Wednesday, July 18, 2012

Why there had to be a Higgs boson

Looking at the history of the weak force from a more inevitable viewpoint

The Atlantic published an article (plus an audio interview) trying to explain why some physicists such as Stephen Hawking made bets against the Higgs boson. I haven't really understood the answer.

The story started in Jáchymov/Joachimstahl, a Sudeten Czech town with lots of pitchblende. Marie Sklodowska ordered some of this stuff. (In the 1950s, the mines in the local uranium mines would host lots of political prisoners of Stalinism. Four centuries earlier, the town minted silver thalers/tolars – which the dollars were later named after.)

Instead, I was offered an explanation why Lawrence Krauss was skeptical about the Higgs. Well, I didn't need any assistance on this puzzle: the reason is that Krauss is an idiot, of course. Instead of trying to clarify the mysterious logic that led to a wrong answer, it may be more useful to analyze the correct logic that led to the correct answer. ;-)

Fine, so why have people like me been certain that there had to be a Higgs boson somewhere in the remaining corners?

In December 2011, it became a sure thing that a Higgs boson of mass \(124-126\GeV\) existed. In fact, one could calculate and I did calculate that it would be 5-sigma-discovered sometimes in Summer 2012 and I have announced this fact in all my popular lectures in Spring 2012. ;-)

However, we've known that there had to be a Higgs boson for decades. How could we know that? Well, you may write the elegant electroweak theory with its electroweak gauge symmetry that is – somewhat elegantly but not too elegantly – spontaneously broken by the Higgs doublet. And you may say that the theory is rather pretty. You don't have competitors so it must be right.
Off-topic: The Pioneer anomaly has been explained away as an artifact of heat radiating in the direction of motion. No new physics.
While this argument isn't as weak as you might think, you may rightfully object that it could have loopholes. A completely different theory could be valid. Is there some more waterproof argument showing that a Higgs boson was needed? Yes, there is. A rather detailed presentation of this logic is given in Dr Jiří Horejší's book "Standard Model From Tree Unitarity" that you may buy via the link on the left side; he was an undergraduate quantum field theory instructor of mine. (His name means George Up-quark-type in Czech; there is another guy in the group who's called George Down-quark-type.)

Fine, so let's start.

A century ago, Marie Sklodowska ordered some pitchblende (uraninite) from Jáchymov, Austria-Hungary. This toxic garbage contained lots of yummy ingredients including radium she became famous for. Radium comes in dozens of isotopes. Their decays include alpha-decays, beta-decays, as well as gamma-decays.

Gamma-decays are about the emission of high-energy photons only. Alpha-decays may be modeled as the tunneling of an alpha-particle (helium-4 nucleus) out of the parent nucleus. Because quantum tunneling has a probability that exponentially depends on the thickness times the height of the barrier, you may get a huge hierarchy of possible lifetimes. Alpha-decaying isotopes may have half-lives between tiny fractions of a second and billions of years.

But we're going to be interested in beta-decays. At the elementary level, beta-decay may always be explained as a decay of a neutron (isolated neutron or a neutron inside a nucleus),\[

n\to p+ e^- + \bar \nu_e

\] or similar reactions in which you move particles from one side of the equation to the other side and change them to antiparticles. Once you know quarks and realize that a neutron is \(n=udd\) while a proton is a \(p=uud\), you may also rewrite the true essence of the beta-decay as\[

d\to u+ e^- + \bar \nu_e

\] but it's really the same thing. It doesn't matter much whether you convert protons and neutrons or quarks. In both cases, they may be organized as doublets. Now, in the 1930s, Enrico Fermi understood that this reaction may be derived from quantum field theory if you add a quartic (fourth-order) term in the quantum fields that are able to create or destroy these particles:\[

\LL_{\rm Fermi} \approx G_F \cdot \psi_d \bar\psi_u \cdot \psi_e \bar\psi_{\nu,e}.

\] The product of the four fermionic fields is able to take one of the four particles, destroy it, and create the remaining three particles at the same point. You may have noticed that the objects \(\psi\) are Dirac (or Weyl/Majorana) spinors and I need to contract the indices in the right way – which must be both Lorentz-invariant as well as compatible with other detailed experiments. So the Lagrangian above is schematic if not sloppy.

Just to be sure, I am introducing you to the weak nuclear force right now.

Fermi and others were able to realize that the objects \(\psi_d\bar\psi_u\) were analogous to the "electric currents" \(j^\mu\) in quantum electrodynamics; note that this current \(\bar\psi \gamma^\mu\psi\) is bilinear in the Dirac fields, too. This current should be contracted – summed over \(\mu\) – with the similar current coming from the leptons, i.e. the electron and its neutrino.

However, it wasn't guaranteed that you should insert exactly a single \(\gamma^\mu\) between \(\bar\psi_u\) and \(\psi_d\). You could also insert nothing (I mean the unit matrix) or other possibilities. Depending on whether you insert \(1\), \(\gamma^\mu\), \(\gamma^\mu\gamma_5\), or \(\gamma^{\mu\nu}\) (the latter is an antisymmetrized product of two gamma matrices), you would call the resulting object scalar, vector, axial vector, or tensor, and the resulting four-fermion interaction would be denoted by the acronym, either S or V or A or T. The same gamma matrices with lower indices are inserted to the other, leptonic bilinear structure and the Lorentz indices (if any) are summed over.

It wasn't clear for quite some time which tensor structure was chosen by Nature to run the weak nuclear interactions and beta-decay. Fortunately, the best and most famous experimenters of the 1960s made the experiments and ended up with a consensus: Nature chooses S or T. Feynman and Gell-Mann declared it was bullshit because they already had a theory in which the interaction was V-A – an equal mixture of vector and axial vector terms. Needless to say, Feynman and Gell-Mann were right and the experimenters were wrong. Feynman had remembered that when he saw the wrong paper for the first time, he noticed that the whole S-T conclusion depended on the last point in a chart – and you shouldn't rely on such points too much because if they were reliable, there would be one more behind them. ;-)

So in the mid 1960s, it was established that a more accurate form of the four-fermion interaction looked like this:\[

\LL_{\rm Fermi} \approx G_F \cdot \bar\psi_u(1-\gamma_5)\gamma^\mu\psi_d \cdot \bar\psi_{\nu,e} (1-\gamma_5)\gamma_\mu \psi_e.

\] The mixture \((1-\gamma_5)\) is what prevents you from assigning a clear name – vector or pseudovector – to the interaction term and that's what violates the left-right symmetry (parity) of the underlying theory. In fact, the interaction above only applies to the left-handed (left-spinning) quarks and leptons and the right-handed (right-spinning) antiquarks and antileptons.

The interaction above is problematic when you try to compute loop corrections. It's because it is nonrenormalizable. Each \(\psi\) has units of \({\rm mass}^{3/2}\) as you can see from the Dirac kinetic term. Four of them have units of \({\rm mass}^6\). So you need to subtract two masses to get the usual Lagrangian density whose units are \({\rm mass}^4\). It means that the dimension of \(G_F\), the coefficient, is \({\rm mass}^{-2}={\rm length}^2\). Such negative mass dimensions mean that in higher-loop diagrams, the higher power of \(G_F\) has to be multiplied by positive powers of the loop momenta and that's what is causing high-energy, ultraviolet divergences that are getting worse as you add additional loops.

So the direct four-fermion interaction is non-renormalizable. More generally, it sucks at higher energies. The founding fathers of the electroweak theory, building on approximate visions that were in place since Oskar Klein's work in the 1930s, realized that one may rebuild the four-fermion interaction into a renormalizable theory in a full analogy with the renormalizable theory of electrodynamics.

The interaction between the four fermionic fields i.e. between the two currents isn't direct, they decided, but it is due to an indirect interaction with an intermediate messenger field \(W_\mu\) which is analogous to the electromagnetic potential \(A_\mu\). So the fundamental Lagrangian doesn't have a direct interaction between the two currents. Much like electromagnetism contains \[

e\cdot \bar\psi \gamma^\mu \psi \cdot A_\mu,

\] the weak interaction includes similar terms such as\[

g\cdot \bar\psi_u (1-\gamma_5) \gamma^\mu \psi_d \cdot W_\mu.

\] The two expressions differ in several respects. First, the coupling constant is called \(g\) and not \(e\). It has a different value but this difference is not exactly profound or conceptual. More importantly, the two \(\psi\) fields in the weak interaction case have different subscripts. It's because unlike the electromagnetism, the weak force is changing the identity of the participating particles: don't forget about the beta-decay which is the exclusive "everyday life phenomenon" that experimentally justifies all this research. Third, there is the extra "chiral" \((1-\gamma_5)\) inserted which makes the weak force left-right-asymmetric.

Finally, the messenger field is called \(W_\mu\) rather than \(A_\mu\). In this case, it's not just a different name. The field \(W_\mu\) has some different properties, too. In particular, it is a field associated with massive particles, the W-bosons, and not with massless particles created by the \(A_\mu\) fields, the photons. The massive character of the W-bosons is what turns the weak interaction into a short-range force.

In electromagnetism, the currents would also "effectively interact" via the electromagnetic field and you would get interaction energies such as \(Q_1 Q_2/r\) where \(r\) is the distance between the charges. Note that each \(Q_1,Q_2\) results from the integral of some \(j^\mu\sim \bar\psi \gamma^\mu\psi\). In the case of the weak force, the \(1/r\) dependence is combined with a quickly, exponentially decreasing factor \(\exp(-m_W r)\) so that the function \(1/r\) effectively turns into a multiple of the delta-function. That's why the original Fermi interaction could have been approximated by a "contact interaction" – all the four participating fermions have to sit at the same spacetime point for the interaction to operate.

Great. So the renormalizability of the four-fermion interaction forces you to introduce a new vector field generalizing the electromagnetic field. It has to be massive. One may also show that for related reasons, the vector bosons have to be associated with a gauge symmetry – they have to be gauge bosons. One realizes that \(SU(2)\times U(1)\) is the minimal group for which the electroweak gauge theory becomes viable. Check the book at the top.

By almost purely theoretical advances, we may argue that Marie Sklodowska's observations of the beta-decay demand the existence of the W-bosons and Z-bosons and indeed, they were discovered at CERN in the early 1980s. Several times earlier, the Nobel prize was already given to the theorists who predicted them – Glashow, Salam, Weinberg. By the way, a week ago, Fox News and others ran a story about Pakistani bigots' success in turning their best scientist Abdus Salam into a taboo heretic. Another reason to thermonuke these breathtaking barbarian fanatics and would-be allies.

Higgs needed for a healthy WW scattering

While I omit the detailed derivation of the existence of the W-bosons and Z-bosons (because we've known that W-bosons and Z-bosons exist from the experiments for 30 years anyway), let me say a few more details about very similar arguments that show that a Higgs boson is necessary as well.

Because the W-bosons exist and are massive, they must have 3 polarizations. That's clear if you go into the W-bosons' rest frame. That's different than for photons that only have 2 transverse polarizations. A photon moving in the \(z\) direction may have its electric field pointing in the \(x\) or \(y\) direction or their combination (or it may be circularly polarized which is still a complex combination of the linear ones) but nothing else. A W-boson may also be longitudinally polarized, i.e. with the field parallel to the direction of motion.

Now, the electroweak theory contains the W-bosons so it must also have a meaningful answer to the following question: What happens if you collide two longitudinally polarized \(W^+\) bosons at high energies? In the leading order in the expansion over the small coupling constants \(g\), the answer is given by the tree-level Feynman diagrams (those without any loops).

(It doesn't matter whether you collide two W-bosons with the same charge or the opposite charge. All the amplitudes are related by the crossing symmetry etc.)

What are the tree-level diagrams contributing to the \(2\to 2\) scattering of \(W^+_L\) bosons? Well, there are three of them. You may connect the 4 external W-boson lines by a direct vertex because there are quartic interactions in Yang-Mills theory. Or you may obtain this interaction between the 4 external lines by inserting an intermediate photon or Z-boson in two possible channels – in \(t\) or \(u\) channel, in my particular formulation of the problem (with two equally charged W-bosons in the initial state).

The resulting scattering amplitude from the three tree-level diagrams is\[

{\mathcal M}_{\rm gauge} = -\frac{g^2}{4 m_W^2} \zav{4-\frac{3}{\rho}} s + {\mathcal O}(u^0).

\] For different assignment of the positive/negative charges, you would get various permutations of \(s,t,u\). Now, \(\rho=m_W^2/m_Z^2\cos^2\theta_W\) is equal to one at the leading (tree-level approximation) so \(4-3/\rho\sim 1\). Nevertheless, you see that a tree-level scattering amplitude increases as \(s\sim E_{c.m.}^2\) at high enough energies. Because the coupling constants are rather small – and run slowly – the higher-order diagrams can't beat this growth.

This growth of the scattering amplitude with energy really means that the probability of scattering exceeds 100 percent for W-boson energies that are comparable to \(1\TeV\). That's too bad because probabilities shouldn't ever surpass 100 percent, regardless of any uncertainty we may have about the detailed laws of physics.

So the correct amplitude for the longitudinal W-boson scattering has to be different. It can't possibly increase with \(s\) so quickly. Something must slow down the growth. And whatever is able to do so may be interpreted as a new diagram contributing to the scattering process. And any new diagram may be interpreted as the "exchange of some new stuff".

It's important to realize that you can't just modify the contributions of individual Feynman diagrams for high energies by any "fixes" you could find convenient. The Feynman diagrams' contributions to scattering amplitudes are completely determined by the spectrum of particles and their interactions. Even in string theory in which the high-energy behavior of the scattering amplitudes starts to "mysteriously soften", you may still prove that this softening is inseparable from new particle species – in fact, a whole infinite tower of excited string modes.

So there have to be new particles below a \(1\TeV\) or so whose interactions with the W-bosons force you to include new diagrams and those new diagrams erase most of the unwelcome growing term proportional to \(s\). The Standard Model Higgs boson does the job in a very straightforward way. If you add the two diagrams with the exchange of a Higgs boson – in the same two channels in which you exchanged the photon or Z-boson above – you obtain a new contribution to the scattering amplitude,\[

{\mathcal M}_{\rm higgs} = \frac{g^2}{4 m_W^2} s + {\mathcal O}(u^0)

\] That's neat because for \(\rho=1\) as promoted above, this exactly cancels the leading term that grew with \(s\). So the leading high-energy behavior is actually constant and you avoid the conflict with the condition that probabilities should never exceed 100 percent.
Directly related breaking news: Five minutes ago, ATLAS released a paper that, for the first time, brings experimental evidence in favor of the interaction HWW between the Higgs boson and a pair of W-boson external lines, exactly the cubic vertex that was used (twice) to derive \({\mathcal M}_{\rm higgs}\) above. The significance of the signal has exceeded 3 sigma.
A single Higgs boson isn't necessarily the only way how to cancel the pathologically growing terms in the scattering amplitude; it's just the simplest one. However, you may think about "qualitatively different" analytic forms of the canceling contributions and you will find out that for them to play the canceling job, each of them has to look like the exchange of a scalar boson. Whenever it does the job, the couplings of the scalar boson are exactly such that you may derive them from a pretty gauge theory with a spontaneously broken symmetry (although the breaking may be done by several Higgs fields, not just one).

You may derive the Feynman rules for the W-bosons, Z-bosons, and the Higgs boson if you start with the "elegant" theory with the electroweak symmetry and you add the symmetry breaking fields, and so on. But as we've seen, you may also start with an ugly theory that describes the dirty experimental data – the four-fermion interaction at the very beginning – and you may keep on refining the theory so that it's well-behaved and cancels an ever greater number of new pathologies. At the end, you end up with a theory that is equivalent to the "pretty" electroweak theory derived from the symmetries – the equivalence boils down to simple field redefinitions. For example, the first approach makes the point \(h=(0,0)\) natural as a starting point (the maximum of the Higgs potential); the second approach is naturally expanded around the physical vacuum \(h=(v,0)\).

One could also try to suppress the high-energy scattering by some more obscure methods, by making the W-bosons composite or by exchanging composite particles. You may get into the muddy waters of compositeness, technicolor, and similar proposals. However, the ability of these theories to cancel the unwanted terms would always look a bit surprising and moreover, all these theories predict some new deviations from the Standard Model physics that would have already been seen. In fact, precision measurements have shown that the compositeness scale couldn't have been below \(10\TeV\) or so.

There are all kinds of reasons why I would have bet that technicolor-like theories couldn't have been right (the natural compatibility of the Higgs mechanism with string theory was a major reason for me) but even if they were right, it would still be true that they must provide us with some new particles whose propagators appear in new diagrams that cancel the pathological growth of the WW scattering amplitudes in the Higgsless Standard Model. I just find a Higgs boson to be the most natural solution – it just "exactly" cancels the term proportional to \(s\).

With a dose of rational thinking and some mathematics, you may "directly see" that there has to exist a Higgs boson just by looking at Marie Curie and her Bohemian pitchblende. Good enough particle physicists have known how to "see" this far for quite some time.

And that's the memo.


  1. I had been under the impression those who "bet against the Higgs" had bet against successfully finding it, not it's existence! Yikes!

  2. Aah, from this nice article I can actually see what the problem with the WW scattering amplitudes was, and HOW the higgs tames them :-)

    I always prefer (slightly, such that I can get it) technical arguments to explanations using just words. The lack of more accurate explanations has even started to annoy me sometimes when reading popular articles, LOL ... :-D

    So thanks for this (for me at an exactly right level ) article, Lumo :-)

  3. The last couple of articles by Lubos are by no means easy. If you are not familiar with field Lagrangian interaction terms, spinors, Dirac matrices, billinear covariants, the methods of renormalization etc - in essence, if you don't know the QFT at good level already - you probably won't understand them. For me, this article is still a little over my head, because I haven't studied the electoweak unification at all and I haven't studied renormalizations properly. But I feel that I am not that far from the level, where I will be able to understand such articles.
    I will still have to convince myself that renormalization rechniques are physics and not some kind of crazy mathematical witchery. Dirac never accepted renormalization as a valid procedure and for Feynman, it was a "dippy process"

    His words are: "The shell game that we play to find n and j is
    technically called renormalization. But no matter how clever the word,
    it is what I would call a dippy process! Having to resort to such
    hocus-pocus has prevented us from proving the theory of quantum
    electrodynamics is mathematically self-consistent. .... I suspect that
    renormalization is not mathematically legitimate."

  4. You guys -- leading theoretical physicists I mean -- really are a superior form of intellectual being. I'm amazed at the conversations you have with each other through the decades. Now if we could only get similar intellect focused on our social, cultural, and historical problems!

  5. Yeah, I`m basically familliar with stuff at the "Demysified" level, the one about SUSY for example:

    explains (reviews) important things such as different spinors, gamma matrices, abelian and non abelian gauge theories, calculation of Feynman diagrams, and many other useful things quite nicely. Apart from introducing SUSY... At the moment it is my favorite DM book and last weekend I`ve just reread the spinor chapter ...

    Concerning the renormalization business I`ll have to dig a bit deeper too ... ;-)


  6. an interesting article

  7. Yup! They were dismissive of the very existence. After all, if you believe that the particle exists and it has the expected properties at least approximately, then it's pretty much guaranteed that after a year or two, the LHC has to find it. So all those people believed that the Higgs had to be literally non-existent - Veltman, Hawking, Krauss, and others.

  8. i am waiting to see how many of them they will find and what the actual mechanism is

  9. Hm Luke, I dont know ...

    I think the main problem ist that most people who are (at powerful positions) in politics are there for egoistic reasons rather than in order to serve the puplic or solve urgent social, cultural, or other problems ...
    Not sure how much this is correlated with a lack of intellectual power.

  10. Hi lumo, could you speculate on what the world would look like without the Higgs boson (or any other electroweak symmetry breaking mechanism)?

  11. Thanks, Lubos, for this fantastic article. Aso for me just at the right level to be able to read it, enjoy it and learn something new on the way. Next thing I will do is to read the article why the Higgs is a "victory for string theory".
    I completely get now what you want to say with the picture.

  12. Hi Lubos,
    did you consider the discover of the Higgs as some form of evidence for string theory? And if so, why?

  13. Great article, it made me open prof. Horejsi's book and read through this again after long time. Wonderful book, I like how the theory is built there. I always liked his lectures, both introduction to electroweak theory and quantum field theory.

  14. I've been waiting for someone to blame America's obesity problem on the
    Higgs, but I never expected to see it first in cartoon format:

    Americans are way overweight because they eat too much and sit
    around too much. But they don't want to hear this because they are too
    undisciplined to cut out their Fritos and BonBons and are too lazy to
    get off their ass and burn some mass.

    Their hopes are probably too high that the anti-Higgs will be
    discovered and made into a mass-reducing pill. I suggest that they cross
    the event horizon of a supermassive black hole, and there they'll find
    more than enough anti-Higgs bosons to be made into a lifetime supply of
    mass-reducing pills. This is one way for the Universe to rid itself of
    some deadweight!

  15. BTW, the resolution of the "pioneer anomaly" without new physics, seems to have taken quite a while to get coverage! Wikipedia already references it, according to which the paper appear on arxiv in April, and the paper appears to have been formally published in June. Also, thermal radiation pressure was apparently the preferred explanation at least as early as last year, also according to Wikipedia. It perhaps speaks to media perpetuation of sensationalist nonsense like our understanding of physics possibly being wrong, that we are only now seeing a story reporting that the pioneer anomaly has been given a mundane explanation.

  16. Yup,

  17. This is my illegitimate (since question was put to Lumo) but not evil answer to Ergil: For a start, you would not be in it. :)

  18. ... everybody who naively and honestly wants to be helpful and improve things for a larger group of people should never ever try to meddle with the muddy business of politicians.
    See how much fun I got for example as I asked if governing (moderating) the SE sites in general should and could be made a bit more (direct) democratic and transparent for ALL (not only the high reputation) members that belong to the different communities, LOL :-D

    I admit that the title of my question probably was a bit ... "catchy" ... :-P

  19. Dear Luke, intellectual rabbits and no match for power hungry wolves. Psychopatic personality traits are a big advantage in politics because without moral and without regards to the wellbeing of others, you have an easier time lying, stealing, promising empty promises and competing with other wolves. The politicians call the intellectuals the "chattering class". And chattering is no match against guns, police, military, courts.

  20. I was hoping for something slightly less obvious...

  21. Dear Ergil, apologies for having given you no answer but I agree with Peter. The world without a Higgs would be a world without you - and a world without your somewhat annoying questions. ;-)

    I don't know what to imagine under the "world without the Higgs". If you just remove the Higgs from the Standard Model, you get an inconsistent theory where probabilities may exceed 0 or 100 percent. It can't work. It makes no sense to think about such a world because it's unphysical.

    So you must mean a different world with something instead of a Higgs. But such a different world isn't unique. I can't give you a unique answer. A different world without Higgs may look like Springfield with two-dimensional Simpsons or a chessboard or the conformal N=4 gauge theory without life or whatever.

  22. What I actually meant was "What would world look like if the electroweak symmetry were unbroken?".

  23. I see.

    The W-bosons and Z-bosons would stay massless, like a photon. The hypercharge U(1) would behave similarly to our electromagnetism. However, quarks and leptons would be massless, too. That's why the fine-structure constant would run all the way to zero at the vanishing energy scale - the electron mass.

    Electrons would always move by the speed of light etc. There could be confined massive hadrons, however, because their mass comes from QCD, not from the electroweak symmetry. There wouldn't be a difference between a proton and neutron, however, because the symmetry relating them would be unbroken.

    Maybe they would form atom-like large nuclei. There could be some new chemistry that would be fully composed of "slightly more complex" nuclear physics.

    Maybe life could exist in such a world, too. I would have to make very many tests and calculations. Someone could probably tell you a clearer verdict on such a thing.

  24. I was thinking about the fact that the Higgs boson restores unitarity to the Standard Model. Unitarity appears central to string theory, so this may be interesting?

  25. Unitarity is central to string theory - as well as any consistent theory. Yes, there are also "directions of would-be research" that just don't care about consistency i.e. unitarity at all. In this sense, Higgs is "just like strings". However, it's not just like strings because unitarity can be discussed and required and has been discussed and required in non-stringy quantum field theory, too. Unitarity isn't really a "purely stringy effect" in any sense.