v.ÉI
Feynman diagrams represent particle interactions. The diagrams obey a few basic rules.
1.
You can draw two kinds of lines, a solid straight line with an arrow or a wiggly line:
va/a/a/ì.^
2.
An intersection is called a vertex. Every vertex must have two arrows and one wiggly line
3.
At each vertex, one of the two arrows must point in and the other arrow must point out.
4.
We will build up to diagrams with 2-3 vertices
5. The diagrams are usually read from left to right. Though,
they are sometimes read from bottom to
top. This is indicated by a "time axis." Each vertex has a "before" side and an "after side."
Time
Before
After
After
Before
Time
6.
In the diagram, label the vertices (4, B, C, D) in the time order that they occur. Identify all flaws.
A'A/, ßrce carctbr
6
o
c
13-L affocil5
c-
6'*)
)¿t
oK
A
Time
D-A
ßrr-t- c^rriu;/ ltÙ affÔ
ù ln
As particles in the Standard Model interact, certain quantities are always conserved.
Conservation of charge: the total charge p is conserved. ('We'll use Q for charge, q for quark.)
Cons'n. of
number : the total lepton count is conserved (antileptons count as negative)
Cons'n. of baryon number: total baryon count is conserved (q count as positive; Q count as negative.¡
a
a
a
Feynman diagrams represent interactions by distinguishing incoming particles from outgoing particles.
.
Solid lines represent free particles.
o Solid arrows that point forward with tinte arc matter particles.
o Solid arrows that point backward again,st time are antimatter partícles.
.
Wavy lines represent exchange particles.
Example
1 lrfo( p'decay
ve
A)
e-
?
B)
BEFORE
State the total charge entering the vertex.
Stäte the total charge
exiting the vertex.
AFTER
U"'
D) State the lepton
Ç7/11
number exiting the vertex.
L-: a
E) State
incoming exchange particle
carries negative charge
vertex.
F)
State
quark/baryon number entering the
o
theliark/baryon number exiting the
vertex.
O
G) Identify (i) the exchange particle and (ii) the force involved
tta/|cac*tb
Example
2 >F
positron entering
the veftex
e*
antineutrino enrerging
from the vertex
anlÌ
¿t¿
l€a
htt¿
A)
State the total charge entering the vertex.
B)
C)
al charge exiting the vertex.
?
State thå lepton number entering the vefiex
E)
State
ve
e!:t 7
State
AFTER
BEFORE
o
the
4'-L
the
aryon number entering the
vertex.
O
F)
State the quatk/baryon number exiting the
veltex.
A positron enters a vertex, emits an exchange
particle and emerges as an antineutrino
ô
G) Identify (i) the exchange particle and (ii) the force involved.
h)*
ûoson',
t 5Ò .axc'k- ì^l^e¡acllon
11^ù Ìa.\lDl
3)
(
Inspect the Feynman diagram below
llrn oêorcôG')
u
A)
vr /flúôn nt.lt'nino
px
Using conservation of charge, determine the
charge of the exchange particle^
m,stl
V. -l
AFTER
BEFORE
B) Identify (i) the exchange particle
and
(ii) the
force involved in this interaction.
fiú61 b< N- $ôsoût
5ô
incoming exchange particle
C) Show that lepton
number is the same entering the vertex as exiting the vertex
ltrt I
D)
1r- ,
I
C)
Show that the baryon number is the same entering the vertex as exiting the vertex.
A ¡¡.lon
4)
-l
þL:
+
n
o
*\eso r
bs a- u0' Øosoa ond
W.¿orns
a.
Inspect the Feynman diagram below.
A) Using conservation of charge,
determine the
electric charge of the exchange particle.
AFTER
å;^e
to
B)
u
rnust þ€- f
This interaction is mediated by the weak
nuclear force. Identify the exchange particle,
(il
a
BEFORE
^fÌ-t
+
boson
C) Explain why (B) could not be answered without being told that the weak force is involved.
I
D)
Show that lepton number is the same enteri
O
E)
),.îr>re
, o "Pfer ,-/
Show that the baryon number is the same entering the vertex as exiting the vertex.
L' 'lZ.
ur r.. C)
/
t -'/z
7
F) This is amore rare form of annihilation. Describe the interaction, in words.
ôú\ ..rg ¿,1-rk
'
gro,Juc-l-
af\¿
o-'^)+
oo\ìùorr¡
Ðæo'.r
n 7'twV
o.nnìh
ì
\o'lt
+o
I
?
Annihilation
õr1ø
V/hen an up quark (u) and up antiquark (u) collide, they annihilate, producing t.tvo photons in their place
Draw this Feynman diagram, using a horizontal time axis.
LL
T
ú
Pair Production
A single photon can spontaneously convert into a particle and its antimatter counterpart (e.g., y -+ u * ü).
If this occurs out in empty space, the resulting matter-antimatter pair are "virtual" particles: they are never
actually separate, and they immediately collide and annihilate, regenerating the single photon. But if this
process occurs near a nucleus, the nucleus can repel the electrically positive particle and attract the
electrically negative particle. This splits up the pair and imparts momentum to them both. Through this
interaction with the nucleus, the matter-antimatter particles become real particles.
This process, wherein a photon interacts with a nucleus to spontaneously convert into a matter particleantimatter particle pair, is called pair production.
(a) The mass of an electron is 9.11 x 10-3r kg. Calculate the minimum energy
produce an electron and antielectron (a positron).
î.
,ne= = ?./,< lo"'?
(sxtoÛ^ 4)
z-
f.A^
= 0.5 /
(b) Using your answer to (a), calculate the minimum frequency
a photon must have
-1'/
t0
r
Ae V/r'
a photon must have
to pair produce an
electron and an antielectron (a positron).
mv?+ lVve
€^ hf *>
to
(c) Draw
r
-a,l
" 8.A, lò
î^ %"ffi,
-lv
for pair production of
positron.
an electron and a
Use a vertical time axis.
a Feynman diagram
4ì'na
+
e-
L
'(
(d) Discuss the differences between annihilation and pair production.
0"oþ
a
Àaåå
à na59
r)
¿
*\^oto.n
to pair
Hz
/.ax l0 "0
5)
In each Feynman diagram below, add the correct directions to each solid line
Time
e+
e-
e'
Time
s
S
ve
d
(¡ I
ã
d
Time
u
a strange quark emits a photon and
remains strange; a down antiquark
absorbs the photon & remains antidown'
they are scattering offeach other
positron and electron
annihilate into a photon
then pair produce back to
a positron and electron
a
an up quark emits a W* boson
& becomes a down quark;the
W* boson becomes a positron
and an electron neutrino
In the middle: how can you tell, from the diagram, that s emits the photon before d absorbs the photon?
ßecuttse, on lhc lirne ttxi.s', lhe,s'ffungc c1uark'sverlex occurs helore lhtt cknt,n uniitltttu'k's verlex.
6) Inspect the interaction below
+3
,/l
u
3
A) Identiff
the exchange particle.
How do you know what it is?
d
BEFORE
AFTER
ut
(^)Qak
ér'ng
emitted exchange particle
carries away positive charge
D)
"Êl-Rr
:
Ò
Describe the interaction with words.
a,y1
7)
number is conserved before &
after the first vertex.
State the lepton number before and after the interaction.
bfare:O
E)
C) Show that the total baryon
up
in/o
7uûL- &cayt
2
Jowrt 2rølz
o¡å ^ Ð+ 5osorl
Inspect the Feynman diagram shown to the right. (a) Add
directions to the solid lines where omitted. (b) On the diagram,
identify the exchange particle. The weak force is the only force
by which a strange quark can decay. Otherwise, the total number
of strange particles is conserved (where s : +l; S: -1).
(c) Describe the diagrammed interaction in words.
A
2"vlC
"r,,å
è
Z"
&.co,y
goso ô t
u
Time
d
t1
o
Ã*
vt¡Àra-\ )
5v
S
nb
ut
8)
Inspect the Feynman diagram below
u
A)
vp
(¿¡t)
b"0"c<
^eLR.f
This interaction is mediated by the
weak nuclear force. Identifu the
exchange particle, and explain your
reasoning.
0on¡e-crlæìrlon
p
ef e\zc*rlr
d
úorzirW. O ,Çru &t)
>" Time
not a muon; how can you reason logically to this conclusion? (There are at least two ways.)
moót VsVe t\ e'\norrp
¡nuál ho"l< f¡.Pho% + -\
particles when writing out an intera
À1" I
9) DrawaFeynmandiagramforthefollowinginteraction:
u * e-
+ d *v".
Useaverticaltimeaxis.
L
u
J
(^
e
10)DrawaFeynmandiagramforthefollowinginteraction:s+u*e-*V".Useahorizontaltimeaxis.
L
(^
5
t^J
-
11) Draw a Feynman diagram
interaction: e+ +
tlme axrs
e-
-+ e+
for the following
* e-. Use a vertical
¿
tì.'t¡
I
V
e
,ft.
¿
e
following
interaction: e- * e- -+ e- t e-, where the two
electrons repel each other electromagnetically
Use a vertical time axis.
12) Draw a Feynman diagram for the
+'\À
e
e
e'
d
¿
13) Draw a Feynman diagram for the following
interaction: F- -+ vu e- tr". Use a
*
*
horizontal time axis.
tìnn,e
ù)-
,"
for the following
interaction: e- * e+ - p- * Lr*.Use a
vertical time axis. The interaction is
in nature.
14) Draw a Feynman diagram
grnsfon
tìr"q
tl'
l
1
L'
g+
15) Draw a Feynman diagram for the
following interaction: s -+ u + u + d.
Use a horizontal time axis.
,1,
L\t^tt
u+%
5
rnvSl
\'>¿
N-' $ 1u'l Y"e'
"ìJh
¿^,
J
/t
hY?
¡
converts into a Poolo ua
(a) State the quarks that make up a proton. State the quarks that make up a neutron.
l6) In beta minus decay ) a nov*"o
çrohov^.
U
d.
rneuhron'.
ß
(b) Write out the complete reaction, in terms of quarks, for a beta minus decay
td+
t,l"+
Ð.*
e'
(c) Draw a Feynman diagram, in terms of quarks, for a beta minus decay:
- e\¿cfcìc chcc¡- rnø-'!nò
- ¿r{-t o\lÁ/- ae! 5o-\o'n -ò {oo 7
Lirn¿
,/3
L
w'
0
u?
e-
17) In beta plus decay
a
)a prÒlon
converts into
v^r¿ufco 4
reaction, in terms of quarks, for a beta plus decay.
(a) Write out the
U- d* /,-+ e+
(b) Draw a Feynman diagram, in terms of quarks, for a beta plus decay:
+Vz
u
èì"r. e
Á
3
e*+
\
/.o
18) (a) Draw a Feynman diagram for the following interaction
5 -+ u
u d. Use a vertical time axis.
* *
ðr'')
z
7
tìme-
ü
+
/3
r^
\)-'t
AJ'
7"
5
,y',
(b) Without balancing the charge, how can you tell that
the interaction involves the weak nuclear force?
*cw'n¡*-rìt$r, (êì.¡.uo
c) ¡.'t
eoa3a-çv4d\
Lö
of
=
5
t ,\
+
+r/l