Standard Model - fma.if.usp.br

Standard Model

Oscar EboliUniversidade de Sao Paulo

Departamento de Fısica [email protected]

December 5, 2006

PBSM-2006 Oscar Eboli

PLAN

➫ I. Low energy processes

➫ II. W Physics

➫ III. Z pole Physics

➫ IV. “Final” remarks

1


I. Low Energy Processes

➡ µ− decay:➡ Fermi’s effective interaction

Leff =4GF√

2(eγαPLνe) (νµγαPLµ)

with GF = 1.166× 10−5 GeV−2

➡ Taking the low energy limit (large MW ) we obtain that

4GF√2≡ 1

2g2

M2W − q2

≈ 12

g2

M2W

=2v2

=⇒ v = (G√

2)−1/2 ' 246 GeV

2


➡ νµ scattering: At low energies we can study the neutral currents andsin θW (= s2

W ) through

νµ e− −→ νµ e− and νµ e− −→ νµ e−

➡ For mf <<√

s << MZ we have

σ(νµe) =G2

Fs

3π

(g2

V + gV gA + g2A

)and σ(νµe) =

G2Fs

3π

(g2

V − gV gA + g2A

)➡ In the SM

gA = −1/2 and gV = −1/2 + 2s2W =⇒ σ(νµe)

σ(νµe)=

3− 12s2W + 16s4

W

1− 4s2W + 16s4

W

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➡ The experimental result is compatible with sin2 θW = 0.2324± 0.0083 CHARM II

➡ Remembering that MW = gv/2 =⇒ MW ' 77 GeV andMZ = MW/cw ' 88 GeV. (not bad)




➡ At low energies the best determination ofsin θW comes from neutrino-nucleon scattering(NC/CC).

sin2 θW = 0.22773± 0.00135± 0.00093




➡ At low energies the best determination ofsin θW comes from neutrino-nucleon scattering(NC/CC).

sin2 θW = 0.22773± 0.00135± 0.00093

➡ At tree level,

sin2 θW = 1− M2W

M2Z

; M2W sin2 θW =

πα√2 GF

; gV = t3` − 2 sin2 θW Q

4


➡ Charged lepton universality can be tested in τ and µ decays

Γτ→ντµ νµ/Γτ→ντe νe Γπ→µ νµ/Γπ→e νe

|gµ/ge| 0.9999± 0.0020 1.0017± 0.0015

Γτ→ντe νe/Γµ→νµe νe Γτ→ντπ/Γπ→µ νµ Γτ→ντK/ΓK→µ νµ

|gτ/gµ| 1.0004± 0.0023 0.9999± 0.0036 0.979± 0.017

Γτ→ντµ νµ/Γµ→νµe νe

|gτ/ge| 1.0002± 0.0022

1τµ

= Γµ =G2

Fm5µ

192 π3f(m2

e/m2µ) (1 + δRC) , f(x) ≡ 1−8x+8x3−x4−12x2 log x

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II. W Physics��

��W decays

➢ The W partial widths are

Γ(W− → νll

−) =GFM3

W

6π√

2,

Γ(W− → uidj

)= NC |Vij|2

GFM3W

6π√

2

➢ The W total width is 2.141± 0.041 GeV and

e µ τ l

Br(W− → νll−) (%) 10.66± 0.17 10.60± 0.15 11.41± 0.22 10.84± 0.09

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PBSM-2006 Oscar Eboli��

��W production in hadron colliders

❆ The reaction is pp→ [qq] →WX→ eνX


��W production in hadron colliders

❆ The reaction is pp→ [qq] →WX→ eνX

❆ We define transverse mass

m2eνT ≡ (EeT + EνT)2 − (~peT + ~pνT)2

❆ In general 0 ≤meνT ≤meν (Proveit!)

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❆ For qq′ → W ∗ → eν there is a Jacobian peak.

dσ

dm2eν,T

∝ ΓWMW

(m2eν −M2

W)2 + Γ2WM2

W

1√m2

eν −m2eν,T

.

❆ rather easy to obtain the W❆ Fitting the data leads to

MW, and, ΓW (hep-ph/0311039)

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� e+e− →W+W−

❇ The W pair production is a nice testof the SM: show WWZ vertex

.

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❇ Angular distributions agree with the SM

10


III. Z pole Physics

➢ At tree level, the Z partial widths are

Γ(Z → ff) = Ncαem

16s2W c2

W

[(gf

V )2 + (gfA)2]MZ

➤ For s2W = 0.23 and MZ = 91.2 GeV we have ΓZ = 2.50 GeV and

Channel Branching ratioνiνi 6.8%`+`− 3.4%

uu (cc) 11.9%dd (ss;bb) 15.1%hadrons 69%

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➤ The born amplitude for e+e− → ff is

Mborn '1sJα

γe ⊗ Jγfα +

1s−M2

Z + isΓZ/MZJα

Ze ⊗ JZfα

➤ The behaviour of the σ(e+e−) is

➤ It is interesting to notice that

σ(e+e− → visible)σQED(e+e− → µ+µ−)

∣∣∣∣peak

' 4300

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➣ LEP I worked for several years with√

s close to MZ. 107 Z ’s wereproduced. Huge impact.

13


➣ The basic observables are

? Mass of the Z MZ

? Hadronic peak cross section σh

? Partial widths into fermions Γf

? Total width ΓZ


➣ The basic observables are

? Mass of the Z MZ

? Hadronic peak cross section σh

? Partial widths into fermions Γf

? Total width ΓZ

? Total invisible width Γinv

? Asymmetries Ae,µ,b,cFB

? Polarization P τ (= σR−σLσR+σL

)

➣ Basic relations/definitions

Γh = Γu + Γd + Γc + Γs + Γb ; σh = 12π ΓeΓh

M2ZΓ2

Z;

Γinv = ΓZ − Γe − Γµ − Γτ − Γh ; R` = ΓhΓ`

; Rb,c = Γb,c

Γh

14


➣ Defining

Af =2gf

V gfA

(gfV )2 + (gf

A)2

the asymmetries on the Z peak are

AfFB =

34AeAf

P f = −Af

➣ gV depends on sin2 θW =⇒ we can measure this parameter.

15


➤ SM is a predictive theory

➤ Large Z sample =⇒ tree level expressions are not enough

➤ Possible set of input parameters

GF = (1.166 37± 0.000 01) · 10−5 GeV−2 ;α−1 = 137.035 999 11± 0.000 000 46 ;MZ = (91.1875± 0.0021) GeV

➤ Direct measurements =⇒ MW = 80.403± 0.029 GeV, however,

sin2 θW = 1−M2W

M2Z

; M2W sin2 θW =

πα√2 GF

; =⇒ MpredictedW = 80.939 GeV

➤ We need to calculate beyond tree level

16


➤ At the Born level, several measurements allow to evaluate sin2 θW

sin2 θW = 1− M2W

M2Z

σ(νµe)σ(νµe)

=3− 12s2

W + 16s4W

1− 4s2W + 16s4

W

πα√2 GF

1M2

W

= s2W

Γ(Z → ff)MZ

= Ncαem

16s2W c2

W

[(gf

V )2 + (gfA)2]

P τ = − 2gτV gτ

A

(gτV )2 + (gτ

A)2

➤ In fact, these observables receive different quantum corrections =⇒ wemust be careful analyzing precision measurements!

17


➤ For instance νµe scattering ismodified by


➤ For instance νµe scattering ismodified by while the shifts in MW and MZ are

due to

but not by the first diagram.

➤ Moreover, radiative correction introduces the masses and couplings of theparticles running in the loop into the game.

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� Basic Parameters

➥ In order to make prediction we must elect a set of independent parameters,for instance the on mass shell scheme:

? Electromagnetic coupling constant αem

? Fermion masses mf

? Boson masses MW , MZ, MH

➥ In this scheme sin2 θW is a derived quantity:

sin2 θW ≡ 1− M2W

M2Z

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��One-loop µ decay

➥ We should add the two-, three-, andfour-point function corrections

➥ The propagator corrections dominatethe quantum corrections.

nada

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➥ Quantum corrections modify the relation between GF , MW , and sW :

GF√2

=πα

2s2WM2

W

[1 + ∆r(α, MW ,MZ,mf)]

➥ It can be shown that

∆r = ∆α− c2W

s2W

∆ρ + ∆rrem

nada


➥ Quantum corrections modify the relation between GF , MW , and sW :

GF√2

=πα

2s2WM2

W

[1 + ∆r(α, MW ,MZ,mf)]

➥ It can be shown that

∆r = ∆α− c2W

s2W

∆ρ + ∆rrem

nada

∆α is the running of α between q2 = 0and M2

Z. “Evaluating”

we get Teubner

α(M2Z) =

α

1−∆α=

1128.914± 0.029

21


➥ The factor ∆ρ is obtained from

−igµνΣW =


➥ The factor ∆ρ is obtained from

−igµνΣW =

∆ρ =ΣZ(0)M2

Z

− ΣW (0)M2

W

' Ncα

16πs2W c2

W

m2t

M2Z

' 0.0094

22


��Z effective couplings nada

➦ Strategy: we compute the decay Z → ff starting from the renormalized1PI functions and then add the correction to the γ and Z propagators.

➦ The final result, after massaging the expressions, is

JµZ =

(√2GM2

Z

)1/2 √ρf ⊗

[(T f

3 − 2Qfs2Wκf

)γµ − T f

3 γµγ5]

➦ Quantum corrections introduce a normalization factor ρf and an effectivemixing angle s2

Wκ.

23


➦ Gauge-boson self-energycontributions are universal. Usuallythey are the largest ones.

nada

➦ The leading universal contributionsare

(∆ρf)univ ' ∆ρ

(∆κf)univ ' c2W

s2W

∆ρ

that are proportional to m2t/M

2Z.

➦ With these form factors the one-loop Z contribution to e+e− → ff (up toboxes) is

Jµe

1k2 + M2

Z + isΓZ/MzJf

µ

details

24


��Comparison with data ➦ SM has been tested with success at 0.1% level

25


➧ The line shape =⇒ MZ, ΓZ, and Nν = 3 (light)


➧ The line shape =⇒ MZ, ΓZ, and Nν = 3 (light)

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➧ The data is consistent with lepton universality and EW loop corrections


➧ The data is consistent with lepton universality and EW loop corrections

27


➧ b and c properties:


➧ b and c properties:

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➨ ρf versus sin2 θefff for b, c and leptons couplings

29


➨ All available EW data is rather consistent

30


➧ However there is some tension on the data


➧ However there is some tension on the data

31


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IV. “Final” Remarks

✪ What we know:

L = Lfkinetic + LGB

kinetic + Lffv + Lvvv + Lvvvv + LEWSB

✪ SU(3)c × SU(2)L ×U(1)Y gauge interaction between fermions and gaugebosons tested at 0.1% level.

✪ Some information on the interactions between the gauge bosons: V V Vcouplings tested at 1–10% level =⇒ SM is doing well . (future)

✪ LEWSB has not been directly tested: origin of masses, flavor physics, . . .

33


✪ SU(2)×U(1) symmetry is broken:

.

• Without EWSB =⇒ fermions are massless

• QCD still confines =⇒ p, n, . . . with some changes

• Mp > Mn (QED corrections)

• rapid decay of p into n changing completely the world: noatoms, etc

✪ The EWSB sector has been elusive, but not for long!

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Limitations of the SM

➲ Even after the Higgs discovery there are unanswered questions:

.

• what is the origin of fermion masses?

• do interactions unify at high energies?

• what is the dark matter?

• why is the cosmological constant so small?

• what is the dark energy?

• what is the origin of baryon asymmetry? . . .

➲ The SM also has some technical problems: (hierarchy problem)

35


.

• Quantum corrections drive scalar masses to high scale∆M2

h ∝ Λ2UV

• This requires new physics in the TeV scale

• There are many solutions, pointing in different directions

• Supersymmetry• Higgs is composite∗ technicolor∗ H is a Goldstone boson (little Higgs)

• Extra spatial dimensions∗ Large ED∗ Warped ED (Randall-Sundrum)∗ Universal extra dimensions . . .

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➤ There are many topics left behind, among them,

❆ Higgs properties [Rosenfeld]

❆ QCD [Rosenfeld]

❆ Neutrino Physics [Nunokawa]

❆ Flavor Physics [Lavignac]

37


� References

➤ A. Pich, hep-ph/0502010

➤ LEPEWWG, hep-ex/0509008,

http://www.cern.ch/LEPEWWG

➤ Matchev, hep-ph/0402031

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��Details: Z effective couplings ponto

➦ The 1PI 3-point function Z–f–f can be written as (neglecting the fermionmass)

VZff = ie

sW cW

[γµ(F f

V − F fAγ5)

]➦ We can also resum the Z (γ) propagators

GZZ(γγ) =1

k2 −M2Z(γ) + ΣZ(γ)

➦ We can also resum the Z–γ 2 point function

GZγ = −GZZ ΣZγ Gγγ

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' −GZZΣZγ

k2

➦ Contracting GZγ with the fermions via em coupling leads to the contribution

TZγ =

(i

e

2sW cW2QfsW cW

ΣZγ

k2γµ

)GZZ

➦ We should also contract GZZ with the fermionic current

TZ =[i

e

2sW cWγµ(gf

V − gfAγ5)

]GZZ

➦ The effective vertex is given by(VZff GZZ + TZ + TZγ

)⊗ Z

−1/2Z

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where ZZ is the Z wave function renormalization factor.

➦ Now writing

k2 −M2Z + ReΣZ = (k2 −M2

Z) (1 + ΠZZ)

we have

ZZ =1

1 + ΠZZ

➦ The final result, after massaging the expressions is

JµZ =

(√2GM2

Z

)1/2 √ρf ⊗

[(T f

3 − 2Qfs2Wκf

)γµ − T f

3 γµγ5]

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➦ The form factors are

ρf = 1 −∆r − ΠZZ +2FF

A

gfA

κf = 1 −cW

sW

ΣZγ

k2+

12s2

WQf

(F f

A

gfV

gfA

− F fV

)

back

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��Future of EW precison measuments nada

❅ Presently mtop = 172.5± 2.3 GeV❅ MW = 80.404± 0.030 GeV

❅ How well should we know MW and mtop?

∆theo δ(∆αhad) = 0.00016 ∆mtop = 2 GeV ∆mtop = 1 GeV

∆MW /MeV 6 3.0 12 6.1

∆ sin2 θlepteff

× 105 4 5.6 6.1 3.1

❅ ∆mtop ' 1 GeV and ∆MW ' 10 MeV is desirable back

❅ MW and mtop similar uncertainties to fits =⇒ ∆MW ' 7× 10−3 ∆Mtop

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