fulltext

Conversion of photons into spinless particles in periodic external

electromagnetic field

HOANG NGOC LONG(1) and DANG VAN SOA(2)(*)

(1) Institute of Theoretical Physics - P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam (2) International Centre for Theoretical Physics - Trieste, Italy

(ricevuto il 9 Settembre 1996; approvato il 14 Novembre 1996)

Summary. — The conversion of photons into axions and dilatons in a periodic external electromagnetic field, namely in the TE10mode, is considered in detail. The differential cross-sections are given.

PACS 14.80.Mz – Axions and other Nambu-Goldstone bosons.

PACS 11.30.Er – Charge conjugation, parity, time reversal, and other discrete symmetries.

PACS 95.30.Cq – Elementary particle processes.

1. – Introduction

Axions induced in Peccei-Quinn symmetry [1] and dilatons, in the Kaluza-Klein theory [2], commonly have two-photon couplings. Therefore they can be created by a photon entering an external electromagnetic (EM) field. The axion properties and their phenomenological consequences have been studied in depth and some experiments trying to discover the axion are under way [3]. Axions might be constituents of the dark matter of the universe and this makes their experimental search even more fascinating.

Dilaton [4] could have arisen in the five-dimensional Kaluza-Klein theory. While the Kaluza-Klein approach has always been one of the most intriguing ideas concerning unification of gauge fields with general relativity, it has dwindled because generally one believes that the theory gives no distinctive and testable predictions.

Almost all experiments so far designed to search for light axions make use of the coupling of the axion to photons. The conversion of axions into cavity was firstly suggested by Sikivie [5]. Various terrestrial experiments to detect invisible axions by

(*) Permanent address: Department of Physics, Hanoi Technical University of Mining and Geology, Dong Ngac, Tu Liem, Hanoi, Vietnam.

making use of their coupling to photons have been proposed [3, 6], and the first results of such experiments appeared recently [7, 8]. Photoproductions of gravitons and dilatons in the external EM fields are considered in detail in ref. [9]. Authors showed that in the present technical scenario the differential cross-sections (DCS) of the creation of high-frequency gravitons and dilatons get observable values.

By applying the Feynman diagram techniques we have considered the conversion of photons into axions in the static EM fields [10]. Based on this result, the laboratory experiment has been proposed. To complete of the experiment, all the possible effects have to be looked for. The purpose of this paper is to consider the conversion of photons into axions and dilatons in a periodic EM field in the TE10 mode.

2. – The conversion of photons into axions in a periodic external EM field

The basic for the conversion of photons into axions is the coupling of the axion to two photons [5]: Lagg4 gg a 4 p fa fa F_mnFAmn_, (1)

where fa is the axion field, a is the fine-structure constant, FAmn4 ( 1 O 2 ) emnrsFrs, and fa is the axion decay constant defined in terms of the axion mass ma by [5, 6]:

fa4 fpmpkmumd[ma(mu1 md) ]21. The coupling constant ggin (1) is model dependent.

In the Dine-Fischler-Srednicki-Zhitnitskii (DFSZ) model [11] gg( DFSZ ) C0.36, and

in the Kim-Shifman-Vainshtein-Zakharov (KSVZ) model [12] gg( KSVZ ) C2 0.97.

Let us consider the conversion of the photon g with momentum q into an axion a with momentum p in the external electromagnetic field with the TE10 mode [13]:

.

`

/

`

´

Hz4 H0cos

g

px

a

h

exp [ikz 2ivt] , Hx4 2

ika

p H0sin

g

px

a

h

exp [ikz 2ivt] , Ey4 i

vam

p H0sin

g

px

a

h

exp [ikz 2ivt] .

(2)

Using the Feynman rules we get the matrix element (3) _{apNMNqb 4} 4 gag ( 2 p)2_kp 0q0

[(

_e3( q K , t) q12 e1( q K , t) q3

)

Fy1 e1( q K , t) q0Fx1 e3( q K , t) q0Fz

]

,

where p0f q01 v, em( q

K

, t) represents the polarization vector of the photon, and [14]:

.

`

`

`

/

`

`

`

´

Fx42 8 kaH0(qx2px) cos

k

1 2 a(qx2px)

l

sin

k

1 2b(qy2py)

l

sin

k

1 2 c(qz2pz1k)

l

p

y

(qx2 px)22 p2 a2

z

(qx2 px)(qy2 py)(qz2 pz1 k) , Fy4 2 mFx, Fz4 2 8 pH0cos

k

1 2a(qx2 px)

l

sin

k

1 2 b(qy2 py)

l

sin

k

1 2c(qz2 pz1 k)

l

a

y

(qx2 px)22 p2 a2

z

(qx2 px)(qy2 py)(qz2 pz1 k) . (4)

Substituting eq. (4) into eq. (3) we find the differential cross-section (DCS) for the conversion of photons into axions as

(5) ds e (g Ka ) dV 4 gag2p0 2( 2 p)2q0

y

(qx21 qz2) Fy21

g

1 2 qx2 q2

h

q0 2_F x21

g

1 2 qz2 q2

h

q0 2_F z22 22 q0qzFxFy2 2

g

q02qxqz q2

h

FxFz1 2 q0qxFyFz

l

.

where gagf gg(a Opfa) 4 (aO2p)(maO 1 eV ) 1027GeV21[15]. Here we use the following

notations: q f N qKN, p f

k

(q 1v)2

2 ma2.

In the following, we use the speed of light as unity and v b q. We also retain the terms to the first order in v in the result. From ( 5 ) it follows that when the momentum of the photon is parallel to the z-axis, DCS, in the context of the above statement, vanishes.

If the momentum of the photon is parallel to the x-axis, i.e. qm

4 (q , q , 0 , 0 ), then eq. (5) gets the final form

(6) ds(g Ka ) d V 8 4 8 gag2H02a2q2 p4

g

1 1 v q

h

y

v(q 2p cos u)2 p2 a2

z

2 3 3

y

cos a 2 (q 2p cos u) sin b

2(p sin u cos W 8) sin

c

2 (2p sin u sin W81k)

y

(q 2p cos u)2

2 p

2

a2

z

p sin u cos W 8(2p sin u sin W81 k)

z

2

where u is the angle between qK and pK_{, W 8 is the angle between the y-axis and the} projection of qK on the (y, z)-plane [10]. From (6) we have

(7) ds(g Ka) dV 8 4 g2 agH02V2q2

[

v

(

q 2

k

(q 1v)22 ma2

)

2 (p2O a2)

]

2 2 p4

_[(

q 2

k

(q 1v)2 2 ma2

)

22 (p2O a2)

]

2

g

1 1 v q

h

3 3cos2

k

a 2

(

q 2

k

(q 1v) 2 2 ma2

)

l

, for u B0, and (8) ds(g Ka ) dV 8 4 2 gag2H 2 0a 2 c2q2_{(vq 2p}2_{O a}2)2 p4 [ (q 1v)2 2 ma2](q22 p2O a2)2

g

1 1 v q

h

3 3cos2

g

a 2q

h

sin 2

g

b 2

k

(q 1v) 2 2 ma2

h

for u 4pO2, W840, and (9) ds(g Ka ) dV 8 4 2 g2 agH02a2b2q2(vq 2p2O a2)2( 1 1vOq) p4(q2 2 p2O a2)2_[

_k

(q 1v)2 2 ma22 v]2 cos2

g

a 2 q

h

3 3sin2

k

c 2

(

k

(q 1v) 2 2 ma22 v

)

l

for u 4pO2, W84pO2.

From (9) it follows that, if q 4p, then DCS is ds(g Ka) dV 8 4 gag2V2H02q2 2 p4

g

1 1 v q

h

. (10)

From (10) we see that DCS in the direction of the axion motion depends quadratically on the intensity H0, the volume V of the external EM field, the photon

momentum q, and then on ( 1 1vOq). Therefore, this direction is the best for the

conversion. In the limit (v Oq) b1 we have a result similar to that of a static field [10].

However, it does not yield the same results, because in this case both electric and magnetic components simultaneously give contributions.

From (8) and (9) it follows that, if q 4p2

O va2, then DCS vanishes. In the limit q K

p Oa eqs. (8) and (9) become, respectively,

(11) ds(g Ka ) dV 8 4 2 g2 agH02a2c2(v 2pOa)2 8 p2[ (p Oa1v)22 ma2]

g

1 1 va p

h

sin 2

u

b 2

o

g

p a 1 v

h

2 2 ma2

v

, and (12) ds(gKa ) dV 8 4 2 g2 agH02a2b2(v2pOa)2( 11vaOp) 8 p2_[

_k

(p Oa1v)2 2 ma22 v]2 sin2

u

b 2

o

g

p a1v

h

2 2ma22v

v

.

In the limit mabp , q, eqs. (8) and (9) become, respectively, ds(g Ka ) dV 8 4 2 g2 agH02a2c2(vq 2p2O a2)2 p4 ( 1 1vOq)(q2 2 p2O a2)2 cos 2

g

a 2 q

h

sin 2

g

b 2(q 1v)

h

, (13) and ds(g Ka ) dV 8 4 2 g2 agH02a2b2( 1 1vOq)(vq2p2O a2)2 p4_(q2 2 p2O a2)2 cos 2

g

a 2 q

h

sin 2

g

c 2q

h

. (14)

For the sake of convenience, however, we can introduce the ratio R between the two DCSs in eqs. (13) and (14) as R 4

g

1 1 v q

h

2 b 2_sin2

g

c 2q

h

c2_sin2

g

b 2 (q 1v)

h

. (15)

If b 4c we expect that axions in the direction u4pO2, W84pO2 are created more than axions in the direction u 4pO2, W840. In the limit vOqb1 the creation of axions in the two directions is the same.

In c.g.s. units, if ma4 1025eV and gg4 0 .36 (the DFSZ model), then (10), (13) and

(14) become, respectively, (16) ds(g Ka ) dV 8 4 2.9310 244V 2_H 02 l2

g

11 l lex

h

, (17) ds(g Ka ) dV 8 4 2.4 3 10 244 H 2 0a2c2 ( 1 1lOlex)

g

4 llex 2 1 a2

h

2 3 3

g

4 l2 2 1 a2

h

22 cos2

g

ap l

h

sin 2

g

bp

g

1 l 1 1 lex

hh

, and (18) ds(g Ka ) dV 8 4 2.4 3 10 244_H2 0a2b2

g

1 1 l lex

h

g

4 llex 2 1 a2

h

2 3 3

g

4 l2 2 1 a2

h

22 cos2

g

ap l

h

sin 2

g

bp l

h

.

To obtain the result in the KSVZ model we only need to note that g2

g( KSVZ ) C7.63

gg2( DFSZ ).

Recently, a photon regeneration experiment using RF photons was described in ref. [15]. That experiment consists of two cavities which are placed a small distance apart. A more or less homogeneous magnetic field exists in both cavities. The emitting cavity 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, the receiving cavity. However, the author considered the problem by using the classical method.

2. – The conversion of photons into dilatons in a periodic external EM field

Now, we move on to the conversion of photons into dilatons. From the Einstein-Hilbert (EH) action in the five-dimensional Kaluza-Klein theory, after some manipulation and performing the fifth coordinate integration, we get [16]

LEH4



k2g

y

R4 16 pG 1 1 2g mn_¯ ms¯ns 2 1 4FmnF mn 2 k_{3 k} 4 sFmnF mn 1 O(k2)

z

dx4. (19)

In the linear approximation, the Lagrangian LEH yields the flollowing

vertices [9]: Lsgg4 2 k_3k 4 hmahnbsF mn_Fab class, (20)

where k 4k_{16 pG, G is the Newton constant.}

Next, let us consider the conversion of the photon g with momentum q into the dilaton s with momentum p in the external electromagnetic field in the TE10 mode

(

eq. (2)

)

. Using the Feynman rules, we get the matrix element

(21) _{apNMNqb 4} 4 2k3k 4( 2 p)2_kp 0q0 el_{( q}K_{, s)[ (h} l2q32 hl3q2) Fx2 hl2q0Fy1 (hl1q22 hl2q1) Fz] .

Because photons and dilatons are massless particles so p 4q1v, and el_{( q}K_{, s)} represents the polarization vector of the photon. Substituting eq. (4) into eq. (21) we find finally the DCS for the conversion of photons into dilatons as

(22) ds(g Ks) dV 4 3 k2_p 0 32( 2 p)2_q 0 [ (qz21 qy2) Fx22 2 q0qzFxFy2 2 qxqzFxFz1 1 (q022 qy2) Fy21 2 q0qxFyFz1 (qx21 qy2) Fz2] . From (22) it follows that when the momentum of the photon is parallel to the z-axis, the DCS vanishes. It implies that, if the momentum of the photon is parallel to the

external field propagation, then there is no conversion of photons into spinless particles.

If the momentum of the photon is parallel to the y-axis, i.e. qm

eq. (22) gets the final form as (23) ds(g Ks) dV9 4 4 3 k2_H 02q2

g

1 1 v q

h

2 a2

y

cosa

2(p sin u sin W 9) sin

b

2(q 2p cos u)

g

p2sin2_{u sinW 92} p

2

a2

h

(q 2p cos u)(2p sin u cos W91k)

z

2

3

3sin2 c

2(2p sin u cos W91k) , where W 9 is the angle between the y-axis and the projection of qK on the (x, z)-plane [14]. From (23) we have

ds(g Ks) dV 9 4 3 k2H02V2q2 32 p4

g

1 1 v q

h

, (24) for u B0, and ds(g Ks) dV 9 4 3 k2H02a2 2 p2_q2

g

1 1 v q

h

sin 2

g

b 2q

h

cos 2

g

c 2q

h

, (25)

for u 4pO2, W940, and

ds(g Ks) dV 9 4 3 k2_H2 0c2

g

1 1 v q

h

8 a2

y

(q 1v)2 2 p 2 a2

z

2cos 2

g

a 2(q 1v)

h

sin 2

g

b 2q

h

, (26)

for u 4pO2, W94pO2.

From eq. (24) we see that this direction is the best for the conversion of photons

into dilatons in the external EM field, and our conclusion is similar to that of

eq. (10).

In c.g.s. units, eqs. (24), (25), and (26) become, respectively, ds(g Ks) dV 9 4 1.6 Q 10 249 V 2 H02 l2

g

1 1 l lex

h

, (27) ds(g Ks) dV 9 4 1.6 Q 10 251_H2 0a2l2

g

1 1 l lex

h

sin2

g

bp l

h

sin 2

g

cp l

h

(28)

and ds(g Ks) dV 9 4 9.1Q10 251 H02c 2

g

11 l lex

h

a2

y

g

1 l1 1 lex

h

2 p 2 4a2

z

2cos 2

g

ap

g

1 l1 1 lex

hh

sin2

g

bp l

h

. (29)

It is easy to show that the cross-section for the reverse process coincides exactly with the above results, so that for the conversion photon-dilaton (or axion)-photon, the cross-section is the square of the previous evaluations.

Note that the gravitational interference effect also could have arisen in the Kaluza-Klein theory in which the conversion of gravitons into photons in the periodic external EM field was considered in ref. [14].

3. – Discussion

Equation shows that in order to get s B10230_cm2_{we need H}

0B 3.4 Q 106lV21, and

from eq. (25) we need H0B 9.1Q109lV21. This means that axions are expected to

produce more in our experiment.

It is known that the cut-off frequency of the TE10is given by v04 p O a and at any

given frequency only a finite number of modes can propagate [13]. It is often convenient to choose the dimensions of the guide such that in the operating frequency only the lowest mode can occur. This is important in order to use these methods for

experiments.

Finally, we see that quantum conversions of photons into spinless particles in an

external EM field exist strongly at high energies.

* * *

One of the authors (DVS) would like to thank UNESCO and the International Atomic Energy Agency for encouragement and hospitality at the International Centre for Theoretical Physics, Trieste, Italy. He would also like to thank Prof. J. TRANTHANHVAN and Rencontres du Vietnam for help and partial support. He thanks P. TAKEOINAMIfor continuous help.

R E F E R E N C E S

[1] PECCEIR. D. and QUINNH., Phys. Rev. Lett., 38 (1977) 1440; Phys. Rev. D, 16 (1977) 1792; WEINBERG S., Phys. Rev. Lett., 40 (1978) 223; WILCZEK F., Phys. Rev. Lett., 40 (1978) 279.

[2] KALUZAT., Sitzungber. Preuss. Acad. Wiss. Berlin Math. Phys. k1, 96; KLEINO., Z. Phys., 37 (1926) 895.

[3] COOPER L. and STEDMAN G. E., Phys. Lett. B, 357 (1995) 464.

[4] DUFF M. J., in General Relativity and Gravitation, edited by M. A. H. MACCALLUM (Cambridge University Press) 1987, p. 18; GERMAN G., Class. Quantum Grav., 2 (1985) 455.

[6] VANBIBBERK., DAGDEVIRENN. R., KOONINS. E., KERMANA. K. and NELSONH. N., Phys.

Rev. Lett., 59 (1987) 759; MINOWAM., INOUE Y., ASANUMAT. and IMAMURAM., Phys. Rev.

Lett., 71 (1993) 4120.

[7] ANSELMA. A., Yad. Fiz., 42 (1985) 1480(Sov. J. Nucl. Phys., 42 (1985) 936); DEPANFILISS., MELISSINOS A. C., MOSKOWITZB. E., ROGERS J. T., SEMERTZIDIS Y. K., WUENSCHW. U., HALAMAH. J., PRODELLA. G., FOWLERW. B. and NEZRICKF. A., Phys. Rev. Lett., 59 (1987) 839; SIKIVIEP., TANNERD. B. and WANGY., Phys. Rev. D, 50 (1994) 4744; HOOGEVEENF. and ZIEGENHAGEN T., Nucl. Phys. B, 358 (1991) 3.

[8] MASSO E. and TOLDRA R., Phys. Rev. D, 52 (1995) 1755.

[9] LONGH. N. and HUONGL. K., Mod. Phys. Lett. A, 6 (1991) 2315. [10] LONGH. N., SOA D. V. and TRAN T. A., Phys. Lett. B, 357 (1995) 469.

[11] DINEM., FISCHLERW. and SREDNICKIM., Phys. Lett. B, 104 (1981) 199; ZHITNITSKIIA. P.,

Yad. Fiz., 31 (1980) 497 (Sov. J. Nucl. Phys., 31 (1980) 260).

[12] KIMJ., Phys. Rev. Lett., 40 (1977) 223; SHIFMANM. A., VAINSHTEINA. I. and ZAKHAROVV. I.,

Nucl. Phys. B, 166 (1980) 493.

[13] JACKSON J. D., Classical Electrodynamics (Wiley, New York, N.Y.) 1975, section 8.4. [14] LONGH. N., SOA D. V. and TRAN T. A., Phys. Lett. A, 186 (1994) 382.

[15] HOOGEVEEN F., Phys. Lett. B, 288 (1992) 195.