Tuesday, October 23, 2007

MSSM and Higgs: introduction

There has been a bump in the data around 160 GeV. The bump is gone. Nevertheless, let us say a few general words about the minimal supersymmetric standard model (MSSM), its Higgs sector, and supersymmetry.

In the early 1970s, it was known that physical states of strings in string theory had to be invariant under all angle-preserving i.e. conformal transformations of the worldsheet. The group of such transformations is infinite-dimensional because they can be identified with complex holomorphic functions and such functions are given by an infinite set of Taylor coefficients, if you wish. A particular generator - the overall scaling - in this "Virasoro algebra" is called L_0. It turned out that the condition that L_0 annihilates a physical state is equivalent to the condition
  • p . p = m^2

i.e. the usual relativistic relationship between energy, momentum, and the rest mass. The squared rest mass m^2 is calculated as a certain total number of internal excitations of the string, in appropriate units. Well, this was how the old bosonic string theory worked. That theory only allowed bosonic excitations of a string i.e. it only predicted bosons in spacetime, including the obnoxious tachyon.

Pierre Ramond wanted to see that fermions were a part of string theory. Fermions are a part of reality and because reality is described by string theory, it was pretty clear to him that they had to be included in string theory, too. So he realized that the condition

  • p . p = m^2

was nothing else than the Klein-Gordon equation for the bosons in spacetime and the natural counterpart for fermions should be the Dirac equation. However, the Dirac wave function has to transform as a spacetime spinor. You may obtain spacetime spinors by quantizing fermionic zero modes on the worldsheet: the spinors are a representation of the algebra of gamma matrices and you obtain the same algebra if you replace the gamma matrices by the zero modes of the fermions in the vector representation.

Because the fermions had to have zero modes - modes that don't contribute to the spacetime mass - they had to be periodic. This periodic sector - a subspace of the Hilbert space of one string - is nowadays called the "Ramond sector" and has to be supplemented with another sector, the "Neveu-Schwarz sector", whose importance for the description of bosons was realized by two physicists whose names can be guessed by the most intelligent readers of this blog. Much like the bosons, Ramond's fermions on the worldsheet transformed as spacetime vectors. In fact, there was a one-to-one correspondence between the worldsheet bosons and worldsheet fermions. Moreover, the Dirac equation

  • p slash = m

that had to be satisfied when acting on the spinorial wave functions followed from the invariance of the physical states of the string - the superstring - under an extended algebra of superconformal transformations: the new key fermionic generator squares to the L_0 generator much like the Dirac operator times a similar one is the Klein Gordon operator. Supersymmetry was born.

At the same time, Russian mathematical physicists discovered supersymmetry by pure algebraic considerations. Shortly afterwards, Wess and Zumino have borrowed Ramond's idea of supersymmetry and applied it to the four-dimensional context to build new interesting models that don't directly depend on string theory. The first interacting model was the Wess-Zumino model - a model with a complex scalar, a Weyl fermion, and the usual renormalizable interactions with parameters related by a constraint that allows for the cool new symmetry to work beyond the level of free particles.

Effect of SUSY on particle spectrum

What does supersymmetry do with the particle content? In four-dimensional theories (or less), it is always possible to visualize supersymmetry as a symmetry acting on an extended version of the normal space, the superspace. Let me simplify a bit. The superspace has the usual coordinates x0,x1,x2,x3 but it also has an anticommuting coordinate theta. In reality, it has two thetas (components of a Weyl spinor) and their two independent complex conjugates but the qualitative conclusions won't change if you do things carefully. So let's call about one theta. What does it mean that it is anticommuting? Well, if you exchange the order of factors, you obtain the same result with a minus sign. It means, for two copies of theta, that

  • theta . theta = - theta . theta

Well, this means that theta squared equals zero! That's funny because any function of theta may be Taylor-expanded and almost all terms may be dropped:

  • f(theta) = a + b.theta

The remaining terms are simply zero! Now imagine that you have a superfield F - which means a field in the superspace. It can be Taylor-expanded in theta:

  • F(x0,x1,x2,x3,theta) = f(x0,x1,x2,x3) + theta g(x0,x1,x2,x3)

You see that one superfield F is equivalent to two ordinary fields f,g that only depend on the usual coordinates x0,x1,x2,x3. Because theta is a fermionic object, f is bosonic while g is fermionic or vice versa: f,g are nothing else than the fields that create particles that are superpartners of each other! One of them is bosonic and the other is fermionic.

So supersymmetry doubles the number of particle species. For each boson, there must exist a fermion, and vice versa. If you want the Standard Model to be a part of a supersymmetric theory, it turns out that no known particle can be a superpartner of another known particle. You must accept that each of them has a superpartner that remains to be seen. They're called sleptons, squarks, selectrons etc. in the case of bosonic partners of known fermions and wino, zino, photino, higgsino, gravitino in the case of fermionic partners of known bosons.


What is the most economical supersymmetric theory with the Standard Model in it? It is called MSSM, the minimal supersymmetric standard model. Besides the usual partners of leptons, quarks, and force messengers, we must double the number of Higgs fields. Why? In the normal Standard Model, a Higgs field is able to make both upper quarks as well as lower quarks massive. It's always the same Higgs doublet, after all: "2" is a pseudoreal representation of SU(2). In supersymmetric theories, the complex conjugate Higgs transforms as a different kind of superfield (that depends on theta-bar as opposed to theta) and can't be included into the "superpotential" - a generalized potential in the superspace that generates the normal potential energy, among other couplings.

So the counting is different. In the normal Standard Model, you have one Weinberg toilet i.e. one Higgs doublet. It is made out of two complex i.e. four real components. Three of them are eaten by the W+, W-, Z bosons that become massive: their family of polarizations is extended from two to three in each case. The remaining fourth component of the Higgs doublet is the field that creates the (sometime in the future) observable Higgs particles.

In the MSSM, the Higgs scalars have eight real components. Again, three of them are eaten by W+, W-, Z bosons. If you can compute 8-3, you may find out that five real components are left. Two of those five fields are CP-even and electrically neutral while the remaining three fields are CP-odd: their charges are 0,+1,-1, respectively.

The CP-odd neutral MSSM Higgs is the particle we were talking about in the context of the CDF bump. Its mass could have been about 150-160 GeV but now it is gone.

1 comment:

  1. So supersymmetry doubles the number of particle species
    To be precise, the number of total degrees of freedom of particle species, so that the fact that For each boson, there must exist a fermion, and vice versa must be considered in this way. For each Weyl fermion, you can expect two spin zero particles. Or you can tell that "particle species" refers to a complete set of observables, including spin state.

    You must accept that each of them has a superpartner that remains to be seen. For the MSSM and traditional extensions, not for the sBootstrap :-) Hey, I felt almost personally addressed with this "You must". Please consider to use the most impersonal "One should accept..." next time

    They're called sleptons, squarks, selectrons etc. in the case of bosonic partners of known fermions I have read somewhere that the s stands for "scalar" but I can not confirm if this was the initial intention. Any clue here?