Exploring novel scenarios of NNEFT

Exploring novel scenarios of NNEFT

Ji-Feng YangEast China Normal University

2009-8-5 KITPC program: EFTs in P&NP 2

OUTLINES

• EFT approach to nuclear forces• Contact potential and LSE• Rigorous solutions• Implications for

renormalization• Various power counting

scenarios• Summary


EFT approach to nuclear forces• Pre-EFT era: Various phenomenological

models

• EFT era (1990-): (a) model independent (guided by chiral symmetry of QCD) and (b) systematic (organized by EFT power counting)

• Status of EFT approach: Successful as a field-theoretical basis for nuclear forces and a bridge between nuclear physics and QCD (A recent review: EHM arXiv: 0811.1338)

• An interesting issue: an understanding of power counting and renormalization of EFT in nonperturbative regime. (Recent efforts: NTvK 2005; LvK 2007; YEP 2007, 2009; BKV 2008; EG 2009; etc.)


Contact potential and LSE

• Weinberg’s proposal for NNEFT in nonperturbative regime (due to IR enhancement from “large” nucleon mass):

• 1) Potentials from CHPT• 2) T-matrices from Lippmann-Schwinger

Equation (LSE) or Schrödinger Equation• 3) EFT power counting (PC) applied to potentials

• In pionless EFT (contact potentials): • 1) The framework is simple• 2) LSE allow for rigorous solutions• 3) Nonperturbative essentials become

transparent


Setups

jiL

jiij

LL qqqqqqV 222/

0,

),(

:, qq

Contact potential in an uncoupled channel (L):

external momentacontact couplings:ij

ijT qqqqU ),,,,()( 420

EFT expansion order

jiij

:

)()(),( qUqUqqqqV TLL

Introducing

In the same fashion,

)()()(),;( qUEqUqqET T Then, LSE reduces to )()(

~)( EEIE

with iMkE

kUkUkdEI

T

/

)()(

2)(

~23

3

assuming all divergences


NOTE: The above formalism does not depend on the specifics of power counting for couplings.

General form of Ĩ(E) reads

)()()()()(~

252132 pUJpUJpUpUEI lLT

with ,MEp ,400 pM

iJ

L

n

nLn

L pJp1

)(212

20

2

021

1)(

p T

dt

tUtUddt

ppU

2

0

12

1)(

pn

n dt

tUddt

ppU

:)0(, 120 nJJ n prescription-dependent !

For example, in 1S0 channel, Ĩ(E) looks like

52

34

032

0

32

00

)(~

JpJpJp

Jp

EI


Rigorous Solutions

1

)(~

1)(

EIE )()()(),;( qUEqUqqET T

LL

L

pJpD

JpN

T 22

2

0 ])[],[;(

])[],[;(1

• Rigorous solutions (uncoupled channels) :

• General form of on-shell T-matrix with L:

• For coupled channels with J:

JJx

JxJJ

JJT

NN

NN

D 2

24

)12(0

01 1

0

0

p

pp

p

“J-”= (J-1,J-1); “J+”= (J+1,J+1); “Jx”= (J-1,J+1) and (J+1,J-1). N..., D..., N... and D…: polynomials again in terms of p squared, [λ] and [J…].• Unitarity: prescription-independent. For

coupled channels:

10

01

2

*11 piM

JJ TT


Examples jiij C

:01S 1

0001 DNT

0 10 N 00 CD 1

001 CT

2 2310 1 JCN 23115

2100 )2( pJCCJCCD

4

2

0

2;00

j

jj pNN

3

0

2;00

j

jj pDD

23207539

23

352

22

732

522522

52310;0~

)2(~

)(~

2~

)1(

JCCJJJJJJCC

JJJCCJCJCJCN

)(~

2)~

1(

952

722

2

72192

252

15200;0

JJJCC

JCCJCJCJCCD

][ 23

2133;02;0 JCJDN

:,, 13

03

11 PPP 21

1101 pDNT

0 11 N 01 D 0T


2 30;1 1 JCN P 0;1 PCD 23

10;0

1 pJCT P

4 21;10;11 pNNN 2

1;10;11 pDDD

732

1;30;2

51;0;1 )1( JJCJCJCN PPP

)2( 51;31;1;1 JCJCN PP 72

1;0;0;1 JCCD PP

)2( 51;1;1;1 JCCD PP ][ 31;11;1 JDN

:13

13 DS

11x

1x1

11T

NN

NN

D 2

24

40

01 1

0

0

p

pp

p

0 0;1111 0 SC NDNN x singular T -11 :

Actually,

010;0

1

11x1- TTCT S

2 21;131;1;11 10 SDSSD CJCC DNN x

23

21;

21;1;5

21;

21;0;1 2 pJCCCJCCC SSDSSDSS N


4 More involved expressions in terms of couplings and [J…].

• The T-matrices obtained above are intrinsically nonperturbative due to their closed form.

• At a finite Δ, only finite many parameters [J…] (or divergences) are present at all, i.e., Rank(Ĩ(E))‹∞.

0 is universally present and ‘disentangled’ from all contact couplings in the inverse on-shell T-matrices.

• N… and D… or N… and D… are 0-independent and ‘perturbative’ in terms of p squared and couplings.

Remarks


• The divergences in Ĩ(E) or [J…] must be so removed that the closed form of the T-matrices be preserved.

• The functional dependence of T-matrix upon momenta is physical and renormalization group (RG) invariant. For closed form T-matrices, RG invariance is consequential.

• The conventional subtraction algorithm could not work for the nonperturbative divergences in the compact T-matrices. The subtraction must be done otherwise.

Implications for renormalization

Let us elaborate on these points below.


Failure of ‘exogenous’ counter-terms

:01S

1

23115

210

231

0)2(

)1(

pJCCJCC

JCT

:11P

Endogenous/exogenous counter-terms: introduced before/after nonperturbative summation is finished.

Thus, for compact T-matrices, exogenous counter-terms could not succeed.

It suffices to examine 1S0 and 1P1 channels at Δ=2.

123

10;0

pJCT P

Evidently, no exogenous counter-terms could remove the divergences in 0, J3 and J5 in the two compact T’s.

A mismatch between λ and Ĩ(E)Since

)(~

)()()(~

)( 11 EIEEEIE


jiEI

Ljiji

ij

ij

,,0)(~

2/,,01

and

:s.t. As

Consequently: The divergences in Ĩ(E) could not be completely absorbed by couplings in a sensible and consistent manner. Thus, beyond the leading order, we must seek for other sources for the endogenous counter-terms beyond couplings.

0. except ...],[ or and between Mismatch JEI )(~1

Then the issue boils down to renormalization of Ĩ(E).

Arising of endogenous counter-terms

In underlying theory viewpoint, EFT’s are built from low-energy projections. Then, divergences arise as projection and loop integration do not commute:

0])(,[ ldLE

C.T.T.

C.T.T. : short-hand notation for commutator


:LE

:

)(ldLE

LE projection from UT onto EFT

Rearranging the commutator:

C.T.T. LELE ldld )()(

:

LEld )(well-defined loop integralill-defined or divergent loop integral

Therefore, the underlying-theory perspective shows that counter-terms naturally arise at the level of loop integrals, hence, ‘endogenous’. So, subtractions should be performed at the level of loop integrals to render [J…] finite, which could not be fully accomplished with counter-terms from couplings within EFT framework due to the reasons given above.The closed form T-matrices will develop nontrivial dependence upon renormalization prescriptions. To remove the nontrivial prescription dependence, it is crucial to impose physical boundary conditions.


From now on, [J...] denote renormalized parameters, which should depend upon scales in the range: (0, Λ], with Λ being the EFT upper scale. In principle, J… may depend on Λ and other physical scales besides a running scale μ, i.e.:

Consequences of RG invariance

,5,3,0 ),,;(4

mM

J mm

Now, for on-shell T-matrices, RG invariance reads, 00 1 T

d

dT

d

d

For the closed form T-matrices,

or :0 ,0])&[],([,00 DNR DNd

d

d

d

R…: appropriate ratios in terms of [N…,D…] or [N…,D…]. So, RG inv beyond LO: [0, R…([N…,D…] or [N…,D…])] 1) J0 (=Re(0)) depends on physical scales only, or it is a physical quantity:


),(4

,0 000

MJJ

d

d

2) A possible form of the ratios R…([N…,D…]) in uncoupled channels:

0

2;

21

2;

00;

;;

0;

;;

11

,, ,

j

jjDL

Li

iiNL

L

jLjDL

L

iLiNL pp

p

Tji

N

D

N

N

R

RRR

The concrete form or value of ρ0 could in principle be determined from physical boundary conditions.

3) An example of nonperturbative running couplings:

:2)(Δ01 S

231

3111;02

31

52

100;0

1

2,

1 JC

JCC

JC

JCCDD

RR

,)(11

)(

)(

)(1

,)(11)(

1

2

31;023

5

31;0

0;00

31;03

1

JJ

J

JC

JJ

C

DD

D

D

RR

R

R

; points fixed IR2

,:0)( 1010;00

;DD CCμ

RR

. :points fixedUV 0,0)( 10 CCμ


4) More physical scales or constraints on couplings and parameters [J...] from RG invariance of [R…]:

:4)(Δ01 S ],[ 2

32

133;02;0 JCJDN

physical, 3;02;03;02;03 // DNDNJ RR

:4)(Δ11 P ],[ 31;11;1 JDN

physical, 1;11;11;11;13 // DNDNJ RR

At leading order, Ĩ(E)=0, T≠0 only for L=0 :

0/1 001 C

d

dT

d

d

invariant, RG is 001

0;0 /1)Re( CJT Rbut J 0 is not!

At this order, we can put

40

MJ

then KSW scaling could be reproduced:

M

MC

/4

/4)(

0;00 R


,4

~),0(4

~;4

~ 012

12122

;

MJn

MJ

MC n

nnL

nL

Due to the nontrivial prescription dependence, the EFT power counting for coupling should be supplemented with that for the prescription paramters [J…]. Below, we explore three different scenarios for comparisons.

•PC B:

,4

~),0(4

~;4

~ 012

121

;

MJn

MJ

MC n

nnLnL

nL

•PC C:,

4~),0(

4~;

4~ 0

1212

122;

M

JnM

JM

C nn

nLnL

•PC A:

(Below we setμ=Λε)

Various power counting scenarios


Qualitative behaviors of T 1S0 (Δ=4)•PC

A

with a fine-tuning of C0 (does not matter for PC A)

...14

...11

4

1

2

235

0

2

2333

pc

M

pM

T

1,0]dim[,

40000 cc

McC

•PC B

2

22

0

2

2

114

11

4

1

pc

M

pM

T

•PC C

Motivations: preserving the conventional EFT power counting while yielding unnatural scattering length. As J0 is RG invariant, we simply set [J…] in the way shown.


...14

...11

4

1

2

235

0

2

2333

pc

M

pM

T

with a finer-tuning of C0

20000 1,0]dim[,

4

cc

McC

ERE(1S0) PC A: PC B: PC C:

1/(a·Λ) (1+ (^3))

natural

(^2+ (^3))

unnatural

(^2+ (^3))

unnatural

re·Λ (1+ (^3))

natural

(1+ ())natural

(1+ (^3))

natural

v2·Λ^3 (1+ (^3))

natural

(1+ ())/ unnatural

(1+ (^3))

natural


ERE PC A: PC B: PC C:

3S1 1/(a·Λ) (1+ ()) N

(^2+ (^3))

U

(^2+ (^3) )

U

3S1 re·Λ (1+ (^3))

N

(1+ ())N

(1+ (^3))

N

3S1 v2·Λ^3 (1+ (^3))

N

(1+ ( ))/U

(1+ ( ))N

3D1 1/(a·Λ^5) (1+ ())N

(^4 + (^5))

U

(^2+ (^3))

U

3D1 re·Λ^3 (+ (^2))

U

(^2+ (^3))

U

(1+ ())N

3D1 v2·Λ (+ (^2))

U

(+ (^2))

U

(1+ ()) N

Results for coupled channel 3S1-3D1 (Δ=4)


Through qualitative analysis, a nonperturbative realization of unnatural scattering lengths, etc., is shown to be possible within EFT approach. In contrast to PC A and B, the simple scenario of PC C yields unnatural scattering lengths only, without ruining the naturalness of the rest ERE parameters. All these result from the nontrivial ‘entanglement’ of the EFT power counting and the nonperturbative prescription structures.


Summary• Within a finite order of EFT expansion, only finite many

nonperturbative divergences appear• EFT expansion of potential is incompatible with the

nonperturbative structures of divergences in LSE• Subtractions should be performed at the level of

integrals, and the resulting nontrivial prescription dependence needs to be removed through physical boundary conditions.

• RG invariance is consequential, which constrains some of the prescription-dependent parameters to be physical or RG invariant

• EFT power counting is ‘entangled’ with prescriptions• Nonperturbative description of origin of unnatural

scattering length is possible within EFT approach


Thank you!

Exploring novel scenarios of NNEFT

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