Nils A' Trnqvist

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Do the light scalars exist
and, do the sigma and the pion mix with the EW
Higgs sector?

• Nils A. Törnqvist
• University of Helsinki

Talk in Marseille CPT February 8-11 2006
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Tentative quarkantiquark mass spectrum for light
mesons

• The states are classified according to their
  total spin J , relative angular momentum L, spin
  multiplicity 2S 1 and radial excitation n. The
  vertical
• Each box represents a flavour nonet containing
  the isovector meson, the two strange isodoublets,
  and the two isoscalar states.
• ,

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Two recent reviews on light scalars
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Why are the scalar mesons important?

• The nature of the lightest scalar mesons has been
  controversial for over 30 years. Are they the
  quark-antiquark, 4-quark states or meson-meson
  bound states, collective excitations, or
• Is the s(600) a Higgs boson of QCD?
• Is there necessarily a glueball among the light
  scalars?
• These are fundamental questions of great
  importance in QCD and particle physics. If we
  would understand the scalars we would probably
  understand nonperturbative QCD
• The mesons with vacuum quantum numbers are known
  to be crucial for a full understanding of the
  symmetry breaking mechanisms in QCD, and
• Presumably also for confinement.

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What is the nature of the light scalars?

• In the review with Frank Close we suggested
• Two nonets and a glueball provide a
  consistent description of data on scalar mesons
  below 1.7 GeV.
• Above 1 GeV the states form a conventional
  quark-antiquark nonet mixed with the glueball of
  lattice QCD.
• Below 1 GeV the states also form a nonet, as
  implied by the attractive forces of QCD, but of a
  more complicated nature.
• Near the centre they are diquark-antidiquark
  in S-wave,
• a la Jaffe, and Maiani et al, with some
  quark-antiquark in P-wave, but further out they
  rearrange as 2 quark-antiquark systems and
  finally as mesonmeson states.

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Recent s(600) pole determinations
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BES collaboration PL B 598 (2004) 149158 Finds
the s pole in J/? ??pp- at (54139)-i(25242)
MeV
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f(1020)?p0p0g
Study of the Decay f(1020)?p0p0g with the KLOE
Detector The KLOE Collaboration Phys.Lett. B 537
(2002) 21-27 (arXivhep-ex/0204013 Apr 2002)
Sigma parameters from
E791
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The two pion invariant mass distribution in D to
ppp decay (dominated by broad low-mass f0(600)),
and (b) the Dalitz plot (from E791).
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The invariant mass distribution in Ds to 3p
decay showing mainly f0(980) and f0(1370). and
Dalitz plot (E791).
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• The D to K- pp Dalitz plot. A broad
  kappa is reported under the dominating K(892)
  bands (E791).

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• Very recently
• I. Caprini, G. Colangelo, H. Leutwyler,
  Hep-ph/05123604 from Roy equation fit get

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Important things to notice in analysis of the
very broad s(600) (and k(800))

• One should have an Adler zero as required by
  chiral symmetry near smp2/2. This means
  spontaneous chiral symmetry breaking in the
  vacuum as in the (linear) sigma model.
  To fit data in detail one should furthermore
  have
• Right analyticity behaviour (dispersion
  relations) at thresholds
• One should include all nearby thresholds (related
  by flavour symmetry) in a coupled channel model.
• One should unitarize
• Have (approximate) flavour symmetric couplings

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The U3xU3 linear sigma model with three flavours
If one fixes the 6 parameters using the well
known pseudoscalar masses and decay constants one
predicts A low mass s(600) at 600-650 MeV with
large (600 MeV) pp width, An a0 near 1030 MeV,
and a very broad 700 MeV kappa near 1120 MeV
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Spontaneous symmetry breaking and the
Mexican hat potential
Cylindrical symmetry mp ms
Cylindrical symmetry mp 0, ms gt 0,
proton massgt0 and constituent quark mass
300MeV
Chosing a vacuum breaks the symmetry spontaneously
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Tilt the potential by hand and the pion gets mass
mp gt 0, ms gt 0
But what tilts the potential? Another
instability?
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Two coupled instabilities breaking the symmetry
If they are coupled, they can tilt each other
spontaneously
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Another way to visualize an instability, An
elastic vertical bar pushed by a force from
above
FgtFcrit
FltFcrit
The cylindrical symmetry broken spontaneously
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Now hang the Mexican hat on the elastic vertical
bar. This illustrates two coupled unstable
systems.
Now there is still cylindrical symmetry for the
whole system, which includes both hat and the
near vertical bar. One has one massless and one
massive near-Goldstone boson.
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To see the anology with the LsM, write the Higgs
doublet in a matrix form

NAT, PLB 619 (2005)145
and a custodial global SU(2) x SU(2) as in
the LsM
L
R
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Compare this with the LsM for p and s in matrix
representation
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The LsM and the Higgs sector are very similar but
with very different vacuum values.

Now add the two models with a small mixing term e
This is like two-Higgs-doublet model, but much
more down to earth.
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The mixing term shifts the vacuum values a little
and mixes the states
And the pseudoscalar mass matrix becomes
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Diagonalizing this matrix one gets a massive pion
and a massless triplet Goldstone
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The pion gets a mass through the mixing mp
Right pion mass if e 2.70 MeV.
The Goldstone triplet is swallowed by the W and Z
in the usual way, but with small corrections from
the scalars.
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Quark loops should mix the scalars of strong and
weak interactions and produce the mixing term e
proportional to quark mass?
2
q
higgs, W
s, p
L
q
Also isospin and other global symmetries schould
be violated by similar graphs
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Conclusions

• We have one extra light scalar nonet of different
  nature, plus heavier conventional quark-antiquark
  states (and glueball).
• It is important to have Adler zeroes, chiral and
  flavour symmetry, unitarity, right analyticity
  and coupled channels to understand the broad
  scalars (s, k) and the whole light nonet, s(600)
  k(800),f0(980),a0(980).
• Unitarization can generate nonperturbative extra
  poles!
• The light scalars can be understood with large
  qqqbar qbar and meson-meson components
• By mixing the E-W Higgs sector and LsM the pion
  gets mass, and global symmetries broken?

Further analyses needed!
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Adler zero in linear sigma model
Destructive interference between resonance and
background
Example resonance constant contact and
exchange terms cancel near s0, Thus pp
scattering is very weak near threshold, but grows
rapidly as one approaches the resonance
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Correct analytic behaviour from dispersion
relation
It is not correct to naively analytically
continue the phase space factor r(s) below
threshold one then gets a spurious anomalous
threshold and a spurious pole at s0.
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Unitarize the basic terms.Example for contact
term resonance graphically
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K-matrix unitarizationF.Q.Wu and B.S.Zou,
hep-ph/0412276
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