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KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association INSTITUTE OF EXPERIMENTAL PARTICLE PHYSICS (IEKP) – PHYSICS FACULTY www.kit.edu Spontaneous Symmetry Breaking and the Higgs Mechanism Roger Wolf 12. Mai 2016
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Page 1: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:

KIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association

INSTITUTE OF EXPERIMENTAL PARTICLE PHYSICS (IEKP) – PHYSICS FACULTY

www.kit.edu

Spontaneous Symmetry Breaking and the Higgs Mechanism

Roger Wolf12. Mai 2016

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Institute of Experimental Particle Physics (IEKP)2

Schedule for today

The problem of masses in the SM

The Higgs mechanism

Spontaneous symmetry breaking

1

3

2

● Is the following statement true: “all parts of the symmetry have a general problem both with mass terms for gauge bosons and fermions”?

● Is the following statement true: “the Higgs boson is a Goldstone boson”?

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Institute of Experimental Particle Physics (IEKP)3

Mass terms in the SM

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Institute of Experimental Particle Physics (IEKP)4

The problem of massive gauge bosons

● Transformation:

● In mass term :

These terms explicitly break local gauge covariance of .

● Example: Abelian gauge field theories (→ see first lecture).

● This is a fundamental problem for all gauge field theories.

● Remember: in Lecture-02 we have explicitly shown that the gauge field naturally emerges as a boson with mass zero.

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Institute of Experimental Particle Physics (IEKP)5

The problem of massive fermions

● Check :

● Transformation:

● In mass term :

● No obvious problem with fermion masses here. So is it a problem of non-abelian gauge symmetries?

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The problem of massive fermions

● Check :

● Transformation:

● In mass term :

● No obvious problem with fermion masses here. So is it a problem of non-abelian gauge symmetries?

Similarly no problem in → no problem of non-Abelian gauge field theories.

● Check :

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Institute of Experimental Particle Physics (IEKP)7

The problem of massive fermions

● Check :

● Transformation:

● In mass term :

● What is the problem of in the SM?

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Institute of Experimental Particle Physics (IEKP)8

The problem of massive fermions

● Check :

● Transformation:

● In mass term :

● What is the problem of in the SM?

● It is the distinction between left- ( ) and right-handed ( ) fermions, with different coupling structure:

singletlower component of doublet.

(1)

(1) check Exercise 9 e).

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Success of the

● Can motivate structure of interactions between elementary particles.

● Gives geometrical interpretation for the presence of gauge bosons (transport of phase information from one space point to another).

● Predicts non-trivial self-interactions and couplings of boson to left- and right-handed fermions.

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Dilemma of the

● Can motivate structure of interactions between elementary particles.

● Gives geometrical interpretation for the presence of gauge bosons (transport of phase information from one space point to another).

● Predicts non-trivial self-interactions and couplings of boson to left- and right-handed fermions.

Explicitly incompatible with (naive) incorporation of mass

terms in Lagrangian density.

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Institute of Experimental Particle Physics (IEKP)11

The remedy

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Spontaneous symmetry breaking

Needle on point: Block in water: Block on stick:

symmetry axis-symmetry symmetry

● Symmetry is present in the system (i.e. in the Lagrangian density ).

● BUT it is broken in the ground state (i.e. in the quantum vacuum).

● Three examples (from classical mechanics):

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Incorporation in particle physics

● invariant under transformations (i.e. symmetric).

● metastable in .

● ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in .

● Goldstone Potential:

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Incorporation in particle physics

● invariant under transformations (i.e. symmetric).

● metastable in .

● ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in .

● Goldstone Potential:

● has radial excitations in the potential .

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Incorporation in particle physics

● invariant under transformations (i.e. symmetric).

● metastable in .

● ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in .

● Goldstone Potential:

● can “move freely” in the circle that corresponds to the minimum of .

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The Goldstone theorem

In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created.

● In particle physics this is formalized in the Goldstone theorem:

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The Goldstone theorem

In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created.

● In particle physics this is formalized in the Goldstone theorem:

● Goldstone Bosons can be:

● Elementary fields, which are already part of .

● Bound states, which are created by the theory (e.g. the H-atom, Cooper-pairs, ...).

● Unphysical or gauge degrees of freedom, which can be removed by appropriate boundary conditions.

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Analyzing the energy ground state

is minimal. This is the case for .

● The energy ground state is where the Hameltonian operator

● To analyze the system in its physical ground state we can make an expansion in an arbitrary point on this cycle:

Top view on Goldstone potential

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Dynamic mass terms (part-1)

const.dynamic mass term

self-couplings

● An expansion in the ground state in cylindrical coordinates leads to:

● Why is there no linear term in ?

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Dynamic mass terms (part-1)

const.dynamic mass term

self-couplings

● We have performed a Taylor expansion in the minimum. By construction there cannot be any linear terms in there.

● An expansion in the ground state in cylindrical coordinates leads to:

● Why is there no linear term in ?

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Dynamic mass terms (part-1)

● An expansion in the ground state in cylindrical coordinates leads to:

● Remarks:

● The mass term is acquired for the field along the radial excitation, which leads out of the minimum of . It is the term at lowest order in the Taylor expansion in the minimum, and therefore independent from the concrete form of in the minimum.

● The field , which does not lead out of the minimum of does not acquire a mass term. It corresponds to the Goldstone boson.

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Extension to a gauge field theory

● For simplicity reasons shown for an Abelian model:

Introduce covariant derivative

Remove by proper gauge:

How does this gauge look like?

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Extension to a gauge field theory

● For simplicity reasons shown for an Abelian model:

Introduce covariant derivative

Remove by proper gauge:

How does this gauge look like?

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Extension to a gauge field theory

Introduce covariant derivative

Mass term for : Quartic and tri-linear couplings with .

● For simplicity reasons shown for an Abelian model:

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Institute of Experimental Particle Physics (IEKP)25

The Higgs mechanism

● The expansion of in the energy ground state of the Goldstone potential has generated a mass term for the gauge field from the bare coupling .

● is a real field, has been absorbed into . It seems as if one degree of freedom were lost. This is not the case:

● as a massive particle it gains one additional degree of freedom (±1-helicity states + 0-helicity state).

● as a massless particle has only two degrees of freedom (±1 helicity states).

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The Higgs mechanism

● The expansion of in the energy ground state of the Goldstone potential has generated a mass term for the gauge field from the bare coupling .

● is a real field, has been absorbed into . It seems as if one degree of freedom were lost. This is not the case:

● as a massive particle it gains one additional degree of freedom (±1-helicity states + 0-helicity state).

● as a massless particle has only two degrees of freedom (±1 helicity states).

One says:

“The gauge boson has eaten up the Goldstone boson and has become fat on it”.

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The Higgs mechanism

● The expansion of in the energy ground state of the Goldstone potential has generated a mass term for the gauge field from the bare coupling .

● is a real field, has been absorbed into . It seems as if one degree of freedom were lost. This is not the case:

● as a massive particle it gains one additional degree of freedom (±1-helicity states + 0-helicity state).

● as a massless particle has only two degrees of freedom (±1 helicity states).

One says:

“The gauge boson has eaten up the Goldstone boson and has become fat on it”. This shuffle

of degrees of freedom fro

m Goldstone boson(s) to gauge

boson(s) is called equivalence prin

ciple of the Higgs mechanism.

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Institute of Experimental Particle Physics (IEKP)28

Notes on the Goldstone potential (part-1)

● The choice of the Goldstone potential has the following properties:

● it leads to spontaneous symmetry breaking.

● it does not distinguish any direction in space (→ i.e. only depends on ).

● it is bound from below and does not lead to infinite negative energies, which is a prerequisite for a stable theory.

● it is the simplest potential with these features.

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Notes on the Goldstone potential (part-2)

● The potential has been chosen to be cut at the order of . This can be motivated by a dimensional analysis:

● What is the dimension of ?

● Due to gauge invariance has to appear in even order (c.f. transformation behavior of objects in Lecture-01).

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Institute of Experimental Particle Physics (IEKP)30

Notes on the Goldstone potential (part-2)

● The potential has been chosen to be cut at the order of . This can be motivated by a dimensional analysis:

● What is the dimension of ?

● Due to gauge invariance has to appear in even order (c.f. transformation behavior of objects in Lecture-01).

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Institute of Experimental Particle Physics (IEKP)31

Notes on the Goldstone potential (part-2)

● The potential has been chosen to be cut at the order of . This can be motivated by a dimensional analysis:

● What is the dimension of ?

● Due to gauge invariance has to appear in even order (c.f. transformation behavior of objects in Lecture-01).

● What is the dimension of ?

● What is the dimension of ?

● What is the dimension of ?

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Institute of Experimental Particle Physics (IEKP)32

Notes on the Goldstone potential (part-2)

● The potential has been chosen to be cut at the order of . This can be motivated by a dimensional analysis:

● What is the dimension of ?

● Due to gauge invariance has to appear in even order (c.f. transformation behavior of objects in Lecture-01).

● What is the dimension of ?

● What is the dimension of ?

● What is the dimension of ?

● NB: it would be possible to extend the potential to higher dimensions of , but couplings with negative dimension will turn the theory non-renormalizable.

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Concluding remarks

● Today we have discussed the problem of mass terms in the SM.

● Prepare “The Higgs Boson Discovery at the Large Hadron Collider” Section 2.4.

● Keep in mind that the SM has two problems of masses with different origin.

● We have introduced the principles of spontaneous symmetry breaking and how it translates into particle physics as the Goldstone theorem.

● Finally we have implemented the concept of spontaneous symmetry breaking into an (Abelian) gauge field theory to see how the Higgs mechanism works.

● Next we will go through the implementation of the Higgs mechanism into the SM. This step will complete the SM as a theory.

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