aa r X i v : . [ phy s i c s . g e n - ph ] M a y Frozen SUSYwith Susyons as the Dark Matter
John A. Dixon ∗ University of Calgary,Calgary, Alberta, Canada
AbstractFrozen SUSY is the maximally suppressed Supersymmetric SU(5) Grand Unified Theory coupled to Super-gravity. In Frozen SUSY, there is only one extra particle in addition to those that appear in the usualnon-supersymmetric SU(5) Grand Unified Theory coupled to gravity. Frozen SUSY also restricts and im-proves the mass predictions, and the cosmological constant (at tree level). As a result, it uses 4 parametersto generate 13 reasonable predicted masses. The one extra particle is an extremely massive gravitino, whichwe call the Susyon. In Frozen SUSY, the Susyon is stable and it interacts purely through gravity. TheSusyon might be a viable candidate for dark matter. . Suppressed SUSY was introduced in [1,2].
The details of the basic mechanism for Suppressed SUSYwere discussed in [1]. Then the application of that mechanism to the SUSY SU(5) Grand Unified Theorycoupled to Supergravity was discussed in [2]. This paper uses the results of those papers, and discusses themaximally suppressed version of [2]. This we call Frozen SUSY, and it has a number of useful features thatwere not considered in [2]. . No superpartners have been found in the many experiments that have looked for them[3,4,5,6].
The methods of Suppressed SUSY allow us to write down a new theory, which we call ‘FrozenSUSY’, in which all possible superpartners (except the gravitino) are ‘frozen’. Frozen SUSY is governed bya Master Equation and BRST cohomology and so it is a ‘genuine theory’, rather than an ‘effective theory’,a distinction explained in [1]. . Because Frozen SUSY does not have any of the usual superpartners for the observed parti-cles, it does not need spontaneous breaking of SUSY, or an invisible sector.
In the Gauge/Higgssector there are only 4 parameters, but the theory predicts 13 masses from those (plus the susyon mass), andthe 13 masses appear to be physically reasonable. These can be found below in paragraph 42. This economyarises because SUSY is still present in Frozen SUSY, through the Master Equation, and the resulting BRSTcohomology. . The Susyon:
This gravitino in Frozen SUSY is stable with a known mass, predicted by the theory.Essentially, its mass comes from the Higgs mass and the Planck mass through a see-saw mechanism † . More-over, its only interaction with other matter is through gravitons. This gravitino will be called the Susyonin this paper. The reason for the stability, and the restricted interactions, of the Susyon, is that all thesuperpartners are replaced by Zinn Sources using the ideas of Suppressed SUSY. This paper discusses thesefeatures below. . Frozen SUSY predicts the Susyon:
The reason for this prediction is that we cannot remove thegravitino from Frozen SUSY. This is because the suppression trick of Suppressed SUSY is not applicable forgauge particles. To see why this is true, we will start by describing how and why it works for scalars andspin fermions. ∗ [email protected] or [email protected] † This is buried in a complicated calculation in [2], and it needs a careful explanation. . Suppressed SUSY and Scalar Particles:
We can focus on the following simple terms in the matterpart of the Master Equation for the Action A of a typical theory with SUSY and Gauge Transformations: M [ A ] = Z d x (cid:26) · · · + δ A δH i δ A δ Γ i + δ A δψ iα δ A δY iα + δ A δλ aα δ A δM aα + · · · (cid:27) (1)In the above expression H i is a Complex Grassmann even Scalar Field, ψ iα is a Grassmann odd Spin field,Γ i is a Grassmann odd Zinn Source for the variation of the Scalar Field H i and Y iα is a Grassmann evenZinn Source for the variation of the Spin field ψ iα . The index i labels different fields, and the index α isa complex Weyl spinor index α = 1 ,
2. We have also added gauginos λ aα and a Zinn Source for them M aα ,where the index a is an index for the gauge group in its adjoint representation. . Now let us look at some simple terms in the action: A Scalar Fields = Z d x (cid:8) H i (cid:3) H i + Γ i (cid:0) C α ψ iα + iT aij H j ω a + · · · (cid:1) + · · · (cid:9) (2)The terms Γ i δH i = Γ i (cid:0) C α ψ iα + iT aij H j ω a + · · · (cid:1) contain the BRST variation δH i of the field H i . TheBRST operator δ is a Grassmann odd, nilpotent, mapping. We have not included interaction terms andgravitational variation terms here because they would not change our conclusions. In (2), C α is a Grassmanneven Spin supersymmetry Ghost field, ω a is a Grassmann odd Spin 0 Faddeev-Popov Ghost field and T aij are group representation matrices. . First we will examine transformations that change the first part of the above equation (1): M [ A ] = Z d x (cid:26) δ A δH i δ A δ Γ i + · · · (cid:27) ⇒ M New [ A New ] = Z d x (cid:26) δ A New δJ i δ A New δη i + · · · (cid:27) (3)We are introducing new variables here. H i was a Grassmann even Scalar field and it gets replaced by J i ,which is a new Grassmann even Scalar Zinn Source and Γ i was a Grassmann odd Zinn Source and it getsreplaced by η i , which is a new Grassmann odd Antighost Scalar Field. This sort of ‘Exchange Transformation’is discussed at length in the papers [1,2]. . The expression (2), after the above change of variables, becomes: A New Theory From Scalars [ J, η ] = A Scalar Fields [ H → J, Γ → η, · · · ] = Z d x (cid:8) J i (cid:3) J i + η i (cid:0) C α ψ iα + T aij J j ω a + · · · (cid:1) + · · · (cid:9) (4) . In the old path integral that defined the original theory, we integrated over H i , but not overΓ i . This was because H i was a quantized scalar field, and Γ i was just a ‘Zinn Source’, originally introducedinto these expressions by Zinn Justin [7,8,9,10,11,12,13,14]. The purpose of introducing the Zinn Sourceswas to allow us to write the Master Equation in a quadratic form like (1). This form is crucial to SuppressedSUSY, because it has some of the properties of a Poisson Bracket. . In the new path integral that will define the new theory with Suppressed SUSY, we chooseto integrate over the new antighost variable η i , and not over the new Zinn Source J i . Note that J i (cid:3) J i in (4)is not quantized, even though it looks like a kinetic term. This is essential because J is a Zinn Source. Thenew field η i is a quantized ‘antighost field’. But η i does not propagate, because it does not appear in anyquadratic term in the action. However a term like η i C α ψ iα in (4) is a perfectly acceptable trilinear interactionterm in the new theory. It just does not get to do anything much in the quantum field theory, because η i does2ot appear in any quadratic term, so it cannot give rise to Feynman diagrams with propagators involving η i . . So in some sense we have ‘frozen out’ the scalar H i degree of freedom, while maintaining theSUSY algebra, and keeping the same old SUSY Master Equation, but with new names and new propertiesfor the scalar field and its Zinn source. . Thus, as far as this part goes, we had, for this piece of the original SU(5) Grand Unified SupergravityTheory: Z [ j, Γ , · · · ] = Π x,i Z δH i ( x ) · · · e i { A + R d yj i H i + ··· } (5)where A contains (2) and j i is a new source inserted to couple to H i , and this piece becomes, for this partof the SU(5) Grand Unified Supergravity Theory with Suppressed SUSY: Z ′ [ J, ζ, · · · ] = Π x,i Z δη i ( x ) · · · e i { A ′ + R d yζ i η i + ··· } (6)where A ′ contains (4), and ζ i is a new source inserted to couple to η i . . We get a new integral and a new generating functional Z . There is a standard and simpleformal derivation involving a Legendre transform and a connected generator of Green’s functions here [7] ‡ .This means that the invariance is such that we will get: M [ G ] = Z d x (cid:26) δ G δH i δ G δ Γ i + · · · (cid:27) (7) ⇒ M New [ G New ] = Z d x (cid:26) δ G New δJ i δ G New δη i + · · · (cid:27) (8)where the G are the 1PI generating functionals for the new and old theories. This is certainly true for thetree level, where the 1PI generating functionals are exactly the action A and A ′ referred to in (5) and (6),which arise in turn from (2) and (4). The case where we go beyond tree level needs examination from manypoints of view, and will not be attempted here. It is an important next step however. . Fermions Work in a Very Similar Way to the Scalars:
Here are the basic terms of the action,by analogy with (2): A Spinor Fields = Z d x (cid:26) ψ iα ∂ α ˙ β ψ ˙ βi + Y iα (cid:16) ∂ α ˙ β H i C ˙ β + F iH C α + iT aij ψ jα ω a + · · · (cid:17) + · · · (cid:27) (9)The terms Y iα δψ iα = Y iα (cid:16) ∂ α ˙ β H i C ˙ β + F iH C α + iT aij ψ jα ω a + · · · (cid:17) contain the BRST variation δψ iα of thefield ψ iα . The new field here is F iH , which is the well known auxiliary F field that goes with the scalar H i .Again we have not included interaction terms and gravitational variation terms here because they would notchange our conclusions. . Now consider the fermionic version of (3) above: M [ A ] = Z d x (cid:26) δ A δψ α δ A δY α + · · · (cid:27) ⇒ M New [ A New ] = Z d x (cid:26) δ A New δ Σ α δ A New δG α + · · · (cid:27) (10) ‡ We will assume that nothing goes wrong with that argument in the present case. But that is not very obvious even in thenormal situation. It requires a proof and a demonstration for this special and unusual kind of case. I expect that the result ofthat is the usual one.
3e are performing exactly the same exercise with the spinors as we did with the scalars above in paragraph8 above. We are again introducing new variables here. ψ iα was a Grassmann odd Spinor field and it getsreplaced by Σ iα , which is a Grassmann odd New Spinor Zinn Source and Y iα was a Grassmann even ZinnSource and it gets replaced by G iα , which is a Grassmann even New Antighost Spinor Field. . The expression (9), after the above change of variables, becomes: A New Spinor Fields = Z d x (cid:26) Σ iα ∂ α ˙ β Σ ˙ βi + G iα (cid:18) ∂ α ˙ β H i C ˙ β + F iH C α + iT aij Σ jα ω a + · · · (cid:19) + · · · (cid:27) (11) . In the new path integral that defines the new theory, instead of integrating over ψ , and not over Y as in the old path integral, we integrate over G , and not over Σ. Note that Σ iα ∂ α ˙ β Σ ˙ βi is not quantizedbecause Σ iα is a Source. The new field G is a quantized ‘antighost spinor field’. But G does not propagate,because it does not appear in any quadratic term in the action. So in some sense we have now ‘frozen out’ the spinor ψ iα degree of freedom, while maintaining the SUSY algebra, and keeping the same old SUSYMaster Equation, but with different names and properties–as to whether some quantities are unquantizedZinn Sources or quantized Fields. This is all completely analogous to paragraph 10 above which dealt withthe scalar case. . The auxiliary field F iH needs special attention: In the case of (11) where we are only changing thespinor field and its Zinn Source, we do not need to change the auxiliary field F iH . But what if we considerthe situation where we introduce the change in paragraph (8)? Here we need to remember that the auxiliaryfields can be integrated explicitly in the path integral, and the result is that one gets expressions that arepolynomials in the other fields in the action from that. For example we might have F iH F Hi + F iH (cid:0) Y iα C α + g ijk H j H k (cid:1) + h.c. ⇒ − (cid:0) Y iα C α + g ijk H j H k (cid:1) (cid:16) Y i ˙ α C ˙ α + g ilm H l H m (cid:17) (12)in the action. The second form is the result of the integration of the first over F iH in the path integral–itgives the quadratic form shown as part of the new action after integration. . The point is that we can eliminate all the auxiliary fields in this way, and the resultingMaster Equation still looks the same.
So when we change the H i variable this gets taken care ofautomatically. We do not need to worry about changing variables, or about having Zinn sources for, theauxiliary variables like F iH . The nilpotence coming from the Master Equation still works too after thisintegration, with the variation being supplied by the variation of the Zinn Sources from the Master Equation,as one can verify. . What if there is spontaneous breaking of supersymmetry?
In this case it follows that someauxiliary fields must have non-zero vacuum expectation values [15,16,17,18,19,20,21,22,23,24,25]. This is acomplication that could cause trouble. But it is exactly the complication that we avoid with Frozen SUSY,at least at tree level, since we assume that supersymmetry is not spontaneously broken in the papers [1] and[2]. Again, there are issues here that need to be looked at carefully at higher orders. Does the cosmologicalconstant reappear at higher orders? That might even be desirable given the experimental results on darkenergy [5,6], and work such as that in [16,17] might still apply in some sense at higher orders. . The same story happens for gauginos λ aα as for chiral spinors given above. The action forgauginos contains an auxiliary D a and the field strength F aαβ . But these do not change the above argumentsat all. . However the story changes for gauge fields. They do not work the same way, becausegauge fields have an inhomogeneous term in their variation.
For example, consider a vector gaugefield: 4. This would yield a quadratic term of the form Z d x Σ µa ∂ µ ω a ⇒ Z d x η µa ∂ µ ω a (13)where we imagine taking a Zinn Source Σ µa and changing it to an antighost field η µa , like we did abovefor the scalar example.2. The problem is that this would wreak havoc with the theory after the Exchange Transformation,because this antighost η µa would propagate because of the quadratic term (13) above. The argumentin paragraphs 11 and 18 above do not work in this case.3. So Suppressed SUSY can only affect the scalars and spin fields, including both chiral spinors andgauginos, but not the vector bosons, and not the gravitino nor the graviton. These all have terms likethe ones in (13).4. This is why we end up with a massive gravitino in Frozen SUSY: the Susyon. But that is not a badfeature: it looks like it could account for the observed dark matter. . Origin of the Mass of the SU(5) Suppressed SUSY Theory Gravitino:
From page 388 of [15],we see that the relevant terms for this are, from the first row of (18.6) and expression (18.7): e − L = 12 κ ψ µ γ µρσ ∂ ρ ψ σ (14)plus, from (18.15): L m, = 12 m ψ µ P R γ µν ψ ν + h . c . (15)where from (18.16) of [15] we have m = κ e κ K / W (16)We note that the expression P L v in (19.1) of [15] is zero for Frozen SUSY. So indeed there is no Goldstonefermion here and no spontaneous breaking of SUSY here.However there is a huge gravitino mass from (16), even though the related cosmological constant is zero attree level. Note that the local supersymmetry invariance is still present in this action, because the MasterEquation is still true by construction. There is work to be done to examine the BRST cohomology of thistheory at one loop (and higher levels). Even writing the Master Equation down in its full form is a majorchore, and we do not propose to do it here. But it needs to be done. . Details of the Mass of the SU(5) Suppressed SUSY Theory Gravitino:
From the paper [2]we have M Gravitino ≡ M ψ = 1 . M SP (17)where M SP = 1 . × g GeV / c ; 1GeV / c = 1 . × − kg . (18) M Gravitino ≡ M ψ = 1 . × . × g × . × − kg . (19) ≈ kg . ≈ ≈ × Mass of a Royal Caribbean Cruiseship (20)We call these Susyons. We have taken g → for no particularly good reason. This g needs to bedetermined with the kind of renormalization group arguments that also explain sin θ W . These can be foundin say [18,19]. I do not know if some problems arise in this context, and they might, given that the masseshere are determined from the scalar potential, whereas in the usual treatment they are determined by thedesire to have the three gauge couplings converge to one.5 . Detailed Reasons for Stability of the Minimal SU(5) Suppressed SUSY Theory Gravitino:
To see that the Susyon is stable, we need to examine the terms in the action that link the Gravitino ψ toother fermions. We shall see that none of these give rise to a decay mode for the Susyon. For the samereason, this analysis shows that the Susyon, in this Frozen SUSY theory, does not interact with anythingother than gravity. From page 388 of [15], we see that these terms are of the following kinds:1. The term with only one Gravitino ψ , linked to only one spin gaugino λ (plus bosons). Here is thisterm § : e − L , = 18 Re f AB ψ µ γ ab (cid:16) F Aab + b F Aab (cid:17) γ µ λ B (21)Since we have transformed all gauginos λ to Zinn sources, this term becomes a Zinn Source term andso it cannot contribute to ψ decay ¶ .2. The term with only one Gravitino ψ , linked to only one spin chiral multiplet fermion χ (plus bosons).Here is this term: e − L , = 1 √ αβ ψ µ ˆ /∂z β γ µ χ α + h . c . (22)Since we have transformed(a) all chiral scalars z α in the chiral Matter (Quark and Lepton) multiplets to Zinn sources, and(b) all chiral spinors χ α in the chiral Higgs multiplets to Zinn sources;every term in (22) becomes a Zinn Source term and so it cannot contribute to ψ decay. We notethat the metric g αβ in (22) connects Squarks to Quarks, Sleptons to Leptons, and Higgsinos to Higgsscalars.3. The term with only one Gravitino, linked to three spin fermions (plus bosons). L , = − √ f ABα ψ.γχ α λ A P L λ B (23)Since we have transformed all gauginos λ to Zinn sources, this term also becomes a source term and itcannot contribute to ψ decay.4. The ‘mixed term’ with only one Gravitino, linked to one spin gaugino. L mix , = ψ.γ iP L λ A P A (24)Since we have transformed all gauginos λ to Zinn sources, this term also becomes a source term and itcannot contribute to ψ decay.5. The other ‘mixed term’ with only one Gravitino, linked to one spin chiral multiplet fermion: L mix , = ψ.γ √ χ α e κ K / ∇ α W + h . c . (25)This last term contains terms that are chiral matter fermions multiplied by chiral matter scalars,which are Zinn Sources, and it also contains Higgs fermions multiplied by Higgsino scalars. The Higgsfermions have been also changed to Zinn Sources. So all terms of this kind are Zinn Source terms andthey cannot contribute to ψ decay. § We label these various terms with subscripts as in L , ¶ The supercovariant gauge curvature b F Aab is given in (18.14) of [15]. It also contains a gaugino, which is also changed to aZinn Source.
6. The fact that all these terms are really Zinn terms also means that there is no coupling between theSusyon and any other matter in the action, except the gravitons. This is a suggestive feature thatmight make the Susyon into a good candidate for the dark matter. . How many Susyons do we need to make up the missing mass of the universe, assumingthat these are the only dark matter?
The estimated local dark matter density [6] is ρ DM ≈ . / cm (26)and this means that we have, for each gravitino, a volume of approximately V Gravitino ≈ GeV . / cm ≈ × cm ≈ { cm } (27) ≈ { m } ≈ { km } ≈ {
10 million km } (28)The distance from the earth to the sun is 93 million miles, or about 150 million km. So this is about times the distance to the sun. . Their number density if they are the dark matter?
There are about 15 = 3375 Susyons in thevolume contained in a sphere whose radius is the distance from the sun to the earth. . Their average velocity if they are the dark matter?
Presumably this is of the order of the escapevelocity from our galaxy. This implies that one susyon hits the earth about every century. Since the sun’sarea as a target is about 10 times that of earth, we could expect that the sun gets hit by a susyon aboutone hundred times a year. . The number of Susyons passing through a typical room on earth?
The probability here issomething like once every million universe lifetimes. . Cross Section and Energy and Parameters of a Collision between a Susyon and somematter on earth?
How hard is this computation? I do not know. It is very complicated for sure. Alsothere is the question what happens in this theory beyond tree level. That is not a simple question, sincethe theory is not renormalizable, and although string theory is obviously relevant, as it must be for anysupergravity theory, it is not at all clear how to implement that with these Exchange Transformations. . Simple First Try:
However at tree level, this is probably a feasible calculation. The Susyon interactsonly through gravity. Can one assume that this can be done classically and non-relativistically? Then it israther like a combination of a simple matter of bending of a path in a gravitational field, with some sortof kinetic transport non-equilibrium calculation of the heating of the medium, and corresponding loss ofmomentum and energy for the Susyon. The Susyon will only affect nearby particles, since gravity is so weak,even for a particle as massive as a Susyon. There is an interesting calculation to do here. Does passagethrough a galaxy, and stars and dust and gas in that galaxy, slow down Susyons at the right rate to make asensible story for dark matter? . If a Susyon goes through the earth near us, would we notice?
Such an event would be extremelyrare, but even when it happens, it seems likely that the effect would be small, because gravity is so weak,and the Susyon is just an elementary particle, even though it has a large mass. If it interacted with othermatter directly that might make a large difference. But these questions need some serious work. . Schwarzlength and Broglielength:
We can define two kinds of length associated with any mass m :Schwarzlength m = Gmc (29)Broglielength m = ~ cm (30)7or example for a proton: Schwarzlength Proton = 1 . × − cm . (31)Broglielength Proton = 2 × − cm . (32)and, for example, for the earth: Schwarzlength Earth = .
442 cm . (33)Broglielength Earth = 5 . × − cm . (34)Note that SchwarzlengthBroglielength = Gm c ~ (35) . The length at which the one becomes larger than the other is at:
SchwarzlengthBroglielength = 1 = Gm c ~ (36)and this happens at the Planck mass: m Planck = r c ~ G = 2 . × − gm . (37)and the Plancklength: Schwarzlength Planck = Broglielength
Planck = 1 . × − cm . (38) . Planck Mass and Length and the Susyon:
So for masses greater than the Planckmass the Schwar-zlength is greater than the Broglielength. Now the Susyon with mass 8 . × gm . has its Schwarzlength =6 . × − cm, much greater than its Broglielength = 3 . × − cm . Since it is an elementary particle, onewonders whether this means it has a tendency to be a black hole, or something like that. Of course, classicalreasoning is not relevant here, presumably. But what is relevant? . The Gauge/Higgs Gravitino Sector of Frozen SUSY:
This was worked out in [2] with the aid ofsome computer code included there k . . There are only four parameters ( g , g , g , M P = κ ) for the Gauge/Higgs Gravitino sector, and one gets 13 boson masses (the graviton, photon, gluons, Higgs, Z, W, X, Y and five very heavy extraHiggs), plus the Susyon mass from them. . The superpotential has the form: W = e − κ ( H iL H Ri + Tr S ) M P q g Tr(
SSS ) − g H iL S ji H Rj (39)where these scalar fields are the 5 , g , g to masses using M = − g M P ; M = g M P (40) . We defined the following ‘Hierarchy Parameters’, and the Bosonic Masses of W, Z andH required these values for g , g : Tiny Parameters f g = M W /M P h = M H /M P . × − g . × − The Superpotential Parameters g r = g /g − . × g − − f / k To bring that code up to date for the current software as of this writing, the extra line $Assumptions = f > . We defined: the Planck and SuperPlanck Masses and the masses required the following VacuumExpectation Values (VEVs): Mass Names (GeV) M P M SP . × . × g VEVs (GeV) M K9 M K1 M K2 − f M P − M P f √ −√ M P (42) . Then we arrived at the following 11 masses (plus the zero mass photon, gluons, andgraviton and the zero value for the cosmological constant):
Electroweak Masses Units of M P M Z = f g M P M W = f g M P M H = hM P (43)Heavy Vector Bosons : Units of M P X Y2 √ g M P √ g M P (44)Super Planck Masses : Units of M SP H Oct H Trip H + H H Gravitino = Susyon2 .
05 0 .
68 2 . .
81 2 .
05 1 .
64 (45) . Conclusion:
This Frozen SUSY theory predicts a Susyon that is terrifically heavy and presumablyterrifically hard to observe. But it also predicts a number of other things, and it raises a number of issues:1. The mass of the
X, Y vector bosons and the lifetime of the Proton are related to the masses of theHiggs, Z and W, and to the Planck mass. The heavier masses for the
X, Y and the five extremelyheavy Higgs multiplets seem to be an improvement over the old models in [27,28].2. The theory allows for the cosmological constant to be naturally zero at tree level, in contrast to theorieswhere there is spontaneous breaking of SUSY.3. The theory is very close to the observed standard model, and it predicts that there will be little moreto discover, as far as new particles are concerned, short of super-Planck masses.4. The theory has a natural way of accounting for gauge symmetry breaking in the usual pattern SU (5) → SU (3) × SU (2) × U (1) → SU (3) × U (1).5. There is no need for an invisible sector, or a messenger sector, or lots of effective coupling parame-ters. These are familiar from the SSM [16,17], where they arise from the hypothesis of spontaneoussupersymmetry breaking.6. In fact, the present theory has only four parameters for the gauge/Higgs sector, and one of them is thePlanck constant. This is why the X, Y vector boson masses and the Susyon mass are predicted, alongwith many other bosonic masses. The quark matter CKM matrices, and their leptonic counterparts,need the usual number of parameters that are familiar from the Standard Model without SUSY.7. The theory is highly constrained by the new Master Equation, which arises from the usual SU(5)SUSY GUT theory, coupled to supergravity, through simple Exchange Transformations. Then loopcorrections are governed by the BRST cohomology as in [26].8. The Master Equation means that the theory is valid for all momenta, and is renormalized in the waythat was worked out for gauge theories before the advent of the idea of ‘effective theories’, whichhave no Master Equation, and which are valid only for low momenta, and which lose control of allsymmetries as a result. Note that although the theory is not renormalizable, the Master Equation stillcontrols its symmetry, no matter how many new terms arise.9. In Frozen SUSY, there is a necessary and simple form for a WIMP, and it might account for dark matter.Can it be understood in terms of cosmological constraints? Does an extremely massive particle of thiskind make any kind of a detectable track as it passes through the earth, on the rare occasions when itdoes so? Does such a particle have any kind of effect like that of a tiny black hole? What is its effectin terms of quantized gravity?10. This theory ought to be derivable from the superstring somehow, but I do not see how.11. What happens, for example, for theories based on SO(10) here? Or SO(32)?12. Presumably it is possible to understand the weak angle using renormalization group arguments, as wasdone for the original SU(5) GUT theory.13. It appears that this theory is chiral anomaly free [27,28].14. There are a number of one loop issues. One important issue is what happens to the cosmologicalconstant at one loop in this theory. . Perhaps there is a lesson to be learned here from the recent revolution in QuantumMechanics, which has now given rise to the entire field of Quantum Information . The relativelyobscure paper by EPR seemed an academic curiosity until Bell showed that it had profound consequencesfor local realism, and this was then tested by experiments such as those by Aspect. The history andexperiments here and these counter-intuitive issues are dealt with nicely in modern texts such as [29]. Whatwe can perhaps learn here is that our notions, and even the instinctive and extremely reasonable ‘localreality’ notions of Einstein, may need revision from experiment and futher thought. . Frozen SUSY suggests that our intuitive notions of invariance may also be up for somerevision.
It seems natural, and indeed obvious, to think that the Zinn sources are merely a convenienceto formulate the BRST identities and the Master Equation. If Frozen SUSY has any validity, it looks likethe Zinn sources may have a more dynamic role, which goes beyond our ideas about symmetry being areflection of the Noether type ideas that symmetry must be contained solely within the fields that we startwith. Without any doubt, Frozen SUSY contains lots of fundamental problems that are not yet apparent,as do all attempts to understand these difficult questions.AcknowledgmentsI thank Peter Scharbach for valuable remarks and insight. I also thank Dylan Harries for help with writingthe code in [2].
References [1] J. A. Dixon, “Genuine and effective actions, the Master Equation and Suppressed SUSY,” Phys. Lett.B (2018) 31. doi:10.1016/j.physletb.2017.12.002[2] J. A. Dixon, “Suppressed SUSY for the SU(5) Grand Unified Supergravity Theory,” arXiv:1706.07796[physics.gen-ph].[3] O. Buchmueller and P. de Jong, The Review of Particle Physics (2017) C. Patrignani et al. (Par-ticle Data Group), SUPERSYMMETRY, PART II (EXPERIMENT) Updated September 2015.http://pdg.lbl.gov/ 104] O. Lahav and A.R. Liddle, The Review of Particle Physics (2017) C. Patrignani et al. (Particle DataGroup), 25. THE COSMOLOGICAL PARAMETERS Updated November 2015. http://pdg.lbl.gov/[5] M. J. Mortonson, D. H. Weinberg, and M. White; revised by D. H. Weinberg and M. White; The Reviewof Particle Physics (2017) C. Patrignani et al. (Particle Data Group), 27. DARK ENERGY RevisedNovember 2013 http://pdg.lbl.gov/[6] M. Tanabashi et al. [Particle Data Group], “Review of Particle Physics,” Phys. Rev. D (2018) no.3,030001. doi:10.1103/PhysRevD.98.030001[7] The Master Equation goes back to [8] and Zinn-Justin’s early contribution is set out in a later textbook[9]. An early and pithy introduction was in [10]. A bit of history and Zinn Justin’s involvement is in[11]. A more recent treatment is in [18].[8] C. Becchi, A. Rouet and R. Stora, “Renormalization of Gauge Theories,” Annals Phys. , 287 (1976).[9] J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena”, Oxford Science Publications,Reprinted 1990.[10] J. C. Taylor, “Gauge Theories of Weak Interactions,” Cambridge 1976, 167p[11] A summary and some history can be found in J. Zinn-Justin, “From Slavnov-Taylor identities to theZJ equation,” Proc. Steklov Inst. Math. , 288 (2011).[12] C. Becchi, “Slavnov–Taylor and Ward identities in the electroweak theory,” Theor. Math. hys. ,no. 1, 52 (2015) [Teor. Mat. Fiz. , 420 (1975).[27] Ross, Graham G., ‘GRAND UNIFIED THEORIES’, Reading, USA: Benjamin/Cummings ( 1984) 497P. ( Frontiers In Physics, 60), (1985)[28] A. Hebecker and J. Hisano, The Review of Particle Physics (2017) C. Patrignani et al. (Particle DataGroup), 16. GRAND UNIFIED THEORIES Revised January 2016 by http://pdg.lbl.gov/[29] Le Bellac, M. (2006). Quantum Physics (P. Forcrand-Millard, Trans.). Cambridge: Cambridge Univer-sity Press. doi:10.1017/CBO97805116164715590 FrozenSUSYpaper.texMay 20, 2019