Physics of vacuum polarization ... Lectures on the physics of vacuum polarization: from GeV to TeV scale [email protected]www-com.physik.hu-berlin.de/∼fjeger/QCD-lectures.pdf F. J EGERLEHNER University of Silesia, Katowice DESY Zeuthen/Humboldt-Universität zu Berlin Lectures , November 9-13, 2009, FNL, INFN, Frascati supported by the Alexander von Humboldt Foundaion through the Foundation for Polish Science and by INFN Laboratori Nazionale di Frascati F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy – November 9-13, 2009 –
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Physics of vacuum polarization ...
Lectures on the physics of vacuum polarization: from GeV to T eV scale
The domain 0 < s < 4m2ℓ is unphysical. A look at the unitarity condition
iT ∗
if − Tfi
= Σ
∫
n
(2π)4 δ(4)(Pn − Pi) T∗nfTni ,
which derives from unitarity of S and the definition of the T –matrix, taking 〈f |S+S|i〉 and inserting a complete
set of intermediate states, tells us that for s < 4m2ℓ there is no physical state |n〉 allowed by energy and
momentum conservation and thus
Tfi = T ∗if for s < 4m2
ℓ ,
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
which means that the current matrix element is hermitian. As the electromagnetic potential Aextµ (x) is real its
Fourier transform satisfies
Aextµ (−q) = A∗ ext
µ (q)
and hence
J µfi = J µ∗
if for s < 4m2ℓ .
If we interchange initial and final state the four–vectors p1 and p2 are interchanged such that q changes sign:
q → −q. The unitarity relation for the form factor decomposition of u2 Πµγℓℓ u1 thus reads (ui = u(pi, ri))
u2
(
γµFE(q2) + iσµν qν2m
FM(q2))
u1
=
u1
(
γµFE(q2)− iσµν qν2m
FM(q2))
u2
∗
= u+2
(
γµ+F ∗E(q2) + iσµν+ qν
2mF ∗
M(q2))
u+1
= u2
(
γµF ∗E(q2) + iσµν qν
2mF ∗
M(q2))
u1 .
The last equality follows using u+2 = u2γ
0, u+1 = γ0u1, γ
+5 = γ5, γ
0γ5γ0 = −γ5, γ
0γµ+γ0 = γµ
and γ0σµν+γ0 = σµν . Unitarity thus implies that the form factors are real
Im F (s)i = 0 for s < 4m2e
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
below the threshold of pair production s = 4m2e. For s ≥ 4m2
e the form factors are complex; they are analytic in
the complex s–plane with a cut along the positive axis starting at s = 4m2e. In the annihilation channel
(p− = p2, p+ = −p1)
〈0|jµem(0)|p−, p+〉 = Σ
∫
n
〈0|jµem(0)|n〉〈n|p−, p+〉 ,
where the lowest state |n〉 contributing to the sum is an e+e− state at threshold : E+ = E− = me and
~p+ = ~p− = 0 such that s = 4m2e. Because of the causal iε–prescription in the time–ordered Green functions
the amplitudes change sign when s→ s∗ and hence
Fi(s∗) = F ∗
i (s) ,
which is the Schwarz reflection principle.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
5. Dispersion Relations
Causality together with unitarity imply analyticity of the form factors in the complex s–plane except for the cut
along the positive real axis starting at s ≥ 4m2ℓ . Cauchy’s integral theorem tells us that the contour integral, for
the contour C shown in the Figure, satisfies
Fi(s) =1
2πi
∮
C
ds′F (s′)
s′ − s .
Im s
Re s
CR
|s0
Analyticity domain and Cauchy contour C for the lepton form factor (vacuum polarization). C is a circle of radius R
with a cut along the positive real axis for s > s0 = 4m2 where m is the mass of the lightest particles which can
be pair–produced
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
Since F ∗(s) = F (s∗) the contribution along the cut may be written as
limε→0
(F (s+ iε)− F (s− iε)) = 2 i Im F (s) ; s real , s > 0
and hence for R→∞
F (s) = limε→0
F (s+ iε) =1
πlimε→0
∞∫
4m2
ds′Im F (s′)
s′ − s− iε+ C∞ .
In all cases where F (s) falls off sufficiently rapidly as |s| → ∞ the boundary term C∞ vanishes and the integral
converges. This may be checked order by order in perturbation theory. In this case the “un–subtracted” dispersion
relation (DR)
F (s) =1
πlimε→0
∞∫
4m2
ds′Im F (s′)
s′ − s− iε
uniquely determines the function by its imaginary part. A technique based on DRs is frequently used for the
calculation of Feynman integrals, because the calculation of the imaginary part is simpler in general. The real part
which actually is the object to be calculated is given by the principal value (P ) integral
Re F (s) =1
πP∞∫
4m2
ds′Im F (s′)
s′ − s ,
which is also known under the name Hilbert transform.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
For our form factors the fall off condition is satisfied for the Pauli form factor FM but not for the Dirac form factor
FE. In the latter case the fall off condition is not satisfied because FE(0) = 1 (charge renormalization condition
= subtraction condition). However, performing a subtraction of FE(0), one finds that (FE(s)− FE(0))/s
satisfies the “subtracted” dispersion relations
F (s)− F (0)
s=
1
πlimε→0
∞∫
4m2
ds′Im F (s′)
s′(s′ − s− iε),
which exhibits one additional power of s′ in the denominator and hence improves the damping of the integrand at
large s′ by one additional power. Order by order in perturbation theory the dispersion integral is convergent for the
Dirac form factor. A very similar relation is satisfied by the vacuum polarization amplitude which we will discuss in
the following section.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
6. Dispersion Relations and the Vacuum Polarization
+ −
+−+
−
+−
+−
+− +−
+−
+−
+−
+−
+−
+−
+−
+−
+ −r
−
−
Vacuum polarization causing charge screening by virtual pair creation and re–annihilation
Dispersion relations play an important role for taking into account the photon propagator contributions. The
related photon self–energy, obtained from the photon propagator by the amputation of the external photon lines, is
given by the correlator of two electromagnetic currents and may be interpreted as vacuum polarization for the
following reason:
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
The em Ward-Takahashi identity (current conservation) in QED reds Zf Γµ(p, p)|on−shell = −eγµ or
−eγµδZf + Γ′µ(p, p)
∣∣∣on−shell
= 0
where the prime denotes the non–trivial part of the vertex function. This relation tells us that some of the diagrams
directly cancel. For example, we have (V = γ)
Vγ
+ 12
V+ 1
2 V= 0
implies that in QED (not in the SM) charge renormalization is caused solely by the photon self–energy correction;
the fundamental electromagnetic fine structure constant α in fact is a function of the energy scale α→ α(E) of
a process due to charge screening. The latter is a result of the fact that a naked charge is surrounded by a cloud
of virtual particle–antiparticle pairs (dipoles mostly) which line up in the field of the central charge and such lead to
a vacuum polarization which screens the central charge. This is illustrated in the figure. From long distances
(classical charge) one thus sees less charge than if one comes closer, such that one sees an increasing charge
with increasing energy.
Diagrammatic representation of a vacuum polarization effect:
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
γ virtual
pairs
γ
µ− µ−
µ− µ−
γ∗ → e+e−, µ+µ−, τ+τ−, uu, dd, · · · → γ∗
Feynman diagram describing the vacuum polarization in muon scattering
The vacuum polarization affects the photon propagator in that the full or dressed propagator is given by the
geometrical progression of self–energy insertions −iΠγ(q2). The corresponding Dyson summation implies that
the free propagator is replaced by the dressed one
iDµνγ (q) =
−igµν
q2 + iε→ iD
′µνγ (q) =
−igµν
q2 + Πγ(q2) + iε
modulo unphysical gauge dependent terms. By U(1)em gauge invariance the photon remains massless and
hence we have Πγ(q2) = Πγ(0) + q2 Π′γ(q2) with Πγ(0) ≡ 0. As a result we obtain
iD′µνγ (q) =
−igµν
q2 (1 + Π′γ(q2))
+ gauge terms
where the “gauge terms” will not contribute to gauge invariant physical quantities, and need not be considered
further.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
Including a factor e2 and considering the renormalized propagator (wave function renormalization factor Zγ ) we
have
i e2 D′µνγ (q) =
−igµν e2 Zγ
q2(1 + Π′
γ(q2)) + gauge terms
which in effect means that the charge has to be replaced by a running charge
e2 → e2(q2) =e2Zγ
1 + Π′γ(q2)
.
The wave function renormalization factor Zγ is fixed by the condition that at q2 → 0 one obtains the classical
charge (charge renormalization in the Thomson limit. Thus the renormalized charge is
e2 → e2(q2) =e2
1 + (Π′γ(q2)−Π′
γ(0))
where the lowest order diagram in perturbation theory which contributes to Π′γ(q2) is
γ γf
f
and describes the virtual creation and re–absorption of fermion pairs
γ∗ → e+e−, µ+µ−, τ+τ−, uu, dd, · · ·→ γ∗.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
In terms of the fine structure constant α = e2
4π reads
α(q2) =α
1−∆α; ∆α = −Re
(Π′
γ(q2)− Π′γ(0)
).
The various contributions to the shift in the fine structure constant come from the leptons (lep = e, µ and τ ) the 5
light quarks (u, d, s, c, and b and the corresponding hadrons = had) and from the top quark:
∆α = ∆αlep + ∆(5)αhad + ∆αtop + · · ·
Also W–pairs contribute at q2 > M2W . While the other contributions can be calculated order by order in
perturbation theory the hadronic contribution ∆(5)αhad exhibits low energy strong interaction effects and hence
cannot be calculated by perturbative means. Here the dispersion relations play a key role.
The leptonic contributions are calculable in perturbation theory. Using our result for the renormalized photon
self–energy, at leading order the free lepton loops yield
∆αlep(q2) =
=∑
ℓ=e,µ,τ
α3π
[
− 53 − yℓ + (1 + yℓ
2 )√
1− yℓ ln(
|√
1−yℓ+1√1−yℓ−1
|)]
=∑
ℓ=e,µ,τ
α3π
[
− 83 + β2
ℓ + 12βℓ(3− β2
ℓ ) ln(
| 1+βℓ
1−βℓ|)]
=∑
ℓ=e,µ,τ
α3π
[ln(|q2|/m2
ℓ
)− 5
3 +O(m2
ℓ/q2)]
for |q2| ≫ m2ℓ
≃ 0.03142 for q2 = M2Z
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
where yℓ = 4m2ℓ/q
2 and βℓ =√
1− yℓ are the lepton velocitiesa. This leading contribution is affected by small
electromagnetic corrections only in the next to leading order. The leptonic contribution is actually known to three
loops at which it takes the value
∆αleptons(M2Z) ≃ 314.98 × 10−4.
As already mentioned, in contrast, the corresponding free quark loop contribution gets substantially modified by
low energy strong interaction effects, which cannot be calculated reliably by perturbative QCD. The evaluation of
the hadronic contribution will be discussed later.
Vacuum polarization effects are large when large scale changes are involved (large logarithms) and because of
the large number of light fermionic degrees of freedom as we infer from the asymptotic form in perturbation theory
∆αpert(q2) ≃ α
3π
∑
f
Q2fNcf
(
ln|q2|m2
f
− 5
3
)
; |q2| ≫ m2f .
The figure illustrates the running of the effective charges at lower energies in the space–like region. Typical values
are ∆α(5GeV) ∼ 3% and ∆α(MZ) ∼ 6%, where about ∼ 50% of the contribution comes from leptons andaThe final result for the renormalized photon vacuum polarization for a lepton of mass m then reads
Π′γ ren(q2) =
α
3π
5
3+ y − 2 (1 +
y
2) (1 − y)G(y)
ff
with y = 4m2/q2 and G(y) = 12√
1−yln
√1−y+1√1−y−1
.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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about ∼ 50% from hadrons. Note the sharp increase of the screening correction at relatively low energies.
Shift of the effective fine structure constant ∆α as a function of the energy scale in the space–like region q2 < 0
(E = −√
−q2). The vertical bars at selected points indicate the uncertainty
Note alternative interpretation of VP: photon self–energy vacuum expectation value of the time ordered
product of two electromagnetic currents:
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
→ ⊗ ⊗
One may represent the current correlator as a Källen-Lehmann representation in terms of spectral densities. To
this end, let us consider first the Fourier transform of the vacuum expectation value of the product of two currents.
Using translation invariance and inserting a complete set of states n of momentum pnb, satisfying the
completeness relation
∫d4pn
(2π)3Σ
∫
n
|n〉〈n| = 1
where∫P
n includes, for fixed total momentum pn, integration over the phase space available to particles of all
bNote that the intermediate states are multi–particle states, in general, and the completeness integral includes an integration over p0n, since
pn is not on the mass shell p0n 6=p
m2n + ~p2n. In general, in addition to a possible discrete part of the spectrum we are dealing with a
continuum of states.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
possible intermediate physical states |n〉, we have
i
∫
d4x eiqx〈0|jµ(x) jν(0)|0〉
= i
∫d4pn
(2π)3
∫
d4x ei(q−pn)x Σ
∫
n
〈0|jµ(0)|n〉〈n|jν(0)|0〉
= i
∫d4pn
(2π)3Σ
∫
n
(2π)4 δ(4)(q − pn)〈0|jµ(0)|n〉〈n|jν(0)|0〉
= i 2π Σ
∫
n
〈0|jµ(0)|n〉〈n|jν(0)|0〉|pn=q .
In our case, for the conserved electromagnetic current, only the transversal amplitude is present and, taking into
account the physical spectrum condition p2 ≥ 0, p0 ≥ 0 we may write the Källen-Lehmann representation
i
∫
d4x eiqx〈0|Tjµem(x) jν
em(0)|0〉
=
∞∫
0
dm2 ρ(m2)(m2 gµν − qµqν
) 1
q2 −m2 + iε
= −(q2gµν − qµqν
)Π′
γ(q2)
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
where Π′γ(q2) up to a factor e2 is the photon vacuum polarization function introduced beforec :
Π′γ(q2) = e2Π′
γ(q2) .
With this bridge to the photon self–energy function Π′γ we can get its imaginary part by substituting
1
q2 −m2 + iε→ −π i δ(q2 −m2)
. Thus contracting with 2Θ(q0)gµν and dividing by gµν(q2 gµν − qµqν) = 3q2 we obtain
2Θ(q0) Im Π′γ(q2) = Θ(q0) 2π ρ(q2)
= − 1
3q22π Σ
∫
n
〈0|jµem(0)|n〉〈n|jµ em(0)|0〉|pn=q .
Again causality implies analyticity and the validity of a dispersion relation. In fact the electromagnetic currentcIn case of a conserved current, where ρ0 ≡ 0, we may formally derive that ρ1(s) is real and positive ρ1(s) ≥ 0. To this end we consider
the element ρ00
ρ00(q) = Σ
Z
n
〈0|j0(0)|n〉〈n|j0(0)|0〉˛
˛
q=pn
= Σ
Z
n
|〈0|j0(0)|n〉|2q=pn≥ 0
= Θ(q0) Θ(q2) ~q2 ρ1(q2)
from which the statement follows.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
correlator exhibits a logarithmic UV singularity and thus requires one subtraction such that we find
Π′γ(q2)− Π′
γ(0) =q2
π
∞∫
0
dsIm Π′
γ(s)
s (s− q2 − iε).
Here, we need the optical theorem which derives from the unitarity of S translated to T –matrix elements
iT ∗
if − Tfi
= Σ
∫
n
(2π)4 δ(4)(Pn − Pi) T∗nfTni ,
and the optical theorem, is obtained from this relation in the limit of elastic forward scattering |f〉 → |i〉 where
2 Im Tii = Σ
∫
n
(2π)4 δ(4)(Pn − Pi) |Tni|2 .
Graphically, this relation may be represented by
=∑
n
2 Im
A, p1
B, p2
A, p1
B, p2
=∑
n
2 ImA, p A, p
Optical theorem for scattering and propagation.
It tells us that the imaginary part of the photon propagator is proportional to the total cross section
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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σtot(e+e− → γ∗ → anything) (“anything” means any possible state). The precise relationship reads
Im Π′γ(s) =
1
12πR(s)
Im Π′γ(s) = e(s)2 Im Π′
γ(s) =s
e(s)2σtot(e
+e− → γ∗ → anything) =α(s)
3R(s)
where
R(s) = σtot/4πα(s)2
3s.
The normalization factor is the point cross section (tree level) σµµ(e+e− → γ∗ → µ+µ−) in the limit
s≫ 4m2µ. Taking into account the mass effects the R(s) which corresponds to the production of a lepton pair
reads
Rℓ(s) =
√
1− 4m2ℓ
s
(
1 +2m2
ℓ
s
)
, (ℓ = e, µ, τ)
which may be read of from the imaginary part. This result provides an alternative way to calculate the
renormalized vacuum polarization function, namely, via the DR which now takes the form
Π′ℓγ ren(q2) =
αq2
3π
∫ ∞
4m2ℓ
dsRℓ(s)
s(s− q2 − iε)
yielding the vacuum polarization due to a lepton–loop.
In contrast to the leptonic part, the hadronic contribution cannot be calculated analytically as a perturbative series,
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
but it can be expressed in terms of the cross section of the reaction e+e− → hadrons, which is known from
experiments. Via
Rhad(s) = σ(e+e− → hadrons)/4πα(s)2
3s.
we obtain the relevant hadronic vacuum polarization
Π′hadγ ren(q2) =
αq2
3π
∫ ∞
4m2π
dsRhad(s)
s(s− q2 − iε).
At low energies, where the final state necessarily consists of two pions, the cross section is given by the square of
the electromagnetic form factor of the pion (undressed from VP effects),
Rhad(s) =1
4
(
1− 4m2π
s
) 32
1|F (0)π (s)|2 , s < 9m2
π ,
which directly follows from the corresponding imaginary part of the photon vacuum polarization. There are three
differences between the pionic loop integral and those belonging to the lepton loops:
the masses are different
the spins are different
the pion is composite – the Standard Model leptons are elementary
The compositeness manifests itself in the occurrence of the form factor Fπ(s), which generates an enhancement:
at the ρ peak, |Fπ(s)|2 reaches values about 45, while the quark parton model would give about 7. The
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
remaining difference in the expressions for the quantities Rℓ(s) and Rh(s), respectively, originates in the fact
that the leptons carry spin 12 , while the spin of the pion vanishes. Near threshold, the angular momentum barrier
suppresses the function Rh(s) by three powers of momentum, while Rℓ(s) is proportional to the first power. The
suppression largely compensates the enhancement by the form factor – by far the most important property is the
mass.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
Compilation of αs measurements (Bethke 04) .
αs(Mτ ) = 0.322± 0.030 at Mτ = 1.78 GeV
pQCD not only fails due to strong
coupling (non-convergence of expansion)
but because of spontaneous chiral symmetry
breaking (100% non-perturbative)
responsible for the existence of pions and quark
condensates, which are missing to all orders in
pQCD, i.e. pQCD fails to correctly describe
the low energy structure of QCD
Must use non-perturbative methods:
Dispersion relations, sum rules
Chiral perturbation theory extended to include spin 1 vecto r states
QCD inspired models: extended Nambu Jona-Lasinio model (EN JL), hidden local
symmetry (HLS) model
large Nc QCD approach (dual to infinite series of narrow vector resona nces)
lattice QCD in future
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
Low energy: the ρ peak
“Outstanding” non-perturbative low energy hadron spectru m: e+e− → π+π−
A precise new KLOE measurement of |Fπ|2 with ISR events
KLOE
SND
CMD-2
10
20
30
40
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|F¼|2
M¼¼
2 2GeV( )
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
ππ event with KLOE detector.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
7. Hadron-production in e+e−–annihilation
at dawn of QCD: 1974 hadron physics in crisis
since 1969: e+e− → hadrons cross-sections with storage ring ADONE at Frascati much too high for
theoretical expectations at that time.
These experiments measured the ratio
R(s) =σtot(e
+e− → γ∗ → hadrons)
σ(mµ=0)µµ
of the hardonic cross section in units of the µ+µ− pair production cross section, and R values in the range from
1 up to 6 were actually measured.
in 1973: R measured at the Cambridge Electron Accelerator (CEA) at Harvard: R ∼ 5[6] at 4[5] GeV
London Conference in 1974: conclusion R raising smoothly form 2 at 2 GeV to about 6 at 5 GeV
on the theory side confusing: about 22 different models were proposed, among them also
QCD [Fritzsch+Leutwyler], as a solution of the “unitarity crisis”.
Unitarity predicts a decrease of the total cross–section like 1/s up to logs at high energies.
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The ratio R as of July 1974.
The solution was QCD, providing a factor 3 from color degrees of freedom, and a new species of quark charm,
which completed the 2nd family of fermions.
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Chronology of e+e− facilities
Year Accelerator Emax (GeV) Experiments Laboratory
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
Some considerations on hadron production at low energy
Certainly, quark-antiquark pair production is far from describing experimental cross–section data at low energies,
where “infrared slavery” (confinement) is the dominating feature.
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Physics of vacuum polarization ...
The Fig. shows a compilation of the low energy data in terms of the pion form factor. The latter is defined by
σππ(s) =πα2
3s(βπ)3|Fπ(s)|2 or |Fπ(s)|2 = 4Rππ(s) (βπ)−3 .
also low energy effective theory of QCD: chiral perturbation theory (CHPT) fails (has no ρ)
looks natural to apply vector-meson dominance (VMD); proper QCD implementation
Resonance Lagrangian Approach (RLA)
in fact photon mixes with hadronic vector–mesons like the ρ0
also nearby resonances like ρ0 and ω are mixing (distorting Breit-Wigner shape)
Pions and Scalar QED
Pions can be seen in a particle detector behave like point particles (relatvistic Wigner state)
on pion level coupling to photon fixed by gauge invariance
charged pion field = complex scalar field ϕ free Lagrangian
L(0)π = (∂µϕ)(∂µϕ)∗ −m2
πϕϕ∗
via minimal substitution ∂µϕ→ Dµϕ = (∂µ + ieAµ(x)) ϕ, which replaces the ordinary by the covariant
derivative, and which implies the scalar QED (sQED) Lagrangian
LsQEDπ = L(0)
π − ie(ϕ∗∂µϕ− ϕ∂µϕ∗)Aµ + e2gµνϕϕ
∗AµAν .
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In sQED the contribution of a pion loop to the photon VP is given by
−i Πµν (π)γ (q) = + .
and the renormalized transversal photon self–energy reads
Π′ (π)γren(q2) =
α
6π
1
3+ (1− y)− (1− y)2 G(y)
where y = 4m2/q2 and G(y) was given before. For q2 > 4m2 there is an imaginary or absorptive part given
by substituting G(y)→ Im G(y) = − π2√
1−ysuch that
Im Π′ (π)γ (q2) =
α
12(1− y)3/2
and for large q2 is 1/4 of the corresponding value for a lepton. According to the optical theorem the absorptive
part may be written in terms of the e+e− → γ∗ → π+π− production cross–section σπ+π−(s) as
Im Π′ hadγ (s) =
s
4πασhad(s)
which hence we can read off to be
σπ+π−(s) =πα2
3sβ3
π
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βπ =√
(1− 4m2π/s) pion velocity sQED predicts
Fπ(s) = 1 independent of s ,
in view of the ρ-resonance sitting there a complete failure up to 50 : 1 results.
Thus sQED only works in conjunction with vector meson dominance model (VDM) or improvements of it
e→ e Fπ(q2), e2 → e2 |Fπ(q2)|2
taking care of bound-state nature of pion (→ form-factor)
Mandatory constraint: electromagnetic current conservation↔ Fπ(0) = 1
Still sQED used in controlling real photon radiation issues: final state radiation (FSR) (Bloch-Nordsieck
prescription), Kinoshita-Lee-Nauenberg theorem and all that.
Urgently need precise experimental investigation of FSR from hadrons (Venanzoni et al.)
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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The Vector Meson Dominance Model
phenomenological fact: photon and ρ meson exhibit direct coupling↔ γ − ρ0–mixing
implementation: VMD model and extensions like RLA
naive VMD model: replacing the photon propagator as
i gµν
q2+ · · · → i gµν
q2+ · · · −
i (gµν − qµqν
q2 )
q2 −m2ρ
=i gµν
q2m2
ρ
m2ρ − q2
+ · · · ,
where the ellipses stand for the gauge terms.
no change as q2 → 0
changes asymptotics 1/q2 → 1/q4
directly exhibits ρ0-resonance
However, the naive VMD model does not respect chiral symmetry properties.
More precisely, VMD relates
hadronic part of the electromagnetic current jhadµ (x)
source density J (ρ)(x) of the neutral vector meson ρ0 by
〈B|jhadµ (0)|A〉 = −
M2ρ
2γρ
1
q2 −M2ρ
〈B|J (ρ)µ (0)|A〉
where q = pB − pA, pA and pB the four momenta of the hadronic states A and B, respectively, Mρ is the
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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mass of the ρ meson. So far our VMD ansatz only accounts for the isovector part, but the isoscalar contributions
mediated by the ω and the φ mesons may be included in exactly the same manner:
=∑
V =ρ0, ω, φ,···
B
A
B
AV
γγ
=M2
V
2γV; =
−1
q2 −M2V
.
The vector meson dominance model. A and B denote hadronic states.
The key idea is to treat the vector meson resonances like the ρ as elementary fields in a first approximation. Free
massive spin 1 vector bosons are described by a Proca field Vµ(x) satisfying the Proca equation
(2 +M2V ) Vµ(x)− ∂µ (∂νV
ν) = 0, which is designed such that it satisfies the Klein-Gordon equation and at
the same time eliminates the unwanted spin 0 component: ∂νVν = 0. In the interacting case this equation is
replaced by a current–field identity (CFI)
(2 +M2V ) Vµ(x)− ∂µ (∂νV
ν) = gV J (V )µ (x)
where the r.h.s. is the source mediating the interaction of the vector meson and gV the coupling strength. The
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
Physics of vacuum polarization ...
current should be conserved ∂µJ(V )µ (x) = 0. The CFI then implies
〈B|Vµ(0)|A〉 = − gV
q2 −M2V
〈B|J (V )µ (0)|A〉
where terms proportional to qµ have dropped due to current conservation. The VMD assumes that the hadronic
electromagnetic current is saturated by vector meson resonancesa
jhadµ (x) =
∑
V =ρ0, ω, φ,···
M2V
2γVVµ(x)
in particular
〈ρ(p)|jhadµ (0)|0〉 = ε(p, λ)µ
M2ρ
2γρ, p2 = M2
ρ .
M2V required for dimensional reasons
γV defines coupling constant (convention)
VMD relation derives from the CFI and ansatz
The VMD model is known to describe the gross features of the electromagnetic properties of hadrons quite well,aIn largeNc QCD all hadrons become infinitely narrow, since all widths are suppressed by powers of 1/Nc, and the VMD model becomes
exact with an infinite number of narrow vector meson states. The large-Nc expansion attempts to approach QCD (Nc = 3) by an expansion
in 1/Nc . In leading approximation in the SU(∞) theory R(s) would have the form
R(s) =9π
α2
∞X
i=0
Γeei Mi δ(s−M2
i ) .
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most prominent example are the nucleon form factors.
A way to incorporate vector–mesons ρ, ω, φ, . . . in accordance with the basic symmetries of QCD is the
Resonance Lagrangian Approach (RLA), an extended version of CHPT, which also implements VMD in a
consistent manner. In the flavor SU(3) sector, similar to the pseudoscalar field Φ(x), the SU(3) gauge bosons
conveniently may be written as a 3× 3 matrix field
Vµ(x) =∑
i
TiVµi =
ρ0
√2
+ ω8√6
ρ+ K∗+
ρ− −ρ0
√2
+ ω8√6
K∗0
K∗− K∗0 −2 ω8√
6
µ
in order to keep track of the appropriate SU(3) weight factors.
ρ0 − γ –Mixing
The unstable spin 1 vector mesons are described by a propagator exhibiting a pole
M2ρ ≡
(q2)
pole= M2
ρ − i Mρ Γρ
in the complex q2-plane with correspondence
physical mass ⇐⇒ real part of location of propagator pole
width ⇐⇒ imaginary part of the location of the pole .
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The simple relation between the full propagator and the irreducible self-energy only holds if there is no mixing, like
for the charged ρ±. In the neutral sector, because of γ − ρ0 mixing, we cannot consider the ρ0 and γ
propagators separately. They form a 2× 2 matrix propagator, so that the pole condition is modified into
sP −m2ρ0 − Πρ0ρ0(sP )−
Π2γρ0(sP )
sP −Πγγ(sP )= 0 ,
with sP = M2ρ0
b.
Note: such mixing is not present in the charged channel (e.g. in τ–decay)
bThe simplest way to treat this problem is to start from the inverse propagator given by the irreducible self-energies (sum of 1pi diagrams).
Again we restrict ourselves to a discussion of the transverse part and we take out a trivial factor −i gµν in order to keep notation as simple as
possible. With this convention we have for the inverse γ − ρ propagator the symmetric matrix
D−1 =
0
@
k2 + Πγγ(k2) Πγρ(k2)
Πγρ(k2) k2 −M2ρ + Πρρ(k2)
1
A
Using 2 × 2 matrix inversion
M =
0
@
a b
b c
1
A ⇒M−1 =1
ac− b2
0
@
c −b
−b a
1
A
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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Vector–Meson Production Cross–Sections
The cross–section for spin 1 boson production in e+e−–annihilation is described in full detail in Sect. 14 of my
Lausanne Lectures. For massive spin 1 boson the polarization vectors ε(p, λ) satisfy the completeness relation
∑
λ
ε(p, λ)µ ε∗(p, λ)ν =
(
−gµν −pµpν
M2ρ
)
and the hadronic tensor reads
hµν =1
3
∑
λ
ε(p, λ)µ
M2ρ
2γρε∗(p, λ)ν
M2ρ
2γρ=
(
−gµν −pµpν
M2ρ
)M4
ρ
2γ2ρ
,
and one easily calculates the spin average |T |2.
The result for the e+e− → ρ0 cross–section and some useful approximations are the following:
we find for the propagators
Dγγ =1
k2 + Πγγ(k2) −Π2
γρ(k2)
k2−M2ρ+Πρρ(k2)
≃1
k2 + Πγγ(k2)
Dγρ =−Πγρ(k2)
(k2 + Πγγ(k2))(k2 −M2ρ + Πρρ(k2)) − Π2
γρ(k2)≃
−Πγρ(k2)
k2 (k2 −M2ρ )
Dρρ =1
k2 −M2ρ + Πρρ(k2) −
Π2γρ(k2)
k2+Πγγ(k2)
≃1
k2 −M2ρ + Πρρ(k2)
.
These expressions sum correctly all the reducible bubbles. The approximations indicated are the one-loop results. The extra terms are higher
order contributions.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Fras cati, Italy – November 9-13, 2009 –
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Breit-Wigner resonance: field theory version
The field theoretic form of a Breit-Wigner resonance obtained by the Dyson summation of a massive spin 1
transversal part of the propagator in the approximation that the imaginary part of the self–energy yields the width
by Im ΠV (M2V ) = MV ΓV near resonance.
σBW (s) =12π
M2R
Γe+e−
Γ
sΓ2
(s−M2R)2 +M2
RΓ2.
Breit-Wigner resonance
The resonance cross–section from a classical non–relativistic Breit-Wigner resonance is given by
σBW (s) =3π
s
ΓΓe+e−
(√s−MR)2 + Γ2
4
.
Narrow width resonance
The narrow width approximation for a zero width resonance reads
σNW(s) =12π2
MRΓe+e−δ(s−M2
R) .
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More realistic refined approaches including isospin breaking from md 6= mu and multiple resonances
Gounaris-Sakurai model
Resonance Lagrangian models
Analytic approach à la Omnès (Colangelo, Leutwyler et al.)
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