Hadronic single inclusive kt distributions inside one jet beyond MLLA

0.18 MB·English

by  François Arleo

Hadronic single inclusive k distributions inside one jet beyond MLLA
⊥
Franc¸ois Arleo∗,1 Redamy P´erez-Ramos,2 and Bruno Machet3
1CERN, PH department, TH division, CH-1211 Geneva 23
2Max-Planck-Institut fu¨r Physik, Werner-Heisenberg-Institut, F¨ohringer Ring 6, D-80805 Mu¨nchen
3 Laboratoire de Physique Th´eorique et Hautes E´nergies †,
BP 126, 4 place Jussieu, F-75252 Paris Cedex 05
(Dated: December 3rd 2007)
Thehadronick -spectruminsideonejetisdeterminedincludingcorrectionsofrelativemagnitude
⊥
(√αs)withrespecttotheModifiedLeadingLogarithmicApproximation(MLLA),atandbeyond
O
8 the limiting spectrum (assuming an infrared cut-off Q0 =Λ and Q0 =Λ ). The agreement
QCD 6 QCD
0 between our results and preliminary measurements by the CDF collaboration is impressive, much
0 better than at MLLA, pointing out very small overall non-perturbativecontributions.
2
PACSnumbers: 12.38.Cy,13.87.-a.,13.87.Fh
n
a
J
Jet production – a collimated bunch of hadrons – in where ΦB(z) denote the DGLAP [6] splitting functions,
7 A
e+e−, e−p andhadroniccollisionsis anidealplayground αs =2π 4Ncβ0(ℓ+y+λ) is the one-loop coupling con-
] for parton evolution in perturbative QCD (pQCD). One stant of(cid:14)QCD [13] and
h
of the major successes of pQCD is the hump-backed
p
- shapeofinclusivespectra,predictedin[1]withinMLLA, ℓ=(1/x) , y =ln k⊥ Q0 , λ=ln Q0/ΛQCD ,
p and later discovered experimentally (see e.g. [2]). Refin- (cid:0) (cid:14) (cid:1) (cid:0) (cid:1)
e ingthecomparisonofpQCDcalculationswithdatataken (Q0 being the collinear cut-off parameter), and where
h
at LEP, Tevatron and LHC will ultimately allow for a
v2 [ chryupcoiathletseisst[o3f]thanedLofocralaPabrettotnerHuanddroenrstDaunadliintyg(oLfPcHoDlo)r QG ≡≡ QG((11))==xxDDGQhh((xx,,EEΘΘ00,,QQ00)),.
neutralizationprocesses. InthisLetter,aclassofnext-to-
1 At small x z, the fragmentation functions behave as
next-to-leading logarithmic (NMLLA) corrections to the
9 ≪
3 single inclusive k -distribution of hadrons inside one jet
3 is determined. U⊥nlike other NMLLA corrections, these B(z) ρh ln z,lnzEΘ0 =ρh (lnz+ℓ,y),
7. terms better accountfor recoileffects andwereshownto ≈ B(cid:18) x Q0 (cid:19) B
0 drastically affect multiplicities and particle correlations
ρh being a slowly varying function of two logarith-
7 injets[4]. WestartbywritingtheMLLAevolutionequa- B
:0 tionsforthefragmentationfunctionDBh x z,zEΘ0,Q0 mbaicckvedar”iapblaletsealun([z1/].x)Inaonrddeyrtthoabtedtteesrcraicbceosutnhtefo“rhruemcopi-l
v of a parton B (energy zE and transv(cid:0)ers(cid:14)e momentum(cid:1)
effects, the strategy followed in this Letter is to perform
Xi k⊥ = zEΘ0) into a gluon (identified as a hadron h with Taylor expansions (first advocatedfor in [7]) of the non-
energy xE according to LPHD) inside a jet of energy
r singular parts of the integrands in (1,2) in powersof lnz
a E. As a consequence of angular ordering in parton cas-
and ln(1 z), both consideredsmall with respect to ℓ in
cading, partonic distributions inside a quark and gluon −
the hard splitting region z 1 z = (1)
jet, Q,G(z)=x z DQh,G x z,zEΘ0,Q0 , obey the sys- ∼ − O
temoftwocoupl(cid:14)edequati(cid:0)on(cid:14)s[5](thesub(cid:1)scripty denotes B(z)=B(1)+B (1)lnz+ ln2z ; z 1 z. (3)
ℓ
∂/∂y) O ↔ −
(cid:0) (cid:1)
1 α Each ℓ-derivative giving an extra √αs factor (see [5]),
Qy =Z0 dz πs Φgq(z)(cid:20)(cid:16)Q(1−z)−Q(cid:17)+G(z)(cid:21), (1) tchoerretcetrimonssBtℓo(1t)hlenzsoaluntdioBnsℓ(o1f)l(n2()1. −Fzro)myie(l3d)NanMdLLthAe
G = 1dz αs Φg(z)(1 z) G(z)+G(1 z) G expressions of the DGLAP splitting functions, one gets
y Z0 π (cid:20) g − (cid:16) − − (cid:17) after some algebra (γ02 =2Ncαs/π) [8]
+nf Φqg(z)(cid:16)2Q(z)−G(cid:17)(cid:21), (2) Q(ℓ,y) = δ(ℓ)+ CF ℓdℓ′ ydy′γ2(ℓ′+y′) (4)
Nc Z0 Z0 0
1 a˜1δ(ℓ′ ℓ)+a˜2δ(ℓ′ ℓ)ψℓ(ℓ′,y′) G(ℓ′,y′),
× h − − − i
∗On leave from Laboratoire d’Annecy-le-Vieux de Physique ℓ y
Th´eorique (LAPTH), Universit´e de Savoie, CNRS, B.P. 110, F- G(ℓ,y) = δ(ℓ)+ dℓ′ dy′γ2(ℓ′+y′) (5)
0
74941Annecy-le-VieuxCedex,France Z0 Z0
†ML.PCTuHriEe,-PUaMriRs67e5t8a`9l’dUunivCeNrsRit´SeDa.sDsoidcie´ereota`-Pl’aUrinsiv7ersit´e P.et 1 a1δ(ℓ′ ℓ)+a2δ(ℓ′ ℓ)ψℓ(ℓ′,y′) G(ℓ′,y′).
× h − − − i
2
with ψ (ℓ,y) = G (ℓ,y)/G(ℓ,y). The MLLA coefficients it into Eq. (9) leads to
ℓ ℓ
a˜1 =3/4 and a1 0.935 are computed in [5] while at
≈
NMLLA, we get [14]: xFAh0 ≈ Z duuDAA0(u,EΘ0,uEΘ)D˜Ah(ℓ,y) (11)
XA
a˜2 = 78 + CNFc (cid:18)58 − π62(cid:19)≈0.42, (6) +XA Z duulnuDAA0(u,EΘ0,uEΘ)dD˜Ahd(ℓℓ,y)
67 π2 13n T C
a2 = 36 − 6 − 18 Nf cR NFc ≈0.06 . (7) +12 (cid:20)Z duuln2uDAA0(u,EΘ0,uEΘ)(cid:21)d2D˜dAhℓ(2ℓ,y).
XA
Computing the NMLLA partonic distributions inside a
The firsttwo terms in Eq.(11)correspondto the MLLA
quark and gluon jet, Q(z) and G(z), is the first step to distribution calculated in [9] when D˜h is evaluated at
determinethedoubledifferentialspectrumd2N/dxdΘof A
NLO and its derivative at LO. NMLLA correctionsarise
a hadron produced with energy xE and at angle Θ with
from their respective calculation at NNLO and NLO,
respectto the jet axisidentifiedwiththe directionofthe
and, mainly in practice, from the third line, which is
energy flow (see [8]). As shown in [9], it is given by
new. Indeed, since x/u is small, the inclusive spectrum
D˜h(ℓ,y) is the solution of the next-to-MLLA evolution
d2N d A
dxdlnΘ = dlnΘFAh0(x,Θ,E,Θ0), (8) eoqfuthaetioconesffi(4c)ieanntda(25()s.eeH(o7w)e),veGr(,ℓb,eyc)asuhsoewosfnthoessigmnaifillcnaensst
differencefromMLLAtoNMLLA.Asaconsequence,we
where FAh0 is givenby the convolutionof two fragmenta- use the MLLA expression for G. It is determined here
tion functions from a representation in terms of a single Mellin trans-
formof confluenthypergeometricfunctions (see Eq.(24)
FAh0 ≡XA Zx1duDAA0(u,EΘ0,uEΘ)DAh (cid:16)ux,uEΘ,Q0(cid:17), oLfL[A10q])u,awrkeldlissutritibedutfioornnQu(mℓ,eyr)iccaalnsttuhdeniesbe[1d5e]d.uTchede fNroMm-
(9) G(ℓ,y) using (4) and (5), which yields
u being the energy fraction of the intermediate parton
C
eAr.gyDuAAE0odffestchreibpeasrttohneAp0ro(bwahbiiclihtyintiotiaetmesitthAe jweti)t,htaenk-- Q(ℓ,y) = NFc hG(ℓ,y)+(cid:16)a1−a˜1(cid:17)Gℓ(ℓ,y) (12)
ing into account the evolution of the jet between Θ0 + a1 a1 a˜1 +a˜2 a2 Gℓℓ(ℓ,y) + (γ02).
and Θ. Dh describes the probability to produce the (cid:16) (cid:16) − (cid:17) − (cid:17) i O
A
hadron h off A with energy fraction x/u and trans- The functions Fh andFh arerelatedto the gluondistri-
g q
verse momentum k⊥ ≈ uEΘ ≥ Q0 (see Fig. 1). As bution via the color currents hCig,q defined as:
C
xFh = h ig,q G(ℓ,y). (13)
g,q N
h c
Θ0 xE C canbe seenas the averagecolorchargecarriedby
A0 E DAA0 A uEΘ DAh (Jet Axis) thheigp,qartonAduetotheDGLAPevolutionfromA0 toA.
Introducing the first and second logarithmic derivatives
of D˜h,
A
1 dD˜ (ℓ,y)
FIG. 1: Inclusiveproduction of hadron h at angle Θ inside a ψ (ℓ,y) = A = (√α ),
high energy jet of total opening angle Θ0 and energy E. A,ℓ D˜Ah(ℓ,y) dℓ O s
1 d2D˜ (ℓ,y)
(ψ2 +ψ )(ℓ,y) = A = (α ),
A,ℓ A,ℓℓ D˜h(ℓ,y) dℓ2 O s
A
discussed in [9], the convolution (9) is dominated by
u ∼ 1 and therefore DAA0(u,EΘ0,uEΘ) is given by Eq. (11) can now be written as
DGLAP evolution [6]. On the contrary, the distribution
rD˜eAdhu≡cesuxDtoAhth(cid:0)eux,huuEmΘp,-bQa0c(cid:1)ke=dDp˜lAha(tℓea+ul,nu,y)atlowx≪u xFAh0 ≈ XA hhuiAA0 +hulnuiAA0ψA,ℓ(ℓ,y)
1
+ uln2u A (ψ2 +ψ )(ℓ,y) D˜h, (14)
D˜Ah(ℓ+lnu,y)x≪≈uρhA(ℓ+lnu,YΘ+lnu), (10) with the notati2ohn iA0 A,ℓ A,ℓℓ i A
ewxipthanYsΘion=ofℓD˜+tyot=helsneEcoΘn/dQo0r.derPienrf(olrnmui)nagntdhpeluTgagyilnogr hulniuiAA0 ≡ Z01du (u lniu) DAA0(u,EΘ0,uEΘ)
3
2 0.6
LO
Y = 6.4
1.8 MLLA 0.4 l = 2
NMLLA
1.6 MLLA〉C Q 0.2
〉CQ 1.4 MLLA〈 ⁄ 0
〈 1.2 NMLLA- Q -0.2 λ = 0.0
1 〉C λ = 0.2
Y = 6.4 δ〈 λ = 0.5
0.8 -0.4 λ = 1.0
l = 2
0.6 -0.6
0 1 2 3 4 0 1 2 3 4
y y
FIG. 2: The color current of a quark jet with YΘ =6.4 as a FIG. 3: NMLLA corrections to the color current of a quark
0
function of y at fixed ℓ=2. jet with YΘ =6.4 and ℓ=2 for various values of λ.
0
1
≈Z0 du (u lniu) DAA0(u,EΘ0,EΘ). (15) Othe(QM0L)L[1A0]r.esuTlhtearNeMdiLspLlAaye(dnoinrmFailgiz.e3df)orcodrirffeecrteinontsvatlo-
ues λ = 0,0.5,1. It clearly indicates that the larger λ,
ThescalingviolationoftheDGLAPfragmentationfunc-
the smaller the NMLLA corrections. In particular, they
tion neglected in the last approximation is a (α ) cor-
O s can be as large as 30% at the limiting spectrum (λ=0)
rection to u . It however never exceeds 5% [8] of the
h i butnomorethan10%forλ=0.5. Thisisnotsurprising
leading term and is thus neglected in the following. Us-
since λ=0 (Q0 =Λ ) reduces the parton emission in
ing (13), the MLLA and NMLLA contributions to the 6 6 QCD
the infrared sector and, thus, higher-order corrections.
leading color current of the parton A0 =g,q read The double differential spectrum d2N/dydℓ, Eq. (8),
δ C MLLA−LO =N ulnu g ψ + C ulnu q ψ , can now be determined from the NMLLA color cur-
h δiAC0 NMLLA−MLLcAh=NiAu0lng2,ℓu g (Fψ2h +ψiA0) q,ℓ rents (16) using the MLLA quark and gluon distribu-
h iA0 + C ulnc2hu q (ψiA20 +ψg,ℓ ).g,ℓℓ (16) tions Integrating it over ℓ leads to the single inclusive
F h iA0 q,ℓ q,ℓℓ y-distribution (or k -distribution) of hadrons inside a
⊥
quark or a gluon jet:
The MLLA correction, √αs , was determined in [9]
O
and the NMLLA contribu(cid:0)tion,(cid:1)O(αs), to the average dN dN YΘ0−y d2N
color current is new. The latter can be obtained from = k = dℓ .
(cid:18) dy (cid:19) (cid:18) ⊥dk (cid:19) Z (cid:18)dℓdy(cid:19)
the Mellin moments of the DGLAP fragmentation func- g,q ⊥ g,q ℓmin g,q
(18)
tions
The MLLA framework does not specify down to which
1
A (j,ξ)= duuj−1DA (u,ξ), values of ℓ (up to which values of x) the double differen-
DA0 Z0 A0 tialspectrumd2N/dydℓshouldbeintegratedover. Since
d2N/dydℓ becomesnegative(non-physical)atsmallval-
leading to
ues of ℓ (see e.g. [9]), we chose the lower bound ℓmin so
d2 as to guarantee the positiveness of d2N/dydℓ over the
huln2uiAA0 = dj2DAA0(j,ξ(EΘ0)−ξ(EΘ))(cid:12)(cid:12)j=2. (17) ℓwqhole ℓ2m)i.n ≤ ℓ ≤ YΘ0 range (in practice, ℓgmin ∼ 1 and
Plugging (17) into (16), the NMLLA color c(cid:12)(cid:12)urrents for mHina∼vingsuccessfullycomputedthesinglek -spectrain-
⊥
gluonandquarkjetsaredeterminedanalytically[8]. For cluding NMLLA corrections, we now compare the result
illustrative purposes, the LO, MLLA, and NMLLA av- with existing data. The CDF collaboration at the Teva-
erage color current of a quark jet with YΘ = 6.4 – cor- tronrecentlyreportedonpreliminarymeasurementsover
0
responding roughly to Tevatron energies – is plotted in a wide range of jet hardness, Q = EΘ0, in pp¯ colli-
Fig. 2 as a function of y, at fixed ℓ = 2. As discussed sions at √s = 1.96 TeV [12]. CDF data, including sys-
in [9], the MLLA correctionsto the LO colorcurrentare tematic errors, are plotted in Fig. 4 together with the
found to be large and negative. As expected, the correc- MLLA predictions of [9] and the present NMLLA calcu-
tion (α )fromMLLAtoNMLLAprovesmuchsmaller; lations,bothatthelimitingspectrum(λ=0)andtaking
s
O
it is negative (positive) at small (large) y. Λ = 250 MeV; the experimental distributions suffer-
QCD
This calculation has also been extended beyond the ing from large normalization errors,data and theory are
limitingspectrum,λ=0,totakeintoaccounthadroniza- normalized to the same bin, ln(k /1GeV) = 0.1. The
6 ⊥ −
tioneffects inthe productionof“massive”hadrons,m= agreementbetweentheCDFresultsandtheNMLLAdis-
4
NMLLA MLLA normalized to bin: ln(k⊥)=-0.1 normalized to bin: ln(k⊥)=-0.1 (N’)
CDF preliminary
CDF preliminary
Q=155 GeV Q=119 GeV 1
⊥
Q=90 GeV Q=68 GeV n k
d l10 -1
N /
d
d ln k⊥ Q=50 GeV Q=37 GeV 1/N’ 10 -2 NMLLA λ = 0
N / Q = 119 GeV λ = 0.5
d
N’ Λ = 250 MeV λ = 1
1/ QCD
Q=27 GeV Q=19 GeV 10 -3
0 1 2 3
ln (k / 1GeV)
⊥
FIG.5: CDFpreliminaryresults(Q=119GeV)forinclusive
k distribution compared with NMLLA predictions beyond
⊥
thelimiting spectrum.
0 1 2 0 1 2 3
ln (k⊥ / 1GeV) ln (k⊥ / 1GeV)
values of λ.0.5, which is not too surprising since these
FIG. 4: CDF preliminary results for the inclusive k distri- inclusivemeasurementsmostlyinvolvepions. Identifying
⊥
bution at various hardness Q in comparison to MLLA and produced hadrons would offer the interesting possibility
NMLLA predictions at the limiting spectrum; the boxes are tocheckadependenceoftheshapeofk -distributionson
⊥
the systematic errors (their lower limits at large k are cut
⊥ the hadron species, such as the one predicted in Fig. 5.
for the sakeof clarity).
To summarize, single inclusive k -spectra inside a jet
⊥
are determined including higher-order (α ) (i.e. NM-
s
O
LLA)correctionsfromtheTaylorexpansionoftheMLLA
tributions over the whole k -range is particularly good. evolution equations and beyond the limiting spectrum,
⊥
In contrast, the MLLA predictions prove reliable in a λ=0. The agreement between NMLLA predictions and
6
muchsmallerk interval. Atfixedjethardness(andthus CDF preliminary data in pp¯collisions at the Tevatron is
⊥
YΘ ),NMLLAcalculationsproveaccordinglytrustablein verygood,indicatingverysmalloverallnon-perturbative
0
a much larger x interval. corrections. TheMLLAevolutionequationsforinclusive
enoughvariablesproveoncemore(seee.g.[6])toinclude
Despite this encouraging agreement with data, the
reliableinformationatahigherprecisionthantheoneat
present calculation still suffers from various theoretical
which they have been deduced.
uncertainties,discussedindetailin[8]. Amongthem,the
Acknowledgments: We gratefully acknowledgeenlight-
variationofΛ –givingNMLLAcorrections–fromthe
QCD ening discussions with Yu.L. Dokshitzer, I.M. Dremin,
defaultvalueΛ =250MeVto150MeVand400MeV
S. Jindariani (CDF), W. Ochs and M. Rubin.
QCD
affects the normalized k -distributions by roughly 20%
⊥
in the largest ln(k /1 GeV) = 3 GeV-bin at Q = 100
⊥
GeV. Also, cutting the integral (18) at small values of ℓ
is somewhatarbitrary. However,we checkedthat chang-
ing ℓg from 1 to 1.5 modifies the NMLLA spectra at [1] Yu.L. Dokshitzer, V.S. Fadin, V.A. Khoze, Phys. Lett.
min B 115(1982)242;Ya.I.Azimov,Yu.L.Dokshitzer,V.A.
largek by 20%only[16]. Finally,thek -distribution
⊥ ∼ ⊥ Khoze,S.I.Troian,Z.Phys.C 31(1986)213;C.P.Fong,
is determined with respect to the jet energy flow from
B.R. Webber, Phys.Lett. B 229 (1989) 289.
2-particlecorrelations(which includes a summationover [2] V.A. Khoze, W. Ochs, Int. J. Mod. Phys. A 12 (1997)
secondary hadrons), while experimentally the jet axis is 2949, and references therein.
determined exclusively from all particles inside the jet. [3] Ya.I.Azimov,Yu.L.Dokshitzer,V.A.Khoze,S.I.Troian,
The question of the matching of these two definitions at Z. PhysC 27 (1985) 65; Yu.L. Dokshitzer, V.A. Khoze,
(α ) accuracy goes beyond the scope of this Letter. S.I. Troian, J. Phys. G 17 (1991) 1585.
s
O [4] F. Cuypers and K. Tesima, Z. Phys. C 54 (1992) 87;
The NMLLA k -spectrum has also been calculated
⊥ Yu.L.Dokshitzer, Phys. Lett. B 305 (1993) 295.
beyond the limiting spectrum, as illustrated in Fig. 5. [5] R. P´erez-Ramos, JHEP 06 (2006) 019.
However, the best description of CDF preliminary data [6] see for example: Yu.L. Dokshitzer, V.A. Khoze, A.H.
is reached at the limiting spectrum, or at least for small Mueller,S.I.Troyan,“BasicsofPerturbativeQCD”,Ed.
5
Fronti`eres, Gif-sur-Yvette,1991, and references therein. would not fit into the present logic of a systematic ex-
[7] I.M. Dremin, Phys.Lett. B 313 (1993) 209. pansion in powers of √αs (see [8]).
[8] F.Arleo, R.P´erez-Ramos, B. Machet, to appear. [14] AssumingQ/G=CF/Nc.Wecheckedthat (√αs)and
[9] R.P´erez-Ramos, B. Machet, JHEP 04 (2006) 043. (αs) corrections affect marginally these coOefficients.
O
[10] Yu.L. Dokshitzer, V.A. Khoze, S.I. Troian, Int. J. Mod. [15] It was also given in [5] a compact Mellin representa-
Phys.A 7 (1992) 1875. tionfromwhichananalyticapproximatedexpressionwas
[11] R.P´erez-Ramos, JHEP 09 (2006) 014. found using thesteepest descent method [11].
[12] S. Jindariani, A. Korytov, A. Pronko, CDF report [16] The effect of varyingℓmin is more dramatic at MLLA.
CDF/ANAL/JET/PUBLIC/8406 (March 2007).
[13] A 2-loop evaluation of the splitting functions and αs