Tuesday, June 4, 2013 PIRE Summer Lecture Series Paul Scherrer
Institute Dan Claes University of Nebraska 1 upersymmetry:
Considerations beyond the Standard Model upersymmetry:
Considerations beyond the Standard Model Slide 2 If youre a
chrystallographer its easy to explain to people the symmetries that
guide your work. 2 Slide 3 Any particle is described by a wave
packet: The square of which localizes a particles position As the
physically significant quantity, this square introduces a
ambiguity. or 3 Slide 4 Any particle is described by a wave packet:
Two particles can have a contact interaction only where their wave
functions overlap The square of which localizes a particles
position As the physically significant quantity, this square
introduces a ambiguity. or the integral of their overlap.
Probability of interacting 4 Slide 5 or The existence and utility
of the two solutions demonstrate a MIRROR SYMMETRY in Schrdingers
equation. 5 The +/- solutions simply result from Schrdingers
equation being a second order differential equation. Slide 6 is a
LINEAR differential equation. which means for any pair of solutions
(r,t), (r,t) 1 2 is also a solution. With which the +/- provides
the symmetric/antisymmetric superpositions 6 Schrdingers Equation
Erwin R. J. A. Schrdinger 1887 - 1961 Slide 7 To which is
insensitive since But the factors + 1 are just the two extremes of
a full phase shift 7 Slide 8 To which is insensitive since But the
factors + 1 are just the two extremes of a full phase shift If we
write: Then for the superposition of two states 8 Slide 9 a real
contribution due only to the RELATIVE PHASE which provides a way to
explain particle interference or exclusion principles the reason
that quantum mechanics came into existence! 9 Slide 10 For a FREE
PARTICLE: We have simple (idealized) plane wave solutions in 1-dim
in 3-dim in terms of physical quantities 10 Slide 11 For a FREE
PARTICLE: We have simple (idealized) plane wave solutions or, if
youre not uncomfortable with using complex expressions in 1-dim in
3-dim in terms of physical quantities 11 and if you really dont
like the complex numbers but neither did Schrdinger Slide 12 The
Fourier transforms used to build wavepackets introduce the complex
factors naturally. 12 Which makes a representative component to any
real wave packet Slide 13 The Fourier transforms used to build
wavepackets introduce the complex factors naturally. Schrdingers
equation assumes complex algebra! The propagation of the wavepacket
is determined by this phase! 13 Which makes a representative
component to any real wave packet Slide 14 We noted was real. So
are other quantities selects out just the real part which sees duty
in the definition of a probability current density: 14 Slide 15 We
noted was real. So are other quantities selects out just the real
part which sees duty in the definition of a probability current
density: velocitydensity According to Quantum Mechanics
Correspondence Principle the -operators eigenvalue momentum so 15
Slide 16 vtvt A NqNq V N per unit cross sectional area per second N
A t = = 16 Slide 17 vtvt A NqNq V N per unit cross sectional area
per second N A t = = (Av t) AtAt velocitydensity 17 Slide 18 ee pp
m proton = 1836 m electron 18 Slide 19 ee pp ee pp m proton = 1836
m electron 19 Slide 20 F = ma Newtons 2 nd Law = v d dt
acceleration, a = v v 0 t t 0 velocity, v = x x 0 t t 0 = x d dt F
= V d dx 20 Slide 21 William Rowan Hamilton (1805-1865) Joseph
Louis Lagrange (1736-1813) 21 Slide 22 often idealized as a
classical systemof mass points The dynamical behavior of any
mechanical system can be derived from its Lagrange function L =
kinetic energy potential energy 22 by applying the Lagrangian to
derive all equations of motion Slide 23 If theres no explicit
dependence on position x,y, or z translation invariance No explicit
t dependencetime translation invariance Conservation of Energy!
Conservation of Momentum! invariance under rotation of the axes -
guarantees Conservation of Angular Momentum! Emmy Noether 1882-1935
Slide 24 If theres no explicit dependence on position x,y, or z
translation invariance No explicit t dependencetime translation
invariance Conservation of Energy! Conservation of Momentum!
invariance under rotation of the axes - guarantees Conservation of
Angular Momentum! Emmy Noether 1882-1935 Invariance to a complex
phase of the wave function Conservation of Electric Charge! Herman
Weyl 1885-1955 Wolfgang Pauli 1900-1958 Slide 25 For linear wave
equations L must involve terms at least quadratic in and 25 L
should be a real scalar (to keep the field operators Hermitian)
guarantees all dynamical variables (energy, momentum, currents) are
real. complex terms must appear as square amplitudes: * vectors
must be part of scalar (dot) products: or any matrices or tensors
must be contracted Slide 26 the simplest Lagrangian with (x ) and
dependence is the Klein-Gordon equation! from which / yields Real
Scalar Field L LL Notice: This simple equation contains no more
than a kinetic energy plus mass term. It must describe a FREE
PARTICLE. 26 Slide 27 The Klein-Gordon equation was offered as a
way to recast Schrdingers equation into a relativistic form:
Starting from the energy-momentum relation: and inserting the
quantum mechanical prescriptions 1920 E. Schrdinger O. Klein W.
Gordon Oskar Klein 1894-1977 Walter Gordon 1893-1939 27 Slide 28
Paul Dirac factored this operator into then looked for solutions to
(either factor, really): Paul Dirac 1892-1984 28 Slide 29 Paul
Dirac factored this operator into then looked for solutions to
(either factor, really): Paul Dirac 1892-1984 (r) = u e i(Et-p r)/h
u(p)u(p) +1/2 1/2 spin up electron spin down electron ?? 1010 cp z
E+mc 2 c(p x +ip y ) E+mc 2 0 10 1 c(p x ip y ) E+mc 2 cp z E mc 2
1010 cp z E mc 2 c(p x +ip y ) E mc 2 0 10 1 c(p x ip y ) E mc 2 cp
z E mc 2 29 Slide 30 a system of particles = L Klein-Gordon + L
DIRAC L (r,t) describes e + e objects describes photons this
describes freely moving non-interacting particles L + L INT What
should an interaction term look like? From free particle
Lagrangians need something like: 30 Slide 31 a system of particles
= L Klein-Gordon + L DIRAC L (r,t) describes e + e objects
describes photons this describes freely moving non-interacting
particles L + L INT What should an interaction term look like? From
free particle Lagrangians need something like: (Limited to
Hermitian Lorentz-invariant products of fields and their
derivatives) Particles interact when their functions overlap, so
expect to include The simplest term would be a bilinear form like:
31 Field equations are derived by the Euler-Lagrange prescription:
Slide 32 Notice: Free particle equations are homogeneous diff. eqs!
When the field is due to a source (electromagnetic (photon!) fields
are due to charges) make the equation inhomogeneous : charged
4-current density thenwould do the trick! 32 Slide 33 Notice: Free
particle equations are homogeneous diff. eqs! When the field is due
to a source (electromagnetic (photon!) fields are due to charges)
make the equation inhomogeneous : charged 4-current density
thenwould do the trick! Dirac electron current exactly the proposed
bilinear form! Or just crack open Jackson: (hermitian conjugate)
field of exiting particle (or antiparticle) field of entering
particle The electromagnetic potential serves as the photon wave
function 33 Slide 34 L Dirac =ic mc 2 Looking more closely at the
FREE PARTICLE Dirac Lagrangian Dirac matrices Dirac spinors
(Iso-vectors, hypercharge) 34 This one parameter unitary U(1)
transformation is a GLOBAL GAUGE TRANSFORMATION. OBVIOUSLY
invariant under the transformation e i (a simple phase change)
because e i and in all pairings this added phase cancels! Slide 35
L Dirac =ic mc 2 Looking more closely at the FREE PARTICLE Dirac
Lagrangian Dirac matrices Dirac spinors (Iso-vectors, hypercharge)
35 Can we GENERALIZE this? Introduce more flexibility to the
transformation? LOCAL GAUGE TRANSFORMATION Is the Lagrangian still
invariant? e i (x) This one parameter unitary U(1) transformation
is a GLOBAL GAUGE TRANSFORMATION. OBVIOUSLY invariant under the
transformation e i (a simple phase change) because e i and in all
pairings this added phase cancels! Slide 36 (e i (x) ) = i( (x))e i
(x) + e i (x) ( ) L Dirac =ic mc 2 36 So: L ' Dirac = c( (x)) ic (
) mc 2 L Dirac ???? Slide 37 (e i (x) ) = i( (x))e i (x) + e i (x)
( ) L Dirac =ic mc 2 37 So: L ' Dirac = c( (x)) ic ( ) mc 2 L Dirac
???? If we identify this new local gauge function with the photon
field: L ' Dirac = e (x) L Dirac Slide 38 L =[ic mc 2 e Demanding
the Lagrangian be invariant under a LOCAL gauge transformation,
forces us to ADD to the free Dirac Lagrangian an extra term! i.e.,
the full Lagrangian MUST include a current-field interaction: 38 )
that couples to This new symmetry suggests free Dirac Lagrangians
are INCOMPLETE. This gauge invariance introduces a new vector field
( Slide 39 To be charged : means the particle is capable of
emitting and absorbing photons e e 39 (hermitian conjugate) field
of exiting particle (or antiparticle) field of entering particle
the photon wave function Slide 40 SU(3) color symmetry of strong
interactions The field is assumed to exist in any of 3 possible
independent color states This same procedure, generalized to
symmetries in new spaces 3-dimensional matrix formed by linear
combinations of 8 independent fundamental matrices SU(3) rotations
occur in an 8-dim space 8-dim vector 8 3 x 3 generators Demanding
invariance of the Lagrangian under SU(3) rotations introduces the
massless gluon fields we believe are responsible for the strong
force. 40 Slide 41 b r u u d dddd rg or gr q q q' 41 ee ee q q a
multi-charged COLOR force is exchanged by gluons Quantum
Chromodynamics Slide 42 hadron decays involve the transmutation of
individual quarks n ee e _ p ?? neutron decay ++ ++ ?? pion decay
ee e _ muon decay ?? in an ambitious effort to explain the
well-known decays: THEN 42 Slide 43 p e+e+ n ?? e _ neutrino
capture by protons ee e as well as the observed inverse of some of
these processes: ?? neutrino capture by muons 43 Chen-Ning Yang
(1922 - ) Robert L. Mills (1927 - 1999) Slide 44 modeled after
photon-mediated charged particle interactions ee ee p p ??? ee e n
p e e W d e W W u e u e e W d required the existence of 3 weakons :
W , W , Z 44 Slide 45 45 Slide 46 Despite the phenomenal success of
the Standard Model, the prevailing theoretical construct consistent
with all high precision measurements to date, purported to describe
all observable matter ~96% of all galaxies (and possibly up to 99%
of all matter in the universe) is DARK non-baryonic matter. 46
Slide 47 47 e e ee e e' e Slide 48 48 e e+ e e+ e e+ e e+ e ee e ee
Slide 49 49 e e+ H u u H d d H s s H t t H + + H Such loop
corrections occur to the Higgs for every massive particle state
(quark and lepton) that exists. Slide 50 Weaknesses Some Standard
Model Weaknesses 19 input parameters Fails to explain the existence
of 3 generations Lagrangian carries a self-interaction Higgs term
-The Higgs couples to itself! - the loop correction to its
self-energy is a quadratically-divergent integral infinite mass HH
50 Slide 51 ~ ~ ~ ~ ~ Particle Name Symbol Spartner Name Symbol
gluon g gluino g charged Higgs H + chargino 1,2,3,4 charged weak
boson light Higgs h neutralino 1,2 heavy Higgs H pseudoscalar Higgs
A neutral weak boson Z photon quark q squark q R,L lepton l slepton
l R,L SUPERSYMMETRY 0 The L ightest S upersymmetric P article
provides E T if the LSP is stable, neutral, colorless &
R-parity is conserved photons and E T if the LSP is a gravitino and
NLSP a neutralino long-lived particles if the LSP decays weakly
SUSY particles are heavy high p T final state objects 51 Each known
particle has a superpartner spin= s 1/2 Bosons have fermionic
partners and vice versa Results from Lie superalgebras Slide 52 e e
W ee ee ee ee electrostatic repulsion nuclear binding u u d d g
weak decays 52 A massless transmits energy and momentum A massless
gluon carries off energy, momentum and color A massive W changes
energy, momentum, and mass! Slide 53 53 e- ~ ??? -- -- ~ q q ~ The
LAST remaining symmetry allowed under S-matrix theory would create
contact interactions transforming fermions into bosons Through new
bilinear products of ~ boson fermion 3 rd field with the final
field factor for the Euler-Lagrange prescription to differentiate
against (and provide the 3rd leg of each of these interactions.
1973 Julius Wess and Bruno Zumino Slide 54 Minimal Supersymmetric
SM (1981) Extension adding the fewest new particles 2 Higgs doublet
h 0 H 0 A 0 H + and described by 4 parameters M 1 SU(1) M 2 SU(2)
gaugino mass parameter at EW scale higgsino mass parameter tan
ratio of VEV of Higgs doublets scalar sector described by MANY mass
parameters different SUSY breaking different class of models MSSM
Assumptions: SUSY particles are pair produced Lightest SUSY
particle (LSP) is stable Lightest SUSY particle is 5 free
parameters m o common scalar mass m 1/2 common squark mass A o
trilinear coupling tan sign( SUSY Symmetry Breaking mSUGRA 54 Super
partner masses expected to between 100 and 1000 GeV Slide 55 Every
allowed Standard Model process is mirrored by a complementary one
with a continuous SUSY line tracing through its history. e e W 55 ~
~ e e W ~ ~ ~ t ~ b ~ W+ WW c ~ u d WW e ee e ee s ~ WW e ee u ~
Slide 56 Provides gauge unification of the 3 fundamental forces
Standard Model MSSM 1/i1/i i = 1 i = 2 i = 3 no SUSY with SUSY
Energy ( GeV ) 60 40 20 0 10 2 10 6 10 10 14 Slide 57 Nuclei can be
fermions or bosons Although composite (with internal structure) and
not fundamental particles Spins add linearly 4 He: 2 protons, 2
neutrons with spin 4 He = 0 (boson!) 57 Fe: 26 protons, 31 neurons
with spin 57 Fe = 1 / 2 (fermion!) 57 Even-Even, Even-Odd, Odd-Odd
theories model complicated nuclei as built of nuclei pairs ( bosons
) and unpaired fermions (1980 Iachello) Slide 58 58 Shell model
Resembles electron shells (works well for light nuclei) Liquid
model (Proposed by Gamow) Works best for heavy nuclei m 100 a.u.
Supported by studies made of nuclear quartets which include a core
of each type of nucleus, all having the same total (bosons +
fermions) A supersymmetric treatment recognizes neutron capture,
and proton as well as two-nucleon transfer reactions as the means
of fermion boson transformations. 196 Au, 195 Au, 195 Pt, 194 Pt
194 Ir, 193 Ir, 193 Os, 192 Os most studied The Interacting boson
model (IBM) Two Popular, Competing Nuclear Models Slide 59 Neutron
Capture Neutron Emission Slide 60 196 Au excited states predicted
from three other nuclei Showed many unknown, but eventually
experimentally confirmed, energy levels (with the proper angular
momentum) 60 have met with success in predicting nuclear reaction
cross sections and the levels (and line intensities) of excitation
spectra. Slide 61 Energy is quantized as a consequence of the wave
nature of matter bounded by a potential only very specific (sine,
cosine, exponential) functions can satisfy the boundary conditions.
The 3d geometry of space forces angular momentum to be conserved!
Spatial descriptions of a system must be symmetric in angle cyclic
in 2 ! Slide 62 Energy is quantized as a consequence of the wave
nature of matter bounded by a potential only very specific (sine,
cosine, exponential) functions can satisfy the boundary conditions.
The 3d geometry of space forces angular momentum to be conserved!
Spatial descriptions of a system must be symmetric in angle cyclic
in 2 ! L z lm ( , )R(r) = m lm ( , )R(r) for m = l, l+1, l 1, l L 2
lm ( , )R(r)= l(l+1) 2 lm ( , )R(r) l = 0, 1, 2, 3,... Slide 63
Spectra of the alkali metals (here Sodium) all show lots of
doublets 1924: Pauli suggested electrons posses some new,
previously un-recognized & non-classical 2-valued property 63
-3.04 -.0021 eV 0.597 nm -5.14 589.0 nm 589.6 nm Energy (eV)
spin-orbit splitting Slide 64 Perhaps the working definition of
angular momentum was too literal too classical Instead consider the
operator relations might be the more fundamental definition
Mathematicians recognize such Commutation Rules as the defining
algebra of a non-abelian (non-commuting) group [ Group Theory;
Matrix Theory ] 64 Slide 65 Perhaps the working definition of
angular momentum was too literal too classical Instead consider the
operator relations might be the more fundamental definition
Mathematicians recognize such Commutation Rules as the defining
algebra of a non-abelian (non-commuting) group [ Group Theory;
Matrix Theory ] Reserving L to represent orbita l angular momentum
a more generic operator J represents any or all angular momentum
study this as an algebraic group Uhlenbeck & Goudsmit find J=0,
, 1, 3 / 2, 2, are actually all allowed! 65 Slide 66 with 0 as the
LSP Stop charm + E T and ~ 66 As long as ~ ~ R-parity pair
production Slide 67 with 0 as the LSP Stop charm + E T and ~ 67 As
long as ~ ~ R-parity pair production Search for: 2 acoplanar jets
plus E T >60 GeV veto on Leptons, isolated tracks to suppress SM
sources of E T and flavor tag (>= 1 jet) Slide 68 Distributions
of jet energies and missing energies in data are compared to the
distributions expected from ordinary phenomena looking for any
excess in regions that would suggest evidence for supersymmetric
decays: 68 Slide 69 With no evidence so far of such particles we
can only identify excluded regions of the SUSY parameter spaces we
search. The enclosed regions at right show the mass ranges in which
t 1 and 1 are known not to exist. ~ ~ 69 Slide 70 Stop charm + E T
Optimizing mass-dependent cuts on H T and P = max + min 70 Jet1
Jet2 ETET min max Slide 71 Stop charm + E T Optimizing
mass-dependent cuts on H T and P = max + min For H T >140 GeV P
Squarks/Gluinos jets + E T At least 3 jets E T > 25 GeV and E T
> 25 GeV Separate 2-jet, 3-jet and >3-jet analysis. 73 Slide
74 Squarks/Gluinos jets + E T Mgluino < 402 GeV/c 2 excluded for
Mg~Mq Mgluino < 309 GeV/c 2 excluded any Mq ~ ~ 0.96 fb -1 A 0
=0 tan = 3
