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Questions about the Electroweak Lagrangian

  1. Aug 16, 2011 #1
    Now bear with me, I'm no expert when it comes to Electroweak Symmetry and Symmetry Breaking; I can only comprehend up to integrating, functions, derivatives, partial derivatives with a small hint of linear algebra and the basic, Hermitian, Hamiltonian, bras and kets.

    So my questions are the following:

    A.) What do all the symbols mu, tau, nu, etc mean when they are on top and/or below the B particle symbol and the W boson symbol?

    B.) What is a covariant derivative?

    C.) What is Yukawa Interaction?
     
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  3. Aug 16, 2011 #2

    WannabeNewton

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    (a)The greek indeces are place holders for the components of the (m, n) tensor in question (for example the faraday tensor [itex]F^{\mu \nu }[/itex] represents the component of the tensor corresponding to the index you place in for the greek indeces such as the [itex]F^{tt}[/itex] component). When you have the SAME indeces repeated with one above and one below then you are using the Einstein summation convention: http://en.wikipedia.org/wiki/Einstein_summation_convention
    where you basically sum over all values the indeces can take. You might not fully grasp it until you yourself actually do a bunch of practice problems involving the summation convention; it definitely becomes second nature once you do it often.
    (b) The covariant derivative is a generalization of the partial derivative to arbitrarily curved manifolds. In simplest terms, it basically uses a connection, which allows one to compare vectors from tangent space to tangent space, (levi - civita connection when the connection is torsion free in which case you can express the covariant derivative in terms of the metric tensor) to account for the relative change of basis vectors from tangent space to tangent space on a manifold. The covariant derivative (in the case of a torsion free connection) can be expressed in terms of components as follows: [tex]\bigtriangledown _{\mu }T^{\alpha_{1} ...\alpha_{n} }_{\beta_{1} ...\beta _{m}} = \partial _{\mu }T^{\alpha _{1} ...\alpha _{n}}_{\beta_{1} ...\beta_{m} } + \Gamma ^{\alpha_{1} }_{\mu \sigma }T^{\sigma ...\alpha _{n}}_{\beta_{1} ...\beta _{m}} + ...\Gamma ^{\alpha _{n}}_{\mu \sigma }T^{\alpha _{1}...\sigma }_{\beta _{1}...\beta _{m}} -\Gamma ^{\sigma }_{\mu \beta_{1} }T^{\alpha_{1} ...\alpha _{n}}_{\sigma ...\beta _{m}} -...\Gamma ^{\sigma }_{\mu \beta _{m}}T^{\alpha _{1}...\alpha _{n}}_{\beta _{1}..\sigma }[/tex]

    EDIT: I forgot to add that tensors are multilinear mappings of vectors AND one - forms to the reals. An (m , n) tensor maps as follows [itex]T:V^{*}\times ...\times V^{*}\times V\times ...\times V \mapsto R[/itex] where the n copies of [itex]V^{*}[/itex] correspond to n copies of the dual vector space and [itex]V[/itex] the m copies of the vector space. On a manifold, the vector space corresponds to the tangent space to the manifold at a point and the dual vector space corresponds to the respective dual tangent space (or cotangent space). The upper indeces correspond to the members of the vector space and the lower indeces correspond to the members of the dual vector space (again on a manifold the upper indeces correspond to the vectors that are members of the tangent space in question and the lower indeces correspond to the one - forms which are members of the respective cotangent space). Sorry for using corresponding so much xD.
     
    Last edited: Aug 16, 2011
  4. Aug 16, 2011 #3
    Sweet! Thank you for the intuition, I'm starting to grasp the concepts better
     
  5. Aug 16, 2011 #4
    c) A yukawa interaction is a type of interaction between a scalar field and a fermion field, i.e. http://en.wikipedia.org/wiki/Yukawa_interaction,

    But I guess you do not know what the difference between scalar fields and fermion fields is, so forget that. As far as the electroweak Lagrangian goes, the thing to know is that Yukawa interactions are the ones that act like mass terms for the fermion fields, i.e. the quarks and leptons.

    e.g.

    [itex]y\Psi\phi\Psi[/itex]

    would be a Yukawa interaction between scalar field [itex]\phi[/itex] and fermion field [itex]\Psi[/itex] (with some indices and conjugates and such left out). The 'y' is the coupling strength of the interaction. The 'matching' mass term describing a fermion field with mass [itex]m[/itex] would be

    [itex]m\Psi\Psi[/itex]

    So if the scalar field (Higgs field) was to stop being a dynamical field, and just adopt a constant value [itex]\phi_{0}[/itex], the Yukawa term would look exactly the same as the mass term, with [itex]m=y\phi_{0}[/itex]. And that's how you can generate mass terms without just sticking them in to your Lagrangian (which you aren't allowed to do in the Standard Model because they break gauge invariance)

    edit: Ok I actually read that previous response now and have this to add:

    a) Perhaps you are looking at a page something like this: http://en.wikipedia.org/wiki/Standard_Model#Lagrangian. The W and B and G in that Lagrangian are field strength tensors, not the fields themselves (although they are functions of the fields). When combined in the Lagrangian in that way they describe the dynamics of the fields they relate to; they more or less generate Maxwell's equations and equivalent things for the other forces. You'll have to learn some special relativity to appreciate those indices properly.

    b) The word 'covariant derivative' is thrown around in different contexts. The previous response talks a little bit about it in terms of a curved manifold, which in general relativity would mean curved spacetime. It is how you do derivatives in curved space.
    However, you are asking about the electroweak Lagrangian, and in this context it means something a bit different. The Standard Model is constructed in flat space, so there are no curved space covariant derivatives to worry about. Instead, there are what are sometimes called 'gauge covariant derivatives', which are how you do derivatives while preserving the gauge symmetries of the model. The two usages are really manifestations of the same underlying math but things are complicated enough without worrying about reformulating the standard model in geometric terms.
    Have a read of the wiki: http://en.wikipedia.org/wiki/Gauge_covariant_derivative

    I am just trying to throw some conceptual ideas out there, because you have a lot more to learn before you can really appreciate what is going on. I do too though, so don't be discouraged if it all seems like gibberish. It will continue to seem like gibberish for a long time yet :).
     
    Last edited: Aug 16, 2011
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