Scaling dimensions in QED3 from the ϵexpansion
HJE
QED3 from
Lorenzo Di Pietro 0 1 3
Emmanuel Stamou 0 1 2
0 Chicago , IL 60637 , U.S.A
1 Waterloo , ON N2L 2Y5 , Canada
2 Enrico Fermi Institute, University of Chicago
3 Perimeter Institute for Theoretical Physics
We study the xed point that controls the IR dynamics of QED in d = 4 dimensions. We derive the scaling dimensions of fourfermion and bilinear operators beyond leading order in the expansion. For the fourfermion operators, this requires the computation of a twoloop mixing that was not known before. We then extrapolate these scaling dimensions to d = 3 to estimate their value at the IR xed point of QED3 as function of the number of fermions Nf . The nexttoleading order result for the fourfermion operators corrects signi cantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of Nf , which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, the expansion provides a convincing estimate for the dimension of the avorsinglet scalar in the full range of Nf .
Conformal Field Theory; Field Theories in Lower Dimensions; Renormaliza

2
tion Group
3.1
3.2
3.3
5.1
5.2
5.3
1 Introduction 2 QED in d = 4 2
2.1 Operator mixing
3 Fourfermion operators in d = 4
2
Operator basis
Renormalizing Green's functions
Evaluation of Feynman diagrams
3.4 Anomalous dimensions at the xed point
4 Bilinear operators in d = 4
2
5 Extrapolation to d = 3
Pade approximants
Bilinears as d ! 3
6 Conclusions and future directions
A Feynman rules
B Renormalization constants
B.1 Flavorsinglet fourfermion operators
B.2 Flavornonsinglet fourfermion operators
C Flavornonsinglet fourfermion operators
persist beyond this largeNf regime, but not much is known about it. Ref. [16] employed
the conformal bootstrap approach to derive bounds on the scaling dimensions of some
monopole operators.
Another method to study the smallNf CFT is the expansion,
which exploits the existence of a xed point of WilsonFisher type [17] in QED continued
to d = 4
2 dimensions.
When
1 we can access observables via a perturbative
expansion in
and subsequently attempt an extrapolation to
. The expansion of
QED was employed to estimate scaling dimensions [18, 19], the free energy F [20], and the
= 12
coe cients CT and CJ [14]. In particular, ref. [18] considered operators constructed out of
gaugeinvariant products of either four or two fermion elds.
Fourfermion operators are interesting because of the dynamical role they can play in
the transition from the conformal to a symmetrybreaking phase, which is conjectured to
exist if Nf is smaller than a certain critical number Nfc [21{26]. In fact, the operators with
the lowest UV dimension that are singlets under the symmetries of the theory are
fourfermion operators. If for small Nf they are dangerously irrelevant, i.e., their anomalous
dimension is large enough for them to
ow to relevant operators in the IR, they may
trigger the aforementioned transition [7, 27, 28].1 The oneloop result of ref. [18] led to the
estimate N c
f
2.
Bilinear operators, i.e., operators with two fermion elds, are interesting because they
are presumably among the operators with lowest dimension. For instance, when continued
to d = 3, the twoform operators
[
]
become the additional conserved currents of the
SU(2Nf ) symmetry, of which only a SU(Nf ) subgroup is visible in d = 4
2 . This leads
to the conjecture that their scaling dimension should approach the value
= 2 as
! 2
1
which was tested at the oneloop level in ref. [18].
In order to assess the reliability of the expansion in QED, and to improve the
estimates from the oneloop extrapolations, it is desirable to extend the calculation of these
anomalous dimensions beyond leading order in . This is the purpose of the present paper.
Let us describe the computations we perform and the signi cance of the results.
We rst consider fourfermion operators. In the UV theory in d = 4
2 , there
are two such operators that, upon continuation to d = 3, match with the singlets of the
SU(2Nf ) symmetry. We compute their anomalous dimension matrix (ADM) at twoloop
level by renormalizing o shell, amputated Green's functions of elementary
elds with a
single operator insertion. As we discuss in detail in a companion paper [32], knowing this
twobytwo ADM is not su cient to obtain the O( 2) scaling dimensions at the IR
xed
point. We also need to take into account the full oneloop mixing with a family of in nitely
many operators that have the same dimension in the free theory. These operators are of
the form
(
n
1::: n )2 ;
(1.1)
1More precisely, when the fourfermion operators are slightly irrelevant in the IR, there is an additional
nearby UV
xed point. As these operators become marginal, the two xed points cross each other, and they
can annihilate and disappear [29]. For a more detailed discussion, see section 5 of ref. [20]. Ref. [
30
] pointed
out that in order to describe properly the conjectured transition, one cannot ignore higherorder terms in
the fourfermion couplings. A study of the RG
ow that employed expansion and included fourfermion
couplings appeared recently in ref. [31].
{ 2 {
HJEP12(07)54
where n is an odd integer, and
n
gamma matrices. All the operators in this family except for the rst two, i.e., n = 1; 3,
vanish for the integer values d = 4 and d = 3, but are nontrivial for intermediate values
3 < d < 4. For this reason, they are called evanescent operators. Taking properly into
account the contribution of the evanescent operators, via the approach described in ref. [32],
we obtain the nexttoleading order (NLO) scaling dimension of the rst two operators.
We then extrapolate to
= 12 using a Pade approximant, leading to the result presented
in subsection 5.2 and summarized in
gure 2. The deviation from the leading order (LO)
scaling dimension is considerable for small Nf , indicating that at this order we cannot
yet obtain a precise estimate for this observable of the threedimensional CFT. Taking,
however, the NLO result at face value, we would conclude that the fourfermion operators
are never dangerously irrelevant.
This resonates with recent results that suggest that
QED3 is conformal in the IR for any value of Nf . Namely, refs. [33{35] argued, based
on 3d bosonization dualities [36{39], that for Nf = 1 the SU(2)
U(1) symmetry is in
fact enhanced to O(4) (this is related to the selfduality present in this theory [40]). Also,
a recent lattice study [41] found no evidence for a symmetrybreaking condensate (for
previous lattice studies see refs. [42{44]).
We then consider the bilinear \tensorcurrent" operators of the form
n
crease the order. As mentioned above, in the limit d ! 3 the operators with n = 1; 2
approach conserved currents of the SU(2Nf ) symmetry. Indeed, we show in subsection 5.3
(see gure 4) that the extrapolated scaling dimension of the twoform operators approaches
the value
= 2 as we increase the order. As d ! 3, the operators with n = 0; 3 approach
scalar bilinears, which are either in the adjoint representation of SU(2Nf ) or are singlets.
For the singlet scalar, which is continued by a bilinear with n = 3, the results of various
extrapolations that we perform are all close to each other (see gure 5), indicating that the
expansion provides a good estimate for this scaling dimension in the full range of Nf . For the
adjoint scalar, di erent components are continued by operators with either n = 0 or n = 3,
giving two independent extrapolations at each order in . As expected, we nd that the
two independent extrapolations approach each other as we increase the order (see gure 5).
The rest of the paper is organized as follows: in section 2 we set up our notation and
describe the
xed point of QED in d = 4
2 ; in section 3 we present the computation
of the twoloop ADM of the fourfermion operators, and then the result for their scaling
dimension at the IR
2 ; in section 4 we present the same result for
the bilinear operators; in section 5 we extrapolate the scaling dimensions to d = 3, and
plot the resulting dimensions as a function of Nf for the various operators we consider;
nally in section 6 we present our conclusions and discuss possible future directions. In
the appendices we collect additional material and some useful intermediate results.
{ 3 {
QED in d = 4
grangian is
with the covariant derivative de ned as
We consider QED with Nf Dirac fermions
a, a = 1; : : : ; Nf , of unit charge. The
La1
4
LQED =
F
F
+
ai
D
a
;
Summation over repeated avor indices is implicit. We work in the R gauge, de ned by
adding the gauge xing term
We collect the Feynman rules in appendix A.
The algebra of the gamma matrices is f ;
g = 2
, with
=
and
= d.
We will employ some useful results on ddimensional Cli ord algebras from ref. [46]. We
normalize the traces by Tr[1] = 4, for any d. For d = 3,
a decomposes as
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
is the
(2.7)
(2.8)
giving 2Nf complex twocomponent 3d fermions
Correspondingly, the gamma matrices decompose as
i, i = 1; : : : ; 2Nf , all with charge 1.
(3)
where f
g =1;2;3 are twobytwo 3d gamma matrices.
In d = 4, the global symmetry preserved by the gaugecoupling is SU(Nf )L
2 , evanescent operators violate the conservation of the nonsinglet axial
currents [47], so only the diagonal subgroup SU(Nf ) is preserved. In d = 3, this symmetry
enhances to SU(2Nf )
U(1).
We de ne
coupling is given by
D
Lg.f. =
1
2
a
d!!3
"
a #
a+Nf
;
d!!3
" 0
(3)
(3)#
0
;
0 = Z
( ) 2 ;
=
2
+ ( ; ) ;
d log Z
d log
:
{ 4 {
where the renormalization constant Z ( ; ) absorbs the poles at
= 0, and
renormalization scale. The beta function reads
where
2
e
16 2 and denote bare quantities with a subscript \0". The renormalized
d
d log
( ; )
where (n) is the Riemann zeta function.
Our convention for renormalizing elds is
=
3
4Nf
+
27
16Nf2
To compute the anomalous dimension of local operators Oi, we add these operators to the
LQED ! LQED + X(C0)i(O0)i ;
i
and compute their renormalized couplings Ci at linear level in the bare ones
In Minimal Subtraction (MS),
depends only on
and not on . The MS QED
function
is known up to fourloop order for generic Nf [48, 49]
( ) =
8
3 Nf 2 + 8Nf 3
898 Nf2 + 4Nf
4
2244634 Nf3 +
16
27
2 the theory has a xed point at
(C0)j = X CiZi j :
i
The Zi j are the mixing renormalization constants from which we obtain the ADM
Like , does not depend on in the MS scheme. We introduce the following notation for
the coe cients of the expansion in
and
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
The most direct way to compute the mixing Zi j is to renormalize amputated
oneparticleirreducible Green's functions with zeromomentum operator insertions and
elementary elds as external legs. Alternatively, one can renormalize the twopoint functions
of the composite operators. The former method has two main advantages. The rst is that
to extract nloop poles only nloop diagrams need to be computed. The second is that we
can insert the operators with zero momentum. This makes higherloop computations more
tractable. The disadvantage is that o shell Green's functions with elementary
elds as
external legs are not gaugeinvariant, so some results in the intermediate steps of the
calculation are dependent, which is why we need to include the dependent wavefunction
renormalization of external fermions. In addition, operators that vanish via the equations
of the fourfermion operators and use it to obtain the O( 2) IR scaling dimension at the
xed point. Next, we employ the already existing results of the threeloop anomalous
dimension of bilinear operators [45] to obtain their IR dimension to O( 3).
3
Fourfermion operators in d = 4
2
In this section, we present the computation of the ADM of the fourfermion operators
En = ( a n
a)2 + anQ1 + bnQ3 ;
at the twoloop level. The antisymmetrization in
ventional normalization factor n1! .
In d = 4, the operators in eq. (3.1) are the only two operators with scaling dimension
six at the free xed point that are singlets under the global symmetry SU(Nf )L
a subset of the diagrams. We report the result for some nonsinglet operators in appendix C.
2 , insertions of Q1 and Q3 in loop diagrams generate additional structures
that are linearly independent to the Feynman rules of Q1 and Q3. To renormalize the
divergences proportional to such structures, we need to enlarge the operator basis. It is
most convenient to de ne the complete basis by adding operators that vanish for
and hence are called evanescent operators, as opposed to Q1 and Q3 that we refer to as
physical operators. There is an in nite set of such evanescent operators. One choice of
basis for them is
5. The terms proportional to the arbitrary constants an and bn
are of the form
times a physical operator; they parametrize di erent possible choices for
the basis of evanescent operators.2
For the computation of the ADM we adopt the subtraction scheme introduced in
refs. [50, 51]. Since this is the most commonly used scheme for applications in
avor
physics, we refer to it as the
avor scheme.
We label indices of the ADM using odd
integers n
1, so that n = 1; 3 correspond to the physical operators, eq. (3.1), and n
5
to the evanescent operators, eq. (3.2). The ADM up to twoloop order is3
the avor scheme. No other twoloop entry enters the prediction of the O( 2) prediction of
the scaling dimensions at the xed point.
2Adding terms with higher powers in
have no e ect in the twoloop computation that we discuss here.
3The oneloop (Q1, Q3) block of the ADM can be found in ref. [52]; it is su
cient to obtain the O( )
prediction of the scaling dimensions [18].
{ 7 {
>
>
>
:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
8 "
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
5 ;
5 ;
Notice that the invariant (Q1; Q3) block of (2;0) depends on the coe cients a5 and
b5, which parametrize our choice of basis. This dependence can be understood as a sign of
schemedependence [53]. Clearly, this implies that the scaling dimensions at O( 2) are not
simply obtained from the eigenvalues of this invariant block, as its eigenvalues depend on a5
and b5 too. The additional contribution that cancels this basisdependence originates from
the O( ) term
(1; 1) in the oneloop ADM. Such O( ) terms are indeed induced in every
scheme that contains nite renormalizations, such as the
avor scheme. For a thorough
discussion of the scheme/basisdependence and its cancellation we refer to ref. [32].
There are a few nontrivial ways of partially testing the correctness of the twoloop
results:
i) We performed all computations in general R gauge. This allowed us to explicitly
check that the mixing of gaugeinvariant operators indeed does not depend on .
ii) We veri ed that all the twoloop counterterms are local, i.e., the local counterterms
from oneloop diagrams subtract all terms proportional to 1 log
in twoloop
diagrams.
iii) We checked that the 12 poles of the twoloop mixing constants satisfy the relation
Z
(2;2) =
2 Z
1 (1;1) (1;1)
Z
2
1 (1;0) (1;1) ;
Z
(3.7)
where (1;0) is the oneloop coe cient of the betafunction. This is equivalent to the
independence of the anomalous dimension [54].
In the next two subsections, we discuss the renormalization of the one and twoloop
Green's functions from which we extract the relevant entries of the mixing matrix Z 
and ultimately the ADM entries in eqs. (3.4), (3.5), and (3.6)  and some technical
aspects of the twoloop computation. A reader more interested in the results for the scaling
dimensions may proceed directly to section 3.4.
3.1
Operator basis
As argued in section 2.1, in general we need to consider also EOMvanishing operators
when renormalizing o shell Green's functions. Moreover, in our computation we adopt an
IR regulator that breaks gaugeinvariance, so we also need to take into account some
gaugevariant operators. Below we list all operators that, together with (Q1; Q3) and fEngn 5,
enter the renormalization of the twoloop Green's functions we consider.
EOMvanishing operators.
There is a single EOMvanishing operator, N1, that a ects
the ADM at the oneloop level and another one, N2, that a ects it at the twoloop level.
They read
1
e
1
e
N1 =
N2 =
a) +
1
{ 8 {
N1 + Q1 ;
N1 + N2 :
(3.8)
(3.9)
HJEP12(07)54
In addition, there are EOMvanishing operators that are only necessary to close the basis
of independent Lorentz structures for certain Green's functions. For completeness, we list
them here
N3 = i a D*= D*= D*= a
;
N4 =
a( D=
(
+
*
D= ) a
F
:
Here D=
D and the arrow indicates on which eld the derivative is acting.
Gaugevariant operators.
Renormalization constants subtract UV poles of Green's
functions. It is thus essential to ensure that no IR poles are mistakenly included in the
renormalization constants. In practice, this means that an energy scale must be present in
dimensionally regularized integrals. Otherwise, UV and IR contributions cancel each other
and the result of the loop integral is zero in dimensional regularization [47].
One possibility to introduce a scale is to keep the external momentum in the loop
integral. However, i) such loop integrals are more involved than integrals obtained by
expanding in powers of external momenta over loop momenta, and ii) keeping external
momenta does not necessarily cure all the IR divergences, e.g., diagrams with gluonic
snails in nonabelian gauge theories. Another possibility for QED would be to introduce a
mass for the Dirac fermions. The drawback in this case is that we would have to consider
many more EOMvanishing operators.
Instead, we apply the method of \Infrared Rearrangement" [55, 56]. This method
consists in rewriting the massless propagators as a sum of a term with a reduced degree of
divergence and a term depending on an arti cial mass, mIRA. Section 3.3 contains more details
about the method. The caveat is that the method violates gauge invariance in intermediate
steps of the computation. All breaking of gauge invariance is proportional to mI2RA and
explicitly cancels in physical quantities. However, to restore gaugeinvariance, also
gaugevariant operators proportional to mI2RA need to be consistently included in the computation.
Fortunately, due to the factor of mI2RA, at each dimension there are only a few of them.
At the dimensionfour level, a single operator is generated, i.e., the photonmass operator:
At the dimensionsix level, there are more operators, but only one, P, enters our ADM
computation because Q1 and Q3 mix into it at oneloop. It reads
mI2RAA A :
P =
1
e mI2RA
X
a
a
aA :
3.2
Renormalizing Green's functions
In this subsection, we highlight the relevant aspects in the computation of the
renormalization constants Zi j , from which we extracted the ADM presented above, via the
renormalization of amputated oneparticle irreducible Green's functions.
For each Green's function we need to specify the operator we insert as well as the
elementary
elds on the external legs. In our case, the external legs are either four
elementary fermions, or two fermions and a photon, or two photons. At treelevel, a Wick
contraction with the elementary
elds de nes a vertex structure for each operator. We
{ 9 {
(3.10)
(3.11)
(3.12)
(3.13)
hN1i(2)
~
S
A
2
4
. . .
. . .
~
S
7 + 6
. . .
5
5
~
S
A
7 + 2Z(1) 6
4
Z(2) + Z(2) + Z(1)Z(1) 6
A
5
7 +
~
S
3
7
5
~
S
eq. (A.10). The squares denote operator insertions and the crosses counterterms. The rst
parenthesis collects twoloop insertions of the operator N1, which is a linear combination of N1 and Q1.
The second collects the oneloop insertions with counterterms on the propagators and the QED
vertices. The third and fourth are oneloop insertions multiplied with the eld and charge
renormalization of the elds and charges composing the N1, see eq. (3.15). The fth are the treelevel
insertions multiplied with the twoloop eld and charge renormalization constant from the N1 .
denote the
^
S. An additional subscript indicates the operator associated to a given structure. The
representation in terms of Feynman diagrams is
structures with S, the
A
ones with S~, and the A A
one with
= iCO S~
O
We collect all structures that enter the computation in appendix A.
In what follows, we refer to
as a sum over a speci c subset of Feynman diagrams: i) All these diagrams have a single
insertion of the operator O. ii) They are dressed with interactions such that they contribute
at O( L). In particular, we include all counterterm diagrams proportional to eld and
charge renormalization constants, but we do not include diagrams that contain mixing
constants. We keep those separate to demonstrate how we extract them. iii) The subscript
S indicates that out of this sum of diagrams we only take the part proportional to the
structure S. In short, the notation of eq. (3.14) denotes the Lloop insertion of O projected
on S, including contributions from
eld and charge renormalization constants.
To illustrate the notation we show in gure 1 a small subset of the Feynman diagrams
for the nontrivial case of hN1i
(2) ~, with S~ any of the structures in eq. (A.10). Note that,
S
since N1 is a linear combination of terms with di erent elds, see eq. (3.8), its eld and
(3.14)
hOi(L)
S
Green's function
Depends on
Constant(s) extracted
Oneloop
Twoloop
A A
A A
A
A
(1)
ZON2
ZO(1N)1, ZO(1P) , ZON2
(1)
(1)
ZOO0, ZON1
(1)
ZQ(2N)2, ZQ(1N) , ZQ(1;)P
ZQ(2N) , ZQ(1O) , ZQ(1N) , ZQP
(1)
ZQ(2Q)0, ZQ(2N)1, ZQ(1O) , ZQ(1N) , ZQP
(1)
(1)
ZON2 (1)
ZO(1N)1, ZOP
(1)
ZOO0
ZQN2
(2)
(2)
ZQN1
(2)
ZQQ0
the mixing renormalization constants that the given Green's function depends on. The last column
contains the ones we extract in each case.
charge renormalizations depend on the part we insert, namely
1
e
(N1)0 = Z 1=2ZA1=2Z
(3.15)
Next we derive the conditions on the Green's functions that determine the mixing
constants. For transparency we frame the constant(s) that we extract from a given condition.
In table 1 we summarize which Green's functions we consider, on which mixing
renormalization constants they depend, and which one we extract in each case. For brevity we use
the following shorthand notation:
Q; Q0 = Q1; Q3 ;
E = En ;
N = N1; N2 ;
O; O0 = Q1; Q3; En :
We collect the results for the renormalization constants in appendix B.
A A
At oneloop there is no insertion of any fourfermion operator that
contributes to the Green's function with only two external photons. Thus
(1)
ZON2
= 0 :
A
at oneloop. Contrarily, oneloop insertions of fourfermion operator contribute
to the
Green's function. By expanding the diagram in the basis of S~ structures, we
determine the mixing into operators with a treelevel projection onto
A , namely N1
and P. For the physical operators the conditions are
hQi(1)
S~N1
(1)
+ ZQN1 hN1i(0)
S~N1
(1)
+ZQN2 hN2i(0)
hQi(1)
(1)
~ + ZQP hPi(0)
SP
S~N1
~
SP
In the rst line we use that ZQN2 = 0, as extracted from the A A
Green's function.
Similarly, we determine the mixing of En into N1
Notice that in this case the mixing constants subtract nite terms, as required by the avor
scheme we adopt.
(3.16)
(3.17)
(3.18)
(3.19)
Next, we compute the oneloop insertions in the
Green's
function. Firstly, we insert physical operators, i.e., Q,
with the only nonvanishing hN1i(0)
SO
(1)
ZQQ1
assumes knowledge of ZQ(1N)1 , which we have previously determined via the
Green's function. Next, we insert evanescent operators. Again, the only di erence here is
A
that their mixing constants into physical operators subtract nite pieces
being the one for O = Q1. We see that extracting
This completes the computation of all oneloop constants required to determine the
mixing of physical operators at the twoloop level. Next, we renormalize the same Green's
functions at the twoloop level.
A A
A A
at twoloop.
At the twoloop order Q1 and Q3 insertions do contribute to the
Green's function. They can thus mix into the operator N2. Even though N2 itself
does not have a treelevel projection on physical operators, we need this mixing to extract
the twoloop mixing of Q1 and Q3 into N1 in the next step. The projection onto the S^
(3.20)
(3.21)
(3.22)
structure results in the condition
level. We only need the twoloop mixing of physical operators into N1, because only N1 has
a treelevel projection onto Q1. To unambiguously determine the projection on the
structure S~N1 , we have to x a basis of linear independent structures, which correspond to
linearly independent operators. At this loop order, we nd that apart from N1 we also need to
include the operators N3 and N4 to project all generated structures. This projection is the
only point in which these operators enter our computation. The niteness of the twoloop
Green's function determines the twoloop mixing of physical operators into N1 via4
at twoloop. Finally, we have collected all results necessary to renormalize the
twoloop
Green's function. The renormalization conditions for the mixing in the
4Note that hN1i(1) S~N1 = hN1 i(1) S~N1 +hQ1i(1) S~N1 ; as N1 has two Feynman rules.
physical sector read
We see here explicitly that, because N1 has a treelevel projection onto Q1, we need ZQN1
Already at the twoloop level the number of Feynman diagrams entering the Green's
functions is quite large. The present computation is thus performed in an automated setup.
Firstly, the program QGRAF [57] generates all diagrams creating a symbolic output for each
diagram. This output is converted to the algebraic structure of a loop diagram and
subsequently computed using selfwritten routines in FORM [58]. The methods for the
computation and extraction of the UV poles of twoloop diagrams are not novel and also widely
used throughout the literature. Here, we shall only sketch the steps and mention parts
speci c to our computation.
One major simpli cation of the computation comes from the fact that we can always
expand the integrand in powers of external momenta over loopmomenta and drop terms
beyond the order we are interested in. For instance, for the
Green's function all
external momenta can be directly set to zero, while for the
A one we need to keep the
external momenta up to second order to obtain the mixing into N1 (see S~N1 in eq. (A.10)).
After the expansion, all propagators are massless so the resulting loopintegrals vanish
in dimensional regularization. To regularize the IR poles and perform the expansion in
external momenta we implement the \Infrared Rearrangement" (IRA) procedure introduced
in refs. [55, 56]. In IRA, an  in our case massless  propagator is replaced using the
identity
1
(p + q)2 =
p2
1
mI2RA
q2 + 2p q + mI2RA
p2
mI2RA
1
(p + q)2
;
(3.27)
where p is the loop momentum, q is a linear combination of external momenta, and mIRA
is an arti cial, unphysical mass. We see that the rst term in the decomposition contains
the scale mIRA and carries no dependence on external momenta in its denominator. In
the second term, the original propagator reappears, but thanks to the additional factor
the overall degree of divergence of the diagram is reduced by one. When we apply the
decomposition recursively, we obtain a sum of terms with only loopmomenta and mIRA
1
in the denominators plus terms proportional to (p+q)2 . These last terms, however, can be
made to have an arbitrary small degree of divergence. Therefore, in a given diagram we
can always perform the decomposition as many times as necessary until terms proportional
1
to (p+q)2 are nite and can thus be dropped if we are only interested in the UV poles.
When applying IRA on photon propagators, the resulting coe cients of the poles are
not gaugeinvariant, because we drop the
nite terms in the expansion of propagators.
This is why some gaugevariant operators/counterterms enter in intermediate stages of the
computation, for instance the operator P. Such operators are always proportional to mI2RA
and so only a small number of them enters at each dimension. For more details on the
prescription we refer to the original work [56].
The IRA procedure results in integrals with denominators that i) are independent of
the external momenta, and ii) contain the arti cial mass mIRA. We can always reduce
these integrals to scalar \vacuum" diagrams by contracting them with metric tensors and
solving the resulting system of linear equations, e.g., see ref. [56]. This tensor reduction
reduces all integrals to one and twoloop scalar integrals of the form
Z
ddp
(p2
m21)n1
and
Z Z
ddp1ddp2
(p21
m21)n1 (p22
m22)n2 (p1
p2)2n3 ;
(3.28)
with the integers n1, n2, n3
1, and m1 6= 0. The oneloop integral can be directly
evaluated, whereas all twoloop integrals can be reduced to a few master integrals using
the recursion relation in ref. [59]. In fact, in our case m1 = m2 = mIRA and the use of
recursion relations is not required.
In the evaluation of the Feynman diagrams, we use the Cli ord algebra in d dimensions
for i) the evaluation of traces with gamma matrices when the diagram in question has closed
fermion loops, and ii) the reduction of the Dirac structures to the operator structures S
or S~ listed in appendix A.
3.4
Anomalous dimensions at the xed point
By substituting the value of the coupling at the xed point, eq. (2.10), in the result of
eq. (3.3), we obtain the ADM at the xed point as an expansion in
>>>> +
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
>
>
:
1
24Nf2
(3.29)
(3.30)
(3.31)
( 2 )nm = <
>>> N24f ( 1) n(n2 1) (n
>
>
for m = 1; n
7:39
46:1
13:4
84:0
3:07
14:1
8:07
23:5
1:72
7:43
6:39
11:6
1:10
4:84
5:60
7:12
0:766
3:51
5:17
4:94
0:562
2:73
4:90
3:70
0:429
2:21
4:71
2:91
0:337
1:86
4:59
2:37
0:272
1:59
4:49
1:99
0:224
1:39
4:42
1:70
for Nf = 1; : : : ; 10. Only three signi cant digits are being displayed.
Note that the physicalphysical block is not invariant at order 2, because there are nonzero
entries ( )n1 and ( )n3 for all n
We are interested in
nding the rst two eigenvalues of
up to order 2
. They
determine the scaling dimensions of the corresponding eigenoperators at the IR xed point.
We denote these scaling dimensions by
( IR)i =
UV( ) + ( 1)i + 2
( 2)i + O( 3) ;
(3.32)
with i = 1; 2 and
UV( ) = 6
4 . To compute the rst two eigenvalues we have truncated
the problem to include a large but nite number of evanescent operators. Taking a su
ciently large truncation, the scheme/basisdependence of the approximated result can be
made negligible at the level of precision we are interested in (for details see ref. [32]). In
table 2, we list the values of ( 1)i and ( 2)i for Nf = 1; : : : ; 10 after we included enough
evanescent operators such that the three signi cant digits listed remain unchanged. The
table is the main result of this section. In section 5, we will use these results as a starting
point to extrapolate the scaling dimensions to d = 3.
4
Bilinear operators in d = 4
2
In this section we consider operators that are bilinear in the fermionic elds. The most
generic bilinear operators without derivatives are
a n
b
;
a n
1::: n 5 b
;
(4.1)
with n
prescription [60, 61]. The indices a; b = 1; : : : ; Nf are indices in the fundamental of the
diagonal \vector" SU(Nf ) subgroup of the SU(Nf )L
SU(Nf )R symmetry of the theory
in d = 4. In d = 4
2 , the conservation of the nonsinglet axial currents is violated
by evanescent operators [47], and thus only the diagonal SU(Nf ) is a symmetry. On the
other hand, the CFT in d = 3 is expected to enjoy the full SU(Nf )L
SU(Nf )R symmetry,
which is actually enhanced to SU(2Nf )
U(1). Therefore, in continuing the operators
of eq. (4.1) to d = 3, we
nd that the ones with
5 are in the same multiplets of the
avor symmetry as those without. So even though their scaling dimensions can di er
as a function of , the enhanced symmetry entails that they should agree when
Since the operators with 5 do not provide new information about the 3d CFT, and the
= 12
't HooftVeltman prescription makes computations technically more involved, we restrict
our discussion here to operators without 5. As a future direction, it would be interesting
to test this prediction of the enhanced symmetry by comparing the scaling dimensions of
operators with 5 after extrapolating to d = 3 at su ciently high order. We also restrict
the discussion to operators with n
3, because the others are evanescent in d = 3.
The anomalous dimension of bilinear operators without 5 has been computed for a
generic gauge group at threeloop accuracy in ref. [45]. For our U(1) gauge theory we
substitute CA = 0 and CF = TF = 1. Moreover, there is a di erence in the normalization
convention for the anomalous dimension, so that
operator decomposes into a singlet and an adjoint component,
here = 2 there. Under SU(Nf ) each
a n
a n
respectively. A priori, the two components can have di erent anomalous dimensions. The
di erence between the singlet and the adjoint originates from diagrams in which the
operator is inserted in a closed fermion loop. When the operator has an even number of gamma
matrices, the closed loop gives a trace with an odd total number of gamma matrices, which
vanishes. So for even n there is no di erence between the singlet and the adjoint, i.e., they
have the same anomalous dimension.
Below we collect the results for n
Scalar:
(Bs(i0n)g) =
(Ba(0d)j) = 3
2
+
140Nf2
9
2Nf
+
60Nf + 135 2
16Nf2
81Nf (16 (3)
5)
32Nf3
Vector: for n = 1 both operators are conserved currents, so they do cannot have an
anomalous dimension, i.e.,
Twoform:
(Bs(i2n)g) =
(Ba(2d)j) = 3
Threeform:
(Bs(i1n)g) =
(Ba(1d)j) = 3
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
3078 3 + O( 4) :
3078 3 + O( 4) :
In d = 4 these threeform operators are Hodgedual to axial currents. Actually, the
fact that they do not get an anomalous dimension at oneloop, as seen from the
equations above, is related to this. However, Hodgeduality cannot be de ned in
d = 4
2 and the anomalous dimensions start to di er from those of the axial
current at the twoloop level.
This exhausts the list of bilinears without 5 that ow to physical operators as d ! 3. In
section 5.3 we discuss which operators of the CFT in d = 3 are continued by the operators
above, and extrapolate the above results to obtain estimates for their scaling dimensions.
5
provides an approximation to the observable, e.g.
the scaling dimension
, in terms of a polynomial
HJEP12(07)54
k
i=1
=
UV( ) + X
i
i + O( k+1) :
Taking
! 12 in this polynomial gives the \ xed order" d = 3 prediction of the
expansion. Typically, the xedorder results show poor convergence as the order is
increased. A standard resummation technique adopted for these kind of extrapolations is to
replace the polynomial with a Pade approximant. The Pade approximant of order (k,l) is
de ned as
The coe cients ci and di are determined by matching the expansion of eq. (5.2) with
eq. (5.1). k + l must equal the order at which we are computing. Another condition comes
from the fact that we are interested in the result for
to smoothly interpolate from
= 0 to
= 12 , an employable Pade approximant should not
! 12 . In order for the expansion
have poles for
2 [0; 12 ] for the values of Nf that we consider. In what follows, we show the
predictions from a Pade approximation only if it does not contain any pole on the positive
axis of for any value of Nf = 1; : : : ; 10.
5.2
Fourfermion operators as d ! 3
In d = 3, the two fourfermion operators in the UV can be rewritten as
Q1
d!!3 ( i
(3) i)2 ;
Q3
d!!3 ( i i)2 ;
where i = 1; : : : ; 2Nf . In this rewriting we see explicitly that these operators are singlets
of SU(2Nf ).
We now evaluate the scaling dimensions ( )1 and ( )2 of the two corresponding IR
eigenoperators, at NLO. For the NLO prediction we employ the Pade approximation of
(5.1)
(5.2)
(5.3)
LO
( )2
LO
NLO Pade (1,1)
4:12 4:23 4:27 4:27 4:26 4:24 4:23 4:21 4:20 4:19
fourfermion operators at d = 3 for various values of Nf . Only three signi cant digits are being
to d = 3, as a function of Nf . In black (lower two lines) ( )1 and in red (upper two lines) ( )2.
Dashed lines are the LO estimate and solid lines the NLO Pade (1,1).
order (1,1). We list the values of the LO and NLO Pade (1,1) predictions for the values of
Nf = 1; : : : ; 10 in table 3.
We visualize the results in gure 2. The dashed lines are the result of the oneloop
expansion computation. Indeed, as discussed in ref. [18], the oneloop approximation
predicts that the lowest eigenvalue becomes relevant for Nf < 3. The twoloop computation
presented here changes this prediction. The two solid lines represent the NLO Pade (1,1)
approximation to the two scaling dimensions. We observe that for no value of Nf does the
lowest eigenvalue reach marginality. We also see that the corrections to the LO result are
signi cant, especially for small Nf , i.e., Nf = 1; 2. This means that for such small values of
Nf , NLO accuracy is not su cient to obtain a precise estimate for this scaling dimension.
Nevertheless, at face value, the result of the twoloop expansion suggests that QED3 is
conformal in the IR for any value of Nf .
Next, we comment on the relation of our result to the 1=Nf expansion in d = 3. At
large Nf , the gauged U(1) current, i
(3) i, is set to zero by the EOM of the gauge eld,
hence the operator Q1 is an EOMvanishing operator. However, besides Q3, there still is
another avorsinglet scalar operator of dimension 4 for Nf = 1, namely F 2 . Q3 and F 2
mix at order 1=Nf [11]. Looking at the expansion result in
gure 2 we see that indeed
only the lowest eigenvalue ( )1 (black lines) approaches 4 for large Nf . The other scaling
dimension (red lines) approaches 6 as Nf !
1, implying that the two eigenoperators
cannot mix at large Nf . This is consistent precisely because there is only one nontrivial
singlet fourfermion operator at large Nf . Its mixing with F 2 cannot be captured within
the expansion, because the UV dimension of F 2 di ers from that of a fourfermion
operator in d = 4
2 . We can, however, test whether for any value of
eigenvalue ( )1, which starts o larger at
levelcrossing would require to revisit the extrapolation to
estimate. The scaling dimension of F 2 in expansion is
= 0, crosses the dimension of F 2 . Such a
= 12 and possibly a ect the
=
( Ba(1d)j )
ij =
i
i
(3) i
(3) j
;
1
2Nf
k
(3) k j
i :
2 d log
=
;
with
given in eq. (2.10) up to O( 4). At three and fourloop order the only Pade
approximation without poles in the positive real axis of
is the order (2,1) and (2,2),
respectively. In gure 3 we plot ( )1;2 and
(F 2) as a function of d for the representative
cases of Nf = 1; 2; and 10. We observe that the only case in which ( )1 crosses
before d = 3 is when Nf = 1 and when we employ N2LO Pade (2,1) to predict
(F 2). The
N3LO Pade (2,2) prediction for Nf = 1 does not cross ( )1 and the same holds for larger
values of Nf . Therefore, at least at this order, F 2 should not play a signi cant role in
obtaining the fourfermion scaling dimension.
5.3
Bilinears as d ! 3
derivatives, the possibilities are
Next we consider bilinear operators in d = 3. In the UV, restricting to the ones without
Bs(i0n)g =
( Ba(0d)j )ij =
i
i
i
j
;
1
2Nf
k
k j
i :
The subscript refers to the representation of SU(2Nf ). The singlet is parityodd.
We can combine parity with an element of the Cartan of SU(2Nf ), in such a way
that one component of the adjoint scalar is parityeven. Since parity squares to the
identity, this Cartan element can only have +1 and
1 along the diagonal, which up
to permutations we can take to be the rst Nf , and the second Nf diagonal entries,
respectively. With this choice, the parityeven bilinear is PaN=f1( a a
This is the candidate to be the \chiral condensate" in QED3 [22].
a+Nf
a+Nf ).
(5.4)
(F 2)
(5.5)
(5.6)
(5.7)
(5.8)
7
6
5
4
3
3
Nf = 1
Nf = 2
4
3
4
3
4
d
d
d
(F 2) : N2LO Pade (2,1)
(F 2) : N3LO Pade (2,2)
HJEP12(07)54
fermion operators (black and red lines) and F 2 (blue lines) as a function of the dimension d, i.e.,
for
Nf = 1; 2; and 10, respectively. We observe that the N3LO Pade (2,2) prediction of
(F 2) never
2 [0; 12 ]. The left, center, and right panel show the result for the representative cases of
crosses the NLO Pade (1,1) prediction of ( )1 in the extrapolation region.
The singlet is the current of the gauged U(1). When the interaction is turned on, it
recombines with the eld strength and does not ow to any primary operator of the
IR CFT. The adjoint is the current that generates the SU(2Nf ) global symmetry.
Therefore, we expect it to remain conserved along the RG and ow to a conserved
current of dimension
= 2 in the IR.
We now identify which d = 4
2 bilinears from section 4 approach the d = 3 bilinears
above. Substituting the decomposition of eqs. (2.4) and (2.5), and also using 3d Hodge
duality, we nd that
We denote by
certain bilinear B
(B) the scaling dimension of the operator in the IR CFT in d = 3 that a
ows to. Therefore, we expect
Bs(i3n)g
Bs(i1n)g
Ba(0d)j
; Ba(3d)j
Ba(1d)j
; Ba(2d)j
d!3
!
d!3
!
d!3
!
d!3
!
Bs(i0n)g ;
Ba(0d)j ;
Bs(i1n)g ;
Ba(1d)j :
(Ba(0d)j);
(Bs(i3n)g)
(Ba(3d)j)
(Ba(2d)j)
d!3
!
d!3
!
d!3
!
(Bs(i0n)g) ;
(Ba(0d)j) ;
2 :
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(Bs(i0n)g)
LO
(Ba(2d)j)
2
2:65
2:49
2:51
1:32
0:238
2
0:500
1:69
1:99
2:75
1:58
1:58
2:00
2
2
2
2
2
2
2
2
2
2:48 2:38 2:32 2:27 2:24 2:21 2:19 2:17 2:16
Ba(0d)j, and the conserved current Ba(1d)j. In cases in which the NLO Pade (1,1) approximant is singular
we list instead the values of the xedorder NLO prediction. Only three signi cant digits are being
The last equation provides a test of the expansion and the rst two provide estimates of
the observables
(Bs(i0n)g) and
(Ba(0d)j). To this end, we employ the viable Pade approximants
for Nf = 1; : : : ; 10. In table 4 we list the expansion predictions at d = 3 for Nf = 1; : : : ; 10.
For the cases in which the order (1,1) Pade approximant is singular, we list the xedorder
NLO prediction.
In gure 4 we plot the extrapolations for the scaling dimension of the conserved
avornonsinglet current Ba(1d)j as a function of Nf . We observe that both N2LO Pade approximants
are closer to 2 than the LO and NLO ones, and they remain close to 2 even for small values
of Nf .
We consider this to be a successful test of the expansion, which supports its
viability as a tool to study QED3.
In gure 5 we plot the various extrapolations for the scaling dimension of the two scalar
operators Bs(i0n)g and Ba(0d)j as a function of Nf . For Bs(i0n)g we nd good convergence behaviour
between the NLO Pade (1,1) and the two N2LO Pade approximations. Therefore, for this
observable we are able to provide a rather convincing estimate. We do stress, however, that
the comparison of the various approximations does not provide rigorous error estimates,
3
3
(1)
Ba(2d)j : LO
Ba(2d)j : NLO
Ba(2d)j : N2LO Pade (2,1)
Ba(2d)j : N2LO Pade (1,2)
5
6
Nf
2
3
4
7
8
9
10
HJEP12(07)54
at d = 3. The operator is associated to the conserved
avornonsinglet current of SU(2Nf ), thus
its scaling dimension is expected to equal 2. We observe that the N2LO Pade approximations are
indeed close to this expectation even for small values of Nf .
(1)
since the error due to the extrapolation is not under control. For Ba(0d)j we have two di erent
operators that provide a continuation to d = 4
2 . It is encouraging that as the order
increases, the two resulting estimates approach each other. Even so, we nd that for small
Nf the N2LO Pade approximations are spread, so the expansion at this order does not
provide a de nite prediction. As Nf increases the situation improves, namely all NLO and
N2LO approximations begin to converge.
In table 4 we list the numerical values for the various estimates of the bilinear scaling
dimensions for Nf = 1; : : : ; 10.
Next, we compare to the largeNf predictions for the scaling dimensions of the bilinears.
The Pade approximants used to estimate the dimensions of Bs(i0n)g and Ba(0d)j do not develop
a pole in the extrapolation region 0
consider the approximants evaluated at
Nf = 1, i.e.,
12 for any value of Nf
1. Therefore, we can
= 12 , as a function of Nf , expand them around
For Bs(i0n)g, the prediction from largeNf is [5, 6]
and compare the coe cient c(k;l) with its exact value obtained from the largeNf expansion,
clargeNf . In what follows we use ' to denote that we display only two signi cant digits.
and the extrapolation obtained from the threeform singlet gives
Nf
+ O(Nf 2) ;
Bs(i0n)g:
(5.16)
(5.17)
(5.18)
2:6
)32:4
(2:2
2:0
5
6
Nf
Bs(i3n)g : LO
Bs(i3n)g : NLO Pade (1,1)
Bs(i3n)g : N2LO Pade (2,1)
Bs(i3n)g : N2LO Pade (1,2)
(0)
(0)
Badj
2:0
1:8
1:6
1:4
1:2
1:0
1
2
3
4
7
8
(left panel), and Ba(0d)j (right panel) at d = 3. The di erent colors for Ba(0d)j correspond to estimates
from di erent continuations of the operator (see legend).
For Ba(0d)j, the prediction from largeNf is [4, 6]
and the extrapolations obtained from the threeform and scalar adjoints give
(0)
Badj:
Ba(3d)j:
Ba(0d)j:
1:4 ;
This suggests that the extrapolation of the threeform may provide a better estimate for
the scaling dimension of the adjoint scalar at this order.
6
Conclusions and future directions
We employed the expansion to compute scaling dimensions of fourfermion and bilinear
operators at the IR xed point of QED in d = 4 2 . We estimated the corresponding value
for the physically interesting case of d = 3. The results seem to con rm the expectations
from the enhancement of the global symmetry as d ! 3 (see gures 4 and 5). Therefore,
going beyond the leading order gave us more con dence that the continuation is sensible.
5
6
Nf
Ba(0d)j : LO
Ba(0d)j : NLO Pade (1,1)
Ba(0d)j : N2LO Pade (2,1)
Ba(3d)j : LO
Ba(3d)j : NLO
Ba(3d)j : N2LO Pade (2,1)
Ba(3d)j : N2LO Pade (1,2)
(0)
(5.19)
(5.20)
(5.21)
At the same time, it appears that  with the exception of the scalarsinglet bilinear
 to obtain precise estimates for the scaling dimensions for small values of Nf requires
even higherorder computations and perhaps more sophisticated resummation techniques
(see for instance chapter 16 of ref. [62] and references therein). The computation of such
higher orders in
via the standard techniques used in the present work would require hard
Feynmandiagram calculations.
In recent years, several authors exploited conformal symmetry to introduce a variety
of novel techniques to compute observables of the
xed point in expansion. Ref. [63]
proposed an approach based on multiplet recombination, further applied and developed in
refs. [64{79]. Another approach is the analytic bootstrap, either together with the
largespin expansion [80], or in its Mellinspace version [81{84]. Finally, ref. [85] aimed at directly
computing the dilatation operator at the WilsonFisher xed point. It would be interesting
to attempt to apply these techniques to QED in d = 4
On a di erent note, ref. [86] recently argued that QCD3 with massless quarks undergoes
a transition from a conformal IR phase, which exists for su ciently large number of avors,
to a symmetrybreaking phase when Nf
f
N c. This is analogous to the longstanding
f
conjecture for QED3, and so fourfermion operators may play the same role. Therefore,
at least for the case of zero ChernSimons level, expansion can be employed in a similar
manner to estimate N c. A LO estimate appeared in ref. [87]. In light of our results for
QED3, it would be worth studying how this estimate is modi ed at NLO.
Acknowledgments
we thank Joachim Brod, Martin Gorbahn, John Gracey, Igor Klebanov, Zohar
Komargodski, and David Stone for their interest and the many helpful discussions. We are also
indebted to the Weizmann Institute of Science, in which this research began. Research at
Perimeter Institute is supported by the Government of Canada through Industry Canada
and by the Province of Ontario through the Ministry of Research & Innovation.
LQED+g.f. =
F
F
ai
D
a
(1
q q
q2
;
(A.1)
(A.2)
(A.3)
(A.4)
A
Feynman rules
From the QED Lagrangian in R gauge,
we obtain the Feynman rules
Ψa
Aμ
Ψa
Ψa
q
→
q
→
1
4
Ψa =
Aν =
1
2
i
q
=
;
i
q2
Aμ
=
ie
:
There is one additional counterterm coupling that we need to specify. It is a relic of the
procedure with which we regulate IR divergences (see section 3.3), which essentially breaks
gauge invariance. For this reason to consistently renormalize Green's function we need to
include a counterterm analogous to a mass for the photon, i.e.,
Only the oneloop value of mI2RA enters our computations. It reads
Aμ
Aν =
i mI2RA
:
mI2RA =
4Nf mI2RA + O( 2) :
To nd the EOMvanishing operators at the nonrenormalizable level we apply the
EOM of the fermion and photon. They read
D
a = 0 ;
D
= 0 ;
+ e a
a = 0 :
(A.7)
For brevity we use the shorthand notation
D
direction in which the derivative in D= acts, i.e. D(= and D=
*
D= .
D= and use an arrow to indicate the
We consider the Lagrangian with additional couplings proportional to the operators
introduced in section 3.1
L = LQED + X CiOi :
To compute the Green's function we need the Feynman rules of the operators we
insert, as well as all the structures that we need to project the amplitude. For instance, to
renormalize the Green's function of
A
with oneloop insertions of Q1 we need not only
the Feynman rule of Q1, but also the
A structure of all operators that Q1 generates
In our case, the Feynman rules for the following three nal states su ce:
Ψa
where the structures SO , S~O , S^O depend on the inserted operator. For the set of operators
Ψa
Ψb
Aμ
q
→
O
O
Ψa
Ψb
O
O
p
→
Aμ
O
q
↓
Ψa
O
(A.5)
(A.6)
(A.8)
HJEP12(07)54
relevant to our computation they read
S
O
O
Q1 : 2
O
N2 :
2
4
q2q q :
In this appendix we list the mixingrenormalization constants of fourfermion operators.
First we list the constants we need to compute the ADM of avorsinglet fourfermion
operators, which we discussed in the main text, and subsequently the constants entering
the computation of the ADM of avornonsinglet fourfermion operators, which we discuss
in appendix C.
and (3.5) via
B.1
Flavorsinglet fourfermion operators
The divergent and nite pieces of the oneloop constants of the mixing between physical and
evanescent operators are directly related to the oneloop anomalous dimension of eqs. (3.4)
with O; O0 any physical or evanescent operator from section 3.1. To extract these constant
from the
Green's function we had to rst compute the oneloop mixing of the
fourfermion operators into the EOMvanishing operator N1. For the physical operators the
corresponding constants are
ZQ(11;1N) 1 =
ZQ(13;1N) 1 =
4
3
8 ;
(2Nf + 1) ;
ZQ(11;0N) 1 = 0 ;
ZQ(13;0N) 1 = 0 ;
and for the evanescent operators they are
ZE(1n;1N)1 = 0 ;
ZE(1n;0N)1 = ( 1) 2
16(n
2)(n
5)!
4
3
(2Nf + 1)an
8bn ;
(A.9)
(A.10)
(A.11)
(B.1)
(B.2)
(B.3)
(B.4)
5. To compute these constants for generic n we used Cli
ordalgebra identities from ref. [46].
As explained in section 3.2, in the computation of the mixing at twoloop level more
operators enter. The only oneloop mixings entering the computation, apart from those
above, is the mixing of the physical fourfermion operators into the EOMvanishing operator
N2, and the gaugevariant operator P. The former vanish, i.e.,
and the latter read
ZQ N2
ZQ(1;P0) = 0 ;
with Q = Q1; Q3.
Finally, the twoloop mixing constants of the two physical operators read
ZQ(2;Q2)0 = " 29 24Nf2 + 20Nf + 103 23 (3Nf + 1)#
;
ZQ(2;Q1)0 =
838 (3Nf + 1)
" 514 (8Nf + 2275)
49 (107Nf + 253)
"
1
2
+ a5 23 (3Nf + 14) 1
22
19 (3Nf + 49)#
56 (8Nf + 9)
0
#
"
0
+ b5 44
+ a7
12 0
0 0 #
1 ;
2
(103Nf + 86)
3 a5(2Nf + 1) 8b5 :
#
with Q = Q1; Q3. We do not list the corresponding constants for the evanescent operators
because they do not enter the twoloop computation of the mixing of physical operators.
In table 1 we summarised on which renormalization constants the Green's functions
we computed depend on. We see that to determine the twoloop mixing of the fourfermion
operators we rst need to determine the twoloop mixing of the physical operators into the
two EOMvanishing operators N1 and N2. The corresponding constants read
(12Nf2 + 10Nf + 11) ;
(Nf + 11) ;
ZQ(21;2N) 2 =
ZQ(21;2N) 1 =
ZQ(23;2N) 2 =
ZQ(23;2N) 1 =
4
9
8
3
8
9 Nf (2Nf + 1) ;
16
3
Nf ;
(24Nf + 11) ;
ZQ(21;1N) 2 =
ZQ(21;1N) 1 =
ZQ(23;1N) 2 =
ZQ(23;1N) 1 =
8
9 Nf ;
4
27
32
9
8
9
4
Nf ;
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
HJEP12(07)54
Flavornonsinglet fourfermion operators
The renormalization of the Green's functions with insertions of avornonsinglet
fourfermion operators is analogous to the one with
avorsinglets but less involved. Their
avoro diagonal structure forbids them to receive contributions from any EOMvanishing
or gaugevariant operator at twoloop order. Therefore, in this case we only need the mixing
constants within the physical and evanescent sectors.
As in the
avorsinglet case, the oneloop mixing is directly related to the oneloop
anomalous dimensions of eqs. (C.4) and (C.5) via
with O; O0 any physical or evanescent avornonsinglet fourfermion operator; the
oneloop anomalous dimensions above are given in appendix C. Finally, the twoloop mixing
constants of the two physical operators read
24Nf 18
81
2
4(11Nf
"
+ a5
+ a7
12 0
9)
1
2
12) 1
19 (Nf + 63) #
16 (32Nf + 3)
0
#
"
0
+ b5 36
"
0 0 #
1 ;
2
#
with Q = Q1; Q3.
C
Flavornonsinglet fourfermion operators
In the main part of this work we investigated bilinear and avorsinglet fourfermion
operators. There exist also fourfermion operators that are not singlets under avor. The ones
we consider in this appendix are spanned by the basis
Q1 = Tbadc( a
Q3 = Tbadc( a 3
b)( c
d) ;
b)( c 3
En = Tbadc( a n 1::: n b)( c n
d) ;
d) + anQ1 + bnQ3 ;
with Tdabc = Tbcda and Taadc = Tbadb = 0. The computation of their ADM at one and twoloop
order entails only a subset of the Feynman diagrams needed for avorsinglet case and is
actually less involved as discussed in appendix B. In this appendix we present their ADM
and their scaling dimensions at the IR
xed point in d = 4
2 , and use this to estimate
the corresponding d = 3 observables.
(B.14)
(B.15)
(C.1)
(C.2)
(C.3)
In the avor scheme, the full oneloop ADM of the physical and evanescent operators
and the twoloop entries required read:
(1;0) = <
nm
8 2n(n
1)(n
5)(n
3)
for m = n
for
8 "
(2;0) = <
nm
176Nf 78
2 0
83 Nf 2
#
+ b5 72
49 Nf # +
634 Nf
"
0
2 #
0
0
not required
for n; m = 1; 3
for n
5 and m = 1; 3
The part of the oneloop result that does not depend on an and bn was rst computed in
ref. [50].
Next we evaluate these ADMs at the xed point
(C.5)
(C.6)
(C.7)
(C.8)
8Nf2 324 + 792Nf
>>>> +
8N3f2 a5
153 + 2Nf
351 + 96Nf
#
" 0
+ 8N3f2 b5 108 4Nf
3 #
5)(n
3)an
an+2 + 36bn) for m = 1; n
1
0
#
729
3 0
4Nf 3
1)(n
1)(n
1)(n
3)bn
the scaling dimension of the
avornonsinglet fourfermion operators for various cases of Nf . To
obtain the twoloop ( 2)i values we implemented the algorithm to include the e ect of evanescent
( 1)1
( 2)1
( 1)2
( 2)2
9:00
35:6
9:00
101
operators [32].
( )1
LO
( )2
LO
NLO
4:50
8:53
4:50
29:3
Nf
NLO Pade (1,1)
3:00
3:63
3:00
14:9
2:25
1:95
2:25
9:40
1:80
1:19
1:80
6:67
1:50
0:782
1:50
5:09
1:29
0:544
1:29
4:08
1:12
0:393
1:12
3:38
1:00
0:292
1:00
2:87
0:900
0:221
0:900
2:49
1
0:500
3:26
8:50
16:7
2
3
4
5
6
7
8
1:75 2:50 2:88 3:10 3:25 3:36 3:44 3:50 3:55
3:17 3:22 3:30 3:37 3:43 3:49 3:53 3:57 3:60
6:25 5:50 5:12 4:90 4:75 4:64 4:56 4:50 4:45
1:09 1:78 2:78 3:23 3:48 3:62 3:72 3:78 3:83
avornonsinglet fourfermion operators at d = 3 for various values of Nf . Only three
signi cant digits are being displayed.
Following ref. [32] we shift to the scheme in which the physicalphysical subblock forms an
invariant subspace. In this scheme we are able to extract the schemeindependent O( 2)
corrections to the scaling dimensions, i.e., the ( 2)is. In table 5 we list the values for
the representative cases of Nf = 1; : : : ; 10 and in table 6 the LO and NLO predictions for
the scaling dimensions at d = 3. The NLO Pade (1,1) prediction of scaling dimension
( )2 contains poles in the extrapolation region
prediction instead.
2 [0; 12 ], so we list the xedorder NLO
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] J.A. Gracey, Electron mass anomalous dimension at O(1=(Nf2) in quantum electrodynamics,
Phys. Lett. B 317 (1993) 415 [hepth/9309092] [INSPIRE].
[2] J.A. Gracey, Computation of critical exponent
at O(1=Nf2) in quantum electrodynamics in
arbitrary dimensions, Nucl. Phys. B 414 (1994) 614 [hepth/9312055] [INSPIRE].
[3] W. Rantner and X.G. Wen, Spin correlations in the algebraic spin liquid: Implications for
highTc superconductors, Phys. Rev. B 66 (2002) 144501 [INSPIRE].
[4] M. Hermele, T. Senthil and M.P.A. Fisher, Algebraic spin liquid as the mother of many
competing orders, Phys. Rev. B 72 (2005) 104404 [condmat/0502215] [INSPIRE].
[5] M. Hermele, T. Senthil and M.P.A. Fisher, Erratum: Algebraic spin liquid as the mother of
[6] R.K. Kaul and S. Sachdev, Quantum criticality of U(1) gauge theories with fermionic and
bosonic matter in two spatial dimensions, Phys. Rev. B 77 (2008) 155105
[arXiv:0801.0723] [INSPIRE].
[7] C. Xu, Renormalization group studies on fourfermion interaction instabilities on algebraic
spin liquids, Phys. Rev. B 78 (2008) 054432.
[8] V. Borokhov, A. Kapustin and X.k. Wu, Topological disorder operators in threedimensional
conformal eld theory, JHEP 11 (2002) 049 [hepth/0206054] [INSPIRE].
HJEP12(07)54
[9] S.S. Pufu, Anomalous dimensions of monopole operators in threedimensional quantum
electrodynamics, Phys. Rev. D 89 (2014) 065016 [arXiv:1303.6125] [INSPIRE].
[10] E. Dyer, M. Mezei and S.S. Pufu, Monopole Taxonomy in ThreeDimensional Conformal
Field Theories, arXiv:1309.1160 [INSPIRE].
(2016) 069 [arXiv:1603.05582] [INSPIRE].
[11] S.M. Chester and S.S. Pufu, Anomalous dimensions of scalar operators in QED3, JHEP 08
[12] Y. Huh and P. Strack, Stress tensor and current correlators of interacting conformal eld
theories in 2 + 1 dimensions: Fermionic Dirac matter coupled to U(1) gauge eld, JHEP 01
(2015) 147 [Erratum ibid. 03 (2016) 054] [arXiv:1410.1902] [INSPIRE].
[13] Y. Huh, P. Strack and S. Sachdev, Conserved current correlators of conformal eld theories
in 2 + 1 dimensions, Phys. Rev. B 88 (2013) 155109 [arXiv:1307.6863] [INSPIRE].
[14] S. Giombi, G. Tarnopolsky and I.R. Klebanov, On CJ and CT in Conformal QED, JHEP 08
(2016) 156 [arXiv:1602.01076] [INSPIRE].
[15] I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement Entropy of 3D
Conformal Gauge Theories with Many Flavors, JHEP 05 (2012) 036 [arXiv:1112.5342]
[INSPIRE].
[arXiv:1601.03476] [INSPIRE].
(1972) 240 [INSPIRE].
[16] S.M. Chester and S.S. Pufu, Towards bootstrapping QED3, JHEP 08 (2016) 019
[17] K.G. Wilson and M.E. Fisher, Critical exponents in 3:99 dimensions, Phys. Rev. Lett. 28
[18] L. Di Pietro, Z. Komargodski, I. Shamir and E. Stamou, Quantum Electrodynamics in D = 3
from the
Expansion, Phys. Rev. Lett. 116 (2016) 131601 [arXiv:1508.06278] [INSPIRE].
[19] S.M. Chester, M. Mezei, S.S. Pufu and I. Yaakov, Monopole operators from the 4
expansion, JHEP 12 (2016) 015 [arXiv:1511.07108] [INSPIRE].
[20] S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QEDd, F Theorem and the
Expansion, J. Phys. A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
[21] R.D. Pisarski, Chiral Symmetry Breaking in ThreeDimensional Electrodynamics, Phys. Rev.
D 29 (1984) 2423 [INSPIRE].
Math. Phys. 95 (1984) 257 [INSPIRE].
QED, Phys. Rev. Lett. 60 (1988) 2575 [INSPIRE].
[22] C. Vafa and E. Witten, Eigenvalue Inequalities for Fermions in Gauge Theories, Commun.
[23] T. Appelquist, D. Nash and L.C.R. Wijewardhana, Critical Behavior in (2 + 1)Dimensional
[INSPIRE].
Abelian Higgs model, hepph/0403250 [INSPIRE].
[25] A.V. Kotikov, V.I. Shilin and S. Teber, Critical behavior of (2 + 1)dimensional QED: 1=Nf
corrections in the Landau gauge, Phys. Rev. D 94 (2016) 056009 [arXiv:1605.01911]
[26] A.V. Kotikov and S. Teber, Critical behavior of (2 + 1)dimensional QED: 1=Nf corrections
in an arbitrary nonlocal gauge, Phys. Rev. D 94 (2016) 114011 [arXiv:1609.06912]
[27] K. Kaveh and I.F. Herbut, Chiral symmetry breaking in QED3 in presence of irrelevant
HJEP12(07)54
interactions: A Renormalization group study, Phys. Rev. B 71 (2005) 184519
[condmat/0411594] [INSPIRE].
Rev. D 90 (2014) 036002 [arXiv:1404.1362] [INSPIRE].
(2009) 125005 [arXiv:0905.4752] [INSPIRE].
[28] J. Braun, H. Gies, L. Janssen and D. Roscher, Phase structure of many avor QED3, Phys.
[29] D.B. Kaplan, J.W. Lee, D.T. Son and M.A. Stephanov, Conformality Lost, Phys. Rev. D 80
[31] L. Janssen and Y.C. He, Critical behavior of the QED3GrossNeveu model: Duality and
decon ned criticality, Phys. Rev. B 96 (2017) 205113 [arXiv:1708.02256] [INSPIRE].
[32] L. Di Pietro and E. Stamou, Operator mixing in expansion: scheme and evanescent
(in)dependence, arXiv:1708.03739 [INSPIRE].
(2017) 017 [arXiv:1609.04012] [INSPIRE].
(2016) 095 [arXiv:1607.07457] [INSPIRE].
[33] A. Karch, B. Robinson and D. Tong, More Abelian Dualities in 2 + 1 Dimensions, JHEP 01
[34] P.S. Hsin and N. Seiberg, Level/rank Duality and ChernSimonsMatter Theories, JHEP 09
[35] F. Benini, P.S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality
in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].
[36] O. Aharony, Baryons, monopoles and dualities in ChernSimonsmatter theories, JHEP 02
(2016) 093 [arXiv:1512.00161] [INSPIRE].
031043 [arXiv:1606.01893] [INSPIRE].
[37] A. Karch and D. Tong, ParticleVortex Duality from 3d Bosonization, Phys. Rev. X 6 (2016)
[38] J. Murugan and H. Nastase, Particlevortex duality in topological insulators and
superconductors, JHEP 05 (2017) 159 [arXiv:1606.01912] [INSPIRE].
[39] N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2 + 1 Dimensions and
Condensed Matter Physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].
[40] C. Xu and Y.Z. You, Selfdual Quantum Electrodynamics as Boundary State of the three
dimensional Bosonic Topological Insulator, Phys. Rev. B 92 (2015) 220416
[arXiv:1510.06032] [INSPIRE].
[41] N. Karthik and R. Narayanan, No evidence for bilinear condensate in parityinvariant
threedimensional QED with massless fermions, Phys. Rev. D 93 (2016) 045020
[arXiv:1512.02993] [INSPIRE].
[42] S.J. Hands, J.B. Kogut, L. Scorzato and C.G. Strouthos, The chiral limit of noncompact
and Nf = 4, Phys. Rev. B 70 (2004) 104501 [heplat/0404013] [INSPIRE].
[44] C. Strouthos and J.B. Kogut, Chiral Symmetry breaking in Three Dimensional QED, J.
Phys. Conf. Ser. 150 (2009) 052247 [arXiv:0808.2714] [INSPIRE].
[45] J.A. Gracey, Three loop MSbar tensor current anomalous dimension in QCD, Phys. Lett. B
488 (2000) 175 [hepph/0007171] [INSPIRE].
HJEP12(07)54
Cambridge University Press, Cambridge (1986).
[48] S.G. Gorishnii, A.L. Kataev and S.A. Larin, Analytical Four Loop Result for function in
QED in Ms and Mom Schemes, Phys. Lett. B 194 (1987) 429 [INSPIRE].
[49] S.G. Gorishnii, A.L. Kataev, S.A. Larin and L.R. Surguladze, The Analytical four loop
corrections to the QED
function in the MS scheme and to the QED psi function: Total
reevaluation, Phys. Lett. B 256 (1991) 81 [INSPIRE].
[50] M.J. Dugan and B. Grinstein, On the vanishing of evanescent operators, Phys. Lett. B 256
(1991) 239 [INSPIRE].
[51] A. Bondi, G. Curci, G. Pa uti and P. Rossi, Ultraviolet Properties of the Generalized
Thirring Model With U(N ) Symmetry, Phys. Lett. B 216 (1989) 345 [INSPIRE].
[52] M. Beneke and V.A. Smirnov, Ultraviolet renormalons in Abelian gauge theories, Nucl. Phys.
[53] S. Herrlich and U. Nierste, Evanescent operators, scheme dependences and double insertions,
B 472 (1996) 529 [hepph/9510437] [INSPIRE].
Nucl. Phys. B 455 (1995) 39 [hepph/9412375] [INSPIRE].
25 (1967) 29 [INSPIRE].
Lett. B 344 (1995) 308 [hepph/9409454] [INSPIRE].
[54] L.D. Faddeev and V.N. Popov, Feynman Diagrams for the YangMills Field, Phys. Lett. B
[55] M. Misiak and M. Munz, Two loop mixing of dimension ve avor changing operators, Phys.
[56] K.G. Chetyrkin, M. Misiak and M. Munz, functions and anomalous dimensions up to three
loops, Nucl. Phys. B 518 (1998) 473 [hepph/9711266] [INSPIRE].
[57] P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279
Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
[58] J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys.
[59] C. Bobeth, M. Misiak and J. Urban, Photonic penguins at two loops and mt dependence of
BR[B ! Xs`+` ], Nucl. Phys. B 574 (2000) 291 [hepph/9910220] [INSPIRE].
[60] G. 't Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl.
Phys. B 44 (1972) 189 [INSPIRE].
[61] P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle,
(2015) 29FT01 [arXiv:1505.00963] [INSPIRE].
JHEP 11 (2015) 040 [arXiv:1506.06616] [INSPIRE].
[64] P. Basu and C. Krishnan, expansions near three dimensions from conformal eld theory,
from conformal eld theory, JHEP 03 (2016) 174 [arXiv:1510.04887] [INSPIRE].
HJEP12(07)54
arXiv:1512.05994 [INSPIRE].
[67] E.D. Skvortsov, On (Un)Broken HigherSpin Symmetry in Vector Models,
[68] S. Giombi and V. Kirilin, Anomalous dimensions in CFT with weakly broken higher spin
symmetry, JHEP 11 (2016) 068 [arXiv:1601.01310] [INSPIRE].
(2016) 107 [arXiv:1605.08868] [INSPIRE].
[69] K. Nii, Classical equation of motion and Anomalous dimensions at leading order, JHEP 07
[70] S. Yamaguchi, The expansion of the codimension two twist defect from conformal eld
theory, PTEP 2016 (2016) 091B01 [arXiv:1607.05551] [INSPIRE].
[71] V. Bashmakov, M. Bertolini and H. Raj, Broken current anomalous dimensions, conformal
manifolds and renormalization group ows, Phys. Rev. D 95 (2017) 066011
[arXiv:1609.09820] [INSPIRE].
[72] C. Hasegawa and Yu. Nakayama, Expansion in Critical 3Theory on Real Projective Space
from Conformal Field Theory, Mod. Phys. Lett. A 32 (2017) 1750045 [arXiv:1611.06373]
[73] K. Roumpedakis, Leading Order Anomalous Dimensions at the WilsonFisher Fixed Point
from CFT, JHEP 07 (2017) 109 [arXiv:1612.08115] [INSPIRE].
JHEP 05 (2017) 041 [arXiv:1701.06997] [INSPIRE].
[74] S. Giombi, V. Kirilin and E. Skvortsov, Notes on Spinning Operators in Fermionic CFT,
[75] F. Gliozzi, A. Guerrieri, A.C. Petkou and C. Wen, Generalized WilsonFisher Critical Points
from the Conformal Operator Product Expansion, Phys. Rev. Lett. 118 (2017) 061601
[arXiv:1611.10344] [INSPIRE].
[76] F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, The analytic structure of conformal
blocks and the generalized WilsonFisher xed points, JHEP 04 (2017) 056
[arXiv:1702.03938] [INSPIRE].
[77] A. Codello, M. Safari, G.P. Vacca and O. Zanusso, Leading CFT constraints on multicritical
models in d > 2, JHEP 04 (2017) 127 [arXiv:1703.04830] [INSPIRE].
[78] C. Behan, L. Rastelli, S. Rychkov and B. Zan, Longrange critical exponents near the
shortrange crossover, Phys. Rev. Lett. 118 (2017) 241601 [arXiv:1703.03430] [INSPIRE].
[79] C. Behan, L. Rastelli, S. Rychkov and B. Zan, A scaling theory for the longrange to
shortrange crossover and an infrared duality, J. Phys. A 50 (2017) 354002
[arXiv:1703.05325] [INSPIRE].
(2016) 445401 [arXiv:1510.07770] [INSPIRE].
bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
HJEP12(07)54
(2017) 019 [arXiv:1612.05032] [INSPIRE].
many competing orders , Phys. Rev. B 76 ( 2007 ) 149906 [arXiv: 0709 .3032].
[30] S. Gukov , RG Flows and Bifurcations, Nucl. Phys. B 919 ( 2017 ) 583 [arXiv: 1608 .06638] QED in threedimensions, Nucl . Phys. Proc. Suppl . 119 ( 2003 ) 974 [ hep lat/0209133] [43] S.J. Hands , J.B. Kogut , L. Scorzato and C.G. Strouthos , Noncompact QED3 with Nf = 1 [46] A.D. Kennedy , Cli ord Algebras in Two ! Dimensions, J. Math. Phys. 22 ( 1981 ) 1330 [47] J.C. Collins , Renormalization, Cambridge Monographs on Mathematical Physics , vol. 26 , [62] H. Kleinert and V. SchulteFrohlinde , Critical properties of 4theories , ( 2001 ) [INSPIRE]. [63] S. Rychkov and Z.M. Tan , The expansion from conformal eld theory , J. Phys. A 48 [65] S. Ghosh , R.K. Gupta , K. Jaswin and A.A. Nizami , Expansion in the GrossNeveu model [80] L.F. Alday , Solving CFTs with Weakly Broken Higher Spin Symmetry , JHEP 10 ( 2017 ) 161 [81] K. Sen and A. Sinha , On critical exponents without Feynman diagrams , J. Phys. A 49 [82] R. Gopakumar , A. Kaviraj , K. Sen and A. Sinha , Conformal Bootstrap in Mellin Space, Phys. Rev. Lett . 118 ( 2017 ) 081601 [arXiv: 1609 .00572] [INSPIRE]. [83] R. Gopakumar , A. Kaviraj , K. Sen and A. Sinha , A Mellin space approach to the conformal [84] P. Dey , A. Kaviraj and A. Sinha , Mellin space bootstrap for global symmetry , JHEP 07