FPP2 (First half) Flashcards
How is the third component of isospin calculated?
1/2 * (N_u - N_d’)
What is the sign convention for quark flavour numbers?
down-type quarks have negative flavour numbers.
i.e: strange quark has -1 strangeness
How can we link electric charge to the third component of isospin?
Q = I_3 + 1/2 * Y
Y is hypercharge
How is hypercharge defined?
Sum of baryon number and all 2nd and 3rd gen quark flavour numbers.
How does parity transformation affect angular momentum?
Angular momentum (and therefore spin) is conserved under P as:
L = r x p
i.e: double negative.
What is the effect of a time-reversal transformation?
Changes the sign of any kinetic property. (including spin)
*NOT charge
How do electric and magnetic fields transform under C P and T transformations?
Consider:
E ~ qr -changes sign under P, C
B ~ qv x r -changes sign under C, T
What are the eigenvalues of C, P and T transformations of QM state vectors?
A phase eigenvalue, parametrised:
eta_X = e^(i epsilon_X)
where X is the type of transformation being applied
What is an eigenstate of C transformation?
A neutral charged state
*don’t forget phase eigenvalue factor
What is the parity transformation eigenvalue of a state of 2 particles.
What about 3 particles?
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
What relationship between parity eigenvalue and total angular momentum J is defined to be “natural”?
What are some examples of “natural” particles?
What about “unnatural”?
parity eigenvalue = (-1)^J is “natural”
1^- is “natural”: photon, rho, phi
0^- is “unnatural”: pions, kaons
*these parity eigenvalues I’m talking about are simply the phase factors associated with a parity transformation of an eigenstate
What proportion of the J/Psi decay cross-section will be leptonic decays?
J/Psi is a c-cbar 1^- meson state.
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?)
How are mesons and anti-mesons defined for neutral meson states?
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)
How can we construct CP eigenstates for neutral mesons?
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.
What are the Sakharov conditions for the baryon asymmetry of the universe?
1: Baryon number violating processes.
2: C and CP symmetry violating processes.
3: These processes must occur out of thermal equilibrium.
What was the finding of the Wu experiment?
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).
What is the GIM formalism?
For a 2 quark generation model, describes the relationship between quark weak (flavour) eigenstates and strong (mass) eigenstates.
What is the form of the GIM matrix?
What experiment could we use to measure the cabibbo angle?
d = Cos() -Sin() d’
s = Sin() Cos() s’
*d’ 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 <d’|d> = cos() and so on.
The ratio of these decay rates can be used to find the cabibbo angle.
How can we show the absence of flavour-changing neutral currents in the GIM formalism?
Consider d’ dbar’ + s’ sbar’
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.
How did the measurement of CP violation in kaons mean there must be a third quark generation?
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.
How many parameters does the CKM matrix have?
4
Three rotation angles and one complex phase.
Or Wolfenstein form.
What is the benefit of the Wolfenstein form of the CKM matrix?
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)
How do we mathematically describe mixing?
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.
What is the nature of mixing transition probability and what parameters is it sensitive to?
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.
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
How can we physically describe mixing?
Box diagrams with two W boson transitions. Intermediate quarks can be of three different flavours, with different supressions.
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.
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)
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.
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.
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.
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.
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.
What do we additionally need to work to study mixing of neutral mesons?
Their flavour at production AND decay.
(and decay time)
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.
