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Semiclassical Derivation of the QCD Coupling
No full textAbstract: The strength of the fundamental nuclear strong force is characterized by the QCD coupling αs. In this paper the measured value of the QCD coupling αs at the energy MZ 0, the variation of αs as a function of energy in QCD, and classical relativistic dynamics are used to investigate virtual pairs of quarks and antiquarks in vacuum fluctuations. For virtual pairs of bottom quarks and antiquarks, the pair lifetime in the classical model agrees with the lifetime from quantum mechanics to good approximation, and the action integral in the classical model agrees as well with the action that follows from the Uncertainty Principle. This suggests that the particles might have small de Broglie wavelengths and behave with well-localized pointlike dynamics. It also permits αs at the mass energy twice the bottom quark mass to be expressed as a simple fraction: 3/16. This is accurate to approximately 10%. The model in this paper predicts the measured value of αs(MZ 0) to be 0.121, which is in agreement with recent measurements within statistical uncertainties.UMUC Working Paper Series – Number 2009-018
A Semiclassical Derivation of the QCD Coupling
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A Semiclassical Derivation of the QCD Coupling
By
David Batchelor
UMUC, SCIP
[email protected]
July 2009
UMUC Working Paper Series – Number 2009-018
Abstract
The strength of the fundamental nuclear strong force is characterized by the QCD
coupling αs. In this paper the measured value of the QCD coupling αs at the energy MZ
0,
the variation of αs as a function of energy in QCD, and classical relativistic dynamics are
used to investigate virtual pairs of quarks and antiquarks in vacuum fluctuations. For
virtual pairs of bottom quarks and antiquarks, the pair lifetime in the classical model
agrees with the lifetime from quantum mechanics to good approximation, and the action
integral in the classical model agrees as well with the action that follows from the
Uncertainty Principle. This suggests that the particles might have small de Broglie
wavelengths and behave with well-localized pointlike dynamics. It also permits αs at the
mass energy twice the bottom quark mass to be expressed as a simple fraction: 3/16.
This is accurate to approximately 10%. The model in this paper predicts the measured
value of αs(MZ
0) to be 0.121, which is in agreement with recent measurements within
statistical uncertainties.
Keywords
Quantum mechanics, quarks, quantum chromodynamics, vacuum fluctuations
This work was supported by the NASA Goddard Space Flight Center
UMUC Working Paper Series – Number 2009-018
A Semiclassical Derivation of the QCD Coupling
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I. INTRODUCTION
Quantum mechanics, with its indeterministic, statistical predictions of physical
phenomena, has inspired attempts to explain its success in terms of classical,
deterministic underlying principles since its beginning. The tension between
deterministic principles and quantum mechanics was exemplified by A. Einstein’s
famous remark that God does not play dice with the Universe. Debate continues today
over whether quantum mechanics arises from “hidden variables” that could explain the
statistical predictions arising from definite dynamical predictions, but which remain
unmeasured as yet.
Recently t’ Hooft and others have investigated the possibility that physics at small
scales might be governed by “basic dynamical Laws” of a deterministic underlying
theory [1]. These laws would not invalidate quantum mechanics, but would yield
quantum mechanics when subjected to statistical analysis. In view of that possibility, this
paper considers a classical dynamical model that might underlie quantum chromo-dynamics
(QCD [2, 3, 4]) at small scales. The model becomes semiclassical in a natural
way.
If a quark and its antiquark are positioned at rest in their center of mass reference
frame and are released, then in general their mutual attraction will draw them to a
collision at the origin where they will annihilate. Photons or other particles would result,
given sufficient energy. However, the quark and antiquark experience a potential energy
U(R) [5] as a function of their separation distance that could reduce the mass-energy of
the system, if the separation R between particles is small enough. The energy
conservation relation
is possible to satisfy, where the first term is the total relativistic energy of the particles,
kinetic plus rest mass-energy. (R = 2r with r the radius of either particle from the origin
at the center of mass.) In classical physics, the collision would not yield any energy and
so no photons or particles could be emitted.
If the time-reversed trajectory occurred, with the vacuum spontaneously creating
a quark-antiquark pair obeying Eq. (1), then the particles could only move apart in one-dimensional
motion to reach turning points separated by Rmax. Continuing this trajectory
so that the particles fall from the turning points back to the origin, they would disappear
back into the vacuum, like a virtual quark-antiquark pair (VQAP; see Fig. 1 for the
Feynman diagram). This is similar to virtual electron-positron pairs discussed by Greiner
(p. 3 of ref. [6]) and Sakurai (p. 139 of ref. [7]). Such virtual pairs of antiparticles are
believed to appear and disappear spontaneously in vacuum fluctuations. Considering
‘t Hooft’s exploration of deterministic physics that might underlie quantum mechanics, it
would make sense to compare the classical treatment of this problem with the quantum-mechanical
VQAP.
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Quantum theory implies that this two-particle system of quarks would obey the
time-energy uncertainty relationship for the energy fluctuation Δε (p. 139 of ref. [7])
The energy fluctuation is Δε = 2 mqc2, since the mass-energy of each quark contributes
mqc2. The quantum-mechanical lifetime of the fluctuation is Δt. Since ħ is the quantum
of action, Eq. (2) establishes an action integral that characterizes a VQAP that has Δε =
2 mqc2.
The first purpose of this paper is to present classical computations of Δt and the
action integral for the trajectory described above, which turn out to give results that
satisfy Eq. (2) remarkably well, provided that the QCD interaction between the particles
is well-described by the potential energy function.
The second purpose of the paper follows from the fact that QCD cannot specify
the value of αs at arbitrary energy or 4-momentum scale Q without an established
measurement of αs at some particular energy μ [8]; but once the renormalized coupling
αs(μ2) is measured, then QCD precisely gives the variation of αs as a function of energy
(the “running” coupling). The present paper offers a theory based on the action integral
that establishes the value of αs at the energy scale twice the bottom quark mass-energy to
good approximation. This enables one to use the QCD running coupling αs(Q2) to
determine the coupling strength in general in the usual way to good approximation, and
suggests that the magnitude of αs is fundamentally established by this underlying
dynamical law.
II. QCD POTENTIAL ENERGY FUNCTION
As discussed in detail by Lucha et al. (especially pp. 161-162 of ref. [5]), a potential
energy function serves to describe the bound states of heavy quarks (charm, bottom, and
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top). For the light quarks the QCD interaction is not satisfactorily described by a
potential energy function and will not be attempted here. Here we apply the standard
potential energy treatment to the intermediate-mass charm and bottom quarks. More
elaborate treatment of the most massive top quark case is necessary, due to the effects of
spin-spin interactions between the top quark and antiquark [9].
We use the standard Cornell potential [10, 11]
Where αs is the dimensionless QCD strong coupling strength and a ≈ 0.25 GeV2. The
second term, aR, is only significant for R > 10-13 cm. We will not need to consider the aR
term, since the first term with the Coulomb-like dependence turns out to strongly
dominate the potential because Rmax « 10-13 cm for VQAPs.
The VQAP is a form of bound state. The standard way to account for the
variation of αs(Q2) in a quark bound state is to let αs depend on the quark masses and use
Q2 = (m1 + m2)2, with the mi the quark masses (see p. 129 of ref. [12]). To model a
VQAP we may then compute αs to leading order (Eq. (6) of ref. [8]). So the QCD
coupling is then given by
where β0 is defined by
nf is the number of quark flavors with masses much less than m1 + m2 , and Λ is the QCD
scale energy
Λ is approximately 0.093 GeV, assuming that μ ≡ MZ
0 = 91.2 GeV (the mass-energy of
the Z0 particle), αs(MZ
0) = 0.119 ± 0.002 (in units defining c ≡ 1). This is a typical value
of Λ for a one-loop approximation ([8] p. R31).
For the running coupling near the charm quark mass nf = 4, and near the bottom
quark mass nf = 5. We now can use m1 = m2 = mq in the expression for Q2 and compute
αs from Eq. (4) for a running coupling. Then with a charm quark current mass mq = mc
of 1.27 GeV/c2 we have αs(2mcc2) = 0.228; with a bottom quark current mass mq = mb of
4.2 GeV/c2 we have αs(2mbc2) = 0.167 [13].
III. CLASSICAL TRAJECTORY AND LIFETIME OF VQAP
Let us calculate the trajectory lifetime tvq for a VQAP. If one solves the energy
equation (1) for the dynamics using V(R), the nonrelativistic potential energy, then the
particle velocities nevertheless exhibit relativistic motion, approaching c asymptotically
at the origin r = 0. Thus it is necessary to correct the potential energy for relativistic
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effects. Jackson ([14], p. 185) demonstrates how this is done by using the relativity
factor γ defined in Eq. (10) below. For this trajectory of linear motion, the transformation
R→ γ R in the expression for V(R) performs the appropriate modification of the potential
energy function (since we are considering the center-of-mass reference frame). The
potential energy function in Eq. (3) becomes
With this U(R) we can solve Eq. (1) for Rmax at the turning point (where γ = 1):
For the charm quark mass mc of 1.27 GeV/c2 ≡ 2.26 · 10-24 g, we find the charm VQAP
has an Rmax = 2.35 · 10-15 cm.
Checking the terms in Eq. (3) for V(Rmax) shows that the first term is about -100
times the second term. This confirms that the quarks are so deep in the potential well that
the aR term of the Cornell potential can be neglected in solving the problem.
Let us define the time from appearance of a quark at the origin r = 0 to the time
that the quark stops at the turning point r = ½ Rmax as ½ tvq. The tvq is the classical
equivalent of the quantum-mechanical Δt that we seek. We note that
and solve for dt, which we shall integrate. We rewrite the energy equation (1) with
ζ ≡ R/Rmax as
The time for the particles to fall from r = ½ Rmax back to r = 0 is also ½ tvq, so we have
The integral is given in ref. ([15], p. 974). The value of tvq is the total time for either
quark to travel from r = 0 to its turning point and back to r = 0.
For the charm quark, with mq = mc ≈ 2.26 ·10-24 g, we find the trajectory lifetime
of the VQAP to be tvq ≈ 1.57 ·10-25 s. In comparison, the standard lifetime of the charm
VQAP, given by the Uncertainty Principle expressed in Eq. (2), is Δt = ħ/(4 mc c2) ≈ 1.20
· 10-25 s. So tvq from the classical computation is approximately 22% larger than Δt. This
is remarkably close agreement of the classical lifetime with the quantum-mechanical
lifetime.
For the bottom quark, with mq = mb ≈ 7.48 ·10-24 g, we find the trajectory lifetime
of the VQAP to be tvq ≈ 3.47 ·10-26 s. In comparison, the standard lifetime of the charm
VQAP, given by the Uncertainty Principle expressed in Eq. (2), is Δt = ħ/(4 mb c2) ≈ 3.90
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· 10-26 s. So tvq from the classical computation is approximately 11% smaller than Δt.
This also is remarkably close agreement of the classical lifetime with the quantum-mechanical
lifetime.
IV. ACTION INTEGRAL FOR THE TRAJECTORY
A key step in quantizing a classical model to make it a semiclassical model of a
quantum system is computation of the action integral. In the present model, that is done
as follows. The expression for the integral of action associated with a potential function
U acting on a particle, in the relativistic case, is given by Lanczos ([16], p. 321):
where ds = c dt/γ. Considering the integrated action of the potential energy field in a
VQAP, we compute the field action integrated over tvq:
For the charm VQAP, αs(2mcc2) = 0.228 and therefore = 0.61 ħ. This action integral is
only 22% larger than the exact VQAP quantum fluctuation action in Eq. (2), ½ ħ.
In the case of the bottom quark, the action integral for the model of the VQAP is
found by substituting αs(2mbc2) = 0.167 into Eq. (15), and we find = 0.45 ħ. This is
10% lower than the quantum-mechanical action for the VQAP, ½ ħ.
This means that the model’s representation of the bottom VQAP inherently is
approximately quantized – a remarkable agreement between a quantum-mechanical
characteristic of a dynamical system and the classical description of it. In comparison,
semiclassical models for mesons, which achieve excellent agreement with measurements
of meson masses [17, 18], need to be formulated with additional quantization conditions
that introduce the factor ħ. We have not imposed any quantization conditions upon the
trajectory in this dynamical model. The model herein achieves approximate quantization
at αs(2mbc2) based upon only the measured value of αs(MZ
0), the QCD theoretical energy
dependence of αs(Q2), and relativistic dynamical theory (Eqs. (1) and (7)).
The discrepancy between tvq and Δt in the case of the charm quark may be
accounted for in part, as the author will show elsewhere [9]: for the charm quark and top
quark, spin-spin interactions which have been ignored here become important and
increase tvq and in such a way as to bring into closer agreement the classical and
quantum results.
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V. CONCLUSIONS
This good agreement between the classical trajectory lifetime and the quantum
uncertainty lifetime at the key mass-energy of the bottom quark is surprising, but it may
have a simple physical explanation: if the vacuum creates these particle in motion at
v ≈ c, then their de Broglie wavelengths λ = h/p should be small, so that the quarks are
pointlike. Dynamics of point masses would then be applicable. It is interesting that the
length scale of any VAP is usually characterized in standard literature by assuming that
v ≈ c [7].
The salient logic in this paper’s result is the following. The measurement of
αs(MZ
0) and the one-loop Λ obtained from the so-called ‘modified minimal subtraction
scheme’ of renormalization theory predict αs(2mbc2) [8]. From this we may use the
classical trajectory lifetime of the VQAP to compute tvq and action , obtaining ≈ ½ ħ
in approximate agreement with quantum mechanics.
Since ħ is a more universal and fundamental parameter than αs, ħ intuitively
would seem to be the governing parameter in Eq. (15).
If αs(2mbc2) equaled 3/16 the would exactly equal ½ ħ. Setting Eq. (4) equal to
3/16, Q2 equal to (2mbc2)2 , and solving for Λ yields Λ = 0.106 GeV instead of the
standard 0.093 GeV. With this value of Λ, Eq. (4) give αs(91.2 GeV ≡ MZ
0) = 0.121.
This is only 2% different from the measured value upon which the accuracy of QCD
depends, and is within the statistical uncertainty in αs(MZ
0) quoted in Ref. [8].
We now have a logical link between the measured αs(MZ
0) and the action integral
from the Uncertainty Principle, ½ ħ. It is reasonable to reverse this logical sequence and
infer that the action integral ½ ħ is what governs the value of αs(MZ
0). This reverse
argument from = ½ ħ through Eqs. (15), (12), and (4) to αs(MZ
0) is the derivation
mentioned in this paper’s title. In this way ‘t Hooft’s vision of an underlying
deterministic Law (classical relativistic dynamics) enables QCD and renormalization
theory to account for the magnitude of αs, not just the variation of αs(Q2) with energy
scale.
The strength of the fundamental strong nuclear force appears to be explained by
this derivation. Such insight into the basis of any one of the four fundamental forces in
nature is new to physics.
Acknowledgements
The author is grateful to NASA’s Goddard Space Flight Center for supporting the
author during this research.
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