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

Thanks for your attention!

• Questions?

Not used

Mass of MPS in DLCQ

mass at resolution n mass at resolution K-n

mass at resolution K

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

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