*No animals were harmed in the writing of this blog post.*

One of the amusing tales in particle physics is the story of how the “penguin diagram” got its name. We won’t go into that here, instead, we’ll make use of some of the tools we’ve developed with Feynman diagrams to understand the *physics* behind these ‘penguin’ diagrams. In doing so, we’ll have a nice playground to really make use of what we’ve learned so far about Feynman rules. (Feel free to review the series if you need a refresher!)

*Caveat: we’ll try to squeeze as much physics as we can out of our diagrams, and occasionally I’ll lapse into some more technical details. Feel free to skip these if you just want a ‘big picture,’ the main idea is independent of the details.*

## Drawing the penguin diagram

In a nutshell, a penguin (in the Feynman diagram sense) is a process where one *flavor *of matter particle changes into another flavor while emitting a photon or gluon: for example, a bottom (*b)* quark converting into a strange (*s)* quark and a photon. These processes are rare in the Standard Model and are interesting because they can get large enhancements from new physics

So let’s jump right in. If I told you that I wanted to study a process where a bottom quark turns into a photon and a strange quark, the simplest diagram to draw would be something like this:

However, this is *wrong*. The problem is that this is not an allowed Feynman rule in the Standard Model—the photon does not connect particles of different flavors. It can talk to a bottom and anti-bottom, or a strange and an anti-strange, but not to a bottom and an anti-strange. (Recall that an anti-strange quark going into the vertex is the same as a strange quark coming out.)

In fact, we know that the only source of flavor-changing in the Standard Model comes from the *W* boson. Thus we conclude that whatever diagram mediates this process, which we succinctly write as *b → sγ* (“bottom quark to strange quark and photon”), must include a *W* boson somewhere. Here’s a simple diagram one could draw:

Great! This actually works. A couple of things to notice right off the bat:

- This is a “
**loop**” diagram, to be distinguished from the “**tree**” diagram we tried to draw above. This demonstrates a principle in the Standard Model: there are no tree-level “*flavor-changing neutral currents*” (FCNC). That is to say, the neutral gauge bosons (photon, gluon,*Z*) do not have direct interactions to fermions which change flavor. The*b → sγ*transition evades this principle by being a loop-level (and hence “more quantum”) process. - The top quark (
*t*) in this diagram could have been replaced by any of the up-type quarks: up, charm, top. In fact, since these internal quarks are*virtual—*they’re not directly observed—quantum mechanically*all three*quarks simultaneously mediate this process. (Mathematically this means that there are three complex numbers that add together.) - In fact, there are several other diagrams one could draw. Since we can learn a lot just by looking at this diagram, I’ll leave it to you to draw these diagrams as homework. (Think of different places to put the photon, and if you’re really slick, think of other bosons that could replace the
*W*.)

**Technical comments**: For those with some more mathematical background, the loop diagrams are higher order in a Taylor expansion. In fact, the whole point of drawing Feynman diagrams is to have a succinct way of writing out a Taylor expansion for a process. The expansion parameters are the coupling constants,

*g,*associated with each vertex: note that a loop diagram contains two extra vertices compared to a tree-level diagram for the same process. It turns out that calculating the loop also involves a 4D integral from including all possible momentum configurations for the virtual particles, and this ends up giving a factor of 1/16π

^{2 }so that the expansion parameter is really

*g*/16π

^{2}^{2}. For the electromagnetic coupling we know that

*e*/4π is 1/137, so that this is indeed a small parameter to expand about.

^{2}## Drawing the diagram using chiral notation

*Edit, 19 March: an earlier version of this post had incorrectly suggested that angular momentum restricts the penguin to be a dipole operator. Thanks to Jack Collins for pointing this out.*

Instead of pushing this diagram for all it’s worth, let’s try to make use of the “chiral nature” of the Standard Model that tells us that left-handed particles are rather different from right-handed particles.

We propose using different notation which makes the “chirality” (handedness) of the matter particles manifest; in other words, we explicitly say if a particles is right- or left-hande, e.g. *b _{R}* for the right-handed bottom. This notation will lead to more complicated diagrams, but it won’t require us to do any math to understand physics that is hidden in the simpler notation.

Angular momentum plays an important role here. What we really mean by a penguin process is one that connects a right-handed fermion with a left-handed fermion, for example *b _{R}*

*→*This may sound little abstract, so here’s a cartoon picture, where the red arrows are carefully drawn to indicate handedness. The green numbers represent the angular momentum in the bottom quark’s direction of motion: right-handed fermions carry half a unit of angular momentum, photons carry 1 unit of angular momentum. (The left-handed strange quark has negative angular momentum since it’s spinning in the opposite direction.)

*s*γ._{L }Okay: so now we’ve learned that penguins are transitions in which a right-handed fermion decays into a photon and left-handed fermion of a different flavor. Of course you can also have left-handed fermions decaying into a photon and a right-handed fermion. Similarly, you can replace the photon with a gluon (“gluonic penguin”). Photonic penguins in which the flavors don’t change have their own fancy name, “electric dipole moments.”

**Technical comment, for physics students**: The identification of the penguin with a “dipole” should make sense from the chirality of the operator. These right-to-left transitions are mediated by a σ

^{μν}. Alternately (and slightly more technically), you can appeal to gauge invariance and observe that the only gauge-invariant operator you can write with one photon that is not a kinetic term must contain a field strength

*F*. The antisymmetry of the Lorentz indices requires that it contracts with a σ

^{μν}^{μν}which necessarily requires a chirality-flip.

You might wonder why we don’t consider same-chirality transitions of the form *b _{L}*

*→*Loop diagrams contribute to this are field-strength renormalizations, they correct the tree-level kinetic term and lead to a redefinition of flavors with respect to the tree-level basis.

*s*γ._{L }Instead of dictating the chiral Feynman rules to you, let’s discover them together. We’ll be sketchy since for our purposes the details won’t buy us much. Let’s naively try to draw a chiral Feynman diagram by just copying the non-chiral diagram above.

We know that the chiral transition must be of the form *b _{R}*

*→*First let’s make the observation that we’ve drawn the

*s*γ._{L }*b*leg with an arrow going in the opposite direction. This is one of the new conventions of the chiral Feynman diagrams:

_{R}Arrows no longer correspond to particle or anti-particle, instead, they correspond to the chirality of the particle.

So right- and left-handed particles have opposite arrows.

Now there’s a clear problem in the diagram above, which we’ve marked with a question mark (?). We know that the *W* boson is biased: it only talks to left-handed particles! Thus we are *not* allowed to have the coupling of a *b _{R}* to a

*. We need to convert the right-handed bottom quark to a left-handed bottom quark (and these are totally different particles!).*

*t*_{L}Fortunately, we can do that! A right-chiral particle can convert into its a left-chiral sibling by bouncing off the Higgs vacuum expectation value. (Once again, electroweak symmetry breaking plays a key role!) We draw the “bounce off of the Higgs vacuum expectation value” as a cross on the fermion that changes the direction of the arrow. We call these crosses “mass insertions” because they are proportional to the mass of the particle.

**Technical details**: for our new “chiral” Feynman diagrams, the arrows no longer represent particle/anti-particle flow. So how do you know if a line corresponds to a particle or anti-particle? Usually it’s clear from angular-momentum conservation. For example, we should have been more precise and said that in the above diagram, the right-chiral *b _{R}* is converted into the anti-

*b*since that’s the guy with same angular momentum as the original

_{L}*b*.

_{R}Why cross on the *b _{R}* (the “bounce of the Higgs vacuum expectation”) physically represent a mass? This makes sense: a massive particle is one which can come in both chiralities since you can always boost into a frame where it’s spinning in the opposite direction.

For students who want to a reference for chiral diagrams, I refer to the encyclopedic “two-component spinor bible.” The results are completely equivalent to what one would obtain using four-component Dirac spinors, but the main benefit is that you hardly have to do any Dirac algebra to see the chiral structure of the amplitude. Further, things like Majorana fermions can be very difficult in Dirac notation but are straightforward using the two-component Weyl spinors.

There! Now we have a diagram that appears to work. Except it doesn’t. The reason is a a bit technical, but it has to do with the fact that not only are there tops (*t _{L}*) running in the loop, but there are also charms (

*c*) and ups (

_{L}*u*). When you sum over all of these contributions, it turns out that the final result is

_{L}*zero*! This result is known as the GIM mechanism.

**Technical detail**: What is the origin of this GIM mechanism cancellation? The *W* boson coupling is actually a 3×3 matrix corresponding to which down-type flavor is being converted into which up-type flavor. These matrices are *unitary*, they encode a change in flavor basis, and the diagram above is proportional to:

This is basically the relation *U ^{†}U = *1 for a unitary matrix

*U*.

The lesson from this is that the previous Feynman diagram is too simple—it needs more internal structure to avoid the GIM cancellation. The particular structure that it needs is something that differentiates the top/charm/up type diagrams. Fortunately, there’s a way to do this: we just add more mass insertions: since the top, charm, and up each have different masses, the sum of the following diagrams (with *t* replaced by *c* and *u* in other diagrams) will not vanish:

Of course, the question mark is our way of pointing out that this still doesn’t work. The mass insertion in the loop converts the left-handed top quark into a right-handed top quark. However, the *W* only couples to left-handed particles, so the *W**t _{R}*

*s*is not allowed. This means that we need

_{L}*another*internal mass insertion to convert the right-handed top back into a left-handed top:

I drew the mass insertion after the photon, but there are other places it could have gone. As an exercise you can draw the other diagrams that contribute to *b _{R}*

*→*These extra mass insertions come at a price: they tend to reduce the size of the diagram.

*s*γ._{L }* *What have we learned?

- One reason that these penguin decays are rare is that there is no tree-level diagram in the Standard Model. It’s a loop-level process which makes it “more quantum.”
- Another reason why the penguin is rare is the “GIM mechanism,” which requires that the diagram picks up additional mass insertions. In order to avoid this, we need additional internal mass insertions which come in pairs and typically suppresses the process. (This also tells us that in the limit where all of the up-type quarks have the same mass, the probability of this process must vanish.)
- The chiral structure of the Standard Model tells us a lot about what’s actually happening in a penguin! We learned that penguins are left-right “dipole” transitions and that (in the Standard Model) they require that the fermions bounce off the Higgs vacuum expectation value a few times due to (a) angular momentum conservation and (2) the coupling of the
*W*to only left-handed fermions.

One of the nice things about the chiral Feynman diagrams is that they’re easier to read when trying to estimate the size of the diagram without doing the nitty gritty details of the calculation. Each mass insertion gives a factor of the fermion mass (or the mass splitting in the case of GIM cancellation) and then we can fudge additional factors by dimensional analysis. This is beyond the scope of this post, but it’s worth explaining why these slightly-more-complicated diagrams are worth their complications. For the above diagram, one can see that the *b _{R}*

*→*penguin is proportional to the mass of the bottom quark and the difference in the squared masses of the internal up-type quarks.

*s*γ_{L }## Bonus: Leptonic penguins

As a final example, let’s quickly go over the story of the leptonic penguin. The prime example is the decay of a muon into a photon and an electron, *μ _{R}*

*→*. Usually the relation between up/down quarks is analogous to that of electrons and neutrinos. This leads us to guess the following diagram:

*e*γ_{L }Here we’ve taken the liberty of introducing a right-handed neutrino into the theory to account for the experimental observation that neutrinos have a very tiny, but non-zero mass. Unfortunately, the above diagram does *not* work since the neutrinos are not electrically charged and so they do not interact with the photon. We need to look for other diagrams. In particular, the photon must not come off of the fermion leg, but perhaps from the *W* leg.

One thus ends up with diagrams of the form:

What is that dashed line? That’s a charged Higgs—one of the Goldstone bosons that was eaten by the *W. *I’ve drawn it here just to show off a little: we can draw diagrams in which we make the interactions with the different components of the *W* manifest. Here we know that the charged Higgs is really the “longitudinal polarization” of the *W*, but we’re drawing it as an independent particle. (We could have “picked a gauge” in which this diagram is absent, but let’s allow ourselves to show off for pedagogical purposes.) Let us use this opportunity to highlight another aspect of the chiral notation:

- Interactions with
*vector*particles (spin 1) preserve fermion chirality. We saw this with the*W*boson above: a left-handed particle stayed left-handed after interacting via a*W*boson. This was also true with the photon coupling, and turns out to be true for all of the gauge bosons. It has nothing to do with the chiral structure of the Standard Model, rather it has to do with conservation of angular momentum. (The chiral structure of the Standard Model shows up when we say the*W*only talks to left-handed particles; compare this to the photon which will talk to pairs of left-handed particles or pairs of right-handed particles, but never a left-handed and a right-handed particle in the same vertex.) - On the other hand, interactions with
*scalars*(spin 0), such as the Higgs vacuum expectation value or any of its components,*do*change chirality. This is just a feature of scalar interactions versus vector interactions.

Now some remarks about the leptonic penguins:

- Note that because there is no photon–neutrino coupling, the set of diagrams for the leptonic penguins are different from those of the quark penguins! (As an exercise, try drawing all the diagrams for the quark–quark–gluon penguins, there are even more since gluons can couple to other gluons.)
- In the diagram above, the charged Higgs coupling to a right-handed muon and a left-handed electron is proportional to the mass of the muon—thus one still picks up a factor of the initial fermion mass.
- We could also draw the reversed diagram where we pick up the mass of the electron. (Exercise: draw this diagram and label all internal states with their chiralities.) However, since the electron is so much lighter than the muon, we can ignore this contribution.
**[technical detail]**For those who know a little field theory, it should be straight forward to do a dimensional analysis on this to determine the dependence of the branching ratio on the internal neutrino masses and the external muon mass. (Let the*W*mass be the dominant mass scale in the diagram, though kinematic factors have to be made up with the mass scale of the process, which is the muon mass.)- The muon mass insertions are
*really*small. We know that these mass insertions are really mass splittings (i.e. proportional to the differences in mass). Since the neutrino mass splittings are experimentally known to be very small, the*μ*_{R}*→*penguin is very rare!*e*γ_{L }

## Punchline: looking for new physics

While this post has been somewhat technical, for the most part we’ve managed to avoid doing any mathematics while still being able to make some fairly quantitative statements about the penguin process. We could, for example, talk about how the *b _{R}*

*→*penguin vanishes if the internal quarks have the same mass, or even guess the dependence of the quantum mechanical amplitude on the masses of the internal and external particles. If we were to do the calculation using the “standard” (rather than chiral) Feynman diagram, these properties would require a little bit of mathematical work to see explicitly.

*s*γ_{L }Now that we’ve really beaten ourselves over the head with these penguins, let me just close by explaining that these penguins are interesting primarily because they are loop level processes where *any* allowed particle may run in the loop, including new particles that aren’t in the Standard Model. This is because such internal particles are *virtual* and don’t need to be on-shell, that is to say that they don’t need to have enough energy to actually exist for long periods of time. In popular books this is often explained with Heisenberg’s uncertainty principle: the internal particles can violate energy conservation for a very short period of time, as long as they decay into states which do respect energy conservation relative to the initial particle. Thus the inside of the penguin can include contributions from exotic new particles. Since the Standard Model contribution is suppressed, there’s a chance that the effect of the exotic new particles might be seen in an enhanced decay rate.