### Flavour physics and you part II: the GIM mechanism

20Jan08

In my previous post I introduced the CKM matrix (which I denote as $V$). I haven’t yet explained why this matters to you, but will take a brief detour to introduce something cute: the suppression of flavour-changing neutral currents by the non-degeneracy of quark masses.

No tree-level FCNCs

Before I go on to the most important feature of the CKM matrix, let’s discuss the GIM mechanism (pronounced like gym’), named after Glashow, Iliopoulos, and Maiani. In the Standard Model, there are no tree-level flavour-changing neutral currents (FCNCs). The neutral’ refers to the electric charge.

Do you see why? If you look at the terms in the Lagrangian corresponding to the Z-bozon current, you have terms of the form $Z^\mu \bar{u_L} \gamma_\mu u_L$. The Dirac matrix doesn’t affect the flavor structure, so when one applies the unitary transformations above, the unitary matrices just cancel out! Hence there are appear to be no transitions between quarks of different flavours.

Loop-induced FCNCs

Well, almost. That’s certainly true at tree level since there are no explicit terms in the Lagrangian. It turns out that FCNCs do occur in the Standard Model at the loop-level (i.e. as manifestly quantum effects). Here’s an example:

One can also swap the charm quark (written as $u_c$) with an up or top.

There are certainly flavour-changing charged currents in the Standard Model. (Just try to perform the above analysis on the W-boson terms.) One can use a [charged] W-boson in a loop to create an overall flavour-changing neutral process. One endpoint of the W gives a factor of $V_{cb}$ while the other gives a factor of $V^\dag_{sc}$. [1]

Following the flavor indices, the amplitude for these diagrams are proportional to $V^\dag_{si} V_{ib}$, where we sum over $i$. In particular, the amplitude is proporional to the s-b element of the product of these two matrices. [2] Note that we are summing over an index of up-type quarks $i$, which corresponds to our observation from the above diagram that there are contributions with each flavour of up-type quark flowing in the loop.

… but the CKM matrix!

We know, however, that the CKM matrix is unitary. This means that the product $V^\dag_{si} V_{ib} = \delta_{bs}$, the unit matrix. Since the amplitude is proportional to an off diagonal element of the unit matrix, it appears to be zero. In which case, we would be led to believe that there are no flavour-changing neutral currents even at the quantum (loop) level. Bummer, that was boring.

What we missed: mass terms.

But, alas! We’ve missed something. In the above argument, we had implicitly assumed that the CKM matrix was the only flavour structure in the amplitude. After all, didn’t I previously suggest that the CKM matrix is the only flavour structure in the Standard Model?

What separates the different generations of quarks? Their masses! (However, see note [3] below for another very interesting difference.)

Recall that we summed together diagrams with each up-type quark passing in the loop. These quarks have different masses, and hence their propagators are different. So when we sum the diagrams together, the flavour structure is actually:

$\sum_{i} V^\dag_{si} V_{ib} (\gamma^\mu p_\mu - m_i)^{-1}$.

Here $p_\mu$ is a loop integration variable. I’ve suppressed the integral for simplicity. The above expression multiplies a flavour-independent factor containing the rest of the amplitude. The up-type masses $m_i$ form a diagonal matrix in flavour space, but is not proportional to the unit matrix. This means that when we perform the sum over $i$ as we do above, the resulting matrix is no longer proportional to one. In particular, the s-b element is no longer zero. Note further that in the case where the quark masses are degenerate between generations, the propagators are flavour-blind and the flavour structure reduces to the original case proportional to $\delta_{sb} = 0$.

To make this a bit more explicit, our naive guess of the flavour structure had the form:

$\sum_{i} V^\dag_{si} V_{ib} = \delta_{sb}$,

whereas the flavour structure with non-degenerate quark masses, on the other hand, takes the form:

$\sum_{ij} V^\dag_{si} M_{ij} V_{jb} = \sum_{i} V^\dag_{si} M_{i} V_{ib} \neq \delta_{sb}$.

Here I’ve used $M$ to denote the general flavour structure from the up-type quark propagator. So we see (if only qualitively for now) that the mass-dependence of the up-quark propagator in the loop allows us to sidestep the unitarity of the CKM matrix and permit loop-level flavour-changing neutral currents.

Neutral Meson Oscillation

Let’s move on to the canonical example and try to get our hands a bit dirtier. Consider $\bar{K} -K$ mixing. (The canonical example is kaon mixing, but B mixing is more timely these days.) The contributing diagrams areas follows:

Here the neutral $K$ meson oscillates with its antiparticle as it propagates. [4] This is a $\Delta s = \pm 2$ flavour-changing neutral current. Using the same argument as above, the amplitude for this process vanishes in the case of degenerate up-quark masses.

Neutral Mesons: Slick Estimate

Let’s do a quick qualitative analysis. (Following the steps of reference [5]) This loop diagram generates a four-point fermion term in the effective lagrangian: $d_s d_d \bar{d_s} \bar{d_d}$. I’ve left out the projection operators for simplicity, so assume all particles are left-handed [6]. We would like to estimate the coefficient of this operator and hence the amplitude for this transition.

First of all, we know the coefficient will have dimensions of inverse mass squared. (On dimensional arguments, the amplitude is estimated by multiplying by the centre of mass energy squared.) The characteristic mass scale comes from the characteristic loop momenta, the W-mass. So let’s stick in a factor of $1/M_W^2$. Each vertex gives a factor of the SU(2) coupling $g$ and the appropriate factor of the CKM matrix, so throw in a factor of $g^4 (V^\dag_{di}V_{is})^2$. Finally let’s attach the familiar factor of $1/4\pi^2$ that comes from the loop. This last factor is almost’ dimensionful [7]. So far we have the coefficient looking like

$\frac{1}{4\pi^2} g^4 \frac{1}{M_W^2}( V^\dag_{di}V_{is})^2$.

We can rewrite this using the Fermi constant $G_F/\sqrt{2} = g^2 / 8 M_W^2$:

$G_f^2 / 16 \pi^2 ( V^\dag_{di}V_{is})^2$.

By dimensional grounds, the amplitude must be proportional to the above factor times a factor of mass-squared. So far so good. This is what we would naively guess the amplitude to be if we didn’t consider the unitarity of the CKM matrix. We haven’t used the unitarity of the CKM matrix yet. We know that only terms with explicit up-quark mass dependence will survive. We know the mass terms arise from the up-type propagators in the loop, so let’s Taylor expand those terms with respect to $m_i^2$. Taking the leading non-trivial term, we can estimate the amplitude to be:

$G_f^2 / 16 \pi^2 ( V^\dag_{di}V_{is})^2 m_i^2$.

$G_f$ contains a hidden factor of $M_W^{-2}$, so the key observation is that the amplitude is proportional to $m_i^2/M_W^2$. That is to say that the unitarity of the CKM matrix decreases our naive estimate of the amplitude by a squared factor of the “smallness of the quark masses relative to the W mass.” (You probably could have guessed this without the hand waving.) This is the GIM mechanism.

You should immediately worry about the top quark, since it is more than twice the mass of the W. Our estimate, however, is saved by the large suppression in $V^\dag_{dt}V_{ts}$. [8] Next, you might worry about cross terms proportional to, say, $V^\dag_{dt}V_{ts}V^\dag_{dc}V_{cs}$. Indeed, for a better approximation one should subtract 2 times the top-charm cross term, whose mass dependence looks like $(m_c^2m_t^2/M_W^2$. Finally, one might be concerned about why the suppression factor is related to the smallness of the up-quark masses’ rather than the degeneracy of the up-quark masses.’ These are essentially the same thing since the up mass is so small. (For this reason one can also safely drop the up-mass terms in a rough estimate.)

Some final thoughts

Our estimate is ultimately validated by actually performing the loop integrals. We work in the rest frame of the meson so that the external momenta are much smaller than the internal loop momenta (which we argued are naturally on the order of the W mass). We can thus set the external momenta to zero in an explicit calculation, making the integrals a bit more manageable. See this earlier post if you need some help with explicitly doing this convergent loop integral. It is worth underlining the importance of the order of magnitude estimate, however. We gleaned a lot of information about the amplitude without actually having to do a nasty integral. One could write down the estimate in fifteen seconds before (or even instead of) spending an hour doing the integral. The estimate then provides a sanity-check for the correctness of a full calculation. This is how actual physics calculations are done.

There’s a slightly deeper way to see how the GIM mechanism works. Instead of working in the mass eigenbasis, one can work in a flavor eigenbasis. Instead of having CKM insertions at the vertices with the W boson, one has chirality– and flavour-mixing mass-insertions along the up-quark propagators.

Also, the end result was dependence on $m_i/M_w$. We made a bit of a heuristic argument about the leading order term being linear, but the amplitude could also depend on other functions of this ratio. Convergence (by power counting) allows us to forget any non-positive powers of this ratio. In general, however, other diagrams may include different dependencies (e.g. logarithmic). See [9] for more details.

Notes and References

[1] I may have accidentally swapped $V$ and $V^\dag$. Don’t worry, the illustration works either way.

[2] Recall that the CKM matrix has an up-type index and a down-type index. When we take the product $V^\dag V$, summing over up-type indices, the result is the unit matrix in the space of down-type flavour (i.e. two down-type indices).

[3] There is another noteable difference between generations that I mentioned in the comments of my last post. The first generation has the +2/3 quark lighter than the -1/3 quark. Because of this, the proton (uud) is lighter than the neutron (ddu) and the proton is the lightest baryon. Thus it is the charge +1 proton, and not the chargeless neutron that is stable. This permits the existence of atoms, such as hydrogen. In the second and third generations, the mass hierarchy is reversed. Hence, in the hypothetical situation where the CKM matrix were diagonal (i.e. equal to the unit matrix), the three quark generations would be decoupled but one would not end up with three copies of the atomic physics. Instead of having two additional stable heavy protons, one would have two stable heavy neutrons and hence no 2nd or 3rd generation atoms. Why is it that the first generation has the mass hierarchy one way and the other two have the reverse hierarchy? This is an open question. The reason may be random (perhaps anthropic) or may come from some yet-undiscovered flavour structure coming from new physics.

[4] Because the meson is neutral it has the same quantum numbers as its antiparticle. The physical propagating state is a superposition of the meson and its antimeson.

[5] Burgess and Moore, The Standard Model: A Primer, section 9.4.2.2. This is a surprisingly delightful book for its physical intuition and its accessibility even to those without a formal quantum field theory background.

[6] To be precise, we need to include the left/right projectors since we are working with a chiral theory. In particular, the weak interaction that mediates this process couples only to left-handed particles. So the term in the effective lagrangian is more correctly written as $\bar{d_s} P_L d_d \bar{d_d} P_L d_s$.

[7] One of my undergraduate advisers explained that $4\pi^2$ is almost dimensionful. In other words, it has dimensions of inverse loop.🙂

[8] In fact, if you think about it, the largeness of the top mass and the smallness of the $V^\dag_{dt}V_{ts}$ are related. They come from the Higgs Yukawa couplings that turn into mass terms. The largeness of the top quark mass is directly related to its small mixing with the other quarks.

[9] Cheng and Li, Gauge theory of elementary particle physics. Page 380-383. Cheng and Li is the grown up’ version of Burgess and Moore. It’s a truly fantastic reference for those interested in phenomenology with clear and pedagogical explanations. Don’t expect them to hold your hands doing calculations, however, you should already know your way around a loop integral. The companion problem book is one of the best for quantum field theory (also focusing on slightly more advanced and applied topics than a first year QFT course). Cheng and Li’s chapters 2-3 also form one of my favourite expositions on renormalisation.