Study of Electroweak Interactions at the Energy Frontier
=========================================================

* The knowledge of the Higgs mass has sharpened the predictions of
  these EWPOs such that the predictions are a factor of 2-4 more
  precise than the experimental measurements.

* In extensions of the SM, which are associated with the
  electroweak symmetry-breaking sector, these EWPOs usually receive corrections
  due to quantum loops (due to e.g.  supersymmetric particles or
  techni-fermions), or due to effective operators (induced for example in
  strongly-interacting light Higgs models), or due to Kaluza-Klein
  modes in extra-dimensional models.
 
* $M_W$ and $\sin^2\theta_\eff^l$ typically have different sensitivities
  to the sources of new physics. This may be demonstrated by the
  parametrisation of new physics in the gauge boson self-energies in
  terms of the $S$, $T$ and $U$ "oblique" corrections. Fixed values of
  $M_W$ and $\sin^2\theta_\eff^l$ correspond to lines in the $S-T$ plane
  with different slopes. Thus, improved measurements of both EWPOs can
  constrain all of the above sources of new physics in a relatively
  model-independent fashion.

* The current world average $M_W$ has a precision of 15 MeV,
  dominated by the combined Tevatron measurement, which has a precision
  of 16 MeV based on the analysis of partial datasets. CDF and D\O\
  have projected that analyses of the full Tevatron statistics can
  yield a 10 MeV measurement, assuming a factor of two improvement in
  the uncertainty due to parton distribution functions, 
  improvement in the calculation of radiative corrections and improved
  understanding of the trackers and calorimeters.

* Studies based on pseudo-data have demonstrated that measurements
  of boson distributions with the 2011-2012 LHC data may be able to
  improve the PDFs relevant for the $M_W$ measurement by a factor of
  two in the near future, enabling the Tevatron potential for $M_W$ to
  be realized.

* Enormous statistics of $W$ bosons and control samples at the LHC
  offer the prospect of higher $M_W$ precision. Studies based on
  pseudo-data have shown that the PDF uncertainty in $M_W$ is about
  twice as big at the LHC as the Tevatron, due mainly to the larger
  fraction of sea quark-initiated production. Thus, further
  improvement by a factor of 2-3 in the PDFs will be required, beyond
  what is needed for the Tevatron.  Furthermore, additional
  improvements in the QED radiative correction calculations and
  NNLO+NNLL generators for $W$ and $Z$ bosons will likely also be
  required. However, considering the 15-year time scale for the
  ultimate $M_W$ measurement from the LHC, we consider a target
  precision of 5 MeV to be appropriate for the LHC.

* Studies of the M_W measurement at the ILC using the threshold scan
  and final state reconstruction have been updated. The ILC TDR
  assuming 100 fb-1 quotes an uncertainty of 6-7 MeV for MW measured
  at the WW threshold.  The updated study projects that the ILC will
  be able to perform the M_W measurement with a precision of 4.5 and
  2.5 MeV for 100 fb-1 and 480 fb-1 respectively.  These estimates
  assume an electron(positron) polarization of 90%(60%) and beam
  energy calibration using final state reconstruction of Z->mu mu and
  J/Psi etc.

* The circular electron-positron TLEP machine, running at the $WW$
  threshold, can produce very high statistics for the $M_W$
  measurement, and is likely to achieve energy calibration at the
  level of 0.1 MeV using teh resonant depolarization technique.  This
  potential motivates further studies of other systematics achievable
  at TLEP. Given an integrated luminosity that can enable a
  statistical precision of $\sim 0.3$ MeV with four detectors, further
  investigations of related issues are clearly warranted. Assuming a 1
  MeV error in the threshold shape calculation the target uncertainty
  is 1.2 MeV.

* The measurement of $\sin^2 \theta_\eff$ from LEP and SLC have
  averaged to a precision of $16 \times 10^{-5}$, albeit with a $\sim
  3 \sigma$ difference between them. Additional, especially improved,
  measurements will be valuable to shed light on this difference.

* A measurement of $\sin^2\theta_\eff^l$ using the full Tevatron dataset
  is projected with a precision of $41 \times 10^{-5}$. This
  measurement will be interesting to compare with LEP and SLC.

* Compared to the Tevatron, measurement of $\sin^2 \theta_\eff^l$ at the
  LHC is handicapped by a larger sensitivity to PDFs due to the
  dilution of the quark and antiquark directions. As with the $M_W$
  measurement, considerable control of the experimental and production
  model uncertainties will be required. Under the condition that a
  factor of 6-7 improvement on PDFs is achieved (a condition also
  required for the $M_W$ target for the LHC), a projected uncertainty
  on $\sin^2\theta_\eff^l$ of $21 \times 10^{-5}$ is obtained. This
  precision is similar to the current LEP and SLC measurements and is
  valuable before the advent of future lepton colliders.

* Considerably more precise measurements of $\sin^2 \theta_\eff^l$ are
  highly desirable for taking the stringency of the SM tests to the
  next order of magnitude. Such measurements are possible at future
  lepton colliders running on the $Z-$pole such as ILC/GigaZ and TLEP.

* The ILC/GigaZ projection for the precision on $\sin^2 \theta_\eff^l$ is
  $1.3 \times 10^{-5}$ without including beam energy and theory systematics.
  This is a factor of 10 improvement on the current
  world average. The corresponding TLEP target uncertainty is $0.3 \times 10^{-5}$
  when extracted from A_LR at the Z peak. 

* TLEP may have the potential to go beyond ILC/GigaZ in the precision
  on $\sin^2\theta_\eff^l$, which also warrants a detailed study.  The
  target precision on $\sin^2 \theta_\eff^l$ is $0.3 \times 10^{-5}$
  when extracted from A_LR at the Z peak without including beam energy
  and theory systematics.  In principle, the precision at TLEP could
  be high enough that all aspects of EWPOs, both theoretical and
  experimental, need to be revisited.

* These improved measurements and predictions of EWPOs will enable
  stringent tests of the SM and of BSM scenarios. Once a discovery is made EWPOs
  will help to test and predict other aspects of the BSM model.

* Measurements of $M_W$ at the few MeV level, and $\sin^2 \theta_\eff^l$
  at the level of $10^{-5}$, require that the parametric uncertainties
  from $m_{top}, M_Z$ and $\alpha_{had}$ (the contribution to the running
  of $\alpha_{EM}$ from hadronic loops) as well as the missing higher
  order calculations be addressed. It is anticipated that calculations
  in the coming years will reduce the effect of missing higher-order
  calculations to a sub-dominant level.  Parametric uncertainties from
  $m_{top}$ and $\alpha_{had}$, if reduced by a factor of two compared
  to current uncertainties, will prevent them from exceeding the total
  precision on $M_W$ and $\sin^2\theta_\eff^l$. A factor of 3-4 improvement
  would be desirable, to $\delta m_{top} \sim 0.3$ GeV and $\delta
  \alpha_{had} \sim 0.3 \times 10^{-5}$. The LHC may be able to
  achieve $\delta m_{top} \sim 0.5$ GeV but further progress at the LHC
  will likely be limited by theoretical uncertainties in the
  non-perturbative QCD effects associated with translating the
  kinematically-reconstructed $m_{top}$ to the pole mass. 
  Further considerable improvement in parametric and theory uncertainties is needed 
  for the target uncertainties at ILC and TLEP.

* The EFT formulation is not limited to specific models; any high
 energy theory can be reduced to a low-energy EFT and the former will
 specify the values of operator coefficients in the latter. Therefore,
 EFT operators provide a general method of parametrizing the effects
 of new physics at a high scale.

* Studies of vector boson scattering and triboson production have
  become possible, for the first time, at the LHC.

* For the next decade, the LHC will continue to be the facility to
  explore these processes at higher levels of precision.

* The HL-LHC is needed to demonstrate that the Higgs couplings to the
  electroweak vector bosons is an essential component of the
  unitarization mechanism for vector boson scattering. The definite
  proof of unitarization via teh Higgs will be difficult at an
  integrated luminosity of 300 fb$^{-1}$.

* The sensitivity to higher-dimension operators improves by a
  factor of 2-3 with the HL-LHC, in comparison with the 300 fb$^{-1}$
  at the LHC.

* Triboson production is also sensitive probe of higher
  dimension operators, complementary to vector boson scattering. 
  This process becomes rapidly more sensitive with increasing beam energy.

* We expect anomalous quartic couplings induced by dimension eight operators
  to be probed with rapidly increasing sensitivity for higher energy colliders.

* Comparison to a 1 TeV ILC sensitivity shows that the LHC is more
  sensitive to quartic anomalous couplings by an order of magnitude.


Conclusions:

A) The W mass measurement at the LHC is competitive with the ILC at 100 fb-1.

B) For all other EWPOs lepton colliders will make significantly more precise measurements of EWPOs than the LHC. 
This is especially the case for sin^2theta_eff^l where ILC will improve upon LEP/SLC by an order of magnitude.
Prelimary estimates of the TLEP potential find that the current LEP measurement of MZ improves by a factor of 10,
the projected ILC accuracy of MW and sin^2theta_eff^l by a factor of about 3 and 4 respectively.

C) Comparison to a 1 TeV ILC sensitivity shows that the LHC is more
  sensitive to quartic anomalous couplings by an order of magnitude.

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