Towards finding the single-particle content of two-
dimensional adjoint QCD
Dr. Uwe TrittmannOtterbein University*
OSAPS Spring 2015 @ Kent State University
March 27, 2015
*Thanks to OSU for hospitality!
Adjoint QCD2 compared to the fundamental QCD2 (‘t Hooft model)
• QCD2A is 2D theory of quarks in the adjoint representation coupled by non-dynamical gluon fields (quarks are “matrices” not “vectors”)
• A richer spectrum: multiple Regge trajectories?
• Adjoint QCD is part of a universality of 2D QCD-like theories (Kutasov-Schwimmer)
• String theory predicts a Hagedorn transition
• The adjoint theory becomes supersymmetric if the quark mass has a specific value
The Problem: all known approaches are riddled with multi-particle states
• We want “the” bound-states, i.e. single-particle states (SPS)
• Get also tensor products of these SPS with relative momentum
• Two types: exact and approximate multi-particle states (MPS)
• Exact MPS can be projected out (bosonization) or thrown out (masses predictable)
NPB 587(2000)PRD 66 (2002)
Group of exact MPS: F1 x F1
Universality: Same Calculation, different parameters = different theory
•DLCQ calculation shown, but typical (see Katz et al JHEP 1405 (2014) 143)
•SPS interact with MPS! (kink in trajectory)
•Trouble: approx. MPS look like weakly bound SPS•Also: Single-Trace States ≠ SPS in adjoint theory
•Idea: Understand MPS to filter out SPS•Do a series of approximations to the theory
•Develop a criterion to distinguish approximate MPS from SPS
Trouble!
Group of approx MPS
‘t Hooft Model(prev slide)Adjoint QCD2
The Hamiltonian of Adjoint QCD
• It has several parts. We can systematically omit some and see how well we are doing
• Hfull = Hm + Hren + HPC,s + HPC,r + HPV + HfiniteN
(mass term, renormalization, parton #conserving (singular/regular), parton# violation, finite N)
• Here: First use Hasymptotic = Hren + HPC,s
• then add HPC,r
• Later do perturbation theory with HPV as disturbance
Asymptotic Theory: Hasympt=Hren+HPC,s
• Since parton number violation is disallowed, the asymptotic theory splits into decoupled sectors of fixed parton number
• Wavefunctions are determined by ‘t Hooft-like integral equations (xi are momentum fractions)
r =3
r =4
Old Solution to Asymptotic Theory
• Use ‘t Hooft’s approximation
• Need to fulfill “boundary conditions” (BCs)– Pseudo-cyclicity:– Hermiticity (if quarks are massive):
• Tricky, but some (bosonic) solutions had been found earlier, with masses:
M2 = 2g2N π2 (n1 +n2+ ..+nk) ; n1>n2> ..>nk ϵ 2Z
New: Algebraic Solution of the Asymptotic Theory
• New take on BCs: more natural to realize vanishing of WFs at xi=0 by
• New solution involves sinusoidal ansatz with correct amount of excitation numbers:
ni ; i = 1…r-1
• ϕr(n1 ,n2 ,…,nr-1) =
-
• “Adjoint t’Hooft eqns” are tricky to solve due to cyclic permutations of momentum fractions xi being added with alternating signs
• But: Simply symmetrize ansatz under
C: (x1, x2, x3,…xr) (x2, x3, …xr ,x1)
• Therefore: ϕr,sym(ni) ϕr(ni)
is an eigenfunction of the asymptotic Hamiltonian
with eigenvalue
New: Algebraic Solution of the Asymptotic Theory (cont’d)
ϕ3,sym(x1, x2, x3) = ϕ3(x1,x2,x3) + ϕ3(x2,x3,x1) + ϕ3(x3,x1,x2) = ϕ3(n1,n2) + ϕ3(-n2,n1-n2) + ϕ3(n2-n1,-n1)
ϕ4,sym(x1, x2, x3, x4) = ϕ4(x1,x2,x3,x4) – ϕ4(x2,x3,x4,x1) + ϕ4(x3,x4,x1,x2) – ϕ4(x4,x1,x2,x3)
It’s as simple as that – and it works! • All follows from the two-
parton (“single-particle”) solution
• Can clean things up with additional symmetrization:
T : bij bji
• Numerical and algebraic solutions are almost identical for r<4
• Caveat: in higher parton sectors additional symmetrization is required
T+
T+
T-
T-
Massless ground state WF is constant! (Not shown)
Generalize: add non-singular operators
• Adding regular operators gives similar eigenfunctions but shifts masses dramatically
• Dashed lines: EFs with just singular terms (from previous slide)
• Here: shift by constant WF of previously massless state
Generalize more: allow parton number violation, phase in “slowly’
Spectrum as function of parton-number violation parameter ϵ•No multi-particle states if ϵ<1
Without parton violation, no MPS
No understanding of how to filter out SPS
Results: Average parton-number as function of parton-number violation parameter
• Hope: SPS are purer in parton-number than MPS (<n>≈ integer)
• Expectation value of parton number in the eigenstates fluctuates a lot
• No SPS-MPS criterion emerges
Results: Convergence of Average Parton-Number with discretization parameter 1/K
• Or does it?!
• Hints of a convergence of <n> with K
• However: too costly!
A look at the bosonized theory• Describe (massless) theory in a more appropriate basis
currents (two quarks J ≈ ψ ψ) bosonization
• Why more appropriate?– Hamiltonian is a multi-quark operator but a two-current operator
• Kutasov-Schwimmer: all SPS come from only 2 sectors: |J..J> and |J…J ψ>
• Simple combinatorics corroborated by DLCQ state-counting yields reason for the fact (GHK
PRD 57 6420) that bosonic SPS do not form MPS: MPS have the form |J..J ψJJ ψ … J ψ>
≈|J..J ψ> |JJ ψ> | …> |J ψ>
Conclusions• Asymptotic theory is solved algebraically
• Better understanding of role of non-singular operators & parton-number violation– No PNV no MPS
• So far no efficient criterion to distinguish SPS from approximate MPS
• Evidence for double Regge trajectory of SPS
• Bosonic SPS do not form MPS
• Next: Use algebraic solution of asymptotic theory to exponentially improve numerical solution
Results: Bosonic Bosonized EFs (Zoom)
• Different basis, but still sinusoidal eigenfunctions
• Shown are 2,3,4 parton sectors
Light-Cone Quantization 10
2
1xxx • Use light-cone coordinates
• Hamiltonian approach: ψ(t) = H ψ(0)• Theory vacuum is physical vacuum - modulo zero modes (D. Robertson)
Results: Fermionic Bosonized EFs (necessarily All-Parton-Sectors calc.)
• Basis (Fock) states have different number of current (J) partons
• Combinations of these current states form the SP or MP eigenstates
Remarks on Finite N theory• Claim of Antonuccio/Pinsky: the spectrum
of the theory does not change with number of colors
• Debunked: probably a convergence problem
Conclusions
• Close, but no cigar (yet)
• Several interesting, but fairly minor results
• Fun project, also for undergrads with some understanding of QM and/or programming
Lingo• The theory is written down in a Fock basis
• The basis states will have several fields in them, representing the fundamental particles (quarks, gluons)
• Call those particles in a state “partons”
• The solution of the theory is a combination of Fock states, i.e. will have a “fuzzy” parton number
• Nevertheless, it will represent a single bound state, or a single- or multi-particle state
What’s the problem?
• Both numerical and asymptotic solutions are approximations
• Some solutions are multi-particle solutions, or at least have masses that are twice (thrice..) the mass of the lowest bound-state
• These are trivial solutions that should not be counted
So what’s the problem? Throw them out!
• These (exact) MPS can indeed be thrown out by hand, or by bosonizing the theory
• The real problem is the existence of approximate MPS: almost the same mass as the exact MPS, but not quite
• Q: Will they become exact MPS in the continuum limit?
• Q: Are they slightly bound states? (Rydberg like)
Lingo II• The theory has states with any number of partons, in
particular, states with odd and even numbers fermions & bosons
• The theory has another (T)-symmetry (flipping the matrix indices) under which the states are odd or even:
T |state> = ± |state>• One can form a current from two quarks, and formulate
the theory with currents. This is called bosonization. • How about an odd fermionic state in the bosonized theory?
Results: Asymptotic EFs without non-singular operators
• The basis consists of states with four (five, six) fermions
• Eigenfunctions are combinations of sinusoidal functions, e.g. ϕ4 (x1, x2, x3, x4)
• Note that the xi are momentum fractions
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