Towards finding the singleparticle content of twodimensional adjoint QCD
Dr. Uwe Trittmann
Otterbein 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
NPB 587(2000)
SPS with relative momentum
PRD 66 (2002)
• Two types: exact and
approximate multi-particle
states (MPS)
• Exact MPS can be projected out
(bosonization) or thrown out
(masses predictable)
Group of exact MPS: F1 x F1
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!
Group of approx MPS
• 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
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
r disallowed,
=3
the asymptotic theory splits into
decoupled sectors of fixed parton number
• Wavefunctions are determined by ‘t Hooftr =4
like integral equations (xi are momentum fractions)
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) =
New: Algebraic Solution of the Asymptotic
Theory (cont’d)
ϕ3,sym(x1, x2,t’Hooft
x3) = ϕ3eqns”
(x1,x2,xare
ϕ3(x2,x3to
,x1solve
) + ϕ3(xdue
3) + tricky
3,x1,x
• “Adjoint
to2)
= ϕ3(n1,n2) + ϕ3(-n2,n1-n2) + ϕ3(n2-n1,-n1)
cyclic permutations of momentum fractions xi
being
added
with
alternating
signs
ϕ4,sym(x1, x2, x3, x4) = ϕ4(x1,x2,x3,x4) – ϕ4(x2,x3,x4,x1)
• But: Simply symmetrize
ansatz
+ ϕ4(x3,x4,x
ϕ4(x4,x1,x2,x3)
1,x2) – 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
It’s as simple as that – and it works!
• All follows from the twoparton (“single-particle”)
solution
T-
• 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
TT+
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 partonnumber 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 PartonNumber 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
Thanks for your attention!
• Questions?
Not used
Mass of MPS in DLCQ
mass at resolution n
mass at resolution K
mass at resolution K-n
Results: Bosonic Bosonized EFs (Zoom)
• Different basis,
but still
sinusoidal
eigenfunctions
• Shown are
2,3,4 parton
sectors
Light-Cone Quantization
• Use light-cone coordinates
• Hamiltonian approach:

1 0 1
x 
x x
2

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
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