Pinning Down the Inner Radiative Correction in Beta Decays Chien-Yeah Seng Helmh mhol oltz-Ins Insti titut tut für Strahl hlen- und und Ke Kernp nphysik and nd Bet ethe Cen enter for or Theor eoretical Physics, Universitä tät t Bonn “Current and Future Status of the First-Row CKM Unitarity” workshop, UMass Amherst, Amherst, USA. 1 17 May, 2019
Out utline 1.The Inner Radiative Correction 2.Dispersive Approach 3.First-Principle Calculation 4.Summary 2
1. T The he Inne nner Ra Radia iativ ive Co Correction
The he I Inne nner Ra Radia iativ ive Co Correctio ion Extraction of V ud ud from be beta d ta decays: • (1) Superallow owed ed be beta ta decay ay • ft values corrected by nuclear structure effects: see Misha’s talk (2) Neutr tron on be beta ta decay • “nucleus-independent” correction “outer” correction: sensitive to electron spectrum: see Leendert’s talk “Inner er r radiativ tive e correc ection tion”: the part of radiativ tive e correc ection tion ( (RC) • which is insensitive to the electron spectrum 4
The he I Inne nner Ra Radia iativ ive Co Correctio ion Main source of uncertainty in inner RC: γ W-box ox diagram • Sensitive to loop momentum q at ALL scales! The “model-dependent” piece involves the axial component of the charged weak current: 5
The he I Inne nner Ra Radia iativ ive Co Correctio ion Previous best determination: Marciano and Sirlin (M&S) S) • Marciano and Sirlin, Phys.Rev.Lett. 96 (2006) 032002 Write the RC as a single-variable integral over Q 2 , and identify the • dominant physics as a function of Q 2 . 1. Short distance: leadin ing OPE + perturbativ tive e QCD 2. Intermediate distance: VM VMD-in inspir ired ed inter erpol olatin ting function tion + 100% uncer 100% erta tainty 3. Long distance: Elastic ic contr trib ibution tion ∆ = Combined: V ( M & S ) 0 . 02361 ( 38 ) R 6
2. Dis Dispersive Ap Approach CYS, M.Gorchtein, H.H.Patel and M.J.Ramsey-Musolf, Phys.Rev.Lett. 121 (2018) no. 24, 241804 CYS, M.Gorchtein and M.J.Ramsey-Musolf, arXiv:1812.03352
Dispersiv Dis ive Ap Approach T 3 depends on virtual al i inte termed ediate e state states: theoretical modeling is less • transparent Disp sper ersi sive e tr treatm atmen ents s to box diagrams are developed since the last • ten years, relating the former to matrix elements of on on-shel ell inter termed ediat ate e state states Hadronic tensor in inclusive scattering: We need only the contribution from the iso sosc scal alar EM curren ent ( (0) • 8
Dispersiv Dis ive Ap Approach Disp sper ersi sion on r relati ation on: • Box diagrams are expressed in terms of the “First st Nac achtma mann • mo momen ment” of F 3 (0) : Central result!!! 9
Dis Dispersiv ive Ap Approach Iso sosp spin sy symmet mmetry: • where the fl flavor or-diagon onal s stru ruct cture re f funct ction ons F F 3 N N are defined through: involving the interference between the FULL electrom romagnet etic c curren rrent and the ISOVECT VECTOR a R axia ial curren ent: 10
Dis Dispersiv ive Ap Approach A “ph “phase e spa pace” e” di diagram f m for F r F 3 (0 (0) Elastic VDM Multi-Hadron States Regge 11
Dis Dispersiv ive Ap Approach Elastic: (isoscalar) magnatic Sach FF and axial FF Z.Ye, J.Arrington, R.J.Hill and G.Lee, Phys.Lett.B777,8 (2018) B.Bhattacharya, R.J.Hill and G.Paz,Phys.Rev.D84,073006 (2011) DIS: polarized Bjorken sum rule +pQCD correction (mere change of integration limit) N π+ Resonance: Negligible (Only I=1/2 intermediate states contributes) 12
Dis Dispersiv ive Ap Approach Multi ti-had hadron on s states es: Regge gge mod odel el + + VDM W W W n p N N 13 (I=1)*(I=0) (I=1)*(I=1)
Dis Dispersiv ive Ap Approach Matching the 1 st Nachtmann moment of the (I=1)*(I=1) piece to ν p/ ν bar p scattering data 14 (I=1)*(I=0) piece is then deduced using Regge model+VDM
Dis Dispersiv ive Ap Approach Significant increase in the multi-hadron contribution compare to M&S result, with reduced uncertainty : 15
Dis Dispersiv ive Ap Approach Reduced hadronic uncertainty in the determin mination tion o of V ud ud : • CYS, M.Gorchtein, H.H.Patel and M.J.Ramsey-Musolf, Phys.Rev.Lett. 121 (2018) no. 24, 241804 CYS, M.Gorchtein and M.J.Ramsey-Musolf, arXiv:1812.03352 (assume nothing else changes; using V us in PDG) Possi ssibl ble e issu ssues: • Quality of the neutrino data? • Residual model-dependence? • 16 which leads to the discussions below.
Dis Dispersiv ive Ap Approach Reduced hadronic uncertainty in the determin mination tion o of V ud ud : • CYS, M.Gorchtein, H.H.Patel and M.J.Ramsey-Musolf, Phys.Rev.Lett. 121 (2018) no. 24, 241804 CYS, M.Gorchtein and M.J.Ramsey-Musolf, arXiv:1812.03352 (assume nothing else changes; using V us in PDG) Possi ssibl ble e issu ssues: • Quality of the neutrino data? • Residual model-dependence? • 17 which leads to the discussions below.
3. F Fir irst-Princ incip iple le Ca Calc lcula ulatio ion CYS and U.G-Meissner, hep-ph/1903.07969 (to appear in PRL)
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Recall the that we are interested in as a function of Q 2 . • Neutrino data helps identifying dominant contri ributor ors a at differ eren ent Q Q 2 : Therefore, to remove the hadronic uncertainties in the box diagrams, we • need to have a good handle of the first N st Nac achtma mann mo mome ment o t of F 3 at at moder erate e Q 2 . Question: is there a way to calculate from FIR FIRST-PR PRIN INCIPLE IPLE? • 19
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Difficult because it involves a a su sum m of al all o on-sh shel ell inter termed mediate te state states. s. • Recently-developed techniques in lattice calculation of PDFs (quasi- • PDF, pseudo-PDF, lattice cross-section etc) do not apply because they rely on OPE that holds only at large Q 2 . We wish to avoi oid direc ect calculation ons of of fou four-poi oint f function tions (noisy • contractions, complicated finite-volume effect…) 20 J. Liang, K-F. Liu and Y-B. Yang, EPJ Web Conf. 175 (2018) 14014
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion A more promising approach is through the Feynman an-Hel ellma lmann • theor eorem em (F (FHT HT): Shif ift in in energy le level el matr matrix elemen ment. Extraction of energy levels on • lattice are more straightforward, avoid complicated contraction diagrams. Momen entum transfer fer could be introduced through period odic ic exter ernal l • poten tenti tial. Shows great potential in studies of: • Nucleon axial charge and sigma term • EM form factors • Compton amplitude • P-even structure functions • Chambers et al., PRL 118, 242001 (2017) Hadron resonances • …… • 21
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion A more promising approach is through the Feynman an-Hel ellma lmann • theor eorem em (F (FHT HT): Shif ift in in energy le level el matr matrix elemen ment. Extraction of energy levels on • lattice are more straightforward, avoid complicated contraction diagrams. Momen entum transfer fer could be introduced through period odic ic exter ernal l • poten tenti tial. Shows great potential in studies of: • Nucleon axial charge and sigma term • EM form factors • Compton amplitude • P-even structure functions • Chambers et al., PRL 118, 242001 (2017) Hadron resonances • …… • 22
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Some warm-up: Kinematics: “Off-shell condition”: Off-shell Consider a period odic ic potentia tial: • The off-shell condition prohibits mixing of degenerate states through • perturbation. Thus, non-degenerate perturbation theory at 1 st -order gives: No f No fir irst-or order er e energ rgy s shift! t! 23
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Our Strategy: Introduce TWO TWO periodic source terms, and study the SECO ECOND ND • ORD RDER ENRG ENRGY SHIFT: v CYS and U.G-Meissner, hep-ph/1903.07969 Plugging it into the dispersion relation of T 3 N : • Central result!!! FHT DR 2 nd order Generalized Forward Structure Function Energy shift Compton tensor 24
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Isolating the inelastic contribution: First Nachtmann moment: Elastic Inelastic Energy shift: Elastic Inelastic Elastic piece fully described by form factors (experiment + lattice): Inelastic contribution starts from the pion production threshold: Small at small Q 2 ! 25
Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Lattice momenta are discret ete: • Requiring Q 2 at the hadronic scale and the off-shell condition imply: • A concret ete e examp ample: impose the • restriction: Liu et al., Phys.Rev.D 96 (2017) 054516 Q 2 ≈0.38GeV 2 Allowed values for ω: Pion production threshold : 26 (assume physical pion mass)
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