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package io.aeron.protocol;

import org.agrona.LangUtil;
import org.agrona.concurrent.UnsafeBuffer;

import java.nio.ByteBuffer;

import static java.lang.Integer.toHexString;
import static java.nio.ByteOrder.LITTLE_ENDIAN;
import static org.agrona.BitUtil.*;

/**
 * Flyweight for Resolution Entry header.
 *

 * Protocol Specification
 * wiki page.
 */
public class ResolutionEntryFlyweight extends HeaderFlyweight
{
    public static final byte RES_TYPE_NAME_TO_IP4_MD = 0x01;
    public static final byte RES_TYPE_NAME_TO_IP6_MD = 0x02;

    public static final int ADDRESS_LENGTH_IP4 = 4;
    public static final int ADDRESS_LENGTH_IP6 = 16;

    /**
     * (S) - Self flag
     */
    public static final short SELF_FLAG = 0x80;

    public static final int RES_TYPE_FIELD_OFFSET = 0;
    public static final int RES_FLAGS_FIELD_OFFSET = 1;
    public static final int UDP_PORT_FIELD_OFFSET = 2;
    public static final int AGE_IN_MS_FIELD_OFFSET = 4;
    public static final int ADDRESS_FIELD_OFFSET = 8;

    public static final int MAX_NAME_LENGTH = 512;

    public ResolutionEntryFlyweight()
    {
    }

    public ResolutionEntryFlyweight(final UnsafeBuffer buffer)
    {
        super(buffer);
    }

    public ResolutionEntryFlyweight(final ByteBuffer buffer)
    {
        super(buffer);
    }

    public ResolutionEntryFlyweight resType(final byte type)
    {
        putByte(RES_TYPE_FIELD_OFFSET, type);
        return this;
    }

    public byte resType()
    {
        return getByte(RES_TYPE_FIELD_OFFSET);
    }

    public ResolutionEntryFlyweight flags(final short flags)
    {
        putByte(RES_FLAGS_FIELD_OFFSET, (byte)flags);
        return this;
    }

    public short flags()
    {
        return (short)(getByte(RES_FLAGS_FIELD_OFFSET) & 0xFF);
    }

    public ResolutionEntryFlyweight udpPort(final int udpPort)
    {
        putShort(UDP_PORT_FIELD_OFFSET, (short)udpPort, LITTLE_ENDIAN);
        return this;
    }

    public int udpPort()
    {
        return getShort(UDP_PORT_FIELD_OFFSET, LITTLE_ENDIAN) & 0xFFFF;
    }

    public ResolutionEntryFlyweight ageInMs(final int ageInMs)
    {
        putInt(AGE_IN_MS_FIELD_OFFSET, ageInMs, LITTLE_ENDIAN);
        return this;
    }

    public int ageInMs()
    {
        return getInt(AGE_IN_MS_FIELD_OFFSET, LITTLE_ENDIAN);
    }

    public ResolutionEntryFlyweight putAddress(final byte[] address)
    {
        switch (resType())
        {
            case RES_TYPE_NAME_TO_IP4_MD:
                putBytes(ADDRESS_FIELD_OFFSET, address, 0, 4);
                return this;

            case RES_TYPE_NAME_TO_IP6_MD:
                putBytes(ADDRESS_FIELD_OFFSET, address, 0, 16);
                return this;
        }

        throw new IllegalStateException("unknown RES_TYPE=" + resType());
    }

    public int getAddress(final byte[] address)
    {
        switch (resType())
        {
            case RES_TYPE_NAME_TO_IP4_MD:
                getBytes(ADDRESS_FIELD_OFFSET, address, 0, 4);
                return 4;

            case RES_TYPE_NAME_TO_IP6_MD:
                getBytes(ADDRESS_FIELD_OFFSET, address, 0, 16);
                return 16;
        }

        throw new IllegalStateException("unknown RES_TYPE=" + resType());
    }

    public void appendAddress(final Appendable appendable)
    {
        try
        {
            switch (resType())
            {
                case RES_TYPE_NAME_TO_IP4_MD:
                {
                    final int i = ADDRESS_FIELD_OFFSET;
                    appendable
                        .append(String.valueOf(getByte(i) & 0xFF))
                        .append('.')
                        .append(String.valueOf(getByte(i + 1) & 0xFF))
                        .append('.')
                        .append(String.valueOf(getByte(i + 2) & 0xFF))
                        .append('.')
                        .append(String.valueOf(getByte(i + 3) & 0xFF));
                    return;
                }

                case RES_TYPE_NAME_TO_IP6_MD:
                {
                    final int i = ADDRESS_FIELD_OFFSET;
                    appendable
                        .append(toHexString(((getByte(i) << 8) & 0xFF00) | getByte(i + 1) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 2) << 8) & 0xFF00) | getByte(i + 3) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 4) << 8) & 0xFF00) | getByte(i + 5) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 6) << 8) & 0xFF00) | getByte(i + 7) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 8) << 8) & 0xFF00) | getByte(i + 9) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 10) << 8) & 0xFF00) | getByte(i + 11) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 12) << 8) & 0xFF00) | getByte(i + 13) & 0xFF))
                        .append(':')
                        .append(toHexString(((getByte(i + 14) << 8) & 0xFF00) | getByte(i + 15) & 0xFF));
                    return;
                }
            }

            appendable
                .append("unknown RES_TYPE=")
                .append(String.valueOf(resType()));
        }
        catch (final Exception ex)
        {
            LangUtil.rethrowUnchecked(ex);
        }
    }

    public ResolutionEntryFlyweight putName(final byte[] name)
    {
        final int nameOffset = nameOffset(resType());

        putShort(nameOffset, (short)name.length, LITTLE_ENDIAN);
        putBytes(nameOffset + SIZE_OF_SHORT, name);
        return this;
    }

    public int getName(final byte[] name)
    {
        final int nameOffset = nameOffset(resType());
        final short nameLength = getShort(nameOffset, LITTLE_ENDIAN);

        getBytes(nameOffset + SIZE_OF_SHORT, name, 0, nameLength);
        return nameLength;
    }

    public void appendName(final Appendable appendable)
    {
        final int nameOffset = nameOffset(resType());
        final short nameLength = getShort(nameOffset, LITTLE_ENDIAN);

        getStringWithoutLengthAscii(nameOffset + SIZE_OF_SHORT, nameLength, appendable);
    }

    public int entryLength()
    {
        final int nameOffset = nameOffset(resType());

        return align(nameOffset + SIZE_OF_SHORT + getShort(nameOffset, LITTLE_ENDIAN), SIZE_OF_LONG);
    }

    public static int nameOffset(final byte type)
    {
        switch (type)
        {
            case RES_TYPE_NAME_TO_IP4_MD:
                return ADDRESS_FIELD_OFFSET + 4;

            case RES_TYPE_NAME_TO_IP6_MD:
                return ADDRESS_FIELD_OFFSET + 16;
        }

        throw new IllegalStateException("unknown RES_TYPE=" + type);
    }

    public static int entryLengthRequired(final byte type, final int nameLength)
    {
        return align(nameOffset(type) + SIZE_OF_SHORT + nameLength, SIZE_OF_LONG);
    }

    public static int addressLength(final byte type)
    {
        switch (type)
        {
            case RES_TYPE_NAME_TO_IP4_MD:
                return 4;

            case RES_TYPE_NAME_TO_IP6_MD:
                return 16;
        }

        throw new IllegalStateException("unknown RES_TYPE=" + type);
    }

    public static boolean isAnyLocalAddress(final byte[] address, final int addressLength)
    {
        if (addressLength == ADDRESS_LENGTH_IP4)
        {
            return 0 == address[0] && 0 == address[1] && 0 == address[2] && 0 == address[3];
        }
        else if (addressLength == ADDRESS_LENGTH_IP6)
        {
            byte val = 0;

            for (int i = 0; i < ADDRESS_LENGTH_IP6; i++)
            {
                val |= address[i];
            }

            return 0 == val;
        }

        return false;
    }
}


Choosing the right end table for your home is an opportunity to enhance the decor while adding to the functionality of any room. There are tables of every color, material and style. Be sure to find one that fits the room, both in presentation, and size. It is also important to measure around existing or planned furniture so that the introduction of a new piece doesn't inhibit the flow of movement through the room. At the same time measure for height so the top of the table provides a comfortable place to set things down, especially if being used as a side table for a couch or chair.
Once the logistics are clear, take into consideration how the material fits in the decor of the room, and if the functionality of the piece is being utilized to the fullest. Nesting tables provide more surface space when needed in less room, while end tables with drawers or shelves allow for storage which can be out of sight in the drawer or a nice basket. Finding the end table, or two or three, that suits your room can be just the right touch to enhance the convenience and comfort of your home.


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The 9 Best Breweries In Las Vegas In 2022
Written by Mariska Lee in Best Of,Nevada
Today, one great industry that's flourishing in Las Vegas is the brewing trade. In the past decade alone, tens of new breweries have popped up across town.
Here are the best breweries that Las Vegas has to offer:
Able Baker Brewing Company
Inspired by the Atomic Age, Able Baker's whole concept comes from Nevada's history of scientific testing and discovery: the first two atomic bombs detonated at the test site in Nevada were called Able and Baker, following the military phonetic alphabet.
They run their own experiments, brewing almost every style of beer imaginable. Able Baker has won awards at the World Beer Cup and offers an exclusive, members-only yearly club.
Their brewery and taproom are located in the Arts District of Las Vegas. The bar offers 30 draft beers, including some guest taps, as well as wine and spirits.
You can choose between a 6oz or 12oz pour for most beers. Arts District Kitchen is also the brewery's very own, in-house cookery. They serve up dishes made from scratch, from burgers and sandwiches to tacos and salads.
Beer to try: Atomic Duck – 7% ABV IPA
Nevada Brew Works
Operating on a very small scale, Nevada Brew Works is a brewpub just down the road from Able Baker. Their size doesn't mean they sacrifice on quality, though.
Their brewers have been trained at the Siebel Institute of Technology to guarantee the best brews. Their brew line-up reads as classic American, with IPAs and lagers predominating their menu. They do offer a schwartzbier, too, though.
As the Brew Works is more bar than brewery, you get a traditional pub experience. Alongside their house brews on draft, they offer a curated cocktail menu of 10 of their own creations, all with witty celebrity-inspired names.
They also have an impressive whiskey collection that is also offered as bourbon flights.
You can line your stomach with some of their pub grub. They serve personal-sized pizzas, burgers, and salads. Imagine finding your new favorite local and learning they brew their own beers!
Beer to try: Ariana – 6.3% Rye IPA * whose proceeds go to charity!
Banger Brewing
Banger Brewing seeks to reflect Vegas in their work by being sexy, classy, and one of a kind. What started among a team of Bellagio employees became an undertaking to produce great, local beer.
The group of founders moved from the casino kitchen to their own to start producing small batches of beer. In the heart of Downtown, they now have a full commercial brewery and taproom to visit.
Sip on a Banger while snooping on their fully-functioning brewery. Behind their barrel-aged beers, a glass window lets you look out onto the brew deck from the bar and see the very place the drink in your cup was made.
Outside, you can spot the brewery thanks to the big, branded conical fermentation tank. Order from close to 20 taps, or go for a flight if you can't make up your mind.
Beer to try: Morning Joseph – 9% ABV Imperial Coffee Kolsch
Hop Nuts Brewing
While the Arts District is now full of delicious, local craft brews, Hop Nuts Brewing has the claim to fame of being the first premium craft microbrewery in the neighborhood.
Presumably, their name comes from their addiction to hops: a love so strong it's clear in their beers. DIPAs, IPAs, and Pale Ales dominate their menu, but they also offer lighter options like a Helles, though that's only on their seasonal line-up. Otherwise you can always drink their core range of 8 beers.
The Hop Nuts taproom started in the Arts District, as we mentioned, and is still home to their production brewhouse.
They are also soon to open a second location in Tivoli Village in Summerlin, September 2022.
Both bars will be open 7 days a week and pouring from 20 taps – 15 of their own draft and 5 guest beers. They don't serve any food at the bar, nor children (it's state law!).
Beer to try: Papa's Red Ale – 6.2% ABV Red Ale
Scenic Brewing Company
A recent venture, Scenic Brewing Company opened in 2019 with a three-fold plan: craft beer, global food, and sports bar. They aim to bring wonderful worldwide elements home to the local community.
Over in the Spring Valley area of Las Vegas, it's a brewery set up for locals more than tourists.
The taproom at Scenic Brewing is everything sports bar. They serve beer from 17 taps, five of which are always SBC's own in-house brews.
You can sip a beer flight while watching the game on their massive TV screens. The global food comes into play for both lunch and dinner – even Sunday brunch.
They also host events at the taproom, like live music, karaoke nights, and beer pairing dinners.
Beer to try: Scenic Sour – 5.5% Pineapple Sour
CraftHaus Brewery
Since 2014, CraftHaus has been pouring their hearts into every pint they brew. Inspired by a friendly trip around Europe, co-owners and couple Wyndee and Dave decided to commit themselves to the craft beer community and work to support Nevada independent brewers, so much so that they helped changed licensing in favor of craft breweries in Las Vegas.
The CraftHaus location in Las Vegas is their second taproom. The actual brewery and original tap are housed in Henderson, NV. CHAD, as the second taproom is called (it's CraftHaus in the Arts District – get it?), has 1800 sq ft of good times in a really chic, art-inspired space. It doesn't look like your usual taproom, giving the craft beer experience a bit of an up-market, chic Las Vegas flair. The taproom serves 24 beers on draft and has light bites inspired by Europe, like pretzel rolls, charcuterie boards, and European sausage.
Beer to try: Evocation – 6.4% Farmhouse Saison
HUDL Brewing Company
HUDL Brewing Company was founded by three friends who went from drinking buddies and homebrew dabblers to full-scale, 15 bbl production craft brewery.
While their brews span many different styles, they seem to have an obsession with the cream ale – ⅓ of their beers on tap are cream ales! They must be good, though, with the number of GABF medals their head brewer has under his belt …
The HUDL brewery and taproom are located in the Arts District, along with most of the other craft breweries in town.
They offer 18 of their beers on draft, plus three guests. You can pop down the brewery for a casual drink or get involved with music bingo or trivia nights.
Their sleek industrial space looks out onto the brew deck, the chrome-top bar and echo of their shiny fermentation tanks in the background. Unfortunately, there's no food at the taproom.
Beer to try: The Attic – 7.3% ABV Cream Ale
Tenaya Creek Brewery
The oldest on the list, Tenaya Creek Brewery first opened in 1999 as a restaurant and brewery. In 2008, they left the cooking to the pros and dropped the restaurant from the business, focusing solely on beer production.
Since then, they've had to move locations thanks to their ever-expanding business. They've won state-wide and national awards for their beer and even took a silver medal in the Australian International Beer Awards in 2013.
Because Tenaya Creek dropped their kitchen, they now invite food trucks to the brewery.
Every single night of the week you can fill up on a wide range of cuisines from local, mouth-watering caterers between 5 and 9 pm.
Want to know what's currently on tap? They have a live cam set up on their tap list so you can stay up-to-date by the second. They also offer 12 different guest beers on draft.
Beer to try: Bonanza – 5.6% ABV Brown Ale
Lovelady Brewing
A family affair, the Lovelady Brewing Company bears the family name. Founded by four brothers, their foray into beer making dates back decades when two of the brothers first made homebrew at a mere 17 years old.
We'll skip the legality of that consumption and go straight to the founding of the brewery that aims to go back to the community – of Henderson, that is. Both locations are in this Las Vegas suburb.
The original taproom and brewery looks like a mid-century cinema (or even a quick place to get hitched, seeing as it's Vegas). Inside is an open-concept, bierhall bar.
They also have a large patio with a mural done by a local artist. Neither location serves food, but is nearby some really great local restaurants.
Beer to try: Love Juice – 5.7% ABV Hazy IPA
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redpajama_set_name: RedPajamaCommonCrawl


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Home/blog/Tucker Joins Dunlop Florida Fall Classic with House of Exclusive Brands
Tucker Joins Dunlop Florida Fall Classic with House of Exclusive Brands
Tucker to Support AMA Amateur Flat Track Event with unique "House of Brands Village."
Center Hill, Florida – The Dunlop Florida Fall Classic, to be held October 19th, 2019 at Volusia Speedway Park, will feature 12 AMA-sanctioned classes running on one of the fastest half miles in the country during the 2019 Daytona Biketoberfest with support from Tucker Powersports.
Tucker brings nine of its twelve recently announced exclusive brands to support the Dunlop Florida Fall Classic, with a unique "House of Brands Village," where spectators, riders, and families can explore 2020 products during the race day activities. The House of Brands Village will showcase brands exclusively distributed by Tucker, including ProTaper, Answer Racing, FirstGear, Speed and Strength, DragonFire, QuadBoss, BikeMaster, Twin Power, and Biker's Choice.
"What's neat about this partnership, is that the brands Tucker is supporting our Flat Track Futures event with aren't all just dirt bike-related companies," stated Flat Track Futures,' Johnny Lewis. "We have ATV and UTV accessories, plus adventure and street products supporting the event. We'll have a diverse crowd of people at our first Flat Track Futures event by joining multiple powersports communities together at one event."
Additional Tucker brands supporting the event include Dunlop Motorcycle Tires as the official title sponsor, and Sena Technologies, for a total of 11 Tucker brands supporting the 2019 Dunlop Florida Fall Classic.
"I feel the best way to introduce flat track racing to people is through the amateur ranks and show it's a sport where at any age, skill level, and bike size you can be part of the community and enjoy the experience in a family-friendly, welcoming environment," explains Lewis. "Tucker is a family of brands, and it shows even more with their evolvement in racing, riding, and community. We're proud to have Tucker be part of our family now."
Racing at the Dunlop Florida Fall Classic kicks off at 6 pm on Saturday, October 19th; but come early from 3 pm till 5:45 pm and take part in the House of Brands Village with live music, demo rides, and a meet-and-greet with youth, amateur, and pro racers. From 6 pm to 9 pm, the area becomes the VIP lounge with special viewing areas, food, and drinks until racing concludes. Following racing the House of Brands Village reopens for an evening of celebration with live music from Little Vicious. General admission tickets for the event start at $20, including the racing action, live music in the House of Brands area, plus free camping for all! Tickets can be purchased at the gate as well as upgraded VIP tickets.
For more information on the Dunlop Florida Fall Classic, including Friday, October 18th open practice, live music details, plus racer details (including sign up) visit www.flattrackfutures.com.
Find your Local Tucker Dealer.


If you are looking for a light and flavorful bread to serve with a bowl of soup or a light salad, this is a great recipe to keep handy. The Asiago cheese adds just the right amount of richness to these biscuits while the generous amount of pepper adds a bit of a bite.
These savory drop biscuits are easy to make because you do not have to roll or cut the dough, you simply drop tablespoons of the dough onto a baking sheet and they are ready to bake. If you prefer a bit more salt, sprinkle Kosher salt or sea salt on the tops of the biscuits before baking. Enjoy!!
Preheat oven to 450 degrees. Combine the flour, baking powder, salt and pepper in a medium bowl and stir. Cut in butter with a pastry blender (or use 2 butter knives) until the mixture is a crumbly texture.
Add the cheese and toss well to combine, then add the buttermilk and stir just until moist.
Coat a baking sheet with cooking spray and drop dough into 8 equal mounds. Bake for 13 minutes or until edges are lightly browned.


The overcoming of oneself, the vitality of landscapes, altitude and the strengthening of the cardio-vascular system, the discipline of trail running is beneficial for its practitioners. Will exceed, reach the summits and develop its muscle mass, trail-running pushes your limits. In order to gain in performance, the watches GPS trail inform you of a glance on your effort athlete to go still higher.
Be in the heart of the nature, paths, overtaking, there are many reasons to put you to the trail. This discipline combines the benefits of the mountain to physical skill, pushing it to self-transcendence. Strengthening the cardio-vascular system, the trail-running and increasing respiratory capacity through the idyllic landscapes.
In order to gain skills, work goals and develop the body, several brands have designed watches of the trail so-called smart. To measure, share and improve the effort sports, they are more than simple accessories trail. Analyzing the races, informing on the heart rate and measuring the terrain, the traileurs access at a glance to valuable information on their career paths. Rider beginner or expert trail-running watches GPS is intended for all athletes seeking to improve their performance.
In order to achieve your goals and boost your efforts, it is essential to have on his wrist a watch cardio GPS connected. Offering many features, they offer important information for the athlete.
Training partner adopted by many athletes, the SUUNTO SPARTAN SPORTS is an expert on the market. With a GPS function, it offers a great visual comfort thanks to its touch screen. Waterproof, it has about 80 modes sports and detailed information visible at a glance. Analyses of training, maps, routes, this watch offers many of the features of where his expertise sports. Finally, his autonomy will allow you to run with confidence and exceed your limits !
Ideal for the trail, the GPS watch, SUUNTO AMBIT3 VERTICAL you will follow to reach the summits. Analyzing your performance in altitude, it is possible to plan your recovery after a big effort in sports. In addition, it saves your progress vertical in order to give you the maximum information on your profile trailer. Through its monitoring of elevation change, prepare effectively your workouts while keeping an eye on your heart rate.
Multisport, SUUNTO AMBIT3 SPORT has been designed for swimming, running or cycling. Offering multiple features such as GPS, speed and distance traveled or the time of recovery, it is the ally of the most reckless ! Indispensable for athletes preparing for a competition, she you will only boost in your performance and you will experience all-new sports experience.
In training or in competition, equip a watch, the trail to track your progress, plan your recovery and gain confidence !


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NewsHounds
We watch Fox so you don't have to!
What's Behind O'Reilly's Fearmongering About Civil War In America?
Posted by Ellen -7842.60pc on November 11, 2016 · Flag
Last night, Bill O'Reilly "just asked" in his Talking Points commentary, "Is there a civil war brewing in the U.S.A.?" However, his immediate goal was not to enlighten anyone but to demonize anti-Trump protesters as dangerous traitors. I think there was another, darker agenda, too.
Here are some excerpts from O'Reilly's commentary (via FoxNews.com's transcript, with my emphases):
O'REILLY: The main beef seems to be that left wing protesters don't respect an honest election. By the way, that's a hallmark around the world. Every communist and socialist takeover from Cuba to Venezuela to Soviet Russia back in the early 20th Century featured violence and assaults on freedom. Here in the U.S.A., we honor protests, but increasingly we are seeing people who want our system destroyed.
[T]here are people, agitators who do want to hurt you. Who do not believe the way you do.
And in San Francisco, there is talk that the State of California should secede from the union.
O'REILLY: That sounds like the south before the civil war, does it not? Finally, the race hustlers are very angry. Chief among them Al Sharpton who despises Donald Trump.
O'Reilly's concern trolling for respecting an honest election might seem a whole lot more authentic and a whole lot less self-serving were it not for his tolerance for his pal Donald Trump's attempts to undermine President Obama's presidency. Can you point to even one instance where O'Reilly condemned Trump's bogus birtherism as a dangerous and anti-American attempt to undermine the election of our first African American president? When O'Reilly brought up the subject in September, his only concern seemed to be that Trump's birtherism might have "hurt" him with the black vote.
And where was O'Reilly when the Tea Party protests were threatening violence in opposition to President Obama's policies? Complaining that they were "not being respected" by the rest of the media.
I also don't recall O'Reilly speaking out against his own network's cheerleading for criminal rancher Cliven Bundy's armed insurrection against the federal government. To his credit, O'Reilly did not join that disgraceful behavior. But I do recall O'Reilly whitewashing his colleagues' support as an example of why Fox News is successful. "We have a wide variety of opinion expressed in a vibrant way," he praised.
So don't tell me that O'Reilly's respect for dissent doesn't depend almost wholly on who's being protested and very little about how.
That's disturbing enough. But I believe O'Reilly is promoting a deeper, more sinister agenda here. And it dovetails all too well with Trump advisor and likely cabinet member Newt Gingrich's desire to resurrect the McCarthy-esque House Un-American Activities Committee. It's hinted at in O'Reilly's conclusion, where he suggests that anyone who "promotes anarchy" should be quickly punished or exiled:
O'REILLY: Summing up, Talking Points respects sincere protests. If you believe Donald Trump is not good for America, I have no problem with you displaying that opinion. However, if you hurt someone, destroy something, or promote anarchy, you then become a danger to the republic. That kind of stuff needs to be punished and quickly. Also, we are living in an amazingly destructive politically correct environment here in America. Just because something offends you doesn't mean you have the right to hurt or destroy.
The new president might want to make that very clear. And if you don't like it, Canada and Mexico are nearby. They might be happy to have you. Then again, they might not. And that's "The Memo."
We can hope that O'Reilly's definition of punishable protest is restricted to actual, serious violence and not, say, the kind of casual talk of a revolution in the clips he showed.
But we can't count on that.
Watch this chilling editorial below, from the November 10, 2016 The O'Reilly Factor.
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John McKee commented 2016-11-13 23:24:35 -0500 · Flag
There's no if about it, Bemused. O'Bully is as hypocritical as they come, and I seriously doubt he has any more true beliefs than Trump himself. Apart from all his everyday crimes and misdemeanors in the service of his own pay-check, O'Reiily has been a leading figure in the Republican demonization of Hillary for nigh on 30 years. They have obviously always seen her as a future candidate and shamelessly smeared and besmirched her in every way possible, doubly so over the past 8 years.
Drip… drip… drip… and the upshot was that this slander quietly penetrated 70 or 80% of the population (even Democrats reflexively doubted her complete honesty) when she really did very little to deserve such a reputation. A slow-motion crucifixion and a damned tragedy for America.
Bemused commented 2016-11-12 13:57:40 -0500 · Flag
O'REILLY said: "The main beef seems to be that left wing protesters don't respect an honest election."
Which candidate refused to say that he would respect the result? Is O'Reilly's memory betraying him? Or is he simply a shill? I vote for the latter.
So true, Dave, So true.
Although I think a civil war situation might have developed had Trump lost: his followers are heavily armed and eager to use their toys. Lefties are more prone to marching and shouting and they are less likely to be armed. The violence at their marches is usually done by people who have nothing to do with the message but a lot to do with wanting to break things. Their intellectual level is that of three-year olds with a stack of porcelain cups.
Dave Wright commented 2016-11-12 08:57:39 -0500 · Flag
Ah, he is just looking for ratings and a desperate bid to stay relevant. Mr. Trump no longer needs him so O'Reilly needs to say something to get attention from the right. He knows about as much about a potential civil war as I do about the mood of the people today in, say, American Samoa.
Kevin Koster commented 2016-11-11 20:51:17 -0500 · Flag
John, I wouldn't be too sure about the right wing not finding a way to blame Dems for their problems. Remember that they've spent the last 8 years trying to blame the Bush Recession on the Dems who finally got majorities in 2006. Count on Fox News to fan these kinds of flames as hard as they can for the next four to eight years, depending on how Pence does as President.
I'm getting a strong feeling from the various personalities on the Right, including the people who are about to be given tremendous power in Trump's White House, that they are in the mood for revenge and punishment. You'd think these guys would be happy that they pulled out the squeaker and that their depress-the-vote negative campaign gambit worked. But no, they're really angry. And the oddest part is that they're swinging at some of their own people first. Republicans who didn't get on board Trump's little ride are now being hung out to dry on Fox News and on AM Radio. Some of them are likely to be primaried from the hard right in 2-4 years.
But if the Far Right is angry at fellow GOP members who didn't step in line fast enough, they're positively furious at the Dems and their supporters. They are clearly bent on tearing up every last piece of the Obama presidency and leaving it on the highway. With this attitude, it shouldn't surprise us that they are warming up to take actions that are likely to cause discomfort for millions of Americans.
I honestly think these guys don't care if there is collateral damage – they just want it done. They don't care if their destruction of the ACA results in people losing their coverage – they just want to be able to say that Barack Obama was a failure and that they are the ones who will take care of things now. If their constituents complain, they'll be told that the ACA was unworkable and was a huge Democrat boondoggle that was costing everyone too much money. They'll be told that the GOP is working on a replacement program but is dealing with so much Democrat obstruction that they will need to vote more GOP legislators in on the next round in 2 years. This is a typically Rovian move, and we should keep these tricks in mind before they come our way unexpectedly.
The harsh nature of the Right's fury is what really concerns me here. If you look at O'Reilly's face in that clip, there's no mistaking the bald, red-faced anger at hand. He's going to use this opportunity to rub every liberal's nose in the mess, and he clearly wants to tell them to like it while he does it. It's that kind of attitude from O'Reilly and Hannity that is scary to watch on Fox News. But it's REALLY scary to see that attitude coming from people like Gingrich and Giuliani and Clarke, all of whom will be in positions with an ability to harm a LOT of people. Gingrich has already gone back to his usually nasty mode in his attacks on fellow GOP members. Just wait until he gets on the notion of punishing Dems. Giuliani may be the scariest one here, as he's playing the card of being genial while saying some incredibly angry things – mainly about finding a way to charge, indict and arrest the Clintons.
There were articles today on CNN about how these guys really won't be able to repeal the ACA and how they really won't be able to indict the Clintons, and I actually burst out laughing at the second article. Because the articles assumed these guys would do what makes sense to a CNN op-ed writer. Which is hysterical on its face. These guys are going for the rich satisfaction of revenge, not anything that would actually help anyone. And unless the Dems show some backbone, the Right will get away with it. Again.
The one silver lining I'm finding here is that I believe we'll see a series of scandals in this White House that will easily dwarf anything we saw in W's two terms. And that mess would be enough to potentially bring some of these guys down – it's just that we'll have to endure at least 4 years of this craziness before we can do much about it.
The silver lining in the gloomiest of clouds hanging over America and the Earth may be that this HAS TO BE the lowest point you can possibly get to.
Five more trillion down the plughole, a god-awful shemozzle over healthcare, job growth destroyed, race and religious relations set back decades and NO CHANCE WHATSOEVER of blaming anything on the Democrats. As it dawns on each subgroup of Trump voters that he's making things worse for them, not better, they must at last reach the tipping point I for one thought they would surely reach before the election.
Rust belt – no new heavy industry.
Coal territory – no re-opened mines.
Poor white workers – no increase in minimum wage.
Mexican haters – no mass expulsion.
AfAm haters – nothing much of anything
Isolationists – plenty of war-mongering
Protectionists – increased prices on everything, including cars and pick-up trucks
Two years of Don the Con and his band of Merry Morons should be plenty.
d d commented 2016-11-11 16:14:29 -0500 · Flag
That was indeed rather chilling and a tad bit too much McCarthy-esque for my liking.
Btw, if BOR is so worried about anarchy, where was he when his milkshake BFF was making comments that could be considered to be promoting anarchy? Just another example of BOR's double standard.
I wonder what O'Reilly's position is on the growing deluge of hate crimes we are now seeing, where white children are attacking non-white children in school, and where whites are attacking non-whites. Does O'Reilly think this behavior is acceptable?
A friend just posted that her neighbors (who are Black and Muslim) have a daughter who was attacked in school on Wednesday for wearing a headscarf. We've all seen the video of white kids in class yelling "Build the Wall" at the non-white kids. Where does O'Reilly stand on the growing evidence of the hatred he has helped encourage?
O'Reilly is in his "triumphal" mode. He went on for several minutes on Wednesday doing what I believe he thought was putting Hillary Clinton in her place. Now he's on the attack against anyone who disagrees that Trump will be great for the country. Granted, O'Reilly has his own criticisms of Trump, but they are not the blanket ones he throws at anyone to the left of him.
That last statement by him was a direct F You to everyone who voted for Clinton and Obama. You could think of it as his attempt to kill the microphone on dissenters.
I'll be curious to see how he handles the upcoming Clinton indictment and prosecution by a Giuliani Justice Dept. My bet is that he cheerleads for it and enjoys the perp walk video when Giuliani provides it next spring.
truman commented 2016-11-11 15:37:45 -0500 · Flag
Deja Vu all over again. Days of Dumbya Bush. When any criticism of Dumbya's insane wars was branded by Bildo as high treason.
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\begin{document}
\begin{titlepage}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{flushright}
     IFUM-573-FT \\
     hep-ph/9707532 \\
     July 1997 \\
\end{flushright}
\par \vskip 10mm
\begin{center}
{\Large \bf
Universality of $1/Q$ corrections to jet-shape observables rescued\footnote{
Research supported in part by MURST, Italy
and by EC Programme ``Human Capital and Mobility",
contract CHRX-CT93-0357 (DG 12 COMA).}}
\end{center}
\par \vskip 2mm
\begin{center}
{\bf
Yu.L. \ Dokshitzer\footnote{
On leave from St. Petersburg Nuclear Institute,
Gatchina, St. Petersburg 188350, Russia},
A.\ Lucenti,  G.\ Marchesini and G.P.\ Salam}\\
\vskip 5 mm
Dipartimento di Fisica, Universit\`a di Milano \\
and INFN, Sezione di Milano, Italy
\end{center}

\par \vskip 2mm
\begin{center} {\large \bf Abstract} \end{center}
\begin{quote}
  We address the problem of potential non-universality of the leading
  $1/Q$ power corrections to jet shapes emerging from the
  non-inclusive character of these observables.
  We consider the thrust distribution as an example and analyse the
  non-inclusive contributions which emerge at the two-loop level.
  Although formally subleading in $\alpha_{\mbox{\scriptsize s}}$,
  they modify the existing na{\"\i}ve one-loop result for the expected
  magnitude of the power term by a factor of order unity.
  Such a promotion of a subleading correction into a numerical factor is
  natural since the non-perturbative power terms are explicitly proportional
  to powers of the QCD scale $\Lambda$ which can be fixed precisely only at
  the two-loop level.
  The ``jet-shape scaling factor'' depends on the observable but remains
  perturbatively calculable.  Therefore it does not undermine the
  universal nature of $1/Q$ power corrections, which remain expressible
  in terms of the universal running coupling and universal
  soft-gluon emission.
\end{quote}
\end{titlepage}

\mysection{Introduction}
The analysis of power-behaved contributions to infrared and collinear safe
characteristics of hard QCD processes has recently been
developed as a method to quantify non-perturbative
effects~[\ref{Web94}--\ref{NS}].
It has been recognised that the leading power corrections to
infrared/collinear safe quantities
are determined by an incomplete cancellation of soft-gluon contributions.
A QCD process at a hard scale $Q\gg\Lambda$ is determined by the small
space-time intervals between the quarks and gluons involved.  In spite
of that, at the level of power suppressed corrections, the cross section
of such a process may acquire a contribution from a soft gluon which
travels a finite distance $1\,{\rm fm}\gg 1/Q$
and thus is sensitive to the non-perturbative interaction domain.

The sensitivity of a given observable to soft-gluon radiation
can be studied perturbatively to determine the
{\em power}\/ $n$ of the expected leading $Q^{-n}$ correction.
Such theoretical input is valuable on its own, especially for jet physics
which deals with intrinsically Minkowskian quantities and therefore
typically receives no hints from the Euclidean-based OPE technology
[\ref{OPE}].
For example, an analysis of large-distance soft-gluon effects predicts
that the properties of jets assembled with the use of the (once
standard) JADE jet finder are contaminated by confinement effects at
the level of
$Q^{-1}$ corrections, while the Durham ($k_\perp$) algorithm produces jets
whose rates will depart from perturbative predictions only at the
$Q^{-2}$ level~[\ref{Web94}].

More challenging is the prediction of the {\em magnitude}\/ of expected
power corrections.
For a single observable this would be impossible given the present
state of the art.
Nevertheless it is natural to expect that soft-gluon universality
implies proportionality between the magnitudes of power-behaved terms
in different observables~[\ref{StKoZakh}].
Along these lines one aims to predict the relative sizes
of power corrections to observables belonging to the same class,
linking together, for example, jet mass and thrust
distributions~[\ref{DW97}]
with $1/M_Q$ effects in heavy quark energy spectra~[\ref{DKT},\ref{NW}]
($n\!=\!1$)
or power corrections to the Drell-Yan
$K$-factor~[\ref{StKoZakh},\ref{BB95}]
with those in the
Gross-Llewellyn-Smith DIS sum rule and $e^+e^-$ fragmentation
functions~[\ref{BPY}] ($n\!=\!2$).

However, some doubt has been expressed as to the universality of power
corrections to jet shapes because the latter are not fully
inclusive with respect to final-state branching of the soft gluon
under focus.  To the best of our knowledge this question was
first considered by Nason and Seymour in~[\ref{NS}].

The notion of soft-gluon universality contains two ingredients.
First, it exploits the universal character of the soft-radiation
matrix element in the spirit of Low's theorem~[\ref{LBK}].
Secondly, it requires universality of the gluon radiation intensity
(running coupling).

The running QCD coupling in hard processes emerges after inclusive
integration over the gluon decay products,
resulting typically in $\alpha_{\mbox{\scriptsize s}}(k^2_\perp)$ with $k_\perp$ the gluon
transverse momentum~[\ref{kt}].
In QED it suffices to consider the virtual photon self-energy
blobs (``bubbles''), while in the QCD context there are also the
corrections to the gluon emission vertex.
The dispersive representation has been used in~[\ref{BPY}]
to give meaning to the QCD running coupling
at low scales~[\ref{DKT},\ref{lowscales}].

The value of a jet-shape variable depends, however, on the kinematics
of all final state particles (hadrons, partons).
As long as the observable is collinear safe the (quasi-)collinear gluon decays,
which are the main contributors to the running $\alpha_{\mbox{\scriptsize s}}$, do not affect
the observable and therefore can be treated inclusively.
However, in the ``large-angle'' gluon decay region, the inclusive
interplay between real and virtual gluon loops gets broken because
a real decay may correspond to a value of the observable different
from that for the case of a virtual loop.
These contributions seem to undermine the above-mentioned universality.

In the case of the thrust distribution, for example, the na{\"\i}ve inclusive
treatment applies as long as the decay products fall into the same
hemisphere.  Then the contribution to thrust, $T$, can be attributed to the
parent gluon, and integration over the virtuality of the latter
results in the running coupling in a standard way.

The existence of the part of the decay phase space
in which the two gluon offspring partons (quarks or gluons)
happen to fly into opposite hemispheres,
has two consequences.
First, this region has to be treated separately as its contribution
to $T$ depends on details of the gluon decay matrix element.
At the same time the effective intensity of emission of the parent gluon
as a whole gets modified.
This is due to the fact that to reconstruct the running coupling one
sums inclusively over the gluon decay products, while in this case one
modifies the final state phase space. Thus the running coupling
(its ``spectral density'') loses part of its standard support.

It has been argued that such a non-inclusive contribution is higher order in
$\alpha_{\mbox{\scriptsize s}}$.
Its quark-antiquark (``Abelian'') part was studied numerically
in [\ref{NS}] and found to be small.

In this paper we analyse the question of (non-)universality of the
leading power corrections to jet shapes and thrust in particular.
We find that the na{\"\i}ve inclusive treatment has to be modified
by two effects.
The first is due to non-inclusive configurations of the parent gluon decay
(non-inclusive correction). The second is due to an incomplete compensation
between real and virtual logarithmic terms coming from the soft-gluon
splitting into gluons (inclusive correction). This correction is
absent in the contribution from the gluon splitting into a
quark-antiquark pair.
We show that both non-inclusive and inclusive corrections,
although formally subleading in $\alpha_{\mbox{\scriptsize s}}$, get promoted to a factor
{\em of order unity}\/  in the magnitude of the $1/Q$ power correction.
As a consequence, previously obtained results
based on the na{\"\i}ve inclusive treatment of an observable $j$
acquire a ``rescaling factor'' $(1+r_j)$
which is {\em calculable}\/ but depends on the observable under consideration.

In this sense one can still consider power terms
in jet observables as universal, i.e. based on the universal coupling
and the universal soft-gluon radiation pattern, although
their relative magnitudes differ from the na{\"\i}ve expectations based on
one-loop off-shell soft-gluon matrix elements.

For thrust values close to 1, only soft-gluon radiation is essential.
For $T$ well below 1, one has contributions both
from multiple soft-gluon emission and from rare $N$-jet configurations
($N>2$).
The first contribution has been found~[\ref{DW97}] to be dominant
at least down to $ T=2/3$,
the minimum $T$ for any three-jet configuration.
As far as power corrections are concerned,
it was shown in~[\ref{NS}] that the order $\alpha_{\mbox{\scriptsize s}}(Q^2)$ three-jet configuration
does not develop a $1/Q$ power correction.

Therefore to analyse the thrust distribution at two loops we take into
consideration [\ref{DMO}] particle ensembles consisting of the primary
quark-antiquark pair
accompanied by partons originating from multiple emission of primary
soft gluons and employ the existing all-order resummed
perturbative prediction~[\ref{CWT}].

We extract the leading $1/Q$ power term in the
radiator, namely the exponent of the Mellin-transformed
$T$-distribution.
We do this in the framework of the treatment of
non-perturbative power corrections to infrared/collinear safe observables
introduced in~[\ref{BPY}], based on the
notion of an infrared-finite strong coupling which
differs from the perturbative form in the infrared region.

At the two-loop level, non-inclusiveness of the thrust reveals itself
and the characteristic QCD scale $\Lambda$ becomes precisely defined
through the fixing of the scheme for the running coupling.
To this end we choose the CMW renormalisation scheme~[\ref{CMW}],
in which $\alpha_{\mbox{\scriptsize s}}$ is defined as the intensity of soft-gluon radiation.

Section 2 contains the necessary ingredients for the two-loop analysis
of the radiator of the thrust distribution: kinematics,
structure of the soft-gluon emission followed by
gluon splitting into $q\bar{q}$ and gluon pairs,
scheme fixing, and thrust resummation.
In section 3 we recall the dispersive method of [\ref{BPY}]
and its application to the thrust distribution in the na{\"\i}ve
inclusive approach [\ref{DW97}].
Section 4 is devoted to the exact treatment of the problem,
the analysis and calculation of the inclusive and
non-inclusive contributions to the thrust rescaling factor $(1+r_T)$.
Section 5 contains a discussion of the results and the prospects
for the future.

\mysection{Two-loop analysis of thrust distribution}
\subsection{Thrust: kinematics}
Consider a QCD hard process in which the final state consists of
a leading quark-antiquark pair with momenta $p$ and $\bar p$
together with a system of $N$ partons (quarks and gluons)
with momenta $k_1, \cdots , k_N$.
It is convenient to cast the cross section in terms of the Sudakov
(light-cone) variables defined with respect to the thrust axis of the event.
To this end we introduce two light-like vectors
$$
 P^\mu\>, \>\> \bar{P}^\mu\>; \quad P^2=\bar{P}^2=0\>; \quad
 P^\mu + \bar{P}^\mu= Q^\mu\>,\>\>
 2(P\bar{P}) = Q^2\>\equiv\>1\>,
$$
and represent the parton 4-momenta as
$$
 k_i^\mu \>=\> \beta_i P^\mu \>+\> \alpha_i \bar{P}^\mu \>+\>
 (k_{i\perp})^\mu,
 \qquad\qquad \alpha_i\beta_i = k_{i \perp}^2
$$
The leading quark momenta become
$$\eqalign{
p&= (1-\sum_{i=1}\beta_i-\beta_{\bar{p}})P\>+\>\alpha_p\bar{P}\>+\>
p_\perp\>, \cr
\bar{p}&=(1-\sum_{i=1}\alpha_i-\alpha_p)\bar{P}
 \>+\> \beta_{\bar{p}} P\>+\> \bar{p}_\perp\>,
}$$
Adding the normalised longitudinal momenta of the secondary partons,
$
\left. k_{i||} \right/Q = \abs{\alpha_i-\beta_i}/2\>,
$
with those of the two quarks,
for the thrust value we obtain
\beql{Ti}\eqalign{
 T &= \sum_i\mbox{\small $\frac{1}{2}$}\abs{\alpha_i-\beta_i}
+ \mbox{\small $\frac{1}{2}$}[\, (1-\sum_{i=1}\beta_i-\beta_{\bar{p}})-\alpha_p\,]
+ \mbox{\small $\frac{1}{2}$}[\, (1-\sum_{i=1}\alpha_i-\alpha_p)-\beta_{\bar{p}}\,] \cr
&= 1-\sum_i \min\{\alpha_i,\beta_i\} - (\alpha_p+\beta_{\bar{p}})\>.
}\end{equation}
This expression implies that $p$ and $\bar{p}$ momenta belong to opposite
hemispheres, which is the dominant configuration
(the correction is numerically small and relatively suppressed
{\em at least}\/ as $(1\!-\!T)$ when $T$ is large).

The values of the ``small components'' of the quark momenta,
$\alpha_p$ and $\beta_{\bar{p}}$ depend on quark transverse momenta
with respect
to the thrust axis through the on-mass-shell conditions,
$p^2=\bar{p}^2=0$.
Transverse quark recoils are typically of order
$$
   p_\perp^2\sim \bar{p}_\perp^2 \sim \sqrt{N}\cdot k_{\perp}^2\>,
$$
with $k_\perp$ characteristic transverse momentum of a secondary parton.
We have kept in the estimate a (logarithmic) enhancement factor due
to the parton multiplicity $N$.

As we shall see the leading power correction comes from the region
$\alpha_i \sim \beta_i$ of the order of the transverse momentum
(in units of $Q$).
Therefore in (\ref{Ti}) we shall neglect $\alpha_p$ and $\beta_{\bar p}$
which are of the order of the transverse momentum squared.

\subsection{Gluon splitting}
\paragraph{Two-body variables and phase space.}
The two-parton phase space is
$$
  d\Gamma_2(k_1,k_2) = (4\pi)^4 \frac{d^4k_1\,d^4k_2}{(2\pi)^6} \,
\delta(k_1^2)\delta(k_2^2)
= \frac{4}{\pi^2}
\left(\frac12 \frac{d\alpha_1}{\alpha_1}\,{d^2k_{1\perp}} \right)
\left(\frac12 \frac{d\alpha_2}{\alpha_2}\,{d^2k_{2\perp}} \right)
\>.
$$
Introducing the relative fraction of $\alpha$-components, $z$, and
the invariant mass squared of the two-parton system $m^2$
we write:
$$
\eqalign{
  \alpha &\equiv\alpha_1+\alpha_2\>, \quad
   \alpha_1=z\alpha\>, \>\> \alpha_2=(1-z)\alpha\>, \cr
  m^2 &= (k_1+k_2)^2 = (\alpha_1\beta_2+\alpha_2\beta_1)-
  2\vec{k}_{1\perp} \cdot \vec{k}_{2\perp}
 = z(1-z)
\left[\,\frac{\vec{k}_{1\perp}}{z} - \frac{\vec{k}_{2\perp}}{1-z}\,\right]^2.
}$$
In terms of the scaled parton transverse momenta,
$$
\vq{1}=\frac{\vkp{1}}{z}\>, \quad \vq{2}=\frac{\vkp{2}}{1\!-\!z}\>,
\qquad \vq{} = \vq{1} - \vq{2},
$$
the phase space takes the form
\beql{d2Gamma}
  d\Gamma_2(k_1,k_2) =
 \frac{d\alpha}{\alpha}\, {dz}\, \frac{d^2q_{1}}{\pi}
  \frac{d^2q_{2}}{\pi}\>  dm^2\,\delta\left(\frac{m^2}{z(1\!-\!z)}-
(\vq1-\vq2)^2 \right).
\end{equation}
For later use we also define
\begin{equation}\eqalign{
 \vec{k}_\perp &= \vkp1 + \vkp2 \>=\> z\vq1 + (1-z)\vq2 \>; \cr
 \beta & = \beta_1+\beta_2 = \frac{k_\perp^2+m^2}{\alpha}\>, \quad
   \beta_1=\frac{z}{\alpha}q_1^2\>; \>\>
   \beta_2=\frac{1\!-\!z}{\alpha}q_2^2 \>.
}\end{equation}

\paragraph{Matrix elements.}
The squared matrix elements for the production of two gluons or a
quark pair
with 4-momenta $k_1$, $k_2$ read, in the soft limit $(Qk_i)\ll Q^2$
(see [\ref{DMO}])
\beql{Mrr2} \eqalign{
M^2(k_1,k_2) &=\>  4C_F^2\;W_1W_2 \>+\>4C_F\left( M^2_{gg}+M^2_{qq}\right) \cr
4\left( M^2_{gg}+M^2_{qq}\right) &= \>
{C_A}\;(2S+H_{g}) \>+\>
{n_f}\, H_q\>.
}\end{equation}
The first term, proportional to $C_F^2$, describes the independent
(Abelian) emission of two soft gluons off the hard $q\bar{q}$, where
$$
W_i \>\equiv\>   W(k_i) \>=\> \frac{p\bar p}{(pk_i)(k_i\bar p)}
= \frac{2}{k^2_{i\perp}}
$$
is the standard dipole radiation cross section.

The second (non-Abelian) term, proportional to $C_F\,C_A$,
contains two pieces.
The $S$ term describes the leading infrared contributions which is singular
in the ratio of gluon energies:
$$
S \equiv W_{12}+W_{21}-W_1\;W_2 \>,
$$
where
$$
W_{ij} \equiv \frac{(p\bar p)}{(pk_i)(k_ik_j)(k_j\bar p)}
\>=\> \frac{4}{m^2}\,\frac{1}{\alpha_i\beta_j}
$$
stands for the ordered 2-dipole emission.

$H_g$ and $H_q$ are responsible for parton configurations with energies of
the same order.
The ``hard'' parts of the two-gluon ($H_g$)
and the two-quark ($H_q$)
contributions read
\bminiG{Hgq}
\label{Hg}
H_g &\equiv& -SR + J^2 - 4\frac{W(k)}{(k_1k_2)} \>, \\
\label{Hq}
H_q &\equiv& -J^2 + 2\frac{W(k)}{(k_1k_2)} \>.
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}
Here
\begin{equation}\eqalign{
\label{J2R}
 J^2 &\>=\> 2 \left(
 \frac{(pk_1)(k_2\bar p)-(pk_2)(k_1\bar p)}{(k_1k_2)(pk)(k\bar p)}
 \right)^2
 = 8\left(\frac{\alpha_1\beta_2-\alpha_2\beta_1}
    {m^2\,\alpha\beta}\right)^2 \>, \cr
R &\>=\> \frac{(pk_1)(k_2\bar p)+(pk_2)(k_1\bar p)}{(pk)(k\bar p)}\>
=\> \frac{\alpha_1\beta_2+\alpha_2\beta_1}{\alpha\beta}\>,
}\end{equation}
and $W(k)= {(p\bar p)}/{(pk)(k\bar p)}$ is
the dipole emission of $k=k_1+k_2$.

We may write the squared matrix elements in the following form:
\beql{cMreduced}
 M^2_{gg}+M^2_{qq} \>=\>
 \frac{{\cal{M}}^2}{m^2(zq_1^2+(1-z)q_2^2)}\>,
\end{equation}
where
\beql{cMsq}
{\cal{M}}^2 \>=\>  2 C_A \left(2{\cal{S}} +{\cal{H}}_g\right) + 2n_f {\cal{H}}_q.
\end{equation}
The distributions determining ${\cal{M}}^2$ are functions of the
dimensionless ratios $u_i^2$ with
$$
  \vec{u}_i = \frac{\vec{q}_i}{|q|}\>.
$$
Namely,
\bminiG{matelms}
\label{matelmgg}
2{\cal{S}} &=& \frac{zu_1^2+(1\!-\!z)u_2^2}{z(1\!-\!z)}\left(\frac1{u_1^2}
 +\frac1{u_2^2}-\frac1{u_1^2u_2^2}\right) \>,\\
{\cal{H}}_g &=& -2 +z(1\!-\!z)\frac{(u_1^2-u_2^2)^2}{zu_1^2+(1\!-\!z)u_2^2}
-\frac{u_1^2+u_2^2}{2}
 \left[\,\frac1{u_1^2}+\frac1{u_2^2}-\frac1{u_1^2u_2^2} \,\right] ,\\
{\cal{H}}_q &=& 1-z(1\!-\!z)\frac{(u_1^2-u_2^2)^2}{zu_1^2+(1\!-\!z)u_2^2}\> .
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}
The collinear limit ($q^2\propto m^2\to 0$) corresponds to taking
$u_1^2\approx u_2^2\to \infty$ while keeping fixed the modulus of the
vector difference,
$$
\left(\vec{u}_1-\vec{u}_2 \right)^2=1\>,
$$
see (\ref{d2Gamma}).
Observing (see Appendix A) that the azimuthal average of the ratio
$$
 \VEV{\frac{(u_1^2-u_2^2)^2}{zu_1^2+(1\!-\!z)u_2^2}} \>\to\> 2\>,
$$
we derive the parton splitting functions
\bminiG{splitfs}
\label{splitgg}
  2C_A(2{\cal{S}}+{\cal{H}}_g) &\to& 4C_A\left(
    \frac{1}{z(1-z)}-2+z(1-z)\right), \\
\label{splitqq}
  2n_f {\cal{H}}_q &\to& 2n_f\left(1-2z(1-z)\right)\>.
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}
Note that the latter expression is twice the standard $g\to q\bar{q}$
splitting probability because thrust as an inclusive observable does not
differentiate between quarks and antiquarks, and the symmetry factor
$1/2!$ will be included in the thrust trigger-function, see (\ref{OT})
below.

\subsection{Gluon decay and the $\beta$-function}
The $\beta$-function is related to
the integrated two-parton production probability.
Keeping fixed the invariant mass and total transverse momentum
of the pair, and integrating $M^2$ over the azimuthal angle
$\phi$ with respect to $\vkp{}$ and over the relative momentum
fraction $z$, it is straightforward to derive
(see Appendix~A)
\beql{beta0}
 \frac1{2!}\,\int_0^1 dz  \int_0^{2\pi} \frac{d\phi}{2\pi}
\left(M_{gg}^2+M_{qq}^2\right)
= \frac{1}{m^2(k_\perp^2+m^2)} \left(-\beta_0
+2 C_A \ln\frac{k_\perp^2(k_\perp^2+m^2)}{m^4}\right)\>,
\end{equation}
where
\beql{betadef}
 \beta_0 = \frac{11}3N_c-\frac{2}{3}n_f \>.
\end{equation}
The logarithmic singularity at $m^2\to0$ comes from the soft
singularity in the $g\to gg$ matrix element (splitting function)
and is cancelled by the corresponding
virtual correction to the one-gluon radiation vertex.
The latter can be written as an integral over an intermediate gluon
virtuality $\mu^2$,
\beql{chidef}
 \chi \>=\> -\int_0^\infty \frac{d\mu^2\> k_\perp^2}
 {\mu^2(k_\perp^2+\mu^2)}
 \left\{ 2C_A \ln\frac{k_\perp^2(k_\perp^2+\mu^2)}{\mu^4}\>+\>
 s\left(\frac{\mu^2}{k_\perp^2}\right) \right\}
 \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2.
\end{equation}
Here we have chosen the form of the logarithmic term so as to cancel
exactly the corresponding real emission contribution in (\ref{beta0}).
The mismatch function $s(x)$ vanishes at the origin, $s(0)=0$, and
depends on the scheme one adopts for calculating the divergent vertex
correction.

\subsection{Fixing the scheme for $\alpha_{\mbox{\scriptsize s}}$}

To fix the scheme for defining the perturbative expansion parameter
$\alpha_{\mbox{\scriptsize s}}$, we have to consider first the relation between particle
production at order $\alpha_{\mbox{\scriptsize s}}^2$ and the coupling.
To this end we address the non-Abelian contribution to an inclusive
quantity such as the quark anomalous dimension.

The inclusive probability for producing a soft gluon or a two-parton system
with gluon quantum numbers reads
\begin{equation}
 dw=4C_F\frac{d\alpha}{\alpha}\, \frac{dk_\perp^2}{k_\perp^2}\> \gamma(\alpha_{\mbox{\scriptsize s}})\,,
\end{equation}
where the ``anomalous dimension'' $\gamma$ becomes
\begin{equation}
 \gamma \>=\>  \frac{\alpha_0}{4\pi}
 + \chi(k_\perp^2)
 + \int_0^\infty \frac{dm^2\> k_\perp^2}{m^2(k_\perp^2+m^2)}
   \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2 \left\{ -\beta_0
 + 2C_A \ln\frac{k_\perp^2(k_\perp^2+m^2)}{m^4} \right\}.
\end{equation}
The singular logarithmic term cancels against that in $\chi(k_\perp^2)$,
and we arrive at
\beql{gammaalpha}
 \gamma \>=\>
 \frac{\alpha_0}{4\pi}
 + c_s   \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2
 + \int_0^\infty \frac{dm^2\> k_\perp^2}{m^2(k_\perp^2+m^2)}
   \left(\frac{\alpha_{\mbox{\scriptsize s}}(m^2)}{4\pi}\right)^2  (-\beta_0)\>.
\end{equation}
Here $c_s$ is a constant given by the integral of the scheme-dependent
piece of ${\cal{R}}_{12}$, see (\ref{chidef}):
$$
   c_s = \int_0^\infty \frac{dx}{x(1+x)}\, s(x)\>.
$$
First we observe that, to the necessary accuracy, $\gamma$ satisfies,
$$
 k_\perp^2\frac{d}{dk_\perp^2}\, \gamma \>=\>
 -\beta_0 \left(\frac{\alpha_{\mbox{\scriptsize s}}(k_\perp^2)}{4\pi}\right)^2 +\ldots \>\simeq\>
  k_\perp^2\frac{d}{dk_\perp^2}\,\frac{\alpha_{\mbox{\scriptsize s}}(k_\perp^2)}{4\pi}\>.
$$
This means that we can choose $\gamma$ to represent the running
coupling $\alpha_{\mbox{\scriptsize s}}$. Equation (\ref{gammaalpha}) then becomes a
dispersive relation for $\alpha_{\mbox{\scriptsize s}}(k_\perp^2)$,
in which the combination of $\alpha_0$ and $c_s\alpha_{\mbox{\scriptsize s}}^2$
defines the coupling constant at $k_\perp^2=0$:
\beql{andim}
\gamma\equiv \frac{\alpha_{\mbox{\scriptsize s}}(k_\perp^2)}{4\pi} \>=\> \frac{1}{4\pi}
\left\{ \alpha_{\mbox{\scriptsize s}}(0) \>+\>
\int_0^\infty \frac{dm^2\> k_\perp^2}{m^2(k_\perp^2+m^2)}\>\rho_s(m^2)\right\}
\>=\> -\frac{1}{4\pi}
\int_0^\infty \frac{dm^2}{k_\perp^2+m^2}\>\rho_s(m^2),
\end{equation}
where $\rho_s(m^2)$ is the spectral density, to be discussed later.

This corresponds to choosing the CMW renormalisation scheme [\ref{CMW}]
in which
$\alpha_{\mbox{\scriptsize s}}$ is defined as the intensity of soft-gluon radiation.
In another scheme, e.g. $\overline{\mbox{MS}}$, the anomalous dimension in the soft
limit would contain both $\alpha_{\mbox{\scriptsize s}}$ and $\alpha_{\mbox{\scriptsize s}}^2$ terms.

\subsection{Thrust: resummation}

For a $q\bar{q}+N$-parton final state the thrust distribution becomes
\begin{equation}
   \frac{d\sigma}{dT} \>=\> \int d\Gamma_N \> M^2(q_1, q_2, \ldots
   q_N)
\> \delta(1-T-\sum_{i=1}^{N} \min\{\alpha_i,\beta_i\})
\>,
\end{equation}
with $M$ the $N$-parton matrix element and $d\Gamma_N$ the
corresponding $N$-body phase space.
In the soft approximation ($1-T\ll 1$) the final partons are either
relatively soft gluons emitted off the quark-antiquark line (primary
gluons) or their decay products.
The matrix element for an ensemble of primary gluons is very simple in
the soft approximation, as it is given by the product of independent
radiation amplitudes (essentially an Abelian pattern).
By taking a Mellin representation for the $\delta$-function,
$$
 \delta\left(1-T-\sum_{i=1}^{N} \min\{\alpha_i,\beta_i\}\right)
 = \int \frac{d\nu}{2\pi i} \exp\{\nu(1-T)\} \prod_{i=1}^{N}
 \exp\{-\nu \min\{\alpha_i,\beta_i\}\}\>,
$$
the thrust distribution takes the form
\begin{equation}
   \frac1{\sigma}
   \frac{d\sigma}{dT} \>=\> \int \frac{d\nu}{2\pi i} \>\mathrm{e}^{\nu(1-T)}
   \exp\{ {\cal{R}}(\nu) \}\>.
\end{equation}
At next-to-leading order, the ``radiator'' contains two terms,
${\cal{R}}(\nu)= {\cal{R}}_1+ {\cal{R}}_2$, due respectively to
one- and two-parton contributions.

\paragraph{One-gluon contribution.}
The one-gluon contribution, ${\cal{R}}_1$, is given by the following formal
expression:
\begin{equation}
\label{cR1}
 {\cal{R}}_{1} \>=\> 8C_F\int_0^1 \frac{d\alpha}{\alpha}
 \int\frac{dk_\perp^2}{k_\perp^2}\>
\left(\mathrm{e}^{-\nu\alpha}-1\right)   \Theta(k_\perp^2-\alpha^2)
\cdot \left( \frac{\alpha_0}{4\pi} \>+\> \chi(k_\perp^2) \right) ,
\end{equation}
where the $\Theta$-function selects the smaller longitudinal component
($\alpha<\beta=k_\perp^2/\alpha$).
The term with $\exp(-\nu\alpha)$ corresponds to the case in which the
gluon is emitted, while $-1$ corresponds to the case in which the
gluon is in a virtual loop.
The function $\chi(k_\perp^2)$ is the order $\alpha_{\mbox{\scriptsize s}}^2$ contribution that
describes the $k_\perp$-dependent one-loop vertex correction to
one-gluon emission, given in (\ref{chidef}).

Using the expression for the virtual correction, (\ref{chidef}),
we can split ${\cal{R}}_1$ into two contributions
$$
   {\cal{R}}_1\>=\>  {\cal{R}}_{11}\>+\> {\cal{R}}_{12} \>,
$$
where
\bminiG{cR1112}
\label{cR11}
 {\cal{R}}_{11} \!\!&=&\!\! 8C_F\int_0^1 \frac{d\alpha}{\alpha}
 \int\frac{dk_\perp^2}{k_\perp^2}\>
\left(\mathrm{e}^{-\nu\alpha}-1\right)   \Theta(k_\perp-\alpha)
\cdot \frac{\alpha_{\mbox{\scriptsize s}}(0)}{4\pi}  , \\
\label{cR12}
 {\cal{R}}_{12} \!\!&=&\!\!-16C_FC_A\int_0^1 \frac{d\alpha}{\alpha}
\left(\mathrm{e}^{-\nu\alpha}-1\right)
\!\!
\int_0^\infty \!\! \frac{d\mu^2\> dk_\perp^2}
 {\mu^2(k_\perp^2+\mu^2)}
 \ln\frac{k_\perp^2(k_\perp^2\!+\!\mu^2)}{\mu^4}
 \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2 \!\!\Theta(k_\perp\!-\!\alpha).
 \quad\qquad { }
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}
The weird factor $\alpha_{\mbox{\scriptsize s}}(0)$ in (\ref{cR11}), which stands for the
two-loop value of the on-mass-shell coupling,
is there simply because a real gluon has been emitted.
The contribution ${\cal{R}}_{12}$ is also ill-defined.

As we have seen above, ${\cal{R}}_{11}$,
is cancelled by the contribution from two-parton gluon splitting
in the collinear limit in (\ref{beta0}), resulting
in a finite answer in which $\alpha_{\mbox{\scriptsize s}}(0)$ gets replaced by the running $\alpha_{\mbox{\scriptsize s}}$.

The second contribution, ${\cal{R}}_{12}$, taken together with the
logarithmic piece ${\cal{R}}_{22}$ of the two-parton emission in ${\cal{R}}_{2}$,
defined below, produces a finite {\em inclusive correction}\/ to the
answer.

\paragraph{Two-parton emission contribution.}
The second part of the radiator describes gluon decay into two quarks
or into two gluons.
The latter is described by the non-Abelian part of
the $q\bar{q}\,g\,g$ matrix element $M_{gg}^2$ of (\ref{Mrr2}),
proportional to $C_FC_A$, see [\ref{DMO}].
We have
\beql{cR2}
 {\cal{R}}_2= 4C_F \int d\Gamma_2\>
\left(\frac{\alpha_{\mbox{\scriptsize s}}(m^2)}{4\pi}\right)^2
\left(M_{gg}^2(k_1,k_2)+M_{qq}^2(k_1,k_2)\right) \Omega_T \>.
\end{equation}
Here essential higher-order virtual corrections have been taken into
account, leading to the running of the coupling with the invariant mass
of the virtual gluon, $|\alpha_{\mbox{\scriptsize s}}(-m^2)|^2=\alpha_{\mbox{\scriptsize s}}^2(m^2)(1+\cO{\alpha_{\mbox{\scriptsize s}}^2})$.

The ``thrust trigger-function'' $\Omega_T$ in (\ref{cR2})
determines the contribution to the thrust of any
two-parton configuration and is given by
\beql{OT}
\Omega_T\>=\> \frac1{2!} \left[\>\exp(-\nu
\min\{\alpha_1,\beta_1\} - \nu\min\{\alpha_2,\beta_2\})
-1 \>\right] .
\end{equation}
We note that having included the symmetry factor $1/2!$ for both
gluons and quarks we have implicitly defined $M^2_{qq}(k_1,k_2)$
to describe the probability of finding {\em either}\/ a quark {\em or}\/
an antiquark with momentum $k_1$.

\paragraph{Splitting the answer.}
Now we are ready to assemble the pieces of the radiator into three
contributions each of which is explicitly finite.

The ``thrust trigger-function'' $\Omega_T$ in (\ref{cR2})
depends on the azimuthal angle between the partons, $\phi$,
and does not allow the integration in (\ref{beta0}) to be performed.
It can however be done,
in a naive (``inclusive'') approximation in which one employs a simplified
version of the $\Omega$-factor, namely,
\beql{Osimp}
  \Omega^{(\mbox{\scriptsize simp})}_T \>=\> \frac1{2!} \left[\>\exp(-\nu
\min\{\alpha,\>\beta\} ) -1 \>\right],
\end{equation}
where $\alpha= \alpha_1 + \alpha_2$ and $\beta = \beta_1 + \beta_2$.
This corresponds to the contribution to the thrust from the parent gluon,
disregarding the possibility that its offspring may go into opposite
hemispheres.
The regions $\alpha>\beta$ and $\beta>\alpha$ contribute equally, so
that it is sufficient to consider only the latter:
\begin{equation}
  \Omega^{(\mbox{\scriptsize simp})}_T \>\Longrightarrow\>
   \left(\mathrm{e}^{-\nu\alpha}-1 \right)\, \Theta(k_\perp^2+m^2-\alpha^2)\>.
\end{equation}

Using $\vq{}$ and $\vkp{}$ as integration variables, we obtain
\beql{R2simp}\eqalign{
 {\cal{R}}_2^{(\mbox{\scriptsize simp})} =& 8C_F
\int \frac{d\alpha}{\alpha}
\int dk^2_{\perp} \int dm^2
\frac{\alpha^2_s(m^2)}{2!\,(4\pi)^2}
\int_0^1 dz\int_0^{2\pi} \frac{d\phi}{2\pi} \left(M_{gg}^2+M_{qq}^2\right)\>
\Omega^{(\mbox{\scriptsize simp})}_T\>,
}\end{equation}
with $\phi$ the angle between $\vq{}$ and $\vkp{}$.

The integral in (\ref{R2simp}) has been calculated above in
(\ref{beta0}).
It consists of two terms
$ {\cal{R}}_2^{(\mbox{\scriptsize simp})}=  {\cal{R}}_{21} + {\cal{R}}_{22}$:
\bminiG{cR2122}
\label{cR21}
 {\cal{R}}_{21} \!\!\!\!&=&\!\! 8C_F
\int \frac{d\alpha}{\alpha}  \left(\mathrm{e}^{-\nu\alpha}\!-\!1\right)
\int \frac{dm^2\>dk_\perp^2}{m^2(k_\perp^2+m^2)}
\cdot (-\beta_0) \> \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2 \>
\Theta(k_\perp^2+m^2-\alpha^2)  \>; \\
\label{cR22}
{\cal{R}}_{22} \!\!\!\!&=&\!\! 16 C_FC_A
\int \frac{d\alpha}{\alpha}  \left(\mathrm{e}^{-\nu\alpha}\!-\!1\right)
\!\!\int\!\! \frac{dm^2\,dk_\perp^2}{m^2(k_\perp^2\!+\!m^2)}
\ln\frac{k_\perp^2(k_\perp^2\!+\!m^2)}{m^4} \left(\frac{\alpha_{\mbox{\scriptsize s}}}{4\pi}\right)^2
\Theta(k_\perp^2\!+\!m^2\!-\!\alpha^2) .
\qquad\quad { }
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}

Thus we may represent the answer as a sum of three finite terms:
\beql{theanswer}
  {\cal{R}}= {\cal{R}}^{(0)} + {\cal{R}}^{(i)} + {\cal{R}}^{(n)}\>,
\end{equation}
where
\bminiG{threeRs}
\label{cR0}
 {\cal{R}}^{(0)} &=& {\cal{R}}_{11} + {\cal{R}}_{21} \>, \\
\label{cRi}
 {\cal{R}}^{(i)} &=& {\cal{R}}_{12}+ {\cal{R}}_{22} \>, \\
\label{cRn}
 {\cal{R}}^{(n)} &=&  {\cal{R}}_2 -{\cal{R}}_2^{(simp)} \>.
\end{eqnarray}\relax\setcounter{equation}{\value{hran}}\renewcommand{\theequation}{\thesection.\arabic{equation}}
In the first contribution (\ref{cR0}) ${\cal{R}}_{11}$ involves $\alpha_{\mbox{\scriptsize s}}(0)$
which is cancelled by ${\cal{R}}_{21}$ to give the standard ``naive''
answer, as we shall see in the next section.
The second contribution, we shall refer to as an ``inclusive
correction''. Here both  ${\cal{R}}_{12}$ and ${\cal{R}}_{22}$ contain $m^2\to0$
logarithmic singularities which, however,
are damped by the difference of the two simplified
``thrust trigger functions''
$$
\Omega^{(\mbox{\scriptsize simp})}_T
- \Omega^{(\mbox{\scriptsize simp})}_T(m^2=0)
\> \propto \>\Theta(k_\perp^2+m^2-\alpha^2)-\Theta(k_\perp^2-\alpha^2)\>.
$$
Similarly, the {\em non-inclusive}\/ correction (\ref{cRn})
is also finite as it involves the difference of the true and the
simplified trigger functions,
\beql{deltaOmega}
\delta\Omega_T = \Omega_T-\Omega^{(\mbox{\scriptsize simp})}_T\>,
\end{equation}
which also vanishes as $m^2\to0$.

Now we are ready to recall the dispersive method and apply it to
the calculation of (\ref{threeRs}), starting from the main term
${\cal{R}}^{(0)}$.

\mysection{Na{\"\i}ve power correction for $T$ in the dispersive method}
\subsection{Dispersive method (recollection)}
In the approach of [\ref{BPY}]
the dispersive representation for $\alpha_{\mbox{\scriptsize s}}$ is implemented,
in which the dispersive variable $m^2$ acts as a ``gluon mass''
in the matrix elements (see \ref{andim}):
\beql{asae}
 \frac{\alpha_{\mbox{\scriptsize s}}(k^2)}{k^2} =
 -\int_0^\infty \frac{dm^2}{k^2(k^2+m^2)} \rho_s(m^2) =
 \int_0^\infty \frac{dm^2}{(m^2+k^2)^2}\alpha_{\mbox{\scriptsize eff}}(m^2)\>.
\end{equation}
At one-loop level one has
\beql{rhodef}
  \rho_s(m^2) =
  \frac{d}{d\ln m^2}\>\alpha_{\mbox{\scriptsize eff}}(m^2) \>=\>
  -\frac{\beta_0}{4\pi} \alpha_{\mbox{\scriptsize s}}^2(m^2)\> + \ldots
\end{equation}
The perturbative answer, $ {\cal{O}}_j(Q^2)$, for an observable $j$
is then formulated in terms of the $m^2$ integral
\beql{calO}
 {\cal{O}}_j(Q^2) = \int_0^\infty\frac{dm^2}{m^2}
\alpha_{\mbox{\scriptsize eff}}(m^2)\cdot \dot{\cal{F}}_j(m^2/Q^2)\>,
\quad \dot{\cal{F}}_j \equiv -\frac{d}{d\ln m^2}{\cal{F}}_j\>,
\end{equation}
with ${\cal{F}}$ the ``characteristic function'' one derives by
computing the $m^2$-dependent soft-gluon radiation matrix element.
For an infrared/collinear safe  observable
the $m^2$ integral converges and is mainly
determined by the region $m^2\sim Q^2$,
the typical hard scale of the process.
This region reproduces the standard
one-loop perturbative prediction
$$
{\cal{O}}^{\mbox{\small PT}}_j=\alpha_{\mbox{\scriptsize eff}}(Q^2)\,{\cal{F}}(1)
\>\approx\> \alpha_{\mbox{\scriptsize s}}(Q^2)\,{\cal{F}}(1)\>.
$$
The genuine non-perturbative component of the answer,
$\delta{\cal{O}}^{\mbox{\small NP}}$, is triggered by the non-perturbative
component of the effective coupling $\delta\alpha_{\mbox{\scriptsize eff}}(m^2)$.
Imposing the ITEP/OPE restriction [\ref{OPE}]
that $\alpha_{\mbox{\scriptsize s}}(k^2)$ in the ultraviolet region
receive no power correction $(k^2)^{-p}$
due to the large-distance interaction domain,
one derives from (\ref{asae}) that the integer moments
of the non-perturbative effective coupling vanish:
$$
 \int_0^\infty \frac{dm^2}{m^2}\delta\alpha_{\mbox{\scriptsize eff}}(m^2)\> (m^2)^i \>=\> 0\>,
 \quad i=1,2, \ldots p\>.
$$
Thus, the power corrections are obtained from the non-analytic
$m^2\to 0$ behaviour of the one-loop matrix element (characteristic function).
For example, the behaviour
$\delta{\cal{O}}_T^{\mbox{\small NP}}\sim Q^{-1}$
in the thrust distribution originates from
\beql{cFT}
 \dot{\cal{F}}_T(m^2/Q^2) \>\to\> \frac{2C_F}{\pi}
\>\sqrt{\frac{m^2}{Q^2}}\>, \qquad m^2\to 0\>.
\end{equation}
The magnitude of the power correction is then expressed in terms
of the $m$-moments of an ``effective charge''  $\delta\alpha_{\mbox{\scriptsize eff}}(m^2)$
describing the intensity of non-perturbative gluon radiation:
\begin{equation}
 \delta{\cal{O}}_j^{\mbox{\small NP}}(Q^2) =
\int_0^\infty\frac{dm^2}{m^2}
\delta\alpha_{\mbox{\scriptsize eff}}(m^2)\> \dot{\cal{F}}_j(m^2/Q^2)
\>\simeq\>  C_j \frac{A_{n,q}}{ Q^{n}}\>,
\end{equation}
where
\beql{momdef}
  A_{2p,q} = \frac{C_F}{2\pi}
  \int_0^\infty \frac{dm^2}{m^2} \>\delta\alpha_{\mbox{\scriptsize eff}}(m^2)\cdot
  (m^2)^{p}\ln^q{m^2}
\end{equation}
(for $p$ half-integer or $q> 0$).
In particular, for the case of thrust from (\ref{cFT}) we deduce:
$p=\mbox{\small $\frac{1}{2}$}$, $q=0$, and the
observable-dependent constant $C_j$ is $C_T=4$.

\subsection{Inclusive treatment of the thrust distribution}
In an inclusive approximation one employs the simplified version of the
$\Omega$-factor (\ref{Osimp}).
The contribution ${\cal{R}}_{21}$  (\ref{cR21}),
proportional to $\beta_0$, is the only one
which has been dealt with in the past~[\ref{DW97}].
Making use of (\ref{rhodef})
it can be written as
\beql{cR21mod}
 {\cal{R}}_{21}\> =\>
 \frac{2C_F}{\pi}
\int \frac{d\alpha}{\alpha}\left(\mathrm{e}^{-\nu\alpha}-1\right)
\int_0^\infty dm^2 \frac{d\,\alpha_{\mbox{\scriptsize eff}}(m^2)}{d m^2}\>
\int {dk^2_{\perp}} \frac{\Theta(k_\perp^2+m^2-\alpha^2)}
{k_\perp^2+m^2}\>.
 \qquad { }
\end{equation}
To reconstruct the known result we integrate by parts.
Since $\alpha_{\mbox{\scriptsize eff}}(0)=\alpha_{\mbox{\scriptsize s}}(0)$, the surface term
cancels ${\cal{R}}_{11}$, given in (\ref{cR11}),
and we obtain
\begin{equation}
{\cal{R}}^{(0)} \>=\>
{\cal{R}}_{11}+{\cal{R}}_{21} \>=\>  \frac{2C_F}{\pi}
\int \frac{d\alpha}{\alpha}\left(\mathrm{e}^{-\nu\alpha}-1\right)
\int_0^\infty \frac{dm^2}{m^2} \>\alpha_{\mbox{\scriptsize eff}}(m^2)\>
\Theta(m^2-\alpha^2) \> .
\end{equation}
The normal perturbative prediction is obtained by integrating over $\alpha$
between $1/\nu \ll 1$ and $1$. In order to examine the
non-perturbative contribution, one substitutes for $\alpha_{\mbox{\scriptsize eff}}$ its
non-perturbative part, $\delta\alpha_{\mbox{\scriptsize eff}}$, which is concentrated
at small $m^2$.
This implies that $\alpha$ is small, allowing us to
expand the exponential term,
\begin{equation}
\delta{\cal{R}}^{(0)}=
 -\nu\frac{2C_F}{\pi}
\int_0^\infty \frac{dm^2}{m^2} \>m\>\delta\alpha_{\mbox{\scriptsize eff}}(m^2)\>
\equiv\> -\nu\> 4A_{1,0}\>,
\end{equation}
resulting in a non-perturbative shift of the thrust distribution,
proportional to the first moment of $\delta\alpha_{\mbox{\scriptsize eff}}$~[\ref{DW97}],
\beql{shift}
\frac{d\sigma(T)}{dT}
=\frac{d\sigma^{(\mbox{\scriptsize PT})}(T\!-\!\Delta T)}{dT}\>,
\quad \Delta T^{(0)} = \frac{4\,A_{1,0}}{Q}\>.
\end{equation}
Taking into account subleading $\alpha_{\mbox{\scriptsize s}}^2$-effects,
the naive inclusive prediction (\ref{shift}) gets modified by the
{\em thrust rescaling factor}\/  $r_T$:
\beql{shiftmod}
  \Delta T = \frac{4\,A_{1,0}}{Q}\, (1+r_T)\>.
\end{equation}
This factor originates from the real and virtual two-loop matrix element
and depends on the observable under consideration.

In what follows we analyse and compute the thrust rescaling factor.
In our framework, it consists of two components:
an {\em inclusive}\/ $r_T^{(i)}$ and a {\em non-inclusive}\/
$r_T^{(n)}$ corrections

\mysection{Exact treatment}
\subsection{Inclusive correction}
As we have seen above,
the logarithmic terms in two-particle production ${\cal{R}}_{22}$
(\ref{cR22}) and
the vertex correction ${\cal{R}}_{12}$ (\ref{cR12}) cancel in the fully
inclusive anomalous dimension.

In a less inclusive quantity, like thrust, the cancellation is no
longer complete: divergences cancel but a finite correction
remains.
Indeed, in the real emission contribution the kinematical restriction
involves the mass of the pair and reads
$$
\Theta\left(\sqrt{k_\perp^2+m^2}-\alpha\right)\>.
$$
In the virtual correction, however, we deal with a real gluon restricted
simply by
$$
\Theta\left(\sqrt{k_\perp^2}-\alpha\right)\>.
$$
As a result the finite correction which emerges reads
\beql{deltacRi}\eqalign{
 {\cal{R}}^{(i)} =& 16C_FC_A
 \int\frac{d\alpha}{\alpha}\left(\mathrm{e}^{-\nu\alpha}-1\right)
\int\frac{ dk_\perp^2 \>dm^2}{m^2(k_\perp^2+m^2)}
 \left(\frac{\alpha_{\mbox{\scriptsize s}}(m^2)}{4\pi}\right)^2  \cr
& \ln\frac{k_\perp^2(k_\perp^2+m^2)}{m^4}
\left[\, \Theta\left(\sqrt{k_\perp^2\!+\!m^2}-\alpha\right)
- \Theta\left(k_\perp-\alpha \right)\,\right]\,.
}\end{equation}
Expanding the exponent and performing the $\alpha$-integration we arrive at
$$\eqalign{
 {\cal{R}}^{(i)} &= -16C_FC_A\nu \int \frac{dm^2}{m^2}
 \left(\frac{\alpha_{\mbox{\scriptsize s}}(m^2)}{4\pi}\right)^2
 \int_0^\infty \frac{dk_\perp^2}{k_\perp^2+m^2}\left(\sqrt{k_\perp^2\!+\!m^2}
 -k_\perp\right)\> \ln \frac{k_\perp^2(k_\perp^2+m^2)}{m^4} \>,
}$$
where we have extended the $k_\perp^2$-integration to infinity since it
converges.
Now we use (\ref{rhodef})
and integrate by parts to arrive at
\begin{equation}\eqalign{
 {\cal{R}}^{(i)} &= -8C_F\frac{C_A}{\beta_0}\nu \int
 {dm^2}\frac{\alpha_{\mbox{\scriptsize eff}}(m^2)}{4\pi}\>
 \frac{d}{d{m^2}} \left\{ 2\,m\> c^{(i)}  \right\}
}\end{equation}
with
\begin{equation}
 c^{(i)}=2 \int_0^\infty \frac{xdx}{1+x^2}\>
 \frac{\ln \,[\,x^2(1+x^2)\,]}{x+\sqrt{1+x^2}}
   = 3.299\>.
\end{equation}
Substituting $\delta\alpha_{\mbox{\scriptsize eff}}$ for $\alpha_{\mbox{\scriptsize eff}}$ we get the power correction
\begin{equation}
  \delta {\cal{R}}^{(i)} \>=\> -\nu\cdot  c^{(i)} \>
\frac{2 C_FC_A}{\pi\,\beta_0}\>
 \int \frac{dm^2}{m^2}\>m\> {\delta\alpha_{\mbox{\scriptsize eff}}(m^2)}\>.
\end{equation}
Recalling the definition of the non-perturbative moment, (\ref{momdef}),
the relative correction to the shift in the thrust distribution
(\ref{shift}) becomes
\beql{DTi}
  \Delta T^{(i)} = \frac{4A_{1,0}}{Q}\cdot r_T^{(i)} \>,\qquad
r_T^{(i)} = \frac{C_A}{\beta_0}\, c^{(i)} \>=1.100 \>(0.900)
\quad \mbox{for}\>\> n_f=3\>(0) \>.
\end{equation}
Notice that the inclusive correction is of the same magnitude as the
na{\"\i}ve term.

\subsection{Non-inclusive correction}
Since the leading power correction originates from the region of
large gluon radiation angles, $\alpha\sim\beta\sim m$, $k_\perp <m$,
the kinematic region of offspring partons moving into opposite hemispheres
will give an essential correction.

This correction is taken into account by ${\cal{R}}^{(n)}$ in
(\ref{cRn}), given by the expression (\ref{cR2}) with the
thrust trigger-function $\Omega_T$ replaced by $\delta \Omega_T$
(\ref{deltaOmega}).
In order to obtain the first power correction it suffices to expand
the exponents in (\ref{OT}) to first order as above.
Straightforward calculation results in
\beql{Otilde}
 \int_0^\infty \frac{d\alpha}{\alpha}\> \delta\Omega
\>\simeq\> -\nu \,\tilde{\Omega}_T\>, \quad
\tilde{\Omega}_T(q_1,q_2)\>=\> zq_1+(1-z)q_2 - \sqrt{zq_1^2+(1-z)q_2^2}\>.
\end{equation}
The expression for ${\cal{R}}^{(n)}$ becomes
\beql{erren}
  {\cal{R}}^{(n)} \>=\> -\nu\> 4C_F\int dm^2
\left(\frac{\alpha_{\mbox{\scriptsize s}}(m^2)}{4\pi}\right)^2
\int dk_\perp^2\int_0^1 dz \int\frac{d\phi}{2\pi}
\left(M_{gg}^2+M_{qq}^2\right) \tilde{\Omega}_T\>.
\end{equation}
Now we use (\ref{rhodef}) and integrate by parts.
Taking into account that at $m\!=\!0$
$$
  \tilde{\Omega}_T(q_1\!=\!q_2) \>=\> 0\>,
$$
the surface term vanishes and we get
\beql{deltacR}
 {\cal{R}}^{(n)} \>=\> -\nu\> \frac{2C_F}{\pi}\int dm^2
 \>{\alpha_{\mbox{\scriptsize eff}}(m^2)}\> \frac{d}{dm^2}\left\{
 \int dk_\perp^2 \int dz \int \frac{d\phi}{2\pi} \>
 m^2 \left(M_{gg}^2+M_{qq}^2\right) \frac{\tilde{\Omega}_T}{2\beta_0}
 \right\}
\end{equation}
The expression in the curly brackets has dimensions of mass.
$Q^2$ enters only in the upper limit
of the $k_\perp^2$ integration which is convergent in the ultra-violet
region.
Therefore we integrate up to infinity so that the result does not depend
on $Q$ and is given by
$$
 \{\mbox{(\ref{deltacR})}\}\>=\>  2\,r_T^{(n)}\,m \>+\> \cO{m^2/Q}\,,
$$
with $r_T^{(n)}$ a number still to be calculated.
In the $m\to0$ limit this leads to
\begin{equation}
  \delta{\cal{R}}^{(n)} \>=\> -\nu\> \frac{4A_{1,0}}{Q}r_T^{(n)}\>.
\end{equation}
As a result, the non-perturbative shift in the thrust distribution
(\ref{shiftmod}) acquires a contribution
\beql{shiftR}
  \Delta T^{(n)} \>=\> \frac{4A_{1,0}}{Q}\>r_T^{(n)}\>.
\end{equation}
In what follows we construct the relevant matrix element
and compute the non-inclusive component of the thrust rescaling factor
$r_T^{(n)}$.

\subsection{Evaluation of $r_T^{(n)}$}
We are now in a position to discuss the determination of the
non-inclusive thrust rescaling factor $r_T$.
From (\ref{deltacR}), we have that
\begin{equation}
 r_T^{(n)} = \frac1{m} \int\frac{d^2k_\perp}{\pi}\frac{d\phi}{2\pi} dz\>
 \frac{{\cal{M}}^2}{zq_1^2 + (1-z)q_2^2} \>
 \frac{\tilde{\Omega}_T(q_1,q_2)}{4\beta_0}\>.
\end{equation}
To express the integration measure in terms of
$u_1$ and $u_2$ we write
$$\eqalign{
& \frac{d^2k_\perp}{\pi}\frac{d\phi}{2\pi}
 =\frac{d^2k_\perp}{\pi}\frac{d^2q}{\pi}\delta(q^2-(\vq{1}-\vq{2})^2)
 = \frac{d^2q_1}{\pi}\frac{d^2q_2}{\pi}\delta(q^2-(\vq{1}-\vq{2})^2)\cr
 &= {q_1dq_1}\,q_2dq_2\> \frac{2}{\pi}\, d\phi \>\delta(q^2-q_1^2-q_2^2
 +2q_1q_2\cos\phi)
 = \frac{4q^2}{\pi}\> \frac{u_1du_1\,u_2du_2 }
 {2u_1u_2|\sin\phi|}
 \>.
}$$
Using the $\delta$-function condition to express $\sin\phi$ we arrive at
\begin{equation}
 \frac{d^2k_\perp}{\pi}\frac{d\phi}{2\pi}
 \>=\> \frac{4q^2}{\pi}\>
 \frac{u_1du_1 \> u_2du_2}{\sqrt{J}} \>,
\end{equation}
where the Jacobian factor $J(u_1,u_2)$ is
$$
 J={((u_1+u_2)^2-1)\,(1-(u_1-u_2)^2)}\>.
$$
Making use of the fact that $\tilde{\Omega}$ in (\ref{Otilde}) satisfies
\begin{equation}
 \tilde{\Omega}_T(q_1,q_2) \>=\> q\> \tilde{\Omega}_T(u_1,u_2)
 \>=\> \frac{m}{\sqrt{z(1-z)}}\> \tilde{\Omega}_T(u_1,u_2)\,,
\end{equation}
we finally obtain
\beql{rTn}\eqalign{
 r_T^{(n)} =& \frac1{\pi\beta_0}
       \int_0^1 \frac{dz}{\sqrt{z(1-z)}}
       \int_0^\infty u_1du_1 \int_0^\infty u_2du_2
       \>\frac{1}{\sqrt{J}}\> \cr
 &\times \frac{{\cal{M}}^2(u_1,u_2)}{zu_1^2 + (1-z)u_2^2} \>\>
  \tilde{\Omega}_T(u_1,u_2)
\>,}
\end{equation}
where the integrals run over the region $J\ge0$.
This region looks simpler in terms of $u_\pm=u_1\pm u_2$:
\beql{d+-}
\int  \frac{du_1\, du_2}{\sqrt{J}}
\>=\> \mbox{\small $\frac{1}{2}$}\, \int_{-1}^1\frac{du_-}{\sqrt{1-u_-^2}}
 \int_{1}^\infty\frac{du_+}{\sqrt{u_+^2-1}}\>.
\end{equation}
\paragraph{Convergence of $r_T^{(n)}$.}
It is necessary to check that the $u_+\to\infty$ and $z(1-z)\to0$
regions do not lead to divergences of the $r_T^{(n)}$ integral.
This is shown in Appendix~B.

Numerical integration results in the value
\begin{equation}
   r_T^{(n)}=\frac{2} {\beta_0}\> (-1.227\,C_A +0.365\, C_A \>-0.052\, n_f)\>,
\end{equation}
where the three terms originate respectively from the soft gluon,
hard gluon and quark matrix elements (\ref{matelms}).
This gives
\begin{equation}
  r_T^{(n)}=
     \> -0.710\>(-0.470)\,, \quad \mbox{for}\>\> n_f=3\>(0) \>.
\end{equation}
We have that the non-inclusive correction alone would lower the ``naive''
expectation for the factor in the leading $1/Q$ power correction to
thrust by about 30\%.
However,
assembling the inclusive and non-inclusive corrections to the thrust
rescaling factor we finally obtain
\beql{numanswer}
 r_T= r_T^{(i)}+ r_T^{(n)}\>=
\> \frac1{\beta_0} \left[\, 1.575 \,C_A
\>-\> 0.104\,n_f\,\right]\>=\> 0.490\>(0.430)\quad \mbox{for}\>\>
n_f=3\>(0)\,.
\end{equation}
The conclusion is that the thrust rescaling factor $1+r_T$
increases the na{\"\i}ve value by 50\%. The most important correction is the
inclusive one (\ref{DTi}).

\mysection{Discussion and conclusions}
Within the dispersive approach of~[\ref{BPY}] power corrections to
perturbative QCD predictions, and to jet-shape observables in
particular, are expressed in terms of moments of the non-perturbative
component of the effective coupling, $\delta\alpha_{\mbox{\scriptsize eff}}(m^2)$.

To fix the absolute normalisation for the leading power corrections to
jet-shape observables a two-loop analysis has to be carried out: since
power corrections are proportional to powers of the QCD scale
$\Lambda$, it is necessary to fix the latter to control the magnitude
of the non-perturbative correction.

In this paper we have examined the thrust distribution in the
high-thrust region to two-loop
accuracy to extract the $1/Q$ power contribution.  To do this it
suffices to consider radiation of soft gluons followed by their
two-parton decays.

To this end we have used the physical CMW scheme~[\ref{CMW}]
in which $\alpha_{\mbox{\scriptsize s}}$ is defined as the strength of inclusive soft-gluon
radiation (the magnitude of the singular part of the two-loop quark
anomalous dimension).

Given $\Delta T^{(0)}$ as the result of the na{\"\i}ve inclusive
one-loop treatment, we have found that the rescaling factor $1+r_T$
for the thrust power correction, $\Delta T=(1+r_T)\Delta T^{(0)}$ is
rather large.
This large correction comes mainly from the gluon splitting into two
gluons.

The correction $r_T$ originates from two sources.
A positive term $r_T^{(i)}$ (the ``inclusive correction'')
comes from an incomplete compensation of the
logarithmic contributions in soft-gluon splitting,
due to the kinematical difference
between virtual (massless) and real (massive) gluon radiation
(\ref{deltacRi}).
This gives the most important correction (see \ref{DTi}).
A negative term $r_T^{(n)}$ (the ``non-inclusive correction'')
is due to the kinematical region of the 4-parton phase space in which the
gluon decay products fly into opposite hemispheres (\ref{erren}).
The contribution to this last term coming from the decay of the
primary gluon into a quark-antiquark pair proves to be positive and
numerically very small.
This agrees with the result of the analysis of Nason and
Seymour~[\ref{NS}].

It is the region of large gluon emission angle, $\theta\simeq \pi/2$,
that triggers the non-analyticity in the gluon virtuality
$m^2$ and thus power corrections.
This makes opposite-hemisphere configurations of the decay
products quite common.
But it so happens that the most important correction is the
inclusive one.

Since the moments of $\delta\alpha_{\mbox{\scriptsize eff}}(m^2)$ are not calculable given the
present state of the art, they must be determined phenomenologically.
Therefore to be of any practical value, the programme of calculating
power corrections at two loops must be extended to other observables
driven by the same (first) moment as the thrust.
In a forthcoming publication we will generalise the present analysis
to energy-energy correlations and jet-broadening in
$e^+e^-$-annihilation~[\ref{futuro}].

For a general observable $j$ we will have a rescaling factor $1+r_j$,
where $r_j$ originates from the primary soft-gluon emission and its
two-parton decay convoluted with the trigger function specific to the
observable $j$. Since $r_j$ depends on the observable, one could argue
that universality of the leading power correction is violated.  On the
other hand, we would prefer to consider universality as still being
valid. Indeed, the ingredients of universality, which are the
universality of soft-gluon radiation and universality of the coupling,
remain intact in the two-loop analysis. Moreover, only at the two-loop
level can universality be given a precise meaning, since only at this
level can one define the QCD scale and thus the coupling.
The fact that the rescaling factor
is observable dependent does not break universality, since $r_j$
remains under perturbative control.

Let us stress that the two-loop analysis has led to a term $r_T$ in
the rescaling factor which does not contain
any small parameter and is just a number.
One may wonder whether higher-loop calculations will give corrections
of the same order as $r_j$.
We argue that this is not the case and that terms higher than
two-loops give ``subleading'' contributions, \hbox{\rm i.e. } with additional
powers of the perturbative coupling.

To see this observe that the $(\ell+2)$-loop matrix
element would lead to the appearance of an $m^2$-integral of
$\alpha_{\mbox{\scriptsize s}}^{\ell+2}(m^2)$.
By using (\ref{rhodef}) and performing the analysis as before, one
finds that the corresponding contribution to the non-perturbative power
correction will be proportional to $\alpha_{\mbox{\scriptsize eff}}^{\ell}\delta\alpha_{\mbox{\scriptsize eff}}$ and thus will be
down, with respect to the two-loop result, by an extra power $\ell$ of
the coupling.
Strictly speaking, such a perturbative argument is far from being
perfect since the coupling here enters at a small scale.
Whether the higher order contributions are actually small depends on the
numerical value of the ``perturbative'' effective coupling $\alpha_{\mbox{\scriptsize eff}}/\pi$
at low scales. The large value of the two-loop correction $r_T$
indicates that $\delta\alpha_{\mbox{\scriptsize eff}}$ is {\it smaller}\/ than was indicated by the
naive treatment.

\vspace{5 mm}
\noindent
{\large\bf Acknowledgements}.  We are grateful to A.H.~Mueller and
B.R.~Webber for illuminating discussions.  We also wish to thank
M.~Dasgupta, L.~Magnea and G.~Smye (the authors of [\ref{DasMagSmy}])
and B.R.~Webber for discussions related to the correction of an error
in the first version of this paper, discovered subsequent to an
independent calculation of the $n_f$-dependent piece in
[\ref{DasMagSmy}].

redpajama_set_name: RedPajamaArXiv


text:
Football Notebook: Six Northwestern players in transfer portal
Joshua Hoffman/Daily Senior Staffer
Ryan Field. Six Northwestern players have entered the transfer portal in the last week.
Patrick Andres and Drew Schott
Within three days of the Wildcats' season-ending 47-14 loss to Illinois, six Northwestern players are in the NCAA transfer portal.
Junior tight end Jason Whittaker was the first, entering on Nov. 22. Whittaker, recruited as a quarterback out of Rockford, Michigan, played two games at tight end for the Wildcats and opted out of the 2020 season. He was a three-star prospect with offers from Indiana, Iowa State and all three of Michigan's Mid-American Conference schools.
Sophomore defensive tackle Te-Rah Edwards and junior defensive lineman Wyatt Blake entered the portal Monday. Edwards, a Groveport, Ohio, native, played in five games and recorded two tackles in his NU career. He had offers out of high school from Michigan, Michigan State and Penn State, among other schools.
Blake, who had three tackles in 14 games as a Cat, is a Crystal Lake, Illinois, native who received offers from Indiana, Minnesota and Vanderbilt. Both players were three-star recruits.
Three other players also entered the portal. Junior offensive lineman Payne He'bert added his name after playing seven career games for NU. In high school, He'bert, who hails from Pearland, Texas, had multiple offers including those from Houston, Rice, Kansas State.
Junior linebacker Jaylen Rivers of Decatur, Georgia, also indicated his intent to transfer. Rivers was a three-star recruit who had offers from Louisville, Mississippi, West Virginia and other schools. Rivers played four games for the Cats in 2021.
On Tuesday, sophomore cornerback Garnett Hollis Jr. entered his name in the portal. The Nashville, Tennessee, native, a three-star recruit, made five tackles in his NU career. Hollis Jr. had offers out of high school from Purdue, Indiana, Rutgers and others.
Entering the portal does not necessarily bind a player to transfer, but it allows coaches from other schools to contact them.
Players earn postseason accolades
Though NU didn't make its third Big Ten Championship in four seasons, the Wildcats earned some recognition following their 3-9 campaign.
Seven Wildcats earned All-Big Ten honors, including sophomore offensive lineman Peter Skoronski, graduate linebacker Chris Bergin, sophomore running back Evan Hull and sophomore safety Brandon Joseph.
Skoronski came to Evanston as a five-star recruit and the highest-ranked NU signee in the history of the 247Sports Composite rankings. One year after replacing Los Angeles Charger Rashawn Slater at left tackle, the Park Ridge, Illinois, native earned a first-team nod by the media and a second-team spot by the conference's coaches.
He has started 21 straight games at left tackle.
Bergin, who leads the country with 141 tackles, earned nods on the second team by the media and the third team by the conference's coaches. Bergin made 10 or more tackles in eight of them, including a career-high 19 against Minnesota, and recorded an interception.
After being thrust into the starting role following a season-ending lower-body injury to sophomore running back Cam Porter, Hull rushed for 1,009 yards and seven touchdowns, becoming the program's first rusher to break 1,000 yards since Justin Jackson in 2017. His 5.15 yards per carry is the sixth-best average for a single season in NU history.
Against Ohio, Hull rushed for a 90-yard score, the Cats' longest run since Bill Swingle raced for a 95-yard touchdown in 1961 versus Boston College.
After winning the Thompson-Randle El Freshman of the Year award and earning a consensus spot on the All-Big Ten first team in 2020, Joseph made the media's second team and earned an honorable mention from the Big Ten's coaches. Despite racking up half his six-interception total from last year, the former All-American finished with 80 tackles, a fumble recovery and a sack.
Junior defensive end Adetomiwa Adebawore had a breakout 2021 campaign, earning him a consensus All-Big Ten honorable mention. He led the Wildcats with eight-and-a-half tackles for loss and four-and-a-half sacks. Junior cornerback A.J. Hampton Jr. earned the same accolade after leading the conference with 13 pass breakups. He scored the first touchdown of his career when he returned a fumble for 49 yards against Wisconsin.
One season after allowing the lowest completion percentage in the Big Ten, sophomore cornerback Cameron Mitchell earned an honorable mention from the Big Ten's coaches after making 48 tackles, breaking up six passes and registering a quarterback hit.
While NU's offense averaged more than 320 yards per game, the Cats' defense took a step back in 2021 under first-year coordinator Jim O'Neil. After having the fifth-best scoring defense in the country last season, NU gave up 429.5 yards per game, a mark that ranked 13th in the conference.
Andrew Clair to return for sixth year
NU running back Andrew Clair will return for his sixth season of college football, the graduate student announced on Instagram on Monday night.
"Look how far I then came and accomplished," Clair wrote. "Got unfinished business in Evanston. Last one best one, one more in this college football ish. See you for one more year."
Clair transferred from Bowling Green in the offseason and rushed for 349 yards for the Wildcats in 12 games in 2021, finishing second on the team behind 1,009-yard sophomore Evan Hull. It was his highest season rushing total since he ran for 702 yards for the Falcons in 2018. He recorded one touchdown, a 3-yard run against Rutgers.
The St. Louis native will join Hull and freshman Anthony Tyus III in a reloaded running back room in 2022. Sophomore Cam Porter, who missed the entire 2021 season with a lower-body injury, is also expected to return.
Twitter: @pandres2001
Twitter: @dschott328
A.J. Hampton Jr.
Adetomiwa Adebawore
Andrew Clair
Brandon Joseph
Chris Bergin
Jason Whittaker
Jaylen Rivers
northwestern football
Payne He'Bert
Te-Rah Edwards
Wyatt Blake
Dr. Jeffrey Sterling discusses NU's racist history, encourages Black leaders in the community to speak up at MLK Candlelight Vigil