Higher-Spin Fermionic Gauge Fields & Their Electromagnetic Coupling

Transcription

1 Higher-Spin Fermionic Gauge Fields & Their Electromagnetic Coupling Rakibur Rahman Université Libre de Bruxelles, Belgium March 28, 2012 CQUeST Workshop on Higher Spins & String Geometry Sogang University, Seoul

2 M. Henneaux G. Lucena Gómez G. Lucena Gómez, M. Henneaux and RR, arxiv:1204.xxxx [hep-th] (to appear shortly). G. Barnich and M. Henneaux, Phys. Lett. B 311, 123 (1993) [hep-th/ ]. R. R. Metsaev, Nucl. Phys. B 859, 13 (2012) [arxiv: [hep-th]]. A. Sagnotti and M. Taronna, Nucl. Phys. B 842, 299 (2011) [arxiv: [hep-th]].

3 Motivation We will consider the coupling of an arbitrary-spin massless fermion to a U(1) gauge field, in flat spacetime with D 4. Such a study is important in that fermionic fields are required by supersymmetry. This fills a gap in the higher-spin literature. No-go theorems prohibit, in flat space, minimal coupling to gravity for s 5/2, and to EM for s 3/2. These particles may still interact through gravitational and EM multipoles. Metsaev's light-cone formulation restricts the possible number of derivatives in generic higher-spin cubic vertices. Sagnotti-Taronna used the tensionless limit of string theory to present generating function for off-shell trilinear cubic vertices. Cohomological methods could reconfirm/check these results.

4 Results Cohomological proof of no minimal EM coupling for s 3/2. Reconfirmation of Metsaev s restriction on the number of derivatives in a cubic 1-s-s vertex, with s = n+1/2. There are Only three allowed values: 2n-1, 2n, and 2n+1. Explicit construction of off-shell cubic vertices for arbitrary s = n+1/2, and presenting them in a very neat form. Non-abelian (2n-1)-derivative vertex containing the (n-1)-curl of the field. 1. Abelian 2n-derivative vertex, for D 5, involving n-curl (curvature tensor). 2. Abelian (2n+1)-derivative vertex of Born-Infeld type (3-curvature). Explicit matching with known results for lower spins. Generic obstruction for the non-abelian cubic vertices.

5 Outline Cohomological reformulation of a gauge theory. The BRST Deformation scheme. Spin 3/2: Simple but nontrivial. Arbitrary spin s = n+1/2. Restrictions on the gauge parameter and the field actually simplify analysis. Comparative study with known results. Comments of second order deformation. Concluding remarks.

6 Consistent Gauge Theory: Cohomological Reformulation Search for consistent interactions in a gauge theory becomes systematic as one as one takes cohomological approach. Corresponding to each gauge parameter an irreducible gauge theory, one introduces a ghost field, with the same algebraic symmetries but opposite Grassmnn parity. The original fields and ghosts are collectively called fields, denoted by Φ A. The configuration space is further enlarged by introducing, for each field and ghost, an antifield Φ* A, that has the same algebraic symmetries but opposite Grassmnn parity. On the space of fields and antifields, one defines an odd symplectic structure, called the antibracket:

7 All the consistency conditions of the gauge theory can be incorporated compactly into the master equation: (S, S) = 0. The solution S of the master equation is an extension of the original gauge-invariant action that includes terms involving ghosts and antifields. It contains the gauge structure functions, and makes use of the Noether identities and higher-order gauge structure equations to solve the master equation. S is the generator of the BRST differential s : s X = (S, X). From the definitions, S is BRST-closed and s 2 = 0.

8 The BRST differential decomposes into two differentials: s = Γ + Δ where Δ is the Koszul-Tate differential, which reproduces the equations of motion, and Γ is the longitudinal derivative along ihe gauge orbits, i.e. it reproduces the gauge transformations. They obey: Γ 2 = Δ 2 = 0, Γ Δ +Δ Γ = 0. The decomposition is related to the various gradings in the algebra generated by fields and antifields: pure ghost number (pgh), antighost number (agh), ghost numer (gh = pgh - agh):

9 An original field has pgh = agh = gh = 0, while its antifield has pgh = 0, agh = 1, gh = -1. A ghost has pgh = 1, agh = 0, gh = 1, while its antifield has pgh = 0, agh = 2, gh = -2. Now the important point is that, the solution S of the master equation belongs to the cohomology of s in the local functionals of the fields, antifields, and their finite number of derivatives. Therefore, one can reformulate the classical problem of introducing consistent interactions in a gauge theory in terms of the BRST differential and the BRST cohomology.

10 The BRST Deformation Scheme Let S 0 be the solution of the master equation of a free gauge theory. Any consistent deformation of the theory corresponds to: S = S 0 + gs 1 + g 2 S 2 + O(g 3 ) where S solves the deformed master equation: (S, S) = 0. Coupling constant expansion gives (S 0, S 0 ) = 0 (S 0, S 1 ) = 0 (S 1, S 1 ) = -2 (S 0, S 2 ) The first equation is fulfilled by assumption, and in fact S 0 is generator of the BRST differential s for the free theory. The second equation says S 1 is BRST-closed: s S 1 = 0.

11 First order nontrivial consistent local deformations: S 1 = a are in 1-to-1 correspondence with elements of H D,0 ( s d ) -- the cohomology of s d, in maximum form degree D, ghost number 0. Therefore one has the cocycle condition: s a D,0 + d b D-1,1 = 0. a D,0 is a top-form with gh=0. b D-1,1 is a (D-1)-form with gh=1. The second-order consistency condition: s S 2 = -(1/2)(S 1, S 1 ), is controlled by the local BRST cohomology group H D,1 ( s d ). A cubic deformation with 0 ghost number cannot have agh>2. Thus one can expand a in antighost number: a = a 0 + a 1 + a 2 (Physically meaningful)

12 Similarly, one can assume: b = b 0 + b 1. Then the cocycle condition reduces, by s = Γ + Δ, to a cascade of relations: Thus cohomology of Γ becomes relevant. It is isomorphic to the space of functions depending on: 1. The curvatures and their derivatives. 2. The antifields and their derivatives. 3. The ghosts and (gamma)traceless part of all possible curls of them. 4. The Fronsdal tensors and their symmetrized derivatives.

13 Massless Rarita-Schwinger Field Coupled to Electromagnetism The set of fields and antifields are: We denote spin-3/2 field-strength (1-curl) as: BRST generator of the free theory is the volume integral of

14 We have the following table: Notice that Γ acts only on the originals fields, while Δ acts only on the antifields.

15 Cohomolgy of Γ isomorphic to the space of functions of: 1. The curvatures and their derivatives. 2. The antifields and their derivatives. 3. Undifferentiated ghosts. ** Note: Fronsdal tensors are already included in group 1. The list of candidates of a 2, for cross-coupling, is: 1. One that contains C: minimal coupling?? 2. One that contains C*: dipole interaction?? Both of them can be straightforwardly lifted to a 1 1. One containing both ghosts: 2. One that contains only 1 ghost:

16 The ambiguity belongs to the cohomogy of Γ Consistency requires that, if the Δ variation of the unambigous piece is not Γ-exact modulo d, the Δ variation of the ambiguity must kill the non-trivial part modulo d, so that one could obtain: But any element of the cohomology of Γ at antighost number 1 contains at least 1 derivative, so that such a cancellation may be possible for the second case, but not the first one. Indeed, the Δ variation of the none of the unambigous pieces is Γ-exact modulo d. Thus minimal coupling is ruled out.

17 The other case gives us To see that this can be lifted to an a 0, we use the identity And also notice all possible forms of the EoMs: This leads us to the following vertex

18 Other vertices come from either a 1 or a 0 itself. In both cases, in view of the EoMs, the vertex can always be written as: Therefore, the most generic form of the vertex is: It is not difficult to see if X contains more

19 But up to terms that are Δ-exact modulo d, this is an abelian 3-curvature (Born-Infeld type) vertex: A 2-derivative vertex can follow from 2 possibilities: But they differ by Δ-exact terms, thanks to the identities:

20 For D > 4, we have an abelian 2-derivative vertex, that is gauge invariant up to a total derivative by Bianchi identity: Comparison of these vertices with those of Sagnotti-Taronna reveal that they differ by terms that are Δ-exact modulo d. The Sagnotti-Taronna vertices, written in the most naïve way, contains many terms. In particular, it not straightforward at all to see that the 2-derivative vertex vanishes for D=4. In the transverse-traceless gauge, our vertices also reduce to known results in the literature.

21 Arbitrary Spin: s = n + 1/2 The set of fields and antifields are: Constrained fermionic field and gauge parameter: A derivative of 0, 1,, (n-2) curls of the gauge parameter is in the cohomology of Γ, but that of (n-1) curl is Γ-exact.

22 The list of candidates of a 2, for cross-coupling, is: 1. A set containing C, i-th curl of the fermionic ghost and i-th curl of its antifield. i=0,1,,n A set containing C*, i-th curl of the fermionic ghost and i-th curl of its Dirac conjugate. i=0,1,,n-1. The second kind cannot be lifted to a 1 unless i=n-1. For the first kind, all can be lifted to a 1. While i=0 corresponds to minimal coupling: ruled out like in s=3/2, other possibilities are also ruled out, because of different natures of the unambiguous piece and the ambiguity in a 1. The rest of the story is like in spin 3/2. (n-1)-curl of the fermion appears in non-abelian vertex, and n-curl in the others.

23 Remarks & Future Perspectives Gravitational coupling of fermions. Mixed Symmetry fields. Similarities with bosonic 1-s-s and 2-s-s results! Scaling limit of massive theory in 4D. Comparison with BCFW results in 4D. Hint of non-locality.