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A little more moderation would be good.

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Donald Trump (1946)

28

Nonabelian Gauge Theory of Strong Interactions

Elementary particles which interact strongly with each other are called hadrons (re-call the classification in Section 24.1). According to their statistics, one distinguishesbaryons and mesons. The most prominent among these are the spin-1/2 particlesin nuclear matter, protons and neutrons. The forces between them arise, to lowestapproximation, from the exchange of a spin-0 meson called pion, and further mesons(middle-heavy particles).

Hadrons exhibit rich mass spectra, as we have seen in Chapters 24 and 25.These spectra were explained to a good approximation by quark models [1]. Just asnuclei are composed of protons and neutrons, baryons are composed of three quarks,mesons of quarks-antiquarks pairs [2].

28.1 Local Color Symmetry

In spite of its success, the initial quark model exhibited several fundamental incon-sistencies. Most importantly, it was not compatible with the spin-statistics theoremderived in Section 7.10. For any reasonable potential between quarks, the groundstate orbital wave function should always be without zeros implying vanishing rel-ative angular momenta for each pair of quarks. Any higher angular momentumwould have at least one zero in the wave function which would increase the gradientand thus the kinetic energy of the Schrodinger field via the centrifugal barrier. Theorbital wave function of the ground state of three quarks, the proton, must thereforebe symmetric under the exchange of two quarks. On the other hand, the SU(6) wavefunction involving internal SU(3) and spin has the Yang tableau . But thisimplies that the nucleon wave function is completely symmetric under exchange ofall positions, all SU(3), and all spin variables. This seems to contradict the factthat quarks have spin 1/2 so that, by the spin-statistics theorem, they should befermions and therefore have a completely antisymmetric wave function.

1486

28.1 Local Color Symmetry 1487

To remedy this contradiction, Han and Nambu suggested that quarks should bea triplet under a further SU(3) group of transformations now called color SU(3) [3].Color appears as a further label to the quark fields:

q(x) =

u(x)d(x)s(x)c(x)t(x)b(x)

, (28.1)

which may thus be written as

qα(x) =

uα(x)dα(x)sα(x)cα(x)tα(x)bα(x)

, (28.2)

with a label α = 1, 2, 3 specifying the three colors. By postulating that all hadronstates are completely antisymmetric in the color indices, so that they are color singletstates, the contradiction disappears and the spin-statistic relation is again valid.

After this somewhat artificial postulate the question arose how nature managesto enforce the color antisymmetry, i.e., how it prevents color non-singlet states tobe excited. The answer suggested by Fritzsch and Gell-Mann was that color SU(3)was a local gauge symmetry of hadronic physics. The action had to be invariantunder arbitrary nonabelian SU(3)-transformations of the quark fields

q(x) → e−iαa(x)λa/2q(x), (28.3)

where the matrices λa now act on the three color labels of the quark field q(x) inSU(3). The tripling of the quarks saved not only the validity of the spin-statisticsrelation for quarks. It also led to the correct rate of the particle decay

π0 → γγ

which is observed experimentally at a rate Γ = 8.4 × 10−17s. In addition, it gavethe correct total interaction cross section observed in e+ e− collisions, which requirethat each quark occurs in three color versions.

Once it was postulated that the quark action is invariant under local symmetrytransformations, it became necessary to introduce a gauge field to maintain theinvariance of the gradient term in the action. This led to the quark Lagrangian

L(x) = q(x)i/D q(x)−Mq(x)q(x) (28.4)

1488 28 Nonabelian Gauge Theory of Strong Interactions

where q(x) carries flavor indices distinguishing the quarks u, d, c, s, . . . and threecolor indices. The gradient term contains the covariant derivative

/D = γµDµ = γµ (∂µ + igAµ) , (28.5)

where Aµ is a 3 × 3 matrix in color space to guarantee that Dµ transforms underlocal color SU(3) transformations in the same way as the field ψ(x) itself:

Dµq(x) → U(x)Dµq(x) = e−iαa(x)λa/2Dµq(x). (28.6)

The covariance property is a bit harder to show than in electromagnetism. Thederivative of the field goes over into:

∂µq(x) → ∂µU(x)q(x) = U(x)[U−1(x)∂µU(x)q(x)]

= U(x)∂µq(x) + U(x)[U−1(x)∂µU(x)]q(x). (28.7)

In order to remove the second term, the gauge field Aµ has to transform like

Aµ(x) → U(x)Aµ(x)U−1(x)− 1

ig[∂µU(x)]U

−1(x)

= U(x)Aµ(x)U−1(x) +

1

igU(x)∂µU

−1(x). (28.8)

Then

[∂µ + igAµ(x)] q(x) → U(x)[

∂µ + iqAµ(x) + U−1(x)∂µU(x)]

= U(x) [∂µ + igAµ(x)] q(x), (28.9)

which is the desired covariant transformation law.

28.2 Gluon Action

After the introduction of such a gauge field, an interaction has to be found describingthe dynamics of this gauge field itself. If color is never observed, the action of thegauge field should be locally SU(3)-invariant as well. The only Lagrangian whichhas this property and contains, at most, first derivatives in Aµ is given by

Lgluon = −1

2tr (FµνF

µν) (28.10)

where

Fµν = ∂µAµ − ∂νAµ + ig[Aµ, Aν ] (28.11)

is the field tensor, which is the nonabelian version of the covariant curl (4.807) of thevector potential Aµ. Just as Aµ, the field tensor is a 3× 3 matrix in color space. Itis easy to verify that the matrix Fµν transforms under SU(3) covariantly as follows

Fµν(x) → U(x)Fµν(x)U(x). (28.12)

28.3 Quantization in the Coulomb Gauge 1489

The theory is described by the Lagrangian

L(x) = q(x)i [∂µ + igAµ(x)] q(x)−1

2tr [Fµν(x)F

µν(x)] . (28.13)

This is a complete nonabelian analog of the gauge field Lagrangian (12.1)–(12.3)in quantum electrodynamics. For this reason it has been given the similar namequantum chromodynamics (QCD).

Instead of 3× 3 matrices Aµ, Fµν , one can also use an octet of vector and tensorfields defined by

Aµ =λa

2Aµ

a, Fµν =λa

2F aµν . (28.14)

Thus we can write the Lagrangian of QCD as

L = q(x)iγµ[

∂µ + igλa

2Aa

µ(x)

]

q(x)− 1

4F aµν(x)F

aµν(x), (28.15)

where

F aµν(x) = ∂µA

aν(x)− ∂νA

aµ(x)− gfabcAb

µ(x)Acν(x). (28.16)

In terms of the eight components Aaµ(x), the Lagrangian (28.10) reads

LFgluon = −1

4F aµνF

aµν . (28.17)

There exists an equivalent way of expressing it in terms of two independent fieldsAa

µ(x) and Faµν , often used by Schwinger:

Lgluon =1

4F aµνF

aµν − 1

2F aµν

(

∂µAaν − ∂νA

aµ

)

. (28.18)

The spacetime integral over this is the canonical action corresponding to the me-chanical action of a free particle [recall (1.14)]:

A =∫

dt

(

−p2

2+ pq

)

. (28.19)

In the past, evidence has accumulated that the gauge field Aaµ is indeed capable

of describing the forces which bind together the quarks inside hadrons. The fieldquanta carried by Aa

µ are called gluons.

28.3 Quantization in the Coulomb Gauge

Let us quantize the theory for the simplest nonabelian symmetry SU(2), where thegluons Aa

µ (a = 1, 2, 3) are vectors in color space. We shall follow the description of


the theory in Ref. [4, 5]. In SU(2)-symmetry, the structure constants fabc reduce toǫabc and the covariant curl (28.16) can be written in vector notation as

Fµν(x) = ∂µAν(x)− ∂νAµ(x)− gAµ(x)×Aν(x), (28.20)

and the Lagrangian (28.18) becomes

LFgluon =

1

4Fµν · Fµν − 1

2Fµν · (∂µAν − ∂νAµ − gAµ ×Aν) . (28.21)

This is invariant under the nonabelian gauge transformations (28.8) whose infinitesi-mal form is a rotation in isospace by a small angle ǫ around the direction n. If u ≡ ǫndenotes the associated infinitesimal rotation vector, the gauge transformations takethe form

Aµ(x) → Auµ(x) = Aµ(x) + u(x)×Aµ(x) +

1

g∂µu(x)

Fµν → Fuµν = Fµν + u× Fµν . (28.22)

The first Euler-Lagrange equation

∂LFgluon

∂F aµν

= 0 (28.23)

reproduces the relation (28.20) between the curl and the auxiliary tensor field F aµν .

The second Euler-Lagrange equation

∂µ∂LF

gluon

∂(∂µAaν)

=∂L∂Aa

ν

(28.24)

yields the field equation

DµFµν ≡ ∂µFµν − gAµ × Fµν = 0. (28.25)

The combination of Eqs. (28.20) and (28.25) coincides with the field equation thatwould be obtained from the second-order formulation with the Lagrangian Lgluon ofEq. (28.17) expressed directly in terms of ∂µAν and Aµ via Eq. (28.11).

In the present first-order formulation, one is given an initial configuration offields Ai and F0i at some time t. From these one determines the fields at any latertime by solving the first-order equations of motion

∂0Ai = F0i + (∇i − gAi×)A0, (28.26)

∂0F0i = (∂j − gAj×)Fji + gA0 × F0i. (28.27)

As in the abelian case of Maxwell electromagnetism [recall Eq. (7.337)], the field A0

is not a dynamical variable since the canonical momenta

∂L/∂ (∂0Aµ) = −F0µ = Fµ0 (28.28)


vanish for µ = 0. There are only three independent field momenta. The Eulerequation for A0 is not an equation of motion but a constraint equation analogousto the Coulomb law (7.339):

(∇k − gAk×)Fk0 = 0. (28.29)

As in the Abelian case, it tells us that not all of the conjugate momenta F0k areindependent.

Note that in the first-order formulation, the field equation obtained from varyingFij ,

Fij = ∂iAj − ∂jAi − gAi ×Aj , (28.30)

is also a constraint equation which allows us to calculate Fij for given Ai at thesame time. In addition we see from Eq. (28.30) that not all field components Ak

can be treated as independent.In order to remove the redundancy we choose the Coulomb gauge

∇kAk = 0. (28.31)

This is always possible because of the gauge invariance of the second kind of theLagrangian density [recall (4.255)]. The gauge (28.31) implies that the vector fieldAi must be transverse. Therefore, the longitudinal components FL

0i of the canonicalmomentum F0i are not independent, but they depend on the other degrees of free-dom through the constraint (28.29). The splitting of F0i into longitudinal FL

0i andtransverse parts FT

0i is defined by the equations

F0i = FT0i + FL

0i, ∇iF0i = ∇iFL0i, ǫijk∇jF

L0k = 0. (28.32)

Our task is now to express A0 and FL0i in terms of the independent fields and

construct the Hamiltonian. As usual we identify the transverse components FT0i as

the electric field strengths Ei. By rewriting the longitudinal components FL0i as a

gradient of a scalar isovector f ,

FL0i = −∇if , (28.33)

we have

∇iF0i = −∇2f . (28.34)

The independent variables are the transverse vector fields Ai and its canonicallyconjugate field momentum Ei.

Inserting (28.33) into the constraint (28.29), we obtain the differential equationfor f :

(

∇2 + gAk ×∇k

)

f = gAi ×Ei. (28.35)


This equation can be solved formally by introducing a Green function Dab(x,x′;A),defined as a solution of the inhomogeneous differential equation

(

∇2δab + gǫacbAc

k∇k

)

Dbd(x,x′;A) = δadδ(3)(x− x′). (28.36)

With the help of this Green function, Eq. (28.35) can be solved for the componentsof f by the integral

fa(x, t) = g∫

d3x′ Dab(x,x′;A)ǫbcdAck(x

′, t)Edk(x

′, t). (28.37)

This may be abbreviated as

f = gD[ATk ,E

Tk ]. (28.38)

The solution Dbd(x,x′;A) of Eq. (28.36) cannot be found explicitly, but only viaa perturbation expansion. The lowest approximation coincides with δab times theGreen function −1/∇2 of electrostatics. A first iteration yields additional terms upto the order g:

Dab(x,x′;A)=δab

4π|x− x′| + g∫

d3x′′1

4π|x− x′′|ǫacbAc

k∇k1

4π|x′ − x′′| + . . . . (28.39)

This first-order approximation is easily verified by inserting it into Eq. (28.36).A similar equation for A0 is obtained by taking the divergence of Eq. (28.26)

and using (28.31) and (28.32) to find

(

∇2 + gAi ×∇i

)

A0 = ∇2f . (28.40)

This can be solved using once more Dab(x,x′;A), since the operator in brackets isthe same as in Eq. (28.35):

Aa0(x, t) =

∫

d3x′ Dab(x,x′;A)∇2f b(x′, t), (28.41)

or in short form:

A0 = D∇2f . (28.42)

We can now construct the Hamiltonian densityH by the Legendre transformationof the Lagrangian density (28.18):

H = Ei ·∂Ai

∂t− LF

gluon. (28.43)

From (28.26), (28.32), (28.33), and (28.42), we find that

∂tAi = Ei −[

∇i − (∇i + gAi×)D ·∇2]

f . (28.44)


Because of (28.36), the operator in brackets acting on f is explicitly transverse.Combining further (28.44) and (28.35) or (28.37), we obtain that

∫

d3xEi · ∂tAi =∫

d3x[

E2i + g(Ei ×Ai) · D ·∇2f

]

=∫

d3x[

E2i − f ·∇2f

]

=∫

d3x[

E2i + (∇if)

2]

. (28.45)

If we now rewrite the Lagrangian density as

LFgluon =

1

4Fµν · Fµν − 1

2Fµν · (∂µAµ + gAµ ×Aν)

=1

2(F2

0k −B2i ) =

1

2(Ek −∇kf)

2 − 1

2B2

k, (28.46)

and identify the magnetic field strength as

Bi =1

2ǫijkFjk, (28.47)

we find the Hamiltonian

H =1

2

∫

d3x[

E2i +B2

i + (∇if)2]

. (28.48)

The last term is like the familiar instantaneous Coulomb interaction discussed inQED in Section 12.3 [see the last term in Eq. (12.81)].

We now express the generating functional WC [j] in the Coulomb gauge in termsof the independent coordinates and momenta, Ai and Ei, as the functional integral

WC [j] =∫

DETi DAT

i exp{

i∫

d4x[

Ek · Ak −1

2E2

k −1

2B2

k −Ak · jk]}

(28.49)

where the superscript T indicates the transverse spatial components of the field, andf is the functional (28.38) depending on ET

i and ATi as specified in Eq. (28.37). Note

that the spatial source term at the end has a negative sign, so that the covariantfunctional to be derived below in Eq. (28.63) will contain the four-dimensional scalarcoupling of the vector field to the source with a positive sign: Aµ · jµ.

The transverse field ETi is somewhat awkward to handle. Therefore we introduce

an initially dummy variable EL by

∫

DETi =

∫

DETi DELδ(3)[EL], (28.50)

where δ(3)[EL] ≡ ∏

x δ(3)(EL(x)) is the δ-functional in four-dimensional spacetime.

Then we define three independent transverse and longitudinal components Ei by

ETi =

(

δij −∇i1

∇2∇j

)

Ei, EL ≡ ∇i1

∇2∇jEj, (28.51)


in terms of which we can rewrite the measure (28.50) as∫

DETi =

∫

DEi Jδ[∇jEj], (28.52)

where J is a field-independent, and thus irrelevant, Jacobian of the transformationfrom the three Ei to ET

i ,EL, and

DEi ≡∏

x

3∏

i=1

3∏

a=1

dEai (x).

The same decomposition is applied to the gauge fields ATi , so that the generating

functional has the functional integral representation

WC [j] = const×∫

DEiDAiδ[∇kEk]δ[∇kAk]

× exp{

i∫

d4x[

Ek · Ak −1

2E2

k −1

2B2

k −1

2(∇kf)

2 −Ak · jk]}

. (28.53)

From this expression we can derive the Feynman diagram rules in the Coulombgauge. As in the abelian case, these are not covariant, and the Lorentz covariance ofthe emerging S-matrix is not obvious. The Coulomb gauge is only useful to derivethe generating functional from the canonical formalism.

In order to find the Feynman rules for the covariant and gauge invariant S-matrix,one should start out with a covariant-looking form of the generating functionalinstead of (28.53). There, f is a function of E and A given by Eq. (28.37). It ispossible to introduce f as a variable of integration and fix its value to satisfy (28.37)by a δ-functional

∫

Df δ [f − gD ·Ak × Ek] = 1. (28.54)

If the generating functional (28.53) is multiplied by this expression, it obviouslyremains unchanged. Let DetM be the Jacobian of the transformation from f to(∇2 + gAi ×∇i)f . The factor in front of f is a functional matrix in isospin space:

Mab(x, x′) =(

∇2δab + gǫabcAc

i(y)∇i

)

δ(4)(x− x′)

= ∇2[

δabδ(3)(x− x′) + gǫabcG(x, x′)Aci(x

′)∇i

]

δ(x0 − x′0), (28.55)

where G(x, y) is the Green function satifying ∇2G(x, y) = δ(3)(x − x′). With the

help of Eq. (28.35), we can now rewrite (28.54) as∫

Df δ [f − gD ·Ak × Ek] = DetM∫

Df δ[(∇2 + gAi ×∇i)f − gAi ×Ei], (28.56)

and the generating functional (28.53) becomes

WC [j] = DetM∫

DAiDEiDf δ[∇iAi] δ[∇iEi]δ[(∇2 + gAi ×∇i)f − gAi × Ei]

× exp[

i∫ {

Ek ·Ak −1

2

[

f2k +B2k + (∇kf)

]2 − ji ·Ai

}

d4x]

. (28.57)


Next we change variables from Ei to

F0i = Ei −∇if , (28.58)

using (28.32) and (28.33). Then we rewrite the measure of integration in (28.57) as

DEiDf δ[∇iEi] δ[(∇2 + gAi ×∇i)f − gAi ×Ei]

= DF0i Df δ[∇iF0i +∇2f ] δ[∇2f − gAi × F0i]

= DF0i Df δ[∇iF0i + gAi × F0i] δ[∇2f − gAi × F0i]. (28.59)

Now we perform the integration over Df using the last δ-function in (28.59). TheJacobian is just Det∇2, i.e., an irrelevant infinite constant which shall be absorbedinto the definition of M. Thus we obtain

WC [j] = DetM∫

DAiDF0i δ[∇iAi] δ[∇iF0i + gAi × F0i] (28.60)

× exp{

i∫

d4x[

F0i · ∂0Ai −1

2F2

0i −1

4(∇iAj−∇jAj+gAi ×Aj)

2 − ji ·Ai

)}

.

The exponent has been found by setting in the exponent of (28.57)

E2k + (∇kf)

2 = (Ek −∇kf)2 = F2

0i,

and omitting the mixed term since it vanishes upon integration over x, due to thetransversality of Ek.

Next we express the last factor in the measure of (28.60) as a functional integralover a dummy field variable A0:

δ[∇iF0i + gAi × F0i] =∏

x

∫ dA0

2πexp {iA0 · (∇0F0i − gAi × F0i)}

=const.×∫

DA0 exp{

i∫

d4xF0i(gA0 ×Ai −∇iA0)}

.(28.61)

Finally, we write the term 14(∇iAj −∇jAi + gAi ×Aj)

2 in the exponent of (28.60)as

∫

DFij exp{

i[

1

4Fij · Fij −

1

2Fij · (∇iAj −∇jAi + gAi ×Aj)

]}

, (28.62)

which is a standard Gaussian integral. Inserting (28.62) and (28.61) into (28.60),we obtain

WC [j] = DetM∫

DAµDFµνδ[∇iAi]

× exp[

i∫

d4x{

−1

2F0i · F0i +

1

4Fij · Fij

−1

2Fij · (∇iAj −∇jAi + gAi ×Aj) + F0i(∂0Ai −∇iA0 + gA0 ×Ai)

]]

= DetM∫

DAµDFµνδ[∇iAi] exp{

i∫

d4x[LFgluon + jµ ·Aµ]

}

. (28.63)


Were it not for the factor DetM, this would directly define covariant Feynmanrules. In order to calculate the effect of thei factor, it is useful to reexpress it interms of an effective Lagrangian density. Recalling the explicit form of the functionalmatrix (28.55), we factorize it as

DetM = Det∇2 ·Det [ 1 + M ], (28.64)

where

M = gǫabcG(x, y)Aci(y) · ∇iδ(x0 − y0), 1 = δabδ4(x− y). (28.65)

The determinant Det∇2 is again an irrelevant infinite constant. The second factoris expanded as

Det ( 1+M) = expTr log( 1 + M) (28.66)

= exp∞∑

n=0

(−1)n−1

n

∫

d4x1 . . . d4xn tr

[

M(x1, x2)M(x2, x3) . . . M(xn, x1)]

.

The trace symbol tr runs only over isospin indices. Inserting (28.65), this becomes

Det ( 1+M) = exp

[

δ(0)

{

−∞∑

n=0

gn

n

∫

d3x1 . . . d3xn

∫

dt tr [T ·Ai1(x1, t)∇i1G(x1,x2)

× T ·Ai2(x2, t)∇i2G(x2,x3) · · · T ·Ain(xn, t)∇inG(xn,x1)]}]

,(28.67)

where (T a)bc = ǫabc and Tr includes the trace over isospin indices.Since (28.67) is a power series in the exponent, it is an effective correction in

each order to the Feynman rules obtained from LFgluon alone.

28.4 General Functional Quantization of Gauge Fields

Equation (28.63) can be further simplified. We can perform the functional integra-tion over F a

µν and obtain

W [j] = DetM∫

DAµδ [∇iAi(x)] exp{

i∫

d4x[Lgluon(x) + jµ(x) ·Aµ(x)]}

, (28.68)

where L(x) is the second-order Lagrangian density (28.17). Except for the fac-tor DetM δ [∇iAi(x)], this expression looks the same as for a standard scalar fieldtheory:

W [j] ∼∫

Dφ exp{

i∫

d4x[L(x) + j(x)φ(x)]}

. (28.69)

For the abelian gauge theory QED we have shown in Section 14.16 how to derivesuch a factor following an intuitive argument due to Faddeev and Popov. Here wemay do the same for the nonabelian case. Recall how the argument went in the

28.4 General Functional Quantization of Gauge Fields 1497

abelian case. There we expressed the quadratic part of the action in the bilocalform

A0 = −∫

d4x1

4(∂µAν − ∂νAµ)

2 =∫

d4xd4x′1

2Aµ(x)D

µν(x, x′)Aν(y), (28.70)

with a functional matrix

Dµν(x, x′) = −(

∂2gµν − ∂µ∂ν)

δ4(x− x′).

This cannot be inverted, since it involves only the transverse components of Aµ,while ignoring the longitudinal components. Hence the Euclidean version of thefunctional integral of Eq. (28.68) contains no Gaussian exponential involving thelongitudinal components of Aµ, and therefore the functional integral over thesediverges. In the abelian case this was a consequence of the invariance of the exponentin (14.347) under the gauge transformation (14.345). Here it is a consequence of theinvariance of the exponent of (28.68) under the gauge transformation

Aµ → Agµ . (28.71)

The right-hand side emerges as a result of applying the element g of the gauge groupG to the field Aµ:

Agµ · L = U(g)

[

Aµ · L+1

igU−1(g)∂µU(g)

]

U−1(g). (28.72)

The exponent in the functional integral (28.72) is constant on the orbits of the gaugegroup, which are formed by all Ag

µ for fixed Aµ, whiled dg runs over the entire groupG. This causes a divergence of the functional integral for the generating functionalW [j]. The amplitude W [j = 0] is therefore proportional to the “volume” of orbits∏

x dg(x), and this factor should be extracted before defining W [j = 0]. Thus thefunctional integral should not be performed over all fluctuations of the gauge fields,but only over the different orbits of Aµ defined by the symmetry transformations ofthe gauge group.

To implement this idea, we choose a “hypersurface” in the manifold of all fieldswhich intersects each orbit only once. Let

fa(Aµ) = 0, a = 1, 2, . . .N (28.73)

describe such hypersurface, where N is the dimension of the group. We shall assumethat the equation

fa(Agµ) = 0

has a unique solution g for any given field Aµ. We are going to integrate over alldifferent hypersurfaces of this kind, instead of integrating over the manifold of allfields. The conditions fa(Aµ) = 0 define a particular gauge, the Coulomb gaugefa(Aµ) = ∇iA

ai being just a particular example.


Before proceeding further, let us recall briefly some simple facts about grouprepresentations. For any two group elements g, g′ ∈ G, the product gg′ is also ∈ G,and their representations satisfy the same multiplication law

U(g)U(g′) = U(gg′).

The invariant Hurwitz measure over the group G is invariant under this operation,so that

dg′ = d(gg′). (28.74)

If we parametrize U(g) in the neighborhood of the identity as

U(g) = 1 + iu · L +O(u2),

then in the neighborhood of the identity we may choose

dg =∏

a

dua, g ∼ 1. (28.75)

Let us define the functional ∆f [Aµ] by

∆f [Aµ]∫

Dg∏

a

δ[fa(Agµ(x))] = 1, (28.76)

where∫

Dg ≡∏

x

[∫

dg(x)]

. (28.77)

Without the gauge invariance, the vacuum-to-vacuum amplitude would be given by

∫

DAµ exp{

i∫

d4xLgluon(x)}

. (28.78)

We now insert the left-hand side of Eq. (28.76) into the functional integral (28.78)without changing the result:

∫

DgDAµ∆f [Aµ]∏

a

δ[fa(Agµ(x))] exp

{

i∫

d4xLgluon(x)}

. (28.79)

Now we perform a gauge transformation Aµ(x) → Agµ(x) defined by (28.72) on

Aµ(x). Under this, the action in (28.78) is invariant. We may also convince ourselvesthat the functional ∆f [Aµ] defined by (28.76) is gauge invariant:

∆−1f [Ag

µ] =∫

∏

x

dg′(x)∏

x,a

δ[fa(Agg′

µ (x))]

=∫

∏

x

d(g(x)g′(x))∏

x,a

δ[fa(Agg′

µ (x))]

=∫

∏

x

dg′′(x)∏

x,a

δ[fa(Ag′′

µ (x))],


so that indeed

∆f [Agµ] = ∆f [Aµ]. (28.80)

Hence we may write Eq. (28.79) also as∫

Dg∫

DAµ∆f [Aµ]∏

x,a

δ[fa(Aµ)] exp{

i∫

d4xLgluon(x)}

,

and we find that the integrand of the group integration is independent of g(x). Thiswas the observation of Faddeev and Popov, who saw that the functional integral

∫ Dgis simply an infinite factor independent of the fields. It can therefore be droppedfrom the amplitude, so that the generating functional W [j] may be defined as

Wf [j] =∫

DAµ∆f [Aµ]∏

a

δ[fa(A)] exp{

i∫

d4x[L(x) + jµ(x) ·Aµ(x)]}

. (28.81)

Faddeev and Popov also gave the canonical derivation of Eq. (28.63) as discussed inthe preceeding section.

Before demonstrating the equivalence of Eqs. (28.68) and (28.81), we shallcompute ∆f [Aµ]. Since the factor ∆f [Aµ] is multiplied by

∏

a δ[fa(Aµ(x))] in

Eq. (28.81), it suffices to compute ∆f [Aµ] only for vector fields Aµ which satisfyEq. (28.73). Let us define the functional matrix Mf by

fa(Agµ(x)) = fa(Aµ(x)) +

∫

d4y∑

b

[Mf (x, y)]abub(y) +O(u2). (28.82)

Then we find from Eq. (28.76) that

∆−1f [Af ] =

∫

∏

x,a

{

dua(x)δ[fa(Ag

µ(x))]}

=∫

∏

x,a

{dua(x)δ[Mfu]} .

The integral receives a contribution only from A-fields satisfying fa(Aµ) = 0, sothat

∆f [Aµ] = DetMf = exp {Tr logMf} . (28.83)

The hypersurface equation fa = 0 is just the gauge condition, and for theCoulomb gauge adopted in the preceeding section, we had

fa(Aµ) = ∇iAai = 0,

which becomes, after an infinitesimal gauge transformation,

fa(Agµ) = ∇iA

ai +

1

g

(

∇2δab − gǫabcAc

i∇i

)

ub(x) +O(u2). (28.84)

From this we identify

[Mf (x, x′)]ab ∼

1

g∇

2(

δab − gabc1

∇2A

ci∇i

)

δ4(x− x′) ∼ [M(x, x′)]ab, (28.85)


which shows that Eq. (28.68) is indeed a special case of Eq. (28.81) for fa = ∇iAai .

Being in possession of Eq. (28.81), we are free to use many different gauges otherthan the Coulomb gauge. If we choose, for example, the manifestly covariant Landaugauge condition ∂µAµ = 0, then Eq. (28.82) takes the form

∂µAgµ(x) = ∂µAµ(x) +

1

g[∂2u+ g∂µ(Aµ × u)] +O(u2), (28.86)

so that Mf is given by

[ML(x, x′)]ab =

1

g(∂2δab − gǫabcA

cµ∂

µ)δ4(x− x′) (28.87)

when Aµ is restricted to ∂µAµ = 0. Removing the trivial factor (1/g)∂2 fromML(x, x

′), we have

∆L ≡ DetML ∼ exp{

Tr ln(1 + ML)}

, (28.88)

where

MabL (x, x′) = gǫabc

∫

DF (x− z)Acµ(z)

∂

∂zµδ4(z − y)d4z, (28.89)

and DF (x−z) is the usual Feynman propagator satisfying −∂2DF (x−y) = δ4(x−y).More explicitly we can write

∆L = exp

{

−∞∑

n=0

(−g)nn

∫

d4x1 . . . d4xntr

[

∂λDF (x1 − x2)T ·Aµ(x2)∂µDF (x2 − x3)

× . . .DF (xn − x1)T ·Aλ(x1)]} . (28.90)

The need to have the extra factor ∆f [Aµ]∏

a δ[fa(Aµ(x)] was first noted by Feynman.

We can write Eq. (28.81) as

Wf [j] =∫

DAµ

∏

a

δ[fa(Aµ(x))] exp{

i[

Seff +∫

d4x jµ(x) ·Aµ(x)]}

, (28.91)

whereAeff =

∫

d4xLgluon(x)− iTr lnML. (28.92)

At this point it is useful to observe that the additional term −iTr lnML in theeffective action may be thought of as arising from loops generated by a fictitiousisotriplet of complex scalar fields c obeying Fermi statistics, whose presence andinteractions can be described by the following action

Ac = −∫

d4x[

∂µc†(x) ·Aµ(x)× c(x)]

∼∫

d4xd4x′∑

a,b

c†a(x) [ML(x.x′)]ab cb(x

′).

(28.93)


With this, Eq. (28.91) may be written as

WL[j] =∫

DAµ

∏

a

δ [fa(Aµ(x))]∫

Dc†Dc exp{

i[

A+Ac +∫

d4x jµ ·Aµ(x)]}

.

(28.94)

Since c and c† appear quadratically in the action (28.93), the path integral∫ Dc†Dc

can trivially be done with the result∫

Dc†Dc exp(iAc) ∼ DetML = exp {Tr lnML} , (28.95)

and we can expand

exp {Tr lnML} ∼ exp[

Tr ln(1 + ML)]

= exp{

TrML − 1

2TrM2

L + . . .− (−)n1

nTrMn

L + . . .}

. (28.96)

The terms in the exponent may arise from loop diagrams of the Fermi fields c.Let us now specify the Feynman rules resulting from WL[j] of Eq. (28.94), fol-

lowing from the general rules desribed in Chapter 14. The gauge boson propagatoris determined from the free-field functional

W 0L[j] =

∫

DAµδ[∂µAµ]e

i∫

d4x{− 1

4(∂µAν−∂νAµ)2 + jµ(x)·Aµ(x)}. (28.97)

A convenient way of evaluating Eq. (28.97) is to express the δ-functional as

δ[∂µAµ(x)] ∼ limα→0

exp{

i

2α

∫

d4x[∂µAµ(x)]2}

, (28.98)

where we have discarded an irrelevant infinite constant∏

x

√2πα. Then we arrive

at the generating functional of free vector boson Green functions:

W 0L[j] = lim

α→0

∫

DAµ exp{

− i

2

∫

d4xAµ(x) ·[

−∂2gµν + ∂µ∂ν(

1− 1

α

)]

Aν(x)

+ i∫

d4x jµ(x) ·Aµ(x)}

= limα→0

exp[

− i

2

∫

d4xd4x′ jµ(x)DµνF (x− x′;α) jν(x

′)]

, (28.99)

where DµνF (x− x′;α) is the free vector boson propagator

DµνF (x− x′;α)=

∫ d4k

(2π)4exp {ik · (x− x′)} 1

k2 + iǫ

[

−gµν +kµkνk2

(1− α)

]

. (28.100)

In the limit α→ 0, this becomes

DµνF (x− x′) =

∫

d4k

(2π)4exp {ik · (x− x′)} 1

k2 + iǫ

(

−gµν +kµkνk2

)

, (28.101)

which is transverse in spacetime. The rest of the Feynman rules can be derived asusual. They are recorded in Figs. 28.1 and 28.2. In addition, the following rule mustbe kept in mind: the ghost-ghost-vector vertex may carry a “dot” which indicatesthat a ghost line is differentiated. Note that a ghost line cannot be dotted at bothends and that a ghost loop carries an extra minus sign.


Figure 28.1 Propagators in the Yang-Mills theory. Wavy lines are vector mesons. Dashed

lines are scalar ghosts.

Figure 28.2 Vertices in the Yang-Mills theory. Note that the ghost with index c in the

last diagram is pictured by a dotted line.

28.5 Equivalence of Landau and Coulomb Gauges

Formally, the S-matrix computed in the Landau gauge is the same as that com-puted in the Coulomb gauge [5]. An element of the unrenormalized S-matrix isobtained from the corresponding Green functions by removing single particle propa-gators corresponding to external lines, taking the Fourier transform of the resulting“amputated” Green function, and placing external momenta on the mass shell. Thedemonstration to be presented is basically correct, except that the S-matrix of agauge theory is plagued by infrared divergences and may not even be defined. Infact, this may be the reason why massless Yang-Mills particles are not seen in na-ture. The point of presenting this demonstration is at this point purely pedagogical.The technique will be useful in the discussion of spontaneously broken versions ofgauge theories.

Let us first establish a relation between WC [j] and WL[j]. From Eq. (28.68) weobtain for the first:

WC [j] =∫

DAµ∆C [Aµ]δ[∇iAi] exp{

iA[Aµ] + i∫

d4x jµ ·Aµ

}

, (28.102)

28.5 Equivalence of Landau and Coulomb Gauges 1503

where ∆C = DetM satisfies

∆C [Aµ]∫

Dgδ[∂µAgµ(x)] = 1. (28.103)

Inserting the left-hand side of Eq. (28.103) into the integrand of the functionalintegration in Eq. (28.102), we write

WC [j] =∫

Dg∫

DAµ∆C [Aµ]∆C [Aµ]δ[∇iAi]

× δ[∂µAgµ] exp

{

iA[Aµ] + i∫

d4xjµ ·Aµ

}

.

Now we perform a gauge transformation of the integration variables Aµ(x):

Aµ(x) → Agµ(x).

Recalling the gauge invariance of the action A, the functional ∆f , and the metricDAµ(x), we find that

WC [j] =∫

DAµ∆C [Aµ]δ[∂µAµ] exp(iA[Aµ]) (28.104)

×∆C [Aµ]∫

∏

x

dg(x)∏

x

δ(∇iAg−1

i ) exp{

i∫

d4xjµ ·Ag−1

µ

}

=∫

DAµ∆L[Aµ]∏

x

δ(∂µAµ(x)) exp{

iA[Aµ] + i∫

d4xjµ ·Ag0µ

}

.

As before, Ag0µ is the gauge transform of Aµ, which satisfies ∂µAµ = 0, such that

L · ∇iAg0i = ∇i

{

U(g0)

[

L ·Ai +1

igU−1(g0)∇iU(g0)

]

U−1(g0)

}

= 0. (28.105)

When deriving Eq. (28.104), we have used the fact that

∆C [Aµ]∫

∏

x

dg(x)∏

x

δ(∇iAg−1

i = ∆C [Aµ]∫

{

∏

x

∏

a

dua(x)

}

×∏

x

δ

(

∇iAg0i − 1

gM[Ag0

µ ]u

)

∼ ∆C [Aµ]∆−1C [Ag0

µ ] = 1.

It is possible to solve Eq. (28.105) for Ag0µ as a power series in Aµ. This can be done

beginning with

Ag0i =

(

δij −∇i1

∇2∇j

)

Aj +O(A2µ).

The source jµ in the Coulomb gauge will be restricted to

j0 = 0, ∇i ji = 0, (28.106)

so that we may write∫

d4x jµ ·Ag0µ =

∫

d4x jµ · Fµ(x;Aλ),


where

Fµ(x;Aλ) = Aµ(x) +O(A2λ). (28.107)

Carrying this construction to high orders we can finally express WC in terms of WL

as follows:

WC [j] =

[

exp

{

i∫

d4xjµ(x) · Fµ

(

x;1

i

δ

δjλ

)}]

WL[j]|j=0. (28.108)

It is helpful to visualize the generating functionals (28.104) or (28.108) with thehelp of Feynman diagrams. The two expressions imply that the Green functionsin the Coulomb gauge are the same as those in the Landau gauge, if the sourceis suitably restricted by equations like those in (28.106). We only must take intoaccount extra vertices between source and field, represented by the term

∫

d4x jµ · (Fµ −Aµ). (28.109)

Then one may construct Green functions in the Coulomb gauge from the Feynmanrules of the Landau gauge. This relationship becomes much simpler if we go tothe mass shell. In this case, we ought to compare only the terms having a polein each of the external momentum square p2i . Of all the diagrams generated bythe extra couplings of (28.109), only those survive in this limit in which the wholeeffect of the extra vertices can be reduced to a type of self-energy insertion into thecorresponding external line. The other corrections introduced by (28.109) do notcontribute to poles of the Green functions at p2i = 0, and therefore do not contributeto the S-matrix. Hence in the limit p2i → 0, the unrenormalized S-matrix elementsin Coulomb gauge C and Landau gauge L with propagators differ by coming fromthe different propagators

limp2→0

D′µν(p;L) =

ZL

p2 + iǫ(gµν + . . .), lim

p2→0D′

µν(p;C) =ZC

p2 + iǫ(gµν + . . .).

In particular, the ratioσ2 = ZC/ZL (28.110)

is different from unity. In general, the unrenormalized S-matrix elements in the twogauges C and L are related to each order by

SC = σnSL = (ZC/ZL)n/2SL,

so that the renormalized S-matrix element

Sren ≡ Z−n/2C SC = Z

−n/2L SL

is independent of the gauge in which the calculation was done.As a consequence, WC [j] is equal to the expression (28.104), and thus it is be

equal to WL[j] except that the coefficient of jµ is the gauge-transformed vector field

28.6 Perturbative QCD 1505

Ag0µ , instead of Aµ itself. For the S-matrix, the only consequence of this difference

is that the renormalization constants attached to each external line depend on thegauge. Thus we have shown that ultimately the S-matrix can be calculated fromWL[j].

As pointed out earlier, the only flaw in the above argument is that the singularityat p2i = 0 is not in general a simple pole.

28.6 Perturbative QCD

For scattering processes at high energy, the strong interaction becomes effectivelyso weak that quarks behave approximately like free pointlike particles. This phe-nomenon of strong interactions is called asymptotic freedom. It was discovered ex-perimentally in deep-inelastic scattering at the Stanford Linear Accelerator and ledFeynman to his famous parton hypothesis [11, 12, 13]. By combining this hypothesiswith Gell-Mann’s field theory of quarks held together by a color octet of non-abeliangauge fields, the present-day quantum field theory of strong interactions was born.It was called Quantum Chromo Dynamics (QCD) [3]. Theoretically the asymptoticfreedom was first observed for charged vector bosons by V.S. Vanyashin and M.V.Terantjev as early as 1965 [8], and for the nonabelian gauge theory with local SU(2)-symmetry by I.B. Khriplovich [9]. But it was not until 1972 that it was noted inthe western physics community, first by ’t Hooft who failed to publish it and toldit only to colleagues, and later by Politzer, Gross, and Wilczek [14, 15] who earnedthe Nobel prize of 1973 for it.

For a sufficiently small number of quark flavors Nf < 33/2, the behavior ofthe beta function in QCD is opposite to that in Eq. (20.31) of φ4-theory and inEq. (27.30) for QED. It starts out with negative slope and remains negative atleast for a certain range. This is what causes the asymptotic freedom when theenergy-momenta of the particles in all Feynman diagrams grow large. To study thisbehavior analytically, we introduce a small number ǫ and imagine living in 4 + ǫdimensions with less than Nf = 17 quark flavors. Then we define an analogue ofthe fine-structure constant α = e2/4π for strong interactions, namely the constant“alpha-strong” αs ≡ g2/(4π)2. By a perturbative calculation of the scale dependenceof the coupling constant, one finds the β-function of the theory as β(g) ≡ µ∂µαs

[recall (20.59)]. To lowest order, this is

β(αs) = ǫαs − bα2s, with b = 11− 2Nf/3. (28.111)

Then the solution of the differential equation for the scale-dependent coupling con-stant becomes, by analogy with (20.126):

log µ =∫ αs(µ)

αs

dαs

ǫαs − bα2s

. (28.112)

Because of the sign change with respect to (20.126), the movement of αs for increas-ing mass scale µ is opposite to that in the φ4-theory and in QED. The couplingconstant αs = g2/(4π)2 goes towards a fixed point g∗ = ǫ/b in the ultraviolet limit.


By comparison with (20.289), we find the solution after changing the signs of ǫand b to satisfy

µ−ǫ

αs(µ0)=

1

αs(µ)− b

ǫ. (28.113)

This shows that αs tends to the fixed point α∗s = ǫ/b in the limit of large µ, i.e., in

the UV-limit. In four spacetime dimensions, we let the auxiliary parameter ǫ go tozero and see that the solution (28.113) tends to

1

αs(µ0)=

1

αs(µ)− b

ǫln

µ

µ0. (28.114)

This is solved for αs(µ) by

αs(µ) =1

α−1s (µ0) + b log

µ

µ0

. (28.115)

The first term in the denominator can be absorbed in the logarithm, and we canrewrite αs(µ) as

αs(µ) =1

12b log

µ2

Λ2QCD

=4π

(

11− 2

3Nf

)

logµ2

Λ2QCD

. (28.116)

The mass ΛQCD is the dimensionally transmuted coupling constant of QCD. Forµ→ ∞, the coupling constant αs(µ) goes to zero. This is why the nonabelian gaugetheory with less than Nf < 33/2 flavors is ultraviolet-free.

If higher loops are included in the calculations of QCD, the β-function has theexpansion [18]

β(g) = ǫαs−2(β0α2s + β1α

3s + β2α

4s + β3α

5s ) + . . . , (28.117)

where

β0 = 11− 2

3Nf =

b

2,

β1 = 102− 38Nf

3,

β2 =325Nf

2

54− 5033Nf

18+

2857

2, (28.118)

β3 =1093Nf

3

729+Nf

2

(

6472ζ(3)

81+

50065

162

)

−Nf

(

6508ζ(3)

27+

1078361

162

)

+ 3564ζ(3) +149753

6.

The various partial sums are plotted in Fig. 28.3.

28.7 Approximate Chiral Symmetry 1507

0.1 0.2 0.3 0.4 0.5 0.6

-5

-4

-3

-2

-1

β(αs) αs

234 5

Figure 28.3 Flow of the coupling constant αs towards the origin as the scale parameter

µ approaches infinity (ultraviolet limit). For the opposite flow direction (infrared limit),

the arrows are reversed and the coupling strength becomes large leading to the formation

of a fat attractive flux bundle between quarks.

If only the term α3s is included in the differential equation for the scale-

dependence of the coupling constant, the result changes from (28.116) to

αs(µ)

4π=

1

b0 lnµ2

Λ2QCD

− b1b30

ln lnµ2

Λ2QCD

ln2 µ2

Λ2QCD

. (28.119)

The presently best fit to the experimental coupling constants yields the value

ΛQCD ≈ 217± 25MeV. (28.120)

Since the theory is asymptotically free, many properties of it can be studied inperturbation theory. The results can be compared quite well with the experimentaldata observed in high-energy collisions. Details are amply available in the literature[16, 17].

28.7 Approximate Chiral Symmetry

At first, the masses of the quarks are approximated to be equal to zero. Thenthe Lagrangian density has an additional invariance. If we restrict our attentiononly to the three quarks u, d, s, the flavor group is SU(3). For massless quarks, thesymmetry is extended to SU(3) × SU(3), where the extension involves SU(3) flavor.Its Noether current densities are

jµa (x) = q(x)γµλa2q(x),

jµ5a(x) = q(x)γµγ5λa2q(x), (28.121)


where the λa matrices act on the flavor-SU(3) indices. The current densities arecolor singlets. The corresponding color octet currents are not observable, due tolocal SU(3) color invariance.

After field quantization the field components satisfy the local SU(3) × SU(3)commutation rules derived in (8.281) and discussed in Section 25.3. The charges Qa

and axial charges Q5a, defined by

Qa =∫

d3xj0,

Q5a =∫

d3xj05a, (28.122)

form the Lie algebra

[Qa, Qb] = ifabcQc, (28.123)

[Qa, Q5b] = ifabcQ5c, (28.124)

[Q5a, Q5b] = ifabcQc. (28.125)

From these we may form the chiral charges

QLa = (Qa −Q5a)/2, QR

a = (Qa +Q5a)/2, (28.126)

which generate the two commuting groups SU(3)L and SU(3)R, respectively. Dueto the chiral invariance of the massless quark gluon Lagrangian density (28.15), thecurrents are conserved:

∂µjµ(x) = 0,

∂µjµ5 (x) = 0. (28.127)

The ground state of the theory breaks the axial part of this symmetry spontaneously.This gives rise to non-zero quark masses. The spontaneous breakdown is accompa-nied by massless pseudoscalar Nambu-Goldstone bosons, which are identified withthe pion and its flavor octet partners.

In nature, quarks are not massless and the axial charges are not conserved. Themasses of non-strange quarks are, however, very small so that the axial chargeswith the SU(2)-indices a = 1, 2 are approximately conserved. This is the basis ofthe PCAC hypothesis (Partial Conservation of Axial vector Current). The nonzeromasses ofmu, md, ms . . . raise the mass of the pion and the other pseudoscalar mesonsto the experimental nonzero values. The quark masses which give a consistentpicture of experimental data are [7]

mu

md

ms

mc

mt

mb

=

4.5 ± 1.47.9 ± 2.4155 ± 501270 ± 5040000 ± 100004250 ± 100

MeV. (28.128)

Notes and References 1509

In recent years, much insight into the theory has been gained from computersimulations of lattice models of the theory. They have confirmed that the theoryhas the desired properties to explain the many strongly interacting particles observedin the laboratory.

Notes and References

[1] If the confining forces are derived from Monte Carlo simulations of gluons in an SU(3)-latticegauge theory, the hadronic mass spectrum of the quark model was calculated byR.D. Loft and T.A. DeGrand, Phys. Rev. D 39, 2678 (1989).If the b-quarks are included, the corresponding hyperons are discussed byC. Alexandrou, A. Borelli, S. Gusken, F. Jegerlehner, K. Schilling, G. Siegert, R. Sommer,Phys. Lett. B 337, 340 (1994) (hep-lat/9407027).

[2] H. Kleinert, Phys. Lett. B 59, 163 (1975) (http://klnrt.de/50); Phys. Lett. B 62, 77(1976) (http://klnrt.de/51).

[3] H. Fritzsch, M. Gell-Mann, Intern. Conf. on Duality and Symmetry in Hadron Physics,Weizmann Science Press, 1971;H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B 47, 365 (1973).

[4] V.N. Popov and L.D. Faddeev, Phys. Letters B 25 (1967) 29;B.S. DeWitt, Phys. Rev. 162, 1195 (1967);V.N. Gribov, Nucl. Phys. B 139, 1 (1978).

[5] The presentation in this section follows closely the excellent review article by

E.S. Abers and T.D. Lee, Phys. Rep. C 9, 1 (1973).

[6] In addition to the references in [4] seeN.P. Konopleva and V.N. Popov, Kalibrovochnye Polya (Atomizdat, Moscow, 1972), inRussian.R.P. Feynman, Acta Phys. Polonica 26, 697 (1963);B.S. DeWitt, Phys. Rev. 162, 1195, 1239 (1967);S. Mandelstam, Phys. Rev. 175, 1580 (1968);E.S. Fradkin and I.V. Tuytin, Phys. Letters B 30, 562 (1969); Phys. Rev. D 2, 2841 (1970);M.T. Veltman, Nucl. Phys. B 21, 288 (1970);G. ’t Hooft, Nucl. Phys. B 33, 173 (1971).

[7] H. Kleinert, Collective Quantum Fields, Lectures presented at the First Erice Sum-mer School on Low-Temperature Physics, 1977, in Fortschr. Physik 26, 565-671 (1978)(http://klnrt.de/55); J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982);G. Harrison et al. Phys. Lett. B 47, 493 (1984).

[8] V.S. Vanyashin and M.V. Terantjev, Sov. Phys. (JETP) 48, 565 (1965).

[9] I.B. Khriplovich, Sov. J. Nucl. Phys. 10, 235 (1969) [Yad. Fiz. 10. 409 (1969).

[10] H.D. Politzer, Phys. Rev. Lett. 30, 134 (1973);D.J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973).

[11] R.P. Feynman, Phys. Rev. Lett. 23, 1415 (1969); See also his lecture on The behavior of

hadron collisions at extreme energies in Proceedings of the 1969 Stony Brook conferenceedited by C.N. Yang et al., Gordon and Breach, New York, p. 237. The name “parton” firstappeared here. See also the textbook:R.P. Feynman, Photon-Hadron Interactions, Benjamin, Reading, 1972.

[12] J.D. Bjorken, Phys. Rev. 179, 1527 (1969).


[13] F.E. Close, An Introduction to Quarks and Partons, Academic Press, London, 1980.

[14] H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).

[15] D.J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973); D 9, 980 (1974).

[16] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, WestviewPress, N.Y., 1995. See especially Chapter 16.

[17] J.C. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics,Nuclear Physics and Cosmology, 2013;T. Muta, Foundations of Quantum Chromodynamics, World Scientific Lecture Notes inPhysics-Vol. 78, Third edition, 2010;Y.L. Dokshitzer, V.A. Khoze, A.H. Mueller, S.I. Troyan, Basics of Perturbative QCD, Edi-tion Frontieres, Paris, 1991;G. Sterman, J. Smith, J.C. Collins, J. Whitmore, R. Brock, J. Huston, J. Pumplin, Wu-Ki Tung, H. Weerts, Chien-Peng Yuan, S. Kuhlmann, S. Mishra, J.G. Morfin, F. Olness,J. Owens, Jianwei Qui, D.E. Soper, Handbook of perturbative QCD, Reviews of ModernPhysics, 67, 157 (1995).

[18] T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, Phys. Lett. B 400, 379 (1997).

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