"Since we're asking you to do some calculations, it's up to you whether you write the \"heavy\" calculations in the notebook or not: it's fine if you don't write them down in the notebook, but show us on paper."
"Let's imagine a parallel universe in which the photon, instead of being a massless vector particle (spin-1), is a massive scalar particle (spin-0). The QED vertex in the Feynman rules in this theory would be $-i g_{e} I$, where $I$ is the $4 \\times 4$ unit matrix (to compare with $-i g_{e} \\gamma^{\\mu}$ for the massless vector photon). There would also be no factors for the outer photon lines, since there is no photon polarisation."
]
},
{
"cell_type": "markdown",
"id": "8d20c0c4",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Assuming that this \"photon\" is heavy enough to decay into a pair of Standard Model particles, calculate the decay rate $\\Gamma$ for $\\gamma \\rightarrow e^{+}e^{-}$ using the helicity spinors (in the centre-of-mass frame). Neglect the electron/positron mass in this calculation.</span>\n",
"</div>"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2c6b3759",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "809505cb",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Is helicity \"conserved\" in high-energy interactions in this theory? Explain how this differs from QED.</span>\n",
"</div>"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5ab2e03e",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "9941f5ee",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Do the calculation again, but this time include the electron/positron mass.</span>\n",
"</div>"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0fb4e658",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "49e604a5",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"If $m_{\\gamma} = 3$~GeV, calculate the lifetime of this \"photon\" in seconds.</span>\n",
"Let's consider the muon decay $\\mu^- \\to \\nu_\\mu \\bar{\\nu_e} e^-$, which is described by the S=standard model's electroweak interaction lagrangian."
]
},
{
"cell_type": "markdown",
"id": "d68a4fdf",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Draw the first-order Feynman diagram describing muon decay; consider $m_W \\gg m_\\mu$ and derive the effective lagrangian (also called the Fermi lagrangian), showing the relationship between the electroweak coupling constant $g_W$ and the Fermi constant $G_F$.</span>\n",
"</div>"
]
},
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"execution_count": null,
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"metadata": {},
"outputs": [],
"source": []
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{
"cell_type": "markdown",
"id": "fcc1c2ae",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Using the Fermi lagrangian, calculate the decay rate of the muon and its lifetime.</span>\n",
"</div>"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6c67f8f0",
"metadata": {},
"outputs": [],
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},
{
"cell_type": "markdown",
"id": "cbc7ad7c",
"metadata": {},
"source": [
"<div class=\"alert alert-info\">\n",
"<strong>Exercise:</strong> \n",
"Using the derived formula for muon decay, calculate the lifetime of the $\\tau$ lepton and compare it with the measured value (check the most recent value on PDG). Are there any differences between the calculated and the measured value? Why are there differences?</span>\n",
"</div>"
]
},
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%% Cell type:markdown id:a1c50278 tags:
# Übungen zu Teilchenphysik I
## Exercise 03 - QED and EW theory
%% Cell type:markdown id:b82039ad tags:
## Preamble
Since we're asking you to do some calculations, it's up to you whether you write the "heavy" calculations in the notebook or not: it's fine if you don't write them down in the notebook, but show us on paper.
Let's imagine a parallel universe in which the photon, instead of being a massless vector particle (spin-1), is a massive scalar particle (spin-0). The QED vertex in the Feynman rules in this theory would be $-i g_{e} I$, where $I$ is the $4 \times 4$ unit matrix (to compare with $-i g_{e} \gamma^{\mu}$ for the massless vector photon). There would also be no factors for the outer photon lines, since there is no photon polarisation.
%% Cell type:markdown id:8d20c0c4 tags:
<divclass="alert alert-info">
<strong>Exercise:</strong>
Assuming that this "photon" is heavy enough to decay into a pair of Standard Model particles, calculate the decay rate $\Gamma$ for $\gamma \rightarrow e^{+}e^{-}$ using the helicity spinors (in the centre-of-mass frame). Neglect the electron/positron mass in this calculation.</span>
Let's consider the muon decay $\mu^- \to \nu_\mu \bar{\nu_e} e^-$, which is described by the S=standard model's electroweak interaction lagrangian.
%% Cell type:markdown id:d68a4fdf tags:
<div class="alert alert-info">
<strong>Exercise:</strong>
Draw the first-order Feynman diagram describing muon decay; consider $m_W \gg m_\mu$ and derive the effective lagrangian (also called the Fermi lagrangian), showing the relationship between the electroweak coupling constant $g_W$ and the Fermi constant $G_F$.</span>
</div>
%% Cell type:code id:88173131 tags:
``` python
```
%% Cell type:markdown id:fcc1c2ae tags:
<div class="alert alert-info">
<strong>Exercise:</strong>
Using the Fermi lagrangian, calculate the decay rate of the muon and its lifetime.</span>
</div>
%% Cell type:code id:6c67f8f0 tags:
``` python
```
%% Cell type:markdown id:cbc7ad7c tags:
<div class="alert alert-info">
<strong>Exercise:</strong>
Using the derived formula for muon decay, calculate the lifetime of the $\tau$ lepton and compare it with the measured value (check the most recent value on PDG). Are there any differences between the calculated and the measured value? Why are there differences?</span>
