Concepts of the Standard Model : r enormalisation

Concepts of the Standard Model: renormalisation

FK8022, Lecture 2

Core text:Quarks and leptons: an introductory course in modern physics, Halzen and MartinFurther reading:Basics of QCD perturbation theory, D.E. Soper (hep-ph/9702203)

Lecture outline

• Infinities arising from loops– Regularisation– Reparameterisation of charge – Renormalisation

finite charge, amplitude Renormalisation scaleRunning of couplings

A simple QED process

23 1 4 22

2 21 3 0.

Consider scattering. Apply Feynman rules to obtain amplitude:

where and

e

ge u p u p u p u p

Q

q p p Q q

M

We are using perturbation theory so the above Feynman diagram should give the approximately correct amplitude withhigher order processes changing the results slightly.

e e

p1

p2 p4

p3

q

2

2 23 1 4 22

2 2

2

22

2

1 ( )

1 ( )

( )

6 (1 ) ln 13

Simplify:

integral over internal loop momenta to divergent contribution of loop

m

ge u p u p I Q u p u p

Q

e I Q B

I Q

dz QI Q z z zz m

M

M

1

0

2

1

4 mass,

z dz

em e

Adding a higher order process

+e e

e e

e e

2

2

2 2

1 22

20

2

2

2

22

2

6 (1 ) ln 1 13

ln

ln ln3

Regularise divergent integral:

Can send to later. Solve first and second integrals:

m

M

m m

dz QI Q z z z z dzz m

dz dz Mz z m

M

M QI Qm

2 2

2 2

2 2

ln3

Mm Q

Q m

Renormalised charge

22

2

2 2 22 2 2 2

2 2 2

2

-4

-4 1 ( ) 4 1 ln3

1 ln 1 ln3 12

-4

,

Redefine coupling, charge:

- finite!

renormalised charge/co

r r

r

r r

B

MI q B BQ

M e MQ e Q eQ Q

Q B

e

M

M

M =

2.,

upling i.e. measured in lab, varies with unobservable "bare" charge/coupling of electron

Finite finite if is infinitesimal and cancels divergence from .r

Qe

e e M

M

Bare and renormalised charges

"bare" charge of electron whenstripped of loops : infinitesimal. The charge which would be seenfor an interaction at zero distance or infinite energy unobservable!

e

e

re2.

.

renormalised charge which is

measured in lab and varies with Divergent loops "absorbed" into the

definition of

r

r

e

Q

e

We were using the wrong charge.

Defining the coupling at a scale

22

2

2

222

2 2

2

2 22

22

2

-4 1 ( ) 4 1 ln3

1 ln3

1 ln3

4 1 ln 43

1 ln3

Define at scale

Have freedom to define at a

r

rr

rr

r

MI Q B BQ

Q

QMQQ M

Q

Q M B QQM

Q

M

M

2

22 22

2 22

2

1 ln 4 1 ln3 3

1 ln3

ny arbitrary scale

Need to rewrite in terms of finite quantities

rr

M M BQM

M

2 2

22

2

2 2 22

22

2

2 2 22

2

4 1 ln3

1 ln3

ln3

1 ln3

ln3

higher orders

higher orders

Estimate t

r

r rr

rr

M BQM

MM

M

M

2

2 22 22

2 2

2 22

2

( ) :

-4 1 ln 1 ln3 3

- 4 1 ln3

o

Finite amplitude with finite terms but depending on an arbitrary number!

r

r rr

rr

M M BQ

Q B

M

M

O

22 22

2 2

2 22 22

2 2

4 1 ln - 4 1 ln3 3

-4 1 ln ln3 3

(A) Photon exchange - no loops

(B) Photon exchange - one loop at m

rr

r rr

M QB BQ

M M BQ

M

2 2

2 2

2 2 2 2

-

,

omentum transfer =

(C) Photon exchange - one loop at momentum transfer =

Amplitude

renormalisation scale

scale at which "infinities are removed"+ charge/coupling set: r r

q

A B C

e

Renormalisation scale

[ ]- - + higher orders

2At Q 2At Q 2 2At Q

(A) (B) (C)

2 2

22 2

2

22 2

1 ( ) , -4 1 ( )

-4 1 ( ) ( ) ..

11 ...1

1-41 1

One loop.

Infinite number of loops :

Taylor expansion: +

,

r

r

I Q I Q B

I Q I Q B

X XX

Q BI Q I Q

M

M

M

Higher orders

}

Scale dependence of amplitude

renormalisation scale. It is arbitrary!No observable can depend on an arbitrary parameter ! Any scale will eventually give the right answer if all orders are calculated. Large variation at lower or

2 2.,

ders.

For QED the optimal scale is known: For QCD finding the right scale is an open question.

Q

r(Q2)2 20 Q

2 21 Q 2 21 2 2 22 3

Order of calculation

e scatteringSingle photon exchange

22

22

2

22 2

2 2

2 2

2 22 2

1

1 ln3

11 11 ln ln

3 3

1 1 1 1 1 1ln ln3 3

At scale :

;

r

r

r

r r

QI Q

QMQ

c cM Mc c

M MQ cQ c

Running coupling

2 2 2

2 2 22 2

22

2 2 2

22 2

1 1 1 1ln ln ln3 3

11 1 ln 1 ln3 3

We can't predict from first principles but we can predict

it's value at one momen

r r

rr

r

r

r

M M QQ c cQ c

cQ

Q c Qcc c

2 2 !tum scale if we know it another value

The coupling changes with momentum transfer scale.

The above is an approximation. Loops from other particles, eg muons,will also play some role.

q c

Consistency check

2

2 22

2

2 2 2

2 22 2

2

-4 1 ln3

-4

1 ln3

Consider scattering.

At renormalisation scale

At "correct" QED scale

Amplitude must be independent of scale.

rr

r

rr r

e

Q B

Q B Q

QQ

M

M

2 2 2

2 222 2

2

1 ln3

1 ln3

Check consistency with formula for running:

The "running" of ensures that an observable does not depend on the choice of

r

r rr r

r

r

QQQ

renormalisation scale.

Divergence in amplitude from

higher order loop

Make finite by redefining coupling/charge and remove infinity

For a complete expansion the amplitude is independent of r

Renormalisation

Finite expression for running of coupling/charge with r

Coupling/charge depends on an arbitrary kinematic scale r for a given interaction

Renormalisation works: electromagnetic coupling running

Interaction distance

(momentum transfer)2 /GeV2

, , , , , , , ,e u d s c b t

(5)

2

2

(5) 2

01

0.031497686

0.00007

0.0275 0.0

emem

e top had

e Z

top Z

had Z

M

M

M

Vacuum polarisation contributions from all leptons and quarks.

Calculable:

Not calculable (pQCD not valid):03 e e - from data.

Large uncertainty - affects precision measurements.

Tiny shift

Physical interpretation of the running coupling

+

e-

The "vacuum" consists of virtual particles fluctuating into and out of existence.An electron is surrounded by virtual particles which act to shield the charge asin a polarised dielectric.

e-e-

e- e-

e- e-

e+,e-

e+,e-

e+,e-

Distance scale

Coupling

Renormalisation of other quantities• Different loop diagrams cause other physical

quantities to be renormalised.• Eg mass

• Masses also run:

What about the strong force ?

q q

q q

q q

q q

q q

q qq q

q q

q,q

Similar story as for electromagnetism except that gluonscan self-interact (they carry colour - the photon carries no charge!)This turns out to be critical.....

q,q

q,q + other higher order diagrams

Asymptotic freedom

s

Momentum transfer /GeV

Interaction distance

12

2 2 22

33 21 ln

6

0.118 0.002

number of quark flavours

Varies strongly with momentum!!

fs s s

f

s Z

N QQ

N

M

Electromagnetic force (QED)Electric charge screened: distance > Compton wavelength

Strong force (Quantum chromodynamics: QCD)Effective colour charge grows at larger distances.Distances < fm quasi

c

-free quarks Asymptotic freedom! Nobel prize (2004) for Gross, Politzer and Wilczek.

Renormalisation scale setting

-

22

2

2

Process:

4 1

centre-of-mass energy ; 3 (no. of colours)

sum over squared charges of quarks which can be produced.

= contribution from pQCD proces

c i QCDi

c

ii

QCD

e e hadrons

N es

s N

e

ses

22

222 2

1.4092 1.9167 ln

12.805 7.18179ln 3.674 ln ...

s sQCD

s

s

s s

2

ln

Coefficients and depend on renormalisation scale.

Scales usually chosen to have a value close to the kinematic hard

scattering scale (eg ). This avoids large logs and

ensures faster

s

ss

convergence in a perturbative series.

Renormalisation scale setting

Scale variation

1

12

2 2 22

22 2 2

2

21 1

Change in scale from ' for leading order calculation:

33 2 '' 1 ln 6

33 2 ' ln 6

33 2

6

s

fs s s

fs s

fs s

c

N

N

Nc c

M

M 2

2 22

'ln

scale variations higher orders in . scale variations offer some sensitivity to missing higher orders.

Typical choices , 22

(More in lecture 13 - simulation technique

s

s

s).

Summary and discussion (1)

General:When including loops the Feynman diagram machinery can break down owing to divergences related to unknown physicsat high energy (short distance). The divergences can be removed by absorbing th

2

2

.

em into thedefinition of a renormalised physical quantity.

Specific for our example process:

The divergent term ln emerges and is cancelled by the

"bare" charge This gives the renormalise

Mm

e

d coupling .

re

Summary and discussion (2)

General:The physical quantity receives a finite correction in addition to the divergence removal.

Specific for our example process:The renormalised coupling/charge varies with the momentum transfer/scale.

Summary and discussion (3)

Renormalisation originally seen as a mysterious stop-gap measure.It is now better understood Nobel prizes (eg Wilson, 't Hooft).

SM integrals with shouldn't necessarily be problem-free !We know th

p

e SM must break down eg, at .

Short distance stuff beyond the SM *should* be absorbed away. Successful theories do not couple short and long distance scales.Eg planetary motion with Newton's l

PlanckM

aws does not need quarks. QCD generates visible mass : absorbed in the planets' masses.