Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes...

31
Baryon χPT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 1 / 31

Transcript of Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes...

Page 1: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Baryon χPT for Two-Boson Exchange Graph

Vadim Lensky

Johannes Gutenberg Universität Mainz

September 30, 2017

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 1 / 31

Page 2: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Motivation

Consider the γγ box at low energies (important corrections to epscattering, µ atoms)χPT — the low-energy EFT of QCD — is a suitable tool in this regimeRemove the lepton line =⇒ proton Compton scattering (CS)— wealth of exp. dataCalculate CS in χPT, confront data, make predictions for γγ box

Extend to γZ and γW at low energies?

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 2 / 31

Page 3: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 3 / 31

Page 4: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 4 / 31

Page 5: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

χPT framework

inlcude nucleons, photons, pions Weinberg, Gasser and Leutwyler, ...

count powers of small momenta p ∼ mπ

numerically e ∼ mπ/MN — count as p

also include the Delta isobar Hemmert, Holstein, ...

∆ = M∆ −MN is a new energy scale =⇒ δ-counting: Pascalutsa, Phillips (2002)

numerically δ = ∆/MN ∼√

mπ/MN , count ∆ ∼ p1/2 if p ∼ mπ

complications due to the spin-3/2 field (consistent couplings etc.)

two energy regimes:ω ∼ mπ:

n = 4L− 2Nπ − NN −12

N∆ +∑

kVk

ω ∼ ∆:n = 4L− 2Nπ − NN − N∆ − 2N1∆R +

∑kVk

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 5 / 31

Page 6: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

NLO: Born and πN Loops

Born graphs

responsible for low energy (Thomson) limit; point-like nucleonO(p2) and O(p3) (a.m.m. coupling)

π0 anomaly and πN loops

leading-order contribution to polarisabilities: O(p3)

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 6 / 31

Page 7: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

NNLO: Delta pole and π∆ loops

Delta pole and π∆ loops

different counting in different energy regimes

Delta pole: O(p4/∆) = O(p7/2) at ω ∼ mπ ; O(p) at ω ∼ ∆π∆ loops: O(p4/∆) = O(p7/2) at ω ∼ mπ ; O(p3) at ω ∼ ∆

at ω ∼ ∆ one needs to dress the 1∆R propagator

iΣ =

at ω ∼ ∆ corrections to γN∆ vertex are O(p2)

+= +

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 7 / 31

Page 8: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 8 / 31

Page 9: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 9 / 31

Page 10: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Results: RCS observables

covariant baryon calculationVL, McGovern, Pascalutsa (2015)

VL, Pascalutsa (2009)

NNLO (O(p7/2)) at ω ∼ mπ

NLO (O(p2)) at ω ∼ ∆

prediction of ChPT

polarisabilities are seenstarting at ∼ 50 MeVpion loops are important at lowenergies and around pionproduction thresholdDelta pole dominates in theresonance region

dW

@nb�sr

D

0 50 100 150

Θlab=180°

0 50 100 150EΓ @MeVD

Θlab=135°

0 50 100 1500

10

20

30

40Θlab=120°

Θlab=90°Θlab=60°

0

5

10

15

20 Θlab=45°

dW

@nb�sr

D

0 100 200 300

Θlab=180°

0 100 200 300EΓ @MeVD

Θlab=135°

0 100 200 3000

50

100

150

200 Θlab=120°

Θlab=90°Θlab=60°

0

100

200

300

400Θlab=45°

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 10 / 31

Page 11: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Scalar polarisabilities: status

��

��

��

��

��

��

�� �� ��� ��� ��� ���

������

�����

���������

����

����������

���

� ����������

�����

� �

������������������

����

��������

��

��

��

��

��

��

�� �� ��� ��� ��� ���

� ����������� ���

� �

�����������������

��������

�������

�����

����

���

��

��

��

��

��

��

�� �� ��� ��� ��� ���

� ����������� ���

� �

�����������������

��������

�������

�����

����

���

��

��

��

��

��

��

�� �� ��� ��� ��� ���

� ����������� ���

� �

�����������������

��������

�������

�����

����

���

��

��

��

��

��

��

�� �� ��� ��� ��� ���

� ����������� ���

� �

�����������������

��������

�������

�����

����

���

��

��

��

��

��

��

�� �� ��� ��� ��� ���

� ����������� ���

� �

�����������������

��������

�������

�����

����

���

for the proton, χPT calculations somewhat differ from other extractions (inparticular, TAPS value)

there are hints that the issue might be due to exp. dataKrupina, VL, Pascalutsa, in preparation

neutron polarisabilities are less well constrained — there is no freeneutron target

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 11 / 31

Page 12: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 12 / 31

Page 13: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

VVCS

forward VVCS amplitude

T (ν,Q2) =fL(ν,Q2) + (~ε ′∗ · ~ε ) fT (ν,Q2)

+ i~σ · (~ε ′∗ × ~ε ) gTT (ν,Q2)− i~σ · [(~ε ′∗ − ~ε )× q̂] gLT (ν,Q2)

low-energy expansion of the amplitude is

fT (ν,Q2) = f BT (ν,Q2) + 4π

[Q2βM1 + (αE1 + βM1)ν2] + . . .

fL(ν,Q2) = f BL (ν,Q2) + 4π(αE1 + αLν

2)Q2 + . . .

gTT (ν,Q2) = gBTT (ν,Q2) + 4πγ0ν

3 + . . .

gLT (ν,Q2) = gBLT (ν,Q2) + 4πδLTν

2Q + . . .

ν-dependent terms can be treated as functions of Q2 and related tomoments of nucleon structure functions

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 13 / 31

Page 14: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

VVCS: results

NLO/LO [O(p4/∆)/O(p3)]VL, Alarcón, Pascalutsa (2014)

HB χPT O(p4)Kao, Spitzenberg, Vanderhaeghen (2003)

IR O(p4)Bernard, Hemmert, Meissner (2003)

covariant χPT O(ε3)Bernard, Epelbaum, Krebs, Meissner (2013)

MAID

HB and IR do not provide adequatedescription

covariant χEFT works much better,especially in γ0 (HB is off the scale there)

δLT puzzle: difference between the twocovariant calculations (the one of Bernardet al. contains π∆ loops subleading in ourcounting)data on δLT from JLab expected

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

ΑE

1+Β

M1

H10-

4fm

3 L

Proton

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20Neutron

0.00 0.05 0.10 0.15 0.20 0.25 0.300

1

2

3

4

Q2 HGeV2L

ΑL

H10-

4fm

5 L

0.00 0.05 0.10 0.15 0.20 0.25 0.300

1

2

3

4

Q2 HGeV2L

0.00 0.05 0.10 0.15 0.20 0.25 0.30-4

-3

-2

-1

0

1

2

Γ0

H10-

4fm

4 L

Proton

0.00 0.05 0.10 0.15 0.20 0.25 0.30-3

-2

-1

0

1

2

3Neutron

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.5

1.0

1.5

2.0

2.5

Q2 HGeV2L

∆L

TH1

0-4

fm4 L

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.5

1.0

1.5

2.0

2.5

3.0

Q2 HGeV2L

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 14 / 31

Page 15: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 15 / 31

Page 16: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

VCS: response functions

++=

VCS Born VCS non-BornBethe-Heitler

e

γ

e′

p

e

p

e

e

p

p

pp

e

e

p′

Im ...Ν(p) (p’)Ν

γ∗(q) γ (q’)

Ν

γπ

N

Ν

γ∗

low-energy expansion (small ω′):Guichon, Liu, Thomas (1995)

d5σVCS =d5σBH+Born

+ ω′Φ Ψ0(Q2, ε, θ, φ) +O(ω′2);

Ψ0(Q2, ε, θ, φ) = V1

[PLL(Q2)−

PTT

ε

]+ V2

√ε(1 + ε)PLT (Q2)

at Q2 = 0:

PLL(0) =6Mαem

αE1

PLT (0) = −3M

4αemβM1

response functions ofVCS in covariant χETVL, Pascalutsa, Vanderhaeghen

(2016)

0.0 0.1 0.2 0.3 0.4 0.5

Q2 @GeV2D0

25

50

75

PL

L-

PT

T�¶@

GeV

-2 D

0.0 0.1 0.2 0.3 0.4 0.5

Q2 @GeV2D

-20

-10

0

PL

T@G

eV-

2 Dmore data from MAMI expected soonlow-Q2 expts. at MESA?

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 16 / 31

Page 17: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 17 / 31

Page 18: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Two-photon corrections

TPE — all TPE — elastic

polarisability = all − elastic

elastic is calculated from formfactors(empirical data)

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

dddddddddd

ddddddddddd

ddddddddddddddddddddddddd

µdddddddµdd

ddddddddddd

ddddddddddddddddddddddddddddd

dddddd

ddddddd

ddddd

ddddddddddd

ddddddddd

dddddddddd

ddddddddddddddddddddddddd

dddddd ddddd dddd dd ddd dddd dddddddddd

review by Hagelstein, Miskimen, Pascalutsa (2016)

muonic hydrogen Lamb shift in theory [energies in meV, Rp in fm]:

∆ELS = 206.0336(15)− 5.2275(10)R2p + ETPE Antognini et al (2013)

ETPE = 0.0332(20); E (pol) = 0.0085(11) Birse, McGovern (2012)

ETPE is an important source of th. uncertaintypolarisability corrections depend on the VVCS amplitude

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 18 / 31

Page 19: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Lamb shift

πN loops give the leading contributionDelta pole strongly suppressedπ∆ loops not included

(b) (c)(a)

(d) (e) (f )

(g) (h) (j)

the O(p3) result is∆E (pol)

2S = −8.2(+1.2−2.5) µeV

Alarcon, VL, Pascalutsa (2014)

consistent with othercalculations

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 19 / 31

Page 20: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

HFS (preliminary)

πN loops give the leading contributionDelta pole importantπ∆ loops not included

(b) (c)(a)

(d) (e) (f )

(g) (h) (j)

the O(p3) result isE (pol)

2S,HFS = −1.4(+1.6−1.2) µeV

Hagelstein, VL, Pascalutsa in preparation

large cancellationsdoes not agree with dispersivecalculations

dd dd dd dd dd dd dd ddd ddd

dχdddddd

ddddddddddddddddddddddd

dddddddddd

ddddddddddddddddddddddd

dddddddddddddddddddd

dddddddddddddddddddd

dddddddddddddµddd

dd dd dd dd dd dd dd ddd ddd

dχdddddd

ddddddddddddddddddddddd

dddddddddd

ddddddddddddddddddddddd

dddddddddddddddddddd

dddddddddddddddddddd

dddddddddddddµddd

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 20 / 31

Page 21: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Lamb shift: HB vs. covariant

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

ddd ddd ddd ddd ddd ddd dd dd

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddχddddddddddddddddddddddddddχdddddddddddddddddddddddddχddddddddddddddddddddddddd

∆ddddddddµddd

∆E (pol)nS =

αem

πφ2

n

∫ Qmax

0

dQQ2

w(τ`)[T (NB)

1 (0,Q2)− T (NB)2 (0,Q2)

]

one can expect larger error in HB sincethe integral converges more slowly thereneither HB nor covariant (nor otherresults), however, can explain the missing∼ 300 µeV

HBΧPT

BΧPT

-17.9

-8.2

0 0.2 0.4 0.6 0.8 1

-5

-10

-15

Qmax2HGeV2L

DE2 S

HpolLHΜeVL

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 21 / 31

Page 22: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Outline

1 χPT framework

2 Results for nucleon Compton scattering and µHRCSVVCSVCSLamb shift and HFS

3 Some thoughts on extension to γZ and γW

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 22 / 31

Page 23: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Some thoughts (and summary)

covariant baryon χEFT works well in nucleon CS (RCS/VCS/VVCS) andgives a O(p3) prediction for µH Lamb shift and hyperfine splittingby connecting γγ box to nucleon CS, we cross-check our χPT calculationand better understand these processeshow can χPT be applied also to γZ and γW boxes?

naïvely: can work at low energies (P2@MESA, β-decays)could follow a route similar to γγ box:

calculate the analog of CS (“vector boson production”)calculate the boxes

could provide a systematic evaluation of γZ and γW boxesfeedback and suggestions welcome!

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 23 / 31

Page 24: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Backup

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 24 / 31

Page 25: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

χPT for nucleon CS and µH

real CS in χPTprediction at O(p4/∆)compare with datapredict nucleon polarizabilities

virtual and doubly-virtual CS in χPTprediction at O(p4/∆)nucleon generalized polarizabilities (different in VCS and VVCS!)

we can calculate µH (Lamb shift and HFS) in χPTprediction at O(p3)

=⇒ RCS, VCS, VVCS and Lamb shift calculated in a single χPT frameworkcomplementary information about properties of the nucleon(verify sum rules in χPT etc.)

our choice is to use the covariant formulation of χPT

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 25 / 31

Page 26: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Lamb shift and Compton scattering

Compton scattering amplitude (forward VVCS)

Tµν(q, p) =

(−gµν +

qµqν

q2

)T1(ν,Q2) +

1M2

(pµ −

p · qq2

qµ)(

pν −p · qq2

qν)

T2(ν,Q2)

−1Mγµναqα S1(ν,Q2)−

1M2

(γµνq2 + qµγναqα − qνγµαqα

)S2(ν,Q2)

spin-dependent terms contribute to hyperfine splitting

nth S-level shift is given by

∆E (pol)nS =

αem

πφ2

n

∫ ∞0

dQQ2

w(τ`)[T (NB)

1 (0,Q2)− T (NB)2 (0,Q2)

]

w(τ`): the lepton weighting function

w(τ`) =√

1 + τ` −√τ`, τ` =

Q2

4m2`

weighted at low virtualities

wHΤΜL

wHΤeL

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.2

0.4

0.6

0.8

1.0

Q2 HGeV2L

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 26 / 31

Page 27: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Compton amplitudes

T1(ν,Q2) and T2(ν,Q2) can be related, via dispersive integrals, withnucleon structure functions (ν̃ = 2Mν/Q2):

T1(ν,Q2) =8παM

1∫0

dxx

f1(x ,Q2)

1− x2ν̃2 − i0, T2(ν,Q2) =

16παMQ2

1∫0

dxf2(x ,Q2)

1− x2ν̃2 − i0

the integral for T1 needs a subtraction: unknown function T1(0,Q2)

high-Q2 behaviour of T1(0,Q2) needs to be modelledformfactors Pachucki (1999), Martynenko (2006)

χPT-inspired formfactors Carlson, Vanderhaeghen (2011), Birse, McGovern (2012)

empirical fits Tomalak, Vanderhaeghen (2016)

something we know about T1(0,Q2): low-energy theorem

T NB1 (0,Q2) = 4πβM1Q2 + · · · , T NB

2 (0,Q2) = 4π(αE1 + βM1)Q2 + · · ·

=⇒ nucleon polarisabilities

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 27 / 31

Page 28: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Polarisabilities

point particle (or low energies) =⇒ charge, mass, a.m.m.higher energies: response of the nucleon to external e.m. field

=⇒ static polarisabilities: low-energy constants of effective γN interaction

H(2)eff = −

12

4π(αE1~E2 + βM1

~H2),

H(3)eff = −

12

4π(γE1E1~σ · ~E × ~̇E + γM1M1~σ · ~H × ~̇H − 2γM1E2Eijσi Hj + 2γE1M2Hijσi Ej

),

H(4)eff = −

12

4π(αE1ν~̇E2 + βM1ν

~̇H2)−1

124π(αE2E2

ij + βM2H2ij ), . . .

Aij =12

(∇i Aj +∇j Ai ), A = ~E , ~H

this EFT breaks down around the pion production thresholdwe can calculate the polarisabilities from our more high-energy theory —

χPT... or find from fits to data (with some help of χPT)

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 28 / 31

Page 29: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

One more step: fit at O(p4) (partial)

add a dipole polarisabilities contact term; fit δαE1 and δβM1 to data

L(4)πN = πe2N

(δβM1FµρFµρ + 2

M2 (δαE1 + δβM1)∂µFµρF νρ∂ν

)N

results:VL, McGovern (2014)

αE1 = 10.6± 0.5

βM1 = 3.2± 0.5

Baldin constrainedχ2/d.o.f. = 112.5/136

Θ lab=60°

200 220 240 260 280 300 320 3400

50

100

150

200

250

300

0 50 100 150 2000

10

20

30

40

50

Θ lab=85°

200 220 240 260 280 300 320 3400

50

100

150

200

0 50 100 150 2000

10

20

30

40

50

Θ lab=133 °

200 220 240 260 280 300 320 3400

50

100

150

0 50 100 150 2000

10

20

30

40

50

Θ lab=155 °

200 220 240 260 280 300 320 3400

50

100

150

0 50 100 150 2000

10

20

30

40

50

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 29 / 31

Page 30: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Sum rules

sum rules connecting VCS, RCS, and VVCS:Pascalutsa, Vanderhaeghen (2015)

δLT = −γE1E1 + 3Mαem

[P′ (M1M1)1(0)− P′ (L1L1)1(0)

],

I′1(0) =κ2

N12〈r2

2 〉+M2

2

{1αem

γE1M2 − 3M[P′ (M1M1)1(0) + P′ (L1L1)1(0)

]}

verified in covariant and HB χPT VL, Pascalutsa, Vanderhaeghen, Kao (2017)

connect experimentally accessible quantitiesallow to obtain complementary information on, e.g., static spinpolarisabilitieshigher-order scalar sum rules VL, Pascalutsa, Vanderhaeghen, Hagelstein in preparation

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 30 / 31

Page 31: Baryon PT for Two-Boson Exchange GraphBaryon ˜PT for Two-Boson Exchange Graph Vadim Lensky Johannes Gutenberg Universität Mainz September 30, 2017 Vadim Lensky (U. Mainz) Electroweak

Sum rules and δLT puzzle

δLT = −γE1E1 + 3Mαem

[P′ (M1M1)1(0)− P′ (L1L1)1(0)

]

δLT puzzle shows here

the result of Bernard et al. for δLTseems to be in contradiction withMAID

new JLab data for the proton δLTare expected to shed light on thispuzzle

Vadim Lensky (U. Mainz) Electroweak Box Workshop September 30, 2017 31 / 31