Ann. Henri Poincaré 4, Suppl. 1 (2003) S247 – S258
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S247-12
DOI 10.1007/s00023-003-0920-3
Annales Henri Poincaré
Progress in Study of Renormalization of Theories
with Nontrivial Internal Symmetry
A.A. Slavnov
Abstract. Problems arising in renormalization of theories with nontrivial internal
symmetry are reviewed. Different approaches to renormalization are discussed. A
new method which provides automatically gauge invariance of renormalized theory
independently of regularization used is proposed.
1 Introduction
Renormalization is an essential part of quantum field theory (QFT). Historically
renormalization procedure for QFT was invented to deal with ultraviolet infinities
which appear in calculations of radiative corrections in quantum electrodynamics
(QED). The early development of renormalization theory is related to the works
of V.Weisskopf and H.Bethe who pointed out that at least some of these divergencies may be reabsorbed in unobservable ”bare” parameters (mass, charge) which
characterize fictitious noninteracting particles. This idea was developed into consistent theory by J.Schwinger, I.Tomonaga, R.Feynman and F.Dyson. Finally the
complete and mathematically rigorous renormalization procedure was constructed
by N.N.Bogoliubov and O.S.Parasiuk. They showed that renormalization is a way
to define a product of distributions describing the particle propagation and constructed the recurrent subtraction procedure (R-operation), which allows to write
for an arbitrary Feynman diagram a well defined convergent expression by subtracting the first terms of Taylor series in external momenta for the corresponding
integrals.
In more intuitive terms renormalization may be described as follows. One
firstly introduces some intermediate regularization (ultraviolet cut-off Λ) to make
all the integrals convergent. At the same time one changes the parameters of ”bare”
particles by introducing Λ-dependent counterterms into Lagrangian
L = −Z3Λ Fµν Fµν + iZ2Λ ψ̄γµ (∂µ − ieAµ )ψ + Z2Λ (m + δmΛ )ψ̄ψ .
(1)
Renormalization theory states that for a special choice of dependence of Z2 , Z3 , δm
on Λ the limit Λ → ∞ exists for all integrals relevant for calculations of observables.
This limit is defined up to certain finite terms, which are fixed by assigning to
particles physical masses, charges and normalizing properly the wave functions.
Having an analogous procedure for the Standard Model describing weak,
electromagnetic and strong interactions, one can calculate unambiguously all ra-
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diative corrections using as the input the renormalized Lagrangian with physical
masses and charges taken from experiment (for the Standard Model 18 parameters
are needed).
One may ask, what is the physical reason for renormalization in QFT? It is
related to impossibility to decouple naively the regions of low and high energies.
Indeed, if the theory were free of ultraviolet divergencies, the influence of very
heavy states on low energy processes would be practically absent. The contribution
of such states to arbitrary Feynman diagram would be proportional to
(2)
f (k1 . . . kn )(ki2 − Λ2 )−1 dk1 . . . dkn
and if the integral converges, for Λ big enough it is negligible. Although decoupling
of different energy regions is not uncommon in physics it does not work in QFT
as in this case the physical ground state is a complicated superposition of virtual
particle states, including heavy excitations. Renormalization theory allows to decouple the low and high energy regions at the expence of redefining the ”bare”
parameters and taking physical values of masses and charges from experiment.
One could think that renormalization is a specific feature of perturbation
theory and may be avoided in nonperturbative calculations. However this hope is
false. Nonperturbative calculations in lattice field theories also require renormalization and tuning the parameters of the Lagrangian.
Renormalization is intrinsically connected with our unability to describe in a
proper way the region of very high energies. Hence a renormalized local quantum
field theory is an effective theory, which is designed to describe physics below
certain ultraviolet scale Λ. The influence of the energies higher than Λ is taken
into account by means of renormalization of bare parameters.
The only hope to avoid ultraviolet infinities and renormalization is to construct a theory which is valid for all energies. Such a theory must include some dimensional parameters and naturally cannot be strictly local. All previous attempts
to construct a consistent nonlocal theory failed leading to violation of basic physical principles like causality or unitarity. At present great expectations are related
to string models and their generalizations leading to hypothetical M -theory, which
is supposed to be the theory of everything. In string and brain models nonlocality
is build in in a natural way and on the other hand the low energy limit of these
models reproduces correctly the relevant field theory models with reasonable particle spectrum. It is possible that if a consistent M -theory is developed it would
allow to avoid renormalization and calculate unambiguously all physical quantities. However at present these hopes are far from being well founded and there
are much more questions than answers in this approach. In particular in my opinion nobody has studied seriously the problem of causality in string models. Most
probably local quantum field theory and its necessary ingredient, renormalization
procedure, in a foreseen future will remain the main tool of studying elementary
particle physics.
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2 Symmetry and renormalization
One of the main problems in renormalization theory is a compatibility of renormalization procedure with the symmetry of the theory. At present the main guiding
principle for the choice of a particular model of interactions is a symmetry, especially gauge invariance. Following the classical example of electrodynamics and
pioneering works of Yang and Mills, practically all modern models are chosen
on the basis of requirement of gauge invariance. Requirement of invariance with
respect to local phase transformations of matter fields
ψ(x) → Ω(x)ψ(x) = exp{iT aαa (x)}ψ(x)
(3)
leads to existence of the vector field Bµ (x) = T a Bµa (x), where T a are generators
of the corresponding Lie algebra, with the transformation law
Bµ (x) → Ω(x)Bµ (x)Ω−1 (x) + ∂µ ΩΩ−1 (x) .
(4)
The interaction with the matter field is introduced via replacement of derivatives
in the free action by covariant derivatives
Lψ = iψ̄γµ (∂µ − igΓ(Bµ ))ψ .
(5)
The gauge invariant Yang-Mills Lagrangian is constructed in terms of the curvature
tensor
1
1
LY M = − Tr F µν Fµν = − Tr ( ∂µ Aν − ∂ν Aµ + g[Aµ , Aν ])2 .
8
8
(6)
The construction presented above may be described by the ”equivalence principle”
similar to the equivalence principle in general relativity: A local change of the basis
in the internal charge space, described by the eq.(3), is equivalent to appearance
of additional gauge field Bµ . If one accepts this principle as a guide for model
building, it fixes practically uniquely the interaction Lagrangian.
The physical meaning of gauge invariance is simple: some components of vector fields which mediate the interactions are unphysical and may be chosen at will
in accordance with gauge invariance. In particular, using the gauge freedom one
may eliminate unphysical components completely. Such ”physical” gauges have
a transparent physical interpretation, however they are usually inconvenient for
practical calculations due to lack of manifest Lorentz invariance. In QED observable electromagnetic field is described by the stress tensor Fµν
Fi0 = Ei ;
ijk F jk = H i
(7)
which depends only on transversal components of vector potential. The longitudinal components ∂i Ai are unphysical. The physical gauge in this case is the
Coulomb gauge ∂i Ai = 0, eliminating unphysical longitudinal components.
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Gauge invariance guarantees that observables, i.e. scattering matrix elements
do not depend on a gauge. Instead of the Coulomb gauge one can choose some
Lorentz invariant gauge condition like ∂µ Aµ = 0, or modify the classical action as
follows
1
1
(∂µ Aµ )2 + iψ̄γµ (∂µ − ieAµ )ψ .
Lα = − Fµν Fµν −
(8)
4
2α
Due to gauge invariance
SC = Sα .
(9)
The r.h.s. of this equation is manifestly Lorentz invariant, whereas the l.h.s. depends only on physical variables and is manifestly unitary. Hence we conclude that
observables, like S-matrix share both these properties. Gauge invariance is essential for consistency of QED and non-Abelian gauge models. Its breaking would
lead to violation of unitarity, Lorentz invariance or some other important physical
properties.
The question arises: does the renormalization procedure respect gauge invariance? As was discussed above, the renormalization is equivalent to some modification of a classical Lagrangian. So it is not obvious that a renormalized theory
possesses necessary symmetry. Moreover it was found that not all symmetries of
a classical theory survive renormalization. For some models it is impossible to
carry out renormalization preserving gauge invariance and these models are physically inconsistent. So the question of a gauge invariant renormalization is of great
importance.
Renormalization procedure cannot be formulated only in terms of observables. It necessary includes correlation functions
< T ψ̄(x)ψ(y)Aµ (z) . . . >. These functions are not gauge invariant. However gauge
invariance imposes the conditions on these functions which are necessary and sufficient for invariance of observables. The physical meaning of these conditions is
rather transparent: the correlators including unphysical degrees of freedom, like
∂µ Aµ may be expressed in terms of correlators depending only on physical components. In QED such relations were obtained long ago by J.Ward [1], and in a
more general form by E.S.Fradkin [2] and Y.Takahashi [3].
The simplest identities are:
α−1 ∂µx
∂µx < T Aµ (x)Aν (y) > =
0
< T Aµ (x)ψ̄(y)ψ(z) > =
e < T ψ̄(x)ψ(z) > δ(x − y)
−
e < T ψ̄(y)ψ(x) > δ(x − z) .
(10)
One sees that the correlators including unphysical longitudinal photons are expressed in terms of physical correlators.
The eqs (10) in QED are a consequence of the classical conservation law for
the electric current
∂µ (ψ̄γµ ψ) = 0 .
(11)
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In non-Abelian gauge theories this simple interpretation does not work. The reason
is that the gauge fixing in Yang-Mills theory breaks the conservation of the classical
current. Moreover the consistent quantization of Yang-Mills theory requires further
modification of the classical action: new unphysical fields, Faddeev-Popov ghosts,
appear. So the complete quantized action looks as follows
1
1 a a
(∂µ Aµ )2 − i∂µ c̄Dµ c
Fµν −
L = − Fµν
4
2α
(12)
Here c̄, c are anticommuting scalar fields. Nevertheless the relations which express
gauge invariance in terms of correlators may be written in this case as well [4],
[5]. These relations have the same physical meaning, expressing the correlators
including unphysical components in terms of physical ones. However in this case
unphysical components include not only longitudinal polarizations of vector fields
but also Faddeev-Popov ghosts. In particular the identity for the three point function including the longitudinal component of the Yang-Mills field expresses it in
terms of the two-point Yang-Mills correlator, two-point correlator of ghost fields
and ghost-Yang-Mills field vertex:
iα−1 < T Bµa (x)Bνb (y)∂ρ Bρc (z) > = < T ∂µx c̄a (x)cc (z)Bνb (y) >
+ gtade < T Bµd (x)c̄e (x)cc (z)Bνb (y) > + . . .(13)
where . . . denotes analogous terms with x → y, µ → ν, a → b. This identity
together with the corresponding identities for two and four point correlators guarantees gauge independence of renormalized S-matrix.
Contrary to the QED case the identity (13) does not correspond to the conservation of the classical current. Nevertheless it is related to some global symmetry
of the effective action, including a gauge fixing term and ghost fields. For the
Yang-Mills theory the symmetry transformations look as follows [6], [7]:
δAaµ
=
(Dµ c)a δca
=
tabd cb cd a
=
(∂µ Aµ )a .
δc̄
(14)
Here is a constant with odd Grassmanian parity, anticommuting with the ghost
fields. The transformation (14) is therefore a supersymmetry transformation.
The identities (13)may be interpreted as a quantum analogue of the conservation law corresponding to the BRST invariance of the effective action.
The identities (13) or, alternatively, BRST-invariance are the necessary and
sufficient conditions of gauge invariant renormalizability and therefore must be
respected by a renormalization procedure for the gauge theory to be consistent.
This raises some problems, both technical and conceptual.
A renormalization procedure usually includes two elements. One firstly introduces an intermediate regularization which makes all the integrals describing
radiative corrections finite. Then the renormalized correlators are constructed by
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subtracting according to R-operation the first terms of Taylor expansion of corresponding integrals, or, equivalently, by introducing suitable Λ-dependent counterterms Zi (Λ). The finite renormalized correlators obtained in this way must satisfy
the Generalized Ward Identities (GWI) (13).
Obviously these identities can be violated by an ultraviolet regularization. If
these violations may be compensated by a suitable choice of local counterterms
it does not lead to inconsistency. However in some models it is impossible. For
example in ”chiral QED”, described by the Lagrangian
1
L = − fµν fµν + iψ̄γµ (∂µ − ieγ5 Aµ )ψ
4
(15)
the three point correlation function must satisfy the identity
∂µ < T Aµ (x)Aν (y)Aρ (z) >= 0 .
(16)
Explicit calculation gives for the Fourier transform of this correlator a finite
nonzero result
g 3 νραβ α β
p k .
(17)
(p + k)µ Γµνρ (p, k) =
8π
This is a simplest example of a quantum anomaly. Classical chiral symmetry is
destroyed by quantum corrections. Similar anomalies are present in conformally
invariant theories and some other models whose symmetry is related to the absence
of dimensional parameters. Renormalization introduces a dimensional parameter
which causes the violation of the classical symmetry.
Breaking of the gauge invariance leads to inconsistency of the theory. Unphysical degrees of freedom do not decouple from physical ones, resulting in violation
of unitarity.
There exists a possibility that new degrees of freedom which appear in anomalous models may have a physical meaning and the quantum theory is consistent,
but has more degrees of freedom than the classical one. It was demonstrated explicitly in some simple model. However no nontrivial four-dimensional models of
this type are known. At present a physical meaning of quantum anomalies is not
completely understood and their absence is considered as a criterion for choosing
self consistent models.
3 Different renormalization schemes.
Even for anomaly free models gauge invariant renormalization is not a trivial problem and its efficient solution is important in particular for practical calculations
in the framework of the Standard model and for the complete analysis of different
supersymmetric models.
There are two main approaches to renormalization of theories with nontrivial
internal symmetry.
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The first approach involves a gauge invariant intermediate regularization,
so that regularized correlators satisfy GWI. In this case one can prove that the
counterterms needed to eliminate ultraviolet divergencies also preserve the structure of the effective action and the gauge invariance is manifest at all stages of
calculations.
The most natural and conceptually simple method of regularization is provided by a lattice regularization. Continuum space-time is replaced by a discrete
lattice with the lattice spacing a. The inverse of a, Λ = a−1 plays a role of ultraviolet cut-off. Wilson [8] showed that discretization can be introduced in a
way compatible with gauge invariance. This approach is successfully used for nonperturbative calculations in the Standard Model with the help of Monte-Carlo
method. However for perturbative calculations it is not convenient due to a complicated structure of the regularized action and breaking of Lorentz invariance.
Additional complications arise in the theories with chiral fermions, in particular
in Salam-Weinberg model.
The second gauge invariant method is the regularization by means of higher
covariant derivatives [9], [10]. The classical action is modified by introducing the
terms with higher covariant derivatives
1 a a
Fµν + iψ̄γµ Dµ ψ −
LR = − Fµν
4
1
i
− 2 Dα Fµν Dα Fµν + 2 Dα ψ̄γµ Dµ Dα ψ .
4Λ
Λ
(18)
Introduction of derivatives into kinetic part suppresses the ultraviolet asymptotics
of propagators
1
1
→ 2
(19)
k2
k − Λ−2 k 4
improving the convergence of Feynman integrals. However requirement of gauge
invariance forces us to introduce covariant derivatives resulting in appearance of
new vertices. The analysis shows that this regularization is not complete. It makes
convergent all the diagrams except for a finite number of one loop diagrams, thus
reducing the problem to analysis of a finite number of relatively simple integrals.
This regularization, although incomplete, has an advantage of being applicable to
any gauge invariant theory. Its existence proves in a trivial way the absence of
anomalies in multiloop diagrams: if the anomalies are absent in one loop diagrams
they cannot appear in multiloop diagrams which allow universal gauge invariant
regularization.
The one loop diagrams require a special treatment. For the Yang-Mills theory
a complete manifestly gauge invariant regularization was constructed by using an
additional Pauli-Villars type regularization for one loop diagrams [11], [12].
This method may be extended to regularize in a gauge invariant way anomaly
free models with chiral fermions, in particular Weinberg-Salam model. PauliVillars type regularization cannot be applied directly to such models as the fermion
mass term breaks chiral invariance. However compensation of anomalies in lepton
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and quark sectors leads to the absence of γ5 matrix in all divergent one-loop diagrams, which coincide up to the factor 1/2 with the corresponding diagrams in
a vectorial theory [13]. It allows to use for their regularization the Pauli-Villars
procedure. This procedure may be also implemented as a modification of the Lagrangian by introducing an infinite series of auxiliary fields [13]. This approach is
closely related to treating chiral fermions with the help of auxiliary fifth dimension
[14]. It got an important development in lattice gauge theories, resulting in the
construction of the overlap method [15] used for nonperturbative calculations on
a lattice.
The method of higher covariant derivatives is a powerful tool for a general
analysis of renormalized gauge theories, but calculations in perturbation theory
are rather complicated due to the presence of additional interaction vertices.
The most economical method of gauge invariant regularization in the framework of perturbation theory is the dimensional regularization [16]. This method is
based on the observation that for some dimensions of the space-time the Feynman
integrals are convergent and the Lorentz invariant amplitudes may be considered
as meromorphic functions of space-time dimension d having poles at d = 4. For
the values of d where all integrals are convergent the GWI are fulfilled and the
poles at d = 4 form a gauge invariant structure. To renormalize the theory in a
gauge invariant way it is sufficient to drop the pole terms when taking the limit
∞. The dimensional regularization
is equivalent to the formal change of
d →
d4 k1 . . . d4 kl f (k1 . . . kl ) by dn k1 . . . dn kl f (k1 . . . kl ), and it preserves the structure of Feynman integrals which makes it convenient for practical calculations.
It requires however a proper definition of tensor objects like Dirac γ-matrices.
There is no consistent definition of γ5 matrix in arbitrary dimension, compatible
with chiral invariance. So the dimensional regularization breaks gauge invariance
in models with chiral fermions, which include such important theories as SalamWeinberg model and supersymmetric theories.
The second approach to renormalization of theories with nontrivial internal
symmetry is the algebraic renormalization [6]. It is a two step procedure. One
firstly defines finite correlation functions by using some, not necessary symmetry
preserving, subtraction scheme. At the second step the freedom in the choice of
finite counterterms is used to satisfy GWI or similar symmetry relations. The
big advantage of this method is its universality. It may be applied to arbitrary
quantum field theory model and used with arbitrary regularization procedure.
In particular some supersymmetric models for which the problem of invariant
regularization is not solved were considered in this framework [17], [18]. It also was
applied to the analysis of the Standard Model [19], [20]. However, from the point
of view of practical calculations this method is rather complicated. It requires
introduction of most general noninvariant counterterms which in some models
(e.g. supersymmetric) have a complicated structure. Some method which shares
universality of algebraic renormalization and manifest symmetry of gauge invariant
regularization schemes would be welcome.
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Renormalization of Theories with Nontrivial Internal Symmetry
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4 Universal invariant renormalization
In this section I describe a special subtraction procedure which may be used with
arbitrary regularization scheme and incorporates automatically Generalized Ward
Identities, hence guaranteeing the symmetry of renormalized theory [21], [22]. The
main idea can be explained using as an example QED.
The correlation functions in QED must satisfy the Ward Identities
pµ Πrµν (p) = 0 ,
(20)
∂Σr (p)
,
∂pµ
(21)
pµ Πrµνρσ (p, q, k) = 0 .
(22)
Γrµ (p, 0) = e
Here Π denote purely photonic amplitudes, and Γ is the electron-photon vertex.
Our goal is to write down explicit expressions for renormalized correlators which
satisfy WI for arbitrary intermediate regularization.
For the vertex function such expression looks as follows
Λ
Γrµ (p, q) = lim [ΓΛ
µ (p, k) − Γµ (p, 0) + e
Λ→∞
∂Σr (p)
].
∂pµ
(23)
Here Σr (p) is the electron self energy renormalized at will. The WI is satisfied
trivially
∂Σr
Γrµ (p, 0) = e
.
(24)
∂pµ
Λ
As ΓΛ
µ (p, k) in the limit Λ → ∞ diverges logarithmically, the difference Γµ (p, k) −
Λ
Γµ (p, 0) is finite. Finally, in spite of apparent nonlocality of subtracted terms the
subtraction is in fact local. Indeed, the subtracted terms may be writen as follows
lim [ΓΛ
µ (p, 0) − e
Λ→∞
∂Σr (p)
] =
∂pµ
=
lim [Γrµ (p, 0) + (Z1 (Λ) − 1)γµ − e
Λ→∞
lim (Z1 (Λ) − 1)γµ
Λ→∞
∂Σr
]
∂pµ
(25)
where Z1 (Λ) is a constant, logarithmically divergent in the limit Λ → ∞.
In a similar way the local subtraction providing automatically WI for four
photon vertex is
Λ
Πrµνρσ (p, q, k) = lim [ΠΛ
µνρσ (p, q, k) − Πµνρσ (0, q, k)] .
Λ→∞
(26)
Obviously Ward Identities are not sensitive to subtraction of local gauge
invariant counterterms. In QED such gauge invariant structure is the transversal
vacuum polarization, which may be renormalized at will, by subtracting
(Z3 − 1)(k 2 gµν − kµ kν ) .
(27)
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Ann. Henri Poincaré
The procedure described above reduces the problem of renormalization in
arbitrary regularization to the renormalization in invariant regularization scheme,
where only gauge invariant counterterms are needed. Noninvariant counterterms
do not appear at all.
In arbitrary gauge theory the corresponding scheme looks as follows. Generalized Ward Identities may be written in the form
pµ Γrµν... (p, k, . . .) = F (Γrν... (p, . . .)) .
(28)
Here the functions Γr at the r.h.s have one external gauge field line less than the
function at the l.h.s. The corresponding differential identities are
Γrµν... (0, k . . .) =
∂F (Γrν... )
|p=0
∂pµ
∂Γrµν... (p, k . . .)
∂Γrαν... (p, k . . .)
|p=0 +
|p=0
∂pα
∂pµ
=
(29)
∂ 2 F (Γrν... (p, k . . .)
|p=0 . (30)
∂pµ ∂pα
For arbitrary regularization and subtraction scheme in anomaly free theory these
identities may be violated by local polynomials. So the renormalized correlators
satisfying GWI are given by the equation
Γµν... (p, k . . .) = lim [ΓΛ
µ... (p, . . .) −
Λ→∞
∂F (Γrν... (p, . . .))
Γµν... (0, k . . .) +
] + g.i.c. .
∂pµ
p=0
(31)
where g.i.c. denotes gauge invariant local terms which obviously have no influence
on GWI and may be fixed at will.
The eq.(31) is written for the case of logarithmic divergency. For linearly divergent integrals the next differential identity (30) should be subtracted. Quadratic
divergencies are treated in a similar way. If necessary one can subtract in addition a gauge invariant counterterm (like photon wave function renormalization in
QED). Such terms are not fixed by GWI and as discussed above, are present also
in gauge invariant regularization schemes.
Therefore we have the renormalization procedure, which is universal, may be
used with arbitrary intermediate regularization and requires only gauge invariant
counterterms. In particular it may be applied to the Standard Model in combination with dimensional regularization, providing a simple computational scheme
preserving chiral gauge invariance.
Another important application of this method is renormalization of supersymmetric gauge theories for which no general invariant regularization is known.
The universal renormalization of supersymmetric theories was considered in a recent paper [23].
Vol. 4, 2003
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Acknowledgments
I am grateful to D.Iagolnitzer and J.Zinn-Justin for invitation and kind support.
This research was supported in part by Russian Basic Research Fund under grant
02-01-00126 and by the president grant for support of leading scientific schools.
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A.A. Slavnov
Steklov Mathematical Institute
Russian Academy of Sciences
Gubkina st.8, GSP-1
117966, Moscow
Russia
Ann. Henri Poincaré