Introduction Free Fermions Spin Weight THE END. Nonminimal SME. Matt Mewes. Cal Poly SME Summer School

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1 Nonminimal SME Matt Mewes Cal Poly 2015 SME Summer School

2 Outline introduction motivation msme vs SME what s been done free fermions construction field redefinitions dispersion relation hamiltonian spin weight

3 Introduction Introduction

4 Introduction: Why Lorentz violation? Newton s Laws E. & M., Atoms, Molecules,... Standard Model Cosmology, Astrophysics,... Newtonian Gravity General Relativity Quantum Mechanics Lorentz Symmetry Curved Spacetime Theory of Everything

5 Introduction: Why Lorentz violation? SME = general description of LV and low energies Newton s Laws E. & M., Atoms, Molecules,... Standard Model Cosmology, Astrophysics,... Newtonian Gravity General Relativity Quantum Mechanics Lorentz Symmetry Curved Spacetime Theory of Everything

6 Introduction: minimal SME What is the minimal SME? power-counting renormalizable (d 4) flat spacetime gauge invariance translation invariance Lorentz violating leading-order (renormalizable?) remnants known physics SM + GR = quantum gravity higher-order nonrenormalizable remnants

7 Introduction: nonminimal SME Why consider nonrenormalizable (d > 4) operators? phenomenologically interesting (new stuff to measure) theoretically plausible/desirable gravity is not renormalizable ideas exist that lead to nonrenormalizable terms (e.g., noncommutative geometry d = 6) natural S = d 4 x(lv coefficient) (dimension d operator) mass dimension of coefficient = 4 d coefficient 1 (M Planck ) d 4

8 Introduction: where we stand nonminimal (nonrenormalizable) extensions: photons neutrinos Astrophysical tests of Lorentz and CPT violation with photons, Kostelecký & Mewes, Astrophys. J. 689, L1, Electrodynamics with Lorentz-violating operators of arbitrary dimension, Kostelecký & Mewes, Phys. Rev. D 80, , Neutrinos with Lorentz-violating operators of arbitrary dimension, Kostelecký & Mewes, Phys. Rev. D 85, , noninteracting Dirac fermions Fermions with Lorentz-violating operators of arbitrary dimension, Kostelecký & Mewes, Phys. Rev. D 88, , 2013.

9 Introduction: where we re going interacting fermions gravity. Short-range gravity and Lorentz violation, Bailey & Kostelecký, Phys. Rev. D 91, , 2015.

10 Free Fermions Free Fermions

11 Free Fermions Phys. Rev. D 88, (2013) [arxiv: ] Fermions with Lorentz-violating operators of arbitrary dimension [hep-ph] 22 Aug 2013 V. Alan Kostelecký 1 and Matthew Mewes 2 1 Physics Department, Indiana University, Bloomington, Indiana 47405, USA 2 Physics Department, Swarthmore College, Swarthmore, Pennsylvania 19081, USA (Dated: IUHET 577, August 2013) The theoretical description of fermions in the presence of Lorentz and CPT violation is developed. We classify all Lorentz- and CPT-violating and invariant terms in the quadratic Lagrange density for a Dirac fermion, including operators of arbitrary mass dimension. The exact dispersion relation is obtained in closed and compact form, and projection operators for the spinors are derived. The Pauli hamiltonians for particles and antiparticles are extracted, and observable combinations of operators are identified. We characterize and enumerate the coefficients for Lorentz violation for any operator mass dimension via a decomposition using spin-weighted spherical harmonics. The restriction of the general theory to various special cases is presented, including isotropic models, the nonrelativistic and ultrarelativistic limits, and the minimal Standard-Model Extension. Expressions are derived in several limits for the fermion dispersion relation, the associated fermion group velocity, and the fermion spin-precession frequency. We connect the analysis to some other formalisms and use the results to extract constraints from astrophysical observations on isotropic ultrarelativistic spherical coefficients for Lorentz violation. I. INTRODUCTION The invariance of the laws of nature under Lorentz transformations is well established, being based on an extensive series of investigations originating in classic tests such as the Michelson-Morley, Kennedy-Thorndike, Ives- Stilwell, and Hughes-Drever experiments [1 4]. Interest in precision tests of relativity has experienced a renewal renormalizable dimensions d 4. To date, the minimal SME has been adopted as the theoretical framework in searches for Lorentz violation in the fermion sector involving electrons [10], protons and neutrons [11], muons [12], neutrinos [13], quarks [14], and gravitational couplings of various species [8, 15]. Discussions in the literature of the nonminimal SME fermion sector are more limited. The general structure and properties of the non-

12 Free Fermions: goals of the paper 1 construct a general theory describing a free fermions single species of noninteracting Dirac fermions field theoretic linear hermitian translationally invariant (energy-momentum conservation) 2 catalog various operators based on dimension d of operator CPT parity tensor structure 3 identify experimental signatures dispersion birefringence kinematics...

13 Free Fermions: construction usual lagrangian: Lorentz-violating modifications: Q operator: 4 4 matrix depends on p µ = i µ L = 1 2 ψ(γµ i µ m ψ )ψ +h.c. δl = 1 2 ψ Qψ +h.c. dimension of Q = 1 (dimension of δl = 4) modified Dirac Eq: (γ µ i µ m ψ + Q)ψ = 0

14 Free Fermions: construction expand in γ matrices: Q = Ŝ +i Pγ 5 + V µ γ µ +µ γ 5 γ µ T µν σ µν break each operator into CPT-even and CPT-odd part: scalar: pseudoscalar: Ŝ = ê m P = f m5 vector: Vµ = ĉ µ â µ pseudovector:  µ = d µ b µ tensor: T µν µν = ĝ Ĥµν

15 Free Fermions: construction expand in momentum: example: ĉ µ = d even c (d)µα 1...α d 3 p α1...p αd 3 coefficients Operator Type d CPT Cartesian coefficients Number m scalar odd, 5 even m (d)α 1...α d 3 d(d 1)(d 2)/6 m 5 pseudoscalar odd, 5 even m (d)α 1...α d 3 5 d(d 1)(d 2)/6 â µ vector odd, 3 odd a (d)µα 1...α d 3 2d(d 1)(d 2)/3 bµ pseudovector odd, 3 odd b (d)µα 1...α d 3 2d(d 1)(d 2)/3 ĉ µ vector even, 4 even c (d)µα 1...α d 3 2d(d 1)(d 2)/3 d µ pseudovector even, 4 even d (d)µα 1...α d 3 2d(d 1)(d 2)/3 ê scalar even, 4 odd e (d)α 1...α d 3 d(d 1)(d 2)/6 f pseudoscalar even, 4 odd f (d)α 1...α d 3 d(d 1)(d 2)/6 ĝ µν tensor even, 4 odd g (d)µνα 1...α d 3 d(d 1)(d 2) Ĥ µν tensor odd, 3 even H (d)µνα 1...α d 3 d(d 1)(d 2)

16 QUIZ scalar: pseudoscalar: Ŝ = ê m P = f m5 vector: Vµ = ĉ µ â µ pseudovector: Â µ = d µ b µ tensor: T µν µν = ĝ Ĥµν Question: Who am I? δl = 1 2 ψkµαβ p α p β γ 5 γ µ ψ Answer: K µαβ = b (5)µαβ

17 Free Fermions: field redefinitions A map ψ = f(ψ) describes same physics provided f is nonsingular. Are the LV terms real? Can they be mapped to the usual case? Can a LV term be absorbed by another? consider tiny transformation: ψ = (1 X iŷ)ψ Q = Q 2m ψ X +pµ {γ µ, X}+ip µ [γ µ,ŷ] example: X = Xµ A γ 5γ µ  µ = µ 2m ψ Xµ A, T µν = T µν 2ǫ µνρσ p ρ XAσ  operators can be absorbed in T

18 Free Fermions: field redefinitions Using field redefinitions we can remove/absorb scalar, pseudoscalar, pseudovector operators and some symmetry components of vector and tensor operators. Ŝ eff = 0, Peff = 0,  µ eff = 0, V µ eff = ( Vµ + 1 m ψ p µ Ŝ ) [0] T µν eff = ( T µν + 1 m ψ p [u  ν]) [2] ( ) [0] = ( ) [2] = Only effective vector and tensor operators affect free fermions. â µ eff = ( â µ 1 m ψ p µ ê ) [0] ĉ µ eff = (ĉ µ 1 m ψ p µ m) [0] ĝµν eff = ( ĝ µν 1 m ψ p [µ bν] ) [2] Ĥµν eff = ( Ĥ µν 1 m ψ p [µ dν] ) [2]

19 Free Fermions: field redefinitions effective coefficients Operator Type d CPT Cartesian coefficients Number â µ eff vector odd, 3 odd a (d)µα 1...α d 3 eff ĉ µ eff vector even, 4 even c (d)µα 1...α d 3 eff ĝµν eff tensor even, 4 odd g (d)µνα 1...α d 3 eff Ĥµν eff tensor odd, 3 even H(d)µνα 1...α d 3 eff (d+1)d(d 1)/6 (d+1)d(d 1)/6 (d+1)d(d 2)/2 (d+1)d(d 2)/2 vector: â µ eff = d a(d)µα 1...α d 3 eff p α1...p αd 3 vector: ĉ µ eff = d c(d)µα 1...α d 3 eff p α1...p αd 3 µν tensor: ĝ eff = d g(d)µνα 1...α d 3 eff p α1...p αd 3 tensor: Ĥµν eff = H (d)µνα 1...α d 3 d eff p α1...p αd 3 (other coeffs may contribute in interacting case, in curved spacetime,...)

20 QUIZ msme coefficient e (4)µ can be moved into the â µ eff â µ eff 1 m ψ p µ p ν e (4)ν Question: What dimension d is the corresponding a eff coefficient? Hint: δl e(4)ν 2m ψ ψp µ p ν γ µ ψ Answer: d = 5

21 Free Fermions: dispersion relation p 2 = m 2 ψ 2p (ĉ eff â eff ) ±2p ( ĝ eff Ĥ eff ) ( ĝ eff Ĥ eff ) p (usual) (nonbirefringent) (birefringent) nonbirefringent spin-independent propagation birefringent spin-dependent propagation note: dispersion relation only depends on effective coefficients, note: as expected

22 Free Fermions: hamiltonian find transformation U so that H is block diagonal Uγ 0 (p γ m ψ + Q)U = (E H) Foldy-Wouthuysen-like calculation gives ( ) h 0 H = 0 h 2 2 relativistic particle hamiltonian h = h 0 +δh δh = + Σ σ E 0

23 Free Fermions: hamiltonian δh = + Σ σ E 0 spin-independent part: = p ĉ eff +p â eff spin-dependent part: Σ j = E 0( ĝ eff Ĥ eff ) 0j ( ĝ eff Ĥ eff ) jk eff pk +( ĝ eff Ĥ eff ) 0k effp k p j /(E 0 +m ψ ) for antiparticles â â, ĝ ĝ

24 Free Fermions: hamiltonian δh = + Σ σ E 0, Σ j are complicated combinations of coeffs for LV complicated boosts and rotations bookkeeping headache expand in a rotation basis (spherical harmonics) makes rotations simple, boosts difficult (many/most experiments focus on rotations) provides systematic classification

25 Free Fermions: spherical expansion spin-independent parts = scalar functions of p expand in usual spherical harmonics (CPT-odd) (CPT-even) h a = dnjm h c = dnjm E 0 d 3 n p n 0Y jm (ˆp)a (d) njm E 0 d 3 n p n 0Y jm (ˆp)c (d) njm d = operator dimension n = energy/momentum dependence j = total angular momentum m = z angular momentum

26 Free Fermions: spherical expansion spin-dependent parts = vector functions of p need tensor spherical harmonics (spin-weighted harmonics: sy jm) (CPT-odd) (CPT-even) h g = h j gσ j h H = h j H σj spin-weight s = 0 parts (scalar radial components): (CPT-odd) (CPT-even) ˆp k h k g = m ψ d 4 n E0 p n 0Y jm (ˆp)(n+1)g (d)(0b) njm ˆp k h k H = m ψ d 4 n E0 p n 0Y jm (ˆp)(n+1)H (d)(0b) njm

27 Free Fermions: spherical expansion ˆp p ˆθ ˆφ ˆǫ r = ˆp ˆǫ ± = 1 2 (ˆθ±iˆφ) for a vector V: V r = ˆǫ r V = (radial component) V ± = ˆǫ ± V = (spin-weighted components) spin-weighted components (expand in spin-weighted harmonics ±1 Y jm ) ˆǫ k ± hk g = [ d 3 n n E 0 p ±1 Y jm (ˆp) ± ˆǫ k [ ± hk H = d 3 n n E 0 p ±1 Y jm (ˆp) ± j(j+1) 2 j(j+1) 2 g (d)(0b) njm H (d)(0b) njm ± g(d)(1b) njm ± H (d)(1b) njm ] + ig(d)(1e) njm + ih(d)(1e) njm ]

28 Free Fermions: spherical expansion how to find spherical expansion: 1 expand each spin-weighted component in s Y jm 2 use p µ -expansion and coefficient symmetries to find limits n,j,m 3 use symmetries to find relations between different expansion coeffs 4 use linear algebra to find minimal set of coefficients

29 Free Fermions: spherical expansion spherical coefficients Coefficient CPT Parity type d n j Number a (d) 1 njm odd E odd, 3 0,1,...,d 2 n,n 2,n 4, (d+1)d(d 1) c (d) 1 njm even E even, 4 0,1,...,d 2 n,n 2,n 4, (d+1)d(d 1) g (d)(0b) 1 njm odd B even, 4 0,1,...,d 3 n+1x,n 1,n 3, (d+1)d(d 1) 1 g (d)(1b) 1 njm odd B even, 4 2,3,...,d 2 n 1,n 3,n 5, (d 2)(d2 d 3) g (d)(1e) 1 njm odd E even, 4 1,2,...,d 2 n,n 2,n 4, (d+2)d(d 2) H (d)(0b) 1 njm even B odd, 3 0,1,...,d 3 n+1,n 1,n 3, (d+1)d(d 1) 1 H (d)(1b) 1 njm even B odd, 5 2,3,...,d 2 n 1,n 3,n 5, (d+1)(d 1)(d 3) H (d)(1e) 1 njm even E odd, 3 1,2,...,d 2 n,n 2,n 4, (d 1)(d2 +d 3) h = h a +h c +(h g) + σ +(h g) rσ r +(h g) σ + +(h H ) + σ +(h H ) rσ r +(h H ) σ + h a = E 0 d 3 n p n 0 Y jm (ˆp)a (d) njm, hc = E 0 d 3 n p n 0 Y jm (ˆp)c (d) njm (h g) r = m ψ E0 d 4 n p n 0 Y jm (ˆp)(n + 1)g (d)(0b) njm (h H ) r = m ψ E0 d 4 n p n 0 Y jm (ˆp)(n + 1)H (d)(0b) njm (h g) ± = [ d 3 n n E 0 p ±1 Y jm (ˆp) ± (h H ) ± = [ d 3 n n E 0 p ±1 Y jm (ˆp) ± j(j+1) 2 j(j+1) 2 g (d)(0b) njm H (d)(0b) njm ± g(d)(1b) njm ± H (d)(1b) njm ] + ig(d)(1e) njm ] + ih(d)(1e) njm

30 Free Fermions: limiting models example: h a = dnjm E 0 d 3 n p n 0Y jm (ˆp)a (d) njm (general case) ha = dn E 0 d 3 n p n å (d) n (isotropic limit:j,m,s = 0) h NR a = p n 0Y jm (ˆp)a NR njm njm (nonrelativistic limit) h UR a = p d 3 0Y jm (ˆp)a UR(d) jm djm (ultrarelativistic limit)

31 Free Fermions: applications PRD 90, (2014) Laboratory tests of Lorentz and CPT symmetry with muons 8v2 [hep-ph] 22 Oct 2014 André H. Gomes, 1 V. Alan Kostelecký, 2 and Arnaldo J. Vargas 2 1 Departamento de Física, Universidade Federal de Viçosa, Viçosa, MG, Brazil 2 Physics Department, Indiana University, Bloomington, Indiana 47405, USA (Dated: IUHET 586, July 2014; published as Phys. Rev. D 90, (2014)) The prospectsare exploredfor testinglorentzandcptsymmetryinthemuonsector viathespectroscopy of muonium and various muonic atoms, and via measurements of the anomalous magnetic moments of the muon and antimuon. The effects of Lorentz-violating operators of both renormalizable and nonrenormalizable dimensions are included. We derive observable signals, extract first constraints from existing data on a variety of coefficients for Lorentz and CPT violation, and estimate sensitivities attainable in forthcoming experiments. The potential of Lorentz violation to resolve the proton radius puzzle and the muon anomaly discrepancy is discussed. I. INTRODUCTION Muons have played a significant role in testing relativity since their discovery in the 1930s [1]. Indeed, the first demonstration of time dilation was the Rossi-Hall experiment studying muons originating from cosmic rays [2]. As another example, the clock hypothesis that acceleration per se has no affect on a clock s ticking rate has been verified using muons in a ring accelerator [3]. In recent years, the prospect of tiny deviations from relativity has emerged as a promising candidate signal for new physics coming from the Planck scale MP GeV, following the demonstration that Lorentz invariance can naturally be broken in a unified framework of quantum gravity such as string theory [4]. Driven by this prospect, many high-precision tests of relativity in differexperiments are of particular interest in this context because they offer excellent prospects for a sensitive study of Lorentz and CPT violation in second-generation matter. However, given the extensive historical impact of research with muons and their comparatively widespread availability, surprisingly little is known about the SME muon sector on both the theoretical and the experimental fronts. For example, inspection of the Data Tables [5] reveals that existing constraints on Lorentz and CPT violation involving muons comprise only a small fraction of the available limits. The effects of minimal-sme coefficients on the behavior of muons [9] have been studied at impressive sensitivities in the laboratory via muonium hyperfine spectroscopy [10] and via measurements of the anomalous magnetic moments of the muon and antimuon [11, 12]. The latter have also been used to place limits on nonminimal interaction terms with d = 5 [13], while

32 ¼½½ ¼½½ ¾ ½ ¾½½ ¾½½ ½½ ½½ ¼½½ ¾ ½ ¼½½ ½½ ¾ ½ ½ ¾½½ ¾½½ ¾½½ ½½ ½½ ¾½½ ½ Free Fermions: applications Ì Ð ½ º ÆÓÒÑ Ò Ñ Ð ÑÙÓÒ ØÓÖ ÓÑ Ò Ø ÓÒ Ê ÙÐØ ËÝ Ø Ñ Ê º ÆÊ ¾ ½¼ Î ½ ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ ÆÊ ½¼ Î ½ ¾¼ ¾ Ê À ÆÊ ¼ µ ¾½½ ÁÑ À ÆÊ ¼ µ Ê ÆÊ ¼ µ ÁÑ ÆÊ ¼ µ ¾½½ ½ ½¼ ½½ Î ½ ¾¼ Ê À ÆÊ ½ µ ¾½½ ÁÑ À ÆÊ ½ µ Ê ÆÊ ½ µ ÁÑ ÆÊ ½ µ ¾½½ ½¼ ½¾ Î ½ ¾¼ ÍÊ µ Ñ ÍÊ µ ½ ØÓ ½µ ½¼ Î ½ ØÖÓÔ Ý ½ Ê À µ ¼ µ ¼½½ ÁÑ À µ ¼ µ ¼½½ ½¼ ¾½ Î ½ ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ ¾ Ê À ÁÑ À Ê À ÁÑ À µ ¾½½ ¾ ½ ½¼ Î ÆÄ ¾ ¾¼ µ µ µ ½ ¾ Ê À µ ÁÑ À ½ ½¼ Î ¾¼ ¾ ½ µ ½ À ¼½¼ À ¾½¼ µ µ ½ ½ µ ½¼ ¾ Î ½ ÆÄ ÊÆ ¾ Ø ¾¼ À ¾ ¼ ¾ ¼µ ½¼ ¾ Î ¾¼ µ ½ ÍÊ µ ØÓ ¼ ¼¼¾ µ ½¼ ¾¼ Î ¾ ØÖÓÔ Ý ½ Ê µ ¼ µ ¼½½ ÁÑ µ ¼ µ ¼½½ ½¼ ¾¼ Î ¾ ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ Ê µ ¼½½ ÁÑ µ ¼½½ Ê µ ¾½½ ÁÑ µ ¾½½ ½¼ Î ÆÄ ¾ ¾¼ ¾ ¾ Ê µ ¾ ½ ÁÑ µ ¾ ½ ½¼ Î ¾¼ ¾ ¾ µ ¼½¼ µ ¾½¼ ¾ ¾ µ ½¼ ¾ Î ¾ ¾¼ µ ¾ ¼ ¾ ¾ µ ½¼ ¾ Î ¾ ¾¼ ÆÊ ½ ½¼ Î ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ ÆÊ ½ ½¼ Î ¾¼ ÆÊ ½ ½¼ Î ¾¼ ÆÊ ½ ½¼ Î ¾¼ Ê À ÆÊ ¼ µ ½½ ÁÑ À ÆÊ ¼ µ Ê ÆÊ ¼ µ ÁÑ ÆÊ ¼ µ ½½ ¾ ½¼ ½ Î ¾¼ Ê À ÆÊ ½ µ ½½ ÁÑ À ÆÊ ½ µ Ê ÆÊ ½ µ ÁÑ ÆÊ ½ µ ½½ ½¼ ¾ Î ¾¼ Ê À µ Ê À µ Ê À µ ¼ µ ¼½½ ÁÑ À µ ¼ µ ¼½½ ½¼ ½ Î ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ À µ µ µ ¼½¼ À ¾½¼ À ½¼ ½ ½ µ ½¼ ¾ Î ÆÄ ÊÆ ¾ Ø ¾¼ À ¾ ¼ À ¼ ¼ ½µ ½¼ ¾ Î ¾¼ µ µ À ¼ ¾ ¾ µ ½¼ ¾ Î ¾¼ µ ÁÑ À Ê À ÁÑ À µ ¾½½ ¾ ¾ ½¼ ¾ Î ÆÄ ¾¼ µ µ ¾ ¾ Ê À µ ÁÑ À ½½ ¾ ¾ ½¼ Î ¾¼ µ ÁÑ À Ê À ÁÑ À µ ½ ½ ½¼ ¾ Î ¾¼ µ µ ¾ Ê À µ ÁÑ À ½ ½ ½¼ Î ¾¼ ½ µ Ê µ ¼ µ ¼½½ ÁÑ µ ¼ µ ¼½½ ½¼ ½ Î ÅÙÓÒ ÙÑ Ô ØÖÓ ÓÔÝ ¾¼ µ ¼½¼ µ ¾½¼ µ ½¼ ¾ ¾ µ ½¼ ¾ Î ÆÄ ¾ ¾¼ µ ¾ ¼ µ ¼ ¾ ¾ µ ½¼ ¾ Î ¾¼ µ ¼ ½ ½ µ ½¼ ¾ Î ¾¼ Ê µ ¼½½ ÁÑ µ ¼½½ Ê µ ¾½½ ÁÑ µ ¾½½ ½ ½¼ Î ¾¼ ¾ Ê µ ½½ ÁÑ µ ½½ ½ ½¼ Î ¾¼ ¾ Ê µ ¾ ½ ÁÑ µ ¾ ½ Ê µ ½ ÁÑ µ ½ ½¼ Î ¾¼ ¾ Ê µ ½ ÁÑ µ ½ ½¼ Î ¾¼ ¾ À µ µ µ ¼½¼ À ¾½¼ À ½¼ À µ ½¼ ½ ½ µ ½¼ ¾ Î ÆÄ ÊÆ ¾ Ø ¾¼ À ¾ ¼ À ¼ À ¼ ¾ µ ½¼ ¾ Î ¾¼ µ µ µ À ¼ À ¼ ¾ ¾ µ ½¼ ¾ Î ¾¼ µ µ À ¼ ½ ½ ½ ½µ ½¼ ¾ Î ¾¼ µ ½¼µ ¼½¼ ½¼µ ¾½¼ ½¼µ ½¼ ½¼µ ½¼ ¾ ¾ µ ½¼ ¾ Î ÆÄ ¾ ¾¼ ½¼µ ¾ ¼ ½¼µ ¼ ½¼µ ¼ ¾ ¾ µ ½¼ ¾ Î ¾¼ ½¼µ ¼ ½¼µ ¼ ½ ½ µ ½¼ ¾ Î ¾¼ ½¼µ ¼ ½ ½ µ ½¼ ¾ Î ¾¼

33 Free Fermions: applications (Arnaldo Vargas, poster session, Tues 8pm) arxiv: Lorentz and CPT tests with hydrogen, antihydrogen, and related systems v1 [hep-ph] 4 Jun 2015 V. Alan Kostelecký and Arnaldo J. Vargas Physics Department, Indiana University, Bloomington, Indiana 47405, USA (Dated: IUHET 592, June 2015) The potential of precision spectroscopy as a tool in systematic searches for effects of Lorentz and CPT violation is investigated. Systems considered include hydrogen, antihydrogen, deuterium, positronium, and hydrogen molecules and molecular ions. Perturbative shifts in energy levels and key transition frequencies are derived, allowing for Lorentz-violating operators of arbitrary mass dimensions. Observable effects are deduced from various direct measurements, sidereal and annual variations, comparisons among species, and gravitational responses. We use existing data to place new and improved constraints on nonrelativistic coefficients for Lorentz and CPT violation, and we provide estimates for the future attainable reach in direct spectroscopy of the various systems or tests with hydrogen and deuterium masers. The results reveal prospective sensitivities to many coefficients unmeasured to date, along with potential improvements of a billionfold or more over certain existing results. I. INTRODUCTION Hydrogen spectroscopy has been intimately linked with precision tests of the foundations of relativity since the exact solution of the Dirac equation for hydrogen [1, 2] matched relativistic quantum mechanics with experiments. Indeed, a famous classic test of special relativity, the Ives-Stilwell experiment confirming time dilation [3], was first performed using a hydrogen clock. Another classic experiment, Gravity Probe A [4], verified the relativistic frequency shift in a gravitational field using a hydrogen maser launched on a suborbital rocket. The minimal-sme terms generate striking effects in the spectra of hydrogen and antihydrogen, including CPT-violating signals and shifts in the hyperfine and 1S- 2S transitions that depend on sidereal time [13]. Published searches for these effects have measured the hyperfine splitting using a hydrogen maser [14 16] and compared the 1S-2S transition in atomic hydrogen to a cesium fountain clock [17, 18]. Related experiments with antihydrogen are being developed [19 22], and experiments with hydrogen molecules and molecular ions have been proposed as well [23]. In the context of the minimal SME, theoretical modifications to the spectra of hydrogen and antihydrogen have been widely studied [13, 24

34 Spin Weight Spin Weight

35 Spin Weight: the math tensor functions have spin ( S) and orbital ( L) angular momentum J = S + L generates rotations possible choices of commuting variables: {S 2,L 2,S z,l z } possible choices of commuting variables: {S 2,L 2,J 2,J z } possible choices of commuting variables: {S 2,J 2,J z,j r } J r = ˆp J = ˆp S = helicity = spin in ˆp direction s = spin weight = helicity = J r spin-weighted harmonics: sy jm (ˆp) spin weight J 2 = j(j +1) J z = m

36 Spin Weight: QUIZ sy jm (ˆp) spin weight J 2 = j(j +1) J z = m j 0 j m j Question: What are the allowed values of spin weight s? Answer: j s j usually thought of as limit on j: j s

37 Spin Weight: example consider a vector: V = V rˆp+v θˆθ +Vφˆφ spin-weighted components: rotation ( about ) ˆp V = θ V φ V ± = ˆǫ ± V = 1 2 (V θ ±iv φ ) ( cosδ sinδ sinδ cosδ V ± = e±iδ V ± )( Vθ V φ ) p ˆθ ˆp ˆφ V ± have s = ±1 V ± (θ,φ) = jm V (±) jm ±1Y jm (θ,φ) V r = V r s = 0 V r (θ,φ) = jm V (0) jm 0Y jm (θ,φ)

38 Spin Weight: definition ˆp ˆφ p ˆθ function s f(ˆp) has spin weight s under local ˆp rotation: s f = e isδ sf spherical-harmonic expansion: s f = jm f jm sy jm

39 Spin Weight: example 2-tensor: T ab (9 components in total) three s = 0 components: T rr,t +,T + two s = +1 components: T r+,t +r two s = 1 components: T r,t r one s = +2 component: T ++ one s = 2 component: T QUIZ Question: How many s = +1 components does a general 3-tensor have? Answer: six T +rr,t r+r,t rr+,t ++,T + +,T ++

40 Spin Weight: rotations QUIZ sy jm J 2 = j(j +1), J z = m, J r = s Question: Which indices are frame dependent? Answer: m general rotations: sf(θ,φ) = sf jm s Y jm (θ,φ) rotated coefficients = s f jm = m D (j) mm ( γ, β, α) s f jm D (j) mm = Wigner matrices α,β,γ = Euler angles relating two frames independent of s

41 Spin Weight: rotations Wigner rotation matrices D (j) mm (α,β,γ) = e iαm e iγm d (j) mm (β) d (j) mm = little Wigner matrices

42 Spin Weight: rotations how to get the Euler angles: 1. start with both frames F and F aligned 2. rotate F about z = z by α 3. rotate F about y by β 4. rotate F about z by γ example: Sun & lab frames α = ω T = local right ascension β = χ = local colatitude γ = φ = angle between lab x and south sfjm lab = e imφ e im ω T d (j) mm ( χ) s fjm Sun m things that change are simple phases

43 Spin Weight: parity spherical expansions can be split into E and B parity E parity : P = ( 1) j B parity : P = ( 1) j examples: scalar: vector: S(ˆp) = E jm 0 Y jm (ˆp) V + (ˆp) = (+E jm +ib jm ) +1 Y jm (ˆp) V (ˆp) = ( E jm +ib jm ) 1 Y jm (ˆp) tensor: T ++ (ˆp) = (E jm +ib jm ) +2 Y jm (ˆp) T (ˆp) = (E jm ib jm ) 2 Y jm (ˆp)

44 Spin Weight: parity example: s = 0 E 00 E 10 E 11 E 20 E 21 E 22

45 Spin Weight: parity for vector fields on the sphere (s = ±1): V E (ˆp) = S(ˆp) = gradient of scalar V B (ˆp) = ˆpP(ˆp) = curl of radial pseudovector example: s = 1,j = 1,m = 0 E 10 B 10

46 Spin Weight: parity for vector fields on the sphere (s = ±1): V E (ˆp) = S(ˆp) = gradient of scalar V B (ˆp) = ˆpP(ˆp) = curl of radial pseudovector example: s = 1,j = 1,m = 1 E 11 B 11

47 Spin Weight: parity for vector fields on the sphere (s = ±1): V E (ˆp) = S(ˆp) = gradient of scalar V B (ˆp) = ˆpP(ˆp) = curl of radial pseudovector example: s = 1,j = 2,m = 0 E 20 B 20

48 Spin Weight: QUIZ Who am I? E 21

49 Spin Weight: QUIZ Who am I? B 22

50 Spin Weight: parity for symmetric traceless tensor fields on the sphere (s = ±2): T = v v w w, v, w = principal axes E 20 B 20

51 Spin Weight: parity for symmetric traceless tensor fields on the sphere (s = ±2): T = v v w w, v, w = principal axes E 21 B 21

52 Spin Weight: parity Who am I? B 22

53 Spin Weight: parity Who am I? E 22

54 THE END THE END