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Why can the Euler beta function be interpreted as a scattering amplitude?

+ 7 like - 0 dislike
2101 views

The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this?

In another word, when the Euler beta function is interpreted as a scattering amplitude, what features does it have that make it able to explain strong force of mesons?

What properties of (or why) the Euler beta function (when interpreted as scattering amplitude) has string behavior?

This post imported from StackExchange Physics at 2014-06-11 07:31 (UCT), posted by SE-user Achmed
asked Jul 5, 2013 in Theoretical Physics by Achmed (35 points) [ no revision ]
Susskind's lecture 6 on string theory discusses the scattering interpretation semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes.

This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user ccorn
Susskind's lecture 6 on string theory discusses the scattering interpretation semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes.

This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user ccorn
I refer you to Schwarz and Witten's textbook 'Superstring Theory,' which provides an excellent analysis of the beta function, and its interpretation as a scattering amplitude.

This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user JamalS
I refer you to Schwarz and Witten's textbook 'Superstring Theory,' which provides an excellent analysis of the beta function, and its interpretation as a scattering amplitude.

This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user JamalS
Some more details about the calculation of for example the scattering amplitude of four open string tachyons and why it corresponds to the Euler-Beta function are outlined here.

This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user Dilaton
@JamalS No love for Green?

This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user 0celo7
@0celo7 Couldn't be bothered typing :)

This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user JamalS

4 Answers

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First I would like to give some more details about what is involved in obtaining scattering amplitudes in string theory and the specific one for the scattering of four open string tachyons.

Generally, for calculating string scattering amplitudes, one starts by drawing the string diagrams (worldsheets) that correspond to the process one is interested in, as Riemann surfaces in the complex plain and brings them by means of conformal transformations into a convenient standard representation. The scattering amplitude of the process is then obtained by integrating over the moduli space that parameterizes all worldsheets with the same initial and outgoing state.

The worldsheet for the scattering of 4 open string tachyons can for example be considered as a disk in the compex plain with four punctures at the locations $P_1$, $P_2$, $P_3$, $P_4$, where the incoming and outgoing particles are inserted. By a conformal transformation, this worldsheet can be mapped to the upper half plain such that the points of insertion appear on the real line

\(P_1 \rightarrow x_1 = 0\)
\(P_2 \rightarrow x_x = \lambda\)
\(P_3 \rightarrow x_3 = 1\)
\(P_4 \rightarrow x_4 = \infty\)

To define the conformal (or linear fractional) transformation only 3 points are needed, $\lambda$ is left as a free parameter and its range $0 < \lambda < 1$ therefore corresponds to the moduli space of the $2D$ Riemann surfaces with 4 punctures to be integrated over to obtain the scattering amplitude of the process.

The Ansatz for the scattering amplitude depends on the momenta of the 4 incoming and outgoing particles $p_i$  and can be written as

\(A(p_1,p_2,p_3,p_4) = g_0^2\int d\mu\)

It is proportional to the coupling constant $g_0$ squared because two interaction vertices are involved, and it contains $d\lambda$ to integrate over the moduli space of the relevant worldsheets. Considering momentum conservation and the on-shell condition of the incoming and outgoing particles, and taking into account that the integration measure has to be conformal invariant, the scattering amplitude can be rewritten as ($\alpha'$ is proportional to the string length squared in natural units)

\(A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1 d\lambda\lambda^{2\alpha'(p_1,p_2)}(1-\lambda)^{2\alpha'(p_2,p_3)}\)

Transforming to the Mandelstem variables $s$, $t$, and $u$, and defining the expressions

\(\alpha(s) = \alpha's +1\)
\(\alpha(t) = \alpha' t +1\)

the Veneziano amplitude can finally be written as

\(A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1d\lambda \lambda^{-\alpha(s)-1}(1-\lambda)^{-\alpha(t)-1}\)

which exactly corresponds to the definition of the Euler beta function as given in the Wikipedia link in the question for example. The symmetry of this amplitude in exchanging s and t (I think it is called channel duality?) is a specific stringy feature.

Somewhat handwavingly, to model the strong interaction by strings mesons are viewed as a pair of quarks sitting at the end of a string, that is represented by a thin tube of color flux lines.  The confinement of the strong interaction can then be explained that the fact that when trying to separate the two quarks more and more mesons are produced such that the "binding energy" linearly increases with the separation $E_B = T L$ and $T$ is the string tension.

In addition, it has been observed for hadrons when plotting the angular momentum against the energy (or mass) squared, they appear as points on  lines $J = \alpha' E^2$ which can be modelled by a classical rotating string. The discrete mass spectrum (the points on the line) of the hadrons including the offset of the linear relationship can be explained by a quantized open rotating string.

answered Jun 14, 2014 by Dilaton (4,295 points) [ revision history ]
edited Jun 15, 2014 by Dilaton
+ 0 like - 0 dislike

I certainly don't have a complete answer, but what I do know is the following. The Veneziano amplitude is the following: $$ \mathcal{A}^{(4)} \propto \lambda\left(B(-\kappa s -1, -\kappa t -1) + B(-\kappa s -1, -\kappa u -1) + B(-\kappa t -1, -\kappa u -1)\right), $$ where $s, t, u$ are the Mandelstem variables and $B(x,y)$ the Euler-Beta function. Note that the above formula is symmetric under exchange of $s,t,u$. I think I read somewhere that physicists expected the strong interaction to be symmetric under such momentum exchanges, but I do not know precisely why. Note that the above formula is what one obtains in string theory for the scattering of open string tachyons, with $\lambda = g_c$ (closed string coupling constant) and $\kappa = \alpha'$.

This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user Funzies
answered Jul 5, 2013 by Funzies (5 points) [ no revision ]
+ 0 like - 0 dislike

I certainly don't have a complete answer, but what I do know is the following. The Veneziano amplitude is the following: $$ \mathcal{A}^{(4)} \propto \lambda\left(B(-\kappa s -1, -\kappa t -1) + B(-\kappa s -1, -\kappa u -1) + B(-\kappa t -1, -\kappa u -1)\right), $$ where $s, t, u$ are the Mandelstem variables and $B(x,y)$ the Euler-Beta function. Note that the above formula is symmetric under exchange of $s,t,u$. I think I read somewhere that physicists expected the strong interaction to be symmetric under such momentum exchanges, but I do not know precisely why. Note that the above formula is what one obtains in string theory for the scattering of open string tachyons, with $\lambda = g_c$ (closed string coupling constant) and $\kappa = \alpha'$.

This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user Funzies
answered Jul 5, 2013 by Funzies (5 points) [ no revision ]
+ 0 like - 0 dislike

A function can be interpretable as a scattering amplitude if that function satisfies the axioms of relativistic S-matrix theory [1]:

  1. Lorentz invariance
  2. Unitarity (Not realized by the beta function, but may be dropped if the function is interpreted as a Born approximation to the exact amplitude)
  3. T, C, P invariance (only for strong nucl. interactions)
  4. Analyticity: singularities in invariant energy complex plane correspond to particle poles or thresholds, in a way that doesn't violate causality.
  5. Crossing symmetry.
  6. Power boundedness
  7. Well-behaved particle poles (stable particle masses are positive, and their residues should be negative)
  8. Analyticity of the second kind: analytic in the complex angular momentum plane.

Then the function should also match some of the empirical facts gathered by experimentalists. When the beta function was proposed the following list would be conjured:

  1. All poles in the complex angular momentum plane move to the right linearly with increasing energy at a universal rate.
  2. Diffractive scattering (not realized by the Euler beta function, but Virasoro's amplitude does realize this)
  3. Inclusive high energy reaction exhibit scaling (also not realized by Euler beta)
  4. Particle spectrum (determined by the poles) in agreement with the quark model (not realized either, but Chan and Paton's method get close)
  5. Reproduces data from weak and electromagnetic processes (not realized.)

References: John Schwarz "Dual Resonance Theory" Phys Rep 8, no. 4, (1973) 269-335

This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user QuantumDot
answered Aug 12, 2014 by QuantumDot (195 points) [ no revision ]

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