Towards finding the single- particle content of two- dimensional adjoint QCD Dr. Uwe Trittmann Otterbein University* OSAPS Spring Kent State University.

Slides:



Advertisements
Similar presentations
Matrix Representation
Advertisements

Summing planar diagrams
Lecture 7: Basis Functions & Fourier Series
Announcements 11/14 Today: 9.6, 9.8 Friday: 10A – 10E Monday: 10F – 10H 9.6 Only do differential cross-section See problem 7.7 to do most of the work for.
Integrals over Operators
Chiral freedom and the scale of weak interactions.
Introduction to Molecular Orbitals
Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.
Categorizing Approaches to the Cosmological Constant Problem
No friction. No air resistance. Perfect Spring Two normal modes. Coupled Pendulums Weak spring Time Dependent Two State Problem Copyright – Michael D.

Chiral freedom and the scale of weak interactions.
4 th Generation Leptons in Minimal Walking Technicolor Theory Matti Heikinheimo University of Jyväskylä.
Chiral freedom and the scale of weak interactions.
Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues
Large N c QCD Towards a Holographic Dual of David Mateos Perimeter Institute ECT, Trento, July 2004.
Planar diagrams in light-cone gauge hep-th/ M. Kruczenski Purdue University Based on:
Chiral freedom and the scale of weak interactions.
Quantum Mechanics from Classical Statistics. what is an atom ? quantum mechanics : isolated object quantum mechanics : isolated object quantum field theory.
Geometric characterization of nodal domains Y. Elon, C. Joas, S. Gnutzman and U. Smilansky Non-regular surfaces and random wave ensembles General scope.
The 2d gravity coupled to a dilaton field with the action This action ( CGHS ) arises in a low-energy asymptotic of string theory models and in certain.
States, operators and matrices Starting with the most basic form of the Schrödinger equation, and the wave function (  ): The state of a quantum mechanical.
Vibrational Spectroscopy
Integrability and Bethe Ansatz in the AdS/CFT correspondence Konstantin Zarembo (Uppsala U.) Nordic Network Meeting Helsinki, Thanks to: Niklas.
Monday, Apr. 2, 2007PHYS 5326, Spring 2007 Jae Yu 1 PHYS 5326 – Lecture #12, 13, 14 Monday, Apr. 2, 2007 Dr. Jae Yu 1.Local Gauge Invariance 2.U(1) Gauge.
Integrability in Superconformal Chern-Simons Theories Konstantin Zarembo Ecole Normale Supérieure “Symposium on Theoretical and Mathematical Physics”,
Chang-Kui Duan, Institute of Modern Physics, CUPT 1 Harmonic oscillator and coherent states Reading materials: 1.Chapter 7 of Shankar’s PQM.
Constraining theories with higher spin symmetry Juan Maldacena Institute for Advanced Study Based on: and by J. M. and A. Zhiboedov.
Wednesday, Apr. 23, 2003PHYS 5326, Spring 2003 Jae Yu 1 PHYS 5326 – Lecture #24 Wednesday, Apr. 23, 2003 Dr. Jae Yu Issues with SM picture Introduction.
Constraining theories with higher spin symmetry Juan Maldacena Institute for Advanced Study Based on & to appearhttp://arxiv.org/abs/
Matrix Cosmology Miao Li Institute of Theoretical Physics Chinese Academy of Science.
Gauge invariant Lagrangian for Massive bosonic higher spin field Hiroyuki Takata Tomsk state pedagogical university(ТГПУ) Tomsk, Russia Hep-th
Family Symmetry Solution to the SUSY Flavour and CP Problems Plan of talk: I.Family Symmetry II.Solving SUSY Flavour and CP Problems Work with and Michal.
1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments The variational method.
Bethe ansatz in String Theory Konstantin Zarembo (Uppsala U.) Integrable Models and Applications, Lyon,
On the ghost sector of OSFT Carlo Maccaferri SFT09, Moscow Collaborators: Loriano Bonora, Driba Tolla.
PHYS 773: Quantum Mechanics February 6th, 2012

Two-dimensional SYM theory with fundamental mass and Chern-Simons terms * Uwe Trittmann Otterbein College OSAPS Spring Meeting at ONU, Ada April 25, 2009.
Vibrational Motion Harmonic motion occurs when a particle experiences a restoring force that is proportional to its displacement. F=-kx Where k is the.
Second Quantization of Conserved Particles Electrons, 3He, 4He, etc. And of Non-Conserved Particles Phonons, Magnons, Rotons…
2. Time Independent Schrodinger Equation
Central potential problem and angular momentum What is a central potential? Separating the Angular and Radial wave equations Asymptotics of the radial.
Relating e+e- annihilation to high energy scattering at weak and strong coupling Yoshitaka Hatta (U. Tsukuba) JHEP 11 (2008) 057; arXiv: [hep-ph]
Hadrons from a hard wall AdS/QCD model Ulugbek Yakhshiev (Inha University & National University of Uzbekistan) Collaboration Hyun-Chul Kim (Inha University)
LORENTZ AND GAUGE INVARIANT SELF-LOCALIZED SOLUTION OF THE QED EQUATIONS I.D.Feranchuk and S.I.Feranchuk Belarusian University, Minsk 10 th International.
Quantum Mechanical Models for Near Extremal Black Holes
Dr. Uwe Trittmann Otterbein University*
Gauge/String Duality and Integrable Systems
Institut d’Astrophysique de Paris
Time Dependent Two State Problem
NGB and their parameters
Chapter 6 Angular Momentum.
Handout 9 : The Weak Interaction and V-A
Section VI - Weak Interactions
Quantum One.
Spin and Magnetic Moments
On a harmonic solution for two-dimensional adjoint QCD
The Harmonic Oscillator
Quantum Two.
Uwe Trittmann Otterbein University* OSAPS Fall Meeting 2018, Toledo
The symmetry of interactions
Time-Dependent Perturbation Theory
Adaptive Perturbation Theory: QM and Field Theory
Sense and nonsense in photon structure
Physics 319 Classical Mechanics
QM2 Concept test 3.1 Choose all of the following statements that are correct about bosons. (1) The spin of a boson is an integer. (2) The overall wavefunction.
American Physical Society
Quantum One.
Presentation transcript:

Towards finding the single- particle content of two- dimensional adjoint QCD Dr. Uwe Trittmann Otterbein University* OSAPS Spring Kent State University March 27, 2015 *Thanks to OSU for hospitality!

Adjoint QCD 2 compared to the fundamental QCD 2 (‘t Hooft model) QCD 2A 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: F 1 x F 1

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 QCD 2

The Hamiltonian of Adjoint QCD It has several parts. We can systematically omit some and see how well we are doing H full = H m + H ren + H PC,s + H PC,r + H PV + H finiteN (mass term, renormalization, parton #conserving (singular/regular), parton# violation, finite N) Here: First use H asymptotic = H ren + H PC,s then add H PC,r Later do perturbation theory with H PV as disturbance

Asymptotic Theory: H asympt =H ren +H PC,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 (x i 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: M 2 = 2g 2 N π 2 (n 1 +n n k ) ; n 1 >n 2 >..>n k 2Z

New: Algebraic Solution of the Asymptotic Theory New take on BCs: more natural to realize vanishing of WFs at x i =0 by New solution involves sinusoidal ansatz with correct amount of excitation numbers: n i ; i = 1…r-1 ϕ r (n 1,n 2,…,n r-1 ) = -

“Adjoint t’Hooft eqns” are tricky to solve due to cyclic permutations of momentum fractions x i being added with alternating signs But: Simply symmetrize ansatz under C : (x 1, x 2, x 3,…x r )  (x 2, x 3, …x r,x 1 ) Therefore: ϕ r,sym (n i ) ϕ r (n i ) is an eigenfunction of the asymptotic Hamiltonian with eigenvalue New: Algebraic Solution of the Asymptotic Theory (cont’d) ϕ 3,sym (x 1, x 2, x 3 ) = ϕ 3 (x 1,x 2,x 3 ) + ϕ 3 (x 2,x 3,x 1 ) + ϕ 3 (x 3,x 1,x 2 ) = ϕ 3 (n 1,n 2 ) + ϕ 3 (-n 2,n 1 -n 2 ) + ϕ 3 (n 2 -n 1,-n 1 ) ϕ 4,sym (x 1, x 2, x 3, x 4 ) = ϕ 4 (x 1,x 2,x 3,x 4 ) – ϕ 4 (x 2,x 3,x 4,x 1 ) + ϕ 4 (x 3,x 4,x 1,x 2 ) – ϕ 4 (x 4,x 1,x 2,x 3 )

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 : b ij  b ji Numerical and algebraic solutions are almost identical for r<4 Caveat: in higher parton sectors additional symmetrization is required 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 No multi-particle states if <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 ( ≈ 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 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 ) 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 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. ϕ 4 (x 1, x 2, x 3, x 4 ) Note that the x i 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=mc 2 ) Still looks sinusoidal