INR-TH/2015-031

Decaying light particles in the SHiP experiment. III.Signal rate estimates for scalar and pseudoscalar sgoldstinos

K. O. Astapov1, 2, ∗ and D. S. Gorbunov1, 3, †

1Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia2Physics Department, Moscow State University, Vorobievy Gory, Moscow 119991, Russia

3Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

For supersymmetric extensions of the Standard Model with light sgoldstinos—scalar and pseu-doscalar superpartners of goldstino—we estimate the signal rate anticipated at the lately proposedfixed target experiment SHiP utilizing CERN SPS beam of 400 GeV protons. We also place newlimits on the model parameters from the similar analysis of the published results of CHARM exper-iment.

I. INTRODUCTION

Probably, the low energy supersymmetry (SUSY) isthe mostly developed extension of the Standard Modelof particle physics (SM) [1, 2]. While, inherent in the su-persymmetry, a technically natural solution to the gaugehierarchy problem implies the SM superpartners to be ator below the TeV energy scale, other and much lighternew particles can exist as well. In particular, if supersym-metry is spontaneously broken at not very high energyscale (see models with gauge mediation of supersymme-try breaking [3, 4] as an example), the particles fromSUSY breaking sector may show up at quite low ener-gies. Their effective couplings to the SM particles areanticipated to be rather weak, therefore a high intensitybeam is required to test the model via production of thenew particles. The CERN Super Proton Synchrotron(SPS) provides with high intensity beam of 400 GeVprotons, and recently proposed beam-dump experimentSHiP (Search for Hidden Particles) [5] (see also [6, 7])can perform the task (see Ref. [8] for a comprehensivediscussion of the SHiP physics case).

The purpose of this paper is to estimate the signalrate expected at the SHiP experiment in supersymmet-ric models with sufficiently light particles of the Gold-stino supermultiplet. The latter contains goldstino (theNambu–Goldstone field, fermion) and its superpartners,scalar and pseudoscalar sgoldstinos. While goldstino isR-odd, sgoldstinos are R-even and hence can be singlyproduced in scatterings of the SM particles and can sub-sequently decay into the SM particles. Of a particularinterest are the sgoldstino decays into two electricallycharged SM particles. These decays yield the signaturewell recognizable at SHiP [5]: two charged tracks froma single vertex supplemented with a peak in the invari-ant mass of outgoing particles. Sgoldstino couplings tothe SM fields are inversely proportional to the param-eter of the order of squared scale of SUSY breaking inthe whole model. This unique feature of the Goldstinosupermultiplet allows to probe the SUSY breaking scale

∗ astapov@ms2.inr.ac.ru† gorby@ms2.inr.ac.ru

by hunting for the light sgoldstinos. A preliminary esti-mate of the sgoldstino signal for a particular productionmechanism and decay channel can be found in Ref. [8].Here we significantly extend that study by consideringboth scalar and pseudoscalar sgoldstinos and by investi-gating both flavor conserving and flavor violating sgold-stino couplings to the SM fermions, which cover varioussgoldstino production mechanisms and decay modes.

The paper is organized as follows. Sec. II containssgoldstino effective lagrangian. Sec. III and Sec. IV aredevoted to SHiP phenomenology of scalar and pseu-doscalar sgoldstinos, respectively. Here we discuss direct(III A, IV A) and indirect (III B, IV B) production mecha-nisms for flavor conserving and flavor violating sgoldstinocoupling patterns. We consider various decay channelsof scalar (III C) and pseudoscalar (IV C) sgoldstinos. Wepresent our results (III D, III D, IV B, IV C) as estimatesof the SHiP sensitivity to the SUSY breaking scale inparticular supersymmetric variants of the SM. In Sec. IIIand Sec. IV we also put new limits on the model param-eters by extending our analysis to the case of CHARMexperiment. We conclude in Sec. V by summarizing theobtained results.

II. SGOLDSTINO LAGRANGIAN

If supersymmetry exists, it is spontaneously broken inorder to be phenomenologically viable [1]. The breakinghappens when a hidden sector dynamics gives a nonzerovacuum expectation value F to an auxiliary componentof a superfield. Dimension of this parameter is masssquared, and

√F is of order of the SUSY breaking scale.

The fermion component of this superfield, goldstino G̃,becomes a longitudinal component of gravitino as a resultof the super-Higgs mechanism making gravitino massive.Goldstino superpartners, scalar S and pseudoscalar Psgoldstinos, remain massless at tree level and gain massesmS,P due to higher order corrections. The masses arelargely model-dependent and we keep them as free pa-rameters in (sub)GeV range for this study.

Interaction of the Goldstino supermultiplet with otherfields is suppressed by parameter F [9]. To the lead-ing order in 1/F sgoldstino coupling to SM gauge fields

arX

iv:1

511.

0540

3v1

[he

p-ph

] 1

7 N

ov 2

015

2

(photons Fµν , gluons Gaµν , where index a = 1, ...8 runs SU(3) color group) and matter fields (leptons fL, up-and down-quarks fU and fD) reads [10, 11]

Leff = − 1

2√

2F

(m2SS

¯̃GG̃+ im2PP

¯̃Gγ5G̃)− 1

2√

2

Mγγ

FSFµνFµν +

1

4√

2

Mγγ

FPεµνρσFµνFρσ −

1

2√

2

M3

FSGµν aGaµν

+1

4√

2

M3

FPεµνρσGaµνG

aρσ −

m̃LR 2Dij√2F

Sf̄DifDj− i

m̃LR 2Dij√2F

P f̄Diγ5fDj

−m̃LR 2Uij√2F

Sf̄UifUj− i

m̃LR 2Uij√2F

P f̄Uiγ5fUj

−m̃LR 2Lij√2F

Sf̄LifLj− i

m̃LR 2Lij√2F

P f̄Liγ5fLj

. (1)

Here M3 is gluino mass, Mγγ = M1 sin2 θW +M2 cos2 θWwith M1 and M2 being U(1)Y - and SU(2)W -gauginomasses and θW is the weak mixing angle, m̃LR 2

Uijand

m̃LR 2Dij

are left-right up and down squark soft mass terms.

Lagrangian (1) includes only single-sgoldstino terms;considered in Refs. [10, 12–14] two-sgoldstino terms aresuppressed by 1/F 2 and are less promising for testing atthe SHiP experiment. In the interesting here range ofF gravitino is very light and can be safely replaced byits goldstino component G̃, entering (1), whenever sgold-stino phenomenology at the beam-dump experiment isconsidered. Hence, sgoldstino couplings to the SM fieldsare proportional to the soft supersymmetry breaking pa-rameters of the Minimal Supersymmetric extension of theSM (MSSM).

Sgoldstinos also mix with neutral Higgs bosons as de-scribed in Ref. [15–17]: scalar sgoldstino S mixes withneutral light h and heavy H Higgs bosons, while pseu-doscalar P mixes with axial Higgs A. In what followswe account only for the first mixing, since the other twodo not change the sgoldstino phenomenology at SHiP forthe set of models we investigate. Mixing of sgoldstinoand lightest MSSM Higgs boson (SM-like Higgs) h canbe written as [17]

Lmixing =X

FSh , (2)

where the mixing parameter X is related to the hig-gsino mixing mass parameter µ, Higgs vacuum expecta-tion value (vev) v = 174 GeV, parameter tanβ describingthe Higgs vev’s ratio, SU(2)W and U(1)Y gauge couplingconstants g2 and g1 as follows

X = 2µ3v sin 2β +1

2v3(g21M1 + g22M2) cos2 2β . (3)

We consider sgoldstino S as to be much lighter than theSM-like Higgs boson of mass mh ≈ 125 GeV. Therefore,at low energies the above mixing ensures the Higgs-likecouplings between the scalar sgoldstino and all the otherSM fields. All the couplings are suppressed by the mixing

angle

θ = − X

Fm2h

. (4)

To illustrate the sensitivity of the SHiP experiment tosgoldstino couplings, in the next Sections we present nu-merical results for the set of values of MSSM parameters(the benchmark point in the MSSM parameter space)shown in Table I. It is an arbitrary choice, except we

M1, GeV M2, GeV M3, GeV µ, GeV100 250 1500 1000

tanβ Al, GeV AQ, GeV mQ, GeV6 1000 2800 1000

TABLE I. MSSM benchmark point.

suppose that all the model parameters take experimen-tally allowed values and the lightest Higgs boson mass is125 GeV. Trilinear soft supersymmetry breaking param-eters Al,Q are defined by the relations m̃LR 2

Dii≡ mDiAQ,

m̃LR 2Uij

≡ mUiAU , m̃LR 2

Lij≡ mLi

Al, where we use SM

fermion masses mDi,Ui,Li .

III. SCALAR SGOLDSTINO

In this Section we consider two different productionmechanisms of scalar sgoldstino relevant for the SHiPsetup. The first one is the direct production via hardgluon fusion in proton scatterings off the target material.The second one is the production in decays of mesonsemerging due to the proton scattering.

A. Gluon fusion

If sgoldstino is much heavier than QCD energy scaleof 100 MeV, it can be produced directly via gluon fu-sion. The relevant parts of the sgoldstino interaction la-

3

grangians (1), (2), (4) read

LSgg=

(θ g1−loophgg (mS)− αs(mS)β(αs(M3))

β(αs(mS))αs(M3)

M3

2√

2F

)×SGµν aGaµν ,

(5)

where the first term, associated with Higgs-sgoldstinomixing (2), is proportional to the Higgs effective couplingto gluons (appearing at 1-loop level via virtual quark ex-changes) [18],

g1−loophgg =3

4

αs(mS)

6√

2πv

(A1/2(τt) +A1/2(τb)+

+A1/2(τc) +A1/2(τs)). (6)

Here τi =4m2

i

m2h

and loop formfactors read

A1/2 = 2τ (1 + (1− τ)f(τ)) (7)

with

f(τ) =

{arcsin2 (1/

√τ) , τ ≥ 1 ,

− 14 log 1+

√1−τ

1−√1−τ , τ < 1 .

(8)

The factor in front of the second term in eq. (5) accountsfor the renormalization group evolution with β(αs) beingthe QCD β-function. Note that both terms in (5) are in-versely proportional to supersymmetry breaking param-eter F .

To obtain a reliable estimate of the direct scalar sgold-stino cross section σpp→S we properly rescale the resultsof Ref. [19], where the coupling similar to (5) is respon-sible for the light inflaton production at the fixed targetexperiment with 400 GeV proton beam. For the MSSMparamaters from Table I we find the following numericalapproximation to the cross section as a function of sgold-stino mass and supersymmetry breaking parameter,

log10(σpp→Sσpp,total

) = −15.8666− 0.93934×( mS

1GeV

)+ 0.02025×

( mS

1GeV

)2+ 0.00052×

( mS

1GeV

)3− 4 log10

( √F

100TeV

), (9)

where σpp,total is the total pp cross section for 400 GeVproton beam. This approximation is illustrated in Fig. 1for two reference values of

√F . Given the expected at

SHiP number of protons on target, about 2 × 1020 [5],one concludes from Fig. 1 that the direct production canprovide with sgoldstinos only in the models with super-symmetry breaking scale below 1000 TeV, if MSSM su-perpartner scale is in TeV range.

B. B meson decays

Scalar sgoldstinos of masses in GeV range are domi-nantly produced by decays of heavy mesons appearing

2 4 6 8 10

10-21

10-18

10-15

mS (P) , GeV

σpp

→S(P)/σpp,total

F = 1000 TeV F = 100 TeV

FIG. 1. Cross sections of scalar and pseudoscalar sgoldstinoproductions in gluon fusion as functions of sgoldstino mass.No difference between scalar and pseudoscalar cases is ex-pected.

by proton scatterings off target. In the context of SHiPexperiment the main source of sgoldstinos is decays of B-mesons. The process is described by the triangle diagramwith t-quark and W -bosons running in the loop. Sgold-stino is emitted by the virtual t-quark through sgoldstino-top-top coupling (1) and sgoldstino-Higgs mixing (2) (thelatter dominates for the values shown in Table I). Adopt-ing the same logic as one used in [19] for light inflatonwe calculate the branching ratio of B-meson decay intoS,

Br(B → XsS) =

= 3.4× 10−6 ×(

1− m2S

m2b

)2(100 TeV√

F

)4, (10)

where Xs stands for the strange meson channel mostlysaturated by a sum of pseudoscalar and vector kaons; mb

stands for b-quark mass. Scalar sgoldstino productioncross section is then a product of the branching ratioabove and the beauty cross section evaluated at the SHiPenergy scale as 1.6× 10−7 × σpp,total [5].

In Fig. 2 we compare the sgoldstino production crosssections provided by the direct and the indirect mecha-nisms for the same value of supersymmetry breaking pa-rameter,

√F = 100 TeV. One can observe from eqs. (9),

(10), that the both cross sections scale as ∝ 1/F 2. Thus,we conclude from the plot in Fig. 2 that the meson chan-nel dominates sgoldstino production when the kinematicsallows. In what follows we concentrate on this case andcomment on prospects of searches for the heavier sgold-stinos, MS(P ) & 4 GeV, (available only via the directproduction) in due course.

4

2.0 2.5 3.0 3.5 4.0 4.5 5.010-19

10-17

10-15

10-13

mS (P) , GeV

σpp

→S(P)/σpp,total

Gluon fusion B meson decays

FIG. 2. Cross sections of scalar sgoldstino production ingluon fusion and in B-meson decays for the model with√F = 100 TeV. The same results are valid for the pseu-

doscalar sgoldstino production.

C. Sgoldstino decay pattern

Sgoldstino is R-even and can decay into pairs of SMparticles, if it is kinematically allowed. For sgoldstino of(sub)GeV mass-range the main decay channels are γγ,e+e−, µ+µ−, π0π0, π+π−, K+K−, K0K̄0, see Ref. [10]for details.

The rate of sgoldstino decay into photons is describedby the following expression

Γ(S → γγ) =

(α(mS)β(α(Mγγ))

β(α(mS))α(Mγγ)

)2 m3S(P )M

2γγ

32πF 2. (11)

Here the dimensionless multiplicative factor accounts forthe renormalization group evolution of the photonic op-erator at different mass scales. For sgoldstino decays intoleptons one finds

Γ(S → l+l−) =m3SA

2l

16πF 2

m2l

m2S

(1− 4m2

l

m2S

)3/2

. (12)

If the scalar is light (mS < 1.5 GeV) then its decay intolight mesons is described by the effective interaction in-volving gluonic operator at low energy scale, as explainedin Ref. [10]. It yields the rates of sgoldstino decays intomesons, e.g.,

Γ(S → π0π0) =α2s(M3)

β2(αs(M3))

πmS

4

m2SM

23

F 2(1− β(αs(M3))

αs(M3)

9

4π

B0

mS

mu +md

mS

AQM3

)2√1−

4m2π0

m2S

, (13)

where β(αs) is QCD beta function, αs(M3) is strong cou-pling constant evaluated at the scale of M3, and param-eter B0 can be expressed via masses of kaon and quarksas follows, B0 = M2

K/(md +ms).

It turns out that for our benchmark point (see Table I)the contribution to (13) from the gluonic operator dom-inates over the quark operator contribution, so we have

Γ(S → π0π0) ≈ α2s(M3)

β2(αs(M3))

πm3SM

23

4F 2

√1−

4m2π0

m2S

, (14)

Γ(S → π+π−) = 2Γ(S → π0π0) , (15)

and analogously for kaons:

Γ(S → K0K̄0) ≈ 4α2s(M3)

β2(αs(M3))

πm3SM

23

4F 2

√1−

4m2K

m2S

, (16)

Γ(S → K+K−) = Γ(S → K0K̄0) . (17)

For light sgoldstinos we disregard multimeson final states.For the sgoldstino masses well above the QCD mass

scale (mS � 1 GeV) the decay into hadrons can be de-scribed as decay into gluon pair, which then hadronise.Its rate reads

Γ(S → gg) =

(αs(mS)β(αs(M3))

β(αs(mS))αs(M3)

)2m3SM

23

4πF 2. (18)

The multiplicative factor is added above to correct for therenormalization group evolution of the gluonic operatorwith mass scale.

The branching ratios of the scalar sgoldstino decaysfor the values of MSSM soft parameters chosen in Ta-ble I are shown in Fig. 3. Hadronic channels, ππ, KK,

0.0 0.5 1.0 1.5 2.010-6

10-5

10-4

0.001

0.010

0.100

1

mS , GeV

Br S

γγ e+e- μ+μ- ππ KK

FIG. 3. Branching ratios of scalar sgoldstino.

naturally dominate (if kinematically allowed), while γγand µ+µ− give small but noticeable contributions. Decayrates (11)–(17) ensure that each branching ratio scalesas the square of the corresponding soft supersymmetrybreaking parameter, e.g., Br(S → µ+µ−) ∝ A2

l .

5

Heavier sgoldstino decays mostly into gluons. The in-visible decay mode, S → G̃G̃ is always negligible, be-cause the corresponding coupling in (1) is proportionalto the sgoldstino mass squared rather than the MSSMsoft terms. The sgoldstino lifetime for a set of valuesof F is presented in Fig. 4. To reach the main detector

0.5 1.0 1.5 2.0

10-14

10-11

10-8

10-5

0.01

10

mS , GeV

τ,sec

F = 1000 TeV F = 100 TeV

FIG. 4. Lifetime of scalar sgoldstino as a function of its mass.

of the SHiP experiment, sgoldstino has to cover a dis-tance of about hundred meters [5]. The results in Fig. 4suggest that SHiP can be sensitive mostly to the modelswith supersymmetry breaking scale of about 100 TeV andhigher. The lifetime scales as τ ∝ F 2 and as τ ∝ 1/M2

3 ,since the hadron channel dominates.

D. Sgoldstino signal event rate at the SHiP

Now we collect all the ingredients required to achievethe main goal of this article, the estimate of the num-ber of sgoldstino decay events inside the fiducial volumeof the SHiP experiment. The SHiP construction is out-lined in Ref. [5]. The 400 GeV proton beam fueled bythe SPS hits the target and produces bunches of mesons,which can decay into new particles (sgoldstinos in thecase at hand). The latter can also appear directly fromthe proton-proton collisions (through the gluon fusion).The detector is placed at the distance of lsh = 63.8 mfrom the target. The vacuum vessel length is aboutldet = 60 m. It forms a cylinder along the beam axiswith the elliptical base of 5 m×10 m. The trajectoriesof electrically charged particles emerging from the newparticle decays can be traced in the detector volume andtheir energies and types can be determined by the regis-tration system utilizing devices arranged at the far endof the detector.

Differential production cross section of sgoldstinos,originated from the B-meson decays, has the following

form,

d3σpp→S(P )

dpdθpdφp=

∫d3~k f(~p,~k)

d3σBdkdθkdφk

, (19)

where f(~p,~k) is the sgoldstino momentum distribution

normalized to the branching ratio (10) and d3σB

dkdθkdφkis

the differential production cross section of B-mesons inproton-proton collisions, that is evaluated along the linesof Ref. [20]. In the integral (19) the total value of 3-momenta and escaping angle of the outgoing particles arespecifically constrained to ensure that the sgoldstino tra-jectory crosses the rear end of the SHiP detector, whichdefines the fiducial volume.

With the above approximations we estimate the num-ber of signal events as

Nsignal =NPOT

σpp,total

∫wdet

dσpp→S(P )

dpdθpdφpd3~p , (20)

where the expected number of protons on the target isNPOT = 2 × 1020 [8] and wdet denotes the probabilityfor sgoldstino to decay inside the fiducial volume of thedetector,

wdet(ES(P ),mS(P ),√F ) = exp(−lsh/γcτS(P ))×

×[1− exp(−ldet/γ(ES(P ))cτS(P ))

], (21)

with the sgoldstino gamma factor γ(ES(P )) =ES(P )/mS(P ).

In Fig 5 we indicate the region in the model parameter

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.2

0.4

0.6

0.8

1.0

mS , GeV

F-1/2,(100TeV

)-1

FIG. 5. The shaded region will be probed at the SHiP exper-iment.

space (mS(P ), 1/√F )), where the number of sgoldstino

decays inside the SHiP fiducial volume exceeds three,

6

Nsignal > 3. That is if no events were observed (the back-ground for the two-body decays into charged SM parti-cles is zero [5]) the region is excluded at the confidencelevel of 95%, in accordance with the Poisson statistics.The upper boundary in Fig. 5 is the region where thesgoldstino coupling constants ∝ 1/F are large enoughto initiate very fast decay of sgoldstino before it reachesthe detector. The lower boundary in Fig. 5 is the re-gion where the couplings are so small that let sgoldstinoescape from the detector without decay. The numberof signal events here scales with the model parametersas Nsignal ∝ M2

3µ6/F 4. The region in Fig. 5 of heaviest

sgoldstino reachable at SHiP, mS ≈ 3.6 GeV, is the meet-ing point of the lower and the upper boundaries. Heresgoldstino decay length is about 100 m, that is the scaleof both the SHiP detector length and the distance fromthe target, γcτ ∼ ldet ∼ lsh. In this case the number ofsignal events scales as Nsignal ∝ µ6/F 2. The scalings ofthe signal events imply that models with higher (as com-pared to that presented in Fig. 5) scale of supersymmetrybreaking can be tested if MSSM parameters µ, M3 areappropriately bigger (as compared to those presented inTable I).

Sgoldstinos of masses 3.6–4.2 GeV, which can be pro-duced through B-meson decays, see Figs. 2, 5, seem to bebeyond the SHiP’s grip for our choice of MSSM param-eters presented in Table I. However the explained abovesignal scaling with model parameters suggests that sgold-stinos of masses above 3.6 GeV can be tested at SHiPin models with higher scale of SM superpartners. Fi-nally, from the results presented in Figs. 2, 4 one canconclude that sgoldstinos of masses above 4 GeV, whichcan appear only via direct production, can not be testedat SHiP. Both sgoldstino production and decay are gov-erned by the same ratio M2

3 /F2, and sgoldstino lifetime

τ ∝ 1/m3S is too short for the reasonably high sgoldstino

production. One needs much higher intensity of the pro-ton beam at SPS to probe this region of model parameterspace.

E. Flavor violating

In supersymmetric models with non-diagonal sfermionleft-right mass terms, m̃LR 2

Dij∝/ δij , etc, sgoldstino cou-

plings (1) violate flavor symmetry. These flavor violatingterms are additional sources of sgoldstino production in abeam-target experiment. To illustrate the point, here weconsider flavor-violating decays of (produced by protonson target) B- and Ds-mesons into kaons and light scalarsgoldstino. These processes are governed by the soft pa-rameters m̃LR 2

D23and m̃LR 2

U12. Observation of oscillations

in the B0 − B0 and D0 − D0 systems and searches forvery rare (within the SM) decays like B → µ+µ− con-strain [21, 22] possible flavor violation in squark sector,which for our reference point given in Table I imposes the

following upper limits on the off-diagonal entries,

m̃LR 2D23

< 0.02 TeV2, m̃LR 2U12

< 0.016 TeV2. (22)

The above flavor violating interaction terms yield thedecay rates:

Γ(B → KS) =

= F 2B→K

(m2B −m2

K

mb +mu

)2λB→KS16π2m3

B

m̃LR 4D23

F 2(23)

and

Γ(Ds → KS) =

= F 2Ds→K

(m2Ds−m2

K

mc +mu

)2λDs→KS

16π2m3Ds

m̃LR 4U12

F 2, (24)

where

λB(Ds)→KS =√

(m2B(Ds)

−m2K −m2

S)2 − 4m2B(Ds)

m2K ,

and dimensionless form factors FB→K and FDs→K aregiven in Ref. [23].

To illustrate the sensitivity of the SHiP experiment tothe flavor violating sgoldstino couplings we take the softparameters m̃LR 2

U12and m̃LR 2

D23to be equal to their upper

bounds (22) and other MSSM parameters as in Table I.Then, treating the flavor violating meson decays as themain sources of sgoldstinos and repeating the analysis ofSec. III D we obtain the expected exclusion plots for theSHiP experiment, see Fig. 6. Similar to Fig. 5, the lowermargins refer to the limit of tiny couplings of sgoldstinoto the SM fields. The numbers of sgoldstino decays herescale with the relevant model parameters as Nsignal ∝m̃LR 4D23(U12)

M23 /F

4.

Note that sgoldstinos can be directly searched for inthe appropriate decay modes of the heavy mesons, forexamples see Refs. [10, 14, 27, 28]. In particular, theprocess B → Ks + S(P ) with sgoldstino subsequentlydecaying outside the detector can mimic the process B →h(∗) + missing, which branching is presently constrainedas [24]

Br(B → K0Sνν̄) < 9.7× 10−5 . (25)

At the chosen value of m̃LR 2D23

sgoldstino’s contribution (ifany) must be smaller than this limit. Similar requirementexists for the twin process with charged mesons. Thus,the region above the horizontal line in Fig. 6 (top panel)is excluded by the constraint (25). This upper limit on

F−1/2 scales as ∝ Br1/4/m̃LRD23

.Comparing Fig. 6 with Fig. 5 one concludes that SHiP

exhibits higher sensitivity to the supersymmetric modelswith flavor violation. We extend the performed analysisto the CHARM experiment, which operated on the sameproton beam at CERN, but collected much lower statis-tics and was placed at much larger distance from the tar-get as compared to the SHiP. Nevertheless, we find thatin the case of flavor-violating sgoldstino coupling we canexclude a part of the model parameter space, see Fig. 6,given the negative results of searches at CHARM [25, 26].

7

1 2 3 4 5

0.005

0.010

0.050

0.100

mS , GeV

F-1/2,(100TeV

)-1

0.4 0.6 0.8 1.0 1.2 1.4 1.60.001

0.005

0.010

0.050

0.100

0.500

1

mS , GeV

F-1/2,(100TeV

)-1

FIG. 6. The shaded regions are expected to be excluded (95%C.L.) at the SHiP experiment, if the flavor violating soft pa-rameters m̃LR 2

D23(top panel) and m̃LR 2

U12(bottom panel) take

their present upper limits for the benchmark point in Table I.The region above the solid horizontal line on top panel isexcluded due to the negative result in searches for the three-body decay B → K0

sνν̄ [24]. The light shaded regions areexcluded by our analysis of the results of the CHARM exper-iment [25, 26].

IV. PSEUDOSCALAR SGOLDSTINO

If parity is (strongly) violated in the sfermion sector ofMSSM, sgoldstino couplings to the SM fermions violateit too. Then scalar and pseudoscalar sgoldstinos are verysimilar as regards the SHiP phenomenology. However,if sgoldstino couplings conserve parity (that takes place,e.g., in left-right extensions of MSSM), phenomenology ofpseudoscalar and scalar sgoldstino are different in some

aspects, see Refs. [10, 27] for details. In this Section weinvestigate SHiP sensitivity to the pseudoscalar sgold-stino couplings.

A. Light pseudoscalar production

Pseudoscalar sgoldstino P can be directly producedvia the gluon fusion with the same cross section as scalarsgoldstino, hence Fig. 1 is valid for both cases. Howeverits production through the meson decays is somewhatdifferent from that of the scalar sgoldstino because of theabsence of mixing with light MSSM Higgs.1

Light pseudoscalar can be produced in B-meson de-cays. The corresponding 1-loop diagram is very simi-lar to that in the case of scalar sgoldstino discussed inSec. III B, but only sgoldstino-top-top pseudoscalar cou-pling (1) contributes. Given the pseudoscalar nature ofP , for the two-body decay it is accompanied by the vectorkaon K∗. The decay rate can be obtained by replacingproperly the coupling constants in the result presentedin Ref. [29] for the case of decay into axion,

Γ(B → K∗P ) =G2Fm

2t

213π3

mRLU33

4

2F 2X̂2A

20λ

3B→K∗

m3B

, (26)

where

λB→K∗ =√

(m2B −m2

P −m2K∗)2 − 4m2

Pm2K∗ , (27)

and

X̂ = 4− 3m2W

m2t −m2

W

− 3m2W (m2

W − 2m2t )

(m2t −m2

W )2log

(m2t

m2W

), (28)

the dimensionless form factor A0 can be found inRef. [30]. Though the formula for the decay rate some-what differs from that in the case of scalar, they givevery close numerical results, so the production rates arealmost the same, and Fig. 2 refers to the both cases.

B. Decay

In contrast to scalar S pseudoscalar sgoldstino P doesnot decay into a meson pair. However it mixes with pseu-doscalar mesons π and η, as explained in Ref. [10]. Sincemesons exhibit four-meson coupling, pseudoscalar sgold-stino can decay into three mesons through the virtualmeson state, P → π0∗/η∗ → 3 mesons.

1 There is a mixing with CP-odd Higgs A0, which is negligiblysmall for our choice of the benchmark point in Table I.

8

Inherent in the Chiral perturbation theory the four-meson interaction reads [31]

Leff =1

12f2πTr(Φ

←→∂ ΦΦ

←→∂ Φ), (29)

where

Φ =

1√2π0 + 1√

6η π+ K+

π− − 1√2π0 + 1√

6η K0

K− K̄0 − 2√6η

. (30)

It induces 4-meson operators Oπ0(η)

4 and Oη4 responsible

for the interesting transitions π0∗/η∗ → 3 mesons.Matrix elements of the initial off-shell π0 meson (with

squared 4-momentum m2π0∗) and three on-shell mesons

in the final state are following,

〈π0∗|Oπ0

4 |π0π0π0〉 =1

6f2(5m2

π0∗ − 3m2π0) , (31)

〈π0∗|Oπ0

4 |π0ηη〉 =2

3f2(2m2

π0∗ +m2π0 − 2m2

η) , (32)

〈π0∗|Oπ0

4 |π0KK〉 =2

3f2(2m2

π0∗ +m2π0 − 2m2

K) , (33)

〈π0∗|Oπ0

4 |π0π+π−〉 =2

3f2(2m2

π0∗ +m2π0 − 2m2

π+) . (34)

One obtains similar expressions for the initial virtual η-meson.

Finally we find for the pseudoscalar decay rate into 3mesons:

Γ(P → π0∗/η∗ → 3 mesons) =

=f2ππ

2m4πε

2

4(m2P −mπ)2

M23

F 2Γ(π∗ → 3 mesons)+

+f2ππ

2m4η

4(m2P −m2

η)2M2

3

F 2Γ(η∗ → 3 mesons), (35)

where 3-body decay widths Γ(π∗(η∗) → 3 mesons) arecalculated using the matrix elements (31) - (34) (andsimilar for η-meson) assuming that squared 4-momentaof the off-shell π∗ and η∗ equal m2

P ; we also make use ofε = (mu −md)/(mu +md).

Pseudoscalar sgoldstino decay rates into photons andleptons have the forms (11) and (12), respectively, withthe obvious replacement mS → mP .

The pseudoscalar sgoldstino lifetime and its relevantdecay branching ratios are presented in Fig. 7 and Fig. 8,correspondingly. Hadronic channels, mainly 3π, ηKK,ηππ and 3η dominate sgoldstino decay. Given its geome-try, the SHiP experiment is sensitive to the supersymme-try breaking scales of about

√F ∼ 100 TeV and above.

Performing for the pseudoscalar case the same proce-dure as that adopted in Sec. III D for the scalar sgold-stino we estimate the SHiP sensitivity to the pseudoscalarsgoldstino interaction. In Fig. 9 we present the plot dis-playing the region in the parameter space (F−1/2,mS)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

10-13

10-10

10-7

10-4

0.1

100.0

mP , GeV

τ,sec

F = 1000 TeV F = 100 TeV

FIG. 7. Pseudoscalar sgoldstino lifetime for√F =

100, 1000 TeV (lines from bottom to top).

0.0 0.5 1.0 1.5 2.010-6

10-5

10-4

0.001

0.010

0.100

1

mP , GeV

Br P

γγ e+e- μ+μ- 3π 0 π 0ηη

π 0π +π - π 0KK 3η 2π 0η ηKK

FIG. 8. Branchings ratios of pseudoscalar sgoldstino.

expected to be explored with the SHiP experiment. Oneobserves, that in the case of pseudoscalar sgoldstino theSHiP is sensitive to models with lower SUSY breakingscale

√F as compared to the scalar sgoldstino, cf. Figs.5,

9.

C. Flavor violation

The study of the flavor violating pseudoscalar sgold-stino coupling is very similar to that of the scalar sgold-stino. These couplings induce two-body heavy mesondecays into the pseudoscalar sgoldstino and a light vec-tor meson (it replaces pseudoscalar meson in the case ofscalar sgoldstino in the final state). For numerical esti-mates we adopt the same patterns of the flavor violatingcouplings as used in the case of scalar sgoldstino. The fi-nal estimates of the SHiP sensitivity to the pseudoscalarsgoldstino are presented in Fig. 10.

9

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.2

0.4

0.6

0.8

1.0

mP , GeV

F-1/2,(100TeV

)-1

FIG. 9. The shaded region can be explored at the SHiP ex-periment.

V. CONCLUSIONS

We have estimated sensitivity of the SHiP experimentto supersymmetric extensions of the SM where sgoldsti-nos are light. Proposed experiment will be able to probethe supersymmetry breaking scale

√F up to 103 TeV for

the model without flavor violation and up 105 TeV forthe model with flavor violating parameters as large asthe corresponding present experimental upper bounds.We have also compared the regions of the sgoldstino pa-rameter space to be probed at SHiP experiment with theregions, which we have excluded from the analysis of theresults of CHARM experiment [25, 26]. The regions areoutlined in Figs. 6 and Fig. 10: as one can see the SHiPexperiment will significantly extend our ability in testingsupersymmetric models with light sgoldstinos.

In this paper we concentrate mostly on the models withsgoldstino masses in the range 0.4–4 GeV, where sgold-stino can be kinematically produced in decays of charmand beauty mesons. Lighter sgoldstinos can in additionbe produced at the SHiP by strange meson decays. Thispossibility deserves a special study beyond the scope ofthis paper. Sgoldstinos can also be produced in decaysof secondary mesons from the hadronic cascade develop-ing in the target. These mesons are numerous, but lessenergetic. The proper account of this contribution re-quires numerical simulations of the hadronic cascade anda more accurate treatment of the SHiP geometry.Acknowledgments We thank I. Timiryasov and

S. Demidov for valuable discussions. The work was sup-ported by the RSF grant 14-22-00161.

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0.4 0.6 0.8 1.0 1.2

0.005

0.010

0.050

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0.500

1

mP , GeV

F-1/2,(100TeV

)-1

1 2 3 4 5

0.005

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mP , GeV

F-1/2,(100TeV

)-1

FIG. 10. The shaded regions can be explored with the SHiPexperiment in the case of nonzero soft parameters m̃LR 2

U12(up-

per panel) and m̃LR 2D23

(lower panel). The region above thesolid horizontal line on the lower panel is excluded due to neg-ative result in searches for the three-body decay B → K0

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