New Evaluation of the W-Box Correction to Neutron and Nuclear - - PowerPoint PPT Presentation

New Evaluation of the γW-Box Correction to Neutron and Nuclear β-Decay

Misha Gorchtein

arXiv: 1807.10197 arXiv: 1812.03352 arXiv: 1812.04229

Johannes Gutenberg-Universität Mainz Collaborators: Chien-Yeah Seng, Hiren Patel, Michael Ramsey-Musolf Top Row CKM Unitarity Workshop - John Hardy’s Career Celebration January 7-8, 2019 - The Mitchell Institute, Texas A&M U., College Station, TX USA Based on 3 papers:

Current status of Vud and top-row CKM unitarity

2

CKM unitarity: Vud the main contributor to the sum and to the uncertainty

|Vud|2 = 0.94906 ± 0.00041

|Vub|2 = 0.00002 |Vus|2 = 0.05031 ± 0.00022

0+-0+ nuclear decays K decays B decays

|Vud|2 + |Vus|2 + |Vub|2 = 0.9994 ± 0.0005

• * Slide stolen from one of John’s talks

Why are superallowed decays special?

3

Superallowed 0+-0+ nuclear decays:

• only conserved vector current (unlike the neutron decay and other mirror decays)
• many decays (unlike pion decay)
• all decay rates should be the same modulo phase space

Experiment: f - phase space (Q value + spectrum profile with Coulomb distortion) and t - partial half-life (t1/2, branching ratio)

• ft values: same within ~2% but not exactly!

Reason: SU(2) slightly broken

• a. RC (e.m. interaction does not conserve isospin)
• b. Nuclear WF are not SU(2) symmetric

(proton and neutron distribution not the same)

Why are superallowed decays special?

4

Modified ft-values to include these effects

Ft = ft(1 + δ0

R)[1 − (δC − δNS)]

Ft = 3072.1 ± 0.7

Average

• δ’R - “outer” correction (depends on e-energy) - QED

δC - SU(2) breaking in the nuclear matrix elements

• mismatch of radial WF in parent-daughter
• mixing of different isospin states

δNS - RC depending on the nuclear structure δC,δNS - energy independent Hardy, Towner 1973 - 2018

Vud from free neutron decay

5

Free neutron decay: axial coupling

• requires additional measurements

|Vud|2 = 5024.49(30) s τn(1 + 3λ2)(1 + ∆V

R)

λ=gA/gV

Vud from free neutron decay

5

Free neutron decay: axial coupling

• requires additional measurements

|Vud|2 = 5024.49(30) s τn(1 + 3λ2)(1 + ∆V

R)

λ=gA/gV

• τ beam

n

= 888.0(2.0)s

τ trap

n

= 879.4(6)s

• λpre2002 = 1.2637(21)

λpost2002 = 1.2755(11)

If using bottle τn + post-2002 λ: consistent but 7 times less precise Unfortunate discrepancy between decay in flight vs. trapped UCN

|V n

ud| = 0.9743(15)

Vud from free neutron decay

5

Free neutron decay: axial coupling

• requires additional measurements

|Vud|2 = 5024.49(30) s τn(1 + 3λ2)(1 + ∆V

R)

λ=gA/gV

• τ beam

n

= 888.0(2.0)s

τ trap

n

= 879.4(6)s

• λpre2002 = 1.2637(21)

λpost2002 = 1.2755(11)

If using bottle τn + post-2002 λ: consistent but 7 times less precise Unfortunate discrepancy between decay in flight vs. trapped UCN

|V n

ud| = 0.9743(15)

May be combined with Vud from 0+-0+ decays eliminate RC (the same for n and nuclei) λ and lifetime are correlated Czarnecki, Marciano, Sirlin, PRL ‘18 τn(1 + 3λ2) = 5172.0(1.1) s

See Bill’s talk

Vud from free neutron decay

5

Free neutron decay: axial coupling

• requires additional measurements

|Vud|2 = 5024.49(30) s τn(1 + 3λ2)(1 + ∆V

R)

λ=gA/gV

• τ beam

n

= 888.0(2.0)s

τ trap

n

= 879.4(6)s

• λpre2002 = 1.2637(21)

λpost2002 = 1.2755(11)

If using bottle τn + post-2002 λ: consistent but 7 times less precise Unfortunate discrepancy between decay in flight vs. trapped UCN

|V n

ud| = 0.9743(15)

May be combined with Vud from 0+-0+ decays eliminate RC (the same for n and nuclei) λ and lifetime are correlated Czarnecki, Marciano, Sirlin, PRL ‘18 τn(1 + 3λ2) = 5172.0(1.1) s

See Bill’s talk

Recently: PERKEO-III halved the uncertainty of λ λ = − 1.27641(45)stat(33)sys Märkisch et al, arXiv:1812.04666 Ongoing effort worldwide on improving τn to 0.2-0.3s

See Albert’s talk

Outline: RC to Beta Decay

6

|Vud|2 = 2984.432(3) s Ft(1 + ∆V

R)

Ft = ft(1 + δ0

R)[1 − (δC − δNS)]

Three caveats:

• 1. Calculation of the universal free-neutron RC ΔRV
• 2. Splitting the full nuclear RC into free-neutron ΔRV and nuclear modification δNS
• 3. Splitting the full RC into “outer” (energy-dependent but pure QED: no hadron structure)

and “inner” (hadron&nuclear structure-dependent but energy-independent)

• nucleon and nuclear case

Will address each point

|Vud| = 0.97420(10)Ft(18)ΔV

R

• 1. Check radiative corrections

to the free neutron decay

7

C-Y Seng, MG, H Patel, M J Ramsey-Musolf, arXiv: 1807.10197

RC on the free neutron

W γ , Z b = ν

e

h ' h W

W γ , ,W Z b = ν

e

h ' h Z ν

e

h ' h W W ν

e

h ' h Z γ ν

e

n p W

( )

( )

ν ν ν π π − − − =

∫

⋅ = ν

( )

ν ν ε π

β α µναβ ν µ

=

∫

⋅

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

q q

W,Z-exchange: UV-sensitive, pQCD; model-independent Outer (depend on e-energy): retain only IR divergent pieces Inner (energy-independent - take E=0) When 𝛿 involved: sensitive to long range physics; model-dependent!

• f ∆V

R = 0.02361(38)

• and Sirlin [5] (in

Until recently: best determination Marciano & Sirlin 2006

8

Separation due to scale hierarchy: me = 0.511 MeV, Q = Mn - Mp = 1.3 MeV; Q/me not small, need to account for exactly. Coulomb distortion: resummation of (Z𝛽)n —> Dirac equation in the Coulomb field IR finite piece: can set me=0 —> if energy-dependent ~ (𝛽/2𝜌) x (E/Λhad) Hadronic structure: relevant scale ~ mπ = 140 MeV - on top of α/2𝜌 ∼ 10-3 —> 10-5 effect <<

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 ,

T µν

γW =

Z dxeiqxhp|T[Jµ

em(x)Jν W (0)]|ni

γ ν

e

n p W

( )

( )

ν ν ν π π − − − =

∫

⋅ = ν

( )

ν ν ε π

β α µναβ ν µ

=

∫

⋅

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

q q

Hadronic tensor: two-current correlator Box at zero energy and momentum transfer

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 (spin-independent) V-V correlator T1,2: conserved vector-isovector current - model-independent Sirlin 1967 - current algebra Axial current not conserved -> A-V correlator T3 - model-dependent

9

⇤V A

γW = 4⇡↵Re

Z d4q (2⇡)4 M 2

W

M 2

W + Q2

Q2 + ⌫2 Q4 T3(⌫, Q2) M⌫

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 amplitude T3 - unknown; Its absorptive part can be related to production of on-shell intermediate states a 𝛿W-analog of the SF F3 Im T γW

3

(ν, Q2) = 2πF γW

3

(ν, Q2)

𝛿W-box from Dispersion Relations

γ γ

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

T3 - analytic function inside the contour C in the complex ν-plane determined by its singularities

• n the real axis - poles + cuts

T3(ν, Q2) = 1 2πi I

C

T3(z, Q2)dz z − ν ν ∈ C

γ ν

e

n p W

( )

( )

ν ν ν π π − − − =

∫

⋅ = ν

( )

ν ν ε π

β α µναβ ν µ

=

∫

⋅

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

q q

Q2 = -q2 ν = (pq)/M

Check MS result + uncertainty independently

10

𝛿W-box from Dispersion Relations

T γW

3

= T (0)

3

+ T (3)

3

Crossing behavior: photon is isoscalar or isovector T (0)

3

(−ν, Q2) = −T (0)

3

(ν, Q2), T (3)

3

(−ν, Q2) = +T (3)

3

(ν, Q2) Different isospin channels behave differently Dispersion representation of the 𝛿W-box correction at zero energy

γ γ

ν π ν =

( )

ν ν ε δ π π

β α µναβ ν µ

= − +

∑

ν

11

□VA(0)

γW

= α πM ∫

∞

dQ2M2

W

M2

W + Q2 ∫ ∞

dν(ν + 2q) ν(ν + q)2 F(0)

3 (ν, Q2)

□VA(3)

γW

= 0

q = ν2 + Q2

First Nachtmann moment of F3 M(0)

3 (1,Q2) = 4

3 ∫

1

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

3 (x, Q2)

□VA(0)

γW

= 3α 2π ∫

∞

dQ2M2

W

M2

W + Q2 M(0) 3 (1,Q2)

(Nachtmann moment = Mellin moment + kinematical higher twist)

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 in DR related (directly or indirectly) to experimentally accessible data

12

W2 = M2 + 2Mν − Q2

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

13

⇤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)

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

• 14

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 ~

Parameters of the Regge model from neutrino scattering

15

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

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

This work MS

M3

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

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

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

Universal γW-box

• 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 reduce the uncertainty in ΔRV by almost factor of 2 due to the use of neutrino data But the shift is more significant than anticipated from the uncertainty estimate by MS

16

Before After Tension with CKM unitarity |Vud| = 0.97420(10)Ft(18)ΔV

R

|Vud| = 0.97370(10)Ft(10)ΔV

R

|Vud|2 + |Vus|2 + |Vub|2 = 0.9994 ± 0.0005 |Vud|2 + |Vus|2 + |Vub|2 = 0.9984 ± 0.0004

2.Radiative corrections to nuclear decays: Nuclear structure modification of the free-n RC

17

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

General structure of RC for nuclear decay

ft(1 + RC) = Ft(1 + δ0

R)(1 − δC + δNS)(1 + ∆V R)

Splitting the γW-box into Universal and Nuclear Parts

18

⇤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 Splitting the full RC into “universal” and “nuclear structure” is natural if the two pieces are treated differently. In the dispersion framework this splitting is unnatural: just calculate the RC on a nucleus δquenched Born

NS

∼ (qS

MqA − 1)CB

Towner 1994: elastic box is quenched in nuclei; size of quenching - from quenching of 1-body spin operators

Modification of CB in a nucleus - QE

19

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: Exploratory QE calculation in free Fermi gas model with Pauli blocking; Disregard fine detail; estimate the bulk effect averaged over all superallowed decays New <δQENS> ~ - 0.10(1)% to be compared to the H&T quenched result averaged over 20 decays <δNSquenched> ~ -0.055(5)%

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

QE calculation - effect on Ft values and Vud

20

Ft = 3072.07(63)s ! [Ft]new = 3070.65(63)(28)s,

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

| | | | | | ± ! |Vud|2 + |Vus|2 + |Vub|2 = 0.9988 ± 0.0004 V new

ud

= 0.97370(14) ! V new, QE

ud

= 0.97392(14)(04)

V old

ud = 0.97420(21) →

Compensates for a part of the shift due to a new evaluation of ΔVR Brings the first row a little closer to the unitarity (4σ → 3σ) Important message: Dispersion relations as a unified tool for treating hadronic and nuclear parts of RC Further work to make the model adequate for the needed precision

3.Splitting of the RC into inner and outer

21

MG, arXiv: 1812.04229

Splitting the RC into “inner” and “outer”

22

Radiative corrections ~ α/2𝜌 ~ 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/2𝜌)x(Ee/ΛNucl) ~ 10-3 Nuclear structure may distort the electron spectrum With dispersion relations can be checked straightforwardly!

23

Nuclear structure distorts the β-spectrum!

With DR: can include energy dependence explicitly Even and odd powers of energy - leading terms

Re ⇤even

γW =

α πN

∞

Z dQ2

∞

Z

νthr

dν F (0)

3

Mν ν + 2q (ν + q)2 + O(E2) 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)

E-dependent correction from dimensional analysis: nuclear radii and polarizabilities Nuclear excitations live at few MeV —> large nuclear polarizabilities New energy scale: polarizability/radius2 RCh ∼ 1.2fmA1/3 αE ∼ (2.2 × 10−3 fm)A5/3 Expect Re □odd

γW ∼ ∓ 2EαE

πNR2

Ch

αE = 2αem M ∫ dω ω3 F1(ω, Q2 = 0) = 2αem∫ dω ω2 ∂ ∂Q2 F2(ω, Q2 = 0) Assume Q2 dependence to follow that of charge form factor ∼ Exp[−R2

ChQ2/6]

Re □odd

γW ∼ ∓ 1 × 10−5

E MeV A N

24

Nuclear structure distorts the β-spectrum!

∆NS

E

= R Em

me dEEp(Q − E)2∆R(E)

R Em

me dEEp(Q − E)2

Correction to Ft values: integrate over spectrum (only total rate measured)

F ˜ Ft = ft(1 + 0

R)(1 − C + NS + ∆NS E )

E-dependent correction from free Fermi gas model (A-V piece)

• same model as was used for the E-independent piece

Re □odd

γW = (1.4 ± 0.2) × 10−4

E MeV Almost an o.o.m. larger than the estimate with polarizabilities (large isovector magnetic moment) How reliable? Strong dependence on fine details of nuclear structure Rough estimate (bound) on the E-dependent correction to diff. spectrum ΔNS

R (E) = 2Re □odd γW = (1.6 ± 1.6) × 10−4

E MeV

25

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)

as Ft = Ft × ∆NS

E

easured superallow

Absolute shift in Ft values

F ˜ Ft = ft(1 + 0

R)(1 − C + NS + ∆NS E )

Shift due to ΔENS: comparable to precision of 7 best-known decays

Ft = 3072.07(63)s ! [Ft]new = 3070.65(63)(28)s,

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 No evidence of a net shift of the Ft value when combined together; uncertainty?

CKM first-row unitarity constraint is low. Solutions: SM or beyond?

26

Discrepancy - BSM?

27

BSM explanation: non-standard CC interactions —> new V,A,S(PS),T(PT) terms

HS+V = (ψpψn)(CSφeφνe + C′

Sφeγ5φνe) +

• ψpγµψn

CV φeγµ(1 + γ5)φνe

• Scalar and Tensor interactions: distort the beta decay spectra

Complementarity to LHC searches!

• Exp. high precision measurement of 6He spectrum (O. Naviliat-Cuncic, A. Garcia, …)

N(E)dE = peE(Em − E)2 [1 + C1E + b me E ] C1 = 0.00650(7) MeV-1 - effect of weak magnetism - positive slope b ~ +- 0.001 - negative slope Energy-dep. polarizability correction —> C’1 ~ 0.00020(20) MeV-1 — at the level 3σ of C1

Conclusions & Outlook

28

Hadronic correction ΔRV Neutrino data at low Q2 used in this analysis are not precise DUNE@Fermilab will provide better data for F3 - direct check Moments M3(0)(N,Q2) at ~1 GeV2 on the lattice - doable! Nuclear correction δNS DR allow to address hadronic and nuclear parts of the calculation on the same footing Better calculations than free Fermi gas are needed The full nuclear correction should be calculated (not just QE) Decay spectra and nuclear polarizabilities This novel effect needs a confirmation in more sophisticated models Can contaminate the extraction of Fierz interference from precise spectra!

• The γW-box was evaluated in a new dispersion relation framework
• Confirmed dominant features of previous calculations but corrected subdominant ones
• Related the model-dependent contribution to neutrino data - systematically improvable!
• Hadronic and nuclear corrections in a unified framework
• Nuclear structure leaks in the outer correction, distorts the beta decay spectrum
• Nuclear uncertainties shift the emphasis on free neutron decay
• Tensions with CKM unitarity: Σi=d,s,b |Vui|2 - 1 = -0.0016(4-6)

Bulk nuclear properties: Fermi momentum and break-up threshold

✏1 = MA−p + Mn − MA ✏2 = MA0−n + Mn − MA

✏ = √✏1✏2 20 decays: 10C -> 10B through 74Rb -> 74Kr

Decay ✏2 (MeV) ✏1 (MeV) ✏ (MeV)

10C →10 B

8.44 4.79 6.36

14O →14 N

10.55 5.41 7.55

18Ne →18 F

9.15 4.71 6.56

22Mg →22 Na

11.07 6.28 8.34

26Si →26 Al

11.36 6.30 8.46

30S →30 P

11.32 5.18 7.66

34Ar →34 Cl

11.51 5.44 7.91

38Ca →38 K

12.07 5.33 8.02

42Ti →42 Sc

11.55 4.55 7.25

26mAl →26 Mg

11.09 6.86 8.72

34Cl →34 S

11.42 5.92 8.22

38mK →38 Ar

11.84 5.79 8.28

42Sc →42 Ca

11.48 5.05 7.61

46V a →46 Ti

13.19 6.14 9.00

50Mn →50 Cr

13.00 5.37 8.35

54Co →54 Fe

13.38 5.13 8.28

62Ga →62 Zn

12.90 3.72 6.94

66As →66 Ge

13.29 3.16 6.48

70Br →70 Se

13.82 3.20 6.65

74Rb →74 Kr

13.85 3.44 6.90

Effective removal energies - all in a small range Fermi momentum also not too different for all A kF (A = 10) = 228 MeV, kF (A = 74) = 245 MeV Can define a universal correction that correctly represents bulk nuclear effect! Further ingredients: Free Fermi gas model (or superscaling) + Pauli blocking

29

¯ ϵ = 7.5 ± 1.5 MeV

QE contribution to the γW-box