1. Hadrons and Nuclei : Nuclear Physics
1/1/2019
Hadrons and Nuclei : Nuclear Physics
Lattice Summer School
Make sure that the toolbars are open
Standard
Formatting
Drawing
Martin Savage
Summer 2007
University of Washington

2. A Grand Challenge for QCD and Nuclear Physics
How do the properties and interactions
of nuclei depend upon the fundamental
parameters of nature ? Q C D ; m u d s e ( M Z ) g
Fundamental physics and intellectual curiosity
Fine-tunings

3. Energy Scales and QCD Energy levels are … D o t h e l v s a r y i m d
4.44
7.654
8Be + a
7.367 Q C D f 2 m u ; d s e Q C D f 1 m u ; d s e Q C D f m u ; d s e D o t h e l v s a r y i m d p n c q ? H w M V f G - u

4. Fine -Tunings in Nuclear Physics are Fine-Tunings of QCD
8Be + a
7.367
10.31
Large splitting
16O
7.12
7.654
3 a
7.275
subthreshold
+ a g
4.44
Thermal Equilibrium around T8 g
Fine-Tuning ~ 130 keV
Anthropic Principle
12C

5. Nuclear Phenomenology from QCD ????
1/1/2019
Nuclear Phenomenology from QCD ????
Vibrational and rotational
excitations
Spin-pairing
Shell-structure

6. Nuclear Phenomenology -- e.g. Collective Excitations
keV
Two L=2 Phonons, L=0,2,4
Degenerate if non-interacting
One Quadrupole (L=2) Phonon

7. The Strategy : QCD to Nuclei
Lattice QCD
Exact methods A≤12 GFMC, NCSM,
Lattice EFT
Shell Model,
Coupled Cluster, … A<100
Density Functional Theory ??
A>100
QCD Lagrangian
N, 2N , 3N, 4N, …
Effective Interactions
Low-Momentum Interactions
For Nuclei
EFT and RG
(RIA web-site)

8. Light Nuclei from NN, NNN Phenomenological Potentials (1)
Ab initio nuclear structure: Green Function Monte Carlo
(ANL/LANL/UIUC)
Since 1992:
algorithms
Variational MC
AV18 (2-body)
Computing
3-body interaction
2-body
3-body T h r e - n u c l o i t a s m p !

9. Light Nuclei from NN, NNN Phenomenological Potentials (2)
For A=10, each state takes
1.5 Tflop-hours

10. NN Potential – not unique !!
r,w,rp,-exchange
—anything you can imagine
p-exchange
pp-exchange
To get large scattering length requires conspiracy between long,
medium and short-range components of NN potential
-- that conspiracy is, of course, QCD

11. Phenomenology of NN Forces
(Bogner, Kuo, Schwenk, Entem, Machleidt,…) R G a n l y s i o f u c e r p t d V w k
Phenomenological NN Potentials consistent with Chiral EFT’s

12. Chiral Symmetry and the 2N, 3N and 4N Interactions
Low energy EFT of QCD
Quark mass dependence
Novel RG behaviour
Softer interaction
Counterterms !!!!

13. Input to NN Theoretical Construction
pp data up to ~ 3 GeV (~ 40,000 data points) of sufficient quality to allow phase-shift analysis (PSA) ( Arndt, Strakowski, Workman, GWU/VPI)
np data up to 1.3 GeV (~ 20,000 data) of sufficient quality for PSA.
Below 350 MeV, “high-precision” PSA (χ²/datum ≈ 1) by Nijmegen group (PWA93).
Since 1994, Nijmegen has announced new PWA up to 500 MeV, but not published.
PWA93 is used in all NN potential constructions and is good enough. However, for χ² calculations, 1992 data base outdated and a newer data base should be used. Â 2 = d o f

14. Nuclear Effective Field Theory (NEFT)
What do we want ?
Want to be able to compute any nuclear process (elastic or inelastic)
Want to be able to reliably estimate theoretical errors
Want to be able to make reliable predictions where expts not possible
Want to know what happens when quarks masses are changed
Nuclear Effective Field Theory (NEFT)

15. 1 = a Nuclear Physics Scales l o g Q Q f , 1 = r m M , ¤ = 4 ¼ f ¼
perturbative QCD
~1 GeV M N , Â = 4 f
hadronic theory with chiral symm
~100 MeV f , 1 = r m
~30 MeV
Pionless theory 1 = a
no small coupling
expansion in Q

16. Calculational Regimes. g n p ! d

17. Very Low Energies – pionless (1)F o r j p
< m i n s , d y a c l e t u h \ E T ( = )
Reproduces NR QM
Relativity included perturbatively
Gauge Invariant operators unrelated to scattering included
Many-body calculations
Quark masses not explicit … dependence in coefficients
Isospin and not chiral symmetry D = ( @ + i e Q A ; r ) , L ( O ) = N y i D + 2 M C T P N !

18. Very Low Energies – pionless (1b)= N y i D + 2 M C T P N !
At threshold in HB limit I = i 2 4 n Z d q ( ) + 1
Which is ill-defined

19. Very Low Energies – pionless (2)A = [ C + I : ] 1 I P D S ! M 4 ( + i j k ) 4 M C = 1 a d [ C ( ) ] = A = 4 M 1 C + i j k W i t h P D S s c e m , C n a u r l z w .

20. Scale-Invariance at Low Energies (1)n p = 2 3 : 7 f m ; r + 5 4 a
> r A = 4 M 1 a + : i j k !
Low-energy NN theory is close to being scale-invariant at LO

21. Scale-Invariance at Low Energies (2)
How do we understand this from QCD ???
Who would have guessed that the quark masses and
strong scale conspire to give a nearly
scale-invariant NN sector
I have no real idea, but here are some thoughts…
The landscape picture of many universes with
different gauge groups and couplings is one plausible scenario.
The question is what is the selection mechanism
Production of carbon in stars is one requirement, but why does this
require a finely-tuned, near scale invariant NN sector? Nuclear physics is not advanced enough to determine if all
nuclear potentials that give rise to sufficient carbon require
a finely tuned NN sector.

22. Very Low Energies – pionless (3)2 = N T P a y 1 O + h : c 4 O a 2 = 1 4 h P ! D + Ã i 6 3

23. Very Low Energies – pionless (4)1 S L O a - o n l y a n d r 2 N L O E R T N L O N i j m e g n a n d r 1 j k M e V

24. Very Low Energies – pionless (4) Deuteron Quadrupole Moment
Interaction
Qd (fm2)
EXPT
AV18
0.2696
Nijm93
0.2706
CD-Bonn
Nijm1
0.2719
What is missing ???

25. Very Low Energies – pionless (5) Deuteron Quadrupole Moment
Obtained from NN
phase-shift analysis
Gauge Invariant
Local Operator O q u a d N T P i y j r 1 ( n ) 2 A C a n o t p r e d i c Q N L O E F T f m s g l .

26. Very Low Energies – pionless (5) Photo-disintegration= 4 2 + j P 3 M N h ~ X 1 E i T e s o v c t r V a n d m p l u ( ) Z L 5 6 C
M1-Counterterm

27. Very Low Energies – pionless (6) Photo-disintegrationt h e r

28. Very Low Energies – pionless (7) Photo-disintegration

29. ElectroWeak Matrix Elements
(William Detmold and MJS)
Solve pionless theory at finite volume with background fields to
extract gauge-invariant counterterms
L X L/10 X L/10 X T lattice

30. Three-Body calculations (1)
nd scattering
length
3-body interaction
``slides’’ along line
Triton binding energy

31. Three-Body calculations (2)LO
NLO
NNLO
AV18 calculations

32. Four-Body calculations (3)
3-body interaction
``slides’’ along line

33. » Dynamical pions » 1 f Weinberg (1990) cPT for the potential
Insert potential into the Schrodinger eqn 1 f 2 1 f 2 q + m

34. Particle Physicists always get this wrong !!
Dynamical pions (2) N a i v e l y , t h d m - 6 o p r s u b n g c w 4 H !
Particle Physicists always get this wrong !!

35. The Hierarchy of Nuclear Forces in Weinberg Counting
2N forces
3N forces
Leading Order
The Hierarchy of Nuclear Forces in Weinberg Counting
The Hierarchy of Nuclear Forces
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order

36. Weinberg For V(q) F E a c h v e r t x f o m L u s Q ; p i l g .
One puts NN irreducible diagrams into V(q)
NN reducible are generated by Schrodinger eqn
For V(q) F E a c h v e r t x f o m L ( n ) u s Q ; p i l 2 4 g 1 .

37. Problems with Weinberg Power-Counting : KSW Power-Counting
However, Weinbergs power-counting has formal problems at the two-loop level.
In order for Schrodinger equation to contribute terms all of the same order in the power-counting, MN ~ 1/Q
This gives rise to divergences that contain mq BUT at LO 1 g 2 A m M 8 f C

38. Q0 Q-1 NLO LO » Dynamical pions » 1 ¹ 1 f ¾ ¢ q + m KSW (1998) 2
cPT for the amplitude
LO is inserted into Schrodinger equation, while everything else is treated in perturbation theory
Formally consistent, i.e. RG invariant order-by-order Q0
NLO
Q-1 LO 1 2 1 f 2 q + m

39. KSW – Perturbative Pions
Correct for spin-singlet channel
Incorrect for spin-triplet channel

40. BBSvK – Expand about Chiral Limit
Chiral limit is different in different spin channels !! J = 1 ( S 2 ; L ) ! t e n s o r D e u t r o n w a v f c i s m b L = d 2 V T = g 2 A 1 6 f m e r + 3 ! O ( 4 )
NO 1/r2 interaction : can be renormalized C h i r a l m t o f N p e n s - c x 3 ( ) d 1 =

41. BBSvK – Solving triplet channel
Unable to Dim Reg !! V L = M N C 2 p T ( + ) 6 r S C o r e c t i n s a 1 = + . m l z b h w y  P T ( r ) = u x w + k 2 V L ; R S

42. BBSvK – Partial-Perturbation Seriesu ( L O ) r ; k = A 3 4 c o s à 2 6 M 5 + Á 1 ! R e s u m V O P E ( r ; ) P r a c t i l p o n : k e V L O ( ; m ) d s , u b

43. BBSvK – Partial-Perturbation Seriesu ( L O ) r ; k = A 3 4 c o s à 2 6 M 5 + Á 1 !
This looks like Weinberg .. Chiral expansion of V(r) ?? h j ^ V O P E i s n Z 1 d r e m = l o g ! 3 2 p N o n - p e r t u b a i v h d 1 = 3 l c y g s w . S o t h e p n i a l D O E v c r x s w f ( m W b g ) B u N T

44. BBSvK – Partial-Perturbation Seriesh j ^ V O P E i s n Z 1 d r e m = l o g ! 3 2 p
Fractional power-dependence of cut-off !

45. Partial Summary of NN (1)
Weinbergs Counting ( chiral expansion of V(r) ) fails in all channels !!!!
1S0 -- not RG invariant
3S1-3D1 – S,D wave counterterms both LO
KSW Counting (perturbative pions) fails in channels with tensor potential (1/r3)
BBSvK Counting –expand potential about chiral limit -- consistent

46. Partial Summary of NN (2)
Convergence is quite slow
Scale set by roughly mr/2 – breakdown of ERE for short-distance component
Practical Implementation
Work to a given order in any formulation
if you work to high enough order then any violence that you might have done, e.g. formal inconsistency, become small as long as RG scale is roughly less than the cut-off
This is because you will be consistent at some lower order – which will be established numerically

47. Phase-Shifts
Epelbaum et al.

48. Summary: χ²/datum
NLO: ≈ 100
NNLO: ≈ 10
N3LO: ≈ 1
Good convergence !

49. Neutron Polarizabilities from gD Compton Scattering
Isoscalar Polarizabilities
(Beane et al, nucl-th/ )

50. pD Scattering and np Scattering Lengthspb pa N T b a N = 4 1 + m M i c

51. Bernard, Epelbaum, Meissner
The 3NF at N3LO
Bernard, Epelbaum, Meissner

52. Larger Nuclei : Three-nucleon forces at NNLO
TPE-3NF
No new parameters!
OPE-3NF
One new parameter, D.
Contact-3NF
One new parameter, E.
Strategy: Adjust D and E to two few-nucleon observables, e.g., the triton and alpha-particle binding energies. Then predict properties of other light nuclei.

53. Navratil et al, nucl-th/0701038
Larger Nuclei (2)
Navratil et al, nucl-th/

54. Photo-Pion Production
Beane et al.
The theory point was predicted (1997), subsequently verified by
data (2000), and disagreed with non-chiral phenomenological models + d ! Â P T

55. Mq-dependence (1) O » N m T r [ ]
Short-distance mq-dependent operators
Corresponds to mq-dependent r,w,… couplings and masses O 1 N y m q + 2 T r [ ]

56. Mq-dependence (2) Lattice QCD will be the ONLY way L » C N + D m :N y + D ( 1 ) 2 m q : T h e D ( i ) 2 a b s o r t n m l z - c d p g f O P E q u w C
Turns out to be very difficult experimentally, as LO pion-exchanges
are always much larger than those from the D2’s
Lattice QCD will be the ONLY way

57. NN Scattering (1) 1S0 : pp , pn , nn 3S1-3D1 : pn : deuteron
(S. Beane, P. Bedaque, K. Orginos, mjs ; PRL97, (2006))
a ~ 1/mp
1S0 : pp , pn , nn
3S1-3D1 : pn : deuteron
Could be scale-invariant in chiral limit … who knew!
Also, see later work by Epelbaum+Meissner

58. NN Scattering (2) P r e s n t l y , c o a i x p w h m d . I n f u t rd . I n f u t r e , s o l y a i c d x p h .

59. Fine -Tunings in Nuclear Physics are Fine-Tunings of QCD … or are they ??
8Be + a
7.367
10.31
Large splitting
16O
7.12
7.654
3 a
7.275
subthreshold
+ a g
4.44
Thermal Equilibrium around T8 g
Fine-Tuning ~ 130 keV
Anthropic Principle
12C

60. Infrared Limit Cycles in QCD and what might have been
Infrared Limit Cycles in QCD and what might have been ! ..The Braaten-Hammer Conjecture..
Conjecture : Can independently tune Mu and Md for scale invariance in both the 1S0 and 3S1-3D1 channels.
Infinite number of triton bound states with
En+1 En
= 515 2 8 a=
Scale invariance in 2N sector
Regulation in 3N system
Discrete Scale invariance in 3N sector
--- Infrared Limit Cycle
Lattice QCD 3 H ( ) = s i n l o g + a r c t 1
Lattice QCD will be able to explore this
(Epelbaum, Meissner)

61. Lattice Nuclear Physics
(D.Lee, B.Borasoy and T. Schafer)
Seki, Muller,
van Kolck,
Triton :
Krebs, Hammer,
Nogga, Meissner…

62. Nuclear Matter, Large Nuclei

63. Summary (1) How does Nuclear Physics emerge from QCD ?
Fine-Tunings in NP, fine-tunings of QCD parameters …. why?
EFT consistent with c-sym established for NP with consistent power-counting
Effective Field Theories provide the link between QCD and the multi-nucleon systems…
nuclei, elastic and inelastic processes

64. Summary (2)
To improve upon the century of nuclear phenomenology, Lattice QCD (when combined with all available rigorous tools) must be able to calculate NEW quantities reliably…
there are a lot of checks along the way