Nuclear Physics from First Principles · Nuclear Physics from First Principles Symmetries in Nuclei from Lattice QCD and Eﬀective Theories! Thorsten Kurth and Evan Berkowitz for

Nuclear Physics from First Principles

Symmetries in Nuclei from Lattice QCD and Effective Theories!Thorsten Kurth and Evan Berkowitz for the Collaboration

Team

• LLNL: Robert Falgout, Ron Soltz, Pavlos Vranas, Chris Schroeder, Evan Berkowitz, Enrico Rinaldi, Joe Wasem

• LBL/UCB: Wick Haxton, Thorsten Kurth, Ken McElvain, Mark Strother

• SDSU: Calvin Johnson

• nVidia: Michael Clark

• BNL: Sergey Syritsyn

• JLAB/W&M: André Walker-Loud

• JSC/Bonn University: Tom Luu

Motivation

• Historically, nuclear physics has relied on models.

• Nuclei are made of protons and neutrons, which in turn are made of quarks and gluons.

• We can use QCD to extract nucleon properties and interactions,

• connecting model parameters to the Standard Model.

• giving first-principles understanding.

Liquid Drop Shell

Bakken at en.wikipedia CC-BY-SA-3.0

Mean Field Potentials QCD

L = (i /D +m) � 1

4Tr(F 2)

Nuclei as Laboratories

• There are observables that are hard to measure experimentally.

• Nuclei provide a low-energy environment where high-precision “experiments” happen all the time.

• Can test many-body physics.

• Hadronic electroweak neutral current responsible for parity violation (PV) is the least constrained observable in the SM.

• Weak interaction is compared to strong interaction .

• Nuclear data show enhanced isoscalar and suppressed isovector PV.

PV PV≠

⇠GFF2⇡=O(10�7)

⇠O(1)

What is QCD?

• Quantum Chromodynamics is the fundamental theory of strong/nuclear interactions (fusion, fission, α-decay, …)

• QCD describes interactions between Quarks and Gluons

• QCD features confinement (small energies) and asymptotic freedom (large energies)

g g g2

Quark-Gluon coupling (QED-like)

Gluon self-interaction

9. Quantum chromodynamics 33

QCD !s(Mz) = 0.1185 ± 0.0006

Z pole fit

0.1

0.2

0.3

!s (Q)

1 10 100Q [GeV]

Heavy Quarkonia (NLO)

e+e– jets & shapes (res. NNLO)

DIS jets (NLO)

Sept. 2013

Lattice QCD (NNLO)

(N3LO)

" decays (N3LO)

1000

pp –> jets (NLO)(–)

Figure 9.4: Summary of measurements of !s as a function of the energy scale Q.The respective degree of QCD perturbation theory used in the extraction of !s isindicated in brackets (NLO: next-to-leading order; NNLO: next-to-next-to leadingorder; res. NNLO: NNLO matched with resummed next-to-leading logs; N3LO:next-to-NNLO).

9.4. Acknowledgments

We are grateful to J.-F. Arguin, G. Altarelli, J. Butterworth, M. Cacciari, L. delDebbio, D. d’Enterria, P. Gambino, C. Glasman Kuguel, N. Glover, M. Grazzini, A.Kronfeld, K. Kousouris, M. Luscher, M. d’Onofrio, S. Sharpe, G. Sterman, D. Treille,N. Varelas, M. Wobisch, W.M. Yao, C.P. Yuan, and G. Zanderighi for discussions,suggestions and comments on this and earlier versions of this review.

References:1. R.K. Ellis, W.J. Stirling, and B.R. Webber, “QCD and collider physics,” Camb.

Monogr. Part. Phys. Nucl. Phys. Cosmol. 81 (1996).2. C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006).3. H. -Y. Cheng, Phys. Reports 158, 1 (1988).4. G. Dissertori, I.G. Knowles, and M. Schmelling, “High energy experiments and

theory,” Oxford, UK: Clarendon (2003).5. R. Brock et al., [CTEQ Collab.], Rev. Mod. Phys. 67, 157 (1995), see also

http://www.phys.psu.edu/~cteq/handbook/v1.1/handbook.pdf.6. A. S. Kronfeld and C. Quigg, Am. J. Phys. 78, 1081 (2010).7. R. Stock (Ed.), Relativistic Heavy Ion Physics, Springer-Verlag Berlin, Heidelberg,

2010.8. Proceedings of the XXIII International Conference on Ultrarelativistic Nucleus–

Nucleus Collisions, Quark Matter 2012, Nucl. Phys. A, volumes 904–905.9. Special Issue: Physics of Hot and Dense QCD in High-Energy Nuclear

Collisions, Ed. C. Salgado. Int. J. Mod. Phys. A, volume 28, number 11.[http://www.worldscientific.com/toc/ijmpa/28/11].

December 18, 2013 12:00

long distance short distance

u d

d

neutron

uu

d

proton

How to solve QCD?

• QCD is described by infinite dimensional integrals

ZE =

ZDUµ D D

¯

exp

0

@�Sg[U ]�Z

R4

d

4x

¯

(x)D[U ] (x)

1

A

How to solve QCD?


• use (euclidian) 4D lattice as regulator

ZE =

ZDUµ D D

¯

exp

0

@�Sg[U ]�Z

R4

d

4x

¯

(x)D[U ] (x)

1

A

a

L

Uµ

How to solve QCD?



• generate configurations and extract observables

ZE =

ZDUµ D D

¯

exp

0

@�Sg[U ]�Z

R4

d

4x

¯

(x)D[U ] (x)

1

A

N

NN

N!

hOi⇡ 1

N

NX

n=1

O[Un, D[Un]�1]

How to solve QCD?




• perform limit a→0, L→∞ to recover continuum QCD

ZE =

ZDUµ D D

¯

exp

0

@�Sg[U ]�Z

R4

d

4x

¯

(x)D[U ] (x)

1

A

N

NN

N!

hOi⇡ 1

N

NX

n=1

O[Un, D[Un]�1]

How to solve QCD?





• but: tCPU~a-5

ZE =

ZDUµ D D

¯

exp

0

@�Sg[U ]�Z

R4

d

4x

¯

(x)D[U ] (x)

1

A

=

ZDUµ D�D�

†exp

0

@�Sg[U ]�Z

R4

d

4x�

†(x)D[U ]

� 12�(x)

1

A

expensive matrix inversions

S(x, y) = (x) (y) = D[U ]�1(x, y)

How to solve QCD?





• but: tCPU~a-5

• big computers needed

!

NN

NN

P P

PP

+ . . .

Nuclear Physics from Bottom Up

• compute multi-nucleon correlation functions

N

NN

N!

single Nucleon 2!1!=2

deuteron 3!3!=36

He3-scattering 5!4!=2880



• measure energy difference between interacting and non-interacting system

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14

6E e

ff(t/a

)

t/a

1S03P03P13P2

�E�e↵(t) =

1

2

log

✓C�

NN (t� 1)

C�NN (t+ 1)

C2N (t+ 1)

C2N (t� 1)

◆




• use Lüscher’s finite volume formula to relate measured energies to phase shifts δ(k)

2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!

free theory





2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!

free theory

2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!interacting theory





2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!

free theory

2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!interacting theory

�E ! k2LFV�! �(k2)





• compute low energy observables from effective range expansion

2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!

free theory

2 3 4 5 6 7 8M

!L

2

2.5

3

3.5

4

E/M

!interacting theory

�E ! k2LFV�! �(k2)

k2l+1cot �l(k

2) = � 1

al+

1

2

rl k2+

1X

n=0

(�)

n+1P (n)l k2n+4

Challenges

• number of contractions/diagrams grows combinatorially

Challenges{xi1, !, b}

{xi2, ", d}

{xi3,#, f}

Bµ


• Baryon blocks

Challenges

B!

Bµ


• Baryon blocks

Challenges

B!

L

Bµ


• Baryon blocks

Challenges

B!

L

Bµ

Pµ!


• Baryon blocks

Challenges

B!

L

Bµ

Pµ!

recompute for every new measurement

compute once, store to disk


• Baryon blocks

Challenges


• Baryon blocks

• use automatic code generation

Challenges


• Baryon blocks


Challenges


• Baryon blocks


• calculating the contractions is expensive: use GPUs

X

µ,⌫,A,B

Pµ⌫ BµA B⌫

B LAB = vTMvvector-Matrix-vector product

Challenges


• Baryon blocks



X

µ,⌫,A,B

Pµ⌫ BµA B⌫

B LAB = vTMvvector-Matrix-vector product

p[tid] = v[I1[tid]] * M[tid] * v[I2[tid]];

p

Challenges


• Baryon blocks



• excited states contributions: sink/source engineering

C(t, t0) =1X

n=0

Z⇤nZn e

�mn(t�t0)

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

0 2 4 6 8 10 12 14 16

E eff(

t/a)

t/ause sources/sinks with spatially displaced nuclei

Challenges


• Baryon blocks



• excited states contributions: sink/source engineering

• Lepage argument

C(t, t0) =1X

n=0

Z⇤nZn e

�mn(t�t0)

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

0 2 4 6 8 10 12 14 16

E eff(

t/a)

t/ause sources/sinks with spatially displaced nuclei

Lepage Argument

• signal-to-noise ratio of correlation functions

SNR ⇠ hNNiph(NN)(NN)†i � hNNi2

Lepage Argument


• numerator


⇠ exp(�mN t)

Lepage Argument


• numerator

• denominator


N

N

⇠ exp(�mN t)

Lepage Argument


• numerator

• denominator


π

π

π

⇠ exp(�mN t)

⇠ exp

✓�3

2

m⇡t

◆

Lepage Argument


• numerator

• denominator

• time-dependence of SNR


π

π

π

⇠ exp(�mN t)

⇠ exp

✓�3

2

m⇡t

◆

huge statistics needed (MG)huge amount of storage needed (HDF5)

large pion mass⇠

pN exp

�A

✓mN � 3

2

m⇡

◆t

�

Parity Violation - Contractions Δ!=2

• 2 Baryon s-wave source



• EW vertices ⇒ 4-quark operator insertion







• 2 Baryon p-wave sink





• In total there are 4896 contractions.

• Degenerate light quarks reduce this to 2208.

• Requires 15 tensors.


















































• 2 baryon blocks & 1 four-quark object

Source and Sink Construction

• use of momentum sources and sinks (non-local in coordinate space) would be optimal

FIG. 20: The momentum-space representations (left) and position-space representations (right) oftwo-body relative states in the T�

1

representation with Lz = 0 for the |n|2 = 1, 2, 3-shells.

39



FIG. 20: The momentum-space representations (left) and position-space representations (right) oftwo-body relative states in the T�

1

representation with Lz = 0 for the |n|2 = 1, 2, 3-shells.

39N(p) =X

x

e

ipx

q1(x) q2(x) q3(x)we wish to use:

we are limited to:N

0(p; p1, p2) =X

x1,x2,x3

e

ip1x1q1(x1) e

ip2x2q2(x2) e

i(p�p1�p2)x3q3(x3)



• we use coordinate-space sources (and sinks) instead

L=0, Lz=0

L=1, Lz=+1,0,-1




• mix different lattice irreducible representations ⇒ excited states

L=0, Lz=0

L=1, Lz=+1,0,-1




• mix different lattice irreducible representations ⇒ excited states

• rotational symmetry breaking: mixing of different continuum multiplets

L=0, Lz=0

L=1, Lz=+1,0,-1

L=4,6, …

L=3,5, …

Preliminary Results: Partial Wave Scattering at mπ=800 MeV

1.5

2

2.5

3

3.5

4

4.5

0 2 4 6 8 10 12 14 16

E eff(

t/a)

t/a

1S03P0Preliminary! mπ=800 MeV

Preliminary Results: Parity Violation Matrix Element at mπ=800 MeV

Preliminary! mπ=800 MeV

0

2e-20

4e-20

6e-20

8e-20

1e-19

1.2e-19

1.4e-19

1.6e-19

0 2 4 6 8 10 12

CPS

(t;t f,

t i)

t/a

Raw Correlator

Preliminary Results: Parity Violation Matrix Element at mπ=800 MeV

Preliminary! mπ=800 MeV

0

5

10

15

20

0 2 4 6 8 10 12

R(t;

t f,t i)

t/a

Normalized Ratio

Reach

Reach

Reach

!

!

!LQCD

Reach

HOBET

!

!

!LQCD

HOBET: Connecting a non-relativistic effective theory to the lattice

• Idea: Build an effective theory that can utilize lattice input to fix parametersthat are unknown, while taking other parameters from experiment

• This includes utilizing either experiment or lattice input on the NN interaction to predict properties of light nuclei

• HOBET is a true ET: the effective interaction is constructed directly in thesoft — or “included” — P-space

• Usual nuclear physics approach:QCD ➝ singular NN potential ➝ P-space effective interaction

• HOBET: QCD ➝ P-space effective interaction

• The difficult effective interactions problem is avoided

• How is this done? HOBET’s unique infra-red/ultraviolet separation allowsus to utilize NN phase shifts to fix ET parameters

Simplified analytic example: hard-core s-wave interaction, chosen to reproduce deuteron binding energy

0 ∼ -50 MeV

∼ 2 GeV

finely tuned: deuteron is bound by 2.22 MeV

P is the soft ET space Q=1-P H = T+V !use the Haxton/Luu factorization of the Bloch-Horowitz equation !

HET = PE

E � TQ

T � T

Q

ET + V + V�

�E

E �QTP


0 ∼ -50 MeV

∼ 2 GeV



HET = PE

E � TQ

T � T

Q

ET + V + V�

�E

E �QTP

sums kinetic energy T to all orders !outgoing solution for a specific E depends on the phase shift δ(E)


0 ∼ -50 MeV

∼ 2 GeV



HET = PE

E � TQ

T � T

Q

ET + V + V�

�E

E �QTP

short-range operator that corrects for the effects of V acting outside P: can be efficiently expanded in terms of contact operators

The BH equation must be solved self-consistently: !So one picks an E, supplies δ(E), calculates , diagonalizes in P !A solution must exist at every E>0 !But the diagonalization does not give an eigenvalue at E !Thus we “dial” - this can be done order-by-order to reproduce δ(E) over a range of E !Through NNLO: !!!!The larger the range in E over which δ(E) is fit, the more terms needed !Phase shifts yield the deuteron binding energy, in a calculation limited to P

HET = HET (E)

HET (E)

V�

V� = aLO�(~r) + aNLO

✓ r

2�(~r) + �(~r)

!r

2◆+ a1NNLO

r

2�(~r)

!r

2+a2NNLO

✓ r

4�(~r) + �(~r)

!r

4◆

Low-energy constants become constant, as higher orders are included

For = 0 the predicted binding energy is -0.68 MeVV�

CalLat is applying these techniques to the parity-violating asymmetry in !Ongoing SNS experiment to measure the neutral current in the hadronic weak interaction (see poster) !We will use HOBET + lattice to fully determine this system, and to relate this observable to other parity-violating observables !Experimental strong phase shifts: fix all strong interaction low-energy constants !Lattice: provides the experimentally unknown parameter — the parity-violating weak phase shift

n+ p ! D + �

Conclusions

• Nuclear physics can be done on the Lattice

• Baryon blocks and tensors tame Wick Contractions ⇒ automatic code generation, parallelization (GPUs)

• sophisticated, expensive non-local sources/sinks for higher partial waves ⇒ treating excited states effects

• exponential decrease of SNR ⇒ large mπ, hugestatistics (MG), huge amount of data to store (HDF5)

• preliminary higher partial wave and PV results look promising, but might need to disentangle angular momentum multiplets

Conclusions

• Expected computational costs:

• mπ↓: inversions⬆, SNR⬇

• L↑: inversions↑, FFT⬆, contractions↑, SNR↑

• feed lattice results into HOBET ⇒ solve complex nuclear systems

Thank You

Documents

Nuclear Physics from First Principles · Nuclear Physics from First Principles Symmetries in Nuclei from Lattice QCD and Eﬀective Theories! Thorsten Kurth and Evan Berkowitz for