Fundamental Interactions - Syracuse...

PHY771, 2/4/2016 Tomasz Skwarnicki 1

Fundamental Interactions

Tomasz SkwarnickiSyracuse University

• Griffiths, 2nd ed., Chapter 2

• Do problem 2.7– important exercise for use of basic conservation laws

– check conservations of Q,E in decays, A,Le,Lµ,Lτ,S

– particle properties listed in the tables in Griffiths or at pdg.lbl.gov

• Do handed out assignment on Feynman Diagrams (also psoted in the web)


Fundamental Forces recap

Force Earlier Theories Present Theory Mediator

i.e. intermediate (gauge) boson

Charge

Gravitational Newton (1686) General Relativity

Einstein (1916)

No good theory of Quantum Gravity

[ graviton (spin 2) ]

[mass]

Electromagnetic Coulomb (1784) Amper (1821) Faraday-Henry

(1835)

Maxwell (1864)

QED:

Quantum Electrodynamics

Tomonaga,Feynman,Schwinger (1940s)

Photon

γ (spin 1)

mγ=0

Electric

Weak Fermi (1933) Electroweak

Glashow,Weinberg,Salam (1960s)

W±,Z0+γ (spin 1)

mw=80.4 GeV

mz=91.2 GeV

Higgs H0 (spin 0)

mH=125.7 GeV

Weak isospin

and hypercharge

Strong Yukawa (1934), Heisenberg (1934), Gell-Mann (1964)

QCD:

Quantum Chromodynamics

Gross,Wilczek,Politzer (1973)

Gluon

g (spin 1)

mg=0

Color

(Many more physicists contributed to the development of the theories than mentioned here)


Quantum Field Theories and Feynman diagrams

• Feynman diagrams (1948) are pictorial representations of the mathematical expressions describing particle interactions

• Even without going into mathematics behind them, they allow intuitive (qualitative) understanding of various processes

• We have been using them already. Today explain them more systematically, but still at a cartoon level.

• Later in the semester we will go into mathematics behind them (though short of complete discussion of QFT)

• Use QED for examples, later discuss QCD and electroweak interactions

Richard Feynman

1918-1988

Nobel Prize 1965


Interaction “vertex”

• Gauge bosons couple to particles carrying charge of related interactions:

e.g. in QED

γ

-e-e

• Each interaction vertex contributes a term proportional to a product of the charge (qe) and of the coupling constant (ge) to quantum mechanical amplitude describing the process

• Total amplitude is a sum over all graphs which have the same incoming and outgoing particles. Probability of the process is proportional to the modulus of the complex amplitude squared (terms can interfere!)

• If the coupling constant is small, then graphs with more interaction vertices represent terms which are corrections (“perturbations”) to the graphs with fewer vertices. The sum is actually over infinite number of terms, but higher order contributions can be neglected – “perturbative calculations”.

• The graph with a fewest vertices for a given process represents the

dominant term, and usually this is the only one that we draw

qege

1

4

1

137

0.03

e

e

e

q

g

g

α

α

π

= −

=

≈

≈

“fine structure

constant”

time


Antiparticles

• Antiparticles are represented as particles going

backwards in time:

γ

time

e+

e+

Notice that we could label all electron/positron lines as “e” and use the arrows on

the electron lines to identify sign of the electron charge (Griffiths does that!)

-e γ

e+


Virtual particles

• Particle which is “exchanged” between two interactions

vertices is in “virtual” i.e. “off mass shell” state:-e γ

e+

µ+

µ−Photon is virtual here

“on mass shell”: Q2 = E2 - p2 = m2

Q=(E,px,py,pz) particle four-momentumAttention!

I use convention in which c=1

thus c is dropped from the formulae

“off mass shell”: Q2 = E2 - p2 = m2

• Virtual particles contribute to the amplitude term its

“propagator” which is proportional to:

2 2

1

Q m−

(all incoming and outgoing particles)

(all exchanged particles)

• The more virtual the particle the smaller the amplitude (thus

the probability for the process)


Example

• Lowest order cross-section for e+e- annihilation to µ+µ−

-e γ

e+

µ+

µ−qe ge qµ ge2 2

1

0

Q m

m

γ γ

γ

−

=

2 2 2( ,0) ( )e e e e

E E E EQ sγ + − + −= =+ + ≡

In the center-of-mass of the collision (total momentum is zero):

4 22

22 1 e

e es sg

q qs

gs s

µ

ασ ∝ Μ ∝ = ∝

“phase-space” factori.e. available “density” of states

Amplitude i.e. “matrix element”

Cross-sectioni.e. specially normalized probability

Exact formula:

24

3 s

πσ

α=


Example – higher order correction:

• Compared to the leading graph they are all suppressed by:

-e γ

e+

µ+

µ−γ

-e γ

e+

µ+

µ−

γ

-e

e+

µ+

µ−

2

2 1~ 0.00005

137α

=

f =e,µ,u,d,…

f(“vacuum polarization diagram”)


Other examples of QED processes

• Photon can be an external particle: -e

γ

e+

γ

(virtual electron in these graphs)

e+e- → γγ

(single photon annihilation to on-shell photon does not conserve momentum thus is not possible)

γe- →γe-

Compton

scattering

γ γ

-e -e

Note:Crossing symmetry between

these two processes!


QED interactions between quarks

• Dominant decay γ

γ

π0 → γγu

u

2

3uq =

quge

quge

π0

γ

γ

π0 → γe+e-u

u

2

3uq =

quge

quge -e

e+

qege

Suppressed by α ∼ 1%


QED process with two leading order graphs

• Bhabha scattering

γ

-e

e+e- → e+e-

-e

e+ e

+

-e γ

e+

-e

e+


QCD vs QED

• Leptons don’t have color charge, they don’t

participate in strong interactions. Only quarks and

gluons do.

gs

• Self-coupling of gluons is a new element:

gs

gs

(q=1)

• Coupling constant (gs or respective αs) “runs” a lot

and can be large


Diagrams with gluon self-coupling

• If the Q2 of the gluon is small (“soft gluon”), then αs(Q2) is large and

the higher order diagrams are no longer “corrections” – the perturbative approach breaks down

Q2

g

g g

gg

g

g g

q

q


Running of ααααs

Short distancesLong distances

Small

energies

Large

energies

αEM~0.007

Strong interactions are in fact much stronger than electromagnetic at presently reachable Q2s

“soft gluons” “hard gluons”

Perturbative QCDNon-perturbative QCD

(e.g. Lattice QCD)


Why coupling constants run?

• Screening of charge

In QED:

Vacuum fluctuations make it act like dielectric

+

Large Q2

Small Q2

Small r

Large r

1h

mc Qλ = ∝

Small effect in QED

In QCD vacuum polarization is much stronger

since there are 3 colors (vs 1 electric charge) and

coupling constant is larger – larger screening effect

– larger running of αs


Constituent and current quark masses

• Quark mass is not well defined since quark is

never free

• Quarks “dress” themselves into cloud of gluons

and virtual quark pairs


Confinement

• Quarks and gluons are colored.

• According to the confinement hypothesis, objects must be white (color neutral) at distances larger than hadron size (~1 fm).

• Therefore, free quark and gluons don’t exist. “External” quark and gluon lines in QCD Feynman diagrams never represent truly free particles.

• In addition to short range (perturbative) QCD interactions represented by a Feynman diagram, there are always long range non-perturbative, confining interactions involved. The latter are often subject of phenomenological modeling – QCD is not as precise as QED is.

c

c

c

d

c

d

ψ

D−

D+

“Fake” Feynman diagram:In reality this entire decay involves a lot of soft gluons everywhere (we might as well omit all of them!)

ψc

c

u

ud

d

d

d

π+

π−

π0

hadro

ns

Long range QCD:

Models

Short range QCD:

perturbative

Long range QCD:

Models

…

OZI rule related to αs running


Weak neutral interactions

• Mediated by Z0

• All quark and leptons (including neutrinos) carry

weak hypercharge coupling to Z0

• Left handed (sz= ½) fermions have smaller weak

hypercharge than right handed (sz= - ½) YWgz

2.4sin cosW W

ez e

gg g

θ θ= ≈

θW~30o

Weinberg’s angle (from

spontaneous symmetry

breaking via Higgs

mechanism)


Why Z0 interactions are weak at low energy

• Z0 coupling constant larger than for photon

• However, Z0 has a large mass (91.2 GeV) which suppresses weak rates via the value of the propagator for small Q2

• At large Q2 weak interactions are stronger than electromagnetic!

2 22

1 1

Z ZQ m m≈

−

sqrt(s)

e+e- → hadrons(log scale!)

EM

γ


Charged weak interactions

T3 gz

2.1sin

ew e

W

gg g

θ= ≈

W• Mediated by W±

• All left-handed quark and leptons (including neutrinos) carry weak isospin coupling to W±

• Right-handed fermions are weak isospin singlets and they do not feel charged weak interactions (maximal violation of parity)

• W is heavy (mw=80.4 GeV) which makes weak decays weak at low

energies. At high energies they are stronger than electromagnetic.

• Neutral weak interactions conserve quark flavors (no FCNC at tree level).

• However, charged weak interactions mix flavors via CKM mechanism [discussed last time]


Self-coupling of weak bosons

• Less dramatic consequences than for QCD since the

coupling constant is not as big


More than one force in one diagram

• This can happen for higher order processes

e.g.

“glouonic penguin diagram”


Grand Unification Theories

• There are many ideas for GUT models; SUSY is the best known model

• They usually predict new forces (with very heavy intermediate bosons) and often new fermions too


GUT dream


Complex world reduced to simple fundamental forces

Gravitational

Binding force144444444424443 1424443

Electromagnetic

Quantum Mechanics

Us

Typical size

1m

Planet

107

Star with planets

1013

Galaxy

1021

Universe

Group of galaxies

General Relativity

10231026

Molecules

10-9

Atom with

electronsand nucleus

10-10

123

Strong

Nucleus

Nuclear Physics

10-14

Proton with

quarks inside

High Energy Physics

10-15


Evidence for Beyond Standard Model physics

• Unknown particles and forces exist, likely hiding at higher energy scales

Mass of thedark matterin galaxies is~6 times the mass of visible matter

Visible

Dark

Matter ?

Dark

Energy ?

~3 times energy of everything else in the universe

Higgs boson?

time

anti-fermionboson boson

fermion

almost all fermions

Baryogensis ?

today

Big Bang

Q=-1 Q= 0 Q=+1 Q=+2

S= 0 ∆− ∆0 ∆+

∆++

S= -1 Σ∗− Σ∗0

Σ∗+

S= -2 Ξ∗− Ξ∗0

S= -3 Ω−

Q= -1 Q= 0 Q=+1

S= 0 n p

S= -1 Σ−

Σ0 ,Λ Σ

+

S= -2 Ξ+ Ξ0

Q= -1 Q= 0 Q=+1

S=+1 K0

K+

S= 0 π+ π0 ,η π+

S= -1 K+

K0

mid 20th centaury → quarks, QCD

now → ?

end of 19th centaury → atoms, QED

Generation

problem ?

Hierarchy

problem ?MH << MPlanck

GUT?How does gravity fits in?


Evidence for Beyond Standard Model physics

12 orders of magnitude differences not

explained; t quark as heavy as Tungsten

Origin of hierarchy in masses and mixing of fermions?

Lo

g-s

ca

le !

Why these values? Are the two

related? Are they related to masses?

Area ~V2

Pontecorvo–

Maki–

Nakagawa–

Sakata neutrino

mixing matrix

Cabibbo-

Kobyashi-

Maskawa- quark

mixing matrix


Conclusion

• Much work remains to be done to understand the

fundamental fabric of the Universe

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