Covariant amplitude decomposition in relativistic fermion scattering problems

Covariant amplitude decomposition in relativistic fermion scattering problems

A parameterization of on-mass-shell relativistic fermion scattering amplitudes by a set of 4 covariant amplitudes is proposed, which in the non-relativistic limit turn to coefficients of the matrix amplitude decomposition over the unity and Pauli matrices, and in the ultra-relativistic limit – to sy...

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Veröffentlicht in:Вопросы атомной науки и техники
Datum:2007
1. Verfasser: Bondarenko, M.V.
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Sprache:English
Veröffentlicht: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
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Elementary particle theory
Online Zugang:https://nasplib.isofts.kiev.ua/handle/123456789/110941
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Zitieren:Covariant amplitude decomposition in relativistic fermion scattering problems / M.V. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 104-110. — Бібліогр.: 11 назв. — англ.

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spelling Bondarenko, M.V.
2017-01-07T09:25:30Z
2017-01-07T09:25:30Z
2007
Covariant amplitude decomposition in relativistic fermion scattering problems / M.V. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 104-110. — Бібліогр.: 11 назв. — англ.
1562-6016
PACS: 11.80.Cr, 13.60.-r, 13.66.-a, 13.75.Cs
https://nasplib.isofts.kiev.ua/handle/123456789/110941
A parameterization of on-mass-shell relativistic fermion scattering amplitudes by a set of 4 covariant amplitudes is proposed, which in the non-relativistic limit turn to coefficients of the matrix amplitude decomposition over the unity and Pauli matrices, and in the ultra-relativistic limit – to symmetrized helicity amplitudes. In the general rela-tivistic case, the covariant amplitudes express as spurs of the matrix amplitude supplemented by γ-matrix factors not exceeding 3-rd degree. Algebraic computation of such spurs provides a comparatively short and fully covariant approach for calculation of fermion scattering processes, allowing account for all polarization observables. For extension of the method to problems of two-fermion scattering, when permitted are both ways of transition 1→3, 2→4 and 1→4, 2→3, relativistic on-mass-shell Fierz relations interconnecting the two possible definitions of transition amplitudes are derived, under simplifying assumptions of equal fermion masses and scattering elasticity. Eigenfunctions of the on-shell Fierz relations are constructed, and advantages of their use for automatic account for contributions from cross-diagrams are demonstrated with the example of Møller scattering.
Запропоновано параметризацію амплітуди розсіяння релятивістського ферміона на масовій поверхні набором 4 коваріантних амплітуд, які в нерелятивістському наближенні переходять у коефіцієнти розкладення матричної амплітуди по одиничній матриці та матрицям Паулі, а в ультререлятивістській границі – у симетризовані спіральні амплітуди. В загальному релятивістському випадку коваріантні амплітуди виражаються через шпури від матричної амплітуди з додатковими γ-матричними факторами, ступінь яких не перевищує 3. Алгебраїчне обчислення таких шпурів дає порівняно короткий та повністю коваріантний шлях розрахунку процесів розсіяння ферміонов, з урахуванням усіх поляризацій. Для узагальнення методу до задач розсіяння двох ферміонів, в умовах коли можливі обидва шляхи переходів 1→3, 2→4 та 1→4, 2→3, виведено релятивістські тотожності Фірца на масовій поверхні, які зв’язують визначення амплітуд переходу, за спрощуючих припущеннях рівності мас усіх ферміонів та пружності розсіяння. Побудовано власні функції тотожностей Фірца, та продемонстровано переваги їх використання для автоматичного врахування вкладів від перехресних діаграм на прикладі Мелеровського розсіяння.
Предложена параметризация амплитуды рассеяния релятивистского фермиона на массовой поверхности набором 4 ковариантных амплитуд, которые в нерелятивистском пределе переходят в коэффициенты разложения матричной амплитуды по единичной матрице и матрицам Паули, а в ультрарелятивистском пределе – в симметризованные спиральные амплитуды. В общем релятивистском случае ковариантные амплитуды выражаются через шпуры от матричной амплитуды, помноженной на γ-матричные факторы степени не выше 3. Алгебраическое вычисление таких шпуров дает сравнительно короткий и полностью ковариантный путь расчета процессов рассеяния фермионов, с учетом всех поляризационных наблюдаемых. Для обобщения метода на задачи рассеяния двух фермионов, в условиях когда допустимы оба пути переходов 1→3, 2→4 и 1→4, 2→3, выведены релятивистские тождества Фирца на массовой поверхности, связывающие два возможных определения амплитуд перехода, при упрощающих предположениях равенства масс всех фермионов и упругости рассеяния. Построены собственные функции тождеств Фирца, и продемонстрированы преимущества их использования для автоматического учета вкладов от кросс-диаграмм на примере Меллеровского рассеяния.
en
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
Вопросы атомной науки и техники
Elementary particle theory
Covariant amplitude decomposition in relativistic fermion scattering problems
Метод коваріантних амплітуд у задачах розсіяння релятивістських ферміонів
Метод ковариантных амплитуд в задачах рассеяния релятивистских фермионов
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Covariant amplitude decomposition in relativistic fermion scattering problems
spellingShingle Covariant amplitude decomposition in relativistic fermion scattering problems
Bondarenko, M.V.
Elementary particle theory
title_short Covariant amplitude decomposition in relativistic fermion scattering problems
title_full Covariant amplitude decomposition in relativistic fermion scattering problems
title_fullStr Covariant amplitude decomposition in relativistic fermion scattering problems
title_full_unstemmed Covariant amplitude decomposition in relativistic fermion scattering problems
title_sort covariant amplitude decomposition in relativistic fermion scattering problems
author Bondarenko, M.V.
author_facet Bondarenko, M.V.
topic Elementary particle theory
topic_facet Elementary particle theory
publishDate 2007
language English
container_title Вопросы атомной науки и техники
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format Article
title_alt Метод коваріантних амплітуд у задачах розсіяння релятивістських ферміонів
Метод ковариантных амплитуд в задачах рассеяния релятивистских фермионов
description A parameterization of on-mass-shell relativistic fermion scattering amplitudes by a set of 4 covariant amplitudes is proposed, which in the non-relativistic limit turn to coefficients of the matrix amplitude decomposition over the unity and Pauli matrices, and in the ultra-relativistic limit – to symmetrized helicity amplitudes. In the general rela-tivistic case, the covariant amplitudes express as spurs of the matrix amplitude supplemented by γ-matrix factors not exceeding 3-rd degree. Algebraic computation of such spurs provides a comparatively short and fully covariant approach for calculation of fermion scattering processes, allowing account for all polarization observables. For extension of the method to problems of two-fermion scattering, when permitted are both ways of transition 1→3, 2→4 and 1→4, 2→3, relativistic on-mass-shell Fierz relations interconnecting the two possible definitions of transition amplitudes are derived, under simplifying assumptions of equal fermion masses and scattering elasticity. Eigenfunctions of the on-shell Fierz relations are constructed, and advantages of their use for automatic account for contributions from cross-diagrams are demonstrated with the example of Møller scattering. Запропоновано параметризацію амплітуди розсіяння релятивістського ферміона на масовій поверхні набором 4 коваріантних амплітуд, які в нерелятивістському наближенні переходять у коефіцієнти розкладення матричної амплітуди по одиничній матриці та матрицям Паулі, а в ультререлятивістській границі – у симетризовані спіральні амплітуди. В загальному релятивістському випадку коваріантні амплітуди виражаються через шпури від матричної амплітуди з додатковими γ-матричними факторами, ступінь яких не перевищує 3. Алгебраїчне обчислення таких шпурів дає порівняно короткий та повністю коваріантний шлях розрахунку процесів розсіяння ферміонов, з урахуванням усіх поляризацій. Для узагальнення методу до задач розсіяння двох ферміонів, в умовах коли можливі обидва шляхи переходів 1→3, 2→4 та 1→4, 2→3, виведено релятивістські тотожності Фірца на масовій поверхні, які зв’язують визначення амплітуд переходу, за спрощуючих припущеннях рівності мас усіх ферміонів та пружності розсіяння. Побудовано власні функції тотожностей Фірца, та продемонстровано переваги їх використання для автоматичного врахування вкладів від перехресних діаграм на прикладі Мелеровського розсіяння. Предложена параметризация амплитуды рассеяния релятивистского фермиона на массовой поверхности набором 4 ковариантных амплитуд, которые в нерелятивистском пределе переходят в коэффициенты разложения матричной амплитуды по единичной матрице и матрицам Паули, а в ультрарелятивистском пределе – в симметризованные спиральные амплитуды. В общем релятивистском случае ковариантные амплитуды выражаются через шпуры от матричной амплитуды, помноженной на γ-матричные факторы степени не выше 3. Алгебраическое вычисление таких шпуров дает сравнительно короткий и полностью ковариантный путь расчета процессов рассеяния фермионов, с учетом всех поляризационных наблюдаемых. Для обобщения метода на задачи рассеяния двух фермионов, в условиях когда допустимы оба пути переходов 1→3, 2→4 и 1→4, 2→3, выведены релятивистские тождества Фирца на массовой поверхности, связывающие два возможных определения амплитуд перехода, при упрощающих предположениях равенства масс всех фермионов и упругости рассеяния. Построены собственные функции тождеств Фирца, и продемонстрированы преимущества их использования для автоматического учета вкладов от кросс-диаграмм на примере Меллеровского рассеяния.
issn 1562-6016
url https://nasplib.isofts.kiev.ua/handle/123456789/110941
citation_txt Covariant amplitude decomposition in relativistic fermion scattering problems / M.V. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 104-110. — Бібліогр.: 11 назв. — англ.
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fulltext COVARIANT AMPLITUDE DECOMPOSITION IN RELATIVISTIC FERMION SCATTERING PROBLEMS M.V. Bondarenco National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; e-mail: bon@kipt.kharkov.ua A parameterization of on-mass-shell relativistic fermion scattering amplitudes by a set of 4 covariant amplitudes is proposed, which in the non-relativistic limit turn to coefficients of the matrix amplitude decomposition over the unity and Pauli matrices, and in the ultra-relativistic limit – to symmetrized helicity amplitudes. In the general rela- tivistic case, the covariant amplitudes express as spurs of the matrix amplitude supplemented by -matrix factors not exceeding 3-rd degree. Algebraic computation of such spurs provides a comparatively short and fully covariant approach for calculation of fermion scattering processes, allowing account for all polarization observables. For ex- tension of the method to problems of two-fermion scattering, when permitted are both ways of transition 1→3, 2→4 and 1→4, 2→3, relativistic on-mass-shell Fierz relations interconnecting the two possible definitions of transition amplitudes are derived, under simplifying assumptions of equal fermion masses and scattering elasticity. Eigenfunc- tions of the on-shell Fierz relations are constructed, and advantages of their use for automatic account for contribu- tions from cross-diagrams are demonstrated with the example of Møller scattering. γ PACS: 11.80.Cr, 13.60.-r, 13.66.-a, 13.75.Cs 1. INTRODUCTION In computations of scattering processes of relativis- tic fermions, when those emit or absorb several quanta, evaluation of spin matrix products consumes consider- able effort, especially if the goal is pursued to bring the result to a visual form. To that objective, there is no common approach, which might afford to completely rely on symbolic calculation with the aid of computer. Since the result of symbolic computation of a -matrix product (basing on algebraic relations, such as those of anticommutation) depends on the number of -matrices in the product, virtually, in a factorial manner, it is ob- vious, that the computation must be carried out not on the level of cross-sections, but on the level of ampli- tudes. But in the letter case, there arise questions, how to preserve covariance of the analysis and the correct number of spin degrees of freedom on the mass shell. For matrix element computation, a formidable number of approaches has been proposed [1-6], which footed on special forms of a basis for bispinors or/and Lorentz- vectors, or, as in [7,8], the bispinors were augmented to the form of density matrices, at the expense of appear- ing polarization-dependent denominators. This contribution offers formulation of an approach, in which no information on bispinor realizations is inv γ γ olved. Beginning with scattering of one fermion, the on- shell matrix element may always be parameterized as ( )uASIuMuuM fi 5γγ+′=′= µµ , (1) given that bispinors u , u ′ obey Dirac equations1 ( ) 0=−γ ump , ( ) 0=′−′′ mpu γ , and so are allowed for 2 spin degrees of freedom. The 1 The account for difference between initial and final fer- mion masses may be essential in problems of baryon excita- tion or change of flavour. component of vector , parallel to µA mpmp ′′+ µµ , does not contribute to (1) on account of the identity ( ) 05 ≡γγ′′+′ µµµ umpmpu , so the set of , contains the correct number of 4 linearly independent components. The choice of , as basic matrices on-shell is not the only possible one, but it is favored because in the non-relativistic limit the spatial part of is equivalent to Pauli matrices. S µA 5γγ I 5γγ The explicit formulas for evaluating covariant am- plitudes introduced in (1) read2 ( ) ( ; 4 11 mpMmpSp mmpp S +′+′ ′+′ = γγ ) (2) ( ) ( ) , 4 11 5 νµνµ γγγγ mpMmpSp mmpp GA +′+′ ′+′ = (3) where3 ( )( ) ( ) ( ) , 4 1 νµµννµ ννµµµνµν αα mpmpmpmp mpmpmpmpgG +′′++′′+ −′′−′′−= with arbitrary vector . The choice of α will further be referred to as “acceptance condition”. Two choices important for practice are µα µ         −= 2m ppgGp νµ µνµν ,         ′ ′′ −=′ 2m ppgp νµ µνµνG . (4) Also, if in some reference frame the scattering is elastic ( , ), it may be convenient to make mm =′ Epp ==′ 00 2 As is straightforward to check by substituting to (2), (3). 5γγASIM += 3 The vector coefficient at mpmp νν +′′ µA in (3) has no effect on the form of the vector and is chosen so that ten- sor is symmetric. µνG PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 104-110. 104 µνG have only spatial components non-zero. Then, G pr = = = = 2′− a ( )       Π+Π+′+Π= µνµνµνµν qrNE mE pp 22 2 4 1 4 1 , (5) where means projector on the vector indicated at its subscript ( µνΠ 2xxxx νµµν ≡Π p′ N q r ), , is the spatial part of vector (i. e., r , ), – a vector, orthogonal to , and to the time axis. According to (5), tensor G is diagonal in the orthogonal basis of , , . µµµ ppq −′= µp+ q r µν N µr 0 =µ 0 p+′ In the non-relativistic limit, at any of the accep- tances (4), (5), reduces to the vector coefficient of expansion of a 2 amplitude over Pauli matrices. µA 2× Amplitudes and carry the full information about the scattering, and have a “minimal” structure from the viewpoint of algebraic covariant computation. These are also well-suited for expression of scattering observables, as will be shown shortly. In what follows, we shall for simplicity put . S µA m m=′ With the given definitions, it is straightforward to find that the cross-section averaged over initial and final fermion spin states expresses only through squares of the covariant amplitudes: ( ) ( ) ( ) ( ) ( ) ( )     −+′     −+′     −+′ +γ+γ′=′ νµµν− ′ *122 222 222 2 2 1 2 1 EEE p p AAGSmpp ASmpp ASmpp MmpMmpSpMuu (6) (positive definiteness of final expressions is secured by vectors , , being space-like). If, further, fermion polarization effects are concerned, a correspon- dence of the covariant amplitudes with mean polariza- tion vectors or spin asymmetries referred to some quan- tization axes has to be established, as is done below. pA pA ′ EA 2. EXPRESSION OF ONE-FERMION POLARIZATION CHARACTERISTICS 2.1. The generic quantity carrying information about all polarization effects in terms of particle polarization vectors is the scattering amplitude square averaged over the density matrices with arbitrary polarization 4-vectors , ( pa , , , ): µa µa′ 0= 12 ≤− a 0=′′ap 1≤ aa Muu ′ ′ , 2 ( )( ) ( )( ) ( ) ,1 1 2 11 2 1 2 55 νµ ′→ µν ′ νν ′ µ ′→ µ ′ ′+′−−′= γγ++γγγ′++γ′= aahagafMuu MampMampSp ppppp (7) where coefficients , , are assumed to be orthogonal to the momentum , and notation µ pf ′ ν pg ′ µν ph ′ ′p ( )µµµµ pp mpp apaa p ′+ +′ ′ −=′→ 2 , ( ) (8) 0=′′→ pa p is utilized for the initial polarization 4-vector a , trans- formed by a straight Lorentz boost to position . In terms of coefficients , and , the final polariza- tion 4-vector may be retrieved as p′⊥ f g h ( ) p pppaa a fin fa ahg a Muu Muu a ′→ µ ′→ µν ′ ν ′ ν ′ν − − = ′δ ′δ ′ −=′ 1 1 , 2 2 . For expression of coefficients (7) in terms of covari- ant amplitudes, it is natural to adopt acceptance ⊥ . Computation from the definition (7) then gives p′ 4 ( ) ; 1Re2 22 ** p ppp p AS AApi m AS f ′ ′′′ ′ − ′− = λκρρκλµµ µ ε (9) ( ) ; 1Re2 22 ** p ppp p AS AApi m AS g ′ ′′′ ′ − ′+ = λκρρκλνν ν ε (10) ( ) ( ) .*Im2Re2 * 22 122   ′ε−−          +      −= λ ′ κκλµνν ′ µ ′ µν ′′ − ′ µν ′ ppp pppp ASp m AA gASASh (11) In appearance, the above expressions are straightfor- ward generalizations of known non-relativistic ones. Since the number of the observables 2Muu ′ µA , , , is 16, exceeding the number of real pa- rameters in the 4 complex amplitudes S , , measur- able, moreover, up to an overall phase, the polarization observables must obey some identities. A possible way to express those is to write µ pf ′ ν pg ′ µν ph ′ ( )( ) ( )( )λ ′ λ ′ λµ ′ µλ ′ µ ′ µ ′ λλ ′ +−=−+ ppppppp gfhhgfh1 , ( ) ( )( ) ( )( ) ( ) .0 1 1 1 2 2 2 2 2 = + ′ε′ε+++ + + ′′−−+ +        ′′ − λλ ′ βγ ′ ααβγνστ ′ ρρστµν ′ ν ′ µ ′ µ ′ λλ ′ νµµνλλ ′ νµ ′ µν ′νµ µν p pppppp p ppp h hphp m gfgf h mppghhh m ppg µ In turn, from the measured analyzing powers , and spin correlation coefficients , covariant ampli- tudes in acceptance can be extracted up to their overall phase as follows: pf ′ ν pg ′ µν ph ′ p⊥ 4 1 2 2 2 λλ ′+ +′ ′ = ph mpp Muu S , 4 The calculation is a simple matter when using the identity ( )( )( )( ) ( )( )( )525 121 γγγγγγγγ pampmppmpampmp ′→++′+′=+′+++′ . 105 4 1 2 2 * στ ′ ρρστµµ ′ µ ′µ ′ ′ε++ +′ ′ = ppp p hpi m gf mpp Muu AS ; ( )( ) 2*** SASSAAA pppp ν ′ µ ′ ν ′ µ ′ = . 2.2. Among spin bases employed for description of relativistic fermion polarization phenomena, spiral one is among most widespread, especially in the high- energy limit of gauge theories, where helicity is virtu- ally con-served. To express helicity amplitudes in terms of covariant ones, it is appropriate to adopt elastic ac- ceptance (5) and the “low-energy” representation for - matrices and bispinors: γ       − + =γγ+ σA σA E E E S S ASI 0 05 ; ;      − += λ λ λ λχ χ mE mEu ( λ ); ±=λ′, ( )+λ′+ λ′λ′ χ′λ′−−χ′+=′ mEmEu , . For definition of Pauli spinor components and overall phases, choose a spatial coordinate frame with axis along vector r , and along . Then, x pp ′+= y ppq −′=         −+ −=σ+σ+σ= Nqr qrN Nqr E AiAA iAAAAAA 321σA , and in the phase convention of [9], ; 2 1 4 4 ∗ +−+ ′=     = χχ θ θ i i e e ∗ −θ− θ − χ′=       −=χ 4 4 2 1 i i e e ( θ is the scattering angle, being, in the chosen frame, the azimuthal angle with respect to -axis). Computing matrix elements with the given entries yields z ( ) ( ) ;22 ,22 q r mAiMM EAMM =− =− −++− −−++ (12a) ( ) ( ) . 2 cos2 2 sin22 ; 2 sin2 2 cos22 N N AmSiEMM AiESmMM θθ θθ −−=+ +=+ +−−+ −−++ (12b) In view of (12a), vector amplitude components , admit direct interpretation as P- and T-asymmetries of helicity amplitudes rA qA 5, but and are related with S NA ( ) 2−−+ + M+M and ( ) 2+ MM S +−−+ via a matrix, unitary up to a common factor, which gets diagonal in the massless limit only (then, is a helicity-flip, – helicity non-flip amplitude). NA 5 At a different choice of basis helicity states, the helicity amplitudes would enter (12a,b) with additional phases. For example, in the convention of [10], ∗ +−+ ′=      = χχ θ 2 1 2 1 ie , ; 12 1 2 ∗ −− ′=     −= χχ θie ( ) ( ) . 2 sin2 2 cos22 ;22 22 22 Nii rii AiESmeMeM EAeMeM θθθθ θθ +=+ =− − −−++ − −−++ 2.3. In the simplest physical situation, when the fer- mion scatters due to absorption of a single (virtual) pho- ton with polarization , by virtue of parity and gauge invariance of electromagnetic interaction, the matrix amplitude admits parameterization νe ( ) ( ) ( ) γ+ ′+ ′+ −= e pp eppmM mme FFF2 2 . , (13) where , are functions of , real in the domain and referred to as charge and magnetic form-factors. Such not a self-evident fashion of parameterization proves advantageous because and appear to not interfere in the cross-section av- eraged over fermion spins – in contrast to form-factors introduced in a fashion suggested by perturbation theory eF p m mF ( −′ p ( 2pp −′ ) ) 02 < eF F λκκλ ⋅ σ−γ= qe m eM 21 F 2 1F . In this regard, it is instructive to compute the covariant amplitudes by substituting (13) into (2), (3). One finds: ( ) ;F 2 empp eppmS +′ ′+ = m mpp eppiA F2+′ ′ε = νβαµαβν µ (in any of acceptances (4) or (5)). Recalling (6), the non-interference of and in the averaged cross- section should be regarded as natural result. eF mF If polarization is linear, the direction of vector serves as a physical quantization axis. Manifestly, if designates a unit real vector in the space-like direc- tion of (so that ), the matrix commutes with both νe A µN µA µµ = NAA N 5γγN γp and γp′ I= , and on account of the identity ( , its eigenvalues equal ±1. Thus, it is possible to pick an initial spin basis u and a final basis u so that Nγγ ′ 25 ) σ σ′ σσ σ=γγ uuN 5 , σ′σ′ ′σ′=γγ′ uNu 5 , ( ) ±=σ′σ, and ( ) +−−+−+ ′==γγ−γγ′=′ uuuNNuuu 055 . Then, ( ) ( )NASuuuASIu σ+′δ=γγ+′ σσσσ′σ µµ σ′ 5 . Owing to reality of the form-factors, thereat , so the polarizing effect is absent and the vector amplitude acts merely as an axis of spin precession. However, for the problem of scattering in a strong, centrally-symmetric electrostatic field, the struc- ture of invariant amplitudes remains similar (with substituted by the time axis direction), but these become energy-dependent and are no longer real. Under such conditions , so the polarizing effect may be essential. |||| NN ASAS −=+ || NAS + νe || NAS −≠ If, moreover, the field is not centrally-symmetric, or the process of scattering is accompanied by radiation with substantial recoil, the direction of the vector ampli- tude is no longer fixed by kinematics alone. In this event, the quantization axis direction can be straight- forwardly computed via formula (3). 106 3. TWO-FERMION COVARIANT AMPLITUDES. RELATIVISTIC ON-SHELL FIERZ RELATIONS Generalization of the covariant amplitude approach to the case of collision between two fermions can be made by introducing parameterizati n of the on-shell matrix element like o ( ){ ( ) ( ) ( ) } 1242 5 31 5 42,3142 5 31424231 5 31423142,31341234 uuBIAIAIISuuuMuuuM fi γγγγ+γγ+γγ+== νµµνννµµ , (14) where subscript 31 at matrices , indicates that those act from spin state of the particle with momentum into spin state of the particle with momentum , and similarly for subscript 42. I 5γγ 1p 3p Convenience of actual use of the representation (14) depends on structure of the matrix . If , where is a shorthand for all variables of intermediate particles exchanged between fer- mions, then covariant amplitudes in (14) are evaluated by formulae trivially generalizing those of one-particle theory: M ( ) ( )∫ ΞΞΞ= 4231 MMdM Ξ ( )( ) ( ) ( )( ) ( ) ( )( 224244113133 24241313 42,31 4 1 4 11 mpMmpSpmpMmpSpd mmppmmpp S +γΞ+γ⋅+γΞ+γΞ ++ = ∫ µ ν µν ) r r (15) and etc. for , , . The cross-section is correspondingly formed by a formula generalizing (6). 31A 42A 4231,B However, if there are two cross-subprocesses interfering in the matrix scattering amplitude: , (16) ( ) ( ) ( ) ( )∫∫ ΞΞΞ+ΞΞΞ= 32414231 MMdMMdM direct calculation of coefficients in (14) with the second term of (16) yields spurs which do not factorize, unlike (15), and straightforward algebraic expansion of such formidable spurs leads to improperly extensive results. In those cir- cumstances, it is more practical to compute contribution from this cross-subprocess in its natural ep esentation ( ){ ( ) ( ) ( ) } 1232 5 41 5 32,4132 5 41323241 5 41324132,41341234 uuBIAIAIISuuuMuuu γγγγ+γγ+γγ+= νµµνννµµ , (14a) whereas contribution from the first part of (15) – be still computed in representation (14). However, when doing so, to analyse quantitatively the sum of such amplitudes, it should be brought to some common basis. It is not obvious to propose any third basis except (14) and (14a), so there must be derived a kind of Fierz relations connecting the two possible representations. In discussion of Fierz relations between on-mass-shell relativistic fermions, we confine ourselves to the case of elastic scattering of two equal-mass particles6, i. e. assume identities µµµµµ Ppppp def =+=+ 4321 , . mmmmm def ==== 4321 When interrelating the amplitudes, it is expedient to adopt for them some common acceptance, a natural choice of which, by symmetry reasons, is (analog of (5)). The resulting spin crossing relations are quoted below: P⊥ ( ) ( )κκγβααβγκ +ε+      −=+ 32412343241 2 31 2 41 22 4231 4 13 4 2 12 AApppmiSppPmSpp ,, ; 2 12 32,41 2 31 2 41 22 κλκλκλ BppGPm NP       Π−− ( ) ( ) 32412344231 4 13 8 ,SpppmiAApp γβαµαβγµµ ε=++ ( )( )κκµκµκ +Π−+ 3241 2 31 2 41 224 AAppGPm NP ( );4 32,4132,41234 λκκλγβαλαβγµκ ε BBpppimGP +− ( ) ( ) ( ) ( )[ ] 3241 2 13 2 14 2 31 2 4142314231 4 13 2 ,,, SGppppppBBpp PN µνµννµµν ++−Π=++ ( )( )κκγβαµαβγνκγβαναβγµκ +ε+ε+ 32412342344 AApppiGpppiGm PP ( )[ ] ;4 32,41 2 31 2 41 22 κλκλµνκλµνκνλµλνκµ BGppGGGGGGPm NPPPPPPP Π−−++ , ( ) ( ) ( )( ) κλδµδκλκκκµκµµµ ε+−−−=−+ 32413241413131414231 2 13 2 ,BPmiAAppppAApp , ( ) ( ) ( ) ( )κκγβααβγκνµνµτµνκτνµµν −    ε−+ε=−+ 324123441313141242,3142,31 2 13 12 AApppipppp mP PmiBBpp ( ).1 32,4132,4131412 λκκλτσρρστκδδλµν −εε+ BBppPP P Here , , and 1331 ppp −= 1441 ppp −= ( )       Π+Π++Π= αβαβαβαβ 412312 2 13 1 4 1 Pm ppNPG (cf. (5)). 6 It is worth reminding, that understood in approximate sense, equality of masses does not necessarily imply identity of parti- cles. The typical example is proton and neutron, and, to a lesser accuracy, Λ -hyperon. 107 Before addressing specific models for particle inter- action, the Fierz relations allow one to establish conse- quences from the suggested symmetry properties. First of all, it is apparent, that amplitudes 4231,S , , (17) µµ 4231 AA + νµµν 42314231 ,, BB + and µµ 4231 AA − , (18) νµµν 42314231 ,, BB − constitute Fierz subgroups. The physical meaning of combinations (17, 18) is that amplitudes (17) are sym- metric under the simultaneous permutation (1↔2, 3↔4), and conserving the initial wave function permu- tational symmetry, whereas amplitudes (18) are anti- symmetric and permutational symmetry changing. For collision of identical fermions, the permutational sym- metry of initial and final states is obliged to be the same (negative), then, only amplitudes (17) remain. Consequences from discrete space-time symmetries are also easy to deduce. Р-invariance of the matrix am- plitude permits covariant amplitudes to have only com- ponents 4231,S , , , , , (19) NA31 NA42 NN ,B 4231 N,N ,B ⊥⊥ 4231 which upon the cross-transformation turn to a set of similar structure: , , , , . 3241,S NA41 NA32 NNB 3241, NNB ⊥⊥ , ,3241 As for T-invariance, in analogy with the non- relativistic case [11], one can infer that a scattering sub- process, during which continuously 1→3, 2→4 (the first term of (15), contributes only to amplitudes 4231,S , , , , . (20) 31 31 ⊥A 31 42 ⊥A 3131 4231 , ,B 3131 4231 ⊥⊥ , ,B A cross-subprocess (the second term of (15)), accord- ingly, yields amplitudes , , , , , which upon crossing transformation turn to 3241,S 41 41 ⊥A 41 32 ⊥A 4141 3241 , ,B 4141 3241 ⊥⊥ , ,B 4231,S , , ,41 42 41 31 ⊥⊥ + AA 4141 4231 , ,B ( ) 4141 42314231 ⊥⊥+ , ,, BB νµµν , 41 42 41 31 AA − , . (21) 4141 4231 4141 4231 , , , , BB ⊥⊥ − − αβδµε mP In the sum of (20) and (21), forbidden are only 3 ampli- tudes, namely, , and . 31 42 31 31 AA − 31,41 42,31 41,31 42,31 BB + 31, 42,31 ,31 42,31 NN BB − In the maximally restricted case, when both P- and T-invariance apply and the colliding fermions are iden- tical, there remain only 5 non-zero amplitudes: S , , , , (direct analogs of Wolfenstein’s parameters known in the non-relativistic case [11]). In their terms, the averaged over all fermion spins matrix element square is obtained to be 4231, NN AA 4231 + NNB , ,4231 3131 4231 , ,B 4141 4231 , ,B ( )       ++++= 2 4231 2 4231 2 4231 22 13 2 1234 2 1 NNNN BAASmpp uMuuu ,, . (22) BPBm , , , , 24141 4231 423131 4231 4 4 4 ++ 4. FIERZ-INVARIANT AMPLITUDES AND ACCOUNT FOR CONTRIBUTIONS FROM CROSS-DIAGRAMS Besides amplitudes (17-18), symmetric and antisym- metric under the simultaneous permutation (1↔2, 3↔4), i. e. physical permutation of particles, of further use can be amplitudes of definite symmetry with respect to separate permutations 1↔2 and 3↔4. At determining such a symmetry, it should be minded that permutation 1↔2 or 3↔4 brings covariant amplitudes to a cross- representation, so to compare the result with the initial expression, it needs to be turned back to the initial rep- resentation, with the aid of Fierz identities. Note, that in the non-relativistic case, amplitudes of definite permuta- tional symmetry both in the initial and in the final state admit simple physical interpretation – those are transi- tion amplitudes between states with definite values of the two-fermion total spin. In terms of covariant ampli- tudes, those classes, accordingly, come as a scalar, two vector and a tensor combinations7 – with constant coef- ficients, granting that non-relativistic spin does not de- pend on momentum. In the relativistic case, inception of dependence on momenta deprives total spin of its absolute sense as a vector, but concepts of permutational symmetry and correspondent state multiplicity remain valid. Based on on-shell Fierz identities, one can pick a full set of “minimal” cross-symmetric and antisymmetric covari- ant amplitude combinations, securing them to be non- interfering in the spin-averaged cross-section. Below quoted are expressions for singlet-singlet and triplet- triplet amplitudes, those which stay non-zero under Р-, Т-invariance and identity of colliding particles: ( ) ( ) ( ) ( ) ;4 4 41,41 32,41 231,31 32,41 2 32,4132,41 2 14 41,41 42,31 231,31 42,31 2 42,3142,31 2 13 BmBPBSpp BPBmBSppU NN NN −−++−≡ ++++= (23a) ( ) { ( ) ( ) { ( ) ;1 1 32412342 32,4132,41 2 14 42312342 42,3142,31 2 13   ++ −+≡   + −+= κκδβααβδκ µµδβα ε AAippp mP BSpp AAippp BSppW NN NN N (23b) ( ) { ( ) ( ) { ( ) ,4 4 32412342 31 2 41 32,4132,41 2 14 42312342 31 2 41 42,3142,31 2 13     +ε− −+−≡     +ε+ −+= κκδβααβδκ µµδβααβδµ AAippp pp m BSpp AAippp pp m BSppW NN NN A (23c) 7 The latter tensor part virtually appears in terms of a sca- lar, a vector (equivalent to antisymmetric tensor), and a sym- metric tensor with zero trace. 108 ( ) ( ) ( ) ( ) ;4 4 41,41 32,41 231,31 32,41 2 32,4132,41 2 14 41,41 42,31 231,31 42,31 2 42,3142,31 2 13 BmBPBSpp BPBmBSppW NN NN NQ −−++≡ −−++= W= (23d) 4141 3241 23131 3241 24141 4231 23131 4231 2 44 , , , , , , , , BmBPBPBmWQ −≡−= . (23e) The spin-averaged cross-section (22) rewrites through the above amplitudes as ( ) ( ) . 42 1 8 1 16 1 16 1 22 31 2 41222 2 13 2 14 2222 1234         + ++ + ++= AN QNQ WppWPm pppp WWUuMuuu (24) It appears, that Fierz-invariant amplitudes can sup- ply an efficient tool for performing calculations, be- cause contribution to them from any Feynman diagram with continuous fermionic lines may be calculated via that representation of covariant amplitudes, which cor- responds to the order of fermionic transitions in this diagram. Moreover, diagrams differing only by crossing of fermion ends need not actually be calculated both – with contributions from one diagram evaluated, it suf- fices to make momentum and charge substitutions to recover that from the other. To illustrate efficiency of this way of conducting calculations, it may be applied to scattering of two elec- trons via one-photon exchange (Møller scattering). In Feynman’s gauge, the corresponding matrix amplitude is [9,10]: ( ) ( ) ( ) ( ) 2 41 3241 2 31 4231 pp M µµµµ γγγγ −= . (25) Computing Fierz-invariant amplitudes for the first term of (25) and deducing those from the other, one finds ;114 2 41 2 31 12         += pp ppU ;11444 2 41 2 31 122 41 13 2 31 14         −=−= pp pp p pp p ppWNQ 211 =+=QW ; ( ) ( ) ( ) ( ) ;114 22 2 41 2 31 12 2 41 2 14 2 13 2 31 2 13 2 14 NQ N W pp pp p pppp p pppp W =         −= +−+ − +−+ = ( ) ( ) ( ) ( ) ( ) . pp Pm p pppp p ppppWA                 +++−= +++ − +++ −= 2 41 2 31 22 2 41 2 13 2 14 2 31 2 13 2 14 11212 22 Note that all those quantities stay finite in the limit , whereas contribution to (24) from W in this limit is vanishing. In the opposite, non-relativistic limit, achieved through , the main contribution to (24) comes from U (singlet Coulomb scattering) and from W (triplet Coulomb scattering). 0→m N 04131 →p,p γ NNQ As is observed, expressions for cross-invariant am- plitudes (even those resulting from individual diagrams) prove to be more simple then the covariant amplitudes they add up of in (23). Computation of loop and weak corrections to such expressions can also be facilitated. 5. SUMMARY The main objective of this note was to demonstrate, that reducing the problem of calculation of relativistic massive fermion scattering observables to evaluation of the introduced covariant amplitudes, in a proper accep- tance, is advantageous from the most points of view. In the one-fermion case, it requires computation of spurs of the matrix amplitude multiplied by polynomials of -matrices not exceeding 3 degree. Such spurs may be calculated using conventional algebraic techniques. The relation with the scattering observables is also well suited – the scattering cross-section averaged over ini- tial and final fermion polarizations expresses through the scalar and the vector amplitude squares. In case if the vector amplitude is proportional to a real vector, the latter serves as a physical quantization axis. In most important limiting cases, the covariant amplitudes re- duce to conventional basis sets: in the non-relativistic limit the 4-vector amplitude turns to the 3-vector one, in the ultra-relativistic limit it turns to spin-non-flip ampli- tudes, and the scalar amplitude – to the spin-flip one. γ For two fermion scattering, the generalization of the covariant approach is rather straightforward, however, a complication arises if the amplitude is an interference of two cross-running subprocesses. Then, it is preferred to compute contribution from each cross-diagram in the spin representation, corresponding to the order of con- tinuous fermionic transitions in it. For bringing the quantities so evaluated to a common basis, relativistic on-mass-shell Fierz relations must be derived (herein presented for simplifying conditions of equal fermion masses and absence of accompanying hard radiation). At practice, performing computations may turn out to be more simple via cross-invariant amplitudes – eigen- functions of the on-shell Fierz relations. In the consid- ered example of Møller scattering, expressions for the cross-invariant amplitudes in arbitrary kinematics ac- quire very simple structure. In applications, it should be minded, that specific composition of the matrix amplitude , as it results from Feynman diagrams or non-perturbative dynamical solutions, may suggest basic -matrix structures, dif- ferent from (1) (or (14)). Some of such self-suggesting bases may even occur to be mutually orthogonal, so that the cross-section averaged over the fermion spins ex- presses through squares of those amplitudes, as good as through (6). If that degree of knowledge about scat- tering is sufficient, there is no actual need for transfor- mation to amplitudes of the form (1). However, when M 109 there is also a need to estimate fermion polarization effects, the amplitudes (1) are convenient in view of their simple and transparent connection to polarization observables, so using them to encode the information about scattering may be worth doing. REFERENCES 1. A.A. Sokolov and I.M. Ternov. Radiation from Relativistic Electrons. New York: AIP, 1986, 312 p. 2. R. Gastmans and T.T. Wu. The Ubiquitous Photon: Helicity Method for QED and QCD, Oxford: Oxford University Press, 1990, 664 p. 3. H.E. Haber. Spin Formalism and Applications to New Physics Searches: Preprint SCIPP 93/49, (hep-ph/9405376), 1994, 83 p. 4. A.L. Bondarev. Methods of minimization of calcu- lations in high energy physics. hep-ph/9710398, 1997, 38 p. 5. M.V. Galynsky, S.M. Sikach. The diagonal spin basis and calculation of processes involving polar- ized particles. // Phys. Part. & Nucl. 1998, v.29, p. 469-518 (hep-ph/9910284). 6. V.V. Andreev. Spinor techniques for massive fer- mions with arbitrary polarization // Phys.Rev. D. 2000, v. 62, p. 014029-014042. 7. E. Bellomo. Sull’uso degli operatori di proiezione per ottenere gli elementi di matrice per particelle di spin ½. // Nuovo Cim. 1961, v. 21, p. 730-739. 8. H.W. Fearing and R.R. Silbar. Method for Expres- sing Dirac Spinor Amplitudes in Terms of Invariants and Application to the Calculation of Cross Sections // Phys. Rev. D. 1972, v. 6, p. 471-477. 9. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii. Quantum Electrodynamics. Oxford: Pergamon Press, 1982, 667 p. 10. J.D. Bjorken, S.D. Drell. Relativistic Quantum Me- chanics. New York, McGraw Hill, 1964, 304 p. 11. L. Wolfenstein and J. Ashkin. Invariance Conditions on the Scattering Amplitudes for Spin ½ Particles. // Phys. Rev. 1952, v. 85, p. 947-949. МЕТОД КОВАРИАНТНЫХ АМПЛИТУД В ЗАДАЧАХ РАССЕЯНИЯ РЕЛЯТИВИСТСКИХ ФЕРМИОНОВ Н.В. Бондаренко Предложена параметризация амплитуды рассеяния релятивистского фермиона на массовой поверхности набором 4 ковариантных амплитуд, которые в нерелятивистском пределе переходят в коэффициенты раз- ложения матричной амплитуды по единичной матрице и матрицам Паули, а в ультрарелятивистском преде- ле – в симметризованные спиральные амплитуды. В общем релятивистском случае ковариантные амплиту- ды выражаются через шпуры от матричной амплитуды, помноженной на γ-матричные факторы степени не выше 3. Алгебраическое вычисление таких шпуров дает сравнительно короткий и полностью ковариантный путь расчета процессов рассеяния фермионов, с учетом всех поляризационных наблюдаемых. Для обобще- ния метода на задачи рассеяния двух фермионов, в условиях когда допустимы оба пути переходов 1→3, 2→4 и 1→4, 2→3, выведены релятивистские тождества Фирца на массовой поверхности, связывающие два возможных определения амплитуд перехода, при упрощающих предположениях равенства масс всех фермионов и упругости рассеяния. Построены собственные функции тождеств Фирца, и продемонстрированы преимущества их использования для автоматического учета вкладов от кросс- диаграмм на примере Меллеровского рассеяния. МЕТОД КОВАРІАНТНИХ АМПЛІТУД У ЗАДАЧАХ РОЗСІЯННЯ РЕЛЯТИВІСТСЬКИХ ФЕРМІОНІВ М.В. Бондаренко Запропоновано параметризацію амплітуди розсіяння релятивістського ферміона на масовій поверхні на- бором 4 коваріантних амплітуд, які в нерелятивістському наближенні переходять у коефіцієнти розкладення матричної амплітуди по одиничній матриці та матрицям Паулі, а в ультререлятивістській границі – у симет- ризовані спіральні амплітуди. В загальному релятивістському випадку коваріантні амплітуди виражаються через шпури від матричної амплітуди з додатковими γ-матричними факторами, ступінь яких не переви- щує 3. Алгебраїчне обчислення таких шпурів дає порівняно короткий та повністю коваріантний шлях розрахунку процесів розсіяння ферміонов, з урахуванням усіх поляризацій. Для узагальнення методу до за- дач розсіяння двох ферміонів, в умовах коли можливі обидва шляхи переходів 1→3, 2→4 та 1→4, 2→3, ви- ведено релятивістські тотожності Фірца на масовій поверхні, які зв’язують визначення амплітуд переходу, за спрощуючих припущеннях рівності мас усіх ферміонів та пружності розсіяння. Побудовано власні функ- ції тотожностей Фірца, та продемонстровано переваги їх використання для автоматичного врахування вкла- дів від перехресних діаграм на прикладі Мелеровського розсіяння. 110 http://arxiv.org/find/hep-ph/1/au:+Galynsky_M/0/1/0/all/0/1 http://arxiv.org/find/hep-ph/1/au:+Sikach_S/0/1/0/all/0/1

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