Dispersion Theory of the W-Box For free and bound neutron -decay - PowerPoint PPT Presentation

Dispersion Theory of the γ W-Box For free and bound neutron β -decay Misha Gorshteyn Johannes Gutenberg-Universität Mainz Collaborators: Based on 3 papers: Chien-Yeah Seng (U. Bonn) arXiv: 1807.10197 Hiren Patel (UC Santa Cruz) arXiv: 1812.03352 Michael Ramsey-Musolf (UMass) arXiv: 1812.04229

Outline Superallowed nuclear decays: Free neutron decay: γ− physics at hadronic scale ν 5099 . 34 s e : | V ud | 2 = 2984 . 43 s : | V ud | 2 = q q ⌧ n (1 + 3 � 2 )(1 + ∆ R ) W γ F t (1 + ∆ V R ) ⋅ ν = n (“m.d”: model-dependent) is: ν − ν − ( ) ∫ = π ( ) F t = ft (1 + δ 0 R )[1 − ( δ C − δ NS )] π − ν ε µναβ ( ) ⋅ ∫ α β µ ν = ν π ν 1. Dispersion formalism for the 𝛿 W-box 2. Calculation of the universal free-neutron RC Δ RV 3. Splitting the full nuclear RC into free-neutron Δ RV and nuclear modification δ NS 4. Splitting the full RC into “outer” and “inner” � 2

1. 𝛿 W-box from dispersion relations � 3

⃗ 𝛿 W-box γ− physics at hadronic scale ν e q q Box at zero momentum transfer * (but with energy dependence) γ W ⋅ ν = n p d 4 q M 2 u e � µ ( k / + m e ) � ν (1 � � 5 ) v ν Z ¯ / � q (“m.d”: model-dependent) is: 2 e 2 G F V ud W T γ W T γ W = µ ν , q 2 [( k � q ) 2 � m 2 q 2 � M 2 ν − ν − (2 ⇡ ) 4 ( ) e ] ∫ = π ( ) W π − ν * Precision goal: 10 -4 ; RC ~ 𝛽 /2 𝜌 ~ 10 -3 ; recoil on top - negligible γ W = ∫ dxe iqx ⟨ f | T [ J μ µναβ ε ( ) ∫ ⋅ α β µ ν = ν em ( x ) J ν ,± T μν π ν W (0)] | i ⟩ Hadronic tensor: two-current correlator General gauge-invariant decomposition of a spin-independent tensor ◆ µ ✓ ◆ ν � g µ ν + q µ q ν T 2 + i ✏ µ ναβ p α q β ✓ ◆ ✓ 1 p � ( p · q ) p � ( p · q ) T µ ν γ W = T 1 + q q T 3 q 2 q 2 q 2 ( p · q ) 2( p · q ) Loop integral with generally unknown forward amplitudes p μ = ( M , 0 ) d 4 qM 2 T γ W = − α 2 π G F V ud ∫ W u e γ β (1 − γ 5 ) u ν ∑ C β i ( E , ν , q 2 ) T γ W ( ν , q 2 ) W − q 2 ) ¯ E = ( pk )/ M i q 2 ( M 2 i ν = ( pq )/ M Known algebraic functions of external energy E and loop variables 𝜉 , q 2 � 4

𝛿 W-box from Dispersion Relations γ γ T 1,2,3 - analytic functions inside the contour C in the complex ν -plane determined by their singularities on the real axis - poles + cuts 2 π i ∮ dz T γ W ( z , Q 2 ) ( ν , Q 2 ) = 1 i T γ W , ν ∈ C i z − ν ν γ γ W W Forward amplitudes T i - unknown; q q q q ν = π ν Their absorptive parts can be related to ε µναβ ( ) production of on-shell intermediate states ∑ µ ν α β π δ + − = ν π ν —> a 𝛿 W-analog of structure functions F 1,2,3 p p p p X X = inclusive strongly-interacting on-shell physical states ν ν = π ν Structure functions F i 𝛿 W are NOT data Im T γ W ( ν , Q 2 ) = 2 π F γ W ( ν , Q 2 ) µναβ ε ( ) ∑ α β π δ + − µ ν = ν i i But they can be related to data π ν � 5

𝛿 W-box from Dispersion Relations γ γ Crossing behavior: relate the left and right hand cut Mismatch between the initial and final states - asymmetric; Symmetrize - 𝛿 is a mix of I=0 and I=1 T ( I ) i ( − ν , Q 2 ) = ξ ( I ) i T ( I ) i ( ν , Q 2 ) 1 i τ a + T ( − ) T γ W , a = T (0) 2 [ τ 3 , τ a ] i i ξ (0) = + 1, ξ (0) ξ ( − ) = − ξ (0) 2,3 = − 1; 1 i i Two types of dispersion relations for scalar amplitudes ν ν = π ν d ν ′ � [ ν ′ � − ν − i ϵ ] F ( I ) ∞ ξ ( I ) i ( ν , Q 2 ) = 2 ∫ 1 i T ( I ) i ( ν ′ � , Q 2 ) ε µναβ ( ) ν ′ � − ν − i ϵ + ∑ µ ν α β π δ + − = ν π ν 0 Substitute into the loop and calculate leading energy dependence ∞ ∞ dQ 2 M 2 W + Q 2 ∫ π MN ∫ α d ν ν + 2 q W ( ν + q ) 2 F (0) Re □ even 3 ( ν , Q 2 ) + O ( E 2 ) γ W = M 2 ν 0 0 ◆ M ∞ ∞ 8 α E  ✓ 3 ν ( ν + q ) + ν + 3 q � Z Z d ν ⌥ F (0) ν F (0) F ( − ) dQ 2 + O ( E 3 ) Re ⇤ odd γ W ( E ) = + 1 ⌥ 1 2 3 ( ν + q ) 3 2 Q 2 3 π NM 4 ν 0 ν thr ν 2 + Q 2 q = � 6

2. Universal inner RC Δ RV � 7

Inner universal RC from DR ∞ ∞ dQ 2 M 2 W + Q 2 ∫ γ W = α π M ∫ d ν ν + 2 q W ( ν + q ) 2 F (0) Re □ even 3 ( ν , Q 2 ) 𝛿 W-box at zero energy M 2 ν 0 0 Re □ odd γ W ( E = 0) = 0 Connection to MS: rewrite in terms of the first Nachtmann moment of F 3 1 + 4 M 2 x 2 / Q 2 1 + 2 x = Q 2 1 3 (1, Q 2 ) = 4 3 ∫ M (0) 1 + 4 M 2 x 2 / Q 2 ) 2 F (0) 3 ( x , Q 2 ) dx 2 M ν (1 + 0 ∞ dQ 2 M 2 ∞ dQ 2 M 2 γ W = 3 α 2 π ∫ 3 (1, Q 2 ) = α 8 π ∫ W W W + Q 2 ) M (0) Re □ even W + Q 2 F MS ( Q 2 ) Q 2 ( M 2 M 2 0 0 F MS ( Q 2 ) = 12 Q 2 M (0) 3 (1, Q 2 ) MS loop fn. F(Q 2 ) directly related to M 3(0) SF F 3 - commutator of em and weak currents - insert complete set of on-shell hadronic states ∝ ∫ dxe iqx ⟨ p | [ J μ ,(0) W (0)] | n ⟩ ∼ ∫ dxe iqx ∑ F (0) ⟨ p | J μ ,(0) em ( x ), J ν ,+ em ( x ) | X ⟩⟨ X | J ν ,+ W (0) | n ⟩ 3 X � 8

Input into dispersion integral γ γ W 2 = M 2 + 2 M ν − Q 2 Dispersion in energy: scanning hadronic intermediate states Dispersion in Q 2 : ν scanning dominant physics pictures ν = π ν ε µναβ ( ) ∑ 2 α β Q π δ + − µ ν = ν π ν Parton + pQCD 2 Boundaries between regions - approximate ~ 2 GeV Input to DR related (directly or indirectly) Born Res. Regge N π to experimentally accessible data +B.G +VMD 2 W 2 2 2 M ~ 5 GeV ( ) M + m π � 9

Input into dispersion integral 2 Q Our parametrization of the needed SF follows from this diagram Parton + pQCD 2 ~ 2 GeV 8 Q 2 & 2 GeV 2 F pQCD , Born Res. Regge N π < F (0) 3 = F Born + +B.G +VMD F π N + F res + F R , Q 2 . 2 GeV 2 : 2 W 2 2 2 M ~ 5 GeV ( M + m ) π Born: elastic FF from e - , ν scattering data 4 M 2 + Q 2 + Q Z ∞ p dQ 2 = � α ⇤ V A, Born ⌘ 2 G A ( Q 2 ) G S M ( Q 2 ) γ W π ⇣p 4 M 2 + Q 2 + Q 0 π N: relativistic ChPT calculation plus nucleon FF Resonances: axial excitation from PCAC (Lalakulich et al 2006) - neutrino scattering isoscalar photo-excitation from MAID and PDG - electron and γ inelastic scattering Above resonance region: multiparticle continuum economically described by Regge exchanges � 10

Inelastic states - low Q 2 , high W Scattering at high energy can be very effectively described by Regge exchanges ✓ ν ◆ α ρ F (0) , Regge ( ν , Q 2 ) = C R ( Q 2 ) 3 ν 0 Regge behavior in EW processes: hadron-like behavior of HE electroweak probes - Vector/Axial Vector Dominance is the proper language γ W-box: conversion of W ± (charged, I=1, axial) to γ (neutral, vector, I=0) requires charged vector exchange w. I=1 - ρ ± effective a 1 - ρ - ω vertex Inclusive ν scattering: conversion of W ± (charged, I=1, axial) to W ± (charged, I=1, axial) requires neutral vector exchange w. I=0 - ω effective a 1 - ω - ρ vertex Minimal model for both reactions - check with data. VM propagators 1/(M a2 +Q 2 )/(M ρ 2 +Q 2 ) ~ 1/Q 4 , more natural for hadronic amplitudes Compare to Bill’s F(Q 2 ) ~ 1/Q 2 at high-Q 2 � 11

Input into dispersion integral F γ W (0) Unfortunately, no data can be obtained for 3 Data exist for the pure CC processes d 2 σ ν (¯ ν ) = G 2 y − y 2 F ME  ✓ 1 − y − Mxy ◆ ✓ ◆ � xy 2 F 1 + F 2 ± x F 3 dxdy 2 E 2 π + F ¯ σ ν p − σ ¯ ν p ∼ F ν p ν p = u p v ( x ) + d p v ( x ) 3 3 Z 1 dx ( u p v ( x ) + d p v ( x )) = 3 Gross-Llewellyn-Smith sum rule 0 Validate the model for CC process; apply an isospin rotation to obtain γ W F ν p +¯ 3 , low − Q 2 = F ν p +¯ + F ν p +¯ + F ν p +¯ + F ν p +¯ ν p ν p ν p ν p ν p 3 , el. 3 , π N 3 , R 3 , Regge 2 Q Low-W part of spectrum: Parton + pQCD neutrino data from MiniBooNE, Minerva, … 2 ~ 2 GeV - axial FF, resonance contributions, pi-N continuum Born Res. Regge N π +B.G +VMD High-W: Regge behavior F 3 ∼ q 𝓌 ∼ x - 𝛽 , 𝛽 ∼ 0.5-0.7 2 W 2 2 2 M ( ) ~ 5 GeV M + m π � 12

Parameters of the Regge model from neutrino scattering Low Q 2 < 0.1 GeV 2 : Born + Δ (1232) dominate 3.5 Not fitted: modern data more precise but cover only limited energy range 3 Fit driven by 4 data points between 0.2 and 2 GeV 2 2.5 GLS SR Model & Uncertainty fully specified 2 - compare M&S vs This work 1.5 WA25 CCFR BEBC/GGM-PS 1 Regge + Born + Δ M 3WW (1,Q 2 ) pQCD MS: INT + Born + Δ 0.5 Isospin symmetry 0 0.01 0.1 1 10 100 Q ² (GeV ² ) M 3 γ W (1,Q 2 ) 0.08 Total 2 ) (0) (1,Q 2 ) / (1 + Q 2 / M w No Born MS M&S: integrand discontinuous at Q 2 = 2.25 GeV 2 0.06 Log scale for x-axis: integral = surface under the curve 0.04 □ (0) γ W = 0.00324 ± 0.00018 MS Total : □ (0) γ W = 0.00379 ± 0.00010 M 3 New Total : 0.02 Uncertainty reduced by almost factor 2; ~ 3-5 sigma shift from the old value 10 ⁻⁵ 10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵ � 13 Q ² (GeV ² )

2. Nuclear structure modification of Δ RV C-Y Seng, MG, M J Ramsey-Musolf, arXiv: 1812.03352 � 14