On integrating out short-distance physics

I would like to explain how short-distance (or high-energy) physics is “integrated out” in a reasonably constructed theory. Speaking roughly and briefly, it is integrated out automatically. I propose to build QFT in a similar way.

Phenomena to describe

Let us consider a two-electron Helium atom in the following state: one electron is in the “ground” state and the other one is in a high orbit. The total wave function of this system  \Psi(\mathbf{r}_{\rm{Nucl}},\mathbf{r}_{\rm{e}_{1}},\mathbf{r}_{\rm{e}_2},t) depending on  the absolute coordinates is conveniently presented as a product of a plane wave  {\rm{e}} ^{{\rm{i}}(\mathbf{P}_{\rm{A}} \mathbf{R}_{\rm{A}} - E_{P_{\rm{A}}} t)/\hbar} describing the atomic center of mass and a wave function of the relative or internal collective motion of constituents \psi_n (\mathbf{r}_1,\mathbf{r}_2)e^{-i E_n t/\hbar} where \mathbf{R}_{\rm{A}}= \left[ M_{\rm{Nucl}}\mathbf{r}_{\rm{Nucl}}+m_{\rm{e}}(\mathbf{r}_{\rm{e}_{1}}+\mathbf{r}_{\rm{e}_{2}})\right ]/(M_{\rm{Nucl}}+2m_{\rm{e}}) and \mathbf{r}_a are the electron coordinates relative to the nucleus \mathbf{r}_a = \mathbf{r}_{\rm{e}_a}-\mathbf{r}_{\rm{Nucl}},\; a=1,2 (see Fig.1).

Coordinates_2

Figure 1. Coordinates in question.

Normally, this wave function is still a complicated thing and the coordinates \mathbf{r}_1 and \mathbf{r}_2 are not separated (the interacting constituents are in mixed states). What can be separated in \psi_n are normal (independent) modes of the collective motion (or “quasi-particles”). Normally it is their properties (proper frequencies, for example) who are observed.

However, in case of one highly excited electron (n\gg 1), the wave function of internal motion, for our numerical estimations and qualitative analysis, can be quite accurately approximated with a product of two hydrogen-like wave functions \psi_n (\mathbf{r}_1,\mathbf{r}_2) \approx \psi_0 (\mathbf{r}_1)\cdot \phi_n (\mathbf{r}_2) where \psi_0 (\mathbf{r}_1) is a wave function of He^+ ion (Z_{\rm{A}}=2) and \phi_n (\mathbf{r}_2) is a wave function of Hydrogen in a highly excited state (n\gg1,\; (Z_{\rm{eff}})_{\rm{A}}=1).

The system is at rest as a whole and serves as a target for a fast charged projectile. I want to consider large angle scattering, i.e., scattering from the atomic nucleus rather than from the atomic electrons. The projectile-nucleus interaction V(\mathbf{r}_{\rm{pr}}-\mathbf{r}_{\rm{Nucl}}) is expressed via “collective” coordinates thanks to the relationship \mathbf{r}_{\rm{Nucl}}=\mathbf{R}_{\rm{A}}-m_{\rm{e}}(\mathbf{r}_1+\mathbf{r}_2)/M_{\rm{A}}. I take a non-relativistic proton with v \gg v_n as a projectile and I will consider such transferred momentum values q=|\mathbf{q}| that are insufficient to excite the inner electron levels by “hitting” the nucleus. Below I will precise these conditions. Thus, for the outer electron the proton is sufficiently fast to be reasonably treated by the perturbation theory in the first Born approximation, and for the inner electron the proton scattering is such that cannot cause its transitions. This two-electron system will model a target with soft and hard excitations.

Now, let us look at the Born amplitude of scattering from such a target. The general formula for the cross section is the following (all notations are from [1]):

d\sigma_{np}^{n'p'}(\mathbf{q}) = \frac{4m^2 e^4}{(\hbar q)^4} \frac{p'}{p} \cdot \left | Z_{\rm{A}}\cdot f_n^{n'}(\mathbf{q}) - F_n^{n'}(\mathbf{q})\right |^2 d\Omega\qquad (1)

F_n^{n'}(\mathbf{q})=\int\psi_{n'}^*(\mathbf{r}_1 , \mathbf{r}_2)\psi_{n}(\mathbf{r}_1 , \mathbf{r}_2)\left (\sum_a {\rm{e}}^{ -{ \rm{i} } \mathbf{q} \mathbf{r}_a }\right ) \exp\left ({ \rm{i} }\frac{ m_{\rm{e}} }{ M_{\rm{A}} }\mathbf{q}\sum_b \mathbf{r}_b \right ){\rm{d}}^3 r_1 {\rm{d}}^3 r_2 \, (2)

f_n^{n'}(\mathbf{q})=\int\psi_{n'}^*(\mathbf{r}_1 , \mathbf{r}_2)\psi_{n}(\mathbf{r}_1 , \mathbf{r}_2) \exp\left ({\rm{i}}\frac{m_{\rm{e}}}{M_{\rm{A}}}\mathbf{q}\sum_a \mathbf{r}_a \right ){\rm{d}}^3 r_1 {\rm{d}}^3 r_2 \qquad (3)

The usual atomic form-factor (2) describes scattering from atomic electrons and it becomes relatively small for large scattering angles \langle(\mathbf{q}\mathbf{r}_a)^2\rangle_n\gg1. It is so because, roughly speaking, the atomic electrons are light compared to the heavy projectile and they cannot cause large-angle scattering for a kinematic reason. I can consider scattering angles superior to those determined with the direct projectile-electron interactions (\theta\gg \frac{ m_{\rm{e}} }{ M_{ \rm{pr} } }\frac{2v_0}{v}) or, even better, I may exclude the direct projectile-electron interactions V(\mathbf{r}_{\rm{pr}}-\mathbf{r}_{{\rm{e}}_{\rm{a}}}) in order not to involve F_n^{n'}(\mathbf{q}) into calculations any more. Then no “screening” due to atomic electrons exists for the projectile nor atomic excitations due to direct projectile-electron interaction at any scattering angle.

Let us analyze the second atomic form-factor (3) in the elastic channel. With our assumptions on the wave function, it can be easily calculated if the corresponding wave functions are injected in (3):

f_n^{n}(\mathbf{q}) \approx \int \left | \psi_{0}(\mathbf{r}_1)\right |^2\left |\phi_n(\mathbf{r}_2) \right |^2 {\rm{e}}^{ {\rm{i}}\frac{m_{\rm{e}}}{M_{\rm{A}}}\mathbf{q}(\mathbf{r}_1+\mathbf{r}_2)}{\rm{d}}^3 r_1 {\rm{d}}^3 r_2\qquad(4)

It factorizes into two Hydrogen-like form-factors:

f_n^{n}(\mathbf{q})\approx f1_0^{0}(\mathbf{q}) \cdot f2_n^{n}(\mathbf{q}) \qquad (5)

Form-factor \left|f1_0^{0}(\mathbf{q})\right| describes quantum mechanical smearing of the nucleus charge (a “positive charge cloud”) due to nucleus coupling to the first atomic electron. This form-factor may be close to unity (smearing may not be “visible” because of its small size \propto (m_{\rm{e}} /M_{\rm{A}})\cdot a_0/2). Form-factor \left|f2_n^{n}(\mathbf{q})\right| describes quantum mechanical smearing of the nucleus charge (another “positive charge cloud”) due to nucleus coupling to the second atomic electron. In our conditions \left|f2_n^{n}(\mathbf{q})\right| is rather small because the corresponding smearing size \propto (m_{\rm{e}}/M_{\rm{A}})\cdot a_n,\; a_n\propto n^2 is much larger. In our problem setup the projectile “probes” these positive charge clouds and do not interact directly with the electrons.

Thus, the projectile may “see” a big “positive charge cloud” created with the motion of the atomic nucleus in its “high” orbit (i.e., with the motion of He^+ ion thanks to the second electron, but with full charge Z_{\rm{A}}=2 seen with the projectile), and at the same time it may not see the additional small positive cloud of the nucleus “rotating” also in the ground state of He^+ ion. The complicated short-distance structure (the small cloud within the large one) is integrated out in (4) and results in the elastic from-factor \left|f1_0^{0}\right| tending to unity, as if this short-distance physics were absent. We can pick up such a proton energy E_{\rm{pr}}, such a scattering angle \theta, and such an excited state |n\rangle, that $\left|f1_0^{0}\right| may be equal to unity even at the largest transferred momentum, i.e., at \theta=\pi.

In order to see to what extent this is physically possible in our problem, let us analyze the “characteristic” angle \theta 1_0 for the inner electron state [1]. (I remind that q_{\rm{elastic}}(\theta)=p\cdot2\sin\frac{\theta}{2}.) \theta 1_0 is an angle at which the inelastic processes become relatively essential (the probability of not exciting the target “internal” states is |f1_0^0|^2 and that of exciting any “internal” state is described with the factor \left[1-|f1_0^0|^2\right]):

\theta 1_0=2 \arcsin\left(\frac{2v_0}{2v}\cdot 5\right)\qquad (6)

Here, instead of v_0 stands 2v_0 for the He^+ ion due to Z_{\rm{A}} = 2 and factor 5 originates from the expression \left(1+\frac{M_{\rm{A}}}{M_{\rm{pr}}}\right). So, \theta 1_0=\pi for v=5 v_0=2.5\cdot2v_0. Fig. 2 shows just such a case (the red line) together with the other form-factor – for a third excited state of the other electron (the blue line) for demonstrating a strong impact of n.

Figure  2. Helium form-factors f1_0^0 and f2_3^3 at v=5v_0. f1_0^0(q_{v=5v_0}(\pi))=0.64

We see that for scattering angles \theta\ll\theta 1_0 (v) form-factor |f1_0^0| becomes very close to unity (only elastic channel is open for the inner electron state) whereas form-factor \left|f2_n^n\right| may be still very small if \theta\ge\theta 2_n\ll 1. The latter form-factor describes a large and soft “positive charge cloud” in the elastic channel, and for inelastic scattering (n'\ne n) it describes the soft target excitations energetically accessible when hitting the heavy nucleus.

The inner electron level excitations due to hitting the nucleus can also be suppressed not only for \theta\ll\theta 1_0 (v), but also for any angle in case of relatively small projectile velocities (Fig. 3).

Figure 3. Helium form-factors f1_0^0 and f2_5^5 at v=2v_0.

By the way, a light electron as a projectile does not see the additional small smearing even at v=10\cdot 2v_0 because its energy is way insufficient (its de Broglie wavelength is too large for that). The incident electron should be rather relativistic to be able to probe such short-distance details [1].

Let us note that for small velocities the first Born approximation may become somewhat inaccurate: a “slow” projectile may “polarize” the atomic “core” (more exactly, the nucleus may have enough time to make several quick turns during interaction) and this effect influences numerically the exact elastic cross section. Higher-order perturbative corrections of the Born series take care of this effect, but the short-distance physics will still not intervene in a harmful way in our calculations. Instead of simply dropping out (i.e., producing a unity factor in the cross section (1)), it will be taken into account (“integrated out”) more precisely, when necessary.

Hence, whatever the true internal structure is (the true high-energy physics, the true high-energy excitations), the projectile in our “two-electron” theory cannot factually probe it when it lacks energy. The soft excitations are accessible and the hard ones are not. It is comprehensible physically and is rather natural – the projectile, as a long wave, only sees large things. Small details are somehow averaged or integrated out. In our calculation, however, this “integrating out” (factually, “taking into account”) the short-distance physics occurs automatically rather than “by hands”. We do not introduce a cut-off and do not discard (absorb) the harmful corrections in order to obtain something physical. We do not have harmful corrections at all. It convinces me in possibility of constructing a physically reasonable QFT where no cut-off and discarding are necessary.

The first Born approximation (3) in the elastic channel gives a “photo” of the atomic positive charge distribution as if the atom was internally unperturbed during scattering, a photo with a certain resolution, though.

Inelastic processes n'\ne n give possible final target states different from the initial one (different could configurations).

The fully inclusive cross section (i.e., the sum of the elastic and all inelastic ones) reduces to a great extent to a Rutherford scattering formula for a free and still point-like target nucleus (no clouds at all!) [1]. The inclusive picture is another kind of averaging over the whole variety of events, averaging often encountered in experiments and resulting in a deceptive simplification. One has to keep this in mind because usually it is not mentioned while speaking of short-distance physics, as if there were no difference between elastic, inelastic, and inclusive pictures.

Increasing the projectile energy (decreasing its de Broglie wavelength), increasing the scattering angles and resolution at experiment helps reveal the short-distance physics in more detail. Doing so, we may discover high-energy excitations inaccessible at lower energies/angles. Thus, we may learn that our knowledge (for example, about pointlikeness of the core) was not really precise, “microscopic”.

Discussion

Above we did not encounter any mathematical difficulties. It was a banal calculation, as it should be in physics. We may therefore say that out theory is physically reasonable.

What makes our theory physically reasonable? The permanent interactions of the atomic constituents taken into account exactly both via their wave function and via the relationships between their absolute and the relative (or collective) coordinates, (namely, \mathbf{r}_{\rm{Nucl}} involved in V(\mathbf{r}_{\rm{pr}}-\mathbf{r}_{\rm{Nucl}}) was expressed via \mathbf{R}_{\rm{A}} and \mathbf{r}_a). The rest was a perturbation theory in this or that approximation. For scattering processes it calculated the occupation number evolutions – the transition probabilities between different target states. It is an ideal in the scattering physics description.

Now, let us imagine for instance that this our “two-electron” theory is a “Theory of Everything” (or a true “underlying theory”) unknown to us so far. Low-energy experiments outlined above would not reveal the “core” structure, but would present it as a point-like nucleus smeared only due to the second electron. Such experiments would then be well described with a simpler, “one-electron” theory, a theory of a hydrogen-like atom with \phi_n (\mathbf{r}_2) and M_{\rm{A}}. The presence of the first electron would not be necessary in such a theory: the latter would work fine and without difficulties – it would reproduce low-energy target excitations.

May we call the “one-electron” theory an effective one? Maybe. I prefer the term “incomplete” – it does not include and predict all target excitations existing in Nature, but it has no mathematical problems (catastrophes) as a model even outside its domain of validity. The projectile energy E_{\rm{pr}} (or a characteristic transferred momentum |\mathbf{q}|) is not a “scale” in our theory in a Wilsonian sense.

Thus, the absence of the true physics of short distances in the “one-electron” theory does not make it ill-defined or fail mathematically. And this is so because the one-electron theory is also constructed correctly – what is know to be coupled permanently and determines the soft spectrum is already taken into account in it via the wave function \phi_n (\mathbf{r}_2) and via the coordinate relationships. That is why when people say that a given theory has mathematical problems “because not everything in it is taken into account”, I remain skeptic. I think the problem is in its erroneous formulation. It is a problem of formulation or modeling (see, for example, unnecessary and harmful “electron self-induction effect” discussed in [2] and an equation coupling error discussed in [3]). And I do not believe that when everything else is taken into account, the difficulties will disappear automatically. Especially if “new physics” is taken into account in the same way – erroneously. Instead of excuses, we need a correct formulation of incomplete theories on each level of our knowledge.

Now, let us consider a one-electron state in QED. According to QED equations, “everything is permanently coupled with everything”, in particular, even one-electron state, as a target, contains possibilities of exciting high-energy states like creating hard photons and electron-positron pairs. It is certainly so in experiments, but the standard QED suffers from calculation difficulties (catastrophes) of obtaining them in a natural way because of its awkward formulation. A great deal of QED calculations consists in correcting its initial wrongness. That is why “guessing right equations” is still an important physical and mathematical task.

Electronium and all that

My electronium model [1] is an attempt to take into account a low-energy QED physics, like in the “one-electron” incomplete atomic model mentioned briefly above. The non relativistic electronium model does not include all possible QED excitations but soft photons; however, and this is important, it works fine in a low-energy region. Colliding two electroniums produces soft excitations (radiation) immediately, in the first Born approximation. (It looks like colliding two complex atoms – in the final state one naturally obtains excited atoms.) There is no background for the infrared problem there because the soft modes are taken into account “exactly” rather than “perturbatively”. Perturbative treatment of soft modes gives a divergent series due to “strongness” of soft mode contributions into the calculated probabilities [4]:

QED_IR

Picture 4. Extraction form [4].

It is easy to understand in case of expanding our second form-factors f_n^n(\mathbf{q}) in powers of “small coupling parameter” m_{\rm{e}}/M_{\rm{A}} in the exponential (3): f_n^n(\mathbf{q})\approx 1-\frac{m_{\rm{e}}^2}{M_{\rm{A}}^2}\langle(\mathbf{q}\mathbf{r}_a)^2\rangle_n. For the first electron (i.e., for the hard excitations) the term \frac{m_{\rm{e}}^2}{M_{\rm{A}}^2}\langle(\mathbf{q}\mathbf{r}_1)^2\rangle_0 may be small (see Fig. 3) whereas for the second one \frac{m_{\rm{e}}^2}{M_{\rm{A}}^2}\langle(\mathbf{q}\mathbf{r}_2)^2\rangle_n is rather large and diverges in the soft limit n\to\infty. In QED the hard and soft photon modes are treated perturbatively because the corresponding electron-field interaction is factually written in the so called “mixed variables” [5] and the corresponding series are similar to expansions of our inelastic form-factors f_n^{n'}(\mathbf{q}) in powers of m_{\rm{e}}/M_{\rm{A}}.

By the way, the photons are those normal modes of the collective motions whose variables in the corresponding \psi_n are separated.

How would I complete my electronium model, if given a chance? I would add all QED excitations in a similar way – I would add a product of the other possible “normal modes” to the soft photon wave function and I would express the constituent electron coordinates via the center of mass and relative motion coordinates, like in the non relativistic electronium or in atom. Such a completion would work as fine as my actual (primitive) electronium model, but it would produce the whole spectrum of possible QED excitations in a natural way. Of course, I have not done it yet (due to lack of funds) and it might be technically very difficult to do, but in principle such a reformulated QED model would be free from mathematical and conceptual difficulties by construction. Yes, it would be still an “incomplete” QFT, but no references to the absence of the other particles (excitations) existing in Nature would be necessary. I would not introduce a cut-off and running constants in order to get rid of initial wrongness, as it is carried out today in the frame of Wilsonian RG exercise.

Conclusions

In a “complete” reformulated QFT (or “Theory of Everything”) non-accessible at a given energy E excitations would not contribute (with some reservations). Roughly speaking, they would be integrated out (taken into account) automatically, like in my “two-electron” target model given above, reducing naturally to a unity factor.

But this property of “insensibility to short-distance physics” does not exclusively belong to the “complete” reformulated QFT. “Incomplete” theories can also be formulated in such a way that this property will hold. It means the short-distance physics, present in an “incomplete theory” and different from reality, cannot be and will not be harmful for calculations technically, as it was eloquently demonstrated in this article. When the time arrives, the new high-energy excitations could be taken into account in a natural way described primitively above as a transition from a “one-electron” to “two-electron” target model. I propose to think over this way of constructing QFT. I feel it is a promising direction of building physical theories.

References

[1] Kalitvianski V 2009 Atom as a “Dressed” Nucleus Cent. Eur. J. Phys. 7(1) 1–11 (Preprint arXiv:0806.2635 [physics.atom-ph])

[2] Feynman R 1964 The Feynman Lectures on Physics vol. 2 (Reading, Massachusetts: Addison-Wesley Publishing Company, Inc.) pp 28-4–28-6

[3] Kalitvianski V 2013 A Toy Model of Renormalization and Reformulation Int. J. Phys. 1(4) 84–93
(Preprint arXiv:1110.3702 [physics.gen-ph])

[4] Akhiezer A I, Berestetskii V B 1965 Quantum Electrodynamics (New York, USA: Interscience Publishers) p 413

[5] Kalitvianski V 2008 Reformulation Instead of Renormalization Preprint arXiv:0811.4416 [physics.gen-ph]

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