Using string theory to study the quark-gluon plasma: progress and perils
PPUPT-2307
Using string theory to study the quark-gluon plasma: progress andperils
Steven S. Gubser
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
Abstract
I outline some of the progress over the past few years in applying ideas from string theory tostudy the quark-gluon plasma, including the computation of the drag force on heavy quarks andestimates of total multiplicity from black hole formation. I also indicate some of the main perilsof the string theory approach.
1. Introduction The gauge-string duality [1, 2, 3] provides a powerful computational framework for studying gauge theories at strong coupling. Recent years have seen a focused e ff ort to use this duality to understand the quark-gluon plasma, despite the obvious di ff erences between the best understood string theory constructions and quantum chromodynamics (QCD). In the gauge-string duality, a thermal state analogous to the quark-gluon plasma is represented as a black hole. Five-dimensional calculations based on the properties of the black hole horizon are translated into statements about strongly coupled thermal gauge theories, using the gauge- string duality. These statements can then be compared with measured properties of the quark- gluon plasma. In this contribution I will highlight a selection of encouraging results from the string theory approach, and at the same time point out the weak points of the calculations behind these results. The organization of the rest of this contribution is as follows. In section 2 I will introduce the gauge-string duality. In section 3 I will discuss predictions related to hard probes. And in section 4 I will discuss predictions for bulk physics. Broader reviews of the gauge-string duality and its applications to QCD include [4, 5, 6, 7]. For a recent and useful review of string theory constructions more specifically tailored to fit QCD than SYM, see [8].
2. The gauge-string duality N = super-Yang-Mills is dual to AdS × S [1, 2, 3] This duality, the simplest example of AdS / CFT, comes from two apparently di ff erent waysof describing D3-branes, which are locations in ten-dimensional spacetime where strings canend (see figure 1). Low-energy excitations of D3-branes are governed by N = L SYM − g YM tr F + (superpartners) . (1) July 2009 a r X i v : . [ h e p - t h ] S e p extradimensions D3D3D3 weak coupling A Μ GB x , x , x D3D3 D3 R , A d S ! S e x t r a d i m e n s i on s strong coupling L t , x , x , x Figure 1: (Color online.) D3-branes at weak coupling and at strong coupling. At weak coupling, strings stretchedbetween the branes behave as nearly free gluons in a 3 + N = AdS × S captures the gauge theory dynamics. β ( g YM ) ≡ perilous . At weak coupling, the open strings ending on the D3-branes behave as gluons, and superpart-ners of gluons, in SYM. At strong coupling, the simplest description of D3-branes is in terms ofnear-horizon geometry,
AdS × S . The five-sphere, S , plays essentially no role in the discussionto follow. The metric of AdS is ds = L z ( − dt + d (cid:126) x + dz ) ( z > , (2)and the characteristic length scale L is related to the ’t Hooft coupling by λ ≡ g YM N = L α (cid:48) , (3)where α (cid:48) is the inverse string tension. Strong coupling means λ (cid:29)
1, which is equivalent to L (cid:29) α (cid:48) . This is the statement that strings are typically much smaller than the radius of curvature of AdS , making geometrical notions like the metric reliable. (cid:15)(cid:15) free = + . λ / + . . . [9, 10, 11] One can introduce finite temperature by replacing
AdS by the AdS -Schwarzschild metric: ds = L z (cid:32) − hdt + d (cid:126) x + dz h (cid:33) h = − z z H . (4)The horizon at z = z H has Hawking temperature T = / ( π z H ), and its Bekenstein-Hawking entropy is related to its area by S = A / G N . SYM has considerably more degrees of freedom than QCD: free field counting gives (cid:15)
S YM free ≈ T (cid:15) QCD free ≈ T (3 massless flavors) . (5)It is perilous to directly compare theories with (cid:15)/ T so di ff erent. On the other hand, it is intrigu- ing that lattice results for (cid:15)/(cid:15) free in QCD are fairly close to the SYM values. q r horizon R AdS −Schwarzschildhorizon r R xz AdS −Schwarzschild fundamentalstring qq Figure 2: (Color online.) Left: a quark and anti-quark are described on the string theory side by strings hanging fromthe boundary of
AdS down to the D3-branes. Right: A lower-energy configuration is for the two strings to join into oneU-shaped string. The relation between quark-anti-quark pairs and hanging strings in
AdS is illustrated in figure 2. The quarks are infinitely massive because there is actually an infinite length of string “close” to the boundary. By calculating on the string theory side the energy gained by passing from the disjoint stringsto the U-shaped string, one finds a Coulombic force, screened in the infrared when the tempera-ture is finite. Equating this force to the quark-anti-quark calculated from the lattice at a separation r ≈ .
25 fm and an energy density corresponding in QCD to T ≈
240 MeV leads to λ S YM = . + . − (6)in SYM [14]. This is surprising because then α S YM ≈ .
15. The match between lattice calcu- lations and SYM is conspicuously imperfect because SYM doesn’t confine. The leading order string theory curve isn’t even fully understood for r > ∼ .
25 fm; but see [15]. These points il- lustrate some the perils of comparing SYM and lattice QCD; however, I will stick with (6) as a physically motivated range of couplings, and also continue to compare SYM and QCD at fixed energy density rather than fixed temperature, as a way of correcting for the larger number of degrees of freedom in SYM.
3. Hard probes = − p τ Q + stochastic [16, 17, 18] The physical picture of quark energy loss in string theory is sketched in figure 3. The trailingstring describes the response of the color-electric fields produced by the quark to the thermalmedium. From the shape of the string, determined through classical equations of motion instring theory, one can deduce that d pdt = − π √ λ T S YM v √ − v = − p τ Q τ Q = m Q π T S YM √ λ (7) τ charm ≈ τ bottom ≈ T QCD =
250 MeV . (8)3 R AdS −Schwarzschild vq f u n d a m e n t a l s t r i n g T mnmn hhorizon Figure 3: (Color online.) A quark drags a trailing string behind it. The string encodes energy loss in the dual gaugetheory.
A recent study shows that these equilibration times are at least roughly consistent with R AA of non-photonic electrons [19]. The stochastic forces on the quark are reflected on the string theory side by oscillations of the trailing string [17, 20, 21, 22]. ff orts to characterize gluon energy loss [23, 24, 25, 26, 27, 28] The first method, originally proposed in [23], is based on using a Wilson loop describing twoquarks separated by a length L and propagating along a null separation L − , measured in the restframe of the plasma. In this approach one findsˆ q LRW = π / Γ (3 / Γ (5 / √ λ T ≈ . fm (cid:18) T S YM
280 MeV (cid:19) . (9)In the last approximate equality, I used λ = π , as preferred by the authors of [23]. In comparing with QCD, a reduction of ˆ q by about a factor of 2 is probably in order to account for fewer degrees of freedom in QCD. Criticisms of the choice of worldsheet [29] suggest that the approach of [23] is not without its perils . The second method is based on a representation of an o ff -shell gluon as a string falling intothe horizon, first proposed in [25]. This method leads to a stopping distance x stop < ∼ π T S YM (cid:32) √ λ ET S YM (cid:33) / . (10)It is perilous to compare (10) (or better estimates of x stop based on worldsheet and spacetimegeodesics) with more standard approaches based on ˆ q , simply because the picture of a gluonadvocated in [25] is significantly di ff erent from the perturbative picture. Nevertheless, one canmake a rough translation of x stop into a value for the jet-quenching parameter:ˆ q rough ≡ E α s x ≈
21 GeV fm , (11)where in the approximate equality we set α s = / λ S YM = . as the usual scheme of matching the energy density of QCD and SYM, with T QCD =
280 MeV. We also assumed that the energy of gluons is between 5 to 25 GeV. perils of this discussion are the crudeness of (11), the sensitivity to the tempera- ture, and the obvious tension between the proposals of [23] and [25].
4. Bulk physics ζ/ s < ∼ / π [30, 31, 32] In [30] it was shown that starting from the gravitational action S − dimensional = (cid:90) d x √ g (cid:34) R −
12 ( ∂φ ) − V ( φ ) (cid:35) , (12)one can adjust the scalar potential V ( φ ) to mimic the equation of state of QCD. Having done this, one can calculate the bulk viscosity from the probability for an appropriate superposition of gravitons and scalars to be absorbed by the black hole. Results of such computations, shown in figure 4, indicate that the peak in ζ/ s is present, but not strong, for a realistic equation of state. A peril in using the action (12) is that it doesn’t come from a first principles calculation. ááááááááááááááááááááááááááá ççççççççççççççççççççççççççççççççççç T (cid:144) T c c s Type III BHType II BHType I BHlattice, pure glue á lattice, 2 + ç QPM ààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò æææææææææææææææææææææææææææææææææææææææææææææææææ æææææææææææææææææææææææææææææææ T (cid:144) T c Ζ (cid:144) s sum rule, 2 + à Type IIIBH ò Type IIBH æ Type I BH
Figure 4: (Color online.) Left: The speed of sound as a function of temperature for several constructions based on (12).Right: The corresponding bulk viscosity. The lattice points, from [33], are for pure glue. Both plots are from [31]. [34, 35, 36] The idea of colliding shocks is to replace a heavy ion with a boosted conformal soliton. Inthe limit of infinite boost, the only non-zero component of the gauge theory stress tensor is (cid:104) T −− (cid:105) = EL π (cid:2) ( x ) + ( x ) + L (cid:3) δ ( x − ) L ≈ . = rms radius of gold , (13)where x − = x − x . The fall-o ff of (cid:104) T −− (cid:105) at large x ⊥ is as a power, perilously di ff erent fromthe exponential fall-o ff of the Woods-Saxon profile. We nevertheless work with (13) because itsgravitational dual is simple: it is a point-sourced gravitational shock wave in AdS . As sketchedin figure 5, a non-spherical event horizon forms when such shocks collide. Following standardbut non-rigorous methods, one can estimate S > ∼ π (cid:32) L G (cid:33) / (2 EL ) / ≈ (cid:32) √ s NN
200 GeV (cid:33) / . (14)5 H z=L CS S H R xx z Landau Hydrodynamics & RHIC Phenomenology 3 (GeV)s ) ! / p a r t N " ( / ! c h N " /2sp) Data @ ppp( Data - e + e - e + Fit to ePHOBOSPHOBOS interp.NA49E895 B µ Landau + Landau
Fig. 1.
Charged particle multiplicities for A+A, p+p (with leading particle e ff ectremoved) and e + e − . Theoretical curves are pQCD (dotted line) [ 12], baryon-freeLandau (solid line) [ 6], and Landau including the baryochemical potential e ff ect(dot-dashed) [ 19].region of phase space) in all collisions involving nuclei, from p + A to Au + Au collisions [ 13, 14].
3. Thermal Phenomenology and Hadrochemistry
In the Landau scenario, freezeout is not expected to occur immediately, as Fermiassumed, but rather when the temperature reaches the limit of the pion Comptonwavelength T = m π . This was based on a suggestion by Pomeranchuk [ 15] to avoidFermi’s prediction that nucleons would outnumber pions by virtue of their largerstatistical weight. This assumption leads to predictions for the relative populationof various particle states similar to those made in the Hagedorn approach [ 16, 17].A+A collisions clearly deviate from the Fermi-Landau formula at low energies.An obvious suspect is the phenomenon of baryon stopping, which is absent in p+pcollisions but is substantial in A+A [ 18]. If one puts back the − µ B N B term into thefirst law of thermodynamics, we immediately see how the presence of a conservedquantity associated with a substantial mass (i.e. the proton mass) will naturallysuppress the total entropy: S = ( E + pV − µ B N B ) /T . Using an existing thermalmodel code, Cleymans and Stankiewicz [ 19] calculated the entropy density as afunction of √ s . It rises to limiting value where µ B → T → T , the Hagedorntemperature. If we then assume that the total multiplicity scales linearly with the Figure 5: (Color online) Left: A trapped surface forms around the point-sources of two gravitational shock wavescolliding head-on in
AdS . Right (from [37]): Total multiplicity scales approximately as E / (as predicted by theLandau model) over a wide range of energies. It may just be starting to roll over to a slower scaling near top RHICenergies. Using the phenomenological estimate S ≈ . N charged , one gets N charged > ∼ gold collisions at top RHIC energies. This compares quite favorably with the data, which gives N charged ≈ serious peril is the E / scaling in (14), which will bring the string theory prediction into conflict with data only slightly above top RHIC energies unless multiplicities rise unexpectedly quickly. A crude resolution, which I nevertheless think is on the right track, is to discard the entropy of the part of the trapped surface above some ultraviolet cuto ff in AdS . This mimics the e ff ect of asymptotic freedom and changes the scaling of S trapped from E / to E / , which is roughly in line with CGC predictions. An infrared cuto ff is probably also necessary. With reasonable choices for the cuto ff s, predictions for total multiplicity at LHC energies come out around N charged = , This extrapolation is perilous because it depends significantly on the cuto ff s.
5. Conclusions The string theory approach to the quark-gluon plasma has made impressive progress. AdS / CFT provides many calculations of strongly coupled phenomena that can be compared to heavy-ion collisions. Such comparisons often come out surprisingly well, among them shear viscosity, the drag force on heavy quarks, jet splitting, total multiplicity at √ s NN =
200 GeV, and perhaps also thermalization and bulk viscosity. (Due to lack of space I have been unable to include discussions of jet splitting and thermalization.) When calculations in AdS / CFT compare poorly with QCD, we often understand why: Usually, it is the strong coupling limit and / or conformal invariance which distorts the results. But the string theory approach is a ffl icted with significant perils . We are too often limited to the N (cid:29) λ (cid:29) N = beyond N = for theoretical fudging. Essentially, this means that the onus is on theorists to create models that are as clean and predictive as possible while capturing the essentials of QCD. Further e ff ort on the theoretical side, and even better measurements of both hard probe and bulk physics observables, are clearly in order to clarify the extent to which the heavy-ion pro- grams at RHIC and the LHC probe experimental predictions of string theory. cknowledgments I am particularly indebted to my collaborators: J. Friess, G. Michalogiorgakis, S. Pufu, and A. Yarom. I thank J. Casalderrey-Solana, M. Gyulassy, B. Jacak, J. Nagle, J. Noronha, K. Ra- jagopal, D. Teaney, and W. Zajc for useful discussions. This work was supported in part in part by the DOE under Grant No. DE-FG02-91ER40671 and by the NSF under award number PHY-0652782. References [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. (1998) 231–252, hep-th/9711200 . [2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett.
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