A. F. Grillo

Tensor representations of conformal algebra and conformally covariant operator product expansion

Abstract A conformally covariant formulation of operator product expansion is discussed as an expansion of the product of two representations into a direct sum of irreducible representations. The basic irreducible representations are analyzed and classified. The isomorphism between the conformal algebra and the O (4, 2) algebra is used to obtain a manifestly covariant formalism. The implications of the isomorphism in the derivation of the representations is discussed. The covariant O (4, 2) formalism directly relates dominant terms to nondominant terms in the light-cone limit. The essential coincidence of the problem of a conformal covariant operator product expansion to the problem of determining the form of the three-point function is stressed, together with the relevance of a selection rule for two-point functions following from exact (not spontaneously broken) conformal covariance. The role of Ward identities in a conformal covariant scheme is pointed out, and the mathematical implications on the n -point functions from causality are described.

Conformal algebra in two space-time dimensions and the thirring model

SummaryThe-infinitesimal generators of the infinite-dimensional conformal algebra in two space-time dimensions are constructed from the stress tensor. The conformal covariance of the operator product expansion in the Thirring model is verified, but the vacuum state is found to be invariant only under theO2,2 subalgebra of the infinite conformal algebra.RiassuntoSi costruiscono i generatori infinitesimi dell’algebra conforme in due dimensioni spazio-temporali, algebra che risulta essere ad infiniti generatori, a partire dal tensore energia-impulso della teoria. Si verifica esplicitamente la covarianza conforme dell’espansione operatoriale nel modello di Thirring. Però si trova che lo stato di vuoto è invariante soltanto sotto la sottoalgebraO2,2 della completa algebra conforme.РеэюмеИэ тенэора напряжений конструируются бесконечно малые генераторы бесконечномерной конформной алгебры в двух пространственно-временных иэмерениях. Проверяется конформная ковариантность раэложения проиэведения операторов в модели Тирринга. Однако, обнаружено, что вакуумное состояние является инвариантным только относительно суб’алгебрыO2, 2 бесконечной конформной алгебры.

Analyticity properties and asymptotic expansions of conformal covariant Green's functions

SummaryConformal covariant Greens functions and operator-product expansions for more than two operators are discussed. Analyticity properties are investigated and asymptotic expressions derived. Extension inD-dimensional space-time is studied and the peculiar role ofD=2, both group-theoretically and as connected to infinite-momentum frames, is pointed out. The role of shadow singularities in relation to Euclidean metrics is discussed.RiassuntoSi discutono le funzioni di Green covarianti conformi e gli sviluppi operatoriali per più di due operatori. Si studiano le proprietà di analiticità e si danno espressioni asintotiche. Si costruisce la generalizzazione per spazio-tempo aD dimensioni, mettendo così in luce il ruolo peculiare diD=2, sia da un punto di vista gruppale sia in relazione a sistemi di riferimento di impulso infinito. Si discute il ruolo delle singolarità ombra in relazione alla metrica euclidea.РезюмеОбсуждаутся конформно-ковариантные функции Грина и разложения операторных произведений более чем двух операторов. Исследуются аналитические свойства и выводятся асимптотические выражения. Исследуется расширение вD-мерном пространстве-времени и отмечается особая рольD=2, как с точки зрения теории групп, так и в системах бесконечного импульса. Обсуждается роль теневых сингулярностей в связи с эвклидовой метрикой.

Properties of partial-wave amplitudes in conformal invariant field theories

SummaryAnalyticity properties of partial-wave amplitudes of the conformal groupOD,2, (D not necessarily integer) in configuration space are investigated. The presence of Euclidean singularities in the Wilson expansion in conformai invariant field theories is discussed, especially in connection with the program of formulating dynamical bootstrap conditions coming from the requirement of causality. The exceptional case ofD = 2 is discussed in detail.RiassuntoSi studiano le proprietà di analiticità delle ampiezze di onda parziale del gruppo conformeOD,2 (con D anche non intero). Si discute la presenza di singolarità euclidee sull’espansione di Wilson in teorie invarianti rispetto al gruppo conforme, in special modo in relazione alla possibile formulazione di condizioni di bootstrap dinamico che seguono dalla causalità. Si esamina in dettaglio il caso eccezionaleD = 2.РезюмеИсследуются свойств а аналитичности парциальных амплиту д конформной группы ОD,2 (причем, D не яв ляется обязательно ц елым числом) в конфигураци онном пространстве. Обсужд ается наличие эвклид овых сингулярностей в разложении Вильсона в конформно инвариан тных теориях поля. Особое внимание уделяется формулиро вке условий динамиче ского бутстрапа, возникающ их из требования причинности. Подробн о исследуется слулай D=2.

Canonical scaling and conformal invariance

Abstract Arguments are given to show that a strictly conformal invariant skeleton theory is compatible with the observed scaling only if it possesses an infinite number of local tensors of spin n = 2, 4, … and scale dimension l n = 2+ n , which are all conserved. The conclusion suggests that conformal invariance is spontaneously broken.

Improved light-cone expansion

Abstract An improved light-cone expansion is obtained by explicitly summing up towers of derivative operators which transform irreducibly under the conformal algebra. The improved expansion explicitly exhibits the causality restrictions and conformal invariance on the light cone.

Logarithmic scaling and spontaneous breaking

Abstract Indecomposable representations of dilatations allow for logarithms in scale invariant operator product expansion. We prove that in absence of spontaneous breaking, they are incompatible with conformal invariance and positivity.

Conformal symmetry at lightlike distances and asymptotic behaviour of electromagnetic form factors

SummaryUsing conformal symmetry for the operator product of the hadronic electromagnetic current and the pion interpolating field, at lightlike distances, we connect the behaviour of the structure function in the scaling limit to the on-mass-shell asymptotic behaviour of the electromagnetic pion form factor. Our results are based on the analysis of the dimension spectrum of local operators appearing in the operator product expansion and on the assumption of precocious scaling. As an application of our results, predictions on the asymptotic behaviour of the e.m. pion form factor as well as for the threshold behaviour of the structure function for the singly inclusive π production in e+e− annihilation, are given.RiassuntoUsando la simmetria conforme per il prodotto sul cono di luce della corrente elettromagnetica degli adroni e del campo interpolante del pione, si connette l’andamento asintotico del fattore di forma elettromagnetico con la corrispondente funzione di struttura. Il nostro risultato è basato sull’analisi dello spettro delle dimensioni degli operatori locali che compaiono nell’espansione operatoriale e sull’ipotesi di precocità delle leggi di scala. Come applicazione del risultato si fanno alcune previsioni sull’andamento asintotico del fattore di forma del pione e sull’andamento di soglia della funzione di struttura relativa al processo inclusivo e+e−→π+adroni.РеэюмеИспольэуя конформную симметрию для проиэведенипя операторов адронного злектромагнитного тока и пионного интерполяционного поля на свето-подобных расстояниях, мы свяэываем поведение структурной функции в масщтабном пределе с асимптотическим поведением на массовой поверхности злектромагнитного форм-фактора пиона. Нащи реэультаты основаны на аналиэе спектра раэмерности локальных операторов, появляюшихся при раэложении проиэведения операторов, и на предположении преждевременного подобия. Как применение нащих реэультатов, приводятся предскаэания для асимптотического поведения злектромагнитного форм-фактора пиона, а также и для порогового поведения структурной функции для π-рождения при аннигиляции е+е−.

Unitarity and Veneziano-like pion form factor

SummaryWe study a unitarization method in connection with the recently proposed Veneziano-like models for the pion electromagnetic form factor. Using as input the model proposed by Suura, good agreement is obtained with the experimental data in the timelike region.RiassuntoSi studia un metodo di unitarizzazione in connessione coi modelli alla Veneziano recentemente proposti per il fattore di forma elettromagnetico del pione. Usando come punto di partenza il modello proposto da Suura si trova un buon accordo con i dati sperimentali nella regione temporale.РезюмеМы исследуем метод унитаризации в связи с недавно предложенными моделями, подобными модели Венециано, для электромагнитного формфактора. Используя, как исходное, модель, предложенную Суура, получается хорошее согласие с экспериментальными данными во времени-подобной области.

TOWARD A UNITARIZATION OF THE VENEZIANO MODEL.

Abstract The N D method is used to constrain the parameters of the Veneziano model. A more refined procedure, preserving exact crossing, is noted.