Post on 10-Aug-2019
Effective theory of 3H and 3He
Sebastian Konigin collaboration with H.W. Grießhammer, H.-W. Hammer, and U. van Kolck
INT Program 16-1, University of Washington
Seattle, WA
April 15, 2016
SK et al., J. Phys. G 43 055106 (2016), 1508.05085 [nucl-th]
Effective theory of 3H and 3He – p. 1
Removing pions
Effective theory of 3H and 3He – p. 2
Removing pions
q2
≪ m2
π −→
Effective theory of 3H and 3He – p. 2
Removing pions
q2
≪ m2
π −→
pions are not resolved at sufficiently low energies!
Effective theory of 3H and 3He – p. 2
Motivation
scaling of Coulomb contributions
nonperturbative → perturbative
three-body forces
H(Λ) =
nd + pd︷ ︸︸ ︷
H0,0(Λ) +H0,1 (Λ) +
pd only︷ ︸︸ ︷
H(α)0,1 (Λ)
Vanasse, Egolf, Kerin, SK, Springer, PRC 89 064003 (2014)
Effective theory of 3H and 3He – p. 3
Motivation
Why pionless EFT?
conceptually clean and (reasonably) simple
allows for a fully perturbative treatment of higher orders
cutoff can be made arbitrarily large
still clearly connected to QCD!
Why talk about it here?
applied to analyze and extrapolate lattice resultsBarnea et al. PRL 114 052501 (2015)
Kirscher et al. PRC 92 054002 (2015)
lattice QCD used to extract extract pionless LECs: np → dγBeane et al. (NPLQCD) PRL 115 132001 (2015)
study general EFT questions (renormalization with Coulomb)
Effective theory of 3H and 3He – p. 4
Outline
Introduction
Coulomb effects in 3He
Divergences
A new expansion
Summary
Effective theory of 3H and 3He – p. 5
Two-body sector
Introduce dibaryon fields. . .
3S1 −→
Kaplan (1996/97), Bedaque, van Kolck (1997/98)
Effective theory of 3H and 3He – p. 6
Two-body sector
Introduce dibaryon fields. . .
3S1 −→
Kaplan (1996/97), Bedaque, van Kolck (1997/98)
. . . and resum bubble-insertions to all orders!
Full dibaryon propagators3S1 : ∆d = = + + + · · ·
1S0 : ∆t = = + + + · · ·
∆d(k) ∼ i
k cot δd︸ ︷︷ ︸
=−γd+ρd2
(k2+γ2d
)+···
− ik =k
Effective theory of 3H and 3He – p. 6
Two-body sector
Introduce dibaryon fields. . .
3S1 −→
Kaplan (1996/97), Bedaque, van Kolck (1997/98)
. . . and resum bubble-insertions to all orders!
Full dibaryon propagators3S1 : ∆d = = + + + · · ·
1S0 : ∆t = = + + + · · ·
∆d(k) ∼ i
k cot δd︸ ︷︷ ︸
=−γd+ρd2
(k2+γ2d
)+···
− ik γdρd ∼ Q/Λ/π = O(1/3)
Effective theory of 3H and 3He – p. 6
Propagator renormalization
Full dibaryon propagators3S1 : ∆d = = + + + · · ·
1S0 : ∆t = = + + + · · ·
i∆(0)d (p0,p) =
−i
σ(0)d + y2
dI0(p0,p)
I0(p0,p) = MN
∫ Λ
d3q1
MNp0 − p2/4 − q2 + iε
= −MN
4π
(
2Λ
π+
√
p2
4−MNp0 − iε
︸ ︷︷ ︸
→ −ik
)
+ O(1/Λ)
absorb linear divergence: σ(0)d =
2Λ
π− γd
1/Λ effects are neglected (equivalent: PDS, 2Λ/π → µR)
without dibaryons: resummation of ∼ C0 = −4π/(MNσ)
Effective theory of 3H and 3He – p. 7
Three-body sector
Nucleon
spin 1/2
isospin 1/2
Deuteron
spin 1
isospin 0
→ two S-wave channels:
1 ⊗ 1
2=
3
2
(
∼)
⊕
spin doublet →3H,3He
︷ ︸︸ ︷
1
2
(
∼ + · · ·)
Effective theory of 3H and 3He – p. 8
Three-body sector
Nucleon
spin 1/2
isospin 1/2
Deuteron
spin 1
isospin 0
→ two S-wave channels:
1 ⊗ 1
2=
3
2
(
∼)
⊕
spin doublet →3H,3He
︷ ︸︸ ︷
1
2
(
∼ + · · ·)
—quartet channel—
Effective theory of 3H and 3He – p. 8
Three-body sector
Nucleon
spin 1/2
isospin 1/2
Deuteron
spin 1
isospin 0
→ two S-wave channels:
1 ⊗ 1
2=
3
2
(
∼)
⊕
spin doublet →3H,3He
︷ ︸︸ ︷
1
2
(
∼ + · · ·)
—quartet channel—
→ solve integral equations to get phase shifts and binding energies
Effective theory of 3H and 3He – p. 8
Three-body sector
Nucleon
spin 1/2
isospin 1/2
Deuteron
spin 1
isospin 0
→ two S-wave channels:
1 ⊗ 1
2=
3
2
(
∼)
⊕
spin doublet →3H,3He
︷ ︸︸ ︷
1
2
(
∼ + · · ·)
—quartet channel— —doublet channel—
= + +
= + +
→ solve integral equations to get phase shifts and binding energies
Effective theory of 3H and 3He – p. 8
The triton
= + +
= + +
3S1
1S0
Λ
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Λ = 40000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Λ = 40000 MeV
Λ = 50000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
Bedaque et al., NPA 676 357 (2000)
amplitude has no well-defined limit!10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Λ = 40000 MeV
Λ = 50000 MeV
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
Bedaque et al., NPA 676 357 (2000)
amplitude has no well-defined limit!10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Λ = 40000 MeV
Λ = 50000 MeV
Three-body force promotion
already at LO: → + , ∼
103
104
105
-10
-5
0
5
10
Λ (MeV)
H(Λ
)
independent of spin and isospin→ SU(4)-symmetry
coupling runs in RG limit cycle
makes amplitude cutoff-independent
Effective theory of 3H and 3He – p. 9
The triton
= + +
= + +
3S1
1S0
Λ
Bedaque et al., NPA 676 357 (2000)
amplitude has no well-defined limit!10 100 1000 10000
-0.1
-0.05
0
0.05
0.1
p (MeV)
p·T(k,p)(a.u.)
Λ = 10000 MeV
Λ = 20000 MeV
Λ = 30000 MeV
Λ = 40000 MeV
Λ = 50000 MeV
Three-body force promotion
already at LO: → + , ∼
103
104
105
-10
-5
0
5
10
Λ (MeV)
H(Λ
)
independent of spin and isospin→ SU(4)-symmetry
coupling runs in RG limit cycle
makes amplitude cutoff-independent
Effective theory of 3H and 3He – p. 9
Universality
Braaten, Hammer
Phys. Rept. 428 259 (2006)
largescattering
length
Effective theory of 3H and 3He – p. 10
Universality
Braaten, Hammer
Phys. Rept. 428 259 (2006)
halo EFT
nucleus by Cam-Ann, Wikimedia Commons
van Kolck, NPA 645, 273 (1999)
Bertulani et al., NPA 712, 37 (2002)...
Zhang et al., 1507.07239 [nucl-th]
largescattering
length
Effective theory of 3H and 3He – p. 10
Universality
Braaten, Hammer
Phys. Rept. 428 259 (2006)
halo EFT
nucleus by Cam-Ann, Wikimedia Commons
van Kolck, NPA 645, 273 (1999)
Bertulani et al., NPA 712, 37 (2002)...
Zhang et al., 1507.07239 [nucl-th]
largescattering
length
Effective theory of 3H and 3He – p. 10
Universality
Braaten, Hammer
Phys. Rept. 428 259 (2006)
halo EFT
nucleus by Cam-Ann, Wikimedia Commons
van Kolck, NPA 645, 273 (1999)
Bertulani et al., NPA 712, 37 (2002)...
Zhang et al., 1507.07239 [nucl-th]
largescattering
length
cold atoms
tunable scattering length!
Efimov effect!
Effective theory of 3H and 3He – p. 10
Coulomb contributions
Coulomb photons: ∼ (ie)i
q2(ie) −→ (ie)
i
q2+λ2(ie)
O(α) diagrams
k p
Effective theory of 3H and 3He – p. 11
Coulomb contributions
Coulomb photons: ∼ (ie)i
q2(ie) −→ (ie)
i
q2+λ2(ie)
O(α) diagrams
k p
only generated by dibaryon kinetic term!
→ higher-order correction ∼ ρd
Effective theory of 3H and 3He – p. 11
Coulomb subtraction
Coulomb photons: ∼ (ie)i
q2(ie) −→ (ie)
i
q2+λ2(ie)
long (infinite) range → very strong at small momentum transfer
Coulomb nonperturbative for η ∼ 1
important for p-d scattering length!→ SK, Hammer, PRC 90 034005 (2014)
C2η k cot δdiff(k) + αµh(η) = − 1
ap-d+ · · ·
(modified ERE)
Effective theory of 3H and 3He – p. 12
Coulomb subtraction
Coulomb photons: ∼ (ie)i
q2(ie) −→ (ie)
i
q2+λ2(ie)
long (infinite) range → very strong at small momentum transfer
Coulomb nonperturbative for η ∼ 1
important for p-d scattering length!→ SK, Hammer, PRC 90 034005 (2014)
C2η k cot δdiff(k) + αµh(η) = − 1
ap-d+ · · ·
(modified ERE)
2 2.5 3 3.5 4-3000
-2500
-2000
-1500
-1000
-500
0
k (MeV)
k·cotδ d
iff(k)(fm
−1)
×
2 2.5 3 3.5 40
0.001
0.002
0.003
0.004
0.005
0.006
0.007
k (MeV)
C2 η(k)
C2η =
2πη
e2πη − 1
= finite value
Effective theory of 3H and 3He – p. 12
The helion and the counterterm
Effective theory of 3H and 3He – p. 13
He-3 binding energy
bound-state ↔ pole! ∼
E + EB
+ regular terms
→ calculate 3He binding energy!
= + + + ×
(
+ +)
+ ×
(
+)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+ +)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+)
cf. Kok et al., Groningen Reports (1979/81)
Ando+Birse (2010)
Effective theory of 3H and 3He – p. 14
Coulomb effects in the proton–proton channel
Coulomb-dressed propagator
= + + + · · ·
= + + + · · ·∆t,pp(p) ∼ 1
−1/ap–p − αMNH(η), η = αMN/
(2i√
p2/4 −MNp0 − iε)
Kong, Ravndal (1999)
→ Coulomb-modified effective range expansion Bethe (1949)
cf. Ando, Birse (2010)
Effective theory of 3H and 3He – p. 15
Coulomb effects in the proton–proton channel
Coulomb-dressed propagator
= + + + · · ·
= + + + · · ·∆t,pp(p) ∼ 1
−1/ap–p − αMNH(η), η = αMN/
(2i√
p2/4 −MNp0 − iε)
Kong, Ravndal (1999)
→ Coulomb-modified effective range expansion Bethe (1949)
cf. Ando, Birse (2010)
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
− αMNH(η)
Note: two divergences absorbed into a single parameter!
Effective theory of 3H and 3He – p. 15
He-3 binding energy
bound-state ↔ pole! ∼
E + EB
+ regular terms
→ calculate 3He binding energy!
= + + + ×
(
+ +)
+ ×
(
+)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+ +)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+)
cf. Kok et al., Groningen Reports (1979/81)
Ando+Birse (2010)
Effective theory of 3H and 3He – p. 16
He-3 beyond leading order
100 1000 10000
7.2
7.6
8
8.4
8.8
9.2
9.6
10
Λ (MeV)
EB(M
eV)
LO
exp. EB(3He)
exp. EB(3H)
Effective theory of 3H and 3He – p. 17
He-3 beyond leading order
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)
NLO corrections
effective ranges
∼ ρd, ∼ r0t
dibaryon-photon coupling
∼ αρd
Effective theory of 3H and 3He – p. 17
He-3 beyond leading order
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)
NLO corrections
effective ranges
∼ ρd, ∼ r0t
dibaryon-photon coupling
∼ αρd
NLO result is not cutoff stable → incomplete renormalization!
refitting the three-body force to EB(3He) gives stable p-d phase shifts!SK, Ph.D. thesis (2013)
SK et al., JPG 42 045101 (2015)
form of new p-d specific counterterm can be derived analytically!
three body-force H(Λ) = H0,0(Λ) +H0,1(Λ) +H(α)0,1 (Λ)
Vanasse, Egolf, Kerin, SK, Springer, PRC 89 064003 (2014)
Effective theory of 3H and 3He – p. 17
Counterterm controversy
A recent paper does not find a new counterterm at NLO!Kirscher+Gazit, PLB 755 (2016) 253, 1510.00118 [nucl-th]
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∗
Effective theory of 3H and 3He – p. 18
Counterterm controversy
A recent paper does not find a new counterterm at NLO!Kirscher+Gazit, PLB 755 (2016) 253, 1510.00118 [nucl-th]
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
Effective theory of 3H and 3He – p. 18
Counterterm controversy
A recent paper does not find a new counterterm at NLO!Kirscher+Gazit, PLB 755 (2016) 253, 1510.00118 [nucl-th]
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?Effective theory of 3H and 3He – p. 18
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?
Effective theory of 3H and 3He – p. 19
Nonperturbative vs. perturbative and helium
= + + + ×
(
+ +)
+ ×
(
+)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+ +)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+)
iterate O(α) diagrams. . .
get 3He pole directly
Effective theory of 3H and 3He – p. 20
Nonperturbative vs. perturbative and helium
= + + + ×
(
+ +)
+ ×
(
+)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+ +)
+ ×
(
+)
= + + ×
(
+)
+ ×
(
+)
iterate O(α) diagrams. . .
get 3He pole directly
Effective theory of 3H and 3He – p. 20
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〉
Effective theory of 3H and 3He – p. 20
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
Effective theory of 3H and 3He – p. 21
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
Effective theory of 3H and 3He – p. 21
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
→ as before: determine counterterm from p–p scattering!
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
−αMNH(η) cf. Kong, Ravndal (1999)
Effective theory of 3H and 3He – p. 21
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
→ as before: determine counterterm from p–p scattering!
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
−αMNH(η) cf. Kong, Ravndal (1999)
Important to isolate divergence for consistent renormalization!
Dressed bubble integral
J0(k)= GC(k2/MN ; 0,0)
= +
= + + · · ·
GC = G(+)0 + G
(+)0 VC GC = G
(+)0 + G
(+)0 TC G
(+)0
TC = VC + VC G(+)0 TC
Effective theory of 3H and 3He – p. 21
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
→ as before: determine counterterm from p–p scattering!
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
−αMNH(η) cf. Kong, Ravndal (1999)
Important to isolate divergence for consistent renormalization!
Dressed bubble integral
J0(k)= GC(k2/MN ; 0,0)
= +
= + + · · ·
GC = G(+)0 + G
(+)0 VC GC = G
(+)0 + G
(+)0 TC G
(+)0
TC = VC + VC G(+)0 TC
Effective theory of 3H and 3He – p. 21
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
→ as before: determine counterterm from p–p scattering!
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
−αMNH(η) cf. Kong, Ravndal (1999)
Important to isolate divergence for consistent renormalization!
Dressed bubble integral
J0(k)= GC(k2/MN ; 0,0)
= +
= + + · · ·
GC = G(+)0 + G
(+)0 VC GC = G
(+)0 + G
(+)0 TC G
(+)0
TC = VC + VC G(+)0 TC
Effective theory of 3H and 3He – p. 21
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
Effective theory of 3H and 3He – p. 22
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
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)
Effective theory of 3H and 3He – p. 22
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
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
Effective theory of 3H and 3He – p. 22
Phillips line
-0.4 0 0.4 0.8 1.2 1.6 26
7
8
9
10
11
_
0.6 0.65 0.7
8.0
8.1
8.2
EB(3H)(M
eV)
LO aphys.t = −23.7 fm
LO at = ∞
exp. EB(3H)
exp. 2an–d
2an–d (fm)
Effective theory of 3H and 3He – p. 23
Phillips line
-0.4 0 0.4 0.8 1.2 1.6 2
6
7
8
9
10
11
0.6 0.65 0.7
8.0
8.1
8.2
EB(3H)(M
eV)
LO aphys.t = −23.7 fm
LO at = ∞
LO at = ∞ + NLO aphys.t
exp. EB(3H)
exp. 2an–d
2an–d (fm)
Effective theory of 3H and 3He – p. 23
Phillips line
-0.4 0 0.4 0.8 1.2 1.6 2
6
7
8
9
10
11
0.6 0.65 0.7
8.0
8.1
8.2
EB(3H)(M
eV)
LO aphys.t = −23.7 fm
LO aphys.t + NLO rt
LO at = ∞
LO at = ∞ + NLO aphys.t
LO at = ∞ + NLO aphys.t , rt
exp. EB(3H)
exp. 2an–d
2an–d (fm)
Effective theory of 3H and 3He – p. 23
Doublet-channel phase shift
0 10 20 30 40 50
-30
-25
-20
-15
-10
-5
0
40 42 44 46 48 50-22
-20
-18
-16
k (MeV)
δ(deg)
LO aphys.t = −23.7 fm
LO aphys.t + NLO rt
LO at = ∞
LO at = ∞ + NLO aphys.t
LO at = ∞ + NLO aphys.t , rt
Effective theory of 3H and 3He – p. 24
New perturbative scheme
Leading order
standard NN spin-triplet (pionless) amplitude (parameter γd)
unitary NN spin-singlet amplitude (parameter-free)
contact three-body force (parameter Λ∗ → EB(3H) or 2an−d)
Effective theory of 3H and 3He – p. 25
New perturbative scheme
Leading order
standard NN spin-triplet (pionless) amplitude (parameter γd)
unitary NN spin-singlet amplitude (parameter-free)
contact three-body force (parameter Λ∗ → EB(3H) or 2an−d)
Next-to-leading order
effective range in the NN spin-triplet channel (parameter ρd)
isospin-symmetric range in the NN spin-singlet channel(parameter rt)
scattering-length correction to unitarity in the NN spin-singletnp and nn channels (parameter at),
scattering-length correction to unitarity in the NN pp channel(parameter ap–p)
one-photon exchange (parameter α = 1/137)
Effective theory of 3H and 3He – p. 25
Two-body sector
Consider two regimes in the p–p sector:
1 Q ∼ αMN → Coulomb non-perturbative! = + + · · ·
2 αMN . 1/ap–p ≪ Q ≪ Λ/π → Coulomb perturbative!
determine counterterm in non-pert. regime (match to modified ERE)
important to use consistent regularization scheme!
Effective theory of 3H and 3He – p. 26
Two-body sector
Consider two regimes in the p–p sector:
1 Q ∼ αMN → Coulomb non-perturbative! = + + · · ·
2 αMN . 1/ap–p ≪ Q ≪ Λ/π → Coulomb perturbative!
determine counterterm in non-pert. regime (match to modified ERE)
important to use consistent regularization scheme!
Coulomb regularization
single photon︷ ︸︸ ︷
two and more︷ ︸︸ ︷
δI0(k) ∼[
log2Λ
αMN− Cζ + log iη
]
, δJ0(k) ∼ −[
ψ(iη) +1
2iη︸ ︷︷ ︸
H(η)
+C∆
]
+MN
4πik
Effective theory of 3H and 3He – p. 26
Coulomb bubble divergence
an additional diagram is logarithmically divergent. . .
. . . but this divergence comes from the photon-bubblesubdiagram!
→ as before: determine counterterm from p–p scattering!
∆t,pp(p0,p) =−i
σt,pp − 2Λ
π+ αMN
(
log2Λ
αMN− CE
)
︸ ︷︷ ︸
=1/aC
−αMNH(η) cf. Kong, Ravndal (1999)
Important to isolate divergence for consistent renormalization!
Dressed bubble integral
J0(k)= GC(k2/MN ; 0,0)
= +
= + + · · ·
δI0(k)
δJ0(k)
GC = G(+)0 + G
(+)0 VC GC = G
(+)0 + G
(+)0 TC G
(+)0
TC = VC + VC G(+)0 TC
Effective theory of 3H and 3He – p. 27
Two-body sector
Consider two regimes in the p–p sector:
1 Q ∼ αMN → Coulomb non-perturbative! = + + · · ·
2 αMN . 1/ap–p ≪ Q ≪ Λ/π → Coulomb perturbative!
determine counterterm in non-pert. regime (match to modified ERE)
important to use consistent regularization scheme!
Coulomb regularization
single photon︷ ︸︸ ︷
two and more︷ ︸︸ ︷
δI0(k) ∼[
log2Λ
αMN− Cζ + log iη
]
, δJ0(k) ∼ −[
ψ(iη) +1
2iη︸ ︷︷ ︸
H(η)
+C∆
]
+MN
4πik
Effective theory of 3H and 3He – p. 28
Two-body sector
Consider two regimes in the p–p sector:
1 Q ∼ αMN → Coulomb non-perturbative! = + + · · ·
2 αMN . 1/ap–p ≪ Q ≪ Λ/π → Coulomb perturbative!
determine counterterm in non-pert. regime (match to modified ERE)
important to use consistent regularization scheme!
Coulomb regularization
single photon︷ ︸︸ ︷
two and more︷ ︸︸ ︷
δI0(k) ∼[
log2Λ
αMN− Cζ + log iη
]
, δJ0(k) ∼ −[
ψ(iη) +1
2iη︸ ︷︷ ︸
H(η)
+C∆
]
+MN
4πik
perturbative prediction for p–p phase shift:
k cot δt,pp(k) = − 1
ap–p+ αMNC∆ +
rt
2k2 + αMN log
(αMN
2k
)
+ · · ·
Effective theory of 3H and 3He – p. 28
Three-body sector: perturbative 3He
New expansion ensures renormalized NLO result for 3He!
1000 10000
6.4
6.8
7.2
7.6
8
8.4
Λ (MeV)
EB(M
eV)
pert. (incomplete)
nonpert. calc.
exp. EB(3He)
exp. EB(3H)
+ +
✦
Effective theory of 3H and 3He – p. 29
Three-body sector: perturbative 3He
New expansion ensures renormalized NLO result for 3He!
1000 10000
6.4
6.8
7.2
7.6
8
8.4
Λ (MeV)
EB(M
eV)
pert. (incomplete)
pert. (full, unit. LO)
nonpert. calc.
exp. EB(3He)
exp. EB(3H)
+ +
+ +
∆E = (−0.86 ± 0.17) MeV
✦
Effective theory of 3H and 3He – p. 29
Three-body sector: perturbative 3He
New expansion ensures renormalized NLO result for 3He!
1000 10000
6.4
6.8
7.2
7.6
8
8.4
Λ (MeV)
EB(M
eV)
pert. (incomplete)
pert. (full, unit. LO)
nonpert. calc.
exp. EB(3He)
exp. EB(3H)
+ +
+ +
∆E = (−0.86 ± 0.17) MeV
Coulomb indeed perturbative in 3He ✦
consistent renormalization crucial to achieve this!
Effective theory of 3H and 3He – p. 29
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!
but an otherwise remaining log-divergence is absent!
Effective theory of 3H and 3He – p. 30
Lessons learned and open questions
1 Coulomb perturbative in 3He ✦
2 consistent renormalization crucial to achieve this!
3 1S0 NN channel can be expanded around unitarity limit
4 unnecessary iteration can cause spurious divergences
5 one-photon exchange generates logarithmic divergences:
∼ log Λ , just like !
Effective theory of 3H and 3He – p. 31
Lessons learned and open questions
1 Coulomb perturbative in 3He ✦
2 consistent renormalization crucial to achieve this!
3 1S0 NN channel can be expanded around unitarity limit
4 unnecessary iteration can cause spurious divergences
5 one-photon exchange generates logarithmic divergences:
∼ log Λ , just like !
Questions
reason for discrepancy with Kirscher and Gazit?(more generally: regulator dependence?)
how much Coulomb has to be iterated in low-energy p-dscattering? (yet another counting above d-breakup threshold?)
Effective theory of 3H and 3He – p. 31