Greiner W., Schrammm S., Stein E. Quantum Chromodynamics

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542 Appendix

The deﬁnitions (A.2) and (A.3) as well as the completeness relation can be used

to derive useful identities:

−1

N=3

, (A.7)

tr t

−1

N=3

= 4 , (A.8)

tr t

−1)

N=3

, (A.9)

tr t

1 − N

N=3

=−

, (A.10)

ABC

tr t

N(N

−1)

N=3

= 6i , (A.11)

where we also have given the numerically important case of N = 3. Useful

relations can also be given for the structure constants:

ACD

BC D

= Nδ

, (A.12)

ABC

= N(N

−1). (A.13)

For the special case of N = 3 the generators t

are given by the Gell-Mann

matrices,

SU(3) : t

, (A.14)

and for the case of N = 2 we get the Pauli matrices,

SU(2) : t

. (A.15)

The structure constants of the SU(2) are given by the antisymmetric tensor

abc

=−ε

acb

= 1(a = 1, 2, 3):

SU(2) :





=iε

abc

(A.16)

and the anticommutator in this case reads

SU(2) :

. (A.17)

For SU(2) structure constants we have in addition

abc

ad f

= δ

−δ

, (A.18)

acd

bcd

= 2δ

, (A.19)

abc

= 6 . (A.20)

B Dirac Algebra in Dimension d 543

B Dirac Algebra in Dimension d

The procedure of dimensional regularization continues the four-dimensional

space–time into d dimensions. Accordingly, also the four-dimensional γ ma-

trices have to be continued to d dimensions. Therefore we have to deﬁne d

γ matrices

,γ

,... ,γ

. (B.21)

is Hermitian, γ

, i > 0 anti-Hermitian:

µ†

= γ

. (B.22)

The algebra is deﬁned through the usual anticommutation relations

,γ

=2g

µν

1 . (B.23)

The trace of the metric tensor is given by

= d (B.24)

and therefore

= d

1 , (B.25)

while tr

1 = 4. Every other choice, for instance tr

1 = d, leads only to a ﬁnite

renormalization which is irrelevant. By convention one sticks to the ﬁrst choice.

Using the relations above, one can derive some useful identities:

=(2 −d)γ

, (B.26)

=(d −4)γ

+4g

αβ

, (B.27)

=−2γ

−(d −4)γ

, (B.28)

αβ

=(d −4)σ

αβ

, (B.29)

µα

=i(d −1)γ

, (B.30)

µα

=i(1 −d)γ

, (B.31)

wherewehaveintheusualway

µν



,γ



, (B.32)

= g

µν

−iσ

µν

. (B.33)

The trace over an odd number of γ matrices vanishes:

tr γ

=0 , (B.34)

tr γ

=4g

µν

, (B.35)

tr γ



µν

αβ

−g

µα

νβ

µβ

να



. (B.36)

544 Appendix

The deﬁnition of the matrix γ

in d dimensions is more complicated. In d = 4

dimensions γ

is deﬁned as

=γ

=iγ

=−iγ

= γ

†

=−

4 !

µνσ



(B.37)

with the completely antisymmetric tensor ε

µνσ

, ε

0123

=−ε

0123

= 1. In d = 4

dimensions we have

d = 4 :

,γ

= 0 . (B.38)

For arbitrary dimensions these deﬁnitions cannot be carried through consistently.

To demonstrate the inconsistencies arising we calculate the following traces,

tr (γ

),tr(γ

),andtr(γ



) in d dimensions, making use of the

anticommutation relation (B.38):

d tr γ

= tr γ

=−tr γ

=−d tr γ

(B.39)

from which it follows that tr γ

=0foralld =0.Inthesamewaywehave

d tr γ

= tr γ

=−tr γ

= (4 −d) tr γ

−4g

µν

tr γ

= (4 −d) tr γ

, (B.40)

so that

(2 −d) tr γ

= 0(B.41)

and therefore tr γ

=0foralld =2. Also, it is easy to see that

(4 −d) tr γ



=0 . (B.42)

These relations allow us to deﬁne a trace that is different from zero for d =4:

d = 4 : tr γ



=−4iε

µνσ

. (B.43)

However, it still requires a vanishing trace for arbitrary d. Obviously, the con-

sistent application of the anticommutation relation (B.38) requires a vanishing

trace over γ

multiplied with any string of γ matrices. Therefore, it was con-

cluded, that either dimensional regularization has to be abandoned for γ

or the

anticommutation relation has to be changed. In the ’t Hooft–Veltman

scheme

full Lorentz covariance in d dimensions will be abandoned, i.e. γ

in4andin

d −4 dimensions is treated differently. The idea is to deﬁne γ

also in d dimen-

sions with the help of the ε

µνσ

, which lives only in 4 dimensions as in (B.37).

Each contraction with the ε

µνσ

tensor therefore projects onto 4 dimensions. The

deﬁnition of γ

in the ’t Hooft–Veltman scheme therefore is

=iγ

=−

4 !

µνσ



(B.44)

G. ’t Hooft and M. Veltman: Nucl. Phys. B 44, 189 (1977)

B Dirac Algebra in Dimension d 545

with the ε

µνσ

tensor

µνσ

⎧

⎪

⎨

⎪

⎩

1 (µνσ) even permutation of (0123)

−1 (µνσ) odd permutation of (0123)

0otherwise

(B.45)

From this one gets the following commutation relations:

,γ

=0 : µ = 0, 1, 2, 3 ,



,γ



=0 : otherwise . (B.46)

anticommutes with the other γ matrices in the physical 4 dimensions and

commutes with them in d −4 dimensions. This procedure may appear unsat-

isfactory, but still it works in the sense that it reproduces the correct anoma-

lous divergence of the axial current operator. To understand the appearance of

a potential contribution to the anomaly we introduce the following notations for

γ matrices in d, d −4, and 4 dimensions:

. (B.47)

From (B.46) we ﬁnd that the anticommutator with γ

does not vanish, instead

a contributions is retained which exists in d −4 dimensions

,γ

. (B.48)

In the limit d → 4 the original expression is recovered. This, however, is no

longer true when in a calculation of some diagram divergencies, which mani-

fest as poles at d = 4, appear. This has as consequence that after renormalization,

contributions proportional to d −4 no longer vanish and give a ﬁnite contribution

lim

d→4



d −4

,γ



= lim

d→4



d −4

·2(d −4)γ



=2γ

. (B.49)

For massless quarks, the divergence of the axial current in the ’t Hooft–Veltman

scheme is given by

∂

Ψγ

Ψ, (B.50)

which means that, as in (B.49), this term will contribute only anomalously.

As basis for the gamma matrices in d = 4 dimensions one can choose the

16 matrices

1,γ

,γ

,σ

µν

.Ford > 4 that basis has to be correspondingly

larger. The following relations therefore are valid only in d = 4 dimensions:

d = 4 : γ





αβ

δ

αδ

β

−g

α

βδ



−iε

αβδ

. (B.51)

Also, it holds that

d = 4 : γ

σ

=iσ

αβ

αβσ

. (B.52)

546 Appendix

As another important relation we give the Fierz transformation of the direct

product of two quark ﬁeld operators:

d = 4 : q

=−

αβ

−

(γ

)

αβ

−

µν

(σ

µν

)

αβ

−

(γ

)

αβ

−

(γ

)

αβ

. (B.53)

C Some Useful Integrals

The master formula for Feynman integrals is

(2π)





−M

)

= i

(

−1

)

a−m

(4π)

d/2

Γ(a +d/2)Γ(m −a −d/2)

Γ(d/2)Γ(m)(M

)

m−a−d/2

, (C.54)

where k is a d-dimensional vector in Minkowski space. For symmetric integra-

tion we have

f(k

) =

µν

f(k

), (C.55)

f(k

) =

µν

λσ

µλ

νσ

µσ

νλ

d(d +2)

f(k

(C.56)

For an odd number of k

(antisymmetric integration) the corresponding integrals

vanish.

The Feynman parameters are

;

i=1

Γ(A)

i=1

Γ(A

)



;

i=1

−1



δ(1 −



i=1

)





i=1



where A =



i=1

. In the simplest case this reduces to

Γ(A +B)

Γ(A)Γ(B)

A−1

B −1

(ua+

ub)

A+B

; u +

u = 1 . (C.57)

For integration over Feynman parameters, the following representation of the

beta function is useful:

duu

A−1

B −1

Γ(A)Γ(B)

Γ(A +B)

. (C.58)

C Some Useful Integrals 547

By suitable substitution one recovers a different representation

∞

A−1

(1 +t)

A+B

dt =

Γ(A)Γ(B)

Γ(A +B)

. (C.59)

For evaluations of Fourier transforms the Schwinger parameterization might be

useful:

(−x

)

Γ(α)

∞

duu

α−1

−u(−x

)

. (C.60)

Knowledge of the d-dimensional Gauss integral

+∞

−∞

ξe

−aξ





d/2

(C.61)

and the deﬁnition of the gamma function

Γ(A) =

∞

dtt

A−1

−t

(C.62)

then allows us to derive the following formula:

i px

(−x

)

=−i2

d−2α

d/2

(−p

)

(α−d/2)

Γ(d/2−α)

Γ(α)

. (C.63)

Note that x is a d-dimensional Minkowski vector in coordinate space, while p is

a d-dimensional Minkowski vector in momentum space. The distinction is im-

portant since the Wick rotations to Euclidean coordinate or momentum space

have different signs:

x =−i

, (C.64)

p =+i

. (C.65)

Here the index E marks an Euclidean vector. This convention leaves the prod-

uct p

invariant under Wick rotation. This has to be kept in mind when using

any of the given integrals in either momentum or coordinate space. For instance,

(C.63) would read

−i px

(−p

)

=+i2

d−2α

d/2

(−x

)

(α−d/2)

Γ(d/2−α)

Γ(α)

. (C.66)

548 Appendix

Note that the overall sign has changed compared to (C.63). The Fourier trans-

form (C.63) can be used to derive many more formulas by taking derivatives with

respect to p. Differentiating twice, one obtains the following formula:

x e

i px

(−x

)

iπ

d/2

d−2α+1

Γ(α)

(−p

)

(α−d/2−2)



Γ(d/2−α +2) +g

µν

(−p

)Γ(d/2−α +1)



(C.67)

Subject Index

adjoint spinor, 22

Aharonov–Bohm effect, 156

anomalies, 168, 220 ff., 231

– axial, 224, 235, 360

– scale, 237

anomalous dimensions, 281 ff., 283,

314, 315, 436

approximation, one-photon, 419

asymptotic expansion, 371, 383

asymptotic freedom, 310

asymptotic series, 381

axial

– anomaly, 224, 235, 360

– current, 222, 228

– gauge, 415

– transformation, 222

background ﬁeld, 235

– technique, 229

bag pressure, 129

Balitsky–Fadin–Kuraev–Lipatov

(BFKL) equation, 437 ff.

β function, 217 ff., 218, 307 ff., 308–

310, 376

– third order, 219

Bjorken limit, 292, 297, 298, 431

Bjorken sum rule, 362, 370, 380, 381

Bjorken variable, 347, 366

Bloch–Nordsieck theorem, 255, 391,

397

Borel

– plane, 373, 374

– representation, 372

– summable, 372

– transform, 384

– transformation, 378, 509

Breit system, 103, 248

bremsstrahlung, 251, 260

Callan–Gross relation, 252, 300, 330,

331

Callan–Symanzik equations, 309

canonical dimensions, 367

Casimir energy, 139

charge screening, 183

chiral condensate, 486

chiral symmetry, 494

– Goldstone boson, 496

– PCAC, 497

coefﬁcient functions, 318, 327

collinear expansion, 424, 427

color, 167

– string, 407

Compton forward scattering, 328

– amplitude, 242, 375

Compton scattering, 240

– amplitude, 318

– – high-energy behavior, 431

conservation laws, 1

contour integration, 325

corrections, higher-twist, 366

correlation function, 447

coupling

– axial vector, 296

– renormalized, 203 ff., 216, 309

– running, 180 ff., 181, 218, 310, 319,

377

– vector, 296

CP violation, 163

cross section

– Drell–Yan, 389, 394, 409

–e

−

pair annihilation, 166

cross-section method, 346

– gluonic contribution to F

, 354

550 Subject Index

CTEQ, 285

current factorization, 73

Cutkosky, 429

d-dimensional

– Fourier transform, 198

– Gaussian integral, 189

– integral, 186, 196

deep inelastic scattering, 239 ff., 366

– lepton–nucleon, 427

DESY, 264, 387, 430

detailed balance, 465

diagrams

– “cat ear”, 321

– “hand bag”, 321

dilogarithm, 396

dimensional regularization, 184 ff.,

192, 213

Dirac equation, 17

– covariant normalization, 23

– current density, 20

– Gordon decomposition, 81

Dirac sea, 222, 227

distribution function, 256, 262, 426

– parametrizations, 281

– singlet, 262

divergence, logarithmic, 185, 192

double logarithmic approximation,

435

Drell–Yan

– heavy photon, 411

– one-gluon contribution, 410

Drell–Yan factorization, 419

Drell–Yan process, 387 ff.

– phenomenology, 405

Ellis–Jaffe sum rule, 362

EMC effect, 257

evolution equations, 313

factorial growth, 374

factorization, 317

– Drell–Yan, 419

Fadeev–Popov, 178

Fermi constant, 96

Fermi’s theory, 95

fermion bubbles, 375, 400

fermion propagator, 458

Feynman

– parameters, 196, 204, 332, 334

– parametrization, 201 ff.

– propagator, 320

– rules for QCD, 172 ff.

Fierz transformation, 421

– color space, 422

and F

, 336

ﬂavor SU(3),3

form factors, 71

Froissart bound, 433

fundamental representation, 5

gamma function, 195, 197

gauge

– axial, 415

– Feynman, 179, 242, 415

– ﬁxing, 179

– Fock–Schwinger, 233, 235

– Landau, 176, 179, 218, 377

– light-like, 245

– Lorentz, 244

– Lorentz condition, 33, 51

– parallel transport, 159

– transverse, 176

gauge invariance, 24, 56, 71, 212, 223

– minimal coupling, 24

gauge principle, 449

gauge theory, 165

– covariant derivative, 158

– renormalizability, 160

– spontaneous symmetry breaking,

160

– standard model, 155

–SU(2)× U(1), 162

– symmetry, 156

Gaussian integral, 190

Gell-Mann matrices, 331, 338, 422

Gell-Mann–Nishijima formula, 164

Gell-Mann–Nishijima relation, 9

generating function, 372

ghost

– ﬁelds, 175, 178 ff., 212

– loop, 212

GLAP equation, 239, 258, 259 ff.,

261, 263, 264, 265 ff., 313

Subject Index 551

gluon

– effective propagator, 377

– loop, 207

– screening mass, 538

Grassmann algebra, 461

Gribov–Levin–Ryskin (GLR)

equation, 438

groups

– adjoint representation, 7

– Casimir operator, 5

– hypercharge, 9

– ladder operators, 8

– Schur’s lemma, 15

–SU(3), 12, 363

Gürsey–Radicati mass formula, 77

hadron models

– hybrid bag models, 142

– MIT bag model, 77, 124, 125

– potential models, 77

– Skyrmion bag model, 77

– SU(6) wave function, 149

hadron tensor, 61

heavy photon decay, 413

heavy-ion collision

– Bjorken picture, 536

– shock waves, 536

HERA, 264, 430

high-temperature phase

– phase transition, 530

hypercharge, 169

inelasticity parameter, 98

infrared slavery, 310

instanton–antiinstanton, singularities,

373

invariant scattering amplitude, 59

K factor, 399, 405

Klein–Gordon equation, 26

– current density, 27

– Euclidean, 456

lambda parameter, 219

lattice

– pseudo fermion, 470

– space, 444

– time, 443

leading-logarithm approximation, 400–

402, 430

lepton tensor, 61

Lie algebra, 4

light-cone expansion, 293

lightlike vectors, 422, 427

link variable, 450

– average, 473

longitudinal structure function F

331

Lyapunov exponent, 539

Mandelstam variable s,46

Mandelstam variable t,45

Mandelstam variables, 37, 102, 243,

247, 350, 389

Markov chain, 465

Matsubara frequencies, 483

minimal coupling, 29

Monte Carlo methods, 464

– Langevin algorithm, 466

– Metropolis algorithm, 466

– microcanonical algorithm, 468

MS scheme, 377

multigluon emission, 401

Nachtmann variable, 316, 366

naive non-Abelianization, 376

one-photon approximation, 419

OPE, 318, 320

operator, 367, 369

– axial vector, 323

– bilocal, 297, 301, 322

– higher twist, 327, 369

– local, 297

– matrix element, 297, 303, 305

– spin, 367

– twist-2, 303, 369

– vector, 323

operator product expansion, 292 ff.,

295, 300, 317, 366, 490

optical theorem, 312, 318, 326

parallel transport, 449

partition function, 445