Exact solution to planar δ -potential using EFT

1. Exact solution to planar δ-potential using EFT • Yu Jia • Inst. High Energy Phys., Beijing ( based onhep-th/0401171 ) Effective field theories for particle and nuclear physics, Aug. 3-Sept. 11, KITPC

2. Outline • Two-dimensional contact interaction is an interesting problem in condensed matter physics (scale invariance and anomaly) • Conventional method: solving Schrödinger equation using regularized delta-potential • Modern (and more powerful) method: using nonrelativistic effective field theory (EFT) describing short-range interaction • Analogous to (pionless) nuclear EFT for few nucleon system in 3+1 dimension • J.-F. Yang, U. van Kolck, J.-W. Chen’stalks in this program

3. Outline (cont’) • Obtain exact Lorentz-invariant S-wave scattering amplitude (relativistic effect fully incorporated) • RGE analysis to bound state pole • Show how relativistic corrections will qualitatively change the RG flow in the small momentum limit

4. Outline (cont’) • For concreteness, I also show pick up a microscopic theory: λф4 theory as example • Illustrating the procedure of perturbative matching • very much like QCDHQET, NRQCD. • Able to say something nontrivial about the nonrelativistic limit of this theory in various dimensions • ``triviality”, and effective range in 3+1 dimension

5. bound state V(x)= - C0δ(x) ψ(x) ∝e -mC0|x|/2 To warm up, let us begin with one dimensional attractiveδ-potential: it can host a bound state Even-parity bound state

6. Recalling textbook solution to one-dimensional δ-potential problem • Schrödinger equation can be arranged into • Define • Integrating over an infinitesimal amount of x: •  discontinuity in ψ’(x) • Trial wave function: • Binding energy:

7. Reformulation of problem in terms of NREFT • NR Effective Lagrangian describing short-range force: • Contact interactions encoded in the 4-boson operators • Lagrangian organized by powers ofk2/m2 • (only the leading operator C0is shown in above) • This NR EFT is only valid for k ＜＜ Λ∽ m (UV cutoff ) • Lagrangian constrained by the Symmetry: • particle # conservation, Galilean invariance, time reversal and parity

8. Pionful (pionless) NNEFT – modern approach to study nuclear force • Employing field-theoretical machinery to tackle physics of few-nucleon system in 3+1 D • S. Weinberg (1990, 1991) • C. Ordonez and U. van Kolck (1992) • U. van Kolck (1997,1999) • D. Kaplan, M. Savage and M. Wise (1998) • J.-F. Yang, U. van Kolck, J.-W. Chen’stalks in this program

9. Two-particle scattering amplitude • Infrared catastrophe at fixed order (diverges as k→ 0) • Fixed-order calculation does not make sense. One must resum the infinite number of bubble diagrams. • This is indeed feasible for contact interactions.

10. Bubble diagram sum forms a geometric series – closed form can be reached • The resummed amplitude now reads • Amplitude → 4ik/m as k→ 0, sensible answer achieved • Bound-state pole can be easily inferred by letting • pole of scattering amplitude • Binding energy: • Find the location of pole is: • Agrees with what is obtained from Schrödinger equation

11. Now we move to 2+1 Dimension • Mass is a passive parameter, redefine Lagrangian to make the coupling C0 dimensionless • This theory is classicallyscale-invariant • But acquire the scale anomaly at quantum level • O. Bergman PRD (1992) • Coupled to Chern-Simons field, fractional statistics: N-anyon system • R. Jackiw and S. Y.Pi, PRD (1990)

12. δ-potential in 2+1 D confronts UV divergence • Unlike 1+1D, loop diagrams in general induce UV divergence, therefore renders regularization and renormalization necessary. • In 2+1D, we have • Logarithmic UV divergence

13. Including higher-derivative operators and relativistic correction in 2+1D NREFT • Breaks scale invariance explicitly • Also recover Lorentz invariance in kinetic term • This leads to rewrite the ``relativistic” propagator as • treat as perturb.

14. Another way to incorporate the relativistic correction in NREFT • Upon a field redefinition, Luke and Savage (1997) • one may get more familiar form for relativistic correction: • More familiar, but infinite number of vertices. Practically, this is much more cumbersome than the ``relativistic” one

15. Though our NREFT is applicable to any short-range interaction, it is good to have an explicit microscopic theory at hand • We choose λф4 theory to be the ``fundamental theory” • In 2+1 D, the coupling λ has mass dimension 1, this theory is super-renormalizable • In below we attempt to illustrate the procedure of perturbative matching

16. In general, the cutoff of NREFT Λ is much less than the particle mass: m • However, for the relativistic quantum field theoryλф4 theory, the cutoff scale Λ can be extended about Λ≤m. • The matching scale should also be chosen around the scalar mass, to avoid large logarithm.

17. Matching λф4 theory to NREFT in 2+1D through O(k2) • Matching the amplitude in both theories up to 1-loop • rel. insertion ( ) C2

18. Full theory calculation • The amplitude in the full theory • It is UV finite • Contains terms that diverge in k→ 0 limit • Contains terms non-analytic in k

19. NREFT calculation • One can write down the amplitude as • In 2+1D, we have

20. NREFT calculation (cont’) • Finally we obtain the amplitude in EFT sector • It is logarithmically UV divergent (using MSbar scheme) • Also contains terms that diverge in k→ 0 limit • Also contains terms non-analytic in k, as in full theory

21. Counter-term (MSbar) • Note the counter-term to C2 is needed to absorb the UV divergence that is generated from leading relativistic correction piece.

22. Wilson coefficients • Matching both sides, we obtain • Nonanalytic terms absent/ infrared finite • -- guaranteed by the built-in feature of EFT matching • To get sensible Wilson coefficients at O(k2), consistently including relativistic correction ( ) is crucial. • Gomes, Malbouisson, da Silva (1996) missed this point, and invented two ad hoc 4-boson operators to mimic relativistic effects.

23. Digression: It may be instructive to rederive Wilson coefficients using alternative approach • Method of regionBeneke and Smirnov (1998) • For the problem at hand, loop integral can be partitioned into “hard” and “potential” region. • Calculating short-distance coefficients amounts to extracting the hard-region contribution

24. Now see how far one can proceed starting from 2+1D NREFT • Consider a generic short-distance interactions in 2+1D • Our goal: • Resumming contribution of C0 to all orders • Iterating contributions of C2 and higher-order vertices • Including relativistic corrections exactly • Thus we will obtain an exact 2-body scattering amplitude • We then can say something interesting and nontrivial

25. Bubble sum involving only C0 vertex • Resummed amplitude: O. Bergman PRD (1992) • infrared regular • Renormalized coupling C0(μ): • Λ: UV cutoff

26. Renormalization group equation for C0 • Expressing the bare coupling in term of renormalized one: • absence of sub-leading • poles at any loop order • Deduce the exactβfunction for C0 : • positive; C0 = 0 IR fixed point

27. Dimensional transmutation • Define an integration constant, RG-invariant: • ρplays the role of ΛQCD in QCD • positive provided that μ small • Amplitude now reads:

28. The scaleρcan only be determined if the microscopic dynamics is understood • Take the λф4 theoryas the fundamental theory. If we assumeλ= 4πm, one then finds • A gigantic “extrinsic” scale in non-relativistic context ! • As is understood, the bound state pole corresponding to repulsiveC0(Λ) is a spurious one, and cannot be endowed with any physical significance.

29. Bound state pole for C0(Λ)<0 • Bound state pole • κ=ρ • Binding energy • Again take λф4 theoryas the fundamental theory. If one assumes λ= - 4πm, one then finds • An exponentially shallow bound state • (In repulsive case, the pole ρ>> Λunphysical)

30. Generalization: Including higher derivative C2n terms in bubble sum • Needs evaluate following integrals • The following relation holds in any dimension: • factor of q inside loop converted to external momentum k

31. Improved expression for the resummed amplitude in 2+1 D • The improved bubble chain sum reads • This is very analogous to the respective generalized formula in 3+1 D, as given by KSW (1998) or suggested by the well-known effective range expansion • We have verified this pattern holds by explicit calculation

32. RG equation for C2(a shortcut) • First expand the terms in the resummed amplitude • Recall 1/C0 combine with ln(μ) to form RG invariant, • so the remaining terms must be RG invariant. • C2(k) diverges as C0(k)2 in the limit k→ 0

33. RG equation for C2(direct calculation) • Expressing the bare coupling in term of renormalized one: • Deduce the exactβfunction for C2 : • Will lead to the same solution as previous slide

34. Up to now, we have not implemented the relativistic correction yet. What is its impact? • We rederive the RG equation for C2, this time by including effects of relativistic correction. • Working out the full counter-terms to C2, by computing all the bubble diagrams contributing at O(k2). • Have C0, δC0 or lower-order δC2 induced by relativistic correction, as vertices, and may need one relativistic vertex insertions in loop.

35. RG equation for C2(direct calculation including relativistic correction) • Expressing the bare coupling in term of renormalized ones • already knownNew contribution! • Curiously enough, these new pieces of relativity-induced counter-terms can also be cast into geometric series.

36. We then obtain the relativity-corrected βfunction for C2 : • New piece • Put in another way: no longer 0! • The solution is: • In the μ→0 limit, relativitistic correction dominates RG flow

37. Incorporating relativity qualitatively change the RG flow of C2n in the infrared limit • Recall without relativistic correction: • C2(μ) approaches 0 as C0(μ)2 in the limit μ → 0 • In the μ→0 limit, relativitistic correction dominates RG flow • C2(μ) approaches 0 at the same speed as C0(μ) asμ → 0

38. Similarly, RG evolution for C4are also qualitatively changed when relativistic effect incorporated • The relativity-corrected βfunction for C4 : • due to rel. corr. • And • In the limit μ→0, we find

39. The exact Lorentz-invariant amplitude may be conjectured • Dilation factor • Where • Check: RGE for C2n can be confirmed from this expression • also by explicit loop computation

40. Quick way to understand RGE flow for C2n • In the limit k→0, let us chooseμ=k,we have approximately • Asum = - ∑C2n (k) k2n • Physical observable does not depend on μ. If we choose μ=ρ

41. Quick way to understand RGE flow forC2n • Matching these two expressions, we then reproduce • recall • RG flow at infrared limit fixed by Lorentz dilation factor

42. Corrected bound-state pole • When relativistic correction included, the pole shifts from ρ by an amount of • RG invariant • The corresponding binding energy then becomes:

43. Another application of RG: efficient tool to resum large logarithms in λф4 theory • At O(k0) • Tree-level matching → resum leading logarithms (LL) • One-loop level matching → resum NLL

44. Another application of RG: efficient tool to resum large logarithms in λф4 theory • At O(k2), • Tree-level matching → resum leading logarithms (LL) • One-loop level matching → resum NLL • difficult to get these in full theory without calculation

45. Some remarks on non-relativistic limit of λф4 theory in 3+1 Dimension • M.A.Beg and R.C. Furlong PRD (1985) claimed the triviality of this theory can be proved by looking at nonrelativistic limit • There argument goes as follows • No matter what bare coupling is chosen, the renormalized coupling vanishes as Λ→ ∞

46. Beg and Furlong’s assertion is diametrically against the philosophy of EFT • According to them, so the two-body scattering amplitude of this theory in NR limit also vanishes • Since → 0 • This cannot be incorrect, since Λin EFT can never be sent to infinity. EFT has always a finite validity range. • Conclusion: whatsoever the cause for the triviality of λф4 theory is, it cannot be substantiated in the NR limit

47. Effective range expansion for λф4 theory in 3+1 Dimension • Analogous to 2+1 D, taking into account relativistic correction, we get a resummed S-wave amplitude: • Comparing with the effective range expansion: • We can deduce the scattering length and effective range

48. Looking into deeply this simple theory Through the one-loop order matching [Using on-shell renormalization for full theory, MSbar for EFT], we get The effective range approximately equals Compton length, consistent with uncertainty principle. For the coupling in perturbative range (λ≤ 16π2), we always have a0 ≤ r0

49. Summary We have explored the application of the nonrelativistic EFT to 2D δ-potential. Techniques of renormalization are heavily employed, which will be difficult to achieve from Schrödinger equation. We have derived and exact Lorentz-invariant S-wave scattering amplitude. We are able to make some nonperturbative statement in a nontrivial fashion. It is shown that counter-intuitively, relativistic correction qualitatively change the renormalization flow of various 4-boson operators in the zero-momentum limit.

50. Thanks!