Download 3. Quantum Field Theory (QFT) — Renormalisation How do we

3. Quantum Field Theory (QFT)
—
Renormalisation
How do we measure particles
• We can measure tracks of individual charged particles with
– pixel detectors
– wire chambers
⇒ we measure the 3-momentum p
~
• We can measure the energy of particles in
– calorimeters
⇒ we measure the energy E
• We can distinguish different kinds of particles by
– sets of Cherenkov counters
– comparing calorimeter- and track-data
⇒ reconstructing the 4-momentum: E 2 = p
~2 + m2
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
How do we measure intermediate particles
• masses are measured for instance with ”invariant-mass distributions ”
• example:
e+ e− → Z → µ+µ−
– the 4-momenta of the muons are reconstructed
– these 4-momenta are summed up and squared: M 2
– events with 2 muons are distributed according to M 2:
∗ this gives a Breit-Wigner distribution (on top of a background)
– measured mass = position of the peak in the Breit-Wigner curve
• couplings are measured with the strength of the interaction
• example:
H → µ+µ− versus
H → bb̄
– identifying the decaying particle in all events
#events(H→µ+ µ− )
– relative coupling strength = #events(H→bb̄)
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
How do we calculate
• we generate all Feynman diagrams {f } for the process
• we calculate the amplitude for each Feynman diagram Mf
P
• we add up all amplitudes M = f Mf
• we square the modulus |M|2
• we integrate over the Lorentz invariant phase space dLips
µ
µ
2
.
.
.
p
|M|
dLips(p
dσ = symmetry
n)
1
f lux
µ
µ
– momentum-conservation P µ = Pin = i pi
– and each particle i in the final state contributes, so
X
Y
d3p
~i
µ
µ
4
4
dLips(p1 . . . pn ) = (2π) δ (P −
pi)
3 2E
(2π)
p
~i
i
i
P
• in perturbation theory, we expand in orders of the coupling constant g:
M=
X
k
g M
(k)
where
M
(k)
=
k
Thomas Gajdosik – Concepts of Modern Theoretical Physics
X
(k)
Mf
f
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
How do we calculate
• When perturbation theory is applicable: |g nM(n)| > |g n+1M(n+1)|
– the leading order (LO) is the smallest n with |M(n)| > 0
– the next-to leading order (NLO) is then n + 1 or n + 2.
• the cross section is also a sum over powers of the coupling constant g:
P
where
dσ (2n) = |g nM(n)|2dLips
and
dσ = k=2n g k dσ (k)
h
i
dσ (2n+1) = (g nM(n))†(g n+1M(n+1)) + (g nM(n))(g n+1M(n+1))† dLips
• we identified the ”physical” coupling g from the measurement of σ
– LO:
– NLO:
R
LO
σ
= dσ (2n)
R
NLO
(2n)
σ
= (dσ
+ dσ (2n+1))
• if σ LO 6= σ NLO . . . what is now the right σ ?
Thomas Gajdosik – Concepts of Modern Theoretical Physics
⇒
13.09.2012
Renormalisation
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• we have 3 real scalar fields A, B, and C and 1 coupling g:
1 Φ (∂ 2 + m2 )Φ + g A B C
L0 = − 2
0
0 0 0 0
0
0,Φ
where
Φ = A, B, C
−1
– masses are the poles in the propagators i × [p2 − m2
0,Φ + iǫ]
– the coupling connects all three different fields
B
C
.
• having the physical masses as mA > mB + mC ,
.A
– A will decay into B and C:
– at LO there is only one diagram, which gives g 1M(1) = g
3
3
d p
~B
d p
~C
dΩ κ
– dLips = (2π)4δ 4(pA − pB − pC ) (2π)
=
3 2E
3
(4π)2 ABC
p
~B (2π) 2Ep
~C
1
σ LO(A → BC) =
2mA
Z
g 2 κABC
κABC =
|g|
2
(4π)
4π 2mA
2 dΩ
g2
• g or α = 4π can be determined from the decay!
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
we calculate now σ NLO(A → BC)
B
.
C
B
.
C
B
.C
• there is no diagram of order g 2
• there are 4 diagrams of order g 3
.
A
.
A
.
A
– 3 of them have a ”bubble” on an external line
⇒
we will have to renormalise also the propagators!
∗ as it turns out, they will not contribute (on-shell scheme)
B
– the 4th diagram is just a triangle
∗ which is usually not zero
∗ but depends on incoming and outgoing momenta
.
.C
A
– so g 3M(3) = g 3tABC
h
i dΩ
1
2
3
†
†
3
κABC
|g| + g × (g tABC ) + g × (g tABC )
2mA
(4π)2
Z
g 2 κABC 2
1 + 2g TABC
with TABC = dΩ
=
4π Re[tABC ]
4π 2mA
NLO
σ(A→BC)
=
Z
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• but we calculated with g0, not with the measured g !
• the same happens to the other parameters of the theory
• writing
g0 = g + δg ,
2
2
m2
0,Φ = mΦ + δmΦ , and
Φ0 =
q
ZΦ Φ = Φ + 1
2 δZΦ Φ
we get a new Lagrangian
q
2
2
2
1
L0 = − 2 ZΦΦ(∂ + mΦ + δmΦ)Φ + (g + δg) ZAZB ZC ABC = L + δL
with
1 Φ(∂ 2 + m2 )Φ + gABC
L = −2
Φ
1 Φ[δZ (∂ 2 + m2 ) + δm2 ]Φ + [ g (δZ + δZ + δZ ) + δg] ABC
δL = − 2
Φ
A
B
C
Φ
Φ
2
where only terms linear in the expansion of the parameters are kept.
• We assume only one-loop accuracy for now.
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• Introducing abbreviations for the loops
pi
• The self energies iΠ(p2) =
.
pf
g2
2
= i (4π)2 B0(p2; m2
0, m1 )
.
• Using the Feynman rules we get to one loop
iΠ(p2i )(2π)4δ 4(pi − pf )
Z
d4 k1 d4k2 (ig)(2π)4δ 4 (pi − k1 − k2)i (ig)(2π)4δ 4(k1 + k2 − pf )i
=
(2π)4 (2π)4
k12 − m21
k22 − m20
Z
2
1
g
2
2 −1
4
2
2 −1
]
[(q
+
p
)
−
m
d
q
[q
−
m
= i(2π)4δ 4 (pi − pf )
i
1]
0
(4π)2 iπ 2
2
g
2
2
2
=: i(2π)4δ 4 (pi − pf ) (4π)
2 B0 (pi ; m0 , m1 )
pB
2 2
• In a similar way the vertex correction iT (p2
A , pB , pC )
is written with the function
=
g3
).
, m2
, m2
; m2
, p2
, p2
C0(p2
2
C
B
A
B
A
C
(4π)
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
p. C
pA
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• Now we have 7 counterterms that we should detemine: δm2
Φ , δZΦ , δg
• they also give rise to new diagrams:
– six counterterms are used by iδZΦ
(p2
−
m2Φ )
−
iδm2Φ
p
.p
=
.
pB
p. C
– the 7th counterterm by iCg = ig2 (δZA + δZB + δZC ) + iδg =
.
pA
• We have to choose Renormalisation conditions for them.
We can take (without proof):
1. the pole of the full propagator should be at m2
Φ
⇒
2
δm2
Φ = Π(mΦ )
2. the residuum of the full propagator should be 1
⇒
d Π(x)|
′ (m2 )
=
−Π
δZΦ = − dx
2
Φ
x=m
Φ
3. the full decay width of A should be like in the tree level.
⇒
2 , m2 ) − g (δZ + δZ + δZ )
,
m
δg = −T (m2
A
B
C
C
B
A
2
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• Calculating the NLO matrix element for the decay of A → B + C,
we have to add 9 diagrams:
B
B
C
.
.
+
b
A
C
.
.
C
+
B
.
.
C
+
c
A
B
d
.
C
.
+
A
B
C
.
B
.
C
.
B
C
.
+
h
A
A
.
+
g
A
e
A
.
+
f
.
B
C
.
+
a
B
.
i
A
.
A
2
• adding c + g and Taylor expanding around p2
A = mA gives
2
2
2
2
2
′
2
2
2 ) − δm2
Π(mA) + Π (mA)(pA − mA) + O (pA − mA)
δZA(p2
−
m
A
A
A
−
g
−g
2
2
p2
p2
A − mA
A − mA
2
2
= O (pA − mA)
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Example of Renormalisation: ABC-theory
• a gives now g 1M(1) = g
2
2
• c + g , d + h , and e + i are all O (pA − mA)
• so g 3M(3) = limp2 →m2 ( b + · · · + i ) = limp2 →m2 ( b + f )
Φ
Φ
Φ
Φ
g
2
2
)
+
,
m
,
m
= T (m2
C
B
A
2 (δZA + δZB + δZC ) + δg = 0
• So now σ LO = σ NLO for this decay !
• g can be determined unambiguously.
• but where do come new predictions?
– A → 3B + C or A → B + 3C or . . .
– scattering B + C → B + C: dσ(BC → BC)
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
s
Example of Renormalisation: ABC-theory, dσ(BC → BC)
• with the same argument as before: no corrections on external legs
• contributing diagrams to the s-channel,
initial momenta k, final momenta p:
C
B.
C
B.
+
.
B
C
a
C
B.
+
.
B
C
b
C
B.
+
.
B
C
c
C
B.
+
.
B
C
d
C
B.
+
.
B
C
e
C
B.
+
.
B
C
f
C
B.
+
.
B
C
g
.
B
C
h
2 = p2 = m2 , k2 = p2 = m2 , and (k + k )2 = (p + p )2 = s.
• kB
B
C
B
C
C
C
C
B
B
• a = (ig)2
g2
i
2
(2)
so g M
=−
s−m2
s−m2
A
A
• .b + c + d + e
=
2 , k2 )
i3gT (s, p2B , p2C )
i3gT (s, kB
i3gCg
i3gCg
C
+
+
+
2
2
2
s − mA
s − mA
s − mA
s − m2A
T (s, m2B , m2C ) − T (m2A , m2B , m2C )
2 0
= −2ig
= O (s − mA )
s − m2A
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
s
Example of Renormalisation: ABC-theory, dσ(BC → BC)
• .f + g
=
(ig)2i2iδZA (s − m2A ) − δm2A
(ig)2i2iΠ(s)
+
(s − m2A )2
(s − m2A )2
= ig
2 Π(s)
− δm2A + δZA (s − m2A )
2 0
2
)
−
m
=
O
(p
A
A
(s − m2A)2
2 , k 2 ; (−p − p )2 , (−p + k )2 ; m2 , m2 , m2 , m2 )
• h = g 4Dbox (p2B , p2C , kC
B
C
C
C
B
A
C
A
B
4
(4)
• .g M
• so
g 2[Π(s) − Π(m2A )] 2g[T (s) − T (m2A )] + g 2Π′ (m2A )
g2
−
+ Dbox =:
∆M
=
(s − m2A )2
s − m2A
s − m2A
dσ LO
s
(BC →BC)
• and dσ NLOs
(BC →BC)
1
=
2κsBC
2 2
−g 2 2 dΩ κsBC
g
dΩ
=
s − m2 (4π)2 2s
4π
4s(s − m2 )2
A
A
−g 2
g4
dΩ κsBC
1
4
(4)
+
·
2Re[g
M
]
=
2κsBC (s − m2A )2
s − m2A
(4π)2 2s
2 2
dΩ
g
LO
−
2Re[∆M])
=
6
dσ
=
(1
s
(BC →BC)
4π
4s(s − m2A )2
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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3. Quantum Field Theory (QFT)
—
Renormalisation
Regularisation
• is the prescription, how to deal with undefined integrals
• examples:
– only integrating up to p2 < Λ
– subtracting the integrand with a regulator mass (Pauli-Villars)
– making the dimension continuous (dimensional regularisation)
• the physical result cannot depend on the regularisation
• the counterterms can ”absorb” the regularisation constants
⇒
renormalisable theory
Regularisation in ABC
• only the selfenergies had ill defined integrals
• δZΦ and δm2
Φ contain regularisation constants
• but they drop out in all physical processes
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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