Our thanks to Robert Coles and Ralph Warfield who provided their photographs to supplement those we took this year.
This year we had a lot more advanced publicity in the media because it was the 70th anniversary, and more people were there than in previous years.
Also 1st Take had a camera there because they intend to update their product "Bath at War". There were TV cameras from ITV, who broadcast a snippet from the service in the evening bulletin of local news; and from the BBC who recorded and broadcast an interview with one of the organisers.
The Mayor of Bath attended but did not take an active part in the service. He was due to speak at the event at the Museum of Bath at Work which would follow the service.
However, the Mayor did mingle with and chat to the the members of the public near him.
.Because of the number of people attending, which meant that some of them were a long way away from the presenters, we used microphones this year.
Brian Vowles, the Chairman of the Project welcomed everybody and gave a few words of introduction.
For copyright reasons we cannot reproduce the text of the poem here.
For those who want to read it for themselves, it can be found in the book "Bath In Quotes".
Prayers were then offered for the victims of the Bath Blitz: for those who suffered yet survived, for those who died, and for their families and friends who mourn them.
The story of Alan Reid.
John Penny told the story of Alan Reid, a schoolboy from London visiting his Grandmother in Bath at the time of the Bath blitz.
Molly is the Head Girl of St Mark's School, and she laid a wreath in memory of the ordinary citizens killed in the Bath Blitz.
St Mark's School kindly donated the wreath.
ATC Cadet Oliver Hemming laid a wreath in memory of those civilians killed in the course of their duties. These included the Police, the Fire Service, the Home Guard among the uniformed civilians, and the Civil Defence Volunteers who provided First Aid, Fire Watching, Air Raid Precautions, Messenger and numerous other services.
Joining Oliver in the respectful salute after the wreath was laid was Army Cadet Toby Symonds.
The second wreath was kindly donated by the Bus Station Florists in Widcombe.
It is understood that St Mark's School also obtained their wreath from the same florist. The fact that the same florist could provide two distinctly different but high quality wreaths is very much appreciated.
Click here for information about the plans for the 2013 ceremony.
Up until this year the Bath Blitz Memorial Project has provided its old A0 sized exhibition material which was originally prepared for the 60th anniversary. Over the intervening 10 years it had become worn, marked, and in some cases outdated. It had also been on occasions a problematic size, being a difficult fit on some display panels, including those available at the Museum of Bath at Work.
Thanks to the Oldfield Ward Councillors who generously offered some funding from their Ward Councillor's Initiative budget, some donations, and a guarantor to cover any residual shortfall (which in the event was not needed), 40 new A1 sized exhibition boards had been designed, printed and laminated (for longevity) especially for the 70th Anniversary. In the exhibition space on the top floor of the Museum of Bath at Work were just 20 of the A1 exhibits, in order to leave room for a seated audience alongside the cabinets of physical relics of the war on loan from their owners.
With the assistance of the staff in the Mayor of Bath's office, the content of the A1 exhibition boards was reduced to A3 which once printed was mounted in the Hudson Room, near the entrance of the Museum.
Thus the complete set of exhibits was available to the public but in two different sizes spread over two display spaces.
These exhibits had been available to all visitors to the Museum since the beginning of April (and had proved popular, as we were informed and the pages of the Museum Visitor's Book confirmed) and therefore the timetable for those arriving at the Museum from the service allowed an interval for refreshments and browsing before the programmed events started.
Peter Blackwood played the piano in the background as those attending took refreshments, enjoyed the exhibits and generally mingled.
Nick Burgess opened the event formally by welcoming everybody and then introducing the Mayor of Bath.
"My apologies for being unable to attend this unique event today. I am sure it is quite an occasion and a fitting memorial to those that lived, and died, during the Bath Blitz.
It's Commemoration has come a long way. When I first did research on the topic, as a callow 6th former from King Edward's School in 1975 for an A Level project, the main records of the Blitz were kept in the Guildhall in a series of what looked like shoe boxes and in the main corridor outside the Records room.
As to actual memorials there were very few. There were a very few plaques: a small one in Queen Square, and references to individual incidents such as at the Gas Works and a few other places, and nothing else. There were still gaps in roads where bombs had fallen - New King Street, Wellsway - and the open ground opposite the Scala supermarket in Oldfield Park was fenced off grass and nothing else.
When my project grew into a book in the early 1980s I concluded that we were in danger of forgetting the whole incident and that the victims deserved better. I am glad to say that my worries proved unfounded. Sterling work by Jim Warren and others has led to the names of the dead being added to an official war memorial, a further memorial in Oldfield Park, annual commemorations and a website that is one of the best that there is devoted to a city's Blitz.
My best wishes to all and congratulations to such an event as this.
The mayor continued with an outline of his Mayor's Theme of Industrial Heritage, and pointed out the connection with the wartime industries and the Bath blitz, and he thanked us for the contribution the Project made to his Heritage Days.
"I know today is the anniversary of the 70 years since the blitz on Bath, and I would have so much liked to have been able to be there to commemorate the event with the remaining survivors and their family, but due to the constraints of work and study here in Finland, this has not been possible.
If you would, please extend my best wishes to all those who remember me and how the project began and the collective effort which has bore its fruit in many good ways bringing healing and reconciliation to many.
I hope to be able to visit Bath again in the future and when I do, I would like to meet if you are free, and also perhaps some of the other project members too.
The audience were treated to a session of Geoff White singing to the accompaniment of Peter Blackwood.
Many old-time favourites were included, among which were Underneath The Arches, Bless 'Em All, Run Rabbit Run, Roll Out The Barrel, White Cliffs Of Dover and several others. Audience participation was inevitable (and encouraged).
Martin Wainwright, the author of the book "The Bath Blitz" was introduced by Brian Vowles.
Martin gave an entertaining and wide ranging speech. It is not possible to reproduce it here, but if we report that it made references to journalism, The Pavilion, the IRA, Lord Asquith, Vera Lynn, Bath's Labour Exchange, Lady Hamilton, the blackout, childhood, Leeds, delphiniums, the Sack Of Bath, the Chronicle, and Cyril Beazer as well as the Bath Blitz, then some impression of the range of ideas can be provided.
Brian Vowles gave a vote of thanks to the organisers and participants.
Project members mingled with and chatted to those enjoying the exhibits until it was time to leave.
Martin left some copies of his book "The Bath Blitz" with the museum for sale to the public.
The Project similarly left copies of its first video, now converted to DVD, with the museum for sale to the public.
The 417 people who died in the Bath Blitz have been remembered in a 70th anniversary service at the city's Royal Victoria Park.
Bombing raids on April 25-27, 1942, left more than 400 people dead and caused terrible damage, with churches and whole streets destroyed.
The annual service, organised by the Bath Blitz Memorial Project, drew not only survivors and descendants of those killed but also representatives from emergency services.
One survivor, Pat Connett, represented the Home Guard at the service. Her father John was among those guarding the water tower at Lansdown in April 1942.
The WRVS was represented by Doreen Williams, the main organiser of the service.
A wreath in memory of all the 390 civilians who died was laid, as well as one to commemorate volunteers.
A poem, "The Bombed City", was read out by Kathryn Pond-Barrett, before prayers and a minute's silence. The Last Post and other musical tributes were played by members of the Bath Spa Band.
People from all generations attended a special memorial service in Royal Victoria Park yesterday to mark the 70th anniversary of the Bath Blitz.
Bombing raids by the German Luftwaffe on the nights of April 25, 26 and 27 1942 left more than 400 people dead and caused terrible damage to the city's Georgian architecture, with churches and whole streets destroyed.
The annual service at the park's war memorial is organised by the Bath Blitz Memorial Project, and brought together not only survivors and descendants of those killed but also representatives from emergency services, including police, firefighters and St John Ambulance.
One survivor of the raids, Pat Connett, represented the Home Guard at the service. Her father John was among those guarding the water tower at Lansdown in April 1942.
The Women's Voluntary Service was represented by Doreen Williams, the main organiser of the service.
Her mother Florence was a WVS member in charge of 200 volunteers running the forces canteen in Bath, where they issued blankets, hot tea and other comforts to bombed-out families.
Mrs Williams's father, Leonard Smith, served as a part-time fireman and was off duty when the raids started, but went out to help his comrades and was killed instantly by a bomb while pumping water at the Circus.
Now in her 80s, she was just 15 when her father became the only firefighter to die in the Bath Blitz.
The civil defence volunteers and messengers were represented at the service by 82-year-old Harry Hemming, from Oldfield Park.
His wife Eileen lost four members of her family when a bomb hit their house at Snow Hill on the Monday of the attacks.
A wreath in memory of all the 390 civilians who died was laid by Molly Crane, head girl of St Mark's School.
In addition, for this special anniversary, there was a second wreath to commemorate the volunteers who lost their lives.
It was laid by Mr Hemming's grandson, 14-year-old Oliver, who is an ATC cadet. He was accompanied by an Army cadet, Toby Symonds.
Oliver said afterwards: "I feel really proud to be representing my family, my grandfather and the air cadets.
I didn't know much about what happened to my family in the bombings until recently and this service has brought it all home to me even more."
Oliver's father Roger added: "I never knew any of my mother's family because they were all killed. It's amazing and fantastic to have three generations of us all here today to remember them."
Heavy showers relented long enough for the service to take place in sunshine.
A poem, 'The Bombed City', was read out by Kathryn Pond-Barrett, before prayers and a minute's silence to remember all 417 people who lost their lives. The Last Post and other musical tributes were played by members of the Bath Spa Band.
Those attending included Wendy Preece, who travelled from Leicester for the occasion.
Her grandfather Seymour Buckingham worked for the Admiralty in Bath and was killed by a bomb falling on his home in Victoria Terrace.
Wendy said: "I was born in 1945, three years after my grandfather's death, so I never knew him. I've visited his grave in Haycombe Cemetery many times, but had never been to the memorial service.
Then I saw on the Bath Blitz Memorial Project website that it was the 70th anniversary service and I just had to come. Today has been really poignant for me."
It is 70 years since a series of terrifying raids were inflicted on the mostly undefended city of Bath.
Today, visitors and residents alike are shocked to learn that Bath was bombed. Although now restored, on close inspection evidence of the devastation can be found – shrapnel scars and scorch marks on heritage buildings and, equally, the areas of "packing case" architecture of the 1960s and 1970s that filled those left derelict as a consequence of post-war building blight.
In March 1942, Britain's Bomber Command, as part of a new policy of "area bombing", achieved its first successful attack on a German city. Their prey, the mostly undefended north German port of Lübeck. Britain had been unable to match the Luftwaffe's strength but until 1942 both sides had attacked legitimate military targets.
Lübeck did manufacture armaments and the port was used for supplying the German armies in Northern Russia. These industrial areas, however, were far from the tightly packed medieval city centre where the RAF offensive resulted in extensive damage. Although Lübeck was a "soft" target, its destruction boosted Allied morale and seriously hurt the German High Command.
As a direct consequence, over the weekend of April 25-27, 1942, Bath suffered three horrifying reprisal raids.
Eighty Luftwaffe planes took off from Nazi-occupied northern France on a moonlit night. As the sirens wailed, few people took cover; even when the first pathfinder flares fell the people of Bath still believed the attack was destined for Bristol. During the first four months of that year, Bristol had been hit almost every night. It would be unfair to suggest that the people of Bath were blasé, but it is understandable that they really did not expect the bombs to fall on them.
The first raid struck just before 11pm on the Saturday and lasted until 1am. Fear of further raids forced people to literally take to the hills, in what was known as "trekking". It is estimated that 10,000 people left the city during and immediately after the raids. Many slept under the stars within close proximity to Bath at Sally in the Woods, and Bathampton Rocks.
Officials, however, required people to stay home. There was a real threat that if houses were unoccupied, fires from incendiaries could take hold rather than being quickly snuffed out and made safe.
In the meantime, enemy aircraft had returned to France; refuelled, rearmed and returned at 4.35am. Bath was still ablaze from the first raid, making it easier for the German bombers to pick out their targets.
The third raid, which only lasted two hours but caused extensive damage, arrived in the early hours of Monday morning. The bombers flew low to drop their high explosives and incendiaries and then returned to rain the streets with machine-gun fire. Their purpose was to terrorise and thereby hamper any attempts to put out the fires or make safe the incendiaries.
Harry Patch, of Combe Down and the last surviving "Tommy" from the First World War, spent the Second World War as a member of the Auxiliary Fire Service. While working on the Wellsway during the blitz, Mr Patch and his crew were machine gunned as they tried to douse the flames created by the illumination flares. He wrote that the raids on Bath were so horrific that they reminded him of his time in Ypres in 1917.
The Bath blitz killed 417 people and injured another 1,000. All loss of life is deplorable, but knowing that at least 50 people were killed when public air-raid shelters were hit is heart breaking.
More than 19,000 buildings were affected, of which 1,100 were seriously damaged or destroyed including 218 of architectural or historic interest. The loss of the Assembly Rooms was recorded as one of Britain's chief cultural wartime losses and evidence of the intense heat can still be seen in the scorched pink stonework surrounding the windows to the left of the entrance. People's homes, places of work and transportation obviously took priority on the repair lists and thus the restoration of the Assembly Rooms was not completed until 1963.
St Bartholomew's in Oldfield Park has the misfortune to be known as the unluckiest church in Bath. In November 1936, the foundation stone for this new church, designed by A Taylor, was laid. It had taken 34 years to raise the money for the scheme, the balance of which was financed with a bank loan.
Fundraising efforts ensured that the mortgage was eventually paid off on March 15, 1942 – St Bartholomew's was finally debt free.
Six weeks later, the church suffered a direct hit from a high explosive during the second raid. The destruction to the church was catastrophic, and very little remained structurally sound. The Luftwaffe targeted churches to undermine civilian morale and, perhaps in defiance, services continued to be held within the ruins of the church.
Along with the attacks on Exeter, York, Canterbury, and Norwich, the raids on Bath have become known as the Baedeker Blitz. The term originated with Baron Gustav Braun von Stumm, deputy head of the German foreign press division. He broadcast that the Luftwaffe had been instructed to attack every building rated highly in the well-known travel guides published by the firm of Karl Baedeker. Although Goebbels was reputedly furious with von Stumm for openly admitting that the Nazi's had adopted a policy of bombing cultural and historic targets, this announcement spread quickly and brought terror to Britain. The cities of the Baedecker raids were never legitimate military targets, but the raids created a momentum of fear because now no where was safe from enemy air attack.
The people of Bath were sitting targets. Survivors believed that Bath was finished and could never recover. A German journalist reported that the Lutfwaffe had truly shown their strength; that the terrible raids of destruction and "fire of annihilation" would go down in history. In reality Bathonions would not allow their spirits to be broken. Through a mixture of stubborn denial, public morale and Government censorship the blitz on Bath was widely forgotten. And, until 2003, there was no city centre memorial to the hundreds of civilians who had lost their lives.
On this the 70th anniversary of the Bath Blitz there are a number of commemorative events taking place including exhibitions at Bath at Work Museum and the Building of Bath Collection. The exhibition at Bath Central Library (April 16-28) will showcase a number of the photographs featured in a new publication from The History Press: Bath in the Blitz: Then and Now by Cathryn Spence and Dan Brown.


text:
\section{Introduction}
\label{sec:intro}

With the Higgs discovery in hand and the Standard Model (SM) field content complete, one of the primary goals of the LHC is
to make precise measurements of SM processes, with the hope of testing its limitations.
As the search for new particles has been unsuccessful as yet, much attention has shifted towards precision measurements of Higgs processes.
In this direction, the SMEFT (SM effective field theory) framework becomes very useful. In the SMEFT, deviations from the SM are parameterized by a tower of $SU(3)\times SU(2)_L\times U(1)_Y$ invariant higher dimension operators containing only SM fields.
The SMEFT is useful because it provides a consistent, gauge-invariant theoretical interpretation of the data, in which higher order corrections can be included, and connects Higgs data with electroweak precision observables at the $Z$ and $W$ poles, diboson data, and top quark measurements.
There have been numerous global fits to LHC and LEP data, yielding limits on the allowed values of the SMEFT coefficients~\cite{Biekotter:2018rhp, Grojean:2018dqj, Ellis:2018gqa, Almeida:2018cld, DiVita:2017eyz}, but thus far all of these fits are totally consistent with the Standard Model, further demonstrating the robustness of the SM.

Ultimately, however, one hopes that the SMEFT is parameterizing some high scale   physics beyond the Standard Model (BSM), and it is of interest to understand how these global fits should be interpreted in this context.
The goal of this work is to consider just such an interpretation in detail for a set of simple benchmark models.
We consider a sample set of UV complete models where the BSM physics contains particles at mass scales above the weak scale, $\Lambda \gg M_Z$.
Each model makes a prediction for the SMEFT coefficients at the high scale, $C_i(\Lambda)$, and typically only a small subset of dimension-6 operators are generated~\cite{deBlas:2017xtg, Henning:2014wua, Bakshi:2018ics}.
As we will see, all of our models predict particular relationships between the different coefficients generated, and we will explore the differences in fitting with these particular patterns as opposed to general values of the coefficients.
We will mostly restrict ourselves to tree-level matching between the BSM physics and the SMEFT at the scale $\Lambda$, but we will also consider the renormalization group running to evolve the coefficients at leading logarithm to the weak scale~\cite{Grojean:2013kd, Chen:2013kfa}, where the predictions can be compared with fits to the data.
Since many of our benchmark models generate operators that are well constrained by electroweak precision observables, including these leading logarithmic effects will significantly change the interpretations of the fits.
These considerations are an important first step in understanding what we can learn about possible UV complete models, and, if a deviation from the Standard Model is observed, how we can discriminate between them. This goal is sometimes referred to as the ``Higgs Inverse Problem''.

Understanding the sensitivity to BSM physics through the extraction of SMEFT coefficients is also useful for comparing the reach of future accelerators~\cite{deBlas:2019rxi} and our calculations are  part of an extended effort to understand the complementarity of the direct observation of new particles with precision measurements~\cite{Gupta:2012mi}.
In addition to Higgs signal strength data, we include theory predictions with NLO QCD corrections for $VV$ and $VH$ ($V = W, Z$)
production~\cite{Baglio:2020oqu}, and the leading logarithmic NLO QCD and electroweak  corrections to $Z$ and $W$ pole observables computed in the full SMEFT~\cite{Dawson:2019clf}.
As a by-product of our study, we obtain an update of the global fit to SMEFT coefficients in the Warsaw basis.

We begin by describing how we match our benchmark models to the SMEFT, including the effects of
renormalization group evolution  (RGE) of the Wilson coefficients down to the weak scale. We then summarize each of the models in turn in Section~\ref{sec:modelsConsidered}. The operators generated in each model are summarized in Section~\ref{sec:summary}.
We then perform a series of fits customized for each model in Section~\ref{sec:results}, and discuss how the SMEFT fit results can differ depending on the correlations present in the underlying high scale model.
Here we will further emphasize the role of the renormalization group evolution of the coefficients in the interpretation of SMEFT fit results.
Finally, we conclude with our updated global fit  in Section~\ref{sec:global} and a discussion of future directions in this type of study in Section~\ref{sec:conc}.

\section{Matching Models to the SMEFT}
\label{sec:models}

In this section, we lay out our benchmark models, and tabulate the relevant SMEFT coefficients obtained in the decoupling limit of these models. The models are chosen to be simple but representative BSM models with new particles at the $\sim$ few TeV scale with only a small set of unknown parameters.
The full set of dimension-6 operators that we will  consider, ignoring flavor,  is given in Table~\ref{tab:opdef}.

\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c||c|c||c|c|}
\hline
${{\cal O}_{ll}}$                   & $(\bar l_L \gamma_\mu l_L)(\bar l_L \gamma^\mu l)_L$  &
  ${{\cal O}}_{H W B}$
 &$ (\vp^\dag \tau^a \vp)\, W^a_{\mu\nu} B^{\mu\nu}$  &
${\cal O}_{\vp D}$   & $\left(\vp^\dag D^\mu\vp\right)^* \left(\vp^\dag D_\mu\vp\right)$
\\
\hline
   ${{\cal O}}_{ He}$  &   $(\vpj) (\overline {e}_R\gamma^\mu e_R)$  &
   ${{\cal O}}_{H u}$ & $(\vpj) (\overline {u}_R\gamma^\mu u_R)$ &
  ${{\cal O}}_{H d}$
       & $(\vpj) (\overline {d}_R\gamma^\mu d_R)$
  \\ \hline
             ${{\cal O}}_{ Hq}^{(3)}$ & $(\vpjt)(\bar q_L \tau^a \gamma^\mu q_L)$  &${{\cal O}}_{H q}^{(1)}$
      &$(\vpj)(\bar q_L \gamma^\mu q_L)$  &
 ${{\cal O}_{\vp l}^{(3)}}$      & $(\vpjt)(\bar l_L \tau^a \gamma^\mu l_L)$
 \\
\hline
${{\cal O}}_{H l}^{(1)}$ &
$(\vpj)(\bar l_L  \gamma^\mu l_L)$ &
${{\cal O}}_{H\square}$ & $(H^\dagger H)\square (H^\dagger H)$  &
${{\cal O}}_{eH}$ & $(H^\dagger H) \bar l_L {\tilde H} e_R$
\\
\hline
${{\cal O}}_{HG}$ & $(H^\dagger H) G_{\mu\nu}^A G^{\mu\nu,A}$  &
${{\cal O}}_{uH}$ &   $(H^\dagger H)({\overline q}_L{\tilde {H}}u_R)$
 &
${{\cal O}}_{dH}$ & $(H^\dagger H)({\overline q}_L{ {H}}d_R)$
\\
\hline
${{\cal O}}_{HB}$ & $(H^\dagger H) B_{\mu\nu} B^{\mu\nu}$ &
${{\cal O}}_{HW}$  & $(H^\dagger H) W_{\mu\nu}^a W^{\mu\nu,a}$  &
${{\cal O}_W}$  & $\epsilon_{abc}W_\mu^{\nu,a}W_\nu^{\rho,b}W_\rho^{\mu,c}$
\\
\hline
${{\cal O}}_H$ & $(H^\dagger H)^3$ &
& &
 &
\\
\hline
\end{tabular}'
\caption{Dimension-6 operators in the Warsaw basis that are considered in this study.  The fermion labels $u, d, e$ refer generically  to all $3$ generations and similarly for the quark, $q$, and lepton, $l$, doublets.\label{tab:opdef}}
\end{table}

\subsection{Matching Procedure and Renormalization Group}
\label{sec:matching}

Here we lay out our benchmark models, and tabulate the relevant SMEFT coefficients obtained in
the decoupling limits of the models.  The models are chosen to be well studied  BSM models with new particles at
the UV scale and are quite simple models, involving a small set of unknown parameters.
We assume that the new particles are at a high mass scale and integrate them out of the theory using the equations of motion at tree level, although for future studies it would be of interest to perform the matching at one-loop, since  one-loop matching typically generates a much
richer spectrum of SMEFT operators than does the tree level matching~\cite{Gorbahn:2015gxa, Henning:2014wua, Jiang:2018pbd}.
This  procedure generates predictions for SMEFT coefficients at the mass scale $\Lambda$ corresponding to the new physics,
\begin{equation}
\mathcal{L} = \mathcal{L}_{SM}+\sum_i {C_i\over \Lambda^2}{\cal O}_i^6+ \dots\, ,
\end{equation}
where we include the dimension-6 operators in the Warsaw basis~\cite{Dedes:2017zog, Grzadkowski:2010es}.   The operators
consist of all  of the $SU(3)\times SU(2)_L\times U(1)_Y$ operators that can be constructed from SM fields.
Since we assume $\Lambda \gg M_Z$, we only consider the dimension-6 operators, $O_i^6$.
Some of the
models we consider generate effects only for third generation quarks, so we do not always assume flavor universality in the
quark sector.
The importance of the assumptions about flavor in the results of the global fits has been emphasized in Refs.~\cite{Falkowski:2019hvp, Grojean:2018dqj}.

We fit data from Higgs processes, diboson $WW$ and $WZ$  production, and electroweak precision observables (EWPOs), including
the $W$ mass and width, to the
patterns of SMEFT coefficients generated in our examples.
For completeness, we define the operators appearing in this work in Table~\ref{tab:opdef} and we neglect flavor indices in this table, although we will include them in some of the models.  We define $H$ to be the $SU(2)_L$ doublet Higgs field with neutral component ${h+v\over\sqrt{2}}$ and,  in terms of the first generation,
$q_L^T=(u_L, d_L)$, $l_L^T=(\nu_L, e_L)$.   Our notation follows that of Ref.~\cite{Dedes:2017zog}.

At tree level, the $Z$ and $W$ pole observables depend on,
\begin{equation}
{{\cal O}_{ll}},\quad
  {{\cal O}}_{H W B}, \quad
{\cal O}_{\vp D},\quad
 {{\cal O}}_{H e},\quad
  {{\cal O}}_{H d}, \quad
  {{\cal O}}_{H u},\
  {{\cal O}}_{H q}^{(3)}, \quad
  {{\cal O}}_{H  q}^{(1)}, \quad
 {{\cal O}_{\vp l}^{(3)}}, \quad
 {{\cal O}}_{H  l}^{(1)},
\end{equation}
and the EWPOs are sensitive to eight combinations of these operators~\cite{Dawson:2019clf, Corbett:2017qgl, Falkowski:2014tna, Berthier:2016tkq}, (at NLO they are sensitive to a combination
of $10$ operators).  We  also include the $2-$ loop contribution to $M_W$ generated by ${\cal O}_H$~\cite{Degrassi:2014sxa, Kribs:2017znd}.

We consider tree level contributions from the  following operators to Higgs data,
\begin{eqnarray}
{\cal O}_{HW},\quad
{\cal O}_{HB},\quad
{\cal O}_{HWB},\quad
{\cal O}_{HD},\quad
{\cal O}_{HG},\quad
{\cal O}_{H\square},\quad
{\cal O}_{Hl}^{(3)},\quad
{\cal O}_{He},\quad  \nonumber \\
{\cal O}_{Hu},\quad
{\cal O}_{Hd},\quad
{\cal O}_{ll},\quad
{\cal O}_{Hq}^{(1)}, \quad
{\cal O}_{Hq}^{(3)},\quad
{\cal O}_{eH},\quad
{\cal O}_{uH},\quad
{\cal O}_{dH}\, .
\end{eqnarray}
We also include the loop contributions to Higgs production and decay from ${\cal O}_H$~\cite{Degrassi:2016wml, Gorbahn:2016uoy, Bizon:2016wgr}.

Finally, the diboson $WW$ and $WZ$ data depend on $7$ effective couplings\footnote{See~\cite{Baglio:2018bkm} for
a convenient mapping from the effective interactions to the Warsaw basis.},
which involve the operators
\begin{equation}
{\cal O}_{HWB},\quad
{\cal O}_{HD},\quad
{\cal O}_W,\quad
{\cal O}_{Hl}^{(3)},\quad
{\cal O}_{ll},\quad
{\cal O}_{Hq}^{(1)},\quad
{\cal O}_{Hq}^{(3)},\quad
{{\cal O}}_{Hu},\quad
{\cal O}_{Hd} ,
\end{equation}
for a total of 19 operators involved in our study when we neglect flavor. For the vector-like quark models we consider,
only contributions to operators involving third generation quark interactions  are generated.  Most of the operators that contribute to the global fits  that we list above do not arise in
 the models we consider and we comment on the importance of this in  the concluding discussion.

The models we consider are chosen to illustrate various types of new physics and to demonstrate uncertainties and assumptions that are made when forming physics conclusions from the global fits.
The models fall in two categories.
\begin{itemize}
\item{Models with high scale scalar resonances:
We consider a real scalar  singlet model, both with and without a $Z_2$ symmetry, and a $2$ Higgs doublet model (2HDM)  in the decoupling limit.}

\item{Models with new particles in loops:
We consider two  models with vector-like quarks (VLQs): One with a color triplet fermion with charge $Q={2\over 3}$, and one with a color triplet,
$SU(2)_L$ doublet of quarks with
charge $Q=({2\over 3}, -{1\over 3})$.
We also briefly compare the results of the models with  vector-like quarks with  a model containing
a heavy color triplet scalar.}

\end{itemize}
There have been extensive studies in the literature computing SMEFT coefficients in these models.
We summarize the models we consider below and the reader is referred to the original literature for further details.

The fits are performed in two different manners.  In the first approach, we match the coefficients at the UV scale $\Lambda$  to the model predictions and perform the global fit.
These fits are only sensitive to the ratios ${C_i\over \Lambda^2}$ and give no independent information about the UV scale.
We always make the identification that $\Lambda$ is the mass of the heavy particle that has been integrated out.
In the second set of fits, we match the coefficients to the model predictions at $\Lambda$ and then use the renormalization group to evolve the coefficients to $M_Z$ before performing the fits.
The coefficients at the weak scale are then,
\begin{equation}
C_i(M_Z)=C_i(\Lambda) -{1\over 16\pi^2}{\dot C}_i\log\biggl({\Lambda\over M_Z}\biggr)\, , ~{\dot C}_i\equiv \gamma_{ij}C_j\, .
\end{equation}
A complete set of the relevant anomalous dimensions  in the Warsaw basis is in Refs.~\cite{Jenkins:2013zja,Jenkins:2013wua,Alonso:2013hga}.
In many cases, the RGE has a dramatic effect on the interpretation of the fits.
The EWPO and diboson data place strong constraints on ${\cal O}_{HD}$, ${\cal O}_{H\square}$,  and ${\cal O}_{HWB}$.
The contributions involving the first and second generation quarks to ${\cal O}_{Hq}^{(3)}$, and ${\cal O}_{Hq}^{(1)}$
are strongly constrained by Higgstrahlung data, while the $Z\rightarrow b {\overline b}$ data constrain these operators with $3^{rd}$ generation quarks.  In models where these coefficients are generated by RGE (even when they are not present at the matching scale), the constraints and the interpretations change dramatically~\cite{Chen:2013kfa}.

To illustrate the importance of including the RGE when fitting to UV complete models, we consider the
strongly constrained operators,  ${\cal O}_{HD}$, ${\cal O}_{H\square}$,   ${\cal O}_{Hq}^{(1)}$ and ${\cal O}_{Hq}^{(3)}$ ~\cite{Baglio:2020oqu}.
${\cal O}_{HD}$ is not generated at tree level in any of the models we consider, while  ${\cal O}_{H\square}$ arises at tree level in the singlet model.
Including only contributions from terms that are generated at tree level in the models we consider (see Table~\ref{tab:sumlag}),
\begin{eqnarray}
{\dot C}_{HD} &=&
{8\over 3}g^{\prime~2}
\biggl[2C_{Ht}-C_{Hb} +\big(C_{Hq}^{(1)}\big)_{33} \biggr] +{20\over 3}g^{\prime ~2}C_{H\square} \nonumber \\
&& -24\biggl[Y_t^2C_{Ht}-Y_b^2C_{Hb}+Y_bY_tC_{Htb}\biggr]\nonumber \\
&&+24\biggl(Y_t^2-Y_b^2\biggr)\big(C_{Hq}^{(1)}\big)_{33}  \nonumber \\
{\dot C_{H\square}}
&=&
6 g^2 \big(C_{Hq}^{(3)}\big)_{33} +{2\over 3}g^{\prime~2}\biggl[2C_{Ht}-C_{Hb}+\big(C_{Hq}^{(1)}\big)_{33} \biggr]
\nonumber \\
&&+\biggl[
-{4\over 3}g^{\prime~2}-4g^2+12\biggl(Y_t^2+Y_b^2\biggr)+4Y_\tau^2\biggr] C_{H\square}
\nonumber \\ &&
- 6\biggl[(Y_b^2-Y_t^2)\big(C_{Hq}^{(1)}\big)_{33}
 +3(Y_b^2+Y_t^2)\big(C_{Hq}^{(3)}\big)_{33}+Y_t^2 C_{Ht}-Y_b^2 C_{Hb}-2Y_bY_tC_{Htb}\biggr]\nonumber \\
\nonumber \\
\big({\dot C}_{Hq}^{(3)}\big)_{33}&=&
3 \biggl[ Y_b^2 - Y_t^2 \biggr] \big(C_{Hq}^{(1)}\big)_{33}
+ \biggl[ -{11\over 3} g^2 + 8 Y_t^2 + 8 Y_b^2 + 2 Y_\tau^2  \biggr] \big(C_{Hq}^{(3)}\big)_{33} \nonumber \\ &&
-{1\over 6}\biggl[3 Y_t^2 + 3Y_b^2 -g^2 \biggr] C_{H\square}
\nonumber \\
\big({\dot C}_{Hq}^{(1)}\big)_{33}
&=&
 \biggl[ {5 \over 9}g^{\prime ~2}  + 10 Y_t^2 + 10 Y_b^2 + 2 Y_\tau^2  \biggr] \big(C_{Hq}^{(1)}\big)_{33}
- 9 \bigg[Y_t^2-Y_b^2\bigg] \big(C_{Hq}^{(3)}\big)_{33}
\nonumber \\ &&
-{1\over 2}\biggl[{ g^{\prime ~2} \over 9}+ Y_b^2 - Y_t^2 \biggr] C_{H\square}
-(Y_t^2 + {4\over9}g^{\prime ~2} ) C_{Ht}
-(Y_b^2 + {2\over9}g^{\prime ~2} ) C_{Hb} \, ,
\label{eq:regsum}
\end{eqnarray}
where $C_{Ht}, C_{Hb}, C_{Htb}, C_{H\tau},   (C_{Hq}^{(1)})_{33}$ and $(C_{Hq}^{(3)})_{33}$ correspond to operators including only the third generation fermions.
See~\cite{Jenkins:2013zja,Jenkins:2013wua,Alonso:2013hga} for a full expression.
We note that ${\cal O}_{HD}$ arises from the RGE in the $(TB)$ VLQ model and in  the singlet model, and so we expect  quite different results in these models when  the RGE is included.  Similarly,
$({\cal O}_{Hq}^{(1)})_{33}$ and $({\cal O}_{HQ}^{(3)})_{33} $ occur at tree level in the $T$ VLQ model, and RGE generates ${\cal O}_{H\square}$ at the weak scale which is strongly constrained.  On the other hand, the operators generated at tree level  in the 2HDM do not contribute to the RG evolution of  $C_{HD}$ or $C_{H\square}$ and we will see that RGE has a relatively minor effect on the interpretation of this model.  In our numerical
results, we include the complete RGE of all the operators that contribute to our fits.

\subsection{Models}
\label{sec:modelsConsidered}

\paragraph{Singlet Scalars\\}
One of the simplest extensions to the Standard Model is obtained by adding an additional scalar that is a  singlet under the SM gauge group.
The scalar  potential can be constructed  both with and without a $Z_2$ symmetry.
The case without a $Z_2$ symmetry  is particularly interesting because it can accommodate a first order electroweak phase transition for some values of the parameters~\cite{Profumo:2014opa,Chen:2017qcz}.
Using the  classical equations of motion~\cite{Henning:2014wua,deBlas:2014mba,Gorbahn:2015gxa,Dawson:2017vgm},
the heavy scalar can be integrated out at tree level, generating  the SMEFT operators,
${\cal O}_H$ and ${\cal O}_{H\square}$ with coefficients,
\begin{eqnarray}
 {v^2\over \Lambda^2} C_{H\square}&=& -{1\over 2}\tan^2\theta\, ,\nonumber \\
 C_H &=& -C_{H\square}\biggl(\tan\theta{m\over 3 v}-\kappa\biggr)\, ,
\end{eqnarray}
where $\theta$ is the mixing angle between the SM-like Higgs boson, $h$, and the new heavy scalar and $\kappa$ and $m$ are Lagrangian parameters that are limited by the requirement that the electroweak minimum be the lowest minimum of the potential~\cite{Chen:2014ask}.
  In this model, the SM-like Higgs couplings to SM particles are uniformly suppressed by  a factor of $\cos\theta$ and
for the case with a $Z_2$ symmetry, there is a cancellation implying $C_H=0$~\cite{Dawson:2017vgm,Gorbahn:2015gxa}.
Details of the model are in Appendix~\ref{sec:scalars}.\\

\paragraph{A Second Higgs Doublet\\}
The 2 Higgs doublet model (2HDM)  has been extensively studied in the literature, and in the limit that the new Higgs bosons are
much heavier than the SM-like Higgs bosons, the Higgs couplings approach those of the SM~\cite{Gunion:2002zf}.   This is the alignment
limit, $\cos(\beta-\alpha)\rightarrow 0$.  In the exact alignment limit, SMEFT operators are not generated at tree level, but
first appear at 1-loop.  Away from the alignment limit, $\cos(\beta-\alpha) \ll 0$, tree level contributions to the Higgs-Yukawa
couplings are generated, along with a correction to the Higgs tri-linear coupling.
To linear
order in $\cos(\beta-\alpha)$, the SMEFT coefficients that affect  Higgs couplings
to fermions, $f$, are~\cite{Gorbahn:2015gxa,Brehmer:2015rna,Belusca-Maito:2016dqe,deBlas:2017xtg},
\begin{equation}
 {v^2 C_{fH}\over \Lambda^2} =-Y_f\eta_f{\cos(\beta-\alpha)\over\tan\beta} \, ,
\end{equation}
where $Y_f=\sqrt{2}m_f/v$ and $\eta_f$ distinguishes the type of 2HDM and is defined in Appendix~\ref{sec:2hdm}.
The corrections to the $WWh$ vertex  are ${\cal{O}}[\cos^2(\beta-\alpha)]$ and are neglected in
our approximation, since they are formally of dimension-8.
The correction to the ${\cal O}_H$ operator scales slightly differently,
\begin{equation}
{v^2 C_H\over \Lambda^2}={\cos^2(\beta-\alpha) M^2\over v^2}\, ,
\end{equation}
where $M$ is the common mass of the heavy decoupled scalars near the alignment limit.
We stress that  our results are only valid near  the alignment limit, where $\cos(\beta-\alpha) \ll 1$and terms of
${\cal  O}[\cos^2(\beta-\alpha)]$ are systematically neglected.
Further details of the model are found in Appendix~\ref{sec:2hdm}.\\

\paragraph{Colored Extensions of the SM: an $SU(3)$ Triplet, $SU(2)_L$ Singlet Fermion\\}
We consider a charge $Q={2\over 3}$ color triplet, $SU(2)_L$ singlet fermion, $T$, and call this the $T$ VLQ  (vector-like quark) model
and  assume that this new quark only couples to the top quark, but not to the lighter quarks.  The model is parameterized by
$3$ parameters:  $m_t$ and $M_T$ are the masses of the  physical top and  new heavy charge  $Q={2\over 3}$  fermion respectively, and $s_L^t$ is the sine of a mixing angle
that defines the mixing between the  left-handed charge ${2\over 3}$ quarks.
Integrating out the heavy fermion generates the SMEFT coefficients involving the third generation quarks only~\cite{Chen:2017hak,Dawson:2012di,delAguila:2000aa,deBlas:2017xtg,Chen:2014xwa}
\begin{eqnarray}
{v^2\over \Lambda^2} \big(C_{Hq}^{(1)}\big)_{33}&=& - {v^2\over \Lambda^2}\big(C_{Hq}^{(3)}\big)_{33}=
\biggl({1\over 2 Y_t} \biggr){v^2\over \Lambda^2}C_{tH} = {(s_L^t)^2\over 2}
\label{eq:topsing}
\end{eqnarray}
where
\begin{eqnarray}
\big({\cal O}_{Hq}^{(1)}\big)_{33}&=& (\vpj)(\bar q_L^3  \gamma^\mu q_L^3)  \nonumber \\
\big({\cal O}_{Hq}^{(3)}\big)_{33}&=& (\vpjt)(\bar q_L^3 \tau^a \gamma^\mu q_L^3)  \nonumber\\
{\cal O}_{tH}&=& (H^\dagger H)({\overline q}_L^3{\tilde {H}}t_R)\, ,
\end{eqnarray}
 ${\overline q}_L^3=({\overline t}_L, {\overline b}_L)$, and the scale $\Lambda$ is identified with $M_T$.
The corresponding coefficients for the first 2 generations are zero in this model.

Although we perform the matching at tree level, we also include ${\cal O}_{HG}$ since it could potentially make a significant
contribution to Higgs production through gluon fusion,
\begin{equation}\label{eq:vlqt_chg}
{v^2\over \Lambda^2} C_{HG}={\alpha_s\over 8\pi}(s_L^t)^2\biggl(F_{1/2}(M_T)-F_{1/2}(m_t)\biggr)\sim {\cal{O}}\biggl({\alpha_s (s_L^t)^2m_t^2\over M_T^2}\biggr)\,,
\end{equation}
where $F_{1/2}$ is defined in Appendix~\ref{sec:vlqsing}.
For $M_T=1\,\textrm{TeV} ~(2\,\textrm{TeV}) $,  $F_{1/2}(M_T)-F_{1/2}(m_t)\sim- .010~ (-.011)$ and the top and $T$ contributions approximately cancel in
the large $m_t$ limit. A summary of the model is in Appendix~\ref{sec:vlqsing}.\\

\paragraph{Colored Extensions of the SM: an $SU(3)$ Triplet, $SU(2)_L$ Doublet Fermion\\}
We next consider a model with an $SU(2)_L$ doublet, color triplet pair of vector-like fermions.
We term this the $(TB) $ VLQ model.
At tree level, the $(TB)$ doublet generates $O_{Ht},~O_{Hb},~O_{Htb},~O_{tH},$ and $O_{bH}$, where the operators are defined in Table~\ref{tab:opdef} in terms of $3^{rd}$ generation fermions. In the decoupling limit, $M_T, M_B \gg M_Z$,
\begin{eqnarray}
 {v^2\over \Lambda^2}C_{Ht}&=&- {v^2\over \Lambda^2}{C_{tH}\over Y_t}=-(s_R^t)^2\nonumber \\
 {v^2\over \Lambda^2} C_{Hb}&=&{v^2\over \Lambda^2}{C_{bH}\over Y_b}=(s_R^b)^2 \nonumber \\
  {v^2\over \Lambda^2} C_{Htb}&=& 2s_R^t s_R^b,
\end{eqnarray}
where $s_R^t$ and $s_R^b$ define the   mixing between  the top and bottom quarks with T and B respectively in the right-handed sector.
At one-loop, ${\cal O}_{HG}$ is also generated,
\begin{eqnarray}\label{eq:vlqtb_chg}
 {v^2\over \Lambda^2}  C_{HG}&=&{\alpha_s\over 8 \pi} \biggl((s_R^t)^2\biggl[F_{1/2}(M_T)-F_{1/2}(m_t)\biggr]+(s_R^b)^2F_{1/2}(M_B)\biggr)
 \nonumber \\
 &\sim &{\alpha_s\over 8 \pi}(s_R^b)^2 (.32)\, ,
\end{eqnarray}
where $M_T=M_B=1\,\textrm{TeV}$ in the last equation.    The approximate cancellation between the $t$ and $T$
contributions found in the $T$ VLQ model remains, but there is an additional contribution from $M_B$.
Details can be found in Appendix~\ref{sec:vlqdoub}.\\

\paragraph{Colored Extensions of the SM: an $SU(3)$ Triplet Scalar\\}
Finally, we consider a model with a color triplet complex scalar, $s$,  with charge $Q={2\over 3}$ and mass, $m_s$. At tree level, four fermion operators that
do not contribute to our global fit are generated, but no dimension-6 EFT operators arise.
At one loop, the colored scalar generates ${\cal O}_{HG}$,
\begin{equation}
{v^2C_{HG}\over \Lambda^2}= -{\alpha_s\kappa v^2 \over 96 \pi m_s^2}\, ,
\end{equation}
where  $\kappa$ is the portal coupling,
$(s\,s^*)\big( H^{\dagger}H\big)$,
and is defined in Appendix~\ref{sec:colorscalar}.
This is an example of a model where the only effect on single Higgs production is to rescale the rate and the SMEFT formalism is not necessary.
The indirect consequences  of the colored scalar and the corresponding SMEFT effects can be searched for in Higgs plus jet or double Higgs production~\cite{Dawson:2015oha,Dawson:2015gka}.

\newpage
\subsection{Summary of Models}
\label{sec:summary}

In Table~\ref{tab:sumlag}, we summarize the coefficients generated by our benchmark models as described in the previous sections, expressing the results in terms of the physical parameters of these models when possible.
The scale $\Lambda$ is consistently identified with the mass of the heavy particle in the model.
More precise definitions of these parameters are given in the Appendices.

\begin{table}[h]
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{| c | c | c | c | c | c | c | }
\hline
& ~${\text{Singlet}}_{\slashed{\mathbb{Z}}_2}$~
&~${\text{Singlet}}_{\mathbb{Z}_2}$~
&~2HDM~
&~T VLQ~
&~(TB) VLQ~
&~~~~~$s$~~~~~
\\
\hline\hline
 ${v^2 C_H  \over \Lambda^2}$ &
${\tan^2\theta\over 2}(\tan\theta{m\over 3 v}-\kappa)$ &
&
${\cos^2(\beta-\alpha) M^2\over v^2}$ &
&
&
\\
\hline
  ${v^2 C_{H\Box} \over \Lambda^2}$ &
  $- {\tan^2\theta\over 2}$   &
  $- {\tan^2\theta\over 2}$ &
  &
  &
  &
  \\
  \hline
 ${v^2 C_{bH}\over \Lambda^2}$ &
&
&
$-Y_b\eta_b{\cos(\beta-\alpha)\over \tan\beta} $&
&
$ Y_b (s_R^b)^2$ &
\\
\hline
${v^2 C_{tH}\over \Lambda^2}$
&
&
& - $Y_t\eta_t{\cos(\beta-\alpha)\over \tan\beta} $
& ${Y_t (s_L^t)^2 }$
& $Y_t (s_R^t)^2$
&
\\
\hline
${v^2 C_{\tau H}\over \Lambda^2}$ &
&
&
$-Y_\tau\eta_\tau{\cos(\beta-\alpha)\over \tan\beta} $&
&
 &
\\
\hline
 ${v^2 (C_{Hq}^{(1)})_{33}\over \Lambda^2}$
&
&
&
&
${(s_L^t)^2 \over 2 } $
&
&
\\
\hline
 ${v^2 (C_{Hq}^{(3)})_{33}\over \Lambda^2}$
 &
 &
 &
 & $- {(s_L^t)^2\over 2} $
 &
 &
 \\
 \hline
 ${v^2 C_{Hb}\over \Lambda^2}$
 &
 &
&
&
& ${ ( s_R^b)^2 } $
 &
 \\
 \hline
${v^2 C_{Ht}\over \Lambda^2}$
&
&
&
&
&
$- {(s_R^t)^2 } $
&
\\
\hline
 ${v^2 C_{Htb}\over \Lambda^2}$
&
&
&
&
&
$2{s_R^t s_R^b}  $
&
\\
\hline
${v^2 C_{HG}\over \Lambda^2}$
&
&
&
&
$-{\alpha_s (s_L^t)^2\over 8 \pi }(.02)$
&${\alpha_s (s_R^b)^2\over 8 \pi }(.65)$
& $ -{\alpha_s \kappa v^2 \over 96\pi m_s^2}$ \\
\hline
\end{tabular}
\caption{Tree level SMEFT coefficients.
We also list $C_{HG}$ which is generated at 1-loop in some models and give numerical values for $C_{HG}$ for heavy masses of $1\,\textrm{TeV}$.
In all cases, we assume the decoupling limit of the models and the parameters are defined in the appendices.
Empty spaces correspond to operators not generated at tree level.
\label{tab:sumlag} }
\end{table}

\newpage
\section{Results}
\label{sec:results}

\subsection{Methodology}
\label{sec:direct}

We perform a series of fits to Higgs, diboson, and EWPO data with prior assumptions about the relationships
between SMEFT coefficients that are motivated by our example models.
We take as non-zero only those coefficients generated in a particular model and examine how that choice changes the fits and the interpretations of the fit results.
The underlying goal is to see how the fits can potentially constrain the high scale models.
All of our fits are performed to linear order in the SMEFT coefficients.

The EWPO fits use the data given in Table III of~\cite{Dawson:2019clf}.
This table uses the most accurate SM theoretical predictions available, typically at NNLO accuracy,  along with the tree level SMEFT contributions.
The LEP-2 data from $e^+e^-\rightarrow W^+W^-$ as parameterized in Ref.~\cite{Ellis:2018gqa} is included in the global fits.
Using the results in the appendix of Ref.~\cite{Dawson:2019clf}, it is straightforward to generalize the leading order SMEFT contributions to the $\chi^2$ given there to allow for the quark operators to have different (but diagonal) couplings to the generations.
In particular the coefficients involving the $b$ quark can be different from those of the lighter fermions, which is relevant for the $T$ and ($TB$) VLQ models.
We assume that the lepton generations all couple identically.
The contribution of the operator ${\cal O}_H$ occurs at 2-loops in the SM and is included using the results from Ref.~\cite{Degrassi:2014sxa}.

The diboson ($WW$ and $WZ$)  and Higgstrahlung ($WH$ and $ZH$) fits use the data from Table IV of~\cite{Baglio:2020oqu} and we include NLO QCD.
As shown in~\cite{Baglio:2020oqu}, the diboson and Higgstrahlung fits are extremely sensitive to whether the fit is performed at linear or quadratic order, with the $WZ$ contribution being particularly sensitive to the inclusion of NLO QCD effects.
The diboson and VH data contribute to the fits predominantly through the high $p_T$ regime, and so it is important to
understand whether the EFT is valid in the regime where it is being applied.
An extensive study was performed in Ref.~\cite{Baglio:2020oqu}, where it was shown that a large portion of the allowed region in the fits corresponds to a strongly coupled theory where higher dimension operators need to be included in future studies (see in particular, Fig.~7 of that paper).
This is not true in the  fits to EWPO, where the fits fall within the weakly coupled regime.
We further consider this question of EFT validity in Ref.~\cite {Baglio:2019uty} for the case of WZ production.
In that study, we remove the high $m_{T}^{WZ}$ bins and show the effects on the fit.
Removing the bin with $m_{T}^{WZ}> 600~\textrm{GeV}$ from the fit, all points satisfy $m_{T}^{WZ} < \Lambda=1\,\textrm{TeV}$ and the EFT is a valid expansion.
In this case, the fits to the coefficients affecting the $WWZ$ vertex are degraded by roughly $30\%$.

Finally, the Higgs predictions use the $80\,\textrm{fb}^{-1}~ 13\,\textrm{TeV}$ LHC data from ATLAS~\cite{Aad:2019mbh} and the $36-137\,\textrm{fb}^{-1}~ 13\,\textrm{TeV}$ LHC data from CMS~\cite{CMS:2020gsy}, along with the $8\,\textrm{TeV}$ data given in Tables 2 and 3 of~\cite{Ellis:2018gqa}.
The contribution to Higgs production and decay from ${\cal O}_H$ occurs at loop order and is included following the prescription of Ref.~\cite{Degrassi:2016wml}.
The identification of the observables with the SMEFT predictions is made using tree- level calculations (except for $C_{HG}$ and
$C_H$) and compared with the results of Refs.~\cite{ATLAS:2019dhi, Ellis:2018gqa}  and reasonable agreement is found.

In the following  subsections, we present results for our test models and discuss the use of the global EFT fits for extracting information about the underlying models.

\newpage
\subsection{Model Dependent Results}

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{singlet_comb_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{singlet_physical_limits.pdf}
\vskip -0.25cm
\caption{\label{fig:singfig}
LHS: $95\%$~C.L. limits on the Wilson coefficients $C_{H\square}$ and $C_H$ generated at the matching scale.
The magenta, cyan, and yellow curves show combinations generated by the singlet model assuming a $Z_2$ symmetry, and with $\kappa = 1, m=0$, and $\kappa = 0, m = 3v$, respectively. The open circles indicate the point along the curve with $\sin\theta = 0.25, 0.50$.
RHS: The SMEFT coefficient limits reinterpreted in terms of a limit on the mixing angle for the $Z_2$ symmetric case as a function of the matching scale, $M$.
In both figures, the complete RGE of $C_{H\square}$ and $C_H$ from $\Lambda=1\,\textrm{TeV}$ to the weak scale is included.
}
\end{figure}

\subsubsection{Singlet Model}
\vskip -0.25cm

The singlet model generates the two operators, ${\cal O}_{H\square}$ and ${\cal O}_{H}$, at the matching scale $\Lambda$, which we take to be $M$, the mass of the heavier Higgs.
We first consider the results of a fit to arbitrary values of  the coefficients corresponding to these operators, assuming only that they are generated at the same scale, and taking all other operators to vanish. The results of this fit are shown on the LHS of Fig.~\ref{fig:singfig}, where the axes show the coefficients evaluated at the matching scale, $\Lambda = 1\,\mathrm{TeV}$.
${\cal O}_{H\square}$ leads to a shift in the Higgs-gauge couplings, and a universal change in the Higgs couplings to fermions, and these shifts are constrained by LHC data.
In addition, via Eq.~\ref{eq:regsum}, a non-zero ${\cal O}_{H\square}$ at the matching scale generates the operators
${\cal O}_{HD}$, ${\cal O}_{Hq}^{(1)}$,~ and ${\cal O}_{Hq}^{(3)}$ at the weak scale, yielding shifts to the EWPOs proportional to $\log(\Lambda / M_Z)$.
Numerically, the most important of these operators is ${\cal O}_{HD}$, which generates the $T$ oblique parameter
\footnote{
The $S$ parameter depends on $C_{HWB}$ which is not generated from RGE in the singlet model.
As demonstrated in Ref.~\cite{Wells:2015cre}, in the Warsaw basis there are additional contributions from $4-$fermion operators at dimension-$6$ that  are needed to obtain a basis independent result for $S$.
It is also interesting to note from Ref.~\cite{deBlas:2017xtg}, that only a very small class of models generate ${\cal O}_{HWB}$ from dimension-$6$ operators and none of the models considered here fall into this class.}.
The limits from EWPOs are shown in Fig.~\ref{fig:singfig} LHS as a purple dashed contour.
As anticipated, the limit on $C_{H}$ is very weak~\cite{Kribs:2017znd,Degrassi:2014sxa}.
The constraint from the  combination of LHC Higgs and diboson data is shown as a solid blue line, and we see that measurements of the Higgs couplings provide a bound on $C_{H\square}$ of the same order as the EWPO fit. The combination of LHC data and EWPOs is shown as the solid green curve.
In single Higgs data, the operators ${\cal O}_{H\square}$ and ${\cal O}_{H}$ do not generate any momentum dependence, but
${\cal O}_{H\square}$ produces momentum-dependent effects in di-Higgs data which could potentially be of use for the discrimination between models~\cite{Brehmer:2015rna, Capozi:2019xsi, DiVita:2017eyz}.

While we consider first general values of $C_{H}$ and $C_{H\square}$, the singlet model generates only a subset of these coefficients.
In the $Z_2$ symmetric case, $C_{H} = 0$, and $C_{H\square}$ is always less than zero (see Appendix~\ref{sec:scalars}), so this class of models generates a vertical ray emanating from the origin, shown as a magenta curve in Fig.~\ref{fig:singfig} LHS, with the open circles indicating values of the mixing angle, $\sin\theta = 0.25, 0.50$.
  The global fit clearly excludes values of $\sin\theta$ above $\sim 0.25$.
Furthermore, in order for the SMEFT to be valid, the coefficients cannot be too large.  If we require ${v^2\over\Lambda^2} C_{H\square}\mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} {1\over 2}$, this would correspond to the requirement that
$\sin\theta \mbox{\raisebox{-.6ex}{~$\stackrel{<}{\sim}$~}} 0.7$, implying that our global fit limits are well within the range of validity of the global fit.
 Limits on the mixing angle from unitarity and perturbativity of the couplings are considerably weaker than those from the global fits~\cite{Robens:2015gla}.

In the absence of the $Z_2$ symmetry, the additional couplings lead to more general combinations of $C_H$ and $C_{H\square}$ at the matching scale, and we show two such combinations, varying the mixing angle, as yellow and cyan rays in Fig.~\ref{fig:singfig} LHS. The values chosen saturate the vacuum stability bounds, Eq.~\ref{eq:vacans},  for $\kappa$ and $m = 0$, respectively. We see that only a small slice of the 2 dimensional SMEFT parameter space at the matching scale can be consistently generated in the singlet model, and that this slice covers only a small part of the allowed range.

With the other coefficients held fixed, the SMEFT limits can be translated into limits on the physical parameter, $\sin\theta$ (defined in Appendix~\ref{sec:scalars}).
In the $Z_2$ symmetric case, it is apparent from the LHS that $\sin\theta \lesssim 0.25$ for $\Lambda = 1\,\mathrm{TeV}$.
More generally, we can interpret limits on the coefficients as limits on the largest allowed mixing angle as a function of the scale, $\Lambda$.
For the $Z_2$ symmetric case, $v^2 C_{H\square} / \Lambda^2 = -{1\over 2}\tan^2\theta$, so the only scale dependence is the weak logarithmic dependence from RG evolution, and this is shown on the RHS of Fig.~\ref{fig:singfig}.
The EWPO limits obtained by fitting the SMEFT coefficients are quite similar to those extracted directly from the EWPOs in the full singlet model~\cite{Falkowski:2015iwa, Lopez-Val:2014jva}
as shown by the magenta curve on the RHS  of Fig.~\ref{fig:singfig},
although there is a roughly $20\%$ difference that can be attributed to the non-logarithmic terms in the complete
model calculation~\cite{Dawson:2009yx}.
While $C_{HD}$  also appears in the LHC Higgs and diboson data, the bounds on $C_{H\square}$ arise at tree level, and there are thus different effects in the scaling on the RHS of Fig.~\ref{fig:singfig}.

\subsubsection{2HDM}
\vskip -0.25cm

\begin{figure}
 \centering
\includegraphics[width=0.49\linewidth]{2hdm_eft_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{2hdm_eft_contours_ch.pdf}
\vskip -0.25cm
\caption{$95\%$ confidence level limits in the 2HDM from LHC data.  All other coefficients are set to zero.
The yellow curve corresponds to a Type I model with $\tan\beta = 1$, while the cyan and magenta curves correspond to Type II models with $\tan\beta = 1, 5$ respectively.
Open circles along the curve indicate points with $\cos(\beta - \alpha) = \pm 0.05, 0.10$.
The coefficients on the axes are evaluated at the matching scale $\Lambda$ and the curves labelled ``Higgs'' do not
include the RGE of the operators.
 }
\label{fg:2hdmfig}
\end{figure}

The decoupling limit of the 2HDM has been extensively discussed in the literature~\cite{Gunion:2002zf, Haber:2015pua}, but here we revisit the question of what information is in the SMEFT fits in the limit where the new scalars are too heavy to be observed.
In Appendix~\ref{sec:2hdm}, we see that the 2HDM generates ${\cal O}_H$, ${\cal O}_{uH}$, ${\cal O}_{dH}$ and ${\cal O}_{eH}$ at tree-level, assuming a small deviation from the alignment limit, $\mid \cos(\beta - \alpha)\mid \ll 1$.
In general, these operators can have arbitrary coefficients for each generation, but to avoid flavor-changing neutral currents, they are usually assumed to be proportional to the SM Yukawa matrices, so --- in the limit that only the third generation Yukawas are non-zero --- we can consider only the $33$ components of these fermionic operators, and label them ${\cal O}_{tH}$, ${\cal O}_{bH}$, and ${\cal O}_{\tau H}$
\footnote{Consistent models with large Yukawa couplings to the first and second generations can also be constructed~\cite{Bishara:2016jga,Alasfar:2019pmn,Egana-Ugrinovic:2019dqu}.}.

\begin{figure}
 \centering
\includegraphics[width=0.49\linewidth]{2hdm_physical_limits_type1.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{2hdm_physical_limits_type2.pdf}
\vskip -0.25cm
\caption{
$95\%$ C.L. limits using the SMEFT fit  on the Type I and Type II 2HDM from LHC data in the $\cos(\beta - \alpha)$ vs. $\tan\beta$ plane.
 The curves labelled ``Higgs'' do not
include the RGE of the operators. Note that our results are formally only valid in the $\mid \cos(\beta-\alpha)\mid \ll1$ limit.
 }
\label{fg:2hdmphysfig}
\end{figure}

None of these operators contribute to the EWPOs at tree level, nor do they generate any of the operators in Eq.~\ref{eq:regsum} at
 leading logarithm, so we consider only LHC constraints.
In Fig.~\ref{fg:2hdmfig}, we consider the limits in the $C_{tH}$--$C_{bH}$ plane (LHS) and the $C_H$--$C_{bH}$ plane (RHS), in each case setting all other operators to zero.
The dashed line shows a fit to Higgs data with the scale-dependence of the operators neglected, while the solid blue curves show the results of a fit where the operators are generated at $\Lambda = 1\,\mathrm{TeV}$ and then RG evolved down to $M_Z$.
On the LHS, we see that the RGE has a minor effect, running the Yukawa-like operators to smaller values in the UV, and resulting in slightly weaker bounds on the coefficients.
As in the singlet model case, we see that $C_H$ is only very weakly constrained, and note that the effect of the RGE on
the RHS is significant in this case.

In both panels of Fig.~\ref{fg:2hdmfig}, we show the patterns of coefficients generated by different choices of the Glashow-Weinberg conditions~\cite{Glashow:1976nt} for the proportionality constants in the fermion-Higgs operators.
The yellow curve corresponds to a Type I model with $\tan\beta = 1$, while the cyan and magenta curves correspond to Type II models with $\tan\beta = 1, 5$ respectively. Note that the $\tan\beta = 1$ curves for the Type I and Type II models are indistinguishable on the RHS.
The distances along these curves correspond to different values of the alignment parameter, and we indicate values of $\cos(\beta - \alpha) = \pm 0.05$ and $\pm 0.1$ by open circles.
We see that most of the allowed parameter space in the $C_{tH}$ vs. $C_{bH}$ plane can be generated by considering different classes of 2HDMs with different values of $\cos(\beta - \alpha)$ and $\tan\beta$.

The fit to SMEFT coefficients is re-interpreted in terms of the parameters of the 2HDM in Fig.~\ref{fg:2hdmphysfig}, for both the Type I and Type II models  near the alignment limit with $M = 1\,\mathrm{TeV}$.
These fits show good agreement with the fits in the full, UV complete 2HDM~\cite{Belusca-Maito:2016dqe}.
In the Type I 2HDM, the effects of the RGE reduce the value of $C_{tH}$ when scaling from $M_Z$ to $\Lambda$  as observed in Fig.~\ref{fg:2hdmfig} and are manifest in the difference between the solid and dashed line in Fig.~\ref{fg:2hdmphysfig} as well.

\subsubsection{Heavy Colored particles in Loops: $T$ VLQ}
\vskip -0.25cm

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{tvlq_ewpo_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{tvlq_lhc_contours.pdf}
\vskip -0.25cm
\caption{
LHS: $95\%$ confidence level limits from EWPOs (LHS) and
RHS: Limits from Higgs and diboson data with SMEFT coefficients in the pattern generated in the $T$ VLQ model.  The curve labelled ``Higgs'' does not include RGE effects.
The coefficients on the axes are evaluated at $\Lambda=1\,\mathrm{TeV}$.
}
\label{fg:tvlqfig}
\end{figure}

The case of a heavy $Q = {2\over 3}$ vector-like quark coupling only to the third generation leads to the operators ${\cal O}_{tH}$, $\big({\cal O}_{Hq}^{(1)}\big)_{33}$ and $\big({\cal O}_{Hq}^{(3)}\big)_{33}$ at tree-level, as well as ${\cal O}_{HG}$ at one loop.
Importantly, we note that only the third generation operators appear, so fits treating these fermion operators universally are impossible to interpret in the context of these models.

We first consider the constraints on only $\big(C_{Hq}^{(1)}\big)_{33}$ and $\big(C_{Hq}^{(3)}\big)_{33}$ from EWPOs.
The dominant effect at tree-level is in the $Z\to b\bar{b}$ decay, which depends only on the combination $\big(C_{Hq}^{(1)} + C_{Hq}^{(3)}\big)_{33}$. This is indicated by the horizontal dashed black lines in Fig.~\ref{fg:tvlqfig} LHS.
At leading logarithm, this degeneracy is broken  and both $\big(C_{Hq}^{(1)}\big)_{33}$ and $\big(C_{Hq}^{(3)}\big)_{33}$ contribute to the anomalous dimension of ${\cal O}_{HD}$. Lines of constant $C_{HD}(M_Z)$ generated from the RGE are shown as dashed gray lines in Fig.~\ref{fg:tvlqfig} LHS, and the resulting limit from the EWPOs including the RGEs, shown as a solid purple curve, clearly follows this shape.
It is  clear that to draw any conclusions about this model from the EWPOs including both the RGE effects and the non-universality of the fermion operators is crucial.

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{tvlq_comb_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{vlqt_physical_limits.pdf}
\vskip -0.25cm
\caption{LHS: Global fit limits with coefficients in the pattern generated in the $T$ VLQ.
The green line is the combination of EWPOs, Higgs, and diboson data, with all RGE effects included.
The magenta line demonstrates  the dependence on the mixing angle corresponding to the SMEFT fits with $s_L^t\equiv \sin \theta_L$.
The coefficients on the axes are evaluated at $\Lambda=1\,\textrm{TeV}$.
RHS:  A translation of the limits of the LHS to the parameters of the $T$ VLQ model.
}
\label{fg:tvlqmodel}
\end{figure}

On the RHS, we consider the limits from Higgs and diboson data. As is apparent from Table~\ref{tab:sumlag}, in the context of the $T$ VLQ, the Wilson coefficients that are generated all come with a fixed pattern, since there is only one independent physical parameter.
To illustrate the importance of this, we thus fix all the relationships between the coefficients except for $C_{HG}$ and plot in the $\big(C_{Hq}^{(3)}\big)_{33} = - \big(C_{Hq}^{(1)}\big)_{33} = - C_{tH}/Y_t$ vs. $C_{HG}$ plane.
The resulting limits are highly correlated, and we see that including the effects of the RGE on the fit with only Higgs data (going from dotted black to dashed red lines) has a mild effect on the fits.
Most of the limit in the vertical direction, however, comes from including the diboson data,
and this dependence arises because  the RGE generates ${\cal O}_{H\square}$, which is well constrained by $VH$ and $VV$ fits~\cite{Baglio:2020oqu,Brehmer:2019gmn}.

Fig.~\ref{fg:tvlqmodel} LHS uses the same set of correlations for the Wilson coefficients, and compares the EWPOs constraint (purple, dashed) with the Higgs plus  diboson constraint (blue, solid), all with the RGE included. The EWPOs still set a superior limit on the fermionic operators from the RGE induced ${\cal O}_{HD}$, while the Higgs data sets the bound on ${\cal O}_{HG}$. The combined limit is shown in solid green.
In magenta, we show the full prediction of the model, with ${\cal O}_{HG}$ related to the other operators via Eq.~\ref{eq:vlqt_chg} and open circles indicating a mixing angle of $\sin\theta_L\equiv s_L^t = 0.1, 0.2$.
As noted above, the $T$ VLQ generates only a 1- dimensional slice of the parameter space, and while a small value of $C_{HG}$ can be generated, the values corresponding to the model are still nearly orthogonal to the EWPO constraints so that the LHC data is less important in setting a bound on the physical parameter.
It is clear that for $M_T = 1\,\textrm{TeV}$, $s^t_L$ is restricted to $< 0.1$.
The RHS of Fig~\ref{fg:tvlqmodel} interprets the SMEFT results as limits on this mixing angle as a function of the $T$ VLQ mass.
As in our other examples, the dependence on the scale is only logarithmic, from the RG evolution.

 It is of interest to consider how well the global fit limits using the SMEFT coefficients reproduce the constraints found in the complete $T$ VLQ model.  A simple estimate can can found from considering the  oblique parameter $\Delta T$.  The $T$ VLQ result is
given in Eq. \ref{eq:tuv}, while the SMEFT limit is derived in Eq. \ref{eq:teft} of Appendix \ref{sec:colored}.  The SMEFT construction reproduces the $\log({M_T\over m_t})$ terms, but not the constant terms,  in $\Delta T$.
The  ratio of the $\Delta T$ parameter in the SMEFT limit to the full result is shown in Fig. \ref{fg:tvlq_dt}.
Even for quite small values of the mixing parameter, there is  a roughly $\sim 20~\%$ difference between the results obtained in the SMEFT  with tree level matching and one-loop RG running, as compared to the full theory for a fixed  value of  $\sin\theta_L^t$.
However, in order for the $T$ VLQ model to remain weakly coupled, the parameters of the Lagrangian need to scale
as $\sin\theta_L^t\sim {1\over M_T}$~\cite{Dawson:2012di}. This limit is also shown in the figure, and again the SMEFT limits differ by $\sim 20~\%$ from those of
the full theory.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\linewidth]{tvlq_deltat.pdf}
\vskip -0.25cm
\caption{The ratio of the oblique parameter $\Delta T$ in the SMEFT limit with RGE to the result in the full $T$ VLQ model for
various values of the mixing parameter.
}
\label{fg:tvlq_dt}
\end{figure}

\subsubsection{Heavy Colored particles in Loops: $(T\,B)$ VLQ}
\vskip -0.25cm

In the $(T\, B)$ VLQ model, the operators ${\cal O}_{Ht},~ {\cal O}_{Hb},~ {\cal O}_{tH},~ {\cal O}_{bH}$ and ${\cal O}_{Htb}$ are generated at tree level, and we emphasize again that only the third generation fermion operators appear.
As in the $T$ VLQ model, the only tree-level contribution to EWPOs is through the $Z\to b\bar{b}$ decay, which now directly constrains $C_{Hb}$.
This is shown as the dashed black lines on the LHS of Fig.~\ref{fg:tbvlqmodelone}, where we show the SMEFT limits in the $C_{Hb}$ vs. $C_{Ht}$ plane, with  the coefficients of all other operators fixed to zero.
At tree level, the EWPOs are independent of $C_{Ht}$, but both $C_{Hb}$ and $C_{Ht}$ generate $C_{HD}$ via the RGE at leading
 logarithm. The contours of constant $C_{HD}$ (with the slope fixed by the relative values of $Y_t$ and $Y_b$, as expected from Eq.~\ref{eq:regsum}) are shown in dashed gray, and we again see that the EWPO limit including the RGEs (purple, solid) follows this pattern.

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{vlqdoub_lep_eft_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{vlqdoub_lhc_eft_contours.pdf}
\vskip -0.25cm
\caption{LHS:  $95\%$ confidence level  limits on $C_{Ht}$ and $C_{Hb}$ when all other coefficients are set to zero from
EWPOs.  RHS: Same, but including the RG evolution from the matching to the weak scale.  The coefficients
on the axes are evaluated at $\Lambda=1\,\textrm{TeV}$.
}
\label{fg:tbvlqmodelone}
\end{figure}

The limits coming from Higgs data and diboson data are illustrated on the RHS of Fig.~\ref{fg:tbvlqmodelone}.
Here, we include the operators ${\cal O}_{tH}$, ${\cal O}_{Ht}$ and ${\cal O}_{HG}$ at the matching scale, with the
coefficients of other operators taken to be zero, and then project onto the plane where $C_{tH} = -Y_t\,C_{Ht}$, as predicted by the $(T\,B)$ VLQ.
Similarly to the $T$ VLQ case, the importance of the diboson data is apparent when the RGE of the  coefficients from the matching scale  to the weak scale is included, as a nonzero $C_{Ht}$ at the matching scale leads to values of $C_{HD}$ at the weak scale that are well constrained by diboson data.
A similar story pervades three of our example models: if a $C_{HD}$ or $C_{H\square}$ appear at leading logarithm in the anomalous dimensions for the operators  generated at the matching scale, the diboson data can place important constraints that may not be apparent in the tree-level matching.

In Fig.~\ref{fg:tbvlqmodel} LHS, we directly compare the EWPO and Higgs plus  diboson constraints by considering the SMEFT fits in the $C_{Ht}$ vs. $C_{Hb}$ plane. To include the effects of all the operators, we again set $C_{tH} = -Y_t\,C_{Ht}$, and similarly set $C_{bH} = -Y_b\, C_{Hb}$.
We also include ${\cal O}_{HG}$ with $C_{HG} = 0.65\,\big(\alpha_s / 8\pi\big)\,C_{Hb}$, as implied by Eq.~\ref{eq:vlqtb_chg} for $M_T=M_B=1\,\textrm{TeV}$.
We see that, even including all of these correlations, the EWPO constraint still sets a superior bound to the Higgs plus  diboson data.

In contrast with the $T$ VLQ, the $SU(2)_L$ doublet VLQ  model has two independent parameters in the decoupling limit, the two mixing angles $\sin\theta_R^b\equiv s_R^b$ and $\sin\theta_R^t\equiv s_R^t$, or equivalently, $s_R^b$ and the mass splitting, $\delta M_{TB} = M_T - M_B$ (see Appendix~\ref{sec:vlqdoub} for details).
In the limit $\delta M_{TB} = 0$, the mixing angles are identical, and $C_{Hb} = C_{Ht}$. This is indicated by the magenta line in Fig.~\ref{fg:tbvlqmodel} LHS.
Allowing for a nonzero mass splitting, however, shifts this relation, as can be seen by the yellow line in Fig.~\ref{fg:tbvlqmodel} LHS for $\delta M_{TB} = 10\,\textrm{GeV}$.
It is obvious from this shift that the EWPOs set a strong constraint on the mass splitting for fixed $s_R^b$ via the RGE induced $C_{HD}$.
This is simply a manifestation of the strong constraint on custodial symmetry violation, which we comment more on in Appendix~\ref{sec:vlqdoub}.
The behavior in Fig.~\ref{fg:tbvlqmodel} also illustrates that varying the two mixing angles sweeps out a region in the $C_{Ht}$ vs. $C_{Hb}$ plane, but that there is still a very tight relationship between all five operators generated  by the model, which changes the interpretation of the global fits in this context significantly.

On the RHS of Fig.~\ref{fg:tbvlqmodel}, we reinterpret the SMEFT bounds in the $\delta M_{TB}$ vs. $s_R^b$ plane, showing both the EWPO constraint (purple) and the Higgs plus  diboson constraint (blue).
We show the results for both $M_T = 1\,\textrm{TeV}$ (solid) and $M_T = 5\,\textrm{TeV}$ (dashed), and note that the logarithmic dependences on $\delta M_{TB}$ and $s_R^b$ have opposite signs.

\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{vlqdoub_comb_eft_contours.pdf}
\hfill
\includegraphics[width=0.49\linewidth]{vlqdoub_physical_limits.pdf}
\vskip -0.25cm
\caption{LHS:  $95\%$ confidence level  limits on $C_{Ht}$ and $C_{Hb}$ when the other coefficients are set to the correlated values of the $(T\,B)$ VLQ model with $M_T=M_B$  at the matching scale and including RG  evolution of the coefficients to $M_Z$.  The magenta
and yellow lines correspond to predictions for the coefficients with $\delta M_{TB}=0$ and $\delta M_{TB}=10~\textrm{GeV}$.
The coefficients on the axes are evaluated at $\Lambda=1\,\textrm{TeV}$.
RHS: Limits on the mass splitting of the $(T\,B)$ VLQ using the model correlations given in Table \ref{tab:sumlag}.
}
\label{fg:tbvlqmodel}
\end{figure}

\subsection{Global Fit to SMEFT Coefficients}
\label{sec:global}

\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{limits_proj.pdf}
\caption{95\% C.L. limits on the  coefficients of operators, with all other coefficients set to zero. The bounds on operators involving fermions
assume universal coefficients, except for $C_{bH}$, $C_{tH}$, and $C_{\tau H}$, which modify only the third-generation Yukawa couplings.}
\label{fig:warproj}
\end{figure}

\begin{figure}[hbp!]
\centering
\includegraphics[width=0.9\linewidth]{limits_marg.pdf}
\caption{
95\% C.L. limits on each the coefficients of each operator, marginalizing over all other operators. We assume universal coefficients for operators involving fermions, except for $C_{bH}$, $C_{tH}$, and $C_{\tau H}$, which modify only the third-generation Yukawa couplings.}
\label{fig:warmarg}
\end{figure}

As a by-product of our study, we present an updated global fit to the 19 SMEFT coefficients considered here in the Warsaw basis.
In comparison to Ref.~\cite{Ellis:2018gqa}, this fit includes higher integrated luminosity data from ATLAS~\cite{Aad:2019mbh} and CMS~\cite{CMS:2020gsy}, as well as the NLO QCD corrections to $VV$ and $VH$ production with the full distributions as in Ref.~\cite{Baglio:2020oqu}.
The results are shown in Fig.~\ref{fig:warproj} with each coefficient treated individually and in Fig.~\ref{fig:warmarg} when marginalizing over all the couplings.
The result for $C_{Hq}^{(3)}$ is considerably strengthened from that  of Ref.~\cite{Ellis:2018gqa}, due to the inclusion of the $WZ$ data not available in that work. .
In general, the inclusion of the  higher luminosity Higgs data strengthens the limits  that are not dominated by EWPO data by ${\cal O}(20\%)$.
Note that, in contrast to many of our particular model fits, here we assume universal couplings to the quark operators $C_{Hq}^{(1)}$, $C_{Hq}^{(3)}$, $C_{Hu}$ and $C_{Hd}$.
The assumption that the quark couplings are flavor blind has a major effect on the fits, as the constraints now have a significant contribution from diboson  and Higgstrahlung production from first-generation quarks.
Numerical values for the fits can be found at
\url{https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/smeft_20}.

\section{Discussion}
\label{sec:conc}

A major goal of precision measurements at the LHC is to uncover hints of new physics through patterns of deviations from the SM.  In this work,
we examine how fits to SMEFT coefficients that are predicated on patterns of coefficients generated in different UV complete models give
information about the high scale physics. Of particular  interest to us are the assumptions made when forming inferences about the source of
new physics from SMEFT fits.

Only two of our models, the $Z_2$ non-symmetric singlet model and the 2HDM generate a shift in the Higgs tri-linear coupling $C_H$.    In the singlet model, this shift is correlated with a non-zero $C_{H\square}$ term that can be observed in $VV$ and $VH$ production.  In the 2HDM, the non-zero $C_H$ is directly proportional to the $C_{fH}$ interaction and a weak limit on $C_H$ is obtained.   The 2HDM and the $(T\,B)$ VLQ models generate $C_{fH}$ terms that can be directly measured in Higgs production at the LHC.  The $(T\,B)$ model also generates $C_{Hf}$ couplings
that are limited by precision $Z$ measurements.   When the fits are performed using the patterns of coefficients predicted in each model, the
results are quite different from the global fit results and  in all cases, the correlated fits deviate significantly from the single parameter fits.

An interesting feature of our work is the importance of the RGE on the interpretation of the fits.
This suggests that redoing the study with complete one-loop matching would be of interest.
If a model predicts coefficients at the matching scale that generate operators through RGE that are constrained by EWPOs or diboson data, then these coefficients are strongly constrained.
The inclusion of RGE completely changes the interpretation of the fits in these cases.

In Fig.~\ref{fig:res_sum}, we summarize our SMEFT results in terms of the physical parameters of the models and show the maximum
allowed mixing angle from the global fits in each model  as a function of scale.
We note that these are the limits in the  SMEFT where the heavy particles have been integrated out of the UV complete model.
The fits are sensitive to  the ratios $C_i/\Lambda^2$,  modulo the logarithmic dependence from the RG running.

\begin{figure}
\includegraphics[width=0.8\linewidth]{thetamaxplot1.jpg}
\vskip -0.8cm
\caption{Maximum allowed mixing angles as a function of the heavy particle mass from the SMEFT global fit. Note that the
2HDM limits are valid in the $\mid\cos(\beta-\alpha)\mid\ll 1$ limit.
}\label{fig:res_sum}
\end{figure}

Our study is just the beginning of an understanding of the discrimination between UV theories from SMEFT fits~\cite{Barklow:2017suo}.
Follow-up work could include information from top physics~\cite{Durieux:2018ekg,Buckley:2015lku,Cirigliano:2016nyn,Hartland:2019bjb,Durieux:2019rbz,Brivio:2019ius}, a consideration of the importance of the quadratic versus linear SMEFT approximation~\cite{Baglio:2020oqu}, and complete 1-loop matching.  The complete 1-loop matching contributions exist for the singlet
model~\cite{Ellis:2017jns,Jiang:2018pbd,Haisch:2020ahr}, but not for the VLQ
models where there are missing heavy-light contributions~\cite{delAguila:2016zcb,Angelescu:2020yzf}.
The inclusion of flavor observables (such as $B$ and $K$ data) into the fit could further restrict the allowed coefficients in the various
models~\cite{Alasfar:2020mne,Descotes-Genon:2018foz,Aoude:2020dwv,Aebischer:2020ls,Bobeth:2016llm}.
It is of considerable interest to expand our study by examining further concrete example models
and performing detailed comparisons of the SMEFT fits with the global fits in the complete models.

\begin{acknowledgments}
SD is supported by the United States Department of Energy under Grant
Contract DE-SC0012704.
The work of SH was supported in part by the National Science
Foundation grant PHY-1915093.
SL is supported by the State of Kansas EPSCoR grant program and the
U.S. Department of Energy, Office of Science, Office of Workforce
Development for Teachers and Scientists, Office of Science Graduate
Student Research (SCGSR) program.
The SCGSR program is administered by the Oak Ridge Institute for
Science and Education (ORISE) for the DOE.
ORISE is managed by ORAU under contract number DE-SC0014664.  Digital data can be found at
\url{https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/smeft_20}.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Geek News Weekly (04.25-05.02)
Comics Games Movies News TV series
5th May 2016 by: Chris 0
Welcome to our second Geek News Weekly, where we collect the hottest geek news from last week.
We have ourselves a new Lara Croft
The wait is over – we now know which actress will become the famous female archeologist in the upcoming live-action reboot of the series. It will be Alicia Vikander, whom you may know from last year's "Ex Machina", "The Man from U.N.C.L.E." and "The Danish Girl" (she got an Oscar for this last one). She seems an ideal choice and it will be interesting to see if she makes a better Lara than Angelina Jolie did. We can't wait to see!
Source: The Hollywood Reporter
Potential actresses eyed for "Battle Angel Alita"
There are three actresses eyed for the main role in the live-action adaptation of Japanese manga series "Battle Angel Alita" (originally known as "Gunm" – a portmanteau of "gun" and "dream"; don't ask us why the English translator chose to change it), currently prepared by James Cameron and Robert Rodriguez. According to The Hollywood Reporter the actresses are: Maika Monroe, who starred in "It Follows" Rosa Salazar, who appeared in "Maze Runner: The Scorch Trials" and Zendaya, the Disney Channel star of "K.C. Undercover". It seems that out of these three Zendaya is the most likely to get the job, although there might be some scheduling conflicts getting in the way. Cameron and Rodriguez want "Battle Angel Alita" to be a star-making platform for whoever takes the role – a bold statement, given how Hollywood live-action manga adaptations did in the past. That being said – we keep our fingers crossed.
Sylvia Hoeks joins "Blade Runner" Sequel
As Deadline reports, Dutch actress, Sylvia Hoeks (known from "The Best Offer") will appear in "Blade Runner" sequel, alongside returning Harrison Ford and a bunch of newbies, such as Ryan Gosling, Robin Wright, Ana de Armas and Dave Bautista. We don't know the details of Hoeks' role yet, but we can say that the cast looks really amazing.
Source: Deadline
Disney confirms movies and claims dates
(c) Walt Disney Company
With a successful line of live-action fairy-tales (with "Alice in the Wonderland", "Maleficent", Cinderella" and "The Jungle Book"), now wonder Disney wants more. So here's what we will get:
– "Cruella" – a "101 Dalmatians" spin-off with Emma Stone,
– sequels to "Maleficent" and "Jungle Book" with more or less the same creative teams,
– "Dumbo" directed by Tim Burton (just please, don't make Dumbo a big weirdo, Tim) from Ehren Kruger's script,
– "A Wrinkle in Time" – an adaptaion of Madelein L'Engle's book directed by Ava DuVernay from Jennifer Lee's script,
– "The Nutcracker and the Four Realms" directed by Lasse Hallstrom from Ashleigh Powell's script,
– a Tinker Bell live-action project with Reese Witherspoon starring and Victoria Strouse writing,
– "Mary Poppins sequel with Emily Blunt (!) and Lin Manuel-Miranda set to star and Rob Marshall directing.
Additionally, Dwayne Johnsonn will star in "Jungle Cruise" based on a Disneyland attraction. Pretty impressive, isn't it? And we can be sure some of these movies will hit on the following dates, booked by Disney:
– UNTITLED DISNEY FAIRY TALE (Live Action) previously dated on 12/22/17 moves up to 7/28/17,
– UNTITLED DISNEY FAIRY TALE (Live Action) now dated on 4/6/18,
– UNTITLED DISNEY LIVE ACTION now dated on 8/3/18,
– UNTITLED DISNEY LIVE ACTION now dated on 12/25/18,
– UNTITLED DISNEY FAIRY TALE (Live Action) now dated on 12/20/19.
We can't say we're not looking forward to seeing all these movies!
Mads Mikkelsen reveals his role in "Rogue One"
In a recent interview Mads "Hannibal Lecter" Mikkelsen revealed his role in the upcoming "Star Wars" spin-off movie, "Rogue One: A Star Wars Story". He said:
I read the script and it was very beautiful and Felicity is playing this lovely, strong woman and I play her father.
Quite an interesting piece of information and we wonder how exactly Mads' character will fit into the "Rogue One" puzzle. But we'll probably have to wait until the movie comes out, which will happen towards the end of this year.
Source: Empire
"The Flash" has just lost director
Seth Grahame-Smith left the cinematic rendition of "The Flash" citing "creative differences", as The Hollywood Reporter reports. This might suggest there are some internal problem in Warner Bros as to the further development of DC Extended Universe, but it also may be that after not entirely satisfactory box-office results of "Batman V Superman" the studio execs didn't want a debutante (it would be a first movie for Grahame-Smith as director) to helm an important movie. Either way, we hope "The Flash" will get a new, amazing director in a… well, flash.
"Sherlock Holmes 3" has just got a screenwriter
(c) Warner Bros. Entertainment Inc.
Remember last week, when we told you "Sherlock Holmes 3" is at least planned? Turns out these plans are far more advanced than we first thought. As Deadline reports, James Coyne will rewrite the screenplay for the movie, which means something's happening and we might see Sherlock reappearing rather sooner than later. Good.
"Goosebumps" sequel is a go
Though the adaptation of R.L. Stine's "Goosebumps" wasn't a tremendous hit it did quite good in financial terms. Well, good enough for Sony to want a sequel, as The Hollywood Reporter writes. The studio would like to work with the creators of the original and to have Jack Black on board (as he's a kind of an audience magnet, I suppose). If the movie turns out to be a as enjoyable as its predecessor, then why not?
The Punisher getting his own Netflix series
Netflix and Marvel have just announced that The Punisher, seen this year in "Daredevil's" second season, will get his own Netflix series. See the teaser below!
The Punisher is back. Locked and loaded.https://t.co/nGKCa2taEV
— Netflix US (@netflix) April 29, 2016
Can't be more excited!
Source: Netflix Twitter
John Krasinski will be the new Jack Ryan
John Krasinski, whom you may remember as Jim Halpert in "The Office", will star in the Amazon series about Jack Ryan. The series, created by former "Lost" showrunner Carlton Cuse and Graham Roland will be a contemporary take on the character known from Tom Clancy's books. By playing the character, Krasinski will join quite a nice company of actors who had the pleasure to do that earlier: Harrison Ford, Alec Baldwin, Ben Affleck and Chris Pine. We'll hopefully see how he will do quite soon.
"24" revival and Lee Daniel's "Star" with series order
Fox already orders two shows straight to series, even before the Upfronts have begun. That means we will for sure see "24" revival with Corey Hawkins taking the spotlight from Kiefer Sutherland (his Jack Bauer will be missed, though Hawkins' character, Eric Carter, looks promising) and Lee Daniel's musical series "Star". These two shows were pretty certain to be picked up, so this particular piece of news doesn't come as a surprise, but it will be interesting to see how they will do on air.
Source: TVLine
"Ajin" TV Anime with a second season
(c) POLYGON PICTURES Inc.
If you are watching "Ajin", which is available on Netflix, here is a good piece of news for you – the anime is getting a second season, which will be released in October.
"Ajin" is based on a manga of the same title, which tells the story of Nagai Kei who is killed in a traffic accident on his way home from school, but ultimately he gets revived, becoming "Ajin" – a demi-human. As others of his kind are hunted, a price is placed on his head…
Source: ANN
Lionhead closes
April 29th was a historical day which marked end of an era. That day, after nearly 20 years and a host of industry-changing titles, British game developer Lionhead closed its doors. Founded by Peter Molyneaux in 1996 it released "Black and White", "Fable" and "The Movies" among others. It for sure will be missed greatly.
Source: IGN
DC Rebirth redesigns
DC Comics introduced the characters' redesigns via Comic Book Resources. Most of the costumes are altered only slightly, but there are some greater changes too. You can see the pictures below (click to enlarge).
rebirth-batman-jim-lee-8d549
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rebirth-designs-7-f3517
rebirth-designs-10-850c3
rebirth-designs-9-976d0
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rebirth-designs-4-3b9cd
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How do you like them?
Source: Comic Book Resources
New Robin Hood interpretation
Robin Hood is a very popular character among different creators and from time to time we are getting a new take on him. The person to propose one now is Robert Rodi, who is writing a comic book called "Merry Men". Per Newsarama:
Set in the 12th century, Merry Men centers on Robert Godwin – a.k.a. Robin Hood – who is discovered to be romantically involved with King Ricard, and quickly banished by Prince John as part of a new law criminalizing homosexuality. Godwin and his friends come to live in Sherwood Forest and are comfortable with their lives in hiding until a mysterious stranger comes asking for help against the Sheriff of Nottingham.
Sounds really interesting and original. The comic book will hit shelves June 1 and we're very curious.
Source: Newsarama
This week we've got "Dead Summer" teaser from "Once Upon a Time" creators, trailer for the season 2 of MTV's "Scream", Oliver Stone's "Snowden" with Joseph Gordon-Levitt in the titular role (might be an interesting movie to watch), Laika's beatufil stop-motion "Kubo and the Two Strings" and a true cherry on top: "The Killing Joke". Check the trailers out!
That is all for this week. Let us know what you think in comments section below or on Facebook and Twitter.
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A letter ought to be concise. however, it will contain every particulars of a candidate. Your letter should incorporate a concise summary that fits together with your abilities and experiences to the job description. Thus, you must make sure the letter is ideal. An address letter is almost any letter that functions as leading page of a larger deal or the initial page introducing another topic that addresses legal transactions whether it's in business or perhaps a personal letter. At one other times you will have an address letter. Sometimes you are going to be requested to send your cover letter being an email rather than someone document. Other cover letters used online to promote an item is by employing a testimonial. Discovering more about an organization is another excellent method to work through how you can tailor your cover letter. Effective letters will yield just excellent outcomes. In the event that you see, cover letter is thoroughly utilized in every on-line enterprise. A successful cover letter will boost your application and raise your odds of landing an interview.


This study appraised the impact of state behaviour on rebellion in La Côte d'Ivoire. It was occasioned by the continued recurrence of rebellion in that country, despite efforts at ensuring peace. Rebellion undermines political and socio-economic development; hence the need to investigate the extent to which state behaviour has contributed to rebellion in La Côte d'Ivoire. The research set out three objectives. First, it sought to examine the relationship between state behaviour and rebellion. Secondly, it was to identify the role of political leadership in the rebellion in La Côte d'Ivoire. Finally, it was to propose sustainable strategies to control rebellion in La Côte d'Ivoire. The scope of the study centered on La Côte d'Ivoire and focuses on rebellion as a consequence of dreadful state behaviour. The time frame was restricted to 1993 and 2011. The research adopted the content analysis method, using primary and secondary data from individuals, the internet, and libraries. Its major finding was that undemocratic behaviour of the political elites such as the introduction of "Ivoirite", redrafting of the electoral code, election malpractices and bad governance led to the rebellion. The study looked at the peaceful and stable atmosphere the Houphouet-Boigny-led government sustained from independence to 1993. Of particular interest was his ability to maintain stability with the policy of "reconciliation" during the latter part of his reign, when the country encountered political turbulence occasioned by economic recession. The research held that the rebellion in La Côte d'Ivoire was a direct consequence of the behaviour of succeeding political leaders towards their citizens. Accordingly, the work viewed state behaviour in La Côte d'Ivoire as those tendencies or traits that generated conflict between governments and their citizens, mainly engendered by incompetent leadership. It was also discovered that, states confronted with incidents of rebellion tackled them by adopting autocratic tendencies. The study concluded that rather than adopt coercive force against aggrieved citizens, the Ivorian Government should institute policies aimed at uniting the different ethnic, religious, political, and interest groups within the country by developing politics of multicultural integration. It should also adopt enhancement strategies like conflict prevention, peace promotion, the consolidation of good behaviour, and democratic practice aimed at eliminating conflicts to facilitate socio-political and economic development. The Government of La Côte d'Ivoire should also ensure that power and resources are equitably distributed to avoid perceived marginalisation of minority ethnic or interest groups. The ECOWAS Secretariat should ensure capacity-building for early warning and conflict management strategies of the sub-region in order to eliminate conflicts and promote development. Finally, the International Criminal Court (ICC) should indict key political figures who lead atrocities against civilians to serve as deterrent to others.


text:
Yes, But How Will It Affect His Friendship With Michael Irvin?
We have quite a bit of history with John Rocker and his PR agent Debi Curzio, but we'll confess — and we're not sure why — we simply never thought to ask either of them about steroids. Maybe it's because we don't care about steroids all that much, or maybe it's just because asking a guy who has been retired for four years about steroids seems beyond the point. Or maybe it's just because we've always assumed half of baseball's roided anyway, and we just don't get too worked up about it.
Our Interview With John Rocker
Two days before Thanksgiving, at Local 138, one of our favorite watering holes in New York City, we
Anyway, it appears Rocker received some HGH from that lab everyone's talking about. The collective jaws of a sports public are dropped.
"That was a growth hormone that was prescribed by a doctor in relation to his rotator cuff surgery in 2003, so I don't really think there is anything to the story," said Debi Curzio, Rocker's publicist.
That's Curzio in this morning's New York Daily News. Doctors say that HGH isn't supposed to help with recovery from rotator cuff surgery, but that's beside the point: If Rocker has been using HGH, for whatever reason ... well, it certainly does explain a lot.
More Rocker Dope [New York Daily News]
Now There's A Shocker [Rumors And Rants]
Our Interview With John Rocker [Deadspin]
John Rocker Rocks Slowly [New York Magazine]
(UPDATE: Ms. Curzio writes in with a statement:
"In 2003, several doctors had recommended that my client, John Rocker, take growth hormones in an effort to continue pitching right up until he had rotator cuff surgery. Growth hormone supplements can be purchased over the counter at any vitamin store across the country, and they were recommended to help heal John Rocker's shoulder and to potentially avoid surgery. Reports by the media trying to link my client to the use of illegal substances are incorrect, and this is nothing more than another example of irresponsible journalism.")