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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
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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
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➡ 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)
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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
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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 tree level,
sin2 θW = 1− M2W
M2Z
; M2W sin2 θW =
πα√2 GF
; gV = t3` − 2 sin2 θW Q
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➡ 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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��W production in hadron colliders
❆ The reaction is pp→ [qq] →WX→ eνX
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��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
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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.
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➣ The basic observables are
? Mass of the Z MZ
? Hadronic peak cross section σh
? Partial widths into fermions Γf
? Total width ΓZ
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➣ 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
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➣ 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.
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➤ 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
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➤ 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!
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➤ For instance νµe scattering ismodified by
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➤ 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
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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
∆α is the running of α between q2 = 0and M2
Z. “Evaluating”
we get Teubner
α(M2Z) =
α
1−∆α=
1128.914± 0.029
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➥ The factor ∆ρ is obtained from
−igµνΣW =
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➥ The factor ∆ρ is obtained from
−igµνΣW =
∆ρ =ΣZ(0)M2
Z
− ΣW (0)M2
W
' Ncα
16πs2W c2
W
m2t
M2Z
' 0.0094
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��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.
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➦ 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
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��Comparison with data ➦ SM has been tested with success at 0.1% level
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➧ The line shape =⇒ MZ, ΓZ, and Nν = 3 (light)
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➧ The line shape =⇒ MZ, ΓZ, and Nν = 3 (light)
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➧ The data is consistent with lepton universality and EW loop corrections
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➧ The data is consistent with lepton universality and EW loop corrections
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➧ b and c properties:
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➧ b and c properties:
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➨ ρf versus sin2 θefff for b, c and leptons couplings
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➨ All available EW data is rather consistent
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➧ However there is some tension on the data
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➧ However there is some tension on the data
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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, . . .
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✪ 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)
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.
• 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]
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� 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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