1. Shock Treatment: Heavy Quark Energy Loss in a Novel Geometry
William Horowitz
The Ohio State University
April 3, 2009
With many thanks to Yuri Kovchegov and Ulrich Heinz
Quark Matter 2009

2. pQCD Success in High-pT at RHIC:
(circa 2005)
Y. Akiba for the PHENIX collaboration, hep-ex/
Consistency: RAA(h)~RAA(p)
Null Control: RAA(g)~1
GLV Calculation: Theory~Data for reasonable fixed L~5 fm and dNg/dy~dNp/dy
Quark Matter 2009

3. Trouble for High-pT wQGP Picture
v2 too small
NPE supp. too large
p0 v2
WHDG
C. Vale, QM09 Plenary (analysis by R. Wei)
NPE v2
STAR, Phys. Rev. Lett. 98, (2007)
Pert. at LHC energies?
PHENIX, Phys. Rev. Lett. 98, (2007)
Quark Matter 2009

4. Motivation for High-pT AdS
Why study AdS E-loss models?
Many calculations vastly simpler
Complicated in unusual ways
Data difficult to reconcile with pQCD
pQCD quasiparticle picture leads to dominant q ~ m ~ .5 GeV mom. transfers
=> Nonperturbatively large as
Use data to learn about E-loss mechanism, plasma properties
Domains of self-consistency crucial for understanding
Quark Matter 2009

5. 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 strongly-coupled QCD in classical SUGRA
Quark Matter 2009

6. AdS/CFT Energy Loss Models I
Langevin Diffusion
Collisional energy loss for heavy quarks
Restricted to low pT
pQCD vs. AdS/CFT computation of D, the diffusion coefficient
ASW/LRW model
Radiative energy loss model for all parton species
pQCD vs. AdS/CFT computation of
Debate over its predicted magnitude
Moore and Teaney, Phys.Rev.C71:064904,2005
Casalderrey-Solana and Teaney, Phys.Rev.D74:085012,2006; JHEP 0704:039,2007
BDMPS, Nucl.Phys.B484: ,1997
Armesto, Salgado, and Wiedemann, Phys. Rev. D69 (2004)
Liu, Ragagopal, Wiedemann, PRL 97:182301,2006; JHEP 0703:066,2007
Quark Matter 2009

7. AdS/CFT Energy Loss Models II
String Drag calculation
Embed string rep. quark/gluon in AdS geom.
Includes all E-loss modes (difficult to interpret)
Gluons and light quarks
Empty space HQ calculation
Previous HQ: thermalized QGP plasma, temp. T,
Gubser, Gulotta, Pufu, Rocha, JHEP 0810:052, 2008
Chesler, Jensen, Karch, Yaffe, arXiv: [hep-th]
Kharzeev, arXiv: [hep-ph]
Gubser, Phys.Rev.D74:126005,2006
Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013, 2006
Quark Matter 2009

8. Energy Loss Comparison
D7 Probe Brane t
z = 0 x v
AdS/CFT Drag:
dpT/dt ~ -(T2/Mq) pT
Q, m
zm = l1/2/2pm
3+1D Brane
Boundary
D3 Black Brane
(horizon)
zh = 1/pT
Black Hole
z = ¥
Similar to Bethe-Heitler
dpT/dt ~ -(T3/Mq2) pT
Very different from LPM
dpT/dt ~ -LT3 log(pT/Mq)
Quark Matter 2009

9. LHC RcAA(pT)/RbAA(pT) Prediction
Individual c and b RAA(pT) predictions:
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
Taking the ratio cancels most normalization differences seen previously
pQCD ratio asymptotically approaches 1, and more slowly so for increased quenching (until quenching saturates)
AdS/CFT ratio is flat and many times smaller than pQCD at only moderate pT
Distinguish rad and el contributions?
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
Quark Matter 2009

10. Universality and Applicability
How universal are th. HQ drag results?
Examine different theories
Investigate alternate geometries
Other AdS geometries
Bjorken expanding hydro
Shock metric
Warm-up to Bj. hydro
Can represent both hot and cold nuclear matter
Quark Matter 2009

11. New Geometries Constant T Thermal Black Brane Shock Geometries
Nucleus as Shock
J Friess, et al., PRD75:106003, 2007
DIS
Embedded String in Shock
Before
After
Albacete, Kovchegov, Taliotis,
JHEP 0807, 074 (2008) Q
vshock x z
vshock x z Q
Bjorken-Expanding Medium
Quark Matter 2009

12. Standard Method of Attack
Parameterize string worldsheet
Xm(t, s)
Plug into Nambu-Goto action
Varying SNG yields EOM for Xm
Canonical momentum flow (in t, s)
Quark Matter 2009

13. New in the Shock Find string solutions in HQ rest frame
vHQ = 0
Assume static case (not new)
Shock wave exists for all time
String dragged for all time
Xm = (t, x(z), 0,0, z)
Simple analytic solutions:
x(z) = x0, x0 ± m ½ z3/3
Quark Matter 2009

14. Shock Geometry Results
Three t-ind. solutions (static gauge):
Xm = (t, x(z), 0,0, z)
x(z) = x0, x0 ± m ½ z3/3
Constant solution unstable
Time-reversed negative x solution unphysical
Sim. to x ~ z3/3, z << 1, for const. T BH geom.
x0 - m ½ z3/3
x0 + m ½ z3/3 x0
vshock Q
z = 0
z = ¥ x
Quark Matter 2009

15. HQ Momentum Loss x(z) = m ½ z3/3 => Relate m to nuclear properties
Use AdS dictionary
Metric in Fefferman-Graham form: m ~ T--/Nc2
T’00 ~ Nc2 L4
Nc2 gluons per nucleon in shock
L is typical mom. scale; L-1 typical dist. scale
Quark Matter 2009

16. Frame Dragging HQ Rest Frame Shock Rest FrameMq L
vsh
vq = -vsh Mq
1/L
vq = 0 i i
vsh = 0
Change coords, boost Tmn into HQ rest frame:
T-- ~ Nc2 L4 g2 ~ Nc2 L4 (p’/M)2
p’ ~ gM: HQ mom. in rest frame of shock
Boost mom. loss into shock rest frame
p0t = 0:
Quark Matter 2009

17. Put Together This leads to
We’ve generalized the BH solution to both cold and hot nuclear matter E-loss
Recall for BH:
Shock gives exactly the same drag as BH for L = p T
Quark Matter 2009

18. Shock Metric Speed Limit
Local speed of light (in HQ rest frame)
Demand reality of point-particle action
Solve for v = 0 for finite mass HQ
z = zM = l½/2pMq
Same speed limit as for BH metric when L = pT
Quark Matter 2009

19. Conclusions and Outlook
Use data to test E-loss mechanism
RcAA(pT)/RbAA(pT) wonderful tool
Calculated HQ drag in shock geometry
For L = p T, drag and speed limit identical to BH
Generalizes HQ drag to hot and cold nuclear matter
Unlike BH, quark mass unaffected by shock
Quark always heavy from strong coupling dressing?
BH thermal adjustment from plasma screening IR?
Future work:
Time-dependent shock treatment
AdS E-loss in Bjorken expanding medium
Quark Matter 2009

20. Backup Slides
Quark Matter 2009

21. Canonical Momenta
Quark Matter 2009

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
We can approximate RAA(pT):
RAA ~ (1-e(pT))n(pT),
where pf = (1-e)pi (i.e. e = 1-pf/pi)
y=0
RHIC
LHC
Quark Matter 2009

23. 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:
String theory drag momentum loss:
Independent of pT and strongly dependent on Mq!
T2 dependence in exponent makes for a very sensitive probe
Expect: epQCD vs. eAdS indep of pT!!
dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST
erad ~ as L2 log(pT/Mq)/pT
eST ~ 1 - Exp(-m L), m = pl1/2 T2/2Mq
S. Gubser, Phys.Rev.D74: (2006); C. Herzog et al. JHEP 0607:013,2006
Quark Matter 2009

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)
Quark Matter 2009

25. LHC c, b RAA pT Dependence
WH and M. Gyulassy,
Phys. Lett. B 666, 320 (2008)
Naïve expectations met in full numerical calculation:
dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST
Significant rise in RAA(pT) for pQCD Rad+El
Large suppression leads to flattening
Use of realistic geometry and Bjorken expansion allows saturation below .2
LHC Prediction Zoo: What a Mess!
Let’s go through step by step
Unfortunately, large suppression pQCD similar to AdS/CFT
Quark Matter 2009

26. 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
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
RcbpQCD(pT) ~ 1 - as n(pT) L2 log(Mb/Mc) ( /pT)
Quark Matter 2009

27. LHC RcAA(pT)/RbAA(pT) Prediction
Recall the Zoo:
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
Taking the ratio cancels most normalization differences seen previously
pQCD ratio asymptotically approaches 1, and more slowly so for increased quenching (until quenching saturates)
AdS/CFT ratio is flat and many times smaller than pQCD at only moderate pT
Distinguish rad and el contributions?
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
Quark Matter 2009

28. Additional Discerning Power
Consider ratio for ALICE pT reach
mc = mb = 0
Adil-Vitev in-medium fragmentation rapidly approaches, and then broaches, 1
Does not include partonic E-loss, which will be nonnegligable as ratio goes to unity
Higgs (non)mechanism => Rc/Rb ~ 1 ind. of pT
Quark Matter 2009

29. Worldsheet boundary Spacelike if g > gcrit
Not So Fast!
D7 Probe Brane Q
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
Worldsheet boundary Spacelike if g > gcrit z
Trailing
String
“Brachistochrone” x
D3 Black Brane
Quark Matter 2009

30. LHC RcAA(pT)/RbAA(pT) Prediction (with speed limits)
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
T(t0): (, highest T—corrections unlikely for smaller momenta
Tc: ], lowest T—corrections likely for higher momenta
Quark Matter 2009

31. Derivation of BH Speed Limit I
Constant HQ velocity
Assume const. v kept by F.v
Critical field strength Ec = M2/l½
E > Ec: Schwinger pair prod.
Limits g < gc ~ T2/lM2
Alleviated by allowing var. v
Drag similar to const. v
Minkowski Boundary
z = 0
F.v = dp/dt E Q v
zM = l½ / 2pM D7
dp/dt
J. Casalderrey-Solana and D. Teaney, JHEP 0704, 039 (2007) D3
zh = 1/pT
Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013 (2006)
z = ¥
Quark Matter 2009

32. Derivation of BH Speed Limit II
Local speed of light
BH Metric => varies with depth z
v(z)2 < 1 – (z/zh)4
HQ located at zM = l½/2pM
Limits g < gc ~ T2/lM2
Same limit as from const. v
Mass a strange beast
Mtherm < Mrest
Mrest ¹ Mkin
Note that M >> T
Minkowski Boundary
z = 0
F.v = dp/dt E Q v
zM = l½ / 2pM D7
S. S. Gubser, Nucl. Phys. B 790, 175 (2008)
dp/dt D3
zh = 1/pT
z = ¥
Quark Matter 2009

33. Trouble for High-pT wQGP Picture
v2 too small
NPE supp. too large
p0 v2
WHDG dN/dy = 1400
C. Vale, QM09 Plenary (analysis by R. Wei)
NPE v2
STAR, Phys. Rev. Lett. 98, (2007)
Pert. at LHC energies?
PHENIX, Phys. Rev. Lett. 98, (2007)
Quark Matter 2009

34. Measurement at RHIC
Future detector upgrades will allow for identified c and b quark measurements
RHIC production spectrum significantly harder than LHC
y=0
RHIC
LHC
NOT slowly varying
No longer expect
pQCD dRAA/dpT > 0
Large n requires corrections to naïve
Rcb ~ Mc/Mb
Quark Matter 2009

35. RHIC c, b RAA pT Dependence
WH, M. Gyulassy, arXiv: [nucl-th]
Large increase in n(pT) overcomes reduction in E-loss and makes pQCD dRAA/dpT < 0, as well
Quark Matter 2009

36. RHIC Rcb Ratio pQCD pQCD AdS/CFT AdS/CFT
WH, M. Gyulassy, arXiv: [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
Quark Matter 2009

37. HQ Momentum Loss in the Shock
Must boost into shock rest frame:
Relate m to nuclear properties
Use AdS dictionary
Metric in Fefferman-Graham form: m ~ T--/Nc2
T00 ~ Nc2 L4
Nc2 gluons per nucleon in shock
L is typical mom. scale; L-1 typical dist. Scale
Change coords, boost into HQ rest frame:
T-- ~ Nc2 L4 (p/M)2
=> m = L4 (p/M)2
x(z) = m ½ z3/3 =>
Quark Matter 2009

38. HQ Momentum Loss in the Shock
x(z) = m ½ z3/3 =>
Relate m to nuclear properties
Use AdS dictionary: m ~ T--/Nc2
T-- = (boosted den. of scatterers) x (mom.)
T-- = Nc2 (L3 p+/L) x (p+)
Nc2 gluons per nucleon in shock
L is typical mom. scale; L-1 typical dist. scale
p+: mom. of shock gluons as seen by HQ
p: mom. of HQ as seen by shock
=> m = L2p+2
Quark Matter 2009

39. HQ Drag in the Shock HQ Rest Frame Shock Rest Frame Recall for BH:Mq L
vsh
vq = -vsh Mq
1/L
vq = 0 i i
vsh = 0
Recall for BH:
Shock gives exactly the same drag as BH for L = p T
Quark Matter 2009

40. HQ Momentum Loss x(z) = m ½ z3/3 => Relate m to nuclear properties
Use AdS dictionary
Metric in Fefferman-Graham form: m ~ T--/Nc2
T’00 ~ Nc2 L4
Nc2 gluons per nucleon in shock
L is typical mom. scale; L-1 typical dist. scale
Change coords, boost into HQ rest frame:
T-- ~ Nc2 L4 g2 ~ Nc2 L4 (p’/M)2
p’ ~ gM: HQ mom. in rest frame of shock
Quark Matter 2009

41. Shocking Drag Boost mom. loss into shock rest frame Therefore
HQ Rest Frame
Shock Rest Frame Mq L
vsh
vq = -vsh Mq
1/L
vq = 0 i i
vsh = 0
Boost mom. loss into shock rest frame
Therefore
p0t = 0:
Recall for BH:
Shock gives exactly the same drag as BH for L = p T
Quark Matter 2009