Extending the reach of axion-photon regeneration experiments towards larger masses with phase shift plates
aa r X i v : . [ h e p - ph ] J un IPPP/07/28; DCPT/07/56; DESY 07-081
Extending the reach of axion-photonregeneration experiments towards largermasses with phase shift plates
Joerg JaeckelCentre for Particle Theory, Durham University,Durham, DH1 3LE, United KingdomAndreas RingwaldDeutsches Elektronen-Synchrotron DESY, Notkestrasse 85,D-22607 Hamburg, GermanyNovember 2, 2018
Abstract
We present a scheme to extend the sensitivity of axion-photon regenerationexperiments towards larger masses with the help of properly chosen and placedphase shift plates. any proposals to embedd the standard model of particle physics into a more general,unified framework predict a number of new very light particles which are very weaklycoupled to ordinary matter. Typically, such light particles arise if there is a global con-tinuous symmetry that is spontaneously broken in the vacuum – a notable example beingthe axion [1, 2], a pseudoscalar particle arising from the breaking of a U(1) Peccei-Quinnsymmetry [3] introduced to explain the absence of CP violation in strong interactions.Other examples of light spin-zero bosons beyond the standard model are familons [4],Majorons [5, 6], the dilaton, and moduli, to name just a few. We will call them axion-likeparticles, ALPs, in the following.At low energies, the coupling of such an ALP, whose corresponding quantum field wedenote by φ , to photons is described by an effective Lagrangian, L = − F µν F µν + 12 ∂ µ φ∂ µ φ − m φ φ − gφF µν ˜ F µν , (1)where F µν ( ˜ F µν ) is the (dual) electromagnetic field strength tensor and m φ is the massof the ALP. Correspondingly, in the presence of an external magnetic field, a photon ofenergy ω may oscillate into an ALP of small mass m φ < ω , and vice versa [9, 10]. PSfrag replacements γ laser γ laser −→ B −→ B φ
Figure 1: Schematic view of ALP production through photon conversion in a magneticfield (left), subsequent travel through an optical barrier, and final detection throughphoton regeneration (right).The exploitation of this mechanism is the basic idea behind ALP-photon regeneration –sometimes also called “light shining through a wall” – experiments [11–13] (cf. Fig. 1).Namely, if a beam of photons is shone across a magnetic field, a fraction of these photonswill turn into ALPs. This ALP beam could then propagate freely through an optical bar-rier without being absorbed, and finally another magnetic field located on the other sideof the wall could transform some of these ALPs into photons – apparently regeneratingthese photons out of nothing.A pioneering experiment of this type was carried out in Brookhaven by the Brookhaven-Fermilab-Rochester-Trieste (BFRT) collaboration, using two prototype magnets for theColliding Beam Accelerator [14, 15]. Presently, there are worldwide several second gener-ation ALP-photon regeneration experiments under construction or serious consideration(cf. Table 1; for a review, see Refs. [21, 22]). These efforts are partially motivated by The effective Lagrangian (1) applies for a pseudoscalar ALP, i.e. a spin-zero boson with negativeparity. In the case of a scalar ALP, the F µν ˜ F µν in Eq. (1) is replaced by F µν F µν . For the more generalcase where φ ceases to be an eigenstate of parity [7], see Ref. [8]. B i and their length ℓ i on production ( i = 1) and regeneration ( i = 2) side (cf.Fig. 1). Name Laboratory Magnets Laser ALPS [16] DESY/D B = B = 5 T ℓ = ℓ = 4 .
21 m ω = 2 .
34 eV
BMV [17] LULI/F B = B = 11 T ℓ = ℓ = 0 .
25 m ω = 1 .
17 eV
LIPSS [18] Jlab/USA B = B = 1 . ℓ = ℓ = 1 m ω = 1 .
17 eV
OSQAR [19] CERN/CH B = B = 11 T ℓ = ℓ = 7 m ω = 1 .
17 eV B = 5 T PVLAS [20] Legnaro/I ℓ = 1 m ω = 1 .
17 eV B = 2 . ℓ = 0 . m φ ∼ meV and a coupling g ∼ − GeV − [28]. Although these parameter values seemto be in serious conflict with bounds coming from astrophysical considerations, there arevarious ways to evade them [29–35]. Therefore, it is extremely important to check theALP interpretation of PVLAS by purely laboratory experiments [32]. Moreover, it wouldbe nice if in this way one might ultimately extend the laboratory search for ALPS topreviously unexplored parameter values (see also Ref. [42]). In this letter, we proposea method to extend the sensitivity of the planned photon-regeneration experiments tohigher ALP masses.Let us start with an outline of the calculation of the photon → ALP conversion probability P γ → φ , to lowest order in the coupling g . As emphasized in Ref. [13], this calculationamounts to solving the classical field equations following from Eq. (1), ∂ µ F µν = g∂ µ (cid:16) φ ˜ F µν (cid:17) ; (cid:0) ∂ µ ∂ µ + m φ (cid:1) φ = g ~E · ~B , (2)to lowest order in gBℓ , where ℓ is the linear dimension associated with the extent of themagnetic field . This can be done by neglecting the modification of the electromagneticfield due to the presence of the pseudoscalar field (through the right hand side of the first In the case of a scalar ALP, the term ~E · ~B in Eqs. (2), (3), and (4) is replaced by ( ~E − ~B ). g of the (pseudo-)scalar versus its mass m φ . Iso-contourof the regeneration probability P γ → φ → γ = P γ → φ P φ → γ , for the parameters of the ALPSexperiment, i.e. magnetic fields B = B = 5 T, over a length ℓ = ℓ = 4 .
21 m, exploitinga green ( λ = 532 nm) photon beam, corresponding to ω = 2 .
34 eV, in vacuum. Alsoshown in red are the 5 sigma allowed regions [28] from PVLAS data on rotation [23] plusBFRT data on rotation, ellipticity, and regeneration [15] plus Q&A data on rotation [38].equation above). Solving for φ in the second equation yields [9, 13] φ ( ± ) ( ~x, t ) = e − i ωt Z d x ′ π e ± i k φ | ~x − ~x ′ | | ~x − ~x ′ | g ~E ( ~x ′ ) · ~B ( ~x ′ ) , (3)where the energy ω and the modulus of the three-momentum k φ are related by k φ = q ω − m φ . This solution simplifies even more if we specialize to the usual experimentalconfiguration of a laser photon beam send along the x -axis with fixed linear polarizationin the z direction. If the transverse extent of the magnetic field is much larger than that ofthe laser beam, the problem is effectively one-dimensional. In one dimension and takinginto account only ALPs that propagate into the positive x -direction, Eq. (3) becomes, φ (+) ( x, t ) = e − i( ωt − k φ x ) i g k φ Z d x ′ ~E ( x ′ ) · ~B ( x ′ ) . (4)Inserting in Eq. (4) furthermore the appropriate plane wave form ~E ( ~x, t ) = ~e z E e i ω ( x − t ) for the electric field of the laser beam and assuming, as realized in all the proposedexperiments, a magnetic field with fixed direction along the z -axis and possibly variable(as a function of x ) magnitude, ~B ( ~x ) = ~e z B ( x ), one ends up with the solution φ ( ± ) ( ~x, t ) = i g k φ E e − i( ωt − k φ x ) Z d x ′ e i qx ′ B ( x ′ ) , (5) The solution (5) applies also in the case of a scalar ALP, if the magnetic field direction is chosen topoint into the y direction, ~B ( ~x ) = ~e y B ( x ) (or, alternatively, if the polarization of the laser is chosento point in the y direction). q = k γ − k φ = ω − q ω − m φ ≈ m φ ω (6)is the momentum transfer to the magnetic field, i.e. the modulus of the momentumdifference between the photon and the ALP. The probability that a photon converts intoan axion-like particle and vice versa can be read off from Eq. (5) and reads [9, 13] P γ → φ = P φ → γ = 14 ωk φ g (cid:12)(cid:12)(cid:12)(cid:12)Z d x ′ e i qx ′ B ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) , (7)which reduces, for a constant magnetic field, B ( x ′ ) = const, of linear extension ℓ , to P γ → φ ≈ g B sin ( qℓ/ /q . (8)Clearly, in the experimental setup considered, the maximum conversion probability, P γ → φ ≈ g B ℓ , is attained at small momentum transfer, q = m φ / (2 ω ) ≪
1, corre-sponding to a small ALP mass. For this mass range, the best limits are obtained in astraightforward manner by exploiting strong and long dipole magnets, as they are usedfor storage rings such as HERA [36] or LHC [37], cf. the experiments ALPS [16] andOSQAR [19], respectively (see Table 1). However, for larger masses, the sensitivity ofthis setup rapidly diminishes.We illustrate this in Fig. 2, which displays an iso-contour of the light shining through awall probability in the g - m φ plane, exploiting the experimental parameters of the ALPSexperiment [16]. Clearly, for this setup, the parameter region in g vs. m φ suggested bythe combination of BFRT plus Q&A exclusion and PVLAS evidence can not be probed.This is even more dramatic for the OSQAR experiment, which exploits an LHC magnet.Moreover, increasing the refraction index by filling in buffer gas does not help since itworks in the wrong direction (contrary to the claim in the ALPS letter of intent [16]).A simple possibility to probe the meV region in the ALPS setup is to reduce the effectivelength of the magnetic field region both on the production and detection side of themagnet by shortening the beam pipe on both sides. As can be seen in Fig. 3, thispossibility enables to extend the mass region probed by the experiment, however at theexpense of sensitivity: one looses about one order of magnitude in the light shiningthrough a wall probability.Another idea to extend the sensitivity towards larger ALP masses was introduced inRef. [13]. There, it was shown that a segmentation of the magnetic field into regions In a refractive medium, the laser beam has a phase velocity 1 /n ≡ v ≡ ω/k γ . The momentumtransfer (6) reads then q = n ω − q ω − m φ ≈ m φ ω + ( n − ω . The second term in this expression hasthe opposite sign as the corresponding term in Ref. [16]. Correspondingly, one would need a buffer gaswith refraction index less than unity, i.e. a plasma, in order to decrease q (and thereby maximize theconversion probability (8)) rather than to increase it. A.R. would like to thank Aaron Chou for pointingout the correct sign. Another possibility to probe larger ALP masses even with a long magnet would be to exploit VUVor X-ray free-electron laser beams [39–41]. However, at the moment conventional lasers seem to offerbetter prospects (see also Ref. [42]) R d x ′ exp(i qx ′ ) B ( x ′ ) that peaks at a nonzerovalue of q , thereby giving sensitivity to higher-mass pseudoscalars. In fact, the conversionprobability (7) reads [13, 43], in a magnet with N segments of alternating field direction(but the same magnitude B ), P γ → φ ≈ g B sin ( qd/ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 ( − k exp { i(2 k − qd/ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (9)= g B q (cid:26) sin ( qℓ/
2) tan ( qℓ/ (2 N )) for N evencos ( qℓ/
2) tan ( qℓ/ (2 N )) for N odd , where d = ℓ/N is the length of each of the N segments. For N >
1, this indeed gives riseto more sensitivity at non-zero values of q .In this letter, we will introduce a similar, but more practical possibility based on the useof phase shift plates. The idea is very simple. From our starting point, Eq. (4), we cansee that what counts is actually ~E ( x ′ ) · ~B ( x ′ ). The configuration based on N alternatingmagnetic fields is therefore equivalent to a configuration with non-alternating magneticfield, however with N − π (“ λ/
2” plates) invertingthe sign of the electric field, placed equidistantly over the length ℓ of the magnet. In thiscase we have alternating signs of cos θ , where θ is the angle between ~E and ~B , insteadof alternating signs of the magnetic field. But both cases have an identical profile of ~E ( x ′ ) · ~B ( x ′ ). In Fig. 4, we show that with a proper choice of the number and positionsof such phase shifters, ALPS should easily cover the region of parameter space suggestedby PVLAS + BFRT + Q&A. The same applies for OSQAR.Let us now get a more intuitive understanding of how this works and see how we can doeven better. The crucial part in Eq. (5) is the integral f ( q ) = Z d x ′ e i qx ′ B ( x ′ ) . (10)6igure 4: Iso-contour of the regeneration probability, as in Fig. 2. Here, we used onephase shift (“ λ/ ℓ the oscillating factor e i qx ′ suppresses the integralcompared to the massless case with q = 0, where the integral is simply | f ( q ) | < | f (0) | = (cid:12)(cid:12)(cid:12)(cid:12)Z ℓ d x ′ B (cid:12)(cid:12)(cid:12)(cid:12) = B ℓ, for B ( x ) = constant . (11)This suppression arises because coherent production of ALPs works only if the ALP andthe photon are in phase. The factor e i qx ′ accounts for the phase difference between ALPand photon.To improve the situation one would want to bring photon and ALP back into phase witheach other. This can be achieved in a simple way by the introduction of phase shift plates.A simplified picture of the phase correction process is given in Fig. 5. At the beginning,photon (red) and ALP (black) are in phase. However, due to its mass, the ALP has aslightly larger wavelength than the photon. After a few oscillations photon and ALP aremore and more out of phase. Then we insert the phase shift plate (turquoise). Withrefractive indices n > increase the phase difference between ALP and photon such that it is exactly 2 π (thephoton does an extra wiggle in Fig. 5.). Now, a phase shift of 2 π is exactly equivalent toa phase shift of 0. Photon and ALP are in phase again. Therefore, we can keep photonand ALP in phase over quite long distances simply by inserting a suitable phase shiftplate whenever the phase difference becomes too large, and we get coherent productionover the whole length of the magnet.Let us now understand more quantitatively how this works. To derive Eqs. (5) and(10) we have assumed that the photon is a plane wave. Therefore, we can identify qx ′ = ( k γ − k φ ) x ′ as the phase difference between the photon wave and the ALP wave atthe point x ′ . In general we should write (cf. Eq. (4)) f ( q ) = Z d x ′ e i ( ϕ γ ( x ′ ) − ϕ φ ( x ′ )) B ( x ′ ) , (12)7 - - - - Figure 5: Illustration of the effect of a properly chosen and placed phase shift plate onthe phase relation between photon and ALP (this simplified picture shows only the phaserelation; the amplitudes of photon and ALP are not correct in this picture). Photon (red)and ALP (black) start in phase. Due to their different wavelength they are, however,somewhat out of phase after several oscillations - say by an amount ζ . This is correctedby introduction of a phase shift plate that causes the photon to get an extra phase 2 π − ζ .In other words the plate causes the photon to complete the extra wiggle.where ϕ γ,φ are the phases of the photon and the ALP fields, respectively.Let us imagine a situation where we insert N − , non-reflective plates that ac-celerate the photon phase by κ at equidistant places sℓ/N = s ∆ x , s = 1 . . . N −
1, in aconstant magnetic field of length ℓ . The plates affect only the photon. The ALP phaseremains unaffected. Therefore, we have, ϕ γ ( x ) = k γ x + sκ for s ∆ x < x ≤ ( s + 1)∆ x, ∆ x = ℓN , (13) ϕ φ ( x ) = k φ x. Inserting this into Eq. (12), we find f ( q ) = B N − X s − Z ( s +1)∆ xs ∆ x d x ′ e i( qx ′ + sκ ) = B e i2 ( qℓ +( N − α )
2i sin (cid:0) q ∆ x (cid:1) q sin (cid:0) N ( q ∆ x + κ ) (cid:1) sin (cid:0) ( q ∆ x + κ ) (cid:1) . (14) One might ask what happens to the photon-ALP system inside the material of the plate. One cancheck (cf., e.g., Ref. [10]) that for sufficiently large refractive index of the material, n − ≫ m φ / (2 ω ),the mixing between photon and ALP is effectively switched off compared to the mixing in vacuum.Photon and ALP simply propagate through the plate without changing their amplitudes (the phaseschange, of course). In other words, the thickness of the plates has to be subtracted from the total lengthof the production or regeneration region. That is why we require thin plates. For practical purposes,this is a rather mild constraint. For n − ∼ .
1, the thickness of the plates required for a phase shiftof the order of 2 π is only d ∼ λ ∼ µm , which is tiny compared to the typical lengths of theproduction/regeneration regions which are of the order of a few m. Reflected photons are effectively lost. κ = π (green), and one plate withthe optimal choice of κ according to Eq. (16) for m φ = 1 . κ . In the right figure, we have the same butwith 3 plates for the green and blue curves.We can now choose the number of plates N and the phase shift κ according to the recipedescribed above. First we choose N large enough such that12 q ∆ x ≪ κ such that the phase difference that has accumulated over ∆ x is“completed” to 2 π , κ = 2 π − q ∆ x. (16)Evaluating Eq. (14) in the limit N ( q ∆ x + κ ) / → | f ( q ) | = B ∆ x N = B ℓ. (17)And we have coherent production over the whole length ℓ .The potential of this approach is demonstrated in Fig. 6 for the example of the ALPS ex-periment. In the optimized mass region we get more than ten times as many regeneratedphotons as we would get if the length of the magnet is reduced as in Fig. 3.Another practical advantage of this method is that we can scan through a whole massrange. Performing several measurements with different phase shift plates we can alwayschoose for each q , i.e. for each m φ , plates with an appropriate κ such that it is closeenough to its optimal value (16), 12 | q ( m φ ) ℓ − N κ | ≪ . (18)For an infinite number of plates this would allow to extend the mass range all the way tothe frequency ω of the photons . In practice, we can insert only a finite number of phase Above the photon frequency, ALP production is energetically forbidden and q becomes imaginary. κ for each m φ . The red curve is the sensitivity without phase correction. The blackcurve is obtained by using three plates but scanning through a whole range of κ . In otherwords to obtain this curve one would insert the plates. Measure. Change the plates to aslightly different value of κ and measure again. This is repeated for all values of κ in therange [0 , π ].shift plates and Eq. (18) cannot be fulfilled for too large masses. But, already a smallnumber of plates leads to a remarkable increase of the sensitivity for higher masses, aswe can see from Fig. 7. In summary:
So called “light shining through a wall” experiments are a promising toolto search for light particles coupled to photons. In this note we have shown how thereach of such an experiment can be extended towards larger masses by inserting properlychosen phase shift plates. Although our explicit discussion is for the case of spin-0 axion-like particles the method works in general for particles exhibiting photon-particle-photonoscillations.
Acknowledgments
We would like to thank Giovanni Cantatore, Aaron Chou, Marin Karuza, Axel Lindner,Giuseppe Ruoso, Pierre Sikivie, and Karl van Bibber for interesting discussions.
References [1] S. Weinberg, Phys. Rev. Lett. (1978) 223.[2] F. Wilczek, Phys. Rev. Lett. (1978) 279.[3] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. (1977) 1440.104] F. Wilczek, Phys. Rev. Lett. (1982) 1549.[5] Y. Chikashige, R. N. Mohapatra and R. D. Peccei, Phys. Lett. B (1981) 265.[6] G. B. Gelmini and M. Roncadelli, Phys. Lett. B (1981) 411.[7] C. T. Hill and G. G. Ross, Nucl. Phys. B (1988) 253.[8] Y. Liao, arXiv:0704.1961 [hep-ph].[9] P. Sikivie, Phys. Rev. Lett. (1983) 1415 [Erratum-ibid. (1984) 695].[10] G. Raffelt and L. Stodolsky, Phys. Rev. D (1988) 1237.[11] A. A. Anselm, Yad. Fiz. (1985) 1480.[12] M. Gasperini, Phys. Rev. Lett. (1987) 396.[13] K. Van Bibber, N. R. Dagdeviren, S. E. Koonin, A. Kerman and H. N. Nelson, Phys.Rev. Lett. (1987) 759.[14] G. Ruoso et al. [BFRT Collaboration], Z. Phys. C (1992) 505.[15] R. Cameron et al. [BFRT Collaboration], Phys. Rev. D (1993) 3707.[16] K. Ehret et al. [ALPS collaboration], “Production and detection of axion-likeparticles in a HERA dipole magnet: Letter-of-intent for the ALPS experiment,”arXiv:hep-ex/0702023.[17] C. Rizzo for the [BMV Collaboration], 2nd ILIAS-CERN-CAST Axion AcademicTraining 2006, http://cast.mppmu.mpg.de/[18] K. Baker for the [LIPSS Collaboration], 2nd ILIAS-CERN-CAST Axion AcademicTraining 2006, http://cast.mppmu.mpg.de/[19] P. Pugnat et al. [OSQAR Collaboration], CERN-SPSC-2006-035, CERN-SPSC-P-331.[20] G. Cantatore for the [PVLAS Collaboration], 2nd ILIAS-CERN-CAST Axion Aca-demic Training 2006, http://cast.mppmu.mpg.de/[21] A. Ringwald, arXiv:hep-ph/0612127.[22] R. Battesti et al. , arXiv:0705.0615 [hep-ex].[23] E. Zavattini et al. [PVLAS Collaboration], Phys. Rev. Lett. (2006) 110406[arXiv:hep-ex/0507107].[24] S. L. Adler, Annals Phys. (1971) 599.[25] S. L. Adler, J. Phys. A (2007) F143 [arXiv:hep-ph/0611267].1126] S. Biswas and K. Melnikov, Phys. Rev. D (2007) 053003 [arXiv:hep-ph/0611345].[27] L. Maiani, R. Petronzio and E. Zavattini, Phys. Lett. B (1986) 359.[28] M. Ahlers, H. Gies, J. Jaeckel and A. Ringwald, Phys. Rev. D (2007) 035011[arXiv:hep-ph/0612098].[29] E. Masso and J. Redondo, JCAP (2005) 015 [arXiv:hep-ph/0504202].[30] P. Jain and S. Mandal, Int. J. Mod. Phys. D (2006) 2095[arXiv:astro-ph/0512155].[31] E. Masso and J. Redondo, Phys. Rev. Lett. (2006) 151802[arXiv:hep-ph/0606163].[32] J. Jaeckel, E. Masso, J. Redondo, A. Ringwald and F. Takahashi, Phys. Rev. D (2007) 013004 [arXiv:hep-ph/0610203].[33] R. N. Mohapatra and S. Nasri, Phys. Rev. Lett. (2007) 050402[arXiv:hep-ph/0610068].[34] P. Jain and S. Stokes, arXiv:hep-ph/0611006.[35] P. Brax, C. van de Bruck and A. C. Davis, arXiv:hep-ph/0703243.[36] A. Ringwald, Phys. Lett. B (2003) 51 [arXiv:hep-ph/0306106].[37] P. Pugnat et al. , Czech. J. Phys. (2005) A389; (2006) C193.[38] S. J. Chen, H. H. Mei and W. T. Ni [Q&A Collaboration], arXiv:hep-ex/0611050.[39] A. Ringwald, arXiv:hep-ph/0112254.[40] R. Rabadan, A. Ringwald and K. Sigurdson, Phys. Rev. Lett. (2006) 110407[arXiv:hep-ph/0511103].[41] U. K¨otz, A. Ringwald and T. Tschentscher, arXiv:hep-ex/0606058.[42] P. Sikivie, D. B. Tanner and K. van Bibber, Phys. Rev. Lett.98