Are mini black holes dangerous?
Gerard Westendorp
what if we did have a small black hole somewhere on earth. What would
happen?
Most non-physicists I talk to have heard that black holes suck up all
matter, and once captured by a black hole, there is no way out. So they
seem pretty dangerous.
But this is not really so obvious. A black hole formed from nearby
matter will have the same mass as as the matter it was formed from. So
from a distance, it will also have the same gravitational attraction.
Remember F=M1*M2/r^2, there is no mention of how compact the arrangement
of the source is.
So if a black hole was formed in Geneva, no one would feel it at least
initially outside a very small radius around it.
A LHC black hole would not weigh much, something like 1 Tev/c^2. This is
not much more than a couple of lead atoms. And since gravitation is a
very weak force compared to electromagnetism, even atoms close by will
not be very impressed by its gravitational pull, unless they get pretty
close by.
Even if they come in reach, they would more likely go into orbit rather
than crash straight in.
So maybe mini black holes are completely harmless?
Gerard
Uncle Al
>
> Like many physicists, I am not too worried about black holes at LHC. But
> what if we did have a small black hole somewhere on earth. What would
> happen?
Did you ever see water swirling into a drain? Angular momentum is
conserved. It is slow going down - limited by dissipation of angular
momentum by friction.
An LHC-grade black hole is way less than a proton's diameter. How
well do you expect a sink filled with water will swirl down that hole?
> So maybe mini black holes are completely harmless?
Wear sunscreen.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
Francis Litterio
> Like many physicists, I am not too worried about black holes at LHC. But what if
> we did have a small black hole somewhere on earth. What would happen?
> A LHC black hole would not weigh much, something like 1 Tev/c^2. This is not
> much more than a couple of lead atoms. And since gravitation is a very weak
> force compared to electromagnetism, even atoms close by will not be very
> impressed by its gravitational pull, unless they get pretty close by.
>
> Even if they come in reach, they would more likely go into orbit rather than
> crash straight in.
I see three options:
1. It evaporates due to Hawking radiation.
2. It speeds off at greater than escape velocity, leaving the Earth
behind (consuming a small amount of matter in the process).
3. It falls into the Earth and oscillates back and forth through the
Earth, slowly growing in mass.
--
Fran
Igor
Black holes evaporate and have a lifetime proportional to the cube of
their mass. So the small ones wouldn't stick around too long.
More info can be found here:
Phillip Helbig---remove CLOTHES to reply
<em...@not.available> writes:
> Gerard Westendorp wrote:
>
> > Like many physicists, I am not too worried about black holes at LHC. But what if
> > we did have a small black hole somewhere on earth. What would happen?
> I see three options:
>
> 1. It evaporates due to Hawking radiation.
>
> 2. It speeds off at greater than escape velocity, leaving the Earth
> behind (consuming a small amount of matter in the process).
>
> 3. It falls into the Earth and oscillates back and forth through the
> Earth, slowly growing in mass.
> --
> Fran
Some folks have pointed out that BHs formed in cosmic-ray collisions
might speed off with escape velocity, but those in the LHC will have
practically zero velocity, and thus could stick around. The argument
"if it happens, it happens all the time as a result of cosmic rays,
which are more energetic than anything in the LHC, so since we are still
here, there is nothing to worry about" is then moot.
What is the probability that a BH formed in cosmic-ray collisions would
have a net electrical charge? For such a BH, would the fact that it is
charged keep it from escaping?
johnd...@iprimus.com.au
> well do you expect a sink filled with water will swirl down that hole?
If we assume a non-evaporating mini black hole.
Is it smaller than a neutrino?
Could you put it into perspective with some maths and size ratios to
other particles ?
Would dark matter fall into mini black holes?
When when matter falls in does it fall as a particle or a wave?
Could mini black holes absorb energy from the bonds between atoms as
it moves through them?
Christoffer Karlsson
> Like many physicists, I am not too worried about black holes at LHC. But
> what if we did have a small black hole somewhere on earth. What would
> happen?
If you enjoy science fiction, you might like the book 'Singularity' by
Bill DeSmedt (http://www.billdesmedt.com/thebooks/singularity/) which
is based on that very question. It is available as a podcast on
podiobooks.com together with a companion podcast about the physics of
the book ('Dr. Jack's Soapbox', if I remember correctly).
Christoffer
Craig Markwardt
>
> Some folks have pointed out that BHs formed in cosmic-ray collisions
> might speed off with escape velocity, but those in the LHC will have
> practically zero velocity, and thus could stick around.
"practically zero" should be defined. I believe that even in the
center-of-momentum frame -- where the *total* momentum is zero -- the
*individual* collision products are traveling at relativistic speeds
(gotta conserve energy as well as momentum), so they are essentially
travelling at the "same" speeds as cosmic rays.
CM
--
--------------------------------------------------------------------------
Craig B. Markwardt, Ph.D. EMAIL: cbmarkwar...@gmail.com
--------------------------------------------------------------------------
Francis Litterio
> An LHC-grade black hole is way less than a proton's diameter. How
> well do you expect a sink filled with water will swirl down that hole?
Not well at all. But, over time, the diameter increases as its mass
increases. If the black hole doesn't evaporate or leave the Earth at
greater than escape velocity, isn't monotonic growth inevitable?
--
fran
Boo
> Earth, slowly growing in mass.
As a matter if interest, given that the total amount of matter isn't changed
(and so presumeably the gravitational field at the surface), would the fact
that the earth was being eaten from the inside have any appreciable effects on
the crust ? I know the micro black holes couldn't do this but what about a mini
black hole ?
--
Boo
Zarkov
wrote in message news:gadjub$44o$1...@online.de...
> In article <i5i6hc8m...@not.available>, Francis Litterio
>
> might speed off with escape velocity, but those in the LHC will have
> practically zero velocity, and thus could stick around. The argument
> "if it happens, it happens all the time as a result of cosmic rays,
> which are more energetic than anything in the LHC, so since we are still
> here, there is nothing to worry about" is then moot.
>
Due to direction of the momentum vector some black holes would "speed off"
toward the earth.
I totally disagree with you and "some folks" moot point.
Tom Roberts
> What is the probability that a BH formed in cosmic-ray collisions would
> have a net electrical charge?
_IF_ such a "black hole" were produced, and if it includes most of the
energy available in the collision, then it seems to me that it would
likely also contain most of the charge in the collision, and that is
definitely nonzero for most cosmic ray interactions with the atmosphere.
For objects produced with much less than the total energy, charge would
presumably be distributed randomly, which implies that a significant
fraction of such objects would have nonzero charge, unless there is some
unknown selection rule involved.
Until we have a theory for this supposed production we won't know for sure.
As I have said before, this whole imbroglio was caused
by some people stringing a bunch of words together
without understanding, causing a media event.
> For such a BH, would the fact that it is
> charged keep it from escaping?
This depends on the energy of the black hole, and several other things.
Let me assume:
a) it has unit charge (larger charge would range out sooner)
b) it behaves electromagnetically like particles we know (no reason
it shouldn't, at least in GR)
c) it does not interact strongly (no reason it should)
d) it has an energy <= 14 TeV (the upper limit of the LHC) relative
to the earth (ECI) frame
e) it intersects the earth (others essentially always escape); this
is a large fraction of the cosmic rays that hit the atmosphere
Ordinary minimum-ionizing particles lose on the order of 1 GeV per meter
of rock or iron [#]. So a 14 TeV singly-charged particle passing through
the earth would range out (stop) in a distance on the order of 14
kilometers (or less). So with these assumptions a reasonable fraction of
all such charged "black holes" produced in the upper atmosphere would
surely stop inside the earth. This is not necessarily so for the most
energetic cosmic rays [@], but is so for charged objects in the energy
range reachable by the LHC.
[#] this is a broad minimum; for protons it occurs from
about 1 to about 10 GeV/c momentum; both lower and higher
momentum protons lose more energy per meter, so this
is a worst case.
[@] the energy loss is rising above 100 TeV. This could
well be sufficient to stop even the highest-energy cosmic
rays in the earth, I don't know.
Craig Markwardt wrote:
> I believe that even in the
> center-of-momentum frame -- where the *total* momentum is zero -- the
> *individual* collision products are traveling at relativistic speeds
> (gotta conserve energy as well as momentum), so they are essentially
> travelling at the "same" speeds as cosmic rays.
The LHC has a nonzero crossing angle, and an object produced at rest in
the center of mass frame moves with a speed on the order of 0.001 c in
the lab. That is well above the gravitational escape speed of the earth
(11 km/s = 0.00004 c). It is unlikely that a collision at the LHC would
produce an object moving with less than 0.00004 c. But not impossible.
Tom Roberts
Gerry Quinn
reply_to_group_not_me@spam_me_no_spam.net says...
If a significant amount of the Earth's volume were eaten, the Earth
would start shrinking, and much tectonic activity would be observed.
Also, if a lot of the eaten mass was released as energy, the Earth would
start heating up even before any noticeable shrinkage occurred, leading
to volcanic eruptions.
But it might be that (say) a cannonball-sized black hole settled at the
centre of the Earth would grow so slowly that it would be a long time
before any effects were noticeable at the Earth's surface. The energy
release would have to be sufficient to hold back the pressure from
surrounding rock. I don't know if anyone has done any simulations or
calculations on this subject.
- Gerry Quinn
Dirk Bruere at NeoPax
Hasn't anyone actually done the calculations?
Without numbers all this is pointless.
--
Dirk
http://www.transcendence.me.uk/ - Transcendence UK
http://www.theconsensus.org/ - A UK political party
http://www.onetribe.me.uk/wordpress/?cat=5 - Our podcasts on weird stuff
Boo
> well do you expect a sink filled with water will swirl down that hole?
As a matter of interest, presumeably a microBH of the right size could
swallow a single quark that was part of e.g. a baryon ? Is this
actually forbidden by anything ?
Since quarks are confined at large distances, would it be possible for
the quark to fall within the BH's event horizon but still close enough
to the other quarks to which it is bound for it still to be "free" (so
they wouldn't notice it was mssing) ? If so then the BH would
presumeably end up with non-integral charge.
Otherwise, presumeably, by the time it reaches the event horizon, the
gravitational energy imparted by its fall must be enough to create a new
pair of quarks, one to fill its place in the baryon and the other to
prevent it being a bare quark inside the BH. Is there anything in our
theorys of QM and QCD that guarantees that the strength of the
gravitational field of a uBH is such that there will always be enough
energy to create these broken quark pairs ? If not then what happens
instead ?
Thanks,
--
Boo
Gerard Westendorp
[..]
> Could you put it into perspective with some maths and size ratios to
> other particles ?
I did a quick calculation.
Suppose an electron feels the same attractive force from a black hole as
from a proton. How heavy would the black hole be?
I get 4E12 kg. This is the mass of an asteroid of about 3.7 km with a
density of water!
So the force a TeV black hole exerts on neighboring particles is
extremely tiny.
[..]
> When when matter falls in does it fall as a particle or a wave?
That is also an interesting question. You could form a gravitational
atom from a black hole and an electron. Let's put the electron in its
lowest orbital. In an ordinary atom, the electron cannot crash into the
nucleus because there is no lower orbital, and there is no reaction allowed
e + p -> ?
(Reverse beta decay is not possible due to lepton number conservation
and due to energy conservation)
But with the black hole (BH) you can have
e + BH_m1 -> BM_m2
The black hole BM_m2 would be heavier than BM_m1, and have an extra charge.
Note that lepton number conservation would be violated.
Because the wave function of the electron would be non-zero at the black
hole volume, this interaction would have a finite probability.
As the black hole moves through matter, it would move through electron
wave functions, so should be possible. It would probably become
negatively charged, because interaction with protons would be less likely.
Gerard
Oh No
>In article <gaeqi7$djl$1$8300...@news.demon.co.uk>,
>reply_to_group_not_me@spam_me_no_spam.net says...
>> > 3. It falls into the Earth and oscillates back and forth through the
>> > Earth, slowly growing in mass.
>>
>> As a matter if interest, given that the total amount of matter isn't
>> changed (and so presumeably the gravitational field at the surface),
>> would the fact that the earth was being eaten from the inside have any
>> appreciable effects on the crust ? I know the micro black holes
>> couldn't do this but what about a mini black hole ?
>
>If a significant amount of the Earth's volume were eaten, the Earth
>would start shrinking, and much tectonic activity would be observed.
>
>Also, if a lot of the eaten mass was released as energy, the Earth would
>start heating up even before any noticeable shrinkage occurred, leading
>to volcanic eruptions.
>
>But it might be that (say) a cannonball-sized black hole settled at the
>centre of the Earth would grow so slowly that it would be a long time
>before any effects were noticeable at the Earth's surface.
A cannonball sized black hole would have a very noticeable effect on
gravity at the earth surface! The Schwarzschild radius of the Earth is
9mm, peanut sized.
Seems to me, that even if a micro black hole produced by the LHC
(phenomenally unlikely) did turn out to be stable (even more
phenomenally unlikely), its gravitational effect would be so small that
it would have to get very close indeed to other particles in order to
start eating it away. The process would therefore be incredibly slow,
probably on the scale of the life of the galaxy.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
Tom Roberts
> As a matter of interest, presumeably a microBH of the right size could
> swallow a single quark that was part of e.g. a baryon ? Is this
> actually forbidden by anything ?
It's rather risky to reify the components of Feynman diagrams.
Physicists who ought to know better do it all the time. Professors tell
students that quantum objects are neither particles nor waves, but all
too often we gloss over that ourselves. Remember that Feynman diagrams
appear in the perturbative APPROXIMATION to the theory, and are
themselves not fundamental at all; ditto for the symbols that appear in
them.
What would it mean for a black hole to "swallow" a wave that extends
throughout all of space? Or even a wave localized to a region much
larger than the size of the black hole's horizon? Remember that quantum
objects are never localized precisely....
Or more poignantly: what would it mean for a black
hole to "swallow" a diagram? Or a vertex of a diagram?
> [...]
All your questions inherently involve relationships between quantum
objects (quarks and gluons) and a black hole. We simply do not know what
happens. Understanding this will require a full theory of quantum
gravity (and even that might not include its relationship to QCD). One
might guess what seems likely, but guesses about quantum theory are
notoriously difficult and error prone. So are guesses about black holes,
and the combination is well beyond me.
Tom Roberts
Boris Borcic
> Francis Litterio wrote:
>> Uncle Al wrote:
>>
>>> An LHC-grade black hole is way less than a proton's diameter. How
>>> well do you expect a sink filled with water will swirl down that hole?
>> Not well at all. But, over time, the diameter increases as its mass
>> increases. If the black hole doesn't evaporate or leave the Earth at
>> greater than escape velocity, isn't monotonic growth inevitable?
>> --
>> fran
>>
>
> Hasn't anyone actually done the calculations?
> Without numbers all this is pointless.
>
Already cited in sibling thread :
http://arxiv.org/abs/0806.3381
http://link.aps.org/abstract/PRD/v78/e035009
Boris Borcic
> atom from a black hole and an electron.
What about a star and a neutrino (or many) ?
BB
Jeremy Henty
> But it might be that (say) a cannonball-sized black hole settled at
> the centre of the Earth would grow so slowly that it would be a long
> time before any effects were noticeable at the Earth's surface.
Cannonball-sized? A black hole that consumed the entire Earth would
still be no larger than a marble.
Regards,
Jeremy Henty
Gerry Quinn
No...@charlesfrancis.wanadoo.co.uk says...
> Thus spake Gerry Quinn <ger...@indigo.ie>
> >In article <gaeqi7$djl$1$8300...@news.demon.co.uk>,
> >reply_to_group_not_me@spam_me_no_spam.net says...
> >> > 3. It falls into the Earth and oscillates back and forth through the
> >> > Earth, slowly growing in mass.
> >>
> >> As a matter if interest, given that the total amount of matter isn't
> >> changed (and so presumeably the gravitational field at the surface),
> >> would the fact that the earth was being eaten from the inside have any
> >> appreciable effects on the crust ? I know the micro black holes
> >> couldn't do this but what about a mini black hole ?
> >
> >If a significant amount of the Earth's volume were eaten, the Earth
> >would start shrinking, and much tectonic activity would be observed.
> >
> >Also, if a lot of the eaten mass was released as energy, the Earth would
> >start heating up even before any noticeable shrinkage occurred, leading
> >to volcanic eruptions.
> >
> >But it might be that (say) a cannonball-sized black hole settled at the
> >centre of the Earth would grow so slowly that it would be a long time
> >before any effects were noticeable at the Earth's surface.
>
> A cannonball sized black hole would have a very noticeable effect on
> gravity at the earth surface! The Schwarzschild radius of the Earth is
> 9mm, peanut sized.
Oops, my intuition failed me on that one! Despite knowing the formula,
I guess I was instinctively assuming the radius would be some fractional
power of the mass...
- Gerry Quinn
Rich L.
wrote:
> Francis Litterio wrote:
..
>
> > Hasn't anyone actually done the calculations?
> > Without numbers all this is pointless.
>
>
> http://arxiv.org/abs/0806.3381http://link.aps.org/abstract/PRD/v78/e035009
I just did a somewhat speculative calculation, based on quantum
mechanics and general relativity, that suggests that the LHC could not
produce black holes. I'd be interested in comments on this:
Consider a small box, a cube with sides 'a'. Consider a wave inside
the box, and assume some reasonable boundary condition (i.e. psi=0 at
the boundaries) and a speed for the wave (i.e. 'c'), then there is a
lowest frequency mode that has a wavelength of 2a and frequency c/2a.
Quantum mechanics tells us that the energy of this mode is hc/2a, and
we can consider this energy to be, in some sense, confined to the
volume of this box. This energy will contribute to the mass of the
box. If the walls of the box are themselves massless, then the mass
of this model particle is the energy of this lowest mode divided by
c^2.
Clearly with large 'a' the energy is quite small and the volume of the
box is quite large. As the size of the box is made smaller, however,
the energy of this mode increases. At some point the "mass"
represented by this mode will exceed the critical mass for a black
hole of comparable size. This would then represent the minimum mass
for a black hole. This will happen when the size of the box is
approximately equal to twice the Schwartzchild radius of a black hole
with the equivalent mass of the lowest mode:
lambda_mode = h/(mc)
r_bh= 2Gm/c^2
putting these together gives
m=sqrt(hc/4G)
If you calculate this mass you get m=38*10^-9 kg. In terms of energy
this is 21*10^27 electron volts.
In comparison, the maximum center of mass energy of the LHC is 14TEV
(14*10^12 electron volts). Even when the Pb ion collider is
operational the center of mass energy will still be only 1,150 TEV,
still 10^-12 of the energy of this minimal black hole. Adding the
mass energy of the nucleons to this kinetic energy still does not come
close to the minimum mass/energy required to make a black hole.
I realize this theory is a bit speculative. I'd be interested in
others opinions.
Rich L.
Ian Parker
> On Sep 16, 4:48 am, Dirk Bruere at NeoPax <dirk.bru...@gmail.com>
> wrote:
>
> > Francis Litterio wrote:
> ..
>
> > > Hasn't anyone actually done the calculations?
> > > Without numbers all this is pointless.
>
> > Already cited in sibling thread :
>
reason CERN might be able to produce black holes is enshrined in M
Theory, or the theory of multiple dimensions. According to MT there is
NOT an inverse square law but an inverse 8th power law. This makes
gravity a lot stronger at short ranges. CERN BHs can only be produced
with a particular view of M theory and a gravitational force which
goes to 8th power at just above current energies.
As I have stated in previous postings sch a BH would be completely
unlike anything of a mass (say) 3 times the Sun where strict GTR
applies. We have 11*11 tensors NOT 4*4.
- Ian Parker
Rich L.
> On 20 Sep, 16:16, "Rich L." <ralivings...@sbcglobal.net> wrote:
>
>
>
> > On Sep 16, 4:48 am, Dirk Bruere at NeoPax <dirk.bru...@gmail.com>
> > wrote:
>
> > > Francis Litterio wrote:
> > ..
>
> > > > Hasn't anyone actually done the calculations?
> > > > Without numbers all this is pointless.
>
> > > Already cited in sibling thread :
>
> > >http://arxiv.org/abs/0806.3381http://link.aps.org/abstract/PRD/v78/e0..
>
>
> - Show quoted text -
That is interesting! I know nothing about M Theory. Would Hawking's
radiation still cause such a black hole to evaporate rapidly? It
seems to me that if the force law is even faster than 1/r^2 near the
event horizon, then the Hawking's radiation might be even stronger.
Rich L.
Ian Parker
> > reason CERN might be able to produce black holes is enshrined in M
> > Theory, or the theory of multiple dimensions. According to MT there is
> > NOT an inverse square law but an inverse 8th power law. This makes
> > gravity a lot stronger at short ranges. CERN BHs can only be produced
> > with a particular view of M theory and a gravitational force which
> > goes to 8th power at just above current energies.
>
> > As I have stated in previous postings sch a BH would be completely
> > unlike anything of a mass (say) 3 times the Sun where strict GTR
> > applies. We have 11*11 tensors NOT 4*4.
> That is interesting! I know nothing about M Theory. Would Hawking's
> radiation still cause such a black hole to evaporate rapidly? It
> seems to me that if the force law is even faster than 1/r^2 near the
> event horizon, then the Hawking's radiation might be even stronger.
[Moderator's note: Quoted text trimmed. -P.H.]
Yes emphatically. I would like to add just one thing to Tom Roberts'
excellent contribution. If you read what he says carefully you will
conclude that without Hawking Radiation a BH cannot swallow anything.
Swallowing a quark is a quantum mechanical process. Paradoxically a
swallowed quark leads to MORE energy not less. Take two quarks and we
get 3 very quickly. Hawking radiation arises because virtual particles
get sucked into the BH. The pair particle is then emitted. Swallowing a
quark therefore is very similar to Hawking radiation.
If we have 3 times the mass of the Sun, Hawking radiation is purely
photons. As soon as the size of our BH is in the range of the strong and
weak forces particles of all types are generated - quarks which generate
other quarks and turn into hadrons as well as photons.
One thing which Tom has promted me to say is this. If it does not emit
HR it cannot swallow. What it does swallow (if a quark) is energy LOSS
and is a form of HR. BTW - Even in things like netron stars the density
of matter is tenous in BH terms. Even in a Neutron Star millions of
times more HR is emitted because of virtual particles than from
swallowing.
- Ian Parker
Gerry Quinn
@sbcglobal.net says...
> It's rather risky to reify the components of Feynman diagrams.
> Physicists who ought to know better do it all the time. Professors tell
> students that quantum objects are neither particles nor waves, but all
> too often we gloss over that ourselves. Remember that Feynman diagrams
> appear in the perturbative APPROXIMATION to the theory, and are
> themselves not fundamental at all; ditto for the symbols that appear in
> them.
Well, we don't know what's fundamental, do we? Some people think that
mathematics is fundamental, and if so they should not object to the
notion that Feynman diagrams represent something fundamental... Feynman
diagrams are quite mathematical after all! Perhaps if they were taken
to the limit, they would express a true theory, which might not
necessarily have a closed-form solution.
And some of the entities appearing in Feynman diagrams, such as
electrons, seem at least as tangible, or more so, than any alternatives
I know of.
- Gerry Quinn
Tom Roberts
> In article <QfUzk.403$W06...@flpi148.ffdc.sbc.com>, tjroberts137
> @sbcglobal.net says...
>> Remember that Feynman diagrams
>> appear in the perturbative APPROXIMATION to the theory, and are
>> themselves not fundamental at all; ditto for the symbols that appear in
>> them.
>
> Well, we don't know what's fundamental, do we?
We don't know what is fundamental in the world we inhabit. But we DO
know what is fundamental in our current theories, and Feynman diagrams
and their components most definitely are not.
> Some people think that
> mathematics is fundamental, and if so they should not object to the
> notion that Feynman diagrams represent something fundamental... Feynman
> diagrams are quite mathematical after all!
Physics is not mathematics, and not all mathematics appears at the same
level in physics. Certainly to me the math of an approximation seems to
be subsidiary to the mathematics of the theory itself. As I said before,
it's not at all clear that the "fundamental entities" (your words) of an
approximation are necessarily "fundamental" in the theory itself, or
ESPECIALLY in the world being modeled -- that's why it is rather risky
to reify the components of Feynman diagrams.
> And some of the entities appearing in Feynman diagrams, such as
> electrons, seem at least as tangible, or more so, than any alternatives
> I know of.
The electrons of an experiment are not well modeled by the fermion legs
in QED -- there's an enormous and complex manipulation called
"renormalization" that is involved in relating them. And there is also
the mandatory anti-symmetrization over external fermion legs and
symmetrization over external boson legs -- this prevents the
identification of any real-world object with any specific feature of any
particular diagram....
Tom Roberts