1 Relation to Feynman's parton model

1 Relation to Feynman’s parton model

Remember (No. 16):

The cross section for inelastic electron-proton scattering was expressed in Eq.(1.1) of No. 16, in terms of the hadronic tensor W^µν (defined in Eq.(1.3) of No. 16). Then W^µν was parametrized in terms of structure functions, see Eqs.(1.11), (1.12) of No. 16, to get more explicit forms for the cross section.

(1) What is the form of the hadronic tensor for a point particle?

Use Eq.(1.3) of No. 16 for the spin averaged case, and insert the currentJ_{F I}^µ of a point particle.

We get 1 2

X

s

W^µν

!

point

= E_p 2

Z

d³p⁰δ⁽⁴⁾(p⁰−p−q) M² E_pE_p⁰

X

s,s⁰

(u(~p⁰, s⁰)γ^µu(~p, s)) (u(~p⁰, s⁰)γ^νu(~p, s))^∗(1.1) To perform the integration, use the identity used in No. 15:

Z d³p⁰ 2E_p⁰ =

Z

d⁴p⁰δ

p⁰²−M² θ(p⁰₀)

By using the definition of the Bjorken variablex=−q²/(2p·q), and performing the spin sums as in No. 16, Eq.(1.1) becomes

1 2

X

s

W^µν

!

point

= M²

2p·qδ(x−1) X

s,s⁰

(u(~p⁰, s⁰)γ^µu(~p, s)) (u(~p⁰, s⁰)γ^νu(~p, s))^∗

= 1

2p·qδ(x−1)

p^0µp^ν+p^0νp^µ−g^µν p⁰·p−M²

= 1

2δ(x−1)

−g^µν+ p^0µp^ν+p^0νp^µ p·q

(1.2) where we used p⁰ ·p−M² = p·q in the last step. (Note that the scattering on a point particle must be elastic, therefore p² = M².) Now compare (1.2) with the parametrization in terms of structure functions, see Eq.(1.15) of No. 16:

1 2

X

s

W^µν

!

point

=F₁^point(x, Q²)

−g^µν +q^µq^ν q²

+ F₂^point(x, Q²) p·q

p^µ−p·q

q² q^µ p^ν − p·q q² q^ν

Using here p·q =−q²/2 (because the scattering on point particle is elastic), and q^µ =p^0µ−p^µ, this becomes

1 2

X

s

W^µν

!

point

= −g^µνF₁^point(x, Q²) + p^0µp^ν +p^0νp^µ 2(p·q)

F₁^point(x, Q²) + 1

2F₂(x, Q²)

+ p^0µp^0ν +p^µp^ν 2(p·q)

−F₁^point(x, Q²) + 1

2F₂(x, Q²)

(1.3)

Comparing (1.2) and (1.3), we obtain for the structure functions of a point particle ¹: F₁^point(x, Q²) = 1

2δ(x−1), F₂^point(x, Q²) =δ(x−1) = 2x F₁^point(x, Q²) (1.4) Note: The delta-function δ(x −1) just expressed the condition of elastic scattering on the point particle. Because a point particle has no structure, the structure functions are independent of Q².

The same calculation as above can be done for the full hadronic tensorW_point^µν of a point particle (not spin averaged):

(W^µν)_point = E_p Z

d³p⁰δ⁽⁴⁾(p⁰−p−q) M² EpEp⁰

X

s⁰

(u(~p⁰, s⁰)γ^µu(~p, s)) (u(~p⁰, s⁰)γ^νu(~p, s))^∗

= M²

p·qδ(x−1)X

s⁰

(u(~p⁰, s⁰)γ^µu(~p, s)) (u(~p, s)γ^νu(~p⁰, s⁰))

= 1

2δ(x−1)

−g^µν+p^0µp^ν +p^0νp^µ

p·q +iM ^µνσλq_σS_λ p·q

(1.5) We compare this with the parametrization in terms of the structure functions F₁^point, F₂^point, g₁^point, g₂^point, see Eq.(1.15) and (1.16) of No. 16. The results for F₁^point and F₂^point are given in Eq.(1.4), and for g₁^point, g₂^point we obtain

g^point₁ (x, Q²) = 1

2δ(x−1), g₂^point(x, Q²) = 0 (1.6)

1The relationF₂= 2x F₁expresses that the “point particle” considered here has spin 1/2.

(2) Experimental data on the proton structure function F₂(x, Q²) for Q² >1.5 GeV²:

We see:F2 is almost independent ofQ², i.e., F2(x, Q²)'F2(x). This is called the Bjorken scaling.

The data for F₁ (not shown here) indicate that also F₁ depends only on x for large Q², and that there is the following relation between F₁(x) and F₂(x):

F₂(x)'2x F₁(x) (1.7)

We will show in lecture No. 18 that scaling and relation (1.7) can be explained naturally in the Bjorken limit: The Bjorken limit (B.L.) is defined by

Q² → ∞, p·q → ∞, x= Q²

2(p·q) fixed (1.8)

The B.L. is only a “theoretical tool” to derive Feynman’s parton model, which gives simple expres- sions and physical interpretations of structure functions. The experimental data show scaling and the relation (1.7) already forQ² >2 GeV² and p·q >1 GeV². For very small values of x, however, the data show a strong violation of scaling.

(3) Idea of Feynman’s parton model (most naive version):

The dominant process for inelastic electron-hadron scattering at large Q² is elastic scattering on point-like constituents (partons).

electron hadron

q

k k’

(p) +q

p_i

p_i

hadrons (p+q)

For elastic scattering on quarki (with 4-momentum pi and mass p²_i =m²_i) we have ² (p_i+q)² =m²_i ⇒ Q²

2(pi·q) = 1 (1.9)

This means: The Bjorken variable of the quark is equal to 1.

Ifz_i is the 4-momentum fraction of partoni, thenp_i =z_ip, wherep=P

ip_i is the total 4-momentum of the hadron. Then we get from (1.9)

Q²

2z_i(p·q) = 1⇒z_i = Q²

2(p·q) =x (1.10)

This means: Only the quark with momentum fraction zi =x can interact with the electron!

Because the quark is point-like, the structure function (F₂) of quarki with charge e_i is (see (1.4)) F_2i =e²_i δ

Q²

2(pi·q)−1

=e²_i δ x

zi

−1

=e²_i z_iδ(x−z_i) = x e²_i δ(x−z_i) (1.11)

2Hereilabels the flavor of the quark: i=u, d, . . ..

Then the total structure function F₂ of the hadron must be given by F₂(x, Q²) =X

i

Z 1

0

dz_iq_i(z_i)F_2i (1.12)

where q_i(z) is the probability to find quark i with momentum fraction z inside the proton, i.e., the

“momentum distribution function” of the quark. By using (1.11), we get for the structure function of the proton:

F₂ =X

i

Z 1

0

dz q_i(z)x e²_i δ(x−z) =x X

i

e²_i q_i(x) (1.13)

The same calculation can be done also for the structure function F₁: From Eq.(1.4), for quark i we haveF_1i = _2z¹

iF_2i = ¹₂e²_i δ(x−z_i), and therefore F₁ of the proton is given by F₁(x, Q²) =X

i

Z 1

0

dz_iq_i(z_i)F_1i = 1 2

X

i

e²_i q_i(x) (1.14)

We see: The structure functions in the parton model, given by Eqs. (1.13) and (1.14), are independent of Q², which means Bjorken scaling, and satisfy the relation

F2(x) = 2x F1(x) (1.15)

Extension of the naive parton model:

First, let us denote theu-quark distribution simply by u(x)≡q_u(x) and the d-quark distribution by d(x)≡qd(x).

The proton (p) consists of two “valence (v)” u-quarks³with chargee_u = 2/3, and one “valence d-quark”

with charge e_d =−1/3. We then get the following structure function of the proton in the extended parton model:

F₂^(p)(x) = x 4

9u^(p)_v (x) + 1

9d^(p)_v (x)

+ (sea quark contributions) (1.16) whereu^(p)v (x) is the momentum distribution of the valenceu- quark in the proton, and d^(p)v (x) is the momentum distribution of the valence d - quark in the proton. The normalizations are

Z 1

0

dx u^(p)_v (x) = 2,

Z 1

0

dx d^(p)_v (x) = 1 (1.17)

Note: “Valence” quark contributions are dominant for large x (x > 0.5)), while “sea” quark and gluon contributions are dominant for small x.

3In addition to the three “valence” quarks (momentum distributionqv(x)), there are also “sea” quarks and anti- quarks in the proton (momentum distributionqs(x) andq_s(x)), because of the spontaneous creation and annihilation of quark-antiquark pairs from the vacuum. (Generally,q_s(x) =q_s(x).) Therefore, the total quark distributions are q(x) =q_v(x) +q_s(x), and the antiquark distributions areq(x) =q_s(x). We will see later that the functionsq(x) and q(x) are defined in field theory (in terms of field operators), and thereforeq_v(x) is defined byq_v =q(x)−q(x).

Relations (1.17) are called the “number sum rules”: Sea quarks and gluons do not contribute to baryon number and charge, and therefore the relations (1.17) are satisfied only by the valence quark distributions. However, (sea quarks + gluons) carry a part of the total momentum of the proton, i.e., they contribute to the “momentum sum rule”:

Z 1

0

dx x u^(p)_v (x) +d^(p)_v (x)

+ (sea quark contributions) + (gluon contributions) = 1 (1.18) For the neutron (one valence u-quark and two valence d-quarks) we obtain

F₂⁽ⁿ⁾(x) = x 4

9u⁽ⁿ⁾_v (x) + 1

9d⁽ⁿ⁾_v (x)

+ (sea quark contributions)

= x 4

9d^(p)_v (x) + 1

9u^(p)_v (x)

+ (sea quark contributions) (1.19) In the last relation above, we assumed “flavor symmetry”:

u^(p)_v (x) =d⁽ⁿ⁾_v (x), d^(p)_v (x) =u⁽ⁿ⁾_v (x) (1.20) The contributions of “sea” quarks to Eq.(1.16) and Eq.(1.19) are almost the same, and cancel if we take the differenceF₂^(p)(x)−F₂⁽ⁿ⁾(x):

The left graph clearly shows that the valence quark distributions have a peak around x ' 1/3, as expected.

From the experimental data ofF₂(x) for the proton and neutron (and other data), one can extract all quark distribution functions (right graph). The gluon distribution function (g(x)) is still very uncertain, but seems to be very large at small values ofx.