Post on 03-Jan-2016
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
OutlineTwo-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: λф4 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)= - C0 δ(x) ψ(x) ∝ e -mC0|x|/2
Even-parity bound state
Recalling textbook solution to one-dimensional δ-potential
problem
Schrödinger equation can be arranged into
DefineIntegrating 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 operatorsLagrangian organized by powers of k2/m2 (only the leading operator C0 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 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 Schrödinger 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 level
O. Bergman PRD (1992)
Coupled to Chern-Simons field, fractional statistics: N-anyon systemR. 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 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
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.
Matching λф4 theory to NREFT in 2+1D through O(k2)
Matching the amplitude in both theories up to 1-loop
rel. insertion ( ) C2
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 (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 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 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 vertex
Resummed amplitude: O. Bergman PRD (1992)
infrared regular
Renormalized coupling C0(μ):
Λ: UV cutoff
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
Dimensional transmutation
Define an integration constant, RG-invariant:
ρplays the role of ΛQCD in QCD
positive provided that μ small
Amplitude now reads:
The scaleρcan only be determined if the microscopic dynamics is understood
Take the λф4 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 C0(Λ) is a spurious one, and cannot be endowed with any physical significance.
Bound state pole for C0(Λ)<0
Bound state pole κ=ρBinding energy
Again take λф4 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 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
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 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
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
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.
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 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
Incorporating relativity qualitatively change the RG flow of C2n in the infrared limitRecall 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
Similarly, RG evolution for C4 are 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
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 k→0, let us choose μ=k, we have approximately Asum = - ∑ C2n (k) k2n
Physical observable does not depend on μ. If we choose μ=ρ
Quick way to understand RGE flow for C2n
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 λф4 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 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
Some remarks on non-relativistic limit of λф4 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 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 λф4 theory is, it cannot be substantiated in the NR limit
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
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
SummaryWe 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.