s. dawson (bnl) february, 2009
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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 =MW
2/(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
• Dominant contributions from 2-point functions
)1(
12 22
2 SM
WZ
W
r
MMM
G
e
e
Wr is a physical parameter
Part A: Triplet ModelModels with 1 at tree level are different
from the SM
• 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
=MW2/(MZ
2c2)1
Higgs Triplet Model
• Add a real triplet
– vSM2=(246 GeV)2=v2+4v’2
– Real triplet doesn’t contribute to MZ
• At tree level, =1+4v’2/v21• PDG: v’ < 12 GeV
2
2222 '4
14 v
vvgMW
)(2
1 00
ihvH
0'v
Simplest extension of SM with 1
Motivated by Little Higgs modelsNeglects effects of scalar loops
Scalar Potential
aaHHHHHV 4
223424
1
222
221 24
0
0
0
0
hcs
sc
K
H
• 4 has dimensions of mass → doesn’t decouple
• Mass Eigenstates:
cs
sc
H
G
• 6 parameters in scalar sector: Take them to be:
MH0, MK0, MH+, v, ,
small since it is related to parameter
tan = 2 v’/v
Forbidden by T-parity
Decoupling at Tree Level
• Require no mixing between doublet-triplet sectors for decoupling
• v’→0 requires 4 →0 (custodial symmetry), or 3→ (invalidating perturbation theory)
• v’→0 implies MK0 MH+
3
4
v
32
222
22
2
1120 vvM
K
432
222
22 2
2
14 vvvM
H
Heavy Scalars → Small Mass Splittings
• Plots are restriction 2 < (4 )2
Forshaw, Vera, & White, hep-ph.0302256
MH+ (GeV)
MK
0 -
MH
+ (
Ge
V)
=0
Allowed region is between curves
Renormalization of Triplet Model
• At tree level, W mass related to input parameters:
• One loop radiative corrections included in parameter r
)1(2 2
2 rGs
MW
122
2
Z
W
Mc
M
For 1, 4 input parameters
Input 4 Measured Quantities(MZ, , G, sin eff)
• Use effective leptonic mixing angle at Z resonance as 4th parameter
eZaveiL ee )( 5
2
1,2
2
1 2 eeff
e asv
• Could equally well have used or MW as 4th parameter
• At tree level, SM and triplet model are identical in seff
scheme (SM inputs , G, sin eff here )
This scheme discussed by: Chen, Dawson, Krupovnickas, hep-ph/0604102; Blank and Hollik hep-ph/9703392
rGs
MeffW 1
2 2
2
Triplet Results
• Compare with SM in effective mixing angle scheme
• Input parameters: MZ, sineff, , 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
(...)sin(...)sin24
~2
22
2
0
H
HKSMtriplet
M
MM
srr
Toussaint, PRD18 (1978) 1626
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+
MW(SM)-MW(Triplet)
• For heavy H+, perturbativity requires MH+~MK0, and predictions of triplet model approach SM
• No large effects in perturbative regime
• SM not exactly recovered at large MH+ due to different MZ
inputs for 1-loop correctionsSimilar conclusions from Chivukula, Christensen, Simmons: arXiv:0712.0546
MH+ - MK=0, 10, 20 Gev
MH+ (GeV)
MW (
Me
V)
=.1,
MH0=120 GeV
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 2-point 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
Peskin & Takeuchi, PRD46 (1992) 381
Taylor expand 2-point functions:
)()0()(
)()0()(
)()0()(
)()0()(
222
222
222
222
qqq
qqq
qqq
qqq
ZZZZZZ
WWWWWW
ZZZ
Vanishes by EM gauge invariance
Fermion & scalar contributions vanish; gauge boson contributions non-zero
6 unknown functions to this order in MZ/
STU Definitions
22
2
2
2
22
2
22
222
222
222
22
)0()0()(2
)0()()0()(4
)0(2
)0()0(
)0(0)()()0()(4
ZW
Z
ZWZWW
Z
ZZWZZW
W
WWWWWW
Z
Z
W
W
Z
ZZ
W
WW
ZZZWW
WWZZZZZZZ
Z
WW
Ms
M
Msc
M
Mc
M
MsU
Mc
s
MMT
Msc
scMM
M
csS
Peskin & Takeuchi, PRD46 (1992) 381
6 unknowns:
3 fixed by SM renormalization, 3 free parameters
S is scaling of Z 2-point function from q2=0 to MZ2
T is isospin violation
U contributes mostly to MW
3 and 4 Site Higgsless ModelsPDG Fits
• Data are 1 constraints with MH=117 GeV
• Ovals are 90% CL contours
)09.0(09.002.0
)07.0(09.004.0
T
S
MH,ref=117 GeV (300 GeV)
S
T
What if There is No Higgs?
• Simplest possibility: No Higgs / No new light particles / No expanded gauge symmetry at EW scale– Electroweak chiral Lagrangian
• 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
inlSMeff LLL
Remember Han talk
No Higgs → Unitarity Violation
• Consider W+W-→W+W-
• Unitarity conservation requires• MH→
• If all resonances (Higgs, vector mesons…etc) much heavier than ~ few TeV
2
1)Re( la
200 32 v
sa
1.7 TeV
New physics at the TeV scale
4
2
2)(
v
sO
v
tsWWWWA
Electroweak Chiral Lagrangian
• Terms with 2 derivatives:
• Unitary gauge: =1– SM masses for W/Z gauge bosons
• This is SM without Higgs– SM W/Z/ interactions
DDTr
vL
4
2
2)/exp( vi
322
B
giW
igD ii
General Framework for studying BSM physics without a Higgs
E4 Terms in Chiral Lagrangian
• 3 operators contribute at tree level to gauge boson 2-point functions
22
88
11
211
4
12
14
1
TWTrgL
TWBTrggL
TVTrL
)(
2 3
DV
TT
Also contribute to gauge boson 3-point functions
Limits from LEP2/Tevatron
Gives tree level isospin violation
Apologies: my normalization is different from Han… ~ l(v/)2
E4 Terms continued
• Contribute to WW, WWZ vertices (but not to 2-point functions)
• Only contribute to quartic interactions
• Conserves CP, violates P
VVTTrTWTrgi
L
VVWTrigL
VVTTrBgi
L
,2
,
,2
99
33
22
2
1010
77
2
1
TVTrTVTrTrL
TVTrTVTrVVTrL
WVTrTVTrgL 1111
TVTrTVTrVVTrL
VVTrL
VVTrL
66
2
55
244
Appelquist & Longhitano
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
• SM fit assumes a value for MH
– Contribution from heavy Higgs:
• Scale theory from to MZ, add back in contribution from MH(ref)– Approach assumes logarithms dominate
82
1
12
4
2
4
eU
T
eS
tree
tree
tree
Z
HH
Z
HH M
M
cT
M
MS ln
8
3,ln
6
12
82
1,
2
12
,
4
2ln8
3
4ln6
eU
McT
eM
S
tree
refHtree
refHtree
0074.0034.3 1 TeV
Extended Gauge Symmetries
• General model with gauged SU(2) x SU(2)N x U(1)
• After electroweak symmetry breaking, massless photon, plus tower of massive Wn
, Zn vector bosons
• Fermions:
• Calculate for general couplings, then apply to specific model
1
1
,,4
1
4
1 N
i
ai
aiG WWBBL
1
1
,,N
nnini WaW
1
1000
N
nnnZbbB
1
10,3
N
nninii ZbbW
0,
1
15
,
1
15
)(
.)1(22
00
0
ni
N
ni
VA
VViijV
nj
N
nij i
ijWf
Vggg
chWg
L
n
i
n
ij
n
3-Site Higgsless Model (aka BESS)
• Global SU(2) x SU(2) x SU(2) →SU(2)V symmetry
• Gauged SU(2)1 x SU(2)2 x U(1)
• Model looks like SM in limit
222
2
21
2
20
2
1
2
2 2
1
2
1
2
1
4
RTrg
VTrg
LTrg
DDTrf
Li
ii
RiiVD
ViiLD
2222
1111
1
1
1
2
1
0
g
gy
g
gx
222
220
4
4
W
W
cg
sg
1 2
L V R
g0 g1 g2
Scales
~10 TeV
MW’ ~ 1 TeV
MW
Strong Coupling
Weakly coupled non-linear model
Calculate log-enhanced contributions to S
3-Site Higgsless Model
• At tree level,
• Get around this by delocalizing fermions
• Ideal delocalization: pick x1 to make S vanish at tree level
2'
22
21
44
W
WWtree M
Ms
gS
LLf DixL 111 )(
2
2'1
2'
22
21
4
W
W
W
WWtree M
Mx
M
MsS
BJVxLxJL YLf 111
What happens at 1-loop?
Requires MW’> 3 TeV
Problem with unitarity
The Problem with Gauge Boson Loops
22222
222
222
1)(
)(
)(
VVV
GG
SS
Mq
qqg
Mq
iqD
Mq
igqD
Mq
igqD
i
i
=1 (Feynman), =0 (Landau), → (Unitary)
Independent of
+ …. Depends on
Pinch Technique
• 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
Individually gauge dependent
=
Gauge independent
Pinch Technique
...
)())((
1)(...)...()(
2
)())((
1)()...)(...()(
2
)())((
1)(...)...()(
2
221
2212
12
221
2211112
221
22112
qkpkMkpupkpu
kdqkpkMk
puppkpkpukd
qkpkMkpukpkpu
kd
Wn
nW
n
nW
n
n
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:
Pinchloop
q
ABqVq
q
AA
2
042
2
01 2
Unknown Terms
• Unknown higher-dimensional operators
– Generalization of 1 term
• 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
)()(8 22
210 ccS
11012222114
VLTrggcBVTrggcL
Leading Chiral Logarithm
• To leading order in x=2MW/MW’
– Landau and also Feynman gauge (Chivukula, Simmons….)
• Unitary gauge, keeping subleading terms in x
2
'
21
2'
2
2
2'
1 ln4
3ln
24
41ln
12 WWW
Wloop Mx
x
MM
MS
02
2'
'2
2'
3 lnln SM
MAA
M
MASS
W
WSW
SW
W
WSWtreesite
Large Corrections in 3-Site Model at 1-loop
Ideal localization only fixes S problem at tree level
4-Site Model
• Gauge symmetry: SU(2)L x SU(2)V1 x SU(2)V2 x U(1)Y
• Why?– With f2f1 can “mimic” warped RS models
– Gauge couplings approximately SM strength if:
– Gauge sector: SM + 2 sets of heavy gauge bosons,
i, 0, i=1,2
1~1~
g
gy
g
gx
Accomando et al, arXiv:0807.5051; Chivukula, Simmons, arXiv: 0808.2017
f1 f2 f1
g g’g̃̃��g̃̃��
22
22 4
,4
cg
sg
L V1 V2 R
See talk by de Curtis
Compute Masses & Mixings
• Compute masses as expansion in x
• 3-site limit: f2→, z→0, 2 decouples
422
22
12
2
221
22
22
22
1
22
21
22
21
4
2~2
14
~
124
1
1
zxffg
M
xfgM
zxff
ffgM wW
22
21
1
2 ff
fz
412
1zzW
RgiVgiD
VgiVgiD
VgiigLD
33233
222122
11111
~
~~
~
3
1
2
4iii
i DDTrf
L
Delocalizing Fermions in 4-Site Model
• One-Site delocalization
• Contribution to S at tree level
• Pick x1 to minimize Stree
L
aaL
aaRRLLf PVTxgPLTxgBPYPYgL ,
11,
1~1
)(12
14
41
4
222
212
2
22
21
2
2
12
2
2
22
1
1
01
02
01
01
xOzM
Mxz
M
Ms
fg
Mx
M
M
M
MsS
W
WW
WWtree
LPDixL 111
Calculate S Using Pinch Technique
Just like 3-Site calculation, but more of it….
S at 1-Loop in 4-Site Model
)(1)()(18
ln14
3
24
331683
ln14
3
24
171741
ln12
12
14
34
212
2
22
21
42
2
2
221
642
2,
2
22
212
2
22
4
2
1
2
11
1
czccz
Mz
x
xzz
M
Mz
x
xzzz
M
Mz
M
Mxz
M
MsS
refHW
WWSite
Note scaling between different energy regimes
Dawson & Jackson, arXIv:0810.5068
One-Loop Results in 4-Site Model
Large fine tuning needed at 1-loop
2
1
M
Mz
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