1 Derivation of the parton model

1 Derivation of the parton model

Remember (No. 16):

The cross section for inelastic inclusive electron-proton scattering was expressed in Eq.(1.1) of No.

16, in terms of the hadronic tensorW^µν, which is given by the current - current correlation function, see Eq.(1.9) of No. 16.

Here we calculate W^µν in the quark model, and derive the parton model described in No. 17. We will use the method of Feynman diagrams, and therefore we first express the usual product of cur- rent operators in Eq.(1.9) of No. 16 by the time-ordered product (T - product) of current operators.

Then we can use Wick’s theorem and Feynman diagrams ¹.

(1) We first show the following identity:

Z

d⁴z e^iq·zhp|J^µ(z)J^ν(0)|pi= 2 Im

i Z

d⁴z e^iq·zhp|T(J^µ(z)J^ν(0))|pi

(1.1) Here theJ’s are current operators ( ˆJ of No. 16), and the T -product is defined by

T (J^µ(z)J^ν(0)) = Θ(z⁰)J^µ(z)J^ν(0) + Θ(−z⁰)J^ν(0)J^µ(z) (1.2) Proof of (1.1): (i) Consider the l.h.s. of (1.1). We insert a complete set of hadronic states (P

n|nihn|= 1) between the current operators, and use the z - dependence of the transition matrix elements (see also Eq.(1.4) of No. 16), like

hp|J^µ(z)|ni=e^i(p−pn)·zhp|J^µ(0)|ni (1.3) Then we can do thez -integration, and get

Z

d⁴z e^iq·zhp|J^µ(z)J^ν(0)|pi=X

n

(2π)⁴ δ⁽⁴⁾(p+q−pn)hp|J^µ(0)|ni hn|J^ν(0)|pi

(1.4)

1A direct evaluation of Eq.(1.9) of No. 16, without Feynman diagrams, is also possible: See lecture of R.L. Jaffe,

“Deep inelastic scattering with application to nuclear targets”, in: Relativistic dynamics and quark-nuclear physics, ed. M.B. Johnson, A. Picklesimer, Wiley, New York, 1986, p. 537.

(ii) Consider the r.h.s. of (1.1). We use the definition of theT-product Eq.(1.2), and insert a complete set of hadronic states between the current operators. Using (1.3) we obtain

2 Im

i Z

d⁴z e^iq·zhp|T(J^µ(z)J^ν(0))|pi

= 2 Im{i Z

d⁴z e^iq·z

×X

n

Θ(z0)e^i(p−pn)·ze^−z0hp|J^µ(0)|ni hn|J^ν(0)|pi+ Θ(−z0)e^{−i(p−pn)·z}e^z0hp|J^ν(0)|ni hn|J^µ(0)|pi }

(1.5) Here we introduced convergence factors ( → 0⁺) for the z⁰ integrals. The R

d³z integration gives 3-momentum conserving δ - functions, and (1.5) becomes

2 (2π)³ Im iX

n

{[δ⁽³⁾(~q+~p−p~_n) Z ∞

0

dz⁰e^{i(Ep−En+q0+i)z0}hp|J^µ(0)|ni hn|J^ν(0)|pi +δ⁽³⁾(−~q+~p−~p_n)

Z 0

−∞

dz⁰e^{−i(Ep−En−q0+i)z0}hp|J^ν(0)|ni hn|J^µ(0)|pi]}

Performing the z⁰ integral, this becomes:

−2 ImX

n

(2π)³ [δ⁽³⁾(~q+~p−~pn)hp|J^µ(0)|ni hn|J^ν(0)|pi E_p −E_n+q₀+i +δ⁽³⁾(−~q+~p−~pn)hp|J^ν(0)|ni hn|J^µ(0)|pi

E_p−E_n−q₀ +i ] (1.6)

We now use the formula (for→0⁺) 1

A+i =P 1

A −iπδ(A) (1.7)

whereP means the principal value. The imaginary part of (1.6) comes from the second term in (1.7).

Noting that En−Ep >0 (because the state |pi is the lowest energy state for given baryon number) and q₀ > 0, we see that the denominator of the second term in (1.6) cannot vanish, and therefore this term gives no contribution to the imaginary part. Then (1.6) becomes finally

X

n

(2π)⁴ δ⁽⁴⁾(p+q−p_n)hp|J^µ(0)|ni hn|J^ν(0)|pi (1.8) which is equal to (1.4). This concludes the proof of Eq.(1.1).

The amplitude for forward Compton scattering on the proton (forward scattering amplitude of pro- ton (momentum p) and virtual photon (momentumq)) is defined as ²

T^µν(p, q) =i2E_p Z

d⁴z e^iq·zhp|T (J^µ(z)J^ν(0))|pi (1.9)

p p

q q

.^{ν µ}. i

Then the hadronic tensor can be expressed by the imaginary part of T^µν: W^µν = 1

2πImT^µν (1.10)

(2) Now we calculate the Compton amplitude in the quark model. It is expressed by the following diagram (“handbag diagram”):

N(p) N(p)

q q

.^ν .^µ

i

Q(k) Q(k)

Q(k+q)

T^{µ ν}(p,q) =

[Note for specialists: According to Feynman rules, each quark line is translated to iS, where S is the usual Feynman propagator. Becausei⁴= 1, we do not attach ito propagators in the following.]

T^µν(p, q) =

Z d⁴k (2π)⁴

X

i=u,d

Tr [Mi(p, k)t^µν_i (k, q)] (1.11) Here i = u, d labels the flavor of the quark, Tr denotes the trace over Dirac indices, and M_i(p, k) corresponds to the lower part of the handbag diagram:

2The factor 2E_p arises from our non-covariant normalization of state vectors, as explained in No. 15.

N(p) N(p) .

Q(k) Q(k)

M (p,k) = _Q

.

^β

.

^α

βα

In this diagram, the external quark lines (momentumk) represent Feynman propagators, and the external nucleon line (momentump) represents the Dirac spinor of the nucleon (u(~p, S)). Therefore, M_i(p, k) is the propagator (2-point function) of the quark i=u, d inside the nucleon. It is defined by (see the previous footnote for the origin of the factor 2E_p)

M_i^βα(p, k) = 2E_p Z

d⁴z e^ik·zhp|T ψ_α(0)ψ_β(z)

|pi (1.12)

We will show later that this quantity is real. - The quantityt^µν_i (p, k) in Eq.(1.11) corresponds to the upper part of the handbag diagram:

q q

.^ν .^µ

Q(k) Q(k)

Q(k+q)

t_Q^{µ ν}(k,q) =

Here the external quark propagators (momentumk) are not included (because they are included in M_i). t^µν_i (k, q) is the amplitude for forward Compton scattering on the quark i=u, d. It is given by

t^µν_i (k, q) = (ie_i)² γ^µS(k+q)γ^ν =−e²_i γ^µ 6k+6q+m_i

(k+q)²−m²_i +iγ^ν (1.13) [Note for specialists: According to Feynman rules, a quark-photon vertex is translated into (−ie_iγ^µ).] Its imaginary part is given by (see Eq.(1.7)

Imt^µν_i (k, q) = π e²_i δ (k+q)²−m²_i)

γ^µ(6k+6q+m_i)γ^ν (1.14) Here we use

(k+q)²−m²_i =k²+q²+ 2k·q−m²_i = 2p·q

−x+k·q

p·q + k²−m²_i 2p·q

wherex=−q²/(2p·q) is the Bjorken variable for the nucleon. Then (1.14) becomes Imt^µν_i (k, q) = π e²_i

2p·qδ

x− k·q

p·q +k²−m²_i 2p·q

γ^µ(6k+6q+m_i)γ^ν (1.15) The Bjorken limit (B.L.) means Q² =−q² → ∞ and p·q → ∞ such that the ratio x=Q²/(2p·q) is fixed. (See No. 17, and note that 0< x <1). In this limit (1.15) simplifies to

Imt^µν_i (k, q)−→^B.L. π e²_i 2p·qδ

x− k·q p·q

γ^µ6q γ^ν (1.16)

The hadronic tensor is given by W^µν = _2π¹ ImT^µν (see Eq.(1.10)), and from (1.11) and (1.16) we obtain in the quark model

W^µν(p, q) = 1 4p·q

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

Tr [Mi(p, k)γ^µ6q γ^ν]

(1.17) Use6q =γ^λq_λ, and the following formula for the product of three Dirac γ-matrices:

γ^µγ^λγ^ν = g^µλg^νσ+g^µσg^νλ −g^µνg^λσ

γ_σ−i^µλνσγ_σγ₅ (1.18) Then we can separate the symmetric and antisymmetric parts of the hadronic tensor (1.17):

W^µν(p, q) = W_s^µν+W_a^µν (1.19)

where

W_s^µν(p, q) = q_λ 4p·q

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

g^µλg^νσ +g^µσg^νλ−g^µνg^λσ

× Tr [M_i(p, k)γ_σ] (1.20)

W_a^µν(p, q) = i^µνλσ q_λ 4p·q

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

Tr [M_i(p, k)γ_σγ₅] (1.21) We now consider each part separately:

• The symmetric partW_s^µν has been parametrized in terms of the structure functionsF₁ and F₂, see Eq.(1.15) of No. 16:

W_s^µν(p, q) = F₁(x, Q²)

−g^µν+ q^µq^ν q²

+F₂(x, Q²)p˜^µp˜^ν

p·q (1.22)

where we defined the 4-vector ˜p^µ by

˜

p^µ ≡p^µ− p·q

q² q^µ (1.23)

such that ˜p·q = 0. In order to extract F₁ and F₂, we (i) take the trace of (1.22), and (ii) contract (1.22) with ˜p_µp˜_ν/˜p². This gives

(W_s)^µ_µ = −3F₁(x, Q²) + p˜²

p·qF₂(x, Q²) (1.24)

˜ p_µp˜_ν

˜

p² W_s^µν = −F₁(x, Q²) + p˜²

p·qF₂(x, Q²) (1.25) In the Bjorken limit, it follows from the definition of ˜p^µ (see Eq.(1.23)) and the definition of the variable x=−q²/(2p·q), that ˜p²/(p·q) = 1/(2x). Then (1.24) and (1.25) become

(W_s)^µ_µ = −3F₁(x, Q²) + F₂(x, Q²)

2x (1.26)

˜ pµp˜ν

˜

p² W_s^µν = −F₁(x, Q²) + F2(x, Q²)

2x (1.27)

One can show from the quark model formula (1.20) in the Bjorken limit³ that

˜ p_µp˜_ν

˜

p² W_s^µν = 0 (1.28)

Then we see from Eq.(1.27) that in the Bjorken limit the structure functions F1 and F2 are related by

F₂(x, Q²) = 2xF₁(x, Q²) (1.29)

in agreements with the observations discussed in No. 17. Then we can calculate F₂ from Eq.(1.26) and (1.20):

F₂(x, Q²) = −x (W_s)^µ_µ

= x

2(p·q)

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

Tr [M_i(p, k)6q]≡x X

i

e²_i q_i(x) (1.30) where we defined the “spin independent quark distribution function”q_i(x) by

qi(x) = 1 2(p·q)

Z d⁴k (2π)⁴δ

x− k·q p·q

Tr [Mi(p, k)6q] (1.31)

• The antisymmetic partW_a^µν has been parametrized in terms of the structure functions g₁ and g₂, see Eq.(1.16) of No. 16:

W_a^µν = M

p·q i^µνλσq_λ

g₁(x, Q²)S_σ +g₂(x, Q²)

S_σ − (S·q)p_σ p·q

(1.32)

3The derivation of (1.28) from (1.20) in the Bjorken limit is delicate, and we postpone the proof to a later lecture, after introducing light cone variables.

Comparison with the quark model expression Eq.(1.21) gives S_σg₁(x, Q²) = 1

4M

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

Tr [M_i(p, k)γ_σγ₅]

≡ S_σ 1 2

X

i

e²_i ∆q_i(x) (1.33)

g₂(x, Q²) = 0 (1.34)

In (1.33) we defined the “spin-dependent quark distribution function” by the following relation:

S^µ∆qi(x) = 1 2M

Z d⁴k (2π)⁴ δ

x− k·q p·q

Tr [Mi(p, k)γ^µγ5] (1.35) In order to express the result (1.35) similar to Eq.(1.31), we contract both sides of Eq.(1.35) with q_µ, and divide by (p·q):

MS·q p·q

∆qi(x) = 1 2(p·q)

Z d⁴k (2π)⁴

X

i

e²_i δ

x− k·q p·q

Tr [Mi(p, k)6q γ5]

(1.36) Note that both functions (1.31) and (1.36) are defined in the Bjorken limit. In the next lecture, we will see how to calculate the Bjorken limit, and how to interpret (1.31) and (1.36).

Here we explain only the result:

Consider a nucleon with 4-momentump^µand spin direction parallel (positive helicity) or anti-parallel (negative helicity) to its 3-momentum. Denote the probability to find a quark, with flavor i=u, d, momentum fractionx, and spin direction parallel (↑) or anti-parallel (↓) to the nucleon spin, byq^↑_i(x) and q^↓_i(x). Then the functions (1.31) and (1.36) can be expressed as

q_i(x) = q_i^↑(x) +q_i^↓(x) (1.37)

∆q_i(x) = q_i^↑(x)−q_i^↓(x) (1.38)

From this interpretation, and the (naive) assumption that the nucleon consists only of 3 valence quarks, it follows that

Z 1

0

dx X

i

q_i(x) = 3,

Z 1

0

dx x X

i

q_i(x) = 1,

Z 1

0

dx X

i

∆q_i(x) = 1 (1.39) Here the first relation (“number sum rule”) means the the nucleon consists of 3 valence quarks, the second relation (“momentum sum rule”) means that the 3 valence quark carry 100% of the nucleon

momentum, and the third relation (“spin sum rule”) means that they carry 100% of the nucleon spin.

However, in the nucleon there are also “sea quarks” (qq pairs) and gluons, which carry no baryon number but can carry momentum and spin. Therefore, in the extended parton model, the number sum rule remains true, but the momentum sum rule and the spin sum rule will be modified.