Some concepts to nuclear physics
Transcript of 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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The structure of Feynman rules
#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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The structure of Feynman rules
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$there can be maximally fourgluons in one fundamentalvertexThis is true for allgauge bosons
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The structure of Feynman rules
Self-#nteractions of the gauge bosons:apart from the interaction with the fermions$ 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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The structure of Feynman rules
#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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How to bui ld Feynman diagrams
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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How to bui ld Feynman diagrams
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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How to bui ld Feynman diagrams
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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Four-vectors and phase space
The delta-function:
8roperties:
#ntegration over volumes:
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Four-vectors and phase space
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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Four-vectors and phase space
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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Four-vectors and phase space
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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Four-vectors and phase space
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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Examples:Higgs decay into fermions
The width is given by
sing the decay 'inematics and four-momentum conservation
yields
E
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Examples:Higgs decay into fermions
/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>