kinematics in the proton rest frame. E k and E k 0 are the energies of the initial and final electrons, and θ is the laboratory scattering angle

energy transfer to the hadrons are

Q²≡ −q²=−(k−k⁰)²−−→_lab 2kk⁰(1−cosθ) ν ≡p·q

M −−→_lab E_k−E_k⁰ ∼k−k⁰. (5.41) The invariant mass-square of the unobserved final hadrons is

W²≡P_X² = (p+q)²=M²+ 2M ν−Q²

−−→_lab M²+ 2M(k−k⁰)−2kk⁰(1−cosθ). (5.42) Elastic scattering X = p is a special case with W² = M² and Q² = 2M ν. The next hadronic threshold is for a nucleon plus one pion, i.e.,X =p+π⁰orn+π⁺, corresponding to W²≥(M+mπ)². Still largerW²can involve more complicated many-particle states, which are summed over. The three independent kinematic variables are (s, Q², ν), or equivalently (k, k⁰, θ) (only two are independent in the special case of elastic scattering). However, the hadronic part of the process can only depend on the two Lorentz invariants Q² and ν. For a fixed initial energy k, the variablesQ² andν (and thereforeW²) can be varied and determined from k⁰ andθ.

It is convenient to define dimensionless variables x≡ Q²

2M ν, y≡ ν

E_k =E_k−E_k0

E_k ∼ k−k⁰

k . (5.43)

xis defined kinematically. However, in the SPMxwill be interpreted as the fraction of the proton momentum carried by the scattered parton.¹⁰ y is the fraction of the e⁻ energy in the lab frame that is transferred to the hadrons. Their ranges are

0≤x≤1, 0≤y≤ 1

1 + ^xM_2k −−−−→_kM 1. (5.44) The relation ofx,W², and θtoν andQ² is shown inFigure 5.12.

10Early papers often used the variableω≡1/x.

Q²

2M ν θ1

θ2

θ3

x = 1

x = 0.5

x = 0.25

Figure 5.12

Kinematic variables for deep inelastic scattering for fixed initial energy

E_k ∼ k m_e

. The horizontal and vertical axes are, respectively, 2M ν = 2M y/k and

Q²

= 2M xy/k, each running from 0 to 2M k. The sloping solid lines are for fixed 0

≤x≡Q²/2M ν ≤

1. The dashed lines are for fixed hadronic invariant mass-square

W²

=

M²

+ 2M ν

−Q²

, where

W²

=

M²

along the

= 1 line, increasing as one moves down and to the right. The dotted lines are for fixed laboratory scattering angle

θ, with θ3> θ2> θ1

.

5.5.2 The Cross Section and Structure Functions

First consider elastic scatteringe⁻p→e⁻pfor a hypothetical point-like proton. The spin-averaged cross section from the first diagram inFigure 5.13(with a point vertex) is

dσ¯= (2π)⁴δ⁴(k⁰+p⁰−k−p) 4M k

d³k⁰ (2π)³2Ek⁰

d³p⁰ (2π)³2Ep⁰

e⁴

q⁴L^µν_e L_pµν, (5.45) whereq=k−k⁰ =p⁰−p, and the traces are collected in the leptonic and hadronic tensors

L^µν_e = 1

2Tr [γ^µ(6k+me)γ^ν(6k⁰+me)] = 2

k^µk^0ν+k^0µk^ν+g^µνq² 2

L_pµν = 1

2Tr [γµ(6p+M)γν(6p⁰+M)] = 2

p_µp⁰_ν+p⁰_µp_ν+g_µνq² 2

(5.46)

The cross section can be rewritten d¯σ dk⁰dΩ =α²

q⁴ k⁰

kL^µν_e W_µν, (5.47)

wheredΩ =dcosθdϕ, and W_µν≡ 1

4πM 1 2

ss⁰

Z d³p⁰(2π)⁴δ⁴(p⁰−p−q)

(2π)³2Ep⁰ hps|J_Qµ^† (0)|p⁰s⁰ihp⁰s⁰|J_Qν(0)|psi. (5.48) The current in (5.48) is the proton part of the electromagnetic current

J_Q^µ = ¯ψpγ^µψp=J_Q^µ†, (5.49)

γ(q)

e⁻(k) e⁻(k)

p p

γ(q)

p e⁻(k)

X e⁻(k)

Figure 5.13

Left: elastic scattering

e⁻p → e⁻p. The shaded circle represents the

effects of the strong interactions. Right: deep inelastic scattering

e⁻p→e⁻X.

with

hp⁰s⁰|J_Q^ν(0)|psi= ¯u(p⁰, s⁰)γ^νu(p, s). (5.50) Of course,

hps|J_Q^µ†|p⁰s⁰i=hp⁰s⁰|J_Q^µ|psi^∗. (5.51) J_Q^µ is actually Hermitian, but it is useful to write (5.48) and (5.50) in a more general form for a later extension to the weak interactions. The tensorW_µν contains all of the information about the hadrons.

The cross section expression in (5.47) can be immediately extended to elastic scattering from a physical (strongly-interacting) proton, provided one replaces (5.50) by

hp⁰s⁰|J_Q^ν(0)|psi →u¯(p⁰, s⁰)Γ^ν_Qu(p, s), (5.52) where Γ^ν_Q(p⁰, p) is the vertex function that includes strong corrections. As discussed in Section2.12.4, the combination of Lorentz covariance, electromagnetic current conservation, and the observed reflection invariance of the strong interactions restricts Γ^ν_Q(p⁰, p) to the form

Γ^ν_Q=γ^νF₁^p(q²) +iσ^νρqρ

2M F₂^p(q²), (5.53)

where F_1,2^p (q²) are form factors that can depend on q² with F₁^p(0) = 1 and F₂^p(0) =κp, whereκ_p∼1.79 is the anomalous magnetic moment of the proton. For elastic scatteringk⁰ and cosθare not independent, but are related by (2.392) on page 77, i.e.,

k⁰

k = 1

1 + _M^k(1−cosθ), (5.54)

(enforced by an energy-conserving delta function inW_µν), with the correspondence of no-tation (k1, k2)→ (k, k⁰), θL → θ, and m_p → M. From (5.47) and (5.53) one obtains the Rosenbluth cross section formula for elastic scattering given in (2.400).

Expression (5.47) continues to hold for the inelastic case provided the hadronic tensor

is redefined as

whereX_N is anN particle state that may contain both fermions and bosons. In the inelastic casek⁰ and cosθare independent variables related to the invariant massW² by (5.42). For unpolarized protons the tensor W_µν can only depend on the four-vectors p and q (it can also depend on the spin vectorsin the polarized case). The only tensors one can construct are

symmetric: pµpν, qµqν, pµqν+qµpν, gµν

antisymmetric: p_µq_ν−q_µp_ν, _µνρσp^ρq^σ, (5.56) each of which can be multiplied by a function of the Lorentz invariantsQ² andν. One can use the reflection invariance of the strong interactions to show that the_µνρσterm is absent.

In any case, the leptonic tensorL^µν_e is symmetric,¹¹so one can keep just the symmetric part ofW_µν. Furthermore, the electromagnetic current is conserved,∂^µJ_Qµ= 0, which implies

q^µWµν = 0, q^νWµν = 0. (5.57)

Thus, only two linear combinations survive, and the most general form is Wµν = proton structure functions. They generalize the form factors F_1,2(q²) of elastic scattering, and contain all of the information about the strong interactions effects. One can combine (5.47) and (5.58) to obtain for the deep inelastic cross section in the proton rest frame. The solid angle element is usually integrated over the azimuthal angle,dΩ =dϕdcosθ→2πdcosθ, and the kinematic variables can be rewritten

11Both the leptonic and hadronic tensors haveµνρσterms if there is polarization, or for parity-violating weak processes such as νN →νX and νN →µX due to the interference between the vector and axial currents.

W₁ and W₂ can be determined separately from the data by varying θ, k, and k⁰ for fixed Q² andν.

An alternative notation is to use the variablesxandQ² and to define F1 x, Q²

=M W1 Q², ν

, F2 x, Q²

=νW2 Q², ν

. (5.62)

This is useful because the simple parton model (QCD) predicts that F_i is independent of (slowly varying with)Q²forQ² large. Then,

d²σ¯ In the second form, the last term vanishes for k M, while the middle term vanishes in the simple quark parton model.

The deep inelastic limit is defined asQ², ν→ ∞withx=Q²/2M νfixed. If the proton were an extended fuzzy object, one would expectF_i(x, Q²)→0 in this limit. However, the MIT-SLAC experiments (Friedman and Kendall, 1972; Mishra and Sciulli, 1989) circa 1970 showed instead that

F_i(x, Q²)−−−−−→

Q²→∞ F_i(x)6= 0, (5.64)

a property known as Bjorkenscaling(Bjorken, 1969). This scaling can be understood in the Feynman parton model (e.g., Field and Feynman, 1977; Drell et al., 1969), in which the pro-ton is made up of hard point-like parpro-ton constituents. The observedy distribution showed that the partons have spin-¹₂, consistent with QCD and asymptotic freedom. Subsequent experiments established that the scaling is only approximate, and in fact the slow (loga-rithmic) variation of theF_i withQ² is what one expects from the higher-order corrections in QCD.¹²

5.5.3 The Simple Quark Parton Model (SPM)

The cross section for elastic scatteringe⁻p→e⁻pfrom a point proton is given in (2.391) on page 77, with the kinematic constraint fork⁰ given in (5.54) (see the subsequent comment on notation). It is convenient to rewrite (2.391) as

d¯σ

12Scaling also breaks down at lowQ², where strong coupling effects are important. In particular, hadronic resonances in theγpchannel for fixedW² are important. The scaling behavior of the structure functions smoothly interpolates these resonances (Bloom and Gilman, 1971; Melnitchouk et al., 2005), as can be understood fromfinite energy sum rules(FESR), derivable from analyticity.

This is a special case of the general formula (5.61) provided we identify the structure

Now assume that the proton is a bound state of point-like quarks. Consider the contribution to the structure functions from an individual quarkqi with electric chargeei, which carries four-momentum xip, where pis the proton momentum. It is plausible that (5.67) applies for that quark, provided one replacesM →x_iM and multiplies by e²_i, i.e., (This result can be better derived in the infinite momentum frame, where the proton and quark masses and the transverse momentum of the proton relative to the electron direction are negligible.) The contribution of that quark to the structure functions is

F₁ⁱ(x, Q²) =M W₁ⁱ(Q², ν) =1

To find F_1,2(x, Q²) one must sum over the quark types and integrate over their possible momenta. This is done in the cross section (i.e., inF_1,2) because the differentiandx_ilead to different (incoherent) final states. Introduce theparton distribution function (PDF)q_i(xi) as the probability density (the absolute square of the momentum space wave function) for finding quarkq_iin the proton with momentum fractionx_i. Then from (5.69) the predicted structure functions are The simple parton model therefore predicts that the F_i are independent of Q² for large Q² (Bjorken scaling). The quantityxq_i(x) is interpreted as the momentum distribution for partonq_i.

The predicted relationF2= 2xF1, theCallan-Gross relation (Callan and Gross, 1969), is a signature of spin-¹₂ constituents. Scattering from spin-0 constituents would lead to F1 = 0, F2 6= 0 and therefore a different angular distribution. An interpretation of this result is that one can show (see, e.g., Renton, 1990) that

R(x, Q²)≡σ_L

σT ∼ F₂−2xF1

2xF1

, (5.71)

whereσ_L andσ_T are, respectively, the total cross sections forγ_L,T^∗ p, whereγ^∗_L,T is a virtual photon with momentumq, andLandT refer to longitudinal (photon helicity 0) and trans-verse (photon helicities ±1) polarizations. In the Breit frame (discussed in Section 2.3.4), the virtual photon has four-momentum q = (0,0,0,−Q), while the incident [final] parton

has momentum ¹₂(Q,0,0, Q) [¹₂(Q,0,0,−Q)], as inFigure 2.4.For spin-0 partons one would haveσ_T = 0 (R=∞) by angular momentum conservation, since there is no orbital angular momentum along the direction of the photon and parton momenta. For spin-¹₂ partons, on the other hand,σ_L=R= 0 using the fact that helicity is conserved for vector transitions of massless spin-¹₂ particles, e.g., (2.214) on page 42.

Even to the extent that the SPM is valid, a real proton is expected to consist of not only the three valence quarks uud of the quark model, but also a sea of q¯q pairs and of gluons produced by soft (low momentum) processes such as illustrated in Figure 5.14.The