# Presentation: 124575

## Theories to resolve hierarchy

- Despite the worry that maybe we have it all wrong, I must still bow to the 25 years of theory that have been predicated on the idea that the standard model must be changed at or around 1 TeV
- I will briefly touch on the principles behind
- Extra Dimensions
- Technicolor
- Little Higgs Models
- Supersymmetry
-  and say a few words about their phenomenology

## Extra Dimensions

*v / M**pl* ~ 10-16
- Perhaps the problem with the hierarchy is that we are using the wrong *M**pl*
- Perhaps four-dimensional quantum field theory breaks down at the TeV scale.
*If there are extra dimensions, the M**pl** we measure is not the “fundamental” one of the full D+1 = (3+1) + (D-3 extra) dimensional theory.*

*Volume of extra dimensions *

*Determinant of metric*
*Scalar Curvature*

*Einstein Lagrangian in D+1 dimensions*

*D -1*

*D -1 *

*D -1 *

*  **D -1    *

## Possible scenarios

- Thus the measured *M**pl,3* may be large because
- The *M**pl,D* dimensional Planck scale is at 1018 GeV and the extra dimensional volume is around the same scale
- The *M**pl,D* Planck scale is of order 1 TeV, and the extra-dimensional volume is very large, of order 1030 GeV(*3-D*)
- Some combination of both; perhaps the Planck scale is at 100 TeV, out of immediate reach
- What would happen if the second case were true?
- A big volume means low energy effects --- so why haven’t we seen evidence for them already?

*  **D -1    *

## How to allow Mpl = TeV

- A simple scenario: we’re trapped on a 3+1 dimensional subspace in the extra dimensions.  (‘Brane’)
- The only way to find out about the extra dimensions that are larger than .01 fm* *is through gravity itself
- If extra bulk dimensions have length L, gravity will look 3+1 dimensional at distances larger than L
- At shorter distances, Newton’s law changes
- But most collider physics does not change until energy reaches ~ *M**pl* .
- _Analogy: _
***  ******electrons trapped in a wire inside of a waveguide***

*The Bulk *

*We are here*

*Extra Dimensions*
*Of Finite Extent*

*Our infinite(?) 3 dimensions*

## Phenomena: Mpl = TeV

- As scattering energy increases, easier to radiate higher-dimensional graviton
-  *q q **** ** + invisible graviton    ,   g g **** g + invisible graviton*
- Gravitons in s-channel, either as many weakly-coupled light particles or as isolated strong resonances, can affect
*q q **** *etc.
- At very high energies, could even see non-linear **quantum gravitational** processes, including black holes or even excited strings from string theory
*q q **** ****boom***
-             multiple particles, crudely thermal spectrum with *T* ~ *M**pl*  , or *M**string*
- CAUTION: Statements in literature that gravity is **universal,** and leptons and photons will be produced abundantly along with quarks and gluons by decaying strings or black holes, are *only true at ultra-high energies *; they need not be the case at LHC!

## Second case: Rextra = TeV

- Suppose *M**pl* somewhat higher, so we may not see gravitational effects, but...
- There are new TeV-scale dimensions in which not all SM fields are points
- Those fields in the extra dimensions have a new feature: Kaluza-Klein states (`KK modes’) with masses ~ 1/ *R**extra*
- The principle behind KK modes is similar to harmonics on violin string
- String modes labelled *n*,  frequency *n**= n* *1** *, wave function sin [*ny/L*]* *

- Higher modes have higher energy.

_*Better analogy:*_
- KK modes are more like
- the modes of photons
-  in a wave guide
*See Jackson E+M book*

- *(y)=* *n* *n* sin [*ny/L*]*     n >0*

## Field in extra dimensions

- Massless scalar field in 4+1 dimensions (with **periodic** extra dimension *y*) has similar mode expansion
- *(x,y)=* *n* *n**(x)* cos [*2**ny/L*]*     n *_***any***_* integer      *[* *sin *also *]
- There are an infinite number of 3+1 dimensional scalar fields, with masses *m**n** = n/L* (and **m*****0***** = 0**, the massless 3+1 dimensional scalar)
- Similar tower of states for gluon (or photon or top quark or Higgs or graviton) in 4+1; a massless ordinary gluon plus a tower of KK gluons of increasing mass, *m**n** = n m**1* .
*But equal spacing of KK mode masses is not a robust prediction, nor is equal spin *
- Not equally spaced:
- suppose 2 extra dimensions of length *R**1**, R**2*; modes specified by integers *n**1**, n**2*; masses are
- Not necessarily same-spin;
- gluon in 5+1 dims  tower of vectors (*A**0*, *A**1*, *A**2*, *A**3*) and scalars (*A**4*, *A**5*) in 3+1 dims
- A spin-1/2 fermion in 3+1 dims can descend from a higher-spin fermion in higher dims

## Spectrum not predicted

- Suppose dimensions are not uniform
*Consider nonuniform violin string*
- Can get any spectrum, depending on density profile
- Only prediction is a tower of states
- Nonuniform extra dimensions have similar possibilities
- One fairly reliable prediction:  towers of repeaters ---
-        including new particles of the same color, electric charge, weak isospin, possibly spin, but higher mass
- In *many* models, repeaters for *W, Z, h, t (b)*.
- Repeaters for the gluon, graviton, other matter etc. possible too.
-   *“*_**Universal Extra Dimensions**_*”:   all SM particles have a tower with similar splittings*

_*Better analogy:*_
- A wave guide of arbitrary cross
-  can have a nearly arbitrary
- spectrum of photon modes

## Physics of Repeaters

- Production modes:
- Consider any 21 SM process, put a repeater in the final state
- Consider any SM 22 production process, put one or two repeaters in final state
- Decay modes: Consider any 12 SM process, put a repeater in the initial state
- Couplings *will not be identical* to those of SM,
- typically some are of similar order while others are highly suppressed.

* **but these couplings may only be large **for some quarks and not others**; likely either to be **flavor-universal** or **heavy-flavor-weighted** to avoid big flavor-changing neutral currents*

- Repeater interactions from the *Zqq* interaction

## Physics of Repeaters

- Production modes:
- Consider any 21 SM process, put a repeater in the final state
- Consider any SM 22 production process, put one or two repeaters in final state
- Decay modes: Consider any 12 SM process, put a repeater in the initial state
- Couplings *will not be identical* to those of SM,
- typically some are of similar order while others are highly suppressed.

- Repeater interactions from the *htt* interaction

## A few examples

*If the couplings of a Z’ are flavor universal, we may produce it directly via qq, see it as lepton-pair resonance*

*If the couplings of a W’ are heavy-flavor weighted, we may produce it in Vector Boson Fusion, *
*see it in heavy flavor or diboson final states*
*(resonance in 4 jets with b-tags,2 leptons + 2 jets)*

*If a t’ repeater has a small coupling to gluon plus top, it may be mainly pair-produced and decay to Wb, Zt, ht .  Up to 10 jets!  *
*But large energy release, so W,Z,t are boosted; look for four fat jets with substructure and two b tags, or three fat jets and a lepton near a b, etc.  *
*Two equal-mass resonances to reconstruct.*

## Back to Hierarchy Problem

**Maybe the hierarchy is due to nonperturbative strongly-interacting dynamics in quantum field theory**
- Not all mass scales have a hierarchy problem.
- QCD has taught us a great deal
- Review
- Abelian Higgs Model
- Standard Model Higgs Model
- A third Higgs-like model: Chiral symmetry breaking in QCD.
- Using this last model to break electroweak symmetry
-         technicolor – solution to the hierarchy problem – problematic
- Generalizing this model and playing some games
-         little Higgs –mitigation of the hierarchy problem – less problematic

## No Hierarchy Problem in QCD

- *QCD* doesn’t appear in Lagrangian;
- What appears is *s*, but unless *s**(M)* ~ 1, *QCD* << *M*
- Thus it is _*easy*_ to obtain *QCD* << *M* ! the *opposite* of the situation with *m**h*
**There is no mystery why the rho meson (770 MeV) is light.**
*Can we get a composite Higgs boson created through new strong forces? With a mass that is light for the same reason that the rho meson is light?*

*One-loop beta function coefficient*
*~ 5 in QCD*

- If *v / M**pl* << 1 is a “problem”,  why isn’t *QCD** / M**pl*  << 1 a “problem”?

## Abelian Higgs

- Abelian Higgs model: U(1) global symmetry or gauge ‘symmetry’
- Complex scalar , potential V( || )
- Minimum at  *  =*  *v e**i***                      ( any angle  )
- This breaks the U(1), since under U(1) rotation would shift.
- If U(1) symmetry is *global*, fluctuations of  are a _*massless Goldstone boson*_
- If U(1) symmetry is *gauged*, fluctuations of  are absorbed by _*photon*_, which becomes _*massive*_
- In either case, fluctuations of | around *v*, the “Higgs boson”, are a _*physical massive particle*_

## Standard Higgs

- Standard Higgs model SU(2) x U(1) global symmetry or gauge ‘symmetry’
- Complex doublet scalar , potential V( |†| )
- Minimum at  *  =*  (*v sin ** e**i***, *v cos ** e**i***) ( any angle  )
- This breaks the U(1), since under U(1) rotation would shift.
- This breaks the SU(2), since under SU(2) rotation would shift.
- But a diagonal U(1) is unbroken:
- to see this, first rotate vev to
-                           *  **=*  (*0***, *v*)
-   Then clearly the following rotation leaves the vev < unchanged.
-      first rotate by SU(2):   ((*e**i*** *e**-i***
-      then rotate by U(1):    ((*e**i*** *e**i***
-                                                        which altogether is     ((*e**2i*** 
- If SU(2) x U(1) symmetry is *global*, fluctuations of  are _*massless Goldstone bosons: three pions.*_
- If SU(2) x U(1) symmetry is *gauged*, fluctuations of  are absorbed by three of the four gauge bosons, which become _*massive W,Z*_*, *and a _*massless photon*_ remains.
- In either case, fluctuations of | around *v*, the “Higgs boson”, are a _*physical massive particle*_

## Chiral Symmetry Breaking

- In QCD, and technicolor, these ideas are generalized; the field  is composite,and its vev is generated by complicated quantum effects.
- In QCD with *N**f* massless quarks, the theory has a *SU(**N**f **)**L** x SU(**N**f **)**R* chiral symmetry
- This chiral symmetry is spontaneously broken at low energy because of strong nonperturbative dynamics
- A *composite* operator, a quark-antiquark bilinear = *q**i**q**j* , an *N**f **x N**f  *matrix scalar
*                                                                             **for two flavors  * =
-                 develops a vev proportional to the unit matrix
- It thereby breaks the *SU(**N**f **)**L** x SU(**N**f **)**R* chiral symmetry to the diagonal
*       **SU(**N**f **)**L+R* , generating *N**f**2* *- 1* massless Goldstone bosons.
- Real QCD has quarks with small masses, *m**u**,m**d** << m**s** < **QCD* , but the fact that
-  22 - 1 = 3 particles must be massless when *m**u**,m**d ** 0 , and*
-  32 - 1 = 8 particles must be massless when *m**u**,m**d**,m**s** 0 *
-            ensures that the s* **K*s*, * are all light, with the s the lightest of all.

## QCD-based Higgs-like Model

*N**f* = 2 QCD model: *SU(2)**L** x SU(2)**R* global symmetry or gauge ‘symmetry’
- 2 x 2 matrix scalar  
- Potential V( |†| ), with minimum at  * =*  *v* **M **which we can always rotate to * =*  *v* **1 **(unit matrix)
- This vev preserves the *diagonal *subgroup *SU(2)**L+R* , with rotations **U****R ***= ***U****L****-1**
- If *SU(2) x SU(2)* symmetry is *global*, fluctuations of **U****R ***= ***U****L** , broken combination of the SU(2)’s, are _*massless Goldstone bosons – three pions*_
- If *SU(2) x** ****U(1)*** symmetry is *gauged*, fluctuations **U****R ***= ***U****L** are absorbed by three of the four gauge bosons, which become _*massive W,Z, and a massless photon*_ remains*.  Nothing remains of the rest of **SU(2)**R* *; it is not a symmetry.*
- In all cases, fluctuations of | around *v*, the “Higgs boson”, are a physical massive particle, known in QCD as the  or the *f**0, *; (very wide in QCD because  very rapidly.)

## QCD-based Higgs-like Model

*N**f* = 2 QCD model: *SU(2)**L** x SU(2)**R* global symmetry or gauge ‘symmetry’
- 2 x 2 matrix scalar  
- Potential V( |†| ), with minimum at  * =*  *v* **M **which we can always rotate to * =*  *v* **1 **(unit matrix)
- This vev preserves the *diagonal *subgroup *SU(2)**L+R* , with rotations **U****R ***= ***U****L****-1**
- If *SU(2) x SU(2)* symmetry is *global*, fluctuations of **U****R ***= ***U****L** , broken combination of the SU(2)’s, are _*massless Goldstone bosons – three pions*_
- If *SU(2) x** ****U(1)*** symmetry is *gauged*, fluctuations **U****R ***= ***U****L** are absorbed by three of the four gauge bosons, which become _*massive W,Z, and a massless photon*_ remains*.  Nothing remains of the rest of **SU(2)**R* *; it is not a symmetry.*
- In all cases, fluctuations of | around *v*, the “Higgs boson”, are a physical massive particle, known in QCD as the  or the *f**0, *; (very wide in QCD because  very rapidly.)

## Technicolor – Scaled-up QCD

- Add a new QCD-like group – technicolor -- to SM
-     with two massless flavors of techniquarks,
-     symmetry group *SU(2)**L** x SU(2)**R** x U(1)**B**  , *and
-     strong coupling scale* *TC* ~ *1 TeV
- Gauge part of symmetry group: *SU(2)**w** x U(1)**Y*
- Then the strong technicolor interactions cause a composite operator, a techniquark-antitechniquark bilinear, to break
*                 **SU(2)**w** x U(1)**Y** **U(1)**em*
- Three TC-pions are absorbed into *W**+ **,W** -**,Z ; *photon remains massless
-  TC= Higgs boson: heavy, wide resonance; decays very rapidly to *WW, ZZ*
- This _solves_ the hierarchy problem, because TC / *M**pl* is naturally small

_*What if there were no Higgs vev in the real world? *_
*- the up and down quarks would have been massless; *
*- SU(2) x SU(2) would have been exact, with SU(2)xU(1) gauged; *
*- the QCD vev would have broken electroweak SU(2)xU(1) **** U(1) in the standard way; *
*- the W and Z bosons would have eaten the pions and would have had mass of tens of ****MeV****; *
*- and the* ***would**** have been the Higgs boson!*

## Why doesn’t this work easily?

- If TC = 1 TeV, expect quantum corrections to classical SM of order (*m**W* /TC)*2 *~ 1 %
*but SM precision tests work to 0.1 % – 1 %*
- Composite Higgs must couple to SM fermions and strongly distinguish different flavors
*but get large flavor changing neutral currents, disallowed by experiment*
- Composite Higgs boson must couple strongly to top quark
*Possibly make top quark [and bottom quark?] composite too? Or at least ****strongly coupled to technicolor sector****.*
*Resolve most problems with non-QCD-like “Walking Technicolor” (some dynamical assumptions necessary)*
*See for example*
*            *Walking** **technicolor** **and electroweak radiative corrections.
-                 _[Raman Sundrum](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Sundrum,Raman%22)_, _[Stephen D.H. Hsu](http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Hsu,%20StephenD.H.%22)_
**                ****Nucl.Phys.B391:127-146,1993** , **hep-ph/9206225**

## Technicolor spectrum

- The spectrum of QCD has far more than just the ,.
- Similarly technicolor has much more than just the *W*, *Z*, *h*
**Repeaters** for the *W*, *Z*, *h*
**Repeaters** for top? bottom?
- Other spins, other group representations.
- Repeaters are expected for any composite system (think of atomic physics)

***M******2 ******/ (770 MeV)******2***

*Spin 1*

*Spin 2*

*Spin 3*

*Spin 4*

*Spin 0*

*TC-Rho mesons are strongly-interacting W,Z repeaters*

- QCD spectrum
- (not complete)

## Detecting Technicolor?

- Unfortunately precision electroweak constraints imply large  TC
- Only lowest states above W,Z probably in reach: TC mesons, 2-3 TeV
- Produce TC in *WW, WZ* Vector Boson Fusion, though very challenging to detect and study at LHC
- More work needed to understand best techniques, reach

## TC vs. Extra Dims

- Both  technicolor and fields in TeV extra dimensions predict towers of repeaters at ~ TeV
- How do we tell the difference between these classes of models?
_***WE CAN’T TELL THE DIFFERENCE! Even in principle!***_
***More precisely, we can’t ******always****** tell the difference, even in principle.***
- Some technicolor models really don’t look extra-dimensional
- Some extra-dimensional models really don’t look QCD-like
- But there is an overlap region where the models are quantum mechanically identical, with buzzwords: “holographic theories”, string-gauge theory duality, Randall-Sundrum TeV-brane models.
*Caution: the statement that RS and technicolor models are identical is glib; the precise relation is known! but beware naïve statements in the literature!*
- It is true that those RS TeV-brane models that solve the hierarchy problem do so in effectively the same way technicolor does

## Technicolor vs. TeV xtra dims

**And we may not be able to tell the difference in practice**
- To properly identify a technicolor or TeV-scale extra-dimensional model, need to study its towers of states
*But LHC will give us only the first one or two repeaters --- probably insufficient, at least in early stages, for distinguishing models*
- Still, production mechanisms and branching fractions tell us much about repeaters –
- Which SM fermions do they mainly couple?
*(Quarks only? L-handed fermions only? Heavy-flavor only? All fermions equally?)*
- Which SM vector bosons do they mainly couple to?
*(Couple to Ws and Zs? To Zs only?  To gluons?)*
- This information will be very helpful in separating models; but –
- We need to measure rates and branching ratios with moderate precision.*     Tough at LHC!*

## Evade constraints?

- All of the models we have just considered have drastically new physics at the weak scale
- All are highly constrained by electroweak precision measurements, which typically demand new physics be above a few TeV
- [but see 3rd lecture for caveats...]
*This is bad for LHC, but it is also bad for keeping the Higgs boson relatively light without large corrections from loops or from breakdown of four-dimensional quantum field theory*
- That suggests that perhaps nature has a light Higgs along with new *weakly-coupled* *four-dimensional* *field-theory* physics at a TeV
- Could we make the Higgs a Goldstone boson?  This would be naturally light and, at low enough energy, weakly interacting (since all Goldstone-boson interactions must vanish at zero momentum)
- The Little Higgs --- pushes the strongly interacting physics up to 10 TeV at the expense of weakly-interacting physics at 1 TeV.

## QCD-based Higgs-like Model

*N**f* = 2 QCD model: *SU(2)**L** x SU(2)**R* global symmetry or gauge ‘symmetry’
- 2 x 2 matrix scalar  
- Potential V( |†| ), with minimum at  * =*  *v* **M **which we can always rotate to * =*  *v* **1 **(unit matrix)
- This vev preserves the *diagonal *subgroup *SU(2)**L+R* , with rotations **U****R ***= ***U****L****-1**
- If *SU(2) x SU(2)* symmetry is *global*, fluctuations of **U****R ***= ***U****L** , broken combination of the SU(2)’s, are massless particles – _three massless GBs_
- If *SU(2) x** ****U(1)*** symmetry is *gauged*, fluctuations **U****R ***= ***U****L** are absorbed by three of the four gauge bosons, which become _massive __*W,Z, *__and a massless photon_ remains*.  Nothing remains of the rest of **SU(2)**R* *; it is not a symmetry.*
- In all cases, fluctuations of | around *v*, the “Higgs boson”, are a physical massive particle, known in QCD as the  or the *f**0, *; (very wide in QCD because  very rapidly.)

## QCD-based Higgs-like Model

*N**f* = 3 QCD model: *SU(3)**L** x SU(3)**R* global symmetry or gauge ‘symmetry’
- 3 x 3 matrix scalar  
- Potential V( |†| ), with minimum at  * =*  *v* **M **which we can always rotate to * =*  *v* **1 **(unit matrix)
- This vev preserves the *diagonal *subgroup *SU(3)**L+R* , with rotations **U****R ***= ***U****L****-1**
- If *SU(3) x SU(3)* symmetry is *global*, fluctuations of **U****R ***= ***U****L** , broken combination of the SU(3)’s, are massless particles – _eight__ massless GBs: _
- _what would be pions, Kaons, and the eta in QCD._
- If ***SU(2)******L+R*** *x ****U(1)******L+R**** *(subgroup of ***SU(3)******L+R***) symmetry is *gauged*,
- The vev does **not** break it, so the _*four gauge bosons remain massless*_.
- However, n*othing remains of the rest of **SU(3)**L** x SU(3)**R* *; *there is no broken global symmetry.
- The fluctuations **U****R ***= ***U****L** are _*massive pseudo-GBs*_, with masses generated at one loop by the gauge bosons.
- Let’s consider this case more carefully.

## Higgs as PGB?

|  |  |  |
| --- | --- | --- |
|  |  |  |
|  |  |  |

*U(1)**R**  **[+1/2, +1/2, -1]*

*SU(2)**L*

- The Eight Pseudo-Goldstone Bosons

*Could Higgs doublets be the “Kaons” of a new strong dynamics? These doublets would naturally be very light. Then we could have  v / ******<<1 ****** smaller loop corrections than TC*

*SU(2)**R*

*U(1)**L** **[-1/2, -1/2, +1]*

| particle | SU(2)L+R | U(1)L+R |
| --- | --- | --- |
|  | 3 | 0 |
|  | 2 | 1⁄2  |
|  | 2 | -1⁄2  |
|  | 1 | 0 |

## Little Higgs vs TC

- Technicolor
*SU(2)xSU(2)* *SU(2)* global breaking by quark bilinear at scale  _breaks_ electroweak *SU(2)xU(1)*  *U(1)*
- Higgs is , fluctuations of vev
- absorbed by *W******W******Z***
- Problem to get small loop corrections
- Proto-Little Higgs
*SU(3)xSU(3)* *SU(3)* global breaking by quark bilinear at scale  leaves electroweak *SU(2)xU(1)* _unbroken_
- Higgs is *K**0 **+ K**0*, develops vev *v* <<  
*K******K******K**0 **- K**0*absorbed by *W******W******Z***
- Problem to keep Kaons light enough, arrange *v* <<  
*In both cases, getting flavor physics, especially large top quark mass, is a challenge*

## Not quite enough yet

- What mass do the SM gauge bosons give to the Kaon-like states?
- This isn’t light enough.
-  still needs to be of order a couple of TeV; we still get pretty big corrections to precision electroweak tests of the SM.  We want  bigger while keeping the Higgs light.
- But there is a trick to make the PGBs lighter!

## Collective Symmetry Breaking

- Simple case: ***enlarge*** SM electroweak group to *SU(2)xU(1)xG’, *with couplings* g**2**, g**1**, g’*
- At scale  ~ 10 TeV, a **new** **strong interaction** causes *SU(2)xU(1)xG’ *to _break_ to* SM SU(2)xU(1).  *The ***resulting***** *****massive gauge bosons*** get masses ~* g *~ 1 TeV SM gauge bosons are still massless

- if all gauge couplings are zero, a large global symmetry group is present and is spontaneously broken by the new strong interaction; the Higgs is a massless GB

- if *g**1**= g**2**= 0 < g’*,
- part of the global symmetry group is broken explicitly by the G’ gauge group;
- a smaller global symmetry group remains and is broken spontaneously
- the Higgs is still a massless GB

- if *g**1**, g**2 **> 0 = g’*,
- a different part of the global symmetry group is broken explicitly by the *SU(2)xU(1) *gauge interaction;
- a smaller global symmetry group remains and is broken spontaneously ;
- the Higgs is again still a massless GB

- If all three gauge couplings are non-zero
- all of the global symmetries broken by the gauge interactions;
- the strong interaction does not spontaneously break any global symmetries, and
- the Higgs boson acquires a mass.

## From one loop to two loops

- ONLY if both *g’* _and_ the SM gauge couplings are non-zero does the Higgs boson acquire a mass.
- Since no one coupling is enough to generate this mass, no one-loop diagram (which always involves squares of one coupling) can be the cause.
***The formula for the Higgs mass must involve a product of two couplings at once (so that it vanishes if either coupling is zero) which can only happen in a two-loop graph.***
- Two-loop corrections are small enough to avoid large electroweak corrections.

*Same coupling*

*g**2**2 **g’**2 **2*

*g**2 **g’ *

## How does this happen?

- Magically, the one-loop graphs must cancel.  What’s doing the canceling?
- At one level, the magic is in the group theory: symmetry.
- On the other hand, there is an explicit cancellation going on.
- The miracle is in the masses and couplings of the **new heavy gauge bosons** , mass about 1 TeV, from the breaking of *SU(2)xU(1)xG’ *,
- They are precisely what is needed to cancel the one-loop contributions from the W and Z bosons.
- The cancellation typically requires **repeaters for *****W, Z, h, t, b******(?)***!
-          *(however there are loopholes here)*

## More Model-Building to do

- We’re not even close to done: need
- to ensure Kaons get vev *v* = 246 GeV << – more structure, other variants – more states in Higgs sector
- to ensure large top quark Yukawa coupling generated without giving large Higgs mass, need additional physics in top quark sector  top quark repeaters *too *
- Repeaters for *W,Z,h,t,[b]* are common feature in successful models but
- Other non-repeater states are often generated
- Not all repeaters are always generated
- Some models with no repeaters exist; at least one exists [original twin Higgs] with no visible states! **(scary!)**
- LHC Phenomenology – as we have discussed for earlier repeaters (perhaps plus exotic higgs and quark-like states) but with weaker couplings at the TeV scale than in TC and some extra dim models
- In the end,
- = 10 TeV, repeaters at 1 TeV, *v *= 246 GeV appears possible
- But this is controversial; no fully convincing model exists
- See Schmaltz and Tucker-Smith review for additional reading on models.
- Ann.Rev.Nucl.Part.Sci.55:229-270,2005 ; e-Print Archive: hep-ph/0502182

## Extra Fermionic Dimensions

- Can we do better? Can we move  , where the Little Higgs is strongly coupled, much higher than 10 TeV?
- Is there a weakly coupled model valid up to the Planck scale with a light Higgs and no hierarchy problem?
*Yes! Extra fermionic dimensions do the trick*.
- This is also called “supersymmetry”.  *I will not study this model in detail – see Polesello 2006 SLAC summer school lectures, Luty TASI 2004 lectures, and many other reviews*
- Just as with ordinary bosonic dimensions, there is a tower of states for each standard model particle
- But there are some key differences
- There need be only one repeater for each particle
- The spin of the repeater differs from the original’s spin by 1/2
- Every particle in the SM *must* have a repeater at scales that cannot differ too much from the weak scale