text:
A Jobcentre worker
In the months between February and June 2020, the number of claims for Universal Credit doubled from 2.7 million to 5.4 million. Universal Credit is the primary benefit available to the unemployed, those too ill to work, and those who are in low-paid work.
Millions who have never claimed benefits before were forced to do so, in response to the economic crisis. This increase doesn't capture the full scale of the crisis, as some people will have claimed other benefits, such as 'new-style' jobseeker's allowance, but it is a good indicator.
When people made their claims, evidence was gathered remotely and processed with the focus on getting the right money to the right people at the right time. Evidence was taken on trust and the whole process of claiming benefits was speedily adapted to deal with the Covid crisis.
Terms like 'sanction' and 'claimant commitment', which have embedded vicious Tory rhetoric about 'scroungers' into the benefit system, were abandoned. For a shining moment, the focus of the welfare system was ensuring that people were paid.
Under huge political pressure, Tory chancellor Rishi Sunak raised the amount payable under Universal Credit by around £80 per month - not much, but the biggest rise in benefits in living memory. Why?
It is possible the government realised that all the years of dirty propaganda by their friends in the media were about to be exposed to millions more people.
They might all see the reality of the punitive, inhuman treatment of the self-employed, the low-waged, those in unstable employment, and those too ill to work - threatening the 'national unity' line the capitalists have been peddling.
Sunak's increase is a tacit recognition of the potential social explosion that Universal Credit has been stoking up since the Tory-Liberal coalition introduced it in 2013.
Charities, NGOs involved in the welfare system, and local government organisations have all been critical of Universal Credit. The Joseph Rowntree Foundation, Citizens Advice, the Child Poverty Action Group and others have pointed to it as driving homelessness and poverty.
The housing component of Universal Credit does not cover full rent. Claimants have to top it up with money from the rest of their benefit payment, which is meant for food and other necessities.
If the government's emergency measures preventing landlords from expelling tenants are lifted - and landlords' organisations are actively demanding this - tens, even hundreds of thousands of people will be at risk of homelessness. This is thanks to the inadequacies of the benefit system, not to mention the 30-year-long decimation of council housing.
Calm before the welfare storm
Tory measures to get through the Covid-19 crisis have subdued angry opposition, but the crisis is far from over. Even as the R-number ebbs a little, the economic crisis which was waiting in the wings before Covid-19 gathers pace.
This is aimed directly at a working class which never recovered from a shabby decade-plus of cuts to wages, jobs and public services. Yet this hasn't stopped Tory attempts to get back to attacking benefits claimants.
The Department for Work and Pensions (DWP), under sharp orders from the secretary of state, has reinstituted the "claimant commitment." That's 35 hours a week of job searching, and the risk of being denied benefits if you don't do 'enough'.
What 'enough' means in the context of a global economic crisis is something that has the ordinary staff in DWP shaking our heads in despair at the stupidity of the bosses. The majority of DWP staff are trade union members and have little truck with government sermonising when they see the reality of Tory pension cuts and below-inflation wage 'rises'.
During the coronavirus crisis, DWP staff have been prepared to walk out over health and safety. Unfortunately, the leadership of our union PCS, under general secretary Mark Serwotka, has not lived up to this militancy. Instead, the majority of the union leadership has succumbed to Tory 'national unity' rhetoric, even watering down the union's own pay claim.
Now, PCS is balloting against the employer's proposals to keep DWP offices open up to 8pm on weekdays and start opening on Saturdays, despite the Covid-19 crisis. Socialist Party members and others will be fighting hard to win this consultative ballot and build towards serious action as the best way to put pressure on the government.
The leadership does not seem to have the same campaigning emphasis, writing to branches: "This is not a strike ballot - this is a consultative ballot to get your views. A yes vote will help the union in negotiations to protect your safety."
A consultative ballot must be followed by a second, statutory ballot to authorise a strike. How will playing this down, instead emphasising a simple poll of views, help the union mobilise members to achieve their demands?
This lacklustre approach has meant that only the difficulties posed by social distancing have prevented the rolling out of the whole battery of anti-claimant 'conditionality' measures introduced since 2012:
Mandatory work activity, where claimants are forced to work unpaid for 13 weeks
Unpaid 'sector-based work academies', where claimants are inducted into the joys of shelf-stacking (with a side helping of corporate propaganda)
Job preparation and 'training' delivered by private providers - who were exposed as not even helping claimants come up with a usable CV, despite being paid millions of taxpayer pounds
Mandatory daily attendance at Jobcentres - the Tory intent being to harass claimants into signing off benefits
And last but not least, a toughened sanctions regime, with a legion of new 'offences' that could be used to deny people Universal Credit payments
Eventually, profit pressure from the capitalist class is likely to outweigh the caution of more far-sighted Tories who understand the potential depths of the anger facing them in the aftermath of their handling of the Covid-19 crisis. They will attempt to bring back all the punitive apparatus of the last ten years, to drive down claimant numbers regardless of whether or not people can find work.
If the Tories are successful in doing this, it will not stop there. It could be worse, reflecting the scale of the crisis. They will try to go further and hit harder.
Civil servants working in DWP will also be hit. Despite the announcement of 13,500 new staff to help prop up a disastrously underfunded benefit system, the Tories will roll out plans for further privatisation, job cuts, wage freezes, and weakening of union rights.
Just last week, the patron saint of public sector union-busting, Francis Maude, who was cabinet secretary under David Cameron, was brought back in to run a review of the Cabinet Office. The outline of future Tory attacks can already be seen.
Once the focus comes off the heroic efforts of key workers, including those in DWP, the establishment media will turn to talk of the national debt, 'lazy civil servants' and 'benefit scroungers'. So we need to be ready.
Unions need fighting programme
Jobs, wages and welfare are all interconnected. Recruitment freezes and redundancies are the first line of attack for bosses seeking to protect their profits. 'Destroy jobs, withhold investment until conditions improve' will be the mantra of the capitalist class.
The future offered by capitalism involves skyrocketing youth unemployment rates and millions thrown onto the benefit system. The first battle, therefore, must be fighting to save every single job, fighting for the nationalisation of businesses threatening job losses, and for an economic plan that creates jobs.
Alongside destroying jobs, the capitalists will attempt to hold down wages to protect their profits. Fighting for fair wages - of at least £12 per hour, and £15 in London, as a starting point - must be a major focus of a national campaign across the trade unions.
This would mobilise not just those who work in poorly paid, poorly unionised sectors like retail and care, but also the young, unemployed workers who will be pushed towards unpaid labour in those sectors if the Tories get their way.
Rishi Sunak has so far soft-pedalled the attacks on young workers. The government has created "traineeships" for 18 to 24-year-olds. When traineeships were first created in 2014, they were unpaid.
The new traineeships are paid at the national minimum wage, which can be topped up by the employer, who also gets a bonus if they keep the young worker on past the end of the traineeship.
These measures make it cheaper to hire young workers, but don't alter the overall number of jobs available - so all they do is move the burden of unemployment around. But it shows the government is worried about a youth revolt.
Having created the unemployment, the capitalist class seeks to keep unemployment benefits low in order that workers will agree to take jobs for the lowest possible wages. If millions of workers were mobilised in a campaign to fight for jobs and wages, the unemployed would inevitably be drawn into the battle too.
As with the recent environmental school strikes, 'claimant strikes' could develop, allied to a militant trade union leadership that was prepared to stand up for low-paid workers - a group which overlaps with the unemployed by virtue of the shared benefit they claim: Universal Credit.
It doesn't take much to imagine the demands posed by the unemployed. A living wage for those out of work. The end of sanctions. The end of mandatory powers for Jobcentre work coaches so that supporting claimants is the one and only focus of Jobcentres.
Improvements to the support available to find work. Proper retraining and education for workers who need to transition between sectors of the economy. A system where it is easy to contact staff for help, and which is resourced to give this help. It would not take much to get DWP staff onside with all this.
Civil servants in the lower grades have found their pay so restricted that they often get (meagre) pay rises now just by the national minimum wage increasing. Thousands of workers in DWP and other departments claim Universal Credit because their pay is so low.
DWP workers and benefit claimants are therefore natural allies, separated only by the anti-trade union laws and the anti-claimant ideology spouted by the government. Both can be defeated, and a mass trade union campaign fighting for the rights of workers in the current economic crisis is the straightest route to achieving this.
Abolish for-profit assessments
For years, in cities around the UK, groups like Disabled People Against Cuts have demonstrated outside the assessment centres where health assessments are held to determine eligibility for Universal Credit, Employment Support Allowance (ESA) and Personal Independence Payment (PIP). These assessments are delivered for profit, by privateers Capita, Atos and CHDA - meaning the whole focus is on keeping costs down.
So reliable are these assessments that in cases disputed up to tribunal level, two-thirds of the decisions based on them are overturned in claimants' favour. It is clear that this privatised system of health assessments has to go.
Just before the crisis, the government announced plans to merge the assessments for Universal Credit and ESA with those for PIP. This is as much about salvaging a wholly discredited process as it is about cutting costs. It shows they are already under pressure.
A working class mobilised to fight on jobs, wages and welfare could sweep away these assessments in favour of something more reliable, more humane and less onerous for claimants. Disability benefits, like other benefits - and indeed the minimum wage - must be based on a real living income.
Such a campaign would also inevitably recognise that the money paid out through benefits must be supplemented with high-quality public services, and rights in both the community and the workplace, to ensure equal access and a good quality of life for all.
All workers into the unions, all unions into the struggle!
Socialists do not have illusions in the tops of the trade union movement, which tend to be heavily influenced by the capitalist class.
In respect of the traineeships for 18 to 24-year-olds discussed above, the Trades Union Congress, Britain's union federation, put out a joint statement with the bosses' Confederation of British Industry when they were first developed in 2014 - welcoming them, even though they were unpaid! We therefore do not expect more than fine words from most of the current crop of union leaders.
Socialists must actively build the unions - recruiting workers in their workplaces, linking up with the advanced layers of the working class, both in other workplaces covered by the same unions, and with the advanced layers of the class in other unions.
A union membership card should be held by every socialist, and they should participate in their union's 'broad left' grouping, if a properly functioning one exists, to form a united front that can mobilise ordinary members and force even right-wing union leaders to act.
Working-class anger can be mobilised. The capitalists can be forced back, and concessions wrested from them directly and from their state - even as socialists campaign to supplant it with a democratic workers' state that would give real meaning to welfare from cradle to grave.
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redpajama_set_name: RedPajamaCommonCrawl


text:
Nikola Radic Lucati
/ artist site
Ethnographers argument
Ethnomimicry
Radio witness
The insufficiency
Predictable outcomes
The fever (numismatic value)
Communal dwellings
Photography, engraving, steel, quicklime, office desk. Dimensions variable.
The site-specific installation "Communal dwellings" was shown at the Cultural centre "Parobrod", 2013, part of the "Duplex" exhibition program, Curated by Mihael Milunović.
Restriction or prohibition of movement, Article 5:
"A foreigner shall be restricted or prohibited from moving and staying in a given area in the Republic of Serbia if so required by the reasons for protecting the public order or security of the Republic of Serbia and its citizens, or on the basis of an international treaty."
Law on Foreigners ("Official Gazette of the Republic of Serbia", No. 97/2008)
"Fairy: So? What is that? What are you doing in that corner? It's like you're cooking-up a conspiracy. It's time to go. I decided that the Light will be your chief. You will all follow her as you are obeying me, and I entrust her with my stick! The children will visit their grandfather and their grandmother who are dead tonight. Out of discretion, you will not follow them. Let them spend the night in the circle of their deceased family. During this time, you will save everything you need for tomorrow's journey, which will be long. Now, get on your way, everybody in your place! "
Maurice Maeterlinck, Blue Bird
Emigration is one of the triggers of traumatic cycles that form the character of those societies that have caused it, as well as those that were formed by migrations. Ethical attitudes towards the forced displacement of civilians in peace and war reveal not only the relation to history, but also the character of contemporary societies and their political programs. The societies that cyclically cause migration, but are also self-renewing through migrations, are associated in this work with the treatment of emigrants from the Reich, a tragedy that left a sufficiently deep trail in order to regain the empathic perspective of the victims, as well as the differences in their presentation in a contemporary, interpretative debate. Decisions, statements and documents related to the emigration of the European Jews via Danube through the Kingdom of Yugoslavia, on the eve of its accession to the Tripartite Pact, have opened the view on the genesis of the "Semi-dependent state" that worked before the official breakdown – the propaganda striving to "nazify" the society, the adoption of Nazi laws and laws, the formation of "communal dwellings" for refugees, as well as profiting on the economic and political changes that these developments have created. The destination to which only a few have fled, was then formed on the growing dicrepancy of the emancipatory promises of Herzlian "Altneuland", and the eugenic selection barring access to a new society. The social hierarchy of the part of Israel's state-owned apparatus has already been realized in the formation of a new society, but also the empathic distance from the endangered, even before the official start of mass destruction in Europe. One of the results of the "new man" and its society was de-facto capital of the future state – Tel Aviv, a brand new city, built on the fulcrum of utopias and conflicts, whose forms were celebrating life by receiving immigrants. Each approach to the subject leads to the sharpening of the similarity of the outcome of empathic dissociation from victims and their humanitarian situation, and in societies on nominally opposite ethical postulates. However, to attempt at juxtaposition is necessary, precisely because art does not share the history's distance from the subject, nor its attachment to its militaristic-national, ideological interpretation. It can still point to the loss of texture and details of life, erased by the apropriative destruction of education and commemoration of the Holocaust and Genocide, a culture positioned for the "day after witnesses". Art, unlike history and cultural policy, could still be the driving force of discourse on the civil history of war.
As an artist, I'm interested in places that are part of the repetitive process. Permanent residence in familiar locations results in a cumulative exposure to subjects that, although deepening the possibility of analysis, pardoxically, narrows the field of work. The initial speed of recognition and embedding into the subject is part of the methodological framework that suppresses the context in the field of research and analytical processing of archival material, avoiding the mediation of the immediacy of experience. In this methodological condition, the thematic dealing with the history of the Holocaust and Genocide through the current state of architectural heritage and the historical narrative in post-transitional normalization, in fact, documents the process of constant de-politicization of the speech of minority memories, under the pressure that by accepting the projections of the dominant majority they will secure their further survival.
Nikola Radic Lucati © 2023. All Rights Reserved.


text:
Supermax Corporation Berhad () started as a trader and exporter of latex gloves in 1987 before venturing into manufacturing in 1989. It is Malaysia's largest Own Brand Manufacturer and the world's second largest producer of rubber gloves.

The Supermax Group was founded by Dato' Seri Stanley Thai and his wife Datin Seri Cheryl Tan in 1987 as a trading business distributing latex gloves.

Supermax has succeeded in establishing its own brands with a strong presence in Canada, the U.S., the United Kingdom and Brazil. Almost 100% of its production is exported to medical and dental buyers.

Its main competitors are Kimberly-Clark, Ansell, Allegiance and Microflex.

In 2020 during the COVID-19 pandemic, strong demand for rubber gloves sent Supermax's share prices on a five-fold jump. Founder Thai Kim Sim recorded an estimated net worth of $1 billion at the stock high in early June 2020.

In November 2020, the company donated a total of RM75 million to the Government's COVID-19 fund set up to battle the pandemic.

In October 2021, United States banned imports from Supermax citing forced labour. A month after, Canada terminated its contracts.

On 15 June 2022, the Government Pension Fund of Norway placed Supermax under observation due to "unacceptable risk that the company contributes to serious violations of human rights".

Controversies
In November 2021, Canada ends contract with Malaysia's Supermax over labor allegations. Based on the seriousness of the allegations and expected timelines for the final audit results, the government of Canada has decided, and Supermax Healthcare Canada has agreed, to terminate by mutual consent the two existing contracts for the supply of nitrile gloves.

In June 2022, Norway's sovereign wealth fund, Norges Bank Investment Management, put Supermax under observation for two years, citing allegations of unacceptable risk that the company contributes to serious violations of human rights.

Subsidiaries
Supermax Glove Manufacturing Sdn Bhd
Maxter Glove Manufacturing Sdn Bhd
Maxwell Glove Manufacturing Bhd
Supermax Latex Products Sdn Bhd
Supermax International Sdn Bhd
Supermax Energy Sdn Bhd
Supermax Healthcare Canada
Supermax Global Ltd (Bermuda)
White Oak Global Property Ltd (USA)

References

External links
Official Website

1987 establishments in Malaysia
Companies listed on Bursa Malaysia

redpajama_set_name: RedPajamaWikipedia


Aggregate Resources Inc is a Merchandise Brokers business in Galesburg, MI.
Aggregate Resources Inc has been operational since 0 and provides B2B services in the industry of Merchandise Brokers from its main office in Michigan. The company functions with 1 to 4 employees and conducts business from its branch with a flagship found at 14500 East Michigan Avenue in Galesburg, MI 49053.
From its inception in 0, Aggregate Resources Inc has seen growth reaching revenues of Unknown per annum. This company of Merchandise Brokers is classified under SIC code 7389 and NAICS number 561499, which can be examined for more business details.
For inquiries or more information about , B2B market variables, and professional , please contact representative Dan Balkema Full Name Report by phone number (269) 746-4275 Full Phone Report. To access Aggregate Resources Inc's online presence, visit its website at aggregateresources.net or connect through social media via on Twitter or on Facebook.


In the construction industry, there has been an evolution and progression that has happened from the first brick and building to the materials that have been used. The past has often been an inspiration and dictator to the designs of buildings of today – but they have also been the key helping both contractors, architects and builders improve on the flaws of the past. One thing that has been a certainty is the continual evolution of improving buildings to be environmentally sustainable. Whether it be in the materials such as glass and steel to the overall design and layout of the inside which might include access panels and doors.
To best understand this, we examine some trends that are taking place in construction and architect and in a way dictating decisions made by contractors and builders alike.
Builders and designers of today are moving towards promoting and marketing green and sustainable practices. This movement of green and sustainable building is often propelled by both activists and politicians – which can also dictate the costs of a project. To help and establish this, contractors, builders and architects look at ways of incorporating the existing environment into the actual plan. For some, this could be integrating the building or materials into the new project.
if you look outside your window, there is an excellent chance that many of the buildings and structures you see are ones that were inspired by the past. From Roman influences in the columns to the English arches – many past influences continue into the present. History has also influenced the future in that many structures and buildings have evolved from their original purposes to now new uses. For example, buildings that may have once been used for factories or businesses have now been transformed into condos, homes or boutiques. In parts of Europe, specifically Amsterdam, they have converted their prisons into boutique hotels.
The past allows builders and contractors to let their imagination roam free but more importantly, it will enable contractors to be cleaner with their approach in the design. For example, some builders have utilized unique building materials such as access panels to install extra storage in limited spaces.
One way of promoting sustainability is sometimes simply refurbishing or updating an existing aspect of the project with newer materials or products. Sustainability and creating an environmentally sustainable environment are sometimes about taking what is there and becoming innovative with space and the elements. A structure can be upgraded with minor or semi-major updates, installing insulation or maybe an access panel or simply using new materials. A challenge that comes from enhancing and improving an area can sometimes be hard – as one has to weight the pros and cons between determining which is the best cost-effective decision.
In today's growing population and significant cities becoming denser, contractors and investors are investing in various properties and turning them into multi-functional properties. This is where the designers, builder or contractor takes the space and sees more than one function. A building or tower can hold business offices, shopping centres, and residences – this is considered a multi-functional project.
Whether it's a mixed-use project or an upgrade, contractors and builders are faced with the unique challenge of combining sustainable practices in an area that typically doesn't often consider its importance. However, something as simple as using the proper materials, flooring, even insulation or installing various access panels – these are attempts to creating a sustainable project.


text:
Ultimate Guide to the La Rioja Wine Region
by Libby Little
La Rioja Wine Region History and Culture
Like many regions in Spain, La Rioja has been home to many groups of people over the centuries. The Phoenicians, Carthagians and the Moors all made their way to La Rioja. But the most notable group was the Romans. The Romans started wineries throughout the region, and many of these wineries have withstood the test of time [source: Cellar Tours].
After the Roman Empire collapsed, the Moors did little to keep the La Rioja wine tradition going. But when Spain became its own country in 1512, winemaking was reborn. During this time, Benedictine monks began cultivating the land and developing numerous wines [source: Cellar Tours]. Slowly but surely, Spanish viticulture was making its mark on the world.
In the 1870s, a phylloxera infestation essentially wiped out vineyards in France. Vintners from Bordeaux decided to move to La Rioja to start over with fresh fruit. The French brought their techniques and money to the region and infused La Rioja with new life and improved wine production. By the time the phylloxera came to the La Rioja in 1890, the area was able to survive based on what the French had already learned about these dangerous grape pests [source: Hand Picked Selections].
Today, La Rioja offers wine routes, vineyard excursions and festivals to celebrate its historical wine culture. Visitors can explore monuments, ancient buildings and modern wineries in any number of tourist packages.
Read on to learn about La Rioja's agriculture and viticulture.
Phylloxera Demystified
How can one little bug cause so many problems for wine producers worldwide? It's all about the eggs. If you've ever seen an aphid, you can imagine what phylloxera look like. They are tiny -- and almost invisible. Female phylloxera are capable of producing up to 600 eggs at the roots of grapevines. With females producing between four to seven generations during the summer growing months, vineyards make for a phylloxera heaven. Vintners have learned how to develop bug-resistant vines, but phylloxera still create havoc in vineyards to this day [source: Wine Mag].


text:
\section{Introduction}

Dimesnional regularization \cite{dimreg} is a wonderful tool for
perturbative computations in quantum field theory but, like all
tools, it can be misused. One way for this to happen involves the
infamous {\it automatic subtraction}: dimensional regularization
registers only logarithmic divergences; it sets power law
divergences to zero. When dimensional regularization is employed
to control an ultraviolet divergence the automatic subtraction
is no problem because the right way to deal with ultraviolet
divergences is by subtracting them with counterterms. In that
case the automatic subtraction merely spares one the labor of
explicitly working out the counterterms required to subtract off
any power law divergences. However, dimensional regularization
can also be used to control infrared divergences, and this
leads to errors if one fails to recall that the technique
automatically sets power law divergences to zero.

The appearance of an infrared divergence in the answer to a
quantum field theoretic question means that something about
the question is unphysical. The right thing to do in that case
is to revise the question so as to make it more physical. The
classic example of this is the infrared divergences one encounters
when computing exclusive scattering amplitudes in quantum
electrodynamics. Because the photon is massless and all real
detectors have finite resolution, one can never exclude the
possibility that the final state contains an extra, very low
energy photon. Including arbitrary numbers of soft photons in the
final state eliminates the infrared divergence \cite{BN}. Another
example is when the vacuum decays, as it does for a massless
scalar with a cubic interaction. Veneziano showed that even
inclusive scattering amplitudes harbor infrared divergences for
this system \cite{GV}. The problem in this case is that the free
scalar vacuum centered around $\phi = 0$ decays, so one cannot
assume the final state is even stationary, much less centered
about $\phi = 0$. Infrared finite results can be obtained by
instead releasing the system at a finite time in a prepared state
centered around $\phi = 0$ and then following its evolution
\cite{TW1}.

A peculiar situation arises in curved backgrounds when parameters
of the background geometry change the infrared properties of the
particle. For example, consider a homogeneous, isotropic and
spatially flat background,
\begin{equation}
ds^2 = -dt^2 + a^2(t) d\vec{x} \!\cdot\! d\vec{x} \; .
\end{equation}
The Hubble parameter $H(t)$ and the deceleration parameter $q(t)$ are,
\begin{equation}
H \equiv \frac{\dot{a}}{a} \qquad , \qquad q \equiv -1 -
\frac{\dot{H}}{H^2} \; .
\end{equation}
The spatial plane wave mode functions for a massless, minimally
coupled scalar are quite complicated for general $q(t)$ \cite{TW2}
but they take a simple form when $q(t)$ is any constant $q_1$,
\begin{equation}
u(t,k) = \sqrt{\frac{-\pi}{4 q_1 H a}} \, a^{1-\frac{D}2}
H^{(1)}_{\nu}\Bigl(\frac{-k}{q_1 H a}\Bigr) \qquad {\rm where} \qquad
\nu = \frac12 - \Bigl(\frac{D \!-\! 2}{2 q_1}\Bigr) \; .
\end{equation}
The naive mode sum for the scalar propagator between $(t,\vec{x})$ and
$(t',\vec{x}')$ is,
\begin{equation}
\int \!\! \frac{d^{D-1}k}{(2\pi)^{D-1}} \, e^{i \vec{k} \cdot (\vec{x}
- \vec{x}')} \Biggl\{ \theta(t \!-\! t') u(t,k) u^*(t',k)
+ \theta(t' \!-\! t) u^*(t,k) u(t',k) \Biggr\} \; . \label{naive}
\end{equation}
From the small $k$ behavior of the mode functions,
\begin{eqnarray}
-1 \leq q_1 < 0 & \!\!\! \Longrightarrow \!\!\! &
u u^* =  \frac{4^{\nu-1} \Gamma^2(\nu)}{\pi (-q_1)^{1-2\nu}}
\frac{(a a')^{\nu -\frac{D-1}2}}{(H H^{'})^{\frac12 -\nu}} \times
\frac1{k^{2\nu}} \Biggl\{ 1 \!+\! O(k^2)\Biggr\} , \label{accel} \\
0 < q_1 & \!\!\! \Longrightarrow \!\!\! &
u u^* =  \frac{4^{-\nu-1} \Gamma^2(-\nu)}{\pi (-q_1)^{1+2\nu}}
\frac{(a a')^{-\nu -\frac{D-1}2}}{(H H^{'})^{\frac12 +\nu}} \times
\frac1{k^{-2\nu}} \Biggl\{ 1 \!+\! O(k^2)\Biggr\} , \qquad \label{decel}
\end{eqnarray}
one can easily recognize that the naive mode sum (\ref{naive}) possesses
infrared divergences for all values of $q_1$ in the range \cite{FP,JMPW1},
\begin{equation}
-1 \leq q_1 \leq \Bigl(\frac{D \!-\! 2}{D}\Bigr) \qquad \Longrightarrow
\qquad {\rm Infrared\ Divergences} \; . \label{bad}
\end{equation}
However, most of these infrared divergences are of the power law type
which are set to zero by dimensional regularization or any other analytic
continuation method. One only encounters {\it logarithmic} infrared
divergences (either in the leading term or one of the $k^{\mp 2\nu + 2N}$
corrections) for the cases \cite{JMPW1},
\begin{eqnarray}
-1 \leq q_1 < 0 & \Longrightarrow & q_1 = - \Bigl( \frac{D \!-\! 2}{D \!-\!
2 \!+\! 2 N} \Bigr) \qquad {\rm for\ Log\ Divergence} \; , \label{bad1} \\
0 < q & \Longrightarrow & q_1 = \Bigl( \frac{D \!-\!2}{D \!+\! 2N} \Bigr)
\qquad {\rm for\ Log\ Divergence} \; . \qquad \label{bad2}
\end{eqnarray}

If one were to incorrectly employ dimensional regularization (or any
other analytic continuation method) to control the infrared divergence
of the naive mode sum (\ref{naive}) it would seem to give a finite
result for the propagator except at the discrete $q_1$ values
(\ref{bad1}-\ref{bad2}) for which there happens to be a logarithmic
infrared divergence \cite{early,recent}. In fact the mode sum is
infrared divergent for all values of $q_1$ in the range (\ref{bad}).
The right way of dealing with these infrared divergences is not
to subtract them but rather to correct whatever unphysical assumption
about the system produced them. In this case the problem derives from
incorrectly assuming that all modes of the initial state are in
coherent Bunch-Davies vacuum, even the ones with super-horizon
wavelengths which cannot be controlled by a local observer. The
system can be made infrared finite either by starting with the
super-horizon modes in some less singular vacuum \cite{AV}, or else
by working on a spatially compact manifold which has no initially
super-horizon modes \cite{TW3}. Both of these procedures augment the
naive propagator (\ref{naive}) with extra terms which can mediate
important effects \cite{JMPW2}.

The purpose of this paper is to point out that a similar situation exists
on de Sitter background ($a(t) = e^{H t}$, with $H$ constant) if one
considers different values of the mass-squared $M^2$. This has important
consequences for the construction of de Sitter invariant propagators. We
show that the formally de Sitter invariant mode sums are infrared singular
for minimally coupled
scalars with $M_S^2 \leq 0$, and for transverse vectors with $M_V^2 \leq
-2(D\!-\!1) H^2$. However, one only encounters logarithmic infrared
divergences for the special cases,
\begin{equation}
M_S^2 = -N (D \!-\! 1 \!+\! N) H^2 \qquad {\rm and} \qquad
M_V^2 = -(N \!+\! 2) (D \!-\! 1 \!+\! N) H^2 \; . \label{probs}
\end{equation}
Using dimensional regularization (or any other analytic continuation
technique) to evaluate the naive mode sums leads to the incorrect
conclusion that a de Sitter invariant propagator exists, except for
the ``problematic'' cases (\ref{probs}). The correct result is rather
that the naive mode sums diverge for any scalar with $M_S^2 \leq 0$
and for any transverse vector with $M_V^2 \leq -2(D \!-\!1) H^2$. We
make the system infrared finite by working on the compact spatial
manifold $T^{D-1}$, and we obtain explicit results for the leading
infrared corrections to the propagator. These corrections break de
Sitter invariance, just as has long been known occur for the massless,
minimally coupled scalar \cite{AF}.

In section 2 we review our conventions for the de Sitter geometry.
Section 3 treats minimally coupled scalars, and section 4 is devoted
to transverse vectors. Section 5 summarizes and discusses our results.

\section{The de Sitter Geometry}

We work on the open conformal coordinate submanifold of $D$-dimensional de
Sitter space. A spacetime point $x^{\mu}$ can be decomposed into its temporal
($x^0$) and spatial $x^i$ components which take values in the ranges,
\begin{equation}
-\infty < x^0 < 0 \qquad {\rm and} \qquad {\rm and} -\infty < x^i <
+\infty \; .
\end{equation}
In these coordinates the invariant element is,
\begin{equation}
ds^2 \equiv g_{\mu\nu} dx^{\mu} dx^{\nu} = a_x^2 \eta_{\mu\nu} dx^{\mu}
dx^{\nu} \; ,
\end{equation}
where $\eta_{\mu\nu}$ is the Lorentz metric and $a_x = -1/Hx^0$ is the
scale factor. The parameter $H$ is known as the ``Hubble constant''.

Most of the various propagators between points $x^{\mu}$ and
$z^{\mu}$ can be expressed in terms of the de Sitter length function
$y(x;z)$,
\begin{equation}
y(x;z) \equiv a_x a_z H^2 \Biggl[ \Bigl\Vert \vec{x} \!-\! \vec{z}
\Bigr\Vert^2 - \Bigl(\vert x^0 \!-\! z^0\vert \!-\! i \epsilon\Bigr)^2
\Biggr]\; . \label{ydef}
\end{equation}
Except for the factor of $i\epsilon$ (whose purpose is to enforce Feynman
boundary conditions) the function $y(x;z)$ is closely related to the
invariant length $\ell(x;z)$ from $x^{\mu}$ to $z^{\mu}$,
\begin{equation}
y(x;z) = 4 \sin^2\Bigl( \frac12 H \ell(x;z)\Bigr) \; .
\end{equation}

Because $y(x;z)$ is a de Sitter invariant, so too are covariant
derivatives of it. With the metrics $g_{\mu\nu}(x)$ and $g_{\mu\nu}(z)$,
the first three derivatives of $y(x;z)$ furnish a convenient basis of
de Sitter invariant bi-tensors \cite{KW1},
\begin{eqnarray}
\frac{\partial y(x;z)}{\partial x^{\mu}} & = & H a_x \Bigl(y \delta^0_{\mu}
\!+\! 2 a_z H \Delta x_{\mu} \Bigr) \; , \\
\frac{\partial y(x;z)}{\partial z^{\nu}} & = & H a_z \Bigl(y \delta^0_{\nu}
\!-\! 2 a_x H \Delta x_{\nu} \Bigr) \; , \\
\frac{\partial^2 y(x;z)}{\partial x^{\mu} \partial z^{\nu}} & = & H^2 a_x a_z
\Bigl(y \delta^0_{\mu} \delta^0_{\nu} \!+\! 2 a_z H \Delta x_{\mu}
\delta^0_{\nu} \!-\! 2 a_x \delta^0_{\mu} H \Delta x_{\nu} \!-\! 2
\eta_{\mu\nu}\Bigr) \; . \qquad
\end{eqnarray}
Here and subsequently $\Delta x_{\mu} \equiv \eta_{\mu\nu} (x \!-\! z)^{\nu}$.
Acting more covariant derivatives just gives back the basis tensors, for
example \cite{KW1},
\begin{equation}
\frac{D^2 y(x;z)}{Dx^{\mu} Dx^{\nu}} = H^2 (2 \!-\!y) g_{\mu\nu}(x) \qquad ,
\qquad \frac{D^2 y(x;z)}{Dz^{\mu} Dz^{\nu}} = H^2 (2 \!-\!y) g_{\mu\nu}(z) \; .
\end{equation}
Similarly, the contraction of any pair of the basis tensors produces
more basis tensors \cite{KW1},
\begin{eqnarray}
g^{\mu\nu}(x) \frac{\partial y}{\partial x^{\mu}} \frac{\partial y}{\partial
x^{\nu}} & = & H^2 \Bigl(4 y - y^2\Bigr) = g^{\mu\nu}(z) \frac{\partial y}{
\partial z^{\mu}} \frac{\partial y}{\partial z^{\nu}} \; , \\
g^{\mu\nu}(x) \frac{\partial y}{\partial x^{\nu}} \frac{\partial^2 y}{
\partial x^{\mu} \partial z^{\sigma}} & = & H^2 (2-y) \frac{\partial y}{
\partial z^{\sigma}} \; , \\
g^{\rho\sigma}(z) \frac{\partial y}{\partial z^{\sigma}}
\frac{\partial^2 y}{\partial x^{\mu} \partial z^{\rho}} & = & H^2 (2-y)
\frac{\partial y}{\partial x^{\mu}} \; , \\
g^{\mu\nu}(x) \frac{\partial^2 y}{\partial x^{\mu} \partial z^{\rho}}
\frac{\partial^2 y}{\partial x^{\nu} \partial z^{\sigma}} & = & 4 H^4
g_{\rho\sigma}(z) - H^2 \frac{\partial y}{\partial z^{\rho}}
\frac{\partial y}{\partial z^{\sigma}} \; , \\
g^{\rho\sigma}(z) \frac{\partial^2 y}{\partial x^{\mu}\partial z^{\rho}}
\frac{\partial^2 y}{\partial x^{\nu} \partial z^{\sigma}} & = & 4 H^4
g_{\mu\nu}(x) - H^2 \frac{\partial y}{\partial x^{\mu}} \frac{\partial y}{
\partial x^{\nu}} \; .
\end{eqnarray}

\section{Scalars}

The purpose of this section is to demonstrate that the de Sitter
invariant propagator equation has no de Sitter invariant solution for
a scalar with $M_s^2 \leq 0$, and then to construct the leading de
Sitter breaking correction terms. We begin by giving the propagator
equation and the plane wave mode functions for Bunch-Davies vacuum.
The associated mode sum can be evaluated formally to give a de Sitter
invariant hypergeometric function which diverges for isolated values
of the scalar mass. By studying the infrared behavior of the mode sum
we show that these isolated values are those for which one of the
infrared divergences, which are actually present for all $M_s^2 \leq 0$,
happens to become logarithmic. Except at the isolated values of $M_S^2$,
all the infrared divergences are of the power law type which dimensional
regularization (or any analytic continuation method) incorrectly sets
to zero. We fix the problem by working on a compact spatial manifold
which has no initially super-horizon modes, and we derive the leading
infrared corrections.

The propagator of a minimally coupled scalar with mass $M_S$
obeys the equation,
\begin{equation}
\sqrt{-g(x)} \, \Bigl[ \square_x - M_S^2\Bigr] i\Delta(x;z)
= i \delta^D(x \!-\! z) \; . \label{nuprop}
\end{equation}
The plane wave mode function corresponding to Bunch-Davies vacuum is,
\begin{equation}
u(x^0,k) \equiv \sqrt{\frac{\pi}{4 H}} \; a_x^{-\frac{D-1}2} \,
H^{(1)}_{\nu}(-k x^0) \quad {\rm where} \quad \nu =
\sqrt{\Bigl(\frac{D \!-\!1}2\Bigr)^2 \!-\! \frac{M_S^2}{H^2}} \; . \label{unu}
\end{equation}
The Fourier mode sum for the propagator on infinite space is \cite{JMPW1},
\begin{eqnarray}
\lefteqn{i\Delta^{\rm dS}(x;z) = \int \!\! \frac{d^{D-1}k}{(2\pi)^{D-1}} \,
e^{i \vec{k} \cdot (\vec{x} - \vec{z})} \Biggl\{ \theta(x^0 \!-\! z^0)
u(x^0,k) u^*(z^0,k) } \nonumber \\
& & \hspace{6.4cm} + \theta(z^0 \!-\! x^0) u(x^0,k) u(z^0,k) \Biggr\} .
\qquad \label{modesum}
\end{eqnarray}
When this sum exists the result is de Sitter invariant \cite{CT},
\begin{eqnarray}
\lefteqn{i\Delta^{\rm dS}(x;z) } \nonumber \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}}
\frac{\Gamma(\frac{D-1}2 \!+\! \nu) \Gamma(\frac{D-1}2 \!-\! \nu)}{
\Gamma(\frac{D}2)} \, \mbox{}_2 F_1\Bigl( \frac{D-1}2 \!+\! \nu,
\frac{D-1}2 \!-\! \nu; \frac{D}2;1 \!-\! \frac{y}4\Bigr) \; , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2} \Gamma(\frac{D}2 \!-\! 1)}{(4
\pi)^{\frac{D}2}} \Biggl\{ \Bigl( \frac{4}{y}\Bigr)^{\frac{D}2 -1}
\mbox{}_2 F_1\Bigl(\frac12 \!+\!\nu,\frac12 \!-\! \nu ; 2 \!-\!
\frac{D}2 ; \frac{y}4\Bigr) \qquad \nonumber \\
& & + \frac{\Gamma(\frac{D-1}2 \!+\!\nu) \Gamma(\frac{D-1}2 \!-\!
\nu) \Gamma(1 \!-\! \frac{D}2)}{\Gamma(\frac12 \!+\! \nu)
\Gamma(\frac12 \!-\! \nu) \Gamma(\frac{D}2 \!-\! 1)} \,
\mbox{}_2 F_1\Bigl(\frac{D-1}2 \!+\! \nu , \frac{D-1}2 \!-\! \nu ;
\frac{D}2 ; \frac{y}4\Bigr) \Biggr\} , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{(4 \pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\! 1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-1}
\nonumber \\
& & \hspace{2cm} - \frac{\Gamma(\frac{D}2) \Gamma(1 \!-\! \frac{D}2)}{
\Gamma(\frac12 \!+\! \nu) \Gamma(\frac12 \!-\! \nu)} \sum_{n=0}^{\infty}
\Biggl[
\frac{\Gamma(\frac32 \!+\! \nu \!+\! n) \Gamma(\frac32 \!-\! \nu \!+\! n)}{
\Gamma(3 \!-\! \frac{D}2 \!+\! n) \, (n \!+\! 1)!} \Bigl(\frac{y}4
\Bigr)^{n - \frac{D}2 +2} \nonumber \\
& & \hspace{5cm} - \frac{\Gamma(\frac{D-1}2 \!+\! \nu \!+\! n)
\Gamma(\frac{D-1}2 \!-\! \nu \!+\! n)}{\Gamma(\frac{D}2 \!+\! n) \,
n!} \Bigl(\frac{y}4\Bigr)^n \Biggr] \Biggr\} . \qquad \label{expansion}
\end{eqnarray}

The gamma function $\Gamma(\frac{D-1}2 -\nu + n)$ on the final line of
(\ref{expansion}) diverges for,
\begin{equation}
\nu = \Bigl(\frac{D\!-\!1}{2}\Bigr) + N \qquad \Longleftrightarrow \qquad
M_S^2 = - N(D \!-\!1 \!+\! N) H^2 \; . \label{logprob}
\end{equation}
Its origin can be understood by performing the angular integration in the
naive mode sum (\ref{modesum}) and then changing to the dimensionless
variable $\tau \equiv k/H\sqrt{a_x a_z}$,
\begin{eqnarray}
\lefteqn{ i\Delta^{\rm dS}(x;z) = \frac{ (a_x a_z)^{-(\frac{D-1}2)}}{
2^D \pi^{\frac{D-3}2} H} \int_0^{\infty} \!\! dk \, k^{D-2}
\Bigl( \frac12 k \Delta x\Bigr)^{-(\frac{D-3}2)}
J_{\frac{D-3}2}(k \Delta x) } \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}(-k x^0)
H_{\nu}^{(1)}(-k z^0)^* + \theta(z^0 \!-\! x^0) \Bigl({\rm conjugate}
\Bigr) \Biggr\} , \qquad \\
& & \hspace{-.5cm} = \frac{H^{D-2}}{2^D \pi^{\frac{D-3}2}} \int_0^{\infty}
\!\! d\tau \, \tau^{D-2} \Bigl( \frac12 \sqrt{a_x a_z} \, H \Delta x \tau
\Bigr)^{-(\frac{D-3}2)} J_{\frac{D-3}2}\Bigl( \sqrt{a_x a_z} \,
H \Delta x \tau\Bigr) \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}\Bigl(
\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr) H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}}
\, \tau\Bigr)^* + \theta(z^0 \!-\! x^0) \Bigl({\rm conjugate}\Bigr)
\Biggr\} . \qquad \label{integral}
\end{eqnarray}
In these and subsequent expressions we define $\Delta x \equiv \Vert
\vec{x} \!-\! \vec{z}\Vert$. That the divergence at (\ref{logprob})
is infrared can be seen from the small argument expansion of the Bessel
function and from its relation to the Hankel function,
\begin{eqnarray}
J_{\nu}(x) & = & \sum_{n=0}^{\infty} \frac{ (-1)^n (\frac12 x)^{\nu +2n}}{
n! \Gamma(\nu \!+\! n \!+\! 1)} \; , \\
H_{\nu}^{(1)}(x) & = & \frac{i \Gamma(\nu) \Gamma(1 \!-\! \nu)}{\pi}
\Bigl\{ e^{-i\nu \pi} J_{\nu}(x) \!-\! J_{-\nu}(x)\Bigr\} \; .
\end{eqnarray}
The small $\tau$ behavior of the integrand (\ref{integral}) derives
from three factors, the first of which is $\tau^{D-2}$. The second
factor from the Bessel function takes the form,
\begin{equation}
\Bigl( \frac12 \sqrt{a_x a_z} \, H \Delta x \tau\Bigr)^{-(\frac{D-3}2)}
J_{\frac{D-3}2}\Bigl( \sqrt{a_x a_z} \, H \Delta x \tau\Bigr) =
\frac1{\Gamma(\frac{D-1}2)} \sum_{n=0}^{\infty} C_1(n) \tau^{2n} \; .
\end{equation}
And the final factor from the Hankel functions is,
\begin{equation}
H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr)
H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr)^* =
\frac{2 \Gamma(\nu) \Gamma(2 \nu)}{\pi^{\frac32} \Gamma(\nu \!+\! \frac12)
\tau^{2\nu} } \sum_{n=0}^{\infty} C_2(n) \tau^{2n} \; .
\end{equation}
One does not need the coefficients $C_1(n)$ and $C_2(n)$ to see that the
small $\tau$ expansion of the integrand takes the form,
\begin{eqnarray}
\lefteqn{\tau^{D-2} \times \frac1{\Gamma(\frac{D-1}2)} \sum_{k=0}^{\infty}
C_1(k) \tau^{2k} \times \frac{\Gamma^2(\nu) 2^{2\nu}}{\pi^2 \tau^{2\nu} }
\sum_{\ell=0}^{\infty} C_2(\ell) \tau^{2\ell} } \nonumber \\
& & \hspace{4cm} = \frac{2 \Gamma(\nu) \Gamma(2\nu)}{\pi^{\frac32}
\Gamma(\frac{D-1}2) \Gamma(\nu \!+\! \frac12)} \, \tau^{D-2-2\nu}
\sum_{n=0}^{\infty} C_3(n) \tau^{2n} \; . \qquad \label{IR}
\end{eqnarray}
Hence the naive mode sum (\ref{modesum}) is infrared divergent for
\begin{equation}
D - 2 - 2\nu \leq -1 \qquad \Longleftrightarrow \qquad M_S^2 \leq 0 \; .
\label{truediv}
\end{equation}
However, there will only be a {\it logarithmic} infrared divergence,
either from the leading term in (\ref{IR}) or from one of the series
corrections at $n = N$, if one has,
\begin{equation}
D - 2 - 2\nu + 2N = -1 \qquad \Longleftrightarrow \qquad M_S^2 = -N
(D \!-\! 1 \!+\! N) H^2 \; .
\end{equation}
This is precisely the condition (\ref{logprob}) for the formal, de Sitter
invariant mode sum (\ref{expansion}) to diverge.

As emphasized in the Introduction, the appeance of an infrared
divergence signals that something is unphysical about the quantity
being computed. The correct response to an infrared divergence is
not to subtract it off, either explicitly or implicitly with the
automatic subtraction of some analytic regularization technique.
One must instead understand the physical problem which caused the
divergence and then to fix the problem.

The divergence (\ref{truediv}) occurs because of the way the
Bunch-Davies mode functions (\ref{unu}) depend upon $k$ for small $k$.
The unphysical thing about having Bunch-Davies vacuum for arbitrarily
small $k$ is that no experimentalist can causally enforce it (or any
other condition) for super-horizon modes. This has led to two fixes:
\begin{enumerate}
\item{One can continue to work on the spatial manifold $R^{D-1}$
but assume the initial state is released with its super-horizon modes
in some less singular condition \cite{AV}; or}
\item{One can work on the compact spatial manifold $T^{D-1}$
with its coordinate radius chosen such that the initial state has no
super-horizon modes \cite{TW3}.}
\end{enumerate}
We will adopt the latter fix. Of course this makes the mode sum
discrete but the integral approximation should be excellent, and
gives a simple expression for the propagator which differs from
(\ref{modesum}) only by having an infrared cutoff at
$k = H$.\footnote{Making the integral approximation does not
alter the renormalization of various $M_s^2 = 0$ scalar models at
one loop order \cite{vacpol,fermself,KW1,KW2}, or even at two loops
\cite{OW,BOW,PTW1,PTW2}. Because the physical graviton polarizations
have the same mode functions as scalars with $M_S^2 = 0$
\cite{Grishchuk}, one can also test the integral approximation with
the graviton propagator. There is no disruption of powerful
consistency checks such as the Ward identity at tree order and one
loop \cite{TW4}, or the nature of allowed one loop counterterms
\cite{TW5,MW1,KW3}.}

From the preceding discussion we see that the infrared corrected
propagator $i\Delta(x;z)$  is just (\ref{integral}) with the lower
limit cutoff at $\tau = 1/\sqrt{a_x a_z}$,
\begin{eqnarray}
\lefteqn{i\Delta(x;z) = \frac{H^{D-2}}{2^D \pi^{\frac{D-3}2}}
\int_{\frac1{\sqrt{a_x a_z}}}^{\infty} \!\!\!\!\! d\tau \, \tau^{D-2}
\frac{J_{\frac{D-3}2}( \sqrt{a_x a_z} \, H \Delta x \tau)}{(\frac12
\sqrt{a_x a_z} \, H \Delta x \tau)^{\frac{D-3}2} } } \nonumber \\
& & \times \Biggl\{ \theta(x^0 \!-\! z^0) H_{\nu}^{(1)}\Bigl(
\sqrt{\frac{a_z}{a_x}} \, \tau\Bigr) H_{\nu}^{(1)}\Bigl(\sqrt{\frac{a_x}{a_z}}
\, \tau\Bigr)^* + \theta(z^0 \!-\! x^0) \Bigl({\rm conjugate}\Bigr)
\Biggr\} . \qquad
\end{eqnarray}
Of course we can express the truncated integral as the full one
minus an integral over just the infrared,
\begin{equation}
\int_{\frac1{\sqrt{a_x a_z}}}^{\infty} \!\!\!\!\! d\tau = \int_0^{\infty} \!\!
d\tau - \int_0^{\frac1{\sqrt{a_x a_z}}} \!\!\! d\tau \quad
\Longleftrightarrow \quad i\Delta(x;z) \equiv i\Delta^{\rm dS}(x;z) +
\Delta^{\rm IR}(x;z) \; .
\end{equation}
In this case it does not matter if dimensional regularization is used to
evaluate both $i\Delta^{\rm dS}(x;z)$ and $\Delta^{\rm IR}(x;z)$ because
the errors we make at the lower limits will cancel.

A further simplification is that $\Delta^{\rm IR}(x;z)$ only needs to include
the infrared singular terms which grow as $a_x a_z$ increases. These terms
come entirely from the $J_{-\nu}$ parts of the Hankel function and they
are entirely real,
\begin{eqnarray}
\lefteqn{\Delta^{\rm IR}(x;z) = -\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{2
\Gamma(\nu) \Gamma(2\nu)}{\Gamma(\nu \!+\! \frac12)}
\int_0^{\frac1{\sqrt{a_x a_z}}} \!\!\! d\tau \, \tau^{D-2}
\frac{J_{\frac{D-3}2}( \sqrt{a_x a_z} \, H \Delta x \tau)}{(\frac12
\sqrt{a_x a_z} \, H \Delta x \tau)^{\frac{D-3}2} } } \nonumber \\
& & \hspace{5.5cm} \times \frac{\Gamma^2(1\!-\!\nu)}{2^{2\nu}} \,
J_{-\nu}\Bigl( \sqrt{\frac{a_z}{a_x}} \, \tau\Bigr) J_{-\nu}\Bigl(
\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr) \; . \qquad \label{homo}
\end{eqnarray}
Before giving the general result for $\Delta^{\rm IR}(x;z)$ it is
instructive to work out the first two terms in the small $\tau$
expansion of the integrand,
\begin{eqnarray}
\lefteqn{\tau^{D-2} \, \frac{J_{\frac{D-3}2}( \sqrt{a_x a_z} \, H
\Delta x \tau)}{(\frac12 \sqrt{a_x a_z} \, H \Delta x \tau)^{\frac{D-3}2} }
\frac{\Gamma^2(1 \!-\! \nu)}{2^{2\nu}} J_{-\nu}\Bigl( \sqrt{\frac{a_z}{a_x}}
\, \tau\Bigr) J_{-\nu}\Bigl(\sqrt{\frac{a_x}{a_z}} \, \tau\Bigr) } \nonumber \\
& & \hspace{-.2cm} = \frac{\tau^{D-2-2\nu}}{\Gamma(\frac{D-1}2)} \Biggl\{1
- \frac{a_x a_z H^2 \Delta x^2 \tau^2}{2 (D \!-\! 1)} + O(\tau^4)\Biggr\}
\Biggl\{1 + \frac{(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}) \tau^2}{4
(\nu\!-\!1)} + O(\tau^4)\Biggr\} . \qquad
\end{eqnarray}
Now use the definition (\ref{ydef}) of the de Sitter length function
to infer,
\begin{equation}
y = a_x a_z H^2 \Bigl[ \Delta x^2 - \Bigl(\frac1{H a_x} \!-\!
\frac1{H a_z}\Bigr)^2 \Bigr] \; \Longrightarrow \; a_x a_z H^2
\Delta x^2 = (y \!-\! 2) + \Bigl(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}
\Bigr) \; .
\end{equation}
Hence we have,
\begin{eqnarray}
\lefteqn{\Delta^{\rm IR}(x;z) = -\frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{2
\Gamma(\nu) \Gamma(2\nu)}{\Gamma(\frac{D-1}2) \Gamma(\nu \!+\! \frac12)}
\int_0^{\frac1{\sqrt{a_x a_z}}} \!\!\! d\tau \, \tau^{D-2-2\nu} } \nonumber \\
& & \hspace{1.5cm} \times \Biggl\{1 - \Biggl[ \frac{y \!-\! 2}{2 (D\!-\!1)}
\!+\! \frac{ \nu \!-\! (\frac{D+1}2)}{2 (D \!-\!1) (\nu \!-\! 1)} \Bigl(
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) \Biggr] \tau^2 + O(\tau^4)
\Biggr\} , \qquad \\
& & = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \frac{\Gamma(\nu) \Gamma(2\nu)}{
\Gamma(\frac{D-1}2) \Gamma(\nu \!+\! \frac12)} \Biggl\{
\frac{(a_x a_z)^{\nu - (\frac{D-1}2)}}{\nu \!-\! (\frac{D-1}2)} \nonumber \\
& & \hspace{1cm} - \Biggl[ \frac{y \!-\! 2}{2 (D\!-\!1)} \!+\!
\frac{ \nu \!-\! (\frac{D+1}2)}{2 (D \!-\!1) (\nu \!-\! 1)} \Bigl(
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) \Biggr] \frac{(a_x a_z)^{\nu
- (\frac{D+1}2)}}{\nu - (\frac{D+1}2)} + \dots \Biggr\} . \qquad \label{exp}
\end{eqnarray}
One can see that the first and second terms of (\ref{exp}) respectively
cancel the $N=0$ and $N=1$ divergences in the naive mode sum
(\ref{expansion}).

To find the general form of $\Delta^{\rm IR}(x;z)$ we first note (from its
expression as an integral over $k$) that it is annihilated by $\square
- M_S^2$. We next note from (\ref{exp}) that $\Delta^{\rm IR}(x;z)$ consists
of a series of terms, each one of which has the form $(a_x a_z)^{\nu -
(\frac{D-1}2) - N}$ times a series involving powers of $(y-2)$ and
$(\frac{a_x}{a_z} + \frac{a_z}{a_x})$. The contributions at fixed $N$
must be separately annihilated by $\square - M_s^2$, and the coefficient
of the highest power of $(y-2)$ at fixed $N$ derives enitrely from the
$N$th order term in the expansion of the Bessel function $J_{\frac{D-3}2}$.
These two facts imply,
\begin{eqnarray}
\lefteqn{\Delta^{\rm IR}(x;z) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}}
\frac{\Gamma(\nu) \Gamma(2\nu)}{\Gamma(\frac{D-1}2)
\Gamma(\nu \!+\! \frac12)} } \nonumber \\
& & \hspace{.7cm} \times \sum_{N=0}^{\infty} \frac{(a_x a_z)^{\nu -
(\frac{D-1}2) - N}}{\nu \!-\! (\frac{D-1}2) \!-\! N} \sum_{n=0}^N
\Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n \sum_{m=0}^{
[\frac{N-n}2]} C_{Nnm} (y \!-\!2)^{N-n-2m} \; , \qquad \label{series}
\end{eqnarray}
where the coefficients $C_{Nnm}$ are,
\begin{eqnarray}
\lefteqn{C_{Nnm} = \frac{(-\frac14)^N}{m! n! (N \!-\!n \!-\! 2m)!} \times
\frac{\Gamma(\frac{D-1}2 \!+\! N \!+\! n \!-\! \nu)}{\Gamma(\frac{D-1}2
\!+\! N \!-\! \nu)} } \nonumber \\
& & \hspace{2cm} \times \frac{\Gamma(\frac{D-1}2)}{\Gamma(\frac{D-1}2
\!+\! N\!-\! 2m)} \times \frac{\Gamma(1 \!-\!\nu)}{\Gamma(1 \!-\! \nu
\!+\! n \!+\! 2m)} \times \frac{\Gamma(1 \!-\! \nu)}{\Gamma(1 \!-\! \nu
\!+\! m)} \; . \qquad \label{cdef}
\end{eqnarray}
Of course there is no point in extending the sum over $N$ to values
$N > \nu -\frac{D-1}2)$ for which the exponent of $a_x a_z$ becomes
negative. Those terms rapidly approach zero, and they can be dropped
without affecting the propagator equation because they are separately
annihilated by $\square - M_s^2$.

We conclude this section by discussing three special cases which occur
with such frequency as to merit special notation. These are
\begin{eqnarray}
M_S^2 = (D\!-\!2) H^2 & \!\!\! \Longrightarrow \!\!\! & \nu = \frac{D\!-\!3}2
\Rightarrow i\Delta_B(x;z) = B(y) \; , \\
M_S^2 = 0 & \!\!\! \Longrightarrow \!\!\! & \nu = \frac{D\!-\!1}2
\Rightarrow i\Delta_A(x;z) = A(y) + \delta A(a_x,a_z,y) \; , \qquad \\
M_S^2 = -D H^2 & \!\!\! \Longrightarrow \!\!\! & \nu = \frac{D\!+\!1}2
\Rightarrow i\Delta_W(x;z) = W(y) + \delta W(a_x,a_z,y) \; . \qquad
\end{eqnarray}
Although the $B$-type propagator is de Sitter invariant, its $A$-type
and $W$-type cousins have de Sitter breaking parts,
\begin{eqnarray}
\delta A & = & k \ln(a_x a_z) \; , \label{dsbA} \\
\delta W & = & k \Biggl\{ (D\!-\!1)^2 a_x a_z - \Bigl(\frac{D\!-\!1}2\Bigr)
\ln(a_x a_z) (y \!-\! 2) - \Bigl(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)
\Biggr\} . \qquad
\end{eqnarray}
The constant $k$ is,
\begin{equation}
k \equiv \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \, \frac{\Gamma(D \!-\! 1)}{
\Gamma(\frac{D}2)} \; .
\end{equation}
The main, de Sitter invariant parts of each propagator consist of a few,
potentially ultraviolet divergent terms (at $y=0$), plus an infinite series,
\begin{eqnarray}
\lefteqn{B(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2 -1} }
\nonumber \\
& & \hspace{1.5cm} + \sum_{n=0}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2)}{(n \!+\! 1)!}
\Bigl(\frac{y}4 \Bigr)^{n - \frac{D}2 +2} \!\!\!\!\! -
\frac{\Gamma(n \!+\! D \!-\! 2)}{\Gamma(n \!+\! \frac{D}2)}
\Bigl(\frac{y}4 \Bigr)^n \Biggr] \Biggr\} , \qquad \label{DeltaB} \\
\lefteqn{A(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{
\frac{D}2 -1} \!+\! \frac{\Gamma(\frac{D}2 \!+\! 1)}{\frac{D}2 \!-\! 2}
\Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-2} \!+\! A_1 } \nonumber \\
& & \hspace{1.5cm} - \sum_{n=1}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2\!+\!1)}{(n\!-\!\frac{D}2\!+\!2) (n \!+\! 1)!}
\Bigl(\frac{y}4 \Bigr)^{n - \frac{D}2 +2} \!\!\!\!\! -
\frac{\Gamma(n \!+\! D \!-\! 1)}{n \Gamma(n \!+\! \frac{D}2)}
\Bigl(\frac{y}4 \Bigr)^n \Biggr] \Biggr\} , \qquad \label{DeltaA} \\
\lefteqn{W(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Gamma\Bigl(\frac{D}2 \!-\!1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2 -1}
\!+\! \frac{\Gamma(\frac{D}2 \!+\! 2)}{(\frac{D}2 \!-\! 2) (\frac{D}2 \!-\!1)}
\Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-2} } \nonumber \\
& & \hspace{3cm} \!+\! \frac{\Gamma(\frac{D}2 \!+\! 3)}{2 (\frac{D}2 \!-\! 3)
(\frac{D}2\!-\!2)} \Bigl(\frac{4}{y} \Bigr)^{\frac{D}2-3} \!+\! W_1 \!+\!
W_2 \Bigl(\frac{y \!-\!2}4\Bigr) \nonumber \\
& & \hspace{1.5cm} + \sum_{n=2}^{\infty} \Biggl[
\frac{\Gamma(n\!+\!\frac{D}2\!+\!2) (\frac{y}4)^{n-\frac{D}2+2}}{(n\!-\!
\frac{D}2\!+\!2) (n \!-\! \frac{D}2 \!+\!1) (n \!+\! 1)!} - \frac{\Gamma(n
\!+\! D) (\frac{y}4)^n }{n (n \!-\!1) \Gamma(n \!+\! \frac{D}2)}
\Biggr] \Biggr\} . \qquad \label{DeltaW}
\end{eqnarray}
And the $D$-depdendent constants $A_1$, $W_1$ and $W_2$ are,
\begin{eqnarray}
A_1 & = & \frac{\Gamma(D\!-\!1)}{\Gamma(\frac{D}2)} \Biggl\{
-\psi\Bigl(1 \!-\! \frac{D}2\Bigr) + \psi\Bigl(\frac{D\!-\!1}2\Bigr) +
\psi(D \!-\!1) + \psi(1) \Biggr\} , \\
W_1 & = & \frac{\Gamma(D\!+\!1)}{\Gamma(\frac{D}2 \!+\!1)} \Biggl\{
\frac{D \!+\!1}{2 D} \Biggr\} , \\
W_2 & = & \frac{\Gamma(D\!+\!1)}{\Gamma(\frac{D}2 \!+\!1)} \Biggl\{
\psi\Bigl(-\frac{D}2\Bigr) - \psi\Bigl(\frac{D\!+\!1}2\Bigr) -
\psi(D \!+\!1) - \psi(1) \Biggr\} .
\end{eqnarray}

The infinite series terms of $B(y)$, $A(y)$ and $W(y)$ makes expressions
(\ref{DeltaB}-\ref{DeltaW}) seem intimidating. However, note that each
pair of terms in the infinite sums cancels for $D=4$, so they only need
to be retained when multiplying a potentially divergent quantity. Further,
because $y^n$ vanishes more and more strongly at coincidence as $n$
increases, only a handful of the smallest $n$ terms ever need to be
included. This makes loop computations manageable. For a massless,
minimally coupled scalar with a quartic self-interaction, two loop
results have been obtained for the expectation value of the stress tensor
\cite{OW}, for the scalar self-mass-squared \cite{BOW} and for the
quantum-corrected mode functions \cite{KO}. In Yukawa theory it has been
used to compute the expectation value of the coincident vertex function
at two loop order \cite{MW2}, and it has been used for a variety of two
loop computations in scalar quantum electrodynamics \cite{PTW1,PTW2}.

The need for de Sitter breaking terms in $i\Delta_A(x;z)$ has long
been recognized \cite{AF}, and ours reproduces the classic and well
known result for the coincidence limit of the propagator \cite{classic}.
The de Sitter breaking terms also show up in the differential equations
obeyed by the de Sitter invariant parts of the various propagators,
\begin{eqnarray}
(4 y \!-\! y^2) B'' + D (2 \!-\! y) B' - (D\!-\!2) B & = & 0 \; ,\label{Beqn}\\
(4 y \!-\! y^2) A'' + D (2 \!-\! y) A' & = & (D \!-\! 1) k \; , \\
(4 y \!-\! y^2) W'' + D (2 \!-\! y) W' + D W & = & \frac12 (D \!+\!1)
(D\!-\!1) k (2 \!-\! y) \; .
\end{eqnarray}
Whereas the equation for $B(y)$ is homogeneous, the equations for
$A(y)$ and $W(y)$ both possess inhomgeneous terms which are cancelled
by $\square - M_S^2$ acting on the de Sitter breaking terms
$\delta A(a_x,a_z,y)$ and $\delta W(a_x,a_z,y)$. Finally, we give some
differential relations between the de Sitter invariant parts which
follow from the series expansions (\ref{DeltaB}-\ref{DeltaW}) \cite{MTW},
\begin{eqnarray}
(2 \!-\! y) A' - k & = & 2 B' \; , \label{rel1} \\
(2 \!-\! y) W'' +\frac12 (D \!-\!1) k & = & 2 A'' \; .
\end{eqnarray}

\section{Vectors}

The purpose of this section is to demonstrate that the de Sitter invariant
propagator equation possesses no de Sitter invariant solution for a massive
vector with $M_V^2 \leq -2(D-1) H^2$. We begin by explaining how the full
vector propagator (longitudinal plus transverse) can be written as the
transverse vector propagator plus a double gradient of the difference of
two known scalar propagators. We then derive a formal, de Sitter invariant
solution for the transverse vector propagator in terms of scalar propagators.
One of these scalar propagators possesses the infrared divergences we found
in the previous section. We show how the problem can be corrected, and we
derive the leading infrared correction.

The full propagator for a massive vector obeys the equation,
\begin{equation}
\sqrt{-g(x)} \, \Bigl[ \square_x - (D\!-\!1) H^2 - M_V^2\Bigr] \,
i\Bigl[\mbox{}_{\mu} \Delta_{\nu}\Bigr](x;z) = g_{\mu\nu} \, i
\delta^D(x \!-\!z) \; . \label{FPeqn}
\end{equation}
Although this object does appear in certain projection operators, there
is greater physical interest in its transverse part which obeys,
\begin{equation}
g^{\mu\nu}(x) \frac{D}{D x^{\mu}} i\Bigl[\mbox{}_{\nu} \Delta^{\rm T}_{\rho}
\Bigr](x;z) = 0 = g^{\rho\sigma}(z) \frac{D}{D z^{\rho}} i\Bigl[\mbox{}_{\mu}
\Delta^{\rm T}_{\sigma}\Bigr](x;z) \; . \label{transcon}
\end{equation}
An excellent early study of the massive, transverse vector propagator was
carried out by Allen and Jacobson \cite{AJ}. A minor error in their work
is that the source term is not transverse. When this is corrected the
propagator equation reads \cite{TW6},
\begin{eqnarray}
\lefteqn{\sqrt{-g(x)} \Bigl[\square_x - (D\!-\!1) H^2 - M_V^2\Bigr]
i\Bigl[\mbox{}_{\mu} \Delta^{\rm T}_{\nu}\Bigr](x;z) } \nonumber \\
& & \hspace{4cm} = g_{\mu\nu} i\delta^D(x \!-\! z) + \sqrt{-g(x)} \,
\frac{\partial}{\partial x^{\mu}} \frac{\partial}{\partial z^{\nu}}
i\Delta_A(x;z) \; . \qquad \label{VPeqn}
\end{eqnarray}
Although $i\Delta_A(x;z)$ contains a de Sitter breaking part,
$k \ln(a_x a_z)$, this makes no contribution when the propagator
is differentiated on both of its arguments. Therefore, equation
(\ref{VPeqn}) is fully de Sitter invariant.

Given the transverse vector propagator one can construct the full vector
propagator by adding the double gradient of a longitudinal part,
\begin{equation}
i\Bigl[\mbox{}_{\mu} \Delta_{\nu}\Bigr](x;z) =
i\Bigl[\mbox{}_{\mu} \Delta^{\rm T}_{\nu}\Bigr](x;z) +
\frac{\partial}{\partial x^{\mu}} \frac{\partial}{\partial z^{\nu}}
\, i\Delta^{\rm L}(x;z) \; . \label{fullprop}
\end{equation}
The equation obeyed by $i\Delta^{\rm L}(x;z)$ follows from substituting
(\ref{fullprop}) in (\ref{FPeqn}), commuting some derivatives and using
(\ref{VPeqn}) to conclude,
\begin{equation}
\frac{\partial}{\partial x^{\mu}} \frac{\partial}{\partial z^{\nu}}
\Biggl\{ i\Delta_A(x;z) + \Bigl[ \square_x - M_V^2\Bigr] \,
i \Delta^{\rm L}(x;z) \Biggr\} = 0 . \label{Leqn}
\end{equation}
A solution to (\ref{Leqn}) is \cite{MTW},
\begin{equation}
i\Delta^{\rm L}(x;z) = \frac1{M_V^2} \Bigl[ i\Delta_A(x;z) - i\Delta_M(x;z)
\Bigr] \; , \label{DeltaL}
\end{equation}
where $i\Delta_M(x;z)$ is the scalar propagator with mass $M_S^2 = M_V^2$.

Because we can always construct the full vector propagator from its
transverse part by the procedure just described, we will henceforth
concentrate on the transverse vector propagator. The most general de
Sitter invariant, symmetric bi-tensor which obeys the transversality
condition (\ref{transcon}) can be expressed using an arbitrary function
$\gamma(y)$ and the basis tensors of section 2 \cite{TW6},
\begin{eqnarray}
\lefteqn{ i\Bigl[\mbox{}_{\mu} \Delta^{\rm dS}_{\nu}\Bigr](x;z) =
\Biggl[\frac{-(4y \!-\! y^2) \gamma' \!-\! (D\!-\!1) (2\!-\!y) \gamma}{
4 (D\!-\!1) H^2} \Biggr] \frac{\partial^2 y(x;z)}{\partial x^{\mu}
\partial z^{\nu}} } \nonumber \\
& & \hspace{6cm} + \Biggl[ \frac{(2 \!-\!y) \gamma' \!-\! (D\!-\!1) \gamma}{
4 (D\!-\!1) H^2} \Biggr] \frac{\partial y}{\partial x^{\mu}}
\frac{\partial y}{\partial z^{\nu}} \; . \qquad \label{dSprop}
\end{eqnarray}
Substituting (\ref{dSprop}) into (\ref{VPeqn}) gives the following
differential equation for $\gamma(y)$ away from coincidence (that is,
away from $x^{\mu} = z^{\mu}$) \cite{TW6},
\begin{equation}
(4y \!-\! y^2) \gamma'' + (D\!+\!2) (2\!-\!y) \gamma' -
\Bigl[2 (D\!-\!1) + \frac{M_V^2}{H^2}\Bigr] \gamma
= 2 (D\!-\!1) B'(y) \; . \qquad \label{gameqn}
\end{equation}
Recovering the delta function in (\ref{VPeqn}) additionally requires that
the most singular term for $y \rightarrow 0$ must be \cite{TW6},
\begin{equation}
\gamma(y) = \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{
\Bigl(\frac{D \!-\! 1}2\Bigr) \Gamma\Bigl(\frac{D}2 \!-\!1\Bigr)
\Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-1} + O(y^{2-\frac{D}2}) \Biggr\} .
\label{sing}
\end{equation}

At this point it helps to note that the left hand side of relation
(\ref{gameqn}) can be expressed as the derivative of a scalar kinetic
operator,
\begin{eqnarray}
\lefteqn{(4y \!-\! y^2) \gamma'' + (D\!+\!2) (2\!-\!y) \gamma' -
\Bigl[2 (D\!-\!1) + \frac{M_V^2}{H^2}\Bigr] \gamma } \nonumber \\
& & \hspace{5cm} = \frac1{H^2} \frac{\partial}{\partial y} \Bigl[
\square_x - (D\!-\!2) H^2 - M_V^2\Bigr] I[\gamma] \; , \qquad
\end{eqnarray}
where $I[y]$ stands for the indefinite integral of $\gamma(y)$ with
respect to $y$. Hence $I[\gamma]$ obeys the scalar propagator equation,
\begin{equation}
\Bigl[\square - (D\!-\!2) H^2 - M_V^2\Bigr] I[\gamma] =
2 (D\!-\!1) H^2 B \; .
\end{equation}
This is very like the equation we just solved for the longitudinal
part of the vector propagator so it should not seem surprising that
the unique solution for relations (\ref{gameqn}-\ref{sing}) is,
\begin{equation}
\gamma(y) = 2 (D\!-\!1) \frac{H^2}{M_V^2} \frac{\partial}{\partial y}
\Bigl[ E(y) - B(y)\Bigr] \; , \label{ansatz}
\end{equation}
where $B(y)$ is the scalar propagator with mass $M_S^2 = (D-2) H^2$ and
$E(y)$ is the scalar propagator with mass $M_S^2 = (D-2) H^2 + M_V^2$.
To verify (\ref{ansatz}), first note the two leading terms in the series
expansions of $B(y)$ and $E(y)$, from expressions (\ref{DeltaB}) and
(\ref{expansion}) with $\nu^2 = (\frac{D-3}2)^2 - \frac{M_V^2}{H^2}$,
\begin{eqnarray}
B(y) & \!\!=\!\! & \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{ \Gamma(\frac{D}2
\!-\! 1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-1} + \Gamma\Bigl(\frac{D}2
\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-2} + \dots \Biggr\} , \\
E(y) & \!\!=\!\! & \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \Biggl\{ \Gamma(\frac{D}2
\!-\! 1\Bigr) \Bigl(\frac{4}{y}\Bigr)^{\frac{D}2-1} + \Bigl(\nu^2 \!-\!
\frac14\Bigr) \Gamma\Bigl(\frac{D}2 \!-\! 2\Bigr) \Bigl(\frac{4}{y}
\Bigr)^{\frac{D}2-2} + \dots \Biggr\} . \qquad
\end{eqnarray}
This establishes that (\ref{ansatz}) has the correct singularity
(\ref{sing}) near coincidence. To check the off-coincidence condition
(\ref{gameqn}), note that $B(y)$ obeys (\ref{Beqn}) and $E(y)$ obeys,
\begin{equation}
(4 y \!-\! y^2) E'' + D (2 \!-\! y) E' - \Bigl[(D \!-\! 2) \!+\!
\frac{M_V^2}{H^2}\Bigr] E = 0 \; . \label{Eeqn}
\end{equation}
Differentiating (\ref{Beqn}) and (\ref{Eeqn}) with respect to $y$ gives,
\begin{eqnarray}
(4 y \!-\! y^2) B''' + (D \!+\! 2) (2 \!-\! y) B'' - \Bigl[2 (D \!-\! 1) \!+\!
\frac{M_V^2}{H^2}\Bigr] B' & = & -\frac{M_V^2}{H^2} \, B' \; , \qquad
\label{G1} \\
(4 y \!-\! y^2) E''' + (D \!+\! 2) (2 \!-\! y) E'' - \Bigl[2 (D \!-\! 1) \!+\!
\frac{M_V^2}{H^2}\Bigr] E' & = & 0 \; . \label{G2}
\end{eqnarray}
Subtracting (\ref{G1}) from (\ref{G2}) and multiplying by $2(D-1)
\frac{H^2}{M_V^2}$ shows that our ansatz (\ref{ansatz}) indeed obeys
equation (\ref{gameqn}).

Of course using our ansatz (\ref{ansatz}) in (\ref{dSprop}) only makes
sense if the two scalar propagators $B(y)$ and $E(y)$ exist! Recall
that they correspond to masses $M_S^2 = (D-2)H^2$ and $M_S^2 = (D-2)H^2
+ M_V^2$, respectively. For $D > 2$ the $B$-type propagator is safe but
the $E$-type propagator can have problems if $M_V^2$ is sufficiently
negative. Now consider its formal series expansion (\ref{expansion})
with $\nu^2 = (\frac{D-3}2)^2 - \frac{M_V^2}{H^2}$, and in particular,
the problematic gamma function, $\Gamma(\frac{D-1}2 \!-\! \nu \!+\! n)$.
Because it is really the derivative $E'(y)$ which appears in our ansatz
(\ref{ansatz}), the problem at $n=0$ drops out, but there are divergences
at $n = N+1$ for all non-negative integers $N$. This corresponds to the
following vector masses,
\begin{equation}
\nu = \Bigl(\frac{D-1}2\Bigr) + N + 1 \qquad \Longleftrightarrow \qquad
M_V^2 = - (N \!+\! 2) (N \!+\! D \!-\! 1) H^2 \; . \label{Vlogs}
\end{equation}
As emphasized in the previous section, condition (\ref{Vlogs}) only
gives the {\it logarithmic} infrared divergences. For $M_V^2 \leq -2
(D-1) H^2$ there will be power law infrared divergences which analytic
continuation techniques incorrectly subtract off. We turn now to
fixing this problem.

Because the infrared divergences we have found for $M_V^2 \leq -2(D-1) H^2$
arise from assuming a de Sitter invariant solution to the propagator
equation (\ref{VPeqn}), they can only be corrected by abandoning that
assumption. However, there is no need to discard the formal de Sitter
invariant solution $i[\mbox{}_{\mu}\Delta^{\rm dS}_{\nu}](x;z)$. Just
like the scalar of the preceding section, we need only to add to it a
de Sitter breaking, infrared correction,
\begin{equation}
i \Bigl[\mbox{}_{\mu} \Delta^{\rm T}_{\nu}\Bigr](x;z) \equiv
i \Bigl[\mbox{}_{\mu} \Delta^{\rm dS}_{\nu}\Bigr](x;z) +
\Bigl[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}\Bigr](x;z) \; .
\end{equation}
Of course the infrared correction must be symmetric and it must obey
the transversality condition (\ref{transcon}). In analogy with the
scalar, we also want it to be annihilated by the kinetic operator
$[\square - (D\!-\!1) H^2 - M_V^2]$.

Abandoning de Sitter invariance in $[\mbox{}_{\mu} \Delta^{\rm
IR}_{\nu}](x;z)$ affects the tensor structure as well as the scalar
coefficient functions. If we preserve spatial homogeneity and isotropy
(as we did for the scalars of the previous section) then the only extra
basis tensors we require are the derivatives of $u(x;z) \equiv
\ln(a_x a_z)$ with respect to $x^{\mu}$ and $z^{\nu}$,
\begin{equation}
\frac{\partial u}{\partial x^{\mu}} = H a_x \delta^0_{\mu} \qquad , \qquad
\frac{\partial u}{\partial z^{\nu}} = H a_z \delta^0_{\nu} \; .
\end{equation}
The most general homogeneous and isotropic tensor with the correct
symmetries is,
\begin{eqnarray}
\lefteqn{ \Bigl[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}\Bigr](x;z) =
F_1(a_x,a_z,y) \frac{\partial^2 y}{\partial x^{\mu} \partial z^{\nu}}
+ F_2(a_x,a_z,y) \frac{\partial y}{\partial x^{\mu}}
\frac{\partial y}{\partial z^{\nu}} } \nonumber \\
& & \hspace{-.3cm} + F_3(a_x,a_z,y) \frac{\partial u}{\partial x^{\mu}}
\frac{\partial y}{\partial z^{\nu}} + F_3(a_z,a_x,y)
\frac{\partial y}{\partial x^{\mu}} \frac{\partial u}{\partial z^{\nu}} +
F_4(a_x,a_z,y) \frac{\partial u}{\partial x^{\mu}}
\frac{\partial u}{\partial z^{\nu}} \; . \qquad \label{vecans}
\end{eqnarray}
The coefficient functions $F_1$, $F_2$ and $F_4$ must be symmetric under
interchange of $a_x$ and $a_z$. $F_3$ does not have this symmetry; when
the argument lists are suppressed (to save space) we will indicate
the interchange of $a_x$ and $a_z$ by a superscript $T$,
\begin{equation}
F_3^T(a_x,a_z,y) \equiv F_3(a_z,a_x,y) \; .
\end{equation}
We will also use a prime to denote differentiation with respect to $y$,
\begin{equation}
F_i'(a_x,a_z,y) \equiv \frac{\partial}{\partial y} \, F_i(a_x,a_z,y) \; .
\end{equation}
The right way to think about the coefficient functions $F_i(a_x,a_z,y)$
is that enforcing transversality determines $F_3$ and $F_4$ in terms of
$F_1$ and $F_2$. Then $F_1$ and $F_2$ are fixed (up to normalization of
each independent term) by demanding that $[\square - (D\!-\!1) H^2 - M_V^2]$
annihilates $[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}](x;z)$.

Covariant derivatives of the new tensor involve some extra identities in
addition to those of section 2,
\begin{equation}
\frac{D^2 u}{D x^{\mu} D x^{\nu}} = -H^2 g_{\mu\nu}(x) -
\frac{\partial u}{\partial x^{\mu}} \frac{\partial u}{\partial x^{\nu}}
\quad , \quad \frac{D^2 u}{D z^{\mu} D z^{\nu}} = -H^2 g_{\mu\nu}(z) -
\frac{\partial u}{\partial z^{\mu}} \frac{\partial u}{\partial z^{\nu}} \; .
\end{equation}
There are also some new contraction identities,
\begin{eqnarray}
g^{\mu\nu}(x) \frac{\partial u}{\partial x^{\mu}}
\frac{\partial u}{\partial x^{\nu}} & = & - H^2 \; , \\
g^{\mu\nu}(x) \frac{\partial u}{\partial x^{\mu}}
\frac{\partial y}{\partial x^{\nu}} & = & - H^2 \Bigl[ y \!-\! 2
+ 2 \frac{a_z}{a_x} \Bigr] \; , \\
g^{\mu\nu}(x) \frac{\partial u}{\partial x^{\mu}}
\frac{\partial^2 y}{\partial x^{\nu} \partial z^{\rho}} & = & - H^2
\Bigl[ \frac{\partial y}{\partial z^{\rho}} + 2 \frac{a_z}{a_x}
\frac{\partial u}{\partial z^{\rho}} \Bigr] \; .
\end{eqnarray}
The covariant divergence $g^{\mu\nu}(x) D_{\mu} i[\mbox{}_{\nu}
\Delta^{\rm IR}_{\rho}](x;z)$ produces one term proportional to
$\partial y/\partial z^{\rho}$ and another proportional to
$\partial u/\partial z^{\rho}$. The term proportional to
$\partial y/\partial z^{\rho}$ gives an equation for the
coefficient function $F_3(a_x,a_z,y)$ in terms of $F_1(a_x,a_z,y)$
and $F_2(a_x,a_z,y)$,
\begin{eqnarray}
\lefteqn{-(2\!-\!y) F_3' + D F_3 + a_x \frac{\partial F_3}{\partial a_x}
+ 2 \frac{a_z}{a_x} F_3' = (2 \!-\!y) F_1' - D F_1
- a_x \frac{\partial F_1}{\partial a_x} } \nonumber \\
& & \hspace{1.7cm} + (4y \!-\!y^2) F_2' + (D\!+\!1) (2 \!-\! y) F_2
+ (2 \!-\!y) a_x \frac{\partial F_2}{\partial a_x} - 2 a_z
\frac{\partial F_2}{\partial a_x} \; . \qquad \label{F3}
\end{eqnarray}
Because the equation involves only first derivatives with respect to
$y$ and $a_x$, the general solution can be given using the Method of
Characteristics. Once $F_3(a_x,a_z,y)$ is known, the transversality
relation proportional to $\partial u/\partial z^{\rho}$ gives an
equation for the remaining coefficient function $F_4(a_x,a_z,y)$,
\begin{eqnarray}
\lefteqn{-(2\!-\!y) F_4' + (D\!-\!1) F_4+ a_x \frac{\partial F_4}{\partial a_x}
+ 2 \frac{a_z}{a_x} F_4' = -2 a_z \frac{\partial F_1}{\partial a_x} -
2 \frac{a_z}{a_x} F_3 } \nonumber \\
& & \hspace{2cm} + (4y \!-\!y^2) {F_3^T}' + D (2 \!-\! y) F_3^T
+ (2 \!-\!y) a_x \frac{\partial F_3^T}{\partial a_x} -
2 a_z \frac{\partial F_3^T}{\partial a_x} \; . \qquad \label{F4}
\end{eqnarray}
This equation can also be solved by the Method of Characteristics.

Once one has determined the coefficient functions $F_3$ and $F_4$ from
equations (\ref{F3}-\ref{F4}), $F_1$ and $F_2$ are obtained, up to
normalization of each independent piece, by requiring
$[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}](x;z)$ to solve the homogeneous
propgator equation,
\begin{equation}
\Bigl[ \square_x - (D\!-\!1) H^2 - M_V^2\Bigr] \, \Bigl[\mbox{}_{\mu}
\Delta^{\rm IR}_{\nu}\Bigr](x;z) = 0 \; .
\end{equation}
Of course this gives a relation for each of the five tensor structures
present in $[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}](x;z)$, however, only
two of these five relations are independent. The one proportional to
$\partial^2 y/\partial x^{\mu} \partial z^{\nu}$ constrains
$F_1(a_x,a_z,y)$,
\begin{eqnarray}
\lefteqn{(4y \!-\! y^2) F_1'' + D (2 \!-\!y) F_1' - \Bigl[D \!+\!
\frac{M_V^2}{H^2} \Bigr] F_1 + 2 (2 \!-\!y) F_2  + 2(2 \!-\!y) a_x
\frac{\partial F_1'}{\partial a_x} } \nonumber \\
& & \hspace{2cm} - 4 a_z \frac{\partial F_1'}{\partial a_x} - (D\!-\!1)
a_x \frac{\partial F_1}{\partial a_x} - a_x \frac{\partial}{\partial a_x}
\Bigl[ a_x \frac{\partial F_1}{\partial a_x} \Bigr] - 2 F_3 = 0 \; .
\qquad \label{F1}
\end{eqnarray}
And $F_2(a_x,a_z,y)$ is constrained by the term proportional to
$\partial y/\partial x^{\mu} \, \partial y/\partial z^{\nu}$,
\begin{eqnarray}
\lefteqn{(4y \!-\! y^2) F_2'' + (D\!+\!4) (2 \!-\!y) F_2' - \Bigl[2D \!+\!
\frac{M_V^2}{H^2} \Bigr] F_2 - 2 F_1' + 2(2 \!-\!y) a_x
\frac{\partial F_2'}{\partial a_x} } \nonumber \\
& & \hspace{2cm} - 4 a_z \frac{\partial F_2'}{\partial a_x} - (D\!+\!1)
a_x \frac{\partial F_2}{\partial a_x} - a_x \frac{\partial}{\partial a_x}
\Bigl[ a_x \frac{\partial F_2}{\partial a_x} \Bigr] - 2 F_3' = 0 \; .
\qquad \label{F2}
\end{eqnarray}

The normalization comes from differentiating (\ref{F1}) with respect to
$y$ and then subtracting (\ref{F2}) to obtain an equation for the
combination $F_1' - F_2$. By making the definition,
\begin{equation}
F_1'(a_x,a_z,y) - F_2(a_x,a_z,y) \equiv \mathcal{E}'(a_x,a_z,y) \; ,
\end{equation}
we can identify this relation as the homogeneous equation for an
$E$-type scalar with mass $M_s^2 = (D\!-\!2) H^2 + M_V^2$,
\begin{equation}
\frac{\partial}{\partial y} \, (\ref{F1}) - (\ref{F2}) = H^{-2}
\frac{\partial}{\partial y} \Bigl[\square_x - (D\!-\!2)H^2 - M_V^2\Bigr]
\, \mathcal{E} = 0 \; .
\end{equation}
Now write the de Sitter invariant part of the propagator in a form
similar to the de Sitter breaking part,
\begin{equation}
i\Bigl[\mbox{}_{\mu} \Delta^{\rm dS}_{\nu}\Bigr](x;z) \equiv
\mathcal{F}_1(y) \frac{\partial^2 y}{\partial x^{\mu} \partial z^{\nu}}
+ \mathcal{F}_2(y) \frac{\partial y}{\partial x^{\mu}} \frac{\partial y}{
\partial z^{\nu}} \; .
\end{equation}
From relations (\ref{dSprop}-\ref{gameqn}) and (\ref{ansatz}) we infer,
\begin{equation}
\mathcal{F}_1' - \mathcal{F}_2 = -\Biggl[ \frac{(4 y \!-\! y^2) \gamma''
\!+\! (D \!+\!2) (2 \!-\!y) \gamma' \!-\! 2(D\!-\!1) \gamma}{4 (D\!-\!1) H^2}
\Biggr] = -\frac{ E'(y)}{2 H^2} \; .
\end{equation}
Hence we infer,
\begin{eqnarray}
\lefteqn{F_1'(a_x,a_z,y) - F_2(a_x,a_z,y) = -\frac1{2 H^2}
\frac{\partial}{\partial y} \, \Delta^{\rm IR}(x;z) \; , } \\
& & = -\frac1{2 H^2} \, \frac{H^{D-2}}{(4\pi)^{\frac{D}2}}
\frac{\Gamma(\nu) \Gamma(2\nu)}{\Gamma(\frac{D-1}2)
\Gamma(\nu \!+\! \frac12)} \sum_{N=0}^{\infty} \frac{(a_x a_z)^{\nu -
(\frac{D+1}2) - N}}{\nu \!-\! (\frac{D+1}2) \!-\! N} \nonumber \\
& & \hspace{.9cm} \times \sum_{n=0}^{N+1} \Bigl( \frac{a_x}{a_z} \!+\!
\frac{a_z}{a_x}\Bigr)^n \sum_{m=0}^{[\frac{N+1-n}2]} (N\!+\!1\!-\!n \!-\! 2m)
C_{N+1 \, nm} (y \!-\!2)^{N-n-2m} , \qquad \label{normalization}
\end{eqnarray}
where $\Delta^{\rm IR}(x;z)$ was defined in equation (\ref{series}), the
coefficients $C_{Nnm}$ are given in (\ref{cdef}), and the index $\nu$ obeys
$\nu^2 = (\frac{D-3}2)^2 - \frac{M_V^2}{H^2}$.

It is instructive to give the first ($N=0$) term in the series expansions
of the four coefficient functions $F_i(a_x,a_z,y)$,
\begin{eqnarray}
(F_1)_{0} & \!=\! & \mathcal{F} \Biggl[ (y \!-\! 2) + \frac{(\nu \!-\!
\frac{D+1}2) (\nu \!+\! \frac{D-1}2)}{(\nu \!+\! \frac{D-3}2) (\nu \!-\! 1)}
\Bigl(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) \Biggr]
\frac{(a_x a_z)^{\nu - \frac{D+1}2}}{\nu \!-\! \frac{D+1}2} \; , \qquad \\
(F_2)_{0} & \!=\! & -\mathcal{F} \frac{(a_x a_z)^{\nu - \frac{D+1}2}}{
\nu \!-\! \frac{D+1}2} \; , \qquad \\
(F_3)_{0} & \!=\! & -\mathcal{F} \frac{(\nu \!-\! \frac{D+1}2)}{(\nu \!+\!
\frac{D-3}2)} \Biggl[ \frac{(\frac{D+1}2)}{(\nu \!-\! 1)} \Bigl(
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) + \Bigl(\frac{a_x}{a_z} \!-\!
\frac{a_z}{a_x}\Bigr)\Biggr] \frac{(a_x a_z)^{\nu - \frac{D+1}2}}{\nu \!-\!
\frac{D+1}2} \; , \qquad \\
(F_4)_{0} & \!=\! & -\mathcal{F} \frac{(\nu \!-\! \frac{D+1}2)}{(\nu \!+\!
\frac{D-3}2)} \Biggl[ \frac{(\nu \!-\! \frac{D+3}2)}{(\nu \!-\! 1)} \Bigl(
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) (y \!-\! 2) \nonumber \\
& & \hspace{5.8cm} + 4 \frac{(\nu \!-\! \frac{D+1}2)}{(\nu \!+\! \frac{D-3}2)}
\Biggr] \frac{(a_x a_z)^{\nu-\frac{D+1}2}}{\nu \!-\! \frac{D+1}2} \; . \qquad
\end{eqnarray}
Here the index $\nu$ obeys $\nu^2 = (\frac{D-3}2)^2 - \frac{M_V^2}{H^2}$ and
the constant $\mathcal{F}$ is,
\begin{equation}
\mathcal{F} \equiv \frac1{16 H^2} \, \frac{H^{D-2}}{(4\pi)^{\frac{D}2}} \,
\frac{\Gamma(\nu) \Gamma(2\nu)}{\Gamma(\frac{D+1}2) \Gamma(\nu \!+\! \frac12)}
\; . \label{Fmath}
\end{equation}
The general series expansion for $F_1(a_x,a_z,y)$ has the form,
\begin{eqnarray}
\lefteqn{F_1 = \mathcal{F} \sum_{N=0}^{\infty} \frac{(a_x a_z)^{\nu-
(\frac{D+1}2)-N}}{\nu \!-\! (\frac{D+1}2) \!-\! N} } \nonumber \\
& & \hspace{3.5cm} \times
\sum_{n=0}^{N+1} \Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n
\sum_{m=0}^{[\frac{N+1-n}2]} F^1_{Nnm} (y \!-\!2)^{N+1-n-2m} \; .
\qquad \label{F1exp}
\end{eqnarray}
As with the de Sitter breaking corrections for the scalar, there is no
point in extending the series for $N > \nu - \frac{D+1}2$. The other
coefficient functions have the same first line as (\ref{F1exp}) so we
give only their subsequent forms,
\begin{eqnarray}
F_2 & \longrightarrow &
\sum_{n=0}^{N} \Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n
\sum_{m=0}^{[\frac{N-n}2]} F^2_{Nnm} (y \!-\!2)^{N-n-2m} \; , \qquad \\
F_3 & \longrightarrow &
\sum_{n=0}^{N} \Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n
\nonumber \\
& & \hspace{.5cm} \times \sum_{m=0}^{[\frac{N-n}2]} \Biggl[
F^{3a}_{Nnm} \Bigl(\frac{a_x}{a_z} \!+\!  \frac{a_z}{a_x}\Bigr) +
F^{3b}_{Nnm} \Bigl(\frac{a_x}{a_z} \!-\!  \frac{a_z}{a_x}\Bigr) \Biggr]
(y \!-\!2)^{N-n-2m} \; , \qquad \\
F_4 & \longrightarrow &
\sum_{n=0}^{N+1} \Bigl( \frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)^n
\sum_{m=0}^{[\frac{N+2-n}2]} F^4_{Nnm} (y \!-\!2)^{N+2-n-2m} \; . \qquad
\label{F4exp}
\end{eqnarray}

We close this section by giving the transverse vector propagator for
the important special case of, $M_V^2 = -2(D\!-\!1) H^2$. For that mass
the associated scalar has $M_S^2 = -D H^2$, corresponding to the $W$-type
propagator considered at the end of section 3. For this reason we give
the vector propagator a subscript $W$, and we decompose it into a de
Sitter invariant part and a de Sitter breaking part,
\begin{equation}
i\Bigl[\mbox{}_{\mu} \Delta^{\rm T}_{\nu}\Bigr]_{W}(x;z) =
\Bigl[\mbox{}_{\mu} W_{\nu}\Bigr](x;z) +
\Bigl[\mbox{}_{\mu} \delta W_{\nu}\Bigr](x;z) \; .
\end{equation}
The de Sitter invariant part is,
\begin{eqnarray}
\lefteqn{\Bigl[\mbox{}_{\mu} W_{\nu}\Bigr](x;z) = \Biggl[ \frac{ (4y\!-\!y^2)
(W\!-\!B)'' \!+\! (D\!-\!1) (2 \!-\! y) (W\!-\!B)'}{4 (D\!-\!1) H^2} \Biggr]
\frac{\partial^2 y}{\partial x^{\mu} \partial z^{\nu}} } \nonumber \\
& & \hspace{3cm} + \Biggl[ \frac{ -(2 \!-\! y) (W\!-\!B)'' \!+\! (D\!-\!1)
(W\!-\!B)'}{4 (D\!-\!1) H^2} \Biggr] \frac{\partial y}{\partial x^{\mu}}
\frac{\partial y}{\partial z^{\nu}} \; . \qquad
\end{eqnarray}
And the de Sitter breaking part is,
\begin{eqnarray}
\lefteqn{\Bigl[\mbox{}_{\mu} \delta W_{\nu}\Bigr] = \frac{k}{4 (D\!-\!1)
H^2} \Biggl\{\Biggl[\frac{(D\!-\!1)^2}2 \ln(a_x a_z) (y \!-\!2) \!+\! D \Bigl(
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr)\Biggr]
\frac{\partial^2 y}{\partial x^{\mu} \partial z^{\nu}} } \nonumber \\
& & \hspace{2cm} - \frac{(D\!-\!1)^2}2 \ln(a_x a_z) \frac{\partial y}{
\partial x^{\mu}} \frac{\partial y}{\partial z^{\nu}} \!-\! \Biggl[ D
\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x} \Biggr] \frac{\partial u}{\partial
x^{\mu}} \frac{\partial y}{\partial z^{\nu}} \nonumber \\
& & \hspace{3cm} - \Biggl[ \frac{a_x}{a_z} \!+\! D \frac{a_z}{a_x} \Biggr]
\frac{\partial y}{\partial x^{\mu}} \frac{\partial u}{\partial z^{\nu}} \!+\!
\Bigl(\frac{a_x}{a_z} \!+\! \frac{a_z}{a_x}\Bigr) (y \!-\! 2)
\frac{\partial u}{\partial x^{\mu}} \frac{\partial u}{\partial z^{\nu}}
\Biggr\} . \qquad
\end{eqnarray}

\section{Discussion}

We have shown that infrared divergences preclude de Sitter invariant
solutions for the propagators of either a minimally coupled scalar
with mass $M_S^2 \leq 0$, or for a transverse vector with mass
$M_V^2 \leq -2(D\!-\!1) H^2$. (If one includes the longitudinal
part of the vector propagator then infrared divergences occur for
$M_V^2 \leq 0$.) However, in most cases these infrared divergences are of
the power law type which is automatically subtracted by any regularization
which is based upon analytic continuation. (We stress that these
considerations apply as well to the standard technique of continuation from
Euclidean de Sitter space.) Only the special values of $M_S^2$ and $M_V^2$
given in equation (\ref{probs}) result in logarithmic infrared divergences
which show up in analytic regularization techniques.  Thus one might reach
the incorrect conclusion that de Sitter invariant propagators exist for all
scalar and vector masses, except for a few ``singular'' cases.

That conclusion is wrong because infrared divergences should not be
renormalized the way one treats an ultraviolet divergence. The appearnace
of an infrared divergence signals that an unphysical assumption has been
made, and the right response is to identify the problematic assumption
and modify it. In our case the unphysical assumption is that the universe
can have been prepared in coherent Bunch-Davies vacuum for arbitrarily
long wavelength modes. There is no causal process by which this can be
accomplished in the de Sitter geometry. When one assumes either that the
initially super-horizon modes are in some less singular state \cite{AV},
or else that the spatial manifold is compact \cite{TW3}, the resulting
propagators become infrared finite, but not de Sitter invariant.

In each case the true propagator can be written as the naive, de Sitter
invariant result (defined by dimensional regularization) plus a de Sitter
breaking infrared correction which is real and obeys the homogeneous
propagator equation. For the scalar propagator we have,
\begin{equation}
i\Delta(x;z) = i\Delta^{\rm dS}(x;z) + \Delta^{\rm IR}(x;z) \; ,
\end{equation}
where $i\Delta^{\rm dS}(x;z)$ is expression (\ref{expansion}) and
$\Delta^{\rm IR}(x;z)$ is given by relations (\ref{series}-\ref{cdef}).
The analogous expression for the transverse vector is,
\begin{equation}
i\Bigl[\mbox{}_{\mu} \Delta^{\rm T}_{\nu}\Bigr](x;z) =
i\Bigl[\mbox{}_{\mu} \Delta^{\rm dS}_{\nu}\Bigr](x;z) +
\Bigl[\mbox{}_{\mu} \Delta^{\rm IR}_{\nu}\Bigr](x;z) \; ,
\end{equation}
where the de Sitter invariant part is defined by expressions (\ref{dSprop})
and (\ref{ansatz}), and the de Sitter breaking terms are given by
equations (\ref{vecans}) and (\ref{Fmath}-\ref{F4exp}). The full vector
propagator, including the logitudinal part, is given by equations
(\ref{fullprop}) and (\ref{DeltaL}).

It might be wondered what physical sense it makes to consider the
propagators of particles with tachyonic masses. First, there is the
mathematical point that they don't possess de Sitter invariant
propagators, despite what one might conclude by erroneously defining
these propagators with some analytic continuation technique. Second,
there is the important issue of following the time dependent vacuum
decay which must occur when symmetry breaking takes place during a
phase of inflation. In this respect the infrared correction terms
may be quite important, as they sometimes are for the analogous case
of FRW geometries with constant deceleration \cite{JMPW2}.

Another application for our propagators is the projection operators
for higher spin propagators, in which case there are no physical particles
with tachyonic masses to worry about. For example, consider the
graviton $h_{\mu\nu}$ in exact de Donder gauge,
\begin{equation}
\Bigl(\delta^{\rho}_{\mu} D^{\sigma} - \frac12 D_{\mu} g^{\rho\sigma}\Bigr)
h_{\rho\sigma} = 0 \; . \label{deDonder}
\end{equation}
Just as the source term for the transverse vector propagator
equation (\ref{VPeqn}) must be consistent with Lorentz gauge
(\ref{transcon}), so too the source term of the graviton propagator
equation must be consistent with (\ref{deDonder}). The resulting
projection operator turns out to involve the full vector propagator,
\begin{eqnarray}
\lefteqn{i\Bigl[\mbox{}_{\mu\nu} \mathcal{P}_{\rho\sigma}\Bigr](x;x') =
g_{\mu (\rho} g_{\sigma) \nu} i \delta^D(x \!-\! x') - \frac1{D \!-\! 2}
\, g_{\mu\nu} g_{\rho\sigma} i\delta^D(x \!-\! x') \nonumber } \\
& & \hspace{2cm} + \frac12 \sqrt{-g(x)} \left\{ \matrix{
D_{\mu} D_{\rho}' \, i [\mbox{}_{\nu} \Delta_{\sigma}](x;x') +
D_{\mu} D_{\sigma}' \, i [\mbox{}_{\nu} \Delta_{\rho}](x;x') \cr
D_{\nu} D_{\rho}' \, i [\mbox{}_{\mu} \Delta_{\sigma}](x;x') +
D_{\nu} D_{\sigma}' \, i [\mbox{}_{\mu} \Delta_{\rho}](x;x') }
\right\} . \qquad \label{proj}
\end{eqnarray}
One can easily check that the de Donder gauge condition,
\begin{equation}
\Bigl[ \delta_{\alpha}^{\rho} D^{\prime \sigma} - \frac12 D_{\alpha}'
g^{\prime \rho\sigma}\Bigr] \, i \Bigl[ \mbox{}_{\mu\nu}
\mathcal{P}_{\rho\sigma}\Bigr](x;x') = 0 \; .
\end{equation}
requires the vector to have mass $M_V^2 = -2(D\!-\!1) H^2$,
\begin{equation}
\sqrt{-g(x)} \, \Bigl[ \square_x + (D\!-\!1) H^2\Bigr] \,
i\Bigl[\mbox{}_{\mu} \Delta_{\nu}\Bigr](x;x') = g_{\mu\nu}
i \delta^D(x \!-\!x') \; . \label{vec1}
\end{equation}
This is not only tachyonic, it actually corresponds to the first of
the special cases (\ref{Vlogs}) for which the transverse part harbors
a logarithmic infrared divergence, so the problem would show up even
in an analytic regularization technique.

This is all highly relevant to the debate concerning the de Sitter
invariance of free gravitons \cite{RPW}. It has long been obvious to
cosmologists that free gravitons cannot be de Sitter invariant because
they share the same mode functions as massless, minimally coupled scalars
\cite{Grishchuk}. Indeed, the tensor contribution to the primordial
anisotropies of the cosmic ray microwave background derives from
precisely the same infrared singular dependence of these mode
functions \cite{Starobinsky}. On the other hand, some relativists
insist that free gravitons must be de Sitter invariant because de
Sitter invariant solutions exist for the propagator when a gauge
fixing term is added to the action \cite{math}. In previous work we have
shown that adding these gauge fixing terms is not valid, owing to the
linearization instability \cite{MTW,TW7}. Imposing an exact gauge
condition such as (\ref{deDonder}) should still be all right, but we
have just seen that it leads to an inevitable breaking of de Sitter
invariance through the projection operator. (This same breaking can
be seen as well in noninvariant gauges by adding the appropriate
compensating gauge transformations \cite{Kleppe}.) Antoniadis and
Mottola long ago discovered a similar problem in another gauge \cite{IAEM}.
Before our current work one might have dismissed these examples as
``spurious IR divergences for the Feynman propagator in the sense that
the IR divergences are absent for other values of gauge parameters''
\cite{Higuchi}. They now appear as just those cases for which infrared
divergences, that are always present and which always break de Sitter
invariance, just happen to go from being of the power law type to
logarithmic, and hence become visible to analytic continuation techniques.
And the correct procedure is not to ignore them or subtract them but
rather to remove the erroneous assumption of de Sitter invariance.

\newpage
\centerline{\bf Acknowledgements}

This work was partially supported by FQXi Mini Grant \#MGB-08-008,
by FONDECYT grant 3100041, by European Union grant MRTN-CT-2004-512194,
by Hellenic grant INTERREG IIIA, by European Union Grant
FP-7-REGPOT-2008-1-CreteHEPCosmo-228644, by NSF grants PHY-0653085 and
PHY-0855021, and by the Institute for Fundamental Theory at the
University of Florida.

redpajama_set_name: RedPajamaArXiv