Transcript Slide 1
String Field Theory Patras 2007 Non-Abelian Tensor Gauge Fields and Possible Extension of SM George Savvidy Demokritos National Research Center Athens Phys. Lett. B625 (2005) 341 Int.J.Mod.Phys. A21 (2006) 4959 Int.J.Mod.Phys. A21 (2006) 4931 Fortschr. Phys. 54 (2006) 472 Prog. Theor. Phys.117 (2007) 729 ------------------------ Takuya Tsukioka Hep-th/0604118 Hep-th/ 0704.3164 ------------------------ Jessica Barrett Hep-th/ 0706.0762 ------------------------ Spyros Konitopoulos • String Field Theory • Extended Non-Abelian gauge transformations • Field strength tensors • Extended current algebra as a gauge group • Invariant Lagrangian and interaction vertices • Propagating modes • Higher-spin extension of the Standard Model String Field • The multiplicity of tensor fields in string theory grows exponentially • Lagrangian and field equations for these tensor fields ? • Search for the unbroken phase ? Witten’s generalization of gauge theories Open string field takes values in non-commutative associative algebra The gauge transformations are defined as: for any parameter of degree zero. Field strength tensor is and transforms “homogeneously” under gauge transformations The gauge invariant Lagrangian is topological invariant - the star product is similar to the wedge product ! There is no analogue of the usual Yang-Mills action, as there is no analogue of raising and lowering indices within the axioms of this algebra. The other possibility is the integral of the Chern-Simons form which is invariant under infinitesimal gauge transformations J.Schwinger. Particles, Sourses, and Fields (Addison-Wesley, Reading, MA, 1970) 1. L. P. S. Singh and C. R. Hagen. Lagrangian formulation for arbitrary spin. I. The boson case. Phys. Rev. D9 (1974) 898 2. L. P. S. Singh and C. R. Hagen. Lagrangian formulation for arbitrary spin. II. The fermion case. Phys. Rev. D9 (1974) 898, 910 3. C.Fronsdal. Massless fields with integer spin, Phys.Rev. D18 (1978) 3624 4. J.Fang and C.Fronsdal. Massless fields with half-integral spin, Phys. Rev. D18 (1978) 3630 Free field Lagrangian and the corresponding equations describe massless particles of helicity The Lagrangian and equations are invariant with respect to the gauge transformation: Free field theories exhibit reach symmetries. Which one of them can be elevated to the level of symmetries of interacting field theory? In our approach the gauge fields are defined as rank-(s+1) tensors and are totally symmetric with respect to the indices A priory the tensor fields have no symmetries with respect to the index the Yang-Mills field with 4 space-time components the non-symmetric tensor gauge field with 4x4=16 space-time components the non-symmetric tensor gauge field with 4x10=40 space-time components The extended non-Abelian gauge transformation of the tensor gauge fields we shall define by the following equations: The infinitesimal gauge parameters are totally symmetric rank-s tensors All tensor gauge bosons carry the same charges as there are no traceless conditions on the gauge fields. , Gauge Algebra In general case we shall get and is again an extended gauge transformation with gauge parameters Extended gauge algebra Difference with K-K spectrum The field strength tensors we shall define as: The inhomogeneous extended gauge transformation induces the homogeneous gauge transformation of the corresponding field strength tensors Yang-Mills Fields First rank gauge fields It is invariant with respect to the non-Abelian gauge transformation The homogeneous transformation of the field strength is where The invariance of the Lagrangian Its variation is The first three terms of the Lagrangian are: The Lagrangian for the rank-s gauge fields is (s=0,1,2,…) and the coefficient is The gauge variation of the Lagrangian is zero: The Lagrangian is a linear sum of all invariant forms It is important that: • Every term in the sum is fully gauge invariant • Coupling constants g_s remain undefined • Lagrangian does not contain higher derivatives of tensor gauge fields • All interactions take place through the three- and four-particle exchanges with dimensionless coupling constant g • The Lagrangian contains all higher rank tensor gauge fields and should not be truncated It is invariant with respect to gauge transformation Equation of motion is The Free Field Equations For symmetric tensor fields the equation reduces to Einstein equation for antisymmetric tensor fields it reduces to the Kalb-Ramond equation In momentum representation the equation has the form: where 16x16 matrix has the form The rank of this matrix depends on momentum Within the 16 fields of non-symmetric tensor gauge field of the rank-2 only three positive norm polarizations are propagating and the rest of them are pure gauge fields. On the non-interacting level, when we consider only the kinetic term of the full Lagrangian, these polarizations are similar to the polarizations of the graviton and of the Abelian anti-symmetric B field. But the interaction of these gauge bosons carrying non-commutative internal charges is uniquely defined by the full Lagrangian and cannot bedirectly identified with the interactions of gravitons or B field. Interaction Vertices The VVV vertex The VTT vertex Interaction Vertices The VVVV and VVTT vertices Higher-Spin Extension of the Standard Model S – parity conservation S=1 Beyond the SM spin 2 3/2 0 S=0 Standard Model spin Masses: 1/2 1 Creation channel in LLC or LHC S – parity conservation standard leptons s=1/2 vector gauge boson tensor lepnos tensor boson Interaction of Fermions Rarita-Schwinger spin tensor fields Vertices Interaction of bosons