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005

CERN–PH–TH/2005–232DESY 05–242

FERMILAB–PUB–05–524–TKEK–TH–1054

SLAC–PUB–11579

Supersymmetry Parameter Analysis: SPA Conventionand Project

J.A. Aguilar-Saavedra1, A. Ali2, B.C. Allanach3, R. Arnowitt4, H.A. Baer5, J.A. Bagger6, C. Balazs7a,V. Barger8, M. Barnett9, A. Bartl10, M. Battaglia9, P. Bechtle11, G. Belanger12, A. Belyaev13, E.L. Berger7,G. Blair14, E. Boos15, M. Carena16, S.Y. Choi17, F. Deppisch2, A. De Roeck18, K. Desch19, M.A. Diaz20,A. Djouadi21, B. Dutta4, S. Dutta22,11, H. Eberl23, J. Ellis18, J. Erler24 b, H. Fraas25, A. Freitas26,T. Fritzsche27, R.M. Godbole28, G.J. Gounaris29, J. Guasch30, J. Gunion31, N. Haba32, H.E. Haber33,K. Hagiwara34, L. Han35, T. Han8, H.-J. He36, S. Heinemeyer18, S. Hesselbach37, K. Hidaka38, I. Hinchliffe9,M. Hirsch39, K. Hohenwarter-Sodek10, W. Hollik27, W.S. Hou40, T. Hurth18,11 c, I. Jack41, Y. Jiang35,D.R.T. Jones41, J. Kalinowski42d, T. Kamon4, G. Kane43, S.K. Kang44, T. Kernreiter10, W. Kilian2,C.S. Kim45, S.F. King46, O. Kittel47, M. Klasen48, J.-L. Kneur49, K. Kovarik23, M. Kramer50, S. Kraml18,R. Lafaye51, P. Langacker52, H.E. Logan53, W.-G. Ma35, W. Majerotto23, H.-U. Martyn54,2, K.Matchev55,D.J. Miller56, M. Mondragon24 b, G. Moortgat-Pick18, S. Moretti46, T. Mori57, G. Moultaka49, S. Muanza58,M.M. Muhlleitner12, B. Mukhopadhyaya59, U. Nauenberg60, M.M. Nojiri61, D. Nomura13, H. Nowak62,N. Okada34, K.A. Olive63, W. Oller23, M. Peskin11, T. Plehn27 c, G. Polesello64, W. Porod39,26e, F. Quevedo3,D. Rainwater65, J. Reuter2, P. Richardson66, K. Rolbiecki42 d, P. Roy67, R. Ruckl25, H. Rzehak68, P. Schleper69,K. Siyeon70, P. Skands16, P. Slavich12, D. Stockinger66, P. Sphicas18, M. Spira68, T. Tait7, D.R. Tovey71,J.W.F. Valle39, C.E.M. Wagner72,7, Ch. Weber23, G. Weiglein66, P. Wienemann19, Z.-Z. Xing73, Y. Yamada74,J.M. Yang73, D. Zerwas21, P.M. Zerwas2, R.-Y. Zhang35, X. Zhang73, S.-H. Zhu75

1 Departamento de Fisica and CFTP, Instituto Superior Tecnico, Lisbon, Portugal2 Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany3 DAMTP, University of Cambridge, Cambridge, UK4 Department of Physics, Texas A&M University, College Station, TX, USA5 Department of Physics, Florida State University, Tallahassee, FL, USA6 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA7 High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA8 Department of Physics, University of Wisconsin, Madison, WI, USA9 Lawrence Berkeley National Laboratory, Berkeley, CA, USA

10 Institut fur Theoretische Physik, Universitat Wien, Wien, Austria11 Stanford Linear Accelerator Center, Stanford, CA, USA12 Laboratoire de Physique Theorique, Annecy-le-Vieux, France13 Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA14 Royal Holloway University of London, Egham, Surrey, UK15 Skobeltsyn Institute of Nuclear Physics, MSU, Moscow, Russia16 Fermi National Accelerator Laboratory, Batavia, IL, USA17 Department of Physics, Chonbuk National University, Chonju, Korea18 PH Department, CERN, Geneva, Switzerland19 Physikalisches Institut, Universitat Freiburg, Freiburg, Germany20 Physics Department, Universidad Catolica de Chile, Santiago, Chile21 LAL, Universite de Paris-Sud, IN2P3-CNRS, Orsay, France22 University of Delhi, Delhi, India23 Institut fur Hochenergiephysik, Osterreichische Akademie der Wissenschaften, Wien, Austria24 Instituto de Fisica, UNAM, Mexico, Mexico25 Institut fur Theoretische Physik und Astrophysik, Universitat Wurzburg, Wurzburg, Germany26 Institut fur Theoretische Physik, Universitat Zurich, Zurich, Switzerland27 Max-Planck-Institut fur Physik, Munchen, Germany28 Centre for High Energy Physics, Indian Institute of Science, Bangalore, India29 Department of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece30 Facultat de Fisica, Universitat de Barcelona, Barcelona, Spain31 Department of Physics, University of California, Davis, CA, USA32 Institute of Theoretical Physics, University of Tokushima, Tokushima, Japan33 Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA, USA34 Theory Division, KEK, Tsukuba, Japan

2

35 Department of Modern Physics, University of Science and Technology of China, Hefei, China36 Center for High Energy Physics and Institute of Modern Physics, Tsinghua University, Beijing, China37 High Energy Physics, Uppsala University, Uppsala, Sweden38 Department of Physics, Tokyo Gakugei University, Tokyo, Jpan39 Instituto de Fısica Corpuscular, CSIC, Valencia, Spain40 Department of Physics, National Taiwan University, Taipei, Taiwan41 Department of Mathematical Sciences, University of Liverpool, Liverpool, UK42 Institute of Theoretical Physics, Warsaw Univerity, Warsaw, Poland43 MCTP, University of Michigan, Ann Arbor, MI, USA44 School of Physics, Seoul National University, Seoul, Korea45 Department of Physics, Yonsei University, Seoul, Korea46 School of Physics and Astronomy, University of Southampton, Southampton, UK47 Physikalisches Institut der Universitat Bonn, Bonn, Germany48 Laboratoire de Physique Subatomique et de Cosmologie, Universite Grenoble I, Grenoble, France49 LPTA, Universite Montpellier II, CNRS-IN2P3, Montpellier, France50 Institut fur Theoretische Physik, RWTH Aachen, Aachen, Germany51 Laboratoire de Physique des Particules, Annecy-le-Vieux, France52 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA53 Department of Physics, Carleton University, Ottawa, ON, Canada54 I. Physikalisches Institut der RWTH Aachen, Aachen, Germany55 Department of Physics, University of Florida, Gainesville, FL, USA56 Department of Physics and Astronomy, University of Glasgow, Glasgow, UK57 ICEPP, University of Tokyo, Tokyo, Japan58 IPN Universite Lyon, IN2P3-CNRS, Lyon, France59 Harish-Chandra Research Institute, Allahabad, India60 University of Colorado, Boulder, CO, USA61 YITP, Kyoto Universty, Kyoto, Japan62 Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany63 William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN, USA64 INFN, Sezione di Pavia, Pavia, Italy65 Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA66 IPPP, University of Durham, Durham, UK67 Tata Institute of Fundamental Research, Mumbai, India68 Paul Scherrer Institut, Villigen, Switzerland69 Institut fur Experimentalphysik, Universitat Hamburg, Hamburg, Germany70 Department of Physics, Chung-Ang University, Seoul, Korea71 Department of Physics and Astronomy, University of Sheffield, Sheffield, UK72 Enrico Fermi Institute, University of Chicago, Chicago, IL, USA73 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China74 Department of Physics, Tohoku University, Sendai, Japan75 ITP, School of Physics, Peking University, Beijing, China

a Supported in part by US DOE, Div. of HEP, contract W-31-109-ENG-38b Supported in part by UNAM grant PAPIIT-IN116202 and Conacyt grant 42026-Fc Heisenberg Fellowd Supported by grant KBN 2 P03B 040 24e Supported by a MCyT Ramon y Cajal contract

February 2, 2008

Abstract. High-precision analyses of supersymmetry parameters aim at reconstructing the funda-mental supersymmetric theory and its breaking mechanism. A well defined theoretical frameworkis needed when higher-order corrections are included. We propose such a scheme, SupersymmetryParameter Analysis SPA, based on a consistent set of conventions and input parameters. A repos-itory for computer programs is provided which connect parameters in different schemes and relatethe Lagrangian parameters to physical observables at LHC and high energy e+e− linear colliderexperiments, i.e., masses, mixings, decay widths and production cross sections for supersymmetricparticles. In addition, programs for calculating high-precision low energy observables, the densityof cold dark matter (CDM) in the universe as well as the cross sections for CDM search exper-iments are included. The SPA scheme still requires extended efforts on both the theoretical andexperimental side before data can be evaluated in the future at the level of the desired precision.We take here an initial step of testing the SPA scheme by applying the techniques involved to aspecific supersymmetry reference point.

J.A. Aguilar-Saavedra et al. 1

1 INTRODUCTION

At future colliders, experiments can be performed inthe supersymmetric particle sector [1,2,3,4], if realizedin Nature, with very high precision. While the LargeHadron Collider LHC can provide us with a set of well-determined observables [5,6], in particular masses ofcolored particles and precise mass differences of var-ious particle combinations, experiments at the Inter-national e+e− Linear Collider ILC [7,8,9] offer high-precision determination of the non-colored supersym-metry sector. Combining the information from LHC onthe generally heavy colored particles with the informa-tion from ILC on the generally lighter non-colored par-ticle sector (and later from the Compact Linear Col-lider CLIC [10] on heavier states) will generate a com-prehensive high-precision picture of supersymmetry atthe TeV scale [11]. Such an analysis can be performedindependently of specific model assumptions and forany supersymmetric scenario that can be tested in lab-oratory experiments. It may subsequently serve as asolid base for the reconstruction of the fundamental su-persymmetric theory at a high scale, potentially closeto the Planck scale, and for the analysis of the micro-scopic mechanism of supersymmetry breaking [12,13].

The analyses will be based on experimental accura-cies expected at the percent down to the per-mil level[9,14]. These experimental accuracies must be matchedon the theoretical side. This demands a well-definedframework for the calculational schemes in perturba-tion theory as well as for the input parameters. Theproposed Supersymmetry Parameter Analysis Conven-tion (SPA) [Sect.2] provides a clear base for calculatingmasses, mixings, decay widths and production crosssections. They will serve to extract the fundamentalsupersymmetric Lagrangian parameters and the super-symmetry-breaking parameters from future data. Inaddition, the renormalization group techniques mustbe developed for all the scenarios to determine thehigh-scale parameters of the supersymmetric theoryand its microscopic breaking mechanism.

By constructing such a coherent and unified basis,the comparison between results from different calcula-tions can be streamlined, eliminating ambiguous pro-cedures and reducing confusion to a minimum whencross-checking results.

A program repository [Sect.3] has therefore beenbuilt in which a series of programs has been collectedthat will be expanded continuously in the future. Theprograms relate parameters defined in different schemeswith each other, e.g. pole masses with DR masses, andthey calculate decay widths and cross sections from thebasic Lagrangian parameters. An additional set of pro-grams predicts the values of high-precision low-energyobservables of Standard Model (SM) particles in su-persymmetric theories. The program repository alsoincludes global fit programs by which the entire setof Lagrangian parameters, incorporating higher-ordercorrections, can be extracted from the experimental

observables. In addition, the solutions of the renormal-ization group equations are included by which extrapo-lations from the laboratory energies to the Grand Uni-fication (GUT) and Planck scales can be performedand vice versa. Another category contains programswhich relate the supersymmetry (SUSY) parameterswith the predictions of cold dark matter in the uni-verse and the corresponding cross sections for searchexperiments of cold dark matter (CDM) particles.

It is strongly recommended that the programs avail-able in the repository adopt the structure of Ref. [15]for the Lagrangian, including flavor mixing and CPphases, and follow the generally accepted Supersym-metry Les Houches Accord, SLHA, for communicationbetween different programs [16]. For definiteness, wereproduce from [16] the superpotential (omitting R-parity violating terms), in terms of superfields,

W = ǫab

[

(YE)ijHad L

biˆEj + (YD)ijH

ad Q

biˆDj

+ (YU )ijHbuQ

aiˆU j − µHa

d Hbu

]

, (1)

where the chiral superfields of the Minimal Supersym-metric Standard Model (MSSM) have the followingSU(3)C ⊗ SU(2)L ⊗ U(1)Y quantum numbers

L : (1, 2,− 12 ), ˆE : (1, 1, 1), Q : (3, 2, 1

6 ), ˆU : (3, 1,− 23 )

ˆD : (3, 1, 13 ), Hd : (1, 2,− 1

2 ), Hu : (1, 2, 12 ) .

The indices of the SU(2)L fundamental representationare denoted by a, b = 1, 2 and the generation indices byi, j = 1, 2, 3. Color indices are everywhere suppressed,since only trivial contractions are involved. ǫab is thetotally antisymmetric tensor, with ǫ12 = ǫ12 = 1.

The soft SUSY breaking part is written as

−Lsoft = ǫab

[

(TE)ijHad L

biLe∗jR

+ (TD)ijHad Q

biLd∗jR

+(TU )ijHbuQ

aiLu∗jR

]

+ h.c.

+m2HdH∗

d aHad +m2

HuH∗

uaHau − (m2

3ǫabHadH

bu + h.c.)

+ Q∗

iLa(m2Q

)ijQajL

+ L∗

iLa(m2L)ij L

ajL

+ uiR(m2

u)ij u∗

jR+ diR

(m2d)ij d

∗

jR+ eiR

(m2e)ij e

∗

jR

+1

2

(

M1bb+M2wAwA +M3g

X gX)

+ h.c. , (2)

where the Hi are the scalar Higgs fields, the fields witha tilde are the scalar components of the superfield withthe identical capital letter; the bino is denoted as b,the unbroken SU(2)L gauginos as wA=1,2,3, and thegluinos as gX=1...8, in 2-component notation. The Tmatrices will be decomposed as Tij = AijYij , where Yare the Yukawa matrices and A the soft supersymmetrybreaking trilinear couplings.

Much work on both the theoretical and the exper-imental side is still needed before data could be eval-uated in the future at the desired level of accuracy.

2 Supersymmetry Parameter Analysis: SPA Convention and Project

SPA CONVENTION

– The masses of the SUSY particles and Higgs bosons are defined as pole masses.– All SUSY Lagrangian parameters, mass parameters and couplings, including tan β, are given

in the DR scheme and defined at the scale M = 1 TeV.– Gaugino/higgsino and scalar mass matrices, rotation matrices and the corresponding angles

are defined in the DR scheme at M , except for the Higgs system in which the mixing matrix isdefined in the on-shell scheme, the momentum scale chosen as the light Higgs mass.

– The Standard Model input parameters of the gauge sector are chosen as GF , α, MZ and

αMSs (MZ). All lepton masses are defined on-shell. The t quark mass is defined on-shell; the

b, c quark masses are introduced in MS at the scale of the masses themselves while taken at arenormalization scale of 2 GeV for the light u, d, s quarks.

– Decay widths/branching ratios and production cross sections are calculated for the set of pa-rameters specified above.

Table 1. Definition of the supersymmetry parameter convention SPA

These tasks of the SPA Project will be defined in de-tail in Sect.4.

In Sect.5 we introduce the SUSY reference pointSPS1a′ as a general setup for testing these tools inpractice. This reference point is defined at a charac-teristic scale of 1 TeV in the Minimal SupersymmetricStandard Model with roots in minimal supergravity(mSUGRA). The point is a derivative of the Snow-mass point SPS1a [17]; its parameters are identicalexcept for a small shift of the scalar mass parameterand a change of the trilinear coupling to comply withthe measured dark matter density [18]. Note, that theSPS1a′ parameters are compatible with all the avail-able high- and low-energy data. The parameters areclose to point B′ of Ref. [19]. The masses are fairlylight so that stringent tests of all aspects in the pro-gram can be performed for LHC and ILC experiments.The final target are predictions on the accuracies ofthe fundamental supersymmetry parameters that canbe expected from a common set of information whenLHC and ILC experiments are analyzed coherently.

Additional benchmark points within and beyondmSUGRA, representing characteristics of different sce-narios, should complement the specific choice of SPS1a′.

2 SPA CONVENTION

Extending the experience collected in analyzing Stan-dard Model parameters at the former e+e− collidersLEP and SLC, we propose the set of conventions de-fined in Table 1. These conventions conform with thegeneral SLHA scheme[16] but they are more specific inseveral points.

Though largely accepted as standard, some of thedefinitions proposed in this SPA Convention should beexplained in a few comments.

For the SUSY Lagrangian parameters the DR sche-me [20,21] is most useful. It is based on regularizationby dimensional reduction together with modified min-imal subtraction. This scheme is designed to preserve

supersymmetry by maintaining the number of degreesof freedom of all fields in D dimensions, and it is tech-nically very convenient. The β-functions for SUSY pa-rameters in this scheme are known up to 3-loop order[22]. It has recently been shown [23] that inconsisten-cies of the original scheme [24] can be overcome andthat the DR scheme can be formulated in a mathe-matically consistent way. The ambiguities associatedwith the treatment of the Levi-Civita tensor can beparameterized as renormalization scheme dependenceas was argued in [25]. Checks by explicit evaluationof the supersymmetric Slavnov-Taylor identities at theone-loop level have shown that the DR method gen-erates the correct counter terms [26]. [We will use theversion of the DR scheme as given in [21], there re-

ferred to as DR′

scheme.] To make use of the highlydeveloped infrastructure for proton colliders, which isbased on the MS factorization scheme [27], a dictionaryis given in Sect.3.2 for the translation between the DRand MS schemes, as well as the on-shell renormaliza-tion schemes.

The SUSY scale is chosen M = 1 TeV to avoidlarge threshold corrections in running the mass pa-rameters by renormalization group techniques from thehigh scale down to the low scale. Fixing the scale Mindependent of parameters within the supersymmetryscenarios is preferable over choices relating to specificparameters, such as squark masses, that can be fixedonly at the very end. By definition, this point canalso be used to characterize uniquely multiple-scale ap-proaches.

Mixing parameters, in particular tanβ, could havebeen introduced in different ways [29]; however, choos-ing the DR definitions proposed above has proven veryconvenient in practical calculations.

The masses of Higgs bosons [30], in the MSSM ofthe charged H±, of the neutral CP-odd A, and of thetwo CP-even h,H particles, are understood as polemasses, MH±,A,H,h. For given MA, the pole massesMH,h of the CP-even Higgs bosons are obtained as

J.A. Aguilar-Saavedra et al. 3

poles q2 = M2H,h of the dressed propagator matrix,

∆Hh(q2) =

(

q2−m2H+ΣHH(q2) ΣhH(q2)

ΣhH(q2) q2−m2h+Σhh(q2)

)−1

involving the tree-level masses mH,h and the diagonaland non-diagonal on-shell-renormalized self-energiesΣ.In the on-shell scheme, the input parameters are renor-malized on-shell quantities, in particular the A-bosonmass, with accordingly defined counter terms.

Owing to the momentum dependence of the self-energies, there is no unique mixing angle (α) for theneutral CP-even Higgs system beyond the tree level,and the SPA choice can be understood as a conventionfor an “improved Born approximation”. A convenientchoice for q2 in the self-energies which minimizes thedifference of such an approximation with respect to cal-culations involving the proper self-energies in physicalmatrix elements, is given by q2 = M2

h .The physical on-shell masses are introduced in the

decay widths and production cross sections such thatthe phase space is treated in the observables closest toexperimental on-shell kinematics. This applies to theheavy particles while the masses of the light particlescan generally be neglected in high energy processes.

In the chargino/neutralino sector the number of ob-servable masses exceeds the number of free parametersin the system, gaugino/higgsino mass parameters andtanβ. The most convenient set of input chargino/neu-tralino masses is dictated by experiment [the three low-est mass states in this sector, for example] while theadditional masses are subsequently predicted uniquely.Similar procedures need to be followed in the sfermionsector.

3 PROGRAM BASE

3.1 PROGRAM CATEGORIES

The computational tasks that are involved in the SPAProject can be broken down to several categories. Eachof the codes that will be collected in the SPA programrepository is included in one or more of these cate-gories. It is understood that in each case the theoreticalstate-of-the-art precision is implemented. For commu-nication between codes SLHA [16] is strongly recom-mended, which is extended in a suitable way whereappropriate.

1) Scheme translation tools:The communication between codes that employ dif-ferent calculational schemes requires a set of trans-lation rules. In the SPA program repository we there-fore collect tools that implement, in particular, thedefinitions and relations between on-shell, DR andMS parameters in the Lagrangian as listed in Sect.3.2 below.

2) Spectrum calculators:This category includes codes of the transition fromthe Lagrangian parameters to a basis of physical

particle masses and the related mixing matrices.This task mainly consists of deriving the on-shellparticle masses (including higher-order corrections)and of diagonalizing the mixing matrices in a con-sistent scheme, making use of the abovementionedtools as needed.

3) Calculation of other observables:3A) Decay tables:

compute the experimentally measurable widthsand branching fractions.

3B) Cross sections:calculate SUSY cross sections and distributionsfor LHC and ILC.

3C) Low-energy observables:compute the values of those low-energy, high-precision observables [e.g., b → sγ, Bs → µµ,gµ − 2] that are sensitive to SUSY effects.

3D) Cosmological and astrophysical aspects:this category of programs covers the derivationof cold dark matter (CDM) relic density in theuniverse, cross sections for CDM particle search-es, astrophysical cross sections, etc. in the SUSYcontext.

4) Event generators:Programs that generate event samples for SUSYand background processes in realistic collider envi-ronments.

5) Analysis programs:These codes make use of some or all of the aboveto extract the Lagrangian parameters from experi-mental data by means of global analyses.

6) RGE programs:By solving the renormalization-group equations, theprograms connect the values of the parameters ofthe low-energy effective Lagrangian to those at thehigh-scale where the model is supposed to match toa more fundamental theory. High-scale constraintsare implemented on the basis of well-defined theo-retical assumptions: gauge coupling unification,mSUGRA, GMSB, AMSB scenarios, etc.

7) Auxiliary programs and libraries:Structure functions, beamstrahlung, numericalmethods, SM backgrounds, etc.

This is an open system and the responsibility forall these programs remains with the authors. SPA pro-vides the translation tables and the links to the com-puter codes on the web-page

http://spa.desy.de/spa/

Conveners responsible for specific tasks of the SPAProject will be listed on this web-page; the informa-tion will be routinely updated to reflect the momentarystate of the project at any time.

3.2 SCHEME TRANSLATION

This subsection presents a few characteristic examplesof relations between on-shell observables and DR, MSquantities at the electroweak scale MZ and the SUSY

4 Supersymmetry Parameter Analysis: SPA Convention and Project

scale M . For brevity, here only the approximate one-loop results are given [31]; it is understood that thecodes in the program repository include the most up-to-date higher-loop results.

(a) Couplings:

• gauge couplings:

gMSi = gDR

i

(

1 −(gDR

i )2

96π2Ci

)

(3)

• Yukawa couplings between the gaugino λi, the chi-ral fermion ψk and the scalar φk:

gMSik = gDR

i

(

1 +(gDR

i )2

32π2Ci −

3∑

l=1

(gDRl )2

32π2C

rk

l

)

(4)

• Yukawa couplings between the scalar φi and thetwo chiral fermions ψj and ψk:

Y MSijk = Y DR

ijk

(

1+

3∑

l=1

(gDRl )2

32π2

[

Crj

l −2Cri

l +Crk

l

]

)

(5)

• trilinear scalar couplings:

These couplings do not differ in the two schemes.

Ci and Cri are the quadratic Casimir invariants of

the adjoint representation and the matter represen-tation r of the gauge group Gi, respectively. Theyare given by Ci = [3, 2, 0] for [SU(3), SU(2), U(1)]and Cr

i = [4/3, 3/4, 3/5× Y 2r ] for the fundamental

representations of SU(3), SU(2), and the U(1) hy-percharge Yr.

(b) SUSY DR, MS and pole masses:

• gaugino mass parameters

MMSi = MDR

i

(

1 +(gDR

i )2

16π2Ci

)

(6)

• higgsino mass parameter:

µMS = µDR

(

1 +

2∑

l=1

(gDRl )2

16π2CH

l

)

(7)

CHl denoting the SU(2) and U(1) Casimir invari-

ants of the Higgs fields.

• sfermion mass parameters:

These parameters do not differ in the DR and MSschemes.

• fermion pole masses:

The pole masses can be written schematically as

mi, pole = MDRi − ReΣ (/q = mi, pole) (8)

where Σ denotes the fermion self-energy renormal-ized according to the DR-scheme at the scale M .As an explicit example we note the one-loop re-lation between the SU(3) gaugino mass parameter

M3(M)DR and the gluino pole mass mg [withoutsfermion mixing] at the one-loop order:

mg = MDR3 (M) (9)

+αDR

s (M)

4π

[

mg

(

15 + 9 lnM2

m2g

)

+∑

q

2∑

i=1

mgB1

(

m2g,m

2q,m

2qi

)

]

where B1 is the finite part of one of the one-looptwo-point functions at the scale in the DR schemeM (and analogously A0, B0 to be used later), cf.Ref. [32].

• scalar pole masses:

A similar relation holds for the squared scalar masses

m2i, pole = M2, DR

i −Σ(q2 = m2i, pole) (10)

The one-loop QCD corrections for the left squarksof the first two generations in the limit of vanishingquark masses may serve as a simple example:

m2q = M2, DR

Q(M) (11)

− 2αDRs (M)

3π

[

(m2q −m2

g)B0(m2q ,m

2g, 0)

− 2m2qB0(m

2q,m

2q , 0) +A0(m

2q) −A0(m

2g)

]

(c) SM parameters:

The following paragraphs summarize the SM inputvalues for the analysis. Only approximate formulaeare presented for brevity, while the complete set ofrelations is available on the program repository.In a few cases the evolution from the scale MZ

to M is carried out by means of RGEs instead offixed-order perturbation theory because they haveproven, presently, more accurate; this may changeonce the necessary multi-loop calculations will becompleted.

• α, αDR(MZ), αDR1,2 (M):

αDR(MZ) =α

1 −∆αSM −∆αSUSY(12)

∆αSUSY = − α

6π

[

lnmH+

MZ+ 4

2∑

i=1

lnmχ+

i

MZ

+∑

f

2∑

i=1

NcQ2f ln

mfi

MZ

]

J.A. Aguilar-Saavedra et al. 5

∆αSM summarizes the SM contributions from theleptons, quarks and the W -boson. In the SUSYcontributions, ∆αSUSY, f sums over all chargedsfermions,Nc is the color factor andQf the (s)ferm-ion charge.

αDR1 (M) =

αDR(MZ)

cos2 θDR(MZ)

1 +1

4π

αDR(MZ)

cos2 θDR(MZ)lnM2

Z

M2

(13)

αDR2 (M) =

αDR(MZ)

sin2 θDR(MZ)

1 +1

4π

αDR(MZ)

sin2 θDR(MZ)lnM2

Z

M2

(14)

• sin2 θDR at MZ and at M :

The electroweak mixing parameter sin2 θDR(MZ) isgiven by

sin2 θDR(MZ)[

1 − sin2 θDR(MZ)]

=παDR(MZ)√

2M2ZGF (1 −∆r)

(15)

where the contributions from loops of SM and SUSYparticles are denoted by ∆r [33,34]. At the scale Mthe electroweak mixing parameter can be calculatedsubsequently from

tan2 θDR(M) = αDR1 (M)/αDR

2 (M) (16)

by making use of the couplings αDRi (M) given in

the preceeding paragraph.

• sin2 θDR and sin2 θeff at MZ :

The electroweak mixing angle in the effective lep-tonic (electronic) vertex of the Z boson is definedas

sin2 θeff ≡ sin2 θ(e)eff (MZ) =

1

4

(

1 − Rege

V

geA

)

(17)

in terms of the effective vector and axial vector cou-plings ge

V,A of the Z to electrons. The relation to

sin2 θDR(MZ) is given by (at one-loop order)

sin2 θDR(MZ) = sin2 θeff (18)

+ sin 2 θeffΠγZ(M2

Z) +ΠγZ(0)

2 M2Z

− f e ,

involving the photon–Z non-diagonal self-energyΠγZ(q2) and the non-universal electron–Z vertexcorrection form factors f e

V,A(q2),

f e = 12 f

eV (M2

Z) − (12 − 2 sin2 θeff) f e

A(M2Z), (19)

with all the loop quantities renormalized in the DRscheme at the scale MZ . For explicit expressionssee [33,34].

• αDRs at MZ and M , related to αMS

s (MZ):

αDRs (MZ) =

αMSs (MZ)

1 −∆αs(20)

∆αs =αs(MZ)

2π

[

1

2− 2

3ln

mt

MZ

− 2 lnmg

MZ− 1

6

∑

q

2∑

i=1

lnmqi

MZ

]

αDRs (M) =

αDRs (MZ)

1 − 34πα

DRs (MZ) ln

M2Z

M2

(21)

• W, Z bosons, pole and DR masses:

The pole masses MV (V = W,Z) and the DRmasses at MZ are related by

M2V = M2, DR

V (MZ) − ReΠTV V (p2 = M2

V ) (22)

involving the renormalized transverse vector-bosonself-energies in the DR scheme at the scaleMZ . TheZ pole mass is a direct input parameter, whereasthe W pole mass is derived from the relation to thelow-energy parameters α and Fermi constant GF

according to the SPA Convention:

M2W

(

1 − M2W

M2Z

)

=πα√

2GF (1 −∆r), (23)

∆r summarizes the loop contributions from the SMand SUSY particles as given explicitly in [33,34,35].

The self-energies at the scale M can be writtensymbolically as

16π2ΠTZZ = 16π2ΠT

ZZ, SM+Higgs (24)

−∑

f

4Nfc v

2fZ,ijB22(M

2Z ,m

2fi,m2

fj)

+∑

χ0,χ+

[

fijZH(M2Z ,mχi

,mχj)

+ 2gijZB0(M2Z ,mχi

,mχj)]

16π2ΠTWW = 16π2ΠT

WW, SM+Higgs (25)

−∑

f

2Nfc v

2fW,ijB22(M

2W ,m2

fi,m2

f ′j

)

+∑

i,j

[

fijWH(M2W ,mχ0

i,mχ+

j

)

+ 2gijWB0(M2W ,mχ0

i,mχ+

j

)]

where vfV,ij are the couplings of the gauge bosonto sfermions and fijV and gijV are combinations ofleft- and right-couplings to charginos and neutrali-nos; B22 and H are combinations of the Bi and Ai

6 Supersymmetry Parameter Analysis: SPA Convention and Project

loop functions. Detailed formulae are given in [36].

• charm and bottom running MS mass at mc,b and

DR mass at MZ , cf. [37,38]:

mDRb, SM(MZ) = mMS

b (mb)

[

αMSs (MZ)

αMSs (mb)

]1223

×[

1 − αDRs

3π− 23α2,DR

s

72π

]

(26)

mDRb (MZ) =

mDRb, SM(MZ) + ReΣ′

b(MZ)

1 −∆mb(MZ)(27)

∆mb(MZ) =2αs

3πmgµ tanβ I(m2

b1,m2

b2,m2

g)

+Y 2

t

16π2Atµ tanβ I(m2

t1,m2

t2, µ2)

− g2

16π2M2µ tanβ

×[

cos2 θt I(m2t1,M2

2 , µ2) +

1

2t→ b

+ cos → sin; Q1 → Q2]

I(a2, b2, c2) =a2b2 log a2/b2 + cyclic

(a2 − b2)(b2 − c2)(a2 − c2)

with Σ′b(MZ) = Σb(MZ) − mDR

b (MZ)∆mb(MZ)and Σb(MZ) being the self-energy of the bottomquark due to supersymmetric particles and heavySM particles and ∆mb(MZ) including the large fi-nite terms proportional to tanβ which have beenresummed [38]. In the case of the charm quark theadditional running between mc and mb has to beincluded. The SUSY contributions are in generalsmall and no resummation is necessary. The massesare evolved from the scaleMZ to M by means of theRGEs for the Yukawa couplings as described below.

• top quark pole mass and DR mass at MZ :

mDRt (MZ) = mt

[

1 − 5αDRs

3π− αDR

s

πlog

(

M2Z

m2t

)

−ct(αDR

s

π

)2

−Σ

]

(28)

where ct(M2Z/m

2t ) is the gluonic two-loop contribu-

tion and Σ accounts for the electroweak as well asthe SUSY contributions. The mass is evolved to thescale M by means of the Yukawa RGEs; see next.

• Yukawa couplings and running masses of SM par-ticles at M :

The vacuum expectation values vDRu and vDR

d areinitially given by:

M2W (MZ) =

1

4g2,DR(MZ) (29)

×[

v2, DRu (MZ) + v2, DR

d (MZ)]

vDRu (MZ)/vDR

d (MZ) = tanβDR(MZ) (30)

tanβDR(MZ) must be evolved down from the con-

ventional parameter tanβDR(M) by means of RGE.From the DR masses at MZ the Yukawa couplingsare calculated:

Y DRt (MZ) =

√2mDR

t (MZ)/vDRu (MZ) (31)

Y DRb,τ (MZ) =

√2mDR

b,τ (MZ)/vDRd (MZ) (32)

In a second step, they are evolved together withthe gauge couplings and the vacuum expectationvalues to M via RGEs. At this scale the runningSM fermion masses and gauge boson masses are re-lated to the Lagrangian parameters by the usualtree-level relations. This is, presently, a better ap-proach for the evolution of the Yukawa couplingsthan fixed-order perturbation theory.

3.3 WIDTHS AND CROSS SECTIONS

(a) Decay widths:

The decay widths are defined as inclusive quanti-ties including all radiative corrections; the massesof the heavy particles are taken on-shell, light par-ticle masses are set zero.

(b) Cross sections for e+e− collisions:

Cross sections, σ(e+e− → ˜F), for the productionof a set of supersymmetric particles/Higgs bosons

F are defined at the experimental level in e+e−

collisions including up-to-date radiative correctionsexcept hard γ bremsstrahlung to exclude large con-tributions from radiative return.In general, large QED-type photonic correctionscannot be disentangled from genuine SUSY-specificparts, and in the comparison of theoretical predic-tions with experimental data all higher-order termshave to be included. To elucidate the role of the spe-cific supersymmetric loop corrections, a reasonableand consistent prescription for cut-independent re-duced cross sections shall therefore be defined. Sincethe leading QED terms arising from virtual and realphoton contributions that contain large logarithmscan be identified and isolated, the “reduced” gen-uine SUSY cross sections are defined, at the theo-retical level, by subtracting the logarithmic termslog 4∆E2/s in the soft-photon energy cut-off ∆Eand in log s/m2

f from non-collinear and collinearsoft γ radiation off light fermions f = e, µ, . . . andvirtual QED corrections. In this definition of re-duced cross sections [see also [39]], the logarithmi-cally large QED radiative corrections are consis-tently eliminated in a gauge-invariant way. By thesame token, the reduced cross sections are definedwithout taking into account beamstrahlung.

J.A. Aguilar-Saavedra et al. 7

(c) Cross sections for hadron collisions:

Cross sections for proton collisions at Tevatron andLHC, σ(pp → F), include all QCD and otheravailable corrections, with infrared and collinearsingularities tamed by defining inclusive observables,or properly defined jet characteristics, and intro-ducing the renormalized parton densities, providedparametrically by the PDF collaborations [40,41].

4 TASKS OF THE SPA PROJECT

A successful reconstruction of the fundamental struc-ture of the supersymmetric theory at the high scaleand the proper understanding of the nature of colddark matter from experimental data require the preciseanalysis of all information that will become availablefrom collider experiments, low-energy experiments, as-trophysical and cosmological observations. Preliminarystudies [see Sect.5], initiating this SPA Project, haveshown that while this aim can in principle be achieved,it still needs much additional work both on the theoret-ical as well as on the experimental side. In particular,we identify the following areas of research as centraltasks of the SPA Project:

Higher-order calculations

While the precision of SUSY calculations has graduallyshifted from leading-order (LO) to next-to-leading or-der (NLO) accuracy [and, in some areas, beyond], thepresent level still does not match the expected exper-imental precision, particularly in coherent LHC+ILCanalyses. The experimental precision, however, has tobe fully exploited in order to draw firm conclusions onthe fundamental theory. To close this gap, the SPAProject foresees new efforts to push the frontier inhigher-order SUSY calculations to the line necessaryfor the proper interpretation of experimental analyses.

Improving the understanding of the DR scheme

The DR scheme recommended for higher-order calcula-tions can be formulated in a mathematically consistentway [23] and is technically most convenient. Many ex-plicit checks at the one-loop level have shown that theDR method generates the correct counter terms. How-ever, there is no complete proof yet that it preserves su-persymmetry and gauge invariance in all cases. There-fore, as the precision of SUSY calculations is pushedto higher orders, the SPA Project also requires fur-ther investigation of the symmetry identities in the DRscheme.

Moreover, there is an obvious dichotomy betweenthe DR scheme, which is convenient for the definition ofSUSY parameters and their renormalization group evo-lution, and the MS scheme, which is generally adoptedfor the calculation of hadronic processes [27]. While, asargued before, the MS scheme requires ad-hoc counterterms to restore supersymmetry, in the DR scheme a

finite shift from the commonly used MS density func-tions to the DR density functions has to be carriedout [42]. Moreover, for massive final state particles spu-rious density functions for the (4 − D) gluon compo-nents have to be introduced to comply with the factor-ization theorem, see [43,44] for details. Formulating anefficient combination of the most attractive elements ofboth schemes in describing hadronic processes is there-fore an important task of the project.

Improving experimental and theoretical precision

The set of observables that has been included so farin experimental analyses, by no means exhausts theopportunities which data at LHC and at ILC are ex-pected to provide in the future. SPA Project studieswill aim to identify any new channels that can give ad-ditional information, either independent or redundant[improving fit results], and they will include them ina unified framework. In connection with realistic es-timates of theoretical uncertainties, a solid accountof error sources and correlations has to be achieved.Furthermore, the sophistication of the experimentalresults will be refined by including more precise sig-nal and background calculations, and improved simu-lations as mandatory for the analysis of real data.

Coherent LHC + ILC analyses

We put particular emphasis on the coherent combina-tion of future data obtained at LHC and ILC. Whilethe LHC will most likely discover SUSY particles, ifthey exist, and will allow for the first tests of the SUSYparadigm, e+e− data make possible high-precision in-vestigations of the weakly-interacting sector. Feedbackand coherently combined analyses, which will greatlybenefit from a concurrent running of both colliders, areindispensable for a meaningful answer to the questionsraised in the present context. Studies as initiated bythe LHC/LC Study Group [45] are a vital part of theSPA Project.

Determining SUSY Lagrangian parameters

While at leading order the Lagrangian parameters con-nected with different supersymmetric particle sectorscan in general be isolated and extracted analyticallyfrom closely associated observables, the analysis is muchmore complex at higher orders. Higher orders intro-duce the interdependence of all sectors in the observ-ables. The development of consistent analyses for theglobal determination of the Lagrangian parameters inthis complex situation has started and, conform withgeneral expectations for iterative steps in perturbativeexpansions, they can be carried out consistently withas few assumptions as possible. The set of Lagrangianparameters and their experimental error matrix can bedetermined, including higher-order corrections. How-ever, the experimental procedure must still be supple-mented by corresponding theoretical errors and theircorrelations.

8 Supersymmetry Parameter Analysis: SPA Convention and Project

Cold dark matter

As the precision is refined, astrophysical data play anincreasingly important role in confronting supersym-metry with experiments. The class of models conserv-ing R-parity predict a weakly interacting, massive, sta-ble particle. The relic abundance of this particle im-poses crucial limits on supersymmetric scenarios [46].While among the supersymmetry breaking models ver-sions of mSUGRA and of gaugino mediation [47] havebeen analyzed in detail, the analyses have to be ex-tended systematically to other scenarios. In modelsthat account for the relic density, specific requirementson the accuracies must be achieved when the CDMparticle is studied in high-energy physics laboratoryexperiments [48]. In turn, predictions based on thecomprehensive parameter analysis of high-energy ex-periments determine the cross sections for astrophysi-cal scattering experiments by which the nature of thecold dark matter particles can be established. The SPAProject provides a platform for a systematic and con-tinuous interplay between the astrophysics and high-energy physics disciplines and the mutual refinementof their programs in the future.

Extended SUSY scenarios

The MSSM, in particular the parameter set SPS1a′

that we suggest for a first study, provides a benchmarkscenario for developing and testing the tools needed fora successful analysis of future SUSY data. However,neither this specific point nor the MSSM itself may bethe correct model for low-scale SUSY. Various param-eter sets [for instance other representative mSUGRApoints as well as non-universal SUGRA, GMSB, AMSB,and other scenarios, see Ref. [49] for a brief summary]and extended models have therefore to be investigatedwithin the SPA Project. In particular, models whichincorporate the right-handed neutrino sector, must beanalyzed extensively [50]. Furthermore, CP violation,R-parity violation, flavor violation, NMSSM and ex-tended gauge groups are among the roads that naturemay have taken in the SUSY sector. The SPA con-ventions are formulated so generally that they can beapplied to all these scenarios. The goal of deriving thefundamental structure from data will also to be pur-sued for many facets in this more general context.

5 EXAMPLE: REF POINT SPS1a′

To test the internal consistency of the SPA scheme andto explore the potential of such extended experimentaland theoretical analyses we have defined, as an exam-ple, the CP and R-parity invariant MSSM referencepoint SPS1a′. Of course, the SPA Convention is set upto cover also more general scenarios.

The results for SPS1a′ presented below are based onpreliminary experimental simulations. In some cases,however, extrapolations from earlier analyses for SPS1aand other reference points have been used in orderto substitute missing information necessary for a first

Parameter SM input Parameter SM input

me 5.110 · 10−4 mpolet 172.7

mµ 0.1057 mb(mb) 4.2

mτ 1.777 mZ 91.1876

mu(Q) 3 · 10−3 GF 1.1664 · 10−5

md(Q) 7 · 10−3 1/α 137.036

ms(Q) 0.12 ∆α(5)had 0.02769

mc(mc) 1.2 αMSs (mZ) 0.119

Table 2. Numerical values of the SM input to SPS1a′.Masses are given in GeV, for the leptons and the t quarkthe pole masses, for the lighter quarks the MS masses eitherat the mass scale itself, for c, b, or, for u, d, s, at the scaleQ = 2 GeV.

comprehensive test of all aspects of the SPA Project. Itis obvious that many detailed simulations are neededto demonstrate the full power of predicting the funda-mental supersymmetric parameters from future sets ofLHC and ILC data.

In e+e− annihilation experimental progress is ex-pected for the heavy chargino and neutralinos. Com-bining the results of such studies with LHC data ap-pear very promising and lead to improved mass deter-minations [51]. New techniques to determine sleptonmasses from cascade decays as very narrow resonances[52,53] should be applied. For cross section measure-ments and other sparticle properties methods to deter-mine the decay branching ratios should be developed.At the LHC a recently proposed mass relation methodoffers substantial improvements in the reconstructionof squark and gluino masses [54].

Analysis of SUSY Lagrangian parameters

The roots defining the Reference Point SPS1a′ are themSUGRA parameters [in the conventional notation forCMSSM – see [55] for the tighter original definition] inthe set

M1/2 = 250 GeV sign(µ) = +1

M0 = 70 GeV tanβ(M) = 10

A0 = −300 GeV

The left column, listing the universal gaugino massM1/2, the scalar mass M0 and the trilinear couplingA0 [Yukawa couplings factored out], is defined at theGUT scale MGUT. The point is close to the originalSnowmass point SPS1a [17]; the scalar mass param-eter M0 is lowered slightly at the GUT scale from100 GeV to 70 GeV and A0 is changed from −100 GeVto −300 GeV. The values of the SM input parametersare collected in Table 2. Extrapolation of the abovemSUGRA parameters down to the M = 1 TeV scalegenerates the MSSM Lagrangian parameters. Table 3displays the couplings and mass parameters after beingevolved from MGUT to M using the RGE part of theprogram SPheno [56] which is based on two-loop anal-

J.A. Aguilar-Saavedra et al. 9

Parameter SPS1a′ value Parameter SPS1a′ value

g′ 0.3636 M1 103.3

g 0.6479 M2 193.2

gs 1.0844 M3 571.7

Yτ 0.1034 Aτ −445.2

Yt 0.8678 At −565.1

Yb 0.1354 Ab −943.4

µ 396.0 tanβ 10.0

MHd159.8 |MHu | 378.3

ML1181.0 ML3

179.3

ME1115.7 ME3

110.0

MQ1525.8 MQ3

471.4

MU1507.2 MU3

387.5

MD1505.0 MD3

500.9

Table 3. The DR SUSY Lagrangian parameters at the scaleM = 1 TeV in SPS1a′ from [56] [mass unit in GeV; M2

Hu

negative]. In addition, gauge and Yukawa couplings at thisscale are given in the DR scheme.

Particle Mass [GeV] δscale [GeV]

h0 116.0 1.3

H0 425.0 0.7

χ01 97.7 0.4

χ02 183.9 1.2

χ04 413.9 1.2

χ±1 183.7 1.3

eR 125.3 1.2

eL 189.9 0.4

τ1 107.9 0.5

qR 547.2 9.4

qL 564.7 10.2

t1 366.5 5.4

b1 506.3 8.0

g 607.1 1.4

Table 4. Supersymmetric masses for the SUSY scale M =1 TeV, and their variation if M is shifted to 0.1 TeV.

yses of the β-functions as well as the other evolutioncoefficients (other codes can be used equally well).

This SPS1a′ set is compatible with all high-energymass bounds and with the low-energy precision data,as well as with the observed CDM data, calculated asB(b→ sγ) = 3.0·10−4 [57],∆[g−2]µ/2 = 34·10−10 [58],∆ρSUSY = 2.1 · 10−4 [58], and ΩCDMh

2 = 0.10 [57].

The physical [pole] masses of the supersymmetricparticles are presented in Table 5. The connection be-tween the Lagrangian parameters and the physical polemasses is presently encoded at the one-loop level forthe masses of the SUSY particles, and at the two-looplevel for the Higgs masses. QCD effects on the heavyquark masses are accounted for to two-loop accuracy.

A systematic comparison with the other public pro-grams ISAJET [59], SOFTSUSY [60] and SuSpect [61] hasbeen performed in [62] to estimate the technical accu-racy that can presently be reached in the evolution.The codes include full two-loop RGEs for all parame-ters as well as one-loop formulas for threshold correc-tions. The agreement between the actual versions ofthese calculations is in general within one percent. Aspecial case are the on-shell masses of the Higgs bosonswhich have been calculated by FeynHiggs [58] start-ing from the SPheno Lagrangian parameters as input.Here, discrepancies for the mass of the lightest Higgsboson amount to 2% or more which can be attributedto different renormalization schemes (see also [63] fordetailed discussions).

Besides the comparison between different codes forspectrum calculations, a crude internal estimate of thetheoretical errors at the present level of the loop calcu-lations may be obtained by shifting the matching pointM from 1 TeV down to 0.1 TeV. A sample of parti-cle mass shifts associated with such a variation of theSUSY scale parameter is displayed in Table 4. With er-rors at the percent level, the experimental precision atLHC can be matched in general. However, it is obviousthat another order of magnitude, the per-mil level, isrequired in the theoretical precision to match the ex-pected experimental precision at ILC and in coherentLHC/ILC analyses – i.e., calculations of the next loopare called for1.

To perform experimental simulations, the branch-ing ratios of the decay modes are crucial: these havebeen calculated using FeynHiggs [58] and SDECAY [65];similar results may be obtained using CPSuperH [66].The most important decay channels of the supersym-metric particles and Higgs bosons in SPS1a′ are col-lected in the Appendix, while the complete set is avail-able from the SPA web-site. Cross sections for the pro-duction of squarks, gluinos, gauginos and sleptons atthe LHC are presented as a function of mass includingthe point SPS1a′. Typical cross sections for pair pro-duction of charginos, neutralinos and sleptons at theILC are presented for the point SPS1a′ as a functionof the collider energy.

If SPS1a′, or a SUSY parameter set in the range ofsimilar mass scales, is realized in nature, a plethora ofinteresting channels can be exploited to extract the ba-sic supersymmetry parameters when combining exper-imental information from sharp edges in mass distribu-tions at LHC with measurements of decay spectra andthreshold excitation curves at an e+e− collider with en-ergy up to 1 TeV [11]. From the simulated experimentalerrors the data analysis performed coherently for thetwo machines gives rise to a very precise picture of thesupersymmetric particle spectrum as demonstrated inTable 6.

1 With β functions and evolution coefficients in the RGEsalready available to third order [22], the calculation of thetwo-loop order for the relation between the Lagrangian pa-rameters and the physical pole masses have been carriedout in the approximation of massless vector bosons [64]

10 Supersymmetry Parameter Analysis: SPA Convention and Project

0

100

200

300

400

500

600

700

m [GeV]SPS1a′ mass spectrum

lR

lLνl

τ1

τ2ντ

χ01

χ02

χ03

χ04

χ±1

χ±2

qR

qL

g

t1

t2

b1

b2

h0

H0, A0 H±

Particle Mass [GeV] Particle Mass [GeV]

h0 116.0 τ1 107.9

H0 425.0 τ2 194.9

A0 424.9 ντ 170.5

H+ 432.7 uR 547.2

χ01 97.7 uL 564.7

χ02 183.9 dR 546.9

χ03 400.5 dL 570.1

χ04 413.9 t1 366.5

χ+1 183.7 t2 585.5

χ+2 415.4 b1 506.3

eR 125.3 b2 545.7

eL 189.9 g 607.1

νe 172.5

Table 5. Mass spectrum of supersymmetric particles [56] and Higgs bosons [58] in the reference point SPS1a′. Themasses in the second generation coincide with the first generation.

Particle Mass “LHC” “ILC” “LHC+ILC”

h0 116.0 0.25 0.05 0.05

H0 425.0 1.5 1.5

χ01 97.7 4.8 0.05 0.05

χ02 183.9 4.7 1.2 0.08

χ04 413.9 5.1 3 − 5 2.5

χ±1 183.7 0.55 0.55

eR 125.3 4.8 0.05 0.05

eL 189.9 5.0 0.18 0.18

τ1 107.9 5 − 8 0.24 0.24

qR 547.2 7 − 12 − 5 − 11

qL 564.7 8.7 − 4.9

t1 366.5 1.9 1.9

b1 506.3 7.5 − 5.7

g 607.1 8.0 − 6.5

Table 6. Accuracies for representative mass measurementsof SUSY particles in individual LHC, ILC and coherent“LHC+ILC” analyses for the reference point SPS1a′ [massunits in GeV]. qR and qL represent the flavors q = u, d, c, s.[Errors presently extrapolated from SPS1a simulations.]

While the picture so far had been based on evaluat-ing the experimental observables channel by channel,global analysis programs have become available [67,68] in which the whole set of data, masses, cross sec-tions, branching ratios, etc. is exploited coherently toextract the Lagrangian parameters in the optimal wayafter including the available radiative corrections formasses and cross sections. With increasing numbers ofobservables the analyses can be expanded and refinedin a systematic way. The present quality of such an

analysis [68] can be judged from the results shown inTable 7. These errors are purely experimental and donot include the theoretical counterpart which must beimproved considerably before matching the experimen-tal standards.

Extrapolation to the GUT scale

Based on the parameters extracted at the scale M , wecan approach the reconstruction of the fundamental su-persymmetric theory and the related microscopic pic-ture of the mechanism breaking supersymmetry. Theexperimental information is exploited to the maximumextent possible in the bottom-up approach [12] in which

the extrapolation from M to the GUT/Planck scaleis performed by the renormalization group evolutionfor all parameters, with the GUT scale defined by theunification point of the two electroweak couplings. Inthis approach the calculation of loops and β functionsgoverning the extrapolation to the high scale is basedon nothing but experimentally measured parameters.Typical examples for the evolution of the gaugino andscalar mass parameters are presented in Fig. 1. Whilethe determination of the high-scale parameters in thegaugino/higgsino sector, as well as in the non-coloredslepton sector, is very precise, the picture of the col-ored scalar and Higgs sectors is still coarse, and strongefforts should be made to refine it considerably.

On the other hand, if the structure of the theory atthe high scale was known a priori and merely the ex-perimental determination of the high-scale parameterswere lacking, then the top-down approach would leadto a very precise parametric picture at the high scale.This is apparent from the fit of the mSUGRA parame-ters in SPS1a′ displayed in Table 8 [67]. A high-qualityfit of the parameters is a necessary condition, of course,

J.A. Aguilar-Saavedra et al. 11

1/M

i[G

eV−

1]

M2 f

[103

GeV

2]

Q [GeV] Q [GeV]

M−1

3

M−1

2

M−1

1

Fig. 1. Running of the gaugino and scalar mass parameters as a function of the scale Q in SPS1a′ [56]. Only experimentalerrors are taken into account; theoretical errors are assumed to be reduced to the same size in the future.

Parameter SPS1a′ value Fit error [exp]

M1 103.3 0.1

M2 193.2 0.1

M3 571.7 7.8

µ 396.0 1.1

ML1181.0 0.2

ME1115.7 0.4

ML3179.3 1.2

MQ1525.8 5.2

MD1505.0 17.3

MQ3471.4 4.9

mA 372.0 0.8

At –565.1 24.6

tanβ 10.0 0.3

Table 7. Excerpt of extracted SUSY Lagrangian mass andHiggs parameters at the supersymmetry scale M = 1 TeVin the reference point SPS1a′ [mass units in GeV].

for the theory to be correct – however it is not a suffi-cient condition; deviations from the theory may hide inlarge errors of some observables which do not spoil thequality of the fit in the top-down approach but whichare manifest in the bottom-up approach.

Cold dark matter

Constraints on SUSY cold dark matter can be obtainedat LHC by specifying the underlying scenario and ana-lyzing all data simultaneously within the given bench-mark model. From a study of the SPS1a point, basedon very large statistics [69], one may expect that therelic density can be determined to ∼ 6% for the SPS1a′

scenario. For SPS1a′, the relic density depends on the

Parameter SPS1a′ value Experimental error

MGUT 2.47 · 1016 GeV 0.02 · 1016 GeV

α−1GUT 24.17 0.06

M 12

250 GeV 0.2 GeV

M0 70 GeV 0.2 GeV

A0 -300 GeV 13.0 GeV

µ 396.0 GeV 0.3 GeV

tan β 10 0.3

Table 8. Comparison of the ideal parameters with the ex-perimental expectations in the top-down approach [68].

parameters of the neutralino and sfermion sector as thedominant channels are annihilation of neutralinos intofermion pairs and coannihilation with staus. In partic-ular, for the most sensitive component, coannihilationprocesses, the relic density is essentially given by themass difference between the lightest slepton τ1 and theLSP χ0

1, which can be directly measured at the ILC.Studies of τ1 production at threshold [70] and decayspectra to χ0

1 in the continuum [71] suggest that forSPS1a′, even with moderate luminosity, a precision of∼ 2% on the cold dark matter abundance is achievable.A systematic analysis of various scenarios is being car-ried out in the LCC project [72] as well as by othergroups.

6 SUMMARY AND OUTLOOK

If supersymmetry is realized in Nature, future experi-ments at the LHC and the ILC will provide very precisemeasurements of supersymmetric particle spectra andcouplings. On the theoretical side these measurements

12 Supersymmetry Parameter Analysis: SPA Convention and Project

must be matched by equally precise theoretical calcu-lations and numerical analysis tools. The SPA Project,a joint theoretical and experimental effort, aims at pro-viding

– a well-defined framework for SUSY calculations anddata analyses,

– all necessary theoretical and computational tools,– a testground scenario SPS1a′,– a platform for future extensions and developments.

On this basis coherent analyses of experimental datacan be performed and the fundamental supersymmet-ric Lagrangian parameters can be extracted. They canserve as a firm base for extrapolations to high scalesso that the ultimate supersymmetric theory and the su-persymmetry breaking mechanism can be reconstructedfrom future data.

Much work is still needed on the experimental andtheoretical side to achieve these goals at the desiredlevel of accuracy. Some of the short- and long-termsubprojects have been identified and should be pursuedin the near future.

The SPA Project is a dynamical system expectedto evolve continuously. The current status of the SPAProject, names of the conveners responsible for spe-cific tasks as well as links to the available calculationaltools, can be found at the SPA home pagehttp://spa.desy.de/spa/.

APPENDIX

(a) Decays of Higgs and SUSY particles in SPS1a′

The branching ratios of Higgs bosons and SUSY par-ticles exceeding 2% are presented in Tables 9–12. Thecomplete listing including all decays is available on theSPA web-site http://spa.desy.de/spa/.

Higgs m,Γ [GeV] decay B decay B

h0 116.0 τ−τ+ 0.077 WW ∗ 0.067

4 × 10−3 bb 0.773 gg 0.055

cc 0.021

H0 425.0 τ−τ+ 0.076 χ01χ

02 0.038

1.2 bb 0.694 χ02χ

02 0.020

tt 0.052 χ+1 χ−

1 0.050

τ±1 τ∓

2 0.030

A0 424.9 τ−τ+ 0.057 χ01χ

02 0.054

1.6 bb 0.521 χ02χ

02 0.060

tt 0.094 χ+1 χ−

1 0.163

τ±1 τ∓

2 0.036

H+ 432.7 νττ+ 0.104 χ+1 χ0

1 0.143

0.9 tb 0.672 ντ τ+1 0.071

Table 9. Higgs masses and branching ratios B > 2% inSPS1a′ from [58].

χ m, Γ [GeV] decay B decay B

χ01 97.7

χ02 183.9 e±Re∓ 0.025 νeνe 0.116

0.083 τ±1 τ∓ 0.578 ντντ 0.152

χ03 400.5 χ±

1 W∓ 0.582 χ01Z

0 0.104

2.4 χ02Z

0 0.224

χ04 413.9 τ±

2 τ∓ 0.033 χ±1 W∓ 0.511

2.9 νeνe 0.042 χ01Z

0 0.022

ντντ 0.042 χ02Z

0 0.024

χ01h

0 0.070

χ02h

0 0.165

χ+1 183.7 τ+

1 ντ 0.536 νττ+ 0.185

0.077 νee+ 0.133

χ+2 415.4 e+

Lνe 0.041 χ01W

+ 0.063

3.1 τ+2 ντ 0.046 χ0

2W+ 0.252

t1b 0.109 χ+1 Z0 0.221

χ+1 h0 0.181

Table 10. Neutralino and chargino masses, widths andbranching ratios B > 2% in SPS1a′ from [65]; branchingratios for the second generation are the same as for thefirst generation.

J.A. Aguilar-Saavedra et al. 13

ℓ m, Γ [GeV] decay B decay B

eR 125.3 χ01e

− 1.000

0.10

eL 189.9 χ01e

− 0.925 χ−1 νe 0.049

0.12 χ02e

− 0.026

νe 172.5 χ01νe 1.000

0.12

τ1 107.9 χ01τ

− 1.000

0.016

τ2 194.9 χ01τ

− 0.868 χ−1 ντ 0.086

0.18 χ02τ

− 0.046

ντ 170.5 χ01ντ 1.000

0.12

Table 11. Slepton masses, widths and branching ratios B >2% in SPS1a′ from [65]; branching ratios for the secondgeneration are the same as for the first generation.

q m,Γ [GeV] decay B decay B

uR 547.2 χ01u 0.990

1.2

uL 564.7 χ02u 0.322 χ+

1 d 0.656

5.5

dR 546.9 χ01d 0.990

0.3

dL 570.1 χ02d 0.316 χ−

1 u 0.625

5.4

t1 366.5 χ01t 0.219 χ+

1 b 0.719

1.5 χ02t 0.062

t2 585.5 χ01t 0.042 χ+

1 b 0.265

6.3 χ02t 0.103 χ+

2 b 0.168

t1Z0 0.354

t1h0 0.059

b1 506.3 χ01b 0.037 χ−

1 t 0.381

4.4 χ02b 0.295 t1W

− 0.281

b2 545.7 χ01b 0.222 χ−

1 t 0.178

1.0 χ02b 0.131 t1W

− 0.401

χ03b 0.028

χ04b 0.038

g 607.1 uRu 0.086 t1t 0.189

5.5 uLu 0.044 b1b 0.214

dRd 0.087 b2b 0.096

dLd 0.034

Table 12. Masses, widths and branching ratios B > 2%of colored SUSY particles in SPS1a′ from [65]; branchingratios for the second generation are the same as for the firstgeneration.

(b) LHC and ILC cross sections in SPS1a′

Total cross sections are presented in Figs. 2 – 6 forSUSY particle production at the LHC and the ILC.

500 1000 1500 2000 2500 300010-4

10-3

10-2

10-1

1

10

102

103pp → qq, qq, qg, gg, titi + X

gg

qg

qq

qq

t1t1

t2t2

mq [GeV]σ

[pb]

500 1000 1500 2000 2500 300010-4

10-3

10-2

10-1

1

10

102

103pp → qq, qq, qg, gg + X

qq

qg

qq

gg

mg [GeV]

σ[p

b]

500 1000 1500 2000 2500 300010-4

10-3

10-2

10-1

1

10

102

103pp → qq, qq, qg, gg + X

qq qg

qq

gg

mq [GeV]

σ[p

b]

Fig. 2. Total cross sections for squark and gluino pair pro-duction at the LHC [27,28] for fixed gluino mass (top),squark mass (center), and gluino/squark mass ratio (bot-tom) [fixed parameters corresponding to SPS1a′ values].Black circles indicate the SPS1a′ mass values. The Borncross sections (broken lines) are shown for some channels.

14 Supersymmetry Parameter Analysis: SPA Convention and Project

100 200 300 400 500 600 70010-4

10-3

10-2

10-1

1

10pp → e+

L e−L , χχ, gχ, qχ + X

m [GeV]

σ[p

b]

gχ02

qχ02

(LO)

χ+1 χ0

2

χ01χ0

2

e+

Le

−

L

Fig. 3. Generic examples of total cross sections (Drell-Yan and Compton production) as a function of the averagemass for production of sleptons, charginos and neutralinosat the LHC [27,28]. The Born cross sections (broken line)are shown for comparison.

200 400 600 800 1000 12000

50

100

150

200

e+e− → χ+i χ−

j

χ+2 χ

−

2

χ+1 χ

−

2

χ+1 χ

−

1

√s [GeV]

σ[fb]

200 400 600 800 1000 1200

25

50

75

100

e+e− → χ0i χ0

j

χ03

χ04

χ02

χ03

χ01

χ02

χ02

χ02

√s [GeV]

σ[fb]

Fig. 4. Total cross section sections for chargino and neu-tralino pair production in e+e− annihilation [73]. The Borncross sections (broken lines) are shown for a few channels.

200 400 600 800 1000 12000

25

50

75

100e+e− → µ+

i µ−i

µ+

Lµ

−

L

µ−

Rµ

−

R

√s [GeV]

σ[fb]

200 400 600 800 1000 1200

200

400

600

800

1000

1200e±e− → e±

Re−R

e+

Re

−

R

e−

Re

−

R

√s [GeV]

σ[fb]

Fig. 5. Total cross sections for smuon and selectron pairproduction in e±e− annihilation [74]. The Born cross sec-tion (broken lines) is shown for comparison.

800 900 1000 1100 12000

10

20

e+e− → t1t1

e−

Le+

R

?e−

Re+

L

?

?

√s [GeV]

σ[fb]

Fig. 6. Total cross sections for t1¯t1 pair production in

e+e− annihilation for left- and right-handed polarized elec-tron (Pe− = 0.8) and positron (Pe+ = 0.6) beams [75]. TheBorn cross section (broken line) is shown for comparison.

J.A. Aguilar-Saavedra et al. 15

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