Standard Model Lagrangian (density)

Extracted and typed by T.D. Gutierrez from a series of appendices in Diagrammatica by Nobel Laureate Martinus Veltman.


As far as I'm concerned, do what thou wilt with the equation as I've presented it: reformat it using your own favorite LaTeX methods, have fun inventing your own exam questions, or use it as is in your talks, web pages, and presentations. And, of course, there's the ever popular parlor game Find the Sign Error. But first, a few things you should know...

The version you see here is a "post-electroweak symmetry breaking" form of the Lagrangian and, although not inaccurate, it may not be its most elegant or transparent form.
By this I mean it includes some specific form for the Higgs fields and Fadeev-Popov ghosts, making things quite messy (and model-dependent; although it's probably more "honest", depending on your point of view).

A cleaner version and description (written sans Higgs, and sans ghosts) can be found here. An organized Veltman-like presentation can be found here.


Standard what? The Standard Model is the current Theory which governs all known strong and electroweak interactions. This includes basically everything that has EVER been measured by humans (sans gravity). The term *model* in this context is really a misnomer (a carry over from the days when it WAS just a bunch of ad hoc models). It really is the best Theory that we have. It has been fantastically successful in predicting and explaining data; there are no obvious or overt violations of this Theory. That isn't to say there aren't some dark corners with big question marks -- which are, of course, the current targets for illuminating research. It also isn't to say that the Theory is entirely satisfactory or complete (from an aesthetic point of view). Qualitatively, it seems too "hodge podge". Historically, "hodge podginess" has usually meant there is "more going on" and that the Theory is incomplete (in this case, gravity's absence and the lack of unified forces are particularly unsatisfactory elements). Most physicists believe the Standard Model is just a stepping stone to bigger and better things (thus don't feel too guilty calling it a *Model*) and so concentrate much effort into "Beyond Standard Model" research (superstrings, supersymmetry, grand unified theories, quantum gravity, etc.). But, given that it doesn't claim to be a "theory of everything" but simply "the best theory we have", the Standard Model is really quite amazing at predicting and explaining data. See this nice Wikipedia article on the Standard Model for more details and links.

LagraHUH? If you don't already know what a Lagrangian is, I probably won't be of much help here. In short, if you know The Lagrangian, in principle you know everything there is to know about a system -- even if you might have to work hard to get it. It is essentially an ultra-compressed way to write A LOT of information.

Where did I get it? It was compiled from the appendices of Veltman's book Diagramatica (thus the quasi-standard notions). However, aside from somewhat irritating non-standard notion, the book is quite readable and pedagogically useful. Also, kudos to Veltman for having the complete SML at all, even if it is spread out over many pages in an appendix.

How long did it take to type? In it's current form, the actual equation took me about four hours (spread over almost a week) to write into a file using LaTeX. Sadly, I do think there is a sign error somewhere, but I just don't have the energy to hunt it down... :(

Why bother? Admittedly, it isn't particularly *useful* in its current form (especially since I haven't mentioned what the notation means!). It is meant to be a kind of "visual treat", if you will, seeing all the different sectors written together. If you stare at it long enough (squinting helps) you can start to induce nearly hypnotic hallucinations. What do YOU see in the Standard Model Lagrangian? (oddly, "a bunny" seems to be the most common response)...And who says our current best Theory can't fit on a T-Shirt (I've just ordered the first batch!)...

I am familiar with quantum field theory and the Standard Model, but many parts of the Lagrangian look unfamiliar; what the hell are all the terms? I'll be putting a legend up soon. Again, Veltman uses somewhat non-standard notation and compression (e.g "u" means "up-like quarks"). If you are already familiar with the Standard Model then you will probably recognize many parts of the Lagrangian already. The Higgs ghost fields (in the electroweak sector) were particularly unfamiliar to me and seem to take up quite a bit of the bulk of the equation.

Posted Circa 1999
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