Hadron Electric Polarizability from Lattice QCD

Hadron electric polarizability from lattice QCD

Andrei Alexandru Michael Lujan, Walter Freeman, Frank Lee

6th Workshop of the APS Topical Group on Hadronic Physics — Baltimore — April 2015 Outline

• E&M polarizabilities and lattice QCD • Background ﬁeld method • Neutral kaon and pion electric polarizability • Neutron electric polarizability • Future directions

2 Preliminaries

• The hadron mass changes when placed in a electromagnetic ﬁeld 1 1 H = µ B E2 ⇥B2 em · 2 2 1 1 ⇤ E E˙ ⇤ B B˙ 2 E1 · ⇥ 2 M1 · ⇥ + ⇤ ⌅ E B ⇤ ⌅ B E E2 i ij j M2 i ij j 1 1 E2 ⇥ B2 + ... 12 E2 ij 12 M2 ij

• μ - the magnetic dipole • α & β - the electric & magnetic polarizability • γ - spin polarizabilities

3 Electric polarizability

• The polarizability measures the dipole moment induced by the ﬁeld

The polarizabilities are measured in Compton • p = E scattering experiments for proton and from elastic and quasi-elastic Compton scattering on deuteron, lead, etc. for neutron

1 E = E2 2

4 Lattice QCD primer

• The basic degrees of freedom are quarks and gluons interacting according to Quantum Chromodynamics.

• Quark and gluon ﬁelds are sampled on a discrete lattice: quarks at sites and glue on links. a

• The action in Euclidean time is o expressed in terms of the discrete ﬁelds. , ¯ • For numerical simulations the spatial volume and “temporal” extent is ﬁnite. Lattice spacing is tuned by changing interaction strength. iagGµ Uµ = e • The calculation is non-perturbative, at least in QCD sector.

5 Sea and valence fermions

• Hadrons are created using quark composite functions.

• Hadron mass is extracted from correlation of the composite ﬁelds.

• Quark masses can be adjusted independently for valence and sea quarks.

• Quenched simulations: sea quarks inﬁnitely heavy. Partially quenched: sea and valence quark masses are diﬀerent.

• In lattice simulations quark masses are larger than physical ones for numerical reasons.

• Since quark masses are not observable, we use pion mass to determine how close we are to the physical point. 6 Electric ﬁeld on the lattice Background ﬁeld method

• Introduce a background electric ﬁeld

! D = igG iqA µ µ µ µ • The U(1) ﬁeld Aμ is static • On the lattice this amounts to changing the links

iqaA ! U e µ U µ µ • The polarizability is extracted from the mass shift

8 Extracting polarizability

• To introduce an electric ﬁeld on the lattice we need to use a real phase factor in the exponential form

aqEt 1 2 ! U U e m = E 1 ⇥ 1 ⇤ 2 • The imaginary phase factor can also be used if we remember to ﬂip the sign

iaqEt 1 2 ! U U e m =+ E 1 1 ⇥ 2 • Magnetic ﬁeld is introduced using an imaginary phase factor

iaqBx 1 2 U U e 3 m = B 2 ⇥ 2 ⇤ 2

AA and F. X. Lee, PoS LATTICE2008 (2008) 145. 9 Boundary conditions

Lattice QCD is formulated in Euclidean time. • Á 100 Á Á Á Á Á Á Á Á Á T Á 50 Á This requires that the Hamiltonian of the Z Á Á • Á Á

Tr Á ÁÊ Ê ÊÁ ÊÁ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê system is bounded from below, i.e. there is a 0 Log Á Á 1 vacuum state of lowest energy. T Á Á Á = Á 0 Á Á

E Á -50 Á Á periodic Á Á Á Á Á Ê Dirichlet Á Á In the presence of a real electric ﬁeld, the Á • -100 Á vacuum is no longer stable -- Schwinger 0 50 100 150 200 instability against pair creation. T

• In a ﬁnite volume box we can make the system stable by limiting the maximal distance E between charges. +q q p • We use Dirichlet boundary conditions in space to stabilize the system.

• Note that this instability exists even in a ﬁnite L volume box if we use periodic boundary conditions. 10 Dirichlet boundary conditions

• Dirichlet boundary conditions are equivalent to a hard wall in the direction of the electric ﬁeld.

• The lowest energy for one-particle states p = corresponds to a non-zero momentum. L • The momentum is related to the size of the box and its magnitude decreases slowly with the k=0.1567 k=0.1565 size of the box. 0.4 0.4

Finite volume eﬀects are thus expected to be 0.3 0.3

• p p m m

important and vanish slowly as we increase the a 0.2 a 0.2 size of the system. We can study the inﬁnite volume limit by varying the dimension of the 0.1 0.1 box only in the ﬁeld direction. 0.0 0.0 20 30 40 50 60 70 20 30 40 50 60 70

Nx Nx • Energy shift due to polarizability is corrected k=0.1562 k=0.1546 due to hadron motion: δm = (E/m)δE. 0.4 2 2 0.4 E⇡(L)= m +(⇡/Lx) 0.3 ⇡ 0.3 p p

m 11m a 0.2 p a 0.2 0.1 0.1

0.0 0.0 20 30 40 50 60 70 20 30 40 50 60 70

Nx Nx Compton polarizability

e e er2 = N¯ (i Dµ M) N + F N¯µ⌫ N + E @⌫ F N¯ N + L µ 4M µ⌫ 4M 6 µ⌫ µ ⇣ ⌘ ✓ ◆ ¯ ↵¯ + ¯ + F F µ⌫ NN¯ F F ⌫↵N¯µi D! N + ... 4 µ⌫ 4M µ↵ ⌫ ✓ ◆ A. L’vov, Int.J.Mod.Phys. A8 (1993) 5267

If we set the polarizabilities to zero in the Lagrangean above and use it to compute the energy of a non-relativistic neutron we have p~2 p~ µ2 E(p~)= µ~ (E~ )+ E~ 2 2M · ⇥ M 2M When using Dirichlet boundary conditions the momentum is aligned with the electric ﬁeld and the second term doesn’t contribute. The last term contributes even when the momentum is zero and we have

µ2 W. Detmold et al, Phys.Rev. D81 (2010) 054502 ↵¯ = ↵ + M 12 Numerical results Lattice parameters

• 243 × 48 nHYP clover ensemble -- pion mass 306MeV • 300 configurations -- 25 source points per config • Lattice spacing a=0.1245(16) fm • 243 × 64 nHYP clover ensemble -- pion mass 227MeV • 450 configurations -- 18 source points per config • Lattice spacing a=0.1215(11) fm • Electric field ~ 1021 V/m - η=a2qE = 0.0001 -- imaginary

M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 14 “Neutral Pion”

• The physical π0 correlator has disconnected contributions in the presence of a background ﬁeld

• The disconnected diagrams cancel only in the isospin limit -- the electric ﬁeld breaks isospin symmetry

• Our calculation doesn’t include disconnected contributions • The particle we study is more like du ¯ when u and d have the same charge

• In this version of QCD the pions are all uncharged and χPT predicts a ﬂat behavior (to leading order).

• The “neutral pion” polarizability is expected to be small and positive.

W. Detmold, B. C. Tiburzi, and A. Walker-Loud, Phys.Rev. D79 (2009) 094505 15 Neutral pion polarizability

0.4

0.2

0.0 D 3 -0.2 fm 4 - 10 @ -0.4 p a 306 MeVnHYP -0.6 227 MeVnHYP

-0.8 Quenched

-1.0 200 250 300 350 400 450 500 mp MeV

@ D 0 1 AA and F. X. Lee, PoS LATTICE2009 144 ⇡ = (¯u5u d¯5d) p2 M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 16 Neutral pion polarizability

• We computed the neutral pion polarizability, neglecting the disconnected contributions, using dynamical quark backgrounds.

• We ﬁnd that the presence of the dynamical quarks changes only slightly the results. The polarizability varies very little with the sea quark mass, but depends rather strongly on the valence mass.

• For mπ=227 MeV the value we ﬁnd is about -0.5 × 10-4 fm3.

• This seems to disagree with χPT expectations that the value is small, slightly positive and has little dependence on the quark mass.

• We identified two possible sources for this discrepancy: finite volume effects and neglecting the electric charge of the sea quarks. We found that the negativity does not go away in the infinite volume limit.

AA and F. X. Lee, PoS LATTICE2010 131 17 Neutral kaon polarizability

0.35

0.30

0.25 D 3 0.20 fm 4 - 10 @ 0.15 K a

0.10 306 MeV nHYP 0.05 227 MeV nHYP

0.00 200 250 300 350 400 mp MeV

@ D

M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 18 Neutral kaon polarizability

0.4

306 MeVnHYP

227 MeVnHYP 0.3 Detmold

D extrapolation 3 fm 4

- 0.2 10 @ K a

0.1

0.0 150 200 250 300 350 400 mp MeV

@ D 4 3 M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 ↵K =0.269(43) 10 fm W. Detmold et al, Phys.Rev. D79 (2009) 094505 ⇥ 19 Neutral kaon polarizability

• We computed the neutral kaon polarizability using dynamical quark backgrounds.

• The polarizability changes only slightly when we change the mass of the light quarks, both for valence and sea quarks.

• Since this parameter varies slowly with respect to the quark mass, we believe that the extrapolation to the physical point can be done with high conﬁdence; we ﬁnd a value of 0.27(4) × 10-4 fm3.

• The small value is in agreement with χPT expectations. This quantity receives no contribution to O(p4), the ﬁrst non-zero contributions are at O(p6).

20 Neutron polarizability

Ê Quenched Wilson ˜ Engelhardt ‡ Detmold 6 nHYP 306 MeV nHYP 227 MeV

D Ê 3 fm

4 4 - 10

@ ‡

n Ê a Ê Ê Ê Ê 2

100 200 300 400 500 600

mp MeV

@ D AA and F. Lee, PoS LAT2010 (2010) 131 M. Engelhardt, PoS LAT2009 (2009) 128 M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 W. Detmold, B. Tiburzi, and A. Walker-Loud, Phys.Rev. D81 (2010) 054502 21 Neutron polarizability

Á nHYP 306 MeV

· nHYP 227 MeV 6 Unitary points D 3 fm

4 4 - 10 @ n

a · · · · · Á Á Á 2 Á

100 200 300 400 500 600

mp MeV

@ D J. McGovern, D. Phillips, and H. Griesshammer, Eur. Phys.J. A49 (2013) 12 M. Lujan, AA, W. Freeman, F. X. Lee, Phys.Rev. D89 (2014) 074506 V. Lensky and V. Pascalutsa, Eur. Phys.J. C65 (2010) 195 22 Neutron polarizability

• We computed the neutron polarizability using dynamical quark backgrounds.

• We ﬁnd that the presence of the dynamical quarks increases slightly the polarizability, but the results are in rough agreement with the ones in the quenched study.

• It is clear that our results are not in agreement with χPT expectations: at 227 MeV pion mass the compute value is only 60% of the one predicted from χPT and more than 3 standard deviations away.

• We identified two possible sources for this discrepancy: finite volume effects and neglecting the electric charge of the sea quarks.

23 Finite volume eﬀects

k=0.1567 k=0.1565

0.4 0.4

0.3 0.3 p p m m

a 0.2 a 0.2

0.1 0.1

0.0 0.0 20 30 40 50 60 70 20 30 40 50 60 70

Nx Nx k=0.1562 k=0.1546

0.4 0.4

0.3 0.3 p p m m

a 0.2 a 0.2

0.1 0.1

0.0 0.0 20 30 40 50 60 70 20 30 40 50 60 70

Nx Nx

2 2 m(L)= m +(/Lx) 24 p Inﬁnite volume extrapolation

mp=306 MeV 4 D 3 3 fm 4 - 10 @

n 2 a

0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 a L

Four diﬀerent lattice sizes: 163 × 32, 243 × 48, 30 × 242 × 48 , 48 × 242 × 48 ê 25 Inﬁnite volume extrapolation

7 m =227 MeV 6 p

5 D 3

fm 4 4 - 10

@ 3 n a 2

0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 a L

Four diﬀerent lattice sizes: 163 × 32, 243 × 64, 28 × 242 × 64 ,32 × 242 × 64 ê 26 Inﬁnite volume extrapolation

M. Engelhardt (LHPC Collaboration), Phys.Rev. D76 (2007) 114502 W. Detmold, B. Tiburzi, and A. Walker-Loud, Phys.Rev. D81 (2010) 054502 27 Future directions Charging the sea quarks

• There are two types of dynamical effects: • quark-antiquark loop • background field effects on the sea quarks • Include the dynamical effects of the electric field -- they become important at smaller quark masses

• use the electric ﬁeld in the generation of conﬁgurations -- not really feasible

det ME GE(t) det M 0 0 • use reweighting GE(t) E = ⇥ det ME ⇥ det M 0 0 ⇥ 29 Reweighting

• We need to “charge” the sea det ME GE(t) det M quarks 0 0 GE(t) E = ⇥ det ME ⇥ det M 0 0 • We use reweighting to exploit ⇥ the correlation between the det ME 2 correlators with and without the w(E)= eD1E+D2E det M ⇡ ﬁeld 0

1 @ME D = TrM 1 0 @E It turns out that standard E=0 • estimators for the reweighting 2 2 1 1 @ ME 1 1 @ME factor do not work D = TrM Tr M 2 2 0 @E2 2 0 @E E=0 ✓ E=0◆ • We use a perturbative expansion

W. Freeman, AA, F. X. Lee, M. Lujan , PoS LAT2013 (2013) 288 W. Freeman, AA, M. Lujan, and F. X. Lee, Phys.Rev. D90 (2014) 054507 30 Neutron polarizability

3 D 3 fm 4 - 10

@ 2 n a

1st 2nd 1 Dynamical nHYP

Charged sea nHYP

0 Engelhardt

250 300 350 400

mp MeV

@ D W. Freeman, AA, M. Lujan, and F. X. Lee, Phys.Rev. D90 (2014) 054507 31 Charged hadrons

0.47

0.46

0.45

0.44 eff m a 0.43

0.42 lattice QCD

0.41 fit model

0.40 0 10 20 30 40 t/a

AA, M. Lujan, W. Freeman, and F. Lee, arXiv:1501.06516 W. Detmold, B. C. Tiburzi, and A. Walker-Loud, Phys.Rev. D79, 094505 Conclusions and outlook

• We presented results for electric polarizability of neutral kaon, pion and neutron.

• We focused on electrical polarizability for neutron as a benchmark calculation.

• We presented lattice QCD results for pion mass as light as 227 MeV and box sizes of 3fm.

• After removing finite volume effects, we find that the polarizability rises as we approach the physical point at a rate similar to the one predicted by chiral perturbation theory.

• We need to include the eﬀects of the electric ﬁeld on the virtual quark-antiquark pairs using better estimators to reduce the errors.

• There are many challenges that need to be addressed as we move to charged hadrons, magnetic polarizability, spin polarizabilities, etc.