Outline – III, The Higgs Boson• W-W Scattering• Vacuum Fields• Higgs Mass

– W vs. Top Mass– Direct Searches @ LEP– Upper Limits

• Higgs Couplings to Bosons and Fermions – Decays and Production

• Vector Boson Fusion• Higgs Quantum Numbers• Higgs Pairs?

W+W Scattering, sans Higgs

The EW interactions diverge and thus violate unitarity. One mechanism proposed to solve this problem is to postulate the “Higgs boson” which cancels the divergences. It also has a vacuum expectation value for the field which is ~ 1/ ~ 300 GeV

G

Heavy Higgs and EW Cross Section

In COMPHEP let the Higgs mass get large. Cross section diverges. As before, H mass should be < 1 TeV

Higgs Potential, Self Coupling422 ||||)( V

2/22 Postulate forces described by a potential with 2 unknown parameters. Postulate a non-zero vacuum expectation value for the field and an excitation of the field, The resulting potential has a ‘cosmological term

a Higgs mass term

No prediction in the SM for the Higgs mass. There are also triplet plus quartic Higgs particle couplings

. Lagrangian density is dimension [M4]So [μ] = M and [λ] = 1

H 4~ 2 2~ H

3 4~ ,H H

.2462 GeVM H

( )V

2 22

How the W and Z get their Mass

•Covariant derivative contains EW gauge fields W,Z. Suppose an additional scaler field exists and has a vacuum expectation value. Quartic couplings give mass to the W and Z, as required by the data [ V(r) ~e(exp(-r/)/r) ] - weak at large r, strength ~e at small r].

2 2 2 2 2 22 1 2

( )( ) ( )( ),

0~

( )( ) ~ /2 ( ) /2 (0)W W Z Z

D D D ieA

D D g g g e

WWZ

W

MggM

gM

M

cos/2/

2/

0

22

21

2

Add gauge fields and look at mass terms

Photon, W and Z masses are induced by vev and predicted numerically

G and Decay Widths

mMm

mG

WW42

352

)/(~

192/

3

2

/ 8 2

( /16)( / ) ~ 1.76t t

W t W t

Gm

m M m GeV

Numerical value of G 2 body EW

3 body EW

Numerical W, Z Mass Prediction

•The masses for the W and Z are specified by the coupling constants. G comes from beta decays or muon decay.

2 2 5 2

2

/ 2 / 8 , 10

/ / 2 2 / 8

2 / 4 , 174

W W

W W

G g M G GeV

M g G

G GeV

2

2

sin ~ 0.231, ~ 28.7 , sin 0.481

~ 1/137, / sin ~ 1/ 31.6, ~ 0.63

oW W W

W W Wg

/ 2 ~ 80/ cos ~ 91

W W

Z W W

M g GeVM M GeV

Extend to Fermion Masses

][~ fg

],[][~ ff mg

2/)/(

]/2[

WfWf

WWfff

Mmgg

gMggm

The Higgs coupling to W,Z are fixed by the gauge symmetry. For fermions we exchange one mass parameter for an unknown coupling constant. Still, it is a compact choice. Except for the top quark these couplings are weak and are proportional to the fermion mass.

~ ( )m Free Dirac density

Yukawa interaction

~ 1tg

W Mass Corrections Due to Top, Higgs

We must simply assert that the propagators for fermions (Dirac equation) and bosons (Klein-Gordon equation) are different, 21/ , 1/q q respectively, for massless quanta. The propagator for massless bosons can be thought of as the Fourier transform of the Coulomb interaction potential. The propagator for fermions follows from a study of the Dirac equation.

2 4 2 3 2 2

2 4 2 2 3 4

~ /( ) ~ / ~ ~

~ /( ) ~ / ~ / ~ ln( )

m

M

M d q q q dq q qdq m

M d q q q dq q dq q M

2 2( ) 0( ) 0P MP M

Klein-GordonDirac

W mass shift due to top (m) and Higgs (M). Note quadratic (strong) divergence due to fermion loop.

2 - What is MH and How Do We Measure It?

• The Higgs mass is a free parameter in the current “Standard Model” (SM).• Precision data taken on the Z resonance constrains the Higgs mass, along

with precision top and W masses. Lowest order SM predicts that MZ = MW/cosW.. Radiative corrections due to loops.

•

• Note the opposite signs of contributions to mass from fermion and boson loops. Crucial for SUSY and radiative stability.

W

W

W

W

b

t

H

W

2 2 2

2

2

cos (1 )

~ [3 ( / ) ] /16

[11 tan / 24 ]ln( / )

W Z W

t W t W

H W W H W

M M

m M

M M

tWtWW dmMmdM )/)(16/3( 2/ [ 11 tan / 48 ]( / )W W W W H HdM M dM M

LEP,CDF D0 Data Indicate Light Higgs – 2011 and Beyond

W t W

b

W H W

W

Quantum mechanics: traces of higher mass states are seen in radiative corrections due to virtual quantum loops, e.g. Lamb shift in atomic spectrum due to virtual e pairs. Note sign – fermion, boson (Quantum Amplitude – phase matters) - SUSY.

Snapshot of H Constraints

Limits are quoted at 95% exclusion. This is ~ ½ the deviation needed to claim a discovery.

Higgs Mass - Upper Limit

•The couplings are a function of the mass scale at which they are probed. We require that (Q) is well behaved from = 174 GeV up to a scale , with 1/ () = 0 (strong coupling at ), the running of includes loops with H and t - with opposite sign.

2 2 2 2 2 2

2 2 2 2 2

( ) ( ) /[1 (3 ( ) /8 ) ln( / 2 )]

1/ ( ) 1/ ( ) (3/8 )[ln( / 2 )]

Q Q

Q Q

Running coupling constants – Lecture IV

Inverse coupling decreases as Q increases. Therefore coupling blows up at some Q

Higgs Mass - Upper Limit

100

105

1010

1015

1020

100

200

300

400

500

600

700

800Upper Limit on Higgs Mass

(GeV)

Hig

gs M

ass(

GeV

)

In quantum field theories the constants are altered in higher order processes (e.g. loops). Asking that the Higgs interactions be well behaved up to a high mass scale (no new Physics) implies a low mass Higgs. Is a high scale plausible?(GUTs ?) Lecture IV

H H H

H

Higgs Decays to Bosons - Direct

•Field excitations interactions with gauge bosons VVH, VVHH, VVV, VVVV due to W, Z fields in the covariant derivative

2( ) / ~ ( /16)( / )H W H WH WW M M M

Higgs couples to mass. Photons and gluons are massless to preserve gauge symmetry unbroken. Thus there is no direct gluon or photon coupling. This is an extension of the W,Z mass with the vev to interactions with the Higgs excitation.

2 2 2 2 22 1 2

0

( )( ) ( ) / 2 ( )( ) / 2

H

H W W H Z ZD D g g g

, ~ gW2 <>[ W W H ] ~ gWMW [ W W H ].

As with W, Z masses and vev, now look at excitation in the Higgs Lagrangian density “kinetic term” and EW gauge replacement

Higgs - Loop Decays to Bosons• There are decay modes that are not accessible through diagrams

with a single coupling constant - zero mass states.

• The coupling is to the heaviest quark in the loop. For a light Higgs, (H --> )/MH ~ 2W/2(MH/MW)2(mt/MH)8 [sin-1(MH/2mt)]4

• (H --> gg)/MH can be estimated by replacing by s. This diagram is the major production mode at the LHC, g + g from p + p fusing into a Higgs boson. The loop integral |I| is ~ O(1).

• For a light Higgs, (H --> gg)/MH ~ [s2W/722](MH/MW)2.

2~ ( / 9)( / )gg W s

COMPHEP does not do loops

Higgs - Production via gg Fusion

• The formation cross section is (Lecture I),

• Using the expression for (H-->gg) and normalizing the gluon distribution with a = 6,

• Note that the MH3 behavior of cancels the 1/ MH

3 behavior of d/dy , leaving a roughly constant cross section,

d/dy ~ 2(Hgg)/(MH3)[xg(x)]x1[xg(x)]x2

d/dy ~ 492[(H gg)/(4MH3)][(1 - MH/s)12] ~ 492(H

gg)/(4MH3)d/dy ~ 49|I|2s

2W/[288MW2].

Higgs Cross SectionCDF and D0 successfully found the top quark, which has a cross section ~ 10-10 the total cross section.

A 500 GeV Higgs has a cross section ratio of ~ 10-11, which requires great rejection power against backgrounds and a high luminosity.

Rate = luminosity * cross sectionLHC at design has ~ 30 times the luminosity of the Tevatron and 7 times the energy.

Higgs Production ModesThe Higgs cross section has as largest contribution g+g with an internal top loop. Note that qqH is quite large, followed by associated production modes including DY production of W, Z with H bremmstrahlung and H with top pairs.

2 2 2( ) 0.4 * 400 *c mb GeV pb TeV

Higgs Decay Rates - Direct and Loop

colortoduellHqqH ),(3)(

2/)()( WWHZZH

2 2 2

2 2 2

( ) ~ / 9( / ) ( / ) | | / 8

( ) ~ / 9( / ) ( / ) | | / 8

W H W s H

W H W H

H gg M M I M

H M M I M

Direct:Quarks and

Leptons2( ) / 8( / )W q W HH qq m M M

Gauge Bosons

2( ) /16( / )W H W HH WW M M M

Loop Decays - Gauge Bosons:

Higgs couples to mass, with no direct H or Hgg coupling. Gauge – Higgs mass squared coupling. fermion coupling – quark mass squared coupling

Higgs Decay Widths

Higgs Branching Ratios

below ZZ “threshold” there is a lZl mode with an “off shell Z”, conventionally called ZZ*. The decay width, Z ~ 2.5 GeV and the Breit-Wigner resonant mass distribution,

2 2 2/ ~ ( / 2) /[( ) ( / 2) ]od dM M M means that the ZZ* decay rate is suppressed by a factor of 2~ [( / 2) /( )]Z ZM M with respect to ZZ decays

Note that q,l width ~ M while W,Z width ~ M3. Hence bb dominates below WW “threshold”. is down by ~ 9 due to coupling to mass, and 1/3 color factor.

Similarly for WW*

Higgs Decay, H -> ZZ ->4

Muons should be the cleanest signal at the LHC:Momentum in tracker * momentum in CSC * match in ( , )

Multiple redundant measurements for rare processes

Use H --> ZZ --> 4e

H natural width is < energy resolution ->Fully active crystals are the best resolution possible -needed for 2 photon decays of the Higgs and Z+Z-> 4 e decays.Z

Z

Higgs couples to mass -> most strongly to W and Z. Next strongest is to heavy quarks such as t, b.

Higgs Strategy and BRH--> is a clean decay mode for low mass Higgs. The ZZ --> 4l mode is clean when it is above threshold at ~ 150 GeV. The dip in ZZ is due to WW rise above threshold at ~ 160 GeV. The WW decay mode does not have a mass peak and is unused save just at threshold, except in the VBF mode where it is a discovery mode.

LEP-II

VBF, H ->WW* -> 2l

VBF

Higgs Decay into Di-PhotonsW,Z

Yang’s theorum, J=1 Cannot decay into 2 photons

t

Higgs Discovery Limits

5

The main final state is ZZ --> 4l.At high masses larger branching ratios are needed.At lower masses the ZZ*, qqWW* and final states are used.LEP II has set a limit ~ 113 GeV.LHC will cover the full range from LEPII to 1 TeV.

1/20 year at design luminosity – CMS and ATLAS are designed to find the Higgs.

Higgs Quantum NumbersIf the Higgs is seen in the 2 photon decay mode , it cannot be a J=1 state (Yang’s theorem). Recall that the 2 photon state is a quantum number filter. A Higgs must have the quantum numbers of the vacuum

Suppose the Higgs is found in the WW decay mode. Look at the spin correlations expected for a 0++ state. The emission of the 2 leptons is then preferentially in the same direction, with small mass.

0PCJ

H --> ZZ --> 4l, Spin and Parity

• Recall the classical pion parity in

• for J = 0 into ZZ, CP requires that S = 0 for the ZZ, with a longitudinal and transverse Z polarization

eeeeo

L

T

2

2

sin

cos1

1 2

1 2

0, n

,

for Jfor P decay pla es aligned

x for P decay planes orthogonal

2

22

)/2(

)2/1/()2/(~)(/)(

HW

LLTT

MM

ZZHZZH

Pion Parity - 1962

Spin of an Enhancement -> Z+Z?

Measure decay angular distribution and extract L and T components

Full Monte Carlo Results0- - decay planes are perpendicular

0+ - decay planes are parallel.

Used to determine the “Higgs” parity in Z+Z decays.

Higgs Partial WidthsFor a light Higgs several partial widths can be determined at the 10-20% level. If the VBF method is successful with the WW and ZZ final state, then gHWW can be determined unambiguously.

EW – W Emission2 2

/

/

~ / 8 (1/ ) ln(4 / )

~ / 4 (1/ )(1 )T

L

q W W W

q W W

f x E M

f x x

2 2 2 2/

2/

2

ˆ( / ) ~ ( / 8 ) (1/ )[ln( / ) ][(2 ) ln(1/ ) 2(1 )(3 )]

( / ) ~ ( / 4 ) (1/ )[(1 ) ln(1/ ) 2(1 )]

ˆ/

T T

L L

qq W W W W

qq W W W

d d s M

d d

M s

Weizacher-Williams approx – virtual W. Source function has coupling strength, EW, and a radiative 1/x behavior. Transverse virtual W dominate. This is like 2 photon physics in electron-positron colliders

Luminosity of transverse W >> that for longitudinal W – but H couples preferentially to longitudinal W. Luminosity of WW in quark- antiquark pair, WW mass M

WW in pp

min

1 1

/ /

1

/

( / ) ( / ) ( / ) ( ) ( / )( / )

/

( ) ( / ) ( )

pp WW q q qq WW

pp WW X pp WW WW X

d d d dx x f x f x d d

s d d d s

Luminosity of WW in pp system and cross section to produce X through VBF in pp reactions.

2 3/

2 2/

~ 16 ( / )( / )

~ / ( / )pp H WW pp WW

pp H W W pp WW

M d d

M d d

VBF of H has a WW width which grows a cube of H mass – cancels the cross section falloff as M cubed. Falloff of H cross section via VBF with energy is slow. At high enough energy VBF is the dominant process.

VBF and Other H Cross Sections

Estimated for 400 GeV Higgs. Some numerical work needed – use the plots in “The Higgs Hunter’s Guide”

“Tag” JetsForward calorimetry needed at the LHC in order to have sufficient hermeticity. Also for good efficiency to detect the “tag” jets – forward going quarks recoiling against an emitted virtual W

Tag Jets and Parity – HWW Coupling

Azimuthal correlation between tag jets

VBF and H Quantum Numbers

The H is a scalar object. That means, in VBF that the “tag jets” reflect the W, and thought of as inverse decays to W+W, their azimuthal correlation makes them back-to back. Other quantum numbers give other patterns.

WW Scattering

Use VBF to study WW scattering at all WW masses? Find HWW coupling constant. If H mass is large, effectively study strong WW scattering as unitarity limit is approached . Can we use VBF to explore strong WW scattering? SM quartic background and t channel H exchange for example.

Higgs Pair Production at SLHC422)( V

2 / 2

2 2 3 4( ) ~ [ ]H H H HV

2 2 22 ( ), (1/ 2) /HM V M

Higgs potentialvev at potential minimumExpand about the minimumAside from mass term for H there are triplet and quartic self couplings

Therefore the Higgs self couplings in the SM are completely specified because the Higgs mass specifies the parameter λ. There are 2 parameters in the potential and μ is fixed by G. Verify the SM by exploring the triple and quartic Higgs self-couplings?

2

2 2 2/ 2

M

M

Higgs – Self-Couplings at the SLHC

Cross section is ~ 20 fb @ 160 GeV H mass. If SLHC is 1000 fb-1/yr, then 20,000 HH produced/yr. There are ~ 400 events with 2 leptons, missing energy and 4 jets(2 W mass peaks in the quark pair spectra).

Final State for Higgs Pairs

Some final states are clean but rare. The ability to definitively find Higgs pairs will require large luminosities – perhaps SLHC….

Problems - III

1. Evaluate the ratio of decay widths of a Higgs boson into W and quark pairs.

2. Use Calchep to compute a “tag jet” cross section - process: u+d->d+u+H.

3. Use Calchep to explore branching fractions, H->2*x for several different masses.

Higgs Coupling to Fermions

][~ fg ],[][~ ff mg

2/)/(

]/2[

WfWf

WWfff

Mmgg

gMggm

• The fermions are left handed weak doublets and right handed singlets. A mass term in the Lagrangian, mff = m(fLfR + fRfL) is then not a weak singlet as is required.

• A Higgs weak doublet is needed, with Yukawa coupling, L ~ gf[(fL)fR + h.c.].

Yukawa

Mass from Dirac Lagrangian density

Fermion weak coupling constant