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

• 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

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:

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

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)

and the corresponding equations describe massless particles of helicity

The Lagrangian and equations are invariant with respect to the gauge transformation:

Free field Lagrangian

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 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.

In general case we shall get

and is again an extended gauge transformation with gauge parameters

Gauge Algebra

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

Standard Model

Beyond the SM

spin 1/2 1

23/2spin 0

Masses:

S=1

S=0

S – parity conservation

Creation channel in LLC or LHC

standard leptons s=1/2

vector gauge boson tensor boson

tensor lepnos

S – parity conservation

Interaction of Fermions

Rarita-Schwinger spin tensor fields

Vertices

Interaction of bosons