s. dawson trieste, 2011 lecture 2users.ictp.it/~smr2244/dawson2.pdf4 leptons • include an su(2)...
TRANSCRIPT
1
EW Symmetry Breaking: Where are we?
S. DawsonTrieste, 2011
Lecture 2
2
Fermi Model
• Current-current interaction of 4 fermions
• Consider just leptonic current
• Only left-handed fermions feel charged current weak interactions
• This induces muon decay
JJGL FFERMI 22
hceJ elept
21
21 55
e
eGF
=1.16637 x 10-5 GeV-2
This structure known since Fermi
3
Fermion Multiplet Structure
• L couples to W
(cf Fermi theory)– Put in SU(2) doublets
• R doesn’t couple to W
– Put in SU(2) singlets• Fix weak hypercharge to get correct couplings to
photon
Put this in by hand
4
Leptons
• Include an SU(2) doublet of left-handed leptons
• Right-handed electron is SU(2) singlet, eR =(1+5 )e/2– No right-handed neutrino
• Define weak hypercharge, Y, such that Qem =(I3 +Y)/2– YeL =-1– YeR =-2
eeL
L
L
)1(21
)1(21
5
5
I3 =1
To make charge come out right
*Standard Model has massless neutrinos—discovery of non-zero neutrino mass evidence for physics beyond the SM
5
Fermions come in generations
RL
RRL
ee
dudu
,, ,
RL
RRL
scsc
,, ,
RL
RRL
btbt
,, ,
Except for masses, the generations are identical
2
1 5,
RL
6
Muon decay
• Consider
e
e
• Fermi Theory:
e
e
• EW Theory:
eF uuuugGie
2
12
122 55
e
W
uuuugMk
ige
21
211
255
22
2
For k<< MW , 22GF =g2/2MW2
e
e
W
7
Higgs Parameters
• GF measured precisely
• Higgs potential has 2 free parameters, 2,
• Trade 2,
for v2, Mh2
– Large Mh strong Higgs self-coupling– A priori, Higgs mass can be anything
22 )( V
42
23
22
2
822h
vMh
vMhMV hhh
22
22
22
vM
v
h
22
2
21
82 vMgG
W
F 212 )246()2( GeVGv F
8
What about fermion masses?• Fermion mass term:
• Left-handed fermions are SU(2) doublets
• Scalar couplings to fermions:
• Effective Higgs-fermion coupling
• Mass term for down quark:
LRRLmmL
..chdQL RLdd L
L du
Q
..0
),(2
1 chdhv
duL RLLdd
vM d
d2
Forbidden by SU(2)xU(1) gauge invariance
9
Fermion Masses, 2
• Mu from c =i2 * (not allowed in SUSY)
• For 3 generations, , =1,2,3 (flavor indices)
0
c
hcuQL RcLu v
Muu
2
,
..2
)( chdduuhvL RLdRLuY
10
Fermion masses, 3
• Unitary matrices diagonalize mass matrices
– Yukawa couplings are diagonal in mass basis– No flavor changing effects in Higgs sector– Not necessarily true in models with extended
Higgs sectors
mRdR
mRuR
mLdL
mLuL
dVduVudUduUu
11
Review of Higgs Couplings
• Higgs couples to fermion mass– Largest coupling is to heaviest fermion
– Top-Higgs coupling plays special role?– No Higgs coupling to neutrinos
• Higgs couples to gauge boson masses
• Only free parameter is Higgs mass!
hffffv
mfhf
vm
L LRRLff
....cos
hZZgMhWWgMLW
ZW
12
Review of Higgs Boson Feynman Rules
• Higgs couples to heavy particles
• No tree level coupling to photons () or gluons (g)
• Mh2=2v2large Mh is strong coupling regime
13
Recap
• Standard model is predictive theory• Only missing piece is Higgs boson• Can test predictions experimentally• Bottom line: everything hangs together
14
Basics
• Four free parameters in gauge-Higgs sector (g, g’, , )– 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
Predicts MW
15
Inadequacy of Tree Level Calculations
• Mixing angle is predicted quantity– On-shell definition cos2W =MW
2/MZ2
– Predict MW
– Plug in numbers: • MW predicted =80.939 GeV• MW experimental =80.399
0.023 GeV
– Need to calculate beyond tree level
222
ZFWW MG
cs
1
22
24112
ZFFW MGG
M
16
Modification of tree level relations
rMG
WWF
11
sin2 22
W
WtFt mGr
2
2
2
2
sincos
283
• Contributions to r from top quark and Higgs loops
2
2
2
2
ln224
11
W
hWFh
MMMGr
• r is a physical quantity which incorporates 1-loop corrections
Extreme sensitivity of precision measurements to mt
17
MW vs mt
• Logarithmic dependence on Mh
Direct measurement of W and top masses
Masses inferred from precision measurements
Higgs boson wants to be light
mt (GeV)
MW
(GeV
)
18
Higgs Boson
• Standard Model Higgs expected to be light
• This assumes the Standard Model!
Standard Model prefers light (Mh < 158 GeV) Higgs boson
Note importance of theory uncertainties
Mh (GeV)
19
Quantum Corrections
• Relate tree level to one-loop corrected masses
• Majority of corrections at one-loop are from 2-point functions
)( 2220 VVVVV MMM
)()()( 222 kBkkkgk XYXYXY
XYi
20
Example of Quantum Corrections
• Example: 1cos22
2
WZ
W
MM
22
)0()0(
Z
ZZ
W
WW
MM
Experimentally,
~ 1
21
Higgs Contribution to
2
2
2 log16
3
W
h
W MM
c
• Higgs contributions have divergences which are cancelled by contributions of gauge boson loops
• Higgs contributions alone aren’t gauge invariant
• Keep only terms which depend on Mh
ZZZZ
h h
22
Looking for the Higgs Boson
• Standard Model Higgs boson expected to be light, < 150 GeV
• But…. Limits assume Standard Model, so look for Higgs boson in all mass regions
• Higgs couplings are absolute prediction, so for a given Higgs boson mass, we can calculate everything
23
Higgs Decays
• hff proportional to mf2
• Identifying b quarks important for Higgs searches
2
2
)()(
mmN
hBRbbhBR b
c
For Mh <2MW , decays to bb most important
h
24
)()( ppgMA W
Higgs Decays to W/Z
• Tree level decay
• Below threshold, h→WW* with branching ratio W* →ff' implied
• Final state has both transverse and longitudinal polarizations
2
2
3
2 4311
16)( WWW
W
h
W
xxxMM
sWWh
2
2
4h
WW M
Mx
h
W+
W-
hW+
W-
ff'
25
Higgs Decays to Gluons
2
2
2
3
22
2
72)(
t
h
W
h
W
s
MMO
MM
sggh
•Top quark contribution most important
•Doesn’t decouple for large mt
•Decoupling theorem doesn’t apply to particles which couple to mass (ie Higgs!)
•Decay sensitive to extra generations
hg
g
26
Higgs Decays to Photons
• Dominant contribution is W loops• Contribution from top is small
2
2
3
22
3
...9
167256
)( W
h
W MM
sh
topWh h h
27
Higgs decays to gauge bosons
For any given Mh, not all decay modes observable
28
Higgs Branching Ratios
Mh (GeV)
29
Total Higgs Width
• Small Mh , Higgs is narrower than detector resolution
• As Mh becomes large, width also increases– No clear resonance– For Mh 1.4 TeV,tot Mh
3
2
3
2
1330
sin16)(
TeVMGeV
MMWWh
h
W
h
W
30
Higgs Searches at LEP2• LEP2 searched for e+e-Zh• Rate turns on rapidly after threshold, peaks just above
threshold, 3/s• Measure recoil mass of Higgs; result independent of
Higgs decay pattern– Pe- =s/2(1,0,0,1)– Pe+ =s/2(1,0,0,-1)– PZ =(EZ , pZ )
• Momentum conservation:– (Pe- +Pe+ -PZ )2=Ph2=Mh2
– s-2 s EZ +MZ2= Mh2
LEP2 limit, Mh > 114.1 GeV
h
31
Higgs production at Hadron Colliders
• Many possible production mechanisms; Importance depends on: – Size of production cross section– Size of branching ratios to observable channels– Size of background
• Importance varies with Higgs mass• Need to see more than one channel to establish
Higgs properties and verify that it is a Higgs boson
32
Production Mechanisms in Hadron Colliders
• Gluon fusion– Largest rate for all Mh at LHC and Tevatron– Gluon-gluon initial state– Sensitive to top quark Yukawa t
Largest contribution is top loop
In Standard Model, b-quark loop contribution small
h
33
Gluon Fusion (in detail!)
)()()()2(
)()( 32 qpiD
TkdvmiTTTrigiA n
n
BAs
)))(())((( 222222 mqkmpkmkD
1
0
1
03))1(1(
21 x
yxCByAxdydx
ABC
322 ))(2(
121qypxkmk
dxdyD
qypxkk
3222 )(
121xyMmk
dxdyD h
Combine denominators
Shift momentum
h
p
q
2
2hMqp
34
Gluon Fusion• Numerator
• Shift momenta, drop linear terms, use
• Dimension regularization:
))(()( mqkmpkmkTrT
qpkkMkmgmT h 4)
2(4
222
rrn
Akkg
nAkkkkd
)(1
)( 2
2
2
)()()(
241
2114
)2(2
3222
22222
qpMmk
dxdyxyqp
xyMn
kmgdxdykdvmgiA
h
hn
n
ABs
12232
2232
2
)1()4(32)(
1)2(
)2()1()4(32)()2(
AiAk
kd
AiAk
kkd
n
n
n
n
35
Gluon Fusion
• Answer:
• Gauge invariant form:
)()(4122
2
22
222
qpxyMm
xydxdy
qpMgv
mhggA
h
hAB
s
)()(2
44
)(2
2
2
2/1 qpqpMgMmF
vhggA h
hAB
s
xymM
xydxdyMmF
hh2
22
2
2/1
1
4144
2 diagrams
36
Gluon Fusion
• Turn it into a partonic cross section
12
ˆ21
6441)ˆ(ˆ
dAs
shgg
sM
Md h
h ˆ12 2
21
s
MMmF
vs h
h
Rshgg ˆ
141024
)()ˆ(ˆ22
2
2
2/12
2
Spin & color average
37
Gluon Fusion• Lowest order cross section:
– q =4mq2/Mh2
– Light Quarks: F1/2 (mb /Mh )2log2(mb /Mh )– Heavy Quarks: F1/2 -4/3
)ˆ
1()(1024
)()ˆ(ˆ22
2/12
2
sMF
vs h
Rshgg
• Rapid approach to heavy quark limit: Counts number of heavy fermions
• NNLO corrections calculated in heavy top limit