The pion form factor

Pion wave function

The spin structure of the pion wave function as a system consisting of quark and antiquark in the J^π = 0⁻ state is identical to the np wave function in J^π = 0⁺ state, eq.(5.33). The negative parity is generated automatically by opposite internal parities of the quark and antiquark and therefore does not change the spin structure. The pion wave function has thus the form:

The representation of this wave function in terms of the variables ~k and ~n is almost identical to (5.34):

with the following relations between the invariant functions:

A₁ = − m 2εk

(g₁+ m

kg₂), A₂ = εk

kg₂ . (7.12)

The normalization condition is a particular case of (3.23): Substituting in eq.(7.13) the wave function represented by (7.10) and (7.11), we get:

N2 =

where z = cos(~k~n). Obviously, in the case where the interaction potential between quarks^c is independent of M, as often assumed in phenomenological approaches, N₂ should be equal to one.

Pion form factor

The diagrammatical representation of the pion electromagnetic current in the impulse approximation almost coincides with fig. 18. For the interaction of a virtual photon with a quark, the horizontal line has to be replaced by the double line of antiquark. This current has the form:

J˜_ρ^γq =

Z

T r[−O^′(ˆk^′₂+ m)j_ρ(ˆk₂+ m)O(m − ˆk1)] 1

(1 − x)D , (7.16) where O depends on the initial momenta, and O^′ depends on the final momenta. The quark current jρ is taken as:

j_ρ= f₁γ_ρ+ if2

2mσ_ρν q^ν . (7.17)

In this equation f1 and f2 are the quark form factors. The minus sign in T r[−O^′· · ·]

is from the fermion loop. We remind that both lines in the loop, single and double, corresponding to the quark and antiquark, are in the same direction. But since, according to the rules of the graph technique, they are followed in the same and opposite orientations respectively, one makes a loop when going through them.

The pion form factor for the γq interaction is given by:

F^γq(Q²) =

Z

T r[−O^′(ˆk₂^′ + m)(ω·j)(ˆk2+ m)O(m − ˆk1)] 1

2ω·p(1 − x)D . (7.18) The pion form factor for the γ ¯q interaction is found similarly:

F^{γ ¯q}(Q²) =

Z

T r[−O(m − ˆk²)(ω·j)(m − ˆk2^′)O^′(m + ˆk1)] 1

2ω·p(1 − x)D . (7.19)

The momenta k2, k₂^′ in (7.16) are the quark momenta, whereas they are the antiquark momenta in (7.19).

After a trivial generalization for the case of different masses of the quark and anti-quark, these formulas can be applied to the K-meson form factors. The final form factor is expressed through the momentum transfer ~∆ (with Q² = ~∆²) and the integration variables ~R⊥, x with the use of the invariants indicated in appendix D.

Let us for a moment put A₂ = 0. We thus have: With A1 expressed through g1 by (7.12) the form factor (7.20) exactly coincides with the expressions given in ref. [109]. We emphasize that the calculation of (7.20), and more generally of any form factor or transition amplitude is, due to the covariance of our approach, a simple routine for analytical computer calculations.

When A2 is different from zero, but for pointlike quarks, i.e. with f1 = 1, f2 = 0 in the current (7.17), the calculation of (7.18) gives:

F^γq(Q²) = the asymptotical behaviour of the pion form factor.

Note that for the interaction of the virtual photon with the antiquark, in eq.(7.19), we get: F^{γ ¯q} = −F^γq. Hence, the full form factor of a system consisting of identical q ¯q is zero: F^γq(t) + F^{γ ¯q} = 0. We get opposite sign of F^{γ ¯q} automatically, without introducing negative charge of antiquark. This negative charge is automatically taken into account in this formalism. It can be obtained from the antiquark contribution, multiplying it by a sign factor depending on the process under consideration.

For the ρ − π transition, the quark and antiquark contributions have the same sign.

The transition amplitude ˜Fµρ can be obtained in this case from the π elastic amplitude J_ρ by replacing in eqs.(7.16) the initial pion wave function O by the wave function of the vector meson φµ. The form of this wave function coincides with the deuteron wave function (5.1).

Asymptotical behaviour of the pion form factor

It is well known that QCD provides a 1/Q²asymptotical behaviour for the pion form factor [40]. As it was shown above, the pion wave function contains in general two components.

The first one leads to the usual non-relativistic wave function, while the second one is of purely relativistic origin and ω-dependent. We shall show in this section that the asymptotical 1/Q² behaviour of the pion form factor is entirely determined by this last

ω-dependent component of the pion wave function [110]. This piece is essential in order to explain the difference with the form factor of the J = 0 state in the Wick-Cutkosky model, which asymptotical behaviour is 1/Q⁴.

The asymptotical behaviour of the form factor can be related to the asymptotical behaviour of the wave function. We shall calculate the latter in a q ¯q model with a one-gluon exchange kernel. The equation for the wave function corresponding to the diagram indicated in fig. 11 has the form:

ψ(k₁, k₂, p, ωτ ) = − 1 s − M²

Z 1

(2π)³δ⁽⁴⁾(p + ωτ^′− k^′1− k2^′)2(ω·p)dτ^′

ׯu(k2)γ_µ(ˆk₂^′ + m)θ(ω·k^′2)δ(k₂^′2− m²)d⁴k^′₂O^′(m − ˆk1^′)θ(ω·k^′1)δ(k₁^′2− m²)d⁴k^′₁γ^µv(k₁)

×K(k^′1, k₂^′, ωτ^′; k1, k2, ωτ ). (7.22)

Here M is the pion mass. We use the gluon propagator in the Feynman gauge and K is the scalar part of the gluon propagator (after separation of −g^µν). It coincides with the scalar t-channel exchange amplitude, given by eq.(2.67) (where one should put µ = 0).

The matrix O^′ is the “inner” part of the pion wave function as defined in eq.(7.10). Prime means that it depends on the momenta in the loop. For simplicity, we omit here any color degrees of freedom which are irrelevant for our consideration.

To find the asymptotical behavior of the wave function, we shall use the iterative procedure already explained in chapter 5 to find the deuteron wave function. We suppose that O^′ is concentrated in a finite region of the quark momenta and the integral (7.22) is dominated by this domain. These momenta are negligible relative to the asymptotical ones we are interested in. This means that we can evaluate all the factors except for O^′ and the delta-functions at zero relative quark momentum. This corresponds to ~R^′_⊥ = 0 and x^′₁ = x^′₂ = 1/2. In that case, the quark four-momenta are equal to each other:

k₁^′ = k^′₂ = (p + ωτ0)/2,

where τ₀ = (4m²− M²)/(2ω·p). We thus get the following replacement in (7.22):

Z 1

(2π)³δ⁽⁴⁾(p + ωτ^′− k^′1− k^′2)2(ω·p)dτ^′

×θ(ω·k2^′)δ(k^′2₂ − m²)d⁴k₂^′ O^′ θ(ω·k1^′)δ(k^′2₁ − m²)d⁴k₁^′ ⇒ O⁰, (7.23) where O0 is the integrated value of the wave function.

We substitute in (7.22) the one-component wave function obtained from (7.10) with A2 = 0. Together with the replacement (7.23), this gives in (7.22) O^′ = A0γ5/m, where A₀ is a constant. The second component A₂ of the wave function is then automatically generated by eq.(7.22).

We thus find for the pion wave function:

ψ = − 2A0

s − M²u(k¯ 2)(ˆp + ˆωτ0− 4m)γ⁵v(k1)K . (7.24)

Substituting here p from the relation p + ωτ = k1+ k2 and using the Dirac equation, we finally obtain:

ψ = 2A0

s − M²u(k¯ 2)(ˆω(τ − τ⁰) + 2m)γ5v(k1)K . (7.25) The pion wave function (7.25) has the form of eq.(7.10) with the following two components:

A1 = 4m²

s − M²A0K, A² ≈ A⁰K (7.26)

with s = R~²_⊥+ m²

x(1 − x). The kernel K given in (2.67) can be rewritten in terms of the variables R~⊥, x. For x ≤ x^′, it reads:

We will see below that this wave function enters in the form factor at x = 1/2. Then s ≈ 4 ~R²_⊥. As follows from (7.27), the kernel K at ~R^′_⊥ = 0, x^′ = 1/2, x = 1/2 obtains the simple form: K = g²/ ~R²_⊥. The components of the wave function which dominate in the asymptotical region are then given by:

A1 = g²m²A0

R~_⊥⁴ , A2 = g²A0

R~²_⊥ . (7.28)

In this region, the component A2 decreases more slowly than A1.

The form factor is now easily calculated through the components A1, A2 according to eq.(7.21). The integral for the form factor is dominated by the region where one of the two wave functions (initial or final ones) takes its non-relativistic value, whereas the other wave function corresponds to the asymptotical one. This is equivalent to the standard calculation discussed in the literature in the usual formulation of LFD [40]. Hence, similarly to eq.(7.22), we can replace A^′₁, A^′₂ in (7.21) by

A^′₁ = A₀γ₅δ⁽²⁾( ~R_⊥)δ(x − 1/2), A^′2 = 0.

and take the asymptotical values (7.28) for A1 and A2. We thus find:

F (Q²) = A0

π³(2A₁+ A₂) ≈ A0

π³A₂. (7.29)

The substitution of A2 from (7.28) in (7.29) with ~R⊥= ~∆/2 and ~∆² = Q² gives:

F (Q²) ≈ 4g²A²₀ π³

1

Q². (7.30)

From (7.29) and (7.30) we see that the asymptotical behaviour ∝ 1/Q² of the pion form factor is indeed determined by the extra component A2 of the pion wave function, related to the ω-dependent spin structure. This originates from the slower decrease of A2 ∝ 1/ ~R²_⊥ in comparison to A1 ∝ 1/ ~R⁴_⊥ (see eq.(7.28)). The enhancement of A2 is determined by the factor τ (ω·p) = (s − M²)/2 ≈ 2 ~R²_⊥ in (7.25). This extra factor originates from the part ωτ in the conservation law p + ωτ = k1 + k2 used to derive eq.(7.25) and therefore accompanies namely the ω-dependent spin structure.