Introduction to QCD

Wolfram Weise

(Notes taken by Ying Cui and Youngshin Kwon )

Winter Semester 2007∼2008

Further Readings

• Introductory:

– Quarks and Leptons: An Introductory Course in Modern Particle Physics

by F.Halzen and A.D.Martin

• Advanced:

– An Introduction to Quantum Field Theory

by M.E. Peskin and D. V. Schroeder

– Foundations of Quantum Chromodynamics

by T.Muta

– The Theory of Quark and Gluon Interactions

by F. J. Y ndurian

– Dynamics of the Standard Model

by J. F.Donoghue, E.Golowich and B.R.Holstein

– The Structure of the Nucleon

by A.W. Thomas and W.Weise

Preparations and Conventions

• We use units such that “ c = ~ = 1 ”

• Space-time four vector x:

contravariant form: xµ = (t, ~x)T

covariant form: xµ = (t, −~x)T = gµνxν

with µ = 0, 1, 2, 3 and metric tensor defined as

gµν = gµν =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

• Inner product:

a · b = aµbµ = a0b

0 − ~a ·~b

where, and also from now on, we use the Einstein summation convention.

• Four-gradient:

∂

∂xµ≡ ∂µ =

(

∂

∂t, ~∇)T

∂

∂xµ

≡ ∂µ =

(

∂

∂t, −~∇

)T

• d’Alembert operator:

≡ ∂µ∂µ =

∂2

∂t2− ~∇2

• Four-momenta:

pµ = (p0, ~p )T = (E, ~p )T

p2 ≡ pµpµ = E2 − ~p 2

• Dirac- and Pauli-matrices:

γµ = (γ0, ~γ )T ; γ0 =

1 0

0 −1 , ~γ =

0 ~σ

−~σ 0

2

1 =

1 0

0 1

, σx =

0 1

1 0

, σy =

0 −ii 0

, σz =

1 0

0 −1

γ5 = γ5 ≡ iγ0γ1γ2γ3 =

0 11 0

• Useful properties of Dirac- and Pauli-matrices:

σµν ≡ i

2

[γµ, γν

]=i

2

(γµγν − γνγµ

)

γµ, γν

= γµγν + γνγµ = 2gµν

σi · σj = i ǫijk σk

[σi, σj

]= 2i ǫijk σk

σi, σj

= 2δij

where ǫijk is the totally antisymmetric tensor with ǫ123 = +1.

3

CHAPTER 1. PRELUDE

1. 1. Quarks in Hadrons and Concept of “Color”

• e+e− annihilation into hadrons

_e

Hadrons

Quark

+e

γ

Antiquark

FIG. 1.1: Feynman diagram of e+e− → hadrons: The reaction is considered to be proceed through

pair production of quark-antiquark as indicated inside the dashed box.

Comparing the total cross section with that of the elementary process e+e− → µ+µ−

which is the analogue in pure QED:

R =σ(e+e− → Hadrons

)

σ(e+e− → µ+µ−

) =∑

q

σ(e+e− → qq

)

σ(e+e− → µ+µ−

) = Nc

∑

q

Z2q (1.1)

with

σ(e+e− → µ+µ−) =

4πα2e

3s

σ(e+e− → qq

)=

4πα2e

3sNcZ

2q

(1.2)

where s is the center of mass energy squared and αe = e2

4π≃ 1

137is the fine struc-

ture constant. The hypothetical introduction of the “color” freedom (Nc = 3) gives

consistent explanation of experimental results as shown in Fig. 1.2.

q u d c s t b

Zq23 −1

323 −1

323 −1

3

TABLE 1.1: The electric charge Zq of each quark flavor in unit e

4

FIG. 1.2: The prediction of the quark model on the total cross section (left) and the ratio R (right)

with Nc = 3. The typical vector meson resonances are represented.

• Spectroscopy and quark models

According to the quark theory ∆++, a particle of spin 3/2, should consist of three u

quarks with parallel spins if in a state of maximal spin projection:

|∆++, mJ = 3/2〉 = | u ↑ u ↑ u ↑〉, (1.3)

while the Pauli exclusion principle forbids three identical fermions in the same ground

state. Therefore it was suggested that each quark has an additional degree of freedom

(three “colors”), thus avoiding violation of the Pauli exclusion principle.

|∆++, mJ = 3/2〉 =1√6

∑

ijk

ǫijk | ui ↑ uj ↑ uk ↑〉 (1.4)

1. 2. Quarks as Dirac-Fields

• Quarks are spin 1/2 particles.

• They exist in 6 species called flavors.

• Each quark carries a 3-fold intrinsic degree of freedom (color).

5

• Representation of quarks in terms of fields

ψ(x) ≡(ψαi(x)

)=

ψui(x)

ψdi(x)

ψsi(x)...

ψαi(x)

(1.5)

where α = u, d, s, c, b, t and i = 1, 2, 3 are flavor and color indices respectively.

• Each of the ψαi(x) satisfies a Dirac equation in case of free quarks

[iγµ∂

µ −m]ψ(x) = 0 (1.6)

with the mass matrix

m =

mu 0 0 0 0 0

0 md 0 0 0 0

0 0 ms 0 0 0

0 0 0 mc 0 0

0 0 0 0 mb 0

0 0 0 0 0 mt

(1.7)

• Explicit representation (spin projection s = ±12)

ψ(x) =∑

s

∫d3p

(2π)3

1

2Ep

[

b(p, s) us(p) e−ip·x + d†(p, s) vs(p) e

ip·x]

(1.8)

where Ep =√~p 2 +m2

q.

• Definition: State vector of a given quark with spin s = ±12

and 4-momentum pµ;

quark : |p, s〉αi = b†αi(p, s)|0〉

antiquark : |p, s〉αi = d†αi(p, s)|0〉(1.9)

6

where the vacuum |0〉 is defined as: b|0〉 = d|0〉 = 0 with:

b† : creation operator for a quark

b : annihilation operator for a quark

d† : creation operator for an antiquark

d : annihilation operator for an antiquark.

• Anticommutation rules for creation and annihilation operators

bαi(p, s), b†βj(p

′, s′)

=

dαi(p, s), d†βj(p

′, s′)

= 2Ep(2π)3δ3(~p− ~p ′)δαβδijδss′ (1.10)

otherwise vanish, e.g.

b†, b†

=b, b

=d†, d†

=d, d

= 0 (1.11)

• Normalization of state vector

〈 p′ s′ | p s 〉 = 2Ep(2π)3δ3(~p− ~p ′)δss′ (1.12)

• Digression: Lorentz invariant phase space∫

d4p

(2π)42π δ(p2 −m2) =

∫dE

2π

d3p

(2π)32π δ(E2 − ~p 2 −m2)

=

∫dE

2π

∫d3p

(2π)32π

δ(E −√

~p 2 +m2)

2E

=

∫d3p

(2π)32Ep

(1.13)

where Ep =√

~p 2 +m2

• Dirac equations for particle and antiparticle

(γµp

µ −m)us(p) = 0

(γµp

µ +m)vs(p) = 0

(1.14)

Free Dirac spinors

us(p) =√

Ep +m

χs

~σ · ~pEp+m

χs

vs(p) = η (−1)12−s

︸ ︷︷ ︸

phase free

√

Ep +m

~σ · ~pEp+m

χ−s

χ−s

,

(1.15)

7

where χs= 12

=

1

0

and χs=− 12

=

0

1

respectively.

• Normalization of spinors

u†s(p) us′(p) = 2Ep δss′

v†s(p) vs′(p) = 2Ep δss′.(1.16)

1. 3. Quark Currents

• Dirac current density of quarks:

Jµ(x) = ψ(x) γµ ψ(x)

J0(x) = ψ(x) γ0 ψ(x) = ψ†(x)ψ(x) ≡ ρ(x)

~J(x) = ψ(x)~γ ψ(x) = ψ†(x) γ0~γ ψ(x) = ψ†(x) ~α ψ(x)

(1.17)

where ψ = ψ†γ0 and ~α =

0 ~σ

~σ 0

• Continuity equation:

∂µJµ(x) =

∂ρ

∂t+ ~∇ · ~J = 0 (1.18)

• Electromagnetic quark current:

Jµe.m.(x) = ψ(x)Qγµψ(x) (1.19)

with quark charges, Q =

±23e for u, c, t or u, c, t

∓13e for d, s, b or d, s, b

.

1. 4. Lagrangian Density (Lagrangian) of free quarks

L0(x) = ψ(x)[iγµ∂

µ −m]ψ(x) (1.20)

• Generalized variables: fields ψ, ∂µψ, ψ and ∂µψ.

8

• Action for free quark:

S0 =

∫

d4xL0(x) = S0

[ψ, ∂µψ ; · · ·

](1.21)

⊲ Stationary action principle: δ S0 = 0.

• Euler-Lagrange equations:

∂L0

∂ψ− ∂µ ∂L0

∂(∂µψ)= 0

∂L0

∂ψ− ∂µ ∂L0

∂(∂µψ

) = 0.(1.22)

• Dirac equations from Euler-Lagrange equations:

∂L0

∂ψ= 0 ⇒

[iγµ∂

µ −m]ψ(x) = 0

∂L0

∂ψ= ∂µ ∂L0

∂(∂µψ

) ⇒ ψ(x)[iγµ∂

µ −m]

= 0

1. 5. Hamiltonian Density (Hamiltonian)

• Canonical conjugate field:

π =∂L∂ψ

(1.23)

L0 = ψ[

iγ0∂

∂t+ i~γ · ~∇−m

]

ψ ⇒ π = iψγ0 = iψ† (1.24)

• Canonical form of Hamiltonian as Legendre transform from Lagrangian:

H(x) = πψ − L(x)

= ψ†[− i~α · ~∇+ βm]ψ

= ψ†i∂

∂tψ,

(1.25)

where ~α ≡ γ0~γ and β ≡ γ0.

• Dirac equation in Hamiltonian form

[− i~α · ~∇+ βm

]ψ(x) = i

∂

∂tψ(x) (1.26)

9

CHAPTER 2. THE QCD LAGRANGIAN

2. 1. Preparation: Gauge invariance for QED

• Consider electrons represented by Dirac field ψ(x). Gauge transformation:

ψ(x)→ Uψ(x) with U = e−iθ (2.1)

– Local gauge transformation, if θ = θ(x)

– Global gauge transformation, if θ = const.

Hypothesis : Local gauge transformations, U = e−iθ(x), leave the physics invariant.

• Current is invariant under local gauge transformation.

ψ(x)γµψ(x)G.T.−→ ψ†γ0U

†γµUψ (2.2)

• Not invariant:

ψiγµ∂µψ → ψiγµU

†∂µ(Uψ)

= ψiγµU†U(∂µψ) + ψiγµψ (U †i∂µU)

︸ ︷︷ ︸

∂µθ(x)

(2.3)

• Introduction of gauge field Aµ(x):

Definition of gauge covariant derivative: Dµ = ∂µ − ieAµ(x) (e > 0)

• Requirement: Under local gauge transformation

Dµψ = U(Dµψ)

then L′ = ψ(iγµD

µ −m)ψ gauge invariant.

U(Dµψ

)= ∂µψ − ieAµ(x)ψ = ∂µ

(Uψ(x)

)− ieAµ(x)Uψ(x)

=(∂µU

)ψ + U

(∂µψ

)− ieAµ(x)Uψ

= U[∂µ − ieAµ(x)

]ψ(x)

(2.4)

⇒ − ieAµUψ = −ieUAµψ −(∂µU

)ψ

⇒ AµU = UAµ − i

e∂µU

10

Aµ = UAµU † − i

e

(∂µU

)U †

= U[

Aµ − i

eU †∂µU

]

U †(2.5)

• Gauge field ↔ Potentials: Aµ(x) =(φ(x), ~A(x)

)T.

• Electromagnetic fields:

~E = −~∇φ− ∂ ~A

∂t

~B = ~∇× ~A

• Electromagnetic field tensor:

F µν = ∂µAν(x)− ∂νAµ(x) =

0 −Ex −Ey −Ez

Ex 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(2.6)

• Lagrangian density of electromagnetic fields

Lγ = −1

4Fµν(x)F

µν(x) = −1

2

(~E 2 − ~B 2

)(2.7)

• Equations of motions for free photon: Aµ(x) = 0

Aµ(x) =∑

λ

∫d3k

(2π)3 2ωk

[

a(k, λ) ǫµ(λ) e−ik·x + a†(k, λ) ǫµ

∗

(λ) eik·x]

(2.8)

where ωk = |~k| and ǫµ(λ) represents the polarization vector.

• State vector of photon:

|k, λ 〉 = a†(k, λ)|0〉

a(k, λ)|k, λ 〉 = |0〉(2.9)

• Lagrangian density of QED:

LQED = ψ(x)[

iγµDµ −m

]

ψ(x)− 1

4Fµν(x)F

µν(x) (2.10)

where Dµ = ∂µ − ieAµ(x)

• Gauge transformations form a group: U = e−iθ(x) (QED), U ∈ Group U(1).

11

2. 2. Local SU(3) Gauge transformations

• Starting point: Quark fields ψ =(ψαi

)

α = u, d, s, c, b, t (flavor index) Nf = 6 ←→ SU(Nf )

i = 1, 2, 3 (color index) Nc = 3 ←→ SU(3)c

where ψαi is a 4-component Dirac-spinor.

Consider Quark fields with color degree of freedom and their free Lagrangian:

ψ =

ψ1

ψ2

ψ3

, L0 = ψ

[iγµ∂

µ −m]ψ (2.11)

• Local SU(3)c gauge transformations

ψ(x) −→ ψ(x) = U ψ(x) (2.12)

with U = exp

[

− i θa(x)λa

2

]

where θa(x) is a real function with a = 1, 2, · · · , 8.

Hypothesis : Physics of strong interaction of quarks is invariant under gauge

transformation: ψ(x)→ U(x)ψ(x).

SU(3)c is a non-abelian gauge group.

• Gauge covariant derivative:

Dµ = ∂µ − i g Aµ(x) (2.13)

where g is a dimensionless coupling strength analogous to e in QED.

Aµ(x) =

8∑

a=1

taAaµ(x) (2.14)

Introducing Aaµ(x), SU(3)c gauge fields “gluons”,

L1 = ψ(x)[iγµD

µ −m]ψ(x) (2.15)

Lagrangian L1 becomes gauge invariant.

Dµψ ≡ ∂µψ − i g Aµψ = U(DµU

)

Aµ = U[Aµ − i

gU † ∂µU

]U †

(2.16)

12

• Infinitesimal gauge transformation

U = exp[− i θa(x) ta

]≃ 1− i θa(x) ta + · · · (2.17)

transformation of gauge field up to terms linear in θa(x)

Aµa(x) → Aµ

a(x) = Aµa(x)− 1

g∂µθa(x) + fabcθb(x)A

µc (x) (2.18)

• Gluons are massless (a mass term mgAµaA

aµ would not be gauge invariant).

• Gluonic field tensors:

If one would take the form analogous to QED,

F aµν(x) = ∂µA

aν(x)− ∂νA

aµ(x), (2.19)

not gauge invariant in QCD.

Introduce additional term to obtain gauge invariant Gluonc field tensor.

Gaµν(x) = ∂µA

aν(x)− ∂νA

aµ(x) + g fabc A

bµ(x)Ac

ν(x) (2.20)

Gµν ≡ ta Gaµν =

i

g

[Dµ , Dν

](2.21)

• Gluonic Lagrangian:

Lglue = −1

4Ga

µν(x)Gµνa (x) = −1

2trGµν G

µν

(2.22)

2. 3. QCD Lagrangian

• QCD Lagrangian:

LQCD = ψ(iγµD

µ −m)ψ − 1

2trGµν G

µν

(2.23)

with Dµ = ∂µ − igAµ(x).1

1 Remark : frequently Aµ → gAµ

⇒ LQCD = ψ(iγµ(∂µ − iAµ)−m

)ψ − 1

2g2trGµν G

µν

13

• Gluonic field tensor of LQCD generates non-linear gluon interactions:

– 3-gluon interaction

L(3) = −g2fabc

(∂µAν

a − ∂νAµa

)Ab

µAcν

∼ g (2.24)

– 4-gluon interaction

L(4) = −g2

4fabc fcdeAaµAbνA

µcA

νd ∼ g2 (2.25)

2. 4. Classical QCD equation of motion

• Euler-Lagrange equations derived from LQCD(ψ, ∂µψ, Aµ, · · · )

∂LQCD

∂qi− ∂µ

∂LQCD

∂(∂µqi)= 0 (2.26)

– Equations of motion for quark field:

[iγµ

(∂µ − igAµ(x)

)−m

]ψ = 0 (2.27)

– Equations of motion for gluon field:

∂µGaµ(x) + g fabc A

µb (x)Gc

µν(x) = −g Jaν (x) (2.28)

with color currents of quarks

Jaν (x) = ψ(x) γν ta ψ(x) = ψ γν

λa

2ψ (2.29)

which are conserved: ∂µJaµ(x) = 0.

2. 5. Gauge fixing

• Digression on gauge fixing in electrodynamics:

Lγ = −1

4FµνF

µν (2.30)

14

Corresponding equation of motion:

∂µFµν(x) = ∂µ(∂µAν − ∂νAµ

)

= Aν − ∂ν(∂µAµ)

= 0

(2.31)

Gauge theories have a certain freedom in defining the gauge field, Aµ(x).

In order to remove the problem, eliminate the gauge freedom by setting constraints

for the field Aµ(x).

For example,

∂µAµ(x) = 0 (2.32)

which is called “Lorenz gauge” (covariant constraint).

• Introduce extra term λ(∂µA

µa(x)

)2with Lagrange multiplier parameter λ = − 1

2ξ

Lγ = −1

4Fµν(x)F

µν(x)− 1

2ξ(∂µAµ(x))2

(2.33)

Equation of motion

Aµ −(

1− 1

ξ

)

∂µ(∂λAλ) = 0 (2.34)

• Gauge fixing choices

ξ = 1 ; Feynman gauge

ξ = 0 ; Landau gauge

Other options:

~∇ · ~Aa = 0 ; Coulomb gauge

A3a = 0 ; Axial gauge

A0a = 0 ; Temporal gauge

15

Appendix: SU(N)-Group and Lie algebra

Short mathematical appendix about groups:

• Group: G = g, h, k, · · ·

– For g, h ∈ G, gh ∈ G

– There exists a “unit” element e such that eg = ge = g.

– For each g ∈ G, there exists an inverse g−1 ∈ G ; g−1g = gg−1 = e.

• Linear group:

Elements g, h, · · · (transformations/operators) with the following property:

For each g, h ∈ G exists αg + βh ∈ G with α, β ∈ C

• Representations of a linear group:

Mapping: g ∈ G→ (aij) ∈ space of complex valued matrices with aij ∈ C.

• Adjoint operator:

Let g ∈ G (linear), then there exists a unique g† with the representation (aij)† = (a∗ji).

• Unitary transformations/operators: U ∈ G

U † = U−1 ⇒ U †U = UU † = 1. (2.35)

Consequently a unitary transformation can be written as follows:

U = exp[ iH ] = 1+ iH +i 2

2H2 + · · · (2.36)

with Hermitian operator H , i.e. H† = H .

Example-1. Group U(1) with elements U = exp[iα] where α ∈ R

U † = e−iα , UU † = U †U = 1Group of gauge transformation in QED

16

Example-2. Group SU(N)

Group of unitary transformations represented by unitary N ×N matrices

U = exp

[

i∑

a

αaXa

]

with | detU |2 = 1

where αa are real parameters with a = 1, · · · , N2 − 1. The hermitian operators Xa

are the generators of the SU(N) group.

Generators form Lie-algebra:

[Xa , Xb

]= i fabc Xc

(2.37)

where fabc are the structure constants of the group.

⊲ For N = 2, SU(2) generators Xa = σa/2 (a = 1, 2, 3)

Pauil matrices:

σ1 =

0 1

1 0

, σ2 =

0 −ii 0

, σ3 =

1 0

0 −1

(2.38)

trσa = 0

trσa σb = 2 δab

(2.39)

Structure constants: fabc = ǫabc.

⊲ For N = 3, SU(3) generators Xa = λa/2 (a = 1, · · · , 8)

Gell-Mann matrices:

λ1 =

0 1 0

1 0 0

0 0 0

, λ2 =

0 −i 0

i 0 0

0 0 0

, λ3 =

1 0 0

0 −1 0

0 0 0

,

λ4 =

0 0 1

0 0 0

1 0 0

, λ5 =

0 0 −i0 0 0

i 0 0

, λ6 =

0 0 0

0 0 1

0 1 0

,

λ7 =

0 0 0

0 0 −i0 i 0

, λ8 = 1√

3

1 0 0

0 1 0

0 0 −2

(2.40)

17

trλa = 0

trλa λb = 2 δab

(2.41)

Lie-algebra:

[λa , λb

]= 2 i fabc λc

(2.42)

Structure constants:

fabc = −i tr([

λa

2,λb

2

]

λc

)

(2.43)

fabc is totally antisymmetric with nonvanishing members,

f123 = 1

f147 = −f156 = f246 = f257 = f345 = −f367 =1

2

f458 = f678 =

√

3

2

(2.44)

• Irreducible representations of SU(2):

Xa ≡ Ja =σa

2(a = 1, 2, 3)

– Casimir operator of SU(2): J2 = J21 + J2

2 + J23

which commutes with all generators

[J2 , Ja

]= 0 (a = 1, 2, 3). (2.45)

– Ladder (raising and lowering) operators:

J± = J1 ± iJ2

J2 =1

2

(J+J− + J−J+

)+ J2

3

[J+ , J−

]= 2 J3 ,

[J3 , J±

]= ±J±

(2.46)

– Eigenstates of J2 and J3 :

J2 |λ, M〉 = λ |λ, M〉 , J3 |λ, M〉 = M |λ, M〉 (2.47)

J2 − J23 = J2

1 + J22 ≥ 0 =⇒ λ−M2 ≥ 0 (2.48)

18

– Let j be the largest M : J+ |λ, j〉 = 0

J−J+ |λ, j〉 =(

J2 − 1

2

[J+ , J−

]− J2

3

)

|λ, j〉

=(J2 − J3 − J2

3

)|λ, j〉

=(λ− j2 − j

)|λ, j〉

= 0.

(2.49)

Therefore

λ = j(j + 1) ≥ 0. (2.50)

– Relabeling the states |λ, M〉 ≡ | j, M〉, Eq. (2.47) becomes

J2 | j, M〉 = j(j + 1) | j, M〉 , J3 | j, M〉 = M | j, M〉. (2.51)

– Let j′ be the smallest M : J− | j, j′〉 = 0

J+J− | j, j′〉 =(J2 + J3 − J2

3

)| j, j′〉

=(j2 + j + j′ − j′ 2

)| j, j′〉

= 0.

(2.52)

Hence

j(j + 1) = j′(j′ − 1) =⇒ j′ = −j. (2.53)

– Basis states:| j, M〉 with M = j, j − 1, · · · , −j, dimension: dj = 2j + 1

.

• Product of representations of SU(2):

J = J (1) + J (2) , J3 = J(1)3 + J

(2)3 (2.54)

J (i)2 | j(i), M (i)〉 = j(i)(j(i) + 1) | j(i), M (i)〉

J(i)3 | j(i), M (i)〉 = M (i) | j(i), M (i)〉.

(2.55)

To look for | j, M〉 with J2 | j, M〉 = j(j + 1) | j, M〉 and J3 | j, M〉 = M | j, M〉, in

general, we form appropriate linear combinations of product states:

| j, M〉 =∑

M (1), M (2)

j(1)M (1)j(2)M (2)| jM

| j(1), M (1)〉 | j(2), M (2)〉 (2.56)

where the quantities

j(1)M (1)j(2)M (2)| jM

are called Clebsch-Gordan coefficients.

19

Example. Coupling of two states in “fundamental” representation of SU(2); basis states| j(i) = 1

2, M (i) = ±1

2〉

i) Start with | j = 1, M = 1〉 = |12, 1

2〉 |1

2, 1

2〉

ii) Successively apply J− to get to all other states

|1, 0〉 =1√2

(

|12, −1

2〉 |1

2, 1

2〉+ |1

2, 1

2〉 |1

2, −1

2〉)

|1, −1〉 = |12, −1

2〉 |1

2, −1

2〉

(2.57)

iii) Find the orthogonal combination to | jmax, M = jmax − 1〉:

|0, 0〉 =1√2

(

|12, −1

2〉 |1

2, 1

2〉 − |1

2, 1

2〉 |1

2, −1

2〉)

(2.58)

• Rules for coupling SU(2) representations

j = 0 [ 1 ] Singlet 12 ⊗ 1

2 : [ 2 ] ⊗ [ 2 ] = [ 1 ]⊕ [ 3 ]

j = 12 [ 2 ] Doublet 1

2 ⊗ 1 : [ 2 ] ⊗ [ 3 ] = [ 2 ]⊕ [ 4 ]

j = 1 [ 3 ] Triplet 1⊗ 1 : [ 3 ] ⊗ [ 3 ] = [ 1 ]⊕ [ 3 ] ⊕ [ 5 ]

j = 32 [ 4 ] Quartet

...

...

j [ 2j + 1 ] Multiplet

• Graphical illustration in terms of weight diagrams:

j = 12

−12

12

[ 2 ]

j = 1−1 10

[ 3 ]

j = 32

−32

32−1

212

[ 4 ]

FIG. 2.1: Graphical representation of SU(2) multiplets.

20

• Building product representations in terms of weight diagrams

[ 2 ]⊗ [ 2 ] =− 1

212 ⊗

− 12

12

=

= = [ 1 ]⊕ [ 3 ]

[ 2 ]⊗ [ 3 ] =− 1

212

⊗−1 10

=

= = [ 2 ]⊕ [ 4 ]

• Irreducible representations of SU(3) group: U = exp[iαata]

ta =λa

2(a = 1, · · · , 8) (2.59)

– Lie-algebra[ta , tb

]= i fabc tc (2.60)

where fabc is the structure constants of SU(3).

– Anticommutation relations:

ta , tb

=

1

3δab + dabc tc (2.61)

where dabc is called “symmetric” structure constants of SU(3).

– Casimir operator in SU(3):

C =8∑

a=1

t2a

T 2 =

3∑

i=1

t2i

T3 = t3

Isospin

Y =2√3t8

Hypercharge

(2.62)

– Raising and lowering operators:

T± = t1 ± i t2︸ ︷︷ ︸

Iso−spin

, U± = t6 ± i t7︸ ︷︷ ︸

U−spin

, V± = t4 ± i t5︸ ︷︷ ︸

V −spin

(2.63)

21

– SU(3) commutation relations:

[T3 , T±

]= ±T±

[T3 , U±

]= ∓1

2U±

[T3 , V±

]= ±1

2V±

[Y , T±

]= 0

[Y , U±

]= ±U±

[Y , V±

]= ±V±

(2.64)

[T+ , T−

]= 2T3

[U+ , U−

]=

3

2Y − T3 ≡ 2U3

[V+ , V−

]=

3

2Y + T3 ≡ 2 V3

(2.65)

[T+ , V+

]=[T+ , U−

]=[U+ , V+

]= 0

[T+ , V−

]= −U−

[U+ , V−

]= T− (2.66)

[T+ , U+

]= V+

[T3 , Y

]= 0

• Weight diagrams of irreducible representations of SU(3)

-1 -

12

12 1

t3

-1

-

23

1

yFundamental Triplet @ 3 D

1

3

-1 -

12

12 1

t3

-1

23

1

yFundamental Anti-triplet @ 3 D

-

1

3

-1 -

12

12 1

t3

yOctet @ 8 D

1

-1

• Product representations and Clebsch-Gordan coefficients of SU(3)

22

– Basis states:∣∣ [α ] t , t3 , y

⟩,

where [α ] denote representations e.g., [ 3 ], [ 8 ] etc.

– 1st step :

∣∣∣∣∣∣

T , T3

[α ] t y , [ β ] t′ y′

⟩

=∑

t3t′3

t t3 t′t′3|TT3

∣∣ [α ] t , t3 , y

⟩∣∣ [ β ] t′ , t′3 , y

′ ⟩ (2.67)

– 2nd step:

∣∣ [ γ ] T , T3 , Y

⟩=∑

t y t′y′

[α ] t y[ γ ]T Y

[ β ] t′ y′

︸ ︷︷ ︸

Isoscalar SU(3) factors

∣∣∣∣∣∣

T , T3

[α ] t y , [ β ] t′ y′

⟩

(2.68)

• Product representations and rules in terms of weight diagrams:

Take “center of gravity” of one representation and place it on all parts of the second

representation

Example. [ 3 ]⊗ [ 3 ] = [ 8 ]⊕ [ 1 ]

-1 -

12

12 1

-1

-

23

1Triplet @ 3 D

1

3

⊗-1 -

12

12 1

-1

23

1Anti-Triplet @ 3 D

-

1

3

=-1 -

12

12 1

-

23

1

1

3

-1

=-1 -

12

12 1

Octet @ 8 D

1

-1

⊕Singlet @ 1 D

• Eigenvalues of Casimir operators

C =8∑

a=1

t2a =1

4

8∑

a=1

λ2a = ~t 2 =

1

4~λ 2 (2.69)

23

Representations Eigenvalues of C

Singlet [ 1 ] 0

Triplet [ 3 ] 43

Anti-triplet [ 3 ] 43

Sextet [ 6 ] 103

Octet [ 8 ] 3

24

CHAPTER 3. GREEN’S FUNCTIONS AND PROPAGATORS

3. 1. Introduction: Scalar field

• Lagrangian with real scalar field φ(x) and “potential” U(φ).

L =1

2

(∂µφ

)(∂µφ

)− 1

2m2φ2 − U(φ) (3.1)

– Field equations

(¤ + m2

)φ +

∂U

∂φ= 0

⇒ (¤ + m2

)φ(x) = −J (x) with J (x) =

∂U

∂φ

(3.2)

– Free field equation: Klein-Gordon equation

(¤ + m2

)φ(x) = 0 (3.3)

– Solution:

φ(x) =

∫d3k

(2π)3 2ωk

(ak e−ik·x + a†k eik·x

)(3.4)

– Commutation relations for ak and a†k:

[ak, a†k′

]= (2π)3 2ωk δ3

(~k − ~k ′

)[ak, ak′

]=

[a†k, a†k′

]= 0

(3.5)

• Definition of time-ordered product:

T φ(x′)φ(x) = θ(t′ − t) φ(x′)φ(x) + θ(t− t′) φ(x)φ(x′) (3.6)

• Correlation function:

i∆F (x′ − x) = 〈0|T φ(x′)φ(x)|0〉 (3.7)

Feynman propagator: Green’s function ↔ time-ordered correlation function

• Apply ¤x′ + m2 to ∆F :

where

¤x′ ≡ ∂2

∂t′ 2− ~∇ 2

~x ′

25

∂

∂t′T φ(x′)φ(x) =

∂

∂t′[θ(t′ − t)φ(x′)φ(x) + θ(t− t′) φ(x)φ(x′)

]

= T ∂φ(x′)∂t′

φ(x) + δ(t′ − t)φ(x′)φ(x)− δ(t′ − t)φ(x)φ(x′)︸ ︷︷ ︸δ(t′−t) [φ(x′), φ(x)]=0

(3.8)

∂2

∂t′ 2T φ(x′)φ(x) = T ∂2φ(x′)

∂t′ 2φ(x) + δ(t′ − t)

[∂φ(x′)

∂t′, φ(x)

]

︸ ︷︷ ︸−iδ4(x′−x)

= T (~∇ 2

~x ′ −m2)φ(x′)φ(x)− iδ4(x′ − x)

(3.9)

⇒ (¤x′ + m2

)T φ(x′)φ(x) = −iδ4(x′ − x)

⇒ (¤x′ + m2

)∆F (x′ − x) = −δ4(x′ − x)

(3.10)

• Solution of inhomogeneous wave equation:

φ(x) =

∫d4x′ ∆F (x′ − x)J (x′) (3.11)

• Propagator ∆F is the Green’s function of the Klein-Gordan equation.

• Fourier representation:

∆F (x′ − x) =

∫d4k

(2π)4

e−ik·(x′−x)

k2 −m2 + iε(3.12)

(¤x′ + m2)∆F (x′ − x) =

∫d4k

(2π)4

(¤x′ + m2)e−ik·(x′−x)

k2 −m2(+iε)

=

∫d4k

(2π)4

(−k2 + m2)e−ik·(x′−x)

k2 −m2(+iε)

= −∫

d4k

(2π)4e−ik·(x′−x)

= −δ4(x′ − x)

(3.13)

• Poles at k2 = m2:

ωk =

√~k 2 + m2 ⇒ k0 = ±ωk

26

• Cauchy’s integral formula:

∆F (x′ − x) =

∫d3k

(2π)3

∫ ∞

−∞

dk0

2π

e−ik0(t′−t)

k20 − ~k 2 −m2 + iε

e−i~k·(~x ′−~x) (3.14)

Ä

Ä

-Ωk

+Ωk

t > t'

t < t'

Re k0

Im k0

Using

∮f(z)

z − a= 2πi f(a)

ωk =

√~k 2 + m2 − iδ

• t′ > t ; ∆F (x′ − x) = −i

∫d3k

(2π)3

e−iωk(t′−t)

2ωk

ei~k·(~x ′−~x)

• t′ < t ; ∆F (x′ − x) = i

∫d3k

(2π)3

eiωk(t′−t)

−2ωk

ei~k·(~x ′−~x)

• For all t, t′:

∆F (x′ − x) = −i

∫d3k

(2π)3ei~k·(~x ′−~x) 1

2ωk

[θ(t′ − t)e−iωk(t′−t) + θ(t− t′)eiωk(t′−t)

](3.15)

– The first term on the r.h.s. of (3.15) describes a particle running forward time

with positive energy ωk and t′ − t > 0.

– The second term describes a “particle” running backward time with negative ωk

and t− t′ > 0: antiparticle.

x x'space

t

t'

time@ Particle D

x x'space

t'

t

time@ Antiparticle D

3. 2. Dirac propagator

• Time ordered product of Dirac fields: T ψα(x′)ψβ(x)

27

B Dirac propagator:

iSF (x′ − x)αβ = 〈0|T ψα(x′)ψβ(0)|0〉 (3.16)

This is the Green’s function of the free Dirac equation:

(iγµ∂µx′ −m)SF (x′ − x)αβ = δ4(x′ − x)δαβ (3.17)

• Fourier representation

SF (x′ − x) =

∫d4p

(2π)3e−ip·(x′−x) γµp

µ + m

p2 −m2 + iε(3.18)

(iγµ∂µx′ −m)SF (x′ − x) =

∫d4p

(2π)4(iγµ∂

µx′ −m) e−ip·(x′−x) γνp

ν + m

p2 −m2 + iε

=

∫d4p

(2π)4

(γµpµ −m)(γνp

ν + m)

p2 −m2 + iεe−ip·(x′−x)

=

∫d4p

(2π)4

p2 −m2

p2 −m2 + iεe−ip·(x′−x)

= δ4(x′ − x)

(3.19)

• Feynman propagator of spin-12

particle in momentum space:

SF (p) =

∫d4x eip·(x′−x) SF (x′ − x)

=/p + m

p2 −m2 + iε

(/p ≡ γµp

µ) (3.20)

3. 3. Free gluon propagator

• Free gluon Green’s function:

iDµνab (x′ − x) = 〈0|T Aµ

a(x′)Aνb (x)|0〉 (3.21)

Dµνab (x′ − x) = δab

∫d4q

(2π)4

dµν(q)

q2

with dµν(q) = −gµν + (1− ξ)qµqν

q2 + iε

(3.22)

ξ = 1 : Feynman gauge

ξ = 0 : Landau gauge

28

CHAPTER 4. S-MATRIX AND FEYNMAN RULES

4. 1. Definition: S-matrix, T-matrix and cross section

• S-matrix

SBA = 〈B, t→∞|A, t→ −∞〉 (4.1)

|A〉 and |B〉 are asymptotic states:

|A, t〉 = eiHt|A, t = 0〉

• T-matrix

〈B|S|A〉 ≡ SBA = δBA + i(2π)4δ4(pA − pB)TBA

〈B|T |A〉 ≡ TBA = −MBA

(4.2)

• Differential cross section for A→ B

– Prototype: two particles colliding in initial state: A = a1 + a2

dσ(a1 + a2 → B) =W(a1 + a2 → B)

JA

dNB (4.3)

– W(a1 + a2 → B): Transition probability for A→ B per unit time.

– dNB: Phase space element in the final state B.

– JA: Flux of incoming particles in state A.

JA =number of particles

time× unit area

• Assume n particles in final state:

dNB =n∏

i=1

d3pi

(2π)3 2Ei

(

Ei =√

~p 2i +m2

i

)

(4.4)

4. 2. Feynman rules (for the calculation of invariant amplitude TBA)

• Factors to be applied for each external lines

29

A

B

BAT

a ) Incoming quark lines

in A : u(p, s)

in B : v(p, s)

b ) Outgoing quark lines

in A : v(p, s)

in B : u(p, s)

c ) External gluon lines: Polarization vector: ǫµ

• Remember QCD Lagrangian

LQCD = ψ(x)

iγµ

(

∂µ − igλa

2Aµ

a(x))

−m

ψ(x)

− 1

4Gµν

a (x)Gaµν(x)

− 1

2ξ(∂µA

µa(x))2

(4.5)

Gµνa (x) = ∂µAν

a(x)− ∂νAµa(x) + gfabcA

µb (x)Aν

c (x)

m: quark mass matrix

m =

mu 0 0 0 0 0

0 md 0 0 0 0

0 0 ms 0 0 0

0 0 0 mc 0 0

0 0 0 0 mb 0

0 0 0 0 0 mt

(4.6)

• Interaction vertices

∼ g

Quark-gluon vertex

∼ g

3-gluon vertex

∼ g2

4-gluon vertex

• Consider QCD in its perturbative domain (“Perturbative QCD”):

αs =g2

4π≪ 1 ⇒ Perturbative expansion of observables in powers of αs.

30

mu md ms mc mb mt

4 (±2) MeV 7 (±2) MeV 120 (±5) MeV 1.3 (±0.1) GeV 4.3 (±0.1) GeV 174 (±5) GeV

TABLE 4.1: Values of quark masses

• Internal lines

a ) Quark: (a, b: color indices; i, j: flavor indices)

p

i, a

j, b

[iSF (p)

]ij

ab= δab δ

ij i

/p−m+ iǫ

= δab δij i(/p+m)

p2 −m2 + iǫ

b ) Gluon: (a, b: color indices; µ, ν: Lorentz indices)

p

µ, a

ν, b

= δab

[

− gµν +(1− ξ

)pµpν

p2

]i

p2 + iǫ

c ) Quark-gluon vertex:

a, µ

factor: igγµta

d ) 3-gluon vertex:

p1

p2

p3a1, µ1

a2, µ2

a3, µ3

factor: gfa1a2a3

[gµ1µ2(p1 − p2)

µ3

+ gµ2µ3(p2 − p3)µ1

+ gµ3µ1(p3 − p1)µ2]

31

e ) 4-gluon vertex:

a2, µ2

a1, µ1

a3, µ3

a4, µ4

factor: − g2[fa1a2afa3a4a(g

µ1µ3gµ2µ4 − gµ1µ4gµ2µ3)

+ fa1a3afa2a4a(gµ1µ2gµ3µ4 − gµ1µ4gµ2µ3)

+ fa1a4afa2a3a(gµ1µ2gµ3µ4 − gµ1µ3gµ2µ4)

4. 3. Examples: Quark-quark and quark-antiquark scattering

in one gluon exchange approximation

a ) T-matrix for qq-scattering:

q

p1

p′1

p2

p′2

Feynman gauge (ξ = 1)

q = p1 − p′1 = p′2 − p2

iT = (ig)2[u(p′1) γµta u(p1)

] −igµνδab

q2 + iǫ

[u(p′2) γνtb u(p2)

](4.7)

b ) T-matrix for qq-scattering to order αs:

p1 − p′1

−p1

−p′1

p2

p′2

− p1 + p2

p′2 −p′1

p2 −p1

iT = (ig)2[v(p1) γµta v(p

′1)] −igµνδab

(p1 − p′1)2 + iǫ

[u(p′2) γνtb u(p2)

]

− (ig)2[u(p′2) γµta v(p

′1)] −igµνδab

(p1 + p2)2 + iǫ

[v(p1) γνtb u(p2)

](4.8)

32

4. 4. Sketch of path integrals (Functional integrals)

Systematic method for derivation of Feynman rules

• Illustration: example of scalar field theory

L(φ, ∂µφ) =1

2(∂µφ∂

µφ−m2φ2)− V (φ) (4.9)

• Action functional:

S =

∫

d4xL(φ, ∂µφ) = S[φ, ∂µφ] (4.10)

• Basic relation for calculating n-point Green’s functions (correlation function)

〈0|T φ(x1)φ(x2) · · ·φ(xn)|0〉 =

∫Dφφ(x1) · · ·φ(xn) eiS[φ,∂µφ ]

∫Dφ eiS[φ,∂µφ ] (4.11)

For n = 2: “2-point function”

x1

x2

“Propagator”

For n = 3: “3-point function”

x3

x1 x2“Vertex”

For n = 4: “4-point function”

x4 x3

x1 x2

“Scattering amplitude”

33

• Definition of functional (path) integral:

xÓ

t

Φ

Infinitesimal volume

Consider infinitesimal volume in space-time is

∆v = δxi δyj δzk δtl

attached to a point (xi, yj, zk, tl) with field

φ(xi, yj, zk, tl) and its differential dφ defined

at that point.

∫

Dφ = lim∆v→0

∏

ijkl

∫ +∞

−∞dφ(xi yj zk tl) (4.12)

• Starting point: Generating functional

Z[J ] =

∫

Dφ eiS eiR

d4x φ(x)J(x) (4.13)

J(x): auxiliary source function.

Then the n-point function (4.11) becomes

G(n)(x1x2 · · ·xn) ≡ 〈0|T φ(x1)φ(x2) · · ·φ(xn)|0〉

=(−i)n

Z[0]

δZ[J ]

δJ(x1) · · · δJ(xn)

∣∣∣∣∣J=0

(4.14)

with the functional derivative:

δZ[J(x)]

δJ(y)= lim

ǫ→0+

Z[J(x) + ǫδ4(x− y)]−Z[J(x)]

ǫ(4.15)

(

in particular:δJ(x)

δJ(y)= δ4(x− y)

)

Example. Free scalar field

L0 =1

2

(∂µφ ∂

µφ−m2φ2)

(4.16)

34

Z0[J ] =

∫

Dφ exp

[

i

∫

d4x(L0 + φ(x)J(x)

)]

(4.17)

Using∫

d4x ∂µφ∂µφ =

∫

d4x ∂µ(φ∂µφ)︸ ︷︷ ︸

surface integral =0

−∫

d4xφφ (4.18)

it follows that

⇒ Z0[J ] =

∫

Dφ exp

[

− i∫

d4x[ 1

2φ( +m2

)φ− Jφ

]]

(4.19)

equation of motion:( +m2

)φ(x) = −J(x)

φ(x) = −∫

d4y∆F (x− y)J(y)

with ∆F (x− y) =

∫d4k

(2π)4

e−ik·(x−y)

k2 −m2 + iǫ

⇒ Z0[J ] = exp

[

− i

2

∫

d4x

∫

d4y J(x) ∆F (x− y) J(y)

]

×∫

Dφ exp

[

− i

2

∫

d4xφ(x)( +m2)φ(x)

]

(4.20)

Now calculate 2-point function as example:

G(2)(x1, x2) = 〈0|T φ(x1)φ(x2)|0〉

= − 1

Z0[0]

δ2Z0[J ]

δJ(x1)δJ(x2)

∣∣∣∣∣J=0

= i∆F (x1 − x2)

(4.21)

Analogous procedures for n-point functions ⇒ Feynman rules for scalar field theory.

4. 5. Appendix: Useful relations

When dealing with path integrals, some basic formulae:

(1) Important matrix identity: let M be a diagonalizable matrix

ln detM = tr lnM

⇒ detM = exp[tr lnM

] (4.22)

35

(2) Gaussian integral:∫ +∞

−∞

dx√2π

exp[

− 1

2ax2]

=1√a

(4.23)

• Let M be real, symmetric N ×N matrix and XT = (x1, · · · , xN)

⇒ Generalization of Gaussian integral

∫ +∞

−∞

dx1√2π· · ·∫ +∞

−∞

dxN√2π

exp[

− 1

2XTMX

]

=1√

detM(4.24)

• In functional integrals: often encounter

∫

Dφ exp

[

− 1

2

∫

d4x

∫

d4x′ φ(x′)M(x′, x)φ(x)

]

(4.25)

Approximate

∫

d4x→∑

i

∆vi by sum over finite number N =

(L

ǫ

)4

of little cubes

and use Eq. (4.24):

∫

Dφ exp

[

− 1

2

∫

d4x′∫

d4xφ(x′)M(x′, x)φ(x)

]

=1√

detM(4.26)

• Complex scalar fields

∫

Dφ∫

Dφ∗ exp

[

− i

2

∫

d4x

∫

d4x′ φ∗(x′)M(x′, x)φ(x)

]

≃ 1

detM

= exp[− tr lnM

](4.27)

4. 6. Fermion fields

ψ(x) =∑

s=± 12

∫d3p

(2π)3 2Ep

[

ap,sus(p)e−ip·x + b∗p,svs(p)e

ip·x]

(4.28)

⊲ Grassmann-Algebra:

ai, aj = bi, bj = · · · = 0 ; ani = 0, n > 1 (4.29)

⊲ Most general form of function of two Grassmann variables

f(a1, a2) = c0 + c1a1 + c2a2 + c3a1a2

= c0 + c1a1 + c2a2 − c3a2a1

(4.30)

36

• Derivative:

∂f

a∂a1= c1 + c3a2

∂f

a∂a2= c2 − c3a1

(4.31)

∂2

∂a1∂a2= − ∂2

∂a2∂a1(4.32)

• Integration:∫

da1da2 F ≡∫

da1

(∫

da2 F

)

(4.33)

∫

da = 0 because

(∫

da

)2

= −(∫

da

)2

= 0 (4.34)

Definition:

∫

da a = 1 as a normalization

• Path integrals with fermion fields:

Given an antisymmetric matrix A with aT = (a1, · · · , aN)

∫

da1 · · ·∫

daN exp

[

− 1

2aTAa

]

=√

detA (4.35)

ani = 0 for n > 1,

∫

dai = 0;

∫

dai ai = 1

• Complex fermion fields:

∫

da1

∫

da∗1 · · ·∫

daN

∫

da∗N exp

[

− 1

2a†Aa

]

= detA (4.36)

• Functional integrals involving fermion fields:

∫

Dψ∫

Dψ∗ exp

[

−∫

d4xd4x′ ψ∗(x′)A(x′, x)ψ(x)

]

= detA

= exp[tr lnA

](4.37)

4. 7. Generating functional of QCD

⊲ Lagrangian density (without gauge fixing):

LQCD = ψ[iγµD

µ −m]ψ − 1

4Ga

µνGµνa (4.38)

37

⊲ Generating functional:

ZQCD[J, η, η] =

∫

DA∫

Dψ∫

Dψ

× exp

[

i

∫

d4x(LQCD(x) + Aa

µ(x)Jµa (x) + ψ(x)η(x) + η(x)ψ(x)

)] (4.39)

• Generate n-point functions by taking functional derivatives with respect to source

fields J(x), η(x) and η(x).

Example-1. 2-point functions:

- Quark propagator:δ2

δη(x)δη(x)ZQCD

∣∣∣∣∣η, η=0

Quark= 〈0|T ψ(x)ψ(y)|0〉 = iSF (y − x)

= i

∫d4p

(2π)4e−ip·(y−x) /p+m

p2 −m2 + iǫ

(4.40)

- Gluon propagator:δ2

δJaµ(x)δJ b

ν(y)ZQCD

∣∣∣∣∣J=0

Gluon= 〈0|T Aµ

a(x)Aνb (y)|0〉 = iDµν

ab (y − x)

= iδab

∫d4q

(2π)4

dµν(q)

q2 + iǫ

(4.41)

Example-2. 3-point functions:

- Quark-gluon vertex:δ3

δJaµ(x) δη(x) δη(x)

ZQCD

∣∣∣∣∣η,η,J=0

= igγµta (4.42)

- 3-gluon vertex:δ3

δJaµ(x) δJ b

ν(x) δJcλ(x)ZQCD

∣∣∣∣∣J=0

38

p1

p3

p2

= gfabc

[gµν(p1 − p2)

λ + cycl.perm.]

(4.43)

4. 8. Gauge invariance and gauge fixing (Sketch)

• Pure gluon theory:

LG = −1

4Ga

µνGµνa (4.44)

where Gaµν = ∂µA

aν − ∂νA

aµ + gfabcA

bµA

cν .

• Action functional:

SG[A, ∂A] =

∫

d4xLG(A, ∂A) (4.45)

• Generating functional:

ZG[J ] =

∫

DA exp

[

i

∫

d4x(LG + Aµ

aJaµ

)]

(4.46)

• Functional integral covers arbitrarily many gauge-equivalent field configurations.

Aµ = U[

Aµ − i

gU †∂µU

]

U †

Aµ ≡ Aµa

λa

2

U = exp(

− iθa(x)λa

2

) (4.47)

• Gauge fixing needs a constraint.

∂µAaµ(x) = Ba(x) (4.48)

(In particular Lorenz condition Ba(x) ≡ 0)

• Insert “unity”: 1 = detM8∏

a=1

∫

Dθa δ(∂µAa

µ(x)− Ba(x))

(4.49)

with Jacobian of gauge transformation:

Mab(x, y) =δ(∂µAa

µ(x))

δθb(y)

• Problem: to calculate Jacobian detM

39

4. 9. Fadeev-Popov method (Sketch)

• Introduce a set of (unphysical) auxiliary fields: χa(x), χ∗ a(x): anticommuting Bose

fields (“ghost fields”)

detM = i

∫

Dχ∫

Dχ∗ exp

[

i

∫

d4x ∂µχ∗a(x)D

abµ χb(y)

]

(4.50)

with gauge covariant derivative:

Dabµ = δab∂µ − gfabcAc

µ

• Result: Gauge fixing condition ⇒ extra term in Lagrangian density.

⊲ QCD Lagrangian including gauge fixing:

LQCD = −1

4Ga

µν(x)Gµνa (x)− 1

2ξ

(∂µAa

µ

)2+ LFP (4.51)

with Fadeev-Popov term:

LFP = ∂µχ∗ a(x)Dabµ χ

b(x) (4.52)

4. 10. Complete generating functional of QCD

(including gauge fixing)

ZQCD[J, η, η; j, j∗]

=

∫

DA∫

Dψ∫

Dψ∫

Dχ∫

Dχ∗

× exp

[

i

∫

d4x(LQCD + Aa

µJµa + ψη + ηψ + χ∗ aja + j∗aχ

a)]

with LQCD = ψ[iγµD

µ −m]ψ − 1

4Ga

µνGµνa −

1

2ξ

(∂µA

µa

)2+ LFP

(4.53)

• Additional Feynman rules associated with ghost:

– Ghost propagator in momentum space:

pa b =

iδab

p2 + iǫ(4.54)

40

– Ghost-gluon vertex:

b

c, µ

a

= gfabc pµ (4.55)

41

CHAPTER 5. RENORMALIZATION

5. 1. Loops: Self energy and vertex corrections (Radiative corrections)

• Leading order loop diagrams:

a ) Quark self-energy:

b ) Gluon self-energy (incl. vacuum polarization):

+ +

c ) Quark-gluon vertex corrections:

+

d ) 3-gluon vertex correction:

• All of these loop diagrams have divergent momentum space integrals.

• Renormalization programme:

Step 1. Regularize loop integrals.

Step 2. Define a subtraction procedure for infinities (divergences) such that finite

results have a physical (predictive) meaning.

42

Example. Quark self-energy:

= +

O(αs)

+

O(α2s)

+ · · ·

=i

/p−m+ iǫ+

i

/p−m[− iΣ(p)

] i

/p−m + · · ·

=i

/p−m[

1 + Σ(p)1

/p−m + · · ·]

(geometric series)

=i

(/p−m)(

1− Σ(p) 1

/p−m

)

=i

/p−m− Σ(p)

In general,

Σ(p) = A(p2)γµpµ +B(q2)1 ⇒ i

(1−A(p2)

)/p−m−B(q2)

iS(0)F (p) =

1

/p−m+ iǫ⇒ iSF (p) =

Z(p2)

/p−M(p2)︸ ︷︷ ︸

renormalized quark propagator

(5.1)

where M(p2) ≡ B(p2) +m and Z(p2) =B(p2)

1− A(p2).

⊲ Calculation of quark self energy (O(αs)):

p p− k p

k

↑ta = λa

2

↑tb = λb

2

Using Feynman gauge (ξ = 1),

−iΣab(p2) = (ig)2

∫d4k

(2π)4taγ

µ i(/p− /k +m)

(p− k)2 −m2 + iǫ

( −igµνk2 + iǫ

)

γνtb (5.2)

Examine integrand for |k| → ∞:

∫d4k

(2π)4

γµkµ

k4diverges.

43

5. 2. Dimensional regularization

• Start in 4 dimensions and generalize to D dimensions:∫

d4k

(2π)4⇒∫

dDk

(2π)D(5.3)

– Analytic continuation to D = 4: introduce ǫ = 4−D.

– Divergencies show as logarithms and inverse powers of ǫ = 4−D.

• Renormalization: introduce “counterterms” ∆Lǫ such that

Lorginal = Lrenormalized + ∆Lǫ (5.4)

5. 3. Renormalization of the gluon propagator

• Study influence of vacuum polarization process on the quark-gluon vertex and coupling

strength g.

quark

antiquark

g +

gluon

g +

ghost

g

• Gluon self energy: qq (vacuum polarization) loop correction.

qq

p

p− q

a, µ b, ν = iΠabµν(q)

= −(ig)2

∫d4p

(2π)4tr

[

γµtai

/p−m+ iδγνtb

i

(/p− /q)−m+ iδ

]

(5.5)

• Dimensional regularization:∫

d4p

(2π)4⇒ µǫ

∫dDp

(2π)D(ǫ = 4−D)

µ is an arbitrary mass scale called renormalization scale.

At the end of the procedure: take limD→4

= limǫ→0

44

• Useful formulae:

– Integration in D dimensions: volume element dDp = pD−1 dp dΩD

with

∫

dΩD =2πD/2

Γ(D2

)

-2 -1Re z

Im z

0

Γ(z + 1) = z Γ(z) = z !

Expansion around poles:

Γ(z) =1

z− γ +O(z) (Re z > 0)

Γ(1/2) =√π

where γ = 0.5772 · · · (Euler-Mascheroni constant)

– Basic loop integrals:

ID(m) =

∫dDp

(2π)D1

p2 −m2 + iǫ

= −imD−2 (4π)D−2 Γ(1− D

2

)(5.6)

by taking derivative with respective to m2:

∫dDp

(2π)D1

(p2 −m2 + iǫ)n= (−1)n i

(m2)D/2−n

(4π)D/2Γ(n− D

2

)

Γ(n)(5.7)

Therefore

Πabµν(q) = 2i δab g2 µǫ

∫dDp

(2π)Dfµν(p, q)

(p2 −m2 + iǫ

)((p− q)2 −m2 + iǫ

)

fµν(q) = pµ(p− q)ν + pν(p− q)µ + gµν(m2 − p · (p− q)

)(5.8)

• Using Feynman parametrization:

a−n b−m =Γ(m+ n)

Γ(m)Γ(n)

∫ 1

0

dzzn−1(1− z)m−1

[az + b(z − 1)

]n+m (5.9)

Eq. (5.8) becomes

Πabµν = δab

[qµqν − q2gµν

]Π(q) (5.10)

Π(q) =g2

4π2

Γ(ǫ/2)

(4π)−ǫ/2µǫ∫ 1

0

dzz(1− z)

(m2 − q2z(1− z)

)ǫ/2(5.11)

45

• Expand around ǫ→ 0

Γ(ǫ/2)

(4π)−ǫ/2=

2

ǫ− γ + ln 4π +O(ǫ) (5.12)

Π(q) =g2

12π2

[1

ǫ+ ln√

4π − γ

2− 3

∫ 1

0

dz (1− z) lnm2 − q2z(1− z)

µ2+O(ǫ)

]

=g2

12π2

1

ǫ+ ln√

4π − γ

2+

5

6− ln

−q2

m2+ · · ·

(|q2| ≫ m2

)

1

ǫ+ ln√

4π − γ

2− 1

2lnm2

µ2+

q2

10m2+ · · ·

(|q2| ≪ m2

)

(5.13)

• Renormalization constant:

“minimal subtraction” (MS) scheme

︷︸︸︷

Z = 1 +g2

12π2

(1

ǫ+ ln√

4π − γ

2

)

(5.14)

︸ ︷︷ ︸

“modified minimal subtraction”(MS)

scheme

• Gluon self energy: gluon loop correction.

qq

k − q

k

= iΠµνab (q)

∣∣gluonic

= −1

2C g2 µǫ

∫dDk

(2π)DNµν(q, k)

(k2 + iδ

)((q − k)2 + iδ

)

(5.15)

Nµν =(−5q2 +2q ·k−2k2

)gµν +

(6−D

)qµqν +

(2D−3

)(qµkν + qνkµ

)+(6−4D

)kµkν

⇒ Πabµν

∣∣gluonic

= − g2

16π23δab

(µ2

−q2

)ǫ/2 [11

3qµqν −

19

6q2gµν

]1

ǫ+ · · · (5.16)

46

5. 4. Renormalization constants

• Divergences in loop integrals ⇒ absorbed in renormalization constant (for fields,

masses and coupling strength)

• Renormalization constants for fields:

Aaµ = Z−1/2A Aa (0)

µ

ψ = Z−1/2ψ ψ(0)

χa = Z−1/2χ χa (0)

(5.17)

• Renormalization of quark mass, coupling strength, gauge parameter:

m = Z−1m Zψm(0)

g = Zψ Z1/2A Z−1

2 g(0)

ξ = Zξ Z−1A ξ(0)

(5.18)

• Vertex renormalization:

Gluon-ghost vertex : Z1

Quark-gluon vertex : Z2

3-gluon vertex : Z3

4-gluon vertex : Z4

(5.19)

• Slavnov-Taylor identities: gauge invariance (color current conservation) implies:

Zξ Z−11 = Zψ Z−1

2 = ZAZ−13 = Z1/2

A Z−1/24 (5.20)

5. 5. Renormalization scale

• Action S =∫

d4xL(x)→∫

dDxL(x) must be dimensionless in units ~ = 1.

• In terms of “mass dimension”:

dim[L]

= D

dim[ψ]

=1

2(D − 1)

dim[Aµ]

=1

2(D − 2)

47

dim[ψγµAµψ

]= D − 1 +

1

2− 1 =

3

2D − 2

dim[g]

= D − 3D

2+ 2 = 2− D

2=

4−D2

=ǫ

2

• Introducing a dimensionless coupling strength g,

g ≡ g(µ) = g µǫ/2

where µ is called renormalization scale.

(5.21)

5. 6. Renormalization of the QCD coupling strength

• Starting point: Quark-gluon vertex

g = g(0) + +

︸ ︷︷ ︸

vertex correction

+ + +

︸ ︷︷ ︸

vacuum polarization

+

︸ ︷︷ ︸

quark self energy correction

+ · · ·

Physical (renormalized) coupling strength:

g = Zψ Z1/2A Z−1

2 g(0) (5.22)

48

• Determination of g up to O(g3)

g(0) =(1−ΔZψ − 1

2ΔZA +ΔZ2

)g (5.23)

with Zα = 1 +ΔZα.

• After sophisticated calculations (in Feynman gauge):

ΔZψ = − 2g 2

(4π)2ε

N2c − 1

2Nc(5.24)

ΔZA = − g 2

(4π)2ε

(4

3Nf − 19

6Nc − 1

6Nc

)(5.25)

ΔZ2 = − 2g 2

(4π)2ε

(N2c − 1

2Nc

− Nc

2

)(5.26)

g(0) = g

(1− 33− 2Nc

48π2εg 2

)

= g − 11− 23Nc

(4π)2εg3 μ−ε +O(g5)

(5.27)

5. 7. The β function of QCD

• Definition of β-function:

β(g) =∂g

∂ lnμ= μ

∂g

∂μ(5.28)

which shows how the coupling strength depends on scales.

49

)

• g(0) is independent of µ in D = 4 dimension.

µ∂g(0)

∂µ= 0

= µ∂g

∂µ− 11− 2

3Nf

(4π)2 ǫ

[

µ−ǫ+1 3g2 ∂g

∂µ− g3ǫµ−ǫ

]

ǫ→0= µ

∂g

∂µ

(

1− 33− 2Nf

(4π)2ǫg2

)

+11− 2

3Nf

(4π)2g3

(5.29)

After renormalization,

β(g) = − β0

(4π)2g3 (5.30)

with β0 = 11− 23Nf .

Theories with β0 > 0(β(g) < 0

)→ “asymptotic freedom”.

cf. QED with β0 < 0(β(e) > 0

)→ coupling grows with renormalization scale.

5. 8. Renormalization group equation

• Dimensionless observable R probed at some given (space-like) 4-momentum(Q2 =

−q2 = ~q 2 − q20

)depends on αs = g2

4π.

R ≡ R

(Q2

µ2, αs(µ)

)

• Any observable R must be independent of the renormalization scale µ.

µ2dR(Q2

µ2 , αs)

dµ2= µ2 ∂R

∂µ2+ µ2∂αs

∂µ2

∂R

∂αs= 0 (5.31)

• Introducing:

t = lnQ2

µ2

β(αs) = µ2∂αs∂µ2

=µ2

4π

∂g2

∂µ2=

g

4πµ∂g

∂µ=

g

4πβ(g),

Simplest version of “renormalization group equation”

−∂R∂t

+ β(αs)∂R

∂αs= 0

(5.32)

50

• Infinitesimal (finite) difference:[

− 1

∆t+ β(αs)

1

∆αs

]

∆R = 0 (5.33)

∆αs∆t

= β(αs)lim ∆t→0−−−−−→ ∂αs

∂t= β(αs) =

g

4πβ(g) (5.34)

• To leading order(O(g3)

):

β(g) = − β0

(4π)2g3 ,

(

β0 = 11− 2

3Nf

)

∂αs∂t

= −β0α2s

4π(5.35)

• Separation of variables:

dαsα2s

= −β0

4πdt ⇒

∫ αs(Q2)

αs(µ2)

dαsα2s

= −β0

4π

∫ lnQ2

lnµ2

dt

⇒ −(

1

αs(Q2)− 1

αs(µ2)

)

= −β0

4πlnQ2

µ2

(5.36)

• Running coupling strength of QCD:

αs(Q2) =

αs(µ2)

1 + αs(µ2)4π

β0 ln Q2

µ2

(5.37)

Using the QCD scale parameter: ΛQCD ≡ µ exp

[

− 2π

β0 αs(µ2)

]

αs(Q2) =

4π

β0 ln Q2

Λ2QCD

(5.38)

• Next to leading order (NLO):

µ∂αs∂µ

= −β0

2πα2s −

β1

(2π)2α3s + · · · (5.39)

where β0 = 11− 2

3Nf and β1 = 51− 19

3Nf .

αs up to NLO (Λ ≡ ΛQCD):

αs(Q2) =

4π

β0 ln Q2

Λ2

(

1− 2β1

β20

ln[

ln Q2

Λ2

]

ln Q2

Λ2

)

(5.40)

51

FIG. 5.1: Summary of measurements of αs(Q2) from Prog.Part.Nucl.Phys.58 : 351 (2007)

52

CHAPTER 6. QCD AND STRUCTURE FUNCTIONS OF THE NUCLEON

6. 1. Deep inelastic lepton scattering (DIS) on the proton

E, ~p

Pµ = (P 0, ~P )lab. frame−−−−−−→ (MN , 0)

E′, ~p ′

ν, ~q

γ∗

lepton

proton

• Energy and momentum transfer carried by virtual photon:

ν = E − E ′, ~q = ~p− ~p ′; qµ = (ν, ~q )

q2 = ν2 − ~q 2 = −Q2 (Q2 > 0)(6.1)

Q2 determines “resolution”:

Large Q2 corresponds to small distance scales probed inside of proton.

• Deep inelastic scattering (DIS):

ν ≫MN , Q2 ≫M2N (6.2)

• Bjorken (scaling) variable:

x =Q2

2P · qlab. frame−−−−−→ x =

Q2

2MN ν(6.3)

• Interpretation of Bjorken x in infinite momentum frame:

Variables: x =Q2

2P · q , q = −2xP ⇒ q2 = −2xP · q (see Fig. 6.1)

x : fraction of total proton momentum carried by a single parton (quark) when struck

by the virtual photon.

• Associated variable: y =q · Pp · P

lab. frame−−−−−→ ν

E

53

proton

parton

lepton

γparton

proton

lepton1−x P( )

xPγ

−xPq

P

FIG. 6.1: DIS in lab. frame (left) and in infinite momentum frame (right).

• Observable (lab. frame): differential cross section:

d2σ

dΩlab dE ′

⇒ Invariant differential cross section:

x (s−M2N )

d2σ

dx dQ2=

2πMN ν

E ′d2σ

dΩlab dE ′ with s = (p+ P )2 (6.4)

d2σ

dx dQ2=

4πα2

Q2

(1− yx− M2

N y

s−M2N

)

F2(x,Q2) + y2F1(x,Q

2)

where F1,2(x,Q2) are called proton structure functions.

(6.5)

6. 2. The parton model

• Assumption: pointlike spin-1/2 particles (partons) inside the proton

⊲ F1,2(x) independent on Q2 (scaling behavior);

Q2-dependence of F1,2(x,Q2) ⇔ QCD corrections.

⊲ Spin-1/2 ⇒ F2(x) = 2xF1(x) (Callan-Gross relation)

approximately observed in experiment.

• Master formula of the parton model:

2xF1(x)spin- 1

2= F2(x) =∑

i

e2i x fi(x) (6.6)

54

fi(x) : parton (quark) distribution function

ei : quark electric charges

Using notations for fi(x) : u(x), d(x), s(x), u(x), d(x), s(x), · · ·

1

xF2(x) =

(2

3

)2[u(x) + u(x)

]+

(1

3

)2[d(x) + d(x) + s(x) + s(x)

]+ · · ·

• Valence quarks: qv(x)

uv(x) = u(x)− u(x)dv(x) = d(x)− d(x)

• Sea quarks: qs(x)

us(x) = u(x) + u(x)− uv(x)ds(x) = d(x) + d(x)− dv(x)

FIG. 6.2: The total valence and sea quark contributions to the structure of the proton.

55

6. 3. QCD and structure functions of the nucleon (DIS)

6. 3. 1. Gluon emission

γ∗

=

γ∗

+

γ∗

+

γ∗

• Elementary subprocess: γ∗ + quark → quark + gluon

p

q

kp+ q

p′

+

q

p k

p′ − qp′

T1 T2

⊲ Mandelstam variables:

s = (p+ q)2 = (p ′ + k)2

t = (q − p ′)2 = (p− k)2

u = (q − k)2 = (p− p ′)2

• All quark masses neglected (mq ≪ Q,√s)

s+ t+ u = q2 , s+ t+ u+Q2 = 0

• Feynman rules:

T a1,j = u(p′)

[

ǫ′ ν(

igγνλa

2

)iej

/p+ /q −mq

(ieγµ) ǫµ

]

u(p)

T a2,j = u(p′)

[

ǫµ (ieγµ)iej

/p ′ − /q −mq

(

igγνλa

2

)

ǫ′ ν]

u(p)

(6.7)

56

• |q2| ≫ m2q (let mq → 0)

• 1

/p+ /q=

/p+ /q

(p+ q)2=

/p + /q

s;

1

/p ′ − /q=

/p ′ − /q

(p′ − q)2=

/p ′ − /q

t

• Spin-averaged square |T |2 (T = T1 + T2): including∑

i

ǫ(i)µ ǫ(i)ν = −gµν

|T |2 ≡∑

spinpolarization

|T1 + T2|2 =1

2tr(T † T

)

= 32π2 e2j ααs4

3

(

− ts− s

t+

2uQ2

st

)(6.8)

with α =e2

4πand αs =

g2

4π.

• Center of mass frame:

p’

q

k

θ

p

transverse momentum:

pT = |~p ′| sin θ

p2T = − st

s +Q2(t≪ s)

(6.9)

• Differential cross section for the elementary process (γ∗ + quark → quark + gluon):

dσ

dp2T

(γ∗ + q → q + g) =|T |2

16πs2(6.10)

dσ

dp2T

t≪s≃ 8π

3e2qααss2

(

−1

t

)[

s+2(s+Q2)Q2

s

]

= σ0

e2qp2T

αs2π

[s

s +Q2+

2Q2

s

] (6.11)

with −st = p2T (s+Q2) and σ0 =

4π2α

s

57

⊲ Define: splitting function

Pqq(z) ≡4

3

1 + z2

1− z (6.12)

dσ

dp2T

= σ0 e2q

αs2πp2

T

Pqq(z) (6.13)

• Back to structure function:

F2 (x,Q2)

x=

1

2

∑

i

e2i fi(x,Q2

); x =

Q2

2P · qlab. frame−−−−−→ Q2

2Mν

F2

x=∑

f

γ∗

proton

parton 2

f

=

γ∗

all f

γ∗2

Optical theorem : Imf(γ∗P → γ∗P ) ∝ σtot(γ∗P )

F2 (x,Q2)

x=σ(γ∗P )

σ0(6.14)

σ(γ∗q → qg) =

∫ p2T,max

µ2

dp2T

d σ

dp2T

= σ0 e2q

αs2π

ln

(p2T,max

µ2

)

Pqq(z)(6.15)

with p2T,max =

s

4=

1− z4z

Q2.

lnp2T,max

µ2= ln

[1− z4z

Q2

µ2

]

= ln

(1− z4z

)

+ ln

(Q2

µ2

)

≃ ln

(Q2

µ2

)

Thereforeσ

σ0≃ e2q

αs(Q)

2πPqq(z) ln

(Q2

µ2

)

(6.16)

x =Q2

2P · q ; p = yP

z =Q2

2p · q =Q2

2P · qP · qp · q = x

P · qp · q =

x

y

58

• Deep inelastic lepton scattering on proton

F2 (x,Q2)

x=∑

q

e2q

∫ 1

x

dy

yq(y)

[

δ

(

1− x

y

)

+αs(Q)

2πPqq

(x

y

)

ln

(Q2

µ2

)

︸ ︷︷ ︸

∆q(x,Q2)

]

(6.17)

∆q(x,Q2

)=αs2π

ln

(Q2

µ2

)∫ 1

x

dy

yq(y)Pqq

(x

y

)

(6.18)

take difference at two neighboring values of Q2

q(x,Q2

)− q

(x,Q2

0

)=αs2π

(

ln

(Q2

µ2

)

− ln

(Q2

0

µ2

))∫ 1

x

dy

yq(y)Pqq

(x

y

)

(6.19)

then take limit (Q20 → Q2)

• Altarelli-Parisi evolution equation:

d

d lnQ2q(x,Q2

)=αs(Q)

2π

∫ 1

x

dy

yq(x,Q2

)Pqq

(x

y

)

(6.20)

6. 3. 2. Quark-antiquark pair production

gluon

γ∗

q

q

• Introduce gluon distribution function: g(x)

• Interchange: pµ ↔ −kµ

⇒ s→ t , t→ u , u→ s

• Result:∆F2 (x,Q2)

x

∣∣∣∣∣γ∗g→qq

=∑

q

e2q

∫ 1

x

dy

yg(y)

αs2πPqg

(x

y

)

ln

(Q2

µ2

)

(6.21)

with splitting function

k

zk

gluon q

q

Pqg(z) =1

2

[z2 + (1− z)2

]

59

6. 4. Complete set of evolution equations for quark and gluon distribution functions

(DGLAP equation)

• Quark distributions: qi (x,Q2), qi (x,Q

2) (i = u, d, s, · · · )

• Gluon distributions: g (x,Q2)

d qi (x,Q2)

d lnQ2=αs(Q)

2π

∫ 1

x

dy

y

[

qi(y,Q2

)Pqq

(x

y

)

+ g(x,Q2

)Pqg

(x

y

)]

d g (x,Q2)

d lnQ2=αs(Q)

2π

∫ 1

x

dy

y

[∑

i

qi(y,Q2

)Pgq

(x

y

)

+ g(x,Q2

)Pgg

(x

y

)] (6.22)

• Interpretation:

qi

qi

Pqq(xy

)

g

qi

Pqg(xy

)

∑

i

qi

g

Pgq(xy

)

g

g

Pgg(xy

)

• Splitting functions

Pqq(z) =4

3

1 + z2

1− zPqg(z) =

1

2

[z2 + (1− z)2]

Pgq(z) =4

3

1 + (1− z)2

z

Pgg(z) = 6

[1− zz

+z

1− z + z (1− z)]

(6.23)

• DGLAP equation (Dokshitzer, Gribov, Lipatov, Altarelli and Parisi)

⊲ Results:

60

– Quark- and gluon-distribution function

– Evolution of F2 (x,Q2) in comparison with experiments

⊲ Note: regularization at z = 1:1

1− z →1

(1− z)+

∫ 1

0

dzf(z)

(1− z)+=

∫ 1

0

dzf(z)− f(1)

1− z

(1

(1− z)+=

1

1− z for z < 1

)

(6.24)

For example,

Pqq(z) =4

3

1 + z2

(1− z)+

+ 2 δ(1− z) (6.25)

Pgg(z) = 6

(1− zz

+z

(1− z)+

+ z (1− z))

+1

2

(

11− 2

3Nf

)

δ(1− z) (6.26)

and also,

Pqq(z) = Pgq(1− z)

Pqg(z) = Pqg(1− z)

Pgg(z) = Pgg(1− z)

(6.27)

61

i='t.3x t 0''x=2x i 0J

x=3.2x l0-r Hl 95+96 datah . a l ; m i h . ^ ,

- . -x=)x I u

^ . ^ - )(=5X t U

x= L3x l0'3

.r=2x 10'3

x=3.2x10-l

x=5x l0'3

x=8x10'3

x= l .3x I 0-2

x=2x10'2

x=3.2x

x=5x 10'2

T-

x=8x 10'2

x=3.2x l0' '

1= l . l x l 0 ' l

Y ' Y ' V O O Q

I

c\l

-+-

o.l

X

f T

J

t 0 r o 2 . . 1 0 3 l o l

Q ' (GeV' )

Figure 7: Comparison of the F! predictions of the 1VIRST partons rvith the preliminary 1995

and 1996 nominal vertex data of H1 [2i]. For display purposes rve acld 0.2(19 - i) to Fl each

time the value of r is decreased, where i = I, i9. These data are not used in the global analysis.

T d

Deep Inelastic Scattering: Structure Function of the

Proton

experimental datavs.

QCD

Q-dependence fromDGLAP EQUATIONS

16. Structure functions 1

16. STRUCTURE FUNCTIONSUpdated September 2007 by B. Foster (University of Oxford), A.D. Martin (University ofDurham), and M.G. Vincter (Carleton University).

16.1. Deep inelastic scattering

High-energy lepton-nucleon scattering (deep inelastic scattering) plays a key role indetermining the partonic structure of the proton. The process !N ! !!X is illustrated inFig. 16.1. The filled circle in this figure represents the internal structure of the protonwhich can be expressed in terms of structure functions.

k

k

q

P, M W

Figure 16.1: Kinematic quantities for the description of deep inelastic scattering.The quantities k and k! are the four-momenta of the incoming and outgoingleptons, P is the four-momentum of a nucleon with mass M , and W is the massof the recoiling system X . The exchanged particle is a ", W±, or Z; it transfersfour-momentum q = k " k! to the nucleon.

Invariant quantities:

# =q · PM

= E " E! is the lepton’s energy loss in the nucleon rest frame (in earlierliterature sometimes # = q · P ). Here, E and E! are the initial and finallepton energies in the nucleon rest frame.

Q2 = "q2 = 2(EE!""!k ·"!k !)"m2

! "m2!! where m!(m!!) is the initial (final) lepton mass.

If EE! sin2($/2) # m2! , m2

!! , then

$ 4EE! sin2($/2), where $ is the lepton’s scattering angle with respect to the leptonbeam direction.

x =Q2

2M#where, in the parton model, x is the fraction of the nucleon’s momentum

carried by the struck quark.

y =q · Pk · P

=#

Eis the fraction of the lepton’s energy lost in the nucleon rest frame.

W 2 = (P + q)2 = M2 + 2M# " Q2 is the mass squared of the system X recoiling againstthe scattered lepton.

s = (k + P )2 =Q2

xy+ M2 + m2

! is the center-of-mass energy squared of the lepton-nucleonsystem.

CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006)

available on the PDG WWW pages (URL: http://pdg.lbl.gov/) November 29, 2007 14:50

10 16. Structure functions

Table 16.1: Lepton-nucleon and related hard-scattering processes and theirprimary sensitivity to the parton distributions that are probed.

Main PDFsProcess Subprocess Probed

!±N ! !±X "!q ! q g(x 0.01), q, q!+(!")N ! #(#)X W !q ! q#

#(#)N ! !"(!+)X W !q ! q#

# N ! µ+µ"X W !s ! c ! µ+ s

pp ! "X qg ! "q g(x " 0.4)pN ! µ+µ"X qq ! "! q

pp, pn ! µ+µ"X uu, dd ! "! u # d

ud, du ! "!

ep, en ! e$X "!q ! q

pp ! W ! !±X ud ! W u, d, u/d

pp ! jet +X gg, qg, qq ! 2j q, g(0.01 x 0.5)

all polarized PDFs. These polarized PDFs may be fully accessed via flavor tagging insemi-inclusive deep inelastic scattering. Fig. 16.5 shows several global analyses at a scaleof 2.5 GeV2 along with the data from semi-inclusive DIS.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

x f(x

)

Figure 16.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, g) using the MRST2001 parameterization [29,13](withuncertainties for uv, dv, and g) at a scale µ2 = 10 GeV2.

June 16, 2004 14:04

ud

sc

d

guv

v

fraction of total proton momentum

distributions of QUARKS and GLUONS

in the PROTON

experimental data +

QCD

SNAPSHOTS of the NUCLEON´S INTERIOR

Deep Inelastic Scattering

4 16. Structure functions

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'"%

'!(%

'!()

'!($

'!(*

'!(#

'!('

'+

,#-+./#0

1234

5'

4678

98:;4

<;8

!

'

#

*

$

)

%

=

'!(=

'!(%

'!()

'!($

'!(*

'!(#

'!('

'+

,#>??-+./#0>@>?-/0

5'

1234

2;8

!

!"!#)

!"!)

!"!=)

!"'

!"'#)

!"')

!"'=)

!"#

'!(%

'!()

'!($

'!(*

'!(#

'!('

+

,#>AA-+./#0>@>B-/0

Figure 16.10: a) The proton structure function F p2 mostly at small x and Q2, measured in electromagnetic

scattering of positrons (H1, ZEUS), electrons (SLAC), and muons (BCDMS, NMC) on protons. Lines areZEUS and H1 parameterizations for lower (Regge) and higher (QCD) Q2. The width of the bins can be up to10% of the stated Q2. Some points have been slightly o!set in x for clarity. References: ZEUS—J. Breitweget al., Phys. Lett. B407, 432 (1997); J. Breitweg et al., Eur. Phys. J. C7, 609 (1999); J. Breitweg et al.,Phys. Lett. B487, 53 (2000) (both data and ZEUS Regge parameterization); S. Chekanov et al., Eur. Phys.J. C21, 443 (2001); S. Chekanov et al., Phys. Rev. D70, 052001 (2004); H1—C. Adlo! et al., Nucl. Phys.B497, 3 (1997); C. Adlo! et al., Eur. Phys. J. C21, 33 (2001) (both data and H1 QCD parameterization);C. Adlo! et al., Eur. Phys. J. C30, 1 (2003); A. Aktas et al., Phys. Lett. B598, 159 (2004); BCDMS, NMC,SLAC—same references as Fig. 16.7.

b) The charm structure function F cc2 (x), i.e. that part of the inclusive structure function F p

2 arising fromthe production of charm quarks, measured in electromagnetic scattering of positrons on protons (H1, ZEUS)and muons on iron (EMC). The H1 points have been slightly o!set in x for clarity. For the purpose ofplotting, a constant c(Q) = 0.05i2Q is added to F cc

2 where iQ is the number of the Q2 bin, ranging from 1(Q2 = 1.8 GeV2) to 11 (Q2 = 650 GeV2). References: ZEUS—J. Breitweg et al., Eur. Phys. J. C12, 35(2000); S. Chekanov et al., Phys. Rev. D69, 012004 (2004); H1—C. Adlo! et al., Z. Phys. C72, 593 (1996);C. Adlo! et al., Phys. Lett. B528, 199 (2002); A. Aktas et al., Eur. Phys. J. C40, 349 (2005); A. Aktas et al.,Eur. Phys. J. C45, 23 (2006); EMC—J.J. Aubert et al., Nucl. Phys. B213, 31 (1983).

Inset: The bottom quark structure function F bb2 (x). For the purpose of plotting, a constant k(Q) = 0.01i1.7

Q

is added to F bb2 where iQ is the number of the Q2 bin, ranging from 1 (Q2 = 12 GeV2) to 5 (Q2 = 650 GeV2).

References: H1—A. Aktas et al., Eur. Phys. J. C40, 349 (2005); A. Aktas et al., Eur. Phys. J. C45, 23(2006).

Statistical and systematic errors added in quadrature are shown for both plots. The data are given as afunction of x in bins of Q2.

Deep Inelastic Scattering: Structure Function of the

Proton

experimental datavs.

QCD

Q-dependence fromDGLAP EQUATIONS

16. Structure functions 7

!

"#$!%&#'

()

*+,-

./01-

21/

-34/

+556

78#

789

785

78:

)

)8#

)89

)7;9

)7;<

)7;#

)7;)

)

Figure 16.2: The proton structure function F p2 given at two Q2 values (3.5 GeV2

and 90 GeV2), which exhibit scaling at the ‘pivot’ point x ! 0.14. See the captionsin Fig. 16.7 and Fig. 16.10 for the references of the data. Also shown is theMRST2006 parameterization [13] given at the same scales.

functions) can be expressed as a convolution of calculable, process-dependent coe!cientfunctions and these universal parton distributions, e.g. Eq. (16.21).

It is often convenient to write the evolution equations in terms of the gluon, non-singlet(qNS) and singlet (qS) quark distributions, such that

qNS = qi " qi (or qi " qj), qS =!

i

(qi + qi) . (16.24)

The non-singlet distributions have non-zero values of flavor quantum numbers, such asisospin and baryon number. The DGLAP evolution equations then take the form

!qNS

! lnµ2 ="s(µ2)

2#Pqq # qNS ,

!

! ln µ2

"qS

g

#=

"s(µ2)2#

"Pqq 2nf PqgPgq Pgg

##

"qS

g

#, (16.25)

November 29, 2007 14:50

16. Structure functions 7

!

"#$!%&#'

()

*+,-

./01-

21/

-34/

+556

78#

789

785

78:

)

)8#

)89

)7;9

)7;<

)7;#

)7;)

)

Figure 16.2: The proton structure function F p2 given at two Q2 values (3.5 GeV2

and 90 GeV2), which exhibit scaling at the ‘pivot’ point x ! 0.14. See the captionsin Fig. 16.7 and Fig. 16.10 for the references of the data. Also shown is theMRST2006 parameterization [13] given at the same scales.

functions) can be expressed as a convolution of calculable, process-dependent coe!cientfunctions and these universal parton distributions, e.g. Eq. (16.21).

It is often convenient to write the evolution equations in terms of the gluon, non-singlet(qNS) and singlet (qS) quark distributions, such that

qNS = qi " qi (or qi " qj), qS =!

i

(qi + qi) . (16.24)

The non-singlet distributions have non-zero values of flavor quantum numbers, such asisospin and baryon number. The DGLAP evolution equations then take the form

!qNS

! lnµ2 ="s(µ2)

2#Pqq # qNS ,

!

! ln µ2

"qS

g

#=

"s(µ2)2#

"Pqq 2nf PqgPgq Pgg

##

"qS

g

#, (16.25)

November 29, 2007 14:50

16. Structure functions 11

enough observables to determine all polarized PDFs. These polarized PDFs may be fullyaccessed via flavor tagging in semi-inclusive deep inelastic scattering. Fig. 16.5 showsseveral global analyses at a scale of 2.5 GeV2 along with the data from semi-inclusiveDIS.

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'!($

'!()

'!(#

'!('

*

*+,-*.

!

!"#

!"$

!"%

!"&

'

'"#

'"$

'!($

'!()

'!(#

'!('

*

*+,-*.

Figure 16.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, b, g) and their associated uncertainties using the NNLOMRST2006 parameterization [13] at a scale µ2 = 20 GeV2 and µ2 = 10, 000 GeV2.

Comprehensive sets of PDFs available as program-callable functions can be obtainedfrom several sources e.g., Refs. [55,56]. As a result of a Les Houches Accord, a PDFpackage (LHAPDF) exists [57] which facilitates the inclusion of recent PDFs in MonteCarlo/Matrix Element programs in a very compact and e!cient format.

16.4. DIS determinations of !s

Table 16.2 shows the values of !s(M2Z) found in recent fits to DIS and related data in

which the coupling is left as a free parameter. There have been several other studies of!s using subsets of inclusive DIS data, and also from measurements of spin-dependentstructure functions, see the Quantum Chromodynamics section of this Review.

November 29, 2007 14:50

DGLAP EQUATIONS

16. Structure functions 1

16. STRUCTURE FUNCTIONSUpdated September 2007 by B. Foster (University of Oxford), A.D. Martin (University ofDurham), and M.G. Vincter (Carleton University).

16.1. Deep inelastic scattering

High-energy lepton-nucleon scattering (deep inelastic scattering) plays a key role indetermining the partonic structure of the proton. The process !N ! !!X is illustrated inFig. 16.1. The filled circle in this figure represents the internal structure of the protonwhich can be expressed in terms of structure functions.

k

k

q

P, M W

Figure 16.1: Kinematic quantities for the description of deep inelastic scattering.The quantities k and k! are the four-momenta of the incoming and outgoingleptons, P is the four-momentum of a nucleon with mass M , and W is the massof the recoiling system X . The exchanged particle is a ", W±, or Z; it transfersfour-momentum q = k " k! to the nucleon.

Invariant quantities:

# =q · PM

= E " E! is the lepton’s energy loss in the nucleon rest frame (in earlierliterature sometimes # = q · P ). Here, E and E! are the initial and finallepton energies in the nucleon rest frame.

Q2 = "q2 = 2(EE!""!k ·"!k !)"m2

! "m2!! where m!(m!!) is the initial (final) lepton mass.

If EE! sin2($/2) # m2! , m2

!! , then

$ 4EE! sin2($/2), where $ is the lepton’s scattering angle with respect to the leptonbeam direction.

x =Q2

2M#where, in the parton model, x is the fraction of the nucleon’s momentum

carried by the struck quark.

y =q · Pk · P

=#

Eis the fraction of the lepton’s energy lost in the nucleon rest frame.

W 2 = (P + q)2 = M2 + 2M# " Q2 is the mass squared of the system X recoiling againstthe scattered lepton.

s = (k + P )2 =Q2

xy+ M2 + m2

! is the center-of-mass energy squared of the lepton-nucleonsystem.

CITATION: W.-M. Yao et al., Journal of Physics G 33, 1 (2006)

available on the PDG WWW pages (URL: http://pdg.lbl.gov/) November 29, 2007 14:50

Deep Inelastic Scattering

Parton (Quark and Gluon)

Distributions

CHAPTER 7. QCD ON THE LATTICE (LQCD)

7. 1. Field theories on a lattice

• Lagrangian L(Φ, ∂µΦ); Fields Φ(x), xµ = (x0, ~x ) = (t, ~x ).

Φ(x) stands generically for quark or gluon fields.

• Green’s function (n-point function):

G(n)(x1, · · · , xn) = N 〈0|T [Φ(x1) · · ·Φ(xn)]|0〉 (7.1)

• Action functional:

S[Φ] =

∫d4xL (Φ(x), ∂µΦ(x)) (7.2)

• Path integrals:

G(n)(x1, · · · , xn) =

∫ DΦ Φ(x1) · · ·Φ(xn) eiS[Φ]

∫ DΦ eiS[Φ](7.3)

• Goal of field theory on a lattice: perform path integrals for G(n) on a discretized

Euclidean space-time lattice numerically.

• Field theories involve renormalization and ultraviolet regularizations.

B Regularization automatically “built-in” in lattice field theory.

• Continuum limit (a → 0).

• Infrared (“long wavelength”) cutoff ↔ finite volume V .

• Minkowski space NOT appropriate (eiS complex and potentially rapidly oscillating).

B Euclidean space: t ≡ x0 → −ix4 ≡ −iτ (τ : Euclidean time)

62

a) Minkowski-metric: aµ = (a0, ~a)

(a · b)M = aµbµ = a0b0 − ~a ·~b

b) Euclidean-metric: aµ = (~a, a4)

(a · b)E = ~a ·~b + a4b4

• Wick rotation

´

´

HMinkowskianL

HEuclideanL

tRe x0

-ix4

Im x0

0

• Definition: Euclidean hypercube

ΩE =

x;xµ

a∈ Z, µ = 1, · · · , 4

(a : lattice constant) (7.4)

7. 2. Euclidean action

• Starting point: fields Φ(x) with x ∈ ΩE (Euclidean fields)

• Kinetic term of Lagrangian in Euclidean space

L(0)E (x) =

1

2∂µΦ(x)∂µΦ(x) =

1

2

4∑i=1

(∂iΦ(x)) (∂iΦ(x)) (7.5)

• Action functional

S =

∫d4xL =

∫dx0

∫d3xL = i

∫d3x

∫dx4 LE

≡ iSE

(7.6)

63

∂0 ≡ ∂

∂x0=

∂

−i∂x4

= i∂

∂x4

∂0∂0 − ~∇ 2 = − ∂

∂x4

∂

∂x4

− ~∇ 2 = −(

~∇ 2 +∂2

∂x24

) (7.7)

eiS = e−SE (7.8)

7. 3. Euclidean Green’s function

G(n)E (x1, · · · , xn) =

∫ DΦ Φ(x1) · Φ(xn) e−SE∫ DΦ e−SE(7.9)

• Example: two-point function(“one particle propagator”)

GE(x, y) =

∫ DΦ Φ(x)Φ(y) e−SE[Φ]

∫ DΦ e−SE[Φ] (7.10)

• Let Φ be a scalar field in Minkowski space:

G(x, y) = i∆F (x− y; m)

= i

∫d4k

(2π)4

e−ik·(x−y)

k2 −m2 + iε

(7.11)

• Translate to Euclidean space (Euclidean propagator):

∫dk0 → −i

∫dk4 ; (−ik · x)M → (ik · x)E

(k2 −m2

)M→ − (

m2 + k2)E

∆F (x; m2) → ∆E(x; m2) = i

∫d4kE

(2π)4

ei(k·x)E

k2E + m2

(7.12)

• Spectral representation of Green’s function:

G ; =G(0); free

+ + · · ·

64

G(x, y) = i

∫ ∞

m20

dm2 ρ(m2)∆F (x− y; m2) (7.13)

• Typical form of spectral function ρ(m2):

∆ Hm2-m02L

m2

• Green’s function in Euclidean space: take volume integral

∫d3xGE(x, y)

!=

∫ ∞

m0

dmρ(m2)e−m|x4−y4| (7.14)

proof) GE(x, y) = −i

∫ ∞

m20

dm2 ρ(m2)∆E(x− y; m2)

∫d3xGE(x, y) =

∫d3x

∫ ∞

m20

dm2 ρ(m2)

∫d4k

(2π)4

eik·(x−y)

k2 + m2

=

∫d3x

∫ ∞

m20

dm2 ρ(m2)

∫dk4

2π

∫d3k

(2π)3

eik4(x4−y4)+i~k·(~x−~y)

~k 2 + m2 + k24

(7.15)

∫d3x ei~k·~x = (2π)3δ3

(~k

)

´

´

im

-im

Τ > 0

Τ < 0

Re k4

Im k4

65

∫d3xGE(x, y) =

∫dm2 ρ(m2)

∫dk4

2π

eik4

τ︷ ︸︸ ︷(x4 − y4)

k24 + m2

︸ ︷︷ ︸(k4+im)(k4−im)

=

∫dm2 ρ(m2)

2πi

2π

e−m|x4−y4|

2im

=

∫ ∞

m0

dm ρ(m2)e−m|x4−y4|

(7.16)

• For large Euclidean time intervals τ = |x4 − y4|:

lowest mass m in spectral distribution ρ(m2) dominates.

m0 = − limτ→∞

1

τln

∫d3xGE(~x, τ ; ~y, 0) (7.17)

B in discretized form (Na = τ):

m0a = − 1

Nln

∫d4xGE(~x,Na; ~y, 0) (7.18)

7. 4. Elements of Lattice QCD

• Degrees of freedom (active fields): quark fields ψ(x), gauge fields (gluons) Aiµ(x)

• Quarks: start from massive quark (infinitely heavy, static).

Quarks “sit” on lattice sites

ψ =

ψ1

...

ψn

(color, flavor, spin indices)

7. 4. 1. Gauge fields on the Lattice

Aµ(x) = −ig Akµ(x)

λk

2(µ = 1, · · · , 4) (7.19)

66

• Cyx: curve on Euclidean space-time Lattice

x y

• With a curve on the Lattice, associate a matrix quantity:

U(Cx+dx, x) = 1−Aµ(x)dxµ (7.20)

• On Euclidean Lattice ΩE, smallest distance for “parallel transport” defined by lattice

constant a.

eµ (µ = 1, · · · , 4) Basis in ΩE

⇒ introduce (x + dx, x) → (x + aeµ, x) = c

e

1

2

e

• c is the elementary connection between two lattice points:

U(c) ≡ U(x + aeµ, x) : Link variable

• Alternative forms:

U(c) ≡ Uµ(a) = P exp

[−

∫ x+aeµ

x

dx′Aµ(x′)]

' 1−Aµ(x)dxµ

= 1−Aµ(x)aeµ

(7.21)

• Path ordering: consider a curve C ∈ ΩE

divide into infinitesimal segments

67

• Introduce: dxl = xl − xl−1

then U(C) = UCyx0

= P exp

[−

∫ y

x0

dxµAµ(x)

]

C

= limdxl→0

[1−Aµ(x0) dx1 µ] · · · [1−Aν(xn−1) dxn ν ]︸ ︷︷ ︸path ordered product of infinitesimal steps

• Closed loops on the lattice: Plaquette variable U(p);

4

2

dy

3

dx1

U(p) = U4→1ν (a) U3→4

µ (a) U2→3ν U1→2

µ (a) (7.22)

smallest possible closed path on the lattice: “Plaquette”.

• In general: U(Cx,x) = 1− Gµν(x) dxµ dxν + · · ·

Gµν(x) = [Dµ, Dν ] , Dµ = ∂µ +Aµ(x) (gauge covariant derivative) (7.23)

• Field strength tensor: Gµν(x) = ∂µAν(x)− ∂νAµ(x) + [Aµ(x), Aν(x)]

7. 4. 2. Action of Lattice gauge fields

• Recall: in Minkowski space Aµi (x)

Field strength tensor: Gµνi = ∂µAν

i (x)− ∂νAµi (x) + g fijk Aµ

j (x) Aνk(x)

• Lagrangian density:

L = −1

4Gµν

i (x)Giµν(x) = −1

2tr GµνG

µν ;

(Gµν =

λi

2Gµν

i

)(7.24)

Action Sg =

∫d4xL (in Minkowski space)

• In Euclidean space: Aµ = −ig Akµ(x)

λk

2

⇒ on Euclidean space-time lattice ΩE.

Lattice gauge field action

Sg = − 1

2g2

∫d4xE tr (Gµν(x)Gµν(x))

discretize−−−−−→ − 1

2g2

∑x∈ΩE

a4 tr [Gµν(x)]2(7.25)

68

Wilson-Wegner action

Sg = const.×∑

p

[1− 1

2tr

(U(p) + U †(p)

)](7.26)

7. 5. Wilson-loop and Potential between Heavy Quarks

• Consider pure gauge field theory (mquark →∞) on the lattice

• Wilson loop:

W (C) = 〈trU(C)〉 =

∫ DU tr U e−Sg

∫ DU e−Sg(7.27)

U(C) = P exp

[−

∮

C

dxµAµ(x)

](7.28)

• Take rectangular loop

• Asymptotically for large Euclidean time T

W (C)τ→∞−−−→ const.× exp (−TV (R)) (7.29)

• V (R): potential between two color sources at space distance R.

V (R) = limT→∞

1

Tln W (C) (7.30)

69

• String tension: Def. σ ≡ limR→∞

V (R)

R= lim

R→∞lim

T→∞1

TRln W (C)

• Result: parametrize as V (R) = σR− c

R

Σ R

-

cR

R

V

• Interpretation: V (R) = σR ← confinement (very different from electric dipole!)

QQ

• Note: this picture is for infinitely heavy (static) quarks (approximately correct for c,

b and t)

• For light quark (u, d and s) ⇒ potential picture NOT valid!

7. 6. Applications: Quarkonium systems

• Charmonium J/ψ, ψ′, ψ′′, · · · : mc ' 1.3 GeV

• Bottomonium Υ, Υ′, · · · : mb ' 4.2 GeV

• “Non-relativistic QCD”: Schrodinger equation

Hamiltonian: H = −~p 2

2µ+ V (r) + Vspin

– Vspin: fine- and hyperfine-splittings

– V (r) = σr − 4

3

αs(r)

r: confining potential and Coulomb like potential

– αs(r) =2π

β0 ln 1Λr

with β0 = 11− 2

3Nf

–√

σ = 0.4 GeV, σ ' 0.8 GeV · fm−1

70

Ψ'

JΨ

r

V

7. 7. Quarks/Fermions on the Lattice

• Free quarks in Minkowski space:

Lagrangian density: Lq = ψ(x) [iγµ∂µ −m] ψ(x) (µ = 0, · · · , 3)

Dirac equation: (iγµ∂µ −m) ψ(x) = 0

Action: Sq

[ψ, ψ

]=

∫d4x ψ (iγµ∂

µ −m) ψ(x)

• Euclidean Action:

SEq

[ψ, ψ

]=

∫d4xE ψ(xE)

(γE

µ ∂µ + m)ψ(xE) (µ = 1, · · · , 4) (7.31)

Euclidean Dirac matrices:γE

µ , γEν

= 2δµν

• Derivative ∂µ ≡ ∂

∂xµ

on the lattice:

∂µψ(x) = lima→0

ψ(x + aeµ)− ψ(x− aeµ)

2a+O(a2) (7.32)

• Action for quarks on the lattice:

Sq = a4∑

n

4∑µ=1

1

2a

(ψ(xn)γµ [ψ(xn + aeµ)− ψ(xn − aeµ)]

)+ ψ(xn)mψ(xn) (7.33)

• Redefine (dimensionless) quantities:

ψα(xn) → 1

a3/2ψα(xn)

m → 1

am

(7.34)

Therefore

Sq =∑

n, l

¯ψα(xn) Qαβ(n, l) ψβ(xl) (7.35)

71

with quark matrix defined as:

Qαβ(n, l) =4∑

µ=1

1

2(γµ)αβ [δl, n+µ − δl, n−µ] + mδlnδαβ (7.36)

• Free Dirac equation on the lattice:

Qαβ(n, m)ψβ(xm) = 0 (7.37)

• Compute Green’s function (n-point function) involving quarks.

• 2-point function in continuum version:

G(x, y) = 〈T ψ(x)ψ(y)〉 =

∫ DψDψ ψ(x)ψ(x)e−Sq

∫ DψDψ e−Sq(7.38)

⇒ Euclidean lattice version:

Gq = 〈ψα(xn)¯ψβ(xl)〉 =

∫ DψD ¯ψ ψα(xn)

¯ψβ(xl)e

−Sq

∫ DψD ¯ψ e−Sq

(7.39)

⇒ Result: Gq = Q−1αβ(n, l))

• Continuum limit:

〈ψα(x)ψβ(y)〉 = lima→0

G(0)q

a3(7.40)

• Reminder: Dirac propagator in Minkowski space-time

〈T ψα(x)ψβ(y)〉 = i

∫d4p

(2π)4

[ γµpµ + m ]αβ

p2 −m2 + iε(7.41)

• Consider now fermions in discrete Euclidean space (3-dim) with volume V = L3 =

(Na)3:

72

⇒ momentum spectrum: discrete

pk =2π

Lνk , (k = 1, 2, 3) , 0 ≤ νk ≤ N − 1, L = Na (7.42)

⇒ energy: Ep = ±√√√√m2 +

3∑

k=1

p2k

pk =

1

asin pka

Brillouin zone: − π ≤ pka ≤ +π

G(0)q

a3=

∫ +π/a

−π/a

d4p

(2π)4

[−i

∑4µ=1 γµpµ + m

]αβ∑4

µ=1 p2µ + m2

(7.43)

• Energy-momentum relation: p2µ = ~p 2 + p2

4 = −m2

p24 = −E2 = −(~p 2 + m2)

• Problem:

zeros of sin pµa at pµ = ±πa

destroy continuum limit;

“fermion doubling problem” for each dimension

⇒ 2 spurious modes with no physical meaning.

• Therapy: Wilson fermions:

S(W )q = Sq − r

2

∑n

¯ψ(xn) ¤ ψ(xn) (7.44)

where r is a dimensionless control parameter.

¤ = a2 ∂µ∂µ

¤ψ(x) =1

a2[ ψ(x + aeµ)− sψ(x) + ψ(x− aeµ) ]

73

• Replacing mass (in mass matrix) by momentum dependent mass

m(p) = m +2r

a

4∑µ=1

sin2(pµa

2

)

removes spurious modes. (⇒ Wilson fermions)

7. 8. Complete Action of Lattice QCD (Wilson fermion)

⇒ Gluonic action:

• Continuum version: ScontG =

1

4

∫d4xF i

µνFiµν

• Lattice representation: SlatticeG =

2Nc

g2

∑p

[1− 1

2Nc

tr(U(p) + U †(p)

) ]

⇒ Quark action including gauge field coupling:

• Continuum: ScontQ =

∫d4xψα(x) Qαβ ψβ(x)

with Quark matrix Qαβ = [ γµ∂µ −M − γµAµ ]αβ

• Quark masses:

mu,d ' 5 MeV

ms ' (100 ∼ 150) MeV

mc ' (1.0 ∼ 1.5) GeV

mb ' (4.0 ∼ 4.5) GeV

mt ' (170 ∼ 180) GeV

(7.45)

• Lattice representation (W is the extra terms in the Wilson construction)

S(W )Q =

∑n

¯ψ(xn)

(M + W

)ψ(xn)

− 1

2

∑µ,n

[¯ψ (r − γµ) Uµ(n) ψ(xn + aeµ)

+¯ψ(x + aeµ) (r + γµ) U †

µ(n) ψ(xn)]

(7.46)

Uµ(n) ≡ U(xn + aeµ , xn)

U †µ(n) ≡ U−1(xn + aeµ , xn) = U(xn , xn + aeµ)

(7.47)

74

• Lattice QCD action:

SQCD[U, ψ, ψ] = SG[U ] + SQ[U, ψ, ψ] (7.48)

• Application: computing observables

Operators F given in matrix representation in discrete Euclidean space

〈F 〉 =

∫ DU DψDψ F e−SQCD

∫ DU DψDψ e−SQCD(7.49)

• “Integrate out” quark fields:

SQ =∑n,m

ψα(xn) Qnmαβ [U ] ψβ(xm) (7.50)

• Effective action:

Seff [U ] ≡ SG[U ]− ln det Q[U ] (7.51)

〈F 〉 =

∫ DU Feff e−Seff [U ]

∫ DU e−Seff [U ](7.52)

• “Quenched approximation”: det Q[U ] → constant

⇒ Suppress large class of quark loops.

7. 9. Examples

(1) Masses of Hadrons: (e.g. mass of nucleon)

GN(~x, τ ; 0) = 〈N(~x, τ)N(0)〉

=

∫ DU DψDψ N(x)N(0) e−SQCD

∫ DU DψDψ e−SQCD

(7.53)

• Nucleon mass from lattice QCD

N(x) = [ q1(x) q2(x) q3(x) ]spin 12

e.g. proton: [ u ↑ u ↓ d ↑ ]spin= 12, Ms=+ 1

2

⇒ Mass of nucleon

MN = limτ→∞

1

τln

∫d3x GN(~x, τ ; 0) (7.54)

75

FIG. 7.1: Two-flavor, dynamical fermion lattice QCD data for ∆, N and vector meson (ρ) mass

data from UKQCD (open circles) and CP-PACS (filled circles). The solid lines are the continuum

limit, finite volume predictions. The squares (barely discernable from the data) are the predicted

masses on a lattice of the same dimensions as the data at the pion mass.

FIG. 7.2: Extrapolation of lattice QCD magnetic moments for the proton (upper) and neutron

(lower) to chiral limit. The experimentally measured moments are indicated by asterisks.

76

CHAPTER 8. SYMMETRIES OF QCD

• QCD is based on local SU(3)c gauge symmetry

• In addition: global symmetries

8. 1. Nother’s Theorem

LQCD(x) = ψ(x) [ iγµDµ −m ] ψ(x) +

1

4Ga

µν(x)Gµνa (x) (a = 1, · · · , 8) (8.1)

where Dµ = ∂µ − igAµ(x), Aµ(x) = Aµa(x)

λa

2and ψ(x) =

u

d

s

.

• Let LQCD be invariant under a global transformation of the quark fields:

ψ(x) → ψ′(x) = exp [iΓaΘa] ψ(x) = 1 + iΓaΘaψ ± · · · (8.2)

with Γa: generators of U(N) or SU(N), Θa independent of x.

• Define: Nother current

Jaµ(x) = − ∂LQCD

∂(∂µψ)

∂ψ′

∂Θa

= ψ(x)γµΓaψ(x) (8.3)

• If LQCD is invariant under the global transformation, then Nother current Jaµ is con-

served:

∂µJµa (x) = 0 (8.4)

• If current is localized in space, then Nother current has conserved charge.

Qa =

∫d3x J0

a(x) =

∫d3xψ†(x)Γaψ(x)

Qa =dQa

dt= 0

(because

∫

V

d3x ~∇ · ~J a =

∮

∂V

d~f · ~J a = 0

) (8.5)

77

8. 2. Baryon Number and Flavor Currents

a ) Global U(1) symmetry:

ψ(x) → eiθψ(x) ⇒ JµB(x) = ψ(x)γµψ(x) (8.6)

conserved charge: B =

∫d3xψ†(x)ψ(x) ⇔ Baryon number

b ) Isospin current: ψ(x) =

u(x)

d(x)

Isospin doublet (Nf = 2)

- Assume: equal masses mu = md.

- SU(2)f transformation: ψ(x) → ψ′(x) = exp[iτi

2θi

]

-τi

2: SU(2) generators (i = 1, 2, 3) → Pauli matrices

- LQCD with mq ≡ mu = md is invariant under SU(2)f

⇒ conserved current: V µi (x) = ψ(x)γµ τi

2ψ(x)

⇒ conserved isospin charge: Qi =

∫d3x V 0

i (x) =

∫d3x ψ†(x)

τi

2ψ(x)

Qi = 0 ⇔ [ H, Qi ] = 0

c ) Flavor current in SU(3)f : ψ =

u

d

s

(Nf = 3)

- Assume: mu = md = ms

- SU(3)f transformation: ψ(x) → ψ′(x) = exp

[iλj

2θj

](j = 1, · · · , 8)

⇒ conserved current: V µi (x) = ψ(x)γµ λi

2ψ(x)

⇒ conserved charge: Qi =

∫d3x ψ†(x)

λi

2ψ(x)

d ) Symmetry breaking: ms 6= mu,d = m

Lmass = ψ(x)

m 0 0

0 m 0

0 0 ms

ψ(x) ⇒ ∂µVµi ∝ (ms −m)

78

8. 3. QCD with Massless Quarks: Chiral Symmetry

• Start with Nf = 2:

LQCD(x) = ψ(x)iγµDµψ(x)− 1

4Ga

µν(x)Gµνa (x)

︸ ︷︷ ︸L0

QCD

+Lmass (8.7)

where Lmass = ψ(x)

mu 0

0 md

ψ(x)

• QCD in the limit of massless quarks: L0QCD = ψ(x)iγµ∂

µψ(x) + Lquark-gluon + Lglue

• Left- and right-handed quark fields: ψ = ψR + ψL

ψR =1

2(1 + γ5)ψ

ψL =1

2(1− γ5)ψ

(8.8)

where γ5 = γ5 = iγ0γ1γ2γ3 =

0 1

1 0

with 1 =

1 0

0 1

γ5, γµ = 0, γ25 =

1 0

0 1

• Quark field:

ψ(x) =∑

s

∫d3p

(2π)32Ep

[bs(p)us(p)e−ip·x + d†s(p)vs(p)eip·x ]

(8.9)

us(p) = N χs

~σ·~pEp+m

χs

Ep=|~p | , m=0−−−−−−−−→ N

χs

~σ·~p|~p | χs

(8.10)

~σ · ~p|~p | = h = ±1

“right”-

“left”-handed.

79

(1± γ5)us(p) = N

χs

hχs

±

hχs

χs

= N (1± h)χs

±(1± h)χs

(8.11)

⇒ 12(1 + γ5) projects on h = +1 (right handed).

⇒ 12(1− γ5) projects on h = −1 (left handed).

• Massless QCD:

L0QCD = ψL(x)iγµD

µψL(x) + ψR(x)iγµDµψR(x) + Lglue (8.12)

where

ψLiγµDµψL =

1

4ψ†(1− γ5)γ0γµD

µ(1− γ5)ψ

=1

4ψ(1 + γ5)iγµD

µ(1− γµ)ψ

=1

2

[ψiγµD

µψ − ψiγµγ5Dµψ

]

ψRiγµDµψR =

1

4ψ†(1 + γ5)γ0γµD

µ(1 + γ5)ψ

=1

2

[ψiγµD

µψ + ψiγµγ5Dµψ

]

(8.13)

• Global transformation: chiral SU(2)R × SU(2)L symmetry

ψR(x) → exp[iτj

2θj

R

]ψR(x)

ψL(x) → exp[iτk

2θk

L

]ψL(x)

(8.14)

with τi: Pauli matrices (i = 1, 2, 3)

• Mass term breaks this symmetry explicitly.

Lmass = ψ(x)

mu 0

0 md

ψ(x) = ψ mψ

= ψR mψL + ψL mψR

(8.15)

⇒ Quark mass term mixes left- and right-handed quarks.

80

• In the limit mu,d → 0: conserved currents:

JµR,i(x) = ψR(x)γµ τi

2ψR(x)

JµL,i(x) = ψL(x)γµ τi

2ψL(x)

(8.16)

• Convenient to introduce vector and axial vector current:

V µi (x) = Jµ

R,i(x) + JµL,i(x) = ψ(x)γµ τi

2ψ(x)

Aµi (x) = Jµ

R,i(x)− JµL,i(x) = ψ(x)γµγ5

τi

2ψ(x)

(8.17)

( ∂µVµi = 0 , ∂µA

µi = 0 )

• Conserved charge:

QVi (t) =

∫d3x V 0

i (x) =

∫d3x ψ†(x)

τi

2ψ(x) (Vector charge)

QAi (t) =

∫d3x A0

i (x) =

∫d3x ψ†(x)γ5

τi

2ψ(x) (Axial charge)

(8.18)

d

dtQV

i (t) = i[H, QV

i

]= 0 ,

d

dtQA

i (t) = i[H, QA

i

]= 0 (8.19)

• Generalization to 3 flavor (Nf = 3) ⇒ SU(3)R × SU(3)L symmetry

replace τi → λi: Gell-Mann matrices (i = 1, · · · , 8)

• Lie algebra of the vector and axial charges:

[QV

i (t), QVj (t)

]= ifijk QV

k (t)[QV

i (t), QAj (t)

]= ifijk QA

k (t)[QA

i (t), QAj (t)

]= ifijk QV

k (t)

(8.20)

with fijk: structure constant of SU(3).

8. 4. Realizations of Chiral Symmetry

• Wigner-Weyl realization:

Ground state (“vacuum”): QVi |0〉 = 0, QA

i |0〉 = 0

⇒ Total symmetry between positive and negative parity.

81

6

?

0.5

1.0

π

η

ρ, ω

η′

φ

K

K∗

N

∆

Λ, Σ

···

···

··

···

Mass[GeV]

­ ª

® ©

PseudoscalarMesons(Jp = 0−)

­ ª

® ©

“Gap”Λ ∼ 1GeV

• Spectrum of states in Wigner-Weyl realization

⇒ Parity doublets: for each state of positive parity, there must be a state of equal

mass with negative parity. But:

a ) For nucleon with Jp = 12

+, there is no equal mass partner with Jp = 1

2

−.

b ) For pseudoscalar mesons with Jp = 0−, there is no chiral partner with Jp = 0+.

c ) Vector- and Axialvector-mesons:

– Vector mesons: Jp = 1−

– Axial vector mesons: Jp = 1+

• Current correlation function:

ΠµνV (q) = i

∫d4x eiq·x〈0|T [V µ(x)V ν(0)]|0〉

ΠµνA (q) = i

∫d4x eiq·x〈0|T [Aµ(x)Aν(0)]|0〉

(8.21)

ΠµνV,A(q) =

(qµqν − q2gµν

)ΠV,A(q2) (8.22)

• In Wigner-Weyl realization:

QV |0〉 = 0 , QA |0〉 = 0 ⇒ ΠV (q2) ≡ ΠA(q2) (8.23)

• Spectral functions: ηV,A(s) = 4π ImΠV,A(q2 = s)

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• But empirically:

0.0 0.5 1.0 1.5 2.0s @GeV2D

Η

Ρ-meson

a1-meson

ΗA

ΗV

• These observations lead to the Nambu-Goldstone realization of chiral symmetry:

Ground state does not have all the symmetries of Lagrangian density.

QVi |0〉 = 0︸ ︷︷ ︸

Isospin symmetry

, QAi |0〉 6= 0 (8.24)

⇒ Axial symmetry is spontaneously broken.

8. 5. Goldstone’s Theorem

For every spontaneously broken global symmetry, there exists a massless state that

carries the quantum numbers of the corresponding symmetry charge.

QAi |0〉 6= 0 , H |0〉 = E0 |0〉

Define |Φi〉 ≡ QAi |0〉

then: H |Φi〉 = HQAi |0〉 = QA

i H |0〉 = QAi E0 |0〉 = E0 |Φi〉

|Φi〉 energetically degenerate with ground state (vacuum) ⇒ Massless Goldstone Boson.

|Φi〉 are states with spin/parity Jp = 0− “Pseudoscalar”.

For Nf = 2; i = 1, 2, 3; Isospin I = 1 ⇒ Pions (π+, π0, π−)

• Goldstone’s theorem:

In the Nambu-Goldstone realization of (spontaneously broken) chiral symmetry, the

Goldstone bosons are weakly interacting at low energies.

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Proof: Consider a state of n Goldstone bosons |(Φ)n〉 = (QA)n |0〉.

H |(Φ)n〉 = H (Qi · · ·Qk)︸ ︷︷ ︸n-times

|0〉 = (Qi · · ·Qk)H |0〉 = E0 |(Φ)n〉

⇒ Each Goldstone boson has energy-momentum relation ε = |~q |. Since n (massless)

Goldstone bosons are degenerate with vacuum, it follows that

⇒ Goldstone bosons do not interact in the limit |~q | → 0.

• Low energy QCD is realized in the form of an effective field theory of weakly interacting

Goldstone bosons.

(Pions for Nf = 2; Pseudoscalar meson octet (π, K, K, η) for Nf = 3)

8. 6. Spontaneous Symmetry Breaking

• Another standard example of spontaneous symmetry breaking: Ferro-magnet

Spin system: Hamiltonian H = H0 +∑i<j

Gij ~σi · ~σj

Invariant under rotational symmetry in R3 (O(3) symmetry)

• Low temperature: Magnetization has non-zero expectation value

〈 ~M〉 6= 0 , T = 0

preferred direction in space ⇒ O(3) symmetry is spontaneously broken (Nambu-

Goldstone realization).

Order parameter:

TcT

< M>

At high temperature T > Tc: O(3) symmetry restored in Wigner-Weyl realization.

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• Goldstone boson: Magnon “Spin wave”

in QCD: 〈 ~M〉 ↔ Chiral (quark) condensation 〈qq〉

• Chiral condensate 〈qq〉 is the order parameter of spontaneously broken chiral symmetry

in QCD.

8. 7. Chiral Condensate (Quark Condensate)

• “Perturbative” and “non-perturbative” vacuum

• Quark field operator:

ψ(x) =

∫d3p

(2π)32Ep

[bpu(p)e−ip·x + d†pv(p)eip·x ]

= ψ(+)(x) + ψ(−)(x)

(8.25)

ψ+(x) =

∫d3p

(2π)32Ep

[b†pu

†(p)eip·x + dpv†(p)e−ip·x ]

(8.26)

⇒ Perturbative vacuum: bp |0〉 = 0, dp |0〉 = 0

⇒ Non-perturbative vacuum: |Ω〉: bp |Ω〉 6= 0, dp |Ω〉 6= 0

ψ(+)(x) |Ω〉 6= 0 , ψ(−)(x) |Ω〉 6= 0

• Wick’s theorem: T ψ(x)ψ(y) = : ψ(x)ψ(y) :︸ ︷︷ ︸“normal product”

+ 〈0|T ψ(x)ψ(y)|0〉︸ ︷︷ ︸iSF (x, y)

• Definition of normal product:

: bpd†q :≡ −d†qbp etc. (8.27)

In the perturbative vacuum:

〈0| : ψ(x)ψ(y) : |0〉 = 0

and the standard Feynman propagator is SF (x, y) = −i〈0|T ψ(x)ψ(y)|0〉.

In the non-perturbative vacuum:

〈Ω|T ψ(x)ψ(y)|Ω〉︸ ︷︷ ︸iSF (x, y)

= 〈Ω| : ψ(x)ψ(y) : |Ω〉+ iSF (x, y)

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• Definition of quark condensate:

〈ψψ〉 = itr limy→x+

[SF (x, y)− SF (x, y)

]

= −tr limy→x+

〈Ω| : ψ(x)ψ(y) : |Ω〉(8.28)

• For Nf = 2, flavor with ψ =

u

d

〈ψψ〉 = 〈uu〉+ 〈dd〉 ; 〈qq〉 with q = u, d

8. 8. Quark Condensate and Spontaneously Broken Chiral Symmetry

• Spontaneous breaking of chiral symmetry (Nambu-Goldstone realization) implies non-

trivial vacuum characterized by non-vanishing chiral condensate:

QAj |0〉 6= 0 ⇔ 〈ψψ〉 6= 0

• Sketch of proof: introduce Pj(x) = ψ(x)iγ5τj

2ψ(x)

Relation:[QA

j (t), Pk(~x, t)]

= − i

2δjkψ(x)ψ(x) (8.29)

Use: QAi (t) =

∫d3x A0

i (~x, t);

ψα(~x, t), ψ†β(~y, t)

= δαβδ3(~x− ~y )

ψα(~x, t), ψβ(~y, t)

= 0

ψ†α(~x, t), ψβ(~y, t)

= 0

(8.30)

• Take expectation value of (8.29):

〈0|QAj Pk − PkQ

Aj |0〉 = − i

2δjk〈ψψ〉 (8.31)

⇒ If QAj |0〉 6= 0 ⇔ 〈ψψ〉 6= 0

• Chiral condensate 〈ψψ〉 = 〈ψRψL + ψLψR〉:

Order parameter of spontaneously broken chiral symmetry.

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8. 9. Thermodynamics of the Chiral Condensate

(and its realization in Lattice QCD)

• QCD at temperature T , volume V :

〈ψψ〉T,V = −tr limy→x+

∫ DADψDψ ψ(x)ψ(y)e−SE(T,V )

∫ DADψDψ e−SE(T,V )(8.32)

• Euclidean action: SE(T, V ) =

∫ β

0

dτ

∫

V

d3x LQCD

β ≡ 1

T= Nτa, V = L3 = (Na)3

• Result for temperature dependence of 〈ψψ〉

Critical temperature Tc ' 190 MeV ∼ ΛQCD

TcT

È< Ψ Ψ>È

mq=0 mq¹0

– for mq = 0 (chiral limit): 2nd order phase

transition (Nf = 2)

– for mq 6= 0: crossover transition

8. 10. Pion Decay Constant fπ

• Starting point: SU(2)R × SU(2)L chiral symmetry

spontaneously broken ⇒ (π+, π0, π−) pions as Goldstone bosons.

⇒ Introduce |πi(p)〉 quantum state of pion,

Normalization 〈πi(p)|πj(p′)〉 = 2Epδij(2π)3δ3(~p− ~p ′ ) where Ep =

√~p 2 + m2.

〈0|Aµj (x)|πk(p)〉 = iδjk fπ pµe−ip·x (8.33)

87

8. 11. PCAC and the Gell-Mann, Oakes, Renner Relation

• Small quark masses mu,d 6= 0 ⇒ Explicit breaking of chiral symmetry

⇔ Partially Conserved Axial-Vector Current (PCAC)

∂µAjµ(x) = iψ(x)

m,

τj

2

γ5ψ(x) (8.34)

• Consider the case j = 1, τ1 =

0 1

1 0

; ∂µA1

µ = (mu + md)ψiγ5τ1

2ψ

– Combine with (8.29):

[QA

1 (t), ψ(~x, t)iγ5τ1

2ψ(~x, t)

]= − i

2ψ(x)ψ(x)

= − i

2

(uu + dd

) (8.35)

– Take expectation value:

〈0| [ QA1 , ∂µA1

µ

] |0〉 = − i

2(mu + md) 〈ψψ〉 (8.36)

• Assume that pion, as Goldstone boson, dominates spectrum of pseudoscalar isovector

excitations

1 =3∑

j=1

∫d3p

(2π)32Ep

|πj(p)〉〈πj(p)| (8.37)

〈0|QAj (t = 0)|πk(p)〉 = iδjk fπ Ep (2π)3δ3(~p )

and 〈0|∂µAjµ(x)|πk(p)〉 = δjk fπ m2

π e−ip·x ⇒ Gell-Mann, Oakes, Renner relation:

m2πf 2

π = −1

2(mu + md) 〈ψψ〉 (8.38)

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For mu + md ' 12 MeV (at renormalization scale µ ∼ 1 GeV)

mπ = 139.6 MeV for π±

fπ = 92.4 MeV from π− → µ− + νµ

〈ψψ〉 ' −(0.3 GeV)3

〈uu〉 ' 〈dd〉 ' −(0.24 GeV)3 ' −1.8 fm−3

⇒ Compare magnitude to baryon number density in center of atomic nucleus:

ρBaryon =Z + N

V= 0.16 fm−3

89