some concepts to nuclear physics

7/24/2019 Some concepts to nuclear physics

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ome Examples

Summary: The structure of Feynman rules and how

to build Feynman diagramsSummary: Four-vectors and two-particle phase spaceExample 1: Higgs decay into fermionsExample !: Electron-positron annihilation into fermions


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The structure of Feynman rules

"eneral structure

#n all interactions$ charges are conserved

#n all interactions$ total four-momentum is conserved

#n all interactions$ baryon and lepton number areconserved %&uar's and leptons must come in pairs(

)epton number is conserved per family in the

Standard *odel

There are never more than four particles entering afundamental vertex %funny enough$ two fermions

count as three particles(


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#nteractions of the fermions:+n the level of fundamental particles %&uar's$leptons($ there are only two types of interactions or

vertices that are allowed for fermions:"auge interactions The specific interaction occurs only$ when the

fermions have a corresponding charge:Electromagnetic charge for photons,olour charge for gluons

ea' charge %isospin( for bosonsThe . boson couples to both electric chargeand wea' isospin

/ll interactions are proportional to thecorresponding coupling and the 0charge


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Self-#nteractions of the gauge bosons:apart from the interaction with the fermions$ thegauge bosons may interact among themselves$ but

only$ if they carry corresponding chargesThe self-interactions of the gluons

The gluons carry a colourcharge therefore they

0see each other However$there can be maximally fourgluons in one fundamentalvertexThis is true for allgauge bosons


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Self-#nteractions of the gauge bosons:apart from the interaction with the fermions$ thegauge bosons may interact among themselves$ but

only$ if they carry corresponding chargesThe interactions of the bosons The photon and . boson

0see the wea' andelectromagnetic charge of the bosons For the 2-boson

vertex the combinations with2 and ! bosons existThere are no fundamentalself-interactions without the-bosons3


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#nteractions of the gauge with the Higgs bosons:)eaving aside self-interactions of the Higgs boson$there is only one more type of interaction$ namely

between gauge and Higgs bosons They reflect themechanism that gives mass to the gauge bosons

The H and ..Hinteractions are proprtional

to the mass of the gaugeboson The HH and..HH vertices are 0gaugeinteractions$ proportional to awea' coupling


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How to bui ld Feynman diagrams

0,oo'boo' recipe for tree-level

)abel the external particles %legs( and their momenta in a uni&ue way%eg p

1to p

4( and group them in all permutations of 4

Ta'e pairs of neighbouring particles and chec' whether there is anyvertex mapping 5oining them #f so$ draw the line of the particle andrepeat this step until all particles are connected

*a'e sure the arrows on the fermions are correct %direction forparticles6anti-particles( #n the Standard *odel$ there are no 0clashing

arrows: /ny fermion-line can have arrows only pointing along onedirection of the line

4ote: For meson decays it is sometimes useful to decompose the mesoninto its &uar' content one of the &uar's may be a non-interactingspectator


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Examples:There are two diagrams$ both in the s-channel$where the electron-positron pair fuses into a gaugeboson either the photon or the .-boson The

Higgs boson has been omitted here$ due to thesmall electron and muon mass )epton flavourconservation disallows vertices where an electron

and a muon are 5oined by a gauge boson

This time$ there is only one diagram$ because theneutrinos have no electrical charge and because inthe Standard *odel$ they are massless The catchhere will be the treatment of colour more later


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Examples:

There are three diagrams$ onewith an exchange of a neutrinoin the t-channel and two witha three-gauge boson vertex TheHiggs boson has again beenomitted here$ due to the smallelectron mass


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Examples:

This is a tric'y case$ since there is no way that

flavour changes when coupling to a photon

therefore such a transition$ if allowed$ has to bemediated by loop-level diagrams$ one of which isshown here Typically$ in such cases$ two instancesof the ,7* matrix are needed Such processeshave a very high phenomenological significance

This time$ the tric' is to 'now the flavourcomposition of the mesons and to guess6reconstructthe transition ta'ing place on the &uar' levelTypically$ there is then one spectator involved$ inthis case the d-&uar'


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Four-vectors and phase space

Four-vectors:

Summary of calculation rules index free:

Energy-momentum relation %a'a 0on-shell condition(

Four-momentum conservation %for (

often the indices are omitted in the delta-function


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The delta-function:

8roperties:

#ntegration over volumes:


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The phase space integration$ revisited:

First step: 9o the 2-d volume integral over p!with the 2-d

delta-function This implies that


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9isentangle energy and three momentum for the remainingfour-momentum se the Theta-functions for the energyinterval se polar coordinates for the three-momentum:

rite

This yields


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9o the integral over the three-momentum with help of thefirst delta-function$ leading to: se this inthe second delta function

fixing the energy to read


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The final expression yields

The 'inematics in the rest frame is entirely fixed by:


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idths

From the phase space$ the final expressions for decay widthscan be written as:

where the s&uared matrix element is summed over alloutgoing spins and colours and averaged over the incomingspins and colours

Typically the 'inematics is fixed in the cm frame of thedecaying particle in the massless case this reads:


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,ross sections

From the phase space$ the final expressions for scatteringcross sections can be written as:

where the s&uared matrix element is summed over alloutgoing spins and colours and averaged over the incomingspins and colours

Typically the 'inematics is fixed in the cm frame of theincoming particles in the massless case this reads:


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Examples:Higgs decay into fermions

There is only one Feynman diagram:

The full amplitude s&uared$ summed over all spins %but notover colours$ if f is a &uar'( is given by


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The width is given by

sing the decay 'inematics and four-momentum conservation

yields

E


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/dding the phase space and doing the angle integral gives:

The extra factor 4ccomes into play to ta'e into account

the fact that the fermions may be coloured:

For &uar's$ $ for leptons$

For a Higgs boson with m ; 1!< "e=

E l


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Examples:ee-annih i lation into fermions

+nly the final state consisting of a massless fermion pairwill be considered There are two Feynman diagrams

The two internal lines$ on the amplitude level$ behave li'e

if * is the mass of the particle and is its width+bviously$ both are < for photons

E l


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Examples:ee-annih i lation into fermions

#n the limit of low centre-of-mass energies around or less>

some concepts to nuclear physics

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