feynman diagrams in particle physics · 1 feynman diagrams in particle physics dr juan rojo vu...

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Feynman diagrams in Particle Physics

Dr Juan Rojo VU Amsterdam and Nikhef Theory group

[email protected]

Juan Rojo Introduction to Particle Particles, 18/03/20201

https://juanrojo.com/teaching/additional material:

mailto:[email protected]

https://juanrojo.com/teaching/

22

Motivation: what Feynman diagrams are useful for?

Feynman diagrams in Quantum Electrodynamics

Feynman diagrams in Quantum Chromodynamics

Feynman diagrams in Electroweak Theory

Effective Theories and Feynman diagrams

Outline


33

Motivation


At the quantum level, fundamental interactions look very different that at the classical level

4

Feynman diagrams

Elementary particles interact by exchanging force carriers among them

For example, the photon is the force carried particle of Quantum Electrodynamics, the quantum version of classical electromagnetic theory

A powerful tool to visualise interactions between elementary particles is known as Feynman diagrams, that represent the trajectories in space and time of the

particles involved in a scattering reaction



5


Feynman diagrams



6


Feynman diagrams


7

Drawing Feynman diagrams

If the scattering reaction involves composite particles (hadrons) first of all determine their quark decomposition making sure all quantum numbers add up consistently


8



Then put at the left of the diagram the initial-state particles and at the right of the diagram the final-state particles


9




Attempt to connect the initial and final state particles among them. Note that some particles will not interact and will be just spectators in the reaction


10




Attempt to connect the initial and final state particles among them. Note that some particles will not interact and will be just spectators in the reaction

Make sure that all interaction vertices conserve the corresponding quantum numbers: for example, if gluons or photons are conserved, then Q, B, S, C, b, … should be conserved


11

Why are Feynman diagrams useful?

Determine whether a given scattering process is possible or impossible


12



μ+ + μ− → e+ + e− allowed??


13



μ+ + μ− → e+ + e− allowed??

Yes since I can draw a Feynman diagram with a known particle mediating the interaction and where all conservation laws are satisfied!

Z0


14



Determine which of two processes has a higher likelihood of taking place based on power counting of coupling constants


15





π0 + p → n + π+

gstrong

gstrong

16





π0 + p → n + π+

gstrong

gstrongℳ ∝ g2

strong

17





π0 + p → n + π+

gweak

gweakℳ ∝ g2

weak

18




ℳ ∝ g2weak

The scattering amplitude mediated by a Z boson has amplitude

ℳ ∝ g2strong

The scattering amplitude mediated by a gluon has amplitude

Since gstrong >> gweak the reaction mediated by a gluon is much more likely!


19




Exploit similarities between apparently different scattering reactions


20


Seem to be very different reactions …


21


Seem to be very different reactions …

until we express them in terms of Feynman diagrams!

μ+

νμ

The two processes take place at very similar rates


22




Exploit similarities between apparently different scattering reactions


Quantum Electrodynamics

2323Juan Rojo Introduction to Particle Particles, 18/03/2020

In QED there is a unique interaction vertex:

24

Quantum Electromagnetism

This fact implies the following important properties about the electromagnetic interaction:

Electric charge is always conserved because the photon does not carry electric charge



25




Being electrically neutral, the photon cannot interact with itself



26





Flavour is conserved by QED interactions: automatic conservation of leptonic and baryonic numbers, as well as strangeness, charmness, and bottomness



27





Flavour is conserved by QED interactions: automatic conservation of leptonic and baryonic numbers, as well as strangeness, charmness, and bottomness

Since the photon is exactly massless, electromagnetism is a long-range force


28

μ+ + μ− → e+ + e−

μ+

μ−

e+

e−

γ

Quantum ElectromagnetismQuantum Electrodynamics can mediate interactions between

any particles with electric charge


Quantum Chromodynamics


It is useful to enumerate the properties of the strong interaction by comparing them with those of the electromagnetic interactions

30

Strong force vs electromagnetism

Electromagnetism Strong interactions

Three different types of colour charge exist: blue, green, red, with their own sign and magnitude

A single type of electric charge exists: the only thing that varies is its sign and magnitude

Electromagnetism is transmitted by photons, which are massless and charge-neutral

The strong interaction is transmitted by gluons, which are massless but charged under color

The strength of the electromagnetic interaction is always small: electromagnetism looks the same at all energies/distances

The strength of the strong interaction varies with the energy/distance: very different behaviour depending on energy/distance


Let us summarise what we have learned about the quantum theory of the strong interactions: Quantum Chromodynamics (QCD)

31


Flavour is always conserved by strong interactions: automatic conservation of leptonic and baryonic numbers, as well as strangeness, charmness, and bottomness

This is a consequence of the fact that the only possible interaction vertices are:

And that gluons do not carry flavour quantum numbers



32



Gluons are charged under color so they can interact with themselves. They are however electrically neutral to they don’t affect the electric charge in strongly interacting processes



33




up quark

up quark

gluonIs this reaction

possible?



34




up quark

up quark

gluon

It is not possible since:

Qin =23

+23

=43

≠ Qfin = 0


gluon


35




up quark

up anti-quark This reaction is instead possible:

Qin =23

−23

= 0 = Qfin

gluon


gluon


36




up quark

up anti-quark This reaction is instead possible:

Qin =23

−23

= 0 = Qfinby convention, in Feynman diagrams antiparticles are indicated by an arrow with opposite sign as particles

gluon



37




The strength of the strong force is not constant: it is more at low energies / large distances (leading to quark confinement into hadrons) but less at high energies / low distances (where it behaves like electromagnetism)



38




The strength of the strong force is not constant: it is more at low energies / large distances (leading to quark confinement into hadrons) but less at high energies / low distances (where it behaves like electromagnetism)

While quarks have fractional electric charge and baryon number, only hadrons with integer electric charge and baryon number are physically allowed


39

Scattering reactions in QCDLet’s try to understand some strong-interacting scattering processes in terms of QCD

π0 + p → n + π+

Write the corresponding Feynman diagram using only quarks and gluons

π0 = (u u) π+ = (u d)Note also how Q, B, S, C, … are conserved in this reaction


40


π0 + p → n + π+


41


ρ+ → π+ + π0

Write the corresponding Feynman diagram using only quarks and gluons

π0 = (u u) π+ = (u d)


42


ρ+ → π+ + π0

Note also how Q, B, S, C, … are conserved in this reaction


ElectroweakTheory


It is useful to enumerate the properties of the weak interaction by comparing them with those of the electromagnetic interactions

44

Weak force vs electromagnetism

Electromagnetism Weak interactions

All particles in the SM are carry a weak charge, and the specific values depend on the matter particle

A single type of electric charge exists: the only thing that varies is its sign and magnitude

Electromagnetism is transmitted by photons, which are massless and charge-neutral

The strength of the electromagnetic interaction is always small: electromagnetism looks the same at all energies/distances

The weak interaction is always weak and confined to small scales (large value of mW,Z)

The strong interaction is transmitted by the W and Z bosons, which are massive and charged under the weak force

range : Δr ∼ m−1Δr ≃ ∞ (EM, QCD)

Δr ≃ 10−18 m (weak)Juan Rojo Introduction to Particle Particles, 18/03/2020

The weak boson WThe weak interactions are mediated by three massive bosons: W+, W+, Z0

45

The main properties of the W bosons are:

As opposed to the massless gluons and photons, the W boson is very massive, around 80 times the proton mass

As in the case of the gluons (but not the photons), the W boson is charged under both electric and weak charges, and therefore can interact with itself

When interacting with quarks, the W boson will change its charge by one unit and therefore also its flavour (including possibly across generations)

W+

du

Qin = + 1 + (−1/3) = + 2/3 = Qfin



46





W+

dc

Cin = 0 ≠ Cfin = + 1

Qin = + 1 + (−1/3) = + 2/3 = Qfin



47





In weak interaction processes mediated by the W boson, the flavour quantum numbers (strangeness, charmness, botomness) are not conserved quantities


The weak boson W

48

Taking into account these properties, some of the physically allowed reactions involving quarks and W bosons will be:

If a given reaction is allowed, the corresponding reaction involving the antiparticles is also physically allowed

You can always replace a given quark by the corresponding quark of a different generation: for example a down antiquark by a strange antiquark

Electric charge is always conserved

u + W+ → s ⇒ u + W− → s


The weak boson W

49

Taking into account these properties, some of the physically allowed reactions involving leptons and W bosons will be:

Electric charge is always conserved

Each interaction vertex involves a charged and a neutral lepton that belong to the same lepton generation

You can always replace the two leptons of a given generation for the corresponding two leptons of another generation

The individual leptonic quantum numbers are always conserved in weak reactions


The weak boson W

50

Draw the Feynman diagram for the following process

π+ → μ+ + νμ π+ = (u d)


The weak boson W

51

π+ → μ+ + νμ π+ = (u d)

Quarks and leptons only interact indirectly via either photons or W, Z bosons

Since the electric charge is Q=+-1, then a positively charged W boson is involved

We know what vertices are allowed involving quarks or leptons and a W boson

Draw the Feynman diagram for the following process

We have a neutrino in the final state: the weak interaction must be involved


The weak boson W

52

π+ → μ+ + νμ π+ = (u d)Draw the Feynman diagram for the following process

You can check that all relevant quantum numbers are conserved: L, B, Q, …


Weak coupling between generations

53

We have seen that in processes mediated by the weak gauge boson W the flavour of the quarks will change

W+

du



54


Moreover we can always replace a given quark by the corresponding quark of a different generation

W+

dc



55



W+

bc



56



W+

bu

The weak interactions mediates transitions between quarks of different generations



57

The strength of the weak coupling is similar between quarks of the same generation

u

W+d

W+s≃

c



58

The strength of the weak coupling is similar between quarks of the same generation

u

u

W+d

W+

s≫

The strength of the weak coupling is smaller between quarks of different generation

W+d

W+s≃

u

c

≫u

W+

b

Weak coupling between gens 1 and 2 bigger than between gens 1 and 3


Heavy hadron decays

59

This hierarchy of the weak couplings between quark generations is particularly important in order to understand the decays of hadrons that contain heavy quarks

B0 → D− + μ+ + νμ B0 = (d b) D− = (dc)what is the corresponding Feynman diagram?


Heavy hadron decays

60


what is the corresponding Feynman diagram?

μ+

W+

νμ

B0 → D− + μ+ + νμ B0 = (d b) D− = (dc)


Heavy hadron decays

61



B0 → π− + μ+ + νμ B0 = (d b) π− = (du)


Heavy hadron decays

62



μ+

W+

νμ

B0 → π− + μ+ + νμ B0 = (d b) π− = (du)

π−

u


Heavy hadron decays

63

μ+

W+

νμ

π−

u

μ+

W+

νμ

Which of the two reactions is most likely to take place?


Heavy hadron decays

64

μ+

W+

νμ

π−

u

μ+

W+

νμ

Which of the two reactions is most likely to take place?

The decay into a pion is more suppressed, since the coupling

Wub (gens 1 and 3) is smaller than the coupling Wcb (gens 1 and 2)


Heavy hadron decays

65

Note that some reaction processes might look very different from the outside, but their similarities become apparent at the Feynman diagram level

B0 → D− + μ+ + νμ

B0 → D− + π+

How do these two decay models relate to each other?


Heavy hadron decays

66


B0 → D− + μ+ + νμ

B0 → D− + π+



Heavy hadron decays

67


B0 → D− + μ+ + νμ

B0 → D− + π+


These two processes have a very similar probability to happen!


The weak boson ZThe weak interactions are mediated by three massive bosons: W+, W+, Z0

68

The main properties of the Z bosons are:

As opposed to the massless gluons and photons, the Z boson is very massive, around 91 times the proton mass (similar to W boson)

As in the case of the gluons (but not the photons), the Z boson is charged under the weak charges, and therefore can interact with itself. It is electrically neutral so it cannot interact via electromagnetism

When interacting with quarks, the Z boson does not change the quark flavour

Z0

dd


The weak boson ZIn terms of its interactions, the weak boson Z is a kind of ``heavy photon’’

69

In diagrams involving quarks and charged leptons, and where the photon mediates the interaction, one can replace the photon by a Z boson

μ+ + μ− → e+ + e−



70


μ+ + μ− → e+ + e−

μ+

μ−

e+

e−

γ



71


μ+ + μ− → e+ + e−

μ+

μ−

e+

e−

Z0



72

The Z boson also mediates processes involving neutrinos

μ+ + μ− → νe + νe

μ+

μ−

νe

νe

Z0


Feynman diagrams& Effective Theories


7474

Why Effective Theories?


We don't need to use General Relativity to compute planetary orbits (vplanet << c)

We don’t need to account for the strong or weak interaction to compute electronic transitions in atoms ( rnucleus , rweak << ratom )

We can model the large-scale of the universe treating galaxies as points (rgalaxy << runiverse)

How does this principle work in particle physics?

heavyparticle

7575

2. Research proposal2a. A description of the proposed research

(Max. 1200 words on max. 3 pages)

2a1. Overall aim and key objectivesParticle physics in the LHC precision era. Following the momentous discovery of the Higgs boson at Large Hadron Col-lider (LHC), high-energy physics finds itself at an historical cross-road. On the one hand, we are faced with pressingquestions that cannot be explained within the (otherwise highly successful) Standard Model (SM) of particle physics,such as the origin of the Higgs electroweak symmetry breaking mechanism, the nature of the mysterious Dark Matter,and the puzzling imbalance between matter and antimatter in the Universe. On the other hand, we lack obvious targetsin our searches for new particles and interactions that could address the structural limitations of the Standard Model.This state of affairs demands the deployment of an extensive program of model-independent searches for new physicsbeyond the SM based on the combination of high-precision measurements and cutting-edge theoretical calculationsfrom the LHC and other facilities. With an unprecedented amount of collisions from the LHC to come in the next years,the time is truly now tomap the high-energy frontier using theoretical particle physics as a quantummicroscope in orderto tackle fundamental open questions both within the Standard Model and beyond it.

Effective Theories: the ultimate quantum microscopeE

ne

rgy

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2

ℒint ⊃c4(λ, MΦ)ϕ4

Full Theory

Effective Theory

In physical systems, the dynamical laws governing widely different energyor distance scales often become effectively decoupled. For instance, knowl-edge about the internal structure of protons is not required to evaluate elec-tronic transition energies. An Effective Field Theory (EFT) is an implementa-tion of this paradigm in the framework of quantum field theories involvingmass hierarchies. Consider a theory composed by a light field � with massm� and a heavy field � with massM� (with m� ⌧ M�), coupled via a��2�2 interaction. At low energies, E ⌧ M�, the heavy field becomesnon-dynamical and can be integrated out. The resulting EFT is composed bythe light field �with a new interaction c4�4, where the value of c4 dependson the full theory parameters � andM�. Therefore, we can infer indirectlythe existence and properties of the heavy particle � even at low energies,via its modifications of the properties of the light degrees of freedom.

We now describe the three objectives of this project and their corresponding methodologies and output, see also Fig. 1.O1: A global model-independent search for new physics using Effective Theories. The Standard Model can also beunderstood as an Effective Field Theory, known as the SMEFT [1], where the effects of eventual new heavy particlesand interactions are encoded in a model-independent way by means of adding complete bases of higher-dimensionaloperators to the SM Lagrangian, LSMEFT = LSM +

PciOi/⇤2, where ⇤ is the scale of sought-for new physics, Oi

are operators of mass-dimension six and {ci} are Wilson coefficients. As explained in the “Effective Theories” box, thecoefficients ci are determined by the properties of the full theory (beyond the SM) valid at high energiesE � ⇤.Here I will carry out the most exhaustive model-independent search for new particles to date by means of the SMEFTinterpretation of measurements from the LHC and other experiments. Building on the aJ16Bh framework that I devel-oped [2], I will constrain the Wilson coefficients {ci} and the new physics scale ⇤ covering a plethora of new physicsscenarios. Key aspects are (i) using a wide range of LHC measurements from Higgs and top quark to flavour data andtheir combination with low-energy measurements; (ii) deploying new Machine Learning (ML) methods to efficientlysample the� 2000 directions of its parameter space; and (iii) constrain the SMEFT directly at the level of LHC detectorevents.Output: to either uncover deviations within the fundamental laws of the SMor to provide themost stringent constraintsto whatever theories might lie beyond it.O2: The ultimate determination of the proton structure with machine learning. A detailed knowledge of the proton’squark and gluon substructure (encoded by Parton Distributions, see box below) is essential for the interpretation ofproton-proton collisions at the LHC. Here Iwill performa next-generation determination of the proton structure usingMLmethods within the successful NNPDF framework [3–6] that I pioneered. Key aspects are (i)ML hyper-optimisation [7],

3

EFTs in particle physicsRelate physical theories at very different scales

e.g. consider a theory with twoparticles of very different mass




Effective Theories: the ultimate quantum microscope

En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3

light particle


7676





ne

rgy

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3

EFTs in particle physics

at high energies we can produceboth the light and the heavy particles





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3


7777





ne

rgy

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3


but at low energies only thelight particle is present!





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3

EFTs: rules to relate theories of particle physics at different scales


7878





ne

rgy

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3


but at low energies only thelight particle is present!





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3





En

erg

y

ϕ (mϕ) , Φ (MΦ)

E ≃ MΦ ≫ mϕ

ϕ (mϕ)

ℒint ⊃λϕ2Φ2


Full Theory

Effective Theory





3

EFTs: rules to relate theories of particle physics at different scales


The Effective Theory is simpler: only includes particles relevant at low energies

The structure of the Effective Theory is dictated by matching from full theory: e.g. the strengths of the EFT interactions depend on properties of heavy particles

We can also construct EFTs bottom up in the absence of a Full Theory

7979

EFTs in beta decays

Fermi Theory describes beta decay of neutrons with good accuracy

however is Fermi Theory valid at all energy scales?


8080

EFTs in beta decaysFermi Theory of neutron decay modelled by interaction between four fermions

ℒfermi ∝ GFψψψψ

n

p

e-

νeit then predicts the following scattering reaction

n p

e-νe


∝ GF

∝ GF

8181

EFTs in beta decays

n p

e-νe

using perturbation theory, the cross-section (probability) for this processas a function of the collision energy E is

σ(E) ∝ c1G2FE2 + c2G3

FE4 + … c1, c2 = 𝒪(1)

for a series to converge the second term should be smaller than the first one …

c2G3FE4

c1G2FE2

≤ 1 → E ≤ G−1/2F ≃ 300 GeV

what happens at such scales, where Fermi theory breaks down?


∝ GF

8282

EFTs in beta decays

n p

e-νe

Fermi theory is an Effective Theory! the full theory is of course the SM

EFT

Full Theory

n p

e-νe


integrating out the W boson at low energies brings us from

SM to Fermi theory

Case study: Scalar QED


84

Feynman rules in scalar QED Scalar Quantum Electrodynamics is a theory that contains a charged scalar particle (possibly with self-interactions) and a photon

Formally it is a Quantum Field Theory defined by the following Lagrangian

ℒ = −14

FμνFμν + (Dμψ)*

(Dμψ) − m2ψ*ψ

where we have defined:

Fμν ≡ ∂μAν − ∂νAμ , Dμ ≡ ∂μ + ieAμ

photon

charged scalar

electric charge

scalar mass

How can we compute the probability that a given scattering reaction in scalar QED takes place?


85

Feynman rules in scalar QED

How can we compute the probability that a given scattering reaction in scalar QED takes place?


In

86

Feynman rules in scalar QED First we need to construct the scattering amplitude following the Feynman rules of the theory

Dr Juan Rojo Quantum Field Theory Extension: Lecture Notes January 28, 2019

Figure 12. The photon-scalar interaction vertices in scalar QED, see Eq. (3.75) and the corresponding discussion

in the text more more details.

not satisfy gauge invariance. Interestingly, from dimensional reasons one could have considered the two

interaction terms separately each with an associated coupling constant,

ig1Aµ [(@µ

⇤) � ⇤ (@µ )] , �g2AµAµ ⇤ , (3.78)

but if we impose gauge invariance to be a symmetry of our theory then these two coupling constants cannot

be independent and must satisfy the relation g21= g2.

From the discussion of the symmetry properties of scalar QED in Sect. 3.2, we found that a global gauge

transformation had associated a conserved current (via Noether’s theorem) of the form of Eq. (3.50), which

can also be expressed in terms of the covariant derivative as

jµ = i ( ⇤Dµ � (Dµ )⇤ ) . (3.79)

The opposite sign between the first and second term implies that the charged scalar field contains states

with both positive and negative charge, as opposed to states with only one type of charge. With some abuse

of notation we can write the scalar field in terms of creation and annihilation operators as

(x, t) =

Zd3p

(2⇡)32!

�as(k)e

ikx + b†s(k)e�ikx

�, (3.80)

so the state created by a†s will be called the “positron” e+ and the state created by b†s will be called the

“electron” e�. Needless to say, electron and positron are the antiparticles of each other. Recall that

represents scalar fields in sQED, so it would be more appropriate to denote the particles created by it

“scalar electron” and “scalar positron”.

Combining the various ingredients together, is it possible to derive the Feynman rules of scalar QED

allowing to construct the corresponding S matrix elements. Skipping due to lack of time their formal

derivation, here we state the results and then study that are their implications at the level of scattering

amplitudes.

• For an incoming positron or electron e+ or e�, one uses the same rules as in the free scalar theory.

Page 61 of 88


k1 k2

k3

k1 k2

k3 k4

Figure 13. The Feynman rules for the interaction vertices in scalar QED.

With these Feynman rules, it is possible to evaluate any process in scalar QED, first at tree level, and then

eventually at the loop level using a generalization of the renormalization procedure that has been described

in Sect. 2 in the case of ��4 theory. Indeed, no major conceptual di↵erences arise when computing one-loop

diagrams in scalar QED as compared to what we have studied for ��4 theory.

The R⇠ gauge. The photon propagator in Eq. (3.83) has been derived using the so-called generalized

Feynman gauge, also known as the R⇠ gauge. This family of gauges are a generalisation of the Lorenz gauge

that we have been using so far, namely @µAµ = 0. Setting ⇠ = 1 leads to the Feynman gauge, while setting

⇠ = 0 gives the Lorenz gauge (also known as the Landau gauge). Given that the choice for the value of

⇠ is gauge-dependent, it should drop from all physical quantities, such as scattering amplitudes. We will

demonstrate that this is actually the case in explicit calculations.

The basic idea of the R⇠ gauge fixing method is to replace the auxiliary equation that determines the

specific gauge condition by adding an explicitly gauge-fixing term to the Lagrangian

�L = � (@µAµ)2

2⇠. (3.85)

If we take the limit ⇠ ! 0 the recover the Lorenz gauge, since then minimizing the action is equivalent to

requiring @µAµ = 0. In this case, the expression for the photon propagator in the Lorentz is given by

�i

k2 + i✏

✓⌘µ⌫ � kµk⌫

k2

◆, (3.86)

which is related to the commutation relations for the photon that were derived above. For calculational

reasons, the Feynman gauge ⇠ = 1 is most useful, the reason is that there the second term of the propagator

cancels out and one ends up with a simplified expression for the photon propagator.

�i⌘µ⌫

k2 + i✏. (3.87)

Page 63 of 88

each of the interaction terms that appear in the Lagrangian of the theory ….

…. corresponds to specific rules in the scattering amplitude

k3

k1 k2

k3

k1 k2

k4


In

87

Feynman rules in scalar QED First we need to construct the scattering amplitude following the Feynman rules of the theory


Figure 12. The photon-scalar interaction vertices in scalar QED, see Eq. (3.75) and the corresponding discussion

in the text more more details.

not satisfy gauge invariance. Interestingly, from dimensional reasons one could have considered the two

interaction terms separately each with an associated coupling constant,

ig1Aµ [(@µ

⇤) � ⇤ (@µ )] , �g2AµAµ ⇤ , (3.78)

but if we impose gauge invariance to be a symmetry of our theory then these two coupling constants cannot

be independent and must satisfy the relation g21= g2.

From the discussion of the symmetry properties of scalar QED in Sect. 3.2, we found that a global gauge

transformation had associated a conserved current (via Noether’s theorem) of the form of Eq. (3.50), which

can also be expressed in terms of the covariant derivative as

jµ = i ( ⇤Dµ � (Dµ )⇤ ) . (3.79)

The opposite sign between the first and second term implies that the charged scalar field contains states

with both positive and negative charge, as opposed to states with only one type of charge. With some abuse

of notation we can write the scalar field in terms of creation and annihilation operators as

(x, t) =

Zd3p

(2⇡)32!

�as(k)e

ikx + b†s(k)e�ikx

�, (3.80)

so the state created by a†s will be called the “positron” e+ and the state created by b†s will be called the

“electron” e�. Needless to say, electron and positron are the antiparticles of each other. Recall that

represents scalar fields in sQED, so it would be more appropriate to denote the particles created by it

“scalar electron” and “scalar positron”.

Combining the various ingredients together, is it possible to derive the Feynman rules of scalar QED

allowing to construct the corresponding S matrix elements. Skipping due to lack of time their formal

derivation, here we state the results and then study that are their implications at the level of scattering

amplitudes.

• For an incoming positron or electron e+ or e�, one uses the same rules as in the free scalar theory.

Page 61 of 88

In Feynman diagrams we also have in general propagators (internal lines)

…. which also have specific rules in the scattering amplitude

DrJu

anRojo

Quan

tum

Field

Theory

Extension

:Lecture

Notes

Janu

ary28,2019

Figure

12.Thephoton-scalarinteractionverticesin

scalarQED,seeEq.(3.75)andthecorrespondingdiscussion

inthetextmoremoredetails.

not

satisfygauge

invarian

ce.

Interestingly,

from

dim

ension

alreason

son

ecould

haveconsidered

thetw

o

interactionterm

sseparatelyeach

withan

associated

couplingconstan

t,

ig1A

µ[(@ µ ⇤ ) � ⇤(@

µ )],

�g 2A

µA

µ ⇤

,(3.78)

butifweim

posegauge

invarian

ceto

beasymmetry

ofou

rtheory

then

thesetw

ocouplingconstan

tscannot

beindep

endentan

dmust

satisfytherelation

g2 1=

g 2.

From

thediscussionof

thesymmetry

properties

ofscalar

QED

inSect.3.2,

wefoundthat

aglob

algauge

tran

sformationhad

associated

aconserved

current(via

Noether’stheorem)of

theform

ofEq.

(3.50),which

canalso

beexpressed

interm

sof

thecovarian

tderivativeas

jµ=

i( ⇤ D

µ �

(Dµ )⇤ ).

(3.79)

Theop

positesign

betweenthefirstan

dsecondterm

implies

that

thechargedscalar

field

containsstates

withbothpositive

andnegativecharge,as

opposed

tostates

withon

lyon

etypeof

charge.W

ithsomeab

use

ofnotationwecanwrite

thescalar

fieldin

term

sof

creation

andan

nihilationop

eratorsas

(x,t)=

Zd3p

(2⇡)32!

� a s(k)e

ikx+

b† s(k)e

�ik

x�,

(3.80)

sothestatecreatedby

a† swillbecalled

the“p

ositron”e+

andthestatecreatedby

b† swillbecalled

the

“electron”e�

.Needless

tosay,

electron

andpositronarethean

tiparticles

ofeach

other.

Recallthat

representsscalar

fieldsin

sQED,so

itwou

ldbemoreap

propriateto

denotetheparticles

created

byit

“scalarelectron

”an

d“scalarpositron”.

Com

biningthevariou

singredientstogether,is

itpossible

toderivetheFeynman

rulesof

scalar

QED

allowingto

construct

thecorrespon

dingS

matrix

elem

ents.

Skippingdueto

lack

oftimetheirform

al

derivation,herewestatetheresultsan

dthen

studythat

aretheirim

plication

sat

thelevelof

scattering

amplitudes.

•For

anincomingpositronor

electron

e+or

e�,on

eusesthesamerulesas

inthefree

scalar

theory.

Page61

of88

Scalar propagator Photon propagator

kμkμ


88

Born level (leading order) Compute the scattering amplitude for Moeller scattering in scalar QED

ℳ (e− + e− → e− + e−)


89

One-loop (next-to-leading order) Compute the scattering amplitude for Moeller scattering in scalar QED at one loop level

ℳNLO ∝ e4 ∫d4kk4

∝ e4 ln Λ → ∞

ℳ (e− + e− → e− + e−)

The amplitude at NLO is infinite! What does it mean? We need to renormalise the theory ….


90

Feynman diagrams: summary Feynman diagrams provide a powerful too to understand what goes on in

reactions between elementary particles

To being with, with Feynman diagrams we can determine whether a given scattering process is possible or impossible

Using them we can also determine which of two processes has a higher likelihood of taking place based on power counting of coupling constants

Moreover, with Feynman diagrams the similarities between apparently different scattering reactions become transparent

Most importantly: they allow us to provide precise predictions for the rates of high-energy scattering reactions, which are key to find possible new particles and interactions beyond the Standard Model!


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