Interactions of Neutrinos
at High and Low Energies
ν
Kevin McFarland
University of Rochester
Neutrinos at SUSSP
12-15 August 2006
ν
Neutrino Interaction Outline
• Motivations for and History of Measuring
Neutrino Interactions
• Weak interactions and neutrinos
ƒ Elastic and quasi-elastic processes, e.g., νe scattering
ƒ Deep inelastic scattering, (νq scattering)
ƒ The difficulties of being in near thresholds…
• Current & future cross-section knowledge
ƒ What we need to learn and how to learn it
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
2
ν
Tone of These Lectures
• Focus will be on
ƒ Cross-sections useful for experiments
ƒ Estimating cross-sections
ƒ Understanding qualitatively the key effects
• Therefore, it should therefore go without saying
that I am the second experimentalist lecturing
at SUSSP…
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
3
The Birth of the Neutrino
ν
Wolfgang Pauli
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
4
Translation from the German,
Please?
ν
4th December 1930
Dear Radioactive Ladies and Gentlemen,
As the bearer of these lines, to whom I graciously ask you to listen, will explain to you in
more detail, how because of the ”wrong” statistics of the N and 6Li nuclei and the
continuous beta spectrum, I have hit upon a desperate remedy to save the ”exchange
theorem” of statistics and the law of conservation of energy. Namely, the possibility that
there could exist in the nuclei electrically neutral particles, that I wish to call neutrons, which
have spin and obey the exclusion principle and which further differ from light quanta in that
they do not travel with the velocity of light. The mass of the neutrons should be of the same
order of magnitude as the electron mass (and in any event not larger than 0.01 proton
masses). The continuous beta spectrum would then become understandable by the
assumption that in beta decay a neutron is emitted in addition to the electron such that the
sum of the energies of the neutron and the electron is constant...
From now on, every solution to the issue must be discussed. Thus, dear radioactive people,
look and judge. Unfortunately I will not be able to appear in Tübingen personally, because I
am indispensable here due to a ball which will take place in Zürich during the night from
December 6 to 7….
Your humble servant,
W. Pauli
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
5
The True Source of Slow
Progress in Neutrino Physics
ν
From now on, every solution to the issue must be discussed. Thus, dear radioactive people,
look and judge. Unfortunately I will not be able to appear in Tübingen personally, because I
am indispensable here due to a ball which will take place in Zürich during the night from
December 6 to 7….
Your humble servant,
W. Pauli
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
6
Translation from the Archaic
Physics Terms, Please?
ν
• To save the law of conservation of energy?
β-decay
The Energy of the “β”
• If the above picture is complete, conservation of energy in
this two body decay predicts monochromatic β
ƒ but a continuous spectrum had been observed (since 1914)
• Pauli suggests “neutron” takes away energy!
•
“The exchange theorem of statistics”, by the way, refers to the fact
that a spin½ neutron can’t decay to an spin½ proton + spin½ electron
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
7
Weak Interactions
ν
• Current-current interaction H = GF J μJ
μ
w
2
Fermi, Z. Physik, 88, 161 (1934)
ƒ Paper rejected by Nature because
“it contains speculations too remote
from reality to be of interest to the reader”
• Prediction for neutrino interactions
ƒ If n → pe ν , then ν p → e n
ƒ Better yet, it is robustly predicted by Fermi theory
−
+
o Bethe and Peirels, Nature 133, 532 (1934)
ƒ For neutrinos of a few MeV from a reactor, a typical
cross-section was found to be
−44
2
σν p ∼ 5 ×10 cm
ƒ (Actually wrong by a factor of two (parity violation)
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
8
How Weak is This?
ν
• σ~5x10-44cm2 compared with
ƒ σγp~10-25 cm2 at similar energies, for example
• The cross-section of these few MeV neutrinos is
such that the mean free path in steel would be
10 light-years
“I have done something very bad today
by proposing a particle that cannot be
detected; it is something no theorist
should ever do.”
Wolfgang Pauli
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
9
Extreme Measures to Overcome
Weakness (Reines and Cowan, 1946)
ν
ν p → e+ n
•
Why inverse neutron beta
decay?
ƒ clean prediction of Fermi
weak theory
ƒ clean signature of prompt
gammas from e+ plus
delayed neutron signal.
o Latter not as useful with
bomb source.
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
10
Discovery of the Neutrino
ν
• Reines and Cowan (1955)
ƒ Chose a constant source,
nuclear reactor (Savannah River)
ƒ 1956 message to Paul: ”We are
happy to inform you [Pauli] that we
have definitely detected neutrinos…”
ƒ 1995 Nobel Prize for Reines
ν p → e+ n
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
11
Better than the Nobel Prize?
ν
Thanks for the message. Everything
comes to him who knows how to wait.
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
12
Interactions and Flavor
•
ν
1962 Lederman, Schwartz, Steinberger at Brookhaven Nat’l Lab
• One neutrino was known (beta decay)
ƒ Question: if μ+→e+νν, why not μ+→e+γ ?
• First accelerator neutrino beam
ƒ 5 GeV protons on Be Target (3.5x1017 of them)
ƒ π+→μ+νμ in a 21m decay region
ƒ Found 34 single-μ events, 5 background,
but NO e-like events!
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
1988 Nobel citation: “for
the neutrino beam method
and the demonstration of
the doublet structure of the
leptons through the
discovery of the muonneutrino”
13
Another Flavor Example
ν
• Radiochemical Solar Neutrino Detector
Ray Davis (Nobel prize, 2002)
ƒ ν+nÆp+e- (stimulated β-decay)
ƒ Use this to produce an unstable isotope,
ν+37ClÆ37Ar+e- , which has 35 day half-life
ƒ Put 615 tons of
Perchloroethylene
in a gold mine
o expect one 37Ar atom
every 17 hours.
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
14
Another Flavor (cont’d)
ν
• Confirmed that sun shines
from fusion, but 1/3 of ν !
• Of course this is oscillation
and flavor selection of
interaction ν+37ClÆ37Ar+e-
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
15
Another Neutrino
Interaction Discovery
ν
• Neutrinos only feel the weak force
ƒ a great way to study the weak force!
• Search for neutral current
ν
ƒ arguably the most famous neutrino
interaction ever observed is shown at right
ν μ e− → ν μ e−
Gargamelle, event from
neutral weak force
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
16
ν
An Illuminating Aside
• The “discovery signal” for the neutral current
was really neutrino scattering from nuclei
ƒ usually quoted as a ratio of muon-less interactions to
events containing muons
σ (ν N → ν X )
μ
ν
R =
•
12-15 August 2006
μ
σ (ν μ N → μ − X )
But this discovery was complicated for 1218 months by a lack of understanding of
neutrino interactions
ƒ backgrounds from neutrons induced by
neutrino interactions outside the detector
ƒ not understanding probability of
fragmentation to high E hadrons which
then “punched through” to fake muons
Kevin McFarland: Interactions of Neutrinos
17
The Future: Interactions and
Oscillation Experiments
ν
• Boris has elegantly described a situation where muon
oscillation appearance experiments at L/E~300 km/GeV
have rich physics potential
ƒ mass hierarchy, CP violation, τ appearance (sterile ν’s)
• What Boris hasn’t worried about (at least in front of you)
ƒ transition probabilities are small, must be precisely measured
for mass hierarchy and CP violation
ƒ the neutrinos must be at difficult energies of 1-few GeV for
electron appearance, many GeV (> charm threshold) for τ
• We are not looking for neutrino flavor measurements in which
distinguishing 1 from 0 or 1 from 1/3 buys a ticket to Stockholm
ƒ Difficulties are akin to neutral current experiments
ƒ Is there a message for us here?
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
18
ν
Present View of
Weak Interactions
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
19
Weak Interactions Revisited
ν
GF μ
J Jμ
• Current-current interaction Hw =
2
(Fermi 1934)
ƒ Paper rejected by Nature because “it contains
speculations too remote from reality to be of
interest to the reader”
• Modern version:
GF
⎡⎣l γ μ (1 − γ 5 )ν ⎤⎦ ⎡⎣ f γ μ (V − Aγ 5 ) f ⎤⎦ +h.c.
H weak =
2
• PL = 1/ 2 (1 − γ 5 ) is a projection operator onto
left-handed states for fermions and righthanded states for anti-fermions
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
20
Helicity and Chirality
ν
• However, chirality
(“handedness”) is Lorentzinvariant
ƒ Frame dependent (if massive)
• Helicity is projection of spin
along the particles direction
– Only same as helicity for
massless particles.
left-helicity
right-helicity
• Neutrinos only interact weakly
with a (V-A) interaction
ƒ All neutrinos are left-handed
ƒ All antineutrinos are righthanded
o because of production!
• If neutrinos have mass then
left-handed neutrino is:
– Mainly left-helicity
– But also small right-helicity
component ∝ m/E
• Only left-handed charged-leptons
(e−,μ−,τ−) interact weakly but
mass brings in right-helicity:
ƒ Weak interaction maximally
violates parity
π + ( J = 0) → μ + ( J = 12 )ν μ ( J = 12 )
μ+
12-15 August 2006
ν
←⎯
⎯⎯ • ⎯⎯→
⇐
⇐
Kevin McFarland: Interactions of Neutrinos
22
Two Weak Interactions
ν
• W exchange gives Charged-Current (CC) events and
Z exchange gives Neutral-Current (NC) events
In charged-current events,
Flavor of outgoing lepton
tags flavor of neutrino
Charge of outgoing lepton
determines if neutrino or
antineutrino
l− ⇒νl
l+ ⇒ν l
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
23
Electroweak Theory
ν
• Standard Model
ƒ SU(2) ⊗ U(1) gauge theory unifying weak/EM
⇒ weak NC follows from EM, Weak CC
ƒ Measured physical parameters related to mixing
parameter for the couplings, g’=g tanθW
Charged-Current
Neutral-Current
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
24
ν
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
25
Electroweak Theory
ν
• Standard Model
ƒ SU(2) ⊗ U(1) gauge theory unifying weak/EM
⇒ weak NC follows from EM, Weak CC
ƒ Measured physical parameters related to mixing
parameter for the couplings, g’=g tanθW
Z Couplings
gL
gR
νe , νμ , ντ
1/2
0
e,μ,τ
−1/2 + sin2θW
sin2θW
u,c,t
1/2 − 2/3 sin2θW
− 2/3 sin2θW
d,s,b
2
−1/2 + 1/3 sin θW
g 2 2 MW
,
= cosθW
e = g sin θW , GF =
8M W2 M Z
2
Charged-Current
1/3 sin θW
• Neutrinos are special in SM
ƒ Right-handed neutrino has NO
interactions!
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
Neutral-Current
26
Why “Weak”?
ν
• Weak interactions are weak because of the
massive W and Z bosons exchange
1
dσ ∝
dq 2 (q 2 − M 2 ) 2
q is 4-momentum carried by exchange particle
M is mass of exchange particle
At HERA see W and Z
propagator effects
- Also weak ~ EM strength
• Explains dimensions of Fermi “constant”
GF =
2 ⎛ gW
⎜⎜
8 ⎝ MW
⎞
⎟⎟
⎠
2
= 1.166 × 10 −5 / GeV 2 ( gW ≈ 0.7)
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
27
Neutrino-Electron Scattering
μ
ν
−
• Inverse μ−decay:
νμ + e− → μ− + νe
νμ
ƒ Total spin J=0
(Assuming massless
muon, helicity=chirality)
e
νe
2
Q max
σ TOT
1
∝ ∫ dQ
2 2
2
Q
+
M
(
0
W )
2
≈
12-15 August 2006
2
Q max
MW
Kevin McFarland: Interactions of Neutrinos
4
28
Touchstone Question #1
What is Q2max?
νμ + e− → μ− + νe
Q ≡−( e −ν e )
2
νμ
ν
μ−
e
2
νe
Let’s work in the center-ofmass frame. Assume, for now,
we can neglect the masses
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
29
Neutrino-Electron (cont’d)
σ TOT ∝ Q 2 = s
ν
μ−
max
σ TOT =
νμ
GF2 s
π
= 17.2 ×10−42 cm 2 / GeV ⋅ Eν (GeV )
e
νe
• Why is it proportional to
beam energy?
s = ( pν μ + pe ) 2 = me2 + 2me Eν (e - rest frame)
• Proportionality to energy is a generic
feature of point-like scattering!
ƒ because dσ/dQ2 is constant (at these energies)
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
31
Neutrino-Electron (cont’d)
ν
• Elastic scattering:
νμ + e − → ν μ + e −
ƒ Coupling to left or righthanded electron
ƒ Total spin, J=0,1
• Electron-Z0 coupling
ƒ (LH, V-A): -1/2 +
sin2θ
ƒ (RH, V+A): sin2θW
12-15 August 2006
W
G s⎛1
⎞
2
4
σ∝
⎜ − sin θW + sin θW ⎟
π ⎝4
⎠
2
F
σ∝
GF2 s
π
Kevin McFarland: Interactions of Neutrinos
(sin
4
θW )
32
Neutrino-Electron (cont’d)
ν
• What are relative
contributions of left and
right-handed scattering
from electron?
ν
ν
f LH
ν
ν
θ
θ
fLH
dσ
= const
d cos θ
12-15 August 2006
ff RH
ff RH
dσ
⎛ 1 + cos θ ⎞
= const × ⎜
⎟
d cos θ
2
⎝
⎠
Kevin McFarland: Interactions of Neutrinos
2
33
Neutrino-Electron (cont’d)
ν
2
G
⎞
2
4
Fs ⎛ 1
0
• Electron-Z coupling σ ∝
⎜ − sin θW + sin θW ⎟
π ⎝4
⎠
ƒ (LH, V-A): -1/2 + sin2θW
ƒ (RH, V+A): sin2θW
Let y denote inelasticity.
Recoil energy is related to
CM scattering angle by
Ee
y=
≈ 1 − 12 (1 − cos θ )
Eν
σ TOT
σ∝
GF2 s
π
(sin
4
θW )
⎧
dy = 1
∫
dσ ⎪LH:
∫ dy dy = ⎨ RH: (1 − y)2 dy = 1
⎪⎩
∫
3
GF2 s ⎛ 1
4 4 ⎞
2
2
−42
cm
θ
θ
sin
sin
1.4
10
/ GeV ⋅ Eν (GeV )
=
−
+
=
×
W
W ⎟
⎜
π ⎝4
3
⎠
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
34
Touchstone Question #2:
Flavors and νe Scattering
ν
The reaction
νμ + e − → ν μ + e −
has a much smaller cross-section than
ν e + e− → ν e + e−
Why?
)
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
35
ν
Touchstone Question #2:
Flavors and νe Scattering
The reaction
νμ +
e−
→ νμ +
νe
e−
has a much smaller cross-section than
νe +
e−
→ νe +
νe
Z
e−
e
e
Why?
νe +
e−
→ νe +
e−
has a second contributing
reaction, charged current
12-15 August 2006
νe
e
W
e
Kevin McFarland: Interactions of Neutrinos
νe
36
ν
Touchstone Question #2:
Flavors and νe Scattering
Show that this increases the rate
(Recall from the previous pages…
σ TOT
dσ
dy
=∫
dy
σ
⎡ dσ
dσ ⎤
= ∫ dy ⎢
+
⎥
dy
dy
⎦
⎣
LH
RH
= σ TOT
+ 13 σ TOT
LH
RH
LH
TOT
∝ total coupling
LH 2
e-
For electron…
LH coupling
RH coupling
Weak NC
-1/2+ sin2θW
sin2θW
Weak CC
-1/2
0
)
We have to show the interference between CC and NC is constructive.
The total RH coupling is unchanged by addition of CC because there is no
RH weak CC coupling
There are two LH couplings: NC coupling is -1/2+sin2θW ≈ -1/4 and the CC
coupling is -1/2. We add the associated amplitudes… and get -1+sin2θW ≈ -3/4
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
37
ν
Lepton Mass Effects
• Let’s return to
Inverse μ−decay:
νμ + e− → μ− + νe
ƒ What changes in the presence
of final state mass?
o pure CC so always left-handed
o BUT there must be finite Q2 to
create muon in final state!
2
Qmax
σ TOT ∝
2
Qmin
≈
2
Qmin
= mμ2
ƒ see a suppression scaling with
(mass/CM energy)2
o can be generalized…
12-15 August 2006
∫
σ TOT =
1
dQ
(Q 2 + M W 2 ) 2
2
2
2
Qmax
− Qmin
MW 4
GF2 ( s − mμ2 )
π
2
⎛
m
= ⎡⎣σ TOT (massless) ⎤⎦ ⎜1- μ
⎜
s
⎝
Kevin McFarland: Interactions of Neutrinos
⎞
⎟⎟
⎠
38
What about other targets?
ν
νany
• Imagine now a proton target
ƒ Neutrino-proton elastic scattering:
νe + p → ν e + p
ƒ “Inverse beta-decay” (IBD):
ν e + p → e+ + n
ƒ and its close cousin:
νany
Z
p
p
νe
e+
W
p
n
ν e + n → e- + p
ƒ Recall that IBD
was the Reines and
Cowan discovery signal
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
39
ν
Proton Structure
• How is a proton different from an electron?
ƒ anomalous magnetic moment, κ ≡
ƒ “form factors” related to finite size
g −2
≠1
2
Determined
proton RMS
charge radius
to be
(0.7±0.2)
x10-13 cm
McAllister and Hofstadter 1956
188 MeV and 236 MeV electron beam
from linear accelerator at Stanford
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
40
Final State Mass Effects
ν
• In IBD, νe + p → e+ + n, have to pay a mass
+
e
ν
e
penalty twice
ƒ Mn-Mp≈1.3 MeV, Me≈0.5 MeV
W
• What is the threshold?
p
n
ƒ kinematics are simple, at least to zeroth order in Me/Mn
Æ heavy nucleon kinetic energy is zero
sinitial = ( pν + p p ) 2 = M p2 + 2 M p Eν (proton rest frame)
sfinal = ( pe + pn ) 2 ≈ M n2 + me2 + 2 M n Eν − ( M n − M p )
(
2
M
+
m
−
M
( n e)
p
)
2
• Solving… Eν min ≈
12-15 August 2006
2M p
Kevin McFarland: Interactions of Neutrinos
≈ 1.806 MeV
41
Final State Mass Effects
(cont’d)
ν
• Define δE as Eν-Eνmin, then
(
sinitial = M p2 + 2 M p δ E + Eν min
)
= M p2 + 2δ E × M p + ( M n + me ) − M p2
2
= 2δ E × M p + ( M n + me )
2
• Remember the suppression generally goes as
2
M n + me )
(
mfinal 2
= 1−
ξ mass = 1 −
2
s
( M n + me ) + 2M p × δ E
12-15 August 2006
2M p
⎧
low energy
⎪ δE×
2
( M n + me )
2M p × δ E
⎪
=
≈⎨
2
( M n + me ) + 2M p × δ E ⎪ ( M n + me )2 M p
high energy
⎪1 − 2 M 2
δE
p
⎩
Kevin McFarland: Interactions of Neutrinos
42
ν
Putting it all together…
σ TOT =
GF2 s
π
(
× cos ϑCabibbo × (ξ mass ) × gV + 3 g A
2
quark mixing!
final state mass
suppression
• mass suppression is proportional to
δE at low Eν, so quadratic near threshold
• vector and axial-vector
form factors (for IBD usually
2
2
)
proton form
factors (vector,
axial) +
νe
e
W
p
n
referred to as f and g, respectively)
gV, gA ≈ 1, 1.26.
ƒ FFs, θCabibbo, best known
from τn
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
43
Touchstone Question #3:
Quantitative Lepton Mass Effect
ν
• Which is closest to the minimum
beam energy in which the reaction
νμ + e− → μ− + νe
can be observed?
(a) 100 MeV
(c) 10 GeV
(b) 1 GeV
(It might help you to remember that Qmin = mμ
or you might just want to think about the total CM energy required
to produce the particles in the final state.)
2
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
2
44
Summary and Outlook
ν
• We know νe- scattering and IBD cross-sections!
• In point-like weak interactions, key features are:
ƒ dσ/dQ2 is ≈ constant.
o Integrating gives σ∝Eν
ƒ LH coupling enters w/ dσ/dy∝1, RH w/ dσ/dy∝(1-y)2
o Integrating these gives 1 and 1/3, respectively
ƒ Lepton mass effect gives minimum Q2
o Integrating gives correction factor in σ of (1-Q2min/s)
ƒ Structure of target can add form factors
• Deep Inelastic Scattering is also a point-like
limit where interaction is ν-quark scattering
12-15 August 2006
Kevin McFarland: Interactions of Neutrinos
46