Precision Tests of QED and Non-Standard Models by Searching Photon-Photon Scattering in Vacuum with High Power Lasers

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Precision tests of QED and non-standard models by searching photon-photon scattering in vacuum with high power lasers

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Please note that terms and conditions apply. JHEP11(2009)043 c , erties, Stan- m at present October 19, 2009 : November 10, 2009 September 25, 2009 : : olarisations. This result Accepted at a rate as small as that ters, improving by several and Marcos Seco eters appearing in the low Received Published a m Electrodynamics and non- e improved at future exawatt rther enhance the sensitivity, t photon-photon scattering as et of experiments that can be 10.1088/1126-6708/2009/11/043 tly by PVLAS collaboration. ving minicharged particles or produced when an ultra-intense on tests of the Standard Model ` Optica, Universitat de València, [email protected] doi: , ntiago de Compostela, t d’ Published by IOP Publishing for SISSA Humberto Michinel b [email protected] , Albert Ferrando, a Beyond Standard Model, Electromagnetic Processes and Prop We study how to search for photon-photon scattering in vacuu [email protected] SISSA 2009 [email protected] Departamento de Aplicada, F´ısica Universidade deAs Vigo, Lagoas E-32004 Ourense, Spain Interdisciplinary Modeling Group, InterTech. Departamen Dr. Moliner, 50. E-46100Departamento Burjassot, Spain de de València, F´ısica Universidade Part´ıculas, de Sa 15706 Santiago de Compostela, Spain E-mail: b c a c Abstract: Daniele Tommasini, Precision tests of QEDsearching and photon-photon non-standard scattering models in by high vacuum power with lasers petawatt laser facilities such as HERCULES,standard and models test Quantu like Born-Infeldaxion-like bosons. theory First, or we compute scenarioslaser the phase beam invol shift crosses that is ais low then power used beam, inenergy order in effective to the photonic design case Lagrangian. a of Inperformed complete fact, arbitrary at test HERCULES, we p of eventually propose allowing all either adue to the s detec to param new physics, ororders to of set magnitude new the limitsWe also on current describe the constraints a relevant obtained multi-cross parame enabling optical recen HERCULES mechanism to that can detectpredicted fu photon-photon by scattering QED. Finally, even we discussfacilities how such these as results can ELI, b thusand providing beyond. a new class of precisi Keywords:

dard Model JHEP11(2009)043 ]. 1 2 4 5 7 9 3 10 , 2 ], and are 6 ]. Most of 11 – 8 ED and in boration [ s at present facilities olarised beams performed at present oach will be to perform . In the last few years, hoton-photon cross sec- γγ n. However, even in these laser. Two exceptions are σ ng photons [ ility to detect PPS at future ller cross sections with a very med prediction of both Quan- etect PPS of QED origin only ew physics scenarios involving ], based on a free electron laser 7 cross section for the process will , that will eventually be available ]. First, we perform a new theoretical ], respectively. 15 13 , 14 – 1 – ] or VIRGO+ [ 12 ] particles. However, all the experiments that have been ] and non-standard models such as Born Infeld theory [ 5 1 ] or axion-like [ 4 ], that discussed the possibility of performing experiment 11 . The best constraints were obtained recently by PVLAS colla , γγ 10 σ Here, we will propose a set of experiments that can already be non-standard models tum Electrodynamics (QED) [ 1 Introduction Photon-Photon Scattering (PPS) in vacuum is a still unconfir 5 Proposal of experiments 6 Improving the sensitivity with multiple7 crossing Conclusions 3 Present constraints 4 Approximate solution for the scattering of orthogonally p Contents 1 Introduction 2 The effective Lagrangian for the electromagnetic fields in Q petawatt laser facilities, such as HERCULES [ Additional contributions to the processminicharged can [ also appear in n to improve the PVLAScases limit the on predicted the sensitivity photon-photon wasat cross found future to sectio facilities, be such sufficient to as d ELI [ performed by now could only be used to set upper limits on the p tion still seven orders of magnitude above the QED prediction for producing two beams of photonsexperiments in at the optical MeV wavelengths, and range.high compensate A the density second sma and/or appr athese long proposals require path the ofin construction interaction of the of new future, facilities the suchrefs. collidi as [ a free electron laser and/or an exawatt there has been anfacilities, increasing using interest two possible in strategies.be studying maximum On the at one possib a hand, possible the future photon-photon collider [ JHEP11(2009)043 · ], = in E 18 ( T (2.2) (2.1) c ξ BI T 0 ξ ], that ǫ 19 pectively. and = d with the L ED and in G ξ Moreover, it can ] would provide an 4 , being value. ξ ], we propose a set of ≡ ss how these results can he interchange of virtual 17 , or to set new limits on -cross optical mechanism , ive than our previous pro- MCP) [ ndard models. iments, eventually enabling inicharged (MCP) or axion- QED T ull test of all the parameters s result turns out to provide 16 reshold for the production of the photons in non-standard ξ sion that can be achieved in hift that is produced by PPS the parameters Lagrangian density [ nitude the current constraints , . oviding a new class of precision n-positron box diagram. If the 6 n for the electromagnetic fields = 3 J , m 2 G 30 QED L T − ξ ξ 7 4 ]. Besides other interesting effects [ 10 ], we would obtain the relation . Therefore, they will appear as the first 1 0 2 × + L 7 2 0 . L 6 L ξ – 2 – ≃ 5 + 3 c 0 ~ 4 e 2 L m α 8 = 45 L = the Lagrangian density of the linear theory, ξ ) with the identification 2 2.1 B 2 c − 2 the dielectric constant and the speed of light in vacuum, res E c 0 2 ǫ ), are the only two Lorentz-covariant terms that can be forme and ], in which the two crossing beams had the same polarisation. 2.1 = of the form ], in general without a definite prediction for the numerical 0 10 ǫ 3 0 , B L 9 7 [ / On the other hand, in Born-Infeld theory [ The presence of a minicharged (or milli-charged) particle ( In fact, in QED photons can interact with each other through t non-standard models and BI L ) and ξ B 4 The additional, non-linear terms, that appear multiplying when two orthogonally polarised beamsa cross test each for other. PPSposal Thi of [ QED origin which is significantly more sensit computation to get a quantitative expression for the phase s being equation ( additional contribution analogous to that from the electro such an interaction leads to the Euler-Heisenberg effective electromagnetic fields at the lowest order above coincides with equation ( be used in combination withappearing our in previous result the to lowmodels, provide energy a such effective f as Lagrangian Born describing like Infeld (ALP) theory and particles. scenariosthe involving In m measurement fact, of taking opticalexperiments into phase that account shifts will the allow and preci the either ellipticities relevant to parameters, [ detect improving by PPSobtained several at by orders HERCULES the of PVLAS mag that collaboration. can We then further proposeHERCULES improve a to the multi detect PPS sensitivity as ofbe predicted improved by this at QED. future set Finally, exawatt we facilities oftests discu such of exper as the ELI, thus Standard pr Model and beyond. 2 The effective Lagrangian for the electromagnetic fields in Q We will consider the caseelectron-positron of photon pairs, energies and well assume belowE an the th effective Lagrangia charged particles running in loop box diagrams [ correction to the linear evolution both in QED and in non-sta JHEP11(2009)043 = S (2.5) (2.3) (2.6) (2.7) (2.8) (2.4) L optical and ective La- en improved G P Φ P c g lectron charge. The on. Assuming again ~ n to the effective La- is still larger than the √ ling with the photons, ]. We obtain of the photons (the eV ǫ he experiments that we − ]. A larger contribution scale, that can be cast in m e already stronger limits, 20 4 be described simply by an = Φ , this limits can be read as ]. Taking masses above the s. [ we would obtain e a different treatment which 4 m ]. This can be a Light Pseu- P [ 5 ξ, L 5 4 ξ ξ − ξ ξ. 4 4 4 e 4 10 ǫ e e m e ǫ ǫ × e ǫm ǫ ǫ 8 2 2 Φ P m m m ǫm ǫm m g ǫm ). ǫm 3 . = cm ~ 7 2 ǫ 2.1 ], we would obtain 7 1 16 16 28 28 243 261 20 = – 3 – MCP T ξ = T = = = ξ ∆ =∆ MCP0 L MCP0 T MCP1 MCP1 T L ξ ξ ξ ξ ∆ ∆ MCP L ∆ ∆ ξ ∆ ) with an additional contribution given by 2.1 is still larger than the energy of the photons (the eV scale in ). As we shall see in the next sections, this constraint has be ǫ ξ , respectively. We can find the leading contribution to the eff 6 m 0 L (10 S , from astrophysical and cosmological observations [ o Φ 15 is the ratio of the charge of the particle with respect to the e . S − ǫ c g 10 Let us now discuss the case of an axion-like particle [ Similar considerations apply if the new MCP is a spinless bos ~ MCP L . ξ √ energy of the photons (the eV scale in optical experiments), the form of equation ( existing laboratory bounds in this regime is On the other hand,grangian would if be the larger, MCP as is computed using a the spin result 1 of boson, ref the contributio where − and new MCP are spin 1/2 fermions, and assuming that their mass doscalar Boson orthat a is Light described Scalar in Boson, the depending Lagrangian density on by the the coup terms might be obtained ifscale the in the MCP present are paper).goes lighter beyond However, than the this the case purposes energyeffective would of Lagrangian scale deserv the of present the work, form since of it equation cannot ( that its mass eV scale, in order∆ to apply the effective Lagrangian approach grangian when the photon energy is much smaller than the experiments), and using the results of ref. [ and by PVLAS collaboration,propose and can in the be presentǫ further paper. strengthened by Of t course, in this case there ar JHEP11(2009)043 ] , | ], 2 5 < L ξ − 22 ) (7 ) as 4 (3.1) (2.9) exp w for − A ξ 2.2 , T GeV is more 0 ], is still ξ 6 5 7 A L nts on the | 10 ξ . Expressing × = ( T ξ 4 iform magnetic µ 1KeV [ A . , therefore PVLAS and ) . L 2 ξ L c hich it only implies a Φ ξ 7 4 Φ combination 1eV, the Cristal Ball [ m utation of ∆ m = s smaller than the order ly by the PVLAS collab- view in the next section. taken into account. The ( & / T ld theory. uge field . This constraint has been ξ P an be applied to any theory e best laboratory constraints Φ esence of a constant external g ave been obtained in ref. [ /J m . . en 3 3 T ξ m J m , valid for 25 1 26 − − and − ], as we shall see in the next section. 10 L 10 6 2 Φ 2 S ξ × g GeV × 3 2 9 cm . 2 ~ is the effective action. Such equations are − . 2 2 ). 3 x 10 = 4 γγ . – 4 – < d σ × gives the contraint L | T L ξ ξ 7 L 1 . . However, this case lies beyond the scope of the ξ ). With our notation, their 95 % C.L. limit reads R ∆ − 2 ), as parametized by artitrary 4 times higher than the QED value of equation ( S,P , this gives the equations of motion as the variational ≡ 2.1 3 . − 3 2.1 g A GeV ∂t T ∂ P 10 3 ξ g − 7 × − | and 6 0 10 . Φ as in QED, their results can be translated in the limit A × m ∇ 2 c . exp 4 − ξ ], and combined with the measurement of ellipticity may allo = 0, where Γ ≤ ]. Their negative result was used to set the current constrai = ≡ 21 µ P 21 T E , g , which is 4 ξ 6 /δA /J = 3 Γ and δ L m ξ ] by searching evidence of birefringence of the vacuum in a un A = 0 in the case of scalars. On the other hand, for = 0 in the case of pseudoscalars, or 26 6 − T L ξ ξ 10 ∇∧ Similar considerations apply for scalar boson, for which th Note however that PVLAS experiment was only sensitive to the × = 2 . B the electromagnetic fields in terms of the four-component ga 3 Present constraints The current limits onoration PPS in [ vacuum have been obtained recent complicated, and the productionlatter can of be real expected axions tomagnetic produce has field dichroism, also just [ to as in be the pr improved recently by the PVLAS consideration [ Assuming derivatives and ∆ and ∆ are also those thatWe were also recently recall set that byof PVLAS, our the that approximations energy we do will of not re the apply for colliding masse photons. In this case, the comp the only difference being the distinction between laboratory limit much more stringent than any laboratory bound. a determination of both parameters appearing in equation ( orders of magnitude for the cross section Again, the astrophysical limits 3 similar to the modified nonlinear Maxwell’s equations that h experiment is unable to set any constraint on a pure Born Infe which can be converted in the limit ∆ present paper, that usesthat a goes phenomenological approach beyond that thedifferent c contribution Standard to equation Model, ( in the energy regime in w of the parameters. In particular, this quantity vanishes wh field background [ JHEP11(2009)043 ) ) ), 2, z z / ( ( ϕ 2 α α (4.1) + A of the 2 of ωt = 0 are y ω + n 0 ) z ǫ kz and = /δA olarised d/dz ρ x Γ cos( ). Therefore, in δ ≡ A t, z y ( = ρ y g waves that are po- waves be polarised in A ], will be confirmed by iori. We then compute lin igh power wave, taking 9 ) , tion (neglecting photon- ) at the non-linear effect is tion for the problem of the = 0 and with will not be generated em up. Therefore, we will t, z t of PPS was to produce a ), and average out the fast 2 and ωt ( ar equations, that was also t ork [ / z ) and d by the non-linear terms in y l to the energy density of the 2 . Their energy density, when − A 2.1 n a perturbative approach, we z: 0 tional approximation, and was A e direction as the original wave, ϕ t, z α /δA ( ]. We then need to chose a good kz 2 Γ x 9 multiplied by the intensity of the ω δ and ting waves, one of which represents A 0 . When these two waves are made t L ) and ǫ x ξ A ) sin( ρ ωt = z ( − x β ≫ ρ . = 0, keeping the lowest order terms in the kz . This justifies a perturbatively-motivated ) y , assuming that the envelop functions T ρ ϕ )+ ξ = 0 is maintained by the non-linear evolution, /δβ cos( + ωt π/k z – 5 – 0 Γ δ A − α ωt and ). In principle, each of the two components can get ) in the Lagrangian ( = L = + kz ξ t t, z 4.1 lin A ( kz ) y ]. A ) cos( = 0 and t, z 9 z cos( ( ( x A α A /δα ) and Γ = = δ and neglecting the higher order space derivatives ( y x t, z dependence, the components A ( A T x ξ y A and x and we allow for an initial phase difference over distances of the order 2 ). First, we note that the assumption of no dependence on z directions respectively, so that their linear (free) evolu kc 2.1 y = ], we have studied the scattering of two counter-propagatin ) will show a much slower variation, as we will verify a poster 9 ω z and ( β We now substitute the ansatz ( Here, we will apply a similar variational method to find a solu beams x and photon scattering) would be where the variational equations a transmitted wave contribution, propagating alongand the sam a reflected wave propagatingcan in the compute opposite these direction. differentfirst I effects neglect separately the and reflected then waves, sum and use th the following ansat 4 Approximate solution for the scattering of orthogonally p ansatz for the fields the result that we will obtain below. variation in larised in the samephase direction, shift and in we have eachother found wave, wave. which that That was the result proportional effec wasshown obtained to by to an agree analytical, varia withobtained a in numerical the second solution of of refs. the [ full non-line scattering of two orthogonally polarised counter-propaga an ultra-high power beam.the Let the low power and the high power In ref. [ Here, we have neglected theinto account effect that of such the anlow low effect power is power beam. expected wave on to This be expectation, the proportiona inspired h by our previous w expansion parameter each of the two waves is taken alone, would be fields guarantees that the condition since we have checked that in this case the equations if they are notdriven present by from the the very beginning. small parameters Second, we note th the absence of respectively. Hereafter, we will assume that variational approach, similar to that introduced in refs. [ automatically satisfied, independently on the values of to scatter, they will affectequation each ( other due to PPS, as describe JHEP11(2009)043 8 − 10 (4.2) (4.4) (4.3) (4.5) (4.6) ), we ] that × 15 4.1 . Taking 2 2 at least by − . , 0 ) . ρ , as described α ) Jm y T ωt ξ A ωt 17 + (0) = 0, in such a − 10 β , kz × 0 kz 7 . Note that this result α . ects. As a result, the z 6 ) sin( ∆ enerated in the component z ) sin( ∼ omponent ( z ρk is the energy density of the th will be smaller than the δ wing variational solution ρ ( (0) = his case, the peak intensity is T σ d. , . Now even in the multi-cross α , the phase of the orthogonal, ρ ξ z , m )+ 1] ) at of the linear problem, we find ≃ 1 )+ = 7 ωt − . ωt 2 ) z/ , , ) . ωt z / 5 0 for all the practical purposes, and + ) ϕ − 2 T . , and assuming that it can be applied ∆ T − 0 z ω + ≃ T 2 kz T /J πρξ χ . 3 ) kz ) = 0 ) = 0 χ A πρξ ωt z | z z 0 between the two crossing waves. m ( 0 ). After substituting in equation ( = ( ( ǫ + + γ z β as for the wavelengths of e.g. the HERCULES α ) cos( 26 ϕ T T /α z T T 1 − ) cos( kz ) φ ( χ kz – 6 – χ χ z − [cos(14 sin(14 z γ ( 10 ( in the last section), we find that , m η − δ ) | cos( 6 kz kz × π π sin( cos( ) )+ 2 0 0 )+ 0 4 4 3 0 z z ωt 10 α α ( ( )+ α A α ′ ′ ωt < × ϕ − ) α β − 8 = = ϕ − z . + implies that . After a long but straightforward algebra, we find the 7 ) = ) = y T x exp z n + z z ξ δ χ T A A ( ( ) = k ωt ∼ δ χ γ z ωt + ) will be smaller than the low power amplitude ( k + can be estimated for the petawatt laser HERCULES [ z + β ( and 2. Assuming the initial condition + kz δ T / kz ), γ ], corresponding to an energy density kz T kz z ξ 3 T 15 πρξ cos( [ cos( k χ 0 cos( 2 cos( 2 A α 0 2. Taking − ), we obtain that the A A α 2 cos( = = c 0 0 = = kz/ nm y x ǫ α T y 7 x A A W cm A ρξ A 22 ≡ 0 ) = = 800 α ), as compared with 10 T z 7 z ( χ λ ( at least roughly (see also figure × α β ≃ Let us now introduce the possibility that a reflected wave is g In other words, taking into account that Finally, let us introduce the possible reflected wave in the c 2 T ) ξ , as described by the ansatz z ∼ x ( into account the PVLAS limit where way that the corrected solutionthen coincides initially with th After repeating the same procedure as above, we get the follo and high power wave, we find that after a crossing distance ∆ we will consider below forI our proposals of experiments. In t finally get the following variational solution: to A low power wave is shifted by an amount ∆ following equations: is independent of the initial phase difference Now, the quantity 14 for the productvariational giving solution the for importance of the nonlinear QED eff laser ( δ configuration that willcentimetre be scale, discussed so below that the crossing leng two orders of magnitude. For this reason, it will be neglecte by the ansatz JHEP11(2009)043 L = φ L ding (4.8) (4.7) φ ∆ − T 4 than ∆ φ ]. Moreover, / 9 arly polarised n will be close f the ellipticity power beam. experiment that =∆ b φ , which is the noise eam splitter in two is its time length. If ], applies then to our ) (0) = 0. We then see other crosses a contra- 17 σ ] provides a technique ωt rad , ct to ref. [ 8 z/c will permit a full analysis 17 ame direction. The phase − − bove, we find the following 16 , T z rresponding phase shifts are ξ L = ∆ 16 χ ase shift ∆ τ shifted both in the polarisations + and which is equal to that obtained in L Ikτ, kz Ikτ, ξ z T L ξ ξ ∆ ) can be used to search PPS in vacuum cos( L , for the sensitivity in the measurement , thus allowing them to set a 95% C.L. 0 χ ) also implies that the high power pulse = 7 = 4 4.8 , η is more sensitive by a factor 7 ) z z = rad and we assume that T 4.8 rad 8 ωt ∆ ∆ 8 L φ )+ 0 − polarisation only exists if it is present from the − φ η ϕ − ρk – 7 – ρk 10 y z 10 T L . Hereafter, to be definite and for simplicity, we + ξ ξ T × × χ ωt 8 rad . (0) = 1 8 = 7 = 4 . + η + − , L T kz φ φ 10 kz ) on both parameters = 1 ρk ), distinguishing between QED and other models such as ∆ ∆ L b × 4.8 ξ cos( σ 2.1 8 cos( . 0 2 α A = 4 < = = L ], the PVLAS collaboration was able to resolve the correspon ξ b y 6 x 3 φ ]. k A A 2 as in QED, we see that ∆ 17 ) implies that, after crossing a counter-propagating, line A T , taking into account that the actual experimental precisio , 2 ξ corresponding to birefringence. In particular, in the same between the transverse an parallel polarisations of the low c T 4.7 b 0 16 φ . A similar sensitivity can be obtained for the measurement o is the intensity of the high power beam and = ǫ φ T L Ikτ φ ρc ξ ) = 2 L = ξ ]. This precision, that holds for ultra-short laser pulses [ L and ∆ 4 χ I 17 L ], as could be expected. branches. One of them propagatespropagating freely ultra-high in power vacuum, while laser the pulse polarised in the s and ∆ with a statistical error − 9 φ We can now propose a set of three experiments: Equation ( L b T 1) A linearly polarised low power laser pulse is divided by a b ξ φ φ to the effect of PPS. This is already an improvement with respe where we assume ultra-intense laser pulse, an ordinaryparallel and laser orthogonal pulse to is that of phase the high power beam. The co to this choice [ of ∆ will use this same numerical value, 2 experimental limit ∆ limit [ of the effective Lagrangian ( By repeating the same kind of computations and arguments as a solution that allows for the measurement of phase∆ shifts as small as 10 ∆ by measuring phase shifts and ellipticities. In fact, ref. [ the dependence of equations ( Born Infeld theories.behaves Finally, like we a note birefringent medium, that producing ( a relative ph ref. [ that the counter-propagating wave in the beginning, and that it gets a phase shift ∆ induced by ∆ where (7 we have cited above [ 5 Proposal of experiments We will now discuss how the result of equations ( JHEP11(2009)043 . , ], b T φ ξ 15 and (5.3) (5.1) (5.2) s =∆ and 14 as small − L L T ξ φ ξ . As far as 10 ∆ by measuring × − . Even in the Ikτ e experimental and 3 r beam, and on T T that corresponds φ L φ = 3 then measured by ξ rallel and another FIkτ ∆ F 7 J/m τ he high power beam although even in this 19 fts ∆ r pulse. Due to equa- = riment will then allow he technique described se. Ellipticity measure- 10 T plete exploration of the ξ e than two orders of mag- have the same sign, as in he product × , thus allowing either to de- T 7 . , or to set an upper limit on ermination of both ξ gent than the current PVLAS LAS. . arameter space, including the e than the combination of the b mation. On the other hand, if L that of HERCULES laser [ 8 φ ξ − ∆ FIkτ = 4 3 and . . 10 8 8 L = × − − ξ FIkτ Ikτ L 3 8 ξ 10 . 10 , for a time length 4 2 2 3 − × × FIkτ FIkτ T < 7 4 8 8 ξ . . 7 | 2 2 L ξ W/cm – 8 – < < 4 ), this configuration can be used to measure the , respectively, thus allowing either to detect PPS 22 L T ), we see that the sensitivity depends on the com- 3 − ξ ξ 4.8 /J . This gives 10 T 3 5.3 ξ m × m 7 7 | − 29 − 10 = 2 ) and ( 10 I = 1), such a facility will be able to resolve × , where we have introduced a gain factor × 1 5.2 F . 6 L . This will allow either to detect PPS or to set the upper limit . φ T ), ( FIkτ ∆ = 8 φ 4 5.1 and 8 = ]. Due to equation ( π/k L /J ξ 17 that will be discussed later. Therefore, the most favourabl 3 , = 2 ), this setup can be used to obtain the parameter m of the experimental parameters of the ultra-high power lase F 16 λ 28 4.8 − have opposite sign, experiment 3) is the most sensitive one, Ikτ 10 T ξ × this parameter as tect PPS, or to set the upper limit is polarised in ation direction ( orthogonal to that of the low powe the phase shift ∆ orthogonal to that ofments the can contra-propagating then be high used power to pul deduce the difference of the phase shi to a multi-cross configurationeither as to discussed detect below. PPS by This measuring expe a non-vanishing comparing with the pulsein that ref. has [ propagated freely,parameter using t shift suffered by the low power pulse as a consequence of PPS is allowing to determine the combination 5 From equations ( . and 2) The configuration is the same as in case 1), except that now t 3) The low power beam polarisation has two components, one pa L as 1 wavelength of non QED origin, or set limits on the parameters that are mor nitude (5 order of magnitudelimits, in in the addition cross section) tocase more the strin of fact Born-Infeld that theories, they that constrain were the unconstrained by full PV p The combination ofparameter these space. three experiments Actually, it will is permit easy a to com see that if absence of any gain factor ( bination a gain factor that reaches peak intensities QED and Born Infeld theories,others experiment and 3) can is be lessξ discarded sensitiv without losing significant infor case the other two measurements would be useful for a full det configuration will be thatwe allowing know, for the the highest maximum value value achieved of at t present facilities is JHEP11(2009)043 π µm and 9 , say s of the is very with a ≃ 2 14 θ 2 − − mirror close to cτ d 10 as small as θ ], that in its T ≃ W cm ξ 12 W/cm τ 22 19 , so that 3 10 10 and g laser pulses. − ry precise alignment × . On the other hand, × L with two figures. . Even in the absence 10 2 ξ at ELI [ ossing points the high 3 2 ξ × µm ≃ r 2 ≃ by making the two beams I 50 J/m sisting of two parallel series rming an angle much larger than r of magnitude greater than 21 ≃ ions in the beam. We assume ≃− e multiplied by the number of ements may be achieved using R 10 mirror π I × − , for a time length mirror ) 8 2 d , with ≃ /R half a way to the path leading to the

R implies that the pulse length and intensity 2 R W/cm Ikτ ≃ mirror fs 25 . In order to avoid diﬀractive distortions, we d µm 8 . 10 0 – 9 – µm = 30 ≃ . An advantage of using parabolic mirrors is that, ≃ 25 τ I 1 d ≃ , respectively. In particular, ELI would allow for the . This gives θ = 2 arccos( /J m π ]. In this case, the high power beam at the mirror should 3 7 mirror − − m 23

θ I/I 31 10

− 98 [ p × . d 10 0 8 = 1), such a facility will be able to resolve × ≃ ≃ r F ], that can work e.g. at the intensity π/k 23 and 5 . UProposed setup for multiple crossing of the two scatterin = 2 /J λ 3 m . A serious technological challenge to be faced will be the ve , in such a way that π 31 Figure 1 − cm A significant improvement will be obtained in the near future 10 = 5 × in order to maximizeplasma their mirrors superposiposition. [ These requir along the direction ofits propagation transversal is width, approximately therefore an the orde two beams must cross fo reflection coefficient HERCULES beam. The time duration will use parabolic mirrors of, say, a double diameter, have a diameter equal to first stage will achieve peak intensities in the paraxial approximation, theythat do the not laser generate pulses aberrat are localised at a distance power laser is focused to the diameter next mirror. To be concrete, we will also assume that at the cr the two planes are assumed to be at a distance wavelength close to R of the mirrors, since any uncertainty in the direction will b of any gain factor ( 6 Improving the sensitivityThe with sensitivity multiple of crossing ourcross proposed each experiments other several can times,of be parabolic using enhanced mirrors a as kind shown of in wave figure guide con detection of PPS of QED origin and for measuring the paramete 9 JHEP11(2009)043 , , t ), z + φ /J k ′ 3 t ′ t (or . D, or = m ω 3 βy/c +∆ y. As a 30 ′ z − 10 − k − ωt ′ t 2). Taking z ∼ ( 10 the vertical − ′ z γ ) k θ/ y × y θ 2) taking into y 0 = . ∆ k 0 and θ/ ′ sin( t R and + ( k ≡ ~ z z ]), the gain factor is d/ ) z 1 = k . the reference system in z p eam that crosses an high βω/c N s are antiparallel. In fact, itional contributions from Lagrangian that describes olving MCPs or axion-like ion of the two experiments a, cility HERCULES, without d being as small as 3 − reflections, this will produce , we have: ection [ ng for the detection of PPS of y ts 1) and 2) that we proposed shows how the PVLAS limits sitron pairs. 1 T N of the mirrors can currently be k nt by PVLAS. The predictions ξ 2 ( d proposed a set of experiments γ ar bosons, can sum with them and and = 98. Moreover, the phase shift is due L . ′ y ξ 0 ]. After k 24 ≃ [ r that we have obtained above, we can find a = 1), figure of this configuration, we note that after each rad ), and , showing the 95% C.L. exclusion regions that F F 8 plane, as discussed in section 2. y 2 – 10 – − F T ξ 10 βck ∼ . It is then easy to see that the phase − 2 and θ ω 2) in this configuration. This result can also be obtained β ( L γ ξ θ/ − 1 = , where we have included a further factor sin( sin( p ′ . A similar result can be obtained in the case of orthogonally n π ω r on the position of the spot at the focus. This uncertainty mus cτ = = , θ θ z +1 γ =0 φ 50, that in principle can be achieved with present technolog respectively, thus allowing to detect PPS as predicted by QE ∆ N n appears to the third power in the expression of the phase shif = ≃ P ′ /J 2)∆ k and NR 3 z 2) , when translated to the laboratory system, gives becomes θ/ ′ m max ( ∼ ), z z /ω = 1000 in the expression of θ/ 4 F 30 y ( ∆ 4 − βct N L ck ξ 10 3 − = sin = ) × ′ z y = sin φ ( β k 7 ( . γ F 2 In order to compute the gain factor Even with the single-crossing version ( Our results are summarised in figure Taking A = 0 ′ ǫ equivalently that it appears to the sixth power in the cross s into account that to the counter-propagating components of the photon moment reflection the intensity is reduced by a factor result, the measurement of theabove phase with shifts such in a gain the factor experimen will be able to resolve limiting value y times the beams are reflected. In principle, the orientation fixed with a precision asan small uncertainty as ∆ 2 find a signal ofwill non-standard also physics. be Note ableparticles, that taking the to into combinat test account Born the discussion Infeld of theory section and 2. scenarios7 inv Conclusions We have computed the phasepower shifts laser affecting pulse a with lowto general power transverse completely laser polarisation, b determine an thePPS well parameter below space the of threshold for the the effective creation of electron-po can be substantially improved at HERCULES, possibly allowi can be obtained withor this with set multi-crossing, of as experiments comparedof at to QED the the present and current fa constrai Born-Infeldminicharged particles, theory or are axion-like scalar explicitlyproduce or a indicated. pseudoscal different point Add in the then with ∆ and 1 be smaller than the diameter of the beams at the focus, so that and horizontal directions in thelaboratory system of figure polarized waves. account that ∆ in more elegant and rigorous way by making the computation in which the total momentum isby zero indicating with and a the prime two the colliding quantities photon in such a system, an where JHEP11(2009)043 2 = 44 aligned N e result of figure w class of precision 25 are measured in units of - 239155PR, and by the T 10 ξ eory or scenarios involving for useful discussions. D.T. on-like particles, and Frank and IMPA Groups for their ss mechanism, HERCULES 26 and um value that we have found - 7-29090-E and FIS2008-0100. bed at HERCULES with single the point is the QED prediction. nt. The next two inner regions L e parameters. 10 ξ lencia. This work was partially hat can be achieved in the mea- ossing. ar bosons. 27 . We are also grateful to Heinrich - 10 28 L - Ξ 10 = 30), respectively. Finally, the last inner region = 30, corresponding to just F 29 – 11 – F - 10 30 - 10 31 - 27 28 29 30 31 25 26 , we also see how the sensitivity will be improved at ELI in the 10 ------2

10 10 10 10 10 10 10 T Ξ = 1) or multiple crossing (choosing F . Exclusion plot for the search of PPS. The parameters . The diagonal line corresponds to Born Infeld theory, while We think that this proposal can eventually contribute to a ne /J 3 represents the range that can be reached at ELI with singlenon-standard cr origin. On thewould other already hand, be by using able a to multi-cro detect PPS of QED origin. Note that th m The darkest regioncorrespond is to excluded the bycrossing parts the ( of current the PVLAS parameter constrai space that can be pro mirrors, which is a moreabove. realistic assumption Finally, than in the figure maxim is obtained using the conservative value Figure 2 future, thus allowing for a more precise determination of th tests of QED and non-standardminicharged models, particles such or as axion-like, Born-Infeld scalar Th or pseudoscal Acknowledgments We thank Eduard Masso for explainingWise us and the Guido contraints Zavattini on for axi surements clarifying of the phase sensitivities shifts t Hora and and Luis ellipticities, Plaja respectively for helpfulthanks remarks Pedro de and and Fernández to Córdoba the Bernardo Adeva wholevery InterTech nice hospitality atsupported the Universidad by de Politécnica Xunta Va deGovernment of Galicia, Spain, by by contracts contract N. 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