NRQCD as an Effective Field Theory

Ron HorganDAMTP, University of Cambridge

Royal Society Meeting Chicheley

Outline

Wilson flow and idea of effective field theory.

Lattice action and NRQCD as an effective field theory

Radiative improvement of NRQCD using background field approach.

Some results.

How successful are our calculations in Quantum Field Theory?

What sort of questions do we ask?

I believe in QED. I believe that, in some sense, it is true.

QED is an effective theory by which I mean that at low energies I can calculate sufficiently accurately to answer any question for all practical purposes.

Certainly it is part of the Standard Model which, itself, is incomplete:

the origin of parameters such as quark masses and the CKM matrix ( ~ 20); the fundamental nature of the Higgs particle: triviality, naturalness, fine tuning. Where is gravity?

QED is successful because we can do perturbation theory

We are good at doing complicated Gaussian integrals

Current QED calculation (Aoyama et al.) is to order

THEORY:

EXPERIMENT (muon g-2 Collaboration):

Toichiro Kinoshita has spent a lifetime computing theanomalous magnetic moment of the muon.

I believe in the Standard Model and particularly in QCD

Different to QED.

is the dimensionless coupling which runs with the typicalenergy scale, , of the experimental probe.

QED:

Diverges for (enormously) large : Landau ghost.

Probably invalid if taken too literally but indication of trouble and that QED is an effective field theory only.

QCD:

(as long a number quark flavours not too large)

Low energy QED is solvable at some level.

Low energy is problematic and non-perturbative. QCD

High energy is under control perturbatively but only up to non-perturbative contributions. However, predictions do factorize

QCD

Need a method to calculate the non-perturbative quantities.

Renormalization Group ideas.How to think about solving Quantum Field Theory non-perturbatively

is the short-distance or UV cut-off

are operators of increasing dimension in energy or momentum units.

In quantum field theory the renormalized mass is defined by

We need to take the limit keeping fixed.

This means that we must tune in this case so that

This is the signature for a continuous phase transition inthe lattice theory. The correlation length diverges in lattice units

As we take the limit

(1) We must tune the couplings so that

This means that we must deal with a very large latticeindeed. It must be much bigger than sites. Thiscosts money! Need big computers.

(2) In QFT we impose conditions on the measurements from experiment. This means that as the couplings in the action must change with so that

the result of measuring a physical quantity does not change

(3) The outcome is that flows with to keep the physics invariant according to

The important features are the fixed points of

On dimensional grounds there are only a few repulsive directionscorresponding to relevant couplings. All other directions and couplingsare termed irrelevant.

As increases (think of like a kind of time):

In QCD there is only one relevant coupling: . It obeys

A fixed point at where also and hence in physical units .

This is the manifestation of asymptotic freedom which says that at small length scales (here given by lattice spacing ) the coupling becomes small.

Really want to be very small but hope that it is small enough that effects of the lattice – lattice artifacts – do not contaminate the answers to questions.

The check is to compute dimensionless ratios of observable masses etc. These must be independent of -- they must scale.But usually

One option is to choose a smaller value for so that we get a larger but that needs a bigger lattice and we run out of computer power.

This means adding irrelevant operators whose job is to cancel off the artifacts; these are counter-terms which need not be of the same form as operators already present. This means calculating using perturbation theory or, in some cases, non-perturbativemethods. This is a hard job.

An alternative to decreasing (i.e., decreasing the spatial cut-off ) is to improve the definition of the action on the lattice.

All theories on a given trajectory predict the same physics

NRQCD and Radiative Improvement

Evolve heavy quark Green’s function with kernel:

where

• At tree level

• Radiatively improve to 1-loop using Background Field Method

• Also include certain four-fermion operators in NRQCD action:

• NRQCD is an effective theory containing (irrelevant) operators with D > 4 which, at tree level, can be restricted to be gauge-covariant.

• Fit at non-zero a to

• Vital to use formulation where no non-covariant operators are generated by radiative processes.

• Gauge invariance is retained by the method of background field gauge, and ensures gauge invariance of the effective action.

• All counter-terms are FINITE in BFG => can compute ALL matching, both continuum relativistic and non-relativistic, using lattice regularization: QED-like Ward Identities.

• Derive 1PI gauge-invariant effective potential. Match (on-shell) S-matrix.

• Implemented in HiPPY and HPsrc for automated lattice perturbation theory.

To use an abuse of notation due to Weinberg we write

• Only meaningful in graphical expansion.

• Expand RHS and keep only terms in quadratic or higher in .

• At one-loop need only quadratic terms.

• Cannot guarantee that can be expanded only on gauge-covariant operator basis.

Gauge invariance implies the Ward Identities

In BFG:

• 1PI vertex functions are finite.

• There is an explicit extra symmetry which means that the effective action is expansible on a basis only of gauge invariant operators.

The background field B defines the function for gauge fixing – it is a gaugeparameter:

Derive general effective potential .

• is the required gauge-covariant effective action

• A theorem by Abbott states that all S-matrix elements, which include both 1PI and 1PR graphs can be built using

Important results:

On the lattice all results go through. Write the link as the ordered product

• The Feynman rules are modified because the ordering matters.• Code the ghost and gauge fixing terms by hand.

Construct effective action (1PI diagrams) using action (Luescher-Weisz)

HiPPy PYTHON and HPsrc FORTRAN codes compute the and NRQCD vertexfunctions and HPsrc builds the graphs with automatic differentiation.

Matching process:

In particular, evaluate the spin-dependent diagrams vital for accurateevaluation of hyperfine structure:

Must also improve operators that are written in terms of effective fields:

• Currents for decays

• Wilson operators for mixing

Example:

From continuumInfraRed log

The hard part

Already , so improvement to spin-dependent NRQCD operators vital.

Comparison of unimproved withimproved from 2013.

Bottomonium system:

Hyperfine Splittings

Set 1Set 2Set 4Set 5Set 7

B-meson system:

Decay Constants

B-meson decay constants,

First LQCD results for with physical light quarks:

LQCD predicts:

Experiment within 1-s.d. of lattice . Error mainly experimental, but also some uncertainty in .

Using world-average, HPQCD, results for find

crucial to decay, and hitherto major source of error:

Second error, from , now competitive

Decay constantSummary plot

b-quark mass

• known to 3-loop order

• is the energy of the meson at rest using NRQCD on the lattice

• is computed fully to 2-loops in perturbation theory as follows:

1. Measure on high- quenched gluon configurations using heavy quark propagator in Landau gauge with t’Hooft twisted boundary conditions.

2. Fit to 3rd-order series in and extract quenched 2-loop coefficient.

3. Compute 2-loop contribution using automated perturbation theory for b-quark self-energy at p = 0.

Similar theory using meson instead.

• Fit to consistent with known 1-loop automated pert. th. result

• Include 3-loop quenched coefficient in

• Error dominated by 3-loop contribution.

for two lattice spacings using both

• Most accurate to use and correlator method.

• Applicable for HISQ and NRQCD valence quarks.

• For HISQ valence need extrapolation in some cases from

Since is not renormalized.

• RHS from 3-loop calculation of Chetyrkin et al.

• LHS from LQCD.

• Use moments .

• Fit to extract .

For this method gives

Weighted lattice average for measurementson configs with 3,4 sea quarks and then runto :

mixing

First LQCD calculation of mixing parameters with physical lightquark masses at three lattice spacings: MILC HISQ sets 3,6,8

Compute matrix elements of effective 4-quark operators derivedfrom box diagrams using LQCD:

• needed for oscillations• All three appear in width difference

• Use radiatively improved NRQCD

Bag parameters defined by

Similarly for with , respectively

Operators: NRQCD b-quark and HISQ light quark matched to continuum:

Results preliminary: more accurate results at physical point imminent.

Using expt., current HPQCD analysis gives:

(PDG: )

(PDG: )

New Calculation

• spin-independent scattering.

• Need ‘tHooft gauge-twisted boundary conditions as IR regulator.

• Use BFG and BF gluons in 1PI exchange diagram. Important

• Match S-matrix elements.

+

Coulomb exchange

moving NRQCD

The Future: NRQCD and HISQ

Selected quantities:

• NRQCD: Have radiatively improved coefficents and operators.

• HISQ: Fully relativistic and applicable for .

• Next generation configs: physical sea quarks; incorporate QED effects.

• Improve QCD parameters: , quark masses and hadronic matrix elements.• See 1404.0319 for relevance of accurate , h=c,b, to higgs physics and future collider programmes.

C. Davies, J. Koponen, B. Chakraborty, B. Colquhoun, G. Donald, B. Galloway (Glasgow)G.P. Lepage (Cornell) G. von Hippel (Mainz)C. Monahan (William and Mary)A. Hart (Edinburgh)C. McNeile (Plymouth)RRH, R. Dowdall, T. Hammant, A. Lee (Cambridge)J. Shigemitsu (Ohio State)K. Hornbostel (Southern Methodist Univ.)H. Trottier (Simon Fraser)E. Follana (Zaragoza)E. Gamiz (CAFPE, Granada)

HPQCD: recent past and present