Transcript powerpoint

PROTON DECAY IN
STRING THEORY
Edward Witten
Sulakfest
Boston University
October 22, 2005
Why do we believe
Baryon number is violated?
Three reasons:
I.
Instantons in the standard model
q
△B = 3
q
q
q
l
q
l
q
l
q
q
q
II.
Black holes violate Baryon number
RADIATION
OUT
MATTER
IN
BLACK HOLE
III. Unification seems to require it
…Quarks and leptons in one
representation of a gauge group – in
four or more dimensions – leads to
proton decay
But it takes a more specific framework
to motivate a proton lifetime near
what we might observe…
Original motivation for the pioneering
IMB experiment – the original SU(5)like GUT’s plus the famous “running
of the three couplings”
The proton, sadly, didn’t decay at IMB
and Kamioka …
However, the same motivation
survives in a slightly modified form
with “Supersymmetry” … raises the
proton lifetime above our current
bounds and also, in the context of
GUT’s, gives an excellent fit to sin2θW
SUSY GUT’s give a relatively modelindependent mechanism for proton decay
Tp = 2x1036 years x (MGUT/2 x 1016 GeV)4
Too long! But close enough that any
enhancement would be interesting, even a
factor of 2π in the amplitude
But supersymmetry also gives proton decay
mechanisms that are more model
dependent – dimension five operators that
actually are a bit embarrassing
Potentially even dimension four operators,
this being part of a larger problem than
includes flavor changing processes, CP
violation, maybe even the Higgs being a
little heavy
The drawbacks seem large, but so are
the virtues -- let us not forget the
neutrino masses, whose order of
magnitude has turned out to be close
to what one would guess from GUT’s
– hinting that this picture is on the
right track
“Split supersymmetry” is a recent
attempt to have our cake and eat
it too – keeping only the virtues of
SUSY – and whatever its merits, it
is potentially interesting for proton
decay
As my title indicates, I am focusing
today on proton decay in string theory,
but I am really going to narrow it down
a lot to consider only GUT-like models
derived from strings.
Apart from the general attractions of this
framework, the reason is that otherwise
there is no reason for the proton lifetime
to be near what we might observe.
To me, a model is “GUT-like” if it naturally
leads to the same fermion representations
as the usual GUT’s, and the same
prediction for sin2θW
I’ll be talking a little about proton decay
in GUT-like string models – but I have
no miracles to offer and the result
may be an anticlimax!
Now part of what made string theory
popular 20 years ago …is how neatly
GUT-like phenomenology CAN be
achieved in string theory
originally via the heterotic string on a
Calabi-Yau manifold
More recently … many related models
In the heterotic string context, one just starts
with the numbers 3,4, or 5
And out pop the right gauge groups and
chiral fermions
Input
3…
Gauge group
E6
4…
SO(10)
5…
SU(5)
Fermion Rep
27
16
5̄+10
Nowadays there is a much larger zoo of
possibilities to get GUT-like models from
strings …
Type I superstrings
Strongly coupled heterotic string
Intersecting D-branes
M-theory on a manifold of G2 holonomy
…..
Often dual to each other
Now one important fact about all
these models is that they
satisfy my definition of a GUTlike model but they are not
GUT’s in the strict sense of
four-dimensional unified gauge
theories …
They lead to similar results for quantum
numbers of chiral fermions –
hopefully, the light quarks and leptons
we know – and the same input values
for the strong, weak, and
electromagnetic couplings.
But, because there isn’t fourdimensional unification, other things
are different.
The most dramatic difference is
that the usual quantization of
electric charge does not hold –
typically, though not always, GUTlike models derived from strings
have unconfined fractional electric
charges at a very high scale. …
Maybe even a dark matter
candidate, if inflation didn’t
get in the way!
The usual proton decay amplitude has a key
factor
g2
Mx2
where g is the unified gauge coupling and
MX is the mass of the X boson, which is the
GUT partner of the ordinary SU(3) x SU(2) x
U(1) gauge bosons.
In the string models, there is no fourdimensional unification, and so there is no
X or Y boson.
Instead there is an infinite tower of KaluzaKlein states, or string states, that have the
same quantum numbers and mediate
proton decay.
So instead of a simple X or Y boson
propagator
g2
Mx2
there is a whole infinite sum
gst
2
Σ
i
2
ci
2
Mx
i
In the original models, based on the
perturbative heterotic string, the sum
is over Kaluza-Klein harmonics. The
sum converges, meaning that you
can do the calculation in tendimensional field theory…
No need for recourse to the full string
theory
The answer is qualitatively the same as in
four-dimensional GUT’s …
But this is a problem where that isn’t precise
enough …
A factor of 2π in the amplitude would make
all the difference…
There is no hope of getting a real answer
right now, because even if string theory is
correct and even if nature is based on one
of its GUT-like realizations, there are far
too many possibilities.
A couple years ago, Igor Klebanov and I did
an illustrative calculation
(following work with T. Friedmann in a
related case)
We wanted a GUT-like string model in which
we could calculate all of the factors
relevant to proton decay, in a way that
would not be too complicated, and would
apply to a whole class of models – without
depending too many details.
The framework we picked involved
“intersecting D-branes”
Further advantage: the relation of the Planck
and GUT scales comes out more or less
correctly in this class of models
M
10
5
¯
5
¯
5
Q
Now there is an interesting difference in this
case from the perturbative heterotic string:
One can again try to calculate the proton
decay amplitude via a sum over KaluzaKlein harmonics
But this time the sum over Kaluza-Klein
harmonics diverges!
This means that one cannot do the
calculation in field theory – not even higher
dimensional field theory – one has to use
the full string theory to get the result for
the proton decay amplitude.
It is this divergence that causes the result to
be independent of many details of the
model …
The amplitude is dominated by the behavior
near the points where a 10 of SU(5) is
inserted.
One might say that in this kind of model,
proton decay is a stringy effect, not just a
GUT effect
There is even a consequence that is
observable in principle …
To the extent that the divergent term
dominates, in the decay of a proton to a
e+ or µ+, the final state lepton will be
left-handed
(The divergence only affects the 10 of
SU(5), not the 5̄ .)
In the case of the µ+, this possibly could be
observed in a next generation experiment
after proton decay is discovered…
But this reminds me to tell you that I cannot
say what fraction of the proton decays do
go to µ+ rather than e+ or τ+ as this does
depend on a lot more details
We had some fun with this computation
because of the enhancement coming from
the way the field theory divergence is cut
off…
There also is another nice factor in these
models coming from the GUT scale
“threshold corrections.” These are
calculable (TF and EW) and often give a
nice enhancement.
For a while, it seemed we might get a
prediction for proton decay by dimension
six operators that would be significantly
more favorable than the usual case.
But sadly, another factor (coming from tendimensional kinematics) ultimately
intruded to spoil the fun, as well as the
punchline of this talk.
Our eventual result for these models was
surprisingly close to the usual answer
computed in four dimensions
Tp = 2x1036 years x (MGUT/2 x 1016 GeV)4
though any of three non 4d factors
(amplitude, threshold correction,
kinematics) would individually make a very
significant difference
I am sure that in this picture
there are models that would make
us happier…
I can only conclude by hoping that
nature is based on one of them …
and wishing “Happy Birthday, Larry”