Minicharged particles search by strong laser pulse-induced vacuum polarization effects
aa r X i v : . [ h e p - ph ] N ov Minicharged particles search by strong laser pulse-induced vacuum polarization e ff ects S. Villalba-Ch´avez a , S. Meuren b , C. M¨uller a a Institut f¨ur Theoretische Physik I, Heinrich-Heine-Universit¨at D¨usseldorf, Universit¨atsstr. 1, 40225 D¨usseldorf, Germany b Max-Planck-Institut f¨ur Kernhysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Abstract
Laser-based searches of the yet unobserved vacuum birefringence might be sensitive for very light hypothetical particles carrying atiny fraction of the electron charge. We show that, with the help of contemporary techniques, polarimetric investigations driven byan optical laser pulse of moderate intensity might allow for excluding regions of the parameter space of these particle candidateswhich have not been discarded so far by laboratory measurement data. Particular attention is paid to the role of a Gaussian waveprofile. It is argued that, at energy regimes in which the vacuum becomes dichroic due to these minicharges, the transmissionprobability of a probe beam through an analyzer set crossed to the initial polarization direction will depend on both the inducedellipticity as well as the rotation of the initial polarization plane. The weak and strong field regimes, relative to the attributes ofthese minicharged particles, and the relevance of the polarization of the strong field are investigated.
Keywords:
Beyond the Standard Model, Minicharged Particles, Vacuum polarization, Laser Fields.
PACS:
1. Introduction
The Standard Model of particle physics is currently under-stood as an e ff ective theory, where charge quantization seemsto be conceived as a fundamental principle. Standard Modelextensions–which are required for other reasons–can be foundeither by enforcing the mentioned quantization through highergauge groups or by incorporating carriers of a tiny charge q ǫ = ǫ | e | , with ǫ denoting the parameter relative to the absolute valueof the electron charge e < ǫ ≪ ff ects in the stellarevolution [13] [ ǫ . − for masses m ǫ below a few keV]and the analysis of the big bang nucleosynthesis [ ǫ < − for m ǫ < testsfor modifications in Coulomb’s law [26, 27] or through highprecision experiments looking for magnetically-induced vac-uum birefringence and vacuum dichroism [28–32]. In the last
Email addresses: [email protected] (S.Villalba-Ch´avez), [email protected] (S. Meuren), [email protected] (C. M¨uller) An alternative regeneration setup based on static magnetic fields has beenproposed in Ref. [25]. A more extended phenomenological overview on MCPs as well as otherweakly interacting particles can be found in the reviews [33–36]. scenario the bound is the more stringent the greater the fieldstrength and its spatial extension are. However, in laboratories,the highest constant magnetic fields do not exceed values of theorder of ∼ G, which are extended over e ff ective distancesof upto 10 −
100 kilometers using Fabry-P´erot cavities.Fields generated from high-intensity lasers might be bene-ficial for these laboratory searches. Indeed, the chirped-pulseamplification technique has enabled us to reach very strongmagnetic field strengths, at the expense of being distributed in-homogeneously over regions of only a few micrometers [37].Strengths as large as ∼ G are accessible nowadays andwill likely exceed values of the order of ∼ G at forthcom-ing laser systems such as ELI and XCELS [38, 39]. This factalso justifies why high-intensity laser pulses are currently con-sidered as valuable instruments for detecting various nonlinearphenomena that have eluded their observation so far. Notably,to measure vacuum birefringence [40–44], the HIBEF consor-tium has proposed a laser-based polarimetric experiment whichcombines a Petawatt optical laser with a x-ray free electronlaser [45, 46]. Meanwhile, alternative setups are being pro-posed for improving the levels of sensitivity necessary for thedetection of this elusive phenomenon [47–50]. Clearly, exper-iments of this nature might also constitute sensitive probes foraxion-like particles [51–55], MCPs and paraphotons [56–59].This forms the main motivation for this work. In this Letterwe show that a polarimetric probe driven by the field of a highintensity linearly polarized Gaussian laser pulse might notablyimprove the existing laboratory limits in some regions of theparameter space of MCPs.Our investigation relies on the one-loop representation ofthe polarization tensor in a plane-wave background [60–62] inwhich the two-point correlation function for MCPs incorporates
Preprint submitted to Elsevier November 11, 2016 he field of the laser pulse in a nonpertubative way [Furry pic-ture]. The weak and strong field regimes, relative to the at-tributes of these degrees of freedom, are investigated and asymp-totic expressions for the observables are derived [see Sec. 3for more details]. In the weak field case, dispersive e ff ectsare found to be maximized at the threshold of pair produc-tion of MCPs, in agreement with the cross section of light-by-light scattering. Finally, a comparison between the present re-sults and those previously obtained for a circularly polarizedmonochromatic plane-wave background [57, 58] is established.
2. Photon propagation in MCPs vacuum
We wish to evaluate the e ff ects induced by quantum vac-uum fluctuations dominated by Dirac fields characterized by amass m ǫ and a tiny fraction of the absolute value of the electroncharge q ǫ ≡ ǫ | e | . As long as such fields are minimally coupledto an electromagnetic field and the corresponding functionalaction preserves the formal invariance properties of QuantumElectrodynamics (QED), the underlying theory would resem-ble the corresponding phenomenology. Accordingly, the equa-tion of motion–up to linear terms in the small-amplitude wave a µ ( x )–has the form (cid:3) a µ ( x ) + Z d x ′ Π µν ( x , x ′ ) a ν ( x ′ ) = , (1)provided the Lorenz gauge ∂ µ a µ = (cid:3) ≡ ∂ µ ∂ µ = ∂ /∂ t − ∇ , whereas the second term in Eq. (1) in-troduces the vacuum polarization tensor Π µν ( x , x ′ ). This ob-ject is basically the same as in QED, with the positron param-eters ( | e | , m ) substituted by the respective quantities associatedwith the MCP ( q ǫ , m ǫ ). It constitutes the lowest nontrivial one-particle irreducible vertex from which the gauge sector of QEDcan acquire a dependence on the external background field. Itsfour-potential is taken hereafter as A µ ( x ) = a µ ψ ( ϕ ) + a µ ψ ( ϕ ) , (2)where a , are two orthogonal amplitude vectors [ a a = ψ , ( ϕ ) arbitrary functions of the strong plane-wave phase ϕ = κ x . The external potential is chosen in the Lorenz gauge ∂ µ A µ = κ µ = ( κ , κκκ ) with κ = a µ , satisfy the constraints κ a , = Λ µ , ( q ) = − F µν , q ν κ q q − a , , Λ µ , ( q , ) = κ µ q , − q µ , ( q κ ) κ q q q , , (3)which are built up from the amplitudes of the external fieldmodes F µν i = κ µ a ν i − κ ν a µ i [ i = , q and q as well as the wave four-vector κ . We note that the short-hand notation q in Eq. (3) may stand for either q or q due From now on “natural” and Gaussian units c = ~ = πǫ = to momentum conservation. The set of four-vectors q , Λ ( q ), Λ ( q ) and Λ ( q ), form a complete orthonormalized basis, i.e., Λ µ i ( q ) Λ j µ ( q ) = − δ i j , g µν = q µ q ν / q − P i = Λ µ i ( q ) Λ ν i ( q ) with g µν = diag( + , − , − , −
1) denoting the metric tensor. A simi-lar statement applies to the set of four-vectors q , Λ ( q ) , Λ ( q )and Λ ( q ).Let us proceed by Fourier transforming Eq. (1). In thefollowing we will seek the solutions of the resulting equationin the form of a superposition of transverse waves a µ ( q ) = P i = , Λ µ i ( q ) f i ( q ). Correspondingly, q f i ( q ) = − X j = , Z ¯ d q Λ µ i ( q ) Π µν ( − q , − q ) Λ ν j ( q ) f j ( q ) , Π µν ( q , q ) = ¯ δ q , q κ + Z d ϕ P µν ( ϕ, q , q ) exp " i ( q − q ) + κ + ϕ , (4)where we have introduced the shorthand notations ¯ d ≡ d / (2 π )and ¯ δ q , q ≡ (2 π ) δ ( ⊥ ) ( q − q ) δ ( − ) ( q − q ). Note that quantitieswith subindices ± and ⊥ refer to light-cone coordinates. Wechoose the reference frame in such a way that the direction ofpropagation of our external plane wave [see Eq. (2)] is alongthe positive direction of the third axis. As a consequence, thestrong field only depends on x − = ( x − x ) / √ ϕ = κ + x − with κ + = ( κ + κ ) / √ = √ κ > x + = ( x + x ) / √ xxx ⊥ = ( x , x ) can beintegrated out without complications.Although the expression above holds for arbitrary externalfield profiles, it still requires a transversely homogeneous field.As a consequence, qqq ⊥ is conserved [see the associated Diracdelta in Eq. (4)], which constitutes a good approximation when-ever the Compton wavelength of the MCP ¯ λ ǫ = / m ǫ becomesmuch smaller than the transverse length scale over which thefield is homogeneous. For a focused laser beam this scale is setby the waist size of the pulse w . Therefore, the plane-waveapproximation is valid in the regime m ǫ ≫ w − . The study ofthe regime m ǫ . w − , where spatial focusing e ff ects becomeimportant, is beyond the scope of the present investigation.The tensorial structure of P µν ( ϕ, q , q ) can be determinedon the basis of symmetry principles, independently of any ap-proximation used to compute the polarization tensor [60, 62]. Itreads P µν ( ϕ, q , q ) = c Λ µ Λ ν + c Λ µ Λ ν + c Λ µ Λ ν + c Λ µ Λ ν + c Λ µ Λ ν . (5)As q − q ∼ κ this decomposition does not depend on whichchoice of q is taken; see also Eq. (3). The form factors c i inEq. (5) depend–among other parameters–on the phase of theexternal field ϕ , q and q . In the one-loop approximation–which is adopted from now on–they turn out to be representedby two-fold parametric integrals in the variables τ ∈ [0 , ∞ )and v ∈ [0 , − im ǫ τ + i µ q ] × (cid:16) Regular Function in q , q and κ q (cid:17) (6)2ith µ = τ (1 − v ). After a suitable integration by parts the reg-ular function becomes independent of q [see Ref. [63], App. Dfor more details], which is assumed in the following.When polarization e ff ects do not dramatically modify thephoton dispersion law in vacuum [ q = f i ( q ) ≈ f i ( q ) + δ f i ( q ). In the following,we suppose a head-on collision between the strong laser pulseand the probe beam characterized by the four-momentum k µ = ( ω kkk , kkk ), so that κ + k − = ω kkk κ and kkk ⊥ = f i ( q ) = | q − | a i ¯ δ ( q ) ¯ δ ( ⊥ ) ( q ) ¯ δ ( − ) ( q − k ),corresponding to f i ( x ) = a i e − i φ with φ = kx = k − x + and a i the amplitude of mode- i . Then, it follows from Eq. (4) that theperturbative contribution is given by δ f i ( q ) = − [2 q + q − − q ⊥ + i − × X j = , a j Λ µ j ( k ) Π µν ( k , q ) Λ ν i ( q ) , (7)where it must be understood that the only nonvanishing light-cone component of the four-vector k µ is k − . Besides, in obtain-ing the expression above we have used the symmetry property Π µν ( − q , − q ) = Π νµ ( q , q ). Here, the poles in the function1 / q have been shifted infinitesimally into the complex planeby an i f i ( ±∞ , xxx ) are implemented. In this case,the solution of Eq. (1) is given by a µ ( x ) = P i = , Λ µ i ( k ) f i ( x ) [seeabove Eq. (4)] with f i ( x ) ≈ f i ( x ) − κ + k − X j = , f j ( x ) Z d ˜ ϕ Z ¯ dq + × e iq + κ + ( ˜ ϕ − ϕ ) Λ µ j ( k ) P µν ( ˜ ϕ, k , q ) q + + i Λ ν i ( q ) . (8)Here, q − = k − , qqq ⊥ = kkk ⊥ =
000 and k + =
0. In orderto provide a more concise expression for f i ( x ), we integrate out q + . This can be carried out by applying Cauchy’s theorem andthe residue theorem, depending upon whether the contour ofintegration is chosen in the upper or lower half of the complexplane. Taking into account the structure of the integrand withrespect to q + [see Eq. (6) and the discussion below], we obtain Z ¯ dq + . . . = − i Λ µ j ( k ) P µν ( ˜ ϕ, k , k ) Λ ν i ( k ) Θ ( ϕ − ˜ ϕ ) , (9)where Θ ( x ) denotes the unit step function. Its emergence re-stricts the integral over ˜ ϕ to ( −∞ , ϕ ] instead of ( −∞ , ∞ ), as re-quired by causality. However, we are only interested in asymp-totically large spacetime distances [ ϕ → ∞ ], i.e., when thehigh-intensity laser field is turned o ff , which restores the origi-nal integration limits. Therefore, inserting this expression intoEq. (8) and taking into account the tensorial decomposition ofthe polarization tensor [see Eq. (5)], we end up with f i ( x ) ≈ f i ( x ) + i κ + k − f ( x ) Z ϕ −∞ d ˜ ϕ (cid:2) c ( ˜ ϕ ) δ i + c ( ˜ ϕ ) δ i (cid:3) + i κ + k − f ( x ) Z ϕ −∞ d ˜ ϕ (cid:2) c ( ˜ ϕ ) δ i + c ( ˜ ϕ ) δ i (cid:3) . (10) The expression above constitutes the starting point for furtherconsiderations. It holds for arbitrary strength and polarizationof the background field, as long as the vacuum polarization issmall. When specifying Eq. (10) to the case of a linearly po-larized plane-wave background, i.e. Eq. (2) with ψ ( ϕ ) = c , vanish [60, 62] and the resulting expres-sion agrees with Eq. (16) in Ref. [63], provided the involvedexponential function is expanded to leading order. However,we emphasize that the aforementioned solution has been estab-lished for the field regime in which the laser intensity parameter ξ = | e | √− a / m with a µ ≡ a µ is very large [ ξ ≫ εεε ( x ) = − ∂ aaa /∂ x with a =
0] as a superposition of plane-waves εεε ( x ) ≈ ε cos( ϑ ) ΛΛΛ Re e − i φ + i κ + k − R ϕ −∞ d ˜ ϕ c ( ˜ ϕ ) + ε sin( ϑ ) ΛΛΛ Re e − i φ + i κ + k − R ϕ −∞ d ˜ ϕ c ( ˜ ϕ ) . (11)Here, ε refers to the initial electric field amplitude, ΛΛΛ , = aaa , / | aaa , | , whereas 0 ϑ < π is the corresponding initialpolarization angle of the probe with respect to ΛΛΛ , i.e., the po-larization axis of the external pulse. Observe that the appear-ance of the phase is due to the approximation 1 + ix ≈ exp( ix )as in Ref. [42].The P µν − form factors are, in general, complex functions c , = Re c , + i Im c , . Correspondingly, the exponents inEq. (11) contain real and imaginary contributions. The latter areconnected to the photo-production of MCP pairs via the opticaltheorem [63, 64], a phenomenon which damps the intensity ofthe probe, I ( ϕ ) = ε π cos ( ϑ ) exp( − κ ) + ε π sin ( ϑ ) exp( − κ ),as it propagates in the pulse. As such, the analytic properties ofthe factors κ , ≡ κ , ( ϕ ) = κ + k − Im R ϕ −∞ d ˜ ϕ c , ( ˜ ϕ ), responsiblefor the damping di ff er from each other, leading to a nontriv-ial di ff erence δκ ( ϕ ) = κ + k − Im ∆ ( ϕ ), where we introduced thefunction ∆ ( ϕ ) = Z ϕ −∞ d ˜ ϕ (cid:2) c ( ˜ ϕ ) − c ( ˜ ϕ ) (cid:3) . (12)Therefore, the vacuum behaves like a dichroic medium, induc-ing a rotation of the probe polarization from the initial angle ϑ to ϑ + δϑ , where δϑ is expected to be tiny. At asymptoticallylarge spacetime distances [ ϕ → ∞ ], we find | δϑ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im ∆ ( ∞ )2 κ + k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ . (13)As the phase di ff erence between the two propagating modes, δφ ( ϕ ) = κ + k − Re ∆ ( ϕ ), does not vanish either [see Eq. (11)], thevacuum is also predicted to be birefringent. Hence, when thestrong field is turned o ff [ ϕ → ∞ ], the outgoing probe shouldbe elliptically polarized and its ellipticity is given by [65] [notethat in this reference a di ff erent notation is used] | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re ∆ ( ∞ )2 κ + k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ . (14)In the case of optical probes, isolated detections of the rotatione ff ect [see Eq. (13)] and the ellipticity [see Eq. (14)] could be3arried out depending on whether a quarter wave plate is in-serted or not in the path of the outgoing probe beam in frontof a Faraday cell and an analyzer [28, 29, 32]. The latter isset crossed to the initial direction of polarization so that thetransmitted photons are polarized orthogonally. Correspond-ingly, no photons are detected in the absence of birefringenceand dichroism. Using high-purity polarimetric techniques for x-rays [66, 67] (QED) vacuum birefringence could also be mea-sured with a similar setup by combining a x-ray probe and astrong optical field [QED-induced dichroism is exponentiallysmall, thus δϑ QED = T = i e ( i ) µ h Π µν QED ( k , k ) + Π µν ( k , k ) i e ( f ) ν / [2 V ( ω kkk ω kkk ) / ]. The expres-sion above includes both, the polarization tensor associated withQED Π µν QED ( k , k ) and the one related to the MCPs. Besides, V denotes the normalization volume, whereas e ( i ) µ and e ( f ) µ arethe initial and final polarization states, respectively. Follow-ing Eq. (11), we suppose that the former is of the form e ( i ) = cos( ϑ ) Λ + sin( ϑ ) Λ . In contrast, the polarization state trans-mitted by the analyzer is e ( f ) = ± sin( ϑ ) Λ ∓ cos( ϑ ) Λ , so that e ( i ) e ( f ) =
0. Finally, we establish the following expression forthe transmission probability [ δϑ QED = P = (cid:2) ψ QED + ψ ( ǫ, m ǫ ) (cid:3) + δϑ ( ǫ, m ǫ ) . (15)This expression indicates that the described setup is not suitableto probe the signals separately. However, one could achieve thisgoal by determining the local minimum of the count rate behindthe analyzer, which is no longer perpendicular to the incomingpolarization direction but shifted by δϑ ( ǫ, m ǫ ) [68]. We indeedfind that in such a configuration, the transmission probability P min = | eee · εεε | / | ε | with eee = ± sin( ϑ + δϑ ) ΛΛΛ ∓ cos( ϑ + δϑ ) ΛΛΛ ,is given by the first term on the right-hand side of Eq. (15). Inconnection, the number of photons transmitted through the ana-lyzer reads N ≈ N in N shot T h ψ + ψ QED ψ ( ǫ, m ǫ ) i , providedthat QED e ff ects are dominant [ ψ QED > ψ ( ǫ, m ǫ )]. Here, N shot counts the number of laser shots used for a measurement, T denotes the transmission coe ffi cient of all optical componentsand N in is the number of incoming x-ray probe photons, re-spectively.
3. Asymptotic regimes
We wish to investigate the optical observables [Eq. (13) and(14)] induced by a plausible existence of MCPs. Since bothdepend on ∆ ( ∞ ) [see Eq. (12)], we will focus on determiningthis function. Indeed, a suitable expression can be inferred fromthe literature [60, 62]. In the one-loop approximation we find: ∆ ( ∞ ) = α ǫ π m ǫ ξ ǫ Z ∞−∞ d ϕ Z − dv × Z ∞ d ττ X ( ϕ ) exp h − im ∗ ( ϕ ) τ i , (16) where α ǫ ≡ ǫ e ≈ ǫ /
137 denotes the fine structure constantrelative to the MCPs, whereas ξ ǫ = ǫ m ξ/ m ǫ is the relative in-tensity parameter. The remaining functions involved in this ex-pression can be conveniently written in the following form X ( ϕ ) = µ (2 κ + k − ) Z dy Z d ˜ y y (˜ y − ψ ′ ( ϕ y ) ψ ′ ( ϕ ˜ y ) , m ∗ ( ϕ ) = m ǫ ( − ξ ǫ µ (2 κ + k − ) Z dy y ψ ′ ( ϕ y ) × "Z d ˜ y ˜ y ψ ′ ( ϕ ˜ y ) − Z y d ˜ y ψ ′ ( ϕ ˜ y ) , (17)where µ = τ (1 − v ) and ϕ y = ϕ − κ + k − ) µ y . These ex-pressions apply for a linearly polarized plane-wave background[ ψ ( ϕ ) ≡ ψ ( ϕ ) and ψ ( ϕ ) = ∆ ( ∞ ) [see Eq. (12)] is quite di ffi cult to perform. Therefore, weconsider now some asymptotic expressions of interest. ξ ǫ ≫ ξ ǫ ≫ τ = ρ/ [ | κ + k − | (1 − v )]. The resulting integration over ρ is divided into two contributions whose domains run from 0to ρ and from ρ to ∞ . The dimensionless parameter ρ > ξ − ǫ ≪ ρ ≪ η ǫ /ξ ǫ ) / ≪ ρ with η ǫ = κ + k − / m ǫ . Inthe former integral we Taylor expand the functions given inEq. (17): X ( ϕ ) ≈ − ρ [ ψ ′ ( ϕ )] and m ∗ ( ϕ ) ≈ m ǫ (cid:20) + ξ ǫ ρ [ ψ ′ ( ϕ )] (cid:21) .Afterward, we perform the change of variable s = ρξ ǫ and ex-tend the resulting integration limit ρ ξ ǫ → ∞ . No relevant con-tribution comes from the integral defined in [ ρ , ∞ ). Therefore,in the strong field regime ξ ǫ ≫ η ǫ ≪ ξ ǫ ], the function ∆ ( ∞ )[see Eq. (16)] is well approximated by ∆ ( ∞ ) = − α ǫ m ǫ Z ∞−∞ d ϕ Z − dv " Gi ′ ( x ) x + i Ai ′ ( x ) x . (18)Here, x = (cid:16) / [ | ζ ǫ ( ϕ ) | (1 − v )] (cid:17) / , Gi( x ) and Ai( x ) are the Scorerand Airy functions of first kind [69], respectively. In this con-text, ζ ǫ ( ϕ ) = χ ǫ ψ ′ ( ϕ ) /
2, with χ ǫ = ξ ǫ η ǫ , refers to the pulse-modulated nonlinear parameter associated with the MCP vac-uum.We proceed our analysis by inserting the imaginary part ofEq. (18) into Eq. (13). As a consequence of the relation Ai ′ ( z ) = − z π √ K / (cid:16) z / (cid:17) , with a modified Bessel function K ν ( z ) [69], thefollowing representation for the rotation angle is found | δϑ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) α ǫ m ǫ √ π ( κ + k − ) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ d ϕ Z dv K / | ζ ǫ ( ϕ ) | − v !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)Likewise, by substituting the real part of Eq. (18) into Eq. (14),4e find for the ellipticity | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) α ǫ m ǫ / ( κ + k − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ d ϕ | ζ ǫ ( ϕ ) | / × Z dv (1 − v ) / Gi ′ | ζ ǫ ( ϕ ) | − v ! / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (20)Eqs. (19) and (20) are used in the next section to estimate theprojected bounds in the parameter space of MCPs. Note that anumerical comparison between these expressions and the cor-responding ones resulting from Eqs. (13), (14) and (16) agreeswithin a few percent whenever ξ ǫ ≫ ζ / ǫ ≪ ξ ǫ , in agree-ment with the conditions imposed above Eq. (18).In addition, further insights can be gained by restricting ζ ǫ = χ ǫ / ζ ǫ ≪
1. To be consistent with ξ ǫ ≫ η ǫ mustbe restricted to η ǫ ≪ / (3 ξ ǫ ). In this limit we can exploit theasymptotes K ν ( z ) ∼ q π z e − z and Gi( z ) ∼ π z [69]. With these ap-proximations, the integrations over v can be performed in bothobservables. The expression for the ellipticity becomes partic-ularly simple and can be computed exactly. Conversely, thecalculation of the integral contained in the rotation angle re-quires additional approximations. To this end, we first applythe change of variable w = (cid:16) − v (cid:17) − and note that the region w ∼ | δϑ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) α ǫ m ǫ √ κ + k − ) Z ∞−∞ d ϕ | ζ ǫ ( ϕ ) | e − | ζǫ ( ϕ ) | , | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) 2 α ǫ m ǫ π ( κ + k − ) Z ∞−∞ d ϕ | ζ ǫ ( ϕ ) | . (21)The situation is di ff erent when ξ ǫ ≫ ζ / ǫ ≫
1. In this case, K ν ( z ) ∼ Γ ( ν )2 (cid:16) z (cid:17) ν and Gi( z ) ∼ π / Γ (cid:16) (cid:17) + π / Γ (cid:16) (cid:17) z applies[69]: | δϑ ( ǫ, m ǫ ) | ≈ √ | ψ ( ǫ, m ǫ ) | , | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) 2 / α ǫ m ǫ Γ ( )7 √ π ( κ + k − ) Γ ( ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ d ϕ | ζ ǫ ( ϕ ) | / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (22)where Γ ( x ) denotes the Gamma function. We remark that, ifthe external background is a constant crossed field [ ψ ′ ( ϕ ) =
1] which extends over ∆ x − , the ellipticity in Eq. (21) agreeswith Eq. (50) in Ref. [42], provided the distance traveled by theprobe is given by d = √ ∆ x − and ϑ = π/ ψ ′ ( ϕ ). To proceed further, we take it of the form ψ ′ ( ϕ ) = e − ϕ ∆ ϕ sin( ϕ ) . (23)Here, ∆ ϕ = π N / √ N referring to the number ofoscillation cycles within the Gaussian envelop (FWHM). Weinsert this function into the expression for the ellipticity [seeEq. (21)] to establish | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) α ǫ m ǫ ζ ǫ ∆ ϕ √ π ( κ + k − ) h − e − ∆ ϕ i . (24) The expression given in Eq. (24) is valid if simultaneously ξ ǫ ≫ ζ ǫ ≪
1. For ξ ǫ = ζ ǫ = .
15 [ ζ ǫ = /
2] and ∆ ϕ = π , itdi ff ers from the exact formula Eq. (14)–with Eqs. (16) and (17)included–by only 0 .
2% [13%].The integrals which remain in | δϑ ( ǫ, m ǫ ) | [see Eq. (21)] can-not be computed analytically. To approximate them, we write Z ∞−∞ d ϕ . . . = ζ ǫ ∞ X n = ( − n − Z n π ( n − π d ϕ ψ ′ ( ϕ ) e ζǫ ( − n ψ ′ ( ϕ ) , assume that ζ ǫ ≪
1, and apply the Laplace method. To this endwe first note that the integrands vanish at the boundaries andthat the main contributions in the series arise from those valuesof ϕ which satisfy the condition ( n − π < ϕ < n π < √ ∆ ϕ .Therefore, the series can be cut o ff at N max = ⌊ + N / ln(2) ⌋ ,where ⌊ x ⌋ refers to the integer value of x . In addition, for thestationary points the condition ∆ ϕ /ϕ ≫ ϕ ≈ (2 n − π/ n ∈ N . As aconsequence, Z ∞−∞ d ϕ . . . ≈ ζ / ǫ r π N max X n = γ n e − ζǫ γ n , (25)with the parameter γ n = exp[(2 n − π / (8 ∆ ϕ )]. We insertthis approximation into | δϑ ( ǫ, m ǫ ) | [see Eq. (21)] and assume N ≈
5. Then, the main contribution arises from the first termof the series above. Explicitly, | δϑ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) 18 α ǫ m ǫ ζ / ǫ ( κ + k − ) r π γ e − ζǫ γ . (26)This result provides evidence that the photo-production proba-bility of a pair of MCPs is suppressed as ∼ exp( − γ /ζ ǫ ), when-ever ξ ǫ ≫ ζ ǫ ≪
1. This is expected because the damp-ing factors of the probe κ , [see above Eq. (12)] represent theprobability of producing a pair from the respective propagatingmode [63].The integration which remains in Eq. (22) can be estimatedby replacing the periodic term | sin( ϕ ) | / by its average value, h| sin( ϕ ) | / i = q π Γ ( ) Γ ( ) . Correspondingly, the ellipticity [rota-tion angle] acquires the form | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) 187 √ π α ǫ m ǫ ( κ + k − ) Γ ( ) Γ ( ) (cid:18) ζ ǫ (cid:19) / ∆ ϕ, | δϑ ( ǫ, m ǫ ) | ≈ √ | ψ ( ǫ, m ǫ ) | . (27)These analytical results were derived by assuming that ξ ǫ ≫ ζ / ǫ ≫
1. The expression for the ellipticity [rotation angle]given in Eq. (27) agrees with Eq. (20) [Eq. (19)] within an ac-curacy of <
19% [ < ζ ǫ > for ∆ ϕ = π .Some comments are in order. First of all, while Eq. (24) isexact with respect to the integration over ϕ , the approximationsused to obtain Eqs. (26) and (27) prevent us from taking themonochromatic limit [ ∆ ϕ → ∞ ] directly. Instead, this limitingcase can be derived by noting that the integrands in | δϑ ( ǫ, m ǫ ) | π -periodic. In this situation, wehave R ∞−∞ d ϕ . . . = N R π d ϕ . . . with N → ∞ and thus, Z ∞−∞ d ϕ . . . = π N √ π ζ / ǫ e − ζǫ ζ ǫ ≪ D | sin( ϕ ) | / E ζ ǫ ≫ , where the result for ζ ǫ ≪ − γ /ζ ǫ ) /γ → Ne − ζǫ and ∆ ϕ → N √ π/ | δϑ ( ǫ, m ǫ ) | and | ψ ( ǫ, m ǫ ) | in the monochromatic limit. ξ ǫ ≪ ξ ǫ ≪
1, the pulse [see Eq. (2) with ψ ( ϕ ) = ∼ ξ ǫ in the polarization tensor Π µν ( x , x ′ ) describes the scattering of a probe photon by a photonof the laser pulse [photon-photon scattering]. Since the light-by-light scattering cross section is maximized in the vicinityof the pair creation threshold [ n ∗ = m ǫ / | κ + k − | ≈ ff ect around the threshold massfor MCPs m ≡ q | κ + k − | . This is understandable because,for such energies [ ω kkk ≈ m ǫ / κ ± δω with m ǫ / κ ≫ δω > q + ǫ q − ǫ field. In contrast, far from the threshold [ n ∗ → ∞ and n ∗ → ff ects are predicted to be much lesspronounced. Accordingly, we can expect less stringent boundsfor masses far away from the threshold mass.Above the pair production threshold 1 > n ∗ the imaginarypart of the polarization operator is di ff erent from zero and thevacuum becomes dichroic. Below threshold, absorptive phe-nomena may also occur, but such processes are less likely sincethey are linked to higher order Feynman diagrams involving–atleast–two photons of the external pulse. Contributions of higherorder processes k + n κ → q + ǫ + q − ǫ with n > ξ ǫ ≪
1. As before, we apply the change ofvariable τ = ρ/ [ | κ + k − | (1 − v )]. The resulting dressing factorin the e ff ective mass m ∗ − m ǫ ∼ ξ ǫ [see Eq. (17)] becomes verysmall in comparison with the leading order term m ǫ , allowing usto make an expansion in ξ ǫ which turns out to be valid whenever n ∗ ≪ ξ − ǫ . Afterward, the variable ϕ is integrated out using thepulse profile function [see Eq. (23)]. Correspondingly, ∆ ( ∞ ) = α ǫ π m ǫ ξ ǫ Z − dv Z ∞ d ρρ Z ∞−∞ d ϕ X ( ϕ ) exp " − in ∗ ρ − v , (28)where Z ∞−∞ d ϕ X ( ϕ ) = √ πρ ∆ ϕ Z dy Z dy ′ e − ρ y − y ′ )2 ∆ ϕ × ( y ′ − y n cos(2 ρσ [ y − y ′ ]) − exp (cid:16) − ∆ ϕ (cid:17)o . (29)Here, we introduced the parameter σ = κ + k − / | κ + k − | . Threeout of the four integrations can be carried out analytically. To this end, we first introduce two new variables s − = y − y ′ and z = y + y ′ and carry out the integrations over z and ρ . Withhelp of the shorthand notation ℓ s = n ∗ s / [ σ (1 − v )], we find atwo-fold integral representation for the real and the imaginarypart [see Eq. (28)]Im ∆ ( ∞ ) = α ǫ ( κ + k − ) ξ ǫ ∆ ϕ Z dv (1 − v ) Z ∞ dss × n e − ∆ ϕ (1 + ℓ s ) + e − ∆ ϕ (1 − ℓ s ) − e − ∆ ϕ ( + ℓ s ) o , (30)Re ∆ ( ∞ ) = √ π α ǫ ( κ + k − ) ξ ǫ ∆ ϕ Z dv (1 − v ) Z ∞ dss × (cid:26) D F ( ∆ ϕ [1 + ℓ s ]) − sig(1 − ℓ s ) D F ( ∆ ϕ | − ℓ s | ) − e − ∆ ϕ D F ( ∆ ϕℓ s ) o , (31)where D F ( x ) = e − x R x dte t is the Dawson function [69]. Now,we perform in Eqs. (30) and (31) the changes of variables x =∆ ϕ [1 + ℓ s ], x = ∆ ϕ [1 − ℓ s ] and x = ∆ ϕℓ s in the first, sec-ond and third contribution, respectively. After an integration byparts with respect to v , the integral over s is eliminated and weend up with the following expression for the rotation angle | δϑ ( ǫ, m ǫ ) | =
14 sin(2 ϑ ) α ǫ ξ ǫ ∆ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dv v (1 − v ) × " − v + v − v ! + v e − ∆ ϕ (1 + ℓ ) sinh (cid:16) ∆ ϕ ℓ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (32)and the induced ellipticity | ψ ( ǫ, m ǫ ) | =
12 sin(2 ϑ ) 14 √ π α ǫ ξ ǫ ∆ ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z dv v (1 − v ) × " − v + v − v ! + v D F ( ∆ ϕ [1 + ℓ ]) − D F ( ∆ ϕ [1 − ℓ ]) − e − ∆ ϕ D F ( ∆ ϕℓ ) (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (33)The expressions in Eqs. (32) and (33) hold for the pulse shapegiven in Eq. (23) and apply whenever ξ ǫ ≪ n ∗ ≪ ξ − ǫ .The numerical values provided by both expressions agree withthe exact results calculated from Eqs. (13) and (14), includingEqs. (16) and (17), within a few percent.It is interesting to deal with some special cases. Let usconsider first the rotation angle [see Eq. (32)]. Assuming thecondition ∆ ϕ > ∆ ϕ n ∗ ≫
1, one can use the approximationsinh ( ∆ ϕ ℓ ) ≈ exp[2 ∆ ϕ ℓ ] and apply the Laplace method.Finally, Eq. (13) leads to the expression | δϑ ( ǫ, m ǫ ) | ≈
14 sin(2 ϑ ) 18 α ǫ ξ ǫ ∆ ϕ √ π (1 − v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − v × ln + v − v ! + v +
12 Erf (cid:16) ∆ ϕ v (cid:17))(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (34)with Erf( x ) = √ π R x dt exp[ − t ] denoting the error function[69]. This formula applies as long as the condition ∆ ϕ − ≪ n ∗ < v = (1 − ∗ ) / defines the relative speed of the final particle states in thecenter–of–mass frame. In the monochromatic limit [ ∆ ϕ → ∞ ],the expression in Eq. (34) contained within the curly bracketsreduces to the unit step function Θ ( v ). We note that, for thetest parameters ξ ǫ = . n ∗ = .
02 and ∆ ϕ = π , the rela-tive di ff erence between this expression and the exact formulaEq. (13)–with Eqs. (16) and (17) included–is smaller than 3%.As ∆ ϕ n ∗ ≪ < ∆ ϕ implies sinh( ∆ ϕ ℓ ) ≈ ∆ ϕ ℓ [seeEq. (32)], we find that | δϑ ( ǫ, m ǫ ) | ∼ n ∗ ∆ ϕ exp( − ∆ ϕ ) is expo-nentially suppressed, which indicates that in this regime vac-uum dichroism tends to vanish.We point out that Eq. (24) also applies if ξ ǫ ≪ ≪ ∆ ϕ n ∗ . To show this, we use D F ( ∆ ϕ (1 ± ℓ )) ≈ ± D F ( ∆ ϕℓ ) ≈± / (2 ∆ ϕℓ ), implying R dv . . . ≈ (1 − e − ∆ ϕ ) in Eq. (33). Inthe regime ∆ ϕ n ∗ ≪ t = − v and introduce a splitting parameter t with ∆ ϕ n ∗ ≪ t ≪ t integration is divided into ranges from 0 to t and from t to 1. In the first region, we have t ≪ Z t dt . . . ≈ t ∆ ϕ n ∗ n D ′ F ( ∆ ϕ ) − e − ∆ ϕ o . (35)Since in the second range ∆ ϕ n ∗ ≪ t , we can expand the expres-sion contained in the curly brackets [see | ψ ( ǫ, m ǫ ) | in Eq. (33)]in ∆ ϕ n ∗ / t . Hence, Z t dt . . . ≈ ∆ ϕ n ∗ n D ′ F ( ∆ ϕ ) − e − ∆ ϕ o × Z t dt t √ − tt +
12 ln + √ − t − √ − t (36)To leading order, the remaining integral reads R t . . . ≈ (1 − t / | ψ ( ǫ, m ǫ ) | ≈
12 sin(2 ϑ ) α ǫ ξ ǫ n ∗ ∆ ϕ √ π × (cid:12)(cid:12)(cid:12)(cid:12) − ∆ ϕ D F ( ∆ ϕ ) − e − ∆ ϕ (cid:12)(cid:12)(cid:12)(cid:12) , (37)where D ′ F ( ∆ ϕ ) = − ∆ ϕ D F ( ∆ ϕ ) has been used [69]. Themonochromatic limit [ ∆ ϕ → ∞ ] can be investigated through D F ( ∆ ϕ ) ≈ ∆ ϕ − ∆ ϕ , in which case the induced ellipticity reads | ψ ( ǫ, m ǫ ) | ≈ sin(2 ϑ ) √ π α ǫ ξ ǫ ∆ ϕ n ∗ . Finally, as a check, wefound that for ξ ǫ = . n ∗ = .
02 and ∆ ϕ = π , the outcomesfrom Eq. (37) and the exact formula Eq. (14)–with Eqs. (16)and (17) included–agree within an accuracy of 0 .
4. Experimental prospects
We start by analyzing the HIBEF experiment proposed in[46], which is based on a Petawatt laser with κ ≈ .
55 eV[ λ =
800 nm], a repetition rate of 1 Hz, a temporal pulselength of about 30 fs [ ∆ ϕ ≈ π ], and a peak intensity I ≈ × W / cm corresponding to ξ ≈
69. The probe beam willbe produced by the European x-ray free electron laser [ ω kkk = . N in ≈ × photons per shot], the transmissioncoe ffi cient of the optics is T = . ϑ = π/
4] an ellipticity | ψ QED | = (9 . ± . × − rad wouldbe detectable [46]. Using Eq. (20), we infer that MCPs withrelative coupling constant ǫ < . × − would not be ruled outwhenever m ǫ .
100 eV. We have arrived at this limit by assum-ing that the induced ellipticity due to MCPs does not overpassthe upper bound set by the QED signal.As discussed below Eq. (4), the energy scale 1 / w associ-ated with the waist size of the pulse w limits the applicabilityof our method to the regime m ǫ ≫ w − [ w ≈ λ ≈ (0 .
12 eV) − for HIBEF]. For the detection of QED birefringence a detailedanalysis of focussing e ff ects has recently been carried out inRef. [48] based on an expression for the polarization opera-tor which was obtained from the Euler-Heisenberg Lagrangian[see also [40, 41]]. It was shown there that focussing e ff ectscould notably improve the signal-to-noise ratio if probe pho-tons which are scattered slightly away from the forward direc-tion are analyzed. Certainly, this fact might be beneficial in thesearch of MCPs as well. However, we point out that such astudy would require to incorporate transverse focusing e ff ectsin the polarization tensor. This computation is challenging inthe energy regimes considered here. Conversely, at low ener-gies ω kkk κ ≪ m ǫ , the Euler-Heisenberg Lagrangian could beused, but this calculation is beyond the scope of this work.Next, let us estimate the projected limits resulting from atechnically feasible experiment in which the rotation of the po-larization plane [see Eq. (13)] and the ellipticity [see Eq. (14)]are probed with an optical laser beam, but none of them is de-tected. In practice, the absence of these signals provides certainupper limits ψ CL% , δϑ CL% which are understood within certainconfidence levels, frequently corresponding to 2 σ . Hereafter,we take ψ CL% , δϑ
CL% ∼ − rad. This choice is in agreementwith the experimental accuracies with which both observablescan nowadays be measured in the optical regime. Here, theprojected sensitivities result from the inequalities 10 − rad > | ψ ( ǫ, m ǫ ) | and 10 − rad > | δϑ ( ǫ, m ǫ ) | . Firstly, we considerthe nanosecond front-end of the PHELIX laser [70], [ τ ≈
20 ns, κ ≈ .
17 eV implying ∆ ϕ ≈ × π , I max ≈ W / cm , ξ ≈ . × − , w ≈ − µ m] combined with a frequencydoubled probe beam [ ω kkk = κ = .
34 eV], having a waist sizeand an intensity much smaller than the corresponding ones ofthe strong laser field.The projected exclusion regions associated with this lasersetup are shaded in Fig. 1 in green and red. These should betrustworthy as long as the limits lie much below the curve cor-responding to ξ ǫ = ǫ m ξ/ m ǫ =
1, i.e. the white dashed line inthe upper left corner. We remark that our potential exclusionbounds are valid whenever the condition m ǫ ≫ w − is satisfied.This translates into m ǫ ≫ . τ ≫ κ − ] and, furthermore, satis-fies the condition w ≫ λ . Therefore, the electromagnetic fieldproduced by this laser system can be treated theoretically as amonochromatic plane wave. It is also worth observing that thesquare of the intensity parameter associated with the PHELIXbeam is much smaller than unity ξ ≪ ξ ǫ ≪ igure 1: Estimates of constraints for MCPs of mass m ǫ and relative coupling constant ǫ derived from the absence of signals in a plausible polarimetric setupassisted by a linearly polarized Gaussian laser pulse. In both panels, the white dashed line correspond to the expression ξ ǫ = vant parameter space]. Under these circumstances, the observ-ables [see Eqs. (13) and (14)] are dominated by a dependenceof the form ∝ ξ ∆ ϕ , as can be read o ff from Eqs. (32) and (33).This fact indicates that–for ω kkk ∼ ∆ ϕ compensates for the relative smallness of ξ . As we anticipated in Sec. 3.2, this enhancement is particu-larly large in the vicinity of the threshold mass m ≈ .
64 eVbecause the cross section for photon-photon scattering is max-imized nearby the pair creation threshold. Here, the projectedbound coming from a search of the induced ellipticity turns outto be ǫ < . × − .We note that the exclusion plot exhibits a discontinuity atthe threshold mass [see discussion below Eq. (34)]. Upperbounds for large masses can be derived when higher order proce-sses–such as the three photon reaction–are taken into account[57, 58]. The e ff ects resulting from this phenomenon are sum-marized in the right panel of Fig. 1 [orange area]. This outcomeas well as the one in darker cyan for the rotation angle were ob-tained previously by assuming the strong field as a circularlypolarized wave and considering a procedure beyond the Bornapproximation [57, 58]. We note that in the case of circular po-larization a slightly more stringent bound of ǫ < . × − at m ≈ .
64 eV results from the induced ellipticity.Both panels include regions colored in purple and black la-beled by PHELIX1000. These excluded areas have been deter-mined by using the PHELIX parameters given above but sup-posing that the signals gain sensitivity by a factor of ∼ ∼ W / cm and would require a collision angle very close to π .Besides, the mirrors should exceed the waist size of the pulsein order to avoid di ff ractive distortions; for further details see[47]. Using the same sensitivity of ∼ − as above, the ex-clusion limit is pushed down to ǫ < . × − at the thresholdmass m ≈ .
64 eV [for all projected sensitivities we assumea counter propagating geometry κ + k − = κ ω kkk and an initialpolarization angle ϑ = π/ τ ≈
13 fs, κ ≈ .
55 eV [ λ =
800 nm] correspond-ing to ∆ ϕ ≈ π , I ≈ W / cm , ξ ≈ . × . Here, weanalyze the results taking the probe beam with doubled fre-quency ω kkk = κ = . ψ CL% , δϑ
CL% ∼ − rad. Furthermore, a single-crossing ge-ometry is assumed again. The projected exclusion areas areshaded in the left panel of Fig. 1 in cyan and blue. Since thefield of the pulse at ELI is expected to be strongly focused[ w ∼ λ ], the estimates associated with this setup are expectedto be reasonable as long as m ǫ ≫ . ǫ lies much above the curves corresponding to ξ ǫ = ζ / ǫ = ξ ǫ . [Note that these curves lie far below the region en-compassed by the figure.] We observe that, in the ELI scenario,the path of the projected exclusion bounds resembles those es-tablished from experiments driven by constant magnetic fields[10–12].
5. Conclusions
We have studied the prospects that laser-based experiments,designed to detect vacuum birefringence, o ff er for probing hy-8othetical degrees of freedom with a tiny fraction of the electroncharge. Throughout this investigation, we have indicated thatthe vacuum of MCPs might induce ellipticity and rotation onthe incoming polarization plane, even though the probe photonenergy is much below the threshold of electron-positron pairproduction. In such a scenario, the transmission probabilitythrough an analyzer set crossed to the initial polarization direc-tion would not be determined solely by the QED ellipticity butalso by the ellipticity and the rotation angle induced by MCPs.We have argued that a slightly modified version of the proposedpolarimeter for a x-ray probe would allow for measuring bothsignals separately. The projected bounds resulting from thisanalysis will depend on the choice of the wave profile. In con-trast to previous studies, the treatment presented here has takeninto account the e ff ects resulting from a Gaussian envelop. Withthe help of contemporary techniques based on plasma mirrors,polarimetric studies driven by an optical laser pulse of moder-ate intensity [ ∼ W / cm ] might allow for excluding MCPswith ǫ > × − and masses 0 . m ǫ < . Acknowledgments
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