Now we can begin to see how the deep inelastic *IN* scattering cross-sections can be factorized into two parts-one being momentum-independent and the other scaling in a way controlled by the renormalization group, where the cross-section is related to the absorptive part of the forward current-nucleon scattering amplitude for which one then makes a light-cone expansion; the maxtrix elements of the local operators will then give rise to the momentum-independent part and the c-number Wilson coefficients satisfy the renormalization group equation. We shall first show that these Wilson coefficients are related to the integrated moments of the *IN* structure functions.

(B) **Moments of structure functions and Wilson coefficients.** In order that the principal festures of our manipulations not be obscured by complicated Lorentz structures we shall first illustrate our procedure with scalar currents *J(x).* Consider the forward scattering amplitude (Fig. 10.9)

FIG. 10.9.

where v = *p* · *q/ M.* (Details of the kinematics may be found in §7.1.) Writing the operator-product expansion

$$T\left(J\left(\frac{\chi}{2}\right)J\left(-\frac{\chi}{2}\right)\right) \approx \sum\_{\mu^2=0} C\_l^j(\mathbf{x}^2) \mathbf{x}^{\mu\_1} \dots \mathbf{x}^{\mu\_j} \mathcal{O}\_{\mu\_1 \dots \mu\_j}^{(j,l)}(0) \tag{10.100}$$

(with the index i ranging over all twist-2 operators), then the amplitude in (10.99) becomes

$$T(q^2, \boldsymbol{\nu}) \approx \sum\_{i,j} \int \mathbf{d}^4 \mathbf{x} \, \mathbf{e}^{-iq \cdot \boldsymbol{\nu}} \mathbf{x}^{\mu\_1} \dots \mathbf{x}^{\mu\_j} C\_i^j(\boldsymbol{\kappa}^2) \langle p | \mathcal{O}\_{\mu\_1 \dots \mu\_j}^{(j,i)}(\mathbf{0}) | p \rangle$$

$$= \sum\_{i,j} (\mathbf{i})^j \frac{\partial}{\partial q\_{\mu\_1}} \frac{\partial}{\partial q\_{\mu\_2}} \dots \frac{\partial}{\partial q\_{\mu\_j}}$$

$$\times \left[ \int \mathbf{d}^4 \mathbf{x} \, \mathbf{e}^{-iq \cdot \boldsymbol{\nu}} C\_i^j(\mathbf{x}^2) \right] \langle p | \mathcal{O}\_{\mu\_1 \dots \mu\_j}^{(j,0)}(\mathbf{0}) | p \rangle \,. \tag{10.101}$$

### 10.3 *Renormalization group analysis of scaling and scaling violation* 301

The matrix element of the local operator which is symmetric and traceless can be parametrized as

$$<\langle p|\mathcal{O}\_{\mu\_1...\mu\_f}^{(f,l)}(0)|p\rangle = O\langle ^0\mathbb{I}p\_{\mu\_1}p\_{\mu\_2}...p\_{\mu\_f} - \text{trace terms}\rangle \tag{10.102}$$

where *oy>* is a constant and the trace term will contain at least one *gµmµ,*  factor. Replacing

$$\frac{\partial}{\partial q\_{\mu\_1}} \frac{\partial}{\partial q\_{\mu\_2}} \dots \frac{\partial}{\partial q\_{\mu\_f}} = 2^j q^{\mu\_1} q^{\mu\_2} \dots q^{\mu\_f} \left(\frac{\partial}{\partial q^2}\right)^j$$

$$+ \text{trace terms},\tag{10.103}$$

we obtain for large - *q2 ,* with - *q2 /2Mv* fixed,

$$T(q^2, \mathbf{v}) \underset{-q^2 \to \infty}{\approx} \sum\_{l,j} (\mathbf{2i})^j (p \cdot q)^j \left(\frac{\partial}{\partial q^2}\right)^j \left[\int \mathbf{d}^4 \mathbf{x} \, e^{-\mathbf{q} \cdot \mathbf{x}} C\_l^j(\mathbf{x}^2)\right] O\_l^{(j)}$$

$$= \sum\_j \sum\_i \left(\frac{2p \cdot q}{-q^2}\right)^j \mathcal{C}\_i^j(q^2) O\_i^{(j)} \tag{10.104}$$

where

$$\tilde{C}\_{l}^{l}(q^{2}) = (-iq^{2})^{l} \left(\frac{\partial}{\partial q^{2}}\right)^{l} \int \mathbf{d}^{4} \mathbf{x} \, \mathbf{e}^{-i\mathbf{q}\cdot\mathbf{x}} C\_{l}^{(l)}(\mathbf{x}^{2}) \tag{10.105}$$

which is essentially the Fourier transform of *xµ1 ... xµiC{(x2).* Note that the trace terms in (10.102) and (10.103) will have lower powers of *(2p* · q) and can be safely neglected in the scaling limit. Thus for the amplitude *T(q2 ,* v), decomposed in terms of spin projections

$$T(q^2, \mathbb{v}) = \sum\_{j} T\_j(q^2, \mathbb{v}),\tag{10.106}$$

we have from (10.104) that

$$T\_j(q^2, \nu) \underset{-q^2 \to \infty}{\approx} \varkappa^{-j} \sum\_i \tilde{C}\_i^{\varphi\_0} O\_i^{\psi} \tag{10.107}$$

for *x* = *-q2 /2Mv.* This implies that to isolate an operator of a given spinj we need just expand *T(q2 ,* v) in powers of x- 1 for large *-q2 •* In deep inelastic scattering, since one actually measures the absorptive part of the forward amplitude *T(q2 ,* v),

$$W(q^2, \nu) = \frac{1}{\pi} \operatorname{Im} \ T(q^2, \nu). \tag{10.108}$$

The amplitude can be reconstructed from the measured quantities by using the dispersion relation

$$\begin{aligned} \text{relation} \\ T(q^2, \boldsymbol{\nu}) = \int \frac{\boldsymbol{\nu}^s \, \mathrm{d} \boldsymbol{\nu}'}{\mathbf{v}'^s (\boldsymbol{\nu}' - \boldsymbol{\nu})} \, \boldsymbol{W}(q^2, \boldsymbol{\nu}') + P\_{s-1}(q^2, \boldsymbol{\nu}) \end{aligned} \tag{10.109}$$

where we have assumed *s* number of subtractions with *P5 \_ 1(q2 ,* v) being a polynomial in *v* of order *s* - 1 for fixed *q2 •* If we further assume that, for

large *-q2 ,* we have *Ps-i(q2 ,v)-+P5 \_i(x),* a polynomial in *x-i* of order *s* - 1, then

$$T(q^2, \mathbf{v}) = \int\_{-1}^{+1} \frac{(\mathbf{x}')^{s-1} W(q^2, \mathbf{x}') \, \mathrm{d}\mathbf{x}'}{(\mathbf{x}')^{s-1} (\mathbf{x}' - \mathbf{x})} + P\_{s-1}(\mathbf{x})$$

$$\approx \underset{\mathbf{x} \to \mathbf{x}}{\text{arg}} P\_{s-1}(\mathbf{x}) + \sum\_{J=s}^{\infty} \underset{-1}{\mathbf{x}'}^{+1} \mathbf{d}\mathbf{x}'(\mathbf{x}')^{J-1} W(q^2, \mathbf{x}'). \quad \text{(10.110)}$$

The first *s* terms in this expansion are undetermined because of the unknown subtraction constants. However for *J?. s* (comparing eqn (10.110) to eqns (10.106) and (10.107)), we have

$$\int\_{-1}^{+1} \mathrm{d}x \boldsymbol{x}^{\prime -1} \boldsymbol{W}(\boldsymbol{q}^{2}, \boldsymbol{x}) \approx \sum\_{-q^{2} \to \infty} \tilde{\mathcal{C}}\_{i}^{(\prime)}(q^{2}) \boldsymbol{O}\_{i}^{(\prime)} \quad \text{for} \quad \boldsymbol{J} \ge \mathrm{s}. \tag{10.111}$$

Thus the moments of the structure functions measure the (Fourier transforms of) Wilson coefficient functions C/)(Q2 ).

For the more realistic case of the electromagnetic current *J---+* we can make a similar analysis. With the usual assumption about the highenergy behaviour of the forward Compton amplitude, the relation (I 0.111) will hold for all *J?.* 2, when the t-channel of the Compton scattering (i.e. the current x current channel) has the quantum number of the vacuum, and for *J?.* 1 in the non-vacuum channels. When decomposed in terms of the two invariant eN inelastic structure functions *Fi.* 2 (x, Q2) as in eqn (7.30) (we have changed notation from that used in §7.1: *G;(x, q2 )* ---+ *F;(x,* Q2 )), the result corresponding to eqn (10.111) reads

$$\int\_{-1}^{1} \mathrm{d}x \boldsymbol{\chi}^{\prime -2} F\_2(\boldsymbol{\chi}, \boldsymbol{\mathcal{Q}}^2) \approx \frac{1}{4} \sum\_{l} \tilde{C}\_l^{(\prime)}(\boldsymbol{\mathcal{Q}}^2) O\_l^{(\prime)}\tag{10.112a}$$

$$\int\_{-1}^{1} \mathrm{d}x \boldsymbol{x}^{\prime -1} F\_1(\boldsymbol{x}, \boldsymbol{Q}^2) \approx \frac{1}{2} \sum\_{i} \tilde{C}\_i^{(\prime)}(\boldsymbol{Q}^2) O\_i^{(\prime)}.\tag{10.112b}$$

We have succeeded in isolating from the cross-section, which has two mass scales *p2* = *M* 2 and *q2 ,* a factor which depends only on *q2 ,* to which we can apply the scale transformation *q11* ---+ *A.q11* and renormalization group analysis. We note that exact Bjorken scaling (eqn (7.32)) *F;(x, q2 )---+ F;(x)* corresponds to free-field behaviour

$$
\tilde{C}\_i^{(J)}(\mathcal{Q}^2) \to \text{constant as } \mathcal{Q}^2 \to \infty. \tag{10.113}
$$

In general we expect this to be modified by the interaction. The simplest possible deviation from this scaling behaviour would be such that the ln(Q2 */m 2 )* powers in every order of the perturbation are summed up into

some Q2 powers as

$$
\tilde{C}\_{\gamma}^{(\prime)}(\mathcal{Q}^2) \sim \left(\frac{1}{\mathcal{Q}^2}\right)^{\gamma\nu^2} \cdot \tag{10.114}
$$

This can be interpreted as an anomalous dimension *y J* acquired by the operator ... *flj* due to the interaction. As conserved quantities are finite, their renormalization constants do not depend on cut-off and their anomalous dimension vanishes. The electromagnetic currents and the energy momentum tensors are such quantities. If they can appear on the right-hand side of the operator-product expansion, then (the Fourier transforms of) their Wilson coefficients will scale as in free-field theory without anomalous dimensions. For the general moments, eqn (10.114) however implies that

$$\int\_{0}^{1} F\_{2}(\mathbf{x}, \mathcal{Q}^{2}) \mathbf{x}^{J-2} \, d\mathbf{x} \underset{\mathcal{Q}^{1} \to \infty}{\approx} \left(\frac{1}{\mathcal{Q}^{2}}\right)^{\chi \mu 2} \cdot \tag{10.115}$$

Since the structure functions *F;* are positive definite, the anomalous dimension *y J* must monotonically increase with *J* and, since the energymomentum tensor *eµv* does appear in vacuum channel of the current operator-product expansion, we have *y2* = 0 and

$$
\gamma\_\rangle \ge 0 \quad \text{for} \quad J \ge 2. \tag{10.116}
$$

This means that the moments of the structure functions should in the vacuum channel decrease with increasing Q2 . Also experimentally observed approximate scaling implies that the anomalous dimensions must be very small.

(C) **Renormalization group equations for the Wilson coefficients.** In Chapter 3 we introduced the renormalization group equation for a general Green's function. We can obtain similar equations for the Wilson coefficients by comparing the renormalization group equations satisfied by Green's functions containing the operator product itself and containing the local operators appearing in the expansion of the operator product (Christ, Hasslacher, and Mueller 1972).

Schematically the operator-product expansion is of the form

$$A(\mathbf{x})B(\mathbf{0}) \underset{\mathbf{x} \to \mathbf{0}}{\approx} \sum\_{l} C\_{l}(\mathbf{x}, g, \mu) \mathcal{O}\_{l}(\mathbf{0})\tag{10.117}$$

where *g* is the coupling constant andµ is the reference (subtraction) point for the renormalization. Or, in terms of n-point Green's functions with insertions of *AB* and@;,

$$
\Gamma^{\{\mathfrak{v}\}}\_{\mathcal{A}\mathfrak{g}} \underset{\mathfrak{x} \twoheadrightarrow \mathfrak{o}}{\operatorname{\sum}} \, \mathrm{C}\_{l}(\mathfrak{x}, g, \mu) \Gamma^{\{\mathfrak{v}\}}\_{\mathbb{R}\_{l}} \tag{10.118}
$$

where

$$\Gamma\_{A\mathcal{B}}^{(n)} = \langle 0|T(\mathcal{A}(\mathbf{x})\mathcal{B}(\mathbf{0})\prod\_{k=1}^{n} \phi\_k(\boldsymbol{\nu}\_k)|0\rangle \tag{10.118a}$$

$$\Gamma\_{\mathcal{C}\_i}^{\{\mathfrak{n}\}} = \langle 0 | T(\mathcal{C}\_i(\mathbf{0}) \prod\_{k=1}^n \phi\_k(\boldsymbol{\wp}\_k)) | 0 \rangle. \tag{10.118b}$$

The Green's functions separately satisfy the renormalization group equations (see eqn (3.58))

$$\left[\mathcal{D} + \sum\_{k} \gamma\_k(g) - \gamma\_A(g) - \gamma\_B(g)\right] \Gamma\_{AB}^{(n)}(g, \mu) = 0$$

and

$$\left[\mathbf{D} + \sum\_{\mathbf{k}} \boldsymbol{\eta}\_{\mathbf{k}}(g) - \boldsymbol{\eta}\_{l}(g)\right] \Gamma\_{\mathbb{S}\_{\boldsymbol{\zeta}}}^{(\mathfrak{s})}(g, \boldsymbol{\mu}) = 0$$

where

$$\mathbf{D} = \mu \frac{\partial}{\partial \mu} + \beta(g) \frac{\partial}{\partial g} \tag{10.119}$$

and *YA• y8 ,* and *Yi* are the anomalous dimensions of the operators of *A, B,* and *(!Ji,* respectively; we have assumed that they do not mix under renormalization. Using (10.118), clearly we have

$$\mathbb{E}\left[\mathcal{D} + \gamma\_A(g) + \gamma\_B(g) - \gamma\_i(g)\right] \mathcal{C}\_i(\mathbf{x}, g, \mu) = \mathbf{0}.\tag{10.120}$$

Thus, the Wilson coefficient Ci behaves as if it were a Green's function of the operators *A, B,* and&;. The solution as given by eqn (3.68) takes on the form

$$C\_{i}(\mathbf{e}^{-t}\chi\_{0},g,\mu) = \mathbf{e}^{(d\mu+d\mathfrak{s}-d)}\exp\left\{\int\_{0}^{t} \mathrm{d}t'[\chi\_{A}(\bar{g}(t')) + \chi\_{B}(\bar{g}(t')) ] \right\}$$

$$-\chi\_{i}(\bar{g}(t')) \Big| \begin{cases} C\_{i}(\chi\_{0},g(t),\mu) \\ \end{cases} \tag{10.121}$$

where *dA> d8 ,* and *di* are the naive dimensions of *A, B,* and *(!Ji·* Similar equations and solutions of course hold for the Fourier transforms of the Wilson coefficients *C1(q2 , g, µ).* For the case of deep inelastic scattering we have the light-cone expansion which can be turned into sums of infinite towers of local operators of increasing spin *n* with Wilson coefficients *Ci(q2 , g, µ),* which are related to the moments of the structure functions by (10.112)

$$M\_n(\mathcal{Q}^2) = \int\_0^1 \mathrm{d}x \mathbf{x}^{n-2} F\_2(\mathbf{x}, \mathcal{Q}^2) \approx \frac{1}{8} \sum\_l \mathcal{E}\_l^n(\mathcal{Q}^2, g, \mu) \mathcal{O}\_l^n$$

$$= \frac{1}{8} \sum\_l \mathcal{E}\_l^n(\mathcal{Q}\_0^2, \bar{g}(t), \mu) \exp\left[-\int\_0^t \gamma\_n^i(\bar{g}(t')) \, \mathrm{d}t'\right] \mathcal{O}\_l^n \qquad \text{(10.122)}$$

where is the anomalous dimension of the

(D) **Deep inelastic scattering in QCD.** We will now apply this analysis of deep inelastic scattering in QCD (Gross and Wilczek 1974; Georgi and

### 10.3 *Renormalization group analysis of scaling and scaling violation* 305

Politzer 1974), where the electromagnetic current is given by

$$J\_{\mu}(\mathbf{x}) = \sum\_{k=1}^{n\_f} :\bar{q}\_k(\mathbf{x})\gamma\_{\mu}e\_k q\_k(\mathbf{x}):\tag{10.123}$$

where *qk(x)* is the quark field operator with flavour index *k, ek* is the charge carried by *qk,* and the sum over colour is implicit. Consider the forward Compton scattering amplitude (averaging over nucleon spin is understood)

$$T^{(p,q)}\_{\mu\nu} = \frac{i}{2M} \int \frac{\mathbf{d}^4 y}{2\pi} \,\mathrm{e}^{ip\cdot p} \langle p | T(J\_\mu(y)J\_\nu(0)) | p \rangle \tag{10.124}$$

$$= \left( -g\_{\mu\nu} + \frac{q\_{\mu}q\_{\nu}}{q^2} \right) T\_1 + \frac{1}{M^2} \left( p\_{\mu} - \frac{p \cdot q}{q^2} q\_{\mu} \right) \left( p\_{\nu} - \frac{p \cdot q}{q^2} q\_{\nu} \right) T\_2. \tag{10.125}$$

The absorptive parts of the invariant amplitudes *T;* are just the structure functions *W;* measured in deep inelastic scattering

$$W\_{1,2}(\mathbf{v}, q\_2) = \frac{1}{\pi} \operatorname{Im} \ T\_{1,2}(\mathbf{v}, q^2). \tag{10.126}$$

(D 1) *Operator-product expansion and moments of structure functions.* The operator-product expansion on the light cone is of the form

$$\mathrm{i}T(J\_{\mu}(\boldsymbol{\nu})J\_{\boldsymbol{\nu}}(\mathbf{0})) = \sum\_{\boldsymbol{n},\boldsymbol{l}} \{-g\_{\mu\nu}\boldsymbol{\nu}\_{\mu\_{1}}\boldsymbol{\nu}\_{\mu\_{2}}\dots\boldsymbol{\nu}\_{\mu\_{n}}\mathbf{i}^{\boldsymbol{n}}C\_{1,i}^{(\boldsymbol{n})}(\boldsymbol{\nu}^{2},g,\boldsymbol{\mu})$$

$$+g\_{\mu\mu\_{1}}g\_{\nu\mu\_{2}}\boldsymbol{\nu}\_{\mu\_{3}}\dots\boldsymbol{\nu}\_{\mu\_{n}}\mathbf{i}^{\boldsymbol{n}-2}C\_{2,i}^{(\boldsymbol{n})}(\boldsymbol{\nu}^{2},g,\boldsymbol{\mu})\}\mathcal{O}\_{i}^{(\boldsymbol{n})\mu\_{1}\mu\_{2}\dots\mu\_{n}}(\mathbf{0}).\tag{10.127}$$

In QCD there are three sets of gauge-invariant twist-2 operators (see eqns (10.128a, b, c) below) which dominate expansion near the light cone *y 2*0

$$\mathcal{V}\xi^{\mu\_1\dots,\mu\_n}(\mathbf{x}) = \frac{1}{2} \frac{i^{n-1}}{n!} \left\{ \bar{q}(\mathbf{x}) \frac{\lambda^a}{2} \gamma^{\mu\_1} \mathbf{D}^{\mu\_2} \dots \mathbf{D}^{\mu\_q} q(\mathbf{x}) \right. $$

$$+ \text{ permutations of vector indices} \right\} \qquad \text{(10.128a)}$$

where the ;.as are the standard nr x nr hermitian traceless matrices in the flavour group SU(nr); thus for a theory with three flavours u, d, and s they are just the familiar SU(3) Gell-Mann matrices. This set of operators will contribute to operator-product expansions for the flavour non-singlet combinations of structure functions such as *Pi* - or *F;A,* for neutrino scattering off an isoscalar target A. Here we will devote most of our effort to the study of the more involved case of the flavour-singlet combinations. They can receive contributions from two sets of operators

$$\begin{aligned} \mathcal{O}\_q^{(n)\mu\_1\dots} &\dots \text{(x)}\\ &= \frac{1}{2} \frac{\mathbf{i}^{n-1}}{n!} \left\{ \bar{q}(\mathbf{x}) \eta^{\mu\_1} \mathbf{D}^{\mu\_2} \dots \mathbf{D}^{\mu\_n} q(\mathbf{x}) + \text{permutations} \right\} \quad \text{(10.128b)}\\ \mathcal{O}\_\mathbf{G}^{(n)\mu\_1\dots\mu\_n} &\text{(x)}\\ &\mathbf{j}^{n-2} \end{aligned}$$

$$=\frac{\dot{\mathbf{u}}^{\pi-2}}{n!}\operatorname{tr}\{G^{\mu\_1\mathbf{v}}\mathbf{D}^{\mu\_2}\dots\mathbf{D}^{\mu\_{\pi-1}}G^{\mu\_n}\_{\mathbf{v}}+\text{permutations}\}.\tag{10.128c}$$

Because and have identical quantum numbers, they can mix under renormalization (see §2.4).

Substituting (10.127) into (10.124), we obtain

$$T\_{\mu\nu}(p,q) = \frac{\mathrm{i}}{2M} \int \frac{\mathrm{d}^4 \mathcal{V}}{2\pi} \,\mathrm{e}^{i\mathfrak{q}\cdot\mathcal{V}} \sum\_{n,i} \left\{-g\_{\mu\nu}\mathcal{V}\_{\mu\_1}\mathcal{V}\_{\mu\_2}\dots\mathcal{V}\_{\mu\_n}\mathrm{i}^n C^{(n)}\_{1,i}(\mathcal{V}^2,g,\mu)\right\} $$

$$+ g\_{\mu\mu\_1}g\_{\nu\mu\_2}\mathrm{i}^{n-2}(\mathcal{V}\_{\mu\_3}\dots\mathcal{V}\_{\mu\_n})C^{(n)}\_{2,i}(\mathcal{V}^2,g,\mu)\} $$

$$\times \left\langle p|\mathcal{C}^{(n)\*}\_{\mathrm{i}}\,\dots\,\mu\_n(0)|p\right\rangle + \dots \,\tag{10.129}$$

Writing the spin-averaged matrix element as

$$<|\mathcal{O}|^{(\mathfrak{n})\mu\_1\dots\mu\_{\mathfrak{s}}}(0)|p\rangle = A\_i^{(\mathfrak{n})}(p^{\mu\_1}p^{\mu\_2}\dots p^{\mu\_{\mathfrak{s}}}+\text{trace terms})\qquad(10.130)$$

where Alnl is a constant, eqn (10.129) becomes

$$T\_{\mu\nu}(p,q) \sim \frac{1}{2M} \sum\_{n,l} \left\{-g\_{\mu\nu}(2p\cdot q)^{n} \left(\frac{\partial}{\partial q^{2}}\right)^{n} \int \mathrm{d}^{4}y \,\mathrm{e}^{iq\cdot\nu}C\_{1,l}^{(n)}(y^{2},g,\mu)\right\}A\_{l}^{(n)} + \dots$$

$$\qquad + p\_{\mu}p\_{\nu}(2p\cdot q)^{n-2} \left(\frac{\partial}{\partial q^{2}}\right)^{n} \int \mathrm{d}^{4}y \,\mathrm{e}^{iq\cdot\nu}C\_{2,l}^{(n)}(y^{2},g,\mu)\right\}A\_{l}^{(n)} + \dots$$

$$= \frac{1}{2M} \sum\_{n,l} \left\{-g\_{\mu\nu} \left(\frac{2p\cdot q}{-q^{2}}\right)^{n} \mathcal{C}\_{1,l}^{(n)}(q^{2},g,\mu)\right.$$

$$\qquad\qquad + p\_{\mu}p\_{\nu} \left(\frac{2p\cdot q}{-q^{2}}\right)^{n-1} \frac{1}{2p\cdot q} \mathcal{C}\_{2,l}^{(n)}(q^{2},g,\mu)\right\}A\_{l}^{(n)} + \dots \tag{10.131}$$
 where

$$\begin{split} \mathcal{K}^{(\mathfrak{n})}\_{1,i}(\mathcal{Q}^{2},g,\mu) &= (\mathcal{Q}^{2})^{\mathfrak{n}} \Big( \frac{\partial}{\partial q^{2}} \Big)^{\mathfrak{n}} \int \mathsf{d}^{4}y \,\mathrm{e}^{\mathfrak{n}\cdot y} C^{(\mathfrak{n})}\_{1,i}(\mathcal{Y}^{2},g,\mu) \\ \mathcal{K}^{(\mathfrak{n})}\_{2,i}(\mathcal{Q}^{2},g,\mu) &= (\mathcal{Q}^{2})^{\mathfrak{n}-1} \Big( \frac{\partial}{\partial q^{2}} \Big)^{\mathfrak{n}-2} \int \mathsf{d}^{4}y \,\mathrm{e}^{\mathfrak{n}\cdot y} C^{(\mathfrak{n})}\_{2,i}(\mathcal{Y}^{2},g,\mu) . \end{split} \tag{10.132}$$

From (I 0.131) we can immediately read out the invariant amplitudes

$$T\_1(\mathbf{x}, \mathcal{Q}^2) = \frac{1}{2M} \sum\_{\mathbf{n}, \mathbf{i}} \mathbf{x}^{-\mathbf{n}} \tilde{C}\_{1, \mathbf{i}}^{(\mathbf{n})}(\mathcal{Q}^2, g, \mu) \mathcal{A}\_{\mathbf{i}}^{(\mathbf{n})} \tag{10.133a}$$

$$\mathcal{L}^{\mathcal{M}}\,\_{n,i}$$

$$T\_2(\mathbf{x}, \mathcal{Q}^2) = \frac{1}{2M} \sum\_{n,i} \mathbf{x}^{-n+1} \mathcal{C}^{(n)}\_{2,i}(\mathcal{Q}^2, g, \mu) \mathcal{A}^{(n)}\_i. \qquad (10.133b)$$

By the same route we took going from (10.106) to (10.112), we can obtain relation between the moments of the structure functions and the (Fourier transforms of) Wilson coefficients

$$\int\_0^1 \mathrm{d}x \mathrm{x}^{n-2} F\_2(\mathbf{x}, \mathcal{Q}^2) \approx \frac{1}{8} \sum\_i \tilde{C}\_{1,i}^{(n)}(\mathcal{Q}^2, g, \mu) \mathcal{A}\_i^{(n)} \tag{10.134a}$$

$$\int\_0^1 \mathrm{d}x \chi^{n-1} F\_1(x, \mathcal{Q}^2) \approx \frac{1}{4} \sum\_i \tilde{C}\_{2,i}^{(\mathfrak{v})}(\mathcal{Q}^2, g, \mu) A\_i^{(\mathfrak{v})}.\tag{10.134b}$$

(02) *Renormalization group analysis and anomalous dimension matrix for the singlet case.* We shall concentrate on the flavour singlet case. Because of mixing among twist-2 operators, the renormalization group equation for the Wilson coefficients takes on the matrix form

$$(\mathcal{D}\,\,\delta\_{jk} - \gamma\_{kj}^{(\mathfrak{n})})\tilde{C}\_{\mathfrak{a},j}^{(\mathfrak{n})}(q^2, g, \mu) = 0, \quad \mathfrak{a} = \mathfrak{l}, \mathcal{D} \tag{10.135}$$

where *y<n>* is the 2 x 2 anomalous dimension matrix for the flavour singlet operators and The solution to eqn (10.135) is

$$\tilde{C}^{(n)}\_{\mathfrak{a},l}(Q^2/\mu^2,g) \sim \sum\_{i} \tilde{C}^{(n)}\_{\mathfrak{a},l}(1,\bar{g}(t)) \exp\left\{-\int\_{0}^{l} \mathrm{d}t' \gamma^{(n)}\_{jl}(\bar{g}(t'))\right\} \quad \text{(10.136)}$$

where

$$t = \frac{1}{2} \ln(Q^2/\mu^2). \tag{10.137}$$

In asymptotically free QCD one can calculate *y<n>* perturbatively according to the one-loop diagrams of Fig. 10.10

$$
\gamma\_{ij}^{(\mathfrak{n})} = d\_{ij}^{(\mathfrak{n})} g^2 + O(g^3) \tag{10.138}
$$

FIG. 10.10. Graphs contributing to (a) (b) (c) and (d)

with

$$d\_{\rm ij}^{(n)} = \begin{pmatrix} d\_{\rm qq}^{(n)} & d\_{\rm Gq}^{(n)} \\ d\_{\rm qG}^{(n)} & d\_{\rm GG}^{(n)} \end{pmatrix}.$$

where

$$\begin{aligned} d\_{\rm qq}^{\langle n\rangle} &= \frac{1}{16\pi^2} \left[ \frac{8}{3} \left( 1 - \frac{2}{n(n+1)} + 4 \sum\_{j=2}^{n} \frac{1}{j} \right) \right] \\\ d\_{\rm qG}^{\langle n\rangle} &= \frac{1}{16\pi^2} \left[ -\frac{16}{3} \frac{n^2 + n + 2}{n(n^2 - 1)} \right] \\\ d\_{\rm qq}^{\langle n\rangle} &= \frac{1}{16\pi^2} \left[ -4n\_{\rm l} \frac{n^2 + n + 2}{n(n+1)(n+2)} \right] \\\ d\_{\rm GG}^{\langle n\rangle} &= \frac{1}{16\pi^2} \left\{ 6 \left[ \frac{1}{3} - \frac{4}{n(n-1)} - \frac{4}{(n+1)(n+2)} + 4 \sum\_{j=2}^{n} \frac{1}{j} \right] + \frac{4}{3} n\_{\rm l} \right\} \end{aligned} \tag{10.139}$$

nr is the number of quark flavours. This matrix can be diagonalized by taking appropriate linear combinations of and and its eigenvalues correspond to the anomalous dimensions of the linear combinations that are multiplicatively renormalizable. Using the lowest-order expression (10.80) of the effective coupling constant in (10.138), we have

$$-\int\_{0}^{t} \mathrm{d}t' \eta\_{jl}^{\{n\}}(\bar{g}(t')) = -\int\_{0}^{t} \frac{g^{2}d\_{jl}^{\{n\}} \, \mathrm{d}t'}{1 + 2bg^{2}t'} $$

$$= \frac{-d\_{jl}^{\{n\}}}{2b} \ln(1 + 2bg^{2}t) \to \frac{-d\_{jl}^{\{n\}}}{2b} \ln t. \qquad (10.140)$$

The Wilson coefficient functions have the large-q2 behaviour

$$\tilde{C}^{(n)}\_{\mathfrak{a},i}(\mathcal{Q}^2/\mu^2,g) \sim \sum\_{j} \tilde{C}^{(n)}\_{\mathfrak{a},j}(1,0) \left[ \ln \left( \frac{\mathcal{Q}^2}{\mu^2} \right) \right]^{-d^{m\_j}\_{j\_l}/2\mathfrak{a}} \tag{10.141}$$

where the ctj( -1, O)s are the Wilson coefficients in free-field theory. For large *Q2 ,* (10.134) now reads

$$\begin{split} M\_{2}^{(\eta)}(Q^{2}) &= \int\_{0}^{1} \mathrm{d} \mathbf{x} \pi^{n-2} F\_{2}(\mathbf{x}, \mathcal{Q}^{2}/\mu^{2}) \\ &\approx \frac{1}{8} \sum\_{i} \bar{C}\_{2,i}^{(\eta)}(1,0) \mathcal{A}\_{i}^{(\eta)} [\ln(Q^{2}/\mu^{2})]^{-d\_{H}^{(\eta)}/2b} \\ M\_{1}^{(\eta)}(Q^{2}) &= \int\_{0}^{1} \mathrm{d} \mathbf{x} \pi^{n-1} F\_{1}(\mathbf{x}, \mathcal{Q}^{2}/\mu^{2}) \\ &\approx \frac{1}{4} \sum\_{i} \bar{C}\_{1,i}^{(\eta)}(1,0) \mathcal{A}\_{i}^{(\eta)} [\ln(Q^{2}/\mu^{2})]^{-d\_{H}^{(\eta)}/2b} . \tag{10.142b} \end{split}$$

These are the principal QCD results on deep inelastic scattering. We have obtained them by factoring the inclusive cross-section into a momentumindependent part (the local operator) and a part that scales according to the renormalization group (the Wilson coefficients). In the asymptotically free QCD the leading singular (Q2 --+ oo) behaviour of the Wilson coefficients can be calculated in terms of the renormalization group P-function and the anomalous dimensions YI.fl, while the matrix elements *Al">* of the local operator cannot be obtained without solving the (long-distance) bound-state problem in QCD. While we cannot calculate the scaling functions themselves, nevertheless we do have enough information on the pattern of scaling and scaling violation. QCD predicts that moments of the structure functions have a very weak dependence on *q2* as in eqn (10.142). So we have approximate Bjorken scaling with logarithmic violations.

(03) *Momentum sum rule revisited.* As an illustration we shall work out the flavour-singlet second moment *(n* = 2) of the structure function, say

*F 2 (x,* Q2 ). In this case one of the spin-2 and twist-2 local operators will be the energy-momentum tensor *(}µv·* Its conservation implies no renormalization *Z6* = 1; hence we should find a vanishing anomalous dimension *y6* = 0. (For a related discussion on the renormalization of the conserved vector current operator ZJ = 1, see §6.1.) For *n* = 2, eqn (10.128) indicates that there are two twist-2 operators

$$\{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\/\!\!\}\_{\mu\nu} = \frac{\text{i}}{\text{4}} \sum\_{k=1}^{m} \bar{q}\_{k} (\chi\_{\mu} \stackrel{\leftarrow}{\partial}\_{\text{v}} + \chi\_{\text{v}} \stackrel{\leftarrow}{\partial}\_{\mu}) q\_{k} \tag{10.143a}$$

$$\mathbb{T}\mathbb{G}\_{\mathbb{G}}^{(2)}\mathbb{J}^{\mu\nu} = \text{tr}(G^{\mu\lambda}G\_{\lambda}^{\nu}).\tag{10.143b}$$

We are taking only the lowest-order terms; thus *Gµv* = *oµAv* - *o,Aµ.* The matrix in (10.139) takes on the value

$$d\_{\rm{ij}}^{(2)} = \frac{1}{36\pi^2} \begin{pmatrix} 16 & -3\eta\_{\rm{f}} \\ -16 & 3\eta\_{\rm{f}} \end{pmatrix} . \tag{10.144}$$

It clearly has a zero eigenvalue *a6* = 0 corresponding to the left eigenvector (1, 1) = (1, 0) + (0, 1), i.e. the combination

$$\boldsymbol{\theta}\_{\mu\nu} = \mathbb{E}\mathcal{O}\_{\mathbf{q}}^{(2)} + \mathcal{O}\_{\mathbf{G}}^{(2)} \mathbf{J}\_{\mu\nu} \tag{10.145}$$

which is just the energy-momentum tensor. The other eigenvalue is *J6•* = (16 + *3nr)/36n2 •* We can in general express *d!j>* in terms of its eigenvalues ii; and their projection operators *P;* with respect to the eigenvectors

$$d\_{ij}^{\{\eta\}} = \sum\_{k} \tilde{d}\_k (P\_k)\_{ij}, \quad P\_k P\_l = \delta\_{kl} P\_k, \quad \sum\_k P\_k = 1. \tag{10.146}$$

It is easy to see that for our *n* = 2 case, we have the projection operators (for left vectors)

$$P\_{\theta} = \frac{1}{16 + 3n\_{\text{f}}} \begin{pmatrix} 3n\_{\text{f}} & 3n\_{\text{f}} \\ 16 & 16 \end{pmatrix}, \quad P\_{\theta'} = \frac{1}{16 + 3n\_{\text{f}}} \begin{pmatrix} 16 & -3n\_{\text{f}} \\ -16 & 3n\_{\text{f}} \end{pmatrix}. \tag{10.147}$$

The second moment of the flavour singlet structure function can be written as

$$\begin{split} \int\_{0}^{1} \mathrm{d}x F\_{2}(\mathsf{x}, \mathsf{Q}^{2}) &= \frac{1}{8} \sum\_{l,j} \langle \tilde{C}^{(2)}\_{2,l}(1,0)(P\_{\theta})\_{lj} A^{(2)}\_{j} \\ &+ \tilde{C}^{(2)}\_{2,l}(1,0)(P\_{\theta})\_{lj} A^{(2)}\_{j} \langle \mathrm{ln}(\mathsf{Q}^{2}/\mu^{2}) \rangle^{-\delta\_{\theta}/2b} \,. \end{split} \tag{10.148}$$

Since *J9,* > 0, we have for *Q2*-+ oo

$$\int\_0^1 \mathrm{d}x F\_2(\mathbf{x}, \mathcal{Q}^2) = \frac{1}{8} \sum\_{i,j} \left\{ \tilde{C}\_{2,i}^{(2)}(1, \mathbf{0}) (P\_\theta)\_{ij} A\_j^{(2)} \right\}. \tag{10.149}$$

To calculate the right-hand side we note that, to lowest-order in *g(t),* only the quarks contribute to the (current) operator-product expansion; hence the free-field values of the Wilson coefficients are

$$
\tilde{C}\_{2,\mathbf{G}}^{(2)}(1,0) = 0, \quad \tilde{C}\_{2,\mathbf{q}}^{(2)}(1,0) = \langle e\_{\mathbf{q}}^2 \rangle \tag{10.150}
$$

where

$$
\langle e\_{\mathbf{q}}^2 \rangle = \frac{1}{n\_{\mathbf{f}}} \sum\_{\mathbf{k}} e\_{\mathbf{k}}^2 \tag{10.151}
$$

is the average quark charge squared. Thus the right-hand side involves only *I:i (P6 )qiA)2>.* To evaluate this we first note

$$(1,0)\frac{1}{16+3n\_{\rm f}}\begin{pmatrix} 3n\_{\rm f} & 3n\_{\rm f} \\ 16 & 16 \end{pmatrix}\begin{pmatrix} \mathcal{O}\_{\rm q}^{(2)} \\ \mathcal{O}\_{\rm G}^{(2)} \end{pmatrix}$$

$$=\frac{3n\_{\rm f}}{16+3n\_{\rm f}}\left(\mathcal{O}\_{\rm q}^{(2)} + \mathcal{O}\_{\rm G}^{(2)}\right) \equiv r\theta\tag{10.152}$$

where() is the energy-momentum tensor operator (10.145) and where

$$r = \frac{3n\_{\text{f}}}{16 + 3n\_{\text{f}}} \tag{10.153}$$

can be interpreted as the fraction of momentum carried by quarks. For three flavours of quarks, *r* = 9/25 which is in accord with the experimental results when interpreted in the quark-parton model (momentum sum rule) of §7.2. Unlike in the general case, the matrix element of *e,,.* is known

$$
\langle p|\theta\_{\mu\nu}|p\rangle = p\_{\mu}p\_{\nu} - g\_{\mu\nu}p^2. \tag{10.154}
$$

When eqn (10.154) is compared with (10.128) and (10.130), we have *(P6)qjA}2>* = *Sr;* in this way eqn (10.149) becomes

$$\left[\int\_{0}^{1} \mathrm{d}x F\_{2}(\mathbf{x}, q^{2})\right]\_{\mathrm{singlet}} = r \langle e\_{q}^{2} \rangle. \tag{10.155}$$

This means that the area under the structure-function curve scales without deviation.

(04) *Pattern of scaling violation.* For general *n,* the diagonalized d!"'s have the property of increasing slowly with *n.* For *n* > 2, where the large *x*  ("' 1) region is important, the moment M<"l(Q2) decreases as [ln(Q2)]-ld•l/2b; for *n* < 2, where the small-x ("' 0) region is important, the moment increases as [ln(Q2)] + *ld•IJ2b.* This implies the following pattern of scaling violation; as *Q2*--+ oo, the large-x part of *Fi(x, Q2 )* decreases while the small*x* part increases, while the area under the curve remains unchanged throughout (see Fig. 10.ll). In the infinite *Q2* limit, the structure function approaches a sharp spike at *x* = 0.

In concluding this section we will briefly discuss a type of experimental check on QCD predictions that has been performed. To this end we will consider the simpler case of a *non-singlet* combination of structure functions

FIG. I 0.11. The pattern of scaling violation of *F2(x,* Q2 ) for > Qi.

where there is no complication due to operator mixing. Then the result corresponding to (10.142) reads

$$\mathbb{E}\left[M\_2^{(\bullet)}(\underline{Q}^2)\right]\_{\text{non-singlet}} = B^{(\bullet)}\mathbb{E}[\ln(\underline{Q}^2/\mu^2)]^{-d^{(\bullet)}/2b} \tag{10.156}$$

where *s<n>* is an unpredicted constant and d(nJ is the *g2* coefficient of the anomalous dimension for the operator in (10.128),

$$d^{(n)} = \frac{1}{6\pi^2} \left[ 1 - \frac{2}{n(n+1)} + 4 \sum\_{j=2}^{n} \frac{1}{j} \right]. \tag{10.157}$$

which is the same as as one would expect. Now we take logarithms of the moments

$$
\ln M^{(\mathfrak{n})}(\mathcal{Q}^2) = \frac{-d^{(\mathfrak{n})}}{2b} \ln \ln(\mathcal{Q}^2/\mu^2) + \dots
$$

$$
\ln M^{(\mathfrak{n}')}(\mathcal{Q}^2) = \frac{-d^{(\mathfrak{n}')}}{2b} \ln \ln(\mathcal{Q}^2/\mu^2) + \dots. \tag{10.158}
$$

The dots on the right-hand sides of (10.158) represent non-leading terms (which could still be significant at the present energy level).

Ifwe plot these two logarithms against each other, we should get a straight line with slope *d<n>;d<n'l.* Such experimental plots have been found to agree with the QCD prediction within limits of experimental error, see, for example, Bosetti *et al.* (1978) and de Groot *et al.* (1979).

# **10.4 The parton model and perturbative QCD**

In this section we present a brief introduction to the study of perturbative QCD. We shall concentrate again on the prototype processes of deep inelastic lepton-hadron scattering. First we present the parton-model probabilistic interpretation of the QCD result obtained in the previous section. We then show in what ways the same result can be recovered by summing the leading logarithms in perturbation theory.

Let us first recapitulate the QCD result for deep inelastic *lN* scatterings obtained by using the operator-product expansion and the renormalization group equation. For the *nth* moment of the flavour non-singlet combinations of the structure functions *F* 2 (x, Q2 ), the result (10.156) may be written

$$\text{ms}\,\,r\_2(\lambda,\,\,\zeta\,\,), \text{ one result (10.150) may be written}$$

$$M^{(n)}(t) = M^{(n)}(0) \left[\frac{\mathfrak{a}\_s(t)}{\mathfrak{a}\_s(0)}\right]^{d^{n\prime}/2b} \tag{10.159}$$

where *t* =t In *Q2* /Q6, a,(t) is the effective strong gauge-coupling constant squared (I 0.82) and band *d<n>* are the leading coefficients of the renormalization group {3-function (I 0. 77) and anomalous dimension (I 0.157), respectively. Thus QCD predicts 'almost scaling' with a logarithmical violation. This pattern is illustrated in Fig. 10.11.

#### **The parton picture of scaling violations**

How are we to understand this QCD pattern of scaling violation in the parton model or, for that matter, patterns of scaling violation in general? As we have discussed in Chapter 7, strict Bjorken scaling can be obtained if we are allowed to make an impulse approximation and the reaction may be viewed as an incoherent sum of scatterings off the (free) constituent of the target. Thus this picture is applicable to any weakly bound system, and can be viewed as a process in which the virtual photon probes the structure of the target. From the uncertainty principle a virtual photon with mass (Q2 ) 1i 2 will resolve structure on the length scale of (Q2 )- 112 • When we increase the virtual photon mass, then the structure at shorter distances will be revealed. Typically this picture of matter as 'a box within a box, etc.' (i.e. discrete levels) will lead to scaling violations at particular intervals of *Q2* (when the next level is reached). Thus we would generally observe scaling and rescaling as layers of matter are unravelled. At these scaling violation junctures, the structure function will change with *Q2* and get redistributed towards the region corresponding to smaller values of the scaling variable *x,* as each constituent will carry a smaller fraction of the target momentum.

In such a parton picture, the QCD result can be interpreted in terms of hadrons having a continuous set of constituent layers. At any particular *Q2* the shape of the nucleon structure function can be understood schematically as in Figs. 7.4--7.6. As we increase *Q2* and penetrate deeper into the dressed quark we will find more virtual quarks and gluons and the valence quarks will have to share the original nucleon momentum more and more with the gluons and sea quarks. Ultimately the structure function approaches a delta function at *x* = 0

$$F\_2(\mathbf{x}, \mathcal{Q}^2) \underset{\mathcal{Q}^1 \hookrightarrow \mathcal{Q}}{\approx} \delta(\mathbf{x}). \tag{10.160}$$

#### **The Altarelli-Parisi equation**

Thus, according to the above picture, the QCD scaling violation comes from the fact that the effective strong coupling *a,(t)* does not vanish fast enough as *Q2* --+ oo. Even though the theory is asymptotically free there are still some residual interactions at short distances. All this can be cast in well-defined quantitative expressions as first suggested by Altarelli and Parisi (1977). For a simple presentation see (Close 1979).

At *t* = t In *Q2* /Q6, let the probability of a quark carrying the fraction *x* of nucleon momentum by *q(x* ). As we increase the virtual photon mass *t* to *t* + *bt* an additional probability *bq(x)* may be revealed corresponding to the

possibility of another quark with momentum fraction *y* > *x* radiating away a gluon and reducing its momentum from *y* to *x* (Fig. I0.12(a)). Thus we can think of this quark as being contained in the one originally observed at a lower value of *t.* This suggests that the quark distribution is actually *!* dependent and that we can define a quantity Pqq(z) corresponding to the variation of the probability distribution (per unit *t)* of finding a quark in a quark with *z* = *x/y* fraction of its momentum (y). This variation is clearly an order-g2 effect; consequently we have the following evolution equation

$$\frac{\mathrm{d}q(\mathbf{x},t)}{\mathrm{d}\mathbf{r}} = \bar{g}^2(t) \int\_{\mathbf{x}}^{1} \mathrm{d}y \int\_{0}^{1} \mathrm{d}z \,\delta(\mathbf{x} - \mathbf{y}z) P\_{\mathrm{qq}}(z) q(\mathbf{y},t)$$

$$= \bar{g}^2(t) \int\_{\mathbf{x}}^{1} \frac{\mathrm{d}y}{\mathrm{y}} P\_{\mathrm{qq}}(\mathbf{x}|\mathbf{y}) q(\mathbf{y},t). \tag{10.161}$$

This convolution integral can be disentangled by a Mellin transform as the moment of the product is the product of the moments of the functions. Define the moments of the distribution (i.e. structure) function

$$M^{(\mathfrak{n})}(t) = \int\_0^1 \mathrm{d}x \lambda^\mathfrak{n} q(\mathfrak{x}, t) . \tag{10.162}$$

Eqn (10.161) then reads

$$\begin{split} \frac{\mathrm{d}M^{(n)}(t)}{\mathrm{d}t} &= \bar{g}^2(t) \int\_0^1 \mathrm{d}x \, x^n \int\_x^1 \frac{\mathrm{d}y}{y} \, P\_{\mathrm{qq}}(\chi/\nu) q(\mathcal{y}, t) \\ &= \bar{g}^2(t) \int\_0^1 \mathrm{d}\chi \mathcal{y}^n q(\mathcal{y}, t) \int\_0^1 \mathrm{d}z \, z^n P\_{\mathrm{qq}}(z) \\ &= \bar{g}^2(t) M^{(n)}(t) D^{(n)} \end{split} \tag{10.163}$$

where

$$D^{\langle \eta \rangle} = \int\_0^1 \mathrm{d}z z^{\eta} P\_{\mathrm{qq}}(z) . \tag{10.164}$$

Thus the parton picture of scaling violations leads to this differential equation of the moment of the structure function; its evolution is seen to be controlled by the t-dependent quark-gluon coupling *oc,(t).* The solution to (10.163) can be easily obtained by integration

$$
\frac{\text{d}M^{(n)}(t)}{M^{(n)}(t)} = \frac{D^{(n)}}{2b} \frac{\text{d}t}{t}
$$

$$
\frac{M''(t)}{M''(t\_0)} = \left[\frac{\alpha\_i(t)}{\alpha\_i(t\_0)}\right]^{-D''/2b}.\tag{10.165}
$$

This is precisely the QCD result for non-singlet moments ( 10.159) if we make the identification

$$D^{(\mathfrak{n})} = -d^{(\mathfrak{n})}.\tag{10.166}$$

Thus the moments of the Altarelli-Parisi function Pqq are simply the nonsinglet anomalous dimension coefficients

When we extend the above analysis to the flavour-singlet combination of structure functions, we have the additional feature that the (singlet) gluon distribution function *G(x, t)* also contributes (see Fig. 10.12(b)). Corresponding to (10.161), we now have

$$\frac{\mathrm{d}q(\mathbf{x},t)}{\mathrm{d}t} = \bar{g}^2(t) \int\_{\mathbf{x}}^{1} \frac{\mathrm{d}y}{y} \left[ \, \! \! \! \! \! / \! \! \! / \! \! \! / \! \! \! / \! \! \! / \! \! \! / \! \! \! / \! \! \! \! / \! \! \! \! / \! \! \! \! / \! \! \! \! \! \! \! / \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

$$\frac{\mathrm{d}G(\mathbf{x},t)}{\mathrm{d}t} = \bar{g}^2(t) \int\_{\mathbf{x}}^{\mathbf{t}} \frac{\mathrm{d}\mathbf{y}}{\mathrm{y}} \left[ \, \!\!P\_{\mathrm{qG}}(\mathbf{x}|\mathbf{y}) q(\mathbf{y},t) \right] $$

$$+ \, \!\!\!P\_{\mathrm{GG}}(\mathbf{x}|\mathbf{y}) G(\mathbf{y},t) ]. \tag{10.168}$$

(Appropriate summation over the quark flavour indices to obtain the singlet combination is assumed.) Eqn (10.167) shows that the singlet quark distribution varies with *t,* not only because of gluon bremsstrahlung, but also because gluons can convert into quark and antiquark pairs in a flavourindependent way, etc. The coupled evolution equations then correspond to operator-mixing in the more formal approach of using the operator product expansion and the renormalization group equation. The moments Pqq• PqG• PGq• and PGG can be identified with (negative) anomalous dimension coefficients and

#### **Perturbation theory and the parton model**

Now we see that QCD results such as (10.159) have a simple interpretation in the parton model with quarks and gluons. This suggests that they can be obtained directly in perturbation theory without invoking the formal

or

apparatus of the operator-product expansion and the renormalization group equation. Before performing this perturbative calculation we shall give an overview of the problem of obtaining the parton-model result in perturbation theory.

As we mentioned in the last section, although QCD is asymptotically free with a decreasing effective coupling constant *i/(q2 )* "" (In *q2 )-* 1, it is still difficult to apply perturbation theory directly to many high-energy processes. There are at least two reasons for this situation.

(l) Perturbation theory deals with quarks and gluons, which are not physical asymptotic states. Thus, by and large, straightforward applications of asymptotic freedom are restricted to calculating gross features that are independent of the detailed final-state properties. Quantities such as high-energy e + e - hadronic total cross-sections and deep inelastic *lN*  inclusive cross-sections are determined by the initial response of the system at short distance-time and are insensitive to the complicated nonperturbative process which turns quarks and gluons into hadrons with unit probability.

(2) Physical quantities usually depend on, in addition to the effective coupling *g(q2 ),* some mass parameter *m.* Thus if one attempts to expand physical quantities in powers of *g(q2 )* in the large-q2 limit, one generally also encounters terms of the form [q2(q2 ) In *q2/m2 ]"* which spoil the expansion. Physically, sensitivity *tom* indicates that the large-distance properties of the theory are involved. Thus the use of perturbation theory in QCD requires that either of the following conditions be satisfied.

(A) In *q2/m2* terms do not occur. This is the case for u101(e+e- --+ hadrons), and u(e+e- --+jets) where groups of particles within some narrow cones are summed (see §7.2). They are finite in them --+ 0 limit and therefore are free of In *q2 /m2* factors.

(B) In *q2/m2* terms can be summed up and the sensitivity to *m* can be somehow factored out. This 'factorization of mass singularities' has already been briefly mentioned in the previous section and it occurs in the case of deep inelastic lepton scatterings. To see the usefulness of the factorization property more explicitly, consider a dimensionless observable *'¥(g(µ 2 ),* Q2 / *µ2 , m2* / *µ2 ).* Since the value of\P cannot depend on the scale *µ* chosen to define the coupling constant *g,* we have

$$
\mu \frac{\mathbf{d} \Psi}{\mathbf{d} \mu} = \left( \mu \frac{\partial}{\partial \mu} + \mu \frac{\partial g}{\partial \mu} \frac{\partial}{\partial g} + \mu \frac{\partial m}{\partial \mu} \frac{\partial}{\partial m} \right) \Psi = 0. \tag{10.169}
$$

This is the renormalization group equation for the observable. If we can establish that *'¥* factorizes as *Q2* --+ oo

$$\Psi\_{\mathcal{Q}^1 \to \infty} \hookrightarrow C(g(\mu^2), \mathcal{Q}^2/\mu^2) \\ D(g(\mu^2), m^2/\mu^2) + O(m^2/\mathcal{Q}^2), \tag{10.170}$$

then (10.169) can be used to determine the Q2-dependence

$$
\mu \frac{\mathbf{d} \Psi}{\mathbf{d} \mu} = \left( \mu \frac{\mathbf{d} \mathbf{C}}{\mathbf{d} \mu} \right) D + \left( \mu \frac{\mathbf{d} D}{\mathbf{d} \mu} \right) \mathbf{C} = \mathbf{0} \tag{10.171}
$$

or

$$\frac{\mu}{C} \frac{\text{d}C}{\text{d}\mu} = -\frac{\mu}{D} \frac{\text{d}D}{\text{d}\mu} \equiv \gamma(g) \tag{10.172}$$

where the separation constant can depend only on *g* and *C* satisfies the renormalization group equation in the standard form

$$
\left[\mu \frac{\partial}{\partial \mu} + \beta(g) \frac{\partial}{\partial g} - \gamma(g)\right] C(g(\mu^2), \mathcal{Q}^2/\mu^2) = 0. \tag{10.173}
$$

From this and *y(g)* = *g2d* we obtain the familiar solution (e.g. see eqn (10.159))

$$C(g(\mu), \mathcal{Q}^2 / \mu^2) = C(0, 1) \left[ \frac{g^2(\mathcal{Q}^2)}{g^2(\mu^2)} \right]^{d/2b}$$

$$= C(0, 1) [1 + b g^2(\mu^2) \ln(\mathcal{Q}^2 / \mu^2)]^{-d/2b} \quad \text{(10.174)}$$

where we have used the explicit form of *g2 (Q2 )* as given in eqn (10.80). We note that the function *C(g(µ), Q2/µ 2)* depends on *g(µ)* and *Q2/µ 2* only through the combination *g2(µ)* 1n(Q2/µ2). That each factor of *g2(µ 2)* is accompanied by ln(Q2/µ2) clearly indicates a correspondence to a summation in perturbation theory of the leading logarithmical terms, which appear as g2n(µ 2)(ln(Q2/µ2)f.

We saw in the previous section that factorization of mass singularities is accomplished using the operator-product expansion; the c-number singular functions then satisfy the renormalization group equations. The present discussion then suggests the following correspondence between the formal field-theoretical apparatus and the simpler perturbative procedures

*Formal apparatus Perturbative procedures*  operator product expansion +-+ factorization of mass singular diagrams renormalization group equation+-+ summation of leading logarithms

It should be pointed out that, while this translation table may be helpful in clarifying the meaning of our calculational procedures, in most cases no such simple separation of steps is possible. In fact factorization is usually not achieved until summation of logarithms have been performed.

Before further discussion of perturbation theory and the parton-model results we will first carry out an explicit calculation illustrating the points made above.

#### **Perturbative calculation of deep inelastic scattering**

We are interested in the forward Compton scattering amplitude of a virtual photon (with momentum *q,,)* by a quark (momentum *p,,).* Corresponding to the operator-product expansion, here we need to identify and factorize diagrams containing powers of In *p2 •* Thus we are looking for diagrams that are divergent as the mass *p 2*= *m2* \_\_,. 0. There are two types of divergences, which are often called *infrared divergences* and *mass singularities.*

#### 10.4 *The parton model and perturbative QCD* 317

The first type of divergence is brought about in the phase-space region when the momentum of the (real or virtual) massless particles vanishes. The Bloch-Nordsieck theorem (1937) assures us that such infrared divergences cancel in the inclusive cross-section. But all cross-sections are in fact inclusive because of the finiteness of energy resolution in all experiments. Essentially this situation still holds for the deep inelastic scattering.

We will concentrate on the divergence resulting from mass singularities. These mass singularities occur in theories with coupled massless particles, and are due simply to the kinematical fact that two massless particles (with momenta *p* and *p',* say) which are moving parallel to each other have combined invariant mass equal to zero

$$k^2 = (p+p')^2 = 2EE'(1-\cos\theta) \to 0 \quad \text{as} \quad \theta \to 0 \tag{10.175}$$

even though neither *Pµ* nor are soft. This is sometimes called *collinear divergence.* There is also a theorem, by Kinoshita (1962) and by Lee and Nauenberg (1964), which can be roughly stated as follows. For inclusively enough cross-sections the mass singularities also cancel. The physical basis of this theorem is very similar to that of Bloch and Nordsieck; i.e. in physical measurements the angular resolution is not perfect and we sum over all states within the finite angular resolution. Thus in our calculation we will not be concerned with mass singularities coming from undetected final-state particles moving parallel to each other, since they cancel. In deep inelastic scattering, however, there are still mass singularities left over. They arise from regions of phase space where the internal momentum of a massless particle is parallel to that of an external massless particle to which it is coupled. Consider the one-loop diagrams in Fig. 10.13. Let *k* = *p* + *p'* be the

FIG. 10.13. Mass singularities are present in (a) but not in (b). *IJ* is the angle between p and p'. The heavy-lined propagator diverges as IJ- 2 in the limit *IJ-+* 0.

loop momentum. Hence we have a factor of *e- 2*from each of the 1/k2 propagators, and *() d()* from the d(cos 0) factor in d4 k, and finally each massless particle vertex contributes a power *of().* Thus Fig. 10.13(a) has mass singularity *d()/(),..,* lnp2 and Fig. 10.13(b) is finite. We shall therefore concentrate on Fig. 10.13(a) and its generalizations to higher orders.

**One-loop diagrams.** It is convenient to use the axial gauge, eqns (9.41) and (9.108),

$$
\eta\_{\mu}A^{\mu} = 0\tag{10.176}
$$

where *Y/µ* is some four-vector. The gluon propagator in this gauge is given by

$$D\_{\mu\nu}(k) = \frac{d\_{\mu\nu}(k)}{k^2 + \text{i.c.}}$$

with

$$d\_{\mu\mathbf{v}}(k) = g\_{\mu\mathbf{v}} - (k\_{\mu}\eta\_{\mathbf{v}} + k\_{\nu}\eta\_{\mu})(k\cdot\eta)^{-1} + \eta^{2}k\_{\mu}k\_{\nu}(k\cdot\eta)^{-2}.\tag{10.177}$$

There is no Faddeev-Popov ghost in this gauge, and the calculation is greatly simplified. To take the absorptive part of the amplitude we replace the propagator *(k2* + ii::)- 1 by n<5(k2 ). Thus the diagram in Fig. 10.14(a)

FIG. 10.14. The absorptive part of the forward Compton scattering amplitudes.

contributes to the hadronic tensor of the inelastic lepton scattering crosssection (eqn (7 .10)) as

$$W\_{\mu\nu}(p,q) = \frac{g^2 s\_2(\mathsf{V})}{2M} \int \frac{\mathsf{d}^4 k}{(2\pi)^3} \frac{\delta((p-k)^2)}{(k^2)^2} \frac{\delta((q+k)^2)}{(k^2)^2} d^{\rho\sigma}(p-k) T\_{\rho\mu\nu\sigma}$$

where

$$T\_{\rho\mu\mathbf{v}\sigma} = \frac{1}{2} \operatorname{tr} \mathbb{E} \{ \mathfrak{p} \gamma\_{\rho} \not\!k \gamma\_{\mu} (\not\!k + \not\!q) \gamma\_{\text{v}} \not\!k \gamma\_{\sigma} \} \tag{10.178}$$

and si(V) is the usual colour factor of eqn (10.88) and equal 4/3 for SU(3)colour· It is convenient to use the Sudakov variables (1956)

$$k\_{\mu} = \xi p\_{\mu} + \beta q\_{\mu}^{\prime} + k\_{\perp \mu} \tag{10.179}$$

with

$$q'=q+\chi p \quad \text{and} \quad k\_{\perp} \cdot p = k\_{\perp} \cdot q'=0. \tag{10.180}$$

We assume that *q' 2 -::::, p2 -::::,* 0 and that *p* · *q'-::::, p* · *q,* together with *-q2 ,* are large.

$$\mathbf{d}^4 k = 2\pi (p \cdot q) \,\mathrm{d}\beta \,\mathrm{d}\xi \,\mathrm{d}k\_\perp^2 \tag{10.181}$$

As we shall see, the final ln( - *q2 /p2 )* term comes from regions of phase space corresponding to *{3-::::,* 0 and *e-::::, x.* Thus the variable *x* introduced in (10.180) has the usual parton-model interpretation as the fraction of the longitudinal

#### 10.4 *The parton model and perturbative QCD* 319

target momentum. (This also *a posteriori* justifies taking *q'2* 0.) In terms of these variables we have

$$\int \mathrm{d}^4 k \,\,\delta((p-k)^2) \,\,\delta((q+k)^2) = 2\pi p \cdot q \int \mathrm{d}\beta \,\,\mathrm{d}\xi \,\,\mathrm{d}k^2\_\perp$$

$$\times \frac{1}{2p \cdot q(1-\xi)} \,\delta\left(\beta - \frac{k\_\perp^2}{2p \cdot q(1-\xi)}\right) \frac{1}{2p \cdot q} \,\delta\left(\xi - \chi + \frac{1-\chi}{1-\xi} \frac{k\_\perp^2}{2p \cdot q}\right) \,\,\mathrm{d}\chi\,\,\,\mathrm{d}k^2\_\perp \tag{10.182}$$

Squaring (10.179) we obtain

$$k^2 = k\_\perp^2 + 2\beta\xi(p \cdot q) = k\_\perp^2 + \frac{\xi}{1 - \xi}k\_\perp^2 = k\_\perp^2/(1 - \xi). \tag{10.183}$$

$$\begin{aligned} \text{Therefore (10.182) can be rewritten} \\ \int \mathbf{d}^4 k \,\delta((p-k)^2) \,\delta((q+k)^2) &= \frac{\pi}{2p \cdot q} \int \mathbf{d}\boldsymbol{\beta} \,\mathbf{d}\boldsymbol{\xi} \,\mathrm{d}k^2 \,\delta\left(\boldsymbol{\beta} - \frac{k^2}{2p \cdot q}\right) \\ &\quad \times \delta\left(\boldsymbol{\xi} - \boldsymbol{\varkappa} + (1-\boldsymbol{\varkappa})\frac{k^2}{2p \cdot q}\right) . \end{aligned} (10.184)$$

Contracting the gluon polarization tensor *dP"(p* - *k)* with the trace term *Tpµva* (taking *17* = *q')* and doing the Dirac· algebra yields a leading term proportional to the tree graph (eqn (7.53))

$$d^{\rho\sigma}(p-k)T\_{\rho\mu\nu\sigma} = (4k^2)\frac{1+\not\xi^2}{1-\not\xi}\frac{1}{2}\operatorname{tr}[\chi p\gamma\_\mu(q+\chi p)\gamma\_\nu].\tag{10.185}$$

This results from the relative contributions of the first and second terms of (10.177) ((1 - *e)k2* and *2ek2* /(1 - *e),* respectively) and from dropping the contribution of the third term which is of the order of *(k2 ) 2*and thus does not contribute to the final *ln(Q2/p2 ).* Collecting terms (10.184) and (10.185),

$$W\_{\mu\nu}(p,q) = \frac{g^2}{6\pi^2} \left[ \frac{\text{d}k^2}{k^2} \left( \frac{1+\chi^2}{1-\chi} \right) \left[ \frac{\chi}{\text{v}} \frac{p\_\mu p\_\nu}{p^2} + \dots \right] \right] \tag{10.186}$$

Comparing (10.186) with the simple parton-model result of eqn (7.55), we see that the one-loop diagram makes the following contribution to the parton distribution function/(x) (which is one originally);

$$\mathrm{d}f(\mathbf{x}) = \frac{1}{2}g^2 P(\mathbf{x}) \frac{\mathrm{d}k^2}{k^2} \tag{10.187}$$

with

$$P(\mathbf{x}) = \left(\frac{1+\mathbf{x}^2}{1-\mathbf{x}}\right) \frac{1}{3\pi^2} \cdot \tag{10.188}$$

We choose to write it in a form that will be more useful for interpretation and for higher-order calculations

$$\mathrm{d}f = \left[\frac{1}{2}g^2 \frac{\mathrm{d}k^2}{k^2}\right] \left\{ \int \frac{\mathrm{d}\xi}{\xi} \,\delta\left(1 - \frac{\chi}{\xi}\right) P(\chi) \right\}.\tag{10.189}$$

Thus the diagram in Fig. 10.14(a) with its gluon bremsstrahlung introduces a momentum dependence in the quark distribution function having an *Altarelli-Parisifunction* of *P(x).* Ifwe have been careful in keeping track of *p2* terms, we will find the limits of integration for the *dk2* integral to be

$$\int\_{p^2}^{Q^2} \frac{d\mathbf{k}^2}{k^2} = \ln(Q^2/m^2) \quad \text{for} \quad p^2 \approx m^2. \tag{10.190}$$

Thus the one-loop diagram (Fig. 10.14(a)) has a mass-singular form

$$f(\mathbf{x}, \mathcal{Q}^2) \sim \left[ g^2 \ln(\mathcal{Q}^2 / m^2) \right] \left( \frac{1 + \chi^2}{1 - \chi} \right) \tag{10.191}$$

**Higher-order diagrams.** Features similar to those in one-loop diagrams also appear in higher-order contributions. The general feature (Gribov and Lipatov 1972) is that the dominant contributions in the axial gauge come from the ladder diagrams (Fig. 10.14(b)) from the region of phase space corresponding to

$$p^2 \ll k\_1^2 \ll k\_2^2 \dots \ll k\_N^2 \ll \mathcal{Q}^2. \tag{10.192}$$

It is understood that in Fig. 10.14(b) the vertex and self-energy insertions are included. One of the effects of these insertions is that at each vertex we should use the momentum-dependent effective couplings *g(kt).*

For each rung of the ladder we have a box diagram similar to the one we have just evaluated with the result written in the form of (10.189). Now for all the transverse momentum integrals *dkt,* corresponding to the square brackets of (10.189), we have

$$I\_{\Gamma}^{(N)}(\mathcal{Q}^2/m^2) = \int\_{\mathfrak{m}^\*}^{\mathcal{Q}^\circ} \frac{\mathrm{d}k\_1^2}{2k\_1^2} \, g^2(k\_1^2) \int\_{\mathfrak{m}^\*}^{k\_1^2} \frac{\mathrm{d}k\_2^2}{2k\_2^2} \, g^2(k\_2^2) \dots$$

$$\times \int \frac{\mathrm{d}k\_N^2}{2k\_N^2} \, g^2(k\_N^2)$$

$$\approx \frac{1}{N!} \left[ \int\_0^{\mathcal{Q}^\circ} \frac{\mathrm{d}k^2}{2k^2} \, g^2(k^2) \right]^N \equiv \frac{1}{N!} \, \rho^N. \tag{10.193}$$

Using the asymptotic form of *g2(k2 ),* we have

$$\rho = \int\_{m^2}^{Q^2} \frac{\mathrm{d}\, \ln(k^2)}{2b \, \ln k^2/\mu^2} = \frac{1}{2b} \left[ \ln \ln(Q^2/\mu^2) - \ln \ln(m^2/\mu^2) \right]. \tag{10.194}$$

#### 10.4 *The parton model and perturbative QCD* 321

The integral over the longitudinal components of momentum, corresponding to the curly brackets of (I 0.189), is of the form

$$I\_{\mathcal{L}}^{(N)} = \int\_{\xi\_2}^{1} \frac{\mathbf{d}\xi\_1}{\xi\_1} \int\_{\xi\_3}^{1} \frac{\mathbf{d}\xi\_2}{\xi\_2} \dots \int\_{0}^{1} \frac{\mathbf{d}\xi\_N}{\xi\_N} \,\delta\left(1 - \frac{\chi}{\xi\_N}\right)$$

$$\times \, P(\xi\_1) P(\xi\_2/\xi\_1) \dots P(\xi\_N/\xi\_{N-1}) . \tag{10.195}$$

The fraction of momentum *p* carried by quark i is and that by quark i + I is 1 . The argument of the Altarelli-Parisi function is the ratio of these two fractions This equation is in a multiple-convolution form and we can take moments to simplify the result

$$\int\_0^1 \mathrm{d}x \chi^{n-1} I\_L^{(N)}(x) = \left[ \int\_0^1 \mathrm{d}z z^n P(z) \right]^N \equiv \left[ d^{(n)} \right]^N. \tag{10.196}$$

Then the moment of the structure function coming from the N-rung ladder diagram is

(10.197)

Summing over *N* we get

$$\begin{split} M^{(\mathfrak{n})}(\mathcal{Q}^2) &= \sum\_{N} M\_N^{(\mathfrak{n})}(\mathcal{Q}^2) = \exp[\rho d^{(\mathfrak{n})}] \\ &= \exp\left\{ \frac{d^{(\mathfrak{n})}}{2b} \left[ \ln \ln(\mathcal{Q}^2/\mu^2) - \ln \ln(m^2/\mu^2) \right] \right\} \\ &= \left[ \ln(\mathcal{Q}^2/\mu^2) \right]^{d^{(\mathfrak{n})/2b}} \left[ \ln(m^2/\mu^2) \right]^{-d^{(\mathfrak{n})/2b}}. \end{split} \tag{10.198}$$

This shows the factorization property that M<nl(Q2) decomposes into a function of *Q2/µ 2* multiplied by a function of *m2/µ 2.* Note that this factorization takes place only after diagram summation; individual terms do not factorize.

Finally let us check to see whether *d<n>* agrees with the result obtained previously. From (10.196) and (10.188),

$$d^{(\mathfrak{n})} = \int\_0^1 \mathrm{d}z z^{\mathfrak{n}} P(z) = \frac{\mathrm{l}}{3\pi^2} \int\_0^1 \mathrm{d}z \frac{z^{\mathfrak{n}} (\mathrm{l} + z^2)}{\mathrm{l} - z} \tag{10.199}$$

which actually diverges as *z* -+ 1. More carefully analysis shows that when we did the calculations for Fig. 10.1 S(a) we should also have included the vertex correction diagram (Fig. 10. l S(b )) for the Bloch-Nordsieck infrared divergence cancellation to take place. This corresponds to the replacement

$$\int\_0^1 \frac{f(z)}{1-z} \,\mathrm{d}z \to \int\_0^1 \frac{f(z) - f(1)}{1-z} \,\mathrm{d}z \,. \tag{10.200}$$

Thus *d<n>* can be calculated

$$d^{(n)} = \frac{1}{3\pi^2} \int\_0^1 \mathrm{d}z \, \frac{z^n(1+z^2)-2}{1-z}$$

$$= \frac{1}{3\pi^2} \int\_0^1 \frac{\mathrm{d}\chi}{\chi} \left[ (1-\chi)^n (2-2\chi+\chi^2) - 2 \right]$$

$$= \frac{1}{6\pi^2} \left[ 1 - \frac{2}{n(n+1)} + 4 \sum\_{j=2}^n \frac{1}{j} \right]. \tag{10.201}$$

Thus by summing leading logarithms we obtain exactly the same results as obtained by using the more formal procedures of renormalization group equations.

Once we understand the ideas and calculations for the familiar deep inelastic case, it is conceptually fairly straightforward to generalize the scheme to other hard scattering processes. The real advantage is that in most such hard scattering reactions a direct application of the operator-product expansion is not feasible. Thus we can use the perturbative QCD approach to calculate the Drell-Yan process of inclusive production of lepton pairs with large invariant mass in hadron collision (see §7.3). Here factorization is achieved and one finds the same parton distribution functions there with momentum dependences as those in the deep inelastic case. Another outstanding example of the perturbative QCD calculation of physical quantities that are not 'infrared sensitive' is the prediction of the final-state angular distribution (jets) (Sternman and Weinberg 1977) and energy flows in high-energy e+e- annihilations into hadrons (Brown and Ellis 1981).

# **10.5 Lattice gauge theory and colour confinement**

Quantum chromodynamics has the remarkable property of being asymptotically free. The vanishing for short distances of the effective coupling gives the correct description of Bjorken scaling (and its violation by logarithms). The same statement also suggests that the effective coupling increases for long distances and this points towards a possible resolution of the central paradox in the phenomenological quark picture, i.e. quarks must behave like free particles for short distances while they must also be completely confined on long time and length scales. Of course strong coupling itself is not enough to explain quark confinement. To do so one must show that in QCD the particle spectrum is realized in terms of bound states of quarks and gluons, etc. In the following we shall first give a brief qualitative discussion of how quark confinement might come about in the asymptotically free gauge theory. This should motivate the lattice method of defining such a theory and we will then demonstrate that the confinement property can be obtained in a strong coupling approximation of the lattice gauge theory.

#### **Qualitative picture of confinement**

Qualitative ideas about the nature of confinement that have some correct physical consequences tend to picture quarks as being bound by 'strings' (Nambu 1974) or tubes of colour flux. It has been suggested that the QCD vacuum is a condensate of gluons and as well as light quark-antiquark pairs. (For a review see, for example, Mandelstam 1979.) This is somewhat analogous to the ground state of a superconductor. There the condensate of paired electrons gives rise to the Meissner effect of magnetic flux exclusion from the condensate unless the energy balance favours a local breakdown to the normal phase. And one can imagine placing a pair of magnetic monopole and antimonopole into this superconducting medium; the magnetic flux will be confined to a string-like configuration joining the pair of monopoles. Analogous to this situation the energetically favoured configuration of, say, a quark and an antiquark, has a strongly localized normal region connecting the pair in which the colour (electric) flux lines are restricted (see Fig. 10.16). By translational invariance the energy density of

FIG. 10.16. Lines of force between a quark and an antiquark. When the quarks move away from each other, the breaking of the string is accompanied by further quark-antiquark pair production.

the gluon field along the flux tube is a constant. Hence the total field energy is linearly proportional to distance. This means the quarks are confined by a linearly rising long-distance potential

$$E(r) \to Kr \tag{10.202}$$

where *r* is the separation between the sources and *K* is a constant referred to as the *string tension.* Thus after the production of a quark-antiquark pair (say, in e+e- annihilation) it becomes energetically favourable with increasing separation for the string to break and produce another quark-antiquark pair, and so on. This proceeds until the original string is broken down into several strings of length typical of the hadronic size of ,.., 1 fm (,.., *5* Ge v- 1) corresponding to the original quark pair being converted into a whole set of hadrons having a typical energy-mass of 1 GeV. Thus we can crudely estimate the string tension to be

$$K \approx \frac{1}{5} \,\text{GeV}^2.\tag{10.203}$$

The string model for hadrons has another consequence in agreement with observation: hadrons lie on *Regge trajectories* (Regge 1959) with slopes ;:::: 1 Ge v- 2 • A decade of intense study of hadron dynamics through their pattern of particle exchange in which unitarity and the analyticity property of the scattering amplitudes play a crucial role has led to many insights into particle interactions. One of the important discoveries in this S-matrix approach (Chew 1962) was that hadrons of a given internal symmetry quantum number but different spins obey a simple spin(J) - mass(MJ) relation (Chew and Frautschi 1961); we say they lie on Regge trajectory

$$J = \mathfrak{a}\_0 + \mathfrak{a}' M\_J^2,$$

with

$$\mathbf{a}' \simeq \mathbf{l} \,\,\mathbf{G} \mathbf{e} \mathbf{V}^{-2}.\tag{10.204}$$

Now imagine two massless (and, for simplicity, spinless) quarks, connected by a string of length *d,* rotating with the speed of light (Gasiorowicz and Rosner 1981). Thus each point, at a distance *r* from the centre, has the local velocity *v/c* = *2r/d.* The total mass is then

$$\mathcal{M} = 2 \int\_0^{d/2} \frac{K \,\mathrm{d}r}{(1 - v^2/c^2)^{1/2}} = \frac{\pi K \,\mathrm{d}}{2} \tag{10.205}$$

and the total angular momentum is

$$J = 2 \int\_0^{d/2} \frac{Krv \,\mathrm{d}r}{(1 - v^2/c^2)^{1/2}} = \frac{\pi K \,\mathrm{d}^2}{8} \cdot \tag{10.206}$$

Thus the string tension *K* of (10.202) can be expressed directly in terms of the Regge slope oc' of (10.204)

$$K = \frac{1}{2\pi\alpha'}.\tag{10.207}$$

The experimental value oc' in (10.204) leads to a *K;::::* 0.2 GeV2 in qualitative agreement with the rough estimate of (10.203).

#### **Field theories on the lattice**

In order to study these long-distance properties of QCD we need to regularize the theory that is independent of the usual Feynman diagram expansion, which is appropriate for weak couplings. For this purpose Wilson (1974, 1975) introduced the lattice gauge theory in which the space-time

continuum is discretized. This provides a natural cut-off scheme as wavelengths shorter than twice the lattice spacing, *a,* have no meaning and this restricts the domain of momenta to a region bounded by *n/a.* Also this formulation of the field theory allows for a close analogy with a statistical mechanics system. So we can call upon all our experience and intuition of statistical mechanics to solve problems in quantum field theory. With a cutoff on high momenta the kinetic energy is bounded; we can then treat it as a perturbation in the strong coupling limit. This corresponds to the method of high-temperature expansion in statistical mechanics. With a finite lattice there are a finite number of variables. It is then possible to study various physically interesting quantities (e.g. energy spectrum, correlation functions, etc.) in the path-integral formalism by computer simulation based on the Monte Carlo method.

As with any cut-off prescription, considerable freedom remains in the lattice formulation. In the limit of vanishing lattice spacing, the physics of a renormalizable field theory should be independent of the details of the regulator; it should, so to speak, lose the memory of the lattice spacing. This means that in this limit the coherence length of the theory should be infinite when compared to the lattice spacing. In the language of statistical mechanics the divergence of the correlation length corresponds to a secondorder, or continuous, *phase transition.* If a model has only a first-order transition, the coherence length never becomes infinite and the desired continuum theory does not exist. Furthermore, analytic results in the strong and weak coupling regimes can easily be established. We need to ascertain whether these properties are connected continuously in the theory, i.e. whether there are phase transitions at intermediate couplings. Thus it is important to study the phase structure of lattice field theory. At present most of our knowledge of the phases of QCD is obtained in various numerical studies of the theory.

There are two popular methods for introducing the lattice in field theory. In the Euclidean lattice formulation both space and time are discretized (Wilson 1974). Here after a Wick rotation (to Euclidean space) the quantization is performed via the path-integral formalism. This method has the advantage of keeping some vestige of the original Lorentz symmetry and this allows for a particularly elegant formulation. Also, as we shall see in the case of lattice gauge theory, there is no need to introduce the gaugefixing terms. The other method is the Hamiltonian lattice formulation in which only the spatial dimensions are discretized in a Minkowski space-time (Kogut and Susskind 1975a). The theory can be canonically quantized with the usual Hamiltonian formalism. This has the advantage that some physical quantities (especially the mass spectrum) are more directly accessible to computation. Our discussion will follow the Euclidean formulation with a space-time lattice spacing *a.* The lattice site will be labelled by a four-vector *n.* The four-dimensional integration will then be replaced by a sum

$$\int \mathbf{d}^4 \mathbf{x} \to a^4 \sum\_{\mathbf{n}} \cdot \tag{10.208}$$

**Scalar fields.** We first study the simplest case of scalar field cfa(x). The continuum field theory in Euclidean space has the action

$$S(\phi) = \int \mathrm{d}^4 x [\![\partial\_\mu \phi] \!]^2 + V(\phi) \!] \, \tag{10.209}$$

where

$$V(\phi) = \frac{1}{2} \, m^2 \phi^2 + \frac{\lambda}{4} \, \phi^4. \tag{10.210}$$

The scalar field *cfa* exists at every lattice site *n,*

$$
\phi(\mathbf{x}) \to \phi\_{\mathbf{n}}.\tag{10.211}
$$

The derivative is replaced by

$$
\partial\_{\mu}\phi(\mathbf{x}) \to \frac{1}{a} (\phi\_{n+\hat{\mu}} - \phi\_n) \tag{10.212}
$$

where *µ* is a four-vector of length *a* in the direction *ofµ.* The lattice action is then

$$S(\phi) = \sum\_{\mathfrak{n}} \left\{ \frac{a^2}{2} \sum\_{\mu=1}^4 (\phi\_{\mathfrak{n}+\hat{\mu}} - \phi\_{\mathfrak{n}})^2 + a^4 \left( \frac{m^2}{2} \phi\_{\mathfrak{n}}^2 + \frac{\lambda}{4} \phi\_{\mathfrak{n}}^4 \right) \right\}. \tag{10.213}$$

It is instructive to go over to momentum space to see the spectrum of the *(A.=* 0) free-field theory. For this we take the Fourier transform

$$\phi\_{\mathfrak{n}} = \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\mathrm{e}^{i\mathbf{k}\cdot\mathfrak{n}} \phi(k) . \tag{10.214}$$

Since it is meaningless to consider wavelengths less than twice the lattice spacing, the above integral is taken over only one *'Brillouin zone'* of the reciprocal lattice. Thus,

$$-\frac{\pi}{a} \le k\_{\mu} \le \frac{\pi}{a} \quad \text{for each } \mu \tag{10.215}$$

where *kµ* = *k* · *µ.* After substituting eqn (10.214) into eqn (10.213), we have factors such as those coming from the kinetic energy term

$$\text{is those coming from the kinetic energy term}$$

$$a^4 \sum\_{\boldsymbol{\mu}} \left\{ \frac{\mathbf{d}^4 k}{(2\pi)^4} \frac{\mathbf{d}^4 k'}{(2\pi)^4} \mathbf{e}^{(\boldsymbol{k}+\boldsymbol{k}') \cdot \boldsymbol{\eta}} (\mathbf{e}^{\boldsymbol{k}k\_{\mu}} - 1)(\mathbf{e}^{\boldsymbol{k}k'\_{\mu}} - 1) \right\}$$

$$= \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \left( \mathbf{e}^{\boldsymbol{k}k\_{\mu}} - 1 \right) (\mathbf{e}^{-\boldsymbol{k}k\_{\mu}} - 1)$$

$$= 4 \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \sin^2(\boldsymbol{k} k\_{\mu}/2).$$

The free field action can then be written

$$\begin{aligned} \text{Heu action can then be written}\\ S\_0(\phi) = \frac{1}{2} \int \frac{d^4k}{(2\pi)^4} \left[ \sum\_{\mu} \frac{4}{a^2} \sin^2 \left(\frac{ak\_\mu}{2}\right) + m^2 \right] \phi(-k)\phi(k). \end{aligned} \quad (10.216)$$

### 10.5 *Lattice gauge theory and.r:olot.(r corifinement* 327

Thus each mode contributes to the action in the momentum space a quantity

$$S(k) \equiv m^2 + \sum\_{\mu} \frac{4}{a^2} \sin^2 \left(\frac{ak\_\mu}{2}\right)^2$$

rather than the standard *m2* + *k2.* Nevertheless, these two expressions have the same continuum limit as they coincide at the minimum value of *k* = 0 (see Fig. 10. l 7(a)).

FIG. 10.17. The dispersion relation *S(k)* for free (a) scalar and (b) fermion systems. The dashed curves *k 2*are for familiar continuum theories. The solid curves correspond to the latticized systems.

The lattice action (10.213) can be quantized by using the Feynman pathintegral formalism in which the expectation value of a product of fields is given by

$$\langle 0|\phi\_{n\_1}\phi\_{n\_2}\dots\phi\_{n\_l}|0\rangle = \frac{1}{Z} \int \prod\_{\mathfrak{n}} \{ \mathbf{d}\phi\_{\mathfrak{n}} \} (\phi\_{n\_1}\phi\_{n\_2}\dots\phi\_{n\_l}) \, \mathbf{e}^{-S(\phi)} \quad \text{(10.217)}$$

where

$$Z = \int \prod\_{n} \left[ \mathbf{d} \phi\_{n} \right] \mathbf{e}^{-S(\phi)}.\tag{10.218}$$

The meaning of the integrals should be clear as we may recall that the usual functional integrals are actually defined on a discretized space-time lattice and an appropriate continuum limit is taken at the end. If we rescale the fields as

$$
\phi'\_{\mathfrak{n}} = \sqrt{\lambda} \phi\_{\mathfrak{n}},
\tag{10.219}
$$

the lattice action becomes

$$\mathcal{S}(\phi) = \frac{1}{\lambda} \mathcal{S}'(\phi') \tag{10.220}$$

where

$$S'(\phi') = \sum\_{n} \left\{ \frac{a^2}{2} \sum\_{\mu} (\phi'\_{n+\hat{\mu}} - \phi'\_n)^2 + a^4 \left( \frac{m^2}{2} \phi'^2\_n + \frac{1}{4} \phi'^4\_n \right) \right\}, \tag{10.221}$$

i.e. the coupling constant *A.* has become an overall factor in the action. In this

way eqns (10.217) and (10.218) may be written as

$$<\langle \cdots \rangle \ldots \langle \cdots \rangle \ldots \rangle \ldots \quad \text{---} \qquad \text{---}$$

$$\langle 0 | \phi\_{n\_1}^{\prime} \phi\_{n\_2}^{\prime} \ldots \phi\_{n\_l}^{\prime} | 0 \rangle = \frac{1}{Z'} \int \prod\_{\mathfrak{n}} \left[ \mathrm{d} \phi\_{\mathfrak{n}}^{\prime} \right] (\phi\_{n\_1}^{\prime} \phi\_{n\_2}^{\prime} \ldots \phi\_{n\_l}^{\prime})$$

$$\times \exp \left\{ -\frac{1}{2} \mathcal{S}'(\phi^{\prime}) \right\} \tag{10.222}$$

$$Z' = \int \prod\_{n} \left[ \mathbf{d} \phi'\_{n} \right] \exp\left\{ -\frac{1}{\lambda} S'(\phi') \right\}.\tag{10.223}$$

We note that (10.223) has the same structure as the partition function in statistical mechanics if we make the identification

$$\frac{1}{\lambda} \to \beta \equiv \frac{1}{kT} \cdot \tag{10.224}$$

Thus the strong coupling expansion (in powers of *A* - i) will correspond to the high-temperature expansion in statistical systems.

**Fermion fields.** We now consider fermion fields on a lattice. The same procedure will yield a Euclidean lattice action for the free-fermion system

$$S\_0(\psi) = \sum\_{\pi} \left\{ \frac{a^3}{2} \sum\_{\mu=1}^4 \bar{\psi}\_{\pi} \gamma\_{\mu} (\psi\_{n+\hat{\mu}} - \psi\_{n-\hat{\mu}}) + ma^4 \bar{\psi}\_{\pi} \psi\_{\pi} \right\}. \tag{10.225}$$

The *y* matrices are Euclidean, that is

$$\{\boldsymbol{\gamma}\_{\mu}, \boldsymbol{\gamma}\_{\nu}\} = \mathcal{Z} \,\delta\_{\mu\nu}.\tag{10.226}$$

Eqn (10.225) has the momentum space form

$$\mathcal{L}S\_0(\psi) = \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\tilde{\psi}(-k) \left\{ \mathbf{i} \,\sum\_{\mu} \gamma\_{\mu} \frac{\sin a k\_{\mu}}{a} + m \right\} \psi(k) \tag{10.227}$$

which yields a dispersion relation

$$S(k) = \frac{\sin^2 ak\_\mu}{a^2} + m^2 \tag{10.228}$$

shown in Fig. 1O.l7(b ). There are now two equal minima in a Brillouin zone. One is located at *k* = 0 and gives the correct continuum limit. The other mode at *k* = *±n/a* carries an infinite momentum as *a-+* 0 and yet can be excited for finite *a.* The fermion degeneracy, i.e. the doubling of the number of fermionic states, must be suppressed with some appropriate modification of the latticized theory. This is permissible as long as the continuum limit is not affected. Clearly many degeneracy regularization procedures are possible; we will present the one due to Wilson, who proposed the simple addition of a non-local factor of

$$
\Delta \mathcal{E}' = \frac{1}{2a} \bar{\psi}\_n (\psi\_{n+\hat{\mu}} + \psi\_{n-\hat{\mu}} - 2\psi\_n) \tag{10.229}
$$

#### 10.5 *Lattice gauge theory and colour corifinement* 329

to the lattice Lagrange density. Thus the free fermion action takes on the form in Euclidean space

$$\begin{split} S\_0(\psi) &= \sum\_{\boldsymbol{\mu}} \left\{ \frac{\boldsymbol{a}^\nu}{2} \sum\_{\mu} \bar{\psi}\_{\boldsymbol{\mu}} [(1+\gamma\_{\mu})\psi\_{\boldsymbol{n}+\hat{\mu}} \\ &\quad + (1-\gamma\_{\mu})\psi\_{\boldsymbol{n}-\hat{\mu}} - 2\psi\_{\boldsymbol{n}} \right\} + \boldsymbol{m}\boldsymbol{a}^4 \bar{\psi}\_{\boldsymbol{n}} \psi\_{\boldsymbol{n}} \end{split} \tag{10.230}$$

which has the momentum-space representation

$$S\_0(\psi) = \left[\frac{\mathbf{d}^4 k}{(2\pi)^4} \sqrt{} (-k) \left\{ \mathbf{i} \sum\_{\mu} \mathcal{y}\_{\mu} \frac{\sin a k\_{\mu}}{a} \right. \right.$$

$$+ m - \sum\_{\mu} \frac{\cos a k\_{\mu} - 1}{a} \!\!/ \psi(k). \tag{10.231}$$

This increases the unwanted minimum without affecting the small-k behaviour and the *k* = 0 minimum will be only one to survive on the continuum limit.

#### **Local gauge invariance and the QCD action**

As we have illustrated above, considerable freedom exists in lattice formulation. One is free to add to the Lagrangian terms which will not contribute in the continuum limit. Using this freedom, Wilson has presented a particularly elegant lattice formulation for gauge theories. His prescription keeps local gauge invariance as an exact symmetry in a mathematically welldefined system.

Recall our discussion in §8.2 of the geometric interpretation of gauge invariance. When a material particle undergoes a parallel transport along a world line C from *x* to *x',* it can be represented by a 'non-integrable phase factor' of the wavefunction (i.e. for every path we can associate a group element as in eqn (8.65))

$$U(\mathbf{x}',\mathbf{x}) = \exp\left\{\mathrm{i}g \int\_{\mathbf{C}} \mathbf{T} \cdot \mathbf{A}\_{\mu}(\mathbf{y}) \, \mathrm{d}y''\right\}.\tag{10.232}$$

Thus, for a gauge transformation, specified by the gauge function

$$\Phi(\theta\_x) = \exp\{i\mathbf{T} \cdot \mathbf{\hat{\theta}}(x)\}\tag{10.233}$$

we have

$$
\psi(\mathbf{x}) \to \Phi(\theta\_{\mathbf{x}}) \psi(\mathbf{x}) \tag{10.234}
$$

$$
\tilde{\psi}(\mathbf{x}) \to \tilde{\psi}(\mathbf{x}) \Phi^\dagger(\theta\_\mathbf{x})
$$

and

$$U(\mathbf{x}',\mathbf{x}) \to \Phi(\theta\_{\mathbf{x}'})U(\mathbf{x}',\mathbf{x})\Phi^\dagger(\theta\_{\mathbf{x}}).\tag{10.235}$$

(Please note the notation changes for the gauge transformation and parallel transport matrices from the notation used in §8.2.)

The lattice version of gauge transformation in (10.234) and (10.235) may be written as

$$
\Psi\_n \to \Phi\_n \Psi\_n$$

$$
\Psi\_n \to \Psi\_n \Phi\_n^\dagger \tag{10.236}$$

and

$$U(n+\hat{\mu}, n) \to \Phi\_{n+\hat{\mu}} U(n+\hat{\mu}, n) \Phi\_n^\dagger \tag{10.237}$$

where, for the SU(3) gauge symmetry,

$$\Phi\_{\mathfrak{n}} = \exp\left\{ \mathrm{i} \, \frac{\lambda^{i}}{2} \, \theta\_{\mathfrak{n}}^{i} \right\} \tag{10.238}$$

with *A\_i, i* = 1, 2, .. ., 8, being the usual Gell-Mann matrices and

$$U(n+\hat{\mu},n) = \exp\left\{\log\frac{\lambda^l}{2}\mathcal{A}\_{n,\mu}^l\right\}.\tag{10.239}$$

This is the lattice version of the parallel transport matrix between adjacent sites *n* -+ *n* + *µ* and is usually called a *link variable.* From (10.236) and (10.237) we have *ifinU(n, n* + *µ}l/ln+µ* as a gauge-invariant combination. This suggests the modification of (10.230) to obtain the quark part of the SU(3) gauge-invariant QCD action

$$S\_{\rm QCD} = S(q) + S(\mathcal{A}) \tag{10.240}$$

$$S(q) = \sum\_{n} \left\{ \frac{a^3}{2} \sum\_{\mu} \bar{\psi}\_n \mathbb{I}(1 + \gamma\_{\mu}) U(n, n + \hat{\mu}) \psi\_{n + \hat{\mu}}.$$

$$+ (1 - \gamma\_{\mu}) U(n, n - \hat{\mu}) \psi\_{n - \hat{\mu}} + 2 \psi\_n \mathbb{I} - m a^4 \bar{\psi}\_n \psi\_n \right\}.\tag{10.241}$$

What about the lattice action for the gluon field *S(A)?* Clearly this term must be composed of link variables only. The simplest gauge-invariant combination will be four-link variables. (As they are matrices in SU(3) space twoand three-link combinations are trivial because of unitarity and determinantal constraints.) This suggests that

$$S(\mathcal{A}) = \frac{-1}{2g^2} \sum\_{p} \text{tr } U\_p \tag{10.242}$$

with

$$U\_p = U(n, n + \hat{\mu})U(n + \hat{\mu}, n + \hat{\mu} + \hat{\eta})U(n + \hat{\mu} + \hat{\eta}, n + \hat{\eta})U(n + \hat{\eta}, n) \tag{10.243}$$

which is a product of four-link variables around an elementary square (called a *plaquette;* see Fig. 10.18). The sum in (10.242) is over all plaquettes of the lattice. From our discussion of a parallel transport around a square given in §8.2, it follows that (10.242) may be written as

$$S(\mathcal{A}) = \frac{-1}{2g^2} \sum\_{\mathfrak{p}} \text{tr} \{ \exp(\text{i}a^4 g^2 F\_{\mathfrak{n}, \mathfrak{p}}) \} \tag{10.244}$$

FIG. 10.18. A lattice plaquette.

with

$$F\_{\mathfrak{n},\mu\mathbf{v}} = \partial\_{\mu}A\_{\mathfrak{n},\mathbf{v}} - \partial\_{\mathbf{v}}A\_{\mathfrak{n},\mu} - \mathrm{i}g \Box A\_{\mathfrak{n},\mu} \lrcorner \mathbf{A}\_{\mathfrak{n},\mathbf{v}} \lrcorner$$

$$\partial\_{\mu}A\_{\mathfrak{n},\mathbf{v}} \equiv \frac{1}{g} \left( A\_{\mathfrak{n}+\tilde{\mu},\mathbf{v}} - A\_{\mathfrak{n},\mathbf{v}} \right) \tag{10.245}$$

where *An,,,* is the gluon field at the site *n.* The continuum limit can be recovered immediately as

$$S(\mathcal{A}) = \frac{-1}{2g^2} \sum\_{p} \left\{ 1 - \frac{a^4}{2} \, g^2 F^i\_{\mu\nu} F^{i\mu\nu} + \dots \right\}$$

$$\rightarrow \frac{1}{4} \int \mathbf{d}^4 \mathbf{x} F^i\_{\mu\nu} F^{i\mu\nu} \tag{10.246}$$

where we have used the fact that the A.s are traceless (so the linear *F,,,* term disappears) and *tr(A.iA.i)* = 2 *t5ii.*

#### **Confinement criteria, the Wilson loop**

To see whether QCD has confinement one can study the energy of a system composed of a quark at *x* = *(t,* **0)** and an antiquark at *x* = *(t,* **R).** If there is no confinement, we expect

$$E(R) \to 2m \quad \text{for} \quad R \to \infty \tag{10.247}$$

where mis the quark mass. Confinement implies that the interquark potential energy grows without bound,

$$E(\mathcal{R}) \to \infty \quad \text{for} \quad \mathcal{R} \to \infty. \tag{10.248}$$

We can represent the qq state at time *t* as (see, for example, Bander 1981)

$$\left| \mathbf{q}(t, \mathbf{0}) \bar{\mathbf{q}}(t, \mathbf{R}) \right\rangle = \sum\_{\mathbf{C}} f(\mathbf{C}) \Gamma[(t, \mathbf{R}), (t, \mathbf{0}); \mathbf{C}] \left| 0 \right\rangle \tag{10.249}$$

where r[x', *x;* C] is the gauge-invariant qq operator

$$
\Gamma[\mathbf{x}', \mathbf{x}; \mathbf{C}] = \overline{\mathbf{q}}(\mathbf{x}') U(\mathbf{x}', \mathbf{x}; \mathbf{C}) \mathbf{q}(\mathbf{x}) \tag{10.250}
$$

with

$$U(\mathbf{x'}, \mathbf{x}; \mathbf{C}) = \exp\left\{ \mathrm{i}g \int\_{\mathbf{x}}^{\mathbf{x'}} \frac{1}{2} \lambda^t \mathcal{A}^t(\mathbf{y}) \, \mathrm{d}y^\mu \right\} \tag{10.251}$$

and C is a path joining *x* and *x'.* Now consider the overlap between the qq

state at *t* = 0 and the qq state at *t* = *T,*

*Q(T, R)* = (Olrl(O, 0), (0, **R);** C]r[(T, 0), *(T,* **R);** CJIO). (10.252)

Inserting a complete set of energy eigenstates, we obtain in the Euclidean space

$$\Omega(T,\mathsf{R}) = \sum |\langle 0|\Gamma^{\dagger}[(0,\mathsf{0}),(0,\mathsf{R});\mathsf{C}]|n\rangle|^{2}\,\mathsf{e}^{-E\_{\mathsf{a}}T}.\tag{10.253}$$

For large *T,* the smallest *En* will dominate. This smallest energy eigenvalue corresponds to the potential energy of the qq system separated by a distance *R*

$$\lim\_{r\_{-}\to} \Omega(T, \mathcal{R}) \sim \text{e}^{-\mathcal{E}(\mathcal{R})T}.\tag{10.254}$$

In terms of the quark fields the overlap function Q(T, R) may be written as

$$\Omega(T,\mathsf{R}) = \langle 0|\bar{\mathsf{q}}(0,\mathsf{R})U[(0,\mathsf{R}),(0,\mathsf{0});\mathsf{C}]\mathsf{q}(0,\mathsf{0})$$

$$= \iota\pi\mathsf{T}\cdot\mathsf{q}\,\mathsf{III}\mathsf{T}\cdot\mathsf{q}\,\mathsf{V}\cdot(\mathsf{T},\mathsf{D})\cdot\mathsf{C}\mathsf{T}\cdot(\mathsf{T},\mathsf{D})\,\mathsf{V}\,\mathsf{V}$$

$$\bar{\mathbf{q}}(T,\mathbf{0})U\{ (T,\mathbf{0}),(T,\mathbf{R});\mathbf{C}\}\mathbf{q}(T,\mathbf{R})|0\rangle. \quad (10.255)$$

lfwe treat the quark fields as external sources (as in the case of heavy quark states), then the quark propagator (in a background of gluon fields) can be expressed as t'

$$
\langle 0|\mathbf{q}^{\boldsymbol{\theta}}(t',\mathbf{x})\bar{\mathbf{q}}^{\boldsymbol{\omega}}(t,\mathbf{x})|0\rangle = \exp\left[\mathrm{i}\int\_{t}^{t'} A\_{0}(\mathbf{r},\mathbf{x})\,\mathrm{d}\tau\right] \langle 0|\mathbf{q}^{\boldsymbol{\theta}}(t',\mathbf{x})\bar{\mathbf{q}}^{\boldsymbol{\omega}}(t,\mathbf{x})|0\rangle\_{\text{free}}
$$

$$
\sim U\left[(t',\mathbf{x}),(t,\mathbf{x});\mathbf{C}\right]\delta^{\mathbf{a}\boldsymbol{\theta}}\,\mathrm{e}^{-n|t'-t|}.\qquad(10.256)
$$

Combining (10.256) and (10.255), we have

$$
\Omega(T, R) \sim \text{e}^{-2\pi T} W(\text{C}) \tag{10.257}
$$

where

$$W(\mathbf{C}) = \langle 0|\text{tr }U\{\mathbf{x}, \mathbf{x}; \mathbf{C}\} |0\rangle. \tag{10.258}$$

Here C is the rectangular path of Fig. 10.19. The correlation function *W(C)*

is called the *Wilson loop* and its behaviour determines the confinement property, as a comparison of (10.257) with (10.254) shows that

$$\lim\_{T \to \infty} W(\mathbf{C}) \sim \mathbf{e}^{-T[E(\boldsymbol{R}) - 2\boldsymbol{m}]}.\tag{10.259}$$

As we shall illustrate presently, in the strong coupling limit, the Wilson loop of lattice gauge theory obeys an *area law* so that for a large contour

$$W(\mathbf{C}) \sim \exp\{-K\mathcal{A}(\mathbf{C})\}\tag{10.260}$$

where *K* is a constant and *A(C)* is the area of the surface enclosed by path C.

Thus for the rectangular path of Fig. l 0.19

$$\mathcal{A}(\mathbf{C}) = TR.\tag{10.261}$$

Substituting eqns (10.260) and (10.261) into eqn (10.259) we have the linearly rising potential (10.202) for the qq system. Also, we can identify the constant *Kin* (10.260) a:. the string tension. Note that we treat the quarks as a mere external colour source. Thus using the Wilson loop it is possible to study the question of colour confinement in a pure gluon theory without quark fields.

#### **The area law in the strong coupling limit**

In eqns (10.222) and (10.223) we displayed the path-intergral formulation of the correlation function in a scalar field theory. The Wilson loop may be similarly expressed in a pure gluon QCD; it will be a functional integral over gluon fields. Since the (infinitesimal) link variables are directly related to the gauge fields, we can choose to work with the link variables as the basic dynamic degrees of freedom in a lattice gauge theory. As we shall see, this allows for a simpler formulation of eqn (10.258)

$$W(\mathbf{C}) = \frac{1}{Z} \left[ \prod\_{n,\mu} \text{d}U(n, n + \hat{\mu}) \text{ tr } U(\mathbf{x}, \mathbf{x}; \mathbf{C}) \exp\left\{ -\frac{1}{2g^2} \sum\_{\mathbf{p}} \text{tr } U\_{\mathbf{p}} \right\} . \right] \tag{10.262}$$

where

$$Z = \int\_{\mathbb{T}^{n,\mu}} \mathrm{d}U(n, n + \hat{\mu}) \exp\left\{-\frac{1}{2g^2} \sum\_{p} \mathrm{tr}\, U\_p\right\}.\tag{10.263}$$

Note that no gauge-fixing term has been added to the action because the link variable (i.e. lattice gauge field) has only finite range. The volume in the path-integral space generated by all possible gauge transformations is finite. Hence no convergence factor corresponding to the gauge-fixing term will be necessary before passing on to the continuum limit (see §9.1). The link variables are group elements of SU(3). We may set up a generalized Euler angle representation for these SU(3) unitary matrices and determine explicitly the form of the group integral in terms of the eight Euler angles. In any case we have the orthogonality properties

$$\int \mathrm{d}U(n, n + \hat{\mu}) [\![U(n, n + \hat{\mu})] \!]\_{ij} = 0 \tag{10.264}$$

$$\int \mathrm{d}U(n,n+\hat{\mu})\mathbb{I}\,U(n,n+\hat{\mu})\mathbb{I}\_{\mathbb{II}}[\![U^{\dagger}(n,n+\hat{\mu})\!]\_{kl} = \frac{1}{3}\,\delta\_{ll}\,\delta\_{jk} \tag{10.265}$$

$$\int \mathrm{d}U(n,n+\hat{\mu}) \|\, U(n,n+\hat{\mu})\|\_{ij} \|\, U(n,n+\hat{\mu})\|\_{kl} = 0. \tag{10.266}$$

The identity (10.265) implies that two links going in opposite directions will cancel each other after integration. Thus, if we have two adjacent plaquettes oriented in the same direction, they will merge into a rectangular path after integrating over the common link as illustrated in Fig. 10.20. In the strong

coupling limit we have l/g2 as a small parameter and the exponential in (10.262) can be expanded

$$\begin{split} W(\mathbf{C}) &= \frac{1}{Z} \int \mathbf{d}U(\mathbf{n}, \mathbf{n} + \hat{\mu}) \text{tr } U(\mathbf{x}, \mathbf{x}; \mathbf{C}) \\ &\times \left[ 1 - \frac{1}{2g^2} \sum\_{\mathbf{p}} \text{tr } U\_{\mathbf{p}} + \frac{1}{2!} \left( \frac{1}{2g^2} \right)^2 \sum\_{\mathbf{p}} \sum\_{\mathbf{p}'} \text{tr } U\_{\mathbf{p}} \text{tr } U\_{\mathbf{p}'}, + \dots \right]. \end{split} \tag{10.267}$$

For simplicity we will take a planar rectangular path C as in Fig. 10.19. It is clear that in this strong coupling limit the orgothonality properties of the link variables given in eqns (10.264}-(10.266) imply that the lowest-order nonvanishing contribution to *W(C)* is the (l/g2)Np term where NP is the minimal number of plaquettes required to cover the area enclosed by the path C,

$$W(\mathbf{C}) \sim \left(\frac{1}{g^2}\right)^{\aleph\_0}.\tag{10.268}$$

This corresponds to the area law since the area enclosed by the path C is given by

$$\mathcal{A}(\mathbf{C}) = a^2 N\_\mathbf{p}. \tag{10.269}$$

Hence,

$$W(\mathbf{C}) \sim (g^2)^{-\mathcal{A}(\mathbf{C})a^2} = \exp\{- (TR\ln g^2)/a^2\} \tag{10.270}$$

or, comparing this with (10.260) we have the linearly arising potential

$$E(R) = KR \tag{10.271}$$

with

$$K = \frac{\ln g^2}{a^2} \cdot \tag{10.272}$$

The weak coupling expansion of the Wilson loop can also be considered by first passing to the continuum limit and replacing the action by the Gaussian approximation. We then obtain a *perimeter law.* It turns out that this is the familiar Coulomb's law, *E(R)* "" *R-* 1, in disguise. With the appropriate renormalization effects taken into account the property of asymptotic freedom is recovered.

Does this mean that we have proven that QCD possesses both properties of asymptotic freedom and colour confinement? No, not yet, as we note that the U(l) lattice gauge theory also has both the perimeter (i.e. Coulomb's) law and the area law (confinement)-the above strong-coupling result is derived without using the non-Abelian nature of the theory. The catch is that the weak and strong coupling regimes may be separated by one or more discontinuous phase transitions. It has been proven that in four dimensions the Abelian gauge theory has a non-trivial phase structure, but similar analytic proof that QCD does not undergo a phase transition at some finite couplings has not been obtained. However this problem can be investigated using numerical analysis. In particular two approaches have been very successful. The first involves the use of Pade techniques to extrapolate from the strong coupling expansion to a regime where weak coupling predictions become valid (Kogut, Pearson, and Shigemitsu 1979). The second approach is Monte Carlo simulation in which the path integral (10.263) is considered as a partition function for a statistical system (Creutz 1979). Various correlations can be calculated by first generating configurations typical of the system in thermal equalibrium. The results from both methods indicate that no phase transitions occur in the intermediate coupling region. The strong coupling behaviour (10.272) *g 2(a)* - *eKa2*does go into the weak coupling *g 2 (a)* - 1/(ln a- 1) as *a* --+ 0. It is found that the transition is smooth and rapid (see also Kogut *et al.* 1981 ), just as a similar calculation for U(l) theory shows convincing evidence for a phase transition. Furthermore, these methods are able to produce satisfactory numerical relations between the long-distance parameter string tension and the gauge coupling at short distance *(or* the conventional QCD scale parameter, 100 MeV 300 MeV). Thus these numerical results are encouraging in indicating that asymptotic freedom and colour confinement do coexist in a single phase ofQCD.

# **11 Standard electroweak theory I· basic structure**

IN this chapter and the following one we shall present the standard gauge theory of weak and electromagnetic interactions. It combines in one framework quantum electrodynamics and the low-energy V-A theory of weak interactions for charged currents. The unified theory is renormalizable. It also predicts a new set of neutral currents; its successful experimental confrontation in recent years strengthens our confidence as to the correctness of the theory. Chapter 11 will serve as an introduction and will emphasize the theoretical structure of the model. In Chapter 12 the phenomenological implications of the model will be presented. Possible extension and modification of the standard electroweak theory will be discussed in Chapter 13.

# **11 .1 Weak interactions before gauge theory**

In this section we shall provide a brief review of weak-interaction theories before the advent of gauge models: the four-fermion theory as well as the intermediate vector boson theory. We shall discuss in particular the difficulties encountered if they were taken to be self-contained theories.

#### **Four-fermion interactions**

Soon after Pauli's neutrino postulate, Fermi (1934a,b) proposed his theory for the n p e v:

$$\mathcal{A}\_{\rm F}^{\rho}(\mathbf{x}) = -\frac{G\_{\rm F}}{\sqrt{2}} \left[ \mathbf{\bar{p}}(\mathbf{x}) \boldsymbol{\upchi}\_{\lambda} \mathbf{n}(\mathbf{x}) \right] \left[ \mathbf{\bar{e}}(\mathbf{x}) \boldsymbol{\upchi}^{\lambda} \mathbf{v}(\mathbf{x}) \right] + \text{h.c.} \tag{11.1}$$

where the fermion field operators are denoted by their particle names, and

$$G\_{\rm F} \simeq 10^{-5}/m\_{\rm p}^2\tag{11.2}$$

is the Fermi coupling constant with mP being the proton mass.

In the ensuing years other processes such as then-µ and µ-e decays have been discovered and found, like to have a comparatively long lifetime. The concept of a distinctive class of interactions, the weak interactions, began to emerge. The surprising discovery of parity nonconservation (Lee and Yang 1956; Wu *et al.* 1957) stimulated a great deal of research and the eventual formulation of the V-A theory (Feynman and Gell-Mann 1958; Sudarshan and Marshak 1958; Sakurai 1958). It is suggested that an effective Lagrangian, very much like the one in eqn (11.1),

#### 11. I *Weak interactions before gauge theory* 337

describes the weak interactions

$$\mathcal{Q}\_{\rm eff}(\mathbf{x}) = -\frac{G\_{\rm F}}{\sqrt{2}} J\_{\lambda}^{\dagger}(\mathbf{x}) J^{\lambda}(\mathbf{x}) + \text{h.c.} \tag{11.3}$$

with the weak current *J;.(x)* being of the vector-minus-axial-vector form.

If we separate in the current the leptonic and hadronic parts,

$$J\_{\lambda}(\mathbf{x}) = J\_{l\lambda}(\mathbf{x}) + J\_{h\lambda}(\mathbf{x}) \tag{11.4}$$

the leptonic current *Jf(x)* can be written directly in terms of lepton fields

$$J\_I^{\lambda}(\mathbf{x}) = \bar{\mathbf{v}}\_e \boldsymbol{\gamma}^{\lambda} (1 - \boldsymbol{\gamma}\_5) \, \mathbf{e} + \bar{\mathbf{v}}\_\mu \boldsymbol{\gamma}^{\lambda} (1 - \boldsymbol{\gamma}\_5) \boldsymbol{\mu} \tag{11.5}$$

and the hadronic current can be decomposed into parts having definite flavour SU(3) transformation properties as in eqn (5.79) and can be neatly summarized by writing the current directly in terms of quark fields

$$J^{\lambda}\_{\hbar}(\mathbf{x}) = \bar{\mathbf{u}} \gamma^{\lambda} (1 - \gamma\_{5}) \, \mathbf{d}\_{\theta} \tag{11.6}$$

with

$$\mathbf{d}\_{\theta} = \cos \theta\_{\mathbf{c}} \,\mathrm{d} + \sin \theta\_{\mathbf{c}} \,\mathrm{s} \tag{11.7}$$

where *(Jc* is the Cabibbo angle 13°. When compared eqn (l 1.6) to (11.5), lepton-quark symmetry then suggests the generalization of (11.6) to

$$J^{\lambda}\_{\hbar}(\mathbf{x}) = \mathbf{\bar{u}}\gamma^{\lambda}(1-\gamma\_{5})\,\mathbf{d}\_{\theta} + \mathbf{\bar{c}}\gamma^{\lambda}(1-\gamma\_{5})\mathbf{s}\_{\theta} \tag{11.8}$$

with

$$\mathbf{s}\_{\theta} = \cos \theta\_{\mathbf{c}} \,\mathrm{s} - \sin \theta\_{\mathbf{c}} \,\mathrm{d} \tag{11.9}$$

where the c-field is the postulated new heavy quark, the charmed quark (Bjorken and Glashow 1964). More importantly, it has been shown that any sensible weak-interaction theory must have this extra hadronic current in order to suppress to an acceptable level the induced strangeness-changing neutral-current effects (Glashow, Iliopoulos and Maiani 1970). This suppression mechanism, although invented before the general acceptance of gauge theories, can best be explained in this new context and we shall do so in §11.3 and in more detail in § 12.2.

We should take note of some of the common properties of the weak currents as given by eqns (11.5) and (11.8). They are charged currents, all with one unit of charge; in the lowest order there are no neutral-current processes such as the reaction v11N-+ v11N. They are bilinear in the fundamental fermion fields involving the helicity projection operator (1 - *y5 ).* We can rephrase this by saying· that only the left-handed (LH) fermions are present in the weak currents. Thus,

$$
\sqrt{\gamma}\gamma^{\lambda}(1-\gamma\_{5})\psi = 2\sqrt{\iota}\gamma^{\lambda}\psi\_{\perp} \tag{11.10}
$$

with

$$
\psi\_{\mathsf{L}} = \frac{1}{2}(1 - \gamma\_{\mathsf{S}})\psi, \quad \psi\_{\mathsf{R}} = \frac{1}{2}(1 + \gamma\_{\mathsf{S}})\psi \tag{11.11}
$$

and

$$
\psi = \psi\_{\sf L} + \psi\_{\sf R}.\tag{11.12}
$$

The rule that 'parity violation is maximal in weak interactions' has a simple interpretation in terms of the lepton and quark fields: in charged-current weak interactions the fundamental dynamical degrees of freedom for matter are the two-component left-handed fermion fields.

If we restrict our applications to the leading order in GF, the V-A Lagrangian of (11.3) is very successful in describing a vast amount of lowenergy weak-interaction experimental data. The exceptions are few. It is not clear whether nonleptonic weak decays are correctly described by a coupling as required by (11.3) because of our inability to do reliable strong-interaction calculations. The CP violation discovered in the neutral K-meson system (Christenson, Cronin, Fitch, and Turlay 1964) cannot be incorporated in any simple fashion. The most successful phenomenological theory of CP nonconservation postulates the existence of a new super-weak interaction (Wolfenstein 1964). In short, the lowest-order V-A theory correctly describes the domain of weak-interaction phenomena in which one believes it should be applicable.

Nevertheless the Lagrangian in (11.3) cannot be taken as a self-consistent quantum field theory of weak interactions: it is not renormalizable and even in the lowest order of GF it violates unitarity at high energies.

(1) *Lack ofrenormalizability.* The interaction in (11.3) is not renormalizable. It is a dimension-six operator, or, more transparently, the coupling constant *GF* has dimension (mass)- 2 • Thus the higher-order contributions are increasingly divergent and they cannot be organized in such a way as to be absorbed in a few 'bare' quantities (see Chapter 2, especially §2.4). In a nonrenormalizable theory, even though the coupling may be small, there is no guarantee that the higher-order terms are not large. Then the whole corpus of selection rules (e.g. the absence of AS> 1, AS= -AQ chargedcurrent processes) which is the basis of the successful V-A theory will be made meaningless if the lowest-order Born terms do not dominate.

(2) *Violation of unitarity.* Even if we restrict ourselves to the Born approximation there are certain processes which will violate unitarity. Consider the reaction of vµ e -+ µv. as described by the effective Lagrangian of (11.3): the amplitude has only the *J* = 1 partial wave and the high-energy cross-section *11* ,...., withs = *2m.E (E* being the v µenergy in the laboratory frame). However unitarity requires that *11( J* = 1) be bounded by *s-* 1 • Thus for energies above *J s* ,...., *Gi 112* 300 Ge V the theoretical cross-section from (11.3) would violate unitarity. (For details see §11.2.)

It turns out that these two problems are closely related. If the lower-order diagrams have bad high-energy behaviour to violate unitarity, the higherorder contributions, which are integrals over lower-order diagrams, cannot be renormalized.

#### Intermediate vector boson theory (IVB)

Like the electromagnetic current, the weak current in !l'.rr (eqn (11.3)) transforms as a four-vector under the Lorentz transformation and one may

#### 11.2 *Construction of the standard SU(2) x U(l) theory* 339

imitate the successful QED by introducing a new massive field W *µ* and write the basic interaction as

$$\mathcal{Q}\_{\text{l}} = g(J\_{\mu}\mathbf{W}^{\mu} + \text{h.c.}). \tag{11.13}$$

Then the four-fermion Lagrangian (11.3) can be viewed as the effective lowenergy theory generated by 2 1 in second order (see Fig. 11.l) with the

FIG. 11.1. The four-fermion interaction as the low-energy approximation to the *g2* diagram in IVB theory.

identification *g2M;/* = *r* 1' 2Gf, as the massive IVB propagator contributes the M;/ factor. Since QED is a gauge theory, one may interpret this interaction Lagrangian as resulting from a Yang-Mills constructionbut with gauge bosons being massive.

Let us now examine the problems ofunitarity and renormalizability in IVB theory. The problem with unitarity remains: but it is shifted to some other processes. For example the reaction vv-+ w+w- with longitudinally polarized Ws can be shown (see §11.2 below) in the IVB theory to have a highenergy amplitude ,.., *GFE2(E* being the neutrino energy) in the pure J = 1 partial wave, which unitarity requires to be bounded by a constant.

Even though the coupling *g* is now dimensionless, this theory is still not renormalizable. The free massive vector boson Lagrangian is

$$\mathcal{L}\_{\mathbf{W}} = -\frac{1}{4} (\partial\_{\mu} \mathbf{W}\_{\mathbf{v}}^{\dagger} - \partial\_{\mathbf{v}} \mathbf{W}\_{\mu}^{\dagger})(\partial^{\mu} \mathbf{W}^{\star} - \partial^{\mathbf{v}} \mathbf{W}^{\mu}) + M\_{\mathbf{W}}^{2} \mathbf{W}\_{\mu}^{\dagger} \mathbf{W}^{\mu} \tag{11.14}$$

which gives the propagator in momentum space

$$\mathrm{i}\Delta\_{\mu\nu}(k) = -\mathrm{i}\,\frac{g\_{\mu\nu} - k\_{\mu}k\_{\nu}/M\_{\text{W}}^2}{k^2 - M\_{\text{W}}^2 + \mathrm{i}\varepsilon} \,\text{d}\tag{11.15}$$

This propagator behaves like a constant as *k* -+ oo and the interaction is not renormalizable by power counting. The problem of course lies in the IVB mass, which gives the term *kµk./* in the propagator. However just this massiveness of W is required in order to yield the desired low-energy fourfermion theory. The key problem is then how to introduce gauge-boson mass terms in the Yang-Mills theory without spoiling renormalizability.

# **1.2 Construction of the standard SU(2) x U(1) theory**

As discussed in the last section, weak interactions must involve massive intermediate vector bosons and yet their mass terms spoil renormalizability. This serious contradiction was finally resolved with the emergence of the spontaneously broken gauge theory. Also, the correct renormalizable theory displays the unity of weak interactions with electromagnetism. The required

gauge symmetry is SU(2) x U(l). This model is now the standard theory of the electroweak interaction.

Noting the vectorial nature of both interactions, Schwinger (1957) was the first to advance the idea of weak and electromagnetic unification. Glashow (1958) suggested that the desired renormalizable theory of weak interactions would involve this unification, and later he (Glashow 1961) proposed such a model which has the SU(2) x U(l) gauge symmetry. Renormalizability is not preserved in his theory as the IVB masses were inserted by hand. A similar attempt was also made by Salam and Ward (1964). Finally the renormalizable theory with IVB masses generated by the Higgs mechanism was proposed by Weinberg (1967) and also discussed independently by Salam a year later (1968). Thus the standard theory is often referred to as the Weinberg-Salam model or the Glashow-Weinberg-Salam model. However, the importance of this approach was not recognized by the general community of high-energy theoretical physicists until 't Hooft *(l97la,b)*  transformed the subject by proving the renormalizability of gauge theories, with and without spontaneous symmetry breaking.

In the following discussion we shall present the Weinberg-Salam theory. For simplicity we shall restrict our initial consideration to the lightest fermion 'family' or 'generation' (v., e, u, d). As we shall show, a theory in the one-family approximation is completely self-contained and self-consistent. The heavier families (vµ, µ, c, s) and (v,, t, t, b) will be given structures identical to the light one. Their incorporation into the standard theory is discussed in §11.3.

#### Choice of the group SU(2) x U(l)

The algebraic approach. To motivate the choice of the gauge group we need only to consider an IVB theory with an electron and its neutrino. The weakinteraction Lagrangian is given by eqns (11.13) and (11.5)

$$\mathcal{A}\_{\Psi} = g(J\_{\lambda}\mathbf{W}^{\lambda} + \text{h.c.})\tag{11.16}$$

where

$$J\_{\lambda} = \bar{\mathbf{v}}\_{\mathbf{e}} \mathbf{y}\_{\lambda} (1 - \mathbf{y}\_{\mathbf{s}}) \, \mathbf{e} \tag{11.17}$$

is the V-A charged current. On the other hand the electromagnetic interaction of these leptons is given by

$$\mathcal{A}\_{\rm em}^{\rho} = \mathbf{e} \, J\_{\lambda}^{\rm em} \mathcal{A}^{\lambda} \tag{11.18}$$

where

$$J\_{\lambda}^{\text{em}} = -\overline{\mathfrak{e}} \,\, \eta\_{\lambda} \,\, \mathbf{e} \tag{11.19}$$

is the electromagnetic current. In a unified gauge theory of weak and electromagnetic interactions, we need at least three vector gauge bosons (W± and the photon) to couple to the currents *J,* Jt, and rm. The simplest group with three generators is SU(2). However, as we shall demonstrate imme-

#### 11.2 *Construction of the standard SU(2)* x *U* (1) *theory* 341

diately, the three currents listed in eqns (11.17) and (11.19) do not close (to form an algebra) under commutation. Define the weak and electric charges as

$$T\_{+}(t) = \frac{1}{2} \int \mathrm{d}^{3} \mathbf{x} J\_{0}(\mathbf{x}) = \frac{1}{2} \int \mathrm{d}^{3} \mathbf{x} \mathbf{v}\_{\ast}^{\dagger}(1 - \boldsymbol{\gamma}\_{5}) \,\mathrm{e}$$

$$T\_{-}(t) = T\_{+}^{\dagger}(t)$$

$$\mathcal{Q}(t) = \int \mathrm{d}^{3} \mathbf{x} J\_{0}^{\mathrm{em}}(\mathbf{x}) = -\int \mathrm{d}^{3} \mathbf{x} \,\mathrm{e}^{\dagger} \mathbf{e}.\tag{11.20}$$

Using the canonical commutation relations for fermions

$$\{\psi\_i^\dagger(\mathbf{x},t), \psi\_f(\mathbf{x'},t)\} = \delta\_{ij}\delta^3(\mathbf{x}-\mathbf{x'}),$$

we can show that

$$\left[\left.T\_{+}(t), T\_{-}(t)\right] = \mathcal{Z}T\_{3}(t)\right.\tag{11.21}$$

with

$$T\_3(t) = \frac{1}{4} \int \mathrm{d}^3 \mathbf{x} \Big[ \mathbf{v}\_\mathbf{e}^\dagger (1 - \mathbf{y}\_\mathbf{s}) \mathbf{v}\_\mathbf{e} - \mathbf{e}^\dagger (1 - \mathbf{y}\_\mathbf{s}) \,\mathrm{e} \Big].\tag{11.22}$$

T3 *=F Q* and thus *T±, Q* do not form a closed algebra. The reason behind this is not difficult to see. In order for *Q* to be a generator of SU(2) the charges of a complete multiplet must add up to zero, corresponding to the requirement that the generators for SU(2) must be traceless. In the case at hand we are attempting to form a doublet out of v. and e which clearly do not satisfy this condition. Also, *T±(t)* are of the V-A form while *Q* is purely vector.

There are two alternatives at this point.

(i) We can introduce another gauge boson coupled to T3 as given in eqn (11.21). These four generators can now form the group SU(2) x U(l). This is the choice we shall adopt eventually.

(ii) We can add new fermions to the multiplet and thus modify the currents in order that the new set of *T* ± and *Q* will be closed to form SU(2) under commutation. In our case we may attempt to form a triplet withe, v., and a new charged heavy lepton E +. Such a theory has in fact been constructed by Georgi and Glashow (1972a). (They also introduced a neutral heavy lepton Nin order to obtain the V-A form for the weak current at low energies.) In this model,

$$
\begin{pmatrix}
\mathbf{E}^+ \\
\frac{1}{2}(\mathbf{l} - \mathbf{\dot{\chi}}\_3) \begin{pmatrix}
\mathbf{E}^+ \\
\mathbf{v}\_\star \cos \mathfrak{a} + \mathbf{N} \sin \mathfrak{a} \\
\mathbf{e}
\end{pmatrix}, \\
\mathbf{e}
$$

$$
\frac{1}{2}(\mathbf{l} + \mathbf{\dot{\chi}}\_3) \begin{pmatrix}
\mathbf{E}^+ \\
\mathbf{N} \\
\mathbf{e}
\end{pmatrix},
$$

and

!(l + Ys)(N cos *IX* - Ye sin *IX)*

are the two triplets and one singlet in the model. This yields the weak charge

$$T\_{+}(t) = \frac{1}{2} \int \mathrm{d}^{3} \mathbf{x} \{ \mathrm{E}^{+} (1 - \boldsymbol{\gamma}\_{s}) (\mathbf{v}\_{\epsilon} \cos \boldsymbol{\omega} + \mathbf{N} \sin \boldsymbol{\omega}) \} $$

$$+ (\mathbf{v}\_{\epsilon}^{\dagger} \cos \boldsymbol{\omega} + \mathbf{N}^{\dagger} \sin \boldsymbol{\omega}) (1 - \boldsymbol{\gamma}\_{s}) \, \mathbf{e} $$

$$+ \mathrm{E}^{\dagger} (1 + \boldsymbol{\gamma}\_{s}) \mathrm{N} + \mathrm{N}^{\dagger} (1 + \boldsymbol{\gamma}\_{s}) \, \mathbf{e} \} . \tag{11.23}$$

It is then straightforward to see that

$$[[T\_+(t), T\_-(t)] = 2Q(t)]$$

with

$$\mathcal{Q} = \int \mathbf{d}^3 \mathbf{x} \{ \mathbf{E}^\dagger \mathbf{E} - \mathbf{e}^\dagger \mathbf{e} \}. \tag{11.24}$$

Clearly the only neutral current in this case is the electromagnetic current and the model was ruled out by the discovery of weak neutral-current effects in 1973. Furthermore, it is difficult to incorporate fractionally charged quarks into such a model.

**The unitarity argument.** Equivalently we can argue from unitarity that it is necessary to introduce either a new charged lepton or a new neutral gaugeboson. Consider the reaction VY--+ w+w- with both Ws being longitudinally polarized. The lowest-order amplitude is given by Fig. 11.2

FIG. 11.2. vv--> w+w- with at-channel exchange of leptons.

$$T\_i(\mathbf{v}\bar{\mathbf{v}}\rightarrow\mathbf{W}^+\mathbf{W}^-) = -\mathbf{i}\bar{v}(p')(-\mathbf{i}g\mathbf{s}')(1-\gamma\_5)$$

$$\times\frac{\mathbf{i}}{p-k-m\_\epsilon}(-\mathbf{i}g\mathbf{s})(1-\gamma\_5)u(p)$$

$$=-2g^2\bar{v}(p')\frac{\mathcal{S}(p-k)\mathbf{e}(1-\gamma\_5)}{(p-k)^2-m\_\epsilon^2}u(p). \tag{11.25}$$

The polarization vectors with s<il · *sUl* = *-l5ii* and *k* · s<il = 0 may be chosen in the rest frame of the W boson as s8l = 0 and = *l5ii.* To obtain s<il for a moving W boson: *k11 =(E,0,* 0, k) with *k* = *(E 2*- Ma,)t we can make

#### 11.2 *Construction of the standard SU(2)* x *U(J) theory* 343

the appropriate Lorentz boost along the z-axis. The transverse polarizations will not change under such a Lorentz transformation while the longitudinal polarization vector becomes = *M*w 1*(k,* 0, 0, *E* ). In the high-energy limit with *k* = *E* - *M* a,;2E + ... the vector can be approximated as

$$
\varepsilon\_{\mu}^{(3)} = k\_{\mu}/M\_{\Psi} + O(M\_{\Psi}/E). \tag{11.26}
$$

Thus, substituting (11.26) into (11.25), we have

$$T\_l \simeq -2g^2/(k^2 - 2p \cdot k)\bar{v}(p')(k'/M\_W)$$

$$\times (p - k)(k/M\_W)(1 - \gamma\_5)u(p)$$

$$\simeq \frac{2g^2}{M\_W^2}\bar{v}(p')k'(1 - \gamma\_5)u(p) \tag{11.27}$$

where we have used the result *(p- k)ku(p)* = *(2p* · *k* - k2 )u(p). To show more explicitly that this amplitude is a pure *J* = 1 partial wave, we can choose the following momentum configurations

$$p\_{\mu} = (E, \mathbf{0}, \mathbf{0}, E), \qquad p'\_{\mu} = (E, \mathbf{0}, \mathbf{0}, -E)$$

$$k\_{\mu} = (E, k\mathbf{e}), \qquad \qquad k'\_{\mu} = (E, -k\mathbf{e})$$

with

$$\mathbf{e} = (\sin \theta, 0, \cos \theta). \tag{11.28}$$

Since v and v have opposite helicities, we have

$$u(p) = \sqrt{E} \begin{pmatrix} 1 \\ \frac{\boldsymbol{\sigma} \cdot \mathbf{p}}{E} \end{pmatrix} \boldsymbol{\chi}\_{-1/2} = \sqrt{E} \begin{pmatrix} 1 \\ \sigma\_z \end{pmatrix} \boldsymbol{\chi}\_{-1/2} \tag{11.29}$$

$$
\vec{v}(p') = \sqrt{E} \,\chi\_{1/2}^\dagger \left(\frac{\mathbf{o} \cdot \mathbf{p'}}{E}, -\mathbf{l}\right) = \sqrt{E} \,\chi\_{1/2}^\dagger (-\sigma\_z, -\mathbf{l}) \tag{11.30}
$$

where

$$\chi\_{1/2} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \chi\_{-1/2} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

and thus the combination in eqn (11.27) becomes

$$\begin{split} \vec{v}(p')k'(1-\chi\_{\sf S})\vec{\mu}(p) &= E\chi\_{1/2}^{\dagger}(-1,-1) \\ &\times \begin{pmatrix} E & k\boldsymbol{\sigma}\cdot\mathbf{e} \\ -k\boldsymbol{\sigma}\cdot\mathbf{e} & -E \end{pmatrix} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ -1 \end{pmatrix} \chi\_{-1/2} \\ &= -4E\chi\_{1/2}^{\dagger}(E-k\boldsymbol{\sigma}\cdot\mathbf{e})\chi\_{-1/2} = 4Ek\sin\theta. \end{split} \tag{11.31}$$

We have Tt *GFE 2*sin() as *E->* oo. The partial-wave expansion for the helicity amplitude is (Jacob and Wick 1959; see also Frazer 1966) in this case av

$$T\_{\lambda\_3 \lambda\_4, \lambda\_1 \lambda\_2}(E, \theta) = \sum\_{J=M}^{\infty} (2J+1) T\_{\lambda\_3 \lambda\_4, \lambda\_1 \lambda\_2}^{J}(E) d\_{\mu \lambda}^{J}(\theta)$$

where A. 1 = *-A.2* = 1/2 and A. 3 = A.4 = 0 are the helicities of the initial and final particles with *A.=* A. 1 - *A.2* = 1, *µ* = A. 3 - A.4 = 0 and *M* = max(A., *µ)*  =1. is the usual rotation matrix with *di 0 (())* = sine. It is clear that *T1*  corresponds to a pure *J* = 1 partial wave and violates the unitarity bound of *rJ=* 1(£) <constant. To cancel this bad high-energy behaviour, we need other diagrams for this reaction. There are two possibilities: s-channel or *u*channel exchange diagrams. (The t-channel diagram will not help as it gives the similar contribution as before with the same sign.)

The heavy-lepton alternative; the u-channel exchange diagram in Fig. l l.3(a) yields the amplitude

$$T\_{\mathbf{u}}(\mathbf{v}\bar{\mathbf{v}}\to\mathbf{W}^{+}\mathbf{W}^{-}) = -2g'^2\bar{v}(p')\frac{\mathfrak{s}(p-k')\mathfrak{s}(1-\chi\_{5})}{(p-k')^2 - m\_{\mathbb{E}}^2}u(p)$$

$$\simeq \frac{-2g'^2}{M\_{\mathbb{W}}^2}\bar{v}(p')\mathsf{k}'(1-\chi\_{5})u(p). \tag{11.32}$$

F1a. 11.3. vv--+ w+w- with (a) u-channel and (b) s-channel exchanges.

If *g2* = *g' 2 ,* this will cancel the bad high-energy behaviour given in eqn (11.27).

The neutral vector boson alternative; the s-channel exchange diagram in Fig. 1 l.3(b) yields the amplitude

$$T\_s(\mathbf{v}\bar{\mathbf{v}}\rightarrow\mathbf{W}^+\mathbf{W}^-) = -\mathrm{i}\bar{\mathbf{v}}(p')(-\mathrm{i}f\gamma\_\theta)(1-\gamma\_3)\mathrm{i}(p)L\_{u\mu\nu}\varepsilon^\nu(k')\varepsilon^\nu(k)$$

$$\times\mathrm{i}\mathbb{[}-g^{\mu\theta}+(k+k')^\mu(k+k')^\theta/M\_z^2]\mathbb{I}[(k+k')^2-M\_z^2].\tag{11.33}$$

Choosing the ZWW coupling to have the Yang-Mills structure

$$L\_{\mathfrak{a}\mu\upsilon} \equiv -\mathrm{i}f'\mathbb{I}(k'-k)\_{\mathfrak{a}}g\_{\mu\upsilon} - (2k'+k)\_{\mathfrak{v}}g\_{\mathfrak{u}\mu} + (2k+k')\_{\mathfrak{v}}g\_{\mathfrak{v}\mathfrak{v}}\mathbb{I},\tag{11.34}$$

we get

$$
\varepsilon^{\prime\prime\mu}\varepsilon^{\prime}L\_{\mathfrak{a}\mu\upsilon} = -\dot{\mathfrak{y}}^{\prime}\left[ (k^{\prime}-k)\_{\mathfrak{s}}(\varepsilon^{\prime}\cdot\varepsilon) - \mathfrak{z}(k^{\prime}\cdot\varepsilon)\varepsilon\_{\mathfrak{a}}^{\prime} + \mathfrak{z}(k\cdot\mathfrak{s}^{\prime})\varepsilon\_{\mathfrak{a}} \right]
$$

$$
\simeq -\frac{\dot{\mathfrak{y}}^{\prime}}{M\_{\mathfrak{w}}^{2}}\left[ (k-k^{\prime})\_{\mathfrak{s}}(k\cdot k^{\prime}) \right].\tag{11.35}
$$

Hence

$$T\_\* \simeq \frac{-ff'}{M\_W^2} \,\bar{v}(p')\mathbb{K}'(1-\gamma\_5)u(p). \tag{11.36}$$

If we choose *ff'* = *2g2 ,* this will also cancel the amplitude in (11.27).

#### 11.2 *Construction of the standard SU(2)* x *U(l) theory* 345

In fact if one demands that all the amplitudes which violate unitarity be cancelled out, one ends up with a renormalizable Lagrangian which is the same as the one derived formally from the algebraic approach. (For details of such a construction see Llewellyn Smith 1973; Bell 1973; Cornwall, Levin, and Tiktopoulous 1974, 1975.)

With the choice of the group SU(2) *x* U(l), it is straightforward to write down the gauge-invariant Lagrangian

$$\mathcal{H}\_1 = -\frac{1}{4} F^{i}\_{\mu\nu} F^{i\mu\nu} - \frac{1}{4} G\_{\mu\nu} G^{\mu\nu} \tag{11.37}$$

where

$$F^{i}\_{\mu\nu} = \partial\_{\mu}A^{i}\_{\nu} - \partial\_{\nu}A^{i}\_{\mu} + g\varepsilon^{ijk}A^{j}\_{\mu}A^{k}\_{\nu}, \quad i = 1, 2, 3 \tag{11.38}$$

and

$$\mathbf{G}\_{\mu\mathbf{v}} = \hat{\boldsymbol{\mathcal{O}}}\_{\mu} \mathbf{B}\_{\mathbf{v}} - \hat{\boldsymbol{\mathcal{O}}}\_{\mathbf{v}} \mathbf{B}\_{\mu} \tag{11.39}$$

are the SU(2) and U(l) gauge-field tensors, respectively. Clearly before spontaneous symmetry breakdown this corresponds to four massless gauge bosons.

#### Fermions (in the one-family approximation)

In this section we shall study the standard model with its fermionic sector composed of e, v. leptons and u, d quarks only. As it turns out such a simplified theory is completely self-contained. Heavier fermions such as µ, s, c, ... , etc. will be incorporated in §11.3.

As we have already mentioned in §11.1 the basic fermion dynamic degrees of freedom are the two-component fields with definite helicities. More pertinently, since gauge interactions conserve helicity, we can and should make independent choices for left-handed and right-handed fermions. Thus the first family is composed of the following 15 two-component fermions

$$\boldsymbol{\psi} = \mathbf{v}\_{\text{eL}}, \mathbf{e}\_{\text{L}}, \mathbf{e}\_{\text{R}}, \mathbf{u}\_{\text{L}}, \mathbf{u}\_{\text{R}}, \mathbf{d}\_{\text{L}}, \mathbf{d}\_{\text{R}} \tag{11.40}$$

with

$$\mathbf{e}\_{\mathsf{L}} = \frac{1}{2} (\mathbf{1} - \boldsymbol{\gamma}\_{\mathsf{s}}) \cdot \mathbf{e}$$

$$\mathbf{e}\_{\mathsf{R}} = \frac{1}{2} (\mathbf{1} + \boldsymbol{\gamma}\_{\mathsf{s}}) \cdot \mathbf{e}, \text{ etc.} \tag{11.41}$$

The colour indices oc = 1, 2, 3 on the quark fields have been suppressed.

SU(2) x U(l) quantum number assignment. From eqns (11.5), (11.7), (11.20), and (11.22) we have learned that the SU(2) group is generated by the weak charges

$$T\_{+} = \int (\mathbf{v}\_{e\mathbf{L}}^{\dagger} \mathbf{e}\_{\mathbf{L}} + \mathbf{u}\_{\mathbf{L}}^{\dagger} \mathbf{d}\_{\mathbf{L}}) \, \mathbf{d}^{3} \mathbf{x}$$

$$T\_{-} = (T\_{+})^{\dagger}$$

$$T\_{3} = \frac{1}{2} \int (\mathbf{v}\_{e\mathbf{L}}^{\dagger} \mathbf{v}\_{e\mathbf{L}} - \mathbf{e}\_{\mathbf{L}}^{\dagger} \mathbf{e}\_{\mathbf{L}} + \mathbf{u}\_{\mathbf{L}}^{\dagger} \mathbf{u}\_{\mathbf{L}} - \mathbf{d}\_{\mathbf{L}}^{\dagger} \mathbf{d}\_{\mathbf{L}}) \, \mathbf{d}^{3} \mathbf{x}.\tag{11.42}$$

From these expressions for the SU(2) generators, it is clear that

$$l\_{\mathbf{L}} \equiv \begin{pmatrix} \mathbf{v}\_{\mathrm{eL}} \\ \mathbf{e}\_{\mathrm{L}} \end{pmatrix} \quad \text{and} \quad \mathbf{q}\_{\mathrm{L}} \equiv \begin{pmatrix} \mathbf{u}\_{\mathrm{L}} \\ \mathbf{d}\_{\mathrm{L}} \end{pmatrix} \tag{11.43}$$

are SU(2) doublets and eR, uR, and dR are singlets. The U(l) group should be chosen in such a way that the electric charge *Q*

$$\begin{aligned} \mathcal{Q} &= \int \left( -\mathbf{e}^{\dagger} \mathbf{c} + \frac{2}{3} \mathbf{u}^{\dagger} \mathbf{u} - \frac{1}{3} \mathbf{d}^{\dagger} \mathbf{d} \right) \mathbf{d}^{3} \mathbf{x} \\ &= \int \left( -\mathbf{e}\_{\mathrm{L}}^{\dagger} \mathbf{c}\_{\mathrm{L}} - \mathbf{e}\_{\mathrm{R}}^{\dagger} \mathbf{c}\_{\mathrm{R}} + \frac{2}{3} \mathbf{u}\_{\mathrm{L}}^{\dagger} \mathbf{u}\_{\mathrm{L}} + \frac{2}{3} \mathbf{u}\_{\mathrm{R}}^{\dagger} \mathbf{u}\_{\mathrm{R}} - \frac{1}{3} \mathbf{d}\_{\mathrm{L}}^{\dagger} \mathbf{d}\_{\mathrm{L}} \right. \end{aligned}$$

$$-\frac{1}{3} \, \mathrm{d}\_{\mathrm{R}}^{\dagger} \mathrm{d}\_{\mathrm{R}} \right) \, \mathrm{d}^{3} \mathbf{x} \tag{11.44}$$

can be a linear combination of the U(l) generator and T3 of SU(2) in eqn (l l.42). We observe that the combination

$$\mathcal{Q} - T\_3 = \int \left[ -\frac{1}{2} (\mathbf{v}\_{\mathrm{el}}^{\dagger} \mathbf{v}\_{\mathrm{el}.} + \mathbf{e}\_{\mathrm{L}}^{\dagger} \mathbf{e}\_{\mathrm{L}}) + \frac{1}{8} (\mathbf{u}\_{\mathrm{L}}^{\dagger} \mathbf{u}\_{\mathrm{L}} + \mathbf{d}\_{\mathrm{L}}^{\dagger} \mathbf{d}\_{\mathrm{L}}) \right. $$

$$-\, \mathbf{e}\_{\mathrm{R}}^{\dagger} \mathbf{e}\_{\mathrm{R}} + \frac{1}{2} \mathbf{u}\_{\mathrm{R}}^{\dagger} \mathbf{u}\_{\mathrm{R}} - \frac{1}{3} \mathbf{d}\_{\mathrm{R}}^{\dagger} \mathbf{d}\_{\mathrm{R}} \right] \mathrm{d}^{3} \mathrm{x} \tag{11.45}$$

has the property of giving the same quantum number to all members of an SU(2) doublet in (l l.43). Clearly it commutes with all the SU(2) generators, i.e.

$$[\![Q-T\_3,\!],\!] = 0, \quad i = 1,2,3. \tag{11.46}$$

We then choose

$$Y = \mathcal{Z}(\mathcal{Q} - T\_{\mathcal{J}}) \tag{11.47}$$

as the generator of the U(l) group and refer to *Y* as the *weak hypercharge.*  Unlike the T;s, *Y* does not obey any nonlinear commutation relations. Its scale and hence the proportional constant between it and *(Q* - T3) is strictly a convention. To obtain the correct electric charges for particles we must use eqns (1 l.45) and (1 l.47) and make the assignments

$$Y(l\_{\mathcal{L}}) = -1, \quad Y(\mathbf{c\_{\mathcal{R}}}) = -2,$$

$$Y(\mathbf{q\_{\mathcal{L}}}) = 1/3, \quad Y(\mathbf{u\_{\mathcal{R}}}) = 4/3, \quad Y(\mathbf{d\_{\mathcal{R}}}) = -2/3. \tag{11.48}$$

These hypercharge values may be remembered as twice the average charges of each multiplet, as the average T3 value is always zero. We should note that the group structure allows for any hypercharge assignment. Thus 'charge quantization' (i.e. particle electric charges are integral multiples of some basic unit) is not automatic in the SU(2)L x U(l)1 theory. This can be obtained only if the gauge group is semi-simple. We shall see in Chapter 14 that this feature will be obtained when we combine this electroweak theory with SU(3) quantum chromodynamics into a 'grand unified gauge theory'.

#### 11.2 *Construction of the standard SU(2)* x *U* (1) *theory* 347

**Anomaly cancellation.** As we have discussed in §§6.2 and 9.3, the ABJ anomaly spoils the renormalizability of a gauge theory (Gross and Jackiw 1972; Georgi and Glashow 1972b; Bouchiat, Iliopoulos, and Meyer 1972). The fermionic gauge couplings must not introduce anomalous Ward identities. Thus, for the fermion representation *R* with representation matrix *T0 (R),* the trace *tr({T0 (R), Tb(R)}r(R))* in eqn (6.60) must vanish.

In the present case, the fermions are either doublets or singlets under SU(2). The matrix *T0* will be either the Pauli matrix *"C0*or the U(l) hypercharge *Y.* Since the group SU(2) is anomaly-free

$$\text{tr}(\{\tau^i, \tau^j\} \tau^k) = 2 \,\delta^{ij} \,\text{tr}(\tau^k) = 0,\tag{11.49}$$

we will consider cases where at least one of the *Ts* is the hypercharge *Y.*  Because every member of a given SU(2) multiplet has the same hypercharge value, for the case of two *Ts* being a *Y* we have,

$$\text{tr}(\mathfrak{r}^i Y Y) \propto \text{tr } \mathfrak{r}^i = 0,$$

and for the case of one *T* being a *Y* we have,

$$\text{tr}(\{\pi^l, \pi^j\} Y) = 2 \text{ } \delta^{lj} \text{tr } Y. \tag{11.49a}$$

Thus this anomaly contribution is proportional to the trace of *Y* (the sum of all fermionic hypercharge values)

$$\text{tr } Y = \sum\_{i} Y\_{i} = \sum\_{\mathsf{k} \mathsf{ptron}} Y + \sum\_{\mathsf{quart}} Y\_{i}$$

But this vanishes by explicit calculation for the fermion assignments in each generation

$$\sum\_{\text{lepton}} Y = -1 \times 2 - 2 = -4$$

$$\sum\_{\text{quark}} Y = \Im(\frac{1}{3} \times 2 + \frac{4}{3} - \frac{2}{3}) = 4.\tag{11.49b}$$

The factor 3 in front of the parenthesis is due to the colour degree of freedom. For the case when all *Ts* are the hypercharge, we have from eqn (11.47),

$$\operatorname{tr}(YYY) = 8\operatorname{tr}(\mathcal{Q}^3 - 3\mathcal{Q}^2T\_3 + 3\mathcal{Q}T\_3^2 - T\_3^3)$$

$$\propto \operatorname{tr}(\mathcal{Q}^2T\_3 - \mathcal{Q}T\_3^2) \tag{11.50a}$$

because tr = 0, and because we can ignore the Q3 term as the electromagnetic current is a vector (V) and the VVV type of triangular fermion loops does not have anomaly. Explicit calculation of the right-hand side of eqn (1 l.50a) yields

$$\sum\_{\text{lepton}} \left( \mathcal{Q}^2 T\_3 - \mathcal{Q} T\_3^2 \right) = -\frac{1}{2} + \frac{1}{4} = -\frac{1}{4}$$

$$\sum\_{\text{quark}} \left( \mathcal{Q}^2 T\_3 - \mathcal{Q} T\_3^2 \right) = \left( \frac{2}{3} - \frac{1}{6} - \frac{1}{2} + \frac{1}{4} \right) = \frac{1}{4}.\tag{11.50b}$$

Thus the anomalies again cancel as in eqn (11.49) and we can conclude that

with the fermion assignments of eqns (11.43) and (11.48) the SU(2) x U(l) theory is free of ABJ anomaly.

In fact there is a convenient way to remember the cancellation displayed by eqns (11.49) and (11.50). Using eqn (11.47) we can express eqns (1 l.49a) and (l l.50a) as

$$\text{tr } Y \propto \text{tr } \mathcal{Q}$$

$$\text{tr}(\mathcal{Q}^2 T\_3 - \mathcal{Q} T\_3^2) \propto \text{tr}(T\_3 \mathcal{Q} Y) \propto \text{tr}(T\_3^2 Y) \propto \text{tr } \mathcal{Q} \tag{11.51}$$

because tr(T3Q2 ) = tr(T3 *Y 2 )* = = 0. Thus the nontrivial contribution to the ABJ anomaly in SU(2) x U(l) theory is proportional to

$$\text{tr}\,\mathcal{Q} = \sum\_{l} \mathcal{Q} = 0 \tag{11.52}$$

Lepton and quark charges cancel when the three colours are taken into account. [Remark: Given (11.52) one can easily show that the entire SU(3)c x SU(2)L x U(l)v standard model is free of anomaly. Namely, the additional triangle diagram contributions involving gluons and electroweak gauge bosons also mutually cancel.]

With the fermions given in (11.40) the gauge-invariant Lagrangian takes on the form

$$\mathcal{H} = \mathcal{H}\_1 + \mathcal{H}\_2$$

where

$$\mathcal{A}\mathcal{C}\_2 = \sqrt{\mathrm{i}}\gamma^\mu \mathbf{D}\_\mu \psi \tag{11.53}$$

with the covariant derivative

$$\mathbf{D}\_{\mu}\psi = \left(\partial\_{\mu} - \mathbf{i}g\mathbf{T} \cdot \mathbf{A}\_{\mu} - \mathbf{i}g'\frac{Y}{2}\,\mathbf{B}\_{\mu}\right)\psi. \tag{11.54}$$

For example,

$$\mathbf{D}\_{\mu}l\_{\mathbf{L}} = \left(\partial\_{\mu} - \mathbf{i}\,\frac{g}{2}\,\mathbf{r}\cdot\mathbf{A}\_{\mu} + \mathbf{i}\,\frac{g'}{2}\,B\_{\mu}\right)l\_{\mathbf{L}}\tag{11.55}$$

and

$$\mathbf{D}\_{\mu}e\_{\mathbb{R}} = (\partial\_{\mu} + \mathrm{i}g'B\_{\mu})e\_{\mathbb{R}}.\tag{11.56}$$

We should note in particular that there are no gauge-invariant terms in *.!£*  that are bilinear in the fermion fields. Hence there are no SU(2) x U(l) symmetric fermion mass terms.

#### Symmetry breaking via the Higgs mechanism

We need to introduce a set of scalar fields  and this set develops a U(I).m symmetric vacuum expectation value (<1>) 0 so that we have the following pattern of symmetry breaking

$$\mathbf{SU(2)\_L \times U(1)\_Y \xrightarrow{\langle \Phi \rangle\_{\mathfrak{g}}} U(1)\_{\mathfrak{m}}}$$

# 11.2 *Construction of the standard SU(2)* x *U(l) theory* 349

Three of the original four SU(2) x U(l) gauge bosons will become massive and one, corresponding to the photons, remains massless.

As we have seen in §8.3, a complex scalar doublet breaks the SU(2) symmetry completely and one member of this doublet must be neutral in order to have possibility of a U(l).m-invariant (<1>) 0 • We need a complex doublet with the charge assignment

$$\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \qquad Y(\Phi) = 1. \tag{11.57}$$

The complete gauge-invariant Lagrangian including the scalar fields is then

$$\mathcal{J} = \mathcal{J}\_1 + \mathcal{J}\_2 + \mathcal{J}\_3 + \mathcal{J}\_4 \cdot \tag{11.58}$$

with

$$\mathcal{A}\_3^{\rho} = (\mathbf{D}\_\mu \Phi)^\dagger (\mathbf{D}^\mu \Phi) - V(\Phi) \tag{11.59}$$

where

$$\mathbf{D}\_{\mu}\Phi = \left(\partial\_{\mu} - \frac{\mathrm{i}}{2}g\boldsymbol{\tau}\cdot\mathbf{A}\_{\mu} - \frac{\mathrm{i}}{2}g'\mathcal{B}\_{\mu}\right)\Phi \tag{11.60}$$

and

$$V(\Phi) = -\mu^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2. \tag{11.61}$$

The most general SU(2) x U(l) Yukawa coupling between scalars and fermions is given by