High energy scattering and emission in QED&QCD media

High energy scattering and emission in QED&QCD media X.G. Feal IGFAE/Instituto Galego de F´ ısica de Altas Enerx´ ıas This work has been done as a PhD thesis for the University of Santiago de Compostela and is partially based on already published research by the author. October 3, 2018 Resumo O estudo dos fen´ omenos de coherencia na emisi´ on de cuantos de enerx´ ıa nun escenario de m´ ultiples interacci´ ons rem´ ontase aos traballos cl´ asicos de Landau, Pomeranchuk e Migdal sobre a supresi´ on da intensidade de fot´ ons en medios condensados, un fen´ omeno amplamente reco˜ necido como efecto LPM. A altas enerx´ ıas, a presenza de m´ ultiples centros de dispersi´ on na direcci´ on de propagaci´ on da part´ ıcula nunha mesma lonxitude de coherencia, trad´ ucese en que todos os diagramas de emisi´ on presentes nesa lonxitude act´ uan de modo coherente entre eles e de xeito incoherente co resto de diagramas. Como resultado, a intensidade de emisi´ on total est´ a fortemente modulada pola fase que regula esta interferencia e, polo tanto, pola enerx´ ıa do fot´ on emitido e polo ´ angulo de colisi´ on acumulado subtendido pola part´ ıcula primixenia con respecto ´ a direcci´ on do fot´ on. Para altas enerx´ ıas do fot´ on, a intensidade debido ´ as colisi´ ons co medio p´ odese entender como a suma totalmente incoherente das intensidades individuais, namentres que para baixas enerx´ ıas do fot´ on, pola contra, o medio p´ odese entender coma unha entidade de radiaci´ on por si mesma, na que a estrutura interna das colisi´ ons ´ e irrelevante. A saturaci´ on, a baixas enerx´ ıas do fot´ on, do efecto LPM para medios condensados finitos, polo tanto, est´ a ditada polas ligaz´ ons impostas polo teorema de fot´ ons brandos de Weinberg. O estudo deste fen´ omeno de coherencia adquiriu recentemente un novo pulo debido a que se pode empregar como mecanismo de predici´ on sobre a creaci´ on de estados non confinados de materia hadr´ onica nas colisi´ ons de alta enerx´ ıa nos grandes aceleradores de hadr´ ons m´ ais recentes, tanto o RHIC como o LHC. Baixo as condici´ ons extremas dunha colisi´ on de i´ ons pesados de moi alta enerx´ ıa, as´ umese hoxe en d´ ıa que a formaci´ on do QGP (o plasma de quarks e gluns) ´ e posible e, polo tanto, m´ etodos indirectos de observaci´ on e de estudo son estritamente necesarios para achar probas das s´ ua existencia, comprender o seu comportamento e extraer as s´ uas propiedades. Entre algunhas destas probas, como os efectos de anisotrop´ ıa na distribuci´ on de part´ ıculas cargadas, relacionadas cun comportamento hidrodin´ amico do QGP, ou a supresi´ on de mes´ ons pesados, o estudo da perda de enerx´ ıa de quarks e glu´ ons durante a s´ ua viaxe por materia hadr´ onica confinada ou deconfinada, resulta fundamental para a entender o comportamento da teor´ ıa das interacci´ ons fortes a temperaturas extremadamente i ii altas, e o devir da materia hadr´ onica, baixo estas circunstancias, nun estado asint´ oticamente pouco ligado como pode ser o QGP. O estudo deste novo estado da materia resulta crucial, non s´ o para entender a natureza complexa e a interesante fenomenolox´ ıa das interacci´ ons fortes, senon tam´ en para estabelecer predici´ ons sobre os estad´ ıos iniciais do universo xusto tralo Big Bang, na co˜ necida como ´ epoca dos quarks. O obxectivo deste traballo consiste en estabelecer un formalismo para as colisi´ ons de alta enerx´ ıa con m´ ultiples corpos, e os procesos de emisi´ on que poidan ocorrer neses escenarios, tanto en electrodin´ amica (QED) como cromodin´ amica cu´ antica (QCD), e que permita unha avaliaci´ on das intensidades resultantes de emisi´ on sen ter que recorrer ´ a ben co˜ necida aproximaci´ on de Fokker-Planck, que admita o c´ alculo para medios finitos ou estruturados, e no que a distribuci´ on angular das part´ ıculas finais poida ser tida en conta. •Para tal fin no primeiro cap´ ıtulo introd´ ucese a integraci´ on de Glauber do estado de alta enerx´ ıa dun fermi´ on. ∂ϕ(n) s(0,x) ∂x3 =ip3−i βgA(n) 0(x)ϕ(n) s(0,x),(1) onde p3´ e a compo¨ nente lonxitudinal e dominante do momento inicial da part´ ıcula, ga constante de acoplamento co campo externo creado polo medio, A(n) 0(x) o campo externo do medio, representado por nfontes de dispersi´ on e ϕs(0,x) unha soluci´ on estacionaria da onda da part´ ıcula baixo a ecuaci´ on de Schr¨ odinger. En termos de ϕs(0,x), o estado da part´ ıcula lese, ψ(n)(0,x)= 1+i 2p0 α·∇+1 2p0 α·p!ϕ(n) s(0,x),(2) onde p0´ e a enerx´ ıa e αi=γiγ0as matrices de Dirac. Estas soluci´ on para as ondas das part´ ıculas de alta enerx´ ıa ser´ an empregadas nos seguintes cap´ ıtulos para avaliar as amplitudes de colisi´ on e emisi´ on tanto en QED como en QCD. A necesidade dunha integraci´ on do estado da part´ ıcula a altas enerx´ ıas ´ e clara, xa que procesos que atinxen un n´ umero arbitrario de colisi´ ons requiren un tratamento non perturbativo. As amplitudes expresadas dun xeito non perturbativo permiten definir, a posteriori, os observables e as intensidades en funci´ on da propia flutuaci´ on cu´ antica no n´ umero de colisi´ ons cos constitu´ ıntes do medio. Existen exemplos ben co˜ necidos deste tipo de proceder, debendo mencionar como tal o traballo de Bethe e Maximon para a avaliaci´ on da intensidade de emisi´ on, usando a aproximaci´ on de Bess e Furry para o estado do fermi´ on baixo o efecto do campo est´ atico e cl´ asico do n´ ucleo. A simplificaci´ on ga˜ nada tras tomar o l´ ımite de altas enerx´ ıas fai posible transformar o problema en termos dunha iii ecuaci´ on de Dirac nun problema de tipo Schr¨ odinger. Os nosos resultados neste senso para part´ ıculas de Dirac reproducen, excepto por correcci´ ons de cambio de esp´ ın, suprimidas de feito a alta enerx´ ıa, os ben co˜ necidos resultados de Glauber para part´ ıculas de tipo Schr¨ odinger. •No segundo cap´ ıtulo pres´ entase unha caracterizaci´ on dos procesos el´ asticos de colisi´ on con m´ ultiples fontes de dispersi´ on, definindo a amplitude de colisi´ on asociada ´ a integraci´ on de altas enerx´ ıas da onda do fermi´ on, M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)βsm p0 f ¯usf(pf)γ0usi(pi)rm p0 i (3) ×Zd2yte−iqt·yt exp "−ig βZ+∞ −∞ dy3A(n) 0(y)#−1!. onde β´ e a velocidade da part´ ıcula, pfepi,sfesi, os 4-momentos e esp´ ıns finais e iniciais, respectivamente, us(p) un espinor de Dirac e ma s´ ua masa. Observaremos que, s´ o a primeira orde perturbativa, a amplitude resultante para ncorpos se pode escribir como unha suma das amplitudes individuais. A orde arbitraria na constante de acoplamento, poderemos definir o o cadrado das amplitudes para un medio promediado sobre unha xeometr´ ıa determinada. Atoparemos que o promedio das amplitudes ao cadrado se pode dividir en d´ uas contribuci´ ons cunha clara interpretaci´ on, M(n) sfsi(pf,pi) 2= Π(n) 2(pf,pi)+ Σ(n) 2(pf,pi).(4) Por unha banda, a amplitude promediada ao cadrado, contendo os efectos cu´ anticos e difractivos de borde e que se corresponde cos termos non diagonais dunha expansi´ on no n´ umero de colisi´ ons, mide as interferencias transversas de colisi´ on con centros distintos. Este t´ ermino ser´ a entendido, ent´ on, coma unha contribuci´ on coherente, e est´ a dado por Π(n) 2(pf,pi)≡S(n) sfsi(pf,pi)−1 2 =M(n) sfsi(pf,pi) 2 .(5) Pola outra banda, a contribuci´ on asociada aos elementos diagonais dunha expansi´ on no n´ umero de colisi´ ons, que representa polo tanto a suma incoherente de todas as nintensidades de colisi´ on ao cadrado cos diferentes centros, estar´ a dada por Σ(n) 2(pf,pi)≡DS(n) sfsi(pf,pi)S(n)∗ sfsi(pf,pi)E−S(n) sfsi(pf,pi)DS(n) sfsi(pf,pi)∗E. (6) iv Veremos que no l´ ımite macrosc´ opico a contribuci´ on coherente red´ ucese a unha contribuci´ on colimada na direcci´ on asint´ otica inicial do fermi´ on, non contribu´ ındo polo tanto a cambios do seu momento, namentres que a contribuci´ on incoherente reproduce o ben co˜ necido resultado de Moliere obtido baixo argumentos markovianos de homoxeneidade, ´ e dicir, a trav´ es dunha ecuaci´ on de transporte, ∂ˆ Σ(n) 2(q,l) ∂l=−n0σ(1) tΣ(n) 2(q,l)+n0Zd2k (2π)2F(1) el (k) 2ˆ Σ(n) 2(q−k,l).(7) onde q´ e o cambio de momento, la profundidade no medio na direcci´ on inicial de propagaci´ on, n0a densidade do medio e F(1) el (k) a amplitude debida a unha fonte de dispersi´ on illada (n=1). Obteremos unha correcci´ on para a definici´ on de Moliere para medios de lonxitude finita e definiremos a aproximaci´ on de Fokker-Planck para a distribuci´ on resultante de momentos. Obteremos tam´ en o valor promedio do cambio de momento baixo unha interacci´ on xeral co medio, e finalmente avaliaremos as amplitudes e as intensidades de colisi´ on m´ ais al´ o da aproximaci´ on puramente de alta enerx´ ıa de Glauber. As correcci´ ons inseridas pola existencia de cambios de momento lonxitudinais crear´ a unha ordenaci´ on das colisi´ ons na direcci´ on de propagaci´ on dominante do fermi´ on que levar´ a, en caso de emisi´ on de part´ ıculas durante as colisi´ ons, a fen´ omenos de interferencia como o xa mencionado efecto LPM. •No terceiro cap´ ıtulo abordaremos o problema da emisi´ on durante escenarios de m´ ultiples colisi´ ons en QED recuperando o traballo do anterior cap´ ıtulo e tentando sentar un procedemento convencional a trav´ es de amplitudes en teor´ ıas cu´ anticas de campos. Ao nivel da amplitude a estructura en diagramas de Feynman t´ ornase m´ ais cristalina e entender os procesos de coherencia entre os diferentes procesos individuais resulta m´ ais sinxelo. Neste proceder, polo tanto, atoparemos que o efecto LPM satura, a moi baixas enerx´ ıas do fot´ on, no plateau de coherencia dictaminado polo teorema do fot´ on brando de Weinberg. Esta particularidade far´ a que para circunstancias de total supresi´ on e para medios finitos, a correcci´ on aos resultados de Migdal e Landau sexa substancial. Definiremos a intensidade de emisi´ on a trav´ es da nosa amplitude, e coma no caso puramente el´ astico atoparemos d´ uas contribuci´ ons, unha relacionada coa contribuci´ on coherente das colisi´ ons, e outra relacionada coa contribuci´ on incoherente. Para a maior´ ıa das situaci´ ons realistas en QED os medios p´ odense considerar macrosc´ opicos, de xeito que a contribuci´ on incoherente ´ e a ´ unica relevante. Para medios microsc´ opicos, sen embargo, os efectos coherentes de colisi´ on son os dominantes e a distribuci´ on de fot´ ons hase ver severamente afectada polas resonancias cos patr´ ons difractivos nese r´ exime de v colisi´ ons. Acharemos unha expresi´ on para a intensidade de emisi´ on que se pode avaliar para unha interacci´ on calquera co medio, que permite obter o espectro angular das part´ ıculas e que no seu l´ ımite de medio semi-infinito e baixo a aproximaci´ on de Fokker-Planck recupera o caso particular da expresi´ on cl´ asica de Migdal, ωdIinc dωdΩ=e2 (2π)2 n Y i=1Zd3pi (2π)3       n Y i=1 φinc(δpi)− n Y i=1 φ(0) inc(δpi)       × hn(y) n X k=1 δn kexp −i k−1 X i=1 kµpµ i p0 0−ωδz 2 (8) +hs(y) n X k=1 δs kexp −i k−1 X i=1 kµpµ i p0 0−ωδz 2 , onde ω´ e a enerx´ ıa do fot´ on, e2a carga do electr´ on, φinc(δpi) as distribuci´ ons el´ asticas para un cambio de 4-momento δpinunha l´ amina de medio de espesura δz,hn(y) e hs(y) os pesos cinem´ aticos das intensidades que involucran conservaci´ on ou cambio de esp´ ın, respectivamente, δn keδs kas correntes de emisi´ on en semellanza ´ a definici´ on da amplitude de BetheHeitler, e kepo 4-momento do fot´ on e o electr´ on. Demostraremos que os formalismos de Zakharov e BDMPS est´ an tam´ en contidos como casos particulares e acharemos que a nosa expresi´ on para a intensidade no l´ ımite continuo xeneraliza os resultados de Wiedemann e Gyulassy, producindo pola contra unha correcta expansi´ on en opacidades no l´ ımite de pouca espesura, admitindo ademais unha avaliaci´ on non gaussiana baixo a interacci´ on de Debye. Os nosos resultados ser´ an comparados con datos experimentais dos aceleradores de SLAC e do CERN, atopando un acordo moi grande. En particular, evidencian a existencia dun plateau de coherencia ditado polo teorema do fot´ on brando de Weinberg, importante para medios de non moi grande lonxitude, e a existencia de fen´ omenos de transici´ on relacionados coa consideraci´ on dos efectos de dispersi´ on, na relaci´ on de enerx´ ıa momento do fot´ on, debido ´ a presenza do medio. •No cuarto cap´ ıtulo procedemos paralelamente ao traballo feito no segundo cap´ ıtulo definindo un formalismo de m´ ultiples colisi´ ons para as interacci´ ons fortes, e supo¨ nendo un medio en QCD non moi fortemente ligado como pode ser o QGP. Os resultados, salvo efectos de carga relacionados coa estrutura non abeliana de QCD, reproducen os mesmos patr´ ons nas distribuci´ ons de momento que os obtidos en QED. En particular, avaliamos a contribuci´ on transversa e coherente de colisi´ on que domina a baixo cambio de momento sobre a distribuci´ on incoherente, e polo tanto debe ser tida vi en conta para medios microsc´ opicos, como ´ e o caso da materia hadr´ onica confinada ou mesmo non confinada, nas distribuci´ ons el´ asticas de colisi´ on para avaliar os procesos de emisi´ on. Do mesmo xeito que no caso de QED, as contribuci´ ons relacionadas cos momentos lonxitudinais levar´ an a efectos de interferencia substanciais no c´ omputo dos procesos de emisi´ on. •No quinto cap´ ıtulo estudamos ditos procesos de interferencia, o efecto LPM, na emisi´ on de glu´ ons baixo m´ ultiples colisi´ ons con materia hadr´ onica confinada ou deconfinada. O esp´ ırito do cap´ ıtulo coincide co caso de QED, tentaremos sentar as bases a trav´ es da definici´ on dunha expresi´ on para a amplitude na que os fen´ omenos de coherencia sexan entendibles dun xeito doado, e na que a emerxencia de todos os casos dispo¨ nibles de emisi´ on permita identificar o proceso a trav´ es de diagramas de Feynman. A consideraci´ on de momentos lonxitudinais, como xa dixemos, leva a unha ordenaci´ on do proceso na direcci´ on de propagaci´ on do glu´ on, o cal en QCD como demostraremos, e debido a que a estrutura non abeliana da teor´ ıa permite interacci´ ons do glu´ on co propio medio, adquire o rol do electr´ on cando se asume o seu l´ ımite de baixa enerx´ ıa. A intensidade de emisi´ on que atoparemos no l´ ımite de glu´ ons brandos, ωdI dωdΩk =g2 sCf (2π)2 n−1 Y i=1 d3ki (2π)3       n Y i=1 φ(ni) g¯g(δki)− n Y i=1 φ(0) g¯g(δki)       × n X k=1 exp −i n−1 X i=k δzi ki µpµ 0 p0 0−ω δn k 2 ,(9) ter´ a polo tanto unha forma moi semellante ´ a de QED. Na anterior expresi´ on ω´ e a enerx´ ıa do glu´ on, gs´ e a constante de acoplamento da interacci´ on forte, Cf=(N2 c−1)/2Ncun Casimir de SU(Nc), Nc=3 o n´ umero de cores, φg¯g(δki) a distribuci´ on el´ astica do glu´ on ap´ os atravesar unha l´ amina do medio de espesura δziepekos 4-momentos do quark e o glu´ on. A emisi´ on est´ a caracterizada por un r´ exime de emisi´ on coherente, que ser´ a o dominante a baixas enerx´ ıas, correspondente aos glu´ ons emitidos dende as patas externas do medio, que se pode considerar como unha entidade de emisi´ on en si mesmo, namentres que a enerx´ ıas m´ ais baixas a emerxencia da estrutura interna de colisi´ ons e o desacoplamento dos diagramas internos favorecer´ a un aumento da intensidade de emisi´ on. Este aumento, como veremos, ser´ a maior a maior lonxitude do medio, a maior coeficiente de transporte e a maior masa de apantallamento. Calcularemos tam´ en a perda de enerx´ ıa baixo un escenario de m´ ultiples interacci´ ons de pouca deflexi´ on, e presentaremos unha aproximaci´ on para o caso de que queiramos avaliar a intensidade despois dunha colisi´ on que produza un severo cambio vii de momento no part´ on. As´ ı mesmo, presentaremos unha ecuaci´ on sinxela que axude a entender cualitativamente o efecto LPM en QCD e produce resultados moi razoables, en particular para os l´ ımites de total coherencia e total incoherencia, nos que se volve exacta. Acharemos que baixo as aproximaci´ ons axeitadas atopamos os resultados de Migdal/Zakharov e os do grupo BDMPS como casos particulares, dentro da aproximaci´ on de Fokker-Planck, e os resultados de Wiedemann e Gyulassy e Salgado como o caso xeral dentro da mesma aproximaci´ on gaussiana. Xa que o noso formalismo permite unha avaliaci´ on m´ ais al´ o da aproximaci´ on de Fokker-Planck, o c´ alculo do espectro de emisi´ on en QCD, baixo o r´ exime de m´ ultiples interacci´ ons, ´ e presentado por primeira vez baixo unha interacci´ on realista entre a part´ ıcula que colisiona e o medio de cor, producindo un aumento substancial da radiaci´ on e da perda de enerx´ ıa. As diferenzas atopadas, polo tanto, entre o caso apantallado real e as amplamente empregadas aproximaci´ ons de Fokker-Planck, suxerir´ a que as interacci´ ons que ocorren entre o part´ on e a materia hadr´ onica observada no RHIC e o LHC non precisar´ ıan estar tan fortemente acopladas, como normalmente se asume, para producir as perdas de enerx´ ıa observadas indirectamente a trav´ es dos factores de modificaci´ on nuclear. Introduction The study of coherence phenomena for the emission of quanta in a multiple interaction scenario dates back to the seminal works of Ter-Mikaelian [1], Landau and Pomeranchuk [2, 3] and Migdal [4] on the bremsstrahlung suppression due to condensed media, the well known Landau-Pomeranchuk-Migdal (LPM) effect. The existence of phases in the elastic amplitudes, which keep track of the placement and momentum change of each of the interactions, lead to interference effects in the squared sum of all the involved amplitudes. At high energies these phases set a coherence length, in the dominant direction of the traveling particle, in which the individual amplitudes can be considered to coherently add in the squared amplitude, but incoherently interfere with the rest of the processes. A manifestation of the coherence between Feynmann diagrams could acquire a more familiar form in analogy with the soft photon theorem [5]. For sufficiently low energy of the emitted photon, the diagrams representing emissions from the internal legs cancel and the total amplitude restricts to the photons coming from the first and last legs. Thus in that limit radiation can be understood as if all the internal interactions coherently emit as a single entity. This immediately addresses the question on how the intensity of photons behaves for larger photon energies and how this energy and the medium modulate the coherence. In other words, when the medium still emits as a single entity, and when, if it occurs, a regime of maximal incoherence between its constituents may be found, in which the total intensity is just the sum of the single [6] intensities. Predictions and observations of this coherence effect have been extensively carried in the past for quantum electrodynamic interactions (QED) and they have suscited a renewed interest in the era of the hadron colliders as a way to indirectly observe the hadronic matter produced in heavy ion collisions. Soon after the birth of quantum chromodynamics (QCD) as the theory of the strong interactions [7–9], it was assumed that a new state of the hadronic matter, consisting in a deconfined phase of quarks and gluons at extremely high temperatures, should exist in nature [10–16]. At temperatures well above the QCD critical temperature ∼170 MeV and/or at densities well above the ordinary nuclear density ∼1 GeV/fm3, collisions involving large momentum change are required in order to consider the interactions between particles significative. The weak value of the ix xContents coupling at such high gluon momenta leads, however, to a relatively small amplitude of these kind of hard interactions to occur. This asymptotically free behavior of QCD at high energies suggests, then, a picture of quasi free, softly interacting quarks and gluons under extreme conditions. In analogy with the weakly coupled QED plasmas, this new state of the matter at such extreme conditions was quickly coined [17] with the name of Quark Gluon Plasma (QGP). The study of the QGP properties results of interest not only as a way of understanding the complex nature and rich phenomenology of QCD, but it also becomes crucial for the understanding of the initial stages of the universe in the quark epoch just after the Big Bang, for example. In the search of evidences of QGP formation at the RHIC and LHC hadron colliders several indirect probes can be considered. Among these QGP signatures, like the observation of anisotropies in the charged particle distributions, related to an hydrodynamical behavior of the QGP, or the suppression of heavy quark mesons, a quantitative prediction of the radiative energy loss of the underlying jet particles, while traveling through the formed QCD medium, should provide an indirect evidence of the QGP formation and its characteristics. For that purpose an accurate study of the LPM effect for the gluon bremsstrahlung spectrum in a multiple collision scenario with the QCD medium becomes, then, indispensable. The structure of this work is approximately guided by the historical developments. We also put some focus in understanding the coherence effects, both in the elastic and the emission processes, following a conventional quantum field theory (QFT) description with amplitudes. In Chapter 1 we lay the foundations for the treatment of high energy scattering under multiple interactions by defining a high energy integration of the fermion state under a general, static and classical external field [18]. In Chapter 2 we present a formalism for multiple scattering at arbitrary perturbative order, and define the medium averaged intensities and observables leading to transverse and longitudinal coherency effects. In Chapter 3 we simply use the former multiple scattering results to build a QFT formalism for high energy emission which can be evaluated for general interactions with mediums of finite size. Under the Fokker-Planck approximation and the semi-infinite medium limit we are able to recover the Migdal result [4] as a particular case. In Chapter 4 we extend the QED multiple scattering results for the QCD scenario and finally, in Chapter 5, we evaluate the gluon bremsstrahlung intensity with these tools. Under the Fokker-Planck approximation, Wiedemann result [19] is found and within some length or mass approximations the well known results of Migdal/Zakharov [4, 20] or the BDMPS group [21, 22] are also recovered. Part I QED 1 1 High energy fermions in an external field The purpose of this chapter is to introduce the Glauber high energy integration of the wave [18] for particles obeying the Dirac equation. These wave solutions will be used in the following chapters to evaluate the QED and QCD scattering and emission amplitudes in a multiple collision scenario. Consider a high energy particle traveling and interacting with a medium, classically characterized as a set of sources distributed in some region of the space. Since the state of the particle is expected to contain an arbitrary number of interactions with the medium, an exact/non-perturbative solution for the traveling wave is required. Processes typically involving the creation or annihilation of other particles will emerge as a result of the interaction with the medium. Their amplitudes, when expressed in terms of the exact wave integrations, correspond to a sum of all the perturbative orders in g, the medium coupling parameter, or, in other words, all the ways in which the process can be represented by means of elementary interactions with medium constituents. This superposition of Feynman diagrams will lead to a quantum fluctuation of the observables in the number of collisions and they may be expressed, on average, as functions of the number of particles, the geometry and the nature and strength of the interaction with the medium. Early efforts in computing amplitudes in this exact approach exist [23]. For photon bremsstrahlung due to a single Coulomb field, for example, an exact Dirac or Klein-Fock-Gordon solution for the electron can be obtained in polar coordinates [24, 25], but at the expense of making the emission intensity calculation an almost impossible task [23]. It is obvious then, that if we try to extend the problem to multiple external fields randomly distributed, some kind of approximations have to be done. Fortunately, as we will see later, two approximations will prove to simplify our problem. First, as already mentioned under the conditions of our interest the medium is considered a classical and static source. The 3 4High energy fermions in an external field validity of this approximation and the form of the interaction is briefly explained in Section 1.1. Second, the high energy solution of the Dirac equation can be given as an operator acting on a Schrodinger like state which is easier to solve in the high energy approximation [18]. A derivation of this result and a discussion of the conditions for the approximation are given in Section 1.2. In Section 1.3 we prove that the perturbative expansion of the found wave agrees with the standard Born approximation, and in Section 1.4 we compute the propagator in momentum and light cone space, showing that except for a negligible spinorial correction, the well known Glauber scattering amplitude [18] is found. This correspondence is a direct consequence of the Schrodinger behavior, except for the spin structure, of the Dirac solution at high energies. 1.1 The medium as a classical source We consider a medium of ordinary solid matter composed, then, of nstatic nuclei of charge Ze and mass mn. When a high energy electron of mass mand charge epasses through this medium a photon field emanates from the induced current, which is going to be considered classical and static. By classical we mean that the nuclei or sources originating the field are completely localized in coordinate space, and by static that these sources have a negligible recoil. Both conditions read for the 4-current J(n) 0(x)=Ze n X i=1 δ(3)(x−ri),J(n)(x)=0,(1.1) where riis the position of each source. In order to take into account screening effects of the electron shell, we will consider that the photon field originated in the interaction has an effective mass µdcanceling photon propagation to distances much larger than rd=1/µd. This screening mass of the nuclei can be estimated as µd=α2mZ1/3, being αthe fine structure constant. In the Lorenz gauge [26], now required for current conservation, the Maxwell equation for the massive photon A(n) µ(x) due to the nsources reads ∂ν∂ν+µ2 dA(n) µ(x)=4πJ(n) µ(x).(1.2) To get the form of A(n) µ(x) we simply propagate the current (1.1) as A(n) µ(x)=Zd4yDF µν(x−y)J(n) ν(y),(1.3) by means of a Feynman-Stueckelberg propagator. By direct inspection of equations (1.2) and (1.3), the propagator, then, has to satisfy the equation ∂η∂η+µ2 dDF µν(x−y)=4πδ(4(x−y)gµν.(1.4) The medium as a classical source 5 By Fourier transforming equation (1.4) we easily find ˆ DF µν(q2)=gµν 4π −q2+µ2 d ,(1.5) With the above equation, and using (1.1) and (1.3) it is straightforward to show that the field does not carry energy, as result of time translation invariance, and it has only one non-vanishing component, namely A(n) 0(x), given by A(n) µ(x)=gµ0Ze n X i=1Zd3q (2π)3 4π q2+µ2 d e−iq·(x−ri).(1.6) Here we notice that the inverse screening radius µdis related to the typical momentum change qin a single collision with one nuclei. Finally, the last integral can be accomplished by using Cauchy’s theorem, Zd3q (2π)3 4π q2+µ2 d e−iq·x=1 π|x|Im Z+∞ −∞ dq q q2+µ2 d eiq|x|=1 |x|e−µd|x|,(1.7) so we find the interaction A(n) 0(x)=Ze n X i=1 DF 00(x−ri)= n X i=1 Ze |x−ri|e−µd|x−ri|.(1.8) This is known as a Yukawa/Debye interaction with screening µd. In the limit µd→0 the field equations transform into the gauge covariant Maxwell equations and correspondingly we find the Coulomb interaction. This simple result derived under the classicity of the current can be justified by doing the pertinent approximations in the quantum current. For a single nuclei we would have, Jµ(x)=rmn p0 b ¯usb(pb)γµusa(pa)rmn p0 a ei(pb−pa)·x,(1.9) where we used spinor conventions given in Appendix A. The amplitude of finding the photon carrying momentum q=pa−pbis, using (1.3), (1.4) and (1.9) Aµ(x)=4πZe −(pa−pb)2+µ2 drmn p0 a ¯usa(pa)γµusb(pb)rmn p0 b e+i(pa−pb)x.(1.10) As usual, this photon has to be joined with the electron current and then the resulting quantity integrated in x. Due to both external constraints of the other nuclei reducing its recoil and because it is much heavier than the typical momentum change produced into a single collision µd, we have pa=0 and pb≈pa, so 6High energy fermions in an external field p0 b≈mn, J0=rmn p0 a ¯usa(pa)γ0usb(pb)rmn p0 b≈1 Jk=rmn p0 a ¯usa(pa)γkusb(pb)rmn p0 b≈(pb)k mn1.(1.11) Similar conclusions can be placed for the electron in the high energy limit. Using then these slow dependences of the nuclei and electron currents on the momentum, using (1.11) integration in time and momentum can be carried independently at (1.10) and we obtain hAµ(x)iq,t≡δµ0Ze Zd4q (2π)4Zdt 4π −q2+µ2 d e+iq·x=δµ0 Ze |x|e−µd|x|,(1.12) in such a way that the coherent superposition of photon amplitudes recovers the classical field limit. The classical approximation is allowed, then, if the inertias of the nuclei and the electron (the rest mass and the energy, respectively) greatly exceed the typical momentum change µdof a single interaction. For moving constituents, like the ones in an ideal gas, it is also sufficient to guarantee that the typical energy of each constituent is much larger than µd, so the medium can be considered classic. And if the traveling fermion energy greatly exceeds also the typical energy of the constituents, the medium can be considered static. 1.2 High energy limit of the Dirac equation We depart, then, from a medium characterized by nsources of the form A(n) 0(x)= n X i=1 A(1) 0(x−ri), ∂0A(n) 0(x)=0.(1.13) The state ψ(n)(x) of the electron is coupled with strength g=e, its charge, to these nfields. Since these fields carry themselves the charge of each target Ze, let us relabel the overall coupling to g=Ze2and leave the interaction A(n) 0(x) coupling independent. The electron obeys the Dirac equation under this field, so it has also to obey the squared Dirac equation, and we write iγµ∂ ∂xµ+m−gγ0A(n) 0(x)! iγµ∂ ∂xµ−m−gγ0A(n) 0(x)!ψ(n)(x)=0.(1.14) Energy operator represents the x0evolution of the states , i∂0≡ ±p0. Accordingly energy positive solutions are the ones represented by ψ(n)(x)=ψ(n)(0,x)e−ip0x0.(1.15) High energy limit of the Dirac equation 7 Using the relations {γµ, γν}=2gµν,αi=γiγ0and taking into account the condition ∂0A(n) 0(x)=0, we find an eigenvalue equation in p0 p2 0−m2+∇2−2gp0A(n) 0(x)ψ(n)(0,x) =igα·∇A(n) 0(x)−gA(n) 02(x)ψ(n)(0,x).(1.16) We take first the infinite momentum limit of (1.16). To do that, we will assume that the energy p0of the fermion greatly exceeds the average magnitude of the interaction hgA(n) 0(x)i, a condition which reads ghA(n) 0(x)i ≡ gZd3x¯ ψ(n)(0,x)γ0A(n) 0(x)ψ(n)(0,x)p0.(1.17) Under this condition, in (1.16) we can drop offthe terms linear in gand g2but not the term in gp0, leading to p2 0−m2+∇2−2gp0A(n) 0(x)ϕ(n) s(0,x)=0,(1.18) where we denoted the solution in this limit ϕ(n) s(x). We state that a free solution with 4-momentum pmust be a good approximation up to a modified wave, which we call φ(n) s(x). Keeping this in mind we write down the ansatz ϕ(n) s(0,x)=rm p0 e−ip·xu(p)φ(n) s(x).(1.19) Solutions in the pure high energy limit ϕs(x) and φs(x) have been labeled with a (s) because they are solutions to an effective Schrodinger equation, as we will see below. Here u(p) stands for a free spinor and N(p)=pm/p0the normalization (see Appendix A). With this ansatz we find the equation for φ(n) s(x) ∇2φ(n) s(x)−2ip·∇φ(n) s(x)−2gp0A(n) 0(x)φ(n) s(x)=0.(1.20) For multiple external fields the above equation does not have a closed exact solution. We investigate, by now, the momentum change induced by A(n) 0(x). The modified wave φs(x) changes asymptotic momentum, which was p, to hϕ(n) s|˜pϕ(n) si=p+Zd3xφ(n),∗ s(x)+i∇φ(n) s(x)=p+δp.(1.21) In order to see how the modification δpcompares to p, one multiplies the last equation by pand uses (1.20), finding phϕ(n) s|˜pϕ(n) si=p2+1 2Zd3xφ(n),∗ s(x)∇2φ(n) s(x) −gp0Zd3xφ(n),∗ s(x)A(n) 0(x)φ(n) s(x),(1.22) 14 High energy fermions in an external field and transforming back to the original system and denoting ˆpis=2βtone finds G(n) s(pf,pi)=Zd3xfe−iq·xfexp −ig βZxl f −∞ ds A(n) 0xt f+ˆpis.(1.61) The particular limit tf→ ∞corresponds to the scattering amplitude if the spinorial part and the low energy corrections of the state are omitted. In this case one finds a conservation of longitudinal momentum change 2πδ(ql) and the integral restricts to the transverse plane to pi. For completeness we may compute the propagator in light-cone variables too. We define light cone variables as customary x+=x0+x3 √2,x−=x0−x3 √2,xt=(x1,x2),(1.62) in such way that the scalar product of two 4-vectors reads xµyµ=x0y0−x3y3−x1x1−x2x2=x+y−+x−y+−xt·yt.(1.63) We depart from a free fermion ψ(0)(x) satisfying the Dirac equation and, thus, the squared Dirac equation (/ p−m)ψ(0)(x)=0→(/ p+m)(/ p−m)ψ(0)(x)=(i∂µ∂ν−m2)ψ(0)(x)=0.(1.64) Now, since the spinorial structure in the squared equation is factorizable, by using ψ(0)(x)=pm/p0u(p)ϕ(0)(x) and the identity ∂0∂0−∂3∂3=1 2 ∂ ∂x+ +∂ ∂x−!2 −1 2 ∂ ∂x+−∂ ∂x−!2 =2∂ ∂x+ ∂ ∂x− ,(1.65) then the squared Dirac equation for ϕ(0)(x) can be rewritten in the light cone variables as 2∂ ∂x+ ∂ ∂x−−∂2 ∂2xt−m2!ϕ(0)(x)=0.(1.66) Let us use p−as the generator of the evolution in x+of the states and define the Fourier transform of ϕ(0)(x) at time point x+ ϕ(0)(x+,x−,xt)=Zdp+ 2πZd2pt (2π)2e−ip+x−+ipt·xt˜ϕ(0)(x+,p+,pt).(1.67) By inserting this expression in equation (1.66) one finds Zdp+ 2πZd2pt (2π)2e−ip+x−+ipt·xt −2ip+ ∂ ∂x+ +p2 t−m2!˜ϕ(0)(x+,p+,pt)=0,(1.68) Propagators in momentum and light cone spaces 15 from which it derives the equation −2ip+ ∂ ∂x+ +p2 t−m2!˜ϕ(0)(x+,p+,pt)=0,(1.69) or i∂ ∂x+ ˜ϕ(0)(x+,p+,pt)= p2 t 2p+−m2 2p+!˜ϕ(0)(x+,p+,pt),(1.70) which is a Schrodinger like equation for a particle of fictitious mass p+moving in the transverse plane to x3. Correspondingly its free Green function is given by G(0) s(x1 +,x1 t;x0 +,x0 t)=exp "+im2 2p+ (x1 +−x0 +)# (p+ 2π(x1 +−x0 +)exp "+ip+ 2(x1 +−x0 +)(x1 t−x0 t)2#).(1.71) This function propagates the free solutions between two space-time points in light cone variables. In order to add an interaction, we simply use the perturbative definition of the interacting Green function and take into account that the propagation is not constrained, in general, to a plane, if the interaction term is not, so G(n) s(x1 +,x0 −,x1 t;x0 +,x0 −,x0 t)=G(0) s(x1 +,x1 t;x0 +,x0 t)+ −ig Zx1 + x0 + dx+Zd3xG(0) s(x1 +,x1 t;x+,xt)A(n) 0(x)G(0) s(x+,xt;x0 +,x0 t)+(...) (1.72) and, since the free propagation does not depend on the x−variable, one finds G(n) s(x1 +,x0 −,x1 t;x0 +,x0 −,x0 t)=G(0) s(x1 +,x1 t;x0 +,x0 t) −ig Zx1 + x0 + dx+Zd2xtG(0) s(x1 +,x1 t;x+,xt) ·(Zdx−A(n) 0(x))G(0) s(x+,xt;x0 +,x0 t)+(...) ≡ZD2rtexp iZx1 + x0 − dx+ p+ 2˙r2 t−gZ+∞ −∞ dx−A(n) 0(x)!(1.73) which is the perturbative expansion of a path integral in the effective potential −ig Zdx−A(n) 0(x) :=−ig Zdx−A(n) 0 xt,x+−x− √2!.(1.74) 16 High energy fermions in an external field Since the interaction terms stops being a function of the x+variable once integrated in x−, consequently, the path integration affects only to the free propagator yielding G(n) s(x1,x0)=G(0) s(x1,x0) exp "−ig Z+∞ −∞ dx−A(n) 0(x)#.(1.75) The above propagator has lost the local dependence of the field with coordinate x3as a result of the time evaluation of x−at x0=±∞. 2 High energy multiple scattering With the development of the wave function in the previous chapter we are now in position to define the scattering amplitude, that is, the amplitude of finding the particle with some momentum and spin due to the effect of the medium [30– 35]. We will find that the obtained result preserves the unitarity of the emerging wave by proving the optical theorem, and that the series expansion in the coupling g=Ze2reproduces the perturbative description of the same problem. As previously stated, the evaluation of the wave or the scattering amplitudes for a particular configuration of the medium constituents is not interesting and we will instead look for averaged squared amplitudes and observables. In this way we will find that these averaged quantities for nscattering sources are always expressible as exponentiations of the single n=1 case. In the process of averaging we will discover also the emergence of interference phenomena related to multiple scattering events which suggests a splitting of the cross sections into two contributions. In a certain limit, namely for macroscopic mediums, the resulting cross sections become totally incoherent and therefore the scattering process will acquire an statistical interpretation in terms of probability distributions, whereas for mediums of microscopic size diffractive phenomena and thus, medium coherence, have a prominent role over the incoherent limit. We will also demonstrate that the incoherent contribution leads to the well-known Moliere’s derivation [36] based on markovian arguments [37] and we will compute the averaged momentum transfer hq2i. Finally we will introduce a beyond eikonal evaluation by considering a non vanishing longitudinal momentum change. We will show that the beyond eikonal squared amplitude, when averaged over mediums of macroscopic dimensions, reduces to the pure eikonal scattering for on-shell particles. However, longitudinal momentum changes at different energy states can produce interferences in the squared amplitudes, ultimately leading to incoherence phe17 18 High energy multiple scattering nomena when creation or annihilation of particles is also considered, as we will see in the next chapter. 2.1 The scattering amplitude We depart from an initial asymptotic free fermion with momentum piand spin si, whose state will be denoted ψ(0) i(x), ψ(0) i(x)=rm p0 i usi(pi)e−ipi·x.(2.1) Due to the effect of the external field (1.13) produced by the medium, in the asymptotic final state one finds the superposition of two states, the former wave itself and an infinite superposition of states of deflected momentum pfand spin sf. This reads ψ(n) f(x)=ψ(0) i(x)+X sf=1,2Zd3pf (2π)3M(n) sfsi(pf,pi)sm p0 f usf(pf)e−ipf·x.(2.2) The weights are referred to as the scattering amplitude or M-matrix, this is, the amplitude to find the wave with deflected momentum pfand spin sf. The asymptotic initial state can be absorbed in the scattering amplitude if we define ψ(n) f(x)=X sf=1,2Zd3pf (2π)3S(n) sfsi(pf,pi)sm p0 f usf(pf)e−ipf·x,(2.3) where the S-matrix includes the no collision distribution plus the collisional Mmatrix distribution, and by direct inspection of (2.2) and (2.3) verifies then S(n) sfsi(pf,pi)≡M(n) sfsi(pf,pi)+(2π)3δ3(pf−pi)δsfsi.(2.4) In order to find the scattering amplitude we rewrite the emerging wave in (2.2) by using the Lippmann-Schwinger recursive equation, ψ(n) f(x)=ψ(0) i(x)+Zd4yS F(x−y)gγ0A(n) 0(y)ψ(n)(y)=ψ(0) i(x)+ψ(n) di f f (x).(2.5) Inserting the result (1.42), valid for the wave in the high energy limit, we find ψ(n) di f f (x)=Zd4yS F(x−y)gγ0A(n) 0(y)·( 1+iγkγ0 2p0 i ∂k!W(n)(y,pi))ψ(0) i(y). (2.6) The scattering amplitude 19 In order to express this diffracted wave as an infinite superposition broadened around pi, we integrate the Feynman-Stueckelberg propagator (1.47) for energy positive solutions getting S(+) F(x)=1 iZd3p (2π)3 / p+m 2p0 e−ip·x=1 iX sZd3p (2π)3 m p0 us(p)⊗¯us(p)e−ip·x,(2.7) where the energy momentum p0(p)=pp2+m2relation arising in the p0integration over the positive pole has been left implicit and the completeness relation X s=1,2 us(p)⊗¯us(p)=/ p+m 2m,(2.8) has been used. By inserting (2.7) in (2.6) one finds ψ(n) di f f (x)=X s1=1,2Zd3p (2π)3sm p0 f usf(pf)e−ipf·xZd4yei(pf−pi)y−igA(n) 0(y) ·sm p0 f ¯usf(pf)γ0( 1+iγkγ0 2p0 i ∂k!W(n)(y,pi))rm p0 i usi(pi) ≡X sf=1,2Zd3pf (2π)3sm p0 f usf(pf)e−ipf·xM(n) sfsi(pf,pi),(2.9) where the integration must be carried on-shell. The expression (2.9) is interpretable as a superposition with amplitude M(n) sfsi(pf,pi) of being at the state (pf,sf) given by ψ(0) f(x)=sm p0 f usf(pf)e−ipfx.(2.10) Thus we define the amplitude, following (2.2), as M(n) sfsi(pf,pi)=Zd4yei(pf−pi)·y−igA(n) 0(y)(2.11) ·sm p0 f ¯usf(pf)γ0( 1+iγkγ0 2p0 i ∂k!W(n)(y,pi))usi(pi)rm p0 i . The quantity in the bottom line of the above expression, which we call J, contains the non perturbative information of the diffracted wave, J=sm p0 f ¯usf(pf)γ0( 1+iγkγ0 2p0 i ∂k!W(n) 0(y,pi))usi(pi)rm p0 i .(2.12) 20 High energy multiple scattering Had we expanded the phase in the coupling we would have found W(n)(y,pi)=1+O g βi!→J=sm p0 f ¯usf(pf)γ0usi(pi)rm p0 i +O g βi!(2.13) which, together the Fourier transform of of A(n) 0(y) would produce the first perturbative term of the scattering matrix, representing all single kicks combinatory with the medium elements. However, as we stated before, we expect that the perturbative processes of high order are the relevant ones, then we have to compute the term non perturbatively. We have J=sm p0 f ¯usf(pf) γ0+i 2p0 i γ† k ∂ ∂xk!usi(pi)rm p0 i W(n)(y,pi).(2.14) Notice that, although the derivative correction is suppressed with an overall factor 1/p0 i, the spinorial terms may introduce p0 icorrections. It is easy to show that for elastic scattering this is not the case, since the above expression produces in the high energy limit, pf≈pi, J≃δsf si 1−iβ 2p0 i ∂3!W(n)(y,pi)=δsf si1−gA(n) 0(y) 2p0 iW(n)(y,pi).(2.15) In this limit spin keeps unchanged sf=sisince the spin-flip amplitudes are suppressed a factor 1/p0 iwith respect to the non-flip case. The term arising in the phase derivative can be neglected so ghA(n) 0(y)i 2p0 i1→J≃sm p0 f ¯usf(pf)γ0usi(pi)rm p0 i W(n)(y,pi) (2.16) where the Glauber condition (1.23) has been used. It can be globally neglected, then, provided that the external interaction is sufficiently smooth and p0 i→ ∞. By performing the time integral we get the energy conservation so βi=βf≡β and M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)sm p0 f ¯usf(pf)γ0usi(pi)rm p0 i (2.17) ×Zd3ye−iq·y−igA(n) 0(y)W(n)(y,p0), where the momentum transfer with the external field is denoted as q=pf−pi. The remaining integral can be performed by placing pialong the z axis and taking The scattering amplitude 21 the high energy limit qzqt. We find the asymptotic value Zd3ye−iq·y−igA(n) 0(y)W(n)(y,p) ≈βZd2yte−iqt·ytZ+∞ −∞ dy3 −ig βA(n) 0(y)!exp "−ig βZy3 −∞ dy0 3A(n) 0(y)# =βZd2yte−iqt·yt exp "−ig βZ+∞ −∞ dy3A(n) 0(y)#−1!.(2.18) Finally, the amplitude of a change of 4-momentum q=p1−p0under the field of nsources, at high energy and at all orders in the coupling is given (2.2) by M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)βsm p0 f ¯usf(pf)γ0usi(pi)rm p0 i (2.19) ×Zd2yte−iqt·yt exp "−ig βZ+∞ −∞ dy3A(n) 0(y)#−1!. Similarly, by including the no collision amplitude (2.3), we write S(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)βsm p0 f ¯usf(pf)γ0usi(pi)rm p0 i (2.20) ×Zd2yte−iqt·ytexp "−ig βZ+∞ −∞ dy3A(n) 0(y)#. It results convenient to extract the energy and spin conservation delta out of the above amplitudes by defining the quantities M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)δsf siβF(n) el (q),S(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)δsf siβS(n) el (q), (2.21) where F(n) el (qt) denotes the integral part of (2.19) F(n) el (qt)≡Zd2yte−iqt·yt exp "−ig βZ+∞ −∞ dy3A(n) 0(y)#−1!.(2.22) and S(n) el (qt) denotes the integral part of (2.20) S(n) el (qt)≡Zd2yte−iqt·ytexp "−ig βZ+∞ −∞ dy3A(n) 0(y)#.(2.23) Since β→1 and energy and spin are preserved F(n)(qt) contains all the relevant physics. In this approximation, as a result of taking the qz→0 limit, the scattering amplitude stops being able to resolve any source structure in the zdirection, 22 High energy multiple scattering since the eikonal phase in (2.22) appears integrated in y3=±∞. This approximation then erased the locality in y3of the wave (1.42), which satisfied that at point y3electron is only affected by sources verifying zi<y3, a sign of a gradual transformation of time causality to y3causality as a result of taking the β→1 limit. Finally, we can connect this result with Glauber’s spinless amplitude f(θ). For a Schrodinger wave the scattered state can be written as ψf(x)=ψi(x)+f(θ)e−i|pi||x| |x|(2.24) where θ=q/βp0 iis the scattering angle. The relation between f(θ) and the Mmatrix is geometric. Since |f(θ)|2is the intensity in the solid angle interval Ωand Ω + dΩ, then dividing by time and the incoming flux, from (2.2) and (2.24) Zf(θ) 2dΩ = Zd3pf (2π)3M(n) sfsi(pf,pi) 2→f(θ)=|pi| 2πiF(n) el (qt).(2.25) Glauber derived the above amplitude for a Schrodinger particle, thus in principleomitting relativistic and spin effects, something which seems contradictory with the high energy limit. His description is, however, completely accurate in the limit β→1 since the high energy limit of the Dirac equation is well approximated by a Schrodinger like equation. 2.2 Perturbative expansion and strong coupling The high energy integration of the scattering amplitude can be expanded in the coupling g=Ze2. Order to order it has to agree with the standard perturbative representation of the elastic scattering with the medium. We note first that the integration of the field in the phase produces for a general interaction (1.13) χ(n) 0(yt)≡Z+∞ −∞ dy3A(n) 0(y)= n X i=1Z+∞ −∞ dy3Zd3q (2π)3eiqt·(yt−ri t)+iq3(y3−ri 3)ˆ A(1) 0(q) = n X i=1Zd2qt (2π)2eiqt·(yt−ri t)ˆ A(1) 0(qt,0).(2.26) The above result means that the y3coordinates of each center are lost, the entire medium is seen as an infinitesimal sheet and any of the single collisions are considered totally eikonal qz=0. For the Debye screened interaction (1.8), in particular, we find χ(n) 0(yt)=Z+∞ −∞ dy3A(n) 0(y)=2 n X i=1 K0(µd|yt−ri t|).(2.27) Perturbative expansion and strong coupling 23 At leading order in g, using (2.26) we obtain from (2.22) F(n) el (qt)=Zd2yte−iqt·yt−ig β n X i=1Zd2kt (2π)2eikt·(yt−ri t)A(1) 0(kt) +O g2 β2! =−ig βˆ A(1) 0(qt) n X i=1 e−iqt·ri t+O g2 β2!(2.28) So amplitude (2.19) at first order in the coupling simplifies to ≡++ + ··· Figure 2.1: Diagrammatic representation of F(1) el (q) 2for a single center up to 3rd order in the coupling g=Ze2. M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)sm p0 f ¯usf(pf)γ0usi(pi)rm p0 i ×(−ig)ˆ A(1) 0(qt) n X i=1 e−iqt·ri t +O g2 β2!(2.29) For the particular case of the Debye screened interaction, using (1.8) or (2.27) we find a superposition of nsingle Mott amplitudes since −ig ˆ A(1) 0(q)=−i4πZe2 q2+µ2 d .(2.30) Only at leading order (l.o.) in gthe total amplitude of each center simply adds with a phase related to its position, M(n) sfsi(pf,pi)l.o.= n X i=1 e−iqt·ri tM(1) sfsi(pf,pi)l.o.(2.31) For larger values of ga numerical integration of (2.19) has to be carried. In Figure 2.2 various cases of the scattering amplitude (2.19) are shown and compared with the leading order approximation (2.29). We see that for large enough qthe 1/q4 tail of the Rutherford scattering is found and the arbitrary coupling evaluation matches the leading order approximation. A closed form of the amplitude (2.19) for a single center can be given if µd→0. In this limit the leading order of 30 High energy multiple scattering volume V=lΩ, with lthe length and Ω = πR2the transverse area. Then the above average is understood as F(n) el (q) 2=1 VnZV d3r1...d3rnF(n) el (q) 2=1 ΩnZV d2rt 1...d2rt nF(n) el (q) 2, (2.60) so we find that it reduces to an average over a single center of the form F(n) el (q) 2=Zd2xtd2yte−iqt·(xt−yt) "1− 1 ΩZΩ d2rtexp "−ig βχ(1) 0(xt−rt)#!n − 1 ΩZΩ d2rtexp "+ig βχ(1) 0(yt−rt)#!n + 1 ΩZΩ d2rtexp "−ig βχ(1) 0(xt−rt)#exp "+ig βχ(1) 0(yt−rt)#!n#.(2.61) The above expression is, however, highly oscillatory and unsuitable for numerical evaluation. One can make a standard approximation, valid for large number of scattering centers n1. By defining the density of sources n0=n/Vwe observe 1 ΩZΩ d2rtexp "−ig βχ(1) 0(xt−rt)#!n = 1+1 ΩZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!!n = 1+n0l nZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!!n .(2.62) Since the interaction vanishes at transverse distances larger than µ−1 dthen the integral is bounded so, for large enough nwe can do lim n→∞ 1 ΩZΩ d2rtexp "−ig βχ(1) 0(xt−rt)#!n =exp "n0lZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!#.(2.63) And finally F(n) el (q) 2=Zd2xtd2yte−iqt·(xt−yt)(2.64) × 1−exp "n0lZΩ d2rtexp −ig βχ(1) 0(xt−rt)−1# −exp "n0lZΩ d2rtexp +ig βχ(1) 0(yt−rt)−1# +exp "n0lZΩ d2rtexp −ig βχ(1) 0(xt−rt)+ig βχ(1) 0(yt−rt)−1#!. Multiple scattering effects 31 The above expression is already suitable for numerical evaluation. We notice, however, that it can be split into two parts which admit a clear physical interpretation. One part is related to the coherent scattering which can be interpreted as the scattering in an averaged medium, and the other to an incoherent contribution. To show this fact we use the relation M=S−1 leading to DM(n) sfsi(pf,pi)M(n) sfsi(pf,pi)∗E=S(n) sfsi(pf,pi)−1 2 DS(n) sfsi(pf,pi)S(n) sfsi(pf,pi)∗E−DS(n) sfsi(pf,pi)EDS(n) sfsi(pf,pi)∗E,(2.65) where we added and subtracted the term ±S(n) sfsi(pf,pi)DS(n) sfsi(pf,pi)∗E,(2.66) in order to find the well known result from statistics hx2i=hxi2+σ2. Two contributions appear with a clear physical interpretation. We will call these contributions the coherent and incoherent average, denoted as M(n) sfsi(pf,pi) 2= Π(n) 2(pf,pi)+ Σ(n) 2(pf,pi),(2.67) with the coherent contribution given by Π(n) 2(pf,pi)≡S(n) sfsi(pf,pi)−1 2 =M(n) sfsi(pf,pi) 2 ,(2.68) and the incoherent contribution being Σ(n) 2(pf,pi)≡DS(n) sfsi(pf,pi)S(n) sfsi(pf,pi)∗E−S(n) sfsi(pf,pi)DS(n) sfsi(pf,pi)∗E. (2.69) The first contribution consists in the square of the averaged amplitude. Correspondingly, as usual by dividing by incoming flux and time and factorizing the conservation deltas we define Π(n) 2(pf,pi)=2πδ(p0 f−p0 i)δsfsiβˆ Π(n) 2(q,l),(2.70) where the relevant coherent average is given by ˆ Π(n) 2(q,l)=DF(n) el (q)E 2 (2.71) =Zd2xte−iqt·xt exp "n0lZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!#−1! 2 . 32 High energy multiple scattering This contribution consists in a coherent superposition of each center at the level of the amplitude. We can integrate (2.71) for a cylinder whose transverse dimensions greatly exceed the dimensions of a single scatterer, R1/µd. In this case ZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!=F(1) el (0).(2.72) Inserting this result in (2.71) we find the larger size approximation of the coherent contribution Π(n) 2(pf,pi)=2πδ(p0 f−p0 i)δsfsiβ (2π)2δ2(qt)exp n0lF(1) el (0)−1 2 .(2.73) Due to symmetry arguments, the infiniteness of the medium transverse direction transforms the coherent part into a pure forward contribution. The values of the single scattering amplitude F(1) el (0) in the forward direction appearing in the last equation must be related to the single elastic cross section F(1) el (0)=1 2−σ(1) tot +iσ(1) osc,(2.74) and can be read from equations (2.54) and (2.55) for arbitrary coupling or from (2.52) and (2.53) at leading order provided that the single collision case is perturbative in g. In any case, we notice that this does not imply the single scattering regime, since n0is arbitrary. For mediums of finite size, however, the coherent term represents the modification of the single scattering matrix due to the border effects. In order to show this fact we write the relation exp −ig βχ(1) 0(xt−rt)=1+1 (2π)2Zd2kte+ikt·(xt−rt)F(1) el (kt),(2.75) which integrated in a finite section Ωproduces an extra window function ZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!=Zd2kt (2π)2e+ikt·xtF(1) el (kt)ZΩ d2rte−ikt·rt. (2.76) The window function for the case of the finite cylinder acquires the form WΩ(kt,R)≡ZΩ d2rte−ikt·rt→Wcyl(kt,R)=2πR |kt|J1(|kt|R).(2.77) where J1(x) is the Bessel function of the first kind. For an arbitrary geometry, the window function momentum domain has an oscillatory behavior in 1/Rmodulating the 1/q4fall off. This can be directly observed at low density by expanding (2.71) in n0yielding Π(n) 2(p1,p0)=2πδ(p0 f−p0 i)δsfsiβ n0lWΩ(qt,R)F(1) el (qt) 2 .(2.78) Multiple scattering effects 33 As expected for a coherent contribution this term is of order n2, so at low density the averaged amplitude squared is n2times the squared elastic amplitude of a single center, with a window function containing information of the medium geometry. This window can be accounted as a border diffractive effect which factorizes from the single elastic amplitude only at low densities. For a cylinder medium of radius R, in particular, we find Π(n) 2(p1,p0)=4n2J2 1(|qt|R) q2 tR2M(1) sfsi(p1,p0) 2 +O(n4).(2.79) We proceed to compute now the incoherent contribution to the scattering amplitude. Using (2.23), the first term is given by S(n) sfsi(p1,p0)S(n) sfsi(p1,p0)∗=2πδ(p0 1−p0 0)δsfsiβDS(n) el (q)S(n),∗ el (q)E,(2.80) where the incoherent average restricts to the integral parts DS(n) el (q)S(n),∗ el (q)E=Zd2xtZd2yte−iqt·(xt−yt)(2.81) exp n0lZΩ d2rt exp −ig βχ(1) 0(xt−rt)+ig βχ(1) 0(yt−rt)−1!. It results convenient to rewrite the terms in the exponent as squared single amplitudes. To do that we notice exp−ig βχ(1) 0(xt−rt)+ig βχ(1) 0(yt−rt)−1 =exp −ig βχ(1) 0(xt−rt)−1+exp +ig βχ(1) 0(yt−rt)−1 +exp −ig βχ(1) 0(xt−rt)−1exp +ig βχ(1) 0(yt−rt)−1.(2.82) By taking the infinite medium limit R1/µdthe first two integrals are just forward single amplitudes, n0lZΩ d2rt exp −ig βχ(1) 0(xt−rt)−1!=n0lF(1) el (0) n0lZΩ d2rt exp +ig βχ(1) 0(yt−rt)−1!=n0lF(1),∗ el (0),(2.83) whereas for the mixed term we find, using (2.75) n0lZΩ d2rtexp −ig βχ(1) 0(xt−rt)−1exp +ig βχ(1) 0(yt−rt)−1 =n0lZd2kt (2π)2e+ikt·(xt−yt)F(1) el (kt) 2.(2.84) 34 High energy multiple scattering The second term of the incoherent contribution is given by S(n) sfsi(pf,pi)S(n),∗ sfsi(pf,pi)=2πδ(p0 f−p0 i)δsfsiβDS(n) el (q)EDS(n),∗ el (q)E,(2.85) where similarly DS(n) el (q)EDS(n),∗ el (q)E=Zd2xtZd2yte−iqt·(xt−yt) ·exp n0lZΩ d2rtexp −ig βχ(1) 0(xt−rt)−1 +n0lZΩ d2rtexp +ig βχ(1) 0(yt−rt)−1.(2.86) Proceeding in the same way, for a medium satisfying Rµ−1 dwe simply have S(n) sfsi(pf,pi)S(n),∗ sfsi(pf,pi)=2πδ(p0 f−p0 i)δsfsiβZd2xtZd2yte−iqt·(xt−yt) ·exp n0lF(1) el (0)+n0lF(1),∗ el (0).(2.87) Correspondingly after summing these two terms one finds Σ(n) 2(pf,pi)=2πδ(q0)δsfsiβˆ Σ(n) 2(q,l),(2.88) where the relevant incoherent average is given by ˆ Σ(n) 2(q,l)=exp h2n0lRe F(1) el (0)iZd2xtd2yte−iqt·(xt−yt) ×(exp "n0lZd2kt (2π)2e+ikt·(xt−yt)F(1) el (kt) 2#−1).(2.89) The above form of the incoherent contribution is related to Moliere’s theory of scattering [36, 37] since by using (2.43), ˆ Σ(n) 2(q,l)= Ω exp −n0lσ(1) tot Zd2xte−iqt·xt ·(exp "n0lZd2kt (2π)2e+ikt·xtF(1) el (kt) 2#−1),(2.90) where the single cross section can be read from equation (2.52) for small coupling or from equation (2.54) for general coupling, and the trivial integral in one of the impact parameters has produced and overall factor Ω = πR2accounting for transverse homogeneity when R→ ∞. Except for the −1, accounting for boundary effects and allowing the integration for screened interactions, (2.90) Multiple scattering effects 35 constitutes a solution of the Moliere equation. The −1 is strictly necessary to integrate in impact parameter for any screened interaction. The Fourier transform of the single amplitude F(1) el (q) in the exponential at (2.90) will be denoted as σ(1) el (x)=Zd2kt (2π)2e+ikt·xtF(1) el (kt) 2.(2.91) and can be numerically computed for arbitrary gusing (2.28) representing the all orders interaction with each single center. The extension to ncenters is just its exponentiation. For low coupling we can use F(1) el (kt)=−i β 4πg k2 t+µ2 d +O(g2)→σ(1) el (xt)=4πg2 β2µ2 d µd|xt|K1(µd|xt|)+O(g3). (2.92) which after inserted in (2.90) produces a suitable form for fast numerical calculations. Observe that we called this function σ(1)(xt) because it is related to the single cross section as σ(1)(0)≡σ(1) tot . An expansion in n0produces Σ(n) 2(pf,pi)= Ω2πδ(p0 f−p0 i)δsfsiβ(2.93) ·Zd2xte−iqt·xt(n0lZd2kt (2π)2e+ikt·xtF(1) el (kt) 2)+O(n2 0) =(n0πR2l)2πδ(p0 f−p0 i)δsfsiβF(1) el (qt) 2+O(n2 0)≡nM(1) sfsi(pf,pi) 2 +O(n2). Only at low densities the probability of changing qtdue to the joint effect of ncenters is just ntimes the probability of changing qtdue to a single center. This relation does not hold for higher densities and, in particular, for saturation densities an asymptotic form is going to be later defined. At low nand R1/µd we can join equations (2.78) and (2.93) leading to M(n) sfsi(pf,pi) 2≃(n+4n2J2 1(|qt|R) q2 tR2)M(1) sfsi(pf,pi) 2.(2.94) The above result may have been obtained from a direct expansion in the number of centers. This approach will give us some insight on the interference pattern and the statistical interpretation of the coherent and incoherent contributions. Let us denote Γi=exp −ig βχ(xt−ri t)−1,(2.95) in such a way that the elastic amplitude can be rewritten as F(n) el (q)=Zd2xte−iqt·xtn Y i=1 (Γi+1) −1 =Zd2xte−iqt·xtn X i=1 Γi+ n X i=1 n X j=i+1 ΓiΓj+···= n X i=1 Ii+ n X i=1 n X j=i+1 Ii j +··· , 36 High energy multiple scattering which is an expansion in the number of collisions. Consider first the terms where only one (Γi) collision appears in the amplitude. We are left with n X i=1 Ii(q)= n X i=1Zd2xe−iq·xexp −ig βχ0(x−ri)−1=n X i=1 e−iq·riF(1) el (q). (2.96) Any other term of higher order can be always written as a convolution with an additional phase factor of the form Ii j(q)=Zd2qi (2π)2 d2qj (2π)2e−iqi·ri−iqj·rjF(1) el (qi)F(1) el (qj)(2π)2δ2(q−qi−qj),(2.97) and so on. Now consider the squared amplitude, in this form it reads F(n) el (q) 2= n X i=1 n X j=1 IiI∗ j+ n X i=1 n X j=1 n X k=j+1IiI∗ jk +I∗ iIjk+··· .(2.98) The first contribution is given by n X i=1 n X j=1 IiI∗ j= n X i=1 n X j=1 e−iq·(ri−rj)F(1) el (q) 2=n+ n X i=1 n X j,i e−iq·(ri−rj)F(1) el (q) 2. (2.99) The interpretation of the two above terms is clear. The first term gives the contribution from nindependent collisions, whereas the second term gives the interference, at this order, between them. Notice that at q=0 we have n X i=1 n X j,i e−iq·(ri−rj)q=0 =n(n−1) (2.100) which is the constructive interference between the ncenters. For q,0 this factor is not positive defined, so that one can not interpret it as a probability density. One can average, however, over center configurations obtaining, for a cylinder of section Ω = πR2 *n X i=1 n X j,i e−iq·(ri−rj)+=n(n−1) Ω2ZΩ d2riZΩ d2rje−iq·(ri−rj)=4n(n−1) J2 1(qR) (qR)2. (2.101) In the R→ ∞ limit we retrieve the forward constraint since the contribution is given by n(n−1)/R2δ(q). Thereby, the contribution to the scattering by decomposition in single scatterings is given by F(n) el (q) 2≃(n+4n(n−1) J2 1(qR) (qR)2)F(1) el (q) 2,(2.102) Multiple scattering effects 37 the first term being the independent (incoherent) sum of the nscattering centers and the second term the (coherent) interference between them, which results in the original scattering amplitude squared modulated by the Fourier transform of the shape of the medium, matching the result (2.94). The coherent term has interference peaks at q=0 and q=5.14/R. For large Rthe second term will be negligible for qlarger than about q=(4πnl/R)1/3. For the terms with two collisions in the amplitude we find n X i=1 n X j,i n X k=1 n X l,k Ii jI∗ kl =Zd2qi (2π)2Zd2qj (2π)2Zd2qk (2π)2Zd2ql (2π)2W2(q) ·F(1) el (qi)F(1) el (qj)F(1),∗ el (qk)F(1),∗ el (ql),(2.103) where the diffractive window function reads now W2(q)= n X i=1 n X j,i n X k=1 n X l,k e−iqi·ri−iqj·rj+iqk·rk+iql·rl = n X k=i=1 n X l=j,i + n X i=1 n X j,i n X k,iX l,k,j e−iqi·ri−iqj·rj+iqk·rk+iql·rl,(2.104) where again we have split the sum into a diagonal and a non diagonal part. The diagonal part is given by Wdg 2(q)=n(n−1) Ω2ZΩ d2riZΩ d2rje−iri·(qi−qk)−irj·(qj−ql) =n(n−1) Ω2 2πRJ1(|qi−qk|R) |qi−qk| 2πRJ1(|qj−ql|R) |qj−ql|,(2.105) which simplifies to δ(qi−qk)δ(qj−ql) in the limit R→ ∞ producing  n X i=1 n X j,i n X k=1 n X l,k Ii jI∗ kl dg =Zd2qi (2π)2Zd2qj (2π)2F(1) el (qi)F(1) el (qj)(2π)2δ2(q−qi−qj), (2.106) that is, a convolution of two independent collisions with different centers leading to a total momentum change q. The non diagonal part will contribute through convolutions of the same type to the forward amplitude. Therefore, the structure of the expansion in nis always split into a contribution which in the large medium limit Rµ−1 dcan be interpreted in probabilistic terms, corresponding to the expansion in n0of equation (2.90), and a contribution accounting for diffractive effects. 38 High energy multiple scattering For general nand gthe probability of finding the fermion with final momentum pfis given necessarily by a numerical evaluation of the full expression (2.64) or by using the coherent (2.71) and incoherent (2.90) terms, easier to compute than (2.64). In Figure 2.4 the probability of finding an electron with transverse momentum qis shown and compared with the leading order in napproximation together with the incoherent contribution. One can check that the optical theorem 1E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 0 5 10 15 20 25 F(n) el (q) 2(KeV−2) q(KeV) n=10 n=100 n=500 Figure 2.4: Squared elastic amplitude F(n) el (q) 2after traversing a medium with Z=7 corresponding to a coupling of g=0.05, Debye mass of µd=71 KeV for increasing values of ncenters distributed in cylinder of radius R=6rdand length l=R/10. Dots are direct evaluation of (2.64), whereas continuum lines are the low density result (2.94) and dot-dashed lines are the incoherent contribution (2.90). is valid also for the medium averaged cross sections. Indeed, X sfZd3pf (2π)3M(n) sfsi(pf,pi) 2=2πδ(p0 i−p0 i)δsisiβZd2q (2π)2F(n) el (q) 2. (2.107) By using (2.64) we find Zd2q (2π)2F(n) el (q) 2=(2.108) 2 Re Zd2xt 1−exp "n0lZΩ d2rtexp −ig βχ(1) 0(xt−rt)−1#!, so finally Zd3pf (2π)3M(n) sfsi(pf,pi) 2=2πδ(p0 i−p0 i)δsisiβ2 Re DF(n) el (0)E ≡ −2 Re DM(n) sisi(pi,pi)E.(2.109) Multiple scattering effects 39 This is remarkable, since the averaging at the level of the amplitudes is rather different than the averaging at the level of the cross sections. Similar result can be obtained by direct integration of the coherent and incoherent contributions (2.73) and (2.90) arriving to σ(n) tot =X sfZd3pf (2π)3hΠ(n) 2(pf,pi)+ Σ(n) 2(pf,pi)i(2.110) =2πR2Re 1−exp n0lF(1) el (0)=2πR21−exp −n0lσ(1) tot 2cos n0lσ(1) osc 2!. where we used the single cross section definitions (2.54) and (2.55). At small density we simply have σ(n) tot =2πR2n0lσ1 tot 2+··· =nσ(1) tot ,(2.111) as expected. In the limit of very large density, instead, we get σ(n) tot =2πR2, which is the correct high energy limit for the cross section of a black disk of radius R. Oscillations of (2.110) are a diffractive effect due to the coherent scattering of the entire medium. Indeed, for the split contributions to the total cross section we find in the incoherent part σ(n) inc ≡X sfZd3pf (2π)3Σ(n) 2(pf,pi)=πR21−exp −n0lσ(1) tot 2,(2.112) whereas for the coherent contribution σ(n) coh ≡X sfZd3pf (2π)3Π(n) 2(pf,pi) =πR21+exp −n0lσ(1) t−2 exp −n0lσ(1) tot 2cos  n0lσ(1) i 2.(2.113) Oscillations in (2.110) with surface density n0ldue to a coherently acting medium have a period n0lσ(1) osc/2. Since σ(1) osc is of order O(g) whereas σ(1) tot is of order O(g2) for small coupling the oscillations will be clearly seen in the total cross section. We see that for n0l≈2/σ(1) tot the total cross section saturates. This defines a saturation scale given by n0l=2 σ(1) t≈µ2 d 2πg2,(2.114) 46 High energy multiple scattering where Θ(y) is the Heaviside step function. The above result says that the wave (1.42) at y3is affected by the set of centers at the left of y3, and changes abruptly at the passage of each center. In this way it preserves the internal structure of the longitudinal organization of the medium although it looses the internal structure of the longitudinal dimensions of each center. We will now consider a set of n1≡n(z1) centers laying in a sheet of vanishing thickness δz→0 at coordinate z. The elastic amplitude of this sheet reads from (2.12) M(n1) s1s0(p1,p0)=2πδ(q0)δs1 s0βZd3ye−iq·y∂ ∂y3 exp −ig βZy3 −∞ ds A(m) 0(y)! =2πδ(q0)δs1 s0βe−iq3zZd2xte−iqt·xtexp −ig β m X i=1 χ(yt−ri t)−1,(2.139) where the integration has been carried by parts. This amplitude reproduces the diffracted part of the wave, so it includes at least one collision. We can write for the total wave the total amplitude S=M+1 S(n1) s1s0(p1,p0)=2πδ(q0)δs1 s0βe−iq3zZd2yte−iqt·ytexp −ig β m X i=1 χ(yt−ri t), (2.140) which leaves opened the possibility of non interacting at all in z. We can consider the medium as a set of nlayers at z1,z2,...,znwith n(zi) scattering centers. The total number of centers in the medium is then n X i=1 n(zi)≡N.(2.141) The amplitude of emerging with momentum pnand spin snafter traversing the n sheets from z1to znis given at high energies by the convolution S(N) sns0(pn,p0)≡ n−1 Y i=1Zd3pi (2π)3 n Y i=0 Sn(zi) si+1si(pi+1,pi),(2.142) sum in repeated indices assumed. From this we simply get M(N) sns0(pn,p0)=S(N) sns0(pn,p0)−S(0) sns0(pn,p0).(2.143) A path integral representation of the above amplitudes can be written. By integrating in internal momenta and summing over intermediating spins we obtain S(N) sns0(pn,p0)=2πδ(p0 n−p0 0)δsn s0β n−1 Y i=iZd2pt i (2π)2  n Y i=1Zd2xt iexp −iqi·xi−ig β n(zi) X j=1 χ(xt i−rt j) ,(2.144) Beyond eikonal scattering 47 where qi=pi−pi−1is a complete 3-momentum change and the energy conservation deltas have been used to fix the longitudinal components as pz≃p0 0− p2 t/(2p0 0). We rewrite the terms according to −i n X i=1 qi·xi=−ipn·xn+i n−1 X i=1 pi·δxi+ip0·x1,(2.145) with δxi≡xi+1−xi. If we also define δzi=zi+1−ziwe then obtain S(N) sns0(pn,p0)=2πδ(p0 n−p0 0)δsn s0β n Y i=1Zd2xt ie−ipn·xn+ip0·x1  n−1 Y i=1Zd2pt i (2π)2exp  +ipt i·δxt i−ipt i2 2p0 0 δzi−ig β n(zi) X j=1 χ(xt i−rt j) .(2.146) Upon performing the momentum integrals and taking the δz→0 limit we find lim δz→0S(N) sns0(pn,p0)=2πδ(p0 n−p0 0)δsn s0βZd2xt nZd2xt 1e−ipn·xn+ip0·x1 Zxt(zn) xt(z1)D2xt(z) exp  iZzn z1 dz p0 0 2˙x2 t(z)−g β n(z) X k=1 χ(1) 0(xt(z)−ri t(z)) ,(2.147) Notice that the evaluation of this amplitude for the most probable path is given by the Euler-Lagrange equation dpt dz =g∇ Z+∞ −∞ dt An(z) 0(x+βt)!,(2.148) which is a classical trajectory approximation assuming a straight propagation. The average of the square of (2.143) over medium configurations can be written as before as the sum of an incoherent and a coherent contribution DM(N)∗ sns0(pn,p0)M(N) sns0(pn,p0)E= Σ(N) 2(pn,p0)+ Π(n) s(pn,p0),(2.149) where the incoherent contribution is given by the quantity Σ(N) 2(pn,p0)=DS(N) sns0(pn,p0)∗S(N) sns0(pn,p0)E−DS(N) sns0(pn,p0)E∗DS(N) sns0(pn,p0)E, (2.150) and the coherent contribution is given by the averaged amplitude squared Π(N) 2(pn,p0)=DS(N) sns0(pn,p0)−S(0) sns0(pn,p0)E 2.(2.151) 48 High energy multiple scattering The cancellation of the longitudinal phases in the macroscopic limit, since the transverse momentum in the conjugated amplitude equals the transverse momentum in the amplitude, leads to an incoherent contribution of the form Σ(N) 2(pn,p0)=2πδ(p0 n−p0 0)βpδsn s0πR2 n−1 Y i=1 Zd2pt i (2π)2!n Y i=1 Zd2xt ie−iδpt i·xt i! ×       n Y i=1 exp δzin0(zi)σ(1) el (xt i)−σ(1) el (0)− n Y i=1 exp −δzin0(zi)σ(1) el (0)       , (2.152) where we divided by time 2πδ(0) ≡Tand incoming flux β. Correspondingly the integration in internal momenta is trivial and the internal structure of the scattering distribution is lost. By taking the δzi→0 limit we obtain Σ(N) 2(pn,p0)=2πδ(p0 n−p0 0)βδsn s0πR2exp −σ(1) el (0)Zzn z1 dz n0(z)! ×Zd2xte−i(pt n−pt 0)·xt exp σ(1) el (xt)Zzn z1 dz n0(z)!−1!.(2.153) The evaluation of the coherent contribution follows the same steps. The average of the amplitude produces DS(N) sns0(pn,p0)E=2πδ(p0 n−p0 0)βδsn s0 n−1 Y i=1 iδzi 2πp0 0!n Y i=1 Zd2xi t!(2.154) ×exp −ipt n·xt n+i n−1 X i=1 p0 0 2 δxt i δzi!2 δzi+ n X i=1 δzin0(zi)π(1) el (xt i)+ipt 0xt 1, where in analogy with the function σ(1) el (x) for the incoherent contribution we defined the Fourier transform of the single elastic amplitude convoluted with the window function of the medium as π(1) el (x)≡Zd2qt (2π)2eiq·xF(1) el (q)WΩ(q,R).(2.155) By taking the δz→0 limit the above average transforms into a path integral in the transverse plane where the time variable is the zposition as DS(N) sns0(pn,p0)E=2πδ(p0 n−p0 0)βpδsn s0Zd2xt ne−ipt n·xt nZd2xt 1e+ipt 0xt 1 ZD2xt(z) exp iZzn z1 dz p0 0 2˙x2 t(z)−in0(z)π(1) el xt(z)!!.(2.156) Beyond eikonal scattering 49 Then the beyond eikonal evaluation of the coherent contribution is given by Π(N) 2(pn,p0)=2πδ(p0 n−p0 0)βδsn s0Zd2xt ne−ipt n·xt nZd2xt 1e+ipt 0xt 1(2.157) ×ZD2xt(z) exp iZzn z1 dz p0 0 2˙x2 t(z)! exp Zzn z1 dz n0(z)π(1) el xt(z)!−1! 2 , where we divided by time and incoming flux. The above beyond eikonal results will become useful when evaluating emission processes occurring in a multiple scattering scenario. As we will see in the next chapter, when an energy gap is considered between the state of the traveling particle and its conjugate, corresponding to an energy carried by the emitted particle, a non vanishing longitudinal phase will lead to coherence effects in the intensity spectrum. 3 High energy emission In the previous chapters we worked out the basic concepts and tools towards an evaluation of the amplitude and the intensity of a high energy fermion transiting from (pα,sα) to (pβ,sβ) due to the effect of the multiple scattering sources in a medium. We will consider now a single photon bremsstrahlung occurring while this multiple scattering develops. For this purpose we take advantage of the fact that, for sufficiently low photon energies, the elastic amplitude has to factorize from the emission amplitude [5]. In consequence, as we will show, our previous results dealing with the pure elastic problem have to be recovered and be still valid for the evaluation of the radiation of soft quanta offhigh energy fermions. The emission intensity in a multiple scattering scenario has been predicted by Ter-Mikaelian [1] and Landau and Pomeranchuk [2] to be suppressed with respect to a naive picture consisting in an incoherent sum of single Bethe-Heitler [6] intensities. Longitudinal phases in the scattering amplitudes regulate the amount of matter which can be considered a single and independent emitter. These phases control the coherence in the sum of all the involved Feynman diagrams and grow with the distance, with the photon energy and with the accumulated angle of the electron-photon pair. When the coherence length extends beyond one scattering length, comprising several collisions on average, the total intensity is not anymore the sum of the single Bethe-Heitler intensities at each collision and, thus, the intensity is substantially reduced. This effect, known as the Landau-Pomeranchuk-Migdal (LPM) suppression, leads to a stopping/energyloss by bremsstrahlung in condensed media significatively smaller than the one expected for a set of unrelated collisions. The implications of this suppression spanned several fields of high energy physics [3, 38–40] and are still open nowadays. Although the first indirect measurements of this suppression were early given through cosmic ray showers [41] soon after its prediction, it was not un51 52 High energy emission til recently that various experiments, first at SLAC [42–44] and then at CERN [45, 46], measured in detail the phenomenon. Recent and renewed interest has also suscited the LPM suppression of gluons in the presence of the QCD media produced in high energy collisions at RHIC and LHC. The formation of new states of the matter, namely the quark-gluon plasma (QGP), are studied, among other probes, through the energy loss pattern of partons while traveling by this colored QCD medium1. Owing to the amount of work behind this subject it becomes necessary to introduce a brief historical remark. The first evaluation of this effect was given by Landau [3] with a classical calculation for a medium of semi-infinite length. In his work, the external field interaction is replaced with the Fokker-Planck approximation, leading to a total squared momentum transfer satisfying a Gaussian distribution. The computation of Landau produces a differential energy intensity which vanishes as √ωin the soft limit ω→0 while approaches the incoherent sum of Bethe-Heitler intensities for larger ω. Landau’s evaluation was later extended by Migdal [4] to the quantum case by means of a Boltzmann transport equation for an averaged target. As it was, maybe, better and later explained in the rederivation by Bell [47], Migdal’s result shown that, except for spinorial corrections in the hard part of the photon spectrum, the LPM suppression is still a classical effect, in correspondence with the classical behavior of the infrared divergence [5, 6]. Following these two seminal works, a finite target evaluation was soon introduced first by Ternovskii [48] by using Migdal’s transport approach, and later by Shul’ga and Fomin [49] through the classical limit. In both works radiation is computed in the softest regime ω→0 and, unlike to the infinite suppression predicted by Landau and Migdal, the resulting intensity is found to saturate into a Bethe-Heitler power-like law in which the medium coherently acts as a single radiation entity. Both results, however, are only given for this coherent limit and in the saturation regime, in which the electron momentum transfer at the end of the process greatly exceeds its mass in magnitude. As we will later see, this coherent regime, which was not considered neither by Landau nor by Migdal, can be treated as part of a more general picture, corresponding to the Weinberg’s soft photon theorem [5]. Very diverse and more recent calculations have appeared since then. We cite the work of Blankenbecler and Drell [50–53], who established a formalism in which the finite target case can be generally computed in the Fokker-Planck approximation. They adopted an approach in which a double integral in longitudinal position accounts for the photon emission point, both in the amplitude and its conjugate, and then the relevant quantity modulating the interference behavior is 1A more detailed study of the LPM effect and its phenomenology in QCD will be given in the next chapters. 53 just the accumulated longitudinal momentum change between these two points. Their result is given for arbitrary lengths, in such a way that, asymptotically for very low ωand in the large length limit, they recover Landau’s and Migdal’s predictions except for constant factors. Simultaneously to their work, and with the scope of developing a bremsstrahlung formalism for QCD matter, Zakharov [20, 54, 55] also treated the semi-infinite case in the Fokker-Planck approximation by reformulating Migdal’s transport approach through path integrals in the transverse plane, leading to the same result. In his approach, however, the finite target implementation [54] is not easy and requires a different treatment. In the same line the BDMPS group [22] following a transport approach very similar to the one by Migdal and, omitting the collinear divergence introduced by the neglection of the electron mass (leading to a fail in reproducing the Bethe-Heitler limit) found a result which was shown [56] to be equivalent to the one by Zakharov in the soft limit. We have to cite also the works of Baier and Katkov [57, 58], who put some focus into accounting the Coulomb corrections to the Fokker-Planck approximation, and also in computing the finite size case, among other interesting situations. Their finite size result is comparable to the one found by Ternovskii, but frequently the various cases of study lack a general formulation and require different treatments. Similar predictions were later given by Wiedemann and Gyulassy [19], extending Zakharov path integral result to account for the angular dependence and the finite size case in a very similar way to the finite size evaluation by Blankenbecler and Drell. For further details on the history behind these works extensive and interesting reviews [59, 60] have been written. A comprehensive non-perturbative QFT description of the emission of quanta for the general case of a finite/structured target, which takes into account not only the LPM effect, but dielectric and transition radiation effects as a particular cases of the same phenomena, admitting an evaluation for general interactions beyond the Fokker-Planck approximation and which takes care of the angular distributions is still missing. Reasons for this situation are diverse and probably related to the complexity and particularities of the vast part of the calculations [59] and the usual approach of considering ab initio the intensity for an averaged target of infinite transverse size, instead of an adequate quantum mechanical definition at the level of the amplitude. As a result the diagrammatic structure is hidden, the transverse coherence effects are never considered and the soft photon limit and length constraints are often misunderstood. Taking these considerations into account, in Section 3.1 we find the emission amplitude and we briefly explain the LPM effect by taking the simpler classical limit. The main purpose of this section is to relate the LPM effect and the Weinberg’s soft photon theorem as part of the same coherence phenomena. With these ideas in mind we define then in Section 3.2 the quantum amplitude [61] 54 High energy emission in terms of the scattering amplitudes found in the previous chapter, with the aim of taking the square of this amplitude in Section 3.3 in order to evaluate the photon intensity. We show that the intensity can be split into a coherent and an incoherent contribution, related to transverse interference effects, just as we did with the pure elastic case. The coherent contribution will be found to be a pure quantum mechanical contribution which does not admit a statistical interpretation, whereas the incoherent contribution results into a mixed contribution which, in the infinite transverse size limit for the medium, reproduces the macroscopic/classical limit and thus admits an statistical interpretation. The resulting expression in the discrete limit, which is the central result of the present work, admits a numerical evaluation by Monte Carlo methods for general interactions, arbitrary medium lengths, angular distributions and medium effects in the photon dispersion relation. In Section 3.4 we take the continuous limit and we find that our result is equivalent to a path integral formulation which can be solved in the Fokker-Planck approximation. This result will be used as a check for the numerical evaluation of our discretized approach. Migdal’s/Zakharov and BDMPS results are recovered as two particular cases. 3.1 Amplitude and the classical LPM effect We consider the amplitude of emission of a photon due to the effect of the multiple scattering sources in a medium. Let the photon be considered free after the emission and let the 4-momentum be denoted by k≡(ω, k). Medium effects in the energy momentum relation or refractive index can be introduced by defining an effective photon mass mγ. Then the photon’s velocity reads βk=q1−m2 γ/ω2,(3.1) and the momentum can be written as k=ω(1, βkˆ k), where ˆ kis the unitary vector in the direction of k. A photon with this momentum kand polarization λis given by Aλ µ(x)=N(k)λ µeik·x,(3.2) where N(k)=√4π/2ωis the normalization. The new electron state under the effect of this emitted photon ψ(n) γ(x) can be always written as the superposition ψ(n) γ(x)=ψ(n) i+eZd4yS (n) F(x−y)γµAλ µ(y)ψ(n) γ(y),(3.3) where ψ(n) i(y) is a solution to the electron state in absence of interaction with the emitted photon Aλ µ(y), but subject to the interaction with the photons from the Amplitude and the classical LPM effect 55 external field (1.13) of the medium. It verifies then iγµ∂µ−gA(n) 0(x)−mψ(n) i(x)=0,(3.4) where the external field A(n) 0(x) is given by (1.8) and S(n) F(x) is the adequate propagator, given in particular by (1.30) at high energies. From the equation (3.3) we easily find the required relation iγµ∂µ−gA(n) 0(x)−mS(n) F(x−y)=δ4(x−y).(3.5) The wave (3.3) can be always expanded in the original basis of scattered states with the (n) sources. Correspondingly, the amplitude of finding the wave ψ(n) γ(x) in the final scattered state ψ(n) f(x) is given, using (3.3), by S(n) em =Zd3xψ(n),† f(x)ψ(n) i(x)+eZd3xψ(n),† f(x)S(n) F(x−y)Aλ µ(y)ψ(n) γ(y).(3.6) The first term is just the pure elastic scattering contribution. Indeed by expanding the scattered states as in (2.3) and taking into account the unitarity of the S-matrix we just find Zd3xψ(n),† f(x)ψ(n) i(x)=S(n) sfsi(pf,pi).(3.7) We are instead interested in the second contribution, which constitutes the radiative correction to the elastic scattering. By using (1.30), we find the conjugate relation ¯ ψ(n) f(y)≡iZd3xψ(n),† f(x)S(n) F(x−y),(3.8) and since γ2 0=1 then S(n) em =S(n) sfsi(pf,pi)−ie Zd4y¯ ψ(n) f(y)γµAλ µ(y)ψ(n) γ(y).(3.9) We then define the amplitude of going from (pi,si) to (pf,sf)while emitting a photon, neglecting the pure elastic contribution, as the contribution M(n) em ≡ −ie Zd4y¯ ψ(n) f(y)γµAλ µ(y)ψ(n) i(y)+O(e2),(3.10) where we expanded in the last step ψ(n) γ(x)=ψ(n) i(x)+O(e), thus the above relation is only valid at leading order in e=√αfor the interaction between the emitted photon and the electron. Evaluating (3.10) for a single source (n=1) 62 High energy emission and continues propagating elastically till znwith energy p0 0−ω, interacting or not, with amplitude S(n) sns(pn,p;zn,z). Finally, we sum over emission points dz, which automatically inserts the adequate poles of the fermion propagators (kµpµ)−1at the phase. This is clearly seen by integrating by parts, in which case one gets from (3.33) the alternative expression M(n) em−M(0) em =−eN(k)X s,s0Zd3p (2π)3Z+∞ −∞ dz exp −iq(p,p+k)z(3.38) ×d dz (S(n) sns(pn,p;zn,z)fλ ss0(p,p+k) q(p,p+k)S(n) s0s0(p+k,p0;z,z1))−(n=0). As an illustrative example we compute the simplest case, a set of ncenters distributed at equal z1coordinate. A perturbative expansion of the beyond eikonal elastic amplitudes as S(n) sasb(pa,pb;za,zb)=δsa sb(2π)3δ3(pa−pb)+2πδ(p0 a−p0 b) ×rme p0 a ¯usa(pa)γ0usb(pb)rme p0 b −4πiZe2 q2+µ2 d n X i=1 e−iq·ri,(3.39) produces for (3.38) with q=p1+k−p0, at leading order in g=Ze2 M(n) em −M(0) em =eN(k)2πδ(p0 1+ω−p0 0)4πiZe2 q2+µ2 d n X i=1 e−iq·ri(3.40) ×       fλ s1s0(p1,p1+k) q(p1,p1+k)e−iq(p1,p1+k)z1rme p0 0 ¯us0(p1+k)γ0us0(p0)rme p0 0 −rme p0 0−ω¯us1(p1)γ0us(p0−k)rme p0 0−ω fλ ss0(p0−k,p0) q(p0−k,p0)e−iq(p0−k,p0)z1     . And with the help of the completeness relation (2.8) and (3.35), (3.36) and (3.37) p0 0−ω kµpµ 0X srme p0 0−ωus(p0−k)¯us(p0−k)rme p0 0−ω=/ p0−/ k+me 2kµpµ 0 , p0 0 kµpµ 1X s0rme p0 0 us0(p1+k)¯us0(p1+k)rme p0 0 =/ p1+/ k+me 2kµpµ 1 ,(3.41) we recover the more familiar form of the Bethe-Heitler amplitude. For mediums of non negligible thickness the above amplitude stops being applicable, since each elastic amplitude at zicarries its own local phase. If we discretize the medium in nsheets and thus pifrom i=1,...,nthe beyond eikonal elastic Amplitude in the quantum approach 63 amplitudes can be read from (2.142) or (2.144). By inserting these amplitudes in (3.33) one gets three terms, so that we write M(n) em =eN(k) n−1 Y i=1Zd3pi (2π)3Mf+Mint +Mi.(3.42) The first of them represents photons emitted in the last leg (+∞,zn). Indeed using (3.35) we get Mf≡fλ sns(pn,pn+k) kµpµ n/p0 0 exp +ikµpµ n p0 0 zn!Sn(zn) ssn−1(pn+k,pn−1) n−1 Y i=1 Sn(zi) si+1si(qi), (3.43) with qi=pi−pi−1and Sn(zi) si+1si(qi)≡Sn(zi) si+1si(pi,pi). The inner term represents photons emitted at any of the (z1,zn) internal legs, Mint ≡ − n−1 X k=1 n Y i=k+1 Sn(zi) sisi−1(qi) fλ sks0 k (pk,pk+k) kµpµ k/p0 0 (3.44) × exp +ikµpµ k p0 0 zk+1!−exp +ikµpµ k p0 0 zk!!Sn(zk) s0 ksk−1(pk+k,pk−1) k−1 Y i=1 Sn(zi) sisi−1(qi). Finally the term consisting in photons emitted in the first leg (z1,−∞), given by Mi=− n Y i=2 Sn(zi) sisi−1(qi)Sn(z1) s1s(p1,p0−k)fλ ss0(p0−k,p0) kµpµ 0/(p0 0−ω)exp +ikµpµ 0 p0 0−ωz1!. (3.45) Terms (3.43) and (3.45) are the only ones surviving when ω→0, since the × × Mk≡ qnqk+2 (···) qk+1 znzk+2 zk+1 pnpk+1 zkzk qkqk pk pk−1−k pk−1pk pk+k pk−1 + kk qk−1qk−2 (· · ·) q1 zk−1zk−2z1 pk−2p0 pk−1 pk Figure 3.2: Diagrammatic representation of the terms in the sum (3.49). phase in (3.44) vanish, leading to the soft photon theorem. They correspond to the classical equation (3.21) leading to the coherent plateau in the squared amplitude. A joint expression of these three terms and its vacuum subtraction can 64 High energy emission be given by reorganizing the sum, which in turns corresponds to the integration by parts at (3.38). This produces M(n) em =eN(k) n−1 Y i=1Zd3pi (2π)3× n X k=1Mk,(3.46) where Mk= n Y i=k+1 Sn(zi) sisi−1(qi)×     fλ sks(pk,pk+k) kµpµ k/p0 0 exp +ikµpµ k p0 0 zk!Sn(zk) ssk−1(pk+k,pk−1) −Sn(zk) sks(pk,pk−1−k)fλ ssk−1(pk−1−k,pk−1) kµpµ k−1/(p0 0−ω)exp +ikµpµ k−1 p0 0−ωzk!    × k−1 Y i=1 Sn(zi) sisi−1(qi). (3.47) The Feynman diagram structure of this joint expression is simple and can be seen in Figure 3.2. We notice that with this notation the vacuum subtracted term has been also discretized as M(0) k= n Y i=k+1 S(0) sisi−1(qi)×     fλ sks(pk,pk+k) kµpµ k/p0 0 exp +ikµpµ k p0 0 zk!S(0) ssk−1(pk+k,pk−1) −S(0) sks(pk,pk−1−k)fλ ssk−1(pk−1−k,pk−1) kµpµ k−1/(p0 0−ω)exp +ikµpµ k−1 p0 0−ωzk!    × k−1 Y i=1 S(0) sisi−1(qi). (3.48) Then we write M(n) em −M(0) em =eN(k) n−1 Y i=1Zd3pi (2π)3× n X k=1 Mk−M(0) k!.(3.49) The classical correspondence of this quantum amplitude has been given in (3.13) and is going to be recovered in the averaged square of (3.49) for a medium of macroscopic transverse size. 3.3 Intensity Relation (3.49) can be squared and then averaged over medium configurations and initial spins, and summed over final states and photon polarizations in order to evaluate the intensity of photons in a particular direction. For that purpose we choose, as with the elastic scattering, a cylinder of transverse area πR2and length l. The unpolarized intensity of photons in the energy interval ωand ω+dωand Intensity 65 in the solid angle Ωkand Ωk+dΩk, per unit of medium transverse area and time, is given by dI ≡ω2dωdΩk (2π)3 1 βpTπR2 1 2X sns0X λZd3pn (2π)3M(n) em −M(0) em 2,(3.50) where we divided by βp, the incoming electron flux, time T, accounting for time translation invariance, and medium transverse area Ω = πR2. In the process of averaging over medium configurations we notice that M(n) em −M(0) em 2=DM(n) em M(n) em∗E−DM(n) emEDM(n) em∗E+DM(n) emE−M(0) em 2. (3.51) Here we added and subtracted the term DM(n) emEDM(n) em∗Eas before. Then we define the incoherent contribution to the emission as Σ(n) em ≡DM(n) em M(n) em∗E−DM(n) emEDM(n) em∗E,(3.52) which as we will shown, in the limit Rµ−1 dcan be interpreted in probabilistic terms and, except for quantum corrections in the hard part of the spectrum, leads to the classical behavior of the infrared divergence and the LPM effect. We also find a coherent contribution Π(n) em ≡DM(n) em −M(0) emE 2,(3.53) which is just the averaged emission amplitude squared, and encodes the quantum diffractive behavior of the LPM effect. As we will later show, this contribution in the Rµ−1 dlimit can be omitted since it represents the negligible high energy transition radiation of the electron. Transverse-coherent contribution The transverse coherent contribution to the intensity consists in the averaged emission amplitude squared. It is given by dIcoh ≡ω2dωdΩk (2π)3 1 βpTπR2 1 2X sns0X λZd3pn (2π)3Π(n) em.(3.54) For microscopic mediums it contains the quantum diffractive behavior of the medium boundaries in the transverse plane, thus it can be interpreted as the contribution related to medium transverse coherence. For macroscopic mediums this diffractive behavior, as we will show, constraints the electron propagation to the 66 High energy emission forward direction, thus this contribution reduces in that limit to the electron transition radiation related to its energy gap and the longitudinal boundaries. Using (3.49) and (3.53) we get Π(n) em(k)=e2N2(k) n−1 Y i=1Zd3pi (2π)3 n X k=1hMki−M(0) k 2 ,(3.55) where the required averages on the right hand side of the above equation affect only to the amplitudes containing centers Mkand are of the form hMki= n Y i=k+1DSn(zi) sisi−1(pi,pi−1)E k−1 Y i=1DSn(zi) sisi−1(pi,pi−1)E(3.56) × fλ sks(pk,pk+k) kµpµ k−1/p0 0 exp +ikµpµ k p0 0 zk!DSn(zk) ssk−1(pk+k,pk−1)E −DSn(zk) sks(pk,pk−1−k)Efλ ssk−1(pk−1−k,pk−1) kµpµ k−1/(p0 0−ω)exp +ikµpµ k−1 p0 0−ωzk!. At this point we note that the elastic amplitudes in (3.56) for the layers between nand k+1 verify p0 i=p0 0−ω, the advanced and retarded elastic amplitudes at the layer kverifies p0 k=p0 0and p0 k=p0 0−ω, respectively, and the elastic amplitudes between kand 1 verify p0 i=p0 0. We briefly review the results of the averaging process, c.f. Section 2.4. A single coherent average at the layer iof n(zi) scattering sources over a cylinder of transverse area πR2and length δz produces DSn(zi) sisi−1(pi,pi−1)E=(2π)3δ(qi)δsi si−1+2πβpδ(q0 i)δsi si−1exp −iqz izi ×Zd2xte−iqt i·xt*exp −ig βp n(zi) X k=1 χk 0(xt)+−1 ,(3.57) where we used the relation Sn(zi) sisi−1(pi,pi−1)=Mn(zi) sisi−1(pi,pi−1)+(2π)3δ3(pi−pi−1)δsi si−1. In the large but finite Rµdlimit this average can be well approximated as Zd2xe−iq·x*exp −ig β n(zi) X k=1 χk 0(x)+≃Zd2xe−iq·xexp n0(zi)δzπ(1) el (x)!.(3.58) The function π(1) el (x) is the Fourier transform of the single elastic amplitude for a collision with a single scattering source coherently distributed at the amplitude level over a cylinder of radius R, and given by π(1) el (x)=Zd2q (2π)2e+iq·xF(1) el (q)Wcyl(q,R),Wcyl(q,R)=2πR |q|J1(|q|R),(3.59) Intensity 67 where Wcyl(q,R) is the window function of the cylinder and J1(x) the Bessel function of the first kind. The function π(1) el (x) has a typical width R, in contrast to dimensions of a single scattering source rd=1/µd. It results convenient to separate the momentum distribution from the spin content and the longitudinal phases. Then we write DSn(zi) sisi−1(pi,pi−1)E=δsi si−1φ(0) coh(δpi, δz)+φ(n) coh(δpi, δz)exp −iδpz izi,(3.60) where we defined the no collision (0) and the collision (n) coherently averaged amplitudes for the matter in δzas φ(0) coh(δpi, δz)≡(2π)3δ(δpi), φ(n) coh(δpi, δz)≡2πβδ(δp0 i)Zd2xe−iδpt i·xexp n0(zi)δzπ(1) el (x)−1.(3.61) These distributions act over the electron wave function and can only be interpreted in probabilistic terms in the squared amplitude. They preserve spin, lead to a momentum distribution of typical width 1/Rand add a local longitudinal phase of the form δpz iziat each step, responsible of modulating the quantum LPM effect. The first section of coherent scatterings produces, by reiterative use of the relation (3.60), k−1 Y i=1DSn(zi) sisi−1(pi,pi−1)E=δsk−1 s0 k−1 Y i=1φ(n) coh(δpi, δz)+φ(0) coh(δpi, δz) ×exp −izk−1pz p0 0 (pt k−1)+i k−2 X i=1 δzpz p0 0 (pt i) +iz1pz p0 0 (pt 0),(3.62) where δz=zi+1−ziand we rearranged the phase by summing by parts. Subindices in the longitudinal momenta denote the energy, read from the energy conservation deltas either at φ(n) coh(δpi) or φ(0) coh(δpi). They fix the longitudinal momentum in terms of the transverse momentum. Similarly the advanced average at the emission layer kproduces, using (3.60), DSn(zk) ssk−1(pk+k,pk−1)E=δs sk−1 φ(n) coh(δpk+k, δz)+φ(0) coh(δpk+k, δz)! ×exp −izkpp0 0(pt k+kt)+izkpp0 0(pt k−1)!,(3.63) 68 High energy emission whereas the average just after the emission, where the electron energy is now p0 0−ω, produces DSn(zk) sks(pk,pk−1−k)E=δsk s φ(n) coh(δpk+k, δz)+φ(0) coh(δpk+k, δz)! ×exp −izkpp0 0−ω(pt k)+izkpz p0 0−ω(pt k−1−kt)!.(3.64) For the last set of coherent scatterings with energy p0 0−ωwe find, using reiteratively (3.60), n Y i=k+1DSn(zi) sisi−1(pi,pi−1)E=δsn sk n Y i=k+1φ(n) coh(δpi, δz)+φ(0) coh(δpi, δz) ×exp −iznpz p0 0−ω(pt n)+i n−1 X i=k+1 δzpz p0 0−ω(pt i) +izk+1pz p0 0−ω(pt k).(3.65) Since ωp0 0we assume that βpcan be taken unaltered in all the process and the photon effect in the elastic amplitude neglected. The leading contribution of the photon to the elastic propagation is enclosed instead in the energy gap of the longitudinal phases and in the loss of ktat the emission layer zk. The insertion of (3.62),(3.63), (3.64) and (3.65) in (3.56) produces then a soft photon factorization of the form hMki= n Y i=1φ(n) coh(δpi, δzi)+φ(0) coh(δpi, δzi)Jk,(3.66) where we shifted the electron momentum variable from the emission point k onwards, as pt i+kt→pt i. The emission current is given by Jk=eiϕk(f(+) k−f(−) k),(3.67) where the phase of each element is given by iϕk≡ −ipz p0 0−ω(pt n−kt)zn+i n−1 X i=k δzipz p0 0−ω(pt i−kt)−ikzzk +i k−1 X i=1 δzipz p0 0 (pt i)+ipz p0 0 (pt 0)z1,(3.68) and we defined the following shorthands for the emission vertex together with the corresponding propagator f(+) k≡fsns0(pk,pk+k) kµpµ k/p0 0 ,f(−) k≡fsns0(pk−1−k,pk−1) kµpµ k−1/(p0 0−ω).(3.69) Intensity 69 The photon longitudinal momentum can be written in the high energy limit as kz≃ω−k2 t/2ω. Then the phase can be rearranged as iϕk= + i ωm2 e 2p0 0(p0 0−ω)+ik2 t 2ω!zk +ipt n−kt2 2(p0 0−ω)zn−i n−1 X i=k δzipt i−kt2 2(p0 0−ω)−i k−1 X i=1 δzi (pt i)2 2p0 0−i(pt 0)2 2p0 0 z1.(3.70) The boundary terms znand z1can be omitted if desired since they will cancel in the squared amplitude. Observe that if a term of the form JkJ∗ lis evaluated for the same electron trajectory in the amplitude and its conjugate we recover the classical phase, JkJ∗ l∝exp  +i k−1 X i=l δzim2 e+ pt i−p0 0 ωkt!2≃exp +iZzk zl dz kµxµ(z)!,(3.71) where we used (3.36) and assumed k>l. It is straightforward to prove that the term containing no interaction with any of the layers can be written in a similar manner to this one. With the replacement φ(n) coh +φ(0) coh →φ(0) coh we find M(0) k= n Y i=1φ(0) coh(δpi, δzi)Jk,(3.72) and we just find the overall vacuum subtraction of the contribution hMki. Since the sum of hMki−M(0) kought to be squared further simplifications can be done. We notice that, c.f. Appendix A, ω2 2X λX sns0f(+) k−f(−) kf(+) l−f(−) l∗=hn(y)δk·δl+hs(y)δs kδs l,(3.73) where the spin flip and non flip currents are given, respectively, by δn k≡k×pk kµpµ k−k×pk−1 kµpµ k−1 , δs k≡ωp0 k kµpµ k−ωp0 k−1 kµpµ k−1 .(3.74) Here y=ω/p0 0is the fraction of energy carried by the photon, p0 k=p0 0−ωand p0 k−1=p0 0and the functions hn(y) and hs(y) are given by hn(y)=1 2(1 +(1 −y)2),hs(y)=1 2y2.(3.75) 70 High energy emission This indicates that we can split the emission current into two independent contributions, Jn kand Js kwhich can be separately squared and then multiplied with hn(y) and hs(y). These currents are given by Jn k≡=1 ω2δn keiϕk,Js k≡1 ω2δs keiϕk,(3.76) where the phase ϕkis given by (3.70). Then we have found a transverse coherent average of the emission intensity given by Π(n) emsn =e2N2 ω2 n−1 Y i=1 d3pi (2π)3 n Y i=1φ(n) coh(δpi, δzi)+φ(0) coh(δpi, δzi) − n Y i=1φ(0) coh(δpi, δzi) n X k=1 δn keiϕk 2 ,(3.77) for the spin no-flip contribution and Π(n) ems f =e2N2 ω2 n−1 Y i=1 d3pi (2π)3 n Y i=1φ(n) coh(δpi, δzi)+φ(0) coh(δpi, δzi) − n Y i=1φ(0) coh(δpi, δzi) n X k=1 δs keiϕk 2 ,(3.78) for the spin flip contribution. The coherent average contribution to the photon intensity is then given by the sum of these two terms once integrated in the final electron momentum ωdIcoh dωdΩk =ω 2π31 πR2TZd3pn (2π)3hn(y)Π(n) emsn +hs(y)Π(n) ems f .(3.79) A particular case of the above result consists in taking the macroscopic R→ ∞ limit. Then Wcyl(q,R)=(2π)2δ2(q), π(1) el (x)=F(1) el (0) and thus φ(n) coh(δpi, δzi)=(2π)3δ3(δpi)exp n0(zi)δziF(1) el (0)−1,(3.80) which leads to a pure forward propagation of the electron at the level of the amplitude. Then the only difference of states in the functions δn iand δs icorresponds to the energy gap of the electron momentum and thus n X k=1 δs keiϕk=(βfsin θ 1−βfcos θ−βisin θ 1−βicos θ)n X k=1 eiϕk,(3.81) Intensity 71 where θis the angle between the photon and the initial electron direction ˆp0 and βfand βiare the final and initial electron velocities. Similarly the spin flip contribution produces n X k=1 δs k=eiϕk(βf 1−βfcos θ−βi 1−βicos θ)n X k=1 eiϕk,(3.82) where the phases are also constrained to the forward direction and given in this case by iϕk=iωm2 e 2p0 0(p0 0−ω)zk+ik2 t 2ωzk≡ikµpµ 0 p0 0 zk,(3.83) as expected. At high energies the negligible energy gap in the velocities βiand βfof the current can be neglected, as we already did in the elastic weights. Then δn k=0 and δs k=0, which means that at high energies in the macroscopic R→ ∞ limit, the coherent average contribution cancels, Π(n) emsn =0 and Π(n) ems f =0, as expected. Transverse-incoherent contribution The transverse-incoherent contribution to the emission intensity corresponds to the averaged squared amplitude. Using (3.49) and (3.52) we get Σ(n) em =e2N2(k) n−1 Y i=1Zd3pi (2π)3 d3ui (2π)3× n X j,k=1hMkM∗ ji−hMkihM∗ ji.(3.84) The averages on the right hand side of the above equation are of the form hMkM∗ ji=An j+1(f+ j)∗A+ j−A− j(f− j)∗Aj−1 k+1f+ kA+ k−A− kf− kAk−1 1,(3.85) hMkihM∗ ji=Bn j+1(f+ j)∗B+ j−B− j(f− j)∗Bj−1 k+1f+ kB+ k−B− kf− kBk−1 1,(3.86) where f+ jand f− jare the emission vertex and propagator shorthands given at (3.69) but with the local phases, thus given by f(+) k≡fλ sks(pk,pk+k) kµpµ k/p0 0 exp +ikµpµ k p0 0!, f(−) k≡fλ ssk−1(pk−1−k,pk−1) kµpµ k−1/(p0 0−ω)exp +ikµpµ k−1 p0 0−ω!,(3.87) and the terms Aj kand Bj kare shorthands for the incoherent averages of the squared amplitudes corresponding to the layers of scatterers from zkto zj. Let δpi= 78 High energy emission of the spectrum agrees with the classical current. Finally, we remove the squared no collision emission diagram, which corresponds to no colliding at any of the layers. Further simplifications can be done for the unpolarized and spin averaged intensity as we did in the coherent contribution. In that case one can define two effective currents, c.f. Appendix A, one corresponding to spin preserving vertices and agreeing with the classical current, the other corresponding to spin flipping vertices, correcting the classical contribution for hard photons y=ω/p0 0∼1, ωdIinc dωdΩ=e2 (2π)2 n Y i=1Zd3pi (2π)3       n Y i=1 φ(n) inc(δpi)+φ(0) inc(δpi)− n Y i=1 φ(0) inc(δpi)       × hn(y) n X k=1 δn kexp −i k−1 X i=1 kµpµ i p0 0−ωδz 2 +hs(y) n X k=1 δs kexp −i k−1 X i=1 kµpµ i p0 0−ωδz 2 , (3.122) where the spin non flip currents agree with the classical result (3.14) δk≡k×pk kµpµ k−pk−1 kµpµ k−1,(3.123) and the spin flip currents are given by δk≡p0 0ω ωp0 0−k·pk−p0 0ω ωp0 0−k·pk−1 .(3.124) The weighting functions of the two contributions are given by hn(y)=(1 +(1 − y)2)/2 and hs(y)=y2/2. The phases the single Bethe-Heitler amplitudes (3.123) and (3.124), agreeing in this incoherent average with the classical phases, are responsible of causing the interferences in their squared sum, leading to the LPM, the dielectric and the transition radiation effects. The interference pattern is essentially the same as the one discussed in the classical correspondence at Section 3.2. The composition of the elastic propagations at each layer produces a squared momentum transfer additive in the traveled length. If δlverifies δl≤λwhere λ=1/n0σ(1) tis the average mean free path, using (3.89) and (2.90) we arrive at (2.134) ∂ ∂lhδp2(l)i=n0σ(1) thδp2(δl)i ≡ 2ˆq,(3.125) where we define the transport coefficient ˆq. This allows to relate the momentum transfer in a length lwith the momentum transfer in a single collision δl, which for the Debye screened interaction produces (2.130) hδp2(δl)i= 2 log 2p0 0 µd!−1!µ2 d≡ηµ2 d,(3.126) Intensity 79 and the long tail introduced a substantial correction ηto the naive expected value µ2 dafter a single collision and a maximum momentum transfer of 2p0 0is allowed due to the energy conservation delta in (3.89). However, since electron momenta appears convoluted in (3.122) together with the Bethe-Heitler functions δkat (3.123) or δkat (3.124), the momentum transfer can be approximated as of the order |δp| ≃ 2.5meand by using µd=αmeZ1/3we obtain instead η=2 log 2.5me µd!−1 2!=2 log 2.5 αZ1/3!−1 2!.(3.127) This can be numerically checked and matches Bethe’s [6, 64] η=2 log(183/Z1/3) prediction within less than a 3% of deviation in the range Z=(1,100). The choice of ηdefines ˆqand thus the Fokker-Planck approximation (2.126) for the scattering distribution ˆ Σ2(δp,z) in (3.89). However, since this fix is only valid for δl≤λthis approximation is valid only in the incoherent plateau, where the single scattering regime holds. A local with ωdefinition of ηand ˆqhas to be employed in order to match the Debye screened intensity in all the spectrum. In this way, a single Fokker-Planck approximation can not be used unless the medium length is assumed infinite, in which case the lower plateau can be neglected and the fail of the Fokker-Planck approximation becomes irrelevant. Intensity (3.122) is suitable for a numerical evaluation for finite size targets under a general interaction. A Monte Carlo code has been developed in which the electron is assumed to describe a piece-wise zig-zag path [47] where the step size is taken as δz=0.1λ. This leads to paths going from ∼1500 steps for the shortest mediums to ∼250000 steps for the largest. The integration in the electron momenta transforms into an average over the paths under the elastic weight (3.89). In a typical run around 104paths had to be computed in order to obtain reasonable precision, spanning 50 photon frequencies and 100 photon angles. Several medium materials and lengths were chosen in order to compare with SLAC [44] and CERN [46] data. We can follow an alternative approach which helps to qualitatively understand the behavior of the found intensity if we observe that (3.122) is just a sum of single Bethe-Heitler amplitudes δkcarrying a phase. These amplitudes are summed, squared and then weighted by the incoherent averages (3.89) of the electron elastic intensity once the photon energy gap effect is neglected in its velocity. This agrees with the classical intensity (3.15) and thus produces crossed terms ∼δkδjeiϕj kinterfering with a phase ϕj k. We define the distance or coherence length in which δl=zj−zkin which ϕj kbecomes larger than unity. Using (3.16) and (3.125) we get ϕj k=1 p0 0−ωZzj zk dz kµpµ(z)≃ω 2p0 0(p0 0−ω)m2 eδl+ˆq(δl)2=1,(3.128) 80 High energy emission 0 0.2 0.4 0.6 0.8 1 0.11 10 100 (a) (ωθ)dI dωdθ ω(MeV) MC (Debye) MC (Fokker-Planck) η=8 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=2 PI (Fokker-Planck) 0 5 10 15 20 25 30 35 40 0.1 1 10 100 (b) (ωθ)dI dωdθ ω(MeV) MC (Debye) MC (Fokker-Planck) η=8 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=2 PI (Fokker-Planck) Figure 3.3: Differential intensity of photons in the angle θ=0.01/γe(a) and θ=0.5/γe(b) radiated from electrons of p0 0=8 GeV, γe=p0 0/m, after traversing an Au sheet of l=0.0023 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown in solid yellow line for the Debye interaction and for the Fokker-Planck approximation with η=8 (purple), η= 4 (dark grey) and η=2 (light grey). Also shown with dot-dashed lines the respective continuous limits of (3.122), leading to the path integral in the Fokker-Planck approximation (3.178). which by inversion produces a coherence length modulated by the photon frequency δl(ω)≡m2 e 2ˆqs1+8ˆqp0 0(p0 0−ω) m4 eω−1 .(3.129) This defines two characteristic frequencies of the LPM interference: the frequency ωcat which the coherence length becomes of the order of the medium length δl(ωc)=lthus ωc≃p0 0(p0 0−ω)/(m2 el+ˆql2) and the frequency ωsat which the coherence length becomes of the order of a mean free path δl(ωs)=λthus ωs≃p0 0(p0 0−ω)/(m2 eλ+ˆqλ2). Since for δl≥lthere are no sources of scattering we further impose to (3.129) δl(ω)=lfor ω≥ωc. In a coherence length the phase can be neglected so that the internal structure of scattering becomes irrel- Intensity 81 evant, c.f. Section 2.1, and the centers in δl(ω) act like a single scattering source with a charge equivalent to the total matter in δl(ω). Since there are l/δl(ω) of these coherence lengths for a given ωwe simply write ωdIinc dω(l)=l δl(ω)e2dΩk (2π)2Zd3δp (2π)3hn(y)|δn 1|2+hs(y)|δs 1|2φ(n) inc(δp, δl(ω)). (3.130) By inserting (3.89) and integrating in the photon solid angle Ωkone gets ωdIinc dω(l)=l δl(ω) e2 π2Zπ 0 dθsin(θ)F(θ)ˆ Σ2(δp, δl(ω)),(3.131) where |δp|=2p0 0βsin(θ) is the electron momentum change and the F(θ) function is given by F(θ)=1−β2cos θ 2βsin(θ/2) p1−β2cos2(θ/2) ×log p1−β2cos2(θ/2) +βsin(θ/2) p1−β2cos2(θ/2) −βsin(θ/2)−1.(3.132) This last integral can be numerically evaluated both for the Debye screened interaction for Σ2(δp, δz) and for its Fokker-Planck approximation (2.126), and the resulting values for ωωsand ωωc, i.e. for the incoherent and coherent plateaus, respectively, are exact. In the Fokker-Planck approximation one can further write an useful interpolating function for these two asymptotic values, ωdIinc dω(l)=l δl(ω) 2e2 π 1+nm(ω) 3A+nm(ω)log 1+Anm(ω),(3.133) where nm(ω)≃2ˆqδl(ω)/m2 eis a measure of the number of transverse masses acquired in a coherence length and A=e−(1+γ)where γis Euler’s constant. Then the qualitative behavior of the LPM effect is as follows. For frequencies lower than ωcthe coherence length extends beyond the medium length and then the radiation intensity consists in the difference of the first and last photon diagrams squared. In this coherent plateau the vanishing phase causes trivial convolutions of the elastic distributions at each layer and it can be shown that the total distribution is then given by Σ(n) 2(δp,l), as stated in (3.130). For frequencies larger than ωcusing (3.129) the number of independent emitters l/δl(ω) grows with √ωwhereas the charge of each emitter logarithmically decreases with log(1/√ω). This enhancement from the coherence plateau stops around ωs, when the coherence length acquires the minimum thickness of matter λ=1/n0σ(1) tto produce radiation, 82 High energy emission since in absence of collision both (3.123) and (3.124) vanish. In this incoherence plateau the radiation consists in the incoherent superposition of the nc=l/λ single Bethe-Heitler intensities, where ncis the average number of collisions. This maximal decoupling of the intensity can be diagrammatically understood as if the phase difference introduced by the internal fermion lines is so big that the electron emerging of an emission after a collision is real and incoherent with the next emission diagram. 0 5 10 15 20 25 30 0.1 1 10 100 (a) (ωθ)dI dωdθ ω(MeV) MC (Debye) MC (Fokker-Planck) η=8 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=2 PI (Fokker-Planck) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.11 10 100 (b) (ωθ)dI dωdθ ω(MeV) MC (Debye) MC (Fokker-Planck) η=8 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=2 PI (Fokker-Planck) Figure 3.4: Differential intensity of photons in the angle θ=2/γe(a) and θ=10/γe(b) radiated from electrons of p0 0=8 GeV, γe=p0 0/m, after traversing a Gold sheet of l=0.0023 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown in solid yellow line for the Debye interaction and for the Fokker-Planck approximation with η=8 (purple), η=4 (dark grey) and η=2 (light grey). Also shown with dot-dashed lines the respective continuous limits of (3.122), leading to the path integral in the Fokker-Planck approximation (3.178). The radiation intensity coming from electrons of p0 0=8 GeV for several photon angles is depicted in Fig. 3.3 and Fig. 3.4 both for the Debye screened interaction and the Fokker-Planck evaluations of (3.122). A target of Gold of l=0.0023 cm is chosen which corresponds to an average of nc=862 collisions. For this element we obtain a screening mass estimate of µd=16 KeV and a transport parameter of ˆq=(η/2)×1.89 KeV3with η≃8 in order to Intensity 83 match the angle-integrated incoherent plateau and comparable to our estimate from (3.127) η≃7.76. As it can be clearly seen the Fokker-Planck approximation which matches the incoherent (single) emission integrated spectrum, mismatches the unintegrated spectrum. At the lower angles the Fokker-Planck approximation overestimates the intensity by a ∼20% while at the larger angles the Fokker-Planck approximation underestimates the intensity. For the larger angle θ=10γ−1 ein particular we find that only half of the real emission is taken into account. In Fig. 3.5, Fig. 3.6 and Fig. 3.7 we show the angle-integrated 0 5.0E-4 1.0E-3 1.5E-3 2.0E-3 2.5E-3 0.1 1 10 100 (a) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 0 5.0E-4 1.0E-3 1.5E-3 2.0E-3 2.5E-3 0.1 1 10 100 (b) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 Figure 3.5: Differential intensity of photons radiated from electrons of p0 0=8 GeV (a) and p0 0=25 GeV (b) after traversing a Gold sheet of l=0.00038 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown for the Debye interaction (yellow and squares) and for the Fokker-Planck approximation with η=8 (purple and circles), η=4 (dark grey and diamonds) and η=2 (light grey and triangles). Also shown Migdal prediction (3.122) (dot-dashed line) with η=8 and our heuristic formula for finite size targets in the Fokker-Planck approximation (dashed line). spectrum for Gold targets of l=0.00038 cm, l=0.0023 cm and l=0.02 cm, which correspond to an average number of nc=142, nc=862 and nc=7502 collisions, respectively. Two electron energies are shown, p0 0=8 Gev and p0 0= 84 High energy emission 0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 1.0E-2 1.2E-2 1.4E-2 0.1 1 10 100 (a) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 0 2.0E-3 4.0E-3 6.0E-3 8.0E-3 1.0E-2 1.2E-2 1.4E-2 0.1 1 10 100 (b) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 Figure 3.6: Differential intensity of photons radiated from electrons of p0 0=8 GeV (a) and p0 0=25 GeV (b) after traversing a Gold sheet of l=0.0023 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown for the Debye interaction (yellow and squares) and for the Fokker-Planck approximation with η=8 (purple and circles), η=4 (dark grey and diamonds) and η=2 (light grey and triangles). Also shown Migdal prediction (3.122) with η=8 (dot-dashed line) and our heuristic formula for finite size targets in the Fokker-Planck approximation (dashed line). 25 GeV, and we present the Debye screened interaction and the Fokker-Planck evaluations of (3.122). For l=0.00038 cm the predicted characteristic frequencies are ωc=8 MeV and ωs=1.1 GeV for electrons of p0=8GeV, and ωc= 80 MeV and ωs=11 GeV for electrons of p0=25 GeV. Since a small number of collisions is occurring and the medium finiteness is taken into account, the difference between the coherent plateau and the incoherent plateau is small. For l=0.0023 cm the characteristic frequencies (3.19) and (3.22) are given by ωc= 0.48 MeV and ωs=418 MeV for electrons of p0=8 GeV, and ωc=4.7 MeV and ωs=4 GeV for electrons of p0=25 GeV. For l=0.02 cm we obtain ωc= 8 KeV and ωs=60 MeV for electrons of p0=8 GeV, and ωc=80 keV and ωs =588 MeV for electrons of p0=25 GeV. We also show our heuristic formula Intensity 85 0 2.0E-2 4.0E-2 6.0E-2 8.0E-2 1.0E-1 1.2E-1 1.4E-1 0.1 1 10 100 (a) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 0 2.0E-2 4.0E-2 6.0E-2 8.0E-2 1.0E-1 1.2E-1 1.4E-1 0.1 1 10 100 (b) ωdI dω ω(MeV) MC (Fokker-Planck) η=2 MC (Fokker-Planck) η=4 MC (Fokker-Planck) η=8 MC (Debye) Heuristic η=8 Migdal η=8 Figure 3.7: Differential intensity of photons radiated from electrons of p0 0=8 GeV (a) and p0 0=25 GeV (b) after traversing a Gold sheet of l=0.0023 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown for the Debye interaction (yellow and squares) and for the Fokker-Planck approximation with η=8 (purple and circles), η=4 (dark grey and diamonds) and η=2 (light grey and triangles). Also shown Migdal prediction (3.122) with η=8 (dot-dashed line) and our heuristic formula for finite size targets in the Fokker-Planck approximation (dashed line). (3.130) in the Fokker-Planck approximation, producing a reasonable agreement with (3.122), in particular in the coherence and incoherence plateaus, where it becomes exact. In Fig. 3.8 we show the angle-integrated spectrum for a Carbon target of l=0.41 cm which produces an average number of nc=9521 collisions, for electron energies of p0 0=8 GeV and p0 0=25 GeV. Estimates of µd=6.8 KeV, ˆq=(η/2) ×2.10 ·10−2KeV3and η=11 are obtained. The evaluation of (3.122) in the Debye screened interaction is shown together with its Fokker-Planck approximation. The characteristic frequencies are given by ωc=1 KeV and ωs =11 MeV for electrons of p0 0=8 GeV, and ωc=11 KeV and ωs=108 MeV for electrons of p0 0=25 GeV. The introduction of medium effects in the photon dispersion relation leads to the dielectric and the transition radiation effects, c.f. 86 High energy emission 0 5.0E-3 1.0E-2 1.5E-2 2.0E-2 2.5E-2 3.0E-2 3.5E-2 4.0E-2 0.1 1 10 100 (a) ωdI dω ω(MeV) MC (Debye) MC (Fokker-Planck) η=11 Heuristic η=11 Migdal η=11 0 5.0E-3 1.0E-2 1.5E-2 2.0E-2 2.5E-2 3.0E-2 3.5E-2 4.0E-2 0.1 1 10 100 (b) ωdI dω ω(MeV) MC (Debye) MC (Fokker-Planck) η=11 Heuristic η=11 Migdal η=11 Figure 3.8: Differential intensity of photons radiated from electrons of p0 0=8 GeV (a) and p0 0=25 GeV (b) after traversing a Carbon sheet of l=0.41 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown for the Debye interaction (yellow and squares) and for the Fokker-Planck approximation with η=11 (purple and circles). Also shown Migdal prediction (3.122) with η=11 (dot-dashed line) and our heuristic formula for finite size targets in the Fokker-Planck approximation (dashed line). Section 3.2. The photon plasma frequency ωpcan be thought as an effective photon mass mγwhich introduces an extra term m2 γ/2ωin the resonances kµpµ(z) at the phases and propagators. Then, a strong suppression occurs for frequencies lower than ωde, which is defined as the frequency at which the extra term becomes of the same order than the phase evaluated at mγ=0, i.e. ω2 de =m2 γlωc. In the particular case that the medium is finite and the photon mass cannot be assumed global for all the emission diagrams, the last photon verifies mγ=0 and thus the intensity is instead dramatically enhanced for frequencies ω≤ωde. In Fig.3.9 we show this dielectric and transition radiation effects of (3.122) for a Gold target of l=0.0023 cm, which introduces an effective photon mass of mγ= 0.08 KeV for all photons except for the one in the last leg mγ=0. The predicted characteristic frequencies are ωde =0.6 MeV for p0 0=8 GeV and ωde =1.9 MeV Intensity 87 0 0.01 0.02 0.03 0.04 0.05 0.1 1 10 100 (a) ωdI dω ω(MeV) Migdal Prediction η=8 MC (Fokker-Planck) η=8 MC (Debye) SLAC data 0 0.01 0.02 0.03 0.04 0.05 0.1 1 10 100 (b) ωdI dω ω(MeV) Migdal Prediction η=8 MC (Fokker-Planck) η=8 MC (Debye) SLAC data Figure 3.9: Differential intensity of photons radiated from electrons of p0 0=8 GeV (a) and p0 0=25 GeV (b) after traversing a Gold sheet of l=0.0023 cm, as a function of the photon energy. Monte Carlo evaluation of (3.122) is shown for the Debye interaction (yellow and squares) and for the Fokker-Planck approximation with η=8 (purple and circles). Also shown Migdal prediction (3.122) with η=8 (dot-dashed line) and SLAC data. for p0 0=25 GeV. As it can be seen, for frequencies lower than ωde a dramatic enhancement of the coherent plateau occurs, since the last leg diagram stops to be compensated due to the dielectric suppression of the first leg diagram and the phase interference between them grows as m2 γ/2ω. Also shown is the SLAC experimental data [44] for the same target, being in very good agreement with our evaluation. In Fig. 3.10 we show our results for an Iridium target of l=0.0128 cm and electrons of p0 0=149 GeV and a Copper target of l=0.063 and electrons of p0 0=207 GeV. Also shown is the CERN data [46] for the same scenario, which covers only a small part of the suppression zone. Slight differences are found in the LPM between theoretical predictions and the experimental data. 94 High energy emission the path integrals, if it were possible, since it guarantees the convergency at large impact parameters. The Fokker-Planck approximation, however, does not require such a regularizator, since in the large number of collisions the elastic incoherent averages Σ(n) 2(q, δz) rapidly converge at large x. We present now the FokkerPlanck approximation of this continuous limit. Since the amplitude has been split in three zones in the zintegration, the intensity leads to nine zones. Six of them, however, are related by conjugation with the interchange of zjand zk. So we find, Z∞ lZ∞ l +Z0 −∞ Z0 −∞ + Z∞ lZ0 −∞ +Z0 −∞ Z∞ l!+Zl 0Zl 0 + Zl 0Z0 −∞ +Z0 −∞ Zl 0!+ Zl 0Z∞ l +Z∞ lZl 0!.(3.157) From here onwards we take the soft photon limit thus p0 0−ω≃p0 0and hn(y)≃1 for simplicity. We also set the zaxis in the initial electron direction so pt 0=0. Using (3.155) the first term (a), representing the square of the photon emitted at the last leg, is given by ωdI(n) a dωdΩk =e 2π2 ω p0 0!2Z∞ l dzjZ∞ l dzkexp −iωm2 e 2(p0 0)2(zj−zk)! ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 .(3.158) We will assume constant density n0(z)≡n0. In the Fokker-Planck approximation for ˆ P(n) inc(xt k) and ˆ P(n) γ(xt j,xt k) we truncate at leading order the interactions in x exp −n0δzσ(1) el (0)+n0δzσ(1) el (x)≈exp −1 2ˆqx2!,(3.159) and the fall offof the interaction at large xis now guaranteed due to the screening neglection. The resulting momentum distribution is Gaussian. Similarly in the propagation with phase ˆ P(n) γ(xj,xk) we find a Gaussian path integral exp iZzj zk dz (p0 0)2 2ω˙xt2(z)+in0σ(1) el (0)−σ(1) el (xt(z))!! ≈exp iZl 0 dz 1 2me f ˙x2(z)−1 2me f Ω2x2!!, whose effective mass can be read in the kinetic term me f =(p0 0)2/ω and then the harmonic oscillator frequency Ωhas to be defined as me f Ω2=−iˆq→Ω = 1−i √2sˆqω (p0 0)2.(3.160) Continuous limit: a path integral 95 With these definitions we simply outline the result of integrating (3.158), ωdI(n) a dωdΩk =e 2π2Z∞ 0 dz z exp −ω 2 me p0 0!2 η(1) a−k2 t 2ωη(2) ak2 tη(3) a+η(4) a, (3.161) where the functions η(i) aare defined as η(1) a≡z, η(2) a≡z 1+izΩ2l, η(3) a≡1 (1 +izΩ2l)3, η(4) a≡2iωΩ (1 +izΩ2l)2.(3.162) Similarly to this term we find the contribution (b) corresponding to the photon emerging from the first leg squared. It is given by ωdI(n) b dωdΩk =e 2π2 ω p0 0!2Z0 −∞ dzjZ0 −∞ dzkexp −iωm2 e 2(p0 0)2(zj−zk)! ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 .(3.163) It produces ωdI(n) b dωdΩk =e 2π2Z∞ 0 dz z exp −ω 2 me p0 0!2 η(1) b−k2 t 2ωη(2) bk2 tη(3) b+η(4) b, (3.164) where as a consistency check the functions η(i) bhave to be given by the ˆq=0 evaluation of the previous functions η(i) a. Indeed η(1) b=z, η(2) b=z, η(3) b=1, η(4) b=0.(3.165) Correspondingly we can integrate in the longitudinal space to obtain ωdI(n) b dωdΩk =e 2π2 k2 tZ∞ 0 dz z exp − ω 2 me p0 0!2 +k2 t 2ωz=e 2π2k2 t ω 2me p0 02 +k2 2ω!2. Observe that this is just (k×p0)2/(kµpµ 0)2, as expected. The next term (c) corresponds to the interference between the photon emitted in the last leg and the photon emitted in the first leg. It is given by ωdI(n) c dωdΩk =e 2π2 ω p0 0!2Z∞ l dzjZ0 −∞ dzkexp −iωm2 e 2(p0 0)2(zj−zk)! ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 +c.c., (3.166) 96 High energy emission and produces ωdI(n) c dωdΩk =e 2π2 2 Re Z∞ 0 dzjZ∞ 0 dzkexp −ω 2 me p0 0!2 η(1) c−k2 t 2ωη(2) c ×k2 tη(3) c+η(4) c,(3.167) where the functions η(i) care given by η(1) c=il +zj+zk, η(2) c=isin(Ωl)+ Ω cos(Ωl)(zj+zk)+izjzkΩ2sin(Ωl) Ωcos(Ωl)+izjΩ2sin(Ωl) η(3) c=Ω2 Ωcos(Ωl)+izjΩ2sin(Ωl)2, η(4) c=0.(3.168) The term (d) contains all the photons emitted from the internal legs and their respective interferences. It is given by ωdI(n) d dωdΩk =e 2π2 ω p0 0!2Zl 0 dzjZl 0 dzkexp −iωm2 e 2(p0 0)2(zj−zk)! ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 +c.c. . (3.169) The integration produces, using δz≡zj−zk, ωdI(n) d dωdΩk =e 2π2 2 Re Zl 0 dzjZzj 0 dzkexp −ω 2 me p0 0!2 η(1) d−k2 t 2ωη(2) d ×k2 tη(3) d+η(4) d,(3.170) where the functions η(i) dare given by η(1) d=i(zj−zk) and η(2) d= isin Ω(zj−zk) Ωcos Ω(zj−zk)−zkΩ2sin Ω(zj−zk), η(3) d= cos Ω(zj−zk) cos Ω(zj−zk)−zkΩsin Ω(zj−zk)3, η(4) d=2iωΩ2zk cos Ω(zj−zk)−zkΩsin Ω(zj−zk)2.(3.171) Continuous limit: a path integral 97 The term (e) containing the interference between the internal photons and the photon emitted in the first leg is given by ωdI(n) e dωdΩk =e 2π2 ω p0 0!2Zl 0 dzjZ0 −∞ dzkexp −iωm2 e 2(p0 0)2(zj−zk)!(3.172) ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 +c.c. . This term produces ωdI(n) e dωdΩk =−e 2π2 2 Re Zl 0 dzjZ∞ 0 dzkexp −ω 2 me p0 0!2 η(1) e−k2 t 2ωη(2) e ×k2 tη(3) e+η(4) e,(3.173) where the functions η(i) eare given by η(1) e=i(zj−zk), η(2) e=−izk+isin(ωzj) Ωcos(Ωzj), η(3) e=1 cos2(Ωzj), η(4) e=0.(3.174) And finally the term ( f) containing the interferences between the photon emitted in the last leg with the internal photons. It is given by ω dI(n) f dωdΩk =e 2π2 ω p0 0!2Zl 0 dzjZ∞ l dzkexp −iωm2 e 2(p0 0)2(zj−zk)!(3.175) ×Zd2xt kˆ P(n) inc(xt k) exp −ip0 0 ωkt·xt k! ∂ ∂xt j·∂ ∂xt k ˆ P(n) γ(xt j,xt k)xj=0 +c.c., and produces using t=(l−zk) ω dI(n) f dωdΩk =e 2π2 2 Im Zl 0 dzjZ∞ 0 dzkexp −ω 2 me p0 0!2 η(1) f−k2 t 2ωη(2) f ×k2 tη(3) f+η(4) f,(3.176) where the functions η(i) fare given by η(1) f=iδ+zkwhere δ=l−zjand η(2) f=isin (Ωδ)+zkΩcos (Ωδ) Ωcos (Ωδ)−zjΩ2sin (Ωδ)+izkΩ2sin (Ωδ)+zjΩcos (Ωδ) η(3) f=cos(Ωδ)+izksin(Ωδ) cos (Ωδ)−zjΩsin (Ωδ)+izkΩsin (Ωδ)+zjΩcos (Ωδ)3 η(3) f=2iωΩ2zj cos (Ωδ)−zjΩsin (Ωδ)+izkΩsin (Ωδ)+zjΩcos (Ωδ)2.(3.177) 98 High energy emission The sum of the above six contributions constitute the path integral version of the intensity in the Fokker-Planck approximation for mediums of arbitrary length ωdI(n) inc dωdΩk−ωdI(0) inc dωdΩk = f X i=a ωdI(n) i dωdΩk .(3.178) In Fig. 3.3 and Fig. 3.4 we show the evaluation of these six path integrals (3.178) for several photon angles and a Gold target of l=0.0023 cm and an electron energy of p0 0=8 GeV. We find a complete agreement with the direct numerical evaluation of the discretized approach (3.122), as expected since we have chosen for that purpose δzλ=1/n0σ(1) t. If the semi-infinite medium length is to be considered the separation of the space in three zones stops having sense. We can define in that case the quantity ωdI(n) inc dωdΩk≡Zl 0Zl 0 +Z0 −∞ Z0 −∞ =ωdI(n) d dωdΩk +ωdI(n) b dωdΩk .(3.179) The integration in ktwithout kinematical restrictions can be easily be done using ω2dΩk=d2kt, producing 1 ω2Zd2kt (2π)2exp −k2 t 2ωη(2) d!k2 tη(3) d+η(4) d=−1 π Ω2 sin2Ω(zj−zk),(3.180) and 1 ω2Zd2kt (2π)2exp −k2 t 2ωη(2) b!k2 tη(3) b+η(4) b=1 π 1 (zj−zk)2.(3.181) Correspondingly for the particular case l→ ∞ we find ωdI(n) inc dω=2e2 πRe Z∞ 0 dzjZzj 0 dzkexp −iω 2 me p0 0!2 (zj−zk) × 1 (zj−zk)2−Ω2 sin2Ω(zj−zk) ,(3.182) which is Migdal’s result [4, 47] for the φ(s) function. By performing one of the trivial integrals, in which a factor proportional to larises, and by rotating the contour of integration to avoid oscillations, we find a suitable expression for the intensity for a medium of length l→ ∞, of the form ωdI(n) inc dω=l2e2 π|Ω| √2Z∞ 0 dz exp −z √2s! × sin z √2s!+cos z √2s!! 1 z2−1 sinh2(z)!,(3.183) Continuous limit: a path integral 99 where the parameter sis given by s≡2 ω p0 0 me!2 |Ω|=2p0 0 m2 erˆq ω.(3.184) An useful approximant to the above integral within less than a 1% of deviation in all the range is given by ωdI(n) inc dω≃l2e2 3πsˆqω (p0 0)2s1−1.52s4+5.8s5 1+2.44s5+2.73s6.(3.185) The other relevant prescription for the infinite length approximation is alternatively given by the quantity Zl −∞ 2 −Z0 −∞ 2 =Zl 0Zl 0 +2 Re Z0 −∞ Zl 0≡ωdI(n) d dωdΩk +ωdI(n) e dωdΩk ,(3.186) As with the prescription leading to Migdal solution, since the final leg photon disappears when l→ ∞ and the initial photon keeps impaired, it has been removed in order to avoid a log(ω) divergence in the angular integration. The angle integration of the second term (e) on the right hand side produces ωdI(n) e dω=2e2 πRe Zl 0 dzjZ∞ 0 dzkexp −iω 2 me p0 0!2 (zj−zk) ×Ω2 sin Ωzj−Ωzkcos Ωzj2.(3.187) Then by taking l→ ∞ we find ωdI(n) dω=2e2 πRe Z∞ 0 dzjZ∞ 0 dzkexp −iω 2 me p0 0!2 (zj−zk) × Ω2 sin Ωzj−Ωzkcos Ωzj2−Ω2 sin2Ω(zj−zk) . (3.188) For the particular albeit unrealistic case in which the fermion mass can be neglected the integral in zkcan be performed and we find ωdI(n) d dω=2e2 πΩRe Zl 0 dz cos(Ωz) sin(Ωz),(3.189) 100 High energy emission for the first term and ωdI(n) e dω=−2e2 πΩRe Zl 0 dz 1 cos (Ωz)sin (Ωz).(3.190) Correspondingly ωdI(n) d dω+ωdI(n) e dω=2e2 πΩRe Zl 0 dz cos(Ωz) sin(Ωz)−1 cos (Ωz)sin (Ωz)! =2e2 πRe log cos Ωl,(3.191) which is the BDMPS result [22]. Unfortunately, the neglection of the projectile mass leads to a continuous enhancement of the intensity in the regime of maximal interference, i.e. ω1, instead of the expected Bethe-Heitler incoherent plateau (3.18). Then the BDMPS result does not reproduce neither Weinberg’s soft photon theorem nor the Bethe-Heitler cross section. In Figures 3.5, 3.6, 3.7, 3.8, 3.9 and 3.10 the approximant (3.185) to the Migdal prediction (3.183) is shown together our evaluations in the Debye screened interaction and the Fokker-Planck approximation for (3.122). For small size targets, compromising less than nc=104collisions on average the Migdal prediction is not adequate, since Weinberg’s soft photon theorem is not considered. Transverse coherent average A reorganization of the coherent average currents in (3.77) and (3.78) can be done. We will restrict to the spin non flip contribution, since the spin flipping contribution follows the same steps. We also consider the limit ω→0 in all the quantities except in the phases. Then we notice that n X k=1 δn keiϕk=k×pn kµpµ n eiϕn+ n−1 X k=1 k×pk kµpµ keiϕk−eiϕk+1−k×p0 kµpµ 0 eiϕ1.(3.192) In the continuous limit the above relation is just an integration by parts, where the two boundary terms correspond to an extension of the integration to (−∞,z1] and [zn,∞) of the interior term. This interior term is given by n−1 X k=1 k×pk kµpµ keiϕk−eiϕk+1= n−1 X k=1 k×pk kµpµ k eiϕk1−eiϕk+1−iϕk.(3.193) Following (3.70) and (3.36) the phase difference is found to be iϕk+1−iϕk=iω(zk+1−zk) 2p0 0(p0 0−ω)m2 e+ pt k−p0 0 ωkt!2 =ikµpµ k p0 0−ωδz.(3.194) Continuous limit: a path integral 101 By taking the δz→0 limit then we find n−1 X k=1 k×pk kµpµ k eiϕ 1−exp ikµpµ k p0 0−ωδz!!≃ − i p0 0 n−1 X k=1 δzk×pkeiϕk.(3.195) The quantity to evaluate at (3.77) is just the convolution of the above current with the elastic weights. For the elements containing interaction this quantity can be always reorganized as  n−1 Y i=1 d3pi (2π)3 n Y i=1φ(n) coh(δpi, δzi)+φ(0) coh(δpi, δzi) n X k=1 δn keiϕk =−i p0 0 exp iωm2 ezk 2p0 0(p0 0−ω)+ik2 tzk 2ω!Zd3pk (2π)3P(n) coh(pn,pk)k×pkP(n) coh(pk,p0), (3.196) where the functions P(n) coh(pa,pb) act at the level of the amplitude and thus cannot be interpreted in probabilistic terms. If we define pz n=−(pt n−kt)2 2(p0 0−ω),pz k=−(pt k−k)2 2(p0 0−ω),pz 0=−(pt 0)2 2p0 0 ,(3.197) and complete the boundary terms to total three momenta as pt·xt+pzz≡p·x the propagators are given by Fourier transforms as follows P(n) coh(pk,p0)≡2πβpδ(p0 k−p0 0) k−1 Y i=1 d2pt i (2π)2 k Y i=1 d2xt i (2π)2e−ipk·xk+ip0·x1 ×exp −i k−1 X i=1 δz (pt i)2 2p0 0−pt i·δxt i δz!+ k X i=1 δzn0(zi)π(1) el (xt i), (3.198) and P(n) coh(pn,pk)≡2πβpδ(p0 n−p0 k) n−1 Y i=k+1 d2pt i (2π)2 n Y i=k+1 d2xt i (2π)2e−ipn·xn+ipk·xk+1 ×exp −i n−1 X i=k+1 δz (pt i−kt)2 2(p0 0−ω)−pt i·δxt i δz!+ n X i=k+1 δzn0(zi)π(1) el (xt i), (3.199) 102 High energy emission where δxt i≡xt i+1−xt i. The integration in internal momenta of the first passage produces  k−1 Y i=1 ip0 0 2πδz k−1 Y i=2 d2xt i (2π)2exp  +i k−1 X i=1 δzi p0 0 2 δxt i δzi!2 + k X i=1 δzin0(zi)π(1) el (xt i) →ZD2xt(z) exp iZzk z1 dz p0 0 2˙x2 t(z)−in0(z)π(1) el (xt(z))!! (3.200) whereas the second passage produces  n−1 Y i=k+1 i(p0 0−ω) 2πδz n−1 Y i=k+2 d3xt i (2π)2exp  +i n−1 X i=k+1 δzi p0 0−ω 2 δxt i δzi!2 +kt·δxt i δzi + n X i=k+1 δzin0(zi)π(1) el (xt i)→exp +ikt·(xt n−xt k+1) ×ZD2xt(z) exp iZzn zk+1 dz p0 0−ω 2˙x2 t(z)−in0(z)π(1) el (xt(z))!!.(3.201) The remaining integration in pkcan be done with some care. We notice that Zd2pt k (2π)2 pt−p0 0 ωkt!exp −ikt·xt k+1+ipt k·(xt k+1−xt k)−iδz(pt k)2 2p0 0! = i∂ ∂xt k−ip0 0 ω ∂ ∂xt k+1 +∂ ∂xt k!!e−ikt·xt k+1 ip0 0 2πδzexp +i(xt k+1−xt k)2 2p0 0δz!.(3.202) In the limit δz→0 we find the representation of the Dirac delta function and then Zd3pk (2π)3P(n) coh(pn,pk)k×pkP(n) coh(pk,p0)=iω2πβpδ(p0 n−p0 0)Zd2xt nei(kt−pt n)·xt n ×Zd2xt 1e+ipt 0·xt 1Zd2xt ke−ikt·xt k(ˆ P(n) coh(xt n,xt k) p0 0 ω−1!∂ ∂xt k ˆ P(n) coh(xt k,xt 1)! + p0 0 ω ∂ ∂xt k ˆ P(n) coh(xt n,xt k)!ˆ P(n) coh(xt k,xt 1)),(3.203) or alternatively iω2πβpδ(p0 n−p0 0)Zd2xt ne−i(pt n−kt)·xt nZd2xt 1e+ipt 0·xt 1(3.204) ×Zd2xt ke−ikt·xt kˆ P(n) coh(xt n,xt k)(−ktp0 0 ω−i∂ ∂xt k)ˆ P(n) coh(xt k,xt 1), Continuous limit: a path integral 103 where the functions ˆ P(n) coh(xt k,xt 1) an ˆ P(n) coh(xt n,xt k) are the path integrals ˆ P(n) coh(xt n,xt k)≡ZD2xt(z) exp iZzn zk dz p0 0−ω 2˙x2 t(z)−in0(z)π(1) el (xt(z))!!, (3.205) and ˆ P(n) coh(xt k,xt 1)≡ZD2xt(z) exp iZzk z1 dz p0 0 2˙x2 t(z)−in0(z)π(1) el (xt(z))!!.(3.206) It is easy to note that the term involving no collisions corresponds to making n=0 in the above expressions. Then using (3.77), (3.79), (3.196) and (3.203), we find the non flip transverse coherent contribution in the continuous limit ωdIcoh dωdΩk =e 2π2 ω p0 0!21 πR2Zd2pt n (2π)2 ×Zdzkexp iωm2 ezk 2p0 0(p0 0−ω)+ik2 tzk 2ω!Zd2xt ne−i(pt n−kt)·xt nZd2xt 1e+ipt 0·xt 1 ×Zd2xt ke−ikt·xt kˆ P(n) coh(xt n,xt k)(−ktp0 0 ω−i∂ ∂xt k)ˆ P(n) coh(xt k,xt 1)−(n=0) 2 . (3.207) Since the Fourier transform of π(1) el (xt) is an even function of qt, the function has a saddle point in xt=0. By assuming a constant density n0(z)=n0a series truncation at high number of collisions holds and −in0π(1) el (xt)≃ −1 2ˆqx2 t,(3.208) where by using (3.59) we find ˆq=−∂2 ∂2xt π(1) el (xt)xt=0 =4πgR Z∞ 0 dq q2 q2+µ2 d J1(qR)=4πgµdRK1(µdR), (3.209) where K1(x) is the modified Bessel function. We observe that the transverse coherent average leads to an equivalent charge given by the average interaction potential. In this Fokker-Planck approximation the path integrals have a Gaussian form and thus they are solvable, ˆ P(n) coh(xt n,xt k)≡ZD2xt(z) exp iZzn zk dz p0 0−ω 2˙x2 t(z)−1 2(p0 0−ω)Ω2 fx2 t(z)!!, (3.210) 110 High energy multiple scattering in QCD |pi,aiihpj, aj| (tα)al akAµ α(x) gs |pk, akihpl, al| gs |pi,αiihpj, αj| (tα)al akAµ α(x) gs |pk, akihpl, al| gs |pi,αiihpj, αj| (Tα)αl αkAµ α(x) gs |pk, αkihpl, αl| gs Figure 4.1: Left: The qq interaction. Center: The qg interaction. Right: The gg interaction. Cqq R≡1 dX ajai 1 dX alak (tα)aj ai(t† β)ai aj(tα)al ak(t† β)ak al=1 d2Tr2(tαtβ)=N2 c−1 4N2 c =2 9.(4.10) Here we used the hermiticity of the tαmatrices and the normalization tα=λα/2, where λαare the Gell-Mann matrices [65], so Tr(tαtβ)=Tfδαβ where Tf=1/2 is the first invariant or Casimir. Similarly, for a qg the gluon emitted by a target gluon is given by the replacement (tα)aj ai→(Tα)αj αi=−i f αj ααi. We find an averaged charge of Cqg R≡1 dX αjαi 1 dAX alak (tα)al ak(t† β)ak al(Tα)αj αi(T† β)αi αj=1 ddA Tr(tαtβ) Tr(TαTβ)=Tf=1 2, (4.11) where we used the adjoint representation normalization Tr(TαTβ)=CAδαβ, with CA=Nc. And finally the case of a gluon scattered by a target gluon produces using the same procedure Cgg R≡1 dAX αjαi 1 dAX αkαl (Tα)αj αi(T† β)αi αj(Tα)αl αk(T† β)αk αl=1 d2 A Tr2(TαTβ)=N2 c N2 c−1=9 8. (4.12) In Fig. 4.1 these three different diagrams of scattering scattering are shown. The above results can be reproduced also in the classical and static field limit. For target quarks heavier than the average momentum change scale µd, or for moving quarks with a typical energy hp0 iimuch greater than µdbut much smaller than the energy of the traveling parton, quarks in the target can be considered almost at rest, p0 i≃mqor p0 i≃ hp0 iirespectively. The recoil can be considered negligible pj≃piand we find smq p0 j ¯usj(pj)γ0usi(pi)smq p0 i≈1,smq p0 j ¯usj(pj)γkusi(pi)smq p0 i≈(pi)k mq1, (4.13) Scattering amplitude 111 pjpi tαAµ α(x) gs pjpj gs tαAµ α(x) Figure 4.2: Left: The field created by a quantum current hi|ji. Right: The field as a classic and static current where pi≈pj. and then the gluon field simplifies to Aµ α(x)=δµ 0 4πgstα −(pj−pi)2+µ2 d e+i(pj−pi)x.(4.14) In this approximation the slow varying momentum dependence of the spinor terms is assumed. After coherently forming the superposition in momentum and integrating in time we find hAµ α(x)iq,t≡δµ 0gstαZd4q (2π)4Zdt 4π −q2+µ2 d eiq·x=δµ 0gstα 1 |x|e−µd|x|,(4.15) where the last integral has been done with the help of Cauchy’s theorem. So the classical field of a particle of charge gs(tα)aj aiat x=0 is recovered, i.e. J0 α(x)= gstαδ(3(x) and Jα=0. Under these simplifications our colored medium, then, will be characterized from here onwards as the classical gluon field of a set of n static quarks, of the form A0 α(x)= n X k=1 gstα |x−rk|e−µd|x−rk|.(4.16) For a mixed scenario consisting of gluons and quarks the coupling of the gluon content can be simply replaced with the adjoint representation (tα)→(Tα) in this classical approximation. In our following derivations, for simplicity, we will restrict to a medium exclusively composed of quarks. 4.2 Scattering amplitude From (4.16) a medium consisting of quarks and/or gluons will be characterized as an external field with one non vanishing time independent component, namely A0 α(x). The quark state at high energies under this interaction, assuming that the 112 High energy multiple scattering in QCD asymptotic state has momentum piplaced along the x3direction, c.f Section 1.2, is given by (1.29) ψ(n)(x)= 1+i 2p0 i α·∇+1 2p0 i α·p!ϕ(n) s(x),(4.17) where ϕ(n) s(x) is a stationary and high energy, Schrodinger-like, solution to the equation (1.43) with asymptotically free initial condition. We write then ∂ϕ(n) s(x) ∂x3 = ip3 i−i βp gstαA0 α(x)!ϕ(n) s(x),lim x3→−∞ ϕ(n) s(x)=ϕ(0) s(x),(4.18) where βpis the quark velocity and ϕ(0) s(x) a free solution of momentum pi. The solution to this matrix equation (4.18) with that initial condition is given by the ordered exponential ϕ(n) s(x)=Pexp −igs βpZx3 −∞ dx0 3tαA0 α(x)!ϕ(0) s(x).(4.19) Noticing that the term α·pis canceled when the derivative of the free part is taken, we find the quark equivalent of (1.42) ψ(n)(x)=( 1+iγkγ0 2p0 i ∂k!Pexp −igs βpZx3 −∞ dx0 3tαA0 α(x)!)ψ(0)(x).(4.20) We will explicitely write the states with the color vector. For the asymptotic initial quark in particular ψ(0)(x)=e−ipi·xsmq p0 i uai si(pi).(4.21) Let the eikonal phase in the high energy integration of the wave be denoted as W(n)(x,pi)≡ Pexp −igs βpZx3 −∞ dx0 3tαA0 α(x)!.(4.22) From here onwards path ordering is going to be implicitly assumed. The scattering amplitude of finding the quark in a state with momentum pfdue to the external field actuation can be found [66] by using the Lippmann-Schwinger relation ψ(n)(x)=ψ(0)(x)+Zd4yS F(x−y)γ0gstαA0 α(y)ψ(n)(y)=ψ(0)(x)+ψ(n) di f f (x), (4.23) Scattering amplitude 113 where the Feynman-Stueckelberg quark propagator is given by Sq F(x)=Zd4p (2π)4eip·x/ p+mq p2−m2 q =1 iX s,aZd3p (2π)3(4.24) × rmq p0ua s(p)⊗¯us a(p)rmq p0!e−ip·x. Here we restricted to energy positive solutions and we used the completeness relation in spin and in color X s,a ua s(p)⊗¯us a(p)=/ p+mq 2mq .(4.25) Then, the diffracted part in the Lippmann-Schwinger equation produces a superposition of states of momentum pf, spin sfand color afof the form ψ(n) di f f (x)=X sf,afZd3pf (2π)3e−ipf·xsmq p0 f uaf sf(pf)M(n) sfsi(pf,pi)af ai ,(4.26) so that we write for the amplitude of finding the quark in this final state fas M(n) sfsi(pf,pi)af ai≡afZd4ye+i(pf−pi)y(4.27) ×smq p0 f ¯usf(pf)γ0−igstαA0 α(y)( 1+iγkγ0 2p0 i ∂k!W(n)(y,pi))usi(pi)smq p0 i ai. The integration of the above equation in the high energy limit follows exactly the same steps as those presented at Section 2.1. In the high energy limit, provided that the Glauber condition (1.23) holds, we can neglect the 1/2p0 ioperator correction. By defining q=pf−piand neglecting by now the beyond eikonal corrections in qzone easily finds M(n) sfsi(pf,pi)=2πδ(p0 f−p0 i)βpsmq p0 f ¯usf(pf)γ0usi(pi)smq p0 i (4.28) ×Zd2yte−iqt·yt exp "−igs βpZ+∞ −∞ dy3tαA0 α(y)#−1!. The spinorial product can be reduced to a spin conservation delta in the high energy limit smq p0 f ¯usf(pf)γ0usi(pi)smq p0 i≃δsf si(4.29) 114 High energy multiple scattering in QCD For convenience we define the integral part of (4.28) as F(n) el (qt)≡Zd2yte−iqt·yt exp "−igs βpZ+∞ −∞ dy3tαA0 α(y)#−1!,(4.30) since spin and energy are preserved and F(n) el (q) contains the relevant momentum distribution after the scattering. We introduce now the notation shorthands igs βpZ+∞ −∞ dy3tαA0 α(y)=ig2 s βp tαtα n X k=1 χ(1) 0(yt−rk t)=ig2 s βp tαtα n X k=1 χk 0(yt).(4.31) The color transitions of the target and traveling quarks are expected to be evaluated in F(n) el (q) and usually color-averaged in its square. For a color averaged single target quark (n=1) performing the color transition bi→bfwe obtain the following useful optical theorem 1 NcZd2qt (2π)2Tr F(1)† el (qt)F(1) el (qt)=1 NcZd2xt exp +ig2 s βp t† βt† βχ1 0(xt)!−1!bi bf × exp −ig2 s βp tαtαχ1 0(xt)!−1!bf bi =−1 Nc Tr F(1)† el (0)+F(1) el (0),(4.32) where the trace refers from here onwards to the colors in the target space only. A symmetrical relation can be written for the color average in the projectile or for both averages together, in general. We notice that the color average of |F(1) el (q)|2 does not admit, however, a reexponentiation. In order to see this fact we expand the amplitude using (4.30) and (4.16). We get F(1) el (q)=−ig2 s βp tα1tα1ˆ A(1) 0(q)−g4 s 2β2 p tα2tα2tα1tα1Zd2k (2π)2ˆ A(1) 0(k)ˆ A(1) 0(q−k)+. . . , (4.33) where the Fourier transform of the single field satisfies ˆ A(1) 0(q)=4π/(q2+µ2 d) for the Debye screened interaction (4.16). The total color average of the squared amplitude produces F(1)† el (qt)aibi afbfF(1) el (qt)afbf aibi =g4 s βp 2 C(2) Rˆ A(1) 0(q) 2+g6 s 2β3 p C(3) Rˆ A(1) 0(q) (4.34) ×Zd2k (2π)2ˆ A(1) 0(k)ˆ A(1) 0(q−k)+g8 s 4β4 p C(4) R Zd2k (2π)2ˆ A(1) 0(k)ˆ A(1) 0(q−k)!2 +. . . . Scattering amplitude 115 The required color charges are given by operations like (4.10). Indeed for the simplest diagram we find the aforementioned result (4.10) C(2) R=1 N2 c Tr(tα1tβ1)=N2 c−1 4N2 c =2 9.(4.35) The interference term C(3) Rhas to cancel by symmetry under conjugation. We obtain, of course C(3) R=1 N2 cTr2(tα1tβ2tβ1)−Tr2(tα2tα1tβ1)=0,(4.36) as expected. The next term requires a little more of work. It is given by C(4) R=1 N2 c 1 2δaf b1δbf a1−1 6δaf a1δbf b1! 1 2δa1 biδb1 ai−1 6δa1 ai!(tβ2)b1 bi(tβ2)ai a1(tβ1)a1 af(tβ1)b1 bf =1 N2 c 10 36 Tr2(tβ1tβ2)−2 12 Tr(tβ1tβ2tβ1tβ2)!=2 27.(4.37) It can be shown that the next order squared-diagram would produce C(6) R= 22/729. So that the average color charge of two kicks with the same center 2/27 is not anymore the square of the averaged color charge of two single kicks (2/9)2, and so on. Since a reexponentiation does not hold, in the high energy formalism we will restrict to leading order evaluations in the coupling of the single amplitudes. On the other hand, the scattering amplitude for the gluon is found by tracing back the same approach with the gluon propagator in the Feynman gauge. By using the completeness and orthogonality relations X λ λ µ(k)λ∗ ν(k)=gµν, λf µ(kf)µ λi(ki)=−δλf λi,(4.38) one arrives at a very similar form to (4.28) M(n) λfλi(kf,ki)=2πδ(δk0)βkλf∗ µ(kf)µ λi(ki) ×Zd2yte−iδkt·yt exp "−igs βpZ+∞ −∞ dy3TαA0 α(y)#−1!.(4.39) Similarly to the quark case, in the high energy limit we find kf≃kiand thus we obtain a polarization conservation delta. The gluon dynamics in this sense is similar to the quark except for the coupling definitions, and contained in the function F(n) el (q). 116 High energy multiple scattering in QCD 4.3 Multiple scattering effects Amplitude (4.28) can be squared and averaged over multiple scatterers configurations. As in the QED case we find an incoherent and a coherent contribution, but with some differences related to the color behavior. It results illustrative to obtain this transverse interference behavior by means of an expansion in the number of collisions prior to a direct evaluation of the full expression. An inspection of (4.30) suggests defining Γk≡exp −ig2 s βp tαtαχk 0(yt)!−1.(4.40) Let us denote by bk fand bk ithe final and initial color, respectively, of the target quark k, then omitting the indices in the left hand side F(n) el (q)=Zd2xe−iq·x n Y k=1(Γk)bk f bk i +δbk f bk i− n Y k=1 δbk f bk i.(4.41) It is now easy to expand the amplitude in the number of collisions. We notice F(n) el (q)=Zd2xe−iq·x n X k=1 (Γk)bk f bk iY j,k δbj f bj i + n X k=1 n X j=k+1 (Γk)bk f bk i (Γj)bj f bj iY l,j,k δbl f bl i +. . . = n X k=1 Ik+ n X k=1 n X j=k+1 Ik j +··· .(4.42) The single collision contribution can be rewritten as n X k=1 Ik= n X k=1Y j,k δbj f bj iZd2xte−iqt·xt exp −ig2 s βp tαtαχk 0(yt)!−1!bk f bk i = n X k=1Y j,k δbj f bj i e−iqt·rk tF(1) el (qt)bk f bk i ,(4.43) which is the expected result, consisting in the transition of the target quark k causing the collision and the no transition of the rest of quarks. Due to the high energy limit any term of higher order can be always written as a convolution Ik j =Y l,j,k δbl f bk iZd2kt (2π)2e−ikt·rk t−i(qt−kt)·rj tF(1) el (kt)bk f bk iF(1) el (qt−kt)bj f bj i ,(4.44) Multiple scattering effects 117 which is easily interpretable also in terms of single processes. We can consider now the squared amplitude, which with this notation is given by F(n)† el (q)F(n) el (q)= n X k=1 n X j=1 I† jIk+ n X k=1 n X j=1 n X l=j+1I† jlIk+I† kIjl+··· .(4.45) The first contribution is given by the square of the single collision amplitudes, which is given by terms as I† jIk=Y l,k δbl f bl iY m,j δbm i bm f e+iqt·rj tF(1)† el (qt)bj i bj f!×e−iqt·rk tF(1) el (qt)bk f bk i.(4.46) The sum of this terms can be always arranged into a diagonal and a non diagonal contribution following n X k=1 n X j=1 I† jIk= n X k=1 I† kIk+ n X k=1 n X j,k I† jIk,(4.47) which produces for the diagonal contribution n X k=1 I† kIk=n n Y j,k δbj f bj i δbj i bj fF(1)† el (qt)bk i bk fF(1) el (qt)bk f bk i =nNn−1 cTr F(1)† el (qt)F(1) el (qt), (4.48) where trace refers to the target color space. By dividing by the dimension of the target color space and by perturbatively expanding the amplitudes up to first order in the coupling (4.33) we get 1 Nn c n X k=1 I† kIk≃n Nc g4 s β2 p tαtβTr(tαtβ)ˆ A(1) 0(q)2=ng4 s β2 p N2 c−1 4N2 cA(1) 0(q)2,(4.49) which is n, the number of independent collisions, times the squared amplitude of a single collision. Notice that the color average in the target produces an average color conservation in the projectile δaf ai. For the non-diagonal contribution we find n X k=1 n X j,k I† jIk= n X k=1 n X j,k n Y l,k δbl f bl i n Y m,j δbm i bm f e−iqt·(rk t−rj t)F(1)† el (qt)bj i bj fF(1) el (qt)bk f bk i =Nn−2 cTr F(1)† el (qt)Tr F(1) el (qt) n X k=1 n X j,k e−iqt·(rk t−rj t) .(4.50) 118 High energy multiple scattering in QCD Unlike to the QED case, by using (4.33) we get that the first order in the coupling vanishes due to the traceless tαmatrices, so at leading order in the coupling Tr F(1)† el (qt)Tr F(1) el (qt)= −1 2 g4 s β2 p tαtβTr(tαtβ)Zd2k (2π)2ˆ A(1) 0(k)ˆ A(1) 0(q−k)!2 . (4.51) By averaging over target spatial configurations, for simplicity over a solid cylinder of section Ω = πR2we have *n X k=1 n X j,k e−iqt·(rk t−rj t)+Ω =n(n−1) 1 πR2ZΩ d2rte−iqt·rt!2 =n(n−1) w(q,R) πR2!2 . (4.52) where w(q,R) is the window function given at (2.77). Then we get 1 Nn c n X k=1 n X j,kDI† jIkEΩ≃n2 1 2 g4 s β2 p N2 c−1 4N2 c w(q,R) πR2Zd2k (2π)2ˆ A(1) 0(k)ˆ A(1) 0(q−k)!2 , (4.53) where nwas assumed large so that n(n−1) ≃n2. Observe that the diagonal contribution produced a term of order nand α2 s, the number of independent collisions times the leading order of a color averaged single collision, whereas the non diagonal contribution produces a term of order n2and α4 s, which measures the n2different interferences between single collisions with different scattering centers. By performing the last integral in momentums the final result can be written explicitely as F(n) el (q) 2nN2 c−1 4N2 c αs βp 4π q2+µ2 d!2 + nN2 c−1 4N2 c w(q,R) πR2 1 2 αs βp!216πarcsinh q 2µd qqq2+4µ2 d  2 .(4.54) For convenience we give the window functions for a solid cylinder and for a cylinder with normalized Gaussian decaying density, of a typical width R2, wc(q,R) πR2≡2J1(qR) qR ,wg(q,R) πR2≡exp −q2R2 2!.(4.55) Notice that in the limit R→ ∞we recover always the forward propagation δ2(q) in the non-diagonal contribution, as expected due to transverse homogeneity. In Multiple scattering effects 119 Figure 4.3 the above result is shown for a typical QCD medium. As it can be seen, although the non-diagonal contribution is a higher order correction in the coupling to the diagonal contribution, for n=10 and higher number of constituents its contribution in the soft scattering zone q∼1/Ris the dominant one. For mediums of large length the above discussion is not enough, since the scattering typically involves more than one collision. For the next order evaluations we refer to the procedure at Section 2.4. Since the expansion in the number of collisions is an alternated series, a functional form is required. We define then the shorthand notations 1e2 1e3 1e4 1e5 0 0.1 0.2 0.3 0.4 0.5 F(n) el (q) 2 q(GeV) (I)+(C) Solid Cyl. n=100 (I)+(C) Gaussian Cyl. n=100 (I) n=100 Same n=50 Same n=25 Same n=10 Figure 4.3: Averaged squared elastic amplitude at leading order in the coupling and in the number of collisions as a function of the momentum change, of a quark emerging from a medium with n =100 quarks (black lines), with Debye screening mass of µd=0.5 GeV and a medium radius of R=10 rd. Incoherent (I) contribution (diagonal term) is shown in dotted lines, and incoherent (I) and coherent (C) contributions (diagonal and non diagonal term) are shown for a cylinder with Gaussian decaying density (dot-dashed lines) and a solid cylinder (solid lines). Same results for n=50 (dark grey), n=25 (medium grey) and n=10 (lightest grey). W(n) 0(y)≡exp −i n X k=1 g2 s βtαtβχk 0(y) = n Y k=1 exp −ig2 s βp tαtαχk 0(x)!!= n Y k=1 Wk 0(y). (4.56) Then the average over target colors and positions can be formally written as 1 Nn c Tr DM(n)† sfsi(pf,pi)M(n) sfsi(pf,pi)E=2πδ(q0)βpδsf si 1 Nn c Tr DF(n)† el (q)F(n) el (q)E, (4.57) 126 High energy multiple scattering in QCD where the eikonal phase reads, using (4.16), Zy3 −∞ dy0 3tαA0 α(y)=gstαtα n X i=kZy3 −∞ dy0 3Zd3q (2π)3 4π q2+µ2 d e−iq·(y−rk).(4.89) We will assume that q2≃q2 tso that the longitudinal structure of each center is neglected, but we keep qzin the phase since y3can be arbitrarily large and we want to keep the longitudinal structure of the medium. This approximation produces Zy3 −∞ dy0 3Zd3q (2π)3 4π q2+µ2 d e−iq·(y−rk)≃Zy3 −∞ dy0 3δ(y0 3−rk 3) (4.90) ×Zd2qt (2π)2 4π q2 t+µ2 d e−iqt·(yt−rk t)= Θ(y3−rk 3)χ(1) 0(yt−rk t)≡Θk(y3)χk 0(yt). Since we placed the quarks at the same zcoordinate we can easily integrate by parts the amplitude, which takes the simpler form M(n) sfsi(pf,pi)=S(n) sfsi(pf,pi)−S(0) sfsi(pf,pi),(4.91) where the eikonal amplitude, including the no collision amplitude, is given as usual by S(n) sfsi(pf,pi)≡2πβpδ(δp0 i)δsf siZd2yte−iq·xexp −ig2 s βp tαtα n X k=1 χk 0(y),(4.92) where the total three momentum change q·xappears now in the phase. We can consider the medium as a set of nlayers of quarks at z1,z2... zn, of n(zi) quarks respectively. The total number of quarks in the medium is defined as n X i=1 n(zi)≡N.(4.93) At high energies the amplitude of finding the quark with momentum pnafter the passage through these layers is given by the convolution of single layer amplitudes, then S(N) sns0(pn,p0)=X s n−1 Y i=1Zd3pi (2π)3 n Y i=1 Sn(zi) sisi−1(pi,pi−1),(4.94) where sum over intermediate spins and ordering is assumed. So that we find M(N) sns0(pn,p0)=S(N) sns0(pn,p0)−S(0) sns0(pn,p0).(4.95) Beyond eikonal scattering 127 At this point we notice that a path integral expression for the amplitude exists by taking the separation between layers δzk→0. The resulting expression is, however, uninteresting out of a formal context, since the interaction carries the matrix structure in the traveling and target color spaces. We instead take the intensity and average over target colors, which leads to color preservation in the projectile. By splitting as before in a incoherent and a coherent contribution we obtain DM(N)† sns0(pn,p0)M(N) sns0(pn,p0)E= Σ(N) 2(pn,p0)+ Π(n) s(pn,p0),(4.96) where the coherent contribution is given by the averaged amplitudes squared Π(N) 2(pn,p0)=DS(N)† sns0(pn,p0)−S(0)† sns0(pn,p0)EDS(N) sns0(pn,p0)−S(0) sns0(pn,p0)E, (4.97) and the incoherent contribution instead by Σ(N) 2(pn,p0)=DS(N)† sns0(pn,p0)S(N) sns0(pn,p0)E−DS(N)† sns0(pn,p0)EDS(N) sns0(pn,p0)E. (4.98) The first term of the incoherent contribution can be written, after dividing by the incoming quark flux and the infinite factor T=2πδ(0) accounting for time translation invariance, as the product DS(N)† sns0(pn,p0)S(N) sns0(pn,p0)E=2πβpδ(p0 n−p0 0)δsn s0(4.99) × n−1 Y i=1Zd2pt i (2π)2Zd2ut i (2π)2exp −ipt i2−ut i2 2p0 0 δzi × n Y i=1Zd2xt iZd2yt ie−iδpt i·xt i+iδut i·yt i n(zi) Y k=1 1 Nc Tr DWk† 0(yt)Wk 0(xt)E! , where δzi≡zi+1−ziand δpi=pi−pi−1and the internal momentum in the conjugated amplitude has been denoted by ui. The bottom line is just a product of averages at each layer of length δziand density n0(zi) like the one computed in the previous section, thus we write Zd2xt iZd2yt ie−iδpt i·xt i+iδut i·yt i n(zi) Y k=1 1 Nc Tr DWk† 0(yt)Wk 0(xt)E!(4.100) =(2π)2δ2(δpt i−δut i)Zd2xi te−iδpt i·xt iexp δzin0(zi)σ(1) el (xt i)−σ(1) el (0). 128 High energy multiple scattering in QCD Similarly for the second term in the incoherent contribution we find DS(N)† sns0(pn,p0)EDS(N) sns0(pn,p0)E=2πβpδ(p0 n−p0 0)δsn s0(4.101) × n−1 Y i=1Zd2pt i (2π)2Zd2ut i (2π)2exp −ipt i2−ut i2 2p0 0 δzi × n Y i=1Zd2xt iZd2yt ie−iδpt i·xt i+iδut i·yt i n(zi) Y k=1 1 Nc Tr DWk† 0(yt)ETr DWk 0(xt)E! , where as before by using the results of the preceding section we obtain for each of the single layer averages Zd2xt iZd2yt ie−iδpt i·xt i+iδut i·yt i n(zi) Y k=1 1 Nc Tr DWk† 0(yt)EDWk 0(xt)E!(4.102) =(2π)2δ2(δpt i−δut i)Zd2xi te−iδpt i·xt iexp −δzin0(zi)σ(1) el (0). We can now proceed to integrate in the conjugate momentums ut iobserving that since ut 0≡pt 0and, by using the transverse momentum deltas, the longitudinal phase vanishes n−1 Y i=1Zd2ut i (2π)2exp −ipt i2−ut i2 2p0 0 δzi n Y i=1(2π)2δ2(δpt i−δut i)=(2π)2δ2(0), (4.103) as expected by symmetry arguments for a medium of infinite transverse size. Then the incoherent contribution is given simply by Σ(N) 2(pn,p0)=2πδ(p0 n−p0 0)βpδsn s0πR2 n−1 Y i=1 Zd2pt i (2π)2!n Y i=1 Zd2xt ie−iδpt i·xt i! ×       n Y i=1 exp δzin0(zi)σ(1) el (xt i)−σ(1) el (0)− n Y i=1 exp −δzin0(zi)σ(1) el (0)       . (4.104) We notice that the integration in the internal momentum variables is trivial and the internal scattering structure is lost. The joint action of the squared averaged amplitudes at each layer simply convolute without phase and the pure eikonal limit remains valid. By taking the δzi→0 limit we obtain Σ(N) 2(pn,p0)=2πδ(p0 n−p0 0)βpδsn s0πR2exp −σ(1) el (0)Zzn z1 dz n0(z)! ×Zd2xte−i(pt n−pt 0)·xt exp σ(1) el (xt)Zzn z1 dz n0(z)!−1!,(4.105) Beyond eikonal scattering 129 which except for the varying local density is our previous result. For the coherent contribution the scenario is however different if we do not take the R→ ∞limit. We have DS(N) sns0(pn,p0)E=2πδ(p0 n−p0 0)βpδsn s0 × n−1 Y i=1 Zd2pt i (2π)2!n Y i=1Zd2xt ie−iδpt i·xt i n0(zi) Y k=1 1 Nc Tr DWk 0(xt k)E! ×exp  +i(pt n)2 2p0 0 zn−i n−1 X i=1 δzipt i2 2p0 0−ipt 02 2p0 0 z1.(4.106) The total kinematical phase must be rearranged as follows −ipt n·xt n+i(pt n)2 2p0 0 zn+i n−1 X i=1 pt i·δxt i δzi δzi−i n X i=1 (pt i)2 2p0 0 δzi−ipt 02 2p0 0 z1+ipt 0xt 1, (4.107) and the medium contribution is given by the results of the previous section n0(zi) Y k=1 1 Nc Tr DWk 0(xt k)E!=exp δzin0(zi)π(1) el (xt i).(4.108) Then we can perform now the integral in internal momenta. If we omit the two boundary terms in znand z0, which otherwise will cancel when taking the square of the amplitude, we find DS(N) sns0(pn,p0)E=2πδ(p0 n−p0 0)βpδsn s0 n−1 Y i=1 iδzi 2πp0 0!n Y i=1 Zd2xi t!(4.109) ×exp −ipt n·xt n+i n−1 X i=1 p0 0 2 δxt i δzi!2 δzi+ n X i=1 δzin0(zi)π(1) el (xt i)+ipt 0xt 1, which by taking the δz→0 limit transforms into a path integral in the transverse plane with time variable the zposition as DS(N) sns0(pn,p0)E=2πδ(p0 n−p0 0)βpδsn s0Zd2xt ne−ipt n·xt nZd2xt 1e+ipt 0xt 1 ZD2xt(z) exp iZzn z1 dz p0 0 2˙x2 t(z)−in0(z)π(1) el xt(z)!!.(4.110) Similarly the second term in the coherent contribution is simply found by doing N=0 in the above relation, thus by dividing by the incoming quark flux and a 130 High energy multiple scattering in QCD factor T=2πδ(0) accounting time translation invariance, we finally write Π(N) 2(pn,p0)=2πδ(p0 n−p0 0)βpδsn s0Zd2xt ne−ipt n·xt nZd2xt 1e+ipt 0xt 1(4.111) ×ZD2xt(z) exp iZzn z1 dz p0 0 2˙x2 t(z)! exp Zzn z1 dz n0(z)π(1) el xt(z)!−1! 2 , which either in the p0 0→ ∞ or the R→ ∞ limits transforms into our previous eikonal result. 5 High energy emission in QCD The LPM effect in a QCD multiple scattering scenario has become an useful tool of indirectly observing the properties of the hadronic matter at extremely high temperatures. High energy collisions of heavy nuclei at the RHIC [67–70] and the LHC [71–73] hadron colliders are expected to reproduce the required extreme conditions for the QGP formation. Among other probes, the energy loss of hard jets constituents [74–78] produced in these collisions can be used to trace back the relevant characteristics of the formed QCD medium. While traveling through the QGP, a high energetic quark or gluon may undergo a collisional process with the medium constituents. The leading contribution to the energy loss suffered in this multiple collision scenario is of radiative nature [77]. As we have already seen, the radiation of quanta leading to this energy loss is going to be severely modified by the existence of multiple sources of scattering [1, 2, 4], compared to an emission scenario consisting in an incoherent sum of single collision intensities [6, 79]. Hence a systematic study of the intensity of gluon bremsstrahlung in media and the related parton energy loss shall provide indicative signs of the QGP formation and its characteristics. Early studies of the intensity of gluon bremsstrahlung in a multiple scattering context exist [39, 75, 76]. The main difficulties introduced with respect to the QED case we have reviewed in Chapter 3 are related to the non abelian nature of QCD, namely the color structure and the gluon ability to reinteract with the medium. As we will show, these circumstances lead to a LPM interference in the soft limit dominated by the gluon rescattering, in analogy with the dielectric and transition radiation effects occurring in QED and shown at Figure 3.2. A first attempt for a QCD evaluation was given at [39] directly using the Migdal prediction [4] but without including gluon rescattering, and a more detailed evaluation was then given at [40, 80] without gluon rescattering effect in the longitudinal 131 132 High energy emission in QCD phase. Semi-infinite medium calculations in the Fokker-Planck approximation for the angle integrated intensity, either following a transport approach [21] or a path integral formalism [20, 55] were soon presented including the full gluon rescattering effect. These works can be shown to be the QCD equivalent to the Boltzmann transport approach [47] used by Migdal [4] with the adequate approximations [56]. Further extensions including the angular dependence, finite, cold or expanding target scenarios were later developed [54, 81–87] although usually lacking a general formulation, until a finite size evaluation of the angle dependence of the spectrum was given [19, 88–91] generalizing the previous results. The intensity of gluon bremsstrahlung within this framework [91–116] was then widely used to make predictions of the quenching behavior of the hadron spectra due to a multiple collision process with confined or deconfined QCD matter. Several other distinct frameworks were also developed at the same time or soon after the previous works, among them a reaction operator formalism for finite media [117–120] and for QGP [121, 122], a high twist formalism for finite nuclei [123, 124] and a finite field theory framework [125–127]. Applications of these results were also presented to make finite plasma, cold nuclear matter and heavy flavor suppression predictions at RHIC and LHC, see for instance [128– 130]. An excellent comparison of all these frameworks was given at [131] and very good and detailed reviews on the subject have been written [132–137]. The aim of this chapter is to develop a formalism [138] in which the intensity of gluons in a multiple scattering scenario can be evaluated for a general interaction with the constituents of a finite or structured medium, and in which the angular spectrum is also considered. We will basically follow the steps we took at Chapter 3 for the QED case, to construct a non-perturbative description of the emission amplitude. At the level of the amplitude the diagrammatic structure becomes clear and the soft photon theorem [5, 139, 140] for finite media can be understood, together with the LPM effect, as part of the same coherence phenomena. The arising gluon intensity, which is the central result of this chapter, will lead, under the adequate color averaged interaction of the gluon and its Fokker-Planck approximation, to the well known results at [19–21]. In order to make a direct connection with the previous results, the results obtained for the multiple soft scattering case are easily adapted to implement an approximation for the emission scenario after a first hard collision. In Section 5.1 we define amplitude of emission of a gluon under the effect of the medium as a set of static and classical Debye screened sources, as we explained at Section 4.1. The diagrammatic structure of the bremsstrahlung amplitude for multiple collisions is explained and the Bertsch-Gunion limit [79] recovered for the single collision scenario. In Section 5.2 we compute the intensity of gluons for a color and space averaged target with finite length lbut for simplicity with macroscopical transverse size R→ ∞. In this approxima- Amplitude 133 tion the transverse coherent contributions to the multiple scattering distributions are neglected. A color averaged effective interaction for the gluon is used which translates into Debye screened single interactions agreeing with previous results [19, 20, 141]. The resulting intensity is shown to satisfy a QED like form, where the LPM interfering phase, however, is governed by the gluon rescattering, and then the role of the traveling fermion is played by the gluon. Emission in a multiple soft scattering scenario will be shown to be dominated by a longitudinal coherent contribution, corresponding to the first and last leg gluons of the process, and lightly enhanced for small gluon energies as a result of the LPM effect, and then suppressed in the soft regime due to the mass suppression effect. An energy-loss formalism for that scenario will be developed using energy conservation constraints. Finally, the radiation due to a multiple soft collision scenario after a hard collision will be approximated as a particular subset of terms. The intensity in that case, except for the suppression in the soft limit related to a non vanishing gluon mass, will be shown to reproduce the well known results [19, 21, 91] if the Fokker-Planck approximation is used. 5.1 Amplitude Let us consider an ideal scenario for an on-shell quark coming from the infinity and going to the infinity, while undergoing a multiple scattering process with a QCD medium. The medium will be composed of Nstatic and classical sources characterized by (4.16), extending from z1to znin the initial quark direction. In QCD the amplitude of emission (3.10) of a single gluon, c.f. Sections 3.1 and 4.2, is given by M(N) em ≡ −igsZd4xψ(N) f(x)A(N) µ,α(x)tαγµψ(N) i(x),(5.1) where ψ(N) f(x), A(N) µ,α(x) and ψ(N) i(x) are the states of the outcoming quark, the emitted gluon and the incoming quark, respectively. The initial and final quark asymptotic states are denoted by ψ(0) i(x) and ψ(0) f(x), with momentum, spin and color (p0,s0,a0) and (pn,sn,an), respectively ψ(0) i(x)≡smq p0 0 ua0 s0(p0)e−ip0·x, ψ(0) f(x)≡rmq p0 n uan sn(pn)e−ipn·x,(5.2) where we used spinor conventions given at Appendix A and mqis the quark mass. The gluon final asymptotic state is written A(0) µ,αn(x), with momentum, polarization and color (kn, λn, αn) A(0) µ,αn(x)=Nkλn µ,αn(kn)e−ikn·x,(5.3) 134 High energy emission in QCD where the normalization, given in Gaussian units, reads Nk=√4π/2ω,ωis the gluon energy and the polarization vector reads λn(kn). The high energy states at any coordinate xunder the effect of the Nsources of the medium can be written as a superposition of diffracted states of the form ψ(N) i(x)=ψ(0) i(x)+Zd3pk (2π)3smq p0 k uak sk(pk)e−ipk·xM(N) q(pk,p0;zk,z1)aksk a0s0 ,(5.4) sum over repeated indices assumed. The matrix elements of M(N) q(pk,p0;zk,z1) are the beyond eikonal elastic amplitude (4.91) of finding the quark in the state (pk,sk,ak) after traversing the colored matter in the interval (z1,zk). These amplitudes carry local and ordered longitudinal phases, responsible of regulating the coherence in the squared amplitude leading to the LPM effect. Similarly for the final quark we write ψ(N) f(x)=ψ(0) f(x)+Zd3pk (2π)3smq p0 k uak sk(pk)e−ipk·xM(N) q(pk,pn;zk,zn)aksk ansn .(5.5) On contrary to QED case (3.10), in QCD the gluon is allowed to reinteract with the medium once it is emitted, then we write also for its state A(N) µ,α(x)=A(0) µ(x)δαn α+Zd3kk (2π)3Nkλk µ(k)e−ik·xM(N) g(kk,kn;zk,zn)akλk afλf ,(5.6) where the matrix elements of M(N) g(kk,kn;zk,zn) are the beyond eikonal elastic amplitude (4.91) of finding the gluon in the state (kn, λn, αn) after traversing the colored matter in the interval (zk,zn). The emitted gluon will be considered massive, with a plasma frequency given by ωp=mgto account for medium effects in the dispersion relation. Then the gluon 4-momentum is denoted by k=ω(1, βkk), where the gluon velocity is given by βk=q1−m2 g/ω2.(5.7) The longitudinal polarization related to mg,0 will be, however, neglected. The effective gluon mass can be considered of the order of µd, the effective screening mass of the external field of the medium [132, 142–145]. As with the QED analogous (3.10), the insertion of (5.4), (5.5) and (5.6) in (5.1) produces a term containing the three unscattered states, representing the emission amplitude in the vacuum M(0) em =−igsNkZd4xψ(0) f(x)A(0) µ,α(x)tαγµψ(0) i(x) (5.8) =−igsNk(2π)4δ4(pn+kn−pi)tαnan a0rmq p0 n ¯usn(pn)γµλn µ(k)us0(p0)smq p0 0 , Amplitude 135 which vanishes due to energy momentum conservation. This diagram has to be subtracted from the full amplitude in order to avoid spurious divergences in the evaluation of the non-perturbative scattering amplitudes. Then we define the amplitude of emission while interacting with the medium. By using S=1+M and inserting (5.4), (5.5) and (5.6) in (5.1) and subtracting (5.8) we obtain M(N) em −M(0) em =−igsNkZd3pk (2π)3Zd3kk (2π)3Z+∞ −∞ dz exp −iq(p,p+k)zk ×fλk s0 ksk(pk,pk+kk)S(N) g(kn,k;zn,zk)αn αk (5.9) ×S(N) q(pn,p;zn,zk)ansn a0 ks0 ktαka0 k akS(N) q(pk+kk,p0;zk,z1)aksk a0s0−M(0) em, where the quark and gluon elastic amplitudes Sqand Sgare the respective beyond eikonal and ordered evaluations outlined at (4.94). The shorthand notation fλk s0 ksk(pk,pk+kk) refers to the emission vertex, which factorizes and reads fλk s0 ksk(pk,pk+kk)≡smq p0 0−ω¯us0 k(pk)λk µ(kk)γµusk(pk+kk)smq p0 0 .(5.10) If pz p0 0 (pt) denotes the longitudinal momentum of a parton state with energy p0 0 and transverse momentum pt, the local longitudinal phase arising at the emission point is given by q(pk,pk+kk)≡pz p0 0−ω(pt k)+kz ω(kt k)−pz p0 0 (pt k+kt k), q(pk−kk,pk)≡pz p0 0−ω(pt k−kt k)+kz ω(kt k)−pz p0 0 (pt k),(5.11) depending on which of the intermediate quark momenta pk, either at (5.4) or (5.5), is chosen to perform the remaining integral in (5.9). The relation of (5.11) with the pole of an off-shell quark propagator for the leg after or before the emission can be seen by taking the high energy limit, βk1 and βp1, respectively, where βpis the velocity of the quark in the QCD medium. Indeed we find qz(p,p+k)≃ − ω 2p0 0(p0 0−ω)m2 q+ pt−p0 0−ω ωkt!2−m2 g 2ω≃ −kµpµ p0 0 ,(5.12) if we carry the integration with p, the momentum of the leg after the emission, of modulus βpp0=βp(p0 0−ω), or qz(p−k,p)≃ − ω 2p0 0(p0 0−ω)m2 q+ pt−p0 0 ωkt!2−m2 g 2ω≃ − kµpµ p0 0−ω,(5.13)