Journal Physics: Conference Series PAPER OPEN ACCESS Jets light hadrons via AdS/CFT correspondence To cite this article: R Morad and W A Horowitz 2015 J. Phys.: Conf. Ser. 645 012007 Related content - Light hadrons in 2 + 1 flavor lattice QCD Urs M Heller and MILC Collaboration - Hadron Beams For Hadron Physics Winston Roberts - Three Lectures on Hadron Physics Craig D. Roberts View article online for updates and enhancements. This content was downloaded from IP address 148.251.232.83 on 03/10/2018 at 23:12
Jets light hadrons via AdS/CFT correspondence R Morad 1 and W A Horowitz 2 Department Physics, University Cape Town, Private Bag X3, Rondebosch 7701, South Africa E-mail: 1 razieh.morad@uct.ac.za, 2 wa.horowitz@uct.ac.za Abstract. Within expanding fireball formed at heavy ion collision, jets are produced which probes QGP. Analysis energy loss se energetic partons as y travel throw QGP may reveal extremely valuable information about dynamics plasma and exhibit distinctive properties such as jet-quenching. The AdS/CFT correspondence which imposes duality between gauge ory and gravity is a novel tool provides valuable insight into strongly coupled plasma. The most important result AdS/CFT is calculating value shear viscosity to entropy density ratio which is in remarkable agreement with hydrodynamics predictions. We study energy loss rate light quarks via AdS/CFT correspondence in both static and expanding plasma. In hope making contact with QGP physics, we propose a novel jet prescription based on separation hard and st modes in dual ory and test AdS/CFT approach with latest light hadron suppression data from CMS. 1. Introduction The spectacular measurements from Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) provide compelling evidence that matter produced in heavy ion collision is a deconfined state QCD, Quark-Gluon Plasma (QGP) [1, 2, 3, 4], at temperatures above 160 MeV which appears to be nearly perfect, with an extremely low viscosity-to-entropy ratio η/s 1/4π [5, 6]. While lattice QCD is proper tool for understanding static equilibrium rmodynamics such strongly coupled plasma, it does not allow us to calculate its dynamics evolution on heavy-ion collision. Recently, a novel tool called AdS/CFT correspondence [7, 8, 9, 10, 11] provide valuable insight into strongly coupled plasma. This conjecture state that N = 4 SYM ory in large N c and large t Hot coupling is dual to classical supergravity on ten-dimensional AdS 5 S 5 geometry [7]. In order to study ory at finite temperature, one can add black hole (BH) to geometry [8] which yields to AdS-Sch metric. Fundamental quarks are described by open strings moving in 10d geometry. In large λ limit, quantum fluctuation string world sheet are suppressed and dynamics string are described by classical string ory. Jets are produced within expanding fireball and probe QGP. Analyses energy loss se energetic partons as y travel throw QGP may reveal extremely valuable information about dynamics plasma and exhibit distinctive properties such as jet-quenching which can clearly be observed at RHIC [1, 2, 3, 4] and more recently LHC [12, 13, 14]. In this paper, we study light quark jet energy loss in both gravitational background dual to static and expanding plasma. We propose a new prescription jets in string ory based Content from this work may be used under terms Creative Commons Attribution 3.0 licence. Any furr distribution this work must maintain attribution to author(s) and title work, journal citation and DOI. Published under licence by Ltd 1
on separation hard and st sectors. We demonstrate that light quark jet energy loss reveal Bragg peak at late times. Then we compute nuclear modification factor jets, renormalise quantity, and compare results with preliminary CMS data [15]. R jet 2. Light quark jet energy loss According to AdS/CFT correspondence [10], N = 4 SYM ory at finite temperature is dual to a 10d black hole geometry with AdS-Schwarzschild (AdS-Sch) metric as follows, ds 2 = L2 u 2 [ ] f(u) dt 2 + dx 2 + du2, (1) f(u) where f(u) 1 (u/u h ) 4 is blackening factor and L is AdS curvature radius. Four dimensional Minkowski coordinates are denoted by x µ and coordinate u is an inverse radial coordinate. So, boundary AdS-Sch spacetime is at u = 0 and event horizon is located at u = u h. The temperature equilibrium SYM plasma relates to event horizon as T 1 (πu h ). World sheet coordinates are as σa where τ σ 0 is denoted as timelike world sheet coordinate, while spatial coordinate is σ σ 1. Fundamental representation quarks added to N = 4 SYM ory are dual to open strings moving in 10d geometry. Addition a N = 2 hypermultiplet to N = 4 SYM ory is performed by adding D7 branes to 10d geometry [16]. These branes extend along radial coordinate from boundary at u = 0 down to maximal coordinate at u = u m as well as fill whole 4d Minkowski space. Also, y wrap on S 3 from S 5 sphere. The bare mass M quark is proportional to 1/u m [17], so for massless quarks D7 brane should fill whole radial direction. Open strings that are attached to D7 brane are dual to quark-anti quark pairs on field ory side. In 5d geometry se strings can fall unimpeded toward event horizon until ir end points reach radial coordinate u m where D7 brane ends. Since for sufficiently light or massless quarks u m > u h, open string end points can fall into horizon. We are interested in studying back-to-back jets so we consider configurations in which two endpoints string move away from each or as total spatial momentum string vanishes. By choosing appropriate frame, one half string has a large spatial momentum in x direction, and or half string carries a large negative spatial momentum. We will limit our attention to strings which move in one direction in R 3 space, x direction, so embedding function string X µ (τ, σ) will be a map to (t(τ, σ), x(τ, σ), u(τ, σ)). So, prile an open string which is created at a point in space at time t = t c is given by t(0, σ) = t c, x(0, σ) = 0, u(0, σ) = u c. (2) By se conditions, string created at time t c and by time evolution, string evolves from a point into an extended object and string endpoints fall toward horizon. The dynamics string is governed by Polyakov action S P = T 0 2 d 2 σ η η ab a X µ b X ν G µν. (3) Variation Polyakov action with respect to embedding functions X µ lead to equation motion as a [ η Π a µ ] = 1 η η ab G νρ 2 X µ ax ν b X ρ, (4) where Π a µ are canonical momentum densities associated to string obtained from varying action with respect to derivatives embedding functions. 2
In order to study jets in AdS/CFT we need to define proper objects in dual string ory that corresponds to a jet, a slippery object even in field ory; jets are truly only defined by algorithm used to measure m. Presumably ideal way to compute jet observables in dual ory is to compute energy momentum tensor associated with a high-momentum probe and run a jet finding algorithm on result. However, one can define a jet prescription in AdS/CFT and calculate rate energy loss from string prile itself. The authors [18] are motivated by localization baryon density on boundary which is scale order x 1/π T and defined jet as a part string which is in x spatial distance from endpoint. We called this as x prescription jet [18]. Figure 1. Illustration x and u prescriptions a jet in string ory; see text for details. In this paper, motivated by separation energy scales in, e.g., rmal field ory, we propose rar a u prescription which we believe will ultimately provide a closer approximation to result a more complete calculation, figure 1. Since radial coordinate in string ory sets an energy scale in field ory, in our u prescription portion string above some cutf u = u in radial direction is considered part jet; portion string below cutf is considered part rmalized medium. By choosing any value u above black hole horizon as cutf, we regain natural result that a jet that is rmalized no longer has detectable energy or momentum. We evaluate energy loss rate jet in radial σ = u parametrization for both prescriptions jet and plot in figure 2(a,b). In order to define jet using x prescription, we choose σ κ (t) as x = 0.3/π T. Our result on energy loss rate light quark using u prescription shows a Bragg peak at late times which means explosive transfer quark energy to plasma at late times and is consistence with previous works [18]. Since quark-gluon plasma produced at ultra-relativistic heavy ion collision is an expanding and cooling medium, we study light quark jet energy loss in a time dependent gravity dual to boost invariant flow [19]. This geometry is similar to static black hole geometry, but location horizon moves in bulk as τ 1/3 where τ is proper time. Also temperature plasma cools as T (τ) τ 1/3. We calculate energy loss rate light quark using both x prescription and u prescription jet. We consider initial temperature plasma in JP metric T c same 3
to around 500 MeV because se low-p T modes could not be measured by detectors and lost. Two jet prescription are illustrated at Fig. (1) (b). We believe that this definition describe jet more physically. We evaluate energy loss rate jet in radial = u parametrization for both prescriptions jet and plot in Fig. (2) (a,b). In order to define jet using x prescription, High weenergy chooseparticle apple(t) Physics as x Workshop =0.3/ T (HEPPW2015). Our result on energy loss rate light quark IOPusing Publishing Journal u prescription Physics: Conference shows aseries Bragg 645 peak (2015) at 012007 late times which means explosive transfer quark energy to plasma at late times and is consistence with previous works [16]. (a) (b) (c) (d) Figure 2: The instantaneous energy loss light quark as a function time in both AdS-BH Figure metric2. using The instantaneous (a) x prescription energy loss jet a light and (b) quarkx jetprescription as a functionjet time andinjp metric AdS- Sch using (a,b) and (c) JP xmetrics prescription (c,d) in jet xand prescription (d) x and prescription u prescription. jet. The The normalization constant constant E q E= q is 100 GeV energy is half initial energy string andjet, T =1/( u which has h ) is a virtuality temperature 175 GeV plasma. 2, and T Our = 350 jet s MeV definition is temperature reconstruct Bragg plasma. peak in energy loss rate light quark in both static and expanding plasma. as temperature plasma in AdS-Sch metric. The result is shown in figure 2(c,d) which demonstrates Since that quark-gluon behavior plasma produced light quark atenergy ultra-relativistic loss in heavy JP metric ion collision sameisasan expanding AdS-Sch metric, and cooling but medium, distance we that study quark traveled light quark before jet energy rmalizing loss in a increases. time dependent gravity dual to boost invariant flow [46]. This geometry is similar to static black hole geometry, but location horizon moves in bulk as 1/3 where is proper time. Also 3. temperature Nuclear Modification plasma cools Factor as T ( ) Jet 1/3. We calculate an approximation nuclear modification factor R for jet using our energy loss model jet. We consider contributions both quark and gluon jets. We assume that produced parton with initial energy p i T loses a fraction its energy ɛ with probability P ( ɛ p i T, L, T ) as final energy parton is given by p f T = (1 ɛ)pi T. Also, we suppose that AdS energy loss is approximately independent initial energy [20], and gluons loss ir energy by factor 2 in large N c limit respect to quarks. We suppose that production spectrum can be approximately by a power law [20], with slowly varying with respect to p T, n we may find a simple equation for jet nuclear modification factor as follows, R R jet (p T ) = dɛ P (ɛ p T, L, T ) ( 1 ɛ R) n R (p T ) 1. (5) For an absolutely uniform nucleus that is a 1D line, geometric average is carried out as an integral over a line production points with a parton that propagates through line. In this case, R R jet (p T ) is obtained from below line integral [20] R R jet Lmax (p T ) = 0 dl ( 1 ɛ R nr (p T ) 1 (p T, l, T )). (6) L max We calculate R jet by using our results jet energy loss in both AdS-Sch and JP metric. Our results are shown in figure 3. The purple curve shows R obtained from AdS-Sch metric, while blue curve is R obtained from JP metric. The AdS/CFT results jet energy loss show an over suppression jets in both static and expanding plasma. The point-like initial condition falling string that we consider here is dual to creation a pair quark-antiquark which fly away each or in strongly coupled plasma, interact and loss ir energy. We expect that jets produced in pp collision do not loss ir energy. So, we consider falling string with same initial conditions in AdS 5 metric. Our results show that string falls in empty AdS 5. So, jet loses its energy in vacuum! We calculate R for a falling string in AdS 5 metric and show in figure 3 (a) in red curve. 4
(a) (b) Figure 3: (a) Jet as function p T in most central Pb-Pb collision obtained via Figure AdS/CFT 3. (a) in AdS Jet R 5 (red), as JP a function (blue) and AdS-Sch p T in (purple) most central metrics. Pb-Pb (b) AdS/CFT collision jet obtained R as via AdS/CFT a function in AdS p 5 T compared (red), JP with (blue) and preliminary AdS-Sch CMS (purple) data in metrics. di erent (b) e ective AdS/CFT cone jet sizes R for as a function anti-k T jets using Bayesian unfolding method for most central Pb-Pb collision at LHC with p p T compared with preliminary CMS data in different effective cone sizes for anti-k T jets s =2.76 usingtev per Bayesian nucleon unfolding [43]. Themethod results for ourmost calculations central in Pb-Pb AdS-Sch collision and JP atmetrics LHC with are shown s = 2.76 by TeVpurple per nucleon and blue [15]. curves, respectively. The vertical lines indicate uncorrelated statistical uncertainty, and wide band systematic uncertainty for Bayesian unfolding R=0.3. The green box above 300 GeV/c represents overall combined uncertainty from T and Since luminosities. definition R measured difference between medium and vacuum effects on jet, we define a renormalized R in AdS/CFT as radial coordinate u? near event horizon. R Our motivation is defining jet as hard partons. renorm = Rmedium The instantaneous energy loss light quark is identified R AdS (7) 5 with energy flux from point at u(, apple) =u?. We have shown that using u prescription jet, light quark energy We loss plot exhibit renormalized Bragg peak R again renorm at late for times jets in in both both static AdS-Sch and expanding and JP metric plasmas. in This figure late 3 (b) time andbehavior compare with jet energy CMS lost implies preliminary that after data traveling for substantial most central distances Pb-Pb through collision at snn plasma, = 2.76rmalization GeV [15]. Asuppression light quark ends factor with a0.5 large foramount high p T energy jets is transferring observed into central Pb-Pb plasma collision which is insimilar comparison to to energy pp loss collisions. rate a fast Ourcharge resultsparticle also show moving surprisingly through ordinary agreement with matter. CMS preliminary results on jet R. We considered a brick plasma and calculated nuclear modification factor jet in both 4. AdS-Sch Conclusions and JP metric. We assumed that temperature plasma around 350 MeV at AdS- InSch thismetric paperand we at have initial purposed time in ajp novel metric. prescription Our results show jet inan avercontext suppression string jetory order and AdS/CFT ten respect correspondence. to data. We have investigated definedthat jet as it ais part because a falling falling string string whichsetup lies above at AdS radial space. coordinate In fact, Ru near jet using event horizon. falling string Our motivation in AdS 5 which is defining is dual to jet jetsasin hard vacuum partons. is The notinstantaneous one, even though energy it isloss less than light one. quark We introduced is identifieda renormalized with energy R renorm flux from by dividing point at u(τ, R σ medium to R in vacuum. Surprisingly, our ratio shows good agreement κ ) = u. We have shown that using with experimental data on R jet u prescription jet, light quark energy most central Pb-Pb collision at LHC Fig. (3) (b). loss exhibit Bragg peak again at late times in both static and expanding plasmas. This late On or hand, light quark energy loss is highly depends on initial conditions time behavior jet energy lost implies that after traveling substantial distances through falling string Fig. (1) (a). The only way to determine energy loss a jet precisely in plasma, rmalization light quark ends with a large amount energy transferring to strongly coupled regime is solving gravitational bulk-to-boundary problem. One can solve plasma Einstein s which equations is similar for to perturbation energy loss rate in 5d a fast geometry charge due particle to moving presence through ordinary string matter. and according to bulk to boundary map interpret near boundary behavior metric perturbation We considered as a brick perturbation plasmain and calculated SYM energy-momentum nuclear modification tensor by factor presence jet injet both AdS-Sch which will andbe JPleft metric. for We future assumed work. that temperature plasma around 350 MeV at AdS- Sch metric at initial time in JP metric. Our results show an aver suppression jet order ten respect to data. We investigated that it is because falling string setup at AdS space. In fact, R jet using falling string in AdS 5 which is dual to jets in vacuum is not one, even though it is less than one. We introduced a renormalized R renorm by dividing 5
= = ( ) > < ( ) Figure 4. The maximum stopping distance jet versus virtuality quark Q 2. Our result show a nontrivial dependency rmalization time to initial condition string. R in medium to R in vacuum. Surprisingly, our ratio shows good agreement with experimental data on R jet most central Pb-Pb collision at LHC figure 3 (b). On or hand, light quark energy loss is highly depends on initial conditions falling string figure 4. The only way to determine energy loss a jet precisely in strongly coupled regime is solving gravitational bulk-to-boundary problem. One can solve Einstein s equations for perturbation in 5d geometry due to presence string and according to bulk to boundary map interpret near boundary behavior metric perturbation as perturbation in SYM energy-momentum tensor by presence jet which will be left for future work. References [1] Adams J et al. (STAR Collaboration) 2005 Nucl.Phys. A757 102 183 (Preprint nucl-ex/0501009) [2] Adcox K et al. (PHENIX Collaboration) 2005 Nucl.Phys. A757 184 283 (Preprint nucl-ex/0410003) [3] Arsene I et al. (BRAHMS Collaboration) 2005 Nucl.Phys. A757 1 27 (Preprint nucl-ex/0410020) [4] Back B, Baker M, Ballintijn M, Barton D, Becker B et al. 2005 Nucl.Phys. A757 28 101 (Preprint nucl-ex/0410022) [5] Policastro G, Son D T and Starinets A O 2001 Phys.Rev.Lett. 87 081601 (Preprint hep-th/0104066) [6] Kovtun P, Son D T and Starinets A O 2005 Phys.Rev.Lett. 94 111601 (Preprint hep-th/0405231) [7] Maldacena J M 1999 Int.J.Theor.Phys. 38 1113 1133 (Preprint hep-th/9711200) [8] Witten E 1998 Adv.Theor.Math.Phys. 2 253 291 (Preprint hep-th/9802150) [9] Gubser S, Klebanov I R and Polyakov A M 1998 Phys.Lett. B428 105 114 (Preprint hep-th/9802109) [10] Aharony O, Gubser S S, Maldacena J M, Ooguri H and Oz Y 2000 Phys.Rept. 323 183 386 (Preprint hep-th/9905111) [11] Casalderrey-Solana J, Liu H, Mateos D, Rajagopal K and Wiedemann U A 2011 (Preprint 1101.0618) [12] Yin Z B (ALICE Collaboration) 2013 Acta Phys.Polon.Supp. 6 479 484 [13] Aad G et al. (ATLAS Collaboration) 2010 Phys.Rev.Lett. 105 252303 (Preprint 1011.6182) [14] Chatrchyan S et al. (CMS Collaboration) 2011 Phys.Rev. C84 024906 (Preprint 1102.1957) [15] Collaboration C (CMS Collaboration) 2012 [16] Karch A and Katz E 2002 JHEP 0206 043 (Preprint hep-th/0205236) [17] Herzog C, Karch A, Kovtun P, Kozcaz C and Yaffe L 2006 JHEP 0607 013 (Preprint hep-th/0605158) [18] Chesler P M, Jensen K, Karch A and Yaffe L G 2009 Phys.Rev. D79 125015 (Preprint 0810.1985) [19] Janik R A and Peschanski R B 2006 Phys.Rev. D73 045013 (Preprint hep-th/0512162) [20] Horowitz W A 2010 (Preprint 1011.4316) 6