Renormalization in Lorentz-violating field theory: external states

Transcription

1 Renormalization in Lorentz-violating field theory: external states Robertus Potting CENTRA and Physics Department Faculdade de Ciências e Tecnologia Universidade do Algarve Faro, Portugal SME2015 Summer School, June 2015, Bloomington, IN

2 Outline Introduction The Kallen-Lehmann representation Motivation Lorentz-violating scalar field theory A model with a fermion field Renormalization and asymptotic states The model Lagrangian The two-point function One-loop calculation of modified propagator Feyman rules for scattering amplitudes

3 Motivation Consider a general time-ordered correlation function for scalar field φ 1 0 Tφ 1 (x 1 )...φ 1 (x n ) 0 : External two-point functions are dressed: (example of cubic two-field coupling λφ 2 1 φ 2)

4 Motivation To get physical S-matrix elements we take the external momenta on the mass shell, and consider the amputated diagram (LSZ reduction formula). The only information needed from the two-point function is the one-particle pole, defined (for scalar field theory) by Dispersion relation: Lorentz invariance, if applicable, dictates p 2 m 2 = 0 for some m; Residue (wavefunction renormalization constant) Källén and Lehmann showed nonperturbatively in the 1950 s that in interacting quantum field theory the two-point function can always be written as a sum of free propagators.

5 The Feynman propagator and the spectral density Feynman propagator: D F (x y) = ds d 4 p (2π) 4 i ρ(s) p 2 s +iǫ e ip (x y) (1) Spectral density ρ/(2π) 3 α δ (4) (p p α ) 0 φ(0) α 2 (2) ρ is positive and vanishes when p is not inside the forward lightcone. By Lorentz invariance it can only depend on p 2 and θ(p 0 ).

6 One-particle pole One-particle pole The spectral density can be split into contributions of a stable one-particle state (if present) and a continuum of multi-particle states: ρ(s) = ρ 1-part (s)+σ(s) (3) mass shell One-particle state should satisfy an equation of motion. In Fourier space: existence of a mass shell defining p 0 in terms of p. Consequently, we expect: ρ 1-part (s) = Zδ(s m 2 ).

7 graphical representation of ρ(s) m 2 M 2 s

8 Generalization to Lorentz-violating case? Can the Källén-Lehmann representation be generalized to Lorentz-violating field theories? If yes, what form does it take?

9 The vacuum (based on PRD 85, (2012) ) Consider the scalar-field action S 1 = d 4 x 1 ( ( µ φ) 2 +c µν µ φ ν φ m 2 φ 2). (4) 2 Use the notation for the vacuum 0 c µν. The vacuum is translation invariant: P µ 0 c µν = 0. Under (observer) Lorentz transformations Λ: 0 c µν 0 c µν with c µν = Λ µ αλ ν βc αβ.

10 Scalar field action The Lorentz-violating coefficient c µν can be transformed away by an appropriate linear transformation of x µ. Therefore, introduce additional scalar field: S 2 = d 4 x 1 ( ( µ φ 1 ) 2 +c µν µ φ 1 ν φ 1 +( µ φ 2 ) 2 2 ) m1φ m2φ V(φ 1,φ 2 ) (5)

11 Two-point function Two-point function D(x y) = 0 φ(x)φ(y) 0 (6) (writing φ(x) φ 1 (x) and suppressing the index c µν of the vacuum state). Write φ(x) = e ip x φ(0)e ip x and insert a complete set of states { n }: D(x y) = n e ipn (x y) 0 φ(0) n 2

12 Two-point function If we now insert the identity 1 = ds d 4 pθ(p 0 )δ(p 2 s)δ (4) (p p n ) It follows D(x y) = ds d 4 pθ(p 0 )δ(p 2 s)e ip (x y) n δ (4) (p p n ) 0 φ(0) n 2. (7) The spectral density ρ/(2π) 3 n δ (4) (p p n ) 0 φ(0) n 2 is a real, non-negative observer Lorentz scalar.

13 The spectral density ρ can depend on p 2, (c n ) p p, n = 1,2,3, where p c n p = p µ η µα1 c α 1β 1 η β1 α 2 c α 2β 2...η βnνp ν (c n ) p p. We conclude that ρ is a function of four independent observer scalars: ρ = ρ [ p 2,cp p,(c2 ) p p,(c3 ) p ] [ p ρ p 2 ;(c i ) p ] p. Thus the two-point function in (7) can be written as D(x y) = d 4 p ds (2π) 3θ(p0 )δ(p 2 s) e ip (x y) ρ [ s;(c i ) p ] p. (8)

14 The Feynman propagator From (8) we find the Feynman propagator: D F (x y) = 0 Tφ(x)φ(y) 0 = θ(x 0 y 0 )D(x y)+θ(y 0 x 0 )D(y x) d 4 p = ds (2π) 3 θ(p0 )δ(p 2 s)ρ [ s;(c i ) p ] p ( θ(x 0 y 0 )e ip (x y) +θ(y 0 x 0 )e ip (x y)). (9) The theta- and delta-functions can be traded for a momentum pole with the usual i ǫ prescription: [ d 4 p i ρ s;(c i ) p ] p D F (x y) = ds e ip (x y) (2π) 4 p 2 (10) s +iǫ

15 One-particle and multi-particle contribution Can split into contributions of a stable one-particle state (if present) and a continuum of multi-particle states: ρ [ s;(c i ) p ] [ p = ρ 1-part s;(c i ) p ] [ p +σ s;(c i ) p ] p mass shell One-particle state should satisfy an equation of motion mass shell defining p 0 ( p). Thus: ρ 1-part[ s;(c i ) p ] [ p = f (c i ) p ] ( [ p δ s g (c i ) p ]) p.

16 Källén-Lehmann form of propagator Useful to express f and g as a Taylor series: f [ (c i ) p ] p = Z (1+ 1 i 1...i r 3 r ) f i1...i r (c i j ) p p j=1 (11) g [ (c i ) p ] p = m 2 r g i1...i r (c i j ) p p, (12) 1 i 1...i r 3 j=1 generalizing the LI case f [ (c i ) p ] [ p = Z and g (c i ) p ] p = m 2. The propagator now takes the Källén-Lehmann-like form d 4 p D F (x y) = i e ip (x y) (2π) ( 4 f [ (c i ) p ] p p 2 g [ (c i ) p ] + ds σ [ s;(c i ) p ]) p p +iǫ M 2 p 2. (13) s +iǫ

17 Equation of motion To first order in c µν one finds the one-particle equation of motion: ( +m 2 +g 1 c +g 2(c 2 ) +g ( ) ) 11 c φ 1,in (x) = 0. Here c cµν µ ν. The coefficients g i can be computed perturbatively. Thus, equation of motion satisfied by external states includes higher-dimensional terms not present in the original Lagrangian!

18 The wave-function renormalization The wave-function renormalization constant Z becomes a momentum-dependent quantity: Z Z(1+f 1 c p p +...) The coefficients f i can be computed perturbatively.

19 Sum rule Using the canonical commutation relations that follow from the action (5): i η 0µ x µ[φ(x),φ(y)] x 0 =y 0 = δ(3) ( x y) (14) (where η µν = η µν +c µν ). Taking vacuum expectation value, one derives the sum rule ( η 00 ) 1 = [ dsf s ( c i) ] 00 δ + For small c µν it follows that 0 Z 1+O(c). ( s g [s ( c i) ]) 00 M 2 dsσ [ s;s(c i ) 00]. (15)

20 Unstable particles Figure: Support of spectral density function. The tensor c µν is taken diagonal, traceless and rotationally symmetric: c 00 = 3c and c ii = c, with c = 1/10;m 1 = m 2 1. We assume the interaction potential V(φ 1,φ 2 ) λφ 2 1 φ 2 with weak coupling which can induce Cherenkov-type decay of the (type-1) particle into the two-particle (type-1 + type-2) state p 1, p 2. See also V.A. Kostelecky and R. Lehnert, PRD 63, (2001)

21 Fermion field action Second example: action with Dirac fermion and scalar field: S 3 = d 4 x( ψ(i µ γ µ +ic µν γ µ ν m)ψ Fermion two-point amplitude S(x y) = 0 ψ α (x) ψ β (y) ( µφ) 2 1 ) 2 (m ) 2 φ 2 +λφ ψψ = e ipn (x y) 0 ψ α (0) n n ψ β (0) 0. n = ds d 4 pθ(p 0 )δ(p 2 s)e ip (x y) (16) n δ (4) (p p n ) 0 ψ α (0) n n ψ β (0) 0. (17)

22 The quantity Fermion spectral density Q αβ n δ (4) (p p n ) 0 ψ α (0) n n ψ β (0) 0 (18) transforms as an (observer) Lorentz scalar built out of η µν, p µ, c µν and γ µ ; satisfies γ 0 Q γ 0 = Q. It follows that: Q = (2π) 3[ ρ ] (c k ) p γ ρ k+1 k=0 with (c k ) p γ = γ c k [ p and ρ k ρ k p 2 ;cp,(c p 2 ) p p(c 3 ) p ] p (19)

23 Fermion two-point function Time-ordered product: S F (x y) αβ = 0 Tψ α (x) ψ β (y) 0 A careful analysis shows: S F (x y) = = θ(x 0 y 0 )S(x y) αβ θ(y 0 x 0 ) S(x y) αβ (20) ds d 4 p i.e ip (x y) ( (2π) 4 p 2 ρ 0 + s +iǫ 3 (c k ) p γ ρ k+1 k=0 ). (21)

24 One-particle pole Suppose spectrum includes a stable one-particle state, then: [ ρ k s;(c j ) p ] [ [ ] p = ρ 1-part k s;(c k ) p p ]+σ k s;(c k ) p p. Analogously to the scalar case [ ρ 1-part k s;(c i ) p ] [ p = fk (c i ) p ] ( [ p δ s g (c i ) p ]) p (22) defines mass-shell condition. One can expand: [ f k (c i ) p ] p = Zk (1+ r ) f k,i1...i r (c i j ) p p 1 i 1...i r 3 j=1 (23)

25 Fermion propagator The fermion propagator S F (x y) becomes [ f d 4 p 0 (c i ) p ] 3 [ p + f k+1 (c i ) p p] (c k ) p γ i e ip (x y) k=0 (2π) 4 p 2 + r [ g i1...i r (c i j ) p ] p m 2 +iǫ + M 2 ds 1 i 1...i r 3 j=1 [ σ 0 s;(c i ) p ] 3 [ p + σ k+1 s;(c i ) p ] p (c k ) p γ k=0 p 2 s +iǫ (24)

26 Equation of motion and wave-function renormalization The one-particle contribution can be cast into the form Z [ (c i ) p ] p /p m [ (c i ) p [ p] + x1 (c i ) p p] c p [ γ + x 2 (c i ) p p] (c 2 ) p γ +... It follows that, once again, the equation of motion satisfied by external states includes higher-dimensional terms not present in the original Lagrangian, and the wave-function renormalization constant Z becomes a momentum-dependent quantity. One can also derive sum rules for the Z k.

27 Model Lagrangian Lagrange density for QED within the minimal SME: L SME = 1 2 i ψγµ Dµψ ψmψ 1 4 F2 1 4 (k F) µνρσ F µν F ρσ +(k AF ) µ A ν Fµν. Here D µ = µ +iea µ, F µν = 1 2 εµνρσ F ρσ and Γ µ = γ µ +c µν γ ν +d µν γ 5 γ ν +if µ gλνµ σ λν +e µ, M = m+a µ γ µ +b µ γ 5 γ µ Hµν σ µν. We simplify: d µν = f µ = g λνµ = e µ = a µ = b µ = H µν = (k AF ) µ = 0 as well as (k F ) µνρσ = 1 2 (ηµρ kνσ η νρ kµσ η µσ kνρ +η νσ kµρ ),

28 Model Lagrangian (2) 2c k In a fermion-photon system only difference 2c µν k µν can be measured at leading order physically observable radiative effects should only depend on 2c µν k µν! Yields independent check on consistency. IR divergences Introduce photon mass to regulate IR divergences. Use modified Stueckelberg procedure for consistent quantization: with η µν = η µν + k µν. L = 1 2 ξ 1 ( µ η µν A ν ) m2 γ A µ η µν A ν,

29 Stueckelberg Lagrangian Introduce a (Stueckelberg) scalar field with lagrangian term δl m = 1 2 ( µφ m γ A µ ) η µν ( ν φ m γ A ν ). which is invariant under δa µ = µ ǫ(x), δφ = m γ ǫ(x) For fixing the gauge, take L g.f. = 1 2ξ ( µ η µν A ν +ξm γ φ) 2.

30 Stueckelberg Lagrangian We also need a Faddeev-Popov ghost term: L F.P. = c( µ η µν ν +ξmγ)c 2, Total photon lagrangian: L = 1 4 F m2 γ A µ η µν A ν 1 2ξ ( µ η µν A ν ) φ( µ η µν ν +ξmγ)φ c( 2 µ η µν ν +ξmγ)c ηκµ η λν F κλ F µν. No φ-a mixing! The fields φ and c, c decouple.

31 Renormalization Quantization: Loops yield UV divergences. Use dimensional regularization Need to do add divergent counterterms to lagrangian Divergences can be absorbed in Lagrangian parameters/fields ψ ψ B, A µ A µ B, m m B, e e B, c µν c µν B, kµν k µν B

32 Bare Lagrange density: Bare lagrangian L = ψ [ ( B i γ µ +c µν B γ ν)( µ +ie B A B ] µ) mb ψb 1 4 ηµν B ηαβ B FB µαfνβ B 1 ( 2ξ µ η µν ) 2 B AB ν + m2 γ 2 AB µ η µν B AB ν. Multiplicative renormalization: ψ B = Z ψ ψ, A µ B = Z A A µ, m B = Z m m, e B = Z e e, c µν B = (Z c) µν αβc αβ (Z c c) µν, k µν B = (Z k) µν αβ k αβ (Z k k) µν.

33 Renormalization functions Adopting Feynman gauge (ξ = 1), 4 ǫ dimensions, minimal subtraction, at one-loop order: Z m = 1 3e2 8π 2 ǫ, Z e = 1+ e2 12π 2 ǫ, Z ψ = 1 e2 8π 2 ǫ Z A = 1 e2 6π 2 ǫ, (Z cc) µν = c µν + e2 ) (2c µν 6π 2 k µν ǫ (Z k k) µν = k µν + e2 6π 2 ǫ ( kµν ) 2c µν. Compatible with the usual QED Ward Takahashi identity Z e ZA = 1.

34 Final model Lagrangian Lagrange density becomes: [ ) L c k = Z ψ ψ i (γ µ +(Z c c) µν γ ν ) ] ( µ +iz e ZA ea µ Z m m ψ Z A 4 Z A 2 ( η µν +(Z k k) µν)( η αβ +(Z k k) αβ) F µα F νβ ( (η µν µν +(Z ) ) 2 k k) µ A ν + Z Amγ 2 A µ (η ) µν µν +(Z k k) A ν. 2

35 Perturbation scheme Perturbation scheme: two alternatives A. Conventional perturbation scheme: k µν insertions (c µν = 0). + B. Resummed perturbation scheme including all tree-level k effects.

36 Two perturbation schemes Scheme A: energy-momentum kinematics same as Lorentz-symmetric case Scheme B: modified energy-momentum cinematics internal photon and fermion lines could go onshell simultaneously at ultra-high momenta: vacuum Cherenkov radiation diagrams in 1-1 correspondence with Lorentz-symmetric case Disregarding instabilities, two schemes are equivalent Scheme B computationally simpler

37 Perturbation scheme B where L c k = L 0 +L 1 +L 2, L 0 = ψ[i (γ µ +c µν γ ν ) µ m]ψ 1 ( η µν + 4 k µν)( η αβ + k αβ) F µα F νβ 1 ( µ A µ + 2 k ) 2 µν m 2 ( γ µ A ν + 2 A µ η µν + k µν) A ν, L 1 = ψ[ e(γ µ +c µν γ ν )A µ ]ψ,

38 and 1 4 Perturbation scheme B (cont) ψ[ ((Zψ L 2 = 1)η µν +Z ψ (Z c c) µν c µν) iγ ν ( µ +iea µ ) ] (Z ψ Z m 1)m ψ [ ( (Z A 1)η µν η αβ +2 Z A (Z k k) µν k µν) η αβ )] + (Z A (Z k k) µν (Z k k) αβ k µν kαβ (F µα F νβ +2( µ A ν )( α A β )) + m2 [ γ 2 A µ (Z A 1)η µν +Z A (Z k k) µν k µν] A ν.

39 Feynman rules scheme B p = i( /p +c p γ +m) p 2 +2c p p m 2, µ ν q = i(ηµν k µν ) q 2 + k q q m 2 γ, µ = ie(γ µ +c µν γ ν ),

40 with Here: Two-point function Γ (2) (p) = /p +c p γ m Σ(pµ ). Σ = Σ LI (/p)+σ LV (p 2,c p γ, k p γ )+δσ(pµ,c µν, k µν ). Σ LI (/p) = f 0 (p 2 )m+f 1 (p 2 )/p, Σ LV (p 2,c p γ, k p γ ) = f c 2 (p2 )c p γ +f k 2 (p 2 ) k p γ, δσ(p µ,cp p, k p p ) = f 3 c (p2 ) cp p pc m +fc 4 (p2 ) / p p m 2 +f k 3 (p 2 ) k p p m +f k 4 (p 2 ) / p k p p m 2.

41 Two-point function (2) General structure of Γ (2) (p): Γ (2) = A(p 2,c p p, k p p)/p +C(p 2 )c p γ +K(p 2 ) k p γ M(p 2,c p p, k p p). At leading order in Lorentz violation: A = 1 f 1 (p 2 ) f c 4 (p 2 ) cp p C = 1 f c 2 (p2 ), K = f k 2 (p 2 ), m 2 f k 4 (p 2 p ) k p m 2, M = m [ 1+f 0 (p 2 ) ] +f c 3 (p2 ) cp p m +f k 3 (p 2 ) k p p m.

42 Extraction of the pole Inverse momentum space two-point function Γ (2) (p) 1 = Z R P 1 R. with P: Dirac operator with all LV effects; Z R normalization function; R: general term, regular at P = 0. Most general form for P: P = /p m+ xc γ p +ȳ k γ p. m, x and p: scalar functions that depend on c p p, k p p, but not on p 2.

43 Extraction of the pole Expanding Γ (2) (p) around P = 0 one finds at first order in LV x = C(m2 ph ) A(m 2 ph ), ȳ = K(m2 ph ) A(m 2 ph ) m = m ph +m c c p p +m k k p p, where m ph, m c and m k are fixed by 0 = m ph A(m 2 ph ) M(m2 ph ), 0 = c p p [ ma( β,c p p, k p p ) M( β,c p p, k p p )] c p p, k p p=0, 0 = k p p[ ma( β,c p p, k p p) M( β,c p p, k p p) ] c p p, k p p=0. and β = m 2 2 xc p p 2ȳ k p p.

44 Extraction of the pole This yields explicit expressions for Z R and R, explicitly Z 1 R = A( β,c p p, k p p)+2 m [ ma ( β,c p p, k p p) M ( β,c p p, k p p) ] +2 [ C (m 2 ph ) xa (m 2 ph )] c p p +2 [ K (m 2 ph ) ȳa (m 2 ph )] k p p.

45 Feynman diagram Single one-loop Feynman diagram for fermion propagator: p k iσ loop (p) = p k will determine functions f 0 (p 2 ), f 1 (p 2 ), f2 c(p2 ), f k 2 (p 2 ), f3 c(p2 ), f4 c(p2 ), f k 3 (p 2 ), and f k 4 (p 2 ).

46 One loop corrections We only need their values and lowest derivatives at p 2 = m 2. In particular: x = 1+f 1 (m 2 ) f c 2 (m 2 ) ȳ = f k 2 (m 2 ) m ph = m [ 1+f 0 (m 2 )+f 1 (m 2 ) ] m c = 2m [ f 0 (m2 )+f m k = 1 m [ f k 3 (m 2 )+f k 4 (m 2 ) ] 1 (m2 ) ] + 1 m [ f c 3 (m 2 )+f c 4 (m2 ) ]

47 One loop corrections and for the wave-function renormalization at one-loop order: Z 1 R = 1 f 1(m 2 ) 2m 2[ ] f 0(m 2 )+f 1(m 2 ) [ +2cp p 2f 1(m 2 )+2m 2 f 0 (m 2 )+2m 2 f 1 (m 2 ) f2 c (m 2 ) f3 c (m 2 ) f4 c (m 2 ) f 4 c ] (m2 ) 2m 2 [ 2 k p p f k 2 (m 2 )+f k 3 (m 2 )+f k 4 (m 2 )+ f k 4 (m 2 ] ) 2m 2 ).

48 One-loop calculation yields: x = 1+2 α π ȳ = α π One loop corrections [ 29 m ph = m+m α π m c = 2α 3πm, [ γ E ( m 2 3 ln 4πµ 2 ( )] m γ E ln [ 1 3γ E ln m k = α 3πm. )], 4πµ 2, ( )] m 2 4πµ 2, Note: infrared divergences cancel in one-loop corrections of LV parameters.

49 One loop corrections and the wave-function renormalization: Z 1 R = 1 α [ ( ) m ln 1+ γ E π m γ ( )] m 2 4 ln 4πµ 2 2α 3πm 2(2cp p k p). p Note: infrared divergence only survives in Lorentz-symmetric term. This divergence cancels in the final scattering cross section with the contribution of soft photons, just like in the Lorentz symmetric case.

50 Physical c coefficient Loop-corrected physical c coefficient: c µν ph = xcµν +ȳ k µν = c µν α [ ( )] 29 m 2 3π 12 γ E ln 4πµ 2 (2c µν k µν ) Loop-corrected Dirac operator can be written as while P 1 = /p +(c ph ) p γ m ph + α [ 2(cph ) p p 3πm ( k ph ) p ] p, P tree (p) = /p +c p γ m.

51 Scattering amplitude built from: Feynman rules a momentum-conserving delta-function; the truncated Green s function; a momentum-dependent wave-function renormalization factor ±iz 1 ( 2 R (c, k) p p) for every external leg; a Dirac spinor for every external leg: uin s ( p) for an incoming fermion; ūin s ( p) for an outgoing fermion; vin s ( p) for an outgoing anti-fermion; v in s ( p) for an incoming anti-fermion. External spinors satisfy modified Dirac equation defined by P 1. Nontrivial normalization conditions apply, arising from canonical quantization.

52 Thanks for your attention!