18.6. S-matrix of the GrossNeveu model The GrossNeveu model is a model of n-component neutral Fermifield k(x);

18.6. S-matrix of the Gross–Neveu model The Gross–Neveu model is a model of n-component neutral Fermi–field ψk(x); k =

1, 2, . . . ,n (n ≥ 3) with four-fermion interaction L = i 2

n



k=1

¯ψ

kγ μ∂μψk + g 8

 n



k=1

¯ψ

kψk



2 where ¯ψk = ψkγ 0 and the 2×2 γ μ matrices satisfy the anti-commutation relation

{γ μ,γ ν} = 2gμν. Like the bosonic O(n) σ model, the Gross–Neveu model is massive, renormalizable, asymptotically free and explicitly O(n) symmetric. It is also integrable.

With the notation of Section 18.9, the exact S-matrix of the Gross–Neveu model can be obtained solving the unitarity and crossing equations for S2(θ)

S2(θ)S2(−θ) = θ2

θ2 +λ2 , S2(θ) = S2(iπ −θ)

with the initial seed Q(θ) = θ

θ−iλ where λ = 2π/(n−2).

a. With the notation of eqn. (18.9.8), show that in this case we end up in U(−)(θ) =

− λ

2π

−i θ

2π 12

−i θ

2π 

12

− λ

2π

−i θ

2π  −i θ

2π

.

b. Prove that the amplitudes U(±) are related as U(−)(θ) = sinhθ +i sinλ

sinhθ −i sinλ

U(+)(θ).

c. Consider the amplitudes with definite isospin channel Sisoscalar = NS1 +S2 +S3, Santisym = S2 −S3, Ssym = S2 +S3.
Bound states exist only in isoscalar and anti-symmetric isospin channels. Denote these new particles B and Bij and show that their masses are mB = mBij = m sin 2π
n−2
sin π
n−2
.
where m is the mass of the elementary fermion.