A Wilsonian viewonthe nuclear effective field theory
Koji Harada
Dept. of Physics, Kyushu Univ.
based on the works in collaboration with
K. Inoue, H. Kubo, A. Ninomiya, T. Sakaeda,
and Y. Yamamoto(Phys. Lett. B636 (‘06), Nucl. Phys. B758 (‘06),
Int. J. Mod. Phys. 24 (‘09),Phys. Rev. C83 (‘11), and work in progress)

2011年9月1日木曜日

Plan
Introduction to NEFT
Power counting and (Wilsonian) RG
NEFT without pions
NEFT with pions
Summary

2011年9月1日木曜日

Introduction to NEFT
NEFT: Nuclear Effective Field Theory
describes interactions among nucleons
is not a model
based on general principles of EFT, systematic,
related to QCD through chiral symmetry (cf.
phenomenological NN potentials)
An extension of chiral perturbation theory(Nonrelativistic) nucleons (and pions) are explicit
degrees of freedom

2011年9月1日木曜日

As an EFT
has a finite range of validity,
determined by a physical cutoff Λ0 (≈400 MeV for NEFT with pions, ≈ mπ for
NEFT without pions)has an infinite number of operators
which should be organized by power
counting rules(We limit ourselves to the NN systems.)
2011年9月1日木曜日

Peculiarity of S-waves
Fine-tuned!
1S scattering length : -23.7 fm
0
3S scattering length : 5.4 fm
1pion Compton wavelength: 1.4fm
inherently nonperturbative
Existence of deuteron

2011年9月1日木曜日

Interactions
local 4-nucleon operators withan arbitrary number ofderivatives (contact operators)
All of them are irrelevant
naively.
pion exchange

2011年9月1日木曜日

Nonperturbative features
The existence of the bound state(deuteron) suggests that a naive power
counting (for the amplitudes) breaks
down somewhere.
The fine-tuning suggests the existenceof a nontrivial fixed point.

2011年9月1日木曜日

Power counting
Power counting is important, and isnontrivial in the NEFT.
There are mainly two power counting
schemes
Weinberg(’90)
Kaplan-Savage-Wise(’98), Van
Kolck (’99)

2011年9月1日木曜日

Weinberg’s power counting
Construct an “effective potential” (sumof 2-nucleon irreducible parts)
perturbatively on the basis of the naive
dimensional analysis
Substitute it into the LippmannSchwinger (or Schrödinger) equation(i.e., Solve nonperturbatively)
2011年9月1日木曜日

Pros and cons
Very successful numerically. There exist N3LO
calculations(Machleidt and Entem (’05),Epelbaum,
Glockle, and Meißner(’05) ), which accurately
reproduce NN scattering data.
Power counting is inconsistent. Higher ordercounterterms are necessary to cancel thedivergences in lower orders.
The fine-tuning is not taken into account.
Nonperturbative divergences (cutoff
dependence) are not cancelled.
2011年9月1日木曜日

KSW power counting
Only the contact operator without
derivatives is iterated.
Other interactions, even the pionexchanges are treated as perturbation.
LO
NLO
perturbative
2011年9月1日木曜日

Pros and cons
Analytic expressions for the amplitudes (andthe phase shifts) can be obtained.
Dimensional regularization with PDS (power
divergence subtraction) may be used.
The fine-tuning is taken into account.
There is no inconsistency in the power
counting.
The EFT expansion however does not converge!

2011年9月1日木曜日

To be perturbative, or nonperturbative?
That is a question!?
NNLO

150

100

<
b0(deg)Fleming et al. (’00)
explicitly show that the
EFT expansion with KSW
power counting does notconverge in several
channels, including 3S1
at the NNLO, due to thetensor force.

2011年9月1日木曜日

LO
50
3

S1

NLO

0
0

100

200

p(MeV)

300

Why it fails?
The tensor part of pion-exchange issingular (∼ 1/r3) at short distances.
Perhaps it must be treated
nonperturbatively...?
Beane, Bedaque, Savage, and van Kolck(’02) propose that only the 1/r3 part of OPE
should be treated nonperturbatively.
What determines the correct power counting?

2011年9月1日木曜日

From “trials and errors”
to a systematic way
Usually, a power counting scheme is
proposed and the justification of it comesfrom a good fit to the data.
It is sometimes hard to find the correct
one when it fails.
We would like to have a theoreticalframework which determines possible power
counting schemes.
2011年9月1日木曜日

Power counting
and (Wilsonian) RG
What is power counting?
Order of magnitude estimate on the basis ofdimensional analysis.
The importance of an operator is determinedby its dimension.
Quantum theoretically, the scaling
dimension should be considered.
It is obtained by a (Wilsonian) RG analysis.
2011年9月1日木曜日

Wilsonian RG
for the NN systems
Simplification for a nonrelativistic fermion system
Anti-fermions are not excited: States are divided
into sectors parametrized by the nucleon number.
(We are interested in the two-nucleon sector.)
Rotational symmetry: Each sectors are further
divided into spin, (isospin, ) and partial wave
sectors. (We concentrate on the S-waves.)
Pauli principle: The number of operators with adefinite number of derivatives are very limited.

2011年9月1日木曜日

Wilsonian RG
for the NN systems
Galilean invariance: The usual
implementation of a cutoff
function in the Eucledean
propagators does not work.
Rather, we cutoff the relative 3momenta in the loops.
A sharp cutoff thus does notproduce terms which arenonanalytic in external
momenta.
2011年9月1日木曜日(E’, p’)k(E, p )

(E’, p’)k(E, p)

Redundant operators
A RG transformation generates all kinds of local
operators which are consistent with symmetry.
Operators such asarealso considered, though they may be eliminatedby the use of field redefinition.
They are necessary for the cutoff independence ofoff-shell amplitudes.
They may be viewed as “energy-dependent
potential.”

2011年9月1日木曜日

NEFT without pions
Valid for p ≤ mπ.
Nucleons are the only explicit degrees of
freedom. Pions (and other mesons, Deltas,etc.) are “heavy,” so that their effects are
represented as contact operators.
A simplest field theory with a nontrivial
fixed point...

2011年9月1日木曜日

Averaged action (up to O(p ))
2

(✁
π)
ΓΛ =

�

� �
�
2
∇
4
†
d x N i∂t +
N
2M


�

(S) (S)(S) (S)(S) (SB)

−C
O
+
C
O
+
2B
O2
,

0
0
2
2

(S)
O0 =

�

�

(T ) (T )(T ) (T )(T B)(SD) (SD)

+ C2
O2
,
 −C0 O0 + C2 O2 + 2B (T ) O2
T

Pa(S) N�† �

T

Pa(S) N

�

N
N
,
��
�
�† �
�
←
→
(S)
O2 = N T Pa(S) N
N T Pa(S) ∇ 2 N + h.c. ,���� �† �
�
�
2
∇
(SB)
O2
N
N T Pa(S) N + h.c. ,=N T Pa(S) i∂t +2M

2011年9月1日木曜日

(T )
O0 =

�

T

(T )
Pi N

�† �

T

(T )
Pi N

(1 S0 channel)(3 S1 –3 D1 channel)
�

N
N
,
��
�
�† �
�
←
→
(T )
(T
)
(T
)
O2 = N T Pi N
N T Pi ∇ 2 N + h.c. ,
��
�
�
�
�
�† �
←
→
←
→
←
→
1
(SD)
(T )
(T )
O2
= N T Pi N
N T ∇ i ∇ j − δij ∇ 2 Pj N + h.c. ,3����� �† �
�
2
∇
(T B)
(T )
(T )
O2
= N T Pii∂t +N
N T Pi N + h.c. ,2M

1RGEs for S0 ( up to O(p2))
dx= − x − (x + y + z)2
/Λ)
t
=
ln(Λ
0
dt
�
�
dy
1 2
3 2
1 2
= − 3y −x + 2xy + y + yz − z
dt
2
2
2
�
�
dz
1 2
1 2
3 2
= − 3z +x + xy + y − xz − yz − z
dt
2
2
2

Operators mix.

We have examined the case with operators
up to O(p4), but the results do notchange very much.
2011年9月1日木曜日

RG flow in a subspace
1

y

0.5

trivial f.p.

0

nontrivial
f.p.

-0.5

-1
-1.5

2011年9月1日木曜日

-1

-0.5

x

0

0.5

1

3
3
RGEs for S1- D1

t = ln(Λ0 /Λ)

dx�
�
�
�
� 2
�2
= − x − (x + y + z ) − 2w
dt
�
�
�
dy
1 �2
3 �2
1 �2
�
� �
� �
�2
= − 3y −x + 2x y + y + y z − z
−w
dt
2
2
2
�
�
�
dz
1 �2
1 �2
3 �2
�
� �
� �
� �
�2
= − 3z +x +xy + y −xz −y z − z
+w
dt
2
2
2
dw�
= − 3w� − (x� + y � + z � )w�
dt

Very similar to the 1S0 case2011年9月1日木曜日

RG flow in a subspace1

trivial f.p.

y’

0.5

0

nontrivial
f.p.

-0.5

-1
-1.5

2011年9月1日木曜日

-1

-0.5

x’

0

0.5

1

Fixed points and scaling
dimensions
Nontrivial fixed point:
1S : (x*,y*,z*)=(-1,-1/2,1/2)
0
3S -3D : (x’*,y’*,z’*,w’*)=(-1,-1/2,1/2,0)1
1

Scaling dimensions:
1S : (+1,-1,-2)
0
3S -3D : (+1,-1,-2,-2)
1
1

Only one combination of operators is relevant.
Anomalous dimensions are +2. (cf. Birse et al. (’99))

2011年9月1日木曜日

What we found:
1

0.5

y

Our Wilsonian RG analysis implies a power
counting very similar to the KSW’s.
0-0.5

It is useful to use the language of criticalphenomena for the NN system in the S-waves:
x
-1

-1.5

-1

-0.5

0

0.5

There are strong-coupling and weak-couplingphases. 1S0 channel is in the weak-couplingphase, whereas 3S1-3D1 in the strong-couplingphase.
The inverse of the scattering length is the orderparameter.

2011年9月1日木曜日

1

NEFT with pions
valid up to Λ0 ≈ 400 MeV
Nucleons and pions are explicit degrees
of freedom. Their interactions areconstrained by chiral symmetry.
Pions propagate as relativistic particles.

2011年9月1日木曜日

Previous Wilsonian RG
analysis by M. Birse (’06)
Using the distorted-wave RG formalism,
treats pion exchanges nonperturbatively.
does not introduce a cutoff(separation
scale) for pion exchanges.
finds that the scaling dimensions shifts
by ±½ from those of the pionless case, evenat the trivial fixed point.

2011年9月1日木曜日

Nonlinearly realized chiralsymmetry
We spent a lot of time for formulating a
Wilsonian RG framework with exact chiralsymmetry, but did not succeed...
The nonlinearly realized (global) symmetry
is similar to local (gauge) symmetry.
We are interested in the flow itself.
We eventually gave it up, and consider theleading order in the (p/M) expansion
qualitatively.
2011年9月1日木曜日

IR enhancement
Consider one-loop diagrams with the 3momentum integral being restricted tothe shell-modes.
The energy integration is dominatedeither by the nucleon poles or by thepion poles.
The contribution of the nucleon poles in
the 2-nucleon reducible diagrams get
the enhancement factor (M/Λ). (cf.
Weinberg (’90))
2011年9月1日木曜日

IR enhancement
The IR enhancement is a consequence of thenonrelativistic kinematics. It is not directly
related to the fine-tuning.
The nucleon pole makes the pion propagator“instantaneous”, i.e., the Yukawa potential.
In the leading order in (M/Λ) expansion, thepions appear only as the instantaneous
Yukawa potential.
i2[−p0 + (p + k)2 /2M ] − ωk2

2011年9月1日木曜日

p0 −p2 /2M �Λ2 /M �Λ

−→−ik2 + m2π

Reduced theory
Because at leading order in (Λ/M),
pions only contribute as theinstantaneous Yukawa potential, andonly the two-nucleon reducible one-loop
diagrams should be considered, we may
start with a reduced action.

2011年9月1日木曜日

Averaged action (up to O(p ))
2

�

�
�
(π)
(c)
(c)
ΓΛ =ΓΛ✁ + d4 x −D2 m2π O0
�
�
2 �
gA
+ 2
dt d3 x d3 y O(S) (x, y)∇2x4f(T )+ Oii (x, y)∇2x

�
��
1
(T )
x x
− 6Oij (x, y) ∂i ∂j − δij ∇2x Y (|x − y|)
3
�
�†�
�
O(S) (x, y) = N T (x)Pa(S) N (y) N T (y)Pa(S) N (x) ,�(T )Oij (x, y) =2011年9月1日木曜日

N

T

�†�

(T )
(x)Pi N (y)

N

T

(T )
(y)Pj N (x)�,L-OPE and S-OPE
It is a general principle to represent physics
beyond the cutoff as contact interactions.
It usually refers to the effects of heavy
particles.
Pion is light, but its short-distance effects(OPE with momentum transfer larger
than the cutoff) should also be represented
as contact interactions.

2011年9月1日木曜日

L-OPE and S-OPE
We will show :A part of the short-distance part of OPE (S-OPE) isrelevant, and should be treated nonperturbatively.
The long-distance part of OPE (L-OPE) should betreated as perturbation.
The separation is important, since they behave
differently.
The introduction of the separation scale is discussed
by Beane, Kaplan, and Vourinen (’09). But thephilosophy is probably different.

2011年9月1日木曜日

1
RGEs for S0 (assuming p, mπ < Λ)
�
� 2
dx
2
2
= − x − x + 2xy + y + 2xz + 2yz + zdt− 2(x + y + z)γ − γ 2 ,
�
�
dy
1 2
3 2
1 2
= − 3y − x + 2xy + y + yz − z
dt
2
2
2
1 2
− (x + 2y)γ − γ ,
2
�
�
dz
1 2
1 2
3 2
= − 3z + x + xy + y − xz − yz − z
dt
2
2
2
1 2
+ (x + y − z)γ + γ ,2
du= − 3u − 2(x + y + z)(u − γ) − 2uγ + 2γ 2 ,dt

γ is a measure of
the strength of OPE
2011年9月1日木曜日

3
3
RGEs for S1- D1 (assuming p, mπ < Λ)
�

dx�
2
2
2
2
= − x� − x� + 2x� y � + y � + 2x� z � + 2y � z � + z � + 2w�
dt�

− 2(x� + y � + z � − 4w� )γ − 9γ 2 ,
�
�
�
dy
1 �2
3 2
1 2
2
= − 3y � −x + 2x� y � + y � + y � z � − z � + w�
dt
2
2
2
7 2
− (x + 2y )γ + γ ,
2
�
�
�
dz
1 �2
1 �2
3 �2
2
�
� �
� �
� �
= − 3z +x + x y + y − x z − y z − z + w�
dt
2
2
2
�

�

9
+ (x� + y � − z � − 4w� )γ + γ 2 ,
2 �
�

dw�
= − 3w� − x� w� + y � w� + z � w�dt

1
+ (2x� + 2y � + 2z � − 9w� )γ + 2γ 2 ,5Large coefficients!

du�= − 3u� − 2(x� + y � + z � )u� + 2(x� + y � + z � − 4w� )γ − 2u� γ + 18γ 2 .
dt

2011年9月1日木曜日

Dimensionless coupling
constant for pion exchange
MΛ
γ≡
2π 2

!

gA
2f

"2

There are other ways of defining thecorresponding dimensionless coupling.
The self-similarity property of RGEs (thecutoff should not appear in β functions)
fixes this ambiguity.
dγ = −γdt
2011年9月1日木曜日

RG flows

1.2
1

in subspaces

0.8
0.6

spin
singlet

0.4
0.2
01

2
0.5

Y

0

-0.5

-2
-1 -4

0

X

1.2
1
0.8

spin
triplet2011年9月1日木曜日

0.6
0.4
0.2
0
1.5

2
1

0.5

0

Y'

0

-0.5

-2
-1 -4

X'

Fixed points and scaling
dimensions
Nontrivial fixed point:
1S : (x*,y*,z*, u*,γ*)=(-1,-1/2,1/2,0,0)
03S -3D : (x’*,y’*,z’*, u’*, w’*,γ*)=(-1,-1/2,1/2,0,0,0)
1
1

Scaling dimensions:
1S : (+1,-1,-1,-2)
0
3S -3D : (+1,-1,-1,-2,-2)
1
1

The nontrivial fixed points are identified with those in the
NEFT without pions.
The scaling dimensions are essentially the same as those in the
NEFT without pions.

2011年9月1日木曜日

It means:
The inclusion of pions (L-OPE) does notaffect the existence (and the location) of
the nontrivial fixed point, nor changes thescaling dimensions.
The tensor force affects only the details of
the flow (strong dragging).
Pions (L-OPE) become less and lessimportant at lower energies.

2011年9月1日木曜日

For Λ less than mπ
We do not expand the contributions in powersof (mπ/Λ) anymore. Instead, we now expand
them in powers of (Λ/mπ) and (p/mπ).
The diagrams are essentially the same, but
the RGEs changes.
The dimensionless coupling for the pionexchange is also changed in accordance withthe self-similarity property.

2011年9月1日木曜日

1
RGEs for S0(assuming p < Λ < mπ)Λ2γ̃ = 2 γ,mπdχ=(first line of the pionless one with x → χ)dt
− 2(χ + y + z)γ̃ − γ̃ 2 ,dy=(first line of the pionless one with x → χ)
dt
3 2
− (2χ + 3y + z)γ̃ − γ̃ ,2dz=(first line of the pionless one with x → χ)
dt
1 2
+ (χ + y − z)γ̃ + γ̃ ,2

2011年9月1日木曜日

m2πũ = 2 u.
Λχ ≡ x + ũdγ̃= −3γ̃
dt

Smoothly connected to the
NEFT without pions
The nontrivial fixed point can be
identified with that in the NEFT without
pions, with x being replace with χ.
The pion exchange (L-OPE) is much
more irrelevant.

2011年9月1日木曜日

Summary
We have performed Wilsonian RG
analyses of the NEFT with and without
pions in the NN systems in the S-waves.
There is only one relevant operator, though
the other operators also get large
anomalous dimensions.
Our analysis suggests a power counting
very similar to KSW’s.
2011年9月1日木曜日

Summary
The separation of pion exchanges into twoparts, S-OPEs and L-OPEs is necessary forthe consistent Wilsonian treatment. It is also
important, because they behave differently.
The inclusion of pions does not affect theexistence (and the location) of the nontrivial
fixed point, nor the scaling dimensions.
Our results are very different from thoseobtained by M. Birse.
2011年9月1日木曜日

Complementary (or, hybrid)
regularization
Dimensional regularization is useful
because operator mixing is minimal.
But it does not have a separation scale.
A regularization with an explicit
separation scale is necessary to separatepion exchanges into two parts, butcauses operator mixing, and makes thepractical calculations intractable.
2011年9月1日木曜日

Complementary (or, hybrid)
regularization
A new regularization scheme
PDS for contact interactionsμGaussian damping for L-OPE

λSchematicallyk2m2π−k2 /λ2−→ 1PDS − 2e222
k + mπk + mπS-OPE
2011年9月1日木曜日

L-OPE

We thinkµ≈λ