Post on 23-Mar-2019
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…]
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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.
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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