(Quantum) Field Theory and the Electroweak Standard Model ...€¦ · A. Bednyakov (JINR) QFT & EW SM 2 / 32. An Ode to Symmetry... With Action you can also studySymmetries... The

(Quantum) Field Theoryand

the Electroweak Standard ModelLecture II

Alexander Bednyakov

Bogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear Research

The CERN-JINR European School of High-Energy Physics,St. Petersburg, Russia, 4 - 17 September 2019

A. Bednyakov (JINR) QFT & EW SM 1 / 32

Outline

Lecture I

What is the Standard Model?Introducing Quantum FieldsInteractions and Perturbation TheoryRenormalizable or Non-Renormalizable?

Lecture II

An Ode to SymmetryGlobal Symmetries (and Conserved Quantities)Local Symmetries (and Gauge interactions)From Fermi Model to EW theory

Lecture III

Finalizing the EW SM (a bit of Higgsing)“Features” of the SMExperimental tests of the EW SMIssues and Prospects of the EW SM


An Ode to Symmetry...With Action you can alsostudy Symmetries...

The latter are intimatelyconnected withtransformations, which leavesomething invariant...

Symmetries are not onlybeautiful but also very useful:

An architect can design only half of thebuilding (parity x → −x)

And winter decoration willtake much less time(rotation by a finite angle)


Field Theory: Symmetries

Transformations can be discrete, e.g.,

Parity : 𝜑′(x, t) = P𝜑(x, t) = 𝜑(−x, t),

Time-reversal : 𝜑′(x, t) = T𝜑(x, t) = 𝜑(x,−t),

Charge-conjugation : 𝜑′(x, t) = C𝜑(x, t) = 𝜑†(x, t),

or depend on continuous parameters, e.g.,

𝜑(x) 𝜑′(x)

x x + a

𝛿𝜑

Re 𝜑(x)

Im 𝜑(x)

𝜑(x)

𝜑′(x)

𝛼

𝜑′(x + a) = 𝜑(x) 𝜑′(x) = e i𝛼𝜑(x)

We can distinguish space-time and internal symmetries.

For x-dependent parameters we have local (gauge) transformations.








𝜑(x) 𝜑′(x)

x x + a

𝛿𝜑

Re 𝜑(x)

Im 𝜑(x)

𝜑(x)

𝜑′(x)

𝛼

𝜑′(x + a) = 𝜑(x) 𝜑′(x) = e i𝛼𝜑(x)










𝜑(x) 𝜑′(x)

x x + a

𝛿𝜑

Re 𝜑(x)

Im 𝜑(x)

𝜑(x)

𝜑′(x)

𝛼

𝜑′(x + a) = 𝜑(x) 𝜑′(x) = e i𝛼𝜑(x)




Global Continuous Symmetries: Noether TheoremGiven 𝒮[𝜑] one can find its symmetries, i.e., particular infinitesimalvariations 𝛿𝜑(x) that for any 𝜑 leave 𝒮[𝜑] invariant up to a surface term

𝒮[𝜑′(x)]− 𝒮[𝜑(x)] =∫

d4x 𝜕𝜇𝒦𝜇, 𝜑′(x) ≡ 𝜑(x) + 𝛿𝜑(x).

We compare this with

𝒮[𝜑′(x)]− 𝒮[𝜑(x)] =∫

d4x

[(𝜕𝜇

𝜕ℒ𝜕𝜕𝜇𝜑

− 𝜕ℒ𝜕𝜑

)𝛿𝜑+ 𝜕𝜇

(𝜕ℒ𝜕𝜕𝜇𝜑

𝛿𝜑

)].

and require 𝜑(x) to satisfy EOMs. This results in a local conservation law:

𝜕𝜇J𝜇 = 0, J𝜇 ≡ 𝒦𝜇 −𝜕ℒ𝜕𝜕𝜇𝜑

𝛿𝜑

Integration over space leads to the conserved charge

d

dtQ = 0, Q =

∫dxJ0

NB: If 𝛿𝜑 = 𝜌i𝛿i𝜑 depends on parameters 𝜌i , we have a conservation lawfor every 𝜌i . For Global symmetries 𝜌i do not depend on coordinates.


Global Continuous Symmetries: Noether TheoremGiven 𝒮[𝜑] one can find its symmetries, i.e., particular infinitesimalvariations 𝛿𝜑(x) that for any 𝜑 leave 𝒮[𝜑] invariant up to a surface term

𝒮[𝜑′(x)]− 𝒮[𝜑(x)] =∫

d4x 𝜕𝜇𝒦𝜇, 𝜑′(x) ≡ 𝜑(x) + 𝛿𝜑(x).

We compare this with

𝒮[𝜑′(x)]− 𝒮[𝜑(x)] =∫

d4x

[��

��XXXXXXXXX

(𝜕𝜇

𝜕ℒ𝜕𝜕𝜇𝜑

− 𝜕ℒ𝜕𝜑

)𝛿𝜑+ 𝜕𝜇

(𝜕ℒ𝜕𝜕𝜇𝜑

𝛿𝜑

)].

and require 𝜑(x) to satisfy EOMs. This results in a local conservation law:

𝜕𝜇J𝜇 = 0, J𝜇 ≡ 𝒦𝜇 −𝜕ℒ𝜕𝜕𝜇𝜑

𝛿𝜑

Integration over space leads to the conserved charge

d

dtQ = 0, Q =

∫dxJ0

NB: If 𝛿𝜑 = 𝜌i𝛿i𝜑 depends on parameters 𝜌i , we have a conservation lawfor every 𝜌i . For Global symmetries 𝜌i do not depend on coordinates.


The Noether Theorem: Space-time symmetriesConsider space-time translations

𝜑′(x + a) = 𝜑(x)

expand in a ⇒ 𝛿𝜑(x) = −a𝜈𝜕𝜈𝜑(x),

𝛿ℒ(𝜑(x), 𝜕𝜇𝜑(x)) = 𝜕𝜈 (−a𝜈ℒ)

𝜑(x) 𝜑′(x)

x x + a

𝛿𝜑

Local conservation of Energy-Momentum Tensor T𝜇𝜈 :

J𝜇 = −a𝜇ℒ+ a𝜈𝜕ℒ𝜕𝜕𝜇𝜑

𝜕𝜈𝜑 = a𝜈T𝜇𝜈 , 𝜕𝜇T𝜇𝜈 = 0

leads to time-independent “charges”

P𝜈 =

∫dxT0𝜈

Ex: For ℒ = |𝜕𝜇𝜑|2 +m2|𝜑|2 find P𝜇 = (ℋ,P). Substitute 𝜑(x) by itsexpansion in terms of operators a±p and b±p and get the expressions

(modulo operator ordering ambiguities) for the Hamiltonian ℋ and3-momentum P operators.


The Noether Theorem: Internal symmetries

There is an additional symmetry of

ℒ = 𝜕𝜇𝜑†𝜕𝜇𝜑−m2𝜑†𝜑

𝜑′(x) = e i𝛼𝜑(x)

𝛿𝜑(x) = i𝛼𝜑(x)

J𝜇 = i(𝜑†𝜕𝜇𝜑− 𝜑𝜕𝜇𝜑†)

Re 𝜑(x1)

Im 𝜑(x1)

𝜑(x1)

𝜑′(x1)

𝛼Re 𝜑(x2)

Im 𝜑(x2)

𝜑(x2)

𝜑′(x2)

𝛼

It is a U(1) symmetry:

It acts in internal space (“rotates” complex number 𝜑(x) at every x)

It is a global symmetry (rotation angle 𝛼 does not depend on x).

Ex: Express the corresponding charge operator Q in terms of a± and b±.


Symmetries in Quantum Field TheoryAfter quantisation the operators corresponding to the conserved quantities

can be used to define a convenient basis of states (quantumnumbers), e.g.,:

|p⟩ ≡ |p,+1⟩, |p⟩ ≡ |p,−1⟩ ⇒ Q|p, q⟩ = q|p, q⟩, P|p, q⟩ = p|p, q⟩

act as generators of symmetries, which are represented by unitary†

operators, e.g., for the space-time translations:

U(a) = exp(i P𝜇a𝜇

), 𝜑(x + a) = U(a)𝜑(x)U†(a)

NB: A symmetry 𝒮 guarantees that transition probability between statesdo not change upon transformation

|Ai ⟩𝒮→ |A′

i ⟩, 𝒫(Ai → Aj) = 𝒫(A′i → A′

j), |⟨Ai |Aj⟩|2 = |⟨A′i |A′

j⟩|2

and represented by (anti) unitary operators:

|A′i ⟩ = U|Aj⟩, ⟨A′

i |A′j⟩ = ⟨Ai |U†U⏟ ⏞

1

|Aj⟩

†or anti-unitary (time-reversal)A. Bednyakov (JINR) QFT & EW SM 10 / 32

Local (Gauge) SymmetriesLet’s consider the free Dirac Lagrangian:

ℒ0 = 𝜓(i𝜕 −m

)𝜓

and make global U(1)-symmetry

𝜓 → 𝜓′ = e ie𝜔𝜓

local, i.e., 𝜔 → 𝜔(x)

𝛿ℒ0 = 𝜕𝜇𝜔 · J𝜇, J𝜇 = −e𝜓𝛾𝜇𝜓, a short-cut to Noether current

To compensate 𝛿ℒ0 we add an interaction of the current J𝜇 with field A𝜇:

ℒ0 → ℒ = ℒ0 + A𝜇J𝜇 = 𝜓[i(𝜕 + ieA)−m

]𝜓, A𝜇 → A′

𝜇 = A𝜇 − 𝜕𝜇𝜔

NB: Symmetries not only restrict possible interactions, but can also forceinteractions to be introduced.


QED: Local U(1) Symmetry

We constructed Quantum ElectroDynamics:

ℒQED = 𝜓(i D −m

)𝜓 − 1

4F 2𝜇𝜈

D𝜇 = 𝜕𝜇 + ieA𝜇, covariant derivative,

F𝜇𝜈 = 𝜕𝜇A𝜈 − 𝜕𝜈A𝜇, field strength tensor.

𝜓 → 𝜓′ = e ie𝜔(x)𝜓

A𝜇 → A′𝜇 = A𝜇 − 𝜕𝜇𝜔

D𝜇𝜓 → D ′𝜇𝜓

′ = e ie𝜔(x)D𝜇𝜓,

Covariant derivative makes the combination 𝜓†𝛼D𝜇𝜓𝛽 gauge-invariant.

(Ex: make it also Lorentz-invariant, 𝛼, 𝛽 are Dirac indices:)


QED: Quantization IssuesThe second Noether theorem says that theories possessing gaugesymmetries are redundant (some degrees of freedom are not physical).Let’s add a gauge-fixing term to the free vector-field Lagrangian:

ℒ0(A) = −1

4F 2𝜇𝜈 −

1

2𝜉(𝜕𝜇A𝜇)

2 ≡ −1

2A𝜇K𝜇𝜈A𝜈 .

This term allows one to obtain the photon propagator by inverting K𝜇𝜈 :

⟨0|TA𝜇(x)A𝜈(y)|0⟩ =∫

d4p

(2𝜋)4−i[g𝜇𝜈 − (1− 𝜉)p𝜇p𝜈/p

2]

p2 + i𝜖e−ip(x−y)

The propagator now involves an auxiliary parameter 𝜉. It controlspropagation of unphysical longitudinal polarization 𝜖L𝜇 ∝ p𝜇. But thecorresponding terms drop out of physical quantities*

eL𝜇J𝜇 ∝ p𝜇J𝜇 = 0, we have no source for unphysical 𝛾

NB: The propagator drops down for large p as 1/p2.*serves as a good cross-check of lengthy computations

So far we managed to describe half of the EW theory - QED...What about the Weak part?


QED: Quantization IssuesThe second Noether theorem says that theories possessing gaugesymmetries are redundant (some degrees of freedom are not physical).Let’s add a gauge-fixing term to the free vector-field Lagrangian:

ℒ0(A) = −1

4F 2𝜇𝜈 −

1

2𝜉(𝜕𝜇A𝜇)

2 ≡ −1

2A𝜇K𝜇𝜈A𝜈 .

This term allows one to obtain the photon propagator by inverting K𝜇𝜈 :

⟨0|TA𝜇(x)A𝜈(y)|0⟩ =∫

d4p

(2𝜋)4−i[g𝜇𝜈 − (1− 𝜉)p𝜇p𝜈/p

2]

p2 + i𝜖e−ip(x−y)

The propagator now involves an auxiliary parameter 𝜉. It controlspropagation of unphysical longitudinal polarization 𝜖L𝜇 ∝ p𝜇. But thecorresponding terms drop out of physical quantities*

eL𝜇J𝜇 ∝ p𝜇J𝜇 = 0, we have no source for unphysical 𝛾

NB: The propagator drops down for large p as 1/p2.*serves as a good cross-check of lengthy computationsSo far we managed to describe half of the EW theory - QED...What about the Weak part?


Fermi Model: Harbinger of the EW theoryIn 1957 R. Marshak & G.Sudarshan, R. Feynman & M. Gell-Mannmodified the original Fermi theory of beta-decay to incorporate 100 %violation of Parity discovered by C.S. Wu in 1956

ℒFermi =GF

2√2(J+𝜇 J

−𝜇 + h.c.),

J−𝜌 = (V − A)nucleons𝜌 +Ψe𝛾𝜌 (1− 𝛾5)Ψ𝜈e +Ψ𝜇𝛾𝜌 (1− 𝛾5)Ψ𝜈𝜇 + ...

This is current-current interactions with GF ≃ 10−5 GeV−1.

Under Parity:

V 0 P→ V 0, VP→ −V,

A0 P→ −A0, AP→ A.

NB: Parity P is conserved for both a pure vectoror axial interactions! Only V 𝜇A𝜇 plays a role!NB: Charge-conjugation C is also violated!

Ex: Having in mind that 𝜓P→ 𝛾0𝜓, prove that 𝜓𝛾𝜇𝜓 ( 𝜓𝛾𝜇𝛾5𝜓) transform as V (A).


From Fermi Model to EW theoryFrom dimensional grounds we can estimate

𝜎(𝜈ee → 𝜈ee) ∝ G 2F s, s = (pe + p𝜈)

2.

This is another manifestation of self-inconsistency of Non-Renormalizablemodels: eventually they violate of unitarity!

The modern view on the Fermi model treats it as an effective theorywith certain limits of applicability:

Warning!

around G−1/2F ∼ 102 − 103 GeV there should be some “New Physics”.

QED is renormalizable. By analogy we introduce mediators -electrically charged vector fields W±

𝜇 :

ℒFermi =GF

2√2(J+𝜇 J

−𝜇 + h.c.)

→ ℒI = − g

2√2(W+

𝜇 J−𝜇 + h.c.).

e− 𝜈e

𝜈e e−

W+


Fermi Model vs QED: Chiral structureThe QED vertex conserves chirality and treat 𝜓L and 𝜓R on equal footing:

ℒI ∋ −eA𝜇 · 𝜓𝛾𝜇𝜓 = −eA𝜇[𝜓L𝛾𝜇𝜓L + 𝜓R𝛾𝜇𝜓R +��

�XXXX𝜓L𝛾𝜇𝜓R +��XXXX𝜓R𝛾𝜇𝜓L

]In the limit (m → 0) we have two non-zero helicity combinations:

e−R e−R

𝛾

e−L e−L

𝛾

e+L e+L

𝛾

e+R e−R

𝛾

The Weak vertex* also conserve chirality but involve only 𝜓L:

ℒI ∋ − g

2√2W+𝜇 · 𝜓e𝛾𝜇(1− 𝛾5)𝜓𝜈e = − g√

2W+𝜇

[𝜓eL𝛾𝜇𝜓𝜈L

]e−L 𝜈eL

W−

e−R 𝜈eR

W−

e+R 𝜈eR

W−

e+L 𝜈eL

W−


From Fermi Model to EW theory

The field W𝜇 in

ℒint = − g

2√2(J+𝜇 W

−𝜇 + h.c.).

should be massive to account for short-range weak interactions.

The scattering amplitude

T = i(2𝜋)4g2

8J+𝛼

[g𝛼𝛽 − p𝛼p𝛽/M

2W

p2 −M2W

]J−𝛽

reproduces the result due to the current-current interaction in thelimit |p| ≪ MW if we identify (“match”)

(effective theory)GF√2=

g2

8MW2

(more fundamental theory)

However, we have to be more clever, since the behavior of theamplitude in the opposite limit (p ≫ MW ) is still the same.


On the WW-production

Another manifestation of the same problem can be observed in the processof W -production:

We know that W± interact with fermions and are electrically charged.

We can consider e+e− → W+W− process at a lepton-antileptoncollider (e.g., LEP:):

e+

e−

W+

W−

𝜈e +

e+

e−

W+

W−

𝛾

and make predictions...

The calculated cross-section for longitudinal W− is found to increasewith center-of-mass energy s without limits (scales with s).

The solution to these problems is to utilize gauge symmetry...


Abelian vs Non-Abelian

The U(1) internal symmetry is Abelian (the order of twotransformations (rotations in 2d plane) is not relevant) :

Re 𝜑(x)

Im 𝜑(x)

𝛼1

Re 𝜑(x)

Im 𝜑(x)

𝛼2 e i𝛼1 · e i𝛼2 = e i𝛼2 · e i𝛼1

Re 𝜑(x)

Im 𝜑(x)

𝛼2

Re 𝜑(x)

Im 𝜑(x)

𝛼1

U(1) transformations commute with each other.

NB: Abelian symmetry is not sufficient to account for W±𝜇 . Why?



We can generalize U(1) to theNon-Abelian case. Consider SU(n)group, i.e., unitary n × n matrices Uij

depending on n2 − 1 parameters 𝜔a andhaving detU = 1:

𝜓i → 𝜓′i = Uij(𝜔)𝜓j , U(𝜔) = e igt

a𝜔a (B. Crowel, General Relativity)

Generators ta of the group obey su(n)-algebra

[ta, tb] = if abctc , f abc − structure constants

For constant 𝜔a this transformation is a symmetry of

ℒ0 = 𝜓i

(i𝜕 −m

)𝜓i , i = 1, ..., n

describing n free fermions in fundamental representation of SU(n).


http://www.lightandmatter.com/genrel/


For space-time dependent 𝜔a(x) we introduce (matrix) covariantderivative depending on n2 − 1 gauge fields W a

𝜇 :

(D𝜇)ij = 𝜕𝜇𝛿ij − igtaijWa𝜇

The transformation properties of W a𝜇 can be deduced from

D ′𝜇𝜓

′ = U(𝜔)(D𝜇𝜓), U(𝜔) = e igta𝜔a

⇒ W′a𝜇 = W a

𝜇 + 𝜕𝜇𝜔a + gf abcW b

𝜇𝜔c

= W a𝜇 + (D𝜇)

ab𝜔b, (D𝜇)ab ≡ 𝜕𝜇𝛿

ab − ig(−if abc)W c𝜇

The field-strength tensor is the commutator of the covariantderivatives (Ex: check this also for U(1) symmetry)

[D𝜇,D𝜈 ] = −igtaℱa𝜇𝜈 ,

ℱa𝜇𝜈 = 𝜕𝜇W

a𝜈 − 𝜕𝜇W

a𝜈 + gf abcW a

𝜇Wb𝜈



Non-Abelian gauge symmetry predicts not only interactions betweenfermions 𝜓 (more generally, fields in fundamental representation ofthe gauge group) and gauge fields W a

𝜇 , but also self-interactions ofthe latter (gauge-fields are also “charged” under the group) :

ℒ = 𝜓(i D −m

)𝜓 − 1

4ℱa𝜇𝜈ℱa

𝜇𝜈 = ℒ0 + ℒI

ℒ0 = 𝜓(i𝜕 −m

)− 1

4F a𝜇𝜈F

a𝜇𝜈 , F a

𝜇𝜈 = 𝜕𝜇Wa𝜈 − 𝜕𝜈W

a𝜇

ℒI = g𝜓i𝛼𝛾

𝜇𝛼𝛽t

aij𝜓

j𝛽W

a𝜇

− g

2f abcW b

𝜇Wc𝜈 F

a𝜇𝜈 −

g2

4f abc f adeW a

𝜇Wb𝜈W

d𝜇W

e𝜈

NB1: The strength of interaction is governed by a single coupling g .NB2: The mass term m2W a

𝜇Wa𝜇/2 violates gauge symmetry...

NB3: Non-Abelian gauge theory is also called Yang-Mills theory.


YM: Quantization Issues

To quantize YM theory we add

ℒgf = − 1

2𝜉(F a)2 , F a = 𝜕𝜇W

a𝜇 ⇐ gauge-fixing function

Contrary to the case of QED, fermionic current Ja𝜇 = g𝜓ta𝛾𝜇𝜓 is notconserved and can produce longitudinal W a

𝜇 .

Self-interactions of W a𝜇 guarantee that at the tree-level all amplitudes

involving unphysical polarizations (for external states) vanish.

𝜓 𝜓

WL WT

+

𝜓 𝜓

WL WT

+

𝜓 𝜓

WL WT

= 0


Fadeev-Popov Ghosts

The cancellation does not work for loops.

Add Faddev-Popov ghosts ca and ca — anticommuting “scalars”.

The Lagrangian for ghosts is related to the gauge-fixing functionFa(= 𝜕𝜇W𝜇)

ℒghosts = −ca𝜕Fa(W

𝜔)

𝜕𝜔bcb = −ca𝜕𝜇D

ab𝜇 cb

= −ca𝜕2ca − gf abc(𝜕𝜇ca)cbAb

𝜇

W W

W

− W W

c

NB: This is the price to pay for explicitly covariant formalism.


Gauge Theory of Electroweak InteractionsTo describe both electromagnetic and weak interactions in the SM we use

SU(2)L ⊗ U(1)Y

The group has four generators = gauge bosons. Three (W𝜇) belongsto SU(2)L, while photon-like B𝜇 mediates U(1)Y .

Weak Hypercharge U(1)Y Weak Isospin SU(2)L

NB: To explain beta-decay and electromagnetism we need only 3 gaugebosons W± and 𝛾. But we predict an additional one! Where is it hiding?

From J. Maldacena, “The symmetry and simplicity of the laws of physics and the Higgs boson”.


https://arxiv.org/abs/1410.6753

Gauge Theory of Electroweak InteractionsThe SM fermions are “charged” under SU(2)⊗ U(1).

Leptons(𝜈ee

)L

eR

(𝜈𝜇𝜇

)L

𝜇R

(𝜈𝜏𝜏

)L

𝜏R

Quarks(ud

)L

uR , dR

(cs

)L

cR , sR

(tb

)L

tR , bR

To account for (V − A) pattern only left fermions interact with W𝜇. Thecovariant derivative for left SU(2) doublets

DL𝜇 =

(𝜕𝜇 − i

2

(gW 3

𝜇 + g ′Y fLB𝜇

)−i g√

2W+𝜇

−i g√2W−𝜇 𝜕𝜇 +

i2

(gW 3

𝜇 − g ′Y fLB𝜇

)) ,while for the right SU(2) singlets we have

DR𝜇 = 𝜕𝜇 − ig ′Y

fR

2B𝜇

Here g and g ′ are the SU(2) and U(1) couplings. But what is Y fL/R?


Gauge Theory of Electroweak InteractionsLet us consider the interactions mediated by the introduced gauge bosonsin more details. We have “charged-current” interactions

ℒlCC =

g√2𝜈eL𝛾𝜇W

+𝜇 eL + h.c. =

g

2√2𝜈e𝛾𝜇W

+𝜇 (1− 𝛾5) e + h.c.

and “neutral-current” interactions

ℒlNC = 𝜈eL𝛾𝜇

(+1

2gW 3

𝜇 +Y lL

2g ′B𝜇

)𝜈eL

+ eL𝛾𝜇

(−1

2gW 3

𝜇 +Y lL

2g ′B𝜇

)eL

+ g ′eR𝛾𝜇Y eR

2B𝜇eR

Q: Where is QED photon A𝜇?

W 3𝜇 = Z𝜇 cos 𝜃W + A𝜇sin 𝜃W

B𝜇 = −Z𝜇 sin 𝜃W + A𝜇cos 𝜃W

The photon A𝜇 should notinteract with 𝜈L, but couple to eLand eR with the same strength!

Weinberg angle

Ex: Derive similar expressions for quarks.A. Bednyakov (JINR) QFT & EW SM 27 / 32

Gauge Theory of Electroweak InteractionsIt is a simple exercise to get the following relations between fermionelectric charges Qf and weak hypercharges Y f :

g sin 𝜃W = e(Q𝜈 − Qe) = e(Qu − Qd)

g ′Y lLcos 𝜃W = e(Q𝜈 + Qe) = −e

g ′Y QL cos 𝜃W = e(Qu + Qd) =

1

3e

g ′Y fRcos 𝜃W = 2eQf , f = e, u, d

and the elementary charge e.

From these relations we obtain e = g sin 𝜃W and, e.g., e = −g ′Y lL cos 𝜃W ,

so that

Y QL = −1

3Y lL, Y e

R = 2Y lL, Y u

R = −4

3Y lL, Y d

R = +2

3Y lL,

are fixed in terms of Y lL. It is convenient to normalize g ′ so that Y l

L = −1.Sorry, too many e’s.


Gauge Theory of Electroweak Interactions: Neutral CurrentThe neutral-current Lagrangian ℒNC takes the form:

ℒNC = eJA𝜇A𝜇 +

g

cos 𝜃WJZ𝜇 Z𝜇,

where

JA𝜇 =∑f

Qf f 𝛾𝜇f , JZ𝜇 =1

4

∑f

f 𝛾𝜇 (vf − af 𝛾5) f ,

vf = 2T f3 − 4Qf sin

2 𝜃W , af = 2T f3

and T f3 = ±1

2 for up-type/down-type fermions.

Example: Qu = 2/3, T u3 = 1/2, so

vu = 1− 8

3sin2 𝜃W , au = 1

Ex: Deduce vf /af for other fermions.NB: The couplings does not depend on generation (Universality).

Gargamelle,1972-73

10

102

103

104

105

0 20 40 60 80 100 120 140 160 180 200 220

Centre-of-mass energy (GeV)

Cro

ss-s

ecti

on

(p

b)

CESRDORIS

PEP

PETRATRISTAN

KEKBPEP-II

SLC

LEP I LEP II

Z

W+W

-

e+e−→hadrons


On Axial Anomalies and Charge Assignment in the SMAnomalies correspond to situations when a symmetry of the classical Lagrangian

is violated at the quantum level. A well-known example is Axial/Chiral Anomaly,

when classical conservation law for the current is modified at quantum level:

JA𝜇 = Ψ𝛾𝜇𝛾5Ψ, 𝜕𝜇JA𝜇 = 2imΨ𝛾5Ψ+

𝛼

2𝜋F𝜇𝜈 F𝜇𝜈 , F𝜇𝜈 = 1/2𝜖𝜇𝜈𝜌𝜎F𝜌𝜎

If JA𝜇 couples to a gauge field, anomalies render the model inconsistent!

The EW interactions in the SM do distinguish left from right.

Anom ∼ Tr[ta, {tb, tc}]L − Tr[ta, {tb, tc}]R

a

b

c

a

b

cIn the SM contributions to anomaliesmiraculously cancel each other separatelyfor each generation.

Ex: Check that all anomalies* are also zero.

TrY 3 =3

[(1

3)3 + (

1

3)3 − (

4

3)3 − (−

2

3)3]

↑ ↑ ↑ ↑ ↑colour uL dL uR dR

+(−1)3 + (−1)3 − (−2)3 = 0.

↑ ↑ ↑𝜈L eL eR

From ArXiv:0901.2208 by Kazakov D.I.


https://arxiv.org/pdf/0901.2208.pdf

The EW interactions: Lecture II summaryX We use gauge principle to introduce EW interactions.X We account for (V − A) structure for charged currents

ℒCC =g

2√2𝜈e𝛾𝜇(1− 𝛾5)W

+𝜇 e + u𝛾𝜇(1− 𝛾5)W

+𝜇 d + h.c.

X We reproduce electromagnetic interactions:

ℒANC = eJA𝜇A

𝜇

X We predict additional neutral boson

ℒANC = gzJ

Z𝜇 Z𝜇, gz ≡ g

cos 𝜃W- Z -boson should be massive and lead to Fermi-like current-currentinteractions JZ𝜇 J

Z𝜇 .

- Relative strength of charged and neutral current-current interactions isparametrized by

𝜌 ≡ M2W

M2Z cos2 𝜃W

, Up to now, no relation between MZ and MW ...


The EW interactions: Lecture II summary

X Non-Abelian SU(2)L predicts also gauge-field triple and quarticself-interactions!

𝛾,Z

W+

W−

𝛾,Z

𝛾,Z

W+

W−

W+

W−

W+

W−

NB: ZWW coupling cures bad behavior of ee → WW .

So far so good, but there is an inconsistency in our reasoning...

× All gauge bosons should be massless! A mass term like M2WW+

𝜇 W−𝜇

is forbidden by symmetry W𝜇 → W𝜇 + 𝜕𝜇𝜔 + ...

× All fermions are massless! A mass term like me ee mixes left and right.But eL belongs to a SU(2) doublet, while eR is an SU(2) singlet!

A solution is to be provided via the Higgs mechanism...A. Bednyakov (JINR) QFT & EW SM 32 / 32

Lecture III?


Break the symmetry?Q: How to make W and Z massive (keeping the nice features of the gaugetheory intact)?

Explicit breakingvia the mass terms

ℒ ∋ m2WW+

𝜇 W 𝜇− +

m2Z

2Z𝜇Z𝜇

leads to inconsistencies...

We have to do something more clever...Hidden symmetry?


Spontaneous Symmetry Breaking

ℒ =1

2(𝜕𝜇𝜌)

2 +e2𝜌2

2B𝜇B𝜇 − V (𝜌2/2)− 1

4F 2𝜇𝜈(B)

Here 𝜌(x) is a dynamical field. We get mass term if 𝜌(x) → v = const.

𝜑

V (𝜑)

𝜇 > 0

V = 𝜇2𝜑†𝜑+ 𝜆(𝜑†𝜑)2

It is invariant underglobal phase-shift

symmetry 𝜑→ e i𝛼𝜑. 𝜑

V (𝜑)

𝜇 < 0

|𝜑0|

The case 𝜇2 > 0 is trivial.

For 𝜇2 < 0 we have a valley of degenerate minima:

𝜕V

𝜕𝜑†= 0 ⇒ 𝜑†0𝜑0 = −𝜇

2

2𝜆=

v2

2> 0 ⇒ 𝜑0 =

v√2e i𝛽


The Brout-Englert-Higgs mechanismNon-zero 𝜑 in the minimum of the potential is interpreted as the vacuumexpectation value (vev) of the quantum field:

v√2= ⟨0|𝜑(x)|0⟩, 𝛽 = 0 (Why?)

To introduce particles as excitations we have to shift the field:

𝜑(x) =v + h(x)√

2e i𝜁(x)/v , ⟨0|h(x)|0⟩ = 0, ⟨0|𝜁(x)|0⟩ = 0

ℒ =1

2(𝜕𝜇h)

2 + e2v2

2 B𝜇B𝜇 + evhB𝜇B𝜇 +e2

2 B𝜇B𝜇h2 − V − 1

4F 2𝜇𝜈(B)

V = −|𝜇|2

2(v + h)2 +

𝜆

4(v + h)4 = 2𝜆v2

2 h2 + 𝜆vh3 + 𝜆4h

4 − 𝜆

4v4

Massive field B𝜇 without explicit symmetry breaking! This is the essenceof Brout-Englert-Higgs-Hagen-Guralnik-Kibble mechanism. The symmetryis hidden..


The Brout-Englert-Higgs mechanism: Counting DOFsWe start with

ℒ1 = 𝜕𝜇𝜑†𝜕𝜇𝜑+ ie

(𝜑†𝜕𝜇𝜑− 𝜑𝜕𝜇𝜑

†)A𝜇 + e2A𝜇A𝜇𝜑

†𝜑− V − 1

4F 2𝜇𝜈(A),

and end with [√2𝜑 = (v + h) exp(i𝜁(x)/v), B𝜇 = A𝜇 − 𝜕𝜇𝜁/(ev)]

ℒ2 =1

2(𝜕𝜇h)

2 − e2v2

2

(1 +

h2

v2

)B𝜇B𝜇 + evhB𝜇B𝜇 − V − 1

4F 2𝜇𝜈(B).

ℒ1: 2 DOFs (complex scalar 𝜑, 𝜑†) + 2 DOFs (massless vector A𝜇).

ℒ2: 1 DOFs (real scalar h) + 3 DOFs (massive vector field B𝜇).

One scalar DOF was “eaten” by the gauge field to become massive.

Q: Which one?

A: The (would-be) Nambu-Goldstone (boson)!

V (𝜑)

𝜁


SSB and Renormalizability

Both ℒ1 and ℒ2 has some issues:

ℒ1 : Manifestly gauge-invariant, but not suitable for PT (imaginary mass);

ℒ2 : Hidden gauge symmetry, only physical DOFs, but non-renormalizableby power counting.

There is another, explicitly renormalizable version of the Lagrangian withshifted 𝜑 written in cartesian coordinates: 𝜑 = 1√

2(v + 𝜂 + i𝜉)

ℒ3 =v4𝜆

4− 1

4F𝜇𝜈F𝜇𝜈 +

e2v2

2A𝜇A𝜇 +

1

2𝜕𝜇𝜉 𝜕𝜇𝜉 − evA𝜇𝜕𝜇𝜉

+1

2𝜕𝜇𝜂 𝜕𝜇𝜂 −

2v2𝜆

2𝜂2 + eA𝜇𝜉𝜕𝜇𝜂 − eA𝜇𝜂𝜕𝜇𝜉 − v𝜆𝜂(𝜂2 + 𝜉2)

− 𝜆

4(𝜂2 + 𝜉2)2 +

e2

2A𝜇A𝜇(2v𝜂 + 𝜂2 + 𝜉2).

NB: Now we have massless unphysical field 𝜉 in the spectrum, but itmixes with longitudinal component of A𝜇 (“partially eaten”).


A Remark on the Goldstone Theorem

V (𝜑)

𝜁

The Goldstone theorem states that if thevacuum breaks a global continuoussymmetry there is a massless boson(Nambu-Goldstone) in the spectrum:any non-derivative interactions violates

𝜁 → 𝜁 + ev𝜔, 𝜔 = const

Fortunately, we have local symmetry with hungry A𝜇...

We need 3 massive bosons W±,Z𝜇.

3 symmetries out of SU(2)L × U(1)Yhas to be spontaneously broken to get 3victims (would-be) Nambu-Goldstonebosons.

NB: The Higgs boson (see lect. by J. Ellis) was a by-product of themass-generation mechanism (the main task was to “exorcise” masslessfields).


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(Quantum) Field Theory and the Electroweak Standard Model ...€¦ · A. Bednyakov (JINR) QFT & EW SM 2 / 32. An Ode to Symmetry... With Action you can also studySymmetries... The