Neutrino interactions with quarks For right now lets assume that the - - PowerPoint PPT Presentation

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Heidi Schellman Northwestern Neutrino interactions with quarks For right now lets assume that the proton (which like mass eigenstates) and the W boson (which wants weak eigenstates) agree that a d quark is a d quark and an s quark is an s quark.

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Heidi Schellman Northwestern

Neutrino interactions with quarks

For right now lets assume that the proton (which like mass eigenstates) and the W boson (which wants weak eigenstates) agree that a d quark is a d quark and an s quark is an s quark. Note that the W boson treats a quark just like an electron, there is no charge suppression.

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Heidi Schellman Northwestern

Charge conservation means that some weak interactions can’t happen.

ν

µ

d µ u W

νµ u µ d W

νµ → W +µ− and νµ → W −µ+ so νµdL → µ−uL and νµuR → µ−dR νµuL → µ+dL and νµdR → µ+uR

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telling dL from uR

ν μ d μ− u J = 0 ν μ u μ+ d J = 1 ν μ u μ− d J = 1 ν μ d μ+ u J = 0

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Heidi Schellman Northwestern

telling dL from uR

ν μ d μ− u J = 0 ν μ u μ+ d J = 1

The νµd → uµ− scattering cross section is isotropic because a left-handed neutrino interacting with a left-handed d-quark has total spin 0. This will also be true for νµd → uµ+ which has a right handed anti-neutrino scattering off

f a right handed u quark.

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ν μ u μ− d J = 1 ν μ d μ+ u J = 0

This is not true for the νµu → dµ+ cross section which has a right-handed anti-neutrino interacting with a left-handed u quark or for νu → dµ− which has a left handed neutrino scattering off of a right handed d quark. In this case forward scattering is allowed but going completely backwards is not. There is a spin suppression factor of ((1 + cos θ)/2)2 = (1 − y)2. y = (P · q) P· = Elab − E′

lab

Elab = (1 − cos θcm) 2 .

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How do you measure your observables

Neutrinos are invisible, so instead of measuring k, you measure E′

k and

ν = Ehad then work backwards to the neutrino energy Ek and hope it is coming in at an angle you understand.

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Where neutrinos and muons come from

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Strangeness and Charm

Of course, we already know that the weak interactions have different ideas about what an s and d quark are than the mass eigenstates. The weak s′ quark is a superposition of the d and s quark mass eigenstates. So a weak interaction involving an ”s” quark can produce either a u or a c quark. An initial state u quark decays weakly to a W + d′ weak eigenstate. You actually observe a superposition of 3 mass eigenstates - which you can distinguish. |W + d′ >= Vdu|W + d > +Vsu|W + s > +Vbu|W + b >

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Unitarity check In the limit where E >> mb, the 3 diagrams with different mass eigenstates will add incoherently since they have different final states. |M|2 ∝ |Uud|2 + |Uus|2 + |Uub|2 = 1 if there are no other unknown flavors. Interestingly, this check was off by around 1% until the early 2000’s.

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However if s ≃ m2

c you run into mass suppression effects and some of the

diagrams are suppressed. In that case you can revert to the 2 × 2 formalism using only the u, d, s, c quarks where   d′ s′     cos θC sin θC − sin θC cos θC     d s   where the Cabibbo angle θC is such that Vud ≃ Vcs = cos θC ≃ 0.973 and Vus = −Vdc = sin θC ≃ 0.227 Normally the b and t quark masses are too big to be important in deep inelastic scattering so the Cabibbo formalism is sufficient for us.

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Heidi Schellman Northwestern

One can then write the cross sections for all the neutrino processes as follows d2σ dxdy (νµe → µ−νe) = G2

F M 4 W

(M 2

W + Q2)2

s π ≡ σ0 d2σ dxdy (νd → µ−u) = |Uud|2σ0xd(x) d2σ dxdy (νu → µ−d) = |Uud|2σ0xu(x)(1 − y)2 d2σ dxdy (νu → µ+d) = |Uud|2σ0xu(x)(1 − y)2 d2σ dxdy (νd → µ+u) = |Uud|2σ0xd(x) d2σ dxdy (νs → µ−u) = |Uus|2σ0xs(x) d2σ dxdy (νs → µ+u) = |Uus|2σ0xs(x)

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Cross section scales with s ≡ 2MElab

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ν + s → µ− + c → µ+ + νµ + X

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Charm production is suppressed by the need to have a 1.7 GeV quark in the final state

d2σ dxdy (νs → µ−c) = |Ucs|2σ0xs(x) ∗ f(mc) d2σ dxdy (νs → µ+c) = |Ucs|2σ0xs(x) ∗ f(mc) d2σ dxdy (νd → µ−c) = |Ucd|2σ0xs(x) ∗ f(mc) d2σ dxdy (νd → µ+c) = |Ucd|2σ0xs(x) ∗ f(mc) f(mc) is the possible suppression factor for charm production.

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If we take proton and neutron targets and the parton probabilities we get: dσ dxdy (νp → µ−X) = σ0x[d(x) + s(x) + |Uud|2u(x)(1 − y)2] dσ dxdy (νp → µ+X) = σ0x[d(x) + s(x) + |Uud|2u(x)(1 − y)2] dσ dxdy (νn → µ−X) = σ0x[up(x) + s(x) + |Uud|2dp(x)(1 − y)2] dσ dxdy (νn → µ+X) = σ0x[up(x) + s(x) + |Uud|2dp(x)(1 − y)2] If you take σν

p + σν n + σν p + σν n you get something close to

σ0x(q(x) + q(x))(1 + (1 − y)2) where q are all of the quarks. If you take σν

p + σν n − (σν p + σν n) you get something close to

σ0x(q(x) − q(x))(1 − (1 − y)2).

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If you study the y dependence, you can solve for u(x), d(x), s(x), u(x), d(x), s(x).

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