Electroweak and Strong Interactions

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General information

This course it taught at VUB, but is frequently followed by students from Leuven (and from Ghent). It is a succesor to Quantum Field Theory; also given by prof. Sevrin and in the same style. You have to write a little paper (5-10 pages) about some topic related to the course which is due in the exam period.

The exam is completely analagous to the exam of QFT. Make sure to give a complete (i.e. say everything you know) answer to his questions on the oral part.

It consisted of two questions. The maximum time was 4,5 hours to solve them, but in the end the exam ended up during 5-5.5 hours.

Old exams

Please note that in 2021, the format of the exam was changed due to the corona pandemic; it was oral and lasted only one hour.

23 Juni 2021

Same questions as below. New questions:

• What is the experimental status of CP violation in the leptonic sector?
• Why does the negative pion decay almost exclusively to the muon and anti-muon neutrino?
• To introduce the BEH mechanism in extensions of the standard model, an additional scalar has to be added. Which degrees of freedom does one obtain?

31 May 2021

All four oral defences were the same as the 28th of may.

28 May 2021

• Explain hierarchy problem
- Is it also a problem for the electron mass?
- …
• Explain why W decays into 2/3 hadrons and only 1/3 leptons?
• What experimental observation let to the introduction of non-diagonal Yukawa couplings?
• Explain how LEP calculated there are only 3 generations of neutrinos.
• Explain running coupling constants
- What’s the difference for abelian vs non-abelian theory?
- …

Discuss the figure about decay channels of Higgs Boson:

- At low Higgs-mass: why is the branching ratio to bb- about 10 times bigger than tau+tau-?
- How can a Higgs-particle decay into 2 photons?
- How is it possible that Higgs decays into WW starting from 80GeV, even though their mass is bigger than that of Higgs?
- Explain the dip of ZZ at +-160GeV?

And many more little questions I’ve already forgotten…

27 May 2021

Credit to Students of UGent Media:Examenvragen-Electroweak-and-strong-interaction.pdf

Question 1

Consider the figure below:

• Give a qualitative explanation for relative strength of the branching ratio to bb with respect to ${\displaystyle \tau \tau }$ for low BEH masses
• Give an explanation with Feynman diagrams of leading order decay of a scalar boson to a gluon-gluon pair and a photon-photon pair.
• Define the concepts of partial decay width, total decay width and branching ratio of a decay process and explain the dip in the ZZ producing process branching ratio
• Discuss the production of a WW and ZZ boson pair from the decay of scalar boson

and explain when these modes are possible and give the respective decay process. Justify why the WW branching ratio is significantly higher than the ZZ branching ratio.

Question 2

Explain what the Hierarchy problem is for the Higgs Boson.

Question 3

What are the running couplings? And why does one constant increases with increasing energy and the other constant become smaller?

28 juni 2019

The exam had similar questions as the exams of the previous years. Professor Sevrin considered this as the easiest exam he made during this examination period, compared to the exams in Gent and Brussels and the other exam in Leuven.

Question 1

Consider a Higgs triplet

${\displaystyle \Phi (x)=\left({\begin{matrix}\phi _{1}(x)\\\phi _{2}(x)\\\phi _{3}(x)\end{matrix}}\right),}$

where the lagrangian density in the scalar sector is given by

${\displaystyle {\mathcal {L}}=\partial _{\mu }\Phi ^{\dagger }\partial ^{\mu }\Phi +m^{2}\Phi ^{\dagger }\Phi -\lambda (\Phi ^{\dagger }\Phi )^{2},}$

and where ${\displaystyle \Phi }$ transforms under the three dimensional representation of ${\displaystyle SU(2)}$ with the ${\displaystyle SU(2)\times U(1)}$ gauge transformation given by

${\displaystyle \Phi (x)\to \Phi '(x)=\exp(ig\,t_{a}\alpha ^{a}(x))\exp(ig'q_{S}\beta (x))\Phi (x).}$

The matrices ${\displaystyle t_{a}}$ obey the usual ${\displaystyle SU(2)}$ commutation relations and are given by

${\displaystyle t_{1}={\frac {1}{\sqrt {2}}}\left({\begin{matrix}0&1&0\\1&0&1\\0&1&0\end{matrix}}\right),\qquad t_{2}={\frac {i}{\sqrt {2}}}\left({\begin{matrix}0&-1&0\\1&0&-1\\0&1&0\end{matrix}}\right),\qquad t_{3}=\left({\begin{matrix}1&0&0\\0&0&0\\0&0&-1\end{matrix}}\right).}$

1) Discuss the groundstate(s) of the system.

2) Take as groundstate for the system

${\displaystyle \Phi ={\frac {1}{\sqrt {2}}}\left({\begin{matrix}0\\0\\v\end{matrix}}\right),}$

with ${\displaystyle v={\sqrt {(}}m^{2}/\lambda )}$. Determine the weak hypercharge ${\displaystyle q_{S}}$ such that the photon remains massless.

3) Give the Higgs multiplet in the unitairy gauge. An extra question on the oral part of the exam was to determine the electrical charge of the ${\displaystyle \phi _{1}}$ part of the the Higgs multiplet.

4) Determine the masses of the W and the Z gauge bosons.

5) Can you write down Yukawa couplings in this representation?

Question 2

We consider now the theory as it is used in the standard model and as we built it during the lectures. The coupling of the Brout-Englert-Higgs particle ${\displaystyle \sigma }$ to the leptons and the quarks ${\displaystyle \psi }$ is given by

${\displaystyle {\mathcal {L}}_{HF}=-{\frac {1}{v}}m_{\psi }\sigma {\bar {\psi }}\psi ,}$

and the coupling to the W- and Z-bosons is given by

${\displaystyle {\mathcal {L}}_{HVB}={\frac {vg^{2}}{2}}\sigma W_{\mu }^{\dagger }W^{\mu }+{\frac {g^{2}}{4}}\sigma ^{2}W_{\mu }^{\dagger }W^{\mu }+{\frac {vg^{2}}{4\cos ^{2}\theta _{W}}}\sigma Z_{\mu }Z^{\mu }+{\frac {g^{2}}{8\cos ^{2}\theta _{W}}}\sigma ^{2}Z_{\mu }Z^{\mu }.}$

The fermion masses are given by

${\displaystyle m_{e}=0.51\cdot 10^{-3}\quad m_{\mu }=0.11\quad m_{\tau }=1.8\quad m_{u}=1-5\cdot 10^{-3}\quad m_{d}=3-9\cdot 10^{-3}}$ ${\displaystyle m_{c}=1.15-1.35\quad m_{s}=0.075-0.17\quad m_{t}=170\quad m_{b}=4.0-4.4}$

A plot of the braching ratios of the decay of the Higgs particle was given similar to that on the last page of the exam of 2011.

1) Compare the Higgs-fermion coupling strenght to the electromagnetic coupling strength of the fermions. Use ${\displaystyle g\sin \theta _{W}=e=g'\cos \theta _{W}}$.

2) Consider the ${\displaystyle b{\overline {b}}}$ and ${\displaystyle \tau \tau }$ decay channels and explain.

3) Consider the ${\displaystyle gg}$ and ${\displaystyle \gamma \gamma }$ decay channels and explain. It is not necessary to explain the detailed form of the curves but a few Feynman diagrams would be nice.

4) Explain the form of the curves of the decay to WW and ZZ. The masses are approximately ${\displaystyle m_{W}=80\,GeV/c^{2}}$ and ${\displaystyle m_{Z}=90\,GeV/c^{2}}$. An extra question on the oral part was to explain why the branching ratio of the Higgs decaying to the W boson is bigger than the branching ratio of the decay to the Z-boson, since the later is heavier so one could expect the opposite situation.

31 May 2019

Very similar questions as the previous years. An extra question on the oral part was to explain why the branching ratio of the Higgs decaying to the W boson is bigger than the branching ratio of the decay to the Z-boson, since the later is heavier so one could expect the opposite situation.

8 juni 2015

Identiek hetzelfde examen als 11 juni 2012.

22 juni 2009

Vraag 1

Toepassing van het Higgs mechanisme.

Gegeven zijn twee scalaire velden ${\displaystyle \Phi _{1}}$ en ${\displaystyle \Phi _{2}}$ die als doubletten onder ${\displaystyle SU(2)_{L}}$ transformeren en beide hebben zwakke hyperlading ${\displaystyle Q_{Y}=1/2}$. We gebruiken de notatie: ${\displaystyle \Phi _{1}=\left(\phi _{11}\quad \phi _{12}\right)^{T}\quad \Phi _{2}=\left(\phi _{21}\quad \phi _{22}\right)^{T}}$ met alle ${\displaystyle \phi }$kes complexe getallen.

• Stel dat de vacuumsverwachtingswaarden van de scalaire velden door

${\displaystyle \Phi _{1}=1/{\sqrt {2}}\left(0\quad v_{1}\right)^{T}}$ en ${\displaystyle \Phi _{2}=1/{\sqrt {2}}\left(0\quad v_{2}\right)^{T}}$ gegeven worden met v1 en v2 reeel. Wat zijn de massa's van de W, de Z, en het foton?

• Veronderstel dat de Lagrange dichtheid voor de scalairen door

${\displaystyle {\mathcal {L}}=\partial _{\mu }\Phi _{1}^{\dagger }\partial ^{\mu }\Phi _{1}+\partial _{\mu }\Phi _{2}^{\dagger }\partial ^{\mu }\Phi _{2}+m_{1}^{2}\Phi _{1}^{\dagger }\Phi _{1}+m_{2}^{2}\Phi _{2}^{\dagger }\Phi _{2}-\lambda _{1}(\Phi _{1}^{\dagger }\Phi _{1})^{2}-\lambda _{2}(\Phi _{2}^{\dagger }\Phi _{2})^{2}-\lambda _{3}(\Phi _{1}^{\dagger }\Phi _{1})(\Phi _{2}^{\dagger }\Phi _{2})}$ gegeven is waar m1, m2 reeel en lamda_j>0 voor j in {1,2,3}. Bepaal v1 en v2 als functie van de diverse koppelingsconstantes.

• Bespreek de fysische vrijheidsgraden van de scalaire velden (maw geef een bondige analyse van de unitaire ijk). Toon bv aan dat het fysische elektrisch geladen veld door ${\displaystyle -\sin \beta \phi _{11}+\cos \beta \phi _{21}}$ gegeven wordt met ${\displaystyle \tan \beta =v_{2}/v_{1}}$.

Vraag 2

Op zoek naar het Brout-Englert-Higgs deeltje. (een grote grafiek van de verschillende branching ratios is gegeven: [1], fig. 3 op p.8 - de figuur voor het exaam was ietsje anders, maar kwam op hetzelfde neer.)

De koppeling van het Higgs (${\displaystyle \sigma }$) aan leptonen of quarks (${\displaystyle \psi }$) heeft de vorm ${\displaystyle {\mathcal {L}}_{HF}=-(1/v)m_{\psi }\sigma {\bar {\psi }}\psi }$ en deze van de Higgs aan de W en Z velden ${\displaystyle {\mathcal {L}}_{HVB}=(vg^{2}/2)\sigma W_{\mu }^{\dagger }W^{\mu }+(g^{2}/4)\sigma ^{2}W_{\mu }^{\dagger }W^{\mu }+(vg^{2}/4\cos ^{2}\theta _{W})\sigma Z_{\mu }Z^{\mu }+(g^{2}/8\cos ^{2}\theta _{W})\sigma ^{2}Z_{\mu }Z^{\mu }}$ Hierin worden de massa's gegeven door (alle massa's werden uitgedrukt in GeV/c^2 en komen van PDG) ${\displaystyle m_{e}=0.51\cdot 10^{-3}\quad m_{\mu }=0.11\quad m_{\tau }=1.8\quad m_{u}=1-5\cdot 10^{-3}\quad m_{d}=3-9\cdot 10^{-3}}$ ${\displaystyle m_{c}=1.15-1.35\quad m_{s}=0.075-0.17\quad m_{t}=170\quad m_{b}=4.0-4.4}$

• Op het volgend blad vind je de branching ratios voor het Higgs verval als functie van de Higgsmassa (${\displaystyle m_{H}=v{\sqrt {2\lambda }}}$). Geef een kwalitatieve bespreking van de bb, tautau, gammagamma, gg, WW en ZZ kanalen.