Introduction to Particle Physics

Risto Orava Spring 2016

strong interactions

2

the standard model - strong interactions

•  properties of the Strong Interaction •  colour •  the basic process •  comparison between QED and QCD •  quarks and hadrons •  the OZI rule, hadronization and jets

3

strong interaction - QCD

•  properties of the strong interaction –  exchange of gluons –  conserves all quantities

–  flavour – parity –  charge ⊗ parity

•  does not involve leptons (no colour) •  can only produce quark-antiquark

pairs

4

the strong interaction - examples

+

−

−

+−

++→+

+→Λ

Λ+→+

+∑→+

π

π

π

π

nppp

p

Kp

nK

0

0

5

colour

•  each quark appears in three colours: red, green and blue. •  gluon has two colour indices; there are eight gluons (23) •  a gluon couples to itself (three gluon vertex) •  each observed hadron is colourless; i.e. a colour singlet combination of quark-antiquark pairs (mesons) or three quarks (baryons) •  A colourless (white) combination can be formed by: (1) Equal mix of red, green and blue (2) Equal mix of anti-red, anti-green and anti-blue (3) Equal mix of color and its complement

BGRBBGGRR ,,

RGB

6

colour – the charge of the strong interaction

examples: proton = antiproton = π-meson = Note: the wavefunctions of these baryons and a meson have to be properly symmetrized and normalized Combination is then asymmetric under interchange of a pair of colour labels as is required by the Fermi statistics of the quarks.

RGBRGB

BBGGRR ++

RGB

7

how to measure the electron charge? - a long distance probe

-

+

-

+ -

+ -

+

-

+ -

+ R

test charge

electron is surrounded by the electron-positron pairs – the electron charge is screened from the long distance probe - Coulomb force

8

how to measure the electron charge? -a short distance probe

-

+

-

+ -

+ -

+

-

+ -

+ R

Test charge

the closer the test charge is placed, the larger the measured charge

at a distance, the charge appears smaller – it is shielded by the vacuum polarization the effect takes place in QED because the vacuum can be polarized some of the time, a photon is a virtual fermion-antifermion pair - as one gets closer to the charge, it gets larger since there is less screening. by the Uncertainty Principle, to probe a small distance, one needs a high momentum (small wave length) probe

9

how to measure the quark charge? - a long distance probe

R Test charge

a red quark is surrounded by the - antiquark pairs – the quark colour is surrounded by the predominantly red colour as seen by a long distance probe.

RR

R

R

RR

R

RR

R

R

RR

R

10

how to measure the quark charge? -a long distance probe

R Test charge

a red quark is surrounded by the pairs – the quark colour decreases when the probe is placed closer – colour is anti-screened from the short distance probe.

- much larger screening by the fermion loops in QCD and with the opposite sign! - due to the gluons that carry colour charges - virtual emissions and absorbtions spreads the colour over a large volume -and a high-energy probe sees a smaller charge. - these effects are proportional to: 2nflavours - 11ncolours - known as asymptotic freedom - a side effect is known as infrared slavery (At low energies, the colour force is very strong and the perturbation series breaks down - it is easy for many gluons to be exchanged. Empirically, the strong force tends to a constant as the distance becomes large.)

RR

R

R

R

RR

R

R R

R R

RR

11

•  the basic process of the strong interactions, Quantum Chromo- Dynamics (QCD): •  the charge is now colour - looks identical to the EM interactions, and indeed it almost is.. three important differences: (1)   three colour charges and eight gluons; since all the hadrons are

colourless, this is not special by itself - the colour quantum number is not an observable

(2) the colour charge is larger than the electric charge: 4παs ≈ 1 and often

the perturbation theory cannot be used -the only other calculational technique is iterative computer calculations using lattice of discrete space-time points instead of continuous space-time.

q

q

g

the strong interaction - QCD...

a quark emits a gluon

12

the strong interaction - QCD... (3)  unlike the photon, the gluon does carry a charge - the colour charge and a basic interaction could look like: -  profound consequences: (1) gluons can interact with each other - two additional fundamental vertices:

and - in 1974 it was realized that the effective coupling constant, or charge, varied with energy and became weak at high energy. i.e. perturbation theory could be used

q

blue-antigreen

green

q g

blue

13

comparison of QCD and QED

•  Only 1 Charge has values + and –

•  1 photon γ

does not carry electric

charge

•  3 Charges values red, green, blue

also anti-colours

•  8 gluons g

carry colour and anti-colour charge

QED QCD

( )

( )bb2ggrr61

,ggrr21

,rg,bg,rb,gb,gr,br

−+

−

14

comparison of QCD and QED...

•  represented by the U(1) abelian symmetry group

•  represented by the SU(3) non-abelian symmetry group

QED

Abelian Group:

mathematical group of transformations in which the order of the transformations is unimportant

QCD

15

comparison of QCD and QED...

QED QCD

•  Interaction between fermion-(anti)fermion pairs

•  Interaction between quarks (antiquarks) only

QED QCD

16

comparison of QCD and QED...

•  αEM = 1/137

•  C1, C2 = r,g,b

•  ≈ 0.1-1.

21CCgγ

√α

f

f

√αS

2Cq

1Cq

0

2

4 επα

cgs

S !=

comparison of QCD and QED...

forbidden allowed

QED QCD

17

18

comparison of QCD and QED...

•  QCD vacuum states produce quark-antiquark pairs and gluon loops where EM only produces fermion-antifermion pairs.

QED QCD

f

f

q

q

comparison of QCD and QED...

•  coupling only increases for very small separations < 10-16.

•  coupling increases at large separations and decreases at small r.

QED QCD Q

r

Q

r

asymptotic freedom: as the distance becomes asymptotically small, the interactions between quarks becomes asymptotically free - 2004 Nobel prize!

19

20

22

quarks and hadrons •  consider a quark antiquark pair moving apart with a large energy

due to an interaction with a photon:

g proton

gp → nπ+π+π-π-π+

23

quarks and hadrons...

u

d u proton neutron

d u d

d -

u

photon

the confinement: no free quarks or gluons.

24

quarks and hadrons...

q

q

γ

γ

- πo → γγ

•  this is not a Feynman diagram since it is not possible to calculate the decay width of π0 from it - the two quarks making up the πο are in a bound state continuously exchanging gluons •  for calculating the decay width, we need to know the πο wave function, specifically Ψ(0) •  nevertheless, these "quark diagrams" are used all the time for describing the basic quark level processes in particle physics.

25

the OZI rule the Ψ decay width is about 1000 times narrower than other strong decays, why?? - the effect is known as the OZI rule (after Okubo, Zweig and Iizuka) ⇒ ϕ → π+π-πo much slower (has a larger width) than expected:

mϕ = 1019 MeV/c2

mK± = 494 MeV/c2 Q = mϕ - 2mK±= 31 MeV B(ϕ→K+K-) = 83.5%

mπ± = 140 MeV/c2

mπo = 135 MeV/c2 Q = mϕ - 2mπ± - mπo= 604 MeV B(ϕ→3π) = 15.3%

ϕ

ϕ

ss

s−K

uus

+K

s

s

+π−π0π

26

the OZI rule... the difference between the two decays is that in ϕ→K+K-, the gluons can be soft (of low energy), but in the decay ϕ→3π, they must carry all of the mass of the ϕ, because if we cut the diagram at the right point in time, there are only gluons present. Because of asymptotic freedom, the couplings of those gluons will be weaker. The same thing happens for the and the , except in these cases, the decays, to the analogue of , are kinematically forbidden since 2mD > mΨ and 2mB > mΥ.

)( ccΨ )( bbΥBBDDKK and ,

27

hadronization

•  consider e-e+ → hadrons at high energy •  have two high energy quarks moving apart •  hadrons will be produced between the

quarks until the final state is colourless

x

t

28

jets •  jets are formed as

asymptotic states of the scattered partons

•  QCD requires that only colourless objects are observable (hadrons) e.g..: π, K, η, etc.

•  a jet is defined to consist of the particles within the cone,

or in a cluster defined by an algorithm…

7.022 =+= φηR

29

jets...

30

31

electron-positron annihilation

32

Heavy Ion Collision

Proton-Proton Collision

Electron-Proton Collision

Introduction to High Energy Physics Hard Scattering R. Orava Spring 2014

33

•  QCD looks much like QED, the basic diagram is •  three main differences (mathematically QED and QCD differ because in QED the gauge

transformation is a simple phase rotation, while in QCD it involves 3x3 matrices): (1)  the charge gs = √(4παs) is about 4-5 times larger than ge = √(4πα) ; αs will vary with q2 because of asymptotic freedom è the perturbation series will not converge & there is uncertainty because calculations only at the parton level - the measurements involve hadrons, their composite (”asymptotic”) states.

•  a typical QCD calculation might be pp → t t + anything, i.e. to calculate:

g g

t

t

p

p

X

X

t

t

p

p

X

X - -

- -

QCD vs. QED

-

)()( xGxG pp∝ )()( xqxq pp∝

For reference see Halzen&Martin Chapter 10

-

34

(2)  the quarks have colour and the gluons have colour and anti-colour and there are three types of quarks and eight types of gluons.

•  quark colour is given by a vector c:

c = for red for blue for green.

the colours form a 3 representation of SU(3); a colour-anticolour combination will form 3×3 = 8×1 - the gluons form a colour octet (an 8) - use the following combinations:

1 0 0

0 1 0

0 0 1

-

QCD

)2(6

18 )(2

7

)(2

16 )(2

5

)(2

14 )(2

13

)(2

12 )(2

11

ggbbrrbggbi

bggbrggri

rggrbbrr

rbbrrbbr

−+=−−=

+=−−=

−=−=

−=+=

35

•  these assignements are not unique - any linear combination of these would work as well. •  the 9th orthogonal combination:

is the singlet, and not a gluon state.

•  to specify the Feynman rules, have to give the SU(3) λ matrices (analogues of the Pauli spin matrices for SU(2))

•  the structure functions fabg are defined by the commutators of the λ matrices:

QCD

)(31 ggbbrr ++

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ −

=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ −

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

200010001

31

0000

000

010100000

00000

00

001000100

000010001

0000000

000001010

8765

4321

λλλλ

λλλλ

ii

i

i

ii

[ ] ∑=

=8

1

2,γ

γαβγβα λλλ fi

36

•  of the 83 = 512 structure functions, only 54 are non-zero:

f123 = 1

f147 = f246 = f257 = f345 = f516 = f637 = ½

f458 = f678 = √3/2

•  even permutations of these have the same value, odd permutations have negative value and all others are zero

QCD

37

•  the Feynman rules are: QED QCD

a b

=ge =gs

QCD

αµµ

αµµ

αµµ

λγπαγπα

γ

εε

εεγ

4 2i- 4

or

or

or

or

s

22

2222

*

a

αβµνµν

*

iQ

q

δ-ig

q

-ig g

-mqmq

-mqm)qi( qq

agγout

agin

vcvqoutcvvqin

cuuqout

ucuqin

−

+/+/

+

+(c is a 3-component vector)

(a is a 8-component vector)

external lines

Vertex

internal lines

38

note: except for the multi-gluon diagrams, the Lorentz structure is the same, and for a simple diagram, such as the dependence on momenta and angles will be the same as in QED. the only change will be due to the colour factor.

α,µγ,λ

β,ν

k1

k2

k3

β,νγ,λ

α,µ δ,ρ

QCD [ ]

[])(

)()(

)()()(

2

133221

λρµννλµρδβηαγη

νρµλλρµνβγηαδν

νλµρνρµνγδηαβη

νλµµνλλµναβγ

ggggff

ggggffggggffig

kkgkkgkkgfg

s

s

−+

+−+−−

−+−+−−

39

•  look at the colour factor for a q and q attracting or repelling each other by gluon exchange:

•  a ”Coulomb’s law” for the strong force, it will be •  the quark-antiquark pair can be either in an octet or singlet state. •  to calculate the octet potential, we take the quark to be red and the

antiquark anti-blue - since this state is orthogonal to the singlet, it must be octet

= colour factor ≡ f

q q

q q

a

b

1

2

3

4

-

- -

QCD

[ ] [ ] [ ] [ ]

[ ])(41

)4(2

)2( )1( 2

)3(

2

2

4213

42213

+−+−++

++

→−−

=

⎥⎦

⎤⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−=−

µµλδλ

γλδ

γλ

βαβα

νβαβ

µνµα

eeiMggCCCC

CvgiCvq

giCugiCuiM

e

s

ss

rcfrV s

qq!α

−=)(

-

40

•  since colour is conserved:

•  the only matrices with non-zero diagonal elements are λ3 and λ8, i.e.

QCD ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

==⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

==

010

and 001

4231 CCCC

( ) ( )

∑=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

== ++

α

αα

αααα

λλ

λλλλ

2211

4213

41

010

0,1,0001

0,0,141

41 CCCCf

[ ]

rc

f

s!α

λλλλ

61V

61

311

41

41

qq,8

822

811

322

311

=⇒

−=⎥⎦

⎤⎢⎣

⎡+−=+=

repulsive

41

•  for a colour singlet state of:

QCD )(

31, ggbbrrqq ++

{ }

rcV

f

sqq

!α

λλλλλλλλλλλλ

λλλλ αααα

34

341111

311

4141

100

)0,0,1(001

)1,0,0(001

)0,0,1(001

)0,0,1(331

31

41

1,

513

531

413

431

212

221

112

121

811

811

311

311

−=⇒

=⎭⎬⎫

⎩⎨⎧

+++++=

+++++=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

attractive

42

•  in the quark-quark case •  the antitriplet turns out to be attractive, f = -2/3, and the sextet

repulsive, f = 1/3 - forming a baryon:

all pairs repulsive some pairs attractive some repulsive

all pairs attractive

QCD

[ ] 18810336333 ⊕⊕⊕=⊗⊕=⊗⊗

3633 ⊕=⊗

43

summary: QED vs. QCD

•  QCD is a non-abelian gauge theory with SU(3) symmetry:

•  Both are relativistic quantum field theories that can be described by Lagrangians:

uvuv

uu

uu FFAemiL

41)( −+−∂= ψγψψγψ

uva

auv

aujka

ujkjku

ujk GGGqqgqmiqL

41)()( −λγ+−∂γ=

QED:

m = electron mass Ψ = electron spinor

electron-g interaction

Aυ= photon field (1) Fυϖ=∂υAϖ-∂ϖAυ

m=quark mass j=color (1,2,3) k=quark type (1-6) q=quark spinor

quark-gluon interaction

Gau=gluon field (a=1-8)

Gauv=¶u Ga

v -¶v Gau-gfabcGb

u Gcv

QCD:

la’s (a=1-8) are the generators of SU(3). la’s are 3x3 traceless hermitian matrices.

[la, lb]=ifabclc

fabc are real constants (256) fabc structure constants of the group

gluon-gluon interaction (3g and 4g)

),(),( ),( txetx txief ψ=ψʹ −

),(),(

8

1 2),(

txetx i

ii txig

ψ∑

=ψʹ =

ωλ−

•  QED is an abelian gauge theory with U(1) symmetry: