Some General Results in Noncovariant Gauges

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Some General Results in Non-covariant Gauges

• S.D. Joglekar
• IIT Kanpur
• Talk given at THEP-I, held at IIT Roorkee from
  16/3/0520/3/05

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Plan of Talk

• Preliminary
• Brief Statement of The Problem and approaches
  towards solving it
• Importance of The Boundary Term and the FFBRS
  solution
• Properties of the naïve path-integral IRGT
  WT-identities Danger inherent in imposing a
  prescription by hand
• A General Approach to the study of non-covariant
  gauges Exact BRST WT-identities and its new
  features
• A simple illustration of how these unusual
  results work
• Interpretation of Results
• Additional restrictions placed by IRGT on
  Renormalization
• A study of the additional restrictions
• References
• 1. S. D. Joglekar, Euro. Phys. Journal-direct
  C12 3, 1-18 (2001). hep-th/0106264
• 2. S. D. Joglekar, Mod. Phys. Letts. A 17,
  2581-2596 (2002). hep-th/0205045
• S. D. Joglekar, Mod. Phys. Letts. A 18, 843
  (2003). hep-th/0209073
• Background
• 1. Bassetto et al
• 2. Leibbrandt

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Preliminary

• Non-covariant gauges Axial, temporal,
  light-cone, Coulomb, planar etc have been widely
  used in standard model calculations as well as in
  formal arguments in gauge theories and string
  theories.
• For example,
• axial and light-cone gauges are used widely in
  QCD calculations because of its formal freedom
  from ghosts.
• Axial gauge is also useful in the treatment of
  the Chern-Simon theories.
• Light-cone gauge useful in N4 supersymmetric
  Y-M theory.
• Coulomb gauge in the discussion of confinement
• Light-cone gauge Cancellation of anomalies in
  superstring theories

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Problems with non-covariant gauges

• In axial gauges, it arises as the problem of
  spurious singularities in the propagator
• In Coulomb gauges, it arises as the problem of
  ill-defined energy-integrals as the propagator
  for the time-like component is
• is poorly damped as k0 ? 8 compared to the
  Lorentz gauges.
• Problem is intimately related with the residual
  gauge transformation.

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Various Approaches a sketch

• 1. Prescriptions
• Based on the general hope that a simple solution
  should work
• Simplicity in momentum space leading to an ease
  of calculation.
• Examples
• CPV 1/h.k ? ½ 1/(h.kie) 1/(h.k-ie)
• LM 1/h.k ? h .k/(h.k h .kie) h20 h
  (h0, - h)
• Possibility of Wick rotation
• Finally, agreement/ disagreement with the Lorentz
  gauge results only known after a calculation.
• 2. Attempts at derivations via canonical
  quantization
• Derivations not unambiguous
• Exist arguments for variety of conflicting
  prescriptions
• Gauge independence of results not obvious.
• 3. Attempts via interpolating gauges
• Construct a gauge that has a variable parameter a
  which connects Lorentz and a non-covariant gauge
• Gauge-independence proves to be trickier than
  expected SDJ EPJ C01

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A Non-trivial Problem
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Importance of the boundary term

• A prescription such as CPV or LM amounts to
  giving the boundary condition for the unphysical
  degrees of freedom. E.g.
• L-M requires causal BC for all degrees of
  freedom
• CPV (h0 ? 0 ) amounts to requiring ½(CA) for
  unphysical d.f.
• A natural question How do you know that the
  chosen BC will produce a result compatible with
  the Lorentz gauges?
• Prompted us (SDJ, Misra, Bandhu) to devise an
  independent path-integral formalism that takes
  into account the boundary term carefully.
• The approach uses Lorentz gauge path-integral
  together with the BC term
• and performs a field-transformation to
  construct an axial-gauge path-integral together
  with a transformed BC term that tells one how
  the axial gauge poles should be treated.

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FFBRS transformation approach

• One then constructs a field transformation in the
  gauge-ghost sector (a field-dependent BRS-type)
  that converts the path-integral from the Lorentz
  gauge to axial gauge. SDJ,Mandal, Bandhu
• In this process, the e-term
  transforms to another term
• which decides how the unphysical poles of the
  axial (or any other gauge) are to be treated.
• Procedure is very general and has worked for
  axial and Coulomb gauges

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Study of the naïve path-integral

• The comparison of this approach with other
  attempts in the literature evoked many questions
  that lead to this general approach and work on
  interpolating gauges to be discussed below. We
  wish to look at the general problem of
  non-covariant gauges and problems associated with
  them in a general setting suggested by our SDJ,
  Misra earlier work.
• We consider first consider the general
  path-integral of the type

• and study its properties.

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Some General Remarks

• We recall that the path-integral (without any
  further modifications) is often used in the
  formal manipulations and for WT identities. The
  latter are important while discussing
  renormalization.
• We should recall that the path-integral has a
  residual gauge invariance, (without the e-term)
  and this results in the problem associated with
  the unphysical poles.
• It is generally not recognized, how badly behaved
  is the path-integral, without the e-term.
• We then introduce a general form of corrective
  measure We shall study some properties of the
  path-integral with this general corrective
  measure.
• We shall demonstrate several unusual features of
  both the path-integrals.

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IRGT Infinitesimal Residual Gauge Transformations

• We generalize residual gauge transformation to
  BRS space so as to leave the entire effective
  action invariant

• Here, s is an infinitesimal parameter. This
  leaves the gauge-fixing term invariant.
• Sgh is invariant under the above, combined with
  local vector transformations on ghosts

And analogous transformations on matter fields.
These are NOT special cases of the BRS
transformations
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IRGT (CONTD)

• We also require that the above transformations do
  not alter the boundary conditions on the
  path-integrals
• These transformations lead to relations between
  Greens functions. These relations will be called
  the IRGT WT-identities.

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EXAMPLES (contd)
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Consequences of IRGT

• Theorem I The Minkowski space Lagrangian
  path-integral of (i) for "type-R gauges" leads to
  physically unacceptable results from the IRGT
  WT-identities

• The propagator for the scalar field vanishes
  D(x,ty,t)0 for x? y, t? t Differentiate
  w.r.t. K (x,t) and K(y,t)
• Evidently this conclusion incompatible with the
  corresponding one for the Lorentz gauges.

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Consequences of IRGT (contd)

• Dmn(x,ty,t)0 for x? y, t? t
  Differentiate w.r.t. Jm (x,t) and Jn (y,t)
• Differentiate w.r.t. J0 (y,t) ? d/dt d(t-t) 0
• The path-integral manipulations along the lines
  of derivation of WT-identities lead to absurd
  results
• It becomes apparent that the role of any
  corrective measure is very important and not
  peripheral.
• We shall later see how some of these results get
  corrected.

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We next study properties of a general
path-integral in which a general corrective
measure is introduced. We consider
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Earlier instances of risky situations (i)
involving e (ii) Dangerous consequences

• Non-covariant gauge literature has several
  specific instances where it has been observed
  that it may be risky to ignore terms with e in
  the numerator
• Works of Cheng and Tsai 1986-7
• Works of Andrasi and Taylor 1988-92
• Landshoff and P. van Niewenhuizen
• In addition, a specific instance of a dangerous
  consequence from path-integral having residual
  gauge invariance has appeared in a work by
  Baulieu and Zwanziger in 1999.
• The present derivation gives a coherent approach
  and a rational for such possibilities. In
  addition, it brings out a number of further
  results and the full IRGT identity which leads to
  such possibilities.

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A general formulation

• where, OA,c,c- is some operator, which
  necessarily breaks the residual gauge-invariance,
  and study its properties.
• The operator is supposed to
  take care of the unphysical poles in some way,
  not necessarily the correct one.
• Various prescriptions are special cases of this
  form.
• To see this, consider the propagator without the
  e-term and propagator with a prescription. The
  latter necessarily contains some small parameter,
  which we define in terms of e. Thus the inverses
  of these will differ by a term e.
• Even when this is not possible, we shall see that
  analogous conclusions should be expected.

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IRGT v/s BRST

• The IRGT are not a special case of BRS
  transformations.
• A natural question Are these IRGT identities
  something extra? Something spurious?
• Answer No, the correct versions of these are
  contained in the correct BRST WT-identities.
• Theorem The exact IRGT WT-identities are
  derivable from the exact BRST identities.
• Then, can we forget the IRGT identities, being a
  part of BRST?
• We have already seen the crucial fact about IRGT,
  that their mathematical validity depends
  crucially on the presence of the e-term.
• Consequence BRST WT-identities can receive
  contributions from the e-term as e ?0.

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An illustration
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Interpretation of Results

• A non-covariant gauge is defined by two things
• An BRST invariant action
• A prescription to deal with the singularities,
  or an e-term
• The e-dependent WT identity contains
  consequences of both
• BRST invariance of Seff
• The specific eO-term
• The e-term must contribute (as e ?0) to avoid
  absurd relations exhibited earlier.
  Renormalization must deal with both of these.
• The additional IRGT identities must also be dealt
  with under renormalization

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Additional consequences due to IRGT

• These relations crucially depend on what we take
  for the operator O.
• We shall illustrate this with a specific form of
  O and for the Coulomb gauge
• The result is
• Theorem III The local quadratic e-term
  implies additional constraints on Green's
  functions that are derivable from IRGT for the
  type-R gauges.
• Now we shall illustrate the proof for the Coulomb
  gauge. The e-term then leads to a term in the
  IRGT WT-identity of depending on

And is
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Additional consequences due to IRGT

• In particular, it leads to an identity

• It is the corrected version of the absurd
  relation d/dt d(t-t) 0
• It puts a constraint on the un-renormalized
  propagator to all orders

• In particular, this gives non-trivial
  information about the exact propagator at p 0,
  p0 ? 0.

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Additional considerations in renormalization a
partial analysis

• How do these additional equations affect the
  renormalization of gauge theories ?
• To study this, we consider for simplicity, the
  axial gauge A3 0 and the following
  path-integral

With,
We perform the following IRGT (with qa
qa(x0,x1,x2))
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Additional considerations in renormalization
(contd)

• Under IRGT, Seff is invariant. This leads to the
  IRGT WT-identity

The above IRGT identity is over and above the
formal BRS identity. We note that the last term
can have a finite limit as e ?0. Without it we
would again land with absurd relations.
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Additional considerations in renormalization An
analysis of IRGT identity

• How do these additional considerations affect the
  renormalization of gauge theories ?
• To study this, it convenient to recall the usual
  set-up of the axial gauges and their expected
  renormalization and see if these extra relations
  agree with them.
• We associate with an axial gauge A3 0, the
  following
• freedom from ghosts,
• A multiplicative renormalization scheme

We ask under what conditions on O are these
relations compatible with the above scheme? We
shall assume, without loss of generality, that O
has a general quadratic form in A
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A Spectator prescription term

• It is usually believed that the prescription for
  treating axial gauge poles is unaffected by
  renormalization. This amounts to assuming that
  the eO term remains unchanged during
  renormalization process. We shall call this a
  spectator prescription term.
• A local O is not compatible with these relations.
  To see this, recall DO ? ?mAm. This leads to
  the IRGT WT-identity

Each term in the above is multiplicatively
renormalizable. But the multiplicative scales do
not agree.
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Renormalization of e-term

• In the more general case, we have to entertain
  the possibility that the prescription term
  receives renormalization.
• In fact, in the present picture this possibility
  becomes transparent and natural because we are
  looking at the prescription as coming from just
  another term in the total action Seff eO.
• This possibility has been analyzed partially
  under certain assumptions and restrictions on O
  have been spelt out.

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A comment on Wick Rotation

• This comment applies to a subset of non-covariant
  gauges which are such that the Wick-rotated
  gauge-function F EA is either purely real or
  purely imaginary. Example
• Coulomb gauge
• Temporal gaugeA0 ?iA4 axial gauge h.A ?h.A
• But not light-cone gauge A0-A3 ?iA4--A3
• The underlying question is whether there will
  exist an Euclidean formulation (with no e-term)
  from which the correct Minkowski formulation with
  the prescription term can be obtained from
  Wick-rotation.
• In covariant gauges, we know that this is always
  possible.
• This question is important, because often
  possibility of Wick rotation has been considered
  a desirable criterion for a prescription.

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A comment on Wick Rotation contd

• An analysis along the present lines seems to
  indicate that for these non-covariant gauges,
  this is unlikely.

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Conclusions

• We have constructed a generalization if residual
  gauge transformation to the BRS space and shown
  that the naïve path-integral leads to
  physically/mathematically unacceptable results by
  performing IRGT.
• This brings out the fact that even formal
  manipulations require careful treatment.
• We constructed a general framework in which to
  study the non-covariant gauges.
• We showed that the exact IRGT identities are
  contained in the exact BRST.
• Unlike the covariant gauges, the e-term can
  contribute to the BRST WT-identities.
• The IRGT identities lead to additional
  consequences that have to be taken into account
  while discussing renormalization.
• We have partially analyzed these conditions.

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AN ANALOGY WITH SYMMETRIES AND ANOMALY
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FAQ1 A Criticism Fault of PI-formulation?

• It is sometimes believed that the path-integral
  is a ill-defined object and illegitimate
  operations of path-integrals are often the cause
  of absurd results.
• This can be resolved in a simple manner We can
  always think of the Field theory as defined in
  terms Feynman rules alone. We can then evaluate
  the quantities on the left hand side and evaluate
  the result. There is a one-to-one correspondence
  between path-integral manipulations and Feynman
  diagrammatic approach.
• These identities are valid even in tree
  approximation, where the e-term is needed in an
  essential way for its validity.

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FAQ2 Divergence structure and prescription
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FAQ2(Contd)

• ILM has no divergences at all.

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FAQ3 FFBRS approach
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FAQ3 FFBRS approach (contd.)
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FAQ4 Absurd conclusion from PI with residual
gauge symmetry