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Flip Tanedo | USLHC | USA

View Blog | Read Bio

“Known knowns” of the Standard Model

This is the tenth (or so) post about Feynman diagrams, there’s an index to the entire series in the first post.

There is a famous quote by former Secretary of Defense Donald Rumsfeld that really applies to particle physicists:

There are known knowns.
These are things we know that we know.
There are known unknowns.
That is to say, there are things that we know we don’t know.
But there are also unknown unknowns.
There are things we don’t know we don’t know.

Ignoring originally intended context, this statement describes not only the current status of the Standard Model, but accurately captures all of our hopes and dreams about the LHC.

  • We have “known knowns” for which our theories have remarkable agreement with experiment. In this post I’d like to summarize some of these in the language of Feynman diagrams.
  • There are also “known unknowns” where our theories break down and we need something new. This is what most of my research focuses on and what I’d like to write about in the near future.
  • Finally, what’s most exciting for us is the chance to trek into unexplored territory and find something completely unexpected—“unknown unknowns.”

Today let’s focus on the “known knowns,” the things that we’re pretty sure we understand. There’s a very important caveat that we need to make regarding what we mean by “pretty sure,” but we’ll get to that at the bottom. The “known knowns” are what we call the Standard Model of particle physics*, a name that says much about its repeated experimental confirmations.

* — a small caveat: there’s actually one “known unknown” that is assumed to be part of the Standard Model, that’s the Higgs boson. The Higgs is currently one of the most famous yet-to-be-discovered particle and will be the focus of a future post. In the meanwhile, Burton managed to take a few charming photos of the elusive boson in his recent post.

First, let’s start by reviewing the matter particles of the Standard Model. These are called fermions and they are the “nouns” of our story.

Matter particles: the fermions

We can arrange these in a handy little chart, something like a periodic table for particle physics:

Let’s focus on only the highlighted first column. This contains all of the ‘normal’ matter particles that make up nearly all matter in the universe and whose interactions explain everything we need to know about chemistry (and arguably everything built on it).

The top two particles are the up and down quarks. These are the guys which make up the proton (uud) and neutron (udd). As indicated in the chart, both the up and down quarks come in three “colors.” These aren’t literally colors of the electromagnetic spectrum, but a handy mnemonic for different copies of the quarks.

Below the up and down we have the electron and the electron-neutrino (?e), these are collectively known as leptons.  The electron is the usual particle whose “cloud” surrounds an atom and whose interactions is largely responsible for most of chemistry. The electron-neutrino is the electron’s ghostly cousin; it only interacts very weakly and is nearly massless.

As we said, this first column (u, d, e, and ?e) is enough to explain just about all atomic phenomena. It’s something of a surprise, then, that we have two more columns of particles that have nearly identical properties as their horizontal neighbors. The only difference is that as you move to the right on the chart above, the particles become heavier. Thus the charm quark (c) is a copy of the up quark that turns out to be 500 times heavier. The top quark (t) is heavier still; weighing in at over 172 GeV, it is the heaviest known elementary particle. The siblings of the down quark are the strange (s) and bottom (b) quarks; these have historically played a key role in flavor physics, a field which will soon benefit from the LHCb experiment. Each of these quarks all come in three colors, for a total of 2 types x 3 colors x 3 columns = 18 fundamental quarks. Finally, the electrons and neutrinos come with copies named muon (?) and tau (?). It’s worth remarking that we don’t yet know if the muon and tau neutrinos are heavier than the electron-neutrino. (Neutrino physics has become one of Fermilab’s major research areas.)

So those are all of the particles. As we mentioned in our first post, we can draw these as solid lines with an arrow going through them.  You can see that there are two types of leptons (e.g. electron-like and neutrino) and two types of quarks (up-like and down-like), as well as several copies of these particles. In addition, each particle comes with an antiparticle of opposite electric charge. I won’t go into details about antimatter, but see this previous post for a very thorough (but hopefully still accessible) description.

You can think of them as nouns. We now want to give them  verbs to describe how they can interact with one another. To do this, we introduce force particles (bosons) and provide the Feynman rules to describe how the particles interact with one another. By stringing together various particle lines to the vertices describing interactions, we end up with a Feynman diagram that tells the story of a particle interaction. (This is the “sentence” formed from the fermion nouns and the boson verbs.)

We will refer to these forces by the ‘theories’ that describe them, but they are all part of the larger Standard Model framework.

Quantum Electrodynamics

The simplest force to describe is QED, the theory of electricity and magnetism as mediated by the photon. (Yes, this is just the “particle” of light!) Like all force particles, we draw the photon as a wiggly line. We drew the fundamental vertex describing the coupling of an electron to the photon in one of our earliest Feynman diagram posts,

For historical reasons, physicists often write the photons as a gamma, ?. Photons are massless, which means they can travel long distances and large numbers of them can set up macroscopic electromagnetic fields. As we described in our first post, you are free to move the endpoints of the vertex freely. At the end of the day, however, you must have one arrowed line coming into the vertex and one arrowed line coming out. This is just electric charge conservation.

In addition to the electron, however, all charged particles interact with the photon the same vertex. This means that all of the particles above, except the neutrinos, have this vertex. For example, we can have an “uu?” vertex where we just replace the e‘s above by u‘s.

QED is responsible for electricity and magnetism and all of the good stuff that comes along with it (like… electronics, computers, and the US LHC blog).

Quantum Flavordynamics

This is a somewhat antiquated name for the weak force which is responsible for radioactivity (among other things). There are two types of force particle associated with the weak force: the Z boson and the W boson. Z bosons are heavier copies of photons, so we can just take the Feynman rule above and change the ? to a Z. Unlike photons, however, the Z boson can also interact with neutrinos. The presence of the Z plays an important role in the mathematical consistency of the Standard Model, but for our present purposes they’re a little bit boring since they seem like chubby photon wanna-be’s.

On the other hand, the W boson is something different. The W carries electric charge and will connect particles of different types (in such a way that conserves overall charge at each vertex). We can draw the lepton vertices as:

We have written a curly-L to mean a charged lepton (e, ?, ?) and ?i to mean any neutrino (?e, ??, ??). An explicit set of rules can be found here. In addition to these, the quarks also couple to the W in precisely the same way: just replace the charged lepton and neutrino by an up-type quark and a down-type quark respectively. The different copies of the up, down, electron, and electron-neutrino are called flavors. The W boson is special because it mediates interactions between different particle flavors. Note that it does not mix quarks with leptons.

Because the W is charged, it also couples to photons:

It also couples to the Z, since the Z just wants to be a copy-cat photon:

Finally, the W also participates in some four-boson interactions (which will not be so important to us):

Quantum Chromodynamics

Finally, we arrive at QCD: the theory of “strong force.” QCD is responsible for binding quarks together into baryons (e.g. protons and neutrons) and mesons (quark–anti-quark pairs). The strong force is mediated by gluons, which we draw as curly lines. Gluons couple to particles with color, so they only interact with the quarks. The fundamental quark-gluon interaction takes the form

The quarks must be of the same flavor (e.g. the vertex may look like up-up-gluon but not up-down gluon) but may be of different colors. Just as the photon vertex had to be charge-neutral, the gluon vertex must also be color-neutral. Thus we say that the gluon carries a color and an anti-color; e.g. red/anti-blue. For reasons related to group theory, there are a total of eight gluons rather than the nine that one might expect. Further, because gluons carry color, they interact with themselves:

QCD—besides holding matter together and being a rich topic in itself—is responsible for all sorts of head aches from both theoretical and experimental particle physicists. On the experimental side it means that individual quarks and gluons appear as complicated hadronic jets in particle colliders (see, e.g. Jim’s latest post). On the theoretical side the issue of strong coupling (and the related idea of confinement) means that the usual ‘perturbative’ techniques to actually calculate the rate for a process quickly becomes messy and intractable. Fortunately, there are clever techniques on both fronts that we can use to make progress.

The missing piece: The Higgs Boson

Everything we’ve reviewed so far are known knowns, these are parts of our theory that have been tested and retested and give good agreement with all known experiments. There are a few unknown parameters such as the precise masses of the neutrinos, but these are essentially just numbers that have to be measured and plugged into the existing theory.

There’s one missing piece that we know must either show up, or something like it must show up: the Higgs boson. I’d like to dedicate an entire post to the Higgs later, so suffice it to say for now that the Higgs is an integral part of the Standard Model. In fact, it is intimately related to the weak sector. The importance of the Higgs boson is something called electroweak symmetry breaking. This is a process that explains why particles have the masses that they do and  why the W, Z, and photon should be so interwoven. More importantly, the entire structure of the Standard Model breaks down unless something like the Higgs boson exists to induce electroweak symmetry breaking: the mathematical machinery behind these diagrams end up giving nonsensical results like probabilities that are larger than 100%. Incidentally, this catastrophic nonsenical behavior begins at roughly the TeV scale—precisely the reason why this is the energy scale that the LHC is probing, and precisely the reason why we expect it to find something.

A fancy way of describing the Standard Model is that there are actually four Higgs bosons, but three of them are “eaten” by the W and Z bosons when they become massive. (This is called the Goldstone mechanism, but you can think of it as the Grimm’s Fairy Tale of particle physics.) This has led snarky physicists to say things like, “Higgs boson? We’ve already found three of them!”

Theories and Effective Theories

By specifying the above particles and stating how the Higgs induces electroweak symmetry breaking, one specifies everything about the theory up to particular numbers that just have to be measured. This is not actually that much information; the structure of quantum mechanics and special relativity fixes everything else: how to write down predictions for different kinds of processes between these particles.

But now something seems weird: we’ve been able to check and cross-check the Standard Model in several different ways. Now, however, I’m telling you that there’s this one last missing piece—the Higgs boson—which is really really important… but we haven’t found it yet. If that’s true, how the heck can we be so sure about our tests of the Standard Model? How can these be “known knowns” when we’re missing the most important part of the theory?

More generally, it should seem funny to say that we “know” anything with any certainty in science! After all, part of the excitement of the LHC is the hope that the data will contradict the Standard Model and force us to search for a more fundamental description of Nature. The basis of the scientific method is that a theory is only as good as the last experiment which checked it, and there are good reasons to believe that the Standard Model breaks down at some scale. If this is the case, then how can we actually “know” anything within the soon-to-be-overthrown Standard Model paradigm?

The key point here is that the Standard Model is something called an effective theory. It captures almost everything we need to know about physics below, say, 200 GeV, but doesn’t necessarily make any promises about what is above that scale. In fact, the sicknesses that the Standard Model suffers from when we remove the Higgs boson (or something like it) are just the theory’s way of telling us, “hey, I’m no longer valid here!”

This is not as weird as one might think. Consider the classical electromagnetic field of a point particle: it is a well known curiosity to any high school student that the potential at the exact location of the point source is infinity. Does that mean that an electron has infinite energy? No! In fact, this seemingly nonsensical prediction is classical electromagnetism telling us that something new has to fix it. That something new is quantum mechanics and the existence of antiparticles, as we previously discussed.

This doesn’t mean that the effective theory is no good, it only means that it breaks down above some region of validity. Despite the existence of quantum mechanics, the lessons we learn from high school physics were still enough for us to  navigate space probes to explore the solar system. We just shouldn’t expect to trust Newtonian mechanics when describing subatomic particles. There’s actually a rather precise sense in which a quantum field theory is “effective,” but that’s a technical matter that shouldn’t obfuscate the physical intuition presented here.

For physicists: the theory of the Standard Model without a Higgs is a type of non-linear sigma model (NL?M). This accurately describes a theory of massive vector bosons but suffers from a breakdown of unitarity. The Higgs is the linear completion of the NL?M that increases the theory’s cutoff. In fact, this makes the theory manifestly unitary, but does not address the hierarchy problem. For an excellent pedagogical discussion, see Nima Arkani-Hamed’s PiTP 2010 lectures.

Where we go from here

The particles and interactions we’ve described here (except the Higgs) are objects and processes that we have actually produced and observed in the lab. We have a theory that describes all of it in a nice and compact way, and that theory requires something like the Higgs boson to make sense at high energies.

That doesn’t mean that there aren’t lots of open questions. We said that the Higgs is related to something called “electroweak symmetry breaking.” It is still unknown why this happens. Further, we have good reason to expect the Higgs to appear in the 115 – 200 GeV range, but theoretically it takes a “natural” value at the Planck mass (1019 GeV!). Why should the Higgs be so much lighter than its “natural” value? What particle explains dark matter? Why is there more matter than anti-matter in the universe?

While the Higgs might be the last piece of the Standard Model, discovering the Higgs (or something like it!) is just the beginning of an even longer and more exciting story. This is at the heart of my own research interests, and involves really neat-sounding ideas like supersymmetry and extra dimensions.

  • Hi…


    Hmm…from this articles “But What Are Quarks Made Of Part 2” & this article its looks like “Extended SUSY” is the best explaination for “light & matter question” which show The Higg’s mechanism pattern…

    Am…I wrong ???

  • Hello Fujimia,

    I’m not quit sure where you’re finding a reference to extended supersymmetry. This is a kind of highly symmetric theory which is unlikely to be relevant at LHC energies. (The reason is that it too symmetric; the fancy phrase is that it doesn’t allow chiral matter, but let’s not worry about what this means for now.)

    The mystery of the Higgs mechanism is why the Higgs wants to have a “vacuum expectation value,” i.e. why it wants to behave like a superconductor. Further, another mystery is why the Higgs mass is so light compared to the Planck scale. Hopefully I can get to these soon, but I haven’t yet thought very carefully about how to best present it.


  • The Dirac equation is beautiful and the truncating its mass term and replace it by the dubious Higgs mechanism is not good idea.

    G. Quznetsov, It is not Higgs. Prespacetime Journal| May 2010 | Vol. 1 | Issue 3 | Page 314-343.

  • Dear Fujimia, as Flip correctly mentions, “extended supersymmetry” is a type of supersymmetry that cannot account for observable phenomena.

    A theory with extended supersymmetry has “at least 8 supercharges”: here, 8 is twice 4, and 4 is the number of supercharges in an ordinary, “minimal supersymmetry” (not extended supersymmetry) which is compatible with the real world, and actually favored over the Standard Model.

    Extended supersymmetry implies that each particle has superpartners with spins that differ by up to 1 unit. Moreover, extended supersymmetry implies that the world would have to be left-right symmetric, but it’s not.

    Extended supersymmetry is important conceptually, for the calculation of phenomena in “nice” theories in field theory and string theory. This high degree of supersymmetry often allows things to be calculated exactly, and statements to be made with certainty, but the price we pay is that these theories – and “vacua” of string theory – cannot be relevant for the observable world which is compatible at most with the minimal supersymmetry.

    Dear Gunn, quite on the contrary: it is must more beautiful and consistent to produce the fermionic masses out of their interactions with a Higgs field than from a direct mass term. In particular, this method of giving fermions their masses is necessary to avoid negative probabilities in any theory in which fermions also interact via weak interactions – through intermediate massive gauge bosons.

  • But the Higgs mass do not have anything to do with gravity, and the Dirac equation with mass terms gives gravity (see link in my previous post — the negative probabilities is also no here).

  • Dear Gunn, the negative probabilities – loss of unitarity – are absent as long as you don’t include any weak nuclear interactions. If you do, you need a Higgs to avoid the negative probabilities.

    *Everything* that carries energy or momentum has “something to do with gravity”. Every form of energy gravitates – equally, in fact: this principle is known as the equivalence principle. So be sure that if a fermion acquires its mass through the Higgs mechanism, this mass may be measured by the fermion’s inertia as well as by the fermion’s own gravitational field.

    It’s not hard to say that for low-energy processes, well below the typical energy scale of the Higgs mechanism – the electroweak scale – the mass induced by the Higgs mechanism will behave just like the explicit mass from the Dirac equation in all measurable respects.

  • No, if the Dirac equation to add one more term with a mass matrix which anti-commutes with all four matrices are already contained there, then no violation of unitarity is not.

    One of these mass terms corresponds to an electron and the other to neutrinos. And electroweak transformation behaves decently.

    Besides, as Higgs define masses than it should much closer to be connected with gravitation than *Everything* that carries energy or momentum has “something to do with gravity”.

  • Dear Gunn,

    the equivalence principle guarantees that nothing can be “more connected with gravity” than any other form of energy – because the well-established principle states that every form of momentum and energy is *equally* connected with gravity – so your last paragraph is an oxymoron.

    The equation you propose in the first two paragraphs – with a gamma_5 added as an extra coefficient after “m” in the mass term (and indeed, I can use the symbol gamma_5 for a matrix that anticommutes with the four gamma matrices) – is equivalent to the normal Dirac equation.

    In your equation
    (p_mu gamma^mu – im gamma_5) psi = 0,

    we can write
    psi = M.chi

    M = (1+i.gamma_5)/sqrt(2)

    to get
    (p_mu gamma^mu – im gamma_5).M.chi = 0,

    Multiply it by M from the left:
    (p_mu.M.gamma^mu.M – m) chi = 0.

    But M.gamma^mu.M is simply gamma^mu again, so chi satisfies the normal Dirac equation. It’s not hard to see that you can’t get anything new – because by inserting gamma_5, you only changed the relative sign between the two two-component spinors.

    If the “i” were missing before the gamma_5 term in your equation, you would obtain a tachyonic particle which is too bad.

    To summarize, it’s clear that your “invention” doesn’t change anything whatsoever about the divergences unless the mass term is produced by the Higgs mechanism.


  • Thanks you very much.

    But I see equation of the following form: (-d_t + beta_k.d^k + im_1.beta_4 + im_2.beta_5)psi = 0.
    Here k into {1,2,3}. Here, your argument does not pass since this equation contains one term without the matrix.

  • Hello! Sorry to interrupt this discussion, but Lubos has written a very nice post on this matter on his blog: http://motls.blogspot.com/2010/12/beauty-of-dirac-equation-and-its.html

    I think the discussion so far has now diverged from this post and can find Lubos’ post as a reasonable fixed point.


  • Don M.

    Hi Flip. You guys sure spend the time and effort on these blogs, sure wished I understood them better, but anyway thanks. Questions- What are chances the that even if the “Higgs” is found, “UNKNOWN UNKOWNS” will still exist in the Standard Model? Is it possible that “dark matter” is the missing “anti-matter” clumped by some unkown force, or by “gravity” in some form?

    Thanks, Don Meares

  • Hi Don! Thanks for the kind words.

    Regarding your question: The Higgs is a “known unknown,” and we’re pretty sure that something is going on there that really *should* be observable at the LHC. In fact, if we *don’t* find anything at all then that would be a pretty big deal. (Arguably that would already be an “unknown unknown” situation!)

    As with anything, it’s hard to predict the appearance of “unknown unknowns,” since by definition you don’t see them coming. When dealing with experiments with lots of often-complicated data, there are always some suspicious results that appear to be unexpected but often turn out to be statistical fluctuations (e.g. if you flip a coin five times in a row and it always ends up heads).

    For some examples of current possible “unknown unknowns,” see these posts by Jester on the blog Resonaances:



    To be honest, most (probably all) of these things will pan out to be nothing, but that’s just an uninformed personal opinion. The LHC, however, represents a machine to test *new* frontiers in energy and it is reasonable to expect that we might see new things that we didn’t know would be there.

    As far as dark matter as anti-matter, this is somewhat unlikely because whenever it came in contact with normal matter it would annihilate into photons which would then be observable to our telescopes. (In fact, many people suspect that some of these gamma ray signals may come from dark matter decays.)

    In fact, we know that dark matter is all around us. It provided the gravitational potential for our galaxy to clump together in the first place. However, as you imply there are several theories where dark matter interacts through some new force.

    Theories where anti-matter is clumped away from us are much more difficult to make viable, primarily because of the issue of the lack of very obvious photon signals. Another question is how could something separate anti-matter and matter so far from one another without having them touch? For example, we don’t see that the universe is divided into an anti-matter half and a normal-matter half. If that were true, then we’d see all sorts of photons coming from the part where the two regions touch.

    Hope that helps,

  • Dear Gunn,

    your patience is appreciated. 😉 But even if you write a mass-like term that is a combination of a constant and “gamma5”, I can still find a field redefinition – writing “psi” as “(a+b.gamma5)chi” for appropriate coefficients “a,b” – that will map your equation to the conventional form. If you need it, I can calculate the general field redefinition for you.

    Again, in the 2-component spinor formalism, it is manifest that you can’t get anything qualitatively new by adding mixtures of “gamma5” in the equations. The purpose of “gamma5” is to separate a 4-component spinor into the 2-component pieces (which have the opposite, +1,-1, eigenvalues of “gamma5”). In the 2-component formalism, the mass term is just a multiple of the other 2-component spinor that appears in the differential equation for the first one, and vice versa. There are a priori two “mass” coefficients in these two equations – being simple combinations of “a,b” above. But one of them may be removed by rescaling one of the spinors while another combination of these two coefficients determines the actual physical mass. No option for new physics here.

    Thanks, Flip, for your compliment. Please know that I like your intros, too. When you mention Jester, Flip, one of Jester’s unknown unknowns – directly related to the dark matter that Don M. mentioned – is the neutralino. It’s apparently an unknown unknown for Jester that will make him lose ten thousand dollars in a bet he entered against myself because of this “unknown unknown” in a year or so. 😉 (I will pay him one hundred is SUSY doesn’t show up.)

    It may be fair to say that theorists are thinking – and have to be thinking – in a less variable framework, so what is called “known unknown” by Flip – e.g. the existence of at least one Higgs boson – is a “known known” for most theorists. Superpartners, probably counted as “unknown unknowns” by experimenters, are “known unknowns” for many theorists and there can be some totally different “unknown unknowns”.

    Of course, a very radical conceivable implication of the LHC would be the emergence of the fourth possibility omitted by Donald Rumself, the “unknown knowns”, more precisely the things we think we know but they don’t exist. 🙂 It’s this category omitted by Rumsfeld that has sparked many previous revolutions in physics – such as relativity (non-existence of the luminiferous aether).

    Best wishes

  • Sorry, one more notice: Let the following equation:


    here s into {0,1,2,3,4,5},
    beta_0 = -1 (the 4time 4 identity matrix),
    psi = psi_1.exp(-im_1.x_4)+psi_2.exp(-im_2.x_5)

  • Dear Gunn,

    first, you can’t have six (0,1,2,3,4,5) 4×4 gamma matrices that anticommute with each other. The minimum dimension is 8×8 because they would be gamma matrices for spinors in a 5+1-dimensional space.

    Second, if you do so, then you can indeed get an equation of the simple type you write. It’s just a Dirac equation in the 5+1-dimensional spacetime. The two additional dimensions will produce something that looks like massive from a 4D viewpoint, indeed.

    There will be a whole tower of possible solutions, depending on the momenta of the fermion in the additional two (compact) dimensions – in your case, the integers you named m1, m2. The theory of extra dimensions that studies it is called the Kaluza-Klein theory. The mass, as seen from the 4 dimensional viewpoint, will be something like sqrt(m1^2+m2^2), with coefficients inserted if necessary. If the dimensions x4, x5 are circular, the integers m1,m2 – the new momenta – wil lbehave as new kinds of U(1) charges in 3+1 dimensions.

    If you artificially restrict the field to be 3+1-dimensional, i.e. not to depend on the two new coordinates, then the two new terms in your equation vanish and you’re back in 4D Dirac equation.

    Best wishes

  • Thanks you again.

    Yes, I wrote that “beta_0 = -1 (the 4time 4 identity matrix)”. That is beta_0 commutes with all other beta_s. Here only
    these other matricies beta_s (1,2,3,4,5) anticommute.

    Now vector psi is double-vector [psi_1, psi_2] in space with basis . And SU(2) acts in
    this 2-dimensionsl space without any negative probabilities.

  • Excuse me: basis is:

    variables x_4 and x_5 are not coordinates any physics events.

  • Excuse me again — bracets missed

    basis is the following: (exp(-im_1.x_4, exp(-im_2.x_5))

  • Dear Gunn, fine, beta0 commutes, but if you don’t also have a gamma0 that anticommutes with the spatial components of gamma (gamma0.beta) – and consequently also with beta_i, then you cannot construct a Lorentz-invariant Lagrangian. You need the gamma0 for the Dirac conjugation.

    For a left-right-symmetric (Dirac spinor) Lorentz-invariant theory in D spacetime dimensions, you need D anticommuting gamma matrices (and the beta’s are calculated from them). And there is a minimum size of such matrices that is possible. In 4D, the Dirac matrices are 4×4, in 6D, they are 8×8.

    I don’t know why you think that x4 and x5 are “not coordinates [of] any physics events”. On the top, you wrote that the field or wave functions depend on x4 and x5, so they definitely *are* coordinates of physics events. They’re just extra dimensions. How could they not be? If they’re just fixed constants, and not coordinates, why wouldn’t you write their value, like 1, instead of x4 and x5?

    Your latest comment fixing the parentheses has 2 opening and 3 closing brackets 🙂 but all of us know which one is missing (after “4”).

    Best wishes

  • Thanks very much to you.

    Equation (beta_s*d^s)psi=0 is invariant under Lorentz transformation
    (x’_0=x_0*cosh(2*alpha)+x_k*sinh(2*alpha) ,
    (here k into {1,2,3})
    if U(alpha)=cosh(alpha)*beta_0+sinh(alpha)*beta_k
    then U'(alpha)*beta_0*U(alpha)=cosh(2*alpha)*beta_0+sinh(2*alpha)*beta_k ,
    U'(alpha)*beta_k*U(alpha)=cosh(2*alpha)*beta_k+sinh(2*alpha)*beta_0 ,
    and for all other beta_s:
    Therefore, since equation of moving is Lorentz-invariant then there a corresponding Lagrangian exists.

  • Leon

    Hi Flip, hi everyone,

    maybe I’m just cruelly overlooking something (in this case, please tell me what), but in your post, Flip, you show some diagrams saying:
    “Finally, the W also participates in some four-boson interactions (which will not be so important to us):” and then there are said four-boson diagrams.

    Aren’t they (just as the two diagrams above) violating conservation of charge. For example, how can a W- turn into a W+ by just interacting with a non-charged particle, the Z-Boson?
    For me, this sums up to a nonconserved charge of 2 elementary charges. Similar argumentation holds for the four-boson diagrams.

    Best wishes from Germany


    PS: Please continue writing your posts about particle physic’s interactions. I really enjoy them 🙂

  • Dear Gunn,

    surprisingly, there doesn’t exist any Lagrangian underlying your Lorentz-invariant equation even though your equation is Lorentz-invariant, indeed.

    It’s because you can’t construct any Lorentz-invariant scalars out of your “Dirac field” if you don’t have any gamma_0.

    The Lorentz-invariant scalars are constructed as psibar.psi where psibar, the Dirac conjugate (overlined psi), is defined as

    psi^{dagger} gamma_{0}.

    It’s the Hermitean conjugate but also multiplied by gamma_0. If you don’t have any gamma_0, you can’t construct the invariants. Also, you can’t construct the electromagnetic current


    and any other bilinear expressions that transform as vectors. Gamma0 is necessary because beta0 is Hermitean while the other betas are anti-Hermitean, and ordinary Hermitean conjugation also flips the signs of the spatial and temporal components of beta’s differently. You need to compensate for it by adding a gamma0 to the ordinary hermitean conjugates of psi.

    Best wishes

  • Dear Leon,

    Flip would surely answer the same thing, so you may get an earlier glimpse of his answer. 😉

    In the W+ W- Z cubic vertex, as Flip drew it, the labels correspond to three incoming particles (or three outgoing particles, but whether they’re incoming or outgoing is chosen in the same way for all three external lines).

    That’s why the cubic vertex, much like all other vertices Flip drew and all others that he didn’t drawn, preserve the electric charge.

    If you want to read the vertex as a diagram in which one W particle is incoming and the other W particle is outgoing, then of course, you would have to change the labels from “incoming W+ outgoing W+” or “incoming W- outgoing W-” on two external legs.

    But an incoming W+ is the same thing as an outgoing W-. In fact, antiparticles – and indeed, W+ is the antiparticle of W- – are just particles with the same charges but negative energies moving in the opposite direction through the Feynman diagrams.

    At any rate, whatever method to “visualize the process” behind the diagram you choose, it will preserve the electric charge. Flip chose the usual convention in which you have the process in which three particles appear in the initial state, and they annihilate into nothing. In that case, one of the W is positive and one of them is negative.

    Of course, at least one of them has to have a negative energy for the energy conservation to hold, but it is not a problem if at least one of the particles is virtual – or interpreted as the antiparticle. An antiparticle is the particle moving backward in time with the opposite sign of its energy. W+ is the antiparticle of W- but Z0 is its own antiparticle.

    Best wishes

  • Corrections:

    “didn’t drawn” should have been “didn’t draw”.
    “labels from” should have been “labels to”.

    I wanted to add that (-1)+(+1)+0 = 0 which is the check of the charge conservation.

  • Dear Lubos,

    May I choose beta_s the following?:

    beta_s := [sigma_s 0, 0 -sigma_s] for s into {1,2,3}, sigma_s are Pauli matrices,
    beta_4 := [0_2 1_2, 1_2 0_2] here 0_2 is zero matrix 2 times 2 and 1_2 is identity matrix 2 time 2,
    beta_5 := [0_2 i*1_2, -i*1_2 0_2],
    beta_0 := -1_4 is identity matrix 4 times 4.

    All these matrices are Hermitean and if s and k into {1,2,3,4,5} then beta_s and beta_k anticommute.

    Lagrangian: L := pci’*beta_s*d^s*psi ? where psi’ is Hermitean conjugated psi.


  • Hi Leon—yep, it’s exactly as Lubos says. The W+ and W- are anti-particles of one another. In other words, an incoming W+ is the same as an outgoing W-.

    (As a check of this, remember that a current of positive charge going one way is the same as a current of negative charge going the opposite way.)

    I’ve implicitly drawn all of my diagrams with the bosons going into the vertex, so that the charges of the W+ and W- cancel.

    Thanks for the kind words,

  • Thanks, Flip, for the confirmation.

    Dear Gunn, your actions and transformations are perfectly OK if “s” is only summed from 0 to 3 in the action – in that case, they are the standard Lorentz-invariant action, as one can see if he rewrites beta_i as gamma_i.gamma_0, and adds the gamma_0 appearing near psi’ to produce the Dirac conjugate.

    The problem is that you don’t have six independent coordinates because of several reasons: first, it is not true that your beta’s anticommute with each other. In particular, beta0 commutes with everyone else (unit matrix) and beta4 as well as beta5 commute with any of beta1,beta2,beta3 (while beta4 anticommutes with beta 5, and pairs within beta1,beta2,beta3 anticommute with each other). Because they fail to anticommute, you won’t be able to derive the (6D) wave equation in which the wrong terms cancel etc. by squaring the Dirac operator. In fact, because of the chaotic rules for the (anti)commuting of the matrix pairs, the addition of the beta4 and beta5 terms breaks even the Lorentz invariance in the 3+1 dimensions; see the end of this text for a simple way to see it.

    Second, your definition implies that beta1.beta2.beta3=beta4.beta5 (or the same with a factor of +-1 or +-i, I don’t want to calculate it now). So their products are not independent as expected from gamma/beta matrices from independent dimensions.

    Equivalently speaking, you won’t be able to write simple variations of psi’. Imagine that the Lorentz transformation is given by

    delta psi = M . psi

    where M is an infinitesimal matrix being a combination of beta’s and their products. Then

    delta psi’ = psi’ . M’

    where M’ is the Hermitean conjugate of M. However, without gamma_0, you won’t find any universal formula to write M’ in terms of M. In the usual picture, M’ is written as

    M’ = gamma_0.M.gamma_0,

    i.e. as the conjugation by gamma_0. This is true because if gamma0 is Hermitean, the remaining ones are anti-Hermitean, and this sign difference can be exactly mimicked by the conjugation of a gamma matrix by gamma_0. That allows you to define the Dirac conjugate. However, no formula of this kind exists for six 4×4 “beta” matrices exactly becausse you can’t find 6 gamma matrices that anticommute with each other.

    The text above should have convinced you that the beta4,beta5 matrices are qualitatively different from beta1,beta2,beta3 in your framework. It’s illogical to use these misleading new symbols because there are no new coordinates. In fact, all 4×4 matrices may be written as linear combinations of gamma_mu for mu=0,1,2,3 and their products, including the identity – 16 different basis vectors in total. So in any basis, all your beta matrices from 0 to 5 may be written in terms of gamma0…gamma3. It’s clear that the kinetic terms have to be the same as in the normal Dirac equation – you copied the standard form of s=0,1,2,3 terms in the kinetic term, too. Effectively, you’re trying to add new “mass-like terms” but all such terms may be written in terms of products of the normal four gamma matrices. It’s not hard to see that only mass terms proportional to 1 and gamma5 = i.product of 4 gamma matrices are Lorentz-invariant, and we’ve already discussed those. You’re just creating havoc by renaming products of 4 gamma matrices as a “new direction of beta” because there are no new directions here.

    Best wishes

  • Dear Lubos,

    Exquse me, but is not true that “beta4 as well as beta5 commute with any of beta1,beta2,beta3”. All of beta_s except beta_0 anticommute among themselves. On beta_0 I wrote in previous posts. I shall rewrite beta_4 and beta_5 in the following way;

    || 0 0 1 0 ||
    beta_4:=|| 0 0 0 1 ||
    || 1 0 0 0 ||
    || 0 1 0 0 || ,

    || 0 0 i 0 ||
    beta_5:=|| 0 0 0 i ||
    ||-i 0 0 0 ||
    || 0 -i 0 0 || ,

    Then you can check that if s and k into {1,2,3,4,5} then beta_s and beta_k anticommute.

  • Sorry, the formulas are shifted.

  • beta_s :=

    || sigma_s 0_2 ||
    || 0_2 – sigma_s ||

    0_2 is zeros 2 times 2 matrix.

  • Phmeonenal breakdown of the topic, you should write for me too!

  • Andrew

    It appears that the diagram for the interaction between two Ws and a Z and the diagram for the interaction between two Ws and a photon are mislabeled.

  • Hi Andrew—I should have mentioned that I labeled those Feynman rules to implicitly have all particles running into the vertex. But your intuition is correct, if we were to read those as Feynman diagrams from left to right the labels would not be correct.


  • Oscar

    Hey flip. Sadly I’m too young to understand all the awesome group theory etc
    at the heart of QM, or really understand the maths of it at all. But I really like learning about it through things like your blog, and I like to think I have a reasonably good understanding by now. Just one final things I want to clear up with standard model pre-Higgs interactions. Can you rotate all feynman diagrams 90 degrees (and the reactions they correspond to) as long as they obey the basic rules. For example, can W and Z bosons scatter the relavent particles without changing the particles involved?

    Secondly, is it just chance that all fermions have mass (although this was not predicted by the standard model of course)

  • Hi Oscar. You can indeed rotate Feynman diagrams (and even pull legs in different directions). As long as the resulting process conserves energy and momentum, it’s a valid diagram.

    The Standard Model fermions get masses from the Higgs boson.