1. How to extract the single-particle content of two-dimensional adjoint QCD
Dr. Uwe Trittmann
Otterbein University*
OSAPS Spring University of Dayton
April 9, 2016
*Thanks to OSU for hospitality!

2. 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

3. The Problem: all known approaches are cluttered 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

4. Universality: Same Calculation, different parameters = different theory
‘t Hooft Model
(prev slide)
Adjoint QCD2
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

5. 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)
Last Year: (Phys.Rev. D92 (2015) no.8, )
First use Hasymptotic = Hren + HPC,s
then add HPC,r
Later do perturbation theory with HPV as disturbance
Didn’t lead anywhere, so ignore

6. 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

7. “Pre-2015” 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

8. PRD92 (2015): 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 number of excitation numbers: ni ; i = 1…r-1
ϕr(n1 ,n2 ,…,nr-1) = =±

9. PRD92: 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)
“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
3 parton WF characterized by 2 excitation numbers

10. 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 (full!) and algebraic (asymptotic!) 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)

11. Higher Parton Sectors: New Thoughts
For r>3 the ansatz is overcomplete on the physical Hilbert space
Need to symmetrize
Subset of correctly
symmetrized states is known:
with mass (squared):
Caveat: Have already shown that there must be other states!
Cyclic symmetry and fixed total momentum means that the integration volume (phys. HS) is greatly reduced:

12. Which Symmetry?
The bound-state masses only depend on differences of excitation numbers
r=4: M2=|n|+|n-m|+|m-l|+|l|
Degenerate states: M2(n,m,l)=M2(n,m±2,l)
“Hidden” symmetry of the Hamiltonian
We can find linear combinations characterized by an additional quantum number replacing a “naïve” excitation number, say m

13. Outlook for Single-Particle spectrum
Complete symmetrization may lead to
Full algebraic solution
Construction of a ONB
First case: Done, treat omitted operators perturbatively
Second case: use ONB to solve full theory numerically using basis state approximation
Get exponential convergence!

14. Conclusions
Asymptotic theory was solved algebraically in low parton sectors
Better understanding of role of non-singular operators & parton-number violation
No Parton number violation no MPS
Higher parton sectors: Need to understand symmetries of Hamiltonian
Next: Use fully symmetrized solution of asymptotic theory to exponentially improve numerical solution

15. Thanks for your attention!
Questions?

16. Not used

17. 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

18. 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

19. 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 

20. 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!

21. 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 ) that bosonic SPS do not form MPS: MPS have the form |J..J ψJJ ψ … J ψ>
≈|J..J ψ> |JJ ψ> | …> |J ψ>

22. Mass of MPS in DLCQ
mass at resolution n mass at resolution K-n mass at resolution K

23. Results: Bosonic Bosonized EFs (Zoom)
Different basis, but still sinusoidal eigenfunctions
Shown are 2,3,4 parton sectors

24. Light-Cone Quantization
Use light-cone coordinates
Hamiltonian approach: ψ(t) = H ψ(0)
Theory vacuum is physical vacuum
- modulo zero modes (D. Robertson)

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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)

31. 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? 

32. 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

33. General (All-Parton-Sector) Solution
The wavefunctions are combinations of states with a different number of fermions in them
Necessary since Hamiltonian does not preserve parton number (E=mc2)
Still looks sinusoidal