String Field Theory Non-Abelian Tensor Gauge Fields and Possible Extension of SM
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String Field Theory
Non-Abelian Tensor Gauge Fields and
Possible Extension of SM
George Savvidy Demokritos National Research Center Athens
Phys. Lett. B625 (2005) 341Int.J.Mod.Phys. A21 (2006) 4959 Int.J.Mod.Phys. A21 (2006) 4931Fortschr. Phys. 54 (2006) 472 Prog. Theor. Phys.117 (2007) 729 ------------------------ Takuya TsukiokaHep-th/0604118 Hep-th/ 0704.3164 ------------------------ Jessica BarrettHep-th/ 0706.0762 ------------------------ Spyros Konitopoulos
Patras 2007
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• 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
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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 ?
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Witten’s generalization of gauge theories
Field strength tensor is
and transforms “homogeneously” under gauge transformations
for any parameter of degree zero.
The gauge invariant Lagrangian
is topological invariant - the star product is similar to the wedge product !
Open string field takes values in non-commutative associative algebra The gauge transformations are defined as:
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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
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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
J.Schwinger. Particles, Sourses, and Fields (Addison-Wesley, Reading, MA, 1970)
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and the corresponding equations describe massless particles of helicity
The Lagrangian and equations are invariant with respect to the gauge transformation:
Free field Lagrangian
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Free field theories exhibit reach symmetries.
Which one of them can be elevated to the level
of symmetries of interacting field theory?
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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
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The extended non-Abelian gauge transformation of the tensor gauge fields weshall 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.
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In general case we shall get
and is again an extended gauge transformation with gauge parameters
Gauge Algebra
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Extended gauge algebra
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Difference with K-K spectrum
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The field strength tensors we shall define as:
The inhomogeneous extended gauge transformation induces the homogeneous gauge transformation of the corresponding field strength tensors
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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
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where
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The invariance of the Lagrangian
Its variation is
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The first three terms of the Lagrangian are:
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The Lagrangian for the rank-s gauge fields is (s=0,1,2,…)
and the coefficient is
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The gauge variation of the Lagrangian is zero:
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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
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It is invariant with respect to gauge transformation
Equation of motion is
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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
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In momentum representation the equation has the form:
where 16x16 matrix has the form
The rank of this matrix depends on momentum
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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.
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Interaction Vertices
The VVV vertex
The VTT vertex
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Interaction Vertices
The VVVV and VVTT vertices
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Higher-Spin Extension of the Standard Model
Standard Model
Beyond the SM
spin 1/2 1
23/2spin 0
Masses:
S=1
S=0
S – parity conservation
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Creation channel in LLC or LHC
standard leptons s=1/2
vector gauge boson tensor boson
tensor lepnos
S – parity conservation
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Interaction of Fermions
Rarita-Schwinger spin tensor fields
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Vertices
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Interaction of bosons
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