Wednesday, June 26, 2013

An anomaly-like argument in favor of SUSY

A new Higgs tadpole cancellation condition reformulating the hierarchy problem
Strings 2013 [talks] is underway.
The first hep-ph paper today probably got to that exclusive place because the authors were excited and wanted to grab the spot. Andre de Gouvea, Jennifer Kile, and Roberto Vega-Morales of Illinois chose the title
\(H\rightarrow \gamma\gamma\) as a Triangle Anomaly: Possible Implications for the Hierarchy Problem
They point out a curious feature of the diagrams calculating the Higgs boson decay to two photons (yes, it's the process that seemed to have a minor excess at the LHC but this excess went away): while the diagram is finite, one actually gets different results according to the choice of the regularization.

\({\Huge \Rightarrow}\)

In particular, the \(d=4\) direct calculation leads to a finite result but it actually violates the gauge invariance so it can't be right. It should be disturbing for you that wrong results may arise from quantum field theory calculations even if you don't encounter any divergence.

However, the right fix is known: work with a regularization, typically dimensional regularization, that automatically respects the gauge invariance. Using dim reg, in \(d=4-\epsilon\) dimensions, one automatically gets the right result. However, it still disagrees with the wrong result computed directly in \(d=4\).

While this episode doesn't mean that QFT is ill-defined or inconsistent and we actually know how to do things correctly, the finite-yet-wrong result in \(d=4\) surely sounds bizarre. The authors propose a new condition on quantum field theories: this strangeness shouldn't be there. In other words, the wrong, Ward-identity-violating terms in the \(d=4\) calculation should cancel. When they cancel, the \(d=4\) calculation will agree with the correct dim reg \(d=4-\epsilon\) calculation.

The paper suggests that this cancellation is a new general principle of physics that constrains the allowed spectra of particles and fields and that should be added next to the usual triangle diagram gauge anomaly cancellation conditions in the Standard Model and similar gauge theories.

Note that the triangle anomaly diagrams may be blamed on linear divergences in the integrals. Here, the new type of an "anomaly" that should be canceled is also related to the linearly divergent part of certain integrals because they behave differently under the shift of momenta. So even the computational origin of their new "anomaly" resembles the case of the chiral anomaly. In some sense, the "new" anomaly only differs from the well-known triangle anomaly by its replacement of one external gauge boson with the Higgs boson.

These diagrams that have to cancel are close to some Higgs tadpole diagrams – Feynman diagrams you would use to compute the shift of the Higgs vacuum expectation value (vev). The "tadpole cancellation conditions" are well-known to string theorists but they weren't really discussed in the context of ordinary 4D quantum field theories yet. I suppose that there should be a more natural way to phrase and justify the Higgs tadpole cancellation condition. The condition looks like eqn (36)\[

3ge^2M_W + \frac{e^2 g m_H^2}{2M_W} +\sum_{\rm scalars} 2\lambda_S v e_s^2-\!\!\sum_{\rm fermions}\!\! 2\lambda_f^2 v e_f^2 = 0

\] Supersymmetry seems to be the only known natural principle that cancels the new "anomaly". The authors have only checked it by some uninspiring brute force calculation in the MSSM as a function of several parameters. I guess that there's a simple proof that supersymmetry – unbroken or broken at an arbitrary scale – cancels the new "anomaly" condition.

It's probably true and they probably realize that the new condition is mostly equivalent to the usual unbearable lightness and naturalness of the Higgs' being. However, if you might phrase the condition for naturalness as a version of an anomaly cancellation condition, it would probably be (or at least look) much more inevitable than the usual arguments discussing the hierarchy problem.


  1. This is how trace anomalies always arise, and it has been known for decades.

  2. Hi, thanks for your comment. Trace anomaly may use similar diagrams and concepts but what they say about these things isn't quite the same as what's been said for decades, is it? Trace anomaly is a quantum violation of the scaling symmetry but the MSSM isn't scale-invariant despite the fact that it cancels their anomaly-like diagrams, is it?

  3. Removing obviously wrong terms became a common practice in your well-defined and correct QFT, didn't it?

  4. Thanks for this nice very interesting and even for me quite readable article, I like this :-)

  5. Dear Lubos,

    First of all, look at Deser-Schwimmer. They explain that trace anomalies always arise from subtle effects where finite terms survive in dim-reg and if one calculates directly in 4d one gets the wrong answer despite the fact that everything is finite. (Epsilon/Epsilon effect.)

    As to the H->gamma gamma process in the SM. This process is complicated but a major part of it can be captured by making the following two assumptions:

    1.) We just concentrate on one-loop
    2.) We neglect loops of the HIggs field itself

    Under these assumptions, the beta functions of the SM are subleading and thus the rules of CFTs apply. Then, the SM looks like spontaneously broken CFT and the rules of how to compute anomalies via Deser-Schwimmer apply.

  6. Lubos, I wish I had your competence to read through the paper and make an intellectual judgement--but I can't. So instead I have to rely on looking at the academic institutions that employ these authors:

    Northwestern University

  7. Thanks, it makes sense. But wouldn't you agree that it's a new proposal that this cancellation should take place as a consistency condition?

  8. Gauge-dependent terms are wrong, it is clear. Dim-reg procedure gives a gauge-invariant result. But is there a guarantee that this result is correct physically? That's the question!

  9. This article reminded me of that I've for a long time argued intuitively and metaphorically (though mainly inside myself against my greedily knowledge-grabbing self) that one should not grasp a slippery soap too firmly, especially not a wet and oval-shaped one :~< since such a soap - or Nature - is 'ultimately disinclined' to be reliably grasped.

    The QM-aligned string/M-type TOE-like method - preferably to be loosely and peripherally married with an Evolutionary Psycho[physiol]logical Type take is, from where I stand, just the right way of grasping all relevant aspects on What Is going on (grasping them as if by FOOT).

  10. An argument of section 4 of the paper could be used in the eq (9) of (substraction of the two divergent integrals using the symmetry argument).

  11. I am probably not following what you're saying. The equation 9 of the new paper does exactly the d=4 cancellation described in section 4 of the older paper, doesn't it? But this cancellation based on SO(3,1) or SO(4) symmetrization is claimed to be a wrong part of the d=4 calculation in both papers because this treatment violates the gauge symmetry. This much would be agreed for years but the new 2013 paper says new things - namely that the vanishing of the discrepancy is a consistency requirement that puts the hierarchy problem on a firmer ground.

  12. I am hope you understand now better what I mean by reformulation - it is a formulation from other physical ideas to obtain unambiguously the right results.