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

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

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 µ m u ; d s e ¶ ¤ Q C D f 1 µ m u ; d s e ¶ ¤ Q C D f µ 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

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

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

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

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)

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 !

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

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

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

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

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

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)

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 , ½ ¤ Â = 4 ¼ f hadronic theory with chiral symm ~100 MeV f ¼ , 1 = r m ~30 MeV Pionless theory 1 = a no small coupling expansion in Q

Calculational Regimes . g n p ! d °

Very Low Energies – pionless (1) F o r j p < m ¼ 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 ¹ = ( @ + i e Q A ; ¡ r ) , L ( O ) = N y · i D + 2 M ¸ ¡ C T P ¢ N !

Very Low Energies – pionless (1b) = N y · i D + 2 M ¸ ¡ C T P ¢ N ! At threshold in HB limit I = ¡ i ³ ¹ 2 ´ 4 n Z d q ( ¼ ) µ + ² ¶ 1 Which is ill-defined

Very Low Energies – pionless (2) A = ¡ [ C + I : ] 1 I P D S ¡ ! µ M 4 ¼ ¶ ( ¹ + i j k ) 4 ¼ M C = 1 a ¡ ¹ ¹ d [ C ( ) ] = A = ¡ 4 ¼ M 1 C + ¹ i j k W i t h P D S s c e m , C n a u r l z w ¹ » ¼ .

Scale-Invariance at Low Energies (1) n p = ¡ 2 3 : 7 § f m ; r + 5 4 a > r A = ¡ 4 ¼ M 1 a + : i j k ! Low-energy NN theory is close to being scale-invariant at LO

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.

Very Low Energies – pionless (3) 2 = ¡ ³ N T P a ´ y 1 · O ¢ + h : c ¸ 4 O a 2 = ¡ 1 4 h P ! D + Ã i 6 3

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

Very Low Energies – pionless (4) Deuteron Quadrupole Moment Interaction Qd (fm2) EXPT 0.2859 +- 0.0003 AV18 0.2696 Nijm93 0.2706 CD-Bonn Nijm1 0.2719 What is missing ???

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 µ 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 .

Very Low Energies – pionless (5) Photo-disintegration ¾ = 4 ¼ ® ¡ ° 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 · ³ ´ ( ) Z L 5 6 µ ¶ C ¸ M1-Counterterm

Very Low Energies – pionless (6) Photo-disintegration ¯ t h e r

Very Low Energies – pionless (7) Photo-disintegration

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

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

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

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

» Dynamical pions » 1 f Weinberg (1990) cPT for the potential Insert potential into the Schrodinger eqn » 1 f 2 ¼ » 1 f 2 ¼ ¾ ¢ q + m

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 ¾ ¢ ¼ H ! Particle Physicists always get this wrong !!

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

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 .

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 ² g 2 A m ¼ M 8 f C

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 ¹ 2 » 1 f 2 ¼ ¾ ¢ q + m

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

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 ¼ f m e r µ + 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 =

BBSvK – Solving triplet channel Unable to Dim Reg !! V L = ¡ M N µ 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 ª ( r ) = µ u x w ¶ + ¡ k 2 V L ; R S ¢

BBSvK – Partial-Perturbation Series ¹ u ( L O ) r ; k = A 3 4 c o s à 2 6 ® ¼ M ¡ 5 + Á 1 ! R e s u m V O P E ( r ; ¼ ) ¡ P r a c t i l p o n : k e V L O ( ; m ¼ ) d s , u b ¡

BBSvK – Partial-Perturbation Series ¹ u ( L O ) r ; k = A 3 4 c o s à 2 6 ® ¼ M ¡ 5 + Á 1 ! This looks like Weinberg .. Chiral expansion of V(r) ?? h ª j ^ V O P E i » s n ± Z 1 d r e ¡ m ¼ ¹ = l o g µ ¶ ! 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

BBSvK – Partial-Perturbation Series h ª j ^ V O P E i » s n ± Z 1 d r e ¡ m ¼ ¹ = l o g µ ¶ ! 3 2 p Fractional power-dependence of cut-off !

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

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

Phase-Shifts Epelbaum et al.

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

Neutron Polarizabilities from gD Compton Scattering Isoscalar Polarizabilities (Beane et al, nucl-th/0403088)

pD Scattering and np Scattering Lengths pb pa N T b a ¼ N = 4 µ 1 + m M ¶ £ ± i ² c ¿ ¡ ¤

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

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.

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

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 ! ¼ Â P T

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 [ ]

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 ¹ 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

NN Scattering (1) 1S0 : pp , pn , nn 3S1-3D1 : pn : deuteron (S. Beane, P. Bedaque, K. Orginos, mjs ; PRL97, 012001 (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

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 r ¼ d . I n f u t r e , s o l y a i c d x p h .

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

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 ³ l o g ¡ ¢ + a r c t 1 Lattice QCD will be able to explore this (Epelbaum, Meissner)

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

Nuclear Matter, Large Nuclei

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

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