Pinning Down the Inner Radiative Correction in Beta Decays - - PowerPoint PPT Presentation

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Pinning Down the Inner Radiative Correction in Beta Decays

Chien-Yeah Seng

Helmh mhol

• ltz-Ins

Insti titut tut für Strahl hlen- und und Ke Kernp nphysik and nd Bet ethe Cen enter for

• r Theor

eoretical Physics, Universitä tät t Bonn

17 May, 2019

“Current and Future Status of the First-Row CKM Unitarity” workshop, UMass Amherst, Amherst, USA.

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Out utline

1.The Inner Radiative Correction 2.Dispersive Approach 3.First-Principle Calculation 4.Summary

• 1. T

The he Inne nner Ra Radia iativ ive Co Correction

• Extraction of Vud

ud from be

beta d ta decays:

• (1) Superallow
• wed

ed be beta ta decay ay

• (2) Neutr

tron

• n be

beta ta decay

• “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

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The he I Inne nner Ra Radia iativ ive Co Correctio ion

“outer” correction: sensitive to electron spectrum: see Leendert’s talk “nucleus-independent” correction ft values corrected by nuclear structure effects: see Misha’s talk

• Main source of uncertainty in inner RC: γW-box
• x diagram

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Sensitive to loop momentum q at ALL scales! The “model-dependent” piece involves the axial component of the charged weak current:

The he I Inne nner Ra Radia iativ ive Co Correctio ion

• Previous best determination: Marciano and Sirlin (M&S)

S)

• Write the RC as a single-variable integral over Q2, and identify the

dominant physics as a function of Q2. 1. Short distance: leadin ing OPE + perturbativ tive e QCD 2. Intermediate distance: VM VMD-in inspir ired ed inter erpol

• latin

ting function tion + 100% 100% uncer erta tainty 3. Long distance: Elastic ic contr trib ibution tion

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Marciano and Sirlin, Phys.Rev.Lett. 96 (2006) 032002

Combined:

) 38 ( 02361 . ) S & M ( = ∆

V R

The he I Inne nner Ra Radia iativ ive Co Correctio ion

• 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

• T3 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

• n-shel

ell inter termed ediat ate e state states

• We need only the contribution from the iso

sosc scal alar EM curren ent ( (0)

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Dis Dispersiv ive Ap Approach

Hadronic tensor in inclusive scattering:

• Disp

sper ersi sion

• n r

relati ation

• n:
• Box diagrams are expressed in terms of the “First

st Nac achtma mann mo momen ment” of F3

(0) :

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Dis Dispersiv ive Ap Approach

Central result!!!

• Iso

sosp spin sy symmet mmetry: where the fl flavor

• r-diagon
• nal s

stru ruct cture re f funct ction

• ns F

F3

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:

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Dis Dispersiv ive Ap Approach

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Regge

VDM

Dis Dispersiv ive Ap Approach Elastic

Multi-Hadron States

A “ph “phase e spa pace” e” di diagram f m for F r F3

(0 (0)

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Dis Dispersiv ive Ap Approach

Elastic: (isoscalar) magnatic Sach FF and axial FF DIS: polarized Bjorken sum rule +pQCD correction Nπ+ Resonance: Negligible

(Only I=1/2 intermediate states contributes) (mere change of integration limit)

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)

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Dis Dispersiv ive Ap Approach

n p N N W W

(I=1)*(I=0) (I=1)*(I=1)

Multi ti-had hadron

• n s

states es: Regge gge mod

• del

el + + VDM W

Matching the 1st Nachtmann moment of the (I=1)*(I=1) piece to ν p/νbar p scattering data

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Dis Dispersiv ive Ap Approach

(I=1)*(I=0) piece is then deduced using Regge model+VDM

Significant increase in the multi-hadron contribution compare to M&S result, with reduced uncertainty:

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Dis Dispersiv ive Ap Approach

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Dis Dispersiv ive Ap Approach

• Reduced hadronic uncertainty in the determin

mination tion o

• f Vud

ud:

• Possi

ssibl ble e issu ssues:

• Quality of the neutrino data?
• Residual model-dependence?

which leads to the discussions below.

(assume nothing else changes; using Vus in PDG)

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

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Dis Dispersiv ive Ap Approach

• Reduced hadronic uncertainty in the determin

mination tion o

• f Vud

ud:

• Possi

ssibl ble e issu ssues:

• Quality of the neutrino data?
• Residual model-dependence?

which leads to the discussions below.

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 Vus in PDG)

• 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)

• Recall the that we are interested in as a function of Q2 .

Neutrino data helps identifying dominant contri ributor

• rs a

at differ eren ent Q Q2 :

• 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 F3 at at moder erate e Q2.

• Question: is there a way to calculate from FIR

FIRST-PR PRIN INCIPLE IPLE?

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

• Difficult because it involves a

a su sum m of al all o

• n-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 Q2.

• We wish to avoi
• id direc

ect calculation

• ns of
• f fou

four-poi

• int f

function tions (noisy contractions, complicated finite-volume effect…)

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

• J. Liang, K-F. Liu and Y-B. Yang, EPJ Web Conf. 175 (2018) 14014

• 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

• dic

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
• Hadron resonances
• ……

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

Chambers et al., PRL 118, 242001 (2017)

• 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

• dic

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
• Hadron resonances
• ……

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

Chambers et al., PRL 118, 242001 (2017)

• Consider a period
• dic

ic potentia tial:

• The off-shell condition prohibits mixing of degenerate states through
• perturbation. Thus, non-degenerate perturbation theory at 1st-order

gives:

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion Some warm-up:

Kinematics: “Off-shell condition”: Off-shell

No f No fir irst-or

• rder

er e energ rgy s shift! t!

• Introduce TWO

TWO periodic source terms, and study the SECO ECOND ND ORD RDER ENRG ENRGY SHIFT:

• Plugging it into the dispersion relation of T3

N:

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

2nd order Energy shift Generalized Forward Compton tensor Structure Function

FHT DR

CYS and U.G-Meissner, hep-ph/1903.07969

v

Central result!!!

Our Strategy:

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

Isolating the inelastic contribution:

Elastic Inelastic First Nachtmann moment: Energy shift: Elastic Inelastic Elastic piece fully described by form factors (experiment + lattice): Inelastic contribution starts from the pion production threshold:

Small at small Q2 !

• Lattice momenta are discret

ete:

• Requiring Q2 at the hadronic scale and the off-shell condition imply:
• A concret

ete e examp ample: impose the restriction:

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

Q2≈0.38GeV2

Allowed values for ω:

Liu et al., Phys.Rev.D 96 (2017) 054516

Pion production threshold:

(assume physical pion mass)

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

A very good reconstruction!

Reconstruc ucti ting ng the first Nachtm htmann n moment nt from energy shifts ts

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

A very good reconstruction!

Reconstruc ucti ting ng the first Nachtm htmann n moment nt from energy shifts ts

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Fir irst-Princ incip iple le Ca Calc lcul ulatio ion

Reconstruc ucti ting ng the first Nachtm htmann n moment nt from energy shifts ts

• What about smaller Q2?
• The available values of ω is less so the reconstruction of Λ(x,Q2)

will be less satisfactory.

• However, ω=0

0 (zero proton momentum) is ALWAYS an accessible point; that gives the first Mellin lin moment of F3

N,

which sets important constraints on its model parameterization.

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1. The axial γW box diagram is one of the main sources of theoretical uncertainty in the extraction of Vud through neutron and superallowed beta decay. 2. The application of dispersion relation utilizing ν p/ νbar p scattering data reduces the uncertainty in the γW box diagram by a half, but at the same time raises tension with the first row CKM unitarity. 3. A recent proposal that hybridizes the dispersion relation and computations of shifted energy levels on lattice may, for the first time, lead to a first-principle theoretical calculation of the γW box.

Sum ummary