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

qq

Rshgg

• Rapid approach to heavy quark limit: Counts number of heavy fermions

• NNLO corrections calculated in heavy top limit