Virtual particles - the ultimate source of any force
Lecture notes
Jan Rak
Jyväskylä University, HIP, Finland
May 8, 2015
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
1 / 21
High Energy experiments
1
Linear vs circular accelerators
linear less rad losses, no
bending magnets
cir. less space
2
fixed target vs colliders
√
Fixed T. - less s for physics
due to the loss for C.M.S
rap.
√
collider: s = 2Ebeam
smaller lumi L
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
2 / 21
High Energy experiments
3
Colliding species:
e+ + e− or (µ+ + µ− )
no bck, but Lp small
√
√
ŝ = s
p + p̄
Larger partonic lumi Lp
bck from many simult. partonic
interactions.
p
√
hŝi = hx1 x2 i s
A+A
huge Lp but bck enormous
Phase Trans. QGP only here.
Nucl. geometry important.
mixtures: (e− , µ± , p, A) + (p, A)
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
3 / 21
High Energy experiments
3
Colliding species:
e+ + e− or (µ+ + µ− )
no bck, but Lp small
√
√
ŝ = s
p + p̄
Larger partonic lumi Lp
bck from many simult. partonic
interactions.
p
√
hŝi = hx1 x2 i s
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
4 / 21
Special Relativity - time/space like 4-vectors
The Lorentz-invariant square of a
4-vector x 2 = x µ xν is said to be
light-like,
M2 = 0
space-like,
M 2 < 0.
x0
x0
( x )2 - (x 1)2
time-like
light-like
x+
(sinh(y), cosh(y))
co
ht
M2 > 0
lig
time-like,
ne
x+’= ey x+
ξM=y
100
space-like
ξE
x1
50
0
-50
-100
10
8
6
4
2
0
-2
-4
-6
-2 0
-8
-6 -4
-10 -10 -8
2 4
10
6 8
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
5 / 21
Negative/positive invariant mass
E [GeV]
10
ll
l
hel she
on
Time-like
β<1
8
M>0
6
When P is the
timelike
4-momentum, then
the virtual particle
has positive mass,
as well as the real
particle. E.g.
Drell-Yan (DY)
process.
s
offπ0
4
Space-like
β>1
2
M<0
0
t-channel
-2
-4
-6
Virtual particle →
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
8
10
M 6= m0
p [GeV/c]
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
6 / 21
Drell-Yan process.
q + q̄ → l + + l −
Time-like (s-channel) virtual photon
propagates in space (β < 1) and
decays into l ± . How “real” is the
virtual?
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
7 / 21
Hard Scattering
We speak about t-channel process mediated by purely space-like
quanta. SL transfers no energy, only momentum.
E [GeV]
10
ll ell
she h
off- on-s
Time-like
β<1
8
M>0
6
P3=(E , p , p
3
1
π0
4
T,3
)
2
3
4
Space-like
β>1
2
x1PA
M<0
0
||,3
p +p → p +p
M2inv=x1x2 s =-p2 = -Q2
t-channel
T
-2
t=(0,0,p )
T,t
-4
x2PB
-6
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
8
10
p [GeV/c]
P4=(E , p , p
4
||,4
T,4
)
T -channel → t = (0, 0, ~pT ,t ) where pT ,t = pT ,3 + pT ,4 Since
pT ,3 = pT ,4 ≡ pT
⇒
2 = −Q 2 = 4p 2
Minv
T
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
8 / 21
What is the lifetime of the virtual particle?
The lifetime T of the virtual particle can be defined on the basis of the
uncertainty principle:
1
T =
|∆E|
where |∆E| is the “distance from reality”
q
q
2
2
∆E = P + m − P 2 + M 2
is the difference of energies of the real (m) and virtual (M) particles
with the same 3-momentum P. Then the lifetime is
T =
E − Ẽ
|m2 − M 2 |
p
P 2 − M 2 is the of the virtual particle, and
where
E
=
√
Ẽ = E 2 + m2 − M 2 is the energy of the real particle.
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
9 / 21
What is the lifetime of the virtual particle?
You can rewrite the previous equation as:
!
r
E
m2 − M 2
T =
1+
+1
|m2 − M 2 |
E2
(1)
Analogously, the space dimensions L of the region of propagation of
the virtual particle can be defined on the basis of the uncertainty
principle√for momenta√and coordinates: L = 1/|∆P| , where
|∆P| = E 2 − m2 − E 2 − M 2 is the difference of momenta of the
real and virtual particles moving along the direction of the momentum
vector P with the same energy E. Then
L=
here again, |P̃| =
particle.
|P| + |P̃|
|m2 − M 2 |
p
P 2 − m2 − M 2 is the momentum of the real
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
10 / 21
What is the path length of the virtual particle?

s
2 − M2
|P|
m
 1+
L=
+ 1
|m2 − M 2 |
P2
(2)
At small differences of masses of the virtual and real particles, when
|m2 − M 2 | E 2 ,
follows
T =
|m2
2E
,
− M 2|
|m2 − M 2 | P 2
L=
2|P|
− M 2|
|m2
and one can construct the 4-vector
{T , L} =
|m2
2P
− M 2|
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
11 / 21
What is the path length of the virtual particle?
2P
|m2 − M 2 |
valid for any 4-momentum of virtual particle timelike (β = P/E < 1)
and spacelike (β = P/E > 1).
{T , L} =
k=(ω,k)
k=(ω,k)
PI=(E , p +k)
1
Θ
Time-Like
Θs
∼ PII=(E2, p1-k) p = (E , p )
Θ
2
2
2
p = (E , p )
2
∆r
2
2
2
p =(E , p )
1
1
1
p =(E , p )
1
1
1
q=(0,q) Space-Like
N
N
q=(0,q) Space--Like
N
I.
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
N
II.
May 8, 2015
12 / 21
T and L of virtual e± in Bremsstrahlung
k=(ω,k)
k=(ω,k)
PI=(E , p +k)
1
Θ
2
Time-Like
Θs ∆ r
∼ PII=(E2, p1-k) p = (E , p )
Θ
2
2
2
p = (E , p )
2
2
2
p =(E , p )
1
1
1
p =(E , p )
1
1
1
q=(0,q) Space-Like
N
N
q=(0,q) Space--Like
N
I.
N
II.
4-momentum of the virtual electron in diagram I, is
PI = p2 + k = {E1 , p2 + k }
and diagram II
PII = p1 − k = {E2 , p1 − k }
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
13 / 21
T and L of virtual e± in Bremsstrahlung
It is evident that the square of mass of the virtual electron for diagram I
has the form
MI2 = (p2 + k )2 = me2 + 2p2 k = me2 + 2E2 ω(1 − β2 cos θ)
and hence
TI =
E1
E2 ω(1 − β2 cos θ)
LI =
|p 2 − k |
E2 ω(1 − β2 cos θ)
TII =
E2
E1 ω(1 − β1 cos θ)
LII =
|p 1 − k |
E1 ω(1 − β1 cos θ)
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
14 / 21
T and L of virtual e± in Bremsstrahlung
At ultrarelativistic energies E1 me , E2 me and small angles
θ 1, θ̃ 1, one can write
1 − β2 cos θ ≈ 1 − β2 + β2
θ2
1
≈
(1 + γ22 θ2 )
2
2γ22
1 − β1 cos θ̃ ≈ 1 − β1 + β1
1
θ̃2
(1 + γ12 θ̃2 )
≈
2
2γ12
where γ1,2 = E1,2 /me . Furthermore, in the limit γ1,2 1 then
|p 2 + k | ≈ E1 and |p 1 − k | ≈ E2 . Then
TI ≈ LI ≈
2γ1 γ2
ω(1 + γ22 θ2 )
and
TII ≈ LII ≈
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
2γ1 γ2
ω(1 + γ12 θ̃2 )
May 8, 2015
15 / 21
T and L of virtual e± in Bremsstrahlung
The main contribution to the process of the bremsstrahlung of an
ultrarelativistic electron in the Coulomb field of a heavy nucleus arises
<
from the region of small angles θ<
∼1/γ2 , θ̃ ∼1/γ1 . As a result, the
effective path length of an ultrarelativistic electron in the
bremsstrahlung in the Coulomb field has a magnitude of the order of
high energy limit
L ∼ LI ∼ LII ∼ ~c
γ1 γ2
ω
For example
for an electron with energy E1 ≈ E2 ≈ 100 GeV and photon energy
ω = 10 MeV, the effective path length L of the virtual electron amounts
to L ∼ 8 × 10−2 cm ∼ 1 mm (the virtual electron life-time T equals
∼ 3 × 10−12 s).
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
16 / 21
Special Relativity - time/space like
t-channel q+q→ q+q
q
q
beam
FSR jet
out
space-like gluon
q
beam
s= s
qout
q
ISR jet
q
out
qout
q
ISR jet
FSR jet
Figure: Schematics of the t-channel parton elastic scattering event.
Momentum is transferred but energy is not (in CM).
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
17 / 21
Formation time and ISR/FSR jets
γ*
e-
γ*
- - -
- - -
- +++ +
+
-+
+- ++ + ++ -
- ++ + +
+
-+ - +- ++ + ++ -
ISR jet
e-
- - -
- - -
- ++
- + - +++
- -
FSR jet
- - - +++ +
+
-+ - ++
+
- +++ - - -
∆E∆t ∼ ∆p∆t ≥
~
2
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
18 / 21
Special Relativity - 4-vectors
With the metric tensor notation the 4-vector P of a particle with
momentum p, energy E and invariant mass m is denoted as
P = (E, p)
or
P = (E, px , py , pz )
The four-dot product of two 4-momentum vectors P1 and P2 is then
P1 · P2 = gµν P1µ P2ν = E1 E2 − p1 · p2
= E1 E2 − p1 p2 cos θ ∼ E1 E2 (1 − cos θ)
(3)
where p1 · p2 is the dot product of the 3-momentum vectors, p1 and p2
are the moduli of the 3-vectors and θ is the angle between them. The
squared modulus of a 4-vector gives a famous Lorenz invariant
P 2 = E 2 − p 2 = m2 .
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
(4)
May 8, 2015
19 / 21
The source of transverse momentum is the Invariant
Mass
This is the answer to my first question - how come that the two
point-like object with no transverse dimension could ever scatter.
x1 PA
Minv= x1x2. s
x2 PB
It is the invariant mass
p
p
p
Minv = P1 · P2 = E1 E2 − p1 p2 cos θ ∼ E1 E2 (1 − cos θ)
the source of any spatial >1D propagation (opening angle).
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
20 / 21
Hard Scattering
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
21 / 21
Thanks for your attention!
Slides for this talk will be available at:
https:
//trac.cc.jyu.fi/projects/alice/wiki/lectures14
Virtual particles - the ultimate source of any force
Lecture notes
Jan Rak
Jyväskylä University, HIP, Finland
May 8, 2015
Jan Rak (Jyväskylä University, HIP, Finland) Virtual particles - the ultimate source of any force
May 8, 2015
22 / 21