Pictures Paths Particles Processes

November 7, 2013 303
rank-2, spin-2 :
rank-2, spin-1 :
rank-3, spin-3 :
(
)
1
∇ µ α∇ ν β + ∇ µ β∇ ν α − 1 2
2 ∇µν ∇ αβ ,
1
2
1
6
(
)
∇ µ α∇ ν β − ∇ µ β∇ ν α ,
(
∇ µ α∇ ν β∇ ρ γ + ∇ µ β∇ ν γ∇ ρ α + ∇ µ γ∇ ν α∇ ρ β
+ ∇ µ β∇ ν α∇ ρ γ + ∇ µ α∇ ν γ∇ ρ β + ∇ µ γ∇ ν β∇ ρ α
− 1 (
∇ µν ( ∇ ρ α∇ βγ + ∇ ρ β∇ γα + ∇ ρ )
γ∇ αβ
12
+ ∇ νρ ( ∇ µ α∇ βγ + ∇ µ β∇ γα + ∇ µ )
γ∇ αβ
)
+ ∇ ρµ ( ∇ ν α∇ βγ + ∇ ν β∇ γα + ∇ ν γ∇ αβ
) ) (10.230)
Compared to the massive case, some coefficients are different : -1/2 rather than
-1/3 in the spin-2 case, and -1/12 instead of -1/15 for spin-3. This is due, of
course, to the different traces of ∆ and ∇. The spin sum for the massless vector
particle (rank-1, spin-1) is in fact that of the axial gauge discussed in Chapter
6, with the gauge vector r chosen to be ¯p. Note that, whatever r µ , we can
always move to the centre-of-mass frame of p µ and r µ , and in that frame we
have precisely r µ = ¯p µ .
10.11.7 Spin of the Kalb-Ramond state
Concerning the Kalb-Ramond (KR) state, there may be some controversy. For
a massless particle in this state, the spin along the axis of motion must, under
measurement, always come out zero. It is not easy to see how such a particle can
be distinguished from a scalar one. Indeed, in string theory where the KR state
comes up naturally, it is considered to describe a (pseudo)scalar particle called
the axion. In order to talk sensibly about the spin of the KR state it is useful
to consider how it may be measured, for instance using fermions. We therefore
consider the coupling of a rank-2, spin-1 state to fermions. The interaction
vertex must have the properties that (a) it is an antisymmetric rank-2 tensor,
and (b) it is current-conserving, in order to make sense in the massless limit.
Denoting the two fermions by ψ and ψ the simplest choice appears to be
ψ ɛ µνρσ p ρ (A + Bγ 5 ) γ σ ψ
where p is the momentum of the antisymmetric tensor state, and A and B
are constants. This interaction vertex vanishes trivially under the handlebar
operation. For the process
¯f(p 1 ) f(p 2 ) → f(p 3 ) ¯f(p 4 )
by the exchange of a KR state of mass M, we then have the amplitude
M = i¯h v(p 1 ) ɛ µνρσ p ρ (A + Bγ 5 ) γ σ u(p 2 )