On the road to N=2 supersymmetric Born- Infeld action
description
Transcript of On the road to N=2 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 arXiv:1212.1902
1. Born-Infeld theory and duality2. Supersymmetrization of Born-Infeld theorya) N=1b) Approaches to deal with N=23. Ketov equation and setup4. Description of the approach: perturbative
expansion5. “Quantum” and “classic” aspects6. Problems with the approach7. Conclusions
S. Bellucci LNF INFN Italy 2Frontiers in Mathematical Physics Dubna 2012
Non-linear electrodynamics
Introduced to remove the divergence of self-energy of a charged point-like particle
S. Bellucci LNF INFN Italy 3Frontiers in Mathematical Physics Dubna 2012
M. Born, L. InfeldFoundations of the new field theory
Proc.Roy.Soc.Lond. A144 (1934) 425-451
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
S. Bellucci LNF INFN Italy 4Frontiers in Mathematical Physics Dubna 2012
E. Schrodinger Die gegenwartige Situation in
der Quantenmechanik Naturwiss. 23 (1935) 807-812
N=1 SUSY:Relies on PBGS from N=2 down to N=1
―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 N=1 superfield
S. Bellucci LNF INFN Italy 5Frontiers in Mathematical Physics Dubna 2012
J. Bagger, A. GalperinA new Goldstone multiplet for partially broken
supersymmetryPhys. Rev. D55 (1997) 1091-1098
M. Rocek, A. TseytlinPartial breaking of global D = 4 supersymmetry,
constrained superfields, and three-brane actionsPhys. Rev. D59 (1999) 106001
For a theory described by action S[W,W] to be duality invariant, the following must hold
where Ma is an antichiral N=1 superfield, dual to Wa
S. Bellucci LNF INFN Italy 6Frontiers in Mathematical Physics, Dubna 2012
S.Kuzenko, S. TheisenSupersymmetric Duality Rotations
arXiv: hep-th/0001068
A non-trivial solution to the duality constraint has a form
where N=1 chiral superfield Lagrangian is a solution to equation
Due to the anticommutativity of Wa, this equation can be solved.
S. Bellucci LNF INFN Italy 7Frontiers in Mathematical Physics, Dubna 2012
J. Bagger, A. GalperinA new Goldstone multiplet for partially
broken supersymmetryPhys. Rev. D55 (1997) 1091-1098
J. Bagger, A. GalperinA new Goldstone multiplet for partially
broken supersymmetryPhys. Rev. D55 (1997) 1091-1098
The solution is then given in terms of
and has the following form
so that the theory is described by action
S. Bellucci LNF INFN Italy 8Frontiers in Mathematical Physics, Dubna 2012
S. Bellucci, E.Ivanov, S. Krivonos•N=2 and N=4 supersymmetric Born-Infeld theories from nonlinear realizations•Towards the complete N=2 superfield Born-Infeld action with partially broken N=4 supersymmetry•Superbranes and Super Born-Infeld Theories from Nonlinear Realizations
S. Kuzenko, S. TheisenSupersymmetric Duality Rotations
Different approaches:—require the presence of another N=2 SUSY which is spontaneously broken—require self-duality along with non-linear shifts of the vector superfield—try to find an N=2 analog of N=1 equation
S. Bellucci LNF INFN Italy 9Frontiers in Mathematical Physics, Dubna 2012
S. KetovA manifestly N=2 supersymmetric Born-Infeld action
Resulting actions are equivalent
The basic object is a chiral complex scalar N=2 off-shell superfield strength W subjected to Bianchi identity
The hidden SUSY (along with central charge transformations) is realized as
where
S. Bellucci LNF INFN Italy 10Frontiers in Mathematical Physics, Dubna 2012
parameters of central charge trsf
parameters of broken SUSY trsf
How does A0 transform?
Again, how does A0 transform?
These fields turn out to be lower components of infinite dimensional supermultiplet:
S. Bellucci LNF INFN Italy 11Frontiers in Mathematical Physics, Dubna 2012
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?S. Bellucci LNF INFN Italy 12Frontiers in Mathematical Physics, Dubna 2012
Making perturbation theory, one can find that
Therefore, up to this order, the action reads
S. Bellucci LNF INFN Italy 13Frontiers in Mathematical Physics, Dubna 2012
It was claimed that in N=2 case the theory is described by the action
where A is chiral superfield obeying N=2 equation
S. Bellucci LNF INFN Italy 14Frontiers in Mathematical Physics, Dubna 2012
S. KetovA manifestly N=2 supersymmetric Born-Infeld action
Mod.Phys.Lett. A14 (1999) 501-510
Inspired by lower terms in the series expansion, it was suggested that the solution to Ketov equation yields the following action
where
S. Bellucci LNF INFN Italy 15Frontiers in Mathematical Physics, Dubna 2012
1. Reproduces correct N=1 limit.2. Contains only W, D4W and their conjugate.3. Being defined as follows
the action is duality invariant.4. The exact expression is wrong:
S. Bellucci LNF INFN Italy 16Frontiers in Mathematical Physics, Dubna 2012
So, if there exists another hidden N=2 SUSY, the chiral superfield Lagrangian is constrained as follows
Corresponding N=2 Born-Infeld action
How to find A0?
S. Bellucci LNF INFN Italy 17Frontiers in Mathematical Physics, Dubna 2012
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.
S. Bellucci LNF INFN Italy 18Frontiers in Mathematical Physics, Dubna 2012
This approximation is just a truncation
after which a little can be said about the hidden N=2 SUSY.
S. Bellucci LNF INFN Italy 19Frontiers in Mathematical Physics, Dubna 2012
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
S. Bellucci LNF INFN Italy 20Frontiers in Mathematical Physics, Dubna 2012
Series expansion
Solution to Ketov equation, term by term:
S. Bellucci LNF INFN Italy 21Frontiers in Mathematical Physics, Dubna 2012
Some lower orders:
S. Bellucci LNF INFN Italy 22Frontiers in Mathematical Physics, Dubna 2012
new structures, not present in Ketov solution, appear
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
S. Bellucci LNF INFN Italy 23Frontiers in Mathematical Physics, Dubna 2012
One can guess that to have a complete set of variables, one should add new objects
to those in terms of which Ketov’s solution is written:
Indeed, B12 contains only these four structures:
S. Bellucci LNF INFN Italy 24Frontiers in Mathematical Physics, Dubna 2012
The next term B14 introduces new structures:
This chain of appearance of new structures seems to never end.
S. Bellucci LNF INFN Italy 25Frontiers in Mathematical Physics, Dubna 2012
S. Bellucci LNF INFN Italy 26Frontiers in Mathematical Physics, Dubna 2012
Message learned from doing perturbative expansion:
Higher orders in the perturbative expansion contain terms of the following form:
written in terms of operatorsetc.
and
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
S. Bellucci LNF INFN Italy 27Frontiers in Mathematical Physics, Dubna 2012
and
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
S. Bellucci LNF INFN Italy 28Frontiers in Mathematical Physics, Dubna 2012
and
and
This equation can immediately be solved as
Curiously enough, this is exactly the expression proposed by Ketov as a solution to Ketov equation!
S. Bellucci LNF INFN Italy 29Frontiers in Mathematical Physics, Dubna 2012
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.
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.
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.
S. Bellucci LNF INFN Italy 30Frontiers in Mathematical Physics, Dubna 2012
to emphasize the quantum nature
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.
S. Bellucci LNF INFN Italy 31Frontiers in Mathematical Physics, Dubna 2012
With the help of operators X X and X3 one can reproduce B2n+4 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.
S. Bellucci LNF INFN Italy 32Frontiers in Mathematical Physics, Dubna 2012
the highest order that we were able to check
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
S. Bellucci LNF INFN Italy 33Frontiers in Mathematical Physics, Dubna 2012
1. We investigated the structure of the exact solution of Ketov equation which contains important information about N=2 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.
S. Bellucci LNF INFN Italy 34Frontiers in Mathematical Physics, Dubna 2012