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IntroductiontoInteractionPhysicsIntroductiontoNeutrinoInteractionPhysicsIntroductiontoNeutrinoInteraction NUFACT08SummerSchoolNUFACT08SummerSchoolNUFACT08SummerSchool

1113June2008 Benasque,Spain PaulSoler ReferencesReferences

1. K Zuber, “Neutrino Physics”, Institute of Physics Publishing, 2004. 2. C.H. Kim & A. Pevsner “ in Physics and ”, Harwood Academic Publishers, 1993. 3. K. Winter (editor), “Neutrino Physics”, Cambridge University Press (2 nd edition), 2000. 4. .N. Mohapatra, P.B. Pal, “Massive Neutrinos in Physics and Astrophysics”, World Scientific (2 nd edition), 1998. 5. H.V. Klapdor-Kleingrothaus & K. Zuber, “ Astrophysics”, Institute of Physics Publishing, 1997. 6. K. McFarland, Neutrino Interaction Physics, lectures at SUSSP61, St Andrews, June 2006, (To be published in Proceedings SUSSP61, Taylor & Francis, 2008).

Neutrino Interaction Physics 2 NUFACT08 Summer School ContentsContents

1.HistoryandIntroduction 2.Neutrino 3.NeutrinoDeepInelasticScattering 4.Quasielastic,Resonant,Coherentand DiffractiveScattering 5.NuclearEffects

Neutrino Interaction Physics 3 NUFACT08 Summer School 1.HistoryandIntroduction 1.1 Fermi Theory 1.2 Neutrino discovery 1.3 violation and V-A theory 1.4 Neutral currents 1.5 neutrino interactions

Neutrino Interaction Physics 4 NUFACT08 Summer School 1.1Fermitheoryofdecay(1932)1.1Fermitheoryofbetadecay(1932)

 Existence of a point-like four interaction (Fermi , 1932): − n e

− n → p + e +ν e

p ν e  Lagrangian of the interaction: G L() = − F [φ (x)γ µφ (x)][φ (x)γ φ (x)] 2 p n e µ ν −5 −2 GF = Fermi constant = .1( 16637 ± .0 00001 )×10 GeV

 Gamow-Teller interaction when final different to initial nucleus:

GF i L(x) = − ∑[φ p (x)Γ φn (x)][φe (x)Γiφν (x)]+ h c.. 2 i Possible interactions: Γi = ,1 γ 5 ,γ µ ,γ 5γ µ ,σ µν = S, P,V , A,T Neutrino Interaction Physics 5 NUFACT08 Summer School FirstneutrinocrossFirstneutrinocross sectioncalculationsectioncalculation

 Bethe-Peierls (1934): calculation of first cross-section for inverse beta reaction + or − using Fermi theory ν e + p → n + e ν e + n → p + e −44 2 νe σ ≈ 5×10 cm for E(ν ) = 2 MeV Accurate to factor 2 σ b Conversion from natural units: hc =197 3. MeV ⋅ fm p 2 −27 2 2 Cross-section: multiply by (hc) = .0 3894 ×10 GeV ⋅cm

 Mean free path of antineutrino in water: 1 λ = ≈ 5.1 ×10 21 cm ≈1600 − years nσ num. free N n = ≈ 2 A ρ volume A 2 × 6 ×10 23 In water: n = = 7.6 ×10 22 cm −3 18  Probability of interaction:  L  L P =1− exp −  ≈ = 7.6 ×10 −20 (m water )−1  λ  λ Need very intense source of antineutrinos to detect inverse beta reaction. Neutrino Interaction Physics 6 NUFACT08 Summer School 1.2Neutrinodiscovery(1956)1.2Neutrinodiscovery(1956) +  Reines and Cowan detect ν e + p → n + e in 1953 (Hanford) (discovery confirmed 1956 in Savannah River): – Detection of two back-to-back γs from prompt signal e +e- ->γγ at t=0. – thermalization: neutron capture in Cd, emission of late γs ~ 20 ms γ γ γ

γ n e + H2O + CdCl 2 γ Scintillator ν Publication 1956: σσσ= 6 x10 -44 cm 2 ±±± 25% (within 5% expected) 1956: parity violation discovery increases theory cross-section: σσσ=(10 ±±±1.7)x10 -44 cm 2 Reanalysis data in 1960: σσσ= (12+7-4) x10 -44 cm 2 Reines 1995 4200 l scintillator Neutrino Interaction Physics 7 NUFACT08 Summer School 1.3ParityviolationandV1.3ParityviolationandV AA

 Parity violation in weak decays postulated by Lee & Yang in 1950  Parity violation confirmed through forward-backward of polarized 60 Co ( Wu , 1957).

60 60 * − Co → Ni + e +ν e More emitted in direction opposite to 60 Co spins, implying maximal parity violation r r r r σ ⋅ p P σ ⋅(− p)  Helicity operator: H = r → r = −H p p Projects spin along direction of

Neutrino Interaction Physics 8 NUFACT08 Summer School 1.3ParityviolationandV1.3ParityviolationandV AA

 Goldhaber, Grodzins, Sunyar (1958) measure helicity of neutrino from K capture of 152 Eu:

152 − − 152 * − Eu 0( ) + e → Sm 1( ) +ν e 152 Sm * 1( − )→152 Sm * 0( + ) +γ

pγ=-pν

σγ=-σν

Hγ=Hν

Asymmetry of spectrum in magnetic determines helicity of νe:

H (ν e )= −1⇒ H (ν e )= +1 Neutrinos are “left-handed” Antineutrinos are “right-handed”

Neutrino Interaction Physics 9 NUFACT08 Summer School 1.3ParityviolationandV1.3ParityviolationandV AA

 Left and right handed projection operators: 1 1 ν = Pν = 1( −γ )ν ν = P ν = 1( +γ )ν L L 2 5 R R 2 5 0 1 2 3  operator: γ 5 = iγ γ γ γ same as helicity operator for massless neutrinos ( E=p ).

γ 5ν L = Hν L = −ν L γ 5ν R = Hν R = +ν R

 If only νL interact and ν_____R do not interact, then Γi have to transform as: eΓiν → ()()PLe Γi PLν = ePRΓi PLν 1 1 V : P γ µP = γ µ ()1− γ A :P γ µγ P = − γ µ ()1− γ R L 2 5 R 5 L 2 5  The only possible coupling is V-A, due to maximal parity violation in weak interactions ( Feynman, Gell-Mann , 1958):

GF µ LV −A = − [φ pγ 1( − g Aγ 5 )φn ][φeγ µ 1( −γ 5 )φν ]with g A = − .1 2573 ± .0 0028 2 (determined empirically) Neutrino Interaction Physics 10 NUFACT08 Summer School 1.4Neutralcurrents1.4Neutralcurrents  Two types of : (CC) and (NC) from electroweak theory of Glashow, Weinberg, Salam.

0 W ± Z

 First example of NC observed in 1973, inside the

filled with freon (CF 3Br): no !

ν µ ν µ

Z 0 N X

ν µ + N →ν µ + X Neutrino Interaction Physics 11 NUFACT08 Summer School 1.5StandardModelNeutrinoInteractions1.5StandardModelNeutrinoInteractions

 Lagrangian for electroweak interactions: g L = i [j(+)W µ + j (−)W µ + ]+ i[g cos θ j )3( − g sin' θ j (Y )2/ ]Z µ + int 2 µ µ W µ W µ )3( (Y )2/ µ + i[]g sin θW jµ + g'cos θW jµ A

 1st term: charged current interactions (W +, W - exchange)  2nd term: neutral current interactions (Z 0 exchange)  3rd term:electromagnetic interactions (photon exchange)

 Electron : e = g sin θW = g'cos θW

e.m. )3( (Y )2/ rd  3 term: ej µ = e( jµ + jµ ) (neutrinos only couple to W ± and Z 0)

Neutrino Interaction Physics 12 NUFACT08 Summer School S.M.interactions(cont)S.M.interactions(cont)

 A) Neutrino electron interaction g (+) µ (−) µ + g (Z ) µ e.m Lint = i [jµ W + jµ W ]+ i jµ Z + iej µ 2 2cos θW 1 j (+) =ν γ e = ν γ 1( −γ )e  Where: µ e,L µ L 2 e µ 5 1 j (−) = e γ ν = eγ 1( −γ )ν µ L µ e,L 2 µ 5 e (Z ) )3( 2 e.m jµ = (2 jµ − sin θW jµ ) = 2 =ν e,Lγ µν e,L − eLγ µ eL + 2sin θW eγ µ e = 1 1 = ν γ 1( −γ )ν − eγ 1( −γ )e + 2sin 2 θ eγ e 2 e µ 5 e 2 µ 5 W µ 1 ⇒ j(Z ) = ν γ 1( −γ )ν + eγ (g − g γ )e µ 2 e µ 5 e µ V A 5 1 1  With: g = − + 2sin 2 θ g = − V 2 W A 2 Neutrino Interaction Physics 13 NUFACT08 Summer School S.M.interactions(cont)S.M.interactions(cont)

 B) weak interactions

g (+) µ (−) µ + g (Z ) µ e.m Lint = i [jµ W + jµ W ]+ i jµ Z + iej µ 2 2cos θW 1  Where: j (+) = uγ 1( −γ )d µ 2 µ 5 1 j (−) = dγ 1( −γ )u µ 2 µ 5 (Z ) jµ = uγ µ (Au − Buγ 5 )u + dγ µ (Ad − Bdγ 5 )d

 With: 1 4 1 A = − sin 2 θ B = u 2 3 W u 2 1 1 2 2 A = − + sin θ Bd = − d 2 3 W 2

Neutrino Interaction Physics 14 NUFACT08 Summer School S.M.interactions(cont)S.M.interactions(cont)

 After introducing Higgs field and spontaneous symmetry breaking: 2 2 4 LHiggs = − Dµφ − µ φ − λ φ

 : mH = 2λv 2 gv   2 m ± g 2 m ± = W W   = = cos θ 2   2 2 W  mZ 0  g + g' g 2 + g'2 m 0 = v Z 2 − /1 2  expectation value: v = ( 2GF ) ≈ 246 GeV  Effective Hamiltonian: g 2 g 2 H = []j (+)µ j (−) + h.c. + j (Z )µ j (Z ) = eff 2 µ 2 2 µ 4mW 8mZ cos θW G = F [2 j(+)µ j (−) + h c.. + j(Z )µ j(Z ) ] 2 µ µ Neutrino Interaction Physics 15 NUFACT08 Summer School S.M.interactions(cont)S.M.interactions(cont)

 The vector masses are then predicted:

G g 2 e2 4πα F = = = α = /1 137 .036 2 2 2 2 2 2 8mW 8mW sin θW 8mW sin θW

mW = 80 .450 ± .0 058 GeV  Masses: 2   m = 91 .1876 ± .0 0021 GeV 2  37 .2805  Z mW =   sin θ 2  W  sin θW = .0 22280 ± .0 00035

 Need radiative corrections: 37 .2805 mW = /1 2 sin θW 1( − ∆r )

with ∆r ≈ .0 03630 ± .0 0011 for m t = 172 7. GeV mH = 117 GeV

yields : mW = 80 51. ± 11.0 GeV

Neutrino Interaction Physics 16 NUFACT08 Summer School 2.NeutrinoElectronScattering 2.1 Charged current 2.2 Neutral current 2.3 Interference charged and neutral current 2.4 suppression 2.5 Number of neutrinos

Neutrino Interaction Physics 17 NUFACT08 Summer School 2.1Neutrino2.1Neutrino electronCCscatteringelectronCCscattering

− −  Only charged current: ν µ + e →ν e + µ (Inverse Muon Decay) − GF µ ν µ µ H = [ν γ 1( − γ )µ][eγ 1( − γ )ν ] eff 2 µ 5 µ 5 e − s = (p + p )2= 2m E (in LAB ) W ν µ e e ν µ t = q 2 = −Q2 = (p − p )2 − ν µ µ Inelasticity variable e ν e p ⋅ (p − p ) E − E (0

− − 1 2 1 1 2 e e gV = − + 2sin θW g = (g + g ) = − + sin θ 2 L 2 V A 2 W 1 1 2 Couples to e and e : J=0,1 g A = − g = (g − g ) = sin θ L R 2 R V A W 2 1+ cos θ * Right handed current suppressed in backward direction: 1− y = 2 dσ (ν e− ) G 2s m4  1 2  NC µ = F z − + sin 2 θ  + sin 4 θ 1( − y)2  2 2 2 W W dy π (Qmax + mZ )  2   dσ (ν e− ) G 2s m4  1 2  NC µ = F Z − + sin 2 θ  1( − y)2 + sin 4 θ  2 2 2 W W dy π (Qmax + mZ )  2   Neutrino Interaction Physics 19 NUFACT08 Summer School 2.2Neutrino2.2Neutrino electronNCscatteringelectronNCscattering (−) (−) − −  Only neutral current (total cross-section): ν µ + e → ν µ + e

2 G 2s  1  1   E  − F 2 4 −41  ν  2 σ NC (ν µe ) = − + sin θW  + sin θW  = 16.0 ×10   cm π  2  3  1GeV  2 G 2s 1  1    E  − F 2 4 −41  ν  2 σ NC (ν µe ) =  − + sin θW  + sin θW  = 13.0 ×10   cm π 3  2   1GeV 

2  Can obtain value of sin θW from neutrino electron elastic scattering (CHARM II): 2 sin θW = .0 2324 ± .0 0058 ± .0 0059

gV = − .0 035 ± .0 017

g A = − .0 503 ± .0 017 E Θ2 = 2m 1( − y) e e Neutrino Interaction Physics 20 NUFACT08 Summer School 2.3InterferenceCCandNC2.3InterferenceCCandNC

 Tree level Feynman diagrams: both neutral and charged currents − − ν e + e →ν e + e − ν ν ν e e e e

0 W − Z − − e− e e ν e

 Effective Hamiltonian: G H = F {[ν γ µ 1( −γ )e][eγ 1( −γ )ν ]+ [ν γ µ 1( −γ )ν ][eγ (g − g γ )e]} eff 2 e 5 µ 5 e e 5 e µ V A 5 G = F {[ν γ µ 1( −γ )ν ][eγ 1( + g − 1( + g )γ )e]} 2 e 5 e µ V A 5 (through a Fierz transformation) Neutrino Interaction Physics 21 NUFACT08 Summer School 2.3InterferenceCCandNC2.3InterferenceCCandNC

 Rearranging terms in charged and neutral current contributions for: − − ν e + e →ν e + e 1 1 1 g = 1( + g +1+ g ) = − + sin 2 θ +1 = + sin 2 θ L 2 V A 2 W 2 W 1 2 gR = 1( + gV − 1( + g A )) = sin θW 2 − 2  2  dσ (ν ee ) GF s  1 2  4 2 Then: =  + sin θW  + sin θW ()1− y  dy π  2   2 G 2s  1  1   E  ⇒ − F 2 4 −41  ν  2 σ (ν ee ) =  + sin θW  + sin θW  = 96.0 ×10   cm π  2  3  1GeV  2 G 2s 1  1    E  − F 2 4 −41  ν  2 Also :σ (ν ee ) =   + sin θW  + sin θW  = 40.0 ×10   cm π 3  2   1GeV  These cross-sections are a consequence of the interference of the charged and neutral current diagrams.

Neutrino Interaction Physics 22 NUFACT08 Summer School 2.3InterferenceCCandNC2.3InterferenceCCandNC

+ −  Neutrino : e + e →ν e +ν e Contribution from both W and Z graphs.

+ + ν e e ν e e

− W e Z ν e

− e ν e Caveat: below the Z pole! 2  2  + − GF s  1 2  1 Then: σ (e e →ν eν e ) =  + 2sin θW  +  12 π  2  4 + −  Only neutral current contribution to: e + e →ν µ +ν µ 2  2  + − GF s  1 2  1 σ (e e →ν µν µ ) =  − 2sin θW  +  12 π  2  4 Neutrino Interaction Physics 23 NUFACT08 Summer School NeutrinoNeutrino electronscatteringsummaryelectronscatteringsummary

 Summary neutrino processes: Process Total cross-section

− − G 2s ν µ + e → µ +ν e F π 2 − − G s  2 4  ν + e →ν + e F 2 4 e e ()2sin θW −1 + sin θW  4π  3  2 − − G s 1 2  ν + e →ν + e F 2 4 e e  ()2sin θW +1 + 4sin θW  4π 3  2 − − G s  2 4  ν + e →ν + e F 2 4 µ µ ()2sin θW −1 + sin θW  4π  3  − − 2 ν µ + e →ν µ + e GF s 1 2 2 4   ()2sin θW −1 + 4sin θW  4π 3  + − G 2s 1  e + e →ν +ν F 2 4 e e  + 2sin θW + 4sin θW  12 π 2  2 + − GF s 1 2 4  e + e →ν +ν  − 2sin θW + 4sin θW  µ µ 12 π 2 

s = 2meEν (in the LAB frame)

Neutrino Interaction Physics 24 NUFACT08 Summer School 2.4Masssuppression2.4Masssuppression

 We have not taken into account the effect of initial and final state masses yet − −  For example: ν µ + e →ν e + µ m 2 − m 2 Threshold s = m 2 + 2m E ≥ m 2 ⇒ E ≥ µ e ≈ 11 GeV e e ν µ ν 2m  Cross-section modification: e 2 Qmax G 2 m 4 G 2 m 4 σ (ν e− ) = F W dQ 2 = F W ()Q2 −Q2 ≈ CC µ ∫ 2 2 2 2 2 2 2 max min 2 π (Q + m ) π (Qmax + mW )( Qmin + mW ) Qmin W G 2 G 2 ≈ F ()Q2 −Q2 = F ()s − m 2 π max min π µ

 Therefore: 2 2 G 2s  m   m  σ (ν e− ) = F 1− µ  = σ massless (ν e− )1− µ  CC µ   CC µ   π  s   s  Neutrino Interaction Physics 25 NUFACT08 Summer School 2.5Numberofneutrinos2.5Numberofneutrinos

 Width of the Z-pole : Breit-Wigner distribution 12 π (hc)2 sΓ Γ σ (e+e− → f ) = e f 2 2 2 2 M Z (s − M Z ) + s ΓZ / M Z 2 2 + − 12 π (hc) ΓeΓf 12 π (hc) 0 + − 0 σ peak (e e → f ) = 2 = B(Z → e e )B(Z → f f ) M Z ΓZ M Z e+ + + + ν e e l e q − 0 − 0 − − 0 e Z ν e e Z l e Z q 2 neutrinos ΓZ = Γhad + 3Γl +l − + Nν Γνν = 2490 MeV

Γhad = Γu + Γd + Γc + Γs + Γb = 1741 MeV

Γ + − = 83 9. MeV l l 3 neutrinos

Γνν = 167 1. MeV

⇒ Nν = .2 9841 ± .0 0083

 Only 3 neutrinos with mass less 4 neutrinos than the Z mass Neutrino Interaction Physics 26 NUFACT08 Summer School 3.NeutrinoNucleonDeepInelasticScattering 3.1 Definition and variables 3.2 Charged current 3.3 Quark content of 3.4 Sum rules 3.5 Neutral current 2 3.6 A case study: sin θW from neutrino interactions 3.7 Charm production in neutrino interactions

Neutrino Interaction Physics 27 NUFACT08 Summer School 3.1DefinitionandVariables3.1DefinitionandVariables

 Deep inelastic neutrino-nucleon scattering reactions have large q2 − ν l ( p) + N → l ( p′) + X 2 2 (q >> mN ,E ν>>mN):  Quark-parton model valid due to of QCD, which makes behave as free point-like .  Infinite frame: a parton takes a fraction x (0> mq (0

u(x) = uV (x) + uS (x) d(x) = dV (x) + dS (x)

In the sea: uS (x) = u(x) dS (x) = d (x) For proton (uud): 1 1 uV (x)dx = []u(x) − u(x) dx = 2 ∫0 ∫0 1 1 dV (x)dx = [d(x) − d (x)]dx =1 ∫0 ∫0  Neutrino-quark/antineutrino-antiquark scattering: 2 dσ (ν q) dσ (ν q) 2G m E E 1 CC µ = CC µ = F q withy = 1− = ()1− cos θ dy dy π E′ 2

 Neutrino-antiquark/antineutrino-quark scattering: dσ (ν q) dσ (ν q) 2G 2m E CC µ = CC µ = F q ()1− y 2 dy dy π Neutrino Interaction Physics 29 NUFACT08 Summer School 3.2Chargedcurrent3.2Chargedcurrent

 Scattering off proton: proton = uud + (uu ) + (dd ) + (ss ) + (cc ) dσ (ν p) G 2ME CC µ = F 2x{[]d(x) + s(x) + []u(x) + c(x) 1( − y)2 } dxdy π dσ (ν p) G 2ME CC µ = F 2x{[]u(x) + c(x) 1( − y)2 + []d (x) + s(x) } dxdy π  Neutron ( symmetry): neutron = uud + (uu ) + (dd ) + (ss ) + (cc ) u (x) = d (x) ≡ d(x) − + n p ν µ µ ν µ µ

dn (x) = up (x) ≡ u(x) W + W − sn (x) = sp (x) ≡ s(x)

cn (x) = cp (x) ≡ c(x) d u u d dσ (ν n) G 2ME CC µ = F 2x{}[]u(x) + s(x) + []d (x) + c(x) 1( − y)2 dxdy π dσ (ν n) G 2ME CC µ = F 2x{[]d(x) + c(x) 1( − y)2 + []u(x) + s(x) } dxdy π Neutrino Interaction Physics 30 NUFACT08 Summer School 3.2Chargedcurrent3.2Chargedcurrent

 Structure functions: 2 ν ,ν 2   d σ GF s 2 2  Mxy  2  y  2 = y 2xF 1(x,Q ) + 21− y − F2 (x,Q ) ± 2y1− xF 3 (x,Q ) dxdy 2π   2E   2   2 Fi (x,Q ) are the structure functions, which depend on the helicity structure of q-W interactions. For massless spin-1/2 partons, we have the Callan- Gross relationship*: 2xF 1(x) = F2 (x)

2 ν ,ν 2 d σ G s  2  Mxy   y   F   2 2 = ()1− y + 1− F2 (x,Q ) ± 2y1− xF 3 (x,Q ) = dxdy 2π   2E   2  

2 Assuming massless GF s 2 2 2 2 = [()1+ ()1− y F (x,Q ) ± ()1− ()1− y xF (x,Q )] target 2π 2 3 * Deviations from the Callan-Gross relation are parameterised in terms of the “longitudinal” cross-section (ie. splittting g→→→qq): σ F (x)  4Mx 2  L 2   RL = = 1+ 2  σT 2xF 1(x)  Q  Neutrino Interaction Physics 31 NUFACT08 Summer School 3.2Chargedcurrent3.2Chargedcurrent

 Comparing the y distribution of both cross-sections we can compare the parton distribution functions to the proton structure functions: ν p F2 (x) = x[d(x) + u(x) + s(x) + c(x)] νp xF 3 (x) = x[d(x) − u(x) + s(x) − c(x)] νp F2 (x) = x[u(x) + c(x) + d(x) + s(x)] νp xF 3 (x) = x[u(x) + c(x) − d (x) − s(x)]

 Also, the neutron structure functions:

ν p F2 (x) = x[u(x) + d (x) + s(x) + c(x)] νn xF 3 (x) = x[u(x) − d (x) + s(x) − c(x)] νn F2 (x) = x[d(x) + c(x) + u(x) + s(x)] νn xF 3 (x) = x[d(x) + c(x) − u(x) − s(x)]

Neutrino Interaction Physics 32 NUFACT08 Summer School 3.2Chargedcurrent3.2Chargedcurrent

 Scattering off isoscalar target (equal number and protons): q ≡ u + d + s + c q ≡ u + d + s + c νN F2 (x) = x[q(x) + q(x)] νN xF 3 (x) = x[q(x) − q(x) + 2(s(x) − c(x))] νN xF 3 (x) = x[q(x) − q(x) − 2(s(x) − c(x))] dσ (ν N) G 2 2ME CC µ = F x{q(x) + q(x 1() − y)2 } dxdy 2π dσ (ν N) G 2 2ME CC µ = F x{q(x)( 1− y)2 + q(x)} dxdy 2π

 Total cross-section: 2 GF s  1  −38 2 σ CC (ν µN) =  Q + Q  = ().0 677 ± .0 014 ×10 cm /GeV × E(GeV ) 2π  3  2 GF s 1  −38 2 σ CC (ν µN) =  Q + Q  = ().0 334 ± .0 008 ×10 cm /GeV × E(GeV ) 2π 3  Neutrino Interaction Physics 33 NUFACT08 Summer School 3.2Chargedcurrent3.2Chargedcurrent

 Structure functions:

Scaling violations

Neutrino Interaction Physics 34 NUFACT08 Summer School 3.3Quarkcontentofnucleons3.3Quarkcontentofnucleons

 Quark content of nucleons from CC cross-sections 1  Define: U = xu (x)dx , etc . ∫0

 Experimental values from y distribution of cross-sections yields: σ (νN) Since r ≡ CC = .0 493 ± .0 016 (measured) σ CC (νN) then: Q 3r −1 = ≈ .0 191 ⇒ Q = .0 405 and Q = .0 078 Q 3 − r

therefore: Q QV = Q −Q ≈ 33.0 = 16.0 ± 03.0 1 Q + Q νN F2 (x)dx = Q + Q ≈ 48.0 ∫0

 Quarks and antiquarks carry 48% of proton momentum, quarks only 33% and sea quarks only 7.8% (u and d sea quarks carry 6%, s quarks carry 1.3% and c quarks 0.5%). Neutrino Interaction Physics 35 NUFACT08 Summer School 3.3Quarkcontentofnucleons3.3Quarkcontentofnucleons

 Parton distribution functions as a function of x, fitted from structure functions: u(x)dx = number of u-quarks in proton between x and x+dx

u(x) = uV (x) + uS (x) d(x) = dV (x) + dS (x)

In the sea: dS (x) = d (x) uS (x) = u(x) For proton (uud): 1 1 uV (x)dx = []u(x) − u(x) dx = 2 ∫0 ∫0 1 1 dV (x)dx = [d(x) − d (x)]dx = 1 ∫0 ∫0

Neutrino Interaction Physics 36 NUFACT08 Summer School 3.4Sumrules3.4Sumrules

 Sum rules: 1 1 – Gross-Llewellyn Smith: ν ν S GLS = ∫ (F3 (x) + F3 (x)) dx 2 0 2 3 1   α s α s  α s  S GLS = ∫ (q(x) − q(x)) dx = 31− − a  − b   = 64.2 ± 06.0 0  π  π   π   – Adler: 1 1 1 1 νn νp S A= ∫ (F2 (x) + F2 (x)) dx = ∫ (uV (x) − dV (x)) dx =1 2 0 x 0 – Gottfried:

1 1 1 1 µn µp 1 1 S G= ∫ (F2 (x) + F2 (x)) dx = ∫ (u(x) + u(x) − d(x) − d (x)) dx = 2 0 x 3 0 3

S G = .0 235 ± .0 026 Maybe isospin asymmetry: u(x) ≠ d (x)

– Bjorken: 2 1 νp νp 2α s (Q ) S B= ∫ (F1 (x) + F1 (x)) dx =1− 0 3π Neutrino Interaction Physics 37 NUFACT08 Summer School 3.5Neutralcurrent3.5Neutralcurrent (−) (−)  Neutral currents: ν µ + p → ν µ + X dσ (ν q) dσ (ν q ) NC µ = NC µ = dy dy 2 GF mqEν  2 2 2 mq 2 2   (gV + g A ) + (gV − g A ) 1( − y) + (g A − gV )y 2π  Eν  dσ (ν q) dσ (ν q ) NC µ = NC µ = dy dy 2 GF mqEν  2 2 2 mq 2 2   (gV − g A ) + (gV + g A ) 1( − y) + (g A − gV )y 2π  Eν   Coupling constants:  1 1 4 2 1 g L = ()gV + g A gV = − sin θW g A = for q=u,c  2 3 2  2 1 1 2 2 1 for q=d,s  g′ = − + sin θ g′ = − g R = ()gV − g A V 2 3 W A 2  2 Neutrino Interaction Physics 38 NUFACT08 Summer School 3.5Neutralcurrent3.5Neutralcurrent

 Neutral currents off nucleons (neglecting c and s quark contributions): (−) (−) ν µ + N → ν µ + X dσ (ν N) G 2ME NC µ = F x{(g 2 + g′ 2 )[]q + q 1( − y)2 + (g 2 + g′ 2 )[]q + q 1( − y)2 } dxdy π L L R R dσ (ν N) G 2ME NC µ = F x{(g 2 + g′ 2 )[]q + q 1( − y)2 + (g 2 + g′ 2 )[]q + q 1( − y)2 } dxdy π R R L L σ (νN) σ (νN) σ (νN)  Defining: R ≡ NC R ≡ NC r ≡ CC ν σ (νN) ν σ (νN) σ (νN) CC 2 CC CC 2 2 R − r R r(R − R ) ′ ν ν 2 2 ν ν yields: g L + g L = 2 gR + g′R = 1− r 1− r 2 2 2 2 2 1 2 5 4 Rν = (g L + g′L ) + r(g R + g′R ) = − sin θW + 1( + r) sin θW 2 9 (Llewelyn-Smith 1 1  1  5 R = (g 2 + g′ 2 ) + (g 2 + g′ 2 ) = − sin 2 θ + 1+  sin 4 θ relationships) ν L L r R R 2 W  r  9 W Neutrino Interaction Physics 39 NUFACT08 Summer School 3.5Neutralcurrents3.5Neutralcurrents

 More relationships from the combination of neutrino and antineutrino tagged interactions: dσ (ν N) dσ (ν N) NC µ + NC µ dy dy 1 10 R + = = − sin 2 θ + sin 4 θ dσ (ν N) dσ (ν N) 2 W 9 W CC µ + CC µ dy dy dσ (ν N) dσ (ν N) NC µ − NC µ dy dy R − rR 1 R − = = ν ν = − sin 2 θ dσ (ν N) dσ (ν N) 1− r 2 W CC µ − CC µ (Paschos-Wolfenstein dy dy relationship)  Paschos-Wolfenstein relation removes the effects of sea quark differences (especially at low x) since the neutrino and antineutrino cross-sections are equal. It would also remove error from c quark  All of these relationships can be used in neutrino experiments to test the 2 electroweak theory and measure sin θW Neutrino Interaction Physics 40 NUFACT08 Summer School 2 3.6sin3.6sin θθW

2  Llewellyn-Smith relationship used to measure sin θW by performing ratios of charged current to neutral current of neutrino nucleon scattering.

σ NC (νN) 1 2 5 4 Rν = = − sin θW + 1( + r ) sin θW σ CC (νN) 2 9 (Llewelyn-Smith relationships) σ NC (νN) 1 2  1 5 4 Rν = = − sin θW + 1+  sin θW σ CC (νN) 2  r  9

 CHARM, CDHS and CCFR and NuTeV are all large sampling calorimeters that can measure large statistics CC and NC data: CCFR/NuTeV

Neutrino Interaction Physics 41 NUFACT08 Summer School 2 3.6sin3.6sin θθW

 The ratio of NC to CC data from an average of different experiments 2 (CDHS, CHARM, CCFR, NUTEV) gives a value of sin θW  This on-shell value relates to the W and Z boson masses: 2 2 on −shell MW CHARM data sin θW = 1− 2 MZ CC For example, the CDHS experiment at CERN obtained:

Rν = .0 3072 ± .0 0033 Rν = .0 382 ± .0 016 2 ⇒ sin θW = .0 233 ± .0 003 ± .0 005  The world average value is: 2 World average : sin θW = .0 2227 ± .0 00037 NC

Example of data from the CHARM experiment

Neutrino Interaction Physics 42 NUFACT08 Summer School 2 3.6sin3.6sin θθW  NuTeV experiment at uses Paschos-Wolfenstein relationship and obtains reduced systematic errors but their result is >3 σ away from world average:

NUTEV : Rν = .0 3916 ± .0 0013 Rν = .0 4050 ± .0 0027 2 ⇒ sin θW = .0 22773 ± .0 00135 ± .0 00095

World average : 2 sin θW = .0 2227 ± .0 00037

 Charged current events had a muon (µ- from neutrinos and µ+ from antineutrinos) and neutral current events were “short” events.  Sign-selected neutrino beam, tags neutrino and antineutrino interactions (selected by decay of π+ and π-).  Allows use of Paschos-Wolfenstein formula to Neutrino Interaction Physics 43 NUFACT08reduce Summer systematics. School 3.7Charmproduction3.7Charmproduction

 Production of charm can be carried out from deep inelastic neutrino scattering from d or s quarks: 2 2 2 2 2 2 (q + ξp) = q + 2ξp ⋅q + ξ M = p' = mc Therefore: − q 2 + m2 Q2 + m2 Q2 + m2 Q2 + m2  m2  c c c c  c  ξ ≈ = = = 2 = x1+ 2  2p ⋅ q 2Mν 2Mν Q / x  Q 

 Slow rescaling model (LO): effect of a heavy quark threshold Q2  m2   Replace:  c  x = → ξ = x1+ 2  2Mν  Q   Cross-section: 3 ν 2 d σ G ME ξ 2 2  xy  F 2 2 2   = {[]u(ξ,Q ) + d(ξ,Q ) Vcd + 2s(ξ,Q )Vcs }1− y + D(z) dξdydz π ξ   −2 1  1 εp  D(z) ∝ 1− −  Fragmentation of charmNeutrino quark Interaction into Physicshadrons: 44 NUFACT08 Summer School z  z 1− z  (Petersen function, but there are others) 3.7Charmproduction3.7Charmproduction

 Production of opposite sign dimuon events is signal of charm production because of of charm:

NOMAD  Charm production can probe strange sea,

measure charm mass and Vcd

CCFR/NUTEV

 High statistics opposite sign dimuon samples were acquired by CDHS, CCFR, NOMAD, CHORUS, NUTEV Neutrino Interaction Physics 45 NUFACT08 Summer School 3.7Charmproduction3.7Charmproduction

 Some results from opposite sign dimuons: Cross-section: between 0.2%-1% depending on energy LO Measurement charm mass (average): mc = 43.1 ± 10.0

Strange sea asymmetry NLO mc = 70.1 ± .0 019 (NUTEV ) Measurement Vcd (average): m NLO = 58.1 ± 09.0 + 04.0 (NOMAD ) LO c − 09.0 Vcd = .0 232 ± .0 010 NLO Vcd = .0 246 ± .0 016

CHORUS

Review: G di Lellis et al, Phys. Rep. 399, 2004, 227. Neutrino Interaction Physics 46 NUFACT08 Summer School