Introduction to SME and Scattering Theory. Don Colladay. New College of Florida Sarasota, FL, 34243, U.S.A.

Transcription

1 June 2012 Introduction to SME and Scattering Theory Don Colladay New College of Florida Sarasota, FL, 34243, U.S.A. This lecture was given at the IUCSS summer school during June of It contains a brief introduction to the Standard Model Extension (SME), followed by an overview of scattering theory in this framework. This lecture is part of a coherent week-long program at IUCSS and therefore may rely on notation and concepts covered in prior lectures.

2 I. INTRODUCTION TO SME The Standard Model Extension [1, 2] (SME) contains all possible couplings between standard model fields and background vectors and tensors that may arise through a generic spontaneous symmetry breaking mechanism. A minimal version is obtained by restricting the terms according to Standard Model gauge invariance, power- counting renormalizability, and translational symmetry. Gauge invariance and power-counting limitations are extremely useful in establishing one-loop renormalizability of the theory. Translational symmetry ensures the existence of a conserved energy and momentum by Noether s theorem. As an example, consider the lepton-higgs sector of the Standard Model (prior to SU(2) U(1) breaking). The Standard Model lepton fields form left-handed doublets and right-handed singlets L A = ( νa l A ) L, R A = (l A ) R, (1) where A = 1, 2, 3 labels flavour and the Higgs field can be written in unitary gauge as φ = 1 2 ( 0 r φ ). (2) Left- and right-handed fields are constructed using the usual helicity projection operators ψ L 1 2 (1 γ 5)ψ, and ψ R 1 2 (1 + γ 5)ψ. The conventional terms take the form for the lepton kinetic terms and L lepton = 1 2 il Aγ µ D µ L A ir Aγ µ D µ R A, (3) L Yukawa = (G L ) AB L A φr B + h.c., (4) for the Yukawa (mass-generating) terms. There is also a Higgs potential and corresponding Lorentz-violating corrections, but these are not shown here for simplicity. The only CPT-even terms compatible with the assumptions are L CPT even lepton = 1 2 i(c L) µνab L A γ µ D ν L B i(c R) µνab R A γ µ D ν R B, (5) 1

3 where c L and c R are tensors (in both coordinate and flavor spaces) that parameterize the violation. The CPT-odd terms compatible with assumptions are L CPT odd lepton = (a L ) µab L A γ µ L B (a R ) µab R A γ µ R B, (6) where a L and a R are vectors in coordinate space and tensors in flavor space. Note that the violation parameters do not imply any relation between the various flavors, they are all independent parameters. This implies that bounds on Lorentz violation for the electron have no particular implication for corresponding bounds on the muon, for example. There also may be flavor-mixing couplings due to off-diagonal terms in flavor space. The next step in the construction is to impose SU(2) U(1) breaking by finding the minimum of the static potential of the electroweak sector. The details are given in [2]. In summary, the W and photon particles have zero expectation value, while the Z µ gains a nontrivial expectation value related to the Higgs lorentz-violation parameters. The r φ component of the Higgs field is perturbed slightly from the conventional case, but remains Lorentz invariant under both particle and observer transformations. This key fact ensures that the fermion masses remain lorentz invariant quantities. Once the symmetry is broken, the terms are re-labeled (a R and a L a and b) and (c L and c R c and d). For example, the electron sector CPT-even terms become and the CPT-odd terms L CPT even electron = 1 2 H µνψσ µν ψ ic µνψγ µ D ν ψ id µνψγ 5 γ µ D ν ψ, (7) L CPT odd electron = a µ ψγ µ ψ b µ ψγ 5 γ µ ψ. (8) Note that letters e, f, and g are missing in the above analysis. These parameters can be included at the level of a strictly U(1) theory, but cannot arise from gauge invariant couplings in the full Standard Model. They can be written down explicitly as L extra electron = 1 2 ie νψ D ν ψ 1 2 f νψγ 5 D ν ψ ig λµνψσ λµ D ν ψ, (9) 2

4 however, it is unlikely that the full standard model will remain renormalizable in their presence due to their explicit gauge invariance violation. A commonly used expression for the above Lagrangian is L electron = i 2 ψγ ν D ν ψ ψmψ, (10) where Γ ν and M are general matrices containing all of the correction terms. II. SCATTERING THEORY The fundamental observable in scattering theory is the scattering cross section. In the present context, it is important to identify properly the parameters that are actually observed in an experiment. The main complication involves the nontrivial relationship between the velocity and momentum for particles in the SME. The differential scattering cross section is constructed as a product of several pieces dσ = 1 Flux Factor τ fi 2 (F.S. Phase Space Factors) (11) all multiplied by an overall four-momentum conserving delta function. The general problem as well as the specific example considered here can be found in [?]. The flux factor is essentially a counting factor that takes into account the incoming flux of the beam. For colinear, opposing velocities, it takes the form F = N 1 N 2 v 1 v 2, (12) where N 1 and N 2 are the beam densities, dependent on the normalization of the incident beam wave functions. With conventional state normalization, N = 2E( p), even in the presence of Lorentz violation, recalling that the energy momentum relation is modified. It is useful to keep the normalizations arbitrary so that it can be verified that it in fact drops out of the calculation in the end, as it must. It is important that the velocities be used in the flux factor rather than the momentum since it is the velocity that determines the counting rate. The velocity may be found by computing the group velocity of a wave packet in the usual way v g = p E( p). (13) 3

5 Another complication arises when it is not easy to solve the dispersion relation for the energy, Finsler-space methods discussed by Neil Russell in a later lecture at this school may be useful for this purpose. The transition matrix element τ fi 2 can be found by application of SME-extension Feynman rules. The propagators are given by the two-point Green s functions for the appropriate kinetic term. For example, the modified fermion propagator is given by S F (p) = 1 γ 0 E A ΓA p AMA. (14) A subtlety in the above expression involves a field redefinition that converts Γ 0 to γ 0 using ψ = Aχ such that A γ 0 Γ 0 A = I making the resulting hamiltonian hermitian. This redefinition is also useful since the conserved charge takes the form j 0 = χγ 0 χ = χ χ, (15) leading to the conventional probability interpretation of the single particle states. Similarly, the canonical energy-momentum tensor leads to Θ 0µ = i 2 χγ0 µ χ, (16) which also takes the conventional form. For practical application on internal lines it is useful to expand the above result using the following geometric series for operators 1 Σ 0 B = 1 Σ Σ 0 B 1 Σ Σ 0 B 1 Σ 0 B 1 Σ 0 +, (17) where Σ 0 = p m is the conventional inverse propagator. Note that this expansion only applies when on-shell contributions are not important as Σ 0 ψ 0 clearly causes a problem in the above formula. In particular, such an expansion cannot be used to include Lorentz-violating contributions to external lines. Exact spinor solutions must be used instead. When spins are summed over, this leads to the expression (for particle fermions) u α ( p)u α ( p) = N( p + m) + L.V corrections, (18) α where N is a normalization factor and the Lorentz-violating corrections depend on the background fields. 4

6 Modified vertices also appear, they can be handled easily by making the replacement γ µ Γ µ using the conventional rule. Translationlal invariance ensures that energy and momentum are conserved at each vertex as in the usual case. Final state phase space factors can also include Lorentz-violating contributions due to the normalization factors that appear in them. For example, a single fermion in the final state requires the inclusion of a factor dπ = d3 p (2π) 3 N, (19) derived by imposing periodic boundary conditions on the spatial part of the wave function as in the usual case. As an explicit example, consider the process e +e + 2γ in the highly-relativistic regime. Lorentz violation in the final state may be neglected as photon bounds are very stringent using other methods. At high energies only the derivative couplings are relevant. Moreover, a sum over initial polarizations eliminates d µν to lowest order. The relevant lagrangian takes the form with the definitions L = 1 2 i η µνχγ µ D ν χ mχχ, (20) m m(1 c 00 ), η µν η µν + C µν, C µν c µν c µ0 η 0ν + c ν0 η 0µ c 00 η µν, (21) where χ is the field after the appropriate field redefinition The dispersion relation is easily obtained as ψ Aχ = (1 1 2 c µ0γ 0 γ µ )χ. (22) λ 2 m 2 = 0, (23) where λ µ η µν λ ν. Defining the positive root as λ 0 ( p) = p 0 ( p) = E( p), to leading order in c µν the energy E is given by E( p) = p 2 + m 2 pj C jk p k p2 + m 2 C0 jp j. (24) 5

7 This expression leads to the group velocity v j g = 1 Ẽ ( pj + C j µ p µ ), (25) to lowest order in c µν. Note that the velocity and momentum are not generally in the same direction. Calculation of the flux factor in the center of momentum frame yields F = 2N( p)n( p)(1 ˆpC ˆp), (26) where ˆpC ˆp indicates a spatial contraction of the relevant indices. Combining this with the transition probability and using the spin sums 2 α=1 u (α) ( p)u (α) ( p) = 2Ẽ( p)( N( p) 2 p + m), α=1 yields the cross section (integrated over φ) of dσ d cos θ = 2π 0 = πα2 2 p 2 dφ dσ [ dω (1 + cos 2 θ sin 2 θ v (α) ( p)v (α) ( p) = 2Ẽ( p)( N( p) p m), (27) ) ] (1 + c 00 + c 33 ) 2 cot 2 θ(c 11 + c 22 2c 33 ), where cos θ = ˆp ˆk is the angle of the outgoing photon relative to the incoming momentum of the electron beam. (28) Note that the first term represents an overall scaling of the conventional cross section while the second term gives a new angular dependence as a correction. As a final step in the analysis, coordinates are transformed to the standard solar celestial frame. The general result is more complicated, but can be integrated over time to yield dσ d cos θ av 1 T T dσ dt d cos θ {( 1 + cos 2 θ 0 = πα2 2 p 2 sin 2 θ ) [1 + c00 + c ZZ (c XX + c Y Y 2c ZZ ) sin 2 χ ] 2 cot 2 θ(c XX + c Y Y 2c ZZ )( 3 2 cos2 χ 1 2 ) }. (29) The capitol letter indices denote the parameters as defined in the solar celestial frame. REFERENCES 6

8 1. D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997). 2. D. Colladay and V.A. Kostelecký, Phys. Rev. D 58, (1998). 3. D. Colladay and V.A. Kostelecký, Phys. Lett. B511, 209, (2001). 7