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
Explanation in two parts
•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)
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
Irreducible photon processes
quark
Photon (g)
Standard Model shape:Number of photon pairs vs. energy
time
B. Todd Huffman, Oxford University
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)
B. Todd Huffman, Oxford University
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?
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
2 years
B. Todd Huffman, Oxford University
Glad I did not book a flight to Stockholm.
15 years
Last Paper for TheoristPrior to Managing a Hedge Fund
B. Todd Huffman, Oxford University
ATLAS
• Features:o Standalone muon spectrometer (air-core toroid).o Conventional EM calorimeter (Pb/LAr).
B. Todd Huffman, Oxford University
CMS
• Some Powerful detectors (e.g. tracker).• Less demanding on muon chamber technology.
B. Todd Huffman, Oxford University
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.
B. Todd Huffman, Oxford University
Geometric exploitspo g g
Fine strip segmentation
Very Useful!
B. Todd Huffman, Oxford University
B. Todd Huffman, Oxford University
The Data – g gCMS ATLAS
B. Todd Huffman, Oxford University
The Data - ZZ*ATLAS CMS
Next: Detector Resolution
B. Todd Huffman, Oxford University
Accurate Measurement
Much Pain: to obtain track resolutions less than ten microns.
To measure m and eenergy as accuratelyas possible
B. Todd Huffman, Oxford University
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?
B. Todd Huffman, Oxford University
B. Todd Huffman, Oxford University
Would have published earlier.
B. Todd Huffman, Oxford University
How do we know this is real?
“The Data were inconclusive, so we applied Statistics” (A quote taken from Louis Lyons’ book)
B. Todd Huffman, Oxford University
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.
B. Todd Huffman, Oxford University
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.
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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!
B. Todd Huffman, Oxford University
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.
B. Todd Huffman, Oxford University
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)
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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)
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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!!
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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.
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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
B. Todd Huffman, Oxford University
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)
B. Todd Huffman, Oxford University
Nobel Choices
P. Higgs F. Englert T. HagenG. Guralnik
R. Brout(deceased)
Who will win the prize?
T. Kibble
Any other questions?
B. Todd Huffman, Oxford University
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.
Explanation in two parts
• 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