Lowenergy effective action in twodimensional SQED: a twoloop analysis
HJE
Lowenergy e ective action in twodimensional SQED:
I.B. Samsonov 0 1
Lower Dimensions, Superspaces
0 141980 Dubna , Moscow region , Russia
1 Bogoliubov Laboratory of Theoretical Physics , JINR
We study twoloop quantum corrections to the lowenergy e ective actions in N = (2; 2) and N = (4; 4) SQED on the Coulomb branch. In the latter model, the lowenergy e ective action is described by a generalized Kahler potential which depends on both chiral and twisted chiral super elds. We demonstrate that this generalized Kahler potential is oneloop exact and corresponds to the N = (4; 4) sigmamodel with torsion presented by Rocek, Schoutens and Sevrin [1]. In the N = (2; 2) SQED, the e ective Kahler potential is not protected against higherloop quantum corrections. The twoloop quantum corrections to this potential and the corresponding sigmamodel metric are explicitly found.
Extended Supersymmetry; Supersymmetric Gauge Theory; Field Theories in

1 Introduction and summary
2
Exact propagators on constant vector multiplet background
Classical action and loop expansion of the e ective action
Oneloop e ective action and the WessZumino term
Vanishing of twoloop corrections to generalized Kahler potential
5
A
Conclusions
N = (2; 2) superspace conventions
2.1
2.2
2.3
2.4
2.5
3.1
3.2
4.1
4.2
4.3
3.2.1
3.2.2
3.2.3
3.3.1
3.3.2
3.3.3
1
Introduction and summary
Twodimensional supersymmetric gauge theories have a wide range of applications in
physics and geometry. In
eld theory, 2d gauged linear sigmamodels in N = (2; 2)
superspace serve as canonical examples which provide very useful insights on lowenergy
dynamics of fourdimensional supersymmetric gauge theories [2, 3]. Geometrically,
twodimensional nonlinear sigmamodels with extended supersymmetry appear very reach
because of existence of numerous twistedchiral multiplets [4] which possess no analogs in
generalized Kahler geometry. It is natural to expect that some of these geometries may arise
as lowenergy e ective actions in twodimensional gauge theories in N = (2; 2) superspace.
The study of lowenergy e ective action in Abelian gauge theories in N = (2; 2)
superspace was initiated long ago [2]. The authors of this work showed that the eld strength
of N
by
= (2; 2) vector multiplet is given by a twisted chiral super eld which we denote
throughout this work. The e ective action for
may have a superpotential W ( )
and a Kahler potential K( ; ), as well as higherderivative terms which form together the
EulerHeisenbergtype e ective action in twodimensional SQED. The structure of oneloop
quantum corrections to these potentials was found in [2]:
W (1)( ) /
ln
K(1)( ; ) / ln ln :
;
(1.1)
(1.2)
HJEP07(21)46
At leading order, one can discard higherderivative terms in the e ective action and treat
the lowenergy theory as a (2; 2) sigmamodel with the Kahler potential (1.2) and
superpotential (1.1).
The superpotential (1.1) is known to be oneloop exact and its form is completely
determined by the anomaly of U(1)
U(1) Rsymmetry [2]. The Kahler potential can,
however, receive higherloop quantum corrections. This paper aims to trigger the study of
quantum corrections to the e ective Kahler potential K( ; ) and corresponding
sigmamodel geometry beyond oneloop order.
We consider twoloop e ective action in N = (2; 2) and (4; 4) SQED on the Coulomb
branch. In the (2; 2) case, the Coulomb branch is known to exist only when the U(1)
charges of chiral multiplets sum to zero [3, 5]. This is typically satis ed for the SQED
with two chiral multiplets which carry opposite charges with respect to the gauge group.
For this theory we explicitly compute twoloop quantum corrections to the e ective Kahler
potential K(2)( ; ).
An important feature of twodimensional gauge theories is that Feynman graphs with
internal (super)photon lines su er from IR divergencies. We show that for supersymmetric
gauge theories in the N = (2; 2) superspace it is possible to introduce gauge invariant mass
term for the vector multiplet which naturally regulates such IR divergencies. This mass
term may be obtained by the dimensional reduction from the threedimensional (super)
ChernSimons action which is also known to be responsible for the gaugeinvariant mass
of the vector multiplet in three dimensions. In our case, the twoloop quantum corrections
to the e ective action explicitly depend on the vector multiplet mass and are singular in
the limit when this mass vanishes.
The (4; 4) vector multiplet in the N
= (2; 2) superspace is described by the pair
( ; ) where
is a twisted chiral multiplet [4]. At leading order in the
derivative expansion, the lowenergy e ective action in the N = (4; 4) SQED is described by
a generalized Kahler potential K( ; ; ; ). Performing explicit quantum computations
we demonstrate that this potential does not receive twoloop quantum corrections and is
oneloop exact. At oneloop order, this function coincides with the potential for the (4; 4)
{ 2 {
sigmamodel with torsion studied in [1]1
1
4
loop exact and quantizes (see, e.g., [9]). This con rms the nonrenormalization of the
potential (1.3) claimed in [10].
Qualitatively, the presence of the WessZumino term in the lowenergy e ective action
out the massive chiral multiplets and consider e ective action for the light vector multiplet.
However, the total contribution to the anomaly should be the same at low and high energies
since the anomaly cannot depend on the energy scale. Thus, the lowenergy e ective action
must include the WessZumino term compensating the contribution to the anomaly from
the fermions that were integrated out. This statement is well known as the 't Hooft anomaly
matching argument [11].
It is pertinent to mention here the amazing analogy of the e ective
potentials (1.1), (1.2) and (1.3) with certain terms in lowenergy e ective actions of
fourdimensional N = 2 and N = 4 gauge theories. Recall that the 4d N = 2 gauge
multiplet may be described by an N
= 2 chiral super eld W. The superpotential (1.1)
is somewhat similar to the socalled holomorphic potential [
12
] F (W) / W2 ln W while
the Kahler potential (1.2) formally coincides with the nonholomorphic potential [13]
H(W; W) / ln W ln W. This analogy is not accidental: both F (W) and the
superpotential (1.1) appear as a result of integration of the anomaly of U(1) Rsymmetry (see [
12
]
and [2], correspondingly). Surprisingly, the potential (1.3) nicely correlates with the
lowenergy e ective action of 4d N = 4 SYM e ective action in the N = 2 superspace [14].
Indeed, the
rst term in the righthand side of (1.3) formally coincides with the
nonholomorphic potential H(W; W) while the last term in (1.3) is very similar to the
hypermultiplet completion of the nonholomorphic potential that was constructed in [14].
This analogy is even more striking. Indeed, in [15] it was demonstrated that the
lowenergy e ective action in N = 4 SYM theory contains the WessZumino term for scalar
elds which originates from the 't Hooft anomaly matching for the Rsymmetry. This
WessZumino term implies the nonrenormalization of the coe cient in front of the
nonholomorphic potential beyond one loop. As we show in this paper, the potential (1.3) is
also responsible for the WessZumino term for twodimensional scalars, and exactly the
same arguments provide its nonrenormalization.
One of the results of this paper is the illustration of the deep interplay between the
twodimensional N = (4; 4) SQED and 4d N = 4 SYM theory at low energies, although
they are very di erent in general.
1This sigmamodel can be considered as a particular case of the N = (4; 4) superLiouville theory which
a short review of the gauge theory in N = (2; 2) superspace and consider basic
properties of the parallel displacement propagator which is a key ingredient of the technique of
gaugecovariant perturbative computations (for 4d gauge theories in N = 1 superspace this
technique was developed in [16{18] and for eld theory on the supergravity background
in [19{22]). Making use of this propagator, we construct exact Green's functions for chiral
super elds on covariantly constant vector multiplet background. In section 3, we compute
the lowenergy e ective action in N = (2; 2) SQED with di erent numbers of chiral
multiplets. We start with a review of old results [2] of oneloop quantum contributions to the
e ective action and show how they can be naturally reproduced by taking advantage of
the technique of covariant perturbative computations in the N = (2; 2) superspace. This
technique is then applied to compute twoloop quantum corrections to the e ective
action of N = (2; 2) SQED on the Coulomb branch. In section 4, we study the structure
of lowenergy e ective action in N = (4; 4) SQED to the twoloop order in perturbation
theory and discuss its interplay with the 4d N = 2 and N = 4 SYM e ective actions. The
Conclusions section is devoted to discussions of possible extensions of the results of this
work. In appendix we give a summary of our superspace conventions.
2
Exact propagators on constant vector multiplet
background
In this section we consider twodimensional nonAbelian gauge theory in N = (2; 2)
superspace and, following [18], we introduce parallel displacement propagator which is a key
ingredient of gaugecovariant technique of multiloop quantum computations. Using the
properties of this propagator we construct exact heat kernels for basic Green's functions
on covariantly constant vector multiplet background. In the Abelian case, we apply these
heat kernels in the subsequent sections to compute twoloop quantum correction to the
e ective action. We hope that the results of this section will be of use also for the study of
e ective action in nonAbelian gauge theories which will be considered elsewhere. We keep
the structure of this section close to the corresponding presentation in [23] to facilitate the
comparison with the threedimensional gauge theory in N = 2 superspace.
2.1
Gauge theory in N = (2; 2) superspace
We consider the twodimensional N = (2; 2) superspace with coordinates zA = (xm; ; ),
where xm, m = 0; 1, are the Minkowski space coordinates,
= 1; 2, are Grassmann
coordinates and
= ( ) are their complex conjugate. Our superspace conventions are
summarized in appendix. They are chosen to be close to the ones employed in the series
of papers [23{27] devoted to the study of super eld theories in threedimensional N = 2
superspace.
The (nonAbelian) gauge theory in the N = (2; 2) superspace is described by the set
of gaugecovariant superspace derivatives
rA = (rm; r ; r ) = DA + VA ;
(2.1)
{ 4 {
where DA
= (@m; D ; D ) are supercovariant derivatives, see (A.3), and VA
=
(Vm; V ; V ) are gauge connections subject to the constraints
fr ; r g =
[r ; rm] =
2i( m
)
( m)
[rm; rn] = iFmn :
rm + 2i"
G + 2 3 H ;
W ;
[r ; rm] = ( m)
W ;
Here G, H, W
and Fmn are super eld strengths with the following conjugation properties
Gy = G ;
Hy = H ; (W )y = W ; (Fmn)y = Fmn :
HJEP07(21)46
In its turn, the tensor eld strength Fmn is expressed via W
and W ,
Fmn
"mnf =
1
4 "mn( 3) (r W
r W ) :
Another important relation appears by commuting (2.2a) with the super eld G and
applying properties (2.5)
transformations
r W
+ r W
=
2i m
rmG + 2 3 [H; G] + "
r W :
The algebra of covariant derivatives (2.2) is invariant under the
gauge
rA ! ei (z)
rAe i (z) ;
with (z) being real gauge super eld parameter, y = .
The gauge connections VA may be expressed via a prepotential. In this paper we
will use the real super eld prepotential V which is introduced in such a way that the
gaugecovariant spinor derivatives acquire the form (chiral representation)
In two dimensions, the antisymmetric tensor Fmn has only one independent component,
Fmn = "mnf , where for the antisymmetric "tensor "mn =
"nm with vector indices we use
the convention "01 =
r W
= r W :
r
= e 2V D e2V ;
r
= D :
{ 5 {
(2.2a)
(2.2b)
(2.2c)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
i
4
1
4
In this case, as a consequence of (2.2a), the scalar super eld strengths are expressed via
the prepotential as
G =
D (e 2V D e2V ) ;
H =
The expressions of the other super eld strengths in terms of V can be obtained using (2.5)
and (2.8). Note that all these super eld strengths transform covariantly under the gauge
transformation of the prepotential
e2V
! ei e2V e i ;
with a chiral .
The super eld strengths G and H can be considered as the real and imaginary parts
of a complex super eld
and its (Hermitian) conjugate
= G + iH ;
= G
iH :
From (2.5) it is easy to deduce twisted chirality properties of these super elds
r+
= r
HJEP07(21)46
(2.14)
(2.15)
where (r+; r )
(r1; r2). The existence of such twisted chiral super elds is an
important feature of twodimensional gauge theory in superspace as compared with the
higherdimensional cases. These super eld strengths play central role in super eld description of
gauge theories in the N = (2; 2) superspace.
2.2
Parallel displacement propagator in N = (2; 2) superspace
In superspace, the parallel displacement propagator was introduced in the work [18] as a
key ingredient which provides correct transformation properties of Green's functions and
corresponding heat kernels under gauge transformations. This allowed the authors of [18]
to develop a gaugecovariant procedure of perturbative computations of e ective actions
in supersymmetric gauge theories. In particular, this technique appeared very fruitful in
the study of lowenergy e ective actions in di erent fourdimensional gauge theories in
N = 1 and N = 2 superspaces [28{34]. For threedimensional gauge theories this method
was generalized in [23, 27]. This section is aimed at extending the basic concepts of the
procedure of covariant perturbative computations to twodimensional gauge theories in the
N = (2; 2) superspace.
Let us consider a super eld
in some representation R of the gauge group, and its
Hermitian conjugate
transforming in the representation R,
(z) !
0(z) = ei (z) (z) ;
(z) !
0(z) =
(z)e i (z) ;
where
=
y is Hermitian, but otherwise arbitrary gauge super eld parameter.
Correspondingly, Green's function for these elds G(z; z0) = ih (z) (z0)i has the transformation
property
G(z; z0) ! ei (z)G(z; z0)e i (z0) :
(2.16)
(2.17)
{ 6 {
In a similar way, the parallel displacement propagator I(z; z0) is, by de nition, a
twopoint superspace function which transforms under the gauge group as
Moreover, it is required to obey the di erential equation
and the boundary condition
I(z; z0) ! ei (z)I(z; z0)e i (z0) :
A
rAI(z; z0) =
A(DA + VA)I(z; z0) = 0 ;
The latter means that at coincident superspace points I(z; z0) reduces to the identity
operator in the gauge group. In eq. (2.19), A
interval with the components
( m; ; ) is the N = (2; 2) supersymmetric
m = (x
x0)m
i(
0)
m 0 + i 0
m (
0) ;
= (
0) ;
= (
It is possible to show that the properties (2.18) and (2.20) imply the important relation
I(z; z) = 1 :
The algebra of covariant derivatives (2.2) can be represented in the condensed form
[rA; rBg = TABC rC + iFAB ;
where TABC is the supertorsion and FAB is the eld strength for gauge super eld
connection (2.1). The nonvanishing components of these tensors can be read o from (2.2). They
appear in the following important relation for the derivative of the parallel displacement
propagator [18]
rBI(z; z0) = i X
1
( 1)n
n=1 (n + 1)!
An : : : A1
rA1 : : : rAn 1 FAnB(z)
+
(n
2
1) An TAnB
C An 1 : : : A1
rA1 : : : rAn 2 FAn 1C (z) I(z; z0) :
This identity shows that any covariant derivative of the parallel displacement propagator
may be expressed in terms of the parallel displacement propagator itself and covariant
derivatives of the super eld strength together with the torsion tensor. This identity appears
crucial in perturbative computations of lowenergy e ective action which is a functional of
these tensors.
In general, (2.25) is an in nite series over covariant derivatives of the eld strength
FAB. It is natural to expect that for certain eld con gurations this series terminates.
{ 7 {
In particular, it is possible to show that for the covariantly constant vector multiplet
the identity (2.25) reduces to
rm
= rm
rmW
= rmW
Green's function can be represented as a propertime integral of the corresponding heat
kernel Kv(z; z0js)
i
Z 1
0
As we will show in the following subsections, these identities appear very useful in
computing heat kernels of Green's functions of various operators in the N = (2; 2) superspace.
2.3
Real super eld Green's function and its heat kernel
The real super eld d'Alembertian is de ned by either expression
v =
=
1
1
By making use of the algebra (2.2), this operator may be brought to the form
Green's function Gv(z; z0) of this operator is de ned as a solution of the equation
where m is a mass parameter and 2j4(z
z0) is the full superspace delta function,
( v + m2)Gv(z; z0) =
2j4(z
z0) ;
2j4(z
z0) = 2(x
x0) 4(
0) :
where
! +0 implements standard boundary condition for the propagator. The equation
for the propagator (2.30) is satis ed when the heat kernel obeys the conditions
(i
d
ds
v)Kv(z; z0js) = 0 ;
s!0
lim Kv(z; z0js) = 2j4(z
z0) :
(2.33)
In general, it is very hard to solve these equations explicitly. Nevertheless, it is possible
to nd the exact solution for the heat kernel when the background gauge super eld obeys
the following two constraints:
i) Gauge multiplet obeys super YangMills equations of motion (onshell background)
HJEP07(21)46
ii) Field strengths are covariantly constant
r W
= 0 ;
rm
= rm
rmW
= rmW
= 0 :
W (s)
(s)
(s)
m(s)
O(s)W
O( s) = W (esN )
;
O(s)
O(s)
O( s) =
O( s) =
O(s) mO( s) =
m + i( m
)
+ W ((esN
1)N 1
)
W ((e sN
1)N 1
)
;
;
dt W (t) (t) + W (t) (t) ;
It is important to note that the compatibility condition for the constraint (2.35) requires
that the background gauge super eld belongs to the Cartan subalgebra of the Lie algebra of
the gauge group. This means that the background gauge super elds are (anti)commuting.
The procedure of solving the heat kernel equation for the covariantly constant vector
multiplet background was developed in the fourdimensional case in [17{20] and successfully
applied for threedimensional gauge theories in [23{25]. In the twodimensional case the
same procedure yields
where 2 =
, 2 =
Kv(z; z0js) =
1
sf
and O(s) is the `shift' operator
O(s) = es(W r
W r ) :
Within quantum loop computations, it is often necessary to know the value of the
heat kernel at coincident superspace points. For this aim, it is useful to have such a
representation for the heat kernel (2.36) where the operator O(s) appears on the right and
hits the parallel displacement propagator,
Kv(z; z0js) =
1
sf
e 4i (f coth sf) m(s) m(s) 2(s) 2(s)I(z; z0js) :
Here the operator O(s) is used to de ne the sdependent super eld strengths and
components of the superspace interval
{ 9 {
where
and (z; z0) solves for
In formulas (2.39) we have introduced the notation
The sdependent parallel displacement propagator (2.40) can be represented in the
Making use of (2.27a) and (2.27b) we nd
(z; z0) = i(
W
+
W )(
G + i( 3) H)
ih (z) T(z0)i =
( + + m2)G+(z; z0) =
(
+ m2)G (z; z0) =
+2j2(z; z0) ;
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
(2.46)
(2.47a)
(2.47b)
(2.48a)
(2.48b)
(z; z0js) can be found from the above formula just by replacing all
super eld strengths and components of the superspace interval by the corresponding
sdependent quantities from (2.39).
2.4
Heat kernel for chiral super eld Green's function
Consider gaugecovariant chiral super eld , r
= 0, and its Hermitian conjugate . The
d'Alembertian operators acting in the space of such elds are de ned in the standard way
+
=
1
16 r r
sentations for these operators
+ =
=
1
1
16 r r
2 2 = r
16 r r
2 2 = r
m
m
rm +
rm +
1
2
1
2
i
i
f ; g + 2 (r W ) + iW
f ; g
2j2(z; z0) are (anti)chiral deltafunctions which are related to the full superspace
For Green's functions (2.48) there are the associated heat kernels
G (z; z0) =
ds K (z; z0js)e s( +im2) ;
! +0 :
It is known [18, 28] that for the onshell vector multiplet background (2.34) the chiral
It should be noted that the identities (2.53) hold only for the onshell vector multiplet
background (2.34). The equations (2.51) imply similar relations for the corresponding
heat kernels
K+(z; z0js) =
41 r2Kv(z; z0js) ;
K (z; z0js) =
14 r2Kv(z; z0js) :
Thus, the computation of the heat kernels K
is reduced to nding the result of the action
of the operators r
2 and r
2 on the heat kernel (2.36).
It is possible to show that upon acting by r
2 2I(z; z0) since the factor in front of this function originates from e is v . The latter
operator commutes with r2 owing to the identities (2.53). Thus, for K+ we have
K+(z; z0js) =
1
sf
Applying (2.27b) we compute the action of the operator r
2 on the parallel displacement
propagator
2 on (2.36), this operator hits only
1 2r2( 2I(z; z0)) = 2
e 21 ( m)
m
W I(z; z0) :
It is easy to check these relations using the identities
G+(z; z0) =
41 r2Gv(z; z0) ;
G (z; z0) =
41 r2Gv(z; z0) :
r
2
+ = r
2
r
2
+ =
v =
2
r ;
e is
e 4i (f coth sf) m(s) m(s) 12 ( m)
m(s) (s)W (s)
Substituting this identity into (2.55) we nd
K+(z; z0js) =
e is
e 4i (f coth sf) m(s) m(s) 12 ( m)
m(s) (s)W (s)
Here we pushed the operator O(s) through on the right that resulted in making all objects
sdependent according to (2.39) and (2.40). In a similar way we nd the antichiral heat
kernel
K (z; z0js) =
(2.50)
(2.51)
(2.52)
(2.53)
(2.54)
(2.56)
(2.57)
(2.58)
We point out that the expressions for the (anti)chiral heat kernels are very similar to the
ones in the fourdimensional supersymmetric gauge theory [28].
Finally, we consider the propagators among chiral and antichiral super elds
With Green's functions (2.59) are associated the corresponding heat kernels
41 r2G+ (z; z0) + m2G (z; z0) =
41 r2G +(z; z0) + m2G+(z; z0) =
ds K+ (z; z0js)e s( +im2) ;
ds K +(z; z0js)e s( +im2) :
This subsection aims to nd explicit solutions for these heat kernels on the covariantly
constant vector multiplet background.
First of all, we point out that, as a consequence of the de nitions of covariantly
(anti)chiral d'Alembertian operators (2.47), Green's functions (2.59) are related to the
(anti)chiral ones (2.48) as
G+ (z; z0) =
41 r2G (z; z0) ;
G +(z; z0) =
Analogous relations hold for the corresponding heat kernels
Thus, the problem is reduced to nding the action of the operators r2 and r2 on the heat
kernels (2.57) and (2.58).
Let us consider the derivation of the heat kernel K+
in some details. It appears
upon acting by the operator r
operator commutes with the expression e is
from e is v . Thus, we need to
properties of the parallel displacement operator (2.27b)
2 on the heat kernel (2.58). Note that, owing to (2.53), this
e 4i (f coth sf) m m since the latter originates
nd the action of this operator on the rest using the
4 r
1 2 e 21 ( m)
m
W
2I(z; z0) = eR(z;z0)I(z; z0) ;
(2.59)
(2.60a)
(2.60b)
(2.61a)
(2.61b)
(2.62)
(2.63)
(2.64)
(2.65)
where
R(z; z0) =
i
i 2
6
G +
( 3) H
W
+
( m)
~
m
1
2
1
2
( m)
~m(
r
W :
W
+
W ) +
W
2i 2
3
is a modi cation of the supersymmetric interval which is chiral with respect to the rst
argument and antichiral with respect to the other
~
m =
m + i
m
D0 ~m = D ~
Given the function R(z; z0) in the form (2.65), we have the following representation for the
heat kernel K+
HJEP07(21)46
As the nal step, in (2.68) we have to push the operator O(s) through on the right and
hit the parallel displacement propagator according to eq. (2.40). This procedure e ectively
makes the super eld strengths and components of supersymmetric interval sdependent
according to (2.39)
We point out that, as follows from (2.65), the function R(z; z0) vanishes at coincident
Grassmann coordinates, R(z; z0)j !0 ! 0. However, the contribution from (2.71) is
nontrivial at coincident points.
3
3.1
Lowenergy e ective action in N
= (2; 2) SQED
General remarks
In general, Abelian gauge theories in N = (2; 2) superspace may include the following
terms in the classical action:
where R(z; z0js) = O(s)R(z; z0)O( s), and (s) is given by (2.45). For practical
computations, it is useful to represent the heat kernel (2.69) in the equivalent form
Here, the function R0(t) can be found explicitly from (2.65) using R0(t) = O(t)[W
r
r ; R]O( t) and combined with (2.45):
R0(t) + (t) = O(t) 2i
W
G
2( 3)
W
H +
W
W
5i
3
i 2W 2 + 2i( 3) ( 3)
W
W
+2i
W
W
11i 2
12
6
1
2
r W 2 + ( m)
~
m
r W 2
O( t) :
(2.71)
The kinetic term for the vector multiplet V
SV =
1 Z
2e2
d2j4z
:
Here e is the dimensional gauge coupling, [e] = 1, and d2j4z is the measure in the full
N = (2; 2) superspace (see appendix for our superspace conventions).
The mass term for the vector multiplet
Sm =
i m Z
where the integration goes over the twisted chiral subspace and m is, in general,
complex mass parameter. Without loss of generality, we can set it to be real, m = m,
just to simplify some formulas below. Note that the sum of actions (3.1) and (3.2)
amounts to the massive WessZumino model for the twisted chiral multiplet
. It
should be noted that the mass term (3.2) may be obtained by the dimensional
reduction from the 3d N = 2 ChernSimons action which plays role of the topological
mass term in threedimensional electrodynamics.
FayetIliopoulos (FI) term
where
SFI =
In (3.3), the real part of the FI parameter r couples with the auxiliary eld D of the
vector multiplet while the imaginary part
corresponds to the topological f term
and quantizes [3, 35].
N charged chiral multiplets Qi with charges qi and mass matrix mij
SQ =
d2j4z Qie2qiV Qi
d2j2z mij QiQj + c:c: :
(3.5)
XN Z
i=1
N
X
i;j=1
Z
Needless to say that the mass matrix mij should be such that the gauge invariance
is preserved. Chiral multiplets may also have real mass which originates from vevs
of scalars in the vector multiplet V .
More generally, it is also possible to study quantum dynamics of twisted chiral
multiplets as well as semichiral ones [4, 8, 36, 37], but such models are beyond the scope of
this paper.
Depending on the number of chiral multiplets and on the values of all mentioned above
parameters, Abelian gauge theories in the N = (2; 2) superspace exhibit di erent phases
which are thoroughly investigated in [3]. In this paper, we are interested in the e ective
action on the Coulomb branch. It is known that the necessary condition for existence of
the Coulomb branch at the quantum level is that the charges of all chiral multiplets should
sum to zero
N
Indeed, when this condition is not satis ed, the following two e ects occur: i) There
are UVdivergent tadpole Feynman graphs which result in the renormalization of the FI
parameter. These quantum corrections lift the Coulomb branch. ii) The e ective twisted
superpotential for the super eld strength
is generated at one loop [2]. This e ective
superpotential may also be interpreted as a functional reproducing correct transformation
properties of the e ective action under anomalous Rsymmetry. Correspondingly, when
the condition (3.6) is satis ed, there are no divergent quantum contributions to the FI
parameter and classical Coulomb branch is preserved at the quantum level. The latter
case is of primary importance for our studies as we are interested in the twoloop quantum
contributions to the e ective action on the Coulomb branch. However, in this section, for
the sake of completeness we will shortly consider a model for which the condition (3.6)
is not satis ed and will give a super eld derivation of the e ective twisted superpotential
obtained originally in [2] by component eld quantum computations.
The typical example of the models for which the constraint (3.6) is violated is the
supersymmetric electrodynamics with one chiral multiplet while the wellknown case when
this constraint is satis ed is the supersymmetric electrodynamics with two chiral multiplets
carrying opposite charges under the U(1) gauge symmetry. The latter will be studied in
section 3.3 while the former is considered just below.
3.2
3.2.1
SQED with one chiral avor
Classical action
In this section, we consider the supersymmetric electrodynamics with one chiral multiplet
carrying charge +1
S =
Z
d2j4z
1
2e2
Qe2V Q
m
2e2
Obviously, in the limit e ! 1 the classical action becomes scale invariant and
superconformal, though this symmetry is known to be broken by quantum corrections [2].
Recall that the N = (2; 2) vector multiplet contains a complex scalar
associated with
the lowest component of the super eld
i Z
2
j
;
'
m
e2 Re
where the barprojection means vanishing variables. Denoting the scalar elds in the
chiral multiplet by
it is not hard to nd the scalar potential which appears after elimination of auxiliary elds
e
2
2
'
Qj ;
Qj ;
2
V =
''
t
+ ''
:
(3.6)
(3.8)
(3.9)
(3.10)
The Coulomb branch is parametrized by the vev of the scalar eld
in the vector
multiplet while the scalars from the chiral multiplet must have vanishing vevs
Coulomb branch:
h i = const;
h'i = 0 :
The vanishing of the scalar potential (3.10) for such values of scalars is possible only for
special value of the FI parameter
t =
m
The even part of the oneloop e ective action
may be found by evaluating trace of
logarithm of square of this operator
Here we have taken into account the de nition (2.46) of the chiral covariant d'Alembertian
in terms of covariant spinor derivatives. Associated with this operator is the Green function
G+(z; z0) de ned in (2.48a) and the corresponding heat kernel K+(z; z0js), see eq. (2.50).
Thus, for the e ective action (3.16) we have the following propertime representation
where
Treatment of these parts in the e ective action requires slightly di erent computational
methods. Therefore, we will consider them separately.
3.2.2
Even part of the oneloop e ective action
Let H be the operator which appears in the matrix of second variational derivatives of S
with respect to the chiral super elds,
In this case the chiral multiplet acquires real mass proportional to the vev of
while the
vector multiplet (`photon') has a small mass m. Naively, one could study the e ective action
for the vector multiplet which appears by integrating out the massive chiral multiplet.
However, the constraint (3.12) appears to be ruined by oneloop quantum corrections and
the Coulomb branch is lifted at the quantum level [3]. Although this scenario is wellknown,
we will demonstrate it explicitly by computing oneloop e ective action in the model (3.7).
The details of these computations will be of use in subsequent sections.
In general, the e ective action for the vector multiplet V may have odd and even parts
with respect to the re ection V !
V ,
H =
i
4
0
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
1
4
0
where Tr+K+(s) means the heat kernel K+(z; z0js) at coincident superspace points, z0 = z,
and integrated over the chiral subspace
Z
Tr+ K+(s) =
d2j2z K+(z; zjs) :
This reduces the problem of computation of the even part of the e ective action to
evaluating the limit of coincident superspace points for the heat kernel, limz0!z K+(z; z0js).
Recall that we consider the lowenergy e ective action for the onshell, constant
vector multiplet background speci ed by the constraints (2.34) and (2.35). For this
background, the heat kernel K+ was found in the form (2.57). This formula involves di erent
HJEP07(21)46
sdependent objets de ned in (2.39) and (2.40). For the oneloop e ective action we need
the values of these objets at coincident superspace points when all components of the
superspace interval vanish, A ! 0. In particular, simple calculations yield
2(s)
A=0
= s2W 2 sinh2 sf
2 ;
(sf =2)2
where f is the component of the super eld strength tensor, Fmn = "mnf , which can
be regarded as a constant for the considered background. It is important to note that
the formula (3.19) contains W 2 that prevents any other contributions from the other
sdepended objets in (2.57). Thus, this kernel acquires simple form at coincident superspace
points
K+(z; zjs) =
sW 2e is
Substituting this expression into (3.17) we nd the even part of the oneloop e ective action
(3.18)
(3.19)
(3.20)
(3.21)
even =
It is instructive to rewrite the functional (3.21) in the full superspace
The term in the rst line here speci es the e ective Kahler potential for the twisted chiral
super eld
. The term in the last line in (3.22) takes into account all higherderivative
corrections with respect to the gauge super eld which can be considered as the
EulerHeisenberg e ective action.
We point out that the e ective action (3.22) was found for the rst time in [2] using
component led oneloop computations and in [38] by means of super eld methods. Here
we just gave a derivation of this e ective action by taking advantage of the super eld heat
kernel technique. This result will be useful in the study of lowenergy e ective action in
the model with two chiral avors which will be considered in section 3.3.
background
hJ i =
2hQe2V Qi = 2iG+ (z; z) ;
where hJ i is the e ective current. In the oneloop approximation, this e ective current
receives contributions only from the chiral super eld propagator in the vector multiplet
HJEP07(21)46
Thus, the computation of (3.23) is reduced to
nding the trace of the heat kernel
K+ (z; z0js).
The problem of evaluating the trace of the heat kernel K+ (z; z0js) is rather technically
involved. However, to
nd the odd part of the e ective action we don't actually need to
now the full expression for K+ (z; zjs). Indeed, the full expression for K+ (z; zjs) contains
di erent terms which are responsible both for odd and even parts of the e ective action.
Since the even part of the e ective action has been fully studied in the previous subsection,
here we have to focus only on possible contributions to
odd from K+ (z; zjs). For this
goal it is su cient to approximate K+ (z; zjs) by the terms with no derivatives of ,
K+ (z; zjs)
e is
:
Substituting (3.26) into (3.25) we have UVdivergent integral over the proper time s.
Introducing a small regularization parameter
this integral may be evaluated
hJ i =
1
2
1
+
ln(
) + O( ) ;
where
is the EulerMascheroni constant. Thus, we see that the odd part of the e ective
action is the sum of divergent and nite contributions
odd =
div +
n :
The divergent part of the e ective action can be immediately read o from the rst
The odd part of the e ective action cannot be found upon squaring of the operator (3.15).
Instead, to catch up the odd contributions we have to consider the general variation of the
e ective action with respect to the vector multiplet
=
Z
d2j4z V hJ i ;
where G+ (z; z0) is de ned in eq. (2.60a), with the mass parameter set to zero, m = 0. It
is useful also to represent this e ective current via the heat kernel K+
using (2.61a)
Z 1
0
hJ i = 2
ds K+ (z; zjs) :
term in (3.27)
of the FI parameter
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
1
4 s
1
2
8
div =
2
1 Z
d2j4z V =
d2j2z~
+ c:c:
This expression, being added to the classical action (3.7), leads to in nite renormalization
This means that even if we switch o the FI parameter classically, it is always generated by
oneloop divergent tadpole diagram. This is the origin of lifting of the classical Coulomb
branch by quantum corrections advocated in [3].
Substituting (3.27) into (3.23) we get the variation of nite terms in the odd part of
the e ective action
Integrating this variation we uncover the e ective twisted potential for
n =
2
d2j4z V ln(
) :
n =
i Z
4
d2j2z~ (ln
1) + c:c:
div =
n =
8
i
i
4
N
X q
i
Z
i=1
N
X qi ln qi
i=1
d2j2z~
This e ective twisted potential was found originally in [2] using component eld
quantum computations. Here we reproduced the same result using the method of covariant
perturbative computations in the N = (2; 2) superspace.
The above results can be readily generalized to the case of electrodynamics with N
chiral avors Qi, i = 1; : : : ; N , with charges qi, see eq. (3.5). For the odd part of the
e ective action we have the following modi cation of formulas (3.29) and (3.32):
This is the su cient condition of existence of the Coulomb branch on the quantum level [5].
To summarize, we have shown that the FI parameter in the model (3.7) receives in
nite oneloop quantum contributions (3.30) which lift the classical Coulomb branch. Such
in nite contributions may cancel among each other in the generalization of the model (3.7)
which involves N charged chiral avors (3.5). This happens when all charges of chiral
The equation (3.33) implies that there is no in nite renormalization of the FI parameter
when the condition (3.6) is satis ed. This is the necessary condition for existence of the
Coulomb branch. This condition is also su cient for vanishing of the e ective twisted
potential in (3.34). However, even when the condition (3.6) is satis ed, the rst term in
the righthand side of (3.34) remains nonvanishing and yields a nite shift of the complex
FI parameter
The main e ect of this nite quantum contribution is the shift of the imaginary part of the
complex FI parameter (3.4). To compensate this shift, one has to add the corresponding
value to the classical FI parameter
t ! t0 = t +
1
2
N
i=1
X qi ln qi :
t =
m
e2 hRe i
1
2
N
i=1
X qi ln qi :
(3.31)
(3.32)
multiplets sum to zero (3.6). However, there is still a nite shift of the imaginary part
of the FI parameter as in (3.35). Therefore, quantum Coulomb branch exists when the
classical FI parameter is tuned to a special value (3.36). Since we are interested in the
e ective action in the Coulomb branch, in subsequent sections we will always assume that
the condition (3.36) is satis ed.
We stress that all results of this subsection are not new; they are wellknown owing
to [2, 3, 5]. Here we just summarized them for the sake of completeness of our consideration.
Let us consider supersymmetric electrodynamics with two chiral multiplets Q+ and Q
carrying charges
1, respectively,
HJEP07(21)46
S = Sgauge[V ] + Smat[Q; V ] ;
Sgauge[V ] =
Smat[Q; V ] =
2e2
Z
i Z
2
d2j4z
where m is the mass of the chiral multiplet while m is the vector multiplet mass. The latter
is assumed to be small as compared to the former,
m
2
m2 + h
i
:
In this regime, we can study the e ective action for the light eld
which appears upon
integrating out the heavy chirals Q . In what follows, without loss of generality we will
assume that both m and m are real, though, in general, they may be complex.
Let '
be scalar elds in the chiral multiplets
(3.37a)
(3.37b)
(3.38)
(3.39)
(3.40)
(3.41)
'
= Q j
;
'
= Q j
:
After elimination of auxiliary elds, one can readily nd the scalar potential
e
2
2
V =
+ m2)('+'+ + ' ' ) :
m
e2 Re )2 + (
Similarly as in the model (3.7), the classical Coulomb branch (3.11) is possible at the
special value of the FI parameter (3.12). However, as is explained in the previous section,
the imaginary part of the FI parameter receives nite oneloop contributions as in eq. (3.35).
To compensate this contribution, we have to set up the corresponding value to the classical
FI parameter
This allows us to study the lowenergy e ective action for the light vector multiplet which
appears by integrating out heavy chiral multiplets beyond oneloop order.
t =
i
2
m
In the framework of the background eld method, we split the gauge super eld V into
background V and quantum v parts2
Upon this splitting, the actions (3.37a) and (3.37c) decompose as
are covariantly (anti)chiral super elds with respect to the background
V ! V + e v :
1
8
Sgauge[V ] ! Sgauge[V ] +
d2j4z v(iD W
+ 2m
+ 2e2t)
+
Z
d2j4z v
D D2D
+ imD D
v ;
Smat[Q; V ] ! Smat[Q; v] ;
Q+ = Q+e2V ;
Q+ = Q+ ;
Q
= Q e 2V ;
Q
= Q :
The operators D D2D and D D in (3.43a) are degenerate and require gauge xing.
The gauge xing is implemented by adding to the action (3.43a) the following term
Z
Sgf =
d2j4z v
1
16 fD2; D2g +
im
4
(D2 + D2) v :
This gauge xing action appears upon inserting the standard deltafunctions [f
[f iD2v] into the functional integral over Dv and averaging them with appropriate weight
(see [39] for details of this procedure in the threedimensional case). After gauge xing, we
get the action for `quantum' elds
Squant = S2 + Sint ;
S2 =
where
This operator obeys the important property
;
d2j4z [e(Q+Q+
Q Q )v + e2(Q+Q+ + Q Q )v2] + O(e3) ;
im
4
H =
(2D D
+ D2 + D2) :
H2 =
m
(3.42)
(3.43a)
(3.43b)
(3.44)
(3.45)
iD2v]
(3.46a)
(3.46b)
(3.46c)
(3.47)
(3.48)
HJEP07(21)46
This identity allows us to represent the propagator for the super eld v in the form
2The background gauge super eld is denoted by the same letter as the original super eld V . This should
not lead to any confusions as the original unsplit gauge super eld does not show up after the
backgroundquantum splitting.
2ihv(z)v(z0)i
(1) = iTr ln( + + m2) ;
Z
sections.
3.3.2
Oneloop e ective action
important features.
to the re ection V !
ihQ+(z)Q (z0)i =
are de ned by the equations (2.48a) and (2.60a),
respectively.
Using the form of cubic and quartic interaction vertices for quantum
elds in (3.46c),
we deduce the formal decomposition of the e ective action up to twoloop order
where
Here m,
of chiral super elds
K0(z; z0js) =
(3.50)
and
are the components of the supersymmetric interval (2.21).
In addition to the photon propagator (3.49), the action (3.46b) yields the propagators
(2) =
2e2
d2j4zd2j4z0[G+ (z; z0)G+ (z0; z) + m2G+(z; z0)G (z; z0)]Gv(z; z0) : (3.52c)
Here (1) is the oneloop e ective action while (2) takes into account twoloop quantum
corrections. These quantum contributions will be calculated separately in the subsequent
The computation of the oneloop e ective action in the model (3.37) is very similar to
the one for SQED with one chiral avor considered in section 3.2. However, it has some
First of all, the e ective action in the model (3.37) possesses no odd part with respect
V . As is demonstrated in section 3.2.3, the odd contributions to
the e ective action cancel against each other in the model where the charges of avors sum
to zero, (3.6). Thus, we have to focus only on the even part of the oneloop e ective action.
The computation of the even part of the e ective action goes along the same lines as in
section 3.2.2. Following these steps, one arrives at the expression (3.21), with two simple
modi cations: (i) The result (3.21) should be multiplied by 2 as we have contributions
from two chiral avors now; (ii) the mass parameter m should be inserted,
(1) =
It is an instructive exercise to rewrite the functional (3.53) in the full superspace. We
give the details of this procedure for the chiral part of (3.53); the antichiral part can be
analyzed in the same way.
(3.51)
HJEP07(21)46
(3.52a)
(3.52b)
At the rst step, we identically rewrite the chiral part of (3.53) as the sum of two terms
=
i Z
i Z
8
8
i Z
8
where, in the last line, we have inserted the unity, 1 = 4f12 D2W 2. In this identity, the
operator D2 can be used to restore the full superspace measure due to (A.8). Then, after
evaluation of the propertime integral in the second line of (3.54), we have
=
+
i Z
8
8
d2j2z
sf =2
1 :
(3.55)
Next, we have to restore the full superspace measure in the rst line of (3.55) using
the operators D
from W
= D G = i( 3) D H, see (2.5). Making use of properties of
the super eld strengths (2.4) and (2.5), one can prove the identity
Z
Z
d2j4z F (X) =
d2j2z W 2[(X
m2)F 00(X) + F 0(X)] ;
for some function F (X) and X
the rst line of (3.55), one nds the following di erential equation for this function
+ m2. Comparing the righthand side of (3.56) with
with the general solution
(X
m2)F 00(X) + F 0(X) =
F (X) = c1 + c2 ln(X
m2) +
1
2
ln2 X
m2
m2
m2
X
m2
1
X
;
+ Li2
where c1 and c2 are arbitrary constants of integration. The terms with these constants drop
out upon integration over the full superspace owing to the properties (2.14) and (2.15).
The remaining two terms in (3.58) allow us to get the fullsuperspace representation for
the rst term in (3.55)
1 Z
8
d2j2z
W 2
+ m2 =
8
1 Z
Note that the last term in (3.59) vanishes in the limit m = 0 owing to the identity
Li2(0) = 0. In this limit, the expression (3.59) coincides with the nonholomorphic
potential in (3.22).
:
(3.56)
(3.57)
(3.58)
(3.59)
i Z
4
+ Li2
sf =2
1 :
(3.60)
The term in the rst line here can be interpreted as the oneloop quantum correction to
the e ective Kahler potential for the twisted chiral super eld
K(1) =
ln ln
+
Li2
m2
:
The second line in (3.60) is responsible for higherderivative corrections in the oneloop
EulerHeisenbergtype action.
Twoloop e ective Kahler potential
In principle, starting from (3.52c) it is possible to determine twoloop quantum corrections
to the EulerHeisenbergtype action.3 However, the form of the resulting expression appears
not very illuminating as it involves numerous propertime integrations and may have very
limited applications. Therefore, in this section we restrict ourself to studying twoloop
quantum corrections only to the e ective Kahler potential for the twisted chiral super eld
. To this aim, it is su cient to consider the gauge super eld background constrained by
W
= 0 ;
W
= 0 ;
while super elds
and
are constant and nonvanishing. In this approximation, the heat
kernels (2.57) and (2.68) reduce to
Recall that we considered here the chiral part of (3.53). It can be shown that the
antichiral part gives the same contribution as (3.55), so that (1) = 2 . Thus,
substituting (3.59) into (3.55), we end up with the representation for the oneloop e ective action
in the full superspace
action was studied in [40] up to the twoloop order. We point out that beyond one loop this e ective action
in the supersymmetric QED cannot be found by simple composition of nonsupersymmetric results in scalar
and spinor electrodynamics.
The twoloop e ective action (3.52c) is given by the sum of two terms, which we denote
by
A and
B, respectively,
K+(z; z0js)
K+ (z; z0js)
1
4 s
1
4 s
e is
e i 42s 2I(z; z0) ;
2
e is
e i 4s I(z; z0) :
(2) =
A =
B =
A +
2e2
Z
B ;
Z
d2j4zd2j4z0 G+ (z; z0)G+ (z0; z)Gv(z; z0) ;
2e2m2
d2j4zd2j4z0 G+(z; z0)G (z; z0)Gv(z; z0) :
(3.61)
(3.62)
(3.63)
(3.64)
(3.65a)
(3.65b)
(3.65c)
Q+
v
Q
u + u
v
Type A
v
u
Q
These two terms correspond to the Feynman graphs of types A and B in gure 1. They
have slightly di erent structure and need to be considered separately. Note that the
twoloop graph of the topology `8' vanishes identically and, thus, does not show up in (3.65).
It is possible to show that the part of e ective action (3.65c) does not contribute
to the e ective Kahler potential. In this formula, we express Green's functions via the
corresponding heat kernels (2.50) and (3.49)
Z
B =
2ie2m2
ds dt du e i(s+t)m2 K+(z; z0js)K (z; z0jt)
K0(z; z0ju) :
The operator H in the last line contains the covariant spinor derivatives (see eq. (3.47)),
which can be integrated by parts
B = 2ie2m2
Z
where we have taken into account the explicit form of the heat kernel (3.50). The last line
in (3.67) contains the terms of the following three types:
K+(z; z0js)K (z; z0jt)
K+(z; z0js)r2K (z; z0jt)
!0
;
!0
;
(3.66)
(3.67)
(3.68a)
(3.68b)
(3.68c)
K+(z; z0js)K (z; z0jt)
r2K+(z; z0js)K (z; z0jt)
!0
;
!0
r K+(z; z0js)r K (z; z0jt)
;
!0
:
The terms involving (3.68a) cannot contribute to the e ective Kahler potential since the
expression (3.20) vanishes in the approximation (3.62). For the same reason there are no
contributions from the terms (3.68b). In a similar way it is easy to argue that the
expressions (3.68c) cannot contribute to the e ective Kahler potential owing to the properties
r K+(z; z0js)j !0 / W ;
r K (z; z0jt)j !0 / W :
(3.69)
These properties follow from the explicit form of the heat kernels (2.57) and (2.58). Thus,
in the approximation (3.62)
Now let us consider the contributions to the e ective Kahler potential from the part
of the e ective action (3.65b). Making use of the identities (2.61) and (3.49), this e ective
action can be cast to the form
(3.70)
Z
A = 2ie2
d2j4z d2j4z0
It is easy to argue that the terms with the operator H in the last line of (3.71) give no
contributions to the e ective Kahler potential. Indeed, covariant spinor derivatives in this
operator can hit the heat kernels yielding the terms
r2K+ (z; z0js)K +(z; z0jt)
r K+ (z; z0js)r K +(z; z0jt)
!0
;
!0
:
For such terms one can derive the following identities
K+ (z; z0js)r2K +(z; z0jt)
!0
;
41 r2K+ (z; z0js)
41 r2K +(z; z0js)
!0
!0
= i
= i
d
ds
d
ds
K (z; z0js)
r K+ (z; z0js)j !0 / W ;
r K +(z; z0jt)j !0 / W :
Thus, contributions from these terms to the e ective action vanish in the
approximation (3.62).
Nontrivial contributions to (3.71) appear only from the terms without the operator H
A =
ie2 Z
d2j4z d2 Z 1 ds dt du
0
s t u
e i(s+t)(m2+
)e ium2 e i42 (s 1+t 1+u 1)
:
Here, the Gaussian integral over d2 can be easily evaluated
A =
ie2 Z
Finally, it is possible to perform integration over one of the parameters, say u, and to
represent the corresponding contribution to the e ective action in the form
s + t
e i(s+t)(m2+
)ei ms2+stt E1
im2s t
s + t
;
(3.72)
(3.73)
(3.74)
(3.75)
(3.76a)
(3.76b)
where E1(z) is the exponential integral
E1(z) =
Z 1 dt
1
t
e tz :
The expression (3.76b) represents the twoloop quantum correction to the e ective
Kahler potential for the twisted chiral super eld . This formula involves integration over
the parameters s and t which are hard to evaluate for generic values of masses m and m.
However, it is possible to
nd explicitly the leading contributions to the e ective Kahler
potential for small photon mass, i.e., in the regime (3.38). In this case, we can use the
asymptotics of the function E1(ix) for small x,
The integrals over s and t reduce to
where a = i(m2 +
). This yields the simple expression for (3.76b)
This formula is a good approximation for the twoloop quantum correction to the e ective
Kahler potential for
in the regime (3.38), i.e., when the photon possesses a small but
nonvanishing mass m. Obviously, (3.80) is singular in the limit m ! 0. It emphasizes a
feature of twodimensional electrodynamics that the quantum loop diagrams with internal
photon lines su er from IR singularities unless the photon possesses a mass.
In conclusion of this section, let us consider the full e ective Kahler potential K( ; )
which contains the classical part 21e2
corrections (3.61) and (3.80)
and includes both one and twoloop quantum
K( ; ) =
+
ln ln
+
1
2e2
+
e
2
4 2 m2 +
8 2 m2 +
+
e
2
1
4
Li2
1
m2
ln
m2 +
m2
:
:
(3.77)
(3.78)
(3.79)
(3.80)
(3.81)
(3.82)
The corresponding sigmamodel metric reads
1
2e2 +
1
1
4 zz + m2
e
2
m2
e
2
zz
m2
8 2 (m2 + zz)3 +
8 2 (m2 + zz)3 ln
m2 + zz
m2
dzdz :
For vanishing mass of the chiral multiplet, m = 0, this metric acquires a simple form
ds2jm=0 =
1
2e2 +
1 1
4 zz
e
2
1
+
8 2 (zz)2 ln
zz
m2
dzdz :
(3.83)
Z 1 ds dt
0
s + t
E1(ix) =
ln x + O(x) :
Z 1 ds dt
s + t
e (s+t)a ln
0
e
2
i
2
e (s+t)a =
s t
s + t
+
=
e
2
1
a
;
2 +
a
1
1
a
ln
ln a ;
m2 +
m2
1
1
4
1
We stress that this metric makes sense for a small but nonvanishing photon mass m.
The twoloop Kahler potential (3.81) and the corresponding metric (3.82) are new
results obtained here by direct quantum computations in the N = (2; 2) superspace. Though
the oneloop quantum corrections to this metric were found long ago in [2], to the best of
our knowledge the twoloop quantum corrections have never been presented before.
4
Lowenergy e ective action in N
= (4; 4) SQED
Classical action and loop expansion of the e ective action
HJEP07(21)46
The (4; 4) vector multiplet may be described by the N = (2; 2) vector multiplet V and a
chiral multiplet . The hypermultiplet is described by the pair of chiral elds (Q+; Q ).
Let us consider the following action for these multiplets
S = Sgauge[V; ] + Smat[Q; V; ] ;
Sguage[V; ] =
2e2
it Z
2
Z
im
4e2
)
Z
Z
Z
+
d2j2z 2
d2j2z~ 2
For m = 0 this action is invariant under `hidden' (2; 2) supersymmetry with anticommuting
parameters
and
(4.1a)
(4.1b)
V =
1
2
= i
=
W ;
) ;
Q ) ;
Q
= i
=
W ;
1 2
4 r (
Q ) ;
(4.2)
where Q
are as in (3.44). For nonvanishing photon mass, m 6= 0, the action (4.1) is
invariant under (4.2) only for the real supersymmetry parameter
=
. This means that
for generic m the model (4.1) describes the N = (3; 3) supersymmetric electrodynamics
while for m = 0 the supersymmetry extends up to N = (4; 4). This scenario is completely
analogous to the threedimensional N = 4 electrodynamics which can have only reduced
N = 3 supersymmetry when the topological ChernSimons mass term is turned on [41,
42]. In our case, in (4.1) we keep nonvanishing photon mass m in order to get rid of
IR singularities of Feynman graphs beyond one loop. The oneloop contributions to the
e ective action, however, are independent of m and have the same form for both N = (3; 3)
and N = (4; 4) cases.
For quantizing the theory, we perform the backgroundquantum splitting
V ! V + e v ;
+ e ;
(4.3)
while the hypermultiplet (Q+; Q ) is considered as the `quantum' super eld which will
be integrated out in the path integral. The background gauge super eld V is constrained
by (2.34) and (2.35) while
is simply constant
D
= 0 ;
D
= 0 :
After adding the gauge xing term (3.45), the `quantum' elds are described by the
action
Squant = S2 + Sint ;
S2 =
d2j2z
im 2
4
Q+ Q
d2j4z[e(Q+Q+
Q Q )v + e2(Q+Q+ + Q Q )v2]
+ e
Z
+ O(e3) :
As compared with (3.46), in (4.5) there are two essential modi cations: (i) in (4.5b) we
have the background chiral super eld
in place of the mass m; (ii) in the last line in (4.5c)
there are two additional vertices with the quantum chiral super eld
and its conjugate
. Taking these features into account, one can readily generalize the N = (2; 2) e ective
action (3.52a) to the N = (4; 4) (or, rather, N = (3; 3)) case
where
=
) ;
(2) =
A +
B +
C +
D ;
A =
B =
D =
2e2
2e2
Z
Z
Z
ie2m Z
+
4
ie2m Z
4
C = 2e2
d2j4zd2j4z0 G+ (z; z0)G+ (z; z0)G0(z; z0) ;
d2j4zd2j4z0 G+ (z; z0)G+ (z0; z)Gv(z; z0) ;
d2j2zd2j2z0 2 G+(z; z0)G+(z0; z)D2G0(z; z0)
d2j2zd2j2z0 2 G (z; z0)G (z0; z)D2G0(z; z0) :
Here Gv(z; z0) is given by (3.49) while G0(z; z0) is simply
G0(z; z0) =
2j4(z
z0) =
1
+ m2
i
Z 1
0
ds e ism2 K0(z; z0js) :
Below, we compute separately the one and twoloop contributions to the e ective
action (4.6).
(4.4)
(4.5a)
(4.5b)
(4.5c)
(4.6a)
(4.6b)
(4.6c)
(4.7a)
(4.7b)
(4.7c)
(4.7d)
(4.8)
Recall that we consider the approximation (4.4) which means that we discard any
derivatives of (anti)chiral super eld
( ). In this case the procedure of computation of the
oneloop e ective action (4.6b) is exactly the same as in section 3.3.2 for the N = (2; 2)
SQED. Thus, we can readily generalize the result (3.53) to the case of N = (4; 4) SQED
(1) =
i Z
8
+
) tanh(sf =2)
sf =2
(4.9)
The e ective action (4.9) is represented as a functional in (anti)chiral superspace. It
HJEP07(21)46
is instructive to rewrite it in the full N = (2; 2) superspace. Following the same procedure
as in section 3.3.2, we nd
4
(1) =
+ Li2
i Z
4
+
) W 2W 2
f 2
sf =2
(4.10)
The terms in the rst line in (4.10) are leading in the derivative expansion of the e ective
action while the terms in the second line correspond to higherderivative corrections. The
leading terms
Z
1
4
leading =
d2j4z K(1)( ; ; ; ) ;
deserve several comments.
First of all, we point out the similarity of the super eld expression (4.11) with the
lowenergy e ective action of fourdimensional N = 4 SYM theory in N = 2 superspace which
was constructed in [14]. Indeed, (4.11b) contains the term ln ln
which is analogous
to the nonholomorphic potential for N = 2 4d super eld strength while the other terms
are very similar to the hypermultiplet completion of the nonholomorphic potential which
was constructed in [14]. Surprisingly, such terms in 2d and 4d cases are described by the
same Li2 function and have very similar form although they are given in very di erent
superspaces and for di erent models. Recall that the N = 4 susycomplete e ective action
in 4d N = 4 SYM theory was derived originally in [14] by imposing the requirement of
invariance under hidden supersymmetry while in subsequent works this e ective action was
found by direct quantum computations in superspace [43{45] (see also [46] for a review). In
our case, we obtained (4.11) as the leading part of the oneloop e ective action in N = (4; 4)
SQED although originally it was found in [1] as a susy completion of the nonholomorphic
potential (1.2).
The mentioned above similarity of (4.11) with the lowenergy e ective action in 4d
N = 4 SYM theory is even deeper. As was demonstrated in [47, 48] (see also [49] for a
review), the structure of leading terms in the lowenergy e ective action in N = 4 SYM
theory can be recovered from the fact that it contains the WessZumino term for scalar
elds. This WessZumino term is known to appear in the lowenergy theory as a result of
't Hooft anomaly matching for SU(4) Rsymmetry of N = 4 SYM theory [15]. Surprisingly,
the e ective action (4.11) may be given exactly the same interpretation. Indeed, in [1] the
action of the form (4.11) was proposed as a super eld generalization of a twodimensional
sigmamodel with the WessZumino term. In our case, the appearance of this term in the
lowenergy e ective action is well understood. Classically, the N = (4; 4)
electrodynamics (4.1) respects the SU(2)
SU(2) symmetry which is the Rsymmetry of N = (4; 4)
Poincare superalgebra. However, because of 't Hooft anomaly, this symmetry cannot be
realized explicitly in the lowenergy theory but is still the symmetry of the e ective
Lagrangian up to full derivative terms. Recall that the e ective action (4.11) is obtained
upon integrating out the hypermultiplet (Q+; Q ) which contains chiral fermions with
respect to the Rsymmetry group. Thus, in the lowenergy theory the WessZumino term
must appear as a response to the change of the number of chiral fermions since the total
contribution to the anomaly should be the same regardless of the energy scale. This is the
essence of the 't Hooft anomaly matching argument [11].
Let us derive the WessZumino term for scalar elds from the super eld action (4.11).
The scalars appear in the component eld expansion of
and
as follows
+ i
i
1 2 2
4
+ : : : ;
1 2 2
4
+ : : : ;
where dots stand for other component elds. Substituting these expressions into (4.11) and
integrating over the Grassmann variables one readily nds in the component eld expansion
the WessZumino term for the scalar elds
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
where
and
are phases of the complex scalars
and
SWZ =
= j jei ;
+
= j jei :
SWZ =
(XAXA)2
{ 31 {
The action (4.14) is explicitly invariant under U(1)
U(1) symmetry which shifts the
phases
and . This symmetry is the subgroup of the full SU(2) SU(2) Rsymmetry group
of the theory. It is possible to show that (4.14) is implicitly invariant under SU(2) SU(2) '
SO(4) since this is the symmetry of the WessZumino term modulo total derivative terms.
To show this, let us introduce real scalars XA = (X1; X2; X3; X4) which transform as a
vector under SO(4)
space
for details)
= X1 + iX2 ;
= X3 + iX4 :
Then, the action (4.14) can be rewritten in the form of integral over a threedimensional
which has standard 2d Minkowski space as its boundary, @
= R1;1 (see e.g. [47]
and h i
.
Type C
Q
u
Q
u +
u
Q+
Q
Type D
u
Q+
where "ABCD and "mnp are antisymmetric tensors.
The WessZumino term in the
form (4.17) has explicit SO(4) symmetry.
One can reverse the arguments: once we know that the WessZumino term (4.17)
appears in the lowenergy e ective action of N = (4; 4) SQED, we can immediately nd (4.11)
as its supersymmetric generalization. However, performing perturbative quantum
computations we uncover not only the leading term (4.11) in the lowenergy e ective action, but
also higherderivative corrections which are encoded in the second line of (4.10).
We point out once more that (4.14) is explicitly invariant under U(1)
U(1) '
SO(2)
SO(2) which is one of the maximal subgroups of the full Rsymmetry group
SO(4). However, there are two more inequivalent maximal subgroups: SO(3) ' SU(2) and
SU(2)
U(1). We speculate that these subgroups may be made manifest in other
supereld descriptions of the N = (4; 4) gauge theory such as the harmonic superspace [50{54].
Recall that in the 4d N = 4 SYM theory the careful account of all maximal subgroups of
the SU(4) Rsymmetry group resulted in di erent but equivalent super eld descriptions of
the lowenergy e ective action [47{49, 55]. It is tempting to develop similar ideas for the
lowenergy e ective action in 2d supersymmetric gauge theories.
4.3
Vanishing of twoloop corrections to generalized Kahler
potential
In the previous section we computed oneloop e ective action which contains the term (4.11)
as the leading part in the derivative expansion. In [10] it was claimed that this potential is
nonrenormalized by higherloop quantum corrections. This section aims to demonstrate
explicitly that twoloop quantum corrections to the generalized Kahler potential (4.11b)
cancel among each other.
In the twoloop expansion of e ective action (4.6c), the terms
A and
B can be
represented by Feynman graphs which have the same structure as those in the N = (2; 2)
SQED given in gure 1. The terms C and
D are new since they involve the propagators
for the (anti)chiral super eld
( ). These terms are represented by the Feynman graphs
in gure 2. To nd the contributions to the e ective action from these terms it is su cient
to consider the vector multiplet background constrained by (3.62) and (4.4).
The details of computations of contributions to the e ective action (4.7a) and (4.7b) are
exactly the same as those in section 3.3.3. We can immediately generalize the results (3.70)
and (3.75) to the N = (4; 4) case
A =
ie2 Z
e ium2 e i(s+t)(
+
It is easy to argue that the contribution to the e ective action (4.7d) vanishes. Indeed,
the propagator G0 (4.8) contains the deltafunction which implies that we need to consider
at coincident points. As follows from (3.20), K+(z; z0js)
D = 0 :
It remains to consider the contribution
C to the e ective action. Substituting here
the heat kernels K+ and K
K+(z0; zjt)j =0 = 0. Thus,
the propagators (3.64) and (4.8) we have
Z 1
0
0
s t u
2ie2 Z
(4 )3
C = 2ie2
)e i 4 (s 1+t 1+u 1)e ium2 :
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
1
2e2
K( ; ; ; ) =
+ Li2
(4.24)
After integration over d2 it becomes evident that this expression contributes to the e
ective action with the opposite sign to (4.18)
C =
ie2 Z
8 2
st + su + tu
e ium2 e i(s+t)(
+
Thus, we conclude that the sum of the terms (4.18), (4.19), (4.20) and (4.22) vanishes
(2) =
A +
B +
C +
D = 0 :
We stress that this does not mean that the complete twoloop e ective action vanishes, but
just implies that there are no twoloop quantum corrections to (4.11b).
The nonrenormalization of the generalized Kahler potential K( ; ; ; ) in (4,4)
gauge theories was claimed in [10]. In this section, we have explicitly demonstrated the
absence of twoloop quantum corrections to this potential. There are also purely
eldtheoretical arguments that all higherloop quantum correction to this function vanish and
K( ; ; ; ) is oneloop exact. Indeed, in the previous section it was demonstrated that
this potential is responsible for the WessZumino term for scalar elds (4.14). It is
wellknown that appearance of WessZumino terms in lowenergy e ective action is strictly
oneloop e ect associated with the 't Hooft anomaly matching [11]. The form of the
WessZumino term (4.17) as well as the coe cient in front of this action are rigidly
xed by
topological arguments (see e.g. [9]). Therefore, the function (4.11b) cannot receive any
higherloop corrections, and we end up with the exact result
+ d d )
:
(4.25)
As is demonstrated in [4], this sigmamodel corresponds to the generalized Kahler geometry
with N = (4; 4) extended supersymmetry. This geometry possesses nontrivial torsion
which can also be read o from (4.11b). Here we have demonstrated that this generalized
Kahler potential naturally arises as the leading term in the lowenergy e ective action in
N = (4; 4) gauge theory on the Coulomb branch.
1
2e2 +
1
+
5
In this paper, we have studied twoloop quantum corrections to the lowenergy e ective
actions in the N = (2; 2) and N = (4; 4) SQED. On the Coulomb branch, leading terms in
the e ective action in N = (2; 2) SQED are represented by a superpotential and a Kahler
potential for super eld strengths described by a twisted chiral super eld . Although, at
oneloop order, these potentials were studied long ago [2], to the best of our knowledge
the twoloop quantum corrections to the e ective Kahler potential (3.80) have not been
presented before. The corresponding sigmamodel metric in the twoloop approximation is
given by (3.83). We point out that this metric depends on the vector multiplet mass m and is
singular in the limit m ! 0. This is a new feature of the twodimensional case as compared
with the lowenergy e ective action of threedimensional [23] and fourdimensional [28, 34]
SQED where it was wellde ned for massless vector multiplet.
In the N = (4; 4) SQED, the leading part of the lowenergy e ective action is described
by the generalized Kahler potential K( ; ; ; ) where
is a chiral and
is a twisted
chiral super elds. We show that this potential is oneloop exact and is given by (4.24).
This potential was introduced for the rst time in [1] in the study of twodimensional sigma
models with torsion which originates from the WessZumino term for scalar elds. In our
case, we demonstrate that the WessZumino term is the integral part of the lowenergy
e ective action associated with the 't Hooft anomaly matching for the SU(2)
SU(2)
Rsymmetry. The from of the potential K appears surprisingly similar to the lowenergy
e ective action in 4d N = 4 SYM theory obtained in [14].
We have studied twoloop e ective action in twodimensional SQED on the Coulomb
branch. In is also very interesting to investigate the structure of the e ective action on the
Higgs branch. Some of the leading terms in this e ective action were discussed in the recent
work [56] but the form of higherloop quantum correction is unknown. Another interesting
problem is the study of quantum aspects of twodimensional gauge theories with semichiral
multiplets [8, 36, 37]. Only some limited results in this direction are available [57], but the
structure of quantum corrections to the generalized Kahler potential remains unknown.
Acknowledgments
I am very grateful to E.A. Ivanov for useful discussions. I acknowledge the support from
the RFBR grant No 150206670.
may be 0 =
i 2, 1 =
the Cli ord algebra
In this paper, we use the twodimensional N = (2; 2) superspace which appears by the
dimensional reduction from the threedimensional N = 2 superspace or from the
fourdimensional standard N = 1 superspace. Therefore, we employ the super eld notation and
conventions which are very close to the ones used in the series of papers [23{27] devoted
to the study of the lowenergy e ective actions in threedimensional super eld theories.
The 2d N = (2; 2) superspace is parametrized by the coordinates zA = (xm; ; ),
where xm = (x0; x1) are the Minkowski space coordinates and
= ( 1; 2) are Grassmann
coordinates (
= ( ) are their complex conjugate). The spinor indices are raised and
lowered by means of the antisymmetric "tensor,
= "
,
= "
, "12 = "21 = 1.
One of the possible choices for the twodimensional gammamatrices ( m) = ( m)
1, where i are the Pauli matrices. The gammamatrices obey
f
m
; n
g =
2 mn12 2 ;
mn = diag(1; 1) ;
and possess the following orthogonality and completeness relations
( m
) ( n) =
2 mn ;
( m
) ( m) = " "
+ ( 3) ( 3) ;
where 3 = 1 0 is the chirality matrix. The chiral projectors are P
The covariant spinor derivatives may be chosen in the form
= 12 (1
3).
with the following anticommutation relation
+ i ( m
i ( m
fD ; D g =
The integration measure in the full N = (2; 2) superspace is de ned as
d2x f (x) =
d2j4z 2 2f (x) ;
for some eld f (x). Here we adopt the following conventions for contractions of spinor
indices
D2 = D D ;
D2 = D D ;
2 =
2 =
The chiral and antichiral subspaces are parametrized by the coordinates z+ = (x+m; )
and z
= (xm;
), correspondingly, where
xm = xm
i m
The integration measure in the chiral subspace d2j2z
d2xd2 is related to the full
superspace measure (A.5) as
d2j4z =
4
1 d2j2zD2 =
4
1 d2j2zD2 ; so that
Z
Z
d2x f (x) =
d2j2z 2f (x+) :
(A.8)
(A.1)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
Given a twocomponent spinor
, we can consider each of its components
independently as they are Lorentzcovariant and appear as the P
projections of ,
( +;
) ;
= P
:
In a similar way, for a spinor with the upper spinor index,
, we have
( +
;
; +). As a consequence, it is possible to introduce twisted chiral coordinates z~+ =
) and the twisted antichiral ones z~ = (x~m;
; +), where
x~m = xm
i( m
The integration measures over these subspaces are denoted by d2j2z~ and d2j2z~. They are
related to the full superspace measure (A.5) as
d2j4z = d2j2z~1 D+D
2
= d2j2z~1 D+D :
2
The existence of twisted chiral subspace in addition to the conventional chiral one is
the crucial feature of the twodimensional superspace as compared with the
higherdimensional story.
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.
(A.9)
) =
(A.10)
(A.11)
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