Long before anorexia, our son had what you might call 'puppy fat' at primary school. He was also much quieter and was bullied by a boy who viewed him as an academic rival. But once at secondary school and away from the bully, he threw himself into lessons, sporting activities and his new circle of friends. He was very happy at school and at home. The weight dropped off, but in a healthy, normal kind of way.
At secondary school, he developed a passion for sport – and was very good at it. He became a star player in the rugby team and also played for his local team. Pre-anorexia, he also enjoyed squash, swimming, walking, cycling, cross-country and athletics. He and his dad even did the grueling Coast2Coast cycle ride.
In those pre-anorexia days, our son was a physically fit and energetic boy, he had a lovely circle of friends, he had an encyclopedic knowledge of history and geography - and he was consistently top of the class in a range of academic subjects. Never, in a million years, could we ever guess that anorexia was about to muscle its way into our lives – especially since anorexia is usually considered to be a 'girls' disease'.
On holiday in France he was swimming 100 lengths a day of the pool (mind you, he'd done that the previous year so we weren't unduly concerned). But he was also going for a run every day, turning down offers of ice cream, refusing to put butter on his toast, making his own pack lunches and increasing his intake of fruit and dried fruit.
When we got home he joined the local gym, went on lengthy and very grueling runs, and started to see his friends less. This concerned us because up to then sleepovers, meals out, cinema visits and hanging out with his mates around town had been a regular part of his life. He also seemed quite subdued. He seemed to have lost his usual zest for life and his sense of fun.
In September 2009 my husband's parents came to stay and my mother-in-law immediately commented on how much weight our son had lost. I guess when you see your child every day it's not so obvious. She hadn't seen him for six months.
Around this time, his obsession with 'healthy eating' reached a height. He cut out all fats and cut down his general food intake severely. He also developed a passion for fruit and dried fruit which he would spend ages ritualistically chopping into little bits before eating slowly.
It was probably mid to late September when I became sufficiently concerned to make the first of three appointments to see our GP. On the first couple of occasions, the GP weighed him and measured his height, asked if he was worried about anything or depressed. Then he told him to go home, eat more and come back in a couple of weeks.
By late October, the school nurse became involved, because our son's behaviour had changed so much – and it wasn't simply our son 'being a teenager'; it went far deeper than that. Anorexia had taken over our son and our family. And anorexia isn't just about eating; it's about a stack of other symptoms - like depression, panic, zero self-esteem, etc. I don't just mean feeling a bit 'down' now and again, I mean deep, dark depression and self-hatred. I mean banging your head against the wall, or thumping your fists against your skull, throwing things around and animal screaming kind of depression.
Other staff were worried. The sports staff commented on his severe loss of weight. From Number 3 in the rugby team, they'd moved him to the less demanding Number 8 and were now questioning whether he should be playing rugby at all. Other staff had noticed him 'punishing himself' in the school gym and on cross country.
Naively we thought we'd get an appointment right away and were horrified to find the waiting list could be 18-22 weeks, crucially NOT from the day of the referral but from the day CAMHS wrote to us to tell us we we'd been added to the list.
It was 23rd November before that letter arrived. A quick calculation told me it could be EASTER before we got help. I wanted to scream!!!
Meanwhile it broke my heart to pick him up from school. I'd watch the other teenagers, joking with each other and looking the picture of health. Then I'd see my son, dragging himself across the car park, skinny as a rake, dark rings round his eyes and looking as miserable as sin.
With the CAMHS treatment so far away, we started to panic. We simply had to do something to stop our son from disappearing in front of our eyes, physically and mentally.
In the end we managed to find a private therapist who provided a bit of 'stop gap' treatment. But it was a bit like trying to stop the Titanic from sinking by sticking a Band Aid across the hole.
And still the CAMHS waiting list loomed ahead. What state would he be in by the time our place came up at Easter? And how quickly would the treatment take effect?
With anorexia it's as if someone else moves into your head. Someone that taunts you all the time, telling you you're fat and unattractive and that you'll never be popular until you get thin.
Anorexia has you pinching the skin on your skinny stomach, taunting you that it's rolls of fat. Anorexia makes you exercise like mad and examine yourself in the mirror critically. Anorexia makes you hate what you see. Anorexia lies to you that it can make you ultra handsome, ultra slim and ultra confident.
Anorexia lies that it can put you in control of your life. And part of this control is to control exactly what goes into your stomach, how much of it and when. The minute you deviate from this rigid eating pattern, anorexia lies to you that you're out of control. Just one serving of dinner that's not the 'right size' and anorexia can have the sufferer banging their head on the fridge and screaming. I know, because that's what our son was doing by February 2010.
Anorexia tempted our son with promises of looking physically perfect. His main aim was to get a 'six pack' to impress the girls. With this 'six pack' would come increasing self-confidence and popularity. By eating less and exercising more he would achieve his aim.
His role models became the Adonis-style men you get in men's fitness magazines in the same way that girls might aim to look like the impossibly gorgeous, size zero celebrities or models you get in women's magazines with their personal trainers, designer clothes and hours of dedication in the gym.
The only problem was that in practice it didn't work out like that. Anorexia robbed our son of his self-confidence, his self-esteem, his social skills and his sense of fun. He even lost his much sought-after 'six pack' as the anorexia began to eat away at the muscles in his body.
By the end of 2009, his body was emaciated due to the anorexia. He had dark rings round his eyes, his face was gaunt, his skin was dry and flaking, his body was skeletal and he was permanently cold.
Anorexia had lied to our son. But that's the nature of anorexia.
Looking at family photographs was a painful, instant reminder of what he used to look like and should look like, but didn't anymore. We had a thin waif for a son who looked like a concentration camp victim and whose mood was so volatile I was terrified every time I picked him up from school for fear of how his day had been.
For any parent, watching your child suffer with anorexia is one of the most excruciating and painful things you'll ever face. I just wished there was a magic pill you could take and - Hey Presto! - our old son would be back. I wept every single day.
Unlike a physical illness, you can't take any medication for anorexia (apart from anti-depressants). You can't have an operation and it goes away. Worse, the wonderful, level-headed, intelligent child you've spent 16 years rearing and getting to know has undergone a total transformation into a volatile stranger whose very sanity seems to have gone AWOL.
You feel frightened – an emotion that quickly changes to fear and terror as you scour the Internet for information about the illness and read the mortality statistics. Then you panic as you realise just how far away the first treatment session still is.
Anorexia also makes you feel very isolated. Okay, there are anorexia help lines you can call and a fabulous forum (see the links page for details) - and the CAMHS team. But it's difficult to talk to a 'lay person' about it; to the outside world it's such a little-known, much misunderstood and even taboo condition. I knew virtually nothing about anorexia until our son contracted it. Over the following 18 months or so I became an anorexia expert! The learning curve was ENORMOUS – and necessary.
It was around February 2010 that I discovered ADTD. Over the next 12 months this forum, and the amazing women I met through it, would become a lifeline.
I read the books they suggested and talked to them at length, often over the phone.
After feeling so isolated, it was incredible to talk to people who had gone through the same thing as us – and to see that our son's symptoms were typical of this horrible illness. But most importantly, to see that there is light at the end of the tunnel, eventually, and to learn the most proven and effective ways of heading towards that light.
In addition, he'd send me umpteen distressing texts about how terrible he was feeling and how alone... He had virtually cut himself off from his friends, retreating to his isolated corner of the common room and munching his way through salad and fruit at lunch. On really bad days, I'd go in to school and bring him home - or not send him at all.
On one particularly bad day I took him back to the GP. Our son walked out shouting "I don't know what I'm doing here!!!!" and stormed off down the road.
Evenings became a lottery where anorexia would usually win and our son would end up in a terrible emotional state. I started to be terrified that he might take his own life; in fact this was my greatest fear at the time.
I was in tears on a daily basis, though I was aware that the 'experts' say you should show a calm front to your anorexic child. Easier said than done!!!
Our son used to love eating at a certain city centre cafe. And that Saturday in February we went into town and decided to go there for lunch.
From the moment we went in, he was frantically examining the packaging for calories. Most were too high. He asked me what I thought he should have, then agitatedly picked up something totally different. The café had started making single sandwiches, so that was his natural choice.
He must have picked up and put down virtually every sandwich in the shop. We even made it to the till on two occasions only to have him rush back and change his mind.
Then he had some kind of breakdown, strode under the staircase and started to weep.
So we tried again to choose lunch without success.
In the end he strode out of the cafe with me frantically following him, chasing him through town. We did manage to get a quick bite to eat elsewhere (amazingly enough). But by then we were both totally frazzled, all thoughts of shopping totally gone.
Our son liked to 'admire' cakes (in a cafe or shop) in the way you or I might admire paintings in a gallery. Of course he would never eat any.
In supermarkets he would examine food packaging meticulously, picking stuff up, then putting it down – or removing items from my trolley. I began to hate the people that decided it was a Good Idea to list the nutritional content on food.
Another anorexic behaviour was the need for food / meal perfection. Anything less than a perfect meal could send him into a freak-out frenzy which resulted in more than one restaurant walk-out – or meals out where he'd just stare at his untouched food, miserably, or break down in tears.
In January 2010, after being on the CAMHS waiting list since November, we received a letter with an appointment for mid February, some six weeks earlier than we'd thought. The relief!
But then in early February I got a call from school. Our son's pulse was just 30, his heart was thumping in a strange way and the school nurse thought I should take him to A&E (ER).
In A&E we were fast-tracked past the usual queue and within minutes he was wired up to an ECG machine. Blood tests were also taken. He was whisked off to another local hospital with a specialist cardio unit. I was whisked off into a living nightmare.
They kept him in overnight, wired up to machines. But thankfully his pulse had stabilised a bit by the next day, enough for him to be discharged. His blood, etc were also fine - so the specialist wasn't unduly worried.
But what he did say was that the heart issue could have been exacerbated by the anorexia. So I got on the phone to CAMHS to ask if we could see them sooner.
Our CAMHS treatment started out as a multidisciplinary team comprising a psychiatrist, nursing specialist and dietitian (who we were to lose in the summer, due to cut-backs).
Naively I thought that treatment would be quick – and it could be just weeks, months at the most, before we got our 'real son' back.
Unlike physical illnesses, you can't just take a course of medication or an operation to 'cure' the anorexia. Because it is a mental illness, the treatment is very different.
I decided that keeping our son in school wasn't doing him any favours. He was spending most of his time in medical or being picked up by me. He was missing lessons and schoolwork. So we decided it was time to have a meeting with the Head. He was brilliant, promising that the school would support our son in whatever way they could.
I took our son out of school the next day and the staff were excellent, emailing work home which we would bring into school once a week for marking and to pick up more work.
Our son spent the rest of the lead up to his GCSEs like this (and went on to get excellent grades!) Since then, the school has continued to go way beyond the 'call of duty' and the wonderful school nurse has become a good friend!
Around this time I stopped taking our son food shopping because he would analyse nutritional content and take ages choosing products. He would also freak out over stuff he found in the fridge because it was impossible to ban him from the kitchen in such a small house as ours.
Throughout March we felt we were making improvements, although we were well aware that it was in the artificial environment of home, with our son out of school. We felt it was important for him to re-establish links with his friends and a good opportunity was a history conference he'd booked onto in March. It was a success, his friends had been excellent and he was very upbeat afterwards.
But then he had another meltdown - a total breakdown where I had to physically restrain him from injuring himself. He confessed to secret exercising and lying about what he'd eat in meals he'd prepared for himself. He said the so-called improvements we'd noticed were lies, even the weekend's normal behaviour was anorexia pretending to be our son. At the conference he'd been secretly exercising in the toilets. He said all the good stuff was fake, the anorexia putting on a front and being a good actor.
He said that his days are just filling in gaps between meals. Food and anorexic thoughts dominate his day and control everything. Even the nice country walks we'd had were only a means of burning off calories and 'earning' an evening meal.
This led to more fights at mealtimes. Typically he'd put down his knife and fork, walk out of the room, start banging his head and shouting in an animal voice trying to get the anorexia 'demon' out of his head. He'd swear at the top of his voice and be horrible. Then he'd return to the table, knuckle down and finish his meal, changing the subject to pleasant 'chit chat' and acting as if nothing had happened. Meanwhile you can imagine what was going on in my head!!!!!
Initially we worked on a strict eating plan comprising 3 meals and 2 snacks a day which came to around 2700 calories. This was easier said than done because our son just wasn't 'buying into' this.
By virtually force-feeding him, he was almost 'weight restored' by June 2010. It was fantastic to see our 'real son' returning physically. However, mentally he had scarcely changed at all.
It was at this point that we started to have problems with our CAMHS team. They felt our son was doing brilliantly and was almost 'cured', but I knew very differently. There began an unsettling period of 'triangulation' as the anorexia deceived the CAMHS team into believing our son was almost 'there'. So much so that they insisted we dispense with calorie counting altogether during our summer holiday in France, they were so confident our son would be able to cope.
Of course the anorexic 'demon' immediately took the side of those that would make its possession of our son even easier and we parents because the 'baddies' of the triangle.
The French holiday was the start of a massive downward spiral, weight and mood-wise.
The holiday itself was a nightmare from Day One when we had a huge row with the anorexia 'demon' on arrival at our holiday cottage. At several points, we almost came home early. It was terrible.
Back in England, our son had lost a huge amount of weight. He also refused to return to the eating plan – and CAMHS allowed him to begin to control his own eating which I instinctively knew was the wrong thing to do at this stage.
Not only did his weight plummet over the summer holidays, his mood hit rock bottom. The depression got worse, as did the resistance to eating, and it was a real uphill battle to keep things afloat. Meanwhile he continued to lose weight; something which continued for the next few months as he headed to his lowest ever weight (but not low enough to warrant hospital admission).
The summer holidays were a nightmare with regular threats of suicide. He even tried to climb out of the loft Velux window on one occasion. Looking back, this period is a king of nightmarish blur as I realised that I almost had to come to 'accept' that I might lose my son.
His psychiatrist suggested starting a course of Prozac to help lift his mood and it wasn't until September that he finally agreed to take it. Prozac takes a few weeks to work, but once it did, it levelled out his mood and proved a true crutch.
Considering he had been away from school since Feb/March, I was really nervous about him starting back in the Lower Sixth Form in September. I was worried about the social problems (he'd found it difficult to walk across the playground or even sit his GCSE exams with his peers). I was worried about school dinners and the fact he'd be eating next to nothing. How would we manage them? I didn't live locally so I couldn't go into school to monitor them.
The first week or so weren't good. He avoided his peers like the plague and spent all his time including breaks, lunchtimes and free periods in the library, swotting. Meanwhile he began a brand new phase of not sleeping due to anxiety - and I mean REALLY not sleeping; at the most a couple of hours a night. Lunches were predictably disastrous which resulted in me sending him in with packed lunches which also proved pretty disastrous as he was still resisting food unless actively encouraged by me being there physically.
After a few crises, we - with the fantastic support of the school and at the suggestion of CAMHS - made a decision. He would go into school mornings only and come home for lunch. School would send home work for the afternoons and friends would lend him their notes.
On days when he didn't sleep, he often missed school altogether. However his sheer determination to keep up with his peers, the fact he's pretty bright and the incredible support of the school has meant that he's not doing too badly at all. But the most important thing to me isn't whether he gets straight A's and gets into university; it's that he eventually recovers - fully.
In late September, something changed at CAMHS. I'm not sure what, but it's as if they suddenly 'got it and almost immediately I felt we had arrived 'at the same page'. It might have been the terrible summer, often with emergency visits to CAMHS, plus the severe weight loss. It was also 'helped along' by our son's readmission to hospital with a dangerously low pulse rate.
It was as if everything suddenly 'clicked' into place all round. CAMHS transformed and accepted that our son wasn't as far advanced in recovery as they had assumed. And our son himself had a total change of attitude.
He started back on the eating plan. He stopped cheating and throwing away food. The resistance all but stopped (although it did / does reappear now and again). More importantly, he started to put on weight and accepting the need to put on weight.
Things gradually began to improve on the social front - no more escaping to the library! At the time of writing (February 2011), he is still in school mornings only and some days he doesn't make it at all. But that's mainly because of the insomnia. These days he loves lessons and being with his friends.
Exercise has always been an issue in the fight against anorexia. From being something our son enjoyed, it became a hated compulsion as he continued to use exercise to lose weight and grow muscle.
At a CAMHS session in January 2011 he was (brutally) honest about his exercising habits (or, more correctly, compulsions). He described his 7-days-a-week exercise regime. Because of a low pulse rate and low body weight, he isn't permitted to do PE at school for the time being. We are all aware that he has a problem with compulsive exercising to compensate for this, and also to ensure he doesn't 'put on enormous amounts of weight' (as he puts it) from week to week. It's a kind of purge, almost like a sufferer of bulimia might vomit to control their weight.
The extent of his compulsive exercising was HUGE - much bigger than any of us imagined. Just when we thought he'd listed all his "100 crunches, 100 sit-ups and 100 press-ups" for any one day he'd interrupt with "I haven't finished yet!", not just once, but several times... And he listed exactly how many of which exercise he did when during any day, school days and home days.
I remember being stunned when he confessed to having exercised in the middle of his GCSE art exam practical when the invigilator was out of the room. Peers told of him asking to be excused to visit the bathroom in lessons only to spend the time running around the school grounds.
At the moment we are trying to break this cycle. But the Good News was that he was brutally honest about his exercise regime (and how much he hates doing it).
Back in November, our son asked CAMHS to come into school to talk to a selected group of his peers and a couple of members of staff (without him being present).
It was extremely successful and CAMHS were impressed with the maturity of his peers, their loyalty towards our son as a good friend and commitment to be there for him throughout his recovery.
Our son's male friends had never talked about his problem, feeling more comfortable ignoring it – and keeping him at a distance. They'd probably never even heard of eating disorders before this happened.
The girls were far more accepting of it - and willing to talk things through with him in depth. In fact at the time of writing, he has more female friends than male friends. And all his friends are really lovely, caring people.
Our son is now 18. Over the past few months, what has helped the most is my own change of attitude. From panic and fear, I managed to become much calmer and more gently supportive /encouraging.
It's not been an easy change, but it has helped enormously. Janet Treasure would describe it as the 'Dolphin Approach'.
I've also found our Recovery Contract massively helpful (introduced 13 months ago - see tab at top of page).
All of this has led to our son being able to talk to me. I mean really talk. Honestly and from the heart. Over the past few months, usually on long and regular walks in the countrywide, he's completely opened up to me about what it's been like to have anorexia and the stage he's at now.
The other day he said how much he appreciated our 'little chats' finding them 'incredibly therapeutic'.
I explained about how, for ages, I felt that I was banging my head against a brick wall. We'd talk about recovery on these walks and he would promise to do X, Y or Z, but he'd rarely keep his word.
As I watched him get thinner and thinner, and more embedded in the anorexia, these walks and talks broke my heart, especially on days when things seemed totally without hope.
But lately, things have been very different. We're going over similar stuff, but the difference is that, since the Autumn, our son has been actively committed to his recovery. We have turned a corner. These days, he follows up suggestions or discussions and we make significant progress. Far from banging my head against a brick wall, I actually feel I am doing good.
We talk about exercising (how he's making a real effort to cut it back, and managing to do this reasonably successfully)... about easing back into school full time (and examining the problems that are making it hard for him - and working out ways to overcome them)... about socialising (the importance of the fantastic friendships he's developing with his supportive friends)... about eating (What has he found difficult this week? What challenges has he made himself? Were they successful? Plus, contrasting how different eating is now compared with several months ago)... about what life will look like without the anorexia (why he really looks forward to having kids when he's older - and also why life without anorexia seems real now, whereas only a few months ago it seemed unobtainable)... and so on.
But as any parent of a recovering anorexic will know, you have to keep your eye firmly on the ball, albeit at a discreet distance. This is because the anorexia will do its utmost to fight its way back in.
The trick is to move on from a 'bad day' or 'bad incident' and start again with a fresh frame of mind, usually having a chat about why things might have gone wrong and how this could be prevented happening next time round.
There is light at the end of the tunnel - and, believe me, when you catch a glimpse of it the light is BRILLIANT!


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Residents Protest Lowrise Height Limits
People line up to express their concerns at a community meeting.
A 2010 change to Seattle's Lowrise 3 (LR3) zoning designation hasn't sparked much public interest until now, when the economy has picked up and developers are taking advantage of increased height limits for new multi-family buildings. Residents in neighborhoods with LR3 zones, especially those living in adjacent single family homes, are concerned about 40-plus-foot tall apartments that alter neighborhood character, create shade and reduce privacy, and increase on-street parking demand. In order to collect citizen comments, on Tuesday night the Seattle Department of Planning and Development (DPD) hosted a community meeting that brought out both enthusiastic supporters and hostile opposition.
The issues stems from four years ago, when the City Council enacted amendments to its lowrise codes that allows apartment buildings in LR3 zones to be up to 40 feet tall within areas designated as 'urban centers', 'urban villages', and 'station area overlay districts'. These terms refer to neighborhoods where growth is being directed because of existing population densities, retail activity, and existing or future light-rail stations. One exception is portions of the property within 50 feet of a single-family zoned lot, where the height limit is 30 feet. The designers and developers of such apartment buildings can increase the height limit, however, if they follow other code provisions: if the bottom level of the building will be partly below ground level, the building can extend its height by 4 feet to compensate but is limited to four stories above ground; if the previous is used, the ridgeline of pitched roofs may extend another 5 feet above the height limit; if at least half the roof is covered with vegetation the height may be an additional 2 feet; elevator penthouses may extend above the height limit up to 16 feet.
Developers can build more dwelling units and earn a larger return on investment when they build taller, so there is an incentive for them to take advantage of the code's exceptions. But since the code changes were enacted, not much lowrise apartment development occurred until the last couple years. City Council President Sally Clark asked DPD to review the changes last October when neighbors began taking notice.
The type of auto-oriented multi-family construction Seattle wants to avoid. (buildlcc.com)
The first public meeting with DPD staff occurred Tuesday night, as residents from across the city converged on Lowell Elementary School in Capitol Hill and packed a cafeteria to the brim. The event started off bitterly almost immediately, as planners Mike Pedowski and Geoffrey Wentland presented background information amid heckles and insults. But the introduction was succinct and explained that lowrise zones cover 12 percent of the city's land area and growth is being directed into them to preserve the lower density neighborhoods. The 2010 zoning update was intended to end the rapid spread of bland auto-court townhouses and increase housing diversity with cottages, rowhouses, and apartments. The update also made standards more flexible, requires better design review, and encourages compatibility with the surrounding neighborhood character. But staff acknowledged that the resulting developments have not been in line with the code's intentions, and DPD will submit revisions to the City Council by March after gathering public comments.
A question-and-testimony session followed, during which many people lined up at a microphone to express utter rage towards the planning department. An early theme was criticism towards the lack of official notice for the meeting and that two weeks for follow-up public comment is too short. Many single family homeowners were dismayed at the tall structures being built next door and the resultant loss of backyard privacy and sunlight, arguing that architects are insensitive to neighboring properties. Others suggested that the movement for affordable housing is misdirected, because the older and smaller apartments buildings that are being replaced have naturally lower rents. Some people noted that the goal of increasing density and reducing parking is foolhardy amid the region's upcoming transit cuts. And several suggested specifically removing the 4 foot increase for sub-grade floors, and to thunderous applause, one man advocated a complete stop on permits for LR3 projects until the code is revised.
The type of housing allowed in LR3 zones currently. (Seattle DPD)
After hearing complaints about the city's focus towards new residents, not current ones, a Capitol Hill resident asked the audience to recall the neighborhood's inclusive spirit and argued growth is good for increasing social diversity. On that point, a supporter of increased density pointed out that the vast majority of the audience was made up of older white homeowners and did not represent the views of young renters. Others were supportive of growth but said it needed to be concentrated on arterial streets and areas with existing multi-family housing, such as Broadway and the Pike-Pine corridor, not near single-family homes.
The meeting became unhinged several times when people in the audience tried to talk over the person who had the floor or yell obscenities, prompting city staff to step in and remind attendees to remain civil. It also veered off topic with complaints about 'aPodments' and the city's unwillingness to impose infrastructure exactions on developers.
Many legitimate concerns were raised by residents, and the meeting revealed a much broader concern for increasing density citywide. Plopping large buildings among small homes can disrupt the urban fabric, so DPD staff should look at downzoning some areas to protect historical districts while upzoning in areas better suited for high-density, like on major streets with retail presence and transit connections. It would also be simpler to remove code exceptions and be more straightforward on what heights the city wants; height bonuses in commercial zones are usually rewarded on the basis of public amenities like open space or public restrooms, so it's unclear how residential lowrise bonuses based only on private design benefit the public. As some attendees mentioned, the public also needs more data on recent development in LR3 zones before a decision is made.
The fact remains that Seattle is an urban city experiencing rapid growth, and because there is no virgin land left, many new residents will need to be accommodated in taller buildings. The long democratic process that Seattle is known for, along with its active citizenry, will likely lead to a delay of the Council vote and plenty of time to flesh out common ground and compromises. Yet if the density-opposition continues being uncivil in public discourse, it is not imaginable they will win many favors.
Video of the meeting can be viewed here.
This entry was posted in Demographics, Government, Housing, Land Use, Policy, Residential and tagged affordability, affordable housing, Capitol Hill, City Council, codes, density, department of planning and development, growth, height, homeowners, housing, lowrise, lowrise 3, LR3, multi-family, neighborhoods, Seattle, single family, zoning. Bookmark the permalink.
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redpajama_set_name: RedPajamaCommonCrawl


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

In the Standard Model~(SM), the flavor-changing neutral current~(FCNC) interactions, which are absent at tree level, are induced by quantum corrections and highly suppressed due to the Glashow-Iliopoulos-Maiani~(GIM) mechanism~\cite{Glashow:1970gm}. Possible new physics~(NP) beyond the SM can manifest itself by altering the expected rates of these FCNC-induced processes. Thus, the study of FCNC interactions plays an important role in testing the SM and probing NP effects.

For the top quark, the FCNC-induced decays $t \to q Z$~(where $q$ denotes either a $c$- or a $u$-flavored quark) are exceedingly rare within the SM, with branching ratios of order of $10^{-10}$~\cite{Eilam:1990zc,AguilarSaavedra:2004wm}. However, these processes could be significantly enhanced by some potential NP models~\cite{Beneke:2000hk}, like supersymmetry, multi-Higgs doublet models and SM extensions with exotic quarks. Any positive signal of these processes at the LHC would therefore imply NP beyond the SM. These top-quark anomalous couplings could also be probed by studying the top-quark production at high-energy colliders~\cite{Beneke:2000hk}. So far, the direct experimental bounds on these anomalous couplings are not so restrictive and the world's best limit is set by the D0 collaboration, with a branching ratio $\mathcal B(t \to qZ)<3.2\%$ at $95\%$ C.L.~\cite{Abazov:2011qf}. The constraint will be improved significantly by the large top-quark sample to be available at the CERN Large Hadron Collider~(LHC), which is expected to produce 80 million top pairs and 34 million single tops annually. For example, with about $10~{\rm fb}^{-1}$ data, the discovery potential of $\mathcal B(t\to qZ)$ at both the ATLAS~\cite{Carvalho:2007yi} and the CMS~\cite{Benucci:2008zz} collaboration is reported to be of the order of $10^{-4}$.

However, if the top-quark anomalous couplings really existed, the low-energy processes with loops involving the top quark may also be affected, and could therefore provide helpful information for a direct search at high-energy colliders~\cite{FCNC-top,arXiv:0802.1413,Han:1995pk,Yuan:2010vk,arXiv:1105.0364}. In this respect, the rare B- and K-meson decays, such as $B_{s,d}\to \mu^+\mu^-$, $B\to X_s \nu \bar\nu$, $B\to K^{(*)}\nu \bar\nu$, $K^+\to \pi^+ \nu \bar\nu$ and $K_L\to \pi^0 \nu \bar\nu$, are particularly interesting. They are all short-distance dominated FCNC processes induced by the Z-penguin and box diagrams, and the calculation of their branching ratios is theoretically rather clean. They are therefore known to be good probes of flavour dynamics within the SM and beyond~\cite{Buras:2009if}. As the anomalous couplings $tqZ$ can enter the Z-penguin diagram, constraints on these couplings can be obtained by studying deviations from the SM predictions for these decays. Although the current experimental upper bounds on these rare decay processes are still weak~\cite{Asner:2010qj}, the measurements will be improved at the LHCb, the future super-B factories, the NA62 at CERN, the KOTO at J-PARC, etc. Thus, closer correlations between the FCNC transition $t\to qZ$ and the rare B- and K-meson decays are expected in the near future.

In our previous works~\cite{Yuan:2010vk,arXiv:1105.0364}, we have studied the anomalous $tq\gamma$ coupling effects in inclusive and exclusive radiative B-meson decays. In this paper, we shall continue to investigate the anomalous $tqZ$ coupling effects on the rare B- and K-meson decays. With the up-to-date experimental data on these decays, we shall first derive constraints on these anomalous couplings, and then discuss the implications for the rare $t \to qZ$ decays at the LHC.

Our paper is organized as follows. In Sec.~\ref{sec-tqZ}, we introduce the effective Lagrangian describing the anomalous interactions $tqZ$, and set the convention used throughout the paper. In Sec.~\ref{sec-theo}, we first recapitulate the basic theoretical formulae for the relevant B- and K-meson decays, and then discuss the anomalous $tqZ$ coupling effects in these decays; the rare decay $t\to qZ$ mediated by the anomalous coupling is also presented in this section. Detailed numerical results and discussions are presented in Sec.~\ref{sec-analysis}. Our conclusions are made in Sec.~\ref{sec-conclusion}. The relevant input parameters are collected in the appendix.

\section{Effective Lagrangian for anomalous $tqZ$ couplings}
\label{sec-tqZ}

In most extensions of the SM, the new degrees of freedom that modify the ultraviolet behavior of the underlying theory appear only at a scale $\Lambda$ which is much higher than the electroweak scale $v=246~{\rm GeV}$. As long as we are only interested in processes occurring much below the scale $\Lambda$, we can always integrate out these new degrees of freedom and describe the NP effects in terms of a few higher-dimensional local operators, which are built out of the SM fields and suppressed by inverse powers of the NP scale $\Lambda$~\cite{Appelquist:1974tg,Grzadkowski:2010es,Buchmuller:1985jz,AguilarSaavedra:2008zc}.

The above effective field theory approach is a powerful theoretical framework for describing the FCNC processes induced by some unknown NP models. Specific to our case, the anomalous coupling $tqZ$ mediating the FCNC transition $t\to qZ$ can be described by the effective Lagrangian~\cite{Grzadkowski:2010es,Buchmuller:1985jz}
\begin{align}
\mathcal L^{\rm eff}_{tqZ}=\mathcal L^{\rm SM}+\sum_i \frac{C_i^{(6)}}{\Lambda^2}\mathcal O_i^{(6)}+\dots\,,
\end{align}
where operators with dimension larger than 6 are neglected. The explicit form of the dimension-6 operators $\mathcal O_i^{(6)}$, which are consistent with the SM gauge symmetries, can be found in Ref.~\cite{Grzadkowski:2010es,Buchmuller:1985jz,AguilarSaavedra:2008zc}. These operators can contribute to the $tqZ$ vertex, resulting an equivalent description by the effective Lagrangian~\cite{AguilarSaavedra:2008zc}
\begin{align}\label{Lagrangian-tqZ}
\mathcal L_{tqZ}&=\frac{g}{2\cos\theta_W}\,\bar q\gamma^\mu(X_{qt}^LP_L+X_{qt}^RP_R)t Z_\mu\nonumber\\
&\quad +\frac{g}{2\cos\theta_W}\,\bar q\frac{i\sigma^{\mu\nu}p_\nu}{M_Z}(\kappa_{qt}^LP_L+\kappa_{qt}^RP_R)\,tZ_\mu+{\rm h.c.}\,,
\end{align}
with $P_{L,R}\equiv (1\mp\gamma^5)/2$. The dimensionless couplings $X_{qt}^{L,R}$ and $\kappa_{qt}^{L,R}$ depend on the unknown Wilson coefficients $C_i^{(6)}$, and are in general complex. The effective Lagrangian given by Eq.~(\ref{Lagrangian-tqZ}) is commonly employed in phenomenological analyses related to top-quark physics~\cite{Beneke:2000hk}.

\section{Theoretical formalism for rare B- and K-meson decays}
\label{sec-theo}

In this section, we shall first recapitulate the basic theoretical formulae for the relevant rare B- and K-meson decays, and then discuss the anomalous $tqZ$ coupling effects on these decays; the rare decay $t\to qZ$ mediated by the anomalous coupling is also presented in this section.

\subsection{$B_{s,d}\to \mu^+\mu^-$}
\label{sec-bsmumu}

The rare decays $B_{s,d}\to \mu^+\mu^-$ are dominated by the Z-penguin and box diagrams involving top-quark exchanges, and the resulting effective Hamiltonian can be written as~\cite{hep-ph/9901288,hep-ph/9901278}
\begin{align}\label{Hamiltonian-b2smumu}
\mathcal H_{\rm eff}&=-\frac{G_{\rm F}}{\sqrt 2}\frac{\alpha}{2\pi\sin^2\theta_W}V_{tb}^*V_{tq}Y(x_t)(\bar b q)_{V-A}(\bar \mu \mu)_{V-A}+{\rm h.c.}\,,
\end{align}
where $(\bar f f^{\prime})_{V-A}\equiv \bar f \gamma^\mu(1-\gamma^5)f^{\prime}$, $q=s(d)$ for $B_s(B_d)$-meson decay, and the gauge-invariant function $Y(x_t)$ is a linear combination of the $V-A$ components of Z-penguin and box diagrams~\cite{hep-ph/9901288,hep-ph/9901278,Inami:1980fz}, with its explicit expression given in Appendix~B. The branching ratio for $B_q \to \mu^+\mu^-$ is then given by~\cite{hep-ph/9901288,hep-ph/9901278}
\begin{align}
\mathcal B(B_q\to \mu^+\mu^-)&=\frac{G_{\rm F}^2}{\pi}\Bigl(\frac{\alpha}{4\pi\sin^2\theta_W}\Bigr)^2 |V_{tb}^*V_{tq}|^2 f_{B_q}^2 m_\mu^2 m_{B_q} \tau_{B_q} \sqrt {1-\frac{4m_\mu^2}{m^2_{B_s}}}\, |Y(x_t)|^2\,,
\end{align}
where $f_{B_q}$ is the $B_q$-meson decay constant. Due to the helicity suppression factor $m_\mu^2$, the branching ratios for these decays are predicted to be very small.

\subsection{Exclusive and inclusive $b\to s\nu\bar\nu$ decays}
\label{sec-bsnunu}

The rare decays $B\to X_s\nu\bar\nu$, $B\to K \nu\bar\nu$ and $B\to K^* \nu\bar\nu$ are all induced by the quark-level $b\to s \nu\bar\nu$ transition, and provide a very good test of modified Z-penguin contributions~\cite{Altmannshofer:2009ma,Bartsch:2009qp,Colangelo:1996ay}. The effective weak Hamiltonian governing the transition $b\to s\nu\bar\nu$ can be written as~\cite{hep-ph/9901288,hep-ph/9901278}
\begin{align} \label{Hamiltonian-b2snunu}
\mathcal H_{\rm eff}=\frac{G_F}{\sqrt 2}\frac{\alpha}{2\pi\sin^2\theta_W}V_{tb}V_{ts}^* X(x_t)(\bar s b)_{V-A}(\bar\nu \nu)_{V-A}+{\rm h.c.}\,,
\end{align}
where the gauge-invariant function $X(x_t)$ is also a linear combination of the $V-A$ components of Z-penguin and box diagrams~\cite{hep-ph/9901288,hep-ph/9901278,Inami:1980fz}. For convenience, we give its explicit expression in Appendix~B.

\subsubsection{$B \to X_s \nu\bar\nu$}

The inclusive decay $B\to X_s\nu\bar\nu$ can be evaluated using heavy-quark expansion and operator product expansion, and is theoretically very clean. Adopting the convention advocated by Ref.~\cite{Altmannshofer:2009ma}, the dineutrino invariant mass distribution can be written as
\begin{align}\label{eq:B2Xsnunubar}
\frac{d\Gamma (B \to X_s\nu\bar\nu)}{ds_b}=&\frac{G_F^2\alpha^2m_b^5}{128\pi^5\sin^4\theta_W}
|V_{tb}V_{ts}^*|^2\kappa(0)|X(x_t)|^2 \nonumber\\
&\qquad \qquad \times\sqrt{\lambda(1,\hat m_s^2,s_b)}\, \left[3s_b\left(1+\hat m_s^2-s_b\right)+\lambda(1,\hat m_s^2,s_b)\right]\,,
\end{align}
with $\lambda(x,y,z)=x^2+y^2+z^2-2(xy+yz+zx)$, $\hat m_i=m_i/m_b$, and $m_b$ denotes the $b$-quark mass in the 1S scheme~\cite{Hoang:1998ng,Bauer:2004ve}. The factor $\kappa(0)=0.83$ contains the virtual and bremsstrahlung QCD corrections to the $b\to s\nu\bar\nu$ matrix element~\cite{Buchalla:1993bv,Grossman:1995gt}. The total branching ratio is then obtained by integrating Eq.~(\ref{eq:B2Xsnunubar}) over the kinematically allowed region $0\le s_b=q^2/m_b^2 \le (1-\hat m_s)^2$, where $q^2$ is the invariant mass of the neutrino-antineutrino pair. In addition, we have also incuded the additional $\mathcal O(\Lambda^2/m_b^2)$ corrections~\cite{Grossman:1995gt,Falk:1995me}, with the HQET parameters $\lambda_1=-0.27\pm 0.04$ and $\lambda_2=0.12\pm 0.01$~\cite{Bauer:2004ve}.

\subsubsection{$B \to K \nu\bar\nu$}

For the exclusive decay $B\to K\nu\bar\nu$, the dineutrino invariant mass distribution can be written as~\cite{Altmannshofer:2009ma,Colangelo:1996ay}
\begin{align}
\frac{d\Gamma(B\to K\nu\bar\nu)}{ds_B}=\frac{G_F^2\alpha^2m_B^5}{256\pi^5\sin^4\theta_W}\, |V_{tb}V_{ts}^*|^2\, \lambda^{3/2}(s_B,\widetilde m_K^2,1)\,[f_+^K(s_B)]^2\, |X(x_t)|^2,
\end{align}
where $\widetilde{m}_K=m_K/m_B$, and $s_B=q^2/m_B^2$ is constrained within the physical region $0 \le s_B  \le (1 -\widetilde{m}_K )^2 \approx 0.82$; $f_+^K(s_B)$ is the $B\to K$ transition form factor, the presence of which results in large theoretical uncertainty and makes the exclusive mode not as clean as the inclusive one. However, significant progress has recently been made by considering simultaneously also the decay mode $B\to K \mu^+ \mu^-$~\cite{Bartsch:2009qp}.

\subsubsection{$B\to K^*\nu\bar\nu$}

For the decay $B\to K^*\nu\bar\nu$, additional information about the polarization of the $K^*$ meson can be extracted from the angular distribution of the $K^*$ decay products. In terms of the three transversity amplitudes~\cite{Altmannshofer:2009ma}
\begin{align}
A_\perp(s_B)&=-2N \sqrt 2 \lambda^{1/2}(1,\widetilde m _{K^*}^2,s_B) X(x_t) \frac{V(s_B)}{(1+\widetilde m_{K^*})}\,, \nonumber\\
A_\parallel(s_B)&=2N \sqrt 2 (1+\widetilde m_{k^*}) X(x_t) A_1(s_B)\,, \nonumber\\
A_0(s_B)&=\frac{N X(x_t)}{\widetilde m_{K^*}\sqrt {s_B}}\left[(1-\widetilde m_{K^*}^2-s_B) (1+\widetilde m_{K^*}) A_1(s_B)-\lambda(1,\widetilde m_{K^*}^2,s_B) \frac{A_2(s_B)}{1+\widetilde m_{K^*}}\right]\,,
\end{align}
with an overall factor
\begin{align}
N = V_{tb}V_{ts}^* \left[\frac{G_F^2 \alpha^2 m_B^3}{3 \cdot 2^{10} \pi^5\sin^4\theta_W} s_B \lambda^{1/2}(1,\widetilde m_{K^*}^2,s_B)\right]^{1/2},
\end{align}
the dineutrino invariant mass spectrum of the decay $B\to K^*\nu\bar\nu$ can be written as~\cite{Altmannshofer:2009ma}
\begin{align}
\frac{d\Gamma}{ds_B}=3m_B^2(|A_\perp(s_B)|^2+|A_\parallel(s_B)|^2+|A_0(s_B)|^2)\,,
\end{align}
where the normalized invariant mass $s_B$ ranges from 0 to the kinematical endpoint $(1-\widetilde{m}_{K^*})^2\approx0.69$. Here the main theoretical uncertainty is due to the normalization and the shape of the three $B\to K^*$ transition form factors $V(q^2)$, $A_1(q^2)$ and $A_2(q^2)$.

\subsection{Rare K-meson decays}
\label{sec-sdnunu}

It is known that, among the many rare B- and K-meson decays, the modes $K^+\to \pi^+\nu\bar\nu$ and $K_L\to \pi^0\nu\bar\nu$ are the theoretically cleanest, and therefore play an important role in the search for the underlying mechanism of flavour mixing and CP violation~\cite{hep-ph/0405132,arXiv:1107.6001}. Especially, these decays are very sensitive to NP contributions in Z-penguin diagrams.

\subsubsection{$K^+\to \pi^+\nu\bar\nu$}

The effective weak Hamiltonian relevant for $K^+\to \pi^+ \nu\bar\nu$ can be written as~\cite{hep-ph/9901288,hep-ph/9308272}
\begin{align}
\mathcal H_{\rm eff}=\frac{G_{\rm F}}{\sqrt 2}\frac{\alpha}{2 \pi \sin^2\theta_W} \sum_{l=e,\mu,\tau} \Bigl(V_{cs}^*V_{cd} X_{NL}^l + V_{ts}^*V_{td} X(x_t)\Bigl)(\bar s d)_{V-A}(\bar\nu_l\nu_l)_{V-A}\,,
\end{align}
where the dependence on the charged lepton mass results from the box diagram. With the help of isospin symmetry, the branching ratio for $K^+\to\pi^+\nu\bar\nu$ is then given as~\cite{hep-ph/0405132,arXiv:1107.6001}
\begin{align}
\mathcal B(K^+\to\pi^+\nu\bar\nu)&=\kappa_+ (1+\Delta_{\rm EM}) \left|\frac{\lambda_t} {\lambda^5} X(x_t) +\frac{\lambda_c} {\lambda}(P_c+\delta P_{c,u})\right|^2,
\end{align}
where $\lambda_q=V_{qs}^*V_{qd}$~(with $q=t,c$), is the product of the Cabibbo-Kobayashi-Maskawa~(CKM) matrix elements~\cite{CKM}, and $\Delta_{\rm EM}=-0.003$ denotes the electromagnetic correction~\cite{arXiv:0705.2025}. The overall factor $\kappa_+$, with $\kappa_+=(5.173 \pm 0.025) \times (\lambda/0.225)^8 \times 10^{-11}$, summarizes the isospin-breaking corrections in relating $K^+\to\pi^+\nu\bar\nu$ to $K^+\to\pi^0 \ell^+ \nu_{\ell}$~\cite{arXiv:0705.2025,428572}. The dimension-six charm operator contribution $P_c$ is given as~\cite{hep-ph/0508165}
\begin{align}
P_c=\frac{1}{\lambda^4} \left(\frac{2}{3} X_{NL}^e + \frac{1}{3}X_{NL}^\tau\right)=0.38\pm 0.04\,,
\end{align}
with error dominated by the charm-quark mass, and $\delta P_{c,u}=0.04 \pm 0.02$ contains the small long-distance~(up quark) and dimension-eight charm operator contributions~\cite{hep-ph/0503107}.

\subsubsection{$K_L\to \pi^0\nu\bar\nu$}

The rare decay $K_L\to\pi^0\nu\bar\nu$, proceeding almost entirely through direct CP violation~\cite{BNL-42227}, is completely dominated by short-distance loop diagrams with top-quark exchanges, and the charm contribution can be fully neglected~\cite{hep-ph/0405132}. Thus, the effective weak Hamiltonian for $K_L\to\pi^0\nu\bar\nu$ can be written as~\cite{hep-ph/9901288,hep-ph/9308272}
\begin{align}
\mathcal H_{\rm eff}=\frac{G_F}{\sqrt 2}\frac{\alpha}{2\pi\sin^2\theta_W}V_{ts}^*V_{td}X(x_t)(\bar s d)_{V-A}(\bar \nu \nu)_{V-A}+{\rm h.c.}\,.
\end{align}
Analogous to the case of $K^+\to\pi^+\nu\bar\nu$, the branching ratio of $K_L\to\pi^0\nu\bar\nu$ can be given as~\cite{hep-ph/0405132,hep-ph/9308272}
\begin{align}
\mathcal B(K_L\to\pi^0\nu\bar\nu)= \kappa_L \left(\frac{{\rm Im}[\lambda_t X(x_t)]} {\lambda^5}\right)^2(1-\delta_\epsilon)\,,
\end{align}
where the overall factor $\kappa_L=(2.231 \pm 0.013)\times (\lambda/0.225)^8 \times 10^{-10}$ encodes the hadronic matrix element related to $K_{\ell3}$ data~\cite{arXiv:0705.2025,428572}, and the parameter $\delta_\epsilon$ denotes the contribution of indirect CP violation to $K_L\to\pi^0\nu\bar\nu$, which is highly suppressed by the small $K^0-\bar{K}^0$ mixing parameter $|\epsilon|$~\cite{hep-ph/9308272}.
For the case of general complex function $X(x_t)$, the expression of $\delta_\epsilon$ can be written as
\begin{align}
\delta_\epsilon =-\sqrt 2 |\epsilon| \frac{{\rm Re}[\lambda_t X(x_t)]+{\rm Re}(\lambda_c) \lambda^4 P_c}{{\rm Im}[\lambda_t X(x_t)]}\,,
\end{align}
which is consistent with the SM result for real $X(x_t)$ given explicitly in Ref.~\cite{hep-ph/9308272}.

\subsection{Anomalous $tqZ$ coupling effects}
\label{sec-tqzeffect}

The anomalous $tqZ$ interactions given by Eq.~(\ref{Lagrangian-tqZ}) affect the rare B- and K-meson decays through the Z-penguin diagrams. As an illustration, in the following we shall consider the effect of anomalous $tqZ$ couplings on the $b\to s\nu\bar\nu$ transition.

\subsubsection{$b\to s\nu\bar\nu$ transition induced by anomalous $tcZ$ coupling}

For the transition $b\to s\nu\bar\nu$, the relevant Feynman diagrams both within the SM~(the first three ones) and with the anomalous $tcZ$ coupling~(the last one) are depicted in Fig.~\ref{fig:b2snunu}.

\begin{figure}[t]
\centering
\subfigure[]{\includegraphics[width=4cm]{b2snunu1.eps}\label{b2snunu1}}
\subfigure[]{\includegraphics[width=4cm]{b2snunu2.eps}\label{b2snunu2}}
\subfigure[]{\includegraphics[width=4cm]{b2snunu3.eps}\label{b2snunu3}}
\subfigure[]{\includegraphics[width=4cm]{b2snunuNP.eps}\label{b2snunuNP}}
\caption{\small Relevant Feynman diagrams for $b\to s\nu\bar\nu$ transition in the unitary gauge, where the first three ones are the one-loop SM contributions, while the last one denotes the contribution induced by the anomalous $tcZ$ coupling.}
\label{fig:b2snunu}
\end{figure}

It should be noted that there exist some other Feynman diagrams induced by the anomalous $tqZ$ couplings, such as the one with the c- replaced by the u-quark line in Fig.~\ref{b2snunuNP}, and the one with exchanges of the t- and c-quark lines in Fig.~\ref{b2snunuNP}. However, analogous to the arguments made in our previous works~\cite{Yuan:2010vk,arXiv:1105.0364}, contributions from these two Feynman diagrams are negligible compared to the one from Fig.~\ref{b2snunuNP}, based on the observation that the associated CKM factors $|V_{cs}|>|V_{us}|$, $|V_{cb}|>|V_{ub}|$, and $|V_{tb}V_{cs}^*| \gg |V_{cb}V_{ts}^*|$. Thus, for the $b\to s\nu\bar\nu$ transition, we need only consider Fig.~\ref{b2snunuNP} with only one anomalous coupling $tcZ$.

It is also observed that the large CKM factor $|V_{tb}V_{cs}^*|\approx 1$ associated with Fig.~\ref{b2snunuNP}, compared to the SM case $|V_{tb}V_{ts}^*|\sim \mathcal O(\lambda^2)$, make the $b\to s\nu\bar\nu$ transition to be very sensitive to the anomalous couplings $tcZ$. It is therefore expected that constraint on the coupling $tcZ$ could be obtained from the precisely measured rare B-meson decays induced by the quark-level $b\to s\nu\bar\nu$ transition.

The calculation of Fig.~\ref{b2snunuNP} could be most conveniently performed in the unitary gauge, where the pseudo-Goldstone components of the SM Higgs doublet are absent. It is noted that, in the unitary gauge, the contribution from the $q^{\alpha}q^{\beta}/m_{W}^2$ part of the $W$-boson propagator is ultraviolet-divergent. Similar to the treatment adopted by Grzadkowski and Misiak~\cite{arXiv:0802.1413}, the divergence could be absorbed by some counterterms served by other dimension-6 operators~\cite{Grzadkowski:2010es,Buchmuller:1985jz}, the ${\rm \overline{MS}}$-renormalized Wilson coefficients of which are assumed to be negligible compared to the ones considered here. Moreover, it is found that the contribution from tensor currents in the effective Lagrangian Eq.~(\ref{Lagrangian-tqZ}) is zero.

Normalized to the effective Hamiltonian Eq.~(\ref{Hamiltonian-b2snunu}), the contribution of anomalous $tcZ$ coupling to the $b\to s\nu\bar\nu$ transition would result in the deviation
\begin{align}
X(x_t) \to X'=X(x_t)+C_{0,b\to s}^{\rm NP}\,,
\end{align}
where the matching coefficient reads
\begin{align}\label{X-b2s}
C_{0,\,b\to s}^{\rm NP}(\mu)=&-\frac{1}{8} \frac{V_{cs}^*}{V_{ts}^*}\biggl[X_{ct}^L \left(-x_t\log\frac{m_W^2}{\mu^2} + \frac{3}{2} + x_t - x_t\log x_t\right) \nonumber\\[0.2cm]
&\qquad\quad + X_{ct}^R \frac{\sqrt{x_{c}x_t}}{2} \left(\log\frac{m_W^2}{\mu^2}-\frac{1}{2}+\frac{x_t-4}{x_t-1}\log x_t\right) \biggr]\,,
\end{align}
with $x_q=\bar m_q^2/m_W^2$. The presence of logarithms $\ln(m_{W}^2/\mu^2)$ results from the ${\rm \overline{MS}}$ prescription for the ultraviolet-divergence. Neglecting the light charm-quark mass $m_{c}$, as done in the SM, the effect of right-handed current $X_{ct}^R$ could be safely neglected.

\subsubsection{$b(s)\to d\nu\bar\nu$ transition induced by anomalous $tuZ$ coupling}

\begin{table}[t]
\begin{center}
\caption{\label{tableCKM} \small The counting of CKM factors for $b\to sZ^*$, $b\to dZ^*$ and $s\to dZ^*$ transitions both within the SM and with the anomalous $tqZ$ couplings. Within the SM, the CKM factors in the Z-penguin and box diagrams are the same.}
\vspace{0.2cm}
\doublerulesep 0.8pt \tabcolsep 0.15in
\begin{tabular}{r l l l}
\hline\hline
& $b\to sZ^*$\,-transition & $b\to dZ^*$\,-transition & $s\to dZ^*$\,-transition \\
\hline
top-sector & $|V_{tb}V_{ts}^*|\sim \mathcal O(\lambda^2)$ & $|V_{tb}V_{td}^*|\sim \mathcal O(\lambda^3)$ & $|V_{ts}V_{td}^*|\sim \mathcal O(\lambda^5)$ \\
$tcZ$-coupling & $|V_{tb}V_{cs}^*|\sim 1$ & $|V_{tb}V_{cd}^*|\sim \mathcal O(\lambda)$ & $|V_{ts}V_{cd}^*|\sim \mathcal O(\lambda^3)$ \\
$tuZ$-coupling & $|V_{tb}V_{us}^*|\sim O(\lambda)$ & $|V_{tb}V_{ud}^*|\sim 1$ & $|V_{ts}V_{ud}^*|\sim \mathcal O(\lambda^2)$ \\
\hline \hline
\end{tabular}
\end{center}
\end{table}

Similar to the case of $b\to s\nu\bar\nu$ transition, it is easily seen that, based on the counting of CKM factors listed in Table~\ref{tableCKM}, the transitions $b\to d\nu\bar\nu$ and $s\to d\nu\bar\nu$ are both dominated by the anomalous $tuZ$ coupling, with the corresponding Feynman diagram obtained from Fig.~\ref{b2snunuNP}, with the c- replaced by the u-quark line and changes of the external quark flavours.

Normalized to the corresponding effective weak Hamiltonian, the contribution of anomalous $tuZ$ coupling to the transitions $b\to d\nu\bar\nu$ and $s\to d\nu\bar\nu$ can be written, respectively, as
\begin{align}
C_{0,\,b\to d}^{\rm NP}(\mu)=& -\frac{1}{8} \frac{V_{ud}^*}{V_{td}^*}\biggl[X_{ut}^L \left(-x_t\log\frac{m_W^2}{\mu^2} + \frac{3}{2} + x_t - x_t\log x_t\right) \nonumber \\[0.2cm]
&\qquad\quad + X_{ut}^R \frac{\sqrt{x_{u}x_t}}{2} \left(\log\frac{m_W^2}{\mu^2}-\frac{1}{2}+\frac{x_t-4}{x_t-1}\log x_t\right) \biggr]\,, \label{C0b2d} \\
C_{0,\,s\to d}^{\rm NP}(\mu)=& C_{0,\,b\to d}^{\rm NP}(\mu)\,, \label{C0s2d}
\end{align}
where the right-handed coupling $X_{ut}^R$ is now more severely suppressed by the light up-quark mass via the factor $\sqrt x_u=\bar m_u/m_W$.

\subsection{Rare $t\to qZ$ decays mediated by anomalous $tqZ$ coupling}
\label{sec-tqzdecays}

Within the SM, the top quark has only one dominant decay channel $t\to bW$, and the branching ratio of $t\to qZ$ decay can be therefore defined as~\cite{Beneke:2000hk,Han:1995pk}
\begin{align}
\mathcal B(t\to qZ)=\frac{\Gamma(t\to qZ)}{\Gamma(t\to bW)}\,,
\end{align}
where the leading-order~(LO) decay width of $t\to b W$ is given explicitly as~\cite{Li:1990qf}
\begin{align}
\Gamma_0(t \to bW) = \frac{G_{\rm F} m_t^3}{8\sqrt{2}\,\pi}|V_{tb}|^2\beta_W^4(3-2\beta_W^2)\,,
\end{align}
with $\beta_W=(1-m_W^2/m_t^2)^{1/2}$, being the velocity of the $W$-boson in the top-quark rest frame. As the SM prediction for $\Gamma(t \to q Z)$ is exceedingly small~\cite{Eilam:1990zc}, we need only consider the decay $t\to qZ$ mediated by the anomalous $tqZ$ interaction. Adopting the convention specified in the effective Lagrangian Eq.~(\ref{Lagrangian-tqZ}), the LO decay width of $t\to qZ$ can be written as~\cite{Beneke:2000hk,Han:1995pk}
\begin{align}\label{eq:t2qZ_LO}
\Gamma_0(t\to qZ) &= \frac{G_{\rm F} m_t^3}{8\sqrt{2}\,\pi}\frac{|X_{ct}^L|^2+|X_{ct}^R|^2}{2}\beta_Z^4(3-2\beta_Z^2)\,,
\end{align}
where $\beta_Z=(1-m_Z^2/m_t^2)^{1/2}$, is the velocity of the $Z$-boson in the top-quark rest frame. Since the NLO QCD corrections to $\Gamma(t\to qZ)$ are found to be negligible~\cite{Zhang:2010bm,Zhang:2008yn,Drobnak:2010by,Drobnak:2010wh}, here we shall only consider the LO result given by Eq.~(\ref{eq:t2qZ_LO}).

\section{Numerical results and discussions}
\label{sec-analysis}

With the theoretical framework presented in previous sections and the input parameters collected in Appendix~A, we shall present our numerical results and discussions in this section.

\subsection{The SM predictions and the experimental data}

\begin{table}[t]
\begin{center}
\caption{\label{table-theoexp} \small The SM predictions for the rare B- and K-meson decays, with the corresponding experimental data given in the third column. Here we give only the world's best existing limits on each decay mode with the reference given in the last column.}
\vspace{0.2cm}
\doublerulesep 0.8pt \tabcolsep 0.20in
\begin{tabular}{l l l r}
\hline\hline
Observables & SM prediction & Experimental data & Ref. \\
\hline
$\mathcal B (B_s \to \mu^+\mu^-)$ & $(3.65^{+0.32}_{-0.35}) \times 10^{-9 }$ & $<0.90 \times 10^{-8}$~(90\% C.L.) & \cite{CMSLHCb}\\
& & $<1.08 \times 10^{-8}$~(95\% C.L.) & \cite{CMSLHCb}\\
$\mathcal B (B_d \to \mu^+\mu^-)$ & $(1.08^{+0.13}_{-0.14}) \times 10^{-10}$ & $<2.6 \times
10^{-9}$~(90\% C.L.) & \cite{:2011zq}\\
& & $<3.2 \times 10^{-9}$~(95\% C.L.) & \cite{:2011zq}\\
$\mathcal B (B \to X_s \nu \bar\nu)$ & $(3.13^{+0.14}_{-0.20}) \times 10^{-5}$ & $<6.4 \times 10^{-4}$~(90\% C.L.) & \cite{hep-ex/0010022}\\
$\mathcal B (B^+ \to K^+ \nu\bar\nu)$ & $(5.29^{+0.76}_{-0.74}) \times 10^{-6}$ & $<1.3 \times
10^{-5}$~(90\% C.L.) & \cite{arXiv:1009.1529}\\
$\mathcal B (B^0 \to K^0 \nu\bar\nu)$ & $(4.91^{+0.70}_{-0.69}) \times 10^{-6}$ & $<5.6 \times
10^{-5}$~(90\% C.L.) & \cite{arXiv:1009.1529}\\
$\mathcal B (B^+ \to K^{*+} \nu\bar\nu)$ & $(1.11^{+0.25}_{-0.23}) \times 10^{-5}$ & $<8.0 \times
10^{-5}$~(90\% C.L.) & \cite{arXiv:0808.1338}\\
$\mathcal B (B^0 \to K^{*0} \nu\bar\nu)$ & $(1.03^{+0.23}_{-0.21}) \times 10^{-5}$ & $<12 \times
10^{-5}$~(90\% C.L.) & \cite{arXiv:0808.1338}\\
\hline
$\mathcal B (K^+\to\pi^+\nu\bar\nu)$ & $(8.52^{+0.70}_{-0.92}) \times 10^{-11}$ & $(1.73^{+1.15}_{-1.05}) \times 10^{-10}$ & \cite{arXiv:0808.2459}\\
$\mathcal B (K_L\to\pi^0\nu\bar\nu)$ & $(2.67^{+0.29}_{-0.36}) \times 10^{-11}$ & $<2.6 \times 10^{-8}$~(90\% C.L.) & \cite{arXiv:0911.4789}\\
\hline
$\mathcal B (t \to qZ )$ & &$<3.2\%$~(95\% C.L.) & \cite{Abazov:2011qf}\\
\hline \hline
\end{tabular}
\end{center}
\end{table}

Within the SM, our predictions for the rare B- and K-meson decays are listed in Table~\ref{table-theoexp}, where the theoretical uncertainties are obtained by varying each input parameter within its respective range and adding the individual uncertainty in quadrature. It can be seen that, with the up-to-date input parameters, the theoretical uncertainties for most of these decays are less than $13\%$ except for $B\to K^*\nu\bar\nu$ decay, which is still about $23\%$ mainly due to the $B\to K^*$ transition form factors.

However, compared to the precise theoretical predictions, the current experimental limits on these decays are still rather weak. At present, only seven events of the decay $K^+\to \pi^+\nu\bar\nu$ have been observed~\cite{arXiv:0808.2459}. For $B_s\to \mu^+ \mu^-$ decay, it is interesting to note that a possible first signal has been recently announced by the CDF collaboration, although with a
low significance~\cite{arXiv:1107.2304}. This result has, unfortunately, not been confirmed by the searches both at the CMS~\cite{CMSLHCb,arXiv:1107.5834} and at the LHCb collaboration~\cite{CMSLHCb,:2011zq,arXiv:1110.2411}. Because of the missing multiple neutrinos in the final state, it is quite difficult to measure the exclusive $b\to s \nu \bar\nu$ decays, and the present experimental limits are only available from the two $e^+ e^-$ B-factories BaBar~\cite{arXiv:1009.1529,arXiv:0808.1338} and Belle~\cite{hep-ex/0507034,arXiv:0707.0138}, both of which have used the hadronic tag technique to reconstruct the accompanying B meson.

To discuss the effects of anomalous $tqZ$ interactions on these rare B- and K-meson decays, we shall use the SM predictions with $2\sigma$ error bars and the experimental upper bounds at 90\% C.L., as listed in  Table~\ref{table-theoexp}. For the decay $K^+\to\pi^+\nu\bar\nu$, on the other hand, the experimental data with $1\sigma$ error bar will be used due to its large uncertainty. In additional, for the exclusive $b\to s\nu\bar\nu$ decays, since the experimental upper bounds on the charged decay modes, $\mathcal B(B^+\to K^+\nu\bar\nu)$ and $\mathcal B(B^+\to K^{*+}\nu\bar\nu)$, are more stringent than their neutral counterparts, $\mathcal B(B^0\to K^0\nu\bar\nu)$ and $\mathcal B(B^0\to K^{*0}\nu\bar\nu)$, we shall only consider constraints from the former in the following discussions.

For the rare top-quark FCNC decay $t\to qZ$, the current world's best limit, $\mathcal B(t\to qZ)<3.2\%$ at $95\%$ C.L., is set by the D0 collaboration~\cite{Abazov:2011qf}. On the other hand, using likelihood-based analyses, the expected branching ratio sensitivity for a $5\sigma$ discovery potential at the LHC could reach $4.4~(1.4)\times 10^{-4}$ with an integrated luminosity of $L=10~(100)~{\rm fb}^{-1}$ at the ATLAS experiment~\cite{Carvalho:2007yi}.

\subsection{$B_s\to\mu^+\mu^-$ and $b\to s\nu\bar\nu$ decays with anomalous coupling $X_{ct}^L$}

\begin{figure}[htbp]
\centering
 \subfigure{\includegraphics [width=7cm]{XctL1.eps}}\hspace{0.5in}
 \subfigure{\includegraphics [width=7cm]{tcZCorrelation1.eps}}
 \subfigure{\includegraphics [width=7cm]{XctL3.eps}}\hspace{0.5in}
 \subfigure{\includegraphics [width=7cm]{tcZCorrelation3.eps}}
 \subfigure{\includegraphics [width=7cm]{XctL7.eps}}\hspace{0.5in}
 \subfigure{\includegraphics [width=7cm]{tcZCorrelation7.eps}}
 \subfigure{\includegraphics [width=7cm]{XctL8.eps}}\hspace{0.5in}
 \subfigure{\includegraphics [width=7cm]{tcZCorrelation8.eps}}
\caption{\small Upper bounds on the anomalous coupling $|X_{ct}^L|$ as a function of $\theta _{ct}^L$, and correlations between the rare B-meson and $t\to c Z$ decays. The allowed regions by rare B-meson decays are shown in dark and green points, with the latter obtained in the case of real coupling $X_{ct}^L$. The horizontal solid and dashed lines denote the D0 bound~\cite{Abazov:2011qf} and the ATLAS $5\sigma$ discovery potential at $L=10~{\rm fb}^{-1}$~\cite{Carvalho:2007yi}, respectively. The vertical solid line is the current experimental bounds on rare B-meson decays.}
\label{figure-Xct}
\end{figure}

For the decay $B_s\to \mu^+\mu^-$, the effect of anomalous $tcZ$ coupling results in a deviation of the function $Y(x)$ from its SM result, and the branching ratio can be formally written as
\begin{align}\label{bs2mumu_numerics}
\mathcal B(B_s\to \mu^+\mu^-) &\sim \Bigl|Y(x_t) + C_{0,b\to s}^{\rm NP}\Bigr|^2 \nonumber \\
&\sim \Bigl|0.96 + 16.91|X_{ct}^L|\,e^{i(\theta_{ct}^L+\beta_s)} - 0.04|X_{ct}^R|\,e^{i(\theta_{ct}^R+\beta_s)}\Bigr|^2\,,
\end{align}
where $\beta_s=-\arg(-\frac{V_{cs}V_{cb}^*}{V_{ts}V_{tb}^*})\simeq 1.04^\circ$, is the phase associated with the CKM matrix element $V_{ts}$. From Eq.~(\ref{bs2mumu_numerics}), one can see that, compared to the left-handed coupling, $X_{ct}^L=|X_{ct}^L|e^{i\theta_{ct}^L}$, the effect of right-handed coupling, $X_{ct}^R=|X_{ct}^R|e^{i\theta_{ct}^R}$, on the branching ratio is quite small, consistent with the observation made in Sec.~\ref{sec-theo}. Thus, we shall neglect the right-handed coupling $X_{ct}^R$ in the following discussions.

For a generic complex coupling $X_{ct}^L$, the branching ratio $\mathcal B(B_s\to \mu^+\mu^-)$ depends also on the phase $\theta_{ct}^L$. It can be seen from Eq.~(\ref{bs2mumu_numerics}) that, for a given value $|X_{ct}^L|$, the NP contribution is constructive to the SM one in the region $\theta_{ct}^L\approx -\beta_s$, whereas in the region $\theta_{ct}^L\approx 180^\circ-\beta_s$, the interference between them becomes destructive. This can be clearly seen from Fig.~\ref{figure-Xct}, where the upper bounds on the anomalous coupling $|X_{ct}^L|$ as a function of $\theta _{ct}^L$, as well as the correlations between rare B-meson and $t\to c Z$ decays are shown.

For the decays $B\to X_s\nu\bar\nu$, $B\to K \nu\bar\nu$ and $B\to K^* \nu\bar\nu$, which are all induced by the quark-level transition $b\to s\nu\bar\nu$, the effect of anomalous $tcZ$ coupling on the branching ratios is to replace the SM function $X(x_t)$ with $X(x_t) + C_{0,b\to s}^{\rm NP}$, and hence we have
\begin{align}\label{b2smumu_numerics}
\mathcal B(B\to K^{(*)} \nu\bar\nu, X_s\nu\bar\nu)
&\sim \Bigl|X(x_t) + C_{0,b\to s}^{\rm NP}\Bigr|^2 \nonumber \\
&\sim \Bigl|1.48 + 16.91|X_{ct}^L|\,e^{i(\theta_{ct}^L+\beta_s)} - 0.04|X_{ct}^R|\,e^{i(\theta_{ct}^R+\beta_s)}\Bigr|^2\,,
\end{align}
where the NP contribution is the same as discussed in $B_s\to \mu^+\mu^-$. However, due to the less stringent experimental bounds and the large theoretical uncertainties, the current constraints on the coupling $|X_{ct}^L|$ from these rare B-meson decays are still rather loose.

\begin{table}[t]
\begin{center}
\caption{\label{table-Xct} \small Bounds on the magnitude $|X_{ct}^L|$ from the purely leptonic $B_s\to\mu^+\mu^-$ decay, with some specific values of the phase $\theta_{ct}^L$. In the last row, we also give the corresponding predicted upper limit on $\mathcal B(t\to cZ)$.}
\vspace{0.2cm}
\doublerulesep 0.8pt \tabcolsep 0.20in
\begin{tabular}{l c c}
\hline\hline
& $\theta_{ct}^L=0^\circ$ & $\theta_{ct}^L=180^\circ$ \\
\hline
$\mathcal B(B_s\to \mu^+\mu^-)$ & $<0.043$ & $<0.16$ \\
D0 bound & $<0.26$ & $<0.26$ \\
\hline
$\mathcal B(t\to cZ)$ & $<8.4 \times 10^{-4}$ & $<0.011$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

As is shown in Fig.~\ref{figure-Xct}, the potentially large top-quark anomalous coupling effect is reflected in the stringent bound on its magnitude $|X_{ct}^L|$, which is currently dominated by the purely leptonic $B_s\to\mu^+\mu^-$ decay. From the numerical results given in Table~\ref{table-Xct}, one can see that, with the bound from $\mathcal B(B_s\to\mu^+\mu^-)$ taken into account, the predicted upper limit of $\mathcal B (t\to cZ)$ is lower than the D0 bound~\cite{Abazov:2011qf}. This is also evident from the correlation plot between $\mathcal B(B_s\to \mu^+\mu^-)$ and $\mathcal B(t\to cZ)$ depicted in Fig.~\ref{figure-Xct}. In particular, the predicted $\mathcal B(t\to cZ)$ is about of the same order as the $5\sigma$ discovery potential of ATLAS with an integrated luminosity of $L=10~{\rm fb}^{-1}$~\cite{Carvalho:2007yi}.

\subsection{$B_d\to\mu^+\mu^-$ and $s\to d\nu\bar\nu$ decays with anomalous coupling $X_{ut}^L$}

For the decay $B_d\to\mu^+\mu^-$, the anomalous $tuZ$ coupling contributes to the branching ratio as
\begin{align}
\mathcal B (B_d\to\mu^+\mu^-) &\sim \Bigl|Y(x_t) + C_{0,b\to d}^{\rm NP}\Bigr|^2 \nonumber \\
&\sim \Bigl|0.96 - 80.08|X_{ut}^L|\,e^{i(\theta_{ut}^L-\beta)}
+ 0.00067|X_{ut}^R|\,e^{i(\theta_{ut}^R-\beta)}\Bigr|^2\,,
\end{align}
where $\beta=\arg(-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*})\simeq 21.78^\circ$, is the phase associated with the CKM matrix element $V_{td}$. The suppression of right-handed coupling is more evident and can be therefore neglected, since its contribution is accompanied by a much smaller factor $\sqrt x_u=m_u/m_W$, see Eq.~(\ref{C0b2d}). Here the interference between the SM and the NP contributions is destructive in the region $\theta_{ut}^L \approx \beta$, whereas constructive in the region $\theta_{ut}^L\approx \beta-180^\circ$. This is clearly shown in Fig.~\ref{figure-Xut}, where the upper bounds on the anomalous coupling $|X_{ut}^L|$ as a function of $\theta _{ut}^L$, as well as the correlations between rare B- and K-meson and $t\to u Z$ decays are depicted.

\begin{figure}[t]
\centering
 \subfigure{\includegraphics[width=7cm]{XutL1.eps}}\hspace{0.5in}
 \subfigure{\includegraphics[width=7cm]{tuZCorrelation1.eps}}
 \subfigure{\includegraphics[width=7cm]{XutL3.eps}}\hspace{0.5in}
 \subfigure{\includegraphics[width=7cm]{tuZCorrelation3.eps}}
 \subfigure{\includegraphics[width=7cm]{XutL2.eps}}\hspace{0.5in}
 \subfigure{\includegraphics[width=7cm]{tuZCorrelation2.eps}}
\caption{\small Upper bounds on the anomalous coupling $|X_{ut}^L|$ as a function of $\theta _{ut}^L$, and correlations between the rare B- and K-meson and the $t\to uZ$ decays. The other captions are the same as in Fig.~\ref{figure-Xct}.}
\label{figure-Xut}
\end{figure}

Although the current experimental bound on $\mathcal B(B_d\to\mu^+\mu^-)$ is still rather weak, it is interesting to note that, with its constraint on the coupling $X_{ut}^L$ taken into account, the predicted upper limit for $\mathcal B(t\to uZ)$ is lower than the D0 bound~\cite{Abazov:2011qf}. However, the limit is still comparable with the $5\sigma$ discovery potential of ATLAS with an integrated luminosity of $L=10~{\rm fb}^{-1}$~\cite{Carvalho:2007yi}, as shown in Fig.~\ref{figure-Xut}.

Due to the charm contribution, the effect of anomalous $tuZ$ coupling on the decay $K^+\to\pi^+\nu\bar\nu$ becomes somewhat complicated, and we have numerically
\begin{align}
\mathcal B(K^+\to\pi^+\nu\bar\nu) & \sim \Bigl|X(x_t) + C_{0,\bar{s}\to \bar{d}}^{\rm NP} + \frac{\lambda_c}{\lambda_t}\lambda^4(P_c+\delta P_{c,u})\Bigr|^2 \nonumber\\
& \sim \Bigl|1.48-80.08|X_{ut}^L|\,e^{-i(\theta_{ut}^L-\beta)}+0.68\,e^{i(\beta+\beta_s)}\Bigr|^2\,,
\end{align}
where $C_{0, \bar{s}\to \bar{d}}^{\rm NP}$ is the CP conjugation of $C_{0, s\to d}^{\rm NP}$ given by Eq.~(\ref{C0s2d}). Compared to the case of $B_d\to\mu^+\mu^-$, the constructive and destructive regions between the SM and the NP contributions are shifted a little bit by the charm sector, which can be seen by comparing the first two plots of the left column in Fig.~\ref{figure-Xut}.

Since the decay $K_L\to\pi^0\nu\bar\nu$ proceeds almost entirely through direct CP violation, only the imaginary part of the SM and the NP contributions affect its branching ratio. The final numerical result reads
\begin{align}
\mathcal B(K_L\to\pi^0\nu\bar\nu) &\sim \Bigl({\rm Im}\Bigl[\lambda_t X(x_t) + \lambda_t\,C_{0,\bar{s}\to \bar{d}}^{\rm NP}\Bigr]\Bigr)^2 \nonumber \\
&\sim \Bigl(1.48\sin (\beta+\beta_s)-80.08|X_{ut}^L|\sin(\theta_{ut}^L+\beta_s)\Bigr)^2.
\end{align}
In this case, the effect of anomalous $tuZ$ coupling depends strongly on its phase $\theta_{ut}^L$. In the regions $\theta_{ut}^L \approx -\beta_s$ or $\theta_{ut}^L \approx 180^\circ -\beta_s$, its effect is quite small and could even be zero. On the other hand, the largest effect comes from the region $\theta_{ut}^L \approx -90^\circ -\beta_s$, where the interference between the SM and the almost purely imaginary NP contributions are constructive. Consequently, the experimental bounds on the magnitude $|X_{ut}^L|$ depend crucially on the phase $\theta_{ut}^L$, as is shown in the last plot of the left column in Fig.~\ref{figure-Xut}.

From Fig.~\ref{figure-Xut}, we can see that, among all the three decay modes discussed here, the $K^+\to\pi^+\nu\bar\nu$ provides the most stringent constraint on the coupling $X_{ut}^L$. With the allowed values for $X_{ut}^L$, the predicted $\mathcal B(t\to uZ)$ is much lower than the D0 bound~\cite{Abazov:2011qf}. Furthermore, since the experimental measurement of $\mathcal B(K^+\to\pi^+\nu\bar\nu)$ is double-bounded, there are actually two solutions for the magnitude $|X_{ut}^L|$ in the destructive region $\theta_{ut}^L \approx \beta \approx 21.78^\circ$, with the larger one corresponding to the sign-flipped solution for the function $X(x_t)$.

Finally, for illustration, we give in Table~\ref{table-Xut} some numerical results for $|X_{ut}^L|$ with some specific values of the phase $\theta_{ut}^L$, where the scenarios S1 and S2 correspond to the same sign and sign-flipped solutions for the function $X(x_t)$, respectively. We can see that the predicted $\mathcal B(t\to uZ)$ is much lower than the D0 bound~\cite{Abazov:2011qf}, but is of the same order as the $5\sigma$ discovery potential of ATLAS with an integrated luminosity of $L=10~{\rm fb}^{-1}$~\cite{Carvalho:2007yi}.

\begin{table}[t]
\begin{center}
\caption{\label{table-Xut} \small Bounds on the magnitude $|X_{ut}^L|$ from the rare $K^+\to\pi^+\nu\bar\nu$ decay, with some specific values of the phase $\theta_{ut}^L$. The two solutions S1 and S2 correspond to the cases in which the sign of the function $X(x_t)$ is not flipped and flipped, respectively. In the last row, we also give the corresponding predicted upper limit on $\mathcal B(t\to cZ)$.}
\vspace{0.2cm}
\doublerulesep 0.8pt \tabcolsep 0.20in
\begin{tabular}{l c c c c c c }
\hline\hline
& $\theta_{ut}^L=0^\circ$~S1 & $\theta_{ut}^L=0^\circ$~S2 & $\theta_{ut}^L=180^\circ$ \\
\hline
$\mathcal B(K^+\to\pi^+\nu\bar\nu)$ & $<0.0048$ & $[0.047, 0.080]$ & $<0.029$ \\
D0 bound & $<0.26$ & $<0.26$ & $<0.26$ \\
\hline
$\mathcal B(t\to u Z)$ & $<1.0\times 10^{-5}$ & $[0.0010, 0.0030]$ & $<3.9\times 10^{-4}$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\section{Conclusions}
\label{sec-conclusion}

In this paper, we have studied the effects of anomalous $tqZ$ couplings on the rare B- and K-meson decays. Among the four operators in the effective Lagrangian given by Eq.~(\ref{Lagrangian-tqZ}), it is found that only the anomalous operator $\bar q \gamma^\mu P_{L} t Z_\mu$ could give a potentially large contribution to these rare decays. With the current experimental data on these decays, bounds on the coupling $X_{qt}^L$ of this operator are then derived. Our main conclusions are summarized as follows.

For the exclusive and inclusive $b\to s\nu\bar\nu$ decays, as well as the purely leptonic $B_s\to\mu^+\mu^-$ decays, we find that the main contribution comes from the anomalous $tcZ$ coupling, solely based on the countering of the associated CKM factors. On the other hand, the anomalous $tuZ$ coupling is found to dominate in the decays $B_d\to\mu^+\mu^-$, $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$. Currently, the most stringent bounds on $tcZ$ and $tuZ$ couplings come from the decays $B_s\to\mu^+\mu^-$ and $K^+\to\pi^+\nu\bar\nu$, respectively.

For generical complex couplings $tcZ$ and $tuZ$, it is found that the interference between the SM and the NP contributions is constructive in the region $\theta_{ct}^L \approx -\beta_s$ for $B_s\to\mu^+\mu^-$ and $\theta_{ut}^L \approx \beta$ for $K^+\to\pi^+\nu\bar\nu$, respectively. Thus, the most stringent constraints on the strength of these anomalous couplings come from these regions.

From the correlations of the rare B- and K-meson decay with the rare $t\to qZ$ decays, we find that, with the constraints from the former taken into account, the predicted upper limit of $\mathcal B(t\to qZ)$ is lower than the D0 bound, but is still of the same order as the $5\sigma$ discovery potential of ATLAS with an integrated luminosity of $10~{\rm fb}^{-1}$.

Although the current experimental upper bounds on these rare decay processes are still rather weak, the measurements will be much improved at the LHCb, the future super-B factories, the NA62 at CERN, the KOTO at J-PARC, etc. Thus, closer correlations between the $t\to qZ$ and the rare B- and K-meson decays are expected in the near future, which will be very
helpful for the searches of  the anomalous top-quark FCNC decays at the LHC.

\section*{Acknowledgements}

The work was supported in part by the National Natural Science Foundation under contract Nos.~11075059, 11005032, 10735080 and 11047165. X.~Q. Li was also supported in part by MEC (Spain) under Grant FPA2007-60323 and by the Spanish Consolider Ingenio 2010 Programme CPAN (CSD2007-00042).

\begin{appendix}

\section*{Appendix A: Theoretical input parameters}
\label{app:input}

In this appendix, we collect all the relevant input parameters used in the numerical analysis of rare B- and K-meson decays, as well as the rare $t\to q Z$ decay.

\subsubsection*{The basic SM parameters}

First, we need some basic SM parameters, which are all taken from the Particle Data Group~\cite{Nakamura:2010zzi}
\begin{align}
& \alpha_s(m_Z)=0.1184\pm 0.0007, \quad \alpha(m_Z)=1/127.925, \quad G_F=1.16637\times 10^{-5}~{\rm GeV}^{-2}, \nonumber \\
& \sin^2\theta_W=0.23146, \quad m_W=80.399~{\rm GeV}, \quad m_Z=91.1876~{\rm GeV}, \nonumber \\
& m_{B^+}=5279.17~{\rm MeV}, \quad m_{B^0}=5279.50~{\rm MeV}, \quad m_{B_s}=5366.3~{\rm MeV},
\nonumber \\
& \tau_{B^+}=1.638~{\rm ps}, \quad \tau_{B^0}=1.525~{\rm ps}, \quad \tau_{B_s}=1.472~{\rm ps}.
\end{align}
We use two-loop running for $\alpha_s$ throughout this paper.

\subsubsection*{The CKM matrix elements}

For the CKM matrix elements, we adopt the Wolfenstein parametrization~\cite{Wolfenstein:1983yz} and choose the four parameters $A$, $\lambda$, $\rho$ and $\eta$ as fitted by the CKMfitter group~\cite{arXiv:1106.4041,Charles:2004jd}
\begin{equation}
A=0.816^{+0.011}_{-0.022}\,, \quad \lambda=0.22518^{+0.00036}_{-0.00077}\,, \quad \overline{\rho}=0.144^{+0.028}_{-0.019}\,, \quad \overline{\eta}=0.342^{+0.015}_{-0.014}\,,
\end{equation}
with $\overline{\rho}=\rho\,(1-\frac{\lambda^2}{2})$ and $\bar{\eta}=\eta\,(1-\frac{\lambda^2}{2})$.

\subsubsection*{The quark masses}

The quark masses given in different schemes are collected below
\begin{align}
 m_t^{\rm pole}&=173.2\pm 0.9~{\rm GeV}~\cite{arXiv:1107.5255}, \qquad m_b^{\rm 1S}=4.68\pm 0.03~{\rm GeV}~\cite{Bauer:2004ve}, \nonumber\\
 m_b(m_b)&=4.164\pm 0.023~{\rm GeV}~\cite{McNeile:2010ji}, \qquad m_c(m_c)=1.273\pm 0.006~{\rm GeV}~\cite{McNeile:2010ji}, \nonumber\\
 m_s(2~{\rm GeV})&=92.2\pm 1.3\,{\rm MeV}~\cite{McNeile:2010ji,Davies:2009ih}, \qquad m_u(2~{\rm GeV})=2.01\pm 0.14~{\rm MeV}~\cite{McNeile:2010ji,Davies:2009ih}.
\end{align}
To get the corresponding running quark masses at different scales, we use the NLO $\overline {\rm MS}$-on-shell conversion and running formulae collected, for example, in Ref.~\cite{Chetyrkin:2000yt}.

\subsubsection*{The nonperturbative hadronic parameters}

For $B_{s,d}\to \mu^+\mu^-$ decays, we need the B-meson decay constants, which are taken from~\cite{arXiv:0910.2928}
\begin{align}
f_{B_s}=238.8\pm 9.5~{\rm MeV} ,\qquad f_B=192.8\pm 9.9~{\rm MeV}.
\end{align}

For the $B\to K^{(*)}$ form factors appearing in $B\to K^{(*)}\nu\bar\nu$ decays, we adopt results obtained with the light-cone sum rule approach~\cite{hep-ph/0412079,hep-ph/0406232}
\begin{align}
V(q^2)&=\frac{r_1}{1-q^2/m_{R}^2}+\frac{r_2}{1-q^2/m_{\rm fit}^2}\,, &\;& \hspace{-0.7cm} {\rm with}\; r_1=0.923\,, r_2=-0.511\,, m_R=5.42\,, m_{\rm fit}^2=49.40\,,
\nonumber\\
A_1(q^2)&=\frac{r_2}{1-q^2/m_{\rm fit}^2}\,, && {\rm with}\; r_2=0.290\,, m_{\rm fit}^2=40.38\,, \nonumber\\
A_2(q^2)&=\frac{r_1}{1-q^2/m_{\rm fit}^2}+\frac{r_2}{(1-q^2/m_{\rm fit}^2)^2}\,, &\;& {\rm with}\; r_1=-0.084\,, r_2=0.342\,, m_{\rm fit}^2=52.00\,,
\nonumber\\
f_+^K(q^2)&=\frac{r_1}{1-q^2/m_{\rm fit}^2}+\frac{r_2}{(1-q^2/m_{\rm fit}^2)^2}\,, &\;& {\rm with}\; r_1=0.162\,, r_2=0.173\,, m_{\rm fit}^2=29.27\,,
\end{align}
which are valid in the full physical region, and the uncertainty is, to be conservative, assigned with the one at $q^2=0$, with $\delta V=\pm 0.033$, $\delta A_1=\pm 0.028$, $\delta A_2=\pm 0.027$, and $\delta f_+^K=\pm 0.041$.

\section*{Appendix B: The Inami-Lim functions X(x) and Y(x)}

The gauge-invariant functions $X(x)$ and $Y(x)$ appearing in rare B- and K-meson decays are given as~\cite{hep-ph/9901288,hep-ph/9901278,Inami:1980fz}
\begin{align}
X(x)=C(x)-4B(x)\,, \qquad Y(x)&=C(x)-B(x)\,,
\end{align}
where the basic Inami-Lim functions $C(x)$ and $B(x)$ correspond to the Z-penguin and the box diagram contribution, respectively. For convenience, their explicit expressions up to the NLO are given below~\cite{hep-ph/9901288,hep-ph/9901278,Inami:1980fz}
\begin{align}
Y(x)&=Y_0(x)+\frac{\alpha_s}{4\pi}Y_1(x),\nonumber\\[0.2cm]
Y_0(x)&=\frac{x}{8}\left[\frac{4-x}{1-x}+\frac{3x}{(1-x)^2}\ln x\right],\nonumber\\[0.2cm]
Y_1(x)&=\frac{10x+10x^2+4x^3}{3(1-x)^2}-\frac{2x-8x^2-x^3-x^4}{(1-x)^3}\ln x + \frac{2x-14x^2+x^3-x^4}{2(1-x)^3}\ln^2 x \nonumber\\
&\quad+\frac{2x+x^3}{(1-x)^2} L_2(1-x) + 8 x \frac{\partial Y_0(x)} {\partial x} \ln x_\mu\,,
\end{align}
and
\begin{align}
X(x)&=X_0(x)+\frac{\alpha_s}{4\pi}X_1(x),\nonumber\\[0.2cm]
X_0(x)&=\frac{x}{8}\left[-\frac{2+x}{1-x}+\frac{3x-6}{(1-x)^2}\ln x\right],\nonumber\\[0.2cm]
X_1(x)&=-\frac{29x-x^2-4x^3}{3(1-x)^2}-\frac{x+9x^2-x^3-x^4}{(1-x)^3}\ln x \nonumber\\
&\quad + \frac{8x+4x^2+x^3-x^4}{2(1-x)^3}\ln^2 x -\frac{4x-x^3}{(1-x)^2} L_2(1-x)+8 x \frac{\partial X_0(x)} {\partial x} \ln x_\mu\,,
\end{align}
with $x_\mu=\mu^2/m_W^2$, and $L_2(1-x)=\int_1^xdt \frac{\ln t}{1-t}$.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


"Policy, Regulations and Economics to Support Total Resource Recovery"
The oil & gas industry in New Mexico is currently facing two major challenges associated with water management that threaten to limit future oil and gas development in the state: acquiring fresh water for drilling and development of new wells; and management of produced water. These challenges are especially critical in southeastern New Mexico, which is arid, has limited fresh water resources, and where produced water disposal by deep well injection is becoming constrained by dwindling geologic disposal capacity and induced seismicity concerns. To help address these issues, New Mexico recently entered into a Memorandum of Understanding (MOU) with the U.S. Environmental Protection Agency (EPA) to clarify the existing regulatory and permitting frameworks related to the way produced water from oil and gas extraction can be re-used, recycled, and renewed for other purposes.
* "New Mexico is currently the third largest oil producer in the United States and that oil is accompanied by even larger quantities of water. Clarifying the state and federal regulatory frameworks associated with its recycling and reuse is of the utmost importance," New Mexico Energy, Minerals and Natural Resources Cabinet Secretary Ken McQueen.
* "Reuse of this water in appropriate applications has the potential to relieve the growing demand on our ground and surface water sources. For that reason alone, this effort makes absolute sense," New Mexico State Engineer Tom Blaine.
* "If there are better, viable uses of this water, it is important to ensure the framework is adequate to both facilitate reuse and protect public health and the environment – this MOU targets these objectives," New Mexico Environment Department Cabinet Secretary Butch Tongate.
The conference is intended for oil and gas, water treatment and management, and economic development professionals. The meeting will help facilitate greater collaboration among stakeholders to synthesize and improve New Mexico and federal regulatory and environmental frameworks; and to foster economically viable opportunities to enhance fresh water conservation, produced water resource recovery, and produced water beneficial use.
The conference will be held in Santa Fe, November 15-16, 2018.
* Administrators from New Mexico, the EPA, and the Department of the Interior discussing produced water issues and trends.
* Private equity, legal, and state infrastructure experts on the requirements, logistics, regional economic development opportunities, and viable cost structures to industry and developers.
The New Mexico Produced Water Conference Planning Committee are working hard and excited about our Keynote Speakers and Sessions. Watch for Further Details!
A block of rooms is reserved at the Hotel Santa Fe, Hacienda & Spa at a special room rate of $119 per night. Please call 1-877-259-3409 and mention New Mexico Produced Water Conference.
The room block and rate is reserved 3 days prior and 3 days after the conference. The room rate includes waived nightly resort fees, complimentary wifi parking. This room block is honored until October 16, 2018. All reservations after October 16, 2018 will be accepted on a space & rate available basis.
The 2018 New Mexico Produced Water Conference is a perfect opportunity to exchange information, update professionals, scientists and regulators on projects, and collaborate with your colleagues on new ideas. We are seeking abstracts for posters from both professionals and students for an evening mixer poster session on Thursday, November 15, 2018.
Posters pertaining to a broad range of Produced Water issues are welcome including treatment, desalination, transportation/storage, economics, environmental issues, alternative or beneficial uses, regulations and management..
For poster approval, please submit your electronic abstract by Monday, October 29, 2018 to: christinekimberling13@gmail.com.
Submittal should be less than 200 words in a MS Word format. They should include title, name of author (s), organization and e-mail address of the primary contact person.


The name of Fowler became so closely tied to the notion of clear and correct writing that the second edition (1965) was published as Fowler's Modern English Usage, even though its eponymous author had died in 1933. His presence continues to hover over the work as it approaches the century mark (Fowler's Modern English Usage, 2004). The content has been almost completely rewritten, but it has never gone out of print.
Similarly, H. M. Robert's Pocket Manual of Rules of Order for Deliberative Assemblies was first published in 1876 and is now in its 11th (highly revised) edition. When a work is in its third, fourth, or (in the case of Robert's Rules of Order) 11th edition, there may not be much left that was actually written by the person who penned the first edition. How should these works be cited? Should we credit the dead hand of the original author or those who carry on the franchise?
The answer follows from one of our basic principles of citation: "Cite what you see." Whose name is on the cover and/or title page? Unless another role is specified (e.g., editor, compiler), that person—dead or alive—is the author. Fowler and Robert can rest in peace while their successors carry on.
Burchfield, R. W. (2004). Fowler's modern English usage (3rd ed. rev.). Oxford, England: Oxford University Press.
Fowler, H. W. (1926). A dictionary of modern English usage. Oxford, England: Oxford University Press.
Fowler, H. W., & Gowers, E. (Ed.). (1965). Fowler's modern English usage (2nd ed.). Oxford, England: Oxford University Press.
Robert, H. M. (1876). Pocket manual of rules of order for deliberative assemblies. Chicago, IL: Griggs.
Robert, H. M., III, Honemann, D. H., & Balch, T. J. (2011). Robert's rules of order newly revised (11th ed.). Philadelphia, PA: Perseus Books.
*E.g., "To shrink with horror from ending [a sentence] with a preposition is no more than foolish superstition; but there are often particular reasons for not choosing that alternative" (Fowler, 1926, p. 635).


"Help! My iPhone X suddenly said that there is a problem detected with the TrueDepth camera and the Face ID has been disabled. I tried to reboot the iPhone, reconfigure Face ID, but none of them worked. Please tell me why it happens and how to fix it. Thanks!"
Some iPhone X/XS/XR users reported that they suddenly received an error notice saying that "Face ID has been disabled" because of something wrong with the TrueDepth camera and they are redirected to Face ID & Passcode settings, where Apple suggests bringing their phone to Apple to fix the problem. So, what does this notice mean? It means that a hardware or software failure in the TrueDepth camera system is detected so the system will be disabled to prevent accidental iPhone unlocks.
No matter how it happens, you don't need to be worried about the safety of your iPhone because you still have the passcode to lock your iPhone screen. But still, you want the Face ID on your iPhone X/XS/XR to work properly, so if you are experiencing the same, follow the steps below to fix Face ID that has been disabled and take your iPhone X/XS/XR back to normal.
Press and hold the Side button. Then the hard reset should be done in several seconds.
It may be the problem of Face ID. You can turn Face ID off and reset it to fix the problem.
Turn Off "Require Attention for Face ID" and "Attention Aware Features".
Toggle off all the items in "Use Face ID For", for example, phone unlock, iTunes & App Store.
Return to Settings > Face ID & Passcode then select "Reset Face ID".
Some users found that after turning off Face ID for phone unlock. iTunes & App Store, etc., they are unable to reconfigure the Face ID because the option is grayed out. If the problem exists after rebooting your iPhone and resetting Face ID, try this one. Just go to General > Reset and select Reset All Settings, which will reset all system preferences on your iPhone X/XS/XR, including Face ID settings.
If you want to know if there is any hardware problem in the TrueDepth camera system, you can check it by taking a selfie in portrait.
Open the Camera app on your iPhone X/XS/XR, choose Portrait Mode, select the Front Camera to take a selfie. If the front camera on your iPhone is not working, there may be a hardware issue and you should contact Apple for help.
Check if there is iOS update for your iPhone by going to Settings > General > Software Update. If there is, get the update and see if it can fix the disabled Face ID.
Also, you can use iTunes to restore your iPhone from the previous backup. If you don't have a backup, make sure you have backed up your iPhone with iTunes before the restore. Check how to restore iPhone without iTunes.
If you have done all the above to try to fix the problem but no one works, it indicates that your iPhone has a hardware problem. In this case, bring your device back to Apple and ask for help. The TrueDepth camera system is very sophisticated, you can't fix it on your own.
All the above are the solutions to "Face ID has been disabled" issue on iPhone X/XS/XR. Does this passage help? We hope that your iPhone is working well right now. Give us a like if you find this article helpful and more iPhone solutions are on fonepaw.com!


text:
10th November 2014 New English, News
The Executive Committee of Eastern Africa Journalists Association (EAJA) successfully held its meeting in Bujumbura, Burundi, on 8 November 2014 and reinstated Omar Faruk Osman as its elected Secretary General. After discussing a wide range of issues raised at its last meeting in January 2014 in Addis Ababa, the Executive Committee agreed by a large majority (5 votes for and 1 against) that Omar Faruk Osman be asked to resume to his post as Secretary General.
In January 2014, the Executive Committee met in Addis Ababa and took the decision to suspend Omar Faruk Osman as EAJA Secretary General and appoint first vice-president Alexandre Niyungeko as Acting Secretary General. This decision was taken under extraordinary circumstances where the Secretary General was not allowed into Ethiopia and the Executive Committee could not hear his views and did not give him a right to reply to allegations raised by some members. Concern was then raised that the suspension decision didn't comply with statutory provisions of EAJA constitution.
EAJA President Dr. Muheldin Ahmed Idris announced to EAJA members as well as IFJ and FAJ leaders attending the regional workshop on working conditions, that Omar Faruk Osman has resumed his duties as Secretary General with effect from November 9, 2014.
"Having given him the opportunity to explain himself on a number of issues, I can confirm that the suspension had been lifted from Omar Faruk Osman and he has been reinstated in office officially as Secretary General" said Dr Idris.
"The return to office of the EAJA Secretary General signifies and confirms our commitment to a democratic, independent, militant, regional organisation that is capable of acting in the best interest of its members" added Dr. Idris. "Now we must see the execution of our collective mandate to ensure that we go forward and implement the decisions of the meeting".
The decision of the EAJA Executive Committee was reported to the workshop and backed by a majority decision. The EAJA Executive Committee also agreed to hold its next meeting in the Sudanese capital Khartoum under the auspices of Sudanese Journalists Union.
The conservation response george write my paper for me wallace established himself in 1963 as one of the nation's leading spokesmen for the defense of segregation.


If you are coming in for your first CT scan, you are probably wondering what actually goes on when you get your test done. At our facility, we have a 16-slice CT scan, which is one of the most complete CT scan images you can have done. Since it is one of the most precise types of imaging available, we get many questions on how it works. Here's a brief overview of how our 16-slice CT scan operates.
A CT scan, or a computed tomography scan, works like many other types of diagnostic imaging. It takes highly-concentrated x-rays around the patient's body which take the slices, or images, that you see with your radiologist. A CT scan uses a large, tubular x-ray that stays in place as the patient is moved through the center of it. The large tube that the patient goes through is also called the gantry, which is what rotates and takes the images inside their body.
These highly-concentrated x-ray images that the gantry takes can be placed on top of each other to form a more detailed image or slice. At our facility, we have a 16-slice CT scan – which takes 16 cross-sectional photos and places them on top of one another. This not only creates an accurate image of the patient's organs, tissues, or bones, but it also goes quicker than lower-slice CT scans as well.
If you are looking for top-of-the-line, digital x-ray imaging in New Jersey, turn to us at Middletown Image. Set up your CT scan with us today at (732) 275-0999.
This entry was posted in Blog on November 14, 2017 by middletownu.


On Wednesday 20th February, two members of Templemore Fire and Rescue Service, James Wixted and Eddie Scully, visited 3rd and 4th classes to teach us about Fire Safety.
the importance of having an escape route in school and at home in case of fire.
We really enjoyed their visit and learned a lot!


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