Transcript Precision EWK - Durham University

The Standard Model
Lecture II
Thomas J. LeCompte
High Energy Physics Division
Argonne National Laboratory
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
2
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
Our next step – fixing these problems
• Breaks gauge invariance
by adding one more piece to the theory.
– Symmetry broken by hand
3
Part IV – The Higgs Mechanism
A Theory with Four Parameters
4
Spontaneous Symmetry Breaking
What is the least amount
of railroad track needed to
connect these 4 cities?
5
One Option
I can connect them this
way at a cost of 4 units.
(length of side = 1 unit)
6
Option Two
I can connect them this
way at a cost of only 3
units.
7
Option Three
This requires only 2
2
8
The Real Optimal Solution
This requires 1
3
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.
9
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.
10
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)
 
 0
 
 

 
 *0 
 


 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
11
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
m = ±1 “transverse”
 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.
m = 0 “longitudinal”
 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.
12
How Cute Is It?
 There’s very little choice involved
in how you write down this theory.
– There’s one free parameter
which determines the Higgs
boson mass
– There’s one sign which
determines if the symmetry
breaks 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
13
Under The Hood
 Let’s explore the “cuteness”
 There is no freedom at all in how
the 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.

V ( )   M    
2
2
4
This is the critical sign

The only freedom we have in the theory is
the strength of the self-interaction  and its
sign.
If  > 0, the potential has a minimum for
positive f2.
Otherwise it’s minimized at f2 =0.
14
The Mysterious Mexican Hat
 We need to make a few observations
– My choice of basis as
f and f0 was terrible.
– Only |f| appears in the potential
– Don’t use rectangular coordinates
in a problem with circular symmetry!
f
 The potential has an infinite number of
minima
– Solution is r = M/, q = anything
– This is an example of sponteneous
V ( )  
symmetry breaking (think ferromagnet)
f0
M 
2
 
2
4

 If I flip the sign of , I have a paraboloid
– Not very interesting
15
C. Hill
The Mysterious Mexican Hat II
S. Carroll
 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/ = 246 GeV
– Excitations about this minimum will become the Higgs boson
16
Misunderstanding the Mexican Hat
 We got one massless boson out. Nature gave us one massless
boson, 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.
 0   *0 
  
– One becomes the Higgs.
  
– The other 3 become Goldstone bosons.
 Where did these Goldstones go?
– They were “eaten” by the w’s.
17
Eaten?
This theory has interactions like this:
w+
f+
w+
 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+.
m = ±1 “transverse”
m = 0 “longitudinal”
18
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 massive
W 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
M
GF


2
4
 This theory predicts one new particle
– a fundamental scalar, the Higgs boson
19
Some Slight of Hand
 I’ve swindled you.
– Twice…the same way.
 I added a Higgs doublet:
  
 0
 
 
 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.
20
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.
21
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.
 dW  Vud
  
 sW    Vcd
 b  V
 W   td
Vus Vub  d 
 
Vcs Vcb  s 
Vts Vtb  b 
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!
 d  Vud
  
 s    Vcd
 b  V
   td
Vus Vub 

Vcs Vcb 
Vts Vtb 
1
 dL 
 
 sL 
b 
 L
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,
22
Let’s Not Go Overboard
 What I described is a theory, not a fact.
 This theory could be wrong on a number of
counts:
– 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.
23
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 YR  YL  2T3L
• 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?
24
Testing the Standard Model
25
Electroweak Radiative Corrections
 Remember, the W mass was calculated to be
77.5 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
26
Stomping Out Nonsense
 One sometimes reads that there are no radiative
corrections to the Z.
– Of course there are – the same sort of loops for
the 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!
27
W and Z Masses
We started with the
tree level prediction of
the GWS theory.
mW  4
mW 
Loops cause a few %
correction to this
MW
MZ 
cosqW
 2a 2
2GF2
4
 2a 2
2G
2
F
1
 77.5 GeV
sin qW
1
sin qW 1  R
MZ 
MW
cosqW
R ~ 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
Z
Z
t
W
W
b
28
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
29
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.
mT  E 2  px2  p y2
Fortunately, the transverse mass
distribution is a function of the true mass.
D0
CDF
Missing ET
(neutrino)
Electron momentum
30
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.
31
The Kind of Thing Experimenters Worry About
Two leptons – do they
see the same field?
To 100 ppm?
Detector’s B field
32
Rapidity – of getting m(W) results published
450
400
m(W) Uncertainty (MeV)
The trend is for
later runs to be on
a curve which
begins lower and
to the right of
earlier runs.
88-89
Run
350
300
250
200
Run I
150
100
50
*
Run II
0
0
500
Tevatron results
1000
1500
2000
2500
3000
3500
No hadron collider
experiment has
published an
uncertainty of 100
MeV in less than
1400 days.
Days to Publication
33
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
34
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
CMS material budget
X~16.5%X0
(red line on lower plots)
ATLAS material budget
35
Difficulty 2 – QCD corrections are more important
q
q
W
g
q
W
q
 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.
36
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?
37
200
200
150
150
150
100
50
Measurement
200
Measurement
Measurement
Systematics – The Good, The Bad, and the Ugly
100
50
0
50
0
0
2
4
6
8
10
12
Some variable
100
0
0
2
4
6
8
10
12
Some variable
Good
Bad
 Masses divided into
several bins in some
variable
 Masses are
consistent within
statistical
uncertainties.
 Clearly there is a
systematic
dependence on this
variable
 Provides a guide as
to what needs to be
checked.
0
2
4
6
8
10
12
Some variable
Ugly
 Point to point the
results are
inconsistent
 There is no
evidence of a trend
 Something is wrong
– but what?
38
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
39
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/ = v.e.v. (246 GeV).
• If M gets large, so does , and now we have a strongly-coupled
theory.
 A number of solutions have been proposed.
– In my opinion, none of them are entirely satisfactory.
40
Another Test of the SM: Multiboson production
Zzzzzzzzzzz
41
What Interactions are in the Standard Model?
g
The (Electroweak)
Standard Model is the
theory that has
interactions like:
W+
W+
Z0
but not
W+
W+
Z0
g
Z0
Z0
&
W-
Z0
g
W-
Z0
Z0
Z0
g
Z0
g
&
but not:
Z0
g
Only three parameters - GF, a and
sin2(qw) - determine all couplings.
42
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
43
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, g, Z
– In the SM, g1Z = kg = kZ = 1 and g = Z = 0
• Often useful to talk about g, k and  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 + g)/2MW
– Electric quadrupole moment = -e(kg - g)/2MW2
– Dimension 4 operators alter g1Z,kg and kZ: grow as s½
– Dimension 6 operators alter g and Z 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
44
Why Center-Of-Mass Energy Is Good For You
Approximate
Run II Tevatron
Reach
Tevatron
kinematic limit
 The open histogram is the
expectation for g = 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
45
Not An Isolated Incident
 Qualitatively, the same thing
happens with other couplings
and processes
 These are from WZ events with
g1Z = 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-1
Insert is from CMS Physics TDR: 1 fb-1
46
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
47
TGC’s – the bottom line
Coupling
Present Value
LHC Sensitivity
(95% CL, 30 fb-1 one experiment)
g1Z
0.005-0.011
 0.01600..022
019
kg
0.03-0.076
 0.02700..044
045
kZ
0.06-0.12
 0.07600..061
064
g
0.0023-0.0035
 0.02800..020
021
Z
0.0055-0.0073
 0.08800..063
061
 Not surprisingly, the LHC does best with the Dimension-6 parameters
 Sensitivities are ranges of predictions given for either experiment
48
Early Running
 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.
49
Why No All-Neutral Couplings?
Z0
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
The w’s don’t carry hypercharge, and the B doesn’t carry
isospin. So the “mixed couplings” are zero
– 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
These are all zero.
– B-B-B-B: zero because U(1)’s are Abelian
Any linear combination
– w3-w3-w3 -w3 : zero in SU(2)
(like the g and Z) of
– All mixed couplings: zero
zeros is still zero.
50
So, Does This Event (and its siblings) Kill the SM?
CDF
pp  ZZ  X , ZZ  eeee
51
No…Experiments Measure Rates, Not Couplings
The experiments are hoping to
see this – evidence for a nonzero ZZZ or gZZ coupling.
Z0
g Z0
Z0
However, the exact same final state
can occur by this (ordinary SM)
process.
Experiments need to look for an
excess of events beyond the SM
prediction, and/or events at high
m(ZZ), where the SM prediction is
small and new physics would be
larger.
Z0
Z0
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What Would an Anomalous Coupling Mean?
 SU(2)L would be only an approximate symmetry
– That doesn’t mean the real group is SU(1.9996) or SU(2.0001)
– It means that SU(2) is a large distance limit of the theory
 Effects that grow as s½ or s are easy to see at large m(VV)
– As m(VV) grows, these effects eventually violate unitarity
• Nice SU(2) cancellations no longer apply
• Of course those are easy to see! They’re impossible to miss!
 It may be useful to think about turning this around
– If there is new physics at high Q2/short distances, it can manifest itself
at lower Q2/longer distances as changes in the couplings: g1Z, kg, kZ,
g and Z
– While the effects will be largest at high Q2, they might not look like
exactly like these predictions – it all depends on what this new
physics is.
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Summary
 We started with QED (a)
 Next, we added the weak force via the
Fermi theory (GF)
– This was an effective theory that broke down
~300 GeV
 We fixed this by adding the W and Z
bosons in an SU(2)L x U(1)Y theory
with the weak boson’s masses put in by hand
– Adds a third parameter: sin2(qw)
– This is also an effective theory, but
good to ~1 TeV
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Summary II
 Finally, we have our four parameter Higgs
theory: a, GF, sin2(qw), m(H)
– This fixes the unitarity problem
– This restores gauge invariance
– Maybe provides some insight to fermion masses
 We touched on some areas of testing the SM
– The SM beyond tree level:
• “Loops” and the W mass
• Exposes a problem with the Higgs
– Triple and Quartic Gauge Couplings
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