Chapter 17. Weak Interactions

Transcription

1 Chapter 17 Weak Interactions The weak interactions are meiate by W ± or (netral) Z exchange. In the case of W ±, this means that the flavors of the qarks interacting with the gage boson can change. W ± coples to qark pairs (,). (c,s), (t,b) with vertices c s t b as well as to leptons (ν e,e). (ν µ,µ), (ν τ,τ) with vertices ν e e ν µ µ ν τ τ Note that in these interactions both qark nmber (baryon nmber) an lepton nmber are conserve. It is this process that is responsible for β-ecay. Netron ecays into a proton becase a -qark in the netron converts into a -qark emitting a which then ecays into an electron an anti-netrino. 125

2 n{ }p e ν e The amplite for sch a ecay is proportional to g 2 W (q 2 M 2 W c2 ), where g W is the strength of the copling of the to the qarks or leptons an q 2 = E 2 q /c2 q 2, where q is the momentm transferre between the netron an proton an E q is the energy transferre. This momentm is of orer 1 MeV/c an so we can neglect it in comparison with M W c which is 80.4 GeV/c. Ths the amplite is proportional to g 2 W M 2 W c2. The copling g W is not so small. In fact it is twice as large as the electron charge e. Weak interactions are weak becase of the large mass term in the enominator. At moern high energy accelerators, it is possible to proce weak interaction processes in which q M W c or even q M W c. In sch cases weak interactions are larger than electromagnetic interactions an almost comparable with strong interactions Cabibbo Theory Particles containing strange qarks, e.g. K ±, K 0, Λ etc. cannot ecay into non-strange harons via the strong interactions, which have to conserve flavor, bt they can ecay via the weak interactions. This is possible becase W ± not only coples a -qark to a -qark bt can also (with a weaker copling) cople a -qark to an s-qark so we have a vertex s g W sinθ c with copling g W sinθ C, whereas the W copling is actally g w cosθ C. θ C is calle the Cabibbo angle an its nmerical vale is sinθ C

3 This copling allows a strange haron to ecay into non-strange harons an(sometimes) leptons. Ths, for example the ecay Λ p + e + ν e occrs when an s-qark converts into a qark an emits a which then ecays into an electron an anti-netrino. The Feynman graph is Λ{ s }p e ν e Likewise, thec-qarkhasacoplingtothes-qarkwithcoplingg W cosθ C anacopling to a -qark with copling g W sinθ C. c s c g W cosθ C g W sinθ C This implies that charm harons are more likely to ecay into harons with strangeness, becase the copling between a c-qark an a s-qark is larger than between a c-qark an a -qark. We can piece this together in a matrix form as follows ( ) ( )( ) cosθ g W s C sinθ C sinθ C cosθ C c This 2 2 matrix is calle the Cabbibo matrix. It is escribe in terms of a single parameter, the Cabibbo angle. Since we know that there are, in fact, three generations of qarks this matrix is extene to a general 3 3 matrix as follows ( ) g W s b V V s V b V c V cs V cb V t V ts V tb c t 127

4 The 3 3 matrix is calle the CKM (Cabibbo, Kobayashi, Maskawa) matrix. Qantmmechanical constraints lea to the conclsion that of the nine elements there are only for inepenent parameters. Comparing the CKM matrix with the Cabibbo matrix we see that to a very goo approximation, V V cs cosθ C an V s V c sinθ C Leptonic, Semi-leptonic an Non-Leptonic Weak Decays Becase the W ± coples either to qarks or to leptons, ecays of strange mesons can either be leptonic, meaning that the final state consists only of leptons, semi-leptonic, meaning that the final state consists of both harons an leptons, or non-leptonic, meaning that the final state consists only of harons. For strange baryons only semi-leptonic an non-leptonic ecays are possible becase baryon nmber is strictly conserve - so there mst be a baryon in the final state. Lepton nmber is also strictly conserve which means that a charge lepton is always accompanie by its anti-netrino (or vice versa) in the final state. For mesons, examples are: Leptonic ecay K µ + ν µ K { s ū ν µ µ As well as converting an s-qark into a -qark to emit a, it is also possible to create a from the annihilation of an s-qark with a ū anti-qark. Semi-leptonic ecay K µ + ν µ + π 0 ū K { s ū }π 0 µ ν µ Non-leptonic ecay K π 0 + π 128

5 K { s ū ū ū }π 0 } π Note that m K > 2m π which is why this non-leptonic ecay moe is energetically allowe. In the case of baryons, we have alreay seen an example of a semi-leptonic ecay, Λ pe ν e. An example of a non-leptonic ecay is Λ p π ū}π Λ{ }p s A is exchange between the s-qark an the -qark in the Λ, converting them into a -qark an a -qark respectively. A ū qark-antiqark pair is create in the process in orer to make p the final state harons of a proton an a negative pion Flavor Selection Rles in Weak Interactions Since in the exchange of a single W ± an s-qark can be converte into a non-strange qark, it is highly nlikely that two strange qarks wol be converte into non-strange qarks in the same ecay process. We therefore have a selection rle for weak ecay processes S = ±1 Therefore, harons with strangeness -2 which ecay weakly mst first ecay into a haron with strangeness -1 (which in trn ecays into non-strange harons). Ths, for example, we have Ξ 0 Λ + π 0 The same selection rles apply for changes in other flavors (charm, bottom). 129

6 17.4 Parity Violation The parity violation observe in β-ecay arises becase the W ± tens to cople to qarks or leptons, which are left-hane (negative helicity), i.e. states in which the component of spin in their irection of motion is 1 2. W ± always cople to left-hane netrinos. For qarks an massive leptons the W ± can cople to positive helicity (right-hane) states, bt the copling is sppresse by a factor mc 2 E, where m is the particle mass an E is its energy. The sppression is mch larger for relativistically moving particles In the case of nclear β-ecay, the ncles is moving non-relativistically, bt the electron typically has energy of a few MeV (an a mass of MeV/c 2 ), so there is a significant sppression of the copling to right-hane electrons. This is what was observe in the experiment by C.S. W on 60 Co. Forthecopling ofw ± toanti-qarksoranti-leptons, thehelicity isreverse -i.e. thew ± always coples to positive helicity anti-netrinos an sally to positive helicity e +, µ +, τ + or to antiqarks, with a sppresse copling to left-hane antileptons or anti-qarks. A striking example of the conseqence of this preferre helicity copling can be seen in the leptonic ecay of K +. K + µ + + ν µ ν K + µ µ + L.H L.H In the rest frame of the K + the momentm is zero, so the µ + an the ν µ mst move in opposite irections. The K + has zero spin, so by conservation of anglar momentm, the two ecay particles mst have opposite spin component in any one chosen irection (e.g. the irection of the µ +. This means that they have the same helicity. This means that the W ± coples to the left-helicity anti-mon, µ + an sch a copling is sppresse by If we look at the ecay moe m µ c 2 E µ K + e + + ν e, the same argment wol lea to a sppression (of the ecay amplite) of m e c 2 E e. 130

7 Since m e m µ we expect the ecay into a positron to be heavily sppresses. In fact we expect the ratio of the partial withs Γ(K + µ + ν µ ) Γ(K + e + ν e ) = m2 µ m 2 e This coincies very closely to the experimentally observe ratio Z-boson interactions As well as exchange of W ± in which flavor is change, the weak interactions are also meiate by a netral gage-boson, Z. This coples to both qarks an leptons bt oes not change flavor. In that sense the interactions of the Z are similar to that of the photon, bt there are some important ifferences. The Z coples to netrinos whereas the photon oes not (netrinos have zero electric charge). The Z has a mass of 91.1 Gev/c 2, so the interactions are short range - like the interactions of the W ±. The Z also has a copling of ifferent strength to left-hane (negative helicity) an right-hane(positive helicity) qarks an leptons an so these interactions also violate parity. Nevertheless, in any process where there can be photon exchange, there can also be Z exchange. In terms of Feynman iagrams for e + e scattering into any pair of final state particles, we have e e + γ bt also e e + Z The first iagram (photon exchange) has a propagator 1/s, 131

8 where s is the centre-of-mass energy, whereas the secon iagram (Z exchange) has a propagator 1 s M 2 Z c4. For relatively low centre-of-mass energies for which s M Z c 2, the secon iagram may be neglecte an the secon iagram gives a negligible contribtion. Bt as s grows to become comparable (or greater than) M Z c 2 both of these iagrams are eqally important. The Z an photon can both cople to W ±, so we get interaction vertices Z an γ The interaction between the photon an W ± is not srprising since the W ± are charge an we wol expect them to interact with photons, with copling e. The interaction of W ± with the Z is similar bt has a ifferent copling. The copling of the Z an photon to the W ± was confirme at the LEPII experiment at CERN where it was possible to accelerate electrons an positrons to sfficient energies to proce a an a in the final state. From the copling of the W to electron an netrino the Feynman iagram for this process is e e + ν bt becase of the copling of the Z an photon to W ± we also have iagrams e e e + γ e + Z The ata from LEPII clearly show that these graphs have to be taken into accont 132

9 It trns ot that the Stanar Moel of weak an electromagnetic ( electroweak ) interactions, evelope in the 1960 s by Glashow, Weinberg, an Salam, gives a relation between the weak copling g W, the (magnite of the ) electron charge, e an the masses of the Z an W ± M W = cosθ W M Z where θ W is known as the weak mixing angle. e = g W sinθ W = g W 1 M2 W MZ 2 This enables s to make an orer of magnite estimate of the rates for weak processes at low energies. At energies M W c 2, the amplite for a W ± exchange process is proportional to so that the rate is proportional to g 2 W 4πǫ 0 M 2 W c4, ( gw 2 ) 2. 4πǫ 0 MW 2 c4 Now for a weak ecay rate we want imensions of inverse time, so we nee to mltiply this by something with imensions of the forth power of energy ivie by time. The only qantity proportional to the energy is the Q vale of the ecay, Q β an to get inverse time we can ivie by so we get an estimate Rate ( g 2 W 4πǫ 0 cm 2 W c4 133 ) 2. Q5 β

10 The pre-factor is actally qite small. For example, for mon ecay Q β m µ c 2, an the mon ecay rate is actally 1 τ µ = ( ) 1 g 2 2 W m 4 µ m µ c 2 768π 3 4πǫ 0 c MW 4. an We know g 2 W 4πǫ 0 c = e 2 4πǫ 0 csin 2 θ W = α sin 2 θ W sin 2 θ W = 1 M2 W. MZ 2 Therefore from the measre masses of the W an Z we can etermine the mon lifetime The Higgs mechanism There is one frther particle preicte by the Stanar Moel of electroweak interactions which has not yet been iscovere. This arises from the mechanism, iscovere by P.Higgs, by which particles acqire their mass. The basic iea is that there exists a fiel, φ calle the Higgs fiel which has a constant non-zero vale everywhere in space. This constant vale is calle the vacm expectation vale, φ. In the absence of this fiel it is assme that all particles wol be massless an wol travel with velocity c. Bt becase of their interaction with the backgron Higgs fiel they are slowe own - thereby acqiring a mass, M g H M = 1 2 ǫ0 c φ, where g H is the copling of the particle to the Higgs fiel ( the enominator factor ǫ 0 c gives it the correct imensions.) This mechanism is part of the Stanar Moel. The Higgs fiel coples to W ± with copling g W so that g W M W = 1 2 ǫ0 c φ. Inserting g W = e/sinθ W with cosθ W = M W /M Z an M W = 80.4GeV/c 2, an M Z = 91.2GeV/c 2, we get the vale of the vacm expectation vale φ = 250GeV/c 2 Other particles cople to the Higgs fiel with coplings that are proportional to their mass. 134

11 In the same way that there are qanta of the electromagnetic fiel which are particles (photons), so there mst be qanta of the Higgs fiel. These are calle Higgs particles. They mst necessarily exist if the Higgs mechanism for generating masses for particles is to be consistent with qantm physics. As it was mentione in the introction, the Higgs boson was iscovere on the 4th of Jly 2012 by ATLAS an CMS collaborations at the LHC, completing the set of particles of the Stanar Moel. With a high confience level this particle is confirme to have the following properties: 1. It has a spin zero. This is consistent with the theoretical preictions since the vacm expectation vale has to be invariant ner Lorentz transformations - so that it is the same in all frames of reference. 2. Higgs boson coples to W ± an Z (which are conseqently massive). 3. It oes not irectly cople to photons (which are massless) so it is ncharge. 4. it oes not not cople irectly to glons (which are massless) an so it oes not take part in the strong interactions. 5. Its copling to massive particles is proportional to the particle mass. 6. Its mass is measre to be abot 125 GeV/c 2. Diagrams for proction mechanisms of the Higgs boson at the LHC are shown below (left) together with the respective cross sections (right). They incle: (a) glon fsion, (b) weak-boson fsion, (c) Higgs-strahlng (or associate proction with a gage boson) an () associate proction with top qarks processes. Note, that the first process is the loopince one: while Higgs boson oes not interact irectly with massless glons, it actally can interact with glons via virtal massive qarks (e.g. top-qarks which the strongest copling to the Higgs boson) in the triangle loop iagram. Actally glon fsion is the main proction process of the Higgs boson, while the weak-boson fsion plays the next to leaing role. Theoretical ncertainties are represente by the withs of the cross section bans. 135

12 Higgs boson ecay is ominate by the most massive particles allowe by its mass becase its copling to particles is proportional to the mass. The t-qark mass is 175 GeV/c 2 so it cannot ecay into a t t pair. The next most massive qark is the b-qark so Higgs boson preominantly ecays into a b b pair, shown in iagram (a) below. Higgs boson is not sfficiently massive to ecay into real or two real Z particles, however it can ecay to one real an another virtal W or Z boson (W an Z ) followe by their ecay into fermion-antifermion pair as shown in iagrams (b) an (c). As in case of glon fsion proction process, Higgs boson can also ecay into photon pair via its interactions with virtal top qark an W-boson as shown in iagram (). Higgs boson also ecays to τ + τ pair, the ominant leptonic ecay channel since τ-lepton is the most massive amongst the leptons. The respective branching ratios for Higgs boson ecay channels are shown in the right frame of the figre below as a fnction of the Higgs boson mass. Higgs boson iscovery was base not on the process with the highest proction an ecay rates, which wol be naively the gg H b b process. Actally it was was base on the processes with optimal signal-to-backgron ratio an the highest signal significance. In particlar, one of the most significant an cleanest signatres comes from H γγ ecay for which Stanar Moel backgron is relatively low. Another very important an significant signatre is base on H ZZ 4leptons ecay which also provie clean 4-lepton signatre. 136