Pictures Paths Particles Processes

270 November 7, 2013
for Φ. If the highest power of interaction in the theory is ϕ m , this equation has
m − 2 complex roots Φ 1 , Φ 2 , . . . , Φ m−2 , and
J p = Φ p − V ′ (Φ p ) , p = 1, 2, . . . , m − 2 . (10.66)
Now, single out that J p that has the smallest absolute value 16 , which we shall
call J 0 , and its corresponding Φ p will be writtten Φ 0 . For J and Φ very close
to the values J 0 and Φ 0 , respectively, we may use Taylor expansion to write
J ≈ J 0 − 1 2 F ′′′ (Φ 0 )(Φ 0 − Φ) 2 , (10.67)
since the linear term vanishes by definition. Hence
Φ ≈ Φ 0 −
(1 − J ) √
1/2
2J 0
J 0 F ′′′ (Φ 0 )
(10.68)
close to the singularity. From the standard Taylor expansion 17
1 − √ 1 − x = ∑ n≥0
(2n)!
(n + 1)!n!2 2n+1 xn+1 (10.69)
we then recover the asymptotic form for N n :
√
(2n − 2)! 1 8J 0
N n ≈
(n − 1)! (4J 0 ) n F ′′′ (Φ 0 ) . (10.70)
This estimate grows roughly as n!, as ought to have been immediately obvious
from the fact that Φ(J) has a finite radius of convergence ; the above, more
careful, treatment gives an estimate that is quite good even for non-huge n. As
an application, we may consider purely gluonic QCD. In this theory, the only
interactions are between 3 or 4 gluons, and the theory is equivalent, as far as
counting is concerned, to the ϕ 3/4 theory, with
F (ϕ) = 1 3! ϕ3 + 1 4! ϕ4 . (10.71)
The solutions of Eq.(10.65) and the corresponding J values are
Φ 1 = −1+ √ 3 , J 1 = − 4 3 +√ 3 ; Φ 2 = −1− √ 3 , J 2 = − 4 3 −√ 3 , (10.72)
so that J 0 = √ 3 − 4/3, Φ 0 = √ 3 − 1, and F ′′′ (Φ 0 ) = √ 3. In the table we
give the exact number N n , and its asymptotic estimate. The approximation is
better than one per cent for n ≥ 3. The non-polynomial (that is, n!) growth of
the number of diagrams with n can be seen as an immediate indication of the
failure of perturbation theory as a convergent series, as discussed in Appendix
1.
16 The case that there are several such values is discussed in the next paragraph.
17 This can be proven by applying the Legendre expansion to the object u = y + u 2 /2 =
1 − √ 1 − 2y, and putting y = x/2.