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EXAMS Summer 2023 > FPP2 (First half) > Flashcards

FPP2 (First half) Flashcards

1
Q

How is the third component of isospin calculated?

A

1/2 * (N_u - N_dā€™)

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2
Q

What is the sign convention for quark flavour numbers?

A

down-type quarks have negative flavour numbers.

i.e: strange quark has -1 strangeness

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3
Q

How can we link electric charge to the third component of isospin?

A

Q = I_3 + 1/2 * Y

Y is hypercharge

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4
Q

How is hypercharge defined?

A

Sum of baryon number and all 2nd and 3rd gen quark flavour numbers.

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5
Q

How does parity transformation affect angular momentum?

A

Angular momentum (and therefore spin) is conserved under P as:

L = r x p

i.e: double negative.

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6
Q

What is the effect of a time-reversal transformation?

A

Changes the sign of any kinetic property. (including spin)
*NOT charge

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7
Q

How do electric and magnetic fields transform under C P and T transformations?

A

Consider:
E ~ qr -changes sign under P, C
B ~ qv x r -changes sign under C, T

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8
Q

What are the eigenvalues of C, P and T transformations of QM state vectors?

A

A phase eigenvalue, parametrised:

eta_X = e^(i epsilon_X)

where X is the type of transformation being applied

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9
Q

What is an eigenstate of C transformation?

A

A neutral charged state

*donā€™t forget phase eigenvalue factor

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10
Q

What is the parity transformation eigenvalue of a state of 2 particles.

What about 3 particles?

A

2 particles: product of their intrinsic parity eigenvalues * (-1)^L.
L is the angular momentum between the two particles.

3 particles: product of their intrinsic parity eigenvalues * (-1)^L * (-1)^l.
L is the angular momentum between two particles, l is the angular momentum between the third particle and the COM of the first two.

-intrinsic parity eigenvalues are the phase factors

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11
Q

What relationship between parity eigenvalue and total angular momentum J is defined to be ā€œnaturalā€?

What are some examples of ā€œnaturalā€ particles?
What about ā€œunnaturalā€?

A

parity eigenvalue = (-1)^J is ā€œnaturalā€

1^- is ā€œnaturalā€: photon, rho, phi
0^- is ā€œunnaturalā€: pions, kaons

*these parity eigenvalues Iā€™m talking about are simply the phase factors associated with a parity transformation of an eigenstate

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12
Q

What proportion of the J/Psi decay cross-section will be leptonic decays?

J/Psi is a c-cbar 1^- meson state.

A

u ubar * 3
d dbar * 3
s sbar * 3
e- e+
mu- mu +

i.e: 2 / 11

(Not really sure what the vibe is with this, just a crude estimate?)

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13
Q

How are mesons and anti-mesons defined for neutral meson states?

A

Neutral mesons have a heavy quark with positive flavour number. (i.e: heavy up-type quark or down-type antiquark)
Neutral anti-mesons have a heavy quark with negative flavour number. (i.e: heavy up-type antiquark or down-type quark)

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14
Q

How can we construct CP eigenstates for neutral mesons?

A

Form a superposition state of the nuetral meson and neutral anti-meson state.
Choosing the phase to be 0 (arbitrary) gives a simple +/- superposition normalised by 1/sqrt(2).

These states are CP eigenstates with eigenvalue +/- 1.

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15
Q

What are the Sakharov conditions for the baryon asymmetry of the universe?

A

1: Baryon number violating processes.
2: C and CP symmetry violating processes.
3: These processes must occur out of thermal equilibrium.

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16
Q

What was the finding of the Wu experiment?

A

That parity is maximally violated in the beta- decay of Cobalt-60.

Electrons are nearly always emitted with negative helicity (as the W boson only couples to LH particles and RH antiparticles).

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17
Q

What is the GIM formalism?

A

For a 2 quark generation model, describes the relationship between quark weak (flavour) eigenstates and strong (mass) eigenstates.

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18
Q

What is the form of the GIM matrix?

What experiment could we use to measure the cabibbo angle?

A

d = Cos() -Sin() dā€™
s = Sin() Cos() sā€™

*dā€™ is a weak (flavour) eigenstate, d is a strong (mass) eigenstate

Consider pi+ (u dbar) decay to W+
and K+ (u sbar) decay to W+. W bosons only couple to weak eigenstates, so consider <dā€™|d> = cos() and so on.

The ratio of these decay rates can be used to find the cabibbo angle.

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19
Q

How can we show the absence of flavour-changing neutral currents in the GIM formalism?

A

Consider dā€™ dbarā€™ + sā€™ sbarā€™

i.e: the coupling of Z bosons to quarks in the lagrangian. Expanding these terms in strong eigenstates leads to d dbar + s sbar, i.e: no d sbar or s dbar couplings.

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20
Q

How did the measurement of CP violation in kaons mean there must be a third quark generation?

A

CP violation requires a complex phase which is not possible in a 2x2 matrix (due to constraints of unitarity and rotation invariance).

Therefore there must be at least 3 quark generations.

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21
Q

How many parameters does the CKM matrix have?

A

4
Three rotation angles and one complex phase.

Or Wolfenstein form.

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22
Q

What is the benefit of the Wolfenstein form of the CKM matrix?

A

Illustrates the hierarchy of the elements. Approximate form.

Parametrised in terms of A~1 and lambda ~ 0.22 (as well as two others).

Diagonal elements ~1
1-2 off-diagonal ~ lambda
2-3 off-diagonal ~ lambda^2
1-3 off-diagonal ~ lambda^3
(factors of lambda relate to cabibbo supression)

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23
Q

How do we mathematically describe mixing?

A

Consider the effective hamiltonian that describes decay into either neutral meson state.

Eigenstates of this effective hamiltonian are superpositions of the flavour eigenstates of the neutral meons, and are not generally orthogonal.

Hamiltonian eigenstates have defined lifetime. However, production and decay are only defined for flavour eigenstates.

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24
Q

What is the nature of mixing transition probability and what parameters is it sensitive to?

A

General exponential decay behaviour.
Sensitive to:

x (mass difference) - sinusoidal modulations of transition rate in time
y (width difference) - modifies decay rate (hyperbolic term)

If either x or y are non-zero (non-degeneracy of neutral meson states) then the mixing transition is possible.

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25
What causes the difference between mixing probabilities we theorize and those we see in experiment?
Amplitude of oscillation decreased. --due to misidentification of meson states as each other. (vertical averaging) --due to decay time resolution (horizontal averaging) Depletion at short decay times. --result of high impact parameter selection requirement
26
How can we physically describe mixing?
Box diagrams with two W boson transitions. Intermediate quarks can be of three different flavours, with different supressions.
27
What three components affect decay width?
The number of available final states. Coupling strength to final states. Available phase space for decay to each final states.
28
What are the main methods of producing flavoured particles?
Fixed-target (Hadron beam, hadron can be chosen by exploiting momentum) e+e- collider (Definite COM energy so can be tuned to produce specific resonances) Hadron collider (High COM energy, used to produce heavy flavours)
29
Why is good temporal resolution needed for fixed-target experiments with heavy flavour hadrons?
The beam is not periodic so there is background from pileup.
30
Which methods of producing flavoured particles give the best cross-sections?
Fixed-target best for low-energy resonances. Hadron colliders also give high cross-sections, especially for heavy flavours compared to other methods. e+e- colliders give low cross-sections.
31
Why must we initially use only single-particle triggers?
If we begin to try and reconstruct composite particles before making tight candidate particle selections, combinatorics result in it becoming far too computationally expensive.
32
How do Cherenkov detectors allow us to measure particle mass?
We can measure particle velocity from the opening angle of the Cherenkov light. Combined with momentum measurement from the tracker this gives mass.
33
What is impact parameter and how is it useful in triggering?
The minimum distance between the interpolated particle track and the primary vertex. Particles from heavy flavour decays will have high impact parameter due to significant lifetime of heavy resonances. As this is a property of individual particles, we can select for composite states without the combinatoric issue.
34
What do we additionally need to work to study mixing of neutral mesons?
Their flavour at production AND decay. (and decay time)
35
How are neutral kaon states distinguished?
Kaon hamiltonian eigenstates are denoted K-long and K-short due to their large difference in lifetime. Detectors must be built to study one or the other, not both.
36
How are neutral D meson states distinguished?
For production via strong decay: D*+ --> D0 and pion. The pion charge determines the D flavour. D0 is a (cu) neutral meson
37
How are neutral B meson states distinguished?
The initial state is always a bbar as bottom flavour number = 0 in initial state. Therefore B meson states are usually produced b2b, with opposite b/anti-b. Using same-side or opposite-side tagging, where the charge of an associated kaon is used to infer B flavour. (Kaon associated with the production of hadronisation of the b-quark) *check in notes Mixing is a confounding factor.
38
What is the relationship between decay rate and decay amplitude?
Decay rate is the modulus squared of the amplitude. (Modulus squared of [sum of individual amplitudes])
39
What happens to weak and strong phases under CP transformation?
Weak phase changes sign. Strong phase does not.
40
What is an approximate form of CP asymmetry of a decay? What can we infer from this?
2r * sin(strong phase difference) * sin(weak phase difference) r is ratio of suppressed / leading amplitudes. Therefore CP violation requires several amplitudes otherwise r = 0. CP violation also requires both strong and weak phase difference.
41
What does the term "colour suppressed" refer to?
When the final state of a decay must be in some colour state: therefore there is a factor of 3 suppression compared to non-colour-suppressed decay. i.e: if both decay products of a meson decay contain quarks that were in the original meson state, then the colours of the decay products are fixed.
42
What kind of amplitudes are important for flavour physics?
Tree-level Exchange Gluonic penguin Annihilation Electroweak penguin
43
What are two examples of detector systematic asymmetries?
B-field deflects oppositely-charged particles to opposite sides of the detector, which may be asymmetric. Matter/antimatter particles have different rates of interaction with the detector medium.
44
How can we try to mitigate systematic sources of asymmetry?
Use a control mode that is affected by the same systematic (production/detection) asymmetries (matched kinematic properties of initial and final state particles). Ideally this mode would have known asymmetry. We can then subtract systematic asymmetries from the signal mode. A_meas. = A_CP + A_prod. + A_det.
45
How are we able to fully show the dynamics of a 3-body decay using a Dalitz plot?
9DOF (momenta) - 4DOF(energy/momentum conservation) - 3DOF(rotation invariance) = 2DOF We usually choose two of the (pairwise) invariant masses of the decay products.
46
What does the banding of resonances in a Dalitz plot tell us?
The spin of the resonance. 0: solid band 1: 1 gap in band 2: 2 gaps in band However, where there are multiple resonances that overlap there will be interference (due to complex amplitudes).
47
What does "interference" CP violation refer to?
CP violation due to the interference of decay and mixing.
48
What does CP violation in mixing refer to?
The transition probability from one neutral meson state to the other is not equal to the reverse transition. i.e: |q/p| != 1
49
Why are semi-leptonic decays usually used for measuring CP violation in mixing?
We ideally want to identify flavour at both production and decay. Semi-leptonic decays allow us to infer the flavour at decay via the charge of the lepton (hence the charge of the W boson).
50
What do we require for CP violation in mixing?
|q/p| != 1 i.e: hamiltonian eigenstates are not equal superpositions of flavour eigenstates. (we also obviously need mixing in the first place - non-degeneracy)
51
What do we require for CP violation in interference?
Non-zero mass difference between hamiltonian eigenstates. The relevant final state must be accessible to both the meson and anti-meson. There is no requirement for CPV in mixing or decay alone.
52
How do we distinguish between CKM matrix elements and their complex conjugates?
CKM matrix elements relate "incoming" down-type quarks to "outgoing" up-type quarks. ("" as is related to the arrow direction not momenta direction) The complex conjugates are the opposite.
53
What does the unitarity of the CKM matrix imply?
V(h.c.) V = 1 --> off-diagonal elements = 0 e.g: V_ud V'_ub + V_cd V'_cb + V_td V'_tb = 0 These can be parametrised as "unitarity triangles" with 3 angles. (primes used to denote * due to silly brainscape)
54
How can we attempt to measure the magnitude of CKM matrix elements?
Using leptonic or semi-leptonic decays (clean signal and easy flavour ID). The decay amplitude for a charged-current interaction will be directly proportional to the relevant CKM matrix element. However, we need to know the other components of the matrix element (e.g: hadronic form factors, kinematic factors...) to extract the CKM element.
55
How can we attempt to measure the angles of CKM unitarity triangles?
These angles are related to the complex phases of the CKM matrix elements, and so we can attempt to measure them via measurement of CP violation. *using 3-body decays allows us to separate out weak phase difference from strong phase difference and r. (strong phase difference varies over phase space but weak does not)
56
What kind of decays are used to measure CKM angle gamma?
B --> DK decays (only tree level as there are no quark/antiquark pairs in final state - no penguin diagrams, also don't want loops as these introduce potential BSM physics) Two possible processes for this decay lead to weak phase difference = gamma.
57
Imagine a diagram that shows how both a strong and weak phase difference between two amplitudes is required for CP asymmetry.
Check notes (CKM angle gamma measurement)
58
How can we work out strong phase difference?
Use multi-body (3 or more) decays.
59
Why are we interested in rare/forbidden decays?
Could be sensitive to BSM physics as BSM effects that would otherwise be negligible (hence why we haven't seen them yet) could be relatively large compared to heavily suppressed SM amplitudes.
60
How can we see flavour-changing neutral currents in the SM?
Via box diagrams with 2 W bosons. Hence at any one point the transition is neutral. These are usually heavily suppressed due to high-order, GIM cancellation, CKM suppression. Considered a rare decay.
61
How could we see lepton-flavour violation within the SM?
Neutrino mixing This is considered a forbidden decay due to the extreme suppression of neutrino mixing at these scales.
62
What kind of BSM/SM model could lead to lepton number violation? Give an example diagram.
Majorana neutrinos (neutrinos are their own antiparticle). *check wall notes
63
What would be the experimentally cleanest signature for a BSM particle decay?
Decay into a di-muon pair
64
What does the sensitivity of a direct search depend on?
S / sigma_S S = # signal candidates sigma_S = error = sqrt(S+B)
65
What is a consequence of CPT symmetry?
Flavour eigenstates (i.e: particles/antiparticles) have equal mass, width, magnetic dipole moment...
66
How can we produce tau leptons?
Resonant production at e+e- colliders. Off-resonance production at e+e- colliders (e.g: B-factories). Pair production via Z decay. Hadron collisions e.g: D decay.
67
How many muon decay processes are there in the standard model?
Only one at first order: electron and two neutrinos. (no hadronic decay as pion mass is greater than muon mass)
68
How can we produce muons?
Best way is via fixed-target pion production. Predominant pion decay is to muon+neutrino.
69
What is electric dipole moment relevant for? How do we measure it?
Relevant to objects with several constituent charges, sensitive to BSM physics. Measure using spin precession.
70
Why is the anomalous magnetic moment of the muon so interesting?
First order g-factor is = 2. Any departure from this is related to higher-order interactions and sensitive to BSM contributions. QED contributions are predicted to very high accuracy. QCD contributions are far less accurate. There is currently a ~4sigma discrepancy in the measured and predicted values of the muon magnetic moment.
71
How do we measure muon magnetic moment?
By measuring both cyclotron frequency and spin precession frequency we can extract the magnetic moment. (in reality there are other complicated EM effects that must be accounted for) The measurement of muon magnetic moment by fermilab is the highest ever precision particle physics experiment.
72
What are the benefits and drawbacks to studying D0 mesons that are produced from B meson decays?
The B meson decays via a W boson to the D0 meson state. If this is leptonic, the D0 meson flavour can be inferred from the lepton charge. However, the missing momentum due to the neutrino increases background and decreases momentum resolution. Another benefit is that the significant lifetime of the B meson allows lots of background to be removed, and also the D0 meson decay can be studied down to very low decay time from the secondary vertex.
73
What kind of flavour-changing neutral current interactions could be observed?
quark transition (between up-type or between down-type, different flavour), producing pair of leptons. e.g: K^0 or B^0_s to mu+mu-
74
Why do charged pions decay to muons 99.99% of the time? ***this also is the case for charged Kaons, although in this case there are also hadronic decays.
Pion has spin zero, so decay products must have antiparallel spins due to L conservation. Momenta must also be in opposite directions due to momentum conservation. As neutrinos are ~massless, we must have coupling of a W boson to a RH charged lepton or LH charged antilepton, which is forbidden. However, massive states are superpositions of chiral states, and as muons are much more massive than electrons, the allowed chiral component can be larger.
75
What is the time-evolution of hamiltonian eigenstates?
*check wall notes e^(-i m t - 0.5 width t) * state
76
What is the quark content of: *pi+, pi- *K+, K-, K0, Kbar0 *D+, D-, D0, Dbar0 *B+,B-,B0, Bbar0 *B_s0, Bbar_s0 *rho? *J/psi? Gamma?
*dbar-u, d-ubar (0 states are superpositions) *sbar-u, s-ubar, sbar-d, s-dbar *c-dbar, cbar-d, c-ubar, cbar-u *bbar-u, b-ubar, bbar-d, b-dbar *bbar-s, b-sbar *rho are just pions but 1^- (pion is 0^-) *c-cbar(1^-), b-bbar(1^-)
77
What are the masses of quarks in ascending order?
up, down, strange, charm, bottom, top
78
What are the masses of these particles? e mu tau pion kaon B meson W Z H
e : 0.5 MeV mu : 0.1 GeV tau : 1.78 GeV pion : ~0.14 GeV kaon : ~0.5 GeV B meson : ~5.3 GeV W : 80 GeV Z : 91 GeV H : 125 GeV
79
Summarise non-degeneracies in these neutral meson states: K0 D0 B0 B_s0
K0 has large differences in both mass (x) and width (y). D0 has no x or y: no mixing. B0 has some x and no y. B_s0 has more x and no y.
80
What essential feature in the weak interaction gives rise to CP-violation in the Standard Model?
CP violation by the weak interaction in the SM arises from the single complex phase in the CKM matrix.
81
Write down the CKM matrix in terms of its elements Vqq0, where q = u, c, t and q 0 = d, s, b. What does the CKM matrix relate?
up-type go along x-axis down-type go along y-axis *check wall notes weak eigenstates = V * strong (mass) eigenstates
82
What is the value of the wolfenstein parameter lambda?
0.22
83
How is decay rate (width) related to amplitudes?
~ amplitudes squared *remember this particularly when considering CKM elements for a Fdiagram
84
What is the difference between a C and CP transformation in defining particles / antiparticles?
C symmetry simply switches the charges. CP switches charges and momenta, so helicities are also switched. This is why both C and CP violation are required for the Sakharov conditions, as if we only have C, the opposite helicity interaction will balance out the asymmetric one.
85
What is notable about neutral kaon mixing?
Kaon hamiltonian (mass) eigenstates are ALMOST CP eigenstates, as p and q are almost 1/sqrt(2).
86
Can you remember all the terms in the Wolfenstein parametrisation of the CKM matrix?
Can you? lol lmao
87
What is interaction rate equal to?
Particle flux * cross section * number target particles
88
Describe a Majorana neutrino LNV Feynman diagram: D+ decay to two mu+ and a pi-
*check wall notes

EXAMS Summer 2023 (6 decks)

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