Spontaneous Symmetry Breaking and the Higgs Mechanism in High Energy Physics

1. The Standard ModelLecture II Thomas J. LeCompte High Energy Physics DivisionArgonne National Laboratory

2. Our Story Thus Far • We started with QED (and a) • We extended this to the Fermi theory of weak interactions • Adding GF • We extended this to Glashow-Weinberg-Salam theory • Adding qW

3. GWS Theory Scorecard • The Good: • Matches every test against data we could think of • Predicts new phenomena, borne out by experiment • W and Z bosons • Neutral Weak Currents • Diboson production at colliders • Explains everything with just three numbers • GF, the strength of the weak force, a, the strength of the EM force, and qw, how they mix • Fixes the 300 GeV Unitarity problem of the Fermi Theory • The Bad: • Theory breaks down above ~1 TeV • Masses put in by hand • Breaks gauge invariance • Symmetry broken by hand Our next step – fixing these problemsby adding one more piece to the theory.

4. Part IV – The Higgs Mechanism A Theory with Four Parameters

5. Spontaneous Symmetry Breaking What is the least amount of railroad track needed to connect these 4 cities?

6. One Option I can connect them this way at a cost of 4 units. (length of side = 1 unit)

7. Option Two I can connect them this way at a cost of only 3 units.

8. Option Three This requires only

9. The Real Optimal Solution This requires Note that the symmetry of the solution is lower than the symmetry of the problem: this is the definition of Spontaneous Symmetry Breaking. + n.b. The sum of the solutions has the same symmetry as the problem.

10. A Pointless Aside One might have guessed at the answer by looking at soap bubbles, which try to minimize their surface area.But that’s not important right now… Another Example of Spontaneous Symmetry Breaking Ferromagnetism: the Hamiltonian is fully spatially symmetric, but the ground state has a non-zero magnetization pointing in some direction.

11. The Higgs Mechanism • Write down a theory of massless weak bosons • The only thing wrong with this theory is that it doesn’t describe the world in which we live • Add a new doublet of spin-0 particles: • This adds four new degrees of freedom(the doublet + their antiparticles) • Write down the interactions between the new doublet and itself, and the new doublet and the weak bosons in just the right way to • Spontaneously break the symmetry: i.e. the Higgs field develops a non-zero vacuum expectation value • Like the magnetization in a ferromagnet • Allow something really cute to happen

12. The Really Cute Thing • The massless w+ and f+ mix. • You get one particle with three spin states • Massive particles have three spin states • The W has acquired a mass • The same thing happens for the w- and f- • In the neutral case, the same thing happens for one neutral combination, and it becomes the massive Z0. • The other neutral combination doesn’t couple to the Higgs, and it gives the massless photon. • That leaves one degree of freedom left, and because of the non zero v.e.v. of the Higgs field, produces a massive Higgs. m = ±1 “transverse” m = 0 “longitudinal”

13. How Cute Is It? • There’s very little choice involvedin how you write down this theory. • There’s one free parameterwhich determines the Higgsboson mass • There’s one sign whichdetermines if the symmetrybreaks or not. • The theory leaves the Standard Model mostly untouched • It adds a new Higgs boson – which we can look for • It adds a new piece to the WW → WW cross-section • This interferes destructively with the piece that was already there and restores unitarity • In this model, the v.e.v. of the Higgs field is the Fermi constant • The sun shines for billions of years because of the Higgs mechanism and the spontaneously broken electroweak symmetry

14. Under The Hood • Let’s explore the “cuteness” • There is no freedom at all in howthe f interacts with the w1,w2 and w3. • We made it a doublet. • End of story. • We do have freedom in deciding how the f interacts with itself. The only freedom we have in the theory is the strength of the self-interaction l and its sign.If l > 0, the potential has a minimum for positive f2. Otherwise it’s minimized at f2 =0. This is the critical sign

15. The Mysterious Mexican Hat • We need to make a few observations • My choice of basis asf+ and f0 was terrible. • Only |f| appears in the potential • Don’t use rectangular coordinatesin a problem with circular symmetry! • The potential has an infinite number ofminima • Solution is r = M/l, q = anything • This is an example of sponteneoussymmetry breaking (think ferromagnet) • If I flip the sign of l, I have a paraboloid • Not very interesting f+ f0

16. C. Hill The Mysterious Mexican Hat II • There is no energy penalty for circular motion • These are called “Goldstone Bosons” and are massless (that’s what “no energy penalty”) means • The potential minimum is at |f| = M/l = 246 GeV • Excitations about this minimum will become the Higgs boson S. Carroll

17. Misunderstanding the Mexican Hat • We got one massless boson out. Nature gave us one masslessboson, the photon. Therefore the circular mode is the photon,right? • No. • The one massless boson nature gave us is not a Goldstone. • The actual potential is a five-dimensional “Mexican hat”. • You have 4 fields: coordinates in the potential. • One becomes the Higgs. • The other 3 become Goldstone bosons. • Where did these Goldstones go? • They were “eaten” by the w’s.

18. Eaten? This theory has interactions like this: • This is bad – we’d like a particle to have one identity and not flip back and forth • This came about because we started with the (unfortunate) “rectangular” basis • If we re-diagonalize to find the correct degrees of freedom, we discover that we’ve gone from two particles with 2 and 1 spin degrees of freedom to a single particle with 3. • The w+ has eaten the f+ and gained a mass to become the physical W+. w+ w+ f+ m = ±1 “transverse” m = 0 “longitudinal”

19. Our Four Parameter Theory • We started with a theory explaining the electroweak interaction with three numbers: GF, a and sin2(qw). • The W mass had to be put in “by hand” • This was not gauge invariant • The Higgs mechanism lets us generate a massiveW 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 • Weinberg and Salam knew all about this • This theory predicts one new particle • a fundamental scalar, the Higgs boson

20. Some Slight of Hand • I’ve swindled you. • Twice…the same way. • I added a Higgs doublet: • Note that one component is charged, the other neutral. • At this point, the theory doesn’t know anything about electric charge. The relevant quantum numbers are weak isospin and hypercharge • Properly, I should have put a more general doublet in: Had we done this, we would have found that one component of the doublet is electrically neutral. We would have learned that the component that gets a v.e.v. is the component that the photon does not couple to. We will end up with a massless photon and an uncharged v.e.v. component.

21. Taking Chirality Seriously • Thus far, we have assigned the chiral nature of the theory to the interaction. • We say the weak interaction is left-handed, and the vertex has a (1-g5) in it. • Instead, we could assign it to the fermion fields: instead of e, we have an eL and an eR. • These are massless fields. One cannot boost into a frame where an eL becomes an eR. • To get a massive physical field, one needs an interaction that mates a left-handed field with a right-handed field through a Yukawa coupling. Only one particle has the right quantum numbers…the Higgs.

22. The CKM Matrix and the Higgs Yesterday, we wrote down the CKM matrix this way – this takes a set of “real” mass eigenstates and tells us how much “theoretical” weak eigenstates are in each one. The Higgs links qL and qR fields just like it links eL and eR. There’s no guarantee that the Higgs interaction eigenstates – which we should call mass eigenstates – are the same as the weak eignestates. We need to include a matrix to rotate the weak states into the mass states. But we already have a matrix that does the inverse – the CKM matrix! Here we have the inverse of the CKM matrix. This then tells us about how the left-handed weak eigenstates link up with their right-handed partners to form mass eigenstates. I switched notation from dW to dL to emphasize that these are the same fields we have been discussing before,

23. Let’s Not Go Overboard • What I described is a theory, not a fact. • This theory could be wrong on a number ofcounts: • 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. I have described one possibility out of of many.

24. A Miracle? • It may seem miraculous that if I introduce a field like eL, I need to add an eR with exactly the right quantum numbers for the Higgs to mate them into a physical electron. • Not really – the theory not only permits this, it mandates it. • Remember anomalies? One of the requirements for a consistent theory is • I need right handed singlets in the theory • Exception: I don’t need right-handed neutrinos (why?) • How does nature know to make right-handed singlets to match left handed doublets? • How does a thermos know to keep hot things hot and cold things cold?

25. Testing the Standard Model

26. Electroweak Radiative Corrections • Remember, the W mass was calculated to be77.5 GeV: about 4% low - why? • “Electroweak radiative corrections” • Or, colloquially, “loops” • Vacuum polarization causes a mass shift of both the W and the Z • Just like there is a QED shift in g-2 • Effect is quadratic in fermion mass • Top quark dominates • Effect is logarithmic in the Higgs mass

27. Stomping Out Nonsense • One sometimes reads that there are no radiativecorrections to the Z. • Of course there are – the same sort of loops forthe W (w1, w2) are there for the Z (w3) • There are several possible choices for the three parameters of the theory. • I used GF, a, and qw, • M(Z) is more popular than GF. • 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!

28. W and Z Masses We started with the tree level prediction of the GWS theory. Loops cause a few % correction to this DR ~ 7-8%: depends mostly on fermion masses Is good to better than ½%. 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 W W Z Z

29. 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. • The relatively poor constraint on the Higgs mass is because the dependence is only logarithmic

30. Measuring the W Mass at a Hadron Collider pz 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. D0 CDF Missing ET (neutrino) Electron momentum

31. Systematic Uncertainties: The Key to the W Mass These systematics are statistically limited. These systematics are not. Today the world average uncertainty is 300 ppm. The best single measurement us good to 600 ppm.To match the top quark mass predictive power will require 100 ppm.

32. The Kind of Thing Experimenters Worry About Two leptons – do they see the same field? To 100 ppm? Detector’s B field

33. 88-89 Run Run I * Run II Tevatron results Rapidity – of getting m(W) results published The trend is for later runs to be on a curve which begins lower and to the right of earlier runs. No hadron collider experiment has published an uncertainty of 100 MeV in less than 1400 days.

34. Measuring m(W) – Why It Takes so Long • Set Momentum Scale • Use known states like Z0, J/y, and U family • As this is done, removing tracking systematic problems: • Misalignments, miscalibrations, twists, distortions, false curvatures, energy loss… • Set Energy Scale • Use electrons and “known” material and momentum scales • Recoil & Underlying Event Characterization • Modeling, Modeling, Modeling • Transverse mass vs. lepton pT vs. missing energy, QCD radiation, QED radiation, production models, underlying event, residual nonlinearities… It’s not unusual for >1000 plots to appear in the (complete) set of internal notes for this analysis

35. Difficulty 1: The LHC Detectors are Thicker • Detector material interferes with the measurement. • You want to know the kinematics of the W decay products at the decay point, not meters later • Material modeling is tested/tuned based on electron E/p • Thicker detector = larger correction = better relative knowledge of correction needed X~16.5%X0 (red line on lower plots) ATLAS material budget CMS material budget

36. No valence antiquarks at the LHC Need sea antiquarks and/or higher order processes NLO contributions are larger at the LHC More energy is available for additional jet radiation At the Tevatron, QCD effects are already ¼ of the systematic uncertainty Reminder: statistical and systematic uncertainties are comparable. q q W W g q q Difficulty 2 – QCD corrections are more important

37. Major Advantage – the W Rate is Enormous • The W/Z cross-sections at the LHC are an order of magnitude greater than the at the Tevatron • The design luminosity of the LHC is ~an order of magnitude greater than at the Tevatron • I don’t want to quibble now about the exact numbers and turn-on profile for the machine, nor things like experimental up/live time • Implications: • The W-to-final-plot rate at ATLAS and CMS will be ~½ Hz • Millions of W’s will be available for study – statistical uncertainties will be negligible • Allows for a new way of understanding systematics – dividing the W sample into N bins (see next slide) • The Z cross-section at the LHC is ~ the W cross-section at the Tevatron • Allows one to test understanding of systematics by measuring m(Z) in the same manner as m(W) • The Tevatron will be in the same situation with their femtobarn measurements: we can see if this can be made to work or not • One can consider “cherry picking” events – is there a subsample of W’s where the systematics are better?

38. Systematics – The Good, The Bad, and the Ugly • Masses divided into several bins in some variable • Masses are consistent within statistical uncertainties. Bad Ugly Good • Point to point the results are inconsistent • There is no evidence of a trend • Something is wrong – but what? • Clearly there is a systematic dependence on this variable • Provides a guide as to what needs to be checked.

39. W Mass Summary • ATLAS and CMS have set themselves some very ambitious goals in a 250 or 200 ppm W mass uncertainty - much less 80 ppm! • This will not be easy • This will not be quick • It might not even be possible • For example, suppose the PDF fits of the time simply have spreads that are inconsistent with better than a 25 MeV uncertainty. • Personal view: given time, the LHC will be competitive with the Tevatron. I wouldn’t want to speculate on how much or how little time this would take. • Even after the Higgs is discovered, this measurement is important • Finding one Higgs is not necessarily the same as finding all of them. • Indirect constraints will be important in interpreting the discovery

40. 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/l = v.e.v. (246 GeV). • If M gets large, so does l, and now we have a strongly-coupled theory. • A number of solutions have been proposed. • In my opinion, none of them are entirely satisfactory.

41. Another Test of the SM: Multiboson production Zzzzzzzzzzz

42. g g W+ Z0 W+ Z0 Z0 W+ W+ g Z0 Z0 Z0 Z0 & & W- Z0 g g Z0 g W- Z0 What Interactions are in the Standard Model? The (Electroweak) Standard Model is the theory that has interactions like: but not but not: Only three parameters - GF, a and sin2(qw) - determine all couplings.

43. The Semiclassical W • Semiclassically, the interaction between the W and the electromagnetic field can be completely determined by three numbers: • The W’s electric charge • Effect on the E-field goes like 1/r2 • The W’s magnetic dipole moment • Effect on the H-field goes like 1/r3 • The W’s electric quadrupole moment • Effect on the E-field goes like 1/r4 • Measuring the Triple Gauge Couplings is equivalent to measuring the 2nd and 3rd numbers • Because of the higher powers of 1/r, these effects are largest at small distances • Small distance = short wavelength = high energy

44. Triple Gauge Couplings • There are 14 possible WWg and WWZ couplings • To simplify, one usually talks about 5 independent, CP conserving, EM gauge invariance preserving couplings: g1Z, kg, kZ, lg, lZ • In the SM, g1Z = kg = kZ = 1 and lg = lZ = 0 • Often useful to talk about Dg, Dk and Dl instead. • Convention on quoting sensitivity is to hold the other 4 couplings at their SM values. • Magnetic dipole moment of the W = e(1 + kg + lg)/2MW • Electric quadrupole moment = -e(kg - lg)/2MW2 • Dimension 4 operators alter Dg1Z,Dkg and DkZ: grow as s½ • Dimension 6 operators alter lg and lZ and grow as s • These can change either because of loop effects (think e or m magnetic moment) or because the couplings themselves are non-SM

45. Why Center-Of-Mass Energy Is Good For You Approximate Run II Tevatron Reach Tevatron kinematic limit • The open histogram is the expectation for lg = 0.01 • This is ½ a standard deviation away from today’s world average fit • If one does just a counting experiment above the Tevatron kinematic limit (red line), one sees a significance of 5.5s • Of course, a full fit is more sensitive; it’s clear that the events above 1.5 TeV have the most distinguishing power From ATLAS Physics TDR: 30 fb-1

46. Not An Isolated Incident • Qualitatively, the same thing happens with other couplings and processes • These are from WZ events with Dg1Z = 0.05 • While not excluded by data today, this is not nearly as conservative as the prior plot • A disadvantage of having an old TDR Plot is from ATLAS Physics TDR: 30 fb-1Insert is from CMS Physics TDR: 1 fb-1

47. Not All W’s Are Created Equal • The reason the inclusive W and Z cross-sections are 10x higher at the LHC is that the corresponding partonic luminosities are 10x higher • No surprise there • Where you want sensitivity to anomalous couplings, the partonic luminosities can be hundreds of times larger. • The strength of the LHC is not just that it makes millions of W’s. It’s that it makes them in the right kinematic region to explore the boson sector couplings. From Claudio Campagnari/CMS

48. TGC’s – the bottom line • Not surprisingly, the LHC does best with the Dimension-6 parameters • Sensitivities are ranges of predictions given for either experiment

49. Reconstructing W’s and Z’s quickly will not be hard Reconstructing photons is harder Convincing you and each other that we understand the efficiencies and jet fake rates is probably the toughest part of this We have a built in check in the events we are interested in The Tevatron tells us what is happening over here. We need to measure out here. At high ET, the problem of jets faking photons goes down. Not because the fake rate is necessarily going down – because the number of jets is going down. Early Running

50. Z0 Why No All-Neutral Couplings? ? Z0 Here’s where thinking about the unbroken symmetries helps. Z0 • Trilinear Coupings • B-B-B: zero because U(1)’s are Abelian, Furry’s Theorem, C, P… • B-B-w3 • B-w3-w3 • w3-w3-w3 • This is where the SU(2) symmetry comes in handy • The Clebsch-Gordon coefficient for (1,0)+(1,0)=(1,0) is zero. • (Recall angular momentum is SU(2) symmetric) • Quartic Couplings • B-B-B-B: zero because U(1)’s are Abelian • w3-w3-w3 -w3 : zero in SU(2) • All mixed couplings: zero The w’s don’t carry hypercharge, and the B doesn’t carry isospin. So the “mixed couplings” are zero These are all zero. Any linear combination (like the g and Z) of zeros is still zero.