Exact symmetries realized on the renormalization group flow
ABSTRACT We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward–Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg–Wilson relation and the Lüscher's symmetry.
- SourceAvailable from: arxiv.org[show abstract] [hide abstract]
ABSTRACT: This lecture note starts with a pedagogical introduction to the theoretical background and properties of perfect actions, gives some details on topology and instanton solutions and ends with a discussion on the recent developments concerning chiral symmetry.04/1998;
arXiv:hep-th/9912262v2 14 Jan 2000
Exact Symmetries realized on the Renormalization Group Flow
Yuji Igarashi, Katsumi Itoh and Hiroto Soa
Faculty of Education, Niigata University, Niigata 950-2181, Japan
aDepartment of Physics, Niigata University, Niigata 950-2181, Japan
We show that symmetries are preserved exactly along the (Wilsonian) renor-
malization group flow, though the IR cutoff deforms concrete forms of the trans-
formations. For a gauge theory the cutoff dependent Ward-Takahashi identity is
written as the master equation in the antifield formalism: one may read off the
renormalized BRS transformation from the master equation. The Maxwell theory
is studied explicitly to see how it works. The renormalized BRS transformation
becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a
generic global symmetry. The master equation considered for the chiral symmetry
provides us with the continuum analog of the Ginsparg-Wilson relation and the
L¨ uscher’s symmetry.
PACS: 11.10Hi; 11.15.Tk; 11.30.-j
Keywords: renormalization group; chiral symmetry; Becchi-Rouet-Stora transformation;
Ward-Takahashi identity; effective action
The Wilsonian Renormalization Group (RG) is one of the most important achievements
in modern physics. In particular, the exact RG equations[2-4] have proved to be power-
ful both in perturbative and non-perturbative studies of field theories1. In a field theory,
quantum fluctuations at shorter distances are integrated out to give an effective action for
longer distances. For the well-defined integration, one needs to introduce some regulariza-
tion procedure, which may be in conflict with symmetries in many important applications:
for example, the presence of gauge symmetry or chiral symmetry is far from trivial. The
incompatibility of symmetries and regularizations is a longstanding problem in the RG
There have been several attempts[10-13] recently to investigate this problem based on
a common recognition: a symmetry is broken at intermediate steps of the RG iteration,
and is recovered only after the IR cutoff k is removed. The breaking of the symmetry
is controlled by the modified Ward-Takahashi (WT) identity, Σk = 0. In practical
calculations, one has to finely tune parameters in an effective action so that it satisfies
the usual WT identity in the limit of k → 0. This viewpoint, recovery of the symmetry
by “fine tuning”, is due to Becchi and extensively studied in refs. ,  and .
Recent development in understanding chiral symmetry on the lattice has brought
another important clue to our problem: L¨ uscher found an exact chiral symmetry on the
lattice, relying on the Ginsparg-Wilson (GW) relation. This provides us with the
first non-trivial example of having an exact symmetry even after the regularization. The
L¨ uscher’s chiral symmetry takes quite different form from that in the continuum limit.
Based on these observations, we shall give in this paper a general method to define
global symmetry along a RG flow. It may be non-local and cutoff dependent, yet exact
symmetry even for k ?= 0. We call this “renormalized symmetry”. Remarkably, our
discussion applies to gauge symmetry as well by considering its global counterpart, the
We begin with a microscopic or UV action which is local and invariant under a sym-
metry transformation. In order to construct the effective action at low momentum, we
consider the continuum analog of the blockspin transformation. This formalism developed
in introduces macroscopic fields (average fields), in terms of which the renormalized
symmetry is realized. The important role of the macroscopic fields is also suggested by
the GW relation and the L¨ uscher’s chiral symmetry. Since the blockspin transformation
is a gaussian integral, we obtain an exact RG flow equation for the effective action of
the macroscopic fields. When expressed by the macroscopic fields and some source fields,
the WT identity Σk= 0 takes the form of a master equation, from which we shall find the
exact symmetry transformation for k ?= 0. We would like to emphasize here that our WT
identity is for the exact renormalized symmetry, not for the broken or modified symmetry.
This is the central issue of our formulation of renormalized symmetry. The flow equation
for Σkholds as a result of the algebraic relation between the operator specifying the RG
1See eg  for non-perturbative studies, [7-9] for reviews of the recent development.
flow and that appeared in the WT identity.
For gauge theories, the master equation is nothing but the one in the antifield for-
malism of Batalin and Vilkovisky. In order to see how the renormalized symmetry
looks like, we give an effective action and a renormalized BRS symmetry for the Maxwell
theory. As another test of our method, we consider chiral symmetry, and show that our
master equation and associated renormalized symmetry are the continuum analog of the
GW relation and the L¨ uscher’s symmetry. In our derivation the GW relation is regarded
as an exact WT identity for the chiral symmetry.
Let ϕAbe a microscopic field2with the Grassmann parity ǫ(ϕA) = ǫAand S[ϕA] a generic
action. The microscopic or UV action is assumed to be invariant δaS[ϕ] = 0 under
an infinitesimal global transformation with parameters εa, ϕA→ ϕA+ δaϕAεa, where
ǫ(δa) = ǫ(εa). The discussion to be given also applies to gauge theory: the action S[ϕ] is
a gauge fixed action and the relevant global transformation is the BRS transformation.
To specify a blockspin transformation, we introduce a function fk(p) with an IR cut-
off k in the Euclidean momentum space, and an invertible matrix [Rk(p)]AB satisfying
ǫ([Rk(p)]AB) = 0, [Rk(p)]AB= (−1)ǫAǫB[Rk(p)]BA. For a boundary condition, we impose
fk(p) → 1, [Rk(p)]AB → ∞ as k → ∞. Possible choices of fk(p) and [Rk(p)]AB were
discussed in , but we do not need to specify them here. Let KA
variations δaϕA: they will play an important role in our symmetry consideration. We
may define an effective action for the macroscopic fields ΦAin the presence of the sources
abe sources for the
Sk[ϕ,Φ,K] = S[ϕ] +1
2(Φ − fkϕ)T
−Rk(Φ − fkϕ)++ KT
where Φ±≡ Φ(±p) and their multiplication implies the integration over momentum as
well as the sum over the index A, eg,
The supertrace, Str, denotes a sum over momenta and indices. Note that fk[Rk]ABΦB, a
linear term of the macroscopic fields, acts as a source term for ϕAin the path integral.
Since only the gaussian term depends on the cutoff k, one obtains the exact RG flow
equation for the macroscopic action Γk[Φ, K] :
k∂kRk1) + Str(∂k(lnfk)1)
2The index A denotes kinds of fields and other indices as a whole, except field momentum.
We consider now the symmetry property of the macroscopic action. Invariance of the
microscopic action under the global transformation can be expressed as
Dϕe−Sk[ϕ,Φ,K]= 0. (2.4)
Assumed here is the translational invariance of the path integral measure, ie, the absence
of anomalies. For each independent parameter εa, the WT identity reads
Σka[Φ,K] ≡ −eΓk[Φ,K]∆ae−Γk[Φ,K]= −?KT
where the expectation value is taken with respect to the action Skand the operator ∆a
is defined by
This takes the form of a master equation in the space of (ΦA, Ka
presently, for the BRS symmetry the source KA(p)/fk(p) can be identified with the anti-
field of the macroscopic field ΦA, and (2.5) becomes the quantum master equation.
A). As will be seen
In oder to obtain the flow equation for Σkain our formulation, we notice that there is
an algebraic relation between operator X in (2.3) and the operator ∆a:
[∆a, X] = (∂k∆a) (2.8)
on any Grassmann even quantity. This leads to the flow equation
∂kΣka= (eΓkXe−Γk)Σka− eΓkX (e−ΓkΣka). (2.9)
It is easily seen that the r.h.s consists of the functional derivatives of Σka.
The above equations (2.5) ∼ (2.9) hold quite generally. They also provide us with the
transformation for the renormalized symmetry. In the following two sections we consider
the BRS and global symmetry separately.
3 Renormalized BRS symmetry
3.1The master equation
For the BRS symmetry, the source KA(p)/fk(p) can be identified with the antifield Φ∗
for the macroscopic field ΦA. Then, the operator ∆ in (2.6) and a bracket defined by
(F, G) ≡
are exactly those in the antifield formalism of Batalin-Vilkovisky. Since the r.h.s of
(2.5) vanishes because of the nilpotency δ2=0, one obtains the condition,
2(Γk[Φ,Φ∗], Γk[Φ,Φ∗]) + ∆Γk[Φ,Φ∗] = 0,(3.2)
which is nothing but the quantum master equation. It is an algebraic equation which
holds for any Φ and Φ∗. The WT flow equation (2.9) tells us then that once it is satisfied
at some cutoff k = k0 it persists along the RG flow. This clearly demonstrates the
presence of a cutoff dependent BRS symmetry, a renormalized BRS symmetry, in the
macroscopic action. If the second term in the master equation vanishes, we may define
the renormalized BRS transformation on Φ and Φ∗by
The cutoff dependent BRS transformation appeared earlier in a different context.
The author took the viewpoint to finely tune the effective action for k ?= 0 with gauge
non-invariant terms so that it satisfies the usual WT identity in k → 0 limit. A series
of papers followed to confirm this point of view perturbatively for various models.
The “modified Slavnov-Taylor identity” and its flow equation are elegantly summarized
in . However the presence of the exact BRS symmetry had not been understood.
Here we have seen that the transformation may be defined with the master equation
in the antifield formalism, and the WT identity Σk= 0 is not a broken but exact identity.
In the next subsection we shall give a simple model of the renormalized BRS symmetry
for the Maxwell theory, where the above stated properties can be confirmed explicitly.
3.2Abelian gauge symmetry
Let us consider the gauge-fixed Maxwell action in D=4 Minkowski space,
4F2+ B(∂ · A +α
2B) + i∂µ¯ c∂µc + ϕ∗Tδϕ
, δϕ =
The microscopic action S0satisfies the (classical) master equation, (S0,S0) = 0, for the
antibracket defined in terms of ϕ and ϕ∗: the ϕ∗is the set of the antifields at the micro-
scopic level.3The macroscopic fields,
3Note that the BRS transformation in (3.4) is defined by the right derivative: δϕA= (ϕA, S0).