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using chiral perturbation theory 2018.9.11-12 ELPH C023 in P-wave pion-nucleus interaction Effects of wavefunction renormalization

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Tokyo Metropolitan University

  • K. Aoki
  • Tokyo Institute of Technology
  • D. Jido
  • Tottori University
  • N. Ikeno
  • Nara Women’s University
  • S. Hirenzaki

2018.9.11-12 ELPH研究会 C023 『原子核中におけるハドロンの性質とカイラル対称性の役割』

カイラル摂動論を用いたπ中間子-原子核相互作用の P波成分に対する波動関数くりこみの影響

『Effects of wavefunction renormalization in P-wave pion-nucleus interaction using chiral perturbation theory』

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2

π-nucleus system: In-medium changes of π-N interaction

π-nucleus S-wave optical potential [linear density approximation] ̶> ・mean free path in nuclear medium is as large as 5-10 fm ★low-energy π-nucleus interaction expected from πN interaction ・single step πN scattering should be dominant

2ωπVopt = −4π

  • 1 + ωπ

MN

  • [b0(ρp + ρn) + b1(ρp − ρn)]

bfree , bfree

1

bfit

0 , bfit 1

πN scattering length

Fits to π-nucleus experimental data

ρeff ∼ 0.6ρ0

pionic atom

  • R = bfree

1

bfit

1

= 0.78 ± 0.05

  • K. Suzuki et al. PRL92, 072302 (2004)

pion-nucleus elastic scattering

  • R = bfree

1

bfit

1

= 0.68

Friedman, et al. PRL93, 122302(2004)

linear density approximation is not valid.

bfreeρ → Xbfreeρ

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SLIDE 3

NG boson ・self-energy ・energy ・momentum

3

Wavefunction renormalization

★Wavefunction renormalization

  • 1. One of the higher order corrections beyond linear density approximation.
  • 2. When self-energy (optical potential) has strong energy-dependence,

wavefunction renormalization is large. For NG bosons, such as pion…… ーChiral perturbation theory (low-energy QCD effective theory) ーNG bosons are written as their energy (momentum) expansion ーNG boson-nucleon interaction has strong energy dependence ーWe expect that NG boson wavefunction renormalization is large

Study in-medium pion properties by the effects of wavefunction renormalization

  • 2mπVopt =
  • 1 + ∂Π

∂ω2

  • Π − ∇

∂Π ∂ω2 + ∂Π ∂q2

  • ∇

★Optical potential in terms of wavefunction renormalization.

q ω Π

Wavefunction renormalization correction to kinetic energy term

Deeply bound pionic atom

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4

Our study

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term

correction to kinetic energy term

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5

Our study

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term Partial wave expansion up to the P-wave

  • Π = ΠS + ΠP ⃗

pcm · ⃗ p ′

cm

  • Z = 1 + ∂Π

∂ω2 = 1 + ∂ΠS ∂ω2 + ∂ΠP ∂ω2 ⃗ pcm · ⃗ p ′

cm

= 1 + zS + zP ⃗ pcm · ⃗ p ′

cm

  • Kolomeitsev, Kaiser, Weise, PRL90, 092501 (2003)

Jido, Hatsuda, Kunihiro, PRD63, 011901 (2001); PLB670, 109 (2008)

★P-wave term S-wave term

  • 1 + ∂Π

∂ω2

  • Π =
  • 1 + zS + zP ⃗

pcm · ⃗ p ′

cm

ΠS + ΠP ⃗ pcm · ⃗ p ′

cm

  • = (1 + zS) ΠS + [(1 + zS) ΠP + zP ΠS] ⃗

pcm · ⃗ p ′

cm + · · ·

correction to kinetic energy term

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SLIDE 6

6

Our study

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term momentum space coordinate space ★P-wave term S-wave term

  • 1 + ∂Π

∂ω2

  • Π =
  • 1 + zS + zP ⃗

pcm · ⃗ p ′

cm

ΠS + ΠP ⃗ pcm · ⃗ p ′

cm

  • = (1 + zS) ΠS + [(1 + zS) ΠP + zP ΠS] ⃗

pcm · ⃗ p ′

cm + · · ·

  • 1 + ∂Π

∂ω2

  • Π(r) = (1 + zS) ΠS + ∇ · [−(1 + zS)ΠP − zP ΠS] ∇

= s(r) + ∇ · p(r)∇

S-wave term ★P-wave term

correction to kinetic energy term

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SLIDE 7

7

Our study

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term

  • ω2 − m2

π + ∇2 − 2mπVopt(r)

  • φ(r) = 0

★Klein-Gordon equation

  • 1 − p(r) + ∂Π

∂ω2 + ∂Π ∂q2

  • ∇2φ(r) + [ω2 − m2

π − s(r) + · · · ]φ(r) = 0

correction to kinetic energy term

correction to kinetic energy term

kinetic energy term

★kinetic term should be >0 ★In case of <0 ̶> instability

  • T. E. O. Ericson and F. Myhrer, Phys. Lett. 74B(1978)163
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8

Our study

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term P-wave phenomenological potential (Ericson-Ericson type)

2µVP (r) = 4π∇ · [c(r) + −1

2 C0ρ2(r)]L(r)∇

c(r) = −1

1

  • c0[ρp(r) + ρn(r)] + c1[ρn(r) − ρp(r)]
  • L(r) =

1 1 + 4

3πλ[c(r) + −1 2 C0ρ2(r)]

★Lorentz-Lorentz correction ★linear density ★kinematical factor

1 = 1 + mπ MN 2 = 1 + mπ 2MN

We compare real part of the potential

  • M. Ericson, T. E. O. Ericson, Ann. Phys. 36(66)496

correction to kinetic energy term

b0 = −0.0283 m−1

π

b1 = −0.12 m−1

π

c0 = 0.223 m−3

π

c1 = 0.25 m−3

π

B0 = 0.042i m−4

π

C0 = 0.10i m−6

π

λ = 1.0

★potential parameters

  • R. Seki and K. Masutani, PRC27(83)2799
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9

Method

①Effects of wavefunction renormalization in P-wave pion-nucleus interaction. ②Correction to kinetic energy term ★How to construct self-energyΠ? STEP1 elementary process of π- -nucleus interactions ̶> πN elastic scattering amplitude STEP2 π-self-energy based onπ-N amplitude STEP3 Wavefunction renormalization Momentum derivative

correction to kinetic energy term

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10

π−p → π−p

Weinberg-Tomozawa term Born term（s-channel） NLO term

gA

fπ c1, c2, c3, c4

χ2 fitting differential cross section at Tπ = 25.8 MeV

π+p → π+p

fπ c1, c2, c3, c4

Weinberg-Tomozawa term NLO term Born term（u-channel）

gA

Our study

STEP1: πN elastic scattering amplitude

Chiral perturbation theory [low-energy QCD effective theory]

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11

STEP2: π-self-energy in nuclear medium based on π-N amplitude

Model of self-energy (optical potential) ̶> linear density approximation ★πN elastic scattering amplitude ★π-self-energy (optical potential) assuming isospin symmetry

Tπ+p = Tπ−n

Our study

2mπVopt = Π = −Tπ−pρp − Tπ−nρn

  • STEP3: Wavefunction renormalization

Momentum derivative

Z = 1 + ∂Π ∂ω2 = 1 − ∂Tπ−p ∂ω2 ρp − ∂Tπ−n ∂ω2 ρn

∂Π ∂q2 = −∂Tπ−p ∂q2 ρp − ∂Tπ−n ∂q2 ρn

Tπ+p Tπ−p

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SLIDE 12

12

Results

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SLIDE 13

13

  • Fig. 1

π−p → π−p differential cross section

0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=25.8 MeV/c

  • H. Dens et al.,

PLB 633 (2006)

Tpi = 25.8 MeV

pion kinetic energy

  • c1 = −0.8 × 10−3 c2 = 2.8 × 10−3 c3 = −4.1 × 10−3 c4 = 3.9 × 10−3 [MeV−1]

χ2/N = 4.2

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SLIDE 14

0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=19.9 MeV/c 0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=32.0 MeV/c 0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=37.3 MeV/c 0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=43.3 MeV/c

Tpi = 19.9 MeV Tpi = 32.0 MeV Tpi = 37.3 MeV Tpi = 43.3 MeV

  • Fig. 2

π−p → π−p differential cross section

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SLIDE 15

0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=19.9 MeV/c 0.5 1 1.5 2 30 60 90 120 150 180 d/d [mb/sr] c.m. [deg] Tpi=25.8 MeV/c

Tpi = 19.9 MeV Tpi = 25.8 MeV

  • Fig. 3

differential cross section

π+p → π+p

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SLIDE 16

0.1 0.2 0.3 0.4 0.5 0.6 2 4 6 8 10 zS r [fm]

★S-wave wavefunction renormalization gives 50% enhancement for in-medium πN interaction.

16

  • Fig. 4 S-wave wavefunction renormalization

121Sn

Tπ=0 threshold

= 1 − ∂Tπ−p ∂ω2 ρp ★P-wave wavefunction renormalization is considerably small.

p − ∂Tπ−n

∂ω2 ρn

∂ω = 1 − ∂Tπ−p ∂ω2 ρp − ∂Tπ−n ∂ω2 ρn

Z = 1 + ∂Π ∂ω2 = 1 − ∂Tπ−p ∂ω2 ρp − ∂Tπ−n ∂ω2 ρn

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SLIDE 17
  • 0.6
  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

2 4 6 8 10 zS

q

r [fm]

17

  • Fig. 5 S-wave momentum derivative

121Sn

Tπ=0 threshold

−∂Tπ−p ∂q2 ρp −

p − ∂Tπ−n

∂q2 ρn

−∂Tπ−p ∂q2 ρp − ∂Tπ−n ∂q2 ρn

★P-wave derivative is considerably small.

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SLIDE 18

0.5 1 1.5 2 2.5 2 4 6 8 10 p(r) r [fm]

18

  • Fig. 6 P-wave term with wavefunction renormalization

Tρ+ renormalization

Wavefunction renormalization Zs

  • 2µVopt = s(r) + ∇ · p(r)∇

121Sn

★Our Tρ potential is consistent with Ericson-Ericson linear potential ★Effect of wavefunction renormalization is opposite direction to phenomenology. linear(Ericson-Ericson) Tρ(Ours)

phenomenology (Ericson-Ericson)

p(r) = −(1 + zS)ΠP − zP ΠS

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SLIDE 19

0.5 1 1.5 2 2.5 2 4 6 8 10 p(r) r [fm]

19

  • Fig. 6 P-wave term with wavefunction renormalization

Tρ+ renormalization

Wavefunction renormalization Zs

  • 2µVopt = s(r) + ∇ · p(r)∇

121Sn

★Our Tρ potential is consistent with Ericson-Ericson linear potential ★Effect of wavefunction renormalization is opposite direction to phenomenology. linear(Ericson-Ericson) Tρ(Ours)

phenomenology (Ericson-Ericson)

p(r) = −(1 + zS)ΠP − zP ΠS

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20

  • Fig. 7 P-wave term with kinetic energy term correction

121Sn

  • 1 − p(r) + ∂Π

∂ω2 + ∂Π ∂q2

  • ∇2φ(r) + [ω2 − m2

π − s(r) + · · · ]φ(r) = 0

[1 − p′(r)] ∇2φ(r) + [ω2 − m2

π − s(r) + · · · ]φ(r) = 0

p′(r) = p(r) − ∂Π ∂ω2 − ∂Π ∂q2

phenomenology (Ericson-Ericson)

linear(Ericson-Ericson) Tρ(Ours)

Tρ+ renormalization

) − ∂Π ∂ω2 − ∂Π ∂q2

p0(r)

★Klein-Gordon equation

0.5 1 1.5 2 2.5 2 4 6 8 10 p(r) r [fm]

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21

Summary

ーEffect of wavefunction renormalization to P-wave interaction ーCorrection to kinetic energy term ★Our study ★Previous study Importance of wavefunction renormalization to S-wave interaction

Kolomeitsev, Kaiser, Weise, PRL90, 092501 (2003) Jido, Hatsuda, Kunihiro, PRD63, 011901 (2001); PLB670, 109 (2008)

Taking into account other in-medium effects ★Outlook Fermi motion of in-medium nucleon