What is Beta function: Definition and 38 Discussions

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral














{\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt}
for complex number inputs x, y such that Re x > 0, Re y > 0.
The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.

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  1. S

    A Relationship between bare and renormalized beta functions

    I'm looking at a proof of beta function universality in ##\phi^4## theory, and at one point they do the following step: after imposing that the renormalized coupling ##\lambda## is independent of the cutoff ##\Lambda##, we have $$0= \Lambda \frac {\text{d} \lambda}{\text{d} \Lambda} = \Lambda...

    I Relation Between Beta and Gamma functions

    So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
  3. PhysicsRock

    Exploring an Unexplored Expression for the Beta Function

    So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the...
  4. A

    Break a Stick Example: Random Variables

    Hello, I would like to confirm my answers to the following random variables question. Would anyone be willing to provide feedback and see if I'm on the right track? Thank you in advance. My attempt:
  5. P

    A $\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

    Hi all, I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback. PART I: I wrote this short summary of what I understand of the beta function: Is this reasoning...
  6. fenyutanchan

    Beta Function for Scalar QCD

    I have calculated $Z$s as $$ \begin{aligned} Z_1 & = 1 + \frac{3g^2}{16\pi^2} \left[ 2 C(R) - \frac12 T(A) \right] \frac1{\epsilon} + \cdots, \\ Z_2 &= 1 + \frac{3g^2}{8\pi^2} C(R) \frac1{\epsilon} + \cdots, \\ Z_3 &= 1 + \frac{g^2}{24\pi^2} \left[ 5 T(A) - T(R) \right] \frac1{\epsilon} +...
  7. DaniV

    RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

    I tried to do a Euler Lagrange equation to our Lagrangian: $$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$ and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
  8. L

    A Integral -- Beta function, Bessel function?

    Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
  9. tomdodd4598

    I QED - running of coupling (beta function)

    Hey there, I am a little confused about the way most textbooks and notes I've read find the beta function for QED. They find it by looking at how the photon propagator varies with momentum ##q##, in particular in the context of a ##2\rightarrow2## scattering process which is proportional to...
  10. J

    Is this manipulation "evil" or simply not intuitive due to unfamiliarity?

    The solution is as follows : The substitution is what nags me , which is as follows : This substitution "trick" to me seems impossibly difficult to arrive at "logically" without pretty much reverse engineering the problem. So is this simply a lack of practice/ familiarisation showing ? I feel...
  11. diegzumillo

    A Conformal Window: Understanding the Basics

    I'm reading about extensions of standard model and this pops up frequently but it's not very clear. I understand it's a region in parameters space so renormalization group naturally becomes relevant and that's about it for my understanding. I can't connect any of this to the beta function of the...
  12. C

    I Two Loop Beta Function for Strong Coupling in QCD

    A one loop expression for the strong coupling can be calculated with the result $$\alpha_s^{(1)}(\mu^2) = \frac{4\pi}{b_o \ln(\mu^2/\Lambda^2)}\,\,\,\,\,(1), $$see e.g in the first few pages of the attached file, where $$Q^2 \frac{\partial \alpha_s(Q^2)}{\partial Q^2} = \beta(\alpha_s) =...
  13. S

    A ΛQCDΛQCD from the beta function of QCD coupling constant

    The beta function for the strong coupling ##g_3## is given by ##\displaystyle{\mu \frac{\partial g_{3}}{\partial\mu}(\mu) = - \frac{23}{3} \frac{g_{3}^{3}(\mu)}{16\pi^{2}},}## with ##\alpha_{3}(\mu = M_{Z}) = 0.118.## We can use separation of variables to solve the beta function equation...
  14. N

    A Perturbative expansion of Beta function - Renormalization

    I am trying to understand the basics of Renormalization. I have read that β encodes the running coupling and can be expanded as a power series as: β(g) = ∂g/(∂ln(μ)) = β0g3 + β1g5 + ... However, I don't understand how this is derived.. I assume that the terms correspond to 1 loop, 2 loops...
  15. dykuma

    Sterling approximation of Beta Function

    Homework Statement Homework Equations The Attempt at a Solution I think this problem is probably a lot simpler than I am making it out to be. However, when I apply sterling's approx., I get a very nasty equation that does not simplify easily. One of the biggest problems I have though is...
  16. S

    Beta function of electroweak theory

    Hi everyone, I have another (probably too) simple question for particle physicists on this forum, but I often realize that my understanding of QFT is still rather poor. Do you know where I can find the electroweak beta function explicitly written down (at one-loop, of course)? I would like to...
  17. nikosbak

    Dimension of interaction in a QFT theory

    The problem statement. When an exercises say " the interaction in a QFT has dimensions Δ" , what does it mean?, it means the field or the Lagrangian has this mass dimension? In this exercise I'm trying to find the classical beta function (β-function) for the assciated couling.
  18. WannabeNewton

    Current conservation beta function

    In section 19.5 of Peskin it is stated that if a scale transformation ##\varphi \rightarrow e^{-\sigma}\varphi(xe^{-\sigma})## is a symmetry of a theory then there is a current ##D^{\mu} = \Theta^{\mu\nu}x_{\nu}## (here ##\Theta^{\mu\nu}## is the Bellifante energy-momentum tensor) with...
  19. J

    Beta Function used in Beta distribution

    Could anyone explain in laymen terms what really the beta function does as said in Beta Function:https://en.wikipedia.org/wiki/Beta_function .I know that gamma function is used to find the factorial of real numbers but when it comes to beta function I can't get what really beta function does.I...
  20. A

    Strong force and beta function

    I heard that one of Euler's Beta/Gamma function identities models the strong force. I was just wondering how it did this. (This might be a stupid question) How do we measure the strong force, and how is it a function of two variables?
  21. A

    Calculating Beta Function for Scalar QCD Theory

    Hi all, I am currently trying to calculate the beta function for scalar QCD theory (one loop for general su(n)). I therefore need to calculate the Feynman rules in order to apply them to the one loop diagrams. Unfortunately I am getting very confused with what the Lagrangian for scalar QCD...
  22. alyafey22

    MHB An integral related to beta function

    Prove the following \int^1_0 \frac{x^{r-1} (1-x)^{s-1}}{\left(ax+b(1-x)+c \right)^{r+s}}\, dx =\frac{\beta(r,s)}{ (a+c)^{r}(b+c)^{s}}
  23. alyafey22

    MHB Exploring the Generalized Beta Function: Integrals and Partial Derivatives

    In this thread we consider the integrals of the form \beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 This is NOT a tutorial , all suggestions are encouraged.
  24. J

    How to derive beta function as pochhammer contour integral?

    Hi, We have: \beta(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}dt,\quad Re(a)>0, Re(b)>0 and according to Wikipedia: http://en.wikipedia.org/wiki/Pochhammer_contour we can write: \left(1-e^{2\pi ia}\right)\left(1-e^{2\pi ib}\right)\beta(a,b)= \int_P t^{a-1}(1-t)^{b-1}dt valid for all a and b where P...
  25. FeDeX_LaTeX

    Can the Beta Function be Extended to Real Powers?

    I was looking at some integration problems the other day and I came across this identity: \int_{0}^{\frac{\pi}{2}} \sin^{p}x \cos^{q}x dx = \frac{1}{2} \mbox{B} \left( \frac{p+1}{2},\frac{q+1}{2}\right) where B(x,y) is the Beta function, for Re(x) and Re(y) > 0. From the way in which the above...
  26. C

    MHB Is it Possible to show this? Beta function

    Hi everyone! I got two versions of one particular function and now I need to show those two versions are equivalent. For that I need to show the follwing, Is it possible to show this by using the properties of Beta functions, Gaussian hypergeometric function etc? Thanks in advance!
  27. T

    Absolite convergence of the beta function

    Determine the values of the complex parameters p and q for which the beta function \int_0^1 t^{p-1} (1-t)^{q-1}dt converges absolutely. The solution says: Split the integral into 2 parts: \displaystyle \int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1...
  28. A

    Integration problem : Beta function related

    Homework Statement the problem is in the figure Homework Equations beta function = ∫u^x-1 * (1-u)^y-1 du " the integral is form 0 to 1" The Attempt at a Solution to use the beta function the integral must be from 0 to 1 but this problem is from 0 to a so i let X^2=a^2 * t and...
  29. M

    Can the Beta Function be Solved? A Demonstration of the Beta Function Problem

    Beta Function Demonstration Problem. I pushed the "Enter" key by accident and the topics name got ruined. Good Day to everyone, I have this problem, (In the context of the Gamma and Beta functions) I have to demonstrate that: Homework Statement Demonstrate that...
  30. M

    Solving Limits Problem with Beta Function

    Homework Statement \int_1^{\infty}\frac{dx}{x^2(x-1)^{1/2}} Homework Equations \int_0^1t^{x-1}(1-t)^{y-1}\,dt \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,dt, The Attempt at a Solution Hi all, I have another beta function problem. This time I'm unsure how to deal with the limits, as the...
  31. M

    Solving the Beta Function Problem: Integrating (2-x)/(2+x)^(1/2) from -2 to 2

    Homework Statement \int_{-2}^{2}\left(\frac{2-x}{2+x}\right)^{1/2}\hspace{1mm}dx Homework Equations B(p,q) = \int_0^{\infty}\frac{y^{p-1}}{(1+y)^{p+q}}\hspace{1mm}dy The Attempt at a Solution I am completely stuck on this one, just a total mental block. The answer in the book...
  32. S

    Beta function in Euclidean and Minkowskian QFT

    Hi! I have a question regarding the renormalization group Beta function, i.e., \beta = \mu \frac{dg_R}{d \mu} where g_R is the renormalized coupling constant and \mu the renormalization scale. My question in a nutshell: are the Beta functions calculated for QFT and, respectively...
  33. N

    Beta Function in String Theory

    I've heard that that origin of String Theory was in Gabriele Veneziano's analysis of the Euler beta function in relation to the strong force. I was wondering if anyone could refer me to a paper or derivation describing how this function ended up describing particles as strings.
  34. R

    Beta Function for QED: 20/9 or 8/3?

    The beta function for QED is given by: \beta=\frac{e^3}{16 \pi^2}*\frac{4}{3}*(Q_i)^2 where (Q_i)^2 represents the sum of the squares of the charges of all Dirac fields. For one generation, for the charge squared you have (2/3)^2 for the up quark, (-1/3)^2 for the down quark, but this is all...
  35. S

    Beta function and changing variables

    Homework Statement \int_{0}^{3}\sqrt[3]{\frac{3-x}{x^2}}\: \mathrm{d}x 2. The attempt at a solution I'm pretty sure I have to use Euler's Beta function, so I tried to change the limits to 0 and 1 by setting x = 3·u (so dx = 3·du). However there must be some mistake when I did it because I...
  36. M

    Beta Function: Prove (m-1)!/n(n+1)...(n+m+1) for m+ve Int

    Homework Statement Show that β(m,n)= (m-1)!/n(n+1)...(n+m+1) where m is a +ve integer ? Homework Equations The Attempt at a Solution Please give me a complete solution as i i asked it before but could not arrive at the solution ?
  37. W

    What was the role of the beta function in Veneziano's discovery?

    I watched the elegant universe and one scienetist noticed the euler beta function seemed to explain what he was doing and it had properties of a vibrating string. So I'm wondering if I could find his original paper, explanation or some additional info.
  38. E

    Calculation of Callan-Symanzik Beta function of QED/QCD to one loop order

    Hi everybody. I was given a project to calculate Callan-Symanzik beta function of QED and QCD (with massless fermions) to one loop order. This problem is actually solved in Peskin, BUT without the needed rigor, plus with funny assumptions and also a few mistakes. I have tried for a long time...