Muon pair creation by two x-ray laser photons in the field of an atomic nucleus
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Muon pair creation by two x-ray laser photons in the field of an atomic nucleus
Carsten M¨uller, Carlus Deneke, and Christoph H. Keitel
Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: October 27, 2018)The generation of muon-antimuon pairs is calculated in the collision of an ultrarelativistic bareion with an intense x-ray laser beam. The reaction proceeds nonlinearly via absorption of two laserphotons. By systematic study throughout the nuclear chart we show that the interplay between thenuclear charge and size, along with the possibility of nuclear excitation leads to saturation of thetotal production rates for high- Z ions, in contrast to the usual Z scaling for pointlike projectiles.The process is experimentally accessible by combining present-day ion accelerators with near-futurelaser sources and in principle allows for the measurement of nuclear form factors. PACS numbers: 12.20.Ds, 21.10.-k, 32.80.Wr, 34.90.+q
In recent years the interest in electron-positron pair cre-ation in combined laser and nuclear Coulomb fields hasbeen stimulated by the large advances in high-power laserdevices and ion accelerators (see [1, 2, 3] and referencestherein). The process is interesting since it proceeds bythe absorption of real photons from the laser field anda virtual photon from the Coulomb field. Most stud-ies have considered the low-frequency tunneling regime,where laser field strengths close to the Schwinger value E ( e ) S ≡ m e c / ~ e = 1 . × V / cm are required. At thisfield intensity, the electric work performed on an elec-tron of charge e along the Compton wavelength λ ( e ) C = ~ /m e c equals its rest energy m e c . In the opposite high-frequency multiphoton regime, pairs are created via thesimultaneous absorption of several high-energy photonswith ~ ω ∼ m e c from a rather weak laser field. In bothcases, the presence of the nucleus is required to guaranteeenergy-momentum conservation.The field strengths and frequencies required for e + e − pair creation in the respective regimes are four ordersof magnitude larger than provided by the most-advancedpresent laser systems. This gap can be bridged in laser-ion collisions, where the ions are counterpropagating thelaser beam at relativistic speed [1, 2, 3]. Then the laserfield strength and frequency as seen by the nucleus in itsrest frame are enhanced by the relativistic Doppler shift.The only experimental observation of e + e − pair produc-tion in laser and Coulomb fields has been achieved atSLAC (Stanford, California) in collisions of ultrarelativis-tic electrons with an intense optical laser pulse [4]. Thepairs were generated indirectly via Compton backscat-tering and a subsequent collision of real photons.In view of the ongoing technical progress the questionarises, whether not only e + e − but also µ + µ − pairs canbe produced with the emerging near-future laser sources.Muon production in the tunneling regime appears ratherhopeless, though, since the required field needs to beclose to E ( µ ) S = ρ E ( e ) S = 5 . × V / cm, with themuon-to-electron mass ratio ρ = m µ /m e ≈ γ -factors which will soon be provided bythe Large-Hadron Collider at CERN (Geneva, Switzer- land) this value seems out of reach ( γ = 7000 for protons[5]). As an alternative, it has recently been proposedthat µ + µ − pairs could be produced in laser and Coulombfields within a two-step process [3]: First, an e + e − pairis created via tunneling in a laser-ion collision, and after-wards driven by the laser field into an annihilating rec-ollision e + e − → µ + µ − . The latter process has also beenstudied in [6], with the electron and positron originatingfrom a positronium atom or an e + e − plasma.In this Letter, we calculate direct µ + µ − production inhighly energetic laser-ion collisions. Rather than the tun-neling regime, we consider multiphoton muon productionvia absorption of few high-frequency laser photons. Thelatter are assumed to stem from an x-ray free electronlaser (XFEL) beam with ~ ω = 12 keV [7], which col-lides head-on with an ultrarelativistic nucleus moving at γ = 7000. In the nuclear rest frame the photon energyamounts to ~ ω ≈ γ ~ ω = 168 MeV, so that the energythreshold ∆ ǫ = 2 m µ c for µ + µ − production can be over-come by two-photon absorption from the XFEL field [8].Corresponding large-scale XFEL facilities are presentlybeing developed at SLAC and DESY (Hamburg, Ger-many), where peak intensities close to 10 W/cm areenvisaged [7]. The projectile nucleus is modelled by anextended charge distribution, whose shape is shown tohave significant impact on the muon creation rates.At first sight, e + e − and µ + µ − pair production in com-bined laser and Coulomb fields seem to be very simi-lar processes since the electron and muon only differ bytheir mass (and lifetime). In this picture, the corre-sponding production probabilities would coincide whenthe laser field strength and frequency are scaled in ac-cordance with the mass ratio, i.e. W µ + µ − ( E ( µ ) , ω ( µ ) ) = W e + e − ( E ( e ) , ω ( e ) ) for E ( µ ) = ρ E ( e ) and ω ( µ ) = ρω ( e ) .This simple scaling argument does not apply, however,as the large muon mass is connected with a correspond-ingly small Compton wavelength λ ( µ ) C = λ ( e ) C /ρ ≈ .
86 fm,which is smaller than the radius of most nuclei. As a re-sult, the nucleus does not look pointlike to the muon andits finite extension must be taken into account. Pro-nounced nuclear size effects have also been found for µ + µ − production by single γ -photon impact on nuclei[9] and in relativistic heavy-ion collisions [10, 11].Following the usual theoretical approach to lepton paircreation in combined laser and Coulomb fields (see, e.g.,[2]), we write the process amplitude in the nuclear restframe as S p + p − = − ie ~ Z dt Z d r Ψ † p − V ( r )Ψ p + . (1)The muons are created with free momenta p ± and de-scribed by relativistic Volkov states Ψ p ± [12] which in-clude their interaction with the laser field to all orders.The nuclear field V ( r ) is taken into account within thefirst-order of perturbation theory. We assume the nucleusto be spherically symmetric with a Gaussian charge dis-tribution ̺ ( r ) = Ze ( √ πa ) e − r /a , (2)where Z is the atomic number and the parameter a is re-lated to the nuclear rms charge radius by r rms = p / a .The charge density (2) will serve us for a systematic studyof the nuclear size effect on the muon creation process.We note that light nuclei with mass number A ∼
10 areGaussian-shaped to a good approximation, whereas thecharge density of heavy isotopes ( A ∼ V ( r ) = ( Ze/r )Φ( r/a ), with the error function Φ( x ). Theamplitude in Eq. (1) can be evaluated analytically by ex-panding its periodic part into a Fourier series. This way,one arrives at the integral˜ V ( q ) = Z d r V ( r ) e i ~ qr = Zeq e − ( qa/ ~ ) , (3)where q ≡ ˜ p + + ˜ p − − n k is the momentum transfer tothe nucleus. Within the multiphoton regime, the laser-dressed momenta ˜ p ± introduced here practically coincidewith the free momenta p ± [12]. Moreover, n = 2 laserphotons of momentum k are absorbed from the field.The first factor on the right-hand side of Eq. (3) is theFourier transform of the Coulomb potential of a point-like nucleus. The second factor is the elastic nuclear formfactor F ( q ), which describes the correction due to the fi-nite nuclear size and leads to substantial reduction of theprocess probability when q ∼ m µ c approaches or exceedsthe inverse nuclear radius. The fully differential rate fortwo-photon muon pair creation is obtained by summingthe square of the amplitude (1) over the lepton spins: dR el = X spins | S p + p − | d p + (2 π ~ ) d p − (2 π ~ ) . (4)Equation (4) accounts for the elastic channel of the pro-cess, where the nucleus remains in its ground state. It adopts the structure dR el = dR Z F ( q ), with dR denoting the differential production rate for a pointlikeproton. This structure is analogous, e.g., to the formfactor-corrected Rutherford cross section or the Rosen-bluth formula for electron-nucleon scattering, which alsofactorize into a product of the result for a point scat-terer and the form factor squared [13]. An additionalcontribution to µ + µ − creation comes from the inelasticchannel, where the nucleus is excited due to the recoilimparted on it [9, 14]. The Z protons inside the nucleusact incoherently here, and one has to a good approxima-tion dR inel ≈ dR Z (1 − F ( q )). The inelastic processbecomes important when F ≪
1. In principle, both pro-duction channels are separately accessible in experimentby observing the final nuclear state in coincidence.Figure 1 demonstrates the nuclear size effect by showingangle-differential rates for elastic muon pair creation inultrarelativistic collisions of various ions with an intenseXFEL beam of linear polarization. The emission anglerefers to the laser wave vector and the rates are scaledby 1 /Z . The laser intensity parameter ξ µ = eEm µ cω re-lates to the muon mass; the value chosen corresponds to2 . × W/cm . We point out that this intensity ex-ceeds the original design values at SLAC and DESY by2-3 orders of magnitude, but can be attained by antic-ipated facility extensions and improved x-ray focussingtechniques [15]. The respective rms charge radii are0.875 fm (proton), 2.470 fm ( C), 3.929 fm ( Zn), and5.851 fm (
U) [16]. For comparison we also show theresult for a pointlike nucleus. With increasing nuclearsize, the production rates are strongly reduced, and themaximum of the angular distributions is shifted towardssmaller angles. The latter is because the nuclear form fac-tor cuts the contributions from large momentum trans-
FIG. 1: Angular spectra of one of the muons produced bytwo-photon absorption from an intense XFEL beam ( ~ ω =12 keV, ξ µ = 6 . × − ) colliding with various ultrarelativisticnuclei ( γ = 7000), as indicated. The rates are taken in thenuclear rest frame and scaled by the nuclear charge. fers, which otherwise give rise to large emission angles.The reduction of the total rates with respect to the resultfor a point charge arises from the fact that the particlesare created at typical distances ¯ r ∼ λ ( µ ) C . When this dis-tance is smaller than the nuclear radius, only a fraction eZ eff ≡ π R ¯ r r ̺ ( r ) dr of the total nuclear charge con-tributes effectively to the process.Figure 2 shows integrated rates of µ + µ − production forseveral projectiles. The nuclear isotope of largest abun-dance was chosen always. The values again refer to thenuclear rest frame. For a point proton, the total ratewould amount to R = ( α / π ) ξ µ ω F ( ω ) ≈ .
58 s − (cp. Eq. (26) in [1]), displaying the typical ξ µ dependenceof a two-photon process. Here, α denotes the finestruc-ture constant and the dimensionless function F ( ω ) is ofthe order of 0.1 in the relevant frequency range (i.e.,1 ≤ ~ ω/m µ c ≤ C impact. For theheavier projectiles, the reduction factors amount to 0.13( Al), 0.087 ( Ca), 0.055 ( Zn), 0.042 ( Kr), 0.026(
Sn), 0.020 (
Nd), 0.010 (
Pb), and 7 . × − ( U), respectively. Since the elastic process probabilityincreases with the projectile charge as Z and decreaseswith its size as ∼ exp[ − ( r rms /λ ( µ ) C ) ], a maximum elas-tic rate arises for atomic numbers around Z ≈
60. Thisis in contrast to e + e − pair creation which increases as Z throughout [1, 2, 3], apart from higher-order correc-tions in αZ which slightly modify this behaviour at high Z . Note that the latter corrections are of minor impor-tance in the present situation, as the muons are producedwell above the energetic threshold. The emergence of themaximum in Fig. 2 can be understood by consideration ofthe effective nuclear charge Z eff contained in a sphere ofradius λ ( µ ) C . The corresponding values 3.66 ( Kr), 3.83(
Sn), 3.96 (
Nd), 3.96 (
Pb), and 3.73 (
U) agreewith the observed location of the maximum. This simplepicture also explains a local minimum for atomic numbersaround Z ≈
70 (not shown), since Z eff ≈ .
72 for
Yb.We stress that the maximum remains at the same posi-tion, when the nuclei are modelled by uniformly chargedspheres, which represents a better approximation thanEq. (2) to the Fermi charge distribution of the relevantheavy isotopes where the maximum occurs [13]. A max-imum also arises for elastic muon production by a singlephoton of twice the energy. Figure 2 moreover displaysthe total rate R tot = R el + R inel , which does not exhibit amaximum but saturates at high Z values since the contri-bution from the inelastic channel increases with nuclearcharge. The total rates are still considerably smaller thanthe point-proton results: The reduction factors amountto, e.g., 0.35 ( C), 0.13 ( Ca), 0.069 ( Kr), 0.036(
Nd), and 0.018 (
U). For e + e − pair creation, theinelastic channel is negligible since q ∼ m e c ≪ ~ /r rms , sothat F ≈ R tot ≈ R Z . FIG. 2: Nuclear-frame rates for muon creation in the XFEL-nucleus collisions as in Fig. 1. The squares and triangles showthe total and elastic rates for various nuclei, respectively, con-nected by fit curves. The dotted line ∝ Z holds for a pointnucleus.FIG. 3: Angular spectra in the laboratory frame for two-photon muon pair production by C nuclei of different shapesand an XFEL beam. The upper (lower) two curves show total(elastic) rates. The collision parameters are as in Fig. 1.
Figure 3 shows angular spectra in the laboratory framefor carbon impact. Apart from the Gaussian charge dis-tribution of Eq. (2), a uniform density within a sphereof radius p / r rms was assumed, which has equal rmsradius. As compared to the result for a Gaussian-shaped C nucleus, the elastic (total) production rate is smallerby 13% (7%) for the hard sphere. The relative differencegrows for heavier nuclei, reaching about 40% (20%) for
U. This demonstrates that the muon production pro-cess is sensitive not only to the nuclear radius, but also tothe nuclear shape. In principle, the process might even beused for determination of the nuclear form factor, whichis traditionally accomplished by electron scattering.According to our results, muon pair creation in XFEL-nucleus collisions is experimentally feasible by employingnear-future technology. In the laboratory frame, the pro-duction rates of Fig. 2 are reduced by a factor 1 /γ due torelativistic time dilation and, accordingly, reach values ofabout R lab ≈ − s − . In the collision of an ion beamcontaining N = 10 particles [5] with an XFEL pulseof τ = 100 fs duration [7], the probability for productionof one muon pair is W µ + µ − ≈ R lab τ N/ ≈ − . Atthe envisaged average XFEL repetition rate of 40 kHz,one muon production event per second is obtained. Themuon yield scales with the laser intensity squared. Thetypical lab-frame energy of the muons ∼
500 GeV ishighly relativistic; the muonic lifetime is accordingly in-creased to ∼
10 ms. e + e − pairs are also produced in thecollision by single-photon absorption in the nuclear field(Bethe-Heitler process [12]). We stress, however, that thesubsequent reaction e + e − → µ + µ − via laser-driven recol-lisions [3] does not occur in the multiphoton regime con-sidered here, because the relative electron-positron mo-mentum ∆ p ∼ m e c satisfies 0 ≪ ∆ p ≪ m µ c . Detectedmuons have thus been produced via the direct process A + nω → A + µ + µ − , with n = 2. Higher photon orders( n ≥
3) are suppressed by an additional factor ξ µ ≪ µ + µ − pairproduction can take place, with the µ − being created ina bound state of the atomic nucleus. In order to esti-mate the corresponding rate, we apply an appropriatlymodified version of the theory developed in [17] for e + e − production with K-shell capture. Assuming the same col-lision parameters as in Fig. 2 and a pointlike proton, weobtain a rest-frame rate of 1 . × − s − . This value isby five orders of magnitude smaller than the correspond-ing rate for free muon pair creation in Fig. 2. For pointprojectiles, the bound-free production channel raises like Z with atomic number [17]. This scaling is also consid-erably damped by the finite nuclear size.Muon pair creation has been studied before in relativis-tic heavy-ion collisions [10]. The process is formally re-lated to muon production by a single photon in the fieldof an atomic nucleus [9, 18] by the Weizs¨acker-Williamsmethod, which describes the transverse electromagneticfield of an ultrarelativistic nucleus by a spectrum of vir-tual photons [12]. In accordance with the present re-sults, the finite nuclear extent was found to reduce theproduction probability substantially. In the high-energylimit, the argument of the famous Bethe-Heitler loga-rithm log(2 ~ ω/m µ c ) is modified by the factor λ ( µ ) C /r rms [10, 18]. This factor is also responsible for the rate sup-pression of two-photon muon pair creation here. Numer-ical calculations of bound-free muon pair production in U- U collisions have found a cross section reductionby five orders of magnitude due to the nuclear extent [19].It is interesting to note that exclusive dilepton produc-tion in hadron collisions also represents a search tool forphysics beyond the standard model [20]. In conclusion, direct production of µ + µ − pairs by two-photon absorption from a high-frequency laser wave col-liding with an atomic nucleus was calculated. The pro-cess is sensitive to the nuclear form factor. It couldbe realized experimentally by combining the radiationfrom upcoming XFEL sources with an ultrarelativisticion beam from the present generation of heavy-ion accel-erators. The highest elastic production rates are reachedby projectiles in the lanthanoid region ( Z ≈ Z values.Useful input by T. J. B¨urvenich, K. Z. Hatsagortsyan,and A. B. Voitkiv is gratefully acknowledged. [1] A. I. Milstein et al. , Phys. Rev. A , 062106 (2006).[2] J. Z. Kaminski, K. Krajewska, and F. Ehlotzky, Phys.Rev. A , 033402 (2006).[3] M. Yu. Kuchiev, Phys. Rev. Lett. , 130404 (2007).[4] D. Burke et al. , Phys. Rev. Lett. , 1626 (1997).[5] W.-M. Yao et al. , J. Phys. G , 1 (2006).[6] C. M¨uller, K. Z. Hatsagortsyan, and C. H. Keitel, Phys.Rev. D , 074017 (2006); Phys. Lett. B , 209 (2008).[7] L. F. DiMauro et al. , J. Phys. Conf. Ser. , 012058(2007); M. Altarelli et al. γ -factor of a pro-jectile electron required for muon pair creation by two-photon absorption is much larger ( γ & ).[9] A. Alberigi-Quaranta et al. , Phys. Rev. Lett. , 226(1962); Y.-S. Tsai, Rev. Mod. Phys. , 815 (1974).[10] C. A. Bertulani and G. Baur, Phys. Rep. , 299 (1988);J. Eichler, ibid. , 165 (1990).[11] We point out that the notion of “two-photon pair produc-tion” is also used in the context of heavy-ion collisions,where it refers to two virtual photons that are absorbedfrom the colliding Coulomb fields.[12] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory (Pergamon, Oxford, 1971).[13] B. Povh, K. Rith, C. Scholz, and F. Zetsche,
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