Wilsonian RG analysis of the P-wave nucleon-nucleon scattering including pions

Koji Haradain collaboration with

Hirofumi Kubo, Tatsuya Sakaeda, and Yuki Yamamoto

Nuclear Effective Field Theory(NEFT)(with pions)

degrees of freedom: nucleons, pions

includes all the operators consistent with the symmetries, i.e., it is not a model.

systematic Power counting isimportant!

Power countingIn the previous papers, we have argued that the power counting should be determined consistently with the scaling dimensions, obtained by the Wilsonian RG analysis.

We have done it in the S waves, and the result leads to the power counting which is very similar to the KSW one.

Harada and Kubo, NPB758 (2006)Harada, Kubo, and Yamamoto, PRC83 (2011)

Kaplan, Savage, and Wise, NPB534 (1998)van Kolck, NPA645 (1999)

Power counting in the P waves

P waves are different from S waves:

No fine tuning seems to be present, i.e., no resummation is necessary.

The leading interaction seems the pion exchange.

There seem no counterterms for the potential-box diagrams.

Strange!

a naive KSW power counting for the P waves(according to Fleming et al.)

No contribution of order

contribution consists solely of single-pion exchange

the order contribution comes from the potential-box diagram

Four-nucleon operators only contribute at higher orders in

Q�1

Q0

Q

Q

no counterterms are provided

Kaplan, Savage, and Wise, NPB534 (1998)Fleming, Mehen, and Stewart, PRC61 (2000)

The results of Fleming et al.

triangles:Nijmegen

long dashed line:NLO

short dashed line:NNLO

They all do not fit well!

The KSW power counting fails !?

Review of the S-wave case

It is important to separate the short-distant part of OPE (S-OPE) from the long-distant part (L-OPE) by using the same cutoff as that which is imposed on contact interactions.

L-OPE is irrelevent, while a part of S-OPE is relevant, which should be treated nonperturbatively.

The resulting RGEs are smoothly connected to the ones without pions.

Harada, Kubo, and Yamamoto, PRC83 (2011)

Wilsonian RGEs

To the order of only the diagrams given above contribute.

To the leading order of pions do not propagate: they are treated as “potential pions.”

O(p2)

�/M

Pion exchangePion exchange in the P waves is different from that in the S waves.

!r13

r13 + m2!

+r14

r14 + m2!

" !m2

!

r13 + m2!

! m2!

r14 + m2!

"

!!ijr13 ! 2pi

13pj13

r13 + m2!

+!ijr14 ! 2pi

14pj14

r14 + m2!

"

S waves P waves

singlet

triplet

rij ⇥ |pi � pj |2

crucial minus signs!

!!ijm2

! + 2pi13p

j13

r13 + m2!

! !ijm2! + 2pi

14pj14

r14 + m2!

"

Pion exchangeIn the P waves, the leading order terms cancel out.

S waves P waves!

M!!

M!3! (momentum factor)

shell mode

This implies that the pion exchange in the P waves is more irrelevant.

Actually, it is of the same order as the leading-order contact interaction.

! ! M!2"2

!gA

2f

"2

Contact operatorsx · · ·� 2�2

M�P

y · · ·� 2�2

M�3P ⇥ r12 + r34

4

z · · · 2�

�3P ⇥

4�

i=1

Si

u · · ·� 2�2

M�3P ⇥m2

�

rij ⇥ |pi � pj |2

Si ⇥ p0i � p2

i /2M

P(1P1) · · ·⇧

34

⌃p12 · p34

�P † ⇥ P

⇥

P(3P0) · · ·⇧

14

⌃pj12 pk

34

�(P k

a )† ⇥ P ja

⇥

P(3P1) · · ·⇧

38

⌃ ⇤p12 · p34 �jk � pk

12 pj34

⌅ �(P k

a )† ⇥ P ja

⇥

P(3P2) · · ·⇧

38

⌃ ⇧p12 · p34 �jk + pk

12 pj34 �

23pj12 pk

34

⌃�(P k

a )† ⇥ P ja

⇥

P =1

2�

2(i�2)(i⇥2)

P ia =

12�

2(i�2�i)(i⇥2⇥a)

RGEs without pionsThe RGEs are the same in all channels

dx

dt= �3x� (x + y + z)2

dy

dt= �5y �

�12x2 + 2xy +

32y2 + yz � 1

2z2

⇥

dz

dt= �5z +

�12x2 + xy � xz +

12y2 � yz � 3

2z2

⇥

There is a trivial fixed point, as well as a nontrivial one (-3,9/2,-9/2).

Harada, Ninomiya, and Kubo, IJMP A24 (2009)

RGEs without pionsHarada, Ninomiya, and Kubo, IJMP A24 (2009)

-2

-1

0

1

2

3

4

5

6

-4 -3 -2 -1 0 1 2

v

u

RG flow for the P-wave

u = x

v = (y � z)/2

couplings

RGEs with pionsThere is an additional contact interaction proportional to the pion mass.

u · · · m2!

!2

!! 2!2

M!

"P

With the contact interactions, it satisfies the following RGE.du

dt= !3u ! 2(x + y + z)u

channelNo tensor force contributes. The effect of L-OPE is just to modify the RGE for .

1P1

u

du

dt= (pionless)+4(x + y + z)�

channelThe tensor force is attractive.

dx

dt= (pionless)�8

3(x + y + z)� � 16

9�2

dy

dt= (pionless)� 4

15(4x + 9y � z)� � 8

15�2

dz

dt= (pionless)+

43(x + y � z)� +

89�2

du

dt= (pionless)+

43(x + y + z � 2u)� +

169

�2

3P0

channelThe tensor force is repulsive.

3P1

dx

dt= (pionless)+

43(x + y + z)� � 4

9�2

dy

dt= (pionless)+

215

(4x + 9y � z)� � 215

�2

dz

dt= (pionless)�2

3(x + y � z)� +

29�2

du

dt= (pionless)+

43u�

channelThe tensor force is attractive.

3P2

dx

dt= (pionless)� 4

15(x + y + z)� � 4

9�2

dy

dt= (pionless)� 2

75(4x + 9y � z)� +

2275

�2

dz

dt= (pionless)+

215

(x + y � z)� +29�2

du

dt= (pionless)+

815

�x + y + z � 1

2u

⇥� +

169

�2

Summary

We have performed the Wilsonian RG analysis for the P waves in the NN sector in the NEFT to the NLO.

We have obtained consistent RGEs for all channels, though a naive KSW power counting suggests that there is no counterterm for the loop diagrams with pions.

Summary

This shows that the L-OPE in the P waves is more irrelevant than that in the S waves.

To our best knowledge, no one has ever suggested that the power counting for the pion exchange depends on the channel.

Future workWe are going to fit the amplitude calculated with the hybrid regularization, which takes into account the results of the Wilsonian RGE analysis, to the experimental values.