Ghost-free vector superfield actions in supersymmetric higher-derivative theories Ryo Yokokura (Keio) 2018. 8. 2 Strings and fields @ YITP with T. Fujimori, M. Nitta, K. Ohashi, Y. Yamada based on JHEP 1709 (2017)143, [arXiv:1708.05129]
What we did Today’s goal: of an Abelian vector superfield in 4D N=1 SUSY We construct general ghost-free higher-derivative actions +h.c. I will use the Wess-Bagger notation. Z Z D 2 ¯ W 2 ) W 2 + D 2 ¯ W 2 , D α W α ) W 2 ¯ d 2 θ F ( ¯ d 4 θ G ( D 2 W 2 , ¯ W 2 L = W α = − 1 D 2 D α V 4 ¯
Contents • Introduction (2 slides) • Ostrogradsky’s ghost and ghost-free actions in SUSY (4 slides) • Ghost-free vector superfield actions (5 slides)
1. Supersymmetric (SUSY) • How about higher derivative interactions? <latexit sha1_base64="S1ev2lcu1ecj+/4Lo5dGn3yXQ=">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</latexit> <latexit sha1_base64="S1ev2lcu1ecj+/4Lo5dGn3yXQ=">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</latexit> <latexit sha1_base64="S1ev2lcu1ecj+/4Lo5dGn3yXQ=">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</latexit> <latexit sha1_base64="S1ev2lcu1ecj+/4Lo5dGn3yXQ=">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</latexit> effective theories (2nd order derivative, global SUSY) A general Lagrangian of chiral+vector superfields • (renormalizability is not always imposed) General non-renormalizable interactions can be included • Determined by IR symmetries • Framework of string effective theories, cosmology, … • (including supergravity) Z Z Z d 4 θ K ( Φ , ¯ d 2 θ W ( Φ ) + d 2 θ f ( Φ ) W α W α + h . c . L = Φ ) +
2. Higher derivative (HD) interactions • Effective theories generally include HD interactions e.g. Dirac-Born-Infeld (DBI) action, α’-corrections, … • How to write down a general HD action in SUSY? • Can it give us stable effective theories? Naive HD extensions cause instability…
Ostrogradsky’s ghost and ghost-free actions in SUSY (4 slides)
1. Ostrogradsky’s ghost negative kinetic sign = ghost <latexit sha1_base64="BWIlk38i+SPg9cWKv80Ds+7g+x8=">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</latexit> <latexit sha1_base64="BWIlk38i+SPg9cWKv80Ds+7g+x8=">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</latexit> <latexit sha1_base64="BWIlk38i+SPg9cWKv80Ds+7g+x8=">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</latexit> <latexit sha1_base64="BWIlk38i+SPg9cWKv80Ds+7g+x8=">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</latexit> Are there ghost-free Lagrangians in SUSY? An example e.g. relativistic particle (Later, we only focus on such instabilities. Other model-dependent instabilities may exist). Negative kinetic mode from HD interactions = unstable mode [Ostrogradsky 1850, Woodard ’06] (not FP ghost) x 2 + 1 L = 1 x 2 2 ˙ 2 α ¨ higher derivative term L 0 = 1 x 2 − α ( ˙ y + 1 2 y 2 ) rewrite by auxiliary field 2 ˙ x ˙ L 0 = 1 x 0 2 − 1 y 0 2 2 ˙ 2 ˙ + potential A sufficient condition for ghost-free Lagrangian : L = L ( x, ˙ x ) p x 2 L = − m 1 − ˙
2. Ghost-free HD in SUSY - chiral superfield - [Khoury et al. (2011)] General ghost-free HD Lagrangian known T: arbitrary function Φ k ∗ ¯ Z Φ ) ijk ∗ l ∗ D α Φ i D α Φ j ¯ α ¯ d 4 θ T ( Φ , ¯ Φ , ∂ Φ , ∂ ¯ α ¯ Φ l ∗ D ˙ L HD = D ˙ • No bosonic ghosts shown by brute force • Fermions expected to be ghost-free by SUSY
3. Ghost-free HD in SUSY - vector superfield - Only some examples known Construct general ghost-free HD actions for a vector superfield Our goal In this talk, I consider a pure U(1) SUSY gauge theory for simplicity. • DBI actions [Cecotti & Ferrara (1985); Bagger & Galperin (1997)] • SUSY Euler-Heisenberg model [Cecotti et al. (1987)] • Non-linearly self-dual actions [Kuzenko & Theisen (2000)] (Explicit Lagrangians given later)
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