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One sentence summarization
For a student initially working on a more phenomenological side of the high energy physics study, what is the recommendation of introductory reading materials for them to dive into a more mathematically rigorous study of the quantum field theory.
"phenomenology side of the high energy physics study" basically means that when this student uploads their article to arXiv, they will, in principle, choose hep-ph as the primary archive. This background also indicates that this student has, in general, already learned at least
a. Undergraduate level physics classes;
b. QFT from the first half of Peskin and Schroeder's classical textbook, or, maybe, Greiner's Field Quantization;
c. Complex analysis like those covered in the first five chapters of Conway's Functions of One Complex Variable I;
d. Some basic knowledge about special functions, like Laguerre Polynomials;
e. Elementary set theory;
f. General topology like those covered in Amstrong's Basic Topology;
g. Elementary vector space theory with no details about the operator algebra;
h. Some elementary conclusions from group theory, with a focus on the Lie group and Lorentz group;
i. Enough particle physics so that they is able to understand what the summary tables of the PDG are talking about, or they at least knows how to find the definition of some of those unknown symbols.
"more mathematically rigorous study of quantum field theory" may have different meanings for different people. Just to give an example, Prof. Tachikawa from IPMU had a talk titled "Mathematics of QFT, by QFT, for QFT" on his website (https://member.ipmu.jp/yuji.tachikawa/transp/qft-tsukuba.pdf), which the OP find really tasteful, of course, from a rather personal point of view. One may agree that all those studies covered there may be considered mathematically rigorous studies of quantum field theory. Or, one may agree on only part of them. Or, maybe the answerer thinks that some other researches also count. That is all OK. But, please specify your altitude before elaborating your answer. The answerer may also want to specify explicitly for which subfield they is recommending references. In the OP's understanding, "more mathematically rigorous study of quantum field theory" basically covers those topics
a. Algebraic/Axiomatic QFT, Constructive QFT or the Yang-Mills millennium problem, which tries to provide a mathematically sounding foundation of QFT.
b. Topological QFT or Cconformal QFT, which utilizes some fancy mathematical techniques to study QFT.
c. Even some math fields stemming from QFT like those fields-awarding work done by Witten.
It is best that those "introductory reading materials" could focus on the mathematical prerequisites of that subfield. It will be really appreciated if the answerer could further explain why one has to know such knowledge before doing research in that subfield. Answers may choose from a really formal vibe or a physics-directing vibe when it comes to the overall taste of their recommendation and may want to point out that difference explicitly. It is also a good recommendation if the answerer lists some review paper, newest progress, or commentary-like article in that subfield so that people wanting to dive into this subfield could get a general impression of that subfield.
In general, this is not a career advice post, in the sense that OP is not asking whether one should embark on this kind of transition. But, if the answerer wants to say something about some real-life issues of both hep-ph or hep-th/math-ph research, or the transition between those two fields, please feel free and don't be shy.
I understand that there already exist some good similar recommendation list posts on this site, but, at least as I know, none of them focuses on the transition from hep-ph to hep-th/math-ph, which is the central point of this post. If there does exist such a post and someone has already offered a quite good answer, please let me know.
Another way of formulating this question is the following. Please give a list of the introductory courses that a graduate student studying mathematical physics should learn, or maybe a list of papers an advisor would recommend to a first-year graduate student studying mathematical physics. But, keep in mind that this student has already had accomplished a Master's level study in hep-th.
This question has also been crossposted on Math SE (https://math.stackexchange.com/questions/4194712/how-should-i-start-to-study-qft-at-a-somewhat-mathematically-rigorous-level-lik) and Math Overflow (https://mathoverflow.net/questions/397143/reading-list-recommendation-for-a-hep-ph-student-to-start-studying-qft-at-a-more)
I'll try to expand this comment into a more descriptive answer later if I get the time, but I found the intro(ish) textbooks by G. Scharf (Finite QED and Gauge Field Theories) and M. Dutsch (From CFT to pQFT) to be excellent for someone with little to no background in the more mathematical side of things. I believe they both have papers, a handful of which are on arxiv, which go more in depth.
Even more mathematically intense would be Advances in Algebraic QFT and related texts. The Simon and Reed quadrilogy of mathematics textbooks (Methods of Modern Mathematical Physics) may be of use if you want to go deeper into the general math background.
If anyone reading this is familiar with the above then they are welcome to elaborate more :)
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