Photon-to-axion conversions in the static electromagnetic fields are
reconsidered in detail by using the Feynman diagram techniques. The
differential cross sections are presented for the conversions in the presence
of the electric field of the flat condenser as well as in the magnetic field of
the solenoid. Based on our results a laboratory experiment for the production
and the detection of the axions is described. This experiment will exploit the
axion decay constant as well as the axion mass.Comment: 7 pages, latex, no figures, to appear in Phys. Lett. 

Anselm

Cheng

Dang Van Soa

DePanfilis

Dine

Hagman

Hoang Ngoc Long

Hoogeveen

Kolb

Long

Minowa

Peccei

Raffelt

Shifman

Sikivie

Tuan A. Tran

Turner

Van Bibber

Weinberg

Wilczek

Zhitnitskii

Physics Letters B

English

arXiv

Crossref

Photon-to-axion conversions in the static electromagnetic fields are reconsidered in detail by using the Feynman diagram techniques. The differential cross sections are presented for the conversions in the presence of the electric field of the flat condenser as well as in the magnetic field of the solenoid. Based on our results a laboratory experiment for the production and the detection of the axions is described. This experiment will exploit the axion decay constant as well as the axion mass

Long, H N

Van Soa, D

Tran, T A

CERN Document Server

ELECTROMAGNETIC DETECTION OF AXIONS
Hoang Ngoc Long1
Institute of Theoretical Physics, National Centre for Natural Science and Technology,
P.O.Box 429, Bo Ho, Hanoi 10000, Vietnam
Dang Van Soa
Department of Physics, Hanoi University of Mining and Geology,
Dong Ngac, Tu Liem, Hanoi, Vietnam.
and
Tuan A. Tran2
International Centre for Theoretical Physics, Trieste, Italy
Abstract
Photon-to-axion conversions in the static electromagnetic elds are reconsidered in
detail by using the Feynman diagram techniques. The dierential cross sections are
presented for the conversions in the presence of the electric eld of the flat condenser as
well as in the magnetic eld of the solenoid. Based on our results a laboratory experi-
ment for the production and the detection of the axions is described. This experiment
will exploit the axion decay constant as well as the axion mass.
PACS numbers : 14.80.Gt, 11.30.Er, 95.30.Cq
1Electronic address:hnlong@bohr.ac.vn
2Electronic address: tatran@ictp.trieste.it
1
In the 1970’s, it was shown that the strong CP problem can be solved [1] by the
introduction of a light pseudoscalar particle, called the axion [2]. However, no positive
indication of its existence has been obtained so far. At present, the parameters of
axions are constrained by laboratory searches [3] and by astrophysical and cosmological
considerations [4].
A particle, if it has a two-photon vertex, may be created by a photon entering
an external electromagnetic (EM) eld. An axion is one of such particles. Conversion
of the axions into EM power in a resonant cavity was rstly suggested by Sikivie [5].
He suggested that this method can be used to detect the hypothetical galactic axion
flux that would exist if axions were the dark matter of the Universe. Various terrestrial
experiments to detect invisible axions by making use of their coupling to photons have
been proposed [6, 7, 8], and the rst result of such experiments appeared recently [9].
Recently, a photon regeneration experiment, using RF photons, was described [10].
That experiment consists of two cavities which are placed a small distance apart. A
more or less homogeneous magnetic eld exists in both cavities. The rst, or emit-
ting is excited by incoming RF radiation. Depending on the axion-photon coupling
constant, a certain amount of RF energy will be deposited in the second, or receiving
cavity. In Ref. [10] the author considered the problem by using the classical method.
By applying the Feynman diagram techniques we have considered the conversion of
the photons into gravitons in the static [11] and periodic [12] EM elds. In this paper
we also apply this method to reconsider the EM conversion of the axions in both the
electric and magnetic elds.
The axion mass and its couplings to ordinary particles are all inversely propor-
tional to the magnitude v of the vacuum expectation value that spontaneously breaks
the UPQ(1) quasisymmetry which was postulated by Peccei and Quinn and of which the
axion is the pseudo-Nambu-Goldstone boson. For the axion-photon system a suitable
Lagrangian density is given by [4, 5]:
L = −
1
4
FF
 + gγ

4
a
fa
F ~F
 +
1
2
@a@
a −
1
2
m2a
2
a[1 + 0(
2
a=v
2)] (1)
where a is the axion eld,ma its mass, ~F =
1
2
"F
, and fa is the axion decay con-
stant and is dened in terms of the axion massma by [5, 9]: fa = fm
p
mumd[ma(mu+
md)]−1. The coupling constant in (1) is model dependent. Interaction of the axions to
the photons arises from the triangle loop diagram, in which two vertices are interac-
tions of the photon to electrically charged fermion and an another vertex is coupling
of the axion with fermion. This coupling is model dependent and is given:
gγ =
1
2

Ne
N
−
5
3
−
md −mu
md +mu

;
where N = Tr(QPQQ2color) and Ne = Tr(QPQQ
2
em). Tr represents the sum over all
left-handed Weyl fermions. QPQ, Qem, and Qcolor are respectively the Peccei-Quinn
2
charge, the electric charge, and one of the generators of SU(3)c. In this paper we
are considering very light, and very weak interacting invisible axions [13], however
our calculation is also valid for the heavy axions case. In this model - the Dine-
Fischler-Srednicki-Zhitnitskii (DFSZ) model, one has Ne =
8
3
N hence the coupling
constant gγ(DFSZ) ’ 0:36. In other model as given by the Kim-Shifman-Vainshtein-
Zakharov [14], where the axions do not couple to light quarks and leptons (hadronic
axion), one has Ne = 0 and hence gγ(KSV Z) ’ −0:97.
Consider the conversion of the photon γ with momentum q into the axion a with
momentum p in an external electromagnetic eld.
For the abovementioned process, the relevant coupling is the second term in (1).
Using the Feynman rules we get the following expression for the matrix element
hpjM jqi = −
gaγ
2(2)2
p
q0p0
"(~q; )"
q
Z
V
ei
~k~rF class d~r (2)
where ~k  ~q − ~p the momentum transfer to the EM eld, gaγ  gγ

fa
= gγma(mu +
md)(fm
p
mumd)−1 and "(~q; ) represents the polarization vector of the photon.
Expression (2) is valid for an arbitrary external EM eld. In the following we
shall use it for two cases, namely conversion in the electric eld of a flat condenser
and in the static magnetic eld of the solenoid. Here we use the following notations:
q  j~qj; p  j~pj = (q2o −m
2
a)
1=2 and  is the angle between ~p and ~q.
Conversion in the presence of an electric eld.{ Now we take the EM eld as a
homogeneous electric eld of a flat condenser of size abc. We shall use the coordinate
system with the x axis parallel to the direction of the eld, i.e., F 10 = −F 01 = E: Then
the matrix element is given by
hpjMejqi =
gaγ
(2)2
p
q0p0
"(~q; )"
01qFe(~k); (3)
where a form factor for the electric region [5, 6]
Fe(~k) =
Z
V
ei
~k~rE(~r)d~r:
The superscript e in Me refers to the process taking place in the presence of an electric
eld.
For a homogeneous eld of intensity E we have [11]
Fe(~k) = 8E sin(
1
2
akx) sin(
1
2
bky) sin(
1
2
ckz)(kxkykz)
−1: (4)
Substituting (4) into (3) we nd nally the dierential cross section (DCS) of
the conversion of the axions in the electric eld of a flat condenser of size a b c
de(γ ! a)
dΩ
=
g2aγE
2
2(2)2
"
sin(1
2
akx) sin(
1
2
bky) sin(
1
2
ckz)
kxkykz
#2
(q2y + q
2
z): (5)
3
From (5) we see that if the photon moves in the direction of the electric eld i.e.,
q = (q; q; 0; 0) then DCS vanishes. If the momentum of photon is parallel to the y
axis, i.e., q =( q, 0, q, 0 ) then Eq. (5) becomes:
de(γ ! a)
dΩ"
=
32g2aγE
2q2
(2)2
"
sin
 
ap sin  sin’"
2
!
sin
 
b
2
(q − p cos )
!
 sin
 
cp sin  cos’"
2
!#2
(p2 sin2  sin’" cos’"(q − p cos ))−2: (6)
where ’" is the angle between the z axis and the projection of ~p on the xz plane [12].
From (6) we have
de(γ ! a)
dΩ"
=
2g2aγE
2a2c2
(2)2q2

1−
r
1− m
2
a
q2
2 sin2
"
qb
2
 
1−
s
1−
m2a
q2
!#
(7)
for   0 and
de(γ ! a)
dΩ"
=
8g2aγa
2E2
(2)2(q2 −m2a)
sin2
 
bq
2
!
sin2
 
cq
2
s
1−
m2a
q2
!
(8)
for  = 
2
; ’" = 0.
In the limit m2a ! 0, Eqs. (7) and (8) become, respectively,
de(γ ! a)
dΩ"
=
g2aγq
2V 2E2
2(2)2
+O(m4a) (9)
and
de(γ ! a)
dΩ"
=
8g2aγa
2E2
(2)2q2
sin2
 
bq
2
!
sin2

cq
2

+O(m4a): (10)
From (9) we see that DCS in the direction of the axion motion depends quadratically
on the intensity E, the volume V of condenser, and the photon momentum q.
For V = 1m  1m  1m, the intensity of the electric eld E = 100kV
m
, the
photon length  = 10−5 cm, and ma ’ 10−5eV [9] then in the DFSZ model the cross
section given by (9) is d(γ!a)
dΩ
’ 8 10−22cm2, while by (10) d(γ!a)
dΩ
’ 8 10−51cm2.
We see that the axion is mainly created in the direction of photon motion. This
coincides with the one dimensional solution which is basics for the experimental setups
in Ref. [6]. To obtain the results in the KSVZ model we only need to note that
g2γ(KSV Z) ’ 7:26 g
2
γ(DFSZ) .
For the case in which q2 ! m2a, Eqs. (7) and (8) become
de(γ ! a)
dΩ
=
2g2aγa
2c2E2
(2)2
sin2
 
bq
2
!
: (11)
4
For a = c = 1m, the intensity of the electric eld E = 100kV
m
, and ma = 1eV [9] then
the cross section is given by (11): d(γ!a)
dΩ
’ 8:1 10−26cm2
Conversion in the presence of a magnetic eld.{ Now we consider the conversion
in a homogeneous magnetic eld of the solenoid with a radius R and a length h, and
without loss of generality suppose that direction of the magnetic eld is parallel to the
z axis, i.e., F 12 = −F 21 = B: After some manipulations we get
dm(γ ! a)
dΩ
=
g2aγF
2
m(~q − ~p)
2(2)2
q2
 
1−
q2z
q2
!
(12)
where Fm is a form factor for the magnetic region [5, 11]:
Fm(~k) =
4BR
kz
q
k2x + k
2
y
J1(R
q
k2x + k
2
y) sin
 
hkz
2
!
: (13)
where J1 is the one-order spherical Bessel function (for a homogeneous magnetic eld
of magnitude B with a size a b c one has formula (4) with replacement E by B).
From (12) it follows that when the momentum of the photon is parallel to the
z axis (the direction of the magnetic eld), DCS vanishes. It implies that if the mo-
mentum of the photon is parallel to the EM eld then there is no conversion. If the
momentum of the photon is parallel to the x axis, i.e., q = (q; q; 0; 0) then Eq. (12)
gets the nal form:
dm(γ ! a)
dΩ0
= 2g2aγR
2B2J21
0B@Rq
vuut 1− cos 
s
1−
m2a
q2
!2
+
 
1−
m2a
q2
!
sin2  cos2 ’0
1CA

24 1− cos 
s
1−
m2a
q2
!2
+
 
1−
m2a
q2
!
sin2  cos2 ’0
35−1 q−2
 sin2
 
hq
2
s
1−
m2a
q2
sin  sin’0
!"
(1−
m2a
q2
) sin2  sin2 ’02
#−1
(14)
where ’0 is the angle between the y axis and the projection of ~p on the yz plane [11].
It is easy to see that
dm(γ ! a)
dΩ0
=
1
2
g2aγR
2h2B2J21
"
Rq
 
1−
s
1−
m2a
q2
!# 
1−
s
1−
m2a
q2
!−2
(15)
for   0 and
dm(γ ! a)
dΩ0
=
1
2
g2aγR
2h2B2J21
 
Rq
s
2−
m2a
q2
! 
2−
m2a
q2
!−1
(16)
5
for  = 
2
; ’0 = 0.
For the limit m2a  q
2 with the notice that
lim
p!q
J1 (R(q − p))
q − p
=
R
2
then Eqs. (15) and (16) become, respectively,
dm(γ ! a)
dΩ0
=
g2aγV
2B2q2
2(2)2
+O(m4a); V  R
2h (17)
and
dm(γ ! a)
dΩ0
=
1
4
g2aγR
2h2B2J21
p
2Rq

+O(m4a): (18)
From (17) we see that DCS in the direction of photon motion depends quadratically
on the magnitude B, the cavity volume V and the photon momentum q.
From (18) it follows that DCS vanishes when pn =
n
R
p
2
with n = 0;1 2...and
has its largest value
dm(γ ! a)
dΩ0
=
1
4
g2aγR
2h2B2J21 (
0
n) (19)
for pn =
0n
R
p
2
, where n and 0n are the roots of J1(n) = 0 and J
0
1(
0
n) = 0.
For the magnetude [9] B = 8T,R = h = 1m,  = 10−5cm and ma ’ 10−5eV by
(17) we have d(γ!a)
dΩ
= 4:7 10−11cm2.
For the limit q2 ! m2a, Eqs. (15) and (16) become:
dm(γ ! a)
dΩ0
=
1
2
g2aγR
2h2B2J21 (Rq) : (20)
For B = 8 T, ma = 1eV and R = h = 1m, we have
d(γ!a)
dΩ
= 1:7 10−16cm2.
In the case of the magnetic eld of size a b c we get a formula similar to (17)
under the same conditions.
It is easy to show that the cross section for the reverse process coincides exactly
with above results, so that for conversion photon-axion-photon, cross section is square
of the previous evaluation.
Based on the above we describe an experiment: The initial photon of energy qo
from the laser (maybe better from X ray) interacts with a virtual photon from the EM
eld to produce the axion of energy qo and momentum p = (q2o −m
2
a)
1=2. The photon
beam is then blocked to eliminate everything except the axions, which penetrate the
wall because of their extremely weak interaction with ordinary matter. (Such shielding
is straightforward for a low-energy laser beam.) The axion then interacts with another
virtual photon in the second EM eld to produce a real photon of energy qo, whose
detection is the signal for the production of the axion. For the details of experimental
setup the reader can see Refs. [6, 15].
6
We note again that here EM eld is understood by not only magnetic eld but
also the electric eld of the condenser. To dier the axion signal from noise the case
in which the EM eld is switched o has to be measured. From the results we can
get limits for fa as well as on the axion mass. The axion-photon coupling constant
in [10] is taken g = 10−8 GeV−1, this coupling constant corresponds to the axion mass
ma  0:1 eV.
We emphasize that the cross section depends quadratically on the momentum of
incoming photons hence high frequencies are preferred despite the technical diculties
discussed in [10].
In conclusion, the axion hypothesis can be tested experimentally. Relatively
simple experiments can provide new information about physics at very high energies.
If the axion exists, we will have new powerful tools to study the galaxy and the sun.
We thank Professor J. Tran Thanh Van for help and partial support. T. A. T.
would like to thank International Centre for Theoretical Physics for nancial support.
D.V.S thanks Prof. N. S. Han for encouragement.
References
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1792 (1977).
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7
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8


Electromagnetic detection of axions

Electromagnetic detection of axions

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