fulltext

D. BERNARD

LPNHE, Ecole Polytechnique, IN2P3 & CNRS - 91128 Palaiseau, France

(ricevuto il 9 Settembre 1997; approvato il 17 Ottobre 1997)

Summary. — An experimental program is in progress, searching for possible non-QED signals in elastic photon-photon scattering in the optical domain [1]. We examine here the potential of this process for the detection of a possible light pseudoscalar such as the axion.

PACS 14.80.Mz – Axions and other Nambu-Goldstone bosons (Majorons, familons, etc.). PACS 12.20 – Quantum electrodynamics.

1. – Introduction

The standard model currently provides a consistent description of elementary particles and of their interactions. Yet, it contains spontaneously broken global symmetries which imply the existence of Goldstone bosons. In the case of the U( 1 ) Peccei-Quinn symmetry introduced to explain the absence of CP violation in QCD [2], this particle is called the axion. These particles have escaped detection up to now.

Originally, the mass scale of the UPQ( 1 ) symmetry breaking was thought to be the

same as the one for weak interactions, which implies an axion mass maB 100 keV. This type of axion has been ruled out by particle decay and beam dump experiments. For this reason, axions with smaller mass and weaker coupling to matter, called invisible axions, have been considered. Various types of axions have been considered. The KSVZ axions [3] are hadronic axions with only induced coupling to leptons. The DFSZ axions [4] naturally couple to all fermions. They couple to two photons via the anomaly of the triangular diagram (fig. 1).

In these models, the mass of the axion maand its coupling constant to two photons g are proportional to each other, with

ma4 g 3 C , (1)

where [5] CKSVZ4 2 .7 Q 1018 eV2and CDFSZ4 7 .4 Q 1018eV2.

Pseudoscalar or scalar particles that couple to two photons but not to fermions have also been considered in ref. [6]. In that case, ma and g are two independent parameters.

Fig. 1. – Axion coupling to two photons.

The actual window for the invisible axion is mainly limited by astrophysical considerations (see the reviews in [7]). Axions can be produced in the core of stars via the Primakoff process. Due to their low interaction with matter, these particles escape almost freely. The star contracts and increases its central temperature to compensate for this additional energy loss. The lifetime of the star is then reduced, which also reduces the number of observed stars in the given part of the sequence. In the case of the KSVZ axion, a limit of the coupling constant is obtained from the helium burning stars g E3Q10210 _GeV21_{, corresponding to m}

aE 2 eV [8]. A more stringent mass upper limit is obtained from the observation of the neutrino pulse from SN1987A. Axions with masses in the range 1023 _{eV Em}

aE 2 eV would have accelerated the cooling of the core and shortened the neutrino burst [9]. A lower bound of the mass is obtained from cosmology, as axions with a mass lower than 1026_{eV would close the universe [10].}

The window for the invisible axion is therefore 1026_–1023_{eV (a recent discussion is} given in [11]). In the lower part of the allowed mass range, axions may constitute a part of the dark matter in the universe.

A less stringent limit, but free from any astrophysical model, of g E3.6Q1027 _GeV21 (for maE 1023 eV) has been obtained recently by the study of the propagation of a laser beam through a transverse magnetic field [12]. The experiment searched for the birefringence induced by the axion coupling to two photons. This limit is expected to improve by a factor 40 with the PVLAS experiment which is in preparation [13].

In this note, we examine the potential of axion discovery in the elastic scattering of photons. An experiment searching for possible non-QED signals in photon-photon elastic scattering in the optical domain has been performed recently by colliding two high-power pulsed lasers beams [1]. The obtained upper limit of the cross-section is 10239 _cm2_{and is expected to improve by 20 orders of magnitude in an experiment which} is in preparation, with the use of high repetition rate lasers, and by stimulating the reaction with a third beam [1].

2 – Scattering cross-section

The interaction of the axion with two photons is described [14] by the interaction Lagrangian LI: LI4 g 4FF mn_FA mn. (2)

Here F is the axion field, FAmn4 ( 1 O2 ) emnlsFls is the dual of the electromagnetic

Fig. 2. – Photon-photon elastic scattering through the exchange of an axion. a) t channel, b) s channel.

smallness of the coupling constant, yields a very small coupling of the axion to two photons. From the antisymmetry of emnls, one obtains

LI4 g 2F¯ n_Am_e mnls¯sAl. (3)

The decay rate of an axion in two photons is obtained from the expression of LI[7]: Ggg4 g2ma3/64 p.

We now turn to gg elastic scattering. The differential cross-section in the center-of-mass system (c.m.s.) reads

ds dV 4 N M N2 64 p2 s , (4)

where the matrix element M is the sum of the terms obtained from the two diagrams in fig. 2: M 4 Ma1 Mb, and s is the c.m.s. energy. Ma (t channel) and Mb (s

channel) are obtained from the expression of LI:

Ma4 2ig 2 4 (emnlsk 81nk1se 81mel1)(emnlsk2nk 82sem2e 82l) (k12 k 81)22 ma2 1 (5) 1 (k1Dk2) 1 (k 81Dk 82) 1 (k1Dk2 and k 81Dk 82) , Mb4 2ig2 4 (emnlsk 81nk 82se 81me 82l)(emnlsk1nk2sem1el2) (k11 k2)22 ma2 1 (6) 1 (k1Dk2) 1 (k 81Dk 82) 1 (k1Dk2 and k 81Dk 82) . Here ki, i 41, 2 (k 8i, i 41, 2) denote the four-momenta of the incoming (outgoing)

photons, and ei, i 41, 2, and e8i, i 41, 2, denote the corresponding polarisation

vectors. The four terms in the expression of Mbare equal to each other. The four terms

in the expression of Ma are equal two by two. We compute the 16 matrix elements

Fig. 3. – Angular variation of the polarized differential cross-sections, in units of g4_k2_{/1024 p}2

(high energy, left) and in units of g4_k6_{/64 p}2_m4

a(low energy, right), as a function of the scattering angle. Polarisations 1122, 1212, 1221 are noted respectively by dashed, dotted, and dashed-dotted line. A solid line indicates the unpolarized differential cross-section.

linear polarisation perpendicular to that plane. We obtain M11224 M22114 ig2k2 2 ( c4 W/2 c2 W/21 ma2/4 k2 1 s4 W/2 s2 W/21 ma2/4 k2 ) , (7) M12124 M21214 2 ig2_k2 2 ( 2 21 1 ma2/4 k2 1 s 4 W/2 sW/22 1 ma2/4 k2 ) , (8) M12214 M21124 2 ig2_k2 2 ( 2 21 1 ma2/4 k2 1 c 4 W/2 cW/22 1 ma2/4 k2 ) , (9)

where W is the scattering angle and cW/24 cos (W/2 ), sW/24 sin (W/2 ). The 10 other terms

are equal to zero. We see that M1212 and M1221 present the usual divergence in the s channel at s 4m2

a.

The polarised differential cross-sections in the high energy (k c ma) and in the low-energy limit, are plotted in fig. 3.

At high energy, the asymptotic behaviour of the polarised total cross-sections is s_{1122 , Q4} g 4_k2 256 p , s1212 , Q4 7 3 g4_k2 256 p , s1221 , Q4 7 3 g4_k2 256 p and for the unpolarised total cross-section: s_{Q4 ( 17 O3 )(g}4

k2_{O256 p). At low energy} we get: s1122 , 04 7 15 g4k6 16 pm4 a , s1212 , 04 83 15 g4k6 16 pm4 a , s1221 , 04 83 15 g4k6 16 pm4 a

sQED4 10125 pa 2 re2

g

h

Here, a is the fine-structure constant and reis the classical radius of the electron. We then get sQED( mb ) 47.3Q10239

(

)

3.1. Low energy limit. – In the low-energy limit, (k b ma) the two cross-sections

grow like k6. The dimensioned axion cross-section reads: s04 173 240 p k6 C4(ˇc) 2 , (11)

that is s0( mb ) 41.7Q10257

(

)

₍

₎

The axion contribution would dominate the cross-section only for C D

k

16 3973

l

1 /4 _(mc2₎2

a ,

(12)

that is C D5.9Q1013_eV2_{, five orders of magnitude below the expected value for the} axion(s).

Fig. 4. – The QED (solid line) and the non-resonant axion cross-section (high-energy: dotted; low energy: dashed), as a function of the c.m.s. photon energy k, for various values of the axion mass.

incompatible with the hypothesis of the high-energy limit.

Therefore the contribution of the axion never dominates (fig. 4). The soft dependence of the axion cross-section and of the QED cross-section [16] on polarisation and on angle does not provide any help.

4. – Upper limits on the coupling constant

In the more general case of a pseudoscalar with independent values of maand g, the upper limit on g is limited by the QED background and is given in the high-energy limit by eq. (13), that is g D ak (mc2₎2

k

l

i.e. g ( GeV21_{) D4Q10}25_{k( eV ), only competitive with respect to the limit obtained by} Cameron et al. [12] for maD 7 Q 1023 eV. The upper limit of the elastic cross-section ob-tained in ref. [1] is 10212_{mb. If improved by twenty orders of magnitude as}

limits and complete references can be found therein.

The value of g that can be attained in a non-resonant photon-photon scattering experiment at k 41 eV, and at the limit of QED background is denoted by NR(QED). The limit that could be obtained in the experiment using a stimulating beam is denoted by NR(18).

The non-resonant photon-photon scattering is clearly of little interest in the search for a light pseudoscalar.

5 – Behaviour close to the pole

Close to the pole, 2 k Bma, the cross-section is described by a Breit-Wigner formula: sBW4 p 4 k2 G2 ( 2 k 2ma)21 G2/4 . (16)

Its maximum value sBW , 04 p/k2, is very large, close to 1018mb—at the unitary limit—but the width is tiny; G 4m5

a/( 64 pC2) i.e. G( eV ) 4 (0.9–6.8 )Q10240

(

)

One limitation of this method is due to the finite mass range that can be tested, that is of the spectrum of the cms energy in the collision. In the experiment described in ref. [1], the spectral width of the beams were about 1 Å, at a cms energy of 3.3 eV, corresponding to a cms relative spectral width Dk/k of only 0 .7 Q 1023 _{at FWHM.}

The useful fraction f of the luminosity is of the order of

f 4 G

ma k Dk , (17)

so that the limit on the coupling constant is glim4

y

z

The corresponding value for ref. [1] is glim4 1 .7 Q 1027 GeV21, but the tested axion mass range is very narrow, 3 .33 60.001 eV.

In the experiment with a stimulating beam which is in preparation, short (35 fs) pulses are used with 40 nm spectral width at FWHM, so that the relative spectral width Dk/k in the c.m.s. is close to 3.6%. We would then obtain a limit of glim4 1 .2 Q 10216GeV21 at ma4 1 .641 6 0 .025 eV. An increased energy range could be tested with given lasers by varying the crossing angle. The exploration of a larger range would require the use of different laser wavelengths (e.g. by frequency multiplying).

Actually the resonant state lives for a time of the order of t 4ˇ/(FG), where F is the stimulating factor. If the stimulating beam is diffraction and Fourier limited, and if its

a

that is g( GeV21_{) 40.02}

₍

a( eV )

)

6. – Conclusion

At the present time, the closing of the high mass (maD 1023 eV) part of the axion window relies on astrophysical arguments. In this range, the limit obtained by direct measurements is provided by decay experiments and amounts to 1024 _GeV21_.

The potential of improvement in this range by non-resonant photon-photon scat-tering is limited by the QED background; it is marginal and amounts to 4 Q 1025 _GeV21_.

The resonant scattering, though interesting in principle, is actually undetectable because the intermediate axion does not decay during the stimulating time.

R E F E R E N C E S

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