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March 16, 2010 Lecture 3 of 4 March 16, 2010 Lecture 3 of 4 @ ORICL Science & Technology Group@ ORICL Science & Technology Group

Yuri Kamyshkov Yuri Kamyshkov University of TennesseeUniversity of Tennessee

@ ORICL Science & Technology Group @ ORICL Science & Technology Group

University of TennesseeUniversity of Tennesseekamyshkov@utk.edukamyshkov@utk.edu

There is a problem with UV logarithmic divergence of the loopThere is a problem with UV logarithmic divergence of the loop radiative corrections (→∞) . These divergences can be hidden inside the renormalized charges (“coupling constants”), masses,and wave functions.and wave functions.

“Vacuum polarization” and other radiative corrections lead to the “running of the coupling constants”, i.e. the strength of interactions changes at small distances or at large energies (or momenta q).

Coupling strength to electric charge is increasing at large momenta q (in QED) due to “vacuum polarization” and other loops. Coupling q ( Q ) p p p gstrengths of weak and strong forces are decreasing at large q’s due to contribution of bosonic loops (self interaction of W,Z-bosons and gluons). The decrease is particularly significant for the strong g ) p y g ginteraction because of the color factor of gluons. Photons have no self interactions.

Distance [m]

1

α

LHC

B l d t lf i t ti f W Z

Color gluon loops W and Z-boson loops

Boson loops due to self-interaction of g, W, Z

Fermion loops

g gq

All possible known fermions will contribute to these loops. Their relative contributions will depend on the fermion mass coupling strength and on the available momentum qthe available momentum q(or collision energy)

If ithi th St d d M d l ld k•• If within the Standard Model we would know more particles (fermions) and/or forces (bosons) will the running behavior of the coupling constants be different?g p g

•• If at some very small distances (if very high energywould be available) we will find that electrons and quarkswould be available) we will find that electrons and quarks have substructure will the behavior of running coupling constants change?

Democritus 460 BC

Distance [m]

Energies or distanceswhere we observe

Forces → 0“asymptotic

freedom”

1

where we observeeffect of running coupling constants

freedom

α

Range of very high

Force →∞

Range of very highenergies or smallest distances the running coupling constants

LHC

are extrapolated to

Fundamental particles are believed to be point‐like objects without internal structure

Potential energy of gravitational interaction of two particles is zero at infinite distance but increasing when distance →0

At very small distance (or at very high energy) the potentialenergy of gravitational interaction can be larger than 2mc2

19Thi i ll d Pl k 1 2 10c

E G V× 19This is called Plank energy 1.2 10

or Plank mass 21

PN

P

E GeVG

M g

G

μ

= ≈ ×

≈

333

or corresponding Plank length = 1.6 10 cm NP

Gl

c−≈ ×

k maxk = Λq q contibution of loop

radiative correction:

⇒

2 20

2 2 ln12lime

mπΛ

Λq k−

12 mπΛ→∞

At Plank energy  ≈ 1019 GeV quantum gravity effects (not yet )probed or understood) should come into play and dominate 

the character of all interactions. This would be a natural valuefor Λ limiting the divergence of virtual loops.g g p

no divergences in SM theory.Beyond there should a different theory

PEE

Λ →Beyond there should a different theoryof "quantum gravity" (not presently known).

PE

Divergences in the Theory are due to our ignorance of very high energy physics E Ehigh energy physics E∼ EP

Renormalized theory is an “effective theory” with the same l b h i th t th ( t t k )low energy behavior as the true theory (not yet known)

Scale dependence is the price for working with an “effective theory’.p p g yAmong the advantages of effective theory we have:(a) high accuracy of SM calculations confirmed by experiment;(b) running coupling constants confirmed by experiments;(c) the understanding that physics beyond the SM is needed.

One of the ideas beyond the SM is Grand Unification Theory (GUT):All SM interactions meet and unify at some high energy (below E )All SM interactions meet and unify at some high energy (below EP)with αGUT ≈ 1/40

αi)

not

1/ α

If no new particles and forces

e of

αi( If no new particles and forces

exist between MW and MX –this is called “Great Desert”

ear s

cale Critical prediction of GUT

the transformation of quarksto leptons (proton decay) is not supported by experiment !

Plank scale

ere

line not supported by experiment !

Not

e h

If one more carefully plots 1/αknown in the measured rangethe coupling constants do p gnot actually meet in the samepoint after running through the Great Desert !

Neutral scalar field (spin=0, one spatial component, ( p , p p ,no direction) is a vital mechanism of the Standard Modelresponsible for masses of all particles.

Higgs field (mechanism) experimentally is not yet established; but theoretically is universally accepted;→ it is the first expected result of the LHC collider→ it is the first expected result of the LHC collider.

Unusual property of Higgs field: it has a “vacuum expectation value” a mean value of the field in theexpectation value a mean value of the field in the absence of any particles or fields (field condensate).VEV of Higgs field has a dynamic origin (due to non-linearHiggs potential)Higgs potential)

Vacuum expectation valueFor comparison: gravitational field in

the absence of planets, starts or

i ti l t th r =∞massive particles, or at the r =∞has zero “vacuum expectation value”

Interaction of fermions (leptons and quarks) withHiggs condensate gives to the former their masses.

e e

h

eg

Each type of particle has its own interaction couplingconstant with Higgs field determining its mass. Higgs field gives mass term to the SM Lagrangian, but it egm

υ=

υh

Higgs field is self-interacting.

does not explain why different types of particles havedifferent masses. 2

em =

Higgs boson HHiggs field is self interacting.

Excitations of Higgs field are called “Higgs boson”Mass of Higgs boson is due to Higgs self interactions.

e Higgs condensateυ υ =

Higgs boson H

hfield

vacuumvacuum

λH H

Potential of Higgs field is non-linear.

λ λ 2Hm υλ= ⋅υυhh2 2 4

2 4V h hυ

λ λ= − +

Self-interaction of Higgs bosons:λ

λυH H H H

HH H

W WWgMass of W and Z bosons: Z ZcosW

W

g

θW W

h h

WgMass of W- and Z-bosons: Z Z

h h

W

1W Wm g υ=

υ υ υ υ2W Wm g υ=

246 G VVacuum expectation value of Higgs field(Mass of the heaviest t-quark is 173 GeV)

246 GeVυ⇒ =

Coupling constant of Higgs to photons (γ) and to gluons (g) equal 0Coupling constant of Higgs to photons (γ) and to gluons (g) equal 0

g0; 0g gγ = =

A quasi-political Explanation of the Higgs Boson; for Mr. Waldegrave, UK Science Minister 1993. from David J. Miller, Physics and Astronomy, University College London. (cartoons courtesy of CERN).

“Condensate of Higgs field of politicians”

“Celebrity particle gets its Higgs mass in the condensate”

“Excitation of the Higgs field can be set”

“Mass of Higgs particle is formed by self interactions”

Strength of Higgs coupling is proportional to the mass:

t

tg tgFor example:tvirtual fermion loop

with t-quarks is more i t t th eimportant than electron-positron loop

eeg eg

Where the mass of Higgs boson can be discovered?

Peter Higgs with his portrait (credit: Callum Bennetts/Maverick Photo Agency)

(d t di t i t i t Hi )(d t di t i t i t Hi )(due to radiative corrections to Higgs mass)(due to radiative corrections to Higgs mass)

( ) ( )2 2 2 2,H H ibareM M gλ= + ⋅ ΛO

bare

19

Quadratically divergent "correction" ! when

If cut-off parameter is then m ( 10 )Pl k H Pm m GeV

Λ → ∞

Λ ≈∼ ∼If cut off parameter is then m ( 10 )

This is called "mass hierarchy problem".Plank H Pm m GeVΛ ≈

in resolution of hierarchy problemin resolution of hierarchy problem

(a) “Fine tuning” : if for unknown reason all possible diagrams cancel each other leaving no Λ2 behavior

(b) No Higgs field, but some new dynamics exists that plays effectively similar role in the SMsimilar role in the SM

(c) New Super Symmetry (SUSY) would exists

All these possibilities hopefully can be sorted at LHCAll these possibilities hopefully can be sorted at LHC

Supersymmetry or SUSY was “discovered” byJ. L. Gervais and B. Sakita (1971)

Yu A Golfand and E P Likhtman (1971)Yu. A. Golfand and E.P. Likhtman (1971)D.V. Volkov and V.P. Akulov (1972)

J. Wess and B. Zumino (1974) and ...

Symmetry between fermions and bosons. Every type of fermion has a similar type boson partner, and vice versa. Number of fundamental particles doubles. yp p p

For every virtual fermion loop now there is a corresponding virtual boson loopcanceling the former contribution. That would resolve the hierarchy problem in a principle way!

Is not perfect. In perfect symmetry masses of particles and super partners are exactly equal and virtual loop cancelation result is exact 0. So far we y q pdid not observed SUSY particles, therefore the only possibilities that theyare heavier than their regular partners.

S t i “b k t ” W k th lSuper-symmetry is a “broken symmetry”. We know many other examples of broken symmetries. Trying to resolve hierarchy problem by introducing SUSY we got a new problem of existence at some high energy scale of a new force that brakes the Super Symmetry.p y y

In SM extended with SUSY the Higgs mechanism becomes perturbative: radiative corrections to the Higgs mass are logarithmic at Λ→∞

( )2

2 22

(with SUSY) ln PH H b

McgM M

M

⎛ ⎞⎛ ⎞⎟⎟⎜ ⎜ ⎟⎟= + ⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎟⎜O( ) 2

( )4

H H bareWMπ⎜ ⎜ ⎟⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠