Random Walks & Path Integrals for SU(N) charges at small x
A CLASSICAL ODDERON IN HIGH ENEGY QCD Raju Venugopalan BNL RBRC Workshop, Sept. 27th-29th, 2005 Outline of talk:
The ground state of a large nucleus at high energies Random walks & Path Integrals for SU(N) quarks A classical Odderon Results Summary and Outlook Work in collaboration with Sangyong Jeon (McGill/RBRC)
A CLASSICAL ODDERON IN QCD AT HIGH ENERGIES. Sangyong Jeon, Raju Venugopalan Phys.Rev.D71:125003,2005 RANDOM WALKS OF PARTONS IN SU(N(C)) AND CLASSICAL REPRESENTATIONS OF COLOR CHARGES IN QCD AT SMALL X Sangyong Jeon, Raju Venugopalan Phys.Rev.D70:105012,2004 Odderon paper inspired by
ODDERON IN THE COLOR GLASS CONDENSATE. Y. Hatta, E. Iancu, K. Itakura, L. McLerran Nucl.Phys.A760:172-207,2005 ground state of a nucleus at high energies Consider large nucleus in the IMF frame: Oppenheimer: separation of large x and small x modes
Dynamical Wee modes Valence modesare static sources for wee modes
The effective action Scale separating sources and fields Generating functional: Gauge invariant weight functional describing distribution of the sources- W obeys RG ( B-JIMWLK) equations with
changing scale with Focus in this talk on weight functional - for large nuclei Coarse grained field theory: # of random quarks in box of size Well defined math problem:
Given k non-interacting quarks belonging the the fundamental SU(N) representation: a) What is the distribution of degenerate irreducible representations? b) What is the most likely representation? c) Is it a classical representation ? N_c = Infinity is classical even for k=1 Random walk problem in space spanned by the n_c -1 Casimirs of SU(N)
QCD IN THE LARGE A ASYMPTOTICS: Evolution effects small => Kinematic region of applicability: Kovchegov: Extend discussion to
An SU(2) Random walk Random walk of spin 1/2 partons: In general, = Multiplicity of representation s when k fundamental reps. are multiplied Binomial Coefficients satisfy:
Multiplicity satisfies: with Using Stirlings formula, k >> s >> 1: # degrees of freedom
Probability: degeneracy of state Mult. of representation Analogous to Maxwell-Boltzmann distribution Casimir:
Average value: PATH INTEGRAL: Coarse graining -> Box of size 1/p_t in transverse plane Sum over spins in box: Classical color/spin density:
=> McLerran, RV Kovchegov Summing over all boxes -> Classical path integral over SU(2) color charge density Color charge squared per unit area- closely related
to saturation scale. For A >> 1, coupling runs as a function of this scale Random walk of SU(3) Color charges Denote SU(3) representations by (m, n) : Recursion relation for SU(3) : =
+ + = multiplicity of (m, n) state in the kth iteration Trinomial coefficients: Solution:
Again, use Stirlings formula Quadratic Casimir Cubic Casimir Probability: Dimension of representation
Note: Prove: Classical color charge: Proof: For any SU(3) representation,
Canonically conjugate Darboux variables Canonical phase space volume of SU(3): Johnson; Marinov; Alexeev,Fadeev, Shatashvilli
Hence, Measure of probability integral has identical argument in m & n to RHS- hence can express in terms of LHS. End of Proof. Path Integrals:
Generates Odderon excitations! As in SU(2), with MV Path integral measure for SU(3): Path integral approach reproduces diagrammatic computations of dipole and baryon C= -1 operators
To summarize Representations of order dominate for k >> 1 These representations are classical - can be represented by an SU(3) classical path integral Can repeat analysis for gluon and quark-anti-quark pairs
- only quadratic Casimir contributes Can add glue representations to quarks-result as for valence quarks - with larger weight. On to the Odderon DIS: IN CGC:
Hatta et al. Dipole Odderon operator: To lowest order Can compute < O > with SU(3) measure To lowest order,
To all orders (in the parton density) Kovchegov, Szymanowski, Wallon Dosch, Ewerz, Schatz 3-quark Baryon state scattering off CGC
To lowest order, All order result feasible, but tedious Conclusions The ground state of a large nucleus contains configurations that generate Odderon excitations - these can be traced to the random walk of valence partons in color space. These results represent a rigorous proof in the large
A asymptotics of QCD at high energies Dipole Odderon and Baryon Odderon operators are computed Phenomenological consequences - to be investigated further Why are higher dim. Reps. classical? riance group
or higher dim. rep. Yaffe r a system prepared in this state, uncertainity in mom position vanishes in this limit -> Coherent States Gitman, Shelepin
SU(N) Can follow same recursion procedure Quadratic Casimir dominates: successive N-2 Casimirs parametrically suppressed by
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