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un $\nu_{\mu}$ en un tiempo $t$ sería \end{equation} \subsubsection*{Neutrinos de Majorana o Dirac} -Las masas de los neutrinos aún no pueden entrar en el modelo estándar, no se entiende la razón de que tengan masas tan pequeñas respecto a las demás partículas elementales. Se propone por un lado la exitencia de un neutrino más masivo derivado de una teoría de gran unificación, en ese esquema las masas de los neutrinos del muón y tau son una razon de la masa de un leptón o el bosósn $W^{\pm}$ sobre la masa de ese neutrino masivo. Pero par que eso suceda se require que los neutrinos sean sus propias anti partículas, es decir, que sean neutrinos de Majorana. +Empezaremos con algunos que yo creo pueden parecer escandalosos, exagero pero es para ponerle emoción al tema. Iniciamos un poco el tema hablando de la ecuación de Dirac, la ampliación relativista a la ecuación de Schrödinger que además posibilita que se pueden definir ecuaciones de onda bien portadas. Derivado de ella vimos que aparecen las antipartículas. Y parece que todas las partículas de las que hemos hablado cumplen con esta ecuación, pero resulta que no existe ninguna prueba experimental de que los neutrinos respondan a esta ecuación. + +En 1937 E. Majrana propuso una nueva ecuación de onda que si puede describir a los neutrinos, con la diferencia de que neutrino y antineutrino son indistinguibles. + +Este poroblema con la ecuación de DIrac se traduce a que las masas de los neutrinos aún no pueden entrar en el modelo estándar, no se entiende la razón de que tengan masas tan pequeñas respecto a las demás partículas elementales. Se propone por un lado la exitencia de un neutrino más masivo derivado de una teoría de gran unificación, en ese esquema las masas de los neutrinos del muón y tau son una razon de la masa de un leptón o el bosósn $W^{\pm}$ sobre la masa de ese neutrino masivo. Pero para que eso suceda se require que los neutrinos sean sus propias anti partículas, es decir, que sean neutrinos de Majorana. Lo que tenemos ahora, donde los neutrinos son distinguibles de sus antipartículas, son los neutrinos de Dirac. @@ -597,17 +601,17 @@ Si experimentalmente pudieramos ver un doble decaimiento beta entonces se compro \subsection*{Interacción fuerte} -A la manera de la electrodinámica cuántica (\emph{QED}) se ha desarrollado una teoría para explicar las interaccions fuertes: la cromodinámica cuántica (\emph{QCD}). NO es que los cuarks estén relacionados con las propiedades ópticas de los materiales, si no que por la peculiaridad de ser una fuerza en la que existen tres cargas, y para llegar a un equlibrio deben estar las tres cargas juntas, se imaginó una similitud con los tres colores básicos que al juntarse dan el color blanco. +A la manera de la electrodinámica cuántica (\emph{QED}) se ha desarrollado una teoría para explicar las interaccions fuertes: la cromodinámica cuántica (\emph{QCD}). No es que los cuarks estén relacionados con las propiedades ópticas de los materiales, si no que por la peculiaridad de ser una fuerza en la que existen tres cargas, y para llegar a un equlibrio deben estar las tres cargas juntas, se imaginó una similitud con los tres colores básicos que al juntarse dan el color blanco. De esta forma las cargas de color son $r$, $g$ y $b$ (\emph{red, green y blue} en inglés). ¿`Porqué tres cargas? Pensemos el caso de un barión, es una partícula compuesta y sus cuarks están en un estado de equilibrio por lo regular (recordemos que hay resonancias y esas no son un estado de equilibrio), como están formadas por tres cuarks cada cuark debe tener una carga, uno tendrá carga $r$, otro carga $g$ y el tercero carga $b$ ?`Cuál tiene que carga? Cualquiera, con que se tengan las tres cargas, entonces el cuark $u$ puede venir en tres presentaciones extras, que corresponden a cada carga. -De acuerdo, en un barion cada cuark tiene una carga distinta de color, así los tres cuarks forman el estado neutro de carga de color ¿`y los mesones? Ahí sólo hay dos cuarks y deben formar un estado neutro por esto de los estados estables. Lo que sucede ahí es que en un mesón el cuark posee una carga de color, digamos $g$, y el antu cuark tiene una carga $\bar{g}$. Entonces hay carga e color y anti-carga de color, de esta forma con dos cuarks se puede llegar al estado neutro. +De acuerdo, en un barion cada cuark tiene una carga distinta de color, así los tres cuarks forman el estado neutro de carga de color ¿`y los mesones? Ahí sólo hay dos cuarks y deben formar un estado neutro por esto de los estados estables. Lo que sucede ahí es que en un mesón el cuark posee una carga de color, digamos $g$, y el anti cuark tiene una carga $\bar{g}$. Entonces hay carga de color y anti-carga de color, de esta forma con dos cuarks se puede llegar al estado neutro. El fotón no posee carga eléctrica, el gluón por el contrario sí tiene carga de color, tiene una carga bicolor de color y anti-color, por ejemplo $r\bar{b}$ o $\bar{g}r$. Esto implica que la teoría de la cromodinámica cuántica sea no Abeliana (no conmutativa). Dado que los gluones tienen carga entonces pueden interactuar gluones con gluones, esto hace que la teoría sea no lineal y la imposibilidad de gluones libres. Que el gluón sea cargado provoca que en las interacciones fuertes haya cambio de color entre cuarks y gluones (nunca cambio de sabor). El confinamiento de color provoca que ninguna partícula con carga de color pueda existir libremente, es decir, los cuarks y gluones no pueden existir libres, deben estar confinados en un hadrón sin color. -Experimentalmente nunca se han visto partículas del tipo cuark-cuark o cuar-cuar-anticuark, por ello sabemos que sólo existen hadrondes de color neutro. Nos hace pensar que las fuerza es atractiva para estados sin color y repulsiva en estados con color. +Experimentalmente nunca se han visto partículas del tipo cuark-cuark o cuark-cuark-anticuark, por ello sabemos que sólo existen hadrondes de color neutro. Nos hace pensar que las fuerza es atractiva para estados sin color y repulsiva en estados con color. \subsubsection*{Cuarks} @@ -687,7 +691,7 @@ El pentaquark es otra de esas posibles partículas hadrónicas, que se dice dete Se cree que todas las fuerzas pueden expresarse como teorías de norma, que a su vez son invariantes con respecto a la norma. Es un hecho que todas las teorías de campo para fuerzas básicas son teorías de norma. La conservación de carga, que es una conservación aditiva, es derivada de una invariancia ante una transformación global de norma, pero al agregar la dependencia para una carga no estática la ecuación de Schrödinger sigue siendo invariante incluso para una transformación local de norma. Si recuerdan esta invariancia se asocia a la libertad parcial de elegir el potencial electromagnético. -LA invariancia de norma dicta la forma de la interacción y campos vectoriales sin masa. Para que la ecuación de Shrödinger se mantenga invariante ante transformaciones de norma requiere de un potencial vectorial $(A_0,mathbf{A})$. Para verlo más claro para una transformación de norma local se define una norma covariante (derivacione covariantes): +LA invariancia de norma dicta la forma de la interacción y campos vectoriales sin masa. Para que la ecuación de Shrödinger se mantenga invariante ante transformaciones de norma requiere de un potencial vectorial $(A_0,\mathbf{A})$. Para verlo más claro para una transformación de norma local se define una norma covariante (derivaciones covariantes): \begin{align} D_{\mu}=& (D_0,\mathbf{D})\\ @@ -748,7 +752,7 @@ De quedarse así habría montones de problemas para el desarrollo de la teoría, \subsection{Mecanismo de Higgs} -Hay dos tipos de rompimiento de simetría, uno es como el caso del isoespín, donde es una \textit{simetría aproximada} que se rompe para interacciones débiles y electromagnéticas, pero si nos quedamos con la parte de interacción fuerte del hamiltoniano no parece romperse nada. El otro tipo es el \textit{rompimiento espontáneo de simetría}, donde el Hamiltoniano completo si mantiene la simetría, pero el estado base la rompe esto sucede si el estado base del hamiltoniano es degenerado. Elegir un estado de estos sobre los demás rompe la simetría. Un ejemplo es el ferromagneto, el hamiltoniano que lo describe es invariante rotacionalmente, pero al darnos cuenta que los espines del ferromagneto podrían apuntar en cualquier dirección y que en la realidad están alienados, la simetría se rompe. A esto se le llama una simetría escondida. +Hay dos tipos de rompimiento de simetría, uno es como el caso del isoespín, donde es una \textit{simetría aproximada} que se rompe para interacciones débiles y electromagnéticas, pero si nos quedamos con la parte de interacción fuerte del hamiltoniano no parece romperse nada. El otro tipo es el \textit{rompimiento espontáneo de simetría}, donde el Hamiltoniano completo si mantiene la simetría, pero el estado base la rompe esto sucede si el estado base del hamiltoniano es degenerado. Elegir un estado de estos sobre los demás rompe la simetría. Un ejemplo es el ferromagneto, el hamiltoniano que lo describe es invariante rotacionalmente, pero al darnos cuenta que los espines del ferromagneto podrían apuntar en cualquier dirección y que en la realidad están alineados, la simetría se rompe. 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