Transcript What Do We Know About EWSB? Sally Dawson (BNL) Lecture 2

What Do We Know About
EWSB?
Sally Dawson (BNL)
Lecture 2
XIII Mexican School of Particles
and Fields, 2008
Basics
• SM is SU(2) x U(1) theory
– Two gauge couplings: g and g’
• Higgs potential is V=-22+4
– Two free parameters
• Four free parameters in gauge-Higgs sector
Basics, #2
• Chose parameters in gauge/Higgs sector
• =1/137.0359895(61)
• GF =1.16637(1) x 10-5 GeV -2
• MZ=91.1875  0.0021 GeV
• MH
Express everything else in terms of these
parameters
GF
g2



2
 M W2
2 8M W
21  2
 MZ
 2
 M W

Inadequacy of Tree Level Calculations
• Mixing angle is predicted quantity
– On-shell definition cos2W=MW2/MZ2
– Predict MW
 
4
M W2   2
1 1
GF 
2GF M Z2




1
sW cW 
2
2
– Plug in numbers:
• MW predicted =80.939 GeV
• MW(exp) =80.399  0.025 GeV
– Need to calculate beyond tree level

GF M Z2
Quantum Corrections
• Relate tree level to one-loop corrected masses
 i 
XY 


 
 XY (k )  g  XY (k )  k k BXY (k )
2
2
2
M V2 0  M V2  VV ( M V2 )
• Majority of corrections at one-loop are from 2point functions
Note sign conventions for 2-point functions
Example of Quantum Corrections
• Example:
M W2
 2
 1  
2
M Z cosW
W W (0)  ZZ (0)
 

2
MW
M Z2
Top quark contributes to W and Z 2-point functions
Top Quark Corrections to  Parameter
• 2-point functions of W, Z
2
g Nc
 ZZ (0) 
32 2 cW2
g 2 Nc
W W (0) 
32 2

 4  mt2
 2 
Rt  Lt
 mt  


 4  2  1 1 
 2  mt   
 2
 mt 

2
Lt=1-4sW2/3
Rt=-4sW2/3
W W (0)  ZZ (0) GF N c  mt2 
 2 
 


2
2
2 
MW
MZ
2 8  M W 
(Neglecting log(mt) pieces and using on-shell definition of sinW)
Heavy Higgs Contribution to 
 M H2 
 3

log  2 
2
16cW
 MW 
•In on-shell scheme, Higgs contributes
logarithmically to quantum corrections & top
quark contributes quadratically
Modification of tree level relations
GF 

1
2M W2 sin 2 W 1  r 
•r is a physical quantity which incorporates 1-loop
corrections
•Contributions to r from top quark and Higgs
loops
2
2

3
G
m
cos
W
t
F t
 2
r  
2 
8 2  sin W
2
11
G
M
F
W
r H 
24 2 2



 M H2 5 
 ln 2  
 MW 6 
Extreme sensitivity of precision
measurements to mt
Understanding Higgs Limit
Theory: Input MZ, GF, 
→ Predict MW
Consistency between direct and indirect measurements
of MW and mt a strong test of theory!
Precision Measurements Limit MH
• LEP EWWG (July, 2008):
• mt=172.4  1.2 GeV
• MH=84+34-26 GeV
• MH < 154 GeV (one-sided
95% cl)
• MH < 185 GeV (Precision
measurements plus direct
search limit)
Best fit in region excluded from direct searches
Caveats
• Low Q2 data not included in fit
– Doesn’t include atomic parity violation in cesium,
parity violation in Moller scattering, & neutrinonucleon scattering (NuTeV)
– Higgs fit not hugely sensitive to low Q2 data
• MH< 185 GeV
– Higgs limit moves around with mt
Higgs limit assumes SM!
EW Measurements test SM
We have a model….
And it works to the 1% level
•Consistency of precision
measurements at multiloop level used to
constrain models with
new physics
•If a new model predicts
some deviation from the
SM, it has to be small
This fit ASSUMES SM
Limits on MH Assume SM
• MH  450-500 GeV allowed with large isospin
violation (T=) and higher dimension
operators
T
We don’t know what
the model is which
produces the
operators which
generate large 
MH (GeV)
S,T,U formalism
• Suppose “new physics” contributes primarily to
gauge boson 2 point functions
– cf r where vertex and box corrections are
small
• Also assume “new physics” is at scale M>>MZ
• Two point functions for  , WW, ZZ, Z
S,T,U, (#2)
• Taylor expand 2-point functions
ij (q )  ij (0)  q ij (0)  ...
2
2
• Keep first two terms
• Remember that QED Ward identity requires
any amplitude involving EM current vanish at
q2=0
  (0)  0
S,T,U (#3)
• To O(q2), there are 6 coefficients:
   q 2   (0)  ...
 W W   W W (0)  q 2  W W (0)  ...
 Z  q 2  Z (0)  ...
 ZZ   ZZ (0)  q 2  ZZ (0)  ...
• Three combinations of parameters absorbed in
, GF, MZ
• In general, 3 independent coefficients which can
be extracted from data
S,T,U, (#4)
WnewW (0)  new
ZZ (0)
T 

M W2
M Z2

2
new
 new
(
M
)

ZZ
Z
ZZ (0)
S

2 2
2
4sW cW
MZ
M Z2
SM contributions
in , GF, and MZ

WnewW ( M W2 ) WnewW (0)
S  U  

2
2
4sW
MW
M W2
Advantages: Easy to calculate
Valid for many models
Experimentalists can give you model
independent fits
Limits on S & T
• A model with a heavy
Higgs requires a source
of large (positive) T
T  
• Fit assumes MH=150 GeV
• S/T/U approach subtracts
off SM contributions
Theoretical Limits on MH
• Unitarity
– If unitarity is violated, interactions grow with
energy (cf longitudinal W’s)
• Perturbativity of couplings
– If perturbativity is violated, loop corrections
may be larger than tree
• Limits tell us where minimal SM is valid
– Unitarity and perturbativity provide strong
limits on beyond the SM physics
– These are model builders tools
• Renormalization of Higgs mass
– What about naturalness?
Unitarity
• Consider 2  2 elastic scattering
d
1
2

A
2
d 64 s
• Partial wave decomposition of amplitude

A  16  (2l  1) Pl (cos  )al
l 0
• al are the spin l partial waves
Unitarity
• Pl(cos) are Legendre polynomials:
1

1
8

s

dxPl ( x) Pl  ( x) 

2 l ,l 
2l  1
 (2l  1) (2l   1)a a  d cos  P (cos  ) P (cos  )
l 0
l 0
16

s
*
l l

 (2l  1) a
l 0
1
1
2
l
• Sum of positive definite terms
l
l
More on Unitarity
1
16
• Optical theorem   ImA(  0) 
s
s
Im( al )  al
2

 (2l  1) a
l 0
Optical theorem derived
assuming only conservation
of probability
• Unitarity requirement:
1
Re( al ) 
2
l
2
More on Unitarity
• Idea: Use unitarity to limit parameters of theory
Cross sections which grow with
energy always violate unitarity at
some energy scale
• Remember WL(p) with L~p/MW
Aside on WW Scattering
A(VL1...VLN  VL1...VLN  )  (i ) N (i ) N  A(1... N  1... N  )
 M W2
O 2
 E



This is a statement about
scattering amplitudes, NOT
individual Feynman diagrams
 , z are Goldstone bosons which are eaten by the
Higgs mechanism to give the W & Z bosons their
longitudinal components
W+W-W+W• Recall scalar potential



M H2 2 M H2
M H2
2
2
 
V
H 
H H  z  2   2 H 2  z 2  2  
2
2v
8v

2
• +-+
2

M
M
iA(      )  2i
   i H
v
v

2
H
2
p
p’
2

 M H2
i

   i
2
v
 t  MH 
2

i

2
 s  MH
s  ( p  q) 2
t  ( p  p' ) 2
q
q’
u  (q  p ' ) 2
+-+• Two interesting limits:
– s, t >> MH2
2
H
2
M
A(     )  2
v




– s, t << MH2
u
A(     )   2
v




Remember: This is a
physical process
2
M
a00   H2
8v
a 
0
0
s
32v
2
Use Unitarity to Bound Higgs
• High energy limit:
2
MH
0
a0  
2
8v
1
Re( al ) 
2
MH < 800 GeV
• Heavy Higgs limit
a 
0
0
s
32v 2
Ec 1.7 TeV
 New physics at the TeV scale
Can get more stringent bound from coupled channel analysis
Another Sort of Limit: Landau Pole
• MH is a free parameter in the Standard Model
• Can we derive limits from consistency?
• Consider a scalar potential:
M H2 2  4
V
H  H
2
4
• This is potential at electroweak scale
• Parameters evolve with energy in a calculable
way
Consider HHHH
• Real scattering, s+t+u=4MH2
• Consider momentum space-like and off-shell:
s=t=u=Q2<0
• Tree level: iA0=-6i
HHHH, #2
• One loop:
n
1
d
k
i
i
iAs  (6i ) 2 
2 (2 ) 2 k 2  M H2 (k  p  q) 2  M H2

92
 2 (42 )( )M H2  Q 2 x(1  x) 
8
• A=A0+As+At+Au

9


2
2
2
A  6 1 
(4 )( ) M H  Q x(1  x)  ...
2
 16



HHHH, #3
• Sum the geometric series to define running
coupling

9
Q2 
A  6 1 
log 2   ...
2
MH 
 16
6
A
 6 (Q)
 Q 
9

1  2 log 
8
 MH 
• (Q) blows up as Q (called Landau pole)
HHHH, #4
• This is independent of starting point
• BUT…. Without 4 interactions, theory is
non-interacting
• Require quartic coupling be finite
1
0
 (Q)
HHHH, #5
• Use =MH2/(2v2) and approximate log(Q/MH) 
log(Q/v)
• Requirement for 1/(Q)>0 gives upper limit on MH
32 2 v 2
M 
 Q2 
9 log  2 
v 
2
H
• Assume theory is valid to 1016 GeV
– Gives upper limit on MH< 180 GeV
• Can add fermions, gauge bosons, etc.
We expect Higgs at electroweak scale
High Energy Behavior of 
• Renormalization group scaling
Q
1
1

 (...) log  
 (Q)  (  )

d
2
2
4
16
 12  12g t  12 g t  ( gauge)
dt
2
 Q2 
t  log  2 
 
Mt
gt 
v
• Large  (Heavy Higgs): self coupling causes  to
grow with scale
• Small  (Light Higgs): coupling to top quark causes
 to become negative
Does Spontaneous Symmetry Breaking
Happen?
• SM requires spontaneous symmetry
• This requires V (v)  V (0)
• For small 
d
16
 16 g t4
dt
2
• Solve
 2 
3g t4
 ()   (v)  2 log  2 
4
v 
Does Spontaneous Symmetry Breaking
Happen? (#2)
• () >0 gives lower bound on MH
2
2


3
v

2
M H  2 log  2 
2
v 
• If Standard Model valid to 1016 GeV
M H  130 GeV
• For any given scale, , there is a theoretically
consistent range for MH
Light Higgs Theoretically Attractive
Extrapolate Higgs potential to high scale 
V=  (+ - v2)2
Forbidden
Allowed
Forbidden
•Standard Model is only
consistent to GUT scale
for small range of Higgs
masses
•Heavy Higgs implies
new physics at some low
scale
Problems with the Higgs Mechanism
• We often say that the SM cannot be the entire
story because of the quadratic divergences of
the Higgs Boson mass
• Whether this is a problem or not is somewhat a
matter of taste
Masses at One-Loop
• First consider a fermion coupled to a massive
complex Higgs scalar
L   (i)     mH   F L R  h.c.
2
2
• Assume symmetry breaking as in SM:
( H  v)
F v
 
mF 
2
2
Masses at One-Loop, #2
• Calculate mass renormalization for 
H
k  mF
  iF  2 d 4 k
 i F ( p)  
 (i) 
4
2
2
2
2
(
2

)
[
k

m
][(
k

p
)

m
2


F
H]
2
Renormalized Fermion Mass
mF   F ( p) p m
F

mF (1  x)
4
i
dx  d k  2
4 
2 2
2
2

32 0
[k  mF x  ms (1  x)]
2
F
1
• Do integral in Euclidean space
k0  ik 4
d 4 k   id 4 k E
2
2
2
2
2
k   k0  k  k 4  k  k E2
2
4
2
2
d
k
f
(
k
)


 E E
 y dy f ( y)
0
Renormalized Fermion Mass, #2
• Renormalization of fermion mass:
2
y dy
 mF
mF  
dx (1  x) 
2 2
2
2
2 
[
y

m
x

m
(
1

x
)]
32 0
F
H
0
2
F
1
 2 
32F mF

log  2   .....
2
32
 mF 
Symmetry and the Fermion Mass
• mF  mF
– mF=0, then quantum corrections vanish
– When mF=0, Lagrangian is invariant under
• LeiLL
• ReiRR
– mF0 increases the symmetry of the threoy
– Yukawa coupling (proportional to mass)
breaks symmetry and so corrections  mF
Scalars are very different
H
  iF  2 d 4 k Tr(k  mF )(( k  p)  mF )
2
 i H ( p )  
 (i) 
(2 ) 4 (k 2  mF2 )[( k  p) 2  mF2 ]
 2 
2
1
2 2
 

2
2
2
2
F
M H   H (mH )  
 mH  mF log 
2
8
 mF
 mH2
mH2 
 ( 2m 
)1  I1  2
2 
 mF
2
F



I1 (a)   dx log 1  ax(1  x) 
0

 1 
   O 2 

 

• MH diverges quadratically!
• Quadratic sensitivity to high mass scales
Scalars (#2)
• MH diverges quadratically!
• Requires large cancellations (hierarchy
problem)
• Can do this in Quantum Field Theory
• H does not obey decoupling theorem
– Effects of heavy particle (H) does not decouple as
MH
• MH0 doesn’t increase symmetry of theory
– Nothing protects Higgs mass from large
corrections
Light Scalars are Unnatural
• Higgs mass grows with scale of new physics, 
• No additional symmetry for MH=0, no protection
from large corrections
H
H
 M H2 
GF
4 2 2
 6M
2
2
W
 3M Z2  M H2  12M t2 
 


200 GeV 
 0.7 TeV

2
MH  200 GeV requires large cancellations
What’s the problem?
• Compute MH in dimensional regularization and
absorb infinities into definition of MH
M M
2
H
2
H0

1

(...)
• Perfectly valid approach
• Except we know there is a high scale
(associated with gravity)
Try to cancel quadratic divergences by
adding new particles
• SUSY models add scalars with same quantum
numbers as fermions, but different spin
• Little Higgs models cancel quadratic
divergences with new particles with same spin
New particles assumed to be at TeV scale
for cancellation of quadratic divergences
We expect something new at the TeV scale
• If it’s a SM Higgs then we have to think hard
about what the quadratic divergences are telling
us
• SM Higgs mass is highly restricted by
requirements of agreement with precision
electroweak data and theoretical consistency