Double Parton Distributions of the Pion

3.9 MB·

by  Christian Zimmermann

Double Parton Distributions of the Pion
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1 Christian Zimmermann∗†
0 UniversityofRegensburg
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E-mail: [email protected]

n
a
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The effects of double hard interactions are no longer negligible at energy scales reached at the
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1 LHC.Doublepartonscattering(DPS)processesareoftendescribedbytakingtheproductoftwo
singlepartonscatteringprocessesassumingthatinterferenceeffectsareverysmall. Wecalculate
]
t fourpointfunctions(4pt-functions),whichappearinthetheDPScrosssection,employinglattice
a
l techniques. Weconsiderapionatrestandtestthevalidityoftheafore-mentionedfactorization
-
p
assumption by convoluting two pion form factors and comparing the result to the 4pt data. For
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h our calculations we use a N =2 gauge ensemble on a 403×64 lattice, with lattice spacing
f
[
a=0.071fmandpionmassmπ =288.8MeV.
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a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speaker.
†forRQCD
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
DPDsofthePion ChristianZimmermann
1. Introduction
Hadron-hadron collisions are an important subject when discovering new physics at high en-
ergyscales. Thereforethecorrectdescriptionoftheunderlyingeventintheseinteractionsisessen-
tial. At energy scales reached in LHC, multi parton interactions (MPIs) are no longer negligible.
The most important MPI processes are double parton scattering (DPS) processes, which are often
naivelydescribedbyneglectinganyinterferenceeffects:
dσ dσ
SPS SPS
dσ = , (1.1)
DPS
Cσ
eff
whereσ isthesinglepartonscattering(SPS)crosssection,andσ issomeeffectivecrosssec-
SPS eff
tion,whichisrelatedwiththehadronsize.C=2or1denotessomesymmetryfactor[1].
A more fundamental description can be done by introducing so-called double parton distributions
(DPDs), which involve two-operator matrix elements. Our intention is to obtain these matrix ele-
mentsfromcalculationsof4pt-functionsonthelattice,startingwiththepionatrest. Wealsowant
tochecktowhatextentthe"naivefactorization"ansatz(1.1)isvalidbycomparingtheconvolution
oftwoformfactorswithourlatticeresultsforthe4pt-functions.
2. DoublePartonScatteringandDoublePartonDistributions
Forthedescriptionofadoublepartonscattering(DPS)process,itisusuallyassumedthatthe
processfactorizesintoasoftandahardpart. Inthatcase,onecanwritethecrosssectionas(inthe
followinglightconecoordinatesareused):
dσ σ σ (cid:90)
= ∑ 1 2 d2y F(x,y )F(x¯,y ).
dx dx¯ dx dx¯ C ⊥ i ⊥ i ⊥ (2.1)
1 1 2 2 polarization
flavor
A derivation from first principles is given in [2]. In the equation above the σ denote the parton
i
scattering cross sections. The variable y can be interpreted as the distance between the scattering
partons. F(x,y ) are the so-called collinear double parton distributions (DPDs), which involve
i ⊥
hadronicmatrixelementsoftwolightconeoperators. ThesecontainWilsonlines,onceweinclude
higherordercontributions.
This is different for Mellin moments of DPDs, where the two parton momentum fractions are
integrated out, which has the consequence that the operators become local and therefore Wilson
linesdonotappearanymore. Latticecalculationsbecomemorefeasibleinthiscase. Explicitlyone
findsforthelowestmoment:
(p+)−1(cid:90)
Mp(y)= dy−Mp(y) (2.2)
2
(cid:12)
Mp(y)= (cid:104)h(p)|Of1f1(cid:48)(0)Of2f2(cid:48)(y)|h(p)(cid:105)(cid:12) (2.3)
1 2 (cid:12)
y+=0
Off(cid:48)(y)=q¯f(y)Γqf(cid:48)(y), (2.4)
1,2
where h(p) denotes a specific hadron with momentum p. Γ is a combination of Dirac matrices,
whichdependsonthepolarizationofthequarkstakingpartontheinteraction. Wewillonlydiscuss
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DPDsofthePion ChristianZimmermann
thevectorandaxialvectorcaseandfurthermorethescalarandpseudoscalarchannels,althoughthe
lattertwocorrespondtohighertwistcontributions.
In the following we will use decompositions of the matrix elements w.r.t. their Lorentz structure
intoinvariantfunctions,e.g.:
Mp (y)=2m2A (py,y2) (2.5)
SS/PP SS/PP
(cid:104) (cid:105)
T Mp,{µν}(y)=(cid:2)2pµpν−1gµνp2(cid:3)A (py,y2)+m2 2p{µyν}−1gµνpy B (py,y2)
VV 2 VV 2 VV
(2.6)
+m4(cid:2)2yµyν−1gµνy2(cid:3)C (py,y2),
2 VV
where T denotes trace subtraction and the {}-notation indicates symmetrization w.r.t. Lorentz
indices.
Asmentionedbeforeweintendtocheckthevalidityofthefactorizationassumption(1.1), i.e.the
factorization of a two-operator matrix element into two one-operator matrix elements. It seems
naturalthatasuitablewayistoinsertacompletesetofintermediatestates,assumingthatstatesof
theobservedhadrondominate:
(cid:90) d4p(cid:48)
(cid:104)h(p)|O O |h(p)(cid:105)≈ (cid:104)h(p)|O |h(p(cid:48))(cid:105)(cid:104)h(p(cid:48))|O |h(p)(cid:105)δ(p(cid:48)2−m2). (2.7)
1 2 (2π)4 1 2
Thisansatzonecanapplytolightconematrixelementsaswellastomatrixelementsoflocaloper-
ators. WewillcheckbothpossibilitiesinSection4.
3. DPDsfromLatticeStudies
C1 A
C2
π+(0) O1(τ,0) π+(t) π+(0) O1(τ,0) π+(t) π+(0) O1(τ,0) π+(t)
O (τ,y)
O (τ,y) 2 O (τ,y)
2 2
D
S1 S2 O (τ,0)
1
π+(0) O (τ,0) π+(t)
1
π+(0) O (τ,0) π+(t)
1
π+(0) O (τ,y) π+(t)
2
O (τ,y)
2
O (τ,y)
2
Figure1: DepictionofthesixpossibleindependentWickcontractions.
We are able to obtain some information about the matrix elements (2.3) by calculating 4pt-
functions on the lattice. We start with the calculation for the pion with zero momentum taking
only channels containing no derivative terms corresponding to the first Mellin moment. These
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DPDsofthePion ChristianZimmermann
Ensemble β a[fm] κ V m [GeV] N(N ) N
π 4pt sm
I 5.29 0.071 0.13632 403×64 0.2888(11) 2025(984) 400
Table1: Detailsoftheensembleusedforthesimulation. N labelsthenumberofconfigurations, which
4pt
areusedforour4pt-calculations. N denotesthenumberofsmearingiterations.
sm
S P V A T
Z 0.4577(18) 0.3538(92) 0.7365(48) 0.76487(64) 0.9141(26)
Z 1.3543 1.3543 1 1 0.93313
conv
Table2: RenormalizationfactorsZandfactorsZ fortheconversiontotheMS-scheme. S,P,V,AandT
conv
denotethedifferentoperatorinsertiontypes.
matrixelementscanbeobtainedbycalculatingthefollowingratio,inalimit,whereexcitationsare
suppressed(thisisanalogoustothecalculationofone-operatormatrixelements,seee.g.[3]):
(cid:12)
M(y)= 2mC4pt(t,τ,y)(cid:12)(cid:12) , (3.1)
C (t) (cid:12)
2pt 0(cid:28)τ(cid:28)t
whereC (t)istheusualpion2pt-functionandC (t,τ,y)denotesthe4pt-function:
2pt 4pt
C (t,τ,y)=(cid:104)O (t)O (0,τ)O (y,τ)O† (0)(cid:105) (3.2)
4pt π+ 1 2 π+
C (t)=(cid:104)O (t)O† (0)(cid:105). (3.3)
2pt π+ π+
O† (t)denotesthepioninterpolator,whereweuse:
π+
1
O† (t)= ∑u¯(x,t)γ d(x,t). (3.4)
π+ V 5
x
For the calculation of the 4pt-function, we have to consider six independent Wick contractions,
which are depicted in Figure 1. The two operator insertions are placed at the same time slice,
whichhasatimeseparationτ tothepionsource. Forthetimeseparationbetweensourceandsink
we chooset =15a. Hence we expect a plateau in the region 6(cid:46)τ (cid:46)9. For the evaluation of the
4pt-graphsweuseanensembleofN =2gaugeconfigurations(seeTable1, sameasensembleV
f
in [4]). The gauge fields are smoothed using APE smearing [5]. For our calculations we use the
One-End-Trick incorporating stochastic Z ⊗Z -sources [6], which are improved by Wuppertal
2 2
smearing [7]. We renormalize our results and convert it to the MS-scheme at a scale µ =2GeV
usingtherenormalizationandconversionfactorslistedinTable2(see[8]).
ForthegraphsC1,C2,S1andAweobtainclearnon-zerosignals,whileforthegraphsS2and
Dthesignalsareverynoisy. ForsomematrixelementstheseresultsareshowninFigure2. Weare
able to project out the invariant functions appearing in (2.5) and (2.6), which are not shown here
(theC1dataofthescalarandvectorchannelsisshowninFigure3comparedwiththeformfactor
convolutions). Notice that disconnected graphs are less suppressed in the (pseudo)scalar channel
thaninthe(axial)vectorchannel.
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DPDsofthePion ChristianZimmermann
Resultsfor𝑀SS(𝐿=40) ResultRsefsourlt𝑀sfSoSr(𝑀logPsPca(l𝐿e,=𝐿4=0)40) Resultsfor𝑀PP(logscale,𝐿=40)
20 15
S1 S1
4−2(|𝑦|)[10GeV]S −1150505 4𝑀(|𝑦|)|[GeV]SS4−2(|𝑦|)[10GeV]P111000−−−−13215005 CCA12 4𝑀(|𝑦|)|[GeV]PP 111000−−−321 CCA12
𝑀S−10 CS11 |𝑀P−10 CS11 |
C2 C2
−15 A 10−−145 A 10−4
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.02.2 0.04.4 0.06.6 0.08.8 11 1.12.2 1.14.4 1.16.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
|𝑦|[fm] |𝑦||𝑦[|f[mfm]] |𝑦|[fm]
2 Resultsfor𝑀V0V0(𝐿=40) 10−12 ResultRsefosrul𝑀tsVfo0Vr0𝑀(lAo0gAs0ca(l𝐿e,=𝐿4=0)40) 10−1 Resultsfor𝑀A0A0(logscale,𝐿=40)
S1 S1
4eV] 1.15 4V]4eV]10−12.5 CCA21 4V] 10−2 CCA21
G eG 1 e
−2(|𝑦|)[10V0−00..055 (|𝑦|)|[GVV−2(|𝑦|)[1000A01100−−043.05 (|𝑦|)|[GAA001100−−43
𝑀V0 −1 CS21 |𝑀𝑀A010−−05.5 CS21 |𝑀 10−5
C1 C1
−1.5 A 10−−61 A 10−6
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.02.20.04.40.06.60.08.8 11 1.12.21.14.41.16.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
|𝑦|[fm] |𝑦||𝑦[f|m[fm] ] |𝑦|[fm]
Figure 2: Results for two-operator matrix elements for the channels SS, PP, V V and A A . Here the
0 0 0 0
resultsofthegraphsC1,C2,S1andAareshownseparately.
4. NaiveFactorization
Finallywewanttocheckthe"naivefactorization"intoone-operatormatrixelements. Apply-
ing(2.7)tomatrixelementsoflightconeoperatorsyieldsthefollowingrelation(test1):
η (cid:90) 1 (cid:18) ζ(cid:19)2 (cid:90) d2r
A (py=0,y2)≈ C dζ 1− (1−ζ)−1 ⊥ e−iy⊥·r⊥F (t(ζ,r )), (4.1)
VV/SS π 2 (2π)2 V/S ⊥
0
where t(ζ,r ) = −(ζ2m2+r2)/(1−ζ) and η = 1 for the scalar case and −1 for the vector
⊥ ⊥ C
case. Ontheotherhand(2.7)maybeinsertedinamatrixelementoflocaloperators,whichcanbe
broughtintothefollowingform(test2):
(cid:90) ∞ d(r2) sin(|y||r|)(cid:0)m+E(r2)(cid:1)2
(cid:104)π+(p)|Ouu(0)Odd(y)|π+(p)(cid:105)≈− F2(t(r2))
V0 V0 4π2|y| 2E(r2) V
0 (4.2)
(cid:90) ∞ d(r2) sin(|y||r|)
(cid:104)π+(p)|Ouu(0)Odd(y)|π+(p)(cid:105)≈ F2(t(r2)),
S S 4π2|y| 2E(r2) S
0
√
whereE(r)= r2+m2 andt(r)=2m2−2mE(r). Bothtestsinvolvetheelectromagnetic(F (t))
V
or scalar form factor (F (t)) of the pion, respectively. These we obtain from 3pt-functions on the
S
lattice. Hereweconsiderforthemomentonlyconnectedgraphsandusecontributionsofmomenta,
√
forwhich|p|≤ 2π 3. Noticethatbothteststriviallyfailforthepseudoscalaroraxialvectorcase,
L
sincethecorrespondingformfactorsareexactlyzeroforsymmetryreasons. Thisis,however,not
thecasefortwo-operatormatrixelements(seeSection3).
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DPDsofthePion ChristianZimmermann
# quantity F M[GeV] p χ2/DOF
0
1 F 1(fixed) 0.777(12) 1(fixed) 6.010
em
2 1(fixed) 0.872(16) 1.173(69) 4.400
3 F 2.222(19)GeV 1.314(39) 1(fixed) 7.886
scal
4 2.212(19)GeV 2.023(50) 2(fixed) 9.877
Table3:Resultsforthefitsontheelectromagneticorscalarpionformfactor. Allfitstakeintoaccounterror
correlations. Thelargestcontributingvalueof|t|inthefitis1.143GeV2.
To perform the integrals in (4.1) and (4.2), we use the following parametrization of the pion form
factor:
(cid:16) t (cid:17)−p
F(t)=F 1+ . (4.3)
0 M2
The parameters F and M may be obtained from a covariant fit on the 3pt data (in the el. m. case
0
F wasfixedto1accordingtochargeconservation). Togetanestimateofoursystematicerror,we
0
performseveralfitsvaryingthefixedvalueof p(fortheel.m.formfactorwealsotreat pasafree
fitparameter,sincethedataqualityishighenough). ThefitresultsareshowninTable3.
Comparisonfor𝐴uVdV:3pt∗3ptto4pt(𝐿=40) Comparisonfor𝐴uSdS:3pt∗3ptto4pt(𝐿=40)
0 3.0
4ptdata
−1 3pt∗3pterror(𝑝=1)
2V] −2 2V] 2.5 33pptt∗∗33pp3ttpcctoo∗nn3vvpootlluuettriirooonnr(((𝑝𝑝𝑝===212)))
Ge −3 Ge 2.0
ud−3𝐴(|𝑦|)[10VV −−−−−87654 3pt∗3p3t3pcpto∗tn∗3v3poptltueertriroornor4r(p((𝑝𝑝𝑝tf=d=rae11tea))) ud−2𝐴(|𝑦|)[10SS 011...505
3pt∗3ptconvolution(𝑝free)
−9 0.0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
𝑚−1 𝑚−1
𝜋 |𝑦|[fm] 𝜋 |𝑦|[fm]
Figure3: Resultsforthefirstnaivefactorizationtestaccordingto(4.1)forourtwofitresults. Theconvolu-
tionsarecomparedwiththe4ptdataforA andA .
VV SS
0 Comparisonfor𝑀V4V4:3pt∗3ptto4pt(𝐿=40) ComparisCoonmfoprar𝑀isVon4Vf4or(l𝑀ogSsSc:al3ep)t:∗33pptt∗3topt4tpot4(𝐿pt=(𝐿40=)40) Comparisonfor𝑀SS(logscale):3pt∗3ptto4pt(𝐿=40)
10−124 4ptd4aptatdata 4ptdata
3pt∗33pptt∗er3rpotre(r𝑝ro=r1(𝑝)=1) 3pt∗3pterror(𝑝=1)
4−3𝑀(|𝑦|)[10GeV]VV44−−−11550 3pt∗3p3tpcto∗n3vpotluetriroonr4p((𝑝𝑝t=d=a11ta)) 4|𝑀(|𝑦|)|[GeV]VV444−2𝑀(|𝑦|)[10GeV]SS111000−−−11543022468 33pptt∗∗3333pppp3ttttp∗∗ccto33o∗nppn33vttvppootccltloouue∗nntrt3irivvpooooontnrlluue(((ttr𝑝𝑝𝑝iiroooff=nnrrree1(((ee𝑝𝑝𝑝)))===212))) 4𝑀(|𝑦|)[GeV]SS 1100−−21 33pptt∗∗33pp3ttpcctoo∗nn3vvpootlluuettriirooonnr(((𝑝𝑝𝑝===212)))
3pt∗3pterror(𝑝free)
−20 3pt∗3ptconvolution(𝑝free) 0 10−3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 00 00..22 00..44 00.6.6 0.08.8 11 1.21.21.41.41.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
𝑚−𝜋1|𝑦|[fm] 𝑚𝑚−𝜋−𝜋11|𝑦|𝑦|[|f[mfm]] 𝑚−𝜋1 |𝑦|[fm]
Figure4: Resultsfortheconvolution(4.2)comparedwiththe4ptdataofthetwo-operatormatrixelements
fortheSSandVVcase(logarithmicscale).
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DPDsofthePion ChristianZimmermann
The integrals (4.1) and (4.2) are evaluated numerically and compared to the corresponding
4pt data, where again only connected contributions are used. The results for test 1 are presented
in Figure 3. In the electromagnetic case one may observe a good agreement between the two
curves for large distances, while for distances smaller than the pion wave length clear deviations
arevisible. Forthescalarcaseasimilarbehaviorcanbefound,althoughtheagreementisslightly
worse. This might be corrected by taking into account disconnected contributions, which are less
suppressedinthescalarcase.
The results of test 2 plotted in Figure 4 are very different. There is no region in y, where the 3pt
data and the 4pt data show the same behavior. The naive factorization ansatz only predicts the
correctorderofmagnitudeforboth,theelectromagneticandthescalarchannel.
5. Conclusion
We have investigated 4pt-functions, which are needed for the description of DPS processes,
by performing lattice calculations. The calculations have been done for all 4pt-graphs, where we
obtainedgoodresultsforatleastfourgraphs. Fortheremainingtwographswestillhavetoincrease
ourstatistics. Wehavecheckedthevalidityofthe"naivefactorization"assumptionbyperforming
two tests, where we have found that this kind of factorization fails for small parton distances, but
alsoforlargedistancesitmightnotbeavalidassumption.
Theefforts,whichhavebeendonesofar,areplannedtobeextendedtofurtheraspects,likehigher
moments and non-zero momenta. Finally we want to investigate other particles, especially the
nucleon,sincethisistheparticlecollidedatLHC.
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