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R. Sekhar ChivukulaMichigan State University
PASI 2012
Buenos Aires, Argentina March 5 & 6, 2012
The Quest for Electroweak Symmetry Breaking
What I will try to avoid:
Your questions are welcome and encouraged at any time!!
Lecture 1 Outline
• Background
• Why do we believe in (effective) field theory?
• In gauge invariance?
• How can vector bosons have mass?
• The Higgs Boson
• Properties
• Limits
• Problems with the Higgs Model
Effective Field Theory
Why do we believe in (effective) field theory?
QFT Reconciles QM with RelativityA local, Lorentz-invariant, Hermitian, QFT
with a finite number of fields yields a unitary, CPT-invariant, S-matrix satisfying cluster decomposition
“They act so cute when they try to understand Quantum
Field Theory”
à la Landau (e.g. superconductivity): the converse is also true!
“Any” S-matrix is derivable from a QFT
References
Example: A Scalar Doublet...
Consider theory valid below UV cutoff Λ:
(Note “scaling dimension” of operators)
Wilsonian Renormalization Group
Wilsonian Renormalization Group
m2(�)
�(�)
�(�)
• Lagrangian and S-matrix are expansions in p2/Λ2 - at any order in p2/Λ2, only a finite number of operators contribute.
• “Renormalizable” theories are a special case, with Λ→∞: S-matrix “exactly” calculable in terms of a few parameters.
m2(�)
�(�)
�(�)
QFT Reinterpreted
Jacob and Wick, 1959
How Large can Λ be?
These formulae apply to the elastic scattering of pairs of particles of fixed helicity
Re a
Im a
Physical Amplitude
Born Amplitude
Unitarity
Bound on
Re a
S†S = I ! |sl|
2= 1
sl = 1 + 2ial ! Im(al) = |al|2
al = ei!l sin !l
Feynman AmplitudeIdentical Particle
Factor
al =1
32(2)⇥
�4k2
s
⇥1/2 ⇤ +1
�1d cos � Pl(�)M(s, �)
Consider Elastic 2-Body Unitarity
Limits of an Effective Theory
L =�
i
�i(�)Oi
�di�4with di>4 leads to M � �i(�)pdi�4
�di�4
Re a
Im a
Physical Amplitude
Born Amplitude
Unitarity
Bound on
Re a
Amplitude “violates” unitarity at scale M, and the (perturbative) effective theory breaks down
M is the (largest) scale at which the description of the theory changes,e.g. the W instead of Fermi Theory
When the QCD coupling becomes strong
• breaks SU(2)L x SU(2)R SU(2)L+R
• pions are the associated Nambu-
Goldstone bosons!
!qLqR" #= 0
(qLqR)
The strong-interaction (QCD) Lagrangian for the u and d quarks (neglecting their small masses)
displays an SU(2)L x SU(2)R global (“chiral”) symmetry
L = iuLD/ uL + idLD/ dL + iuRD/ uR + idRD/ dR
Example: Pions in QCD
Coleman, et. al. Phys. Rev. 177, 2239 (1969)
The Chiral Lagrangian
• Pions encoded in unitary matrix field:
• SU(2)L x SU(2)R symmetry represented nonlinearly
• Lowest order terms consistent with symmetries:
• Cutoff of order 1 GeV
⌃(x) ! L⌃(x)R†
⇤�SB 4⇡f⇡ ⇡ 1GeV
L2 =f2⇡
4Tr
�@µ⌃@µ⌃
†� = 1
2@µ⇡a@µ⇡
a+ interactions
⌃(x) = exp
✓i⇡
a(s)�
a
f⇡
◆
Gauge Invariance and Massive Vector Bosons
Gauge Invariance?
• The only consistent S-matrix for a spin-1 massless particle arises when it couples to a conserved current - e.g., like a gauge-boson! (Weinberg’s theorem)
• Corollary: Given a spin-1 boson of mass m, the only theory consistent up to scale M is, in the limit m/M→0, a gauge theory.
• Coupling constant universality (and custodial symmetry)!
• LEP I/II and Tevatron: SU(2) x U(1) gauge-invariance good to ~ few TeV! e.g.
(�†Dµ�)2
M2 → αT or Δρ
Warnings*
* Things you should know about QFT, but were afraid to ask.
• The QFT description of an S-matrix need not be unique, e.g. QCD and the χLagrangian, ADS/CFT.
• “Gauge Symmetries” are not symmetries: they are redundancies in our description.
• “Coupling constants” are not observables.
• “Fundamental” and “Composite” are in the eye of the calculator ... more important: strong or weak
References
Massive Vectors: The Higgs Mechanism
(Haber)
“Eaten” Goldstone Boson
Massive Gauge Bosons: SU(2) x U(1) @ E4
Sum 0
Massive Gauge Bosons: SU(2) x U(1) @ E2
including (d+e)
O(E2)
Theory with only massive W and Z particles breaks
down at O(1 TeV)!
⇤EWSB 4⇡v
The One-Doublet “Standard” Higgs Model
the Higgs Model
2+1/2 under SU(2) x U(1)
References
Matrix Notation
!!
neatly separates the radial “Higgs boson” from the “pion” modes (Nambu-Goldstone Bosons).
Polar Decomposition
!!
Massive Gauge Bosons: SU(2) x U(1) @ E2
including (d+e)
References
Custodial Symmetry
(i.e. mass differences)
Violations of Custodial Symmetry
0.2%new
Custodial Symmetryis an important part
of any theory of EWSB!
SU(2)V
Properties of the Higgs Boson
Properties of the Higgs Boson
• All masses proportional to ⟨H⟩=v, hence
• and
• Important loop effects
LSM � M2W
✓1 +
H
v
◆2
W+µW�µ +
M2Z
2
✓1 +
H
v
◆2
ZµZµ
LSM � �X
f
mf
✓1 +
H
v
◆¯ Lf Rf + h.c.
�(pp ! H) �(H ! ��)
gluon
gluon
LEP Electroweak Working Group, July 2011
Indirect Higgs Boson Constraints
0
1
2
3
4
5
6
10030 300mH [GeV]
6r2
Excluded
6_had =6_(5)
0.02750±0.000330.02749±0.00010incl. low Q2 data
Theory uncertaintyJuly 2011 mLimit = 161 GeV
80.3
80.4
80.5
155 175 195
mH [GeV]114 300 1000
mt [GeV]
mW
[G
eV]
68% CL
6_
LEP1 and SLDLEP2 and Tevatron
July 2011
Assume(!) Standard Model
Fermilab press release: Mar. 2, 2012
Hot off the press!
Hints of the Higgs?Stay Tuned...
http://www.atlas.ch
ATLAS Searches for the Higgs Boson
!!
CMS Searches for the Higgs Boson
)2Higgs boson mass (GeV/c100 200 300 400 500 600
SMm/
m95
% C
L lim
it on
-110
1
10
Observedm 1±Expected m 2±Expected
LEP excludedTevatron excludedCMS excluded
Observedm 1±Expected m 2±Expected
LEP excludedTevatron excludedCMS excluded
-1 = 4.6-4.7 fbintCombined, L = 7 TeVsCMS Preliminary, Observed
m 1±Expected m 2±Expected
LEP excludedTevatron excludedCMS excluded
-1 = 4.6-4.7 fbintCombined, L = 7 TeVsCMS Preliminary,
http://cms.web.cern.ch
!!
Problems with theHiggs Boson
References
Problems with the Higgs Model• No fundamental scalars observed in nature
• Hierarchy or Naturalness Problem
• No explanation of dynamics responsible for Electroweak Symmetry Breaking
• Triviality and Vacuum Stability Problems...
Problems with the Higgs Model
m2(�)
�(�)
�(�)
T. Hambye and K. Riesselmann, Phys. Rev. D55, 7255 (1997), [hep-ph/9610272].
Triviality and Stability
d�
d log µ=
3
8⇡2
⇥4�2
+ 2�g2t � g4t⇤
Elias-Miro, et. al., arXiv:1112.3022
Updated
Or: other particles (e.g. superpartners) could stablize the potential...
If there is no “Standard” Higgs, then
what?
More next time!
R. Sekhar ChivukulaMichigan State University
PASI 2012
Buenos Aires, Argentina March 5 & 6, 2012
The Quest for Electroweak Symmetry Breaking:
If not The Higgs, What?
A Fork in the Road...
• Make the Higgs Natural: Supersymmetry
• Eliminate the Higgs
• Technicolor
• “Higgsless” Models
• Make the Higgs Composite
• Little Higgs
Lecture 2 Outline
• Multi-Higgs Models
• General Properties
• Symmetries and Flavor
• SUSY
• Strongly Coupled Theories
• Technicolor
• Higgsless Models
• Composite Higgs
Multi-Higgs Models
Two-Higgs Model
Two-Higgs Model
Two-Higgs Model
Couplings to Fermions
Glashow and Weinberg, 1977
Most general quark couplings:X
ij
h�u1ij q
iL�1uj + �u
2ij qiL�2uj + �d
1ij qiL�1dj + �d
2ij qiL�2dj
i+ h.c.
Fermion masses and Higgs couplings not diagonalized at same time!
Model-building solutions:• “Type-I”: λ2u,d=0
One Higgs gives mass only to W and Z
•“Type-II”: λ2u=0 & λ1d=0Each Higgs gives mass to only ups or downsmd∝1/sinβ, could have λt=λb
Multi-Higgs Symmetries
Extended EWSB Sectors
SuperSymmetry
Make the Higgs Boson natural!
• Higgs mass protected by chiral symmetry of partner• Δm2H∝log(M2SUSY)•λ∝g2,g’2, mH bounded by ~130 GeV
• SUSY requires two Higgs bosons to give u- and d-masses• Naturally “type-II” see lectures by Carena
Strongly Coupled
EWSB
Technicolor:Higgsless since 1976!
For a new approach to generating mass, we turn to the strong interactions (QCD) for inspiration
Why is the pion so light?
Consider the hadrons composed of up and down quarks:
Inspiration from QCD
Energy
[coupling]2
Recall that the QCD coupling varies with energy scale, becoming strong at energies ~ 1 GeV
Asymptotic Freedom
When the QCD coupling becomes strong
• breaks SU(2)L x SU(2)R SU(2)L+R
• pions are the associated Nambu-
Goldstone bosons!
!qLqR" #= 0
(qLqR)
The strong-interaction (QCD) Lagrangian for the u and d quarks (neglecting their small masses)
displays an SU(2)L x SU(2)R global (“chiral”) symmetry
L = iuLD/ uL + idLD/ dL + iuRD/ uR + idRD/ dR
Chiral Symmetry Breaking
Bonus: from chiral to electroweak symmetry breaking
• uL,dL form weak doublet; uR,dR are weak singlets
• so also breaks electroweak symmetry
• could QCD pions be our composite Higgs bosons?
!qLqR" #= 0
Not Quite:
• MW = .5g Fπ = 80 GeV requires Fπ ~ 250 GeV
• only supplies fπ~ 0.1 GeV
• need extra source of EW symmetry breaking
!qLqR"
From QCD to EWSB
This line of reasoning inspired Technicolor:
Susskind, Weinberg
introduce new gauge force with symmetry SU(N)TC
force carriers are technigluons, inspired by
QCD gluons
add techniquarks carrying SU(N)TC charge:
matter particles inspired by QCD quarks
• e.g. TL = (UL, DL) forms a weak doublet
UR, DR are weak singlets
• Lagrangian has familiar global (chiral)
symmetry SU(2)L x SU(2)R
Technicolor
If SU(N)TC force were stronger than QCD ... then spontaneous symmetry breaking and pion formation would happen at a higher energy scale... e.g.
• gauge coupling becomes large at
• breaks electroweak symmetry
• `technipions’ become the WL, ZL
• W and Z boson masses are the size seen in experiment!
So far, so good... but what about unitarization?
!TC
!TLTR" # 250 GeV
!TC ! 1000 GeV
Technicolor
What about fermion masses?Lectures by Simmons
(“Low-Energy” Analog)
“Abelian Higgs Model”
Weinberg: “Superconductivity for Particular Theorists”
B C S
⇥���⇤ �= 0
Classic TC @ LHC:
I=J=1 ππ scattering
NB: unlike Higgs!!
Classic TC @ LHC:
WZ Scattering at SLHC
+ forward jets
F. Gianotti, et. al., hep-ph/0204087
Low-ScaleTechnicolor
Limits:• Model Dependent
• Just Reachinginteresting range!
• LHC extend limits substantially
Narain, Womersley, RSCPDG review
Kalanand Mishra, Lattice Meets Experiment, Fermilab, October 15, 2011
ATLAS Limits on Low-Scale TC
CMS Limits on Low-Scale TC
Kalanand Mishra, Lattice Meets Experiment, Fermilab, October 15, 2011
Composite Higgs
Composite HiggsHiggs as (Pseudo-)Goldstone Boson:
Hard to do!
V (h) =Cg2
16!2
!
!"2f2|h|2 + "4
|h|4
2+ . . .
"
Decay Constantg ! 1
!h"2 #!2
!4
f2Yields:
Georgi & Kaplan; Banks Chacko et. al., hep-ph/0510273
But, EWPT: f > 4 ! 5 TeV
Must suppress !2 without suppressing !4
The Little HiggsCollective Symmetry Breaking:
k1
m0 m1 m2 mN mN+1
k2 k3 kN+1kN
For weak springs, masses at end very weakly coupled!
In practice: m2
h !
g2
16!2f2
!2
!4
!
g2
16"2
Meade, hep-ph/0402036Arkani-Hamed, Cohen, Georgi
Little Higgs : The Hierarchy
Schmaltz hep-ph/0210415
Cancellation of divergences by particles of same spin!T-Parity: minimizeZ-pole effects & DM
From Technicolor to Extra-Dimensions ... and Back Again: Higgsless Models
Can Extra-D be related to EWSB?Consider Loss of Unitarity in
Extra-D Theories and Massive Vector Boson Scattering
Expand 5-D gauge bosons in eigenmodes: e.g. for S1/Z2:
Extra-D
KK mode
4-D gauge kinetic term contains1
2
!!
n=1
"
M2
n(Aan
µ )2 ! 2MnAan
µ !µA
an
5 + (!µAan
5 )2# i.e., A
anL ! A
an5
4-D KK Mode Scattering
Cancellation of bad high-energy behavior through
exchange of massive vector particles
RSC, H.J. He, D. Dicus
Can we apply this to W and Z?
Higgsless Models
• Can we use Extra-D/AdS-CFT in EWSB?
• Unitarize TeV-scale WLWL scattering using vector bosons?
• If KK modes exist, MW << MKK!!
• Luckily, unitarization generalizes to a large class of 5-d manifolds and boundary conditions!
Csaki, Grojean, Murayama, Pilo, Terning
Energy Scales and Couplings
g4 =
g5!
!RMn =
n
R!UV !
1
g25
4D EFT
5D EFT
???
1/R
�UV
AdS/CFT DualityConjecture: Equivalence of 5D theory in AdS and 4D CFT
Strong evidence for N=4 SUSY YM string theory on AdSStrongly-coupled CFT ⇔ Weakly-coupled 5D Theory!
NB: Rescaling Invariance!
ds2=
!
R
z
"2#
!µ!dxµdx!! dz2
$
UV IR
R < z < R!
“Walking Technicolor” ⇔ Higgsless Models
• Choose “bulk” gauge group, location of fermions, and boundary conditions
• Choose g(x5)
• Choose metric/manifold: gMN
(x5)
• Calculate spectrum & eigenfunctions
• Calculate fermion couplings
• Compare to Standard Model: S, T, U, ...
Recipe for a Higgsless Model:
Can we do better? Yes, using deconstruction!
Deconstruction
van Gogh
Wolff
x5
xµ
Latticize Fifth Dimension
• Discretize fifth dimension
• 4D gauge group at each site
• Nonlinear sigma model link fields
• To include warping: vary fj
• For spatially dependent coupling: vary gk
• Continuum Limit: take N infinity
• Finite N, a 4D theory!
g1
f1 f2
gN
fN fN+1
g2
f3
g0 gN+1
Arkani-Hamed, Georgi, Cohen & Hill, Pokorski, Wang
... Back Again: Deconstruction
• SU(2)N x U(1); general fj and gk
• Fermions sit on “branes” [sites 0 and N+1]
• Many 4-D/5-D theories are limiting cases... study them all at once!
• e.g., N=1 equivalent to technicolor/one-Higgs
Deconstructed Higgsless Modelsg0 g1
f1 f2
gN gN+1
fN fN+1
g2
f3
Foadi, et. al. & Chivukula et. al.
• by folding, represent SU(2) x SU(2) x U(1) in “bulk”
• modify fermions’ location (brane? bulk?)
Generalizations
g0 g1
f1 f2gN
gN+1fNfN+1
g2
f3 "IR""UV"
gN+2 gN+M gN+M+1fN+2
fN+M+1
“UV”
“IR”
Theory SummaryTheory
WW Scattering
Hierarchy Problem
“Calculable”@ LHC?
Precision EW ΛUV
Fundamental Higgs I=J=0 YES! ✔ ✔
1 TeV - MGUT
SUSY I=J=0 No ✔ ✔ MGUT?
Composite Higgs I=J=0 No ✔ f > 5 TeV 50 TeV
Higgsless I=J=1 No ✔Ideal
fermions 10 TeV
Technicolor I=J=1 No ?? Non-QCD few TeV
Conclusions• What unitarizes WW scattering?
• A Higgs boson, with or without SUSY?
• Two new mechanisms, and one old, to address EWSB
• Technicolor
• Composite/Little/Twin Higgs
• Higgsless Models
• All predict new TeV Scale particles
• Extended Electroweak Gauge Symmetries
• Extra scalars or fermions
Backup Slides
Observations•Our standards have changed
• We are content with a low-energy effective theory valid to ~ few TeV
• This is a good thing in preparation for the LHC ...
• Fine-tuning is in the eye of the beholder
• S=O(1) in QCD-like technicolor; experimental bound O(0.1) - hence need 10% fine-tuning?
• Dynamics matters: Inflation makes fine-tuning of flatness problem irrelevant.
Some References
• PDG, “Strong Dynamics Review”, RSC, Narain, & Womersley
• Technicolor: Hill & Simmons, Phys.Rept.381:235-402,2003
• Little Higgs: Schmaltz, Ann.Rev.Nucl.Part.Sci 55:229-270,2005
• Little Higgs: Perelstein, Prog.Part.Nucl.Phys.58:247-291,2007
• Twin Higgs: Goh and Su, Phys.Rev.D75:075010,2007
• Higgsless Models: H. J. He, et. al., Phys.Rev.D78:031701,2008
• “Holographic TC”: Hirn, Martin, Sanz, arXiv:0807.2465
Conflict of S & Unitarity
too large by a factor of a few!
Independent of warping or gauge couplings chosen...
Heavy resonances must unitarize WW scattering(since there is no Higgs!)
This bounds lightest KK mode mass:
... and yields a value of the S-parameter that is
! S !4s2
Zc2
ZM2
Z
8"v2=
!
2
mZ1<
!
8! v
Technicolor Review:Higgs Mechanism:
Custodial Symmetry:
What about the S-parameter?Why are we still talking about technicolor?
• Technicolor may be there –No “computations” of S in non-QCD like
theories (SQCD∼0.5-1, a few too high)
• Technicolor has interesting experimental signatures–Complementary to other BSM theories
• AdS/CFT Correspondence: –Some 4D strongly-coupled theories “dual” to
weakly-coupled 5D theories–New model building ideas–Address S parameter issues