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