radiative corrections to the standard model s. dawson tasi06, lecture 3
TRANSCRIPT
Radiative Corrections to the Standard Model
S. Dawson
TASI06, Lecture 3
Radiative Corrections
• Good References:– Jim Wells, TASI05, hep-ph/0512342– K. Matchev, TASI04, hep-ph/0402031– M. Peskin and T. Takeuchi, Phys. Rev. D46,
(1992) 381– W. Hollik, TASI96, hep-ph/9602380
Basics
• SM is SU(2) x U(1) theory– Two gauge couplings: g and g’
• Higgs potential is V=-22+4
– Two free parameters
• Four free parameters in gauge-Higgs sector– Conventionally chosen to be
• =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
22
22
2
1282
WZ
WW
F
MMMM
gG
Example: parameter
• At tree level,
• Many possible definitions of ; use
...1cos22
2
WZ
W
M
M
1CC
NC
G
G
JJG
L CCCC
2
ZZNC
NC JJG
L2
At tree level GCC=GNC=GF
parameter (#2)
22
)0()0(
Z
ZZ
W
WW
MM
Top quark contributes to W and Z 2-point functions
qq piece of propagator connects to massless fermions, and doesn’t contribute here
Top Corrections to parameter (Example)
• 2-point function of Z
• Shift momentum, k’=k + px
den
Tkdi
c
igNi
n
n
WcZZ
)2()(
42
2
])][([ 2222tt mpkmkden
nde
Tdx
kdi
c
igNi
n
n
WcZZ
1
0
2
2
)2()(
4
2222 ])1([ tmxxpknde
Nc=number of colors
n=4-2 dimensions
Top Corrections to parameter (#2)
)()()( PLPRmpkPLPRmkTrT tttttt
...162
14 2222
02
gmRLLRkT tttttp
•Shift momenta, keep symmetric pieces
•Only need g pieces
2
1 5P
Top Corrections to parameter (#3)
222
22221
0
2 162
14
)2(4)0(
t
ttttt
n
n
WcZZ
mk
mRLLRkdx
kd
c
gNi
22
2
2222
2
)2()1(
)1(4
16)()2( tt
n
n
mm
i
mk
kkd
• Only need answer at p2=0
• Dimensional regularization gives:
)1(4
16)(
1
)2( 2
2
2222
t
n
n
m
i
mk
kd
Top Corrections to parameter (#4)
• 2-point function of W, Z
22
222
2 4
32)0( tt
t
tW
cZZ LR
m
mc
Ng
2
114
32)0( 2
22
2
tt
cWW m
m
Ng
Lt=1-4sW2/3
Rt=-4sW2/3
2
2
222 82
)0()0(
W
tcF
Z
ZZ
W
WW
M
mNG
MM
Top quark doesn’t decouple
• Longitudinal component of W couples to top mass-- eg, tbW coupling:
• Decoupling theorem doesn’t apply to particles which couple to mass
tbM
mig
tM
pb
ig
tbig
W
t
W
W
)1(22
)1(22
)1(22
5
5
5
For longitudinal W’s: W
WL M
p
Why doesn’t the top quark decouple?
• In QED, running of at scale not affected by heavy particles with M >>
• Decoupling theorem: diagrams with heavy virtual particles don’t contribute at scales << M if – Couplings don’t grow with M – Gauge theory with heavy quark removed is still
renormalizable• Spontaneously broken SU(2) x U(1) theory violates both
conditions– Longitudinal modes of gauge bosons grow with mass
– Theory without top quark is not renormalizable• Effects from top quark grow with mt
2
Expect mt to have large effect on precision observables
Modification of tree level relations
rMG
WW
F
1
1
sin2 22
r is a physical quantity which incorporates 1-loop corrections
2
2
2
2
W
W
W
W
F
FSM M
M
s
s
G
Gr
Corrections to GF
• GF defined in terms of muon lifetime
• Consider 4-point interaction,with QED corrections
• Gives precise value:
• Angular spectrum of decay electrons tests V-A properties
2
2
2
3
52
701.6810.118
1192
1
m
mmGeF
2510)00001.016637.1( GeVGF
GF at one loop
BVW
WW
BV
qWWW
W
F
Mv
Mq
iqi
M
gG
22
0
222
2
2
)0(1
2
1
)(182 2
Vertex and box corrections are small
(but 1/ poles don’t cancel without them)
)log(
2
476
4
41 22
2
22
2
2 WW
W
WWWBV c
s
s
sMs
Renormalizing Masses
• Propagator corrections form geometric series
2
02
22
02
)(V
VVV Mp
igpi
Mp
ig
)( 220
2 pMp
ig
VVV
2220, VVV MMM
)(Re 2VVV M
Running of , 1
(0)=1/137.0359895(61) (p2)= (0)[1+]
)()0()( 22 pp
)()( 222 ppp
ppBgppi )()( 22
Running of , 2
]))[((
)()(
)2()()()()(
222222
mpkmk
mpkmkTrkdiiepi
n
n
pk
k+p
2222
2221
0
2
)1(
...)1(12
)2(4
xxpmk
xxpmkn
g
dxkd
en
n
...)1(2
)1(
4)1(
22
21
0
2
gxxxxpm
dxpi
• Photon is massless:(0)=0
Calculate in n=4-2 dimensions
Running of , 3
)1(1log)1(
2)0()(
2
21
0
2 xxm
pxxdxp
2
22
15)0()( 22
m
pp
pm
Contributions of heavy fermions decouple:
2
2
2
22
3
5log
3)0()( 22
p
mO
m
pp
pm
Contributions of light fermions:
Running of , 4
2
2
2
22
3
5log
3)0()( 22
p
mO
m
pp
pm
00036.002761.0)(
)(Re
3)(
24
2
22
m Z
ZZ
hadronic
Mss
sRds
MM
031.0954.128)( 21 ZM
• From leptons: lept(MZ)=0.0315
• Light quarks require at low p2
•Strong interactions not perturbative
•Optical theorem plus dispersion relation:
)(
)()(
*
*
ee
hadronseesR
Have we seen run?
Langacker, Fermilab Academic Series, 2005
Final Ingredient is sW2
• sW is not an independent parameter
122
2
ZW
W
Mc
M
2
2
2
2
2
2
2
2
2
2
2
2
2
2
64 W
tc
W
W
W
W
Z
Z
W
W
W
W
M
mNg
s
c
M
M
M
M
s
c
s
s
Predict MW
• Use on-shell scheme:
r incorporates the 1-loop radiative corrections and is a function of , s, Mh, mt,…
2
22 1sin
Z
WW M
M
r
AM
W
W
1sin0
GeV
GA
F
28.372
0
2
2
2
2
W
W
W
W
F
FSM M
M
s
s
G
Gr
Data prefer light Higgs
• Low Q2 data not included– Doesn’t include atomic parity
violation in cesium, parity violation in Moller scattering, & neutrino-nucleon scattering (NuTeV)
– Higgs fit not sensitive to low Q2
data
• Mh< 207 GeV– 1-side 95% c.l. upper limit,
including direct search limit
Direct search limit from e+e-Zh
Logarithmic sensitivity to Mh
• Data prefer a light Higgs
2006
Understanding Higgs Limit
1118.0
)(085.0
1172
525.0102761.0
)(5098.0
100ln008.0
100ln0579.0364.80
2)5(
2
Zs
tZhad
hhW
M
GeV
MM
GeV
M
GeV
MM
MW(experiment)=80.404 0.030
Light Higgs and Supersymmetry?
• Adding the extra particles of a SUSY model changes the fit, but a light Higgs is still preferred
How to define sin2W
• At tree level, equivalent expressions
• Can use these as definitions for renormalized sin2W
• They will all be different at the one-loop level
2
22 1sin
Z
WW M
M
WWZ
FM
G
222 cossin2
22
22sin
gg
gW
On-shell
Z-mass
SM
Effective sW(eff)
1
4
1)(2
eA
eV
W g
geffs
sin2w depends on scale
• Moller scattering, e-e-e-e-
-nucleon scattering• Atomic parity violation
in Cesium
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
• Keep first two terms• Remember that QED Ward identity requires
any amplitude involving EM current vanish at q2=0
...)0()0()( 22 ijijij qq
0)0(
S,T,U (#3)
QQWQWW
Z
QQWQWWW
ZZ
WWW
ssc
e
sssc
e
s
e
e
23
2
43
23322
2
112
2
2
2
• Use JZ=J3-sW2JQ
Normalization of JZ differs by -1/2 from Lecture 2 here
S,T,U (#4)• To O(q2), there are 6 coefficients:
• Three combinations of parameters absorbed in , GF, MZ
• In general, 3 independent coefficients which can be extracted from data
...)0()0(
...)0(
...)0()0(
...)0(
332
3333
32
3
112
1111
2
q
q
q
q
QQQQ
S,T,U, (#5)
22
2
2
22
2
22
22
)0()(
4
)0()(
4
)0()0(
W
newWW
W
WnewWW
W
Z
newZZ
Z
ZnewZZ
WW
Z
newZZ
W
newWW
MM
MUS
s
MM
MS
cs
MMT
Advantages: Easy to calculate
Valid for many models
Experimentalists can give you model
independent fits
SM contributions in , GF, and MZ
Limits on S & T
• A model with a heavy Higgs requires a source of large (positive) T
• Fit assumes Mh=150 GeV
T
Higgs can be heavy in models with new physics• Specific examples of heavy Higgs bosons in Little Higgs and
Triplet Models• Mh 450-500 GeV allowed with large isospin violation
(T=) and higher dimension operators
We don’t know what the model is which produces the operators which generate large T
Compute S,T, U from Heavy Higgs
0
log12
3
log12
1
20
2
2
20
2
U
M
M
cT
M
MS
h
h
W
h
h
• SM values of S, T, U are defined for a reference Mh0
Using S,T, & U
• Assume new physics only contributes to gauge boson 2-point functions (Oblique corrections)
• Calculate observables in terms of SM contribution and S, T, and U
U
s
scTcSM
sc
cSMMM
W
WWWZ
WW
WWW 2
2222
22
222
42
1)(
Conclusion
• Radiative corrections necessary to fit LEP and Tevatron data
• Radiative corrections strongly limit new models