A new treatment of mixed virtual and real IR-singularities

0.09 MB·English

by  Janusz Gluza

DESY08–007
8 SFB/CPP–08–08
0
0 HEPTOOLS08–013
2

n
a
J

8
A new treatment of mixed virtual and real
2

] IR-singularities
h
p
-
p
e
h
[

1 JanuszGluza
v DepartmentofFieldTheoryandParticlePhysics,InstituteofPhysics,
8
UniversityofSilesia,Uniwersytecka4,PL-40-007Katowice,Poland
2
E-mail: [email protected]
2
4
Tord Riemann
. ∗
1
DeutschesElektronen-Synchrotron,DESY,Platanenallee6,15738Zeuthen,Germany
0
8 E-mail: [email protected]
0
:
v We discussthedeterminationoftheinfraredsingularitiesofmassiveone-loop5-pointfunctions
i
X with Mellin-Barnes (MB) representations. Massless internal lines may lead to poles in the e
r expansionoftheFeynmandiagram,whileunresolvedmasslessfinalstateparticlesgiveendpoint
a
singularities of the phase space integrals. MB integrals are an elegant tool for their common
treatment.Anevaluationbytakingresiduesleadstoinversebinomialsums.
8thInternationalSymposiumonRadiativeCorrections(RADCOR)
October1-52007
Florence,Italy
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
MixedvirtualandrealIR-singularities TordRiemann
1. Introduction
Westudytheinfrared(IR)singularities ofsomemassiveone-loop n-pointFeynmanintegrals,
eeg E ddk T(k)
I= , (1.1)
ip d/2 Z (q21−m21)n1...(q2i −m2i)n j...(q2n−m2n)nn
by representing them with standard Feynman parameter integrals with characteristic F and U
forms:
I = eeg E(−(cid:213) 1ni)=N1nGG ((cid:0)nNi)n −d2(cid:1)Z01(cid:213)j=n1dxj xnjj−1d (1−i(cid:229)=n1xi)FU(x(x)N)Nn n−−d/d2 P(T),
(1.2)
with Nn =(cid:229) ni=1n i. Forone loop integrals, theU =(cid:229) xi maybe set toone. TheF-form isbilinear
in the x and may be represented in turn by a multiple Mellin-Barnes (MB) integral, using the
i
representation:
i¥ +R
1 1 G ( z)G (n +z)
(A(x)+B(x))n = 2p i dzA(x)z B(x)−n −z −G (n ) , (1.3)
i¥Z+R
−
wheretheintegration contour separates thepolesoftheG -functions.
Afterwards, theFeynmanparameters maybeintegrated outandonehastosolvetheresulting
MBintegral. Thisisingeneralquitenon-trivial. However,thereisaninteresting kindofproblems
where a systematic approach might be developed, namely the evaluation of the IR divergent parts
oftheFeynmanintegrals. Theyareatthebeginofthee -expansion(e =(4 d)/2)oftheFeynman
−
integralandsoofsmallerdimensionalityinthevariablesz. Infact,usuallyonesubtractsthemfrom
therestoftheintegralandtreatsthemseparately.
The MB representation allows to do this in a special way which might be of some practical
usefulness. Wewilldiscuss here only scalar one-loop functions, T(k)=1, but tensors don’t show
additional problems. For basic definitions and formulae we refer to [1–3] and references cited
therein. We use here and in the following the Mathematica packages AMBRE [1] and MB [4]
for the derivations of the MB representations and for the e -expansions. In section 2 we apply the
MB-approach tothemassiveBhabhavertexandboxfunctions andextracttheire -poles. Section3
containsthetreatmentofboththevirtuale -polesandtheendpointsingularitiesfromanunresolved,
massless particles in apentagon diagram of massive Bhabha scattering. Themethod may be gen-
eralizedtomorecomplexcases,includinghigherlooporders,butexplicitevaluationsbecomethen
moreandmorecomplicated.
2. Simplee -poles: MassiveQEDvertex andbox
Wewillsetm=1,ands,t aretheusualMandelstam variables. TheQEDvertexfunction has
theF-form:
F(s)=[X[2]+X[3]]2+[ s]X[2]X[3], (2.1)
−
2
MixedvirtualandrealIR-singularities TordRiemann
leading, without a continuation in e , to a one-dimensional MB representation and a series over
residues [2]:
i¥ 1/2
1 eeg E − − G 2( z)G ( z+e )G (1+z)
V(s) = 2se 2p i dz (−s)−z − G−( 2z)
i¥ Z1/2 −
− −
eeg E (cid:229)¥ sn G (n+1+e )
= . (2.2)
− 2e 2n (2n+1) G (n+1)
n=0 n
Thecompleteseriesmaybesummed(cid:0)dir(cid:1)ectlywithMathematica1,andthevertexbecomes:
eg
e E
V(s)= G (1+e ) F [1,1+e ;3/2;s/4]. (2.3)
− 2e 2 1
Alternatively,onemayderivethee -expansionbyexploitingthewell-knownrelationwithharmonic
numbersS (n)=(cid:229) n 1/ik:
k i=1
G (n+ae ) = G (1+ae )exp (cid:229)¥ (−ae )kS (n 1) . (2.4)
G (n) − k k −
" k=1 #
Theproduct exp(eg )G (1+e )=1+1z [2]e 2+O(e 3)yields expressions withzeta numbers z [n],
E 2
and, taking all terms together, one gets a collection of inverse binomial sums2; the firstof them is
theIRdivergent part:
V (s)
1
V(s) = −e +V0(s)+ (2.5)
···
1 (cid:229)¥ sn 14arcsin(√s/2)
V (s) = = . (2.6)
−1 2 2n (2n+1) 2 √4 s√s
n=0 n −
Thisprocedure applies similarly to(cid:0)the(cid:1)Bhabha box diagram [7]. Wetake for definiteness the
schannelscalarloopintegral. TheF-formis(againwithm=1):
F(s,t)=[X[2]+X[4]]2+[ s]X[1]X[3]+[ t]X[2]X[4], (2.7)
− −
andanMBrepresentation is,aftercontinuation tosmalle ,asumoftwoterms:
+i¥ 7/16
B(s,t) = (−2ss)t−e G [1+G [e ]2G e[−] e ]2e2egp Ei − dz1(−t)−z1G 3[−G z[1]G2[z1]+z1] (2.8)
− i¥ Z7/16 − 1
− −
+i¥ 3/4
+ 1 1 eeg E − dz s z1G [ z ]G [ 2(1+e +z )]G [1+z ]2
t2G [ 2e ](2p i)2 1 t − 1 − 1 1
− i¥ Z3/4 (cid:16) (cid:17)
− −
+i¥ 7/16
− G [ 1 e z z ]2
× dz2(−t)−e−z2G [−z2]G [ −2(1−+e−+1z−+2z )]G [2+e +z1+z2].
i¥ Z7/16 − 1 2
− −
1TheexpressionforV(s)wasalsoderivedin[5];seeadditionally[6].
2Forthefirstfourtermsofthee -expansionintermsofinversebinomialsumsorofpolylogarithmicfunctions,see
[2].
3
MixedvirtualandrealIR-singularities TordRiemann
Duetothepre-factors, G [ e ]2/G [ 2e ]= 2/e +2z [2]e +O(e 2)and1/G [ 2e ]= 2e +4g e 2+
E
− − − − −
O(e 3),onlythefirstintegralcontributes tothefirsttwotermsofthee -expansion,
1 V (t)
B(s,t) = e −ln(−s) (−1s) +O(e ), (2.9)
(cid:20) (cid:21) −
whereV isfrom(2.6),andtheIRdivergency is:
1
−
V (t)
1
B 1(s,t) = − . (2.10)
− ( s)
−
Wereproduceherethewell-knownfactthatIR-divergences ofverticesandboxesarealgebraically
related, seee.g. [3].
3. Mixedvirtual andreal IR-singularities: massiveBhabha pentagon
Things become more interesting for pentagon diagrams (se also [3, 8, 9]). We again use
Bhabhascattering asanexample. AcompactF-formis:
F(s,t,t ,V ,V ) = (x +x +x )2+[ s]x x +[ V ]x x +[ t]x x +[ t ]x x +[ V ]x x .
′ 2 4 2 4 5 1 3 4 3 5 2 4 ′ 2 5 2 1 4
− − − − −
(3.1)
It exhibits a set of five invariants (out of a set of 10 scalar products at choice) describing the
kinematics ofa2 3process (hereassuming afinalstateemission ofanunresolved photon from
→
an s channel box diagram). The s,t,t are the usual Mandelstam variables for the fermions in
′
e+e e+e g ,and:
− −
→
Vi=2pfipg , i=1,...4. (3.2)
TheV areproportional totheenergyofthepotentially unresolved masslessparticle.
i
From a subsequent phase space integration, we have to expect endpoint singularities arising
from terms proportional to 1/V2 1/Eg and 1/V4 1/Eg , so we have to control, for a complete
∼ ∼
treatment of the IR-problem, not only the e -expansion, but also the first terms of theV ,V expan-
2 4
sionsforsmallV ,V .
2 4
Infact,theF-form(3.1),writtenhereinitsshortest form,depends onthosetwoofthefourV
i
whicharerelatedtothephasespaceofthegiventopology.
TheMB-representationisausefultoolforthatproblem. FortheIRlimit,wemayapproximate
t =t,andthescalarpentagon maybewrittenas:
′
12
I5 = (−2peegi)E4 (cid:213) 4 +i¥ +uidzi(−s)z2(−t)z4(−V2)z3(−V4)−3−e−z1−z2−z3−z4G j(cid:213)G=1G G j ,
i=1 i¥Z+ui 0 13 14
−
(3.3)
with u =( 5/8, 7/8,...) and with a normalization G =G [ 1 2e ], and the other G -functions
i 0
− − − −
are:
G = G [ z ], G =G [ z ], G =G [ z ], G =G [1+z ],
1 1 2 2 3 3 4 3
− − −
4
MixedvirtualandrealIR-singularities TordRiemann
G = G [1+z +z ], G =G [ z ], G =G [1+z ], G =G [ 1 e z z ],
5 2 3 6 4 7 4 8 1 2
− − − − −
G = G [ 2 e z z z z ], G =G [ 2 e z z z ],
9 1 2 3 4 10 1 3 4
− − − − − − − − − − −
G = G [ e +z z +z ], G =G [3+e +z +z +z +z ], (3.4)
11 1 2 4 12 1 2 3 4
− −
and,inthedenominator:
G = G [ 1 e z z z ], G =G [ e z z +z ]. (3.5)
13 1 2 4 14 1 2 4
− − − − − − − −
Leavingoutherethedetailsofderivation(see[3]forthat),wejustmentionthatwehavetoconsider,
aftercontinuationine ,elevenMB-integrals,beingatmost4-dimensional(fort =t). Theresulting
′
IR-sensible partis:
IIR = IIR(V )+IIR(V ), (3.6)
5 5 2 5 4
Is (V)
I5IR(Vi) = −1e i +I0s(Vi). (3.7)
Thee -poleisagainproportional tothatofthevertex:
I−s1e(Vi) = 2sV1 e (cid:229)¥ (t)n = Vs−V1(et), (3.8)
i n=0 2n i
(2n+1)
n
!
and:
Is(V) = 1 (cid:229)¥ (t)n [ 2ln( V) 3S (n)+2S (2n+1)], (3.9)
0 i 2sVi n=0 2n − − i − 1 1
(2n+1)
n
!
where wehave tounderstand ln( V)=ln(V/s)+ln[ (s+id )/m2]. Theseries forIs(V)maybe
− i i − 0 i
summed up in terms of polylogarithmic functions with the aid of Table 1 of Appendix D of [10],
seealso[3].
Equations(3.6)–(3.9)arethemainphysicalresultofthestudy. Onemayexpressthecomplete
IR-divergent part of an amplitude with 5-point functions in terms of those expressions, subtract
it from the complete, divergent amplitude and get a matrix element, which is integrable in four
dimensions.
In the rest of this short write-up we would like to demonstrate why we have here besides
the harmonic numbers S (n) also those of the kind S (2n+1). As mentioned, the scalar 5-point
1 1
function maybewrittenasasumofelevenMB-integralsaftere -continuation, beforee -expansion.
Two types of them contribute to the IR-part (there are four such integrals [in an ad-hoc notations
J ,J ,J ,J ],butwithasymmetryV V ). Thefirstoneis:
3 4 7 9 2 4
↔
+i¥ 5/8
J7 = −(V2s/Vs)2e G [−2e ]G [1+2e ]e2egp Ei − dz(−t)−1−zG [e −z]G [2G e[2−e z]G2[z−]z]G [1+z]
4 i¥ Z5/8 −
− −
= (V2/s)2e eeg Ee √p G [−2e ]G [2e ]G [1+2e ] F [1,1+2e ,3/2+e ,t/4]
− sV 22e G [3/2+e ] 2 1
4
5
MixedvirtualandrealIR-singularities TordRiemann
= (V2/s)2e eeg E G [ 2e ]G [1+2e ] (cid:229)¥ tn 1G [e +n]G [2e +n]. (3.10)
− sV − − G [2e +2n]
4 n=1
The J is proportional to 1/V . We have a second integral of the same type, being proportional to
7 4
1/V :
2
J = J (V V ). (3.11)
3 7 4 2
↔
TheothertypeofintegralsJ ,J ,with:
4 9
J = J (V V ), (3.12)
4 9 4 2
↔
aretwo-foldMB-integrals:
+i¥ 5/8 +i¥ 7/8
J9 = G [G [−12e2]e ](2epeg iE)2 − dz1 − dz2(−s)z2(−t)e−z1+z2(−V2)−1−2e−z2(−V4)−2−z2
− − i¥ Z5/8 i¥ Z7/8
− − − −
G [ z ]G [ 1 z ]G [ 2e z ]G [ 1 e z z ]G [ e +z z ]G [ z ]
1 A 2 2 1 2 C 1 2 2
× − − − − − − − − − − − −
G [2+z ] G [1+2e +z ] G [1+e z +z ]
2 B 2 1 2
− . (3.13)
× G [ 2z ] G [ 1 2e 2z ]
1 2
− − − −
The integral looks like being, in the limit V 0, too singular. This limit is an endpoint of the
4
→
phase space integration. Let us close the contour to the left. Weshift now the integration contour
in z to the left, raising in this way the (real part of the) power of ( V ) to a value which makes
2 4
−
thephotonphasespaceintegralexplicitelyintegrableatV 0ind=3 2e spacedimensions. If
4
→ −
singularities oftheintegrand (from G -functions) atsomevalues z =z arecrossed onehastoadd
2 R
thecorresponding residues JR(z ),sogettingone-dimensional MB-integralstobeconsidered:
9 R
J =2p i(cid:229) JR(z )+Jshift. (3.14)
9 9 R 9
zR
shift
The resulting integral J differs from J only by the shifted integration path, but will now not
9 9
contribute to the IR-singular part and will not be considered here any more. We see that only the
residues of crossed singular points in z contain the IR-relevant endpoint singularities in V ,V .
2 2 4
Here, twoof them (at z = 1 [argument of G in (3.13)] and at z = 1 2e [argument of G in
2 A 2 B
− − −
(3.13)])contribute duetoashiftfrom´ z = 7/8to´ z = 7/8 1= 15/8. Thefirstofthem
2 2
− − − −
is:
( V2)−2e G [ 2e ]G B[2e ]eeg E
J = − −
9A − sV G [ 1 2e ] 2p i
4
− −
+i¥ 5/8
− G [ 2e z ]G [ e z ]G [ z ]G [1+z ])
× dz1(−t)−1−z1 − −G [11 −2e−]G [1 2e− 12zC)] 1
i¥ Z5/8 − − − − 1
− −
( V ) 2e 22e e √p G [ 2e ]2G [2e ]
= − 2 − eeg E − B F [1,1 2e ,3/2 e ,t/4]. (3.15)
sV G [ 1 2e ]G [3/2 e ] 2 1 − −
4
− − −
Weperformedhereanirrelevantshiftz z +e inordertomaketheargumentofG independent
1 1 C
→
ofe . ThiswillbeheretheonlyG -functionproducingresiduesinz whenclosingagainthecontour
1
6
MixedvirtualandrealIR-singularities TordRiemann
totheleft. Theexplicite -expansionofthehypergeometricfunctionin(3.15)maybeobtainedwith
the Mathematica package HypExp2 [11, 5]. Alternatively, we may perform the sum of residues
arising fromG (1+z )directly:
C 1
J9A = −(−VsV2)−2e eeg EG G [−[ 21e ]G 2[2ee]] (cid:229)¥ tn−1G [−2G e[+2ne]G+[−2ne]+n], (3.16)
4 − − n=1 −
and apply then (2.4) to it. It is here where we may see why the harmonic numbers S (2n+1)
1
appear, which were not contributing to the vertex or the box: There was no G -function with an
e -shifted, doubled argument in the denominator of the final sums, while here this appears. In the
general massivecase,thiswillusuallyhappen.
The second residue crossed by the contour shifting in the z -plane gives the third kind of
2
contribution tobeadded;afterashiftz z e :
1 1
→ −
+i¥ 5/8
J = (V4/s)2e G [1 2e ]G [ 2e ]G [2e ]eeg E − dz ( t) 1 z1
9B sV − − A 2p i 1 − − −
4 i¥ Z5/8
− −
G [e z ]G [2e z ]G [ z ]G [1+z ]
1 1 1 C 1
− − −
× G [ 1 2e ]G [2e 2z ]
1
− − −
(V /s)2e G [1 2e ]G [ 2e ]G [2e ]2
= 4 eeg Ee √p − − F [1,1+2e ,3/2+e ,t/4]
sV 22e G [ 1 2e ]G [3/2+e ] 2 1
4
− −
= (V4s/Vs)2e eeg EG [1−G2[e ]G1[−22ee ]]G [2e ] (cid:229)¥ tn−1G [e G+[2ne]G+[22en+] n] (3.17)
4 − − n=1
Again,thismaybeexpandedintoane -seriesoverinversebinomialsumsbyuseof(2.4).
Collecting everything together, werediscover (3.6) (plus additional terms ofno relevance for
theIR-treatment):
1
J +J +J +J +J +J = Is (V )+Is (V ) +Is(V )+Is(V )+ (3.18)
3 4A 4B 7 9A 9B e −1 2 −1 4 0 2 0 4 ···
(cid:2) (cid:3)
4. Conclusions
WegaveapedagogicalintroductiontothetreatmentofmixedrealandvirtualIR-singularities.
This kind of problems arises in NNLO problems, where one has to treat the unresolved mass-
less particle phase space for loop integrals. The presented method was exemplified for a scalar
integral, but it may be easily applied to general tensor functions. For the QED pentagon, this
is discussed in [3], which may be considered as an introduction to this presentation. A deriva-
tion of MB-representations for higher n point functions or for multi-loop integrals is more or less
straightforward, although an analytical evaluation will become more and more troublesome. It
is an interesting open question how useful the MB techniques might appear for realistic, so far
unsolved applications.
Acknowledgments
WewouldliketothankJ.Fleischerforusefuldiscussions.
Thepresentworkissupported inpartbytheEuropean Community’sMarie-CurieResearchTrain-
ingNetworksMRTN-CT-2006-035505‘HEPTOOLS’andMRTN-CT-2006-035482‘FLAVIAnet’,
7
MixedvirtualandrealIR-singularities TordRiemann
andbySonderforschungsbereich/Transregio 9–03ofDeutscheForschungsgemeinschaft ‘Comput-
ergestützte Theoretische Teilchenphysik’.
References
[1] J.Gluza,K.Kajda,andT.Riemann,AMBRE-aMathematicapackagefortheconstructionof
Mellin-BarnesrepresentationsforFeynmanintegrals,Comput.Phys.Commun.177(2007)879–893,
[arXiv:0704.2423 [hep-ph]].
[2] J.Gluza,F.Haas,K.Kajda,andT.Riemann,AutomatizingtheapplicationofMellin-Barnes
representationsforFeynmanintegrals,PoS(ACAT)(2007)081,arXiv:0707.3567 [hep-ph].
[3] J.GluzaandT.Riemann,Newresultsfor5-pointfunctions,Proc.LCWS,Hamburg(2007)
[arXiv:0712.2969 [hep-ph]].Seealsoat
http://www-zeuthen.desy.de/ riemanns/LCWS07-Loops.htm.
∼
[4] M.Czakon,AutomatizedanalyticcontinuationofMellin-Barnesintegrals,Comput.Phys.Commun.
175(2006)559–571,[hep-ph/0511200].
[5] T.HuberandD.Maitre,HypExp2,expandinghypergeometricfunctionsabouthalf-integer
parameters,arXiv:0708.2443 [hep-ph].
[6] A.DavydychevandM.Kalmykov,Newresultsforthee -expansionofcertainone-,two-and
three-loopFeynmandiagrams,Nucl.Phys.B605(2001)266–318,
[http://arXiv.org/abs/hep-th/0012189].
[7] J.Fleischer,J.Gluza,A.Lorca,andT.Riemann,FirstorderradiativecorrectionstoBhabha
scatteringind dimensions,Eur.J.Phys.48(2006)35–52,[hep-ph/0606210].
[8] J.Fleischer. ApplicationofMellin-Barnesrepresentationtothecalculationofmassivefive-point
functionsinBhabhascattering,talkgivenattheConferenceonFrontiersinPerturbativeQuantum
FieldTheory,June14-162007,ZiF,Bielefeld.
[9] J.Fleischer,J.Gluza,K.Kajda,andT.Riemann,PentagondiagramsofBhabhascattering,ActaPhys.
Polon.B38(2007)3529,[arXiv:0710.5100 [hep-ph]].
[10] A.DavydychevandM.Kalmykov,MassiveFeynmandiagramsandinversebinomialsums,Nucl.
Phys.B699(2004)3–64,[hep-th/0303162].
[11] T.HuberandD.Maitre,HypExp,aMathematicapackageforexpandinghypergeometricfunctions
aroundinteger-valuedparameters,Comput.Phys.Commun.175(2006)122–144,
[hep-ph/0507094].
8