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