# Exact solution to planar δ -potential using EFT Yu Jia Inst. High Energy Phys., Beijing ( based on...

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Exact solution to planar -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

OutlineTwo-dimensional contact interaction is an interesting problem in condensed matter physics (scale invariance and anomaly) Conventional method: solving Schrdinger 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. Chens 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: 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 -potential: it can host a bound state

bound state

V(x)= - C0 (x)

(x) e -mC0|x|/2 Even-parity bound state

Recalling textbook solution to one-dimensional -potential problemSchrdinger equation can be arranged into

DefineIntegrating over an infinitesimal amount of x:

discontinuity in (x)

Trial wave function:

Binding energy:

Reformulation of problem in terms of NREFTNR Effective Lagrangian describing short-range force:

Contact interactions encoded in the 4-boson operatorsLagrangian organized by powers of k2/m2 (only the leading operator C0 is shown in above)

This NR EFT is only valid for k m (UV cutoff )

Lagrangian constrained by the Symmetry: particle # conservation, Galilean invariance, time reversal and parity

Pionful (pionless) NNEFT modern approach to study nuclear forceEmploying 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. Chens 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 reachedThe 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 Schrdinger equation

Now we move to 2+1 Dimension

Mass is a passive parameter, redefine Lagrangian to make the coupling C0 dimensionless

This theory is classically scale-invariantBut acquire the scale anomaly at quantum levelO. Bergman PRD (1992)

Coupled to Chern-Simons field, fractional statistics: N-anyon systemR. Jackiw and S. Y.Pi, PRD (1990)

-potential in 2+1 D confronts UV divergenceUnlike 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 NREFTBreaks 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 NREFTUpon 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 handWe choose 4 theory to be the ``fundamental theory

In 2+1 D, the coupling 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 is much less than the particle mass: m However, for the relativistic quantum field theory4 theory, the cutoff scale can be extended about m.

The matching scale should also be chosen around the scalar mass, to avoid large logarithm.

Matching 4 theory to NREFT in 2+1D through O(k2)Matching the amplitude in both theories up to 1-loop

rel. insertion ( ) C2

Full theory calculationThe amplitude in the full theory

It is UV finite Contains terms that diverge in k 0 limit Contains terms non-analytic in k

NREFT calculationOne 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 C2 is needed to absorb the UV divergence that is generated from leading relativistic correction piece.

Wilson coefficientsMatching 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.

Digression: It may be instructive to rederive Wilson coefficients using alternative approachMethod 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 NREFTConsider 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 amplitudeWe then can say something interesting and nontrivial

Bubble sum involving only C0 vertexResummed amplitude: O. Bergman PRD (1992)

infrared regular

Renormalized coupling C0():

: UV cutoff

Renormalization group equation for C0Expressing the bare coupling in term of renormalized one:

absence of sub-leading poles at any loop order

Deduce the exactfunction for C0 :

positive; C0 = 0 IR fixed point

Dimensional transmutationDefine an integration constant, RG-invariant:

plays the role of QCD in QCD

positive provided that small

Amplitude now reads:

The scalecan only be determined if the microscopic dynamics is understoodTake the 4 theory as the fundamental theory. If we assume = 4m, one then finds

A gigantic extrinsic scale in non-relativistic context !

As is understood, the bound state pole corresponding to repulsive C0() is a spurious one, and cannot be endowed with any physical significance.

Bound state pole for C0()> unphysical)

Generalization: Including higher derivative C2n terms in bubble sumNeeds 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 DThe 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 C2 (a shortcut)First expand the terms in the resummed amplitude

Recall 1/C0 combine with ln() to form RG invariant,so the remaining terms must be RG invariant.

C2(k) diverges as C0(k)2 in the limit k 0

RG equation for C2 (direct calculation)Expressing the bare coupling in term of renormalized one:

Deduce the exactfunction for C2 :

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 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, C0 or lower-order C2 induced by relativistic correction, as vertices, and may need one relativistic vertex insertions in loop.

RG equation for C2 (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 function for C2 :

New piece

Put in another way: no longer 0!

The solution is:

In the 0 limit, relativitistic correction dominates RG flow

Incorporating relativity qualitatively change the RG flow of C2n in the infrared limitRecall without relativistic correction:

C2() approaches 0 as C0()2 in the limit 0

In the 0 limit, relativitistic correction dominates RG flow

C2() approaches 0 at the same speed as C0() as 0

Similarly, RG evolution for C4 are also qualitatively changed when relativistic effect incorporatedThe relativity-corrected function for C4 :

due to rel. corr.

And

In the limit 0, we find

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

Quick way to understand RGE flow for C2n In the limit k0, let us choose =k, we have approximately Asum = - C2n (k) k2n

Physical observable does not depend on . If we choose =

Quick way to understand RGE flow for C2nMatching these two expressions, we then reproduce

recall

RG flow at infrared limit fixed by Lorentz dilation factor

Corrected bound-state poleWhen relativistic correction included, the pole shifts from by an amount of

RG invariant

The corresponding binding energy then becomes:

Another application of RG: efficient tool to resum large logarithms in 4 theory At O(k0)Tree-level matching resum leading logarithms (LL)

One-loop level matching resum NLL

Another application of RG: efficient tool to resum large logarithms in 4 theoryAt O(k2),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 4 theory in 3+1 DimensionM.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

Beg and Furlongs assertion is diametrically against the philosophy of EFTAccording to them, so the two-body scattering amplitude of this theory in NR limit also vanishes

Since 0

This cannot be incorrect, since in EFT can never be sent to infinity. EFT has always a finite validity range.

Conclusion: whatsoever the cause for the triviality of 4 theory is, it cannot be substantiated in the NR limit

Effective range expansion for 4 theory in 3+1 DimensionAnalogous 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 theoryThrough 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 ( 162), we always have a0 r0

SummaryWe have explored the application of the nonrelativistic EFT to 2D -potential. Techniques of renormalization are heavily employed, which will be difficult to achieve from Schrdinger 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!

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