Holographic energy loss in non-relativistic backgrounds
aa r X i v : . [ h e p - t h ] D ec Holographic energy loss in non-relativistic backgrounds
Mahdi Atashi , Kazem Bitaghsir Fadafan , Mitra Farahbodnia Physics Department, Shahrood University of Technology,P.O.Box 3619995161 Shahrood, Iran
Abstract
In this paper, we study some aspects of energy loss in non-relativistic theories fromholography. We analyze the energy lost by a rotating heavy point particle along acircle of radius l with angular velocity ω in theories with general dynamical exponent z and hyperscaling violation exponent θ . It is shown that this problem provides a novelperspective on the energy loss in such theories. A general computation at zero and finitetemperature is done and it is shown that how the total energy loss rate depends non-trivially on two characteristic exponents ( z, θ ). We find that at zero temperature thereis a special radius l c where the energy loss is independent of different values of ( θ, z ).Also at zero temperature, there is a crossover between a regime in which the energy lossis dominated by the linear drag force and by the radiation because of the acceleration ofthe rotating particle. We find that the energy loss of the particle decreases by increasing θ and z . We note that, unlike in the zero temperature, there is no special radius l c atfinite temperature case. e-mail:[email protected] e-mail:[email protected] e-mail:[email protected] Introduction
From the AdS/CFT correspondence, gravity in the asymptotic AdS geometry is related tothe conformal field theory (CFT) on the boundary. It is well known that the dual quantumfield theory would be a strongly coupled theory with a UV fixed point which is invariantunder the scaling of space ( ~x ) and time ( t ). Recently, the generalization to the field theorieswhich are not conformally invariant and the asymptotic geometries are not AdS has beenstudied. Such field theories are very important in the condensed matter physics. As anexample, consider the Lifshitz fixed point theories with the following anisotropic scalingsymmetry ( t, ~x ) → ( w z t, w ~x ) , (1)here w is constant and z is called dynamical exponent. The case of z = 1 is related to therelativistic scale invariance theory. Then one may consider z = 1 theories as non-relativistictheories because of different scaling of time and space.From the AdS/CFT correspondence, the gravity dual to such theories has been studiedin [1] and [2]. From [3], the gravity dual of the Lifshitz fixed point can be found. To derivesuch non-relativistic geometries from Einstein gravity, one should add other matter fieldslike massive gauge fields. New backgrounds have been found by including both an abeliangauge field and a scalar field with nontrivial potential as ds d +2 = u θd ( − dt u z + du u + dρ u + ρ dφ u + P di =3 dx i u ) . (2)Here, the boundary is located at u = 0 where u is the bulk or radial direction. Also θ is hyperscaling violation exponent and the spatial dimension of the boundary field theoryis given by d . We have written two of the d spatial dimensions using coordinates ( ρ, φ ).Physical conditions leads to the following relations between d , θ and z as( z − d + z − θ ) ≥ , ( d − θ )( d z − θ − d ) ≥ . (3)In this paper we probe the non-relativistic theories by studying how the energy lossof a rotating heavy point particle depends on the different values of θ and z . The studyof energy loss is an interesting and important problem in studying quark-gluon-plasma(QGP) produced at RHIC and LHC [4]. In this case the point particle could be a heavyquark. Study of such problems need non perturbative strongly coupled approaches andtime dependent methods, then using the AdS/CFT correspondence is reliable [5, 6]. Firststudy of the energy loss of heavy quarks has been done in [7, 8]. They studied a movingheavy quark with a constant velocity through the QGP and calculated the energy requiredto keep the heavy quark at constant speed. It is found that in this case the energy loss isproportional to the momentum of the moving quark like the energy loss mechanism in thedrag force. Then one mechanism for energy loss of the particle comes from the drag forcechannel. The other possible mechanism for energy loss of the particle could be radiationbecause of the acceleration of the particle. Study of the energy loss from accelerating objectsis basic and interesting problem of quantum field theories. It is very difficult to describe it inthe strongly coupled systems. Finding a framework to describe radiation in non-relativistictheories would be very interesting.Using the AdS/CFT correspondence, we study the energy loss of accelerated heavy ob-jects in the non-relativistic backgrounds. The moving particle is described by a classical1tring in the bulk. Such string ends on the boundary and the end point corresponds toposition of the particle. We study the dynamics of the heavy point particle rotating with aconstant angular velocity in the non-relativistic backgrounds with fixed values of z and θ .We assume that the particle rotates along a circle of radius l with a constant angular fre-quency ω . Then the constant velocity and acceleration are v = lω and a = l ω , respectively.Here, we discuss how different mechanisms of energy loss, i.e radiation and drag force couldbe accessible.The energy loss of a rotating quark in strongly coupled N = 4 SYM theory has been firststudied in [9]. It was shown that the rotating particle serves as a model system in whichtwo different mechanisms of energy loss is accessible via a classical gravity calculation.This interesting simple model provides a novel perspective on different open questions instudying the energy loss in strongly coupled systems. For example, the radiation pattern ofan accelerating quark in a nonabelian gauge theory at strong coupling was studied in [10].They find the same angular distribution of radiated power in strongly and weakly coupledregimes which propagates at the speed of light and does not show broadening. Such studyat finite temperature plasma has been done in [11]. The absence of the broadening isrelated to the backreaction of the massive particle on the boundary. It means that bulksources with the speed of light do not generate any energy on the boundary field theory. Itwas argued in [12] that the main reason comes from the supergravity approximation. Theexpectation value of the energy density sourced by the massive particle with an arbitrarymotion was studied in [13]. Because of the important role of anisotropy during the initialstage of producing QGP, the energy loss of a rotating quark in an anisotropic stronglycoupled plasma has also been considered in [14]. For studying of the energy loss of arotating particle in confining strongly coupled theories see [15].There is an interesting property of energy loss in non-relativistic theories; it was shownthat even at zero temperature the drag force in non-relativistic theories such as Schrodingeror Lifshitz theories is not zero [16, 17, 18]. We have studied before the drag force forasymptotically Lifshitz space times in [17]. The drag force on heavy object in an effectivetheory with hyperscaling violation has been studied in [19] and [20]. To explore this effect,one should notice that in the relativistic theories when z = 1, the energy and momentumare conserved and because of the invariance under boosts the drag force is zero. In z = 1theories, one finds dissipation and energy and momentum drain into the soft IR modes. Itwas found that for certain values of z a moving particle only travels a finite distance [21].Also this study extended to the case of the response of quantum critical points with thehyperscaling violation to a disturbance caused by a heavy charged particle in [22].In this paper, we start by studying the energy loss of rotating particle at zero temper-ature. Surprisingly, we find a critical radius l c where the total energy loss of the rotatingparticle does not depend on the non-relativistic parameters of the theory z and θ . At thisradius the string end point moves at the speed of light. Also, as we explained unlike therelativistic case, the accelerated heavy particle looses its energy by drag force. Then oneexpects that the rate of the energy loss would be the same as the drag force at constantvelocity and small acceleration where a = vω →
0, meaning ω →
0. One reaches in thislimit by increasing the radius l and decreasing the angular velocity ω . We check this be-havior numerically by drawing ratio of the total energy loss of the rotating particle to thedrag force. Interestingly, plots confirm this statement. Therefore, we study the effect of dy-namical z component and hyperscaling violation parameter θ on this ratio. We summarize2he final statements in the discussion section. We also expect the other channel of energyloss,i.e radiation due to acceleration of the particle. In the limit of decreasing l ( l → ω ( ω → ∞ ), one expects domination of radiation. In the relativistic case,Mikhailov has derived a general result for radiation of an accelerated particle in N=4 SYMvacuum [23]. One may describe the radiation by expansion process as the relativistic caseand consider a systematic expansion [21]. Although, one can not separate out differenteffects from radiation by the rotating particle in our problem, it would be useful to un-derstand the physics qualitatively. As it was discussed in [9], one can not make a sharpdistinction between radiation and drag force. We confirm in this study that there are twodifferent regimes at zero temperature as the acceleration-radiation-dominated regime andthe drag-dominated regime, respectively.We extend our study to the case of the energy loss at finite temperature hyperscalingtheories. We also note that, unlike in the zero temperature, there is no special radius l c atfinite temperature case.This paper is organized as follows. In section two, we will present the details of calcula-tion of energy loss at zero temperature and study two different mechanisms of energy loss,i.e drag force and radiation. We study behavior of different mechanisms as a function ofparameters of non-relativistic theory z and θ . We consider this study at finite temperaturein section three. In the last section we summarize our results. In this section, we consider the rotating particle at zero temperature. The backgroundis given by (2). Based on the AdS/CFT correspondence, the particle is located at theend point of the classical string attached to the boundary of the background geometry.Holographically, the energy loss of rotating quark could be studied by studying the motionof rotating spiral classical string in the geometry (2). The dynamics of the spiraling stringgoverned by the Nambu-Goto action. More details of the calculation can be found in [14].The world-sheet ansatz for a rotating quark is paramerized as follows: X µ = ( t = τ, u = σ, ρ = ρ ( u ) , φ = ωτ + φ ( u ) , x = 0) , (4)The radial and angular profiles of the rotating string is given by ρ ( u ) and φ ( u ), respectively.They obey the following boundary conditions ρ (0) = l, φ (0) = 0 . (5)and the Lagrangian density can be obtained as L = u θd − − z q (1 − ρ ω u z − )(1 + ρ ′ ) + ρ φ ′ , (6)where ” ′ ” shows derivative with respect to ” u ”. We then obtain the equations of motion We would like to thank D. Tong for discussion on radiation in non-relativistic theories. Ω u c Θ= d2 ÈÈ z = Θ= ÈÈ z = Θ= ÈÈ z = Ω u c Θ= ÈÈ z = Θ= ÈÈ z = Figure 1: Behavior of world sheet horizon u c vs ω , θ and z as a function of angular velocity ω . In all plots we fixed Π = 1.from the Lagrangian. It depends on φ ′ but not on φ , so the conjucate momentum, Π is aconstant of motion and can be written as follows:Π = − ∂L∂φ ′ = − u m ρ φ ′ q (1 − ρ ω u n )(1 + ρ ′ ) + ρ φ ′ , (7)where m = θd − − z and n = (2 z − ρ ( u ) is then given by ρ ′′ + ( ρ ( u − mρρ ′ ) u ( u − m Π − ρ ) + (2 − nu n − ω ρ ρ ′ )2 ρ (1 − u n ω ρ ) )(1 + ρ ′ ) = 0 (8)Solving the equation of motion (8) by considering the boundary conditions in (5), thespiral profile of string will be obtained. By imposing the reality condition on φ ′ , one findsthe special value of the radial coordinate u c . The radius of spiraling string at this value isdenoted as ρ c . They can be found as ρ c = (Π ω ) − q m − n ω , u c = (Π ω ) m − n (9)For relativistic case z = 1, it was shown in [9] that the string in the range of u < u c which rotates with a speed slower than the local speed of light is casually disconnected fromthe part of string in u > u c . While here the local velocity of string is fixed by v = ρ c ω andit can change from zero to infinity. This indicates the non-relativistic nature of the dualfield theory. In Fig. 1, we study behavior of u c in terms of changing angular frequency ω and dynamical exponents θ and z .It would be interesting to point out that by rotating the string, a horizon appears onthe world sheet and the special point of u c coincides with a special point on the world-sheet which is denoted as a world sheet horizon. It is very interesting that even at zerotemperature it exists while there is no any black hole in the bulk. The physics of the worldsheet horizon is related to the Brownian motion of the dual particle in the boundary [24].Regarding this study one should consider a gluonic field around the particle in the boundary,also by rotation an ”internal degree of freedom is being excited” [25].Based on (9), if Π ω < ω > z and θ leads to different behavior. Asit is clearly seen from Fig. 1, at fixed Π = 1 the critical value of angular velocity would be4 c = 1. So for ω > ω c effect of increasing z or θ is not the same as ω < ω c . The specialpoint in the bulk direction, u c , is decreasing by increasing ω . Also these two plots showthat, there is a critical value for angular velocity, ω c = 1, so that for ω < ω c , u c increasesby increasing θ while decreases by increasing z , and vice versa for ω > ω c . u Ρ Θ= ÈÈ z = Θ= d2 ÈÈ z = Θ= ÈÈ z = u Ρ Θ= ÈÈ z = Θ= d2 ÈÈ z = Θ= ÈÈ z = u Ρ Θ= d2 ÈÈ z = Θ= d2 ÈÈ z = u Ρ Θ= ÈÈ z = Θ= ÈÈ z = Figure 2: String radius vs radial direction in different θ and z with ω = Π = 1. As shownin the top row, by increasing θ , at fixed u , the string radius is increasing. As shown in thebottom row, at fixed θ and u , the string radius increases by increasing z .Fig 2, shows the behavior of string coiling radius, ρ ( u ), versus radial direction, u , fordifferent values of θ and z with ω = Π = 1. Notice that we set u = 0 as boundary, so ρ (0)is the radius of rotation of the particle. The first row of Fig. 2 shows that at fixed z and ω , the rotation radius for greater value of θ is bigger. Also the second row shows similarbehavior of ρ versus z . The generic features of the spiraling string in geometry (2) wasdiscussed in [19] and the radius of motion ρ as a function of u for the case of z = 2 and θ = d , d , θ , at fixed u , the string radius is increasing. Alsoat fixed θ and u , increasing z leads to increasing the string radius. The energy loss of rotating particle is given by dEdt = 12 πα ′ Π ω (10)We have studied behavior of the energy loss of rotating particle in terms of the radiusof the rotation in the boundary in Fig. 3. One finds that at fixed θ , by increasing angular5elocity ω , the energy loss is increasing for each z . Also interestingly, there is a criticalvalue for rotation radius, l c , so that l c ω = 1. For l > l c , at fixed ω and l , the energy loss isincreasing by decreasing θ , and vice versa for l < l c . As it is clearly seen from this figure, at l = l c , the energy loss does not depend on θ and ω . To distinguish between different valuesof dynamical exponent z , we plot the energy loss versus l for z = 2 and = 3 in Fig. 3.One finds that for l > l c , the energy loss corresponding to z = 2 is greater than energy losscorresponding to z = 3 in all values for θ and ω , and vice versa for l < l c . The energy lossis also z -independent at l = l c . Briefly, the general features of the energy loss of rotatingparticle at zero temperature non-relativistic theories can be summarized as • As the same as the relativistic case, the energy loss is increasing by increasing ofrotation radius l . • By increasing rotation velocity ω , the energy loss is increasing for each z and θ . • There is a critical value for rotation radius, l c , so that l c ω = 1 and interestingly theenergy loss is θ and z independent. l P z = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= d2 ÈÈ Ω= l P z = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω=
10 15 20 25 l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω= = ÈÈ Θ= ÈÈ Ω=
10 15 20 25 l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= d2 ÈÈ Ω= l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= d2 ÈÈ Ω= l P z = ÈÈ Θ= d2 ÈÈ Ω= = ÈÈ Θ= d2 ÈÈ Ω= Figure 3: Energy loss versus rotation raduis of particle for different values of θ , ω and z . Inthe first row, we fixed z = 2 and in each plot from top to down we assumed ω = 5 , . , . θ = 0 , d , d . In the secondrow, we fixed z = 2 and in each plot from top to down we assumed θ = 0 , d , d . Also inthe second row from left plot to right plot we considered ω = 5 , . , .
05. In each plot ofthe third row, we fixed θ = d and from left plot to right plot ω = 0 . , , . z , we plotted the ratio of l l where l and l are therotation radius corresponding to z = 3 and z = 2, respectively. This ratio is plotted versus6nergy loss Π. As shown in Fig. 4, where Π ω = 1 the ratio goes to unity. This pointcorresponds to l = l c where the energy loss does not depend on θ , ω and z . By increasingof θ at fixed ω and z , the ratio increases for Π ω < ω >
10 20 30 40 50 P l l Ω= Θ= d2 Θ= Θ=
10 20 30 40 50 P l l Ω= Θ= d2 Θ= Θ= Figure 4: Ratio of l z =3 l z =2 vs Π for different values of θ and ω . The point that the ratio is 1, isthe same for similar values of ω for each value of θ . This point corresponds to l = l c wherethe energy loss does not depend on θ , ω and z . To find the drag force, one should consider an open string moving in the non-relativisticbackground (2). From the correspondence, it represents an external heavy point particlemoving with constant velocity v in a non-relativistic field theory. The drag force for non-relativistic field theories with Lifshitz symmetries or Schrodinger symmetries and theorieswith hyperscaling violation have been studied before in [16, 17, 18, 19, 21, 20]. Interestingly,it was shown that even at zero temperature the energy loss is non-zero which is expressedas 2 πα ′ dEdt drag ≡ Π drag = v d − θd z − d +2 (11)In this subsection, we compare the total energy loss in (10) with the energy loss from dragforce channel in (11). As we discussed in the introduction, one expects that the rate ofthe energy loss could be the same as (11) at constant velocity and small acceleration where ω →
0. To reach in this limit, we increase the radius l and decrease the angular velocity ω . We checked this behavior numerically by drawing ratio of the total energy loss of therotating particle to (11) in Fig. 5. Interestingly, calculations confirm this statement. Forexample, one finds from Fig. 5 that the ratio at fixed z and ω goes to unity faster for greatervalue of θ . On the other word at fixed l the ratio corresponding to smaller value of θ isgreater. It means that by increasing θ the contribution of radiation becomes smaller. Also,the ratio at fixed θ and ω goes to unity faster for greater value of z . As shown, the ratiocorresponding to smaller z is greater at fixed l for l < l s and vice versa for l > l s , where the l s is a special rotation radius that the ratio for different value of z are equal there.7 l PP drag z = ÈÈ Θ= d (cid:144) È Ω= = ÈÈ Θ= È Ω= = ÈÈ Θ= (cid:144) È Ω=
10 15 20 25 30 l PP drag z = ÈÈ Θ= d (cid:144) È Ω= = ÈÈ Θ= d (cid:144) È Ω= l PP drag z = ÈÈ Θ= d (cid:144) È Ω= = ÈÈ Θ= È Ω= = ÈÈ Θ= (cid:144) È Ω= l PP drag z = ÈÈ Θ= d (cid:144) È Ω= = ÈÈ Θ= d (cid:144) È Ω= Figure 5: Ratio of energy loss over drag force vs rotation radius for different values of θ (left) and z (right) at fixed ω = 0 .
05 and ω = 5. One important advantage of studying energy loss of rotating particle is getting informationabout radiation in strongly coupled non-relativistic theories. As it was discussed before,we expect the other channel of energy loss, i.e radiation due to acceleration of the particle.To dominate this regime, we decrease l and increase ω . For example, see Fig. 5 andconsider ω = 5 cases. If one fixes the radius of rotating particle, concludes that at fixeddynamical exponent z , increasing hyperscaling violation parameter θ leads to decreasingthe energy loss from radiation channel. On the other hand, if one fixes the rate of radiationand increases the parameter θ , finds that l will increase, too. To understand the effect ofchanging dynamical exponent z , we fixed θ = d and increased z = 2 to z = 3. It is clearlyseen from Fig. 3 that at fixed l increasing z leads to decreasing of radiation of the particle.However, at fixed energy loss by increasing z , the radius of rotating particle will increase.It seems that changing z or θ at fixed l has the same effect on the radiation.We compare the radiation in non-relativistic theories with relativistic vacuum radiationproposed in [23]. In the circular motion, the formula is given by dEdt Relativistic = √ λ π a (1 − v ) . (12)Now we define the following ratio to compare the result with vacuum radiation in N = 4SYM theory. ΠΠ Relativistic = Π v ω (1 − v ) . (13)We plot ΠΠ Relativistic in Fig. 6. This ratio shows that the radiation in strongly coupled non-8elativistic theories is smaller than the relativistic case at small velocities. As one expectsfrom (13), the ratio vanishes at speed of light, i.e v = 1. In this figure we fixe ω = 5, then at l = 0 . l PP Relativistic
Figure 6: Ratio of energy loss over radiation in relativistic vacuum vs rotation radius.
So far we have studied the energy loss of particle in quantum field theories with non zerodynamical exponent z and hyperscaling violation θ at zero temperature. We now generalizethe calculations to the case of finite temperature non-relativistic backgrounds. The gravitydual is given by the following metric [26] ds = (cid:18) uu F (cid:19) θd (cid:18) − u − z f ( u ) dt + du u f ( u ) + d −→ x u (cid:19) , f ( u ) = 1 − (cid:18) uu h (cid:19) z + d − θ . (14)The boundary is located at u → u h which can be found bysolving f ( u h ) = 0. The gravity is valid only for u > u F which means that u F is a dynamicalscale and we are follwing an effective holographic description. In order to study finitetemperature case, one needs to consider u F < u h . We rescale the metric (14) by u F in ouranalysis. The temperature is given by T = d + z − θ π u zh (15)First, we should use the asatz in (4) and Nambu-Goto action to study the shape of thespiraling string in the bulk. The lagrangian density is given by L = u θd − − z s(cid:0) f ( u ) − ρ ω u z − (cid:1) (cid:18) f ( u ) + ρ ′ (cid:19) + ρ φ ′ f ( u ) , (16)9hich is exactly matches with (6) when f ( u ) →
1. We assume the constant of motion asΠ, then the equation of motion of φ ( u ) is given by φ ′ = Π (cid:16) − f ( u ) u z + ρ ω u (cid:17) (cid:16) u f ( u ) + ρ ′ u (cid:17)(cid:16) − ρ ( u ) f ( u ) u z (cid:17) (cid:16) − ρ ( u ) f ( u ) u θd − z − + Π (cid:17) (17)The world sheet horizon u t also appears in this case and one finds it by solving thefollowing equations: − f ( u t ) u zt + ρ t ω u t = 0 , − u θd − z − t ρ ( u t ) f ( u t ) + Π = 0 . (18)We can use (17) to eliminate φ ′ from the equation of motion and obtain an equationof motion for ρ ( u ) in terms of the constant Π. The differential equation after the partialdecomposition can be written as the following form ρ ′′ ( u ) + u θd − ρ ( u ) ( − d u + ρ ( u ) ( − f ( u )( d + dz − θ ) + duf ′ ( u )) ρ ′ ( u )) (cid:0) f ( u ) ρ ′ ( u ) (cid:1) d ρ ( u ) (cid:16) − Π u z + u θd f ( u ) ρ ( u ) (cid:17) +2 u + ρ ′ ( u ) (cid:0) u ρ ( u ) h ′ ( u ) + 2 u f ( u ) ρ ′ ( u ) + u z ω ρ ( u ) (2 − z + 2 f ( u ) − zf ( u ) + uf ′ ( u )) ρ ′ ( u ) (cid:1) u ρ ( u ) ( u f ( u ) − u z ω ρ ( u ) ) = 0 . (19)This is an important result as general equation of motion for the rotating particle in thefinite temperature non-relativistic theories whith different values of ( T, d, z, θ ). The samegeneral equation at zero temperature was given in [19].One may check this result by assuming ( d = 3 , θ = 0 , z = 1 , f ( u ) →
1) and finding thedifferential equation for the rotating particle at N = 4 SYM theory as [9] ρ ′′ ( u ) + ρ ( u )(1 + ρ ′ ( u ) ) ( u + 2 ρ ( u ) ρ ′ ( u )) u (Π u − ρ ( u ) ) + 1 + ρ ′ ( u ) ρ ( u ) (1 − ω ρ ( u ) ) = 0 . (20)Also at finite temparure N = 4 SYM theory with ( d = 3 , θ = 0 , z = 1), one finds the sameresult as [10, 9] ρ ′′ ( u ) + ρ ( u ) (cid:0) f ( u ) ρ ′ ( u ) (cid:1) (4 f ( u ) ρ ( u ) ρ ′ ( u ) + u (2 − ρ ( u ) f ( u ) ρ ′ ( u )))2 u ( − Π u + f ( u ) ρ ( u ) ) +2 + ρ ( u ) f ′ ( u ) ρ ′ ( u ) + 2 f ( u ) ρ ′ ( u ) + ω ρ ( u ) f ′ ( u ) ρ ′ ( u ) ρ ( u ) ( f ( u ) − ρ ( u ) ω ) = 0 . (21)One can use the special point ( ρ t , u t ) as initial value for the differential equation of motionand solve it. The second initial condition ρ ′ t can be found from the prescription of [9] byusing an expansion of ρ ( u ) around u = u t . One finds that the differential equation itselfdetermines ρ ′ t , one obtains ρ ′ t by solving the following equation: − p f ( u t ) u zt ( d − θ ) ω + 2 f ( u t ) u zt ( d − θ ) ω ρ ′ ( u t ) + ρ ′ ( u t ) (cid:0) θ f ( u t ) (cid:0) f ′ ( u t ) u t − f ( u t )( z − (cid:1) + d (cid:0) − zu t f ( u t ) f ′ ( u t ) + 2 f ( u t ) ( z −
1) + 2 u zt ω (cid:1)(cid:1) = 0 . (22)10 = 2 u Ρ Z = u Ρ Figure 7: The finite temperature analyziz of the string radius vs radial direction in different θ and z with ω = Π = 1. In each plot from top to down θ = d , d , . As it is clear this is a general equation which depends on the parameters of the non-relativistic theory. In the case of N = 4 SYM theory, it reduces to − u t ω ρ ′ ( u t ) + ρ ′ ( u t ) = 0 . (23)For finite temperature N = 4 SYM theory, one finds − p f ( u t ) ω + (cid:0) − f ( u t ) f ′ ( u t ) + 2 u t ω (cid:1) ρ ′ ( u t ) + 2 f ( u t ) / ω ρ ( u t ) = 0 . (24)Having initial values of ρ ( u t ) and ρ ′ ( u t ), one can solve (19) for different values of ( T, d, z, θ ).We fixe the temperature in our analyziz and consider u h = 1 . Also assume d = 3. Therefore,study how changing of θ and z affect the energy loss of the rotating particle.First like Fig 2 we study radial string configuration for the rotating particle at finitetemperature in Fig 7. This figure shows the behavior of string coiling radius, ρ ( u ), versusradial direction, u , for different values of θ and z with ω = Π = 1 at finite temperature.Again we set u = 0 as boundary, so ρ (0) is the radius of rotation of the particle. In each plotfrom top to down θ = d , d , . One finds that at fixed z and ω , the radius of the particleincreases by turning on θ , i.e increasing θ leads to increasing l , significantly. At fixed θ and ω , by increasing z the radius also increases. Also the string does not bend outward byincreasing z . Then it is clear that the shape of spiraling string depends on the values ofnon-relativist theory, significantly.Next we study the energy loss of the test particle at finite temperaure. The energy lossis given in terms of the the constant Π as dEdt = 12 πα ′ Π ω. (25)Then we choose different values for ( z, θ ) and find the energy loss of particle at finitetemperature. In this way, we should study behavior of u t from (18) and find shape of the11 = 2 Ω = 5 l P z = 2 Ω = 0.5 l P z = 2 Ω = 0.05 l P z = 3 Ω = 5 l P z = 3 Ω = 0.5 l P z = 3 Ω = 0.05 l P Figure 8: Energy loss versus rotation raduis of the rottaing particle for different valuesof θ , ω and z at finite temperature. In the first row we consider z = 2 and change theangular velocity from left plot to right plot ω = 5 , . , .
05. In the second row we increaseddynamical exponent z to z = 3. In each plot from top to down θ = 0 , d , d .spiraling string in the bulk by solving (19). Here, we consider θ = 0 , d , d and changevalues of dynamical exponent z . Fig. 8 presents the energy loss at different values of theangular frequency ω . As it is clear from this figure the energy loss monotically increasesby increasing the radius of the circular rotation. Also by increasing ω , radius l decresessignificantly at fixed Π. In particular, for large values of ω only small values of l is allowded.We also studied the effect of increasing z on the enrgy loss in the second row of this figure.An important result is that at at fixed value of l , increasing z leads to decreasing of theenrgy loss of rotating particle. Increasing hyperscaling violation parameter θ also leads todecreasing of the energy loss of the particle. Briefly, the general features of the energy lossof rotating particle at finite temperature non-relativistic theories can be summarized as • As the same as the zero temparature non-relativistic theory, the energy loss is increas-ing by increasing of rotation radius l , monotonically. • By increasing z and θ , the energy loss of the rotating particle decreses. • Unlike the zero temparature non-relativistic theory, there is a critical value for rotationradius, l c .We checked the last staement by considering different values of z and θ , however, it seemsthat there is no such critical radius at finite temperature non-relativistic theory. Lineardrag force limit,
Because of moving the particle at finite temperaure strongly coupledfield theory, one expects that the particle experiences the linear drag force. Such calculationhas been done in [16, 17]. Now we want to study if the same mechanism exists for the totalenergy loss of the particle, i.e it looses the energy due to the linear drag force.12ne finds the detail of drag force calculations in [17]. The rate of the energy loss ofheavy probe is given by P ≡ πα ′ dEdt = v u − θd c , (26)where u c should be found by solving f ( u c ) u − z +2 c − v = 0 . Comparing with z = 1 case in [9], N=4 SYM at finite temperature, at small angularvelocity ω , where a →
0, the rotating particle looses its energy from linear drag force on aparticle moving in a straight line with a constant velocity v at finite temperature. Here, weshow the ratio of the total energy loss of rotating particle to the linear drag force at finitetemperature as a function of velocity in the Fig. 9. Because of the non-relativistic theorythe velocity can change from zero to infinity. In each plot of this figure, from top to downthe angular velocity ω decreases as ω = 1 , . , .
05. We have changed z and θ to study theratio. It is clearly seen that the enrgy loss from the linear drag force mechanism is dominantwhen ω goes to zero. Increasing the parameters of z and θ also leads to this phenomena.However, the total energy loss exceeds the linear drag by increasing the angular velocity ω or decreasing the parameters of z and θ . z = 2 Θ = 0 v P z = Θ = d v P z = Θ = d v P Figure 9: The ratio of the total energy loss of rotating particle to the linear drag force atfinite temperature as a function of velocity. In each plot from top to down ω = 1 , . , . In this paper, we have studied some aspects of energy loss in non-relativistic theories fromholography. It has been done by analyzing the energy lost by a rotating heavy point particlealong a circle of radius l with angular velocity ω in holographic non-relativistic field theorieswith general dynamical exponent z and hyperscaling violation exponent θ . It was shownthat this problem provides a novel prespective on the energy loss in non-relativistic stronglycoupled field theories. A general computation at zero and finite temperature is done andit is shown that how the total energy loss rate depends non-trivially on two characteristicexponents, ( z, θ ). One should notice that there is a null singularity in such theories whichcould be resolved by adding some stringy corrections. We have ignored such issues andconsidered only some special range of the energies.First, by studying the shape of the spiraling string at zero and finite temperature weshowed that a world-sheet horizon appears at u c . From the boundary theory point of view,one can distinguish between the far field region or the near field region, too. Also, as theparticle radiates it should experience small kicks leading to Brownian motion of the particlein no-relativistic theories. We have shown how the world sheet horizon u c depends on the( θ, z ) in Fig. 1. 13ext, we have studied dEdt , the energy lost by the particle, which is the energy expendedby the external force moving the particle. The calculation has been done at zero and finitetemperature. This problem has been studied in N=4 SYM and although it is not a physicalsituation, however novel preprctives were found by studying it. It was shown that at zerotemperature there is a special radius l c where the energy loss is independent of differentvalues of ( θ, z ). It was shwon in [21, 22] that there is a qualitative difference between non-relativistic theories with different values of z . For example, if one throws a massive particlein one of the boundary theory directions it will travel only a finite distance in the case of z = 2 theory [21]. Following these studies, we expect to find different behaviors for rotatingparticle by changing ( θ, z ). We summarize our results at as • We found that like the relativistic case, the energy loss is increasing by increasing ofrotation radius l . • It was shown that by increasing angular velocity ω , the energy loss is increasing fordifferent values of z and θ . • We found that at zero temperature there is a critical value for rotation radius, l c , sothat l c ω = 1 and interestingly the energy loss is independent of parameters of θ and z . • By increasing z and θ , the energy loss of the rotating particle decreses. • At finite temparature non-relativistic theory, there is no a critical value for rotationradius, l c . • At zero and finite temeparur, it was shown that, like N=4 SYM case at finite temper-ature, there is a crossover between a regime in which the energy loss is dominated bythe linear drag force and by the radiation because of the acceleration a . However, theradiation process in such theories with non zero ( z, θ ) is not the same as the Mikhailovresult proposed in [23].It is worth to remark again that the linear drag force even at zero temperature is notzero. Therefor, two different mechanisms of linear drag force would be linear drag forceat zero and finite temperature. We analysed the total energy loss at finite temperatureby studing its ratio to the drag force at zero and finite temperature and we could observethe limit where the drag force channel dominates. Then we saw the significant advantageoffered by the analysis of the rotating object in the non-relativistic theories.Another feature of the energy loss of the particle at finite temperature that is worthemphasizing is to study the energy loss due to linear drag at zero temperature. It meansthat if particle looses its energy from drag force channel at zero temperature in (11). Inthis case, we considered different values of z and θ and studied ratio of the total energyloss, i.e Π ω to the drag force at zero temperature. To better underestand the physics, westudied also the ratio of the total energy loss to the linear drag force at zero plus the lineardrag force at finite temperature. By changing z and θ , we could find that when the partclerotates very fast this ratio goes to one. This means that the particle looses its energy fromthe linear drag force mechanism at zero and finite temperature, i.e theay have almost samecontribution in the energy loss of the particle.14ased on holography, the rotating particle has a gravitational dual; a semiclassicalstring. The end point of this string corresponds to the particle in the non-relativisticquantum field theory. So the rotating motion of particle corresponds to coiling of theclassical string about the same axis of the rotation. From the correspondence, the stringinduces a 4 d stress tensor on the boundary that is expectation value of stress tensor of theboundary field theory. As a result, we can study quantum effects in the boundary fieldtheory by classical calculation in the bulk. Thus, one can use the qualitative idea behindthe AdS/CFT correspondence and study how the classical depths of the spiraling stringsin the bulk is related to the length-scale in the non-relativistic theory. Because that depthin the radial direction corresponds to length-scale in the field theory. Such study has beendone in the case of N=4 SYM theory in [11]. Surprisingly, they found that this intuitiveresult from the AdS/CFT correspondenceway does not work and the rotating string fallsdeeper and deeper into the radial direction while the thickness of the flux tube of energydensity in the boundar field theory does not change. 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