A.Ereditato SS 2013 1 Elementarteilchenphysik Antonio Ereditato LHEP University of Bern Lesson...

A.Ereditato SS 2013 1

Elementarteilchenphysik

Antonio Ereditato

LHEP University of Bern

Lesson on: Electroweak interaction and SM (8)


Unification of fundamental interactionsUnification of fundamental interactions

•At the moment of Big Bang energy density was very large, interactions happened at really high energy, of the order of the Planck scale (1019 GeV), at such energy the four fundamental forces had the same intensity, the coupling constant of gravitational, strong, electromagnetic and weak interactions were equal.

•When the forces acquired different values? At which scale the unification breaks down?

•The first force to decouple was gravitation, followed by strong force and then electromagnetic and weak forces. Electromagnetic and Weak force decoupled at low energies high q2 (~104 GeV).

•The breaking of the symmetry makes three of the four mediating bosons of the EW theory very massive (W+, W- and Z0) determining the short range of the weak interaction and thus its “weakness”.

•Historically the first “unification” was made by Maxwell unifying electrostatic and magnetic formalisms. By the end of the 60’ Glashow, Salam and Weinberg worked-out a theory predicting the unification of the weak and the electromagnetic interactions into the electro-weak (EW) force

Glashow Salam Weinberg

•Differently from electromagnetic unification,

the GWS theory predicted a unique coupling

constant, the electric charge e and only one

parameter (sin2W), to be measured in

experiments.

•The EW theory is very well established today

and its success contributes to the success of

the so-called Standard Model of particles and

interactions


Energy density affect the coupling constants (i.e. the intensity) of the forces, during time in our universe the coupling constant changed, or as it is said they are “running coupling constants”.

The electroweak unification (1 and 2)The electroweak unification (1 and 2)The EW + strong force unificationThe EW + strong force unification

In many SUSY models all coupling converge to the same value at high energies, one of the goals of LHC and high energy accelerators is to explore energies present at the very first moment after the big bang

Time Time


The Glashow, Weinberg, Salam ModelThe Glashow, Weinberg, Salam Model

As energy rises electromagnetic and weak forces present the same coupling, that is the two forces have

the same intensity. This energy is well within present accelerator reach. GWS theorized a model based on

the SU(2)L x U(1)Y. From the group algebra one would expect four massless mediating bosons, arranged

into a SU(2)L weak isospin triplet (I = 1) and a singlet (I = 0) of the weak hypercharge group U(1)Y.

SU(2)

group of unitary 2 x 2 matrices

U(1)group of unitary 1 x 1 matrices

The three isospin triplet bosons are: W W(1), W

(2) , W(3) while Bis the the isospin singletThe physical

(massive) states are the W +, W -, Z 0 and photon A. The latter (neutral bosons) are related to the

massless bosons via mixing (rotation):

W

W is called the Weinberg angle

Nb: These matrixes operate on spinors!!

Nb: in GWS theory all bosons are massless,For the sake of renormalizability

A.Ereditato SS 2013

Experimental approachExperimental approach

Weak interactions have been always been observed to have a weaker coupling than EM decays, e.g. beta

decays presented longer decay times. Many hypotheses were made. Fermi made a first effective theory

on weak decays.

GF= 10-5 GeV-2

GF is not dimensionless so it is not a fundamental quantityAnd a point like interaction do not allow renormalization. Then the point like interaction should be replaced by a mediator.

GF=g2/MW2

We could substitute a point like interaction with a propagator, the point like interaction would held if q2<<MW

2, the mass is needed to make the propagator weak at low energies.

Massive propagators can not be renormalized, so without any other theory we could not make amplitude calculation (i.e. solve Feynman diagrams) of weak interactions.

Higgs theory (shown later) solve this problem, by adding a new potential to the electroweak lagrangian which make it possible to have massive W and Z bosons, and also gives mass to fermions.


Weak boson massesWeak boson masses

We remember that: Now, since it follows that:

It can be shown that: while, in the most general case of Higgs scalars:

The Weinberg angle measured in weak interaction processes (neutrino interactions) and the masses

of the weak bosons measured at the colliders agree well:

Measured masses of the weak bosons:

MW = 80.398 ± 0.025 GeV

MZ = 91.1876 ± 0.0021 GeV

Measured masses of the weak bosons:

MW = 80.398 ± 0.025 GeV

MZ = 91.1876 ± 0.0021 GeV


We define the weak hypercharge, analogously to the strong hypercharge (apart from a factor 2 difference):

One has then:

Performing the calculations, recalling the mixing relation, and setting g’/g = tanW , one obtains:

weak charged current weak neutral current EM neutral current

i.e. unification of weak and EM interactions:i.e. unification of weak and EM interactions:

Theoretical approachTheoretical approach


In other words, we have expressed the two experimentally observed neutral currents (the EM and the

weak) in terms of the currents belonging to the symmetry groups SU(2)L, and U(1)Y which depend on

the coupling g and a free parameter W

These two coupling constants are replaced by e and by the sin2W parameter, the latter to be

measured in experiments:

Just as Q generates the group U(1)EM , which regulates the EM currents, the Y operator generates the

symmetry group U(1)Y, and takes into account both the EM and neutral weak component. The enlarged

group SU(2)L x U(1)Y takes into account the incorporation of weak and EM interactions.

The SU(2)L x U(1)Y group was introduced by Glasgow (1961) before the discovery of neutral currents and

extended by Weinberg (1967) and Salam (1968).

Another important result obtained with the EW unification theory is that one naturally eliminates the

divergencies occurring in calculating graphs in the original weak interaction theory.


Electroweak couplingsElectroweak couplings

As the EW bosons, also fermions get weak isospin and hypercharge. In doing so the GWS model has to

take into account that the weak charged-current interaction violates parity, while the EM interaction is

parity-conserving. Remember that Y = Q - I3

Fermion multiplets I I3 Q/e Y

Leptons

eL

eL

L

L

L

L

½+ ½

- ½

0

-1-1/2

eR R R 0 0 -1 -1

Quarks

uL

d´L

cL

s´L

tL

b´L

½+ ½

- ½

+2/3

-1/3+1/6

uR cR tR 0 0 +2/3 +2/3

dR sR bR 0 0 -1/3 -1/3

Recalling the expression of the weak neutral current:

one obtains the corresponding relations for the coefficients of the LH and RH couplings of fermions to the Z:


d

Z0

I

I3

I

I3

I3

Examples of couplingsExamples of couplings

(g’/g = tanW )


Spontaneous symmetry breakingSpontaneous symmetry breaking

This transformation is the result of the phenomenon of Spontaneous Symmetry Breaking (SSB). In the case of the electroweak force, it is known as the Higgs Mechanism.

Wμ1,2,3, Bμ W+, W-, Z0, γ

Spontaneous symmetry breaking (SSB) occurs in a situation where, given a symmetry of the equations

of motion, solutions exist which are not invariant under the action of this symmetry without any

explicit asymmetric input (hence the attribute “spontaneous”).

A situation of this type can be illustrated by means of a simple (classical physics) example. Consider the

case of a linear vertical stick with a compression force applied on the top and directed along its axis. The

physical description is obviously invariant for all rotations around this axis. As long as the applied

force is mild enough, the stick does not bend and the equilibrium configuration (the lowest energy

configuration) is invariant under this symmetry. When the force reaches a critical value, the symmetric

equilibrium configuration becomes unstable and an infinite number of equivalent lowest energy stable

states appear, which are no longer rotationally symmetric but are related to each other by a rotation. The

actual breaking of the symmetry may then easily occur by effect of a (however small) external asymmetric

cause, and the stick bends until it reaches one of the infinite possible stable asymmetric equilibrium

configurations.

In substance, what happens is that when some parameter reaches a critical value, the lowest energy

solution respecting the symmetry of the theory ceases to be stable under small perturbations and

new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are

asymmetric but are all related through the action of the symmetry transformations. In other words, there

is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct

asymmetric solutions of identical (lowest) energy, that maintain the symmetry of the theory.


Spontaneous symmetry breaking (cont.)Spontaneous symmetry breaking (cont.)

The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state. This

means that there may exist symmetries of the laws of nature which are not manifest to us because the physical world in

which we live is built on a vacuum state which is not invariant under them. In other words, the physical world of our

experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the

fundamental laws of nature. SSB offers a key for understanding (and utilizing) this physical possibility.

The application of SSB to particle physics in the 1960s and successive years led to profound physical consequences and

played a fundamental role in the edification of the current Standard Model of elementary particles. In the case of a global

continuous symmetry, massless bosons (known as “Goldstone bosons”) appear with the spontaneous breakdown of the

symmetry according to a theorem by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious

problem since no particles of the sort had been observed, was in fact the basis for the solution, by means of the so-called

Higgs mechanism, of another similar problem. This is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields

predicted unobservable massless particles, the gauge bosons.

According to the “mechanism” established in a general way in 1964 independently

by P. Higgs and others, in the case that the internal symmetry is promoted

to a local one, the Goldstone bosons “disappear” and the gauge bosons acquire a

mass. The Goldstone bosons are “eaten up” to give mass to the gauge bosons,

and this happens without breaking the gauge invariance of the theory.

Note that this mechanism for the mass generation for the gauge fields is also what

ensures the renormalizability of theories involving massive gauge fields

(such as the Glashow-Weinberg-Salam electroweak theory.


The Higgs mechanismThe Higgs mechanism

Peter Higgs

In the Higgs model particle masses arise in a beautiful, but complex, progression. It starts with a particle that has only mass, and no other characteristics, such as charge, that distinguish particles from empty space. We can call his particle H. H interacts with other particles; for example if H is near an electron, there is a force between the two. H is of a class of particles called bosons. It is also a scalar particle (s = 0).

The parameters in the equations for the field associated with the particle H can be chosen in such a way that the lowest energy state of that field (empty space) is one with the field not zero. It is surprising that the field is not zero in empty space, but the result is: all particles that can interact with H gain mass from the interaction.

The picture is that of the lowest energy state, "empty" space, having a crown of H particles with no energy of their own. Other particles get their masses by interacting with this collection of zero-energy H particles. The mass (or inertia or resistance to change in motion) of a particle comes from its being "grabbed at" by the Higgs particles.

The Higgs particle (field) permeates the whole Universe at any space-time point


Search for the Higgs as a free particle at the LHCSearch for the Higgs as a free particle at the LHC

The strength of the Higgs coupling is proportional to the mass of the particles involved so its coupling is greatest to the heaviest decay products which have mass < mH/2. For example, if mH > 2Mz then

the couplings for decay to the following particle pairs:

Z0Z0 : W+W- : τ+τ- : pp : μ+μ- : e+e- are in the ratio

1.00 : 0.88 : 0.02 : 0.01 : 0.001 : 5.5 x 10-6

Negative searches have been conducted at LEP and TEVATRON. The present bounds on its mass is [114 GeV, ~1000 GeV].

The Higgs discovery will be one of the main goals of the LHC with the LHC ATLAS and CMS experiments.


Backup


Neutrino scatteringNeutrino scattering

Neutrino electron scattering gave the first evidence for the correctness of the EW theory and allowed the first estimates of sin2W (remember the Gargamelle experiment).

This was followed by a series of neutrino scattering experiments: off hadrons (for the

measurement of structure functions) and off electrons (to study purely leptonic, EW processes).

For example, the CHARM II experiment at CERN (1984-1991), that collected the largest world

statistics of and interactions, for the measurement of sin2W


Neutrino scattering (cont.)Neutrino scattering (cont.)

Where:

s is the cms squared energy

y is the elasticity Ee/E

LL, RR, RL, LR indicate the helicities of the scattering leptons

Lower cross-section for antineutrinos: it would be the same as for neutrinos if the scattering

occurred on RH electrons (forbidden by the V-A structure of the weak charged current)

For neutral current reactions:

For charged current reactions:

For the electroncoupling to the W

Measurement of cross-sections for different reactions (ambiguities) allows determining the coupling constants

Measurement of cross-sections for different reactions (ambiguities) allows determining the coupling constants


Examples of purely leptonic neutrino interactionsExamples of purely leptonic neutrino interactions

Electron-antineutrino off electron scattering can proceed via NC and CC reactions

Muon-neutrino off electron scattering can only proceed via NC reactions

e-

Z0

e-

+

e

e-

Z0

e-

e

e-

e

W-

e

e-

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