The Standard Model Part. II

Transcription

1 Our Story Thus Far The Standard Model Part. II!!We started with QED (and!)!!we extended this to the Fermi theory of weak interactions! Adding G F!!Today we will extended this to Glashow-Weinberg-Salam theory (or EWK Theory) # e µ $ e # µ$! Adding " W 2 Part III: Glashow-Weinberg-Salam Theory A Troublesome Event; Prestigious Discovery # µ$ µ $ A Charged Current (CC) event the outgoing muon has a different charge than the incoming neutrino. A picture of an event from the CERN Gargamelle bubble chamber. # µ$ # µ$ A Theory with 3 Parameters!! They were trying to unify QED with a theory of weak interactions with helps from Higgs mechanism,renormalization (Veltman,t-Hooft, B.Lee)..! Their theory predicted weak neutral currents and the W & Z bosons 3 This incomprehensible plot from 1973 shows Neutral Weak-Currents. A Neutral Current (NC) event the outgoing neutrino has the same charge as the incoming neutrino. 4

2 A Troublesome Event The Roadmap!! We want QED to come out! So let s start by putting it in The Gargamelle bubble chamber, now exhibited at CERN # µ$ # µ$!! We want a left-handed weak theory to come out! So let s start by putting it in! Let s pick a small group that will let us have charged currents! and at least the possibility of neutral currents *** one of the most important discoveries ever made at CERN; the first experimental observation of the weak neutral currents in 1973, shortly after their theoretical prediction. A Neutral Current (NC) event the outgoing neutrino has the same charge as the incoming neutrino. 5!! Once we ve done this, we will match terms to pull QED out!! What s left will be our (new) theory of weak interactions 6 First Ingredient: Weak Hypercharge Second Ingredient: Weak Isospin!! This is where the Lie Algebra formalism starts to help us:! We start with a U(1)! This means that we will end up with a theory just like QED! Instead of a field A µ, we call it B µ! Instead of a charge q, we have a hypercharge Y.! Replace qa µ with YB µ in the Lagrangian and we re done.!! Weak Hypercharge: a conserved quantum number relating the electrical charge (Q) and the third component of weak isospin (T 3 ). It is frequently denoted Y W and corresponds to the gauge symmetry U(1).!! Its associated quantum field B mixes with the W 3 EWK quantum field to produce the observed Z 0 gauge boson and the photon. 7!! In this case, our gauge group is SU(2)! This is the same algebra that governs angular momentum addition! (i.e SU(2) is the symmetry associated with spin)! It s a non-abelian group the fields themselves carry weak isospin!! Weak Isospin is:!! a quantum number relating to the weak interaction!! usually given the symbol T or I with the third component written as T z, T 3, I z or I 3.!! The weak isospin conservation law relates the conservation of T 3 ; all weak interactions must preserve T 3.!! For this reason T 3 is more important than T and often the term "weak isospin" refers to the "3rd component of weak isospin".!! W bosons have a T = 1, with 3 different values of T 3.!! W + (T 3 = +1) is emitted in transitions {(T 3 = +1 2)! (T 3 = "1 2)},!! W " (T 3 = "1) is emitted in transitions {(T 3 = "1 2)! (T 3 = +1 2)}.!! W 0 (T 3 = 0) would be emitted in reactions where T 3 does not change. However, under EWK unification, the W 0 mixes with the weak hypercharge gauge boson B, resulting in the observed Z 0 boson and the photon. 8

3 Second Ingredient: Weak Isospin!! In this case, our gauge group is SU(2)! This is the same algebra that governs angular momentum addition! It s a non-abelian group the fields themselves carry weak isospin Interaction Between these Fields and Matter Weak Hypercharge Weak Isospin!! We have three fields: w 1, w 2 and w 3. (c.f. Aµ in U(1))! They interact among themselves in this way w 2 w 1 w1!! Here g 1 and g 2 are the coupling constants for hypercharge and isospin!! The j s (J Y, J 1, J 2, J 3 ) are the fermion currents w 1!! Our next step is to declare electric charge to be: Q =!Y + T 3.! this specifies which w is neutral and which are charged w 3 w 2 w A Change of Basis Charged Currents: What Exactly Have We Done? We should now cast everything in terms of electric charge we want QED to fall out of this: µ$ #$ #$ e We ve replaced the 4-fermion interaction with two vertices where the fermion current couples to a W field. µ$ #$ W #$ e Then becomes µ$ W We just calculated this vertex factor: it s Now we have everything we need to calculate the same process in the old 4 -fermion theory and the new GWS theory. If we do this, and match the results, we get: #$ Positive Negative Neutral 11 12

4 Neutral Currents We have no guarantee that the w 3 and B are the physical fields (in fact, they aren t) so we introduce a rotation matrix to mix them. (in the new basis) g 1 (or g) = e/cos" w : coupling constant for hypercharge g 2 (or g ) = e/sin" w : coupling constant for isospin where " w : EWK mixing angle Neutral Currents We have no guarantee that the w 3 and B are the physical fields (in fact, they aren t) so we introduce a rotation matrix to mix them. More On Neutral Currents!! Matching up terms gives us a relation between the weak and EM couplings! They are NOT INDEPENDENT (in the new basis)!! We can plug this into the Z term, and get f Z By matching terms, we effectively calculated this vertex factor: it s Note that the weak coupling is larger than the EM coupling:! weak ~ 1/30 vs. 1/137 for! EM. But we already know from QED what this has to be! So we simply match up terms f The weak force is weak not because the coupling is small, but because the W is heavy

5 More On Neutral Currents Some Predictions!! Matching up terms gives us a relation between the weak and EM couplings! They are NOT INDEPENDENT!! We can plug this into the Z term, and get This evaluates to 77.5 GeV. The measurement is 80.4 GeV. Using the measured W mass, this evaluates to 91.6 GeV. The measurement is 91.2 GeV. By matching terms, we effectively f Fermion c A c V Z calculated this vertex factor: it s Neutrinos # # f Charged Leptons -# -# + 2 sin 2 " w (-0.04) Up-type quarks # # - 4/3 sin 2 " w (0.19) Down-type quarks -# -# + 2/3 sin 2 " w (-0.34) Z decays Calculated BF Measured BF Leptons (per flavor) 3.4% 3.4% Invisible 20.5% 20.0% All jets 69.1% 69.9% Bottom quark jets 15.2% 15.5% All these predictions use " W measured independently - from neutrino experiments The W Boson in Pictures The Z Boson In Pictures From the original discovery (UA1/UA2) This is a UA2 event UA1 OPAL DELPHI CDF D0 CDF Missing E T (neutrino) Electron momentum L3 ALEPH 19 20

6 Additional Consequences of SU(2) We originally introduced SU(2) because we wanted the W to be charged ( charged current ). That means it has to couple to the photon: W + W + %$ W+Z Events CDF D0 This is the physical manifestation of the 3-field interaction in the unbroken SU(2) theory. The physical states are projections of w 1, w 2 and w 3. w 3 w 1 w 2 But these aren t the only projections one can make. W + W + Z 0 W electron Z muon Not only does this theory predict the existence of these events, the rate of these events is completely determined. The theory predicts the W couples to the Z as well. Z muon W neutrino A Word on Symmetries!! This theory is often described as SU(2) x U(1)!! The unbroken symmetry is SU(2) Left x U(1) Hypercherge! We broke both symmetries when we declared the w 3 component to be electrically neutral! We will discuss another way to break these symmetries later Fixing the Unitarity Problem DONE?!! The Fermi theory has a problem with unitarity violation above 300 GeV! There must be new physics below that.! We added new physics below that: the W and Z s at 80 and 90 GeV!! We ve fixed that problem! The cross-sections drop like they are supposed to. 23 *** Reminder: Fermi theory: x-sec up with E e.g. &(e# " e#) ~ G 2 F S At sufficient high energies, the probability of some process happening > 1 In the Fermi theory: sqrt(s) ~ 300 GeV What about GWS? -- see next page 24

7 Fixing the Unitarity Problem Almost GWS Theory Scorecard!! Imagine you could collide beams of W bosons: WW! WW!! If you calculate this in GWS theory, it too violates unitarity! Specifically, one piece: W L W L! W L W L! New physics has to enter in below about 1 TeV!! So we haven t so much as solved the problem as moved it to higher energy. Potential Troublemaker!! The Good:! Matches every test against data we could think of! Predicts new phenomena, confirmed by experiments! W and Z bosons! Neutral Weak Currents! Diboson production at colliders! Explains everything with just three numbers! G F, the strength of the weak force,!, the strength of the EM force, and " w, how they mix! Fixes the 300 GeV Unitarity problem of the Fermi Theory!! The Bad:! Theory breaks down above ~1 TeV! Symmetry broken by hand Our next step fixing these problems by adding one more piece to the theory Our Story Thus Far Part IV The Higgs Mechanism!!We started with QED (and!)!!we extended this to the Fermi theory of weak interactions! Adding G F!!We extended this to Glashow-Weinberg-Salam theory! Adding " W A Theory with Four Parameters 27 28

8 Spontaneous Symmetry Breaking (SSB)!! Experimentally, the weak bosons (W/Z) are massive!!!! Why?!! Because we give mass to the gauge bosons through the some mechanism!! Here, we re having a big problem!!!! A mass term for the gauge field is not invariant under the gauge transformation!! so, at some point, we need to break the gauge symmetry!! More precisely, spontaneously break the symmetry!!!! What s the meaning of symmetry breaking? -- new phase with more D.O.F.. Spontaneous Symmetry Breaking (SSB)!! Experimentally, the weak bosons (W/Z) are massive!!!! Why?!! Because we give mass to the gauge bosons through the some mechanism!! Here, we re having a big problem!!!! A mass term for the gauge field is not invariant under the gauge transformation!! so, at some point, we need to break the gauge symmetry!! More precisely, spontaneously break the symmetry!!!! What s the meaning of symmetry breaking? -- new phase with more D.O.F..!! Q: HOW? & what is the sector responsible for the breaking SU(2)? with which dynamics?!! A: introduce a complex scalar doublet (Higgs doublet). generate mass term from the KE term of a scalar doublet field ' that undergoes SSB # this add 4 new D.O.F. (~ 4-real scalar fields) " 1 Higgs boson + 3 Goldstone bosons (W1,W2,W3) EWSB in SM The Mysterious Mexican Hat ) +$ ) 0$!! The potential has an infinite number of minima! This is an example of spontaneous symmetry breaking Q: vacuum? bound state in QFT = minimum point of potential If SSB, several vacuum state possible according to symmetry!! If we flip the sign of (, we have a paraboloid 31 32

9 The Mysterious Mexican Hat II C. Hill Misunderstanding the Mexican Hat S. Carroll!! The actual potential is a multi-dimensional Mexican hat.! We have 4 fields: coordinates in the potential.! One becomes the Higgs.! The other 3 become Goldstone bosons.!! There is no energy penalty for circular motion! These are called Goldstone Bosons and are massless (that s what no energy penalty ) means!! Where did these Goldstones go?! They were eaten by the W(Z) s.!! The potential minimum is at ) = M/( = 246 GeV! Excitations about this minimum will become the Higgs boson Eaten? This theory has interactions like this: w + ) + w +!!So, true story is! When the symmetry is broken by Higgs mechanism "! Create Nambu-Goldstone bosons! so, the W + (no mass yet) has eaten the ) + (Goldstone boson) and gained a mass to become the physical (or massive) W +. Our Four Parameter Theory!! We started with a theory explaining the electroweak interaction with three numbers: G F,! and sin 2 (" w ).! The W mass had to be put in by hand! This was not gauge invariant!! The Higgs mechanism lets us generate a massive W and Z naturally.! This is at a cost of one (not two) extra parameters! Either the Higgs mass or its self-coupling! It leaves the photon massless! It keeps gauge invariance!! This theory predicts one new particle! a fundamental scalar, the Higgs boson 35 36

10 Let s Not Go Overboard!! What I described is a theory, not a fact.!! This theory could be wrong on a number of counts:! The Higgs mechanism might not be the correct theory of EWSB! It could be strong dynamics (e.g. Technicolor ), where resonances between the W s and Z s break the symmetry! It could be a top quark condensate where the top quark plays a special role! There might be multiple Higgs bosons! One gives mass to the W s and Z s, and a totally different one gives mass to the quarks! One could give mass to some quarks, and a different one (or ones) to some other. Electroweak Radiative Corrections!! Remember, the W mass was calculated to be 77.5 GeV: about 4% low - why?! Electroweak radiative corrections! Or, loops!! Vacuum polarization causes a mass shift of both the W and the Z! Effect is quadratic in fermion mass! Top quark dominates! Effect is logarithmic in the Higgs mass Stomping Out Nonsense!! One sometimes reads that there are no radiative corrections to the Z.! Of course there are the same sort of loops for the W (w 1, w 2 ) are there for the Z (w 3 )!! There are several possible choices for the three parameters of the theory.! We used G F,!, and " w,! M(Z) is more popular than G F.!! If the measured mass is an input parameter, one doesn t worry about corrections! The corrections have already been included but that doesn t make them zero! W and Z Masses We started with the tree level prediction of the GWS theory. Loops cause a few % correction to this *R ~ 7-8%: depends mostly on fermion masses Is good to better than #%. That s because there are radiative corrections to both the W and the Z. The differences depend on which particles circulate in the loops. t t Z Z W W 39 40

11 State of the Art!! This is from a fit of all the world s data.! The top mass is from hadron colliders! The W mass is dominated by LEP, but hadron colliders are now overtaking them. Measuring the W Mass at a Hadron Collider p z for the neutrino isn t measured, so we can t measure m(w). The best we can do is the transverse mass. Fortunately, the transverse mass distribution is a function of the true mass.!! The relatively poor constraint on the Higgs mass is because the dependence is only logarithmic D0 Missing E T (neutrino) CDF Electron momentum Loops and the Higgs So loops have a few % impact on the W mass. Do they have any effect on the Higgs mass?!! Excellent question.!! Radiative corrections to the Higgs mass are enormous, and want to push it up to a very high scale. This is a problem.! Remember, the Higgs had to have a mass less than ~1 TeV to restore unitarity to WW scattering.! Also, we know M/( = v.e.v. (246 GeV).! If M gets large, so does (, and now we have a strongly-coupled theory.!! A number of solutions have been proposed. 43