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Wilkinsons Corrections and Role of Pseudoscalar Interactions in Neutron Beta Decays Andrey N. Ivanov TU Wien Atominstitut, Austria Amhest Center for Fundamental Interactions 17 May 2019 /USA A. N. Ivanov Current and Future Status of the

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Wilkinson’s Corrections and Role of Pseudoscalar Interactions in Neutron Beta Decays

Andrey N. Ivanov

TU Wien Atominstitut, Austria

Amhest Center for Fundamental Interactions 17 May 2019 /USA

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 2

Wilkinson’s Corrections to Neutron Beta Decay

  • D. H. Wilkinson, NPA 377, 474 (1982)

Electron-energy and angular distribution d5λ(W)

n

(Ee, ke, k¯

ν,

ξn, ξe) dEedΩedΩ¯

ν

= = Q(Ee, ke, k¯

ν, Z = 1)L(Ee, Z = 1)C(Ee, Z = 1)J(Z = 1)K(α)

×d5λ(JTW)

n

(Ee, ke, k¯

ν,

ξn, ξe) dEedΩedΩ¯

ν

  • J. D. Jackson, S. B. Treiman, and H. W. Wyld,Jr., PR 106,

517 (1957)

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 3

Q(Ee, ke, k¯

ν, Z = 1) - proton recoil correction to Fermi

function F(Ee, Z = 1)

Fermi function F(Ee, Z = 1) F(Ee, Z = 1) =

  • 1 + 1

2γ 4(2rpmeβ)2γ Γ2(3 + 2γ) e πα/β (1 − β2)γ

  • Γ
  • 1 + γ + i α

β

  • 2

γ =

  • 1 − α2 − 1

, rp = 0.841 fm Q(Ee, Z = 1) of order O(α) Q(Ee, Z = 1) = 1 − πα β Ee M − πα β3 E0 − Ee M

  • ke ·

k¯

ν

EeE¯

ν

  • D. H. Wilkinson, NPA 377, 474 (1982)
  • A. N. Ivanov, M. Pitschmann, and N. I. Troitskaya, PRD 88,

073002 (2013); arXiv: 1212.0332 [hep-ph], Appendix H

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 4

Corrections for correlation coefficients caused by proton recoil contribution to Fermi function

Ee = 0.761 MeV δX(Ee)/X(Ee) Ee = 0.966 MeV −2.5 × 10−5 ≥ δζ(Ee)/ζ(Ee) ≥ −2.8 × 10−5 +3.0 × 10−4 ≥ δa(Ee)/a(Ee) ≥ +1.1 × 10−4 −6.3 × 10−7 ≤ δA(Ee)/A(Ee) ≤ −3.5 × 10−7 −6.3 × 10−7 ≤ δB(Ee)/B(Ee) ≤ −3.5 × 10−7 +9.0 × 10−5 ≥ δAW(Ee)/AW(Ee) ≥ +3.5 × 10−5 +5.1 × 10−7 ≥ δG(Ee)/G(Ee) ≥ +1.3 × 10−7 −6.3 × 10−7 ≤ δN(Ee)/N(Ee) ≤ −3.5 × 10−7 −6.2 × 10−7 ≤ δH(Ee)/H(Ee) ≤ −3.3 × 10−7 +5.0 × 10−4 ≥ δKe(Ee)/Ke(Ee) ≥ +1.9 × 10−4

Table: Analytical expressions one may find in our papers: PRD88, 073002 (2013); arXiv: 1212.0332 [hep-ph], Appendix H; PRC95, 055502 (2017); arXiv:1705.07330 [hep-ph]; PRD99, 053004 (2019); arXiv:1905.01178 [hep-ph]. Correction to the neutron decay rate (1 + δζ(Ee)/ζ(Ee)) = 1 − 2.7 × 10−5.

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 5

L(Ee, Z = 1) - correction caused by a finite proton radius, rp = 0.841 fm: Pohl et al., Nature466,213(2010)

L(Ee, Z = 1) = 1 + 13 60 α2 − α rp Ee

  • 1 − 1

2 m2

e

E2

e

  • =

= 1 + 1.154 × 10−5 − 4.183 × 10−5 Ee E0 + 0.827 × 10−5 me Ee Correction to the neutron decay rate L(Ee, Z = 1) = 1 − 0.85 × 10−5 Correction to correlation coefficients δX(Ee) X(Ee) = 1 − L(Ee, Z = 1) The correction, caused by the finite proton radius, is the same for all correlation coefficients and, correspondingly, is not important for asymmetries of the neutron beta decay.

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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C(Ee, Z = 1) - correction caused by lepton - nucleon convolution through a finite nucleon volume

C(Ee, Z = 1) = 1 − 2.854 × 10−5 −1.238 × 10−5 Ee E0 − 0.018 × 10−5 me Ee − 1.361 × 10−5 E2

e

E2 Correction to the neutron decay rate C(Ee, Z = 1) = 1 − 4.09 × 10−5 Correction to correlation coefficients δX(Ee) X(Ee) = 1 − C(Ee, Z = 1) The correction, caused by the lepton–nucleon convolution, is the same for all correlation coefficients and is not important for asymmetries of the neutron beta decay The analytical expression for C(Ee, Z = 1) is given in our paper PRC95, 055502 (2017); arXiv:1705.07330 [hep-ph]

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 7

J(Z = 1) - correction caused by outer radiative corrections of higher order in α

J(Z = 1) = = 1 +

  • α2 ℓn M

me + α3 π

  • 3 ℓn 2 − 3

2 + π2 3

  • ℓn M

me + α3 2π ℓn2 M me

  • ×
  • 1 + α

2π ¯ g(E0) −1 = = 1 + 3.5 × 10−4 Correction to the neutron decay rate J(Z = 1) = 1 + 3.5 × 10−4 Correction to correlation coefficients δX(Ee) X(Ee) = 1 − J(Z = 1) The correction, caused by the outer radiative corrections of higher order in α, is the same for all correlation coefficients and, correspondingly, is not important for asymmetries of the neutron beta decay.

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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K(α) represents various further electromagnetic corrections of order O(α) (H. D. Wilkinson)

Wilkinson did not calculate K(α). However, according to Wilkinson NPA377, 474 (1982), one may expect that K(α) should contain, for example, the radiative correction ∆V

R,

i.e. K(α) = 1 + ∆V

R, calculated by

  • W. J. Marciano and A. Sirlin, PRL56, 22 (1986)
  • A. Czarnecki, W. J. Marciano, and A. Sirlin, PRD70,

093006 (2004)

  • W. J. Marciano and A. Sirlin, PRL96, 032002 (2006)

Ch.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, PRL121, 241804 (2018) Ch.-Y. Seng, M. Gorchtein, and M. J. Ramsey-Musolf, arXiv:1812.03352 [nucl-th]

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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Wilkinson’s Correction to Neutron Decay Rate

Wilkinson’s total correction to the neutron decay rate Q(Ee, ke, k¯

ν, Z = 1)L(Ee, Z = 1)C(Ee, Z = 1)J(Z = 1) =

= 1 + 2.7 × 10−4 Thus, one may assert that Wilkinson’s total correction to the neutron decay rate can, in principle, change a digit at the fourth decimal place of the value of the neutron decay rate

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 10

Role of Pseudoscalar SM and beyond the SM Interactions in Neutron Beta Decay

Amplitude of the neutron beta decay with contributions of pseudoscalar SM and beyond the SM interactions M(n → pe−¯ νe) = = −GF √ 2 Vud

  • ¯

up

  • γµ(1 + λγ5) + κ

2M iσµνqν + 2Mλ qµ m2

π − q2 − i0γ5

un

  • ×

¯ ueγµ(1 − γ5)v¯

ν

  • +

¯ upγ5un ¯ ue(Cp + ¯ CPγ5)v¯

ν

  • The contribution of the one-pion-pole exchange is required

by conservation of the charged axial-vector hadronic current in the chiral limit mπ → 0: Y. Nambu, PRL 4, 380 (1960). The last term is given by J. D. Jackson, S. B. Treiman, and

  • H. W. Wyld,Jr., PR 106, 517 (1957)
  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 11

Corrections to Electron-Energy and Angular Distribution: Ivanov et al.; arXiv:1905.04147 [hep-ph]

d5δλn(Ee, ke, k¯

ν,

ξn, ξe) dEedΩedΩ¯

ν

= = (1 + 3λ2) G2

F|Vud|2

32π5 (E0 − Ee)2

  • E2

e − m2 e Ee F(Ee, Z = 1)

×

  • Cps
  • λ E0 − Ee

E0 me Ee + λ me E0

  • ke ·

k¯

ν

EeE¯

ν

− λ me E0

  • ξe ·

ke Ee + . . .

  • + . . .
  • +C′

ps

  • − Ee

E0

  • ξe · (

ke × k¯

ν)(

ξn · ke) E2

eE¯ ν

+ . . .

  • Cps =

2 1 + 3λ2

  • λ me

m2

π

E0 − E0 4M Re(CP − ¯ CP)

  • =

= −1.47 × 10−5 − 1.17 × 10−4 Re(CP − ¯ CP) C′

ps = −

1 1 + 3λ2 E0 2M Im(CP − ¯ CP) = −1.17 × 10−4 Im(CP − ¯ CP)

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 12

Contribution of Pseudoscalar SM and Beyond the SM Interactions to Neutron Decay Rate

Fierz-like interference term

  • 1 + λ Cps

E0 − Ee E0 me Ee

  • = 1 − 0.33 Cps

Cps = −1.47 × 10−5 − 1.17 × 10−4 Re(CP − ¯ CP) λ = −1.27641(56), B. Märkisch et al.: arXiv: 1812.04666 [nucl-ex] The Fierz-like interference term, caused by the contribution

  • f the pseudoscalar interaction beyond the SM only, was

calculated by M. González-Alonso and J. M. Camalich, PRL112, 042501 (2014); arXiv: 1309.4434 [hep-ph]. A complete set of corrections to all correlation coefficients and the electron-energy and angular distribution one may find in our paper arXiv:1905.04147 [hep-ph]

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 13

CKM Matrix Element Vud from Neutron Beta Decay

Vud from the neutron beta decay |Vud| =

  • 5099.81

τn(1 + ∆R)(1 + 2.7 × 10−4)W(1 − 0.33Cps) ∆R = α 2π ¯ g(E0) + ∆V

R

0.33Cps = −3.86 × 10−5 0.127 + Re(CP − ¯ CP)

  • Neutron beta decay: Vud = 0.97370(14)

Ch.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, PRL121, 241804 (2018) Ch.-Y. Seng, M. Gorchtein, and M. J. Ramsey-Musolf, arXiv:1812.03352 [nucl-th] Thus, one may assert that Wilkinson’s corrections and pseudoscalar interactions can, in principle, change digits at the fourth and fifth decimal places of the Vud value

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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SLIDE 14

Some estimates

x Vud from the rate of the neutron beta decay with Wilkinson’s corrections |Vud| = 0.97370(14) √ 1 + 2.7 × 10−4 = 0.97357(14) Re(CP − ¯ CP) from sensitivity of the Vud definition |Re(CP − ¯ CP)| < 7.13 Note added after this talk: f = 1.68857 × (1 + RC)MS × (1 + 2.7 × 10−4)W = = 1.68857 × (1 + 0.03886) × (1 + 2.7 × 10−4) = = 1.75466, where f is the phase volume and abbreviations MS and W mean Marciano-Sirlin’s and Wilkinson’s corrections. For RC = 0.03992 (Seng et al.) we get f = 1.75645

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity

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Thank You for Attention

  • A. N. Ivanov

Current and Future Status of the First-Row CKM Unitarity