Appendices

Appendix A: r-Algebra in Dimension D

The y matrices are taken to be of dimension 4. We have D yll matrices,

and the matrix l Y5' They verify the anticommutation relations,

with gllv=O, w=l= v, for i = 1, ... , D - 1.

gllv = gilV.

few useful relations are, with SIlV'P = gllvg,P + gllPgv, - gll,gvP'

(odd) Tryll ... t=O,

(odd) TrY5yll ... t=O,

Tr yllyVy'YP = 4SIlv,P = 4{g1lV g'P + gllPgV' _ gll'gVP};

SIlVP'SIlVPP = 3(D - l)gp

~~ = a2; ~~~ = - a2~ + 2(a'b)~,

ylly"yPYIl = 4g'P + (D - 4)y'yP,

yllY'yPybYIl = - 2ybyPy' + (4 - D)y'yPyb.

For D = 4, Y5 = iyOyly2y3, and then, defining the totally antisymmetric tensor f,llvpa by

f,0l23 = - 1, f,Ol23 = + 1,

and cyclically. Also for D = 4,

1 More about Ys may be found in Sections 3.1 and 7.5.

Trysyl'yVyAya = 4iel'VAa;

g~(Je~l'pa e(JVtA = _ gl'v(gPtgaA _ gPAgat) _ gI'A(gPV gpt _ gptgav)

+ gl't(gPVgaA _ gPAgaV);

eI'V~(Je~la = 2(gvPgl'a _ gPPgV~.

Appendix B 293

Moreover, {yP, Ys} = O. In the Pauli or Weyl realizations, Y2YpY2 = - y: and, also, YoY;Yo=YI',yo(iys)+Yo=iys' Finally, ifw1,w2 are spinors and rw·.,rn

are any of the matrices YP' iys,

(WI r 1" .rnW 2)* = W2rn··· r 1 WI'

Appendix B: Some Useful Integrals

In D dimensions,

f dDk (P), . (- ly-m r(r + D/2)r(m - r - D/2)

(2n)D' (P - R2)m = I (16n2)D/4' r(D/2)r(m)(R2t- r - D/2 ;

f dDk _ 1-=0' k2 + iO '

Symmetric integration:

dD kb(1 -Ikl) = --(in Euclidean space). f 2rrp/2

- - r(D/2)

f dDkkPkVf(k2) = g;v f dDkk2f(k2);

fdDkkPkVkAkaf(k2) = gPVgAa + gPAgVa + gPagVA fdDkk4f(k2). D2+2D '

f dDkkP' ... kP2n + 'f(P) == O.

As e-+O.

00 (_ e)n r(l + e) = 1- YEe + L --((n),

n=2 n!

r is Euler's function, ( Riemann's function and YE ~ 0.5772 is the Euler­Mascheroni constant.

Feynman parameters:

1 r(rx + 13) II x~-I(1- X)(J-l AaB(J = r(rx)r(f3) 0 dx {xA + (1 - X)B}d(J'

_----:-_ = dx' x d y 1 2 3 1 r(rx+f3+y)II II U~-IU(J-luy-l

A~B(JcY r(rx)r(f3)r(y) 0 0 {u i A + u2B + U3C}~+(J+y'

294 Appendices

In general,

More formulas may be found in Narison (1982). Some numerical integrals:

t dxlog(1 +x)=2Iog2-1

Ii dx log(1 + x) = 11:2.

o x 12

Many useful integrals can be derived from Euler's formula:

Ii dxx"(1 _ x)P = r(l + ocW(l + {3). o r(2 + oc + {3)

For example, by differentiation, we obtain

Ii dxx"logx = 1; o (oc+ 1)2

il dxx"(l- x)Plogx = [Sl(OC) - Si(l + oc + {3)] r(l + ex)r(1 + {3), o r(2 + ex + {3)

Ii x" - 1 dx-- = - Sl(ex),

o 1-x

Ii dxx"logxlog(1- x) = Sl(1 + oc) + S2(1 + ex) _ 11:2._1_; o (1 + ex)2 1 + ex 6 1 + oc

Ii log2 X dxx"-- = 2,(3) - 2S3(oc),

o 1-x

dx--Iogxlog(1- x) =-Sl(OC) - Sl(ex)S2(ex) - S3(ex) + '(3), il x" 11:2

o 1-x 6

Appendix C 295

f dxxa(1- x)/llog X 10g(1 - X)

r(1 + a)r(1 + 13) { n2 = S2(1+a+f3)--+[Sl(a)-Sl(a+f3+1)J

r(2 + a + 13) 6

x [Sl(f3) - Sl(a + 13 + I)J },

dxxa(1- x)/llog2 X = ([Sl(a) - Sl(a + 13 + I)J2 f 1 r(1 + a)r(1 + 13) o r(2 + a + 13)

etc. Here, Sl(a) = W) - I:,= 1 [1/(k + a)lJ, 1> 1; Sl(a) = IJ= 11// for a = positive

integer, any 1. Note that S2( (0) = n2/6, Sl( (0) = W) where (is Riemann's function. For 1 = 1, the aboveformula for Sl may be replaced by Sl(a) = aI:'= 1 [1/k(k + a)] = IJ=11/j, the last equality valid for a=integer>O. Also, Sl(a)= 1/1 (a + 1)+)'E' with I/I(z) = d log r(z)/dz. For the special functions r,l/I, (, see Abramowicz and Stegun (1965).

Appendix C: Group-Theoretic Quantities

( rri ~). C 0 n c 0 -~} )J= 0 j=I,2,3; A,4 = 0 0 A,5= 0 0

1 0 i 0

A'~G o 0) ).' ~ (~

0 -D; C 0 ~} o 1 ; 0 A,8=~ ~ ~ 1 0 -2

rr1 = (0 1) 1 0 ' rr2=G -~). rr3 =G -~).

We can introduce the matrices ca with elements C:c = - ifabc == - ijabc. The commutation relations of the t and Care

and anti-commutation relations are

{ta, tb} = Idabctc + tbab.

The f are totally anti-symmetric, the dabc == dabc ate totally symmetric, and the only nonzero elements (up to permutations) are as follows:

296 Appendices

1 = 1123 = 2/147 = 2/246 = 2/257 = 2/345

2 2 = - 2/156 = - 2/367 = -/458 = -/678;

J3 J3

For an arbitrary SU(N) group, we define the invariants C A, CF, TF by

b C = Tr caCb = "Iaee'lbee' • A ~, ee'

bab TF = Tr tatb = L tfkt~i' k,i

One has

N 2 -1 C -F---W'

Useful relations are

and other useful invariants are

,,2 40 ~dabe=~3 ' abc

1 TF =-·

2

Appendix D: Feynman Rules for QeD

Ordinary Formalism

The Feynman rules are as follows:

A J k

~,b,q v, e,k

e

• ~ . P k

. .... a,I' k b, v

Appendix D 297

+ f llcefbde( ) g .. ,.gv" - g .. "g,.v

+ faderbe(g"vg,." - g .. ,.g"v)}

i () p-m+iO jk

i () k2 + iO lib

(axial gauge)

An overall (2n)4 {)(P j - P f) is included for global energy-momentum conservation, and (-1) for closed fermion or ghost loops. Statistical factors are as follows:

298 Appendices

for --D-2!

for e ' etc. 3!

Insert vci- D J dDk/(2n)D == J dDj( for each loop integration. Diagrams with dis­connected bubbles are excluded. A diagram is to be read against the directions of the arrows of its oriented lines. To obtain the S-matrix elements, amputate the amplitude and add, for external particles,

P, (T

~ . p,O--<III •

k, ~ --~---

p,er . ~ P, (T -• <III

k, ~ ~

k, ~ --- .....

(2n) - 3/2V(p, 0")

(2n) - 3/2 sfl(k, A)

The spinors and polarization vectors are normalized to

L u(p, a)ii(p, a) = p + m,

I sll(k, 2)* sV(k, 2) = - gllV(Feynman gauge). ~

Appendix D 299

These rules differ from the ones in Bjorken and Drell (1965) by the normalization of the spinors,

_ p+m I UBDUBD = ~-, (J 2m

and the factors (2n) - 3/2 due to our normalization of :T which differs from :T BD by precisely these factors.

Background field formalism

a k b -- -- -- ---

300 Appendices

x

+ iadxixbAgJlvg;.p - gJl;.gv)

+ iacxixbigJlvg;.p - gJlPgv;')]

d, 9 -ig2~[iabxfxcd(9Jl;'9vp-gJlpgv;.+~~gJlvg).p)

In I

P • I

l.. - - - -I'(p\ C J.l - giacb(P + q)Jl i""""""'~ .I

q! I b

/ /

/ b ~

+ iadxfxbc(9Jlvg;'P-gJlAgvP- 1 ~-~gJlPgvA)

+ iacxixbd(gJlvg;.P - gJlPgv;') J 1 0 I

P .. I

~ c,J.l -giacbPJl ,

q. I

'b

Appendix E 301

igt'jk'/'

k

Appendix E: Feynman Rules for Composite Operators

Let 1'+ = 1,1'_ =1'5' and,1 an arbitrary four-vector with ,12=0.

A k k

A k,,", k,v

.~ p,a,,", k,c,>-

q, b,v

N = q(O)yI" . .. iY'ny ± qO)

gJl),1·k)n + e,1Jl,1v(,1·k)"-2

- (kJl,1v + ,1JlkJ(,1·k)n-l

n-2 gt~j,1JlI/l. L (,1·pd(,1·P2)"-j-2 y±

j=O

n-2

j= 1

+ [(gJl).~V - ,1Jlgv).)(,1· k)

+ ,1).(,1Jlkv - ,1vkJl)J(,1·k)n-2 }

+ permutations.

Cf. also Floratos, Ross, and Sachrajda (1977,1979).

302 Appendices

Appendix F: Some Singular Functions

We define x-space causal functions by

~(X" rn2) = __ e-'k·x---::-_-:---_ f d4k. i

, (2n)4 k2 - rn2 + iO'

D1lV(X) = if d4k e-ik-x - gil" + ~k/lk"/(k2 + i~ ~ (2n)4 k2 + iO '

. -'f d4k -ik'x ~+rn sex, rn) - 1 --4 e 2 2 .' (2n) k - rn + 10

We will at times omit the variable rn from ~, S. In terms of time ordered VEV s,

< T4>(x)4> (0» ° = ~(x;rn);

< T B~(x)B~(O) >0 = c5abDnx).

The character of Green's functions of the propagators is exhibited clearly by the equations (a; + rn2)i~(x - y) = c5(x - y), etc. Furthermore,

sex, rn) = (i~ + rn)~(x, rn).

On the light cone,

-1 1 irn28(x2) rn 2 rnlx211/2 ~(x,rn2) ~ -'--+ +-log + ...

x2~0 4n2 x 2 - iO 16n 8n2 2

2ix/y/l Sex) ~ 2 2 2+···,etc.

x2~0(2n) (x -iO)

Additional relations may be found in Bjorken and Drell (1965).2 Fourier trans­forms of distributions are given in Gel'fand and Shilov (1962), pp. 277ff, 316ff. The ones used in the text are

f d4xe-ik.x_l-=4n2_1-· -x 2 ± iO k2 +- iO'

f d4xe-ik-x 2 1. 2 = - n2ilog(k2 +- iO) + constant. (x ± 10)

Equal-time and light-cone commutation relations for fermions:

{q~(x), q~(y)} = 0; c5(xO - yO) {q~(x), q~(y) +} = c5ap c5 ik c5(x - y),

{qa(x), qp(O)} ~ (~- irn)aP {~e(XO)c5(X2) _ _ rn_ 8(x2)e(xO) + ... } X2~0 2n 4np

2 Our causal functions ditTer by i from the Bjorken-Drell (1965) ones: S = iSSD• D = iDsD •. ...

Appendix G 303

Appendix G: Kinematics, Cross Sections, Decay Rates

The states of a particle with helicity 2 and momentum p are normalized according t03

This corresponds to a density of particles per unit volume of

2po p(p) = (2n)3'

We define the scattering amplitude in terms of the S matrix:

S = 1 + iff, <flffli) = f>(Pf - Pi)F(i-l> f).

For Ii) a state of two particles with masses ml,m2' the cross section is then

d . f 2n2 . 2 dp f1 dp fn 0'(1-1> )= 1/2 2 2 f>(Pf-P;)IF(I-I>f)1 -0-"'-0-'

2 (s,ml'm2) 2Pfl 2Pfn

where

2(a,b,c) = a2 + b2 + c2 - 2ab - 2ac - 2bc,

For the case Pl + P2 -I> p~ + p~, one obtains

dO'(i -I> f) n3

d 2( 2 2)IF(i-l>fW, t s,m 1 ,m2

- =-'-IF(i-l>fW, dO'l n2 q' dOcm 4s q

O'(i -I> all) = [4n2 / 2 1/2(S, mi, mD] 1m F(i -I> i).

Here

11/2( 2 2) _1- 1- I\. s,ml'm2 q - Plcm - 2S1/2 .

11/2( 12 12) '_1-' I_I\. s,m1 ,m2 q - Plcm - 2S1/2 '

dO = d cos edf/>.

3 The transformation properties for an arbitrary field are

U(a)<I>(x)U-1(a) = <I>(x + a), U(a) = eiP·a.

s=pf·

304 Appendices

Likewise, the decay rate is4

. f 1 . 2 dp fl dp In dr(l--+ )=-<5(Pi-PI )IF(I--+f)1 -0-"'-0 ' 4nmi 2p 11 2p In

For massless particles,

fdP1".dPn<5(p_I,p;)=(P2)"-2 (n/2)n-1 . 2p~ 2p~ (n - 1)!(n - 2)!

Our units are such that h = c = 1. Some useful formulas in this system are:

1(MeV)-1 = 1.973 x 10- 11 cm = 6.582 x 10- 22 sec.

1(GeV)-2 = 3.894 x 1O- 4 barn.

1 MeV = 1.783 x 10- 27 gr = 1.602 x 10- 6 erg;

1 cm = 5.068 x 1010 (MeV)-l, 1 sec = 1.519 x 1021 (MeV)-l

1 barn = 2.568 x 103 (Ge V) - 2

1 gr = 5.610 x 1026 MeV, 1 erg = 6.242 x 105 MeV.

Appendix H: Functional Derivatives

A functional is an application of the space of sufficiently smooth functions, U(x)}, into the complex numbers:

F:f --+F[f].

Note that F need not be linear. We will treat functionals F[f, g, ... J in the same way. We may consider a functional as a generalization of an ordinary function in the following sense: divide the space of the x values 5 in N cells, and let each Xj lie one in each cell. Then F[fJ is the limit for vanishing cell size of F N(fl>'" ,fj'" .),fj == f(xJ The derivative of N/ofj is

of N(f1,"· ,fj'''') l' F N(f1"" ,fj + t:, ... ) - F N(f1,··· ,fj ,···) -=--~-- = 1m ,

ofj ,"'0 t:

i.e., it may be obtained by shifting fi --+ fi + t:<5ij. So, in the limit, we define

<5F[fJ = lim F[f + t:<5y J - F[fJ, <5f(y) ,"'0 t:

where <5y is the delta function at y: <5 y(x) = <5(x - y). An important case is that

4 All the formulas are valid for indistinguishable or distinguishable particles. When calculating integrated rates, however, we have to divide by the number of redundant permutations. For example, if we integrate over the momenta of j identical bosons or fermions, divide by j!.

5 We take this space to be of finite size. L. Otherwise, an extra limit L--+ (fJ has to be performed.

of integral functionals:

F[f] = f dxKF(x)f(x);

then,

c5F[f] = K ( ). c5f(y) F Y

Appendix I 305

Taylor series may be generalized to functional series. If the kernels Kn(x 1, ... , Xn) are symmetric (anti-symmetric for fermionic f) and we consider the functional

we may easily verify that

c5n F[f] Kn(xl,···,xn)= c5f(x l )···c5f(xn)

A concept related to that of the functional derivatives is that of functional integration. We define

f n df(x)F[f] == lim fdfl" .dfNF NUl"" ,fN)' x N-oo

As in the case of functional differentiation, functional integration obeys rules analogous to those of ordinary integration. Both for differentiation and inte­gration, some modifications are required to accommodate anti-commuting func­tions; they are described in Section 1.3.

Functional derivatives of expressions that do not involve integrals are found by reexpressing them as integrals. For example, the derivative entering Equation (2.5.9) is so evaluated:

c5oBa(x) = _c5_~ "fd4Zc5(Z _ x c5 Bil Z c5B~(y) c5B~(y) axIl ~ ) ac c ( )

o = c5ab OXP c5(x - y).

Appendix I: Gauge-Invariant Operator Product

It is intuitively obvious that in a gauge theory, an expression like those appearing in OPE,

~ XII! ... Xlln _ q(O)q(X) = L , q(O)OIl!" .Ollnq(O),

n.

306 Appendices

should be replaced by another with derivatives substituted by covariant deriva­tives, 01'--+ VI'. Here we sketch a formal proof of how this comes about. When the fields are interacting, the propagators are not free propagators. For example, for a fermion in the presence of the gluonic field, the propagator satisfies the equation, derived directly from the Lagrangian,

(if) - m)Sint(X, y) = i<5(x - y).

Retaining only the more singular (lower twist) terms, the solution to this is

Sint(X, y) ~ {p exp i {X dzl' L ta B~(z) }S(X - y),

where S is the free propagator and P indicates ordering along the path from y to x. If we repeat the OPE taking this into account, we find that operator products q(x)q(y) are replaced by the gauge-invariant combination

q(X){ P exp i {X dzl' L ta B~(z) }q(y),

whose expansion for x ~ y is precisely that with covariant derivatives. The same is of course true for operators built from gluon fields. Additional details may be found in the paper of Wilson (1975) and the review of Efremov and Radyushkin (1980b).

Appendix J: Group Integration

Let us consider a group G. We will assume that the group is topological, which is the case in the particular instances where G is a discrete group or a Lie Group. Then, there exists a measure, called the H aar measure, positive definite, and left invariant: if we denote by dJ-lL(g) to the measure,

fa dJ-lL(g)f(g) = fa dJ-lL(g)f(hg),

for any smooth function f that decreases sufficiently fast, at infinity of the group, and for any group element h. A right invariant measure also exists:

fa dJ-lR(g)f(g) = fa dJ-lR(g)f(gh).

For abelian groups, dJ-lR obviously coincides with dJ-lL; and the same can be proved to be the case for discrete and compact groups (abelian or not). Moreover, dJ-lL = dJ-lR = dJ-l is unique, up to a multiplicative constant.

For discrete groups, the Haar measure is merely the sum over group elements:

fdJ-l(9)f(9)--+ L f(g)· allg

Appendix J 307

For Lie groups, that we take compact to simplify, one can reduce the Haar measure to an ordinary integral as follows. Let (Xl' •.. (Xn be the parameters determining the group elements, 9 = g((Xl, ... , (Xn). If we consider that g((Xl,···' (Xn) is the product of g(Pl' .. · ,Pn) and g(Yl,···, Yn),g((Xl,··· ,(Xn) = g(Pl'··· ,Pn)g(Yl'···' Yn), then obviously the (X are functions of the P and y:

(Xi = ((Ji(P 1, ... , Pn; Y l' ... , Y n)·

We will consider that the unit of the group has the parameters 0, ... ,0. Define then the Jacobian,

J -l( ) = d (iJ((Ji(Pl, ... ,Pn;Yl' ... ' Yn))1 Yl, ... ,Yn et . iJPj /l=O

The expression for the invariant measure is then, writing dg for dJl(g),

f dgf(g) = f d(Xl· •. d(XnJ((X 1 , ... , (Xn)f(g((Xl,· .. , (Xn))·

An elementary proof that this is true may be found in the text of Creutz (1983).

A basic theorem in group integration is the following:

Peter-Weyl Theorem. Let D(1)(g) be a unitary representation of a compact group, and D(2)(g) another, inequivalent to the first. Then,

L dgDl]l(g)Dk7)(9) = 0,

for any matrix elements ij, kl. Adjusting the arbitrary constant in the definition of dg so that

L dg= 1,

we also have

L dgDU)(g)D~})(g)* = bikbj /.

Actually, all integrals over representations of the group (including the last one) can be deduced from the Peter-Weyl theorem as follows.

Consider the integral of an arbitrary product of representations:

L dgDl~],(g)·· .Dl~].(g)· One may decompose this as a sum of irreducible representations, using the

appropriate Clebsch-Gordan coefficients:

Dl~], (g) ... Dl~].(g) = C(O)(il ,jl;···; iv,jv)D(O) + L C(I)(il,jl;···; i .. jvlkl)D~~)(g),

308 Appendices

and D(O) == 1 is the identity (trivial, 9 -+ 1) representation. Because of the Peter­Weyl theorem all terms give zero save the first, so

f d D(1) () D(v) ( ) - C(O)(' '. .' .) G9 i,il9··· i.j,9- ll,h,···,lv,]v·

Useful algorithms for calculating these Clebsch-Gordan coefficients for the important case ofthe groups SU(n) may be found in the book ofCreutz (1983).

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Subject Index

Action d 14,20 Euclidean d 245, 257

Adler sum rule 127 Altarelli-Parisi

equations 123, 124 P functions 123, 124, 125

Anomalous dimension of a Green's function, Yr 67, 68 of nonsinglet moments, YNS' d(n) 106, 111 of operators in deep inelastic

scattering to first order 106, 111 of operators in deep inelastic

scattering to second order 112 of singlet moments, Y, D(n) 106, 111 of the mass, Ym, dm 67, 73, 74 of the quark propagator, dF~ 75

Anomaly Adler-Bell-lackiw, or axial 226,228 of the energy momentum tensor 232 U(I) 232

Anti-commuting c-numbers 16,31 Anticorrelation (energy-energy) 165 Anti-instantons 249 Asymptotic freedom 5, 73 Axial (Adler-Bell-lackiw) anomaly

(see Anomaly)

Background field: method/formalism 40 renormalization 63

Background gauge 41 Bare couplings, guD' muD> A.uD 56 Becchi-Rouet-Stora transformations

in QCD 31 in QED 30

Bianchi identities 245 Bjorken limit 94 Bjorken variables x, Q2 94 BKW (see WKB)

Callan-Gross relation 103, 114 Callan-Symanzik equation 67

Callan-Symanzik functions p, Y, b 67, 72, 73, 74 Chiral

charges L~(t) 216 currents, J"t 216 group, SU;(n) x SU;(n); U;(n) x U;;(n)

213 transformations U ± 216

Chiralities (for U(I)), X 234 Color 2

transformations SU(3), global and local 18 Commutation relations

canonical 21 equal-time 43, 216

Condensate gIuon 81 of heavy quarks 288 quark 81, 221

Constituent mass 207 Contraction 11 Counter-terms 51, 53 Counting rules 184 Covariant

curl, D x 19 derivative, DP 19

Currents conserved and quasi-conserved 43 equal-time commutation relations of 43,216 J:, V:' A: 2, 3,43, 216 U(I) 233 U(I) current 232, 233, 234

Decay constant of kaon fK 220 of pion fa 219

Deep inelastic scattering sum rules 126 Density (or probability) function

of gluons 124 of quarks 97, 118, 125

DilTractive scattering 168 Dimension (of space-time) 46 Dipole moment of the neutron 239 Dirac algebra (in dimension D) 47, 292 Drell-Yan processes/scattering 147

318 Subject Index

Dual field strength tensor, ft, G 228, 244 anti-dual 244 self-dual 244

EEC 164 Energy-energy correlation, see EEC Energy-momentum tensor

Euclidean, ~p, 245 0 P' 42

Equal time commutators, see commutators Equations of motion (Euler-Lagrange

equations) 20

F factor 153 Fermi factor, see F factor Fermion doubling 254 Fermion Lagrangian (Euclidean) 256

Wilson, see Wilson fermion Fine splitting 199 Flavor,J, q=u, d,s,c,b ... , 1 Form factor of the pion, F. 178 Fragmentation function, see Altarelli-Parisi P

function

Gauge axial 28 background field 41 Coulomb 27 Fermi-Feynman 23 fixing 22 Landau, or transverse 23 lightlike 28 Lorentz, or linear covariant 27 parameter, A., a =), -1, ~ = 1-a 22, 23 physical 27 pure gauge field 245 transformations (global and local) 18

Gauge fields B~ (x) 4, 18 pure 245

Generating functional Z[·] 14 Ghosts (Fadeyev-Popov), w, (D 26, 34 Glueball 191, 282, 283 Gluon 4

Lagrangian (Euclidean) 256 Gluon, Wilson action, see Wilson action Gluonium 282, 283 Goldstone-Nambu symmetry 218 Green's function

amputated 55 connected 15 1 PI (One particle irreducible) 17 renormalized, r R, GR 53, 54, 56 unrenormalized, r UD' GUD 54, 56

Gross-Llewellyn Smith sum rule 127 Gupta-Bleuler space, t;GB 21

Hadronization 157 Hamiltonian (density) .Yf' 44 Hyperfine splitting 199

ITEP sum rules, see SVZ sum rules Infinite momentum frame 96 Infrared finite observables 155 Infrared singularities 49 Instantons 249

anti 249 Intercept of a Regge trajectory 132

Jets 157

K factor 152

A, effective A(nJ ) 73,78,285 Au", 270 AMOM 270, 287 Lagrangian

free and interaction .Po, .Pint 4, 7, 8 of QCD, .P, .P QeD' .Po 4, 20, 27, 235 renormalized 52

A-parameter in QCD 73, 78, 285 effective, A(nJ) 78,285

Lattice, cubic 257 Lattice sites, see sites Leading logarithms 123 Light cone expansion 81 Lund model 157,281

Mass, invariant QCD-parameter, A 73, 78, 285 Mass (invariant) of a quark, m 73 Mass, running of a quark, see Running Minimal subtraction scheme, MS 57

modified, MS 57 Moment equations 109 Moments of structure functions Jl(n, Q2) 108 Momentum sum rule 128

n J, effective 77 Nachtmann's variable ~ 139

OZI rule (see Zweig) Operator

product expansion (OPE) 79 renormalization 69

Operators composite 6,68 Normal, or Wick product of, :AB ... Z: 6 time-ordered or T-product of, T A(x) B(y) . ..

Z(z) 6

Parton 97 wave function 181

Peccei-Quinn mechanism 238 Peter-Weyl theorem 307 Plaquette 263 Pontryagin number (see Topological) Potential: "perturbative" 191

Coulombic 194 effective 205 linear 205 and Wilson loop 273

Probability function (quark, or parton density) qf 97, 118, 124, 125

Propagator general definition 11, 15 of the gluon, D~b 22, 23, 28 of the gluon (non-perturbative part) 83

on the lattice 268 of the photon 86 of the quark (non-perturbative part) 81, 82 of the quark, Sij 56, 81

on the lattice 259, 260

QCD sum rules, see SVZ sum rules Quark (field) q, qf, q~, qi I, 118

density functions, see Probability function Quenched approximation 274

Rapidity, y 154 Reduction formulas 10 Regge scattering/trajectory 131, 169 Regularization (dimensional) 45 Renormalization

constants 52 group 66 MS, MS (see Minimal subtraction) of coupling constant Zg 62, 63 of deep inelastic operators, Z:±, Zn 105,109,

III of gauge parameter Z A 60 of gluon field ZB 60 of mass Zm 56, 57 of pion form factor operators, Zn .• 181 of quark field ZF 56,57 of qq, ZM 69,70 Jl-scheme 54

Representations (of SU (3) of color), adjoint and fundamental 18

Running coupling constant, g, 01:, 68, 73 Running gauge parameters, ~, A., ii 68, 74 Running mass, m 68,73,290,291 Running parameters 68

S-matrix 7, 14

Subject Index 319

Scaling 3, 97 ~ 139

Short-distance expansion 79 Sites (lattice) 261 Slavnov-Taylor identities 31 Spectroscopic notation (for quark bound

states) 191 Spherical harmonics with spin '!If 209 Sterman-Weinberg jets 156 String tension, K 276 Strong coupling limit 274 Structure functions,f,,f2,f3,fL 95

longitudinal 95, 113 Sudakov form factor 152 SVZ sum rules 173 Sum rules (see Deep inelastic scattering)

Thrust 163 Tiling 275 Time-ordered product (T-product)

(See Operators) Topological charge 236

operator, QK 236, 250 Topological number, v 250 Trajectory (Regge) 132 Tunnelling 243 Twist 103

higher 104, 142

U(I) current, AI) 232 Ultraviolet singularities 49

V-variable 162 VEV (Vacuum expectation value) 10 Vacuum 19> 235 Veltmann-Sutherland theorem 226 Vertex

ijqB (renormalization of) 61 wwB 38

WKB approximation 241 Ward identities in QED 30 Wick, or normal product (See Operators)

rotation 45 Wigner-Weyl symmetry 217 Wilson action, for abelian fields/gluons 264,266 Wilson coefficients 79 Wilson expansion 79 Wilson fermions 260 Wilson loop 273 Winding number (See Topological)

Yang-Mills (pure) Lagrangian, !l' YM 20

Zweig rule 144