On integrating out short-distance physics

Question

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.

Let us consider a two-electron Helium-3 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 $\Psi(\mathbf{r}_{Nucl},\mathbf{r}_{e_{1}},\mathbf{r}_{e_2},t)$ is a product of a plane wave $e^{i(\mathbf{P}_A\mathbf{R}_A- E_{P_A} t)/\hbar}$ 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 $\mathbf{r}_1$ and $\mathbf{r}_2$ are the electron coordinates relative to the nucleus. Normally, this wave function is a complicated thing and the coordinates $\mathbf{r}_1$ and $\mathbf{r}_2$ are not separated. What can be separated in $\psi_n$ are normal (independent) modes of the collective motion.

However, in case of one highly excited electron, the wave function of internal motion, for numerical estimations, can be quite accurately approximated with a product of two hydrogen-like wave functions $\psi_n (\mathbf{r}_1,\mathbf{r}_2) \approx \psi_0 (\mathbf{r}_1) \phi_n (\mathbf{r}_2)$ where $\psi_0 (\mathbf{r}_1)$ is a wave function of $^{3}He^+$ ion ($Z_A=2$) and $\phi_n (\mathbf{r}_2)$ is a wave function of Hydrogen in a highly excited state ($n\gg1,\; Z_{eff}=1$).

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. I take a non-relativistic proton with $v \gg v_n$ as a projectile and I will consider such transferred momentum values $|\mathbf{q}|$ 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 [1]:

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

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

$$f_n^{n'}(\mathbf{q})=\int\psi_{n'}^*(\mathbf{r}_1 , \mathbf{r}_2)\psi_{n}(\mathbf{r}_1 , \mathbf{r}_2) \exp\left (i\frac{m_e}{M_A}\mathbf{q}\sum_a \mathbf{r}_a \right )d^3 r_1 d^3 r_2\qquad (3)$$ 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 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 ($\theta\gg \frac{m_e}{M_{pr}}\frac{2v_0}{v}$) or, even better, I may exclude the direct projectile-electron interactions in order not to involve $F_n^{n'}(\mathbf{q})$ into calculations any more.

Let us analyze the second atomic form-factor 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):

$$f_n^{n}(\mathbf{q}) = \int \left | \psi_{0}(\mathbf{r}_1)\right |^2\left |\phi_n(\mathbf{r}_2) \right |^2 e^{ i\frac{m_e}{M_A}\mathbf{q}(\mathbf{r}_1+\mathbf{r}_2)}d^3 r_1 d^3 r_2\qquad(4)$$ It factorizes into two Hydrogen-like form-factors: $$f_n^{n}(\mathbf{q})=f1_0^{0}(\mathbf{q}) \cdot f2_n^{n}(\mathbf{q}) \qquad (5)$$ Form-factor $\left|f2_n^{n}(\mathbf{q})\right|$ is small in our conditions and from-factor $\left|f1_0^{0}(\mathbf{q})\right|$ can be close to unity at the same time. The difference between them looks qualitatively like a difference between the red and the black lines at $\theta=0.1$ in Fig. 3 from [1], but even "stronger" and in a wider region of scattering angles.

In other words, the projectile "sees" a big positive charge cloud created with the motion of the atomic "core" (i.e., with $^{3}He^+$ ion), but it does not see the true structure of the atomic "core" consisting of the nucleus and the ground state electron. The complicated short-distance structure is integrated out in (4) and results in an elastic from-factor $\left|f1_0^{0}\right|$ tending to unity. We can pick up such a proton energy $E_{pr}$ 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 that this is physically possible in our problem, let us analyze the "threshold" angle $\theta 1_0$ for the inner electron state [1]: $$\theta 1_0=2 \arcsin\left(\frac{2v_0}{2v}4\right)\qquad (6)$$ Here, instead of $v_0$ stands $2v_0$ for the $^{3}He^+$ ion due to $Z_A = 2$ and the factor 4 originates from the expression $\left(1+\frac{M_A}{M_{pr}}\right)$. So, $\theta 1_0=\pi$ for $v=4 v_0=2(2v_0)$.

We see that for scattering angles smaller than $\theta 1_0 (v)$ form-factor $|f1_0^0|$ becomes very close to unity (only elastic channel is open for the inner electron state) whereas form-factor $\left|f2_n^n\right|$ is still very small ($\theta 2_n \ll 1$). The latter describes a large "positive charge cloud", and for inelastic scattering ($n'\ne n$) it describes the energetically accessible target excitations.

The first Born approximation in the elastic channel gives a "photo" of the atom charge distribution as if the atom was unperturbed (a photo with a certain resolution, though). Inelastic processes give possible final states different from the initial one. Inclusive cross section reduces to a great extent to a Rutherford scattering formula for a still point-like target charge with $Z=2$ and with mass $M_{Nucl}\approx 3M_{pr}$.

Let us note that for small projectile velocities the first Born approximation may become somewhat inaccurate: the projectile "polarizes" the atomic "core" 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), 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 due to lack of energy. The projectile sees it mostly as a point-like charge. It is comprehensible physically and is rather natural. In our calculation, however, this "integration out" (factually, "taking into account") of short-distance physics occurs automatically rather than "by hands", i.e., with introducing a cut-off and discarding the harmful corrections. It convinces me in possibility of constructing a physically reasonable QFT where no cut-off and discarding are necessary.

Discussion

What makes our theory physically reasonable? The permanent interactions of the atomic constituents taken into account exactly both in their wave function and in the relationships between their absolute coordinates and the relative (or collective) coordinates (for example, $\mathbf{r}_{Nucl}$ via $\mathbf{R}_A$ and $\mathbf{r}_a$). The rest is a perturbation theory in some approximation. It calculates the occupation number evolutions.

Now, let us imagine for instance that this our "two-electron" theory is a "theory of everything". Low-energy experiments would not reveal the "core" structure, but would present it as a point-like charge-1 "nucleus". 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 $M_A \approx 3M_{pr}$. The presence of the other electron is not necessary in such a theory - the latter works fine and without difficulties.

May we call a "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 as a model even outside its domain of validity. The projectile energy $E_{pr}$ (or the 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 a "one-electron" theory does not make it fail mathematically. And this is so because the one-electron theory is constructed correctly too - what is know to be coupled permanently is already taken into account exactly in it via the wave function $\phi_n$. Hence, 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 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 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 of obtaining them in a natural way because of its awkward formulation.

Electronium and all that

My electronium model [1] is an attempt to take into account a low-energy physics exactly, like in a "one-electron" incomplete atomic model mentioned briefly above. It 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 would produce soft radiation immediately, in the first Born approximation. By the way, the photons are those normal modes of the collective motions whose variables in $\psi_n$ 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 electron coordinates via the center of mass and relative motion coordinates, like in electronium or in atom. Such a completion would work as fine as my non-relativistic 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 difficulties by construction. It would be an "incomplete" QFT, but no references to the absence of the other particles (excitations) existing in Nature would be necessary to justify manually integrating out the "short-distance physics" in it, 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 automatically, like in my "two-electron" target model given above.

But this property of "insensibility to short-distance physics" does not exclusively belong to the "complete" 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, will not be harmful for calculations technically, as it was 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. I propose to think over this way of constructing QFT. I feel it is a promising direction of building physical theories.

[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])

EDIT: on arXiv there is an updated version of this communication, which is also available in the Review section.