the top quark and precision measurements
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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,
– SM has =MW2/(MZ
2c2)=1 at tree level• s is derived quantity
– Models with =1 at tree level include• MSSM• Models with singlet or doublet Higgs bosons• Models with extra fermion families
2
22
2 ZMGsc
2/122
22
2 24
'4
Gv
cg
sg
We have a model….And it works to the 1% level
EW Measurements test consistency of SM
Consistency of precision measurements at multi-loop level used to constrain models with new physics
2005
Models with 1 at tree level are different
• SM with Higgs Triplet
• Left-Right Symmetric Models
• Little Higgs Models
• …..many more
• These models need additional input parameter
• Decoupling is not so obvious beyond tree level
NEWSM LLL
As the scale of the new physics becomes large, the SM is not always recovered, violating our intuition
i
ii
NEW Oc
L2
Lore: Effects of LNEW become very small as
Muon Decay in the SM
• At tree level, muon decay related to input parameters:
• One loop radiative corrections included in parameter rZ
• Where:
22222 22 WZ MsMcsG
2
2
2
22
2
2
s
s
c
sc
M
M
G
Gr
Z
Z
)1(2 222
rMcs
GZ
e
e
W
122
2
Z
W
Mc
M
If 1, there would be 4 input parameters
Calculate top quark contribution to rZ
(mt2 dependence only)
• Muon decay constant:
• Vertex and box corrections, V-B small neglect
• Vacuum polarization, /, has no quadratic top mass dependence
• Z-boson 2-point function:
BVW
WW
MG
G
2
)0(
2
11)1(
4
32)0( 2
2
2
2
2
tt
cWW mm
QgN
1)1(
4
32 2
2
2
2
22
2
2
2
tZ
tc
Z
Z
m
Q
M
m
c
gN
M
M
Calculate top quark contribution to rZ
(continued)
• Need s2/s2
• From SM relation using on-mass shell definition for s2
2
22 1
Z
W
M
Ms
2
2
2
2
2
2
2
2
2
2
2
2
2
2
64 W
tc
W
W
Z
Z
M
mNg
s
c
M
M
M
M
s
c
s
s
MW and MZ are physical masses
s2/s
2 not independent parameter
Includes all known corrections
2
2
2
282t
cSMt m
s
cNGr
Predict MW in terms of input parameters and mt
2005
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+4v2/v21
• Return to muon decay:
2
2222 4
14 v
vvgMW
Blank & Hollik, hep-ph/9703392
2
2
2
22
2
2
s
s
c
sc
M
M
G
Gr
Z
Ztriplet
)1(2 222
rMcs
GZ
Need Four Input Parameters With Higgs Triplet
• Use effective leptonic mixing angle at Z resonance as 4th parameter
• Variation of s:
241 sa
v
e
e
eZaveiL ee )( 5
2
22
2
222
22
2
2
2
2
log3
4
2
1
3
2
)()(
2)(
)(
t
e
ZzeeA
e
ZzeeVe
eeA
e
ee
Z
ZZ
m
Qs
s
a
M
v
M
cs
vm
a
av
M
M
s
c
s
s
2
1,2
2
1 2 ee asv
This is definition of s:
Proportional to meneglect
Contrast with SM where s2 is proportional to mt
2
* Could equally well have used as 4th parameter
SM with triplet, cont.
2
2
2
22
2
2
s
s
c
sc
M
M
G
Gr
Z
Ztriplett
2
2
2
2
2
2
2
2
s
s
c
s
M
M
M
M
Z
Z
W
W
• Putting it all together:
• Finally,
mt2 dependence cancels
mt2 dependence cancels
rttriplet depends logarithmically on mt
2
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
• Physical Higgs bosons: 4 H0, 2A0, 2H
• Count parameters:
(g, g’, , ’, vR) (e , MW1, MW2, MZ1, MZ2)
'0
00,
2
1,
2
1
Czakon, Zralek, Gluza, hep-ph/9906356
EWSB
0
00)2,0,1(
LL v
0
00)2,1,0(
RR v
SU(2)R x U(1)B-L U(1)Y
Assume vL=0 (could be used to generate neutrino masses)
Assume gL=gR=g
Renormalization of s in LR Model
2
22
2222222
22222
'21
2
1
)'(22
12
1
21
12
12
g
gvgMM
ggvgMM
vgMM
RZZ
RZZ
RWW
22222222
2
1
2
112 RRWW vgvgMM
2cos',
eg
s
eg
• Expand equations to incorporate one-loop corrections:
22222
222222
22222
22222222
2222
222222
))()((
)2()2(
2
1
))()((
))(())((
2
1
)()(
)()(2
1212
212121
1212
12121212
1212
1212
WWZZ
ZZZZZZ
WWZZ
WWZZZZWW
WWZZ
WWZZ
MMMM
MMMMMM
MMMM
MMMMMMMM
MMMM
MMMMcs
etc
• Gauge boson masses after symmetry breaking:
+2=2+’2
• Solve for s2 using
Renormalization of s in LR Model, cont.
22222
222222
22222
22222222
2222
222222
))()((
)2()2(
2
1
))()((
))(())((
2
1
)()(
)()(2
1212
212121
1212
12121212
1212
1212
WWZZ
ZZZZZZ
WWZZ
WWZZZZWW
WWZZ
WWZZ
MMMM
MMMMMM
MMMM
MMMMMMMM
MMMM
MMMMcs
• Scale set by: 222
22
22222
121212 2cos
1
2cos2 WWRWWZZ MMvg
MMMM
• At leading order in MW12/MW2
2 v2/vR2:
)()(
24
)()(2
)()(
)()(2
22
2222
2
2
2
22
222
2
2
2222
2222
2
2
2
2
12
1
12
1
1212
1212
WW
WtcF
WW
W
WWZZ
WWZZ
MM
MmNsc
s
cG
MM
Msc
s
c
MMMM
MMMM
s
c
s
s
Very different from SM!
• As MW22, s2/s2 0
• The SM is not recovered!
Thoughts on Decoupling
Limit MW22, s20
SM 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
Scale fixed to go through data pointAbsolute scale arbitrary
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)]2SU(2) x U(1)SM
is complex Higgs triplet
22
22
1
x
x
I
I
fie /2
2/
2/2/
2/*
Th
hh
h
Littlest Higgs Model, continued
• Model has complex triplet (1) at tree level
– Requires 4 input parameters
• Quadratic divergences cancelled at one-loop by new states• 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!
T T T
T tb
2
2
(...)1f
v
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 1/162
– 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
• In SM:
• Quadratic dependence on Mh0 in LR Model:
• Quadratic dependence also found in little Higgs model
22
2
2
222
22 0
22
114)21(
224
1h
ZW
WFLRh M
M
c
M
scMG
sr
Czakon, Zralek J. Gluza, hep-ph/9906356
2
2
2
2
log192
11
W
hSMh M
Mgr
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
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