Last month had the unique pleasure of making my first trip to CERN (more on that in a later post). I made a point to stop by the CERN gift shop to pick up a snazzy mug to show off to my colleagues back in the US, and am now the proud owner of a new vessel for my tea:
My brand new "Standard Model Lagrangian" mug from CERN.
The equation above is the Standard Model Lagrangian, which you can think of as the origin of all of the Feynman rules that I keep writing about. Each term on the right-hand side of the above equation actually encodes several Feynman rules. Roughly speaking, terms with an F or a D contain gauge fields (photon, W, Z, gluon), terms with a ψ include fermions, and terms with a ϕ include the Higgs boson. Some representative diagrams coming from each of the terms are depicted below:
Representative Feynman rules coming from each term in the Lagrangian.
But alas, there’s a bit of a problem with the design. It appears that there’s an extra term which isn’t included in the usual parametrization of the Standard Model:
This term really shouldn't be here. It's not necessarily "wrong," but it is misleading and doesn't match what is written in textbooks. Technically, it is not `canonically normalized.'
I won’t go so far as to call this a mistake because technically it’s not wrong, but I suspect that whoever designed the mug didn’t mean to write this term. Let me put it this way: if I had written the above expression, my adviser would pretend he didn’t know me. The “h.c.” means Hermitian conjugate, which is a generalization of the complex conjugate of a complex number. In terms of Feynman diagrams, this “+h.c.” term means “the same diagram with antiparticles.”
The problem is that the term above,
already includes its Hermitian conjugate. In physics-speak, we say that the kinetic term is self-conjugate (or Hermitian, or self-adjoint). This just means that there is no additional “+h.c.” necessary. In fact, including the “+h.c.” means that you are writing the same term twice and the equation is no longer “canonically normalized.” This just means that you ought to rescale some of your variables.
I was mulling over this not-quite-correct term on my mug while looking over photos from CERN when I discovered the same ‘error’ in a chalkboard display in the “Universe of Particles” exhibit:
Display at the "Universe of Particles" exhibit in The Globe of Science and Innovation at CERN.
The “+h.c.” on the top right is the same ‘error’ as printed on the CERN mug. I wonder who wrote this?
To be clear: this expression does summarize the basic structure of the Standard Model in the sense that it does give all of the correct Feynman rules. However, the extra “+h.c.” introduces a factor of two that needs to be accounted for by weird conventions elsewhere (that would not match any of the usual literature or textbooks).
Anyone who has done tedious physics calculations is familiar with the frequent agony of being off by a factor of 2. Now when people make remarks about this ‘error’ on my mug, I’m quick to tell them that the factor of 2 mistake just makes it more authentic.
At least, let’s hope that CERN has correctly measured the circumference 27 kilometers and it’s not just 13.5+13.5 where the right circumference is just 13.5 kilometers.
If it is supposed to be the Standard Model, the F-mu-nu-F-mu-nu should also be a trace, or summed over adjoint indices, right?
Hi Lubos. Good point about the trace. I think I’m willing to take this as assumed since none of the other gauge indices are written explicitly.
Maybe you should contact John Ellis about this:
http://www.bnl.gov/bnlweb/pubaf/pr/photos/2010/04/Ellis-HR.jpg
(=;
@ Gregor: *LoL* The conspiracy goes all the way to the top!
-F
What about the Yukawa term? Unless we are asuming all fermions to be Majorana, how does that term have two psis and no pisi-bars?
Hi Innocent Bystander, written in a Weyl basis, the Yukawa term connects two left-chiral particles or two right-chiral particles. You can see this just from the spinor structure: either there are two undotted indices or two dotted indices. Alternately, you can write out the usual Yukawa term with Dirac spinors and then write the Dirac spinor in terms of Weyl spinors.
I think the main confusion is that the Yukawa term has, for example, a left-chiral particle and a the anti-partner of a right-chiral particle. The latter field is a left-chiral field, so that you end up with two unbarred fields. Similarly, the “+h.c.” contains two barred fields.
Great question.
Best,
Flip
Perhaps the “+h.c.” is someone’s initials. You said that it doesn’t make the equation technically wrong. perhaps that was just their idea of a joke. They noticed their initials were short for Hermitian conjugate and added it to the end as an inside joke knowing very few people would actually get it and understand it. Look around for anybody with those initials who works there and would understand it.
I’ve heard from a quite trustworthy source that it is John Ellis’ handwriting on the mug. Perhaps you gotta check this beef with him
Shea is on to something.
+h.c. I would fathom means “plus hot coffee” as well as an in joke of being off by a factor of 2
Nice try Wicked, but why does Globe of Science exhibit have the same “misstep”?
“whoever”, not “whomever.
Good catch, not-an-elephant! Post corrected, thanks.
When may we expect this mug to be discussed on “The Big Bang Theory?”
Never, the writers of the show know diddly squat about physics
The writers may not know much physics, but there’s a physics professor at UCLA who consults on their whiteboard equations and (to the best of my recollection) these are all correct and have (in the past) made their way into the show.
If physics used Tau instead of Pi, perhaps all those awful “off by a factor of 2″ errors would go away
Hi Matson. Ah, tau day. Alas, while historically it may have been nicer to have used tau instead of pi, humans have more or less adapted to this convention particularly well. Unfortunately, the factors of 2 which are puzzling aren’t typically from pi versus 2*pi, but rather they often come from more subtle things like the quantum statistics of identical particles. -F
you’re all so silly and confused – the hc stands for “hot coffee”. after all, it’s a mug.
It seems you’re not the only one who’s noticed that recently. M Shifman posted a similar remark on his Facebook wall, with picture (hopefully his are publicly visible): https://www.facebook.com/photo.php?fbid=244828688867538&set=a.101417873208621.3533.100000212256542&type=1&ref=nf
How about dark matter, doesn’t that probably involve violating the Standard Model? So most of the universe doesn’t follow those rules anyway …
I love reading articles like this.. because for a layman like me every second word could be invented and the whole thing sounds like such an abstraction that its actually hard to tell if this is real or not.
Is there any way to get a mug sent to North America? I googled and couldn’t find a Cern gift shop website but apparently that’s because it is a not for profit so there are some rules around it. Any ideas?
Hi kb, unfortunately I’m also at a loss for this. It seems like ATLAS has an online gift shop, but they don’t seem to have the mug available. Maybe you could try contacting them to see if they can put you in contact with the main gift shop?
Yeah, I saw the ATLAS site too, but good idea to contact them, thanks!
Thanks for this very interesting post Flip !
What the heck are y’all talkin’ about.
Tau, Tau,
Dirk