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A Higgs Theory Primer

A Higgs Theory Primer. Angel M. López University of Puerto Rico – Mayaguez. The “Standard Model”. Force particles. Matter particles. Fundamental Theoretical Elements of the Standard Model. Quantum Field Theory Common construct for both particles and forces Quantum fluctuations

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A Higgs Theory Primer

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  1. A Higgs Theory Primer Angel M. López University of Puerto Rico – Mayaguez

  2. The “Standard Model” Force particles Matter particles

  3. Fundamental Theoretical Elements of theStandard Model • Quantum Field Theory • Common construct for both particles and forces • Quantum fluctuations • Virtual particles • The vacuum is not empty • Gauge (phase) Symmetry ↔ Force • Unification • Local gauge theories are renormalizable • Spontaneous Symmetry Breaking • Theory is symmetric but ground state (vacuum) is not

  4. A Micro Course in Relativistic Quantum Mechanics LagranRelativistic Energy, Momentum Relation for a Free Particle Using energy-momentum four-vector notation LagranObtain Quantum Equation the same way as in Non-Relativistic LagIntroduce a notation to simplify equations We obtain the Klein-Gordon Equation

  5. Relativistic Quantum Mechanics, continued LagranAn alternative was derived by Dirac. He wanted a first order equation. This can be done if the γ are 4x4 matrices. The result is the Dirac Equation The wavefunction now has four components. The Dirac equation is appropriate for dealing with fermions (spin ½ ). The Klein-Gordon equation is appropriate for dealing with bosons (spin 0).

  6. LagranLagrangian Formulation of Classical Mechanics LagranEuler – Lagrange Equation LagranRelativistic Quantum Mechanics – Quantum Field Theory Lawhere LagrF For a massive scalar field, we can use which leads to the Klein – Gordon equation

  7. LagrFor a spin ½ field, we can use which leads to the Dirac equation For a vector field (spin =1), the field, A, has four components. The Lagrangian is which can be written more compactly as by defining The best example of this is the electromagnetic field. It is massless.

  8. Gauge Symmetry and the EM Force The Dirac Lagrangian is invariant under a global phase transformation, i.e. a phase which is the same at every spacetime point. But it is NOT INVARIANT under a LOCAL phase transformation where the phase is an arbitrary function of spacetime. In that case the transformed Lagrangian has an additional term However, one can have a local phase invariant Lagrangian by introducing a vector field which transforms in the following way For the EM field this is the well known property of gauge invariance.

  9. Gauge Symmetry and the EM Force The combined local phase invariant Lagrangian is NOTICE THAT IT DOES NOT INCLUDE A MASS TERM FOR THE VECTOR FIELD. SUCH A TERM WOULD BREAK THE SYMMETRY. This Lagrangian is the starting point for Quantum Electrodynamics (QED), the most precise theory we have. The important points for our discussion are that: THE ELECTROMAGNETIC FORCE CAN BE SEEN AS THE CONSEQUENCE OF REQUIRING LOCAL PHASE INVARIANCE IT APPEARS THAT THE FORCE FIELD CARRIERS HAVE TO BE MASSLESS.

  10. THE HIGGS MECHANISM The Higgs mechanism explains how one can build a theory which has a local gauge invariant Lagrangian where the force fields have mass. The physical manifestation of this is the weak interaction whose carriers, the W and Z particles, are very massive. ELECTROWEAK UNIFICATION As an added bonus, the theory has both the EM and the weak force coming from a common theoretical source.

  11. THE HIGGS MECHANISM Consider a theory with two scalar fields and the following Lagrangian: The last two terms can be considered as the “potential energy function” The ground state (vacuum) will be a state where U is a minimum. Actually there are an infinite number of such states lying in a circle of radius µ/λ.

  12. THE HIGGS MECHANISM To obtain a local gauge invariant theory we must introduce a gauge field and change the derivative into the so called “covariant derivative”. We obtain the following Lagrangian: where we have compacted the notation by combining the pair of real scalar pair of fields into one complex field.

  13. THE HIGGS MECHANISM We can choose the vacuum state arbitrarily. THE PHYSICALLY RELEVANT FIELDS WILL BE FLUCTUATIONS ABOUT THE VACUUM STATE. When we write the Lagrangian in terms of these new fields, we get: THE VECTOR FIELD HAS ACQUIRED MASS!!! The ξ (Goldstone boson) field has disappeared. The η field (massive) is the Higgs field.

  14. THE HIGGS MECHANISM What happened to the Goldstone boson? We were able to get rid of it by choosing a particular gauge. There is some physical content to this. A massless vector field has only two polarization states, e.g., the photon. When the gauge field acquires mass it also acquires an additional degree of freedom, i.e. a third polarization state. This degree of freedom comes from the Goldstone boson which has disappeared.

  15. THE HIGGS MASS From the Lagrangian we get the following expressions for the masses. The Higgs mass The gauge field mass

  16. FERMION MASSES Higgs theory includes the way fermions receive mass through the Higgs. This is based on starting with a Lagrangian with a Yukawa potential as the interaction between the Higgs and the fermion. In the following Lagrangian the bare fermion mass (m1) is zero. When this Lagrangian is written in terms of the Higgs field, a fermion mass term appears which is proportional to the coupling between the Higgs and the fermion (α ).

  17. Electroweak Unification • Higgs theory can be implemented in a two dimensional isospin space where the local gauge transformations are members of an SU(2) group. • There are four gauge fields in that case related to one another. One of them is massless (photon). The others the massive weak gauge bosons.

  18. Invariant “Mass” • For any multiparticle final state, define the total energy and momentum as: • Et = ΣEi= (pi2 + mi2)½ • pt = Σpi • The invariant “mass” (M) is a Lorentz invariant, a property of that state which is independent of the frame where it is calculated. • M = (Et2 - pt2)½ • We measure M experimentally for groups of particles we believe are the final state of the decay of some particle. • If that is in fact the case, M will be equal to the rest mass of the decaying particle within experimental error. We will see a “mass peak” in the invariant mass distribution. (The Heisenberg uncertainty principle will also contribute to the width of this peak for the cases where the decaying particle has a very short lifetime.) • Typically this peak is on top of a smooth distribution which comes from events where the final state is the product of some other production mechanism or where we have misidentified one or more of the particles. These constitute the background in our invariant mass plots.

  19. Calculable Consequences of the Higgs Theory • It predicts the masses of the weak carriers • For W+, W- it predicts 80.4 GeV • For Zo it predicts 91.1 GeV • ¿Is this in accordance with reality? • Zo decays to two muons. We can measure the momenta of the muons and determine the Zo mass. • If we do this for many Zo decay events, we obtain a distribution for the mass values which we can predict with the theory.

  20. Zom+m- We predict this distribution to show up when looking at many thousands of Zom+m-decay events Peak at 91.1 GeV This is what we see Background events with two muons but not necessarily from simply Zom+m-

  21. Higgs Properties • Spin Zero • Production Cross Section • Couplings Proportional to Mass of Decay Products

  22. Standard Model Higgs Decays • The SM Higgs is unstable • Decays “instantly” in a number of ways with very well known probabilities (called Branching Fractions or Ratios that sum up to 1). • Branching ratios change with mass as seen here • Some decay modes are more easily seen than others • If they end with electrons, muons, or photons

  23. How should we see the Higgs Boson? Simulation NB: These old plots correspond to ~50 times more sensitivity than we have now (20x more data, 2x the energy)!

  24. Couplings as Functions of Mass [CMS-PAS-HIG-13-005]

  25. Theory Reference • Introduction to Elementary Particles • David Griffiths • Wiley and Sons • 1987

  26. A Large Appendix of the Experimental Higgs

  27. The Large Hadron Collider

  28. The Large Hadron Collider LHC : 27 km long ~100m underground

  29. Some LHC facts • Tunnel • Diameter 3m, Length 16 miles • 2 billion pounds excavated • Beams • Made up of bunches • 1.2-1.5x1011 protons/bunch • 1404 (2808) maximum bunches in machine for 50 (25) ns separation • 1 ns = 1 billionth of a second • 50 ns separation = 15 m • At Interaction Point (IP) • Bunch length ~ 6 cm • Beam radius ~23 mm • Bunch collision rates • 31.6 MHz (25 ns spacing) • 15.8 MHz (50 ns spacing) • Superconducting dipoles • challenge: magnetic field of 8.33 Tesla in total 1232 magnets, each 15 m long operated at 1.9 K • It’s colder than space • It’s emptier than space • Largest cryogenic system in the world

  30. Colliding Beams • 2 beams circulate in opposite directions • Beams are made up of 1380 bunches • each bunch has 150 billion proton • Bunches cross at 4 places on the 27 km long LHC ring. • ~ 20-30 pairs of protons collide each time bunches cross

  31. The Large Hadron Collider ATLAS General Purpose,pp, heavy ions CMS General Purpose:pp, heavy ions

  32. A typical collision

  33. CMS

  34. Searching for a new particlethrough its decays Daughter 3 H Daughter 2 Daughter 1 • Higgs decay patterns are dictated by its presumed properties • We track and identify the daughters and check to see whether they are coming from a common vertex • Since the Higgs gives mass to all particles it has many decay “channels” and this in itself is evidence that it is the Higgs although some channels are more probable than others • Higgs search channels are chosen on the basis of their relative probability but also on their experimental accessibility

  35. Properties of Detected* Particles *Detected means that it passes through CMS and leaves a signal in some detector.

  36. Basic HEP Search TechniquesDetecting Decays • Tracking • Which particles come from a common vertex? • Momentum magnitude and direction at vertex • Use a magnetic field to measure magnitude • Match tracks to hits in calorimeters • Particle Identification • We expect certain particles in the final state • Use the decay product mass to calculate invariant mass of parent • Calorimetry – Electron and Photon in EM; hadrons in HM • For neutral particles, measure energy and direction to calculate invariant mass of parent • Muon Detector is the furthest from the beam line

  37. The CMS Onion Layers

  38. CMS Barrel Pixel Detector CMS pixel detector barrel • Sixty million channels • Pixel size - 100 µm x 150 µm • Position resolution - 10 µm Kapton cable Module end-ring

  39. Micro-Vertexing with Pixels 9” diameter Light quark (u,d,s) jet b,c,t jet

  40. Quite a camera • CMS is like a camera with 80 Million pixels • But it’s obviously no ordinary camera • It can take up to 40 million pictures per second • The pictures are 3 dimensional • And at 31 million pounds, it’s not very portable • The problem is that we cannot store all the pictures we can take so we have to choose the good ones fast!

  41. Experimental Challenges • Collisions are frequent • Beams cross ~ 16.5 million times per second at present • About 20-30 pairs of protons collide each crossing • Interesting collisions are rare - • less than 1 per 10 billion for some of the most interesting ones • We can record only about 400 events per second. • We must pick the good ones and decide fast! • Decision (‘trigger’) levels • A first analysis is done in a few millionths of a second and temporarily holds 100,000 pictures of the 16,500,000 • A final analysis takes ~ 0.1 second and we use ~10000 computers • We still end up with lots of data

  42. Underground Experiment Cavern Late 2004

  43. Lowering CMS sections ~30 stories Lowering YE+1 (Jan’07)

  44. Insertion of Tracking System Dec 2007 Tracking System200 m2 of Silicon strip detectors

  45. The CMS Detector when it was last opened in 2009

  46. CMS Ready to Close

  47. CMS closed and ready for beams

  48. CERN

  49. Two photon candidate event

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