The Standard Model of Particle Physics: Status & Low-Energy Tests

Abstract

Precision measurements of low-energy observables provide stringent tests of the Standard Model structure and accurate determinations of its parameters. An overview of the present experimental status is presented. The main topics discussed are the muon anomalous magnetic moment, the asymptotic freedom of strong interactions, the lepton universality of gauge couplings, the quark flavour structure and CP violation. Comment: 12 pages, 6 Postscript Figures

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arXiv:hep-ph/0206011v1 3 Jun 2002
The Standard Model of Particle Physics:
Status & Low-Energy Tests
Antonio Pich
IFIC, Universitat de Val` encia – CSIC, Apt. 22085, E-46071 Val` encia, Spain
Abstract. Precision measurements of low-energy observables provide stringent tests
of the Standard Model structure and accurate determinations of its parameters. An
overview of the present experimental status is presented. The main topics discussed are
the muon anomalous magnetic moment, the asymptotic freedom of strong interactions,
the lepton universality of gauge couplings, the quark flavour structure and CP violation.
Invited talk at the ESO–CERN–ESA Symposium on Astronomy, Cosmology
and Fundamental Physics (Garching bei M¨ unchen, Germany, March 2002)
1 Standard Model Structure
The Standard Model (SM) [1,2] is a gauge theory, based on the group SU(3)C⊗
SU(2)L⊗U(1)Y, which describes strong, weak and electromagnetic interactions,
via the exchange of the corresponding spin-1 gauge fields: 8 massless gluons and
1 massless photon for the strong and electromagnetic forces, respectively, and
3 massive bosons, W±and Z, for the weak interaction. The gauge symmetry
determines the dynamics in terms of the three couplings gs, g and g′, associ-
ated with the SU(3)C, SU(2)L and U(1)Y subgroups. Strong interactions are
governed by the first group factor, while the other two provide a unified descrip-
tion of the electroweak forces, their gauge parameters being related through
g sinθW = g′cosθW= e.
The fermionic matter content is given by the known leptons and quarks,
which are organized in a 3-fold family structure:
?νe u
e−d′
?
,
?νµ c
µ−s′
?
,
?ντ t
τ−b′
?
,(1)
where (each quark appears in 3 different colours)
?νl qu
l−qd
?
≡
?νl
l−
?
L
,
?qu
qd
?
L
,l−
R,(qu)R,(qd)R,(2)
plus the corresponding antiparticles. Thus, the left-handed fields are SU(2)L
doublets, while their right-handed partners transform as SU(2)Lsinglets. The
three fermionic families in (1) appear to have identical properties (gauge inter-
actions); they only differ by their mass and their flavour quantum number.
The gauge symmetry is broken by the vacuum, which triggers the Sponta-
neous Symmetry Breaking (SSB) of the electroweak group to the electromagnetic
subgroup:
SU(3)C⊗ SU(2)L⊗ U(1)Y
SSB
−→ SU(3)C⊗ U(1)QED.(3)

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2Antonio Pich
The SSB mechanism generates the masses of the weak gauge bosons, and gives
rise to the appearance of a physical scalar particle, the so-called Higgs. The
fermion masses and mixings are also generated through the SSB mechanism.
The SM constitutes one of the most successful achievements in modern
physics. It provides a very elegant theoretical framework, which is able to de-
scribe all known experimental facts in particle physics. A detailed description of
the SM and its impressive phenomenological success can be found in [3,4].
2 Quantum Corrections
The high accuracy achieved by the most recent experiments allows to make strin-
gent tests of the SM structure at the level of quantum corrections. The follow-
ing discussion concentrates on Quantum Electrodynamics (QED) and Quantum
Chromodynamics (QCD). Electroweak effects are covered in [5].
2.1Running Couplings
Let us consider the electromagnetic interaction between two electrons. At lowest
order, the scattering amplitude T(q2) ∼ α/q2with α = e2/(4π). The leading
quantum corrections are generated by the photon self-energy contribution:
T(Q2) ∼
α
Q2
?1 − Π(Q2) + Π(Q2)2+ ···?
=
α
Q2
1
1 + Π(Q2)
∼
α(Q2)
Q2
.
This defines an effective running coupling,
α(Q2) =
α(Q2
0)
1 −β1
2πα(Q2
0) ln(Q2/Q2
0)
,(4)
where Q2≡ −q2and α(m2
with β1 = 2/3 > 0. Therefore, the effective QED running coupling increases
with the energy scale: α(Q2) > α(Q2
charge decreases at large distances. This can be intuitively understood as the
screening due to virtual e+e−pairs generated, through quantum effects, around
the electron charge. The physical QED vacuum behaves as a polarized dielectric
e) = α. The e+e−loop induces a logarithmic correction
0) if Q2> Q2
0, i.e., the electromagnetic
e–
e–
e–
e–
γ
q
=
+
+
+. . .
Fig.1. Photon self-energy contribution to e−e−scattering

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The Standard Model of Particle Physics: Status & Low-Energy Tests3
12510
µ (GeV)
20 50100200
0.0
0.1
0.2
0.3
0.4
αs(µ)
Fig.2. Energy dependence [10] of the strong coupling αs
medium. The huge difference between the electron and Z mass scales makes this
quantum correction relevant at LEP energies [6,7]:
α(m2
e)−1= 137.03599976(50) > α(M2
Z)−1= 128.95± 0.05 .(5)
The strong interaction between two quarks can be analyzed in a similar way.
Owing to the non-abelian character of the SU(3)C group, QCD leads to cubic
and quartic self-interactions among gluons. This results in a strong running
coupling αs(Q2) with the same Q2dependence (4), but with a negative β1[8]:
β1 =
2Nf− 11NC
6
< 0 .(6)
The contribution proportional to the number of quark flavours Nf is generated
by the q-¯ q loop corrections to the gluon self-energy. The gluonic self-interactions
introduce the additional negative term proportional to the number of quark
colours NC. Since β1 < 0, αs(Q2) decreases at short distances. Thus, QCD
has the required property of asymptotic freedom: quarks behave as free particles
when Q2→ ∞. The predicted running of αs, known to four loops [9], agrees very
well with the experimental determinations at different energies. Normalizing all
measurements at the Z mass scale, the present world average is [10]:
αs(M2
Z) = 0.118 ± 0.002 .(7)
2.2Lepton Anomalous Magnetic Moments
The most stringent QED test [11,12] comes from the high-precision measure-
ments of the e [10] and µ [13] anomalous magnetic moments aγ
l≡ (gl−2)/2:
(Theory)
(Experiment)
aγ
e=
?(115965215.35± 2.40) × 10−11
(115965218.69± 0.41) × 10−11
,(8a)

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4Antonio Pich
(a)?(b)?
(c)?(d)?
ν?
W?W?
γ , Z?
γ?
f?
f?
Fig.3. Some Feynman diagrams contributing to aγ
l
aγ
µ=
?(1165917.9± 1.0) × 10−9
(1165920.3± 1.5) × 10−9
(Theory)
(Experiment)
.(8b)
The impressive agreement between theory and experiment promotes QED to the
level of the best theory ever build by the human mind to describe nature.
To a measurable level, aγ
earises entirely from virtual electrons and photons;
these contributions are known [11] to O(α4). The theoretical error is dominated
by the uncertainty in the input value of the QED coupling α. Turning things
around, aγ
eprovides the most precise determination of the fine structure constant.
The anomalous magnetic moment of the muon is sensitive to virtual contribu-
tions from heavier states; compared to aγ
oretical uncertainty on aγ
µhas a QCD origin. Since quarks have electric charge,
virtual quark-antiquark pairs induce hadronic vacuum polarization corrections
to the photon propagator (Fig. 3.c). Owing to the non-perturbative character of
QCD at low energies, the light-quark contribution cannot be reliably calculated
at present; fortunately, this effect can be extracted from the measurement of the
cross-section σ(e+e−→ hadrons) and from the invariant-mass distribution of
the final hadrons in τ decays [6]. Additional QCD uncertainties stem from the
smaller light-by-light scattering contributions (Fig. 3.d); a recent reevaluation of
these corrections [14] has detected a sign mistake in previous calculations [15],
improving the agreement with the experimental measurement [13].
The Brookhaven E821 experiment [13] is expected to push its sensitivity to
at least 4 × 10−10, and thereby observe the contributions from virtual W±and
Z bosons [12,16]. This would require a better control of the QCD corrections.
e, they scale as m2
µ/m2
e. The main the-
3Lepton Universality
In the SM all lepton doublets have identical couplings to the W boson:
L =
g
2√2W†
µ
?
l
¯ νlγµ(1 − γ5)l + h.c.(l = e, µ, τ) .(9)
Comparing the measured decay widths of leptonic or semileptonic decays which

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The Standard Model of Particle Physics: Status & Low-Energy Tests5
Table 1. Experimental determinations of the ratios gl/gl′
Γτ→ντµ ¯ νµ/ντe ¯ νe
Γπ→µ ¯ νµ/e ¯ νe
ΓW→µ ¯ νµ/e ¯ νe
|gµ/ge| =1.0006 ± 0.00211.0017 ± 0.00151.000 ± 0.011
Γτ→ντe ¯ νe/Γµ→νµe ¯ νe
Γτ→ντπ/Γπ→µ ¯ νµ
Γτ→ντK/ΓK→µ ¯ νµ
ΓW→τ ¯ ντ/µ ¯ νµ
|gτ/gµ| =0.9995 ± 0.00231.005 ± 0.0070.977 ± 0.016 1.026 ± 0.014
Γτ→ντµ ¯ νµ/Γµ→νµe ¯ νe
ΓW→τ ¯ ντ/e ¯ νe
|gτ/ge| =1.0001 ± 0.00231.026 ± 0.014
-0.041
-0.038
-0.035
-0.032
-0.503-0.502-0.501-0.5
gAl
gVl
Preliminary
68% CL
l+l−
e+e−
µ+µ−
τ+τ−
mt
mH
∆α
Fig.4. Contours of 68% probability in the al-vl plane from LEP and SLD measure-
ments [17]. The solid contour assumes lepton universality. The shaded region corre-
sponds to the SM prediction for mt = 174.3 ± 5.1 GeV and mH = 300+700
−186GeV.
only differ by the lepton flavour, one can test experimentally that the W inter-
action is indeed the same, i.e. that ge= gµ = gτ ≡ g. As shown in Table 1,
the present data [4,10,17] verify the universality of the leptonic charged-current
couplings to the 0.2% level.
The interactions of the neutral Z boson are diagonal in flavour. Moreover, all
fermions with equal electric charge have identical axial-vector, af= Tf
3= ±1/2,