All deep inelastic scattering calculations involving distribution amplitudes rely so far on the operator product expansion or OPE. Introduced 1969 by Wilson [31]
into particle physics to face various problems in strong interaction calculations and proven few years later by W. Zimmermann [32] in perturbative quantum field theory, it became the most important tool in quantum field theorical calculations.
Given the importance of this tool, it was proven also for conformal field theories [33, 34, 35], while the formal mathematical proofs based on different axiomatic settings on Minkowski space-time were found in [36, 37, 38, 39]. Recently OPE was also extended to general Lorentzian curved space-time [40].
The operator product expansion states that a product of operators at differ-ent space-time points which is usually singular in quantum field theories can be written as an asymptotic series of coefficient functions times a local operator at a nearby pointy
O1(x1). . .On(xn) =X
k
E1...n,k(x1, . . . , xn)O1...n,k(y). (2.37) For small distances the Wilson coefficients E can be calculated perturbatively, while the local operators on the right-hand-side encodes the nonperturbative con-tent. Therefore, the operator product expansion is used in a small distance region, where the smallness is usually ensured by an “internal reason” like the W-boson
k q
x
2p
z
1y
2p
′x
1p 0 y
1p
′z
2∆
Figure 2.4: Lowest order Feynman diagram with the gluon exchange between two valence quarks of the meson. ∆ = p0 −x2p, ∆2 ≈ x2q2, k = y2p0 −x1p k2 ≈x1y2q2
mass or the heavycandb-quark masses. In this case the leading terms in the oper-ator product expansion are operoper-ators with minimal dimensionality. In our case of exclusive processes in deep inelastic scattering the smallness of the relative dis-tance is ensured by an “external reason”, the large momentum transfer. Then the parameter which determines the importance of the operators in the expansion is the twist which is defined as the dimension of the operator minus its spin.
As the method for obtaining operator expansion for exclusive processes does not follow directly from the Wilson expansion of local operator products we demonstrate it here for a “simple” case, the meson form factor which is defined as matrix element
hp0|Jµ|pi (2.38)
where the quark current Jµ = ¯qγµq is sandwiched between mesonic states with different momenta pand p0. The main idea for the approach we illustrate below was proposed in [41] and applied e.g., in [42].
Mesons are built of a quark and antiquark, therefore the lowest order Feyn-man diagrams describing the coupling of the photon to these quarks have the form given in Figure 2.4. The virtuality of the quark and gluon propagators, h0|¯q(z1)q(0)|0iandh0|G¯ν(z1)Gρ(z2)|0i, is of order∆2 ∼k2 ∼q2, i.e.(z1−0)2 ∼ (z2 −0)2 ∼ (z2 −z1)2 ∼ 1/q2, so the use of perturbation theory for this parts is justified due to the asymptotic freedom of the QCD. On the other hand quarks which are produced close to each other at the distance of order (z2 −z1) ∼ 1/q and stay a long time collinear to each other will interact strongly. Therefore we should not calculate the external quark lines explicitly but should remain with the Heisenberg operator acting in the small virtuality and small momentum transfer
subspace. Then, we can write the meson formfactor as
means a integration over the small space-time volume∼1/q4 around0, i, j, k, l are the colour indices andα, β, γ, δ the spinor indices. Cµ,αβγδijkl dentotes the hard scattering part of the process. The meaning of this expression becomes more clearly as we look at the operator expansion of the current at short distances
J(0)→[¯q(0)q(0)] +g2I
dz1dz2[¯qq]|0iC1h0|[¯qq]
+g3I
dz1dz2dz3[¯qqF]|0iC2h0|[¯qq] +. . . ,
(2.40)
where we have suppressed for simplicity the different indices and F is the field strength tensor. The eq. (2.40) can be understood as follows. Due to quantum fluctuations in the small vicinity of0the photon transforms into two independent systems of quarks and gluons which move then in opposite directions. The ampli-tudeCifor this transition can be calculated perturbatively. After taking the matrix element between the mesonic states the matrix elements of the moving quarks and gluons describe then the transition of the partonic systems to mesons. Due to the increasing dimension of the operators on the right hand side of eq. (2.40) the contributions becomes suppressed by additional powers of1/qimplying that the minimal Fock state with two quarks is the leading one.
However also the leading Fock state operator is still a nonlocal operator but should be seen as a generating functional for local operators required by the OPE which are then obtained by Taylor expansion of the nonlocal operators. Let us exemplify that on the leading bilocal quark-antiquark operator. The leading twist contribution to the bilocal matrix element can be then written as
h0|¯q(z)q(0)|pi= Lorentz indices and the substraction of traces. Hence at largeQ2 only the leading twist of the leading Fock state will contribute to the meson formfactor.
At the end we should stress that we described here only roughly the scheme to obtain the operator expansion for the mesons in the asymptotic case. However, for baryons and for not so largeQ2, the analysis will be obviously more involved because of increased number of valence quarks and contributions from higher twists which are not negligible in this energy region. Additionally, we want to point out that later we will use operator expansion within the framework of the Light Cone sum rules (LCSR), which allow calculation at intermediate values of Q2. Therefore, we will use a version of the operator expansion, which is adapted to the Light Cone sum rules, but the ideas and the line of arguments are basically the same as displayed in this section. Unfortunately the discussion of the different sum rule methods is far beyond the scope of this work and therefore we will restrict ourselves in further discussion of Light Cone sum rule approach only to some basic parts required in our calculations.