A potential problem with supersets of the Standard Model is quantum-mechanical anomalies. Certain diagrams cannot be evaluated consistently.
[0802.0634] Lectures on Anomalies,
PHY 522 Topics in Particle Physics and Cosmology: Anomalies in Quantum Field Theory, like this one on the Standard Model:
Lecture 7: Fri Oct 9
In particular, for the Standard Model and its supersets, it's the "triangle anomaly", a fermion loop with three gauge particles coming out of it. This is for 4 space-time dimensions. For D dimensions, it only exists for D being even, and it has D/2+1 gauge particles. Thus, for 10 dimensions, the anomaly is a hexagon anomaly. In the diagram, even numbers of gauge particles can be replaced with gravitons, giving a gauge-gravitational anomaly or even a pure gravitational anomaly.
The anomaly consists of two parts multipled together: an interaction part and a loop-integral part. It's the loop integral that causes the trouble, so the only way to get rid of anomalies is to make the interaction part cancel. That implies constraints on possible quantum field theories.
Let us see how to evaluate the interaction part. For triangle anomalies, with gauge particles a, b, and c, the interactions with the fermions can be given by matrices Ta, Tb, and Tc. Since we want to sum over all the fermions, we get
Tr(Ta.Tb.Tc + Ta.Tc.Tb)
left - Tr(Ta.Tb.Tc + Ta.Tc.Tb)
right
If (say) b and c are gravitons, then Tb = Tc = I * (some constant), and we get
Tr(Ta)
left - Tr(Ta)
right
The interaction parts are proportional to the "Casimir invariants" of the gauge algebra:
Tr(C(T,p))
left - Tr(C(T,p))
right
for p = 1,3 (4D) and p = 0, 2, 4, 6 (10D)
Every algebra has an infinite number of them, but only a finite number of linearly independent ones, and all of them can be constructed from the smallest linearly-independent ones.
Thus, U(1) has the smallest independent one C(1) = (U(1)-factor value). All the others are C(p) = C(1)
p.
Let's now look at the sizes of the smallest independent ones:
U(1): 1
SU(n) = A(n+1): 2, 3, ..., n
SO(2n) = D(n): 2, 4, ..., 2n-2, n
SO(2n+1) = B(n): 2, 4, ..., 2n
Sp(2n) = C(n): 2, 4, ..., 2n
G2: 2, 6
F4: 2, 6, 8, 12
E6: 2, 5, 6, 8, 9, 12
E7: 2, 6, 8, 10, 12, 14, 18
E8: 2, 8, 12, 14, 18, 20, 24, 30
For all but U(1), the smallest one is 2.
So for the Standard-Model anomalies, the contributors are: C(QCD,3), C(QCD,2)*C(WHC,1), C(WIS,2)*C(WHC,1), C(WHC,1)
3, C(WHC,1) for QCD = SU(3) = quantum chromodynamics, WIS = SU(2) = weak isospin, and WIH = U(1) = weak hypercharge.
In the Standard Model's elementary-fermion multiplets, the triangle anomaly cancels. The Minimal Supersymmetric Standard Model includes a pair of Higgs doublets, the up Higgs and the down Higgs, and the triangle anomaly cancels for them also. One has to evaluate it for the MSSM's
higgsinos, because supersymmetry gives the Higgs particles fermionic counterparts.
Looking at GUT's, we start at Georgi-Glashow SU(5). C(GUT,3) is nonvanishing here, but the interaction part does vanish for both the extrapolated elementary-fermion content and the extrapolated higgsino content.
But for SO(10), E6, E7, and E8, the interaction part automatically vanishes, because C(GUT,3) = 0 for all of them. This counts in favor of such theories.
But in 10D, the hexagon anomaly contains C(GUT,2), C(GUT,4), and C(GUT,6), and these are in general nonzero. Thus, in heterotic superstrings, their cancellation only for SO(32) and E8*E8 is a nontrivial result.
