Of Particular Significance Conversations About Science with Theoretical Physicist Matt Strassler e Standard Model tember 9, 2013] the hierarchy problem.] string theorists mean when they refer to a particular array of particles n’t mean “part of nature”. Everything in the universe is part of nature, meanings. The one that scientists are using in this context isn’t “having pical” or “or “generic” — “just what you’d have expected”, or “the usual” ted screaming when she bumped her head”, or “naturally it costs more hadn’t worn those glasses in months, so naturally they were dusty.” doesn’t scream, when the city center is cheap, and when the glasses are thing unnatural happens, there’s a good reason. ics and related subjects, surprises — big surprises, anyway — are pretty k at a physical system, it usually behaves more or less along lines that, ist, you’d naturally expect. If it doesn’t, then (experience shows) there’s and if that reason isn’t obvious, the unnatural behavior of the system ng profound that you don’t yet know. n the notion of naturalness is so important is that there are two big cle physicists and our friends have to confront. The first is that the erred to as “dark `energy’ ” in public settings] is amazingly small, y expect. The second is that the hierarchy between the strength of other forces is amazingly big, compared to what you’d expect. as follows: the Standard Model (combined with Einstein’s theory of e use to predict the behavior of all the known elementary particles and undly, enormously, spectacularly unnatural theory. There’s only one y one aspect in all of science — that is more unnatural than the Standard cal constant. d “Unnatural” s is best illuminated by a bit of story-telling. Naturalness and the Standard Model | Of Particular Significance http://profmattstrassler.com/articles-and-posts/particle-physics-ba... 1 of 62 10/21/15 10:23 AM college (I’ll call them Ann and Steve) got married, and now have two ir kids were younger — say, 4 and 7 years old — they were pretty wild. d at each other, threw things, and generally needed at lot of supervision. tiful flowers and put them in her favorite glass vase. But before she put e doorbell rang. She ran to the front, carrying the vase, and as she made mindedly put the vase down on the small, rickety table that sits by the d home with the kids, and sent them into the play room to occupy tled in from the day and prepared dinner. They heard the usual sounds: of bouncing balls and falling blocks, yells of “no fair” and “ow! stop hat blissfully stopped almost as soon as it started… hen Ann noticed the vase with the flowers wasn’t on the kitchen table. tchen and dining room, she suddenly realized that she’d put it down and us place in the house. y room, hoping she wasn’t too late. And what do you think she found gure 1). Choose the most plausible. e she’d left it, comfortably placed at the center of the table. d the flowers crushed, down on the floor. he table, right at the edge, within a millimeter of disaster. with the kids playing nearby, where is the vase? On the table? Or right at the edge? We’d all believe the first two before we’d third — unless the third was carefully arranged. was, just hanging there. Naturalness and the Standard Model | Of Particular Significance http://profmattstrassler.com/articles-and-posts/particle-physics-ba... 2 of 62 10/21/15 10:23 AM Fig. 2: Imagine a lot of different possible universes, each one described by equations similar to our own universe, but with small adjustments. ieve me. Or at least, if you do believe me, you probably are assuming explanation that I’m about to give you as to how this happened. It can’t were playing wildly in the room and somehow managed to get the vase osition just by accident, can it? For the vase to end up just so — not f the table, but just in between — that’s … that’s not natural! an explanation. e of the table and the vase stuck to it before falling off? Maybe one of the and holding the vase there as a practical joke on his mom? Maybe her ing around the vase and attached it to the table, or to the ceiling, so that the table and vase are both magnetized somehow…? can’t just end up that way on its own… especially not in a room with two d throwing things around. e Standard Model ard Model, combined with Einstein’s theory of gravity. much like our t of equations — speak — much avity). To keep erse even has all nd forces as our the strengths of h which the nown particles determines how have) are a little r maybe even up L such ed by Standard he strengths with interact with 50%. What will ghtly different ile in Figure 2) we will find three general classes, with the following properties. d’s average value will be zero; in other words, the Higgs field is OFF. In rticle will have a mass as much as ten thousand trillion 0) times larger than it does in our world. All the other known elementary up to small caveats I’ll explain elsewhere). In particular, the electron will be no atoms in these worlds. Naturalness and the Standard Model | Of Particular Significance http://profmattstrassler.com/articles-and-posts/particle-physics-ba... 3 of 62 10/21/15 10:23 AM n Figure 2, whose equations differ just slightly from those that s field is FULL ON. The Higgs field’s average value, and the Higgs ass of all known particles, will be as much as ten thousand trillion 0) times larger than they are in our universe. In such a world, there will toms or the large objects we’re used to. For instance, nothing large like a hout collapsing and forming a black hole. ield is JUST BARELY ON. It’s average value is roughly as small as in imes larger or smaller, but comparable. The masses of the known different from what they are in our world, at least won’t be wildly types of particles that have mass in our own world will be massless. In can even be atoms and planets and other types of structure. In others, we’re not used to. But at least a few basic features of such worlds will be ds are in class 3? Among all the Standard Model-like theories that we’re esemble ours at least a little bit? urdly tiny fraction of them (Figure 3). If you chose a universe at random Model-like worlds, the chance that it would look vaguely like our smaller than the chance that you would put a vase down carelessly on a on the edge of disaster, just by accident. Naturalness and the Standard Model | Of Particular Significance http://profmattstrassler.com/articles-and-posts/particle-physics-ba... 4 of 62 10/21/15 10:23 AM
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Of Particular SignificanceConversations About Science with Theoretical Physicist Matt Strassler
Naturalness and the Standard Model
sler [August 27 – September 9, 2013]
ct is closely related to the hierarchy problem.]
article physicists and string theorists mean when they refer to a particular array of particles
as “natural”? They don’t mean “part of nature”. Everything in the universe is part of nature,
natural” has multiple meanings. The one that scientists are using in this context isn’t “having
nature” but rather “typical” or “or “generic” — “just what you’d have expected”, or “the usual”
aturally the baby started screaming when she bumped her head”, or “naturally it costs more
r the city center”, or “I hadn’t worn those glasses in months, so naturally they were dusty.”
tural is when the baby doesn’t scream, when the city center is cheap, and when the glasses are
sually, when something unnatural happens, there’s a good reason.
ntexts in particle physics and related subjects, surprises — big surprises, anyway — are pretty
means that if you look at a physical system, it usually behaves more or less along lines that,
experience as a scientist, you’d naturally expect. If it doesn’t, then (experience shows) there’s
really good reason… and if that reason isn’t obvious, the unnatural behavior of the system
inting you to something profound that you don’t yet know.
rposes here, the reason the notion of naturalness is so important is that there are two big
n nature that we particle physicists and our friends have to confront. The first is that the
[often referred to as “dark `energy’ ” in public settings] is amazingly small,
to what you’d naturally expect. The second is that the hierarchy between the strength of
the strengths of the other forces is amazingly big, compared to what you’d expect.
one can be restated as follows: the Standard Model (combined with Einstein’s theory of
the set of equations we use to predict the behavior of all the known elementary particles and
— is a profoundly, enormously, spectacularly unnatural theory. There’s only one
hysics — perhaps only one aspect in all of science — that is more unnatural than the Standard
that’s the cosmological constant.
n of “Natural” and “Unnatural”
concept of naturalness is best illuminated by a bit of story-telling.
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f friends of mine from college (I’ll call them Ann and Steve) got married, and now have two
ildren. Back when their kids were younger — say, 4 and 7 years old — they were pretty wild.
played rough, got mad at each other, threw things, and generally needed at lot of supervision.
nn bought some beautiful flowers and put them in her favorite glass vase. But before she put
the kitchen table, the doorbell rang. She ran to the front, carrying the vase, and as she made
the door, she absent-mindedly put the vase down on the small, rickety table that sits by the
ur later, Steve returned home with the kids, and sent them into the play room to occupy
s while he and Ann settled in from the day and prepared dinner. They heard the usual sounds:
crashes, the sounds of bouncing balls and falling blocks, yells of “no fair” and “ow! stop
oment of screaming that blissfully stopped almost as soon as it started…
y-five minutes later when Ann noticed the vase with the flowers wasn’t on the kitchen table.
ment searching the kitchen and dining room, she suddenly realized that she’d put it down and
t in the most dangerous place in the house.
t running into the play room, hoping she wasn’t too late. And what do you think she found
get three options (Figure 1). Choose the most plausible.
ase was exactly where she’d left it, comfortably placed at the center of the table.
ase was smashed, and the flowers crushed, down on the floor.
ase was hanging off the table, right at the edge, within a millimeter of disaster.
: After nearly an hour with the kids playing nearby, where is the vase? On the table?
ashed on the floor? Or right at the edge? We’d all believe the first two before we’d
believe the third — unless the third was carefully arranged.
nswer is #3. There it was, just hanging there.
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Fig. 2: Imagine a lot of different possible
universes, each one described by equations
similar to our own universe, but with small
adjustments.
I suspect you don’t believe me. Or at least, if you do believe me, you probably are assuming
be some complicated explanation that I’m about to give you as to how this happened. It can’t
that two young kids were playing wildly in the room and somehow managed to get the vase
tremely precarious position just by accident, can it? For the vase to end up just so — not
he table, not falling off the table, but just in between — that’s … that’s not natural!
t (mustn’t there?) be an explanation.
re was glue on the side of the table and the vase stuck to it before falling off? Maybe one of the
iding behind the table and holding the vase there as a practical joke on his mom? Maybe her
ad somehow tied a string around the vase and attached it to the table, or to the ceiling, so that
uldn’t fall off? Maybe the table and vase are both magnetized somehow…?
so unnatural as that can’t just end up that way on its own… especially not in a room with two
dren playing rough and throwing things around.
tural Nature of the Standard Model
let’s turn to the Standard Model, combined with Einstein’s theory of gravity.
to imagine a universe much like our
ibed by a complete set of equations —
in theoretical-physics speak — much
ndard Model (plus gravity). To keep
ple, let’s say this universe even has all
lementary particles and forces as our
nly difference is that the strengths of
and the strengths with which the
interacts with other known particles
determines how
s the known particles have) are a little
t, say by 1%, or 5%, or maybe even up
fact, let’s imagine ALL such
all universes described by Standard
equations in which the strengths with
he fields and particles interact with
are changed by up to 50%. What will
described by these slightly different
(shown in a nice big pile in Figure 2)
se imaginary worlds, we will find three general classes, with the following properties.
e class, the Higgs field’s average value will be zero; in other words, the Higgs field is OFF. In
worlds, the Higgs particle will have a mass as much as ten thousand trillion
00,000,000,000,000) times larger than it does in our world. All the other known elementary
cles will be massless (up to small caveats I’ll explain elsewhere). In particular, the electron will
assless, and there will be no atoms in these worlds.
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ig. 3: The universes in Figure 2, whose equations differ just slightly from those that
second class, the Higgs field is FULL ON. The Higgs field’s average value, and the Higgs
cle’s mass, and the mass of all known particles, will be as much as ten thousand trillion
00,000,000,000,000) times larger than they are in our universe. In such a world, there will
be nothing like the atoms or the large objects we’re used to. For instance, nothing large like a
or planet can form without collapsing and forming a black hole.
third class, the Higgs field is JUST BARELY ON. It’s average value is roughly as small as in
orld — maybe a few times larger or smaller, but comparable. The masses of the known
cles, while somewhat different from what they are in our world, at least won’t be wildly
rent. And none of the types of particles that have mass in our own world will be massless. In
of those worlds there can even be atoms and planets and other types of structure. In others,
may be exotic things we’re not used to. But at least a few basic features of such worlds will be
fraction of these worlds are in class 3? Among all the Standard Model-like theories that we’re
g, what fraction will resemble ours at least a little bit?
r? A ridiculously, absurdly tiny fraction of them (Figure 3). If you chose a universe at random
g our set of Standard Model-like worlds, the chance that it would look vaguely like our
ould be spectacularly smaller than the chance that you would put a vase down carelessly on a
nd up putting it right on the edge of disaster, just by accident.
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govern our own, may be divided into three classes. Of these, the first two are very
mmon — natural — but the third class is, relative to the other two, extremely sparsely
ulated. Surprisingly, our own universe — if the Standard Model actually describes all
of its particle physics — is found among this tiny set of unnatural universes.
(and it’s a big “if”) the Standard Model (plus gravity) describes everything that exists
ld, then among all possible worlds, we live in an extraordinarily unusual one — one that is as
as a vase nudged to within an atom’s breadth of falling off the table. Classes 1 and 2 of
are natural — generic — typical; most Standard Model-like theories would give universes in
e classes. Class 3, of which our universe is an example, includes the possible worlds that are
non-generic, non-typical, unnatural. That we should live in such an unusual universe —
since we live, quite naturally, on a rather ordinary planet orbiting a rather ordinary star in a
inary galaxy — is unexpected, shocking, bizarre. And it is deserving, just like the weirdly
e, of an explanation. One certainly has to suspect there might be a subtle mechanism,
about the universe that we don’t yet know, that permits our universe to naturally be one that
is the analogy to the playing children who endanger the vase, and make its balanced condition
implausible? It is quantum mechanics itself — the very basic operating principles of our world.
effects do not coexist well with accidental, unstable balance.
o discuss those quantum effects, and how they make the Standard Model unnatural, in a
But first, although I hope you liked my story, I should point out there’s one important
between the vase on the table and the universe. If somebody bumps the table or the vase, it
ly fall off, or perhaps, if we’re lucky, slide toward the center of the table. In other words, it can
e away from its precarious position if it is disturbed. Our universe, by contrast, is not in
of smoothly shifting its properties, and becoming a universe in Class 1 or Class 2.
s possible that someday it could shift suddenly to become a very different universe, through
nown as tunneling or vacuum decay, this event is likely to be unimaginably far off; this is a
another day, but it’s not something to worry about.] The real issue for the universe is in the
among the vast number of possible universes, did we end up in such an apparently unnatural
re something about our universe that we don’t yet know which makes it not as unnatural as it
perhaps the fact that many (most?) natural universes don’t seem hospitable for life has
to do with it? Or maybe we humans haven’t been clever enough yet, and there some other
ntific explanation? Whatever the reason, either it is due to a timeless fact or due to something
ned very long ago; the universe (or at least the large region we can see with our eyes and
) has been unchangingly unnatural [if the Standard Model fully describes it] for billions of
won’t be changing anytime in the near future.
e, let’s move on now, to understand the quantum physics that makes a universe described by
rd Model (and gravity) so incredibly unusual.
Physics and (un)Naturalness
please read about quantum fluctuations of quantum fields, and the energy carried in those
if you haven’t already done so. Along the way you’ll find out a little about another
s problem: the cosmological constant. After you’ve read that article, you can continue with
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he Higgs (and Other Similar Particles)
fluctuations of fields, and their contribution to the energy density of empty space (the
vacuum energy”) play a big part in our story. But our goal here requires we set the
cal constant problem aside, and focus on the Higgs particle and on why the Standard Model is
This is not because the cosmological constant problem isn’t important, and not because we’re
ain the two problems are completely unrelated. But since the cosmological constant has
to do with gravity, while the problem of the Higgs particle and the naturalness of the
odel doesn’t have anything to do with gravity directly, it’s quite possible they’re solved in
ays. And each of the two problems is enormous on its own; if in fact we need to solve them
usly, then the situation just gets worse. So let’s just send the cosmological constant to a far
ake a little nap. We do need to remember that it’s the elephant in the room that we can’t
t the Higgs field. There are three really important questions about the Higgs field and particle
nt to answer. [I’ll phrase all these questions assuming the Standard Model is right, or close to
f it isn’t, don’t worry: the ideas I’ll explore remain essentially the same, even though slightly
iggs field is “ON” — its average value, everywhere and at all times, at least since the very
Why is it on?
GeV. What sets its value?
iggs particle has a mass of about 125 GeV/c!. What sets this mass?
o explain to you how and why these questions are related to the issue of how the energy of
ce (part of which comes from quantum fluctuations of fields) depends on the Higgs field’s
s Field’s Value and the Energy of Empty Space
ld — not just the Higgs field — how is it determined what the average value of the field is in
se? Answer: a field’s average value must have the following property: if you change the value
it, larger or smaller, then the energy in empty space must increase. In short, the field must
e for which the energy of empty space is at a minimum — not necessarily the minimum, but
(If there is more than one minimum, than which one is selected may depend on the
the universe, or on other more subtle considerations I won’t go into now.)
f illustrative examples of how the energy of empty space in our universe, or in some imaginary
ight depend on the Higgs field, or on some other similar field, are shown in Figure 4. In each
cases I’ve drawn, there happen to be two minima where the Higgs field could sit — but that’s
e. In other cases there could be several minima, or just one. The fact that the Higgs field is
world implies there’s a minimum in the universe’s vacuum energy when the Higgs field has a
6 GeV. While it’s not obvious from what’s I’ve said so far, we are confident, from what we
t nature and about our equations, that there is no minimum when the Higgs field is zero, and
our universe’s Higgs field isn’t OFF. So in our universe, the dependence of the vacuum energy
gs field probably looks more like the left-hand figure than the right-hand one, but, as we’ll see,
look much like either of them. If the Standard Model describes physics at energies much
at distances much shorter than the ones we’re studying now at the Large Hadron Collider
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ig. 4: How the energy density of empty space might depend on a Higgs-like field’s
rage value, in two different possible universes. The Higgs field’s average value must
t a minimum — not necessarily the lowest minimum — of the energy of empty space.
ere is more than one minimum, the one the Higgs field `chooses’ may depend on the
tory of the universe. The Higgs field may be ON (left) or OFF (right); but it can only be
F if the energy has a minimum when the field’s value is zero (as in the right-hand plot).
he mass of the Higgs particle is determined by how sharply curved the minimum is
ere the field’s value lies — if the energy rises slowly away from the minimum (left) the
s particle will have a small mass, while if it rises more rapidly away from the minimum
(right) the Higgs particle will have a larger mass.
n the form of the corresponding curve is much more peculiar — as we’ll see later.
s Particle’s Mass and the Energy of Empty Space
t the Higgs particle’s mass? It is determined (Figure 4) by how quickly the energy of empty
ges as you vary the Higgs field’s value away from where it prefers to be. Why?
rticle is a little ripple in the Higgs field — i.e., as a Higgs particle passes by, the Higgs field has
a little bit, becoming in turn a bit larger and smaller. Well, since we know the Higgs field’s
lue sits at a minimum of the energy of empty space, any small change in that value slightly
hat overall energy a little bit. This extra bit of energy is [actually half of] what gives the Higgs
mass-energy (i.e., it’s E=mc! energy.) If the shape of the curve is very flat near the minimum
4), the energy required to make a Higgs particle is rather small, because the extra energy in
g Higgs field (i.e., in the Higgs particle) is small. But if the shape of the curve is very sharp
inimum, then the Higgs particle has a big mass.
he flatness or sharpness in the curve in the plot, at the point where the Higgs field’s value is
he “curvature at the minimum” — that determines the Higgs particle’s mass.
n’t Easy to Have The Higgs Particle’s Mass Be Small
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is measured to be about 125-126 GeV/c!, about 134 times the proton‘s mass.
an’t we just put that mass into our equations, and be done with this question about where it
m is that the Higgs field’s value, and the Higgs particle’s mass, aren’t things you put directly
quations that we use; instead, you extract them, by a complex calculation, from the equations
d here we run into some difficulty…
se two quantities — the average value and the mass of the field and particle — by looking at
ergy of empty space depends on the Higgs field. And that energy, as in any quantum field
the Standard Model, is a sum of many different things:
gy from the fluctuations of the Higgs field itself
gy from the fluctuations of the top quark field
gy from the fluctuations of the W field
gy from the fluctuations of the Z field
gy from the fluctuations of the bottom quark field
gy from the fluctuations of the tau lepton field
for all the fields of nature that interact directly with the Higgs field… I’ve indicated these —
ally! these are not the actual energies — as blue curves in Figure 5. Each plot indicates one
n to the energy of empty space, and how it varies as the Higgs field’s average value changes
to the maximum value that I dare consider, which I’ve called v .
e of you may have read that these calculations of the energy of empty space give infinite
is is true and yet irrelevant; it is a technicality, true only if you assume v is infinitely
hich it patently is not. I have found that many people, non-scientists and scientists alike,
anks to books by non-experts and by the previous generations of experts — even Feynman
hat these infinities are important and relevant to the discussion of naturalness. This is false.
n to this widespread misunderstanding, which involves mistaking mathematical
ies for physically important effects, at the end of this section.]
? It’s as far as one could can push up the Higgs field’s value and still believe our calculations
Standard Model. What I mean by v is that if the Higgs field’s value were larger than this
uld make the top quark’s mass larger than about v /c ) then the Standard Model would no
urately describe everything that happens in particle physics. In other words, v is the
between where the Standard Model is applicable and where it isn’t.
is… and that ignorance is going to play a role in the discussion. From
appears to be something like 500 GeV or larger. However, for all we
could be as much as 10,000,000,000,000,000 times larger than that. We can’t go beyond
because that’s the (maximum possible) scale at which gravity becomes important; if v
arge, top quarks would be so heavy they’d be tiny black holes! and we know that the Standard
’t describe that kind of phenomenon. A quantum mechanics version of gravity has to be
that point… if not before!
is somewhere between 500 GeV and 1,000,000,000,000,000,000
max
max
max
max2
max
max
max
max
max
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Fig. 5: Summing up the energy from the quantum fluctuations of known fields
schematically shown, upper row) up to the maximum energy scale v (down to the
inimum distance scale) where the Standard Model still applies, and adding to this
contributions from unknown effects from still higher energies and shorter distances
chematically shown, middle row), we must somehow find what experiment tells us is
: that the Higgs field’s average value is 246 GeV and the Higgs particle’s mass is 125
is much larger than 500 GeV, this requires a very precise cancellation
between the known and unknown sources of energy, one that is highly atypical of
quantum theories.
In Figure 5, I’ve assumed it’s quite a bit bigger than 500 GeV; we’ll look in Figure 6 at the case
f the contributions in the upper row of Figure 5 is something we can (in principle, and to a
t in practice) calculate, for any Higgs field value between zero and v , and for all quantum
s with energy less than about v . [I’m oversimplifying somewhat here; really this energy
not be quite the same as v , but let’s not get more complicated than necessary.] If v is
ach one of these contributions is really big — and more importantly, the variation as we
Higgs field’s value from zero to v is big too — something like v /(hc) … where h is
uantum constant and c is the universal speed limit, often called “the speed of light”.
not all. To this we have to add other contributions, shown in the second row of Figure 5, which
physical phenomena that we don’t yet know anything or much about, physics that does not
pear in the Standard Model at all. [Technically, we absorb these effects from unknown
to parameters that define the Standard Model’s equations, as inputs to those equations; but
max
max
max
max max
max max4 3
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puts, rather than something we calculate, precisely because they’re from unknown
In addition to effects from quantum fluctuations of known fields with even higher energies,
uantum mechanics of gravity,
y particles we’ve not yet discovered,
s that are only important at distances far shorter than we can currently measure,
r more exotic contributions from, say, strings or D-branes in string theory or some other
ich may depend, directly or indirectly, on the Higgs field’s value. I’ve drawn these unknown
ed; note that these curves are pure guesswork. We don’t know anything about these effects
exist (and the gravity effects definitely exist), and that some or all of them could
… as big as or bigger than the ones we know about in the upper row. In principle, all these
be zero — but that wouldn’t resolve the naturalness problem, as we’ll see, so
there’s no obvious reason to expect these unknown effects in red
y way connected with the known contributions in blue. After all, why should
ravity effects, or some new force that has nothing to do with the weak nuclear force, have
o do with the energy density of quantum fluctuations of the top quark field or of the W field?
like conceptually separate sources of the energy density of empty space.
the puzzle. When we add up all of these contributions to the energy of empty space [Unsure
d curves like these together? Click here for an explanation…] — each of which is big and many
ary a lot as the Higgs field’s value changes from zero to the maximum that we can consider —
incredibly flat curve, the one shown in green. It’s almost perfectly flat near the vertical
not quite at zero Higgs field; it’s slightly away from zero, at a Higgs field
All of those different contributions in blue and red, which curve up and down in
grees, have almost (but not quite) perfectly canceled each other when added together. It’s as
piled a few mountains from Montana into a deep valley in California and ended up with a
How did that happen?
bad is this problem? How surprising is this cancellation? The answer is that it depends on
is only 500 GeV, then there’s no real cancellation needed at all — see Figure 6. But if v is
ancellation is incredibly precise, as in Figure 5. The larger is v , the more remarkable it is
contributions cancelled.
rkable? The cancellation has to be perfect to something like one part in (v /500 GeV) , give
is close to 500 GeV, that’s no big deal; but if v = 5000 GeV, we need a
n to one part in 100. If it’s 500,000 GeV, we need cancellation to one part in a million.
as high as possible — if the Standard Model describes all non-gravitational particle
we need cancellation of all these different effects to one part in about
0,000,000,000,000,000,000,000,000.
case, the incredible delicacy of the cancellation is particularly disturbing. It means that if you
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is not much much larger than 246 GeV, then no special cancellation is really
uired; the sum of the blue and red curves could easily look something like the green
is much larger do we get the surprising effect seen in Figure 5, where,
mpared to the scale at which the blue and red curves wiggle, the green curve is very
flat.
the W particle’s mass, or the strength of the electromagnetic force, by a tiny amount — say,
a million million — the cancellation would completely fail, and you’d find the theory would
1 or Class 2, with a ultra-heavy Higgs particle and either a large or absent Higgs field value
3). This incredible sensitivity means that the properties of our world have to be, very
— like a radio that is set exactly to the frequency of a desired radio station, finely
ch extreme “fine-tuning” of the properties of a physical system has no precedent in science.
another way: what’s unnatural about the Standard Model — specifically, about the Standard
g valid up to the scale v , if v is much larger than 500 GeV or so — is the cancellation
igure 5. It’s not generic or typical… and the larger is v , the more unnatural it is. If you take
generic curves like those in Figure 5, each of which has minima and maxima at Higgs field
t are either at zero or somewhere around v , and you add those curves together, you will find
m of those curves is a curve that also has its minima and maxima at
[Class 2 theories — see Figure 3],
theories],
ot somewhere non-zero that is much much smaller than v [Class 3 theories].
if the curves are substantially curved near their minima and maxima, their sum will also
ave substantial curvature near their minima and maxima [i.e. the Higgs particle’s mass will be
, as in Class 1 and Class 2 theories], and won’t be extremely flat near any of its minimum
r the Higgs particle to be much lighter than v /c , as occurs in Class 3 theories.] This is
for the addition of just two curves, in Figure 7, where we see the two curves have to have a
l relationship if their sum is to end up very flat.
naturalness problem. It’s not just that the green curve in Figure 5 is remarkably flat, with a
at a small Higgs field value. It’s that this curve is an output, a sum of many large and
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Fig. 7: (Top) If you add together two generic curves, the result will be a
curve that is also generic. (Bottom) Only if the two curves have equal
and opposite curvature in the region near the blue arrows will the
result of adding them together be nearly flat. While this could happen
by pure accident, it is perhaps more likely that there was a hidden
relation between the two curves which assured they were nearly equal
and opposite.
ities seem to be somehow swept under the rug, leading to finite predictions. These infinities,
emoval via renormalization, sometimes lead people — even scientists — to claim that particle
don’t know what they are doing, and that this causes them to see a naturalness problem where
s are badly misguided. These technical issues (which are well understood nowadays, in any
ompletely irrelevant in the present context.
ies that arise in certain calculations of the Higgs particle’s mass, and of the Higgs field’s value,
of the naturalness problem, a mathematical symptom that shows up if you insist on
to infinity, which, though often convenient, is an unphysical thing to do. The infinities are
aturalness problem, nor are they at its heart, nor are they its cause.
any ways to see this, one very easy way is to study the wide variety of finite quantum field
discovered in the 1980s (a list of references can be found in an old paper of mine with Rob
w a professor at the University of Illinois].) These theories have minimal amounts of
, as well as being finite. If you take such a theory (see Figure 8), and you ruin the
, while assuring the theory that remains at lower energies still has
fields like the Higgs field, you do not introduce any infinities. Moreover, there is no need
ially cut the theory off at energies below v (as I have done in Figure 5, separating known
nown) since in this example we know the equations to use at energies above as well as
. The energy of empty space, and its dependence on the various fields, can be calculated
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8: Infinities have nothing to do with naturalness.
amples of finite theories abound; if they have
symmetry, there is no naturalness problem, but if
ymmetry is only applicable above an energy scale
, then the naturalness problem immediately
pears, and no spin-zero Higgs-like particles are
ically found with mass-energy (i.e. E=mc energy)
far below E .
without any ambiguity, infinities, or
infinite renormalization. So — is
there a naturalness problem here
too? Do the spin-zero particles
generically get masses as big as
v /c ? Do the spin-zero fields have
values that are either zero or roughly
as big as v ? You Bet! No
infinities, no sweeping anything
under a rug, no artificial-looking
cutoffs — and a naturalness problem
that’s just as bad as ever.
By the way, there’s an interesting
loophole to this argument, using a
lesson learned from string theory
about quantum field theory. But
though it gives examples of theories
that evade the naturalness problem,
neither I nor anyone else was able (so
far) to use it to really solve the
naturalness problem of the
Standard Model in a concrete way.
Perhaps the best attempt was this
lso repeat this type of calculation within string theory (a technical exercise, which does not
assume string theory really describes nature). String theory calculations have no infinities.
, the energy scale where the Standard Model fails to work, is much larger than 500 GeV, the
s problem is just as bad as before.
etting rid of the infinities that arise in certain Higgs-related calculations does NOT by itself
fect the naturalness problem.
to the Naturalness Problem
logical grounds, a couple of qualitatively different types of solutions to this problem come to
123 Google
2
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NATURALNESS AND THE STANDARD MODEL”
First Stab at Explaining “Naturalness” | Of Particular Significance
August 27, 2013 at 9:23 AM | Reply
le Matt. More please…
ugust 27, 2013 at 9:34 AM | Reply
xplanation. Is there a calculation as to exactly how unnatural our universe is or do
know that it is a small probability and the exact number is dependent on the
f various theoretical frameworks one may use? I want to know whether the
is something on the order of one in one to the googol or something even much
August 27, 2013 at 9:48 AM | Reply
do a precise (or rough) calculation if you make definite (or rough) assumptions.
l problem is that you don’t know the probability “measure”. One way to say this is
roll two dice and I *assume* they are fair dice, then I can calculate the probability
ice showing 9 dots. But if I don’t *know* they are fair dice, I can’t calculate it. If I
ey are roughly fair, I can roughly calculate.
n the situation of having at best an extremely rough guess at the probabilities, so
, at best, an extremely rough estimate. But when you’re dealing with numbers that
small, getting them wrong by a huge amount doesn’t change the qualitative
ion: our universe, no matter how you calculate it, is very unusual, on the face of it.
August 28, 2013 at 10:22 PM | Reply
ught the idea of random physical constants was still speculative. The basic theory,
nderstand it, is that we began as a fluctuation within a multiverse from which
y other universes may also have been born; is that right? But what if our universe
s out to be — as most physicists seemed to believe when I was at Berkeley — the
one? In that case, maybe you will have to wait for more experimental data, or
e mathematical revolution, before you can say anything definite about the values of
ugust 27, 2013 at 9:41 AM | Reply
: sorry meant to write is the probability one in a googol, not one in one to the
August 27, 2013 at 10:30 AM | Reply
ve values for the fractions?
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a hard time estimating the number of balls in the other two figures
August 27, 2013 at 12:00 PM | Reply
remark by the mathematician below, and my (upcoming) reply. We cannot
te the fraction without defining a probability measure. Nevertheless, when you are
with numbers this gigantic, you can see you have a serious problem even if you
ow if the problem is one in a million trillion or one in a trillion trillion or one in a
trillion trillion. The point is that it’s certainly not one in a thousand.
st 27, 2013 at 10:47 AM | Reply
e that this is heading in the same direction as the anthropic principle. Do you
wo ideas, ie anthropic P and Naturalness, are the same, linked, or totally different.
August 27, 2013 at 11:58 AM | Reply
important, before you start talking about solutions to a problem, to make sure
erstand the problem. I cannot answer your question without having gone further
rticle; please be patient.
ugust 27, 2013 at 11:16 AM | Reply
matician (not a physicist) I find this argument rather unconvincing. It depends
tally on the existence of some meaningful measure on the parameter space which is
removed from the scale in which we choose to express the model. After all, if you
nough times, any numbers become the same order of magnitude. Are we certain
ole problem isn’t just caused by a misrepresentation of the parameters? It’s not as
xamine the parameter space experimentally.
there is something important to explain, but the same is true for all the other
s of the model: why do they take precisely the values we measure in experiments?
e is a deeper theory which explains them all (presumably including the very large
e are part of a multiverse, and anthropics explains away anything.
ole problem be reduced to “big numbers are more important than small ones”?
August 27, 2013 at 12:11 PM | Reply
ument is NOT entirely convincing. But it is a strong argument nevertheless,
the numbers involved are so huge — typically something like 10^{-32} or so —
’d have to have a hugely convoluted measure on the parameter space to make it
short: if it is a problem of “a misrepresentation of the parameters”, then it’s a huge
whose solution will likely earn someone a professorship at a major university,
bably a Nobel Prize if it can be shown to be true experimentally. Certainly no one
r proposed a re-representation of the parameters which, without adding new
s accessible at scales fairly close to the Higgs particle’s mass, would bring us even
solving the problem. It’s easy to say it’s just a problem of the measure — but give
example where this would solve a naturalness problem in quantum field
and I’ll be extremely impressed.
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akes the argument much more convincing is that there are solids and fluids to
imilar arguments apply. The equations are of the same type (Quantum Field
) and there are higgs-like scalars (which are massless at phase transitions.) If you
andom solid or fluid system away from a phase transition, and ask if it has scalar
s that are vastly lighter than other massive degrees of freedom in the system, the
is “no”, unless they are Nambu-Goldstone bosons (which the Higgs in the
rd Model is not). The same is true for the few examples in particle physics. The
rd Model (if indeed it is the complete theory of nature) is unique in this regard. I’ll
this more later in the article, I think.
se a selection bias (i.e. anthropic principle or something similar) is one possible
tion… but an incomplete one within the Standard Model, because appealing to a
n bias ***also*** requires you to know a probability measure within the
rse… perhaps the dynamics which causes one set of parameters to be realized more
an another… so it doesn’t resolve the problem you mentioned for the naturalness
our last question (if I understand it correctly) — this kind of thing is under active
ion, of course. The answer may be yes. How will we verify this, however? That is
stion that has to be addressed… otherwise, it will remain speculation.
August 27, 2013 at 12:43 PM | Reply
nks! To expand on my last question, my training is in logic, and to me, any number
ept perhaps 0, 1, e & pi) begs for an explanation. I’m uncomfortable with the idea
small numbers might just be facts of life, but large numbers can’t be. Information
ent is not dependent on magnitude.
August 27, 2013 at 9:27 PM | Reply
ut logic and math is different from physics. For example, you may be 1.24534
imes taller than your wife. Does that require explanation? No. Why not? Because
his ratio is determined by a combination of a dynamical equation, on the one
initial conditions which are not given by pure numbers, on the other.
n general, in physics, dynamical equations (i.e. equations that describe how things
hange) assure that most pure numbers that we measure are of the order that we
ould (with experience) guess, up to factors between, say, 0.1 and 10. Sometimes
ou’ll see something as small as 0.01 and 100, just by accident, or even a bit greater.
o I would claim you’re profoundly misled by thinking about physics as similar to
gic or number theory. It’s not… it’s dynamical evolution, and most results of
hysics problems are not nice numbers like 1 or pi or even e^pi.
he notion that extremely large (and extremely small) numbers require explanation
as a history going back nearly a century and has been extremely profitable for
cientists. No explanation is needed for numbers like 3.2435 or 0.543662 — it’s not
matter of “information content”, it’s a matter of whether the dynamical equations
e know are likely or not to spit out such a number.
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August 28, 2013 at 6:41 AM |
OK, but doesn’t that mean you’re assuming that there is some kind of dynamics
in the parameter space? That it is governed by laws of the same character as the
laws of physics, even though we can never test those laws in any way?
August 28, 2013 at 9:07 AM |
Going back to your vase/table analogy, aren’t you also assuming that we are at a
stable point in the dynamics of parameter space? I.e. that the vase isn’t still
falling (with respect to parameter space dynamics)?
This would make sense if you assume that the dynamic parameter is our own
familiar time, and all this happened before we could measure any of the
parameters of the model, but given the local nature of time in GR, this seems
quite a strong assumption.
August 28, 2013 at 10:39 PM |
That point — about dynamical equations — helps me understand better what you
August 29, 2013 at 7:31 PM | Reply
hen the number represents the odds against an event that has demonstrably
ccurred, then of course bigger numbers demand more of an explanation than small
nes. If you disagree, I’d be interested in hearing your reasoning why over a game of
100 minimum-bet craps. I’ll bring the dice. ;)
August 30, 2013 at 8:42 AM |
This is precisely my point: when you play dice, you know that, however biased
the dice may be, there is still some probability measure governing their fall. Even
if you don’t know what the odds are, you still know that the dice behave
randomly, so you can use both your experience of games and statistical theory to
reason about what’s happening, and to judge when the game is fixed.
When you consider the basic parameters of your model of physics, you don’t
have that. If Einstein was right, and time is part of the universe, then the
question “When were the parameters determined?” doesn’t make sense, because
there wasn’t any such “when”. Even the question “How were the parameters
determined?” assumes that there is some mechanism that operates beyond our
physics to determine them. Saying that large numbers require more explanation
than small ones is making assumptions about this mechanism.
I simply want to know what the assumptions behind the naturality argument are.
I don’t think it follows from just assuming that GR + SM is all there is.
Torbjörn Larsson, OM | September 1, 2013 at 5:11 PM |
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If physics is dynamic, the parameters can be determined anytime. As I
understand it, in some models the particles “freeze out” as the temperature of
the universe dropped. And that could happen differently in different places. (So
called landscapes of, say, string theory with many possibilities for physics.)
September 3, 2013 at 7:03 PM |
It’s actually about whether or not you need any such parameter-forcing
mechanism exists at all. GR+SM allow the parameters to be anything, with no
preferences. The values are arbitrary.
If the numbers were small, as in the ratio of the size of the total parameter space
to the subset that created a recognizable (i.e. has atoms in it) universe was
reasonable, then you could continue to say that the values are arbitrary, i.e. no
mechanism behind them needed, the Null Hypothesis, Occam’s Razor, and
getting a universe like ours is still no surprise.
When the numbers are this ridiculously huge, then you do need some
mechanism to explain how the parameters ended up this way. You either have to
figure out some “mechanism that operates beyond our physics” to force these
parameters to end up in this particular state, or you have to rely on the
Anthropic Principle to beat the long odds.
So the assumption is to *not* assume a bias in the parameter space, *not* to
assume some meta-physics mechanism for forcing parameters or merely biasing
them. That’s why it follows from assuming GR+SM is all there is. But then we
run into the problem which suggests that can’t be true.
September 4, 2013 at 12:15 PM |
I realized might be saying “Well you’re assuming a uniform distribution, and
why’s that assumption any better than any else”, so I wanted to make the point
more clear: The less fine-tuned the parameters need to be, the less need there is
to make any assumption at all. As in: Pick whatever distribution or rule for the
parameters you want. Does it result in a universe vaguely like ours? With the big
numbers we have, you need a very finely specified rule to get the parameters we
have and figuring out which one would be a big problem. With small numbers,
just about any rule would work, so you ultimately don’t need to assume any rule
Torbjörn Larsson, OM | September 1, 2013 at 5:06 PM | Reply
appealing to a selection bias ***also*** requires you to know a probability
measure within the multiverse…
n’t understand this point. Is the claim that despite Weinberg was able to predict
alue of today’s vacuum energy (cosmological constant) it isn’t relevant to selection
? And why would a full measure (as I assume the text is describing) be necessary?
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rect comparison would be biology and its selection bias. The full fitness space is
r known, nor its underlying mapping to physics. To look for ecological niches in
oastal zone you don’t need to know the exact waves, the exact dynamics of waves,
e exact probability distribution function of waves. You need to know the extremes
the optimum of the “niche construction”, what the population deems fit and
uld have assumed a gaussian centered on the average between the habitable limits
nnatural parameters (cc, Higgs mass-squared parameter), a gaussian because of
ribution of many other factors in the SM, would be the expected and needed
ability measure for selection bias. Why wouldn’t that work?