Schrödinger’s “What is Life?” at 75

Rob Phillips∗
Division of Biology and Biological Engineering and Department of Physics,
California Institute of Technology, Pasadena, California, U.S.A

$\ast$ E-mail<EMAIL_ADDRESS>
“These facts are easily the most interesting that science has revealed in our
day.” -Erwin Schrödinger in “What is Life?”, discussing how the “dislocation
of just a few atoms” in a gene can bring about a “well-defined change in the
large-scale hereditary characteristics of the organism.”

## Abstract

2019 marked the 75th anniversary of the publication of Erwin Schrödinger’s
“What is Life?”, a short book described by Roger Penrose in his preface to a
reprint of this classic as “among the most influential scientific writings of
the 20th century.” In this article, I review the long argument made by
Schrödinger as he mused on how the laws of physics could help us understand
“the events in space and time which take place within the spatial boundary of
a living organism.” Though Schrödinger’s book is often hailed for its
influence on some of the titans who founded molecular biology, this article
takes a different tack. Instead of exploring the way the book touched
biologists such as James Watson and Francis Crick, as well as its critical
reception by others such as Linus Pauling and Max Perutz, I argue that
Schrödinger’s classic is a timeless manifesto, rather than a dated historical
curiosity. “What is Life?” is full of timely outlooks and approaches to
understanding the mysterious living world that includes and surrounds us and
can instead be viewed as a call to arms to tackle the great unanswered
challenges in the study of living matter that remain for 21st century science.

##  1 Background: The Author, His Book and its Readers

###  1.1 Schrödinger and the circumstances of his public lectures

Erwin Schrödinger is one of the luminaries of twentieth-century physics. His
insights into the workings of the microscopic world were codified in the famed
wave equation that bears his name and gave rise (among many other things) to
the s, p and d orbitals we all learn about in our first real encounter with
this world in high-school chemistry. Schrödinger was born and raised in
Austria, the only child of a highly intellectual family with a father deeply
interested in botany, leaving Schrödinger himself with a sincere and enduring
interest in the living world. In his Autobiographical Sketches he recounts
that because of discussions with his father, “I had virtually devoured The
Origin of Species” …Of course I soon became an ardent follower of Darwinism
(and still am today) [1].”

In the late 1930s after the Anschluss, Schrödinger had to flee the Nazis and
turn his back on his professorship in Graz, Austria. Fortunately, Irish prime
minister Éamon de Valera was in the process of establishing the Institute for
Advanced Studies in Dublin, and it was there that the great physicist
relocated. Later, speaking of his 17 years in Dublin, Schrödinger called them
“the happiest years of my life,” and it was in the context of his life there
that he offered the public lectures that ultimately became the book we
celebrate here (see Figure 1 for the cover page of the 1944 edition). To get a
deeper sense of the impressive intellect that Schrödinger brought to bear on
topics far and wide such as his gift for languages (including English, Spanish
and ancient Greek), the breadth of his accomplishments in physics and his
personal lifestyle, there are several excellent sources [2, 3, 4]. Noted
physicist Max Born’s autobiography gives us an impression of the regard
Schrödinger commanded among his intellectual peers: “His private life seemed
strange to bourgeois people like ourselves. But all this does not matter. He
was a most lovable person, independent, amusing, temperamental, kind and
generous, and he had a most perfect and efficient brain.”

Figure 1: The 1944 edition of Erwin Schrödinger’s “What is Life?,” published
by Cambridge University Press. The book resulted from a series of public
lectures offered by Schrödinger in Dublin in 1943.

###  1.2 The Book’s Mission: “Accounting” for Life

Schrödinger’s “What is Life?” constitutes a long argument in which he sets
himself the task of nothing less than the search for “unified, all-embracing
knowledge” to describe the natural phenomenon we refer to as life. In the
section entitled “The General Character and Purpose of the Investigation,” he
articulates his quest for universal knowledge more precisely by asking the
oft-quoted question “How can the events in space and time which take place
within the spatial boundary of a living organism be accounted for by physics
and chemistry?” And it is to answering that question that the entirety of the
long argument that follows is devoted. He asserts that the intellectual ledger
sheet was not yet balanced, with the data outpacing corresponding conceptual
understanding that “accounts” for them, words that remain true to this day.
Indeed, to really put forth a critical reading of Schrödinger’s text, we need
to understand what he means by the word “account,” a point that was overlooked
by many of his most ardent critics and which serves as the first key point of
this essay.

###  1.3 Impact on the Founders of Modern Biology

Most contemporary discussions of Schrödinger’s book focus not on its content
but on its influence on a generation of biologists. As a result, it seems
important to touch briefly on that story before turning to the main argument
that Schrödinger’s call for a physics of the living world remains valid today.
Nearly a decade before the publication of the structure of DNA by Watson and
Crick, Schrödinger was already thinking about what came to be known as the
coding problem. As Horace Freeland Judson tells us in his classic history of
modern biology, “The Eighth Day of Creation” [5]: “The earliest mention of
coding that counts was Erwin Schrödinger’s, in 1944 in ‘What is Life?’ …The
fascination of the book lay in the clarity with which Schrödinger approached
the gene not as an algebraic unit but as a physical substance that had to be
almost perfectly stable and yet express immense variety.” The influence of
Schrödinger’s book on some of the founders of modern biology is evidenced by
Figure 2, which shows a letter from the young Francis Crick to the elderly
master.

Figure 2: Letter from Crick and Watson to Schrödinger openly acknowledging the
importance of “What is Life?” to their careers.

With the historical preliminaries behind us, we turn instead to a different
question: to what extent is Schrödinger’s classic relevant to the practice of
biology and physics here and now? I argue that the vast majority of what
Schrödinger had to say remains unfinished business. This makes it an opportune
time at this three-quarters of a century milestone to ask what would it look
like for the entirety of modern science, with no allegiance to any particular
sub-discipline within science, to revisit the question of the “physical aspect
of the living cell?” The remainder of the essay centers on two key points: (i)
Schrödinger argues that the part of the natural world we refer to as life
needs to be “accounted for” in physical terms and (ii) the quest to do such
accounting will lead to new physics, not only helping us make sense of life,
but also enriching physics itself.

##  2 The Meaning of “Accounting for the Living Organism”

Schrödinger’s charge was to find out how the actions within the walls of a
living organism can “be accounted for by physics and chemistry.” When he uses
the words “account for” what I think he had in mind was the idea that our
understanding of some phenomenon of interest can be seen as resulting from an
appeal to some underlying principle, that that appeal is formulated in
mathematical language in such a way that observed phenomena are seen as
quantitative consequences of these underlying principles and that the
resulting insights make it possible to make statements about phenomena not yet
seen or measurements not yet made.

###  2.1 “Accounting For” Inanimate Matter

A fitting analogy for what Schrödinger meant when talking of “accounting for”
some class of phenomenon “by physics and chemistry” is provided by his own
work, for which he is so deservedly famous. In the mid- and late 19th century,
there was an explosion in our factual knowledge about the light given off by
and absorbed by different chemical elements in the form of their atomic
spectra as shown in Figure 3. Just as with our current proliferation of gene
and protein names and the burdensome nomenclature for the pathways that
connect them, the era of factual discovery in atomic spectra saw a
proliferation of fascinating and complicated spectral nomenclature with
concepts such as the D-line of Na and the different series such as the Balmer,
Lyman, Brackett and Paschen series that are observed in the H spectrum (for a
sense of the enormity of the factual diversity of these lines see [6]). But
what set the wavelength of absorbed or emitted light for a given element, or
the number of spectral lines in a series?

Understanding in physics demands that factual knowledge based on experiments
be complemented by conceptual knowledge. Precisely the same kind of unified,
all-embracing knowledge that Schrödinger was speaking of in “What is Life?”
was earlier needed for thinking about the huge diversity of special cases
found in atomic spectra (for an enlightening description of the history of the
study of atomic spectra, see Mehra and Rechenberg [7]). Nearly a century of
struggle followed the discovery and experimental characterization of spectral
lines as evidenced by the words of Max Planck in 1902: “If the question
concerning the nature of white light may thus be regarded as being solved, the
answer to a closely related but no less important question - the question
concerning the nature of light of the spectral lines - seems to belong among
the most difficult and complicated problems which have ever been posed in
optics or electrodynamics.” [8] One outcome of these struggles (see Figure 3)
was the discovery of empirical formulae that gave a phenomenological
mathematical description of the measured wavelengths of various spectral lines
seen in Figure 3(A) of the form

${1\over\lambda}=R\left({1\over p^{2}}-{1\over n^{2}}\right),$ (1)

where $\lambda$ is the wavelength of the spectral line in question, $R$ is a
phenomenological parameter known as the Rydberg constant, later “accounted
for” by physical understanding, and $p$ and $n$ are integers.

Figure 3: Accounting for the spectrum of hydrogen. (A) The spectrum of
hydrogen with the wavelengths of various spectral lines reported in nm. (B)
The Balmer formula for the wavelengths of the spectral lines featuring the
Rydberg constant, $R$. (C) Page from the English translation of Schrödinger’s
paper on wave mechanics in which he shows that the equation that now bears his
name amazingly gives rise to precisely the energy levels found earlier by
Niels Bohr that explain the spectrum of hydrogen.

To account for such phenomenological understanding requires us to construct
theories that explain why spectral lines are described by such a formula and
what determines the phenomenological constant $R$. Niels Bohr’s theory of the
hydrogen atom gave a first tentative conceptual success where he posited that
the electron orbits are restricted to certain quantized values with energies

$E_{n}=-{Z^{2}me^{4}\over 8h^{2}\varepsilon_{0}^{2}n^{2}},$ (2)

with $m$ the mass of the electron, $e$ the elementary charge of the electron,
$h$ the symbol for Planck’s constant and $\varepsilon_{0}$ the permittivity of
free space. Here the steps towards accounting take the form of the stunning
realization that the empirical Rydberg constant can in fact be written as

$R={me^{4}\over 8h^{3}c\varepsilon_{0}^{2}}\approx 1.1\times
10^{7}~{}\mbox{m}^{-1}.$ (3)

Erwin Schrödinger’s great atomic triumph, shown in Figure 3, was the insight
that he could interpret the discrete energy levels of atomic systems in much
the same way that we interpret the musical notes from a guitar string or an
organ pipe using wave equations.

Our digression into the niceties of how theoretical understanding “accounted”
for the many complexities of spectral lines and so much more about the
complicated microscopic world is meant to serve as an invitation to the kind
of physical frameworks that Schrödinger might have had in mind when discussing
living matter and using the words to “account for” a given phenomenon.

###  2.2 Accounting for Living Matter: From Thermal Energy to Shot Noise

Though Schrödinger clearly left the door open for quantum insights into the
phenomenon of life, much of his thinking centers on what is now known as
statistical physics. In the 19th century, statistical physics arose in
response to other phenomena demanding his type of accounting. By then,
mechanics had dominated as an explanatory framework for more than a century,
from Newton’s 1687 presentation of his “System of the World,” until the middle
of the 19th century when questions about the nature of heat began to take
center stage [9, 10]. We can use Schrödinger’s language in that context as:
how can our experience of temperature and heat be accounted for by the
mechanical motions of material particles? The quest to answer that question
took the better part of a century, culminating in the twin edifices of
statistical physics and thermodynamics, but also leaving in its wake
unresolved problems such as the specific heats of crystalline solids that
would have to await the quantum theory for their proper resolution [11].

Statistical physics was clearly much on Schrödinger’s mind in his Dublin years
as evidenced not only by “What is Life?,” which is full of deep insights into
the subject that can be read with great profit with no reference to biology
whatsoever, but also by his 1946 book, “Statistical Thermodynamics,” still a
masterful treatise that one can learn from to this day. Schrödinger was
schooled in the Austrian tradition of Ludwig Boltzmann’s statistical physics,
having only missed having one of the great founders (along with Maxwell and
Gibbs) of statistical mechanics as a professor by a year or two due to
Boltzmann’s tragic suicide. But this did not stop Schrödinger from being
steeped in the tradition of statistical physics that ultimately became central
not only to our scientific understanding of classical physics, but also to the
way we view the quantum world he helped uncover. For Schrödinger as announced
in his Autobiographical Sketches, “no perception in physics has ever seemed
more important to me than that of Boltzmann - despite Planck and Einstein,” a
resounding testament to the statistical physics sensibilities he brought to
the table in his thinking about the nature of life, and the shortcomings of
which led to his belief in the ultimate need for “new physics” to account for
living matter [1].

As an introduction to the shortcomings of the physics of Schrödinger’s time to
account for the living organism, Chapter 1 of “What is Life?” introduces “The
Classical Physicist’s Approach to the Subject.” Schrödinger reminds us that
statistical mechanics is the central conceptual framework used until that time
to interpret the collective properties of matter. To that end, he begins by
providing a beautiful and compelling introduction to the key ideas of
statistical physics. Why? Because, as he points out, “it is in relation to the
statistical point of view that the structure of the vital parts of living
organisms differs so entirely from that of any piece of matter that we
physicists and chemists have ever handled physically in our laboratories or
mentally at our writing desks.” This idea of what Schrödinger dubs a
“difference in statistical structure” between conventional and living matter
is a theme that permeates the entirety of his short book. The first chapter
thus lays the general groundwork for what follows by giving a highly
simplified but profound view of statistical physics. There, he raises many of
the themes that dominate modern biology including the roles played by
stochastic effects, small numbers, adaptation, accuracy and noise in the
reproducible processes of cells and organisms.

In Chapters 2-5, he brings the general arguments of Chapter 1 to bear on his
first big specific question: how is genetic information so stably passed from
one generation to the next, despite how few atoms are implicated in the gene
(as we will discuss in the next section)? Indeed, Schrödinger argues that the
stability of the genetic material is inconsistent with the then-known laws of
statistical physics. Though it is mind-boggling to consider, one could say
that in some sense biological information is more stable than is the Earth
itself. In the 50 million or so years since the Indian subcontinent collided
with Asia, 8000 m mountains have been thrust up from the Earth, while at the
same time, the Hox genes that confer body plans in animals from flies to
humans have passed from one generation to the next such that the family
resemblances are completely evident when we compare sequences from different
organisms. By focusing on the nature of the hereditary molecule now known to
be DNA, Schrödinger foreshadowed the agenda for the DNA-centric perspective
that colors much of modern biological investigation.

After concluding that the stability of genetic information leaves puzzles for
the physics of his (and our) day, Schrödinger then turns to another set of
biological phenomena that defy the physics of his time. Here he focuses on
what in modern guise we might call the physico-chemical basis of the self-
organization of living matter. One of the most disturbing of human tendencies
is how we all become accustomed to the wonders around us, whether in our
science or in our lives. One example that I find remarkable is seen on
transatlantic flights where, as we pass over the glaciers of Greenland, nearly
all passengers have their shades closed without even a thought of what lies
below. I might recast Schrödinger’s question about the physico-chemical basis
of the organization of living cells thus: how do the elements of the periodic
table, when sculpted by the consumption of energy at the nanometer scale
through ATP hydrolysis, come together and form remarkable structures such as
the insect eye or the dynamic and constantly-renewing outer segment of animal
photoreceptors? These amazing rhodopsin-filled structures are a reminder to do
the scientific equivalent of opening the shades over Greenland and marvel at
living cells as collections of atoms from the periodic table that produce
“endless forms most beautiful and most wonderful.”

Let’s look a little more closely at the way that Schrödinger examined the
capacity of statistical physics to account (or not) for living matter. As a
first example of the statistical structure of classical physics, he tackles
the question of paramagnetism (how the tiny individual magnets of single atoms
conspire to give rise to the macroscopic phenomenon of magnetism and how it
depends upon temperature). The magnetic phenomenon results from a competition
between the energy advantage that comes from aligning spins with an applied
magnetic field and the entropy that comes from those spins adopting random
orientations. The key point is the recognition that the thermal energy scale,
$k_{B}T\approx 4.1$ pN nm, as Schrödinger points out, is the natural energy
scale that presides over the atomic and molecular world (and the molecules of
the living world can only find immunity from this energy scale by investing
some other energy such as ATP hydrolysis to avoid it - see the bottom panel of
Figure 4).

With the “order from disorder“ exhibited by magnetic materials in hand, he
then proceeds to a description of Brownian motion. Schrödinger’s little four
page discussion of diffusion culminating in the diffusion equation itself is
as perfect a description of the subject as I have ever seen. The important
point of this discussion is to illustrate the lawful mathematical precision
that emerges from the apparent lawlessness of the flips of a coin as each
molecule randomly chooses to move left or right, up or down. Speaking of the
jostling of these Brownian particles, Schrödinger poetically muses: “Their
movements are determined by the thermic whims of the surrounding medium; they
have no choice. If they had some locomotion of their own, they might
nevertheless succeed in getting from one place to another - but with some
difficulty, since the heat motion tosses them like a small boat in a rough
sea.” Again, the statistical structure of classical physics reveals that
ordered states with the mathematical precision of an exponential function such
as the concentration gradient shown in Figure 4 in fact emerge from the second
law of thermodynamics which tells us that systems evolve towards states of
maximum entropy, yielding a kind of order from disorder. Turing’s paper
showing an even more subtle and beautiful example of order from disorder was
still a decade in the future [12].

Figure 4: Schrödinger’s order from disorder and order from order. In the top
panel, Schrödinger illustrated the way that strictly random motions of atoms
or molecules could give rise to reproducible patterns of concentration (left).
Modern developmental biology has borne that out in the form of morphogen
gradients such as the gradient of the transcription factor Bicoid in the fly
embryo shown here schematically beneath the graph. The length ($L$) of the
embryo is plotted in dimensionless units $x/L$. The profile of Bicoid can be
thought of as emerging from the “disordered” processes of diffusion and
protein degradation. In the bottom panel, the expenditure of energy at very
small scales in the form of ATP hydrolysis leads to order at the cellular
scale, the organismal scale and even the population scale.

Once these foundations have been laid, Schrödinger starts homing in on
biological phenomena by considering the “limits of accuracy of measuring,” the
insight being that there are limits to such measurement due to the natural
fluctuations of the measuring device itself, a phenomenon perhaps even more
important in living organisms than in the experimenter’s apparatus. Indeed, in
my view, in this section, Schrödinger foreshadowed one of the most important
themes of modern physical biology, namely, how living organisms defy the
strictures of equilibrium physics. These questions arise in settings ranging
from the origins of high fidelity polymerization and the emergence of the
concept of kinetic proofreading [13, 14] to the measurement of concentration
differences by cells performing chemotaxis [15] to the emergence of herds with
orientational order in apparent defiance of fundamental physical theorems
[16]. The topic of fidelity in biological polymerization was also brilliantly
undertaken by Linus Pauling [17], but from a molecular perspective rather than
the conceptual point of view adopted by Schrödinger.

Schrödinger starts his discussion on the limits of accuracy of measuring with
an analysis of the kinds of torsional balance apparatus used by Cavendish
(gravity) and Coulomb (electrostatics) to measure the classic inverse-square
laws. His point is that as the apparatus gets smaller and smaller, the
deflections induced by the forces of interest will be comparable in magnitude
to those induced by thermal motion. Questions of biological accuracy and
precision have now become a centerpiece of rigorous thinking in modern biology
in the form of the Berg-Purcell limit, but more generally in the context of
how well living organisms can sense their environments, whether in the context
of hearing or seeing, or in the detection of chemical messengers [18].
Schrödinger himself understood precisely the physics question in play, noting:
“The uncontrollable effect of the heat motion competes with the effect of the
force to be measured and makes the single deflection observed insignificant.
You have to multiply observations, in order to eliminate the effect of the
Brownian movement of your instrument. This example is, I think, particularly
illuminating in our present investigation. For our organs of sense, after all,
are a kind of instrument. We can see how useless they would be if they became
too sensitive.”

But this is all more than pretty words. Schrödinger wants to describe these
physical challenges to the living organism quantitatively. He notes the
monotonous repetition of this same statistical principle in the inorganic
world, but wants his listeners and readers to know that there is a simple
mathematical law, the “so-called $\sqrt{n}$ law,” that tells us the size of
the fluctuations to be expected in a system containing $n$ atoms or molecules.
Schrödinger summarizes that law as: “The laws of physics and physical
chemistry are inaccurate within a probable relative error of the order of
$1/\sqrt{n}$, where $n$ is the number of molecules that co-operate to bring
about the law - to produce its validity with such regions of space or time (or
both) that matter, for some considerations or for some particular experiment.”
These ideas are central to modern biological enquiry. For example, when
photosynthetic bacteria divide, they have to carry with them copies of a
photosynthetic organelle known as the carboxysome, with only a few (3-6)
copies per cell. During the division process, they have special machinery to
prevent these unwanted $1/\sqrt{n}$ partitioning errors [19]. By way of
contrast, partitioning errors in transcription factors are a demonstrable part
of the reason for noisy gene expression [20, 21]. And yet, and here is the
crux of the whole book, “The classical physicist’s expectation, far from being
trivial, is wrong.” As seen by the carboxysome example, living organisms have
found ways to insulate themselves from the all-important $\sqrt{n}$ law and
many of the other strictures of equilibrium statistical physics.

As I will elaborate on below, one way to think of that “wrongness” is
summarized in the bottom panel of Figure 4 which hints at the need for a
statistical physics of the phenomena that emerge when energy is invested to
keep those systems out of equilibrium, an enterprise that has been undertaken
to great effect in recent decades (for a flavor of this new physics, see
[22]). To further elaborate on why the classical physicist’s expectation is
wrong, let’s revisit Schrödinger’s quantitative analysis of the question of
the stability of the genetic information.

##  3 Fermi Problems in Schrödinger Style

Numbers sharpen our questions. Schrödinger appreciated this sharpness and used
quantitative estimates as a key part of his arguments. In the world of
physics, numerical estimates to describe some phenomenon of interest are
sometimes known as Fermi problems and are a model for the power of order-of-
magnitude reasoning to clarify both our questions and our thinking in response
to those questions. Their name refers to a style of thinking brought to an art
form by Italo-American physicist Enrico Fermi, who could estimate his way to
numerical answers to questions ranging from the number of piano tuners in a
big American city, to the heat loss in our homes if we forget to install storm
windows, to the yield of the atomic explosion in the Trinity Test of 1945.
Fermi brought this same approach [23] to critical questions in the science
challenges he faced in his professional life as a physicist. For example, the
design of the first successful nuclear reactor - built upon a string of
systematic preliminary studies, each characterized by numerical estimates on
topics such as neutron diffusion - led Fermi to predict the precise moment
that the Stagg Field nuclear reactor would reach a self-sustained nuclear
reaction [24].

Schrödinger’s book draws from the same Fermi-problem inkwell. In fact,
Schrödinger’s short manifesto gives perhaps the best example of a physicist
working through Fermi problems in plain view that I have seen in written form.
Over and over, Schrödinger poses questions of the form: what sets the scale of
X?, the quintessential framing of questions in order-of-magnitude thinking.
Examples include: what sets the relative scale of organisms and atoms, what is
the size of a gene and how much energy is needed to maintain the stability of
the gene?

Having laid down his statistical mechanical and order-of-magnitude thinking
foundations, Schrödinger was now prepared to take up the question of the size
of a gene. His musing on this question is motivated by the statistical
structure of classical physics and his interest in how biological systems deal
with the $\sqrt{n}$ law introduced above. His argument is that if the number
of atoms associated with a gene is small, then the natural fluctuations of
such systems would impact the stability of genetic information. Concretely, he
wonders, how many atoms are associated with a gene and if that number is small
in the sense of the $\sqrt{n}$ law, then how do these genes safeguard
themselves from the inevitable statistical fluctuations present in any
collection of atoms?

###  3.1 On the Physical Dimensions of Genes

To examine the question of the size of a gene, Schrödinger performs several
different estimates which are illustrated in Figure 5. Note the painstaking
and often misguided detective work involved in answering such a deep question
as the size of the gene before it was even widely accepted that DNA is the
genetic material. Schrödinger begins with two entirely independent estimates
of the size of the gene. The first, shown in Figure 5(A) builds on the
subsection of his book entitled “Crossing-over. Location of properties” in
which he explains how maps like the classic case for Drosophila of Alfred
Sturtevant are discovered. As seen in the figure, when there is a crossing
over event, we can ask the frequency with which two genes end up on the same
strand together. The more likely that occurrence, the closer those two genes
are on the chromosome and from these frequencies an actual map of gene
positions on the chromosome can be divined. Given one of these maps of a
chromosome, Schrödinger proposes a strictly order-of-magnitude upper bound on
the size of a gene based on the size of a chromosome and the number of
“properties” (i.e., genes) per each chromosome. As he notes, the estimate
provides a bound, because at the time of his writing, the entirety of the
genes on a chromosome had not yet been mapped. The concept of the estimate is
reasonable as is indicated for a bacterial genome in Figure 5(D). The second
such estimate is a very clever (though “wrong”) idea of using the banded
patterns of chromosomes as a measure of gene size. As seen in Figure 5(B), if
we consider the 23 chromosomes (Schrödinger refers to 24 pairs of chromosomes,
true for chimps, but not humans) with their banding patterns, this estimate
would say that the size of a gene is much larger than we know to be true.

Figure 5: The size of a gene. The first estimate exploited the beautiful
insights into mapping genes on chromosomes (originally in Drosophila) to
figure out the mean spacing of genes. Given the gene number and the size of
chromosomes this approach would yield helpful insights. Schrödinger’s second
estimate uses banding patterns on chromosomes on the mistaken assumption that
each band is a gene. The Delbrück estimate was based on the mutation rate
induced by x-rays. A modern estimate uses the number of base pairs in a gene
and the very useful rule that the volume of a base pair is 1 nm3. For more
details of the paper of Timoféeff-Ressovsky, Zimmer and Delbrück that measured
the data shown on mutation rate and radiation dosage, see the book “Creating a
Physical Biology” which includes its translation [25].

He really hits his stride when considering a third estimate of the size of a
gene as seen in Figure 5(C). This is the point in the story where
Schrödinger’s reports on the work of Max Delbrück and coworkers, inspiring
Perutz’s remark “In retrospect, the chief merit of “What is Life?” is its
popularization of the Timoféeff, Zimmer and Delbrück paper that would
otherwise have remained unknown outside the circles of geneticists and
radiation biologists” [26]. As seen in the figure, experiments on ionization
of gases allowed for a measure of radiation intensity that could then be
directly translated into a tool for examining radiation damage in DNA. By
measuring the radiation-induced mutation rate, an estimate for the size of a
gene could be made leading to the idea that a gene involves roughly 1000
atoms. Our modern understanding of the DNA double helix tells us that a base
has $few\times 10$ atoms, meaning that a typical thousand nucleotide bacterial
gene would involve on the order of $few\times 10^{4}$ atoms, a bit more than a
factor of 10 larger than Schrödinger’s estimate. Though his estimate is too
low for the size of an entire gene and a little too large for the size of a
basepair, it is an impressive use of indirect reasoning to arrive at molecular
dimensions that conjures images of the way Benjamin Franklin, Lord Rayleigh,
Agnes Pockels and Irving Langmuir attacked the question of the molecular
dimensions of lipid molecules indirectly by looking at the spreading of oil on
water [27, 28].

The outcome of these arguments leaves Schrödinger with a mystery: “We are now
seriously faced with the question: How can we, from the point of view of
statistical physics, reconcile the facts that the gene structure seems to
involve only a comparatively small number of atoms (of the order of 1000 and
possibly much less), and that nevertheless it displays a most regular and
lawful activity - with a durability or permanence that borders upon the
miraculous?” In examining this question, we have to put ourselves in the
mindset of Schrödinger who is once again asking us from the perspective of
what it means for physics (not biology) to account for these phenomena.
Schrödinger then goes farther by noting “Thus we have come to the conclusion
that an organism and all the biologically relevant processes that it
experiences must have an extremely ‘many-atomic’ structure and must be
safeguarded against haphazard, ‘single-atomic’ events attaining too great an
importance.” All of this kind of “what sets the scale of X” thinking brought
Schrödinger to precisely the same place that was arrived at 30 years later in
a different way by John Hopfield and Jacques Ninio when they presented a
theory of just the kind of “safeguarding” called for in “What is Life?” and
that now goes under the name of kinetic proofreading [13, 14].

##  4 The Biological Frontiers of Physics: New Laws to Be Expected in the
Organism

Having made his arguments about the shortcomings of the statistical physics of
his time to account for the stability of living matter, Schrödinger closes the
book with some higher-level thinking on the implications of living matter for
the future of physics. In their great book “The Evolution of Physics,” Albert
Einstein and Leopold Infeld make it very clear how physics has repeatedly been
driven forward by new experimental measurements (think Faraday and
electromagnetic induction) that demand the emergence of new concepts [29]. At
the beginning of the 19th century, the concept of entropy, one of the
foundational ideas of modern science, had not even been conceived or defined,
but an increasingly sophisticated experimental program revealing the character
of temperature and heat demanded it. Similarly, although the idea of “field
theory” was implicit in the development of continuum mechanics by great
thinkers such as Euler, it would have to await the labors of Faraday, Maxwell
and others before it took the proportions that would lead Einstein and Infeld
to say “A new concept appears in physics, the most important invention since
Newton’s time: the field. It needed great scientific imagination to realize
that it is not the charges nor the particles, but the field in the space
between the charges and the particles which is essential for the description
of physical phenomena.” The question is: to deliver on Schrödinger’s call to
“account for” the phenomena of life, what new experiments and concepts does
expanding our domain of physical enquiry into the realm of the living demand?

My reading of Schrödinger’s book is that he is arguing that, just as the
phenomena of heat and electrodynamics forced upon us a myriad of new concepts
such as temperature, entropy, the electric field and radiation pressure, the
phenomena of the living world similarly demand the continued evolution of
physics through new concepts. In his section on “New Laws to Be Expected in
the Organism,” Schrödinger reveals his hand: “What I wish to make clear in
this last chapter is, in short, that from all we have learnt about the
structure of living matter, we must be prepared to find it working in a manner
that cannot be reduced to the ordinary laws of physics. And that not on the
ground that there is any “new force” or what not, directing the behaviour of
the single atoms within a living organism, but because the construction is
different from anything we have yet tested in the physical laboratory.”
Indeed, would anyone be surprised by the idea that, as we subject new classes
of phenomena to quantitative scrutiny in the physical laboratory, yet again,
as has been true every century since the days of Tycho Brahe, new concepts
will be demanded of us?

Schrödinger makes this point beautifully through a colorful analogy for
thinking about how the elements of the periodic table can be exploited to
totally different ends in living matter. “To put it crudely, an engineer,
familiar with heat engines only, will, after inspecting the construction of an
electric motor, be prepared to find it working along principles which he does
not yet understand. He finds the copper familiar to him in kettles used here
in the form of long wires wound in coils; the iron familiar to him in levers
and bars and steam cylinders here filling the interior of those coils of
copper wire. He will be convinced that it is the same copper and the same
iron, subject to the same laws of Nature, and he is right in that. The
difference in construction is enough to prepare him for an entirely different
way of functioning. He will not suspect that an electric motor is driven by a
ghost because it is set spinning by the turn of a switch, without boiler and
steam.”

Schrödinger thus concludes “it needs no poetical imagination but only clear
and sober scientific reflection to recognize that we are here obviously faced
with events whose regular and lawful unfolding is guided by a ‘mechanism’
entirely different from the ‘probability mechanism’ of physics.” One category
of “new laws” that might prove particularly potent in biology and that have
been given short shrift in the molecular biology era are strictly
phenomenological models that make no reference to underlying “mechanism.”
These kinds of laws have formed a centerpiece of physics for more than three
hundred years in contexts ranging from the laws of elasticity to
hydrodynamics. Here what I have in mind is a phenomenological link in
mathematical terms that connects the variables that have emerged from
measurements. Both the familiar Hooke’s law and ideal gas law originally
emerged as highly powerful, and yet, strictly phenomenological reflections of
what had been measured in the laboratory. Examples have already started to
emerge in physical biology as well. Recent high-resolution measurements make
it possible to explore the growth of bacteria resulting in a number of
propositions for bacterial growth laws [30, 31, 32]. With no reference to the
molecular underpinnings, these phenomenological laws engender corresponding
phenomenological hypotheses for how populations of bacterial cells maintain a
narrow distribution of cell sizes, such as adder and timer models in which the
growing cell either waits to divide until it has added a certain fixed
quantity of material or until a fixed amount of time has elapsed [33]. Another
provocative and beautiful set of phenomenological laws has emerged concerning
the character of the entire proteome [34, 35]. Here the idea is that the
growth rate of cells serves as a kind of “state variable” and that depending
upon this growth rate, the fraction of the proteome devoted to protein
production (i.e., ribosomes) takes a very specific value. Schrödinger’s
chapter 6 asks us to think about the emergence of order from order as seen in
Figure 4, a topic that has become central to modern physical biology, whether
in the context of bird flocks [36, 37] or cytoskeleton-motor systems such as
those that partition chromosomes during cell division [38, 39, 40, 41]. In all
of these cases, phenomenological laws have emerged and we are now in a stage
of science when the laws are tested and refined, the implications are further
examined and efforts are set forth to try and derive those laws from some
deeper understanding of the underlying processes.

Some of the many other areas in which I suspect we will find “new physics”
focus on the ways in which biological systems locally defy the tendency
towards equilibrium. Examples include a series of problems such as the
accuracy problem (how biological systems achieve such high fidelity in
comparison with what is implied by the “statistical structure” of classical
physics), the adaptation problem (how biological systems change their physico-
chemical behavior in real time in response to environmental cues), the
reproducibility problem (how living organisms achieve the same outcome such as
the human body plan over and over again in a nearly error-free fashion), the
structure problem (how structures such as the outer segment of photoreceptors
are constructed and maintained in the face of the noisy world that surrounds
them). I also suspect that there are surprises before us in the unfinished
business of dynamics, a subject that started with the great successes of
classical mechanics, then to the incomplete promise of thermodynamics and now
the full-fledged challenge of the out-of-equilibrium aspects of biological
dynamics, whether the separation of chromosomes or the origin of species. Of
course, these “problems” are offered tentatively and subjectively since every
generation has to decide what are its most pressing problems, but my main
point is about the kind of solutions that we should demand in accounting for
biological phenomena.

##  5 A Manifesto: Schrödinger’s Unfinished Business

A perennial question for any scientist is: what to work on? Of course, there
is no one right answer and one of my own favorite answers is: whether young or
old, scientists should work on whatever they are truly most curious about
understanding from the vast array of mysteries presented by the world around
us. In academia, much debate and angst are powered by this question in
disguise as we ask ourselves, should this person be hired or that grant be
funded? One of the ways that people try to sharpen that question is by asking
whether we will find new physics or new biology, depending upon within which
department that question is considered. For a very thoughtful modern
reflection on the question of new physics more broadly by a noted physicist
who worked in many domains of physics, see “Does Astronomy Need ‘New
Physics’?” [42]. Stated most succinctly, Schrödinger’s short work ventures the
guess that indeed, by looking at living matter, those who subscribe to the
definition of understanding demanded in physics will find new physics there.

As already alluded to throughout the article, the last 30 years have seen
enormous progress at the interface between physics and biology. As for whether
or not there is new physics to report, I think it depends on how we take that
question. If we are practicing what Thomas Kuhn referred to as “normal
science,” there is no doubt that there has been an impressive array of results
that surely count as new physics. Examples abound, whether in the context of
cellular motility, detection and adaptation in the context of chemotaxis [43,
15, 44], or in the analysis of population genetics with its beautiful
analogies between the Boltzmann distribution of statistical mechanics and the
distribution of allele frequencies [45], or in the surprising new features
revealed in the study of active matter [16]. In an excellent series of
lectures, John Toner notes: “…the biggest surprise in the entire field of
active matter is that a ‘polar ordered dry active fluid phase’ is even
possible in two dimensions,” technical words behind a fun and interesting
example of the new physics that has emerged in thinking about bird flocks. Yet
another example is offered by the phenomenon of cytoplasmic streaming [46],
observed before the founding of the United States, and yet only in the last
decade has the natural language of dimensionless variables allowed us to
compare the relative importance of diffusion and flow in transporting
materials within large cells and to compute the kinds of flow patterns that
emerge [47, 48, 49, 50]. All of these examples constitute the activities of
normal science and the act of “doing physics” on them has resulted in not only
sharpening our questions, but also in increasing the depth of our
understanding. In a playful turn of the millennium piece entitled “Molecular
Vitalism,” Kirschner, Gerhart, and Mitchison show how many of the issues
raised by Schrödinger about what gives living organisms their distinct
physicochemical attributes remain as fresh now as they were in the 1940s [51].
“We do not question the importance of genetics, nor dispute the role of DNA as
the blueprint for all the components of living systems, but we think it worth
asking to what extent the postgenomic view of modern biology would convince a
nineteenth century vitalist that the nature of life was now understood.”

If we adopt a more sweeping view in which we ask for a revolution in physics
that has come on the heels of investigating biological phenomena, we may need
to exercise a little more patience. We shouldn’t be surprised by the glacial
pace at which we get our revolutions. There was a century between the initial
discovery of spectral lines and properly “accounting for” them in the series
of classic papers by Schrödinger [52]. That said, I am wary of the common
attitude that, because something has not been done, or worse yet, because a
given commentator themself has not done it, that means that it cannot be done.
I suspect that in the physics pedagogy a hundred years hence (if humans are
still teaching each other important ideas by then), there will be courses
whose central ambition will be to explain the harvest of the revolution that
resulted from physicists trying to answer the question “what is life?” to
their own satisfaction, in much the same way that we have physics courses
dedicated to the question of “what is matter?”

Despite both the insight and promise of Schrödinger’s thinking in “What is
Life?”, not all responses to his book were positive [2]. Three greatly
accomplished scientists who came down on the negative side of the ledger were
famed American geneticist HJ Muller, Caltech chemist and visionary Linus
Pauling, and the structural biologist Max Perutz, Schrödinger’s compatriot.
Their reactions strike to the very heart of how different fields view what
questions are interesting and what constitutes acceptable answers to those
questions [53]. Indeed, as is now clear, a major thrust of this essay has been
that in adopting Schrödinger’s physicist definition of understanding, we will
see that the study of living matter will demand new physics. Though
differences in philosophy about what it means to understand something explain
some of the negative reaction to Schrödinger’s classic, I find a marked lack
of generosity given the circumstances of the book’s origin as a written
summary of public lectures.

In a centenary volume celebrating Schrödinger’s contributions to modern
science [54], Pauling wrote “Schrödinger’s discussion of thermodynamics is
vague and superficial to an extent that should not be tolerated even in a
popular lecture.” Writing in the same volume, Perutz is even more scathing,
making comments such as “Sadly, however, a close study of his book and of the
related literature has shown me that what was true in his book was not
original, and most of what was original was known not to be true even when the
book was written….the apparent contradiction between life and the statistical
laws of physics can be resolved by invoking a science largely ignored by
Schrödinger. That science is chemistry.” Hence, as did Crick, Perutz takes
Schrödinger to task for ignoring “chemistry,” which he argues would make the
stability of genetic information clear. Here I part ways with Perutz since as
our ability to routinely melt DNA in our PCR machines shows, the stability of
the genetic material is intimately related to precisely the thermal physics
discussed by Schrödinger. Understanding the high-fidelity of the processes of
the central dogma, to name but one example that falls outside the purview of
both classical statistical physics and the chemistry of the mid 20th century,
demands much more of us than Boltzmann distributions and chemical bonding.

It is intriguing to explore the claim that if Schrödinger had but only
appealed to the science of chemistry, no mysteries would have remained for the
classical physicist trying to “account for” the living organism. Having spent
now nearly a decade trying to come to terms with the field that Perutz was
central in creating, namely, the subject of allostery, I remain more skeptical
than ever of what I will call the salt-bridge argument [55, 56], an example of
the conviction that if we only understand the atomic-level structures of the
macromolecules of the living world, then “mechanism” and understanding will
unfold before our eyes. Really, what we are talking about here is precisely
the kind of polarizing debate that separates our political lives, what Thomas
Sowell christened a “conflict of visions.” Schrödinger was not invalidating or
critiquing the world view of chemists or biologists, he was trying to explain
what it looks like for physicists to “account for” a subject. His views are
echoed forcefully and eloquently by today’s leading biophysical thinkers [18,
57, 58, 59]. The point of my loving review of Schrödinger’s little book “What
is Life?” 75 years on is that, to really understand the book’s meaning, we
have to remember both the question being considered and the audience being
addressed. In my view, it is a mistake to think of his work as a manifesto
about biology for biologists. It is a manifesto about the frontiers of physics
and the way that every time physics tackles new classes of phenomena, it
requires new concepts and ultimately results in the formulation of new laws.
It is also a manifesto about the unity of nature. Nature cares not for the
names of our subjects. Names such as physics and biology are a strictly human
conceit and the understanding of the phenomenon of life might require us to
blur the boundaries between these fields.

Sidney Brenner once quipped that, in research, one should either be 6 months
ahead of the scientific pack or 30 years behind. There is much to that remark
since over and over again, pathbreaking discoveries are made when new
technologies are used to reconsider “old” problems. Nowhere is this more true
than in the case of the hydrogen atom, one of the most remarkable gifts ever
given to science [60]. Hydrogen has served as the quintessential test case for
what it means to “account for” the behavior of atoms (spectral lines), the
nucleus (the deuteron), the coupling between radiation and matter, Bose-
Einstein condensates and beyond. Similar case studies are only waiting to be
exploited in biology once Schrödinger’s notion of what it means to account for
a phenomenon is accepted. The study of living matter needs its hydrogen atoms.
Erwin Schrödinger’s remarkable ‘What is Life?’ makes it clear that Brenner
could have gone even farther and exhorted us to search 75 years into the past
to find an inspiring charge for the future.

Acknowledgments

I am especially grateful to David Booth who kindly suggested that I undertake
this labor of love, and to Christina Hueschen with whom I have repeatedly
discussed Schrödinger’s classic book over the last several years. My biggest
debt in learning about the confluence of biology and physics is to my book co-
authors Jane Kondev, Julie Theriot, Hernan Garcia, Christina Hueschen, Ron
Milo, Wallace Marshall and Thomas Lecuit. In addition, I have learned so much
about physics, biology and their intersection from a veritable who’s who of
deep thinkers on the state of the art in modern science. For either their
direct or indirect help with this article, I want to especially thank Clarice
Aiello, Howard Berg, Bill Bialek, Curt Callan, Anders Carlsson, Griffin Chure,
Ken Dill, Ethan Garner, Bill Gelbart, Lea Goentoro, Ray Goldstein, Stephan
Grill, Christoph Haselwandter, Hopi Hoekstra, Joe Howard, Tony Hyman, Sri
Iyer-Biswas, Quincey Justman, Marc Kirschner, Heun Jin Lee, Jennifer
Lippincott-Schwartz, Niko McCarty, Madhav Mani, Tim Mitchison, Chris Miller,
Andrew Murray, Phil Nelson, Hirosi Ooguri, Mariela Petkova, Molly Phillips,
Steve Quake, Manuel Razo, Udo Seifert, Pierre Sens, Lubert Stryer, Mark Uline,
Ron Vale, Aleks Walczak, Jon Widom, Chris Wiggins, Ned Wingreen and Carl
Zimmer. I am grateful to all of these impressive thinkers for useful
discussions and/or commenting on the manuscript, though the views expressed
here should not be blamed on them. I am also grateful to the NIH for support
through award numbers DP1OD000217 (Director’s Pioneer Award) and R01 GM085286.
The trust and financial support of this great institution make it possible for
today’s scientists to grapple with the endless fascination of trying to answer
Schrödinger’s classic question, “What is Life?”

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[1]Axel Schmidt

1]Department of Physics, The George Washington University, 725 21st St. NW,
Washington, DC 20052, USA

# Radiative Corrections in Super Rosenbluth Experiments

Quinn Stefan<EMAIL_ADDRESS>[

###### Abstract

Super Rosenbluth experiments, elastic electron-proton scattering experiments
that eschew traditional electron detection and opt instead for the detection
of the recoiling proton, have several experimental advantages. One claimed
advantage is that radiative corrections are more favorable, i.e., smaller and
with less kinematic dependence. In this paper, we explore this claim by
conducting Monte Carlo simulations of both Super Rosenbluth and traditional
Rosenbluth experiments with different models of radiative effects. When using
a model that employs the peaking approximation, we indeed confirm that reduced
kinematic dependence of the radiative corrections. However, we find that more
sophisticated models that avoid the peaking approximation are unable to
produce numerically stable results, due to a large enhancement to the cross
section for bremsstrahlung radiation from the proton when the invariant
squared momentum transfer $Q^{2}$ approaches zero. Since this enhancement is
not modelled in the peaking approximation, a more robust approach to radiative
corrections in Super Rosenbluth experiments is needed.

###### keywords:

Electron scattering, elastic scattering, Rosenbluth separation, radiative
corrections

##  1 Introduction

The electromagnetic form factors of the proton, $G_{E}$ and $G_{M}$, which
encode the distributions of charge and magnetism, respectively, have been
mapped from decades worth of elastic electron-proton scattering measurements
(see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9]). The form factors $G_{E}$ and
$G_{M}$, are functions of a single variable, $Q^{2}$, the negative squared
four-momentum transfer; $G_{E}^{2}$ and $G_{M}^{2}$ can be independently
determined for a given value of $Q^{2}$ through the traditional technique of
Rosenbluth separation. By making measurements of the unpolarized elastic
scattering cross section with several combinations of beam energy and
scattering angle, but with $Q^{2}$ fixed, the contributions from the electric
and magnetic form factors can be disentangled. This technique faces several
experimental challenges; it relies on multiple measurements of absolute cross
sections made at different kinematics with starkly different counting rates.
The analysis of the data invariably leads to strong anti-correlation in the
inferred $G_{E}^{2}$ and $G_{M}^{2}$.

These challenges, as well as the emergence of a discrepancy in the early 2000s
between determinations of $G_{E}/G_{M}$ from polarization transfer
measurements [10, 11, 12, 13, 14, 15, 16, 17] and determinations from earlier
Rosenbluth separations, were part of the impetus for new experiments based on
the so-called “Super Rosenbluth” technique. In a typical Rosenbluth
separation, a spectrometer is used to detect the scattered electron. By
contrast, in a Super Rosenbluth experiment, only the recoiling proton is
detected. One major advantage of the Super-Rosenbluth technique is that for
fixed $Q^{2}$, the out-going proton always has the same momentum, regardless
of the initial beam energy; hence there is no need to modify the magnetic
field setting of the spectrometer between measurements, reducing systematic
uncertainty. Two Super Rosenbluth measurements were conducted at Jefferson Lab
[18, 19], and results from one were published [7], confirming the results of
prior, traditional Rosenbluth measurements.

Ref. [7] claims another advantage of Super Rosenbluth measurements is a
reduced sensitivity to radiative effects. Specifically, Ref. [7] writes that
“radiative corrections (mainly electron bremsstrahlung) $\ldots$ have smaller
$\varepsilon$-dependence when the proton is detected,” where $\varepsilon$ is
the virtual photon polarization parameter. This is advantageous because in
Rosenbluth separation, the two form factors are disentangled based on the
$\varepsilon$-dependence of the cross section. Smaller
$\varepsilon$-dependence of extraneous effects, such as radiative corrections,
means a reduced risk of bias in the results. Ref. [7] goes on to write that
for the experiment in question, “Radiative corrections to the cross section
are 20%, with a 5%–10% $\varepsilon$-dependence, smaller than in previous
Rosenbluth separations where the electron was detected.”

The possible advantage of proton detection in terms of reducing sensitivity to
radiative effects comes from the fact that bremsstrahlung radiation is
primarily emitted from the electron (either incoming or outgoing) owing to the
electron’s much smaller mass. By ignoring the electron and detecting the
proton, one need not model the precise shape of the so-called radiative tail
of the electron as it loses energy to radiation. This argument seems sound
until considering that calculating the radiative correction to a proton
detection experiment requires one to integrate over all possible out-going
electron momenta. The electron is a light particle, prone to radiation, and
one must consider this radiation not only when the emitted energy is small,
but also when it is very large.

In this paper, we revisit the claims about radiation in Super-Rosenbluth
experiments made in Ref. [7]. First, we repeat the approach used in Ref. [7]
based on a Monte Carlo simulation of radiative effects in the peaking
approximation, and we confirm that the radiative corrections are in fact
smaller and less $\varepsilon$-dependent than a corresponding electron-
detection experiment. Then, we examine a more sophisticated model of radiative
effects that go beyond the peaking approximation. We find that the more
sophisticated model has enormous variance in event weights due to dramatic
rise in the cross section for hard radiation emitted from the proton leg in
the direction of scattered electron, which is not captured in the peaking
approximation. We conclude that the peaking approximation may be misleadingly
inaccurate for Super Rosenbluth experiments and see a need for numerically-
stable non-peaking approaches for future experiments.

##  2 Methodology

We have evaluated the radiative correction prescribed by different models of
radiative effects by performing Monte Carlo pseudo-experiments. Traditionally,
radiative corrections have been formulated in terms of a correction, $\delta$,
such that:

$d\sigma_{\text{rad.}}=(1+\delta)d\sigma_{\text{Born}},$ (1)

where $d\sigma_{\text{rad.}}$ is the measured cross section which includes
radiative effects and $d\sigma_{\text{Born}}$ is the corresponding cross
section under the Born approximation, i.e., one-photon exchange only. Various
prescriptions for $\delta$ have been suggested (see, e.g., [20, 21, 22, 23]),
which account for effects at the next-to-leading-order level, including vertex
corrections, two-photon exchange, vacuum polarization, and the emission of a
soft bremsstrahlung photon, which is experimentally indistinguishable from
elastic scattering. It was pointed out, first in Ref. [24], that accounting
for soft-bremsstrahlung to all orders can be approximated by exponentiating
the correction, i.e.,

$d\sigma_{\text{rad.}}=e^{\delta}d\sigma_{\text{Born}},$ (2)

As we consider both exponentiated and non-exponentiated models in this paper,
we choose to define the radiative correction factor $f_{\text{RC}}$ as the
ratio of the cross section including radiative effects to the equivalent cross
section under the Born approximation:

$f_{\text{RC}}\equiv\frac{d\sigma_{\text{rad.}}}{d\sigma_{\text{Born}}}.$ (3)

We consider only the elastic electron-proton scattering reaction in the fixed-
target frame, in which the final state has a scattered electron, with
scattering angle $\theta_{e}$ and outgoing momentum $p_{e}$, a recoiling
proton emerging at an angle $\theta_{p}$ and momentum $p_{p}$, and possibly
some energy and momentum radiated in the form of photons. In this reaction,
$Q^{2}$ can be determined from the outgoing lepton kinematics, i.e.,

$Q^{2}=2E_{\text{beam}}E_{e}-2p_{\text{beam}}p_{e}\cos\theta_{e}-2m_{e}^{2},$
(4)

with $E_{\text{beam}}$ and $p_{\text{beam}}$ being the incoming beam energy
and momentum, respectively, $E_{e}$ being the energy of the outgoing electron,
and $m_{e}$ being the lepton mass. Alternatively, $Q^{2}$ can be defined from
the outgoing proton kinematics, i.e.,

$Q^{2}=2E_{p}m_{p}-2m_{p}^{2}$ (5)

where $E_{p}$ is the outgoing proton energy and $m_{p}$ is the proton mass. In
the absence of any energy radiated by photons, these definitions coincide.
Another important variable considered in Rosenbluth separations is
$\varepsilon$, the virtual photon polarization parameter, which, in terms of
electron kinematics, is defined as:

$\varepsilon=\left[1+2(1+\tau)\tan^{2}\frac{\theta_{e}}{2}\right]^{-1}$ (6)

where $\tau\equiv Q^{2}/4m_{p}^{2}$.

In order to calculate $f_{\text{RC}}$, we ran Monte Carlo simulations using
different event generators with different models for incorporating radiative
effects. Simulations were conducted in the form of pseudo-experiments; after
generating samples of Monte Carlo events, events were accepted if they fell
within an acceptance window around a central kinematic point. We used a polar
angle ($\theta$) acceptance of $\pm 1$ mrad, small enough to heavily suppress
effects due to the change in cross section across the acceptance window. This
is substantially smaller than the acceptance of most high-resolution
spectrometers. For example, a Hall A High Resolution Spectrometer (HRS), used
to make the measurement in Ref. [7], has a horizontal angle acceptance of
approximately $\pm 25$ mrad, depending on the desired momentum acceptance
[25]. Our choice was motivated by the goal of evaluating the radiative
correction at specific kinematic points, rather than realistically simulating
any specific experiment. Since the unpolarized cross section has no azimuthal
dependence, i.e., is azimuthally symmetric, the size of the azimuthal
acceptance window has no impact on our results. We therefore assumed a full
$2\pi$ azimuthal acceptance.

For every kinematic point we studied, we conducted two pseudo-experiments: one
in which the spectrometer was positioned to detect the electron, and another
in which the spectrometer was positioned to detect the proton, i.e., a Super-
Rosenbluth configuration. For each kinematic point we studied, we performed
separate simulation runs. We studied 15 kinematic points in total. We
considered the three $Q^{2}$ values of the data in Ref. [7]: 2.64, 3.2, and
4.1 GeV${}^{2}/c^{2}$, and at each we tested five values of $\varepsilon$ from
0.1 to 0.9, opting for a wider $\varepsilon$-coverage, rather than exactly
matching the kinematic points reported in Ref. [7].

To avoid any needless complexity, for all Monte Carlo codes, the proton is
assumed to have dipole form factors, i.e.,

$\displaystyle G_{E}=$
$\displaystyle\left[1+\frac{Q^{2}}{\Lambda^{2}}\right]^{-2},$ (7)
$\displaystyle G_{M}=$ $\displaystyle\mu_{p}G_{E},$ (8)

with $\Lambda=0.71$ GeV${}^{2}/c^{2}$ and $\mu_{p}=2.79$. The dipole model has
well known deficiencies (see, for example, [26]), but we want to avoid the
appearance that some specific form factor model is germane or even responsible
for our findings.

##  3 Peaking Approximation Results

The first radiative correction model we considered was the pure-peaking
approximation described by Ent et al., developed for the NE-18 Experiment
[27]. This approach is widely used, and incorporated in many standard
simulation codes, including simc [28]. This approach was used to determine the
radiative corrections in the experiment of Ref. [7]. In this approach,
radiation is assumed to be emitted, independently, from the incoming electron,
the outgoing electron, and the outgoing proton, purely in the direction of the
particle momentum vectors, i.e., the incoming electron radiates photons in the
direction of the beam, the outgoing electron radiates in the direction
$\theta_{e}$, and the outgoing proton radiates in the direction $\theta_{p}$.
Each radiating particle can emit multiple photons; the only relevant quantity
is the total energy radiated from the particle. The cross section for
scattering reaction is evaluated with an updated beam energy, i.e., the
electron energy remaining after emitting initial state radiation. The effect
of loop diagrams (i.e. vacuum polarization, vertex corrections), which are
non-exponentiated, are also evaluated in the updated kinematics. The radiative
cross section is given in Eq. 66 of Ref. [27].

The peaking approximation is often employed because the angular distribution
of bremsstrahlung radiation is sharply peaked in the directions of the
momentum vectors of the incoming and outgoing particles, especially those of
the electron, due to its low mass. However, because radiation from each
particle leg is treated independently, this approach is inaccurate in its
treatment of interference terms, e.g. the interference of bremsstrahlung
emitted from the incoming electron with bremsstrahlung emitted from the
outgoing electron.

Figure 1: Left: an example radiative tail from the pure-peaking Monte Carlo
generator for electron detection (blue) vs. proton detection (red). Right: the
resulting integral of the radiative tail over the elastic region, i.e., from 0
up to some cut-off in $\Delta p$, as a function of that cut-off. The large
difference in normalization between the two tails comes primarily from the
Jacobian between the electron solid angle and proton solid angle.

Example radiative tails are shown in Fig. 1 for a pseudo experiment conducted
at $Q^{2}=3.2$ GeV${}^{2}/c^{2}$ and $\varepsilon=0.5$, which corresponds to
$E_{\text{beam}}=2.993$ GeV, $\theta_{e}=54.2^{\circ}$, and
$\theta_{p}=25.0^{\circ}$. In both panels, the data in blue correspond to an
electron-detection experiment, while the data in red correspond to an
experiment in which only the proton is detected. The left panel shows the
momentum spectrum that would be measured by a spectrometer in terms of $\Delta
p$, the momentum loss due to radiation. We define this momentum loss as the
magnitude of the difference between its momentum and the momentum a particle
would have at that angle absent any radiation:

$\Delta
p\equiv\begin{cases}\frac{E_{\text{beam}}m_{p}}{m_{p}+E_{\text{beam}}(1-\cos\theta_{e})}-p_{e}&\text{Electron
detection}\\\ &\\\
\frac{2m_{p}E_{\text{beam}}(E_{\text{beam}}+m_{p})\cos\theta_{p}}{E_{\text{beam}}^{2}\sin^{2}\theta_{p}+2E_{\text{beam}}m_{p}+m_{p}^{2}}-p_{p}&\text{Proton
detection}\end{cases}$ (9)

(neglecting the electron mass). The minimum $\Delta p$ is zero, and increasing
$\Delta p$ means detecting a particle with less and less momentum. One should
note that the momentum loss need not come exclusively from final state
radiation from the detected particle. There can also be initial state
radiation as well, reducing the effective beam energy, leading to a lower
momentum scattered particle. The right panel of Fig. 1 shows the integrated
elastic cross section as a function of the cut-off point one uses to define
which events are sufficiently elastic. At that kinematic point, the
differential cross section $d\sigma/d\Omega$ in the proton detection
experiment is approximately four times larger than that in the electron
detection experiment due to the Jacobian between $\Omega_{e}$ and
$\Omega_{p}$. The tails at other kinematic points were qualitatively similar
to those shown in Fig. 1.

Figure 2: The radiative correction corresponding to the tails in Fig. 1 as a
function of the $\Delta p$ momentum cut-off for electron detection (blue) and
proton detection (red). The left plot shows the correction for an absolute
$\Delta p$, while the right plot gives the correction in terms of the relative
cut-off, $\Delta p/p$, which more closely corresponds to the acceptance of a
magnetic spectrometer.

By comparing the cross sections in the right panel of Fig. 1 to the Born cross
section, we can determine the radiative correction factor $f_{\text{RC}}$,
shown in Fig. 2. The left panel shows $f_{\text{RC}}$ as a function of the
$\Delta p$ cut-off. The right panel shows the same correction factor plotted
against a cut off point in $\Delta p/p$, i.e, the relative change in momentum.
This is arguably more representative of experiments using magnetic
spectrometers, whose momentum acceptances typically cover a fraction of the
central momentum. It can be seen in both panels that in the proton detection
experiment, $f_{\text{RC}}$ has a smaller slope with respect to the cut-off
point. The behavior of the radiative corrections are qualitatively similar at
the other kinematic points we studied.

Figure 3: The radiative correction factors predicted by the NE18 prescription
[27] of the peaking approximation at $Q^{2}=4.1$ GeV${}^{2}/c^{2}$ for a
traditional electron detection experiment (left panel) versus a proton
detection experiment (right panel), as a function of several different elastic
cut-off momenta. In the proton-detection experiment, the radiative correction
factors have less $\varepsilon$-dependence.

Fig. 3 shows the radiative correction factor evaluated for a range of
$\varepsilon$ and relative momentum cut-off values for electron detection
(left) and proton detection (right) experiments. The figure shows the results
for the $Q^{2}=4.1$ GeV${}^{2}/c^{2}$ kinematics, but the results are
qualitatively similar for all three $Q^{2}$ values. The radiative correction
factors in the proton detection experiment have reduced dependence on
$\varepsilon$ compared to those in an equivalent electron detection
experiment. Furthermore, at higher values of $\varepsilon$, the proton-
detection $f_{\text{RC}}$ is closer to unity. This result validates the claims
of Ref. [7]. Within the peaking approximation, radiative corrections in proton
detection appear to be less likely to pose a risk of bias on a Rosenbluth
separation.

##  4 Non-Peaking Results

The suitability of the peaking approximation for Super-Rosenbluth experiments
can be validated by calculations using radiative models that do not make the
peaking approximation. We attempted calculations with two such models. The
first, ESEPP [29], was developed with particular attention to calculating
lowest-order bremsstrahlung process exactly, even preserving terms with the
electron mass. This is critical for high-precision experiments at lower
energies or lower values of $Q^{2}$, such as those attempting to determine the
proton radius. ESEPP does not treat multi-photon bremsstrahlung. However its
treatment of single-photon bremsstrahlung is exact, up to knowledge of proton
structure, including all interference terms. Unfortunately, ESEPP, in its
current form, is unable to perform a calculation for a Super Rosenbluth
experiment. Our attempts to run the generator over the full electron phase
space caused the program to crash, a problem we were unable to solve.

The second model we considered was the radiative event generator developed for
the OLYMPUS Experiment [30]. The OLYMPUS Generator (described so far in Refs.
[31, 32], with an article in preparation), treats multi-photon bremsstrahlung
with an exponentiation approach, and avoids the peaking approximation, instead
evaluating the exact bremsstrahlung matrix element, including all interference
terms. This was important for the primary goal of OLYMPUS Experiment;
measuring any deviations from unity in the ratio of positron-proton to
electron-proton elastic scattering cross sections [33]. The interference of
lepton- and proton-bremsshtralung produces an asymmetry between electron and
positron cross sections and was thus a background for the measurement [33].

The OLYMPUS Generator produces events with the kinematics of one-photon
emission, i.e., $ep\rightarrow ep\gamma$, by randomly sampling the electron
angles $\theta_{e}$ and $\phi_{e}$, the momentum loss of the electron, $\Delta
p_{e}$ (as in Eq. 9), and angles of the radiated photon. The proton momentum
vector is determined from energy and momentum conservation. The events are
weighted in order to achieve a model cross section of the form:

$\frac{d^{5}\sigma}{d\Omega_{e}d\Delta
p_{e}d\Omega_{\gamma}}=\frac{d^{5}\sigma}{d\Omega_{e}d\Delta
p_{e}d\Omega_{\gamma}}_{\text{brems.}}\times
e^{\delta(\theta_{e},\phi_{e},\Delta p_{e})},$ (10)

where the $d^{5}\sigma_{\text{brems.}}$ term is the tree-level bremsstrahlung
cross section, and $\delta(\theta_{e},\phi_{e},\Delta p_{e})$ is a standard
radiative correction such as appears in Eqs. 1 and 2. The sampling
distributions for the kinematic variables were designed to approximate the
bremsstrahlung cross section in order to reduce the variance in event weights.
See Ref. [31] for further details. In this approach, the radiated photon is
not _associated_ with any of the charged particles in the reaction. When
calculating, $d^{5}\sigma_{\text{brems.}}$, four diagrams (emission from the
initial state electron, final state electron, initial state proton, and final
state proton) all contribute coherently. It should also be noted that since
the OLYMPUS Experiment used a thin gas target which produced negligible
external radiation, the OLYMPUS generator has no treatment of external
bremsstrahlung.

Figure 4: Radiative tail prediction for a proton-detection experiment using
the OLYMPUS radiative event generator. The full calculation includes
contributions from all bremsstrahlung diagrams, while the lepton-only
calculation neglects diagrams in which bremsstrahlung photons are radiated
from the proton. Radiation from the proton causes the OLYMPUS generator to
have large fluctuations in event weights when integrating over the full
electron phase space.

The OLYMPUS Generator was able to run when full electron phase space was
requested. The radiative tail produced by running the OLYMPUS generator for a
proton detection experiment at $Q^{2}=3.2$ GeV${}^{2}/c^{2}$ and
$\varepsilon=0.5$ are shown in Fig. 4. The results, when running the full
bremsstrahlung calculation (shown by the thin purple histogram), have large
fluctuations in event weights. This indicates a possible divergence or other
unregulated numerical problem when trying to cover the full electron phase
space. By contrast, when limiting the bremsstrahlung calculation purely to
diagrams in which the electron radiates (shown by the thicker orange
histogram, labelled “lepton only”) eliminates this variance. Even though the
electron phase space is being integrated over, it is radiation from the proton
that is causing the problem within the generator.

Figure 5: Left: the electron energy and scattering angle for events in our
proton detection pseudo experiment in which the proton is detected as nearly
elastic, i.e., within 5% of the elastic momentum given its angle. There is a
large enhancement in cross section for low-energy electrons. Right: a
schematic of how this enhancement occurs. The electron can produce a virtual
photon with $Q^{2}=0$, but with nearly the full beam energy. This virtual
photon can Compton scatter from the target proton, giving it an angle and
momentum similar to those of elastic scattering.

To examine this further, we considered events in which the proton would have
been measured in the spectrometer as being “nearly elastic,” i.e., within 5%
of the elastic momentum given the proton’s angle, and looked where the
electrons were going. The results of this study (conducted at $Q^{2}=3.2$
GeV${}^{2}/c^{2}$ and $\varepsilon=0.5$) are shown in the left panel of Fig.
5. The expected elastic peak can be found at approximately 54∘ and 1300
MeV$/c$. However, there is also a large increase in cross section for events
in which the electron has lost nearly all of its momentum. The radiated
bremsstrahlung photon in such events is primarily emitted from the proton but
in the direction of otherwise elastic electrons, i.e., 54∘. A cartoon of the
process that leads to this enhancement is shown on the right side of Fig. 5.
In losing all of its energy, the incoming electron produces a virtual photon
with nearly the same energy and direction as the beam. This virtual photon can
Compton scatter from the proton, the kinematics of which are nearly identical
to elastic scattering. The enhancement in cross section comes from the virtual
photon having $Q^{2}\approx 0$. The virtual photon propagator carries a factor
of $1/Q^{2}$.

It appears that this process does not lead to any divergences ($Q^{2}$ is
never exactly zero), but the enhancement in the cross section is not well
matched by the sampling distribution, leading to large variance in event
weights. Large weight variance by itself is not necessarily cause for alarm;
it merely slows the convergence of any Monte Carlo calculation. However, large
weight variance at a kinematic end point suggests that the generator might be
vulnerable to numerical problems caused by rounding at this end point. Given
how ESEPP evaluates the bremsstrahlung matrix elements, it is likely to suffer
from similar problems. The numerical problems in the OLYMPUS generator could
be repaired by modifying the sampling distribution from which kinematic
variables are randomly chosen, so that the rapid changes in cross section are
matched by similar changes in the sampling distribution. We think this
approach is worth further investigation.

Alarmingly, this process is completely neglected in the peaking approximation.
The peaking approximation allows radiation to be emitted from the outgoing
proton; however, this radiation is restricted to being purely in the direction
of the outgoing proton. There is no mechanism within this approximation to
handle proton radiation in the elastic lepton direction. Radiation from the
proton is also neglected in another previous non-peaked calculation of
radiative corrections in proton detection experiments [34].

##  5 Conclusions

We have investigated radiative corrections to elastic electron-proton
scattering for Super Rosenbluth experiments, in which the recoiling proton is
detected instead of the scattered electron. The Super Rosenbluth technique has
several experimental advantages, and more favorable radiative corrections has
been put forward as one of them [7]. We performed Monte Carlo pseudo-
experiments using two different radiative models to evaluate that claim. When
using the peaking approximation, we find that the radiative corrections are
indeed less $\varepsilon$-dependent than in equivalent electron-detection
experiments, confirming the claims of Ref. [7]. However, in attempting to use
a non-peaking model, we encountered numerical instabilities, which we
attribute to a large enhancement in the cross section near $Q^{2}\approx 0$
when the proton radiates a bremsstrahlung photon in the elastic electron
direction. We see no reason why this process would not contribute to an actual
proton-detection experiment. Nevertheless, the process is neglected in the
peaking approximation, which raises questions about the suitability of the
peaking approximation for Super Rosenbluth experiments.

One argument given for the favorability of radiative corrections in the Super
Rosenbluth technique is that one detects the proton, and the proton radiates
to a much less degree than the lighter electron. This is true. However, the
evaluation of radiative corrections depends as much on the particles being
detected as the particles that are undetected. By not detecting the electron,
the evaluation of the radiative correction requires integrating over all of
the possible phase space where the electron can end up. In that respect,
evaluating the radiative correction in electron detection experiments may be
an easier problem.

Given the other experimental benefits of the Super Rosenbluth technique,
developing a robust radiative corrections procedure that avoids reliance on
the peaking approximation should be pursued.

Acknowledgments

This work was supported by the US Department of Energy Office of Science,
Office of Nuclear Physics, under contract no. DE-SC0016583. Q. S. acknowledges
support from a Luther Rice Fellowship from the George Washington University’s
Columbian College of Arts and Sciences and from a Walker Fellowship from the
Department of Physics.

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