symmetry preserving regularization of nuclear forces and ... · zwei-nukleon-kraft führender...

Hermann KrebsRuhr-Universität-Bochum

Progress in Ab Initio Techniques in Nuclear Physics

TRIUMF, Vancouver BC Canada

February 26, 2019

Symmetry Preserving Regularization of Nuclear Forces and Currents

Outline

Regularization of NN force

Violation of chiral symmetry by regulator in 3NF

Higher-derivative regularization in pion-sector

Respects all symmetries by construction

Application to 3NF and current operators

Deuteron form factors

Nuclear forces and currents in chiral EFT

Nuclear χEFT in the Precision Era Evgeny Epelbaum

Zwei-Nukleon-Kraft

Führender Beitrag

Korrektur 1. Ordnung



Drei-Nukleon-Kraft Vier-Nukleon-KraftTwo-nucleon force Three-nucleon force Four-nucleon force

LO (Q0)

NLO (Q2)

N2LO (Q3)

N3LO (Q4)

N4LO (Q5)

Figure 1: Chiral expansion of the nuclear forces. Solid and dashed lines refer to nucleons andpions, respectively. Solid dots, filled circles, filled rectangles, filled diamonds and open rectanglesrefer to the vertices of dimension ∆i = 0, ∆i = 1, ∆i = 2, ∆i = 3 and ∆i = 4, respectively.

the resulting contributions to the amplitude are enhanced by powers of mN/|p⃗ |, where mN refersto the nucleon mass, as compared to estimates based on dimensional analysis and underlying thederivation of Eq. (2.2). Fortunately, the contributions of the enhanced ladder-like diagrams canbe easily and efficiently resummed by solving the LS integral equation (or its generalizations inthe case of three- and more-nucleon systems) whose kernel involves all possible irreducible graphswhich obey the scaling according to Eq. (2.2) and are derivable in perturbation theory. This is theessence of what is commonly referred to as Weinberg’s approach to nuclear chiral EFT. The set ofall possible irreducible contributions to the scattering amplitude can be viewed as the interactionpart of the nuclear Hamiltonian and comprises two-, three- and more-nucleon forces. The approachoutlined above is straightforwardly generalizable to reactions involving external sources and allowsone to derive exchange currents consistent with the nuclear forces.

It is a simple exercise to enumerate the various diagrams which may contribute to the nu-clear force at a given order ν by looking at Feynman rules for the chiral Lagrangian and applyingEq. (2.2), see Fig. 1. Here, it is understood that the shown diagrams only serve the purpose ofvisualization of the corresponding contributions and do not have the meaning of Feynman graphs.In particular, one needs to separate out the irreducible pieces in order to avoid double counting.Notice further that while one can draw three-nucleon diagrams at next-to-leading order (NLO),the resulting contributions are either reducible or suppressed by one power of Q/mN [25]. As animmediate consequence of the chiral power counting in Eq. (2.2), one observes the suppression ofmany-body forces [26], the feature, that has always been assumed but could be justified only in thecontext of chiral EFT.

4

Weinberg ’90

Ordonez, van Kolck ’92

Ordonez, van Kolck ’92

Kaiser ’00 - ‘02

van Kolck ’94; Epelbaum et al. ’02

Bernard, Epelbaum, HK, Meißner,’08, ’11 Epelbaum ’06

Entem, Kaiser, Machleidt, Nosyk ’15Epelbaum, HK, Meißner ’15

Girlanda, Kievsky, Viviani ’11HK, Gasparyan, Epelbaum ’12,’13

(short-range loop contrib. still missing)

still have to be worked out

[parameter-free] [parameter-free]

Chiral Expansion of the Nuclear Forces

Available matrix elementsLENPIC ´19

Nuclear currents in chiral EFT

Axial-vector source

H[a, v, s, p]

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .

in [10�4fm4] with µ = M⇡/mN

�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT

�pLT (box) = 14 fLT

�nLT (box) = 4 fLT

�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

�0[10�4fm4]

Q2[GeV2]

m� [MeV]

Q2 = 0

�pLT (box)

1

Vector source Scalar source

Pseudoscalar source

Chiral EFT Hamiltonian depends on external sources

H[a, v, s, p]

abµ(x)

abµ(x)

Vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

�0[10�4fm4]

Q2[GeV2]

1

H[a, v, s, p]

abµ(x)

abµ(x)

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

�0[10�4fm4]

Q2[GeV2]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

p

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

p

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

�⇤

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

p

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

�⇤

n

Abµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

�⇤

n

abµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

µ�

⌫µ

�⇤

abµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

µ�

⌫µ

�⇤

abµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

n

vµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



�pLT (box) = 1.32⇥ 10�4fm4

�0[10�4fm4]

1

H[a, v, s, p]

abµ(x)

abµ(x)

e�

µ�

⌫µ

W

abµ

�

�0

�0

�LT

�LT

�pLT

�pLT (box) = 3.35µ2 � 2.78µ3 � 1.08µ4 + 1.67µ5 +

+ 0.10µ6 + 0.05µ7 � 0.01µ8 + . . .


�LT ⇠

Zd⌫

⌫3. . .

�L⇤0+E0+ + 2L⇤

1+(M1+ + 3E1+) + . . .

1

mN

in [10�4fm�4]

(9 + 5⌧3)fLT



1

Electroweak probes on nucleons and nuclei can be described by current formalism

single-nucleon two-nucleon three-nucleon

Q-3

Q-1

Q0

Q1

depend on d8, d9, d18, d21, d22,no 1/m corrections…

parameter-free

depend on C2, C4, C5, C7 + L1, L2; no loop corrections depend on CT

parameter-free static two-pion exchange

Chiral expansion of the electromagnetic current and charge operators

ci

1/m

di

ei

Vector currents in chiral EFT

Chemtob, Rho, Friar, Riska, Adam, Gari, …

Park, Min, Rho, Kubodera, Song, Lazauskas (earlier works, incomplete, TOPT)Pastore, Schiavilla et al. (TOPT), Kölling, Epelbaum, HK, Meißner (UT)

depend on CT

HK, Epelbaum, Meißner (UT)arXiv:1902.06839 [nucl-th]

Up to order Q onlysingle-nucleon current operator does depend on energy-transfer k0

Needed for verificationof continuity equation for OPE part

single-nucleon two-nucleon three-nucleon

Q-3

Q-1

Q0

Q1

depend on d2, d5, d6, d15, d18, d23,no 1/m corrections… parameter-free

depend on z1, z2, z3, z4; no loop corrections1/m corrections

parameter-free static two-pion exchange

depend on CT

Chiral expansion of the axial vector current and charge operators

ci

1/m

di

Axial vector operators in chiral EFT

Ando, Park, Kubodera, Myhrer (2002)

Park, Min, Rho (earlier works, incomplete, TOPT)Baroni et al. (TOPT), HK, Epelbaum, Meißner (UT)

di

1/m2

Bernard, Kaiser, Meißner (HBCHPT)

Regularization of NN Force Regularize one-pion-exchange propagator: (inspired by Rijken ’91)

all 1/Λ-corrections are short-range interactions

Reinert, HK, Epelbaum ’17

Implement similar regularization for two-pion exchange

Jµ

jµ

�,W,Z

p

N

1

q2 +M2⇡

!

exp⇣�

q2+M2

⇡⇤2

⌘

q2 +M2⇡

=1

q2 +M2⇡

�1

⇤2+

q2 +M

2⇡

2⇤4+ . . .

V (q) =2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2! V⇤(q) =

2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2exp

✓�q2 + µ

2

2⇤2

◆

400MeV < ⇤ < 550MeV

X(⌫, t) =1X

m,n=0

xmn⌫mtn

⌫̄µ µ�

W�

mp = 938MeV, mn = 940MeV

pi ⌧ ⇤�

mN ⇠ 1GeV ⇠ ⇤�

Eikin ⌧ mN

Eikin =

p2i

2mN+O(1/m2

N )

h A|

AX

i=1

�E

ikin =

qp2i +m

2N �mN

�| Ai < E

Ab ⌧ AmN

h A|

AX

i=1

p2i

mN| Ai < Eb ⌧ AmN

Q/⇤�

l0

A0(~k = 0, k0 = 0) = A

0LO(~k = 0, k0 = 0)

(g⇢)unquenched = 2.21(8)

�2/dof

T⇡ > 250MeV

d14�15 = �9.88(7)

⇤b ⇠ 600MeV

1

Jµ

jµ

�,W,Z

p

N

1

q2 +M2⇡

!

exp⇣�

q2+M2

⇡⇤2

⌘

q2 +M2⇡

=1

q2 +M2⇡

�1

⇤2+

q2 +M

2⇡

2⇤4+ . . .

V (q) =2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2! V⇤(q) =

2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2exp

✓�q2 + µ

2

2⇤2

◆

400MeV < ⇤ < 550MeV

X(⌫, t) =1X

m,n=0

xmn⌫mtn

⌫̄µ µ�

W�


pi ⌧ ⇤�

mN ⇠ 1GeV ⇠ ⇤�

Eikin ⌧ mN

Eikin =

p2i

2mN+O(1/m2

N )

h A|

AX

i=1

�E

ikin =

qp2i +m

2N �mN

�| Ai < E

Ab ⌧ AmN

h A|

AX

i=1

p2i

mN| Ai < Eb ⌧ AmN

Q/⇤�

l0

A0(~k = 0, k0 = 0) = A

0LO(~k = 0, k0 = 0)


�2/dof

T⇡ > 250MeV

d14�15 = �9.88(7)

⇤b ⇠ 600MeV

1

Compared to simple gaussian regulator πN-coupling gets quenched

Jµ

jµ

�,W,Z

p

N

exp

✓�

q2

⇤2

◆

1

q2 +M2⇡

!

exp⇣�

q2+M2

⇡⇤2

⌘

q2 +M2⇡

=1

q2 +M2⇡

�1

⇤2+

q2 +M

2⇡

2⇤4+ . . .

V (q) =2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2! V⇤(q) =

2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2exp

✓�q2 + µ

2

2⇤2

◆

400MeV < ⇤ < 550MeV

X(⌫, t) =1X

m,n=0

xmn⌫mtn

⌫̄µ µ�

W�


pi ⌧ ⇤�

mN ⇠ 1GeV ⇠ ⇤�

Eikin ⌧ mN

Eikin =

p2i

2mN+O(1/m2

N )

h A|

AX

i=1

�E

ikin =

qp2i +m

2N �mN

�| Ai < E

Ab ⌧ AmN

h A|

AX

i=1

p2i

mN| Ai < Eb ⌧ AmN

Q/⇤�

l0

A0(~k = 0, k0 = 0) = A

0LO(~k = 0, k0 = 0)


�2/dof

T⇡ > 250MeV

1

Jµ

jµ

�,W,Z

p

N

gA ! gA exp

✓�M

2⇡

2⇤2

◆< gA

exp

✓�

q2

⇤2

◆

1

q2 +M2⇡

!

exp⇣�

q2+M2

⇡⇤2

⌘

q2 +M2⇡

=1

q2 +M2⇡

�1

⇤2+

q2 +M

2⇡

2⇤4+ . . .

V (q) =2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2! V⇤(q) =

2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2exp

✓�q2 + µ

2

2⇤2

◆

400MeV < ⇤ < 550MeV

X(⌫, t) =1X

m,n=0

xmn⌫mtn

⌫̄µ µ�

W�


pi ⌧ ⇤�

mN ⇠ 1GeV ⇠ ⇤�

Eikin ⌧ mN

Eikin =

p2i

2mN+O(1/m2

N )

h A|

AX

i=1

�E

ikin =

qp2i +m

2N �mN

�| Ai < E

Ab ⌧ AmN

h A|

AX

i=1

p2i

mN| Ai < Eb ⌧ AmN

Q/⇤�

l0

A0(~k = 0, k0 = 0) = A

0LO(~k = 0, k0 = 0)


1

SMS Regulator & Chiral Symmetry

Chiral Symmetry is preserved by SMS regulator in NN force

NN contact LECs are not constrained by chiral symmetry

Chiral Symmetry is preserved by SMS regulator in 3N force at N2LO

There are no vertices with three or more pions at this order

Chiral Symmetry is violated by SMS regulator in 3N and 4N forces at N3LO

&Without modification of four-pion vertex nuclear forces start to depend on chiral parametrization

Call for Consistent RegularizationViolation of chiral symmetry due to different regularizations: Dim. reg. vs cutoff reg.

Naive local cut-off regularization of the current and potential

1/m - corrections to pion-pole OPE currentproportional to gA

&

First iteration with OPE NN potential

No such D-like term in chiral Lagrangian

To be compensated by two-pion-one-pion-exchange if calculated via cutoff regularization

In dim. reg. is finite

V Q0,⇤1⇡ = � g2A

4F 2⇡

⌧1 · ⌧2~q1 · ~�1~q1 · ~�2

q21 +M2⇡

exp

✓�q21 +M2

⇡

⇤2

◆

<latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit>

BERNARD, EPELBAUM, KREBS, AND MEIßNER PHYSICAL REVIEW C 84, 054001 (2011)

(7)(1) (2) (3) (4) (5) (6)

FIG. 3. Tree diagrams contributing to the two-pion-exchange and one-pion-exchange-contact topology of the 3NF at N3LO. The solidboxes denote insertions of either subsubleading di vertices from the effective pion-nucleon Lagrangian or the leading 1/m corrections. Fornotation see Fig. 1.

there are also no contributions at N3LO from tree diagramsinvolving one insertion of the higher-order di vertices in theeffective Lagrangian [see graphs (6) and (7) in Fig. 3] exceptfor the relativistic corrections which will be considered inSec. IV. As explained in Ref. [4], the contributions from thesediagrams are suppressed by at least one power of Q/m whereQ denotes a genuine soft scale.

We are thus left with Eqs. (2.1) and (2.3) as the onlynonvanishing contributions to the one-pion-exchange-contact3NF topology. We now show that these terms cancel eachother exactly if one takes into account the antisymmetricnature of few-nucleon states. In particular, we use theidentities

(τ 3σ

i3 + τ 2σ

i2

)A23 = 1

4

(τ 3σ

i3 + τ 2σ

i2 − τ 3σ

i2 − τ 2σ

i3

+ τ 2 × τ 3[σ⃗2 × σ⃗3]i)

≡ Bi ,

(τ 2 × τ 3[σ⃗2 × σ⃗3]i)A23 = 2Bi ,(τ 3σ

i2 + τ 2σ

i3

)A23 = −Bi , (2.4)

where the superscript i refers to the Cartesian component ofthe Pauli spin matrices and A23 denotes antisymmetrizationwith respect to nucleons 2 and 3, which, for a momentum-independent operator X, can be written in the form

(X)A23 ≡ 12

(X − 1 + σ⃗2 · σ⃗3

21 + τ 2 · τ 3

2X

). (2.5)

It is easy to see that adding the contribution from interchangingthe nucleons 2 and 3 to Eqs. (2.1) and (2.3) and performingantisymmetrization with respect to these nucleons leads toa vanishing result. Therefore, we conclude that there are noone-pion-exchange-contact terms in the 3NF at N3LO.

III. TWO-PION-EXCHANGE-CONTACT TOPOLOGY

We now turn to the two-pion-exchange-contact diagramsshown in Fig. 4. Evaluating the matrix elements of theoperators listed in Eq. (A1)

for diagrams (1)–(7) in this figure we find the g4ACS and g4

ACT contributions to the two-pion-exchange-contact topology of theform

V2π-cont = g4ACT

8F 4π

∫d3l

(2π )3

[τ 1 · τ 2

{(1

ω4+ω2

−+ 1

ω2+ω4

−

)[q2

1 (q⃗1 · σ⃗2) (q⃗1 · σ⃗3) + 2q21 l2 (σ⃗2 · σ⃗3) − q2

1 (l⃗ · σ⃗2)(l⃗ · σ⃗3)

−q41 (σ⃗2 · σ⃗3) − l2(q⃗1 · σ⃗2)(q⃗1 · σ⃗3) + l2(l⃗ · σ⃗2)(l⃗ · σ⃗3) − l4(σ⃗2 · σ⃗3)

]+

(1

ω4+ω2

−− 1

ω2+ω4

−

)

×[− q2

1 (q⃗1 · σ⃗2)(l⃗ · σ⃗3) + q21 (l⃗ · σ⃗2)(q⃗1 · σ⃗3) + l2(q⃗1 · σ⃗2)(l⃗ · σ⃗3) − l2(l⃗ · σ⃗2)(q⃗1 · σ⃗3)

]}

+2τ 2 · τ 3

(1

ω4+ω2

−+ 1

ω2+ω4

−

) [q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) + (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)

+(q⃗1 · l⃗)(l⃗ · σ⃗1)(q⃗1 · σ⃗2) − (q⃗1 · l⃗)2(σ⃗1 · σ⃗2) − l2(q⃗1 · σ⃗1)(q⃗1 · σ⃗2)]+ 6

(1

ω4+ω2

−+ 1

ω2+ω4

−

)

×[− q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) − (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)

− (q⃗1 · l⃗)(l⃗ · σ⃗1)(q⃗1 · σ⃗2) + (q⃗1 · l⃗)2(σ⃗1 · σ⃗2) + l2(q⃗1 · σ⃗1)(q⃗1 · σ⃗2)]]

DR= g4ACT

48πF 4π

{2τ 1 · τ 2(σ⃗2 · σ⃗3)

[3Mπ − M3

π

4M2π + q2

1

+ 2(2M2

π + q21

)A(q1)

]

+9[(q⃗1 · σ⃗1)(q⃗1 · σ⃗2) − q2

1 (σ⃗1 · σ⃗2)]A(q1)

}, (3.1)

054001-4


(7)(1) (2) (3) (4) (5) (6)




(τ 3σ

i3 + τ 2σ

i2

)A23 = 1

4

(τ 3σ

i3 + τ 2σ

i2 − τ 3σ

i2 − τ 2σ

i3

+ τ 2 × τ 3[σ⃗2 × σ⃗3]i)

≡ Bi ,

(τ 2 × τ 3[σ⃗2 × σ⃗3]i)A23 = 2Bi ,(τ 3σ

i2 + τ 2σ

i3

)A23 = −Bi , (2.4)


(X)A23 ≡ 12

(X − 1 + σ⃗2 · σ⃗3

21 + τ 2 · τ 3

2X

). (2.5)






V2π-cont = g4ACT

8F 4π

∫d3l

(2π )3

[τ 1 · τ 2

{(1

ω4+ω2

−+ 1

ω2+ω4

−

)[q2

1 (q⃗1 · σ⃗2) (q⃗1 · σ⃗3) + 2q21 l2 (σ⃗2 · σ⃗3) − q2

1 (l⃗ · σ⃗2)(l⃗ · σ⃗3)


]+

(1

ω4+ω2

−− 1

ω2+ω4

−

)

×[− q2


]}

+2τ 2 · τ 3

(1

ω4+ω2

−+ 1

ω2+ω4

−

) [q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) + (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)


(1

ω4+ω2

−+ 1

ω2+ω4

−

)

×[− q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) − (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)


DR= g4ACT

48πF 4π

{2τ 1 · τ 2(σ⃗2 · σ⃗3)

[3Mπ − M3

π

4M2π + q2

1

+ 2(2M2

π + q21

)A(q1)

]

+9[(q⃗1 · σ⃗1)(q⃗1 · σ⃗2) − q2

1 (σ⃗1 · σ⃗2)]A(q1)

}, (3.1)

054001-4


(7)(1) (2) (3) (4) (5) (6)




(τ 3σ

i3 + τ 2σ

i2

)A23 = 1

4

(τ 3σ

i3 + τ 2σ

i2 − τ 3σ

i2 − τ 2σ

i3

+ τ 2 × τ 3[σ⃗2 × σ⃗3]i)

≡ Bi ,

(τ 2 × τ 3[σ⃗2 × σ⃗3]i)A23 = 2Bi ,(τ 3σ

i2 + τ 2σ

i3

)A23 = −Bi , (2.4)


(X)A23 ≡ 12

(X − 1 + σ⃗2 · σ⃗3

21 + τ 2 · τ 3

2X

). (2.5)






V2π-cont = g4ACT

8F 4π

∫d3l

(2π )3

[τ 1 · τ 2

{(1

ω4+ω2

−+ 1

ω2+ω4

−

)[q2

1 (q⃗1 · σ⃗2) (q⃗1 · σ⃗3) + 2q21 l2 (σ⃗2 · σ⃗3) − q2

1 (l⃗ · σ⃗2)(l⃗ · σ⃗3)


]+

(1

ω4+ω2

−− 1

ω2+ω4

−

)

×[− q2


]}

+2τ 2 · τ 3

(1

ω4+ω2

−+ 1

ω2+ω4

−

) [q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) + (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)


(1

ω4+ω2

−+ 1

ω2+ω4

−

)

×[− q2

1 l2(σ⃗1 · σ⃗2) + q21 (l⃗ · σ⃗1)(l⃗ · σ⃗2) − (q⃗1 · l⃗)(q⃗1 · σ⃗1)(l⃗ · σ⃗2)


DR= g4ACT

48πF 4π

{2τ 1 · τ 2(σ⃗2 · σ⃗3)

[3Mπ − M3

π

4M2π + q2

1

+ 2(2M2

π + q21

)A(q1)

]

+9[(q⃗1 · σ⃗1)(q⃗1 · σ⃗2) − q2

1 (σ⃗1 · σ⃗2)]A(q1)

}, (3.1)

054001-4

Vg2A

2⇡,1/m = i g2A

32mF 4⇡

~�1·~q1 ~�3·~q3(q21+M2

⇡)(q23+M2

⇡)⌧1 ·(⌧2⇥⌧3)(2~k1 ·~q3+4~k3 ·~q3+i [~q1⇥~q3] ·~�2)

<latexit sha1_base64="keEJ9NVnUWLQEgUdpXC6B5PrHeM=">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</latexit>

Vg2A,⇤

2⇡,1/m = Vg2A

2⇡,1/m exp⇣� q21+M2

⇡⇤2

⌘exp

⇣� q23+M2

⇡⇤2

⌘

<latexit sha1_base64="xQgYWUdABOGw/xXlYDOf88AMIZQ=">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</latexit>

Vg2A,⇤

2⇡,1/m

1

E �H0 + i✏V

Q0,⇤1⇡ + V

Q0,⇤1⇡

1

E �H0 + i✏V

g2A,⇤

2⇡,1/m = ⇤g4A

128p2⇡3/2F 6

⇡

(⌧2 · ⌧3 � ⌧1 · ⌧3)~q2 · ~�2~q3 · ~�3

q23 +M2

⇡

+ . . .

<latexit sha1_base64="dsg9PakGsJG5R9EA31APFcE0x5w=">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</latexit>

V2⇡�1⇡<latexit sha1_base64="bwPbeLAAU+lXvqQmP0Vj6CSDpmw=">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</latexit>

V2⇡�1⇡<latexit sha1_base64="bwPbeLAAU+lXvqQmP0Vj6CSDpmw=">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</latexit>

Call for Consistent RegularizationViolation of chiral symmetry due to different regularizations: Dim. reg. vs cutoff reg.

35

FIG. 13: One-pion exchange diagrams leading to non-vanishing 1/m-contributions to ~A(Q)2N . Open rectangles refer to 1/m-

vertices from L(2)⇡N . For remaining notation see Fig. 10.

current operator have the form

~Aa (Q)

2N: 1⇡, 1/m=

gA

16F 2⇡m

⇢i[⌧ 1 ⇥ ⌧ 2]

a

1

(q21+M2

⇡)2

✓~B1 �

~k~k · ~B1

k2 +M2⇡

◆+

1

q21+M2

⇡

✓~B2

(k2 +M2⇡)2

+~B3

k2 +M2⇡

+ ~B4

◆�

+ ⌧a

1

1

(q21+M2

⇡)2

✓~B5 �

~k~k · ~B5

k2 +M2⇡

◆+

1

q21+M2

⇡

✓~B6

(k2 +M2⇡)2

+~B7

k2 +M2⇡

+ ~B8

◆��+ 1 $ 2 , (5.19)

where the vector-valued quantities ~Bi depend on various momenta and the Pauli spin matrices and are given by

~B1 = g2

A~q1 · ~�1[�2(1 + 2�̄8)~q1 ~k1 · ~q1 � (1� 2�̄8)(2~q1 ~k2 · ~q1 � i ~q1 ⇥ ~�2

~k · ~q1],

~B2 = (1� 2�̄8)g2

A~k~k · ~q1~q1 · ~�1[2~k · ~k2 � i~k · ~q1 ⇥ ~�2],

~B3 = 2~kh� g

2

A((1 + 2�̄9)~k · ~q1

~k1 · ~�1 + (1� 2�̄9)~q1 · ~�1(~k · ~k2 + ~k2 · ~q1))

+ ~q1 · ~�1(~k · ~k2 + i~k · ~q1 ⇥ ~�2 �~k1 · ~q1 + ~k2 · ~q1)

i,

~B4 = g2

A[2(1 + 2�̄9)~q1~k1 · ~�1 + (1� 2�̄9)~q1 · ~�1(2~k2 � i~k ⇥ ~�2)]� 2~q1 · ~�1(i~q1 ⇥ ~�2 � i~k ⇥ ~�2 + 2~k2),

~B5 = g2

A~q1 · ~�1

h(1� 2�̄8)(~q1 ~k · ~q1 � 2i ~q1 ⇥ ~�2

~k2 · ~q1)� 2i(1 + 2�̄8)~q1 ⇥ ~�2~k1 · ~q1

i,

~B6 = �(1� 2�̄8)g2

A~k ~q1 · ~�1[(~k · ~q1)

2� 2i~k · ~k2

~k · ~q1 ⇥ ~�2],

~B7 = g2

A~k

h(1� 2�̄9)~q1 · ~�1(�2i(~k · ~k2 ⇥ ~�2 + ~k2 · ~q1 ⇥ ~�2) + k

2 + q2

1)� 2i(1 + 2�̄9)~k1 · ~�1

~k · ~q1 ⇥ ~�2

i,

~B8 = �g2

A[(1� 2�̄9)~q1 · ~�1(~k � 2i~k2 ⇥ ~�2)� 2i(1 + 2�̄9)~q1 ⇥ ~�2

~k1 · ~�1]. (5.20)

It is not quite straightforward to make a connection between the derived relativistic corrections to the axial currentoperator and the corresponding 1/m-terms appearing in the 3N force at N3LO. This is because the later ones also

Naive local cut-off regularization of the current and potential

1/m - corrections to pion-pole OPE currentproportional to gA

~Aa,(Q: gA)2N: 1⇡,1/m = i [⌧1 ⇥ ⌧2]

a gA8F 2

⇡m

~k

(k2 +M2⇡)(q

21 +M2

⇡)

⇣~k2 · (~k + ~q1)� ~k1 · ~q1 + i~k · (~q1 ⇥ ~�2)

⌘+ 1 $ 2

<latexit sha1_base64="pc/RjRQA2o5x9mOzbr9DQ1mOeQg=">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</latexit><latexit sha1_base64="pc/RjRQA2o5x9mOzbr9DQ1mOeQg=">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</latexit><latexit sha1_base64="pc/RjRQA2o5x9mOzbr9DQ1mOeQg=">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</latexit><latexit sha1_base64="pc/RjRQA2o5x9mOzbr9DQ1mOeQg=">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</latexit>

&

First iteration with OPE NN potential

+ . . .<latexit sha1_base64="O0AJ3UeMWdfERnK6u2yhb0ckIR8=">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</latexit><latexit sha1_base64="O0AJ3UeMWdfERnK6u2yhb0ckIR8=">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</latexit><latexit sha1_base64="O0AJ3UeMWdfERnK6u2yhb0ckIR8=">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</latexit><latexit sha1_base64="O0AJ3UeMWdfERnK6u2yhb0ckIR8=">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</latexit>

No such counter term in chiral Lagrangian

~Aa,(Q: gA,⇤)2N: 1⇡,1/m

1

E �H0 + i✏V

O0,⇤

1⇡ + VO

0,⇤

1⇡

1

E �H0 + i✏

~Aa,(Q: gA,⇤)2N: 1⇡,1/m = ⇤

g3A

32p2⇡3/2F 4

⇡

([⌧1]a � [⌧2]

a)~k

k2 +M2⇡

~q1 · ~�1

<latexit sha1_base64="CKlqMD1cg7/1eQguz3xKnb+Ccm8=">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</latexit><latexit sha1_base64="CKlqMD1cg7/1eQguz3xKnb+Ccm8=">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</latexit><latexit sha1_base64="CKlqMD1cg7/1eQguz3xKnb+Ccm8=">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</latexit><latexit sha1_base64="CKlqMD1cg7/1eQguz3xKnb+Ccm8=">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</latexit>

To be compensated by two-pion-exchange current if calculated via cutoff regularization~Aa (Q)2N: 2⇡

<latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit>

In dim. reg. is finite~Aa (Q)2N: 2⇡

<latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit><latexit sha1_base64="/PLDto8BOfFaYgEyuqDo7EyX0d0=">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</latexit>

V Q0,⇤1⇡ = � g2A

4F 2⇡

⌧1 · ⌧2~q1 · ~�1~q1 · ~�2

q21 +M2⇡

exp

✓�q21 +M2

⇡

⇤2

◆

<latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit><latexit sha1_base64="b/NtLeCY3fRTDNgOToz5PMW07gM=">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</latexit>

~Aa,(Q: gA,⇤)2N: 1⇡,1/m = ~Aa,(Q: gA)

2N: 1⇡,1/m exp

✓�q21 +M2

⇡

⇤2

◆

<latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="I3qzNHs0O153gIiFHHCMQ4HedNo=">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</latexit><latexit sha1_base64="RzKsonPRXQIY85Nx+mm6t4RBpEY=">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</latexit><latexit sha1_base64="RzKsonPRXQIY85Nx+mm6t4RBpEY=">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</latexit><latexit sha1_base64="jYSm0v1vZjHs0E+HpvT11s3uJfM=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">AAADenicnVFba9RAFD7ZeGnrba345EvoIuzSdZssBYsitNiKD1ZacNtCpxsm2dl02NyczC7WkN/k79Hf0ge/jKmgRQpOmJwz3znnO7cgj2WhXfe71bJv3b5zd2l55d79Bw8ftR+vHhXZXIViFGZxpk4CXohYpmKkpY7FSa4ET4JYHAezt7X9eCFUIbP0k77IxVnCo1ROZcg1IL/9jRmOMojnomILEZY7lV+WTCXO8OMr1q88lsu+420k1bjkfad7CNCJ/J2+wz4gzYT3qjf/y9GrmPiSs1hMdfcFmyoelp99bzxc3/cRMR5WZZMDKlMyOtc9v91xB645znXFa5QONecga/8gRhPKKKQ5JSQoJQ09Jk4FvlPyyKUc2BmVwBQ0aeyCKlpB7BxeAh4c6Az/CK/TBk3xrjkLEx0iS4yrEOnQc9x3hjGAd51VQC8gL3G/Giz6Z4bSMNcVXkAGYFw2jPvANZ3D46bIpPG8quXmyLorTVPaMt1I1JcbpO4z/M2zC4sCNjMWh/aMZwSOwLwXmEAKOUIF9ZSvGBzT8QSSGykMS9owcvApyHr6qAdr9v5e6nXlaDjw3IF3uNnZdpuFL9EzWqMutvqStuk9HaCO0HpqvbZ2rb3Wpb1m9+z1X64tq4l5Qn8ce/Mnav3KYg==</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit><latexit sha1_base64="751PY/Rh1VIIF7/+OS6Lo74g0j8=">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</latexit>

= ⇤g3A

32p2⇡3/2F 4

⇡

([⌧1]a � [⌧2]

a)~k

k2 +M2⇡

~q1 · ~�1<latexit sha1_base64="oMo4TOYLd4D9UZR8pasjOu8XhAQ=">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</latexit>

Higher Derivative Regularization

Change leading order pion - Lagrangian (modify free part)

Every derivative should be covariant one

Jµ

jµ

�,W,Z

p

N

1

q2 +M2⇡

!

exp⇣�

q2+M2

⇡⇤2

⌘

q2 +M2⇡

=1

q2 +M2⇡

�1

⇤2+

q2 +M

2⇡

2⇤4+ . . .

V (q) =2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2! V⇤(q) =

2

⇡

Z 1

2M⇡

dµµ⇢(µ)

q2 + µ2exp

✓�q2 + µ

2

2⇤2

◆

400MeV < ⇤ < 550MeV

X(⌫, t) =1X

m,n=0

xmn⌫mtn

⌫̄µ µ�

W�


pi ⌧ ⇤�

mN ⇠ 1GeV ⇠ ⇤�

Eikin ⌧ mN

Eikin =

p2i

2mN+O(1/m2

N )

h A|

AX

i=1

�E

ikin =

qp2i +m

2N �mN

�| Ai < E

Ab ⌧ AmN

h A|

AX

i=1

p2i

mN| Ai < Eb ⌧ AmN

Q/⇤�

l0

A0(~k = 0, k0 = 0) = A

0LO(~k = 0, k0 = 0)


�2/dof

T⇡ > 250MeV

d14�15 = �9.88(7)

⇤b ⇠ 600MeV

1

has to be invariant under L(2)⇡,⇤ SU(2)L ⇥ SU(2)R ⇥U(1)V

Lagrangian should be formulated in terms of L(2)⇡,⇤ U(~⇡(x)) 2 SU(2)

<latexit sha1_base64="PePLN0e4Hxb9UhpWzJWIwNPK6Y4=">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</latexit><latexit sha1_base64="PePLN0e4Hxb9UhpWzJWIwNPK6Y4=">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</latexit><latexit sha1_base64="PePLN0e4Hxb9UhpWzJWIwNPK6Y4=">AAAC53ichVFNT9tAEH2YAuE7tMeqkkWElFwiEyERbpEoVS9IVK0JEkbINktY4i+tnYgQ5cQf4IS49si1/Br6W3ro8+IgAUKstZ7ZN2/ezux4SSDTzLIeJozJD1PTM6XZufmFxaXl8srH/TTuKV/YfhzE6sBzUxHISNiZzAJxkCjhhl4g2l53O4+3+0KlMo5+ZYNEHIVuJ5Kn0nczQsflL3bV6Qt/6CRyVL2o1RwZDR0Vmj/tUbVROy5XrLqll/nkrL90KijWXlz+CwcniOGjhxACETL6AVyk/A6xDgsJsSMMiSl6UscFRphjbo8sQYZLtMt/h6fDAo14zjVTne3zloBbMdPEGvc3reiRnd8q6Ke0/7gvNdZ584ahVs4rHNB6VJzVirvEM5yR8V5mWDDHtbyfmXeV4RRN3Y1kfYlG8j79J52vjChiXR0xsaOZHWp4+tznC0S0NivIX3msYOqOT2hdbYVWiQpFl3qKNn991sMxvxrqa8du1Lfq1o+NSqtZzLuEz1hFlUPdRAvfsccyfFzhDn9wb5wb18aNcftINSaKnE94tozf/wFa3pmL</latexit>

Based on ideas: Slavnov, NPB31 (1971) 301; Djukanovic et al. PRD72 (2005) 045002; Long and Mei PRC93 (2016) 044003

rµU = @µU � i(vµ + aµ)U + iU(vµ � aµ)U<latexit sha1_base64="LUs89hN7uvNs4Z2aj4FNl96mrH4=">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</latexit><latexit sha1_base64="LUs89hN7uvNs4Z2aj4FNl96mrH4=">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</latexit><latexit sha1_base64="LUs89hN7uvNs4Z2aj4FNl96mrH4=">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</latexit>

� = 2B(s+ ip)<latexit sha1_base64="aDruq+JdlBomY/ziT4OIq2kM750=">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</latexit><latexit sha1_base64="aDruq+JdlBomY/ziT4OIq2kM750=">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</latexit><latexit sha1_base64="aDruq+JdlBomY/ziT4OIq2kM750=">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</latexit>

Building blocks

Gasser, Leutwyler ´84, ´85; Bernard, Kaiser, Meißner ´95

S(2)⇡ =

Zd4x

1

2~⇡(x)

��@2 �M2

⇡

�~⇡(x) ! S(2)

⇡,⇤ =

Zd4x

1

2~⇡(x)

��@2 �M2

⇡

�exp

✓@2 +M2

⇡

⇤2

◆~⇡(x)

<latexit sha1_base64="8LmJAjd+YwoL7JwRvuGbLgtuhyw=">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</latexit><latexit sha1_base64="8LmJAjd+YwoL7JwRvuGbLgtuhyw=">AAAEE3iclVHLbtNAFL2uebTm0RSWbCwiUCJo5FhILQukSjzUBZWKSmilThON7Yk7ynhsjSdRiuXf4Df4AVaILUsWsIAfYcH1xGlaKlQxkXPv3HvOuWdmgkzwXHved2vJvnL12vXlFefGzVu3Vxtrd97l6ViFrBemIlUHAc2Z4JL1NNeCHWSK0SQQbD8YPa/6+xOmcp7Kt/okY0cJjSUf8pBqLA3WrG0SsJjLggoey9LZG5CM94uW3y6fES61G/WfuFMyVDQsumXhl2TCwgIxZWvaJoINdWudZFRpTkXfX98xdN8lisfHun0W/JDo1CHGcaFYVO4Nqs5j8hrNRrSczaz7gRizxfz/Hj+fvhhG2DSbwWdaC86jmlMWtRNMz/ONmbNzXSJTOU4CphzCZFRf3aDR9DqeWe5p0v07aUK9dtPGDyAQQQohjCEBBhI05gIo5Pg7hC54kGHtCAqsKcy46TMowUHuGFEMERSrI/yPcXdYVyXuK83csEOcIvBTyHThAX6vjGKA6GoqwzzH+Bu/96YW/3NCYZQrhycYA1RcMYo7WNdwjIjLmEmNnHu5nFmdSsMQNs1pOPrLTKU6Z3iq8wI7Cmsj03HhpUHGqBGY/QRvQGLsoYPqlucKrjlxhJGayIyKrBUp6imM1e2jH3zmC496Men5nacd743f3PLr916Ge3AfWvioG7AF27CLNkLro/XN+mn9sj/Yn+zP9pcZdMmqOXfh3LK//gGgYwyK</latexit><latexit sha1_base64="8LmJAjd+YwoL7JwRvuGbLgtuhyw=">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</latexit>

Higher Derivative RegularizationRegularization of pion - Lagrangian will not affect nucleon Green function

Schrödinger or LS-equations get not modified

Only nuclear forces get affected

We are not going to change pion-nucleon Lagrangian

Not every chiral symmetric higher derivative extension of pion - Lagrangian leads to a regularized theory

L(2)⇡ =

F 2

4Tr

⇥@µU

†@µU⇤! F 2

4Tr

h@µU

† exp⇣�~@ 2/⇤2

⌘@µU

i

<latexit sha1_base64="X0nb2ZxYNUFa0B2R/MkZCfECb6s=">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</latexit><latexit sha1_base64="X0nb2ZxYNUFa0B2R/MkZCfECb6s=">AAADpXicnVHbattAEB1FvSTuzW0f+yJqSh1oE0mExnkoGNqGPiSQgu0EvLZZyWtlsW6s1qap0Ic239KHHm3kQqEh0BWrmT1z5szMbpDHstCu+9Pasu/df/Bwe6f16PGTp8/az1+MimylQjEMszhTFwEvRCxTMdRSx+IiV4InQSzOg+WnOn6+FqqQWTrQV7mYJDxK5UKGXAOata9LFvK4dE6qalayXFbTsuvvVh/ZQvGwPJ76VXlQlUwlzkBVLBYLPWY5V1ryeMaSlTOcsjmPIqGcDTw1MFMyutQTpjOn9R9a4ntuCN33bC3CckNDd+ydX+2zE0w451P/pszuP2vP2h13z3Vdz/Oc2vEOP7hwjo56vtdzvDqE1aFmnWXta2I0p4xCWlFCglLS8GPiVOAbk0cu5cAmVAJT8KSJC6qohdwVWAIMDnSJf4TTuEFTnGvNwmSHqBJjK2Q69Ab72CgGYNdVBfwC9hf2D4NFt1YojXLd4RVsAMUdo3gKXNMlGHdlJg1z08vdmfVUmhbUM9NI9JcbpJ4z/KPzGREFbGkiDn0xzAgagTmvcQMp7BAd1Le8UXDMxHNYbqwwKmmjyKGnYOvbRz945s1bOrc7I3/Pc/e8bwedfq958G16Ra+pi1c9pD59pTP0EVp9a2FlVm6/tU/tgT26oW5ZTc5L+mvZs98YWeDN</latexit><latexit sha1_base64="X0nb2ZxYNUFa0B2R/MkZCfECb6s=">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</latexit><latexit sha1_base64="X0nb2ZxYNUFa0B2R/MkZCfECb6s=">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</latexit>

Unregularization of two propagators

Higher Derivative RegularizationFour-nucleon force as a regularization test

Only two linear combinations of momenta get regularized Unregularized 4NF

Which additional constrain is needed to construct a regularized theory?

All higher derivative terms of the non-linear sigma model Lagrangianin Slavnov, NPB31 (1971) 301 are proportional to equation of motion

Generalize this idea to chiral EFT: all additional terms ~ EOM

classical equation of motion for pions

Higher Derivative LagrangianTo construct a parity-conserving regulator it is convenient to work with building-blocks

Possible ansatz for higher derivative pion Lagrangian

Expand in Lorentz-invariance only perturbatively

Modified VerticesEnhanced by exp

✓p2

⇤2

◆

<latexit sha1_base64="oG18L7Wllw9mKikqm4NZQbnrPGA=">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</latexit><latexit sha1_base64="2ABhjDsrAc4bRI1GBro3OjpjfwI=">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</latexit><latexit sha1_base64="2ABhjDsrAc4bRI1GBro3OjpjfwI=">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</latexit>

Every propagator is suppressed by

Pionic sector becomes unregularized

Use dimensional on top of high derivative regularization

Dimensional regularization will not affect effective potential andSchrödinger or LS equations but will regularize pionic sector

exp

✓� p2

⇤2

◆

<latexit sha1_base64="vpN2DLKLv8v1t99Eodwvoe/ERdw=">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</latexit><latexit sha1_base64="EttVpYTguUmwuM+cAgTyyBPkd4o=">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</latexit><latexit sha1_base64="EttVpYTguUmwuM+cAgTyyBPkd4o=">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</latexit>

exp

✓� p2

⇤2

◆

<latexit sha1_base64="vpN2DLKLv8v1t99Eodwvoe/ERdw=">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</latexit><latexit sha1_base64="EttVpYTguUmwuM+cAgTyyBPkd4o=">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</latexit><latexit sha1_base64="EttVpYTguUmwuM+cAgTyyBPkd4o=">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</latexit>

exp

✓p2

⇤2

◆


exp

✓p2

⇤2

◆


Regularization of 3NF at N3LOModify pion-propagators in all 3NF diagrams

Recalculate 3NF diagrams with four-pion vertices

No four-pion vertices in ring and 2𝛑-1𝛑 topologies

&

Modified four-pion vertex leads to exponential increase in momenta

3NF’s do not depend onchiral parametrization of U - fields by construction

Large cut-off limit of the 3NF: we see linear divergence which we can not be absorbed by rescale of ci’s and CD LECs

Non-renormalizable 3NF at N3LO

Renormalizable 3NF at N3LOIntroduce additional unitary transformations to renormalize 3NF at N3LO⇠ 1/⇤2

<latexit sha1_base64="7/Zb0jb3mOqTpqGSZL4Qb3OYHpc=">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</latexit>

S(1/⇤2) =

↵(1/⇤2)

⇤2⌘H

(1)21

�1

E2⇡

H(2)22 �1E⇡H

(1)21 ⌘ � h.c.

<latexit sha1_base64="TPfp1F6LAkczRMAoH/+Dr4k1gao=">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</latexit>

U = exp⇣S(1/⇤2)

⌘

<latexit sha1_base64="WGVQbcx8tWXvEpBOoZJLQIT2U2M=">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</latexit>

with

U† V3N U<latexit sha1_base64="UQxXOtsQZbN+LCejfcuSyLofClA=">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</latexit>

becomes renormalizable for ↵(1/⇤2) = 24<latexit sha1_base64="5GP0Uz+YoLMpv8PyhAfnHuSUa/4=">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</latexit>

Introduced unitary transformation modifies relativistic corrections

�V (1/⇤2)1/m = i~�1 · ~q1~�3 · ~q3⌧1 · (⌧2 ⇥ ⌧3)

3g2A8F 4

⇡m⇤2(~k1 · (~q1 � ~q2) + ~k3 · (~q3 � ~q2))

exp⇣� q21+q23+2M2

⇡⇤2

⌘

q23 +M2⇡

<latexit sha1_base64="hJEaChm45ORLkyo8dSpvaXjlMYo=">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</latexit>

From naive dimensional analysis this D-like term is of order (N5LO)Q3<latexit sha1_base64="jNt1ThxT1+OE8ZYRQgdpQsRwTnc=">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</latexit>

Neglecting this term, however, will lead to divergent iteration with OPEin limit which can not be absorbed by LECs at any order⇤ ! 1

<latexit sha1_base64="lpZlr+gM6OTudoyngr2qNkfN+UQ=">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</latexit>

Regularization of Vector Current Modify pion-propagators in a vector current

Modify two-pion-photon vertex

Modified two-pion-photon vertex leads to exponential increase in momenta

Regularization of Vector Current Regularization of pion-exchange vector current

Riska prescription: longitudinal part of the current can be derived from continuity equation (Siegert theorem)

Riska, Prog. Part. Nucl. Phys. 11 (1984) 199hHstrong,⇢

i= ~k · ~J

A(�1)0,1

k0/m ⇠ Q4/⇤4

b

H()n,p

n p = d+3

2n+ p+ a� 4

Aµ(k) ! U†Aµ(k)U, P (k) ! U

†P (k)U, Hstrong ! U†HstrongU

Aµ(k) ! U†Aµ(k)U, ~K ! U

† ~KU, Hstrong ! U†HstrongU, Xµ ! U

†XµU

Xµ ! Xµ � i

h~e · ~K,Y µ(~k)

i+ ~e · ~rk

hHstrong,Y µ(~k)

i� Y ?

µ

h~e · ~K,Hstrong

i= i~e · ~P

hi~e · ~K, i

hHstrong, Yµ(~k)

ii= i

hHstrong,

hi~e · ~K,Yµ(~k)

ii� i~e · ~k Yµ(~k)

U [a] = exp

✓i

Zd3xY µ(~x) · aµ(~x, x0)

◆

U [p] = exp

✓i

Zd3xZ(~x) · p(~x, x0)

◆

Aµ(k) ! Aµ(k) + i

hHstrong,Y µ(~k)

i� i k0Y µ(~k)

P (k) ! P (k) + i

hHstrong,Z(~k)

i� i k0Z(~k)

lhs of cont. eq. ! lhs of cont. eq.+ i

hHstrong,

~k · ~Y (~k)i

rhs of cont. eq. ! rhs of cont. eq.+ i

hHstrong,

~k · ~Y (~k)i

lhs of cont. eq. ! lhs of cont. eq.+mq

hHstrong,Z(~k)

i

rhs of cont. eq. ! rhs of cont. eq.+mq

hHstrong,Z(~k)

i

h0out|0inia,v,s,p = exp (i Z[a, v, s, p]) = exp (i Z[a0, v0, s0, p0]) = h0out|0inia0,v0,s0,p0

U(R,L)

i@

@t = He↵ [a, v, s, p]

i@

@tU

†(R,L) = He↵ [a0, v

0, s

0, p

0]U†(R,L)

R = 1 +i

2⌧ · ✏R(x)

L = 1 +i

2⌧ · ✏L(x)

1

Higher orders work in progress

Application to Electromagnetic Charge Electromagnetic charge operators in chiral EFT

4 Hermann Krebs: Nuclear Currents in Chiral E↵ective Field Theory

5 Nuclear currents in chiral EFT

In this section we summarize all expressions of the nu-clear currents up to order Q in chiral expansion. We skiphere the discussion of their construction. All details aboutFeynman diagrams and specification of unitary phases canbe found in [2,6,7].

5.1 Power counting

We adopt here a two-nucleon power counting where weperform power counting of momenta and masses of lightdegrees of freedom (which we universaly denote by Q) inthe two-nucleon system. Even if we look at single nucleoncurrent we consider it in the presence of the second nu-cleon which plays a role of a spectator. Due to the presenceof spectator we have to multiply single nucleon current op-erator by a three-dimensional delta-function which countsas Q�3. A Feynman diagram which contributes to currentoperators counts as Q⌫ where

⌫ = �3 +X

i

Vii. (41)

Here Vi denotes the number of vertices of a type i wherei denotes inverse mass dimension of the coupling constantin the vertex

= d+3

2n+ p+ c� 4. (42)

Here, d is the number of derivatives and/or pion masses.The letters n, p and c denote here the number of nucleon-fields, pion-fields and the number of external sources, re-spectively.

We adopt here a two-nucleon power counting where1/m-contributions count as two powers of Q

Q

m⇠

Q2

⇤2�

. (43)

Here ⇤� ⇠ 700MeV is chiral symmetry breaking scale.

5.2 Vector current up to order Q

5.2.1 Single nucleon current

We start our discussion with electromagnetic vector cur-rent. Leading contribution to the vector current starts atthe order Q�3. At this order there is only a contribution tosingle-nucleon charge operator. It is well known that chi-ral expansion of the single nucleon current does not con-verge well. For moderate virtualities Q2

⇠ 0.3GeV2 phe-nomenological explicit inclusion of ⇢-meson is essential.For this reason the usual practice is to parametrize sin-gle nucleon vector current by e.g. Sachs form factors anduse their phenomenological form extracted from experi-mental data in practical calculations. The general formof the single-nucleon current can be characterized by its

non-relativistic one over the nucleon mass expansion givensymbolicaly by

V01N = V

01N:static + V

01N:1/m + V

01N:1/m2 , (44)

V 1N = V 1N:static + V 1N:1/m + V 1N:1/m2 + V 1N:o↵�shell.

In terms of Sachs formfactors the non-relativistic chargeis parametrized by

V01N:static = eGE(Q

2),

V01N:1/m =

i e

2m2k · (k1 ⇥ �)GM (Q2),

V01N:1/m2 = �

e

8m2

⇥Q

2 + 2 ik · (k1 ⇥ �)⇤GE(Q

2), (45)

and the non-relativistic current is given by

V 1N:static = �i e

2mk ⇥ �GM (Q2),

V 1N:1/m =e

mk1 GE(Q

2),

V 1N:1/m2 =e

16m3

ik ⇥ �(2k 2

1 +Q2) + 2 ik ⇥ k1 k1 · �

+ 2k1(ik · (k1 ⇥ �) +Q2)� 2k k · k1

+ 6 ik1 ⇥ � k · k1

�GM (Q2), (46)

where k is a photon momentum, k1 = (p 0 + p)/2, andp0(p) are outgoing (incoming) momenta of the single nu-cleon current operator. As already briefly explained inSec. 3 (see [8] for more comprehensive discussion) we ap-ply unitary transformations on Hamilton operator whichexplicitely depend on external source and thus on time.These unitary transformation generate o↵-shell contribu-tions to the longitudinal component of the current whichdepend on energy transfer k0 and additional relativistic1/m corrections. This contribution can also be parametrizedby Sachs formfactors via

V 1N:o↵�shell = k

✓k0 �

k · k1

m

◆e

Q2

�GE(Q

2)�GE(0)�

+i

2m2k · (k1 ⇥ �)

�GM (Q2)�GM (0)

��.

(47)

5.2.2 Two-nucleon vector current

We switch now to a discussion of the two-nucleon vectorcurrent operator. Various contributions can be character-ized by number of pion exchanges and/or short range in-teractions

Vµ

2N = Vµ

2N:1⇡ + Vµ

2N:2⇡ + Vµ

cont. (48)

One-pion-exchange vector current Leading contributionto the one-pion-exchange (OPE) current shows up at the

Hermann Krebs: Nuclear Currents in Chiral E↵ective Field Theory 5

order Q�1. At this order we get a well known result for

current operator

V (Q�1)2N:1⇡ = i

eg2A

4F 2⇡

[⌧ 1 ⇥ ⌧ 2]3 q2 · �2

q22 +M2

⇡

✓q1

q1 · �1

q21 +M2

⇡

� �1

◆

+ 1 $ 2, (49)

and for the charge operator

V0,(Q�1)2N:1⇡ = 0. (50)

Here e is electric coupling, gA axial vector coupling tothe nucleon, F⇡ pion decay constant, M⇡ pion mass, �i

and ⌧ i are Pauli spin and isospin matrices with label i =1, 2 labeling a corresponding nucleon. Momenta q1,2 aredefined b

qi = p0i� pi, (51)

where i = 1, 2 and p0ior pi are outgoing or incoming

momenta of the i-th nucleon, respectively.There is no contribution at order Q

0 such that thenext correction starts at the order Q which are leadingone-loop contributions in the static limit and/or leadingrelativistic correction to OPE current

Vµ,(Q)2N:1⇡ = V

µ,(Q)2N:1⇡, static + V

µ,(Q)2N:1⇡,1/m. (52)

Their explicit form of static contributions can be given interms of scalar functions f1...6(k). The vector contributionis given by [7]

V (Q)2N:1⇡, static =

�2 · q2

q22 +M2

⇡

q1 ⇥ q2

⇥[⌧ 2]

3f1(k)

+ ⌧ 1 · ⌧ 2 f2(k)] + [⌧ 1 ⇥ ⌧ 2]3 �2 · q2

q22 +M2

⇡

⇢

k ⇥ [q2 ⇥ �1] f3(k) + k ⇥ [q1 ⇥ �1] f4(k)

+ �1 · q1

✓k

k2�

q1

q21 +M2

⇡

◆f5(k)

+

�1 · q1

q21 +M2

⇡

q1 � �1

�f6(k)

�+1 $ 2 , (53)

where the scalar functions fi(k) are given by

f1 (k) = 2iegA

F 2⇡

d̄8 , f2 (k) = 2iegA

F 2⇡

d̄9 ,

f3 (k) = �iegA

64F 4⇡⇡2

⇥g3A(2L(k)� 1) + 32F 2

⇡⇡2d̄21

⇤,

f4 (k) = �iegA

4F 2⇡

d̄22 ,

f5 (k) = �ieg2A

384F 4⇡⇡2

2(4M2

⇡+ k

2)L(k) +

✓6 l̄6 �

5

3

◆k2

� 8M2⇡

⇤,

f6 (k) = �iegA

F 2⇡

M2⇡d̄18 . (54)

Here d̄i are low-energy constants (LEC) from Q3 pion-

nucleon Lagrangian. l̄6 is a LEC from Q4 pion-Lagrangian.

Their values can be fixed from pion-nucleon scattering andpion-photo- or elektroproduction. The charge contributionis given by

V0,(Q)2N:1⇡, static =

�2 · q2

q22 +M2

⇡

[⌧ 2]3

�1 · k q2 · kf7(k) + �1 · q2f8(k)

�

+ 1 $ 2, (55)

where

f7(k) = eg4A

64F 4⇡⇡

A(k) +

M⇡ � 4M2⇡A(k)

k2

�,

f8(k) = eg4A

64F 4⇡⇡

⇥(4M2

⇡+ k

2)A(k)�M⇡

⇤. (56)

The loop function L(k) and A(k) are defined

L(k) =1

2

pk2 + 4M2

⇡

klog

pk2 + 4M2

⇡+ kp

k2 + 4M2⇡� k

!,

A(k) =1

2karctan

✓k

2M⇡

◆. (57)

Relativistic corrections for the vector operator vanish

V (Q)1⇡:1/m = 0. (58)

Relativistic corrections for the charge operator are

V0,(Q)2N:1⇡,1/m =

eg2A

16F 2⇡mN

1

q22 +M2

⇡

⇢(1� 2�̄9)

⇥�[⌧ ]32 + ⌧ 1 · ⌧ 2

��1 · k�2 · q2 � i(1 + 2�̄9) [⌧ 1 ⇥ ⌧ 2]

3

⇥

�1 · k1�2 · q2 � �2 · k2�1 · q2 � 2

�1 · q1

q21 +M2

⇡

�2 · q2

⇥ q1 · k1

��+

eg2A

16F 2⇡mN

�1 · q2 �2 · q2

(q22 +M2⇡)2

(2�̄8 � 1)

⇥ ([⌧ 2]3 + ⌧1 · ⌧ 2)q2 · k + i [⌧ 1 ⇥ ⌧ 2]

3 ��2�̄8 � 1�q2 · k1

��2�̄8 + 1

�q2 · k2

��+ 1 $ 2. (59)

Here �̄8 and �̄9 are phases from unitary transformationswhich are not fixed. The same phases show up in nuclearforces. Usually they are fixed by requirement of minimalnon-locality of OPE NN potential.

Two-pion-exchange vector current Contributions to two-pion-exchange (TPE) vector current start to show up atorder Q. These contributions are parameter free. Due tothe appearance of three-point functions the expressionsare involved and will not be given here. Their explicitform can be found in App. B of [6].

Short range vector current XXX: I get factor 1/2di↵erence! To be checked XXX The first contribution

Static 2N charge operator does not contribute to deuteron form factors

Apply higher derivative regularizationto relativistic correction of the charge

1N charge operator is parametrized in terms of em form factors

HK, Epelbaum, Meißner arXiv:1902.06839 [nucl-th]

Charge Form Factor of Deuteron

Semilocal momentum space (SMS) regularized NN forceReinert, HK, Epelbaum ’17

Cutoff 450 MeV

Excellent description of the data for regularized charge even at higher momentum transfer Q

1N electromagnetic form factors Belushkin, Hammer, Meißner ’07

Preliminary

0.001

0.01

0.1

1

0 1 2 3 4 5 6

|GC(Q)|

Q [fm-1]

0.6

0.8

1

1.2

0 1 2 3 4

GC(Q) scaled

Q [fm-1]

0.95

1

0 1 20.001

0.01

0.1

1

0 1 2 3 4 5 6

GQ(Q)[fm

2 ]

Q [fm-1]

0.8

1

1.2

1.4

0 1 2 3 4

GQ(Q)/G

Q(0) scaled

Q [fm-1]

Out[737]=

0 1 2 3 4 510-4

0.001

0.010

0.100

1

k in fm-1

GC

One-body SMS (� = 400 to 550 MeV)

+ Two-body SMS (� = 400 to 550 MeV)

+ Two-body SMS (non-regularized current)

One-Body CDBonn2000

Abbott et al. (JLab, 2000)Abbott et al (JLab, 2000)

Truncation error band from Bayesian analysis:68% DoB, ⇤b = 600MeV

<latexit sha1_base64="CqPEVx5A/oaD5MvrFdxqbr4hA+s=">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</latexit>

More on this in Evgeny’s talk

Furnstahl et al.´15

�22

Summary

Local regulator in momentum space chosen not to affect analytic structureof NN force at any order in 1/𝝠-expansion

Naive application of the same regulator violates chiral symmetry in 3NF’s

Higher derivative in combination with dimensional regularization in pion-sectorregularizes 3NF’s and currents respecting all symmetries

Appearance of modified four-pion and pion-current vertices

First applications to deuteron form factors

Quadrupole Form Factor of Deuteron

Semilocal momentum space (SMS) regularized NN forceReinert, HK, Epelbaum ’17

Excellent description of the data for regularized charge even at higher momentum transfer k

1N electromagnetic form factors Belushkin, Hammer, Meißner ’07

Preliminary

0.001

0.01

0.1

1

0 1 2 3 4 5 6

|GC(Q)|

Q [fm-1]

0.6

0.8

1

1.2

0 1 2 3 4

GC(Q) scaled

Q [fm-1]

0.95

1

0 1 20.001

0.01

0.1

1

0 1 2 3 4 5 6

GQ(Q)[fm

2 ]

Q [fm-1]

0.8

1

1.2

1.4

0 1 2 3 4

GQ(Q)/G

Q(0) scaled

Q [fm-1]

Cutoff 450 MeV

Truncation error band from Bayesian analysis:68% DoB, ⇤b = 600MeV

<latexit sha1_base64="CqPEVx5A/oaD5MvrFdxqbr4hA+s=">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</latexit>

More on this in Evgeny’s talk

Out[737]=

0 1 2 3 4 510-4

0.001

0.010

0.100

1

k in fm-1

GC

One-body SMS (� = 400 to 550 MeV)

+ Two-body SMS (� = 400 to 550 MeV)

+ Two-body SMS (non-regularized current)

One-Body CDBonn2000

Abbott et al. (JLab, 2000)Abbott et al (JLab, 2000)

Furnstahl et al.´15

symmetry preserving regularization of nuclear forces and ... · zwei-nukleon-kraft führender...

Documents