A Higgs Theory Primer

Angel M. LópezUniversity of Puerto Rico – Mayaguez

The “Standard Model”M

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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

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

A Micro Course in Relativistic Quantum Mechanics

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).

Relativistic Quantum Mechanics, continued

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

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.

Gauge Symmetry and the EM ForceThe 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.

Gauge Symmetry and the EM ForceThe 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.

THE HIGGS MECHANISMThe 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 UNIFICATIONAs an added bonus, the theory has both the EM and the weak force coming from a common theoretical source.

THE HIGGS MECHANISMConsider 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.

THE HIGGS MECHANISMTo 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.

THE HIGGS MECHANISMWe 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.

THE HIGGS MECHANISMWhat 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.

THE HIGGS MASSFrom the Lagrangian we get the following expressions for the masses.

The Higgs mass

The gauge field mass

FERMION MASSESHiggs 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 ).

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.

Invariant “Mass”For any multiparticle final state, define the total energy and momentum as:

Et = Σ Ei = (pi2 + mi

2)½

pt = Σ pi

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 - pt

2)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.

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.

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

Background events with two muons but not necessarily from simply Zo m+m-

This is what we see

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

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

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)!

Couplings as Functions of Mass

[CMS-PAS-HIG-13-005]

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

A Large Appendix of the Experimental

Higgs

The Large Hadron Collider

LHC : 27 km long~100m underground

The Large Hadron Collider

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)

Some LHC facts

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

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

The Large Hadron Collider

General Purpose,pp, heavy ions

General Purpose,pp, heavy ions

CMS

ATLAS

General Purpose:pp, heavy ions

General Purpose:pp, heavy ions

A typical collision

CMS

Searching for a new particlethrough its decays

H

Daughter 1

Daughter 2

Daughter 3

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

Particle Particle Type

Mass(GeV)

Lifetime(ns)

10 GeV Decay Length (m)

EM? Strong? Absorbed inCalorimeter

γ EM carrier 0 ∞ ∞ x EM

e lepton 0.0005 ∞ ∞ x EM

μ lepton 0.1 2000 60000 x Nowhere

π± hadron 0.1 30 900 x x Hadron

K± hadron 0.5 10 60 x x Hadron

K0 hadron 0.5 50 300 x Hadron

n hadron 0.9 9x1011 3x1012 x Hadron

p hadron 0.9 ∞ ∞ x x Hadron

Properties of Detected* Particles

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

Basic HEP Search TechniquesDetecting Decays

I. 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

II. 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

The CMS Onion Layers

CMS Barrel Pixel Detector

Sixty million channels

Pixel size - 100 µm x 150 µm

Position resolution - 10 µm

CMS pixel detector barrel

Module

Kapton cable

end-ring

Micro-Vertexing with Pixels

9” diameter

Light quark (u,d,s) jet

b,c,t jet

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!

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

Underground Experiment Cavern

Late 2004

Lowering CMS sections ~30 stories

Lowering YE+1 (Jan’07)

Insertion of Tracking System

Tracking System200 m2 of Silicon strip detectors

Dec 2007

The CMS Detector when it was last opened in 2009

CMS Ready to Close

CMS closed and ready for beams

CERN

Two photon candidate event

A CMS ZZ* event with both Z’s decaying to muons.

This is a “candidate” for Higgs decay at low mass.

All channels combined (2011 data)

Expected exclusion 114.5 - 543 GeVObserved exclusion 127.5 - 600 GeV

All channels combined (2011 and 2012 data)

Exclusion info95% CL: 110-117.5, 118.5-122.5, 129-539 Obs)95% CL: 120-555 GeV (Exp)

Not excluded:117.5-118.5, 122.5-129,>539 GeV

Excluded at 99% CL: 130-486 GeV (Obs)

Conclusion

Both CMS and ATLAS observe a new particle state with a mass 125 GeV and consistent with the Higgs boson. The statistical significance of the observation is greater than 7.