By Stefan Scherer, Matthias R. Schindler

Chiral Perturbation concept, as powerful box conception, is a regularly accredited and good tested operating device, approximating quantum chromodynamics at energies good lower than ordinary hadron masses.

This quantity, in keeping with a few lectures and supplemented with extra fabric, offers a pedagogical creation for graduate scholars and beginners getting into the sector from similar parts of nuclear and particle physics.

Starting with the the Lagrangian of the powerful interactions and normal symmetry ideas, the fundamental suggestions of Chiral Perturbation thought within the mesonic and baryonic sectors are constructed. the appliance of those recommendations is then illustrated with a couple of examples. numerous workouts (81, with whole strategies) are integrated to familiarize the reader with precious calculational techniques.

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55. 1 Degenerate Ground States Before discussing the case of a continuous symmetry, we will first have a look at a field theory with a discrete internal symmetry. This will allow us to distinguish between two possibilities: a dynamical system with a unique ground state or a system with a finite number of distinct degenerate ground states. In particular, we will see how, for the second case, an infinitesimal perturbation selects a particular vacuum state. To that end we consider the Lagrangian of a real scalar field UðxÞ [8] 1 m2 2 k 4 LðU; ol UÞ ¼ ol Uol U À U À U ; 2 4 2 ð2:1Þ which is invariant under the discrete transformation R : U 7!

Z d d d À ijà ðxÞ Ã þ oxl 0 ¼ d4xeðxÞ ijðxÞ ð1:141Þ W½ j; jà ; jl Š: djðxÞ dj ðxÞ djl ðxÞ Since Eq. 141 must hold for any eðxÞ we obtain as the master equation for deriving Ward identities, ! d d d À jà ðxÞ Ã À ioxl ð1:142Þ jðxÞ W½j; jà ; jl Š ¼ 0: djðxÞ dj ðxÞ djl ðxÞ We note that Eqs. 142 are equivalent. As an illustration let us re-derive the Ward identity of Eq. 129 using Eq. 142. For that purpose we start from Eq. 134,   d3 W oyl Gl ðx; y; zÞ ¼ ðÀiÞ3 oyl à ; ; dj ðxÞdjl ðyÞdjðzÞ j¼0;jà ¼0;jl ¼0 apply Eq.

8 8 X ka X ka þ" qL eLa þ eL MqR À " qL M eRa þ eR qR 2 2 a¼1 a¼1 "   8 X ka ka ¼ Ài eRa " qL M qR þ eR ð"qR MqL À "qL MqR Þ qR MqL À " 2 2 a¼1 #   8 X ka ka þ eLa " qR M qL þ e L ð " ð1:110Þ qL MqR À " qL MqR À "qR MqL Þ ; 2 2 a¼1 which results in the following divergences,20   odLM ka ka ol Lla ¼ ¼ Ài " qL MqR À "qR M qL ; oeLa 2 2   odL k k M a a ¼ Ài " qR MqL À "qL M qR ; ol Rla ¼ oeRa 2 2 odL M ¼ Àið" qL MqR À "qR MqL Þ; ol Ll ¼ oeL odLM ol Rl ¼ ¼ Àið" qR MqL À "qL MqR Þ: oeR ð1:111Þ The anomaly has not yet been considered.

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