Consistent interactions for high-spin fermion fields The 8th International Workshop on the Physics of Excited Nucleons Thomas Jefferson National Accelerator Facility Newport News, VA Tom Vrancx Ghent University, Belgium arxiv:1105.2688 May 19, 2011 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 1 / 25
Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 2 / 25
The Rarita-Schwinger formalism The Rarita-Schwinger formalism Interacting Rarita-Schwinger fields Rarita-Schwinger equations (i / m)ψ µ1...µ n = 0 γ µ 1 ψ µ1 µ 2...µ n = 0 ψ µ1...µ n : spin-(n + 1 2 ) Rarita-Schwinger (R-S) field Unphysical components are eliminated through R-S constraints Massless theory requires invariance under R-S gauge ψ µ1...µ n ψ µ1...µ n + γ µ 1 ξ µ1 µ 2...µ n 1 = 0 i µ1 ξ µ2...µ n(n 1)! n P (µ) Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 3 / 25
The Rarita-Schwinger formalism Interacting Rarita-Schwinger fields Interacting Rarita-Schwinger fields On-shell case On-shell R-S field is described by R-S spinor Unphysical components of R-S spinor decouple from the interaction Off-shell case Off-shell R-S field is described by R-S propagator Unphysical components of R-S propagator do not decouple a priori Consistent interaction should be invariant under certain local gauge Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 4 / 25
The Rarita-Schwinger formalism Interacting Rarita-Schwinger fields Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 5 / 25
Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories Consistency and locality of the interaction Consistent interaction for ψ µ 1...µ n Γ i P m < n k, l Γ f = Γ i p/ + m p 2 m 2 Pn+ 1 2 Γ i A m+ 1 2 kl P m+ 1 2 kl Γ f = 0 Γ f Interpretation Interaction is mediated purely by physical component of R-S field Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 6 / 25
Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories Consistency and locality of the interaction Preservation of locality P n+ 1 2 µ 1...µ n;ν 1...ν n (p) contains p 2k singularities (k {1,..., n}) Transversality conditions of interaction vertices p µk Γ µ 1...µ n I = 0 k {1,..., n} Equivalent to invariance of interaction under unconstrained R-S gauge ψ µ1...µ n ψ µ1...µ n + i µ1 χ µ2...µ n(n 1)! n P (µ) Unconstrained R-S gauge concurrently guarantees consistent interaction Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 7 / 25
Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories The gauge-invariant Rarita-Schwinger field Gauge-invariant Rarita-Schwinger field G µ1...µ n,ν 1...ν n = O n+ 1 2 (µ 1...µ n,ν 1...ν n)λ 1...λ n ( )ψ λ 1...λ n Spin-(n + 1 2 ) interaction operator O n+ 1 2 (µ 1...µ n,ν 1...ν n)λ 1...λ n ( ) = 1 (n!) 2 O 3 2 (µ1,ν 1 )λ 1 ( ) O 3 2 (µn,νn)λ n ( ) P (ν) P (λ) Pascalutsa s spin- 3 2 interaction operator O 3 2 (µ,ν)λ ( ) = i ( µ g νλ ν g µλ ) Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 8 / 25
Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories The gauge-invariant Rarita-Schwinger field Properties G µ1...µ n,ν 1...ν n totally symmetric in µ and ν indices, and G µ1...µ n,ν 1...ν n = ( 1) n G ν1...ν n,µ 1...µ n Invariance under unconstrained R-S gauge assured by λ k O n+ 1 2 (µ 1...µ n,ν 1...ν n)λ 1...λ n ( ) = 0 Construction of consistent interaction theory L I = G µ1...µ n,ν 1...ν n T µ 1...µ nν 1...ν n + T µ1...µ nν 1...ν n G µ 1...µ n,ν 1...ν n Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 9 / 25
Consistent interaction theories Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories Reduced gauge-invariant Rarita-Schwinger field Ψ µ1...µ n = γ ν1 γ νn G µ1...µ n,ν 1...ν n Properties Ψ µ1...µ n obeys R-S constraints µ 1 Ψ µ1...µ n 0 and γ µ 1 Ψ µ1...µ n 0 Intuitive interaction structure O n+ 1 2 P O n+ 1 2 = p 2n p/ + m p 2 m 2 Pn+ 1 2 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 10 / 25
Consistency and locality of the interaction The gauge-invariant Rarita-Schwinger field Consistent interaction theories 3. Consistent interactions in hadron physics Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 11 / 25
Practical use of consistent interactions: an example Isobar modeling of s channel of K + Λ photoproduction High-spin nucleon resonances are created from the proton γ K + N p Λ Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 12 / 25
σ (µb) 3 10 10 1 10 + 5/2 + 3/2 + 1/2 Toy N m R = 1700 MeV Γ R = 50 MeV J P R = 1 2 +, 3 + 2, 5 + 2 3 10 1500 2000 2500 3000 W (MeV) Hadronic form factor required to suppress high-energy behavior Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 13 / 25
Standard hadronic form factors Dipole form F d (s; m R, Λ R ) = Λ 4 R (s m 2 R )2 + Λ 4 R Gaussian form F G (s; m R, Λ R ) = exp ( (s ) m2 R )2 Λ 4 R Specific s channel contribution to p(γ, K + )Λ: m R = 1685 MeV N = N(1680) F 15 Γ R = 130 MeV JR P = 5 + 2 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 14 / 25
γp N(1680) F 15 K + Λ, F = F G σ G (µb) 0.12 0.10 0.08 0.06 0.04 0.02 (a) 1500 MeV σ G (µb) 2 10 4 10 6 10 8 10 10 10 (b) 1500 MeV 0.00 1500 2000 2500 3000 W (MeV) 12 10 1600 1650 1700 1750 1800 W (MeV) Remarks Artificial bump arises when σ is cut off by F G No resonance peak is observed at W m R Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 15 / 25
σ G (µb) 0.12 0.10 0.08 0.06 0.04 0.02 (a) 1250 MeV 1000 MeV 750 MeV 500 MeV 0.00 1500 2000 2500 W (MeV) σ G (µb) 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 (b) 1250 MeV 1000 MeV 750 MeV 500 MeV 0.00 1500 2000 2500 W (MeV) Remarks Lowering Λ R results in shift of artificial bump towards W 0 Lowering Λ R only effective when Γ R is small practically all N s listed by PDG have large Γ R Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 16 / 25
Cause of unphysical behavior Rise of σ with W and fall of F G with W combines to form artificial bump Fast increase of σ with W causes resonance peak with large Γ R to vanish Remedy High-energy behavior of σ needs to be regulated σ(s) (s 2 ) J R 1 2 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 17 / 25
F mg (s; m R, Λ R, Γ R, J R ) = ( m 2 R Γ 2 R (J R) Multidipole-Gauss form factor (s m 2 R )2 + m 2 R Γ 2 R (J R) ) JR 1 2 exp ( (s ) m2 R )2 Λ 4 R Dipole part of F mg raises multiplicity of propagator pole Modified decay width Γ R (J R ) = Γ R 2 1 2J R 1 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 18 / 25
σ mg (µb) 0.12 0.10 0.08 0.06 0.04 (a) 1250 MeV 1000 MeV 750 MeV 500 MeV σ mg (µb) 0.12 0.10 0.08 0.06 0.04 (b) 1250 MeV 1000 MeV 750 MeV 500 MeV 0.02 0.02 0.00 1600 1700 1800 1900 2000 2100 W (MeV) 0.00 1600 1650 1700 1750 1800 W (MeV) Remarks Artificial bump is removed and resonance peak is restored Threshold effects for m R Γ R 2 W 0 Peak position not at W = m R Peak position and width are function of Λ R Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 19 / 25
Conclusions Summary Invariance under unconstrained Rarita-Schwinger gauge guarantees consistency of the interaction Standard hadronic form factors give rise to unphysical behavior in energy dependence of cross section Multidipole-Gauss hadronic form factor leads to physically acceptable energy dependence of cross section Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 20 / 25
Backup slides Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 21 / 25
Backup slide I(a) P n+ 1 2 Spin-(n + 1 2 ) projection operator µ 1...µ n;ν 1...ν n (p) = ( kmax,1 A n k P µ 1 µ 2 P ν1 ν 2 P µ2k 1 µ 2k P ν2k 1 ν 2k P (µ) P (ν) k=0 k max,2 + P/ µ1 P/ ν1 k=0 B n k P µ 2 µ 3 P ν2 ν 3 P µ2k µ 2k+1 P ν2k ν 2k+1 n i=2k+1 n i=2k+2 P µi ν i P µi ν i ) P µν = g µν 1 p 2 p µp ν }{{} singularity and P/ µ = γ ν P µν = γ µ p/ p 2 p µ }{{} singularity Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 22 / 25
Backup slide I(b) P µ1...µ n;ν 1...ν n (p) = Spin-(n + 1 2 ) propagator p/ + m p 2 m 1 n+ 2 P 2 µ 1...µ n;ν 1...ν n (p) P µν = g µν 1 p 2 p µp ν g µν 1 m 2 p µp ν P/ µ P/ ν = γ µ γ ν + p/ p 2 (γ µp ν γ ν p µ ) 1 p 2 p µp ν γ µ γ ν + 1 m (γ µp ν γ ν p µ ) 1 m 2 p µp ν Substitutions have no effect since p µk Γ µ 1...µ n I = 0 Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 23 / 25
Backup slide II(a) Inconsistent spin- 3 2 interaction theories L φψψ µ = ig 0 m φ ψ µ Θ µν (z 0 )Γψ ν φ L (1) A µψψ µ = ig 1 2m ψ ψ µ Θ µν (z 1 )Γγ λ ψf λν L (2) A = g 2 µψψµ 4m 2 ψ µ Θ µν (z 2 )Γ λ ψf λν ψ L (3) A = g 3 µψψµ 4m 2 ψ µ Θ µν (z 3 )Γψ λ F λν ψ Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 24 / 25
Backup slide II(b) Consistent spin-(n + 1 2 ) interaction theories L φψψ = in g 0 µ1 Ψ µ1...µ...µn n Γψ µ1 µn φ m 2n φ L (1) A µψψ µ 1...µn L (2) A µψψ µ 1...µn L (3) A µψψ µ 1...µn = in g 1 (2m ψ ) 2n Ψ µ 1...µ n 1 µ n Γγ ν µ1 µ n 1 ψf νµn = in+1 g 2 (2m ψ ) 2n+1 Ψ µ 1...µ n 1 µ n Γ ν µ1 µ n 1 ψf νµn = in+1 g 3 (2m ψ ) 2n+1 Ψ µ 1...µ n 1 µ n Γ µ1 µ n 1 ψ ν F νµn Tom Vrancx Consistent interactions for high-spin fermion fields NSTAR 2011, Jefferson Lab, Newport News, VA 25 / 25