GGI, 23 May 2007 Large N & SUSY: Large N & SUSY: some new ideas and results new ideas and results some Gabriele Veneziano Veneziano Gabriele (CERN & CdF CdF) ) (CERN &
Part I: Planar Equivalence Part I: Planar Equivalence Old and new large-N QCD Old and new large-N QCD Orientifold planar equivalence planar equivalence Orientifold Part II: Planar Quantum Mechanics Part II: Planar Quantum Mechanics Hamiltonian Planar QM Hamiltonian Planar QM An intriguing SUSY matrix model An intriguing SUSY matrix model
Part I based on Part I based on A. Armoni Armoni, M. , M. Shifman Shifman, GV, hep-th/0302163, , GV, hep-th/0302163, A. 0307097, 0309013, 0403071, 0412203, 0701229; 0307097, 0309013, 0403071, 0412203, 0701229; A. Armoni Armoni, G.M. Shore, GV, hep-ph/0511143 , G.M. Shore, GV, hep-ph/0511143 A. Part II based on Part II based on J. Wosiek Wosiek & GV & GV hep-th/0512301, 0603045, J. 0607198, 0609210 (cond-mat); E. Onofri Onofri, J. , J.Wosiek Wosiek & GV & GV math-ph/0603082 E.
Large-N expansions in QCD Large-N expansions in QCD Planar & quenched limit ( Planar & quenched limit (‘ ‘t t Hooft Hooft, 1974) , 1974) 1/N c expansion @ fixed λ = g 2 N c and N N f 1/N c expansion @ fixed λ = g 2 N c and f Leading diagrams Leading diagrams Corrections: : O( O(N N f /N c ) from q-loops q-loops, , Corrections f /N c ) from O(1/N c ) from higher genus diagrams O(1/N 2 ) from higher genus diagrams 2 c
Properties at leading order Properties at leading order Resonances have zero width + Resonances have zero width + 1. 1. U(1) problem not solved, WV @ NLO - ? U(1) problem not solved, WV @ NLO - ? 2. 2. Multiparticle production not allowed production not allowed - Multiparticle - 3. 3. Theoretically (if not phenomenologically phenomenologically) ) Theoretically (if not appealing: should give the tree-level tree-level of of appealing: should give the some kind of string theory kind of string theory some Proven hard to solve, except in D=2… …. . Proven hard to solve, except in D=2
Planar unquenched limit Planar unquenched limit = Topological Expansion (GV ‘ ‘74-- 74--’ ’76) 76) = Topological Expansion (GV 1/N expansion at fixed g 2 N and and N N f /N c 1/N expansion at fixed g 2 N f /N c Leading diagrams include “ “empty empty” ” q-loops q-loops Leading diagrams include Corrections: Corrections: O(1/N 2 ) from non-planar diagrams O(1/N 2 ) from non-planar diagrams
Properties Properties Widths are O(1) - Widths are O(1) - 1. 1. U(1) problem solved to leading order, no reason U(1) problem solved to leading order, no reason 2. 2. for WV to be good + ? for WV to be good + ? Multiparticle production allowed + production allowed + Multiparticle 3. 3. => Bare Pomeron Pomeron & & Gribov Gribov’ ’s s RFT RFT => Bare Perhaps phenomenologically phenomenologically more appealing than more appealing than Perhaps ‘t t Hooft Hooft’ ’s s but even harder to solve but even harder to solve… … ‘ But there is a third possibility… … But there is a third possibility
Generalize Generalize QCD to N QCD to N ≠ ≠ 3 (N = 3 (N = N N c hereafter) in ) in c hereafter other ways by by playing with matter rep playing with matter rep. . The The other ways conventional way, , QCD QCD F , is is to to keep the keep the quarks in N + quarks in N + conventional way F , N* rep rep. . N* Another possibility, , called called for for stringy reasons stringy reasons* * ) Another possibility ) QCD OR , is to to assign assign quarks to quarks to the the 2-index-antisymm. 2-index-antisymm. QCD OR , is rep. of SU(N) (+ . of SU(N) (+ its its c.c.)** c.c.)** ) rep ) As in ‘ ‘t Hooft t Hooft’ ’s s exp exp. ( . (and unlike and unlike in TE), N in TE), N f is kept is kept As in f fixed (N (N f < 6, or else else AF AF lost at lost at large N) large N) fixed f < 6, or old QCD! QCD! good old NB: For N = 3 N = 3 this is still this is still good NB: For ------------ ------------ * ) ) see e.g. P.Di Vecchia et al. hep-th/0407038 * ** ) Pioneered by by Corrigan and Ramond Corrigan and Ramond (1979) for (1979) for ** ) Pioneered very different reasons very different reasons
Leading diagrams are planar, include “ “filled filled” ” q- q- Leading diagrams are planar, include loops since there are O(N since there are O(N 2 ) quarks loops 2 ) quarks Widths are zero, U(1) problem solved, no p.pr. Widths are zero, U(1) problem solved, no p.pr. Phenomenologically interesting? interesting? Phenomenologically Don’ ’t know. t know. Don Better manageable? Better manageable? Yes, I claim. Yes, I claim.
Numerology of QCD F vs vs. QCD . QCD OR Numerology of QCD F OR Large-N, Th Th Large-N, YM QCD F YM QCD QCD OR QCD F N f =1 N f =1 OR coeff ( 11N-2(N-2) ) /3 11N/3 11N/3 ( f ) 11N-2(N-2)N N f /3 (11N-2N f )/3 (11N-2N f )/3 3N β 0 3N β 0 17N 17N 2 2 - - N N f f (N-2) x (N-2) x 17N 17N 2 2 - 3N - 3N f f x x 17N 2 17N 9N 2 2 9N 2 ( 5N + 3(N-2)(N+1)/N ) ( 5N + 3(N-2)(N+1)/N ) 3 β β 1 (13N/6 -1/2N) (13N/6 -1/2N) 3 1 3(N 2 -1)/2N 3(N-2)(N+1)/N 3(N 2 -1)/2N 3(N-2)(N+1)/N X X 3N 3N γ 0 γ 0 QCD OR as an interpolating interpolating theory: theory: QCD OR as an Coincides with pure YM YM (AS fermions decouple) @ (AS fermions decouple) @ N=2 N=2 Coincides with pure Coincides with QCD QCD @ @ N=3 N=3 Coincides with … and at and at large N large N? ? …
ASV claim of Planar Equivalence ASV claim of Planar Equivalence QCD of At large-N a bosonic bosonic sector sector of QCD OR is equivalent to At large-N a is equivalent to OR a corresponding sector corresponding sector of of QCD QCD Adj i.e. of QCD with a i.e. of QCD with Adj N f Majorana Majorana fermions in the fermions in the adjoint adjoint representation representation N f If true, important corollary: If true, important corollary: For N N f = 1 and m = 0, QCD = 1 and m = 0, QCD OR is planar-equivalent to For f OR is planar-equivalent to supersymmetric Yang-Mills (SYM) theory Yang-Mills (SYM) theory supersymmetric Some properties of the latter should show up in one- one- Some properties of the latter should show up in flavour QCD QCD … … if N=3 is large enough if N=3 is large enough flavour NB: Expected accuracy 1/N 1/N NB: Expected accuracy ASV gave both perturbative perturbative and NP arguments and NP arguments ASV gave both
Sketch of non-perturbative non-perturbative argument argument Sketch of (ASV ‘ ‘04, A. Patella, 04, A. Patella, ‘ ‘05) 05) (ASV Integrate out fermions (after having included Integrate out fermions (after having included masses, bilinear bilinear sources) sources) masses, Express Express Trlog Trlog(D+m+J) in terms of Wilson-loops (D+m+J) in terms of Wilson-loops using world-line formulation using world-line formulation Use large-N to write Use large-N to write adjoint adjoint and AS Wilson loop as and AS Wilson loop as products of fundamental and/or antifundamental antifundamental products of fundamental and/or Wilson loops (e.g. W W adj = W = W F x W F* +O(1/N 2 )) Wilson loops (e.g. adj F x W F* +O(1/N 2 )) Use symmetry relations Use symmetry relations between F and F* Wilson between F and F* Wilson loops and their connected correlators correlators loops and their connected < W > conn An example: < (2) > An example: W (1) (1) W W (2) conn
W (1) W (1) adj adj SYM SYM W (2) W (2) adj adj W (1) W (1) or or OR OR W (2) W (2) or or
Key ingredient is C! Key ingredient is C! Clear from our NP proof that C-invariance is necessary necessary. . Kovtun, Unsal and Yaffe have argued that it is also sufficient sufficient U&Y (see also Barbon & Hoyos) have also shown that C is spontaneously broken if the theory is put on R 3 xS 1 w/ small enough S 1 . PE doesn’t (was never claimed to) hold in that case Numerical calculations (De Grand and Hoffmann) have confirmed this, but also shown that, as expected on some general grounds (see e.g. ASV), C is restored for large radii and in particular on R 4 Lucini, Patella & Pica have shown (analyt.lly & numer.lly) that SB of C is also related to a non-vanishing Lorentz- breaking F#-current generated at small R but disappearing as well as R is increased Overwhelming evidence for PE on Overwhelming evidence for PE on R R 4 ? 4 ?
An interesting proposal An interesting proposal Kovtun, Unsal and Yaffe (‘07) have also made the claim that QCD adj , unlike QCD F and QCD OR , suffers no phase transition as a volume-reducing process a la Eguchi-Kawai is performed at large-N If this is indeed the case, we could get properties of QCD adj at small volume by numerical methods and use them at large volume where the connection to QCD OR can be established (C being OK there) Finally, one would make semi-quantitative predictions for QCD itself (at different values of N f ) by extrapolating down to N=3 For the moment, we shall try to use instead the connection with a SUSY theory
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