text:
An artist's eye view of the Russian Revolution
1917: The Real October—An animated documentary by Katrin Rothe
By Sybille Fuchs
An animated work about the most important event of the 20th century, the Russian Revolution, could have gone badly wrong. However, German writer and filmmaker Katrin Rothe's 1917: The Real October is an interesting experiment.
Rothe, who has twice won the prestigious Grimme Award (for work in German television, in 2014 and 2017), wanted to know how artists responded to the tumultuous events a century ago.
To this end, she read various historical studies, but she was impressed most of all by a work including several eye-witness accounts. This gave the filmmaker the idea of presenting the events of February to October 1917 in Russia from the subjective standpoint of a number of artists, while supplementing this with documentary footage from the great Herman Axelbank documentary, Tsar to Lenin (1937).
The revolutionary developments are documented by the artist (voiced by Inka Friedrich) in her studio as she cuts out figures and places them on a timeline on the wall, painted in red, and comments on them. The most important events are introduced with clips from Tsar to Lenin, photos and cardboard signs. For example, poet Vladimir Mayakovsky's denunciation of high society is underscored by film images of Tsar Nicholas II and his family.
Rothe uses collages cut out with scissors, animated figures, each created individually, and an ingenious arrangement of various materials (sparkling bubble wrap, corrugated cardboard, felt, etc.). The background and interiors are drawn carefully and sometimes underlaid with colour. Rothe aptly describes the result as "a mixture of two-dimensional animated drawing and three-dimensional puppet animation."
In the imagery, Rothe repeatedly draws on the terse and expressive iconography of the revolution (and post-revolutionary period). In the film her Mayakovsky cites the comment by the anarchist newspaper Russkaya Volnya on March 18, 1917: "If you want your placards and banners to attract more attention, then ask the artists for help." Many of them, including the poet, were willing to provide it.
The scenes of the masses are impressively effective, especially the mass funeral for the victims of the February Revolution with its red coffins. In the words of Alexandre Benois, one of her protagonists, "The purest Breughel."
The five protagonists are the writer Maxim Gorky (voice: Martin Schneider); the out-and-out sceptic, bourgeois lyricist and diary writer Zinaida Gippius (Claudia Michelsen); the painter, stage director and critic Benois (Hans Zischler); the suprematist painter Kazimir Malevich (Arne Fuhrmann); and the enthusiastic revolutionary and futurist Mayakovsky (Maximilian Brauer).
From February to October, the five artists adopt sharply differing attitudes to the ever intensifying struggle for power between the masses and the provisional government.
Gippius, the wife of famed writer Dmitry Merezhkovsky, who lives comfortably opposite the seat of government in the Tauride Palace, observes the mass movement with increased hostility after her initial hopes of a bourgeois democracy in Russia appear less and less likely to be realised.
Gippius is skeptical from the outset. A friend of Alexander Kerensky, she becomes disappointed by his political failures. She represents the bourgeois and petty bourgeois circles who initially welcome the collapse of the autocracy and old society, while simultaneously fearing its consequences.
The poet defines her attitude to the war, "I am for the war. For its quickest possible and dignified end." Over the course of the months, Gippius's comments, especially in regard to the Bolsheviks, become ever more venomous.
In Literature and Revolution (1924), Trotsky draws a biting portrait of the poet, which provides a necessary corrective to the figure in the film, who is portrayed in a rather favorable manner. Trotsky cites one of her verses, "And swiftly you will be driven to the old stable with a club, o people, disrespectful of holy things," and remarks, "What an inimitable slice of life is this effort of the decadent mystic poetess to wield a club (in iambics!)"
In regard to another passage, in which Gippius threatens the people with her "whips," Trotsky comments that "one can see her nature quite clearly. Only yesterday she was a Petrograd lady, languid, decorated with talents, liberal, modern. Suddenly today, this lady, so full of her own subtleties, sees the black outrageous ingratitude on the part of the mob 'in nailed boots,' and, offended in her holy of holies, transforms her impotent rage into a shrill womanish squeak (in iambics). And indeed, if her squeak will not shatter hearts, it will arouse interest. A hundred years hence the historian of the Russian Revolution will perhaps point out how a nailed boot stepped on the lyrical little toe of a Petrograd lady, who immediately showed the real property-owning witch under her decadent-mystic-erotic Christian covering."
It is no accident that later, exiled in Paris, this delicate lady placed all her hopes in Hitler's Third Reich.
Gorky, who always refers to himself as a socialist (although he was never a Marxist, as Rothe asserts), is increasingly unsettled by the mass movement in 1917 and fears the worst. His traumatic experience during the failed revolution of 1905, in which he was a participant, leads him to dark thoughts, "A lot of blood will flow, an endless amount."
Gorky intervenes when he sees art and culture under threat and founds a working group to defend monuments. The artists are divided among themselves, but agree on a resolution calling for the "saving of culture" and present the latter to the provisional government, which graciously accepts it. A commission on culture is founded. Troubled by the situation and dissatisfied by the government and its determination to prosecute the Russian war effort, Gorky launches a newspaper Novaya Zhizn (New Life) and wants to set up his own party, which he believes may only have three members.
The Bolshevik and literary critic Aleksandr Voronsky later characterised Gorky like this: "Man's thought is majestic, free and fearless, but with us in Russia, it is separated and torn away from the original instincts of life. The writer [Gorky] sees in this rift between thinking and feeling the tragedy of our revolution. During the revolution, it emerged that the principle of 'reason'—the intelligentsia—ended up outside of 'the element of the people'." And Gorky, notwithstanding his proletarian origins, belongs, as Rothe's 1917: The Real October makes clear, to the intelligentsia.
Mayakovsky, who is spared from going to the front during World War I thanks to Gorky's intercession and is stationed instead in Petrograd, is the most impassioned figure in the film.
He not only shows up everywhere guns are being fired, but appears during the first days of the February Revolution to author a "poetical timeline" of events. Mayakovsky ardently hopes for the seizure of power by the Bolsheviks, visits the workers and reads them his poems. He founds an all-Russian artists' council and earns money by drawing for newspapers. He cites his two-line satirical verse, "Eat your pineapples, chew your quail's eggs / Be warned, bourgeois, you're on your last legs," with which the film also concludes.
The out-and-out bourgeois Benois, by contrast, is only fleetingly touched by the revolutionary tumult. He keeps his distance, and is concerned by the fate of the Winter Palace and the Hermitage with its artistic treasures. After the seizure of power by the workers and peasants, he is astonished at the lack of damage caused during the overturn.
Benois (who went into exile in 1927 and died in Paris in 1960) increasingly regrets having become involved in the revolution. Resigned, he retreats into isolation. He expresses a complete lack of understanding and sympathy for the needs of the working class. His cook, who has to wait in line for food for hours and can hardly put a meal on the table, is a "really silly goose." She makes "repeated demands, such as having Sundays free and the like."
Malevich, the creator of the "Black Square" (1915), becomes an enthusiastic supporter of the revolution because he sees it as a rebirth of art. He proposes that artists organise museum visits for workers and establish "city halls of art." With art, he wants to come to the aid of the younger generation, which carry the "spark of the new," across the country.
Katrin Rothe explained her notion of the relevance of the film in an interview: "It is a film that looks at the past from today. Artists do not make revolutions, they have no weapons and are not politicians. But they are significant participants in something, which can be described as the 'general mood'. They cannot control what emerges from this, but they can either assume this responsibility or not."
Rothe's summary is: "Art experienced an awakening after 1917, but the sought after freedom was only a brief illusion." This perhaps applies to the personal fates of many individual artists, but what the Soviet and Russian artists created and experimented with in the few years before their suppression by the Stalinist bureaucracy continues to have an impact to this day and continues to provide contemporary artists with an impulse to develop their work.
The director ends her work with the pessimistic conclusion, "I experienced the end of socialism in 1989. I wanted to find the true October, I have found hundreds of truths. What is left? Words and works."
What the director, born in 1972, experienced was the collapse of the Stalinist regime, the gravedigger of the revolution, and not socialism. Her film itself, the fact that she took up this subject, is proof that the ideas and challenge of the Russian Revolution, and the answer it gave to war and exploitation, are posed once again today.
Centenary of the Russian Revolution
100 years on: The significance of the 1917 Russian Revolution for today
I'm Your Woman: The problem of films not genuinely drawn from life
Disney's Soul in Denmark: New York Times disapproves of a white actor dubbing a black actor's voice
Louis van Beethoven: A German film biography of the great composer
Magia Record, Kakushigoto, Fruits Basket and Sword Art Online: A general review of anime in 2020
499 and Summertime at the 10th annual GuadaLAjara Film Festival
Conductor Riccardo Muti expresses solidarity with Metropolitan Opera Orchestra musicians
British government rejected visa-free EU touring for musicians to bolster hostile immigration policy
German federal and state governments reject consistent lockdown measures
Germany's Christian Democrats choose new leader
Neo-Nazi arms networks spreading across Europe
Massive rise in unemployment in Germany during the pandemic
After COVID-19 kills tram driver in Berlin, transit workers in Germany demand more security


text:
Brooklyn Gospel Pastor Injured in PA Car Crash
Dave Hogarty
Modified Jul 6, 2008
Famous gospel singer Rev. Timothy Wright of the Grace Tabernacle Christian Center in Crown Heights was seriously injured in a car accident on I-80 in Pennsylvania Friday evening. His wife Betty did not survive the crash, after a man driving a Chevy Corsica the wrong way on the interstate struck their car as they were driving home from a conference in Detroit. The Wrights' grandson, 14-year-old D.J. Wright, was also fatally injured in the wreck and passed away in a Pennsylvania hospital. The driver of the Corsica, 44-year-old John Pick of Lewisburg, PA, was killed in the crash.
It's unclear whether Wright is aware that his wife and grandson were killed yet. A friend of the family says the news will be devastating. He told the Daily News, "They were like a match made in heaven. She will be a tremendous loss to the community as well as to Christendom."
Rev. Wright is known as the Godfather of Gospel and was nominated for a Grammy award in 1995. His MySpace page lists a full slate of summer tour dates around the U.S. He has released more than a dozen gospel recordings in his career. Currently, Rev. Wright is in critical condition at Geisinger Medical Center in Danville, PA. The Wrights have five sons and live in Roosevelt, Long Island.
d.j. wright
Rev. Timothy Wright

redpajama_set_name: RedPajamaCommonCrawl


text:
by Whatsupin@rt – Roberta Foglino and Paola Zanchi – 2011
Piero Leonardi was born in Rome, Italy in 1957. Starting from the Eighties his passion for photography intercepts the experience of Tazio Secchiaroli – master of social reportage in the years of Fellini's La Dolce Vita, first recognized photoreporter of the Italian cinema: He is the one who, in 1981, selects Piero's shots for his first exhibition – "Irpinia un anno dopo" – a photo-reportage on the reconstruction of the places and the human fabric that come back to life one year after the earthquake; his photographic gaze oscillates between a sensitive participation to the human and social degradation and a caustic irony that captures the paradoxes in the drama.
Between 1981 and 1982 Piero met Ugo Attardi, Giacomo Manzù, Carmelo Martinez and Miguel Ortiz Berrocal, with whom he had a fruitful artistic exchange and a dynamic confrontation that led to the project of a photographic exhibition entitled "Animated Scultur".
Attracted by details – circumscribed forms that, alone, could have narrated the essence of the work – from the beginning his vision of sculpture is instinctively focused on synthesis and matter: its structure, its forms, its language.
To take photographs means to tell the intimate story of what exists: lines, lights and shadows that are able to give back the atmosphere of the sculptor's thought and not only of his work.
The same narrative synthesis is sought, a few years later, when in 1984 Piero turns to macro.
His photography focuses on minimal universes, where the detail, such as a spool of thread, worn out by the everyday life of the consumerist gaze, regains the dignity of a living reality of colors and shapes.
In the 1990s, with the spread of new digital technologies, photographic art underwent a socio-cultural metamorphosis, transforming itself into a so-called "mass phenomenon".
If the wide diffusion allows anyone to become a "photographer", how is it possible then – Piero wonders – for each of us to identify ourselves through the photos we take? The perception of reality is individual, so being able to photograph from this point of view means showing a subjective point of view: our unique vision.
Between 1999 and the early 2000s, the founding intuition of the artist's "photographic thought" was consolidated.
Determined to investigate the new inspirational sensibilities, he develops a technical-artistic approach of a more speculative and systemic nature.
Now he intends to photograph not only the material, but rather the perceptions that arise from the relationship between the photographer and the object-subject. Piero himself defines this type of research as "The Soul of Things".
Penetrating the soul of things means revealing imperceptible links between the world and the observer and therefore capturing aspects that go beyond the common sight, drawing on the essence most hidden to the superficiality of the glance.
It is this intuition that leads him to develop the concept of Perceptive Photography, according to which it is not the object that is photographed, neither as a document nor as an interpretation, but rather the message between the photographer and that object-subject.
In Perceptive Photography, the viewer is led to accept the artist's distortion, at times further transfiguring what the photographer "has seen". After all – Piero maintains – in a photograph there is both what we show and what we make people imagine.
A phase of intense experimentation begins around the concept of pareidolia, at the basis of the artist's intuition.
The instinctive and automatic tendency to re-find familiar or subconscious forms in images or details taken from nature converges, between 2004 and 2005, in the Pepperlife collection: twists, transparencies and colors are explored with a freedom of look and mind that allows the artist to re-read with amused irony the seductive drives of natural forms and organic structures.
Surprises to see Piero in the Roman countryside while he frees some poppies from the constraint of the stem and, no longer anchored to the earth… he makes them fly, even if through his shots; he catches their desire of infinity and freedom and releases their soul in the Butterflowers collection.
At the same time the artist "discovers" the marble quarries of the Apuan Alps, Tuscany, in Italy.
His attention moves into the folds and cracks of the rock to capture, or rather, "dig out" that interiority which is still intact, preserved for millennia in the memory of the material. "If they let you in", he says, "it means that they want to be discovered, understood, scrutinized. I decided to please them, entering more and more into their intimacy: paintings within paintings, or sculptures. A game of Chinese boxes made of natural art."
The artist-photographer enters the geological forms of the rock, through a surface exploration and sudden cores in depth. Simple signs, veils, dripped traces, spots of color, cracks and graffiti, linear or chromatic geometries constitute the informal and abstract alphabet, sometimes connoted by allusive and pareidoleic meanings, at the base of Color on the Rock's research, still in progress.
It emerges with increasing clarity a subjective vision of the photographic operation, where the medium is transformed into an artistic poetics.
Piero's approach aims at focusing and developing the awareness of his own "Photographic Calligraphy": a peculiar visual writing, able to fix the perceptive-artistic observation of the world and, consequently, to transform it into a recognizable visual style, representative of the uniqueness of perspective observation, what he himself defines as "Style of View".
With an act of courage, the artist's photography frees itself from the duty of the representation of an object, of the human document, the original vocation of the photographic language, to investigate rather its autonomous and cognitive potential. He proceeds through fragments torn away from reality, suggesting an "other" dimension, where the change of context makes his shots take on the force of readymades that short-circuit the experience of our material presence in the world.
A demonstration of this is the Purgatory collection, born from the ethereal and evocative landscape of the winter hills and valleys of the Umbria-Marche Apennines.
Observing the artist's photographs, one feels a sense of disorientation. Where is the high and the low? Clouds or patches of light? Air or snow? Sky or earth? The natural signs lose their consistency and become instruments of a disorienting process of dislocation, where "the thing" separates from its anchoring points.
If photography in its ontological essence, according to some, acts through cuts within reality, in Piero the action is even more radical, until it reaches a form of re-writing of reality, where signs fix the trace of something invisible and are transformed into non-natural signs.
In Landsigns the seriality of the cut is a constant: the technical instrument, the camera, is reduced to a pure extension of a gesture that forcefully incises the spatial sequence, expels the relationship with the earth, tears from the material fabric signs of a language where light and shadow become the ink of a visual writing.
The idea of the absent thing is there, indicated by "vectors" and axes that divide the field into light and dark zones which, if they apparently appeal to our sense of orientation, in fact disorient us in the unpredictability of the solutions.
The result is a "double-edged" photograph, where the linguistic sign "doubles" its potential, as in a mirror. However, here, the artist's photographic investigation does not stop only at primary geometrical forms, but explores different values, also linked to the specificity of materials in their opposition of consistency – earth/snow – destined to be consumed by processes of transformation and deterioration of nature to be assumed as artificial signs.
Piero deliberately chooses to move, almost as if to engage in a challenge with a worn-out subject, within a landscape – that of Antelope Canyon – exploited by mass tourist photography, easily involved by the suggestive chromatic characteristic of the underground context. Here, again, he descends into the depths, chiseling in direct contact a plastic writing that distorts the anatomy of the rock and creates involuntary sculptures. In this phase of his research, the echoes of "Animated Sculpture" resurface in Piero.
His images metamorphose at the moment of the shot and are charged with an ambiguous and strongly allusive value: distortions, stratifications, cavities, sliding of matter and light evoke archetypal signs that rise to semiological statutes, suspended between a dreamlike reading – which sometimes borders on erotic symbolism – and an autonomous plastic interest.
In the balance between visual writing and plastic investigation, in a new spatial extension, are the shots of Graphics. Minimal objects of the landscape, isolated within wide non-colored backgrounds, reconstruct a visual syntax that is the result of an autonomous mental operation, through which Piero develops a photo-non-photo, an informal language on the borderline between abstractionism and spatialism, between poor art and surreality.
The artist's intervention amplifies the minimal detail, detaches it from the inert world of things to insert it in an ideal sphere, where it acquires a reversible and interchangeable value.
Piero reinforces the conceptual investigation: the artist's operation becomes exclusionary, it goes through a process of reduction that leads to the focalization of primary structures. Rhythmic extroversions, material surfaces, monochromatic purity, dimensional deviations, imprints and sculptures kinetically drawn by lights and shadows – such as unusual mobiles – create an osmosis between reality and sur-reality, re-capture that "other" dimension, as secret, where time, space and form are synthesized in an abstract suspension.
In recent years, the artist's photographs have brought into focus the univocal sense of his research: his collections reveal coherent transversal elements, indicating the priority of his artistic language and highlighting expressive orientations and narrative choices that give a peculiar character to his production.
Piero's research therefore ranges from the visual excavation within the graphic and sign resonances of matter, to the ethereal and metaphysical evocation of spaces of light; from the washed-out and chiaroscuro mapping of visual traces, to the reconnaissance of sculptural and chromatic form, recovered from visceral depths, and focuses on abstract and minimalist compositions in empty and white spaces. Piero is now mainly interested in photography generally identified as "conceptual", in addition to the new epistemological categories he elaborated.
In what language does Piero's work speak to us? His visual calligraphy has the ability to unveil the object in its presence-absence, in its allusiveness, in order to re-evocate it, re-construct it and present it in its panic evidence.
His poetic perspective rereads backwards, in the sign engraved in the material, in the void of space, in the single element, in the plastic form, the trace of time, not to measure it, but rather to become a medium with its origin, through a manipulation that recreates its memory.
This kind of action takes the artist away from the idea of photography as representation, and strengthens, rather, the conceptuality of his language.
In this sense, the artist's photography seems to collect the inheritance of the experiments of the '60s and '70s, for which photography functions as a medium of a process of relation virtually proposed to the user and becomes the privileged instrument of conceptual intervention on reality, operating a synthesis on the sense of the image, on the act that generated it.
Even more Piero becomes the interpreter of that total liberalization of artistic operation, acquired in the 80s with the conceptual: his photography becomes an object of reflection, a sign of an artistic language to be created.
The result is a "Calligrafia Fotografica" (Photographic Calligraphy) that manifests in an increasingly clear way other relationships, other afferences, reconnecting to linguistic contaminations of contemporary art and that fits well in the connotation of a language on the border between Art-Photography/Photography-Art.


text:
\section{Introduction}

Dark matter as the lightest component of an electroweak multiplet remains a viable explanation for the observed relic abundance.  One feature of this type of dark matter model is the potential for a striking signature in the form of a disappearing charged track in a collider experiment.  This is due to an order $100\,$MeV radiatively-induced mass difference between the neutral multiplet component, and the heavier charged components.  The exact length of such a track is extremely sensitive to the value of this mass difference.

At the lowest order (tree level) in perturbation theory, all components of an electroweak multiplet have the same mass.  After electroweak symmetry breaking, radiative corrections from massive gauge bosons push the physical masses of the charged components slightly above that of the neutral component \cite{Hisano2005,Cheng99}.  In many phenomenological studies, a one-loop calculation of this mass splitting is sufficient to give reasonable constraints on physical observables.  However, as we will show, due to the strong dependence on the mass splitting, two-loop corrections can result in up to a 40\% change in the lifetime of a charged multiplet component, and should be included when comparing theory with experiment.

Here we compute two-loop mass splittings for two phenomenologically-relevant electroweak multiplets.  The first is the wino in the minimal model of $R$-parity conserving supersymmetry, a fermionic electroweak triplet.  We focus specifically on the scenario where the lightest supersymmetric particle (LSP) is a pure wino (neutralino), corresponding to the neutral component of the triplet, and the rest of the supersymmetric spectrum is sufficiently massive to be decoupled. In this case, the next-to-lightest supersymmetric particle (NLSP) is also a pure wino (chargino), corresponding to the charged component of the triplet.  In this limit, a wino of mass $\sim$3\,TeV would give the correct relic abundance \cite{Hisano2007,Hryczuk2011}.  This model and the radiatively-induced mass splitting have been studied extensively, including calculation of radiative corrections to the mass splitting at two-loop order \cite{Cheng1999,Feng1999,Ibe2013}.  We refine the existing calculations by treating light quarks as massive, and by using input parameters computed using a full model spectrum.  We compare to existing results based on massless light quarks and simple threshold corrections.

The second model that we consider is the minimal dark matter (MDM; \cite{Cirelli2006,Cirelli2009}) fermionic quintuplet.  In general, MDM refers to a class of dark matter models, each consisting of the SM plus a different electroweak multiplet with some minimal set of quantum numbers and charges under the SM gauge groups.  Most models in this class have been ruled out \cite{Cai2015}, although the fermionic quintuplet with zero hypercharge is still viable.  This model has a weakly-interacting massive particle, which for a mass of $\sim$9\,TeV gives the expected dark matter relic abundance \cite{Cirelli2007,Cirelli2009}.  This model is also favoured because it stabilises the electroweak vacuum \cite{Chen2012a} by increasing the running of the electroweak gauge coupling.  Although this results in the model becoming non-perturbative at a lower scale than the SM, it at least remains perturbative until only a few orders of magnitude below the Planck scale \cite{DiLuzio2015}.  The quintuplet contains neutral, charged and doubly-charged components.  This is the first two-loop calculation of the splitting between the masses of these components.

The proper lifetime of a charged component, $\tau$, which we will express in units of (mm/c), is on the order of nanoseconds to picoseconds for the models considered here.  This corresponds to disappearing track lengths on the millimetre to centimetre scale, or more precisely about $6\,$cm \cite{Ibe2013} for the wino limit of the MSSM.  This is the motivation for many disappearing-track searches \cite{2009PhLB..672..339A,2011JHEP...05..097B,1999EPJC...11....1D,2008PhLB..664..185A,2007PhLB..644..355I,1999PhRvL..83.1731F}.  Searches with the ATLAS \cite{2013PhRvD..88k2006A} and CMS \cite{2014arXiv1411.6006C} detectors have excluded wino dark matter up to masses of $270\,$GeV and $260\,$GeV respectively.  It has been estimated that a future $100\,$TeV collider could discover pure wino dark matter up to masses of $3\,$TeV \cite{2014JHEP...08..161L,2015JHEP...01..041C}.

Similar search strategies can be applied to the MDM model.  In Ref.~\cite{2015PhRvD..92e5008O}, $8\,$TeV ATLAS and CMS results were used to exclude masses below $267\,$GeV and $293\,$GeV for the neutral components of Majorana and Dirac fermionic quintuplets, respectively.  The same paper also estimated that with a $14\,$TeV high-luminosity LHC run, Majorana MDM with a mass of up to $524$\,GeV could be discovered, as could Dirac MDM with a mass of up to $599$\,GeV.

The calculation of two-loop radiative corrections is a computationally challenging task, which has been significantly simplified with the introduction of modern tools.  Even at the most rudimentary level, determining all possible topologies is non-trivial, let alone simplifying and evaluating the resulting integrals.  Fortunately, \textsf{FeynArts}  \cite{Hahn2001}, \textsf{FeynCalc}  \cite{Mertig1991,Shtabovenko2016}, \textsf{TARCER}  \cite{Mertig1998}, \textsf{FIRE}  \cite{Smirnov2015}, \textsf{FeynHelpers}  \cite{SHTABOVENKO201748} and \textsf{TSIL}  \cite{Martin2006} have made each step of this process far more achievable than in the past.

The computational difficulty of the two-loop mass calculation is significantly greater for the MDM quintuplet model than for a triplet, due to the $\sim$300 additional amplitudes that must be considered compared to the triplet.  We overcome this by using a new computational framework that is almost completely automated.  This framework eventually makes the generalisation from a triplet to quintuplet trivial, and in future can be extended to make two-loop calculations achievable with even more diagrams.

Although precision two-loop self-energy corrections are essential for accurately constraining the lifetimes of charged multiplet components, the values of the input parameters used for these calculations are equally important.  Due to the scale dependence of parameters in perturbative quantum field theory, all quantities entering into a precision mass calculation are subject to potentially large uncertainties.

Computing all masses and couplings in a perturbative quantum field theory such as the MSSM is rather involved.  The physical masses must be correctly matched to corresponding running masses, which depend on the renormalisation scale.  Similarly, the couplings, which appear in the Lagrangian of the theory, are scale-dependent quantities.  Because different quantities of the calculation are defined at different scales, threshold corrections must be applied to match some low-energy theory, such as QCD, to the high energy theory of interest, such as the MSSM.  In our example, input parameters such as the running masses of the light quarks and leptons are defined in the low-energy effective QCD theory, but we are interested in determining the values of running parameters at some higher scale $Q$, so that we can use them as inputs to our two-loop self-energy calculations for the electroweak multiplet components.  To achieve this, it is necessary to numerically solve a set of ordinary differential renormalisation-group equations (RGEs) with boundary conditions defined across a hierarchy of scales, and perform the appropriate matching.

Spectrum generators are software packages that are designed to do all this in a consistent and precise way.  A number exist for the MSSM \cite{Porod:2011nf,2013CoPhC.184..899C,1993pcas.workQ...2B,2007CoPhC.176..426D,2003CoPhC.153..275P,Allanach2002}.  There are also packages intended to compute precision masses for specific states, such as \textsf{FeynHiggs}  \cite{2000CoPhC.124...76H} and \textsf{SUSYHD}  \cite{arXiv:1504.05200}, which compute Higgs masses.  However, these packages are hardcoded to a specific model, and a specific parameterisation of that model.  In this paper, we consider both a specific limit of the MSSM, and a non-supersymmetric theory.  We therefore use tools that can create a spectrum generator from a Lagrangian, providing a consistent approach across both models.  A major part of computing a spectrum is obtaining the analytical forms of the RGE equations and the radiatively-corrected masses, threshold and tadpole corrections.  It is then the part of the spectrum generator to use numerical techniques to solve and evaluate those functions.  We use \textsf{SARAH}   \cite{Staub:2009bi,Staub:2010jh,Staub:2012pb,Staub2014} to produce two-loop RGEs and one-loop masses and threshold corrections, and then use \textsf{FlexibleSUSY} \cite{Athron2015, Athron2017} to generate a spectrum generator for the MDM and MSSM models.  We link the spectrum generator to our self-energy calculations, in order to provide precision running masses and couplings as inputs to our two-loop mass-splitting calculations.

In Section \ref{sec2} of this paper, we define and detail the models that we investigate.  We then describe our calculation methods in Section \ref{sec3}, our results in Section \ref{sec4}, and summarise in Section \ref{sec5}.  We give explicit expressions for the one-loop self energies and counter-term couplings required for computing two-loop mass splittings in Appendix \ref{sec:self_energies}.

\section{Models and parameters}
\label{sec2}

\subsection{The wino limit of the MSSM}

In the wino limit of the MSSM, all supersymmetric particles except the lightest neutralino and chargino decouple from physics at the weak scale.  This corresponds to $M_2 \ll M_1, M_3, \mu$.  This implies that $gv \ll M_1,\mu$, making the lightest neutralino and chargino mass eigenstates pure winos.  Together, they constitute an $SU(2)_L$ triplet $\raisebox{0pt}[1ex][1ex]{$\chi$}$ with hypercharge $Y=0$, coupled to the SM via the electroweak sector.  The $\MSBar$  renormalised Lagrangian is
\begin{align}
\mathcal{L}=\mathcal{L}_{\text{SM}}+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}}\,(i\slashed{\mathcal{D}}-\hat{M})\,\raisebox{0pt}[1ex][1ex]{$\chi$} \label{eqn:wino_limit}
\end{align}
where $\mathcal{L}_{\text{SM}}$ is the SM Lagrangian, $\hat{M}$ is the degenerate $\MSBar$  tree-level mass of the triplet and $\slashed{\mathcal{D}}$ is the $SU(2)_L$ covariant derivative.  Expanding out the covariant derivative gives
\begin{align}
\begin{split}
\mathcal{L}=&\mathcal{L}_{\text{SM}}+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^0}(i\slashed{\partial}-M)\raisebox{0pt}[1ex][1ex]{$\chi$}^0+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}(i\slashed{\partial}-M)\raisebox{0pt}[1ex][1ex]{$\chi$}^{+}\\
&+g\left (\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^+ \right)\left(s_wA_{\mu}+c_wZ_{\mu}\right)+g\left(\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^0\right)W_{\mu}^++\text{h.c.}
\end{split}
\end{align}
The triplet therefore couples to the SM via the electroweak gauge bosons only.  It is clear that at tree level, the charged and neutral components have the same mass, $\hat{M}$.  We will express the physical masses of the neutral and charged components as $M_{\text{pole}}^0$ and $M_{\text{pole}}^+$ respectively.

When implementing the electroweak triplet in \textsf{SARAH}  model files, we express it in matrix form:
\begin{align}
\raisebox{0pt}[1ex][1ex]{$\chi$} = \left( \begin{array}{cc} \raisebox{0pt}[1ex][1ex]{$\chi$}^0 /\sqrt{2} &  \raisebox{0pt}[1ex][1ex]{$\chi$}^+ \\
\raisebox{0pt}[1ex][1ex]{$\chi$}^- & -\raisebox{0pt}[1ex][1ex]{$\chi$}^0/\sqrt{2}
\end{array}\right).
\end{align}

\subsection{The electroweak quintuplet}

In the remaining viable version of MDM, a fermionic $SU(2)_L$ quintuplet $\raisebox{0pt}[1ex][1ex]{$\chi$}$ with hyper-charge $Y=0$ is coupled to the SM via the $SU(2)_L$ gauge sector.  Analogous to the triplet case, the Lagrangian is
\begin{align}
\mathcal{L}=\mathcal{L}_{\text{SM}}+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}}\,(i\slashed{\mathcal{D}}-\hat{M})\,\raisebox{0pt}[1ex][1ex]{$\chi$},
\end{align}
where $\slashed{\mathcal{D}}$ is the $SU(2)_L$ covariant derivative and $\hat{M}$ is the $\MSBar$  renormalised tree-level quintuplet mass.  Expanding the $SU(2)_L$ covariant derivative $\slashed{\mathcal{D}}$, we obtain the interaction terms
\begin{align}
\begin{split}
\mathcal{L}=&\mathcal{L}_{\text{SM}}+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^0}(i\slashed{\partial}-M)\raisebox{0pt}[1ex][1ex]{$\chi$}^0+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}(i\slashed{\partial}-M)\raisebox{0pt}[1ex][1ex]{$\chi$}^{+}\\
&+\frac{1}{2}\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^{++}}(i\slashed{\partial}-M)\raisebox{0pt}[1ex][1ex]{$\chi$}^{++}\\
&+g\left(\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^++2\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^{++}}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^{++}\right)\left(s_wA_{\mu}+c_wZ_{\mu}\right)\\
&+g\left(\sqrt{3}\,\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^+}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^0+\sqrt{2}\,\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^{++}}\gamma_{\mu}\raisebox{0pt}[1ex][1ex]{$\chi$}^+\right)W_{\mu}^++\text{h.c.}
\end{split}
\end{align}
As with the triplet, the quintuplet couples only to the photon and $W$ and $Z$ bosons at tree level.  We will express the physical masses of the neutral, charged and doubly-charged components as $M_{\text{pole}}^0$, $M_{\text{pole}}^+$ and $M_{\text{pole}}^{++}$ respectively.

For implementing the quintuplet $\chi$ in a \textsf{SARAH}  model file, we express it in tensor representation as \cite{2012PhRvD..86a3006K}
\begin{align}
\begin{split}
\raisebox{0pt}[1ex][1ex]{$\chi$}_{1111} &=  \mychi^{++} \ \ , \ \ \raisebox{0pt}[1ex][1ex]{$\chi$}_{1112} = \frac{1}{\sqrt{4}} \mychi^+ \ \ , \ \ \raisebox{0pt}[1ex][1ex]{$\chi$}_{1122} = \frac{1}{\sqrt{6}}\mychi^0 \\
\raisebox{0pt}[1ex][1ex]{$\chi$}_{1222} &= -\frac{1}{\sqrt{4}}  \mychi^- \ \ , \ \ \raisebox{0pt}[1ex][1ex]{$\chi$}_{2222} =  \mychi^{--}
\end{split}
\end{align}
where the relative signs are chosen such that $\raisebox{0pt}[1ex][1ex]{$\chi$}$ is isospin self-conjugate \cite{2015JCAP...10..058G}.  In this representation the mass term is given by
\begin{equation}
\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^C}\raisebox{0pt}[1ex][1ex]{$\chi$}\equiv\overline{\raisebox{0pt}[1ex][1ex]{$\chi$}^C}_{ijkl}\raisebox{0pt}[1ex][1ex]{$\chi$}_{i'j'k'l'}\epsilon^{ii'}\epsilon^{jj'}\epsilon^{kk'}\epsilon^{ll'}.
\end{equation}

\subsection{Input parameters}

In this paper, we use a fully-computed model spectrum to obtain the input parameters for our self-energy calculations.  To generate the spectrum, we therefore require a full set of SM input parameters.  These are given in Table \ref{table:params}.  The central values and experimental uncertainties are from the latest Particle Data Group tables \cite{Patrignani2016}.  We quantify the parametric sensitivity of the mass splitting to each of these uncertainties by varying one parameter at a time, and holding the rest fixed.  We show the results of this exercise in Table \ref{tab:uncertainty_analysis}, at a phenomenologically relevant value of the degenerate mass for each model.

The renormalisation scale $Q$ is an important input parameter in our calculation.  This is the scale to which all mass parameters and couplings are run, and where the self energies, and subsequent pole mass, are evaluated.  The range of this parameter should reflect the scale of missing logarithmic corrections in the calculation, which are of the form $\log(m/Q)$ for some mass $m$.  When using a non-iterative method for computing the multiplet mass splitting, we find that the dominant missing logarithmic corrections come from masses near the electroweak scale.  Contributions from the multiplet itself, with masses around the TeV scale, are cancelled.  See Ref.\ \cite{2017arXiv171001511M} for a detailed discussion.  Therefore, for this study it is sufficient to vary the renormalisation scale around the mass of the top quark.  We therefore choose the range $m_t/2\leq Q \leq 2m_t$.

\begin{table}[tp]
\centering
\caption{Input parameters and uncertainties used for the calculations in this study (unless stated otherwise).  These ranges and central values are taken from the latest Particle Data Group tables \cite{Patrignani2016}.} \label{table:params}
\vspace{0.4cm}
\begin{tabular}{l@{\hspace{4mm}}c@{}c}
\hline
Parameter & & Values \\
\hline
Electromagnetic coupling & $1/\alpha^{\overline{MS}}(m_Z)$        & $127.940(42)$       \\
Top pole mass  & \phantom{$^{\overline{MS}}$}$m_t$\phantom{$^{\overline{MS}}$}  &  $173.34(2.28)$\,GeV\\
Higgs pole mass & \phantom{$^{\overline{MS}}$}$m_h$\phantom{$^{\overline{MS}}$}  & $125.5(1.6)$\,GeV \\
W pole mass & \phantom{$^{\overline{MS}}$}$m_W$\phantom{$^{\overline{MS}}$}  & $80.385(15)$\,GeV \\
Z pole mass & \phantom{$^{\overline{MS}}$}$m_Z$\phantom{$^{\overline{MS}}$}  & $91.1876(21)$\,GeV \\
Electron pole mass & \phantom{$^{\overline{MS}}$}$m_e$\phantom{$^{\overline{MS}}$}  & $0.510 998 9461(31)$\,MeV \\
Muon pole mass & \phantom{$^{\overline{MS}}$}$m_{\mu}$\phantom{$^{\overline{MS}}$}  & $ 105.6583745(24)$\,MeV \\
Tau pole mass & \phantom{$^{\overline{MS}}$}$m_{\tau}$\phantom{$^{\overline{MS}}$}  & $ 1776.86(12)$\,MeV \\
Down quark mass & $m_d^{\overline{MS}}(2\,\text{GeV})$  &   $4.80(96)$\,MeV  \\
Up quark mass & $m_u^{\overline{MS}}(2\,\text{GeV})$     & $2.30(46)$\,MeV \\
Strange quark mass & $m_s^{\overline{MS}}(2\,\text{GeV})$  & $95(15)$\,MeV\\
Charm quark mass & $m_c^{\overline{MS}}(m_c)$ & $1.275(75)$\,GeV \\
Bottom quark mass & $m_b^{\overline{MS}}(m_b)$    & $4.18(9)$\,GeV \\
Strong coupling & $\alpha_S^{\overline{MS}}(m_Z)$  & $0.1181(11)$\\
Renormalisation scale & $Q$    & $m_t/2 - 2m_t$\\
\hline\end{tabular}
\end{table}

\section{Method}
\label{sec3}

To determine the mass splitting we must compute the physical, or \textit{pole}, masses of the multiplet components to a fixed order in perturbation theory.  The definition of a pole mass is the complex pole of the two-point propagator, which for a fermion has a denominator given by the one-particle irreducible effective two-point function
\begin{equation}
\Gamma_2=\slashed{p}-\hat{M}+\Sigma_K(p^2)\slashed{p}+\Sigma_M(p^2).
\label{eqn:propagator}
\end{equation}
Here $p_{\mu}$ is the four-momentum of the particle, $\hat{M}$ is the tree-level $\MSBar$  mass and $\slashed{p}=\gamma^{\mu}p_{\mu}$.  The self energy, $\Sigma(p^2)=\Sigma_M(p^2)+\slashed{p}\Sigma_K(p^2)$, is in general a function of the renormalisation scale and any relevant masses or couplings in the theory.  We will expand the self energy up to second order in a perturbation parameter $\alpha$, for a two-loop result.

The pole mass is obtained by demanding $\Gamma_2=0$.  This can be achieved by setting $p^2=M_{\text{pole}}^2$ (and $\slashed{p}=M_{\text{pole}}$), and solving the resulting implicit expression for the pole mass
\begin{align}
M_{\text{pole}}=\text{Re}\left[\frac{\hat{M}-\Sigma_M(M_{\text{pole}}^2)}{1+\Sigma_K(M_{\text{pole}}^2)}\right] \label{eqn:M_pole_iterative}.
\end{align}

Solving (\ref{eqn:M_pole_iterative}) iteratively results in unwanted scale-dependent logarithms.  Alternatively, one can take advantage of the perturbative nature of this expression to write down an explicit result for the pole mass that preserves a fortunate cancellation of the scale-dependent logarithms.  Working to the two-loop order, this expression is
\begin{align}
\begin{split}
&M_{\text{pole}}=\left[\hat{M}-\Sigma^{(1)}_M-\Sigma^{(2)}_M-\hat{M}\Sigma^{(1)}_K-\hat{M}\Sigma^{(2)}_K\right.\\
&+(\Sigma^{(1)}_M+\hat{M}\Sigma^{(1)}_K)(\Sigma^{(1)}_K+2\hat{M}\dot{\Sigma}^{(1)}_M+2\hat{M}^2\dot{\Sigma}^{(1)}_K)\\
&\left.+\mathcal{O}\left(\alpha^3\right)\right]_{p^2=\hat{M}^2}, \label{eqn:M_pole_explicit}
\end{split}
\end{align}
where $\Sigma^{(n)}_X=\Sigma^{(n)}_X(p^2)$.  For a full review of this method and the implications of the iterative procedure see Ref.\ \cite{2017arXiv171001511M}.

For this study we use the Feynman-'t Hooft ($\xi=1$) gauge for all calculations.  One-loop mass splittings computed in the Landau ($\xi=0$), Feynman-'t Hooft ($\xi=1$) and Fried-Yennie ($\xi=3$) gauges can also be found in Ref.\ \cite{2017arXiv171001511M}.

\tikzset{
    photon/.style={decorate, decoration={snake,segment length=2mm, amplitude=0.8mm}, draw=black},
    wino/.style={draw=black},
    scalar/.style={draw=black,style={dash pattern=on 4pt off 4pt}},
    ghost/.style={draw=black,style={dash pattern=on 1pt off 1pt}}
}

\tikzstyle{vertex} = [circle,fill=black,inner sep=0pt,minimum size=3pt]
\tikzstyle{empty} = [circle,fill=none,inner sep=0pt,minimum size=3pt]
\tikzstyle{SMcorrection} = [circle,fill=black,inner sep=0pt,minimum size=15pt]
\tikzstyle{counterterm} = [circle,fill=black,inner sep=0pt,minimum size=10pt]
\newcommand{\CIRCLE}{$\mathbin{\tikz [x=2.28ex,y=2.28ex,line width=.5ex, black] \draw (0,0) -- (1,1) (0,1) -- (1,0);}$}%
\newcommand{\Cross}{$\mathbin{\tikz [x=2.28ex,y=2.28ex,line width=.5ex, gray] \draw (0,0) -- (1,1) (0,1) -- (1,0);}$}%
\newcommand{\CT}{$\mathbin{\tikz [x=1.28ex,y=1.28ex,line width=.5ex, gray] \draw (0,0) -- (1,1) (0,1) -- (1,0);}$}%

\begin{figure*}[t!]
\center{\includegraphics{figure_1.pdf}}
\caption{Two-loop diagrams involving only the gauge bosons and multiplet fermions.  Solid lines indicate multiplet fermions ($\chi^{0},\, \chi^{\pm},\, \chi^{\pm\pm}$) and wiggly lines electroweak vector bosons ($W^{\pm}$, $Z$, $\gamma$).} \label{fig:Feynman_diagrams}
\end{figure*}

\begin{figure*}[t!]
\center{\includegraphics{figure_2.pdf}}
\caption{Two-loop diagrams formed by reinserting the 1-loop gauge boson self-energy into its own propagator.  Solid lines indicate fermions ($\chi^{0},\, \chi^{\pm},\, \chi^{\pm\pm},\, q,\, l,\, \nu$), wiggly lines electroweak vector bosons ($W^{\pm}$, $Z$, $\gamma$), dashed lines scalars (Higgs and Goldstone bosons) and dotted lines indicate ghosts.} \label{fig:Feynman_diagrams2}
\end{figure*}

\begin{figure*}[t!]
\center{\includegraphics{figure_3.pdf}}
\caption{Two-loop counter-term diagrams.  Small circles with crosses indicate counter-term insertions.  Solid lines indicate multiplet fermions ($\chi^{0},\, \chi^{\pm},\, \chi^{\pm\pm}$) and wiggly lines electroweak vector bosons ($W^{\pm}$, $Z$, $\gamma$).} \label{fig:Feynman_diagramsCT}
\end{figure*}

In the wino limit of the MSSM there are about 200 two-loop diagrams, and about 500 for the MDM quintuplet model.  The generic two-loop topologies are given in Figures \ref{fig:Feynman_diagrams} and \ref{fig:Feynman_diagrams2}, and counter-term diagrams of two-loop order in Figure \ref{fig:Feynman_diagramsCT}.  We determine the counter-term couplings from the one-loop self energies of the electroweak gauge bosons and electroweak multiplets.

\subsection{Details of self-energy calculation}\label{sec:calc_method}

In this subsection, we describe our automated process for calculating self-energies at two loops.

A complete self-energy calculation (at any order) requires the construction of a symbolic amplitude, followed by its numerical evaluation.  In general, interfaces between tools are sufficient for generating symbolic amplitudes at both one and two-loop level.  For one-loop calculations, the evaluation step can be performed with various existing tools: \textsf{FeynHelpers}  \cite{Shtabovenko2016} provides analytic one-loop amplitudes for this purpose, and other codes do this by making use of the \textsf{LoopTools} package \cite{Romao2006} (e.g.\ \textsf{SARAH}  \cite{Staub2014} interfaced to either \textsf{SPheno}  \cite{Porod2003} or \textsf{FlexibleSUSY}  \cite{Athron2015, Athron2017}).

The interface between the tools available for generic two-loop calculations is only complete up to the stage of the symbolic amplitude.  The necessary conversions exist between \textsf{FeynArts}, \textsf{FeynCalc}  and \textsf{TARCER}, but the final step of numerical evaluation requires significant user intervention.  The \textsf{TSIL}  library provides numerical, and in some cases analytical, solutions for the basis integrals that appear in two-loop self-energies.  However, in order to make use of these, one must construct a C\nolinebreak\hspace{-.05em}\raisebox{.4ex}{\tiny\bf +}\nolinebreak\hspace{-.10em}\raisebox{.4ex}{\tiny\bf +}  interface to call the \textsf{TSIL}  libraries and then use them to evaluate the amplitudes.  Although the \textsf{TSIL}  functions are extremely user-friendly, making use of them from a symbolic \textsf{Mathematica}  expression provided by one of the other tools is highly non-trivial. There is therefore no automated method for obtaining numerical implementations of two-loop amplitudes.  Given that there can be hundreds or even thousands of such amplitudes, this makes the final step of the calculation an arduous process.  By completely automating the generation of this C\nolinebreak\hspace{-.05em}\raisebox{.4ex}{\tiny\bf +}\nolinebreak\hspace{-.10em}\raisebox{.4ex}{\tiny\bf +}  interface with a new software framework, we have been able to dramatically simplify the process of computing two-loop self energies.  This framework has already been used to generate two-loop amplitudes used in Ref.\ \cite{2017arXiv171001511M}.

Our method also makes it possible to split the calculation of many loop diagrams into manageable pieces.  Simultaneously computing $\mathcal{O}(10)$ different amplitudes (of distinctly different masses and/or topologies) with symbolic tools like \textsf{FeynCalc}  takes an extremely long time, as \textsf{FeynCalc}  attempts to symbolically simplify the amplitudes.  On the other hand, keeping track of all terms on a diagram-by-diagram basis is a serious task by any manual or even semi-automated method.  By completely automating the whole process, we are instead able to keep track of all terms, and simply evaluate them independently and numerically.  On a modest computing setup, this is the only way to obtain a result in a feasible timeframe without additional user intervention.

We calculate the amplitudes either one diagram at a time, or in selected groups, using \textsf{FeynArts}, \textsf{FeynCalc}  and \textsf{FIRE}, run from C\nolinebreak\hspace{-.05em}\raisebox{.4ex}{\tiny\bf +}\nolinebreak\hspace{-.10em}\raisebox{.4ex}{\tiny\bf +}  via the Wolfram Symbolic Transfer Protocol (WSTP).  We decompose the resultant symbolic amplitudes into lists of coefficients to be applied to basis integrals, and keep a master list of all the basis integrals required.

The algorithm begins by evaluating the finite part of the amplitude $\mathcal{A}$.  It then computes the coefficients $\{C_1,C_2, \ldots  \}$ of every possible basis integral $\{\mathcal{B}_1,\mathcal{B}_2,\ldots \}$.  For the non-zero $C_i$, it then constructs a trial amplitude of the form
\begin{equation}
\mathcal{A}_{\rm trial} = C_1 \mathcal{B}_1 + C_2 \mathcal{B}_2 + \ldots
\end{equation}
and checks the difference $\mathcal{A} - \mathcal{A}_{trial}$ for the presence of basis integrals with non-zero coefficients, in order to identify any cross-terms that have been double-counted in the first step.  From the set of basis integrals $\{\mathcal{B}_i,\mathcal{B}_j,\ldots \}$ with non-zero coefficients at this stage, the algorithm then creates new `compound basis integrals' $\mathcal{B}_{ij} =  \mathcal{B}_i\mathcal{B}_j$, and presents them to \textsf{Mathematica}  as unified objects.  We can then instruct \textsf{Mathematica}  to extract new coefficients $C_{ij}$ for the compound basis integrals.  The final amplitude is then
\begin{align*}
 \mathcal{A}_{trial}\ = \ & \ \ C_1 \mathcal{B}_1 + C_2 \mathcal{B}_2 + \ldots \\
  &-\frac{1}{2} C_{12} (\mathcal{B}_1\mathcal{B}_2)  -\frac{1}{2} C_{21} (\mathcal{B}_2\mathcal{B}_1) - \ldots \\
  & + C_{11} (\mathcal{B}_1\mathcal{B}_1) + C_{22}(\mathcal{B}_2\mathcal{B}_2) + \ldots
\end{align*}
where $ C_{ij}$ is the coefficient of $\mathcal{B}_i\mathcal{B}_j$ in the original amplitude $\mathcal{A}$.  We convert these coefficients into C\nolinebreak\hspace{-.05em}\raisebox{.4ex}{\tiny\bf +}\nolinebreak\hspace{-.10em}\raisebox{.4ex}{\tiny\bf +}  format, and generate numerical routines for evaluating both them and the relevant basis integrals.

This automated framework is fully generic, allowing numerical routines to be generated for two-loop diagrams in almost any \textsf{FeynArts}  model file.  The only limitations are computational: problems involving over $\sim$1000 diagrams require long runtimes to generate the amplitudes, and produce large amounts of generated code.  Other features include: \begin{itemize}
\item automatic determination of one-loop counter-term couplings for two-point diagrams (using the one-loop self-energies),
\item optimisation of the evaluation of the two-loop basis integrals, by automatically determining which integrals can be evaluated in symmetry groups, and
\item flexibility and reusability of precomputed amplitudes (by separating the symbolic calculations from the final code generation).
\end{itemize}
We intend to make an open-source release of the full package in the near future.

In this paper, we use \textsf{FeynCalc}  \textsf{9.2.0} \cite{Mertig1991,Shtabovenko2016} and \textsf{FeynArts}  \textsf{3.9} \cite{Hahn2001} to obtain symbolic amplitudes, and reduce them to basis integrals with \textsf{FIRE}  \textsf{5} \cite{Smirnov2015} (via \textsf{FeynHelpers}  \textsf{1.0.0} \cite{SHTABOVENKO201748}) and \textsf{TARCER}  \textsf{2.0} \cite{Mertig1998}.  We evaluate the basis integrals using \textsf{TSIL}  \textsf{1.41} \cite{Martin2006} and analytical forms from the literature \cite{Pierce1997}.

\subsection{Check for divergence free-result}

It is important to confirm that the pole masses are free of non-physical divergences.  Ultra-violet (UV) divergences can be regulated using dimensional regularisation by computing in $D=4-2\epsilon$ dimensions and using modified minimal subtraction.  Using the generated symbolic amplitudes and our numerical implementation, we have confirmed that the individual pole masses are free from any poles in $\epsilon$ when the appropriate counter-terms are included.

Infra-red divergences arise from the zero mass of the photon.  To regulate these divergences, we retain an explicit mass $m_{\gamma}$ for the photon throughout the calculation, and take the limit $m_{\gamma}\rightarrow0$ in the evaluation.  IR-divergent diagrams exist at two-loop order, but their divergences are cancelled by the derivative of the one-loop self-energies in the two-loop expansion of the pole mass (Eq.\ \ref{eqn:M_pole_explicit}).  The proof of this cancellation is given in Ref.~\cite{Ibe2013} for the wino model.  The analogous result holds identically for the MDM quintuplet, so we do not repeat the details here.

We also encounter `fictitious' IR divergences in our numerical implementation.  These can arise from including a finite photon mass when attempting to evaluate non-IR divergent diagrams.  We nonetheless include this mass for all diagrams, as in some cases, taking a zero photon mass \textit{before} the tensor integral reduction causes the tensor integrals to reduce to basis integrals that are not available in current mathematical libraries. Using a regulator mass enables the reduction to proceed further, giving a result in terms of known basis integrals.  The price to pay for this convenience is an apparent IR singularity in the result: the amplitude picks up $\mathcal{O}(1/m_{\gamma}^2)$ terms.  However, the sum of the coefficients of these terms is numerically equivalent to zero for every diagram (i.e.\ to within a small factor of the floating-point machine accuracy times the largest individual coefficient).  We therefore always see numerically that these terms cancel, even if the integral reduction fails to cancel them symbolically.  We take care in our evaluation step to explicitly check for the numerical cancellation, and to then remove the terms \textit{before} taking the limit $m_{\gamma}\rightarrow 0$, as the latter would otherwise cause numerical cancellation errors between the $\mathcal{O}(1/m_{\gamma}^2)$ terms to blow up and dominate the result.

Also, because the basis integral $T(x,y,z)$ is not defined for small $x$, in the limit of $m_{\gamma}\rightarrow0$ we make the replacement $T(x,y,z) \equiv \bar{T}(x,y,z) - B(y,z)\log(x/Q^2)$ \cite{Martin2006}, where $B$ is another basis integral.  This will cancel with other terms in the amplitude of the form $A(x)B(y,z) = x\left[\log(x/Q^2)-1\right]B(y,z)$, and because $\bar{T}(0,y,z)$ is finite, will give a total that is IR safe.

\subsection{Spectrum calculation}

We use \textsf{FlexibleSUSY} \textsf{1.7.4}\footnote{\textsf{FlexibleSUSY}  also uses some code pieces from \textsf{SOFTSUSY}  \cite{Allanach2002,Allanach:2013kza}.} \cite{Athron2015, Athron2017} to create a spectrum generator, based on output from \textsf{SARAH}  \textsf{4.8.0} \cite{Staub:2009bi,Staub:2010jh,Staub:2012pb,Staub2014}.  This provides two-loop RGEs, one-loop threshold and tadpole corrections and one-loop self energies for all fields.  Because the spectrum generator requires a tree-level parameter prior to computing the loop-corrected EWSB conditions, the Higgs pole mass is an output rather than an input parameter.  Thus we also employ a simple iterative procedure to determine the correct input value for the Lagrangian Higgs mass parameter $\mu$, such that the observed Higgs pole mass is produced.

From the computed spectrum we extract the $\MSBar$  masses for the gauge bosons, Higgs and quarks at a common scale $Q$.  We also extract the running electroweak gauge couplings at $Q$ and use them to compute the value of $\alpha_{\text{EM}}(Q)$ from the relation
\begin{align}
\alpha_{\text{EM}} = \frac{g_1^2g_2^2}{4\pi (g_1^2+g_2^2)}.
\end{align}
From these parameters we compute the Weinberg angle $\theta_W=\arccos(m_W/m_Z)$ and the Higgs vacuum expectation value $v=2\sin(\theta_W)m_W  /  \sqrt{4\pi}$.  This preserves the required tree-level relations that are necessary to retain the proper cancellations between parts of the self-energies of the charged and neutral multiplet components.

It is important to consider threshold corrections when matching the SM to the wino or MDM model at two loops.  These corrections include the determination of $\MSBar$  masses consistent with a specified physical pole mass (particularly important for the $W$ and $Z$ bosons), and matching the $\MSBar$  gauge couplings in the SM to the model containing additional fermions.  The relevant threshold correction for the electroweak coupling is
\begin{align}
\alpha_{\text{EM,\,wino}}(Q) = \alpha_{\text{EM,\,SM}}(Q) \left[ 1- \frac{X\alpha_{\text{EM}}(Q)}{3\pi}\log\left(\frac{\hat{M}(Q)}{Q}\right)\right]^{-1},\label{eq:tc}
\end{align}
where $X=2$ for the wino model and $X=10$ for the MDM quintuplet.  \textsf{FlexibleSUSY}  applies this correction and does the mass matching, iteratively, at $Q=m_Z$.

For the one-loop calculation, we do not need to apply threshold corrections, as they are of the next loop order.  If these corrections are applied, then important cancellations do not occur between the threshold corrections and the self-energies, resulting in a spurious logarithmic increase or decrease in the mass splitting.   When calculating one-loop mass splittings, we therefore use pole masses in place of the $\MSBar$  masses, and neglect the threshold corrections to the gauge couplings.  This is consistent with the method of Ref.\ \cite{Ibe2013}.

\section{Results}
\label{sec4}

\subsection{The wino limit of the MSSM}

\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{difference.pdf}
\caption{The two-loop mass splitting in the wino model for $m_t=173.2\,$GeV, $m_h = 125.5\,$ GeV, $\alpha_S^{\overline{MS}}(m_Z)=0.1184$ and $Q=163.3\,$GeV.  In both panels the black dashed line is the fit given by Eq.\ (17) of Ref.~\cite{Ibe2013}, and the grey region in the lower panel is the stated deviation of this fit from the actual result.  The solid lines indicate our result with one and two-loop RGEs in blue and red respectively, with all light quark masses taken to be zero.  The red and blue dashed lines in the lower panel correspond again to one and two-loop RGEs respectively, but with all light quark masses included.}\label{fig:Ibe_compare}
\end{figure*}

As electroweak mass splittings have already been studied at the two-loop level in the wino limit of the MSSM \cite{Yamada2010,Ibe2013}, we are able to compare our results to the previous ones, and in the process demonstrate the impacts of the improvements that we have made in this paper.  This also serves as a validation of the consistency of our method, in particular the use of a full spectrum generator, before applying our method to the MDM quintuplet.

In Figure \ref{fig:Ibe_compare} we compare our two-loop results for the mass splitting $\Delta M \equiv M_{\text{pole}}^+ - M_{\text{pole}}^0$ to the one given by the polynomial fit in Eq.\ (17) of Ref.\ \cite{Ibe2013}.  For consistency, we use the same top pole mass $m_t=173.2\,$GeV \cite{SatoPrivate} and strong coupling $\alpha(m_Z) = 0.1184$ as in Ref.\ \cite{Ibe2013}.  We compare with both one and two-loop RGEs, and with both finite and zero masses for the light quarks.  The authors of Ref.\ \cite{Ibe2013} state that their polynomial fit gives less than a $0.02\%$ deviation from the true value over the mass range $100-4000$\,GeV, so we expect to be able to achieve a result close to this when comparing with our calculation.  We see that our equivalent result (one-loop RGEs, zero light quark masses) is in good agreement with theirs, with the deviation clearly the result of ringing from the polynomial fit rather than an inconsistency between the methods used.  This ringing is worst at large masses, where we expect the mass splitting to be constant; the polynomial fit fails to properly represent this behaviour.  Over the whole mass range we have no more than a $0.05$\% deviation from this previous calculation.  The impact of the light quark masses is to increase the mass difference by about 0.03--0.04\% across the whole mass range.

The consistency of our result with that of Ref.\ \cite{Ibe2013} can also be seen by making a polynomial fit to the curve computed with 1-loop RGEs, massless light quarks, $Q=163.3\,$GeV, $m_t=173.2\,$GeV and $\alpha(m_Z) = 0.1184$, over the range $100\,\mathrm{GeV}\le M^0_\mathrm{pole} \le 4$\,TeV:
\begin{widetext}
\begin{eqnarray}
\frac{\Delta M}{1~ {\rm MeV} } &=&  - 412.2
+304.7 \left(\ln \frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)
-60.71 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^2
+5.403 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^3\label{eqn:fit1}
-0.181 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^4.
\end{eqnarray}
\end{widetext}
This is in very close agreement with Eq.\ (17) of Ref.\ \cite{Ibe2013}.

\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{deltam_mssm.pdf}\includegraphics[width=0.5\textwidth]{decays_MSSM.pdf}
\caption{The two-loop mass splitting (\textit{left}) and decay lifetime of the chargino (\textit{right}) in the wino model as a function of the degenerate tree-level $\MSBar$  mass.  The green and red bands are the respective ranges of the one and two-loop mass splittings when $Q$ is varied continuously between $m_t/2$ and $2m_t$.  The light green band is the estimated uncertainty on the one-loop result using Eq.~(\ref{eqn:error_estimate}).}\label{fig:wino_two_loop}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{BR_MSSM.pdf}\
\caption{Branching fractions for $\chi^+\rightarrow X\chi^0$ in the wino limit of the MSSM, where $X \in \{e\nu_e,\mu\nu_\mu,\pi\}$.  Solid lines are the branching fractions using the two-loop mass splitting, and dotted use the one-loop result, both evaluated at $Q=m_t=173.34\,$GeV.}\label{fig:BFs}
\end{figure*}

In Figure \ref{fig:wino_two_loop} we present the two-loop mass splitting in the wino model using the parameters in Table \ref{table:params}, two-loop RGEs and non-zero light quark masses.  The dark green and red uncertainty bands are given by the maximum and minimum $\Delta M$ possible for values of $Q$ between $m_t/2$ and $2m_t$.  For the two-loop mass splitting, at values of $\hat{M}\lesssim m_t$ the minimum splitting occurs at $Q=m_t/2$, whereas for $\hat{M} \gtrsim 2m_t$ the minimum occurs at $Q=2m_t$.  For intermediate values $\hat{M}\sim m_t$, around the point where the crossover occurs, we find that the extrema occur at values of $Q$ inside the chosen range.  As a result, although the uncertainty band appears by eye to become very narrow, it does in fact maintain a non-zero width.  At even lower multiplet masses than shown here ($\hat{M}\lesssim100$\,GeV), the two-loop uncertainty from scale variation on the mass splitting is comparable to, and eventually becomes larger than, the equivalent one-loop uncertainty; this is due to the additional electroweak-scale logs introduced at the two-loop level, and their tendency to blow up as $\hat{M}$ drops significantly below $m_t$.

The light green uncertainty band is the naive estimate \cite{Ibe2013} of the missing two-loop contribution expected from loops involving the top quark:
\begin{align}
\frac{\alpha_{\text{EM}}^2 \,m_t}{16\pi \sin^4(\theta_W)} \sim 4 \ \text{MeV}. \label{eqn:error_estimate}
\end{align}
As we can see in Figure \ref{fig:wino_two_loop}, this estimate does indeed give a reasonable rough estimate of the uncertainty on the one-loop result.

\begin{table*}[tp]
\centering
\caption{The effect of uncertainties in input parameters on the mass splitting and decay lifetime in the wino and MDM models.  The effect on the decay lifetime is taken to be the difference between the upper and lower lifetimes normalised by the mean of the upper and lower values, expressed as a percentage.}\label{tab:uncertainty_analysis}
\footnotesize{
\vspace{0.4cm}
\begin{tabular}{cll@{\hspace{4mm}}cll}
 & \multicolumn{2}{c}{Wino model ($\hat{M}=3\,$TeV)} & & \multicolumn{2}{c}{MDM ($\hat{M}=9.6\,$TeV)}  \\
 \cline{2-3}  \cline{5-6}
& Change in & Change in & & Change in & Change in  \\
Parameter & $\Delta M$ (MeV) & lifetime (\%) & & $\Delta M^+$ ($\Delta M^{++})$ (MeV) & lifetime (\%) \\
\hline
$1/\alpha^{\overline{MS}}(m_Z)$   											&$0.0919$     & $0.310$ && $0.101$ ( $0.402$ )  & $0.348$ ( $0.209$ ) \\
 \phantom{$^{\overline{MS}}$}$m_t$\phantom{$^{\overline{MS}}$}  		& $0.192$       &$0.647$ &&  $0.175$ ( $0.699$ ) & $0.604$ ( $0.364$ )  \\
\phantom{$^{\overline{MS}}$}$m_h$\phantom{$^{\overline{MS}}$}  		& $0.0124$     &$ 0.0417$ &&  $0.017$ ( $0.068$ )  &  $0.0588$ ( $0.0354$ )   \\
\phantom{$^{\overline{MS}}$}$m_W$\phantom{$^{\overline{MS}}$}  		& $\num{8.22e-8}$  & $\num{2.77e-7}$&&  $\num{4.6e-9}$ ( $\num{1.85e-8}$ )   &  $\num{1.59e-8}$ ( $\num{9.65e-9}$ )  \\
 \phantom{$^{\overline{MS}}$}$m_Z$\phantom{$^{\overline{MS}}$}  		& $0.00936$   &$0.0316$&& $0.00467$ ( $0.0187$ )  & $0.0162$ ( $0.00974$ )  \\
 \phantom{$^{\overline{MS}}$}$m_e$\phantom{$^{\overline{MS}}$}  		& $\num{8.23e-6}$ & $\num{2.78e-5}$&&  $\num{2.04e-5}$ ( $\num{8.15e-5}$ )   & $\num{7.05e-5}$ ( $\num{4.24e-5}$ )   \\
\phantom{$^{\overline{MS}}$}$m_{\mu}$\phantom{$^{\overline{MS}}$}  	& $\num{3.69e-9}$ & $\num{1.25e-8}$&& $\num{9.87e-9}$ ( $\num{3.95e-8}$ ) & $\num{3.41e-8}$ ( $\num{2.06e-8}$ )   \\
\phantom{$^{\overline{MS}}$}$m_{\tau}$\phantom{$^{\overline{MS}}$}  	&  $\num{3.551e-6}$ & $\num{1.199e-5}$&&  $\num{3.37e-06}$ ( $\num{1.35e-5}$ )  & $\num{1.16e-05}$ ( $\num{7.01e-6}$ )  \\
$m_d^{\overline{MS}}(2\,\text{GeV})$  										& $\num{1.85e-4}$  &$ 0.000623 $&& $0.000845$ ( $0.00338$ )  & $0.00292$ ( $0.00176$ ) \\
$m_u^{\overline{MS}}(2\,\text{GeV})$     										& $\num{3.0927e-4}$    &$ 0.00104$&&   $0.00477$ ( $0.0191$ )  & $0.0165$ ( $0.00994$ ) \\
$m_s^{\overline{MS}}(2\,\text{GeV})$  										& $\num{8.467e-5}$   &$0.000286$&&  $0.001$ ( $0.00402$ ) & $0.00348$ ( $0.00209$ )   \\
$m_c^{\overline{MS}}(m_c)$ 													& $0.00176$    &$0.00595$&& $0.0017$ ( $0.00679$ )  & $0.00587$ ( $0.00354$ ) \\
$m_b^{\overline{MS}}(m_b)$  											     &  $0.0007539$ & $0.00255$&&  $0.00195$ ( $0.0078$ )  &  $0.00674$ ( $0.00406$ )  \\
$\alpha_S^{\overline{MS}}(m_Z)$  										     &  $0.00224$              &$0.00759$&& $0.00436$ ( $0.0174$ )   & $0.0151$ ( $0.00908$ )   \\
$Q$    																		     & $0.304$ 	    & $1.03$&& $0.242$ ( $0.969$ )   & $0.839$ ( $0.505$ )   \\
\hline\end{tabular}
}
\end{table*}

In Table \ref{tab:uncertainty_analysis} we present a detailed analysis of the uncertainties entering into this calculation.  As there are several uncorrelated uncertainties to include, we simply consider the effect of each individually.  The effect of including light quark masses is a $+0.0532\,$MeV change in the mass splitting, resulting in a $0.180$\% decrease in the lifetime.  The parameter with the largest effect on the mass splitting is the renormalisation scale.  Although this uncertainty is greatly reduced at the two-loop level (as seen in Figure \ref{fig:wino_two_loop}), it is still the dominant contribution.  We find that the uncertainties on the top mass and electromagnetic coupling also induce an $\mathcal{O}(0.1)$\,MeV uncertainty in the mass splitting.  All other parameters have negligible impacts on the mass splitting.  Although including the light quarks does slightly increase $\Delta M$, the uncertainties on these masses have almost no impact on the result.  Finally, we note that the strong coupling even has some influence, which is entirely indirect through the calculation of the spectrum, as this coupling is not directly involved in the wino mass calculation.

In the right panel of Figure \ref{fig:wino_two_loop} we present the decay lifetime of the charged component in units of mm/$c$, as a function of the degenerate mass $\hat{M}$.  The charged component decays as $\chi^+\rightarrow X\chi^0$, which is dominated by channels where $X$ is either a pion, an electron+neutrino or muon+neutrino pair.

The decay width for the pion channel in an electroweak multiplet with total weak isospin $j$, with eigenstates $\chi_{I}$ where $I\in\{-j,-j+1,.\,.\,.\,,j-1,j\}$, is given by
\cite{2015JHEP...07..074D}
\begin{align}
\Gamma\left(\chi_{I+1}\rightarrow\chi_{I}\pi^+\right) = T^2_+\frac{G_F^2\Delta M^3 V_{\text{ud}}^2f_{\pi}^3}{\pi}\sqrt{1-\frac{m_{\pi}^2}{\Delta M^2}}, \label{eqn:pi_width}
\end{align}
where $T_+=\sqrt{j(j+1)-I(I+1)}$, $f_\pi=130.2\pm1.7$\,MeV, $|V_{\text{ud}}|= 0.97417\pm0.00021$ \cite{Patrignani2016,2015arXiv150902220R} and $m_{\pi}$ is the pion mass. $T_+^2$ is equivalent to $(n^2-1)/4$ for $I=0$, for a representation of dimension $n$, as given in Ref.~\cite{Cirelli2009}, however for the MDM case we will need this more general expression.  For wino dark matter we have $j=1$ and $I=0$ to give $\Gamma(\chi^+\rightarrow\chi^0\pi^+)$.

For $\Delta M\approx 170\,\mathrm{MeV} > m_{\pi}$ the pion decay is the dominant channel, with a 97.7\% branching fraction \cite{Cirelli2009}.  The other kinematically-allowed channels are the electron-neutrino and muon-neutrino ones, which have widths
\begin{align}
\Gamma^{\chi^+}_{e}=T^2_+\frac{G_F^2\Delta M^5}{15\pi^2} \label{eqn:e_width}
\end{align}
and $\Gamma^{\chi^+}_{\mu}=0.12\Gamma^{\chi^+}_e$.  The expected lifetime of the charged component is thus $\tau=(\Gamma^{\chi^+}_e+\Gamma^{\chi^+}_{\mu}+\Gamma^{\chi^+}_{\pi})^{-1}$.  The large step in the decay lifetime in Figure \ref{fig:wino_two_loop} is where $\Delta M>m_{\pi}$ and the pion channel opens, and the smaller step is due to the muon channel opening.  These can be seen clearly as branching fractions in Figure \ref{fig:BFs}.

The most phenomenologically interesting mass range for pure wino-like neutralino dark matter is $\hat{M}\sim 3\,$TeV, as this would give the correct dark matter relic abundance \cite{Hisano2007,Hryczuk2011}.  For this value and assuming $Q=m_t$, the two-loop mass splitting is $164.5\,$MeV, compared to the one-loop value of $167.5\,$MeV.  This difference in mass splitting represents a $9.7\,$\% increase in the decay lifetime of the chargino when going from the one-loop to the two-loop calculation.  For other masses, this ratio can be larger, depending on the dominant decay channel.  For example, for a wino of $70\,$GeV mass, the one and two-loop mass splittings are $142.3\,$MeV and $145.5\,$MeV respectively, with a increase in the lifetime of 40.1\%.  Thus, although the difference in the mass splittings is approximately the same ($\sim$3\,MeV), as we can see in Figure \ref{fig:BFs}, this mass value is exactly where the pion channel opens up, so the effect on the lifetime in this range is far more significant.

We now offer an updated fit, using the latest values in Table \ref{table:params}, two-loop RGEs and non-zero light quark masses,
\begin{widetext}
\begin{eqnarray}
\frac{\Delta M}{1~ {\rm MeV} } &=&  - 413.7
+305.7 \left(\ln \frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)
-60.96 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^2
 +5.429 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^3\label{eqn:fit1a}
-0.182 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^4.
\end{eqnarray}
\end{widetext}
This fit is valid over the range $100\,\mathrm{GeV} \le M^0_\mathrm{pole} \le 4$\,TeV.  The effect of including light quark masses is a small positive shift in $\Delta M$, and 2-loop RGEs a smaller negative shift, with a total difference of approximately $-0.03\%$.

\subsection{The MDM quintuplet}

The MDM quintuplet has two mass splittings.  The first, $\Delta M^{+}\equiv M_{\text{pole}}^{+}-M_{\text{pole}}^0$, is analogous to $\Delta M$ in the wino model, with a one-loop value of $\mathcal{O}(170)\,$MeV.  The second, $\Delta M^{++}\equiv M_{\text{pole}}^{++}-M_{\text{pole}}^0$, between the neutral and doubly-charged component, has a value of $\mathcal{O}(670)\,$MeV at one loop.  In this section we present the first analysis of these mass splittings at the two-loop level and the subsequent decay lifetimes of the charged components.  In Section \ref{sec:comparison} we discuss the differences between the charged/neutral component mass splitting in the MDM and wino models.

In Figure \ref{fig:deltam_mdm} we present the two-loop mass splittings between the neutral and charged (\textit{left panel}) and the neutral and doubly-charged (\textit{right panel}) components.  The dominant uncertainty, resulting from the choice of renormalisation scale, is indicated by the dark shaded regions at one loop (\textit{dark green}) and two loops (\textit{red}), where $Q$ has been varied continuously between $m_t/2$ and $2m_t$.  Once again we see a significant reduction in the uncertainty at the two-loop level, at least for moderate and large multiplet masses; at lower multiplet masses ($\hat{M}\lesssim100$\,GeV), the two-loop uncertainty grows due to the additional electroweak-scale logs introduced at the two-loop level, just as in the triplet case.  The light-green band is the naive estimate of the missing two-loop contribution, where we use Eq.\ (\ref{eqn:error_estimate}) for $\Delta M^+$, and multiply this by a factor of four for $\Delta M^{++}$, based on the generic charge-dependent pre-factors for one-loop electroweak mass splitting in Eq.\ (4) of Ref.~\cite{Yamada2010}.

In Table \ref{tab:uncertainty_analysis}, we also give a detailed presentation of the uncertainties entering into the two-loop calculation in the MDM model.  Again, we consider the effect of each uncertainty individually.  As in the wino case, the parameter with the largest effect on the mass splitting is the renormalisation scale, but its effect is greatly reduced by going to two loops (Figure \ref{fig:deltam_mdm}).  The top mass and electromagnetic coupling are again responsible for an $\mathcal{O}(0.1)$ MeV uncertainty in the mass splittings.  All other parameters have negligible impacts on the splittings.  Including the masses of light quarks results in a $+0.0125\,$MeV change in $\Delta M^+$ and a $0.0499\,$MeV change in $\Delta M^{++}$, which translate into $0.0432$\% and $0.0258$\% reductions in the respective lifetimes of the charged and doubly-charged states.  As with the triplet, although finite light quark masses affect $\Delta M$, the uncertainties on those masses have little impact -- and the strong coupling has some influence via the calculation of the spectrum (on the order of $0.01$\%).

In Figure \ref{fig:decays_mdm} we plot the decay lifetimes of the charged and doubly-charged components.  The lifetime of the charged component can be computed using Eqs.~(\ref{eqn:e_width}) and (\ref{eqn:pi_width}) with $j=2$ and $I=0$; the calculation is the same for the doubly-charged component, but with $I=1$ instead.  The doubly-charged component has an additional decay channel via the process $\chi^{++}\rightarrow\chi^+K^+$, where $K^+$ is a kaon.   We take the partial decay width to the kaon channel to be
\begin{align}
\Gamma_{K^+} = T^2_+\frac{G_F^2\Delta M^3 V_{\text{us}}^2f_{K^+}^3}{\pi}\sqrt{1-\frac{m_{K^+}^2}{\Delta M^2}}, \label{eqn:k_width}
\end{align}
where $f_{K^+}=155.6\pm0.4$\,MeV, $|V_{\text{us}}|=  0.2248\pm0.0006$ \cite{Patrignani2016,2015arXiv150902220R} and $m_{K^+}$ is the kaon mass.

The most phenomenologically interesting mass range for MDM is $\hat{M}\sim 9.6\,$TeV, as this would give the correct dark matter relic abundance \cite{Cirelli2007}.  For this value the two-loop mass splittings are $\Delta M^+ = 163.6\,$MeV and $\Delta M^{++} =  654.3\,$MeV, which can be compared with the one-loop values of $168.3\,$MeV and  $673.4\,$MeV respectively, for a choice of $Q=m_t$.  This difference in mass splitting represents a 15.5\% change in the decay lifetime of the charged component when going from the one-loop to the two-loop calculation, and a 9.78\% change in the decay lifetime of the doubly-charged component.  Like in the wino model, this ratio will be larger at different mass values, depending on the dominant decay channel.  One important new feature in this calculation is the opening up of the kaon channel, which we indicate with the orange line in the right panel of Figure \ref{fig:BR_mdm}.

\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{deltam_mdm.pdf}\includegraphics[width=0.5\textwidth]{deltam2_mdm.pdf}
\caption{The two-loop mass splittings between the charged and neutral components (\textit{left}) and the doubly-charged and neutral components (\textit{right}) in the MDM model as a function of the degenerate tree-level $\MSBar$  mass.  The dark green and red bands are the range of the one and two-loop mass splittings respectively when $Q$ is varied continuously between $m_t/2$ and $2m_t$.  The light green band is the estimated uncertainty on the one-loop result using Eq.~(\ref{eqn:error_estimate}).}\label{fig:deltam_mdm}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{decays_MDM.pdf}
\caption{The decay lifetimes of the charged and doubly-charged components in the MDM model as a function of the degenerate tree-level $\MSBar$  mass.  The dark green and red bands are the respective ranges of the one and two-loop mass splittings when $Q$ is varied continuously between $m_t/2$ and $2m_t$.  The light green band is the estimated uncertainty on the one-loop result using Eq.~(\ref{eqn:error_estimate}).}\label{fig:decays_mdm}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{BR_MDM.pdf}\includegraphics[width=0.5\textwidth]{BR2_MDM.pdf}
\caption{Branching fractions in the MDM model for the $\chi^+\rightarrow X \chi^0$ (\textit{left}) and $\chi^{++}\rightarrow X \chi^+$ (\textit{right}) processes, where $X \in \{e^+\nu_e,\mu^+\nu_\mu,\pi^+,K^+\}$.  The solid lines are the branching fractions using the two-loop mass splitting and dotted lines are the results using the one-loop result, both evaluated at $Q=m_t=173.34\,$GeV.}\label{fig:BR_mdm}
\end{figure*}

These two-loop mass splitting results can be reproduced using the following fitting formulae.  For $\Delta M^{(+)}$, we have
\begin{widetext}
\begin{eqnarray}
\frac{\Delta M^{(+)}}{1~ {\rm MeV} } &=&  - 328.6
+250.1 \left(\ln \frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)
- 47.7 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^2
 +4.049 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^3
-0.1292 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^4. \label{eqn:fit2}
\end{eqnarray}
and for $\Delta M^{(++)}$, we have
\begin{eqnarray}
\frac{\Delta M^{(++)}}{1~ {\rm MeV} } &=&  - 1314
+1000 \left(\ln \frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)
- 190.7 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^2
+16.18 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^3
-0.5162 \left(\ln\frac{M_{\text{pole}}^0}{1~ {\rm GeV} }\right)^4. \label{eqn:fit3}
\end{eqnarray}
\end{widetext}
These formulae are valid for values of $M^0_\mathrm{pole}$ between $100\,$GeV and $10\,$TeV.

\subsection{Differences between triplet and quintuplet models}\label{sec:comparison}

The two-loop loop mass splitting between the charged and neutral multiplet component is not identical in the triplet and quintuplet models.  At the one-loop level this mass splitting is the same in both representations, yet when we go to the next loop order there are subtle differences.  In this section we discuss these differences and determine which diagrams are responsible.

In the MDM model, for multiplet masses $\gtrsim 1$\,TeV we see a decrease in the two-loop mass splitting.  In the two-loop wino result, and in the one-loop case for both models, we see a constant mass splitting in the limit of large $\hat{M}$.  In the one-loop case, this can be seen directly from the difference of the one-loop self energies (given in Appendix \ref{sec:self_energies}), and the fact that we do not apply threshold corrections (as they are technically of higher loop order).  If we were to include threshold corrections to the one-loop result, we would see a similar decrease in the mass splitting for large $\hat{M}$, as we would be introducing extra logarithmic terms with nothing to cancel them.

In the wino model the constant mass splitting at large $\hat{M}$ is the result of a cancellation between these threshold corrections and one specific set of diagrams.  These are specifically the corrections to the gauge boson propagators coming from the new multiplet fermions.  The diagrams that contribute to the gauge boson propagators are are all those in Figure \ref{fig:Feynman_diagrams2} and the first counter-term diagram in Figure \ref{fig:Feynman_diagramsCT}.  Ref.\ \cite{Yamada2010} asserts that this cancellation occurs exactly for \textit{all} SU(2) multiplets, and therefore goes on to ignore threshold corrections and the influence of the multiplet fermions on the gauge boson propagator.  Our calculations show that this cancellation does indeed occur for the triplet, but that the resulting logs do \textit{not} perfectly cancel in the quintuplet case.  The fact that the mass splitting is almost flat in the large $\hat{M}$ limit indicates that \textit{most} of the logs have cancelled (as e.g.\ neglecting threshold corrections results in a clear logarithmic increase in the splitting with increasing $\hat{M}$) -- but some small residual term of the form $-\log(\hat{M}/Q)$ remains.

\begin{figure*}
\centering
\includegraphics[width=0.5\textwidth]{deltam_AB_mssm.pdf}\includegraphics[width=0.5\textwidth]{deltam_AB_mdm.pdf}
\caption{
The one-loop (green band), partial two-loop (red band) and extended partial two-loop (blue band) mass splittings in the wino limit of the MSSM (\textit{left}) and the MDM quintuplet model (\textit{right}). The partial two-loop mass splitting is computed using self energies constructed from diagrams in Figure \ref{fig:Feynman_diagrams} and all except the left-most diagram in Figure \ref{fig:Feynman_diagramsCT}, that is, all two-loop diagrams except those that include a correction to a gauge boson propagator.  The extended partial two-loop mass splitting is calculated with all two-loop diagrams except those that include a correction to a gauge boson propagator by $\chi$ fermions.  The coloured bands are determined by varying $Q$ continuously between $m_t/2$ and $2m_t$.  For these calculations, we neglect threshold corrections and all running of parameters.}\label{fig:multiplet_compare}
\end{figure*}

To illustrate this point, we can construct a partial two-loop mass-splitting calculation with the terms responsible for the residual logs excluded.  First, we construct two-loop amplitudes by neglecting threshold corrections and excluding all contributions to the gauge boson self-energy, i.e. all diagrams in Figure \ref{fig:Feynman_diagrams2} and the first in Figure \ref{fig:Feynman_diagramsCT}.  In Figure \ref{fig:multiplet_compare}, we plot the resulting partial two-loop mass splittings in each model as `Partial Two-loop no $\Pi_{VV}$', along with the one-loop results.  We see that the results are indeed identical at large $\hat{M}$.  We can also see that this incomplete subset of diagrams misses some important cancellations of scale-dependent logarithmic terms, as the uncertainty from scale dependence in the resulting two-loop splitting is much larger in the partial two-loop amplitude compared to the full two-loop result.

To investigate further, we next exclude \textit{only} those diagrams where the multiplet fermions contribute to the gauge boson propagator, i.e.\ the versions of the second diagram in Figure \ref{fig:Feynman_diagrams2} with a $\chi$ fermion in the upper loop.  Continuing to neglect threshold corrections, we then recompute the corresponding counter-term (the first in Figure \ref{fig:Feynman_diagramsCT}) with the same contributions removed from the gauge boson propagator, and recompute the mass splitting.  We refer to this extended partial amplitude as `Partial Two-loop no $\Pi_{VV,\chi}$' in Figure \ref{fig:multiplet_compare}.  The splitting is still flat at large $\hat{M}$ in both models, albeit with a larger scale dependence in the MDM model due to the contributions of a large number of additional diagrams (relative to the wino model) in Fig.\ \ref{fig:Feynman_diagrams2} with $\chi^{\pm\pm}$ in the lower internal propagator.  The flatness of the extended partial two-loop result at large $\hat{M}$ shows that the uncanceled logarithms in the full quintuplet calculation specifically arise from the failure of the threshold corrections to fully cancel the logs from the contribution of the multiplet fermions to the gauge boson self-energies.  Unlike light quark masses, which increase the mass splitting (Figure \ref{fig:Ibe_compare}), the addition of multiplet fermions reduces the splitting, as the two types of fermions enter into the gauge boson self energies with opposite signs.  The fact that the mass splitting ultimately turns down in the MDM quintuplet therefore indicates that the impacts of the multiplet fermions on the gauge boson self-energies dominate over the threshold corrections in this model.

That the logarithms do not fully cancel in the quintuplet model suggests that they will also not completely cancel for higher-dimensional representations of SU(2).

\section{Conclusion}
\label{sec5}

We have presented a two-loop calculation of mass splitting in electroweak multiplets, in the wino limit of the MSSM and in the MDM fermionic quintuplet model.  In the wino model, we showed that our calculation is in agreement with the previous two-loop calculation.  We improved on the previous calculation by using two-loop RGEs and including finite masses for light quarks.

We also presented the first complete two-loop calculation of the splitting in the MDM quintuplet model, showing that it is not constant in the limit of large multiplet masses.  This is contrary to the triplet case, and the naive expectation from the one-loop result.  This result comes from the influence of the additional heavy fermions on the gauge boson self energies, and subsequently the two-loop self energies of the multiplet.  As the mass of the multiplet increases, so does its effect on the mass splitting through these diagrams.

The two-loop corrections that we present here are phenomenologically relevant, resulting in a $\sim$10\% change in the lifetime of the charged components in both models.  This is in agreement with previous calculations for wino dark matter \cite{Ibe2013}.  It is similarly important to include the two-loop radiative corrections presented here when considering disappearing track searches for MDM.

\begin{acknowledgments}
We would like to thank Ryosuke Sato and Sanjay Bloor for useful discussions, Vladyslav Shtabovenko and Stephen Martin for helpful correspondence regarding technical aspects of two-loop calculations and the use of \textsf{FeynCalc}  and \textsf{TSIL}, respectively, and Peter Athron and Alex Voigt for helpful correspondence about \textsf{FlexibleSUSY}, \textsf{SARAH}  and threshold corrections.  We thank Julian Heeck for helpful comments on the submitted version of this paper. JM is supported by the Imperial College London President's PhD Scholarship and PS by STFC (ST/K00414X/1, ST/N000838/1, ST/P000762/1).
\end{acknowledgments}

\begin{widetext}

redpajama_set_name: RedPajamaArXiv


text:
UK consumers getting more for less in telecoms, but prices are going up
Written by Mary Lennighan
23 July 2021 @ 11:03
Consumers in the UK are paying less for their communications services than they were five years ago, even though data usage is increasing, according to Ofcom's latest Pricing Trends report.
However, the regulator also noted that a number of big names in the industry have hiked prices in recent months and warned telcos that they must "do more" to help low-income users.
Average mobile prices were 22% lower last year than in 2015, while minutes of usage increased by 32% and data usage was up by 369%, according to Ofcom. The average cost of a basket of mobile services, based on average usage and excluding the handset, was £11 per month in 2020, the regulator added, down by 10% on 2019 despite growing usage.
Similarly, average new customer prices for superfast broadband and landline bundles fell by almost 20% last year compared with 2015, but data usage grew by 342% and download speeds rose by 178%.
Telecoms service prices in the UK also compare favourably with other markets analysed by Ofcom. The country comes out on top in the mobile market, offering cheaper services than France, Germany, Italy, Spain and the US. The UK also ranked second and third respectively for average fixed-line and fixed broadband prices, and came second for the price of triple-play bundles. Overall, the UK comes in third of the six, behind perennial leader France, which continues to offer the lowest telecoms prices, and Italy.
Despite those positive figures – for consumers, at least, if not so much for telcos – the regulator sounded a number of notes of caution for communications services customers in the UK.
On the mobile side, it advised consumers that they are better off taking SIM-only deals and buying a handset separately, rather than paying for the phone as part of an operator contract.
We estimate the difference in price is equivalent to customers being charged an APR of up to 23% for their handset 'loan,' Ofcom said. "This can be higher if customers do not move to a SIM-only deal when their contract ends."
That probably doesn't come as a huge surprise to UK consumers, nor does the fact that high-usage customers buying bundles are getting better value for money. People buying bundled services including superfast broadband or mobile products with large amounts of data "tend to have access to a wide range of keenly priced tariffs," Ofcom said.
But the flip side of that is that some lower-use customers are less well served. For example, some standard broadband users who do not keep an eye on the market could find they are paying more than superfast users for a slower service. Meanwhile, in the mobile market, a paucity of sub-1 GB data plans means these tend to be relatively expensive, again hitting low usage customers, while low-end mobile plans with no bundled data actually rose last year.
"In a complex marketplace, shopping around brings considerable savings," Ofcom advised.
This is good advice, particularly given that a number of big telco names have increased prices in recent months. Ofcom listed BT, its mobile arm EE, Three and Vodafone as examples of providers that have introduced above-inflation price rises in the first half of 2021.
It also noted that pay-TV prices have risen over the past year, adding that consumers are more on the ball when it comes to market engagement in broadband and mobile services than they are with pay TV; churn in the pay TV sector was just 7% over 12 months, compared with 15% in mobile, for example. Again, the advice is to engage with the market.
For some customer segments though, engagement only goes so far. The headline finding of Ofcom's report – or at least the point it chose to lead with – is that despite generally falling prices and the introduction of some low-cost tariffs for low-income users, telcos need to do more to help those struggling to afford communications services.
The regulator claims around 2 million UK households find paying for Internet connectivity a struggle. This year BT, Community Fibre, Hyperoptic, KCOM, Virgin Media and VOXI have all introduced low-cost tariffs – ranging from £10 to £20 per month – for people on benefits, or improved their existing offers. However, take-up of those tariffs has been low, with only 40,000 households, or 0.15% of UK homes, having signed up; that equates to just 1% of those in receipt of out-of-work benefits.
"Telecoms firms need to do more," Ofcom declared, without really specifying whether it is looking for more low-cost services or more marketing and education, or indeed something else. It hinted that it will force the issue if it has to though.
"If the telecoms industry does not take sufficient action to address our concerns, we think there would be a strong case for exploring whether mandatory social tariffs would be necessary to fill the gaps in support, alongside other potential options," Ofcom said.
Well, that's cleared things up.
Tags: BSS, Ofcom, pricing, Regulation, tariffs, UK
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DAZN reportedly on cusp of sealing $800m BT Sport deal
Telenor Myanmar to hike prices following regressive tax


text:
Q: Почему такой результат умножения Почему так происходит ?
Код:
unsigned short x = 0 - 1;
unsigned short y = 0 - 1 ;
unsigned long z = x * y;
cout << hex << x << " \t" << y << " \t" << z <<  " \t" << x * y << endl;

Результат:
ffff    ffff    fffffffffffe0001        fffe0001

A: @DUP "Мне интересно было почему последние два результата отличаются? fffffffffffe0001 и fffe0001."
Это для 64-разрядного CPU. long 64-bit, x * y неявно приводится компилятором к int (32-bit), при преобразовании в long знаковый бит размножается. Для 32-bit CPU последние 2 результата будут равны (fffe0001).
Вообще будьте осторожнее с вычислениями для операндов с разной разрядностью и unsigned. Лучше перед вычислениями явно делайте присваивания промежуточным переменным (меньше непоняток будет).
A: Шутки приведения типов. Когда умножаете x*y, получаете 0xfffe0001, приведённое к типу int. Когда это выражение присваиваете его z типа unsigned long, оно приводится к типу long (расширением знака), получается fffffffffffe0001. Если выводите умноженный результат, то получаете fffe0001 в шестнадцатиричном формате (-131071 в десятичном). Вроде так

redpajama_set_name: RedPajamaStackExchange


Billie joined the National Drug and Alcohol Centre (NDARC) in 2018. Billie is currently working with the CESSATE Study Research Team, a study evaluating the cost-effectiveness of two smoking cessation products, Desmoxan and Champix. Previously, Billie's research has focused on Public Health, Indigenous Health, Rural Communities and Qualitative Social Research. Billie's thesis (Hons, 2017) examined nurse memorials in Australian communities, focusing on cultural artefacts to uncover further narratives of nurse influence in rural Australian communities.
Cultural History · Bachelor of Arts (Hons) Univeristy of Wollongong.
Nov 20, 2016 Australian and New Zealand Journal of Public Health, vol. 41, no.1, pp. 27–31.


text:
package com.alcedomoreno.sirme.business.data;

import java.util.Date;

import com.alcedomoreno.sirme.business.transform.PhotoTransform;
import com.alcedomoreno.sirme.business.transform.Transformator;

public class Photo implements BusinessObject, Cloneable {

	private int idPhoto;
	private String path;
	private String comments;
	private Date dateCreated;
	private Work work;

	@Override
	public String toString() {
		return idPhoto + "," + path + "," + comments + "," + dateCreated;
	}

	@Override
	public Class<? extends Transformator> getTransformator() {
		return PhotoTransform.class;
	}

	public int getIdPhoto() {
		return idPhoto;
	}

	public void setIdPhoto(int idPhoto) {
		this.idPhoto = idPhoto;
	}

	public String getPath() {
		return path;
	}

	public void setPath(String path) {
		this.path = path;
	}

	public String getComments() {
		return comments;
	}

	public void setComments(String comments) {
		this.comments = comments;
	}

	public Date getDateCreated() {
		return dateCreated;
	}

	public void setDateCreated(Date dateCreated) {
		this.dateCreated = dateCreated;
	}

	public Work getWork() {
		return work;
	}

	public void setWork(Work work) {
		this.work = work;
	}

	@Override
	public int hashCode() {
		final int prime = 31;
		int result = 1;
		result = prime * result + idPhoto;
		return result;
	}

	@Override
	public boolean equals(Object obj) {
		if (this == obj)
			return true;
		if (obj == null)
			return false;
		if (getClass() != obj.getClass())
			return false;
		Photo other = (Photo) obj;
		if (idPhoto != other.idPhoto)
			return false;
		return true;
	}

}

redpajama_set_name: RedPajamaGithub