Lattice results on QCD-strings

N.D. Hari DassInstitute of Mathematical Sciences

Chennai

In collaboration with Dr Pushan Majumdar, U. Muenster

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• Networking is through 3D torus SCI (Dolphinics) (6x6x4)• Network attached storage 1.5 TB• Sustained Node to Node Bandwidth: 318 MB/s• Bandwidth between processors on same node: 864 MB/s• Latency: 3.8 s between different nodes and 0.7 s on same

node• HPL performance: 1.002 Teraflop sustained (Peak 1382)• Scaling on MILC codes: ks_imp_dyn1 (75%-80%) ; on

pure_gauge (85%)

Understanding nuclear forces

• The challenge in the beginning was to understand the structure and stability of atomic nucleus.

• Initially it was thought that protons, neutrons and pions were the elementary constituents and that the exchange of pions between protons and neutrons gave rise to the attractive nuclear forces.

Proliferation of particles.

• But with higher and higher energy collisions more and more particles other than the protons, neutrons and pions were produced.

• The simple picture of protons, neutrons and pions as the elementary constituents of nuclear matter was not viable.

• Two proposals came to be made almost concurrently to face the situation.

The hadronic string

• Chew and Frautschi observed that all the new particles produced lay on parallel lines (Regge trajectories) when their masses(or squared masses) were plotted against their angular momenta.

• Veneziano proposed a simple formula for the scattering amplitudes for the new particles.

• Nambu, Nielsen and Susskind made the astonishing proposal that both the Veneziano formula and the Regge trajectories could be understood if the particles were states of excitation of a relativistic string.

Problems with hadronic strings

• The full consistency of the hadronic string model required space-time to have 10 dimensions.

• Closer inspection of the spectrum of these strings revealed states that could not be hadronic states e.g massless spin-2 states.

• Such states would violate the so called Froissart bound for scattering amplitudes.

• Subsequently it was proposed not to use string theories to explain hadronic physics but as a means of unifying all interactions including gravity.

Quark model

• At around the same time Gellmann and Zweig independently proposed the quark model.

• According to this protons are constituted by 3 quarks, 2 u-quarks of +2/3 electric charge and one d-quark of -1/3 charge. Likewise neutron constituents were 2 d-quarks and one u-quark.

• Pions were made up of a quark and anti-quark pair.

• The quarks were taken to be spin ½ particles.

Quark model…..

• To overcome problems with Pauli exclusion principle, Nambu proposed that each quark comes with 3 different colours .

• This model was extremely succesful as book-keeping for the myriad of particles and could even explain some features like the observed magnetic moments.

• Attention then turned to constructing a theory for the interaction between quarks.

• Gellmann, Minkowski and Fritzsch proposed QCD with vector interactions motivated by the observed chiral symmetry among hadrons.

What is QCD? – More technically…

• QCD is a non-Abelian Gauge Theory.• The gauge group is SU(3).• The Quarks carry the fundamental representation 3.• The Gluons, which transmit the forces between quarks,

carry the adjoint representation 8.• 3 has triality, 8 has no triality.• The theory in its nonperturbative domain has defied

analytical approach despite the best efforts for more than 25 years.

• It is an outstanding problem of theoretical physics.

Quarks and Gluons of SU(3) QCD

Quark Confinement

• Quarks are not liberated even in very high energy collisions!

• The theory of Quarks must be such that they can never be free!

The dual superconductor idea

Lattice

Lattice QCD

Wilson Loop

Numerical Evidence For Flux Tubes

Our Simulations

• We measure the qqbar-potential of d=4 su(3) pure gauge theory extremely accurately on 24^3x32 and 32^4 lattices at = 5.7

• Lattice spacing a = 1/6 fm so temporal extent is nearly 5.3 fm while spatial extent is 4fm^3.

• We use Polyakov Loop Correlation Function to measure the potential.

• We use the Luscher-Weisz Multilevel algorithm.• We also use the analytical multihit method to

achieve speedup(60%).

Type of accuracies needed

• The polyakov loop correlator is a stochastic variable of nearly unit magnitude.

• At a separation of r=8 the average value of this is around 10 -26

• It roughly falls by two orders of magnitude with every increase in r by unity.

• This too needs to be measured at fraction of a percent if string-like behaviour is to be extracted.

• Without the multihit and multilevel methods this would be a hopeless task.

Polyakov Loop Correlators

Luscher-Weisz Multilevel Algorithm

Some string actions and potentials

Results of Luscher and Weisz

Initial Conclusions of Luscher and Weisz

• The coefficient of the 1/r term in the qqbar potential agrees to within 15% from that of the free bosonic string theory in the distance range 0.5-1.0 fm.

• They argued that the discrepancy could be due to boundary terms and other interaction terms.

• They found that a boundary term with b=.04 could accommadate the data well.

• They found it surprising that even at 0.5fm there was evidence for string behaviour.

The Work of Kuti et al

• Kuti et al have undertaken very detailed studies of the spectrum of string excitations.

• They use the extended Wilson loops for this purpose.

• They use 24^2x30x60 lattice with as =0.2 fm and at = .04 fm so that their longest time extent is 2.4 fm.

Wilson Loops for Spectrum Studies

Results of Kuti et al

Spectrum according to Kuti et al

Summary of Results of Kuti et al

• The ordering of the spectrum does not agree with that of the bosonic string even upto nearly 3fm.

• There is a systematic overshooting of the string results as one goes to larger and larger distances.

• The data fits the free bosonic string ground state energy very well in the range 0.5-1.0 fm.

Our Simulation Results

Force vs Distance

Summary of Our Results

• We do not see any obvious convergence to either Nambu-Goto or free bosonic string to even upto 0.75 fm.

• As r approaches 1 fm we see clear convergence to the Nambu-Goto potential.

• We see very clear discrepancy with the free bosonic string expectations. In fact a fit to this behaviour produces very poor 2

Luscher-weisz Revisited

• How then do we reconcile our conclusions with those of Luscher-Weisz who had suggested onset of string behaviour as early as 0.5 fm by ascribing the differences over free string behaviour to boundary terms?

• In 2004 they showed that Open String – Closed String duality forbids boundary terms of the type considered before.

• It is better to interpret the results as string behaviour not setting so soon.

But Nambu-Goto string is inconsistent in d=4

• Though the data picks the Nambu-Goto potential so accurately, there is a serious problem!

• This string theory can be consistently quantised only in d=26

• The Luscher term is a consequence of such quantisations, so why should one place too much weight on its occurrence?

Effective string theories

• Polchinski and Strominger in 1992 set out to formulate effective string theories and their quantisations.

• This is similar in spirit to effective theories like chiral models for the description of pions.

• Non-polynomial, non-renormalisable….• Designed so that quantisation preserves Lorentz

Invariance.• Leading correction to the linear potential is again exactly

the old Luscher term.• What are the corrections and how well does data fit

them? Tachyons?

Are string and gauge theories dual to each

• Polyakov and others believe that gauge and string descriptions are dual to each other.

• This is also the content of Maldacena’s AdS/CFT correspondence.

• This is expected to hold even for non-supersymmetric theories like QCD.

• Satchi Naik (HRI) showed how all these theories give the conventional Luscher term.

• The d=3 case is more straightforward in this picture.

What Next?

• Push the simulations to larger and larger distances.• Very time consuming! With each r = 1 the simulation time

increases by 2 and already at r=7 it takes a day for 3 meas and for good statistics one needs about 500 meas!

• Probe effective string theories as well as AdS/CFT correspondence deeper.

• Extrinsic curvature strings.• Investigate the Center issue: investigate adjoint strings (large

memory and large simulation time).• Investigate Center-less groups like SO(3), G2 etc• Investigate z(3) gauge theories.• Investigate baryons and effective string coupling. Construct an

effective QCD String Theory. • The eventual goal is solving the Quark Confinement Problem