Model building using Lie-point symmetries

Transcription

1 Model building using Lie-point symmetries Damien P. George Nikhef theory group Amsterdam, The Netherlands CERN LPCC Summer Institute 18 th August 2011 (arxiv: )

2 The Lie point symmetry method Want to systematically find all the symmetries of a model, even if symmetry is spontaneously broken, also derive parameter relationships that give enhanced symmetries. The Lie point symmetry method consists of finding the determining equations, whose solutions describe infinitesimal symmetries, and then solving these equations. Point: transformations depend only on coords and fields, not on derivatives of fields. Overview: The determining equations. Example with 2 scalars. Automation. N interacting scalars. Spin-1 plus N scalars. Spontaneous symmetry breaking. The standard model. D.P. George Model building using Lie-point symmetries 2/13

3 Variation of the action Infinitesimal Lie point symmetries: x µ x µ + η µ (x, φ) φ i φ i + χ i (x, φ) S S + δs should be unchanged. ( ) L Solve for the fields Euler-Lagrange equations: φ i L µ ( µφ i ) = 0. h i Form a divergence Noether s theorem: µ Lη µ + L ( µφ i ) (χi ην νφ i) =0. Solve for the infinitesimals master determining equation: L dηµ dx µ + L x µ ηµ + L χ i + L ( dχi φ i ( µ φ i ) dx µ φ i dη ν ) x ν dx µ = 0 Total derivative: d dx µ x + φ µ i x µ φ i. D.P. George Model building using Lie-point symmetries 3/13

4 Example: two scalars Only field symmetries, φ i φ i + χ i (φ i ). Master determining equation: Apply to Lagrangian L φ i χ i + L φ j χ i ( µ φ i ) x µ = 0. φ j L = 1 2 µ φ 1 µ φ µ φ 2 µ φ m2 1φ m2 2φ 2 2. Determining equation is m 2 1φ 1 χ 1 m 2 2φ 2 χ 2 + µ φ 1 µ φ 1 χ 1 φ 1 + µ φ 1 µ φ 2 χ 1 φ 2 + µ φ 2 µ φ 1 χ 2 φ 1 + µ φ 2 µ φ 2 χ 2 φ 2 = 0. Equate independent terms to zero: m 2 1φ 1 χ 1 m 2 2φ 2 χ 2 = 0, χ 1 φ 1 = 0, χ 1 φ 2 + χ 2 φ 1 = 0, χ 2 φ 2 = 0. D.P. George Model building using Lie-point symmetries 4/13

5 Example: two scalars Determining equations: m 2 1φ 1 χ 1 m 2 2φ 2 χ 2 = 0, χ 1 φ 1 = 0, General solution to last three equations: χ 1 φ 2 + χ 2 φ 1 = 0, χ 2 φ 2 = 0. χ 1 (φ 2 ) = α 1 + βφ 2, χ 2 (φ 1 ) = α 2 βφ 1. Symmetries: α 1 : shift of φ 1. α 2 : shift of φ 2. β: rotation between φ 1 and φ 2. Final determining equation is α 1 m 2 1φ 1 + α 2 m 2 2φ 2 + β(m 2 1 m 2 2)φ 1 φ 2 = 0. the model parameters dictate the symmetries. D.P. George Model building using Lie-point symmetries 5/13

6 Automation of LPS method Two (massive) scalars have algebraic determining equation α 1 m 2 1φ 1 + α 2 m 2 2φ 2 + β(m 2 1 m 2 2)φ 1 φ 2 = 0. Gaussian elimination (with branching) to find null space of m α 1 0 m α 2 = m 2 1 m2 2 β Differential equations generalised Gaussian elimination. Define ordering on η µ and χ i. Sort terms. Arrange as rows. Perform row reduction to diagonal form. c 1 (λ i ) i f + X 1 (f) = 0, c 2 (λ i ) i+j f + X 2 (f) = 0. c 1 (λ i ) = 0: remove i f term. c 1 (λ i ) 0: use i f to eliminate i+j f. D.P. George Model building using Lie-point symmetries 6/13

7 N interacting scalar fields Symmetries dictated by structure of interactions between fields. General Lagrangian for N spin-0 fields Determining equations V µ η µ + V φ i χ i = 0, L = 1 2 µ φ i µ φ i V (φ). µ χ i V ηµ = 0 φ i µ i, (χ = χ(φ)) µ η ν + ν η µ = 0 µ ν, µ ν, (Poincaré) χ i + χ j = 0 φ j φ i i j, i j, (shift, rot.) 1 2 ση σ µ η µ + χī φ ī = 0 µ ī, (scaling) η µ = 0 φ i µ i. (η = η(x)) D.P. George Model building using Lie-point symmetries 7/13

8 N interacting scalar fields General Lagrangian L = 1 2 µ φ i µ φ i V (φ). For D 2 the general coordinate symmetries are (b µν anti-symm) η µ (x) = a µ + b µ νx ν + c x µ. General field symmetries are (β ij anti-symm) χ i (φ) = α i + β ij φ j + 2 D c φ i. 2 Remaining determining equation is DcV + V ( α i + β ij φ j + 2 D ) c φ i = 0. φ i 2 Form of V allowed symmetries. D.P. George Model building using Lie-point symmetries 8/13

9 Spin-1 plus N scalars L = 1 2 µ φ i µ φ i 1 4 F µν F µν + J i A µ µ φ i + K ij A µ φ i µ φ j V (φ, A 2 ) General solution for infinitesimals: η µ (x) = a µ + b µ νx ν + c x µ + 2d ν x ν x µ d µ x ν x ν χ i (x, φ) = α i (x) + β ij (x)φ j + (2 D)( 1 2 c + d νx ν )φ i ξ µ (x, A) = µ Λ(x) + (b µ ν + 2d ν x µ 2d µ x ν )A ν + (2 D)( 1 2 c + d νx ν )A µ E.g. massive U(1): when solving rest of determining equations, demand: gauge symmetry: Λ(x) is arbitrary, V massive vector: A = m 2 A µ µ derive allowed form of L and relations between parameters. 1 field: Stückelberg (J = m), 2 fields: Higgs. D.P. George Model building using Lie-point symmetries 9/13

10 Spontaneously broken symmetries Spontaneously broken scale symmetry: V = λφ 4 has scale symmetry. V = λ(φ + v) 4 has shift-scale-shift symmetry. V = λ(φ φ2 2 v2 ) 2 has U(1). Define φ 2 = v + ϕ. V = λ(φ ϕ2 + 2vϕ) 2 has shift-u(1)-shift. LPS method will find symmetry, no matter how broken/hidden it may be. For example, solve for relationships between c i in V = c 1 + c 2 φ 1 + c 3 φ 2 + c 4 φ c 5 φ 1 φ 2 + c 6 φ c 7 φ c 8 φ 2 1φ 2 + c 9 φ 1 φ c 10 φ c 11 φ c 12 φ 3 1φ 2 + c 13 φ 2 1φ c 14 φ 1 φ c 15 φ 4 2. D.P. George Model building using Lie-point symmetries 10/13

11 The standard model Schematic structure of the standard model: L SM ( φ) 2 + φ 2 φ + φ 2 + φ 4 + ψ ψ + φψ 2. N = 244 real degrees of freedom (with RH neutrinos and Higgs). About 10 7 terms in L SM. Maximum number of determining equations: (but many are duplicated, and many are single term). Apply the LPS method: Find all (continuous) symmetries and prove that there are no more. Use know values of parameters, and run them. Find approximate symmetries. Add new degrees of freedom looking for new symmetries (e.g. GUT). Given measurements of new particles/interactions, can they form part of a new symmetry? D.P. George Model building using Lie-point symmetries 11/13

12 Conclusions Coordinate variation η µ, field variation χ i. Master determining equation: L dηµ dx µ + L x µ ηµ + L φ i χ i + The Lie point symmetry method: Counterpart to the Euler-Lagrange equations. Finds all possible symmetries. L ( dχi ( µ φ i ) dx µ φ i dη ν ) x ν dx µ = 0 Finds all interesting relationships between parameters. Works even for spontaneously broken symmetries. Can be automated; crucial for large systems. Future work: Find all symmetries of the standard model. Allow for discrete symmetries [Hydon (1998)]. Extend to supersymmetry [Grundland, Hariton, Snobl (2008)]. D.P. George Model building using Lie-point symmetries 12/13

13 References Text book: Olver, Applications of Lie Groups to Differential Equations, Reduction to standard form: Reid, J. Phys. A: Math. and General, 23 (1990) L853. Reid, Eur. J. of Appl. Math., 2 (1991) 293. Reid, Proc. ISSAC 92 (1992). LPS method and computation: Hereman, CRC Handbook of Lie Group Analysis of Differential Equations, (1996) 367. Previous work using LPS for field theories: Hereman, Marchildon & Grundland, Proc. XIX Intl. Colloq. Spain, (1992) 402. Marchildon, J. Group Theor. Phys., 3 (1995) 115. Marchildon, J. Nonlin. Math. Phys., 5 (1998) 68. DPG, A systematic approach to model building, arxiv: D.P. George Model building using Lie-point symmetries 13/13

14 Symmetries of one scalar Specialise to N = 1: Four distinct cases: dγv + dv (α + γφ) = 0. dφ V = 0: α and γ free. Independent shift and scale symmetries. Rank associated with field is R χ = (2). V = const: γ = 0 but α is free. Field rank R χ = (1). V = λ(φ + v) d : Solve above differential equation. Given v, relationship between shift and scale symmetry is fixed by v = α/γ. Field rank R χ = (1). V arbitrary: α = γ = 0. No shift or scale symmetry. Field rank R χ = (0). D.P. George Model building using Lie-point symmetries backup 14

15 Symmetries of two scalars dγv + V φ 1 (α 1 + βφ 2 + γφ 1 ) + V φ 2 (α 2 βφ 1 + γφ 2 ) = 0. Go to polar field variables, φ 1 = r cos θ, φ 2 = r sin θ: Determining equation is L = 1 2 µ r µ r + r µ θ µ θ V (r, θ). dγv + V r (α 1 cos θ + α 2 sin θ + γr) V θ A solution: ( V (r, θ) = λ r k ve lθ) m. k and m related by mk = d. Relationship between scale and rotation symmetry fixed by kγ = lβ. Action of the symmetry is r e γ r, θ θ kγ/l and x µ e dγ/d x µ. ( ) sin θ cos θ α 1 α 2 + β = 0. r r D.P. George Model building using Lie-point symmetries backup 15

16 Non-linear symmetries Field (no coordinate) symmetries of L = φ m ( µ φ µ φ) n. m and n 0 are constant exponents. Determining equation mφ m 1 χ + 2nφ m dχ dφ = 0. Solve for χ: χ = aφ m/2n a is integration constant. Non-linear symmetry acts by φ = a φ m/2n, solution φ (φ p + paɛ) 1/p with p = 1 + m/2n. D.P. George Model building using Lie-point symmetries backup 16

17 The action versus the equations of motion Distinction between the symmetries of action and symmetries of corresponding equations of motion. G a symmetry of an action = G also a symmetry of the Euler-Lagrange equations. Converse not necessarily true. Denote the system by j (x µ, φ i, φ i ) = 0. 1 Construct the prolonged symmetry operator pr (k) α. α = η µ x µ + χ i. φ i Prolongation extends α to include all possible combinations of derivatives of φ, to order k. 2 Apply pr (k) α to the system: (pr (k) α ) =0 = 0. 3 Equate all independent coefficients to zero determining equations. D.P. George Model building using Lie-point symmetries backup 17

18 Equations of motion example System defined by Euler-Lagrange equation φ φ + m 2 φ = 0. What are its symmetries? m = 0 has m 0 has η t (t, x) = F + (t + x) + F (t x), η x (t, x) = F + (t + x) F (t x) + f, χ(t, x, φ) = G + (t + x) + G (t x) + g φ(t, x). η t (x) = a t + bx, η x (t) = a x + bt, χ(t, x, φ) = + dk where ω = k 2 + m 2. [ H + (k) e i(ωt+kx) + H (k) e i(ωt kx)] + g φ(t, x), D.P. George Model building using Lie-point symmetries backup 18

19 Degrees of freedom in the standard model N = 244 real degrees of freedom (with RH neutrinos): gauge = 4 real components (1 hyp + 3 weak + 8 strong) = 48, leptons = 8 real components 3 gens (ν + e) = 48, quarks = 8 real components 3 gens 3 cols (u + d) = 144, and Higgs = 2 real components weak-doublet = 4. D.P. George Model building using Lie-point symmetries backup 19