One-Loop Anomaly Mediated Scalar Masses and (g-2)_mu in Pure Gravity Mediation
Jason L. Evans, Masahiro Ibe, Keith A. Olive, Tsutomu T. Yanagida
UUMN–TH–3314/13, FTPI–MINN–13/41, IPMU13-0227
One-Loop Anomaly Mediated Scalar Massesand ( g − µ in Pure Gravity Mediation Jason L. Evans , Masahiro Ibe , Keith A. Olive and Tsutomu T. Yanagida William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455, USA ICRR, University of Tokyo, Kashiwa 277-8582, Japan Kavli IPMU (WPI), TODIAS, University of Tokyo, Kashiwa 277-8583, Japan
Abstract
We consider the effects of non-universalities among sfermion generations in models of Pure GravityMediation (PGM). In PGM models and in many models with strongly stabilized moduli, the grav-itino mass may be O(100) TeV, whereas gaugino masses, generated through anomalies at 1-loop,remain relatively light O(1) TeV. In models with scalar mass universality, input scalar masses aregenerally very heavy ( m (cid:39) m / ) resulting in a mass spectrum resembling that in split super-symmetry. However, if one adopts a no-scale or partial no-scale structure for the K¨ahler manifold,sfermion masses may vanish at the tree level. It is usually assumed that the leading order anomalymediated contribution to scalar masses appears at 2-loops. However, there are at least two possiblesources for 1-loop scalar masses. These may arise if Pauli-Villars fields are introduced as messengersof supersymmetry breaking. We consider the consequences of a spectrum in which the scalar massesassociated with the third generation are heavy (order m / ) with 1-loop scalar masses for the firsttwo generations. A similar spectrum is expected to arise in GUT models based on E /SO (10) wherethe first two generations of scalars act as pseudo-Nambu-Goldstone bosons. Explicit breaking ofthis symmetry by the gauge couplings then generates one-loop masses for the first two generations.In particular, we show that it may be possible to reconcile the g µ − a r X i v : . [ h e p - ph ] D ec Introduction
Although the mass of the recently discovered Higgs boson [1] is light enough that it canbe accommodated in supersymmetry, it is near the upper limit of simple models like theCMSSM [2, 3]. This large Higgs mass and the lack of evidence for supersymmetric particlesat the LHC [4, 5] have put severe constraints on the simplest models of supersymmetry [6, 7]including the CMSSM.Since both the LHC constraints on the superpartners and the observed Higgs mass favorheavier sfermion masses [4, 5], it could be that nature does indeed have a mass splittingamong the supersymmetric particles as is the case in split supersymmetry [8], pure gravitymediation (PGM) [9–12], and models with strongly stabilized moduli [13–15]. In modelsof PGM [9], sfermions get a tree-level mass, as in the CMSSM, while gauginos get a one-loop mass from anomaly mediation [16]. Recently, we showed that models based on PureGravity Mediation, with [11] and without [12] scalar mass universality, could explain virtuallyall experimental constraints with electroweak symmetry breaking generated radiatively. Inthe case of full scalar mass universality, the theory can be described in terms of two freeparameters, the gravitino mass, m / and tan β the ratio of the Higgs vacuum expectationvalues. However, these models placed a rather strict constraint on tan β = 1 . − .
5. TheHiggs mass constraint then restricted the gravitino mass to the range m / = 300 − β is only restricted byperturbativity of the Yukawa couplings and m / can be as low as 80 TeV. However, evenfor a gravitino mass this light all sfermions masses are much larger than the weak scale.If all sleptons have mass of order 80 TeV or more, there is little hope of explaining thediscrepancy in the anomalous magnetic moment of the muon [17] or sfermion detection atthe LHC. As was recently shown in [18], sleptons need to be lighter than about 2 TeV ifthere is to be any hope of explaining ( g − µ . The LHC reach varies greatly dependingon the masses of the first two generation squark masses. If squarks are lighter than 2 TeV,the LHC reach on the gluino can be as high as about 4 TeV [19]. To get sfermion massesthis light in PGM, there must be additional sources of non-universalities in the sfermionboundary masses. Since large stop masses are important in explaining the Higgs mass [20],it will be advantageous to take tree-level masses of order m / for the stops. Furthermore,if the Higgs bi-linear mass term, µ , is much larger than the stau mass, as is often the casein PGM, the stau tends to be tachyonic [18]. This problem can also be evaded by having atree-level stau mass. These arguments suggests that the third generation should have tree-level masses while the first and second generations boundary masses should be suppressed.Phenomenologically viable models can also be found for suppressed third generation masses,however, they tend to be qualitatively similar to the PGM models discussed in [11, 12].A, possibly, more compelling reason to discuss light first and second generation sfermionmasses is the hierarchy in the Yukawa couplings. If the first two generations are pseudoNambu-Goldstone multiplets (pNG) of some broken global symmetry [21], this would natu-rally suppress the sfermion masses. Since the Yukawa couplings are an explicit breaking ofthe global symmetry, the Yukawa couplings of the pNG would also be suppressed. A similarsuppression of the first and second generation sfermion masses can be realize from a no-scale1ike geometry for the K¨ahler potential [22]. This geometry can arise from a brane separationwhere on one brane we have the SUSY breaking fields as well as the Higgs boson and thirdgeneration fields and on the other we have the first and second generations fields. In both ofthese scenarios the Yukawa coupling hierarchies are linked to the sfermion mass hierarchies.Generating hierarchically small soft masses for the first two generations is not so problem-atic. However, because the gauginos are small in comparison to the third generation masses,the RG running of the first two generations will give tachyonic masses for the simplest ofmodels. These tachyonic masses can be evaded if sfermion masses of the first two genera-tions are generated at one-loop. In the case of no-scale like boundary conditions this canbe accomplished if the Pauli-Villars fields, that regularize the low-scale theory, interact withsupersymmetry breaking generating a one-loop soft masses [23]. The Pauli-Villars fields actas the messengers of supersymmetry breaking. In the case of E /SO (10) [12], the preons actas messengers generating a similar one-loop mass much like the Pauli-Villar fields. Thus, itis possible that we can construct a spectrum in which m ˜ u, ˜ c ∼ m ˜ g (cid:28) m ˜ t ∼ m / , where m ˜ u, ˜ c, ˜ t refer to the three generations of sfermion masses, and m ˜ g refers to gaugino masses. As wewill see, this type of mass hierarchy is capable of simultaneously explaining the Higgs massand the deviation in ( g − µ .In section 2, we will discuss our model of PGM which will allow for light first and secondgeneration sfermions. We also describe the mechanism for generating one-loop anomalymediated masses for the first two generation sfermions. As we will see, due to our ignoranceof the precise mechanism for transmitting supersymmetry breaking, we inevitably have threenew parameters associated with the one-loop masses correlated with the three low energygauge groups. In Section 3, we derive results with light first and second generation sfermionsin the context standard grand unified theories in which there is an assumed relation betweenthe new parameters, and in section 4 we discuss the impact of these models on the value ofdeviation in the anomalous magnetic moment of the muon, ∆a µ . In section 5, we will discussalternate grand unified scenarios where the anomalous magnetic moment of the muon canbe more easily explained. Lastly, in section 6 we will conclude. The back bone of our discussion will be the pure gravity mediated models discussed in [11,12]with a K¨ahler potential K = y i y ∗ i + K ( H ) + K ( Z ) + ln | W | , (1)where the K¨ahler potential for the Polonyi-like modulus, Z , which is responsible for super-symmetry breaking, contains a stabilizing term [24] K ( Z ) = ZZ ∗ (cid:18) − ZZ ∗ Λ (cid:19) , (2)and the K¨ahler term for the Higgs fields contains a Giudice-Masiero-like term [11, 25, 26] K ( H ) = | H | + | H | + c H ( H H + c.c ) , (3)2nd y i represent the other MSSM fields. We also assume that the superpotential is separablebetween the matter fields and hidden sector fields: W = W ( Z ) + W (SM) , (4)where W (SM) contains all Standard Model (SM) contributions to the superpotential. Fur-thermore, we assume a simple Polonyi form for W ( Z ) [27], W ( Z ) = ˜ m ( Z + ν ) , (5)It has recently been shown that strongly stabilized models of this type are free from any ofthe cosmological problems normally associated with moduli or gravitinos if Λ (cid:46) × − [28].For this K¨ahler potential, the MSSM scalar fields will have an input mass m ˜ f = m / atthe universality scale which we associate with the Grand Unified Theory (GUT) scale. Inthe absence of a non-trivial gauge kinetic function, the gaugino masses are generated fromanomalies and will have loop suppressed masses given by M i = b i g i π m / , (6)where the b i are the coefficients of the beta function. The tree-level contribution to the A -terms are quite small, A ∼ (Λ /M P ) m / [15]. The leading order contribution to the A -terms are the one-loop anomaly mediated contributions and are effectively zero.For the universal case discussed above, tan β is restricted to the range 1 . − . m / (cid:38)
300 TeV in order to get a sufficiently large Higgs mass [11]. However, ifwe take non-universal Higgs boundary masses [12], tan β is only constrained by the weakerrestrictions of perturbativity of the Yukawa couplings. Non-universality is easily achievedby adding non-minimal couplings of the Higgs fields to the modulus, Z . For example, K ( H ) = (cid:18) a ZZ ∗ M P (cid:19) H H ∗ + (cid:18) b ZZ ∗ M P (cid:19) H H ∗ + ( c H H H + h.c. ) (7)will generate Higgs soft masses which depend on the couplings a and b [12] m = (1 − a ) m / ; m = (1 − b ) m / . (8)In this case, the lower bound on m / is due to the wino mass [29] placing a lower bound ofabout m / (cid:39)
80 TeV.The RG running in these models is rather simple. Since the gaugino masses are small,they do not affect the RG running of the sfermion masses. Because only the third generationYukawa couplings are large, only the third generation masses will run at one-loop. However,the variations of the third generation masses from RG running preserves O ( m / ) masses forthe third generation. If all the sfermion masses are O ( m / ), they cannot be seen at the LHCand will be of no aid in explaining the discrepancy in ( g − µ . To make things worse, if allscalar masses are universal at the GUT scale, their masses need to be or order 300 TeV toget a suitably large Higgs mass. Only by breaking the universality of the Higgs soft masses3an this constraint on tan β be weakened. The lower bound on the scalar masses can thenbe as low as 80 TeV, with this lower bound coming from the constraints on the wino mass.But, even sfermion masses of order 80 TeV can not explain ( g − µ or be detected at theLHC.To have anything other than the vanilla gauginos signals at accelerators for these models,at least some of the scalars need to be light and thus additional non-universalities are neededbeyond the Higgs soft masses. As is well known [16], in the absence of a large tree levelscalar mass, scalar masses are present at least at the two-loop level. However, as we discussin more detail below, it is possible that scalar masses also arise at one-loop. Indeed one canimagine a no-scale construction where all scalar masses vanish at the tree level as in no-scalesupergravity [22]. The K¨ahler potential can be written as K = − (cid:18) − (cid:2) K ( Z ) + K ( H ) + y i y ∗ i (cid:3)(cid:19) + ln | W | , (9)where K ( H ) is given by Eq. (3). If all sfermion masses vanish at the tree level and receiveone-loop contributions, it will be difficult to generate a Higgs mass as large as 125 GeV forgeneric parameters unless m / (cid:38)
150 TeV. Since the sfermions are still rather heavy, thismodel will be qualitatively the same as PGM.Instead, the approach we take below is to suppress only the masses of the first and secondgeneration sfermion masses. Here, we discuss two ways of suppressing scalar masses of thefirst two generation sfermions. The first is to take a similar no-scale like K¨ahler potential ofthe form K = y (3) i y ∗ (3) i − (cid:18) − (cid:104) K ( Z ) + y (1 , i y ∗ (1 , i (cid:105)(cid:19) + K ( H ) + ln | W | , (10)where y (1 , i are first and second generation fields in the MSSM and y (3) i are the third genera-tion fields. Although this K¨ahler potential is capable of suppressing the sfermion masses, itwill be advantageous to also take non-universal Higgs masses coming from a K¨ahler potentialof the form given in Eq. (7). For this model, the bulk of the features of PGM remain butin addition we have very light sfermion masses for the first two generations which are nowgenerated by anomalies.The other possibility for suppressing the first and second generation sfermion masses isto associate these fields with the pNG of the global symmetry E /SO (10). However, in thiscase the gauge and Yukawa couplings act as an explicit breaking of this symmetry. As wewill see below, this is actually an advantage. The gauge and Yukawa couplings break thesymmetry and one-loop masses are generated.In an actual no-scale like model, the sfermion masses would be generated from the one-loop gaugino mass contributions to the RG equations. However, this no-scale like running isbroken by the presence of a heavy third generation. This breaking of the no-scale structurehas a drastic effect on the spectrum and as we will see, we will need to find an additionalsource of mass for the first and second generation sfermions.4 .1 General Features of the Renormalization Group Running In this section we will discuss the bulk features of the running of the first and second genera-tion sfermion masses. As usual, we can take the third generation dominance approximationand will neglect the Yukawa couplings of the first two generations (see appendix A on theSUSY FCNC contributions). In this approximation, the only one-loop contribution to thefirst two generation sfermion mass running comes from gaugino masses and S = 12 Tr (cid:0) Y m (cid:1) , (11)where Y is the hypercharge and m represents the sfermion masses of the particles chargedunder hypercharge. Since this contains contributions from the third generation, it will gen-erally be the dominant contribution to the running of the first two generations. The changein the sfermions masses from S can be easily determined because it has a rather simple RGequation, dS Y dt = g Y π (cid:88) i (cid:18) Y i (cid:19) S Y , (12)with solution, S Y ( Q ) = S Y ( Q ) g Y ( Q ) g Y ( Q ) . (13)After integrating the RG, this contribution to the sfermion masses is of order O ( m / ). Thisis much too large and would typically lead to tachyonic sfermion masses. However, if sfermionmasses are universal or determined by gauge interactions, S Y ( Q ) = 0 and so it remains zeroat one-loop for the entire running . S Y ( Q ) = 0 is unchanged for non-universal Higgs massesas long as m = m , as in the NUHM1 [30]. Since we are considering a combination of thesemodels, we have S Y ( Q ) = 0 and S Y does not play a significant role in the RG running,though it is included in our analysis below.As stated above, the other one-loop contribution to the RG running of the first twogeneration is proportional to the gaugino masses squared. Since the gaugino masses are loopsuppressed relative to m / , their effective contribution to the RG running is of order m / (16 π ) , (14)effectively a three-loop contribution much too small to be important. Thus, the two-loopcontributions which are proportional to third generation masses will have a much moreimportant effect on the masses of the first and second generation sfermions.Since the tree-level sfermion masses of the first two generations are suppressed, terms inthe beta functions proportional to them will not be important. Only contributions involving This relationships is broken at two-loops. However the effect of S Y still tends to be sub-dominant inthis case. D -terms, i.e. RG terms coming from (cid:104) ( D a D a ) (cid:105) or (cid:104) D a D a ( ˜ f i f j f k ) (cid:105) which giveterms like those in Eq. (38)–(41). The rough sizes of these contributions to the RG runningof the first two generations are O (1) g i (16 π ) m / and O (1) g y i (16 π ) m / , (15)where g i are the gauge couplings and y i are the Yukawa couplings. Their exact form can befound in Appendix B. As can be seen there, the RG running from a two-loop contributionin the beta function will diminish the sfermion mass by an amount of order O (1) m / (16 π ) , (16)if we are running down from the GUT scale. Clearly, a one-loop boundary mass is need tooffset the RG contribution to the mass and the two-loop anomaly mediated contribution isinsufficient. In this section, we address the generation of one-loop masses for the sfermions. Since stringtheory is a renormalizable theory, it should provide some mechanism to renormalize itself.The renormalization for the gauge interactions can be parameterized by adjoint Pauli-Villars(PV) fields. Because string theory gives us no indication of how these PV fields interactwith the hidden sector, we cannot say how strongly they feel supersymmetry breaking. Ifthe PV fields do in fact interact with the hidden sector they would act as messengers ofsupersymmetry breaking. As was shown in [23], this gives a one-loop contribution which isproportional to the gauge interactions and Yukawa couplings. In Appendix C, we give a toymodel showing how these one-loop masses are generated in the flat supersymmetric limit.Since there is no way of knowing how the PV fields interact with the hidden sector, the massesof the sfermions are effectively free parameters. However, we make the assumption that thePV fields corresponding to each generation interact with the hidden sector identically. Wefind this a reasonable assumption since gauge symmetries in general do not distinguishbetween generations.Another possibility is to consider a global E /SO (10) which has two generations thatare pNG. To have exact Nambu-Goldstone bosons (NGB), the gauge and Yukawa couplingsneed to be zero. By introducing explicit breaking to the E /SO (10) in the form of gaugeand Yukawa couplings, the masses of the NGB are lifted. These mass corrections should beat the one-loop order. This can be understood by noting that when the gauge interactionsare turned on, they will generate one-loop corrections to the K¨ahler potential. This one-loopcorrection deforms the K¨ahler potential of E /S (10) breaking the cancellation needed to givemassless fields. Since this breaking is at the one-loop order, we expect the sfermion massesto be generated at the one-loop order. As before, we get one-loop masses for the first andsecond generation sfermions. To calculate these masses exactly we need the details of the6nderlying QCD like theory at the preon level. However, we know they are at the one-looplevel and proportional to the gauge and Yukawa couplings. To parameterize our lack of knowledge about the Plank scale dynamics or preon model, wewill define γ i = 18 π g i C ( r ) , (17)where g i is the gauge coupling and C ( r ) is the quadratic Casimir . The soft mass for a givensfermion is then given by m f = (cid:88) i c i γ i m / , (18)where we have made the assumption that the c i are generation independent. Including theseparameters, our full list of free parameters is m / tan β m = m c c c . (19)The boundary masses for the first two generations then take the form m f i ( Q GUT ) = (cid:88) j C j ( r i ) c j g j π m / , (20)where c j is defined above and C j ( r i ) is the quadratic Casimir for ˜ f i from the gauge group j . We are now in a position to examine the simplest realization of this model, namely with c = c = c . This relationship among the c i is what would be expected if the grandunified theory stemmed from a simple SU (5). In this case the PV fields will stem fromcomplete multiplets of the SU (5) gauge group. If SU (5) is broken in a generic fashion, weget c = c = c . In models like these, the lightest sfermion is a squark. Because the gaugecouplings are universal at the GUT scale, where we apply our boundary masses, the squarksare only slightly heavier then the sleptons. However, the RG running of the squarks is muchstronger since g (cid:29) g at the weak scale. This leads to the lightest sfermion being the downsquark as we explain below. The Casimir is important because we have adjoint fields interacting with fundamental fields in thesuperpotential. This will lead to a Casimir when we form loops from these interactions as can be seenin [23].
7t is also important to note that non-universal Higgs soft masses are advantageous. If m , ∼ m / , we have m / (cid:38)
300 TeV [11] and even one-loop sfermion masses will remainout of reach for the LHC since generically m ˜ q would still be rather heavy. Not only wouldtaking non-universal Higgs masses allow us to choose smaller m / , it also has an importanteffect on the running. The non-universalities in the Higgs masses become important, because S (cid:48) (listed in Appendix B ) depends on the Higgs soft masses. If the Higgs soft masses areuniversal, S (cid:48) is suppressed and it has little affect on the running of the sfermion masses.Because universality is not an option, we have a significant contribution to the sfermionmass running from S (cid:48) . This running splits the squark masses.For the simplified model we consider here, the down squark is the lightest sfermionbecause it has the largest positive hypercharge. With non-universal Higgs masses, S (cid:48) is largeand deflects the mass of Q, d down and u up. Since the hypercharge of d is larger thanthat of Q , the down squark is the lightest. A plot of the mass spectra for these modelscan be seen in Fig. (1) which shows the sfermion mass contours in the m = m , c U plane,where c U = c = c = c is the universal coefficient of the one-loop input soft masses.The line type identifications are given in the caption. The shaded regions correspond totheoretically excluded regions for the following reasons: the upper left corner is excludebecause m A <
0, the lower region is excluded because scalar down is tachyonic. Notice thatthe down squark mass gets small near this boundary and the mass squared evolves veryquickly as the boundary is approached, rapidly turning negative. The shaded region on theright is excluded because µ <
0. As can be seen in these figures, the down squark (greendashed curves) is the lightest sfermion.To see the effect of changing m / we display, in Fig. (1), two values of m / , m / = 120TeV (left) and m / = 150 TeV (right) for tan β = 5, the latter is chosen to get an acceptableHiggs mass. The sign for m = m refers to the sign of m . As can be seen from the figures,the region with small down squark masses is shrinking and so it becomes increasingly moredifficult to get a small mass for the down squark as m / becomes larger. Once m / (cid:38) β m f ∼ O (1) g i (16 π ) m / ∼ (2 TeV ) (cid:16) m / TeV (cid:17) . (21)Once the sfermion masses become similar in size to the beta function, the sfermion mass willbe quickly driven to zero . Thus, even if we adjust the boundary mass of the sfermions, itwill be difficult to get a sfermion mass smaller than the size of the beta function.In Fig. (2, we plot the mass contours for m / versus c U . As in Fig. 1, the lower regionin the figure is excluded because the scalar down is tachyonic. As can be seen in this figure In this regime, the typical approximation of setting m ˜ f = 0 in the beta functions once m ˜ f < Q , where Q is the RG scale, is invalid. In the figures, we assume that we can extrapolate between regions where wecan safely integrate out the sfermions to the region where the sfermions become tachyonic, knowing that allpossible sfermion masses should be traversed. -200 -100 0 100 2001.21.31.41.5 . m = 120 TeVtan b = 5 m u = m Q = m d = m L = m e = c U m = m (TeV) ~ ~ ~~~ (a) -200 -100 0 100 2001.11.21.31.41.5 .. m = 150 TeVtan b = 5 m u = m Q = m d = m L = m e = . c U m = m (TeV) ~ ~~~~ (b) Figure 1:
Here we show the contour plots of the sfermion masses of the first and secondgeneration masses in the m = m versus c U plane for (left) m / = 120 TeV and (right) m / = 150 TeV and tan β = 5 . The line types are as follows: left-handed squarks (yel-low solid); right-handed scalar up (light blue dashed); right-handed scalar down (light greendotted); left-handed sleptons (red dot-dashed); and right-handed sleptons (blue double-dot-dashed) . The masses are in TeV. it is rather difficult to get the scalar down to be lighter than 2 TeV as m / increases. InFig. (2b), we have also plotted the gluino mass as well as the ratio of the down squark massto the gluino mass, r dg . Examining r dg in Fig. (2b), we see that the down squark is smallerthan the gluino only for regions close to the lower boundary. These regions correspond toregions where the beta function for the down squark is similar in size to the down squark.This is why this region is somewhat small. Along this edge we see that the down squark isless than 2.5 TeV only if m / (cid:46)
130 TeV. This corresponds to a gluino mass of about 3 TeV.By optimizing the parameters we can get down squarks below 2.5 TeV for a gluino mass ofabout 3.3 TeV. Regardless, this corresponds to an increased reach in the gluino mass andsome interesting prospect for detection at the LHC. ( g − of the Muon One of the persistent problems facing the SM is the deviation of the SM prediction for( g − µ with respect to the experimental value. The current deviation in the muon anomalous9 e e
120 160 c U m (TeV)tan b = 5 m Q = m e = m L = m u = ~ ~~~ (a) e +
80 120 160 m d = m g = r d g = c U m (TeV)tan b = 5 ~~ (b) Figure 2:
We show contours of the sfermion masses in the m / versus c U plane. Thecontours are as in Fig. (1). On the right, the solid red line shows the gluino mass contourand the dashed blue line shows the ratio r dg = m ˜ d /m ˜ g . The masses are in TeV. magnetic moment is [17]∆a µ = ( a µ ) exp − ( a µ ) SM = (26 . ± . × − . (22)As the LHC has pushed the scale of new physics to higher and higher scales, it is becomingincreasingly hard to find explanations for this deviation. In fact, there are few models ofsupersymmetry which predict a large enough ∆a µ .In the mass insertion approximation, the supersymmetric contributions to the anomalousmagnetic moment of the muon take the form ∆a µ = m µ tan βµ (cid:2) g M F ( M , m ˜ µ L , m ˜ µ R ) (23)+ g i M i F i ( M i , µ, m ˜ µ L , m ˜ m R ) + g M F ( M , µ, m ˜ ν ) (cid:3) , where m ˜ µ L,R are smuon soft masses, and m ˜ ν are sneutrino soft masses. For spectra with allSUSY breaking masses and the Higgs bilinear term of similar size, the anomalous magneticmoment of the muon is roughly [31]∆a µ (cid:39) π g tan β m µ m SUSY (cid:39) × − (cid:18)
260 GeV m SUSY (cid:19) (cid:18) tan β (cid:19) . (24) In Eq. (23), F is related to ∆ a N µ , F is related to ∆ a N (2 − µ , and F is related to ∆ a Cµ of [31]. F . This gives a rough estimate of the size of theHiggs bilinear and slepton masses needed to explain ∆a µ . In general it is not easy to getsleptons this light while still getting squark masses larger than the LHC constraints. Forthis reason it is rather difficult to explain ∆a µ in SUSY unless one splits the masses of thefirst two generations from that of the third [32].In PGM, this problem is exacerbated since sfermion masses are pushed to even highermass scales. Since the size of µ is related to the stop masses, µ is also rather large. If how-ever, the masses of the first two generations are suppressed, ∆a µ may increase substantially.Because µ is relatively unaffected by this, F and F i are still suppressed, F i ∼ F ∼ µ m µ . (25)With F independent of µ , it has no residual suppression and we have∆a µ = m µ tan βµg M F ( M i , m ˜ µ L , m ˜ µ R ) . (26)Since this contribution to ∆a µ is proportional to µ , it will grow linearly with µ . To showthis important µ dependence, we have plotted ∆a µ with respect to µ in Fig. (3) for thesample spectrum M = 720 GeV, M = 230 GeV, m ˜ µ L = 660 GeV, and m ˜ µ R = 840GeV and tan β = 25. With this rather large hierarchy between the first two generationsfermion masses and Higgs bilinear mass, it is possible to explain ∆a µ in PGM like modelsfor µ ∼ m / (cid:38)
25 TeV, even if the smuon masses are larger than 600 GeV. Fig. (3) alsoshows the extrapolation between nearly degenerate masses and a hierarchically larger µ . Inthe region of degenerate masses, the F i contribution dominates. For µ increasing, the F quickly becomes the dominant contribution to ( g − µ , as we naively argued above.With heavy third generation masses and light first two generation masses, we also evadeanother possibly problematic constraint, tachyonic staus. In PGM, the mixing of the leftand right sfermions is proportional to µm τ . Since the tau mass is non-trivial, the Higgsbilinear mass can not be too much larger than the diagonal soft masses of the stau. Becausethe third generation masses are also large in the models we are considering, this constraintis irrelevant. There is a much weaker constraint coming from having positive masses for thesmuons. However, the muon is much lighter and so these constraints are much weaker. Thismuch weaker constraint will allow us to push the value of µ up enough in order to explain∆a µ . Because of the difficulty in obtaining small sfermion masses in the 1st two generations withuniversal constants, c U , we next look at models where c (cid:54) = c (cid:54) = c . This equates to con-sidering a non-standard breaking of SU (5) or no gauge coupling unification. One possibilityfor a non-standard breaking of SU (5) is to take the product unification SU (5) × U (2) or SU (5) × U (3) [33]. In these models, there are three c i . Since the gauge fields of the standardmodel do not come solely from the SU (5), but are mixtures of the SU (5) gauge field and the11 -12 -11 -10 -9 -8 D a m D a m m (GeV) g M m tan b m m F totalg i2 M i m tan b m m F i12 g M m tan b m m F -9 Figure 3: µ dependence of the various contributions to ∆a µ . additional gauge fields, the PV fields that regulate the low scale gauge fields will be mixedleading to independent c i . The advantage of considering three c i is the possibility of lightsleptons which can explain the deviation in ( g − µ . Below, we will consider several differ-ent scenarios. Initially, we will scan over generic values of the c i to see what the parameterspace looks like. Then we will focus on some specific and unique examples which have someinteresting results. In this section, we examine the parameter space for the c i . As we will see below, the sleptonmasses are strongly influenced by the Higgs soft masses, m , . For large and negative valuesof m , , the two-loop gauge running from SU (2) and U (1) are reduced. Since the slepton RGrunning is independent of SU (3), these will be the dominant contributions to the runningmaking weak scale sleptons easier to realize.With these relations in mind, we examine m / = 80 TeV, m = m = −
80 TeV, andtan β = 7 and scan over the c i . m / = 80 TeV is needed to get a sufficiently large winomass and tan β = 7 is chosen so the Higgs mass is sufficiently light. As mentioned above, m = m = −
80 TeV is chosen to reduce the beta functions of the sleptons making it easierto realize weak scale sleptons. We then scan over the c i with the results found in the top twopanels of Fig. (4). As can be seen in these figures, the correction to ( g − µ is large enoughto account for the experimental discrepancy, but it does require somewhat special values forthe c i . In each figure, we distinguish between cases for which one (or both) of the Higgs12 D a m m = 80 TeVm = m = -80 TeVtan b = 7 c m + m < 0 or m + m < 0m + m > 0 and m + m > 0 D a m (2s) -0.04
0 0.04 0.0810-8 D a m m = 80 TeVm = m = -80 TeVtan b = 7 c m + m < 0 or m + m < 0m + m > 0 and m + m > 0 D a m (2s)
400 800 1200 D a m m = 80 TeVm = m = -80 TeVtan b = 7 m + m < 0 or m + m < 0m + m > 0 and m + m > 0 D a m (2s) m l (GeV) ~
118 121 124 127 129
400 800 1200 ( G e V ) m = 80 TeVm = m = -80 TeVtan b = 7 m l (GeV) m h m c /2 ~ Figure 4:
The change in the anomalous magnetic moment of the muon, ∆a µ , with respect to c (top left), c (top right), and average slepton mass, m ˜ l ≡ ( m e L + m e R ) / (bottom left) for m / = 80 TeV, m = m = − TeV, and tan β = 7 . The dotted line corresponds to the σ lower limit of ∆a µ . The red +’s have m + µ < or m + µ < and the green × ’s have m , + µ > . The bottom right panel shows the change in the m χ / (green × ’s) and theHiggs mass (red +’s) with respect to the average slepton mass. All four panels are based onthe same data. squared masses is negative at the GUT scale, m i + µ <
0, for which there are potentialcosmological problems [34] and those which are always safe since the Higgs squared massesare both always positive.To better understand the parameter space, we give some additional plots. In bottom leftof Fig. (4), we plot the average slepton mass, m ˜ l = ( m e L = m e R ) /
2, with respect to ∆a µ . Ascan be seen in these plots, the average slepton mass is rather heavy even for points that canexplain g −
2. This is due to a large µ . The other two important parameters for constrainingthese models, the wino mass (for clarity, m χ / µ is very sensitiveto the average slepton mass. Note that while the Higgs masses shown are somewhat high,13 -10 -9 -8 D a m m = 80 TeVm = m = 0 TeVtan b = 7 c m + m > 0 and m + m > 0 D a m (2s) -10 -9 -8 D a m m = 80 TeVm = m = 0 TeVtan b = 7 c m + m > 0 and m + m > 0 D a m (2s) -10 -9 -8
400 600 800 1000 1200 1400 1600 1800 D a m m = 80 TeVm = m = 0 TeVtan b = 7 m + m > 0 and m + m > 0 D a m (2s) m l (GeV) ~
110 112 114 116 118 120 122 124 126 128 130 400 600 800 1000 1200 1400 1600 1800 ( G e V ) m = 80 TeVm = m = 0 TeVtan b = 7 m l (GeV) m h m c /2 ~ Figure 5:
Same as Fig. (4) except with m = m = 0 . we expect that there is a roughly 2 GeV uncertainty in the calculation of its mass (c.f. [11].To portray the sensitivity of the RG running on the Higgs soft masses, we show similarplots for m = m = 0. These plots can be seen in Fig. (5). There are several importantthings to note. First, the sleptons tend to have similar sizes since this is predominantlyset by m / . However, the lighter slepton masses arise for c i which are tuned to a greaterdegree. Another important difference is a large decrease in the wino mass. This is due toa significant change in m A and µ . Because the threshold corrections to the wino dependstrongly on both µ and m A , the wino mass is much lighter for m = m = 0. This is anadditional reason why m = m = −
80 TeV is advantageous. For m = m = 0, we wouldneed to take a larger value of m / making it more difficult to get weak scale sleptons. Next, we examine some special values of the c i which tend to be interesting. In particular, wefirst allow c to vary so that we obtain light first and second generation squarks. Althoughit may seem this has no affect on g −
2, it will have some rather important and unexpectedeffects. In Fig. (6), we examine the Higgs mass for different but fixed values of c , and vary14
127 130 3000 4000 5000 6000 m = 80 TeVm = m = -80 TeVtan b = 7 m Q (GeV) ~ m h ( G e V ) c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20m h (2 s )
125 127 129
131 3000 4000 5000 6000 m = 80 TeVm = m = -80 TeVtan b = 35 m h ( G e V ) c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20m h (2 s ) m Q (GeV) ~ Figure 6:
Here we plot the Higgs mass versus the mass of the left-handed squark mass ofthe first two generations for (left) tan β = 7 and (right) tan β = 35 . The red +’s are for c = 0 . and c = 0 . . The green × ’s are for c = 0 . and c = 0 . . The blue stars arefor c = 0 . and c = 0 . . The magenta boxes are for c = 0 . and c = 0 . . The cyan filledboxes are for c = 0 . and c = 0 . . The gray circles are for c = 0 . and c = 0 . . Thehorizontal dashed line corresponds to the 2 σ lower limit on ∆a µ . c which we will parameterize by the left-handed squark mass, m ˜ Q .In this figure, we see that as soon as m ˜ Q (cid:46) g − µ , because it allows us to push up the value of tan β and still have a sufficiently smallHiggs mass. The Higgs mass is sensitive to the first two generation squark masses throughalterations in the running of the gauge and Yukawa couplings. When the first, second, andthird generation sfermion masses are similar there are effectively two regions of RG running,above and below the sfermion mass scale. However, if the first and second generations aresufficiently separated from the third generation, there is a third region that emerges. Inthis third region, the beta function for SU (2) nearly vanishes. This leads to rather largedeviations in the gauge couplings for the scale where the third generation decouples. Thisdeviation in the coupling leads to a significant change in the Higgs mass. As it turns out, wecan get a light enough Higgs mass even for large tan β . Because of this new found freedomin tan β , we can further enhance ∆a µ in the region where the squark masses are light bytaking tan β large. This enhancement of ∆a µ for regions with light squark masses can beseen in Fig. (7).Since the gauge couplings are deflected by the alteration of the beta functions from lightfirst and second generation sfermions, we will also see changes in the masses of the gauginos.These changes are fairly mild as can be seen in Fig (8), although the scaling on the axismakes it appear somewhat drastic. For completeness, we also plot the slepton masses versusthe anomalous magnetic moment. This is shown in Fig. (9).15 c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20 m = 80 TeVm = m = -80 TeVtan b = 7 m Q (GeV) ~ D a m m = 80 TeVm = m = -80 TeVtan b = 35 m Q (GeV) ~ D a m c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20 D a m (2 s ) Figure 7:
Here we plot the change in the anomalous magnetic moment ∆a µ with respect theleft-handed squark mass for (left) tan β = 7 and (right) tan β = 35 . The symbols used areidentical to that in Fig. (6).
240 250 260 270 3000 4000 5000 6000 m = 80 TeVm = m = -80 TeV m c ( G e V ) m Q (GeV) ~ c = 0.70 c =0.05 tan b = 35c = 0.75 c =0.20 tan b = 35c = 0.75 c =0.20 tan b = 7c = 0.70 c =0.05 tan b = 7 m = 80 TeVm = m = -80 TeV m g ( G e V ) m Q (GeV) ~ c = 0.70 c =0.05 tan b = 35c = 0.75 c =0.20 tan b = 35c = 0.75 c =0.20 tan b = 7c = 0.70 c =0.05 tan b = 7 ~ Figure 8:
Here we plot the change in the neutralino mass, m χ , with respect to the left-handedsquark mass. The red +’s are for c = 0 . and c = 0 . with tan β = 35 . The green × ’s arelikewise for tan β = 7 . The blue stars are for c = 0 . and c = 0 . with tan β = 35 . Themagenta boxes are likewise for tan β = 7 . m = 80 TeVm = m = -80 TeVtan b = 7 D a m c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20 m l (GeV) ~ m = 80 TeVm = m = -80 TeVtan b = 35 D a m c = 0.70 c =0.05c = 0.75 c =0.05c = 0.70 c =0.10c = 0.75 c =0.10c = 0.75 c =0.20c = 0.70 c =0.20 D a m (2 s ) m l (GeV) ~ Figure 9:
Here we plot the change in the anomalous magnetic moment with respect to theaverage slepton masses for (left) tan β = 7 and (right) tan β = 35 . The symbols used areidentical to that in Fig. (6). Finally, we consider some models which can relax the constraint on the wino mass. Sinceit is this constraint which is responsible for pushing up the gravitino mass, relaxing thisconstraint will drastically reduce the fine tuning needed to get light sleptons. There areactually two simple ways to evade the wino mass constraint: increase its mass for a givengravitino mass or change the decay width of the wino. Both of these mechanisms requiredark matter to come from some source other than the lightest supersymmetric particle (LSP).However, since the constraints on wino dark matter are getting ever more stringent [35], itis worth examining the case where the wino is not the dominant source of dark matter.One interesting possibility is to assume that dark matter arises from a PQ like theory. Thefields responsible for PQ symmetry breaking then act as messenger for the gauginos [36]enhancing the gaugino masses for a given value of the gravitino mass. Another option isto allow R -parity violation. This relaxes the constraint on the wino by increasing its decaywidth. In PGM, the LSP is a neutral wino and the charged wino is about 160 MeV heavier.Because these particles are nearly degenerate there is a strong phase space suppression of thedecay. If R-parity violating interactions are included, the charged wino can decay directlyto standard model particles alleviating the phase space suppression .Because of the additional features of these models, a much lighter gravitino mass isallowed. In this case, the two-loop beta functions are much smaller and we can easily geta large enough correction to ( g − µ to explain the experimental discrepancy. Since m / is much smaller, we are free to take large tan β . Here, we will take c = c = 1 / c = 2, m / = 30 TeV, and tan β = 35 and scan over m = m . We repeat this exercise for c = c = 3 /
4. The results of these scans can be seen in Fig. (10). As can be seen from thelower panel of Fig. (10), the anomalous magnetic moment of the muon can be sufficientlyenhanced with out tuning the c , c . Since the only parameters that are changing in these To evade baryon asymmetry washout, some model building is needed. See the review [37] m = m (TeV) ( G e V ) m = 30 TeVc = c = 0.5 c = 2tan b = 35 m L ~ m e ~ m Q ~ m u ~ m d ~ m / -40 -20 0 m = 30 TeVc = c = 0.75 c = 2tan b = 35 m L ~ m e ~ m Q ~ m u ~ m d ~ m / m = m (TeV) ( G e V ) D a m m = m (TeV) m = 30 TeVc = 2tan b = 35c = c = 0.75c = c = 0.5 Figure 10:
In the top left and top right panels we show the mass spectra with respect to m = m for m / = 30 TeV, c = 2 , tan β = 35 , and c = c = 0 . and c = c = 0 . respectively. The red + is for the left-handed slepton. The green × ’s are for the right-handedsleptons. The blue star is for the left-handed squarks. The magenta box is for the right-handed up squark. The cyan filled box is for the right-handed down squark. The yellow circleis for µ/ . In the lower panel, we have plotted a µ for the same sets of parameters. The red+’s are for c = c = 0 . and the green × ’s are for c = c = 0 . . c = c ( G e V ) m = 30 TeVm = m = 0 c = 2tan b = 35 m L ~ m e ~ m Q ~ m u ~ m d ~ m / c = c m = 30 TeVm = m = 0 c = 2tan b = 35 D a m Figure 11:
In the left panel, we show the mass spectra for m / = 30 TeV, c = 2 , tan β = 35 ,and m = m = 0 . The symbols are as in Fig. (10). In the right panel, we have plotted ∆a µ for the same set of parameters. figures are the Higgs boundary masses, the Higgs mass is relatively unchanged and about127 GeV.Lastly, we plot the mass spectra and anomalous magnetic moment of the muon withrespect to c = c , with c = 2, m / = 30 TeV, m = m = 0, and tan β = 35. In Fig. (11),we see that by varying c , c , we can easily get an anomalous magnetic moment consistentwith experiment. The recent discovery of the Higgs boson has placed rather severe constraints on simplemodels like the CMSSM. To get a reasonable Higgs mass ( m h >
124 GeV) in the CMSSM,supersymmetry breaking mass parameters must be pushed to order 1 TeV resulting in squarkand gluino masses of order 2 TeV. If however, there is a hierarchy between the sfermion massesand gaugino masses, such as in split-supersymmetry, PGM, and strong moduli stabilization,the observed Higgs mass can easily be accommodated. Furthermore, models such as PGMcan be made consistent with radiative electroweak symmetry breaking for a limited rangein tan β . In addition, models with strongly stabilized moduli tend to have a much simplercosmology, avoiding the problems of excess entropy production and/or gravitino production[28]. Indeed, for quite some time cosmological model building has suggested this hierarchy.Although simple models like PGM have many advantages, there are some drawbacks toheavy sfermions. If the squarks are heavy, detection at the LHC may be rather difficult.Furthermore, the deviation in ( g − µ has little hope of being explained, in this case. Sinceboth of these experimental difficulties hinge on the masses of the first two generations, whilethe Higgs mass depends primarily on the third generation masses, there may be hope ofsimultaneously getting all of these nice features. In fact, if the first and second generationmasses are generated at one-loop, with respect to m / while the third generation masses19emain at tree-level, both of these difficulties can be resolved. In these scenarios, the downsquark can be pushed below the gluino mass increasing the reach of the LHC for standard SU (5) based models. These models also allow the sfermions to be light while keeping µ oforder m / . If the theory stems from product unification or has no unification at all, ( g − µ can be explained, even for sleptons of order 1 TeV.A nice and simple way to generate these one-loop masses is through anomaly like contri-butions. If the regulated theory has Pauli-Villar fields which interact with the hidden sector,the theory will have one-loop masses generated by the gauge and Yukawa couplings [23]. Theinteractions of the Pauli-Villar fields with the hidden sector may be a natural part of stringtheory and by merely including this additional interaction at the Planck scale, we obtainone-loop masses. Since we are quite ignorant about what the universe is like at the Planckscale, this is an acceptable assumption.Lastly, we comment on the testable signatures of these models. One unique type ofspectrum that can come from the type PGM we considered is a down squark that is lighterthan the gluino. This unique spectra would result in an extended reach for the LHC andHLHC and would be fairly indicative of these types of models. If this form of PGM explainsthe deviation in the experimental value of ( g − µ , we also expect that the wino should beseen at the ILC. Otherwise, the sleptons would be too heavy to give a significant contributionto ( g − µ . Although, these signatures are not necessary, they would be highly suggestiveof this type of PGM model. A Off diagonal sfermion squared masses
In the split family scenarios, the model generically induces the FCNC processes throughthe flavor structure of the Yukawa coupling (see e.g. Ref. [38]). In our model, however,the FCNC contributions are suppressed since the soft masses in the first two generationsare mainly generated by the one-loop anomaly mediated contributions, and hence, are veryclose to each other.In the soft mass diagonalized basis, the mass terms and the supersymmetric Yukawainteraction terms are given by,
L (cid:39) m f ( | ˜ f | + | ˜ f | ) + m f | ˜ f | ,W = ¯ u iR Y uij Q jL H u + ¯ d iR Y dij Q jL H u + ¯ e iR Y eij L jL H u . (27)In this basis, we expect | Y u,d,eij | (cid:28) | Y u,d,e | , ( i (cid:54) = 3 o r j (cid:54) = 3) . (28)when we assume that the sfermion mass hierarchies are linked to the Yukawa couplinghierarchies. It should be noted that the left-right mixing soft masses are safely neglectedsince they are suppressed by the Higgs expectation value and by the small Yukawa couplings.First, let us discuss the flavor mixing effects in the first two generations at the tree-level.For that purpose, it is convenient to rotate the above scalar mass diagonal basis into the20o-called super-CKM basis by, u iL = U iju ˜ u Lj , d iL = U ijd ˜ d Lj , ¯ u iR = V iju ˜¯ u Rj , ¯ d iR = V ijd ˜¯ d Rj , (29)where the supersymmetric Yukawa couplings are diagonalized, V Tu Y u U u = Y u diag , V Td Y d U d = Y d diag . (30)The CKM matrix is given by, V CKM = U † u U d . In this basis, the soft squared masses have offdiagonal elements, m fij = m f δ ij + X ∗ i X j ( m f − m f ) , (31)and similarly (cid:16) m fij (cid:17) − = (cid:16) m f (cid:17) − δ ij + X ∗ i X j (( m f ) − − ( m f ) − ) , (32)where X = U u , U d , V ∗ u , V ∗ d . The mixing angles, X , X , are expected to be of O ( λ ) and O ( λ ) with the Wolfenstein parameter λ (cid:39) .
2, with respectively. Therefore, from the aboveexpression of m − ij , we see that the flavor mixing parameter in the first two generations is ofthe order of | X X | at the tree-level, and hence, is highly suppressed.Next, let us discuss the flavor violating effects from the RGEs. In a general flavor basis,the flavor dependent part of the RGEs of the soft masses are given by ddt m Q ij = 116 π (cid:104) ( m Q ik + 2 m H u δ ik ) Y u † k(cid:96) Y u(cid:96)j + ( m Q ik + 2 m H d δ ik ) Y d † k(cid:96) Y d(cid:96)j +( Y u † ik Y uk(cid:96) + Y d † ik Y dk(cid:96) ) m Q (cid:96)j + 2 Y u † ik m u k(cid:96) Y u(cid:96)j + 2 Y d † ik m d k(cid:96) Y d(cid:96)j (cid:105) , (33) ddt m u ij = 116 π (cid:104) (2 m u ik + 4 m H u δ ik ) Y u † k(cid:96) Y u(cid:96)j + 2 Y u † ik Y uk(cid:96) m u (cid:96)j + 4 Y u † ik m Q k(cid:96) Y u(cid:96)j (cid:105) , (34) ddt m d ij = 116 π (cid:104) (2 m d ik + 4 m H d δ ik ) Y d † k(cid:96) Y d(cid:96)j + 2 Y d † ik Y dk(cid:96) m d (cid:96)j + 4 Y d † ik m Q k(cid:96) Y d(cid:96)j (cid:105) . (35)In the super-CKM basis, by neglecting the Yukawa couplings in the first two generations,the above RGEs are reduced to ddt m Q ij (cid:39) π y t m Q y t m Q y t m Q y t m Q y t ( m H u + m Q + m u ) , Flavor independent RGE contributions are absorbed in the m f and m f , and hence, does not lead toan additional flavor mixing to the tree-level effects. dt m u ij (cid:39) π y t m u y t m u y t m u y t m u y t ( m H u + m Q + m u ) ,ddt m d ij (cid:39) π y b m d y b m d y b m d y b m d y b ( m H d + m Q + m d ) , Therefore, the soft squared mass matrices in Eq. (31) receive flavor-violating radiative cor-rections.By taking the inverse of the radiatively corrected soft mass squared at the low energyscale, we immediately find the radiatively induced flavor mixing parameters( δ d ) LL ∼ y t | X X | π (cid:32) m f m f (cid:33) log M input m ˜ f (cid:39) − (cid:18) X X − (cid:19) (cid:16) m ˜ f
100 TeV (cid:17) (cid:32) m ˜ f (cid:33) , (36)( δ d ) RR ∼ y b | X X | π (cid:32) m f m f (cid:33) log M input m ˜ f (cid:39) − (cid:18) X X − (cid:19) (cid:16) m ˜ f
100 TeV (cid:17) (cid:32) m ˜ f (cid:33) , (37)at the leading order. Here, we have estimated the radiative corrections in the leading logapproximation. We have also used y b (cid:39) . β (cid:39)
10. As a result, the RGEinduced FCNC contributions are also consistent with the constraints from the K − ¯ K mixing(∆ m K and ε ); (( δ d ) LL ( δ d ) RR ) / (cid:46) − ( m ˜ d / δ d ) LL (cid:46) − ( m ˜ d / B Important Contributions to the Beta Function
The important contributions to the RG running of the sfermion masses come from loops of D terms and so are proportional to gauge couplings. The pure gauge contributions are [40]∆ β g Y m f = Y g Y (16 π ) (cid:2) m + m ) + Tr (cid:0) m Q + 3 m L + 8 m u + 2 m d + 6 m e (cid:1)(cid:3) (38)∆ β g m f = 3 g (16 π ) (cid:2) m + m + Tr (cid:0) m Q + m L (cid:1)(cid:3) (39)∆ β g m f = 163 g (16 π ) Tr (cid:2) m Q + m u + m d (cid:3) . (40)The fourth and final important contribution comes from adding a loop to the one-loop D term diagrams of hypercharge. This contribution is [40] S (cid:48) = Tr (cid:104) − (3 m + m Q ) Y † u Y u + 4 Y † u m u Y u + (3 m − m Q ) Y † d Y d − Y † d m d Y d + ( m + m (cid:96) ) Y † e Y e − Y † e m e Y e (cid:3) + (cid:20) g + 310 g (cid:21) (cid:8) m − m − Tr( m (cid:96) ) (cid:9) + (cid:20) g + 32 g + 130 g (cid:21) Tr( m Q ) (41) − (cid:20) g + 1615 g (cid:21) Tr( m u ) + (cid:20) g + 215 g (cid:21) Tr( m d ) + 65 g Tr( m e ) , β S (cid:48) m f = 2 Y g (16 π ) S (cid:48) , (42)where g Y = (3 / g .Here we give a naive estimate of the size of these contributions to the sfermion masses if weassume the third generation masses dominate. We will also assume that the third generationmasses do not run. Although this is not true, it does give us a good order of magnitudeestimate for the size of these contributions. Since the Higgs soft mass running complicatesour approximation, we will focus on the SU (3) contribution. In this approximation, we have643 g (16 π ) m / . (43)Using the RGE for the gauge couplings we find that this give∆ m f = − π (cid:0) g ( µ ) − g ( µ ) (cid:1) m / , (44)which is of order one-loop. A similar but slightly less accurate calculation can be done foreach gauge group with similar results. C Toy Model with One-Loop Masses
Here, we provide a simple example of PV renormalization. This will be more of a sketchthen a detailed calculation since we will not discuss the η i which will multiply each of theloops and are important for the cancellation of infinities. These factors are needed to getthe exact coefficients of the one-loop masses. However, since this is not important to ourconsiderations we will not address this issue and mostly focus on the diagrams themselves.We will also use a supergraph mass insertion method, which is perfectly justified since weassume the supersymmetric masses are much larger than the SUSY breaking masses. Wewill start with the model K = Q † e V Q , (45)where the Q are matter fields, and V represents the gauge fields. and we take no superpo-tential for the physical fields. The K¨alher potential for the PV fields is K P V = Q (cid:48)† e V Q (cid:48) + ¯ Q (cid:48)† e V ¯ Q (cid:48) + Tr (Φ † e V Φ) + Tr ( ¯Φ † e V ¯Φ) , (46)and superpotential W P V = µ (cid:0) Q (cid:48) ¯ Q (cid:48) + Φ ¯Φ (cid:1) + g i √ Q (cid:48) T a Φ a Q , (47)with Φ = Φ a T a and the same for ¯Φ. Now we calculate the one-loop wave function renormal-ization and the PV one-loop contribution that renormalizes it. The important graphs can23e seen in Fig. (12) which is in supergraph notation. We have also suppressed the covariantderivatives. To be clear, we will include an x for mass insertions in the graphs when we arereferring to super propagators of the type (cid:104) Φ ¯Φ (cid:105) and no x when we mean the propagators ofthe type (cid:104) Φ † Φ (cid:105) .If we calculate these graphs, we find a total contribution of∆ K = 2 g i C ( r ) (cid:90) dθ d p (2 π ) Q † ( − p, ¯ θ ) Q ( p, θ ) (cid:0) B (cid:0) p , m Q , (cid:1) − B (cid:0) p , µ, µ (cid:1)(cid:1) , (48)where B (cid:0) p , m , m (cid:1) = i (cid:90) d k (2 π ) k − m p − k ) − m (49)Examining ∆ K , we see that this integral is indeed finite as long as µ is finite. Since we don’tcare about the details of this renormalization we will just leave it at that. Next we need toinclude SUSY breaking. Looking at the graphs in Fig. (12), we can get some insight in tohow the PV fields can act as messengers. First, we comment on our calculation of the one-loop mass for Q . In the example we are considering, ¯ Q (cid:48) is one of the field running in the loopwhich renormalizes the gauge interactions. If ¯ Q (cid:48) feels SUSY breaking differently than Q , thePV renormalization scheme would not work because there would be uncanceled infinities.The field Q (cid:48) , on the other hand, never shows up in the one-loop supergraph. Because of thisfact, the theory can be renormalized nor matter how Q (cid:48) feels supersymmetry breaking. Toshow this, we will calculate two more graphs. To incorporate SUSY breaking, we will modifythe K¨ahler potential to read K P V = (1 + θ m Q (cid:48) ) Q (cid:48)† e V Q (cid:48) + (1 + θ m Q (cid:48) ) ¯ Q (cid:48)† e V ¯ Q (cid:48) (50)+ (1 + θ m (cid:48) ) Tr (Φ † e V Φ) + (1 + θ m (cid:48) ) Tr ( ¯Φ † e V ¯Φ)Using these corrections to the K¨ahler potential as interactions in our graphs we get theadditional diagrams found in Fig. (13), which are again supergraphs. Φ a ¯ Q ′ Q Q † (a) PV Contribution Q QQV (b) SUSY Contribution
Figure 12: The Feynman Diagrams for Renormalizing Yang-Mills theory at one-loop.24ith a mass insertion of m Q (cid:48) , there is an additional contribution to one-loop renormal-ization of the Q of∆ K = 2 g i C ( r ) m Q (cid:48) (cid:90) dθ d p (2 π ) Q † ( − p, ¯ θ ) Q ( p, θ ) (cid:0) C (cid:0) p , µ , µ , µ (cid:1)(cid:1) (51)where C (cid:0) p , m , m , m (cid:1) = i (cid:90) d k (2 π ) k − m k k − m p − k ) − m (52)As can be seen from power counting, this is logarithmical divergent. Now, if Q had a SUSYbreaking mass of m Q (cid:48) , we would get another diagram from the physical fields which wouldcancel this contribution. Clearly, we need these masses to be equal or our PV regularizationdoesn’t work. Since we are considering the mass of the physical fields to be zero, this type ofdiagram will not appear. However, if we include a mass insertion of m Q (cid:48) things are different.Here, we need to change the propagators in the loop from (cid:104) ¯ Q † Q (cid:105) to (cid:104) Q ¯ Q (cid:105)(cid:104) Q (cid:48)† ¯ Q (cid:48)† (cid:105) m Q (cid:48) θ .Doing this we get a mass contribution∆ K = 2 g i C ( r ) m Q (cid:48) (cid:90) dθ d p (2 π ) Q † ( − p, ¯ θ ) θ Q ( p, θ ) (cid:0) C (cid:0) p , µ , µ , µ (cid:1)(cid:1) (53)where C (cid:0) p , m , m , m (cid:1) = i (cid:90) d k (2 π ) k − m m m k − m p − k ) − m (54) C ( p , µ , µ , µ ) is finite even in the limit µ → ∞ and can be easily evaluated giving∆ K = 2 g i C ( r ) m Q (cid:48) π (cid:90) dθ d p (2 π ) Q † ( − p, ¯ θ ) θ Q ( p, θ ) . (55) Φ a ¯ Q ′ Q Q † θ m i (a) If ¯ Q (cid:48) has a soft mass, likewise for Φ a Q Q † θ m i ¯ Q, Φ a Φ a , ¯ Q (b) If Q (cid:48) has a soft mass, likewise for ¯Φ a Figure 13: The Feynman Diagrams which could give mass at one-loop. (left) This Diagramis only allowed if the physical field also has the same soft mass. (right). This diagram givesa finite contribution and is always allowed. 25here is an identical diagram with a mass insertion in the Φ a line. Giving the same resultwith m Q (cid:48) → m . Summing these contributions to the mass of Q we find m Q = g i C ( r ) m Q (cid:48) + m π (56)This means we get a one-loop SUSY breaking soft mass no matter what the messengerscale is. Also, this is the exact one-loop contribution. Any additional insertions of the softmass in the diagram will lead to corrections of order m f /µ , which vanish when we take µ → ∞ . This is good since the PV fields mass should be taken to infinity in the end. Also,since the supersymmetric mass is much larger than the SUSY breaking masses for the PVfields, the sign of the soft mass does not need to be positive. This means that the one-loopmass can be positive or negative. This simple toy model gives results which are consistentwith those found in [23].Although, we have applied this calculation to PV fields, it does have broader implications.For example, if the PV fields were take as just additional GUT scale fields, they would stillgenerate one-loop masses if they interacted with hidden sector. Therefore, we can generateone-loop masses from physical fields at any scale using this type of set up. Acknowledgments
We would like to thank N.Yokozaki for a useful discussion on the FCNC problem. The workof J.E. and K.A.O. was supported in part by DOE grant DE–FG02–94ER–40823 at theUniversity of Minnesota. This work is also supported by Grant-in-Aid for Scientific researchfrom the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 22244021(T.T.Y.), No. 24740151 (M.I.), and also by the World Premier International Research CenterInitiative (WPI Initiative), MEXT, Japan.
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