arXiv:0704.0169v1  [hep-ph]  2 Apr 2007
VERY STRONG AND SLOWLY VARYING MAGNETIC FIELD
AS SOURCE OF AXIONS
Giorgio CALUCCI*
Dipartimento di Fisica Teorica dell’Università di Trieste, Trieste, I 34014 Italy
INFN, Sezione di Trieste, Italy
Abstract
The investigation on the production of particles in slowly varying but
extremely intense magnetic field in extended to the case of axions. The
motivation is, as for some previously considered cases, the possibility that
such kind of magnetic field may exist around very compact astrophysical
objects.
* E-mail: giorgio@ts.infn.it
http://arxiv.org/abs/0704.0169v1
1. Statement of the problem
A magnetic field of huge strength can give rise to real particles even if
its rate of variation is very small: this posibility could be of some interest
from a pure theoretical point of view, but it gains more physical relevance
if one accepts that such kind of field configurations may be present around
some very compact astrophysical objects[1-3]. In this case the time vari-
ation is related to the evolution of the source, by collapse, rotations or
else, and it is therefore very slow, in comparison with the times typical
of elementary-particle processes. We can call the former time the macro-
scopic time and the latter the microscopic one. The production of light
particles in these processes has been analyzed in some detail in some pre-
vious papers[4],with the suggestion that it is one of the mechanisms at
work in the phenomenon of gamma-ray bursts[5]; the typical microscopic
time is related to the electron mass since photons are produced through
real or virtual intermediate states of e− − e+-pairs. The lightest particles
that could be produced are massive neutrinos, but the magnetic-moment
coupling induced by the standard electroweak interactions is extremely
small.
There is, at least in the theoretical realm, another very light parti-
cle, that is the axion[6]: owing to its dynamical characteristics it must
be coupled also to the electromagnetic field[7], even more its electomag-
netic coupling is being actively studied from an experimental side[8] and
the possibility of detecting such particles as coming from nonterrestrial
sources has already been foreseen[9]. It is immediately seen that the pro-
duction of axions by a varying magnetic field must be realized through a
mechanism different from the previously considered one, in fact the axions
are coupled[7] only to the pseudoscalar density E ·B so the presence of
an electric field is necessary as a starting point, but a nonstatic magnetic
field creates always an electric field and even though the rate of variation
is small, the very large magnetic strength makes the electric field not a
tiny one.
In the present paper the coupling of the axions field with a given
E ·B density is written in standard second-quantized formalism, then the
effect of time variation of that density on the axion vacuum is determined
and the consequent production is calculated. The result depends both
on the spatial shape and on the time variation of the magnetic field: in
accordance with the prevailing astrophysical hypotheses[1-3] the magnetic
field is seen as a bundle of lines of force which may safely be considered
straight in comparison with the microscopic scale. The time-variation
could affect both the shape and the strength of the fields, both are effective
in the production process. The calculation procedure is not the standard
adiabatic approximation [10] as used in previous investigations [4], but the
feature that one has to deal with a two-scale problem is still fully relevant.
2. General form of the production probability
The starting point is a second-quantized axion field in presence of
given, classical, magnetic and electric fields. The axion field φ(x) is cou-
pled to the pseudoscalar density * G(x) = E(x) · B(x) and the coupling
constant, of dimension of length, is here indicated by C. We assume that
the interaction lasts from an initial time to until a final time t.
So we have for the axion field the expression:
φ(x) = φo(x) + χ(x) = φo(x) + C
∆R(x− y)G(y)d4y (1)
Here ∆R(x − y) is the standard retarded Green function, the source is a
c−number, so the same holds for χ. The field φ is free before to, where it
has the standard expansion:
φo(x) =
(2π)3/2
ao(k)e
ik·r−iωt + a†o(k)e
−ik·r+iωt
, (2)
then it acquires a contribution from χ, this term has the following actual
expression:
χ(x) =
(2π)3/2
e−iωkt
eiωkτgk(τ)dτe
ik·r − c.c.
(3.1)
gk(t) =
(2π)3/2
G(r, t)e−ik·r (3.2)
The reality condition for G, that gives g∗k = g−k have been used, together
with the initial condition χ̇(to) = 0. The separation into positive and
negative frequencies in χ(x) is unambiguous until the typical frequencies
of G are small.
Having found the time evolution of the field the total production of
axions is calculated in Heisenberg description of motion, i.e. we take an
initial state ao(k)|◦ >= 0 ∀k, i.e. the vacuum in the absence of inter-
action then we express the mean particle number as the time dependent
N (k, t) =< ◦|a†(k, t) a(k, t)|◦ >
* the same kind of coupling is possible also for the neutral pion, but in this case
other channels of production are present [11]
Since all the effect of the interaction is a c−number shift on the opera-
tors a(k, t) = ao(k) + b(k) the calculation is easy, in particular when the
interaction no longer acts we get
Nf (k) = |bf (k)|2 (4)
The expression of bf (k) can be read off from eq.s (2,3.1,3.2), it is:
bf (k) =
eiωkτgk(τ)dτ (5)
3. Detailed calculations
As anticipated in the introduction a more definite model of the mag-
netic field can by a field of uniform direction (at a given time) with some
transverse shape, both the direction and the shape may vary in time, one
possible restriction is the conservation of the total flux. More explicitly
these conditions are realized by giving:
n ∧ r
1− w(µr⊥)
B = − F
n ∂⊥w(µr⊥) . (6)
The unit vector n gives the instantaneous direction of the magnetic field,
r2 − n · r2 and ∂⊥ the corresponding derivative, F is the total
magnetic flux; the parameter µ defines the size of the field in the transverse
directions, w is taken to be cylindrically symmetric and, obviously it must
go to zero at infinity, the requirement w(0) = 1 avoids singularities in E.
Since E = −Ȧ and A ·B = 0 only the terms coming from the time
variation of the direction of A contributes to G = E ·B so we get:
)2J · r
(1− w)∂⊥w
Here J gives the angular velocity of n i.e. J = n ∧ ṅ. In this configuration
the Fourier transform of the source is
gk(t) =
(2π)3/2
C(2F )2δ(n · k)J · kS(k⊥/µ) (8.1)
S(k⊥/µ) = (µ/k⊥)
J1(ρk⊥/µ)
(1− w(ρ))w′(ρ)
dρ/ρ (8.2)
Here J1 is the Bessel function of order one and w
′ indicates the derivative
with respect to the argument. It is useful to remember that, owing to the
presence of the factor δ(n · k) in the expression of S we can substitute k2⊥
simply with k2.
We now remember that the model of magnetic field we have at hand
is such that it is uniform along one direction, but this direction is contin-
uously varying, so a most significant quantity is obtained by an angular
integration
dΩkNf (k) (9)
So we need a quantity like
dΩkgk(τ)g
′) which contains a singularity
due to the presence, in the domain of integration, of a δ-square term which
arises for τ = τ ′. This is clearly due to the unphysical assumption that
at every time there is a direction in which the pseudoscalar density G
is absolutely uniform. Since we are integrating over the direction at the
end the effect of the singularity is mild enough, it results in a logarithmic
divergence, however this fact must be explicitly death with, considering a
finite extension of the fields. A more careful treatment is mathemetically
heavier, so it is presented in an Appendix.
The final result is given by
≈ C2(2F )4 k
Ψ[S(k/µ)]2
2 ln 4kL . (10)
If we give a definite transverse shape to the fields we get, evidently,
a definite answer. In the situation where the fields change direction but
not intensities, i.e. we keep µ constant, we may give, tentatively the form
w(ρ) = e−ρ
. Then the expression for the function S(k/µ) is[12]:
S(k/µ) =
− exp
. (11)
The limit k → 0 of this expression is finite, so the whole production goes
to zero only owing to the phase-space factor k2 in front of the expression in
eq.(10). This result, as it appears from the whole derivation, can be valid
only for axion masses and so energies which are definitely larger than the
typical frequencies of the astrophysical phenomena, no resonant dynamics
is included.
4. Some conclusions
The rate of production of axions by a slowly varying but very strong
magnetic field has been calculated. The conditions are such that the as-
trophysical frequencies are very much lower than the proper frequencies of
the axion field. In fact with an axion mass mA of the order of 1eV[13] the
typical frequencies of the field shall exceed 1015 Hz and so no resonance
conditions appear realistic. The relevant parameter turns out to be the
dimension of the spatial inhomogeneity, in the present model 1/µ, here
also is seems very reasonable to assume that µ << mA. It is also possible
to give an expression for the total number of produced particles.
A = ζ C
2F 4µ3
Ψ lnLµ (12)
Some of the factors owe their origin to the general form of the interaction,
eq(1) and to dimensional requirament, it appers clear the role of the total
rotation of the field (Ψ) in determining the overall production; so when
the rotation is uniform the rate is proportional to the angular velocity.
The numerical factor is more model dependent, in the chosen case it
is ζ = 8
3− 2−
2] = 0.704 . . ..
The transformation from the incoming Heisenberg field to the outgo-
ing field can be implemented by the simple unitary operator
U = exp
d3k[a(k)b∗(k) − a†(k)b(k)]
Ua(k)U† = a(k) + b(k) .
The actual form of the evolution operator gives the further information
that the axion are produced with a Poissonian distribution of multiplicity,
strictly speaking this is true for a production in a totally defined state,
in operations like the one leading to eq. (9) this particular form can be
blurred.
A very short mention on this problem was made at the XXXVI In-
ternational Symposium on Multiparticle Dynamics, Paraty, R.J. - Brazil,
September 2006
Appendix
We want to calculate
dΩkgk(τ)g
′), with the functions g given by
eq.(8.1) and taking care of the finite extension of the magnetic field. This
is implemented by substituting the δ−functions as:
δ(n · k) → Lπ−1/2 exp[−L2(n · k)2] (A.1)
The calculation is performed in the particular case in which the rotation
takes place in a constant plane, so J‖J′. The integration over the angles is
performed in Cartesian coordinates, it is useful to introduce the unit vector
of the three-momentum direction k = kv, then
dΩk = 2δ(v
2 − 1)d3v
and through standard although lengthy calculations the representation is
obtained:
dΩkδ(n · k)J · kδ(n′ · k)J′ · k →
JJ ′×
dλ exp[iλ(w2 − 1)]w2dw
(n ∧ n′)2 − 2iλ/(Lk)2 − λ2/(Lk)4
]−1/2
(A.2)
In the limit L→ ∞ it results
I = JJ ′/|n ∧ n′|
which can be obtained in a simpler way.
Now we must integrate over τ and τ ′, times an oscillating factor
eiωk(τ−τ
′). In the conditions that have been chosen the motion takes
place in a plane, so n is characterized by a unique angle ψ and n′ by
ψ′ hence the integration over time amounts at an angular integration, in
fact (n ∧ n′)2 = (sin(ψ − ψ′))2 and moreover J = ψ̇ and J ′ = ψ̇′. So
we must integrate I in dψ , dψ′ from 0 to some final angle Ψf . Defin-
ing γ = ψ − ψ′ we see that the integrand shows, in the limit L → ∞,
a singularity for γ = 0, so we perform the integration from −1
π to 1
because the domain which do not include zero has no singular behavior,
the oscillating factor is approximated with its value on the singular point
τ = τ ′, so that the exponential factor reduces to 1. Then the integral is a
complete elliptic integral,[12] which can be conveniently expressed in term
of the hypergeometric function. In fact the integration in dγ is:
sin2 γ +Q
with Q = − 2iλ
(Lk)2
(Lk)4
(A.3)
The result of the integration is
and in the limit L→ ∞, which gives Q→ 0 we get;
2 ln 2 + lnLk
Since the dominant term in the limit is independent of the auxiliary pa-
rameter λ the rest of the integration in eq. (A.2) is straightforward and it
gives
I = Ψ
2 ln 4kL (A.4)
A comment: what is excluded is the possibility that the magnetic field
should perform more than a complete rotation, this would destroy the
correspondence ψ = ψ′ ↔ τ = τ ′.
References
1. C. Thompson, R.C. Duncan, Astrophys.J. 408 (1993) 194
2. H. Hanami, Astrophys.J. 491 (1997) 687
3. C. Kouveliotou, R.C. Duncan, C. Thompson, Sci.Am. 288,2 (2003)
4. G. Calucci, Mod.Phys.Lett. A14 (1999) 1183;
A. DiPiazza, G.Calucci, Phys.Rev.D 65 (2002) 125019;
A. DiPiazza, G.Calucci, Phys.Rev.D 66 (2002) 123006;
A. DiPiazza, Eur. Phys. J.C 36 (2004) 25
5. T. Piran, Phys.Rep.314 (1999) 375;
J. van Paradijs, C. Kouveliotou, R.A.M.J. Wijers, Annu.Rev. As-
tron.Astrophys. 38 (2000) 279:
P. Mészáros, Annu.Rev.Astron.Astrophys. 40 (2002)137;
T. Piran, Rev.Mod.Phys. 76 (2004) 1143
R. Ruffini, Nuovo Cimento B 119 (2004) 785
6. R.D.Peccei, H.R.Quinn, Phys.Rev.Lett. 38 (1977) 1440;
R.D.Peccei, H.R.Quinn, Phys.Rev.D 16 (1977) 1791;
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7. L. Maiani, R.Petronzio, E.Zavattini Phys.Lett.B 175 (1986) 359
8. E. Zavattini, G. Zavattini, G. Ruoso, E. Polacco, E. Milotti, M.
Karuza, U. Gastaldi, G. DiDomenico, F. DellaValle, R. Cimino, S.
Carusotto, G.Cantatore, M. Bregant Phys.Rev.Lett. 96 (2006) 110406
S. Lamoreaux Nature 41 (2006) 31
9. The CERN Axion solar telescope (C.E.Asleth et al.) Nucl.Phys.B
(proc.suppl.) 110 (2002) 85
10. A.B. Migdal, V. Krainov Approximation methods in quantum mechan-
ics (Ch. 2) (Benjamin, NewYork 1969)
D.R.Bates ed. Quantum theory, vol 1 (Ch. 8)Academic press, London
11. A. DiPiazza, G.Calucci, Mod.Phys.Lett. A20 (2005), 117
12. M.Abramowitz and I.A.Stegun, Handbook of mathematical functions
(Dover, 1964).
13. S. Hannestad, A. Mirizzi,G. Raffelt, J. Cosm. Astrop. Phys. 07002
(2005)

  The investigation on the production of particles in slowly varying but
extremely intense magnetic field in extended to the case of axions. The
motivation is, as for some previously considered cases, the possibility that
such kind of magnetic field may exist around very compact astrophysical
objects.

arXiv:0704.0169v1  [hep-ph]  2 Apr 2007
VERY STRONG AND SLOWLY VARYING MAGNETIC FIELD
AS SOURCE OF AXIONS
Giorgio CALUCCI*
Dipartimento di Fisica Teorica dell’Università di Trieste, Trieste, I 34014 Italy
INFN, Sezione di Trieste, Italy
Abstract
The investigation on the production of particles in slowly varying but
extremely intense magnetic field in extended to the case of axions. The
motivation is, as for some previously considered cases, the possibility that
such kind of magnetic field may exist around very compact astrophysical
objects.
* E-mail: giorgio@ts.infn.it
http://arxiv.org/abs/0704.0169v1
1. Statement of the problem
A magnetic field of huge strength can give rise to real particles even if
its rate of variation is very small: this posibility could be of some interest
from a pure theoretical point of view, but it gains more physical relevance
if one accepts that such kind of field configurations may be present around
some very compact astrophysical objects[1-3]. In this case the time vari-
ation is related to the evolution of the source, by collapse, rotations or
else, and it is therefore very slow, in comparison with the times typical
of elementary-particle processes. We can call the former time the macro-
scopic time and the latter the microscopic one. The production of light
particles in these processes has been analyzed in some detail in some pre-
vious papers[4],with the suggestion that it is one of the mechanisms at
work in the phenomenon of gamma-ray bursts[5]; the typical microscopic
time is related to the electron mass since photons are produced through
real or virtual intermediate states of e− − e+-pairs. The lightest particles
that could be produced are massive neutrinos, but the magnetic-moment
coupling induced by the standard electroweak interactions is extremely
small.
There is, at least in the theoretical realm, another very light parti-
cle, that is the axion[6]: owing to its dynamical characteristics it must
be coupled also to the electromagnetic field[7], even more its electomag-
netic coupling is being actively studied from an experimental side[8] and
the possibility of detecting such particles as coming from nonterrestrial
sources has already been foreseen[9]. It is immediately seen that the pro-
duction of axions by a varying magnetic field must be realized through a
mechanism different from the previously considered one, in fact the axions
are coupled[7] only to the pseudoscalar density E ·B so the presence of
an electric field is necessary as a starting point, but a nonstatic magnetic
field creates always an electric field and even though the rate of variation
is small, the very large magnetic strength makes the electric field not a
tiny one.
In the present paper the coupling of the axions field with a given
E ·B density is written in standard second-quantized formalism, then the
effect of time variation of that density on the axion vacuum is determined
and the consequent production is calculated. The result depends both
on the spatial shape and on the time variation of the magnetic field: in
accordance with the prevailing astrophysical hypotheses[1-3] the magnetic
field is seen as a bundle of lines of force which may safely be considered
straight in comparison with the microscopic scale. The time-variation
could affect both the shape and the strength of the fields, both are effective
in the production process. The calculation procedure is not the standard
adiabatic approximation [10] as used in previous investigations [4], but the
feature that one has to deal with a two-scale problem is still fully relevant.
2. General form of the production probability
The starting point is a second-quantized axion field in presence of
given, classical, magnetic and electric fields. The axion field φ(x) is cou-
pled to the pseudoscalar density * G(x) = E(x) · B(x) and the coupling
constant, of dimension of length, is here indicated by C. We assume that
the interaction lasts from an initial time to until a final time t.
So we have for the axion field the expression:
φ(x) = φo(x) + χ(x) = φo(x) + C
∆R(x− y)G(y)d4y (1)
Here ∆R(x − y) is the standard retarded Green function, the source is a
c−number, so the same holds for χ. The field φ is free before to, where it
has the standard expansion:
φo(x) =
(2π)3/2
ao(k)e
ik·r−iωt + a†o(k)e
−ik·r+iωt
, (2)
then it acquires a contribution from χ, this term has the following actual
expression:
χ(x) =
(2π)3/2
e−iωkt
eiωkτgk(τ)dτe
ik·r − c.c.
(3.1)
gk(t) =
(2π)3/2
G(r, t)e−ik·r (3.2)
The reality condition for G, that gives g∗k = g−k have been used, together
with the initial condition χ̇(to) = 0. The separation into positive and
negative frequencies in χ(x) is unambiguous until the typical frequencies
of G are small.
Having found the time evolution of the field the total production of
axions is calculated in Heisenberg description of motion, i.e. we take an
initial state ao(k)|◦ >= 0 ∀k, i.e. the vacuum in the absence of inter-
action then we express the mean particle number as the time dependent
N (k, t) =< ◦|a†(k, t) a(k, t)|◦ >
* the same kind of coupling is possible also for the neutral pion, but in this case
other channels of production are present [11]
Since all the effect of the interaction is a c−number shift on the opera-
tors a(k, t) = ao(k) + b(k) the calculation is easy, in particular when the
interaction no longer acts we get
Nf (k) = |bf (k)|2 (4)
The expression of bf (k) can be read off from eq.s (2,3.1,3.2), it is:
bf (k) =
eiωkτgk(τ)dτ (5)
3. Detailed calculations
As anticipated in the introduction a more definite model of the mag-
netic field can by a field of uniform direction (at a given time) with some
transverse shape, both the direction and the shape may vary in time, one
possible restriction is the conservation of the total flux. More explicitly
these conditions are realized by giving:
n ∧ r
1− w(µr⊥)
B = − F
n ∂⊥w(µr⊥) . (6)
The unit vector n gives the instantaneous direction of the magnetic field,
r2 − n · r2 and ∂⊥ the corresponding derivative, F is the total
magnetic flux; the parameter µ defines the size of the field in the transverse
directions, w is taken to be cylindrically symmetric and, obviously it must
go to zero at infinity, the requirement w(0) = 1 avoids singularities in E.
Since E = −Ȧ and A ·B = 0 only the terms coming from the time
variation of the direction of A contributes to G = E ·B so we get:
)2J · r
(1− w)∂⊥w
Here J gives the angular velocity of n i.e. J = n ∧ ṅ. In this configuration
the Fourier transform of the source is
gk(t) =
(2π)3/2
C(2F )2δ(n · k)J · kS(k⊥/µ) (8.1)
S(k⊥/µ) = (µ/k⊥)
J1(ρk⊥/µ)
(1− w(ρ))w′(ρ)
dρ/ρ (8.2)
Here J1 is the Bessel function of order one and w
′ indicates the derivative
with respect to the argument. It is useful to remember that, owing to the
presence of the factor δ(n · k) in the expression of S we can substitute k2⊥
simply with k2.
We now remember that the model of magnetic field we have at hand
is such that it is uniform along one direction, but this direction is contin-
uously varying, so a most significant quantity is obtained by an angular
integration
dΩkNf (k) (9)
So we need a quantity like
dΩkgk(τ)g
′) which contains a singularity
due to the presence, in the domain of integration, of a δ-square term which
arises for τ = τ ′. This is clearly due to the unphysical assumption that
at every time there is a direction in which the pseudoscalar density G
is absolutely uniform. Since we are integrating over the direction at the
end the effect of the singularity is mild enough, it results in a logarithmic
divergence, however this fact must be explicitly death with, considering a
finite extension of the fields. A more careful treatment is mathemetically
heavier, so it is presented in an Appendix.
The final result is given by
≈ C2(2F )4 k
Ψ[S(k/µ)]2
2 ln 4kL . (10)
If we give a definite transverse shape to the fields we get, evidently,
a definite answer. In the situation where the fields change direction but
not intensities, i.e. we keep µ constant, we may give, tentatively the form
w(ρ) = e−ρ
. Then the expression for the function S(k/µ) is[12]:
S(k/µ) =
− exp
. (11)
The limit k → 0 of this expression is finite, so the whole production goes
to zero only owing to the phase-space factor k2 in front of the expression in
eq.(10). This result, as it appears from the whole derivation, can be valid
only for axion masses and so energies which are definitely larger than the
typical frequencies of the astrophysical phenomena, no resonant dynamics
is included.
4. Some conclusions
The rate of production of axions by a slowly varying but very strong
magnetic field has been calculated. The conditions are such that the as-
trophysical frequencies are very much lower than the proper frequencies of
the axion field. In fact with an axion mass mA of the order of 1eV[13] the
typical frequencies of the field shall exceed 1015 Hz and so no resonance
conditions appear realistic. The relevant parameter turns out to be the
dimension of the spatial inhomogeneity, in the present model 1/µ, here
also is seems very reasonable to assume that µ << mA. It is also possible
to give an expression for the total number of produced particles.
A = ζ C
2F 4µ3
Ψ lnLµ (12)
Some of the factors owe their origin to the general form of the interaction,
eq(1) and to dimensional requirament, it appers clear the role of the total
rotation of the field (Ψ) in determining the overall production; so when
the rotation is uniform the rate is proportional to the angular velocity.
The numerical factor is more model dependent, in the chosen case it
is ζ = 8
3− 2−
2] = 0.704 . . ..
The transformation from the incoming Heisenberg field to the outgo-
ing field can be implemented by the simple unitary operator
U = exp
d3k[a(k)b∗(k) − a†(k)b(k)]
Ua(k)U† = a(k) + b(k) .
The actual form of the evolution operator gives the further information
that the axion are produced with a Poissonian distribution of multiplicity,
strictly speaking this is true for a production in a totally defined state,
in operations like the one leading to eq. (9) this particular form can be
blurred.
A very short mention on this problem was made at the XXXVI In-
ternational Symposium on Multiparticle Dynamics, Paraty, R.J. - Brazil,
September 2006
Appendix
We want to calculate
dΩkgk(τ)g
′), with the functions g given by
eq.(8.1) and taking care of the finite extension of the magnetic field. This
is implemented by substituting the δ−functions as:
δ(n · k) → Lπ−1/2 exp[−L2(n · k)2] (A.1)
The calculation is performed in the particular case in which the rotation
takes place in a constant plane, so J‖J′. The integration over the angles is
performed in Cartesian coordinates, it is useful to introduce the unit vector
of the three-momentum direction k = kv, then
dΩk = 2δ(v
2 − 1)d3v
and through standard although lengthy calculations the representation is
obtained:
dΩkδ(n · k)J · kδ(n′ · k)J′ · k →
JJ ′×
dλ exp[iλ(w2 − 1)]w2dw
(n ∧ n′)2 − 2iλ/(Lk)2 − λ2/(Lk)4
]−1/2
(A.2)
In the limit L→ ∞ it results
I = JJ ′/|n ∧ n′|
which can be obtained in a simpler way.
Now we must integrate over τ and τ ′, times an oscillating factor
eiωk(τ−τ
′). In the conditions that have been chosen the motion takes
place in a plane, so n is characterized by a unique angle ψ and n′ by
ψ′ hence the integration over time amounts at an angular integration, in
fact (n ∧ n′)2 = (sin(ψ − ψ′))2 and moreover J = ψ̇ and J ′ = ψ̇′. So
we must integrate I in dψ , dψ′ from 0 to some final angle Ψf . Defin-
ing γ = ψ − ψ′ we see that the integrand shows, in the limit L → ∞,
a singularity for γ = 0, so we perform the integration from −1
π to 1
because the domain which do not include zero has no singular behavior,
the oscillating factor is approximated with its value on the singular point
τ = τ ′, so that the exponential factor reduces to 1. Then the integral is a
complete elliptic integral,[12] which can be conveniently expressed in term
of the hypergeometric function. In fact the integration in dγ is:
sin2 γ +Q
with Q = − 2iλ
(Lk)2
(Lk)4
(A.3)
The result of the integration is
and in the limit L→ ∞, which gives Q→ 0 we get;
2 ln 2 + lnLk
Since the dominant term in the limit is independent of the auxiliary pa-
rameter λ the rest of the integration in eq. (A.2) is straightforward and it
gives
I = Ψ
2 ln 4kL (A.4)
A comment: what is excluded is the possibility that the magnetic field
should perform more than a complete rotation, this would destroy the
correspondence ψ = ψ′ ↔ τ = τ ′.
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