Transcript Electroweak Precision Measurements and BSM Physics: (A) Triplet Models

Electroweak Precision Measurements
and BSM Physics:
(A) Triplet Models
(B) The 3- and 4-Site Models
S. Dawson (BNL)
February, 2009
S. Dawson and C. Jackson, arXiv:0810.5068; hep-ph/0703299
M. Chen, S. Dawson, and C. Jackson, arXiV:0809.4185
WARNING: THIS IS A THEORY TALK
Standard Model Renormalization
• 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/(MZ2c2)=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
Muon Decay in the SM
• At tree level, muon decay related to input parameters:
• One loop radiative corrections included in parameter rSM
G 

 M W2
2 1  2
 MZ
 2
 M W

(1  rSM )
• Dominant contributions from 2-point functions
r is a physical parameter


e
W
e
Part A: Triplet Model
Models with 1 at tree level are different
from the SM
=MW2/(MZ2c2)1
•
•
•
•
•
•
SM with Higgs Triplet
Left-Right Symmetric Models
Little Higgs Models
…..many more
These models need additional input parameter
Decoupling is not always obvious beyond tree
level
Higgs Triplet Model
Simplest extension of SM with 1
• Add a real triplet
  


0
   v' 
  







H  1
0
0 
( v  h  i ) 

 2

– vSM2=(246 GeV)2=v2+4v’2
– Real triplet doesn’t contribute to MZ
g 2 v 2  4v'
M 
1 2
4 
v
2
W
2




• At tree level, =1+4v’2/v21
• PDG: v’ < 12 GeV
Neglects effects of scalar loops
Motivated by Little Higgs models
Scalar Potential
V   H     1 H 
2
1
2
2
2
2
4
2
4
 
4
3
2
H   4 H  a H a
2
2
• 4 has dimensions of mass → doesn’t decouple
• Mass Eigenstates:
 H 0   c
 0
 K   s
   
s  h0 
 0 
c  
 G    c
    
H  s

  
Forbidden
by T-parity
s    
  
c  
• 6 parameters in scalar sector: Take them to be:
MH0, MK0, MH+, v, , 
tan  = 2 v’/v
 small since it is related to  parameter
Decoupling at Tree Level
• Require no mixing between doublet-triplet sectors for
decoupling
v 
4
3
• v’→0 requires 4 →0 (custodial symmetry), or 3→
(invalidating perturbation theory)
M
2
K0
M H2 
1 2
2

   12v 2  v 3
12 2
2
2
  2  4v 2  v 3  2v4
2
• v’→0 implies MK0  MH+
2
2
MK0 - MH+ (GeV)
Heavy Scalars → Small Mass Splittings
=0
Allowed
region is
between
curves
MH+ (GeV)
• Plots are restriction 2 < (4 )2
Forshaw, Vera, & White, hep-ph.0302256
Renormalization of Triplet Model
• At tree level, W mass related to input parameters:
M W2 

2
2s G
(1  r )
M W2
  2 2 1
c M Z
• One loop radiative corrections included in
parameter r
For 1, 4 input parameters
Input 4 Measured Quantities
(MZ, , G, sin eff)
• Use effective leptonic mixing angle at Z resonance as 4th
parameter
L  ie   (ve  ae 5 )eZ
1
v e   2 seff 2 ,
2

1
ae 
2
• Could equally well have used  or MW as 4th parameter
• At tree level, SM and triplet model are identical in seff
scheme (SM inputs , G, sin eff here )
M W2 

2 seff 2G
1  r 
This scheme discussed by: Chen, Dawson, Krupovnickas, hep-ph/0604102;
Blank and Hollik hep-ph/9703392
Triplet Results
• Compare with SM in effective mixing angle scheme
• Input parameters: MZ, sineff, , G, MH0, MK0, MH+,
– =1/cos2 = (MW /MZ cos eff )2 predicts sin  =.07 (v’=9 GeV)
– System is overconstrained (can’t let v’ run)
• Triplet model has extra contributions to r from K0, H+
• SM couplings are modified by factors of cos , cos 
Quadratic dependence on Higgs mass
• Triplet model with MH0 << MK0  MH  and small mixing
r
triplet
 ~
r
SM
2
2
 MK  MH

 sin  (...)  sin  (...)
2
2
24s
MH

0

Inputs different in triplet model and SM
Triplet model: MZ=91.1876 GeV is input
SM (in this scheme): MZ is calculated = 91.453 GeV
Perturbativity requires MK0~MH+ for large MH+
Toussaint, PRD18 (1978) 1626
MW(SM)-MW(Triplet)
• For heavy
perturbativity requires
MH+~MK0, and predictions
of triplet model approach
SM
MH+ - MK=0, 10, 20 Gev
• No large effects in
perturbative regime
MW (MeV)
H+,
=.1,
MH0=120 GeV
MH+ (GeV)
• SM not exactly recovered at large MH+ due to different MZ
inputs for 1-loop corrections
Similar conclusions from Chivukula, Christensen, Simmons: arXiv:0712.0546
Conclusions on Triplets
MW=80.399  0.025 GeV
Triplet model consistent
with experimental data if
MK ~MH+
Small mixing angles
required
• Part B: Higgsless Models and
Effective Lagrangians
• What can we learn from precision
electroweak measurements?
The Usual Approach
• Build the model of the week
• Assume new physics contributes primarily to gauge
boson 2-point functions
• Calculate contributions of new particles to S, T, U
• Extract limits on parameters of model
THIS
NOT THIS
STU Assumptions
• Assume dominant contribution of new physics is to 2point functions
• Assume scale of new physics,  >> MZ
– This means no new low energy particles
– Taylor expand in MZ/
– Symmetry is symmetry of SM
• Assume reference values for MH, Mt
• Assume =1 at tree level
– Otherwise you need 4-input parameters to renormalize
STU Definitions
Taylor expand 2-point functions:
  (0)  q 2  (q 2 ) Vanishes by EM gauge invariance
 Z (q 2 )   Z (0)  q 2 Z (q 2 )
Fermion & scalar
W W (q 2 )  W W (0)  q 2 W W (q 2 ) contributions vanish; gauge
boson contributions non-zero
2
2
2
 ZZ (q )   ZZ (0)  q ZZ (q )
  (q 2 ) 
6 unknown functions to this order in MZ/
Peskin & Takeuchi, PRD46 (1992) 381
STU Definitions
6 unknowns:
3 fixed by SM renormalization, 3 free parameters

4sW2 cW2 
cW2  sW2
2
2
2
S 

(
M
)


(
0
)


(
M
)


(
M
)
0

(
0
)


ZZ
Z
ZZ
Z
Z
Z
Z
Z
M Z2 
cW sW

W W (0)  ZZ (0)
sW  Z (0)
T 


2
M W2
M Z2
cW M Z2
2
2

  Z ( M W2 )   Z (0)  2   (0) 
2   W W ( M W )   W W (0)
2   ZZ ( M W )   ZZ (0) 
  sW
  2cW sW 
U  4sW 
 cW 
2
2
2
2 


MW
M
M
M


Z
Z
Z






S is scaling of Z 2-point function from q2=0 to MZ2
T is isospin violation
U contributes mostly to MW
Peskin & Takeuchi, PRD46 (1992) 381
3 and 4 Site Higgsless Models
PDG Fits
• Data are 1 constraints with MH=117 GeV
• Ovals are 90% CL contours
S  0.04  0.09 (0.07)
T  0.02  0.09 (0.09)
T
MH,ref=117 GeV (300 GeV)
S
What if There is No Higgs?
• Simplest possibility: No Higgs / No new light particles /
No expanded gauge symmetry at EW scale
– Electroweak chiral Lagrangian
Leff  LnlSM   Li
• LSMnl doesn’t include Higgs, so it is non-renormalizable
• Assume global symmetry SU(2) x SU(2) →SU(2)V or
U(1)
– SU(2) x SU(2) →SU(2)V is symmetry of Goldstone Boson sector
of SM
• Li is an expansion in (Energy)2/2
Remember Han talk
No Higgs → Unitarity Violation
• Consider W+W-→W+W1
• Unitarity conservation requires Re( al ) 
s
2
0
a


0
2
• MH→
32v
 1.7 TeV
 New physics at the TeV scale
• If all resonances (Higgs, vector mesons…etc) much
heavier than ~ few TeV
 s2 
st
A(W W  W W )  2  O 4 
v
v 




Electroweak Chiral Lagrangian
• Terms with 2 derivatives:

v2
L2  Tr D   D  
4

 
  exp( i  / v)
ig
ig 
D        i  Wi 
B  3
2
2
• Unitary gauge: =1
– SM masses for W/Z gauge bosons
• This is SM without Higgs
– SM W/Z/ interactions
General Framework for studying BSM
physics without a Higgs
E4 Terms in Chiral Lagrangian
• 3 operators contribute at tree level to gauge boson 2point functions

L1 
L1 
L8 
1
1 Tr TV  2
4
1
1 gg Tr B TW 
2
1
 8 g 2 Tr TW  2
4

T  2 T 3 
V  ( D ) 
Gives tree level isospin violation

Also contribute to
gauge boson 3-point
functions
Limits from
LEP2/Tevatron
Apologies: my normalization is different from Han…  ~ l(v/)2
E4 Terms continued
• Contribute to WW, WWZ vertices (but not to 2-point functions)
ig 
 2 B Tr T V  , V 
2
L3  ig  3Tr W V  , V 
i
L9 
g 9Tr TW Tr T V  , V 
2

 
L2 




• Only contribute to quartic interactions
L4   4 Tr VV 
2

L5   5 Tr VV 
L7 
 7 Tr VV  Tr TV Tr TV  
L10 
1
10 TrTr TV Tr TV 
2
2


 
L6   6 Tr VV Tr TV Tr TV



• Conserves CP, violates P
L11  11g Tr TV Tr V W 
Appelquist & Longhitano



2
12 E4 Operators
• Assume custodial SU(2)V (=1at tree level)
– L1’, L6, L7, L8, L9, L10 vanish
– 6 operators remain
– Assume P conservation, L11 vanishes
Simple format for BSM physics
Estimate coefficients in your favorite model
SU(2)

SU(2)
Tree Level
• 2-point functions
Stree  4e 21
Ttree  21
U tree  4e 2 8
• SM fit assumes a value for MH
– Contribution from heavy Higgs:
S H 
1  MH 
 ,
ln 
6  M Z 
TH 
M
3
ln  H
2
8c  M Z
• Scale theory from  to MZ, add back in contribution
from MH(ref)



– Approach assumes logarithms dominate
Stree
Ttree
U tree
  

ln
6  M H ,ref

  4e 21


3   

ln
 2 1
8c 2  M H ,ref 
 4e 2 8
  3 TeV
.0034  1  .0074
Extended Gauge Symmetries
• General model with gauged SU(2) x SU(2)N x U(1)
1
1 N 1 a

LG   B B   Wi , Wi a , 
4
4 i 1
• After electroweak symmetry breaking, massless photon, plus tower
of massive Wn, Zn vector bosons
Wi
,
N 1
  ainW
n 1
• Fermions:
,
n
N 1
W3,i  bi 0    bin Z n
n 1
N 1
L f  ij
n 1
N 1
g ijW 
n
2 2
n 1
j
n 1
 i  (1   5 ) jWn,   h.c
 i g ijV 0 i ( g

N 1
B  b00   b0 n Z n

Vn0
Vi
 g  5 ) iVn0,
Vn0
Ai
• Calculate for general couplings, then apply to specific model
3-Site Higgsless Model (aka BESS)
L
V
R
1
2
g0
g1
g2
• Global SU(2) x SU(2) x SU(2) →SU(2)V symmetry
• Gauged SU(2)1 x SU(2)2 x U(1)
f2
L2 
4
2


 Tr D i D  i 
i 1
1
1
1
2
2
2




Tr
L

Tr
V

Tr R 


2
2
2
2 g0
2 g1
2g2
D 1    1  iL 1  i1V
D  2     2  iV  2  i 2 R
• Model looks like SM in limit
x
g0
 1
g1
g 02 
4
sW2
y
g2
 1
g1
g 22 
4
cW2
Scales
~10 TeV
Strong Coupling
MW’ ~ 1 TeV
Weakly coupled
non-linear  model
MW
Calculate log-enhanced contributions to S
3-Site Higgsless Model
• At tree level,
4 4sW2 M W2
Stree  2 
g1
M W2 '
Requires MW’> 3 TeV
Problem with unitarity
• Get around this by delocalizing fermions


L f   x1 Li( D1 )1 L
L f  J L  1  x1 L  x1V  J Y B
• Ideal delocalization: pick x1 to make S vanish at tree
level
4sW2 M W2
Stree 
M W2 '
 x1M W2 ' 
1 

2 
2M W 

What happens at 1-loop?
The Problem with Gauge Boson Loops
ig 
DS (q )  2
q  M S2
ig 

2
DGi (q )  2
q  M G2i
 
i
q  q 

2


DV (q )  2
 g  1    2
q  M V2 
q  M V2 

2
=1 (Feynman), =0 (Landau), → (Unitary)
Independent of 
+ ….
Depends on 
Pinch Technique
=
Gauge independent
Individually gauge dependent
• STU extracted from 4-fermion interactions
• Idea:
–
–
–
–
Isolate gauge-dependent terms in vertex/box diagrams
Combine with 2-point diagrams
Result is gauge-independent
q4 and q6 terms cancel in unitary gauge
Degrassi, Kniehl, & Sirlin, PRD48 (1993) 3963;
Papavassiliou & Sirlin, PRD50 (1994) 5951
Pinch Technique
d nk
1



u
(
p
)
...(
k

p
)
k
...
u
(
p
)
1
1
(k 2  M W2 )( k  p1 ) 2 (k  q ) 2
2 n 2
d nk
1

u ( p2 )...(k  p1 )( k  p1  p1 )...u ( p1 ) 2
n
(k  M W2 )( k  p1 ) 2 (k  q ) 2
2 
d nk
1
2

u
(
p
)
...(
k

p
)
...
u
(
p
)
 ...
2
1
1
n
2
2
2
2
(
k

M
)(
k

p
)
(
k

q
)
2 
W
1


Extract terms from vertex/box corrections that
look like 2-point functions
Associate this piece with propagator
Pinch Technique
• Calculate pinch 2-point
functions with SM gauge
sector plus W’,Z’
• Unitary gauge minimizes
number of diagrams
• Neutral gauge boson 2-point
functions:
A1loop 
A0
q
2



 2q 2V  q 4 B 
A0
q
2
 Pinch

Unknown Terms
• Unknown higher-dimensional operators
– Generalization of 1 term



L4  c1 g1 g2Tr V 2 B  2  c2 g1 g0Tr L 1V  1

• L2 gives poles at one loop: 1/→ Log (2/MW2)
– Poles absorbed in redefinition of arbitrary couplings, c1 and c2
• Approach only makes sense if logarithms dominate
• Results have scheme dependence

S 0  8 c1 ( )  c2 ( )
2
2

Leading Chiral Logarithm
• To leading order in x=2MW/MW’
– Landau and also Feynman gauge (Chivukula, Simmons….)
  M W2 '  41  2  3 x1  2 



S1loop 
ln 
ln 
ln 
12  M W2  24  M W2 '  4 x  M W2 ' 
• Unitary gauge, keeping subleading terms in x
 M W2 ' 
 M W2 ' 
S
S
S3 site  Stree A ln  2   AW '  AW ln  2   S0
 MW 
 MW 
S
W


Large Corrections in 3-Site Model at 1-loop
Ideal localization only fixes S problem at tree level
See talk by de Curtis
4-Site Model
g̃̃̃̃
g
f1
L
g̃̃̃̃
f2
V1
g’
f1
V2
R
• Gauge symmetry: SU(2)L x SU(2)V1 x SU(2)V2 x U(1)Y
• Why?
– With f2f1 can “mimic” warped RS models
– Gauge couplings approximately SM strength if:
g
x  ~  1
g
g
y  ~  1
g
g2 
4
,
2
s
g 2 
4
c2
– Gauge sector: SM + 2 sets of heavy gauge bosons,
i, 0, i=1,2
Accomando et al, arXiv:0807.5051; Chivukula, Simmons, arXiv: 0808.2017
Compute Masses & Mixings
3
L
i 1

2
fi
Tr D  i D i
4
D 1    1  igL 1  ig~1V1
D  2     2  ig~V1  2  ig~ 2V2 
D  3     3  ig~V2   3  ig  3 R

• Compute masses as expansion in x
M 
2
W
M 2  
1
M  
2
1


g 2 f12 f 22
1  x 2 zw
2
2
4 f1  2 f 2
g~ 2 f12  x 2 
1  
4 
2
g~ 2 f12  2 f 22  x 2 4 
1  z 
4
2 



• 3-site limit: f2→, z→0, 2 decouples
z
zW 
f1
f12  2 f 22

1
1 z4
2

Delocalizing Fermions in 4-Site Model
• One-Site delocalization
L   x1   iD 1 1 PL

L f  g    YL PL  YR PR B   g 1  x1   T a La, PL  g~x1  T aV1a, PL
• Contribution to S at tree level
Stree 

2
 4 M 2 0 
4 sW2 M W2  M 10
1  2  x1  2 12 
2
 g f1 
M 0  M 0


1
2

2

x1M  
4 sW2 M W2 
2
2
1

1

z

1

z
  O( x 2 )
2
2
M  
2M W


1
• Pick x1 to minimize Stree


Calculate S Using Pinch Technique
Just like 3-Site calculation, but more of it….
S at 1-Loop in 4-Site Model
2
2



x
M
M

1
4s M 
  1
1
2
2 

S 4 Site 
1 z 
1 z  
ln
2
2


M 
2M W
12  M H2 ,ref



1
 M 2 

  41  z 2  17 z 4  17 z 6 3x1
 
 2 1  z 2  ln  22

24
4x
  M 1
 2 
  83  16 z 2  33z 4 3x1
2 
 
 2 1  z  ln  2 

24
4x
  M  2 
 8 1  z 2 c1 ( )  c2 ( )   1  z 4 c3 ( )
2
W
2
W










Note scaling between different energy regimes
Dawson & Jackson, arXIv:0810.5068
One-Loop Results in 4-Site Model
z
M 
1
M 
2
Large fine tuning needed at 1-loop
The Moral of the Story is….
• Triplet models
– Can’t use STU approach in triplet models
– Triplet models can fit EW data if heavy scalars
roughly degenerate
– Minimizing the potential requires small mixing in
neutral and charged sectors
• 2, 3, 4 Site Higgsless models
– Ideal delocalization at tree level doesn’t solve
problem at 1-loop
– Unknown coefficients make predictions problematic