Energy loss by radiation to all orders in 1/N
aa r X i v : . [ h e p - t h ] J un Energy loss by radiation to all orders in 1/N
Bartomeu Fiol and Blai Garolera
Departament de F´ısica Fonamental iInstitut de Ci`encies del Cosmos,Universitat de Barcelona,Mart´ı i Franqu`es 1, 08028 Barcelona, Catalonia, Spain ∗ We use the AdS/CFT correspondence to compute the energy radiated by an infinitely massive half-BPS particle charged under N = 4 SU ( N ) SYM, transforming in the symmetric or antisymmetricrepresentation of the gauge group, and moving in the vacuum, to all orders in 1 /N and for large ’tHooft coupling. For the antisymmetric case we consider D5-branes reaching the boundary of AdS at arbitrary timelike trajectories, while for the symmetric case, we consider a D3-brane in AdS that reaches the boundary at a hyperbola. This D3-brane solution is the analytic continuation ofthe one previously used to compute the expectation value of a circular Wilson loop [1]. We compareour results to the one obtained for the fundamental representation by Mikhailov [2], obtained byconsidering a string in AdS . I. INTRODUCTION
Given a gauge theory, one of the basic questions onecan address is the energy loss of a particle charged un-der such gauge fields, as it follows arbitrary trajectories.For classical electrodynamics this is a settled question,with many practical applications [3]. Much less is knownfor generic quantum field theories, especially in stronglycoupled regimes. This state of affairs has started to im-prove with the advent of the AdS/CFT correspondence[4], which has allowed us to explore the strongly coupledregime of a variety of field theories. Within this frame-work, the particular question of the energy radiated by aparticle charged under a strongly coupled gauge theory -either moving in a medium, or in the vacuum with non-constant velocity - has received a lot of attention (see[5] for relevant reviews). The motivations are manifold,from the more phenomenological ones, like modeling theenergy loss of quarks in the quark-gluon plasma [6] to themore formal ones, like the study of the Unruh effect [7].In most of these studies the heavy particle transforms inthe fundamental representation of the gauge group, andthe dual computation is in terms of a string moving inan asymptotically AdS space. The main purpose of thisnote is to extend this prescription to other representa-tions of the gauge group, which will amount to replacethe fundamental string by D3 and D5 branes (see [8] for aprevious appearance of this idea), in complete analogy tothe prescription developed for the computation of Wilsonloops [1, 9–12].Besides the intrinsic interest of this generalization, ourmain motivation in studying it is that, as it happens inthe computation of certain Wilson loops, the results forthe energy loss obtained with D-branes give an all-ordersseries in 1 /N . Given the paucity of such results for large N
4d gauge theories, this by itself justifies its consider-ation. Furthermore, these 1 /N terms might shed some ∗ bfi[email protected], bgarolera@ffn.ub.es light on some recent results in the study of radiation us-ing the AdS/CFT correspondence. Let’s briefly reviewthem.The case of an infinitely massive particle transformingin the fundamental representation and following an arbi-trary timelike trajectory was addressed by Mikhailov [2],who quite remarkably found a string solution in AdS that solves the Nambu-Goto equations of motion andreaches the boundary at any given particle world-line.Working in Poincar´e coordinates, ds AdS = L y (cid:0) dy + η µν dx µ dx ν (cid:1) (1)it was furthermore shown that the energy of that stringwith respect to the Poincar´e time is given by E = √ λ π (cid:18)Z dt~a − | ~a ∧ ~v | (1 − v ) + γ y | y =0 (cid:19) (2)where the integral is with respect to the world-line timecoordinate, and λ = g Y M N is the ’t Hooft coupling.The second (divergent) term corresponds to the (infi-nite) mass and γ is the Lorentz factor. The first termcorresponds to the radiated energy, so in the supergrav-ity regime the total radiated power by a particle in thefundamental representation is P F = √ λ π a µ a µ (3)which is essentially Lienard’s formula for radiation inclassical electrodynamics [3] with the substitution e → √ λ/ π . This √ λ dependence also appears - and hasthe same origin - in the computation of the vev of Wil-son loops at strong coupling [9, 13].Having computed the total radiated power, a more re-fined question is to determine its angular distribution.For a particle moving in the vacuum, this has been donein [14, 15], who found that this angular distribution is es-sentially like that of classical electrodynamics. This is asomewhat counterintuitive result, as one might have ex-pected that the strong coupling of the gauge fields wouldtend to broaden the radiating pulses and make radiationmore isotropic. In particular, the authors of [15] arguethat these results are an artifact of the supergravity ap-proximation, and might go away once stringy effects aretaken into account (see [16] for alternative interpreta-tions). Here is where considering particles in other rep-resentations might be illuminating, since the 1 /N expan-sion of the radiated power we find can be interpreted ascapturing string loop corrections [17].The plan of the present note is as follows: in the nextsection we introduce D5-branes dual to particles in theantisymmetric representation following arbitrary timeliketrajectories, and evaluate the corresponding energy loss.We then consider a D3-brane dual to a particle in thesymmetric representation following hyperbolic motion,and compute its energy loss. We end by discussing thepossible connection of this result with the similar one forparticles in the fundamental representation, and men-tioning possible extensions of this work. II. D5 BRANES AND THE ANTISYMMETRICREPRESENTATION
Given a string world-sheet that solves the Nambu-Gotoaction in an arbitrary manifold M , there is a quite gen-eral construction due to Hartnoll [18] that provides a so-lution for the D5-brane action in M × S , of the formΣ × S where Σ ֒ → M is the string world-sheet and S ֒ → S . The evaluation of the respective renormalizedactions gives then a universal relation between the vevof Wilson loops in the antisymmetric and fundamentalrepresentations, already observed in particular examples[10, 19]. More recently, this construction has been used toevaluate the energy loss of a particle in the antisymmet-ric representation, moving with constant speed in a ther-mal medium [8]. In this section we combine Mikhailov’sstring world-sheet solution [2] with Hartnoll’s D5-braneconstruction [18] to compute the radiated power for aparticle in the antisymmetric representation.For a given timelike trajectory, we consider a D AdS × S , with world-volume Σ × S where Σ is thecorresponding Mikhailov world-sheet [2]. On Σ there isin addition an electric DBI field strength with k units ofcharge [18]. This D5-brane is identified as the dual toa particle transforming in the k-th antisymmetric repre-sentation, and following the given timelike trajectory. Asshown in [18] the equations of motion force the angle of S in S to besin θ cos θ − θ = π (cid:18) kN − (cid:19) (4)We now proceed to compute the energy with respect tothe Poincar´e time coordinate and the radiated power ofsuch particle. The energy density for the D5-brane is E D = T D L y | γ + F | s p −| γ + F | = T D L y | γ Σ | s sin θ p −| γ Σ | p | γ S | where the subscript s means that the determinant is re-stricted to the spatial directions of the D5-brane or thefundamental string. We have used that in Hartnoll’s solu-tion the DBI field strength is purely electric and the DBIdeterminant is block diagonal. Integrating over the S part of the world-volume one immediately obtains up toconstants the energy density of the fundamental string,so E D = 2 N π sin θ E F This is the same relation as the one found between therenormalized actions of the D5-brane and the fundamen-tal string [18], and in [8] for the relation of drag forcesin a thermal medium. In the regime of validity of su-pergravity, the radiated power of a particle in the k-thantisymmetric representation is therefore related to theradiated power of a particle in the fundamental represen-tation (3) by P A k = 2 N π sin θ P F (5) III. D3 BRANES AND THE SYMMETRICREPRESENTATION
The computation of Wilson loops of half-BPS parti-cles in the symmetric representation is given by eval-uating the renormalized action of D3 branes [11], andanalogously we propose to compute the radiated powerof a half-BPS particle in the symmetric representationby evaluating the energy of a D3 brane that reaches theboundary of
AdS at the given timelike trajectory. Con-trary to what happens for the fundamental or the anti-symmetric representations, we currently don’t have thegeneric D3-brane solution, so we will focus on a particu-lar trajectory. On the other hand, since these D3-branesare fully embedded in
AdS , we don’t use any possibletransverse dimensions, so the results should be valid forother 4d conformal theories with a gravity dual.The particular trajectory we will consider is one-dimensional motion with constant proper acceleration,which in an inertial system corresponds to γ a = 1 /R .The trajectory is hyperbolic, − ( x ) + ( x ) = R . Arelevant feature is that a special conformal transforma-tion applied to a straight world-line (static particle) givesthe two branches of hyperbolic motion [20]. Besides itsprominent role in the study of radiation and the Unruheffect, another reason to choose this trajectory is that therelevant D3-brane is the analytic continuation of the onealready found in [1].The radiated energy of a particle in the fundamentalrepresentation, eq.(2) derived in [2], is written in terms ofthe world-line of the heavy particle. At least in particu-lar cases, it is possible to obtain an alternative derivationthat emphasizes the presence of a horizon in the world-sheet metric, which encodes the split between radiativeand non-radiative gluonic fields, and therefore signals theexistence of energy loss of the dual particle, even in thevacuum [21]. It is convenient to briefly rederive this re-sult for the particular case of hyperbolic motion, sincethe computation of the energy loss using a D3 branethat we will shortly present resembles closely this secondderivation. Working in Poincar´e coordinates, Mikhailov’sstring solution for hyperbolic motion can be rewritten as y = R + ( x ) − ( x ) ; the Euclidean continuation ofthis world-sheet is the one originally used to evaluate thevev of a circular Wilson loop [22] (see also [23]). Thisworld-sheet is locally AdS and has a horizon at y = R ,with temperature T = 1 / πR , which is the Unruh tem-perature measured by an observer following a r = R trajectory in Rindler space. By integrating the energydensity from the horizon to the boundary we obtain, E = √ λ π Z R dyy p R + ( x ) − y = √ λ π (cid:18) − x R + γ y | y =0 (cid:19) (6)The contribution from the boundary is just the (diver-gent) second term, corresponding to the mass of the par-ticle. The first term comes from the horizon contribution,and corresponds to the radiated energy. A. The D3-brane solution
We are interested in a D3-brane that reaches theboundary of
AdS at a single branch of the hyperbola − ( x ) + ( x ) = R . To find it, we change coordinateson the ( x , x ) plane of (1) to Rindler coordinates, so thenew coordinates cover only a Rindler wedge ds = L y (cid:0) dy + dr − r dψ + dr + r dφ (cid:1) (7)In these coordinates the relevant D3-brane solution foundin [1] is given by( r + r + y − R ) + 4 R r = 4 κ R y (8)where κ = k √ λ N Near the
AdS boundary y = 0, this solution goes to r =0 , r = R , so it reaches a circle in Euclidean signatureand the branch of a hyperbola in the Lorentzian one. TheD3-brane also supports a non-trivial Born-Infeld field-strength on its world-volume [1]. By a suitable changeof coordinates, its world-volume metric can be written as[1] ds = L (1+ κ )( dζ − sinh ζdψ )+ L κ ( dθ +sin θdφ )(9)so it is locally AdS × S , with radii L √ κ and Lκ respectively, and it has a horizon at ζ = 0 (i.e. r =0 in the coordinates of (7)). The temperature of this horizon can be computed by requiring that the associatedKilling vector is properly normalized at infinity; this iseasily done in the coordinates of (7) and the resultingtemperature is again T = 12 πR (10) B. Evaluation of the energy
To determine the total radiated power of this solutionwe will evaluate the energy with respect the Poincar´etime coordinate x . The energy density is E = T D L y | γ + F | s p −| γ + F | − L y ! (11)After we substitute the Lorentzian continuation of thesolution of [1] in this expression, the energy density is E = T D L y (1 + κ ) R + ( x ) p κ (1 + κ ) R − κ R r − (1 + κ ) R r − ! (12)The energy is the integral of this energy density from theboundary to the world-volume horizon. A long compu-tation yields E = 2 N κπ (cid:18) − x R p κ + γ y | y =0 (cid:19) (13)Exactly as it happened for the string, eq. (6), the bound-ary contributes only the second term, which is divergent,and is just k times the one for the fundamental string, eq.(6). The first term is the contribution from the horizon,and from it we can read off the total radiated power P S k = 2 N κπ p κ R = k √ λ π r k λ N R This result was found for a particular timelike trajectorywith a µ a µ = 1 /R . Nevertheless, in classical electrody-namics the radiated power depends on the kinematicsonly through the square of the 4-acceleration, a µ a µ andas we have seen, the same is true in theories with gravityduals for particles in the fundamental, eq. (3), and an-tisymmetric representations, eq. (5). It is then naturalto conjecture that in the regime of validity of supergrav-ity, the radiated power by a particle in the symmetricrepresentation following arbitrary timelike motion is P S k = k √ λ π r k λ N a µ a µ (14)It would be interesting to check this conjecture by find-ing D3-branes that reach the AdS boundary at arbitrarytimelike trajectories and evaluating the correspondingenergies.We now discuss the range of validity of this result, andits possible relevance for the case of a particle in the fun-damental ( k = 1) representation. By demanding that theradii of the D3 brane are much larger than l s and that itsbackreaction can be neglected, one can conclude [1] thatthis result can be trusted when N /λ ≫ k ≫ N/λ / .Is is therefore not justified a priori to set k = 1 in our re-sult, eq. (14). Nevertheless, the Euclidean continuationof this very same D3-brane was used in [1] to computethe expectation value of a circular Wilson loop, which for k = 1 is known exactly for all N and λ thanks to a ma-trix model computation [17, 24], and it was found [1] thatthe D3-brane reproduces the correct result in the large N, λ limit with κ fixed, i.e. even for k = 1. This betterthan expected performance of the Euclidean counterpartof this D3-brane in a very similar computation suggeststhe exciting possibility that (14) might capture correctlyall the 1 /N corrections to the radiated power of a parti-cle in the fundamental representation, i.e. for k = 1, inthe limit of validity of supergravity.An obvious question that we are currently pursuing iswhether the angular distribution of the radiated energyobtained from this D3-brane differs qualitatively from the results obtained with fundamental strings [14, 15].Finally, as already mentioned, the Euclidean versionof the D3-brane considered here was used in [1] to eval-uate the vev of a circular Wilson loop. That D3-braneresult is in turn only an approximation to the exact re-sult, available for all N and λ thanks to a matrix modelcomputation [17, 24]. It would be extremely interestingto understand whether the radiated power of a particlecoupled to a conformal gauge theory can be similarlycomputed by a matrix model. IV. ACKNOWLEDGEMENTS
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