Dispersion corrections to QWEAK and PREx

Transcription

1 Dispersion corrections to QWEAK and PREx Mikhail Gorshteyn Nuclear Theory Center & Indiana University, Bloomington PREx Workshop, ECT* Trento, August 3-7, 9

2 Outline Precision electroweak physics: points of interest Elastic electron scattering in forward kinematics: nucleon and spin- nuclei Dispersion calculation of the γz box in forward regime Input for DR - model the (PV) DIS structure functions Results for QWEAK and PREx Model dependence: Isospin rotation of e.-m. data t-dependence Summary & Outlook

3 Precision electroweak physics: test the SM and/or models of nuclear structure Weak mixing angle: central role in SM W µ ± = (A µ ± iaµ ) Z µ = cos θ W A µ 3 + sin θ W B µ A µ = sin θ W A µ 3 + cos θ W B µ g = e sin θ W g = e cos θ W M W M Z = cos θ W sin θ W Weak Mixing Angle Scale dependence in MS scheme including higher orders current future SM APV Q W (p) Q W (e) ed-dis ν-dis Z-pole A FB sin θ W (MZ) =.33 ±.5 PDG sin θ W () =.387 ±.7 Erler et al Q [GeV] d running of the weak mixing angle in the Standard Model, as de Weak charge of the proton in SM Q p W () = 4 sin θ W.5 QWEAK: 4% measurement of proton s weak charge -.3% determination of the weak mixing angle To reduce uncertainties due to hadronic structure effects: small Q² <.5 GeV², beam energy ~ GeV PREx: the same observable, similar kinematics, similar precision; Target = Pb-8: -% measurement of the neutron skin - possible implications for physics of neutron stars

4 Elastic electron scattering in forward direction s = (P + K) u t s+u+t e (k) e (k ) N(p) N(p ) = (P K) = = Q = M m M PK s M + / ν= = M M p + p! k + k! P =, K=, Elastic electron-proton scattering amplitude (massless electron) = k k! = p! p = 6/(e helicity cons.) = 8 (T reversal) = 6 T = fi = fi (ν, t)! " e e α!! µ µα u (k )γµ u(k)n (p ) f γ + f iσ N (p) + f3 u (k! )γµ γ5 u(k)n (p! )γ µ γ5 N (p) t M t! " GF G u (k! )γµ γ5 u(k)n (p! ) f4 γ µ + f5 iσ µα α N (p) F f6 u (k! )γµ u(k)n (p! )γ µ γ5 N (p) M PC PV Near forward direction: further reduction! " µ P α N γ µ N = N + iσ µα N P µ M M N γ µ γ5 N = P µ λz (p) f3 = f3γγ + f3z = O(t)

5 Elastic electron scattering in forward direction Polarized forward electron scattering T (t ) = e t f ū(k )P/ u(k) G F f4 ū(k )P/ γ 5 u(k) + O(t) fi (ν) = f i (ν, t = ) Precisely the same result as for a spin- target (effects of nucleon spin are of order ~t) Unpolarized cross section ( dσ dω lab ) = 4α cos θ t E 3 E f ( + O(t)) Parity violating asymmetry A P V (t ) = σ R σ L σ R + σ L = G F t 4πα Re( f f 4 ) f One-Boson-Exchange Including radiative corrections f () = Z f () 4 = ga[ N e + Z Q p W ] f = f 4 = f () + O(αt) () f 4 [ + δ RC (ν)] + O(αt, G F,... )

6 Elastic electron scattering in forward direction Proton A P V QW EAK = G F t 4πα Qp EAK W [ + ReδQW RC (ν)] + O(t ) SM value Measure deviation from (to -%) Lead: Z=8, N=6 P REx = G F t 4πα A P V [ ] ( N F W Z C (t) Qp W F C (t) ) REx + ReδP RC (ν) + O(t ) For both: theory uncertainties cannot exceed % to keep systematics under control - calculate radiative corrections to % precision

7 Radiative corrections Account for all radiative corrections at relative order QED, soft photons + hadronic FF s QED, soft photons + hadronic FF s Can be constrained using e+e hadrons data Represent substantial theoretical difficulty: non-local product of electroweak currents

8 Radiative corrections Existing estimates: Marciano, Sirlin 83,84 (APV) - Ramsey-Musolf 99 (proton) k l d 4 l... (l M W,Z ) WW, ZZ-box: dominated by large momenta in the loop δ W W = δ ZZ = 7α 4π sin θ W Q p W α 4π sin θ W cos θ W Q p W Purely short-range contribution - calculable perturbatively ~ 6% correction to QWEAK [ ] sin θ W ( 4 sin θ W + 8 sin 4 θ W ) ~ 3% correction to QWEAK

9 Radiative corrections k l γz-box d 4 l... l l M Z no reason to be dominated by large loop momenta Marciano&Sirlin: for APV - cancellation between the box and crossed ū[γ α (k/ + l/ )γ β (g e V + g e Aγ 5 ) + γ β (g e V + g e Aγ 5 )(k/ l/ )γ α ]u T αβ D(l ) iɛ αλβµ l λ ūγ µ (gv e γ 5 + ga)u e T αβ D(l ) + O(k) Large axial term - exact cancellation for l k δ MS γz = 5α π [ ln M Z Λ ] + C γz(λ) 6%.65% QWEAK: E ~ GeV - is it still small? Provide a dispersion calculation to investigate the energy dependence

10 Forward dispersion relations for δᵧz Imaginary part - box only ImT γz = G F e (π) 3 d 3 k E l µν W µν Q ( + Q /M Z ) Leptonic tensor Hadronic tensor - from optical theorem l µν = ū(k )γ ν k/ γ µ (g e V + g e Aγ 5 )u(k) W µν = π { g µν F + P µ P ν P q F + iɛ µναβ P } αq β P q F 3 PV DIS structure functions Imδ γz (ν) = α 4Q p W s W π dw s M Q max dq + Q M Z { ga e P q [ ( P q P k F P k + Q P ) P k ] F gv e (P, k + k ) P q P k F 3 } δ γza δ γzv Analyticity + Crossing -> Dispersion Relations Re δ γz (ν) = π [ dν ν π ν ν ± ] ν Im s δ γz (ν ) ν

11 Forward dispersion relations for δᵧz Crossing: partial CP-transformation e (k) e (k ) CP/ e+(-k ) e+(-k) p p p p C-parity: l µν γz, direct J µ em (J ν V + J ν A) l µν γz, crossed ( J µ em) ( J ν V + J ν A) CP is conserved - different parity behavior for (em,v) and (em,a) parts Two sum rules for the two dispersion corrections Reδ γza (ν) = ν π ν π dν ν ν Imδ γz A (ν ) Reδ γzv (ν) = π ν π ν dν ν ν Imδ γz V (ν ) Reδ γza () = Marciano&Sirlin mechanism for APV: Reδ γzv α ge V Q p W ln M Z Λ Finite energy: these results just indicate the range of convergence of DR

12 Forward dispersion relations for δᵧz Vector NC DIS structure functions over a wide range in Q² and W² - no data available (yet) W µν γz N T [J µ NC V J ν em] N W µν γγ N T [J µ emj ν em] N Vector x Vector - should be similar SM: Photon and Z are related by an isospin rotation - try to isospin-rotate the e.-m. data J em µ = ξ I= J µ I= + ξi= J µ I= + ξs J s µ J µ NC V = ξv I= J µ I= + ξi= V J µ I= + ξs V J s µ ξ I= = 3, ξi= =, ξ s = 3 ξv I= = 4 3 s θ W, ξv I= = ( 4s θ W ), ξv s = s θ W J µ emj ν em = [J µ I= J ν I= + 9 J µ I= J ν I= + 9 J µ s J ν s ] J µ NC V J ν em [J µ I= J ν I= 9 J µ I= J ν I= J µ s J ν s ] End of the story - if quarks were free particles... We do not know the isospin structure of the data in the general case W µν γγ = (π) 3 δ(p + q p X ) X N J µ em X X J ν em N X contains all possible Fock states: X=N, πn, N*, Δ, ππn, VN, qx, qq X etc. The main difficulty: each contribution has to be isospin-rotated individually

13 Modeling e.-m. data: real photons Model e.-m. DIS structure functions first σ T = 4π α P q F σ L = 4π α P q Real photons [( x + M ) ] F F P q Total cross sections: resonance + Regge Match the two components to fit the real photon data σ γp (W, ) = R σ R Γ R Γ γ R M R (W M R ) + M R Γ R + σ Regge γp (W ) Bianchi et al., PRC 96 R = P33(3), D3(55), P(44), S(535), F5(68), F37(95).5 "!" - world data (PDG) Resonance + Regge Regge Fit to high energy data: σ Regge γp (W ) = 45µb (W ) / µb (W ).97 "!" (mb) ! (GeV)

14 Modeling e.-m. data: virtual photons Q² dependence: different for high energy and resonance contributions Resonances: transition form factors ~ dipole form with ~ GeV F T (Q ) = ( + Q /Λ ) Regge: hybrid GVD/CDP approach Cvetic et al. γ V V γ - starts with generalized VDM (DR over non-diagonal V-V) γ V V γ - employs QCD (CDP) to relate the dipole c.s. to the Vp cross section - uses geometric scaling of low-x DIS data to model the dipole cross section Connects nicely low and high Q² regimes

15 Compare models to e.-m. DIS data NMC - x=.5 E665 - x=.5 Model I Model II NMC - x=.35 E665 - x= NMC - x=.9 E665 - x= NMC - x=.75 E665 - x= NMC - x=.5 E665 - x= NMC - x=.5 E665 - x= NMC - x= NMC - x=.8.6 E665 - x= NMC - x= Fixed x NMC - x= NMC - x= Models vs. data for F₂(x,Q²). NMC - x=.7.8 E665 - x= NMC - x= NMC - x= Q² (GeV²) Model I Model II H Q!=.5 GeV! NMC Q!=.5 GeV!.4. e-5 e-4 e-3 e Q!=3.5 GeV! e-5 e-4 e-3 e-..5 Q!=8.5 GeV!.5.5 e-5 e-4 e-3 e-...8 Q!= GeV!.6.4..e-4 e-3 e Fixed Q² Q!=5 GeV! e-5 e-4 e-3 e-..5 Q!= GeV!.5.5 e-5 e-4 e-3 e-. x Model I: GVD/CDP (Cvetic et al. EPJ ) Model II: naive VDM+continuum (Alwall, Ingelman PLB 4).4..8 Q!=.5 GeV!.6.4. e-5 e-4 e-3 e Q!=6.5 GeV! e-4 e-3 e-.

16 Dispersion relations for δᵧz - results for the proton 4 Imaginary part Reδ γza (ν) = ν π ν π dν ν ν Imδ γz A (ν ) Im!$z (%) Dispersion correction to QWEAK ~ 6% Full Resonances Regge 5 5 E!"# (GeV) Resonance Regge Full Re!!z (%) Elab (GeV) MG & CJ Horowitz, PRL 9 The dispersion correction to PVES is energy-dependent - negligibly small for APV, not so small for QWEAK Present estimate ~6% is model-dependent, since no PV DIS data for F₁,₂ are available Has to be studied carefully to fit into.% theory uncertainty for QWEAK

17 Dispersion correction δᵧz for spin- nuclei How does the dispersion correction affect PVES on nuclear target? T (t ) = e t f ū(k )P/ u(k) G F f4 ū(k )P/ γ 5 u(k) + O(t) f OBE = Z f 4 OBE = ga e [Z Q p W N] σ γ A T,L p Aσγ T,L Naive scaling δ γz [A, N] δ p γz A ZQ p W N Qp W 6- times smaller than for the proton δ P REx γz (.85 GeV ) (.5 )% Additional uncertainties: nuclear structure, nuclear form factors, multi-photon exchange?... A A A, GDR Quasi-elastic Inelastic

18 Isospin structure - different kinematical regions Q (GeV ). RESONANCE VALENCE DIS: x >. GVDM REGGE VDM DIFFRACTIVE DIS: x <. W (GeV) Re!!z (%) Full Resonances W<.5 GeV "VDM" Q"<3 GeV" "GVD" Q"<3 GeV" Valence DIS - Q">3 GeV", x>. Diffractive DIS - Q">3 GeV", x<. Resonance: Regge/VDM: Regge/GVD: Valence DIS: Diffractive DIS: W.5 GeV, Q GeV W GeV, Q 3 GeV W GeV, Q 5 GeV x. GeV, Q 3 GeV x. GeV, Q 3 GeV Elab (GeV) These kinematics give <.7%

19 Isospin rotating data in resonance region Resonance contribution - to ~7% dominated by the Δ(3) N -> Δ(3) transition - purely isovector J µ NC p = ( 4 sin θ W )J µ em J µ I= p J µ em p Matsui,Sato,Lee, PRC 5 Second most important resonance - D₁₃(55) ~ % - contains both isoscalar and isovector pieces Once Δ(3) contribution is fixed - one can aim at ~3% uncertainty due to other resonances Re!!z (%) P33(3) D3(55) F5(68) P(44) S(535) F37(95) Elab (GeV) would translate into ~.6 % uncertainty in δᵧz from resonance region Different models for Δ exist - compare all of them to have an idea of model dependence Blunden, Melnitchouk [arxiv:93.759]; Nagata et al. [arxiv:8.3539];...

20 Isospin rotating - VDM regime VDM sum rule: σ tot (γp) = HERA: NPB V =ρ,ω,φ 6π 4πα f V dσ γp V p dt (t = ) ZEUS: Z.Phys. 95, 96, PLB ± 4 (µb) ± 3 (µb) at W = 7 GeV! "*p#vp /! "*p#$p ZEUS VDM sum rule: obeyed to ~ 8% (65% - ρ).6 Exp. data on V production with real and virtual γ s.5 σ γ p ωp (ei= ) σ γ p ρp (e I= ) = 9.4 σ γ p φp σ γ p ρp (e s ) (e I= ) = 9.3. Exact SU(3) - constant ratio for any Q². σ γ ρ : σ γ ω : σ γ φ : 9 : 9 σ γ Z ρ : σ γ Z ω : σ γ Z φ : 9 4s θ W 4s : 3 4s θ W θ W 9 4s θ W Q (GeV ) Work in progress - MJ Ramsey-Musolf, CJH, MG Breitweig et al., [ZEUS]

21 Additional uncertainties: t-dependence Optical theorem - in exact forward direction only To go to finite t - correct the input for off-forward kinematics [ ] dσ dσ e Bt σ γa (t ) σ γa (t = )e Bt/ dt dt t= Experimental Compton slope: proton helium4 B p = 7.5 ±.5 GeV Bauer et al. 78, 79 B4 He = 3.95 ±.9 GeV Aleksanian et al., 87 Applied to analyzing power (proton and He-4 data available) - works quite well Use the same model for the t-dependence of Z - interference Reδ γz (ν, t) = f (γz) 4 (ν, t) f (Z) 4 (t) Q² = Q² =.3 GeV² Q² =.5 GeV² Q² =. GeV² e B pq / F p (Q ) (Q ) % (96 ± ) % (9.5 ±.5) % (84 ± 3) % At most % relative reduction - overall <.5 % effect for QWEAK kinematics f (γz) 4 (ν, ) f (Z) 4 () e Bt/ F W (t)

22 Summary & Outlook Dispersion relation calculation for the γz-box: establish a sum rule for δᵧz Crossing behavior (C+P parity): δᵧz grows with energy (= at zero energy - APV) The size of the dispersion correction is larger than it was thought QWEAK: ~6% PREx: < % How precise can we calculate it? estimates with e.-m. DIS structure functions available isospin structure: isovector dominance main contribution - from shadow region - VDM domain The uncertainty on δᵧz can be constrained by careful isospin analysis - work in progress t-dependence: can be responsible for up to % reduction of dispersion corrections at forward kinematics; can be checked by measuring analyzing power in the same kinematics Any detailed PV DIS data for F₁,₂ are welcome - once available will serve an ultimate cross-check for the model for PVDIS

23 Vector analyzing power (Mott asymmetry, Beam normal spin asymmetry) A n = σ + σ ImT γγtγ σ + + σ T γ ImT γγ = e 4 (π) 3 d 3 k E Q lµνw γγ Q γγ µν An is zero in forward direction A inelast n 4π m e M A e BQ / s E lab Z F C (Q ) tan θ cm Elab dωωσ γp (ω) ln [ Q m e ( ) ] Elab ω Afanasev, Merenkov 4; Gorchtein 6, 7 - for the proton target

24 Vector analyzing power on spin- target High energies: cross section is roughly constant A inelast n A e B X Q n F X (Q ) tan θ cm A n = m ee lab σ R 8π M s A Z [ln Q m e ] { 4ppm, 8 P b and E lab = 85MeV 7ppm, 4 He and E lab = 3GeV σ R µbarn MG and CJH 8 4 He, E lab = 3 GeV 8 P b E = 85 MeV HAPPEX Elastic - Coulomb distortion (Chuck) Inelastic Elastic + Inelastic

25 Contributions from resonance region Resonance parameters: Resonance W r MeV r MeV I r b p d p d p n d P 33 (3) D 3 (55) F 5 (68) P (44) F 37 (95) S (535) dipole form factors P33(3) - similar to Blunden, Melnitchouk [arxiv:93.759] In QWEAK kinematics Full resonance contribution ~ % P33(3).3% D3(55).4% F5(68).6% P(44).% S(535) <.% F37(95) <.% Im!!z (%) Re!!z (%) P33(3) D3(55) F5(68) P(44) S(535) F37(95) 5 5 Elab (GeV) P33(3) D3(55) F5(68) P(44) S(535) F37(95) Elab (GeV)

26 Two sum rules for the two dispersion corrections Reδ γza (ν) = αge A ν πq p W ν π dν ν ν s W π dw s M Q max dq + Q M Z P q [ ( P q P k F P k + Q P ) P k F ] Convergence at energies much below Mz - purely hadronic information Reδ γzv (ν) = αge V πq p W ν π ν dν ν ν s W π dw s M Q max dq + Q M Z (P, k + k ) P q P k F 3 Convergence at energies ~ Mz - gives rise to the large log; perturbative estimates exist

27 Input to dispersion relations for δᵧz Isospin-rotate the e.-m. data to obtain the PV data J em µ = ξ I= J µ I= + ξi= J µ I= + ξs J s µ J µ NC V = ξv I= J µ I= + ξi= V J µ I= + ξs V J s µ J µ I= = (ūγ µ u + dγ µ d) J µ I= = (ūγ µ u dγ µ d) J µ s = sγ µ s ξ I= = 3, ξi= =, ξ s = 3 ξ I= V = 4 3 s θ W, ξ I= V = ( 4s θ W ), ξ s V = s θ W J µ emj ν em = (ξ I= ) J µ I= J ν I= + (ξ I= ) J µ I= J ν I= + (ξ s ) J µ s J ν s J µ NC V Jem ν = (ξ I= ξv I= )J µ I= J I= ν + (ξ I= ξv I= )J µ I= J I= ν + (ξ s ξv s )J s µ Js ν We do not know the isospin structure of the data in the general case! W µν γγ = (π) 3 δ(p + q p X ) X N J µ em X X J ν em N X contains all possible Fock states: X=N, πn, N*, Δ, ππn, VN, qx, qq X etc. The main difficulty: each contribution has to be isospin-rotated individually

28 Modeling e.-m. data: virtual photons Q² dependence: different for high energy and resonance contributions Resonances: transition form factors ~ dipole form with ~ GeV F T (Q ) = ( + Q /Λ ) Q² dependence of the Regge part: VDM σ T = A γ [ V σ L = A γ [ V Alwall, Ingelman 4 ( ) m ] r V m V Q + m + V Q + m r C Q ( ) m ( ) m ξ V r V m V V Q + m + ( V Q ln + Q m ) ] m Q + m ξ C r C Works well at low Q² only ξ = σ L σ T A γ = 45µb (W ) / µb (W ).97 Alternatively: hybrid GVD/CDP approach - starts with the GVD (DR over non-diagonal V-V) - employs QCD (CDP) to relate the dipole c.s. to the Vp cross section - uses geometric scaling of low-x DIS data to model the dipole cross section Cvetic et al. γ V V γ γ V V γ Connects nicely low and high Q² regimes

29 Isospin content: real photons Δ(3) resonance - isovector transition High energy: Naïve Vector Meson Dominance (VDM) γ V V γ γ = V C γ V V σ γp tot = V =ρ,ω,φ V= ϕ 4πα Elastic Vp cross section ~ independent of V σ V p f V Vector meson decay constants - known 4π f V N =.4545,.437,.5435 (ρ, ω, φ) Dominated (to ~ 8-85%) by the isovector channel N VDM (Stodolsky) sum rule: HERA: NPB σ tot (γp) = V =ρ,ω,φ 6π 4πα f V dσ γp V p (t = ) dt ZEUS: Z.Phys. 95, 96, PLB ± 4 (µb) ± 3 (µb) at W = 7 GeV VDM sum rule: obeyed to ~ 8% Naïve generalization of VDM - continuum contribution γ V V σ γp tot = V =ρ,ω,φ 4πα fv σ V p + σ Cp ~ % - unknown isospin content γ