Studies of low-energy effective actions in supersymmetric field theories

2.5 MB·

by  Simon James Tyler

Studies of low-energy effective actions in
supersymmetric field theories
Simon James Tyler
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January 2013
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r This thesis is presented for the degree of
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Doctor of Philosophy
of The University of Western Australia
School of Physics.
Supervisor: Winthrop Professor Sergei M. Kuzenko
Co-supervisor: Winthrop Professor Ian N. McArthur
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Abstract
This thesis examines low-energy effective actions of supersym-
metric quantum field theories. These effective actions contain infor-
mation about the low-energy field content and dynamics of quantum
field theories and are essential for understanding their phenomeno-
logical and theoretical properties.
In chapters 2 to 5, the covariant background field method is used
to investigate quantum corrections to sectors of the low-energy effec-
tiveactionsforavarietyofsupersymmetricfieldtheoriesatone-and
two-loops. We start by looking at the background field quantisation
of a general N = 1 super-Yang-Mills theory and rederiving the well
known one-loop finiteness conditions. This is followed by a reex-
amination of the effective potential of the simplest supersymmetric
quantum field theory, the Wess-Zumino model. Next, the two-loop
Euler-Heisenberg effective action is constructed for N = 1 super-
symmetric quantum electrodynamics. This is a natural object to
study in the progression of such two-loop Euler-Heisenberg calcula-
tionsandisonlythesecondsuchresultusingsuperfields. Thetheory
is renormalised and the self-dual limit of the renormalised effective
action is given explicitly in terms of digamma functions. The final
quantum effective action studied is the two-loop K¨ahler potential for
β-deformed N = 4 super-Yang-Mills theory. This sector of the effec-
tive action is purely a product of the deformation and its finiteness
is demonstrated in a general background before specialising to give
explicit results for two special cases.
Chapter 6 studies spontaneously broken supersymmetry and, in
particular, the pure Goldstino action. This is a universal sector
of the low-energy effective action of any theory with spontaneously
broken supersymmetry. A very general approach to constructing
explicit field redefinitions is used to relate all known models of the
Goldstino and to study both their nonlinear supersymmetries and
their previously unnoticed trivial symmetries. This approach is also
used to construct the most general pure Goldstino action and to
examine its nonlinear supersymmetry transformations. Finally, a
new embedding of the Goldstino into a complex linear superfield is
presented. Its interactions to matter and gravity are examined and
compared to existing Goldstino superfield constructions.

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This thesis is based in part on the following five published papers:
[1] S. M. Kuzenko and S. J. Tyler, “Supersymmetric Euler-Heisenberg
effective action: Two-loop results,” JHEP 2007 05, (2007) 081,
[arXiv:hep-th/0703269].
[2] S. J. Tyler, “Two loop Kahler potential in beta-deformed N=4 SYM
theory,” JHEP 2008 07, (2008) 24, [arXiv:0805.3574].
[3] S. M. Kuzenko and S. J. Tyler, “Relating the Komargodski-Seiberg
and Akulov-Volkov actions: Exact nonlinear field redefinition,” Phys.
Lett. B 689 4, (2010) 319–322, [arXiv:1009.3298].
[4] S. M. Kuzenko and S. J. Tyler, “On the Goldstino actions and their
symmetries,” JHEP 2011 5, (2011) 32, [arXiv:1102.3043].
[5] S. M. Kuzenko and S. J. Tyler, “Complex linear superfield as a model
for Goldstino,” JHEP 2011 4, (2011) 8, [arXiv:1102.3042].
Permission has been granted to include this work:
Simon J. Tyler
Sergei M. Kuzenko

Contents
Title Page
Abstract i
Contents v
Acknowledgements ix
1 Introduction 1
1.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Effective actions . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Supersymmetry algebra . . . . . . . . . . . . . . . . . . . . . 5
1.4 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Quantisation of N = 1 SYM 9
2.1 Classical theory . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Background-field splitting . . . . . . . . . . . . . . . . . . . 12
2.3 Constructing the perturbation theory . . . . . . . . . . . . . 13
2.4 Propagators and Feynman rules . . . . . . . . . . . . . . . . 20
2.5 One-loop effective action . . . . . . . . . . . . . . . . . . . . 22
2.5.1 Ka¨hler potential . . . . . . . . . . . . . . . . . . . . . 23
2.5.2 F2 and F4 corrections . . . . . . . . . . . . . . . . . 24
2.5.3 One-loop finiteness . . . . . . . . . . . . . . . . . . . 26
3 Wess-Zumino model 29
3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Ka¨hler potential . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 One-loop . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Two-loop . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Renormalisation . . . . . . . . . . . . . . . . . . . . . 35
v
vi CONTENTS
3.3 Auxiliary potential . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Four-derivative term via direct expansion . . . . . . . 38
3.3.2 Four-derivative term via the heat kernel . . . . . . . 42
3.3.3 The full, one-loop auxiliary potential . . . . . . . . . 45
3.3.4 Component projections and comparisons . . . . . . . 47
4 Supersymmetric quantum electrodynamics 51
4.1 Classical action and quantisation . . . . . . . . . . . . . . . 52
4.2 One-loop Euler-Heisenberg effective action . . . . . . . . . . 54
4.3 Two-loop Euler-Heisenberg effective action . . . . . . . . . . 55
4.4 Renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 The limit of a self-dual background . . . . . . . . . . . . . . 64
4.6 One-loop matter sector in the Feynman gauge . . . . . . . . 68
4.6.1 Quantisation in Fermi-Feynman gauge . . . . . . . . 69
4.6.2 Two-point function . . . . . . . . . . . . . . . . . . . 70
4.6.3 K¨ahler potential . . . . . . . . . . . . . . . . . . . . . 72
4.6.4 Renormalisation . . . . . . . . . . . . . . . . . . . . . 73
5 Beta-deformed N=4 SYM 75
5.1 Classical action . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.1 Cartan-Weyl basis and the mass operator . . . . . . . 79
5.2 One-loop K¨ahler potential . . . . . . . . . . . . . . . . . . . 81
5.3 Two-loop Ka¨hler potential . . . . . . . . . . . . . . . . . . . 82
5.3.1 Evaluation of Γ . . . . . . . . . . . . . . . . . . . . . 83
I
5.3.2 Evaluation of Γ . . . . . . . . . . . . . . . . . . . . 85
II
5.3.3 Finiteness and conformal invariance . . . . . . . . . . 85
5.4 Special backgrounds and explicit masses . . . . . . . . . . . 87
5.4.1 SU(N) → SU(N −1)×U(1) . . . . . . . . . . . . . 90
5.4.2 SU(3) → U(1)2 . . . . . . . . . . . . . . . . . . . . . 90
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Goldstino actions 95
6.1 Supersymmetry breaking and the supersymmetric sigma model 96
6.1.1 Supersymmetric nonlinear sigma model . . . . . . . . 97
6.1.2 Models with a single chiral superfield . . . . . . . . . 100
6.1.3 The O’Raifeartaigh model . . . . . . . . . . . . . . . 101
6.2 Goldstino actions and their symmetries . . . . . . . . . . . . 102
6.2.1 The Akulov-Volkov model . . . . . . . . . . . . . . . 102
6.2.2 General Goldstino action . . . . . . . . . . . . . . . . 105
6.2.3 Roˇcek’s Goldstino action . . . . . . . . . . . . . . . . 108
CONTENTS vii
6.2.4 Casalbuoni-De Curtis-Dominici-Feruglio-Gatto
and Komargodski-Seiberg action . . . . . . . . . . . . 112
6.2.5 The chiral Alkulov-Volkov action . . . . . . . . . . . 116
6.2.6 The supersymmetric Born-Infeld action . . . . . . . . 118
6.2.7 The chiral-scalar Goldstino action . . . . . . . . . . . 120
6.2.8 Trivial symmetries and field redefinitions . . . . . . . 122
6.3 Goldstino dynamics from a constrained complex linear su-
perfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3.1 Constrained complex linear superfield . . . . . . . . . 125
6.3.2 Comparison to other Goldstino models . . . . . . . . 128
6.3.3 Couplings to matter and supergravity . . . . . . . . . 130
6.3.4 Coupling to a matter sector . . . . . . . . . . . . . . 131
7 Conclusion 137
A Background field propagators 141
A.1 Regularisation by dimensional reduction . . . . . . . . . . . 141
A.2 Parallel displacement propagator . . . . . . . . . . . . . . . 143
A.3 Free bosonic heat kernels . . . . . . . . . . . . . . . . . . . . 144
A.4 Covariantly constant background . . . . . . . . . . . . . . . 145
A.4.1 Chiral/antichiral propagators . . . . . . . . . . . . . 150
A.4.2 Self-dual limit . . . . . . . . . . . . . . . . . . . . . . 152
A.4.3 The simplest non-trivial background . . . . . . . . . 153
A.5 Wess-Zumino propagator . . . . . . . . . . . . . . . . . . . . 153
A.5.1 Calculating the heat kernel . . . . . . . . . . . . . . . 154
Results for the heat kernel . . . . . . . . . . . . . . . . . . . 158
A.5.2 K¨ahler approximation . . . . . . . . . . . . . . . . . 159
A.5.3 Expansion up to four derivatives . . . . . . . . . . . . 160
B Effective potential for WZ 163
B.1 Background field quantisation . . . . . . . . . . . . . . . . . 163
B.2 One-loop effective potential . . . . . . . . . . . . . . . . . . 164
C Integer relation algorithms 167
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
C.2 Application to (S)QED with a self-dual background . . . . . 170
C.2.1 LLL in Mathematica . . . . . . . . . . . . . . . . . . 170
C.2.2 PSLQ in python with mpmath . . . . . . . . . . . . . 173
D Two-loop vacuum integrals 175
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
viii CONTENTS
D.2 Fish diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 176
E Goldstino appendices 183
E.1 Minimal basis for Goldstino actions . . . . . . . . . . . . . . 183
E.2 Composition rule for field redefinitions . . . . . . . . . . . . 184
Bibliography 189