Probing phase of a scattering amplitude beyond the plane-wave approximation
aa r X i v : . [ h e p - ph ] D ec epl draft Probing phase of a scattering amplitude beyond the plane-waveapproximation
Dmitry Karlovets Department of Physics, Tomsk State University, Lenina Ave. 36, 634050 Tomsk, Russia
PACS – Relativistic scattering theory
PACS – Lepton-lepton interactions
PACS – Optical angular momentum and its quantum aspects
Abstract – Within a plane-wave approach, a number of scattering events in a collision is insensi-tive to a general phase of a transition amplitude, although this phase is extremely important for anumber of problems, especially in hadronic physics. In reality the particles are better described aswave packets, and here we show that the observables grow dependent upon this phase if one laysaside the simplified plane-wave model. We discuss two methods for probing how the Coulomb-and hadronic phases change with a transferred momentum t , either by colliding two beams at anon-vanishing impact-parameter or by employing such novel states as the vortex particles carryingorbital angular momentum or the Airy beams. For electron-electron collision, the phase contribu-tion to a cross section can reach the values higher than 10 − − − for well-focused beams withenergies of hundreds of keV. Introduction. –
In a quantum theory of scatteringwith all states being unlocalized plane-waves, the crosssection dσ is known to be independent of the phase ζ fi of a transition amplitude T fi = | T fi | exp { iζ fi } . It is soas long as we neglect finite sizes of the wave packets andbeams, their spreading during the collision, and supposethat the momentum uncertainties σ are vanishing. Butfor several important exceptions [1–6], this approximationworks very well. On the other hand, this phase turns outto be of high importance for hadronic physics, especiallyfor pp and p ¯ p collisions, in which it is extracted from inter-ference between the Coulomb- and the hadronic parts ofthe amplitude (see, for example, [7] and also [8] for a mod-ern review). The function ρ = Re T fi / Im T fi = 1 / tan ζ fi is calculated within different models, including Regge ap-proaches, and it is believed to depend in a complex wayon the energy and on the transferred momentum. Thisphase has been recently extracted from measurements bythe TOTEM collaboration at the LHC at √ s = 7 TeV [9],and it has been found that, in agreement with the unitaryconsiderations, the amplitude becomes mostly imaginaryfor the small scattering angles [10]. A further analysis ofthe data showed, however, that at large transferred mo-menta a real part of the amplitude can dominate [11].Here we show that the observables become sensitive tothe phase ζ fi if one lays aside the simple but unrealis- tic plane-wave approximation and treats all the particlesas spatially- and temporarily localized wave packets. Letin a general 2 → N f scattering process with two iden-tical incoming beams ( ee → X, pp → X , etc.) a ratio λ c /σ b serve as a Lorentz-invariant small parameter, where λ c = ~ / ( mc ) is a particle’s Compton wave length and σ b is a beam’s width. Then the cross section, dσ = dN/L ,represents a series in powers of ( λ c /σ b ) : dσ = dσ ( pw ) + dσ (1) + O (( λ c /σ b ) ) , (1)where dσ (1) = O (( λ c /σ b ) ), dN, L are the numberof events and the luminosity, respectively, dσ ( pw ) =(2 π ) δ (4) ( h p i + h p i − p f ) | T fi | dn f /υ is the standardplane-wave cross section. This first correction to the lat-ter, dσ (1) , vanishes in the plane-wave limit with σ b → ∞ and depends upon the phase ζ fi , on an impact parameter b between the beams’ centers (a head-on collision is im-plied for simplicity), and on the phases ϕ ( p ) , ϕ ( p ) ofthe beams’ wave functions ψ , ( p , ). To be more precise,the cross section depends upon a combination b ϕ − (cid:18) ∂∂ p − ∂∂ p (cid:19) ζ fi , b ϕ = b − ∂ϕ ∂ p + ∂ϕ ∂ p . (2)As we demonstrate below, one can probe the phase ζ fi , orrather its derivative ∂ζ fi ( s, t ) /∂t , with s = ( p + p ) , t =p-1. Karlovets( p − p ) , by comparing cross sections with a flipped signof b ϕ . This can be realized either by swapping the beams,i.e. b → − b (see Fig.1), or by changing signs of the phases ϕ , .Whereas conventional Gaussian beams are needed inthe first scenario, the second one requires more sophis-ticated quantum states. The beams with the phases, i.e. ψ ( p ) ∝ exp { iϕ ( p ) } , are not plane waves, even approxi-mately, and their wave functions in configuration spacemay turn out to be non-Gaussian. Depending on thesephases, such states can represent vortex particles carryingorbital angular momentum (OAM) with respect to theiraverage propagation direction [12–18], the so-called Airybeams [19–22], as well as their generalizations [23–25].These novel states were experimentally realized for pho-tons, for electrons with the energy of 200 −
300 keV and,more recently, for cold neutrons [26]. They have alreadyfound numerous applications – see, for example, Refs.[13, 27–31]. For a vortex electron, for instance, the phaselooks like ϕ = ℓφ with φ being an azimuthal angle and ℓ ≡ ℓ z the OAM. That is why change of the phase’s signcan be achieved by flipping the latter, ℓ → − ℓ .A key quantity of interest in this study is a scatteringasymmetry, which comes into play because the cross sec-tion is neither even nor odd in b ϕ . It is only moderately at-tenuated, A ∝ λ c /σ b , in both the methods we discuss andfor elastic scattering it signifies a lack of an up-down sym-metry in the angular distributions of scattered particles.This asymmetry is a Lorentz scalar and it can reach thevalues higher than 10 − − − for well-focused electronswith the intermediate energies. Thus, one can in princi-ple answer the question of how a Coulomb- or hadronicphase ζ fi changes with the scattering angle or with t bymeasuring the angular distributions of final particles. Thesystem of units ~ = c = 1 is used. The scattering asymmetry. –
Let us consider a2 → N f head-on collision of two beams with the meanmomenta h p i , h p i , their uncertainties σ , σ , the spatialwidths σ b, , σ b, , with overall phases of the wave functions(in momentum representation) ϕ , ϕ , and let the centersof the beams be separated by an impact-parameter b .A quantitative measure for contribution of the phase ζ fi to the number of events dN is the following asymmetry A = dN [ b ϕ ] − dN [ − b ϕ ] dN [ b ϕ ] + dN [ − b ϕ ] = dσ [ b ϕ ] − dσ [ − b ϕ ] dσ [ b ϕ ] + dσ [ − b ϕ ] , (3)which has a simple analytical form in the paraxial regime.Indeed, in the chosen kinematics the asymmetry can de-pend only on the following vectors: ∆ u = u − u , b ϕ ,( ∂∂ p − ∂∂ p ) ζ fi , and it must be a linear function of thetwo latter ones. The only true scalar that satisfies thesecriteria is A = 2Σ Σ Σ + Σ (cid:20) ∆ u | ∆ u | × (cid:20) ∆ u | ∆ u | × h b ϕ i (cid:21)(cid:21) · (cid:18) ∂ζ fi ∂ h p i − ∂ζ fi ∂ h p i (cid:19) + O (cid:18) Σ Σ + Σ (cid:19) ! , (4) where u , = h p i , /ε , ( h p i , ) , ε ( p ) = p p + m , andthe vector h b ϕ i = b − h ∂ϕ /∂ p i + h ∂ϕ /∂ p i is averagedwith the Gaussian distributions, ( √ πσ , ) − exp {− ( p , −h p i , ) /σ , } . This asymmetry is invariant under Lorentzboosts along the collision axis and it vanishes in the plane-wave limit when either σ , → σ b, , → ∞ . Of coursethe main formula of this study, Eq.(4), can also be rigor-ously derived from a general expression for the scatteringevents: see the Appendix.A small parameter of this series,2Σ Σ Σ + Σ ≡ O ( σ − b ) , Σ , = σ , σ , σ b, , ≡ O ( σ − b, , ) , (5)appears thanks to finite overlap of the incoming packets.It is approximately 2 / ( σ b, + σ b, ) for wide beams with σ b, , ≫ /σ , and Σ , ≈ /σ b, , . This approximationis realized in the majority of practical cases. Say, for theLHC proton beam with σ b ∼ µ m and the monochro-maticity of σ/ h p i .
1% [32], we have σσ b > .On the other hand, for non-relativistic beams the sit-uation with σσ b & , ∼ /σ b, , . Thus Σ , represent themomentum widths of the particle beams.The averaging of b ϕ has appeared because some phases ϕ , may not be analytical in the entire p -domain, butcontain a finite number of removable singularities. Say,for vortex beams with ϕ = ℓφ the derivative ∂ϕ∂ p = ℓ ˆ z × pp ⊥ (6)is not analytical for a vanishing transverse momentum.This singularity is removable and the mean value of this, D ˆ z × pp ⊥ E = ˆ z × h p ih p ⊥ i (cid:16) − e −h p ⊥ i /σ (cid:17) , (7)simply vanishes when h p ⊥ i →
0. Note that the number ofevents itself is suppressed as exp {− ℓ Σ , / (2 p ⊥ , , ) } when p ⊥ , , ≪ Σ , .The asymmetry (4) depends on the final particles’ mo-menta and in order to measure it, one should compareoutcomes of the two experiments with a flipped sign of h b ϕ i . This could be realized • Either by swapping the two incoming beams with nophases whatsoever, that is, by b → − b (see Fig.1), • Or by changing the signs of the phases, ϕ , → − ϕ , (say, by ℓ → − ℓ ), with zero impact-parameter.In what follows we shall discuss these means in detail.Note that Eq.(4) was obtained in the lowest order of per-turbation theory with Σ , ≪ |h p i , | ; that is why all cor-rections due to the phases are supposed to be small any-way, that is, |A| ≪ |A| .
1. Otherwisethis expression is inapplicable.p-2robing phase of a scattering amplitude beyond the plane-wave approximation
Fig. 1: For measuring the asymmetry, one compares outcomesof the two experiments with the swapped beams or, when theimpact parameter b is vanishing, with the beams having oppo-site signs of the phases. Alternatively, one can carry out justone experiment and measure the up-down asymmetry in theangular distributions. For a 2 → ≈ /σ b (say, pp → p ′ p ′ or ee → e ′ e ′ ), wefind from Eq.(4) A ≈ − σ b p u × [ u × h b ϕ i ] u ∂ ζ fi ∂t , (8)where we have put h p i ≡ p = u ε = −h p i . From Eq.(8)we infer that for the strictly forward scattering, p → p ,the asymmetry vanishes.1 st scenario: off-center collision of Gaussianbeams. – In a first scenario with the two (in fact, notnecessarily Gaussian) beams with no phases, one can put u ⊥ = 0 and h b ϕ i = b = { b, , } , where b ∼ σ b . In thiscase we find from Eq.(8): A ≈ p σ b sin θ sc cos φ sc ∂ ζ fi ∂t . (9)This asymmetry is only linearly attenuated by σ b and ithas a simple sin θ sc cos φ sc dependence on the scatteringangles θ sc , φ sc . Any deviation of the measured asymmetryfrom this dependence would be an evidence of a non-trivialphase ζ fi ( s, t ).A numerical estimate of the asymmetry can be ob-tained for elastic scattering in the relativistic case with p ≈ p, t ≈ − p θ sc , θ sc ≪ , γ = ε/m ≫
1. Assumingthat the phase is a fast function of the scattering angle θ sc , but a slow one of p , we get A ≈ − pσ b cos φ sc ∂ ζ fi ∂θ sc ≈ − λ c σ b cos φ sc γ ∂ ζ fi ∂θ sc (10)where λ c /σ b and γ − ∂ζ fi /∂θ sc are Lorentz invariant sep-arately. As we have seen, it is the factor λ c σ b ≪ − formoderately relativistic beams focused to a spot of ∼ µ mand it is ∼ − for protons with p ≈ σ b &
10 nm [33]. The estimate (10), however, isinapplicable for such a non-relativistic case. Conversely, in collision of electrons the ratio λ c /σ b be-comes bigger than 10 − for 300-keV beams focused in aspot of the order of 1˚A [27] (regardless of the OAM), eventhough the estimate (10) can be used only for qualitativeanalysis for such intermediate energies. For the Coulombphase on a one-loop level [7]1 γ ∂ ζ fi ∂θ sc ∼ α em γθ sc (12)with α em ≈ /
137 and hence A = O (cid:18) λ c σ b α em γθ sc (cid:19) . (13)This estimate is in accordance with that of the recent pa-per [34] where what we call the 2 nd scenario is studied. Inthe current scheme, we bring two sub-nm-sized electronbeams into collision (note that in this case 1 /σ ∼ σ b ),slightly off-center, and that is why one ought to be ableto control their relative position with the accuracy betterthan 0 . θ sc ∼ − − − is | A | ∼ − − − , (14)which is in principle measurable with high statistics. Onecould further increase it by performing measurements atyet smaller scattering angles or by making the impact pa-rameter very large, b ≫ σ b . In the latter case, however,the price is a drop in the number of events.Returning to scattering of protons, little can be saidindependently of a model, unfortunately, about the fac-tor in the left-hand-side of (12). The TOTEM collabo-ration is able to perform measurements at the scatteringangles smaller than 10 − at √ s = 7 TeV [9], which yields γθ sc ∼ . −
1, and the hadronic (or relative) phase ζ fi it-self, unlike the Coulomb one, is not attenuated by a smallparameter α em → α s , as scattering within a diffractioncone is not described by perturbation theory. This can,at least partly, compensate the lower value of λ c /σ b andlead to a detectable effect for the hadronic phase. Anywayone should strive to make the beam’s width σ b as small aspossible.Note that this asymmetry is a purely quantum effectthat vanishes in the plane-wave limit and might seem tobe counter-intuitive from a classical perspective. Indeed,for a pair of azimuthally symmetric wave packets their sub-stitution clearly does not alter the (classical) cross section.It is violated when either the packets are not-azimuthallysymmetric (the 2 nd scenario) or the particles themselveshave some inner structure (atoms, ions, hadrons). It is thelatter case in which the phase ζ fi comes into play.2 nd scenario: colliding beams with phases. – Within the second scenario, we start with a head-onp-3. Karlovetscollision of two vortex beams with b = 0, the phases ϕ , = ℓ , φ , , ℓ ≡ ℓ z , and the opposite signs of their or-bital helicities [29]. The spatial distribution of such beamsis no longer Gaussian but is a doughnut-shaped one witha minimum on the collision axis (see details in [16–18]).Still working in the frame with h p i ≡ p = −h p i , we findwith the help of Eq.(7): h b ϕ i = − ( ℓ + ℓ ) ˆ z × h p ih p i ⊥ (1 − e −h p i ⊥ /σ ) . (15)This vector vanishes, together with the asymmetry, wheneither the total OAM of the system is zero, ℓ + ℓ = 0, orazimuthally symmetric beams with u ⊥ = 0 collide (say,the so-called pure Bessel beams, which are the simplestmodels of vortex states with an azimuthally symmetricprofile [15]). This takes place because in order to havea non-vanishing A azimuthal symmetry of the problemmust be broken already in the initial state (exactly as inthe previous example).With vanishing impact-parameter, violation of the az-imuthal symmetry can be achieved by shifting the phasevortex off a symmetry axis of the beam. When deal-ing with the holograms (as in Refs. [17, 18]), a shift ofa fork dislocation off the beam center provides a (small)azimuthal asymmetry or, in other words, a non-vanishingtransverse momentum (see details, for example, in [35]).Such a shift is to be small, δρ . σ b , δp ⊥ & /σ b , δθ ∼ / ( pσ b ) and it is made to opposite directions for bothbeams. To put it simply, in this scenario a non-vanishingtransverse momentum plays the same role as does a finiteimpact parameter in the first one.We arrive at the following estimate from Eq.(8): A ≈ − ℓ + ℓ ) p σ b σ sin θ sin( φ − φ ) ∂ ζ fi ∂t (16)The major difference between this expression and Eq.(9) isappearance of the factor ℓ + ℓ , which can be very large. Itmight seem therefore that the second scenario with ℓ , ≫ {− ℓ , / (2 σ b, , p , , ⊥ ) } in Eq.(32) in theAppendix.As the production of twisted hadrons with azimuthallynon-symmetric profiles seems to be more technologicallychallenging than it is for electrons, we turn to elastic scat-tering of the latter. By analogy with Eq.(10), the factor ℓ + ℓ pσ b σ (17)determines the sensitivity to the asymmetry in the rela-tivistic case. The maximum value of the OAM for whichthe number of events is not suppressed is ℓ max ∼ p ⊥ σ b ∼ σσ b , and that is why A = O ( λ c /σ b ) , (18) exactly as in the previous scenario. Taking again 300-keVtwisted electrons focused to σ b ∼ σ/p . σσ b ∼
1, we arrive at the samedependence (18) when ℓ , = ℓ max = 1. In order to mea-sure the asymmetry, the angular distributions of scatteredelectrons are to be compared in the two experiments with ℓ , = 1 and ℓ , = −
1, respectively. As before, one canalternatively carry out only one experiment with ℓ , = 1when comparing angular distributions in the upper- and inthe lower semi-spaces. The numerical estimate (14) staysvalid. Since for such a study we need vortex electrons withazimuthally asymmetric profiles, we would also like to findsuch states for which the requirement of a non-vanishingtransverse momentum can be relaxed.As can be readily seen, it is the case for Airy beams astheir azimuthal distribution itself is not symmetric. Forcollision of two such states with u ⊥ = 0, the phase ϕ =( ξ x p x + ξ y p y ) /
3, and the opposite signs of their parameters ξ = − ξ ≡ ξ = { ξ, , } , we find: h b ϕ i = − σ { ξ , , } , A ≈ − σ σ b ξ p sin θ sc cos φ sc ∂ ζ fi ∂t , (19)where we have used h p x i = h p x i + σ /
2. The typicalvalues of ξ follow from the factor exp {− Σ ( σ ξ / / } in the probability formula. That is why ξ max ∼ σ b when σσ b ∼ , or ξ max ∼ σ b ( σσ b ) / ≪ σ b when σσ b ≫ . (20)In both cases, this yields the same p /σ b factor in theasymmetry as in Eq.(9) and λ c /σ b for the relativistic en-ergies. Therefore the use of Airy beams leads to the verysame predictions for the asymmetry as in the previous ex-amples.Moreover, one could think of such a phase ϕ ( p ) thatmaximizes the asymmetry. Within the paraxial regime,however, the phases are limited by the following inequal-ity: (cid:28)(cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ∂ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) . σ b , (21)which is simply analogous to b . σ b . That is why theasymmetry stays O ( λ c /σ b ) for all other types of the non-plane-wave states as well.The idea of using vortex states for probing the ampli-tude’s phase was put forward by Ivanov [30]. By analogywith his work, let us consider now scattering of a lightparticle by a heavy one (say, ep → X, γp → X ) with σ /σ ≪
1. Working in the frame in which the longitu-dinal momentum of the heavy particle is zero, we assumethe light one to be in the pure Bessel state with u ⊥ , = 0.We obtain that the asymmetry, A ∝ ℓ σ σ σ , (22)p-4robing phase of a scattering amplitude beyond the plane-wave approximationdoes not depend on the OAM ℓ of the light particle and,compared to Eq.(16), has an additional small factor σ /σ ,which is less than 10 − for the available beams. This factoralso appears for the Airy beams when p = − p but σ ≪ σ . That is why the higher values of the asymmetry favorthe case with σ ∼ σ , in accordance with the Ref. [30].The difference between the two methods describedabove can be elucidated by comparing two ways of col-liding two rubber balls. If the balls are pumped up well,they are azimuthally symmetric and in order to violate thissymmetry in scattering we need to collide them slightlyoff-center. Conversely, when the balls are deflated theyare most likely no longer azimuthally symmetric and thatis why they can collide even at a zero impact parameter.One simply needs to imagine a wave packet with a non-trivial wave front instead of such a deflated ball. Summary. –
As we have demonstrated, the cross sec-tion in a general scattering process becomes sensitive tothe overall phase of the scattering amplitude in a morerealistic model with incoming particles described as wavepackets. This phase reveals itself in the up-down angu-lar asymmetry when either impact-parameter between thetwo beams is non-vanishing or the beams have non-trivialwave functions, that is, carry phases. In both these scenar-ios, violation of the azimuthal symmetry of the problem,which does not take place in the plane-wave approxima-tion, yields a non-vanishing contribution of the phase ζ fi .The asymmetry is only linearly attenuated by a smallparameter λ c /σ b , regardless of the scenario. Its numericalestimate for the Coulomb phase is higher than 10 − − − for well-focused electrons with the intermediate energies.Corresponding experiments, albeit being challenging, canbe performed at modern electron microscopes, both withthe Gaussian beams and with the vortex- and/or Airy onesif they are focused to a spot of the order of or less than1˚A in diameter.Predictions for the hadronic (or relative) phase are lesscertain and inevitably model-dependent. Whereas the pa-rameter λ c /σ b does not exceed 10 − even for the mod-erately relativistic proton beams, effects of the hadronicphase per se must be much stronger at small scattering an-gles (within the diffraction cone) than it is for the Coulombphase, as the perturbation theory does not work there.This can improve the chances for detecting the asymmetry.As the twisted- or Airy protons have not been created yet,the corresponding experiments can be carried out withinthe first scenario. On the other hand, generation of thefast but non-relativistic protons with the non-Gaussianwave functions (say, of the Airy ones) could facilitate therealization of these objectives. ∗ ∗ ∗ I am grateful to E. Akhmedov, I. Ginzburg, I. Ivanov,P. Kazinski, G. Kotkin, V. Serbo, O. Skoromnik,A. Zhevlakov and, especially, to A. Di Piazza for many fruitful discussions and criticism. I also would like tothank C. H. Keitel, A. Di Piazza and S. Babacan for theirhospitality during my stay at the Max-Planck-Institutefor Nuclear Physics in Heidelberg. This work is supportedby the Alexander von Humboldt Foundation (Germany)and by the Competitiveness Improvement Program of theTomsk State University.
REFERENCES[1]
Kotkin G. L., Serbo V. G. and
Schiller A. , Int. J.Mod. Phys. A , (1992) 4707.[2] Melnikov K., Kotkin G. L., Serbo V. G. , Phys. Rev.D , (1996) 3289.[3] Melnikov K., Serbo V. G. , Nucl. Phys. B , (1997)67.[4] Akhmedov E. Kh., Smirnov A. Yu. , Phys. Atom.Nucl. , (2009) 1363.[5] Akhmedov E. Kh., Kopp J. , JHEP , (2010) 008.[6] Akhmedov E. K., Smirnov A. Y. , Found. Phys. , (2011) 1279.[7] West G. B., Yennie D. R. , Phys. Rev. , (1968) 1413.[8] Dremin I. M. , Physics-Uspekhi , (2013) 3; (2015)61.[9] Antchev G., et al. (TOTEM Collab.) , Europhys.Lett. , (2011) 21002.[10] Antchev G., et al. (TOTEM Collab.) , Europhys.Lett. , (2013) 21004.[11] Dremin I. M., Nechitailo V. A. , Phys. Rev. D , (2012) 074009.[12] Allen L., Beijersbergen M. W., Spreeuw R. J. C.,et al. , Phys. Rev. A , (1992) 8185.[13] Torres J. P. and
Torner L. (Editors),
Twisted pho-tons: Applications of light with orbital angular momentum (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)2011.[14]
Bliokh K. Yu., Bliokh Yu. P., Savel’ev S., Nori F. , Phys. Rev. Lett. , (2007) 190404.[15] Bliokh K. Yu., Dennis M. R., Nori F. , Phys. Rev.Lett. , (2011) 174802.[16] Uchida M. and
Tonomura A. , Nature , (2010) 737.[17] Verbeeck J., Tian H., Schlattschneider P. , Nature , (2010) 301.[18] McMorran B. J., Agrawal A., Anderson I. M., etal. , Science , (2011) 192.[19] Voloch-Bloch N., Lereah Y., Lilach Y., et al. , Na-ture , (2013) 331.[20] Berry M. V., Balazs N. L. , Am. J. Phys. , (1979)264.[21] Siviloglou G. A., Christodoulides D. N. , Opt. Lett. , (2007) 979.[22] Siviloglou G. A., Broky J., Dogariu A. and
Christodoulides D. N. , Phys. Rev. Lett. , (2007)213901.[23] Cong A., Renninger W. H., Christodoulides D. N.,et al. , Nature Photonics , (2010) 103.[24] Zhao J., Chremmos I. D., Song D., et al. , Sci. Re-ports , (2015) 12086.[25] Zhang P., Hu Y., Li T., et al. , Phys. Rev. Lett. , (2012) 193901. p-5. Karlovets [26] Clark C. W., Barankov R., Huber M. G., et al. , Nature , (2015) 504.[27] Verbeeck J., Schattschneider P., Lazar S., et al. , Appl. Phys. Lett. , (2011) 203109.[28] Jentschura U. D., Serbo V. G. , Eur. Phys. J. C , (2011) 1571; Phys. Rev. Lett. , (2011) 013001.[29] Ivanov I. P. , Phys. Rev. D , (2011) 093001; Ivanov I.P., Serbo V. G. , Phys. Rev. A , (2011) 033804.[30] Ivanov I. P. , Phys. Rev. D , (2012) 076001.[31] Ivanov I. P. , Phys. Rev. A , (2012) 033813.[32] Wenninger J. , Accelerators & Technology Sec-tor Reports , CERN-ATS-2013-040 (2013)http://cdsweb.cern.ch/record/1546734[33]
Watt F., van Kan J. A., Rajta I., et al. , Nucl. Instr.and Meth. B , (2003) 14.[34] Ivanov I. P., Seipt D., Surzhykov A., Fritzsche S. , Phys. Rev. D , (2016) 076001.[35] Mair A., Vaziri A., Weihs G., Zeilinger A. , Nature , (2001) 313. Appendix: derivation of the asymmetry for-mula. –
For a 2 → N f scattering, the probability ina general non-plane-wave case according to Kotkin et al. [1] represents a functional of the (generalized) cross sec-tion dσ ( k , p , ) and of a function that we shall denote L (2) ( k , p , ) and call the particle correlator: dW = | S fi | N f +2 Y f =3 V d p f (2 π ) = Z d p (2 π ) d p (2 π ) d k (2 π ) × dσ ( k , p , ) L (2) ( k , p , ) , dσ ( k , p , ) == (2 π ) δ (cid:16) ε ( p + k /
2) + ε ( p − k / − ε f (cid:17) × δ (3) ( p + p − p f ) × T fi ( p + k / , p − k / T ∗ fi ( p − k / , p + k / × υ ( p , p ) N f +2 Y f =3 d p f (2 π ) , L (2) ( k , p , ) = υ ( p , p ) Z dt d r d R e i kR × n ( r , p , t ) n ( r + R , p , t ) , (23)where T fi = M fi s ε ε N f +2 Q f =3 ε f ,υ = p ( p p ) − m m ε ( p ) ε ( p ) = p ( u − u ) − [ u × u ] ,ε f = N f +2 X i =3 ε i ( p i ) , ε ( p ) = p p + m , p f = N f +2 X i =3 p i , u , = p , /ε , ( p , ) , (24)and n ( r , p , t ) is a particle’s Wigner function with the nor-malization R d p (2 π ) d r n ( r , p , t ) = 1.One can define an effective cross section by dividing the probability by a luminosity L , dσ = dWL , L = Z d p (2 π ) d p (2 π ) d k (2 π ) L (2) ( k , p , ) == Z d p (2 π ) d p (2 π ) dtd r υ n ( r , p , t ) n ( r , p , t ) . (25)It is this quantity that we are going to compare the plane-wave cross section with.For a pure state, the Wigner function is n ( r , p , t ) = Z d k (2 π ) e i kr ψ ∗ ( p − k / , t ) ψ ( p + k / , t ) , (26)where ψ ( p , t ) = ψ ( p ) exp {− it ε ( p ) } . As an example, wetake a Gaussian wave packet with the phase ϕ ( p ): ψ ( p ) = π / (cid:16) σ (cid:17) / exp n − i r p − ( p − h p i ) σ ++ iϕ ( p ) o . (27)where r denotes a center of the wave packet at t = 0.Then the Wigner function is n = 1( √ π σ ) Z d k exp n − ( p − h p i ) σ − k (2 σ ) ++ i k ( r − r ) − it ( ε ( p + k / − ε ( p − k / i (cid:16) ϕ ( p + k / − ϕ ( p − k / (cid:17)o . (28)In a real experiment the beams of N b particles collide,and their widths, σ b , may be many orders of magnitudehigher than a coherence length of one wave packet 1 /σ .Taking as an example two Gaussian beams with N b, , N b, particles and the corresponding distribution functions, π − / σ − b exp {− ( r − r b ) /σ b } with r b pointing to thebeam’s center at t = 0, we perform statistical averagingof the one-particle correlator from Eq.(23). In doing so,we imply that the mean distance between particles in thebeams, ∼ σ b /N b , does not exceed the coherence length1 /σ , i.e. N b & or ≫ σσ b . The result of the averaging is L b = N b, N b, (2 π ) υ ( π σ σ ) δ (cid:16) ε ( p + k / −− ε ( p − k /
2) + ε ( p − k / − ε ( p + k / (cid:17) × exp n − ( p − h p i ) σ − ( p − h p i ) σ −− (cid:18) k (cid:19) (cid:18) + 1Σ (cid:19) − i kb ++ i (cid:16) ϕ ( p + k / − ϕ ( p − k / ϕ ( p − k / − ϕ ( p + k / (cid:17)o . (29)where the relative impact parameter of the two beams, b = r b, − r b, = { b ⊥ , } , is introduced and Σ , are from Eq.(5).p-6robing phase of a scattering amplitude beyond the plane-wave approximation dN = N f +2 Y f =3 d p f (2 π ) N b, N b, (2 π ) ( π σ σ ) Z d p (2 π ) d p (2 π ) d k (2 π ) δ ( ε ( p − k /
2) + ε ( p + k / − ε f ) × δ ( ε ( p + k /
2) + ε ( p − k / − ε f ) δ ( p + p − p f ) T fi ( p + k / , p − k / T ∗ fi ( p − k / , p + k / × exp n − ( p − h p i ) σ − ( p − h p i ) σ − (cid:18) k (cid:19) (cid:18) + 1Σ (cid:19) −− i kb + i ( ϕ ( p + k / − ϕ ( p − k /
2) + ϕ ( p − k / − ϕ ( p + k / o . (30)As a result, we arrive at the formula (30) for the num-ber of scattering events. It can be further simplified inthe paraxial regime with Σ , ≪ h p i , . To this end, weuse integral representations for the energy delta-functions, δ ( ε ) δ ( ε ′ ) = R dτ π dτ ′ π exp { iτ ε + iτ ′ ε ′ } , expand all the func-tions under the integral into k -series up to the 2nd orderinclusive, thus keeping terms not higher than O ( σ − b ), andthen integrate over k . In doing so, we imply that the am-plitude T fi and the phases ϕ , are analytical- and slowlyvarying functions of the momenta and use the followingformulas: Z d k exp (cid:26) − k i A i − B ij k i k j (cid:27) { , k m k n } == (2 π ) / √ det B exp (cid:26) B − ij A i A j (cid:27) n , B − mn ++ B − mi B − nj A i A j o ,T fi ( p + k / , p − k / T ∗ fi ( p − k / , p + k / ≈≈ (cid:16) | T fi | + 14 k i k j C ij + O ( k ) (cid:17) × exp n i k ∂ ∆ p ζ fi + O ( k ) o ,∂ ∆ p = ∂∂ p − ∂∂ p , ∆ u = u − u ,C ij = | T fi | ∂ ∆ p i ∂ ∆ p j | T fi | − ( ∂ ∆ p i | T fi | )( ∂ ∆ p j | T fi | ) B ij = α δ ij + β u ,i u ,j , α = 12Σ + 12Σ −− iτ (cid:18) ε + 1 ε (cid:19) , β = iτ (cid:18) ε + 1 ε (cid:19) , det B = α ( α + β u ) , B − ij = 1 α (cid:16) δ ij −− u ,i u ,j βα + β u (cid:17) . (31)The remaining integral over τ ′ is easily evaluated, andthe final result is: dN = N f +2 Y f =3 d p f (2 π ) N b, N b, (2 π ) ( πσ σ ) × Z d p (2 π ) d p (2 π ) dτ π δ ( p + p − p f ) √ det B ∆ uB − ∆ u × | T fi | + 14 C ij h B − ij − B − im B − jn (cid:16) ˜ b ϕm ˜ b ϕn − A case when ∂ϕ , /∂ p , contains a removable singularity isdiscussed in the main text. − u m ˜ b ϕn ∆ uB − ˜ b ϕ ∆ uB − ∆ u + ∆ u m ∆ u n ∆ uB − ∆ u (cid:16) uB − ˜ b ϕ ) ∆ uB − ∆ u (cid:17)(cid:17)i! exp n iτ ( ε ( p ) + ε ( p ) − ε f ) −− ( p − h p i ) σ − ( p − h p i ) σ + 12 (∆ uB − ˜ b ϕ ) ∆ uB − ∆ u −−
12 ˜ b ϕ B − ˜ b ϕ o , (32)where ˜ b ϕ = b − ∂ϕ ∂ p + ∂ϕ ∂ p − (cid:18) ∂∂ p − ∂∂ p (cid:19) ζ fi (33)is denoted.The resultant integrand in (32) can be expanded intoΣ , ( ≈ /σ b, , )-series when spreading of the beams dur-ing the collision is small (but non-vanishing!), that is, t col ≪ t diff ∼ σ b u ⊥ ∼ σ b ε. (34)Indeed, one can neglect β ( τ ) compared to 1 / (2Σ ) +1 / (2Σ ) in | α | in Eq.(31) exactly when | τ | ≪ σ b ε for two identical beams with σ b, = σ b, = σ b ≫ /σ , , ε = ε = ε . Following this procedure, weget series dN = dN ( pw ) + dN (1) + ... where the firstterm, as can be readily checked, yields the conventionalplane-wave result, dN ( pw ) = L ( pw ) dσ ( pw ) with L ( pw ) =2Σ Σ υ/ (2 π | ∆ u | (Σ + Σ )), and it is the first correction, dN (1) , that contains all dependence on the phases ζ fi and ϕ , that does not take place in the plane-wave approx-imation. A similar expansion can also be made for theluminosity, L ≈ L ( pw ) + L (1) and for the cross section, dσ = dN/L ≈ dσ ( pw ) + dσ (1)(1)