Quarks for Dummies: Modeling (e/ m /n?-N Cross Sections from Low to High Energies: from DIS to Resonance, to Quasielastic Scattering

1
Quarks for Dummies Modeling (e/ m /n?-N Cross
Sections from Low to High Energies from DIS to
Resonance, to Quasielastic Scattering

• Arie Bodek, Univ. of Rochester
• Un-Ki Yang, Univ of Chicago

• Work in progress most recently presented in
  invited talks at
• NuFact02 -Imperial College, London , July, 2002
  (2 talks)
• DPF Meeting, Virginia May 2001
• APS Meeting, New Mexico April, 2001
• NuInt01, KEK Japan Dec. 2001
• Final studies with xw A(W,Q2) in preparation
  for NuInt02, Irvine, Dec. 2002.
• Studies in LO (xw HT scaling) - Being
  written for proceedings of NuFact 02
• Studies in LO (Xw HT scaling) - Bodek and
  Yang hep-ex/0203009 (2002) to appear in
  proceedings of NuInt 01 (Nuclear Physics B)
• Studies in NNLOHT - Yang and Bodek Eur. Phys.
  J. C13, 241 (2000)
• Studies in NLOHT - Yang and Bodek Phys. Rev.
  Lett 82, 2467 (1999)
• Studies in 0th ORDER (QPM Xw HT scaling) -
  Bodek, el al PRD 20, 1471 (1979

2
Neutrino cross sections at low energy

• Neutrino oscillation experiments (K2K, MINOS,
  CNGS, MiniBooNE, and future experiments with
  Superbeams at JHF,NUMI, CERN) are in the few GeV
  region
• Important to correctly model neutrino-nucleon and
  neutrino-nucleus reactions at 0.5 to 4 GeV
  (essential for precise next generation neutrino
  oscillation experiments with super neutrino beams
  ) as well as at the 15-30 GeV (for future
  n?factories)
• The very high energy region in neutrino-nucleon
  scatterings (50-300 GeV) is well understood at
  the few percent level in terms QCD and Parton
  Distributions Functions (PDFs) within the
  framework of the quark-parton model (data from a
  series of e/m/n DIS experiments)
• However, neutrino differential cross sections
  and final states in the few GeV region are poorly
  understood. ( especially, resonance and low Q2
  DIS contributions). In contrast, there is
  enormous amount of e-N data from SLAC and Jlab in
  this region.

3
Examples of Current Low Energy Neutrino Data
Quasi-elastic cross section
?tot/E
4
How are PDFs Extracted from global fits to High
Q2 Deep Inelastic e/m/n Data
Note additional information on Antiquarks from
Drell-Yan and on Gluons from p-pbar jets also
used.

• MRSR2 PDFs

At high x, deuteron binding effects introduce an
uncertainty in the d distribution extracted from
F2d data (but not from the W asymmetry data).
5
Neutrino cross sections

• Neutrino interactions --
• Quasi-Elastic / Elastic (WMp) nm
  n --gt m- p (x 1, WMp) well
  measured and described by form factors (but need
  to account for Fermi Motion/binding effects in
  nucleus) e.g. Bodek and Ritchie (Phys. Rev.
  D23, 1070 (1981)
• Resonance (low Q2, Wlt 2) nm p --gt
  m- p p????????????????Poorly measured and
  only 1st resonance described by Rein and Seghal
• Deep Inelastic
• nm p --gt m- X (high Q2, Wgt 2)
• well measured by high energy experiments and
  well described by quark-parton model (pQCD with
  NLO PDFs), but doesnt work well at low Q2
  region.

GRV94 LO
1st resonance

• (e.g. JLAB data at Q20.22)
• Issues at few GeV
• Resonance production and low Q2 DIS contribution
  meet.
• The challenge is to describe both processes at a
  given neutrino (or electron) energy.

6
Building up a model for all Q2 region..
Challenges

• Can we build up a model to describe all Q2 region
  from high down to very low energies ? resonance,
  DIS, even photo production
• Advantage if we describe it in terms of the
  quark-parton model.
• then it is straightforward to convert
  charged-lepton scattering cross sections into
  neutrino cross section. (just matter of different
  couplings)

• Understanding of high x PDFs at very low Q2?
• There is a of wealth SLAC, JLAB data, but it
  requires understanding of non-perturbative QCD
  effects.
• Need better understanding of resonance scattering
  in terms of the quark-parton model? (duality
  works, many studies by JLAB)

7
What are Higher Twist Effects- page 1

• Higher Twist Effects are terms in the structure
  functions that behave like a power series in
  (1/Q2 ) or Q2/(Q4A), (1/Q4 ) etc.

(a)Higher Twist Interaction between Interacting
and Spectator quarks via gluon exchange at Low
Q2-mostly at low W (b) Interacting quark TM
binding, initial Pt and Missing Higher Order QCD
terms -In DIS region. -gt(1/Q2 ) or Q2/(Q4A),
(1/Q4 ).
?
Pt

• While pQCD predicts terms in as2 ( 1/ln(Q2/ L2
  ) ) as4 etc
• (i.e. LO, NLO, NNLO etc.) In the few GeV
  region, the terms of the two power series cannot
  be distinguished

In NNLO p-QCD additional gluons emission terms
like as2 ( 1/ln(Q2/ L2 ) ) as4 Spectator
quarks are not Involved.
In pQCD high Q2 impulse approximation,
the interacting quark and the spectator quarks
are resolved and do not affect each
other.
?
8
What are Higher Twist Effects - Page 2

• Nature has evolved the high Q2 PDF from the
  low Q2 PDF, therefore, the high Q2 PDF include
  the information about the higher twists .
• High Q2 manifestations of higher twist/non
  perturbative effects include difference between
  u and d, the difference between d-bar, u-bar and
  s-bar etc. High Q2 PDFs remember the higher
  twists, which originate from the
  non-perturbative QCD terms.
• Evolving back the high Q2 PDFs to low Q2 (e.g.
  NLO-QCD) and comparing to low Q2 data is one way
  to check for the effects of higher order terms.
• What do these higher twists come from?
• Kinematic higher twist initial state target
  mass binding (Mp, xTM) initial state and final
  state quark masses (e.g. charm production)- xTM
  important at high x
• Dynamic higher twist correlations between
  quarks in initial or final state.gt Examples
  Initial or final state multiquark correlations
  diquarks, elastic scattering, excitation of
  quarks to higher bound states e.g. resonance
  production, exchange of many gluons important
  at low W
• Non-perturbative effects to satisfy gauge
  invariance and connection to photo-production
  e.g. F2(n ,Q2 0) Q2 / Q2 C 0. important
  at very low Q2.
• Higher Order QCD effects - to e.g. NNLO
  multi-gluon emissionlooks like Power higher
  twist corrections since a LO or NLO calculation
  do not take these into account, also quark
  intrinsic PT (terms like PT2/Q2). Important at
  all x (look like Dynamic Higher Twist)

9
Old Picture of fixed W scattering - form factors

• OLD Picture fixed W Elastic Scattering,
  Resonance Production. Electric and Magnetic
  Form Factors (GE and GM) versus Q2 measure size
  of object (the electric charge and magnetization
  distributions).
• Elastic scattering W Mp M, single final
  state nucleon Form factor measures size of
  nucleon.Matrix element squared ltp f V(r) p
  i gt 2 between initial and final state lepton
  plane waves. Which becomes
• lt e -i k2. r V(r) e i k1 . r gt 2
  
• q k1 - k2 momentum transfer
• GE (q) ? e i q . r r (r) d3r Electric
  form factor is the Fourier transform of the
  charge distribution. Similarly for the
  magnetization distribution for GM Form factors
  are relates to structure function by
• 2xF1(x ,Q2)elastic x2 GM2 elastic (Q2) d
  (x-1)
• Resonance Production, WMR, Measure transition
  form factor between a quark in the ground state
  and a quark in the first excited state. For the
  Delta 1.238 GeV first resonance, we have a
  Breit-Wigner instead of d (x-1).
• 2xF1(x ,Q2) resonance x2 GM2 Res. transition
  (Q2) BW (W-1.238)

e i k2 . r e i k1.r rMp Mp
e i k2 . r

e i k1 . r
q
MR
Mp
10
Duality Parton Model Pictures of Elastic and
Resonance Production at Low W

• Elastic Scattering, Resonance Production
  Scatter from one quark with the correct parton
  momentum x, and the two spectator are just
  right such that a final state interaction Aw (w,
  Q2 ) makes up a proton, or a resonance.
• Elastic scattering W Mp M, single nucleon
  in final state.
• The scattering is from a quark with
  a very high value of x, is such that one
  cannot produce a single pion in the final
  state and the final state interaction makes a
  proton.
• Aw (w,
  Q2 ) d (x-1) (times)
• integral over x, from pion
  threshold to x 1 local duality
• (This is just a check of
  local duality, better to use Ge,Gm)
• Resonance Production, WMR,
  e.g. delta 1.238 resonance. The scattering is
  from a quark with a high value of x, is such
  that that the final state
  interaction makes a low mass
  resonance. Aw (w, Q2 ) includes Breit-Wigners.
  Local duality
• Therefore, with the correct scaling variable, and
  if we account for low W and low Q2 higher twist
  effects, the prediction using QCD PDFs q (x,
  Q2) should give an average of F2 in the elastic
  scattering and in the resonance region.
  (including both resonance and continuum
  contributions). If we modulate the PDFs with a
  final state interaction resonance A (w, Q2 ) we
  could also reproduce the various Breit-Wigners
  continuum.

q
X 1.0 x 0.95

Mp
Mp
X 0.95 x 0.90
MR
Mp
11
Photo-production Limit Q20Non-Perturbative -
QCD evolution freezes

• Photo-production Limit Transverse Virtual and
  Real Photo-production cross sections must be
  equal at Q20. Non-perturbative effect.
• There are no longitudinally polarized photons at
  Q20
• ?L (n, Q2) 0 limit
  as Q2 --gt0
• Implies R (n, Q2) ?L/? T Q2 / Q2 const
  --gt 0 limit as Q2 --gt0
• s(g-proton, n ) ?T (n, Q2) limit as
  Q2 --gt0
• ?implies? s(g-proton, n ) 0.112 mb 2xF1 (n, Q2)
  / (KQ2 ) limit as Q2 --gt0
• ???????????????s(g-proton, n ) 0.112 mb F2
  (n, Q2) D / KQ2 limit as Q2 --gt0
• ???or F2 (n, Q2) Q2 / Q2 C
  --gt 0 limit as Q2 --gt0
• K 1 - Q2/ 2M? ? D (1 Q2/ ? 2 )/(1R)
• If we want PDFs to work down to Q20 where pQCD
  freezes
• The PDFs must be multiplied by a factor Q2 /
  Q2 C (where C is a small number).
• The scaling variable x does not work since
  s(g-proton, n ) ?T (n, Q2)
• At Q2 0 F2 (n, Q2) F2 (x , Q2) with x
  Q2 /( 2Mn) reduces to one point x0
• However, a scaling variable xc (Q2 B) /(
  2Mn) works at Q2 0
• F2 (n, Q2) F2 (xc, Q2) F2 B/ (2Mn), 0
  limit as Q2 --gt0

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How do we measure higher twist (HT)

• Take a set of QCD PDF which were fit to high Q2
  (e/m/n? data (in Leading Order-LO, or NLO, or
  NNLO)
• Evolve to low Q2 (NNLO, NLO to Q21 GeV2) (LO to
  Q20.24)
• Include the known kinematic higher twist from
  initial target mass (proton mass) and final heavy
  quark masses (e.g. charm production).
• Compare to low Q2data in the DIS region (e.g.
  SLAC)
• The difference between data and QCDtarget mass
  predictions is the extracted effective dynamic
  higher twists.
• Describe the extracted effective dynamic higher
  twist within a specific HT model (e.g. QCD
  renormalons, or a purely empirical model).
• Obviously - results will depend on the QCD order
  LO, NLO, NNLO (since in the 1 GeV region 1/Q2and
  1/LnQ2 are similar). In lower orders, the
  effective higher twist will also account for
  missing QCD higher order terms. The question is
  the relative size of the terms.
• Studies in NLO - Yang and Bodek Phys. Rev.
  Lett 82, 2467 (1999) ibid 84, 3456 (2000)
• Studies in NNLO - Yang and Bodek Eur. Phys. J.
  C13, 241 (2000)
• Studies in LO - Bodek and Yang
  hep-ex/0203009 (2002)
• Studies in QPM 0th order - Bodek, el al
  PRD 20, 1471 (1979)

13
Lessons from previous NLO QCD study

• Our NLO study comparing NLO PDFs to DIS SLAC,
  NMC, and BCDMS e/m scattering data on H and D
  targets shows (for Q2 gt 1 GeV2) refYang
  and Bodek Phys. Rev. Lett 82, 2467 (1999)
• Kinematic Higher Twist (target mass ) effects
  are large and important at large x, and must be
  included in the form of Georgi Politzer xTM
  scaling.
• Dynamic Higher Twist effects are smaller, but
  need to be included. (A second NNLO study
  established their origin)
• The ratio of d/u at high x must be increased if
  nuclear binding effects in the deuteron are taken
  into account.
• The Very high x (0.9) region - is described by
  NLO QCD (if target mass and renormalon higher
  twist effects are included) to better than 10.
  SPECTATOR QUARKS modulate A(W,Q2) ONLY.
• Resonance region NLO pQCD Target mass Higher
  Twist describes average F2 in the resonance
  region (duality works). Include Aw (w, Q2 )
  resonance modulating function from spectator
  quarks later.
• A similar NNLO study using NNLO QCD we find that
  the empirically measured effective Dynamic
  Higher Twist Effects in the NLO study come from
  the missing NNLO higher order QCD terms. ref
  Yang and Bodek Eur. Phys. J. C13, 241 (2000)

14
F2, R comparison of QCDTM plot vs. NLO
QCDTMHT (use QCD Renormalon Model for HT)
PDFs and QCD in NLO TM QCD Renormalon Model
for Dynamic HTdescribe the F2 and R data re
well, with only 2 parameters. Dynamic HT effects
are there but small
15
Same study showing the QCD-only Plot vs. NLO
QCDTMHT (use QCD Renormalon Model for HT)
PDFs and QCD in NLO TM QCD Renormalon Model
for Dynamic Higher Twist describe the F2 and R
data reasonably well. TM Effects are LARGE
16
A simlar study of NLO QCDTM vs. QCDTMHT
(use here Empirical Model for Dynamic HT) -
backup slide
PDFs and QCD in NLO TM Empirical Model for
Dynamic HT describe the data for F2 (only)
reasonably well with 3 parameters. Dynamic HT
effects are there but small
Here we used an Empirical form for Dynamic HT.
Three parameters a, b, c. F2 theory (x,Q2)
F2 PQCDTM 1 h(x)/ Q2 f(x) f(x)
floating factor, should be 1.0 if PDFs have the
correct x dependence. h(x) a (xb/(1-x) -c)
17
Kinematic Higher-Twist (GP target massTM)
Georgi and Politzer Phys. Rev. D14, 1829 (1976)
Well known

• x TM 2x / 1 k 1 Mc2 / Q2
• (last term only for heavy charm product)
• k ( 1 4x2 M2 / Q2) 1/2
• (Derivation of x TM in
  Appendix)
• For Q2 large (valence) F22 x F1 x F3
• F2 pQCDTM(x,Q2) F2pQCD (x, Q2) x2 / k3x2
• J1 (6M2x3 / Q2k4 ) J2(12M4x4 / Q4k5 )
• 2F1 pQCDTM(x,Q2) 2F1pQCD (x, Q2) x / kx
• J1 (2M2x2 / Q2k2 ) J2(4M4x4 / Q4k5 )
• F3 pQCDTM(x,Q2) F3pQCD(x, Q2) x / k2x
• J1F3 (4M2x2 / Q2k3 )

Ratio F2 (pQCDTM)/F2pQCD At very large x,
factors of 2-50 increase at Q215 GeV2 from TM


18
Kinematic Higher-Twist (target massTM) x TM
Q2/ Mn (1 (1Q2/n2 ) 1/2 )
Compare complete Target-Mass calculation to
simple rescaling in x TM

• The Target Mass Kinematic Higher Twist effects
  comes from the fact that the quarks are bound in
  the nucleon. They are important at low Q2 and
  high x. They involve change in the scaling
  variable from x to xTM and various kinematic
  factors and convolution integrals in terms of the
  PDFs for xF1, F2 and xF3
• Above x0.9, this effect is mostly explained by a
  simple rescaling in xTM.
• F2pQCDTM(x,Q2)
• F2pQCD(xTM?Q2)

Ratio F2 (pQCDTM)/F2pQCD
Q215 GeV2
19
Dynamic Higher Twist- Renormalon Model

• Use Renormalon QCD model of WebberDasgupta-
  Phys. Lett. B382, 272 (1996), Two parameters a2
  and a4. This model includes the (1/ Q2) and (1/
  Q4) terms from gluon radiation turning into
  virtual quark antiquark fermion loops (from the
  interacting quark only, the spectator quarks are
  not involved).
• F2 theory (x,Q2) F2 PQCDTM 1 D2 (x,Q2)
  D4 (x,Q2)
• D2 (x,Q2) (1/ Q2) a2 / q (x,Q2) ? (dz/z)
  c2(z) q(x/z, Q2)
• D4 (x,Q2) (1/ Q4) a4 times function of x)
  
• In this model, the higher twist effects are
  different for 2xF1, xF3 ,F2. With complicated x
  dependences which are defined by only two
  parameters a2 and a4 . (the D2 (x,Q2) term is
  the same for 2xF1 and , xF3 )
• Fit a2 and a4 to experimental data for F2 and
  RFL/2xF1.
• F2 data (x,Q2) F2 measured l d F2 syst
  ( 1 N ) c2 weighted by errors
• where N is the fitted normalization (within
  errors) and d F2 syst is the is the fitted
  correlated systematic error BCDMS (within
  errors).

q-qbar loops
20
QCD Power Law Corrections

• Renormalon QCD model of WebberDasgupta- Phys.
  Lett. B382, 272 (1996), includes only two
  parameters a2 and a4. This infrared renormalon
  model (only one renormalon chain of bubble
  graphs) leads the (1/ Q2) and (1/ Q4) terms from
  gluon radiation turning into virtual quark
  antiquark fermion loops (from the interacting
  quark), the spectator quarks are not involved).
• QCD is an asymptotic series which gets closer to
  the correct answer if taken up up to a certain N
  and then individual terms begin to factorially
  increase at any fixed x. This introduces an
  ambiguity into the theory (trying to estimate the
  size of the infinite number of remaining terms in
  the series). This ambiguity is referred to as the
  power corrections, or infrared renomalons (while
  ultraviolet renormalons are a power series in
  ?s). Calculations show that the infrared
  renormalons (to estimate the remaining terms in
  the series) have a power law dependence, and
  therefore look like higher twist. Note that these
  power law corrections are from interactions of
  the Interacting Quark only, and have to do with
  the corrections for the missing higher order
  terms in QCD . These terms lead to multiple final
  state gluons (I.e. final state quark effective
  mass), and initial state Pt and effective mass
  from multiple gluon emission.
• What we have shown, is that NNLO QCD (with the
  world average of ?s) is already very good, since
  the extracted power law corrections within the
  renomalon model are very small. We will go back
  later and focus on a LEADING ORDER analysis. We
  can now correct for the missing higher order NLO,
  NNLO etc. within our new physical model for the
  renormalon Higher Twist (for LO) in terms of
  effective initial quark Pt and mass, and
  effective final quark mass, and enhanced Target
  Mass corrections.

Higher Order QCD Corr.
q-qbar loops
Renormalon Power Corr.
21
Very high x F2 proton data (DIS resonance)(not
included in the original fits Q21. 5 to 25 GeV2)
Q2 25 GeV2 Ratio F2data/F2pQCD
F2 resonance Data versus F2pQCDTMHT
Q2 1. 5 GeV2
pQCD ONLY
Q2 3 GeV2
Q2 25 GeV2 Ratio F2data/ F2pQCDTM
pQCDTM
Q2 15 GeV2
Q2 9 GeV2
Q2 25 GeV2 Ratio F2data/F2pQCDTMHT
pQCDTMHT
Q2 25 GeV2
pQCDTMHT
x ????
x ????
Aw (w, Q2 ) will account for interactions with
spectator quarks

• NLO pQCD x TM higher twist describes very
  high x DIS F2 and resonance F2 data well.
  (duality works) Q21. 5 to 25 GeV2

22
Look at Q2 8, 15, 25 GeV2 very high x
data-backup slide
Ratio F2data/F2pQCDTMHT
Q2 9 GeV2

• Pion production threshold Aw (w, Q2 )
• Now Look at lower Q2 (8,15 vs 25) DIS and
  resonance data for the ratio of
• F2 data/( NLO pQCD TM HT
• High x ratio of F2 data to NLO pQCD TM HT
  parameters extracted from lower x data. These
  high x data were not included in the fit.
• The Very high x(0.9) region It is described by
  NLO pQCD (if target mass and higher twist effects
  are included) to better than 10

Q2 15 GeV2
Q2 25 GeV2
23
F2, R comparison with NNLO QCDgt NLO HT mostly
missing NNLO terms
Size of the higher twist effect with NNLO
analysis is really small (but not 0) a2
-0.009 (in NNLO) versus 0.1( in NLO) - gt
factor of 10 smaller, a4 nonzero
24
Converting NLO PDFs to NNLO PDFs backup slide

• f(x) the fitted floating factor, which is the
  fitted ratio of the data to theory . Note f(x)
  1.00 if pQCD PDFs describe the data.
• fNLO Here the theory is pQCD(NLO)TMHT using
  NLO PDFs.
• fNNLO Here the theory is pQCD(NNLO)TMHT also
  using NLO PDFs
• Therefore fNNLO / fNLO is the factor to
  convert NLO PDFs to NNLO PDFs (NNLO PDFs are
  not yet available.
• NNLO PDFs are lower at high x and higher at low
  x.
• Use f(x) in NNLO calculation of QCD processes
  (e.g. hardon colliders)
• Recently MRST did a similar analysis including
  NNLO gluons.
• True NNLO PDFs in a year or two

• Floating factors

NLO
NNLO
25
Lessons from the NNLO pQCD analysis

• For Q2gt1 GeV2 The origin of the empirically
  measured small dynamic higher twist effects in
  NLO is from the missing NNLO QCD terms.
• Both TM and Dynamic higher twists effects should
  be similar in electron and neutrino reactions
  (aside from known mass differences, e.g. charm
  production)

(F2NNL0/F2NLO)-1
The NNLO pQCD corrections and the Dynamic Higher
Twist (e.g. QCD renormalon) effects in NLO both
have the same Q2 dependence at fixed x. Both
involve only gluon and fermion loops off the legs
of the interacting quark (not spectator quarks).
26
Derivation for initial quark mass m I and final
mass m bound in a proton of mass M - INCLUDING
Quark INITIAL P2t
q

• Georgi and Politzer Phys. Rev. D14, 1829
  (1976)- GP did not include Pt
  
• (Pi q)2 m I 2 2qPi q2 m 2 and q
  (n, q3) in lab
• 2qPi 2 n Pi0 q3 Pi3 Q2 m 2 - m
  I 2 (eq. 1)
• ( note sign since q3 and Pi3 are pointing at
  each other)
• x Pi0 Pi3 /P0 P3 --- frame
  invariant definition
• x Pi0 Pi3 /M ---- In lab
  or Pi0
  Pi3 x M --- in lab
• x Pi0 -Pi3 Pi0 Pi3 Pi0 -Pi3 / M --
  multiplied by Pi0 -Pi3
• Pi0 -Pi3 (Pi0 ) 2 - ( Pi3 ) 2 / M (m I 2
  P2t) /x M or
  Pi0 -Pi3 (m I 2 P2t) /x M --- in lab
• 
  Get
  2 Pi0 x M (m I 2 P2t) /x M
• Plug into (eq. 1) and 2
  Pi3 x M - (m I 2 P2t) /x M
• n x M n (m I 2 P2t) /x M q3 x M
  - q3 (m I 2 P2t) 2 /x M - Q2 m 2 - m I
  2 0
• a b c
• x 2 M 2 (n q3) - x M Q2 m 2 - m I 2
  (m I 2 P2t) (n- q3) 0 x -b (b 2 -
  4ac) 1/2 / 2a gt solution
• use (n 2- q3 2) q 2 -Q 2 and (n
  q3) n n 1 Q 2/ n 2 1/2 n n 1
  4M2 x2/ Q 2 1/2
• 
  

Pi Pi0,Pi3,mI
Pf, m
P P0 P3,M

Get x w Q2 B / Mn
(1(1Q2/n2) 1/2 ) A where 2Q2 Q2 m 2 -
m I 2 ( Q2m 2 - m I 2 ) 2 4Q2 (m I 2
P2t) 1/2 or 2Q2 Q2 m F2 -
m I 2 Q4 2 Q2(m F2 m I 2 2P2t ) (m
F2 - m I 2 ) 2 1/2 If mi0 2Q2 Q2
m 2 ( Q2m 2 ) 2 4Q2 (P2t) 1/2
Add B and A account for effects of additional
? m2 from NLO and NNLO effects. (at high Q2
these are current quark masses, but at low Q2
maybe constituent masses?)
27
Pseudo Next to Leading Order Calculations
Use LO PDFs (Xw) times (Q2/Q2C) And PDFs
(x w) times (Q2/Q2C)

• 
  

q
Pi Pi0,Pi3,mI
Xw Q2 B / 2Mn A x w
Q2 B / Mn (1(1Q2/n2) 1/2 ) A
Pf, m
P P0 P3,M
where 2Q2 Q2 m 2 - m I 2 ( Q2m 2 -
m I 2 ) 2 4Q2 (m I 2 P2t) 1/2 or
2Q2 Q2 m F2 - m I 2 Q4 2 Q2(m F2
m I 2 2P2t ) (m F2 - m I 2 ) 2 1/2

• Add B and A account for effects of additional
  ? m2 from NLO and NNLO effects.
• There are many examples of taking Leading Order
  Calculations and correcting them for NLO and NNLO
  effects using external inputs from measurements
  or additional calculations e.g.
• Direct Photon Production - account for initial
  quark intrinsic Pt and Pt due to initial state
  gluon emission in NLO and NNLO processes by
  smearing the calculation with the MEASURED Pt
  extracted from the Pt spectrum of Drell Yan
  dileptons as a function of Q2 (mass).
• W and Z production in hadron colliders.
  Calculate from LO, multiply by K factor to get
  NLO, smear the final state W Pt from fits to Z Pt
  data (within gluon resummation model parameters)
  to account for initial state multi-gluon
  emission.
• K factors to convert Drell-Yan LO calculations to
  NLO cross sections. Measure final state Pt.
• K factors to convert NLO PDFs to NNLO PDFs
• Prediction of 2xF1 from leading order fits to F2
  data , and imputing an empirical parametrization
  of R (since R0 in QCD leading order).
• THIS IS THE APPROACH TAKEN HERE. i.e. a Leading
  Order Calculation with input of effective initial
  quark masses and Pt and final quark masses, all
  from gluon emission.

28
At low x, Q2 NNLO terms look similar to
kinematic final state mass higher twist or
effective final state quark mass -gt enhanced
QCD

• At low Q2, the final state u and d quark
  effective mass is not zero

Charm production s to c quarks in neutrino
scattering-slow rescaling
u
u
M (final state interaction) Production of pions
etc Or gluon emission from the Interacting quark
c
s
Mc (final state quark mass

• 2 x C q.P Q2 Mc2 (Q2
  -q2 )
• 2 x C Mn Q2 Mc2 x C - slow
  re-scaling
• x C Q2Mc2 / 2Mn (final state
  charm mass

• x C Q2M2 / 2Mn (final state M
  mass))
• versus for mass-less quarks 2x q.P Q2
• x Q2 / 2Mn (compared
  to x

• (Pi q)2 Pi2 2q.Pi q2 Pf2 Mc2

• (Pi q)2 Pi2 2q.Pi q2 Pf2 M2

F2
Low x QCD evolution x C slow rescaling looks
like faster evolving QCD Since QCD and slow
rescaling are both present at the same Q2
x
At Low x, low Q2 x C gt x (slow rescaling x
C) (and the PDF is smaller at high x, so the low
Q2 cross section is suppressed - threshold
effect.
Final state mass effect
Lambda QCD
Ln Q2
29
At high x, NNLO QCD terms have a similar form
to the kinematic -Georgi-Politzer x TM TM
effects -gt look like enhanced QCD evolution at
low Q

• x2 TM M 2 x TM q.P - Q2 0 (Q2
  -q2 )
• mnemonic- solve quadratic equation
• x TM Q2/ Mn (1 (1Q2/n2 ) 1/2 ) proton
  target mass effect in Denominator)
• Versus Numerator in
• x C Q2M2 / 2Mn (final state M
  mass)
• Combine both target mass and final state
  mass
• x CTM Q2M2B / Mn (1(1Q2/n2) 1/2
  ) A - includes both
  initial state target proton mass and final state
  M mass effect) - Exact derivation in
  Appendix. Add B and A account for additional ?
  m2 from NLO and NNLO effects.

Final state mass

• Target Mass (G-P) x - tgt mass

• (Pi q)2 Pi2 2qPi Q2 Pf2

Initial state target mass
F2 fixed Q2
x lt x C
x TMlt x
X1
X0
F2
At high x, low Q2 x TM lt x (tgt mass x) (and
the PDF is higher at lower x, so the low Q2 cross
section is enhanced .
x
Target mass effects
RefGeorgi and Politzer Phys. Rev. D14, 1829
(1976)
QCD evolution
High x
Mproton
Ln Q2
30
Towards a unified model
Different scaling variables

• We learned that the NNLO TM describes the DIS
  and resonance data very well.
• Theoretically, this breaks down at low Q2lt1
  Practically, no way to implement it in MC
• HT takes care of the NNLO term - So what about
  NLO TM HT?
• Still, it break down at very low Q2, - No way
  to implement photo-production limit.
• Well, can we do something with LO QCD and LO
  PDFs ? YES

• Resonance, higher twist, and TM

q
X-0.95
x 0.9
M (final state interaction) Or multigluon
emission
M

• (Pi q)2 Pi2 2qPi Q2 M2

• x C Q2M 2 / ( 2Mn) (quark final state M
  mass)
• x TM Q2/Mn (1 (1Q2/n2) 1/2 (initial proton
  mass)
• x Q2M 2 / Mn (1(1Q2/n2) 1/2 )
  combined
• x x 2Q22M 2 / Q2 (Q4 4x2 M2 Q2)
  1/2

F2
B term (M )
Low x

• TRY Xw Q2B /2Mn A x Q2B / Q2
  Ax
• (used in pre-QCD early fits to SLAC data in 1972)
• And then follow up by using the above
• x w Q2B / Mn (1(1Q2/n2) 1/2 ) A
• (xw works better and is theoretically
  motivated)
• Xw worked in 1972 because it approximates xw

Xw
Photoproduction limit- Need to multiply
by Q2/Q2C
A term (tgt mass)
High x
Lambda QCD
Ln Q2
31
Modified LO PDFs for all Q2 region?
Philosophy

• 1. We find that NNLO QCDtgt mass works very
  well for Q2 gt 1 GeV2.
• 2. That target mass and missing NNLO terms
  explain what we extract as higher twists in a
  NLO analysis. i.e. SPECTATOR QUARKS ONLY MODULATE
  THE CROSS SECTION AT LOW W. THEY DO NOT
  CONTRIBUTE TO DIS HT.
• 2. However, we want to go down all the way to
  Q20. All NNLO and NLO terms blow up. However,
  higher twist formalism in terms of initial state
  target mass binding and Pt, and final state mass
  are valid below Q21, and mimic the higher
  order QCD terms for Q2gt1 (in terms of effective
  masses, Pt due to gluon emission).
• While the original approach was to explain the
  empirical higher twists in terms of NNLO QCD at
  low Q2 (and extract NNLO PDFs), we can reverse
  the approach and have higher twist model
  non-perturbative QCD, down to Q20, by using LO
  PDFs and effective target mass and final state
  masses to account for initial target mass, final
  target mass, and missing NLO and NNLO terms.
  I.e. Do a fit with
• F2(x, Q2 ) Q2/ Q2C F2QCD(x w, Q2) A (w, Q2 )
  (set Aw (w, Q2 ) 1 for now - spectator
  quarks)
• x w Q2B / Mn (1(1Q2/n2) 1/2 ) A
  or Xw Q2B /2Mn A
• Beffective final state quark mass. Aenhanced
  TM term, RefBodek and Yang
  hep-ex/0203009

32
Modified LO PDFs for all Q2 (including 0)
Construction
Results

• 1. Start with GRV94 LO (Q2min0.23 GeV2 )
• - describe F2 data at high Q2
• 2A. Replace X with a new scaling, Xw
• x Q2 / 2Mn
• Xw Q2B / 2MnA
• A initial binding/target mass effect plus NLO
  NNLO terms )
• B final state mass effect (but also photo
  production limit)
• 2B. Or Replace X with a new scaling, x w
• x w Q2B / Mn (1(1Q2/n2) 1/2 ) A
• 3. Multiply all PDFs by a factor of Q2/Q2c for
  photo prod. Limitnon-perturbative
• F2(x, Q2 ) Q2/Q2C F2QCD(x w, Q2) A (w, Q2 )
• 4. Freeze the evolution at Q2 0.24 GeV2
• -F2(x, Q2 lt 0.24) Q2/Q2C F2(Xw, Q20.24)
• Do a fit to SLAC/NMC/BCDMS H, D data.- Allow the
  normalization of the experiments and the BCDMS
  major systematic error to float within errors.
• HERE INCLUDE DATA WITH Q2lt1 if it is not in the
  resonance region

• Modified LO GRV94 PDFs with three parameters (a
  new scaling variable, Xw, x w) describe DIS F2
  H, D data (SLAC/BCDMS/NMC) well.
• A1.735, B0.624, and C0.188 Xw (note for Xw, A
  includes the Proton M)
• A0.700, B0.327, and C0.197 x w works better as
  expected
• Keep final state interaction resonance
  modulating function A (w, Q2 )1 for now (will be
  included in the future). Fit DIS Only
• Compare with SLAC/Jlab resonance data (not used
  in our fit) -gtA (w, Q2 )
• Compare with photo production data (not used in
  our fit)-gt check on C
• Compare with medium energy neutrino data (not
  used in our fit)- except to the extent that GRV94
  originally included very high energy data on xF3

RefBodek and Yang hep-ex/0203009
33
Comparison of Xw Fit and xw Fit backup slide
Same construction for Xw and x w fits
Comparison

• Modified LO GRV94 PDFs with three parameters and
  the scaling variable, Xw, describe DIS F2 H, D
  data (SLAC/BCDMS/NMC) reasonably well.
• A1.735, B0.624, and C0.188
  (-0.022) (-0.014) ( -0.004)
  ?2 1555 /958 DOF
• With x w A and B are smaller Modified LO GRV94
  PDFs with three parameters and the scaling
  variable, x w describe DIS F2 H, D data
  (SLAC/BCDMS/NMC) EVEN BETTER
• A0.700, B0.327, and C0.197
  (-0.020) (-0.012) ( -0.004)
  ?2 1351 /958 DOF
• Note No systematic errors (except for
  normalization and BCDMS B field error) were
  included. GRV94 Assumed to be PEFECT (no f(x)
  floating factors). Better fits expected with
  GRV98 and floating factors f(x)

• Xw Q2B / 2MnA used in 1972
• x w Q2B / Mn (1(1Q2/n2) 1/2 ) A
• (theoretically derived)
• Multiply all PDFs by a factor of Q2/Q2C
• Fitted normalizations
• HT fitting with Xw
• p d
• SLAC 0.979 -0.0024 0.967 - 0.0025
• NMC 0.993 -0.0032 0.990 - 0.0028
• BCDMS 0.956 -0.0015 0.974 - 0.0020
• BCDMS Lambda 1.01 -0.156
• HT fitting with xw
• p d
• SLAC 0.982 -0.0024 0.973 - 0.0025
• NMC 0.995 -0.0032 0.994 - 0.0028
• BCDMS 0.958 -0.0015 0.975 - 0.0020

34
LOHT fit Comparison with DIS F2 (H, D) data
These SLAC/BCDMS/NMC are used in this Xw fit ?2
1555 /958 DOF

• Proton

Deuteron
35
LOHT fit Comparison with DIS F2 (H, D)
dataSLAC/BCDMS/NMC ?w works better ?2
1351 /958 DOF

• Proton

Deuteron
36
Comparison with F2 resonance data SLAC/ Jlab
(These data were not included in this Xw fit)

• The modified LO GRV94 PDFs with a new scaling
  variable, Xw describe the SLAC/Jlab resonance
  data very well (on average).
• Even down to Q2 0.07 GeV2
• Duality works The DIS curve describes the
  average over resonance region
• lt---Xw fit.
• For now, lets compare to neutrino data and
  photoproduction
• Next repeat with ?w
• Next repeat with GRV98 and f(x)
• Note QCD evolution between Q20.85 qnd Q20.25
  small. Can use GRV98
• then add the Aw (w, Q2 ) modulating function
  (to account for interaction with spectator quarks
  at low W)
• Also, check the x1 Elasic Scattering Limit.

Q2 0.07 GeV2
Q2 0.25 GeV2
Q2 0.8 5 GeV2
Q2 1. 4 GeV2
Q2 9 GeV2
Q2 3 GeV2
Q2 1 5 GeV2
Q2 2 5 GeV2
37
Comparison with F2 resonance data SLAC/ Jlab
(These data were not included in this ?w fit)

• The modified LO GRV94 PDFs with a new scaling
  variable, ?w describe the SLAC/Jlab resonance
  data very well (on average).
• Even down to Q2 0.07 GeV2
• Duality works The DIS curve describes the
  average over resonance region
• lt--- ?w fit.
• For now, lets compare to neutrino data and
  photoproduction
• Next repeat with GRV98 and f(x)
• Note QCD evolution between Q20.85 qnd Q20.25
  small. Can use GRV98
• then add the Aw (w, Q2 ) modulating function.
  (to account for interaction with spectator quarks
  at low W)
• Also, check the x1 Elasic Scattering Limit.

Q2 0.07 GeV2
Q2 0.25 GeV2
Q2 0.8 5 GeV2
Q2 1. 4 GeV2
Q2 9 GeV2
Q2 3 GeV2
Q2 1 5 GeV2
Q2 2 5 GeV2
38
Comparison of LOHT to neutrino data on Iron
CCFR (not used in this Xw fit)
Construction

• Apply nuclear corrections using e/m scattering
  data.
• Calculate F2 and xF3 from the modified PDFs with
  Xw
• Use RRworld fit to get 2xF1 from F2
• Implement charm mass effect through a slow
  rescaling algorithm, for F2 2xF1, and XF3

Xw fit
The modified GRV94 LO PDFs with a new scaling
variable, Xw describe the CCFR diff. cross
section data (En30300 GeV) well. Will repeat
with ?w
39
Comparison with photo production data(not
included in this Xw fit)
mb

• s?g-proton) s?? Q20, Xw)
• s? 0.112 mb 2xF1/( KQ2 )
• K depends on definition of virtual photon flux
  for usual definition K 1 - Q2/ 2M? ?
• s? 0.112 mb F2(x, Q2) D(? , Q2) /( KQ2 )
• D (1 Q2/ ? 2 )/(1R)
• F2(x, Q2 ) limit as Q2 --gt0
• Q2/(Q20.188) F2-GRV94 (Xw, Q2 0.24)
• Try R 0
• R Q2/ ? 2 ( evaluated at Q2 0.24)
• R Rw (evaluated at Q2 0.24)
• Note Rw0.034 at Q2 0.24(see appendix R
  data figure)

Xw
The modified LO GRV94 PDFs with a new scaling
variable, Xw also describe photo production data
(Q20) to within 25 To get better agreement at
high ?? 100 GeV (very low Xw), the GRV94 need to
be updated to fit latest HERA data at very low x
and low Q2. So will switch to GRV98 or 2002 LO
PDFs. If we include these photoproduction data
in the fit, we will get C of about 0.22, and
agreement at the few percent level. To evaluate D
(1 Q2/ ? 2 )/(1R) more precisely, we also
need to compare measured Jlab R data in the
Resonance Region at Q2 0.24 to the Rw
parametrization.
40
Comparison with photo production data(not
included in this ? w fit)
mb

• s?g-proton) s?? Q20, Xw)
• s? 0.112 mb 2xF1/( KQ2 )
• K depends on definition of virtual photon flux
  for usual definition K 1 - Q2/ 2M? ?
• s? 0.112 mb F2(x, Q2) D(? , Q2) /( KQ2 )
• D (1 Q2/ ? 2 )/(1R)
• F2(x, Q2 ) limit as Q2 --gt0
  Q2/(Q20.188) F2-GRV94 (? w,
  Q2 0.24)
• Try R 0
• R Q2/ ? 2 ( evaluated at Q2 0.24)
• R Rw (evaluated at Q2 0.24)
• Note Rw0.034 at Q2 0.24 is ver small (see
  appendix R data figure)

? w
The modified LO GRV94 PDFs with a new scaling
variable, ? w also describe photo production data
(Q20) to within 15 To get better agreement at
high ?? 100 GeV (very low ? w ), the GRV94 need
to be updated to fit latest HERA data at very low
x and low Q2. So will switch to GRV98 or 2002 LO
PDFs. If we include these photoproduction data
in the fit, we will get C of about 0.22, and
agreement at the few percent level. To evaluate D
(1 Q2/ ? 2 )/(1R) more precisely, we also
need to compare measured Jlab R data in the
Resonance Region at Q2 0.24 to the Rw
parametrization.
41
Comparison of u quark PDF for GRV94 and CTEQ4L
and CTEQ6L (more modern PDFs)
Q210 GeV2
Q21 GeV2
Q20.5 GeV2
CTEQ6L
CTEQ4L
GRV94
CTEQ4L
CTEQ6L
CTEQ6L
CTEQ4L
GRV94
GRV94
X0.01
X0.0001

• GRV94 LO PDFs need to be updated.at very low x,
  but this is not important in the few GeV region

The GRV LO need to be updated to fit latest HERA
data at very low x and low Q2. We used GRV94
since they are the only PDFs to evolve down to
Q20.24 GeV2 . All other PDFs (LO) e.g. GRV98
stop at 1 GeV2 or 0.5 GeV2. Now it looks like we
can freeze at Q20.8 and have no problems. So
switch to modern PDFs.
42
Summary

• Our modified GRV94 LO PDFs with a modified
  scaling variables, Xw and ?w describe all
  SLAC/BCDMS/NMC DIS data. (We will investigate
  further refinements to ?w, and move to more
  modern PDFs GRV98, 2002 MRST and CTTEQ LO PDFs. )
• The modified PDFs also yields the average value
  over the resonance region as expected from
  duality argument, ALL THE WAY TO Q2 0
• Also good agreement with high energy neutrino
  data.
• Therefore, this model should also describe a low
  energy neutrino cross sections reasonably well
• This work is continuing focus on further
  improvement to ?w (although very good already)
  and A(W, Q2) (low W spectator quark modulating
  function).
• What are the further improvement in ?w - Mostly
  to reduce the size of the three free parameters
  a, b, c as more theoretically motivated terms
  are added into the formalism (mostly intellectual
  curiosity, since the model is already good
  enough).

43
Future Work - part 1

• Implement A e/??(W,Q2) resonances into the model
  for F2 with x w scaling.
• For this need to fit all DIS and SLAC and JLAB
  resonance date and Photo-production H and D data
  and CCFR neutrino data.
• Check for local duality between x w scaling curve
  and elastic form factors Ge, Gm in electron
  scattering. - Check method where its
  applicability will break down.
• Check for local duality of x w scaling curve and
  quasielastic form factors GA, GV in quasielastic
  neutrino and antineutrino scattering.- Good
  check on the applicability of the method in
  predicting exclusive production of strange and
  charm hyperons
• Compare our model prediction with the Rein and
  Seghal model for the 1st resonance (in neutrino
  scattering).
• Implement differences between n? and e/?? final
  state resonance masses in terms of A n,n?
  bar?(W,Q2) See Appendix)
• Look at Jlab and SLAC heavy target data for
  possible Q2 dependence of nuclear dependence on
  Iron.
• Implementation for R (and 2xF1) is done exactly -
  use empirical fits to R (agrees with NNLOGP tgt
  mass for Q2gt1) Need to update Rw to include
  Jlab R data in resonance region.
• Compare to low-energy neutrino data (only low
  statistics data, thus new measurements of
  neutrino differential cross sections at low
  energy are important).
• Check other forms of scaling e.g. F2(1 Q2/ n2
  )???n W2 (for low energies)

44
Future Work - part 2

• Investigate different scaling variables for
  different flavor quark masses (u, d, s, uv, dv,
  usea, dsea in initial and final state) for F2. ,
  
• Note x w Q2B / Mn (1 (1Q2/n2) 1/2 )
  A assumes m F m i 0, P2t0
• ?More sophisticated General expression (see
  derivation in Appendix)
• x w Q 2B / Mn (1 (1Q2/n2) 1/2 ) A
  with
• 2Q2 Q2 m F2 - m I 2 ( Q2
  m F2 - m I 2 ) 2 4Q2 (m I 2 P2t) 1/2
• or 2Q2 Q2 m F2 - m I 2 Q4 2 Q2(m
  F2 m I 2 2P2t ) (m F2 - m I 2 ) 2 1/2
  Here B and A account for effects of additional
  ? m2 from NLO and NNLO effects. However, one can
  include P2t, as well as m F , m i as the current
  quark masses (e.g. Charm, production in neutrino
  scattering, strange particle production etc.).
  In x w, B and A account for effective
  massesinitial Pt. When including Pt in the fits,
  constrain the Q2 dependence of Pt to agree with
  the measured mean Pt of Drell Yan data versus Q2.
• Include a floating factor f(x) to change the x
  dependence of the GRV94 PDFs such that they
  provide a good fit all high energy DIS, HERA,
  Drell-Yan, W-asymmetry, CDF Jets etc, for a
  global PDF QCD LO fit to include Pt, quark masses
  A, B for x w scaling and the Q2/(Q2C) factor,
  and A e/??(W,Q2) as a first step towards modern
  PDFs
• Later work with PDF fitters to produce PDFs
  GRV-LO-02-Xsiw, MRST-LO-00-xsiw, CTEQ-LO-02-Xsiw,
  we should good down to Q20, including A (W,Q2),
  A, B, C, Pt, quark masses etc. I.e. fit
  everything all at once.
• Put in fragmentation functions versus W, Q2,
  quark type and nuclear target

45
Future Neutrino Experiments -JHF,NUMI

• Need to know the properties of neutrino
  interactions (both structure functions AND
  detailed final states on nuclear targets (e.g.
  Carbon, Oxygen (Water), Iron).
• Need to understand differences between neutrino
  and electron data for H, D and nuclear effects
  for the structure functions and the final states.
  
• Need to understand neutral current structure
  functions and final states.
• Need to understand implementation of Fermi motion
  for quasielastic scattering and the
  identification of Quasielastic and Inelastic
  processes in neutrino detectors (subject of
  another talk).
• A combined effort in understanding electron,
  muon, photoproduction and neutrino data of all
  these processes within a theoretical framework is
  needed for future precision neutrino oscillations
  experiments in the next decade.

46
Nuclear effects on heavy targets

• F2(iron)/(deuteron)

• F2(deuteron)/(free NP)

What are nuclear effects for F2 versus XF3 what
are they at low Q2 possible differences between
Electron, Neutrino CC and Neutrino NC at low Q2
(Vector dominance effects).
47
Current understanding of R

• We find for R (Q2gt1 GeV2)
• 1. Rw empirical fit works well (down to Q20.35
  GeV2)
• 2. Rqcd (NNLO) tg-tmass also works well (HT
  are small in NNLO)
• 3. Rqcd(NLO) tgt mass HT works well (since HT
  in NLO mimic missing NNLO terms)
• 4. Need to constrain R to zero at Q20.
• gtgt Use Rw for Q2 gt 0.35 GeV2
• For Q2 lt 0.35 use R(x, Q2 )
• 3.207 Q2 / Q4 1) R(x, Q20.35 GeV2 )
• Plan to compare to recent Jlab Data for R in
  the Resonance region at low Q2.

48
Different Scaling variables for u,d,s,c in
initial and u, d, s,c in final states and
valence vs. sea
For further study

• Bodek and Yang hep-ex/0203009 Georgi and
  Politzer Phys. Rev. D14, 1829 (1976)

We Use Xw Q2B / 2Mn A
x Q2B / Q2 Ax Could also try x w
Q2B / Mn (1 (1Q2/n2) 1/2 ) A Or
fitted effective initial and final state quark
masses that mimic higher twist (NLONNLO QCD),
binding effects, final state intractions could
be different for initial and final state u,d,s,
and valence vs sea? Can try ?? x w Q 2B
/ Mn (1 (1Q2/n2) 1/2 ) A
q
x
m I 2
m 2
M

• x Q2m 2 / ( 2Mn) (quark final state m
  mass)
• x Q2/Mn (1 (1Q2/n2) 1/2 (initial proton
  mass)
• x Q2m 2 / Mn (1 (1Q2/n2) 1/2
  combined
• x x Q2m 2 2 / Q2 (Q4 4x2 M2 Q2)
  1/2

• (Pi q)2 Pi2 2qPi q2 m2

In general GP derive for initial quark mass m I
and final mass m bound in a proton of mass
M (at high Q2 these are current quark masses, but
at low Q2maybe constituent masses? )
GP get 2Q2 Q2 m 2 - m I 2 Q4 2
Q2(m 2 m I 2 ) (m 2 - m I 2 ) 2 1/2
x x 2Q2 / Q2 (Q4
4x2 M2 Q2) 1/2 - note masses may depend on
Q2 x Q2 /
Mn (1(1Q2/n2) 1/2 ) (equivalent form)

Can try to models quark masses, binding effects
Higher twist, NLO and NNLO terms- All in terms
of effective initial and final state quark masses
and different target mass M with more complex
form.
49
One Page Derivation In general GP derive for
initial quark mass m I and final mass m bound
in a proton of mass M (neglect quark initial Pt)

• Georgi and Politzer Phys. Rev. D14, 1829
  (1976)- GP
• (Pi q)2 m I 2 2qPi q2 m 2 and q
  (n, q3) in lab
• 2qPi 2 n Pi0 q3 Pi3 Q2 m 2 - m
  I 2 (eq. 1)
• ( note sign since q3 and Pi3 are pointing at
  each other)
• x Pi0 Pi3 /P0 P3 --- frame
  invariant definition
• x Pi0 Pi3 /M ---- In lab
  or Pi0
  Pi3 x M --- in lab
• x Pi0 -Pi3 Pi0 Pi3 Pi0 -Pi3 / M --
  multiplied by Pi0 -Pi3
• Pi0 -Pi3 (Pi0 ) 2 - ( Pi3 ) 2 / M m I 2
  /x M or Pi0
  -Pi3 m I 2 /x M --- in lab
• 
  Get
  2 Pi0 x M m I 2 /x M
• Plug into (eq. 1) and 2
  Pi3 x M - m I 2 /x M
• n x M n m I 2 /x M q3 x M - q3
  m I 2 /x M - Q2 m 2 - m I 2 0
• a b c
• x 2 M 2 (n q3) - x M Q2 m 2 - m I 2
  m I 2 (n- q3) 0 gtgtgt x -b (b 2 - 4ac)
  1/2 / 2a gt solution
• use (n 2- q3 2) q 2 -Q 2 and (n
  q3) n n 1 Q 2/ n 2 1/2 n n 1
  4M2 x2/ Q 2 1/2
• Get x
  Q2 B / Mn (1(1Q2/n2) 1/2 ) A
• 
  x Q2 B / M n (1 1 4M2 x2/ Q 2
  1/2 ) A (equivalent form)

q
Pi Pi0,Pi3,mI
Pf, m
P P0 P3,M

where 2Q2 Q2 m 2 - m I 2 Q4 2 Q2(m
2 m I 2 ) (m 2 - m I 2 ) 2 1/2
or x x 2Q2 2B / Q2 (Q4
4x2 M2 Q2) 1/2 2Ax (equivalent form)
Add B and A account for effects of additional
? m2 from NLO and NNLO effects. (at high Q2
these are current quark masses, but at low Q2
maybe constituent masses?)
50
Derivation for initial quark mass m I and final
mass m bound in a proton of mass M - INCLUDING
Quark INITIAL P2t
q

• Georgi and Politzer Phys. Rev. D14, 1829
  (1976)- GP did not include Pt
  
• (Pi q)2 m I 2 2qPi q2 m 2 and q
  (n, q3) in lab
• 2qPi 2 n Pi0 q3 Pi3 Q2 m 2 - m
  I 2 (eq. 1)
• ( note sign since q3 and Pi3 are pointing at
  each other)
• x Pi0 Pi3 /P0 P3 --- frame
  invariant definition
• x Pi0 Pi3 /M ---- In lab
  or Pi0
  Pi3 x M --- in lab
• x Pi0 -Pi3 Pi0 Pi3 Pi0 -Pi3 / M --
  multiplied by Pi0 -Pi3
• Pi0 -Pi3 (Pi0 ) 2 - ( Pi3 ) 2 / M (m I 2
  P2t) /x M or
  Pi0 -Pi3 (m I 2 P2t) /x M --- in lab
• 
  Get
  2 Pi0 x M (m I 2 P2t) /x M
• Plug into (eq. 1) and 2
  Pi3 x M - (m I 2 P2t) /x M
• n x M n (m I 2 P2t) /x M q3 x M
  - q3 (m I 2 P2t) 2 /x M - Q2 m 2 - m I
  2 0
• a b c
• x 2 M 2 (n q3) - x M Q2 m 2 - m I 2
  (m I 2 P2t) (n- q3) 0 x -b (b 2 -
  4ac) 1/2 / 2a gt solution
• use (n 2- q3 2) q 2 -Q 2 and (n
  q3) n n 1 Q 2/ n 2 1/2 n n 1
  4M2 x2/ Q 2 1/2
• Get x
  Q2 B / Mn (1(1Q2/n2) 1/2 ) A
• 
  

Pi Pi0,Pi3,mI
Pf, m
P P0 P3,M

where 2Q2 Q2 m 2 - m I 2 ( Q2m 2 -
m I 2 ) 2 4Q2 (m I 2 P2t) 1/2 or
2Q2 Q2 m F2 - m I 2 Q4 2 Q2(m F2
m I 2 2P2t ) (m F2 - m I 2 ) 2 1/2 If
mi0 2Q2 Q2 m 2 ( Q2m 2 ) 2
4Q2 (P2t) 1/2
Add B and A account for effects of additional
? m2 from NLO and NNLO effects. (at high Q2
these are current quark masses, but at low Q2
maybe constituent masses?)
51
In general GP derive for initial quark mass m I
and final mass ,mFm bound in a proton of mass
M -- Page 1 (neglect quark initial Pt)
qq3,q0

• x Is the correct variable which is Invariant in
  any frame q3 and P in opposite directions.

• x

PF PF0,PF3,mFm
PF PI0,PI3,mI
P P0 P3,M

52
In general GP derive for initial quark mass m I
and final mass mFm bound in a proton of mass M
-- Page 2 (neglect quark initial Pt)
qq3,q0

• x

PF PF0,PF3,mFm
PF PI0,PI3,mI

• x For the case of non zero mI
• (note P and q3 are opposite)

P P0 P3,M

--------------------------------------------------
--------------------------------------------------
--------------------------------------------------
--------------------------------------------------
--------------------------------------------------
--- Keep terms with mI multiply by xM and
group terms in x qnd x 2 x 2 M 2 (n q3) -
x M Q2 m 2 - m I 2 m I 2 (n- q3) 0
General Equation a b c gt solution
of quadratic equation x -b (b 2 - 4ac) 1/2
/ 2a use (n 2- q3 2) q 2 -Q 2 and (n q3)
n n 1 Q 2/ n 2 1/2 n n 1 4M2
x2/ Q 2 1/2
Get x Q2 B / Mn (1(1Q2/n2) )
1/2 A
x Q2 B / M n (1 1
4M2 x2/ Q 2 1/2) A or
x x 2Q2 2B / Q2 (Q4
4x2 M2 Q2) 1/2 2Ax (equivalent form)
where 2Q2 Q2 m 2 - m I 2 Q4 2 Q2(m
2 m I 2 ) (m 2 - m I 2 ) 2 1/2 (at high
Q2 these are current quark masses, but at low Q2
maybe constituent masses?) Add B and A account
for effects of additional ? m2 from NLO and
NNLO effects.
53
In general GP derive for initial quark mass m I
and final mass mFm bound in a proton of mass M
--- Page 3 - (neglect quark initial Pt)
why did Xw work in 1970
qq3,q0
Plan to try to fit for different initial/final
quark masses for u-I, u-F, d-I,d-F,
s-I,s-F,c-F. and A and B for HT

• x

PF PF0,PF3,mFm
PF PI0,PI3,mI
P P0 P3,M

• Get x Q2 B
  / Mn (1(1Q2/n2) 1/2 ) A
• x
  Q2 B / M n (1 1 4M2 x2/ Q 2
  1/2? A
• or x x 2Q2
  2B / Q2 (Q4 4x2 M2 Q2) 1/2 2Ax
  (equivalent form)
• where 2Q2 Q2 m 2 - m I 2 Q4 2 Q2(m
  2 m I 2 ) (m 2 - m I 2 ) 2 1/2
• Numerator x 2Q2 Special cases for 2Q2
  2B
• m m I 0 2Q2 2B 2 Q2 2B
  current quarks (mi mF 0)
• m m I 2Q2 2B Q2 Q4 4 Q2 m 2
  1/2 2B constituent mass (mi mF 0.3 GeV)
• m I 0 2Q2 2B 2Q2 2 m 2 2B
  final state mass (mi 0,mF charm)
• m 0 2Q2 2B 2Q2 2B -----gt
  initial constituent(mi0.3) final state current
  (mF 0)
• denominator Q2 (Q4 4x2 M2 Q2) 1/2 2Ax
  at large Q2 ---gt 2 Q2 2 M2x2 2Ax
• Or Mn (1(1Q2/n2) 1/2 ) A
  which at small Q2 ---gt 2Mn M2 /x A
• Therefore A M2 x at large Q2 and M2 /x at
  small Q2 -gtAconstant is approximately OK
• versus Xw Q2B / 2Mn A
  x 2Q22B / 2Q2 2Ax
• In future try to include fit using above form
  with floating masses and B and A. Expect A,B to
  be much smaller than for Xw. C is well
  determined if we include photo-production in the
  fit.

54
Initial quark mass m I and final mass mFm
bound in a proton of mass MPage 4
qq3,q0
Plan to try to fit for different intial/final
quark masses for u-I, u-F, d-I,d-F,
s-I,s-F,c-F. and A and B

• x

PF PF0,PF3,mFm
PF PI0,PI3,mI
P P0 P3,M

• At High Q 2, we expect that the initial and final
  state quark masses are the current quark masses
• (e.g. u5 MeV, d9 MeV, s170 MeV, c1.35 GeV,
  b4.4 GeV.
• For massive final state quarks, this is known as
  slow-rescaling
• At low Q 2, Donnachie and Landshoff (Z. Phys. C.
  61, 145 (1994) say that
• The effective final state mass should reflect the
  true threshold conditions as follows
• m2 (Wthreshold) 2 -Mp 2 . Probably not exactly
  true since A(w) should take care of it if target
  mass effects are included. Nonetheless it is
  indicative of the order of the final state
  interaction.
• (This is known as fast rescaling - I.e.
  introducing a ? ?function to account for
  threshold)
• Reaction initial state final state final
  state Wthreshold m2
  m mF1 mF2
• quark qua