D-term Dynamical Supersymmetry BreakingK. Fujiwara and, H.I. and M. SakaguchiarXiv: hep-th/0409060, P. T. P. 113arXiv: hep-th/0503113, N. P. B 723H. I., K. Maruyoshi and S. MinatoarXiv:0909.5486, Nucl. Phys. B 830

cf.

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with N. Maru (Keio U.)• arXiv:1109.2276 • one in preparation

I) Introduction• spontaneous breaking of SUSY is much less frequent compared with that of internal symmetry • most desirable to break SUSY dynamically (DSB) • F term DSB has been popular since mid 80’s, in particular, in the context of instanton generated superpotential • In this talk, we will accomplish D term DSB, DDSB, for short• based on the nonrenormalizable D-gaugino-matter fermion coupling and most natural in the context of SUSY gauge theory spontaneous broken to ala APT-FIS

II) Basic idea• Start from a general lagrangian

: a Kähler potential : a gauge kinetic superfield of the chiral superfield in the adjoint representation: a superpotential.

• bilinears:

where .no bosonic counterpart assume is the 2nd derivative of a trace fn.

the gauginos receive masses of mixed Majorana-Dirac type and are split.

: holomorphic and nonvanishing part of the mass

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• Determination of

stationary condition to

where is the one-loop contribution

and is a counterterm.

In fact, the stationary condition is nothing but the well-known gap equation ofthe theory on-shell which contains four-fermi interactions.

condensation of the Dirac bilinear is responsible for

The rest of my talk

ContentsI) IntroductionII) Basic ideaIII) Illustration by the Theory with vacuum at tree levelIV) Mass spectrum at tree level and supercurrentV) Self-consistent Hartree Fock approximationVI) Vacuum shift and metastability (qualitative)VII) Our work in the context of MSSMVIII) More on the fermion masses in the H. F. (qualitative)

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III) Theory with 　 vacuum at tree level

• U(N) gauge group assumed for definiteness (product gauge group O.K.)

• : prepotential, input function

• superpotential W supplied by the electric and magnetic FI terms, made possible by a particular fixing of rigid SU(2)R symmetry

• should contrast with

• Later, will work with

Action to work with

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Off-shell component lagrangianThe off-shell component lagrangian is

where is the Kähler metric and its derivatives are defined asand .

The gauge part is, in components,

Finally, the superpotential can be written as

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Eq of motion for auxiliary fields

While, from the transformation laws,

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susy of and tree vacua• construction of 2nd susy : Let be

• the form of 　　　 and are derived by imposing

•

• ; vacuum condition

•

• generic breaking pattern of gauge symmetry:

so that follows from

where

2nd susy broken

IV) Mass spectrum at tree level and supercurrent

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a

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vacuum condition

V) Self-consistent Hartree-Fock approximation

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For simplicity, consider the case U(N) unbroken

Recall we hunt for the possibility (up to one-loop):

no such coupling to bosons present

Mixed Maj.-Dirac mass to gaugino,

DSB

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• :

mass matrix (holomorphic and nonvanishing part)

The eigenvalues are

We obtain

where

the entire contribution to the 1PI vertex function

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• :

In order to trade A with in Vc.t. ,

impose, for instance,

we obtain

(some number),

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• gap equation:is a stationary condition to

susy is broken to .

Aside from a trivial solution , a nontrivial transcendental solution

gap eq. In the approximate form

in general exists

vacuum condition of vacuum

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VI) Vacuum shift and metastability (qualitative)

computable

•

e.g.

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• obviously and

the tree vacuum is not lifted.

So the vacuum is metastable.

Estimate of the decay rate:

provided

can be made long lived∴

VII) Our work in the context of MSSM

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Symbolically

•

vector superfields, chiral superfields, their coupling

extend this to the type of actions with s-gluons and adjoint fermions so as not to worry about mirror fermions e.t.s.

• gauge group , the simplest case being

• Due to the non-Lie algebraic nature of the third prepotential derivatives, or , we do not really need messenger superfields.

• transmission of DDSB in to the rest of the theory by higher order loop-corrections

the sfermion masses

the gaugino masses of the quadratic Casimir of representation some function of , which is essentially

Fox, Nelson, Weiner, JHEP(2002)

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• Demanding

We obtain

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VIII) More on the fermion masses in the H. F. (qualitative)

• Back to the general theory with 3 input functions

• H. F. can be made into a systematic expansion by an index loop argument.

• Take to be .

• In the unbroken phase,

The gap eq. is

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• Two sources beyond tree but leading in H. F.

i) Due to the vacuum shift, as well

ii) For U(1) sector, an index loop circulates

These contribute to the masses in the leading order in the H. F.

+

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D-term Dynamical Supersymmetry BreakingK. Fujiwara and, H.I. and M. SakaguchiarXiv: hep-th/0409060, P. T. P. 113arXiv: hep-th/0503113, N. P. B 723H. I., K. Maruyoshi and S. MinatoarXiv:0909.5486, Nucl. Phys. B 830

cf.with N. Maru (Keio U.)• arXiv:1109.2276 • one in preparation

gluino

gluon

𝜓 ′λ ′massive fermion

scalar gluon

-1 -1/2 0 1/2 1

mass

h-1/2 0 1/2

mass

𝑆𝑧

Obserbale (SU(N)) sector