GAUGED SYSTEM MIMICKING GURSEY MODEL

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GAUGED SYSTEM MIMICKING GURSEY MODEL

• Bekir Can LÜTFÜOGLU
• Mahmut HORTAÇSU
• Ferhat TASKIN

ISTANBUL TECHNICAL UNIVERSITY Mugla Akyaka 2006
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A MODEL with INTERACTING COMPOSITES

• Bekir Can LÜTFÜOGLU
• Mahmut HORTAÇSU

I.T.U.
Mugla 2005
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MOTIVATION
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• Fermions are an essential ingredient in nature.
• HEISENBERG Theory of everything
• Ref W.Heisenberg. Z.Naturforschung 9A, 210
  (1954)
• GÜRSEY Conformal Invariant Classical Model
• Ref F.Gürsey, Nuovo Cimento 3, 988 (1956)

• Non polynomial form
• KORTEL solutions of the model
• Ref F.Kortel, Nuovo Cimento 4, 210 (1956)
• AKDENIZ named solutions as instantonic and
  meronic
• Ref K.G.Akdeniz, Lett. Nuovo Cimento 33,40 (1982)

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• HORTAÇSU et al made the quantum sense of the
  model.
• Ref G.Akdeniz, M.Arik, M.Durgut, M.Hortaçsu,
  S.Kaptanoglu, N.K.Pak, Physics Letters B
  116,34,41(1982)
• NAMBU and JONA-LASINIO Another Famous Model
• RefY.Nambu G.Jona-Lasinio, Phys. Rev.
  112,345(1961)

• KOCIC and KOGUT trivial in 4 dim.
• RefA.Kocic and J.B.Kogut, Nucl. Phys. B 422,593
  (1994)

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• BARDEEN et al.If GAUGED there is a certain
  coupl. cons when the dimension of the operator
  goes from 6 to 4.
• Ref W.A.Bardeen, C.N.Leung and S.T.Love,
  Phys.Rev.Lett. 56,1230 (1986)
• Ref C.N.Leung, S.T.Love and W.A.Bardeen, Nucl.
  Phys. B 273(1986)
• REENDERS with a sufficient number of fermion
  flavors the gNJL theory becomes non-trivial.
• RefM.Reenders, Phys. Rev. D 62 025001 (2000).
• Those facts gave us a motivation to reconsider
  Gürsey model and see if we can make a non-trivial
  model out of it.

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EQUIVALENT MODEL in a NAIVE SENSE
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• Start with the Lagrangian where ? is a Lagrange
  multiplier field, ? is a scalar field with no
  kinetic part, g, a are dimensionless coupling
  constants.

• This is an attempt to write the orginal model in
  a polynomial form.
• Have two constraint equations

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• Both Lagrangians have ?5 symmetries.

• This sym. prevents ? from acquiring a finite mass
  in higher orders.
• If we take the original Lagr. with a s
  symmetry operation

where

• Then postulate L should be invariant under s
  operation.
• Get the inital L Faddeev-Popov Term but also
  non polynomial term.
• So this sym. Not RETAINED. THE SECOND MODEL ONLY
  APPROXIMATES the original one without claiming
  Equivalence so we replace the original one in a
  Naive sense.

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CONSTRAINT ANALYSIS à la DIRAC
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• Perform the constraint analysis we get the usual
  spinor constraint plus the additional ones
• The Canonical and Primary Hamiltonian can be
  written as

• If we check the time evolution we get secondary
  constraints.
• The system is closed by solving all the arbitrary
  constants a1,b1,..
• Check the class of the cons. we find all of them
  as Second Class.

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FADDEEV-POPOV FORMALISM
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• The Faddeev-Popov determinant

• The partition function is

• By using the ghost fields c and c, we find the
  resulting Lag.

• Redefine the fields

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• The effective Lagrangian

• ? field decouples from the spinor sector.
• Integration over the spinor fields we get the
  effective action as

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• To kill the tadpole contribution we take the
  first derivative with respect to the fields ?, ?
  and evaluate them to zero.

• -v, s are vacuum expectation values of the
  fields. VEV of ghost fields are set to zero.
• A consistent solution is v s 0
• At least one phase exists which has zero mass as
  dictated by ?5 symmetry.
• Second derivatives give the Inverse Propagators

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• Here used dimensional regularization ? 4 n
• All the other fields ?,c,c propagators vanish.
• No mixing between the fields ?,?.
• Conclusion We have a model only with the spinor
  fields and Composite Field ?.

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DRESSED FERMION PROPAGATOR
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• Cal. the above results for the higher orders
  Dyson Schwinger Eq.
• For Dressed Fermion Propagator

• With simple algebra one can get easily

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• This integral is clearly finite. We get zero for
  the right hand side as ? goes to zero. Since mass
  is equal to zero in the free case we get this
  constant equal to zero.
• Similarly for A with simple algebra

• A is a constant as ? goes to zero. Since the
  integral is finite, it equals unity for the free
  case. We take A1.
• Conclusion No MASS is generated for higher
  orders.

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HIGHER ORDERS
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Yukawa Vertex

• By dimensional analysis one over epsilon times
  epsilon so finite.

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Fermion Scattering

• The first one goes to zero because as power of
  epsilon comes from the intermediate composite
  scalar.
• The second diagram goes to zero as epsilon
  square.
• The third diagram goes to zero as epsilon cube.
• So no fermion scattering.

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Four Composite Scalar Vertex

• The first diagram diverges as one over epsilon.
• The second one is finite.
• The third one goes to zero as power of epsilon.
• The last one diverges as one over epsilon so we
  need renormalization for this type of processes.

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Other type of Composite Scalar Productions

• The odd number of Scalar production is forbidden
  because of the ?5 symmetries as mentioned before.
• Only the even number of composite scalar
  productions are avaliable.

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Fermion Production

• Although the box diverges as 1 over epsilon, but
  also two scalar intermediates have epsilon square
  so this type of spinor production down by factor
  of epsilon.

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BETHE-SALPETER EQUATION
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• Our aim is to check the claims for the processes
  by Bethe-Salpeter equation in the quenched
  ladder approximation.
• For Yukawa vertex equation the BS equation can be
  written as

• S is dressed fermion propagator, K is four-point
  Kernel.

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• Four Fermion interaction the BS equation

• G0(2)(p,qP) is two non-crossing spinor lines,
  G(2)(p,qP) is the proper four point function. K
  is the Bethe-Salpeter Kernel.

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CONCLUSION
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• Only infinite renormalization is needed for
  4-Comp. Scalar vertex.
• The first correction to the tree diagram is box
  diagram (one over ?).
• Since we included the four ? term in our
  original lagrangian, we can renormalize the
  coupling constant of this vertex to incorporate
  this divergence .
• Higher order ladder diagrams give at worst the
  same type of divergence.
• This divergence for the four scalar vertex can be
  renormalized using standart means.

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• As a result of this analysis we end up with a
  model where there is no scattering of the
  fundamental fields, i.e. the spinors, whereas the
  composite fields, the scalar field, can take part
  in a scattering process.
• We find a model which is trivial for the
  constituent spinor fields, whereas finite results
  are obtained for the scattering of the composite
  scalar particles. The coupling constant for the
  scattering of the composite particles run,
  whereas the coupling constant for the
  spinor-scalar interaction does not run.
• In the classical model, one coupling constant g',
  which is divided into two as g and a. We see that
  these two behave differently in the quantum case.
  
• PUBLISHED in MOD.PHY.LETT.A. Vol 21No8 (2006)

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Gauged SYSTEM
with Ferhat Taskin I.T.Ü.
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• Couple an elementary vector field to our model.

• Here A? is the elementary vector field and F?? is
  defined in the usual way.
• Now we have three renormalized coupling
  constants a, g and e instead of two a
  and g.
• The three first order renormalization group
  equations for these three coupling constants.

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• Where b,c,d are numerical constants. Their values
  are

• These processes are shown in the figure below.
  Here dotted lines represent the scalar, wiggly
  lines the vector, solid lines the spinor
  particles.

• ? is inversely proportional to ln .
  Solutions are

where
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• In diagrammatical analysis, two scalar particles
  goes to two scalar particles gets further
  infinite contribution from the box type diagrams
  with vector field insertions, where one part of
  the diagram is connected to the non-adjacent
  part with a vector field as shown in the figure
  below.

• Go as one over epsilon, no higher divergences for
  this process.
• The diagram where the internal photon is
  connecting adjacent sides, as shown above, will
  be a contribution to the coupling constant
  renormalization of one of the vertices.

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• No essential change in the spinor propagator.
• Miransky explains no mass generation for the
  coupling constant .
• Here
• Ref V.A.Miransky, Dynamical Symmetry Breaking in
  Quantum Field Theories World Scientific (1993)
• J.C.R. Bloch states that the quenched and the
  rainbow approximations, used by Miransky, have
  non physical features, namely they are not gauge
  invariant, making the calculated value wildly
  vary depending on the particular gauge used.
• He uses the Ball-Chiu vertex instead of the bare
  one, where the exact longitudinal part of the
  full QED vertex, is uniquely determined by the
  Ward-Takahashi identity relating the vertex with
  the propagator.
• The transverse part of the vertex, however, is
  still arbitrary, uses the Curtis-Pennington
  vertex. He gets rather close values for the
  critical coupling. He also performs numerical
  calculations where the approximations are kept to
  a minimum.
• Ref J.C.R. Bloch, hep-ph/0208074

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• Using on the arguments in the Bloch's thesis,
  also using the results of his numerical
  calculations, we conclude that at least for ?lt
  0.5 we can safely claim that there will be no
  mass generation or the assumed ?5 symmetry will
  be not broken. Since we do not study heavy ion
  processes, the numerical value we have for ?
  will be much smaller than this limit. Hence, our
  results will be valid.
• Additional contribution to the scalar propagator
  by using diagrammatical analysis.
• If we take only the planar diagrams which connect
  non adjacent spinor lines, as shown in the
  diagram given below, the scalar field
  contributions are only of order (1/?) the same as
  the one loop initial contribution. Higher order
  divergences will come from the vector field
  insertions.

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• The diagrams where there are n-1 nonadjacent and
  planar vector field contributions, go as
  where is a numerical
  constant.

• In an odd dimension , if we sum up this series,
  which is a geometric series, as where
  . We get the inverse propagator
  contribution as . If the scalar
  particle were an elementary field, this
  contribution would go as .
• If we could translate these results to our
  case, C goes to infinity and is not less than
  unity, the condition for the convergence of the
  series, our model has a different behaviour
  compared to the model where one uses elementary
  fields.
• RefD.I.Kazakov and G.S. Vartanov, hep-th/0607177

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• The similar type of summation can be performed
  for the scalar-2 spinor vertex.

• The vector- spinor- antispinor vertex do not get
  infinite contributions from our composite scalar
  particle. A typical
• diagram is given in below at the lowest order.

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• The vertex, for the purely electromagnetic case,
  is vastly studied in the literature.
• Ref J.S.Ball and T.W.Chiu, Phys. Rev. D 22
  (1980)
• Ref D.C.Curtis and M.R.Pennington, Phys. Rev. D
  42 (1990)

• An important difference is in spinor spinor
  scattering. Previously it was prohibited but now
  we have due to the presence of the vector field.

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• The figure below is ultraviolet finite from
  dimensional analysis and is calculated in
• Ref Portoles, P.D.Ruiz-Femenia, Eur. Phys.
  Journal. C 25, (2002)

• Another difference is the spinor production from
  the scalar particles by vector intermediaries.

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• The photon propagator also will be the similar as
  the one given in QED, with only additional
  contributions from the scalar particle
  insertions. The lowest order diagram of this
  process is shown in below. The dominant
  contribution will be from the vector insertions,
  which are studied in QED.

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Thanks for your patience.