4 QCD: The Theory of Strong Interactions
113
The connection of these results with the RGE general formalism occurs via
the light cone OPE (recall Eq. (4.66) for W μν and Eq. (4.63) for the OPE of two
currents). In the case of DIS the c-number term I (x 2 ) does not contribute, because
we are interested in the connected part < p| . . . |p > − < 0| . . . |0 >. The relevant
terms are:
J
† (x)J (0) = E(x
2 )
∞
n=0
c n (x
2 )x
μ 1 . . . x
μ n · O
n
μ 1 ...μ n (0) + less sing. terms
(4.84)
A formally intricate but conceptually simple argument (Ref. [6], page 28) based on
the analiticity properties of the forward virtual Compton amplitude shows that the
Mellin moments M n of structure functions are related to the individual terms in
the OPE, precisely to the Fourier transform c n (Q 2 ) (we will write it as c n (t, α)) of
the coefficient c n (x 2 ) times a reduced matrix element h n from the operators O n :
< p|O n
μ 1 ...μ n (0)|p >= h n p μ 1 . . . p μ n :
c n < p|O
n
|p >→ M n =
1
0
dxx
n−1 F (x)
(4.85)
Since the matrix element of the products of currents satisfy the RGE so do the
moments M n . Hence the general form of the Q 2 dependence is given by the RGE
solution (see Eq. (4.33)):
M n (t, α) = c n [0, α(t)] exp
α(t)
α
γ n (α )
β(α )
dα
· h n (α)
(4.86)
In lowest order, identifying in the simplest case M n with q n , we have:
γ n (α) =
P n
2π
α + . . . ,
β(α) = − bα
2
+ . . .
(4.87)
and
q n (t) = q n (0) exp
α(t)
α
γ n (α )
β(α )
dα
= [
α s
α s (t)
]
Pn
2πb · q n (0)
(4.88)
which exactly coincides with Eq. (4.83).
Up to this point we have implicitly restricted our attention to non-singlet (under
the flavour group) structure functions. The Q 2 evolution equations become non
diagonal as soon as we take into account the presence of gluons in the target. In
fact the quark which is seen by the photon can be generated by a gluon in the target
(Fig. 4.12).
113
The connection of these results with the RGE general formalism occurs via
the light cone OPE (recall Eq. (4.66) for W μν and Eq. (4.63) for the OPE of two
currents). In the case of DIS the c-number term I (x 2 ) does not contribute, because
we are interested in the connected part < p| . . . |p > − < 0| . . . |0 >. The relevant
terms are:
J
† (x)J (0) = E(x
2 )
∞
n=0
c n (x
2 )x
μ 1 . . . x
μ n · O
n
μ 1 ...μ n (0) + less sing. terms
(4.84)
A formally intricate but conceptually simple argument (Ref. [6], page 28) based on
the analiticity properties of the forward virtual Compton amplitude shows that the
Mellin moments M n of structure functions are related to the individual terms in
the OPE, precisely to the Fourier transform c n (Q 2 ) (we will write it as c n (t, α)) of
the coefficient c n (x 2 ) times a reduced matrix element h n from the operators O n :
< p|O n
μ 1 ...μ n (0)|p >= h n p μ 1 . . . p μ n :
c n < p|O
n
|p >→ M n =
1
0
dxx
n−1 F (x)
(4.85)
Since the matrix element of the products of currents satisfy the RGE so do the
moments M n . Hence the general form of the Q 2 dependence is given by the RGE
solution (see Eq. (4.33)):
M n (t, α) = c n [0, α(t)] exp
α(t)
α
γ n (α )
β(α )
dα
· h n (α)
(4.86)
In lowest order, identifying in the simplest case M n with q n , we have:
γ n (α) =
P n
2π
α + . . . ,
β(α) = − bα
2
+ . . .
(4.87)
and
q n (t) = q n (0) exp
α(t)
α
γ n (α )
β(α )
dα
= [
α s
α s (t)
]
Pn
2πb · q n (0)
(4.88)
which exactly coincides with Eq. (4.83).
Up to this point we have implicitly restricted our attention to non-singlet (under
the flavour group) structure functions. The Q 2 evolution equations become non
diagonal as soon as we take into account the presence of gluons in the target. In
fact the quark which is seen by the photon can be generated by a gluon in the target
(Fig. 4.12).