Developing non- relativistic effective theory for scalars MATTHEW GONDERINGER – WAYNE STATE UNIVERSITY WITH ALEXEY PETROV, ANDREW BLECHMAN Introduction Fundamental scalars are important Most students of field theory begin by learning about scalar fields Thanks to the LHC we know at least one scalar particle, the Higgs, exists Scalars exist in many models of beyond the Standard Model physics such as SUSY (sleptons and squarks) and dark matter (scalar singlets) Just as NRQED provides a useful framework for studying low energy QED processes, a non-relativistic theory for scalars would have useful applications Dark matter direct detection through scattering off atomic nuclei is a low energy process The study of heavy, non-relativistic colored scalars produced at the LHC (including formation of bound states, e.g. stoponium, and mixing with other scalars) I will describe the progress we’ve made in formulating a non-relativistic effective theory for scalars MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 2 Scalar “spinors” Feshbach and Villars showed that the Klein-Gordon equation for scalars can be recast in a two-component form: 휃 휙 → Φ = where 휃 ∼ 휙 + 휙 and 휒 ∼ 휙 − 휙 휒 We use this form of the scalar field to develop our non-relativistic effective theory The two-component vector mimics the four-component spinors arising from the Dirac equation This will make it clear that we obtain the non-relativistic theory by integrating out the negative energy modes for both complex and real scalars Other studies of non-relativistic scalars (Hill and Solon 2012, Grinstein and Trott 2007, Hoang and Ruiz-Femenia 2006) follow different approaches that are less transparent MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 3 Klein-Gordon equation 휃 Using Φ = , we rewrite the Klein-Gordon equation as 휒 퐷2 푖푣 ⋅ 퐷 − 휎 + 푖휎 ⊥ − 푚휎 Φ = 0 3 2 2푚 3 휇 휇 휇 퐷 is the covariant derivative, 푣 = 1,0,0,0 , and 퐷⊥ = 퐷 − 푣 푣 ⋅ 퐷 2 2 푣 ⋅ 퐷 is a time derivative, and 퐷⊥ ∼ −훻 † The corresponding Lagrangian with Φ = Φ 휎3 is 퐷2 ℒ = Φ 푖푣 ⋅ 퐷 − 휎 + 푖휎 ⊥ − 푚휎 Φ 3 2 2푚 3 This ℒ reproduces the KG Lagrangian for complex scalars † 휇 2 † ℒ퐾퐺 = 퐷휇휙 퐷 휙 − 푚 휙 휙 MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 4 Mode expansion The solutions to the free particle equation of motion lead to the mode expansion 푑3푝 1 푚 + 퐸 푚 − 퐸 −푖푝⋅푥 † +푖푝⋅푥 Φ = 푎 푝푒 + 푏푝푒 2휋 3 퐸 푚 푚 − 퐸 푚 + 퐸 For a complex scalar, 푎 is the particle operator associated with positive energy modes and 푏 is the antiparticle operator associated with negative energy modes 2 2 2 Note that for 푝 ≪ 푚 we can write 푝0 = 퐸 = 푝 + 푚 ≃ 푚 + 푝 /2푚 + ⋯ and so: 푚 + 퐸 ≃ 2푚 푚 − 퐸 ≃ 0 MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 5 NR effective theory We follow the approach of HQET and NRQED to integrate out the negative energy modes In ℒ write Φ = 푒−푖푚푣⋅푥 휂 + 휁 where 푣 = 1,0,0,0 and 1 + 휎 1 − 휎 휂 = 푒푖푚푣⋅푥 3 Φ 휁 = 푒푖푚푣⋅푥 3 Φ 2 2 Applying this to the mode expansion, we see that 1 ± 휎3 acts as a projection operator 푚 + 퐸 푚 − 퐸 0 0 휂 ∼ 푎 푒−푖⋯ + 푏 †푒+푖⋯ 휁 ∼ 푎 푒−푖⋯ + 푏 †푒+푖⋯ 0 0 푚 − 퐸 푚 + 퐸 Thus at leading order in 푝 /푚 with 푚 + 퐸 ≃ 2푚 and 푚 − 퐸 ≃ 0: 휂 represents positive energy modes associated with the particle operator 푎 휁 represents negative energy modes associated with the antiparticle operator 푏 MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 6 NR Klein-Gordon Lagrangian The Lagrangian in terms of 휂 and 휁 is 퐷2 퐷2 퐷2 퐷2 ℒ = 휂† 푖푣 ⋅ 퐷 − ⊥ 휂 − 휁† 푖푣 ⋅ 퐷 + ⊥ + 2푚 휁 − 휂†푖휎 ⊥ 휁 + 휁†푖휎 ⊥ 휂 2푚 2푚 2 2푚 2 2푚 The positive energy mode 휂 is massless and the negative energy mode 휁 has mass 2푚 At energies below the mass 푚 we can integrate out the heavy 휁 modes, leaving an effective Lagrangian for 휂 that is an expansion in 1/푚 1 1 1 ℒ = 휂† 푖푣 ⋅ 퐷 휂 − 휂†퐷2휂 + 휂†퐷4휂 − 휂†퐷2푖푣 ⋅ 퐷퐷2휂 + ⋯ 2푚 ⊥ 2푚 3 ⊥ 2푚 4 ⊥ ⊥ This is the non-relativistic Klein-Gordon scalar Lagrangian MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 7 Additional gauge interactions In principle a Lagrangian for a non-relativistic effective theory could contain many gauge-invariant operators representing all possible interactions between the scalar and gauge bosons 푔2 푔 ℒ ∋ 푐 휂†푣 푣훽퐺휇훼퐺 휂 + 푐 휂† 퐷훼, 퐷훽, 퐺 휂 + ⋯ 퐴 2푚 3 훼 휇훽 푀 2푚 3 ⊥ ⊥ 훼훽 Operators which do not appear in the non-relativistic KG scalar Lagrangian would have coefficients 푐 = 0 at tree level Such operators could arise at loop level Analogous to QED interactions for point particles: 퐹1 = 1 + 풪 훼 and 퐹2 = 0 + 풪 훼 MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 8 Matter interactions Other possible interactions for a non-relativistic scalar: 2 Self-interactions: 휙†휙 , 휙3 + 푐. 푐., … Interactions with other scalars and fermions: 휙†휙퐻†퐻, 휙†휙푞 푞, … We can rewrite 휙†휙 using our two-component formulation of the scalar field 1 1 휙†휙 = Φ 휎 + 푖휎 Φ = 휂 휂 − 휁휁 + 휂 푖휎 휁 + 휁푖 휎 휂 2푚 3 2 2푚 2 2 After integrating out 휁 we’re left with 1 퐷2 1 휙†휙 → 휂†휂 − 휂† ⊥ 휂 + 휂† 퐷2, 푖푣 ⋅ 퐷 휂 + 풪(1/푚5) 2푚 2푚2 2푚 3 ⊥ MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 9 Future plans Continue working out non-relativistic effective scalar theory interactions Formulate a non-relativistic description of a real scalar Apply the theory to problems such as dark matter scattering on nuclei and colored scalar bound states with mixing Perform matching calculations for representative models MATTHEW GONDERINGER - WAYNE STATE UNIVERSITY 10.