### January 2009

Here’s a new installment to the never-ending debate about the best way to draw figures in LaTeX. (A previous suggestion: Adobe Illustrator.)

Stereographic projection image made using TikZ by Thomas Trzeciak, available at TeXample.net.

A promising solution is the combination of PGF (“Portable Graphics Format”) and TikZ (“TikZ ist kein Zeichenprogramm,” or “TikZ is not a drawing program”), both developed by Till Tantau, whom you may know better for creating Beamer.

PGF is the ‘base system’ that provides commands to draw vector images. This layer of the graphics system is tedious to use directly since it only provides the most basic tools. TikZ is a frontend that provies a user-friendly environment for writing commands to draw diagrams.

Like other LaTeX drawing packages (the picture environment, axodraw, etc.), the TikZ figures are a series of in-line commands and so are extremely compact and easy to modify. Unlike other LaTeX packages, though, TikZ provides a powerful layer of abstraction that makes it relatively easy to make fairly complicated diagrams.  Further, the entire system is PDFLaTeX-friendly, which is more than one can say for pstricks-based drawing options.

The cost is that the system has a bit of a learning curve. Like LaTeX itself, there are many commands and techniques that one must gradually become familiar with in order to make figures. Fortunately, there is a very pedagogical and comprehensive manual available. Unfortunately the manual is rather lengthy and many of the examples contain small errors that prevent the code from compiling. TeXample.net comes to the rescue, however, witha nice gallery of TikZ examples (with source code), including those from the manual. If you’re thinking about learning TikZ, go ahead and browse some of the examples right now; the range of possibilities is really impressive.

To properly learn how to use TikZ, I would suggest setting aside a day or two to go through the tutorials (Part I) of the manual. Start from the beginning and work your way through one page at a time. The manual was written in such a way that you can’t just skip to a picture that you like and copy the source code, you need to be sure to include all the libraries and define all the variables that are discussed over the course of each tutorial. (You can always consult the source code at the gallery of TikZ examples in a pinch.)

Let me just mention two really neat things that TikZ can do which sold me onto the system.

TikZ Feynman Diagram by K. Fauske, available at TeXample.net.

The first example is, of course, drawing a Feynman diagram. The TikZ code isn’t necessarily any cleaner than what one would generate using Jaxodraw, but TikZ offers much more control in changing the way things look. For example, one could turn all fermions blue without having to modify each line.

TikZ arrows on a Beamer presentation. Image by K. Fauske, available at TeXample.net

The next example is a solution to one of the most difficult aspects of Beamer: drawing arrows between elements of a frame. This is consistently the feature that PowerPoint, Keynote, and the ‘ol chalkboard always do better than Beamer. No longer!

Check out the source code for the example above. Adding arrows is as easy as defining some nodes and writing one line of code for each line. The lines are curved ‘naturally’ and the trick works with Beamer’s overlays. (Beamer is also built on PGF.)

Anyway, for those with the time to properly work through the tutorials, TikZ has the potential to be a very powerful tool to add to one’s LaTeX arsenal.

Pros

• Tools for creating high-quality vector graphics
• “Node” structure is very useful for drawing charts and Feynman diagrams
• Works with PDFLaTeX
• Images are drawn ‘in-line’ (no need to attach extra files)
• Easy to insert TeX into images

Cons

• A bit of a learning curve to overcome
• No standard GUI interface

Download PGF/TikZ. Installation instructions: place the files into your texmf tree. For Mac OS X users, this means putting everything into a subdirectory of ~/Library/texmf/tex/latex/.

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Today I’ll be reviewing P.M. Stevenson, “Dimensional Analysis in Field Theory,” Annals of Physics 132, 383 (1981). It’s a cute paper that helps provide some insight for the renormalization group.

A theory is a black box that we can shake to make predictions of physical observables.

We’ve already said a few cursory words on dimensional analysis and renormalization. It turns out that we can use simple dimensional analysis to yield some insight on the nature of the renormalization group without having to think about the technical ‘heavy machinery’ required to do actual calculations.

First let us define a theory as a black box that is characterized by a Lagrangian and its corresponding parameters: coupling constants, masses, fields, etc. All these things, however, are contained within the black box and are in some sense abstract objects. One can ask the black box to predict physical observables, which can then be measured experimentally. Such observables could be cross sections, ratios of cross sections, or potentials, as shown in the image above.

Let’s now restrict ourselves to the case of a naively-scale-invariant’ or naively-dimensionless’ theory, i.e. one where there are no couplings with mass dimensions. For example, $\lambda\phi^4$ theory or massless QCD. We shall further restrict to dimensionless observables, such as the ratio of cross sections. Let’s call a general observable $\rho(Q)$, where we have inserted a dependence on the energy $Q$ with foresight that such things renormalize with energy scale.

Dimensional Analysis

But one can immediately take a step back and realize that this is ridiculous. How could a dimensionless observable from a dimensionless theory have a nontrivial dependence on a dimensionful quantity, $Q$? Stevenson makes this more explicit by quoting a theorem of dimensional analysis:

Thm. A function $f(x,y)$ which depends only on two massive variables $x,y$ and which is

1. dimensionless
2. uniquely defined
3. defined without any dimensionful constants

must then be a function of the ratio of $x/y$ only, $f(x,y)=f(x/y)$.

Cor. If $f(x,y)$ is independent of $y$, then $f(x,y)$ is constant.

Then by the corollary, $\rho(Q)$ must be constant. This is a problem, since our experiments show a $Q$ dependence.

Evading Dimensional Analysis

The answer is that the theorem doesn’t hold: the theory inside the black box is not uniquely defined,’ violating condition 2. This is what we meant by the stuff inside the black box being abstract,’ the Lagrangian is actually a one-parameter family of theories with different bare couplings. That is to say that the black box is defined up to a freedom in the renormalization conditions.

Now that we see that it is possible to have $Q$-dependence, it’s a bit of a curiosity how our dimensionless theory manages to define a dimensionful dependence of $\rho$ without any dimensonful quantities to draw upon. The simplest way to do this is to have the theory define the first derivative:

$\frac {d\rho}{dQ} = \frac{1}{Q} \beta(\rho)$,

where $\beta$ is the usual beta function calculated in perturbation theory. It is dimensionless and is uniquely defined by the theory. Another way one can define $Q$ dependence is to do so recursively; one can read Stevenson’s paper to see that this is equivalent to defining the $\beta$ function.

One can integrate the equation for $\beta$ to write,

$\log Q +$ constant $= \int^\rho_{-\infty} \frac{d\rho'}{\beta(\rho)} \equiv K(\rho)$.

The constant of integration now characterizes the one-parameter ambiguity of the theory. (The ambiguity can be mapped onto the lack of a boundary condition.) We may parameterize this ambiguity by writing

constant = $K_0 - \log \mu$,

for some arbitrary $\mu$ of mass dimension 1. (This form is necessary to get a dimensionles logarithm on the left-hand side.) The appearance of this massive constant is something of a `virgin birth’ for the naively-dimensionless theory and is called dimensional transmutation. By setting $Q=\mu$ we see that $K_0 = K(\rho(\mu))$. Thus we see finally that the integral of the $\beta$ equation is

$\log(Q/\mu) + K(\rho(\mu)) = K(\rho(Q))$.

All of the one-parameter ambiguity of the theory is now packaged into the massive parameter $\mu$. $K$ is an integral that comes from the $\beta$ function, which is in turn specified by the Lagrangian of the theory. On the left-hand side we have quantities which depend on the arbitrary scale $\mu$ while the right-hand side contains only quantities that depend on the energy scale $Q$.

If $K(\rho(\mu))$ vanishes for some $\mu=\Lambda$, then we can write our observable in terms of this scale,

$\rho(Q) = K^{-1}(\log(Q/\Lambda))$.

Note that $\mu$ is arbitrary, while $\Lambda$ is fixed for a particular theory. This latter quantity is rather interesting because even though it is an intrinsic property of the black box, it is not predicted by the black box, it must be fixed by explicit measurement of an observable. (more…)

I recently put up my first paper on the arXiv and have been dealing with the torrent of e-mails asking for citations. This is normal and part of the publication process, though I’ve been amused by some of the e-mails I’ve been getting…

• One person decided to e-mail my adviser even though it was my e-mail address that was associated with the paper. There is a reason why my e-mail associated with the paper: the senior collaborators don’t want to have to deal your “please cite me” e-mails! Don’t worry, I discuss everything with my collaborators,  but let’s keep things organized, yes?
• E-mails that start with “I read your paper with great interest…” This is a very nice thing to say, but of course when you send it just 30 minutes after the paper is made public, then I know that you really mean: “I quickly searched your bibliography for my name… with great interest.
• All these e-mails make me wonder if anything I’ve done is original at all.
• There is something to be said about being a competent writer. When I skim some of these papers begging for citations, it is clear why they’re not part of the ‘standard’ set of cited papers: they’re unreadable. Yes, being a native English speaker is a huge advantage here and yes, that’s unfair for those who aren’t native speakers, but that’s the way it is.
• I’ll cite papers even though they don’t really have to be cited. This is partly to avoid confrontation, but also because I can sympathize with other grad students who keep an eye on their citations on SPIRES.