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The Weak Nuclear Force

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The Weak Nuclear Force The Weak Nuclear Force Contents Fermi Theory and Parity Violation Higgs Mechanism Breaking Goldstones theorem massive W s Symmetry ercharge and the Z Hyp Fermion Masses and Cabibb o Mixing Higgs Searches Outcomes Learning Know main exp erimental results on the weak force Be able to explain the U Higgs mechanism in detail T Y Know Q 3 3 Be able to write down W B mass matrix and diagonalize it Be able to explain how the Higgs generates fermion masses Know the Higgs search metho ds Reference Bo oks Intro duction to High Energy Physics Perkins Elementary Particle Physics Kenyon The Weak Force We now turn our attention to the description of the weak nuclear force and the asso ciated gauge theory We will have to explore the Higgs mechanism of symmetry breaking to explain the gauge b oson masses Intro duction The weak nuclear force is resp onsible for decay of nuclei n p e e This was the pro cess which lead Pauli to predict the existence of the neutrino in order to reconcile the exp erimental data with energy and momentum conservation A t the quark level the pro cess is d u e e The force is a weak force in day to day life b ecause it is very short range Fermi successfully describ ed the force by p ostulating a Feynman rule for a four fermion rst interaction at a p oint In mo dern language we would write d u e− _ ν e M u u G u u F p n e 2 where G is the Fermi constant Note it replaces the q typ e bit of the gauge F b oson propagator we have previously seen in this typ e of expression G therefore F 2 2 5 has dimension of mass exp erimentally it is ab out GeV Such a theory on its own though is not a renormalizable theory at lo op level we get many divergences that may not b e absorb ed into parameters of the theory The story will get considerably more complicated but lets start to see the mix of ideas that will enter A Massive Propagator The Feynman propagator for a massless particle controlled by the Klein Gordon 2 2 If we intro duce a mass term 2 m we may equation 2 is iq treat it as a p erturbation to the massless theory The Feynman rule that emerges turns out to b e 2 im Now we can work out the massive propagator by resumming the p erturbation series for this particle i i i i i i 2 2 2 im im im 2 2 2 2 2 2 q q q q q q 2 4 m i m 2 2 4 q q q 1 2 m i 2 2 q q i 2 2 q m 2 Note that at small momentum the propagator b ecomes a constant im This suggest that G in the description of the weak force should corresp ond to the mass of a massive F exchange particle SU There are hints of an SU gauge theory in the weak force decay pro cess SU transformations act on a doublet of wave functions as a a i T e a where the T are the generators of SU which are just the Pauli matrices i 2 1 3 T T T i However to place u d or e in doublets we would require them to b e identical e particles which they manifestly are not in low energy exp eriments A gauge theory would also predict the existence of three massless SU gauge b osons which we do not observe Parity Violation An added confusion to the mix is provided by Mdm Wus exp eriment that shows the weak force violates parity She studied the decay of Cobalt in a magnetic eld 60 60 C o N i e e The spins in the exp eriment align as Bz J=5 J=4 J=1/2 J=1/2 One would exp ect the electrons to emerge at all angles If we take the massless o o limit and think ab out helicity in this exp eriment we can see that at and there could b e pro cesses o e at 0 e ν RH LH o e at 180 e ν LH RH Exp erimentally though one measures an angular distribution of the emerging elec o trons that ts the prole cos In other words at there is no pro duction This means that the weak force violates parity and only couples to left handed par ticles In the massless limit where left and right handed particles decouple in the Dirac equation there is no reason why this should not b e the case The confusion we will have to resolve is how this can b e consistent with the electron having as mass that t helicity states mixes these dieren The Higgs Mechanism The key additional idea we need to make sense of the weak force is the Higgs Mech anism We will now explore these ideas in simple environments b efore returning to the full theory of nature In tro duction Our rst task will b e to understand how to give masses to gauge b osons This seems like a hard task given that the much prized gauge symmetry principle actively forbids such masses The Higgs mechanism intro duces an extra ingredient we will ll space with matter charged under the gauge symmetry A gauge b oson then trying to propagate through this go o will interact continually and we will see that the energy of interaction will b ecome the eective mass of the gauge b oson Mass is a scalar quantity and this tells us the background go o must b e a scalar eld or wave function the Higgs do the basics of the maths in QED where we will give the photon a mass Lets We have seen how such a charged particle with a scalar wave function satisfying the Klein Gordon equation would enter into the QED Maxwell equations 2A J iq D D the current is just the charge times the numb er density current but where we have set D iq A as minimal substitution requires Now imagine that throughout space v we say it has a vacuum exp ectation value vev often written hi v Spatial derivatives of v are just zero and the only surviving bits in the wave equation ab ove are 2 2 2A q v A or 2 2 2 2 2 m A m q v which makes the gauge eld massive as we wanted When we give a vev we are eectively changing the normalization of the wave function so there are a large numb er of particles p er unit volume we can think of as a classical eld in this limit Well well just co ok things Why would have this nonzero vacuum value though so it do es without answering that question Heres an example p otential for that do es the trick V v Φ This lo oks a bit adho c but so far so go o d We will see that since this background go o has electric charge all electrically charged particles will end up coupling to it and get a mass as a result We will generate the electron mass etc in the SU version of this story This all lo oks quite simple but to understand how we have got a mass and gauge invariance together we will need to think more deeply ab out what were doing Breaking Symmetry The reason we can suddenly have a mass for the gauge eld in the Higgs Mechanism is b ecause the symmetry is b eing broken by the vacuum of the theory To b egin to understand this idea think ab out a ferromagnet Ab ove the critical temp erature the atomic spins are randomly p ointing due to thermal excitation Below the critical temp erature the thermal energy is insucien t to disrupt a natural alignment of the spins due to their interactions T > Tc T < Tc spins random spins aligned In the absence of defects the direction in which the spins cho ose to nally lie arranged b elow T is truly random in a quantum theory The nal arrangement c picks out a particular direction in space and therefore breaks rotational symmetry that existed at high temp erature or without the atoms present This is the basic idea of sp ontaneous symmetry breaking Breaking an Internal Symmetry Consider a complex scalar eld with a global U internal symmetry i e We can write a p otential for the eld 2 2 V 2 When is p ositive the p otential lo oks like V Re φ Im φ The minimum is at The p otential clearly exhibits the U symmetry 2 If is negative though the p otential lo oks like V Re φ Im φ 2 4 Close to the origin the jj term dominates whilst at large values the jj term dominates There is a ring of minima To nd the value of jj lo ok on the real axis and require dV 2 3 d so the minima is at 2 2 jj The value of the p otential at the minimum is 4 V min Globally the p otential again exhibits the U symmetry the p otential do es not dep end on the phase of However the minima have a symmetry breaking form there are many degenerate vacua and which one we end up in will b e random When one lo oks lo cally around a minimum there is not a rotational symmetry Theorem Goldstones In the ab ove example there is a massless state We can see this by writing i 2 e r r The p otential do es not dep end on Thus there is no energy cost to a conguration where changes in space V vacuum 1 vacuum 2 Re φ derivative energy Im φ vacuum 2 vacuum 1 The only energy cost is due to the derivative from the changing eld In terms of the Klein Gordon equation though 2 2 m we know that derivatives of are not a mass but momentum So this state is massless This is not true for the r degree of freedom If we change r there is a direct energy cost from the p otential which corresp onds to a mass Goldstones Theorem there is a massless degree of freedom for every broken generator of a group In the ab ove case the U degree of freedom is broken so there is one massless mo de If we were to break an SU symmetry with three generators we would therefore exp ect to nd massless mo des in the theory Is this a blo ck to describing the weak force as a broken symmetry there are no physical massless mo des In fact in a gauge theory something extra happ ens Sup erconductivity The rst example of a broken gauge symmetry to b e understo o d was in the phenom ena of sup erconductivity In a sup erconductor interactions with the lattice of ions pro duces forces that bind electrons into Co op er pairs a charge spinless
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