1. Predictions for high energy neutrino cross-sections from ZEUS-S Global fit analysis S Chekanov et al, Phys Rev D67,012007 (2002) The ZEUS PDFs are sets of PDFs with errors extracted from a global NLO-QCD fit in the DGLAP formalism to ZEUS data and world fixed target data. The DGLAP equations were solved at NLO in the \msbar\ scheme with the renormalisation and factorisation scales chosen to be $Q^2$. The standard ZEUS fit (ZEUS-S) was done using the Thorne Roberts Variable Flavour Number Scheme -TRVFN. The PDFs and their uncertainties, deriving from uncorrelated and correlated experimental systematic uncertainties, are available on http://durpdg.dur.ac.uk/hepdata/zeus2002 For the present analysis this fit has been updated to include ALL the ZEUS inclusive cross-section data from the first phase of running (1994-2000) as specified for the ZEUS-JETS fit S Chekanov et al, Eur.Phys J. C42,1 (2005) and the PDF parametrisation used at Q2_0 has been extended from 11 to 13 parameters (p2 valence and p5 glue are set free) The resulting PDFs and their uncertainties are very similar to those of the published ZEUS-S fit

2. For Q2= 5,100,1000,20000 (~2*M W 2 ) the x axis scale has been extended down to x=10 -8 for this analysis Sea- note difference in y axis scale Gluon The sea and gluon PDFs rise strongly as x decreases. Here are representative examples (central line plus upward and downward uncertainties)

3. d 2  (νN) = G F 2 M W 4 [ Y + F 2 (x,Q 2 ) - y 2 F L (x,Q 2 ) + Y_xF 3 (x,Q 2 )], dxdQ 2 2π x(Q 2 + M W 2 ) 2 Y± = 1 ± (1-y) 2 and in LO QCD F 2 = x (u +d + 2s +2b + ubar +dbar +2cbar) xF 3 = x (u +d + 2s +2b – ubar – dbar – 2cbar) = x(uv +dv +2s +2b -2cbar) F L =0 where the quark distributions are functions of x,Q 2 This is more complicated at NLO, but these expressions give a good idea of the main contributions NOTE b quarks enter this expression, but in practice will be very suppressed until Q2 ~ 150000 (~4M top 2 ) because the b/t coupling is dominant and t quark mass is so large I have assumed a massless treatment (like Quigg) and a more correct heavy quark treatment TRVFN The neutrino double differential cross-section is given by

4. Representative structure functions for the massless treatment F2 xF3 FL Note FL is NOT zero- NLO contributions DO matter! F2 is the dominant structure function – note the change of scale on the other plots

5. Representative structure functions- massive treatment (Thorne-Roberts VFN) F2 xF3 FL For F2 and FL this is not a big effect but it IS important for xF3. This is because u and d valence are small at low-x and strange and cbar almost cancel– so xF3 is all b.. nevertheless because F2 is the dominant structure function this doesn’t affect cross sections strongly

6. Now to look at how these structure functions build the cross-section. Remember there are kinematic restrictions Q 2 = s x y,and y is bounded 0 to 1- (in practice it is also hard to identify events for which y~1), so all x values are not accessed for all Q2. Here’s how it works for s=3.6 10 7, Eν = 1.9 10 7. What is plotted is the reduced cross-section [ Y+ F2(x,Q2) - y 2 FL(x,Q2) + Y- xF3(x,Q2)]/2, The Q 2 values are 1,100, 10000, 100000 GeV 2, to illustrate how the cross-section is suppressed for large Q2. You can also see that Q2 values below M W 2 contribute strongly. masslessmassiveKinematic cut off

7. Plots of [ Y+ F2(x,Q2) - y 2 FL(x,Q2) + Y- xF3(x,Q2)]/2, for higher energy s= 10 10, Eν= 5.3 10 9 Such energies access very low-x for Q 2 ~ 10,000 massivemassless It remains true that Q 2 less than M W 2 contributes quite strongly and that higher Q 2 are suppressed. Note also that the PDF uncertainty on the reduced cross-section is about the same size as difference between the massless and massive treatments

8. Plots of the full differential cross-section d 2 б/dxdQ 2, including the propagator, as a function of x at the same Q 2 values 1,100, 10000, 100000 GeV 2. Note that below the kinematic cut-off I have set the value of the cross-section to 1.0 rather than 0.0, because it’s a log-plot., so ignore these horizontal lines at 1.0. Kinematic cut off s=3.6 10 7, Eν = 1.9 10 7 s= 10 10, Eν = 5.3 10 9 These are both for the massless treatment, you can hardly see the massless/massive difference or the PDF uncertainties on such log-plots

9. To calculate the total cross-section d 2 б/dxdy is integrated This cross-section is shown below, as a function of x at four different values of y, y= 0.01,0.1,0.5, 0.95 It is not a strong function of y, the value of y just affects the position of the kinematic cut-off s=3.6 10 7, Eν = 1.9 10 7 A table of the total cross-section for various νN c.of m. energies is given on the next page together with their PDF uncertainties, the difference from massless/massive treatment and Quigg’s CTEQ4 values

10. s б pb (Mandy) б pb (Quigg) PDF uncertainty Massless/ massive difference 10 7 15311529~3%~4% 3.6 10 7 25512434~3%~4% 10 8 37723528~4%~5% 3.6 10 8 60715671~5%~6% 10 9 87818138~6%~7% 3.6 10 9 1378412956~6%~8% 10 1968318773~7%~10% 3.6 10 10 3010629887~7%~11% 10 11 4271143305~8%~12% 3.6 10 11 6395568941~9%~14% 10 12 8952199895~10%~15%