Shock Treatment: Heavy Quark Energy Loss in a Novel Geometry

Quark Matter 2009 14/3/09

William Horowitz

Shock Treatment: Heavy Quark Energy Loss in a Novel

GeometryWilliam HorowitzThe Ohio State University

April 3, 2009

With many thanks to Yuri Kovchegov and Ulrich Heinz


William Horowitz

pQCD Success in High-pT at RHIC:

– Consistency: RAA()~RAA()

– Null Control: RAA()~1

– GLV Calculation: Theory~Data for reasonable fixed L~5 fm and dNg/dy~dN/dy

Y. Akiba for the PHENIX collaboration, hep-ex/0510008

(circa 2005)


William Horowitz

Trouble for High-pT wQGP Picture– v2 too small – NPE supp. too large

STAR, Phys. Rev. Lett. 98, 192301 (2007)

0 v2

PHENIX, Phys. Rev. Lett. 98, 172301 (2007)

NPE v2

Pert. at LHC energies?

C. Vale, QM09 Plenary (analysis by R. Wei)

WHDG


William Horowitz

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 ~ ~ .5 GeV mom. transfers=> Nonperturbatively large s

• Use data to learn about E-loss mechanism, plasma properties– Domains of self-consistency crucial for

understanding


William Horowitz

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


William Horowitz

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.C71:064904,2005Casalderrey-Solana and Teaney, Phys.Rev.D74:085012,2006; JHEP 0704:039,2007

BDMPS, Nucl.Phys.B484:265-282,1997Armesto, Salgado, and Wiedemann, Phys. Rev. D69 (2004) 114003Liu, Ragagopal, Wiedemann, PRL 97:182301,2006; JHEP 0703:066,2007


William Horowitz

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, Phys.Rev.D74:126005,2006Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013, 2006

Kharzeev, arXiv:0806.0358 [hep-ph]

Gubser, Gulotta, Pufu, Rocha, JHEP 0810:052, 2008Chesler, Jensen, Karch, Yaffe, arXiv:0810.1985 [hep-th]


William Horowitz

Energy Loss Comparison

– AdS/CFT Drag:dpT/dt ~ -(T2/Mq) pT

– Similar to Bethe-HeitlerdpT/dt ~ -(T3/Mq

2) pT

– Very different from LPMdpT/dt ~ -LT3 log(pT/Mq)

tx

Q, m v

D7 Probe Brane

D3 Black Brane(horizon)

3+1D Brane Boundary

Black Holez =

zh = 1/T

zm = 1/2/2m

z = 0


William Horowitz

LHC RcAA(pT)/Rb

AA(pT) Prediction

• Individual c and b RAA(pT) predictions:

– 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)

WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)


William Horowitz

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


William Horowitz

New Geometries

Albacete, Kovchegov, Taliotis,JHEP 0807, 074 (2008)

J Friess, et al., PRD75:106003, 2007

Constant T Thermal Black Brane

Shock GeometriesNucleus as Shock

Embedded String in Shock

DIS

Q

vshock

x

zvshock

x

zQ

Before After

Bjorken-Expanding Medium


William Horowitz

Standard Method of Attack• Parameterize string worldsheet

– X(, )

• Plug into Nambu-Goto action

• Varying SNG yields EOM for X

• Canonical momentum flow (in , )


William Horowitz

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

• X = (t, x(z), 0,0, z)

• Simple analytic solutions:– x(z) = x0, x0 ±

½ z3/3


William Horowitz

Shock Geometry Results• Three t-ind. solutions (static gauge):

X = (t, x(z), 0,0, z)

– x(z) = x0, x0 ± 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 ½ z3/3 x0 ½ z3/3

x0

vshock

Qz = 0

z = x


William Horowitz

HQ Momentum Loss

Relate to nuclear properties– Use AdS dictionary

• Metric in Fefferman-Graham form: ~ T--/Nc2

– T’00 ~ Nc2 4

• Nc2 gluons per nucleon in shock

• is typical mom. scale; typical dist. scale

x(z) = ½ z3/3 =>


William Horowitz

Frame Dragging• HQ Rest Frame • Shock Rest Frame

vshMq

1/

vq = -vsh

Mq

i i vsh = 0vq = 0

– Change coords, boost T into HQ rest frame:

• T-- ~ Nc2 4Nc

2 4 (p’/M)2

• p’ ~ M: HQ mom. in rest frame of shock

– Boost mom. loss into shock rest frame

– 0t = 0:


William Horowitz

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 = T


William Horowitz

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 = ½/2Mq

– Same speed limit as for BH metric when = T


William Horowitz

Conclusions and Outlook– Use data to test E-loss mechanism

• RcAA(pT)/Rb

AA(pT) wonderful tool

– Calculated HQ drag in shock geometry• For = 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


William Horowitz

Backup Slides


William Horowitz

Canonical Momenta


William Horowitz

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-(pT))n(pT),

where pf = (1-)pi (i.e. = 1-pf/pi)

y=0

RHIC

LHC


William Horowitz

– 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: pQCD 0 vs. AdS indep of pT!!

• dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST

rad s L2 log(pT/Mq)/pT

Looking for a Robust, Detectable Signal

ST 1 - Exp(- L), = T2/2Mq

S. Gubser, Phys.Rev.D74:126005 (2006); C. Herzog et al. JHEP 0607:013,2006


William Horowitz

Model Inputs– AdS/CFT Drag: nontrivial mapping of QCD to SYM

• “Obvious”: s = SYM = const., TSYM = TQCD

– D 2T = 3 inspired: s = .05– pQCD/Hydro inspired: s = .3 (D 2T ~ 1)

• “Alternative”: = 5.5, TSYM = TQCD/31/4

• Start loss at thermalization time 0; end loss at Tc

– WHDG convolved radiative and elastic energy loss• s = .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)


William Horowitz

– LHC Prediction Zoo: What a Mess!– Let’s go through step by step

– Unfortunately, large suppression pQCD similar to AdS/CFT– Large suppression leads to flattening– Use of realistic geometry and Bjorken expansion allows saturation below .2– Significant rise in RAA(pT) for pQCD Rad+El– Naïve expectations met in full numerical calculation: dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST

LHC c, b RAA pT Dependence



William Horowitz

• 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

An Enhanced Signal

RcbpQCD(pT) 1 - s n(pT) L2 log(Mb/Mc) ( /pT)


William Horowitz

LHC RcAA(pT)/Rb

AA(pT) Prediction

• Recall the Zoo:

– 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)



William Horowitz

Additional Discerning Power

– 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

– Consider ratio for ALICE pT reachmc = mb = 0


William Horowitz

• Speed limit estimate for applicability of AdS drag– < crit = (1 + 2Mq/1/2 T)2

~ 4Mq2/(T2)

• Limited by Mcharm ~ 1.2 GeV

• Similar to BH LPM– crit ~ Mq/(T)

• No single T for QGP

Not So Fast!Q D7 Probe Brane

Worldsheet boundary Spacelikeif > crit

TrailingString

“Brachistochrone”

z

x

D3 Black Brane


William Horowitz

LHC RcAA(pT)/Rb

AA(pT) Prediction(with speed limits)

– T(0): (, highest T—corrections unlikely for smaller momenta

– Tc: ], lowest T—corrections likely for higher momenta



William Horowitz

Derivation of BH Speed Limit I

• Constant HQ velocity– Assume const. v kept by F.v

– Critical field strength Ec = M2/½

• E > Ec: Schwinger pair prod.

• Limits < c ~ T2/M2

– Alleviated by allowing var. v• Drag similar to const. v

z = 0

zM = ½ / 2M

zh = 1/T

EF.v = dp/dt

dp/dt

Q

Minkowski Boundary

D7

D3

v

J. Casalderrey-Solana and D. Teaney, JHEP 0704, 039 (2007)

Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013 (2006)z =


William Horowitz

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 = ½/2M

– Limits < c ~ T2/M2

• Same limit as from const. v

– Mass a strange beast• Mtherm < Mrest

• Mrest Mkin

– Note that M >> T

z = 0

zM = ½ / 2M

zh = 1/T

EF.v = dp/dt

dp/dt

Q

Minkowski Boundary

D7

D3

v

S. S. Gubser, Nucl. Phys. B 790, 175 (2008)

z =


William Horowitz

Trouble for High-pT wQGP Picture– v2 too small – NPE supp. too large

STAR, Phys. Rev. Lett. 98, 192301 (2007)

0 v2

PHENIX, Phys. Rev. Lett. 98, 172301 (2007)

NPE v2

Pert. at LHC energies?

C. Vale, QM09 Plenary (analysis by R. Wei)

WHDG dN/dy = 1400


William Horowitz

Measurement at RHIC– Future detector upgrades will allow for

identified c and b quark measurements

y=0

RHIC

LHC

• • NOT slowly varying

– No longer expect pQCD dRAA/dpT > 0

• Large n requires corrections to naïve

Rcb ~ Mc/Mb

– RHIC production spectrum significantly harder than LHC


William Horowitz

RHIC c, b RAA pT Dependence

• Large increase in n(pT) overcomes reduction in E-loss and makes pQCD dRAA/dpT < 0, as well

WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]


William Horowitz

RHIC Rcb Ratio

• Wider distribution of AdS/CFT curves due to large n: increased sensitivity to input parameters

• Advantage of RHIC: lower T => higher AdS speed limits

WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]

pQCD

AdS/CFT

pQCD

AdS/CFT


William Horowitz

HQ Momentum Loss in the Shock

• Must boost into shock rest frame:• Relate to nuclear properties

– Use AdS dictionary• Metric in Fefferman-Graham form: ~ T--/Nc

2

– T00 ~ Nc2 4


• is typical mom. scale; typical dist. Scale

– Change coords, boost into HQ rest frame:• T-- ~ Nc

2 4(p/M)2

=> = 4(p/M)2

x(z) = ½ z3/3 =>


William Horowitz

HQ Momentum Loss in the Shock

Relate to nuclear properties– Use AdS dictionary: ~ T--/Nc

2

– T-- = (boosted den. of scatterers) x (mom.)

– T-- = Nc2 (3 p+/) x (p+)


• is typical mom. scale; typical dist. scale• p+: mom. of shock gluons as seen by HQ• p: mom. of HQ as seen by shock

=> = 2p+2

x(z) = ½ z3/3 =>


William Horowitz

HQ Drag in the Shock• HQ Rest Frame • Shock Rest Frame

vshMq

1/

vq = -vsh

Mq

i i vsh = 0vq = 0



William Horowitz

HQ Momentum Loss

Relate to nuclear properties– Use AdS dictionary

• Metric in Fefferman-Graham form: ~ T--/Nc2

– T’00 ~ Nc2 4


• is typical mom. scale; typical dist. scale

– Change coords, boost into HQ rest frame:• T-- ~ Nc

2 4Nc2 4 (p’/M)2

• p’ ~ M: HQ mom. in rest frame of shock

x(z) = ½ z3/3 =>


William Horowitz

Shocking Drag• HQ Rest Frame • Shock Rest Frame

vshMq

1/

vq = -vsh

Mq

i i vsh = 0vq = 0


• Boost mom. loss into shock rest frame

• Therefore– 0t = 0:

Documents

Shock Treatment: Heavy Quark Energy Loss in a Novel Geometry