1

THE STORY OF CHIRAL EFT

U. van Kolck Institut de Physique Nucléaire d’Orsay

and University of Arizona

Supported in part by CNRS, Université Paris Sud, and US DOE

NON-PC ABOUT PC A SERMON

2

Why? Chiral “EFT” potentials based on Weinberg’s power counting

widely used in nuclear physics because of their supposed link to QCD

Problem: Weinberg’s power counting inconsistent with renormalization

The problem doesn’t exist: Renormalization not important

Kaplan, Savage + Wise ’96, …, Nogga, Timmermans + Nogga ’05, …

Epelbaum + Meissner ’06, …, Epelbaum + Gegelia ’09, …

Anyway, there is a solution for the problem that doesn’t exist: Relativity essential in a non-relativistic problem

Epelbaum + Gegelia ’12

(Oh, yeah, this solution doesn’t completely solve the problem that doesn’t exist ---counterterms still need to be promoted--- but that is a detail

which barely needs acknowledgement…)

Solution: Certain counterterms appear at lower order than expected; subleading terms should be treated in perturbation theory

the talk yesterday

VS.

3

Outline

Effective field theory & model spaces Pre-story: ChiPT The story Conclusion & Outlook

4

EFT E

Λ

m

( )0 ( , )

( , ) ( , )d nEFT i i

d i d nMc mO ϕ

∞

=

∂Λ=∑ ∑

0Z∂∂Λ

=

underlying dynamics renormalization-group invariance

local underlying symmetries

M ( )( )

( )

4

4

exp ( )

( )

exp ( )

und

EFT

Z i d x

f

i d x

ϕ δ ϕ

ϕ ϕ

Λ

= Φ Φ

× − Φ

=

∫ ∫∫

∫ ∫

Weinberg, Wilson, ...

most general

5

min

( ), ,( ) ~ ( ) ( ) ;i i

iT T N Q QQ

m mcM

MF

ν

ν νν ν

∞∞

=

= Λ

Λ∑ ∑

0T∂∂Λ

=

),,( ndνν =

( )

( ) 1v

vT Q

T∂ =

ΛΛ Λ ∂

( ) 1 ,T T QM

Qν

Λ = +

For Q ~ m, truncate …

“power counting”

e.g. # loops L

non-analytic, from loops

normalization

… consistently with RG invariance:

controlled model independent

MΛ >

want If so

realistic estimate of errors comes from variation [ ),MΛ∈ ∞

6

E

Λ

m

M

λ

To limit the number of one-particle states,

introduce IR cutoff in addition to UV cutoff

λ Λmomentum

Cutoffs define “model spaces”

( ) 1 , ,M

QT QQ

T ν λ = +

Λ

MΛ >

Qλ To minimize “model space” error (to “converge”), want

7

Harmonic-Oscillator Box

2

1mb

max2

Nmb

“No-Core Shell Model” Lattice Box

2

2mLπ

nuclear matter Stetcu et al. ’06 … Müller et al. ’99 finite nuclei

L aN=2b mω=

2 2

2

NmLπ

few nucleons Lee et al. ’05 … few atoms Stetcu et al. ’07 …

2

2mΛ

2

2mλ

energy

“Lattice Field Theory”

Popular examples

maxL bN

few atoms Kaplan et al. ’10 … atomic matter Bulgac et al. ’06 …

8

Coon, Avetian, Kruse, Maris, Vary + v.K., ‘12

2 1L

λ=Λ

Extrapolations in a HO basis

scaling

for much more see Furnstahl, Hagen + Papenbrock ’12

More et al. ‘13

9

Chiral EFT

• d.o.f.s: pions, nucleons, deltas

• symmetries: Lorentz, P, T, chiral

( ~ 2 )Nm m mπ∆ −

1 GeVQCDMQ mπ

pN

n

= 0

++

+

−

∆ ∆ ∆ = ∆

∆

( )( )

1

2

03

2

2i

π πππ π ππ π

+ −

+ −

+ = = − −

π

212 4f fµ

µπ π

≡ − +

∂ D

2π π

22 f

iµ µ µπ

≡ − ⋅ ×

∂

T Dπ

(chiral) covariant

derivatives

pion

baryon, isospin T

( )( )2u d QCDm m m Mπ +=

chiral invariants

Weinberg ’68 Callan, Coleman, Wess + Zumino ‘69

spontaneously broken: non-linear realization

+ chiral breaking as in quark mass terms

non-derivative interactions proportional to masses

( )92MeV 4QCDf Mπ π=

10

Example: pion sector (similar in one-nucleon sector)

= + + … Tππ + +

Weinberg ’79 Gasser + Leutwyler ’84

Manohar + Georgi ‘84 …

+

current algebra

Weinberg ’66 …

( ) ( ) ( )

22

0

2 2 41 2 3 4

2 22

42 2 2

2

12 12 4

1 1

f

c c c

f mf

mf f mf

c

µπ µ π

π

µ µ ν µ ππ µ π µ ν π µ

π

=

= ⋅ − − +

+ ⋅ + ⋅ ⋅ + ⋅ + + +

+

ππ

π π

D D

D D D D D D D D

Pre-story: ChiPT

quantum corrections

( )( ){ ( )

62 2 4 24

2

22 2

2 41 # # # # # # ln # n4

l k Qk m k m k mmf mf

π π ππ ππ ππ

+ + + + + + +

ΛΛ

Λ Λ

NDA: naïve dimensional

analysis 11

( )2

2 22 2

1 # # Qk mf fππ π

+

( )4

41,2

2 22 23 4

41 # # # ( )iQk m kc c c cm

f fπ ππ π

Λ+ +

( ) ( )( )

2 24

24 2 2 24

, ,(

, ,12 )d l k m l k ml

f lil m k m iπ π

π π ππ ε ε

Λ

− +− − −∫+ …

( ) 2( )# ln

(4 )R

i if mc c

π ππΛ

= − +

Λ

forbidden by chiral sym

absorbed in non-analytic

error not dominant

as long as

6

2 4QCDM

Qfπ

QCDMΛ >

four parameters; if omitted: • cutoff becomes physical • only one parameter = model

( )

4

2 24 fQ

fπ ππ

( ) ( )) 2( 2( )4 Di CR

Qfc Mππ − −= =

( ) 2 2( )# # lnln

( ) ( )4 4R

i im fc

fc

π ππ πα

πα

= + +

ΛΛ

cf.

2 22

{ , , }{ , ,

22

22

}2 2

, ,p f

QCDQCD Q

n

EFT n p fn p f D QCD

N

C

MM M

m m m fM

cf f

ππ

π π

ψ ψ∆+

=

−∑ πD

( )

0

∞∆

∆=

=∑ 2 2 02 2f fn p d∆ ≡ + + − ≡ + − ≥

chiral symmetry

calculated from QCD: lattice, … fitted to data

(1)=(NDA)

,4

επα =

isospin conserving

isospin breaking

Generalizing,

“chiral index”

min2 2 2i ii

A L V Aν ν= − + + ∆ ≥ = −∑# vertices of type i # loops

min

( ), ,( ) ~ ( ) ( ) ;QCD

QCDi i

i

Q QQm m

T T NM

c FMν

ν νν ν π π

∞∞

=

=

ΛΛ∑ ∑

# nucleons = 0,1 12

13

The story* Weinberg ‘90, ’91, ’92

Ordonez + v.K. ’92 v.K. ’94

Ordonez, Ray + v.K. ’94, ’96 Brockmann, Kaiser + Weise ’96

Gerstendoerfer, Kaiser + Weise ’97 Friar ’99

Kaiser ’99 … …

* Not a history, not even Whiggish

The era of the scriptures

1 2

0l

3

3

2 2(2 )Nml

l kd V Vπ

= +−∫

V

V4

0 2 2 04 2 2

1 1(2 ) N N N N

ll k l l km m m

di V Vi m ilπ ε ε+ − − − + − −∫

1 2

2

N

kEm

=

infrared enhancement: no ChiPT expansion for T for A ≥ 2

potential = sum of subdiagrams without IR enhancement: amenable to ChiPT expansion, cutoff absorbed in counterterms of NDA size

Weinberg’s recipe (“W PC”): truncate potential, solve dynamical equation exactly

[and, as always, check assumptions…]

14

min

, ,( ) ~ ( ) ( ) ;i ii

M Q Qm

fmM

V N cν

ν νν ν

∞

=

ΛΛ Λ∑ ∑

min2 2 2i ii

A L V Aν ν= − + + ∆ ≥ = −∑

not an observable: in general depends on cutoff, form of dynamical equation, choice of nucleon fields, etc.

15

2

2 2

1

QCD

Qf Mπ

2

1fπ

3

2 3

1

QCD

Qf Mπ

4

2 4

1

QCD

Qf Mπ

LO

2-body 3-body 4-body

NLO 2

1

QCD

QMfπ

NNLO

NNNLO

(parity violating)

etc.

…

NNNNLO

in German

…

16

Potential to O(Q^3) with and to O(Q^4) without delta isobar derived

Fit of NN phase shifts to O(Q^3) with delta encouraging; similar accuracy (or lack thereof) for three cutoffs from 500 to 1000 MeV

Pions perturbative in F waves and higher

TPE potential to O(Q^3) without delta improves Nijmegen PWA

min

, ,( ) ~ ( ) ( ) ;i ii

M Q Qm

fmM

V N cν

ν νν ν

∞

=

ΛΛ Λ∑ ∑

min2 2 2i ii

A L V Aν ν= − + + ∆ ≥ = −∑

not an observable: in general depends on cutoff, form of dynamical equation, choice of nucleon fields, etc.

17

Also, many processes with external probes:

o pion elastic scattering o electroweak currents o pion photoproduction o pion production o Compton scattering o …

Weinberg ’92 Rho ’93

Park, Min + Rho ’94 … Beane, Lee + v.K. ‘95

…

18

The Reformation Kaplan, Savage + Weise ’96

Cohen + Phillips ’97 Kaplan ‘97

… v.K. ’97

Kaplan, Savage + Weise ’98 Gegelia ’98

Bedaque, Hammer + v.K. ’98, ... …

Amplitude in 1S0 solved in semi-analytic form for W LO:

NDA fails for chiral symmetry-breaking operators: W PC not entirely correct

YT = +

I YT

YT

χ = +

= +

(0) ( ; ) ( ; )( , ; ) ( , ; ) 1 ( )Y

p k p kT p p k T p p kI k

c

χ χ′′ ′= +

−

χ2 24 ( ) # # ln

4N

N

I mm f

m kkf mπ

π ππππ

= + +

ΛΛ

Λ

02

22( )c Cm mDπ π= + +

W PC: LO NNLO

19

Detailed study of renormalization, validity of NDA, perturbativity of subLOs, power counting, etc.

in simpler pionless EFT for

Moral: faced with W PC vs RG, choose RG

Some lessons: 1) fine-tuning necessary for large scattering lengths can be incorporated into PC for amplitude 2) non-perturbative renormalization intrinsically different from renormalization of corresponding perturbative series 3) one gains no understanding of the renormalization of the A-body system by just monkeying around with higher-order terms in the A-1-body system 4) NDA has very limited usefulness; e.g., three-body force of very high order by NDA, but renormalization requires it at LO 5) subleading interactions must be treated in perturbation theory 6) fully consistent theory works well for very low-energy processes involving (at least) light nuclei and cold atoms, incorporating universal properties such as the Efimov effect, Phillips and Tjon lines, Wigner SU(4), …; yet, mostly ignored by nuclear physics community

Q mπ<

20

Proposal for perturbation approach to pion exchange in chiral EFT

(“KSW PC”)

Some Results 1) manifestly consistent PC 2) rescues NDA for chiral symmetry-breaking operators 3) converges only for Q < 100-150 MeV; at that point pion tensor force no longer perturbative

21

4

NQmπ

LO

2-body 3-body 4-body

NLO

NNLO

NNNLO

etc.

…

0l =

0l =

1l = …

?

4

N NNm Mπ

2

4

N NNm MQπ

2

3

4

N NN

Qm Mπ

24NN

N

fMm

ππ≡

22

LO

LO

NLO

NLO

NNLO

NNLO

NNLO + h.o. ct.

Nijmegen PWA

Nijmegen PWA

Fleming, Mehen + Stewart ‘00

23

4Nm Qπ

Weinberg’s IR enhancement

4pi enhancement compared to ChPT

2

1 4

N NNmf Mπ

π≡

4NN

N

f ffMm

ππ π

π≡

Resum when

NNQ M>

3

3

2 2(2 )Nml

l kd V Vπ

= +−∫

V

V2

4N VQmπ

2

N

kEm

=

But, since Weinberg’s PC inconsistent, then what?

( )2

2

4Qπ

instead of

(0)V = + ?

(0)T = (0)V +

(0)T

(0)V

b.s. at 2

10 MeV4N

NNM fBm

π

π

24

The Counter-Reformation Epelbaum, Gloeckle + Meissner ’98 …

Entem + Machleidt ’03 …

Ekstroem et al., last week

Countless improvements under W PC: 1) elimination of redundant operators 2) correction of some mistakes 3) smart choice of regulator (cutoff not on transferred momentum, to decouple effects of short-range interactions on various partial waves) 4) careful treatment of relativistic corrections … N) fits to NN data at O(Q^4) without delta of similar quality as purely phenomenological pots

Chiral “EFT” becomes input of choice for a new generation of ab initio methods for light and medium-mass nuclei

… Goes Viral

Faced with W PC vs RG, choose W’s PC

Elevate cutoff to physical quantity constrained to CNN Q DM M< Λ <

(But also some steps back, e.g., no deltas until recently, different regulators for different loops)

25

The Reckoning? Beane, Bedaque, Savage + v.K. ’02

Nogga, Timmermans + v.K. ’06 Pavon Valderrama + Ruiz Arriola ‘06

Birse ’06 …

Long + v.K. ’08 Yang, Elster + Phillips ‘09 Pavon Valderrama ’10, ’11

Long + Yang ’11, ‘12 …

Conjecture: NNM mπ>

so that one can think of T as an expansion around the chiral limit, only necessary resummation being that of the tensor force: o singlet channels ~ KSW (solves the W problem with chiral symmetry breaking) o triplet channels ~ W (solves the KSW problem of convergence)

However, W’s PC fails also in triplets!

26

W PC at LO

Nogga, Timmermans + v.K. ’05

10(MeV)E

50100

Attractive-tensor channels:

incorrect renormalization…

That means some counterterms deemed to be subLO because of NDA are actually LO!

27

Nogga, Timmermans + v.K. ’05

11, 0 3(2 )l j

cV ppπ= = = ′

Add needed counterterms at this order,

e.g.,

cf.

0, 1 3(2 )t

l jcVπ= = =

28

W PC at NNLO

Yang, Elster + Phillips ’09

inco

rrec

t

reno

rmalizat

ion…

That means some counterterms deemed to be subNNLO because of NDA are actually NNLO or lower!

29

W PC at NNNLO

0.5 GeVΛ =

1GeVΛ =

5 GeVΛ =

Zeoli, Machleidt + Entem ’12

incorrect renormalization…

That means…

YOU ARE USING THE WRONG PC

30

New, emerging PC: LO: OPE plus needed counterterms (one per wave where OPE is non-perturbative, singular, attractive) subLOs: NPE given by ChPT plus counterterms given by NDA with respect to the lowest order they appear at, treated in perturbation theory

Root of the problem: pion exchanges (long-ranged, contribute to waves higher than S)

are singular (sensitive to short-range physics, require counterterms)

This has NOthing to do with relativity… (For the opposite opinion, see Epelbaum + Gegelia ’12)

(contrast with Epelbaum + Gegelia ’09, who suggest: if you cannot take a large cutoff when treating certain subLOs non-perturbatively,

don’t take a large cutoff. )

31

2

4

N QCDm MQπ

2

3

4

N QCDm MQπ

LO

2-body 3-body

NLO 4

N QCDm Mπ

NNLO

NNNLO

etc.

…

in German 1S =2l ≤

3

4

4

N QCDm MQπ

NNNNLO …

1S =2l ≤

?

(Details still being worked out, e.g. at ESNT Saclay workshop two weeks ago)

4

NQmπ

0S =0l =

0S =0l =

0S =0l =

0S =0l =

32

(0)T = + + … (0)V = +

(0)V(0)V

(0)V(0)V

…

… …

…

…

…

… …

…

…

…

…

(1)T (1)V=

(1)V

(0)T

(0)T

(1)V

(0)T

(1)V

(0)T

+ + +

(0)T

… … … …

…

… …

…

…

…

… …

…

…

( )(0) (0)

(0) (0)

T V

E

ψ

ψ

+

=

(0) ((1) 1) (0)VE ψ ψ=

smaller

large enhancement 4 N Qmπ

b.s.s, resonances

33

(0 1)T + = + = +

(1)V(0 1)V +

(0 1)V + (1)V

…

…

…

…

… …

…

…

…

…

(0 1)T +

(0)T…

…

(1)T…

… + … +

( )( )

( ) 1TT

ν

ν

∂=

∂Λ

Λ

( ) ( )T T fM

Tν ν = + Λ

model dependent uncontrolled

(2)T =

(1)V

(1)V…

…

…

… … (0) (1) (0) (0) (1) (0)

(0) (2) (0

(0)

)

( )(2

0) n n

nn E EV V

E

V

ψ ψ ψ ψ

ψ ψ

=−

+

∑+ …

sum even smaller

(2)V+ … + …

…

missed

error estimate???

cf.

34

Fits to data Pavon Valderrama ’10, ’11 Long + Yang ’11, 12

e.g.

Pavon Valderrama ’10

bands (not error estimates): coordinate-space cutoff variation 0.6 – 0.9 fm cyan: NNLO in Weinberg’s scheme

new PC

35

Conclusion & Outlook much has been learned about EFT in a non-perturbative context

non-analytic parts of long-range pots derived a chiral EFT NN amplitude consistent with RG being constructed compared to the NN amplitude obtained with W PC: it contains more counterterms (thus parameters) at a given order but subLOs require perturbation theory (sorry, but that is what physics asks of you) details still being worked out, but first results suggest possibility of better fits to data than W PC; perhaps a “realistic” amplitude emerges at NNNLO? few-body forces and currents remain to be studied; effects could be substantial since they are tied to NN amplitude