Old and New PBB cosmologies

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Large-N, Supersymmetryand QCD G. Veneziano
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• First part
• (w/ A. Armoni, M. Shifman, G. Shore)
• QCDF vs. QCDOR (AA MS)
• Planar equivalence in PT and beyond (AA MS)
• SUSY relics in Nf1 QCD (AA MS)
• lt yy gt in Nf1 (AA MS) Nf3 (AA GS) QCD
• Second part
• (w/ J. Wosiek)
• Planar matrix models a Hamiltonian approach
• A SUSY example (with a sense of beauty?)

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Large-N expansions in QCD

• Planar quenched limit (tHooft, 1974)
• 1/Nc expansion _at_ fixed l g2Nc and Nf
• Leading diagrams

Corrections O(Nf /Nc) from q-loops, O(1/Nc2)
from non-planar diagrams
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• Properties at leading order
• Resonance have zero width
• U(1) problem not solved, WV _at_ NLO -?
• Multiparticle production not allowed -
• Theoretically, if not phenomenologically,
  appealing
• should give the tree-level of some string theory
• (gt accidental discovery of string theory?)
• Proved hard to solve, except in D2.

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• Planar unquenched limit (GV 74--76) TE
• 1/N expansion _at_ fixed l g2N and Nf /Nc
• Corrections
• O(1/N2) from non-planar diagrams
• Leading diagrams include empty q-loops

It all started from a paper by Di Giacomo et al.
1970..
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From my talk at meeting in memory of Sergio
Fubini (May 2005)
Another problem with DRM was the lack of
unitarity (e.g. resonances had zero width). Loops
could be added, but only perturbatively In 1970,
Sergio, A. Di Giacomo, L. Sertorio and I proposed
a non-perturbative way of implementing unitarity
in DRM where the topology of the diagram, not its
order, is what counts This line of thought led
eventually to the topological expansion of QCD
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• Properties
• Widths are O(1) -
• U(1) problem solved to leading order, no reason
  for WV to be good ?
• Multiparticle production allowed
• gt Bare Pomeron Gribovs RFT
• Perhaps phenomenologically more appealing than
  tHoofts but even harder to solve

There is a third possibility
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• Generalize QCD to Ngt3 in other ways by playing
  with matter representation (N Nc hereafter)
• The conventional way, QCDF, is to keep the quarks
  in fundamental antifundamental (N N) rep.
• The one we shall consider is called, for stringy
  reasons, QCDOR (OR for Orientifold see e.g. P.Di
  Vecchia et al. hep-th/0407038)
• Put quarks in the 2-index-antisymmetric
  (AS)-tensor rep. of SU(N) ( its complex
  conjugate)
• As in tHoofts expansion (and unlike in TE) Nf
  fixed
• NB. For N3 this is still ordinary QCD

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• In string construction both S and AS tensor
  reps. are possible, but the former is never QCD
• However, the use of S has been advocated recently
  by Sannino et al. as a way to  resuscitate 
  technicolour (see e.g. hep-ph/0505059)

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• Leading diagrams are planar, include filled
  q-loops since there are O(N2) quarks
• Widths are zero, U(1) problem solved, no p.pr.
• Phenomenologically interesting?
• Theoretically manageable? Yes, I claim.

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QCDF vs. QCDOR
th
Large-N, Nf1
YM
QCDF
QCDOR
coeff
b0
11N/3
(11N-2(N-2)Nf)/3
(11N-2Nf)/3
3N
17N2/3 - Nf (N-2) x (5N 3(N-2)(N1)/N)/3
17N2/3 - Nf (13N/6 - 1/2N)
17N2/3
3N2
b1
3(N-2)(N1)/N
3(N2-1)/2N
X
3N
g0
QCDOR as an interpolating theory Coincides with
pure YM (fermions decouple) _at_ N2 Coincides with
QCD _at_ N3 and at large N?
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ASV claim of Planar Equivalence

• At large-N a bosonic sector of QCDOR is
  equivalent to a corr. sect. of QCDAdj i.e. of QCD
  with Nf Majorana fermions in the adjoint
  representation
• Important corollary
• For Nf 1 and m 0, QCDOR is planar-equivalent
  to supersymmetric Yang-Mills (SYM) theory
• Some properties of the latter should show up in
  Nf 1 QCD if N3 is large enough
• NB Expected accuracy is only 1/N

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Perturbative ArgumentDraw a planar diagram on
sphere
QCDOR
Double-line rep.
QCDAdj
Differ by an even number of - signs
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Sketch of non-perturbative argument (ASV, A.
Patella)

• Integrate out fermions (after having included
  masses, bilinear sources)
• Express Trlog(DmJ) in terms of Wilson-loops
  using world-line formulation ( CGL,BdVH,S, DG)
• Use large-N to write adjoint and OR Wilson loop
  as product of fundamental and/or antifundamental
  Wilson loops (e.g. Wadj WF x WF O(1/N2))
• Use relations between F and F Wilson loops and
  their connected correlators
• An example

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W(1)adj
SYM
W(2)adj
W(1)or
OR
W(2)or
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Before moving to SUSY..

• It would be interesting to check numerically what
  happens to QCDOR and to QCDAdj as we increase N
  even for
• m ? 0, Nf ? 1,
• quenched limit
• The two theories should approach each other
• Another numerical (analytic?) check could be
  comparing fermionic determinants in both theories
  as N is increased

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SUSY relics in QCDOR , QCD(Nf1)

• Approximate parity doublets. Indeed, from mS mP
  mF in SYM we deduce that mS mP ltlt mF
• Looks OK if can we make use of
• i) Experiments for mS (s _at_ 600MeV ) ,
• ii) WV for mP (mP ?2(180)2/95 MeV 480 MeV
  excluding quark masses)
• NB Composite-fermion masses are NOT related. In
  SYM we can have a colour-singlet baryon by
  pairing a gluon with a gluino, in QCDOR (or in
  QCDF) we need O(N) quarks to make a baryon

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• Related to degeneracy approximate absence of
  activity in certain chiral correlators
• ltyRyL (x) yRyL (y) gt constant
• while ltyRyL (x) yLyR (y) gt has much activity
• In fact, in SYM, a WI gives
• Of course the constancy of the former is due to
  an exact cancellation between intermediate scalar
  and pseudoscalar states.

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• One-flavour QCD should be a confining theory with
  a mass gap, no Nambu-Goldstone bosons (only
  continuous axial symmetry broken by anomaly/inst.
  even _at_ large N)
• Should have N O(1) distinct vacua
  characterized by the phase of the quark
  condensate. Indeed one expects N-2 distinct
  vacua.

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• Vanishing cosmological constant at leading order
  in spite of the fact that the planar spectrum of
  the OR theory is purely bosonic
• An analytic estimate of the quark condensate
  (coming next)

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The quark condensate in Nf1 QCD

• Claim (ASV, hep-th/0309013)
• where (all in MS)

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• By observing that _at_ N2 fermion decouples
• gt K(1/N) (1-2/N)k(1/N)
• with k 1 30 _at_ N3 gives the quoted
  result.
• This can also be written as
• where both sides are RGI

K
1
1/N
1/2
1/3
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Comparison with data

• There are of course no real data on Nf 1 QCD
• Unfortunately there are no fake (MC) data
  eitherPLEASE..
• We can try to argue (ASV) about relation between
  Nf 1 condensate and the one of real QCD (from
  phenomenology or quenched lattice calculations).
• New strategy (A-GS-V) extend arguments to Nf gt1

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Extension to Nf gt1 (A,GS,V)

• Take OR theory and add to it nf flavours in NN
• At N2 its nf-QCD, _at_ N3 its (nf1)-QCD.
• At large N it cannot be distinguished from OR
  (fits SYM b-functions even better at nf 2 e.g.
  same b0)
• Vacuum manifold, NG bosons etc. are different!
• Some correlators should still coincide in large-N
  limit. These should include the combination of F
  and AS bilinears that decouples from NG bosons
• If so the result for Nf3 QCD is as follows
  (factor 1/3 comes out automatically)

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Instead of previous
we get, for Nf3,
up to the usual 30..
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Quark condensate (ren. _at_ 2 GeV in MeV3) vs
as(2GeV) (false scale)
lt yy gt2GeV
as(2GeV)
Very encouraging!
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Conclusions,part I

• The orientifold large-N expansion is arguably the
  first example where large-N considerations lead
  to quantitative analytic predictions in non-SUSY,
  D4, strongly coupled gauge theories
• More work is needed, particularly on
• Tightening the NP proof of planar equivalence
• Estimating 1/N corrections
• Providing numerical checks
• Extending the equivalence in various directions

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Part II (with J. Wosiek)

• One of the original motivations of this work was
  to check planar equivalence and compute its
  accuracy at finite N in a simple case
• This has not been done yet.
• However, JW and I stumbled on a rather amusing
  modeland we are still playing with it ..
• (a lesson to learn about beauty?)

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Planar QM a Hamiltonian approach(GV and JW,
hep-th/0512301)

• The idea is simply that, in the large-N limit of
  a theory like YM or SYM, the colour-singlet
  states are obtained from acting on the Fock
  vacuum with single-trace operators
• They have the structure of a ring, or necklace
• In the simplest case of SQM there is a single
  bosonic matrix a and a single fermionic matrix f
  (see MP 1990)
• Generic bra (ket)
• Binary necklaces see e.g.
• http//www.theory.csc.uvic.ca/cos/inf/neck/Neckla
  ceInfo.html

At leading order planar Hamiltonians act simply
on such states. The result (after normalizing the
states) depends only on l
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An intriguing SUSY example

• Take the SUSY charges to be, quite simply
• Q Tr(f A(a)) Tr(f(a g a2)) etc. Q2 0
• H Q, Q , C Q, Q , C2 H2
• Diagonalize, H, C, F Tr(ff)
• Trivial E0 vacuum,
• E gt 0 SUSY doublets of states with same CF
  (-1)FC
• Q are there, at each level (say for g0), as
  many binary necklaces with even and with odd
  fermion number? The naïve answer is no. Example _at_
  E2
• (aa), (ff), (af)(fa) gt 2 bosons, 1 fermion, ..
  but

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• Paulis exclusion principle gives back the
  balance between bosons and fermions (see figure).
  This SUSY pairing of states works at all levels.
  It looks related to a property (apparently
  unknown to mathematicians ?) of
• binary supernecklaces
• The way states pair is non-trivial (see figure).
  It has already been checked numerically (VW to
  appear) for the F2,3 states, e.g. for the 6
  states present at E6
• Two F2 states form SUSY doublets with two
  linear combinations of the four F3 states the
  remaining two F3 states split (checked) and
  should become partners of two of the three F4
  states (not yet checked). The left-over F4 state
  will finally pair with the single F5 state.

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F

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Supernecklaces in a Chew-Frautschi
plot (weak-coupling spectrum)
8
7
4
6
10
3
5
7
3
14
104 states
4
3
5
9
14
5
4
2
3
7
10
4
2
2
3
2
3
1
E
8
9
6
10
7
1
2
0
4
3
5
At E10 there are 56BNL 52FNL but 4 of the
former are Pauli forbidden!
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Strong coupling surprises in F0,1 sectors

• There is a (1st order?) phase transition at l 1
  the weak-coupling energy gap disappears
• The spectrum becomes discrete again for l gt1 and
  the eigenvalues at l are related to those at 1/
  l by a strong-weak duality formula
  
• The spectrum can be computed analytically in
  terms of the zeroes of some (incomplete!)
  Beta-function. Duality and phase transition can
  be studied analytically

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• At l gt 1 a second E0 bosonic ground state pops
  up making Wittens index jump by one unit (within
  F0,1 sectors).
• This was first found numerically. The analytic
  form of the state can be formally given at all l
  and is only normalizable at lgt1

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Lowest bosonic and fermionic states as a funtion
of l for different values of the cutoff B (NB
swapping of SUSY partners at finite cutoff)
l
Witten index and free energy as functions of l
l
Energies related by l2 (E(1/ l)1) E( l)1
l
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Conclusions,part II

• A direct planar-Hamiltonian approach should
  perhaps be tried again, both analytically and
  numerically
• As a warm-up exercise I have presented a simple
  SQM model were one can go a long way towards
  solving the planar theory (as opposed to doing so
  N by N) and uncovering some very non-trivial
  results
• Extending this approach to (semi) realistic QFTs
  could be very rewarding