Electroweak Symmetry Breaking and the Higgs Mechanism
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1 Electroweak Symmetry Breaking and the Higgs Mechanism Roger Wolf 06. Mai 2014 INSTITUTE OF EXPERIMENTAL PARTICLE PHYSICS (IEKP) PHYSICS FACULTY KIT University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz Association
2 Recap from Last Time Charged current interaction: Neutral current interaction: can describe electroweak IA including gauge boson self- couplings. 2
3 Quiz of the Day 3 Is the mass problem the same for boson and fermion masses? Is the following statement true: the Higgs boson couples proportional to the mass to all massive particles? We have seen that QED does not at all have a problem with fermion masses (first lecture). Are fermion masses a problem specific to non-abelian gauge symmetries? Is the Higgs boson a Goldstone boson? If not what is the difference between these types of particles?
4 Schedule for Today 3 The Higgs Mechanism in the SM 2 Spontaneous Symmetry Breaking & Higgs Mechanism 1 The Problem of Masses in the SM 4
5 The Problem of Masses in the SM 5
6 Problem 1: Massive Gauge Bosons Example: Abelian gauge field theories ( first lecture) Transformation: In mass term : These terms explicitly break local gauge covariance of. 6
7 Problem 1: Massive Gauge Bosons Example: Abelian gauge field theories ( first lecture) Transformation: In mass term : These terms explicitly break local gauge covariance of. 7 Fundamental problem for all gauge field theories!
8 Problem 2: Massive Fermions 8 No problem in Transformation: In mass term : :
9 Problem 2: Massive Fermions 9 No problem in Transformation: In mass term : : Similarly no problem in (not specific to non-abelian gauge field theories).
10 Problem 2: Massive Fermions No problem in Transformation: In mass term : : Similarly no problem in So what is the problem of the 10 (not specific to non-abelian gauge field theories). in the SM?!
11 Problem 2: Massive Fermions No problem in Transformation: In mass term : : Similarly no problem in So what is the problem of the (not specific to non-abelian gauge field theories). It is the distinction between left-handed ( 11 in the SM?! ) and right-handed ( ) fermions:
12 Dilemma of the Gauge Field Theory Local gauge invariance can motivate all interactions between elementary particles.... gives a geometrical interpretation for the presence of gauge bosons (propagate info of local phases btw space points).... predicts non trivial self-interactions between gauge bosons!
13 Dilemma of the Gauge Field Theory Local gauge invariance a ss m... can motivate all interactions between elementary particles. e naiv f o on i t a r o presence of gauge bosons... gives a geometrical interpretationcoofrpthe n i with points). (propagate info of local phasesibbtw space e l pat m o c in y l... predicts non trivial self-interactions p e! De in between tgauge erms bosons!
14 The Remedy 14
15 Spontaneous Symmetry Breaking 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): Needle on point: symmetry 15 Block in water: axis-symmetry ). Block on stick: symmetry
16 Application to Particle Physics Goldstone Potential: 16 invariant under (i.e. symmetric). metastable in transformations. ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in.
17 Application to Particle Physics Goldstone Potential: 17 invariant under (i.e. symmetric). metastable in transformations. has radial excitations in the potential. ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in.
18 Application to Particle Physics Goldstone Potential: 18 invariant under (i.e. symmetric). metastable in transformations. ground state breaks symmetry, BUT at the same time all ground states are in-distinguishable in. can move freely in the circle that corresponds to the minimum of.
19 The Goldstone Theorem In particle physics this is formalized in the Goldstone Theorem: In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created. 19
20 The Goldstone Theorem In particle physics this is formalized in the Goldstone Theorem: In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created. Goldstone Bosons can be: 20 Elementary fields, which are already part of.
21 The Goldstone Theorem In particle physics this is formalized in the Goldstone Theorem: In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created. Goldstone Bosons can be: Elementary fields, which are already part of Bound states, which are created by the theory (e.g. the H-atom). 21.
22 The Goldstone Theorem In particle physics this is formalized in the Goldstone Theorem: In a relativistic covariant quantum field theory with spontaneously broken symmetries massless particles (=Goldstone bosons) are created. Goldstone Bosons can be: Elementary fields, which are already part of Bound states, which are created by the theory (e.g. the H-atom). Unphysical or gauge degrees of freedom. 22.
23 Analyzing the Energy Ground State The energy ground state is where the Hameltonian is minimal. This is the case for. To analyze the system in its physical ground state we make an expansion in an arbitrary point on the cycle: Up view on Goldstone potential 23
24 Goldstone Bosons & Dynamic Massive Terms An expansion in the ground state in cylindrical coordinates leads to: const. dynamic mass term self-couplings Why is there no linear term in 24?
25 Goldstone Bosons & Dynamic Massive Terms An expansion in the ground state in cylindrical coordinates leads to: const. dynamic mass term self-couplings Why is there no linear term in? We have performed a Taylor expansion in the minimum. By construction there cannot be any linear terms in there. 25
26 Goldstone Bosons & Dynamic Massive Terms An expansion in the ground state in cylindrical coordinates leads to: Remarks: 26 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.
27 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? 27
28 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? 28
29 Extension to a Gauge Field Theory For simplicity reasons shown for an Abelian model: Introduce covariant derivative Mass term for 29 : Quartic and trilinear couplings with.
30 Higgs Mechanism Goldstone potential and expansion of state has generated a mass term the bare coupling. 30 in the energy ground for the gauge field from
31 Higgs Mechanism was originally complex ( i.e. w/ 2 degrees of freedom). is a real field, has been absorbed into. It seems as if one degree of freedom had been lost. This is not the case: 31 as a massless particle has only two degrees of freedom (2 longitudinal polarizations). as a massive particle it gains one additional degree of freedom (2 longitudinal +1 transverse polarization).
32 Higgs Mechanism was originally complex ( i.e. w/ 2 degrees of freedom). is a real field, has been absorbed into. It seems as if one degree of freedom had been lost. This is not the case: as a massless particle has only two degrees of freedom (2 longitudinal polarizations). as a massive particle it gains one additional degree of freedom (2 longitudinal +1 transverse polarization). One says: The gauge boson has eaten up the Goldstone boson and has become fat on it. 32
33 Higgs Mechanism was originally complex ( i.e. w/ 2 degrees of freedom). e hdegree t is a real field, has been absorbed into. It seems as if one to ) of freedom had been lost. This is not the case: n(s o os b e (2 longitudinal polarizations). as a massless particle has only two degrees of freedom n o st d l o G as a massive particle it gains one additional degree of freedom (2 longitudinal +1 transe h t. verse polarization). m m o fr n i s m ha o c d ee s me r f of igg One says: s e dh e r lle up the Goldstone boson and has become eg ceaten The gauge bosondhas a f o fat on it. ffle (s) is hu son s o is b h T uge ga 33
34 Notes on the Goldstone Potential The choice of the Goldstone potential has the following properties: 34 it leads to spontaneous symmetry breaking.
35 Notes on the Goldstone Potential The choice of the Goldstone potential has the following properties: 35 it leads to spontaneous symmetry breaking. it does not distinguish any direction in space ( i.e. only depends on ).
36 Notes on the Goldstone Potential 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 36 ). it is bound from below and does not lead to infinite negative energies, which is a prerequisite for a stable theory.
37 Notes on the Goldstone Potential The potential has been chosen to be cut at the order of be motivated by a dimensional analysis: 37 Due to gauge invariance What is the dimension of. This can has to appear in even order.?
38 Notes on the Goldstone Potential The potential has been chosen to be cut at the order of be motivated by a dimensional analysis: 38 Due to gauge invariance What is the dimension of. This can has to appear in even order.?
39 Notes on the Goldstone Potential The potential has been chosen to be cut at the order of be motivated by a dimensional analysis: 39 Due to gauge invariance What is the dimension of? What is the dimension of? What is the dimension of? What is the dimension of?. This can has to appear in even order.
40 Notes on the Goldstone Potential The potential has been chosen to be cut at the order of be motivated by a dimensional analysis: 40 Due to gauge invariance What is the dimension of? What is the dimension of? What is the dimension of? What is the dimension of?. This can has to appear in even order.
41 Notes on the Goldstone Potential The potential has been chosen to be cut at the order of be motivated by a dimensional analysis: Due to gauge invariance What is the dimension of? What is the dimension of? What is the dimension of? What is the dimension of?. This can has to appear in even order. It would be possible to extend the potential to higher dimensions of but couplings with negative dimension will turn the theory non-renormalizable. 41
42 Final Construction of the SM 42
43 The SM without mass terms Compilation of the last two lectures: Fermion kinematic 43
44 The SM without mass terms Compilation of the last two lectures: Charged current IA Fermion kinematic 44
45 The SM without mass terms Compilation of the last two lectures: Charged current IA Fermion kinematic 45 Neutral current IA
46 The SM without mass terms Compilation of the last two lectures: Charged current IA Fermion kinematic 46 Neutral current IA Gauge field kinematic
47 Extension by a new field Add as doublet field: Transformation behavior 47 Can you point to the Goldstone bosons?
48 Extension by a new field Add as doublet field: Transformation behavior 48 Can you point to the Goldstone bosons?
49 Extension by a new field Add as doublet field: Transformation behavior 49 is covariant under global Can you point to the Goldstone bosons? transformations.
50 Extension by a new field Add as doublet field: Transformation behavior Introduce covariant derivative Can you point to the Goldstone bosons? to enforce local gauge invariance: (analogue to fermion fields) (Gell-Mann Nischijama) 50
51 Expansion in the Energy Ground State of Develop in its energy ground state at : w/o loss of generality this can be done in the lower component of. Non-zero vacuum expectation value. 51 Radial excitation field. This is the Higgs boson!
52 Expansion in the Energy Ground State of Develop in its energy ground state at : w/o loss of generality this can be done in the lower component of. couples gauge fields to. 52
53 Expansion in the Energy Ground State of Develop in its energy ground state at : w/o loss of generality this can be done in the lower component of. couples gauge fields to. (covariant derivative) 53
54 Expansion in the Energy Ground State of Develop in its energy ground state at : w/o loss of generality this can be done in the lower component of. couples gauge fields to. (covariant derivative) 54
55 Expansion in the Energy Ground State of Develop in its energy ground state at : w/o loss of generality this can be done in the lower component of. couples gauge fields to. (covariant derivative) 55
56 Expansion in the Energy Ground State of Resolve products of Pauli matrices ( 56 ):
57 Expansion in the Energy Ground State of Resolve products of Pauli matrices ( 57 Ascending operator ( ): ) shifts unit vector of the down component up.
58 Expansion in the Energy Ground State of Resolve products of Pauli matrices ( 58 Ascending operator Descending operator ( ): ) shifts unit vector of the down component up. ( ) destroys unit vector of the down component.
59 Expansion in the Energy Ground State of Resolve products of Pauli matrices ( 59 Ascending operator Descending operator Operator ( ): ) shifts unit vector of the down component up. ( ) destroys unit vector of the down component. switches sign for unit vector of down component.
60 Expansion in the Energy Ground State of Evaluate components of absolute value squared: 60
61 Expansion in the Energy Ground State of Evaluate components of absolute value squared: 61
62 Expansion in the Energy Ground State of Evaluate components of absolute value squared: 62
63 Expansion in the Energy Ground State of Evaluate components of absolute value squared: 63
64 Masses for Gauge Bosons By introducing as a state we have obtained: doublet with a non-zero energy ground Dynamic mass terms for the gauge bosons: Characteristic trilinear and quartic couplings of the gauge bosons to the Higgs field. A solid prediction of the SM on the masses of the gauge bosons: 64
65 Gauge Degrees of Freedom We had discussed how gauge bosons obtain mass by a gauge that absorbs the Goldstone bosons in the theory. 65 As a complex doublet four degrees of freedom. has In the final formulation only the radial excitation of did remain. The Goldstone bosons ( ) have been absorbed into the gauge fields &, which have obtained masses from this.
66 Gauge Degrees of Freedom We had discussed how gauge bosons obtain mass by a gauge that absorbs the Goldstone bosons in the theory. As a complex doublet four degrees of freedom. has In the final formulation only the radial excitation of did remain. The Goldstone bosons ( ) have been absorbed into the gauge fields &, which have obtained masses from this. Congratulations - you got it!!! 66
67 Gauge Degrees of Freedom We had discussed how gauge bosons obtain mass by a gauge that absorbs the Goldstone bosons in the theory. As a complex doublet four degrees of freedom. In the final formulation only the radial excitation of did remain. The Goldstone bosons ( ) have been absorbed into the gauge fields &, which have obtained masses from this. Congratulations - you got it!!! Almost has
68 Solution to the Problem of Fermion Masses 68
69 Solution to the Problem of Fermion Masses The Higgs mechanism can also help to obtain mass terms for fermions, by coupling the fermions to. 69 check behavior: singlet singlet
70 Solution to the Problem of Fermion Masses The Higgs mechanism can also help to obtain mass terms for fermions, by coupling the fermions to. 70 check behavior: singlet singlet
71 Solution to the Problem of Fermion Masses The Higgs mechanism can also help to obtain mass terms for fermions, by coupling the fermions to. 71 check behavior: singlet singlet check behavior:
72 Solution to the Problem of Fermion Masses The Higgs mechanism can also help to obtain mass terms for fermions, by coupling the fermions to. check behavior: singlet check behavior: singlet 72 singlet singlet
73 Solution to the Problem of Fermion Masses The Higgs mechanism can also help to obtain mass terms for fermions, by coupling the fermions to. check behavior: singlet check behavior: singlet 73 singlet NB: singlet Manifest gauge invariant. can be chosen real. Residual phases can be re-defined in.
74 Solution to the Problem of Fermion Masses Expand in its energy ground state to obtain the mass terms: We obtained the desired mass term and a coupling to the Higgs boson field, which is proportional to the fermion mass. Check the relation: 74
75 Solution to the Problem of Fermion Masses Expand in its energy ground state to obtain the mass terms: We obtained the desired mass term and a coupling to the Higgs boson field, which is proportional to the fermion mass. Check the relation: Here comes the 64$ question: On slide 11 I told you that terms of type Did I lie to you? When yes, when? 75 break gauge invariance.
76 Solution to the Problem of Fermion Masses Expand in its energy ground state to obtain the mass terms: We obtained the desired mass term and a coupling to the Higgs boson field, which is proportional to the fermion mass. Check the relation: Here comes the 64$ question: On slide 11 I told you that terms of type break gauge invariance. Did I lie to you? When yes, when? Of course I never lie to you. Single terms of this type do indeed break gauge invariance. It is the combination w/ the coupling to the Higgs boson field, which restores gauge invariance. 76
77 Concluding Remarks The Higgs mechanism = incorporation of spontaneous symmetry breaking into a gauge field theory. This leads to the fact that the gauge bosons eat up the Goldstone bosons, which exist in the system and gain mass on this. This mechanism leaves one or more degrees of freedom of radial excitations in the potential behind as Higgs boson(s). The Higgs boson obtains its mass from the Goldstone potential. The gauge bosons obtain their mass from their coupling to via the covariant derivative. The Fermions obtain their mass via a direct Yukawa coupling to. Gauge bosons couple to Higgs like Higgs like. 77, fermion fields couple to
78 Sneak Preview for Next Week Wrap up what we have learned during the last three lectures. Discuss the way from Lagrangian to measurable quantities ( Feynman rules). Discuss loop corrections and higher orders to tree level calculations (pictorially). Constraints on 78 within the theory itself.
79 Backup & Homework Solutions 79
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