Weak Interactions – II · the local gauge symmetry, we ﬁnd that our would-be massless state...

Weak Interactions – II

Susan Gardner

Department of Physics and AstronomyUniversity of Kentucky

Lexington, KY 40506 [email protected]

Lecture 1: A survey of basic principles and features with an emphasis on low-energy probes

Lecture 2: The electroweak Standard Model — and interpreting it as an effective field theory

The Standard Model

describes all known electromagnetic, weak, and strong interactionphenomena in a formalism with predictive power.It has gauge bosons γ,W±,Z 0,g and three generations of quarks(

ud

) (cs

) (tb

)and leptons (

eνe

) (µνµ

) (τντ

)and a fundamental scalar HSM which mediates the generation of mass.

S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 2

Spontaneous Symmetry Breaking

Let’s begin with the example of a Heisenberg ferromagnet: H = −g∑

i,j Si · Sjwith g > 0.At T < Tc the system develops a non-zero magnetizationM 6= 0.The Hamiltonian is rotationally invariant, but its ground state is not – thesymmetry of H is hidden.Spontaneous symmetry breaking (SSB) also operates in QCD.The u,d quarks are very light compared to Mp. If mq = 0

LQCD = LLQCD + LR

QCD

If this chiral symmetry were explicit, one would expect the low-lying hadronicspectrum to contain parity doublets, but this does not occur.Perhaps the axial vector currents are spontaneously broken. [Nambu and Jona-Lasinio,

Phys. Rev. 122, 345 (1961).]

Goldstone’s Theorem: For every spontaneously broken continu-

ous symmetry, the theory must contain a massless particle (Gold-

stone boson). [Goldstone, Nuovo Cim. 19, 154 (1961)]


The Pattern of Low-Lying Meson Masses

Masses of states which differ only in (u,d) are nearly degenerate.

There are eight low-lying 0− states — π±, π0,K 0, K 0,K±, and η— the η′ is much heavier.

We can explain this pattern by invoking symmetries which are, in turn,approximate (isospin), spontaneously broken (chiral), and anomalous(axial U(1)).


Spontaneous Symmetry Breaking

Here’s a class of potentials which can be used to describe the spontaneousbreaking of a continuous symmetry...

A “Mexican Hat” Potential


Secret Symmetry

Let’s see how this can work. Consider the potential for a real field φ. Supposeµ2 > 0, real and λ > 0.

V (φ) =12µ2φ2 +

λ

4!φ4

This is symmetric under φ→ −φ, and the minimum energy state is φ = 0.What if µ2 → −µ2? Then

V (φ) = −12µ2φ2 +

λ

4!φ4

Now the minimum energy state corresponds to φ 6= 0! There are two minima.Expanding φ about one minimum, φ(x) = v + σ(x), e.g., we would find thatthe new potential no longer had σ → −σ manifest.If we had looked at the potential corresponding to the surface of revolution ofthis potential (N = 2 scalar fields), we would have had a continuoussymmetry, and if we had expanded about the vacuum expectation value, wewould have found a massless state.


The Higgs Mechanism

A continuous, local symmetry can be spontaneously broken without yieldingGoldstone bosons; rather, the gauge bosons gain mass. Our by now-familarpotential is that of the Higgs scalar field.

For now, we set aside the questionof why the W± and Z gauge bosonshave the masses that they do; thismechanism provides noexplanation for this, nor for thepattern of fermion masses.

A “Mexican Hat” Potential


Spontaneous Symmetry Breaking without Goldstone Bosons

Here we work in the context of a theory with local gauge invariance –and that makes all the difference!Let’s consider QED with no fermions but with a complex scalar field:

L = −14

FµνFµν + (Dµφ)(Dµφ∗)− V (φ)

This L is invariant under

φ(x)→ eiα(x)φ(x) ; Aµ(x)→ Aµ(x)− 1e∂µα(x)

This is a U(1) symmetry.N.B. φ is coupled to the photon through Dµ = ∂µ − ieAµ.Now when we expand φ about the minimum of V (φ), spontaneously breakingthe local gauge symmetry, we find that our would-be massless state givesmass to the photon!A model for the Meißner effect in Type I superconductors! [Landau-Ginzburg theory!]

This mechanism generalizes to non-Abelian gauge theory.


The Glashow-Salam-Weinberg Model

To build the Standard Model, we must

Pick the gauge group: The electroweak portion of the StandardModel is a quantum field theory based on a SU(2)L×U(1)Y localgauge symmetry.

Choose the particle content and its group representations and chargeassignments:We put a complex scalar field in a SU(2)L doublet. This upon SSB willyield 3 massive gauge bosons: W±,Z 0. Since the W± carry electriccharge, electromagnetism must “lie across” SU(2)L×U(1)Y . N.B.Q = T3 + Y . Note φ has Y = +1/2.We put the fermions in left-handed doublets and right-handed singletsgeneration by generation.E.g., (

νe

e−)

L≡ EL T3 = ±1

2Y = −1

2eR T3 = 0 Y = −1


The Glashow-Salam-Weinberg Model

Fermion MassesWe cannot give the fermions mass as in QED because

meψψ = me(ψLψR + ψRψL)

is forbidden by SU(2)L×U(1) gauge invariance!We must use the Higgs mechanism!

Le mass = −λeELφeR + h.c.

with φ0 = 1√2

(0v)

Le mass = − 1√2λeveLeR + h.c.

and we identify the electron mass me = 1√2λev .

For 3 generations of down-like and up-like quarks:

Lq mass = −λijd Q i

Lφd jR − λ

ijuQ i

Lφ†uj

R + h.c.

CP is broken if λijd,u are complex!

N.B. QL has T3 = ±1/2, Y = +1/6; uR has Y = +2/3; dR has Y = −1/3S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 10

Quark Masses and Mixings

The λijd,u can be anything! They are only constrained by symmetry

before we fit them to experiment.We can simplify by rotating to a basis in which the quark mass matrix isdiagonal

u′ iL = U iju uj

L ; d ′ iL = U ijd d j

L

This complicates the expression, however, for the quark charged weakcurrent:

JµW + =1√2

uiLγµd i

L

=1√2

u′ iL γµ(U†uUd )ijd ′ iL

Enter the Cabibbo-Kobayashi-Maskawa matrix:

VCKM = U †uUd

Uu,d is clearly unitary, and thus VCKM is also.S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 11

The Cabibbo-Kobayashi-Maskawa (CKM) Matrix

The decay K− → µ−νµ occurs: the quark mass eigenstates mix underthe weak interactions. By conventiond ′

s′

b′

weak

= VCKM

dsb

mass

; VCKM =

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

In the Wolfenstein parametrization (1983)

VCKM =

1− λ2

2 λ Aλ3(ρ− iη)

−λ 1− λ2

2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

+O(λ4)

where λ ≡ |Vus| ' 0.22 and is thus “small”. A, ρ, η are real.


Testing the Quark Mixing Matrix

In the Standard Model (SM)

There are three “generations” of particles. Thus, the CKM matrixis unitary.The unitarity of the CKM matrix and the structure of the weakcurrents implies that four parameters capture the CKM matrix.A real, orthogonal 3× 3 matrix is captured by three parameters.The fourth parameter (η) must make VCKM complex.All CP-violating phenomena are encoded in η.

To test the SM picture of CP violation we must test the relation-

ships it entails.


Testing CKM Unitarity – “the” Unitarity Triangle

Interesting anomalies may stand with improved statistics. There is alsosome evidence for the violation of lepton flavor univerality in B decay.


A Precision Test of CKM Unitarity

Using the global fits...The first row of the CKM matrix yields the most precise test of CKM unitarity:[PDG, 2014]

|Vud |2 + |Vus|2 + |Vub|2 = 0.9999± 0.0006whereas

|Vcd |2 + |Vcs|2 + |Vcb|2 = 1.024± 0.032|Vud |2 + |Vcd |2 + |Vtd |2 = 1.000± 0.004|Vus|2 + |Vcs|2 + |Vts|2 = 1.025± 0.032

N.B. the W leptonic width and the Vud unitarity test (1st row) yields

|Vcd |2 + |Vcs|2 + |Vcb|2 = 1.002± 0.027

cf. α + β + γ = (175± 9)◦.[“The CKM quark-mixing matrix,” PDG, 2014.]

Tests to this precision require precise computations of SM radiativecorrections...The precision of the current unitarity tests probes the reliability of thesecomputations to the sub-1% level.


EDMs in the Standard Model

QCD possesses a source of CP violation, namely, LCP = αs θ8π ε

αβµνF aαβF a

µν butthe tiny value of the neutron EDM tells us it does not operate. Why??

Why are they so small in the CKM model?The structure of the CKM matrix guarantees that dq is zero at two-loop order.[Shabilin, Sov. J. Nucl. Phys. 28, 75 (1978)]

The first non-trivial contributions come at 3 loops, the largest involving agluon [Khriplovich, PLB 173, 193 (1986)]

In leading logarithmic order at three loopsdCKM

d ' 10−34 e-cm.[Czarnecki and Krause, PRL 78, 4339 (1997)]

de for massless neutrinosfirst appears at 4-looporder:dCKM

e ≤ 10−38 e-cm[Khriplovich and Pospelov, Sov. J. Nucl. Phys.

53 (1991) 638.]

N.B. ν mixing gives some enhancement

[Ng and Ng, 1996; Archamlault, Czarnecki,

Pospelov, 2004]

Although CP is not a symmetry of the SM, these effects areunobservably small and thus EDMs give us a window on new physics.


EDMs in the Standard Model

Chiral dynamics can give an enhancement

Namely, dCKMn ' 10−32 e-cm.

[Gavela et al., PLB 109, 215 (1982); Khriplovich and Zhitnitsky, PLB 109, 490 (1982).]

Bound-state effects involving charm may also yield an enhancement:

Namely, dCKMn ' 10−31 e-cm. [Mannel and Uraltsev, 2012]

Even with such effects SM EDMs remain unobservably small.S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 17

The SM as an effective theory

That the charged weak current in the SM is universal and respects a “V-A”law is captured by

New physics searches at low energy trade on the ability to test thepattern of the SM through precision measurement


Theoretical Framework for Charged Current Processes

(N.B. v = (2√

2GF )−1/2)S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 19

Theoretical Framework



N.B. neutron decay also probes εT !S. Gardner (Univ. of Kentucky) Weak Interactions – II FNP Summer School, UTK, 6/15 23

What of the “Two-Component Neutrino”?

We have direct empirical evidence from terrestrial experiments for physicsbeyond the Standard Model.The observation of neutrino oscillations [KamLAND, PRL 94, 081801 (2005)]

allows us to conclude ∆m2 ≡ m2i −m2

j 6= 0 with surety.That is, neutrinos have mass.We see then that the Standard Model is incomplete: effectively there is a νR ,which is “sterile” under Standard Model interactions.This is not to say that the effects of neutrino mass are large.Distortions in the shape of the electron energy spectrum in 3H β-decay nearits endpoint bound m2

ν . [KATRIN] Measuring cyclotron radiation can permit aprecise determination of the electron energy (and of a!)! [Project 8, arXiv:1408.5362]

The neutrino mass is still “practically” zero, but the two-component neutrinopicture may fail – the neutrino may not be its own antiparticle!


Themes in the Search for New Physics

Could understanding flavor differences help crack the puzzle?


Weak Interactions – II · the local gauge symmetry, we ﬁnd that our would-be massless state...

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