regulator dependence and isospin breaking - indico.gsi.de fileregulator dependence and isospin...

Regulator dependence and isospin breaking

Sebastian Konig

EMMI RRTF Workshop:

“The systematic treatment of the Coulomb interaction in few-body systems”

GSI, Darmstadt

May 30, 2016

Regulator dependence and isospin breaking – p. 1

Motivation: counterterm controversy

Should there be a new counterterm at NLO!

Yes! Vanasse et al., PRC 89 (2014) 064003 No! Kirscher+Gazit, PLB 755 (2016) 253

100 1000 10000

7.2

7.6

8

8.4

8.8

9.2

9.6

10

Λ (MeV)

EB(M

eV)

LO

NLO part. resum.

NLO fully. pert.

exp. EB(3He)

exp. EB(3H)

7.55

7.6

7.65

7.7

7.75

800 Λ rit.

∼ 1230 1600 1800 2000 2200 2400

B(3

H

e

)

[

M

e

V

℄

Λ [MeV℄

exp

LO

NLO

LO, DΛ∗

NLO, DΛ∗





100 1000 10000

7.2

7.6

8

8.4

8.8

9.2

9.6

10

Λ (MeV)

EB(M

eV)

LO

NLO part. resum.

NLO fully. pert.

exp. EB(3He)

exp. EB(3H)

7.55

7.6

7.65

7.7

7.75

800 Λ rit.

∼ 1230 1600 1800 2000 2200 2400

B(3

H

e

)

[

M

e

V

℄

Λ [MeV℄

exp

LO

NLO

LO, DΛ∗

NLO, DΛ∗

Vanasse et al.

momentum-space formalism

sharp cutoff (non-local)

can take Λ arbitrarily large

Kirscher and Gazit

configuration-space (R)RGM

Gaussian regulators (local)

limited cutoff range





100 1000 10000

7.2

7.6

8

8.4

8.8

9.2

9.6

10

Λ (MeV)

EB(M

eV)

LO

NLO part. resum.

NLO fully. pert.

exp. EB(3He)

exp. EB(3H)

7.55

7.6

7.65

7.7

7.75

800 Λ rit.

∼ 1230 1600 1800 2000 2200 2400

B(3

H

e

)

[

M

e

V

℄

Λ [MeV℄

exp

LO

NLO

LO, DΛ∗

NLO, DΛ∗

Vanasse et al.

momentum-space formalism

sharp cutoff (non-local)

can take Λ arbitrarily large

Kirscher and Gazit

configuration-space (R)RGM

Gaussian regulators (local)

limited cutoff range

power counting ↔ regulators?Regulator dependence and isospin breaking – p. 2

Coulomb corrections for nuclei

How much Coulomb should we really iterate?

Coulomb regimes

trinucleon binding momentum ∼ 80 MeV . . .

→ should Coulomb not be a small perturbative correction?


Nonperturbative vs. perturbative and helium

= + + + ×

(

+ +)

+ ×

(

+)

+ ×

(

+)

= + + ×

(

+)

+ ×

(

+ +)

+ ×

(

+)

= + + ×

(

+)

+ ×

(

+)

iterate O(α) diagrams. . .

get 3He pole directly


Nonperturbative vs. perturbative and helium

= + + + ×

(

+ +)

+ ×

(

+)

+ ×

(

+)

= + + ×

(

+)

+ ×

(

+ +)

+ ×

(

+)

= + + ×

(

+)

+ ×

(

+)

iterate O(α) diagrams. . .

get 3He pole directly

= + +

= + +

SK, Grießhammer, Hammer, J. Phys. G 42 045101 (2015)

use trinucleon wavefunctions

fully perturbative in α!

∆E : −

∆E = 〈ψ|VC |ψ〉


Coulomb bubble divergence

an additional diagram is logarithmically divergent. . .

. . . but this divergence comes from the photon-bubblesubdiagram!


Coulomb bubble divergence

an additional diagram is logarithmically divergent. . .

. . . but this divergence comes from the photon-bubblesubdiagram!

StrategySK, Grießhammer, Hammer, van Kolck, JPG 43 055106 (2016), 1508.05085 [nucl-th]

1 isolate divergence:

2 take the leading-order 1S0 in the unitarity limit!

a1S0= −23.7 ≈ ∞ 1/a1S0

≈ 0

3 include divergent diagram together with finite a1S0

+ = finite


A new expansion

Take the leading-order singlet channel in the unitarity limit!

at = −23.4 ≈ ∞ !

σt = σ(0)t + σ

(1)t

σ(0)t − 2Λ/π = 0

i∆(0)t (p0,p)

−i

σt + y2t I0(p0,p)

→ −i√

p2

4 −MNp0 − iε


A new expansion


at = −23.4 ≈ ∞ !

σt = σ(0)t + σ

(1)t

σ(0)t − 2Λ/π = 0

NLO corrections

scattering length: σ(1)t ∼ −1/at

effective range: c(1)t =

MNr0t

2

∼ −iσ(1)t(,pp) ∼ −ic

(1)t

i∆(1)t

(p0, p) = i∆(0)t

(p0, p) ×

[

−iσ(1)t

− ic(1)t

(

p0 −

p2

4MN

) ]

× i∆(0)t

(p0, p)


A new expansion


at = −23.4 ≈ ∞ !

σt = σ(0)t + σ

(1)t

σ(0)t − 2Λ/π = 0

NLO corrections

scattering length: σ(1)t ∼ −1/at

effective range: c(1)t =

MNr0t

2

∼ −iσ(1)t(,pp) ∼ −ic

(1)t

i∆(1)t

(p0, p) = i∆(0)t

(p0, p) ×

[

−iσ(1)t

− ic(1)t

(

p0 −

p2

4MN

) ]

× i∆(0)t

(p0, p)

new 1S0 LO is isospin-symmetric and parameter-free

allows matching between perturbative and non-perturbative Coulomb regimes


Divergence dissection

Vanasse, Egolf, Kerin, SK, Springer, PRC 89 064003 (2014)

Look at structure of p-d three-body force:

H(α)0,1 (Λ) = h

(α)I (Λ) + h(α)

κ (Λ)

h(α)I (Λ) = −3π(1 + s2

0)

16

{

1

12(rp–p − rt)Λ

[

1 − · · ·]

+ various terms ∼ log Λ, all ∝ (rp–p − rt) or ∝ (γp–p − γt)

}

/ sin2 (· · · )

h(α)κ = −

√3κπ(1 + s2

0)

48

{

various terms ∼ log Λ

that do not vanish in the isospin limit + Ψ(Λ)}

/ sin2 (· · · )

,

pieces associated with rp–p 6= rt and αρd, αrt

these have been relagated to a higher order in the new scheme!SK, Grießhammer, Hammer, van Kolck, JPG 43 055106 (2016), 1508.05085 [nucl-th]

but an otherwise remaining log-divergence is absent!Regulator dependence and isospin breaking – p. 7

Regulator dependence: overview

power counting ↔ regulators?


Regulator dependence: overview

power counting ↔ regulators?

similar issues in chiral EFT!

PNM

Tews et al. 1507.05561 [nucl-th]

non-local, MS

local, CSpionless EFT can study this

question in a clean scenario

without other complications!

chiral EFT with Weinberg counting does not absorb all cutoffdependence in counterterms

pionless EFT formulated with RG invariance

different schemes can still give different runnings for LECs


Pionless two- and three-body regularization

Momentum-space overview

typically used in pionless EFT with dibaryons:dim. reg. (PDS) in two-body sector, cutoff in three-body equation

∆(p) ∼ 1

−1/a︸︷︷︸=σ−µ

+√

p2

4 −MNp0 − iε, T (E; k, p) = K(E; k, p) +

∫ Λ

0

dq2K(E; k, q)D(E; q)T (E; q, p)

as long as everything converges, this should not matterBedaque et al., NPA 676 357 (2000)

to check this: use cutoff regulator for two-body system,keep finite-Λ corrections

Vanasse, Coulomb RRTF Part I (Jan. 2016)

Faddeev calculations use Gaussian regulator functions for two-and three-body sector

Platter et al., PRA 70 052101 (2004), PLB 607 254 (2005), . . .


Three-body force with two-body cutoff regulator

Λ* = 36.8 MeVstandardH

0,0(Λ

)

−7.5

−5

−2.5

0

2.5

5

7.5

Λ (MeV)100 1000 10

410

510

6


Three-body force with two-body cutoff regulator

Λ* = 36.8 MeVΛ* = 26.5 MeV

standardfinite-Λ 2-body

H0,0(Λ

)

−7.5

−5

−2.5

0

2.5

5

7.5

Λ (MeV)100 1000 10

410

510

6

∆(p0,p) ∼ 1

−γ +√

p2

4 −MNp0 − iε− 2π

√p2

4 −MNp0 − iε arctan

(√p2

4 −MNp0 − iε/Λ

)


Three-body force with Gaussian regulators

plot from H. Deleon, Jan. 2016


nd doublet phase shift with two-body cutoff regulator

n-d doublet S-wave

Λ = 140 MeVΛ = 180 MeVΛ = 220 MeVΛ = 300 MeVΛ = 400 MeVΛ = 1600 MeV

Re

δ(k)

(deg

)

−80

−60

−40

−20

0

k (MeV)0 20 40 60 80 100


nd doublet phase shift with two-body cutoff regulator

n-d doublet S-wave

Λ = 140 MeVΛ = 180 MeVΛ = 220 MeVΛ = 300 MeVΛ = 400 MeVΛ = 1600 MeV

Re

δ(k)

(deg

)

−80

−60

−40

−20

0

k (MeV)0 20 40 60 80 100

∆(p0,p) ∼ 1

−γ +√

p2

4 −MNp0 − iε− 2π

√p2

4 −MNp0 − iε arctan

(√p2

4 −MNp0 − iε/Λ

)


Divergence dissection

Vanasse, Egolf, Kerin, SK, Springer, PRC 89 064003 (2014)

Look at structure of p-d three-body force:

H(α)0,1 (Λ) = h

(α)I (Λ) + h(α)

κ (Λ)

h(α)I (Λ) = −3π(1 + s2

0)

16

{

1

12(rp–p − rt)Λ

[

1 − · · ·]

+ various terms ∼ log Λ, all ∝ (rp–p − rt) or ∝ (γp–p − γt)

}

/ sin2 (· · · )

h(α)κ = −

√3κπ(1 + s2

0)

48

{

various terms ∼ log Λ

that do not vanish in the isospin limit + Ψ(Λ)}

/ sin2 (· · · )

,

pieces associated with rp–p 6= rt and αρd, αrt

these have been relagated to a higher order in the new scheme!SK, Grießhammer, Hammer, van Kolck, JPG 43 055106 (2016), 1508.05085 [nucl-th]

but an otherwise remaining log-divergence is absent!Regulator dependence and isospin breaking – p. 13

NLO divergence contributions

exp. 3He

exp. 3H

rp-p ≠ rt

EB (

MeV

)

7

8

9

10

11

12

Λ (MeV)1000 10

4



exp. 3He

exp. 3H

rp-p ≠ rt

rp-p = rt

EB (

MeV

)

7

8

9

10

11

12

Λ (MeV)1000 10

4



exp. 3He

exp. 3H

rp-p ≠ rt

rp-p = rt

rp-p = rt, no dibaryon-photon coupling

EB (

MeV

)

7

8

9

10

11

12

Λ (MeV)1000 10

4



exp. 3He

exp. 3H

rp-p ≠ rt

rp-p = rt

rp-p = rt, no dibaryon-photon coupling

EB (

MeV

)

7

8

9

10

11

12

Λ (MeV)1000 10

410

5


NLO divergence from isospin breaking (no Coulomb)

fit three-nucleon force with isospin-symmetric 1S0 dibaryon

use this in calculation with separate n-n channel

exp. 3H

rn-n = rt

rn-n ≠ rt

EB (

MeV

)

8

9

10

11

12

Λ (MeV)1000 10

4


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