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

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Dispersion Theory of the γW-Box For free and bound neutron β-decay

Misha Gorshteyn

arXiv: 1807.10197 arXiv: 1812.03352 arXiv: 1812.04229

Johannes Gutenberg-Universität Mainz Collaborators: Chien-Yeah Seng (U. Bonn) Hiren Patel (UC Santa Cruz) Michael Ramsey-Musolf (UMass) Based on 3 papers:

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Outline

2

Ft = ft(1 + δ0

R)[1 − (δC − δNS)]

• 1. Dispersion formalism for the 𝛿W-box
• 2. Calculation of the universal free-neutron RC ΔRV
• 3. Splitting the full nuclear RC into free-neutron ΔRV and nuclear modification δNS
• 4. Splitting the full RC into “outer” and “inner”

Superallowed nuclear decays: Free neutron decay:

γ ν

e

n W

( )

( )

ν ν ν π π − − − =

∫

⋅ = ν

( )

ν ν ε π

β α µναβ ν µ

=

∫

⋅

physics at hadronic scale γ− (“m.d”: model-dependent) is:

q q

: |Vud|2 = 5099.34s ⌧n(1 + 32)(1 + ∆R)

: |Vud|2 = 2984.43s Ft(1 + ∆V

R)

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• 1. 𝛿W-box from dispersion relations

3

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𝛿W-box

TγW = p 2e2GF Vud Z d4q (2⇡)4 ¯ ueµ(k / q / + me)ν(1 5)vν q2[(k q)2 m2

e]

M 2

W

q2 M 2

W

T γW

µν ,

Hadronic tensor: two-current correlator Box at zero momentum transfer* (but with energy dependence)

T µν

γW =

✓ gµν + qµqν q2 ◆ T1 + 1 (p · q) ✓ p (p · q) q2 q ◆µ ✓ p (p · q) q2 q ◆ν T2 + i✏µναβpαqβ 2(p · q) T3

General gauge-invariant decomposition of a spin-independent tensor

4

Tμν

γW = ∫ dxeiqx⟨f |T[Jμ em(x)Jν,± W (0)]|i⟩

*Precision goal: 10-4; RC ~ 𝛽/2𝜌 ~ 10-3; recoil on top - negligible

γ ν

e

n W

( )

( )

ν ν ν π π − − − =

∫

⋅ = ν

( )

ν ν ε π

β α µναβ ν µ

=

∫

⋅

physics at hadronic scale γ− (“m.d”: model-dependent) is:

q q

TγW = − α 2π GFVud∫ d4qM2

W

q2(M2

W − q2) ¯

ueγβ(1 − γ5)uν∑

i

Cβ

i (E, ν, q2)TγW i

(ν, q2) Loop integral with generally unknown forward amplitudes Known algebraic functions of external energy E and loop variables 𝜉, q2 pμ = (M, ⃗ 0 ) E = (pk)/M ν = (pq)/M

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W W

γ γ

q q q q p p p p

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

Forward amplitudes Ti - unknown; Their absorptive parts can be related to production of on-shell intermediate states —> a 𝛿W-analog of structure functions F1,2,3

𝛿W-box from Dispersion Relations

γ γ

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

T1,2,3 - analytic functions inside the contour C in the complex ν-plane determined by their singularities

• n the real axis - poles + cuts

5

TγW

i

(ν, Q2) = 1 2πi ∮ dz TγW

i

(z, Q2) z − ν , ν ∈ C

Im TγW

i

(ν, Q2) = 2πFγW

i

(ν, Q2)

Structure functions Fi𝛿W are NOT data But they can be related to data X = inclusive strongly-interacting

• n-shell physical states

X

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𝛿W-box from Dispersion Relations

6

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, Q2) = ξ(I) i T(I) i (ν, Q2)

TγW,a

i

= T(0)

i τa + T(−) i

1 2 [τ3, τa]

ξ(0)

1

= + 1, ξ(0)

2,3 = − 1;

ξ(−)

i

= − ξ(0)

i

γ γ

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

T(I)

i (ν, Q2) = 2∫ ∞

dν′[ 1 ν′− ν − iϵ + ξ(I)

i

ν′− ν − iϵ ] F(I)

i (ν′, Q2)

Re ⇤odd

γW (E) =

8αE 3πNM

∞

Z dQ2

∞

Z

νthr

dν (ν + q)3  ⌥F (0)

1

⌥ ✓3ν(ν + q) 2Q2 + 1 ◆ M ν F (0)

2

+ ν + 3q 4ν F (−)

3

• + O(E3)

Two types of dispersion relations for scalar amplitudes Substitute into the loop and calculate leading energy dependence

q = ν2 + Q2

Re □even

γW =

α πMN ∫

∞

dQ2M2

W

M2

W + Q2 ∫ ∞

dν ν ν + 2q (ν + q)2 F(0)

3 (ν, Q2) + O(E2)

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• 2. Universal inner RC ΔRV

7

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Inner universal RC from DR

𝛿W-box at zero energy

8

Connection to MS: rewrite in terms of the first Nachtmann moment of F3 Re □even

γW = 3α

2π ∫

∞

dQ2M2

W

Q2(M2

W + Q2) M(0) 3 (1,Q2) = α

8π ∫

∞

dQ2M2

W

M2

W + Q2 FMS(Q2)

M(0)

3 (1,Q2) = 4

3 ∫

1

dx 1 + 2 1 + 4M2x2/Q2 (1 + 1 + 4M2x2/Q2)2 F(0)

3 (x, Q2)

FMS(Q2) = 12 Q2 M(0)

3 (1,Q2)

Re □even

γW = α

πM ∫

∞

dQ2M2

W

M2

W + Q2 ∫ ∞

dν ν ν + 2q (ν + q)2 F(0)

3 (ν, Q2)

Re □odd

γW (E = 0) = 0

x = Q2 2Mν MS loop fn. F(Q2) directly related to M3(0)

F(0)

3

∝ ∫ dxeiqx⟨p|[Jμ,(0)

em (x), Jν,+ W (0)]|n⟩ ∼ ∫ dxeiqx

∑

X

⟨p|Jμ,(0)

em (x)|X⟩⟨X|Jν,+ W (0)|n⟩

SF F3 - commutator of em and weak currents - insert complete set of on-shell hadronic states

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Input into dispersion integral

γ γ

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

Dispersion in energy: scanning hadronic intermediate states Dispersion in Q2: scanning dominant physics pictures

2

W

2

Q

( )

2 π

m M +

2

M

Born Parton + pQCD Nπ

Res. +B.G Regge +VMD

2

GeV 2 ~

2

GeV 5 ~

Boundaries between regions - approximate Input to DR related (directly or indirectly) to experimentally accessible data

9

W2 = M2 + 2Mν − Q2

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Input into dispersion integral

2

W

2

Q

( )

2 π

m M +

2

M

Born Parton + pQCD Nπ

Res. +B.G Regge +VMD

2

GeV 2 ~

2

GeV 5 ~

F (0)

3 = FBorn +

8 < : FpQCD, Q2 & 2 GeV2 FπN +Fres+FR, Q2 . 2 GeV2

Our parametrization of the needed SF follows from this diagram Born: elastic FF from e-, ν scattering data πN: 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

⇤V A,Born

γW

= α π Z ∞ dQ 2 p 4M 2 + Q2 + Q ⇣p 4M 2 + Q2 + Q ⌘2 GA(Q2)GS

M(Q2)

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Inelastic states - low Q2, high W

F (0),Regge

3

(ν, Q2) = CR(Q2) ✓ ν ν0 ◆αρ

11

Scattering at high energy can be very effectively described by Regge exchanges γW-box: conversion of W± (charged, I=1, axial) to γ (neutral, vector, I=0) requires charged vector exchange w. I=1 - ρ± effective a1 - ρ - ω vertex Regge behavior in EW processes: hadron-like behavior of HE electroweak probes - Vector/Axial Vector Dominance is the proper language 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 a1 - ω - ρ vertex Minimal model for both reactions - check with data. Compare to Bill’s F(Q2) ~ 1/Q2 at high-Q2 VM propagators 1/(Ma2+Q2)/(Mρ2+Q2) ~ 1/Q4, more natural for hadronic amplitudes

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Input into dispersion integral

Unfortunately, no data can be obtained for Data exist for the pure CC processes F γW (0)

3

F νp+¯

νp 3, low−Q2 = F νp+¯ νp 3, el.

+ F νp+¯

νp 3, πN

+ F νp+¯

νp 3, R

+ F νp+¯

νp 3, Regge

d2σν(¯

ν)

dxdy = G2

F ME

π  xy2F1 + ✓ 1 − y − Mxy 2E ◆ F2 ± x ✓ y − y2 2 ◆ F3

• 12

Low-W part of spectrum: neutrino data from MiniBooNE, Minerva, …

• axial FF, resonance contributions, pi-N continuum

High-W: Regge behavior F3 ∼ q𝓌 ∼ x-𝛽, 𝛽 ∼ 0.5-0.7 Z 1 dx(up

v(x) + dp v(x)) = 3

σνp − σ¯

νp ∼ F νp 3

+ F ¯

νp 3

= up

v(x) + dp v(x)

Gross-Llewellyn-Smith sum rule Validate the model for CC process; apply an isospin rotation to obtain γW

2

W

2

Q

( )

2 π

m M +

2

M

Born Parton + pQCD Nπ

Res. +B.G Regge +VMD

2

GeV 2 ~

2

GeV 5 ~

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Parameters of the Regge model from neutrino scattering

13

Low Q2 < 0.1 GeV2: Born + Δ(1232) dominate Not fitted: modern data more precise but cover only limited energy range Fit driven by 4 data points between 0.2 and 2 GeV2 M3WW (1,Q2) M3γW (1,Q2) Isospin symmetry Model & Uncertainty fully specified

• compare M&S vs This work

Log scale for x-axis: integral = surface under the curve

M&S: integrand discontinuous at Q2 = 2.25 GeV2 Uncertainty reduced by almost factor 2; ~ 3-5 sigma shift from the old value

0.01 0.1 1 10 100

Q² (GeV²)

0.5 1 1.5 2 2.5 3 3.5

GLS SR WA25 CCFR BEBC/GGM-PS Regge + Born + Δ pQCD MS: INT + Born + Δ

MS Total : □(0)

γW = 0.00324 ± 0.00018

New Total : □(0)

γW = 0.00379 ± 0.00010

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

Q² (GeV²)

0.02 0.04 0.06 0.08

Total No Born MS

M3

(0) (1,Q2) / (1 + Q2/ Mw 2)

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• 2. Nuclear structure modification of ΔRV

14

C-Y Seng, MG, M J Ramsey-Musolf, arXiv: 1812.03352

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Splitting the γW-box into Universal and Nuclear Parts

15

⇤VA, Nucl.

γW

= ⇤VA, free n

γW

+ h ⇤VA, Nucl.

γW

− ⇤VA, free n

γW

i

Input in the DR for the universal RC Input in the DR for the RC on a nucleus General structure of RC for nuclear decay

ft(1 + RC) = Ft(1 + δ0

R)(1 − δC + δNS)(1 + ∆V R)

NS correction reflects this extraction of the free box Nuclear modification in the lower part of the spectrum

ΔV

R ∝ Ffree n 3

∝ ∫ dxeiqx ∑

X

⟨p|Jμ,(0)

em (x)|X⟩⟨X|Jν,+ W (0)|n⟩

ΔV

R + δNS ∝ FNucl. 3

∝ ∫ dxeiqx ∑

X′

⟨A′|Jμ,(0)

em (x)|X′⟩⟨X′|Jν,+ W (0)|A⟩

RC on a free neutron RC on a nucleus

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Nuclear γW-box

16

⇤V A, Nucl.

γW

= α NπM

1

Z dQ2M 2

W

M 2

W + Q2 1

Z dν (ν + 2q) ν(ν + q)2×F (0), Nucl.

3, γW

(ν, Q2),

T W nuc

µ⌫

⇠ X

k,`

hf|JW

µ (k) Gnuc JEM ⌫

(`)|ii

Need to know the full nuclear Green’s function indices k, l count the nucleon d.o.f. in a nucleus

T A

µ⌫ =

X

k

hf|JW

µ (k) Gnuc JEM ⌫

(k)|ii T B

µ⌫ =

X

k6=`

hf|W

µ (k) Gnuc JEM ⌫

(`)|ii

Two cases: (A) same active nucleon (B) two nucleons correlated by G Case (A): on-shell neutron propagating between interaction vertices Case (B): all two-nucleon contributions (QE 2p2h and nuclear excitations) Insert on-shell intermediate states:

TA

μν → ∑ k

⟨f |JW

μ (k)[SN F ⊗ GA′′ nuc]JEM ν

(k)|i⟩

The elastic nucleon box is replaced by a single N QE knockout

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Universal vs. Nuclear Corrections

17

Towner 1994 and ever since: Idea: calculate Gamow-Teller and magnetic nuclear transitions in the shell model; Insert the single nucleon spin operators —> predict the strength of nuclear transitions “Quenching of spin operators in nuclei”: shell model overestimates those strengths! Each vertex is suppressed by 10-15% But from dispersion relation perspective it corresponds to a contribution of an excited nuclear state, not to the modified box on a free nucleon! The correct estimate should base on quasielastic knockout with an on-shell N + spectator in the intermediate state Numerically: on average between the 14 superallowed decays δquenched Born

NS

= [q(0)

S qA − 1]2 □free n, Born γW

≈ − 0.058(14) % universal nuclear δNS □Nucl

γW − □free n γW

= [q(0)

S qA − 1] □free n γW

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Modification of CB in a nucleus - QE

18

𝜉 = Q2/2M 𝜉 ≥ Q2/2M + ϵ kF

Reduction for QE: finite threshold ϵ (binding energy) + Fermi momentum kF

F (0), B

3

= −Q2 4 GAGS

Mδ(2Mν − Q2)

Integral is peaked at low 𝜉, Q2

⇤V A, Nucl.

γW

= α NπM

1

Z dQ2M 2

W

M 2

W + Q2 1

Z dν (ν + 2q) ν(ν + q)2×F (0), Nucl.

3, γW

(ν, Q2),

Born on free n: QE calculation in free Fermi gas model with Pauli blocking assign a generous 30% model uncertainty New δQENS ~ - 0.11(3)% instead of the previous estimate δqNS ~-0.058(14)%

C-Y Seng, MG, M J Ramsey-Musolf, arXiv: 1812.03352

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3.Splitting of the RC into inner and outer

19

MG, arXiv: 1812.04229

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Splitting the RC into “inner” and “outer”

20

Radiative corrections ~ α/𝜌 ~ 10-3 Precision goal: ~ 10-4 When does energy dependence matter? Correction ~ Ee/Λ, with Λ ~ relevant mass (me; Mp; MA) Maximal Ee ranges from 1 MeV to 10.5 MeV Electron mass regularizes the IR divergent parts - (Ee/me important) - “outer” correction If Λ of hadronic origin (at least m𝜌) —> Ee/Λ small, correction ~ 10-5 —> negligible

• certainly true for the neutron decay
• hadronic contributions do not distort the spectrum, may only shift it as a whole

However, in nuclei binding energies ~ few MeV — similar to Q-values A scenario is possible when RC ~ (α/𝜌)x(Ee/ΛNucl) ~ 10-3 Nuclear structure may distort the electron spectrum With dispersion relations can be checked straightforwardly!

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21

Nuclear structure distorts the β-spectrum!

Re ⇤odd

γW (E) =

8αE 3πNM

∞

Z dQ2

∞

Z

νthr

dν (ν + q)3  ⌥F (0)

1

⌥ ✓3ν(ν + q) 2Q2 + 1 ◆ M ν F (0)

2

+ ν + 3q 4ν F (−)

3

• + O(E3)

Evaluate the E-dependent contribution Estimate with nuclear polarizabilities and size

↵E = 2↵ M

1

Z

✏

d⌫ ⌫3 F1(⌫, 0) = 2↵

1

Z

✏

d⌫ ⌫2 @ @Q2 F2(⌫, 0).

Photonuclear sum rule: Supplement with the nuclear form factor: αE(Q2) ∼ αE(0) × e−R2

ChQ2/6

Radius and polarizability scale with A: RCh ∼ 1.2 fm A1/3, αE ∼ 2.25 × 10−3 fm3 A5/3 δNS(E) = 2 × 10−5 ( E MeV ) A N Dimensional analysis estimate: Estimate in Fermi gas model (same as for E-independent) δNS(E) = (2.8 ± 0.4 ± 0.8) × 10−4 ( E MeV) Uncertainty: spread in ϵ and kF, plus 30% on model

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22

Correction to Ft values: integrate over spectrum (only total rate measured) Use the two estimates as upper and lower bound of the effect δNS(E) = (1.6 ± 1.6) × 10−4 ( E MeV ) Spectrum distortion due to nuclear polarizabilities ~ 0.016 % per MeV Roughly independent of the nucleus; The total rate will depend on nucleus: different Q-values!

Nuclear structure and E-dependent RC

δE

NS =

∫

Q me dEEp(Q − E)2δNS(E)

∫

Q me dEEp(Q − E)2

δE

NS ≈ (8 ± 8) × 10−5

Q MeV

˜ ℱt = ft(1 + δ′

R)(1 − δC + δNS + δE NS)

Average Q ~ 6 MeV —> expect an effect

⟨δE

NS⟩ ≈ (1.5 ± 1.5) × 10−4

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4.Putting numbers together

23

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• f ∆V

R = 0.02361(38)

|Vud|2 = 2984.432(3) s Ft(1 + ∆V

R)

∆V

R = 0.02467(22)

Marciano & Sirlin 2006 Dispersion relations DR allowed to ~ halve the uncertainty in ΔRV due to the use of neutrino data

24

Universal correction

ℱt = ft(1 + δ′

R)(1 − δC + δNS)

Nuclear corrections - E-dependent and independent

˜ ℱt = ft(1 + δ′

R)(1 − δC + ˜

δNS + δE

NS)

New δNS uses QE estimate instead of the quenched Born estimate δquenched Born

NS

= − 0.058(14) % δQE

NS = − 0.11(3) %

δE

NS = 0

δQE

NS = + 0.05(5) %

No net shift to the Ft central value ℱt = (3072.1 ± 0.7)s → ℱt = (3072 ± 2)s

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• f ∆V

R = 0.02361(38)

∆V

R = 0.02467(22)

MS 2006 + HT 2014-2018 Dispersion relations

25

|Vud| = 0.97420(10)Ft(18)ΔV

R

|Vud| = 0.97370(30)Ft(10)ΔV

R

ℱt = (3072 ± 2)s ℱt = (3072.1 ± 0.7)s

|Vud|2 = 2984.432(3) s Ft(1 + ∆V

R)

|Vud|2 + |Vus|2 + |Vub|2 = 0.9994 ± 0.0005 |Vud|2 + |Vus|2 + |Vub|2 = 0.9984 ± 0.0006 Unitarity in the top row Substantial shift in the central value by two old sigmas (due to ΔRV) Discrepancy with the unitarity mainly depends now on nuclear uncertainties

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Conclusions & Outlook

26

Nuclear correction δNS DR allow to address hadronic and nuclear parts of the calculation on the same footing The full nuclear correction should be calculated (not just QE) - further test of H&T δNS Decay spectra and nuclear polarizabilities Can contaminate the extraction of Fierz interference from precise spectra!

• The γW-box in the forward dispersion relation framework
• Hadronic and nuclear corrections in a unified framework
• Nuclear structure leaks in the outer correction, distorts the beta decay spectrum
• Better calculations than free Fermi gas should be done
• Nuclear uncertainties shift the emphasis on free neutron decay
• Tensions with CKM unitarity: Σi=d,s,b |Vui|2 - 1 = -0.0016(6)

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Conclusions & Outlook

27

However… the largest correction to Ft is ISB δC non-dispersive Range from 0.15% to 1.5% Can its calculation be related to neutron skin calculations for PVES? Which ingredients are common and which are not? MESA@Mainz will measure the weak radius of C-12 to <1% (2023 on) Other nuclei (including symmetric ones) possible in the future Potentially a strong statement between two fields

SLIDE 28

Backup slide

SLIDE 29

29

Nuclear structure distorts the β-spectrum!

F

Decay Q (MeV) ∆NS

E (10−4) δFt(s)

Ft(s) [3]

10C

1.91 1.5 0.5 3078.0(4.5)

14O

2.83 2.3 0.7 3071.4(3.2)

22Mg

4.12 3.3 1.0 3077.9(7.3)

34Ar

6.06 4.8 1.5 3065.6(8.4)

38Ca

6.61 5.3 1.6 3076.4(7.2)

26mAl

4.23 3.4 1.0 3072.9(1.0)

34Cl

5.49 4.4 1.4 3070.7+1.7

−1.8 38mK

6.04 4.8 1.5 3071.6(2.0)

42Sc

6.43 5.1 1.6 3072.4(2.3)

46V

7.05 5.6 1.7 3074.1(2.0)

50Mn

7.63 6.1 1.9 3071.2(2.1)

54Co

8.24 6.6 2.0 3069.8+2.4

−2.6 62Ga

9.18 7.3 2.2 3071.5(6.7)

74Rb

10.42 8.3 2.6 3076(11)

Absolute shift in Ft values Shift due to δENS: comparable to precision of 7 best-known decays Decay electron polarizes the daughter nucleus As a result the spectrum is slightly distorted towards the upper end Positive-definite correction to Ft ~ 0.05% Ft = 3072.07(63)s → Ft = 3073.6(0.6)(1.5)s Previously found: E-independent piece lowers the Ft value by about the same amount Nuclear structure uncertainties might be underestimated

Ft = 3072.07(63)s → [Ft]new = 3070.50(63)(98)s,

˜ ℱt = ft(1 + δ′

R)(1 − δC + δNS + δE NS)

δℱt = ℱt × δE

NS

SLIDE 30

QE calculation - effect on Ft values and Vud

30

Adopting a new estimate of the in-nucleus modification of the free-nucleon Born Ft → Ft(1 + δnew

NS − δold NS)

Shifts the Ft value according to Numerically:

|Vud|2 = 2984.432(3) s Ft(1 + ∆V

R)

Will affect the extracted Vud

|Vud|2 + |Vus|2 + |Vub|2 = 0.9984 ± 0.0004

V old

ud = 0.97420(21) →

Compensates for a part of the shift due to a new evaluation of ΔVR Brings the first row closer to the unitarity (4σ → 2.2σ) Important messages: a nuclear contribution may shift by 2 sigma if evaluated with a different method dispersion relations as a unified tool for treating hadronic and nuclear parts of RC

Ft = 3072.07(63)s → [Ft]new = 3070.50(63)(98)s, |V new

ud | = 0.97370(14) → |V new, QE ud

| = 0.97395(14)(16)

| | | | | | ± → |Vud|2 + |Vus|2 + |Vub|2 = 0.9989 ± 0.0005.

and 1 sigma away from the PDG: 0.9994 ± 0.0005