What's the radius of convergence for 1/N expansion in QCD?

In summary, the question asks if, for the radius of convergence, exact results can be obtained by summing a sufficient number of terms. The conversation discusses the convergence of the 1/N series and potential issues related to large N, such as the limits of temperature and volume not commuting. It also mentions the possibility of non-perturbative corrections in 1/N. Overall, the speaker has no precise statement on the convergence of the 1/N series in N=3 QCD and is aware of statements regarding perturbative results in the 't Hooft coupling being convergent at large N.
  • #1
petergreat
267
4
I'm asking this question as someone who has not studied this topic technically. By radius of convergence, I mean exact results, not just approximations, can be obtained by summing sufficiently many terms. (does this ever happen?) I don't mind if you need 1000 terms, as long as the series is convergent mathematically. Is N=3 QCD within the radius of convergence? And how about N=2, N=1?
 
Physics news on Phys.org
  • #2
Though I know of no precise statement to this effect, I think the answer is that the 1/N series does not converge. I am aware of statements that some perturbative results in the 't Hooft coupling can be convergent exactly at large N, but this is different expansion. There definitely can be non-perturbative in 1/N corrections; one has to worry about such things in AdS/CFT, for example.

There are lots of subtle issues relating to large N. For example, the limits [tex] T \rightarrow 0 [/tex] and [tex] N \rightarrow \infty [/tex] may not commute. More generally, the limits of infinite volume and large N may not commute as in Eguchi-Kawai reduction. The limits [tex] \omega \rightarrow 0 [/tex] and [tex] N \rightarrow \infty [/tex] relevant for response functions do not commute. There is plenty more.

Hope this helps.
 
  • #3
Interesting question; I studied large-N 1+1 dim. QCD up to first loop order (= with mesonic fluctuations) w/o ever wondering about convergence ...
 

1. What is the radius of convergence for 1/N expansion in QCD?

The radius of convergence for 1/N expansion in QCD refers to the maximum value of the parameter N for which the expansion series in quantum chromodynamics (QCD) is valid. This expansion series is based on the number of colors in the theory, with N=3 representing the real world case. The exact value of the radius of convergence is still an active area of research, but it is generally accepted to be between 3 and 5.

2. How is the radius of convergence for 1/N expansion in QCD determined?

The radius of convergence is determined through theoretical calculations and numerical simulations. It involves analyzing the behavior of the expansion series as N increases and looking for patterns and trends. Various techniques have been developed to estimate the radius of convergence, such as the Padé approximant method and the Borel resummation method.

3. What is the significance of the radius of convergence in QCD?

The radius of convergence in QCD is important because it indicates the range of applicability of the 1/N expansion series. If N is within the radius of convergence, the expansion series can be used to make accurate predictions about the behavior of the theory. However, if N is outside the radius of convergence, the expansion breaks down and may lead to incorrect results.

4. Can the radius of convergence for 1/N expansion in QCD change with different physical conditions?

Yes, the radius of convergence can change with different physical conditions. For example, it can depend on the temperature, energy scale, and coupling strength of the system. In certain extreme conditions, such as at high temperatures or densities, the radius of convergence may decrease, indicating that the expansion series is no longer valid.

5. How does the radius of convergence for 1/N expansion in QCD relate to the confinement of quarks and gluons?

The confinement of quarks and gluons is a fundamental aspect of QCD, and the radius of convergence for 1/N expansion plays a role in understanding this phenomenon. It has been proposed that the radius of convergence is related to the critical behavior of the confinement-deconfinement phase transition in QCD. As N approaches the radius of convergence, the theory undergoes a phase transition from a confined state to a deconfined state, similar to the transition from a gas to a liquid. This connection is still an active area of research in theoretical physics.

Similar threads

Addition of power series and radius of convergence
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Radius of Convergence and Power Series Expansion Around Multiple Values
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
What is the radius of convergence for a series with logarithmic terms?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
B Difficulty with understanding whether 1/n converges or diverges
    • Calculus
Replies
11
Views
2K
MHB Finding Radius of Convergence for Series: n/2^n and 1/(4+(-1)^n)^3n
    • Topology and Analysis
Replies
3
Views
2K
Interval of convergence
    • Calculus and Beyond Homework Help
Replies
3
Views
946
I Does the sum of all series 1/n^m, m>1 converge?
    • Calculus
Replies
4
Views
2K
Does the Series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## Converge?
    • Calculus and Beyond Homework Help
Replies
29
Views
4K
Show that the following series converges
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Testing Absolute Convergence of ∑(-2)n+1/n+5n
    • Calculus and Beyond Homework Help
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top