On the road to N=2 supersymmetric Born-Infeld action

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On the road to N2 supersymmetric Born-Infeld
action

• S. Belluccia S. Krivonosb A.Shcherbakova
  A.Sutulinb
• a Istituto Nazionale di Fisica Nucleare,
  Laboratori Nazionali di Frascati , Italy
• b Bogoliubov Laboratory of Theoretical Physics,
  JINR

based on paper arXiv1212.1902
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Brief summary

• Born-Infeld theory and duality
• Supersymmetrization of Born-Infeld theory
• N1
• Approaches to deal with N2
• Ketov equation and setup
• Description of the approach perturbative
  expansion
• Quantum and classic aspects
• Problems with the approach
• Conclusions

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Born-Infeld theory
M. Born, L. Infeld Foundations of the new field
theoryProc.Roy.Soc.Lond. A144 (1934) 425-451

• Non-linear electrodynamics
• Introduced to remove the divergence of
  self-energy of a charged point-like particle

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Born-Infeld theory
E. Schrodinger Die gegenwartige Situation in
der Quantenmechanik Naturwiss. 23 (1935)
807-812

• The theory is duality invariant.
• This duality is related to the so-called
  electro-magnetic duality in supergravity or
  T-duality in string theory.
• Duality constraint

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Supersymmetrization of Born-Infeld
J. Bagger, A. Galperin A new Goldstone multiplet
for partially broken supersymmetry Phys. Rev. D55
(1997) 1091-1098

• N1 SUSY
• Relies on PBGS from N2 down to N1
• supersymmetry is spontaneously broken, so that
  only ½ of them is manifest
• Goldstone fields belong to a vector (i.e.
  Maxwell) supermultiplet
• where V is an unconstraint N1 superfield

M. Rocek, A. Tseytlin Partial breaking of global
D 4 supersymmetry, constrained superfields, and
three-brane actions Phys. Rev. D59 (1999) 106001
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N1 SUSY BI and duality
S.Kuzenko, S. Theisen Supersymmetric Duality
Rotations arXiv hep-th/0001068

• For a theory described by action
• SW,W to be duality invariant, the
• following must hold
• where Ma is an antichiral N1 superfield, dual
  to Wa

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Solution to the duality constraint
J. Bagger, A. Galperin A new Goldstone multiplet
for partially broken supersymmetry Phys. Rev. D55
(1997) 1091-1098

• A non-trivial solution to
• the duality constraint has a form
• where N1 chiral superfield Lagrangian is a
  solution to equation
• Due to the anticommutativity of Wa, this equation
  can be solved.

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Solution to the equation

• The solution is then given in terms of
• and has the following form
• so that the theory is described by action

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N2 supersymmetrization of BI

• Different approaches
• require the presence of another N2 SUSY which is
  spontaneously broken
• require self-duality along with non-linear shifts
  of the vector superfield
• try to find an N2 analog of N1 equation

Resulting actions are equivalent

• S. Bellucci, E.Ivanov, S. Krivonos
• N2 and N4 supersymmetric Born-Infeld theories
  from nonlinear realizations
• Towards the complete N2 superfield Born-Infeld
  action with partially broken N4 supersymmetry
• Superbranes and Super Born-Infeld Theories from
  Nonlinear Realizations

S. Kuzenko, S. Theisen Supersymmetric Duality
Rotations
S. Ketov A manifestly N2 supersymmetric
Born-Infeld action
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N2 BI with another hidden N2

• The basic object is a chiral complex scalar N2
  off-shell superfield strength W subjected to
  Bianchi identity
• The hidden SUSY (along with central charge
  transformations) is realized as
• where

parameters of broken SUSY trsf
parameters of central charge trsf
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N2 BI with another hidden N2

• How does A0 transform?
• Again, how does A0 transform?
• These fields turn out to be lower components of
  infinite dimensional supermultiplet

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Infinitely many constraints

• A0 is good candidate to be the chiral superfield
  Lagrangian. To get an interaction theory, the
  chiral superfields An should be covariantly
  constrained
• What is the solution?

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Finding the solution

• Making perturbation theory, one can find that
• Therefore, up to this order, the action reads

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N2 analog of
S. Ketov A manifestly N2 supersymmetric
Born-Infeld action Mod.Phys.Lett. A14 (1999)
501-510

• It was claimed that
• in N2 case the theory is described by the action
• where A is chiral superfield obeying N2 equation

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Ketov solution to eq.

• Inspired by lower terms in the series expansion,
  it was suggested that the solution to Ketov
  equation yields the following action
• where

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Properties of the action

• Reproduces correct N1 limit.
• Contains only W, D4W and their conjugate.
• Being defined as follows
• the action is duality invariant.
• The exact expression is wrong

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Set up

• So, if there exists another hidden N2 SUSY, the
  chiral superfield Lagrangian is constrained as
  follows
• Corresponding N2 Born-Infeld action
• How to find A0?

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Set up

• Observe that the basic equation
• is a generalization of Ketov equation
• Remind that this equation corresponds to duality
  invariant action. So let us consider this
  equation as an approximation.

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Set up

• This approximation is just a truncation
• after which a little can be said about the hidden
  N2 SUSY.

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Perturbative solution to Ketov eq.

• Equivalent form of Ketov equation
• The full action acquires the form
• Total derivative terms in B are unessential,
  since they do not contribute to the action

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Perturbative solution to Ketov eq.

• Series expansion
• Solution to Ketov equation, term by term

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Perturbative solution to Ketov eq.

• Some lower orders

new structures, not present in Ketov solution,
appear
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Perturbative solution to Ketov eq.

• Due to the irrelevance of total derivative terms
  in B , expression for B8 may be written in form
  that does not contain new structures
• For B10 such a trick does not succeed, it can
  only be simplified to

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Perturbative solution to Ketov eq.

• One can guess that to have a complete set of
  variables, one should add new objects
• to those in terms of which Ketovs solution is
  written
• Indeed, B12 contains only these four structures

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Perturbative solution to Ketov eq.

• The next term B14 introduces new structures
• This chain of appearance of new structures seems
  to never end.

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Perturbative solution to Ketov eq.

• Message learned from doing perturbative
  expansion
• Higher orders in the perturbative expansion
  contain terms of the following form
• written in terms of operators

the full solution can not be represented as some
function depending on finite number of its
arguments
Unfortunately, this type of terms is not the only
one that appears in the higher orders
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Quantum aspects of the pert. sol.

• Introduction of the operators
• is similar to the standard procedure in quantum
  mechanics. By means of these operators, Ketov
  equation
• can be written in operational form

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Classical limit

• Once quantum mechanics is mentioned, one can
  define its classical limit. In case under
  consideration, it consists in replacing operators
  X
• by functions
• In this limit, operational form of Ketov equation
• transforms in an algebraic one

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Classical limit

• This equation can immediately be solved as
• Curiously enough, this is exactly the expression
  proposed by Ketov as a solution to Ketov
  equation!

Clearly, this is not the exact solution to the
equation, but a solution to its classical
limit, obtained by unjustified replacement of
the operators by their classical expressions.
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Operational perturbative expansion

• Inspired by the classical solution, one can try
  to find the full solution using the ansatz
• Up to tenth order, operators X and X are
  enough to reproduce correctly the solution.
• The twelfth order, however, can not be reproduced
  by this ansatz
• so that new ingredients must be introduced.

to emphasize the quantum nature
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Operational perturbative expansion

• The difference btw. quantum and the exact
  solution in 12th order is equal to
• where the new operator is introduced as
• Obviously, since
• it vanishes the classical limit.

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Operational perturbative expansion

• With the help of operators X X and X3 one can
  reproduce B2n4 up to 18th order (included) by
  means of the ansatz
• Unfortunately, in the 20th order a new quantum
  structure is needed. It is not an operator but a
  function
• which, obviously, disappears in the classical
  limit.

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Operational perturbative expansion

• The necessity of this new variable makes all
  analysis quite cumbersome and unpredictable,
  because we cannot forbid the appearance of this
  variable in the lower orders to produce the
  structures already generated by means of
  operators X, bX and

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Conclusions

1. We investigated the structure of the exact
   solution of Ketov equation which contains
   important information about N2 SUSY BI theory.
2. Perturbative analysis reveals that at each order
   new structures arise. Thus, it seems impossible
   to write the exact solution as a function
   depending on finite number of its arguments.
3. We proposed to introduce differential operators
   which could, in principle, generate new
   structures for the Lagrangian density.
4. With the help of these operators, we reproduced
   the corresponding Lagrangian density up to the
   18th order.
5. The highest order that we managed to deal with
   (the 20-th order) asks for new structures which
   cannot be generated by action of generators X and
   X3.