Coulomb Interactions in EFT Gautam Rupak Mississippi State - - PowerPoint PPT Presentation
Coulomb Interactions in EFT Gautam Rupak Mississippi State University EMMI: The Systematic Treatment of the Coulomb Interaction in Few-Body Systems Darmstadt, Jan 11-15, 2016 Motivations Astrophysics Low energy reactions dominate
Gautam Rupak Mississippi State University
EMMI: The Systematic Treatment of the Coulomb Interaction in Few-Body Systems Darmstadt, Jan 11-15, 2016
measure experimentally
calculations important
Use Effective Field Theory
The first item is of paramount importance because you want to make error estimates even when resumming potentially sub-leading pieces for practical reasons. I concentrate on the power-counting.
Hide UV ignorance
IR explicit
Not just Ward-Takahashi identity
L = c0 O(0) +c1 O(1) +c2 O(2) + · · · cn expansion in Q
Λ
+ + −C0 + · · ·
Weinberg ’90 Bedaque, van Kolck ’97 Kaplan, Savage, Wise ’98
EFT non-perturbative
iA(p) = 2π µ i p cot δ0 − ip = 2π µ i −1/a + r
2p2 + · · · − ip
a >> 1/Λ iA(p) ≈ −2π µ i 1/a + ip 1 + 1 2 rp2 1/a + ip + · · ·
−i
1 C0 + i µ 2πp ⇒ C0 = 2πa
µ
large coupling
=
+ + + ⋅ ⋅ ⋅
+ + + ⋅ ⋅ ⋅
In EFT there is no way to relate coupling from n-p and p-p channel.
2π µC0 = 1 a − λ d − 3 − 2αµ[ 1 d − 4 − ln µπ αM − 1 + 3 2CE]
Kong, Ravndal 1999
C0 → g2
∆
Doesn’t work so nicely for p-d. No enhancement of Coulomb ladder: physics missing!
(Ed, ⃗ p) (En, ⃗ p) (Ed,⃗ k) (En,⃗ k) ≃
e2 (⃗ p−⃗ k)2 ∼ e2 p2
(Ed, ⃗ p) (Ed, ⃗ p) (Ed,⃗ k) (En,⃗ k) ⃗ q ⃗ p − ⃗ q −⃗ p + ⃗ q ≃ e4 d3q
1 ET−(q−p)2/(2µ) 1 q2(q−p+k)2
∼ e2
p2 αµ p
nth − loop :∼ e2 p2 ✓αµ p ◆n
It is clear that we will have the right physics if the in the EFT the dimer propagator has the right physical
D(p0, ~ p) = 1 ∆ → 2⇡ µg2 1 − + p p2/4 − µp0
D(Ed + q0, ~ p) ∼ 2 (q2 − p2)/4 − MNq0 ∼
Put nucleon on pole, IR enhancement. Coulomb ladder recovered
= + + + ...
x x put nucleons on-shell
But ... what happens if we also include nucleon-exchange?
In p-d scattering we either have have a deuteron- nucleon or 3-nucleon state. The latter is off-shell by deuteron breakup and for these diagrams the loop momentum need not scale with but with
γ ∼ Q
p
= 4πα p2 Rupak, Kong 2001
This loop is not IR enhanced but larger by 1/Q
Zd×
nucleon lines always scale as M/Q^2
After we put a single nucleon in every loop on-shell, either no no nucleon line or two nucleon lines exists.
(1) Loop : Z d3q ∼ Q3 or p3 (2) Nucleon propagator : MN/Q2 (3) Dressed dimer : Q/q2 (4) photon : 1/q2
Usual one when p~Q
(a) (b) (c) (d) (e) (f) (g) (h)
Rupak, Kong NPA 717, 73 (2003)
On blackboard
(a) (b) (c) = + + + (d) + +
On blackboard
40 60 80 100
δ
(0)
k (MeV)
p-d n-d see König’s and Vanasse’s talk
see van Kolck’s talk
cluster coordinates, spins,etc.
possibilities --- traditional methods, continuum EFT, lattice method
a(b, γ)c
Rupak & Lee, PRL 2013, arXiv:1302.4158 Rupak & Ravi, PLB 2014, arXiv:1411.2436 Pine, Lee, Rupak, EPJA 2013, arXiv:1309.2616 Elhatisari et al, Nature 528, 111 (2015), arXiv:1506.03513
Rupak, Ravi PLB 2014 Kong, Ravndal 1999 ΛEF T (0) ≈2.51 Lattice fit : Λ(0) ≈2.49 ± 0.02
200 400 600 800 1000 1200 0.01 0.02 0.03 0.04 0.05 E (keV) Tfi (MeV−1) Analytic b=1/100 MeV−1 b=1/200 MeV−1
3% fitting error propagates Gamow peak 6 keV
Λ(p) = s γ2 8πC2
η
|Tfi(p)|