3 The spin crisis of the proton (an introduction)

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1 Physical meaning of structure functions and quark distri- bution functions

Remember (No. 18): (1) The spin-independent structure functionsF₁(x),F₂(x) in the Bjorken limit are given by F₁(x) = ¹₂ P

i=u,d e²_i q_i(x) and F₂(x) = x P

i=u,d e²_i q_i(x). Here the spin-independent quark distribution function is defined by¹

q_i(x) = Ep

p·q

Z d⁴k (2π)⁴

Z

d⁴z e^ik·zδ

x− k·q p·q

hp, S|T ψ_i(0)6q ψ_i(z)

|p, Si (1.1) (2) The spin-dependent structure functions g₁(x), g₂(x) in the Bjorken limit are given by g₁(x) =

1 2

P

i=u,d e²_i ∆q_i(x) and g₂(x) = 0. Here the spin-dependent quark distribution function is defined by

MS·q p·q

∆q_i(x) = E_p p·q

Z d⁴k (2π)⁴

Z

d⁴z e^ik·zδ

x− k·q p·q

hp, S|T ψ_i(0)6q γ₅ψ_i(z)

|p, Si (1.2) Why can we call (1.1) and (1.2) “quark distribution functions”? Let us first check the normalizations:

If we integrate over x, the delta functions in (1.1) and (1.2) go away. Then the integral _(2π)^d⁴^k4 gives δ⁽⁴⁾(z), and we obtain

Z 1

0

dx q_i(x) = Ep

p·qhp, S|ψ_i(0)γ^µψ_i(0)|p, Siq_µ (1.3)

MS·q p·q

Z 1

0

dx∆q_i(x) = E_p

p·qhp, S|ψ_i(0)γ^µγ₅ψ_i(0)|p, Siq_µ (1.4) where we left out the time-ordering symbol T, because both quark field operators now appear at the same space-time point z = 0. In our non-covariant normalization of the state |pi we have the following relations² :

hp, S|ψ_i(0)γ^µψi(0)|p, Si = Ni

M

E_pu(~p, S)γ^µu(~p, S)

=Ni

p^µ

E_p (1.5)

hp, S|ψ_i(0)γ^µγ₅ψ_i(0)|p, Si = (∆N_i) M

E_pu(~p, S)γ^µγ₅u(~p, S)

= (∆N_i) M

E_p S^µ (1.6)

1To get Eqs.(1.1) and (1.2), insert Eq.(1.12) of No. 18 forMi(p, k) into Eq.(1.31) and (1.36) of No. 18. In this section, we also attach the flavor label to the quark fields (ψ_i), in order to make the flavor dependence explicit, and denote the spin direction of the nucleon byS.

2To check relation (1.5), consider the caseµ= 0, whereψ_i^†ψ_iis the operator for the number density of quarks with flavori. Thenhp, S|ψ^†_iψ_i|p, Si= ^N_Vⁱhp, S|p, Si=N_i, where we used our non-covariant normalizationhp, S|p, Si=V, see No. 15. In a similar way, to check relation (1.6), consider the caseµ= 1,2,3, where ψ_i~γ γ₅ψ_i is the operator for the spin density of quarks with flavor i, i.e.; the number density of (quarks with spin parallel to the nucleon spin) minus the number density of (quarks with spin anti-parallel to the nucleon spin). Its expectation value is given by hp, S|ψ^†_i~γγ₅ψ_i|p, Si= ^∆N_Vⁱ _E^M

p

S~hp, S|p, Si= ∆N_i _E^M

p

S.~

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where N_i ≡ N_i^↑+N_i^↓ is the number of quarks (of either spin direction) with flavor i = u, d inside the hadron, and ∆N_i ≡ N_i^↑−N_i^↓ is the (number of quarks with spin parallel to the nucleon spin) minus the (number of quarks with spin anti-parallel to the nucleon spin) with flavor i =u, d inside the hadron. We then obtain from Eq.(1.3) and (1.4):

Z 1

0

dx q_i(x) = N_i =N_i^↑+N_i^↓ (1.7)

Z 1

0

dx∆q_i(x) = ∆N_i =N_i^↑−N_i^↓ (1.8) The relation (1.7) suggests thatq_i(x) may be the sum of probability densities to find a quark (flavor i) with spin parallel (↑) and anti-parallel (↓) to the nucleon spin, with some value of x. Similarly, relation(1.8) suggests that ∆q_i(x) may be the difference of probability densities to find a quark (flavor i) with spin parallel (↑) and anti-parallel (↓) , with some value ofx.

In order to see the meaning of the variablex, we define light-cone momentum components as follows:

k^± ≡ 1

√2 k⁰±k³

, ~kT = k¹, k²

(1.9) This is a simple variable transformation from the Minkowski 4-vector componentsk^µ = (k⁰, k¹, k², k³) to the light-cone 4-vector components k^µ = (k⁺, k⁻, k¹, k²). We also define k± ≡ ^√¹

2(k₀±k₃), so that k⁺ =k− and k⁻ =k₊. The scalar product of two 4-vectors a and b can be rewritten in terms of light-cone components by

a·b=a⁰b⁰−~a·~b=a⁺b⁻+a⁻b⁺−~a_T ·~b_T (1.10) We will also define the light-cone plus (+) and minus (-) components of the Diracγ-matrices as

γ^± = 1

√2 γ⁰±γ³

, γ± = 1

√2(γ₀±γ₃) so thatγ⁺ =γ− and γ⁻ =γ₊.

We now choose a frame where the momentum transfer from the electron to the hadron is along the

−ˆz direction: In usual coordinates, q^µ = (q⁰,0,0, q³) with q³ < 0. Then the Bjorken limit (B.L.) corresponds to the limit q⁰ → ∞, q³ → −∞, such that q⁻ = (q⁰ − q³)/√

2 → ∞ and q⁺ = (q⁰+q³)√

2→ −x/, p⁺, because

q² = 2q⁺q⁻ −→ ∞^B.L. , x= −q²

2p·q = −q⁺q⁻ p⁺q⁻+p⁻q⁺

−→ −B.L. q⁺

p⁺ (1.11)

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Therefore, in the Bjorken limit, q⁺ −→ −p^B.L. ⁺x=−^√¹

2(E_p+p³)x. For example, in the rest frame of the hadron, this is equal to −(M x)/√

2.

Now we can express the functions q_i(x) of Eq.(1.1) and ∆q_i(x) of Eq.(1.2) in terms of light-cone coordinates. For this, we note the following most important relation in the Bjorken limit:

k·q

p·q = k⁺q⁻+k⁻q⁺ p⁺q⁻+p⁻q⁺

−→B.L. k⁺

p⁺ =x (1.12)

This means thatxis really a momentum fraction carried by the quark, but not of a usual momentum component, but of a light-cone momentum component: k⁺=p⁺x. For example, in the rest frame of the hadron,k⁺ = (M x)√

2.

For the other factors in Eqs.(1.1) and (1.2), we use the relation 6q

p·q = γ⁺q⁻+γ⁻q⁺ p⁺q⁻+p⁻q⁺

−→B.L. γ⁺ p⁺

Taking also the Bjorken limit on the l.h.s. of Eq.(1.2) gives (S ·q)/(p·q) −→^B.L. S⁺/p⁺. Then the functionsq_i(x) and ∆q_i(x) of Eq.(1.1) and (1.2) can be expressed as

q_i(x) = E_p p⁺

Z d⁴k (2π)⁴δ

x− k⁺ p⁺

Z

d⁴z e^ik·zhp, S|T ψ_i(0)γ⁺ψ_i(z)

|p, Si (1.13) M S⁺

p⁺

∆q_i(x) = E_p p⁺

Z d⁴k (2π)⁴δ

x− k⁺ p⁺

Z

d⁴z e^ik·zhp, S|T ψ_i(0)γ⁺γ₅ψ_i(z)

|p, Si (1.14) Note that the δ-function in those expressions fixes the light-cone momentum component k⁺ of the quark as k⁺=p⁺x. Using here k·z =k⁺z⁻+k⁻z⁺−~k_T ·~z_T, we can integrate over k⁻ and ~k_T:

Z dk⁻d²k_T

(2π)³ e^i(k⁻^z⁺⁻^~^k^T^·~^z^T⁾ =δ(z⁺)δ⁽²⁾(~z_T) (1.15) and Eqs.(1.13) and (1.14) become

q_i(x) = E_p

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|T ψ_i(0)γ⁺ψ_i(z⁻)

|p, Si (1.16) M S⁺

p⁺

∆q_i(x) = E_p

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|T ψ_i(0)γ⁺γ₅ψ_i(z⁻)

|p, Si (1.17) where ψ_i(z⁻) ≡ ψ_i(z⁺ = 0, z⁻, ~z_T = ~0_T). ³ The quark field operators in (1.16) and (1.17) are separated by a distance on the light cone, because z⁺ = 0 means that z⁰ +z³ = 0. Therefore

3We define that the limitz⁺ = 0 is taken first, and then ~z_T = 0. In terms of light-cone momenta, this means that in Eq.(1.15), we first integrate overk⁻, keeping the transverse momenta~k_T fixed, and then integrate over the transverse momenta of the quarks. (It is also possible to keep~k_T finite, and the functionq_i(x, ~k_T) is called a “transverse momentum dependent quark distribution function”. We will not consider this case in the lectures.)

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(1.16) and (1.17) can be called light-cone correlation functions. The time-ordering symbol T in those relations is actually unnecessary. To see this, we note that, forz⁺ = 0 but finite ~z_T, the usual time variable is given byz⁰ =z⁻/√

2. Therefore (denoting the Dirac indices by α, β = 1, . . .4) T

ψ^(α)(0)ψ^(β)(z⁻, ~z_T)

= θ(−z⁻) ψ^(α)(0)ψ^(β)(z⁻, ~z_T)−θ(z⁻)ψ^(β)(z⁻, ~z_T)ψ^(α)(0)

= ψ^(α)(0)ψ^(β)(z⁻, ~z_T)−θ(z⁻){ψ^(α)(0), ψ^(β)(z⁻, ~z_T)}₊

= ψ^(α)(0)ψ^(β)(z⁻, ~z_T)

where {. . .}₊ denotes the anticommutator, and we used that the anticommutator between fermion field operators separated by a space-like distance z² = 2z⁺z⁻−~z²_T =−~z²_T <0 vanishes because of causality. Then (1.16) and (1.17) become finally

q_i(x) = E_p

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|ψ_i(0)γ⁺ψ_i(z⁻)|p, Si (1.18)

MS⁺ p⁺

∆q_i(x) = E_p

Z dz⁻

p S⁺. Therefore the sum rules (1.7) and (1.8) are valid.

In order to get a more explicit expression for ∆q_i(x), we consider the case where the nucleon has definite helicity ⁴ λ_N =±1. This means that the spin 3-vector of the nucleon is given byS~₀ =λ_N~p.ˆ We insert this relation into the expression for the spin 4-vector, which is given in usual components µ= 0,1,2,3 by (see No. 4, Eq.(2.16))

S^µ = p~·S~₀

M , ~S₀+ ~p(~p·S~₀) M(E_p+M)

!

If we insert here S~₀ =λ_N~p, this becomesˆ S^µ=λ_N

p

M, p~ˆE_p M

If we choose the direction ˆ~p along the positive z-direction, we obtain for the usual components µ= 0,1,2,3: S^µ = λ_N

p

M,0,0,^E_M^p

. For the light-cone component this gives S⁺ =λ_Np⁺, and the

4We note that the function ∆q_i(x) itself is independent of the choice for the nucleon spin or momentum components.

The matrix element on the r.h.s. of Eq.(1.19), however, depends on the choice of the nucleon spin and momentum components, and this dependence is expressed by the factor

M^S_p+⁺

on the l.h.s. of (1.19).

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factor on the l.h.s. of Eq.(1.19) becomes the helicity of the nucleon:

M^S_p+⁺

=λ_N. Therefore, for a nucleon with helicity λ_N =±1 moving in the positive z-direction, we obtain from (1.19):

∆qi(x) = λNEp

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|ψ_i(0)γ⁺γ5ψi(z⁻)|p, Si (1.20)

2 Field theory on the light cone (an introduction)

So, why are qi(x) and ∆qi(x) of Eqs.(1.18), (1.20) probability distributions? To understand this, we remember that, in usual Minkowski coordinates, the field quantization is done by imposing conditions one the quark fields like

{ψ^(α)(~z, t), ψ^†(β)(~z⁰, t)}₊ =δ⁽³⁾(~z−~z⁰)δ^αβ, or{ψ^(α)(~z, t), ψ^(β)(~z⁰, t)}₊ = 0 etc, at equal time (t). With the light-cone variables defined by

z^±= 1

√2 z⁰±z³

, ~z_T = z¹, z²

we can impose similar quantization conditions for equal z⁺ orz⁻, so we must decide whether we use z⁺ orz⁻ as the “light-cone time”. Usually one usesz⁺, and performs the field quantization at equal values ofz⁺. The Lagrangian of the Dirac theory can easily be expressed by light-cone variables, for example for the free part:

L=ψ(iγ^µ∂_µ−m)ψ =ψ

iγ⁺∂₊+iγ⁻∂−+i~γ_T ·∇~_T −m

ψ (2.1)

where ∂_µ = _∂z^∂µ. But then one notes that the “time” derivative ∂₊ = _∂z^∂+ acts only on a part of the field ψ, i.e., not all components of ψ are dynamical variables. To see this, one introduces the following decomposition of the field ψ:

ψ = ψ₊+ψ− , where ψ± ≡ P±ψ = 1

√2γ⁰γ^±ψ

By using (γ⁺)² = (γ⁻)² = 0, we see that these operators P_± are projection operators: P₊+P₋ = 1, (P₊)² = P₊, (P−)² = P−, P₊P− = P−P₊ = 0. Then we see from (2.1) that the “time derivative”

∂₊ = _∂z^∂+ does not act on ψ−, because γ⁺ψ− = γ⁺P−ψ = ^√¹

2γ⁺γ⁰γ⁻ψ = ^√¹

2 (γ⁺)² γ⁰ψ = 0. The conclusion is that, in the light-cone field theory, only the components

φ ≡P₊ψ = 1

√2γ⁰γ⁺ψ (2.2)

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are dynamical variables, and the remaining part (P−ψ) should be eliminated by using the field equations, before imposing the quantization conditions.

Home work: (i) Show that the Lagrangian (2.1) can be expressed as follows:

L =√ 2

ψ₊^† i∂₊ψ₊+ψ^†₋i∂−ψ−

−ψ₊^†

i~α_T ·∇~_T +γ⁰m

ψ−−ψ^†₋

i~α_T ·∇~_T +γ⁰m

ψ₊ (2.3) (ii) Use the field equation ∂L/∂ψ^†₋ to eliminate ψ− as follows:

ψ− = 1

√2i∂−

i~α_T ·∇~_T +γ⁰m

ψ₊ (2.4)

(iii) Insert (2.4) into (2.3) to express the Lagrangian only by the dynamical field ψ₊. Returning now to (1.18) and (1.20), we note that

ψ(0)γ⁺ψ(z⁻) = ψ^†(0)γ⁰γ⁺ψ(z⁻) =√

2φ^†(0)φ(z⁻) ψ(0)γ⁺γ₅ψ(z⁻) = ψ^†(0)γ⁰γ⁺γ₅ψ(z⁻) = √

2φ^†(0)γ₅φ(z⁻) (2.5) where we used (2.2) and also P₊^† =P₊, (P₊)² =P₊, and [P₊, γ₅] = 0. We see that q_i(x) and ∆q_i(x) can be expressed only by the dynamical field φ = P₊ψ. In light cone field theory, we must also change the normalization of the hadron state vector |p, Si according to⁵

|p, Si → s

p⁺

E_p |p, Si (2.6)

with hp, S|p, Si=V as before. Then (1.18) and (1.20) become q_i(x) = p⁺√

2

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|φ^†_i(0)φ_i(z⁻)|p, Si (2.7)

∆q_i(x) = p⁺√ 2

Z dz⁻

2π e^(p⁺^x)z⁻hp, S|φ^†_i(0)γ₅φ_i(z⁻)|p, Si (2.8)

5The reason is as follows: Our state|p, Siin “non-covariant normalization” was defined such that|p, Sihp, S|is the residue at the pole (p₀=E_p) of the Feynman propagator. That is, the pole term of the Feynman propagator is given by |p, Sihp, S|

p0−Ep

. This is expressed in terms of light-cone variables as follows:

|p, Sihp, S|

p0−Ep

p₀'Ep

≡ 2Ep

|p, Sihp, S|

p²−M²

= 2E_p |p, Sihp, S|

2p⁺p⁻−~p²_T −M² =E_p p⁺

|p, Sihp, S|

p⁻−p

where_p=^~^p²^T_2p^+M−². Because the variablep⁻plays the role of “energy” (it is conjugate to the “time” variablex⁺), the correctly normalized states in the light-cone theory should be defined by the residues of the Feynman propagator at the polep⁻=_p. This leads to Eq.(2.6).

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Now we introduce the Fourier expansions of the field operators φ(z⁻, ~z_T, z⁺) and φ^†(z⁻, ~z_T, z⁺) for fixed “time” z⁺, in the same way as in the usual second quantization method ⁶:

φ_i(z⁻, ~z_T, z⁺) =

Z dk⁺d²kT

(2π)^3/2

rm k⁺

X

s=±1

b_s(k⁺, ~k_T)u₊(k⁺, ~k_T;s)e^−ik⁺^z⁻e^i~^k^T^·~^z^T + (antiquark term) (2.9) φ^†_i(z⁻, ~z_T, z⁺) =

Z dk⁺d²k_T (2π)^3/2

rm k⁺

X

s=±1

b^†_s(k⁺, ~k_T)u^†₊(k⁺, ~k_T;s)e^ik⁺^z⁻e^−i~^k^T^·~^z^T + (antiquark term) (2.10) where b^†_s(k⁺, ~k_T) creates a quark with light-cone momentum components (k⁺, ~k_T) and (twice the ) spin projection s =±1 along the spin direction of the nucleon, b_s(k⁺, ~k_T) annihilates a quark, and the spinor u₊ ≡P₊u, where u is our usual Dirac spinor (normalized by uu = 1), but expressed by light-cone momentum components. (The explicit form ofu+(k⁺, ~kT;s) is given in the Appendix.) We now introduce the field expansions (2.9) and (2.10) into Eqs.(2.7) and (2.8). In the calculation, we use the following relation which follows from momentum conservation:

hp, S|b^†_s(k⁺⁰, ~k_T⁰ )b_s(k⁺, ~k_T)|p, Si=δ(k⁺⁰ −k⁺)δ⁽²⁾(~k_T⁰ −~k_T)(2π)³

V hp, S|b^†_s(k⁺, ~k_T)b_s(k⁺, ~k_T)|p, Si and the following spinor matrix elements, which are derived in the Appendix:

u^†₊(k⁺, ~k_T;s)u₊(k⁺, ~k_T;s) = k⁺

√2m (2.11)

u^†₊(k⁺, ~k_T;s)γ₅u₊(k⁺, ~k_T;s) = ± k⁺

√2m (for s=±1) (2.12)

The results are as follows:

q_i(x) = p⁺ V

Z

d²k_T hp, S|b^†_↑(p⁺x, ~k_T)b↑(p⁺x, ~k_T) +b^†_↓(p⁺x, ~k_T)b↓(p⁺x, ~k_T)|p, Si

(2.13)

∆q_i(x) = p⁺ V

Z

d²k_T hp, S|b^†_↑(p⁺x, ~k_T)b↑(p⁺x, ~k_T)−b^†_↓(p⁺x, ~k_T)b↓(p⁺x, ~k_T)|p, Si

(2.14) Homework: Derive Eqs.(2.13) and (2.14).

6To simplify the notation, we do not indicate the flavor indexiin the following equations, i.e., all operatorsb_sand b^†_srefer to the fixed flavoriindicated on the l.h.s. of the equations.

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Check the sum rules (1.7) and (1.8): Using k⁺=p⁺x we get Z 1

0

dx q_i(x) = 1 V

Z p⁺

0

dk⁺ Z

d²k_Thp, S|b^†_↑(k⁺, ~k_T)b↑(k⁺, ~k_T) +b^†_↓(k⁺, ~k_T)b↓(k⁺, ~k_T)|p, Si

= N_i^↑+N_i^↓ ≡N_i (2.15)

Z 1

0

dx∆q_i(x) = 1 V

Z p⁺

0

dk⁺ Z

d²k_Thp, S|b^†_↑(k⁺, ~k_T)b↑(k⁺, ~k_T)−b^†_↓(k⁺, ~k_T)b↓(k⁺, ~k_T)|p, Si

= N_i^↑−N_i^↓ ≡∆N_i (2.16)

(Note the normalization of the state vectorhp, S|p, Si=V.)

The results (2.13) and (2.14) clearly show our previous conjectures:

• q_i(x) is the sum of probability densities to find a quark (flavor i) with spin parallel (↑) and anti-parallel (↓) to the nucleon spin, with light-cone momentum fraction x = k⁺/p⁺. The integral (2.15) gives the number of quarks with flavor i in the nucleon, and is therefore called the “number sum rule”. In the naive quark model, we expect N_u = 2, N_d= 1 (N_u+N_d = 3).

• ∆q_i(x) is the difference of probability densities to find a quark (flavor i) with spin parallel (↑) and anti-parallel (↓) to the nucleon spin, with light-cone momentum fraction x=k⁺/p⁺. The integral (2.16) gives the contribution of quarks to the spin of the nucleon, and is therefore called the “spin sum rule”. In the naive quark model, we expect ∆Nu+ ∆Nd = 1, i.e., 100% of the nucleon spin comes from the spin of the quarks⁷.

7In the SU(6) quark model, ∆N_u= ⁴₃ and ∆N_d=−¹₃, which means that the spins of the up quarks are parallel to the nucleon spin, and the spin of the down quark is antiparallel to the nucleon spin.

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3 The spin crisis of the proton (an introduction)

Remember: The spin-dependent structure function of the proton g₁^(p) in the Bjorken limit of the parton model takes the form

g₁^(p)(x) = 1 2

4

9∆u^(p)(x) + 1

9∆d^(p)(x)

(3.17) For the neutron, we have

g⁽ⁿ⁾₁ (x) = 1 2

4

9∆u⁽ⁿ⁾(x) + 1

9∆d⁽ⁿ⁾(x)

= 1

2 4

9∆d^(p)(x) + 1

9∆u^(p)(x)

(3.18) Here we used the flavor symmetry: ∆u^(p)(x) = ∆d⁽ⁿ⁾(x), and ∆d^(p)(x) = ∆u⁽ⁿ⁾(x).

The parton model predicts that g₂^(p)(x) =g⁽ⁿ⁾₂ (x) = 0.

Experimental data:

From these data one can get ∆u^(p)(x) and ∆d^(p)(x). These distributions can be separated into

“valence” (v) quark and “sea” (s) quark contributions: ∆u^(p)(x) = ∆u^(p)v (x) + ∆u^(p)s (x). Because the sea quarks always come in qq pairs, we have ∆u^(p)s (x) = ∆u^(p)(x), etc. In inclusive electron-proton scattering, only the sum ∆u^(p)(x) = ∆u^(p)v (x) + ∆u^(p)s (x) can be measured. But there are other experiments ⁸, from which one can get the antiquark distributions, like ∆u^(p)(x) = ∆u^(p)s (x), etc.

8These are proton-proton collisions, where a lepton pair (`⁺`⁻, called “Drell-Yan pair”) is observed in the final state. In the parton model, the elementary process is: (quark)_{hadron 1} + (antiquark)_{hadron 2}−→`⁺`⁻. A parton model analysis of this process give information on the product of quark and antiquark distribution functions.

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Then the valence quark distributions can be obtained from ∆u^(p)v (x) = ∆u^(p)(x)−∆u^(p)s (x).

The result of such an analysis of parton distributions in the proton looks as follows:

These results are very surprising, because if we integrate ∆u^(p)(x) and ∆d^(p)(x) over x, we find that the contribution of the spin of the quarks to the spin of the proton is very small. Including the small contribution from the strange (s) quark (not shown in the figure), one gets

∆u^(p)+ ∆d^(p)+ ∆s^(p)= 0.29±0.06 (3.19) where we defined

∆q^(p) = Z 1

0

dx ∆q^(p)(x) + ∆q^(p)(x)

= Z 1

0

dx ∆q^(p)_v (x) + 2∆q^(p)_s (x)

(q=u, d, s) (3.20) Therefore, only about 30 % of the proton spin is carried by the spin of the quarks! This is called the

“spin crisis” (or “spin puzzle”): From where comes the rest of about 70 % ?

We note that, to obtain the surprising result (3.19), there are only 3 necessary informations:

1. The integral of the measured structure function (3.17):

Z 1

0

dx g₁^(p)(x) = 1 2

4

9∆u^(p)+1

9∆d^(p)+1 9∆s^(p)

= 0.153 (3.21)

2. The axial vector coupling constant (g_A=G_A/G_V = 1.26) for theβ-decay n→p+e⁻+ν_e (on the quark level: d→u+e⁻+ν_e) is defined by the following matrix element of the axial vector current:

hp, S|X

i

ψ_i(0)γ^µγ₅τ₃ψ_i(0)

|p, Si = hp, S| ψ_u(0)γ^µγ₅ψ_u(0)−ψ_d(0)γ^µγ₅ψ_d(0)

|p, Si

≡g_A M

E_pu(~p, S)γ^µγ₅τ₃u(~p, S)

(3.22)

(11)

(Here τ₃ is a Pauli isospin matrix.) Comparison with Eq.(1.6) gives

∆u^(p)−∆d^(p) =g_A= 1.26 (3.23)

This is called the “Bjorken sum rule”.

3. In a similar way, the axial vector coupling constant for the β-decay Σ⁻ →n+e⁻+ν_e (on the quark level: s→u+e⁻+νe) gives the following sum rule:

∆u^(p)+ ∆d^(p)−2∆s^(p)= 0.59 (3.24)

Then from the three independent relations (3.21), (3.23), (3.24), one obtains

∆u= 0.825, , ∆d=−0.435, , ∆s=−0.1 which gives the surprisingly small value (3.19).

At present, it is not yet clear whether the remaining 70% of the proton spin come from

• orbital angular momentum of quarks

• spin of gluons

• orbital angular momentum of gluons.

The planned electron-ion collider (at Brookhaven Lab or Jefferson Lab in the US) will probably solve this spin puzzle.

(12)

4 Appendix

The form of the positive energy Dirac spinor with normalizationuu was given in No. 3 as follows:

u(~k, s) =

rE_k+m 2m

ϕs

~σ·~k Ek+mϕ_s

!

(4.25) whereϕ_s is a 2-component Pauli spinor. In order to get the spinor u₊=P₊u, we need to multiply

P₊ = 1

√2γ⁰γ⁺ = 1

2γ⁰ γ⁰+γ³

= 1

2 1 +α³

= 1 2

1 σ₃ σ₃ 1

(4.26) This gives

u₊(~k, s) = P₊u(~k, s) = 1 2

rE_k+m 2m

1 + ^σ_E³^(~^σ·^~^k)

k+m

σ₃+_E^(~^σ·^~^k)

k+m

!

ϕ_s (4.27)

We can simplify this by writing it in the following form:

u₊(~k, s) =

rE_k+k³ 4m

U σ₃U

ϕ_s (4.28)

where we introduced the following unitary 2×2 matrix:

U = s

E_k+m

2(E_k+k³) 1 + σ₃(~σ·~k) E_k+m

!

(4.29) Homework: Confirm the relations (4.27), (4.28), and U^†U = 1 (unitarity).

We can now define the following “rotated” 2-component Pauli spinor ⁹ :

χ_s ≡U ϕ_s (4.30)

to express Eq.(4.28) in the simple form u₊(~k, s) =

rE_k+k³ 4m

1 σ₃

χ_s (4.31)

Homework: Noting that E_k + k³ = k⁺√

2, use the spinor (4.31) and the form of the matrix γ₅ =

0 1 1 0

to confirm the relations (2.11) and (2.12).

9The rotation (4.30) is called a “Melosh rotation”.