Skip to content

Appendix D Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

B-1. Determining Mass Dimension (Yukawa Interaction)

The Lagrangian density of the Yukawa interaction is

\[ \mathcal{L}_{\text{int}} = -g\,\bar{\psi}\psi\,\phi \]

where \(\psi\) is a Dirac field and \(\phi\) is a scalar field. Determine the mass dimension \([g]\) of the coupling constant \(g\).

Hint

Use the facts that \([\mathcal{L}] = 4\), \([\psi] = 3/2\), and \([\phi] = 1\).

→ View solution

B-2. Determining Mass Dimension (Scalar Field in 6-Dimensional Spacetime)

The free Lagrangian density of a real scalar field in \(d\)-dimensional spacetime is

\[ \mathcal{L} = \frac{1}{2}(\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2}m^2\phi^2 \]

Find the mass dimension \([\phi]\) of the scalar field \(\phi\) in the case \(d = 6\). Furthermore, find the mass dimension of the coupling constant \(g\) in the \(\phi^3\) interaction \(\mathcal{L}_{\text{int}} = -\frac{g}{3!}\phi^3\).

Hint

In \(d\) dimensions, from \([d^d x] = -d\) and \([S] = 0\), we have \([\mathcal{L}] = d\). Determine \([\phi]\) from the kinetic term, then substitute into the interaction term.

→ View solution

B-3. Direct Calculation of Feynman Parameters

Using the basic Feynman parameter formula (D.24), rewrite the following expression in Feynman parameter representation:

\[ \frac{1}{(k^2 - m^2)((k+q)^2 - m^2)} \]

Furthermore, perform the change of variables \(\ell = k + (1-x)q\) and rearrange the denominator into the form \(\ell^2 - \Delta\). Express \(\Delta\) in terms of \(m\), \(q^2\), and \(x\).

Hint

Set \(A = k^2 - m^2\) and \(B = (k+q)^2 - m^2\), then apply formula (D.24). The key to completing the square is to perform a change of variables "so that the linear term in \(\ell\) vanishes."

→ View solution

B-4. Signs in Wick Rotation

Rewrite the following Minkowski space quantities in terms of Euclidean space quantities using the Wick rotation \(\ell_0 = i\ell_0^E\):

(a) \(\ell^2 = \ell_0^2 - \vec{\ell}^{\,2}\)

(b) \(d^4\ell\)

(c) \(\displaystyle\frac{1}{(\ell^2 - \Delta + i\varepsilon)^3}\)

Make the signs and factors of \(i\) explicit at each step.

Hint

(a) Substitute \(\ell_0 = i\ell_0^E\). (b) Use \(d\ell_0 = i\,d\ell_0^E\). (c) Substitute the result of (a) and handle the \((-1)^3\) factor. The \(i\varepsilon\) becomes unnecessary in Euclidean space.

→ View solution

B-5. Solid Angle of a 4-Dimensional Sphere

The area (solid angle) of a unit sphere in \(d\)-dimensional Euclidean space is given by

\[ \Omega_d = \frac{2\pi^{d/2}}{\Gamma(d/2)} \]

Calculate \(\Omega_d\) for \(d = 2, 3, 4\) respectively, and verify that they agree with the known results (\(2\pi\), \(4\pi\), \(2\pi^2\)).

Hint

Use \(\Gamma(1) = 1\), \(\Gamma(3/2) = \frac{\sqrt{\pi}}{2}\), \(\Gamma(2) = 1\).

→ View solution

B-6. Verification of the \(2\pi\) Factors

Verify that the 4-dimensional Fourier transform convention

\[ f(x) = \int \frac{d^4p}{(2\pi)^4}\;\tilde{f}(p)\,e^{ipx}, \qquad \tilde{f}(p) = \int d^4x\;f(x)\,e^{-ipx} \]

is self-consistent. That is, substitute the expression for \(\tilde{f}(p)\) into the expression for \(f(x)\), and show that \(f(x)\) is identically reproduced.

Hint

Use \(\int d^4x'\,e^{i(p-p')\cdot x'} = (2\pi)^4\delta^{(4)}(p - p')\).

→ View solution

B-7. General Feynman Parameter Formula (\(n = 3\))

Write down the Feynman parameter formula (D.25) for three factors \(A_1, A_2, A_3\), and rewrite it by eliminating \(x_3\) using the delta function. Describe (in words or a sketch) what the integration region looks like.

Hint

Use \(\delta(1 - x_1 - x_2 - x_3)\) to set \(x_3 = 1 - x_1 - x_2\). The region where \(x_1, x_2 \geq 0\) and \(x_1 + x_2 \leq 1\) is a triangle.

→ View solution

B-8. Estimating Divergences from Mass Dimension

In 4-dimensional spacetime, determine the superficial degree of divergence of the following loop integrals from mass dimension arguments:

(a) \(\displaystyle\int \frac{d^4k}{(2\pi)^4}\;\frac{1}{k^2 - m^2}\)

(b) \(\displaystyle\int \frac{d^4k}{(2\pi)^4}\;\frac{1}{(k^2 - m^2)^2}\)

(c) \(\displaystyle\int \frac{d^4k}{(2\pi)^4}\;\frac{k^2}{(k^2 - m^2)^3}\)

Hint

Examine the behavior of the integrand as \(k \to \infty\). Note that \(d^4k \sim k^3 dk\). When the integrand behaves as \(k^n\), the integral diverges if \(n + 3 \geq -1\) (i.e., \(n \geq -4\)).

→ View solution

Medium

M-1. Complete Reduction of a One-Loop Integral Using Feynman Parameters

In \(\phi^3\) theory (scalar field with mass \(m\), coupling constant \(g\)), the one-loop self-energy diagram involves the following integral for external momentum \(p\):

\[ \Sigma(p^2) = \frac{g^2}{2}\int \frac{d^4k}{(2\pi)^4}\;\frac{1}{(k^2 - m^2 + i\varepsilon)((k-p)^2 - m^2 + i\varepsilon)} \]

Carry out the calculation following the steps below:

(a) Introduce a Feynman parameter \(x\) and combine the denominators into a single expression.

(b) Perform the change of variables \(\ell = k - (1-x)p\) and express \(\Delta\) in terms of \(m^2\), \(p^2\), and \(x\).

(c) Perform a Wick rotation and rewrite the integral in Euclidean space.

(d) Using 4-dimensional spherical coordinates, carry out the angular integration and reduce the result to a one-dimensional radial integral. Show that this integral is logarithmically divergent.

Hint

In part (d), evaluate \(\int_0^\Lambda d\ell_E\;\ell_E^3 / (\ell_E^2 + \Delta)^2\). The logarithmic divergence can be seen from the fact that the integrand behaves as \(\sim 1/\ell_E\) for \(\ell_E \gg \sqrt{\Delta}\).

→ View solution

M-2. Verification of the Validity of Wick Rotation

In the \(\ell_0\) complex plane, show the following:

(a) Show that with the Feynman propagator \(i\varepsilon\) prescription, the poles of \(1/(\ell^2 - \Delta + i\varepsilon)\) in \(\ell_0\) are located in the second and fourth quadrants (assume \(\Delta > 0\)).

(b) When rotating the \(\ell_0\) integration contour by 90° counterclockwise from the real axis to the imaginary axis, argue that the integrand decays sufficiently rapidly to zero on the closed contour (quarter-circle arc), so that the contribution from the arc vanishes.

(c) From the above, explain how the Wick rotation is justified as a consequence of Cauchy's integral theorem.

Hint

(a) Solving \(\ell_0^2 = \vec{\ell}^{\,2} + \Delta - i\varepsilon\) gives \(\ell_0 = \pm(\omega - i\varepsilon')\) (\(\omega > 0\)). The positive pole is slightly below the real axis (fourth quadrant side), and the negative pole is slightly above the real axis (second quadrant side). (b) Use the fact that on the quarter-circle arc, when \(|\ell_0| \to \infty\), the denominator grows as \(|\ell_0|^4\).

→ View solution

M-3. Derivation of the Basic Formula for Dimensional Regularization

Derive the following integral formula in \(d\)-dimensional Euclidean space:

\[ \int \frac{d^d\ell_E}{(2\pi)^d}\;\frac{1}{(\ell_E^2 + \Delta)^n} = \frac{1}{(4\pi)^{d/2}}\;\frac{\Gamma(n - d/2)}{\Gamma(n)}\;\frac{1}{\Delta^{n-d/2}} \]

Follow these steps for the derivation:

(a) Perform the angular integration using \(d\)-dimensional spherical coordinates, employing \(\Omega_d = 2\pi^{d/2}/\Gamma(d/2)\).

(b) Reduce the radial integral \(\int_0^\infty d\ell_E\;\ell_E^{d-1}/(\ell_E^2 + \Delta)^n\) to a Beta function \(B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)\) via the substitution \(t = \ell_E^2/\Delta\).

Hint

(b) Setting \(t = \ell_E^2 / \Delta\) gives \(d\ell_E = \frac{\sqrt{\Delta}}{2\sqrt{t}}\,dt\). Rewriting the integrand in terms of \(t\) yields the form \(\int_0^\infty dt\;t^{d/2-1}/(1+t)^n\), which equals the Beta function \(B(d/2, n - d/2)\).

→ View solution

M-4. Restoring Units and Estimating Cross Sections

In QED, the total cross section for \(e^+e^- \to \mu^+\mu^-\) in the limit where the center-of-mass energy \(\sqrt{s} \gg m_\mu\) is given by

\[ \sigma = \frac{4\pi\alpha^2}{3s} \]

where \(\alpha = e^2/(4\pi) \approx 1/137\).

(a) Verify that in natural units, the mass dimension of \(\sigma\) is \(-2\).

(b) When \(\sqrt{s} = 10\ \text{GeV}\), use the conversion factor from Eq. (D.6) to express \(\sigma\) in picobarns (pb).

(c) Estimate the number of events expected per day at a collider with luminosity \(\mathcal{L} = 10^{33}\ \text{cm}^{-2}\text{s}^{-1}\).

Hint

(a) From \([\alpha] = 0\) and \([s] = 2\), we get \([\sigma] = -2\). (b) Compute \(\sigma = \frac{4\pi}{3 \times 137^2 \times 100}\ \text{GeV}^{-2}\) and convert using \(1\ \text{GeV}^{-2} = 0.3894\ \text{mb} = 3.894 \times 10^8\ \text{pb}\). (c) \(N = \sigma \mathcal{L} T\).

→ View solution

Advanced

A-1. The \(\gamma_5\) Problem in Dimensional Regularization and the ABJ Anomaly

The matrix \(\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3\), defined in 4 dimensions, satisfies \(\{\gamma_5, \gamma^\mu\} = 0\) (\(\mu = 0,1,2,3\)). However, when analytically continuing to \(d = 4 - 2\epsilon\) dimensions in dimensional regularization, a contradiction arises between this anticommutation relation and the trace formulas for \(\gamma^\mu\).

(a) Assume that \(\{\gamma_5, \gamma^\mu\} = 0\) holds for all \(\mu\) even in \(d\) dimensions. Show that a contradiction arises when computing \(\text{Tr}[\gamma_5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma]\) by combining \(\gamma^\alpha\gamma_\alpha = d\) with \(\{\gamma_5, \gamma^\alpha\} = 0\).

(b) Explain the basic idea of the 't Hooft–Veltman prescription (where \(\gamma_5\) anticommutes only within the 4-dimensional subspace) that avoids this contradiction, and qualitatively discuss how the trace calculation of the triangle diagram (AVV vertex) is modified under this prescription.

(c) Describe what physical consequences the above discussion has for the computation of the ABJ (Adler–Bell–Jackiw) anomaly, from the perspective of the violation of the conservation law of the axial current.

Hint

(a) Compute \(0 = \text{Tr}[\gamma_5\gamma^\alpha\gamma_\alpha\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma]\) in two ways. Compare the method of moving \(\gamma^\alpha\) to the right and anticommuting it with \(\gamma_5\), with the method of first using \(\gamma^\alpha\gamma_\alpha = d\). (b) In the 't Hooft–Veltman prescription, \(\gamma^\mu\) is decomposed into a 4-dimensional component \(\hat{\gamma}^\mu\) and a \((d-4)\)-dimensional component \(\tilde{\gamma}^\mu\), and \(\gamma_5\) anticommutes only with \(\hat{\gamma}^\mu\). (c) The regularization procedure breaks axial symmetry, and a finite anomaly term remains.

→ View solution

A-2. Relationship between Feynman Parameters and the Mellin–Barnes Representation

In higher-loop calculations, Feynman parameter integrals can become overly complicated. An alternative technique is the Mellin–Barnes representation.

(a) Prove the following identity:

\[ \frac{1}{(A + B)^n} = \frac{1}{\Gamma(n)}\;\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} ds\;\Gamma(-s)\Gamma(n+s)\;\frac{A^s}{B^{n+s}} \]

where the integration contour is chosen to separate the poles of \(\Gamma(-s)\) (at \(s = 0, 1, 2, \ldots\)) to the right and the poles of \(\Gamma(n+s)\) (at \(s = -n, -n-1, \ldots\)) to the left.

(b) Using this representation, express the one-loop integral containing two propagators with different masses

\[ I(p^2) = \int \frac{d^d k}{(2\pi)^d}\;\frac{1}{(k^2 - m_1^2)((k-p)^2 - m_2^2)} \]

as a Mellin–Barnes integral, and verify that it is equivalent to the Feynman parameter representation.

(c) For the case \(m_1 = 0\), \(m_2 = m\), \(p^2 = -Q^2\) (\(Q^2 > 0\): spacelike momentum), obtain the first two terms of the asymptotic expansion of \(I\) in the limit \(Q^2 \gg m^2\) by evaluating residues of the Mellin–Barnes representation.

Hint

(a) Evaluate the right-hand side as a sum of residues in \(s\), and reduce it to the binomial expansion \(\sum_{k=0}^\infty \binom{-n}{k}(A/B)^k \cdot 1/B^n\). (b) After obtaining \(\Delta\) using Feynman parameters, separate the two terms in \(\Delta\) using Mellin–Barnes. (c) In the limit \(m^2/Q^2 \to 0\), the residues at \(s = 0, 1\) are dominant. Residues of \(\Gamma\) function poles and the \(\psi\) function (digamma function) will appear.

→ View solution