spontaneous symmetry breaking and the higgs mechanismekprwolf/teaching/ss16/... · 2016. 10....
TRANSCRIPT
![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:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/1.jpg)
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
![Page 2: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/2.jpg)
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”?
![Page 3: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/3.jpg)
Institute of Experimental Particle Physics (IEKP)3
Mass terms in the SM
![Page 4: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/4.jpg)
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.
![Page 5: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/5.jpg)
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?
![Page 6: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/6.jpg)
Institute of Experimental Particle Physics (IEKP)6
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 :
![Page 7: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/7.jpg)
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?
![Page 8: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/8.jpg)
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).
![Page 9: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/9.jpg)
Institute of Experimental Particle Physics (IEKP)9
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.
![Page 10: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/10.jpg)
Institute of Experimental Particle Physics (IEKP)10
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.
![Page 11: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/11.jpg)
Institute of Experimental Particle Physics (IEKP)11
The remedy
![Page 12: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/12.jpg)
Institute of Experimental Particle Physics (IEKP)12
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):
![Page 13: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/13.jpg)
Institute of Experimental Particle Physics (IEKP)13
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:
![Page 14: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/14.jpg)
Institute of Experimental Particle Physics (IEKP)14
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 .
![Page 15: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/15.jpg)
Institute of Experimental Particle Physics (IEKP)15
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 .
![Page 16: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/16.jpg)
Institute of Experimental Particle Physics (IEKP)16
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:
![Page 17: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/17.jpg)
Institute of Experimental Particle Physics (IEKP)17
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.
![Page 18: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/18.jpg)
Institute of Experimental Particle Physics (IEKP)18
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
![Page 19: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/19.jpg)
Institute of Experimental Particle Physics (IEKP)19
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 ?
![Page 20: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/20.jpg)
Institute of Experimental Particle Physics (IEKP)20
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 ?
![Page 21: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/21.jpg)
Institute of Experimental Particle Physics (IEKP)21
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.
![Page 22: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/22.jpg)
Institute of Experimental Particle Physics (IEKP)22
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?
![Page 23: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/23.jpg)
Institute of Experimental Particle Physics (IEKP)23
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?
![Page 24: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/24.jpg)
Institute of Experimental Particle Physics (IEKP)24
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:
![Page 25: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/25.jpg)
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).
![Page 26: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/26.jpg)
Institute of Experimental Particle Physics (IEKP)26
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”.
![Page 27: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/27.jpg)
Institute of Experimental Particle Physics (IEKP)27
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.
![Page 28: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/28.jpg)
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.
![Page 29: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/29.jpg)
Institute of Experimental Particle Physics (IEKP)29
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).
![Page 30: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/30.jpg)
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).
![Page 31: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/31.jpg)
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 ?
![Page 32: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/32.jpg)
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.
![Page 33: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/33.jpg)
Institute of Experimental Particle Physics (IEKP)33
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.
![Page 34: Spontaneous Symmetry Breaking and the Higgs Mechanismekprwolf/teaching/ss16/... · 2016. 10. 25. · Spontaneous symmetry breaking Needle on point: Block in water: Block on stick:](https://reader035.vdocument.in/reader035/viewer/2022071412/610885c2028fe95f64358254/html5/thumbnails/34.jpg)
Institute of Experimental Particle Physics (IEKP)34
Backup