Ghost-free vector superfield actions in supersymmetric - - PowerPoint PPT Presentation

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:

L = Z d2θF( ¯ D2 ¯ W 2)W 2 + Z d4θG(D2W 2, ¯ D2 ¯ W 2, DαWα)W 2 ¯ W 2

+h.c.

Wα = − 1

4 ¯

D2DαV

We construct general ghost-free higher-derivative actions

• f an Abelian vector superfield in 4D N=1 SUSY

I will use the Wess-Bagger notation.

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)

effective theories

• Framework of string effective theories, cosmology, …
• Determined by IR symmetries
• General non-renormalizable interactions can be included


(renormalizability is not always imposed)

• A general Lagrangian of chiral+vector superfields 


(2nd order derivative, global SUSY)

• L =

Z d4θK(Φ, ¯ Φ) + Z d2θW(Φ) + Z d2θf(Φ)W αWα + h.c.

How about higher derivative interactions?

(including supergravity)

• 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

A sufficient condition for ghost-free Lagrangian: L = L(x, ˙

x)

(Later, we only focus on such instabilities. Other model-dependent instabilities may exist). [Ostrogradsky 1850, Woodard ’06] (not FP ghost)

Negative kinetic mode from HD interactions = unstable mode

L = 1

2 ˙

x2 + 1

2α¨

x2

higher derivative term

L0 = 1

2 ˙

x2 − α( ˙ x ˙ y + 1

2y2)

+ potential

L0 = 1

2 ˙

x02 − 1

2 ˙

y02

negative kinetic sign = ghost rewrite by auxiliary field

An example Are there ghost-free Lagrangians in SUSY?

L = −m p 1 − ˙ x2

e.g. relativistic particle

• 2. Ghost-free HD in SUSY
• chiral superfield -

LHD = Z d4θT(Φ, ¯ Φ, ∂Φ, ∂ ¯ Φ)ijk∗l∗DαΦiDαΦj ¯ D ˙

α ¯

Φk∗ ¯ D ˙

α ¯

Φl∗

General ghost-free HD Lagrangian known

• No bosonic ghosts shown by brute force
• Fermions expected to be ghost-free by SUSY

[Khoury et al. (2011)] T: arbitrary function

• 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)

• 4. SUSY ghost-free actions are

non-trivial

L = 1

4

Z d2θW 2 + Z d4θf(DαWα) + h

higher derivative term (W α

U := − 1 4 ¯

D2DαU)

rewrite by real superfields and

U

L

L0 = 1

4

Z d2θW 2 + 1

2

Z d2θW α

U Wα + · · ·

partial integration

L0 = 1

4

Z d2θW 2 + Z d4θf(L) − Z d4θU(DαWα − L) + h.c.

ghost

L

L0 = 1

4

Z d2θW 02 − 1

4

Z d2θW 002 + · · ·

the cross term causes a ghost

Example [Dudas & Ghilencea (2015); Fujimori, Nitta, Ohashi, Yamada, RY (2017)] How to obtain general ghost-free actions of a vector superfield? Naive HD extensions give rise to ghosts.

Ghost-free vector superfield actions

[Fujimori, Nitta, Ohashi, Yamada, RY (2017)]

L = Z d2θF( ¯ D2 ¯ W 2)W 2 + Z d4θG(D2W 2, ¯ D2 ¯ W 2, DαWα)W 2 ¯ W 2

+h.c.

Wα = − 1

4 ¯

D2DαV

(5 slides)

• 1. How to construct
• 1. Symmetries for a vector superfield: Lorentz, U(1) gauge and SUSY
• 2. Ghost-free condition for boson: Lboson = Lboson(Fµν) , no terms

∂Fµν

Fermions expected to be ghost-free by SUSY

Strategy

The gauge field only appear as field strength by gauge symmetry.

L = Z d2θF( ¯ D2 ¯ W 2)W 2 + Z d4θG(D2W 2, ¯ D2 ¯ W 2, DαWα)W 2 ¯ W 2

+h.c.

(We consider a pure U(1) SUSY gauge theory)

• 2. SUSY+ gauge invariance


+ ghost-free

Lboson = Lboson(DαWβ|, ¯ D ˙

α ¯

W

˙ β|)

DαWβ| ∼ Fµν + D

auxiliary
 field

Condition for boson Lagrangian

• 1. SUSY: action is given by a vector superfield V
• 4. ghost-free: no spacetime derivatives on
• 3. SUSY: field strength embedded into DαWβ|
• 2. gauge invariance: action written by gaugino superfield

Wα = − 1

4 ¯

D2DαV

vertical bar:
 θ=0 projection

DαWβ|

• 3. Lorentz invariance

Bosonic Lagrangian is Lorentz invariant

Lboson = Lboson(D2W 2|, ¯ D2 ¯ W 2|, DαWα|)

Irreducible decomposition Contraction

DαWβ = 1

2✏αβDγWγ + 1 2(DαWβ + DβWα)

D(αW β)D(βWγ) = δα

γ ( 1 2DβW δDβWδ − 1 4(DβWβ)2)

DβW δDβWδ| = D2W 2| + fermions

Rewriting to (anti-)chiral comb.

minimal scalars given by(DαWβ|, ¯

D ˙

α ¯

W

˙ β|)

= Lboson(DαWβ|, ¯ D ˙

α ¯

W

˙ β|)

• 4. SUSY invariance again

chiral real

R d2θ → D2

R d4θ → D2 ¯ D2

and

L = Z d2θF( ¯ D2 ¯ W 2)W 2 + Z d4θG(D2W 2, ¯ D2 ¯ W 2, DαWα)W 2 ¯ W 2

+h.c.

= Lboson(D2W 2|, ¯ D2 ¯ W 2|, DαWα|)

Superspace Lagrangian leading to

W 2 W 2 ¯ W 2

and are needed because

(Maxwell term included into F)

D-type HD F-type HD

• 5. Comparison with known

examples

Known examples written in terms of previous action

• SUSY Euler-Heisenberg model

L = 1

4

Z d2θW 2 + 1

Λ

Z d4θW 2 ¯ W 2

• SUSY DBI action

H = − 1

4 ¯

D2 ¯ W 2

L = µα2

4

Z d2θW 2 + µα2

4

Z d4θW 2 ¯ W 2 ✓ 1 − α2

4 (H + ¯

H) + q 1 − α2

2 (H + ¯

H) + α4

16 (H − ¯

H)2 ◆−1

D-type HD

= µ − µ q − det(ηµν + αFµν)

+fermions

[Cecotti & Ferrara (1985); Bagger & Galperin (1997)] [Cecotti et al. (1985)] Λ: parameter μ, α: parameter

D-type HD

Maxwell

Maxwell

Summary

• We systematically formulate ghost-free actions of 


a U(1) vector superfield.

• Known examples fall into the above class.
• Matter coupling Lagrangians also possible (see our paper).
• Extension: Non-Abelian, p-form, N=2 SUSY, SUGRA, …

L = Z d2θF( ¯ D2 ¯ W 2)W 2 + Z d4θG(D2W 2, ¯ D2 ¯ W 2, DαWα)W 2 ¯ W 2

+h.c.