Finally, we comment very briefly on the role of instantons. For a very small coupling constant *g,* corresponding to very small distances *r,* tunnelling is negligible as indicated by eqn (16.63) and QCD perturbation theory can be used in appropriate circumstances as shown in Chapter 10. For some intermediate values of *gone* can continue to use perturbation theory but with allowance made for vacuum tunnelling. This produces (Callan, Dashen, and Gross 1980) a sudden rapid increase in *g* as *r* is increased from zero to a distance comparable to the hadron size ( 0.5 fm). This is compatible with the phenomenological quark models and with the results of lattice gauge theory calculations discussed at the end of §10.5. For large distances there is at present no reliable method for calculating the effect of instantons (in fact it is not even clear whether they are relevant at all) although they do remind us that the QCD vacuum must be very complicated.

# **Appendix A Notations and conventions**

# **Metric**

$$\begin{array}{ccccc}\hline\\\text{Metric tensor} & g\_{\mu\nu} = g^{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}. & \text{(A.1)}\\\text{Contravariant coordinate} & \mathbf{x}^{\mu} = (\mathbf{x}^0, \mathbf{x}^1, \mathbf{x}^2, \mathbf{x}^3) = (t, \mathbf{x}, \boldsymbol{\uprho}, \boldsymbol{z}) = (t, \mathbf{x}). & \text{(A.2)}\\\text{Covariant coordinate} & \mathbf{x}\_{\mu} = g\_{\mu x} \mathbf{x}^\mu = (t, -\mathbf{x}). & \\\text{Scalar product} & A \cdot B = A\_{\mu} B^{\mu} = A\_{\mu} g^{\mu\nu} B\_{\nu} = A\_0 B\_0 - A \cdot \mathbf{B}. & \text{(A.3)}\\\text{Derivative} & \partial^{\mu} \equiv \frac{\partial}{\partial \mathbf{x}\_{\mu}} = \left(\frac{\partial}{\partial t}, -\mathbf{V}\right) & \partial\_{\mu} \equiv \frac{\partial}{\partial x^{\mu}} = \left(\frac{\partial}{\partial t}, \mathbf{V}\right). & \\\text{where} & \mathbf{V} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right), & \text{(A.4)}\\\\\text{Four-divergences} & \partial^{\mu} A\_{\mu} = \frac{\partial A\_0}{\partial t} + \mathbf{V} \cdot \mathbf{A}. & \text{(A.6)}\\\end{array}$$

## **Pauli matrices**

$$
\mathfrak{o} = (\sigma\_1, \sigma\_2, \sigma\_3). \tag{A.7}
$$

$$
\sigma\_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \qquad \sigma\_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \qquad \sigma\_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \tag{A.8}
$$

*r,iik:* totally antisymmetric tr(cr; *a)* = 2Jii. 8 123 = I. (A.9) (A.IO)

$$\text{Completeness: } \sum (\sigma\_l)\_{ab} (\sigma\_l)\_{cd} = \mathcal{Z} (\delta\_{bc} \, \delta\_{ad} - \frac{1}{2} \, \delta\_{ab} \, \delta\_{cd}). \tag{A.11}$$

# **Dirac matrices**

i

$$\{\gamma^{\mu}, \gamma^{\nu}\} = \mathfrak{Z}g^{\mu \nu} \qquad \qquad \qquad \gamma\_{\mu} = g\_{\mu \nu} \mathfrak{j}^{\nu}. \tag{A.12}$$

$$\gamma^{\mu} = (\mathfrak{y}^{0}, \mathfrak{y}) \qquad \qquad \qquad \qquad \mathfrak{y}\_{\mu} = (\mathfrak{y}\_{0}, -\mathfrak{y}). \tag{A.13}$$

$$\gamma^{\mathfrak{s}} = \gamma\_{\mathfrak{s}} = \dot{\mathfrak{y}}^{0} \gamma^{1} \gamma^{2} \gamma^{3} \qquad \qquad \sigma\_{\mu \mathbf{v}} = \frac{\dot{\mathfrak{s}}}{2} \{\gamma\_{\mu}, \gamma\_{\mathbf{v}}\}. \tag{A.14}$$

*Appendix A Notations and conventions*

$$\text{Spin matrix}\tag{A} = \frac{1}{4}\mathbf{c}^{ij\mathbf{k}}\sigma^{jk}, \qquad i = 1, 2, 3. \tag{A.15}$$

$$\text{Charge conjugation} \qquad \qquad \psi^{\varepsilon} = C\psi^{\dagger} \qquad C\gamma\_{\mu}C^{\dagger} = -\gamma\_{\mu}^{\star} \tag{A.16}$$

$$\mathbf{Identity}\tag{A}\tag{1}\tag{1}$$

$$\mathbf{Identity}\tag{2}$$

### Fierz transformation

$$\text{Let } \Gamma\_{\mathbb{S}} = 1, \,\, \Gamma\_{\mathbb{V}} = \boldsymbol{\gamma}\_{\mu}, \,\, \Gamma\_{\mathbb{T}} = \sigma\_{\mu\nu}, \,\, \Gamma\_{\mathbb{A}} = \boldsymbol{\gamma}\_{\mu}\boldsymbol{\gamma}\_{\mathfrak{s}}, \,\, \Gamma\_{\mathbb{P}} = \boldsymbol{\gamma}\_{\mathfrak{s}}\,. \tag{A.18}$$

$$\text{Then } \sum\_{i} g\_i(\Gamma\_i)\_{\mathfrak{a}\mathfrak{f}}(\Gamma\_i)\_{\gamma\mathfrak{b}} = \sum\_{j} \hat{g}\_j(\Gamma\_j)\_{\mathfrak{a}\mathfrak{b}}(\Gamma\_j)\_{\gamma\mathfrak{f}} \tag{A.19}$$

where the indices *i, j* run over the set S, V, T, A, and P and the *UiS* are related to *Yi* by

$$
\begin{bmatrix}
\hat{g}\_{\mathbf{S}} \\
\hat{g}\_{\mathbf{V}} \\
\hat{g}\_{\mathbf{I}} \\
\hat{g}\_{\mathbf{A}} \\
\hat{g}\_{\mathbf{P}}
\end{bmatrix} = \frac{1}{4} \begin{bmatrix}
1 & 4 & 12 & -4 & 1 \\
1 & -2 & 0 & -2 & -1 \\
\frac{1}{2} & 0 & -2 & 0 & \frac{1}{2} \\
1 & -4 & 12 & 4 & 1
\end{bmatrix} \begin{bmatrix}
g\_{\mathbf{S}} \\
g\_{\mathbf{V}} \\
g\_{\mathbf{T}} \\
g\_{\mathbf{A}} \\
g\_{\mathbf{P}}
\end{bmatrix}.\tag{A.20}
$$

#### **Dirac representation**

$$\boldsymbol{\upbeta}^{\bullet} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \qquad \boldsymbol{\upgamma} = \begin{pmatrix} 0 & \boldsymbol{\sigma} \\ -\boldsymbol{\sigma} & 0 \end{pmatrix} \qquad \boldsymbol{\upgamma}\_{\mathcal{S}} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} . \tag{A.21}$$

Hermitian conjugate

$$(\lambda\_0^{\prime})^{\dagger} = \lambda^0 \quad (\lambda^k)^{\dagger} = -\gamma^k, \quad (k=1,2,3) \quad \sigma^{\dagger}\_{\mu\nu} = \sigma^{\mu\nu} \quad \gamma^{\dagger}\_{\mathfrak{s}} = \gamma\_{\mathfrak{s}}. \tag{A.22}$$

$$\text{Spin matrix } \mathbf{s} = \frac{1}{2} \begin{pmatrix} \sigma & 0\\ 0 & \sigma \end{pmatrix} . \tag{A.23}$$

Charge conjugate C = *iy2 •*  (A.24)

# **Plane wave and Dirac spinor**

Normalization

$$\bar{u}(p,s)u(p,s') = 2m \,\delta\_{\text{ss}'} \qquad \bar{v}(p,s)v(p,s') = -2m \,\delta\_{\text{ss}'} \qquad \text{(A.32)}$$

Projection operators

$$
\sum\_{s} \mu\_{\mathfrak{a}}(p, s) \bar{\mu}\_{\mathfrak{f}}(p, s) = (p + m)\_{\mathfrak{a}\mathfrak{f}}
$$

$$
\sum\_{s} v\_{\mathfrak{a}}(p, s) \bar{v}\_{\mathfrak{f}}(p, s) = (p - m)\_{\mathfrak{a}\mathfrak{f}}.\tag{A.33}
$$

Gordon decomposition

$$\overline{u}(p,s)\gamma^{\mu}u(q,s) = \frac{1}{2m}\overline{u}(p,s)\mathbb{I}(p+q)^{\mu} + \mathrm{i}\sigma^{\mu\nu}(p-q)\_{\nu}\mathbb{I}u(q,s) \quad \text{(A.34)}$$

$$
\bar{u}(p,s)\gamma^{\mu}\gamma\_{5}u(q,s) = \frac{1}{2m}\,\bar{u}(p,s)\mathbb{E}(p-q)^{\mu}\gamma\_{5} + \mathrm{i}\sigma^{\mu\nu}(p+q)\,\gamma\_{5}\mathbb{J}u(q,s).
$$

In the Dirac representation (A.21), Dirac spinors are given by

$$u(p,s) = (E+m)^\natural \begin{pmatrix} 1 \\ \sigma \cdot \mathbf{p} \\ \overline{E+m} \end{pmatrix} \chi\_s \quad v(p,s) = (E+m)^\natural \begin{pmatrix} \sigma \cdot \mathbf{p} \\ \overline{E+m} \\ 1 \end{pmatrix} \chi\_s$$
  $s = 1, 2$

$$\text{where } \chi\_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \qquad \chi\_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \qquad E = (\mathbf{p}^2 + m^2)^{1/2}. \tag{A.36}$$

# **Normalization of states, cross sections and decay rates**

One-particle states are normalized as

$$
\langle \mathbf{p}, \mathfrak{a} | \mathbf{p}', \mathfrak{a}' \rangle = 2E(2\pi)^3 \,\delta^3(\mathbf{p} - \mathbf{p}') \,\delta\_{\mathfrak{a}\mathfrak{a}'} \tag{A.37}
$$

where oc, oc' label the spin and/or internal symmetry indices. The projection for a one-particle state

$$\begin{aligned} \text{parameter } \mathbf{a} \mathbf{a} \mathbf{a} \\ \sum\_{p,a} |p,a\rangle \langle p,a| &= \int \frac{\mathbf{d}^3 p}{(2\pi)^3 2E} \sum\_a |p,a\rangle \langle p,a| \\ &= \int \frac{\mathbf{d}^4 p}{(2\pi)^3} \,\delta(p^2 - m^2) \theta(p\_0) \sum\_a |p,a\rangle \langle p,a|. \end{aligned} \tag{A.38}$$

The transition probability per unit time per unit volume for the process i f is

$$
\alpha(\mathbf{i}\rightarrow\mathbf{f}) = (2\pi)^4 \,\delta^4(p\_i - p\_f) |\langle \mathbf{f}|T|\mathbf{i}\rangle|^2 \tag{A.39}
$$

where (flTli) is the covariant T-matrix and is related to the S-matrix by

$$
\langle \mathbf{f} | \mathbf{S} | \mathbf{i} \rangle = \delta\_{\mathbf{i} \mathbf{j}} + \mathbf{i} (2\pi)^{4} \,\delta^{4} (p\_{\mathbf{i}} - p\_{\mathbf{j}}) \langle \mathbf{f} | T | \mathbf{i} \rangle. \tag{A.40}
$$

#### **Decay rate**

The transition probability per unit time per unit volume to a specific final

#### *Appendix A Notations and conventions*

state f is given by

$$\mathbf{d}\Gamma(\mathbf{i}\to\mathbf{f}) = \frac{\alpha(\mathbf{i}\to\mathbf{f})}{\rho\_{\parallel}}\mathbf{d}N\_{\mathbf{f}} \tag{A.41}$$

where Pi = is the density of the decaying state and

$$\mathbf{d}N\_{\mathbf{f}} = \prod\_{j=1}^{n} \frac{\mathbf{d}^{3}p\_{j}}{(2\pi)^{3}2E\_{\mathbf{f}}} \text{ is the density of the final state.}$$

The total decay rate is

$$\begin{aligned} \text{Total decay rate is} \\ \Gamma(\text{i}\rightarrow\text{f}) &= \frac{1}{2E\_{\text{i}}} \int\_{-1}^{n} \frac{\text{d}^{3}p\_{j}}{(2\pi)^{3}2E\_{\text{j}}} (2\pi)^{4} \left. \delta^{4} \left( p\_{i} - \sum\_{j=1}^{n} p\_{j} \right) \right| \langle \text{f}|T|\text{i}\rangle|^{2} \text{S} \quad (\text{A.42}), \end{aligned}$$

where Sis the statistical factor which is obtained by including a factor *1/m* ! if there are *m* identical particles in the final state,

$$S = \prod\_i \frac{1}{m\_i!} \cdot \tag{A.43}$$

#### **Scattering** cross-sections

The cross-section for a1 + a2 - f is given by

$$\mathbf{d}\sigma(\mathbf{a}\_1 + \mathbf{a}\_2 \to \mathbf{f}) = \frac{\omega(\mathbf{a}\_1 + \mathbf{a}\_2 \to \mathbf{f})}{J\_l} \mathbf{d}N\_l \tag{A.44}$$

where *.f;* is the flux of the incoming particles and is given by

$$J\_l = \rho\_1 \rho\_2 v \tag{A.45}$$

where p1, *p2* are the densities of the initial state and vis the relative velocity between two particles in the initial state. The flux factor *.f;* can be written in the centre-of-mass system as

$$J\_i = 2E\_1 2E\_2 \left| \frac{\mathbf{p}\_1}{E\_1} - \frac{\mathbf{p}\_2}{E\_2} \right| = 4|\mathbf{p}\_1 E\_2 - \mathbf{p}\_2 E\_1| = 4|\mathbf{p}\_1|(E\_1 + E\_2)$$

$$= 4[(p\_1 \cdot p\_2)^2 - m\_1^2 m\_2^2]^{1/2} \tag{A.46}$$

Note that *.f;* in eqn (A.46) is expressed in terms of the Lorentz-invariant quantity. Hence it is valid in any other frame.

The cross-section is given by

$$\mathbf{d}\sigma = \frac{(2\pi)^4}{4[(p\_1 \cdot p\_2)^2 - m\_1^2 m\_2^2]^{1/2}} |\langle \mathbf{f} | T | \mathbf{i} \rangle|^2 \prod\_{j=1}^{\bullet} \frac{\mathbf{d}^3 p\_j}{(2\pi)^3 2E\_j} \mathbf{S} \tag{A.47}$$

497

# **Appendix B Feynman rules**

In field theory, the standard procedure for calculating any physically interesting quantities in perturbation theory is summarized in a set of Feynman rules. Efficient use of these rules will greatly simplify the calculation. The most important parts of the Feynman rules are the expressions for the propagators and vertices, which characterize the structure of the theory. In this appendix, we will outline a practical method for deriving the propagators and vertices for the most general situation. We will also summarize the results for the physically interesting cases at the end.

# **A practical guide to the derivation of Feynman rule propagators and vertices**

### The i..cj>4 case

We will first discuss the simple case of A.¢4 theory for the spin-0 boson and then generalize the result to more complicated cases. When the A.¢4 theory Lagrangian is divided into the free part *!l'* 0 and the interaction part .!l'1

$$\mathcal{P} = \mathcal{P}\_0 + \mathcal{P}\_1 \tag{8.1}$$

$$\mathcal{A}\_0^{\prime}(\phi) = \frac{1}{2}(\partial\_{\mu}\phi)^2 - \frac{1}{2}\mu^2\phi^2 \tag{B.2}$$

$$
\mathcal{H}\_{\mathbb{I}}(\phi) = -\frac{\lambda}{4!} \,\phi^{\star},\tag{\text{B.3}}
$$

the generating functional for the Green's functions is given by

$$W[J] = \int \left[ \mathbf{d}\phi \right] \exp\left\{ \mathbf{i} \int \left[ \mathcal{Q}\_0(\phi) + \mathcal{Q}\_l(\phi) + J(\mathbf{x})\phi(\mathbf{x}) \right] \mathbf{d}^4 \mathbf{x} \right\} $$

$$= \exp\left\{ \mathbf{i} \int \left[ \mathcal{Q}\_1 \left( -\mathbf{i} \frac{\partial}{\partial J} \right) \right] \mathbf{d}^4 \mathbf{x} \right\} W\_0[J] \tag{B.4}$$

where

$$\mathcal{W}\_0 \llbracket \mathcal{J} \rrbracket = \int \llbracket \mathbf{d}\phi \rrbracket \, \exp \left\{ \mathbf{i} \int \mathbf{d}^4 \mathbf{x} \llbracket \mathcal{J}\_0(\mathbf{x}) + \mathbf{J}(\mathbf{x}) \phi(\mathbf{x}) \right\} \cdot \tag{\text{B.S.}}$$

Since the free Lagrangian *!l'* 0 given in (B.2) is quadratic in the fields, the integral in eqn (B.5) can be evaluated as follows. Write (B.5) as

$$\begin{aligned} \,^0W\_0 \mathbf{I} \,^J \mathbf{J} &= \int \mathbf{I} \,^j \mathbf{d}\phi \, \mathbf{J} \exp\left\{ \mathbf{i} \int \left[ \mathbf{I} \frac{1}{2} (\partial\_\mu \phi)^2 - \frac{1}{2} \mu^2 \phi^2 + J(\mathbf{x}) \phi(\mathbf{x}) \right] \mathbf{d}^4 \mathbf{x} \right\} \\ &= \int \mathbf{I} \,^j \mathbf{d}\phi \, \mathbf{J} \exp\left\{ \mathbf{i} \int \left[ \mathbf{I} \frac{1}{2} \phi(\mathbf{x}) P(\mathbf{x}) \phi(\mathbf{x}) + J(\mathbf{x}) \phi(\mathbf{x}) \right] \mathbf{d}^4 \mathbf{x} \right\} \end{aligned} \tag{B.6}$$

#### *Appendix B Feynman rules* 499

where

$$P(\mathbf{x}) = -(\partial^2 + \mu^2 - \mathrm{i}\mathbf{e})\tag{\mathbf{B}.7}$$

is a Hermitian operator. In (B.6) we have inserted a factor exp{-! J *e</>2*d4x}, with *e* > 0, to make the path integral convergent. Let </>c(x) be the solution of the classical free-field equation in the presence of the external source *J* (x)

$$P(\mathbf{x})\phi\_{\mathbf{c}}(\mathbf{x}) = \ -J(\mathbf{x}).\tag{\mathbf{B.8}}$$

Using the usual Green's function technique, this equation can be inverted. Defining AF(x) by

$$P(\mathbf{x})\,\Delta\_{\mathbb{F}}(\mathbf{x}-\mathbf{y})=\delta^{4}(\mathbf{x}-\mathbf{y}),\tag{\mathbf{B.9}}$$

we have

$$\phi\_{\mathbf{c}}(\mathbf{x}) = -\int \Delta\_{\mathbf{F}}(\mathbf{x} - \mathbf{y}) J(\mathbf{y}) \, \mathrm{d}^4 \mathbf{y} \,. \tag{\text{B.10}}$$

It is easy to see that for the *P(x)* given in (B.7), we have

$$\text{Line } F(\lambda) \text{ given in (B.1), we have}$$

$$\Delta\_{\mathbf{F}}(\mathbf{x}) = \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\mathbf{e}^{-\mathbf{k}\cdot\mathbf{x}} \,\Delta\_{\mathbf{F}}(k) \tag{\mathbf{B.11}}$$

with

$$\Delta\_{\rm F}(k) = \frac{1}{k^2 - \mu^2 + i\varepsilon} \cdot \tag{B.12}$$

We now change the integration variable in eqn (B.6) from *<f>(x)* to *</>'(x)*  defined by

$$
\phi(\mathbf{x}) = \phi\_c(\mathbf{x}) + \phi'(\mathbf{x}).\tag{\mathbf{B.13}}
$$

Then (B.6) becomes, as in the case of usual Gaussian integrations (see eqns (l.49) and (1.81))

$$\begin{aligned} \mathcal{W}\_0[J] &= N \exp\left\{-\mathrm{i} \int \mathbb{E}^\mathrm{d} J(\mathbf{x}) \, \Delta\_\mathbf{F}(\mathbf{x} - \mathbf{y}) J(\mathbf{y}) \right\} \, \mathrm{d}^4 \mathbf{x} \, \mathrm{d}^4 \mathbf{y} \end{aligned} \tag{\text{B.14}}$$

where the normalization factor

$$N = \int \mathbb{I}\{\mathbf{d}\phi'\} \exp\left\{\mathrm{i}\int \frac{1}{2}\phi'(\mathbf{x})P(\mathbf{x})\phi'(\mathbf{x})\,\mathrm{d}^4\mathbf{x}\right\}\tag{\text{B.15}}$$

is independent of the source function *J(x)* and is irrelevant for calculating the connected Green's function. The generating functional is then given by

$$W[J] = N \exp\left\{ \mathrm{i} \int \mathrm{d}^4 x \left[ \mathcal{L}\_1 \left( -\mathrm{i} \frac{\delta}{\delta J} \right) \right] \right\} \exp\left\{ -\frac{\mathrm{i}}{2} \int \mathrm{d}^4 x\_1 \mathrm{d}^4 x\_2 J(x\_1) \right\} $$

$$ \times \Delta\_\mathrm{F}(\mathbf{x}\_1 - \mathbf{x}\_2) J(\mathbf{x}\_2) \right\} \tag{\text{B.16}} $$

and the Green's function can be obtained from

$$<\langle 0|T(\phi(\mathbf{x}\_1)\dots\phi(\mathbf{x}\_n))|0\rangle = (-\mathbf{i})^n \left. \frac{\delta^n W[J]}{\delta J(\mathbf{x}\_1)\dots\delta J(\mathbf{x}\_n)} \right|\_{J=0}.\tag{\mathbf{B.17}}$$

Using the rule for functional differentiation,

$$\begin{aligned} \text{For recurrenceal differences,} \\ \frac{\delta J(\mathbf{y})}{\delta J(\mathbf{x})} = \frac{\delta}{\delta J(\mathbf{x})} \int \delta^4(\mathbf{y} - \mathbf{x}) J(\mathbf{x}) \, \mathbf{d}^4 \mathbf{x} = \delta^4(\mathbf{x} - \mathbf{y}) \end{aligned} \tag{\mathbf{B.18}}$$

it is not difficult to see that the functional differentiation in eqns (B.16) and (B.17) will reproduce Wick's theorem.

The connected Green's functions can be obtained by differentiating the ln *W[J]* functional (eqn (1.76))

$$\left. \text{i"} \frac{\delta^\bullet \ln \mathcal{W}[J]}{\delta J(\boldsymbol{\chi}\_1) \dots \delta J(\boldsymbol{\chi}\_n)} \right|\_{J=0} = G^{(\boldsymbol{\eta})}(\boldsymbol{\chi}\_1 \dots \boldsymbol{\chi}\_n) = \langle 0 | T(\phi(\boldsymbol{\chi}\_1) \dots \phi(\boldsymbol{\chi}\_n)) | 0 \rangle\_{\text{conn}}. \tag{B.19}$$

The free propagator, a two-point function, zeroth-order in the coupling *A.,*  can be obtained from ln W0 [ *J]* as

$$<\langle 0|T(\phi(\mathbf{x})\phi(\mathbf{y}))|0\rangle\_{\text{free}} = \left. \mathrm{i}^2 \frac{\delta^2 \ln|W\_0 \mathbb{I} J|}{\delta J(\mathbf{x}) \, \delta J(\mathbf{y})} \right|\_{J=0} = \mathrm{i}\Delta\_\mathbf{F}(\mathbf{x} - \mathbf{y}).\tag{B.20}$$

To get the basic vertex in A.cf>4 theory, we take the four-point function G<4>(x1 ... x4) and keep terms lowest order in *A..* After some algebra we obtain

$$G^{(4)}(\mathbf{x}\_1 \dots \mathbf{x}\_4) = -\mathbf{i}\lambda \int \mathbf{i} \Delta\_\mathbf{F}(\mathbf{x}\_1 - \mathbf{x}) \mathbf{i} \Delta\_\mathbf{F}(\mathbf{x}\_2 - \mathbf{x})$$

$$\times \mathbf{i} \Delta\_\mathbf{F}(\mathbf{x}\_3 - \mathbf{x}) \mathbf{i} \Delta\_\mathbf{F}(\mathbf{x}\_4 - \mathbf{x}) \,\mathrm{d}^4 \mathbf{x} \tag{B.21}$$

or, in momentum space,

$$G^{(4)}(k\_1 \dots k\_4) = \mathrm{i}\Delta\_\mathrm{F}(k\_1)\mathrm{i}\Delta\_\mathrm{F}(k\_2)\mathrm{i}\Delta\_\mathrm{F}(k\_3)\mathrm{i}\Delta\_\mathrm{F}(k\_4)(-\mathrm{i}\lambda) \tag{B.22}$$

where

$$\begin{aligned} (2\pi)^4 \,\delta^4(k\_1 + k\_2 + k\_3 + k\_4) G^{(4)}(k\_1 \dots k\_4) \\ &= \int \prod\_{i=1}^4 \,\mathrm{d}^4 \mathbf{x}\_i \,\mathrm{e}^{-i\mathbf{k}\_i \mathbf{x}\_i} G^{(4)}(\mathbf{x}\_1 \dots \mathbf{x}\_4) . \end{aligned} \tag{\text{B.23}}$$

After removing the propagators for the external lines, the four-point lPI function (the vertex) is

$$
\Gamma^{(4)}(k\_1, k\_2, k\_3, k\_4) = -\mathbf{i}\lambda.\tag{\text{B.24}}
$$

To summarize, in A.cf> 4 theory, the propagator (B.12) and the vertex (B.24) are given by

$$\begin{array}{c} \begin{array}{c} --\stackrel{k}{\rightarrow} ---\stackrel{l}{\ \left( \begin{array}{c} k \end{array} \right)} \\ \end{array} \end{array} \tag{B.25}$$

$$\left(\times\right)^{-\mu}\tag{B.26}$$

Note that the propagator in momentum space is just the Fourier transform of the inverse of the operator *P(x)* which appears in the quadratic term of the

free Lagrangian (B.6). The vertex is just the coefficient of the interaction term multiplied by the factor i and by the number of permutations of identical particles in the interaction term.

#### **Generalization**

To generalize the above result to the case of more complicated interactions, consider the Lagrangian

$$\mathcal{L}^{\varphi}(\mathbf{x}) = \frac{1}{2} \phi\_i(\mathbf{x}) P\_{ij} \phi\_j(\mathbf{x}) + \chi\_i^{\star}(\mathbf{x}) V\_{ij}(\mathbf{x}) \chi\_j(\mathbf{x}) + \psi\_i(\mathbf{x}) X\_{ij}(\mathbf{x}) \psi\_j(\mathbf{x})$$

$$+ \mathcal{R}\_l(\phi, \chi, \chi^\star, \psi, \bar{\psi}) \tag{\text{B.27}}$$

where </J;(X;) denotes a set of real (complex) boson fields that may be scalar or vector fields and t/J; the set of fermion fields. The index *i* stands for any spinor, Lorentz, isospin, etc. index. *P, V,* and *X* are matrix operators that may contain derivatives and must have an inverse. *X* and *V* are taken to be hermitian operators while *P* is taken to be a real symmetric operator

$$X^\dagger = X, \qquad V^\dagger = V, \qquad \text{and } P^\dagger = P. \tag{\text{B.28}}$$

It is understood that they contain the *ie* factor so that the path integral will be convergent. A general term in .P1(x) has the explicit form:

*.P,* = *I* d4x1 d4x2 ... °'•' ,· ,· ,· ,· *(x;x1,X2 ... Xm ... Xn ... XP"'Xq ... )* **1**• • • Ill••• *n* • • • *p* • • • *q•* • • x l{i;,(X1) ... *t/!;,,,(Xm) ... </J;,,(Xn) ... ... X;iXq) ...*  (B.29)

Define the inverses of *P, V, X* by

$$
\sum\_{j} P\_{ij}(\mathbf{x}) P\_{jl}^{-1}(\mathbf{x} - \boldsymbol{\nu}) = \delta\_{ll} \,\delta^4(\mathbf{x} - \boldsymbol{\nu})
$$

$$
\sum\_{j} V\_{ij}(\mathbf{x}) V\_{jl}^{-1}(\mathbf{x} - \boldsymbol{\nu}) = \delta\_{ll} \,\delta^4(\mathbf{x} - \boldsymbol{\nu})
$$

$$
\sum\_{j} X\_{ij}(\mathbf{x}) X\_{jl}^{-1}(\mathbf{x} - \boldsymbol{\nu}) = \delta\_{ll} \,\delta^4(\mathbf{x} - \boldsymbol{\nu})\tag{B.30}
$$

and their Fourier transforms

$$P\_{ij}^{-1}(\mathbf{x}) = \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x}} \,\tilde{\mathcal{P}}\_{ij}^{-1}(k)$$

$$\begin{split} V\_{ij}^{-1}(\mathbf{x}) &= \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x}} \,\tilde{\mathcal{V}}\_{ij}^{-1}(k) \\ X\_{ij}^{-1}(\mathbf{x}) &= \int \frac{\mathbf{d}^4 k}{(2\pi)^4} \,\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x}} \,\tilde{X}\_{ij}^{-1}(k) . \end{split} \tag{\text{B.31}}$$

Then the propagators are given by

. "' *x j--+- M(k);i* = *I* d 4 x eik ·x (01 T(c/>;(x)cf>j(O))JO) = i[.P- 1(k)]ii *,M(k);j* = *I* d4x eik-x(OI *T(xr(x)xj(O))IO)* = i[ *v- 1(k)]ij SF(k)ii* = *I* d4 x eik·x(OJT(r]/;(x)t/li(O))IO) = i[X- 1(k)]ii. (B.32)

For the vertex we define

$$\begin{aligned} \left(\mathbf{a}\_{l\_1l\_2\dots}\right)\left(\mathbf{x};\mathbf{x}\_1,\mathbf{x}\_2\dots\right) &= \int \frac{\mathbf{d}^4k\_1}{(2\pi)^4} \frac{\mathbf{d}^4k\_2}{(2\pi)^4}\dots\mathbf{e}^{\left[\mathbf{k}\_1(\mathbf{x}-\mathbf{x}\_1)+\mathbf{k}\_2(\mathbf{x}-\mathbf{x}\_2)+\dots\right]} \\ &\times \tilde{a}\_{l\_1l\_2\dots}(k\_1,k\_2\dots). \end{aligned} \tag{\mathbf{B.33}}$$

*a.* contains a factor *ikiµ* for every derivative *o/oxiµ* acting on a field with argument *xi.* The vertex is then given by

$$I(k\_1, k\_2, \dots) = \operatorname{i} \sum\_{\{1 \ldots m-1\}} \sum\_{\{m \ldots n-1\}} \sum\_{\{n \ldots p-1\}} \sum\_{\{p \ldots q-1\}} \sum\_{\{q \ldots q-1\}} \dots$$

$$\times (-1)^p \tilde{a}\_{l\_1, l\_2}(k\_1, k\_2 \ldots). \tag{B.34}$$

The summations are over all permutations of indices and momenta as indicated in Fig. (B. l ). The momenta are taken to flow inward. Any field *ill*

(or *x\*)* corresponds to a line with an arrow pointing out; a field t/I (or *x)* has an oppositely directed arrow. The *<P* fields are represented by undirected lines. The factor ( - l *l* is only of importance if several fermion fields occur. There is a factor of ( - 1) for every permutation exchanging two fermion fields.

For example, in ;.q,4 theory the interaction term can be written

$$-\frac{\lambda}{4!} \phi^4(\mathbf{x}) = -\frac{\lambda}{4!} \int \mathrm{d}^4 \mathbf{x}\_1 \dots \,\mathrm{d}^4 \mathbf{x}\_4 \,\delta^4(\mathbf{x} - \mathbf{x}\_1) \,\delta^4(\mathbf{x} - \mathbf{x}\_2)$$

$$\times \,\delta^4(\mathbf{x} - \mathbf{x}\_3) \,\delta^4(\mathbf{x} - \mathbf{x}\_4) \phi(\mathbf{x}\_1) \phi(\mathbf{x}\_2) \phi(\mathbf{x}\_3) \phi(\mathbf{x}\_4) \quad \text{(B.35)}$$

*Appendix B Feynman rules* 503

Then

$$(\mathbf{x}; \mathbf{x}\_1 \dots \mathbf{x}\_4) = -\frac{\lambda}{4!} \delta^4(\mathbf{x} - \mathbf{x}\_1) \dots \delta^4(\mathbf{x} - \mathbf{x}\_4) \tag{\text{B.36}}$$

and

$$
\tilde{a}(k\_1 \dots k\_4) = -\frac{\lambda}{4!} \cdot \tag{\text{B.37}}
$$

The vertex is given by

$$I(k\_1\ldots k\_4) = \operatorname{i} \sum\_{\{1,2,3,4\}} \tilde{\mathfrak{a}}(k\_1, k\_2, k\_3, k\_4)$$

$$= \operatorname{i} \sum\_{\{1,\ldots,4\}} (-\lambda \!/ \!/ \!\!/ \!\!/ ) = -\operatorname{i} \lambda. \tag{\text{B.38}}$$

This agrees with the result (B.24) obtained before.

### **General Feynman rules**

Given the propagator and vertices we can now write down the Feynman rule for the computation of Green's functions.

(a) Draw all topologically distinctive, connected diagrams at desired order;

(b) In each diagram, attach a propagator to each internal line

\_\_ \_,,, \_\_ *k k 2*i 2 . for a spin-0 boson - *µ +le (* i . ) for a spin-1/2 fermion *P \P-m-Ieap*

Spin-1 boson propagators will depend on the theory;

(c) To each vertex, assign a vertex function given in (B.34), derived from the relevant term in the interaction Lagrangian;

(d) For each internal momentum *knot* fixed by the momentum conserv- . . . ., *I* d4k at10n at vertices, give a 1actor -)4 · (2n

(e) Multiply the contribution for each diagram by

(i) a factor ( -1) for each closed fermion loop:

(ii) a factor ( -1) between graphs which differ from each other only by an interchange of two identical external fermion lines;

(iii) a symmetry factor s- 1 with

$$\mathcal{S} = g \prod\_{\mathfrak{n} = 2, 3 \dots} 2^{\mathfrak{f}} (n!)^{\mathfrak{a}\_{\mathfrak{n}}}$$

where *an* is the number of *pairs* of vertices connected by *n* identical selfconjugate lines, *f3* is the number oflines connecting a vertex with itself, and *g* is the number of permutations of vertices which leave the diagram unchanged with fixed external lines.

For example,

$$\begin{array}{c} \longrightarrow \stackrel{\frown}{\longrightarrow} \longrightarrow \qquad g=1, \mathfrak{a}\_{2}=1, \beta=0, \text{ and } \mathcal{S}=2! \\\\ \longleftrightarrow \stackrel{\frown}{\longrightarrow} \qquad g=1, \mathfrak{a}\_{3}=1, \beta=0, \text{ and } \mathcal{S}=3! \\\\ \longrightarrow \stackrel{\frown}{\longrightarrow} \qquad g=1, \mathfrak{a}\_{n}=0, \beta=1, \text{ and } \mathcal{S}=2 \\\\ \longrightarrow \quad g=2, \mathfrak{a}\_{2}=1, \ \beta=0, \text{ and } \mathcal{S}=4 \end{array}$$

(f) The proper (or one-particle irreducible) Green's function r<•>(p1 ... *p.)*  comes from the one-particle irreducible (lPI) diagrams

(i) For the connected Green's function a<•>(p1 ... *p.),* attach propagators for the external lines;

(ii) For the scattering amplitude *T(p 1*... *p.),* put all the external lines on their mass-shell, i.e. *pf* = *ml* and provide external fermion lines with spinors: *u(p)* [or *v(p)]* for fermions [or antifermions] entering with momentum *p; ii(p)* [or *v(p)]* for fermions [or antifermions] leaving with momentum *p.* Provide external vector bosons with polarization vectors: *aµ(k,* Jc) [or a!(k, Jc)] for the vector boson entering [or leaving] with momentum *k.*

# **Summary of** J..<j>4 , **Yukawa, QED and OCD propagators and vertices**

(a) A.<1>4**and Yukawa theory of scalar-fermion interactions.**

$$\mathcal{L}^{\rho} = \frac{1}{2} (\partial\_{\mu}\phi)^{2} - \frac{\mu^{2}}{2}\phi^{2} - \frac{\lambda}{4!} \phi^{4} + \overline{\psi}(\mathrm{i}\gamma \cdot \hat{\boldsymbol{\sigma}} - \boldsymbol{m})\psi + \mathrm{i}g\overline{\psi}\gamma\_{5}\psi\phi^{4}$$

scalar propagator: ---->----

$$\begin{array}{llll}\text{comm} & \text{proagnostic} & \text{a} & \text{if } \text{"} - \mu^2 + \text{ic} \\\\ \text{fermion propagator:} & \begin{array}{ll} \rho \xrightarrow{\begin{array}{l} \text{---} \rho \text{---} \\ \rho \end{array}} & \text{i} \, \text{S}\_{\mathbb{F}}(p)\_{a\#} = \begin{pmatrix} \text{i} \\ \text{p-} \text{-} \text{m} + \text{ic} \end{pmatrix}\_{a} \end{array} \end{array}$$

vertices:

$$\lambda\_{\mu\_{(\lambda)}} \lesssim\_{\mu} \sum\_{\kappa} \lambda\_{\kappa}$$

**(b) QED.**

$$\mathcal{Q}' = -\frac{1}{4} \left( \partial\_{\mu} A\_{\nu} - \partial\_{\nu} A\_{\mu} \right)^{2} - \frac{1}{2\xi} \left( \partial\_{\mu} A^{\mu} \right)^{2} + \overline{\psi} (\text{i}\gamma \cdot \partial - e\gamma \cdot A - m) \psi$$

Fermion propagator: *P*

*Appendix B Feynman rules* 505

$$\begin{aligned} \text{Photon propagator:} & \mathop{\text{ev}\nolimits \mathop{\text{ev}\nolimits \mathop{\text{span}}}}\_{k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{iv}}\_{\text{-}k} \mathop{\text{v}}\_{\text{-}k} \mathop{\text{v}}\_{\text{-}k} \mathop{\text{v}}\_{\text{-}k} \\ & \qquad \begin{aligned} \xi &= 1 & \text{Feymana gauge} \\ \xi &= 0 & \text{Landau gauge} \\ \xi &= 0 & \text{Landau gauge} \\ \xi &= \xi\_{\text{-}el\text{'}\text{'}\text{'}} \end{aligned} \end{aligned}$$

(c) **Scalar QED.**

$$\begin{split} \mathcal{L} &= -\frac{1}{4} (\partial\_{\mu} \mathcal{A}\_{\nu} - \partial\_{\nu} \mathcal{A}\_{\mu})^{2} - \frac{1}{2\xi} (\partial\_{\mu} \mathcal{A}^{\mu})^{2} \\ &+ [ (\partial\_{\mu} + \mathrm{i} e \mathcal{A}\_{\mu}) \phi ]^{\dagger} [ (\partial^{\mu} + \mathrm{i} e \mathcal{A}^{\mu}) \phi ] - \mu^{2} \phi^{\dagger} \phi - \frac{\lambda}{4} (\phi^{\dagger} \phi)^{2} . \end{split}$$

Scalar propagator: --7-- *k*

*e* = I Feynman gauge *e* = 0 Landau gauge

Vertices:

**(d) Non-Abelian gauge theory.**

$$\begin{split} \mathcal{L}^{\mathfrak{r}} &= -\frac{1}{4} \left( \hat{\boldsymbol{\upsilon}}\_{\mu} \boldsymbol{A}\_{\mathrm{v}}^{a} - \hat{\boldsymbol{\upsilon}}\_{\nu} \boldsymbol{A}\_{\mu}^{a} + g \boldsymbol{C}^{abc} \boldsymbol{A}\_{\mu}^{b} \boldsymbol{A}\_{\mathrm{v}}^{c} \right) (\hat{\boldsymbol{\upsilon}}^{\mu} \boldsymbol{A}^{a\boldsymbol{v}} - \hat{\boldsymbol{\upsilon}}^{\boldsymbol{\upsilon}} \boldsymbol{A}^{a\boldsymbol{\mu}} + g \boldsymbol{C}^{abc} \boldsymbol{A}^{d\boldsymbol{\mu}} \boldsymbol{A}^{a\boldsymbol{\upsilon}}) \\ &- \frac{1}{2\xi} \left( \hat{\boldsymbol{\upsilon}}\_{\mu} \boldsymbol{A}^{a\mu} \right)^{2} - \bar{\eta}^{a} \left \hat{\boldsymbol{\upsilon}}\_{\mu} (\hat{\boldsymbol{\upsilon}}^{\mu} \boldsymbol{\delta}^{\boldsymbol{\alpha}} - g \boldsymbol{C}^{\mathrm{abc}} \boldsymbol{A}^{b\boldsymbol{\mu}}) \eta^{c} \\ &+ \bar{\psi} \left[ \mathrm{i} \boldsymbol{\upsilon}\_{\mu} (\hat{\boldsymbol{\upsilon}}^{\mu} - \mathrm{i} g \boldsymbol{A}^{a\boldsymbol{\mu}} \boldsymbol{T}^{a}) - m \right] \psi \\ &+ \left[ (\hat{\boldsymbol{\upsilon}}^{\mu} - \mathrm{i} g \boldsymbol{A}^{a\mu} \boldsymbol{R}^{a}) \phi \right] \mathrm{I} \left[ \left( \boldsymbol{\partial}\_{\mu} - \mathrm{i} g \boldsymbol{A}^{b}\_{\mu} \boldsymbol{R}^{b} \right) \phi \right] - m\_{\phi}^{2} \phi^{\dagger} \phi - \frac{\lambda}{4} \left( \phi^{\dagger} \phi \right)^{2} . \end{split}$$

Vector meson propagator: *b a k* µ *j*  · . *ab* -I *u k /k2* IDF *(k)µv* =*-k2* . [gµv + (e - *l)kµ* v ] *+IC e* = I Feynman-'t Hooft gauge *e* = 0 Landau gauge . *ij lu* Fermion propagator: ---'>--- *1SF(P)ap* = 0 . · ) p *p a -m+rn ap*

For QCD we have

with *a, b, c* = 1, 2, ... , 8

and the representation matrices

$$\begin{aligned} (T^\bullet)\_{ij} &= (\lambda^\bullet / 2)\_{ij} \qquad \text{with } i, j = 1, \text{ 2, 3} \\ (R^\bullet)\_{lm} &= 0 \end{aligned}$$

# **Ri; gauge Feynman rules for the standard SU(2) x U(1) theory**

Since these rules are slightly more complicated we provide some steps in their derivation.

#### **The gauge-fixing and FP ghost terms in the effective Lagrangian**

After spontaneous symmetry breaking the covariant derivatives of the scalar

#### *Appendix B Feynman rules* 507

field give rise to mixing terms between vector bosons and scalar particles

$$\begin{split} (\mathbf{D}\_{\mu}\boldsymbol{\phi})^{\dagger}(\mathbf{D}^{\mu}\boldsymbol{\phi}) &= \left[ \left( \boldsymbol{\partial}\_{\mu} - \mathrm{i}\boldsymbol{g}\,\frac{\mathbf{r}}{2}\cdot\mathbf{A}\_{\mu} - \mathrm{i}\boldsymbol{g}\,'\frac{\mathbf{B}\_{\mu}}{2} \right) (\boldsymbol{\phi}\,' + \langle\boldsymbol{\phi}\rangle\_{0}) \right]^{\dagger} \\ &\times \left[ \left( \boldsymbol{\partial}^{\mu} - \mathrm{i}\boldsymbol{g}\,\frac{\mathbf{r}}{2}\cdot\mathbf{A}^{\mu} - \mathrm{i}\boldsymbol{g}\,'\frac{\mathbf{B}^{\mu}}{2} \right) (\boldsymbol{\phi}\,' + \langle\boldsymbol{\phi}\rangle\_{0}) \right] \\ &= \mathrm{i}g\mathbf{A}^{\mu} \left[ -\boldsymbol{\partial}\_{\mu}\boldsymbol{\phi}\,'^{\dagger}\frac{\mathbf{r}}{2} \langle\boldsymbol{\phi}\rangle\_{0} + \langle\boldsymbol{\phi}\boldsymbol{\phi}\rangle\_{0}\frac{\mathbf{r}}{2}\,\partial\_{\mu}\boldsymbol{\phi}\,' \right] \\ &+ \mathrm{i}g\mathbf{'}\,\frac{\mathbf{B}^{\mu}}{2} \left[ \langle\boldsymbol{\phi}\boldsymbol{\phi}\,'\rangle\_{0}\,\partial\_{\mu}\boldsymbol{\phi}\,' - \partial\_{\mu}\boldsymbol{\phi}\,'\rangle\langle\boldsymbol{\phi}\rangle\_{0} \right] + \dots \,\quad \text{(B.39)} \end{split}$$

For practical calculations, it is more convenient to cancel such mixing terms. For this purpose we choose the gauge functions (9.27) to be

$$f\_l = \partial\_\mu A^\mu\_l + \mathrm{i}g\xi \Big(\phi^{\prime\prime} \frac{\mathsf{r}\_l}{2} \langle \phi \rangle\_0 - \langle \phi^\dagger \rangle\_0 \frac{\mathsf{r}\_l}{2} \phi^\prime \Big) \qquad \text{for } SU(2)$$

$$f = \partial\_\mu B^\mu + \mathrm{i}g^\prime \frac{\xi}{2} \Big(\phi^{\prime\prime} \langle \phi \rangle\_0 - \langle \phi^\prime \rangle\_0 \phi^\prime \Big) \qquad \text{for } U(1) \qquad (\text{B.40})$$

so that the gauge-fixing term (9.68) is of the form

$$\begin{split} \mathcal{L}\_{\mathfrak{g}^{\dagger}} &= -\frac{1}{2\xi} \Bigg[ \partial\_{\mu} \mathbf{A}^{\mu} + \mathrm{i}g\xi \Big( \phi^{\prime \dagger} \frac{\mathfrak{r}}{2} \langle \phi \rangle\_{0} - \langle \phi^{\flat} \rangle\_{0} \frac{\mathfrak{r}}{2} \phi^{\prime} \Big) \Bigg]^{2} \\ &- \frac{1}{2\tilde{\xi}} \Bigg[ \partial\_{\mu} \mathcal{B}^{\mu} + \frac{\mathrm{i}g^{\prime}\xi}{2} \Big( \phi^{\prime \dagger} \langle \phi \rangle\_{0} - \langle \phi^{\flat} \rangle\_{0} \phi^{\prime} \Big) \Bigg]^{2} . \end{split} \tag{\text{B.41}}$$

To calculate the Faddeev-Popov ghost from these gauge conditions, we make the infinitesimal SU(2) gauge transformation, with gauge function u(x),

$$\begin{aligned} \delta A^i\_\mu(\mathbf{x}) &= e^{ijk} u^j(\mathbf{x}) A^k\_\mu(\mathbf{x}) - \frac{1}{g} \partial\_\mu u^i(\mathbf{x}) \\\\ \delta \phi(\mathbf{x}) &= -\mathrm{i} \, \frac{\mathbf{r}}{2} \cdot \mathbf{u}(\mathbf{x}) \phi(\mathbf{x}) \\\\ \delta \phi' &= -\mathrm{i} \, \frac{\mathbf{r}}{2} \cdot \mathbf{u}(\phi' + \langle \phi \rangle\_0). \end{aligned}$$

or

Then

$$\begin{split} \delta f\_{l} &= \mathcal{O}\_{\mu} \left[ \mathfrak{c}^{ijk} \mathfrak{u}^{j} A^{k}\_{\mu} - \frac{1}{g} (\mathcal{O}\_{\mu} \mathfrak{u}^{i}) \right] + \mathrm{i}g\zeta \Big[ (\phi^{\prime \dagger} + \langle \phi^{\dagger} \rangle\_{0}) \mathrm{i} \, \frac{\mathfrak{c}}{2} \cdot \mathfrak{u} \, \frac{\tau\_{i}}{2} \, \langle \phi \rangle\_{0} \, \Big] \\ &+ \langle \phi^{\prime} \rangle\_{0} \, \frac{\tau\_{i}}{2} \, \mathrm{i} \, \frac{\mathfrak{c}}{2} \cdot \mathfrak{u} (\phi^{\prime} + \langle \phi \rangle\_{0}) \, \Big] \\ \delta f &= \mathrm{i} \, \frac{g^{\prime}}{2} \, \xi \Big[ (\phi^{\prime \dagger} + \langle \phi^{\dagger} \rangle\_{0}) \mathrm{i} \, \frac{\mathfrak{c}}{2} \cdot \mathfrak{u} \langle \phi \rangle\_{0} + \langle \phi^{\dagger} \rangle\_{0} \mathrm{i} \, \frac{\mathfrak{c}}{2} \cdot \mathfrak{u} (\phi^{\prime} + \langle \phi \rangle\_{0}) \Big]. \end{split}$$

(B.42)

$$\begin{split} \delta f\_{i} &= \left\{ -\frac{1}{g} \left\langle \partial^{\mu} \!\! \! \! \! / \delta\_{ij} \ \! \! \! / \mu - g \varepsilon\_{ijk} \mathcal{A}\_{\mu}^{k} \!\! \right\} \\ &- g \xi \bigg[ \langle \phi^{\dagger} \rangle\_{0} \frac{1}{2} \langle \phi \rangle\_{0} \,\delta\_{ij} + \phi^{\prime \dagger} \frac{\tau\_{j} \tau\_{i}}{4} \langle \phi \rangle\_{0} + \langle \phi^{\dagger} \rangle\_{0} \frac{\tau\_{i} \tau\_{j}}{4} \phi^{\prime} \bigg] \right\} u\_{j} \\ \delta f &= -\frac{g^{\prime}}{2} \xi \bigg[ \langle \phi^{\dagger} \rangle\_{0} \tau\_{j} \langle \phi \rangle\_{0} + \phi^{\prime \dagger} \frac{\tau\_{j}}{2} \langle \phi \rangle\_{0} + \langle \phi^{\dagger} \rangle\_{0} \frac{\tau\_{j}}{2} \phi^{\prime} \bigg] u\_{j} . \end{split} \tag{B.43}$$

Similarly under the infinitesimal U(l) gauge transformation, with gauge function oc(x ),

$$
\delta \mathcal{B}\_{\mu} = -\frac{1}{g'} \partial\_{\mu} \alpha
$$

$$
\delta \phi' = -\mathrm{i}\frac{\alpha}{2} (\phi' + \langle \phi \rangle\_0),
\tag{B.44}
$$

we have

$$
\delta f\_i = -\frac{g}{4} \,\xi \, [2\langle\phi^\dagger\rangle\_0 \tau\_i \langle\phi\rangle\_0 + \phi^{\prime\dagger} \tau\_i \langle\phi\rangle\_0
$$

$$
+ \langle\phi^\dagger\rangle\_0 \tau\_i \phi^\prime\rangle \alpha
$$

$$
\delta f = -\frac{1}{g^\prime} \Big\{\partial^2 + \frac{g^\prime}{4} \,\xi \, [2\langle\phi^\dagger\rangle\_0 \langle\phi\rangle\_0 + \phi^{\prime\dagger} \langle\phi\rangle\_0]
$$

$$
+ \langle\phi^\dagger\rangle\_0 \phi^\prime\rangle \Big\} \alpha. \tag{B.45}
$$

We can combine (B.43) and (B.45) in matrix notation

$$\begin{array}{ll} \text{(D.4-x) and (D.4-y) in maini notation} \\\\ \delta \binom{f\_l(\mathbf{x})}{f(\mathbf{x})} = \int \mathbf{d}^4 \boldsymbol{\upchi} \begin{pmatrix} M\_f(\mathbf{x}, \boldsymbol{\upchi})\_{lj} & M\_f(\mathbf{x}, \boldsymbol{\upchi})\_i \\ M\_f(\mathbf{x}, \boldsymbol{\upchi})\_j & M\_f(\mathbf{x}, \boldsymbol{\upchi}) \end{pmatrix} \begin{pmatrix} \boldsymbol{u}\_j(\mathbf{y})/g \\ \boldsymbol{a}(\mathbf{y})/g' \end{pmatrix} \end{array} \tag{B.46}$$

where

$$M\_f(\mathbf{x}, \mathbf{y})\_{ij} = -\left\{\partial^\mathbf{s} \left[\delta\_{ij}\,\delta\_\mu - g\varepsilon\_{ijk}A^k\_\mu\right] + g^2\xi \left[\frac{|\langle\phi\rangle\langle\phi|^2}{2}\delta\_{ij} + \phi^{,\uparrow}\frac{\tau\_j\tau\_i}{4}\langle\phi\rangle\_0\right] \right. \tag{8.47}$$

$$+ \langle\phi^{,\uparrow}\rangle\_0 \frac{\tau\_i\tau\_j}{4} \langle\phi^{,\uparrow}\rangle\Big|\int \delta^4(\mathbf{x} - \mathbf{y})$$

$$M\_f(\mathbf{x}, \mathbf{y})\_j = \frac{-gg'}{2}\xi \left[\langle\phi^{,\uparrow}\rangle\_0\tau\_j\langle\phi\rangle\_0 + \phi^{,\uparrow}\frac{\tau\_j}{2}\langle\phi\rangle\_0 + \langle\phi^{,\uparrow}\rangle\_0\frac{\tau\_j}{2}\phi^{,\uparrow}\right]$$

$$\times \delta^4(\mathbf{x} - \mathbf{y})$$

$$M\_f(\mathbf{x}, \mathbf{y}) = -\left\{\partial^2 + \frac{g'\tilde{\xi}}{4}[2\langle\langle\phi\rangle\_0|^2 + \phi^{,\uparrow}\langle\phi\rangle\_0 + \langle\phi^{,\uparrow}\rangle\_0\phi^{,\uparrow}]\right\}\delta^4(\mathbf{x} - \mathbf{y}). \tag{8.47}$$

or

#### *Appendix B Feynman rules*

The FP ghost-field Lagrangian (9.69) is then given by

$$\begin{aligned} \text{IF } \mathsf{ghost-pred} \text{ Lagargian} & \text{ } (\mathsf{g}, \mathsf{b}) \text{ is then given by} \\ \mathcal{L}\_{\mathsf{FPG}} &= \int \mathsf{d}^{4} \mathsf{x} \, \mathsf{d}^{4} \mathsf{y} (\omega\_{i}^{\dagger}(\mathsf{x}), \mathsf{y}^{\dagger}(\mathsf{x})) \begin{pmatrix} M\_{f}(\mathsf{x}, \mathsf{y})\_{lj} & M\_{f}(\mathsf{x}, \mathsf{y})\_{l} \\ M\_{f}(\mathsf{x}, \mathsf{y})\_{j} & M\_{f}(\mathsf{x}, \mathsf{y}) \end{pmatrix} \begin{pmatrix} \alpha\_{j}(\mathsf{y}) \\ \chi(\mathsf{y}) \end{pmatrix} . \end{aligned} \tag{B.48}$$

#### **Propagators and vertices for bosons and FP ghosts**

Define the physical vector bosons as

$$\begin{aligned} W^{\pm}\_{\mu} &= \frac{1}{\sqrt{2}} \left( \mathcal{A}^{1}\_{\mu} \mp \mathrm{i} \mathcal{A}^{2}\_{\mu} \right) \\ Z\_{\mu} &= \cos \theta\_{\mathrm{W}} \mathcal{A}^{3}\_{\mu} - \sin \theta\_{\mathrm{W}} \mathcal{B}\_{\mu} \\ \mathcal{A}\_{\mu} &= \sin \theta\_{\mathrm{W}} \mathcal{A}^{3}\_{\mu} + \cos \theta\_{\mathrm{W}} \mathcal{B}\_{\mu} \end{aligned}$$

and write the scalar mesons as

$$\phi' = \begin{pmatrix} \phi^+ \\ \frac{\phi\_1 + \mathrm{i}\phi\_2}{\sqrt{2}} \end{pmatrix}$$

For the ghost fields we can define similar combinations

$$\begin{aligned} \boldsymbol{\phi}^{\pm} &= \frac{1}{\sqrt{2}} \left( \boldsymbol{\phi}\_1 \mp \mathbf{i} \boldsymbol{\phi}\_2 \right) \\ \boldsymbol{\phi}\_{\mathbf{Z}} &= \cos \theta\_{\mathbf{W}} \boldsymbol{\phi}\_3 - \sin \theta\_{\mathbf{W}} \boldsymbol{\chi} \\ \boldsymbol{\phi}\_{\mathbf{\gamma}} &= \sin \theta\_{\mathbf{W}} \boldsymbol{\phi}\_3 + \cos \theta\_{\mathbf{W}} \boldsymbol{\chi} \end{aligned}$$

With these definitions, we can easily work out the propagators from the quadratic part of the Lagrangian 2 1 + 2 3 +!!'gr+ !EFPG in eqns (11.37), (11.59), **(B.41),** and (B.48)

$$\begin{aligned} \text{W}^{\pm} & \sim \zeta \smile \zeta \smile \zeta \smile \chi \smile \quad \frac{-i}{k^2 - M\_{\text{w}}^2 + i\varepsilon} \left[ \varrho\_{\mu\nu} + (\xi - 1)k\_{\mu}k\_{\nu}l \left(k^2 - \zeta M\_{\text{w}}^2 \right) \right] \\ \text{Z} & \sim \zeta \smile \zeta \smile \zeta \smile \quad \frac{-i}{k^2 - M\_{\text{Z}}^2 + i\varepsilon} \left[ \varrho\_{\mu\nu} + (\xi - 1)k\_{\mu}k\_{\nu}l(k^2 - \zeta M\_{\text{Z}}^2) \right] \\ \text{\pm}^{\pm} & -- --- --- \qquad \frac{i}{k^2 - \zeta M\_{\text{w}}^2 + i\varepsilon} \\ \text{\pm}\_2 & -- ---- \qquad \frac{i}{k^2 - \zeta M\_{\text{Z}}^2 + i\varepsilon} \\ \text{\pm}\_1 &--- --- --- \qquad \frac{i}{k^2 - 2\mu^2 + i\varepsilon} \\ \text{\pm}^{\pm} & --- \cdots &--- \qquad \frac{-i}{k^2 - \zeta M\_{\text{Z}}^2 + i\varepsilon} \\ \text{\pm}\_2 &--- \cdots &--- \qquad \frac{-i}{k^2 - \zeta M\_{\text{Z}}^2 + i\varepsilon} \\ \text{\pm}\_3 &--- \cdots &--- \qquad \frac{-i}{k^2 - \zeta M\_{\text{Z}}^2 + i\varepsilon} \end{aligned}$$

where = **1** 't Hooft-Feynman gauge, = 0 Landau gauge, and = oo unitary gauge.

510 *Appendix B Feynman rules*

The boson vertices are

as well as the AZWW vertex *-iegcos Bw Sµv,J.p* with *Sµv,J.p* = *2gµvg;.pgµ;.gvp-gµpgvJ.·* In graphs below *all* charged boson lines are taken to be entering *into* the vertices.

**511**

512 *Appendix B Feynman rules*

**Inclusion of leptons and quarks**

*p*Propagator: p-m;+ie

Vertices for leptons: *l* = (e, µ, 't), v1 = (v0 , Vµ, v,)

for quarks q: p = (u, c, t), n = (d, s, b) with the CKM mixing matrix *Upn* of eqn (12.39).

# **Bibliography**

We have collected here a set of review articles and monographs pertaining to several of the general topics covered in this book. They are not a complete listing and inevitably some of the important contributions have been left out; we regret such omissions. Nevertheless it is hoped that a reader who wishes to go further will find this informal guide useful.

#### Particle physics

#### Quantum field theory

#### Feynman and path integral

#### Group theory

#### Quark model

#### Current algebra

#### Parton model and light-cone

#### Gauge theory (general)

#### *Bibliography* 515

#### QCD

#### Lattice QCD

#### Weak interactions

#### Standard model of electroweak interactions

1. Taylors, J.C. (1976). *Gauge theories of weak interactions.* Cambridge University Press, Cambridge.

#### GUT

#### Cosmology

# **References**

	-

Abers, E. S. and Lee, B. W. (1973). *Phys. Rep.* 9C, 1.

Adler, S. L. (1965a). *Phys. Rev.* 139, B1638.

- (1965b). *Phys. Rev.* 140, B736.

	-

Sch. Subnucl. Phys. 'Ettore Majorana' (ed. A. Zichichi), p. 358. Editrice Compositori, Bologna.

#### *References*

Georgi, H. and Politzer, H. D. (1974). *Phys. Rev.* D9, 416.

Ginzburg, V. L. and Landau, L. D. (1950). *J. exp/. theoret. Phys. USSR* 20, 1064.

Glashow, S. L. (1958). Ph.D. Thesis. Harvard University.

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Lipkin, H. J. (1973). *Phys. Rep. BC,* 173.

#### *References* 525

# **Index**

A0 = 0 gauge 482 Abelian group 86, 128-9; *see also* U(l) action 3 Yang-Mills theory in covariant gauges 259 lattice field theory 326-31 self-dual Euclidean gauge field 481 Adler consistency conditions 155, 161 Adler current algebra sum rule 137-41, 212-13 Adler-Bardeen theorem 179 Adler-Bell-Jackiw (ABJ) anomaly 173-82; *see also* anomaly Adler-Weisberger sum rule 158-60 Aharonov-Bohm effect 458, 474 algebra charge 130 current 130-4, 136, 139-40 SU(2) x U(I) 340-1 SU(5) 431 Altarelli-Parisi equation 312-14, 319-21 angular momentum and monopole 455-6, 471-2 anomaly 173-82; *see also* axial U(l) problem cancellation SU(3)c x SU(2)L x U(l)v 347-8 SU(5) 432 instantons 488 non-Abelian cases 179-80 non-renormalization theorem 179 and renormalizability 278 triangle diagram 176-9 anomalous dimension 74, 77, 84, 307-11, 314, 322 Appelquist-Carazzone theorem, *see* decoupling of heavy particles asymptotic freedom 84, 280-91, 322, 335 applications e+e- 296-8 *IN* 298-311, 316-22 and confinement 322-3, 335 discovery 280 in non-Abelian gauge theories 284-7 in gauge .theories with scalar mesons 289-90 paramagnetic property of Yang-Mills vacuum 287-9 atomic parity experiments 370 axial gauge 248, 254-7, 271, 317-18 polarization sum of vector particles 271 axial U(l) problem 294, 487-92; *see also* 11-mass, '1 -+ 3n, instantons, Kogut-Susskind pole, 0-vacuum axial-vector current Ward identity anomaly 173- 182; *see also* anomaly axion 493

B, *see* baryon number b-quark 122, 356

B - L 418, 443,451-2 BPH renormalization 39-45, 56, 67 BRS transformation 273-6 baryon masses 119-20, 165-6 baryon number asymmetry in the universe 444-6 baryon minus lepton number, *see* B - L conservation in QCD 292 in standard electroweak theory 355 non-conservation in GUT 442-6 fl-decays 153, 336, 373 bilocal operators 223-5, 299-300 biunitary transformation 65, 293, 357-9 Bjorken scaling, *see* scaling Bloch-Nordsieck theorem 316-17, 321 bottom, *see* b-quark c-quark, *see* charm CP symmetry B-asymmetry in the universe 445-6 electric dipole moment 385-6, 492 Kobayashi-Maskawa theory 359-61 neutral Kaon system 376-8, 383-5 phases 356, 359-61, 442 soft v. hard violation 386, 493 strong CP problem 492-3; *see also* 0-vacuum Cabibbo angle 136, 210, 337, 356, 371, 373; *see also* Kobayashi-Maskawa mixing Cabibbo-Kobayashi-Maskawa (CKM) matrix, *see* Kobayashi-Maskawa mixing Cabibbo universality 373 Callan-Gross relation 207-8 Callan-Symanzik equation 70-7 canonical commutation relations 4, 248 canonical dimension 58-9, 61, 77; *see also* anomalous dimension canonical quantization formalism 3-11, 255 Casimir operator 91, 408 charge conjugation 412, 429, 445-6, 448; *see also*  Majorana fermion charge quantization 346 grand unification 430-1 magnetic monopole 455, 469 charm 121-2, 337; *see also* GIM mechanism, Zweig rule chiral symmetry 125-68, 294, 403, 488-92 algebra 132, 136 breaking 125, 151-2, 160-8 Coleman-Weinberg mechanism, *see* radiative in-

duced spontaneous symmetry breaking collinear singularities 317; *see also* mass singularities

# colour 123-4, 291, 401 anomaly cancellation 348 e + e - hadronic total cross-section 216 *n°-+* 2y 182 fractional quark charge 431 magnetic charges of GUT monopole 474 colour flux tubes 323 composite field operators 62-6, 73, 221 mixing under renormalization 65-6, 307-10, composite Higgs particle 401-2 condensate, *see* vacuum expectation value confinement 124, 279, 322-3 and asymptotic freedom 322-3, 335 criteria 331-3 linearly rising potential 323, 333-4 Wilson loop 331-3 conserved vector current (CVC) hypothesis 136 cosmology; *see also* baryon number asymmetry, cosmological bound on neutrino masses closed, open, flat universe 415--16 critical mass density *Pc* 415-16 Hubble's law and constant 415 primodial blackbody photon radiation 415-16, 445

helium abundance 415, 445 neutrinos 416 coupling constant; *see also* asymptotic freedom, gauge field coupling, pion-nucleon coupling definition 36, 52, 59, 81-2 dependenci: on subtraction point 81-2 dimension and renormalizability 57 effective 75-6, 81-5, 295-6, 437-41 renormalization 36-7, 285 unification 437-41 counterterm BPH renormalization 40-5, 58-60 composite operator 64-6 gauge theory renormalization 267-8 Goldstone boson mass, absence of 186-7 minimal subtraction scheme 78-80 soft symmetry breaking 186-8 covariant derivative 230, 232, 233 as connection 236 SU(2) x U(l) 348 SU(5) 439 covariant gauges 257-67; *see also* Landau gauge, Lorentz gauge, *R,* gauge, 't Hooft-Feynman gauge, unitary gauge currents anomaly 480, 488-9 axial-vector 173-4, 223, 488 commutator 130, 136, 200-1, 223-5 physical electromagnetic 134, 200, 222, 340 weak charged 135-6, 204, 337, 340, 353 neutral 336, 342; flavour-conserving 353- 4, 362, 364-71; strangeness changing 337, 362-3, 371-2, 378-83 symmetry (Noether) 127-8, 134

topological 463

current algebra 125, 180, 279, 488 Adler's sum rule 137-41 soft pion theorems 151-67 Adler's consistency condition 155, 161 Adler-Weisberger sum rule 158-60 *'1* -+ *3n* problem 167, 490 Goldberger-Treiman relation 153-5, 160, 161 *n°* -+ 2y 180-2 *nN* scattering length 159 CT-terms 157-9 *nN* CT-term 161, 165-6 pseudoscalar meson masses 119, 164; Dashen sum rule 167; 11-mass problem 488-92 current commutators 130-2, 136, 201, 223-5; *see also* current algebra current conservation 127, 144, 170, 174, 201 symmetry conservation laws, *see* Noether theorem topological conservation laws 463-4 current--<:urrent interaction baryon number nonconserving current 443 charged weak current, *see* V-A interaction neutral weak current 354, 365 *dabc* symbol 163, 432 Dashen sum rule 167 decoupling of heavy particles 409, 436, 437-41 deep Euclidean limit asymptotic behaviour of Green's functions 73- 7, 81, 280 deep inelastic *IN* scattering 199-214, 218-26 deep inelastic scattering 84, 199 dimensional transmutation 196-7 diagonalization of mass matrices 357-9 diquark 442; *see also* leptoquark Dirac monopole 453 quantization condition 455-60, 469 and angular momentum 455--6 string 456-60, 470 and 't Hooft-Polyakov monopole 469-71 Wu-Yang formulation 460 Dirac quantization condition 455-60, 469 Dirac string 456-60, 470 direct product group 87 dispersion relation 159-60, 301-2 divergences 30, 33 disjoint 45 index of 57, 59, 188 linear 34, 176 logarithmic 33, 34, 176 nested 45 overlapping 45 power counting method 56 primitive 33, 44 quadratic 34, 42, 50, 437 superficial degree of 40-1, 43, 57, 73 double P-decay, neutrinoless 414

dual field 454, 480 self dual field 481

duality transformation of Maxwell equation 454

#### *Index*

314

#### *Index*

Yan processes quark-parton model 216-18 perturbative QCD 322 W and Z production 394 dynamical fermion masses 168, 408; *see also*  constituent v. current quark masses dynamical symmetry breaking 401-9; *see also*  composite Higgs particles, technicolour e+e- annihilation; *see also* forward-backward asymmetry in e + e - --+ µ + µ - , quarkonium jets 216 total hadronic cross-section colour 216 free field singularities 223-4 QCD corrections 296-8 quark-parton model 215-16, 291 ratio *R* 215-16, 224 effective potential formalism 189-91 eightfold way 90, 114, 116-21 electric charge; *see also* charge quantization as gauge coupling 230 in electroweak theory 340-2, 346-8, 353 and Weinberg angle 353 electric dipole moment 385-6, 492; *see also* CP electromagnetic current, *see* currents electroweak unification 336, 339-40 r1-mass problem 488-92; *see also* axial U( l) problem 11--+ 3it 167, 490; *see also* axial U(l) problem, isospin violation by strong interactions energy-momentum tensor 309-10 Euclidean field theories fermion 328 gauge theory 479 scalar 326 Euclidean gamma matrices 328 extended technicolour 406-9 *f"b'* symbol 98, 131-2, 163 factorization of singularities 296, 298, 315-16, 321; *see also* operator product expansion Faddeev-Popov ansatz 254 determinant 253-4, 257-8 ghosts 258, 261-3, 269-73 absence of ghosts in Abelian case 257; in axial gauge 255, 273; in lattice gauge theory 333 BRS transformation 274 loop 263, 270, 285 and unitarity 273 Ward identities 272, 277 Fermi constant 135, 204, 336, 339, 353, 373 gauge boson mass 353, 355 Fermi theory of P-decays 336 Fermion family replication problem 363, 429; *see also* generation Fermion masses; *see also* chiral symmetry, quark masses, vacuum tunnelling Dirac and Majorana types 412-14 mass matrix 357-9; *see also* biunitary transformation, mass eigenstates

in theories with spontaneous symmetry breaking 151 in Weinberg-Salam model 351, 357. 395 in technicolour theories 405-9 in minimal SU(5) model 449-51 ferromagnetism 142-3 Feynman parameters 46 Feynman rules 3-4, 22, 498-512 Abelian *R,* gauge 264-5 covariant gauges 259-63, 504-6 fermion loop 23, 29 A.</J4 10-11, 183, 186, 498, 504 path integral quantization 498-503 QED 504-5 *R,* gauge Weinberg-Salam theory 506 scalar QED 505 Fierz transformation 354, 382, 443, 495 fine structure constant (at *Mw)* 388, 438 fixed point 82-3; *see also* renormalization group form factors 153, 202; *see also* structure functions forward-backward asymmetry in e + e - --+ µ + µ - 368-9 four-fermion effective Lagrangian, *see* currentcurrent interaction functional integration 3, 12, 23 GIM mechanism 337, 262-3, 371-2, 378-83, 420, 427; *see also* weak neutral current, charm mass insertion 381-2 power v. logarithmic suppression 382-3 *YA* 153, 160 GUT, *see* grand unification gauge; *see also* Euclidean gauge theory, Higgs phenomenon, lattice gauge theory. puregauge, instantons, solitons coupling constant 230, 232, 234-5; *see also*  electric charge, coupling constant unification field 230, 232, 234, 238, 240 couplings fermion 231, 263 ghost 262 self 234, 262 transformation gauge 230, 232-4, 479 singular 457-9 topological properties, *see* instantons. solitons fixing conditions 242, 248, 252, 258-9; *see also A0* = 0 gauge, axial gauge, covariant gauge, renormalizable gauges, unitary gauge absence in lattice gauge theories 333 259, 261, 264-7, 274, 285-6, 426, 505-9 symmetries (interactions) Abelian 229 non-Abelian (Yang-Mills), 231 eigenstates 356, 371, 410, 430, 447; *see also*  mass eigenstates theories, *see* QCD, QED, QFD, Weinberg-Salam theory, GUT

history and origin of name 235-6.

	- in er-model 150
	- group parameters 88, 90
	- rank 90

Heisenberg picture 6, 11, 126 helicity amplitude 205, 208, 343 Higgs particle 243, 244, 263 elementary v. composite Higgs 401-2 composite Higgs (technicolour models) 402-9 charged Higgs 400 triplet Higgs 418-19, 466-7 in Weinberg-Salam theory 351, 355, 394-400 couplings to fermions 349-50, 357, 395 couplings to gauge bosons 390, 395 mass 351, 396-8 production 398-400 Higgs phenomenon 240-7 Abelian case 241-3 non-Abelian case 243-7 in Weinberg-Salam theory 348-52 in S0(3) model 466-7 in SU(5) GUT 434--7 homotopic classes 477 S 1 ---> S 1 477-8 s 2 \_\_\_, s 2 467, 476 S 3 ---> S 3 478-9 hypercolour, *see* technicolour hypercharge 115 infinite momentum frame 140 infrared divergences 316-17, 321 instantons 292, 476-93; *see also* axial U(l) problem, axion, anomaly, homotopic classes, *e*vacuum, strong CP problem and path integral formalism 484, 487 size 481 vacuum tunnelling 482-7 interaction picture 6 interfamily couplings, suppression 373-4, 447-8 intermediate vector boson charged (W) 135, 339, 342-4, 351-3, 379, 386- 94 neutral (Z) 344, 351-5, 364--9, 386-94 isospin 90, 114--15, 129; *see also* SU(2) and chiral symmetry 125, 167 structure of er-term 157-8 *nN* amplitude 159 violation by electromagnetism 115, 167 by strong interactions 166-7 *J* /ijl 122; *see also* charm Jacobi identity 89 jets 216, 322 K0 -K0 system 374--85; *see also* GIM, CP violation, charm KL-Ks mass difference 372, 377, 380-1 CP parameters 376-8, 383-5 KM-matrix, *see* Kobayashi-Maskawa matrix kink 462

#### *Index*

CP violation theory 361, 378, 383-6 GIM mechanism 380--3 GUT (kinship hypothesis) 449 leptonic mixings 361, 410 in W-decays 388-90 Kogut-Susskind pole 492 ladder diagrams 320 Landau gauge 196, 263, 505 Landau singularity 84 lattice field theory 324-31 fermion field 328-9 gauge field 329-35 link variable 330 Monte Carlo calculation 325, 335 plaquette 330, 334 scalar field 326-8 lepton flavour number 362, 409; *see also* lepton mixing angles, muon number, µ ---> ey, neutrino masses lepton nucleon scattering; *see also* neutrino nucleon scattering Bjorken scaling and quark-parton model 199- 214 Altarelli-Parisi equation 312-13 light-cone singularities in free field theory 224-6 QCD calculations, operator productexpansion and renormalization group equation 298-311; perturbative 316-22 sum rules, *see* Adler, Gross-Llewellyn-Smith, momentum sum rules current algebra, testing, *see* Adler's sum rule elastic 201-2 inelastic 84, 134-41, 199-214, 224, 298--311, 316-22, 369 polarized eD asymmetry 370-1 weak neutral current structure, 369-71 lepton number 355, 414, 418 leptonic mixing angles 361, 410; *see also* neutrino masses and oscillations leptonic weak charged current 136, 204, 337 leptoquark 442; *see also* diquark Lie algebra 89, 91, 127 Lie groups 87-8, 433 light-cone singularities 218-26, 299-300 Linde-Weinberg bound on Higgs mass 396-8 linearly rising potential 323, 333, 334 loop expansion 191-2 ).<jJ4 193-6 scalar QED 196-8 as expansion in Planck's constant 192 local symmetries, *see* gauge symmetries Lorentz gauge 248, 259 low energy theorems single soft pion 153-5 two soft pion 156-67 soft photon 156

magnetic current 454 magnetic moment of gauge vector particle 288 magnetic monopole 453; *see also* Dirac mo.nopole, grand unification monopole, sohtons, 't Hooft-Polyakov, topological conservation laws and charge quantization 454-5, 469

	- lepton couplings 361, 410; *see also* lepton flavour number
	- neutral vector bosons in SU(2) x U(l) electroweak theory, *see* Weinberg angle ro-<jJ' 120--1

Nambu-Goldstone boson, *see* Goldstone boson neutr!no flavour counting and Z decay 393 neutrmo mass 361, 409-20; *see also* lepton flavour number, neutrino oscillations, neutrinoless double cosmological bound 415-16

neutrino-nucleon scattering

flavour oscillation 410-12

# non-integrable phase factor 329, 332 nonexceptional momenta 73 non-leptonic weak processes 338, 384

non-perturbative effects instanton 476 lattice field theory 322 soliton 460 non-renormalizable interactions 57 normalization points (mass scale parameter) in momentum subtraction schemes 67-70 in dimensional regularization scheme 77 0(4) 352, 481; *see also* SU(2) x SU(2) one-particle-irreducible (I Pl) diagrams 31, 192 Green's functions 31, 62, 192 OJ--<jl mixing 120-1 ideal mixing 121 operator product expansion 298-308 factorization of singularities 298 light-cone expansion 299 short distance expansion 298 PCAC 151-8, 161-2 paramagnetic property of Yang-Mills vacuum 287-9; *see also* asymptotic freedom parity nonconservation 336, 338 in atomic physics 370 partial wave 338, 343-4 partially conserved axial-vector current, *see*  PCAC parton; *see also* parton model, quarks, gluons point-like constituents of hadron 203, 206, 208-9 sea 209, 211 spin 1/2 nature of charged partons 207-8 part on model; *see also* deep inelastic scattering, parton, QCD-parton picture, scaling Drell-Yan process 216-18 e+e- annihilation 214-16 impulse approximation 206 lepton-nucleon scattering 206-14 parton distribution functions 207-12 Paschos-Wolfenstein relation 370 path integral quantization 11-17 scalar 17-22 fermion 23-9 gauge theories 248-67 perturbation expansion; *see also* Feynman rules, non-perturbative effects, renormalization canonical formalism 6-8 interaction picture 6 U-matrix 7 chiral perturbation 161 path integral formalism ).<jJ 4 19-22, 498-501 gauge theories 260 perturbative QCD 311 phase transition 325, 335 *n°* -> *2y* 180-2

anomaly 181 colour 182, 291 pion nucleon coupling 153 pion nucleon (rrN) amplitude, *see* Adler's consistency condition, Adler-Weisberger sum rule, isospin, scattering length, u-term polarizations of vector particles 271, 342-3 polarized eD asymmetry 370-1 Pontryargin number, *see* topological winding number propagator functions 499-509; *see also* Feynman rules fermion 23, 221, 263 ghost 261 Higgs scalar 264 scalar 5, 9-10, 31-2, 34-5, 68, 170-1, 219-20 vector 60-2, 261, 264, 339 *R,* gauge 265 would-be-Goldstone boson 265 proton decay 442-4 SU(5) amplitude 443 model-independent approach 443 pseudo-Goldstone bosons 246 in technicolour theories 405, 407, 409 pseudoscalar meson masses, *see* current algebra u-term, Dashen sum rule, 17-mass problem, Gell-Mann-Okubo formula, Goldstone boson, quark masses pseudo-particles 476; *see also* instantons pure gauge 480 QCD 124, 279; *see also* gluons, quarks effective coupling constant 295-6, 312, 438; *see also* asymptotic freedom, confinement global symmetries 125, 168, 294; *see also* chiral symmetry, strong CP problem Lagrangian 291 operator product expansion and renormalization group calculations 291-311; *see also*  e + e- annihilation, lepton nucleon scattering, Bjorken scaling part on picture 208-9, 214, 312; *see also*  Altarelli-Parisi equation perturbative calculations 311-22; *see also*  lepton nucleon scattering scale parameter 296, 335, 403, 442 vacuum 323, 493; *see also* vacuum in Yang-Mills theories QED gauge theory, as prototype 229 quantization, difficulty of 248 renormalization group 67 P-function 283-4 charge screening 280 renormalization programme 31 Ward identities axial-vector (anomaly) 173-9 vector 174, 267-8, 272 QFD 336-427 quantization canonical formalism 3 scalar 4-11, 170, 219-20

> fermion 22-3, 221 lattice field theory 325

#### *Index*

path integral formalism 3, 498-503 scalar 11-22 fermion 23-9 gauge theories 248-60 lattice field theories 325, 327-8 theories with 111agnetic monopole 454-60 quantum chromodynamics, *see* QCD quantum electrodynamics, *see* QED quantum flavour dynamics, *see* QFD quark currents; *see also* baryon number, colour, GIM mechanism electromagnetic 134-5, 210, 346; fractional charge 117-18, 123, 346-8, 431, 474 weak; charged 136, 210, 359, 371-4; neutral 353, 362, 364-5, 371 masses; *see also* chiral symmetry chiral symmetry breaking 125, 152, 160-8 constituent v. current masses 168 SU(5) relations to lepton masses 449-51 sizes *m.,* md, *m,* 167-8; *m,,* mb 122 strong CP problem 493 model, simple; *see also* Gell-Mann-Okubo formulae, co-<j> mixing, Zweig rule and chiral symmetry 131-4; *see also* axial U(I) problem eightfold way SU(3) flavour symmetry 115- 21 paradoxes 123 quantum numbers 117-18 quarkonium 122-3; *see also J;ljl.* r

```
R; gauge
  Abelian 263-7
  non-Abelian case 267
  standard SU(2) x U(I) theory 420, 506-12
radiation gauge 248
rank 90, 433-4
regularization schemes 33. 45
  covariant cutoff (conventional momentum
       cutoff) 45-52
    momentum subtractions 67-9
  dimensional regularization 52-6, 78, 267, 278
    minimal subtraction 78-9
renormalizable gauge 247. 264; see also R: gauge
renormalizability 45, 56, 60
  multiplicative 39, 64
  renormalizable interaction 57
    non- 57, 61
    super- 57
  soft symmetry breaking 187-8
  vector field 61-2. 263-4. 340
renormalization; see also anomaly. perturbation
       expansion, renormalization group.
       renormalizability
  constants 40, 267-8
    coupling constant 37. 285
    vertex Z;. 37
        285
     wavefunction (field) Z,p 35
     ZA 285
```
composite operator 62-6 mass 34 Goldstone boson 186-7 A</i4 theory 31-45 physical picture 30 programme BPH 39-45 conventional 31-9 SSB theories 182-8. 189-98 Yang-Mills theory 284-6 renormalization group; *see also* anomalous dimension, effective (running) coupling calculations SU(5) GUT coupling constant unification 439 lepton-quark mass relations 449-51 proton decay amplitude 444 QCD corrections e + e- total cross-section 297-8 IN scaling 303-4 equations *p* functions 71--2, 77. 80 *i.<fi 4*72, 80, 82-3, 281-2 QED 283-4 QCD (Yang-Mills) 286-7. 289, 295-6 Yukawa 282-3 SU( SJ 439-40 Callan-Symanzik 70--3 other forms 75, 304, 307 asymptotic solutions 74-7, 81, 304, 307 ;· functions 71-2, 77, 80, 83--4 QCD 307, 311 moments of Altarelli-Parisi functions 314 physical basis 67-70 summation of leading radiative corrections 82. 316 representation 87 adjoint 89. 113. 432, 434 basis states of 87. 90. 92 complex 407-8, 434: *see also* real representation conjugate 89, 102-3. 109. 118. 429 dimension of 87, 92, IOI. 106-7 fundamental (defining) 87. 102. 109. 116, 429 graphic representation of 95-6. 99-102 irreducible 87. 92. 104 product 93-6, 108-9 real 89, 96-7, 434 vector 103. 109, 434 µ-parameter 352, 364. 394 experimental value 387. 419 for weak isospin of Higgs *T=* 1/2 352 *T=* I 419 general *T* 394 right-handed neutrino 418-20 rotational group 89. 91-2 S0(2), *see* U(I) S0(3), *see* SU(2) S0(4). *see* SU(2) x SU(2)

SSB, *see* spontaneous symmetry breaking

#### 533

SU(2) 90-7; *see also* S0(3) gauge theories 231 Georgi-Glashow model 341, 466 Higgs phenomenon 243 quantization (covariant gauges) 259-63 instanton 479 Slavnov-Taylor identity and BRS transformation 273 isospin flavour symmetry 114-15 Noether theorem, 129 SU(2) x U(l) 87, 340-1 SU(2) x SU(2); *see also* 0(4), chiral symmetry u-model, SSB and Goldstone theorem 149 breaking to SU(2) and *p* = 1 352, 404 SU(3) 97-102 colour group 123-4 QCD Lagrangian 291 decouplet 116-20 eightfold way, flavour symmetry 115-23 octet 116-17, 119-21 quarks, free field theory of 131 symmetry breaking 119, 165 weak current transformation properties 136 SU(3) x SU(3) charge algebra 130 current algebra 132 SU(4) 102 flavour symmetry 122 lepton-quark mass relations in SU(5) model 449 tumbling 408-9 SU(5) 90 GUT 428-52 tumbling 407-9 SU(n) 90, 102-13 scaling 199, 203-6, 279, 308, 312 e+e- 214-16, 223-4 Drell-Yan process 216-18 lepton-nucleon (Bjorken) 203-14, 218-19, 224-6, 308 functions 204, 205 variables, *x, y* 203 light-cone singularities 218-19 violations 308, 310-11, 312 Schrodinger picture 6, 11 Schwinger terms 133, 157 self-dual field 481 self-energy 32, 34-5, 49-50, 68-9, 78, 449-50 u-model 149-51, 403 u-term 157, 160-1 itN 157-60, 161, 165-6 pseudoscalar meson masses 162, 164, 166-7 simple group 87, 235, 430, 433, 469, 472 semi-simple group 87, 346, 433 singularities in product of field 219, 298-9 currents 221-3, 300, 305 Slavnov-Taylor identities 267-78

soft photon theorems 156 soft-pion theorems one soft pion 153-5 two soft pions 156-60, 162 soft symmetry breaking 187 and renormalizability 187-8 spontaneous symmetry breaking 188 solar neutrino puzzle 410 solitons 460-9 J.cf>4 in 2 dimensions 460-4 J.cf>4 in 4 dimensions 464-5 gauge theories with SSB ('t Hooft-Polyakov monopole) 466-9 spontaneous symmetry breaking (SSB) 125, 141- 51, 240; *see also* chiral symmetries, Goldstone theorem, Higgs phenomenon, VEV fermion masses 151, 357, 407, 449 pattern of 245-6 radiatively induced 195-8, 396-8 and renormalization 182-8, 264 and solitons 464 standard electroweak theory, *see* Weinberg-Salam theory string, Dirac 456-60, 470 string model of hadrons 323 string tension 323, 333, 335 strong coupling expansion 325, 333-5 structure function 138, 201, 204, 211-14, 305; *see also* form factor, scaling, parton distribution function flavour singlet combination 305, 308-10, 314 flavour non-singlet combination 305, 310-12, 314 helicity 205, 208 moments 212-14, 300-3, 308, 311, 313-14 subgroup 86 invariant subgroup 87 subtraction 34; *see also* renormalization, regularization scheme point (normalization condition) 68-70, 78, 81, 82, 281 schemes momentum (conventional) 67-70; intermediate renormalization 68-9 on-shell renormalization 69 minimal (mass-independent) 78-80 Sudakov variables 318-19 summation of leading logarithm 82, 315-16, 322 supersymmetry 290, 437 symmetry; *see also* baryon number, CP, chiral, currents, global symmetries, gauge, lepton numbers, SSB, Ward identities and group theorem 86 and mass degeneracy 113-14, 125-8, 142 and renormalization 169 symmetry factor 9, 11, 32, 187, 193, 269, 503-4; *see also* Feynman rules t-quark 122, 356 tadpole diagrams 182, 183 and shift in VEV 182-6 and massless Goldstone bosons 186-7 tau-leptons 356 technicolour 401-9; *see also* extended technicolour scale parameter ATc 405

#### *Index*

534

#### *Index*

tensor representation of SU(n) 102-13, 428 e-vacuum 486-7, 490-3; *see also* strong CP problem, axial U(l) problem 't Hooft-Feynman gauge 263, 268, 277, 379, 426 't Hooft-Polyakov monopole 453, 466-75; *see also* magnetic monopole, solitons charge quantization and simple group 469, 472 coupling of spatial and internal symmetries 471 as Dirac monopole 469-71 S0(3) 466-72 SU(5) 472-5 chromomagnetic charges 474 catalyses proton decay 474-5 (3,3\*)EB (3\*,3) model 162; *see also* chiral symmetry breaking top, *see* t-quark topological conservation law 463-4 topological winding number 477-9, 480, 483-7 tumbling 407-9 twist 299-300, 305, 309 U(l) 86; *see also* S0(2) axial U(l) problem 294, 487 gauge symmetry 229-31 Goldstone theorem 147-9 at one-loop level 186 Higgs phenomenon 241-3 Noether theorem 128-9 vector Ward identity, radiative correction 170-- 3 unitarity Abelian gauge theory 263-4, 266 and renormalizability in weak interaction theory 338, 342-5 and Ward identities 268-73, 278 unitary gauge 242, 244, 263, 349-51, 426 unitary group 86 universality Cabibbo 365, 373 gauge theory 231, 23 5 *Y* 122; *see also* b-quark VEV, *see* vacuum expection value V-A theory 135, 204, 336-8, 353, 364 vacuum 18 QCD 323, 493 degenerate vacua in theories with SSB 143, 464 multiple vacua with different winding number 482-6; *see also* e-vacuum vacuum expectation value condition for SSB 143 effective potential formalism 189 one-loop perturbative shift 182-6 sizes in electroweak unification 353, 402, 436 in GUT 436 vacuum polarization 297, 383, 404 vacuum saturation approximation 380 vacuum-to-vacuum amplitude 18, 29, 189, 254, 259, 275, 484, 487; *see also* generating functional

vector-axial-vector ambiguity in neutrino reactions 367

vector meson masses gauge fields 231, 235 hadrons, *see* Gell-Mann-Okubo formula, *w---</J*  mixing Higgs phenomena 240-7 and renormalizability 60-2, 264, 338-9 W and Z 351-2, 355, 387-8 vertex renormalization constant 37, 285 W; *see also* Z in pre-gauge-theory models 338-9, 342-4 in standard SU(2) x U(l) theory 351-3, 379, 384-94 decays, leptonic branching ratio 389 Higgs 390 width 389 mass 387-8 production in Drell-Yan process 394 WKB approximation and imaginary time 484-6 Ward identities 156, 169, 267-8 vector current in ).<f;i4 170--3 axial-vector current in QED 173-9 and unitarity 267, 268-73 non-Abelian gauge theories (Slavnov-Taylor identities) 272-8 weak axial nucleon coupling, *see 9A*  charged currents, *see* currents eigenstate, *see* gauge interaction eigenstate hypercharge 346-8, 431, 439 mixing angles, *see* Cabibbo angle, KM mixings, lepton mixings, Weinberg angle neutral currents, *see* currents radiative correction to QCD Lagrangian 294-5 weak interaction theories Fermi 336 Georgi-Glashow S0(3) 341-2 IVB 338-9 standard SU(2) x U(l), *see* Weinberg-Salam theory V-A 135, 204, 336-8, 353, 354, 364 Weinberg angle definition 352 W-Z mass relation 352, 364, 387 value from neutral current processes 354, 365 pure leptonic 365-9 vN scatterings 369-70 SU(5) prediction 439-41 Weinberg-Salam theory; *see also* CP symmetry, GIM mechanism, generation, Higgs phenomenon, Kobayashi-Maskawa mixing, weak neutral current, Weinberg angle choice of SU(2) x U(l) group 340-5 fermions anomaly cancellation 347-8 neutral current couplings 354, 365, 391 quantum number assignment 345-6 global symmetries 355-63 baryon and lepton numbers 355 CP violation 359-61 muon number 362

#### *Index*

Weinberg-Salam theory *(cont.)*  Higgs particle 348-50, 394-401 *p* = 1 and isodoublet structure 352, 394 massless neutrinos 361, 417-18 SSB 348-50 W and Z gauge bosons 351-4, 386-94 Weinberg's theorem on asymptotic behaviour of Green's functions 73-4 on convergence of Feynman integrals 42-3 Wick rotation 47 Wick theorem 8-10, 21, 62, 221 Wilson coefficient 298, 300-2, 303-4 Wilson loop 331-4 winding number, *see* topological winding number would-be-Goldstone boson 243, 244, 247, 263-7, 351, 420-6 Yang-Mills fields, *see* non-Abelian gauge fields Young tableau 104-9, 117-19; *see also* tensor representation of SU(n) conjugate representation 109 dimension of irreducible representation 106-8 distance to the first box 107 fundamental theorem 106

hook length 107

lattice permutation 108-9 product representation 108-9 standard tableau 105-6 Yukawa couplings 349-50, 357, 395 P-function 282-3, 290 fermion masses in SSB theories u-model 151 Weinberg-Salam theory 349-50, 357, 395 extended technicolour theories 406 neutrino masses 418-19 SU(5) model 448-51 global symmetry properties CP 386 B and L in electroweak theory 355 B - Lin minimal SU(5) 451-2

Z 351-4, 364; *see also* W decays, branching ratios 392 counting neutrino flavours 393 neutrino pair final states 391 Higgs 392, 399 mass 387-8; *see also* p-parameter production in Drell-Yan processes 394 ZWW couplings 393 Zweig rule 121-2, 165

# 536


*Article* **Sustainable Tourism and Renewable Energy's Potential: A Local Development Proposal for the La Florida Community, Huaral, Peru**

**Kiara Riojas-Díaz 1 , Ricardo Jaramillo-Romero 1 , Fátima Calderón-Vargas 1,\* and David Asmat-Campos 2,\***

**Abstract:** Tourism contributes 10% to global gross domestic product (GDP), yet it generates 5% of all anthropogenic CO2 , while 50 to 60% of carbon emissions are indirectly related to the sector. High levels of poverty afflict rural areas in developing countries, and sustainable tourism based on renewable energy is an ideal approach to generate local development. Our objectives are thus to gauge sustainable tourism's influence on local development in the community of La Florida, Huaral, Peru and to evaluate the potential of renewable energy (solar and wind power) to propose an eco-efficient business alternative. A non-experimental, quantitative approach was used, in which 265 local residents completed a survey to ascertain their perspectives on the proposal. Moreover, the potential for solar and wind energy was measured to identify sustainable alternatives that residents might incorporate into local ventures. The results demonstrate a relationship between sustainable tourism and local development, as tourism activity enables community members to improve their quality of life and offers them the opportunity to generate new enterprises. Likewise, the assessment of renewable energy potential confirms its feasibility in this area.

**Keywords:** local community; community development; sustainable community; tourism and renewable energy