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Category Archives: Language
The Lability of Grammatical Categories in Contemporary American English
In the twenty-first century, what used to be rigidly distinct in the grammar of American English––namely, the morphological distinction between verbs and nouns––has loosened so that verbs can easily serve as nouns without any change in form and vice versa. Thus the noun gift has come to be used as a verb (e. g., "X gifted me with Y"), and the verb build as a noun, as in computer lingo (e. g., "Choose how you get Insider builds," from the Microsoft update page). To a linguistic purist like Y-H-B, this completely insouciant crossing of previously impervious categorial boundaries is stylistically anathema and not to be adjudged felicitous. Hoswever, there is evidently a semantic need that is being filled by such innovations, and younger speakers use them without any qualms. O tempora, o mores!
Broad and Flat A in American English Redivivus
A reporter on todays' NPR broadcast of "Morning Edition," in speaking about the political troubles in Spain over Catalonia, was heard pronouncing the word Catalan with two broad a's (as in want) in the first and last syllable instead of the normative flat vowel [æ] (as in fan). The reporter (a woman) had evidently never heard this word pronounced correctly and, therefore, took it to be marked (i.e., conceptually restricted), for which the marked broad vowel occurs de rigueur in contemporary American English.
Apropos, readers of this blog are reminded that among the PDFs on the upper portion of the homepage there is the link to my article, "Broad and Flat A in Marked Words," American Speech 72.4 (1997), 436-439, which to this day provides the definitive explanation of the phenomenon at issue.
Norms of Professional Jargons
Every contemporary language, particularly in the First World, has professional jargons with their own norms that may differ from the standard in some aspect of grammar (usually phonology). Thus in contemporary standard Russian, mariners say kompás 'compass' with stress on the second syllable, whereas the ordinary norm for this word is stress on the initial syllable. A New York plumber may retain the flat vowel instead of the diphthong in the stressed first syllable of the word radiator.
Listening this morning to VPR Classical, Y-H-B heard the host mispronounce the word ballade in announcing one of Chopin's Ballades. Instead of the accepted norm for musical pieces of this genre with stress on the second syllable, this host (evidently of the younger generation) put the stress on the first syllable, as if the word were spelled ballad. This mistake was a flagrant violation. Perhaps he had never heard the word pronounced properly by someone in the know.
Vocabulary as an Indicator of Intelligence
As is his custom, Y-H-B was sitting in his Sunday morning restaurant in Manchester Center, Vermont, "Up for Breakfast," talking to one of the waitresses, a middle-aged woman who grew up in Rhode Island and now lives in Bennington, Vt. We were engaging in the usual early breakfast banter when the waitress (Gwen), a very nice and well-spoken lady, used the word potpourri to refer to a miscellaneous collection of implements in the restaurant. I immediately made note of this usage as a sign of higher-than-expected intelligence and silently congratulated Gwen for using this word.
While it is common for children in every country to be urged to expand their vocabulary in preparation for higher and higher schooling, in America it is not common for ordinary speakers to use high-style words like potpourri (a French borrowing) unless the context calls for their special use. In general, speech in America between people of different social classes and professions rarely goes beyond the boundaries of common vocabulary, although doctors and dentists are well-known for gratuitously engaging in special vocabulary out of habit instead of using common words to describe medical conditions.
Word choice as a stylistic matter is not merely facultative: it is always a sign of intelligence and necessarily has the effect of affecting one's general evaluation of one's interlocutors.
Asserting One's Bona Fides
In contemporary American English, speakers are constantly asserting their good faith to their interlocutors by interpolating (or adding, or prefacing) the phrase "to be honest" as if to assure them that what they saying is valid or authentic, thereby elevating their bona fides as far as the authenticity of their utterances is concerned. However, almost without exception, there is no patent need for this phrase because the speaker's sincerity or the accuracy of the information being conveyed is never in question.
This is yet another instance when Americans resort to utterly otiose verbal warrants in speech in order to prevent their interlocutors from so much as suspecting that what is being said fails to represents the speaker's true thoughts. Media chatter of all kinds, because of its ubiquity and pervasiveness, only exacerbates the situation by providing a typically inauthentic model of ordinary speech.
The Psycholinguistic Pathos of Everyday Life 16: When Words Fail One
Y-H-B turned on his radio this morning to VPR Classical (as usual) and heard Chopin's Barcarolle in F-sharp major, Op. 60, being played by Vladimir Ashkenazy. This is a piece I often heard as a child growing up in Tokyo and later Los Angeles, when my mother of blessed memory (1905-1983), Lydia Ita Shapiro (née Chernetzky [Лидия Абрамовна Шапиро, ур. Чернецкая], practiced it in preparation for one of her solo concerts (including Hibiya Hall in Tokyo). It was also the last piece that my beloved wife, Marianne Shapiro (1940-2003 עליה השלום), was learning before she was no longer able to sit at the piano.
When one hears music that is so close to one's heart, there are no words to describe the emotional power of the sounds.
The Psycholinguistic Pathos of Everyday Life 15: Two Russian Proverbs and a Poem in Prose
As instanced in these posts more than once, Y-H-B is prone to engage in silent autocommentary by repeating material from his fund of Russian lore. Today, under particularly onerous mental circumstances on a cold day in Vermont, the following two sayings/proverbs came to mind:
Не мытьем, так катаньем 'By hook or by crook.'
Попытка не пытка, спрос не беда. 'It can't hurt to try.'
This paroemiac material was then mentally garnished with the following poem in prose by the great Russian writer, Ivan Turgenev:
Иван Тургенев (1818-1883)
Стихотворения в прозе: Русский язык
Во дни сомнений, во дни тягостных раздумий о судьбах моей родины, — ты один мне поддержка и опора, о великий, могучий, правдивый и свободный русский язык! Не будь тебя — как не впасть в отчаяние при виде всего, что совершается дома? Но нельзя верить, чтобы такой язык не был дан великому народу!
[Translation: THE RUSSIAN TONGUE [Constance Garnett]
In days of doubt, in days of dreary musings on my country's fate, thou alone art my stay and support, mighty, true, free Russian speech! But for thee, how not fall into despair, seeing all that is done at home? But who can think that such a tongue is not the gift of a great people!
June 1882.]
One's thoughts do not necessarily have to be original to embody emotional force, especially in sad moments. Vox Populi, Vox Dei.
Syntactic Arbitrariness and Analogical Leveling in Language Change
When speakers of American English make grammatical mistakes like saying *play a factor (instead of be/constitute a factor) and *ask a question to (instead of ask a question of), they are relying on the analogy between similar constructions, viz. play a role and direct a question to. These two departures from normative grammar in present-day speech are frequently heard in media language and point to the arbitrariness of the traditional syntax involved.
The mistakes arise, of course, via the establishment of an implicit dominance relation between the newer (mistaken) form and the older traditional one. The postposition of in the verb phrase ask of represents a use of the postposition that is no longer felt by speakers to have the meaning of directionality, since its contemporary predominant meaning is possessive, i. e., "belonging to" and not "directed at/toward." Similarly, the word factor has been reconstrued as being identical in meaning to role, hence to be governed by the verb play.
Analogical leveling of this sort in language change always has a structure that reposes on a reconstrual of the meaning of the constituents involved in governing their syntax.
The Psycholinguistic Pathos of Everyday Life 14: On the Sociology of Knowledge
Y-H-B is allowing himself to go slightly off-topic in this post, focusing as it does not on language and linguistics but on the SOCIOLOGY OF KNOWLEDGE, which is defined as the study of the relationship between human thought and the social context within which it arises. This field of investigation was pioneered primarily by the French sociologist Émile Durkheim (1885-1917) at beginning of the 20th century.
About three years ago I submitted an article to The New York Times for its Op-Ed page, which was not printed. Here it is:
"Sham Solutions to Real Problems in Education by Michael Shapiro
Educational reform is constantly in the news, driven by the Obama administration's avowed aim of improving test results nationwide. The most prominent nostrums today are the same as before: measuring achievement by standardized testing, awarding merit pay to teachers, and ending tenure. None of these comes to grips with the real problem: recruiting and retaining good teachers. Everything that happens in the classroom––the only genuinely crucial arena where learning takes place–begins and ends with the teacher. Good teaching is the only portal at which knowledge enters and educated students exit. Boards of education, superintendents, principals, department chairs–the entire horde of barnacles on the ship of learning–only impede the ship's progress. Parents and politicians must learn this lesson and act on it if America is ever going to have an excellent system of public education.
Here are two real-life cases known to me personally which illustrate how good teachers are prevented from entering and staying in teaching. Both are drawn from the world of higher education but apply to the problem in general.
Case 1: an assistant professor is hired to teach in a language department but is not told that she is merely a place-holder for the chairman's chosen candidate who cannot assume the position until he satisfies a visa requirement. A year goes by. The place-holder does an excellent job. The fair-haired boy then becomes eligible, so the chairman manufactures an entirely bogus dossier to make it look like the woman is incompetent and get her terminated. The vice chancellor rubber stamps the chairman's action. Learning of this, the woman complains of sham treatment to a university-wide committee, which rules in her favor and censures the chairman. But the vice chancellor [may he rot in hell!] ignores the ruling, saying to the aggrieved scholar: "Do you think I'm going to exchange a chairman for an assistant professor?" The teacher sues the university but is unsuccessful because the federal judge refuses to allow into evidence material from confidential files that would have vindicated the plaintiff and reinstated her. The law suit not only leaves her jobless but renders her unable to secure a tenured position over her entire working life because the chairman and his henchmen repeatedly blackball her out of spite by writing poisoned-pen letters to prospective employers. The woman perseveres as a scholar and dies prematurely, her only revenge being a legacy of written work that marks her as the most accomplished and versatile American specialist in her field of the 20th century.
Case 2: a university professor with forty years' experience, a record of distinguished scholarship and teaching at all levels through the postdoctoral, and an international reputation for pioneering research in his field retires and some time later offers his services–without salary–to the presidents of five colleges near his new domicile in a Northeastern state. Three do not bother to answer his letter at all. Of the two that do respond, only one accepts the offer, the second's response being, "There is currently no appropriate context for your services."
Despite their differences, these two cases––each in its own way––testify to the fact that administrators, who have the responsibility for insuring that the best teachers are recruited and retained, routinely allow considerations unconnected with education to hold sway over their personnel decisions. Their overriding goals, in order of importance, are: 1) keeping themselves in their jobs; 2) keeping the natives (alias their subordinates) from causing trouble. The second conduces to the first. Anything that avoids resentments and recriminations trumps everything else. After all, unlike business or commerce, there is no profit motive in education.
Ending tenure might help but is not a panacea; nor is merit pay, which already exists anyway and is bootless to change the situation fundamentally. In higher education these issues are usually raised in the context of a false dichotomy–between good teaching and research. After more than fifty years spent in the trenches, I can testify that only infinitesimally are professors who produce no lasting scholarship first-rate classroom teachers all the same. That most publications are dross, not gold, changes nothing: it's their habits of thought that scholars and scientists of the first rank pass on to their students. A useful comparison is with music pedagogy. My parents' conservatory instructors in inter-war Germany, the cellist Julius Klengel (Leipzig) and the pianist Leonid Kreutzer (Berlin), were world-renowned pedagogues whose mastery of their instruments was an indispensable ingredient of their success as teachers. My father's University of Freiburg teacher Edmund Husserl, whose philosophical writings will endure for all time, gave lapidary lectures that attracted students and colleagues from all faculties. But for every Husserl, there are always numberless—-tenured!—-time servers. Abolishing tenure won't change that. Nor will merit pay."
Schwa in Unstressed Syllables as between British and Amderican English
Among the many differences between British and American English is the way that vowels are treated in unstressed syllables. For the most part, British English tends to pronounce them with schwa, so that the final vowel in Pentagon and Amazon, which has a relatively full vowel in American speech, has a reduced vowel in British. Occasionally, the results are inverted, so that the final syllable of Maryland and partisan in British speech has the full vowel, whereas a schwa is the norm in American.
The occurrence of a reduced vowel in unstressed position (or in syllables with potentially secondary stress) is to be interpreted as a sign that the word is being interpreted as a unified whole, with a constituent structure. Maryland pronounced by a speaker of American English is thus construed as having no segment structure phonetically, whereas in British speech the maintenance of [a] and secondary stress is a sign that speakers are interpreting the word has having two constituents, Mary– plus –land.
This sort of (tacit) construal of a word's constituent structure has different phonetic outcomes in the speech of newer vs. older generations of American speakers when it comes to items that contain final morphemes like {ent} in student and president, with younger speakers coming increasingly to pronounce them with a full vowel rather than the traditional schwa. This secondary stress in the newer variant seems to be occasioned by the reconstrual of these words as containing a suffix, since the stems (stud– and presid-) also occur elsewhere in verbs.


7) Nazi march beat 2m2B take 112 0'50"
16) 4m2 take 95 0'28"
1 OSCAR (ACADEMY AWARD) : best music, original score.
1 Nomination aux GOLDEN GLOBE AWARDS : best original score.
1 GRAMMY AWARD : instrumental composition for a motion picture or television.


The Wooden Serving Tray coordinates easily with exsisting decor or with coffee, console tables, hotel,etc. Also can be upside down to use as a flat surface.The metal handle make this tray easy to carry .This generously sized tray is great for serving your favorite appetizers or a complete meal in bed, as well as your guests. Serve your guests or storage some treasures in style this rectangle serving tray from SAWTRU.


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In a previous post, I touched on the point of testing and briefly talked about unit testing. In this post, I will be going into more detail about what unit testing is and why it's important to do it. In the previous post, I said that Unit Tests are: low level tests, meaning that they are close to the source of the product. They should be written with the aim of testing individual methods and functions for a given code base, using a unit test framework to support the authoring and execution of a test. As a developer, you would typically author the unit tests in a development tool like Visual Studio; you'd run them locally to ensure that the tests pass; and then they would be executed on a regular basis as a task in a Build Pipeline within Azure DevOps. Unit Tests are cheap to automate and should be quick to run.To expand on this, unit tests are written by a developer to apply to a unit of code. But what do we mean by "unit of code"? A unit of code is the smallest testable part of a solution – verifying that the individual part or component of a solution works as intended, independently from other parts. A unit could be a C# method; a PowerShell function; a T-SQL Stored Proc, and many others. Like most forms of testing, unit tests follow a pattern of:Initialise system under testCall method under testAssert expected outcome against result of methodA best practice would be to write the unit test before the writing any code, but if you've not got to that level of maturity with your test approach - writing tests after code is still good practice. How do you write a good unit test?Keep it simpleA unit test shouldn't replicate the code it is intended to test. You'll be writing lots of them, so make them quick and easy to write.ReadableBy keeping it simple, the test should also be readable. Making it easy to know what method is being tested and the expected behaviour of the method.By making it readable, you can easily address any failures that may surface.Reliable and RepeatableUnit tests should only fail if there are bugs in the system, not because there are bugs in the tests. Keeping it simple and readable will avoid that issue. Unit tests need to be run many times, sometimes multiple times throughout the course of a day, so they need to be executed quickly in a repeatable manner. Keeping it simple helps achieve this aim.How do you write a unit test?We've got an understanding of what a unit test is, but how do we write one? For this example, we'll be writing our code and tests using C#.Our application is a very simple calculator, which adds two numbers together. Simply, to add a new Unit Test, we can right-click on the method and select Create Unit Tests. Because we've not built any unit tests before, we can use it to create a new unit test project using a framework of choice. If we already had a unit test project, we could add the new test to the existing project. Using this method, it creates a skeleton of a unit test from which we can amend for our needs.As you can see, this doesn't contain what we need, so we amend the test so that it reflects our requirements, as in the below.To run a Unit Test, you can either right-click on the test method and click on Run Test(s) or open up the Test Explorer window, navigate to the desired test and click on Run Selected Tests.Unit Tests in Azure DevOpsWe've written our unit tests and have run them locally, but how do we make it repeatable? We utilise the power of Azure DevOps to have repeatable tests run against a changing code base as part of the Build or Continuous Integration process. The process is:Install NuGet on the Build AgentRestore any packages from NuGet that your application requiresBuild solutionRun testsPublish testsCopy successfully built and tested artifacts to a staging directoryPublish those artifactsUsing Azure DevOps, or another CI tool, we can rely upon our tests in a repeatable manner. Additional ReadingThere's a good post by Sergey Kolodiy on the importance of writing good code and how unit testing encourages good behaviour. My colleague Jon has also written a post on the subject: Setup Unit Testing with NUnit and NBi.
In a blog introducing the Data Lake Framework, keen readers will be aware that in the diagram there's a box titled "ARCHIVE" but it has not been brought up since. The reason why the Archive layer in the data lake has not been discussed is because we've been waiting for the Archive Tier in Blob Storage.To remind readers of the framework and where the archive layer sits in it, here it is again with the archive layer highlighted.The Archive BlobThe Archive access tier in blob storage was made generally available today (13th December 2017) and with it comes the final piece in the puzzle to archiving data from the data lake. Where Hot and Cool access tiers can be applied at a storage account level, the Archive access tier can only be applied to a blob storage container. To understand why the Archive access tier can only be applied to a container, you need to understand the features of the Archive access tier. It is intended for data that has no or low SLAs for availability within an organisation and the data is stored offline (Hot and Cool access tiers are online). Therefore, it can take up to 15 hours for data to be made online and available. Brining Archive data online is a process called rehydration (fitting for the data lake). If you have lots of blob containers in a storage account, you can archive them and rehydrate them as required, rather than having to rehydrate the entire storage account. Archive PatternAn intended use for the Archive access tier is to store raw data that must be preserved, even after it has been fully processed, and does not need to be accessed within 180 days. Data gets loaded into the RAW area of the data lake, is fully processed through to CURATED, and a copy of the raw data is archived off to a blob container with a Cool access tier applied to it. When the archive cycle comes about, a new Cool access tiered blob container is created and the now old container has its access tier changed to Archive. For example, our Archive cycle is monthly and we have a Cool access tiered blob container in our storage account called "December 2017". When data has finished being processed in the Azure Data Lake, the Raw data is archived to this blob container. January comes around, we create a new blob container called "January 2018" with Cool access tier settings and change the access tier of "December 2017" from Cool to Archive. This data has now been formally achieved and is only available for disaster recovery, auditing or compliance purposes.
Data Science can fit seamlessly within the ecosystem of the data lake, whether this is through HDInsight or the extensibility of Azure Data Lake Analytics and U-SQL. This blog will give a brief overview of what Data Science is; how to link Data Science toolkits to the Azure Data Lake; and best practices for managing the data output from experiments. Data Science Data Science is the relatively new kid on the block. One way to consider data science is as an evolutionary step in interdisciplinary fields like business analysis that incorporate computer science, modelling, statistics, analytics, and mathematics.At its core, data science involves using automated methods to analyse massive amounts of data and to extract insight from them. Data science is helping to create new branches of science, and influencing areas of social science and the humanities. The trend is expected to accelerate in the coming years as data from mobile sensors, sophisticated instruments, the web, and more, grows. Data Science In The Data Lake The nature of the Azure Data Lake Store lends itself to Data Science in that it can hold any data, which the data scientist will want to access, transform and analyse. HDInsight contains many implementations for data science, such as Spark, R Server and others. Hooking HDInsight to Azure Data Lake Store is pretty simple and follows these steps: In the Azure Portal Marketplace, select HDInsight which will bring up a series of blades. In this blog, I will be using Spark as my cluster type on HDInsight. In the storage settings you can then link your HDInsight cluster to Azure Data Lake Store Confirm your configuration on the next blade and wait around 20 minutes for your cluster to deploy and you're good to go! With Azure Data Lake Analytics, you incorporate data science by extending the capabilities of U-SQL and you do this by installing a series of files. Open up your Azure Data Lake Analytics account and click on Sample Scripts at the top. This will bring forward the following blade From there you'll want to click on the U-SQL Advanced Analytics tab, which will copy about 1.5GB of files to the default Azure Data Lake Store associated to your ADLA account. This will take about 3 minutes to complete. When it's finished copying the files it will then call a job to register the extension, which can be found in the assemblies folder of the master database. More resources about extensibility of U-SQL can be found here: https://blogs.msdn.microsoft.com/azuredatalake/2017/03/10/using-custom-python-libraries-with-u-sql/ https://blogs.msdn.microsoft.com/azuredatalake/2016/11/22/u-sql-advanced-analytics-introducing-python-extensions-for-u-sql/ The Laboratory Within Azure Data Lake Store, Folder and File Management is incredibly important for a well running data lake. See my blogs on Storage and Best Practices and Shaping The Lake for more information on how to set up your Azure Data Lake Store. The Laboratory is an area to be exclusively used by a data scientist. It's an area where they can persist the results of experiments and data sets without impacting the day-to-day operations within the data lake or other data scientists. The laboratory is organised in to two broad area: Desks and Exhibits. Desks contain personal workspaces, the contents of which can be organised however the person wishes. It can be as well organised, or disorganised, as the person themselves. The Exhibit contains data sources produced in the Laboratory which are ready to be consumed by other users or systems. Both of which are laid out below. As always, if you have any feedback or comments do let me know!
The Azure Data Lake Store is an integral component for creating a data lake in Azure as it is where data is physically stored in many implementations of a data lake. Under the hood, the Azure Data Lake Store is the Web implementation of the Hadoop Distributed File System (HDFS). Meaning that files are split up and distributed across an array of cheap storage. What this blog will go into is the physical storage of files in the Azure Data Lake Store and then best practices, which will utilise the framework. Azure Data Lake Store File Storage As mentioned, the Azure Data Lake Store is the Web implementation of HDFS. Each file you place into the store is split into 250MB chunks called extents. This enables parallel read and write. For availability and reliability, extents are replicated into three copies. As files are split into extents, bigger files have more opportunities for parallelism than smaller files. If you have a file smaller than 250MB it is going to be allocated to one extent and one vertex (which is the work load presented to the Azure Data Lake Analytics), whereas a larger file will be split up across many extents and can be accessed by many vertexes. The format of the file has a huge implication for the storage and parallelisation. Splittable formats – files which are row oriented, such as CSV – are parallelizable as data does not span extents. Non-splittable formats, however, – files what are not row oriented and data is often delivered in blocks, such as XML or JSON – cannot be parallelized as data spans extents and can only be processed by a single vertex. In addition to the storage of unstructured data, Azure Data Lake Store also stores structured data in the form of row-oriented, distributed clustered index storage, which can also be partitioned. The data itself is held within the "Catalog" folder of the data lake store, but the metadata is contained in the data lake analytics. For many, working with the structured data in the data lake is very similar to working with SQL databases. Azure Data Lake Store Best Practices The best practices generally involve the framework as outlined in the following blog: http://blogs.adatis.co.uk/ustoldfield/post/Shaping-The-Lake-Data-Lake-Framework The framework allows you to manage and maintain your data lake. So, when setting up your Azure Data Lake Store you will want to initially create the following folders in your Root Raw is where data is landed in directly from source and the underlying structure will be organised ultimately by Source. Source is categorised by Source Type, which reflects the ultimate source of data and the level of trust one should associate with the data. Within the Source Type, data is further organised by Source System. Within the Source System, the folders are organised by Entity and, if possible, further partitioned using the standard Azure Data Factory Partitioning Pattern of Year > Month > Day etc., as this will allow you to achieve partition elimination using file sets. The folder structure of Enriched and Curated is organised by Destination Data Model. Within each Destination Data Model folder is structured by Destination Entity. Enriched or Curated can either be in the folder structure and / or within the Database.


What should I do if the police or detectives want to speak with me?
You should call a lawyer immediately. Police officers, and detectives, are employed to investigate, and solve, crimes. Interviewing witnesses, or targets of an investigation, is an important tool for doing that. Since police officers, and detectives, rarely tell people that they're targets of a criminal investigation, many people speak with the police, and unwittingly say things that can be used against them in a criminal case. Consequently, speaking with the police can have serious far-reaching consequences for some people. But if you speak with a lawyer, first, you can know your rights. Then you can decide whether to speak with the police, or not.
No, you do not have to answer questions asked by the police. Under the U.S. Constitution, and the Constitution of the State of New York, you have the right to remain silent, and to not answer questions. It is not a crime to refuse to answer questions, and you do not have to give a reason for not answering questions. However, if you do speak with the police, or other government agents, anything you say can be used against you in a court of law. Also, if you lie to them, that can be a crime. If you are stopped by the police while driving a car, though, the situation is somewhat different. You are not required to answer questions, but you must hand over your driver's license, and insurance card, if asked by the police to do so.
Are the police allowed to search my car, my home, or myself?
It depends on the situation. Ordinarily, the police are not allowed to search your car, home, or yourself, unless you consent to the search, or they get a search warrant ordered by a court to conduct a search. However, there are times when the police may conduct a search without your consent, and without a warrant. For example, if the police stop you in your car, they have the right to search your car without your consent, and without a warrant, if they have probable cause to believe that you have committed, or are committing a crime. If the police arrest you in your home, they can search the area near where you are arrested without your consent, and without a warrant, but they cannot search the entire residence without a warrant that describes in detail the places to be searched and people or things to be seized. Finally, if the police stop you on the street, they have the right to pat down your clothing if they suspect that you might have a weapon.
An arraignment is the first appearance in Criminal (or District) Court following an arrest. Arraignments usually happen within twenty-four hours of an arrest. Several things happen at arraignment. First, the person who has been arrested, and is referred to as, the "defendant," is formally told of the charges levelled against them. They learn what crimes they have been accused of committing, and the specific allegations that constitute the crimes. Second, the Court conducts a bail hearing. That means that the judge decides whether to release a defendant on his or her own recognizance (ROR), or on bail or bond, or to remand him or her to the custody of the Department of Corrections. Additionally, the District Attorney's office, and your lawyer serve notices upon one another. Notices are legal notifications of important issues.
What is a Desk Appearance Ticket (DAT)?
A Desk Appearance Ticket, or DAT, is a way for certain defendants to appear in Criminal (or District) Court for arraignment. If the police issue a DAT, the defendant will be released from custody, and ordered to appear in Criminal (or District) Court for arraignment, usually three or four weeks after the arrest. DATs are only given for certain kinds of cases, usually misdemeanors that do not entail allegations of violence. While granting DATs for certain low-level, non-violent, felonies is permitted, it does not happen with great frequency. DATs are never given in domestic violence cases.
The most common types of bail are: cash or bond. Usually, at arraignment, judges set bail in an amount that is for cash, and an amount that is for bond. For example, bail could be set at $2,500 (cash) and $5,000 (bond). To get out of jail, a defendant's friend or family must pay one or the other forms of bail, but not both. To post cash bail, a defendant's friend or family would pay cash, or a bank check, on behalf of the defendant to the Criminal (District) Court Clerk immediately after the arraignment. This person is known as the "surety." To bail out a defendant quickly, it is important for the surety to be present in court during the arraignment because defendants are transferred to local jails shortly after arraignment, if bail has not been posted. If a defendant has been taken to a local jail, the surety would have to post bail at the jail. Cash bail is refundable in full, at the end of the case, if it is dismissed. If the case concludes with a conviction, 3% of the amount posted will be deducted by the Court before the balance is returned to the surety. To post bond, the surety would have to hire a bail bondsman to post an insurance company bond on behalf of the defendant. Bail bondsmen usually require some form of collateral, and a non-refundable fee for their service.
No, you do not have to hire a lawyer for a criminal case. But you should hire a lawyer. Otherwise, you may represent yourself, "pro se." You may do this whether, or not, you are a lawyer. However, it is generally unwise for anyone, including a lawyer, to represent himself. In fact, there's an old adage that, "a lawyer who represents himself has a fool for a client." So, representing yourself "pro se" is generally unwise. If you cannot afford to hire a lawyer, the Court will assign a lawyer to represent you, free of charge.
Felonies are serious crimes, while misdemeanors are less serious crimes. In New York, you can go to jail for up to one year on the most serious types of misdemeanors. For felonies, though, you can go to jail for more than one year. Additionally, the length of a probationary sentence is longer for a felony than it is for a misdemeanor.
What is the difference between Criminal (District) Court and Supreme (County) Court?
In New York City, and in Nassau and Suffolk Counties, most criminal cases begin in the lower court, which is the Criminal (District) Court. If the most serious charge is a misdemeanor, the case will remain in the Criminal (District) Court until it has concluded. But if the most serious charge is a felony, and if a Grand Jury votes to indict the case, then it must be transferred to the higher court, which is the Supreme (County) Court, for trial. That's because the Supreme (County) Court is authorized to impose sentences greater than one-year in jail, whereas the Criminal (District) Court is authorized to impose jail sentences of no greater than one-year.
How can a criminal case end?
Generally, there are three ways to resolve a criminal case. First, any defendant, charged with any crime, has the right to a trial. Trial can be by judge, or jury. A relatively small number of criminal cases go to trial. Second, a defendant can plead guilty pursuant to a negotiated plea bargain. Typically, a defendant's lawyer will negotiate the terms of a plea bargain with the District Attorney's Office, and the Judge who is presiding over the case. Generally, in a plea bargain, a defendant will plead guilty to a less serious charge, and receive a sentence that is more lenient than it would be had he or she gone to trial, and been convicted. Most cases are resolved with a plea bargain. Finally, if there is a legal basis, or a negotiated agreement, a case can be dismissed.
A suppression hearing is a legal proceeding conducted by a Court prior to trial. The purpose of a suppression hearing is to determine if a defendant's constitutional rights were violated, and if evidence was seized from a defendant, as a result of the constitutional violation. If the Court makes such a finding, then the evidence in question should be suppressed, or kept out, at trial.
A trial is a formal examination of evidence in a judicial proceeding to decide guilt in a criminal or civil case. In New York, defendants have a right to a trial by jury if they face the possibility of a jail sentence greater than six months. However, a defendant has an absolute right to waive his or her right to a jury trial, and have a judge decide the issue of guilt at what is known as a, "bench" trial.


text:
Ukraine Rejects New Russian Ambassador
Mikhail Babich Kremlin Press Service
Ukraine will not accept the newly appointed Russian ambassador, the Ukrainian Deputy Foreign Minister for European Integration, Olena Zerkal, said Wednesday.
"This issue has been taken off Ukraine's agenda," Zerkal said on Ukraine's TV 5 Kanal.
Zerkal claimed that the absence of an ambassador will not affect the relations between the two countries. "The previous ambassador had a minimal part in shaping those relations," she said.
President spokesperson Dmitry Peskov earlier announced that Russia had requested to send Mikhail Babich to the country as Russian ambassador to Ukraine. Mikhail Babich, the former presidential envoy to Russia's Volga Federal District, was appointed to the role last week. Putin made the move after relieving former ambassador Mikhail Zurabov of his duties. Zurabov had been ambassador to Ukraine since August, 2009.
Putin Fires Russia's Ambassador to Ukraine
Russian President Vladimir Putin signed a decree to release Mikhail Zubarov from his post as ambassador to Ukraine on Thursday, according to a message...
Crimean Map Scandals: Choice Between Offending Russia or Ukraine
Since Russian troops invaded and annexed Ukraine's Crimean peninsula, world topography has struggled to place the international border between the two...
Ukraine Blacklists 243 Russian Companies for Working in the Donbass
The Ukrainian government has moved to suspend imports from 243 Russian companies operating in Ukraine, the Ukrainian Security Services (SBU) said in a...

redpajama_set_name: RedPajamaCommonCrawl


Posted on Monday, July 24, 2017 by "T".
Who knew something fishy can happen with the fish tanks in a casino?
According to a report from a security firm called Darktrace, hackers have managed to gain access to a North American casino via the use of a smart fish tank.


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Home » Programs » Internships » Intern Blogs » The First Week: Why Am I Always Getting Lost Though?
My head is spinning. I haven't been able to function on less than 8 hours of sleep since way back in freshman year (because you know, that was FOREVER ago). For the first time in a while, I feel like I truly belong. This past week has been a whirlwind of new and exciting experiences and as I'm writing to you today is honestly the first time I'm processing it all.
It all started Friday. I left my quiet neighborhood (we call it the old people neighborhood) back in Clayton, NC and began the long journey to our nation's capital. With me, I had my siblings (16 y.o. brother and 11 y.o. sister) and my mother. My dad wanted to be there but he had a conference in Texas (which turned out to be extremely hot) so I didn't fault him for it. Anyway, we drove about 3 hours before we stopped at an old friend's house in Virginia near Richmond. I call her Aunty Nikki because I've known her all my life and she has supported me like any other family member would. I admire her creativity and vigor and was glad to reconnect after way too long.
After that pitstop we headed to visit my other Aunty right outside of DC. It was too short of a visit but it's relieving to know she's nearby. I actually got to experience Maryland blue crabs for the first time ever at a true hole-in-the-wall named Fifer's Seafood. It was an… interesting experience. They were no doubt the best crabs I've ever tasted in my life… so succulent and flavorful with a hint of sweetness. Being the food-lover I am, I enjoyed that night very much (even though I had to get my hands dirty).
At long last, I made it to my home for the summer. It was a wet, dark and busy move-in day but to me it was the best day ever. Despite the leaky shower, busted blinds, and burnt-out bulbs, I loved it. I loved it because it was mine (and those other three people lul). I felt it symbolized a new chapter in my life and I was ready for whatever came my way. After jazzing it up a bit and making three new friends in my suitemates, I finally retired for the night.
I awoke the next morning and literally hopped out of bed. I barely slept the night before because I was way too excited for orientation day. The four of us set out on the street to the metro. While transferring to the next train, we met up with two more awesome people and by the time we reached our destination I was 5 friends in.
Orientation was amazing. I sort of expected it to more like a "Don't do this, and definitely don't do that" type deal but it was more of a "Welcome to the family" situation. Everyone was so excited to meet us and they were all very friendly and interesting people. They fed us, bestowed words of wisdom, and told us all of the fun events they have in store. They genuinely wanted to hear more about us and what we had to say. And it was the same for all the other interns here. But the end of the first day, I had a whole new family.
After orientation, I met up with my mentor for the summer, Dr. Crystal Bailey. It was so great to meet her in person. She is an inspiring and lively woman with a "Let's get this done!" type of attitude I live for. I appreciate that she allows my creativity to flow and is still there for support when I need. Although a bit challenging, I decided to take on a video project. This week I have been mostly doing research and writing the script for it. In short, I'm making a informational video about interesting careers in physics (there are A LOT) and how to go about getting them. Luckily, I have had some filmmaking experience before so I'm not going in blind (sure Elon, sure).
So if you made it this far, congrats! You're finally going to understand the title. It seems I have made a habit of going the wrong way on the train… The first three times I was like "Whoops. Oh well..." but I have to say I'm starting to get a little… perturbed. It turns out in fact I shouldn't feel comfortable solving Lagrangians when my stop is like 5 minutes away. Anywho, that's all for today. Stayed tuned to find out if I ever figure out how to not get lost!


text:
\section{Introduction}

The low-energy behavior of interacting fermion systems can be strongly influenced by collective bosonic degrees of freedom such as critical order parameter fluctuations or emergent gauge fields.
In metallic systems, the scattering of electrons by gapless bosons destroys Landau quasi-particles and thus leads to a breakdown of Fermi liquid theory~\cite{loehneysen07}.
These effects are particularly pronounced in two-dimensional systems, and they have therefore frequently been discussed in the context of layered compounds such as cuprate- or iron-based high-temperature superconductors.
In this paper we focus on systems in which the critical bosons carry a small momentum and couple to the electron charge, leading thus to a singular but spin-conserving {\em forward scattering}\/ of electrons.
This class of systems comprises electron liquids coupled to emergent $U(1)$ gauge fields~\cite{lee06}, as well as quantum critical metals at the onset of nematic or other translation-invariant charge order~\cite{oganesyan01, metzner03, metlitski10}.

The breakdown of Fermi liquid behavior due to singular forward scattering is revealed already by first-order perturbation theory.
One-loop results for the boson and fermion self-energies in two-dimensional metals were first derived for fermions coupled to a $U(1)$ gauge field~\cite{lee89}, and later for systems at a nematic quantum critical point (QCP)~\cite{oganesyan01,metzner03}.
The bosons receive a Landau damping term generated by fermionic particle-hole excitations, and the one-loop fermion self-energy scales as $|\omega|^{2/3}$ at low frequencies $\omega$.
This self-energy dominates over the linear frequency term in the fermion propagator and thus implies a destruction of Landau quasi-particles.
The main contributions to the fermion self-energy at a fixed Fermi momentum ${\bf k}_F$ come from particle-hole excitations near the same ${\bf k}_F$ and its antipode $-{\bf k}_F$, with small momentum transfers ${\bf q}$ tangential to the Fermi surface and excitations energies of the order $|{\bf q}|^3$.
The coupled fermion-boson theory is scale-invariant with a dynamical exponent $z=3$ at the one-loop level~\cite{polchinski94}.

An early analysis of two-loop corrections did not reveal any qualitative changes of the boson and fermion propagators~\cite{altshuler94}.
The loop expansion was expected to be controlled by the inverse fermion flavor number $N_f$, as usual~\cite{polchinski94}.
However, the whole case was opened again when S.-S.\ Lee~\cite{lee09} discovered that the naive $1/N_f$ expansion breaks down and Feynman diagrams of any loop order contribute even in the limit $N_f \to \infty$.
Subsequently, Metlitski and Sachdev~\cite{metlitski10} constructed a general scaling theory for two-dimensional metals at a nematic QCP and the formally similar $U(1)$ gauge field problem.
Symmetry constraints restrict the theory such that only two independent anomalous scaling exponents are possible: an anomalous dimension of the fermion fields $\eta_f$ and an anomalous dynamical scaling exponent $z \neq 3$.
A small contribution to $\eta_f$ was indeed found at {\em three-loop}\/ order,
while the dynamical exponent remained unrenormalized at $z=3$ at that loop level~\cite{metlitski10}.
Most recently it was shown that the absence of renormalizations of $z$ up to three-loop order has special reasons that do not apply to all higher-order contributions. A divergence leading to anomalous dynamical scaling was found in a class of {\em four-loop} diagrams~\cite{holder15}.
Surprisingly, the divergence turned out to be stronger than logarithmic, indicating an anomaly or instability whose nature has not yet been clarified.

Naive power-counting frequently overestimates the actual degree of divergence of perturbative contributions. Cancellations may occur for single Feynman diagrams, due to oscillating integrands, and also between distinct Feynman diagrams.
In this context, the scaling behavior of {\em fermion loops}\/ plays a particularly important role. Most Feynman diagrams contain fermion loops that are connected to each other and/or to open fermion lines by bosons.
All hitherto observed cancellations are related to cancellations within a loop or between loops with permuted vertices.

In this paper we analyze the asymptotic behavior of fermion loops (with $N$ vertices) and sums of loops with permuted vertices in the scaling limit that applies to fermions coupled to bosons in two dimensions as described above.
We perform the analysis for a generalized class of theories with dynamical exponents $z > 2$, as originally introduced by Nayak and Wilczek~\cite{nayak94}.
Systematic cancellations are obtained if one or several of the external loop momenta are much smaller than the others.
As an application, we can assess the degree of divergence of several classes of Feynman diagrams without embarking on a tedious specific calculation of the corresponding integrals.

The paper is structured as follows.
In Sec.~II we introduce the low-energy effective quantum field theory for the systems described above, and we provide the precise definition of the fermion loops.
The result of the elementary integration over the momentum in the loop is presented in Sec.~III.
The asymptotic behavior of fermion loops with three vertices is analyzed in Sec.~IV. An important application is a deeper understanding of the mechanism leading to the absence of a renormalization of the dynamical exponent $z$ in the three-loop calculation by Metlitski and Sachdev~\cite{metlitski10}.
Asymptotic properties of fermion loops with an arbitrary number of vertices $N$ are derived in Sec.~V. In particular, it will be clarified to what extent cancellations obtained at three-loop order can be expected also at higher orders.
In Sec.~VI we provide several examples of improved power-counting.
We finally summarize the main results in Sec.~VII.

\section{Field theory and fermion loops}

The low-energy behavior of two-dimensional fermion systems coupled to critical bosons with small momenta can be described by an effective field theory whose fermionic states are restricted to two Fermi ``patches'' near fixed Fermi momenta ${\bf k}_F$ and $-{\bf k}_F$~\cite{lee09,metlitski10}.
Choosing momentum variables such that the $x$-component is perpendicular and the $y$-component tangential to the Fermi surface at $\pm {\bf k}_F$, the Lagrangian of the field theory can be written in the form~\cite{metlitski10}
\begin{align} \label{lagrangian}
 L &= \sum_{s=\pm} \psi^\dagger_s
 \left( \partial_\tau -
 is \partial_x - \partial^2_y \right) \psi_s \notag\\
 & \quad -\sum_{s=\pm} g_s\phi\psi^\dagger_s\psi_s
 - \frac{N_f}{2e^2}(\partial_y\phi)^2 .
\end{align}
Here $\phi$ is a bosonic scalar field, while $\psi_{\pm}$, $\psi_{\pm}^{\dagger}$ are fermionic fields with $N_f$ flavor components corresponding to states with momenta on the the two patches near $\pm{\bf k}_F$.
In the U(1)-gauge field problem, $\phi$ is the transverse gauge field and $g_+ = -g_-$. For the Ising nematic QCP, $\phi$ is the order parameter field and $g_+ = g_-$. In both cases the physical flavor number is $N_f = 2$.
The derivatives are with respect to real space and imaginary time variables.
Several numerical prefactors have been absorbed by a rescaling of fields and space coordinates. In particular $|g_s| = 1$.
The gradient term in $x$-direction $(\partial_x \phi)^2$ is irrelevant under the scaling that emerges on one-loop level, and has therefore been discarded in the Lagrangian.

In random-phase approximation (RPA), which corresponds to a one-loop calculation of the boson and fermion self-energies, the boson and fermion propagators are obtained as~\cite{lee89}
\begin{align}
 D^{-1}(q) &=
 N_f \left( \frac{q_y^2}{e^2} +
 \gamma \frac{|q_0|}{|q_y|} \right), \\
 G^{-1}_s(k) &=
 s k_x + k_y^2 - i \frac{\kappa}{N_f} \frac{k_0}{|k_0|^{1/3}} ,
\end{align}
where $q = (q_0,{\bf q})$ and $k = (k_0,{\bf k})$ contain momentum and frequency variables, and $\gamma$ and $\kappa$ are positive constants.
These propagators solve the RPA equations also self-consistently~\cite{polchinski94}.
The linear frequency term in the fermion propagator is subleading compared to the fermion self-energy and has therefore been discarded.
Note that $D(q)$ does not depend on $q_x$.

In the following we consider a generalization of the theory where $q_y^2$ is replaced by $q_y^{z-1}$ in the boson propagator, such that
\begin{equation}
 D^{-1}(q) =
 N_f \left( \frac{|q_y|^{z-1}}{e^2} + \gamma \frac{|q_0|}{|q_y|} \right) \, .
\end{equation}
This generalization was introduced by Nayak and Wilczek~\cite{nayak94} for the sake of an expansion in $\epsilon = z-2$. More recently, it was used to define a manageable large-$N_f$ limit of the theory~\cite{mross10}.
For any $z>2$, the one-loop fermion propagator then assumes the generalized form
\begin{equation} \label{Gk}
 G_s(k) = \frac{1}
 {sk_x + k_y^2 - i\{k_0\}} \, ,
\end{equation}
where
\begin{equation}
 \{k_0\} = \frac{\kappa}{N_f} \frac{k_0}{|k_0|^{\alpha}} \quad
 \mbox{with} \quad \alpha = 1- \frac{2}{z} \, .
\end{equation}

The fermion propagator scales homogeneously as $G(k) \mapsto \lambda^{-2} G(k)$ under the scaling of momentum and frequency variables
\begin{equation} \label{scaling}
 k_y \mapsto \lambda k_y , \quad
 k_x \mapsto \lambda^2 k_x , \quad
 k_0 \mapsto \lambda^z k_0 .
\end{equation}
The boson propagator scales as $D(q) \mapsto \lambda^{1-z} D(q)$ under the analogous scaling of $q_0$, $q_x$, and $q_y$.
For $z=3$ we recover $\alpha = 1/3$, and $D(q)$ scales with the same power $\lambda^{-2}$ as $G(k)$.

Higher-order contributions (beyond one-loop) to the fermion and boson self-energies are analyzed in an expansion around the one-loop fixed point, that is, by computing Feynman diagrams with one-loop propagators~\cite{altshuler94,lee09,metlitski10}.
Note that this procedure can be viewed as a simple shift of the expansion point, where the one-loop self-energy is added to the ``bare'' part of the action and subtracted as a counterterm from the interaction part.

The computation and asymptotic analysis of higher-order contributions is done most efficiently by evaluating first the fermion loops in the Feynman diagrams, and then the remaining bosonic momentum integrations.
The {\em $N$-point fermion loop}\/ on patch $s$ is defined as the integrated product of $N$ fermion propagators,
\begin{eqnarray} \label{PiN}
 \Pi_{N,s}(q_1,\dots,q_N) &=& I_{N,s}(p_1,\dots,p_N) \nonumber \\
 &=& \int \frac{dk_0}{2\pi} \int \frac{d^2{\bf k}}{(2\pi)^2}
 \prod_{i=1}^N G_s(k-p_i) \, . \hskip 5mm
\end{eqnarray}
The variables $p_i$ and $q_i$ are related by
\begin{eqnarray} \label{pq}
 q_i &=& p_{i+1} - p_i \quad \mbox{for} \quad i = 1,\dots,N-1
 \nonumber \\
 q_N &=& p_1 - p_N \, .
\end{eqnarray}
Note that $I_{N,s}(p_1,\dots,p_N)$ is invariant under a global shift of momenta $p_i \mapsto p_i + q$, and $q_1 + \dots + q_N = 0$ due to energy and momentum conservation.
$\Pi_{N,s}(q_1,\dots,q_N)$ can be represented graphically as a fermion loop, as shown in Fig.~1.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{fig1.pdf}
\caption{Graphical representation of the $N$-point fermion loop.}
\end{figure}

The sum of Feynman diagrams contributing at a certain order in the loop expansion can be written in terms of {\em symmetrized}\/ fermion loops,
\begin{equation}
 \Pi_{N,s}^{\rm sym}(q_1,\dots,q_N) = \frac{1}{N}
 \sum_{{\cal P}} \Pi_{N,s}({\cal P} q_1,\dots,{\cal P} q_N) \; ,
\end{equation}
with a sum over all permutations of $q_1,\dots,q_N$.
The prefactor $1/N$ compensates the $N$-fold multiplicity arising from cyclic permutations.
In various limits the symmetrized loops are much smaller than the unsymmetrized contributions, due to systematic cancellations.
Strong cancellations between contributions to symmetrized loops are well-known in one-dimensional Luttinger liquids~\cite{dzyaloshinskii74}, and have also been established for two-dimensional Fermi liquids~\cite{neumayr98, kopper01}.

In the remainder of our paper we evaluate the $N$-point loops and analyze their asymptotic behavior in various important limits relevant to quantum critical metals, from which we can derive improved power-counting estimates of Feynman diagrams.
To simplify the notation, we specify to fermion loops defined on the Fermi patch with $s=+$ and drop the patch index.
Results on the other patch ($s=-$) can be obtained from those for $s=+$ by simply switching the sign of the $x$-component of all momenta.
Furthermore, we absorb the prefactor $\kappa/N_f$ in the fermion propagator by a rescaling of the frequency variable. Hence, $\{ k_0 \}$ stands for $k_0/|k_0|^\alpha$ in the following sections.

\section{Integration of fermion loops}

The fermion loop as defined in Eq.~(\ref{PiN}) involves a momentum and a frequency integration. The momentum integral can be performed analytically by using the residue theorem (see Appendix A), yielding
\begin{eqnarray} \label{IN}
 I_N(p_1,\dots,p_N) &=& \sum_{i<j}
 \int_{p_{j0}}^{p_{i0}} \frac{dk_0}{4\pi} \,
 \Theta\Big(\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}}\Big) \nonumber \\
 &\times& (p_{iy} - p_{jy})^{N-3} \prod_{l \neq i,j} J_{ijl}(k_0)
\end{eqnarray}
for any $N \geq 3$. The indices $i,j,l$ run from 1 to $N$ with restrictions as indicated, $\Theta$ is the step function, and
\begin{equation} \label{Jijl}
 J_{ijl}(k_0) = \frac{1}{D_{ijl} + F_{ijl} + i\Omega_{ijl}(k_0)} \; ,
\end{equation}
with
\begin{eqnarray}
 D_{ijl} &=& p_{ix}(p_{ly} - p_{jy}) + {\rm cyc} \, , \label{Dijl} \\
 F_{ijl} &=& (p_{jy} - p_{iy})(p_{ly} - p_{jy})(p_{iy} - p_{ly}) \, ,
 \label{Fijl} \\[2mm]
 \Omega_{ijl}(k_0) &=&
 \{k_0 - p_{i0}\} (p_{ly} - p_{jy}) +
 {\rm cyc} \, , \label{Omijl}
\end{eqnarray}
where ``cyc'' denotes cyclic permutations of the indices $i,j,l$.
Note that the remaining frequency integration is limited to finite intervals and hence convergent. The fermion loop integral does not require any ultraviolet regularization.
In the static limit $q_{i0} = 0$, all frequency variables $p_{i0}$ are equal, such that $I_N(p_1,\dots,p_N) = 0$.

The quantities $D_{ijl}$, $F_{ijl}$, and $\Omega_{ijl}(k_0)$ are antisymmetric in the indices $i,j,l$. Hence, $J_{ijl}(k_0)$ is also antisymmetric in its indices.
Using this antisymmetry, the three-point loop can be written in the particularly simple form
\begin{eqnarray} \label{I3}
 I_3(p_1,p_2,p_3) &=& \sum_{(i,j) = (1,2),(2,3),(3,1)}
 \int_{p_{j0}}^{p_{i0}} \frac{dk_0}{4\pi} \,
 \Theta\Big(\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}}\Big) \nonumber \\
 &\times& J_{123}(k_0) \, .
\end{eqnarray}
Under the scaling Eq.~(\ref{scaling}), the quantities $D_{ijl}$, $F_{ijl}$, and $\Omega_{ijl}(k_0)$ scale homogeneously as
\begin{equation}
 D_{ijl} \mapsto \lambda^3 D_{ijl} , \;
 F_{ijl} \mapsto \lambda^3 F_{ijl} , \;
 \Omega_{ijl}(k_0) \mapsto \lambda^3 \Omega_{ijl}(k_0) \, .
\end{equation}
Hence $J_{ijl}(k_0) \mapsto \lambda^{-3} J_{ijl}(k_0)$ and
\begin{equation}
 I_N(p_1,\dots,p_N) \mapsto \lambda^{3+z-2N} I_N(p_1,\dots,p_N) \, ,
\end{equation}
in agreement with direct power-counting applied to the definition of $I_N(p_1,\dots,p_N)$ as a loop-integral.
The symmetrized fermion loops $\Pi_N^{\rm sym}(q_1,\dots,q_N)$ exhibit the same scaling behavior. As a consequence, effective boson interactions with amplitudes proportional to $\Pi_N^{\rm sym}(q_1,\dots,q_N)$ are marginal for all $N$~\cite{thier11}.

In the special case $\alpha = 0$, corresponding to a bare fermion progagator, the $k_0$-dependence in $\Omega_{ijl}(k_0)$ cancels in the cyclic sum, such that $\Omega_{ijl}(k_0) = \Omega_{ijl} = p_{i0} (p_{jy} - p_{ly}) + {\rm cyc}$.
The frequency integration in Eq.~(\ref{IN}) then becomes trivial, yielding
\begin{eqnarray} \label{IN0int}
 I_N^0(p_1,\dots,p_N) &=& \frac{1}{4\pi} \sum_{i<j}
 \Theta\Big(\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}}\Big)
 (p_{i0} - p_{j0}) \nonumber \\
 &\times& (p_{iy} - p_{jy})^{N-3} \prod_{l \neq i,j} J_{ijl} \; .
\end{eqnarray}
In particular, the three-point loop with bare propagators is obtained as
\begin{eqnarray} \label{I30}
 I_3^0(p_1,p_2,p_3) &=& \frac{1}{4\pi} \sum_{(i,j) = (1,2),(2,3),(3,1)}
 \Theta\Big( \frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}} \Big) \nonumber \\
 &\times& (p_{i0} - p_{j0}) \, J_{123} \, .
\end{eqnarray}
In the infrared scaling limit $p_{i0} \mapsto \lambda^z p_{i0}$, $p_{ix} \mapsto \lambda^2 p_{ix}$, $p_{iy} \mapsto \lambda p_{iy}$ with $z>2$ and $\lambda \to 0$, the term $\Omega_{123}$ is subleading compared to $D_{123}$ and $F_{123}$ in the denominator of $J_{123}$. In that limit, Eq.~(\ref{I30}) is consistent with the result for the $3$-point loop with a full Fermi surface integral obtained previously in Ref.~\onlinecite{thier11}.

\section{Asymptotic behavior of three-point loop}

In this section we derive asymptotic scaling properties of the three-point loop. In particular, we analyze the behavior in a limit that determines the ultraviolet behavior of the Aslamasov-Larkin-type diagrams depicted in Fig.~2.
These diagrams have been computed explicitly in the static limit (external $q_0=0$) by Metlitski and Sachdev~\cite{metlitski10}.
\begin{figure}[b]
\centering
\includegraphics[width=8cm]{fig2.pdf}
\caption{Aslamasov-Larkin type diagrams contributing to the bosonic self-energy in a coupled theory involving fermions (solid lines) and bosons (wiggly lines).}
\end{figure}

Denoting the external frequency-momentum variable in the Aslamasov-Larkin diagrams by $q$, and the bosonic loop integration variable by $q'$, the contributing fermion loops are $\Pi_3(q,q',-q-q') = I_3(0,q,q+q')$ and $\Pi_3(q,-q-q',q') = I_3(0,q,-q') = I_3(0,-q,-q-q')$.
In the static limit $q_0 = 0$, Eq.~(\ref{I3}) yields
\begin{eqnarray}
 I_3(0,q,q+q') &\stackrel{q_0=0}{=}& \int_0^{q'_0} \frac{dk_0}{4\pi}
 \left[ \Theta \Big( \frac{q'_0}{q_y + q'_y} \Big) -
 \Theta \Big( \frac{q'_0}{q'_y} \Big) \right] \nonumber \\
 &\times& J_{123}(k_0) \, ,
\end{eqnarray}
with
\begin{eqnarray}
 J_{123}(k_0) &=& \big[ q'_x q_y - q_x q'_y - q_y q'_y (q_y + q'_y)
 \nonumber \\
 && + i\left( \{k_0\} - \{k_0-q'_0\} \right) q_y \big]^{-1} \, .
\end{eqnarray}
Hence, $I_3(0,q,q+q')$ with $q_0 = 0$ is non-zero only if $|q'_y| < |q_y|$ and ${\rm sgn}(q'_y) = - {\rm sgn}(q_y)$. This {\em kinematic constraint} restricts the $q'_y$ integral to an interval of length $|q_y|$.

Now we can assess the ultraviolet (UV) behavior of the $q'$-integral in the Aslamasov-Larkin diagrams. We introduce an ultraviolet cutoff $\Lambda$ such that $|q'_0| < \Lambda^z$, $|q'_x| < \Lambda^2$, and $|q'_y| < \Lambda$. However, for fixed $q_y$, the $q'_y$-integral is effectively restricted by $q_y$. With this restriction, one has
\begin{equation}
 D_{123} \sim q_y \Lambda^2, \quad F_{123} \sim q_y^3, \quad
 \Omega_{123}(k_0) \sim q_y \Lambda^2 \, ,
\end{equation}
and thus $J_{123}(k_0) \sim (q_y \Lambda^2)^{-1}$, such that the UV behavior of the three-point loop is
\begin{equation}
 I_3(0,q,q+q') \sim \frac{\Lambda^{z-2}}{q_y} \, .
\end{equation}
The Aslamasov-Larkin diagrams contain two three-point fermion loops and two boson propagators. For $q'_y \sim q_y$ and $q'_0 \sim \Lambda^z$ the latter scale as $q_y/\Lambda^z$. The integration measure scales as $\Lambda^z \Lambda^2 q_y = \Lambda^{z+2} q_y$. Hence, the integral diverges in the UV limit as $q_y \Lambda^{z-2}$. This agrees with the result obtained in a more explicit calculation for $z=3$ by Metlitski and Sachdev~\cite{metlitski10}.
Note that the above contribution is obtained only if the two fermion loops are integrated over antipodal Fermi patches. For loops on the same Fermi patch, the $q'_x$-integral vanishes since the integration contour can be closed without encircling any poles in that case~\cite{lee09,metlitski10}.

The sum of all Aslamasov-Larkin diagrams can be expressed by symmetrized loops.
The symmetrized three-point loop can be written as
\begin{equation}
 \Pi_3^{\rm sym}(q_1,q_2,q_3) = I_3(p_1,p_2,p_3) + I_3(-p_1,-p_2,-p_3) \, .
\end{equation}
Under the inversion $p_i \mapsto -p_i$ and $k_0 \mapsto -k_0$, the quantities $D_{ijl}$, $F_{ijl}$ and $\Omega_{ijl}(k_0)$ transform as
\begin{equation}
 D_{ijl} \mapsto D_{ijl} , \quad F_{ijl} \mapsto - F_{ijl} , \quad
 \Omega_{ijl}(k_0) \mapsto \Omega_{ijl}(k_0) \, .
\end{equation}
The symmetrized three-point loop contributing to the Aslamasov-Larkin diagrams in the static limit can thus be written as
\begin{eqnarray}
 I_3^{\rm sym}(0,q,q+q') \stackrel{q_0=0}{=} \int_0^{q'_0} \frac{dk_0}{4\pi}
 \left[ \Theta \Big( \frac{q'_0}{q_y + q'_y} \Big) -
 \Theta \Big( \frac{q'_0}{q'_y} \Big) \right] \nonumber \\
 \times \! \left[
 \frac{1}{D_{123} \!+\! F_{123} \!+\! i\Omega_{123}(k_0)}
 - \frac{1}{D_{123} \!-\! F_{123} \!+\! i\Omega_{123}(k_0)} \right] \! . \hskip 3mm
\end{eqnarray}
In the ultraviolet limit the leading terms in the last set of brackets cancel, leaving a much smaller contribution of order $q_y/\Lambda^4$, such that the symmetrized three-point loop scales as
\begin{equation}
 I_3^{\rm sym}(0,q,q+q') \sim q_y \Lambda^{z-4} \, .
\end{equation}
This is by a factor $(q_y/\Lambda)^2$ smaller than the unsymmetrized three-point loop, that is, the degree of UV divergence has been reduced by two upon symmetrization.
The sum over the two inequivalent Aslamasov-Larkin diagrams is thus ultraviolet-finite at least for $z<4$, as shown previously for $z=3$ by Metlitski and Sachdev~\cite{metlitski10}.

\section{Asymptotic properties of $N$-point loop}

We have seen that the three-point loop is cut off by a kinematic constraint at large momenta, if one of the external momenta stays fixed, and cancellations occur upon symmetrization.
In this section we investigate to what extent these properties can be generalized to $N$-point loops. To this end, we first rewrite $I_N(p_1,\dots,p_N)$ in a form which is more convenient for an asymptotic analysis.

\subsection{Reduction formula for $N$-point loop}

Feldman et al.~\cite{feldman98} derived a {\em reduction formula} expressing the $N$-point loop with bare propagators and a quadratic dispersion relation as a linear combination of three-point loops,
\begin{equation} \label{IN0red}
 I_N^0(p_1,\dots,p_N) = \sum_{i<j<l}
 \Big[ \prod_{\nu \neq i,j,l} f_{ijl,\nu}^{-1} \Big] \,
 I_3^0(p_i,p_j,p_l) \, ,
\end{equation}
where $f_{ijl,\nu}$ is a rational function of $p_i,p_j,p_l,p_{\nu}$~\cite{neumayr99}.

For the $N$-point loop with non-Fermi liquid propagators of the form Eq.~(\ref{Gk}), a similar formula can be derived. In Appendix B we show that $I_N(p_1,\dots,p_N)$ as given by Eq.~(\ref{IN}) can be expressed as
\begin{eqnarray} \label{INred}
 I_N(p_1,\dots,p_N) &=& \sum_{i<j<l}
 \int_{p_{j0}}^{p_{i0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{i0}-p_{j0}}{p_{iy}-p_{jy}} \Big) \nonumber \\
 &\times& \Big[ \prod_{\nu \neq i,j,l} f_{ijl,\nu}^{-1}(k_0) \Big]
 J_{ijl}(k_0) + {\rm cyc} , \hskip 5mm
\end{eqnarray}
where
\begin{equation} \label{fk0}
 f_{ijl,\nu}(k_0) =
 \frac{D_{ijl} J_{ij\nu}^{-1}(k_0) - D_{ij\nu} J_{ijl}^{-1}(k_0)}
 {D_{ijl} (p_{iy} - p_{jy})} \, ,
\end{equation}
with $J_{ijl}(k_0)$ and $D_{ijl}$ from Eqs.~(\ref{Jijl}) and (\ref{Dijl}), respectively. ``cyc'' denotes cyclic permutations of $i,j,l$.
Expressing $J_{ijl}(k_0)$ and $D_{ijl}$ explicitly in terms of momenta and frequencies, one finds that the product $D_{ijl} f_{ijl,\nu}(k_0)$ is an antisymmetric polynomial of the form
\begin{eqnarray}
 D_{ijl} f_{ijl,\nu}(k_0) &=&
 \left| \begin{array}{llll}
 p_{ix} & p_{jx} & p_{lx} & p_{\nu x} \\
 p_{iy} & p_{jy} & p_{ly} & p_{\nu y} \\
 p_{iy}^2 & p_{jy}^2 & p_{ly}^2 & p_{\nu y}^2 \\
 1 & 1 & 1 & 1 \end{array} \right| \nonumber \\
 &-& i \left| \begin{array}{cccc}
 p_{ix} & p_{jx} & p_{lx} & p_{\nu x} \\
 p_{iy} & p_{jy} & p_{ly} & p_{\nu y} \\
 \{k_{i0} \} & \{k_{j0} \} & \{k_{l0} \} & \{ k_{\nu 0} \} \\
 1 & 1 & 1 & 1
 \end{array} \right| , \hskip 5mm
\end{eqnarray}
with $k_{i0} = k_0 - p_{i0}$. Hence, $D_{ijl} f_{ijl,\nu}(k_0)$ is totally antisymmetric in all four indices. Since $D_{ijl}$ is also antisymmetric, it follows that $f_{ijl,\nu}$ is invariant under permutations of $i,j,l$.

For loops constructed with bare propagators $G_0$, the quantities $J_{ijl}$ and $f_{ijl,\nu}$ are independent of $k_0$. The frequency integration in Eq.~(\ref{INred}) can then be carried out and one recovers the reduction formula for $I_N^0(p_1,\dots,p_N)$ in the form Eq.~(\ref{IN0red}).

\subsection{Kinematic constraint for $N$-point loop}

We now analyze the asymptotic behavior of an $N$-point loop traversed by a single large momentum. More precisely, we assume that two of the external momenta, say $q'$ and $q'' \approx -q'$ are large and almost antiparallel, while all the other momenta are kept finite, that is, relatively small.
This implies that the momenta $p_i$ fall in two groups, where momenta within a group are close together, while the distance between momenta in different groups is large (see Fig.~3). Choosing $p_1 = 0$, the momenta in the group containing $p_1$ are all close to zero, that is, relatively small.
\begin{figure}[b]
\centering
\includegraphics[width=5cm]{fig3.pdf}
\caption{$N$-point loop with two large momenta $q'$ and $q'' \approx - q'$. Momenta $p_i$ on the bold lines are large, those on thin lines (relatively) small.}
\end{figure}

Using the generalized reduction formula (\ref{INred}), we can show that the frequency integrations are effectively restricted to small intervals, although the differences $p_{i0} - p_{j0}$ may be large.
To this end, let us fix some $i,j,l$ and write out the cyclic sum:
\begin{eqnarray} \label{cycle1}
 && \int_{p_{j0}}^{p_{i0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}} \Big) g(k_0) \nonumber \\
 &+& \int_{p_{l0}}^{p_{j0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{j0} - p_{l0}}{p_{jy} - p_{ly}} \Big) g(k_0) \nonumber \\
 &+& \int_{p_{i0}}^{p_{l0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{l0} - p_{i0}}{p_{ly} - p_{iy}} \Big) g(k_0) \, ,
\end{eqnarray}
where $g(k_0) =
 \Big[ \prod_{\nu \neq i,j,l} f_{ijl,\nu}^{-1}(k_0) \Big] J_{ijl}(k_0)$.
Note that the latter function is invariant under cyclic permutations of $i,j,l$.

If $p_i$, $p_j$, $p_l$ are in the same group of small or large momenta, all frequency integrations in (\ref{cycle1}) are obviously limited to small intervals. Now assume that they are in two distinct groups, say $p_i$ is large and the others are small. The argument of the first and third step function in (\ref{cycle1}) is then dominated by $p_i$, leaving $\Theta(p_{i0}/p_{iy})$ in both cases, and the whole expression can be simplified to
\begin{equation} \label{cycle2}
 \int_{p_{j0}}^{p_{l0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{i0}}{p_{iy}} \Big) g(k_0) +
 \int_{p_{l0}}^{p_{j0}} \frac{dk_0}{4\pi}
 \Theta\Big( \frac{p_{j0} - p_{l0}}{p_{jy} - p_{ly}} \Big) g(k_0) \, .
\end{equation}
The frequency integration is thus effectively restricted to the small interval between $p_{j0}$ and $p_{l0}$. The large contributions from integrations from $p_{j0}$ to $p_{i0}$ and from $p_{i0}$ to $p_{l0}$ cancel in the cyclic sum.

In naive power-counting for the $N$-point loop, the frequency integration yields a factor $\Lambda^z$. In the specific limit discussed above, the ultraviolet asymptotics is thus reduced by that factor.
For the above argument it was important that both $q'_0$ and $q'_y$ are large. The $x$-component did not matter. If only $q'_0$ is assumed to be large, the $k_0$-integration does extend over a large interval, yielding a factor $\Lambda^z$, but only as long as $q'_y$ is of the order of the other $y$-components. One thus obtains an effective restriction of $q'_y$ as discussed already for the three-point loop in Sec.~IV.

In summary, we have shown that a fermion loop traversed by a {\em single large}\/ momentum is suppressed by cancellations in the cyclic sum in Eq.~(\ref{INred}).
No such cancellation occurs if three or more momenta $q_i$ are large. In that case three momenta $p_i$, $p_j$, $p_l$ can be far apart from each other so that no systematic cancellation occurs in the cyclic sum.

\subsection{Symmetrized N-point loop with one or two small momenta}

We now show that systematic cancellations occur in the {\em symmetrized}\/ N-point loop in the limit of one or two {\em vanishing}\/ momenta.
This generally reduces the degree of divergence of Feynman diagrams with bosonic external legs coupled to a fermion loop.

We first show that the symmetrized $N$-point loop vanishes, if one of the external momenta vanishes.
The symmetrized $N$-point loop is given by a sum over all permutations of external momenta.
For a vanishing external momentum two fermion lines in the loop carry the same internal momentum. By permutations the vertex with the vanishing momentum is cycled around the loop, yielding the sum
\begin{equation}
 \int \frac{dk_0}{2\pi} \int \frac{d^2{\bf k}}{(2\pi)^2} \,
 \sum_{j=1}^{N-1} G(k-p_j) \prod_{i=1}^{N-1} G(k-p_i) \, .
\end{equation}
Since the denominator of $G(k-p_i)$ is linear in $k_x$, the integrand can be written as a $k_x$-derivative,
$- \frac{\partial}{\partial k_x} \prod_{i=1}^{N-1} G(k-p_i)$.
Performing the $k_x$-integration one thus finds that the symmetrized $N$-point loop vanishes if one leg has a vanishing momentum.
By dimensional analysis, the UV scaling of the symmetrized loop with one fixed external momentum is thus reduced by a factor $\Lambda^{-1}$.

Using analyticity and the invariance under $q_i \mapsto -q_i$ one can conclude that symmetrized loops vanish even quadratically, if a vanishing momentum $q$ enters and leaves the same loop at two distinct vertices, provided that the other momenta remain finite. Due to momentum conservation this is possible only for $N \geq 4$. Hence, the UV scaling of symmetrized loops for $N-2$ large momenta and two fixed external momenta $q$ and $-q$ is reduced by a factor $\Lambda^{-2}$.

\section{Improved power-counting}

We now use the above results on the asymptotic behavior of fermion loops to obtain improved power-counting estimates for the ultraviolet behavior of several classes of Feynman diagrams. We consider contributions to the boson and fermion self-energies, the fermion-boson vertex, and the three-boson vertex.

\subsection{Boson self-energy}

Perturbative contributions to the boson self-energy $\Sigma_b(q)$ with $L$ loop-integrations contain $L-1$ boson propagators and $2L$ fermion propagators.
The boson propagators decay as $\Lambda^{1-z}$ for large momenta, the fermion propagators as $\Lambda^{-2}$. Each loop integration contributes a power $\Lambda^{3+z}$. Hence, according to naive power-counting, $L$-loop contributions to the boson self-energy may diverge as $\Lambda^{z-1}$. In particular, for $z=3$ a quadratic UV divergence seems possible for any loop order $L$.

However, systematic cancellations of UV divergences occur between distinct contributions (represented by distinct Feynman diagrams) at any given loop order. Feynman diagrams contributing to the boson self-energy $\Sigma_b(q)$ contain fermion loops but no open fermion lines. Hence, the external boson legs with ingoing and outgoing momentum $q$ connect directly to fermion loops. The sum over all diagrams of a certain order can be written in terms of symmetrized fermion loops.
In Sec.~V.C we have shown that in symmetrized fermion loops with a fixed (relatively small) momentum $q$, the UV divergences are suppressed by a factor $\Lambda^{-1}$ upon symmetrization, and in symmetrized loops with two fixed momenta $q$ and $-q$ by a factor $\Lambda^{-2}$.
In Feynman diagrams contributing to the boson self-energy the external momenta $q$ and $-q$ are either attached to two distinct loops or to just one loop.
In both cases there is a power-counting gain of the order $\Lambda^{-2}$.
Hence, the sum over all $L$-loop contributions to the boson self-energy behaves as $\Lambda^{z-3}$ for large $\Lambda$. For $z<3$ this means that the sum is UV-convergent. For $z=3$, quadratic and linear UV divergences must cancel. This cancellation is also imposed by a Ward identity following from current conservation~\cite{metlitski10}.
However, logarithmic UV divergences are not excluded for $z=3$.

At three-loop order, the only candidates for a logarithmic UV divergence for $z=3$ are the Aslamasov-Larkin diagrams (see Fig.~2). However, as discussed already in Sec.~IV, these diagrams are restricted by a kinematic constraint arising from cancellations in the representation of the three-point fermion loop as a cyclic sum, so that the total contribution is UV-finite for any $z<4$.

As shown in Sec.~V.B, a kinematic constraint from a cancellation of contributions to the $N$-point fermion loops in the UV limit occurs only if only one large momentum traverses the loop.
Contributions to the boson self-energy where this condition is {\em not}\/ satisfied occur at four-loop order. An example is shown in Fig.~4.
\begin{figure}[tb]
\centering
\includegraphics[width=7cm]{fig4.pdf}
\caption{A class of four-loop diagrams with fermion loops connected to three boson propagators with possibly large momenta.}
\end{figure}
In Ref.~\onlinecite{holder15}, those contributions (with symmetrized fermion loops) were computed explicitly for $z=3$ in the static limit $q_0 = 0$, and were indeed found to diverge logarithmically. Surprisingly, the divergence turned out to be of the order $\log^5(\Lambda/|q_y|)$ instead of the expected simple logarithm. Such a divergence is not renormalizable. The implications of that singularity are not clear at the moment.

It is remarkable that the one-loop result receives qualitative corrections only at four-loop order, while the dynamical exponent $z=3$ remains unchanged at two- and three loop order. Such a situation is peculiar but not unprecedented in quantum field theory.
For example, in non-linear $\sigma$-models describing the critical behavior of the Anderson localization transition, only the four-loop contributions lift a degeneracy between distinct symmetry groups and correct the one-loop result for critical localization length and conductivity exponents~\cite{wegner89}.

\subsection{Fermion self-energy}

Perturbative contributions to the fermion self-energy $\Sigma_f(k)$ with $L$ loops contain $L$ boson propagators and $2L-1$ fermion propagators. Hence, $L$ loop contributions to the fermion self-energy diverge as $\Lambda^2$ in the ultraviolet limit for any $L$ and $z$. This divergent contribution is however momentum and frequency independent and can be absorbed by a shift of the Fermi surface. Physically more interesting is the momentum and frequency dependence of the self-energy. In particular, the derivative $\partial\Sigma_f/\partial k_+$ with $k+ = k_x + k_y^2$ determines the anomalous dimension of the fermion fields~\cite{metlitski10}. Since $k_x$ and $k_y^2$ scale as $\Lambda^2$, that derivative scales as $\Lambda^0$ according to the above power-counting. Hence, logarithmic UV divergences are expected, and indeed occur already at three-loop order (see Fig.~5), as discovered for $z=3$ by Metlitski and Sachdev~\cite{metlitski10}.
The three-point fermion loop in these three-loop contributions is not reduced by any of the cancellations we discussed above, since all three boson momenta can be large.

\begin{figure}[htb]
\centering
\includegraphics[width=8cm]{fig5.pdf}
\caption{Logarithmically divergent three-loop contributions to the fermion self-energy.}
\end{figure}

\subsection{Fermion-boson vertex}

We now discuss the fermion-boson vertex $\Gamma(k,q)$ with one boson and two fermion legs. Perturbative $L$-loop vertex corrections contain $L$ boson propagators and $2L$ fermion propagators. Hence, $\Gamma(k,q)$ scales as $\Lambda^0$ in the UV limit, that is, logarithmic divergences are expected.
The vertex corrections can be grouped in two classes. In the first case, the external boson leg couples to an open fermion line, and in the second to a fermion loop. In Fig.~6 we provide a three-loop example for each case.
\begin{figure}[htb]
\centering
\includegraphics[width=8cm]{fig6.pdf}
\caption{Two types of vertex corrections with the external boson leg linked to an open fermion line (left) and to a fermion loop (right).}
\end{figure}
In the first case we do not expect any cancellations, but in the second we have a fermion loop with a fixed boson momentum $q$. Hence, summing all diagrams corresponding to the symmetrized fermion loop, the UV contributions are suppressed by a factor $\Lambda^{-1}$ so that the sum is guaranteed to be finite.

The fermion-boson vertex is related to the fermion self-energy by the usual Ward-identity following from charge conservation~\cite{metlitski10}.
Since the fermion self-energy exhibits a logarithmic divergence already at three-loop order, the fermion-boson vertex has to diverge, too. The above argument shows that the divergent contributions to the vertex come exclusively from those diagrams where the boson leg couples to an open fermion line.

\subsection{Three-boson vertex}

As a final example, we consider the three-boson vertex.
Its lowest order (one loop) contribution is simply the symmetrized three-point fermion loop $\Pi_3^{\rm sym}$, which does not exhibit any UV divergence.
$L$-loop corrections to the three-boson vertex contain $L-1$ boson propagators and $2L+1$ fermion propagators. Naive power-counting thus yields a UV scaling of the form $\Lambda^{z-3}$. For $z=3$ this is marginal.
However, each of the external boson legs is linked directly to a fermion loop
(see, for example, Fig.~7). Upon loop symmetrization possible UV divergences are thus suppressed. Without investigating the exact degree of suppression we can say that in the most interesting physical case $z=3$ all corrections will be UV finite.
The UV convergence of $N$-loop vertices with $N>3$ is also improved by cancellations in the symmetrized fermion loops, but they are already UV convergent even within naive power-counting for $z < 2N-3$.
\begin{figure}[htb]
\centering
\includegraphics[width=5cm]{fig7.pdf}
\caption{Four-loop diagram contributing to the three-boson vertex.}
\end{figure}

\section{Conclusion}

We have analyzed general properties of the perturbative loop expansion for two-dimensional quantum critical metals with singular forward scattering.
Important cases falling in this class of systems are metals at an Ising nematic QCP and metals coupled to a $U(1)$ gauge field. Our analysis is based on the effective field theory for such systems~\cite{lee09,metlitski10}, extended to arbitrary dynamical exponents $z>2$ as proposed earlier by Nayak and Wilczek~\cite{nayak94}.

We have derived asymptotic properties of fermion loops appearing as subdiagrams of Feynman diagrams contributing to the perturbation series.
Substantial cancellations occur when summing over contributions corresponding
to permutations of vertices at fermion loops with a small external momentum $q$ or two small momenta $q$ and $-q$ (while all the other momenta become large).
Furthermore, fermion loops are suppressed below the naive power-counting estimate when they are traversed by only one large bosonic momentum.

Using these properties we have shown in a number of specific examples how one can perform a {\em sharp}\/ ultraviolet power-counting that takes all possible cancellations into account.
Most importantly, for the boson self-energy there is always a gain of order
$\Lambda^2$. As a consequence, perturbative contributions to that quantity are UV convergent for $z<3$. Hence, perturbative self-energy corrections to the boson propagator are finite at all loop orders for $z<3$. In the static limit $q_0 = 0$, the boson self-energy is then fixed by dimensional analysis to be proportional to $q_y^{z-1}$ with a finite prefactor.
A bare $z<3$ is thus not renormalized by fluctuations.
On the other hand, logarithmic divergences are possible in the important physical case with a dynamical exponent $z=3$, and indeed do occur at four-loop order~\cite{holder15}.
For $z>3$ there are UV divergences proportional to $\Lambda^{z-3}$.

Obviously, $z=3$ is a critical value at the boundary between the qualitatively different cases $z<3$ and $z>3$. Interestingly, the same critical value $z=3$ separates two qualitatively different behaviors of the {\em compressibility}, if the chemical potential $\mu$ couples to the bosonic mass appearing away from the QCP. For $z < 3$ the compressibility is finite, while for $z \geq 3$ it diverges upon approaching the QCP~\cite{metlitski10}. Our result that a bare $z < 3$ is not renormalized by fluctuations at any loop order thus implies that the compressibility remains finite in that case.

There are no systematic cancellations reducing the UV divergences of the fermion self-energy. Hence, the fermionic field renormalization obtained from a first momentum derivative of the self-energy diverges logarithmically for any $z$. In contrast to the behavior of the boson self-energy, the degree of divergence of the fermion self-energy is independent of $z$. Logarithmic divergences obtained in perturbation theory can be summed up to yield anomalous power-law scaling, as demonstrated in the three-loop calculation for $z=3$ by Metlitski and Sachdev~\cite{metlitski10}.

At this point, the perturbative singularity structure of the above class of critical metals seems to be clarified for $z<3$. The one-loop fixed point is only modified by an anomalous scaling of the fermion fields. However, the final fate of the theory in the special but physically relevant case $z=3$ remains open.
Organizing the loop expansion in terms of symmetrized fermion loops and exploiting the cancellations described above should be useful for any future work on fluctuation corrections. In particular, a numerical computation of fluctuation corrections is greatly facilitated by incorporating the cancellations directly via symmetrized loops.

\begin{acknowledgments}
We are grateful to A.~Eberlein, M.~Metlitski, D.~Vollhardt, and R.~Zeyher for valuable discussions.
\end{acknowledgments}

\newpage

\begin{appendix}

\section{Integration of loops by residues}

Here we derive Eq.~(\ref{IN}) for the $N$-point fermion loop
\begin{equation}
 I_N(p_1,\dots,p_N) =
 \int_{-\infty}^{\infty} \! \frac{dk_0}{2\pi} \!
 \int_{-\infty}^{\infty} \! \frac{dk_x}{2\pi} \!
 \int_{-\infty}^{\infty} \! \frac{dk_y}{2\pi} \!
 \prod_{i=1}^N G(k-p_i) \, ,
\end{equation}
where $G(k) = \left[ k_x + k_y^2 - i\{k_0\} \right]^{-1}$
with $\{k_0\} = k_0/|k_0|^{\alpha}$.
The $k_x$-integration can be performed by closing the integration contour in the upper complex half-plane and using the residue theorem. There are $N$ simple poles situated at $k_x = i\{k_0 - p_{i0}\} + p_{ix} - (k_y - p_{iy})^2$ with $i=1,\dots,N$.
Summing the contributions from all poles in the upper half plane ($k_0 > p_{i0}$) yields
\begin{widetext}
\begin{eqnarray}
 I_N(p_1,\dots,p_N) &=& i\int_{-\infty}^{\infty} \frac{dk_0}{2\pi}
 \int_{-\infty}^{\infty} \frac{dk_y}{2\pi} \,
 \sum_{i=1}^N \Theta(k_0 - p_{i0}) \nonumber \\
 && \hskip -1cm \times \prod_{j \neq i} \, \frac{1}
 {i\{k_0 - p_{i0}\} - i\{k_0 - p_{j0}\} + p_{ix} - p_{jx} +
 2(p_{iy} - p_{jy})k_y - p_{iy}^2 + p_{jy}^2} \, . \hskip 1cm
\end{eqnarray}
Now the $k_y$-integration can be performed analogously.
For each $i$ there are $N-1$ simple poles at
\begin{equation}
 k_y = \frac{i\{k_0 - p_{j0}\} - i\{k_0 - p_{i0}\} - p_{ix} + p_{jx} +
 p_{iy}^2 - p_{jy}^2}{2(p_{iy} - p_{jy})} \nonumber
\end{equation}
with $j \neq i$. These poles are situated in the upper complex half plane if (and only if) $\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}} > 0$. Closing the integration contour in the upper complex half plane thus yields
\begin{eqnarray} \label{A3}
 I_N(p_1,\dots,p_N) &=&
 - \int_{-\infty}^{\infty} \frac{dk_0}{2\pi}
 \sum_{i} \sum_{j\neq i} \Theta(k_0 - p_{i0})
 \Theta\Big(\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}}\Big)
 \frac{1}{2(p_{iy} - p_{jy})} \nonumber \\
 &\times& \prod_{l \neq i,j} \Big[
 i\{k_0 - p_{i0}\} - i\{k_0 - p_{l0}\} + p_{ix} - p_{lx} - p_{iy}^2 + p_{ly}^2
 \nonumber \\
 &+& \frac{p_{iy} - p_{ly}}{p_{iy} - p_{jy}} (i\{k_0 - p_{j0}\}
 - i\{k_0 - p_{i0}\} - p_{ix} + p_{jx} + p_{iy}^2 - p_{jy}^2) \Big]^{-1}.
 \hskip 5mm
\end{eqnarray}
Defining
\begin{eqnarray}
 D_{ijl} &=& - {\rm det} \left( \begin{array}{cc}
 p_{ix} - p_{jx} & p_{ix} - p_{lx} \\
 p_{iy} - p_{jy} & p_{iy} - p_{ly} \end{array} \right) = p_{ix} (p_{ly} - p_{jy}) + {\rm cyc} \, , \\[2mm]
 F_{ijl} &=& (p_{jy} - p_{iy})(p_{ly} - p_{jy})(p_{iy} - p_{ly})
 \, , \\[2mm]
 \Omega_{ijl}(k_0) &=& \{k_0 - p_{i0}\} (p_{ly} - p_{jy}) + {\rm cyc} \, ,
\end{eqnarray}
Eq.~(\ref{A3}) can be written as
\begin{eqnarray}
 I_N(p_1,\dots,p_N) &=&
 - \int_{-\infty}^{\infty} \frac{dk_0}{4\pi}
 \sum_{i} \sum_{j\neq i} \Theta(k_0 - p_{i0})  \Theta\Big(\frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}}\Big)
 (p_{iy} - p_{jy})^{N-3}  \prod_{l \neq i,j}
 \left[ D_{ijl} + F_{ijl} + i\Omega_{ijl}(k_0) \right]^{-1} . \hskip 8mm
 \end{eqnarray}
Defining
$J_{ijl}(k_0) = \left[ D_{ijl} + F_{ijl} + i\Omega_{ijl}(k_0) \right]^{-1}$,
and using the antisymmetry $J_{ijl}(k_0) = - J_{jil}(k_0)$, one obtains $I_N(p_1,\dots,p_N)$ as a sum over indices $i$ and $j$ restricted to $i<j$ in the form Eq.~(\ref{IN}).

\section{Derivation of reduction formula}

In this Appendix we derive the expression Eq.~(\ref{INred}) for the $N$-point loop from Eq.~(\ref{IN}).
The crucial step is the identity
\begin{eqnarray} \label{redN}
 (p_{iy} - p_{jy})^{N-3} \prod_{l \neq i,j} J_{ijl}(k_0) &=&
 \sum_{l \neq i,j}
 \Big[ \prod_{\nu \neq i,j,l} f_{ijl,\nu}^{-1}(k_0) \Big]  J_{ijl}(k_0) \, ,
\end{eqnarray}
where $f_{ijl,\nu}(k_0)$ is defined as in Eq.~(\ref{fk0}), that is,
\begin{equation}
 f_{ijl,\nu}(k_0) =
 \frac{D_{ijl} J_{ij\nu}^{-1}(k_0) - D_{ij\nu} J_{ijl}^{-1}(k_0)}
 {D_{ijl} (p_{iy} - p_{jy})} \, .
\end{equation}
This can be verified by first considering the cases $N=4$ and $N=5$, and then proceeding by induction.
For $N=4$, Eq.~(\ref{redN}) reads (suppressing the $k_0$-dependence)
\begin{equation} \label{red4}
 (p_{iy} - p_{jy}) J_{ijl_1} J_{ijl_2} =
 f_{ijl_1,l_2}^{-1} J_{ijl_1} + f_{ijl_2,l_1}^{-1} J_{ijl_2} \, ,
\end{equation}
which follows from
\begin{eqnarray}
 J_{ijl_1} J_{ijl_2} &=& J_{ijl_1} J_{ijl_2} \Big(
 \frac{D_{ijl_1} J_{ijl_1}}{D_{ijl_1} J_{ijl_1} - D_{ijl_2} J_{ijl_2}} +
 l_1 \leftrightarrow l_2 \Big)
 =
 \frac{D_{ijl_1}}{D_{ijl_1} J_{ijl_2}^{-1} - D_{ijl_2} J_{ijl_1}^{-1}} J_{ijl_1} +
 l_1 \leftrightarrow l_2 \, .  \nonumber
\end{eqnarray}
For $N=5$, one finds from repeated application of Eq.~(\ref{red4})
\begin{eqnarray}
 (p_{iy} - p_{jy})^2 J_{ijl_1} J_{ijl_2} J_{ijl_3} &=&
 f_{ijl_1,l_2}^{-1} f_{ijl_1,l_3}^{-1} J_{ijl_1}
+ f_{ijl_2,l_1}^{-1} f_{ijl_2,l_3}^{-1} J_{ijl_2}
   + (f_{ijl_2,l_1}^{-1} f_{ijl_3,l_2}^{-1} +
      f_{ijl_1,l_2}^{-1} f_{ijl_3,l_1}^{-1}) J_{ijl_3} \nonumber \\
&=& f_{ijl_1,l_2}^{-1} f_{ijl_1,l_3}^{-1} J_{ijl_1} +
   f_{ijl_2,l_1}^{-1} f_{ijl_2,l_3}^{-1} J_{ijl_2}
+ f_{ijl_3,l_1}^{-1} f_{ijl_3,l_2}^{-1} J_{ijl_3} ,
\end{eqnarray}
where in the last step we have used
$f_{ijl_2,l_1}^{-1} f_{ijl_3,l_2}^{-1} + f_{ijl_1,l_2}^{-1} f_{ijl_3,l_1}^{-1} =
 f_{ijl_3,l_1}^{-1} f_{ijl_3,l_2}^{-1}$, which can be verified by explicit calculation.
It is now clear that Eq.~(\ref{redN}) follows from $(N-3)$-fold iterated application of Eq.~(\ref{red4}).

Inserting Eq.~(\ref{redN}) into Eq.~(\ref{IN}) yields
\begin{eqnarray}
 I_N(p_1,\dots,p_N) &=& \sum_{i<j} \sum_{l \neq i,j}
 \int_{p_{j0}}^{p_{i0}} \frac{dk_0}{4\pi} \,
 \Theta\Big( \frac{p_{i0} - p_{j0}}{p_{iy} - p_{jy}} \Big)
 \Big[ \prod_{\nu \neq i,j,l} f_{ijl,\nu}^{-1}(k_0) \Big]
 J_{ijl}(k_0) \, .
\end{eqnarray}
Using the invariance of $J_{ijl}(k_0)$ and $f_{ijl,\nu}(k_0)$ under cyclic permutations of $i,j,l$ one obtains $I_N(p_1,\dots,p_N)$ in the cyclic form Eq.~(\ref{INred}).

\end{widetext}

\end{appendix}

redpajama_set_name: RedPajamaArXiv


To SSJ's first Alpha for Junior High Students! Alpha is a series of interactive sessions that explore the basics of the Christian faith. The Alpha Youth Film Series is designed to engage youth in conversations about life, faith and meaning.
Each session will begin at 10:15 AM after the 9AM Mass in the Youth House. We will have food, a talk and dicussion at each Alpha. The session will go until 11:30 AM.
Teens are encouraged to invite other 7th and 8th grade students who would like to explore this to attend. You can register your teen by emailing Jen Pitera at jpitera@simonjude.org.
** 2/24 will be our Alpha Day Away and will go from 9AM to 6 PM.
Location information will be given at the beginning of February.


text:
US winter storm: Hundreds of thousands face mass outages on Christmas Day
By Euronews with AP • Updated: 25/12/2022
Many have found themselves stranded as a major snow storm pummeled North America - Copyright AP/Derek Gee/The Buffalo News
Dozens of people have died in the US as an extreme winter storm continues to batter parts of the country, with blizzards causing dangerous travel conditions.
Across the U.S., officials have attributed deaths to exposure, car crashes, a falling tree limb and other effects of the storm.
The storm unleashed its fury on Buffalo, in the state of New York, with hurricane-force winds causing whiteout conditions.
Forecasters said 71 centimetres of snow accumulated as of Saturday in Buffalo.
More than 315,000 homes and businesses were without power on Christmas Eve. Thousands of flights have been cancelled during the festive period.
Blinding blizzards, freezing rain and frigid cold also knocked out power in places from Maine to Seattle, while a major electricity grid operator warned the 65 million people it serves across the eastern U.S. that rolling blackouts might be required.
The storm is nearly unprecedented in its scope, stretching from Canada to the Texas border.
About 60% of the U.S. population faced some sort of winter weather advisory or warning.
How a deadly storm turned a French resort into a climate change laboratory
Tropical Storm Lisa: Hurricane warnings as high winds head for Belize
Hurricane Ian: Dozens confirmed dead as storm heads north after devastating Florida and Carolinas
USA COLD EXTREME WEATHER


text:
YMCA Trinity Group is part of a federation of over 116 YMCAs across England and Wales. Our vision is to help create supportive, inclusive and transforming communities where young people can truly belong, contribute and thrive. At the heart of how we work in communities, and with each other, are strong and committed relationships rooted in our inclusive values, approach and heritage. As a federation, YMCA England and Wales has developed a strategic plan, covering five core areas of work which all YMCAs follow. Our approach is based on having a national presence with local relevance, and is achieved by YMCA Trinity Group in a variety of ways across the five core areas of work as follows:
Our values are based on that:
We believe every young person should have someone they can trust
We believe every young person should have a safe place to stay
We believe every family should have the support they need to develop and lead fulfilling lives
We believe everyone should enjoy the benefits of good health and wellbeing
We believe every young person should be able to fulfill their potential
See how we put our vision and values into action reading our the stories in our news section.
You can get involved in helping us by:
Donate to/ visit our charity shops
YMCA Trinity Group
Cambridge:
Gonville Place
Cambridge CB1 1ND
NCA No: H4179
YMCA Trinity Group is committed to safeguarding and promoting the welfare of vulnerable groups and expects all staff and volunteers to share this commitment.
Website by Adrenaline Creative
YMCA enables people to develop their full potential in mind, body and spirit. Inspired by, and faithful to, our Christian values, we create supportive, inclusive and energising communities, where young people can truly belong, contribute and thrive.


00:10 Cinderbeak transforms into a Fast Air Animal and strafes Apekick. Cinderbeak strafes Apekick. Cinderbeak strafes Apekick. Apekick is slightly damaged (85% power remaining). Apekick is slightly damaged (80% power remaining). Apekick is injured (71% power remaining).
00:17 Apekick attacks paul porsche359 2275 with a Reaver-Annihilation Assault Rifle. paul porsche359 2275 transforms into a Light Rescue Vehicle and avoids the attack!
00:35 Apekick transforms into a Tough Land Animal and rams Cinderbeak. Cinderbeak is injured (71% power remaining).
00:56 Apekick transforms into a Tough Land Animal and rams paul porsche359 2275. paul porsche359 2275 is slightly damaged (76% power remaining).
01:05 Cinderbeak hits Apekick with a Reaver-Annihilation Assault Rifle. Apekick is damaged (46% power remaining).
01:11 Cinderbeak hits Apekick with a Reaver-Annihilation Assault Rifle. Apekick is severely injured (17% power remaining).
01:19 Apekick transforms into a Tough Land Animal and rams Cinderbeak. Cinderbeak is damaged (38% power remaining).
01:26 Cinderbeak misses Apekick with a Reaver-Annihilation Assault Rifle.
01:33 Cinderbeak transforms into a Fast Air Animal and strafes Apekick. Cinderbeak strafes Apekick. Apekick is severely damaged (7% power remaining). Apekick has gone into stasis lock!