Download Radiation of an Electric Charge in a Screened Magnetic Monopole Potential Abstract

Radiation of an Electric Charge in a Screened Magnetic
Monopole Potential
By: Kramer Benjamin
Supervisor: Dr. Michael Lublinsky
Abstract
This project follows the paper "Radiation of an electric charge in the field of a magnetic
monopole" by Michael Lublinsky, Claudia Ratti and Edward Shuryak: Ref. [1]. In this paper, one
can find an analytic description of the scattering trajectory of an electrically charged particle from a
magnetic monopole (which is equivalent to a similar problem in chromodynamics), and the analysis
of the electromagnetic radiation emitted by such a particle.
An upper limit on the impact parameter was used, in order to take into account a finite
density of the magnetic monopoles. This limit is obtained by assuming a crystal-like structure of the
monopoles.
A broader look at this problem suggests that the magnetic field is that of a Yukawa potential,
due to a certain density of magnetic monopoles. The purpose of this project is to retrace the steps in
the above mentioned paper using a screened Yukawa potential - analytically as much as possible,
and numerically beyond that point. This will allow us to take the upper limit on the impact
parameter to infinity, and hopefully receive a better approximation for the E.M. radiation in the
limit of small frequencies.
1 Introduction
The article "Radiation of an electric charge in the field of a magnetic monopole" - Ref. [1]
deals with the collision of particles inside Quark-Gluon Plasma (QGP), and specifically collision of
quarks with heavy and scarce "color-magnetic monopoles". These particles, which are assumed to
exist in QGP within a certain region of temperatures, produce a Coulomb-like chromo-magnetic
field on which quarks can scatter (hereinafter – Magnetic Scattering).
In this article we make an analogy to electrodynamics. The magnetic scatterings mentioned
above are of the same nature as scattering of a light electrically charged particle from a (theoretical)
heavy magnetic monopole.
The article Ref. [1] studied these magnetic scatterings in an attempt to explain an excess in
the production rate of dileptons measured in collisions of heavy ions at Relativistic Heavy Ion
Collider (RHIC).
Such measurements can be partially attributed to radiation emitted from the magnetic
scatterings in "small" frequencies. To be more specific, we focus on the problem of soft photon
radiation and neglect the change in energy throughout the scattering process.
The results are analyzed by comparison to the collisions of quarks and anti-quarks amongst
themselves, which are equivalent to a Coulomb scattering of electrically charged particles
(hereinafter – Coulomb Scattering). The final result that we would like to replicate, and perhaps
1
improve, is the ratio between the total radiation emitted in these two types of collisions.
In this article, as well as in Ref. [1], the product of magnetic and electric charges - q e ⋅q m ,
is taken to be ħ. The calculation is non-perturbative – we deal with trajectories that are governed
completely by the chromo-electric and chromo-magnetic forces.
In order to repeat the calculations we will re-examine the magnetic scattering problem with
the addition of screening. We will continue to solve the equations of motion numerically, and
present the trajectory (Chapter 2).
To maintain a sense of consistency, we will also apply our previous considerations to the Coulomblike collisions, and solve the relevant equations of motion numerically with the addition of
screening (Chapter 3).
Finally, we shall examine the dependency of the ratio mentioned above (between the total
emitted radiation in the magnetic and the Coulomb scatterings) on the initial velocity (Chapter 4)
and on the screening parameter of the Yukawa potential (Chapter 5).
All of the results will be presented using the natural units, namely c = ħ = 1 . All of the quantities
will be expressed in units of energy, or eV to be exact.
2 The Magnetic Scattering Problem
2.1 A Short Analytic Examination
First, we'll need to introduce some of the quantities that'll be used:
- q e , q m are the chromo-electric and chromo-magnetic charges, respectively.
- m is the mass of the quark.
- μ is the screening parameter of the magnetic Yukawa potential.
- ⃗r is the quark's distance from the magnetic monopole.
d ⃗r
d 2 ⃗r
=⃗
v ;
=⃗
a
dt
d t2
We need to solve the scattering problem of an electric charge from the magnetic potential
/ r . This potential produces the magnetic field:
−μ r
qm e
d −e −μ r
1 μ
⃗
B = qm (
) r̂ = qm e −μ r ( 2 + )̂r
dr
r
r
r
(2.1)
The equation of motion for our electrical charge is thus:
1 μ
m⃗
a = qe (⃗v × ⃗
B) = qe qm ( ⃗
v ×̂r ) e−μ r ( 2 + )
r r
(2.2)
a⊥⃗
v , which means that ∣ ⃗v ∣ is constant throughout the
Immediately we notice that ⃗
motion. We also notice that if we multiply Eq. (2.2) by ⃗r we get:
2
⃗r
d 2 ⃗r
=0
2
dt
This expression can be substituted into:
d2 2
d
d ⃗r
d 2 ⃗r
d ⃗r 2
(⃗
r
)
=
(
2
⃗
r
)
=
2
⃗
r
+2
(
)
dt
dt
dt
d t2
d t2
Which finally gives us:
d2 2
(r ) = 2 v 2 = const
d t2
The solution of this differential equation is:
r =√ v 2 t 2+d 2
(2.3)
v r =v 2 t / √ v 2 t 2+d 2 = v 2 t /r
(2.4)
v ⊥ =v d / √ v t +d = v d /r
(2.5)
Using Eq. (2.3), we also find:
And v 2 = v 2r +v 2⊥ ⇒
2 2
2
The equation of motion (2.2) can now be rewritten as:
ma
⃗ = qe B( ⃗v × r̂ ) = qe B(⃗v ⊥ × r̂ )
From which we can deduce:
∣⃗
a∣=
qe B
q q
1 μ
v ⊥ = e m v d e−μ r ( 3 + 2 )
m
m
r r
(2.6)
Later in Chap. 2.3 we will show that these results agree with our numeric calculations.
Unfortunately, this is a far as analytic calculations go. In Appendix A we describe the
complete analytic solution for the problem in the absence of screening ( μ=0 ), and the difficulty
in finding such a solution for general μ .
3
2.2 Solving the Equation of Motion
Eq. (2.2) was solved numerically using the Runge Kutta method (RK4) described in Ref.
[2], as follows:
The acceleration of the particle depends on
equations describing the problem is thus:
⃗r , ⃗v . A system of first order differential
a ( ⃗v , ⃗r ) ; ⃗r ' = ⃗
v
⃗v ' = ⃗
(2.7)
At a time t , we evaluate the following expressions:
v⃗1 = ⃗
a ( ⃗r , ⃗v )⋅Δ t
r⃗1 = ⃗v ⋅ Δ t
v⃗2 = ⃗
a (⃗r +
r⃗1
v⃗
,⃗
v + 1 )⋅ Δ t
2
2
r⃗2 = (⃗v +
v⃗1
)⋅ Δ t
2
v⃗3 = ⃗
a (⃗r +
r⃗2
v⃗
,⃗
v + 2 )⋅ Δ t
2
2
r⃗3 = (⃗
v+
v⃗2
)⋅ Δ t
2
v⃗4 = ⃗a ( ⃗r +r⃗3 , ⃗
v +v⃗3 )⋅Δ t
r⃗4 = ( ⃗v + v⃗3)⋅ Δ t
These allow us to calculate the values of ⃗r , ⃗v at time t+Δ t :
⃗r (t+Δ t) = ⃗r (t )+( r⃗1+2 r⃗2+2 r⃗3+ r⃗4 )/6
(2.8a)
v (t)+( v⃗1+2 v⃗2+2 v⃗3+ v⃗4)/6
⃗v (t+Δ t) = ⃗
(2.8b)
In order to evaluate the radiated intensity we'll need to calculate the Fourier Transform of the
acceleration for continuous impact parameters b∈[b min , bmax ] . To do so, we use the initial
conditions:
⃗r (0) = ( d , 0 , 0)
⃗v (0) = (0 , V 0 , 0)
v 2 is constant throughout the motion, so the trajectory calculated in this manner
corresponds to the scattering of a particle with initial velocity V 0 and impact parameter b .
b can be evaluated at the end of the simulation via:
√
b=lim r 2 (t )−
t→0
1
( ⃗r (t)⋅⃗v (t ))2
2
V0
(2.9)
It turns out that b≈d (They are not generally equal, as is the case with μ=0 ).
The trajectory is symmetric to inversions - t →−t ; y →− y . That is:
x (t ) = x (−t ) ; y (t ) = − y (−t) ; z ( t) = z (−t)
The same is true for the acceleration. Therefore, we write the Fourier Transform of the
acceleration as:
4
∞
∞
∞
F (a i) = ∫ a i (t) e−i ωt dt = ∫ a i (−t)e i ω t dt +
−∞
0
∫ a i (t )e −i ω t dt
⇒
0
∞
F (a x ) = 2∫ a x (t )cos (ω t)dt
(2.10a)
0
∞
F (a y ) = −2 i ∫ a y (t)sin(ω t)dt
(2.10b)
0
∞
F (a z) = 2 ∫ a z (t )cos (ω t) dt
(2.10c)
0
We denote a⃗ω =( F (a x ), F (a y ) , F (a z )) .
The final result of our numeric calculation is the function ∣ a⃗ω(b) ∣ , for some range of
small frequencies. We will show a plot of this function ahead in Chap. 2.4.
2.3 The Trajectory
The trajectory in the absence of screening takes place on the surface of a cone, as described
in Appendix A. As a result of the screening the cone shape is evident only at small distances from
the origin, whereas the motion at larger distances approaches a straight line:
Figure 2.1: The scattering of a quark from a magnetic monopole.
The quark's mass is m=3 [GeV ] . It is released at infinity with an
5
initial velocity of 0.5, and an impact parameter of b=1 [ GeV −1] .
The screening parameter in this case is μ=1/1.8≈0.56 [GeV ] .
During the rest of this section we will examine various quantities related to this trajectory.
In the following three figures, the screening parameter as well as the quark's mass, initial velocity
and impact parameter are the same as in Fig. 2.1.
Eq. (2.3) gives us the distance between the particles as a function of time:
(a)
(b)
Figure 2.2.a: The quark's distance from the magnetic monopole vs. time
(dotted). The result of Eq. (2.3) : r =√ v 2 t 2+d 2 is also presented
(dashed), with the parameters: v =0.5 , d =1 [GeV −1] .
Figure 2.2.b: The ratio between the above mentioned results.
Similarly, we can examine the velocity in the radial direction and that in the direction
perpendicular to r̂ according to Eqs. (2.4) and (2.5). The former is described in the next figure.
(a)
(b)
Figure 2.3.a: The quark's radial velocity (dotted).
The result of Eq. (2.4) : v r =v 2 t / √ v 2 t 2+d 2 is also presented
(dashed), again with the parameters: v =0.5 , d =1 [GeV −1] .
Figure 2.3.b: The ratio between the above mentioned results.
Last, we'd like to examine the acceleration, according to equation (2.6) :
6
(a)
(b)
Figure 2.4.a: The quark's total acceleration (dotted).
q q
1 μ
a ∣ = e m v d e−μ r ( 3 + 2 ) is also
The result of Eq. (2.6) : ∣ ⃗
m
r r
presented (dashed), with the parameters mentioned in Fig. 2.1.
Figure 2.4.b: The ratio between the above mentioned results.
The relative error in these results is at most 0.02%. We discuss the error in length in the
following chapters.
2.4 Evaluation of the Radiated Intensity
If we denote the electric charge of the quark as e em , it's the dipole moment is given by
⃗p =e em ⃗r . Using the approximation of dipole radiation, the radiated intensity into the solid angle
d Ω is, according to Ref. [3]:
dI
1 .. 2
1 .. 2 2
=
p⊥ =
p sin (θ)
dΩ
4π
4π
(2.11)
Integrating Eq. (2.11) gives us the total radiated intensity in all directions:
I=
2 .. 2 2
2
p = (e em ⃗
a)
3
3
(2.12)
Taking a Fourier Transform gives us the energy radiated in the frequency interval
[ω , ω+d ω] :
d εω =
(2.10).
4
dω
(e a⃗ )2
3 em ω 2 π
(2.13)
Here, a⃗ω is the three-dimensional Fourier Transform of the acceleration defined in Eq.
For different impact parameters we have different contributions to d ε ω . A typical
distribution is presented in the following figure:
7
Figure 2.5: Contribution to radiated energy from different
frequencies and impact parameters.
The initial velocity of the quark is 0.5, its mass is 3 [GeV ] , and
the screening parameter μ is taken to be 0.56 [GeV ] .
For an element of frequency d ω , we calculate the total energy due to a uniform
distribution of quarks:
bmax
d κω = ∫ d ε ω 2 πb db
(2.14)
bmin
In our case, b min is the effective size of the magnetic monopole. These particles are very
large compared to the quark's size. In the proceeding calculations we use the value of
−1
b min = 0.78 [GeV ] , according to Ref. [1].
The value of b max is our primary interest in this article:
We denote by n M the density of the magnetic monopoles in the plasma.
In the original model presented in Ref. [1], the authors assume a crystal like structure for
these monopoles. The effective range at which the quark is considered to undergo a collision with a
certain monopole is r < 0.5 nM −1 /3 . This was the original cutoff value used for b max .
However, our interest is in low frequency radiation, which is most prominent at large
distances where the scattering force is weak. This contribution is lost due to the impact parameter
cutoff in the integral.
As a novelty of this project, we use a screened magnetic potential. We set the screening
parameter to be inversely proportional to the above mentioned value: μ=2 nM 1 /3 ≈ 0.56 [GeV ] ,
hence the value of μ used in previous calculations.
8
The value of interest to us is not d κω , but rather the radiated energy per unit volume
associated with the scattering of quarks from the magnetic monopoles. This value is:
d Σω =
2
n n d κω
9 M q
(2.15)
Where n q is the sum of densities of u, d and s quarks, and the factor of 2/9 comes from
the different electric charges of these quarks (respectively) :
2 1
2 2
1 2
1 2
= ⋅(( ) +(− ) +(− ) )
9 3
3
3
3
The values we assume for the quark and monopole densities are:
3
3
n q = 0.12 [GeV ] ; n M = 0.02 [GeV ] (also, according to Ref [1]) .
3 The Coulomb Scattering problem
3.1 A Short Analytic Examination
In the absence of screening, we can find a full analytic solution to the scattering problem of
two electrically charged particles in Ref. [4].
We introduce the following quantities:
- e i , m i ; i = 1,2 are the electric charge and mass of each particle respectively.
- α s is the strong coupling constant.
- C R is a Casimir factor of the color group (explained later in this chapter).
- α = C R αs
m1 ⋅ m2
- λ=
is the reduced mass.
m 1+m2
ωα
- ν=
3
λV 0
√
- ϵ = 1−
- H (1)
λ b 2 V 40
α2
is the Hankel function of the first kind.
For an attractive force, the element of radiated energy is:
d εω =
2
2
2 π α2 ω 2 e1 e2
(1)
2 ϵ −1
2
(
−
)
{H
'
(i
ν
ϵ
)
+
H (1)
iν
i ν (i ν ϵ ) }d ω
4
2
m
m
3V 0
ϵ
1
2
For a repulsive force:
9
(3.1)
2
2 2
2
2 π α ω e1 e2
(1)
2 ϵ −1
2
−2 π ν
d εω =
( − ) {H i ν ' (i ν ϵ ) + 2 H (1)
dω
i ν (i ν ϵ ) }e
4
m1 m2
3V 0
ϵ
(3.2)
In our consideration of the radiated energy in the quark-gluon plasma, we have to consider
collisions of different quarks and anti-quarks. These divide the collisions into different channels
with respect to the SU(3) color group:
The color decomposition of quark anti-quark interactions is:
The color decomposition of quark quark \ anti-quark anti-quark interactions is:
The first type introduces the singlet and octet channels, and the second type introduces the
sextet and anti-triplet channels. The interactions in the different channels vary by the following
Casimir factors:
̄
1
1
1
C 1 = 1 /6 ; C 3 = ⋅(1/3) ; C 6 = ⋅( 2/3) ; C 8 = ⋅( 4/3)
(3.3)
3
6
8
The attractive channels are the singlet and anti-triplet. The repulsive are the sextet and octet.
We define the following expressions:
2
f k = { H (1)
i ν ' (i ν ϵ ) +
ϵ 2−1 (1)
H i ν (i ν ϵ ) 2 } ; ν , ϵ → ν (α k ) , ϵ (α k ) ; α k =α s ⋅C k
2
ϵ
(3.4)
By taking the different channels into consideration, the total radiated energy per unit volume
associated with the collision of quarks is received by:
2
∞
−2 π ν(α6 )
−2 π ν (α8 )
d Σ 2 αem n q ω2 π2
=
[8 α 21 f 1+6 α 2̄3 f ̄3 +3α 26 f 6 e
+α28 f 8 e
] 2 π b db (3.5)
d ω 3 π 18 2 V 40 λ 2 ∫
0
3.2 Reduction from the screened potential problem
With the addition of screening, this problem too becomes difficult to solve analytically, so
we resort to numerical calculations. We use pretty much the same method described in Chap. 2.2,
except that now the equation of motion per channel is:
2
α
d ⃗r
1 μ
= k e−μ r ( 2 + ) r̂
2
λ
dt
r r
(3.6)
Also, the impact parameter now differs from the minimal distance between the particles.
Therefore we calculate this minimal distance prior to the calculation of the Fourier Transform.
We wish to express Eq. (3.5) using the Fourier Transform of the acceleration to . To do so,
10
we first substitute Eq. (3.4) into Eq. (3.1) .
2 π α 2k ω2 e1 e 2 2
d εω =
( − ) fk dω
m1 m2
3V 40
(3.7)
Using Eq. (2.13) we receive a similar expression for the Fourier components:
d εω =
2
4 2 e1 e 1
dω
λ ( − ) a⃗ω2
3
m1 m1
2π
(3.8)
From this we can derive:
2
F k = a⃗
ω,k =
ω2 π2 α 2k
fk
V 40 λ 2
(3.9)
Eq. (3.2) for the repulsive force is modified similarly.
Finally, substituting this into Eq. (3.5) yields:
bmax
2
d Σ 2 α em nq
=
⋅
⋅ ∫ [8 F 1 + 6 F ̄3 + 3 F 6 + F 8 ] 2 π b db
dω
3 π 182 0
(3.10)
Before we proceed, we'd like to test our numeric calculations vs the analytic results.
In the next figure we show the error in the energy distribution of a repulsive and an attractive
channel, respectively:
(a)
(b)
Figure 3:
Relative error in the numeric calculations for an attractive channel (a)
and a repulsive channel (b). Eta is defined as follows:
ω2 π 2 α 2k
ηk = (F k −
f k) / F k
V 40 λ 2
The initial velocity is 0.7. The reduced mass is 3/ 2 [GeV ] .
11
First thing we may notice is the sharp increase of F for attractive forces around zero impact
parameter. This has been verified to be an artifact of a numeric instability: when the impact
parameter is small, the particles can get arbitrarily close (beyond precision we can
achieve/maintain). This can be solved by extrapolating a more accurate value of F using the result
of nearby larger impact parameters ( f actually behaves quite well in this area).
Second issue that might be of interest is a seemingly linear increase of the error with the
frequency, for repulsive forces. Fortunately, we are interested mainly in low frequencies, and thus
this error is acceptable.
4 A Comparison of Coulomb and Magnetic Scatterings
We substitute Eq. (2.13) into (2.14) which in turn is substituted into (2.15). The resulting
expression for the radiated energy per unit volume (per unit frequency) associated with magnetic
scatterings is:
b
2 α em 2 n M n q max 2
d Σ Magnetic
ω
=
⋅
⋅ ∫ a⃗ω 2 π b db
dω
3π
9
bmin
(4.1)
We shall examine the relation of this expression to the corresponding "Coulomb" expression
received in the previous section. The results are presented in Fig. 4.
Comparing the results in Fig. 4 (specifically the 'cutoff' curves) to Figs. 6.b, 6.d, 6.f of Ref.
[1], we notice that for larger frequencies, our results are inaccurate. Their quality improves with
increase in velocity, which is unfortunate since we used a non-relativistic approximation.
Nevertheless, for small frequencies (which are our primary area of interest), the results of
the numeric calculations are very good for all three velocities.
(a)
12
(b)
(c)
Figure 4:
The ratio of energy per unit volume (per frequency) of the Coulomb vs. the Magnetic scatterings.
The curve marked 'cutoff' shows this relation for μ=0 , where a cutoff value was used for the
integrals: 1 [GeV −1 ] for Coulomb scatterings, and 1.8 [GeV −1 ] for magnetic scatterings.
The curve marked 'Yukawa' shows this relation for μCoulomb =1 [GeV ] ; μ Magnetic =0.56 [GeV ] ,
where the integrals' upper bound was taken to infinity.
The quark's mass is 3 [GeV ] for all figures and the reduced mass used for Coulomb scatterings
is half that. The initial velocity is: 0.3 (a), 0.5 (b), 0.7 (c) .
5 Variation of the screening parameter
In our calculations we used certain values for the density of quarks and magnetic monopoles
that were estimated in Ref. [1]. We used the inverse of these values as the screening parameters of
the Yukawa potentials: μCoulomb =1 [GeV ] ; μ Magnetic =0.56 [GeV ] .
In this last part, we changed the screening parameter and repeated some of the calculations.
The result are presented in the next figure:
13
Figure 5:
The ratio of energy per unit volume per frequency of the Coulomb vs. the
Magnetic scatterings.
Yukawa potentials were used with the following screening parameters:
(a) - μCoulomb =0.5 [GeV ] ; μ Magnetic=0.28 [GeV ]
(b) - μCoulomb =1 [GeV ] ; μ Magnetic =0.56 [GeV ]
(c) - μCoulomb =2 [ GeV ] ; μ Magnetic =1.11 [GeV ]
The initial velocity is 0.7.
The quark's mass is 3 [GeV ] and the reduced mass used for Coulomb
scatterings is 3/2 [GeV ] .
We notice that as we increase the scattering parameters by an additional 300%, the results
vary by a maximum of 30%. This gives our model a sense of stability.
6 Outlook and Conclusions
•
•
•
•
•
Despite some discrepancies, we managed to replicate the results of the original article in
the area of small (near zero) frequencies.
The original result suggests that in this area, the energy emitted due to quark-monopole
collisions is larger than that emitted due to quark-quark collisions. The results of our
calculations support these results, and show an additional increase in emission of about
20%-40% in the magnetic collisions.
The resulting ratio between radiated energy from Coulomb and magnetic scatterings has
little dependence on the exact choice of the scattering parameters, as long as the ratio
between these parameters is constant.
The numeric calculations are inaccurate due to a certain numeric instability at the
vicinity of the magnetic monopole. Also, the iteration method was an explicit Runge
Kutta method (of the 4th order), and as an explicit method, it is subject to an
accumulating error. Using an implicit method may greatly improve our results.
The number of iterations we needed to undergo in order to receive a full set of results for
14
•
•
one initial velocity is approximately 2.5 ⋅1010 . Using an implicit method would mean
increasing the total number of calculations in each iteration by up to 2 orders of
magnitude. This will take too long even for our somewhat optimized c++ program.
However, using an implicit method may reduce the total number of iterations needed to
achieve the same accuracy.
We can also improve our program further by utilizing the GPU (graphic processing unit)
which specializes in a large number of independent parallel calculations (just as is the
case in question).
Further improvements of our work should involve:
• Friction by radiation: At small distances the particles experience strong acceleration,
which may cause significant loss of energy to radiation. This may change the
trajectory, and alter or results.
• Relativistic effects.
• Quantum calculation.
There are known models which suggest the existence of dyons. With small modifications
we should be able to study the trajectory and emitted radiation for collisions involving
these particles, which also have relevance to the collision of heavy ions.
7 Appendix A
We write our equation (2.2) of motion in the following form:
a = (⃗v × r̂ )⋅
⃗
qe
m
⋅∣ ⃗
B∣
(7.1)
In spherical coordinates, we consider the general vectors:
⃗r = r r̂
⃗v = ṙ r̂ + r θ̇ θ̂ + r sin [θ ] ϕ̇ ϕ̂
(7.2a)
(7.2b)
a = ( r̈ − r θ̇ 2 − r ϕ̇ 2 sin 2[θ ])̂r + ( r θ̈ + 2 ṙ θ̇ − r ϕ̇ 2 sin[ θ ]cos[θ ])θ̂ +
⃗
(r ϕ̈ sin [θ ] + 2 ṙ ϕ̇ sin [θ ] + 2 r θ̇ ϕ̇ cos [θ ]) ϕ̂
(7.2c)
From (7.2b) we have:
(⃗v × r̂ ) = r ϕ̇ sin [θ ]θ̂ − r θ̇ ϕ̂
(7.3)
By substituting (7.2c) and (7.3) into (7.1), The equations of motion in spherical coordinates
take the form:
r̈ − r θ̇ 2 − r ϕ̇ 2 sin 2 [ θ ] = 0
r θ̈ + 2 ṙ θ̇ − r ϕ̇ 2 sin [θ ]cos [θ ] = r ϕ̇ sin[ θ ]⋅
15
(7.4a)
qe
m
⋅∣ ⃗
B∣
(7.4b)
r ϕ̈ sin [θ ] + 2 ṙ ϕ̇ sin [θ ] + 2 r θ̇ ϕ̇ cos [θ ] = −r θ̇ ⋅
qe
m
⋅∣ ⃗
B∣
(7.4c)
We could not solve this set of equations, in itself.
In the absence of screening, we find that the angular momentum associated with the electromagnetic fields of a pair of electric and magnetic monopoles takes the following form according to
Ref. [5]:
⃗
J EM = −qe qm ̂r
(7.5)
This result is obviously independent of the distance between the particles.
The total "angular momentum" vector is conserved, allowing us to simplify the equations of
motion:
⃗
J = ⃗r × m ⃗
v − qe q m ̂r = const ⇒ ⃗
J ⋅ ̂r = −qe⋅q m = const
(7.6)
When placing the z axis along −⃗J this means that θ =const . Now the solution to the
equations of motion becomes:
ϕ [t ]=
1
tv
arctan [ ]
sin [θ ]
d
(7.7)
In our case, the conventional angular momentum isn't conserved. To see this we only need to
consider two particles approaching each other directly:
At large distances the particles don't “see“ each other and the angular momentum is 0. At
small distances the screening effect can be neglected and the angular momentum once again takes
the form in Eq. (7.5).
Nevertheless, the symmetry of this problem suggests the existence of an additional integral
of motion. We however, failed to find one.
8 References
1. M. Lublinsky, C. Ratti, E. Shuryak Phys.Rev.D81:014008 (2010)
http://arxiv.org/abs/0910.1067
2. Kendall E. Atkinson - An Introduction to Numerical Analysis 2nd ed. - Runge Kutta
Methods p. 420
3. Landau, Lifshitz – The Classical Theory of Fields vol.2 (4th ed. , ButterworthHeinemann, 1994) p. 189: "§ 67. Dipole radiation"
4. Landau, Lifshitz ... p. 195: "§ 70. Radiation in the case of Coulomb interaction"
5. K. A. Milton: Rept.Prog.Phys.69:1637-1712 (2006) http://arxiv.org/abs/hepex/0602040
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