11/13/08
WH, S. Wicks, M. Gyulassy, M. Djordjevic, in preparation
Nuclear Seminar, The Ohio State University
William Horowitz
45
T Renk and K Eskola, Phys. Rev. C 75, 054910 (2007)
WH, S. Wicks, M. Gyulassy, M. Djordjevic, in preparation
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
46
A Closer Look at ASW
The lack of sensitivity needs to be more closely examined
because (a) unrealistic geometry (hard cylinders) and no
expansion and (b) no expansion shown against older data (whose
error bars have subsequently shrunk
(a)
(b)
K. J. Eskola, H. Honkanen, C. A. Salgado, and U. A. Wiedemann,
Nucl. Phys. A747:511:529 (2005)
11/13/08
A. Dainese, C. Loizides, G. Paic, Eur. Phys. J. C38:461-474 (2005)
Nuclear Seminar, The Ohio State University
William Horowitz
47
Surface Bias vs. Surface Emission
– Surface Emission: one phrase explanation of fragility
• All models become surface emitting with infinite E loss
– Surface Bias occurs in all energy loss models
• Expansion + Realistic geometry => model probes a large
portion of medium
A. Majumder, HP2006
11/13/08
S. Wicks, WH, M. Gyulassy, and M. Djordjevic, nucl-th/0512076
Nuclear Seminar, The Ohio State University
William Horowitz
48
A Closer Look at ASW
– Difficult to draw conclusions on
inherent surface bias in AWS
from this for three reasons:
• No Bjorken expansion
• Glue and light quark contributions
not disentangled
• Plotted against Linput (complicated
mapping from Linput to physical
distance)
A. Dainese, C. Loizides, G. Paic, Eur. Phys. J. C38:461-474 (2005)
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
49
Additional Discerning Power
– Adil-Vitev in-medium fragmentation rapidly approaches, and then broaches, 1
» Does not include partonic energy loss, which will be nonnegligable as ratio goes to unity
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
50
Conclusions
• AdS/CFT Drag observables calculated
• Generic differences (pQCD vs.
AdS/CFT Drag) seen in RAA
– Masked by extreme pQCD
• Enhancement from ratio of c to b RAA
– Discovery potential in Year 1 LHC Run
• Understanding regions of selfconsistency crucial
• RHIC measurement possible
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
51
Shameless self-promotion by the presenter
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
52
Geometry of a HI Collision
Medium density and jet production
are wide, smooth distributions
Use of unrealistic geometries strongly
bias results
S. Wicks, WH, M. Djordjevic, M. Gyulassy,
Nucl.Phys.A784:426-442,2007
1D Hubble flow => r(t) ~ 1/t
=> T(t) ~ 1/t1/3
M. Gyulassy and L. McLerran, Nucl.Phys.A750:30-63,2005
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
53
Outline
• Motivation for studying AdS/CFT
• Introduction to Heavy Ion Physics
• pQCD vs. AdS Drag: Expectations,
Results, Limitations
• Conclusions
11/13/08
Nuclear Seminar, The Ohio State University
William Horowitz
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