The  Standard  Model    

of  Par0cle  Physics  

Back  to  Gauge  Symmetry  

Laws  of  physics  are  phase  invariant.  

Probability:

P = ψ (r ,t) 2 =ψ *(r ,t)ψ (r ,t)

Unitary scalar transformation:

U(r ,t) = eiaf (r ,t )     U 2 =U*U = e−iaf (r ,t )eiaf (r ,t ) = 1

ψ (r ,t) ⇒ ′ψ (r ,t) =Uψ (r ,t) =ψ (r ,t)eiaf (r ,t )

′P = ′ψ (r ,t) 2 = ′ψ * (r ,t) ′ψ (r ,t)U*U =ψ *(r ,t)ψ (r ,t) = P

Local Unitary Phase Transformation:

Let ψ (x, t)→ψ (x, t)eiaf (x,t ).

Schroedinger's equation for a free particle:

−2

2m∂2ψ (x, t)

∂x2 = i ∂ψ (x, t)∂t

−∂2 ψ (x, t)eiaf (x,t )( )

∂x2 = i 2m

∂ ψ (x, t)eiaf (x,t )( )∂t

This no longer has the same form of the Schroedinger eq.

Carry out the differentiation. It is straight forward but messy.

Vector and scaler fields: Bx and B0   in one dimension Bx and B0  ( )

Notice, I changed the name from A and V to Bx and B0for a very important reason - to be revealed.

Which transformation as:

Bx → ′Bx = Bx +∂f∂x

B0 → ′B0 = B0 +1c∂f∂t

f is an arbitrary function of x and t.

Need to introduce new fields?

12m

−i ∂∂x

−g1cBx

⎛⎝⎜

⎞⎠⎟2ψ (x, t) = i ∂

∂t− g1B0

⎛⎝⎜

⎞⎠⎟ψ (x, t)

Schroedinger’s  equa0on.  

Charge changing weak interactions only involve left handed doublets.Include them in wave function.

ψ (x, t)→ψ (x, t)χL            χL,l : leptons ,       χL,q : quarks ,

χLl ,  χLq   are column matrices for left handed leptons and quarks respectively.

Examples:              χL,l =νee−

⎛

⎝⎜⎜

⎞

⎠⎟⎟L

          10

⎛⎝⎜

⎞⎠⎟ L

= νLe ,  01

⎛⎝⎜

⎞⎠⎟ L

= eL−        

                              χL,q =du

⎛⎝⎜

⎞⎠⎟ L          1

0⎛⎝⎜

⎞⎠⎟ L

= dL ,  01

⎛⎝⎜

⎞⎠⎟ L

= uL  

SU2 Gauge Transformation!

SU(2) transformation: ψ (x, t)→ψ (x, t)eiε iτ =ψ (x, t)χLe

i εν (x,t )τν1

3∑

τν are 2 × 2 matrices which operate on 2 ×1 column matrix χL = (χLl , χLq )

Requiring invariance under SU(2) phase transformation introduces the

3 SU(2) fields: Wµ+, Wµ

−, Wµ0 with weak coupling charge g2

Compare with ψ (x, t)ξR,L →ψ (x, t)ξR,Leiaf (x,t ).

ξR,L are either right or left singlets, e.g. eR− , eL

−

3 neutral fields : Bµ, Coupling charge : g1

U1 Gauge Transformation!

How  are  these  fields  and  charges  related  to  the  observed  fields  and  charges  of  the  electromagne0c  and  weak  interac0ons?  

Combining U(1)L × SU(2)LU(1)L × SU(2)L transformation: ψ L (x, t)→ψ L (x, t)eiβ+i

ε iτ

Bµ  and Wµ0  act together on charge neutral L  leptons and quarks.

Since SU(2)L only couples to L

U(1)R  transformation: ψ R (x, t)→ψ R (x, t)eiβ(x,t )

Only Bµ couples to R.

e−R → e−Re−Re−R

Bµ

U1R  

g1YLBµ + g2Wµ0 and g2Bµ − g1YLWµ

0     →    Zµ and Aµe−L → e−L

W0    and  B  can  interfere  at  the  amplitude  level    since  they  involve  the  same  two  ini0al  and  final  states..      

SU2L  

e−u

ue−

e−u

ue−

e−u

ue−

e−u

ue−

Zµ

Aµ

Wµ0

Bµ

Electro-­‐Weak  Interference  

Weak  Mixing  Parameter  

Aµ = cosθWWµ0 + sinθW Bµ Zµ = − sinθWWµ

0 + cosθW Bµ

g2g1

= tanθW

Aµ =g2Wµ

0 + g1Bµg22 + g1

2Zµ =

g1Wµ0 − g2Bµg22 + g1

2sin2θW ≈ 0.23

1e2

=1g12 +

1g22 or      1

α=1

αg1+1

αg2

MW

MZ

= cosθW MWc2 ≈ 80 GeV MZc

2 ≈ 91 GeV

The Strong Interaction

What is the quantum of the strong interaction?

The range is finite, ~ 1 fm.

Therefore, it must be a massive boson.

Gauge Transformations!

SU(3) transformation: ψ (x, t)ξ →ψ (x, t)ξeiα iT =ψ (x, t)ξe

i αν (x,t )Tν1

8∑

ξ =rbg

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟        Tν are 3× 3 matrices

Giν 8 strong QCD color fields: rg, rg,rb , rb,gb ,gb, gg,bb ,rr (-1 since they are not all independent)

Color  Force  Field    • The  quantum  of  color  is  the  gluon.  

• Strong  charges  come  in  types  labeled  r,  g,  b  for  red,  green  and  blue.     (E&M  only  has  one  kind  of  charge)    

• Both  quarks  and  gluons  posses  color  charge.     (photons  carry  no  electric  charge.)  

The  Higgs  

The  U(1)XSU(2)  Lagrangian  is  only  gauge  invariant  if  the  masses  of  all  the  Fermions  and  Bosons  in  the  wave  equa0on  have  zero  mass.    

Masses  of  Bosons  (e.g.  MW    and    MZ)    can  be  very  large!  Mass  of  leptons  are  not  zero  (some  are  quite  large,  e.g.  top  quark)  

Need  addi0onal  field  to  make  the  electroweak  wave  func0on  guage  invariant.  

Predict  new  massive  boson  field  ,  i.e.  Higgs  field  φ.            

Higgs  Par0cle  

All  par0cles  had  no  mass  just  a^er  the  Big  Bang.  As  the  Universe  cooled  and  the  temperature  fell  below  a  cri0cal  value,  an  invisible  force  field  called  the  ‘Higgs  field’  was  formed  together  with  the  associated  ‘Higgs  boson’.  The  field  prevails  throughout  the  cosmos:  any  par0cles  that  interact  with  it  are  given  a  mass  via  the  Higgs  boson.  The  more  they  interact,  the  heavier  they  become,  whereas  par0cles  that  never  interact  are  le^  with  no  mass  at  all.  

Standard  Model:        Gauge  Symmetry    Field  quanta    (Bosons)    and  interac0ons  with  par0cles  (Fermions)      

All  masses  (field  Bosons,  par0cle  Fermions)  violate  gauge  symmetry    

CDF  –  Tevatron  (Fermi  Lab)  

MH  >  170  GeV/c2    

1  TeV  1  TeV  

LHC  CERN  

7  TeV  7  TeV  

The  Standard  Model  

Electroweak

L = 14WµνW

µν − 14 BµνB

µν + Lγ µ ∂µ − g 12 τ iWµ − ′g Y

2 Bµ( )L

+Rγ µ i∂µ − ′g Y2 Bµ( )R + i∂µ − g 1

2 τ iWµ − ′g Y2 Bµ( )φ −V φ( )

− G1LφR +G2LφR( ) + h.c.

Strong - QCD

L = q iγ µ∂µ − m( )q − g qγ µTaq( )Gµa −Gµν

a Gaµν

Symmetry breaking → Weak + Electromagnetic

Aµ ∝ ′g Wµ0 + gBµ , Zµ ∝ gWµ

0 − gBµ , Wµ1 , Wµ

2

The  Standard  Model  

Electroweak:

L = 14WµνW

µν − 14 BµνBµν + Lγ µ ∂µ − g 12 τ iWµ − ′g Y

2 Bµ( )L

+Rγ µ i∂µ − ′g Y2 Bµ( )R + i∂µ − g 12 τ iWµ − ′g Y

2 Bµ( )φ −Vφ

− G1LφR +G2LφR( ) + h.c.

Strong:

L = q iγ µ∂µ − m( )q − g qγ µTaq( )Gµa −Gµν

a Gaµν

The Strong Interaction

What is the quantum of the strong interaction?

The range is finite, ~ 1 fm.

Therefore, it must be a massive boson.

Gauge Transformations!

U(1) transformation: ψ (x, t)→ψ (x, t)eiaf (x,t )=ψ (x, t)eiβ(x,t )

⇒A, V Electromagnetic scalar and vector potentials.

SU(2) transformation: ψ (x, t)→ψ (x, t)eiε iτ =ψ (x, t)e

i εν (x,t )τν1

3∑

τν are 2 × 2 matrices.

⇒W +,

W −,

W 0 Weak fields.

SU(3) transformation: ψ (x, t)→ψ (x, t)eiα iT =ψ (x, t)e

i αν (x,t )Tν1

8∑

Tν are 3× 3 matrices

⇒Gν 8 strong QCD color fields: rg, rg,rb , rb,gb ,gb, gg,bb ,rr (-1 since they are not all independent)

•  Strongly  interac0ng  par0cles  are  called  hadrons.  

•  Quarks  are  the  fundamental  objects  of  strong  interac0ons.  

•  Quarks  have  spin  ½    and  are  described  by  the    Dirac    equa0on.  

•  Quark  wave  func0ons  are  quantum  states  of  a  6-­‐dimensional  “flavor”    symmetry  SU(6)  whose    mathema0cal  descrip0on  is  similar  to  the  descrip0on  of  angular  momentum.      The  flavors,  denoted  u,  d,  s,  c,  b  and  t.    are  components  of  a  flavor  vector  in  a  6  dimensional  space.  

•  Perfect  SU(6)  symmetry  would  imply  all  quarks  have  the  same  mass  energy  and  the  magnitude  of  its  “SU(6)-­‐vector”  would  be  independent  of  the  rota0ons  in  flavor  space.  

•  Flavor    is  a  strongly  broken  symmetry!    

Strong  Interac0ons  (Rohlf  Ch.  18.  p502)  

Color  Force  Field    • The  quantum  of  color  is  the  gluon.  

• Strong  charges  come  in  types  labeled  r,  g,  b  for  red,  green  and  blue.     (E&M  only  has  one  kind  of  charge)    

• Both  quarks  and  gluons  posses  color  charge.     (photons  carry  no  electric  charge.)  

V ∝1r

Electrosta0c  interac0on  

q q

qqq q

quark-­‐quark  interac0on  

q q

V ∝Ar+ Br

Energy in a flux tube of volume v:V = ρv = ρar = Br

Large r  

Small r  

q qV ∝

Ar

V ∝

Ar+ Br A .05 GeV-fm B ~ 1 GeV/fm

Note:  when  r~1  fm,  the  energy  is  ~  1  GeV.  

This  is  the  field  energy  in  the  flux  tube  which  accounts  

for  most  of  the  mass  of  the  hadron.  

Mass  of  the  nucleon:  Mc2  ~  1000  MeV.  

Mass  of  quark:    muc2=1.5-­‐4  MeV    mdc

2=4-­‐8  MeV    

Where  does  the  nucleon  mass  come  from?  

modest  resolu0on:  cons0tuent  quarks  

high  resolu0on:  current  quarks,  an0quark  pairs,  and  gluons  

2 / 3

Y=B+S  

Iz  

−2 / 3

−1 / 2 −1 / 2

u

s

ds

du

Y=B+S  

Iz  

−2 / 3

−1 / 2 −1 / 2

2 / 3

The  fundamental  SU3  mul0plets.  Gell-­‐Mann,  Neiman  (1963)  

Ψ =ψ (space)ψ (spin)ψ (color)ψ (flavor)

π 0 ∝ uu − ddπ− ∝ du π + ∝ ud

η ~ η8 ∝ uu + dd − ssK+ ∝ usK 0 ∝ ds

K 0 ∝ dsK− ∝ ds

ψ (color)∝ RR + BB +GG

SU(3)  flavor  mul0plets  and  their  wave  func0ons  in  flavor  for  the  simplest  mesons  

in  which  the  quarks  are  in  a  rela0ve  s  state    (l=0)    and  spins  an0-­‐aligned  (j=0)      

Mesons are composed of quarks-antiquark pairs.

Baryons  are  composed  of  three  quarks.    SU(3)  flavor  mul0plets  and  their  wave  func0ons  in  flavor  for  the  simplest  baryons  in  which  the  quarks  are  in  a  rela0ve  s  state    j=1/2  and  l=0  

p ∝ u ↑ u ↓ d ↑ +u ↓ u ↑ d ↑ −u ↑ u ↑ d ↓ + all permutations.

uud

uus

udd

dds

dss uss

uds

ψ (color)∝ RGB − RBG + BRG − BGR +GBR −GRB

The  Lowest  State  in  SU(4)  u,d,s,c  quarks  

Quark-­‐Quark  Poten0al  Discovery  of  J/Ψ  

BNL p + p→ e+ + e− + X

SLAC e+ + e− → e+ + e− , µ+ + µ−

Charmonium  

Charmonium    Produc0on  

States  of  charmonium  

Construc0ng  hadrons  from  quarks.  

Decay interaction

Decay interaction

weak  

Vacuum  polariza0on.  Running  coupling  constant.  Rohlf  P502  

Running  coupling  constant.  

αS ≈12π

33 − 2n f( ) ln k2

Λ2

⎛

⎝⎜⎞

⎠⎟

Λ ≈ 0.2 GeV/c

Convert to distance:

αS ≈12π

33 − 2n f( ) lnRΛ

2

r2

⎛

⎝⎜

⎞

⎠⎟

RΛ ≈ λΛ = 6 fm.

Compare with electromagnetic: α ~ 0.01 Beginning to converge!

Running  strong  coupling  constant