Exact solution to planar dpotential using EFT

1
Exact solution to planar d-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
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 31 dimension
• J.-F. Yang, U. van Kolck, J.-W. Chens
  talks 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 ??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 31
  dimension

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To warm up, let us begin with one dimensional
attractive d-potential it can host a bound state
Even-parity bound state

• ?(x) ? e -mC0x/2

• bound state
• V(x) - C0 d(x)

6
Recalling textbook solution to one-dimensional
d-potential problem

• Schrödinger equation can be arranged into
• Define
• Integrating over an infinitesimal amount of x
• ?
  discontinuity in ?(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 of k2/m2
• (only the leading operator C0 is shown in
  above)
• This NR EFT is only valid for k ltlt ?? 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 31 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

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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.

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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

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Now we move to 21 Dimension

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

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d-potential in 21 D confronts UV divergence

• Unlike 11D, loop diagrams in general induce UV
  divergence, therefore renders regularization and
  renormalization necessary.
• In 21D, we have
• Logarithmic UV divergence

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Including higher-derivative operators and
relativistic correction in 21D NREFT

• Breaks scale invariance explicitly
• Also recover Lorentz invariance in kinetic term
• This leads to rewrite the relativistic
  propagator as
• 
  treat as perturb.

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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

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

• We choose ??4 theory to be the fundamental
  theory
• In 21 D, the coupling ? has mass dimension 1,
  this theory is super-renormalizable
• In below we attempt to illustrate the procedure
  of perturbative matching

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In general, the cutoff of NREFT ? is much less
than the particle mass m

• However, for the relativistic quantum field
  theory??4 theory, the cutoff scale ? can be
  extended about ?m.
• The matching scale should also be chosen around
  the scalar mass, to avoid large logarithm.

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Matching ??4 theory to NREFT in 21D through O(k2)

• Matching the amplitude in both theories up to
  1-loop
• rel. insertion (
  ) C2

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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

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NREFT calculation

• One can write down the amplitude as
• In 21D, we have

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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

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Counter-term (MSbar)

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

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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.

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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

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Now see how far one can proceed starting from
21D NREFT

• Consider a generic short-distance interactions in
  21D
• 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

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Bubble sum involving only C0 vertex

• Resummed amplitude O. Bergman PRD (1992)
• 
  infrared regular
• Renormalized coupling C0(µ)
• ?
  UV cutoff

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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 exactßfunction for C0
• 
  positive C0 0 IR fixed point

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Dimensional transmutation

• Define an integration constant, RG-invariant
• ?plays the role of ?QCD in QCD
• positive provided
  that µ small
• Amplitude now reads

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The scale?can only be determined if the
microscopic dynamics is understood

• Take the ??4 theory as the fundamental theory. If
  we assume ? 4pm, 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.

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Bound state pole for C0(?)lt0

• Bound state pole
• ??
• Binding energy
• Again take ??4 theory as the fundamental theory.
  If one assumes ? - 4pm, one then finds
• An exponentially shallow bound state
• (In repulsive case, the pole ?gtgt ? unphysical)

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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

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Improved expression for the resummed amplitude in
21 D

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

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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

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RG equation for C2 (direct calculation)

• Expressing the bare coupling in term of
  renormalized one
• Deduce the exactßfunction for C2
• Will lead to the same solution as previous slide

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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, dC0 or lower-order dC2 induced by
  relativistic correction, as vertices, and may
  need one relativistic vertex insertions in loop.

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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.

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We then obtain the relativity-corrected ßfunction
for C2

• 
  New piece
• Put in another way
  no longer 0!
• The solution is
• In the µ?0 limit, relativitistic correction
  dominates RG flow

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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

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Similarly, RG evolution for C4 are also
qualitatively changed when relativistic effect
incorporated

• The relativity-corrected ßfunction for C4
• 
  due to rel. corr.
• And
• In the limit µ?0, we find

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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

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Quick way to understand RGE flow for C2n

• In the limit k?0, let us choose µk, we have
  approximately
• Asum - ? C2n (k) k2n
• Physical observable does not depend on µ. If we
  choose µ?

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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

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Corrected bound-state pole

• When relativistic correction included, the pole
  shifts from ? by an amount of
• 
  RG invariant
• The corresponding binding energy then becomes

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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

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Another application of RG efficient tool to
resum large logarithms in ??4 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
  

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Some remarks on non-relativistic limit of ??4
theory in 31 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 ?? 8

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Beg and Furlongs 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 ?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

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Effective range expansion for ??4 theory in 31
Dimension

• Analogous to 21 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

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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 (? 16p2),
  we always have a0 r0

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Summary
We have explored the application of the
nonrelativistic EFT to 2D d-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.
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Thanks!