Transcript The Top Quark and Precision Measurements S. Dawson BNL April, 2005 M.-C. Chen, S.

The Top Quark and Precision Measurements
S. Dawson
BNL
April, 2005
M.-C. Chen, S. Dawson, and T. Krupovnikas, in preparation
M.-C. Chen and S. Dawson, hep-ph/0311032
Standard Model Case is Well Known
• EW sector of SM is SU(2) x U(1) gauge theory
– 3 inputs needed: g, g’, v, plus fermion/Higgs masses
– Trade g, g’, v for precisely measured G, MZ, 
g 
2
4
g' 
2
2
s
4

v  G 2
2
2
c

1 / 2
– SM has =MW2/(MZ2c2)=1 at tree level
• s is derived quantity
c s 
2
2

2
2G M Z
– Models with =1 at tree level include
• MSSM
• Models with singlet or doublet Higgs bosons
• Models with extra fermion families
EW Measurements test consistency of SM
2005
We have a model….
And it works to the 1% level
Consistency of precision
measurements at multi-loop level
used to constrain models with new
physics
Models with 1 at tree level are different
•
•
•
•
SM with Higgs Triplet
Left-Right Symmetric Models
Little Higgs Models
…..many more
L  LSM  LNEW
LNEW  
i
ci

2
Oi  
Lore: Effects of LNEW become
very small as 
• These models need additional input parameter
• Decoupling is not so obvious beyond tree level
As the scale of the new physics becomes large, the SM
is not always recovered, violating our intuition
Muon Decay in the SM
• At tree level, muon decay related to input parameters:
G 

2 s c M 
2
2
2
Z


2
2 s M
2

2
W
MW
2
c M
2
Z
1
If 1, there would be
4 input parameters
• One loop radiative corrections included in parameter rZ
G 
• Where:
r  
G
G

M Z2
2
MZ
 c2  s2


  c2


2
2
2 s c M
 s2
 2
 s
 
2
Z
(1  r )


e
W
e
Calculate top quark contribution to rZ
(mt2 dependence only)
• Muon decay constant:

WW
G

G
 W W (0)
2
MW
2
2
N c g  4Q

(0)  
2
2
32  mt
 V  B


1 1
 (1   )mt2   

 2

• Vertex and box corrections, V-B small  neglect
• Vacuum polarization, /, has no quadratic top mass dependence
• Z-boson 2-point function:
M Z
2
2
MZ
2
2
mt  4Q


2 2
2
2
32 c M Z  mt
Nc g
2


1
 (1   )



Calculate top quark contribution to rZ
(continued)
• Need s2/s2
• From SM relation using on-mass shell definition for s2
2
s  1 
2
MW
MW and MZ are physical masses
2
MZ
s2/s2 not independent parameter
s2
2
s
rt
2
2
2
c  M Z M W
 2 

2
2
s  M Z
MW
SM
2005
 c2 g 2 N c mt2
 2
 s 64 2 M 2

W

G N c  c2


2  2
2 8  s
 2
mt  


Predict MW in terms of input
parameters and mt
Includes all known corrections
What’s different with a Higgs Triplet?
• SM: SU(2) x U(1)
– Parameters, g, g’, v
• Add a real triplet, (+,0,-), 0=v
– Parameters in gauge sector: g, g’, v, v
– vSM2=(246 GeV)2=v2+4v2
– Real triplet doesn’t contribute to MZ
• At tree level, =1+4v2/v21
• Return to muon decay: G 

r
triplet

G
G

M Z2
2
MZ
 c2  s2


  c2

M

2 s c M Z 
2
2
2
2
W
2
2 2
4 v
g v 
1 

2
4 
v




(1  r )
 s2 
 2 
 s

 
Blank & Hollik, hep-ph/9703392
Need Four Input Parameters With Higgs Triplet
• Use effective leptonic mixing angle at Z resonance as 4th
parameter
L  ie   (ve  ae 5 )eZ

ve
ve
This is definition of s:
ae
1
2
 2 s ,
2
s
c
  
 s

ae 
1
2
 1  4 s
2
Proportional to meneglect
• Variation of s:
s2
2
   Z ( M Z2 ) ve2  ae2 e
ve
2


 A (me ) 
2

ae
2 s c
 M Z
 Q2
2  1 4 2 
 2

s
log

 
2
m
3s  2 3 
 t
2
zee
2
 zee
( M Z )  A ( M Z ) 
 V




ve
ae






Contrast with SM where s2 is proportional to mt2
* Could equally well have used  as 4th parameter
SM with triplet, cont.
• Finally,



M W2
M
2
W

M Z2
2
MZ
 s2
  2
 c
 s2
 2
 s
 
mt2 dependence cancels
• Putting it all together:
rt
triplet

G
G

M Z2
2
MZ
 c2  s2


  c2

 s2 
 2 
 s

 
mt2 dependence cancels
rttriplet depends logarithmically on mt2
If there is no symmetry which forces v=0, then no matter how
small v is, you still need 4 input parameters
v  0 then   1
Triplet mass, M  gv  Two possible limits:
• g fixed, then light scalar in spectrum
• M fixed, then g and theory is non-renormalizable
SU(2)L x SU(2)R x U(1)B-L Model
• Minimal model
 1 1  
   , , 0   
2 2  0
0

 ' 
 0
 L  (1, 0 , 2 )  
 vL
0

0 
 0
 R  ( 0 ,1, 2 )  
 vR
0

0 
 EWSB
Assume gL=gR=g
Assume vL=0 (could be used to generate
neutrino masses)
SU(2)R x U(1)B-L U(1)Y
• Physical Higgs bosons: 4 H0, 2A0, 2H
• Count parameters:
(g, g’, , ’, vR) (e , MW1, MW2, MZ1, MZ2)
Czakon, Zralek, Gluza, hep-ph/9906356
Renormalization of s in LR Model
• Gauge boson masses after symmetry breaking:
M W 2  M W1 
1
M
1
2
M
M
2
2
Z2
Z1
M

Z2

2
Z1
2
1
2
2
g
2


g    vR
2
2
2
+2=2+’2
   2vR ( g  g ' )
2
2
g  vR 1  2
2
2
g'
g
2
2
2
• Expand equations to incorporate one-loop corrections:
M W  M W 
2
2
2
1
1
2
 g    v R  
2
• Solve for s2 using
2
2
2
2
2
2
2
2
2
2
2
2
2
 etc
e
cos 2
2
2
2
2
2
2
2
2
(( M Z 2  M Z1 )  ( M W2  M W1 ))
2
2
2
1 ( M W2  M W1 )(M Z 2  M Z1 )  ( M Z 2  M Z1 )(M W2  M W1 )
2

(( M Z 2  M Z1 )  ( M W2  M W1 ))
2
2

g     v R
g' 
,
1 (2 M Z1  M Z 2 )M Z1  (2M Z 2  M Z1 )M Z 2
2

2
2
1
s
( M Z 2  M Z1 )  ( M W2  M W1 )
2
2
e
g
(M Z 2  M Z1 )  (M W2  M W1 )
2
s2  2c2
2
2
2
2
2
2
2
Renormalization of s in LR Model, cont.
 s  2 c
2
2

( M
2
Z2
(M
1 (2 M
2
 M
2
Z1
)  ( M W 2   M W 1 )
M
2
Z1
)  (M
2
Z2
2
Z1
M
(( M
2
Z2
2
Z2
• Scale set by: M
2
) M
2
Z1
2
W2
2
M
 (2 M
M
2
Z1
)  (M
2
Z2
M
2
Z2
M
2
W1
1 ( M W 2  M W 1 )(  M
2
Z2
 M
2
2
Z2
M
2

)
M
2
W2
2
Z1
2
W1
(( M
2
Z1
) M
))
2
  M
2
2
W2
)  (M
2
Z1
2
Z1
)  (M
2
Z2
2
W2
M
M
2
Z1
2
W1
)(  M W 2   M W 1 )
2
))
2
2
2
Z2
M
2
W1

g
2
vR 
1
2
2 cos 2
2
cos 2
2
M
• At leading order in MW12/MW22  v2/vR2:
s2
2
2
c (M Z 2  M Z1 )  (M W2  M W1 )
2
2
s
2
s

GF
4
2
(M
2
2
Z2
 M )  (M
2
Z1
2
2 s
2
W2
2
M
2
2
c
2
(c  s )
2
2
2
W1
)
 2
2
s
2
N c mt M W1
( M W2  M W1 )
2
2
Very different from SM!
• As MW22, s2/s2 0
• The SM is not recovered!
M W2
2
c
(c  s )
2
2
1
(M
2
W2
 M W1 )
2
2
W2
 M W1
2

Thoughts on Decoupling
 Limit
 SM
MW22, s20
is not recovered
4 input parameters in Left-Right model: 3 input parameters in SM
No continuous limit from Left-Right model to SM
Even if vR is very small, still need 4 input parameters
No continuous limit which takes a theory with
=1 at tree level to 1 at tree level
Results on Top Mass Dependence
Absolute scale arbitrary
Scale fixed to go
through data point
Plots include only mt dependence
Final example: Littlest Higgs Model
• EW precision constraints in SM require Mh light
• To stabilize Mh introduce new states to cancel quadratic
dependence on higher scales
– Classic model of this type is MSSM
• Littlest Higgs model: non-linear  model based on
SU(5)/SO(5)
– Global SU(5)  Global SO(5) with 
– Gauged [SU(2) x U(1)]1 x [SU(2) x U(1)]2SU(2) x U(1)SM
e
2i / f



   
I
 2x2
–  is complex Higgs triplet
1
I2x2 






  h / 2




h /
2

*
h /
T
h /
2



2


Littlest Higgs Model, continued
  1  (...)
• Model has complex triplet (1) at tree level
– Requires 4 input parameters
• Quadratic divergences cancelled at one-loop by new states
v
2
f
2
• W, Z, B  WH, ZH, BH
• t T
• H
• Cancellation between states with same spin statistics
– Naturalness requires f ~ few TeV
• Just like in SM with triplet, dependence of r on charge 2/3
quark, T, is logarithmic!
b
T
t
T
T
T
Littlest Higgs Model, continued
• One loop contributions numerically important
– Tree level corrections (higher order terms in chiral perturbation
theory)  v2/f2
– One loop radiative corrections  1/162
– Large cancellations between tree level and one-loop corrections
– Low cutoff with f  2 TeV is still allowed for some parameters.
– Contributions grow quadratically with scalar masses
Quadratic contributions
cancel between these
Quadratic contribution remains from
mixed diagrams
Fine Tuned set of parameters in LH Model
 Parameters chosen for
large cancellations
Models with triplets have Quadratic dependence
on Higgs mass
• Mh0 is lightest neutral Higgs
 M 
 11 g 
• In SM:

 log 
r

2
SM
h
 192 

2


M

2
h
2
W


• Quadratic dependence on Mh0 in LR Model:
r
LR
h
2
2
2
2
G F M W1 
 c (1  2 s ) 4 c  1 
 2



 M h0
2 2
2
2
24  s
M
M
2 

W2
Z2


1
Czakon, Zralek J. Gluza, hep-ph/9906356
• Quadratic dependence also found in little Higgs model
M.-C. Chen and S. Dawson, hep-ph/0311032
Conclusion
• Models with 1 at tree level require 4 input parameters in
gauge sector for consistent renormalization
– Cannot write models as one-loop SM contribution plus tree level new
physics contribution in general
• Models with extended gauge symmetries can have very
different behaviour of EW quantities from SM beyond tree
level
– Obvious implications for moose models, little Higgs models, LR
models, etc