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 −푖푝⋅푥 푚 − 퐸 † +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

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