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

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’s talks in this program

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

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

To warm up, let us begin with one dimensional attractive δ -potential: it can host a bound state bound state V(x)= - C 0 δ(x) ψ(x) ∝ e -mC 0 |x|/2 Even-parity bound state

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:

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 of k 2 /m 2 (only the leading operator C 0 is 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

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’s talks in this program

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.

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

Now we move to 2+1 Dimension Mass is a passive parameter, redefine Lagrangian to make the coupling C 0 dimensionless This theory is classically scale-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)

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

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.

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

Though our NREFT is applicable to any short-range interaction, it is good to have an explicit microscopic theory at hand We choose λф 4 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

In general, the cutoff of NREFT Λ is much less than the particle mass: m However, for the relativistic quantum field theoryλф 4 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.

Matching λф 4 theory to NREFT in 2+1D through O(k 2 ) Matching the amplitude in both theories up to 1-loop rel. insertion ( ) C 2

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

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

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

Counter-term (MSbar) Note the counter-term to C 2 is needed to absorb the UV divergence that is generated from leading relativistic correction piece.

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(k 2 ), 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.

Digression: It may be instructive to rederive Wilson coefficients using alternative approach Method of region Beneke 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

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 C 0 to all orders Iterating contributions of C 2 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

Bubble sum involving only C 0 vertex Resummed amplitude: O. Bergman PRD (1992) infrared regular Renormalized coupling C 0 (μ): Λ: UV cutoff

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

Dimensional transmutation Define an integration constant, RG-invariant: ρ plays the role of Λ QCD in QCD positive provided that μ small Amplitude now reads:

The scaleρcan only be determined if the microscopic dynamics is understood Take the λф 4 theory as the fundamental theory. If we assume λ= 4πm, one then finds A gigantic “extrinsic” scale in non-relativistic context ! As is understood, the bound state pole corresponding to repulsive C 0 (Λ) is a spurious one, and cannot be endowed with any physical significance.

Bound state pole for C 0 (Λ)<0 Bound state pole κ=ρ Binding energy Again take λф 4 theory as the fundamental theory. If one assumes λ= - 4πm, one then finds An exponentially shallow bound state (In repulsive case, the pole ρ>> Λ unphysical)

Generalization: Including higher derivative C 2n 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

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

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

RG equation for C 2 (direct calculation) Expressing the bare coupling in term of renormalized one: Deduce the exactβfunction for C 2 : Will lead to the same solution as previous slide

Up to now, we have not implemented the relativistic correction yet. What is its impact? We rederive the RG equation for C 2, this time by including effects of relativistic correction. Working out the full counter-terms to C 2, by computing all the bubble diagrams contributing at O(k 2 ). Have C 0, δC 0 or lower-order δC 2 induced by relativistic correction, as vertices, and may need one relativistic vertex insertions in loop.

RG equation for C 2 (direct calculation including relativistic correction) Expressing the bare coupling in term of renormalized ones already known New contribution! Curiously enough, these new pieces of relativity-induced counter-terms can also be cast into geometric series.

We then obtain the relativity-corrected βfunction for C 2 : New piece Put in another way: no longer 0! The solution is: In the μ →0 limit, relativitistic correction dominates RG flow

Incorporating relativity qualitatively change the RG flow of C 2n in the infrared limit Recall without relativistic correction: C 2 ( μ ) approaches 0 as C 0 ( μ ) 2 in the limit μ → 0 In the μ →0 limit, relativitistic correction dominates RG flow C 2 ( μ ) approaches 0 at the same speed as C 0 ( μ ) as μ → 0

Similarly, RG evolution for C 4 are also qualitatively changed when relativistic effect incorporated The relativity-corrected βfunction for C 4 : due to rel. corr. And In the limit μ →0, we find

The exact Lorentz-invariant amplitude may be conjectured Dilation factor Where Check: RGE for C 2n can be confirmed from this expression also by explicit loop computation

Quick way to understand RGE flow for C 2n In the limit k →0, let us choose μ =k, we have approximately A sum = - ∑ C 2n (k) k 2n Physical observable does not depend on μ. If we choose μ = ρ

Quick way to understand RGE flow for C 2n Matching these two expressions, we then reproduce recall RG flow at infrared limit fixed by Lorentz dilation factor

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:

Another application of RG: efficient tool to resum large logarithms in λф 4 theory At O(k 0 ) Tree-level matching → resum leading logarithms (LL) One-loop level matching → resum NLL

Another application of RG: efficient tool to resum large logarithms in λф 4 theory At O(k 2 ), Tree-level matching → resum leading logarithms (LL) One-loop level matching → resum NLL difficult to get these in full theory without calculation

Some remarks on non-relativistic limit of λф 4 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→ ∞

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 λф 4 theory is, it cannot be substantiated in the NR limit

Effective range expansion for λф 4 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

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 ≤ 16π 2 ), we always have a 0 ≤ r 0

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. It is shown that counter-intuitively, relativistic correction qualitatively change the renormalization flow of various 4-boson operators in the zero-momentum limit. We have derived and exact Lorentz-invariant S-wave scattering amplitude. We are able to make some nonperturbative statement in a nontrivial fashion.

Thanks!