finding the higgs boson

FindingThe Higgs

BosonA (hopefully) slightly better explained

version of the events around July 4, 2012

Dr. B. Todd Huffman, Oxford University Dr. A. Weidberg, Oxford University

Explanation in two parts

• Finding the Higgs (part 1)•Standard Model Higgs properties•How to Find the BosonoBump HuntingoSpecial Relativity

• July 4th: the data• Detector performance• CMS and ATLAS results• Stat. Confidence of Discovery


•Why Higgs? (part 2)(Explain why it is needed.)•Conclude

B. Todd Huffman, Oxford University

Standard Model Higgs(part 1)

Standard Model does not predictthe Higgs Mass though.

• Start with Higgs boson as a given.

• Standard Model is a quantitative theory.o Predicts Probability of a Higgs

boson at the LHC• Prediction is the cross-

section (sh) in “barns”

• What does this mean for us?

Cross Section is an area. 10 pb = 10-35 cm2.

Brightness = Lum. # Particles per cm2 per second =

1034 n/(cm2s)


Reason for Radiation Hard Electronics

Mh = 125 GeV/c2 SM Higgs Production Rate = 10-35 cm2 x 1034 cm-2s-1

= 0.1 Hz or one every 10 seconds.

But Hang on!spp ~ 100 mb

that’s “millibarns”With L = 1034 cm-2s-1 random interactions a billion times a second. (not Higgs)

Beam bunches cross once every 50 ns. 50

INTERACTIONS/CROSSING

The LHCOnce the Energy is fixed(ring size)

Then the only thing we can tweak is Luminosity.

This is a hard problem.

More Predictions:Higgs Decays

g g


Irreducible photon processes

quark

Photon (g)

Standard Model shape:Number of photon pairs vs. energy

time


Irreducible ZZ* processes

Standard Model shape:Number of 4-lepton pairs vs. energy

quark

Zo

Zo

l+l- l- l+

Anti-quark

Why did we find it in the decays that are so

rare?Higgs to gg → 100 per year

Higgs to ZZ* → 1000 per year (but Z to ee, mm means ~5 per year)


One Step back: Special Relativity

Two things happen!Explosion A

At time t0 and location x0

Decay B

At time t1 and location x1

But what if we were moving really fast to the left?


V

Two things happen!

Explosion A

At time t0’ and location x0’

Decay B

At time t1’ and location x1’

The order and distance depends on the speed you travel!

Special Relativity

Time t0 ; location x0

t1 ; x1

But this quantity is the same in ALL frames of reference.

(∁ ∆ 𝑡 )2−∆ 𝑥2=𝑐𝑜𝑛𝑠𝑡=(∁ ∆ 𝑡 ′)2−∆𝑥 ′ 2

Invariant Scalar

Special Relativity• Momentum and Energy do this

too!

E2 - P2C2 = M2C4• No momentum, P = 0, then you get E = MC2

• Throw in this fact of nature:o Energy and momentum are conserved. ALWAYS

g0g1Mhiggs


Invariant MassE2-

P2C2=M2C4

Mhiggs

(E0 + E1)2 – (P0 + P1)2c2 = M2higgsc4

Works for any number of particles.Works no matter how fast or slow the Higgs is moving in the lab.

e-

e+

m-

m+

Does not work if they did not come from a Higgs


Irreducible ZZ* processes

Standard Model shape:Number of 4-lepton pairs vs. energy

quark

Zo

Zo

l+l- l- l+

Anti-quark

time

Higgs Bump Hunting

Many events have 4 lepton or two photon

candidates.So just plug E and p of eachone into the formula to find their scalar invariant mass.

Mostly not Higgs. The scalarformula then puts a pip randomly on this histogram

If there really is a parent ALL combinations land at Mhiggs; every time.

6 months


2 years


Glad I did not book a flight to Stockholm.

15 years

Last Paper for TheoristPrior to Managing a Hedge Fund


ATLAS

• Features:o Standalone muon spectrometer (air-core toroid).o Conventional EM calorimeter (Pb/LAr).


CMS

• Some Powerful detectors (e.g. tracker).• Less demanding on muon chamber technology.


Why The Pain is Worth It

• Backgroundso H g g

• Protons have quarks with electric charge. Two photons can result when q-qbar’s annihilate

• Neutral pions decay to photons po g go Bad News; Quark jet could fake a photono CMS and ATLAS detectors built to ID pions this way.

o H Z Z* then Z e+ e- or m+ m-• Proton-Proton Z Z* happens too• No Higgs involved• “Irreducible Background”

• We must deal with Backgrounds• Careful Detector design.


Geometric exploitspo g g

Fine strip segmentation

Very Useful!


The Data – g gCMS ATLAS


The Data - ZZ*ATLAS CMS

Next: Detector Resolution


Accurate Measurement

Much Pain: to obtain track resolutions less than ten microns.

To measure m and eenergy as accuratelyas possible


What would happen if tracking resolution was worse?

LHC 2 years

Meaning: What if the momentum we measure is further away from the true momentum?


Would have published earlier.


How do we know this is real?

“The Data were inconclusive, so we applied Statistics” (A quote taken from Louis Lyons’ book)


15 years

Random events can, occasionally,fake a signal.

Basic Question: What is the probability, if it IS just random, that this “signal” is just a fake?

Discovery!And Limits

ATLAS

CMS

Why bother with a Higgs at all?

(Part 2)

Warning! There will be a lot more math(s).

Bigger Warning! If I do this right, your brain will hurt.(But in a good way)

But Fortune Favours us! Reception just down the road afterward(s) and alcohol really does help.


Maxwell’s Equations(review)

• Note: setting speed of light and Plank’s constant equal to unity (c = = 1)

• (Gauss’s law)

• (Faraday – Lenz laws)

• (no Magnetic mono-poles)

• (Ampere’s law)

Quantum Physics acts on Potentials

• We re-cast Maxwells equations in terms of the electric potential (V) and the vector potential ()

• Put these into and • Result After much tedious vector algebra:

xtVtV ,22

2

-

xtjAtA

,22

2

-

Wave Equations with sources.

Free-space the right sides both zero.


Important thing• Reasons:• The curl of a gradient is

always zero.o B-field unaltered by the

vector potential-plus-gradient

• “x” and “t” are independent so

• Substitutions:

o “q” = constant• E and B fields stay the

same! xttx

22

Quantum Physics(another review)

• Start with Energy equation of a free particle:

• Turn “E” and “p” into operators o and (or just think of it as px -id/dx)

• they operate on the wave function y • the probability of finding the particle between x

and x+dx is given by:

MpE 22

dxtxdxtxP 2),(),( y


Time Dependent SDE• Non-relativistic Schrödinger Equation (SDE)

• E = p2/2m

• the probability of finding the particle between x and x+dx is given by:

• Again, what we measure, P(x,t), has a symmetry. • Y Y’ = exp(ib)Y and BOTH P(x,t) and SDE are

unchanged if b = constant.

dxtxdxtxP 2),(),( y


Invariant and symmetry

• We make a change to something.

• e.g. Substitute in the gradient of a function in addition to the potential

• If the quantity remains unchanged we call this a “symmetry”.

• And the physics is “invariant”

• Substitutions:

o “q” = constant• E and B fields stay the

same!


Summary of things to remember!

• There are hidden symmetries in these equations

• Leaves E and B AND therefore Maxwell’s equations unchanged, whileoY Y’ = e(ib)Y (b=constant)

• Leaves SDE, P(x,t) and Physics unchanged.


Going from “global” to “local” invariance

oY Y’ = e(ib)Y (b=constant)

• What if we make “b b(x,t)” but try to demand the SDE “looks” the same anyway?

• Why do this?o With the standard Schrodinger equation, we have no

interactions!o This is really really boring.

• So let’s play a theorists gameo Dink around by doing the next easiest thing we might do.o b=constant b = b(x,t)


Going from “global” to “local” invariance

oY Y’ = e(ib)Y (b=constant)

• If b b’=b(x,t) … and after MUCH tedious algebra

yy

-

+-tb

tibi

m2

21 yy

tii

m

- 2

21

These do not look the same to me. Clearly NOT “locally” invariant.

Reminder:E and B and their dynamics the same


We Can Fix this!

• Non-relativistic equation for a charged particle in an EM field.

• Invariant when:

• Replace • Replace +iqV• Then:

• Looks like the free SDE but EM is built into it.

yy 02

21 iDDim

-

yy )(21 2

iqVt

iAiqim

+

--

b b’ = b(x,t)


Explain the whole

theoretical game.

• So the modified SDE

• And the equations of the EM wave in free space

• Are all invariant under these transformations

• Non-Rel. QM theory.

Idea is to seek out all kinds of these possible

symmetries.

Look at implications if we make the symmetry “local”, allowing it to be different at

every space-time point.

Does this then hang together with what is

observed? What does this predict?

yy 02

21 iDDim

-

022

2

- VtV 02

2

2

- AtA

“Photons” with Mass

• But we have some massive “photons” like the Z0 or the W± . These would need “vector” equations too.

• What do equations of “photons with mass” look like?

022

2

- VtV

022

2

- AtA

Wave Equations in free space

Massless photons, Relativity included

0222

2

+- VMVtV

Z

0222

2

+- AMAtA

Z


Proca’s EquationsPhotons with mass

•

•

•

Still leaves E and B unchanged.

The equations of motion of the fields are now affected.

“Massive Photons lack our symmetry”

Oh NO!!


This is the problem!• Problem for ANY type of

symmetry you want to make “local” with a massive force carrier.

• I’ve shown “U(1)”• But it is true in “SU(2)”

as well (and others)…• Unfortunate because

SU(2) requires two charged and one neutral boson!o Weak interactions!

• How to solve it?You hire Prof. Peter Higgs


How do you make a mass term, but maintain

local invariance?Higgs mechanism!

Lagrangian L = T - UPrinciple of “least action”…a minimization procedure gives the equations of motion from the Lagrangian.


Lagrange EquationOne abstraction up

• Do not worry about details so much.

• Just note that these are Energy equationso The Kinetic energy minus

the potential energy in classical mechanics. (T – U)

• They have a dynamic term, Derivatives of the field.

• And a mass term (if the particle has mass)

mm

mmmm

mm

AAmAAAAL

mL

2

22

21

41

21

21

+---

-

The above can reproduce equations of motion for spin 0 and spin 1 EM field objects!

Note presence of mass term.We have seen the EM equations are not invariant with a mass term


Higgs Mechanism• Start with a complex scalar “Higgs field” • Goal is to make the equations invariant when

‘ = exp{iq(x)}• Replace • Replace +iqV• Another way of putting this same thing is:• Replace

• Look at the scalar Higgs potential

Higgs Potential 22

21

21 m

m mL -

Potential function for a spin-0 particle.Note terms for dynamics and mass.

mmmm

mm m

iqAD

L

+

-+2*2*2*

41

21

21

Brout, Englert, Guralnik, Hagen, Higgs, Kibble

Postulate a somewhat different potential:

Whaa? Imaginary mass?

Higgs Potential 2*2*2*

41

21

21 m m

m -+L

All perturbative calculations are based on small perturbations about the min.

Zero is not a true minimum. Expand about this point.

),(, txitxv ++

Higgs Potential 2*2*2*

41

21

21 m m

m -+L

A massive photon pops out!

Expand about this point, then apply our trick.(So that “b” can go to “b(x,t)”.)

mmmm iqAD +

You thought previous algebra tedious!!

..

21

21

21

2

22

TO

AAqAAAA

L

+

+--+

+-

mmmm

mm

mm

mm

m

m


Conclusion• The Theorists Game

o Find some constant phase to promote to function status• This “constant” can be far more complicated than what I did

o Alter the derivatives to make it look like a “free” equation again.• U(1) Symmetry

o Showed this for phase eib(x,t) that can vary at every point.o Magically! The interactions with EM dropped out!

• But EM had to be massless Problem when dealing with Z or W bosons!• Need Massive “photons” Enter Higgs mechanism!

o Scalar field is postulated with a very funny “wine bottle” potentialo But a mass appears when potential is expanded about the true minimum!

• The Higgs mechanism gave mass, but kept the local phase invariance. o say a tiny bit about SU(2)


Nobel Choices

P. Higgs F. Englert T. HagenG. Guralnik

R. Brout(deceased)

Who will win the prize?

T. Kibble

Any other questions?


References• David Griffiths, "Introduction to Elementary

Particles, 2nd ed.", Wiley-VCH, Weinheim, Germany, 2008.

• F. Halzen and A. D. Martin, “Quarks & Leptons: An Introductory Course in Modern Particle Physics” John Wiley & Sons.

• I. J. R. Aitchison and A. J. G. Hey, “Guage Theories in Particle Physics, 2nd Ed.”, Adam Hilger, Bristol.


• Why Higgs? (part 2)(Explain why it is needed.)o Maxwell (EM) remindero The key degree of freedom (symmetry)• Quantum Physics review• Makes all of EM and massless photons• But Problems with massive force carriers

o Scalar Higgs and EM• To show how breaking a symmetry makes mass.

o Extending to weak force• Conclude

finding the higgs boson

Documents

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year higgs

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predictthe higgs mass

oxford universityexplanation

higgs beam bunches cross

oxford universityvtwo

standard model higgspart