On the two-loop four-derivative quantum corrections in 4D N=2 superconformal field theories

doi:10.1016/j.nuclphysb.2004.01.034

Nuclear Physics B

Volume 683, Issues 1–2, 5 April 2004, Pages 3-26

https://doi.org/10.1016/j.nuclphysb.2004.01.034 Get rights and content

Abstract

In $N =2,4$ superconformal field theories in four space–time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in $N =1$ superspace, we test the Dine–Seiberg proposal by comparing the two-loop F⁴ quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) $N =4$ SYM (i.e., $N =2$ SYM with a single adjoint hypermultiplet); (ii) $N =2$ SYM with 2N hypermultiplets in the fundamental. According to the Dine–Seiberg conjecture, these theories should yield identical two-loop F⁴ contributions from all the supergraphs involving quantum hypermultiplets, since the pure $N =2$ SYM and ghost sectors are identical provided the same gauge conditions are chosen. We explicitly evaluate the relevant two-loop supergraphs and observe that the F⁴ corrections generated have different large N behaviour in the two theories under consideration. Our results are in conflict with the Dine–Seiberg conjecture.

Introduction

Some time ago, we developed a manifestly covariant approach for evaluating multi-loop quantum corrections to low-energy effective actions within the background field formulation [1]. This approach is applicable to ordinary gauge theories and to supersymmetric Yang–Mills theories formulated in superspace. Its power is not restricted to computing just the counterterms—it is well suited for deriving finite quantum corrections in the framework of the derivative expansion. As a simple application of the techniques developed in [1], we have recently derived [2] the two-loop (Euler–Heisenberg type) effective action for $N =2$ supersymmetric QED formulated in $N =1$ superspace.

The work of [2] has brought a surprising outcome regarding one particular conclusion drawn in [3] on the basis of the background field formulation in $N =2$ harmonic superspace [4]. According to [3], no super F⁴ (four-derivative) quantum corrections occur at two loops in generic $N =2$ super-Yang–Mills theories on the Coulomb branch, in particular in $N =2$ SQED. However, by explicit calculation carried out in [2], it was shown that a non-vanishing two-loop F⁴ correction does occur in $N =2$ SQED. It was also shown in [2] that the analysis in [3] contained a subtle loophole related to the intricate structure of harmonic supergraphs. A more careful treatment of two-loop harmonic supergraphs given in [2] leads to the same non-zero F⁴ term in $N =2$ SQED at two loops as that derived using the $N =1$ superfield formalism.

The work of [3] provided perturbative two-loop support to the famous Dine–Seiberg non-renormalization conjecture [5] that the $N =2$ supersymmetric four-derivative term¹ $∫ d^{4} x d^{8} θ ln W ln W ̄$ is one-loop exact on the Coulomb branch of $N =2,4$ superconformal theories.² It is known that the Dine–Seiberg conjecture is well supported by non-perturbative considerations [12], [13]. But since the two-loop F⁴ conclusion of [3] is no longer valid, it seems important to carry out an independent calculation of the two-loop F⁴ quantum corrections in $N =2$ superconformal theories. It is the aim of the present note to provide such a calculation. As will be demonstrated below, the Dine–Seiberg conjecture is not fully supported at the perturbative level.

To test the Dine–Seiberg conjecture, we consider two different $N =2$ superconformal theories with the same gauge group SU(N): (i) $N =4$ SYM or, equivalently, $N =2$ SYM with a single adjoint hypermultiplet; (ii) $N =2$ SYM with 2N hypermultiplets in the fundamental. At the quantum level, with the same gauge conditions chosen, these theories are identical in the pure $N =2$ SYM and ghost sectors. The difference between them occurs only in the sector involving quantum hypermultiplets. If the Dine–Seiberg conjecture holds, then since the pure $N =2$ SYM and ghost sectors are identical, these theories should yield identical two-loop F⁴ contributions from all the supergraphs with quantum hypermultiplets. However, as will be shown below by direct calculations, the relevant two-loop F⁴ contributions have different large N behaviour in the theories under consideration.³

From the point of view of $N =1$ supersymmetry, the chiral superfield strength W of the $N =2$ vector multiplet is known to consist of a chiral scalar φ and a constrained chiral spinor W_α, the latter being the $N =1$ vector multiplet field strength. When reduced to $N =1$ superspace, the functional (1.1) is given by a sum of several terms, of which the leading (in a derivative expansion) term is $ϒ=∫ d^{8} z W^{α} W_{α} W ̄_{α̇} W ̄^{α̇} φ^{2} φ ̄^{2},$ while the other terms involve derivatives of φ and $φ ̄$ . If one uses the $N =1$ superspace formulation for $N =2$ superconformal field theories, it is typically sufficient to compute quantum corrections of the form (1.2) in order to restore their $N =2$ completion (1.1).

This note is organized as follows. Section 2 contains the necessary setup regarding $N =2$ superconformal field theories and their background field quantization (for supersymmetric 't Hooft gauge) in $N =1$ superspace. In Section 3 we work out a useful functional representation for two-loop supergraphs with quantum hypermultiplets. In Section 4 we describe, following [1], [2], the exact superpropagators in a special $N =2$ vector multiplet background which is extremely simple but perfectly suitable for computing quantum corrections of the form (1.2). 5 , 6 form the (somewhat technical) core of this paper. In Section 5 we evaluate the two-loop F⁴ corrections in $N =2$ SYM with 2N hypermultiplets in the fundamental. This consideration is extended in Section 6 to the case of $N =4$ SYM. Finally, in Section 7 we compare the two-loop corrections in the large N limit for the two theories being studied. Some aspects of the cancellation of divergences are discussed in Appendix A.

Section snippets

$N =2$ SYM setup

The classical action of an $N =2$ superconformal field theory, S_SCFT=S_vector+S_hyper, consists of two parts: (i) the pure $N =2$ SYM action $S_{vector} = 1 g^{2} tr_{F} ∫ d^{8} z Φ^{†} Φ+∫ d^{6} z W^{α} W_{α};$ (ii) the hypermultiplet action $S_{hyper} =∫ d^{8} z Q^{†} Q + Q ̃^{†} Q ̃ − i ∫ d^{6} z Q ̃^{T} Φ Q + i ∫ d^{6} z ̄ Q^{†} Φ^{†} Q ̃ ̄ .$ Here Φ, Q and Q̃ are covariantly chiral superfields which transform, respectively, in the following representations of the gauge group: (1) the adjoint; (2) a representation R; and (3) its conjugate R_c. The covariantly chiral superfield strength $W_{α}$ is

Functional representation for two-loop supergraphs with quantum hypermultiplets

The interactions for the quantum hypermuliplets are: $S_{int} =∫ d^{8} z v^{μ} q^{†} T_{μ} q + 12 ∫ d^{8} z v^{μ} v^{ν} q^{†} T_{μ} T_{ν} q − i 2 ∫ d^{6} z ϕ^{μ} q^{T} 0 T_{μ}^{T} T_{μ} 0 q + i 2 ∫ d^{6} z ̄ ϕ ̄^{μ} q^{†} 0 T_{μ}^{T} T_{μ} 0 q ̄,$ where $T_{μ} = T_{μ} 00 −T_{μ}^{T}$ are the generators of the representation R⊕R_c.

There are four two-loop supergraphs with quantum hypermultiplets, and they are depicted in Fig. 1, Fig. 2, Fig. 3, Fig. 4.

The contributions from the first two supergraphs can be combined in the form $Γ_{I+II} =− i 2^{9} ∫ d^{8} z∫ d^{8} z′ v^{μ} (z)v^{ν} (z′) tr T_{μ} D ̄^{2} D^{2} G^{(R⊕R_{c})} (z,z′) T_{ν} D ̄^{′2}, D^{′2} G^{(R⊕R_{c})} (z′,z),$ where we have used the

Exact superpropagators

For computing quantum corrections of the form (1.2), it is sufficient to consider a very special type of background field configuration specified by the constraint $D_{α} W_{β} =0.$ This is the simplest representative of background vector multiplets for which all Feynman superpropagators are known exactly [1], [2].

For the Green's function G≡G^(R), we introduce the Fock–Schwinger proper-time representation $G(z,z′)= i ∫ 0 ∞ d s K(z,z′|s) e^{−i(|M|^{2}−iε)s},ε→+0.$ The corresponding heat kernel reads $K(z,z′|s)=− i (4πs)^{2} e^{iρ^{2}}$

SU(N) SYM with 2N hypermultiplets in the fundamental

From now on, we choose the gauge group to be SU(N). Lower-case Latin letters from the middle of the alphabet, i,j,… , will be used to denote matrix elements in the fundamental, with the convention $i=0,1,…,N−1≡0, i$ . We choose a Cartan–Weyl basis to consist of the elements: $H_{I} ={H_{0},H_{I}}, I =1,…,N−2, E_{ij},i≠j.$ The basis elements in the fundamental representation are defined similarly to [18], $(E_{ij})_{kl} =δ_{ik} δ_{jl},$ $(H_{I})_{kl} = 1 (N−I)(N−I−1) (N−I)δ_{kI} δ_{lI} − ∑ i=I N−1 δ_{ki} δ_{li},$ and are characterized by the properties $tr_{F} (H_{I} H_{J})=δ_{IJ}$

$N =4$ SYM

We now turn to evaluating to the two-loop supergraphs with quantum hypermultiplets in the $N =4$ super-Yang–Mills theory which is simply $N =2$ SYM with a single hypermultiplet in the adjoint.

Discussion

As pointed out in the Introduction, the two $N =2$ superconformal field theories with gauge group SU(N) considered in this paper differ only in the hypermultiplet sector—one contains a single hypermultiplet in the adjoint representation, the other contains 2N hypermultiplets in the fundamental representation. If the Dine–Seiberg conjecture holds, then the two-loop F⁴ contributions to the effective action must vanish in both theories. This would necessitate a cancellation of the F⁴ corrections

Acknowledgements

We are grateful to Joseph Buchbinder, Jim Gates and Arkady Tseytlin for comments. This work is supported in part by the Australian Research Council and UWA research grants.

References (23)

I.L. Buchbinder et al.
On the D=4, N=2 non-renormalization theorem
Phys. Lett. B
(1998)
I.L. Buchbinder et al.
The background field method for N=2 super-Yang–Mills theories in harmonic superspace
Phys. Lett. B
(1998)
M. Dine et al.
Comments on higher derivative operators in some SUSY field theories
Phys. Lett. B
(1997)
B. de Wit et al.
Nonholomorphic corrections to the one-loop N=2 super-Yang–Mills action
Phys. Lett. B
(1996)
M. Henningson
Extended superspace, higher derivatives and SL(2,Z) duality
Nucl. Phys. B
(1996)
F. Gonzalez-Rey et al.
Nonholomorphic N=2 terms in N=4 SYM: 1-loop calculation in N=2 superspace
Phys. Lett. B
(1998)
F. Gonzalez-Rey et al.
Self-dual effective action of N=4 super-Yang–Mills
Nucl. Phys. B
(1999)
I.L. Buchbinder et al.
Comments on the background field method in harmonic superspace: non-holomorphic corrections in N=4 SYM
Mod. Phys. Lett. A
(1998)
E.I. Buchbinder et al.
Non-holomorphic effective potential in N=4 SU(n) SYM
Phys. Lett. B
(1999)
N. Dorey et al.
Instantons, higher-derivative terms, and nonrenormalization theorems in supersymmetric gauge theories
Phys. Lett. B
(1997)
I.L. Buchbinder et al.
Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace
(1998)
J. Zinn-Justin
Quantum Field Theory and Critical Phenomena
(1989)

S.M. Kuzenko et al.

On the background field method beyond one loop: a manifestly covariant derivative expansion in super-Yang–Mills theories

JHEP

(2003)

Cited by (19)

Low-energy 6D, N=(1,1) SYM effective action beyond the leading approximation
2020, Nuclear Physics B
Integrability in N=2 superconformal gauge theories
2015, Nuclear Physics B
Any $N = 2$ superconformal gauge theory (including $N = 4$ SYM) contains a set of local operators made only out of fields in the $N = 2$ vector multiplet that is closed under renormalization to all loops, namely the $SU (2, 1 | 2)$ sector. For planar $N = 4$ SYM the spectrum of local operators can be obtained by mapping the problem to an integrable model (a spin chain in perturbation theory), in principle for any value of the coupling constant. We present a diagrammatic argument that for any planar $N = 2$ superconformal gauge theory the $SU (2, 1 | 2)$ Hamiltonian acting on infinite spin chains is identical to all loops to that of $N = 4$ SYM, up to a redefinition of the coupling constant. Thus, this sector is integrable and anomalous dimensions can be, in principle, read off from the $N = 4$ ones up to this redefinition.
Two-loop low-energy effective action in Abelian supersymmetric Chern-Simons matter models
2014, Nuclear Physics B
We compute two-loop low-energy effective actions in Abelian Chern–Simons matter models with $N = 2$ and $N = 3$ supersymmetry up to four-derivative order. Calculations are performed with a slowly-varying gauge superfield background. Though the gauge superfield propagator depends on the gauge fixing parameter, it is shown that the obtained results are independent of this parameter. In the massless case the considered models are superconformal. We demonstrate that the superconformal symmetry strongly restricts the form of two-loop quantum corrections to the effective actions such that the obtained terms have simpler structure than the analogous ones in the effective action of three-dimensional supersymmetric electrodynamics (SQED) with vanishing topological mass.
Effective action of β-deformed N = 4 SYM theory: Farewell to two-loop BPS diagrams
2007, Nuclear Physics B
Within the background field approach, all two-loop sunset vacuum diagrams, which occur in the Coulomb branch of $N = 2$ superconformal theories (including $N = 4$ SYM), obey the BPS condition $m_{3} = m_{1} + m_{2}$ , where the masses are generated by the scalars belonging to a background $N = 2$ vector multiplet. These diagrams can be evaluated exactly, and prove to be homogeneous quadratic functions of the one-loop tadpoles $J (m_{1}^{2})$ , $J (m_{2}^{2})$ and $J (m_{3}^{2})$ , with the coefficients being rational functions of the squared masses. We demonstrate that, if one switches on the β-deformation of the $N = 4$ SYM theory, the BPS condition no longer holds, and then generic two-loop sunset vacuum diagrams with three non-vanishing masses prove to be characterized by the following property: $2 (m_{1}^{2} m_{2}^{2} + m_{1}^{2} m_{3}^{2} + m_{2}^{2} m_{3}^{2}) > m_{1}^{4} + m_{2}^{4} + m_{3}^{4}$ . In the literature, there exist several techniques to compute such diagrams. For the β-deformed $N = 4$ SYM theory, we carry out explicit two-loop calculations of the Kähler potential and $F^{4}$ term. Our considerations are restricted to the case of β real.
Relaxed super self-duality and N = 4 SYM at two loops
2004, Nuclear Physics B
A closed-form expression is obtained for a holomorphic sector of the two-loop low-energy effective action for the $N = 4$ super Yang–Mills theory on its Coulomb branch where the gauge group $SU (N)$ is spontaneously broken to $SU (N - 1) \times U (1)$ and the dynamics is described by a single Abelian $N = 2$ vector multiplet. In the framework of the background-field method, this holomorphic sector is singled out by computing the effective action for a background $N = 2$ vector multiplet satisfying a relaxed super self-duality condition. At the two-loop level, the $N = 4$ SYM effective action is shown to possess no $F^{4}$ term (with F the $U (1)$ field strength), in accordance with the Dine–Seiberg non-renormalization conjecture hep-th/9705057 and its generalized form given in hep-th/0310025. An unexpected outcome of our calculation is that no (manifestly supersymmetric invariant generating) $F^{6}$ quantum correction occurs at two loops. This is in conflict with previous results.
Relaxed super self-duality and effective action
2004, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
A closed-form expression is obtained for a holomorphic sector of the two-loop Euler–Heisenberg type effective action for $N =2$ supersymmetric QED derived in hep-th/0308136. In the framework of the background-field method, this sector is singled out by computing the effective action for a background $N =2$ vector multiplet satisfying a relaxed super self-duality condition. The approach advocated in this Letter can be applied, in particular, to the study of the $N =4$ super-Yang–Mills theory on its Coulomb branch.

View all citing articles on Scopus

View full text

On the two-loop four-derivative quantum corrections in 4D N=2 superconformal field theories

Abstract

Introduction

Section snippets

N=2 SYM setup

Functional representation for two-loop supergraphs with quantum hypermultiplets

Exact superpropagators

SU(N) SYM with 2N hypermultiplets in the fundamental

N=4 SYM

Discussion

Acknowledgements

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Nucl. Phys. B

Phys. Lett. B

Nucl. Phys. B

Mod. Phys. Lett. A

Phys. Lett. B

Phys. Lett. B

On the background field method beyond one loop: a manifestly covariant derivative expansion in super-Yang–Mills theories

JHEP

On the two-loop four-derivative quantum corrections in 4D $N =2$ superconformal field theories

$N =2$ SYM setup

$N =4$ SYM