Physics 558 – Lecture 2

Introduction to SuperSymmetry: (See, e.g., Chapter 17 in Rolnick.)

Recall that last quarter we discussed at length the extent to which physics is specified by the underlying symmetries. In the next lecture we will discuss the symmetries apparent in the Standard Model. Before doing that let us take a brief detour and extend our symmetry discussion to include the concepts of supersymmetry.

At the end of last quarter we considered situations (i.e., Lagrangians) where the physics was the same for (i.e., the Lagrangian was symmetric under the interchange of) states of differing I₃ (isospin symmetry), J₃ (rotational symmetry), strangeness (flavor SU(3) symmetry) or color (color symmetry). In each case the physical states appear in degenerate multiplets of the appropriate symmetry. So the question arises, under what conditions do we see degenerate multiplets of states with different spin, e.g., of spin ½ and 1 or spin ½ and 0. Systems that display invariance under the interchange of states with differing spin, in particular, of states whose spins differ by ½, are said to be supersymmetric. The underlying supersymmetric algebra will involve both bosonic and fermionic operators, i.e. operators that satisfy relations based on both commutators and anti-commutators.

The study of such systems is interesting for at least three reasons. The first is simply that this symmetry between bosons and fermions is intrinsically interesting. The second is that string theory studies suggest that supersymmetry, hereafter referred to as SUSY, plays a role in particle physics, at least at very short distances. The third, and most direct, reason is associated with wanting relatively low mass scalar particles, e.g., Higgs bosons, in the theory. This latter point deserves some level of explanation, which will allow us to begin discussing some other concepts important to the way we look at modern particle physics. As we discussed last quarter, both vector particles and spin ½ particles naturally display symmetries that can ensure that they remain massless. In the former case it is a gauge symmetry that plays this role. As long as the gauge symmetry remains unbroken, there can be no interactions in the theory (i.e., terms in the Lagrangian, including a bare mass term) that give the vector gauge boson a mass. The symmetry also ensures that this situation remains true even at higher order in perturbation theory. In particular, radiative corrections in the form of loop diagrams (to be explained more thoroughly below) will not cause the vector gauge boson to acquire a mass. Likewise a Lagrangian with chiral symmetry (i.e., a Lagrangian where the right-handed fermions are treated independently from the left-handed ones) will exhibit massless fermions to all orders in perturbation theory. Recall that the typical terms in the Lagrangian for a gauge theory, except for fermion mass terms, can be written separately for the different chiral components. For example, the gauge covariant derivative is diagonal in the chiral basis,

while a mass term is off-diagonal in the same basis,

While the former allows independent transformations of the chiral states,

the latter does not. Hence the requirement of chiral symmetry means that fermion masses are not allowed in the Lagrangian and, more importantly, will not arise from (perturbative) quantum corrections at higher orders. If the lowest order interactions (those in the Lagrangian) respect the chiral symmetry, the higher order interactions will also. As we will discuss shortly, the symmetry in the Lagrangian can be broken only by “spontaneous” effects (the structure of the vacuum) or by “dynamical” effects (nonperturbative bound-state structure).

On the other hand, scalar particles may have zero “bare” mass (i.e., no mass term in the Lagrangian) but they will generally pick up a mass from their self-interaction at higher orders. Consider a theory with quartic interactions for the scalar field (as in a j⁴ theory). There are radiative corrections to the scalar field inverse propagator (the mass²) of the form (here g² is the strength of the quartic coupling)

where k is the momentum running around the loop and the 1/k² is the propagator for the scalar particle in the loop. (For now, do not worry about the details of how this result is obtained. Dimensional analysis is enough.) This integral is clearly quadratically divergent in the UV (at the upper limit). Here we have simply put in a cutoff L. We interpret this result as saying that, if the underlying theory is meant to be valid up to a scale M_GUT or M_Planck, where a higher symmetry or gravity become relevant, respectively, the “natural” scale for the renormalized mass of the scalar field (modulo possible factors of g²) is M_GUT or M_Planck. Unfortunately, to allow the Higgs mechanism in the Standard Model to work its magic, we want the Higgs mass scale to be around 1 TeV in order to explain the observed electro-weak symmetry breaking scale. Without a fix for this issue it is difficult to see how the Higgs mechanism can work in the context of the Standard Model. As we shall see, SUSY offers at least the possibility of a fix. The basic idea is that, since in a SUSY world every scalar field degree of freedom will have a SUSY related fermion degree of freedom, every divergent diagram like that above with a scalar loop will be matched by a diagram with a fermion loop,

As indicated, the fermion loop is kinematically very similar in that each of the two fermion propagators goes like . Its coefficient, however, will differ by an overall factor of –1, due to essentially to the fermi statistics of the corresponding operators (we will consider this in more detail later). Thus the divergences in the two diagrams will cancel. Since we do not experimentally observe degenerate superpartners at current energies, the world we live in does not respect SUSY at energy scales below 1 TeV. However, the supersymmetric cancellation noted above could still “protect” the mass of the Higgs boson down to 1 TeV, if the scale of SUSY breaking is not much above a TeV. As a result, if SUSY is correct, we should see super-quarks (squarks) and super-leptons (sleptons), etc., at the LHC (if not at Fermilab).

With this somewhat mysterious “motivation”, let us try to understand the structure of a simple example of SUSY. Consider first a supersymmetric harmonic oscillator, i.e., let us start with SUSY quantum mechanics. We define the system to have the usual bosonic excitations with creation operator so that (in natural units)

In the usual way it follows that

Now define also a “fermionic” excitation with creation operator where

Thus we cannot create a state with two identical fermions,

as required by Fermi statistics. The two different operators must commute, i.e., the two kinds of excitations are distinct,

Now we define a Hamiltonian for this simple system in the form

with eigenvalues

where n_B and n_F count the number of bosonic and fermionic excitations.

The SUSY limit corresponds to the choice

Note that, due to Fermi statistics, n_F is either 0 or 1 (but not larger). So the spectrum of these systems has a vacuum state , n_B = n_F = 0, plus a tower of “superpartnered” states, i.e., degenerate pairs of bosons and fermions, for all n > 0:

Boson	Fermion	E
nB = n, nF = 0	nB = n-1, nF = 1	n w

To describe this system we can define a new symmetry operator that has the effect of changing the number of fermion excitations by 1 and changing the number of boson excitations by 1 in the opposite direction. The operator is typically defined to be self-conjugate so that it can take us both from a fermion state to a boson state and vice versa. Note that the conventional symbol for this operator is Q but it is NOT the electric charge,

This is a symmetry operation because

The operator Q does not change the energy of a state (or any other internal quantum number, explaining the choice to define it to be self-conjugate). Rather it changes only the number of bosonic and fermionic excitations. To see this result in detail, let us write it out,

The first term (a bosonic term) yields

while the second term (a fermionic term) yields

Similarly the third and fourth terms yield

and

To obtain the fermionic results we have used the fact that two contiguous fermionic annihilation operators annihilate any state, while acts like ,

Likewise two contiguous fermionic creation operators annihilate any state, while acts like .

Thus we see that Q commutes with H precisely because of a cancellation between the bosonic contribution and the fermionic contribution

As suggested earlier, it is this cancellation between bosonic and fermionic degrees of freedom that is the hallmark of SUSY. Now comes the really interesting relation (and the reason for the choice of normalization for Q). We consider the following anticommutator

Again we have used the fact that and annihilate all states. This result suggests that we should enlarge the concept of an algebra, which up to now has involved only commutators of generators, to include also anticommutators. It also underlines the close connection between the “internal symmetry” generated by Q and space-time symmetries as indicated by H (recall that it was in this context of mixing internal and space-time symmetries that we first mentioned supersymmetry last quarter). In fact, the mathematicians have long studied such mixed algebras. The result is called a graded Lie algebra. An example is the above set of operators, , which close under a set of both [,] and {,},

If we label the usual bosonic operators as “even” operators and the new fermionic operators as “odd” operators, we having the following general structure for a graded Lie algebra

When we move from the algebra to the full group, we expect to exponentiate the operator Q to obtain e^iQa. Since the resulting operator should satisfy commutation relations (i.e., be “even” although Q itself is “odd”), the number a is not an ordinary number. Instead we must introduce the concept of anticommuting numbers. Again the mathematicians have a name for these, Grassmann numbers. For example, if a and b are Grassmann numbers, then ab = -ba.

If we consider “supersymmetrizing” a locally gauge symmetric field theory (initially with chiral fermions and gauge bosons) instead of the quantum mechanical system described above, there will generally be a scalar degree of freedom for every fermionic degree of freedom, e.g., for each chiral state, and a chiral degree of freedom for each gauge boson degree of freedom. With enough SUSY operators one should see spin 0, ½, 1, 3/2 and 2 particles, all in degenerate multiplets.

In realistic applications of SUSY we want to enlarge to algebra to include not just H but rather the full Poincaré algebra with generators (recall the discussion in Lecture 5 of Physic 557)

If we have a typical internal (even) symmetry, whose generators we represent by T_a and whose internal space is disjoint from space-time, we have

These commutation relations, which state that a Lie group containing both the Poincaré group and an internal symmetry group must be a direct product and not mix the dependence on space-time and internal degrees of freedom in a non-trivial way, was demonstrated as a “no-go” theorem in 1967 by Coleman and Mandula under quite general conditions. SUSY evades this theorem by including both commutation and anticommutation relations in the algebra, i.e., by using a graded Lie algebra.

To pursue this further consider a fermionic operator, Q_a (a=1-4), which has the structure of a Majorana operator, i.e., is self conjugate as above,

As we discussed last quarter, such operators are, in some sense, half of a Dirac operator, i.e., represent only two degrees of freedom instead of 4. To see this connection we write a general Dirac field as

The 2 components,

are independent Majorana spinors that satisfy

just as we require of the Q. Recall that the Majorana spinors can have no nonzero quantum numbers (other than that they are fermions).

The fact that these are spinor operators is encoded in the commutator that specifies their properties under a Lorentz transformation,

where we recall that

This last object (an antisymmetric Lorentz tensor and a matrix in spinor space) provides a spinor representation of the Lorentz group.

The (necessarily true) Jacobi Identity of commutators

requires that the operator Q_a is translationally invariant and represents a symmetry of the system,

The graded Lie algebra is then closed by including the anticommutator noted earlier

which exhibits the explicit mixing of space-time with internal symmetries that is intrinsic to SUSY. We will delay a full derivation of this remarkable result until we have a more fully developed grasp of field theory (but we will discuss the Wess-Zumino model briefly in the Appendix and the HW.) In fact, this anticommutator, which is an even generator, a tensor with respect to the spinor indices ab, a Lorentz scalar and must be linear in one of the other generators, can only have the form shown or be proportional to s^mn J_mn. But the latter expression does not lead to a closed algebra. In the homework we will study an explicit example of this structure in the context of massless particles.

If we express the (formerly) 4-component Majorana spinor operator in terms of 2-component left-handed Weyl (or chiral) spinors, W, with our convention for the Dirac matrices, we find

where we used the fact that (in this notation) the conjugation operation looks like

While we use the notation here, you should probably think since we are really discussing operators. [This 2-component notation is used in Peskin and Schroeder, Chapter 22.4]

ASIDE - Recall the following properties of Dirac matrices: In the Appendix of Lecture 9 in Phys. 557 we introduced a specific representation of the Dirac matrices that are useful when using the language of chiral or Weyl fermions as here. We used the convention of the text by Rolnick,

To avoid confusion (or to maximize it) it is useful to note that in several other texts, for example, the Field Theory book by Peskin and Schroeder, a different convention is used. In this second convention the have changed sign so that must also change sign and the matrices have the form

We only see differences when we express results in detail. In particular, when we want to write Dirac or Majorana 4 component spinors in terms of 2 component Weyl spinors, the identifications are different. In our convention

while in the other convention (with the opposite sign for ) we have

When we apply this formalism in the homework to a massless particle traveling in the direction, P_m = (E,0,0,E), we learn that the second component of the Weyl spinor and its conjugate, , act just like the operator Q in the harmonic oscillator case. They commute with the Hamiltonian,

and their anticommutator is

When applied to a left-handed massless particle, the helicity, and thus the spin, is changed by ½. In this case, the other components, , carry no new information.

As applied to a Standard model with massless quarks, leptons, gauge bosons, Higgs bosons and gravitons, we expect SUSY to be realized by superpartners for the “observed” particles: spin 0 squarks and sleptons, spin ½ gauginos (photino, gluino, wino, bino, zino), spin ½ higgsino, and a spin 3/2 gravitino. The charged gauginos and higgsinos are called charginos while the neutral states are called neutralinos. Since some of the (yet to be) observed states will be linear combinations, the labels wiggsino and ziggsino also sometimes used. Note that the super multiplets, for example, (the left-handed multiplet)

must have all other quantum numbers identical within the super multiplet. Thus the massless matter fermions are not connected via SUSY to massless vector particles. The matter fermions are typically in the fundamental representation of the other internal symmetries (U(1), SU(2) and SU(3)) while the only allowed massless vector particles (i.e., the only ones that stay massless when radiative corrections are included and the only ones that yield renormalizable theories) are the gauge particles that are in the adjoint representations of the symmetries. As in our discussion of the neutrino, CPT implies the presence of antiparticles with the opposite helicity.

ASIDE Massive super multiplets: Without derivation we note that for the case of a massive particle in its rest frame, P^m = (M,0,0,0), both components of W have a nonzero anticommutator,

and act as SUSY generators. There are now three distinct “SUSY raising” operators, , that can connect 3 states with a helicity structure suggested by the following figure.

Hence this super multiplet has 1 scalar, 1 fermion and 1 vector and all have the same mass M.

Clearly SUSY is broken in the universe we live in – we do not see degenerate super multiplets. However, if SUSY is a symmetry at short distances and serves to “protect” the Higgs boson from having a mass larger than 1 TeV, we can expect that the masses of the broken-symmetry partners to be of order 1 TeV and therefore accessible at the LHC, if not the Tevatron. After a discussion of the Wess-Zumino Model in the Appendix, we will return to our discussion of the Standard Model.