Excitation of Chandrasekhar-Kendall photons in Quark Gluon Plasma by an ultrarelativitsic quark

0.36 MB·

by  Kirill Tuchin

Excitation of Chandrasekhar-Kendall photons in Quark Gluon Plasma
by ultrarelativitsic quark
Kirill Tuchin1
1Department of Physics and Astronomy, Iowa State University, Ames, Iowa, 50011, USA
(Dated: November 4, 2016)
Aquarkpropagatingthroughthequarkgluonplasmaandscatteringoffthethermalgluons
6 can radiate photons in states with definite angular momentum and magnetic helicity. These
1
0 states, known as the Chandrasekhar-Kendall states, are eigenstates of the curl operator and
2
have a non-trivial topology. I compute the spectrum of these states in the ultrarelativistic

v
o limit and study its properties.
N

3
I. INTRODUCTION

]
h
p
Theelectromagneticradiationisaprecisetooltostudythedynamicsofthequark-gluonplasma
-
p
e (QGP) [1–9]. Significant process has been made over the recent decade [10–16] though some open
h
[ problems still remain. The theoretical calculations usually focus on the momentum spectrum

4
of the radiation and thus treat photons as plane waves [17]. It has recently been pointed out in
v
9
[18,19]thatthesphericalwavesofphotons, i.e.statesofelectromagneticfieldwithdefiniteangular
9
3
momentum, isaninvaluableinstrumentforstudyingthetopologicalpropertiesofmagneticfieldsin
5
0
media with a chiral anomaly, such as QGP. Certain photon spherical waves are states with definite
.
1
0 magnetic helicity, which is a topological invariant proportional to the number of twisted and linked
6
1 flux tubes. These topological states are known as the Chandrasekhar-Kendall (CK) states [20, 21].
:
v
In [19] time-evolution of an initial topological state has been followed using the Chern-Simons-
i
X
r Maxwell model [29–32]. The non-trivial physics of this evolution has been emphasized before in
a
[25–28] in different contexts. The precise form of the initial state does not play an important role
at long enough times. However, the evolution time in QGP is restricted by its lifetime. This is
why the initial condition must play an important role in the dynamics of electromagnetic fields.
These observations motivate the author to address the problem of radiation of topological
spherical waves of photons, or simply, the CK photons. There are many ways to radiate a CK
photon from the QGP. In this paper, which I consider as a benchmark for future studies, I discuss
radiation of the CK photons by an ultrarelativistic quark scattering off thermal gluons: q +g →
q+γ . This process is similar to Compton scattering except that the radiated photon is now in
CK
a topologically non-trivial state. The main goal of this paper is to calculate the spectrum of the
2
CK photons emitted by an ultrarelativistic quark and study its properties.
It is important to realize that the CK photon emission and the time-evolution of the magnetic
field mentioned above do not interfere since they occur at different time scales. At long time scales,
neither energy nor magnetic helicity are conserved due to energy dissipation through the induced
Ohmic electric currents. Moreover, magnetic helicity is non-conserved due to the anomalous (non-
dissipative) electric current generated by chiral imbalance in presence of the chiral anomaly [22,
23]. However, these effects are small on the short time scales inherent for perturbative processes
discussed in this paper.
The paper is structured as follows. In Sec. II the spherical photon waves and the CK photons
are introduced. This is followed by the quantization the electromagnetic field in the basis of the
CK states in Sec. III and discussion of the applicability of the free-field approximation in Sec. IV.
Since calculation of the scattering matrix is most convenient in the momentum space, one needs to
expand wave functions of the CK photons in the plane wave basis. This done in Sec. V. The main
section is Sec. VI where the scattering cross section is computed. Finally, in Sec. VII the results
are discussed and summarized.
II. THE CK STATES IN THE QGP
I am working in the radiation gauge A = 0, ∇·A = 0 in which A satisfies the wave equation
0
∇2A−∂2A = 0. (1)
t
Its positive-energy solutions that have definite values of angular momentum can be written in the
form
Ah (r,t) = hkWh (r)e−iωkt, (2)
klm klm
where Wh (r) are eigenfunctions of the curl operator:
klm
∇×Wh (r) = hkWh (r). (3)
klm klm
Here h = ±1 is magnetic helicity, l is the orbital angular momentum and m = −l,...,l its
projection. The set of functions Wh (r) is complete on a unit sphere at any given k. Their
klm
explicit expressions read [24]
(cid:16) (cid:17)
Wh (r) = N¯ Th (r)−ihPh (r) , (4)
klm k klm klm
3
where
j (kr) i
Th (r) = l L[Y (θ,φ)], Ph (r) = ∇×Th (r), l ≥ 1. (5)
klm (cid:112)l(l+1) lm klm k klm
Here L = −i(r ×∇) is the orbital angular momentum operator. Although functions T and
klm
P also form a complete set on a unit sphere (at fixed k), they do not have definite magnetic
klm
helicity. The normalization constant N¯ is given by
k

1  1 , discrete;
N¯k = √ · R3/2jl+1(kR) (6)
3/2
2ω  k, continuous,
k
forthediscreteandcontinuousspectrarespectively. Itischoseninsuchawaythattheorthogonality
conditions read
(cid:90)
1
Wh(cid:48)∗ (r)·Wh (r)d3r = δ δ δ δ , (7)
k(cid:48)l(cid:48)m(cid:48) klm 2ω3 kk(cid:48) ll(cid:48) mm(cid:48) hh(cid:48)
k
for a discrete spectrum and
(cid:90)
π
Wh(cid:48)∗ (r)·Wh (r)d3r = δ(k−k(cid:48))δ δ δ , (8)
k(cid:48)l(cid:48)m(cid:48) klm 2ω3 ll(cid:48) mm(cid:48) hh(cid:48)
k
for a continuous one, which can be readily verified using the properties of the spherical Bessel
functions
(cid:90) R 1
j (kr)j (k(cid:48)r)r2dr = R3δ j2 (kR), (9)
l l 2 kk(cid:48) l+1
0
(cid:90) ∞ π
j (kr)j (k(cid:48)r)r2dr = δ(k−k(cid:48)). (10)
l l 2k2
0
The normalization conditions (7),(8) are used in the relativistic scattering theory (though they are
different from [19] and [24]).
A discrete spectrum with the quantized k emerges if one imposes a boundary condition on the
magnetic field at some distance r = R [18]. Although there seems to be no physical reason to
impose such a boundary condition, in practical calculations it is sometimes more convenient to
handle a discrete spectrum than a continuous one, and afterwards set R → ∞. Throughout the
paper the same letter k is used to denote both the continuous and discrete variables.
III. QUANTIZATION OF THE FREE FIELD
Substituting (2) into (1) we obtain the dispersion relation of free photons: ω = k. In terms of
k
the normal modes (2), the vector potential reads (in the discrete case)
(cid:88) (cid:16) (cid:17)
A(r,t) = hkah Wh (r)e−iωkt+c.c. , (11)
klm klm
klmh
4
where ah are arbitrary coefficients. The electric and magnetic fields are given by
klm
(cid:88) (cid:16) (cid:17)
E(r,t) = −∂ A(r,t) = ihkω ah Wh (r)e−iωkt+c.c. , (12)
t k klm klm
klmh
(cid:88) (cid:16) (cid:17)
B(r,t) = ∇×A(r,t) = k2ah Wh (r)e−iωkt+c.c. , (13)
klm klm
klmh
Thetotalelectromagneticenergyofthediscretespectrumcanbewrittenasasumovertheenergies
of all CK states as follows:
(cid:90)
1 (cid:88)
E = (E2+B2)d3r = ω ah ah∗ . (14)
2 k klm klm
klm
The normalization in (7) and (8) was chosen so that the energy of a single CK state is ω . For the
k
continuous spectrum, the total electromagnetic energy can be written as∗
(cid:88)(cid:90) ∞ dk
E = |ω |ah ah∗ . (15)
π k klm klm
0
lmh
The magnetic helicity of electromagnetic field reads
(cid:90)
(cid:88)
H = A·Bd3r = hah ah∗ , (16)
klm klm
klmh
indicating that magnetic helicity of a single CK state is h. Clearly, it is a conserved quantity.
Finally, the quantized electromagnetic field can be written down using (11) as
(cid:88) (cid:16) (cid:17)
A(r,t) = hkah Wh (r)e−iωkt+h.c. , (17)
klm klm
klmh
where now ah is an operator obeying the usual bosonic commutation relations.
klm
IV. ROLE OF ELECTRIC CURRENTS
It is important to delineate the region of applicability of the free-field approximation of the
previous section. The perturbation theory that I am employing in this paper, hinges on the
assumption that the electrical currents in the medium can be treated as small perturbations, since
they are proportional to α . For illustration, consider a model of classical electrodynamics with
em
∗ Coefficients ah are normalized differently in (14) and (15).
klm
5
an anomalous current given by the Maxwell-Chern-Simons equations [29–32]:
∇·B = 0, (18)
∇·E = 0, (19)
∇×E = −∂ B, (20)
t
∇×B = ∂ E +σE +σ B. (21)
t χ
In the last equation σE and σ B stand for the Ohmic and anomalous electrical current densities.
χ
In the radiation gauge Eqs. (18)-(21) yield an equation for the vector potential
−∇2A = −∂2A−σ∂ A+σ ∇×A, (22)
t t χ
where for simplicity I treat σ as a positive constant. Substituting (2) into (22) one finds a
χ
dispersion relation
k2−hσ k−ω (ω +iσ) = 0, (23)
χ k k
which has the following two solutions:
iσ (cid:113)
ω = − ± (k2−hσ k)−σ2/4. (24)
k χ
2
Evidently, if k (cid:29) σ, the dissipation effects due to the Ohmic currents can be neglected.
Unlike the dissipative currents that preclude the very notion of definite energy states, the
anomalous currents are non-dissipative [33] and in principle the electromagnetic field could have
been quantized in their presence. However, there a two problems. First, the magnetic helicity is
not conserved due to the chiral anomaly, see e.g. [19]. Second, the dispersion relation (24) contains
an unstable solution at k < σ [25–28]. This is easily seen at σ = 0: when h = 1, ω is purely
χ k
imaginary and positive, hence the corresponding eigenfunction exponentially increases with time.
Taking the time-dependence of σ into account does not resolve the problem [25]. In fact, the
χ
presence of a spatially uniform current σ B in (21) violates causality. It seems possible that a
χ
more realistic model of the anomalous term may cancel the instability. These problems, however,
are rather academic. Indeed, as far as the QGP of a realistic size R ∼ 5−10 fm and electrical
conductivity σ ∼ 5−6 MeV [34–37] is concerned, the requirement that k (cid:29) σ is always satisfied
because k (cid:38) 1/R (cid:29) σ. Since σ is probably of the same order of magnitude as σ, it implies that
χ
k (cid:29) σ and the unstable modes do not contribute to the spectrum of the CK states (and neither
χ
does the anomaly).
6
V. PLANE WAVE EXPANSION OF THE CK STATES
The wave function of a photon with a given momentum q and polarization λ is given by
1
A (r) = √ (cid:15) eiq·r, (25)
qλ qλ
2ωV
where λ = 1,2 are two polarization states and V is the plasma volume. In the next section I will
need the following expansion of the CK states into the plane waves (25):
(cid:88)(cid:90) d3q
Wh (r) = eiq·r(cid:15) wh (q,λ). (26)
klm (2π)3 qλ klm
λ
where
(cid:90) (cid:16) (cid:17)
wh (q,λ) = d3re−iq·r(cid:15)∗ ·Wh (r) = N¯ (cid:15)∗ · Th (q)−ihPh (q) (27)
klm qλ klm k qλ klm klm
with
(cid:90) (cid:90)
Th (q) = d3re−iq·rTh (r), Ph (q) = d3re−iq·rPh (r). (28)
klm klm klm klm
Substituting (5) and denoting by rˆ a unit vector in the r direction yields
(cid:90) (cid:90)
j (kr) j (kr)
Th (q) = d3re−iq·r l L[Y (rˆ)] = − d3rL[e−iq·r] l Y (rˆ) (29)
klm (cid:112) lm (cid:112) lm
l(l+1) l(l+1)
(cid:90)
j (kr)
= L d3re−iq·r l Y (rˆ), (30)
(cid:112) lm
l(l+1)
where in the last line the angular momentum operator is in the momentum representation: L =
−iq×∇ . Using the expansion of the plane wave into the spherical waves
q
(cid:88)
eiq·r = 4π ilj (qr)Y∗ (qˆ)Y (rˆ), (31)
l lm lm
lm
along with the orthonormality relations (9),(10) furnishes
j2 (kr)
Th (q) = 2πR3(−i)l l+1 δ LY (qˆ), (32)
klm (cid:112) kq lm
l(l+1)
for the discrete spectrum and
2π2 1
Th (q) = (−i)l δ(k−q)LY (qˆ) (33)
klm k2 (cid:112)l(l+1) lm
for the continuous one. From (5) and (28) it follows that
1
Ph (q) = − q×Th (q). (34)
klm k klm
Substituting (32)–(34) into (27) we obtain the desired expansion of the CK states into the plane
waves.
7
VI. SPECTRUM OF THE CK PHOTONS
Consider an ultra-relativistic quark traveling through the QGP at temperature T with energy
ε (cid:29) T. As it scatters off a thermal gluon it radiates a CK photon through the process q(pµ)+
g(kµ) → q(p(cid:48)µ) + γ (k(cid:48)lmh). Quantities in parentheses denote the quantum numbers of the
CK
correspondingparticles; inthecaseofquarksandgluonsthesearetheir4-momenta. Thescattering
matrix element for this process reads
(cid:90) (cid:90) (cid:104)
S =e g d4x d4yψ¯ (x) iA/h∗ (x)iS(x−y)(−iA/a (y))
fi q f k(cid:48)lm kλ
(cid:105)
+(−iA/a (x))iS(x−y)(iA/h∗ (y)) ψ (y), (35)
kλ k(cid:48)lm i
where e and g are the electromagnetic and strong coupling constants (assumed to be small). The
q
gluon field potential Aa (y) is normalized as in (25):
kλ
1
Aa (r) = √ ta(cid:15) e−ikµ·xµ, (36)
kλ kλ
2ωV
where ta is the color symmetry generator. The initial and final quark wave functions are states
with given momentum and spin that read
1
ψ (y) = √ u e−ipµ·yµ (37)
i ps
2εV
1
ψ¯f(x) = √ u¯p(cid:48)s(cid:48)eip(cid:48)µ·xµ, (38)
2ε(cid:48)V
where s and s(cid:48) stand for spin projections. The most convenient way to compute the scattering
matrix element (35) is to employ the plane wave representation of the CK photon wave function
derived in the previous section. Substituting (26) into (2) we have
Ahk(cid:48)lm(x) = hk(cid:48)Wkh(cid:48)lm(x)e−iωk(cid:48)x0 = hk(cid:48)(cid:88)(cid:90) (d23πq)(cid:48)3e−iqµ(cid:48)·xµ(cid:15)q(cid:48)λ(cid:48)wkh(cid:48)lm(q(cid:48),λ(cid:48)), (39)
λ(cid:48)
where I denoted q(cid:48)µ = (ω(cid:48),q(cid:48)). Energy conservation, which is explicit in (32),(33), requires q(cid:48) = k(cid:48)
q
implying that ω(cid:48) = ω(cid:48). Substituting (36),(37),(38),(39) into (35) and using the momentum space
q k
representation of the free fermion propagator
(cid:90) d4(cid:96) /(cid:96)+m
S(x−y) = e−i(cid:96)·(x−y) q , (40)
(2π)4 (cid:96)2−m2+i(cid:15)
q
one arrives at the following expression:
−i(2π)hk(cid:48)ta (cid:88)(cid:90)
S = √ √ √ d3q(cid:48)wh∗ (q(cid:48),λ(cid:48))M (qg → qγ)δ4(pµ+kµ−p(cid:48)µ−q(cid:48)µ), (41)
fi 2ε(cid:48) 2ε 2ωV3/2 k(cid:48)lm C
λ(cid:48)
8
where the Compton scattering amplitude is given by:
M = −e gu¯ (cid:20)/(cid:15)∗ k/+p/+mq /(cid:15) +/(cid:15) p/−/q(cid:48)+mq /(cid:15)∗ (cid:21)u (42)
C q p(cid:48)s(cid:48) qλ(cid:48)(k+p)2−m2+i(cid:15) kλ kλ(p−q(cid:48))2−m2+i(cid:15) qλ(cid:48) ps
q q
and describes the Compton scattering q(pµ)+g(kµ) → q(p(cid:48)µ)+γ(q(cid:48)µ) (modulo the color matrix ta
that is extracted for convenience). The scattering matrix in (41) is a convolution of two physical
processes: the Compton scattering yielding an intermediate photon carrying momentum q(cid:48)µ and
polarization λ(cid:48) and the conversion of this photon into a CK state. Integrating over q(cid:48) in (41) and
multiplying by its complex conjugate one finds
(cid:12) (cid:12)2
(2π)2k(cid:48)2tata t (cid:12)(cid:88) (cid:12)
|S |2 = δ(ε+ω−ε(cid:48)−ω(cid:48)) (cid:12) wh∗ (p+k−p(cid:48),λ(cid:48))M (qg → qγ)(cid:12) , (43)
fi 2ε(cid:48)2ε2ωV3 k 2π (cid:12) k(cid:48)lm C (cid:12)
(cid:12) (cid:12)
λ(cid:48)
where t is the total observation time. The production rate of a CK photon is calculated as follows:
(cid:12) (cid:12)2
1 1 (2π)k(cid:48)2 (cid:90) (cid:88)(cid:12)(cid:88) (cid:12)
Rh (k(cid:48)) = (cid:12) wh∗ (p+k−p(cid:48),λ(cid:48))M (qg → qγ)(cid:12)
lm 42N 2ε(cid:48)2ε2ωV (cid:12) k(cid:48)lm C (cid:12)
c (cid:12) (cid:12)
λss(cid:48) λ(cid:48)
d3p(cid:48) d3k
×δ(ε+ω−ε(cid:48)−ω(cid:48)) f(ω) , (44)
k (2π)3 (2π)3
where the Compton color factor is 1 Tr(tata) = 1 . The Bose-Einstein distribution of
Nc(Nc2−1) 2Nc
gluons in the QGP at temperature T is given by
2(N2−1)
f(ω) = c . (45)
eω/T −1
Eq. (44) gives the CK photon production rate in the case of a discrete spectrum. To obtain a
formula for the continuous spectrum, the left-hand-side of (44) should be replaced by the rate
density R → πdR/dk(cid:48). Since |k| = ω one can use the remaining delta-function to take the integral
over the gluon momentum to derive (for a continuous spectrum)
(cid:12) (cid:12)2
πdRhlm = 1 1 (2π)k(cid:48)2 (cid:88)(cid:12)(cid:12)(cid:88)wh∗ (q(cid:48),λ(cid:48))M (qg → qγ)(cid:12)(cid:12) d3q(cid:48) f(ω)dΩkω2 , (46)
dk(cid:48) 42N 2ε(cid:48)2ε2ωV (cid:12) k(cid:48)lm C (cid:12) (2π)3 (2π)3
c (cid:12) (cid:12)
λss(cid:48) λ(cid:48)
where dΩ is an element of the solid angle in the gluon momentum k direction. I also changed the
k
integration variable form p(cid:48) to q(cid:48) = p+k−p(cid:48). To obtain the cross section one has to divide the
rate by the flux density
p·k
j = . (47)
Vεω
In the ultrarelativistic limit the quark mass m can be neglected. In this case the Compton
q
amplitude M is the same for the left- and right-polarized gluons (see e.g. [38]). Thus, it can be
C
9
taken out of the sum over the intermediate photon polarizations λ(cid:48) in (44):
(cid:12) (cid:12)2 (cid:12) (cid:12)2
(cid:12)(cid:88) (cid:12) (cid:12)(cid:88) (cid:16) (cid:17)(cid:12)
(cid:12) wh∗ (q(cid:48),λ(cid:48))M (cid:12) ≈ |N¯ |2|M |2(cid:12) (cid:15) · Th∗ (q(cid:48))+ihPh∗ (q(cid:48)) (cid:12) , (48)
(cid:12) k(cid:48)lm C(cid:12) k(cid:48) C (cid:12) q(cid:48)λ(cid:48) k(cid:48)lm k(cid:48)lm (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
λ(cid:48) λ(cid:48)
where (27) was used. The Compton amplitude is obtained with the standard manipulations [38]:
(cid:88) 1 (cid:88) (cid:18)p·k p·q(cid:48)(cid:19)
|M |2 = |M |2 = 4e2g2 + = 8e2g2, (49)
C 2 C q p·q(cid:48) p·k q
λss(cid:48) λλ(cid:48)ss(cid:48)
where in the last equation I used the conservation law (p(cid:48)µ)2 = (pµ +kµ −q(cid:48)µ)2 that implies that
p·k = p·q(cid:48) for a highly energetic quark ε (cid:29) ω †. Let χ be the angle between the vectors p and q(cid:48)
and θ be the angle between the vectors p and k. Then p·q(cid:48) = p·k implies
ω(cid:48)(1−cosχ) = ω(1−cosθ). (50)
q
Substituting (48) and (49) into (46) and dividing by the flux density (47) one gets the differential
cross section:
(cid:12) (cid:12)2
πdσlhm = e2qg2 k(cid:48)2ω |N¯ |2f(ω)(cid:12)(cid:12)(cid:88)(cid:15) ·(cid:16)Th∗ (q(cid:48))+ihPh∗ (q(cid:48))(cid:17)(cid:12)(cid:12) d3q(cid:48)dΩ . (51)
dk(cid:48) 8(2π)5N ε(cid:48)ε(1−cosθ) k(cid:48) (cid:12) q(cid:48)λ(cid:48) klm klm (cid:12) k
c (cid:12) (cid:12)
λ(cid:48)
To integrate over the directions of k it is helpful to introduce a new variable y = 1/(1−cosθ) and
denote a = ω(cid:48)(1−cosχ). In view of (50) one finds
q
(cid:90) ωdΩ (cid:90) ∞ dy
k = 2πa = −2πT ln(1−e−a/2T). (52)
(eω/T −1)(1−cosθ) eay/T −1
1/2
To sum over the polarizations λ(cid:48) of the intermediate photon introduce a Cartesian coordinate
system ξηζ with ζ-axis in the direction of vector q(cid:48). Since the factorization property (48) holds
only for circularly polarized photons we choose the polarization vectors as (cid:15)q(cid:48)λ(cid:48) = √1 (ξˆ+iλ(cid:48)ηˆ),
2
with λ(cid:48) = ±1. This yields (for a continuous spectrum):
(cid:12) (cid:12)2
(cid:12)(cid:88) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12)2
(cid:12) (cid:15) · Th∗ (q(cid:48))+ihPh∗ (q(cid:48)) (cid:12) = 2(cid:12)ξˆ· Th∗ (q(cid:48))+ihPh∗ (q(cid:48)) (cid:12) (53)
(cid:12) q(cid:48)λ(cid:48) k(cid:48)lm k(cid:48)lm (cid:12) (cid:12) k(cid:48)lm k(cid:48)lm (cid:12)
(cid:12) (cid:12)
λ(cid:48)
=k(cid:48)4l8(πl+4 1)[δ(k(cid:48)−q(cid:48))]2(cid:12)(cid:12)(Lξ +ihLη)Yl∗m(qˆ(cid:48))(cid:12)(cid:12)2 , (54)
where I used (33) and (34). To integrate over q(cid:48), introduce another Cartesian coordinate system
xyz,suchthatz-axisisinthedirectionofvectorp. Thedirectionofq(cid:48) inthisframeischaracterized
† Inprinciple,ifthescatteringangleχisverysmallχ(cid:28)m /ε,thescatteredquarkenergyε(cid:48) canbecomparableto
q
the energy of a CK state ω(cid:48). However, as we will demonstrate below, the main contribution to the cross section
k
comes from ω(cid:48) <T (cid:28)ε.
k
10
by the polar and azimuthal angles χ and φ. Without loss of generality fix ηˆ to be in the plane of
vectors p and q(cid:48). The two coordinate frames are related as follows:
ξˆ= sinφxˆ−cosφyˆ, (55)
ηˆ = cosχcosφxˆ+cosχsinφyˆ−sinχzˆ, (56)
ζˆ= sinχcosφxˆ+sinχsinφyˆ+cosχzˆ. (57)
Using these formulas it is straightforward to derive
1 1
−i(L +ihL ) = e−iφ(1+hcosχ)L + eiφ(−1+hcosχ)L −hsinχL , (58)
ξ η + − z
2 2
where
(cid:112)
L Y = (L ±iL )Y = l(l+1)−m(m±1)Y . (59)
± lm x y lm l,m±1
To write the cross section in a compact form define the F-function
(cid:90)
Flhm(x) = − ln(1−e−x(1−cosχ)/2)l(l+1 1) (cid:12)(cid:12)(Lξ −ihLη)Ylm(qˆ(cid:48))(cid:12)(cid:12)2dΩq(cid:48). (60)
Sinceφ-dependenceofthesphericalharmonicsisgivenbyY ∼ eimφ,itfollowsfrom(58)and(59),
lm
that function (L −ihL )Y contains three terms that depend on φ as ei(m+2)φ, e−i(m+2)φ and eiφ.
ξ η lm
These are mutually orthogonal on a unit circle meaning that the three terms do not interfere once
we integrate over the directions of q(cid:48) in (51). This brings the F-function to the following form:
2π (cid:90) π
Fh (x) = − dχsinχln(1−e−x(1−cosχ)/2)
lm l(l+1)
0
(cid:26)
1
× sin2χm2|Y (χ,0)|2+ (1−hcosχ)2[l(l+1)−m(m−1)]|Y (χ,0)|2
lm l,m−1
4
(cid:27)
1
+ (1+hcosχ)2[l(l+1)−m(m+1)]|Y (χ,0)|2 . (61)
l,m+1
4
Explicit expressions for the the p-wave F-functions are listed in the Appendix. Substituting (54)
and (52) into (51) and replacing one of the delta functions by R/π one obtains for the continuous
spectrum
1 dσlhm = e2qg2CF T Fh (k(cid:48)/T), (62)
R dk(cid:48) 16π2 k(cid:48)ε2 lm
which is the radiation cross section per unit length, while for the discrete spectrum
e2g2 C T
σh = q F Fh (k(cid:48)/T), (63)
lm 16π R2j2 (k(cid:48)R)k(cid:48)3ε2 lm
l+1