260 Appendix

where Ta are the hermitian generators which satisfy

[Ta, Tb] = irbcTc, tr(TaTb) = ~bab

• Strength tensor

Fp.v=8p.Av - 8vAp. + [Ap.,Av] = -igF:vTa

Fa =8 Aa - 8 Aa + gjabcAb A C p.v p. v v p. p. v

• Gauge transformations

B=-iBaTa

1jJ' = eO 1jJ ~ (1 + B)1jJ

bAp. = -8p.B + [B, Ap.]

bFp.v = [B(x), Fp.v]

• Covariant derivative

Dp.1jJ = (8p. + Ap.)1jJ

Dp.Fpcr =8p.Fpa + [Ap., Fpa]

[Dp., Dv] =Fp.v

• Yang-Mills Lagrangian

.c - 1 (F FP.V) - I Fa Fp.va Y M - -2 tr p.v - - - p.v 2g 4

Abelian Gauge Fields

• Strength tensor

Fp.v = 8p.Av - 8vAp.

• Gauge transformations

1jJ' =e-iq0 1jJ ~ (1 - iqB)1jJ 1

bAp.=-8p.B g'

bFp.v=O

• Covariant derivative

• Lagrangian

.c = -~Fp.vFp.v 4

(A.8)

(A. g)

(A. 10)

(A.l1)

(A.12)

(A.13)

(A.14)

(A.15)

(A.16)

A. Useful Formulae and Notation 261

Dirac Spinors

In the chiral representation

1P= (~~), 1PL=PL1P= (~L), 1PR=PR1P= (;R) , (A.17)

(A.18)

Notice that in the path integral formalism with Grassmann variables, 1P and -;jj are considered as independent variables.

In any representation

• Dirac Lagrangian

.cD =-;jj(i JjJ - m)1P = -;jjL(i JjJ)1PL + -;jjR(i JjJ)1PR - m(ih1PR + -;jjR1Pd JjJ="IJL(OJL + AJL) (A.19)

• Effective action and fermion determinant

eir[AJ = J [d1P][d1j;] exp (i J dx-;jj(i JjJ(A) - m + if)1P )

=det(i JjJ - m) .

A.2 Notation in Euclidean Space-Time

Wick Rotation

Objects with a hat corresponds to the Euclidean space-time.

(A.20)

iA = ixo, ~4 = -ioo, ~4 = -iAo, ~4i = -iFoi , 1'4 = "10, . (A.2I) X' = x" Oi = Oi, Ai = Ai, Fij = Fij , 1" = -i"/" 1'5 = "15,

with these definitions all Euclidean Dirac matrices are hermitian. The action transforms as follows:

i J dx.c = J dxL

with L the corresponding Euclidean Lagrangian.

• Metric

gAJLV = diaD"(- ___ ) b , , ,

(A.22)

(A.23)

262 Appendix

• Dirac matrices

{'it, ;yv} = - 2g/l-V, 1'5 = -1'11'21'31'4

• Dirac Lagrangian

£ = -7Jj(1lJ + m)'lf;

with IlJ = 1'/l-(8/l- + ..4/l-)'

(A.24)

(A.25)

In Euclidean space-time 7Jj is no longer 'If; t /0 but transforms as 'If; t. As a Grassman variable is independent of 'If; in the path integral.

• Yang-Mills Lagrangian

£ = _~Fa F/l-va 4 /l-V

• Fermion determinant

e-r[A] = j[d'lf;][d7Jj] exp ( - j dx7Jj(IlJ(A) + m)'lf; )

= det(Q> + m) .

A.3 Useful Formulae

Traces and Products of Dirac Matrices

t ( /l- V) = {4g/l-V Minkowskian space-time r / / -4g/l-VEuclidean space-time

t ( 5 0 (3 (j P) = {4i€0(3{jPMinkowskian space-time r / / / / / 4€o(3{jp Euclidean space-time,

(A.26)

(A.27)

(A.28)

(A.29)

where €0123 = _€0123 = -1 in Minkowski space-time and €1234 = -1 in the Euclidean case.

(A. 30)

in Minkowski space-time, the Euclidean case is obtained simply by replacing g/l-V --+ g/l-v.

(A.31)

Dimensional Regularization Formulae

In several parts of the book we have used

j d- (q2t _i(-lt-mr(r+D/2)r(m-r-D/2)/1/ q [q2 _ R2]m - (47r)D/2 r(D/2)r(m)(R2)m-r-D/2 '

(A.32)

where D = 4 - € and dij is /1/dDq/(27r)D. We have also made use of the following Feynman trick

B. Notes on Differential Geometry 263

1 r1 1 ab = io dX[ax+b(1_x)j2 . (A.33)

The poles appearing in dimensional regularization are customarily parametrized as

2 N. = - + log4n -, , (A.34)

to

where , ~ 0.577 is the Euler constant.

B. Notes on Differential Geometry

In this appendix, we briefly review some concepts of differential geometry that have been used in the text. We assume that the reader is familiar with the use of differential forms, exterior products, etc. Otherwise, we refer to the literature given in the bibliography for a more detailed account. A general review of all these topics, with an special emphasis on physical applications, can be found in [1].

B.l Riemannian Geometry

Semi-Riemannian Manifold

Given an m-dimensional differentiable manifold M, we say that it is semiRiemannian if there is a continuous (2,0) tensor g, called the metric, such that

a) 9 is symmetric: gp(U, V) = gp(V, U), VU, V E TpM b) 9 is a non-degenerated bilinear form, i.e

gp(U, V) = 0, VU E TpM {::} V = 0,

where TpM is the tangent space to M in the point p. Following our conventions in Appendix A, we will say that gp is Riemannian if it is definitely negative.

Let (U, ¢) be a chart in M and {xl'} the coordinates, then the metric tensor reads

(B.1)

Since gfLv is a symmetric tensor its eigenvalues are real. We denote by (i,j) the index of g, i.e, i is the number of positive eigenvalues and j the number of negative ones (in a strict sense, and as far as 9 can have negative eigenvalues, it is just a pseudo-metric, but for brevity we will keep on calling it a metric). Physically, the two more relevant cases are:

• When 9 is Riemannian. Then i = 0 and j Riemannian manifold:

m and (M, g) is called a

264 Appendix

• When i = 1, j = m - 1, 9 is called Lorentzian metric and (M, g) is a Lorentzian manifold. A particular case being the Minkowski metric TJab = diag(l, -1, -1, -1).

In general, for an arbitrary i and j, 9 is called a semi-Riemannian metric.

Coordinate and Non-coordinate Basis

Let {xlL } be the coordinates in a chart around p E Ai. We define the coordinate basis on TpM as {ell} = {a/axIL} and the dual coordinate basis on T; M as {ell} = {dXIL}, which satisfies < ell, ev >= D~. We denote by < ',' >: TpM 0 T; M -+ IR the usual scalar product (TpM, T; M rv IRm).

When there is a metric tensor defined on !vI, we can define the so called non-coordinate basis as

ea = e~ ~a , {e~} E GL(m, IR) uxlL

(B.2)

with det(e~) > 0 (preserving the orientation). We also impose {ea } to be orthonormal, that is

(B.3)

(in the case of Riemannian manifolds we should substitute TJab by -Dab). Here eb' is the inverse matrix of e~, that is, e~ e~ = D~.

A given vector in both basis reads

V = VlL elL = Vaea (B.4)

and therefore

(B.5)

Notice that we are using Greek indices for objects referred to the coordinate basis and Latin indices for those referred to the non-coordinate basis.

The dual basis of {ea} is {ea} so that < ea, eb > = Db' and therefore

(B.6)

In this basis the metric tensor is just Minkowskian

d ILd v Aa Ab 9 = glLv X X = TJabe e (B.7)

{ ea } and {ea } are called the non-coordinate basis and e~ the vielbein (vierbein in four-dimensional manifolds). Notice that there are many orthonormal basis on TpM, all them related by local orthogonal transformations: e~ = (U-l)b aeb, and eta = Uabeb, where U(p) E 80(1, m - 1) or 80(m) for Lorentzian or Riemannian manifolds, respectively. According to the above discussion, the vielbein changes as etalL(p) = Uab(p)eb IL(P). Under these transformations (B.7) remains invariant.


Lie Bracket

The action of a vector V on a smooth function is defined as V (f) = VILeIL(f) = V IL8IL f E IR. We say we have a vector field when, at each point p in M, we have a vector X p , such that X(f) is a smooth function in M. The set of all vector fields is denoted by X(M).

In the book we have used several times the Lie bracket [X, Y] E X(M), which is defined as follows:

Affine Connection

Another structure that we can introduce on a manifold M is that of an affine connection which allows us to define the notion of parallel transport. An affine connection \7 is a map

\7 :X(M) x X(M)----+X(M) X Y ----+\7xY·

(B.9)

The affine connection corresponds, in the context of (semi-) Riemannian geometry, to the usual covariant derivative. It satisfies the following properties:

\7 X (Y + Z) = \7 X Y + \7 X Z

\7x+yZ=\7xZ + \7yZ

\7 f X Y = f\7 X Y

\7x(fY)=X(f) + f\7xf ,

where X, Y, Z E X(M) and f is a smooth function on M.

(B.I0)

In the coordinate basis we define the connection components fILvA as

(B.ll)

using the general properties in (B.IO), it is easy to show that for any arbitrary vector

(B.12)

In the non-coordinate basis, we define the connection components in a similar fashion

(B.13)

Recalling that ea = e~eIL' we obtain the relation with the components in the coordinate basis

(B.14)

Given an affine connection \7 we can define the notion of parallel transport. Let

266 Appendix

"((t) : [a, b]-+M

t-+"((t) (B.IS)

be a curve on M and X a vector field which on the curve reads: XI')'(t) =

XI-L("((t»e}.L" The vector field X is said to be parallel transported along "((t) if

'VvX = 0, 'It E [a,b] , (B.16)

where

(B.17)

is the tangent vector along "((t). When the tangent vector V itself is parallel transported along "((t) then this curve is called a geodesic. In components the geodesic equation 'VvV = 0 reads

d2 xll- A dxY dxA

dt2 + rll-YA dtdt = 0 (B.18)

with xI-L the coordinates on the curve. As far as the geodesics are the curves that transport parallely their own tangent vectors, they can be considered as the straightest possible curves.

Up to the moment, we have defined the covariant derivative acting on vector fields. In order to extend it to arbitrary tensor fields, first we define it as the ordinary derivative for (0,0) tensors (scalar functions). That is

'Vxf = XU)· (B.19)

Then it is enough to use the Leibnitz rule with arbitrary tensors fields Tl and T2 . In other words

(B.20)

Hence, the covariant derivative in components reads as follows:

(B.21)

The Metric Connection

If we have a metric 9 in our manifold M, it seems natural to look for the affine connection that leaves our scalar product invariant under parallel transport along any curve. That amounts to impose that the metric tensor has to be covariantly constant, i.e

(B.22)

or, equivalently, in components

B. Notes on Differential Geometry

8>..g/1-V - T\/1-g",v - T\vg"'/1- = O.

The affine connection is then said to be metric compatible. Defining the torsion tensor as

T '" rA", T'" >"/1-= >"/1-- /1->"

the metric compatibility implies that T\/1- can always be written as

T\/1- = r\/1- + K\/1- '

where

K"'/1-v = ~(T"'/1-v + T/1-'" v + Tv'" /1-)

is called the contorsion tensor and

r"'/1-V = ~g"'>"(8/1-gv>.. + 8vg/1->" - 8>..g/1-v)

267

(B.23)

(B.24)

(B.25)

(B.26)

(B.27)

are known as Christoffel symbols. If apart from being metric compatible, 'V is such that the torsion tensor vanishes, i.e, it is symmetric, then it is called a Levi-Civita connection.

The fundamental theorem of (semi-)Riemannian geometry states that given a (semi-)Riemannian manifold (M, g), there is a unique symmetric connection, i.e. with vanishing torsion, which is compatible with the metric g. This unique Levi-Civita connection is given by the Christoffel symbols. In this case the geodesics are not only the straightest possible curves but also, locally, the shortest ones.

Metric Connection in the Non-coordinate Basis

Defining the Ricci rotation coefficients A Ad

rabe = 'T/adr be (B.28)

for Lorentzian manifolds, (the 'T/ad tensor is substituted by -8ad in Riemannian manifolds), the metric compatibility condition reads

Tabe = - Teba .

In such case, defining lea, eb] = Caged, the torsion free condition is d Ad Ad

Cab = r ab - r ba

since now the torsion tensor is written as

T abd = T abd - T adb - C bd

(B.29)

(B.30)

(B.31)

268 Appendix

Curvature and Torsion

We can define two intrinsic objects on a manifold. The torsion tensor is a (1,2) type tensor defined as follows:

T(X, Y) = \7xY - \7yX - [X, Yj ,

and the curvature Riemann tensor, which is type (1,3), reads

R(X, Y,Z) = \7x\7y Z - \7y\7xZ - \7[X,y]Z .

In the coordinate basis their components are A 'A 'A

T tJ-V =r tJ-V - r VtJ-

R'\tJ-v =OtJ-t"'VA - Ovt"'tJ-A + t'7VAt"'WI - t'7tJ-At"'V'7

with the symmetry properties

TAtJ-v=-TAvtJ-

R"'AtJ-V=-R"'AVtJ- .

(B.32)

(B.33)

(B.34)

(B.35)

From the Riemann tensor we can define the Ricci tensor, which is of (0,2) type, and the scalar curvature

RtJ-v::=RAtJ-AV

R::=gtJ-v RtJ-v . (B.36)

If the connection is Levi-Civita these tensors satisfy further symmetry properties, which in components are

R"'AtJ-V = - RA"'tJ-V

RtJ-v=RvtJ-

R'" AtJ-V + R'" tJ-VA + R'" VAtJ- = 0 First Bianchi identity (B.37)

(\7 ",R)'7 AtJ-V + (\7 tJ-R)'7 AV'" + (\7 vR)'7 A"'tJ- = 0 Second Bianchi identity.

In analogy with gauge fields, we can introduce a connection one-form tab ::= tacbec = tatJ-bdxtJ-, that allows us to define the following two-forms:

dea + tab 1\ eb = ~ Tabceb 1\ eC ::= T a 2

'a ' a A cIa 'c ,d _ a dr b + r c 1\ r b = "2 R bcde 1\ e = R b , (B.38)

where d and 1\ denote the usual external derivative and wedge product, respectively. Under the orthogonal transformations U (p) in TpM, introduced before, these objects transform as follows:

T,a=uabTb

R'b=Uac(U-l)dbRC d . (B.39)


From these transformations, we obtain the change in the connection one-form, which is given by

F'~ = UacFcd(U-l)db - (U-l)db(du)ad . (B.40)

Isometries

Let (M,g) be a (semi-)Riemannian manifold. A diffeomorphism which maps a point of coordinates x into that of coordinates x' is called an isometry if it preserves the metric, i.e

g~V(x) = gILv(x) Vx

or equivalently

ox'p ox,u gILv(x) = OXIL oxv gpu(x').

(B.41)

(B.42)

The identity map, the composition and the inverse of an isometry are also isometries, therefore they form a group.

If we want the infinitesimal transformation

(B.43)

to satisfy (B.41) to first order in €, we find

_ OXIL oXv IL ogpu(x) o - ~gILu(x) + ~gpv(x) + X (x) 0 (B.44) u~ u~ ~

Such vector field X, which generates an isometry, is called a Killing field. This expression, in the case of Levi-Civita connections, can be rewritten as

0= Xu;p + Xp;u . (B.45)

Given any pair of Killing fields X and Y we can obtain another one just from

i) a linear combination (aX + bY), with a, b constant real numbers; ii) the Lie bracket [X, Y].

Therefore, the set of Killing fields on a manifold form a Lie algebra. Finally, X and Yare said to be dependent if there are two constants a and b such that aX(x) + bY(x) = 0, Vx. Indeed, the number of independent Killing vector fields can be larger than the dimension of the manifold, although their maximum number on an m-dimensioIial manifold is m( m + 1) /2 [1]. Those manifolds with the maximum number of Killing fields are called maximally symmetric.

B.2 Homogeneous Spaces

Given a manifold M and a Lie group G, we say that G acts on the left on M when, for any 9 E G, there is a diffeomorphism

270 Appendix

Tg:M -4 M

x -4 Y = Tg(x)

such that Tgg, = TgTg' and y is a smooth function not only of x but also of g. An action of G on M is said to be transitive if

\;fx,y E M :Jg E G so that y = Tg(x).

Whenever there is a manifold M for which there is a transitive action, we say that M is an homogeneous space. In particular, taking M = G and defining the action as the left group inner product, we see that the Lie groups are homogeneous spaces.

The set of transformations leaving a point x E .tv! invariant

Hx = {g E G I Tg(x) = x}

is called the isotropy group of x. For homogeneous spaces, it is possible to show that Hx is isomorphic to Hx" for any x and x' E M. That isomorphism is just h -4 ghg- 1 where g is such that Tg(x) = x'. Thus, from now on we will denote by H the isotropy group of M.

If we now take an arbitrary Xo E M, then

GjH-4M

gH -4 Tg [xol

is a one to one map. That is, any point in M can be obtained from Xo by applying some Tg • Conversely any equivalence class in G j H can be associated to a unique point in M (notice that the above map is independent of the representative g). In such sense we have that M '" G j H and we can represent homogeneous spaces as coset spaces. In particular, we have dimM = dimG j H = dimG - dimH.

Whenever the group G of an homogeneous space M has an automorphism a which is involutive (a2 = 1) and H is a-invariant, we say M is a symmetric space. If, in addition, G is compact, H is the maximal invariant subgroup. Such spaces are physically interesting since they allow us to associate a parity to the G generators. Indeed, it is always possible to assign a positive parity to the H generators and negative to the rest. In this book, the relevant examples of symmetric spaces are SN '" O(N + l)jO(N) and SU(N) x SU(N)jSU(N) '" SU(N).

B.3 The Geometry of Gauge Fields

Definitions

In this appendix we will introduce the gauge fields as geometrical objects. This construction is fundamental in the study of anomalies, non-perturbative effects, instantons, vacuum structure, etc. in gauge theories. We start by considering the space-time as some compact d-dimensional manifold M with


Euclidean signature. In general, this manifold will be described as the union of r patches as follows: M = Dl U D2 U ... U Dr. Anyone of these patches is homeomorphic, i.e. topologically equivalent, to IRd. Let G be some compact Lie group (it will be SU(N) in most of the interesting cases in this book). Within this formalism, our usual gauge fields are nothing but a map where we assign to each point in M a one-form valued on the Lie Algebra

X -? A(x) = -igTa A~(x)dxi' = Ai'(x)dxi' ,

where Ta are the hermitian group generators with

[Ta, TbJ = irbcTc

(B.46)

(B.47)

normalized so that tr(TaTb) = (jab /2. In fact, this is not the complete definition of a gauge field, since we still have to specify how it transforms from patch to patch. Let us then consider some point x belonging to the intersection of two patches Di and Dj . Notice that we have assigned two different gauge fields A (i) (x) and A(j)(x) to each of the points belonging to Di n Dj .

To have a properly defined gauge field in the whole manifold M, the two gauge fields A{i)(x) and A(j)(x) should be related by a gauge transformation gij(X), as follows

(B.4S)

with d = 0/ oxi'dxi'. Mathematically these gauge transformations, which are maps from Di n Dj into the gauge group G, are called transition functions and our gauge field A is called a connection on the principal bundle with base M and fiber G. Consistency requires that in any triple intersection like Di n Dj n Dk the transition functions obey

(B.49)

When dealing with differential forms it is very useful to define a wedge or external product, that allows us to obtain forms of higher order. Its definition is the following

(B.50)

Indeed, the strength tensor, which is a two-form within this formalism, is given by

(B.51)

The product A 1\ B for any A, B will be sometimes abbreviated as AB. It can be easily checked that the F components have the usual form

(B.52)

Some of the properties of the strength tensor follow very easily from the above definitions, as for instance, the Bianchi identity

dF+ [A,F] = 0 (B.53)

272 Appendix

since

dF = d( dA + A 2) = dAA - AdA = FA - AF = [F, A] , (B.54)

where we have used the identity d2 = O. We can also define the covariant differential as

DX = dX + [A,X] , (B. 55)

and thus the Bianchi identity reads DF = o. Once we have defined F, we can now build the Yang-Mills action nmc

tional, which is written as

SYM[A] = ~ r trFp,vFP,v, 2g JM (B.56)

where we have omitted the element of volume, which is defined as usual. The corresponding equations of motion are

(B.57)

Whenever F(x) = 0 't/x, we say we have a pure gauge field since then, in any patch, A can be written as A(x) = g(x)-ldg(x), where g(x) is a map from the corresponding patch into the gauge group C.

A gauge field will be called trivial and the corresponding transition function gij small, whenever gij belongs to the trivial class (that is, gij can be written as gij(X) = eio:(x) ~ 1 +ia(x) for any x, a(x) being some Lie algebra valued function). Such small gauge transformations are the ones considered in perturbation theory.

In applications to four-dimensional quantum field theory one is initially interested on Minkowski space-time. However, the formalism usually requires a Wick rotation to a four-dimensional Euclidean space-time (for instance, to define (anti)self-dual gauge configurations properly, as we will see below). Moreover, it is also convenient to compactify the space-time to the S4 sphere. This manifold is described with two patches as S4 = D+UD_, where D+nD_ is homeomorphic to the S3 sphere (see Fig. B.l). In this figure D+ represents the standard space-time, the equatorial S3 sphere is the space-time infinity and D_ is just an artifact of the compactification.

At this point it is convenient to introduce the dual strength tensor as

- I-F = "2 Fp,vdxP, 1\ dxv , (B.58)

where

p _ 1 p,vpo F p,v - "2E po , (B.59)

and therefore, following our conventions for the Euclidean Levi-Civita symbol in Appendix A,

trF2 = trF 1\ F = -~trFp,vpP,Vdxll\ ... 1\ dx4 . (B.60)


---------

D

Fig. B.1. Pictorial view of the S4 compactification of the Euclidean space-time. D+ represents space-time and its intersection with D-, homeomorphic to the equatorial sphere S3, is the space-time boundary

Self-Dual Fields and the Lower Bound of the Action

A field is called self-dual or antiself-dual when

F/-,v = ±F/-,v, (B.61)

Such configurations can exist in S4, but not in Minkowski space. Now, starting from the inequality

r - 2 iM(F/-'v±F/-,v) ;::::0, (B.62)

it is not difficult to show that

[A] - 1 1 D /-'V 1 1 D - /-'V SYM - -2 trr/-'vF ;:::: -2 trr/-'vF 2g M 2g M

= - :2 1M tr F2 . (B.63)

Or introducing for further convenience the functional

W[A] = ---;. r trF2, 87r iM

(B.64)

we have

(B.65)

Furthermore, using the Bianchi identity in (B.53), it is easy to show that (anti)self-dual fields are solutions of the equation of motion, i.e., DF = O. In addition, for these field configurations we have

87r2 SYJ\.dA] = -2 W[A] ,

9 (B.66)

i.e., they saturate the above inequality. In order to understand better what the above equality means, let us notice that tr F2 is a closed form, that is,

274 Appendix

d(trF2) = O. Then, from the Poincare lemma it follows that there is a 3-form Q3(A) such that trF2 = dQ3(A), at least locally, i.e., in every patch. This is the so called Chern-Simons (CS) form (see below) and its explicit expression is

Q3[A] = tr (AdA + ~A3) Thus we can write

W[A] = -; r trF2 811" JD+UD_

= ~; (1+ dQ3[A+] + 1_ dQ3 [A-])

= 8-; (r Q3[A+] + r Q3[A-]) 11" JaD+ JaD_

= 8-; r (Q3[A+]- Q3[A-]) . 11" JS 3

(B.67)

(B.68)

Observe that A+ and A- are the gauge fields defined on D+ and D_. Therefore A+ = 9'+: (A- +d)9+-, where 9+_ is the transition function defined on 8D+ = -8D_ rv 8 3, i.e. a map 9+_ : 8 3 --+ G. In the third step we have used Stokes theorem. If we take A- = 0 and then A+ = 9.+:d9+-, we find

W[A] = 8-; r Q3[9.+:d9+-] = 241 2 r tr(9.+:d9+_)3. (B.69) 11" JS3 1r JS3

This result remains valid for any A- (see (B.76) below) and then W[A] only depends on the gauge transition function. Moreover, we will show below that W[A] is related with the Dirac operator and the homotopy group 1I"3(G). For example, if G = 8U(N) and N ~ 2 we have 1I"3(8U(N)) = 71.. In that case, the continuous applications 9+_ are labeled by W[A] E 71., which is called the winding number.

Instantons

When dealing with the strong CP problem (see Sect. 5.6), the gauge configurations of the form A- = 0 and A+ = 9'+:d9+- on 8 3, turn out to be very relevant. Indeed, both anti and self-dual solutions of this kind have been explicitly found and are called instantons. One example of such a solution is the 8U(2) gauge field [2]

(B.70)

so that

F - 4iE >.2 /1-// - /1-//[1 x-a 12 +>.2]2 ' (B.71)


where EJ.LV = niJ.Lv (ji/2 for i = 1 2 3 niJ.Lv = _nivJ.L = f.iJ.Lv for 1/ v = 1 2 3 " , , '" '/ t-"'" ,

and 'f}iJ.L4 = {jiJ.L; X are Euclidean cartesian coordinates and a and ). are some parameters, which can be interpreted as the position and size of the instanton. The main properties of this gauge field configuration are the following:

• The tensor EJ.Lv is antisymmetric and self-dual and so is FJ.Lv.

• It is a solution of the Euclidean Yang-Mills equation of motion which in the limit I x I~ 00, i.e., on the 8 3 infinity sphere is a pure gauge.

• The fact that the action of this field is just 8n2 / g2 leads, according to our previous discussion, to the result that the winding number of this field is 1. This is also the label of the homotopy class of the g field defined on the infinity sphere 8 3 (see below).

This field configuration is called a one-instanton solution. Note that the instanton is a solution in Euclidean space-time and not in some point of the physical space at a given time, as it would be the case of standard solitons like skyrmions or monopoles. From (B.70) it is very easy to find the antiinstanton solution with winding number -1. It is also possible to find explicitly N-instanton solutions with winding number N, which correspond to N oneinstantons of different sizes located at different Euclidean space-time points. Instanton gauge configurations are thought to playa very important role in the dynamics of quantum gauge theories (see [3]). In particular, we have seen in Sect. 5.6 that, even though they are classical events in imaginary time, they can provide paths for tunneling between different topological vacua in the semiclassical approximation.

Characteristic Classes

We will now turn to the concept of characteristic classes defined over a spacetime manifold M of even dimension 2n [1, 4]. They are local forms P[F, RJ, where F denotes the gauge strength field tensor in (B.51) and R is the Riemann curvature tensor of the manifold. For the purposes of this book, it is enough to consider the dependence on the gauge curvature F. These forms are defined through the following two properties:

• They are closed, that is, dP[F] = O. • They are not globally exact. In other words, they can only be written as

the exterior derivative of a given form, P = dQi, locally over each patch Di , but not globally over the whole M.

The second property above means that the integral of P[F] over the whole manifold, using the Stokes theorem, only depends on the gauge transformations defined over the intersections among different patches. That implies [1] that the global integrals of characteristic forms are topological invariants. Furthermore, we can choose them so that they are nonvanishing if and only if there are gauge transformations non-homotopically equivalent to the identity map.

276 Appendix

It is not difficult to show that any polynomial in F satisfying gauge invariance, i.e. P[g-l Fg] = P[F] is a characteristic class. Such a polynomial can always be written as a sum of 2m-forms of the type Pm = trFm, with m = 1, ... , n. It can be checked that Pm fulfills the above two properties, since F satisfies the Bianchi identity in (B.53). Let us now consider the (2m - 1)form Q2m-l, related to Pm in each patch via the second property above, that is

trFm = dQ2m-dA, F] , (B.72)

where A is the gauge field. The Q2m-I[A, F] form is known as the CS form. We had already met it for m=2.

In Chap. 4 we have seen that Q5 appears in the formulation of the WessZumino-Witten (WZW) term and that we need to know how it transforms in order to include the gauge fields. In addition, the transformation properties of Q3 have been used above in (B.68) and (B.69). Therefore, it is very useful to obtain how the CS form changes under a gauge transformation. With that purpose, let At and Ft be a one-parameter interpolation of the connection and curvature forms, with t E [0, 1]. That is, At is a continuous function of t that varies between Ao and AI, and the same for Ft. We can then define a differential operator It acting as follows

ltAt =0 ; ltFt = dt(A1 - Ao)

ltCBpAq) = (ltEp)Aq + (-l)PEp(ltAq) , (B.73)

Ep and Aq being two arbitrary forms of degree p and q. Then, from the definition in (B.72) of the CS form and (B.73), it is useful to write

Q2m-dA, F] = 11 lttrFtm (B.74)

with Ft given by

Ft = tdA + t 2 A2 = tF + t(t - 1)A2 . (B.75)

The next step is to write the difference between the CS form for A9 and that for A as it appears in (B.74). Then, after some algebra it is possible to obtain [4] the transformation formula

Q2m+1[A9, F9]

= Q2m+1[A,F] + Q2m+1[g-1dg, 0] +do:2m [A,g-1dg] , (B.76)

where O:2m is the following 2m-form

O:2m[A,g-1dg] = 11ltQ2m+1[At,Ft] , (B.77)

At and Ft being the forms that interpolate between a gauge field A and its gauge transformed A9 = g-1(A + d)g. That is

At =g-ldg + tg- l Ag

Ft =dAt + A; . (B.78)


Moreover, the CS form Q2m+1[g-ldg,0] for a pure gauge configuration can be obtained by taking A = g-ldg, F = ° in (B.74) and performing the t integration, which yields

Q [g-ldg 0] = (_I)m (m + 1)!m! tr(g-ldg)2m+1 2m+1, (2m + I)! . (B.79)

As we have discussed in Chap. 4 in connection with gauge anomalies, it is very useful to introduce the first order variation of the CS form

Q~m[8A,A,F] == Q2m+IlA + 8A,F]- Q2m+1[A,F] ,

where 8A = A9 - A. It can be shown that

Q~m[8A, A, F] = m(m + 1) 101 8t(1 - t)str(8A, dCA, Ftm- 1)) ,

(B.80)

(B.81)

where Ft is given in (B.74) and str means the symmetrized trace, which acts over a set of matrix-valued forms En = E~.>.a as

perm

str(Eb d(E2' E3 , ... ))

= str(Eb dE2, ... ) + (-I)P2 str(E1, E2, dE3 , ... ) + ... , (B.82)

where, in the first formula above, we sum over all the permutations of the .>. matrices, P(i) denoting the permutation index, and Pi the degree of the form E i ·

The Index Theorem

We will analyze here a very powerful topological result, the Atiyah-Singer index theorem [5]. This theorem relates the index (which will be defined below) of a gauged Dirac operator, with topological invariants built from characteristic classes, which depend on the strength tensor F. We will not consider here the manifold curvature contributions to the index theorem (see [4] for further details). Thus, let M be a 2n-dimensional Euclidean space-time manifold and .fJ the Dirac operator with a gauge field A

(B.83)

where the "Ii-' are Euclidean Dirac matrices. Now, let us define the rightchirality Dirac operator i f/JR == i f/JPR, where "15 is defined in arbitrary even dimension as the product of the 2n Dirac matrices. Then, the index of i f/JR is defined as

(B.84)

278 Appendix

where n± is the number of eigenstates ¢o of i .w[A] with zero eigenvalue and chirality ±, namely, r5¢O = ±¢o. The Atiyah-Singer index theorem then states that

ind(i WR)[A] = j)Ch(F)]vol , (B.85)

where ch(F) is known as the Chern character of F, defined as

ch(F) == trexp (~~) . (B.86)

The subindex "vol" in (B.85) means that only the term proportional to the volume 2n-form contributes. That is, if we expand the Chern character (B.86) in powers of the curvature F, only the Fn power should be retained. Thus, we get for the index theorem

(B.87)

Homotopy Groups

We will give here a brief account of some homotopy results that have been used in the text. The general aim of this techniques is to classify the continuous maps f : X ~ Y, where X and Yare topological spaces, in equivalence classes according to homotopy transformations. For detailed reviews we refer the reader, for instance, to [1, 6]. Here we are only interested in applications f : sn ~ G, with G a Lie group. We say that two given applications f and 9 are homotopically connected if there is a continuous map ht : I x sn ~ G, where tEl = [0,1]' such that ho = f and hI = g. An homotopy class is the set of applications homotopically connected. Furthermore, the homotopy classes from sn ~ G form the so-called n-th homotopy group, denoted as 1fn(G). We have seen different physical examples in the book where these groups play a central role. For instance, the non-trivial solutions of the non-linear sigma model (Chap. 3), the WZW term and the non-perturbative anomaly (Chap. 4), instantons (Chap. 5), etc. Next we list the 1f n (G) of the Lie groups that are needed in the text

1fl(SU(N))=1f2(SU(N)) = 0 for N 2: 2

1f3(SU(N)) =71.. for N 2: 2

1f4(SU(2)) =71..2 1f4(SU(3)) = 0

1f5(SU(N)) =71.. for N 2: 3 . (B.88)

If the homotopy group is 71.., the integer that labels a given class is usually called the winding number, in analogy with the maps from SI into U (1) '" SI (1fl (SI) = 71..). In general, the applications belonging to different homotopy classes will be labeled by topological invariants. For instance, if we consider 1f3(SU(N)) = 71.. with N 2: 2, the winding number of a given map

C. Aspects of Quantum Field Theory 279

9 : S3 ---4 SU(N) is nothing but W[g-ldg], given in (B.69). It is not difficult to check that W is indeed the winding number, simply by considering the index theorem in (B.87) with M = S4. Then, from the discussion in (B.68) and (B.69) we have:

W[g-ldg] = ~ r tr(g-ldg)3 = -~ r trp2 = ind i .fl>R[A] , (B.89) 2471" } 83 871" } 84

where A is any gauge field on S4 that has 9 as gauge transition function on S3 = 8D+ = -8D_. On the one hand, W is the index of a Dirac operator and hence it is an integer. On the other hand, according to our previous discussion on characteristic classes, it is a topological invariant and hence only varies when 9 is changed from one homotopy class to another. The same argument can be applied for 7I"5(SU(3)) = 7l as it is done in Chap.4 in the analysis of the WZW anomalous action.

B.4 References

[1] M. Nakahara, Geometry, Topology and Physics, lOP Publishing, 1990 [2] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tupkin, Phys. Lett.

B59 (1975) 85 [3] R. Rajaraman, Solitons and Instantons, North-Holland, 1982 [4] L. Alvarez-Caume and P. Cinsparg, Ann. Phys. 161 (1985) 423 [5] M.F. Atiyah and LM. Singer Ann. Math. 87 (1968) 485 and 546 [6] C.W. Whitehead, Elements of homotopy theory, Springer-Verlag, 1978

c. Aspects of Quantum Field Theory

C.l Renormalization Group Equations

In this section we review the basic aspects of the renormalization program in quantum field theory (QFT) [1]. For the sake of definiteness and simplicity we will consider the example of the ),tJ>4 model described by the Lagrangian

L = ~[8 tJ>8J.LtJ> - m 2tJ>2]_ ~tJ>4 . (C.l) 2 J.L 4!

As it is well known, if one applies the standard methods of perturbation theory to this Lagrangian, one arrives to divergent integrals in the Green functions. However, as far as the theory is renormalizable, it is possible to redefine the parameters and the fields of the original Lagrangian to find finite answers. In order to do so one has to start from what is known as the renormalized Lagrangian

_ _ 1 J.L 1 2 2 ),0 4 LR - Lreg + Let - -8J.LtJ>08 tJ>0 - -motJ>o - ,tJ>0 , (C.2)

2 2 4.

where Lreg is the regularized Lagrangian, i.e. the original one but including some regularization method for the divergent integrals. For instance, to

280 Appendix

introduce a cutoff at some given energy scale A or to apply dimensional regularization, which is nothing but working in D = 4 - ( dimensions [2] . .cct contains the counterterms needed to cancel the infinities of the Green functions when A -> CXl or ( -> 0, up to some given order in perturbation theory. Note that those counterterms are defined to cancel the divergences, so that their finite parts can be chosen arbitrarily. When the theory is renormalizable, as it happens in our AP4 model, .cR has the same form as the original Lagrangian but now in terms ·of the so called bare field, mass and coupling Po, mo and AO which depend on A or (. The renormalized Lagrangian can also be written in terms of the renormalized field, mass and coupling PR, mR

and AR, which are finite, as

1 1 2 2 AR 4 .cR = 28j.LPRfY.LPR - 2 mRPR - 4fPR

1 j.L 1 2 2 AR 4 -113-8j.LPR8 PR + 11o- m RPR + 111-, PR 2 2 4.

1 1 2 2 AR 4 = Z3 2 8j.LPR8j.LPR - Z02mRPR - Zl4fPR , (C.3)

so that we have

Z3=1- 113

Zo=l- 110

Zl =1 - 111

and '" _Z-lj2", '¥R - 3 '1'0

2 Z Z-l 2 mR= 3 0 mo

AR=ziz1lAo.

(C.4)

(C.5)

It is possible then to obtain Green functions from the renormalized Lagrangian in terms of the renormalized field, mass and coupling. The result will be the same found with the original Lagrangian (but now written in terms of the renormalized magnitudes) plus the contribution of the new Feynman rules coming from the extra 11 terms [3]. However, we can now choose, order by order in perturbation theory, the 11i constants in order to absorb the previously found divergences so that the new (renormalized) Green functions are finite when A -> CXl or ( -> O. These renormalized Green functions are thus defined as

C~) (Xl, X2, ... ,Xn ) =< OIT(PR(xd, PR(X2), ... ,PR(Xn )) 10 > Z -n j 2C (n) ( ) = 3 0 Xl, X2, ... ,Xn , (C.6)

and they can be obtained order by order in perturbation theory, using the renormalized coupling AR as the expansion parameter. Notice that with this procedure, we have just determined the divergent part of 11 i , but the finite part remains arbitrary. Every different choice of this finite part gives rise to


what is called a renormalization scheme. For instance, in the Minimal Subtraction (M S) scheme one works in dimensional regularization and defines the L1i just to cancel the poles in E without adding any finite part. In the Modified Minimal Subtraction scheme (M S) one uses the fact that the E poles always appear in the combination

2 N. = - -, + log47l" (C.7)

E

due to the specific way to continue the Green functions to arbitrary dimension D = 4 - E. Thus, in order to cancel the infinities and at the same time to avoid the somewhat artificial, and log 47l" constants, in the ]v! S the various L1i contain not just the E poles but the whole N. factors. For example, in our case, and to lowest order in the coupling, we find the following renormalization constants

MS

Zo = 1 + (4~j2.

Zl = 1 + d;)~.

Zo = 1 + 2(~~)2 N.

Zl = 1 + 2zi:r)2 N •.

In dimensional regularization, apart from the E parameter, we have to introduce an arbitrary energy scale J.L in order to keep the volume element dij = dD QJ1,' / (27l")D with the right dimensions. Therefore, all the renormalized quantities such as the wave function, mass and couplings will depend on J.L; changing its value will produce a change in those magnitudes. Something similar happens when a cutoff 11 is used to regularize the divergent integrals. Nevertheless, the physical observables should not depend on such an arbitrary parameter and, as we will see below, this condition imposes how the couplings, masses, etc. should depend on the scale J.L.

Both the M Sand M S renormalization schemes are examples of massindependent renormalization methods since the Zi are mR independent. Such methods are very useful for calculational purposes but the physical interpretation of the parameters and the Green functions is more obscure. In particular mR is by no means the mass of any physical state, since in fact it is f.L dependent. There are however, other kind of renormalization techniques, called on-shell schemes. The name is due to the fact that the finite additive constants appearing in the definition of the Zi are chosen so that some Green functions can have a physical interpretation when their momenta are on shell. This kind of renormalization schemes is more transparent from the physical point of view but much more involved for making calculations. In practice, the mass-independent schemes are used when dealing with processes where the masses of the particles are not relevant or precisely defined. That is the case of quantum chromo dynamics (QCD), since the masses of the quarks have a difficult interpretation (see Chap. 6). However, when working with

282 Appendix

particles whose masses are directly measured or when they are important for the process, it could be convenient to use an on-shell scheme. That is the case, for instance, in Chap. 7, where we study the radiative corrections to the electroweak theory, since in that case the masses of the W± and Z play a fundamental role.

Hence, for the sake of simplicity, we will use a mass-independent renormalization scheme together with dimensional regularization. Thus the renormalized connected Green functions in momentum space are obtained as

G~r;1 (kb k2,· .. , kn; AR, mR, J.l)

= Z;n/2(AR,€)G~~)(kl,k2, ... ,kn;AO,mO'€)' (C.S)

where G~r;1 is finite in the € -+ 0 limit. As far as we are using a mass

independent renormalization scheme [4], Z;/2 does not depend on mR. Therefore in momentum space we have

C<n) _ Zn/2c<n) cO- 3 cR·

Using the fact that G~~) is J.l independent, we can write

dC<n) -:-;:,......=cco,,--:- = 0 d(logJ.l)

or in other words

( 0 dAR 0 dmR 0) Gn o(logJ.l) + d(logJ.l) OAR + d(logJ.l) omR cR

__ ?'!:Z-l dZ3 en - 2 3 d(logJ.l) cR·

It is then usual to define

dAR (3(AR)=' d(logJ.l)

1 dmR 1'm(AR) =. - del )

mR ogJ.l

( ' )_ld(logZ3) l' AR --

- 2 d(log J.l)

(C.g)

(C.lO)

(C.II)

(C.12)

Thus we arrive to the Callan-Symanzik renormalization group equation (RGE) [5]

( 0 0 0 ) o(logJ.l) + (3(AR) oAR + 1'm(AR)mR omR + n1'(AR)

xG~r;1 (kb k2'···' kn; AR, mR, J.l) = 0 . (C.13)

Notice that in the general case, the (3, l' and 1'm coefficients should also depend on mR/ J.l, but that does not happen with the scheme we are using. The RGE gives a description of the renormalized Green function dependence on the


scale JL. As we will see later, a change in the renormalization scale yields a change in G~";i, )..R and mR in such a way that the physical observables (not the Green functions) will not change.

Let us now solve the RGE. First we consider the homogeneous equation (r = 0) and we define

JL' t = log - .

JL (C.14)

In the following we will consider only renormalized quantities (G~";i, )..R and

mR) and then, for simplicity, we will omit from now on the index R, i.e., (G~n), ).., m) = (G~";i, )..R, mR). We will also take).. = )"(JL) and m = m(JL). For later convenience we will now introduce the functions X(t,)..) and m(t;).., m) which are the solution of the differential equations

aX(t; )..) =(3(X) at

om(t;).., m) = (X) - ( .).. ) at "'1m m t, ,m (C.15)

with the initial conditions X(O,)..) = ).. and m(O;).., m) = m. It is not difficult to show that

(C.16)

is a solution of the homogeneous RGE. For the complete equation we find

G~n) (kl, k2' ... ,kn;)", m, JL)

- G(n) (k k k . '() - () t) n J: "Y().(t'))dt' (C.17) - c 1, 2,"" n, /\ t ,m t ,JLe e 0

However, )"(JL') = X(t) = )..' and m(JL') = m(t) = m' and therefore the above solution can be written in terms of the renormalized quantities).. and m only

G~n)(k1' k2' ... ,kn;)", m, JL) ft I I - G(n)(k k k . \1 I ') n J, -y(t )dt

- c 1,2,···, n,/\,m,JL eo, (C.18)

where)..' and m ' are just the renormalized parameters at the scale JL'. Usually "'I is called the anomalous dimension and X(t) the running coupling constant. It is useful to consider the running mass m(t) as another coupling not to be confused with any physical mass since we are working in a mass-independent renormalization scheme.

Let us now consider some physical magnitude M. When we calculate this magnitude the final answer will be in general a function of ).., m and JL

M = M()..,m,JL) , (C.19)

where, as discussed above, ).. = )"(JL) and m = m(JL). However, M has to be JL independent since it is an observable magnitude. This means that the explicit dependence on JL should be cancelled by the implicit dependence through ).. and m or, in other words

284 Appendix

where we have defined the R operator as the total log p, derivative. Note that this RGE is different from that for Green functions in (C.13), since in that case we also have the ano.malous dimension term, which means that the Green functions are not p, independent, in contrast with physical observables. Let us see how this works in the very important case of 8-matrix elements. To start with, we will consider the renormalized two point connected Green function (the propagator) G~2)(k,A,m,p,). The corresponding RGE is

-(2) [R + 2-Y(A)] Gc (k, A, m, p,) = 0 . (C.21)

However, around the physical mass Mph it is possible to write

-(2) _ R(A,m,p,) Gc (k,A,m,p,)- k2-M2 +L1(k,A,m,p,), (C.22)

ph

where L1(k,A,m,p,) is a regular function at k2 = M;h' From the above equations it is not difficult to show

[R + 2-Y(A)] R = 0 (C.23)

RM;h =0

and therefore, as it should be, the physical mass Mph is p, independent. Let us now introduce the so called amputated Green functions G~n). In momentum space those functions are obtained by removing a propagator from each externalline of the corresponding connected Green function; or in other words, by including an inverse propagator factor for every external line. This definition is quite useful since the Lehman-Symanzik-Zimmermann reduction formula can be written in terms of these amputated Green functions just as

(C.24)

Furthermore, it is very easy to show that the RGE corresponding to these Green functions is similar to that for connected Green functions, although with an opposite sign in the anomalous dimension term. Thus we find

R8 = lim R (Rn/ 2G(n») k~ .... M2 a

• ph

= k~~~~h [R ( Rn/2) G~n) + Rn/2R ( G~n»)] ,

but in addition we have

R (Rn/2) = _n-yRn/2

R ( G~n») =n-yG~n) ,

(C.25)

(C.26)


and therefore we arrive to

(C.27)

Thus, the S-matrix elements and the physical predictions of the theory are J-l independent in spite of the fact that Green functions are not [6]. Indeed this is one of the most important results in renormalization theory. However, in practice one obtains the S-matrix elements approximately and therefore some residual J-l dependence arises in these perturbative results since the RGE only applies to exact Green functions.

The RGE can also be used to extract information on the dependence of the Green functions on external momenta k i . With such purpose, we make the scale transformation

(C.28)

and we define J-l' = etJ-l = aJ-l, i.e., a = et . Then, according with the RGE we have

(C.29)

_ DtG(n)(k.' \1 -t 1 ) n f -y(t')dt' -e c "A,e m,J-le 0 ,

where D is the canonical dimension of G~n) (D = n for scalar fields) and we have used the fact that this Green function has to be homogeneous of degree D on ki' m and J-l. As a consequence, we arrive to

G~n)(aki; >'(J-l) , m(J-l) , J-l)

= G~n) (ki ; >.(aJ-l) , m(aJ-l)ja, J-l)eDt+n J: -y(t')dt' (C.30)

This is an exact equation describing the scaling behaviour of Green functions on external momenta and it is the reason why 'Y is called the anomalous dimension. As we can see the scaling behaviour is controlled by the f3, 'Ym and 'Y. However, in practice one only knows these functions perturbatively and therefore we cannot write and solve the above equation exactly. Nevertheless, as we will see in a moment, there are special regions where perturbation theory can be safely applied and the above equation turns out to be very useful to obtain the asymptotic behaviour of the theory. Let us concentrate on this issue in more detail, but for simplicity we will consider only the case of the (3 function. As discussed above, the evolution of the running coupling constant and the renormalized coupling is given by

d>'J-l d(log(J-l)) = (3(>'(J-l)) (C.31)

with >'(J-lo) = >'0' If the theory is renormalizable we can consider J-l as large as we want (otherwise we would have to introduce some cutoff 11 with J-l « 11).

286 Appendix

Let us remark that if we want the perturbative expansion to make sense, the expansion parameter has to be small. In our case this parameter is the renormalized coupling >., which depends on fl. The values of fL where >. is small are determined dynamically by the theory itself through (3 and that is why the form of this function is so relevant. In particular, the zeros of (3 play an special role and are called fixed points. By defining t = 10g(fL/ flo) we can write

1>. d>.' t-

- >'0 (3(>.') . (C.32)

When t -> +00 or t -> -00 (or what it is the same fL -> 00 or fL -> 0), either >. -> 0 or approaches the nearest zero of the (3 function, which will be called 5., i.e.

(C.33)

Customarily, if limw_+ oo >.(fL) = 5. it is said that 5. is an ultraviolet (UV) stable fixed point. Reciprocally, when limp.-->o >.(fL) = 5., we call 5. a stable infrared (IR) fixed point. In both cases we will say that>. lies in the domain of attraction of the fixed point 5.. Physically, the domains around different fixed points correspond to different phases of the same theory. The sign of the (3 derivative determines the nature of the fixed point: it is UV if (3'(5.) < 0 and IR whenever (3'(5.) > o.

We are interested in the fL regions where we have a fixed point 5. = 0, since then the truncated perturbative expansion will make sense. In principle, (3(0) = 0 in perturbation theory. Consequently in a theory with only one coupling constant we have two possibilities:

• If (3'(5.) < 0 that means that>. = 0 is an UV fixed point. As perturbation theory can be applied close to this fixed point we can use RGE to arrive to the conclusion that the theory becomes non-interacting at high energies. When this happens we say the theory is asymptotically free .

• In case (3'(5.) > 0, something similar happens when the energy tends to zero and we say that the theory is infrared stable.

A typical example of an asymptotically free theory is QCD [7], which thus can only be treated perturbatively at high energies. At low energies it is believed that the coupling constant tends to infinity (infrared slavery) giving rise to confinement and the rich hadronic structure of the strong interactions. In contrast, quantum electrodynamics has an IR fixed point at zero coupling and therefore a perturbative approach can be safely applied at low energies.

Another interesting case is when (3(>.) grows in such a way that the scale evolution factor of >. tends to infinity at some critical fLc. Then, the theory would not be consistent to all energies unless >.(fL) = 0 for all fL, a problem which is known as "triviality" . In this case the theory could only be considered as an effective theory just valid for fL < A < /-Lc, with A playing the role of


some ultraviolet cutoff. For example, for the A<p4 theory we have to one-loop that

(J(A) _ dA - d(logfL)

(C.34)

and integrating this equation

A( ) _ A(fLo) fL - 1 _ 3>'(/-'0) log J:!:... • ~ /-'0

(C.35)

Therefore, if this result were valid to all orders we would have a critical value

(C.36)

and the theory would be consistent only for A(f-lo) = 0, which means A(f-l) = 0 for any f-l value (and therefore trivial) [8J. In fact it is believed that this is the case for the A<p4 theory, even beyond perturbation theory. If this is really true, the only possibility to give sense to the theory is the introduction of some ultraviolet cutoff A and apply the theory only at energies below A, in spite of the fact that it is is renormalizable.

Back to the general case, in order to determine completely a theory with one coupling constant A and to obtain predictions, we need to know the value of A at some energy scale f-lo. Another possibility is to give the value of the energy A' where A reaches some conventional value, like unity. For example, in the A<p4 theory to one loop

(C.37)

and therefore 1

A(fL) = 3 I 1!:... 1 - (4.,,-)2 og A'

(C.38)

In this way, A' completely defines the theory. However, contrary to A, it has energy dimension and it sets a scale. This mechanism is used in Sect. 6.1, when dealing with QCD to define a AQcD of the order of a few hundred MeV, which is precisely the scale where the first hadronic resonances appear. In general, this phenomenon is called "dimensional transmutation" [9J and it is closely related with the trace anomaly explained in Chap. 4.

C.2 Quantization of Gauge Theories and BRS Invariance

Even in the case of QED, which is is nothing but an invariant theory under the Abelian U(l)EM gauge group, some difficulties arise in the canonical quantization formalism. Following the standard procedure, we identify as operators the electromagnetic gauge field A/-,(x) and its canonical conjugate momentum 1f'/-, = 8.cj8(8oA/-,(x», But once this is done, we cannot postulate the canonical commutation relations, since 1f'o(x) and V'7i'(x) are identically

288 Appendix

zero and therefore Ao(x) and V' A(x) commute with any other operator. In other words, both operators are just numbers, which is a consequence of the fact that gauge invariance leaves only two independent degrees of freedom of A/-L(x), namely, its transversal components.

From these arguments, we have two possibilities to quantize the theory: Either we keep the commutation relations and we restrict the description just to transversal fields, thus spoiling the manifest Lorentz invariance, or we make use of spurious degrees of freedom. This last procedure requires the introduction of a Hilbert space containing states of negative norm, together with some gauge-fixing conditions to discard these physically unacceptable states. Both formalisms are equally valid and allow us to quantize consistently any Abelian gauge theory.

It is important to remark that, in presence of an spontaneous symmetry breaking, precisely due to the exceptional features described above, the applicability of the Goldstone theorem is spoilt and thus the N ambu-Goldstone bosons (NGB) disappear from the physical spectrum. Such a behavior, as discussed in Chap. 3, is known as the Higgs mechanism and provides a mass for the gauge bosons without destroying gauge invariance.

Next we will briefly review the gauge field quantization procedure, always referring to the non-Abelian case, since Abelian groups can then be treated as a particular example.

C.2.1 Quantization of Gauge Theories

When dealing perturbatively with non-Abelian gauge theories, the quantization procedure demands the introduction of some auxiliary fields, known as ghost fields, which are not physical. The need for such fields was first observed by Feynman [10], based on unitarity arguments. The functional formalism is the best suited in order to deal with these new variables and so we will use it in what follows. Indeed, we have just commented that a gauge-fixing condition is needed for quantization. It is rather simple to understand this necessity using functional methods. Let us start by writing the usual Yang-Mills term for the non-Abelian gauge field A~ [l1J

(C.39)

where jabe are the structure constants of the group. In order to evaluate perturbatively any process involving gauge fields, the

first object to calculate is the propagator that, in the functional formalism, is obtained from the action for the free fields, which is written as follows

(C.40)

(notice that we have integrated once by parts). The propagator is then obtained as the functional inverse of L1/-LV = (g/-LV fj2 - 8/-L8V). But L1/-LV satisfies

c. Aspects of Quantum Field Theory 289

~Jt>' ~~ = ~Jt/l, which means that it is a projection operator (precisely on the transversal components). However, the projection operators do not have an inverse. That is easy to see in our case, since whenever ~Jt/l is applied to two vectors with the same transversal components it yields the same result, irrespective of their longitudinal parts. Thus, if there is no inverse, we are left without a propagator and, apparently, without a consistent perturbative approach.

The solution to this problem comes from the fact that any QFT is not only defined by the Lagrangian, but also from the measure in functional space. As a matter of fact, in a gauge theory, two field configurations are equivalent if both can be related with a gauge transformation. Thus we are not interested in all the possible states, but only in one representative of each equivalence class or gauge orbit. In the perturbative sense (for small gauge transformations) it is indeed possible to restrict the measure to the representatives and then the kinetic operator will have a unique inverse that we are allowed to use as a propagator.

That is why we are going to build now a measure that will only count once each gauge orbit, or in other words, that will be invariant under the action of the gauge group. Mathematically, this kind of measures are known as Haar measures, although in the context of Yang-Mills theories this procedure as a whole is known as the Faddeev-Popov method [12]. To start with, let us remember that two gauge fields belong to the same orbit if they are related by

A!(x) = g-l(X) (AI' (x) + 81') g(x) g(x) = eiTaea(x) , (C.41)

where AJt(x) = -igA~(x)Ta. Thus, perturbatively, we can choose one orbit representative by means of a function r(AJt) that for each given class vanishes for a unique go(x). The representative will be A~o and f is known as the gauge-fixing function.

Let us now consider the following gauge invariant functional

(C.42)

Its invariance follows from the invariance of [dg] under gauge transformations. Therefore ~fl[AJt] = ~fl[A~]. If we now introduce conveniently the unity in the path integral, we are left with

j[dAJt]eiS[A]= j[dAJt ] ~f[AJt] j[dg]8[fa(A!)] eiS[A] , ~ ..

1

= N j[dAJtPf [AJtl 8 [fa (AJt)]eiS[A] , (C.43)

where, in the last step, we have used that not only the measure, but also the action, are gauge invariant, so that performing a g-1 gauge transformation the whole integrand becomes 9 independent and we can factorize

290 Appendix

N = J[dg]. In principle, this constant N is not finite, since it is the volume of the gauge group; but in the functional formalism this detail is irrelevant since the Green functions are obtained from the normalized generating functional, which amounts to dropping the N factor.

We have already built an invariant measure, but we still have to identify what Ll f [All] is. If we interpret the f function as a change of variables in the group, it is easy to see (simply by changing variables in (C.42)) that Ll is nothing but its determinant. Indeed, expressing the group elements in terms of their parameters ea (see (C.4l)), we can write

Llf[AIl] = det I ~~: I ' (C.44)

which is known as the Faddeev-Popov determinant. All together, our functional integral is then given by

J [dAIl] 8 [fa (All)] det I ~~: I eiS[A] . (C.45)

The measure we have just built ensures the existence of an inverse for the kinetic term, which now has changed its form. Thus we can now obtain consistent perturbative calculations. In practice, however, we have to exponentiate the gauge fixing and the Faddeev-Popov determinant in order to obtain a quantum Lagrangian, so that we can extract the new Feynman rules. There are some standard techniques to exponentiate these contributions:

• Let us see first how to deal with the gauge fixing. If we define a a new function f~(AIl) = fa(AIl) - Ba(x), we can integrate over Ba and, no matter which measure we use, the result is just a change in the normalization which, as we know, yields an equivalent theory. However, we should remember that the purpose of gauge fixing is to provide an invertible kinetic term for the gauge fields and thus we want the gauge-fixing condition to contribute to the quadratic part of the Lagrangian. Usually, the gauge-fixing functions are linear in All' and then there is a simple way to introduce the gauge fixing quadratically in the exponent; we just have to use a gaussian measure as follows

J[dB]e-~ J d4xB2(x)8[Ja(A Il ) - Ba(x)] = e-~ J d4xf2(AI") , (C.46)

where ~ is an arbitrary constant known as the gauge parameter. • In order to exponentiate the Faddeev-Popov determinant, it is enough to

remember the following rule of gaussian integrals over anticommuting or grassmanian variables Ca , Cb

det M = J [dc][dc]ei J d4xd4yca(x)Mab(X,y)q(x) . (C.47)

Therefore, using both results, we are able to exponentiate the complete invariant measure. Introducing the corresponding external currents J~ we can write the generating functional as follows:


W[J] = j[dAJL][dc][dc]exP(ij d4x [-~F:v(X)FaJLV(X) +ca(X)~~:Cb(X)

- 21~f2 (AJL(x)) + JZ(X)AJLa(x)]). (C.48)

The new variables ca , Cb are called ghost fields and although they anticommute, they are not physical fermionic fields (as far as their kinetic term does not look like the Dirac Lagrangian).

There is still left the choice of a gauge-fixing function. Probably the simplest choice of gauge-fixing condition that yields an acceptable propagator is r[A~] = aJL A~. However, for the purposes in this book, which deals with the standard model and therefore very often with spontaneous symmetry breaking, the most convenient are the so called renormalizable or Rf; gauges, that were first introduced by t'Hooft [13]. Their simplest form is f(A JL ,1» = aJL A~ + ~gv1>a /2, where 1>a is the NGB associated to A~. The main advantage of these gauges is that they cancel in the Lagrangian a mixing term gvA~aJ.t1>a, which otherwise would make the perturbative expansion extremely cumbersome (see Chap. 3). The most common choices of the gauge parameter are: ~ = 1, known as Feynman gauge and ~ = 0, or Landau gauge.

Finally, notice that if we want to use a covariant formalism under NGB reparametrizations, we will have to change slightly the gauge-fixing function. This is due to the fact that the 1> fields are coordinates in the NGB manifold and therefore they do not transform covariantly. The solution is to use instead of 1> a covariant function whose first order expansion is rv 1>. The specific choice of this function has been explained in Chap. 3, as well as further complications that appear when using non-linear gauge-fixing functions.

C.2.2 The BRS Symmetry

We have already built an appropriate generating functional for perturbation theory. However, in the process, we have apparently spoilt the main property of the theory, which is gauge invariance. Indeed, the Lagrangian in (C.48) is not gauge invariant, but once we know the explicit form of its terms, it can be easily checked that it is invariant under the following Becchi-Rouet-Stora (BRS) [14] transformations

OBRS[AJ.ta] =1'/(aJ.tca + grbc AJ.tbcc) = 1'/s[AJ.ta] (C.49)

OBRS[Ca]=-g1'/ rbccbcc = 1'/s[ca] 2

OBRS[CU]=-g~1'/ r[AJ.t,w] = 1'/s[CU] ,

where 1'/ is an anticommuting parameter and fa is the gauge-fixing function introduced in the preceeding section. Notice that for the A fields, they are nothing but gauge transformations with parameter ea = -g1'/Ca(x).

These transformations can be generalized to other fields in case we want to include them in our Lagrangian. In particular, for the purposes of this

292 Appendix

book, we are interested in the BRS transformations of fermions and NGB, which can be obtained from their corresponding gauge transformations by substituting oa = -g'f}Ca(x). That is

8BRS[dl<] =-g'f}~~(w)ca == 'f}s[wQ] (C.50)

8BRS['I/J]=-ig'f}Taca'I/J == 'f}s['I/J],

in Sect. 3.2 we saw that the ~~ determine the NGB gauge transformation. With these definitions it is easy to see that s2[A] = 0, s2[w] = 0, s2['I/J] = 0 and s2[c] = o.

As far as the gauge-fixing function is a complicated object, it is convenient to define an auxiliary b field [15] and a new gauge-fixing term

£GF = bar + ~baba . (C.51) 2

Notice that, after a functional gaussian integration we would recover the usual gauge-fixing function, and that ba is basically the r (its equations of motion are ba = - r /~). The advantage is that instead of Green functions containing gauge-fixing functions we can calculate Green functions of ba fields, which are not composite operators. With this definition, the s operator, which is the one we are interested in, will act as follows

s[wQ]=-g~~(w)ca

s['I/J] = -igTa ca'I/J s[AILa] =8ILca + gjabc AILbcc == D~ccc

s[ca] = _Q rbccbcc 2

s[C"]=ba

s[ba ]=0 ,

(C.52)

but now the advantage is that we are treating the gauge-fixing function in the same footing as the normal fields and the s operator has become nilpotent. In other words, S2 = 0 when acting on any field. Our Lagrangian is invariant under these transformations and they are the quantum analog of gauge symmetry.

The relevance of BRS transformations is that they generate a set of Ward identities which, in this context, are called Slavnov-Taylor identities [16]. It can be shown that these relations between Green functions ensure the gauge invariance of physical quantities.

Apart from the BRS symmetry, there is another set of transformations that also leaves our Lagrangian invariant, which is the following

s[wQ] = -~~ (w)C"

s['I/J] = -igTaC"'I/J

s[ AILa] = D~cC"

s[C"] = -~ nctcc


s[C"] = _ba - gfbcc!'cc

s[ba] = -gfbccbcc , (C.53)

which are known as the anti-BRS transformations. The action of both sand s on other field combinations is defined as if they were differential operators obeying the generalized Leibnitz rule: s[XY] = s[X]Y ± X sty], where the minus sign appears whenever in the X field product there is an even number of ghost or antighosts. From (C.52) and (C.53) it is easy to obtain some general nilpotency relations

S2 = ss + ss = S2 = 0 . (C.54)

In the book, we will only make use of the BRS symmetry in order to derive the Slavnov-Taylor identities in Chap. 7. Nevertheless, we would like to add a last remark on the utility of the anti-BRS symmetry. We have already seen that in case one uses a covariant formalism, the gauge-fixing function is not linear on the fields; it can be shown [17] that in such case it is necessary for renormalization to introduce a quartic ghost interaction in the Lagrangian, which is not present in the Faddeev-Popov method.

There is another possibility to quantize the gauge fields which is a generalization of the Faddeev-Popov method and includes such quartic ghost vertices. We will not give the details here, but with that procedure the Lagrangian can be written as follows [18]

(C.55)

working out the action of the sand s operators we obtain again the gaugefixing function and other terms with ghosts. The (anti-)BRS invariance is immediate from the nilpotency relations in (C.54). Observe that F is an arbitrary function that one chooses conveniently to obtain the desired form of the gauge-fixing condition. We get the same physical results either using the Faddeev-Popov method with a quartic ghost interaction introduced by hand or this Lagrangian, although the latter formulation is more natural in this context.

C.3 The Background Field Method

Gauge Theories

In Sect. C.2.1 we have seen how the Faddeev-Popov method can be used to define properly perturbation theory for gauge theories. The Feynman rules are then derived from a Lagrangian which, in addition to the standard classical Yang-Mills term, also has a gauge-fixing and a ghost term. This new Lagrangian is no more gauge invariant, but only BRS. This BRS invariance can be used to derive the so called Slavnov-Taylor identities, which guarantee that the S-matrix elements are gauge invariant, together with many

294 Appendix

other well-known properties of gauge theories like unitarity or renormalizability. However, we remark that Green functions are not gauge invariant and the same happens with the effective action (the generating functional of the one-particle irreducible Green functions). Nevertheless, there is a gaugefixing choice which gives rise to a gauge invariant effective action and allows us to avoid the BRS formalism. This procedure is known as the background field method [19], which is very useful in many cases, such as the calculation of the trace anomaly or in quantum gravity.

Let us start from the standard path integral expression of the generating functional

W[J] = eiZP] = j[dA]ei(S[A]+<JA» , (C.56)

where A is the gauge field and < JA > stands for J dxA~J,..a, with JZ the external sources. B[A] includes the gauge fixing and the Faddeev-Popov terms but the ghost fields have not been displayed explicitly for economy of notation. The effective action r[J] is obtained as usual as the Legendre transform of Z[J]

where Ad is at the same time a field and a functional of J defined by

8Z[J] Acl(X; J] = 8J(x) .

(C.57)

(C.58)

As we have commented above, r[Acz] is not in general gauge invariant. However we can split A as

(C.59)

where A Q and ABare called the quantum and the background fields respectively. Now we define

W[J,AB] = eiZP,AB] == j[dAQ]ei(S[AQ+AB]+<JAQ» (C.60)

with the corresponding effective action

(C.61)

where

(C.62)

Now we choose the so called background field gauge-fixing condition

(C.63)

where jabc are the gauge group structure constants. It is not difficult to show, just by making the transformation


AQa _ AQa + fabc()b AQc (C.64) I-' I-' I-' '

that i[J, AB] is invariant under

8ABa =-!o ()a+rbc()bABc (C.65) I-' 9 I-' I-'

8Ja =fabc()b JC I-' 1-"

when the above gauge-fixing condition is used. Then it follows that r[A~, AB] is invariant under

8ABa =_!o ()a + rbc()b ABc (C.66) I-' 9 I-' I-'

8AQa =fabc()b AQc ell-' ell-'

and, in particular, r[O, AB] is a gauge-invariant functional. Now we can try to find the relation between r[A~, AB] and the standard effective action functional r. With that purpose, we make the following change of path integral variable

~-~-~ (Cm) I-' I-' I-'

in the definition of i[J, AB]. Then, in the background field gauge, we find

W[J, AB] = W[J]e-i<JA B > , (C.68)

where W[J] is the standard generating functional evaluated with the gaugefixing function

(C.69)

Notice that, since we have changed the gauge-fixing function, the FaddeevPopov Lagrangian for W[J, AB] is obtained using the above ga (note that Z[J] depends also on A~), instead of r. Therefore we have

i[J,AB] = Z[J]- < JAB> . (C.70)

By using (C.57) and (C.61), we get

r[A~, AB] = Z[J]- < J(A~ + AB) >= r[A~ + AB] , (c.n)

so that r[A~, AB] is just the original effective action evaluated at Ael = A~ + AB. As a consequence

(C.72)

As we have commented above, the r[AB] is invariant under AB gauge transformations, since r is evaluated in the background field gauge r.

Up to the moment, our manipulations have been purely formal. In order to make calculations we derive the Feynman rules from i[J, AB] with the gauge-fixing function r, to arrive to r[O,AB] (no diagrams with external A~ legs). It is then possible to show [19] that the renormalization of a pure

296 Appendix

Yang-Mills theory only requires one Z constant. That is, in the background field method, we have

A~ = Zi/2 Ait (C.73)

Z -1/2 90 = 3 9R·

a)

/" "-

/ \

b) roooo~ ~ \ /

'- ./

Fig. C.la,b. One loop diagrams contributing to the wave function renormalization in a Yang-Mills theory. (a) Gauge boson loop, (b) Ghost loop

Then, to calculate the (3 function we only need to consider the diagrams corresponding to the wave function renormalization (see Fig. C.l). Therefore, the renormalized Lagrangian reads

(C.74)

In particular, it is immediate to show that the divergence of the anomalous current associated to scale transformations D I-' is given by

81-'DI-' = - 2(3(9R) £YM(AR ) 9R

(C.75)

as it is shown in Chap. 4.

Ward Identities

Let us nOw introduce another method, much more general than the one discussed in Chap. 2, to derive Ward identities from a given invariance group G of the Lagrangian. The method is based on the introduction of some background fields. It is easy to use and can also be applied when the fields appearing in the action do not transform linearly under G or the action depends on higher derivatives of the fields. The method involves several steps: First we define the generalized action

S[n, J, A, K] = So[n, A] + J dx(Jana + K £1) . (C.76)


Where BO[7r, A] is the invariant action under G local transformations and we have denoted by A the AJL = -igTa A~ associated gauge fields. The .c1 term is the one that explicitly breaks the global G invariance. The simplest way to ensure this local invariance is to substitute the standard 7r field derivatives by covariant derivatives (see Sect.3.5 for the details on gauging a global symmetry of a non-linear (J model (NLSM)).

As usual, J is a source for the 7r field but we have also introduced K (called the background field) as a source for .c 1. With these definitions the above action is invariant under the local G transformations

A,a(x) = Aa(x) - iaoa(x) + fabcOb(X)AC(x)

7r'<>(x) = 7r<>(x) + 87r(x)<> .c~(x) = .c1 (x) + 8.c1(x) J~(x) = J<>(x) + 8J<>(x) K'(x) = K(x) + 8K(x)

as long as we define 8J and 15K so that

(8J<> (x) )7r<>(x) = -J<> (x)87r<> (x)

(8K(x)).c 1(x)=-K(x)8.c1(x) .

(C.77)

(C.7S)

Therefore, provided no anomalies are present, the generating functional

W[A, J, K] = j[d7r]eiS [7r,J,A,KJ (C.79)

is invariant under the above local transformations (we assume here that the 7r field measure has been properly defined, i.e. for the case of the NLSM it should include the J9 factor). Therefore one has

W[A,J,K] = W[A',J',K']. (C.SO)

Note that it is not necessary that the 7r field transforms linearly or to make any hypothesis on the form of the original action, in order to obtain the above equation and thus this method is much more general than that described in Chap. 2.

Remarkably, all the information about the Ward identities of the theory is contained in the last equation, which can be seen as follows: just consider an infinitesimal local G transformation which leads us to

j ( 8W 8W A 8W ) dx 8J 8J + 8Aff 8AJL + 8K 8K = O. (C.SI)

This equation generates the different Ward identities by differentiating W with respect to the sources A, J and K and setting A = 0, J = 0 and K = 1 in the end. There is a straightforward generalization to include Ward identities for composite operators. In such case, we only have to introduce their associated background field transforming under the local action of G, so that the generating functional remains G invariant.

298 Appendix

C.4 The Heat-Kernel Method

When working in the path-integral formulation of QFT, the only integrals that can be calculated explicitly are those of gaussian-type. For both scalar and fermion fields, the result of such integrals is formally the determinant of some differential operator (see Appendix A). For instance, as we have seen in several parts of the book, this kind of integral also appears when we expand the action near a classical field configuration. Therefore, we need a consistent definition of functional determinants.

For that purpose, we will use the so called Heat Kernel method. In addition, it is going to be a useful technique to regularize the J acobians of fermionic measures, like those appearing when dealing with anomalies (see Chap.4).

For technical reasons, it is particularly useful to work in Euclidean space. Let us then consider the following general second order elliptic Euclidean differential operator, whose determinant we are going to calculate:

H = '\l J1. '\lJ1. + Y(x)

'\l J1. =aJ1. + XJ1.(x) , (C.82)

where XJ1. and Yare matrix-valued functions, XJ1. being anti-hermitian and Y hermitian. For further convenience, it will be useful to define the following function:

which satisfies the differential equation

arG = -HxG

with the boundary condition

G(x, Yj 0) = 8(x - y)

(C.83)

(C.84)

(C.85)

corresponding to the normalization of the Ix) states. Notice that (C.84) for Y = X = 0 is the heat equation and the G function is called, for this reason, the heat kernel. Now we turn to show the relation between G and det H. We have

det H = e Tr log H = etr J dx(xllog Hlx) , (C.86)

where tr is over internal and Dirac indices and Tr also includes the functional trace. The matrix element (xl log Hlx) will be in general divergent, so that we need first to give it sense by means of some regularization method. We give here two possible alternatives:

On the one hand, we can use

b 100 dT log - = - (e- ar _ e-br ) a 0 T

(C.87)


and then we get

1CXl dT (xllogHlx) = - -G(X,X;T) + C,

o T (C.88)

where C is an infinite constant which is not going to play any physical role, since it is a normalization constant. The expression in (C.88) is particularly useful when working in dimensional regularization, as we will see below.

On the other hand, we can write

(xllog Hlx) = - :s (xIH- S Ix{=o (C.89)

and then

a J.L2s roo i (xllogHlx) = - asT(s) Jo dTT8

-1G(X,X;T) s=o ' (C.90)

where J.L is an arbitrary energy scale, introduced to make the above matrix element dimensionless. This prescription is called the (-function regularization, since we can define a generalized Riemann zeta function as (H ( s) = Ln A;:; S ,

with An the eigenvalues of H and then

10gdetH = - : (H(S)i s 8=0

(C.91)

Both in (C.88) and (C.90) we see that the heat kernel appears in the so called coincidence limit y -+ x. At this point, let us consider the following expansion for the HK in this limit

00 n

G(x, x; T) = L an(x) ( T )D/2 ' n=O 41l"T

(C.92)

where D is the space-time dimension. The above expression is known as the Seeley-DeWitt [20, 21] expansion of the HK. One of its advantages is that usually UV singularities only arise in the first coefficients. For instance, let us introduce some mass parameter through H -+ H + m2 in order to avoid IR singularities. In dimensional regularization, replacing (C.92) in (C.88), we get

~ T(n - D/2) D-2n (xl log Hlx) = C - L...- an (x) ( )D/2 m

n=O 41l" (C.93)

and then we see that all the possible divergences are in the n = 0, ... , D /2 terms, with D even.

The coefficients an(x) can be calculated using several methods. For instance, for the general case x =1= y, the Seeley-DeWitt expansion reads

2 00 n -Ix-yl ~ T

G(x, y; T) = e-4 -T - L...- an(x, y) ( )D/2 n=O 41l"T

(C.94)

300 Appendix

Then, by replacing (C.94) in (C.84), we obtain the following recursion relations

ao(x,x)=l

(x - y)I'-\1~ao(x,y)=O

nan (x, y) - (x - y)I'-\1~an(X, y)= -Hxan-l(X, y), n ~ 1 . (C.95)

The above relations can be used to calculate the an (x, y) coefficients. After that, it is possible to take the coincidence limit y --+ x. However, this method is rather cumbersome and it even requires the use of computer symbolic algebra. Explicit expressions for the an coefficients have been obtained up to n = 5 [22, 23]. However, if one is only interested in the lowest order coefficients, in the coincidence limit (say, for the calculation of the determinant or the anomaly), a more direct method to obtain them consists in changing to a plane wave basis. In fact, this was the way originally used by Fujikawa in the calculation of the axial anomaly (see Chap. 4). We will briefly show how this method works.

Since H is an hermitian operator, we can perform a change of basis from the states Ix) to Ik) in momentum space. With the usual normalization for spinor fields we get:

G(x, Xj 7) = 1 (~;)kD (xlk)e-TH (klx)

1 dDk Ok H ok = (271")D e-t Xe-T et x .

Now, taking into account that

[H, eikx] = eikx (k2 + 2ikl'-\1 1'-) ,

we arrive to

G(x x· 7)=1 dDk e-T(k2 +2ik"V,,+H) , , (271")D

(C.96)

(C.97)

_ 1 1 dDk _k2 ~ (_l)m . r= I'- m - D/2 -( )De ~--,-(-2tV7k\1I'--7H) ,(C.98)

7 271" m=O m.

where it is understood that the operators are acting to the right on the identity. Notice that it has been performed a change of variable k --+ k / ..,fi in the last integral. Now, we can simply expand (C.98) up to a given order n in 7 and then read the corresponding an coefficients by comparing with (C.92). The first Seeley-DeWitt coefficients in the coincidence limit are then

ao=l

al=-Y

a2 = ~ y2 + ~ [\11'-, [\11'-' Yll + 112 [\11'-, \1V ][\1 1'-' \1 v] . (C.99)


We remark that the above method is useful only up to n = 2. For higher order coefficients, it is possible to use the recursion relations in (C.95), as commented above. Finally, we refer to [24] for the extension of these techniques to more general differential operators, which are not relevant for our purposes.

C.5 References

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8] [9] [10] [11] [12] [13] [14]

[15]

[16]

[17]

[18]

[19]

[20]

[21] [22] [23]

R.P. Feynman, Phys. Rev. 74 (1948) 939 and 1430 J. Schwinger, Phys. Rev. 73 (1948) 416, 75 (1949) 898 S. Tomonaga, Phys. Rev. 74 (1948) 224 F.J. Dyson, Phys. Rev. 75 (1949) 486 G. 't Hooft and Veltman, Nucl. Phys. B44 (1972) 189 C.G. Bollini and J.J. Giambiagi, Phys. Lett. 40B (1972) 566 N.N. Bogoliubov and O.S. Parasiuk, Acta Math. 97 (1957) 227 K. Hepp, Comm. Math. Phys. 2 (1966) 301 W. Zimmermann, Lectures on elementary particles and quantum field theory. Proc. Brandeis Summer Institute (ed. S. Deser et al. ). MIT Press, Cambridge, Massachusetts, 1970 G. 't Hooft, Nucl. Phys. B61 (1973)455 S. Weinberg, Phys. Rev. D8 (1973) 3497 C.G. Callan, Phys. Rev. D2 (1970) 1541 K. Symanzik, Comm. Math. Phys. 18 (1970) 227 D.J. Gross, Methods in Field Theory, Les Houches, eds. R. Balian and J. Zinn-Justin, North-Holland, Amsterdam, 1975 D. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343 H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346 D.J.E. Callaway, Phys. Rep. 167 (1988) 241 S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888 R.P. Feynman, Acta Physica Polonica 24 (1963) 697 C.N. Yang and R.L. Mills, Phys. Rev. 96 (1954) 191 L.D. Faddeev and V.N. Popov, Phys. Lett. B25 (1967) 29 G. 't Hooft, Nucl. Phys. B35 (1971) 167 C. Becchi, A. Rouet and R. Stora, Comm. Math. Phys. 42 (1975) 127 LV. Tyutin, Preprint PhIAN N 39, (1975) J. Zinn-Justin, Bonn Lectures, 1974 T. Kugo and I. Ojima, Pr09. Theor. Phys. Suppl. 66 (1979) 324 J.C. Taylor, Nucl. Phys. B33 (1971) 436 A.A. Slavnov, Theor. Math. Phys. 10 (1972) 99 N.K. Nielsen, Nucl. Phys. B140 (1978) 499 R.E. Kallosh, Nucl. Phys. B141 (1978) 141 L. Baulieu, Phys. Rep. 129 (1985) 1 L. Alvarez-Gaume and L. Baulieu, Nucl. Phys. B212 (1985) 255 W. DeWitt, Phys. Rev. 162 (1967) 1195 and 1239 G. 't Hooft, Acta Universitatis Wratislavensis 368 (1976) 345 L.F. Abbott, Nucl. Phys. B185 (1981) 189; Acta Physica Polonica B13 (1982) 33 B. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach, New York, 1965 R. Seeley, Ann. Math. Soc. Proc. Symp Pure Math. 19 (1967) 288 A.E.M. Van de Ven, Nucl. Phys. B250 (1985) 593 R.D. Ball, Phys. Rep. 182 (1989) 1

302 Appendix

[24) P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, eRe press, 1995

D. Unitarity and Partial Waves

D.l Unitarity

Let us now concentrate on the issue of unitarity and review some basic results. The fact that the probability of any process has to be a real number between zero and one, implies that S is a unitary matrix, that is, sst = 1. It is easy then to obtain a bound on the S-matrix elements

L SikSjk = 8ij => ISijl ::; 1 . k

(D.1)

As it is well know, the amplitudes are related with S-matrix elements through the relation

Sfi = 8f i + i(211")484(Pi - Pf )Tfi , (D.2)

where i is the initial state, with total momentum Pi, and f is the final state with total momentum Pf. Therefore, the unitarity condition for the amplitudes is given by

Tif - Tji = i LTinTjn(211")484(pi - Pn) , (D.3) n

where the sum is over all physically accessible intermediate states n. In the following we will concentrate in two-particle elastic processes; in such cases the unitarity constraint can be expressed in a simpler way by using partial wave amplitudes. For elastic Nambu-Goldstone boson (NGB) scattering in chiral perturbation theory (ChPT) the relevant quantum numbers are the isospin I and the angular momentum J, so that we define

1 1 Jl tIJ(s) = 32 K d(cos(})PJ(cos(})TI(S,t,U) , 11" -1

(DA)

PJ being the Jth Legendre polynomial and K = 1,2 depending on whether the two particles are identical or not. As usual, the Mandelstam variables are defined as

(D.5)

where Pk are the momenta of the four particles and () is the angle between the spatial three vectors PI and P3. (D.1) can be now translated into

1 tIJ(s) I::; 1 , (D.6)

which provides us with a definition of an strongly interacting system: whenever t IJ ~ 1 we say that the interaction is strong.

D. Unitarity and Partial Waves 303

However, the most widely used unitarity condition for the elastic scattering of two particles a, b, can be obtained from (D.3), assuming T invariance, that is TiJ = TJi. For physical s and in terms of partial waves it reads

ImtIJ(s) = (Tab(S) 1 tIJ(S) 12 , (D.7)

where

(Tab(S) = J (1- (Ma: Mb)2) (1- (Ma ~ Mb)2) (D.8)

is the integrated two body phase space that comes from the momentum integral of 84 (Pi - Pn). Notice that this unitarity condition is only valid for energies below the inelastic threshold, that is, supposing that the physically accessible intermediate states are precisely those made of the very two same a, b particles.

In case we had a two-particle inelastic process, we could also define partial waves FfJ' where A stands for other relevant quantum numbers of the process. In particular, in Chap. 6 we have analyzed in more detail the II -+ 7r7r

reaction and then A is the total helicity defined as A = Al - A2, where Al and A2 are the helicities of the initial photons (as a consequence 1 AI:::; J). Hence, the partial wave definition now is

FfJ = J J47r /1 d(cosB) r27r dc/>Fl1,)..2(s,cosB,c/»Y.h(cosB,c/» , (D.9) 2 + 1 -1 Jo

where instead of the Legendre polynomials we have to use the spherical harmonics Yj L (cos B, c/» since depending on the helicity there can be some dependence in c/>.

Starting once more from (D.3), and neglecting electromagnetic effects, we can derive again the unitarity condition for the inelastic partial waves that we have just defined. It reads

ImFfJ(s) = (T7r7r(s)Ffj(s)tIJ(s) , (D.lO)

where tIJ is the I J 7r7r elastic scattering partial wave defined in (D.4).

D.2 Dispersion Relations

In this section we will review some very basic notions of dispersion theory that we use in the text. For a more rigorous treatment we refer the reader to the specialized literature [1]. Let us first recall the analytic structure of the amplitudes obtained from a relativistic quantum field theory, in terms of the Mandelstam variables s, t, u. For positive real values of S and a given u = uo, we know that they vanish below the threshold energy, S < Sth, and they have a nonzero value over this threshold. Therefore they have to posses a cut over the right hand side of the real axis. Indeed there is a new cut whenever we cross a new threshold, and thus we find a superposition of cuts over the

304 Appendix

right positive axis. In addition, we should remember that the amplitudes in the u and t channels are related to that in the s channel, since all them are obtained from the same Green function. These other channels also present their respective cuts, which, for the partial waves, become new cuts on the whole left hand side of the real (l..xis.

Once we now the analytic structure of the partial amplitudes we can use the Cauchy integral formula over the contour C represented in Fig. D.l, that is

tIJ(s) = ~ r tI/(s') ds' 27fz Jc s - s

Fig. D.l. Analytic structure of an elastic scattering amplitude

(D.ll)

If we assume that tIJ(s) ----> 0 as lsi ----> 00, then the contribution from the curved part of the contour vanishes when the radius tends to infinity, and we are left with

1 100 ImtIJ(s')ds' 1 1-00 ImtIJ(s')ds' tIJ(S) = - , . + - , .,

7f 8th S - S - zc 7f 0 S - S - u; (D.12)

where we have used that the discontinuity of tIJ across the cuts is given by 2iImtIJ, which is a consequence of (iISIf) = UISli); that is, T invariance. This kind of relations are known among physicists as dispersion relations, and as Hilbert transforms for mathematicians.

In the derivation of (D.12) we have made use of the strong assumption that tIJ(s) ----> 0 as lsi ----> 00, but that will not be the case for amplitudes within the effective Lagrang ian formalism, which are basically polynomials in s that grow like IsI N . When an amplitude presents such a behavior we cannot follow the above derivation, but instead we can apply Cauchy's theorem to the function

tIJ(s) 9IJ(S) = N+l ' (D.13)

s


which now has the desired s ~ 0 limit. The only difference is that now a N + 1 order pole occurs in the 9IJ(S) function, and we have to collect its residues in an N degree polynomial whose coefficients are in principle unknown. The result is called an N + 1 subtracted dispersion relation. It is important to notice that the integrals in the dispersion relations are taken over the whole range of physical s. When using subtracted relations we are damping the contribution at high energies while enhancing the low and intermediate energy effects. The subtraction coefficients of the lowest energy powers in the polynomial can be related with the parameters of the chiral Lagrangian.

In Chap. 6, we study the properties of chiral amplitudes at 0(p4) = 0(s2), therefore we will need at least three subtractions. Then, for elastic scattering of two particles a and b, we are interested in the following dispersion relation, which is used in Sect. 6.6

tIJ(s) = Co + CiS + C2S2 (D.14)

s31°° ImtIJ(s')ds' s3 JO ImtIJ(s')ds' + - ,3 ( , . ) + - ,3 ( , . )

7r (Ma+Mb)2 S S - S - U 7r -00 S S - S - U'

We have seen how to obtain dispersion relations for elastic scattering of two particles, but it is also possible to obtain similar identities for other processes as well as for form factors, although in the last case the left cut does not occur. Related with this kind of integral equations, there is a classical result that we will briefly review for completeness.

The M uskhelishvili-Omnes Problem

That is the name given to the problem of finding those functions f (s) which are analytic except for a cut (Ma + Mb)2 < s < 00, which are real when s < (Ma + Mb)2, and for which f(s) exp( -i8(s)) is real when s approaches the cut from above.

It has been shown [2] that whenever the phase tends to a finite value as s ~ 00 and if f(s) does not grow faster than s, then the solution, in the two particle unitary approximation (below inelastic thresholds), is given by

f(s) = P(s)D(s)

( s 100 8(S')dS') D(s) = exp - "., 7r (Ma+ M b)2 s (s - s - u)

(D.15)

where pes) is a polynomial with P(O) = 1, and D(s) is known as the Omnes function.

Notice that this is not exactly our case, since we are looking for an amplitude which also presents a left cut, but we can apply the same methods to write

t(s) = P(s)D(s) , (D.16)

where now P( s) is a function that carries the left cut and is real in the physical region.

306 Appendix

The Omnes solution therefore justifies why we can obtain the appropriate analytic structure on the right-hand-side real axis once the phase shift is known (see Sect. 6.6.1).

D.3 NGB Amplitudes to O(p4)

In the following, we give the expressions for the chiral Lagrangian amplitudes to O(P4) that have been used in the text, together with some useful relations and definitions of their partial waves.

In Fig. D.2 we have displayed the generic diagram for NGB elastic scattering. Note that, as far as the wa fields have an SU(2) or O(N) symmetry, the different s, u, ant t channels look the same except for permutations of indices and momenta. Indeed, we can use both these chiral and crossing symmetries to derive any amplitude from just a simple function, as follows:

Ta{3"(o(s, t, u)

= A(s, t, U)ba{3b"(o + A(t, s, U)ba"(b{3O + A(u, t, S)baob{3"( , (D.17)

where A(s, t, u) is obtained from the four point Green function, calculated at O(P4). Later these amplitudes are usually projected in partial waves, of definite angular momentum J and isospin I, following (D.4).

Let us now give the actual O(p4) amplitudes for the examples used in the book.

Fig. D.2. Generic diagram for NGB elastic scattering

D.3.1 1m -+ 7r7r in ChPT

For 7r7r scattering we only have three definite isospin states, I = 0,1,2. In that basis, the (D.17) leads to the following amplitudes

To(s, t, u) =3A(s, t, u) + A(t, s, u) + A(u, t, s)

Tl (s, t, u) =A(t, s, u) - A(u, t, s)

T2 (s,t,u)=A(t,s,u) +A(u,t,s). (D.18)


The elastic 7r7r ~ 7r7r scattering was first obtained to 0(p4) within SU(2) ChPT [3]. However, as we have devoted most of Chap. 6, to the SU(3) ChPT formalism, we will use the latter to give the actual expression for the A(s, t, u) function, which is the following:

(s-M;) 3 A(s, t, u) = F2 + B(s, t, u) + C(s, t, u) + O(s )

1r

B(s,t,u) = }j { ~; J;.,(s) + ~(s2 - M;)J;1r(s) + ~S2JKK(S) 1

+"4(t - 2M;)2 J;1r(t) + t(s - u)

x [M;7r(t) + ~MKK(t)] + (t ~ u) }

C(s, t, u) = ;j { (2L~ + L3)(S - 2M;)2

+L;[(t - 2M;)2 + (u - 2M;)2]

+(4L4 + 2L~)M;(s - 2M;) + (8L6 + 4Lg)M; } .

(D.19)

Notice that it only depends on the following combinations of chiral parameters

2L~ + L3 L;

2L4 + L~ 2L6 + Lg . The Jaa and Maa functions, with a = 7r, K, 'T/, that appear in the amplitude

above come from the loop integration and provide the imaginary part as well as the analytic cuts in the complex s plane. Their expression is

r () - () k r () S - 4M; - () 1 k 1 ( ) Jaa s = Jaa s - 2 a, Maa S = 12 Jaa S -"6 a + 288 ' D.20

where

Jaa ( s ) = 16~2 [(T log ( : ~ ~) + 2 ]

ka = 3217r2 [lOg (~;) + 1] their p dependence cancels exactly with that of the £1; (p) parameters.

D.3.2 7rK ~ 7rK in ChPT

In the case of 7r K elastic scattering, there are two possible isospin states, namely I = 1/2 and 3/2. Once more, one single function is enough to determine both channels and incidentally it is nothing but the very I = 3/2

308 Appendix

amplitude. As a matter of fact, the I = 1/2 amplitude can be obtained as follows:

3 1 T1/ 2(S,t,u) = "2T3/2(U,t,s) - "2T3/2(S,t,u) , (D.21)

where the O(p4) result from ChPT is [4]

3/2( ) E7rK - S T( ) p( ) TU( ) O( 3) T . s,t,u = 2F7rFK +T4 s,t,u +T4 s,t,u + 4 s,t,u + s

T 1 T4 (s, t, u) = 16F7rFK Ll7rK(3J.L7r - 2J.LK + J-L.,,) (D.22)

T[(s,t,u) = F2~2 {4L~(t-2M;)(t-2Mk) 7r K

+2L2 [(s - E7rK)2 + (u - E7rK)2]

+L3 [(u - E7rK)2 + (t - 2M;)(t - 2Mk)]

+4L4 [tE7rK - 4M;Mk]

+2L~M;(Ll7rK - s) + 8(2L6 + L~)M;Mk}

U 1 {3 [ T4 (s, t, u) = 4F;Fk "2 (s - t) (L7rK(U) + LK.,,(u)

-u (M;K(U) + MK.,,(u))) - Ll;K (M;K(U) + MK.,,(u)) 1 +t(u - s)[2M;7r(t) + MKK(t)]

1 -"2Ll7rK[K7rK(U)(5u - 2E7rK ) + KK.,,(u)(3u - 2E7rK)]

+~J;K(U) [l1u2 -12uE7rK + 4E;Kl + J;K(s)(S - E7rK)2

+~JK"'(U) (u - ~E7rK ) 2 + ~J;7r(t)t(2t - M;) + ~JKK(t)e

1 r 2( 8 2)} +"2J.".,,(t)M7r t - gMK ,

where Eab = M~ + M;, Llab = M~ - M;, with a, b = 1f, K, 1/. Notice that the loop-integrals provide the imaginary part through the following functions:

J;b ( s ) = Jab ( s) - 2kab ,

r _ Llab -Kab(S)- 2s Jab(S) ,


D.4 References

[1] R.J. Eden, P.V. Landshoff, D.1. Olive and J.C. Polkingorne, The Analytic S-matrix, Cambridge University Press, 1966 A.O. Barut, The Theory of the Scattering Matrix, Macmillan. New York, 1967 K. Nishijima, Fields and Particles: Field Theory and Dispersion Relations. W.A. Benjamin Inc. New York, 1969

[2] R. Omnes, Nuovo Cimento 8 (1958) 1244 [3] J. Gasser and H. Leutwyler, Ann. of Phys. 158 (1984) 142 [4] V. Bernard, N. Kaiser and U.G. Mei1~ner, Phys. Rev. D43 (1991) 2757; Nuc.

Phys. B357 (1991) 129 A. Dobado and J.R. Pelaez, LBL-38645, hep-ph/9604416. To appear in Phys. Rev. D.

Subject Index

anomalous dimension 283 anomaly 59 - 7r0 -+ '"Y'"Y 63, 64, 91 - ABJ 63 - ambiguities 76 - axial 60, 65 - baryonic 109,237 - cancellation 105, 239 - consistency conditions 72, 76 - consffitent 71,73,78 - counterterms 76 - covariant 73, 75, 78 - gauge 68,238 - gravitational 240 - leptonic 109,237 - non-Abelian 78 - non-linear (j model 85 - non-perturbative 83 - QCD 63,86 - regularization methods 73 - Topology 78 - trace 93 asymptotic freedom 126,286

background field method 246, 293 baryon number 55 beta function 112 - QCD 127 Bianchi identities 268, 271 boson - gauge 50 - Higgs 2 - Nambu Goldstone 16,35,36,41 - - dynamics 45,48 - - large-N scattering 224 -- pseudo 37,129 - - scattering 306 -- would be 176 BRS transformations 291 - renormalized 204

Cabibbo angle 105 Cabibbo-Kobayashi-Maskawa matrix

103 characteristic classes 275 charge - conserved 24 - electric 100 charge quantization 242 Chern-Simons form 276 chiral fermions 17 chiral Lagrangian - O(p4) 135, 182 - leading order 130, 179 - renormalization 136,186 Chiral Perturbation Theory 135 - Nt = 2 149 - unitarity 151 conformal group 28 connection - affine 265 - Levi-Civita 267 - metric 266 coset 42 - topology 54 counterterms - non-linear (j model 53 covariant - derivatives - formalism CP problem CP violation - strong 113 - weak 103 current

50 43,185 119

- axial 60, 128 - conserved 24 - Noether 24 - vector 60, 128 curvature 268

312 Subject Index

Dashen Conditions 37,38 decoupling - chiral fermions 17 - spontaneous symmetry breaking 14 - theorem 9 dilatations 27 dimensional regularization 6,47,262 dimensional transmutation 287. Dirac - Lagrangian 231, 261 - matrices 259,261 - spinors 261 dispersion relations 153, 220, 303 divergences 3 - General Relativity 248 dual strength tensor 272

effective action 4 Einstein-Hilbert action 229 electroweak chiral parameters 184 - heavy Higgs model 191 - phenomenological determination

193 - QCD-like model 192 electroweak interactions 98 Equivalence Principle 230,231 Equivalence Theorem 201 - applicability 216 -- O(g2) effects 217 -- energy range 216 - effective Lagrangians 214 - unitary models 212 Euler - Euler-Heisenberg Lagrangian 11 - constant 7,263 - Euler-Lagrange equations 24

Faddeev-Popov term 52,181,290 fermions 98 fixed points 286 Fujikawa method 67 functional measure 47

gauge - R( 52,181,291 - anomaly 68 - boson - - scattering 197,212,218,224 -- trilinear vertex 196 - Feynman 245,291 - fields 50,98,259,270 - fixing 52, 181 - harmonic 247 - invariance 51

- Landau 183 - t'Hooft 52, 181 Gell-Mann-Oakes-Renner formula

130,134 Gell-Mann-Okubo formula 130, 142 General Relativity 229 Generalized Equivalence Theorem

210 generating functional 29,46 geodesic 266 ghost fields 52,291 Goldstone Theorem 33 gravitational anomalies 240 Green functions - amputated 208, 284 - connected 29,49 - definition 29 - Euclidean 29 - renormalized 30,33,280 group - chiral 36 - conformal 28 - isometry 44 - Lorentz 26 - Poincare 26 GUT 112

Heat Kernel method 298 Higgs - boson 103, 190,221 - mass 221,224 - mechanism 4,49, 175 - width 225 homogeneous space 43,269 homotopy - class 54, 278 hypercharge 99 - assignments 99,243 - matrices 99

identities - Bianchi 268, 271 - Slavnov-Taylor 208 - Ward 32,296 index - Dirac operator 68, 277 - theorem 65,68,277 infrared stable theory 286 instantons 117,274 invariance - gauge 51 - reparametrization 48 inverse amplitude method 154,220

isometries 269 isospin 37

Killing vectors 44, 269

Lagrange - Euler-Lagrange equations 24 large-N 161 - linear (j model 221 - pion phenomenology 168 LEP 193 leptons 97 Levi-Civita connection 267 LHC 193,197 Lie bracket 265 Lorentz - group 26 - local transformations 233 low-energy constants 136 - large-Nc 145 - phenomenological estimates 142 - renormalization 139 - theoretical estimates 145 - values 144,151

mass - decoupling 15 - fermion 176 - gauge boson 50, 175, 180 - Higgs boson 190,221,224 - mesons 141 - physical 30 - Planck 229 - quarks 39 - renormalization 9 - skyrmion 55 metric 263 - invariant 44 - tensor 263 - uniqueness 46 minimal coupling of gravity 235 minimal subtraction 281 mixing term 51 model - Fermi Feynman Gell-Mann 2,15 - Heavy Higgs 189 - linear (j 35,38,41 - - decoupling 15 - - large-N 168, 221 - non-linear (j 36,41,134 - - anomalies 85 -- decoupling 16 - - divergences 48 - - dynamics 42

Subject Index 313

- - fermion coupling 45 -- gauged 50,53 -- generalized 45,47 - - geometry 42 -- Lagrangian 44,131 -- large-N 163 - - quantum 46 - QCD-like 192 - standard 97 muon decay 1 Muskhelishvili-Omnes problem 305

Naturalness 177 Newton potential 251 Noether Theorem 24

oblique corrections on-shell conditions

194 187, 191

Palatini formalism 254 parallel transport 265 partial wave 302 PCAC 130 phase shifts 156, 157 pion - 7r K scattering 307 - anomalous decay 63, 64, 91 - Nambu Goldstone boson 36,129 - scattering - - amplitudes 306 --large-N 163,170 - - low-energy 132 -- phase shifts 157 Planck mass 229 Poincare Group 26 Principle of General Covariance 231

QCD 98 - axial anomaly 63 - anomalous processes 86,91 - chiral Lagrangian 130,135 - decoupling 10 - Lagrangian 125 - low energies 128 - skyrmions 54 - two flavor massless 36,39,54 QED - anomalies 60 - decoupling 11 quark condensate 38,39, 130, 134 quarks 97,126

Reduction Formula 28,49,208

314 Subject Index

relations - closure 44, 50 - commutation 42,50 renormalization - on-shell 187 resonance - K* 159 - p 130,159 - Higgs boson 221,223 - techni-p 221 - unitarization 159 Ricci tensor 268 Riemann tensor 268 running coupling constants 112,126

S-matrix - reparametrization invariance 48 - reduction formula 30 saddle point method 4 Seeley-DeWitt expansion 67,299 self-dual configurations 272 skyrmions 54 Slavnov-Taylor Identities 205,208 solitons 54 spinors 261 Standard Model 97 - baryon anomaly 109,237 - lepton anomaly 109,237 - Symmetry Breaking Sector 101 - anomaly cancellation 105, 239 - charge quantization 242 - curved space-time 234 - heavy Higgs 189 - matter 97 - precision tests 194 - symmetries 121 - Symmetry Breaking Sector 175 strong CP problem 113 structure constants 50 symmetry - U(l)A 37 - breaking pattern 42 - BRS 53 - chiral 36 - classical 23 - explicit breaking 37 - gauge 49

quantum 31 - spontaneous breaking 33

(7 models 41 - - decoupling 14

tensor - energy-momentum 27,241,245 - Ricci 268 - Riemann 268 theorem - Weinberg low-energy 48 - decoupling 9 - equivalence 201 - generalized equivalence 210 - Goldstone 33 - index 65, 277 - invariance 133 - Noether 24 theta vacuum 115 torsion 268 transformations - O(N) 16,41 - SU(2)L 99 - SU(2)L x SU(2)R 181,190 - SU(2)L x U(l)y 181,185 - SU(Nc) x SU(2)L x U(l)y 100 - SU(Nc) 98 - SU(Nf)L x SU(Nf)R 128 - U(l)y 99 - BRS 204, 291 - chiral 36 - conformal 28 - Lorentz 26, 232 - scale 27,33 translations 26 triangle diagrams 61 Triviality 177,286

U(l)A problem 65 unitarity 4, 151,302 unitarization 152 - 'Y'Y -+ 1m 159, 162, 170 - inelastic case 159 - inverse amplitude method 154,220 - large-N 161

vacuum - B 115 - choice 41 - classical 47 - degenerate vacua - expectation value - symmetry group vertex

37 17,34

34

- trilinear gauge boson vielbein 264

196

Ward identity 31 - anomalous 62 Weinberg - angle 100, 180, 187 - low-energy theorems 48,132

Wess-Zumino-Witten term 18,57,86, 89

- SU(2) 92 - gauge fields Wick rotation winding number

Subject Index 315

89 261

79,274,278

Yukawa coupling 17,103

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