Does the Higgs Field Give the Higgs Particle Its Mass, or Not?

When I wrote my article last week about the relation between the Higgs and gravity, emphasizing that there really was no relation at all, I said that the Higgs field is not the universal giver of mass. I cited four reasons:

  1. The Higgs field does not give an atomic nucleus all of its mass, and since the nucleus is the vast majority of the mass of an atom, that means it does not provide all of the mass of ordinary matter.
  2. Black holes appear at the centers of galaxies, and they appear to be crucial to galaxy formation; but the Higgs field does not provide all of a black hole’s mass. In fact the Higgs field’s contribution to a black hole’s mass can even be zero, because black holes can in principle be formed from massless objects, such as photons.
  3. There is no reason to think that dark matter, which appears to make up the majority of the masses of galaxies and indeed of all matter in the universe, is made from particles that get all of their mass from the Higgs field.
  4. The Higgs field, though it provides the mass for all other known particles with masses, does not provide the Higgs particle with its mass.

Although it doesn’t matter too much to the main point of the Higgs-and-gravity article (since the first three points are not in question), the editor of a leading physics journal, Robert Garisto, took issue with the fourth point, arguing that I was making a statement that really wasn’t right, or at least is too strong. His argument has some merit, though in the end, I stick with my statement. I think it’s worth describing what he had in mind (as best I understand it) and why I feel strongly that one should think about it differently. There are some semantic aspects to the disagreement, but there are also some interesting and important subtle scientific points.  I don’t want to suggest that this discussion is really that big a deal — the very fact that we can argue about whether the Higgs field does or doesn’t provide the Higgs particle with its mass distinguishes the Higgs particle from, say, the W particle, whose mass indisputably arises from the Higgs field. But there’s something to learn here about quantum field theory and how the Higgs mechanism works.

What Garisto said was basically this: within the Standard Model (the equations for the known particles and forces along with the simplest possible Higgs particle), the formula for the Higgs particle’s mass is simply (as explained in this article, for those who’ve got a little math background)

  • mh = √2 (h/2πc2) b v

where

  • v is the non-zero average value of the Higgs field, equal to 246 GeV
  • b is a quantity that determines how strongly the Higgs field interacts with itself
  • h is Planck’s constant, and
  • c is the universal speed limit (often called “the speed of light.”)

This is to be compared with the formula for the W particle’s mass

  • mW = ½ (gW/c2) v

where gW is the number that determines how strongly the Higgs field interacts with the W field. These formulas look very similar, in that both the W mass and the Higgs mass are proportional to v — and so, what’s the difference? It would appear that the Higgs field’s value v determines both masses.

That this argument has a flaw isn’t instantly obvious, and is subtle, especially if you don’t have some technical experience. I’ve been slow to answer because it isn’t so simple to answer this in a form that non-experts can follow. [Fine point: And then, there’s an additional complication which I haven’t yet explained on this website, involving the effect of the strong nuclear force on certain particle masses, which means that making precise statements requires some very tricky caveats.]

To explain my viewpoint, let me start by making an almost-true statement:

  • If the Higgs field’s value were zero instead of v, nature would have a massless W particle, a massless Z particle, a massless electron, massless quarks, massless neutrinos, and so forth.
  • However, there would be no massless Higgs particle.

This is a crucial indication that there is something fundamentally different about the Higgs particle, as far as its relation to the value v, compared to the others.

A Minor Complication, Requiring a Caveat

Now this statement, though simple and clear, isn’t exactly right, because it turns out the strong nuclear force does something complicated: it creates, spontaneously, a composite (i.e. not elementary) Higgs-like field [which I’ll call the “Sigma field”, or “Σ”].  The non-zero value of Σ is far too small to play the role of the what we call the real “Higgs field” H, but it does mean that if H were zero, the W and Z particle and the others wouldn’t quite be massless.  So to evade having to explain this subtlety every single time I make a remark that’s not quite true, I’m going to call this the Σ caveat, and whenever you see me refer to it, you should understand this to mean that the nearby statement in the text is almost true, and would be precisely true if somehow the strong nuclear force were turned off, in which case Σ would not exist at all.

[Fine point:  The correct version of the statement above about the masslessness of other particles is this: in the presence of the strong nuclear force and its composite Σ field, made from combining quarks and antiquark fields to form a new one, the masses of the various known particles wouldn’t quite be zero, but they would be extremely small compared to their actual masses — smaller than we observe by at least a factor of a thousand and in many cases much more. The exception is the Higgs particle’s mass — the mass of the particle of the field H — which would not be similarly reduced.]

Underlying Principles

What’s at stake here? The issue is connected to very deep issues in particle physics.

For certain types of particles, masses can be forbidden by fundamental obstructions, and there are mathematical theorems that for masses to arise, something has to come along and violate the conditions of those theorems, and remove the obstructions.  In the case of W particles and electrons and the rest of the known particles, the theorems apply, and what violates them is “v”, the non-zero value of the Higgs field.  (Σ caveat: or the non-zero value of Σ.) But there is no such theorem for the mass of a Higgs particle, or indeed for any similar type of particle.  The absence of such a theorem is intimately related to the existence of the hierarchy problem.

It is these theorems, and their limitations, that assure that the Higgs field being zero would force many types of particles to be massless (Σ caveat), but not necessarily all of them… and among the massive ones would be the particles of the H field themselves. (Should we still call them “Higgs particles” even when the Higgs field’s average value is zero? I still would, but Garisto wants to call by a different name.  However, this point is semantic: it doesn’t matter what we call them, the important point is that they aren’t massless.)

No matter how we word the debate, the conclusion is the same: the Higgs field is not, in principle, the universal giver of mass to all the elementary particles of nature.

The Demonstration

Still, we’ve seen above that in the formulas of the Standard Model, it sure does look as though the value of the Higgs field determines both the W particle’s and the Higgs particle’s mass.  Both masses are proportional to v.  So how do I explain that?

It’s an accident.  The Standard Model is so very simple that this accident is unavoidable.   In other theories that are even just a little more complicated than the Standard Model, it isn’t true that the Higgs particle’s mass is proportional to v.

In any theory with a single Higgs field, the W particle’s mass must be proportional to v [even more precisely, must vanish when H=0 (Σ caveat)], because of a theorem; but the fact that the Standard Model Higgs particle’s mass is proportional to v is an accident [and in general it need not vanish when H=0 (and on this point there is no Σ caveat)].

So let me now prove this to you.  Apologies to those of you who don’t do math; you’ll have to take my word for the final conclusion.

First let’s recall what happens in the Standard Model, and its simplest possible Higgs.  The Higgs field’s constant value satisfies the equation of motion

  • 0 = a2 H – b2 H3 = – b2 H (H2  – [a/b]2)

where a and b are positive constants.  This has an unstable solution at H=0; the stable solution is at H = a/b, so we identify v, the equilibrium value of the Higgs field, as simply a/b.  The formula for the Higgs mass mh, obtained in this article, was quoted above in terms of b and v, but we should write it in terms of the parameters a and b that actually appear in the equation of motion,

  1. v = a/b
  2. mh = √2 (h/2πc2) a

Now in this case, because v = a/b and therefore a = b v, we can choose to rewrite the second formula as 

  • mh = √2 (h/2πc2) b v

which seems to imply the Higgs field’s value determines the Higgs particle’s mass the same way it determines all the other masses.

However, suppose we move away from the Standard Model very slightly; we don’t add any additional fields or particles, but we change the equation of motion to read

  • 0 = a2 H – b2 H3 – d2 H5

where d, like a and b, is a positive constant.  If you now follow the same mathematical logic used in this article and this one, you will find that v is given by one formula in terms of a, b and d, while mh is given by a quite different formula — and in this case the formula for mh cannot be written as proportional to v.  Despite this, the formulas for the masses of the W and Z particle, the electron, the top quark and the rest continue to be proportional to v.  [(Σ caveat)

Where does the Higgs mass come from, then?  We can’t say it comes from the Higgs field’s value v; they’re related, but not closely enough for one to say that v is entirely responsible for the Higgs particle’s mass.  All we can say is that it arises in a more complex way from the quantities a, b and d in the equation of motion, and so we have to figure out where they come from — which has not yet been done.

Final Comments

The example given above is far from the only one.  A few comments about others.

In other similar variants, it can happen that mh is zero even when v is not, and it can happen that v is zero even when mh is not.  This is never true of, say, the W particle’s mass [(Σ caveat).].

And there are plenty of other variants, including ones with one or more additional spin-zero fields that are unaffected by the electromagnetic, weak nuclear or strong nuclear forces, with similar features.

Also there are variants where the formula relating v to the mass of, say, the top quark, can become more complicated.  Nevertheless, it always remains true that its mass is zero when the Higgs field is zero [(Σ caveat).]  This is not true of the Higgs particle’s mass.

So the point is that while v and the mass of the Higgs particle (or masses of the various Higgs-like particles that may be in the theory) are always given by functions of the parameters (such as a, b and d) that appear in the equations, only in the Standard Model (and a few of its simple variants) can one can write the formula for the mass of the Higgs particle in a form that naively looks very similar to the formula for the mass of any other known particle, such as a W particle.  And this is a consequence of the underlying logical point: for particles like the quanta of the W field, the Z field, the electron field, the muon field, the quark fields, etc., it is impossible for their particles to have masses unless there is a Higgs field around [(Σ caveat)].  But that’s simply not true for the Higgs field H and its particle.

27 responses to “Does the Higgs Field Give the Higgs Particle Its Mass, or Not?

  1. Marcel van Velzen

    Hello Matt,

    To complicate things even more I have a nice question:

    Can there still exist inertial reference frames if the vacuum expectation value of the Higgs is zero?
    That is, can we build mass from massless particles that can behave as rest mass and not move at the speed of light?
    Or should we use your massive Higgs particles?

    I get the impression that the answer to this question is or extremely simple or extremely complicated.

    Thanks in Advance

  2. This is another interesting article that raises questions. Firstly, is the strong force the only force capable of exerting a ‘sigma effect’, or merely one with a large (but not compared to the Higgs field) effect? I assume the mechanism is similar to that of the Higgs mechanism, involving mixing particles. Do the same particles mix as in the Higgs mechanism? what are the particles of the sigma field? Mesons? Quark-antiquark pairs? Ripples in both quark and antiquark fields, or something else?

  3. Dr. Strassler,

    Since mass is a property that can exist with and without a Higgs field, are we any closer to understanding “fundamentally” what mass “is”? E=mc^2 shows the energy equivalence, but what is it? Is non-Higgs mass a “requirement” of some unknown broken symmetry group as the universe cooled because as we approach the early big bang before quark production, all was massless…right? I know we’re taught that if particles have mass, light speed is unattainable. But is it correct to see it the other way? For particles to go at sublight speed or even stop, mass is a requirement.

  4. I think I might have been thinking about this classically. Mass is a property in CFT. In QFT, masses are field excitations. So my question above is modified somewhat but the same, are the mass excitations a “requirement” or a by-product?

  5. Calm down Professor, calm down …
    While young and energetic folk like yourself are dwelling on the differences among “things”, I, who has only few years left in the “organic state” am interested in the common denominator of Nature. I don’t know if it is still valid to call it the Z particle (not to confuse it with the Z boson), the particle of the fundamental field, call it the “G” field.

    With this preamble, I ask you if you had a chance to look at the link (sorry for the link) and my comment (19-Oct) to your article, “Why the Higgs and Gravity are Unrelated”?

    • Nah Oaktree,

      what Prof. Strassler explains are not just some random pedantic unimportant differences among “things” one can gracefully overlook. They are very deep and important if one really wants to understand where mass comes from.
      And that Prof. Strassler cares so much in setting misunderstood things straigth is a very good thing too. I always like and appreciate this a lot and I learn many things from reading such articles here :-)

      Cheers

  6. Dr Strassler, I take / In QFT, masses are field excitations/ this from Rex Groves.
    For example, a rotating wheel appears stationary in stroboscopic effect, is’nt an analogy of zero value of other fields than Higgs Field giving mass to other unknown things- because it is also having momentum and energy.?

    Higgs field is a relative field because it have a mass?. We feel the excitation(non-zero) only if there is no harmony(mathematical illusion) with zero value?

  7. The Higgs field is not, in principle, the universal giver of mass to all the elementary particles of nature. – because it is relative and mathematically tuned to parameters measured by experiment or Lagrangian, and not to quantum corrections to it.
    But Higgs particle(s)are(is) not relative and universal?. Is that Higgs mechanism is an another Darwin theory bridging the discrepancies in Hierarchy problem between quantum mechanics and classical mechanics like bridging man and the ape, the ape-man ?

  8. Dear Prof Strassler,

    I really like this article. I have 2 points to suggest to perhaps clarify even more:

    1) Another central reason why the higgs is not connected to gravity is: the higgs is not the graviton, and it does not set the value of Newton’s constant.

    2) Another way of explaining why the higgs does not give mass to itself is to consider the early universe, where V(h)=+a h^2+b h^4 (rather than -a h^2). In this case v=0, so the W boson, etc, are massless, but the higgs is evidently massive, with m^2=a.

  9. My question is about the limits of validity of the Standard Model, since you talk about the parameter d in an “on the other hand” kind of way. I know you are basing your articles on the Standard Model which is a lot simpler to understand than the alternatives, but please give us a feeling for its problems. What I know of the alternatives is which not all that much. But the incorrect predictions of important physical constants by string theory for example really make me wonder about about the limits of their validity.

    • The only thing I’ve heared is that, apart from some minor not even 2 sigma deviations, the standard model works quite fine so far.
      So there is no reason to talk about alternatives or other off topic things, that when mentioned here cause nothing but flame wars …

    • An article on the limitations of the Standard Model is needed on this site, I agree. I have been giving public lectures on this, which is a good exercise for me in advance of writing such an article. You might want to listen to the colloquium I gave (at least the first half of it) at the University of Toronto, intended for junior physics majors and above — I think you have the background to follow it, if not let me know. http://profmattstrassler.com/2012/10/19/colloquia-here-and-there/

      The Standard Model is a self-consistent but incomplete story; there are 20 or so parameters put in by hand; we don’t know where they come from. But with those 20 parameters we make 1000s of predictions, so it’s still progress. Many of these parameters are well-measured and it’s known that we have the right ones in the equations, because we have used the equations to make many correct predictions.

      But among these parameters are those of the Higgs field; since we don’t even know whether there is one Higgs field or several, we can’t say yet whether the Standard Model matches the data and whether we even have the right kinds of parameters in the equations. There could certainly be new phenomena which means the simple equation of motion for a single Higgs field isn’t right. So this is the most pressing issue; to determine whether the Higgs field is that of the Standard Model.

      Also important is whether there are other particles that can be discovered at the LHC, in which case they will need to be incorporated into an extension of the Standard Model or of a more complicated theory that has a more complicated set of Higgs fields or something even more exotic.

      String theory hasn’t made any definitive predictions yet (despite what you will read here and there). It’s too rich a theory, with too many possible universes that it could exhibit, and so as of now there’s no way to evaluate its validity. Specific applications of string theory with specific types of possible universes have given wrong predictions, but to say that string theory gives wrong predictions is spectacularly overstating the point. It’s like this: imagine you wanted to predict the actions of President Obama, so you took a proposed theory of human beings… and then when you treated the equations in a particular way and looked for a particular type of solution, hoping to find a description of Obama, what came out of the equations was Frank Zappa. The fact that this happens doesn’t necessarily mean your theory can’t describe Obama, it just may mean that you didn’t use the equations in just the right way; any theory of humans is pretty darned complicated!

      In any case, string theory isn’t an **alternative** to the Standard Model; it’s really something vastly more ambitious, and there’s many steps to go from one to the other.

  10. Pingback: Garisto’s Reply to Previous Post | Of Particular Significance

  11. Pingback: Massive Equation « amaya ellman

  12. Pingback: Minute Physics: Why E=mc² is Incomplete « Ye Olde Soapbox

  13. E=mc2 results in an explosion (hence the atom bomb/big bang theory)
    E/m=c2 gives you fusion
    E/c2=m gives matter its mass
    A.E.I.O.U (Absolute Energy equals Input, Output Utilization)

  14. Pingback: Why the Higgs Matters, In A Few Sentences | Of Particular Significance

  15. Dear Matt, two questions
    1. What if the potential is just lambda*phi^4, no symmetry breaking, no quadratic term? Is this a massless or massive Higgs? What is the mass?
    2. If you add higher order terms to the potential, is that still renormalizable?

  16. Pingback: Understanding what the “God particle” isn’t | The Hexacoto

  17. For proper clarification to the general public, could you plainly specify exactly: what is a ‘point’?; what is a ‘wave’?; what is a ‘field’?; in their physics sense of the words? Please be specific if shape or any geometry can be used to help visualize the three concepts. You have Higgs particles and Higgs fields, how are the not identical?

  18. phd ayhan yuzubenli

    if in quantum field theory self interaction is possible, for example an electron can interact with its photon, then it must be possible a higgs particle can interact with its higgs field, then it must acquire mass,

  19. For an elementary particle having mass means that it curves space. The crucial question is not how the particle obtained its mass. The real crucial question is how the particle curves space! Or with other words, how can space be curved?

    • Siva Prasad Kodukula

      Sir, Space is not alone.When it combines with time it will form a fluid like medium. Every mass will be balanced and associated with certain space-time fluid. It is related by ‘Siva’s classical Equations for space-time’m =7×10^12 (d^1/3).and γ d^(8/3) = 1.69×10^12 . The density of the space time fluid give rise to curvature just similar to fluid mechanics. Also the same space time fluid is cause for the creation of mass and explained as the basic building block of mass named as’K-Suryon’.

      Siva Prasad Kodukula

  20. Siva Prasad Kodukula

    I predicted a particle K-Suryon which is mass less and a singularity point as per ‘siva’s Quantum Theory of Gravity’. When a film chage occurs as per ‘Film theory of science it interact with another k-suryon thus the space or diameter will be formed for that singularity. and space time fluid creation starts. This again combines with another two k-suryons to form a mass. The real particle with combination of four k-suryons forms a mass that can exist in this nature. Its mass will be 4.6×10^-64kg and radius is 10^-134mts.It is the least mass and basic building block of any mass.Not Higgs.The formation of space time fluid will have a curvature to hold its mass which is not yet created thus there is a relation between K-Suryon and gravity.Even this is related to electromagnetic field also. The K-Suryon which is in the form of black hole is dark matter.
    People should have stamina to go deep to find the secrets.The arguments will be lenghty when they are not interested to go deep.

  21. It’s all within never and always. Like infinity one should be very precausious when we get close to ‘nothing’ in calculations. Nothing – or no mass – meassured with what? Guess with what we got in our toolbox today.
    like: 0 = a2 H – b2 H3 = – b2 H (H2 – [a/b]2
    0 could as well be a ? There’s more to that dawn 0 than meet our eyes.
    Good Will Hunting (-;

  22. Please read my article: “Nambu’s Nobel Prize, the $\sigma$ meson and the mass of visible matter”, Martin Schumacher,
    Ann. Phys. (Berlin) 526, 215 (2014), arXiv:1403.7804

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