# On Radiation by a Heavy Quark in𝒩=4 SYM

Authors: Rudolf Baier
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

A short note on radiation by a moving classical particle in𝒩=4 supersymmetric Yang-Mills theory is discussed in this paper.


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## Body

## 1. Introduction

In papers [1–3], radiation by a point-like quark in 𝒩=4 supersymmetric Yang-Mills theory at strong coupling is investigated using the AdS/CFT correspondence in the supergravity approximation [4–6]. In this paper, modifications of the published radiation pattern are suggested, which are consistent with the results in [7]. This analysis is motivated by the description of electrodynamic radiation in classical electrodynamics [8, 9].The important result in the context of radiation by an accelerated chargee is given by the Abraham-Lorentz four-vector force [8, 10, 11] in classical relativistic electrodynamics [12] as
(1.1)fμ=2e23(aνaνvμ+a˙μ),
where the particle velocity is x˙μ=vμ≡dxμ/dτ and the acceleration v˙μ=aμ≡dvμ/dτ with the proper time τ for the particle [13] (the signature used here is (+---)). It is important to note the orthogonality of the force to the velocity as
(1.2)vμfμ=0.
This force vanishes for uniformly accelerated motion, fμ=0 [8].


## 2. 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].Assume sources, restricted to a finite domain, which emit radiation. The four-momentum of the classical electromagnetic field is given in terms of the energy-momentum tensor by integrating on a hyper-surface as follows:(2.1)Pν=∮‍Tμνdσμ.
Gauß and Maxwell allow us to rewrite it as
(2.2)Pν=∫‍∂μTμνd4x=∫‍jμFμνd4x,
in terms of the current jμ=(ρ,j→) and the field-strength Fμν.In the following it is important in order to determine the radiation force that the radiation field tensor and the vector potential are introduced in terms of the retarded and advanced fields [10, 11] as follows:
(2.3)Fradμν=12(Fretμν-Fadvμν).
Replacing Fμν by Fradμν in (2.2) gives Pradν, which for point-like charges satisfies (compare to (1.2))
(2.4)vνdPradνdτ∝vμFradμνvν=0.
Using current conservation ∂μjμ=0 and introducing the vector potential
(2.5)Aradμ=12(Aretμ-Aadvμ)≡(ϕ,A→),
one obtains the power
(2.6)dPrad0dt=∫‍[j→⋅∂A→∂t-ρ∂ϕ∂t]d3x+ddt∫‍ρϕd3x.
In [9, equation (I.17)], Schwinger discards the second term of this formula, which has the form of a total time derivative.The radiation vector potential [9] is expressed by
(2.7)Aradμ(t,x)=i2π∫‍exp⁡[iω(n⋅(x-x′)-(t-t′))]jμ(t′,x′)d3x′dt′ωdωdΩ4π.
A point-particle current is assumed as
(2.8)jμ=e(1,v(t)=dxdt)δ(x-R(t)).
The integrals in (2.6) are as follows:
(2.9)∫‍ρϕd3x=e2∫‍δ′(t′-t+n⋅(R(t)-R(t′)))dt′dΩ=-e2∫‍dΩ4πn⋅aξ3,
with ξ=(1-n·v), a=dv/dt, and from equation (I.41) of [9] one obtains
(2.10)∫‍[j→⋅∂A→∂t-ρ∂ϕ∂t]d3x=e2∫‍dΩ4π[a2ξ3+2n⋅av⋅aξ4-(n⋅a)2γ2ξ5]+ddte2∫‍dΩ4π[-v⋅aξ3+n⋅aγ2ξ4],
with γ=1/1-v2 and vμ=(γ,γv).Finally, using the notion of emitted power and a Schott-type term (e.g., in the notation of [14]), the result of the angular radiation pattern is
(2.11)dPrad0dtdΩ≡Prad(n,t)=Pemitt(n,t)+PSchott(n,t),
where
(2.12)Pemitt(n,t)=e24π[a2ξ3+2n⋅av⋅aξ4-(n⋅a)2γ2ξ5],PSchott(n,t)=e24πddt[-v⋅a+n⋅aξ3+n⋅aγ2ξ4].
Indeed, there are two terms contributing to the radiation power. Schwinger [9] claims that only the first one Pemitt(n,t), the one denoting the emission, should be retained. It has the characteristics of an irreversible energy transfer. The second one in the form of a total time derivative is reversible in nature.Following Jackson [15], to obtain the radiated energy density of a charged particle, one starts from the large distance −1/R contribution of the Liénard-Wiechert electric field as
(2.13)Erad=eRn∧[(n-v)∧a](1-n⋅v)3=eR[-a(1-n⋅v)2+(n⋅a)(n-v)(1-n⋅v)3],
to obtain from
(2.14)Evector=18π(E2+B2),
with |Brad|=|Erad|(2.15)Evector=e24πR2[a2(1-n⋅v)4+2(v⋅a)(n⋅a)(1-n⋅v)5-(n⋅a)2γ2(1-n⋅v)6].
There is agreement between
(2.16)R2(1-n⋅v)Evector=Pemitt(n,t).
Integrating the angular dependence (see, e.g., the useful integrals in Appendix A in [2]), one obtains
(2.17)γdPrad0dt=dPrad0dτ=2e23γ4[a2+γ2(v⋅a)2]γ-2e23ddτ[γ4(v⋅a)],
which, with aμ=(γ4v·a,γ2a+γ4(v·a)v), can be written as
(2.18)dPrad0dτ=-f0=-2e23[aμaμv0+a˙0],
that is, the zero component of the (negative) relativistic Abraham-Lorentz vector force (1.1) [8, 10, 12, 14], as
(2.19)fμ=femittμ+fSchottμ=2e23[(d2xdτ2)2dxμdτ+d3xμdτ3].
The first term represents an irretrievable loss of energy, and the second, the Schott contribution, is a total time differential, which contributes nothing to an integral by dτ, when the initial value of aμ is returned at the end [12].In summary, as Schwinger stated already in 1949, only the spectrum for the irreversible transferPemitt(n,t) is the relevant one for discussing the emitted radiation and, therefore, should be retained. It is consistent with the derivation via the radiative electric field (2.13) as given in [15].


## 3. Classical𝒩=4 SYM Radiation

In order to calculate the radiation power, one has to add to the vector part a contribution due to a massless scalar fieldχ [1, 2] and the replacement e2→eeff2=λ/8π. This leads to
(3.1)∂μTscalarμν=jχ∂νχ,
with the current
(3.2)jχ=ρχ=eeff1-v2δ(x-R(t)).
This scalar contribution leads to
(3.3)Pscalar(n,t)=eeff24π[γ2(v⋅a)2ξ3-2(v⋅a)(n⋅a)ξ4+(n⋅a)2γ2ξ5]+eeff24πddt[v⋅aξ3-n⋅aγ2ξ4],
(see also [1, 2]). Adding the vector parts (2.12) from the previous section, the weak coupling angular spectrum is given by
(3.4)Prad(n,t)=λ32π2a2+γ2(v⋅a)2ξ3-λ32π2ddt[n⋅aξ3].
The term for Pemitt(n,t) may also be expressed as
(3.5)Pemitt(n,t)=λ32π2γ2[a2-(v∧a)2](1-n⋅v)3.
Performing the angular integration gives
(3.6)∫‍γPrad(n,t)dΩ=λ8πγ4[a2+γ2(v⋅a)]γ-λ8πddτ[γ4(v⋅a)].
Up to the coupling, this expression agrees with the one from classical electrodynamics, given by (2.17). The 𝒩= 4 SYM Abraham-Lorentz force [16, 17] in the weak coupling limit reads
(3.7)fSYM,weakμ=λ8π[aνaνvμ+a˙μ],
allowing the same interpretation as in the classical electrodynamics given above. fSYM,weakμ satisfies the constraint (1.2).


## 4. Radiation in𝒩=4 SYM at Strong Coupling

Based on the work by [1], Hatta et al. [2] performed a detailed and transparent calculation of the radiation pattern by a heavy quark in 𝒩= 4 SYM at strong coupling, to be followed rather closely. The result consists of two parts for the energy density, to be identified as
(4.1)Pemitt(n,t)=λ8π2γ2[a2-(v∧a)2](1-n⋅v)3,
and a term in the form of a total time derivative
(4.2)Ptt(n,t)=λ24π2ddt[v⋅aξ3-n⋅aγ2ξ4],
which is, up to the couplings, the same given by (2.10) in the classical electrodynamics [9].In the notation of [2], Pemitt(n,t)=R2ξℰrad(1)(t,r) and Ptt(n,t)=R2ξℰrad(2)(t,r). It is noted in [2] that integrating the sum of these two terms with respect to dΩ does not give a proper Abraham-Lorentz force [16, 17], and the constraint (1.2) is not satisfied, as it is the case in the weak coupling limit, when compared with (3.7).A possible source of this deficiency may be found that only retarded contributions for the radiation are taken into account, instead of following the prescription given, for example, by (2.3) and (2.5) in the previous sections.There is no need to repeat the derivations given in [2], but instead relying on the expressions of the energy density in the gauge theory, that is, on the Minkowski boundary given therein.First consider the following quantity:(4.3)EA=λ4π2∫‍dtqδ(Wq)(A1γ2Ξ2+∂∂tqA0γΞ2),
with the following definitions:
(4.4)Wq≡-(t-tq)2+|r-rq|2,Ξ≡(t-tq)-υq⋅(r-rq)=12dWqdtq.
In [2], the integral is evaluated by the retarded condition: tr=tr(t,r) which denotes the value of tq for which 𝒲q(tq)=0, with
(4.5)t-tr=|r-rq(tr)|=R.
Writing δ(𝒲q)=δ(tq-tr)/2|Ξ|, the result in the large R-limit taken from [2] is
(4.6)EAret=λ8π2R2|ξ|(γ4[a2-(v∧a)2](2-ξ)ξ2)+λ8π2R2|ξ|∂∂tr(n⋅a+γ2(v⋅a)(2-ξ)ξ2).
As a conjecture, let us consider
(4.7)EArad=12(EAret-EAadv),
by performing the integral for ℰArad starting from (4.3), but using the advanced condition with
(4.8)t-tr=-|r-rq(tr)|=-R.
This amounts to the substitutions, when on top n→-n, which do not affect the force as
(4.9)Ξ⟶-R(1-n⋅v),
that is,
(4.10)ξ⟶-ξ,
whereas ∂/|ξ|∂tr remains unchanged.From (4.6), one obtains
(4.11)EAadv=λ8π2R2|ξ|(γ4[a2-(v∧a)2](2+ξ)ξ2)+λ8π2R2|ξ|∂∂tr(-n⋅a+γ2(v⋅a)(2+ξ)ξ2),EArad=-λ8π2γ4[a2-(v∧a)2]R2ξ2+λ8π2R2ξ∂∂tr[n⋅aξ2-γ2v⋅aξ].
In an analogous way, the contribution ℰBrad is evaluated, starting from
(4.12)EBret=-λ8π2R2|ξ|(γ4[a2-(v∧a)2](-(1/γ2)+2ξ-ξ2)ξ3)-λ8π2R2|ξ|∂∂tr(n⋅a(ξ-1)ξ3+γ2v⋅a(2-ξ)ξ2+1|ξ|∂∂tr[16γ2ξ2+1ξ]).
After the substitution (4.10), ℰBadv and then ℰBrad are obtained as
(4.13)EBrad=-λ8π2R2(γ4[a2-(v∧a)2](-(1/γ2)-ξ2)ξ4)-λ8π2R2ξ∂∂tr(n⋅aξ2-γ2v⋅aξ+1|ξ|∂∂tr1ξ).
Finally, adding ℰArad and ℰBrad, the angular radiation power in the strong coupling limit is obtained as follows:
(4.14)Pstrongrad(n,t)=Pemitt(n,t)+PSchott(n,t)=λ8π2γ2[a2-(v∧a)2]ξ3-λ8π2ddt[n⋅aξ3].
The total time derivative term PSchott(n,t) differs from Ptt(n,t) in (4.2).Up to the dependence on the couplingλ, the same angular radiative spectrum is found in the weak as well as in the strong coupling limit of the 𝒩= 4 supersymmetric Yang-Mills theory, that is, λ/4→λ. As in electrodynamics [9], it is suggestive that for strong coupling as well only the emission spectrum Pemitt(n,t)=(λ/8π2)(γ2[a2-(v∧a)2]/ξ3) is the relevant one for radiation, that is, for the irreversible energy transfer [7].Furthermore the force [16, 17] is
(4.15)fSYM,strongμ=λ2π[aνaνvμ+a˙μ].
Up to the coupling dependence, the Abraham-Lorentz forces in classical electrodynamics as well as in the Yang-Mills theory have the same dependence on the acceleration aμ and the velocity vμ, when comparing (1.1), (3.7), and (4.15). All do satisfy the constraint (1.2).As in electrodynamics [8, 18] the forces fSYM,weakμ and fSYM,strongμ vanish in weak and strong coupling 𝒩= 4 SYM, respectively, for uniformly accelerated motion [2, 19], for example, along the x direction, xμ=((1/g)sinh(gτ),(1/g)cosh(gτ)=t2+(1/g2),0,0), that is, aμaμ=-vμa˙μ=-g2, although radiation is emitted.In the spirit of [20], a phenomenological derivation of Pstrongrad (4.14) could be done as follows: retain only Pemitt(n,t) of (4.1) as derived in [2], calculate after the angular integration the force femittμ=(λ/2π)aνaνvμ. Enforce the orthogonality (1.2) to vμ, together with fSYM,strongμ=0 for uniformly accelerated motion, by adding the Schott-type term (λ/2π)a˙μ. This one is consistently obtained after integrating PSchott(n,t) which is assumed to be of the same form in strong as well as in weak coupling (compare with (3.4)).In [3], the time averaged energy density of an oscillating quark with small linear oscillations is derived, vq(t)=ϵΩcosΩT,aq=-ϵΩ2sinΩt and ϵ≪1. It is asymptotically isotropic and, after correcting the numerical coefficient by a factor 6, given by
(4.16)∫-∞+∞‍dt〈T00(t,x→)〉=ϵ2Ω4λ16π2R2∫-∞+∞‍dt,
which is consistent with the result by Mikhailov [7], namely, for
(4.17)Pemitt=λ2πaq2(t).


## 5. Conclusion

The essence of this paper is based on the structure of the Abraham-Lorentz forcefμ (1.1), which holds even for strong coupling, with the properties of the orthogonality to the velocity and its vanishing for uniformly accelerated motion. It is surprising that the force, up to its strength, 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 the𝒩= 4 SYM model in the strong coupling limit, the special case of synchrotron radiation with frequency ω0 is considered in [1]. An independent derivation of the synchrotron radiation in this model is given in [21].In this case, withv·a=0, a2=v2ω02 the energy density (4.14) reads as
(5.1)Pstrongrad(n,t)=λω028π2ξ4[3-(4+γ-2)ξ-3v2sin⁡2Θ+2ξ2],
which differs from the one using Ptt of (4.2) (compare with equation (3.71) in [1] and with equation (6.5) in [2]).A quantity of interest considered in [1] is the time-averaged angular distribution of power, given by
(5.2)dPemitt(n)dΩ=ω02π∫02π/ω0‍dtλ8π2a2(1-n⋅v)3,
with 1-n·v=1-vsinΘsin(ϕ-ω0t).For this periodic motion for synchrotron radiation, the contributions of the total time derivativesPSchott as well as Ptt vanish in the time-averaged distribution. In any case, when emitted radiation is considered, total time derivative terms should not be retained. They do not represent irreversible loss of energy in contrast to Pemitt [9, 12].Integration in (5.2) leads to
(5.3)dPemitt(n)dΩ=λ8π2a2γ51+(v2/2)sin⁡2Θ(γ2cos⁡2Θ+sin⁡2Θ)5/2,
which agrees with the resulting equation (3.72) derived in [1]. The total power emitted,
(5.4)Pemitt=λ2π[γ2vω0]2,
is the same as the result obtained in [1, 2, 7].



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*Source: 592854-2012-08-16.xml*