[SOLVED] Supersymmetry and cosmological constant [Archive]

Mikael Djurfeldt

Jun10-04, 10:54 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In his article "String Theory" (2001) Current Science 81(12):1547-1553,\nJohn Schwarz says that if supersymmetry is broken at the 1 TeV scale,\nthat suggests a vacuum energy resulting in a cosmological constant of\naround 10^{-60}.\n\nIf I would like to do that calculation myself (ending up with 10^{-60}),\nwhich papers should I read?\n\nM\n\n-------\n[Moderator\'s note: You should read the following ten lines. The\ngravitational scale - reduced Planck scale - is 10^{18} GeV = 10^{15} TeV\nwhich means that 1 TeV is 10^{-15} times this fundamental scale. In\nparticle physics, the cosmological constant is usually normalized as the\nenergy density, which is dimensionally [energy^4], and assuming that the\nSUSY breaking scale 1 TeV is the only scale that enters the dimensional\nanalysis, the resulting energy density is [1 TeV]^4 =\n[10^{-15} m_{grav}]^4 = 10^{-60} [m_{grav}]^4, i.e. 10^{-60} in reduced\nPlanck units. It\'s pretty small, but not small enough - we need\nsomething like 10^{-120} to agree with observations. LM]\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In his article "String Theory" (2001) Current Science 81(12):1547-1553,
John Schwarz says that if supersymmetry is broken at the 1 TeV scale,
that suggests a vacuum energy resulting in a cosmological constant of
around 10^{-60}.

If I would like to do that calculation myself (ending up with 10^{-60}),
which papers should I read?

M

-------
[Moderator's note: You should read the following ten lines. The
gravitational scale - reduced Planck scale - is 10^{18} GeV = 10^{15} TeV
which means that 1 TeV is 10^{-15} times this fundamental scale. In
particle physics, the cosmological constant is usually normalized as the
energy density, which is dimensionally [energy^4], and assuming that the
SUSY breaking scale 1 TeV is the only scale that enters the dimensional
analysis, the resulting energy density is [1 TeV]^4 =[10^{-15} m_{grav}]^4 = 10^{-60} [m_{grav}]^4, i.e. 10^{-60} in reduced
Planck units. It's pretty small, but not small enough - we need
something like 10^{-120} to agree with observations. LM]

Lubos Motl

Jun10-04, 01:14 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Mikael Djurfeldt wrote:\n\n> However, I don\'t even understand how to get from the energy of SUSY\n> breaking to the energy density. That is, how can I know that I can use\n> the above dimensional analysis?\n\nThese are, of course, very good questions, and the answer below will be\nincomplete, and I hope that someone will clarify it.\n\nIn 4D field theory, the vacuum energy is counted by vacuum Feynman graphs,\nthose loops with no external legs, and they give you some vacuum energy\ndensity. A particle of mass "m" inevitably contributes vacuum energy of\norder m^4. In this case, you don\'t really have a choice - "m" is the only\ndimensionful parameter in this problem (well, except for a UV cutoff, but\nthis thing is supposed to make things even worse because UV cutoffs are\ncloser to the huge Planck scale), and a specific calculation confirms it.\n\nFor example, you know that what you calculate is the "Hamiltonian\ndensity", and therefore no powers of G_{Newton} can appear there - the\nvacuum energy is independent of its gravitational effects. There are no\ngravitational vertices in this diagrams, and therefore no powers of\nG_{Newton}.\n\nThe SUSY breaking scale near 1 TeV leads to the superpartners\' mass\nsplittings of order 1 TeV. Before SUSY was broken, the selectron\'s\ncontribution (loop of selectron with no external legs) exactly canceled\nthe (negative) contribution of the electron. However, after you break\nSUSY, the masses of electron and selectron differ, and the heavier one\n(the selectron) contributes much more - and it is not canceled. The mass\nof the selectron is of order (SUSY breaking scale i.e.) 1 TeV, and\ntherefore you get (1 TeV)^4.\n\nWell, there may be, in general, cancellations - so that even though the\nselectron contributes roughly (1 TeV)^4, the other particles (maybe, the\nvery heavy ones) conspire in such a way that their sum is much smaller\nthan (1 TeV)^4. Also, you might imagine that the field theoretical\ncalculation of a single contribution will be incorrect. The problem is\nthat the one-loop diagrams in string theory (torus, cylinder, ...) seem to\nconfirm all these rough expectations, at least in the classes of models\nthat the people have studied.\n\nI can personally still imagine that either a yet-undiscovered subtlety in\nthe calculations that must be taken into account, or some special features\nof some "good" models may lead to a naturally small cosmological constant\n(or at least small curvature effects resulting from this energy density),\nwithout any need for the "anthropic landscape", but it seems that my POV\nis gradually becoming a minority\'s opinion - largely because a sequence\nof failed attempts to solve the cosmological constant problem in this way.\n\nBest wishes\nLubos\n___________________________________ ___________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mikael Djurfeldt wrote:

> However, I don't even understand how to get from the energy of SUSY
> breaking to the energy density. That is, how can I know that I can use
> the above dimensional analysis?

These are, of course, very good questions, and the answer below will be
incomplete, and I hope that someone will clarify it.

In 4D field theory, the vacuum energy is counted by vacuum Feynman graphs,
those loops with no external legs, and they give you some vacuum energy
density. A particle of mass "m" inevitably contributes vacuum energy of
order m^4. In this case, you don't really have a choice - "m" is the only
dimensionful parameter in this problem (well, except for a UV cutoff, but
this thing is supposed to make things even worse because UV cutoffs are
closer to the huge Planck scale), and a specific calculation confirms it.

For example, you know that what you calculate is the "Hamiltonian
density", and therefore no powers of G_{Newton} can appear there - the
vacuum energy is independent of its gravitational effects. There are no
gravitational vertices in this diagrams, and therefore no powers of
G_{Newton}.

The SUSY breaking scale near 1 TeV leads to the superpartners' mass
splittings of order 1 TeV. Before SUSY was broken, the selectron's
contribution (loop of selectron with no external legs) exactly canceled
the (negative) contribution of the electron. However, after you break
SUSY, the masses of electron and selectron differ, and the heavier one
(the selectron) contributes much more - and it is not canceled. The mass
of the selectron is of order (SUSY breaking scale i.e.) 1 TeV, and
therefore you get (1 TeV)^4.

Well, there may be, in general, cancellations - so that even though the
selectron contributes roughly (1 TeV)^4, the other particles (maybe, the
very heavy ones) conspire in such a way that their sum is much smaller
than (1 TeV)^4. Also, you might imagine that the field theoretical
calculation of a single contribution will be incorrect. The problem is
that the one-loop diagrams in string theory (torus, cylinder, ...) seem to
confirm all these rough expectations, at least in the classes of models
that the people have studied.

I can personally still imagine that either a yet-undiscovered subtlety in
the calculations that must be taken into account, or some special features
of some "good" models may lead to a naturally small cosmological constant
(or at least small curvature effects resulting from this energy density),
without any need for the "anthropic landscape", but it seems that my POV
is gradually becoming a minority's opinion - largely because a sequence
of failed attempts to solve the cosmological constant problem in this way.

Best wishes
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Urs Schreiber

Jun10-04, 01:28 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 10 Jun 2004, Lubos Motl wrote:\n\n> I can personally still imagine that either a yet-undiscovered subtlety in\n> the calculations that must be taken into account,\n\nI tend to share this feeling, though I may not fully know what I am\ntalking about. Always seems to me that in these kind of field theoretic\narguments possible effects of quantized gravity are completely ignored\n(but I may be wrong). Isn\'t the argument morally based on writing Einstein\'s\nequations G = T and taking a "naive" quantum expectation value on the\nright?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 10 Jun 2004, Lubos Motl wrote:

> I can personally still imagine that either a yet-undiscovered subtlety in
> the calculations that must be taken into account,

I tend to share this feeling, though I may not fully know what I am
talking about. Always seems to me that in these kind of field theoretic
arguments possible effects of quantized gravity are completely ignored
(but I may be wrong). Isn't the argument morally based on writing Einstein's
equations G = T and taking a "naive" quantum expectation value on the
right?

Mikael Djurfeldt

Jun10-04, 02:17 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Many thanks! I especially appreciate your and Urs pointing\nout potential uncertainties in the thinking around these issues.\nNow I feel on track again.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Many thanks! I especially appreciate your and Urs pointing
out potential uncertainties in the thinking around these issues.
Now I feel on track again.

Lubos Motl

Jun10-04, 02:31 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 10 Jun 2004, Urs Schreiber wrote:\n\n> I tend to share this feeling, though I may not fully know what I am\n> talking about. Always seems to me that in these kind of field theoretic\n> arguments possible effects of quantized gravity are completely ignored\n> (but I may be wrong). Isn\'t the argument morally based on writing Einstein\'s\n> equations G = T and taking a "naive" quantum expectation value on the\n> right?\n\nExcept that it is probably less naive than we think. The cosmological\nconstant, measured by its curving effects on space, should be the same\nobject that we can measure by the scattering with very soft gravitons -\nand this size of the vacuum energy seems to be large, and we assume that\nwe know how to compute it from one-loop string diagrams.\n\nNevertheless, let me propose the following example as a solution of the\ncosmological constant problem.\n\nA particle of mass M has usually vacuum energy M^4, etc. But there should\nbe also a UV cutoff, let\'s call it the Planck scale - i.e. terms M_{Pl}^4.\nHow do we construct the full vacuum energy?\n\nWell, how do we compute the total energy E of a particle that has both\nrest mass M as well as momentum P? Well, E=\\sqrt{M^2+P^2}. This is a\npretty usual type of a formula in physics.\n\nSo let\'s apply it to the energy density that should incrase both with the\nmass of the particle, as well as the Planck scale. The natural guess for a\nfermion is\n\n-\\sqrt{M_{Pl}^8 + m^8}\n\nHere, m is the mass of the electron, for example. Similarly, the selectron\ngives\n\n+\\sqrt{M_{Pl}^8 + M^8}\n\nwhere M is the mass of the selectron, of order 1 TeV. Of course, if you\nsum these two things, the leading terms M_{Pl}^4 cancel (by high-energy\nsupersymmetry), and the subleading terms, computed from the Taylor\nexpansion, are of order\n\nM^8 / M_{Pl}^4\n\nwhich is the correct order of magnitude of the observed cosmological\nconstant. All important contributions will be of this order, and let\'s\nhope that the number of species is effectively finite.\n\nOnce again, the only thing that you need to solve the cosmological\nconstant problem is to prove that the natural energy density from a\nparticle of mass M is something like (the relativistic dispersion relation\ntype of formula)\n\n+\\sqrt{M_{Pl}^8 + M^8}\n\nEven if this specific proposal is wrong, I can imagine that there is a\nsimilar proposal that is correct, and all the recent anthropic craziness\nwill once be removed by a one-line argument doing things properly.\n_______________________________________ _______________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 10 Jun 2004, Urs Schreiber wrote:

> I tend to share this feeling, though I may not fully know what I am
> talking about. Always seems to me that in these kind of field theoretic
> arguments possible effects of quantized gravity are completely ignored
> (but I may be wrong). Isn't the argument morally based on writing Einstein's
> equations G = T and taking a "naive" quantum expectation value on the
> right?

Except that it is probably less naive than we think. The cosmological
constant, measured by its curving effects on space, should be the same
object that we can measure by the scattering with very soft gravitons -
and this size of the vacuum energy seems to be large, and we assume that
we know how to compute it from one-loop string diagrams.

Nevertheless, let me propose the following example as a solution of the
cosmological constant problem.

A particle of mass M has usually vacuum energy M^4, etc. But there should
be also a UV cutoff, let's call it the Planck scale - i.e. terms M_{Pl}^4.
How do we construct the full vacuum energy?

Well, how do we compute the total energy E of a particle that has both
rest mass M as well as momentum P? Well, E=\sqrt{M^2+P^2}. This is a
pretty usual type of a formula in physics.

So let's apply it to the energy density that should incrase both with the
mass of the particle, as well as the Planck scale. The natural guess for a
fermion is

-\sqrt{M_{Pl}^8 + m^8}

Here, m is the mass of the electron, for example. Similarly, the selectron
gives

+\sqrt{M_{Pl}^8 + M^8}

where M is the mass of the selectron, of order 1 TeV. Of course, if you
sum these two things, the leading terms M_{Pl}^4 cancel (by high-energy
supersymmetry), and the subleading terms, computed from the Taylor
expansion, are of order

M^8 / M_{Pl}^4

which is the correct order of magnitude of the observed cosmological
constant. All important contributions will be of this order, and let's
hope that the number of species is effectively finite.

Once again, the only thing that you need to solve the cosmological
constant problem is to prove that the natural energy density from a
particle of mass M is something like (the relativistic dispersion relation
type of formula)

+\sqrt{M_{Pl}^8 + M^8}

Even if this specific proposal is wrong, I can imagine that there is a
similar proposal that is correct, and all the recent anthropic craziness
will once be removed by a one-line argument doing things properly.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Peter Woit

Jun10-04, 02:52 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Mikael Djurfeldt wrote:\n\n>In his article "String Theory" (2001) Current Science 81(12):1547-1553,\n>John Schwarz says that if supersymmetry is broken at the 1 TeV scale,\n>that suggests a vacuum energy resulting in a cosmological constant of\n>around 10^{-60}.\n\nOne way to think about this is that in a supersymmetric theory the vacuum\nenergy is the order parameter for supersymmetry breaking (since Q^2=H,\nwhere Q is a supersymmetry generator, H the Hamiltonian). So, if the scale\nof supersymmetry breaking has to be at least 1 TeV (since we don\'t see\nsuperpartners), so does the scale of the vacuum energy.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mikael Djurfeldt wrote:

>In his article "String Theory" (2001) Current Science 81(12):1547-1553,
>John Schwarz says that if supersymmetry is broken at the 1 TeV scale,
>that suggests a vacuum energy resulting in a cosmological constant of
>around 10^{-60}.

One way to think about this is that in a supersymmetric theory the vacuum
energy is the order parameter for supersymmetry breaking (since Q^2=H,
where Q is a supersymmetry generator, H the Hamiltonian). So, if the scale
of supersymmetry breaking has to be at least 1 TeV (since we don't see
superpartners), so does the scale of the vacuum energy.

Ulmo

Jun11-04, 04:24 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Mikael Djurfeldt <djurfeldt@nada.kth.se> wrote in message news:<40c8745e\\$0\\$580\\$b45e6eb0-100000@senator-bedfellow.mit.edu>...\n> In his article "String Theory" (2001) Current Science 81(12):1547-1553,\n> John Schwarz says that if supersymmetry is broken at the 1 TeV scale,\n> that suggests a vacuum energy resulting in a cosmological constant of\n> around 10^{-60}.\n>\n> If I would like to do that calculation myself (ending up with 10^{-60}),\n> which papers should I read?\n>\n> M\n>\n\nAll you have to do is just say that above the supersymmetry breaking\nscale, there is a boson for every fermion, and the bosonic and\nfermionic contributions to the vacuum energy cancel each other.\n\nDavid\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mikael Djurfeldt <djurfeldt@nada.kth.se> wrote in message news:<40c8745e$0$580$b45e6eb0-100000@senator-bedfellow.mit.edu>...
> In his article "String Theory" (2001) Current Science 81(12):1547-1553,
> John Schwarz says that if supersymmetry is broken at the 1 TeV scale,
> that suggests a vacuum energy resulting in a cosmological constant of
> around 10^{-60}.
>
> If I would like to do that calculation myself (ending up with 10^{-60}),
> which papers should I read?
>
> M
>

All you have to do is just say that above the supersymmetry breaking
scale, there is a boson for every fermion, and the bosonic and
fermionic contributions to the vacuum energy cancel each other.

David