1 / 50

Exact solution to planar δ -potential using EFT

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

adora
Download Presentation

Exact solution to planar δ -potential using EFT

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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!

More Related