From Quantum Mechanics, Topology, and Noise, to (Maybe) the LHC Erich Poppitz

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The wonders of supersymmetry: from quantum mechanics, topology, and noise, to (maybe) the LHC Erich Poppitz http://www.slac.stanford.edu/spires/hep/

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ﬁnd title SUPERSYM# or title SUSY# or title MSSM or title NMSSM or title NNMSSM

Nov. 21, 2005 Paper 1 to 25 of 13979

Oct. 18, 2007 Paper 1 to 25 of 15656 what is supersymmetry? what is it good for? why we have studied it for so long?

(will we ever stop?) supersymmetry is a quantum mechanical space-time symmetry, which relates bosons and fermions curiously, it was not discovered in quantum mechanics first, but in string theory and quantum field theory Ramond 1971 Golfand, Likhtman 1972 Wess, Zumino 1974 nevertheless, some of its most important applications to date involve quantum mechanics Witten 1981 moreover, supersymmetric quantum mechanics has all features that make it clear why supersymmetry is interesting to particle physicists, as well as to other fields the harmonic oscillator in quantum mechanics, we know that

implies bosons: commutation relations between creation and annihilation operators

energy spectrum of bosonic oscillator what are fermions? they obey Pauli principle! explicitly:

fermions: anti-commutation relations between creation and annihilation operators the -1/2 may look cooked-up, but it comes from quantizing the Dirac equation+C-symmetry

we’ll simply deﬁne fermionic oscillator that way energy spectrum of fermionic oscillator add them up: the supersymmetric oscillator

Proof: introducing: the supersymmetry generators

this is not as silly as it looks! we just found one of the simplest SUPERALGEBRAS!

superalgebra has even (H) and odd (Q) elements, obeying commutation or anticommutation relations we know that Hamiltonian generates time translations - - took square root of time translations! similar to the Hamiltonian H, the supersharges Q can be thought as of generating translations in a “fermionic dimension” of spacetime, making it thus into a superspace Physica 15D (1985) 289-293 North-Holland, Amsterdam

Physica 15D (1985) 289-293 North-Holland, Amsterdam

STUPERSPACE

STUPERSPACE V. GATES}-, Empty KANGAROO:~, M. ROACHCOCK*, and W.C. GALL* California Institute of Technology, Pasadena, CA 91125, USA Pentember, 1999 V. GATES}-, Empty KANGAROO:~, M. ROACHCOCK*, and W.C. GALL* California Institute of Technology, Pasadena, CA 91125, USA Pentember, 1999 We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, instantons, anomalons, and other phenomenons. Also, they We prove, once and for all, that people who don't use supercsapnac'te hairdee really from ogravityut of it. forever. This in cludes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, instantons, anomalons, and other phenomenons. Also, they can't hide from gravity forever.

Whatever it is, I'm against it.

Groucho Marx Whatever it is, I'm against it.

Groucho Marx Supersymmetry is the wave (particle) of the [11], Reaganomic symmetry [12], tensor arithmetic future in high energy physics [1]. Even people who [13], and maiden forms [14]. It is also a very Supersymmetry is the wave (particle) of the used[11], to d Ro esatguaffn olimkeic I NsyTmEmSeTtrIyN [E1s2 ],a tnedn spoarn aarvitihsimoent ic compact notation: e.g., a superfield equation ~ = future in high energy physics [1]. Even people who are n[13],ow daonidn g msuaipdeernlN fToErmSTs I[14].NEs [I2t ] i(sG aUlsTo sau pvee) ry Dq, (as in, e.g., a Dq~ of relief) would in compo- used to do stuff like INTESTINEs and panavision and csoumpeprapcat nnaovtiastiioonn : [3e].g. .G, ar asvuiptyer fpielodp leeq uaartei ornu n~- = nents be [15] are now doing superlNTESTINEs [2] (GUT supe) ningD oqu, t( aosf inid, eea.sg,. , tao oD. qA~ lotfh oreulgiehf ) twheoyu ldh aivne cpormop-o - and superpanavision [3]. Gravity people are run- nents be [15] gressed from classical calculations to medieval ones ning out of ideas, too. Although they have pro- qt + * ( & % / 1 3 7 . ) f d~ ~ [4], modern calculations are impossible without gressed from classical calculations to medieval ones qt + * ( & % / 1 3 7 . ) f d~ ~ [4], modern calculations are impossible without supergravity due to the well-known X-ray diver- = lira e dx A FAn(J§°Kv/-5 gences [5]. (As we all know, X-rays travel along 13onzo --* college supergravity due to the well-known X-ray diver- = lira e dx A FAn(J§°Kv/-5 gences [5]. (As we all know, X-rays travel along parallel1 3olniznoe --s* ,college wh ich never diverge.) Also, super- • lx"°de'fTb • c ® (x, y, z)Sl/ parallel lines, which never diverge.) Also, supersymmetric theories are the most symmetric, which symmetric theories are the most symmetric, which makes •t lhx"e°mde 'ftThbe • pcr e®t t(ixe,s ty, , szo) Slt/h ey've got to be / _ $ - # ! ? ! . . . . O ? ~ eo. makes them the prettiest, so they've got to be right. C/ _h$an- #ce!s? !a r.e. . .t heyO'll? g i~v eo .c onfeynman, too, right. Chances are they'll give confeynman, too, since with 1073741824 + 1073741824 components Furthermore, component formulations leave out Furthermore, component formulations leave out components needed to make nice even numbers siwhilence abstract,with 1 0superspace73741824 is +quite 1073741824 a useful concept, com pwhenone nhandledts [6] withthe rcare'se no room left for quarks to be free [6] there's no room left for quarks to be free (miccromwapvoen esnetgsr engeaetdioend) t[7].o m ake nice even numbers like 1073741824 [16]. Superfields also allow the use (microwave segregation) [7]. Sulipkeer s1p0a7c3e7 4[188] 2i4s t[h1e6 ]g. rSeuapteersfti eilndvse anltsioo nal lsoinwc eth eth ues e of supergraphs [17], which give amazing cancella- Superspace [8] is the greatest invention since the wheeolf [s9u]p. eIrtg frapr hssu r[p1a7s]s, ewsh aiclhl ogtihve ra mapapzriongac chaensc etlola - tions like the l-loop vacuum bubble: tions like the l-loop vacuum bubble: wheel [9]. It far surpasses all other approaches to supersymmetry [10], like Chevrolet cohomologies sPuhpyseicar s15Dym (1985)me 289-293try [ 10], like Chevrolet cohomologies O = 0 . North-Holland, Amsterdam O = 0 . Physica 15D (1985) 289-293 ]'Jolly Good Fellow, supported in part by Ulysses S. Grant North-Holland, Amsterdam We use the following notation: Greek letters for ]'Jolly Good Fellow, supported in part by Ulysses S. Grant # 10036.W e use the following notation: Greek letters for # 10036. ~On leave of his senses. vile spinor indices, Roman letters for isospin in- ~On leave of his senses. vile spinor indices, Roman letters for isospin in- STUPERSPACE * Work supported by the U.S. Department of Momentum. dices, Cyrillic letters for vector indices, Hebrew * Work supported by the U.S. Department of Momentum. * Pedrimcaense, ntC aydrdirleliscs : l"eYttoeurrs Rfooyra l vHeicgthonre sisn"d. ices, Hebrew * Permanent address: "Your Royal Highness". letters for 10-dimensional vector indices, Roman V. GATES}-, Empty KANGAROO:~, M. ROACHCOCK*, and W.C. GALlLe*t ters for 10-dimensional vector indices, Roman CaliforniSa TInUstitPutEe oRf Technology,SPACE P asadena, CA 91125, USA 0167-2789/85/$03.30 © Elsevier Science Publishers B.V. 0167-2789/85/$03.30 © ElseviePre nStemcbieer, n19c9e9 Publishers B.V. (North-Holland Physics Publishing Division) (North-HVo.l lGaAnTdE SP}h-y, sEimcsp tPyu KbAliNshGiAnRg ODOi:v~i,s Mio.n R) OACHCOCK*, and W.C. GALL* We pCroavleif, oornnciea a nInd sftoitru all,te othfa tTechnology, people who dPona'sta usede nsau,p eCrsAp ac91125,e are really US Aou t of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations).Pentemb eBesides,r, 1999 all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, instantons, anomalons, and other phenomenons. Also, they can't hide from gravity forever.

We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always Whaeteivtehre irt is,w aIv'me atghaienisrt it.ha nds or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, instantons, anomalons, and other phenomenons. Also, they Groucho Marx can't hide from gravity forever.

Supersymmetry is the wave (particle) of the [11], Reaganomic symmetry [12], tensor arithmetic future in high energy physics [1]. Even people who [13], and maiden forms [14]. It is also a very used to do stuff like INTESTINEs and panavision compact notation: e.g., a superfield equation ~ = Whatever it is, I'm against it. are now doing superlNTESTINEs [2] (GUT supe) Dq, (as in, e.g., a Dq~ of relief) would in compo- and superpanavision [3]. Gravity people are run- Groucho Marx nents be [15] ning out of ideas, too. Although they have pro- gressed from classical calculations to medieval ones qt + * ( & % / 1 3 7 . ) f d~ ~ [4], modSeurnp ecraslcyumlamtioentsr ya rei si mtphoes siwblea vwe ith(opuatr ticle) of the [11], Reaganomic symmetry [12], tensor arithmetic superfguratvuirtye dinu eh tiog hth ee nweerlgl-yk npohwyns iXc-sr a[y1 ]d. iEvevr-e n pe=o ple wlirha o e dx[13], A FA n(aJ§n°dK v/m-5 aiden forms [14]. It is also a very 13onzo --* college genceus s[e5d]. t(Ao sd woe satlul fkfn loiwke, XIN-raTyEs StrTavIeNl Ealso nagn d panavision compact notation: e.g., a superfield equation ~ = parallaerl el ineosw, w dhoicihn gn esvuerp edirvleNrgTe.E) SATlsIoN, Esusp e[r2- ] (GUT supe) symmetric theories are the most symmetric, which • lx"°de'fTb • c ®D (qx, y(,a zs) Siln/ , e.g., a Dq~ of relief) would in compo- makeas ntdh emsu ptheer pparentativesits, ioson t[h3e]y.' vGe rgaovti ttyo pbe ople a/r_e $r-u#n! ?- ! . . . . nenOt?s b~e e o[15]. right.n Cinhgan coeus t aroef thideye'alls ,g ivteo oc.o nAfelytnhmoaung, ht ooth, ey have pro- since gwreitshs e1d07 f3r7o41m82 c4l a+ s1073741824sical calc ulcaotmioponnse tnots m ediFeuvrathle ormneosre , component formulations leave out components needeqdt +to *m(a&ke% n/i1c3e 7e.v)e nf nudm~b ~e rs [6] th[4],ere 's mnoo dreoronm cleafltc ufolar tiqounarsk s artoe biem pfroese sible without (microwave segregation) [7]. like 1073741824 [16]. Superfields also allow the use Supseurpspearcger a[8v] iitsy t hed ugree ateos t tihnvee nwtieonll -siknnceo wthne X-roafy s udpievregra-p hs [17=], whichl igriav e amaez idnxg Ac aFncAenll(aJ- §°Kv/-5 wheelg [e9n].c Iets fa[r5 s]u. r(pAasss esw ael l aoltlh ekr naopwpr,o aXch-ersa ytos tratvioenl s alilkoen tgh e l-loop v1a3ocnuzuom --* bcollegeubble : superpsyamramlleetrly l[i1n0e]s, ,l ikwe hCihcehv ronleevt ecro hdomivoelroggeie.s) Also, super- O = 0 . symmetric theories are the most symmetric, which • lx"°de'fTb • c ® (x, y, z)Sl/ ]'Jolly Good Fellow, supported in part by Ulysses S. Grant We use the following notation: Greek letters for # 10036.m akes them the prettiest, so they've got to be / _ $ - # ! ? ! . . . . O ? ~ eo. ~On leave of his senses. vile spinor indices, Roman letters for isospin in- right. Chances are they'll give confeyndmicaens,, Ctoyori,l lic letters for vector indices, Hebrew * Work supported by the U.S. Department of Momentum. Furthermore, component formulations leave out * Permsianncenet awddiretshs: "1Y0o7ur3 R7o4y1al8 H2i4gh n+e ss1073741824". colmetpteorsn efonrt s1 0-dimensional vector indices, Roman [6] there's no room left for quarks to be free components needed to make nice even numbers 0167-2789/85/$03.30 © Elsevier Science Publishers B.V. like 1073741824 [16]. Superfields also allow the use (Nort(hm-Hicorlloawnda vPhey siecgs rPeugbalitsihoinng) D[7].ivi sion) Superspace [8] is the greatest invention since the of supergraphs [17], which give amazing cancella- wheel [9]. It far surpasses all other approaches to tions like the l-loop vacuum bubble: supersymmetry [10], like Chevrolet cohomologies O = 0 .

]'Jolly Good Fellow, supported in part by Ulysses S. Grant We use the following notation: Greek letters for # 10036. ~On leave of his senses. vile spinor indices, Roman letters for isospin in- * Work supported by the U.S. Department of Momentum. dices, Cyrillic letters for vector indices, Hebrew * Permanent address: "Your Royal Highness". letters for 10-dimensional vector indices, Roman 0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) the simplest superalgebra

reveals a very general algebraic structure

from D=1 (QM) to D=11 (M-theory)! so far, we learned that supersymmetry is: quantum mechanical, space-time, relates bosons and fermions...

but what are its implications? generic properties? energy spectrum of supersymmetric oscillator

1 all energy levels positive or zero 2 zero point energy is gone! (true only if unbroken supersymmetry) 3 all nonzero energy levels degenerate, related by Q “supermultiplets”

1,2,3: general features, not just this example! PHY1500F, Fall 2006, Graduate Statistical Mechanics, Homework # 4:

Due on Wednesday, October 25, in class.

energy spectrum of supersymmetric quantum ﬁeld theory

modesΣ of ﬁeld in a box

if nature was supersymmetric, degenerate supermultiplets would imply the existence of superparticles - selectron, photino,... alas “sligthly nondegenerate”- the LHC may ﬁnd them, if they exist...

given we are some time away from studying the LHC data, I. Relativistic gas thermodynamics is there supersymmetry in real physical systems?

Calculate the partition function of an extremely relativistic gas of N particles, i.e. particles with E ! |p!|c (for such a gas the average momenta obey p " mc). Find the free energy and the equation of state. Compare with the nonrelativistic case. For the case of, say, a gas of hydrogen atoms, what are the temperatures where an extreme relativistic treatment is justiﬁed? Do you expect all the gas to be ionized or not? What about a gas of electrons (e.g. in a metal)?

II. A gas of polar molecules in an electric ﬁeld.

Consider a gas of N noninteracting diatomic molecules each having an electric dipole moment µ. The gas is placed in a constant external electric field of strength E and occupies a volume V at temperature T . Consider the gas to be classical. 1. Write down an expression for the energy of a single molecule in the external field E. Neglect the vibrational degrees of freedom (i.e. treat the molecules as rods of fixed length with masses attached at the ends). Let the mass of the diatomic molecule be m and its moment of inertia be I. Then, calculate the partition function for a single molecule. Hint: Be careful to use the proper measure when integrating over the rotational degrees of freedom; unless you can guess the correct answer, the most straightforward route is to start with the usual phase space measure for two atoms and then make an appropriate (for the case at hand) change of variables. Also keep in mind that in this problem E denotes electric field, not energy!

2. Find the average electric dipole moment per unit volume, P , of the gas in the external ﬁeld E and thus determine the dielectric susceptibility χ of the gas (recall that P = χE; insert #0’s where needed if the units you use require it).

3. Find the free energy of the gas as function of the external parameters (N, V, T, E). How does the free energy in electric field differ from the free energy without it? What about the equation of state of the gas? What is the change of free energy in a process when only the electric field E changes by a small amount?

4. Finally, water molecules have an electric dipole moment of about 6.1 × 10−30 C m. Use your results (ignoring the fact that water is not diatomic... explain why it’s OK to do so!) to ﬁnd the dielectric constant of steam at 100oC and atmospheric pressure. Compare with the experimental value 1.00587 coming from tables for #r (equal to #/#0 in SI units). g=2 Landau levels are supersymmetric spin and velocity precess at the same rate tiny difference due to (g-2) from radiative corrections, use to measure it a slightly more fancy (but somewhat less “real-world”) example is that of emergent supersymmetry some critical lattice systems in two space dimensions give rise to a supersymmetric “QFT” system, exploiting the equivalence:

D-dim classical equilibrium stat mech = (D-1)+1-dim quantum ﬁeld theory in Euclidean space, i.e. QFT with t -> it

very general correspondence, only illustrate on 2-dim Ising model example

- magnetic moments - lattice particles

Boltzmann partition function: any spin conﬁguration is completely speciﬁed by

the loops around up-spin islands, so the Ising model Z is really a Boltzmann- weighted sum over all possible closed-loop conﬁgurations closed loops = world lines of virtual particle-antiparticle pairs -

- precisely the stuff the vacuum is made of!

thus: 2-dim Ising = (1+1)-dim Euclidean QFT

a closer look reveals that: 2-dim Ising = 2 (1+1)-dim massive free majorana fermions of spins 1/2, 3/2 now, we know that: low-T - magnetization, mostly large loops high-T - random orientations, mostly small loops in between - Tc - loops of all sizes when loops of all sizes contribute to Z -> no scale in the problem at criticality

spin-1/2 fermion becomes massless -> scale and conformal invariance Onsager, 1944 “emerge” at critical temperature use conformal QFT to compute critical exponents of classical spin systems, lattice gases, etc! Polyakov, 1970 Belavin, Polyakov, Zamolodchikov, 1984

another example of the many

(courtesy of Prof. A. Steinberg) the 2d Ising example shows how QFT with fermions can be relevant for describing classical critical phenomena

since we’ve already got the fermions and the emergent conformal invariance at criticality - could it happen that in a lattice model modiﬁed to also have bosons, a conformal invariance + supersymmetry (= superconformal) would emerge at criticality?

Friedan, Qiu, Shenker 1985 -yes: tricritical Ising model at tricritical point is equivalent to a (1+1)-dim “N=1 supersymmetric” conformal quantum ﬁeld theory

tricritical Ising aka “Blume-Emery-Grifﬁths” model, e.g. phases of He3 -He4 mixtures supersymmetry predicts relations between different critical exponents has a 2-dim system with above H been realized experimentally at tricritical point? answer appears to be NO, so far... perhaps possible via cold atoms? in “real” QFT (Minkowskian) e.g., so-called “N=1 supersymmetry in 4d” we similarly take square root of space-time translations, introduce new “fermionic dimensions” of spacetime, etc., all this most easily done by simply “covariantizing” our QM result:

Haag, Lopushansky, Sohnius 1975: “super-Poincare” is the most general nontrivial extension of Poincare algebra consistent with relativistic S-matrix...

- find that all particles come in supermultiplets - the already mentioned selectron, photino, and other “superpartners” - but of course, can not be exactly degenerate - may be they will be seen at the LHC, may be not,... not my main topic today instead, stick with QM and illustrate some of the uses of supersymmetry one of the great simplifications that supersymmetry brings is in finding ground state wave functions - the first thing you want to know about any theory!

this implies that the ground state energy is zero if and only if the ground state is annihilated by both Q and Q+ then one can look for the ground state by solving not

- a second order diff. equation but by solving a ﬁrst order equation instead! this (or rather, similar simpliﬁcations in more complex cases) has led to some great progress in studying the ground states of supersymmetric systems

from strongly coupled QFT - the “supersymmetric cousins” of QCD

to string theory... also led to great progress in mathematics and mathematical physics related to computing topological properties, such as various topological indices... a concept introduced by Witten plays a crucial role in these developments to explain, go back to QM... QM of a particle of spin-1/2 moving on a line, with potential and “spin-orbit” interaction, both determined by one function: W(x) - the SUPERpotential

consider one hermitean linear combination of supercharges (turns out to be sufﬁcient)

then, because H=Q 2 , ground state (recall, it has E=0) wave function obeys: ->

-> which is immediately solved:

normalizable zero energy state exists iff = unbroken supersymmetry

in other words supersymmetry is unbroken iff W(x) is “even at inﬁnity” equivalently, iff W(x) has odd number of extrema, hence, V(x) has an odd number of zeros since V ~ (W’)2 -in other words supersymmetry is unbroken iff W(x) is “even at inﬁnity” -equivalently, iff W(x) has odd number of extrema, -hence, V(x) has an odd number of zeros since V ~ (W’)2

broken unbroken broken (!) unbroken

- even in QM, analysis of effect of tunneling on zero-energy “tree-level” ground state is complex! - exact answer makes such analysis “unnecessary”... imagine the bonus you get in difﬁcult theories... ( important side remark about third case:

supersymmetry unbroken classically, E 0 = 0

but broken by nonperturbative effects (tunneling = instantons):

VERY relevant for particle theory models and the hierarchy problem. scales: hierarchy problem:

gravity 1019 GeV how does a theory with a characteristic length -33 “UV cutoff scale” scale of 10 cm give rise to structure (the

2 electro-weak interactions + all the stuff we are weak interactions 10 GeV made of) 17 orders (and more) of magnitude larger? previous experience in physics makes one think that there should be a dynamical reason typical answers: supersymmetry, or strong dynamics..., or, these days, (what amounts to) ﬁne tuning how does supersymmetry breaking by quantum nonperturbative effects help? suppose supersymmetry breaking scale and the electroweak scale are somehow related

then selectron, photino,... neutralino, chargino,... masses are all “of order” the electroweak scale

supersymmetry breaking also responsible for W-,Z-boson masses - supersymmetry breaking ultimately responsible for physics as we see it analogy: gravity (Planck) scale - frequency near bottom of one well superparticle masses & weak scale - ground state energy

is M superpar tner ~ M electr o w eak ? ... ask the LHC more on this can be learned from my review for non-experts: “Dynamical supersymmetry breaking: why and how?” arXiv: hep-ph/9710274

during the ten years past the subject has evolved - LEP-2 data indicate supersymmetry more ﬁne-tuned than hoped for - newer theoretical ideas about supersymmetry

both sane ...”s p l i t supersymmetry,” “splat supersymmetry”...

and otherwise... Supersplit Supersymmetry

Patrick J. Fox,1 David E. Kaplan,2 Emanuel Katz,3, 4 Erich Poppitz,5 Veronica Sanz,6 Martin Schmaltz,4 Matthew D. Schwartz,7 and Neal Weiner8 1Santa Cruz Institute for Particle Physics, Santa Cruz, CA, 95064 2Dept. of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 3Stanford Linear Accelerator Center, 2575 Sand Hill Rd. Menlo Park, CA 94309 4Dept. of Physics, Boston University, Boston, MA 02215 5Department of Physics, University of Toronto, 60 St George St, Toronto, ON M5S 1A7, Canada 6Universitat de Granada, Campus de Fuentenueva, Granada, Spain 7University of California, Dept. of Physics, Berkeley, CA 94720-7300 8Center for Cosmology and Particle Physics, Dept. of Physics, New York University, New York , NY 10003 (Dated: April 1, 2005) The possible existence of an exponentially large number of vacua in string theory behooves one to consider possibilities beyond our traditional notions of naturalness. Such an approach to electroweak physics was recently used in “Split Supersymmetry”, a model which shares some successes and cures some ills of traditional weak-scale supersymmetry by raising the masses of scalar superpartners significantly above a TeV. Here we suggest an extension - we raise, in addition to the scalars, the gaugino and higgsino masses to much higher scales. In addition to maintaining many of the successes of Split Supersymmetry - electroweak precision, flavor-changing neutral currents and CP violation, - still impordtantimension -to4 an dw5 orkproton don,ecay -tothe mansodel walsoerallows for natural Plaendnck-sca leimporsupersymmtantetry side remark breaking, solves the gluino-decay problem, and resolves the coincidence problem with respect to is M g a u gi n o a n d~H igMgs m a s s es . T he la c k o f u n?ifi c a...tio naskof cou pthelings su gLHCgests a n atural solution to possible ) superparprobtnerlems from dim enelectrsion-6 porowtoeakn decay. While this model has no weak-scale dark matter candidate, a Peccei-Quinn axion or small black holes can be consistently incorporated in this framework.

I. INTRODUCTION in the string landscape are still being worked out [3, 8]. It is exciting to consider modifications to this model. For a many years, our motivations for considering new However, unlike traditional unwieldy model-building, in physics at the weak scale have been strongly influenced which additional fields are added and their phenomeno- by ideas of naturalness. The difficulty of maintaining logical consequences studied, here we remove fields and light scalars in a theory with a high cutoff has led us their associated phenomenological problems. This has to consider compositeness, supersymmetry, or pseudo- already been proposed in a limited form in [9] where the goldstone boson theories at the weak scale. gauginos were decoupled from the weak scale in addi- Recently, it has been realized that the broad string tion to the scalars (alternatively, the higgsinos could be landscape may have an exponentially large number of decoupled [10]). Here a dark matter candidate remains metastable vacua [1, 2, 3]. With so many vacua, it is in the Higgsinos, but gauge coupling unification occurs possible to appeal to Weinberg’s argument for a solution at a low scale (1014GeV), which would typically induce to the cosmological constant problem based upon a scan unacceptable rates of proton decay. over many possible universes [4]. arXiv:hep-th/0503249 v2 1 Apr 2005 Of course, the presence of such a severe fine-tuning may involve scannings and associated tunings of other parameters. The case that this may impact our expecta- tions of the weak scale, and in particular supersymmetric theories was made by Arkani-Hamed and Dimopoulos [5]. II. THE MODEL In “Split Supersymmetry”, all but one scalar of the many of the minimal supersymmetric standard model (MSSM) are given very large masses. Of the two scalar superpart- The next logical extension would be to decouple one ners of the Higgsinos, one linear combination remains Higgsino, in addition to the scalars and gauginos. Un- light and then acquires a vacuum expectation value (vev) fortunately, due to anomalies, this is not possible, so we which breaks electroweak symmetry and gives masses to take the next simplest possibility, which is to decouple the weak gauge bosons. The fermion masses, which can both Higgsinos. The low energy effective theory consists be protected by symmetries, remain small. of SU(3) × SU(2) × U(1) gauge fields and three genera- While flying in the face of naturalness, unification of tions of quarks and leptons, as well as one scalar (whose gauge couplings [6, 7] and weak scale dark matter - both mass is tuned to be light) which is responsible for elec- often used to compel supersymmetry - are maintained. troweak symmetry breaking. The schematic of this model This makes it a phenomenologically appealing model, in comparison to traditional and Split SUSY is presented even if the notions of what constitutes a natural point in figure 1. most important lesson from QM example was that the criterion whether supersymmetry breaks or not is topological: existence of zero energy ground state depends only on the properties of W(x) “at infinity”... independent on details in between, so long as smooth function!

this can be viewed as an example of a topological index:

Witten index

Witten index is invariant under deformations that do not change the asymptotics of the potential at inﬁnity; if nonzero, then supersymmetry deﬁnitely unbroken!

allowed deformations include changes of potential (i.e., of coupling constants!), so can be calculated at weak coupling but true at any coupling

- a theorist’s dream! generic spectrum not equidistant, but always degenerate for all E>0... upon deformations of the theory (change W, volume, coupling) states can leave E=0 or become E=0 states, but only in B-F pairs, however, such changes do not affect the index... Q.E.D. our example was, of course, a very simple one

more examples are obtained when one considers QM of particles moving on, e.g. general Riemann manifolds

then one can show that the Witten index is equal to the Euler characteristic

χ = 2 - 2g for a genus g Riemann surface g= number of handles: sphere χ=2, torus χ=0...

moreover analytic formulae for the Euler characteristic can be obtained most simply - for a physicist, at any rate - by PfstudyingΩ = (2π)nχ(M) (1) the partition function of the supersymmetric QM: M!

n i1...in PfΩ = (2π) χ(M) PfΩ # Ri1i2 . . . R(1in) 1in (2) ∼ − M! Gauss-Bonnet theorem, classic result in mathematics ! ! x˙ = #W !(x) + η(t) supersymmetry η(t)η(t ) δ(t (2)t ) (3) thus:− physics " mathematics# ∼ − e !W (x) η(t)η(t!) δ(t t!) P (x, t ) = − (3) (4) " # ∼ − !W (x) → ∞ dxe− e !W (x) " − d ˆ P (x, t ) = !W (x) ρ(x, t) = HF P ρ((4x,)t) (5) → ∞ dxe− dt − d " 2 ˆ d 1 2 1 ρ(x, t) = HF P ρ(x, t) Hˆ = + (W !(x)) (5) #W !!(x) (6) dt − F P −dx2 8 − 4 d2 1 1 2 ˆ 2 d 1 2 1 HF P = + (W !(x)) #HWˆ !!(x=) + (W !(x)) (6) #W !!(x) (7) −dx2 8 − 4 F P −dx2 8 − 4 2 d 1 2 1 Hˆ = + (W !(x)) #W !!(x) (7) F P −dx2 8 − 4

1 supersymmetry physics mathematics

beware: 929 pages! supersymmetry physics mathematics indices of massless Dirac operators in external gauge ﬁelds ind(D) = n - n L R to study, one engineers a supersymmetric QM whose Witten index = index of the Dirac operator of interest ...fastest way (for physicists, for sure!) to obtain index theorems ...not how all index theorems were ﬁrst obtained, but some new ones physical relevance?

weak interactions: B+L violation in the SM due to nonperturbative effects, role in generating baryon asymmetry of the universe

strong interactions: U(1) problem, axion mass (strong CP problem)

chiral fermions - L/R asymmetry of nature: chiral fermions in 4d are zero modes of Dirac operators in extra dimensions; nontrivial index guarantees chirality the only way we understand UV origin of chirality!

- nonzero index is the ultimate reason why we exist! MORAL:

supersymmetry has led to some very interesting, beautiful, and important developments in mathematical physics, so

supersymmetry is of interest whether or not weak-scale supersymmetric particles exist

particle physicists hope to prove or disprove the existence of weak-scale supersymmetry at the LHC supersymmetry also shows up, somewhat intriguingly, in another place: the theory of motion driven by random noise and the relaxation to equilibrium high-viscosity limit equation of motion of a “Brownian particle” - Langevin equation

regular “drift force” (W’) + random, Gaussian, delta-correlated “noise”

Langevin equation is a stochastic differential equation - has only a probabilistic solution (probability P(x,t|0,0) that at (x,t) if at (0,0)):

substitution yields Fokker-Planck equation with positive Fokker-Planck “Hamiltonian” note the similarity of supersymmetric quantum mechanics from a few slides back: to the Fokker-Planck Hamiltonian

Now, as t goes to inﬁnity, lowest eigenvalue dominates, so equilibrium probability distribution = “ground state wave function”

limiting P(x) exists iff W(x) is “even at inﬁnity,” hence unbroken “supersymmetry” = existence of equilibrium distribution for the experts: - fermions are “ghosts” introduced to enforce Langevin equation as a constraint while noise-averaging - what really happens is that the BRS symmetry of path-integral representation of probability distribution gets enlarged to a quantum-mechanical supersymmetry --- this example of

unbroken “supersymmetry” = existence of equilibrium distribution

illustrates how supersymmetry appears rather generally in problems involving random noise averages this “Parisi-Sourlas supersymmetry (1979)” arises in many problems involving averaging over random noise (also, slightly different, in averaging over disorder in condensed matter systems) as well as in stochastic quantization it is a “quantum-mechanical” supersymmetry (rather than d+1-dim) even in d>1 theories with random noise it is a beautiful new quantum mechanical symmetry what is supersymmetry? it is rich of properties

it helps physicists better understand complex math, and even contribute to its development! what is it good for? it appears strikingly in all sorts of problems, not all quantum-mechanical (random noise averages & topology & classical critical phenomena) why we have studied it it’s fun for so long? also, it is of great interest in particle physics, but details of that are a subject to a separate talk and near-future experimental developments!