Two-loop low-energy effective actions in N = 2
and N = 4 three-dimensional SQED
Boris Merzlikin
Tomsk state pedagogical university
Supersymmetries and Quantum Symmetries,
Dubna 2013
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
SQS’13, Dubna
1 / 15
Plan
Introduction and overview
General motivation
Three-dimensional N = 2 SQED
Exact superpropagators
Loops quantum corrections
Three-dimensional N = 4 SQED
Conclusion
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
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Introduction and overview
V.I. Ritus (two-loop results in spinor and scalar QED) Sov. Phys. JETP
(1975, 1977)
S.M. Kuzenko and I.N. McArthur (two-loop superfield effective action in
N = 2 d=4 SQED) JHEP (2003)
S.M. Kuzenko and S.J. Tyler (two-loop superfield effective action in
N = 1 d=4 SQED) JHEP (2007)
A.N. Redlich (one-loop effective action in d=3 QED) Phys.Rev.Lett.
(1984) and Phys.Rev.D (1984)
I.L. Buchbinder, N.G. Pletnev, I.B. Samsonov (one-loop effective action
in d=3 N = 2 and N = 4 abelian gauge superfield model) JHEP (2010)
I.L. Buchbinder, I.B. Samsonov, BM (two-loop effective action in d=3
N = 2 and N = 4 SQED without Chern-Simons kinetic term) JHEP (2013)
the talk is based on this work
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
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General motivation
The modern interest to three-dimensional supersymmetric field models is partly
motivated by recent progress in constructing field theories describing multiple M2
branes. These are three-dimensional superconformal field theories of Chern-Simons
gauge fields interacting with matter in a special way such that the superconformal
invariance and N = 6 (or even N = 8) extended supersymmetry are preserved
N = 8 case Bagger-Lambert & Gustavsson (BLG) model Phys.Rev.D
(2007,2008) and Nucl.Phys.B (2009)
N = 6 case Aharony-Bergman-Jafferis-Maldacena (ABJM) model JHEP
(2008)
One of the general problems for the BLG and ABJM models is to study the
effective action. It is the open question.
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
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Three-dimensional N =2 SQED
The classical action for the N = 2 d=3 supersymmetric electrodynamics:
Z
Z
1
7
2
SN =2 =
d z G − d7 z Q̄+ e2V Q+ + Q̄− e−2V Q−
2
e
Z
5
− m d z Q+ Q− + c.c. ,
(1)
where G = 2i D̄α Dα V is a scalar field strength of vector multiplet V .
The gauge multiplet V:
real scalar φ,
complex spinor λα ,
vector field Am ,
real auxiliary scalar D.
The matter multiplet Q± :
complex scalar f± ,
α
complex spinor ψ±
,
complex auxiliary scalar F± .
The additional scalar field φ arises after dimension reduction the gauge vector
field Am from d=4 to d=3.

A0
 A1 

=
 A2 
3
A

d=4
m
A
Boris Merzlikin (TSPU)

A0
1
=  A  + φ ≡ A3 .
A2

⇒
d=3
m
A
Two-loop low-energy effective actions
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5 / 15
Background field setup
We specify the constraints on the background gauge superfield under
considerations:
i) The gauge superfield obeys the N = 2 supersymmetric free Maxwell
equations,
D α Wα = 0 ,
D̄α W̄α = 0 .
(2)
ii) Within the derivative expansion of the effective action we look for the
leading terms without space-time derivatives of the gauge superfields. Such
a long-wave approximation is effectively taken into account by considering
the constant background,
∂m G = 0 ,
∂ m Wα = 0 ,
(3)
∂m W̄α = 0 .
This approximation suffices to study the Euler-Heisenberg-type effective action
which is induced by the N = 2 supersymmetric quantum matter fields.
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
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6 / 15
Diagrams
There are two types of diagrams which contribute to two-loop effective action in
N = 2 d=3 SQED
The propagators
ihQ+ (z)Q− (z 0 )i
=
−mG+ (z, z 0 ) ,
ihQ+ (z)Q̄+ (z )i
=
G+− (z, z 0 ) = G−+ (z 0 , z) ,
2ihv(z) v(z 0 )i
=
G0 (z, z 0 ) ,
0
(4)
0
The photon propagator G0 (z, z ) through the heat kernel reads
G0 (z, z 0 ) = i
∞
Z
ds K0 (z, z 0 |s) e−s ,
K0 (z, z 0 |s) =
0
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
iρm ρm
1
e 4s ζ 2 ζ̄ 2 .
3/2
(4iπs)
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(5)
7 / 15
Exact superpropagators
Following the work S. M. Kuzenko and I. N. McArthur, JHEP 0305 (2003) we
obtain propagators for matter fields.
The chiral propagator G+ (z, z 0 )
G+ (z, z 0 ) = i
∞
Z
ds K+ (z, z 0 |s)ei(m
2
+i)
,
→ +0 ,
(6
0
K+ (z, z 0 |s) =
m n 1 β
γ
2
i
sB
1
eisG O(s)e 4 (F coth(sF ))mn ρ ρ − 2 ζ̄ ρβγ W ζ 2 I(z, z 0 ).
(4iπs)3/2 sinh(sB)
The chiral-antichiral propagator G+ (z, z 0 )
G+− (z, z 0 ) = i
∞
Z
ds K+− (z, z 0 |s)ei(m
2
+i)
,
→ +0 ,
(7)
0
K+− (z, z 0 |s) = −
2
m n
0
i
sB
1
eisG O(s)e 4 (F coth(sF ))mn ρ̃ ρ̃ +R(z,z ) I(z, z 0 ),
(4iπs)3/2 sinh(sB)
α
¯
α
where the operator O(s) = es(W̄ ∇α −W ∇α ) and two point functions ζ α , ζ̄ α and
ρm are the components of the N = 2 d=3 supersymmetric interval
ζ A = {ρm , ζ α , ζ̄α }
ρm
=
(x − x0 )m − iγ m αβ ζ α θ̄0β + iγ m αβ θ0α ζ̄ β ,
α
=
(θ − θ0 )α ,
ζ
Boris Merzlikin (TSPU)
ζ̄ α = (θ̄ − θ̄0 )α .
Two-loop low-energy effective actions
(8)
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Exact superpropagators
The parallel displacement propagator I(z, z 0 ) provides the correct gauge
transformation properties of the heat kernel defines as the solution of the equation
ζ A ∇A I(z, z 0 ) = 0 ,
(9)
I(z, z) = 1 .
¯ α ) satisfy the algebra
The covariant derivatives ∇A = (∇m , ∇α , ∇
¯ β } = −2i(γ m )αβ ∇m + 2iεαβ G ,
{∇α , ∇
¯ α , ∇m ] = (γm )αβ W β
[∇
[∇α , ∇m ] = −(γm )αβ W̄ β ,
[∇m , ∇n ] = iFnm ,
(10)
The two-point function R(z, z 0 ) is defined as
R(z, z 0 )
=
−iζ ζ̄G +
i 2
1
1
7i 2
ζ̄ ζW +
ζ ζ̄ W̄ − ζ̄ α ρ̃αβ W β − ζ α ρ̃αβ W̄ β
12
12
2
2
1 α β γ
ζ ζ̄ [ρ̃β Dα Wγ − 7ρ̃γα Dγ Wβ ] .
12
m β
0 m
ρm + iζ α γαβ
ζ̄ ,
Dα
ρ̃ = D̄α ρ̃m = 0 .
(11)
+
ρ̃ m
=
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Two-loop low-energy effective actions
(12)
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Loop corrections
The one-loop effective action [I. L. Buchbinder et al. JHEP (2010)] is defined by
the coincide point z → z 0 limit of anti-chiral propagator G+− (z, z) (7)
(1)
ΓN =2
=
p
p
d7 z G ln(G + G2 + m2 ) − G2 + m2
Z
Z ∞
ds is(G2 +m2 ) W 2 W̄ 2 tanh(sB/2)
1
√
+
d7 z
e
−
1
, (13)
8π
B2
sB/2
iπs
0
1
2π
Z
where Wα = D̄α G and B 2 = 12 D2 W 2 .
The two-loop corrections to effective action have the following superfield
structure:
Z
h
i
e2
(A)
(B)
(2)
d7 z L1 + W α L2α β W̄β + (L3 + L3 )W 2 W̄ 2 ,
ΓN =2 = ΓA + ΓB =
3
16π
Here L1 , L2 α β and L3 are some functions of G and B to be found from direct
quantum computations. Note that the contributions of the form L1 and L2 are
impossible in the d=4 case.
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
SQS’13, Dubna
10 / 15
Loop corrections
p
2arccosh a/b
ds dt i(s+t)(G2 +m2 )
B2
√ e
p
,
sinh(sB) sinh(tB)
st
a(a − b)
0
p
Z
2arccosh a/b
2iG ∞ ds dt i(s+t)(G2 +m2 )
B2
√ e
p
− 2
B
sinh(sB) sinh(tB)
st
a(a − b)
0
Z
L1
=
L2α β
=
(A)
L3
=
+
−
(B)
L3
=
∞
×(e−sN − 1 + sN + e−tN − 1 + tN )α β ,
−
Z
i ∞ ds dt i(s+t)(G2 +m2 )
B2
1
2F
F+
√ e
−
2 0
sinh(sB) sinh(tB) a − b
b
a
st
Z ∞
+
2
a(F − F − ) − bF +
ds dt i(s+t)(G2 +m2 )
B
i
√ e
(f (s) + f (t) +
sinh sB sinh tB 2
a(a − b)
st
0
p
2
2arccosh
G
2
2
p
((sB
−
sinh
sB
+
tB
−
sinh
tB)
−
(cosh
sB
+
cosh
tB
−
2)
)
B4
a(a −
4m2
B2
∞
Z
0
p
sB
ds dt i(s+t)(G2 +m2 )
tB arccosh a/b
√ e
p
tanh
tanh
.
2
2
st
a(a − b)
where
a(s, t) = B coth(sB) + B coth(tB) ,
Boris Merzlikin (TSPU)
b(s, t) = s−1 + t−1
Two-loop low-energy effective actions
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11 / 15
Three-dimensional N =4 SQED
The classical action for the N = 4 d=3 supersymmetric electrodynamics:
Z
1
1
SN =4 =
(17
d7 z(G2 − Φ̄Φ) + SQ ,
e2
2
Z
Z
Z
SQ = − d7 z(Q̄+ e2V Q+ + Q̄− e−2V Q− ) − d5 z Q+ ΦQ− + d5 z̄ Q̄+ Φ̄Q̄−
The action (17) is invariant under the following ‘hidden’ N = 2 supersymmetry,
δV
δΦ
δQ+
δ Q̄+
1 α
(¯
θ̄α Φ − α θα Φ̄) ,
2
= iα Wα ,
δ Φ̄ = i¯
α W̄α ,
1 ¯2 α
1 ¯2 α
(¯
θ̄α Q̄− ) ,
δQ− = ∇
(¯
θ̄α Q̄+ ) ,
= − ∇
4
4
1
1
= − ∇2 (α θα Q− ) ,
δ Q̄− = ∇2 (α θα Q+ ) ,
4
4
=
(18)
where α and ¯α are the supersymmetry parameters.
Note that the action (17) appears from the N = 2, d=4 electrodynamics by
means of the dimensional reduction.
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
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12 / 15
Diagrams
The two-loop effective action in the N = 4 electrodynamics gets additional
contribution as compared with the N = 2 case which arise from the following
diagram
Here the propagator for the chiral superfield φ,
i
hφ(z)φ̄(z 0 )i = − D̄2 D2 G0 (z, z 0 ) .
8
and the photon propagator G0 (z, z 0 )
Z ∞
G0 (z, z 0 ) = i
ds K0 (z, z 0 |s) e−s ,
K0 (z, z 0 |s) =
0
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
(19)
iρm ρm
1
e 4s ζ 2 ζ̄ 2 .
3/2
(4iπs)
SQS’13, Dubna
13 / 15
Loop corrections
The one-loop effective action in the N = 4 electrodynamics was computed in
[I. L. Buchbinder et al. JHEP (2010)]. It has the same form as in N = 2 case, but
the mass parameter m should now be replaced to the background superfield Φ,
Z
i
h
p
p
1
(1)
d7 z G ln(G + G2 + Φ̄Φ) − G2 + ΦΦ̄
ΓN =4 =
2π
Z
Z ∞
ds is(G2 +ΦΦ̄) W 2 W̄ 2 tanh(sB/2)
1
√
d7 z
−
1
(20)
.
+
e
8π
B2
sB/2
iπs
0
The two-loop corrections to effective action have the following superfield
structure:
Z
h
i
e2
(2)
ΓN =4 =
d7 z (L(A+C) + L(B) )W 2 W̄ 2 ,
(21)
3
16π
Z
i ∞ ds dt i(s+t)(G2 +Φ̄Φ)
stB 2
L(A+C) =
e
3/2
2 0 (st)
sinh sB sinh tB
r −
+
1
2F
F
2a(F + − F − ) − bF +
a
×
−
+
arccosh
.
a−b
b
a
b
(a(a − b))3/2
p
Z
4ΦΦ̄ ∞ ds dt i(s+t)(G2 +Φ̄Φ)
sB
tB arccosh a/b
√ e
p
L(B) =
tanh
tanh
.
B2 0
2
2
st
a(a − b)
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
SQS’13, Dubna
14 / 15
Conclusion
First, we obtain the explicit expressions (6) and (7) for the propagators of matter
superfields using the manifestly N = 2 supersymmetric and gauge-covariant
technique in N = 2 d=3 superspace.
Then, we computed two-loop Euler-Heisenberg effective actions in N = 2 and
N = 4 supersymmetric electrodynamics in the N = 2 d=3 superspace and
discussed some distinctions with d=4 case. We noted, that in both cases the loop
corrections do not contain UV divergencies.
Outlook An important extension of the results of the present work is the
inclusion of the Chern-Simons term into considerations. The two-loop
computations in this case require separate studies.
Boris Merzlikin (TSPU)
Two-loop low-energy effective actions
SQS’13, Dubna
15 / 15