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Best Fit to the Gluino Mass

arXiv:hep-ph/9205237v1 27 May 1992THE UNIVERSITY OF ALABAMAUAHEP924Best Fit to the Gluino MassL. CLAVELLI, P.W.

COULTER, and KAJIA YUANDepartment of Physics and AstronomyThe University of AlabamaBox 870324, Tuscaloosa, AL 35487-0324, USAABSTRACTAssuming that perturbative QCD is the dominant explanation for the narrowness ofthe vector quarkonia, we perform a χ2 minimization analysis of their hadronic decays asa function of two parameters, the mass of the gluino and the value of α3(MZ). A valuebelow 1 GeV for the gluino mass is strongly preferred.

Consequences for SUSY breakingscenarios are discussed.May, 1992

Recently it has been pointed out that the quarkonia data can be made consistent withminimal supersymmetric grand unification if the gluino and photino masses are below halfthe Z0 mass [1]. In addition, the quarkonia data becomes consistent with the world averagemeasurements of the strong fine structure constant, α3(MZ), if these masses are not abovethe Υ region.

The result of Ref. [1] was based on differences in the running of the strongcoupling constant in the presence of gluinos.

If, however the gluino mass is below half theΥ mass a second effect comes into play. Namely, the possibility for a quarkonium state todecay into gluino-containing final states affects the analysis of α3 at the relevant scale forthat decay.

In the current work we seek a best fit value to the gluino mass, m˜g, taking botheffects into account. This allows us to consider the full range of relevant gluino masses ina χ2 analysis and to determine the best fit value of the gluino mass m˜g.This work depends on the following basic assumption: The dominant explanation forthe narrowness of the quarkonia states including the J/ψ and φ is perturbative QCD.Any future models for the quarkonia data that rely on very large non-perturbativeeffects or relativistic corrections should be considered as alternative to the present analysisand judged on the basis of their relative physical plausibility.Small non-perturbativecorrections, of course, would only modify our results by small amounts.We analyze up to second order in QCD the hadronic decay rates of six quarkonia states,φ(1019), J/ψ(3097), J/ψ(3686), Υ(9460), Υ(10020), and Υ(10350).Ignoring possiblehigher order or non-perturbative effects and relativistic corrections each of these defineswithin errors the strong coupling constant at an appropriate scale µS(i)α3(µS(i)) = α3,i ± δi,(i = 1, .

. ., 6).

(1)Assuming the gluino mass, m˜g, lies above half the mass of the quarkonium state, thesevalues are independent of the gluino mass and are given in Ref. [1] using the 1992 branchingratio averages of the Particle Data Group [2].

In the current work we consider also lowervalues of m˜g so that the α3,i become functions of m˜g.If the gluino mass lies below half the quarkonium mass, there are decays into twogluinos (˜g˜g) and into two gluinos plus one gluon (˜g˜gg) which compete with the standardthree gluon (ggg) decay. These gluino-containing final states [3] however are suppressed byfour powers of mq/m˜q and are negligible for currently allowed values of the squark mass m˜q.The dominant gluino-related correction to the quarkonium width is therefore the two-gluonplus two-gluino (gg˜g˜g) final state with no intermediate squarks.

The corresponding decay1

rate as a function of the gluino to quark mass ratio m˜g/mq was written down in Ref. [4]by applying a color+spin correction factor to the rate for the ggq¯q decay as calculated inRef.

[5]. The result isΓ(3S1(¯qq) →gg˜g˜g)Γ(3S1(¯qq) →ggg) = 3α3(µS, m˜g)πR(r),(2)where r = 2m˜g/m(3S1) ≃m˜g/mq.

For small r (but r > 0.1),R(r) = −ln r +932(π2 −9)h−3.56 −r2(8 ln r + 0.95) + 2827π2r3+ r4(−163 ln2 r + 1.1 lnr −8.1) + O(r5)i.(3)Eq. (3) is infrared divergent as r →0.

A correct treatment [3] shows that R(0) ≈1.57.We use the above expression for R in region r > 0.1 and use a quadratic interpolatingpolynomial to join R smoothly to its value at r = 0. The results presented here are notsensitive to the exact form of the cutoff.

R(r) falls rapidly for increasing r. For r > 0.5 itis necessary to use the exact integral expression for R(r). It is possible [1] to choose thescale µS(i) so that the known (first order) standard model corrections to the 3 gluon decayrate vanish identically.

The hadronic decay rate into states of dissimilar quarks is thenΓ(3S1(¯qq) →hadrons) = Γ(3S1(¯qq) →ggg) + Γ(3S1(¯qq) →gg˜g˜g) + · · ·(4)We have thenα33,i(µS, m˜g)1 + 3α3,i(µS, m˜g)πR(r)= α33,i. (5)The gggg and ggq¯q correction to Eq.

(4) is absorbed into the three gluon term by thechoice of scale µS [1]. The coupling α3,i on the right hand side of Eq.

(5) is the resultobtained in Ref. [1] assuming no gluino contribution (m˜g > m(3S1)/2).Our analysis proceeds as follows.

For fixed value of the gluino mass we solve Eq. (5)to obtain six values of the strong coupling constant at six scales appropriate to each of thevector quarkonia.

The extrapolation to these scales from the Z0 mass is done using thethree loop renormalization group equation,Qdα3(Q)dQ= −α23(Q)2πha + b4π α3(Q) +c16π2 α23(Q)i,(6)2

wherea = 11 −23nf(Q)1 + 1110α14π + 92α24π−2n˜g(Q),(7a)b = 102 −383 nf(Q) −48n˜g(Q),(7b)c = 28572−503318 nf(Q) + 32554 n2f(Q). (7c)The effects of scalar quarks and Higgs bosons are neglected in accord with the decouplingtheorem.

The off-diagonal two loop effects are treated as electroweak corrections to theone-loop a coefficient. We neglect the running of the electroweak corrections, using insteadas an average valueαem = 1/133,sin2θW = 0.2333,(8)andα1 = 53αemcos2θW,α2 =αemsin2θW.

(9)The gluino contribution to the 3-loop c coefficient is not known. We keep the zero gluinocontribution (Eq.

(7c)) but monitor the effect of the c term as an estimate of unknownperturbative effects. In the fits of reasonable χ2, the contribution of the c term is small.We take into account the threshold dependence of quarks and gluinos according to theformulanf(Q) =6Xif Q24m2qi,(10a)n˜g(Q) = f Q24m2˜g,(10b)with [6]f(x) = 1 +12px(1 + x)lnh√1 + x −√x√1 + x + √xi.

(11)For each value of α3(MZ) and m˜g we can extrapolate from the Z0 mass down to thequarkonium region using Eq. (6) and calculate a χ2.The basic assumption given above does not require the total absence of non-perturbative effects.

In fact, in the case of the (most accurately measured) 1S quarkoniastates the graphs shown in Ref. [1] do exhibit some scatter in the data around the QCDpredictions at about the 10% level.

To find the preferred values of the gluino mass we3

follow two alternative customary procedures for the treatment of data in such cases. Theresults of these two procedures do not greatly differ.Method 1: discard the data point of worst agreement and minimize the χ2 of theremaining data points.

In the present case this is the J/ψ(1S) decay. The remaining fivevector quarkonia decays, which contain at least one entry from strange, charm, and bottomquarks agree well with the theory and provide a relatively sharp minimum χ2 as a functionof two parameters α3(MZ) and the gluino mass m˜g.Method 2: retain all six vector quarkonia decays but add in quadrature with theexperimental errors a “theoretical error” to take into account possible non-perturbative orbinding effects, i.e.α3(µS(i)) = α3,i(µS, m˜g) ±qδ2i + λ2i α23,i(µS, m˜g).

(12)On general grounds one would expect such corrections to be more important for the lighterquarkonia than for the Υ states. The λi parameterize our ignorance about higher orderand non-perturbative corrections.

They do not, of course, constitute a model for sucheffects (since they do not shift the central values of α3,i) and in fact no reliable modelexists apart from lattice QCD which has not as yet attained sufficient numerical accuracy.Clearly for sufficiently large λi all predictive power is lost. We take our basic assumptionto imply λi ≪1.

An adequate χ2 is found with λi = 0.05 for the bottomonium states,and λi = 0.10 for the charmonium and strangeonium vector states. Our conclusions arequalitatively insensitive to increasing this value for the lighter quarkonia in the sense thatthe gluino mass of minimum χ2 remains low although the χ2 values increase more slowlyaway from the minimum.

Similarly the favored light gluino also persists for smaller λialthough the minimum χ2 is then not a mathematically acceptable fit.For comparison with the best fits allowing a light gluino, we show in Fig. 1 the bestfit to the quarkonia data in method 2 assuming the gluino lies at high mass (400 GeV)where it essentially decouples.

This fit seems surprisingly good to the naked eye speciallyconsidering that it relates pure perturbative QCD to the hadronic decay rates of six vectorquarkonia of three species over mass scales varying by a factor of ten. However it is nota mathematically good fit since it corresponds to a χ2 per degree of freedom (χ2/DoF) of3.7.

In Fig. 2 we show the variation in the χ2/6 for this heavy gluino case as a function ofα3(MZ).

The minimum χ2 of 3.7 is several standard deviations worse than the best fitswith a light gluino. In addition this best fit corresponds to a value of α3(MZ) that is many4

standard deviations away from the world average value and is inconsistent with SUSYunification with a SUSY scale below 10 TeV. A similar attempt to fit the five quarkoniastates as in method 1 but without light gluinos would yield a minimum χ2 per degree offreedom many times larger than the minimum χ2 of Fig.

2.In Fig. 3 we show the χ2 contours for method 1 treating the gluino mass m˜g andα3(MZ) as variable.

The χ2/5 = 1 contours define two acceptable regions:α3(MZ) = 0.1135 ± 0.0005,m˜g = (0.32 ± 0.05) GeV;(13a)orα3(MZ) = 0.1145 +0.0013−0.0006,m˜g =0.01 +0.04−0.01GeV. (13b)The best fit corresponding to the central values of (13a) is shown in Fig.

4. Comparing withFig.

1, it is clear that the fit has improved due to decay of Υ states into gluino containinghadrons and due to the slower falloffof the coupling constant as a function of energy. In thesolution corresponding to (13b), even the φ decay has significant contribution from gluinocontaining final states.

In this case both gluinos must presumably hadronize into a singlepion where they can readily mix with gluon pairs. With light gluinos one must expectthat all hadrons have non-negligible gluino components just as there is a non-negligibleprobability to find strange quarks in the sea of non-strange hadrons.Fig.

5 shows the χ2 contours for method 2. In this method the χ2/6 = 1 contour lieswithin the region:α3(MZ) = 0.1115 +0.0018−0.0013 ,m˜g = (0.44 ± 0.17) GeV.

(14)More conservative values (90% CL) can be read from the χ2/DoF = 2 contours in Fig. 3or Fig.

5. In method 2 there is also a tendency for the χ2 to drop again toward zerogluino mass although in this case the χ2/DoF does not fall below 1 outside of the regionof Eq.

(14). The χ2/DoF < 2 region is defined by gluino masses less than 1.2 GeV.

The fitto the six vector quarkonia assuming the central values of Eq. (14) is shown in Fig.

6. Allthree values of α3(MZ) picked out by the quarkonia data with light gluinos are in excellentagreement with the world average value for this quantity:α3(MZ) = 0.113 ± 0.003(World Average) [7].

(15)5

There is at present no well established theory of supersymmetry breaking that wouldallow the unambiguous prediction of the gluino mass. Nevertheless, in the most realisticmodels that have been extensively studied, supersymmetry is softly broken, triggered by asuper Higgs mechanism in the hidden sector of some minimal N = 1 supergravity theories[8].

Therefore, the possible SUSY breaking scenarios in such models can be parameterizedin terms of only a few constants at the unification scale: the common gaugino mass m1/2,scalar mass m0, and the A and B parameters of dimension mass characterizing the cubicand quadratic soft-breaking terms that often exist as well.Although the analysis wepresented above is independent of any specific SUSY breaking models, it is tempting todiscuss the implication of our results to such models. For simplicity, we now consider sucha SUSY GUTs model which assumes the low energy form of the minimal supersymmetricextension of the standard model (MSSM) [9].

For our purpose, it is enough to considerthree soft-breaking parameters m1/2, m0 and A. It is interesting to note that, in thisframework, the low gluino masses favored by our analysis are natural if the dominantSUSY breaking seed is the universal scalar mass m0, i.e., m1/2 ≪m0.

Such a SUSYbreaking pattern has been supported by recent considerations of proton stability in thecontext of (non-flipped) SU(5) supergravity [10], and also favored by cosmological studies[11]. In fact, our results suggestm1/2 = 0.

(16)Such a model might be theoretically appealing since then SUSY breaking, like electroweakbreaking, finds its origin in the scalar sector. On the other hand, SUSY breaking scenarioswith m1/2 ̸= 0 would lead to quite large gluino masses [12].In the m1/2 = 0 scenario, generally, one expects a supersymmetric spectrum relativelylight compared to what one would get in other scenarios.

Besides the soft-breaking pa-rameters and the currently unknown top quark mass, one needs two more parameters inorder to specify the full spectrum: the ratio of the two Higgs vev’s tanβ ≡v2/v1, and theHiggs mixing parameter µ. In fact, all the gaugino masses then vanish at the tree level andthe gauginos only receive masses through radiative corrections which are naturally smallthough dependent upon the masses of other particles [13].In particular, the one-loopcorrections to the gluino and photino (which is an exact mass eigenstate in this scenario)6

masses are known, with the dominant contribution coming from graphs in which the topquark and its two superpartners circulate around the loop [13],δm˜g = α3(mt)8πmtFm2˜t1m2t,m2˜t2m2t,(17)δm˜γ = αem(mt)3πmtFm2˜t1m2t,m2˜t2m2t. (18)where m˜t1,˜t2 are the masses of the two scalar top quarks (m˜t1 < m˜t2), andF(x, y) = sin2θhx1 −x ln x −y1 −y ln yi.

(19)The actual gluino and photino masses up to one-loop order are given by the absolute valuesof these mass corrections. The θ in (19) is the mixing angle between the scalar partners ofleft- and right-handed top quarks, which rotates the ˜tL,R states into the mass eigenstates˜t1,2.

The overall factor sin2θ was omitted in Ref. [13], corresponding to the case wherethe difference between two diagonal terms of the 2 × 2 mass-squared matrix of the scalartop quarks can be neglected (see Eq.

33 below). The more general result presented herehas been recently calculated by one of us [14], and makes transparent the fact that theseone-loop mass corrections vanish exactly if there is no left-right mixing, even if ˜tL and ˜tR(now mass eigenstates themselves) have non-degenerate masses.

Since we are primarilyinterested here in the gluino mass, we neglect in Eq. (18) a contribution to the photinomass from the W ±-chargino loop diagrams [13].

A discussion of this contribution in itsgeneral form will be given elsewhere [14]. As shown below, with currently favored valuesfor the top quark mass mt and the two stop quark masses m˜t1, m˜t2, the function F is suchthat gluino masses below 1 GeV are quite natural.We now discuss the allowed regions for the relevant parameters in the m1/2 = 0 sce-nario.To simplify our approach we neglect the Yukawa couplings for all the fermionsalthough, strictly speaking, this approximation is only very good for the first two gener-ations.

As a result, the diagonal elements of the sfermion mass-squared matrix can bewritten as [15]m2˜f = m20 + m2f + M 2Zcos2βhT3,f −efsin2θWi. (20)Here we have included the D-term contributions as well.

The average mass-squared of thesfermions is seen to differ from the average mass-squared of the fermions by the parameterm20,Dm2˜fE= m20 +m2f. (21)7

The average mass-squared of the SUSY partners is approximately equal to the effectiveSUSY scale M 2S that enters into grand unification considerations. In Ref.

[1] (Eq. (3.11a))it was shown that the assumption of minimal SUSY unification with a light gluino (belowMZ/2) requires the approximate relationMS = 150 GeV × e−518.5(sin2θW −0.2336)e1.85(α3−1(MZ)−0.113−1).

(22)Equating M 2S withDm2˜fEand substituting the world average values of sin2θW and α3(MZ)from Ref. [7] yields within errors the following range for m0,75 GeV < m0 < 270 GeV(23)For the top quark mass, we will assume [22]92 GeV < mt < 147 GeV(24)In the m1/2 = 0 scenario, the tree-level masses of the two charginos, χ±1,2, are given bym2χ±1,2 = 12h2M 2W + µ2 ∓qµ4 + 4M 2W µ2 + 4M 4W cos22βi.

(25)From Eq. (25) and the requirement that the lightest chargino (χ±1 ) has to be heavier thanabout half the Z0 mass [16], it is found that tanβ is restricted from both sides, i.e.0.441 < tanβ < 2.266.

(26)And for each value of tanβ in the above range there is an upper limit on the absolute valueof the Higgs mixing parameter µ. Furthermore, the lower limit of 41 GeV on the mass ofthe light CP-even Higgs boson yields the additional constraints [17]0.55 < tanβ < 0.65;or1.5 < tanβ.

(27)Combining (26) with (27) yields two allowed ranges for the cos2β factor of Eq. (20),−0.674 < cos2β < −0.385;or0.406 < cos2β < 0.536.

(28)The bounds of Eq. (27) change with the experimental lower limit on the Higgs mass,becoming inconsistent with Eq.

(26) if this mass is required to be above 70 GeV. The8

allowed parameter space of the m1/2 = 0 model also requires that the mass of the lightestchargino be below MW .The mass-squares of the two scalar top quarks entering into Eqs. (17) and (18) aregiven bym2˜t1,˜t2 = 12hm2LL + m2RR ∓q(m2LL −m2RR)2 + 4m4LRi(29)with the diagonal elements (see Eq.

(20))m2LL = m20 + m2t + M 2Zcos2βh12 −23sin2θWi,(30)m2RR = m20 + m2t + M 2Zcos2βh23sin2θWi,(31)and offdiagonal elementm2LR = mtAt +µtanβ≡mtm0 ˜At. (32)Here we have introduced the dimensionless mixing parameter ˜At as a useful combinationof tanβ, µ and the low energy top soft-breaking parameter At.

If there is a single sourceof SUSY breaking corresponding to a single scale m0, we might expect values of ˜At to beeither zero or of the order unity. In terms of ˜At the mixing angle factor in Eq.

(19) is thensin2θ =−2m0mt ˜Atq(m2LL −m2RR)2 + 4m20m2t ˜A2t. (33)Imposing the experimental constraint m˜t1 > MZ/2 together with (23), (24) and (28)requires that ˜At < 3.2.

We also use for α3 the value at the top mass α3(mt) ≃0.1 inevaluating the gluino mass according to Eq. (17).

In Fig. 7 we show the range of gluinomasses predicted by Eq.

(17) as a function of ˜At for the allowed range of values of ˜At,m0, mt, and cos2β described above. The constraint from Eq.

(29) that the stop quarksbe above half the Z0 mass is also required in this allowed range. For each value of ˜Atthere exist an upper and lower limit on the gluino mass m˜g.

Values of ˜At near zero areconsistent with the near zero gluino masses of Eq. (13b).Values of ˜At near unity areconsistent with the χ2 minima of Eqs.

(13a) and (14). The entire range of values of ˜Atassuming Eq.

(23) yields gluino masses below 1.3 GeV in agreement with the quarkoniadata at the two-standard-deviation level. Assuming the result of Eqs.

(13) or (14), detailedpredictions for the individual squark and slepton masses within narrow ranges can be made.Much of the allowed parameter space in Fig. 7 predicts one or more of the scalar quarks9

and leptons to have a mass between MZ/2 and MZ. If the effect of a light gluino anda possible light squark is taken into account the anomalously large quoted values of α3coming from the Z0 hadronic decay can perhaps be reconciled with the world average value[18].

In addition, using α(mt) ≃1/127.9 in Eq. (18) the photino mass m˜γ would then beabout five times smaller than the gluino mass m˜g.

A stable photino of mass about 100 eVcould provide enough dark matter to close the universe [19]. More massive photinos would“overclose” the universe unless they could decay efficiently into photon plus gravitino orannihilate efficiently into photons.

Such ultralight photinos have also been discussed asthe explanation of other astrophysical observations [20]. On the other hand photinos ofsuch mass have been found to be disfavored [21] by data from Supernova 1987A unless thesquark masses are outside the preferred range of 60 GeV to 2.5 TeV.CONCLUSIONSWe have shown that the quarkonia data behave as if there were a gluino octet in theregion below 1 GeV.

Treating the data in either of two ways, such a light gluino is favoredby at least several standard deviations over the best QCD fits without a light gluino. Inaddition the fits without a light gluino are in conflict with SUSY unification and in dis-agreement with the world average values of α3(MZ) as discussed in Ref.

[1]. However, onemust remain aware of the usual possibility that any phenomenological fit to data could becoincidental.

We can not rule out the possibility that the real explanation for the behaviorof the quarkonia data might lie in relativistic binding corrections or other non-perturbativeeffects, although this would contradict the general assumption that the narrowness of thevector quarkonia is due to perturbative QCD and asymptotic freedom. Confirmation fromother independent data will certainly be required before the results presented here couldbe considered compelling.

We are presently investigating the possibility that supportingevidence may be present elsewhere in existing data or that definitive experimental testscan be proposed.In addition we should address the question as to whether such light gluinos are ruledout by current bounds. The strongest bounds on the masses of supersymmetric particlescome from the decay of the Z0.

Any particle with electroweak charge must lie above aboutone half the Z0 mass.Bounds on other particles such as gluinos or bounds from other processes are all toa greater extent model dependent. The status of these bounds is discussed in Ref.

[2].Although many lower bounds on the gluino mass above the mass region indicated here10

have been quoted, all of these are to some extent dependent on untested assumptions. Forexample, the stringent bounds from the hadron colliders on heavy gluinos allow windowsfor light gluinos below 50 GeV.

The low energy windows are well illustrated in Ref. [23].Ref.

[2] confirms the lack of unanimous opinion concerning whether or not very lightgluinos have been ruled out. Many of the purported bounds have obvious loopholes someof which are pointed out in Ref.

[1] and elsewhere [24]. For example the CUSB [25] boundthat disfavors gluino masses between 0.6 GeV and 2.2 GeV from the non-observation ofγ+gluinoball final states in Υ decay is strongly dependent on the value of the wave functionat the origin of the gluinoball for which only models can be made [26].

Although the gluinobehaves like a quark with a different color charge, it does have quartic couplings to gluonsand gluinos that do not affect quarkonia in the same way. The binding of gluinos into newhadrons is therefore more closely related to the binding of gluons into new hadrons whichis a very poorly understood area of hadronic physics at present.

Cosmological constraintson light photinos and gluinos are subject to similar uncertainties.Most of the rangeof photino masses between 100 eV and 2 GeV is disfavored by one or more cosmologicalarguments. However there is a window noted in Ref.

[27] for a photino in the mass rangefrom 4 to 15 MeV. The gluino would then be in the range from 20 to 75 MeV consistentwith Eq.

(13b). It is not clear whether there is sufficient uncertainty in the cosmologicalarguments to stretch this range by a factor of four to accommodate the results of ourEq.

(13a) or (14).Given the current situation we feel that prudence requires the adoption of a conser-vative, non-dogmatic attitude concerning the compatibility of light gluinos with existingdata.AcknowledgmentsK.Y would like to thank Jorge Lopez for a useful discussion. This work has been sup-ported in part by the U.S. Department of Energy under Grant No.

DE-FG05-84ER40141and by the Texas National Laboratory Research Commission under Grant No. RCFY9155.11

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Figure CaptionsFig. 1.Best fit to the 3-gluon decays of six quarkonia states assuming a gluino mass of400 GeV.

The experimental errors are increased according to “method 2”.Fig. 2.χ2/6 as a function of α3(MZ) for the 3-gluon decays of six quarkonia statestreated according to method 2 but assuming a heavy gluino (mass 400 GeV).Fig.

3.χ2 contours for the fit to 5 vector quarkonia according to “method 1” as a functionof α3(MZ) and the gluino mass m˜g.Fig. 4.Fit to the 5 vector quarkonia using the “best fit” values from Eq.

(13a).Fig. 5.χ2 contours for the fit to 6 vector quarkonia according to “method 2” as a functionof α3(MZ) and the gluino mass m˜g.Fig.

6.Best fit to the 6 vector quarkonia with the expanded errors of method 2. The fitparameters are given by the central values of Eq.

(14).Fig. 7.The allowed values of the gluino mass in the soft SUSY breaking picture withm1/2 = 0 are bounded by the closed figure shown here.

The range of allowedgluino masses at given ˜At corresponds to taking the full allowed range of theremaining parameters m0, mt, and tanβ.14

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