Gluonic fields of a static particle to all orders in 1/N
aa r X i v : . [ h e p - t h ] A p r Preprint typeset in JHEP style - PAPER VERSION
Gluonic fields of a static particle to all ordersin 1/N.
Bartomeu Fiol , Blai Garolera and Aitor Lewkowycz Departament de F´ısica Fonamental iInstitut de Ci`encies del Cosmos,Universitat de Barcelona,Mart´ı i Franqu`es 1, 08028 Barcelona, Catalonia, Spain Perimeter Institute for Theoretical Physics,Waterloo, Ontario N2L 2Y5, Canada [email protected], [email protected], [email protected]
Abstract:
We determine the expectation value of the gauge invariant operator
T r [ F + . . . ] for N = 4 SU(N) SYM, in the presence of an infinitely heavy staticparticle in the symmetric representation of SU(N). We carry out the computation inthe context of the AdS/CFT correspondence, by considering the perturbation of thedilaton field caused by the presence of a D3 brane dual to such an external probe.We find that the effective chromo-electric charge of the probe has exactly the sameexpression as the one recently found in the computation of energy loss by radiation. ontents
1. Introduction and conclusions 12. The computation 53. Acknowledgements 7
1. Introduction and conclusions
Among the many applications of the AdS/CFT correspondence, a very broad area ofresearch is the study of the behavior of external probes in strongly coupled field theoriesand the response of the fields to such probes. The first examples of such computationswere the evaluation of the static quark-antiquark potential in [1, 2] by means of aparticular string configuration reaching the boundary of
AdS . Following those seminalworks, the key idea of realizing external heavy quarks by strings in the bulk geometryhas been generalized in many directions. In particular, as we will briefly review, probestransforming under different representations of the gauge group are holographicallyrealized by considering different types of branes in the supergravity background.An external probe in the fundamental representation of the gauge group is dualto a string in the bulk. At least for the simplest implementations of this identification(i.e. in the absence of additional scales like finite mass or non-zero temperature), thecomputed N = 4 SYM observables reveal a common feature: while at weak ’t Hooftcoupling λ we can identify this coupling as the analogous of the charge squared, atstrong ’t Hooft coupling there is a screening of this charge, in the sense that the resultsobtained demand the strong coupling identification e fund ∼ √ λ (1.1)This generic behavior stems from the fact that the Nambu-Goto action evaluated forworld-sheet metrics embedded in AdS goes like S NG = − πα ′ Z d σ p −| g | = − √ λ πL Z d σ p −| g | – 1 –here L is the AdS radius which generically cancels out from this expression whenspecific world-sheet metrics are plugged-in. Some examples of this are the originalquark-antiquark potential [1, 2], the computation of the gauge invariant operators inthe presence of a particle at rest [3, 4] or following arbitrary motion [5, 6], and theformula for energy loss by radiation [7].Moving on to probes transforming in other representations of the gauge group, it isby now well understood that they are realized by D3 and D5 branes. Specifically, on thegravity side, the duals of particles in the symmetric or antisymmetric representations ofthe gauge group are given by D3 and D5 branes respectively, with world-volume fluxesthat encode the rank of the representation [8, 9, 10]. One of the novel features of thisidentification - and the chief reason for our interest in the topic - is that some computedobservables are functions of k/N , where k is the rank of the symmetric/antisymmetricrepresentation. Thus, by means of the AdS/CFT correspondence, the physics of theseprobes of N = 4 SYM can be studied at large N , large λ , beyond the leading term.While the holographic prescription is in principle equally straightforward for thestudy of probes in the symmetric and the antisymmetric representations, when it comesto actual computations we currently face more difficulties in the symmetric case than inthe antisymmetric one. One of the reasons behind this difference comes from the exis-tence of a quite universal result for the embedding of D5 branes in terms of embeddingsof fundamental strings, due to Hartnoll [11]: given a string world-sheet that solves theNambu-Goto action in an arbitrary five-dimensional manifold M with constant nega-tive Ricci tensor, there is a quite general construction that provides a solution for theD5-brane action in M × S , of the form Σ × S where Σ ֒ → M is the string world-sheetand S ֒ → S . This gives a link between the string used to describe a particle in thefundamental representation and the D5 brane used to represent a probe in the anti-symmetric representation. Moreover, this link translates into a quite robust result forthe evaluation of observable quantities; for the k-antisymmetric representation, typi-cally the strong coupling √ λ dependence of observables for probes in the fundamentalrepresentation gets replaced (up to numeric factors) by √ λ → N π sin θ √ λ ∼ e A k (1.2)where θ denotes the angle of S inside S and is the solution of [12]sin θ cos θ − θ = π (cid:18) kN − (cid:19) This identification is supported by explicit computations of Wilson loops [9, 11] whichmatch matrix model computations [13], energy loss by radiation in vacuum [14] and– 2 –n a thermal medium [15], or the impurity entropy in supersymmetric versions of theKondo model [16].On the other hand, for probes in the symmetric representation we currently don’thave a generic construction that links the string that realizes a particle in the funda-mental representation with a D3 brane that realizes a probe in the symmetric repre-sentation. Furthermore, while the observables analyzed so far seem to depend on thecombination κ = k √ λ N (1.3)introduced in [17], they do not display a common function that replaces the √ λ depen-dence of the fundamental representation. For instance, in the computation of the energyloss by radiation in vacuum of a particle moving with constant proper acceleration, itwas found in [14] that √ λ → N κ √ κ = k √ λ r k λ N ∼ e S k (1.4)while for the vev of a circular Wilson loop, it was found that [17] √ λ → N ( κ √ κ + sinh − κ )While both functions expanded as a power series in κ start with the common term k √ λ (i.e. k times the result for the fundamental representation) they are clearly differentbeyond this leading order.The purpose of this note is to shed some light on the issue of observables for probesin the symmetric representation, by computing the expectation value of a particulargauge invariant operator in the presence of an infinitely heavy half-BPS static par-ticle, transforming in the k-symmetric representation of N = 4 SU ( N ) SYM. Morespecifically, this operator is the one sourced by the asymptotic value of the dilaton[18] , O F = 12 g Y M Tr (cid:0) F + [ X I , X J ][ X I , X J ] + fermions (cid:1) On general grounds, in the presence of a static probe placed at the origin, we expectthe one-point function to be of the form < O F ( ~x ) > = f ( k, λ, N ) | ~x | We follow the conventions of [6]. – 3 –nd our objective is to compute the dimensionless function f ( k, λ, N ) when the probetransforms in the k-symmetric representation of SU ( N ). By analogy with the Coulom-bic case, one might refer to f ( k, λ, N ) as the square of the ”chromo-electric charge” ofthe heavy particle. To carry out this computation we will consider a particular half-BPS D3-brane embedded in AdS × S and analyze the linearized perturbation equationfor the dilaton, with the D3-brane acting as source. The advantage of considering thisoperator is that the perturbation equation of the dilaton decouples from the equationsfor metric perturbations, so its study is quite straightforward.The analogous computation for a particle in the fundamental representation wascarried out some time ago, considering in that case the perturbation equation for thedilaton sourced by a fundamental string [3, 4] (see also [6]). For the sake of comparison,let’s quote their final result in our conventions, < O F ( ~x ) > fund = √ λ π | ~x | In the next section we will give details of our computation, but let us jump ahead andpresent the final result, < O F ( ~x ) > S k = N κ √ κ π | ~x | = k √ λ π q k λ N | ~x | As we can see, we obtain again a result organized as a function of κ . Furthermore,the form of the function replacing the √ λ result of the fundamental representation isexactly the same (including the numerical factor) that was found in the computation ofthe energy loss [14]. This supports the identification of the effective charge for probesin the symmetric representation made in eq. (1.4). To us, the complete coincidence ofthese two results was far from a foregone conclusion: while the two D3-brane solutionsused in the computations are related by a conformal transformation, this doesn’t evenimply that the same observable will coincide, as the differing expectation values ofthe corresponding Wilson loops beautifully show [19, 17]. Moreover, the energy losscomputation in [14] captured physics of radiative fields, encoded in the bulk by thepresence of a horizon in the world-volume of the D3-brane, while in the computationto be presented shortly, the physics of static fields is captured by the behavior near the AdS boundary, and the D3-brane world-volume has now no horizon. Let us conclude this introduction by mentioning some of the questions we wouldlike to address in the future. For probes in the fundamental and the antisymmetric Incidentally, the world-volume metric of the D3-brane we will consider is
AdS × S with radii L √ κ and Lκ respectively [17], so κ √ κ (divided by L ) happens to be the product of thesetwo radii. – 4 –epresentations, a variety of observables at strong coupling (even without supersym-metry and at non-zero temperature!) support the identification of effective charges ineqs. (1.1) and (1.2) respectively, for probes following arbitrary world-lines. For thesymmetric representation, a first limitation is that currently we only have at our dis-posal the D3 brane corresponding to a static probe in the vacuum and the conformallyrelated case of hyperbolic motion. It would be highly desirable to find D3 branes thatcorrespond to probes following arbitrary world-lines.Even for this very limited set of world-lines, there are other gauge invariant op-erators whose expectation value would be interesting to compute, e.g. the energy-momentum tensor of the gluonic fields in the presence of the static probe consideredhere, along the lines of [5], or perhaps more interestingly, for the same probe in hyper-bolic motion [14, 17]. We have shown in this note that, while there is no unique functionof κ appearing in different observables, at least for the Li´enard-type formula for energy-loss and the expectation value of < O F > of Coulomb-type fields, the functions of κ coincide. In general, are there relations we can expect or prove for the κ dependence ofdifferent observables? Finally, another interesting issue is the actual range of validityof these different computations. As argued in [17, 14], the a priori range of validity ofthe computations of the vev of the circular Wilson loop and of energy loss by radiationdo not include taking k = 1 (i.e. the fundamental representation). Nevertheless, forthe case of the vev of the circular Wilson loop it was shown in [17] that if one sets k = 1 in the probe D3-brane result, one still correctly recovers (at leading order in1 / √ λ and to all orders in 1 /N ) the gauge theory result, available thanks to a matrixmodel computation [19]. This is presumably due to the large amount of supersymme-try of the configuration, also present in the computation of energy loss in hyperbolicmotion and in the current one for a static half-BPS particle, so it would be importantto understand for which of these computations there are non-renormalization theoremsthat extend their range of validity beyond the region argued in [17].
2. The computation
In this section we will present the details of the computation of < O F > , the expec-tation value of the operator that the dilaton couples to, in the presence of a heavyprobe transforming in the symmetric representation of SU(N). We will first computethe linearized perturbation of the dilaton field caused by the D-brane probe, and fromits behavior near the boundary of AdS we will then read off the expectation value of O F . Our computations will closely follow the ones presented in [3, 4] (see also [6]) forthe case of a probe in the fundamental representation.– 5 –e work in Poincar´e coordinates and take advantage of the spherical symmetry ofthe problem ds AdS = L z (cid:0) dz − dt + dr + r dθ + r sin θdϕ (cid:1) The D3-brane we will be interested in was discussed in [1, 17]. It reaches the boundaryof AdS ( z = 0 in our coordinates) at a straight line r = 0, which is the world-lineof the static dual particle placed at the origin. Since we let the D3 brane reach theboundary, the static particle is infinitely heavy. To describe the D3-brane, identify( t, z, θ, ϕ ) as the world-volume coordinates; then the solution is given by a function r ( z ) and a world-volume electric field r = κz F tz = √ λ π z with κ as defined in eq. (1.3). As shown in detail in [17] this D3-brane is half-BPS.Our next step is to consider at linear level the backreaction that this D3-braneinduces on the AdS × S solution of IIB supergravity. More specifically, since thedilaton is constant in the unperturbed solution, and its stress-energy tensor is quadratic,at the linearized level the equation for the perturbation of the dilaton decouples fromthe rest of linearized supergravity equations. As in [3, 4], we work in Einstein frame,and take as starting point the action S = − Ω L κ Z d x p −| g E | g mnE ∂ m φ∂ n φ − T D Z d ξ q −| G E + e − φ/ πα ′ F | The resulting equation of motion can be written ∂ m (cid:16)p −| g E | g mnE ∂ n φ (cid:17) = J ( x )with the source defined by the D3-brane solution J ( x ) = T D κ Ω L κ sin θz δ ( r − zκ )To compute φ ( x ) we will use its Green function D ( x, x ′ ) φ ( x ) = Z d x ′ D ( x, x ′ ) J ( x ′ )It is convenient to write D ( x, x ′ ) purely in terms of the invariant distance v defined bycos v = 1 − ( t − t ′ ) − ( ~x − ~x ′ ) − ( z − z ′ ) zz ′ (2.1)– 6 –he explicit expression for D ( v ) can be found for instance in [3] D = − π L sin v ddv (cid:18) cos 2 v sin v Θ(1 − | cos v | ) (cid:19) To carry out the integration, we follow the same steps as [4]. We first define a rescaleddilaton field, ˜ φ ≡ Ω L κ φ and use eq. (2.1) to change variables from t ′ to v to obtain, after an integration byparts˜ φ = N κz π Z ∞ dr ′ Z π dθ ′ sin θ ′ Z π dϕ ′ Z ∞ dz ′ δ ( r ′ − z ′ κ )( z + z ′ + ( ~x − ~x ′ ) ) Z π dv cos 2 v (cid:16) − zz ′ cos vz + z ′ +( ~x − ~x ′ ) (cid:17) The integral over v is the same one that appeared in the computation of the per-turbation caused by a string dual to a static probe [4]. The novel ingredient in thecomputation comes from the non-trivial angular dependence in the current case. Whileit might be possible to completely carry out this integral, at this point it is pertinent torecall that to compute the expectation value of the dual field theory operator, we onlyneed the leading behavior of the perturbation of the dilaton field near the boundary of AdS . Specifically [3], < O F > = − z ∂ z ˜ φ | z =0 (2.2)so for our purposes it is enough to expand the integrands in powers of z , and keep onlythe leading z term. This simplifies the task enormously, and reduces it to computingsome straightforward integrals. Skipping some unilluminating steps we arrive at˜ φ = N κ π z ( z + r ) κ ) / (cid:0) − κ κ r r + z (cid:1) + O ( z )which upon differentiation, and setting then z = 0 as required by eq. (2.2), leads toour final result < O F > = N κ √ κ π | ~x | = k √ λ π q k λ N | ~x |
3. Acknowledgements
We would like to thank Mariano Chernicoff for helpful conversations. The research ofBF is supported by a Ram´on y Cajal fellowship, and also by MEC FPA2009-20807-C02-02, CPAN CSD2007-00042, within the Consolider-Ingenio2010 program, and AGAUR– 7 –009SGR00168. The research of BG is supported by an ICC scholarship and by MECFPA2009-20807-C02-02. The research of AL was partly supported by a Spanish MEP-SYD fellowship for undergraduate students. AL further acknowledges support fromFundaci´on Caja Madrid and the Perimeter Scholars International program.
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