
3
a fixed point g∗satisfies β(g=g∗) = 0. In addition, the
critical exponents θIat a fixed point are also universal.
Linearizing the βfunctions near a fixed point, the flow
is governed by the stability matrix B, the eigenvalues of
which determine the critical exponents,
Bij =∂βgi
∂gjg=g∗
, θI=−eig(B).(8)
Since we work in an approximation scheme, our results
for the universal quantities listed in the following also
exhibit only approximate universality and thus depend
on the scheme to some extent. For concrete computa-
tions, we use the partially linear regularization scheme
which is known to be optimized for fast convergence to
universal results in a class of approximations also used
here [82, 83].
Positive (negative) critical exponents θi>0 (θi<0)
are associated with relevant (irrelevant) directions. The
eigendirections associated with the positive exponents
span the UV critical surface of trajectories emanating
from the fixed point. The dimensionality of this surface,
and thus the number of positive exponents, is equal to
the number of physical parameters to be fixed in order to
render the long-range behavior of the theory fully com-
putable. (Eigenvalues θI= 0 denote marginal directions;
here, higher orders beyond the linearized flow determine
relevance or irrelevance. For instance, the QED4gauge
coupling eis marginally irrelevant.)
In addition to gauge symmetry as a local redundancy,
the action has a global U(Nf) flavor symmetry, whereas
an extended U(Nf)L×U(Nf)Rchiral symmetry is present
only in the absence of the mass and the Pauli term, m= 0
and κ= 0. In the general case, the action is also invariant
under a discrete axial rotation of the spinors by an angle
of π/2 and a simultaneous sign flip of κand m, as well as
under charge conjugation and a simultaneous sign flip of
eand κ. These latter Z2symmetries are also visible on
the level of the flow equations which remain invariant un-
der (e, κ, m)→(e, −κ, −m) and (e, κ, m)→(−e, −κ, m)
as well as combinations thereof. Correspondingly, fixed
points can exist in these sign-flip multiplicities, but, of
course, describe one and the same universality class.
In the following, results for fixed-point searches in
various dimensions and for various fermion degrees of
freedom (flavor number, spinor representation) are pre-
sented. As fixed points at finite couplings are an inher-
ently nonperturbative phenomenon, it is important to
discuss consistency criteria in the absence of a generically
small control parameter. Our systematic approximation
scheme represents a combined expansion in operator di-
mension and in derivatives. In this sense, the inclusion of
wave function renormalizations Zψ,A already represents
a next-to-leading order contribution. As a quantitative
control, we can compare to the leading-order result which
is obtained by ignoring wave-function-renormalization ef-
fects in loop terms. In practice, this corresponds to set-
ting ηψ,A = 0 inside the loop terms (technically, inside
the threshold functions, see App. A), but retaining them
in the scaling terms displayed in Eqs. (3)-(5).
We list results only for fixed points where the tran-
sition from leading- to next-to-leading-order results are
quantitatively controlled. This control is implemented by
demanding that the anomalous dimensions remain suffi-
ciently small, |ηψ,A|.O(1). From a technical view-
point, the flow equations (3)-(5) feature rational func-
tions of high order in the couplings on the right-hand
sides. Generically, they exhibit a large number of fixed-
points, most of which do not satisfy our quality criteria
and are thus considered as artifacts of our approxima-
tion. Only a small number fulfills the consistency condi-
tions in a remarkably stable manner. These are the ones
presented in the following sections.
III. PAULI-TERM FIXED POINTS IN d= 4
As a reference point, we start by reviewing the phase
diagram of QED4with a Pauli term for Nf= 1 as has
been found in the literature [9]. Generalizations to differ-
ent dimensions and different flavor numbers can be well
understood by analyzing the similarities and differences
to this reference case.
In addition to the Gaussian fixed point Acharacter-
ized by vanishing couplings, we find two interacting fixed
points that satisfy all our consistency criteria, cf. Tab.I.
Both these fixed points Band Coccur at finite Pauli κ
but vanishing gauge coupling. Fixed point Balso oc-
curs at finite mass parameter mand has full Z2×Z2-fold
multiplicity, whereas for fixed point Conly the charge
conjugation multiplicity is pertinent.
e κ m multiplicity nphys θmax ηψηA
A: 0 0 0 −1 1.00 0.00 0.00
B: 0 5.09 0.328 Z2×Z22 3.10 −1.38 0.53
C: 0 3.82 0 Z23 2.25 −1.00 0.00
TABLE I. Fixed points of d= 4 dimensional QED.
At the Gaussian fixed point A, only the mass is a rel-
evant direction. At fixed point B, the direction towards
finite gauge coupling also represents a relevant direction.
Fixed point Cfeatures 3 relevant directions, implying
that all couplings in the action correspond to physical
parameters that define the long-range behavior of the
system. The flow towards the long-range IR is visual-
ized in the phase diagram of the (κ, m) plane at e= 0
in Fig. 1. Note that the rapid flow near the maxis to-
wards large values of mreflects the fact that mdenotes
a dimensionless mass parameter increasing as m∼1/k
for k→0 if the physical mass approaches a finite value.
As discussed in [9], UV-complete trajectories that
agree with the observed long-range behavior of pure
QED4can be constructed with fixed point Cas a UV fixed
point. Even though UV-complete trajectories emanating
from B, of course, also exist, their long-range behavior