NRQCD as an Effective Field Theory - PowerPoint Presentation

Slide1

NRQCD as an Effective Field Theory

Ron HorganDAMTP, University of Cambridge

Royal Society Meeting

Chicheley

Slide2

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.Slide3

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?Slide4

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 (m

uon

g-2 Collaboration):

Toichiro

Kinoshita has spent a lifetime computing the

anomalous magnetic moment

of the

muon

. Slide5

I believe in the Standard Model and particularly in Q

CDDifferent to QED

.

is the dimensionless coupling which runs with the typical

energy 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.Slide6

QC

D:

(as long a number quark

flavours not too large)

Low energy QED is solvable at some level.

Low energy is problematic and non-perturbative

. QC

D

High energy

is

under control

perturbatively

but

only

up to non-

perturbative

contributions. However, predictions

do

factorize

Q

C

D

Need a method to calculate the non-

perturbative

quantities.Slide7

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.Slide8

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 in

the lattice theory.

The correlation length diverges in lattice unitsSlide9

As we take the limit

(1) We must tune the couplings so that

This means that we must deal with a very large lattice

indeed. It must be much bigger than sites. This

costs money! Need big computers.

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

The outcome is that flows with to keep the physics

invariant according toSlide10

The important features are the fixed points of

On dimensional grounds there are only a few

repulsive

directions

corresponding to relevant couplings. All other directions and couplingsare termed

irrelevant.As increases (think of like a kind of

time):Slide11

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.

R

eally 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. Slide12

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

s

ame form as operators already present. This means calculating

using perturbation theory or, in some cases, non-

perturbative

methods. 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. Slide13

All theories on a given trajectory predict the same physicsSlide14

NRQCD and

Radiative ImprovementEvolve 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:Slide15

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.

Slide16

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.Slide17

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 gauge

p

arameter:

Derive general effective potential . Slide18

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 vertex

f

unctions and

HPsrc

builds the graphs with automatic differentiation. Slide19

Matching process:Slide20

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:

C

urrents for decaysWilson operators for mixing Slide21

Example:

From

continuum

InfraRed

log

The hard partSlide22

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

Comparison of unimproved with

improved from 2013.

Bottomonium

system:

Hyperfine

SplittingsSlide23

Set 1

Set 2

Set 4

Set 5

Set 7

B-meson system:Slide24

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

a

lso some uncertainty in . Slide25

Using world-average, HPQCD, results for find

crucial to decay, and hitherto major source of error:

Second error, from

, now competitive

Decay constant

Summary plotSlide26

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:

Measure on

high-

quenched gluon configurations using

heavy quark propagator in Landau gauge with

t’Hooft

twisted

boundary conditions.

2. Fit to 3

rd

-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.Slide27

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 Slide28

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 measurements

o

n

configs

with 3,4 sea quarks

a

nd then run

to : Slide29

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 derived

from 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 Slide30

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: )Slide31

New Calculation

spin-independent scattering.

Need ‘

tHooft gauge-twisted boundary conditions as IR regulator.

Use BFG and BF gluons in 1PI exchange diagram. ImportantMatch S-matrix

elements.

+

Coulomb exchangeSlide32

moving NRQCDSlide33

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

.Slide34

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)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