YangMills groundstate wavefunctional in 2 1 dimensions

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Yang-Mills ground-state wavefunctional in 21
dimensions

• (in collaboration with Jeff Greensite)
• J. Greensite, O, Phys. Rev. D 77 (2008) 065003,
  arXiv0707.2860 hep-lat

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Motivation

• QCD field theory with six flavors of quarks with
  three colors, each represented by a Dirac spinor
  of four components, and with eight four-vector
  gluons, is a quantum theory of amplitudes for
  configurations each of which is 104 numbers at
  each point in space and time. To visualize all
  this qualitatively is too difficult. The thing to
  do is to take some qualitative feature to try to
  explain, and then to simplify the real situation
  as much as possible by replacing it by a model
  which is likely to have the same qualitative
  feature for analogous physical reasons.
• The feature we try to understand is confinement
  of quarks.
• We simplify the model in a number of ways.
• First, we change from three to two colors as the
  number of colors does not seem to be essential.
• Next we suppose there are no quarks. Our problem
  of the confinement of quarks when there are no
  dynamic quarks can be converted, as Wilson has
  argued, to a question of the expectation of a
  loop integral. Or again even with no quarks,
  there is a confinement problem, namely the
  confinement of gluons.
• The next simplification may be more serious. We
  go from the 31 dimensions of the real world to
  21. There is no good reason to think
  understanding what goes on in 21 can immediately
  be carried by analogy to 31, nor even that the
  two cases behave similarly at all. There is a
  serious risk that in working in 21 dimensions
  you are wasting your time, or even that you are
  getting false impressions of how things work in
  31. Nevertheless, the ease of visualization is
  so much greater that I think it worth the risk.
  So, unfortunately, we describe the situation in
  21 dimensions, and we shall have to leave it to
  future work to see what can be carried over to
  31.

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Introduction

• Confinement is the property of the vacuum of
  quantized non-abelian gauge theories. In the
  hamiltonian formulation in Dd1 dimensions and
  temporal gauge

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• At large distance scales one expects
• Halpern (1979), Greensite (1979)
• Greensite, Iwasaki (1989)
• Kawamura, Maeda, Sakamoto (1997)
• Karabali, Kim, Nair (1998)
• Property of dimensional reduction Computation of
  a spacelike loop in d1 dimensions reduces to the
  calculation of a Wilson loop in Yang-Mills theory
  in d Euclidean dimensions.

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• Strong-coupling lattice-gauge theory systematic
  expansion
• Greensite (1980)
• At weak couplings, one would like to similarly
  expand
• For g!0 one has simply
• Wheeler (1962)

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• A possibility to enforce gauge invariance
• No handle on how to choose fs.

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Suggestion for an approximate vacuum
wavefunctional
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Warm-up example Abelian ED
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Free-field limit (g!0)
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Zero-mode, strong-field limit (D21)

• D. Diakonov (private communication to JG)
• Lets assume we keep only the zero-mode of the
  A-field, i.e. fields constant in space, varying
  in time. The lagrangian is
• and the hamiltonian operator
• Natural choice - 1/V expansion

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• Keeping the leading term in V only
• The equation is solved by
• since

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• Now the proposed vacuum state coincides with this
  solution in the strong-field limit, assuming
• The covariant laplacian is then
• Lets choose color axes so that both color
  vectors lie in, say, (12)-plane

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• The eigenvalues of M are obtained from
• Our wavefunctional becomes
• In the strong-field limit

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D31
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Dimensional reduction and confinement

• What about confinement with such a vacuum state?
• Define slow and fast components using a
  mode-number cutoff
• Then

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• Effectively for slow components
• we then get the probability distribution of a 2D
  YM theory and can compute the string tension
  analytically (in lattice units)
• Non-zero value of m implies non-zero string
  tension ? and confinement!
• Lets revert the logic to get ? with the right
  scaling behavior 1/?2, we need to choose

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Why m02 -?0 m2 ?

• Samuel (1997)

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Non-zero m is energetically preferred

• Take m as a variational parameter and minimize ltH
  gt with respect to m
• Assuming the variation of K with A in the
  neighborhood of thermalized configurations is
  small, and neglecting therefore functional
  derivatives of K w.r.t. A one gets

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• Abelian free-field limit minimum at m2 ?0 ? 0.

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• Non-abelian case Minimum at non-zero m2 ( 0.3),
  though a higher value ( 0.5) would be required
  to get the right string tension.
• Could (and should) be improved!

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Calculation of the mass gap

• To extract the mass gap, one would like to
  compute
• in the probability distribution
• Looks hopeless, KA is highly non-local, not
  even known for arbitrary fields.
• But if - after choosing a gauge - KA does not
  vary a lot among thermalized configurations
  then something can be done.
• Numerical simulation

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Numerical simulation of ?02

• Define
• Hypothesis
• Iterative procedure

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• Practical implementation
• choose e.g. axial A10 gauge, change variables
  from A2 to B. Then
• Spiral gauge
• given A2, set A2A2,
• the probability P AKA is gaussian in B,
  diagonalize KA and generate new B-field (set
  of Bs) stochastically
• from B, calculate A2 in axial gauge, and compute
  everything of interest
• go back to the first step, repeat as many times
  as necessary.
• All this is done on a lattice.
• Of interest
• Eigenspectrum of the adjoint covariant laplacian.
• Connected field-strength correlator, to get the
  mass gap
• For comparison the same computed on 2D slices of
  3D lattices generated by Monte Carlo.

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Eigenspectrum of the adjoint covariant laplacian
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Mass gap
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Summary (of apparent pros)

• Our simple approximate form of the confining YM
  vacuum wavefunctional in 21 dimensions has the
  following properties
• It is a solution of the YM Schrödinger equation
  in the weak-coupling limit
• and also in the zero-mode, strong-field limit.
• Dimensional reduction works There is confinement
  (non-zero string tension) if the free mass
  parameter m is larger than 0.
• m gt 0 seems energetically preferred.
• If the free parameter m is adjusted to give the
  correct string tension at the given coupling,
  then the correct value of the mass gap is also
  obtained.

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Open questions (or contras?)

• Can one improve (systematically) our vacuum
  wavefunctional Ansatz?
• Can one make a more reliable variational estimate
  of m?
• Comparison to other proposals?
• Karabali, Kim, Nair (1998)
• Leigh, Minic, Yelnikov (2007)
• What about N-ality?
• Knowing the (approximate) ground state, can one
  construct an (approximate) flux-tube state,
  estimate its energy as a function of separation,
  and get the right value of the string tension?
• How to go to 31 dimensions?
• Much more challenging (Bianchi identity,
  numerical treatment very CPU time consuming).
• The zero-mode, strong-field limit argument valid
  (in certain approximation) also in D31.
• Comparison to KKN
• N-ality
• Flux-tube state

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Elements of the KKN approach

• Matrix parametrisation

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Comparison to KKN

• Wavefunctional expressed in terms of still
  another variable
• Its argued that the part bilinear in field
  variables has the form
• The KKN string tension following from the above
  differs from string tensions obtained by standard
  MC methods, and the disagreement worsens with
  increasing ?.

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N-ality

• Dimensional reduction form at large distances
  implies area law for large Wilson loops, but also
  Casimir scaling of higher-representation Wilson
  loops.
• How does Casimir scaling turn into N-ality
  dependence, how does color screening enter the
  game?
• A possibility Necessity to introduce additional
  term(s), e.g. a gauge-invariant mass term
• Cornwall (2007)
• but color screening may be contained!
• Strong-coupling
• Greensite (1980)

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• Guo, Chen, Li (1994)

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Flux-tube state

• A variational trial state
• The energy of such a state for a given
  quark-antiquark separation can be computed from
• On a lattice
• Work in progress.

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Epilogue/Apologies

• It is normal for the true physicist not to worry
  too much about mathematical rigor. And why?
  Because one will have a test at the end of the
  day which is the confrontation with experiment.
  This does not mean that sloppiness is admissible
  an experimentalist once told me that they check
  their computations ten times more than the
  theoreticians! However its normal not to be too
  formalist. This goes with a certain attitude of
  physicists towards mathematics loosely speaking,
  they treat mathematics as a kind of prostitute.
  They use it in an absolutely free and shameless
  manner, taking any subject or part of a subject,
  without having the attitude of the mathematician
  who will only use something after some real
  understanding.
• Alain Connes in an interview with C. Goldstein
  and G. Skandalis, EMS Newsletter, March 2008.

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Spiral gauge
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Dimensions