aa r X i v : . [ h e p - t h ] J u l BI-TP 2011/22
On radiation by a heavy quark in N = Rudolf Baier ∗ Physics Department, University of Bielefeld, D-33501 Bielefeld, Germany
A short note on radiation by a moving classical particle in N = 4 supersymmetricYang-Mills theory. I. INTRODUCTION
In the papers [1–3] radiation by a pointlike quark in N = 4 supersymmetric Yang-Mills theory at strong coupling is investigated using the AdS/CFT correspondence in thesupergravity approximation [4–6]. In this note modifications of the published radiationpattern are suggested, which are consistent with the results in [7]. This analysis is motivatedby the description of electrodynamic radiation in classical electrodynamics [8, 9].The important result in the context of radiation by an accelerated charge e is given bythe Abraham-Lorentz four-vector force [8, 10, 11] in classical relativistic electrodynamics[12], f µ = 2 e a ν a ν v µ + ˙ a µ ) , (1)where the particle velocity is ˙ x µ = v µ ≡ dx µ dτ and the acceleration ˙ v µ = a µ ≡ dv µ dτ with theproper time τ for the particle [13] [The signature used here is (+ − −− )]. It is important tonote the orthogonality of the force to the velocity, v µ f µ = 0 . (2)This force vanishes for uniformly accelerated motion, f µ = 0 [8]. II. CLASSICAL RADIATION OF ACCELERATED ELECTRONS
In order to set the framework, it is helpful to discuss radiation in classical electrodynamics.A usefull approach is found in the 1949 paper by Schwinger [9]. ∗ E-mail: [email protected]
Assume sources, restricted to a finite domain, which emit radiation. The four-momentumof the classical electromagnetic field is given in terms of the energy-momentum tensor byintegrating on a hyper-surface P ν = I T µν dσ µ . (3)Gauß and Maxwell allow us to rewrite it as P ν = Z ∂ µ T µν d x = Z j µ F µν d x , (4)in terms of the current j µ = ( ρ,~j ) and the field-strength F µν .In the following it is important in order to determine the radiation force that the radiationfield tensor and the vector potential are introduced in terms of the retarded and advancedfields [10, 11]: F µνrad = 12 ( F µνret − F µνadv ) . (5)Replacing in (4) F µν by F µνrad gives P νrad , which for point-like charges satisfies (compare to(2)) v ν dP νrad dτ ∝ v µ F µνrad v ν = 0 . (6)Using current conservation ∂ µ j µ = 0 and introducing the vector potential A µrad = 12 ( A µret − A µadv ) ≡ ( φ, ~A ) , (7)one obtains the power dP rad dt = Z [ ~j · ∂ ~A∂t − ρ ∂φ∂t ] d x + ddt Z ρ φ d x . (8)In [9], eq. (I.17), Schwinger discards the second term of this formula, which has the form ofa total time derivative.The radiation vector potential [9] is expressed by A µrad ( t, x ) = i π Z exp [ iω ( n · ( x − x ′ ) − ( t − t ′ ))] j µ ( t ′ , x ′ ) d x ′ dt ′ ωdω d Ω4 π . (9)A point-particle current is assumed j µ = e (1 , v ( t ) = d x dt ) δ ( x − R ( t )) . (10)The integrals in (8) are as follows: Z ρ φ d x = e Z δ ′ ( t ′ − t + n · ( R ( t ) − R ( t ′ )) dt ′ d Ω = − e Z d Ω4 π n · a ξ , (11)with ξ = (1 − n · v ), a = d v dt , and from eq. (I.41) of [9] R [ ~j · ∂ ~A∂t − ρ ∂φ∂t ] d x = e Z d Ω4 π [ a ξ + 2 n · a v · a ξ − ( n · a ) γ ξ ]+ ddt e Z d Ω4 π [ − v · a ξ + n · a γ ξ ] , (12)with γ = √ − v , and v µ = ( γ, γ v ).Finally, using the notion of emitted power and a Schott type term (for example, in thenotation of [14]), the result of the angular radiation pattern is dP rad dt d Ω ≡ P rad ( n , t ) = P emitt ( n , t ) + P Schott ( n , t ) , (13)where P emitt ( n , t ) = e π [ a ξ + 2 n · a v · a ξ − ( n · a ) γ ξ ] , (14)and P Schott ( n , t ) = e π ddt [ − v · a + n · a ξ + n · a γ ξ ] . (15)Indeed there are two terms contributing to the radiation power. Schwinger [9] claims thatonly the first one P emitt ( n , t ), the one denoting the emission should be retained. It has thecharacteristics of an irreversible energy transfer. The second one in the form of a total timederivative is reversible in nature.Following Jackson [15] to obtain the radiated energy density of a charged particle onestarts from the large distance - 1 /R contribution of the Li´ e nard-Wiechert electric field E rad = eR n ∧ [( n − v ) ∧ a ](1 − n · v ) = eR (cid:20) − a (1 − n · v ) + ( n · a )( n − v )(1 − n · v ) (cid:21) , (16)to obtain from E vector = 18 π (cid:0) E + B (cid:1) , (17)with | B rad | = | E rad |E vector = e πR (cid:20) a (1 − n · v ) + 2 ( v · a )( n · a )(1 − n · v ) − ( n · a ) γ (1 − n · v ) (cid:21) . (18)There is agreement between R (1 − n · v ) E vector = P emitt ( n , t ) . (19)Integrating the angular dependence (see e.g. the useful integrals in Appendix A in [2]) oneobtains γ dP rad dt = dP rad dτ = 2 e γ [ a + γ ( v · a ) ] γ − e ddτ [ γ ( v · a )] , (20)which, with a µ = ( γ v · a , γ a + γ ( v · a ) v ), can be written as dP rad dτ = − f = − e a µ a µ v + ˙ a ] , (21)i.e. the zero component of the (negative) relativistic Abraham-Lorentz vector force (1)[8, 10, 12, 14], f µ = f µemitt + f µSchott = 2 e d xdτ ) dx µ dτ + d x µ dτ ] . (22)The first term represents an irretrievable loss of energy, the second, the Schott contribution,is a total time differential, which contributes nothing to an integral by dτ , when the initialvalue of a µ is returned to at the end [12].In summary, as Schwinger stated already in 1949, only the spectrum for the irreversibletransfer P emitt ( n , t ) is the relevant one for discussing the emitted radiation, and thereforeshould be retained. It is consistent with the derivation via the radiative electric field (16)as given in [15]. III. CLASSICAL N = In order to calculate the radiation power one has to add to the vector part a contributiondue to a massless scalar field χ [1, 2], and the replacement e → e eff = λ π . This leads to ∂ µ T µνscalar = j χ ∂ ν χ , (23)with the current j χ = ρ χ = e eff √ − v δ ( x − R ( t )) . (24)This scalar contribution leads to P scalar ( n , t ) = e eff π [ γ ( v · a ) ξ − v · a )( n · a ) ξ + ( n · a ) γ ξ ]+ e eff π ddt [ v · a ξ − n · a γ ξ ] , (25)(see also [1, 2]). Adding the vector parts (14) and (15) from the previous section the weakcoupling angular spectrum is given by P rad ( n , t ) = λ π a + γ ( v · a ) ξ − λ π ddt [ n · a ξ ] . (26)The term for P emitt ( n , t ) may also be expressed as P emitt ( n , t ) = λ π γ [ a − ( v ∧ a ) ](1 − n · v ) . (27)Performing the angular integration gives Z γP rad ( n , t ) d Ω = λ π γ [ a + γ ( v · a )] γ − λ π ddτ [ γ ( v · a )] . (28)Up to the coupling this expression agrees with the one from classical electrodynamics, givenby (20). The N = 4 SYM Abraham-Lorentz force [16, 17] in the weak coupling limit reads f µSY M,weak = λ π [ a ν a ν v µ + ˙ a µ ] , (29)allowing the same interpretation as in classical electrodynamics given above. f µSY M,weak satisfies the constraint (2). IV. RADIATION IN N = Based on the work by [1] Hatta et al. [2] performed a detailed and transparent calculationof the radiation pattern by a heavy quark in N = 4 SYM at strong coupling, to be followedrather closely. The result consists of two parts for the energy density, to be identified as P emitt ( n , t ) = √ λ π γ [ a − ( v ∧ a ) ](1 − n · v ) , (30)and a term in form of a total time derivative P tt ( n , t ) = √ λ π ddt [ v · a ξ − n · a γ ξ ] , (31)which is - up to the couplings - the same given by (12) in classical electrodynamics [9].In the notation of [2] P emitt ( n , t ) = R ξ E (1)rad ( t, r ) and P tt ( n , t ) = R ξ E (2)rad ( t, r ). It is noted in[2] that integrating the sum of these two terms with respect to d Ω does not give a properAbraham-Lorentz force [16, 17], and the constraint (2) is not satisfied, as it is the case inthe weak coupling limit, when compared with (29).A possible source of this deficiency may be found that only retarded contributions for theradiation are taken into account, instead of following the prescription given e.g. by (5) and(7) in the previous sections.There is no need to repeat the derivations given in [2], but instead relying on the expressionsof the energy density in the gauge theory, i.e. on the Minkowski boundary given therein.First consider the quantity E A = √ λ π Z d t q δ ( W q ) (cid:18) A γ Ξ + ∂∂t q A γ Ξ (cid:19) , (32)with the definitions W q ≡ − ( t − t q ) + | r − r q | , Ξ ≡ ( t − t q ) − υ q · ( r − r q ) = 12 d W q d t q . (33)In [2] the integral is evaluated by the retarded condition: t r = t r ( t, r ) denotes the value of t q for which W q ( t q ) = 0, with t − t r = | r − r q ( t r ) | = R . (34)Writing δ ( W q ) = δ ( t q − t r ) / | Ξ | the result in the large R -limit taken from [2] is E ret A = √ λ π R | ξ | (cid:18) γ [ a − ( v ∧ a ) ](2 − ξ ) ξ (cid:19) + √ λ π R | ξ | ∂∂t r (cid:18) n · a + γ ( v · a )(2 − ξ ) ξ (cid:19) . (35)As a conjecture let us consider E rad A = 12 ( E ret A − E adv A ) (36)by performing the integral for E rad A starting from (32) but using the advanced condition with t − t r = −| r − r q ( t r ) | = − R . (37)This amounts to the substitutions, when on top n → − n , which does not affect the force,Ξ → − R (1 − n · v ) , (38)i.e. ξ → − ξ , (39)whereas ∂ | ξ | ∂t r remains unchanged.From (35) one obtains E adv A = √ λ π R | ξ | (cid:18) γ [ a − ( v ∧ a ) ](2 + ξ ) ξ (cid:19) + √ λ π R | ξ | ∂∂t r (cid:18) − n · a + γ ( v · a )(2 + ξ ) ξ (cid:19) , (40)and E rad A = − √ λ π γ [ a − ( v ∧ a ) ] R ξ + √ λ π R ξ ∂∂t r (cid:20) n · a ξ − γ v · a ξ (cid:21) . (41)In an analogous way the contribution E rad B is evaluated, starting from E ret B = − √ λ π R | ξ | γ [ a − ( v ∧ a ) ]( − γ + 2 ξ − ξ ) ξ ! − √ λ π R | ξ | ∂∂t r (cid:18) n · a ( ξ − ξ + γ v · a (2 − ξ ) ξ + 1 | ξ | ∂∂t r [ 16 γ ξ + 1 ξ ] (cid:19) . (42)After the substitution (39) E adv B and then E rad B is obtained, E rad B = − √ λ π R γ [ a − ( v ∧ a ) ]( − γ − ξ ) ξ ! − √ λ π R ξ ∂∂t r (cid:18) n · a ξ − γ v · a ξ + 1 | ξ | ∂∂t r ξ (cid:19) . (43)Finally, adding E rad A and E rad B the angular radiation power in the strong coupling limit isobtained P radstrong ( n , t ) = P emitt ( n , t ) + P Schott ( n , t )= √ λ π γ [ a − ( v ∧ a ) ] ξ − √ λ π ddt [ n · a ξ ] . (44)The total time derivative term P Schott ( n , t ) differs from P tt ( n , t ) in (31).Up to the dependence on the coupling λ the same angular radiative spectrum is found inthe weak as well as in the strong coupling limit of the N = 4 supersymmetric Yang-Millstheory, i.e. λ → √ λ . As in electrodynamics [9] it is suggestive that for strong couplingas well only the emission spectrum P emitt ( n , t ) = √ λ π γ [ a − ( v ∧ a ) ] ξ is the relevant one forradiation, i.e. for the irreversible energy transfer [7].Furthermore the force [16, 17] is f µSY M,strong = √ λ π [ a ν a ν v µ + ˙ a µ ] . (45)Up to the coupling dependence the Abraham-Lorentz forces in classical electrodynamics aswell as in the Yang-Mills theory have the same dependence on the acceleration a µ and thevelocity v µ , when comparing (1), (29) and (45). All do satisfy the constraint (2).As in electrodynamics [8, 18] the forces f µSY M,weak and f µSY M,strong vanish in weak andstrong coupling N = 4 SYM, respectively, for uniformly accelerated motion [2, 19], e.g. alongthe x direction, x µ = ( g sinh( gτ ) , g cosh( gτ ) = q t + g , , a µ a µ = − v µ ˙ a µ = − g ,although radiation is emitted.In the spirit of [20] a phenomenological derivation of P radstrong (44) could be done as follows:retain only P emitt ( n , t ) of (30) as derived in [2], calculate after the angular integration theforce f µemitt = √ λ π a ν a ν v µ . Enforce the orthogonality (2) to v µ , together with f µSY M,strong =0 for uniformly accelerated motion, by adding the Schott-type term √ λ π ˙ a µ . This one isconsistently obtained after integrating P Schott ( n , t ) assumed to be of the same from in strongas well as in weak coupling (compare with (26)).In [3] the time averaged energy density of an oscillating quark with small linear oscillationsis derived, v q ( t ) = ǫ Ω cos Ω
T , a q = − ǫ Ω sin Ω t and ǫ <<
1. It is asymptotically isotropicand - after correcting the numerical coefficient by a factor 6 - given by Z + ∞−∞ dt < T ( t, ~x ) > = ǫ Ω √ λ π R Z + ∞−∞ dt , (46)which is consistent with the result by A. Mikhailov [7], namely for P emitt = √ λ π a q ( t ) . (47) V. CONCLUSION
The essence of this note is based on the structure of the Abraham-Lorentz force f µ (1),which holds even for strong coupling, with the properties of the orthogonality to the velocityand its vanishing for uniformly accelerated motion. It is surprising that the force - up to itsstrength - is the same in relativistic electrodynamics, as well as in weak and strong coupling N = 4 SYM, although the underlying angular distributions P ( n , t ) are different.For the N = 4 SYM model in the strong coupling limit the special case of synchrotron ra-diation with frequency ω is considered in [1]. An independent derivation of the synchrotronradiation in this model is given in [21].In this case with v · a = 0, a = v ω the energy density (44) reads P radstrong ( n , t ) = √ λω π ξ (cid:2) − (4 + γ − ) ξ − v sin Θ + 2 ξ (cid:3) , (48)which differs from the one using P tt of (31) (compare with eq. (3.71) in [1] and with eq. (6.5)in [2]).A quantity of interest considered in [1] is the time-averaged angular distribution of power,given by dP emitt ( n ) d Ω = ω π Z π/ω dt √ λ π a (1 − n · v ) , (49)with 1 − n · v = 1 − v sin Θ sin ( φ − ω t ).For this periodic motion for synchrotron radiation the contributions of the total time deriva-tives P Schott , as well as P tt , vanish in the time-averaged distribution. In any case, whenemitted radiation is considered total time derivative terms should not be retained. They donot represent irreversible loss of energy in contrast to P emitt [9, 12].Integration in (49) leads to dP emitt ( n ) d Ω = √ λ π a γ v sin Θ( γ cos Θ + sin Θ) / , (50)which agrees with the result eq. (3.72) derived in [1]. The total power emitted, P emitt = √ λ π [ γ vω ] , (51)is the same as the result obtained in [1, 2, 7].0 Acknowledgments
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