D-term Dynamical Supersymmetry Breaking K. Fujiwara and, H.I. and M. Sakaguchi arXiv: hep-th/ , P. T. P. 113 arXiv: hep-th/ , N. P. B 723 H. I., K. Maruyoshi and S. Minato arXiv: , Nucl. Phys. B 830 cf. 1 with N. Maru (Keio U.) arXiv: 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 2

3 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 of the theory on-shell which contains four-fermi interactions. condensation of the Dirac bilinear is responsible for

The rest of my talk Contents I) Introduction II) Basic idea III) Illustration by the Theory with vacuum at tree level IV) Mass spectrum at tree level and supercurrent V) Self-consistent Hartree Fock approximation VI) Vacuum shift and metastability (qualitative) VII) Our work in the context of MSSM VIII) More on the fermion masses in the H. F. (qualitative) 4

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 5

Off-shell component lagrangian The off-shell component lagrangian is where is the Kähler metric and its derivatives are defined as and. The gauge part is, in components, Finally, the superpotential can be written as 6

Eq of motion for auxiliary fields While, from the transformation laws, 7

8 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 99 a

10 vacuum condition

V) Self-consistent Hartree-Fock approximation 11 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

12 : mass matrix (holomorphic and nonvanishing part) The eigenvalues are We obtain where the entire contribution to the 1PI vertex function

13 : In order to trade A with in V c.t., impose, for instance, we obtain (some number),

14 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

15 VI) Vacuum shift and metastability (qualitative) computable e.g.

16 provided can be made long lived

VII) Our work in the context of MSSM 17 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) 18

19 Demanding We obtain

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

21 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. +

22 D-term Dynamical Supersymmetry Breaking K. Fujiwara and, H.I. and M. Sakaguchi arXiv: hep-th/ , P. T. P. 113 arXiv: hep-th/ , N. P. B 723 H. I., K. Maruyoshi and S. Minato arXiv: , Nucl. Phys. B 830 cf. with N. Maru (Keio U.) arXiv: one in preparation gluino gluon massive fermion scalar gluon -1 -1/2 0 1/2 1 mass -1/2 0 1/2 mass Obserbale (SU(N)) sector