arxiv:hep-ph_0511344.pdf - cern document server

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Supersym m etry Param eterAnalysis:SPA Convention

and 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 .B�elanger12,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.Erler24b,H.Fraas25,A.Freitas26,

T.Fritzsche27,R.M .G odbole28,G .J.G ounaris29,J.G uasch30,J.G union31,N.Haba32,H.E.Haber33,

K .Hagiwara34,L.Han35,T.Han8,H.-J.He36,S.Heinem eyer18,S.Hesselbach37,K .Hidaka38,I.Hinchli�e9,

M .Hirsch39,K .Hohenwarter-Sodek10,W .Hollik27,W .S.Hou40,T.Hurth18;11c,I.Jack41,Y.Jiang35,

D.R.T.Jones41,J.K alinowski42d,T.K am on4,G .K ane43,S.K .K ang44,T.K ernreiter10,W .K ilian2,

C.S.K im 45,S.F.K ing46,O .K ittel47,M .K lasen48,J.-L.K neur49,K .K ovarik23,M .K r�am er50,S.K ram l18,

R.Lafaye51,P.Langacker52,H.E.Logan53,W .-G .M a35,W .M ajerotto23,H.-U.M artyn54;2,K .M atchev55,

D.J.M iller56,M .M ondragon24b,G .M oortgat-Pick18,S.M oretti46,T.M ori57,G .M oultaka49,S.M uanza58,

M .M .M �uhlleitner12,B.M ukhopadhyaya59,U.Nauenberg60,M .M .Nojiri61,D.Nom ura13,H.Nowak62,

N.O kada34,K .A.O live63,W .�O ller23,M .Peskin11,T.Plehn27c,G .Polesello64,W .Porod39;26e,F.Q uevedo3,

D.Rainwater65,J.Reuter2,P.Richardson66,K .Rolbiecki42d,P.Roy67,R.R�uckl25,H.Rzehak68,P.Schleper69,

K .Siyeon70,P.Skands16,P.Slavich12,D.St�ockinger66,P.Sphicas18,M .Spira68,T.Tait7,D.R.Tovey71,

J.W .F.Valle39,C.E.M .W agner72;7,Ch.W eber23,G .W eiglein66,P.W ienem ann19,Z.-Z.Xing73,Y.Yam ada74,

J.M .Yang73,D.Zerwas21,P.M .Zerwas2,R.-Y.Zhang35,X.Zhang73,S.-H.Zhu75

1Departam ento de Fisica and CFTP,Instituto Superior Tecnico,Lisbon,Portugal

2Deutsches Elektronen-Synchrotron DESY,Ham burg,G erm any

3DAM TP,University ofCam bridge,Cam bridge,UK

4 Departm entofPhysics,Texas A& M University,College Station,TX,USA5Departm entofPhysics,Florida State University,Tallahassee,FL,USA

6Departm entofPhysics and Astronom y,Johns Hopkins University,Baltim ore,M D,USA

7High Energy Physics Division,Argonne NationalLaboratory,Argonne,IL,USA

8Departm entofPhysics,University ofW isconsin,M adison,W I,USA

9Lawrence Berkeley NationalLaboratory,Berkeley,CA,USA

10 Institutf�ur Theoretische Physik,Universit�atW ien,W ien,Austria11

Stanford Linear Accelerator Center,Stanford,CA,USA12

Laboratoire de Physique Theorique,Annecy-le-Vieux,France13

Departm entofPhysics and Astronom y,M ichigan State University,EastLansing,M I,USA14

RoyalHolloway University ofLondon,Egham ,Surrey,UK15

Skobeltsyn Institute ofNuclear Physics,M SU,M oscow,Russia16 Ferm iNationalAccelerator Laboratory,Batavia,IL,USA17

Departm entofPhysics,Chonbuk NationalUniversity,Chonju,K orea18

PH Departm ent,CERN,G eneva,Switzerland19

Physikalisches Institut,Universit�atFreiburg,Freiburg,G erm any20

Physics Departm ent,Universidad Catolica de Chile,Santiago,Chile21

LAL,Universit�e de Paris-Sud,IN2P3-CNRS,O rsay,France22 University ofDelhi,Delhi,India23

Institutf�ur Hochenergiephysik, �O sterreichische Akadem ie der W issenschaften,W ien,Austria24

Instituto de F�isica,UNAM ,M �exico,M exico25

Institutf�ur Theoretische Physik und Astrophysik,Universit�atW �urzburg,W �urzburg,G erm any26

Institutf�ur Theoretische Physik,Universit�atZ�urich,Z�urich,Switzerland27

M ax-Planck-Institutf�ur Physik,M �unchen,G erm any28 Centre for High Energy Physics,Indian Institute ofScience,Bangalore,India29

Departm entofTheoreticalPhysics,Aristotle University ofThessaloniki,Thessaloniki,G reece30

Facultatde F�isica,Universitatde Barcelona,Barcelona,Spain31

Departm entofPhysics,University ofCalifornia,Davis,CA,USA32

Institute ofTheoreticalPhysics,University ofTokushim a,Tokushim a,Japan33

Santa Cruz Institute for Particle Physics,University ofCalifornia,Santa Cruz,CA,USA34 Theory Division,K EK ,Tsukuba,Japan

http://arxiv.org/abs/hep-ph/0511344v2

2

35Departm entofM odern Physics,University ofScience and Technology ofChina,Hefei,China

36Center for High Energy Physics and Institute ofM odern Physics,Tsinghua University,Beijing,China

37High Energy Physics,Uppsala University,Uppsala,Sweden

38Departm entofPhysics,Tokyo G akugeiUniversity,Tokyo,Jpan

39 Instituto de F��sica Corpuscular,CSIC,Val�encia,Spain40 Departm entofPhysics,NationalTaiwan University,Taipei,Taiwan41

Departm entofM athem aticalSciences,University ofLiverpool,Liverpool,UK42

Institute ofTheoreticalPhysics,W arsaw Univerity,W arsaw,Poland43

M CTP,University ofM ichigan,Ann Arbor,M I,USA44

SchoolofPhysics,SeoulNationalUniversity,Seoul,K orea45 Departm entofPhysics,YonseiUniversity,Seoul,K orea46 SchoolofPhysics and Astronom y,University ofSoutham pton,Southam pton,UK47

Physikalisches Institutder Universit�atBonn,Bonn,G erm any48

Laboratoire de Physique Subatom ique etde Cosm ologie,Universit�e G renoble I,G renoble,France49

LPTA,Universit�e M ontpellier II,CNRS-IN2P3,M ontpellier,France50

Institutf�ur Theoretische Physik,RW TH Aachen,Aachen,G erm any51 Laboratoire de Physique des Particules,Annecy-le-Vieux,France52 Departm entofPhysics and Astronom y,University ofPennsylvania,Philadelphia,PA,USA53

Departm entofPhysics,Carleton University,O ttawa,O N,Canada54

I.Physikalisches Institutder RW TH Aachen,Aachen,G erm any55

Departm entofPhysics,University ofFlorida,G ainesville,FL,USA56

Departm entofPhysics and Astronom y,University ofG lasgow,G lasgow,UK57 ICEPP,University ofTokyo,Tokyo,Japan58 IPN Universit�e Lyon,IN2P3-CNRS,Lyon,France59

Harish-Chandra Research Institute,Allahabad,India60

University ofColorado,Boulder,CO ,USA61

YITP,K yoto Universty,K yoto,Japan62

Deutsches Elektronen-Synchrotron DESY,Zeuthen,G erm any63 W illiam I.Fine TheoreticalPhysics Institute,University ofM innesota,M inneapolis,M N,USA64 INFN,Sezione diPavia,Pavia,Italy65

Departm entofPhysics and Astronom y,University ofRochester,Rochester,NY,USA66

IPPP,University ofDurham ,Durham ,UK67

Tata Institute ofFundam entalResearch,M um bai,India68

PaulScherrer Institut,Villigen,Switzerland69 Institutf�ur Experim entalphysik,Universit�atHam burg,Ham burg,G erm any70 Departm entofPhysics,Chung-Ang University,Seoul,K orea71

Departm entofPhysics and Astronom y,University ofShe�eld,She�eld,UK72

Enrico Ferm iInstitute,University ofChicago,Chicago,IL,USA73

Institute ofHigh Energy Physics,Chinese Academ y ofSciences,Beijing,China74

Departm entofPhysics,Tohoku University,Sendai,Japan75

ITP,SchoolofPhysics,Peking University,Beijing,China

aSupported in partby US DO E,Div.ofHEP,contractW -31-109-ENG -38

bSupported in partby UNAM grantPAPIIT-IN116202 and Conacytgrant42026-F

cHeisenberg Fellow

dSupported by grantK BN 2 P03B 040 24

eSupported by a M CyT Ram on y Cajalcontract

O ctober22,2013

Abstract. High-precision analyses ofsupersym m etry param eters aim at reconstructing the funda-

m entalsupersym m etric theory and its breaking m echanism .A wellde�ned theoreticalfram ework

is needed when higher-ordercorrections are included.W e propose such a schem e,Supersym m etry

Param eterAnalysisSPA,based on a consistentsetofconventionsand inputparam eters.A repos-

itory forcom puterprogram sisprovided which connectparam etersin di�erentschem esand relate

the Lagrangian param eters to physicalobservables at LHC and high energy e+e�linear collider

experim ents,i.e.,m asses,m ixings,decay widthsand production crosssectionsforsupersym m etric

particles.In addition,program s for calculating high-precision low energy observables,the density

ofcold dark m atter (CD M ) in the universe as wellas the cross sections for CD M search exper-

im ents are included.The SPA schem e stillrequires extended e�orts on both the theoreticaland

experim entalside before data can be evaluated in the future at the levelofthe desired precision.

W e take here an initialstep oftesting the SPA schem e by applying the techniques involved to a

speci�c supersym m etry reference point.

J.A.Aguilar-Saavedra etal. 1

1 IN TRO D UCTIO N

At future colliders,experim ents can be perform ed in

thesupersym m etricparticlesector[1,2,3,4],ifrealized

in Nature,with very high precision.W hile the Large

Hadron ColliderLHC can provideuswith asetofwell-

determ ined observables [5,6],in particular m asses of

colored particles and precise m ass di�erences ofvar-

ious particle com binations,experim ents at the Inter-

nationale+ e� Linear Collider ILC [7,8,9]o�er high-

precision determ ination ofthe non-colored supersym -

m etry sector.Com biningtheinform ation from LHC on

thegenerally heavy colored particleswith theinform a-

tion from ILC on thegenerally lighternon-colored par-

ticle sector (and later from the Com pact Linear Col-

liderCLIC [10]on heavierstates)willgeneratea com -

prehensivehigh-precision pictureofsupersym m etry at

theTeV scale[11].Such an analysiscan be perform ed

independently ofspeci�c m odelassum ptions and for

any supersym m etricscenariothatcan betested in lab-

oratory experim ents.It m ay subsequently serve as a

solid baseforthereconstruction ofthefundam entalsu-

persym m etric theory ata high scale,potentially close

to the Planck scale,and forthe analysisofthe m icro-

scopicm echanism ofsupersym m etry breaking [12,13].

Theanalyseswillbebased on experim entalaccura-

ciesexpected atthe percentdown to the per-m illevel

[9,14].Theseexperim entalaccuraciesm ustbem atched

on the theoreticalside.This dem ands a well-de�ned

fram ework for the calculationalschem es in perturba-

tion theory as wellas for the input param eters.The

proposed Supersym m etryParam eterAnalysisConven-

tion (SPA)[Sect.2]providesaclearbaseforcalculating

m asses,m ixings,decay widths and production cross

sections.They willserve to extract the fundam ental

supersym m etricLagrangianparam etersand thesuper-

sym m etry-breaking param eters from future data.In

addition,the renorm alization group techniques m ust

be developed for all the scenarios to determ ine the

high-scale param eters of the supersym m etric theory

and itsm icroscopicbreaking m echanism .

By constructing such a coherentand uni�ed basis,

thecom parison between resultsfrom di�erentcalcula-

tions can be stream lined,elim inating am biguouspro-

cedures and reducing confusion to a m inim um when

cross-checking results.

A program repository [Sect.3]has therefore been

builtin which a seriesofprogram shasbeen collected

thatwillbe expanded continuously in the future.The

program srelateparam etersde�ned in di�erentschem es

with each other,e.g.polem asseswith DR m asses,and

theycalculatedecaywidthsand crosssectionsfrom the

basicLagrangian param eters.An additionalsetofpro-

gram spredictsthevaluesofhigh-precision low-energy

observables ofStandard M odel(SM ) particles in su-

persym m etric theories.The program repository also

includes global�t program s by which the entire set

ofLagrangian param eters,incorporating higher-order

corrections,can be extracted from the experim ental

observables.In addition,thesolutionsoftherenorm al-

ization group equationsareincluded bywhich extrapo-

lationsfrom thelaboratory energiesto theG rand Uni-

�cation (G UT) and Planck scales can be perform ed

and vice versa.Another category contains program s

which relate the supersym m etry (SUSY) param eters

with the predictions ofcold dark m atter in the uni-

verse and the corresponding cross sections for search

experim entsofcold dark m atter(CDM )particles.

Itisstronglyrecom m endedthattheprogram savail-

able in the repository adoptthe structure ofRef.[15]

for the Lagrangian,including avor m ixing and CP

phases,and follow the generally accepted Supersym -

m etry LesHouchesAccord,SLHA,forcom m unication

between di�erent program s [16].For de�niteness,we

reproduce from [16]the superpotential(om itting R-

parity violating term s),in term sofsuper�elds,

W = �ab

h

(YE )ijHad L

bi�E j + (YD )ijH

ad Q

bi�D j

+ (YU )ijHbu Q

ai�U j � �H

ad H

bu

i

; (1)

wherethechiralsuper�eldsoftheM inim alSupersym -

m etric Standard M odel (M SSM ) have the following

SU (3)C SU (2)L U (1)Y quantum num bers

L :(1;2;� 1

2); �E :(1;1;1);Q :(3;2;1

6); �U :(�3;1;� 2

3)

�D :(�3;1;13);H d :(1;2;�

1

2);H u :(1;2;

1

2):

TheindicesoftheSU (2)L fundam entalrepresentation

aredenoted by a;b= 1;2and thegeneration indicesby

i;j = 1;2;3.Colorindicesare everywhere suppressed,

since only trivialcontractionsare involved.�ab is the

totally antisym m etrictensor,with �12 = �12 = 1.

ThesoftSUSY breaking partiswritten as

� Lsoft = �ab

h

(TE )ijHad~LbiL~e�jR + (TD )ijH

ad~Q biL~d�jR

+ (TU )ijHbu~Q aiL~u�jR

i

+ h:c:

+ m2H dH

�

daH

ad + m

2H uH

�

uaH

au � (m 2

3�abHadH

bu + h:c:)

+ ~Q �

iL a(m 2

~Q)ij ~Q

ajL+ ~L�

iL a(m 2

~L)ij~L

ajL

+ ~uiR (m2~u)ij~u

�

jR+ ~diR (m

2~d)ij ~d

�

jR+ ~eiR (m

2~e)ij~e

�

jR

+1

2

�

M 1~b~b+ M 2 ~w

A ~w A + M 3~gX ~gX

�

+ h:c:; (2)

wheretheH i arethescalarHiggs�elds,the�eldswith

atildearethescalarcom ponentsofthesuper�eld with

the identicalcapitalletter;the bino is denoted as ~b,

the unbroken SU (2)L gauginos as ~w A = 1;2;3,and the

gluinos as ~gX = 1:::8,in 2-com ponent notation.The T

m atriceswillbedecom posed asTij = A ijYij,whereY

aretheYukawam atricesand A thesoftsupersym m etry

breaking trilinearcouplings.

M uch work on both the theoreticaland the exper-

im entalside is stillneeded before data could be eval-

uated in the future at the desired levelofaccuracy.

2 Supersym m etry Param eterAnalysis:SPA Convention and Project

SPA CO NVENTIO N

{ The m assesofthe SUSY particlesand Higgsbosonsare de�ned aspole m asses.

{ AllSUSY Lagrangian param eters,m ass param eters and couplings,including tan�,are given

in the D R schem e and de�ned atthe scale ~M = 1 TeV.

{ G augino/higgsino and scalar m ass m atrices,rotation m atrices and the corresponding angles

arede�ned in theD R schem eat ~M ,exceptfortheHiggssystem in which them ixing m atrix is

de�ned in the on-shellschem e,the m om entum scale chosen asthe lightHiggsm ass.

{ The Standard M odel input param eters of the gauge sector are chosen as G F , �, M Z and

�M Ss (M Z ).Alllepton m asses are de�ned on-shell.The t quark m ass is de�ned on-shell;the

b;cquark m assesare introduced in M S atthe scale ofthe m assesthem selveswhile taken ata

renorm alization scale of2 G eV forthe lightu;d;s quarks.

{ D ecay widths/branching ratios and production cross sections are calculated for the setofpa-

ram etersspeci�ed above.

Table 1.De�nition ofthe supersym m etry param eter convention SPA

These tasksofthe SPA Projectwillbe de�ned in de-

tailin Sect.4.

In Sect.5 we introduce the SUSY reference point

SPS1a0 as a generalsetup for testing these tools in

practice.This reference point is de�ned at a charac-

teristicscaleof1 TeV in theM inim alSupersym m etric

Standard M odelwith roots in m inim alsupergravity

(m SUG RA). The point is a derivative of the Snow-

m ass point SPS1a [17]; its param eters are identical

except for a sm allshift ofthe scalarm ass param eter

and a change ofthe trilinearcoupling to com ply with

the m easured dark m atterdensity [18].Note,thatthe

SPS1a0 param eters are com patible with allthe avail-

able high- and low-energy data.The param eters are

close to point B0 ofRef.[19].The m asses are fairly

light so that stringent tests ofallaspects in the pro-

gram can beperform ed forLHC and ILC experim ents.

The �naltarget are predictions on the accuracies of

the fundam entalsupersym m etry param etersthatcan

be expected from a com m on set ofinform ation when

LHC and ILC experim entsareanalyzed coherently.

Additional benchm ark points within and beyond

m SUG RA,representingcharacteristicsofdi�erentsce-

narios,shouldcom plem entthespeci�cchoiceofSPS1a0.

2 SPA CO N VEN TIO N

Extending the experience collected in analyzing Stan-

dard M odelparam eters at the form er e+ e� colliders

LEP and SLC,we propose the setofconventionsde-

�ned in Table 1.These conventionsconform with the

generalSLHA schem e[16]butthey arem orespeci�cin

severalpoints.

Though largely accepted asstandard,som e ofthe

de�nitionsproposed in thisSPA Convention should be

explained in a few com m ents.

FortheSUSY Lagrangian param eterstheDR sche-

m e[20,21]ism ostuseful.Itisbased on regularization

by dim ensionalreduction togetherwith m odi�ed m in-

im alsubtraction.Thisschem e isdesigned to preserve

supersym m etry by m aintaining the num berofdegrees

offreedom ofall�eldsin D dim ensions,and itistech-

nically very convenient.The�-functionsforSUSY pa-

ram etersin thisschem e are known up to 3-loop order

[22].Ithasrecently been shown [23]thatinconsisten-

cies ofthe originalschem e [24]can be overcom e and

that the DR schem e can be form ulated in a m athe-

m atically consistent way.The am biguities associated

with the treatm ent ofthe Levi-Civita tensor can be

param eterized as renorm alization schem e dependence

as was argued in [25].Checks by explicit evaluation

ofthesupersym m etricSlavnov-Tayloridentitiesatthe

one-loop levelhave shown that the DR m ethod gen-

eratesthe correctcounterterm s[26].[W e willuse the

version ofthe DR schem e as given in [21],there re-

ferred to as DR0

schem e.]To m ake use ofthe highly

developed infrastructure forproton colliders,which is

based on theM S factorizationschem e[27],adictionary

isgiven in Sect.3.2 forthetranslation between theDR

and M S schem es,as wellasthe on-shellrenorm aliza-

tion schem es.

The SUSY scale is chosen ~M = 1 TeV to avoid

large threshold corrections in running the m ass pa-

ram etersbyrenorm alizationgroup techniquesfrom the

high scale down to the low scale.Fixing the scale ~M

independentofparam eterswithin the supersym m etry

scenariosispreferable overchoicesrelating to speci�c

param eters,such as squark m asses,that can be �xed

only at the very end. By de�nition, this point can

alsobeused tocharacterizeuniquely m ultiple-scaleap-

proaches.

M ixing param eters,in particulartan�,could have

been introduced in di�erentways[29];however,choos-

ing theDR de�nitionsproposed abovehasproven very

convenientin practicalcalculations.

The m assesofHiggsbosons [30],in the M SSM of

the charged H � ,ofthe neutralCP-odd A,and ofthe

two CP-even h;H particles, are understood as pole

m asses,M H � ;A ;H ;h. For given M A , the pole m asses

M H ;h of the CP-even Higgs bosons are obtained as


polesq2 = M 2H ;h

ofthe dressed propagatorm atrix,

� H h(q2)=

�q2� m 2

H + � H H (q2) � hH (q

2)

� hH (q2) q2� m 2

h+ � hh(q2)

� � 1

involving the tree-levelm assesm H ;h and the diagonal

andnon-diagonalon-shell-renorm alizedself-energies�.

In theon-shellschem e,theinputparam etersarerenor-

m alized on-shellquantities,in particularthe A-boson

m ass,with accordingly de�ned counterterm s.

O wing to the m om entum dependence ofthe self-

energies,there is no unique m ixing angle (�) for the

neutralCP-even Higgs system beyond the tree level,

and theSPA choicecan beunderstood asa convention

foran \im proved Born approxim ation".A convenient

choice for q2 in the self-energieswhich m inim izes the

di�erenceofsuch an approxim ationwith respecttocal-

culationsinvolving the properself-energiesin physical

m atrix elem ents,isgiven by q2 = M 2h.

The physicalon-shellm assesareintroduced in the

decay widths and production crosssectionssuch that

thephasespaceistreated in theobservablesclosestto

experim entalon-shellkinem atics.This applies to the

heavy particleswhile the m assesofthe lightparticles

can generally be neglected in high energy processes.

In thechargino/neutralinosectorthenum berofob-

servablem assesexceedsthenum beroffreeparam eters

in the system ,gaugino/higgsino m assparam etersand

tan�.Them ostconvenientsetofinputchargino/neu-

tralinom assesisdictated byexperim ent[thethreelow-

est m ass states in this sector,for exam ple]while the

additionalm assesaresubsequently predicted uniquely.

Sim ilarproceduresneed to befollowed in thesferm ion

sector.

3 PRO GRAM BASE

3.1 PRO GRAM CATEGO RIES

Thecom putationaltasksthatareinvolved in theSPA

Projectcan bebroken down toseveralcategories.Each

ofthecodesthatwillbecollected in theSPA program

repository is included in one or m ore of these cate-

gories.Itisunderstood thatin each casethetheoretical

state-of-the-artprecision isim plem ented.Forcom m u-

nication between codes SLHA [16]is strongly recom -

m ended,which is extended in a suitable way where

appropriate.

1) Schem e translation tools:

Thecom m unication between codesthatem ploydif-

ferentcalculationalschem esrequiresa setoftrans-

lationrules.In theSPA program repositorywethere-

forecollecttoolsthatim plem ent,in particular,the

de�nitionsand relationsbetween on-shell,DR and

M S param etersin theLagrangian aslisted in Sect.

3.2 below.

2) Spectrum calculators:

Thiscategory includescodesofthetransition from

the Lagrangian param eters to a basis ofphysical

particle m asses and the related m ixing m atrices.

This task m ainly consists ofderiving the on-shell

particlem asses(includinghigher-ordercorrections)

and ofdiagonalizing the m ixing m atricesin a con-

sistentschem e,m aking use ofthe abovem entioned

toolsasneeded.

3) Calculation ofotherobservables:

3A) Decay tables:

com putetheexperim entally m easurablewidths

and branching fractions.

3B) Crosssections:

calculateSUSY crosssectionsand distributions

forLHC and ILC.

3C) Low-energy observables:

com pute the values ofthose low-energy,high-

precision observables [e.g.,b ! s ,Bs ! ��,

g� � 2]thataresensitiveto SUSY e�ects.

3D) Cosm ologicaland astrophysicalaspects:

thiscategory ofprogram scoversthe derivation

ofcold dark m atter(CDM )relic density in the

universe,crosssectionsforCDM particlesearch-

es,astrophysicalcrosssections,etc.in theSUSY

context.

4) Eventgenerators:

Program s that generate event sam ples for SUSY

and background processesin realisticcolliderenvi-

ronm ents.

5) Analysisprogram s:

These codesm ake use ofsom e orallofthe above

to extractthe Lagrangian param etersfrom experi-

m entaldata by m eansofglobalanalyses.

6) RG E program s:

Bysolvingtherenorm alization-groupequations,the

program sconnect the values ofthe param eters of

thelow-energy e�ectiveLagrangian to thoseatthe

high-scalewherethem odelissupposed tom atch to

a m ore fundam entaltheory.High-scale constraints

are im plem ented on the basisofwell-de�ned theo-

retical assum ptions: gauge coupling uni�cation,

m SUG RA,G M SB,AM SB scenarios,etc.

7) Auxiliary program sand libraries:

Structure functions, beam strahlung, num erical

m ethods,SM backgrounds,etc.

This is an open system and the responsibility for

alltheseprogram srem ainswith theauthors.SPA pro-

videsthe translation tablesand the linksto the com -

putercodeson the web-page

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

Conveners responsible for speci�c tasks of the SPA

Project willbe listed on this web-page;the inform a-

tion willberoutinelyupdated tore ectthem om entary

stateofthe projectatany tim e.

3.2 SCH EM E TRAN SLATIO N

Thissubsection presentsa few characteristicexam ples

ofrelationsbetween on-shellobservablesand DR,M S

quantitiesatthe electroweak scale M Z and the SUSY



scale ~M .Forbrevity,here only the approxim ate one-

loop results are given [31];it is understood that the

codesin the program repository include the m ostup-

to-datehigher-loop results.

(a) Couplings:

� gauge couplings:

gM S

i = gD R

i

1�(g

D Ri )

2

96�2C i

!

(3)

� Yukawa couplingsbetween thegaugino �i,thechi-

ralferm ion k and the scalar �k:

gM S

ik = gD R

i

1+(g

D Ri )

2

32�2C i �

3X

l= 1

(gD R

l )2

32�2C

rkl

!

(4)

� Yukawa couplings between the scalar �i and the

two chiralferm ions j and k:

YM S

ijk = YD R

ijk

�

1+

3X

l= 1

(gD R

l )2

32�2

�C

rj

l� 2C

ril+ C

rkl

��

(5)

� trilinear scalar couplings:

These couplingsdo notdi�erin the two schem es.

Ci and C ri are the quadratic Casim irinvariantsof

theadjointrepresentation and them atterrepresen-

tation r ofthe gauge group G i,respectively.They

are given by Ci = [3;2;0]for[SU (3);SU (2);U (1)]

and C ri = [4=3;3=4;3=5� Y 2

r ]forthe fundam ental

representationsofSU (3);SU (2),and the U (1)hy-

perchargeYr.

(b) SUSY DR,M S and pole m asses:

� gaugino m assparam eters

MM Si = M

D Ri

1+(gD Ri )2

16�2Ci

!

(6)

� higgsino m assparam eter:

�M S = �

D R

1+

2X

l= 1

(gD Rl )2

16�2CHl

!

(7)

C Hl denoting the SU (2)and U (1) Casim ir invari-

antsofthe Higgs�elds.

� sferm ion m assparam eters:

These param etersdo notdi�erin the DR and M S

schem es.

� ferm ion pole m asses:

The pole m assescan be written schem atically as

m i;pole = MD Ri � Re� (=q= m i;pole) (8)

where� denotestheferm ion self-energy renorm al-

ized according to the DR-schem e at the scale ~M .

As an explicit exam ple we note the one-loop re-

lation between the SU(3)gaugino m assparam eter

M 3(~M )D R and the gluino pole m ass m ~g [without

sferm ion m ixing]atthe one-loop order:

m ~g = MD R3 (~M ) (9)

+�D Rs (~M )

4�

�

m ~g

�

15+ 9ln~M 2

m 2~g

�

+X

q

2X

i= 1

m ~gB 1

�m

2~g;m

2q;m

2~qi

��

where B 1 is the �nite part ofone ofthe one-loop

two-pointfunctionsatthe scale in the DR schem e~M (and analogously A 0;B 0 to be used later),cf.

Ref.[32].

� scalar pole m asses:

A sim ilarrelationholdsforthesquaredscalarm asses

m2i;pole = M

2;D R

i � �(q2 = m2i;pole) (10)

The one-loop Q CD correctionsforthe leftsquarks

ofthe�rsttwogenerationsin thelim itofvanishing

quark m assesm ay serveasa sim ple exam ple:

m2~q = M

2;D R

~Q(~M ) (11)

�2�D Rs (~M )

3�

�

(m 2~q � m

2~g)B 0(m

2~q;m

2~g;0)

� 2m 2~qB 0(m

2~q;m

2~q;0)+ A 0(m

2~q)� A 0(m

2~g)

�

(c) SM param eters:

Thefollowing paragraphssum m arizetheSM input

valuesfortheanalysis.O nly approxim ateform ulae

arepresented forbrevity,whilethecom pletesetof

relationsisavailableon the program repository.

In a few cases the evolution from the scale M Z

to ~M is carried out by m eans ofRG Es instead of

�xed-orderperturbation theory because they have

proven,presently,m ore accurate;thism ay change

once the necessary m ulti-loop calculations willbe

com pleted.

� �,�D R(M Z ),�D R1;2 (

~M ):

�D R (M Z )=

�

1� �� SM � �� SU SY

(12)

�� SU SY = ��

6�

"

lnm H +

M Z

+ 4

2X

i= 1

lnm

~�+

i

M Z

+X

f

2X

i= 1

N cQ2f ln

m ~fi

M Z

#


�� SM sum m arizesthe SM contributionsfrom the

leptons, quarks and the W -boson. In the SUSY

contributions, �� SU SY , f sum s over all charged

sferm ions,N c isthecolorfactorandQ f the(s)ferm -

ion charge.

�D R1 (~M )=

�D R (M Z )

cos2 �D R (M Z )

1+1

4�

�D R (M Z )

cos2 �D R (M Z )lnM 2

Z

~M 2

(13)

�D R2 (~M )=

�D R (M Z )

sin2 �D R (M Z )

1+1

4�

�D R (M Z )

sin2 �D R (M Z )lnM 2

Z

~M 2

(14)

� sin2 �D R atM Z and at ~M :

Theelectroweakm ixingparam etersin2 �D R (M Z )is

given by

sin2 �D R (M Z )

h

1� sin2 �D R (M Z )

i

=��D R(M Z )

p2M 2

ZG F (1� �r)

(15)

wherethecontributionsfrom loopsofSM andSUSY

particlesaredenoted by �r[33,34].Atthescale ~M

theelectroweakm ixingparam etercanbecalculated

subsequently from

tan2 �D R (~M )= �D R1 (~M )=�D R2 (~M ) (16)

by m aking use ofthe couplings �D Ri (~M ) given in

the preceeding paragraph.

� sin2 �D R and sin2 �e� atM Z :

The electroweak m ixing angle in the e�ective lep-

tonic (electronic)vertex ofthe Z boson isde�ned

as

sin2 �e� � sin2 �(e)

e�(M Z )=

1

4

�

1� RegeV

geA

�

(17)

in term softhee�ectivevectorand axialvectorcou-

plings geV;A ofthe Z to electrons.The relation to

sin2 �D R (M Z )isgiven by (atone-loop order)

sin2 �D R (M Z )= sin2 �e� (18)

+ sin2 �e�� Z(M

2Z )+ � Z(0)

2 M 2Z

� fe;

involving the photon{Z non-diagonalself-energy

� Z(q2) and the non-universalelectron{Z vertex

correction form factorsfeV;A (q2),

fe = 1

2feV (M

2Z )� (1

2� 2 sin2 �e�)f

eA (M

2Z );(19)

with alltheloop quantitiesrenorm alized in theDR

schem e at the scale M Z .For explicit expressions

see[33,34].

� �D Rs atM Z and ~M ,related to �M Ss (M Z ):

�D Rs (M Z )=

�M Ss (M Z )

1� �� s

(20)

�� s =�s(M Z )

2�

�1

2�

2

3ln

m t

M Z

� 2lnm ~g

M Z

�1

6

X

~q

2X

i= 1

lnm ~qi

M Z

�

�D Rs (~M )=

�D Rs (M Z )

1� 3

4��D Rs (M Z )ln

M 2

Z

~M 2

(21)

� W ;Z bosons,pole and DR m asses:

The pole m asses M V (V = W ;Z) and the DR

m assesatM Z arerelated by

M2V = M

2;D R

V(M Z )� Re� T

V V (p2 = M

2V ) (22)

involving therenorm alized transversevector-boson

self-energiesin theDR schem eatthescaleM Z .The

Z pole m ass is a direct input param eter,whereas

theW polem assisderived from therelation to the

low-energy param eters � and Ferm iconstant G F

according to the SPA Convention:

M2W

�

1�M 2

W

M 2Z

�

=��

p2G F (1� �r)

; (23)

�rsum m arizestheloop contributionsfrom theSM

and SUSY particlesasgiven explicitlyin [33,34,35].

The self-energies at the scale ~M can be written

sym bolically as

16�2� TZ Z = 16�2� T

Z Z;SM + H iggs (24)

�X

f

4N fc v

2fZ;ij

~B 22(M2Z ;m

2~fi;m

2~fj)

+X

~�0;~�+

�fijZ H (M 2

Z ;m ~�i;m ~�j

)

+ 2gijZ B 0(M2Z ;m ~�i

;m ~�j)�

16�2� TW W = 16�2� T

W W ;SM + H iggs (25)

�X

f

2N fc v

2fW ;ij

~B 22(M2W ;m

2~fi;m

2~f0j

)

+X

i;j

�fijW H (M 2

W ;m ~�0

i;m

~�+

j

)

+ 2gijW B 0(M2W ;m ~�0

i;m

~�+

j

)�

where vfV;ij are the couplings ofthe gauge boson

to sferm ionsand fijV and gijV arecom binationsof

left-and right-couplingsto charginosand neutrali-

nos; ~B 22 and H arecom binationsoftheB i and A i


loop functions.Detailed form ulaearegiven in [36].

� charm and bottom running M S m assatmc;b and

DR m assatM Z ,cf.[37,38]:

mD Rb;SM (M Z )= m

M Sb (m b)

"

�M Ss (M Z )

�M Ss (m b)

# 12

23

�

"

1��D Rs

3��23�2;D Rs

72�

#

(26)

mD Rb (M Z )=

m D Rb;SM

(M Z )+ Re� 0

b(M Z )

1� �m b(M Z )(27)

�m b(M Z )=2�s

3�m ~g� tan� I(m

2~b1;m

2~b2;m

2~g)

+Y 2t

16�2A t� tan� I(m

2~t1;m

2~t2;�

2)

�g2

16�2M 2� tan�

��cos2 �~tI(m

2~t1;M

22;�

2)+1

2f~t! ~bg

+ fcos! sin; ~Q 1 ! ~Q 2g�

I(a2;b2;c2)=a2b2 loga2=b2 + cyclic

(a2 � b2)(b2 � c2)(a2 � c2)

with � 0

b(M Z ) = � b(M Z )� m D Rb (M Z )�m b(M Z )

and � b(M Z ) being the self-energy ofthe bottom

quark due to supersym m etric particles and heavy

SM particlesand �m b(M Z )including the large �-

nite term s proportionalto tan� which have been

resum m ed [38].In the caseofthe charm quark the

additionalrunning between m c and m b has to be

included.The SUSY contributions are in general

sm alland noresum m ation isnecessary.Them asses

areevolvedfrom thescaleM Z to ~M bym eansofthe

RG EsfortheYukawacouplingsasdescribed below.

� top quark pole m assand DR m assatMZ :

mD Rt (M Z )= m t

"

1�5�D Rs

3��D Rs

�log

�M 2

Z

m 2t

�

� ct

��D Rs

�

�2

� �

#

(28)

wherect(M2Z =m

2t)isthegluonictwo-loop contribu-

tion and � accountsforthe electroweak aswellas

theSUSY contributions.Them assisevolved tothe

scale ~M by m eansofthe Yukawa RG Es;seenext.

� Yukawa couplingsand runningm assesofSM par-

ticles at ~M :

The vacuum expectation values vD Ru and vD Rd are

initially given by:

M2W (M Z )=

1

4g2;D R (M Z ) (29)

�

h

v2;D Ru (M Z )+ v

2;D R

d(M Z )

i

vD Ru (M Z )=v

D Rd (M Z )= tan�D R (M Z ) (30)

tan�D R (M Z )m ustbe evolved down from thecon-

ventionalparam etertan�D R (~M )bym eansofRG E.

From theDR m assesatM Z the Yukawa couplings

arecalculated:

YD Rt (M Z )=

p2m D R

t (M Z )=vD Ru (M Z ) (31)

YD Rb;� (M Z )=

p2m D R

b;� (M Z )=vD Rd (M Z ) (32)

In a second step,they are evolved together with

the gauge couplings and the vacuum expectation

values to ~M via RG Es.At this scale the running

SM ferm ion m assesand gaugeboson m assesarere-

lated to the Lagrangian param eters by the usual

tree-levelrelations.Thisis,presently,a betterap-

proach for the evolution ofthe Yukawa couplings

than �xed-orderperturbation theory.

3.3 W ID TH S AN D CRO SS SECTIO N S

(a) Decay widths:

The decay widths are de�ned as inclusive quanti-

ties including allradiative corrections;the m asses

oftheheavy particlesaretaken on-shell,lightpar-

ticlem assesaresetzero.

(b) Crosssectionsfore+ e� collisions:

Crosssections,�(e+ e� ! ~fF g),fortheproduction

ofa setofsupersym m etric particles/Higgsbosons

f~Fg are de�ned atthe experim entallevelin e+ e�

collisionsincluding up-to-dateradiativecorrections

excepthard brem sstrahlungto excludelargecon-

tributionsfrom radiativereturn.

In general, large Q ED-type photonic corrections

cannotbedisentangled from genuineSUSY-speci�c

parts,and in the com parison oftheoreticalpredic-

tionswith experim entaldataallhigher-orderterm s

havetobeincluded.Toelucidatetheroleofthespe-

ci�csupersym m etricloop corrections,a reasonable

and consistentprescription forcut-independentre-

ducedcrosssectionsshallthereforebede�ned.Since

theleadingQ ED term sarisingfrom virtualand real

photon contributionsthatcontain largelogarithm s

can be identi�ed and isolated,the \reduced" gen-

uine SUSY crosssectionsare de�ned,atthe theo-

reticallevel,by subtracting the logarithm ic term s

log4�E 2=s in the soft-photon energy cut-o� �E

and in logs=m 2ffrom non-collinear and collinear

soft radiation o� lightferm ionsf = e;�;:::and

virtualQ ED corrections.In this de�nition of re-

duced crosssections[see also [39]],the logarithm i-

cally large Q ED radiative corrections are consis-

tently elim inated in a gauge-invariantway.By the

sam e token,the reduced crosssectionsare de�ned

withouttaking into accountbeam strahlung.


(c) Crosssectionsforhadron collisions:

Crosssectionsforproton collisionsatTevatron and

LHC,�(pp ! f~F g),include allQ CD and other

available corrections, with infrared and collinear

singularitiestam edbyde�ninginclusiveobservables,

or properly de�ned jet characteristics,and intro-

ducing therenorm alized parton densities,provided

param etrically by the PDF collaborations[40,41].

4 TASKS O F TH E SPA PRO JECT

A successfulreconstruction ofthe fundam entalstruc-

ture ofthe supersym m etric theory at the high scale

and the proper understanding of the nature of cold

darkm atterfrom experim entaldatarequiretheprecise

analysis ofallinform ation that willbecom e available

from colliderexperim ents,low-energy experim ents,as-

trophysicaland cosm ologicalobservations.Prelim inary

studies [see Sect.5],initiating this SPA Project,have

shown thatwhilethisaim can in principlebeachieved,

itstillneedsm uch additionalworkboth on thetheoret-

icalaswellason the experim entalside.In particular,

we identify the following areas ofresearch as central

tasksofthe SPA Project:

Higher-ordercalculations

W hiletheprecision ofSUSY calculationshasgradually

shifted from leading-order(LO )to next-to-leading or-

der(NLO )accuracy [and,in som e areas,beyond],the

presentlevelstilldoesnotm atch the expected exper-

im entalprecision,particularly in coherent LHC+ ILC

analyses.The experim entalprecision,however,hasto

befully exploited in orderto draw �rm conclusionson

the fundam entaltheory.To close this gap,the SPA

Project foresees new e�orts to push the frontier in

higher-order SUSY calculations to the line necessary

fortheproperinterpretation ofexperim entalanalyses.

Im proving the understanding ofthe DR schem e

TheDR schem erecom m ended forhigher-ordercalcula-

tionscan beform ulated in am athem atically consistent

way [23]and istechnically m ostconvenient.M any ex-

plicitchecksattheone-loop levelhaveshown thatthe

DR m ethod generatesthecorrectcounterterm s.How-

ever,thereisnocom pleteproofyetthatitpreservessu-

persym m etry and gaugeinvariancein allcases.There-

fore,as the precision ofSUSY calculations is pushed

to higher orders,the SPA Project also requires fur-

therinvestigation ofthesym m etry identitiesin theDR

schem e.

M oreover,there is an obvious dichotom y between

theDR schem e,which isconvenientforthede�nition of

SUSY param etersand theirrenorm alizationgroupevo-

lution,and theM S schem e,which isgenerally adopted

forthecalculation ofhadronicprocesses[27].W hile,as

argued before,the M S schem erequiresad-hoc counter

term s to restore supersym m etry,in the DR schem e a

�nite shiftfrom the com m only used M S density func-

tions to the DR density functions has to be carried

out[42].M oreover,form assive�nalstateparticlesspu-

rious density functions for the (4� D ) gluon com po-

nentshaveto beintroduced to com ply with thefactor-

ization theorem ,see[43,44]fordetails.Form ulatingan

e�cientcom bination ofthem ostattractiveelem entsof

both schem esin describinghadronicprocessesisthere-

forean im portanttask ofthe project.

Im proving experim entaland theoreticalprecision

The set ofobservables that has been included so far

in experim entalanalyses,by no m eans exhausts the

opportunities which data atLHC and atILC are ex-

pected to provide in the future.SPA Project studies

willaim to identify any new channelsthatcan givead-

ditionalinform ation,eitherindependentorredundant

[im proving �t results],and they willinclude them in

a uni�ed fram ework.In connection with realistic es-

tim ates of theoretical uncertainties, a solid account

oferror sources and correlations has to be achieved.

Furtherm ore, the sophistication of the experim ental

results willbe re�ned by including m ore precise sig-

naland background calculations,and im proved sim u-

lationsasm andatory forthe analysisofrealdata.

CoherentLHC + ILC analyses

W e putparticularem phasison the coherentcom bina-

tion offuture data obtained at LHC and ILC.W hile

the LHC willm ost likely discover SUSY particles,if

they exist,and willallow forthe�rsttestsoftheSUSY

paradigm ,e+ e� data m akepossible high-precision in-

vestigationsoftheweakly-interactingsector.Feedback

and coherently com bined analyses,which willgreatly

bene�tfrom aconcurrentrunningofboth colliders,are

indispensablefora m eaningfulanswerto thequestions

raised in the present context.Studies as initiated by

the LHC/LC Study G roup [45]are a vitalpartofthe

SPA Project.

Determ ining SUSY Lagrangian param eters

W hileatleadingordertheLagrangianparam eterscon-

nected with di�erent supersym m etric particle sectors

can in generalbe isolated and extracted analytically

from closelyassociatedobservables,theanalysisism uch

m ore com plex at higher orders.Higher orders intro-

duce the interdependence ofallsectorsin the observ-

ables.The developm entofconsistentanalysesforthe

globaldeterm ination ofthe Lagrangian param etersin

this com plex situation has started and,conform with

generalexpectationsforiterativestepsin perturbative

expansions,they can be carried outconsistently with

asfew assum ptionsaspossible.The setofLagrangian

param etersand theirexperim entalerrorm atrix can be

determ ined,including higher-order corrections.How-

ever,the experim entalprocedurem uststillbe supple-

m ented by corresponding theoreticalerrors and their

correlations.


Cold dark m atter

Asthe precision isre�ned,astrophysicaldata play an

increasingly im portant role in confronting supersym -

m etry with experim ents.The classofm odelsconserv-

ingR-paritypredictaweakly interacting,m assive,sta-

ble particle.The relic abundance ofthis particle im -

posescruciallim its on supersym m etric scenarios[46].

W hileam ongthesupersym m etry breakingm odelsver-

sionsofm SUG RA and ofgaugino m ediation [47]have

been analyzed in detail,the analyses have to be ex-

tended system atically to other scenarios. In m odels

thataccountfortherelicdensity,speci�crequirem ents

on the accuracies m ust be achieved when the CDM

particle is studied in high-energy physics laboratory

experim ents [48]. In turn, predictions based on the

com prehensive param eter analysis ofhigh-energy ex-

perim entsdeterm ine the crosssectionsforastrophysi-

calscattering experim entsby which the nature ofthe

cold darkm atterparticlescan beestablished.TheSPA

Projectprovidesa platform fora system atic and con-

tinuous interplay between the astrophysics and high-

energy physics disciplines and the m utualre�nem ent

oftheirprogram sin the future.

Extended SUSY scenarios

The M SSM ,in particular the param eter set SPS1a0

thatwesuggestfora�rststudy,providesabenchm ark

scenariofordevelopingand testingthetoolsneeded for

a successfulanalysis offuture SUSY data.However,

neitherthisspeci�cpointnortheM SSM itselfm ay be

thecorrectm odelforlow-scaleSUSY.Variousparam -

eter sets [for instance other representative m SUG RA

pointsaswellasnon-universalSUG RA,G M SB,AM SB,

and otherscenarios,seeRef.[49]fora briefsum m ary]

and extended m odelshavethereforeto beinvestigated

within the SPA Project.In particular,m odels which

incorporatetheright-handed neutrino sector,m ustbe

analyzed extensively [50].Furtherm ore,C P violation,

R-parity violation, avor violation,NM SSM and ex-

tended gaugegroupsaream ong theroadsthatnature

m ay have taken in the SUSY sector.The SPA con-

ventionsare form ulated so generally thatthey can be

applied to allthesescenarios.Thegoalofderiving the

fundam entalstructure from data willalso to be pur-

sued form any facetsin thism oregeneralcontext.

5 EXAM PLE:REF PO IN T SPS1a0

Totesttheinternalconsistency oftheSPA schem eand

to explorethepotentialofsuch extended experim ental

and theoreticalanalyseswe havede�ned,asan exam -

ple,the CP and R-parity invariant M SSM reference

pointSPS1a0.O fcourse,theSPA Convention issetup

to coveralso m oregeneralscenarios.

TheresultsforSPS1a0presentedbelow arebasedon

prelim inary experim entalsim ulations.In som e cases,

however,extrapolationsfrom earlieranalysesforSPS1a

and other reference points have been used in order

to substitute m issing inform ation necessary fora �rst

Param eter SM input Param eter SM input

m e 5:110� 10� 4

mpole

t 172:7

m � 0.1057 m b(m b) 4:2

m � 1.777 m Z 91:1876

m u(Q ) 3� 10� 3G F 1:1664� 10� 5

m d(Q ) 7� 10� 3

1=� 137:036

m s(Q ) 0.12 ��(5)

had0:02769

m c(m c) 1.2 �M Ss (m Z ) 0:119

Table 2. Num erical values of the SM input to SPS1a0.

M asses are given in G eV,for the leptons and the t quark

thepolem asses,forthe lighterquarkstheM S m asseseither

atthe m assscale itself,forc,b,or,foru,d,s,atthe scale

Q = 2 G eV.

com prehensivetestofallaspectsoftheSPA Project.It

is obviousthat m any detailed sim ulationsare needed

to dem onstratethefullpowerofpredicting thefunda-

m entalsupersym m etricparam etersfrom futuresetsof

LHC and ILC data.

In e+ e� annihilation experim entalprogress is ex-

pected for the heavy chargino and neutralinos.Com -

bining the results ofsuch studies with LHC data ap-

pearvery prom ising and lead to im proved m assdeter-

m inations [51].New techniques to determ ine slepton

m assesfrom cascadedecaysasvery narrow resonances

[52,53]should be applied.For cross section m easure-

m entsand othersparticlepropertiesm ethodsto deter-

m ine the decay branching ratiosshould be developed.

AttheLHC a recently proposed m assrelation m ethod

o�ers substantialim provem ents in the reconstruction

ofsquark and gluino m asses[54].

AnalysisofSUSY Lagrangian param eters

Therootsde�ning theReferencePointSPS1a0arethe

m SUG RA param eters[in theconventionalnotation for

CM SSM { see[55]forthetighteroriginalde�nition]in

the set

M 1=2 = 250G eV sign(�) = + 1

M 0 = 70G eV tan�(~M )= 10

A 0 = � 300G eV

The left colum n, listing the universalgaugino m ass

M 1=2,the scalar m ass M 0 and the trilinear coupling

A 0 [Yukawa couplings factored out],is de�ned atthe

G UT scale M G U T .The point is close to the original

Snowm ass point SPS1a [17];the scalar m ass param -

eter M 0 is lowered slightly at the G UT scale from

100G eV to70G eV and A 0 ischanged from � 100G eV

to � 300 G eV.Thevaluesofthe SM inputparam eters

are collected in Table 2.Extrapolation ofthe above

m SUG RA param etersdown to the ~M = 1 TeV scale

generatesthe M SSM Lagrangian param eters.Table 3

displaysthecouplingsand m assparam etersafterbeing

evolved from M G U T to ~M using the RG E partofthe

program SPheno [56]which isbased on two-loop anal-


Param eter SPS1a0value Param eter SPS1a

0value

g0

0:3636 M 1 103:3

g 0:6479 M 2 193:2

gs 1:0844 M 3 571:7

Y� 0:1034 A � � 445:2

Yt 0:8678 A t � 565:1

Yb 0:1354 A b � 943:4

� 396:0 tan� 10:0

M H d159:8 jM H u j 378:3

M L 1181:0 M L 3

179:3

M E 1115:7 M E 3

110:0

M Q 1525:8 M Q 3

471:4

M U 1507:2 M U 3

387:5

M D 1505:0 M D 3

500:9

Table 3.TheD R SUSY Lagrangian param etersatthescale~M = 1 TeV in SPS1a

0from [56][m ass unitin G eV;M

2

H u

negative].In addition,gauge and Yukawa couplings atthis

scale are given in the D R schem e.

Particle M ass[G eV ] �scale [G eV ]

h0 116:0 1:3

H0

425:0 0:7

~�01 97:7 0:4

~�02 183:9 1:2

~�04 413:9 1:2

~��

1183:7 1:3

~eR 125:3 1:2

~eL 189:9 0:4

~�1 107:9 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.Supersym m etric m assesforthe SUSY scale ~M =

1 TeV,and their variation if ~M is shifted to 0:1 TeV .

yses ofthe �-functions as wellas the other evolution

coe�cients(othercodescan be used equally well).

ThisSPS1a0 setiscom patiblewith allhigh-energy

m assbounds and with the low-energy precision data,

aswellaswith the observed CDM data,calculated as

B(b! s )= 3:0� 10� 4 [57],�[g� 2]�=2= 34� 10� 10 [58],

��SU SY = 2:1� 10� 4 [58],and C D M h2 = 0:10 [57].

The physical[pole]m asses ofthe supersym m etric

particlesarepresented in Table5.The connection be-

tween theLagrangianparam etersand thephysicalpole

m asses is presently encoded at the one-loop levelfor

them assesoftheSUSY particles,and atthetwo-loop

levelfor the Higgsm asses.Q CD e�ects on the heavy

quark m assesareaccounted forto two-loop accuracy.

A system aticcom parison with theotherpublicpro-

gram sISAJET[59],SOFTSUSY[60]and SuSpect[61]has

been perform ed in [62]to estim atethe technicalaccu-

racy that can presently be reached in the evolution.

The codesinclude fulltwo-loop RG Esforallparam e-

tersaswellasone-loop form ulasforthreshold correc-

tions.The agreem ent between the actualversions of

these calculations is in generalwithin one percent.A

specialcasearetheon-shellm assesoftheHiggsbosons

which have been calculated by FeynHiggs [58]start-

ing from the SPheno Lagrangian param etersasinput.

Here,discrepanciesforthe m assofthe lightestHiggs

boson am ountto 2% orm orewhich can beattributed

to di�erentrenorm alization schem es (see also [63]for

detailed discussions).

Besidesthecom parison between di�erentcodesfor

spectrum calculations,a crudeinternalestim ateofthe

theoreticalerrorsatthepresentleveloftheloop calcu-

lationsm ay beobtained by shiftingthem atchingpoint~M from 1 TeV down to 0.1 TeV.A sam ple ofparti-

cle m assshiftsassociated with such a variation ofthe

SUSY scaleparam eterisdisplayed in Table4.W ith er-

rorsatthepercentlevel,theexperim entalprecision at

LHC can bem atched in general.However,itisobvious

thatanotherorderofm agnitude,the per-m illevel,is

required in the theoreticalprecision to m atch the ex-

pected experim entalprecision atILC and in coherent

LHC/ILC analyses{ i.e.,calculationsofthenextloop

arecalled for1.

To perform experim entalsim ulations,the branch-

ing ratios ofthe decay m odes are crucial:these have

been calculated using FeynHiggs[58]and SDECAY [65];

sim ilar results m ay be obtained using CPSuperH [66].

The m ostim portantdecay channelsofthe supersym -

m etric particles and Higgs bosons in SPS1a0 are col-

lected in theAppendix,whilethecom pletesetisavail-

ablefrom theSPA web-site.Crosssectionsforthepro-

duction ofsquarks,gluinos,gauginosand sleptons at

theLHC arepresented asa function ofm assincluding

the point SPS1a0.Typicalcrosssectionsforpair pro-

duction ofcharginos,neutralinos and sleptons at the

ILC are presented for the pointSPS1a0 asa function

ofthe colliderenergy.

IfSPS1a0,ora SUSY param etersetin therangeof

sim ilarm assscales,isrealized in nature,a plethora of

interestingchannelscan beexploited toextracttheba-

sicsupersym m etry param eterswhen com bining exper-

im entalinform ation from sharp edgesin m assdistribu-

tionsatLHC with m easurem entsofdecay spectra and

threshold excitationcurvesatan e+ e� colliderwith en-

ergyup to1TeV [11].From thesim ulatedexperim ental

errorsthe data analysis perform ed coherently for the

two m achinesgivesriseto a very precisepictureofthe

supersym m etric particle spectrum asdem onstrated in

Table6.

1W ith � functionsand evolution coe�cientsin theRG Es

already available to third order[22],the calculation ofthe

two-loop orderfortherelation between theLagrangian pa-

ram eters and the physicalpole m asses have been carried

outin the approxim ation ofm asslessvectorbosons[64]


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 M ass[G eV] Particle M ass[G eV]

h0

116:0 ~�1 107:9

H0

425:0 ~�2 194:9

A0 424:9 ~�� 170:5

H+

432:7 ~uR 547:2

~�01 97:7 ~uL 564:7

~�02 183:9 ~dR 546:9

~�03 400:5 ~dL 570:1

~�04 413:9 ~t1 366:5

~�+

1183:7 ~t2 585:5

~�+

2415:4 ~b1 506:3

~eR 125:3 ~b2 545:7

~eL 189:9 ~g 607:1

~�e 172:5

Table 5. M ass spectrum of supersym m etric particles [56] and Higgs bosons [58] in the reference point SPS1a0. The

m asses in the second generation coincide with the �rstgeneration.

Particle M ass \LHC" \ILC" \LHC+ ILC"

h0

116:0 0:25 0:05 0:05

H0

425:0 1:5 1:5

~�01 97:7 4:8 0:05 0:05

~�02 183:9 4:7 1:2 0:08

~�04 413:9 5:1 3� 5 2:5

~��

1183:7 0:55 0:55

~eR 125:3 4:8 0:05 0:05

~eL 189:9 5:0 0:18 0:18

~�1 107:9 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. Accuraciesforrepresentative m assm easurem ents

of SUSY particles in individualLHC, ILC and coherent

\LHC+ ILC" analysesforthe reference pointSPS1a0[m ass

unitsin G eV].~qR and ~qL representthe avorsq = u;d;c;s.

[Errors presently extrapolated from SPS1a sim ulations.]

W hilethepicturesofarhad been based on evaluat-

ing the experim entalobservableschannelby channel,

globalanalysis program s have becom e available [67,

68]in which the whole setofdata,m asses,crosssec-

tions,branching ratios,etc.isexploited coherently to

extracttheLagrangian param etersin theoptim alway

after including the available radiative corrections for

m assesand crosssections.W ith increasing num bersof

observablesthe analysescan be expanded and re�ned

in a system atic way.The present quality ofsuch an

analysis[68]can be judged from the resultsshown in

Table 7.These errorsare purely experim entaland do

notincludethetheoreticalcounterpartwhich m ustbe

im proved considerablybeforem atchingtheexperim en-

talstandards.

Extrapolation to the G UT scale

Based on theparam etersextracted atthescale ~M ,we

canapproachthereconstructionofthefundam entalsu-

persym m etric theory and the related m icroscopic pic-

ture ofthe m echanism breaking supersym m etry.The

experim entalinform ation isexploited tothem axim um

extentpossiblein thebottom -upapproach[12]inwhich

the extrapolation from ~M to the G UT/Planck scale

is perform ed by the renorm alization group evolution

forallparam eters,with the G UT scalede�ned by the

uni�cation pointofthe two electroweak couplings.In

thisapproach the calculation ofloopsand � functions

governing the extrapolation to the high scale isbased

on nothing but experim entally m easured param eters.

Typicalexam plesfortheevolution ofthegaugino and

scalarm assparam etersare presented in Fig.1.W hile

the determ ination ofthe high-scale param etersin the

gaugino/higgsino sector,aswellasin the non-colored

slepton sector,is very precise,the picture ofthe col-

ored scalarand Higgssectorsisstillcoarse,and strong

e�ortsshould be m ade to re�ne itconsiderably.

O n theotherhand,ifthestructureofthetheory at

the high scale wasknown a prioriand m erely the ex-

perim entaldeterm ination ofthehigh-scaleparam eters

were lacking,then the top-down approach would lead

to a very precise param etric picture atthe high scale.

Thisisapparentfrom the�tofthem SUG RA param e-

tersin SPS1a0displayed in Table8 [67].A high-quality

�toftheparam etersisanecessarycondition,ofcourse,


1=M

i[GeV�1]

M2 ~ f[103GeV2]

Q [GeV] Q [GeV]

M� 1

3

M� 1

2

M� 1

1

Fig.1.Runningofthegaugino and scalarm assparam etersasa function ofthescaleQ in SPS1a0[56].O nlyexperim ental

errors are taken into account;theoreticalerrors are assum ed to be reduced to the sam e size in the future.

Param eter SPS1a0 value Fiterror[exp]

M 1 103.3 0:1

M 2 193.2 0:1

M 3 571.7 7:8

� 396.0 1:1

M L 1181.0 0:2

M E 1115.7 0:4

M L 3179.3 1:2

M Q 1525.8 5:2

M D 1505.0 17:3

M Q 3471.4 4:9

m A 372.0 0:8

A t {565.1 24:6

tan� 10.0 0:3

Table 7.Excerptofextracted SUSY Lagrangian m assand

Higgs param eters atthe supersym m etry scale ~M = 1 TeV

in the reference pointSPS1a0[m ass units in G eV].

forthe theory to be correct{ howeveritisnota su�-

cientcondition;deviationsfrom thetheorym ay hidein

largeerrorsofsom eobservableswhich do notspoilthe

quality ofthe �tin the top-down approach butwhich

arem anifestin the bottom -up approach.

Cold dark m atter

Constraintson SUSY cold darkm attercan beobtained

atLHC by specifying theunderlying scenarioand ana-

lyzing alldata sim ultaneously within thegiven bench-

m ark m odel.From a study ofthe SPS1a point,based

on very large statistics [69],one m ay expect thatthe

relicdensity can bedeterm ined to� 6% fortheSPS1a0

scenario.ForSPS1a0,the relicdensity dependson the

Param eter SPS1a0 value Experim entalerror

M G U T 2:47� 1016

G eV 0:02� 1016

G eV

�� 1

G U T24.17 0.06

M 1

2

250 G eV 0.2 G eV

M 0 70 G eV 0.2 G eV

A 0 -300 G eV 13.0 G eV

� 396.0 G eV 0.3 G eV

tan� 10 0.3

Table 8.Com parison ofthe idealparam eters with the ex-

perim entalexpectations in the top-down approach [68].

param etersoftheneutralinoand sferm ion sectorasthe

dom inantchannelsareannihilation ofneutralinosinto

ferm ion pairsand coannihilation with staus.In partic-

ular,forthe m ostsensitivecom ponent,coannihilation

processes,the relic density is essentially given by the

m assdi�erencebetween thelightestslepton ~�1 and the

LSP ~�01,which can be directly m easured at the ILC.

Studies of ~�1 production at threshold [70]and decay

spectra to ~�01 in the continuum [71]suggest that for

SPS1a0,even with m oderate lum inosity,a precision of

� 2% on thecold darkm atterabundanceisachievable.

A system aticanalysisofvariousscenariosisbeing car-

ried out in the LCC project [72]as wellas by other

groups.

6 SUM M ARY AN D O UTLO O K

Ifsupersym m etry isrealized in Nature,future experi-

m entsattheLHC and theILC willprovideveryprecise

m easurem entsofsupersym m etricparticle spectra and

couplings.O n thetheoreticalsidethesem easurem ents


m ustbe m atched by equally precise theoreticalcalcu-

lationsand num ericalanalysistools.TheSPA Project,

ajointtheoreticaland experim entale�ort,aim satpro-

viding

{ awell-de�nedfram eworkforSUSY calculationsand

data analyses,

{ allnecessary theoreticaland com putationaltools,

{ a testground scenario SPS1a0,

{ a platform forfutureextensionsand developm ents.

O n thisbasiscoherentanalysesofexperim entaldata

can be perform ed and the fundam entalsupersym m et-

ricLagrangian param eterscan beextracted.They can

serve as a �rm base for extrapolations to high scales

sothattheultim atesupersym m etrictheoryand thesu-

persym m etrybreakingm echanism canbereconstructed

from future data.

M uch work isstillneeded on theexperim entaland

theoreticalside to achieve these goals at the desired

levelof accuracy.Som e of the short- and long-term

subprojectshavebeen identi�ed and should bepursued

in the nearfuture.

The SPA Projectis a dynam icalsystem expected

to evolvecontinuously.The currentstatusofthe SPA

Project,nam es ofthe conveners responsible for spe-

ci�ctasksaswellaslinksto theavailablecalculational

tools,can be found atthe SPA hom e page

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

APPEN D IX

(a) D ecays ofH iggs and SUSY particles in SPS1a0

The branching ratiosofHiggsbosonsand SUSY par-

ticlesexceeding 2% arepresented in Tables9{12.The

com pletelisting including alldecaysisavailableon the

SPA web-sitehttp://spa.desy.de/spa/.

Higgs m ;� [G eV ] decay B decay B

h0 116:0 �

��+ 0:077 W W

� 0:067

4� 10� 3

b�b 0:773 gg 0:055

c�c 0:021

H0

425:0 ��+

0:076 ~�01 ~�

02 0:038

1:2 b�b 0:694 ~�02 ~�

02 0:020

t�t 0:052 ~�+

1~��

10:050

~��

1~��

20:030

A0 424:9 �

��+ 0:057 ~�01 ~�

02 0:054

1:6 b�b 0:521 ~�02 ~�

02 0:060

t�t 0:094 ~�+

1~��

10:163

~��

1~��

20:036

H+

432:7 ��+

0:104 ~�+

1~�01 0:143

0:9 t�b 0:672 ~��~�+

10:071

Table 9. Higgs m asses and branching ratios B > 2% in

SPS1a0from [58].

~� m ;� [G eV ] decay B decay B

~�01 97:7

~�02 183:9 ~e�

Re� 0:025 ~�e�e 0:116

0:083 ~��

1��

0:578 ~�� 0:152

~�03 400:5 ~�

�

1W

�0:582 ~�

01Z

00:104

2:4 ~�02Z0 0:224

~�04 413:9 ~��

2�� 0:033 ~�

�

1W

� 0:511

2:9 ~�e�e 0:042 ~�01Z

00:022

~�� 0:042 ~�02Z

00:024

~�01h

00:070

~�02h

00:165

~�+

1183:7 ~�

+

1�� 0:536 ~��

+ 0:185

0:077 ~�ee+

0:133

~�+

2415:4 ~e

+

L�e 0:041 ~�

01W

+0:063

3:1 ~�+

2�� 0:046 ~�02W

+ 0:252

~t1b 0:109 ~�+

1Z0

0:221

~�+

1h0

0:181

Table 10. Neutralino and chargino m asses, widths and

branching ratios B > 2% in SPS1a0from [65];branching

ratios for the second generation are the sam e as for the

�rstgeneration.




~‘ m ;� [G eV ] decay B decay B

~eR 125:3 ~�01e

�1:000

0:10

~eL 189:9 ~�01e� 0:925 ~�

�

1�e 0:049

0:12 ~�02e

�0:026

~�e 172:5 ~�01�e 1:000

0:12

~�1 107:9 ~�01�� 1:000

0:016

~�2 194:9 ~�01�

�0:868 ~�

�

1�� 0:086

0:18 ~�02�

�0:046

~�� 170:5 ~�01�� 1:000

0:12

Table 11.Slepton m asses,widthsand branchingratiosB >

2% in SPS1a0 from [65]; branching ratios for the second

generation are the sam e as for the �rstgeneration.

~q m ;� [G eV ] decay B decay B

~uR 547:2 ~�01u 0:990

1:2

~uL 564:7 ~�02u 0:322 ~�

+

1�d 0:656

5:5

~dR 546:9 ~�01d 0:990

0:3

~dL 570:1 ~�02d 0:316 ~�

�

1�u 0:625

5:4

~t1 366:5 ~�01t 0:219 ~�+

1b 0:719

1:5 ~�02t 0:062

~t2 585:5 ~�01t 0:042 ~�

+

1b 0:265

6:3 ~�02t 0:103 ~�

+

2b 0:168

~t1Z0

0:354

~t1h0

0:059

~b1 506:3 ~�01b 0:037 ~�

�

1t 0:381

4:4 ~�02b 0:295 ~t1W� 0:281

~b2 545:7 ~�01b 0:222 ~�

�

1t 0:178

1:0 ~�02b 0:131 ~t1W

�0:401

~�03b 0:028

~�04b 0:038

~g 607:1 ~uR �u 0:086 ~t1�t 0:189

5:5 ~uL �u 0:044 ~b1�b 0:214

~dR �d 0:087 ~b2�b 0:096

~dL �d 0:034

Table 12. M asses, widths and branching ratios B > 2%

ofcolored SUSY particles in SPS1a0from [65];branching

ratiosforthesecond generation arethe sam easforthe�rst

generation.

(b) LH C and ILC crosssections in SPS1a0

Totalcross sections are presented in Figs.2 { 6 for

SUSY particleproduction atthe 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.Totalcrosssectionsforsquark and gluino pairpro-

duction at the LHC [27,28] for �xed gluino m ass (top),

squark m ass (center), and gluino/squark m ass ratio (bot-

tom ) [�xed param eters corresponding to SPS1a0values].

Black circles indicate the SPS1a0m ass values. The Born

cross sections (broken lines)are shown for som e channels.


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. G eneric exam ples of total cross sections (Drell-

Yan and Com pton production)asa function ofthe average

m ass for production ofsleptons,charginos and neutralinos

atthe LHC [27,28].The Born cross sections (broken line)

are shown for com parison.

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.Totalcross section sections for chargino and neu-

tralino pairproduction in e+e�annihilation [73].TheBorn

cross 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. Totalcross sections for sm uon and selectron pair

production in e�e�annihilation [74].The Born cross sec-

tion (broken lines) isshown for com parison.

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 forleft-and right-handed polarized elec-

tron (Pe� = 0:8)and positron (Pe+ = 0:6)beam s[75].The

Born cross section (broken line) isshown for com parison.


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