On the rotational invariance and non-invariance of lepton angular distributions in Drell-Yan and quarkonium production

University of Groningen

On the rotational invariance and non-invariance of lepton angular distributions in Drell-Yan

and quarkonium production

Peng, Jen-Chieh; Boer, Daniel; Chang, Wen-Chen; McClellan, Randall Evan; Teryaev, Oleg

Published in:

Physics Letters B

DOI:

10.1016/j.physletb.2018.11.061

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from

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Publication date:

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Citation for published version (APA):

Peng, J-C., Boer, D., Chang, W-C., McClellan, R. E., & Teryaev, O. (2019). On the rotational invariance

and non-invariance of lepton angular distributions in Drell-Yan and quarkonium production. Physics Letters

B, 789, 356-359. https://doi.org/10.1016/j.physletb.2018.11.061

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

On

the

rotational

invariance

and

non-invariance

of

lepton

angular

distributions

in

Drell–Yan

and

quarkonium

production

Jen-Chieh Peng

a

,

∗

,

Daniël Boer

b

,

Wen-Chen Chang

c

,

Randall

Evan McClellan

a

,

d

,

Oleg Teryaev

e

a_{DepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,Urbana,IL 61801,USA}

b_{VanSwinderenInstituteforParticlePhysicsandGravity,UniversityofGroningen,Groningen,theNetherlands} c_{InstituteofPhysics,AcademiaSinica,Taipei11529,Taiwan}

d_{ThomasJeffersonNationalAcceleratorFacility,NewportNews,VA23606,USA} e_{BogoliubovLaboratoryofTheoreticalPhysics,JINR,141980Dubna,Russia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received13August2018

Receivedinrevisedform26October2018 Accepted7November2018

Availableonline21December2018 Editor:D.F.Geesaman Keywords: Drell–Yanprocess Quarkoniumproduction Lam–Tungrelation Rotationalinvariance

SeveralrotationalinvariantquantitiesfortheleptonangulardistributionsinDrell–Yanandquarkonium production were derived several years ago, allowing the comparison between different experiments adopting different reference frames. Using an intuitive picture for describing the lepton angular distribution in these processes, we show how the rotational invariance of these quantities can be obtained. Thisapproach canalsobe used todetermine the rotationalinvariance ornon-invarianceof various quantities specifying the amount ofviolation for the Lam–Tungrelation. While the violation of the Lam–Tungrelationis oftenexpressedby frame-dependentquantities, wenote that alternative frame-independentquantitiesarepreferred.

©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

Theangulardistributionsofleptons producedintheDrell–Yan process [1] andthequarkoniumproductioninhadron–hadron col-lisions [2,3] remain a subject of considerable interest. The polar andazimuthalangulardistributionsofleptons producedin unpo-larizedandpolarizedDrell–Yanprocessallowtheextractionof var-ioustypesoftransverse-momentumdependentdistributions [4,5]. First(leadingorder)resultsontheextractionoftheBoer–Mulders functions [6,7] havebeenobtainedfromunpolarizedDrell–Yan ex-periments using pion [8,9] or proton [10] beams, indicating that thequark transverse spin iscorrelated withthequark transverse momentum inside unpolarized protons. A more precise determi-nation of the amount of quark polarization requires inclusion of higherorderperturbative correctionsbecause gluonradiationcan also affectthe lepton angular distributions [11–14]. Recent mea-surementofDrell–Yanangulardistributions witha pionbeamon a transverselypolarizedproton target provided thefirst informa-tion fromDrell–Yanon the correlation betweenthe quark trans-versemomentumandthespindirectionofatransverselypolarized proton [15]. Forquarkonium production, the lepton angular dis-tributions reveal sensitively the underlying partonic mechanisms,

*

Correspondingauthor.

E-mailaddress:jcpeng@Illinois.edu(J.-C. Peng).

asvarioussubprocessescouldleadtodistinctpolarizationsforthe quarkonium [3,16,17].

The lepton angulardistributions inDrell–Yanand quarkonium productionaregenerallymeasuredintherestframeofthe dilep-tons. Manydifferent choicesof the referenceframesexist in the literature,dependingonhowtheaxesofthecoordinatesystemare chosen.Whileitiscommontodefinethey axistobealongthe di-rectionnormaltothereactionplane(whichistheplanecontaining thebeamaxisandthedilepton’smomentumvector)andthex and z axes lyingon thereactionplane, the specificdirectionofthe z

axisischosendifferentlyfordifferentreferenceframes.In particu-lar,theCollins–Soperframe [18] hasthez axisbisectingthebeam andtargetmomentumvectors,whilethehelicityframealignsthe

z axiswiththedileptonmomentum vector inthecenter-of-mass frame. TheGottfried–Jacksonframe [19] andthe u-channelframe have the z axis parallel to the beam and target momentum di-rection,respectively.Thesevarious referenceframesarerelatedto eachotherbyrotationsalongthe y axisbycertainangles [8,20].

Ageneralexpressionforthelepton angulardistributioninthe Drell–Yanprocessorquarkoniumproductionisgivenas

d

σ

d



∝

1

+ λ

θcos

2

_θ

_{+ λ}

θ φsin 2

θ

cos

φ

+ λ

φsin2

θ

cos 2

φ,

(1) https://doi.org/10.1016/j.physletb.2018.11.061

0370-2693/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

where

θ

and

φ

referto thepolarandazimuthal anglesofl− (e− or

μ

−) in the restframe of the dilepton.While the polarangle dependenceis specified by the parameter

λ

θ, the azimuthal

de-pendencies of the lepton angular distributions are described by the parameters

λ

θ φ and

λ

φ. Note that these parameters are

re-latedtotheparameters

λ, μ, ν

inRef. [21] as

λ

θ

= λ,

λ

θ φ

=

μ

and

λ

φ

=

ν

/

2.The values of

λ

θ

,

λ

θ φ and

λ

φ dependon the choiceof

the coordinate system. While the Collins–Soper frame is chosen bymanyexperimentsforthedataanalysis,otherreferenceframes are alsoutilized by some experiments. Goingfrom one frame to anotheractsasanonlineartransformationonthesethree parame-ters [22],makingithardtoconnecttheresultsindifferentframes. The frame-dependence of the angular distribution parameters couldpotentiallyleadtoconfusionwhencomparingresultsof lep-ton angular distributions or quarkonium polarizations measured in differentexperiments [20,23]. In order to mitigate the confu-sioncaused by the frame dependenceof the parameters

λ

θ,

λ

θ φ

and

λ

φ, Faccioli et al. [24–26] pointed out that various

quanti-tiescan be formed from

λ

θ,

λ

θ φ and

λ

φ with the property that

theyare invariantunderthetransformationsamongdifferent ref-erenceframes. The comparison between measurements obtained withdifferentreferenceframescould beperformed, ifsuch rota-tioninvariantquantitiesareusedratherthantheindividual

λ

θ,

λ

θ φ

and

λ

φ parameters.Examples ofsuch rotationalinvariant

quanti-tiesinclude [25,26]

F

=

1

+ λ

θ

+

2

λ

φ 3

+ λ

θ

,

(2) and

˜λ =

λ

θ

+

3

λ

φ 1

− λ

φ

.

(3)

The reason for considering these particular combinations is not justtherotationalinvariance,butalsothat theyare measuresfor thedeviationoftheLam–Tungrelation [27],1

− λ

θ

=

4

λ

φ,thatis

satisfiedinthe Drell–Yanprocessatorder

α

s incaseofcollinear partondistributions.Its violationresults fromtheacoplanarityof the partonic subprocess, as discussed in detail in Refs. [12,14]. This acoplanarity can arise from intrinsic transverse momentum ofquarksinsidetheproton,butalsofromperturbativegluon radi-ationbeyondorder

α

s.Theyleadtoadeviationof

F

from 1₂ and of

˜λ

from1.Incontrast,thedeviationof1

− λ

θ

−

4

λ

φ fromzero

oftenconsideredinexperimentalandtheoreticalstudies [7–11] is

not arotationallyinvariantquantity,pointedoutfirstinRef. [26], andhenceapotentialsourceofconfusionwhencomparingits val-uesobtainedindifferentframes.

Anotherrotation-invariantquantity invokingall three parame-tersis [3,28]

˜λ



₌

(λ

θ

− λ

φ

)

2

+

4

λ

2_{θ φ}

(

3

+ λ

θ

)

2

.

(4)

Althoughnot immediatelyobvious fromtheir definition interms of

λ

θ,

λ

θ φ and

λ

φ,theabove threequantities,

F,

˜λ

,

˜λ

,are

invari-ant only under rotations around the y axis, which includes the transformations connecting the various references frames in the literature. On the other hand,the quantity

G

is invariant under therotationalongthex axis [26],

G

=

1

+ λ

θ

−

2

λ

φ 3

+ λ

θ

.

(5)

Finally,

λ

θ isinvariantundertherotationalongthez-axis [26].

Fig. 1. DefinitionoftheCollins–Soperframeandvariousanglesandplanesinthe restframeofγ∗/Z oravectorquarkonium.ThehadronplaneisformedbyPBand



PT,themomentumvectorsofthetwointeractinghadrons.Thex andˆ ˆz axesof

theCollins–Soperframebothlieinthehadronplanewithˆz axisbisectingthe PB

and−PTvectors.Thequark(q)andantiquark(q)¯ annihilatecollinearlywithequal

momentatoformγ∗/Z oravectorquarkonium,whilethequarkmomentumvector

ˆ

zandtheˆz axisformthequarkplane.Thepolarandazimuthalanglesofˆzinthe Collins–Soperframeareθ1andφ1.Thel−andl+areemittedback-to-backwithθ

andφasthepolarandazimuthalanglesforl−.

The rotational invariance of

F

,

˜λ

,

˜λ

 and

G

was obtained in Refs. [25,26,28,29] fromtheconsiderationofthecovariance prop-erties ofangular momentum eigenstates of a vector meson. Ina recentstudy [12,14],itwasshownthatsomesalientfeaturesofthe parameters

λ

θ,

λ

θ φ and

λ

φ intheDrell–Yanprocess and Z -boson

productioncanbewelldescribedbyanintuitiveapproach.In par-ticular, the pronounced transverse-momentum dependence of

λ

θ

and

λ

φforZ -bosonproductionandtheclearviolationoftheLam–

Tung relation at the LHC [30,31] can be well described by this approach. In this paper, we show how the rotational invariance propertiesof

F

,

˜λ

,

˜λ

 and

G

can bededuced usingtheapproach of Refs. [12,14]. It is alsoclear fromthe analysis belowthat the rotationalinvarianceornon-invariance ofvariousquantities char-acterizingtheviolationoftheLam–Tungrelationcanbeobtained. Inthedileptonrestframe,wefirstdefinethreedifferentplanes, namely,the hadronplane, thequark plane, andtheleptonplane, shownin Fig. 1.Fordileptons with non-zerotransverse momen-tum, qT, the momenta of the two interacting hadrons, P



B and



PT, are not collinear in the rest frame of

γ

∗

/

Z , and they form the “hadron plane” shown in Fig. 1. Fig. 1 also shows the “lep-ton plane”,formed bythe momentumvector ofthel− andthe

ˆ

z

axis.Inthe restframeofthe dilepton,thel− andl+ areemitted back-to-backwithequalmomenta.

Inthedileptonrestframe,apairofcollinearq andq with

¯

equal momentaannihilateintoa

γ

∗

/

Z oravectorquarkonium,as illus-trated inFig.1.We definethe momentumunit vector ofq as

ˆ

z, and the “quark plane” is formed by the

ˆ

z and

ˆ

z axes. The po-larandazimuthalanglesofthez

ˆ

 axisintheCollins–Soperframe are denoted as

θ

1 and

φ

1.Forthe coplanarcase,

φ

1

=

0 and the

hadronplanecoincideswiththequarkplane.When

φ

1

=

0,

φ

1

sig-nifiestheacoplanarityangle.Withrespecttotheq

− ¯

q axis,called thenaturalaxis [32],thel−hasanazimuthallysymmetricangular distribution,namely,

d

σ

d



∝

1

+

a cos

θ

0

+ λ

0cos

2

_θ

where

θ

0isthepolaranglebetweenthel−momentumvectorand

the

ˆ

zaxis(seeFig.1),anda istheforward–backwardasymmetry originatingfromthe parity-violatingcoupling,which isimportant only when the dileptonmass is close to the Z boson mass.The parameter

λ

0 dependsonthe reactionmechanism. ForDrell–Yan

process in which a virtual photon decays into a lepton pair, we have

λ

0

=

1.This isa consequence ofhelicity conservation

lead-ingto atransversely polarizedvirtualphoton withrespecttothe naturalaxis.Forquarkoniumproduction,thevalueof

λ

0 depends

on the specific mechanism. We note that

λ

0

=

0 for unpolarized

quarkoniumproduction,while

λ

0

= −

1 forproductionof

longitu-dinallypolarizedquarkonium.

Theangles

θ

and

φ

areexperimentalobservables,anditis nec-essarytoexpress

θ

0 intermsof

θ

and

φ

.Thiscanbeaccomplished

usingthefollowingtrigonometric relation:

cos

θ

0

=

cos

θ

cos

θ

1

+

sin

θ

sin

θ

1cos

(φ

− φ

1

).

(7)

SubstitutingEq. (7) intoEq. (6),weobtain

d

σ

d



∝ (

1

+

1 2

λ

0sin 2

_θ

1

)

+ (λ

0

−

3 2

λ

0sin 2

_θ

1

)

cos2

θ

+ (

1

2

λ

0sin 2

θ

1cos

φ

1

)

sin 2

θ

cos

φ

+ (

1

2

λ

0sin 2

_θ

1cos 2

φ

1

)

sin2

θ

cos 2

φ

+ (a sin

θ

1cos

φ

1

)

sin

θ

cos

φ

+ (a cos

θ

1

)

cos

θ

+ (

1

2

λ

0sin 2

_θ

1sin 2

φ

1

)

sin2

θ

sin 2

φ

+ (

1

2

λ

0sin 2

θ

1sin

φ

1

)

sin 2

θ

sin

φ

+ (a sin

θ

1sin

φ

1

)

sin

θ

sin

φ.

(8)

AcomparisonbetweenEq. (1) andEq. (8) showsthat

λ

θ,

λ

θ φ,

and

λ

φ can be expressed as a function of

λ

0,

θ

1 and

φ

1 (cf.

with [32] forzeroacoplanarityangle

φ

1

=

0):

λ

θ

=

2

λ

0

−

3

λ

0sin2

θ

1 2

+ λ

0sin2

θ

1

λ

θ φ

=

λ

0sin 2

θ

1cos

φ

1 2

+ λ

0sin2

θ

1

λ

φ

=

λ

0sin2

θ

1cos 2

φ

1 2

+ λ

0sin2

θ

1

.

(9)

Thetermsproportionaltosin 2

φ

andsin

φ

donotappeardueto Lorentz invariance,provided there are novectors(like transverse polarization) normal to the hadron plane. Such terms in Eq. (8) integrate to zero due to the acoplanarity angle average. Unless one considers polarizedleptons, parityortime-reversal violation, Eq. (8) reducestoEq. (1).

First,we consider thequantity

F

in Eq. (2). From Eq. (9), we obtain

F

=

1

+ λ

0

−

2

λ

0sin2

θ

1sin2

φ

1 3

+ λ

0

=

1

+ λ

0

−

2

λ

0y2₁ 3

+ λ

0

,

(10)

where y1

=

sin

θ

1sin

φ

1 is the component of the unit vector z

ˆ



along the y-axisin the dileptonrest frame. The invarianceof

F

with respect to a rotation along the y axis is clearly shown in Eq. (10), since

λ

0 and y1 are both invariant under such a

rota-tion.ItisinterestingtonotethatfortheDrell–Yanprocess,where

λ

0

=

1,

F

becomes

(

1

−

y12

)/

2. As pointed out in Refs. [12,14], y1,orthe non-coplanarity angle

φ

1 betweenthe hadronandthe

quarkplanesinFig.1,isingeneralnotequaltozero.Forthe spe-cialcaseof

φ

1

=

0 (or y1

=

0),

F =

1

/

2 and

F

isinvariant under

anyarbitraryrotationinthedilepton’srestframe.Asdiscussedin Refs. [12,14], the Lam–Tung relation in the Drell–Yan process is satisfiedwhentheangle

φ

1 vanishes.ThisisverifiedfromEq. (9),

whenthevaluesof

λ

0 and

φ

1aresetat1and0,respectively.

Wenextconsiderthequantity

˜λ

.UsingEq. (9),Eq. (3) becomes

˜λ =

λ

0

+

3

λ

0sin2

θ

1sin2

φ

1 1

+ λ

0sin2

θ

1sin2

φ

1

=

λ

0

+

3

λ

0y21 1

+ λ

0y21

.

(11)

Again,

˜λ

mustbeinvariantunderarotationalongthe y axis,since

λ

0 and y1 are both invariant undersuch rotation.In the special

caseofcoplanaritybetweenthehadronplaneandthequarkplane, we have y1

=

0, andEq. (11) becomes

˜λ = λ

0.In that case,

˜λ

is

invariant under rotationalong any axis.However,

˜λ

is ingeneral not the same as

λ

0, and

˜λ

is in general not invariant under an

arbitraryrotation.

We turn our attention next to the quantity

˜λ

 in Eq. (4). All threeparameters,

λ

θ,

λ

θ φ,and

λ

φareinvolvedin

˜λ

.UsingEq. (9),

weobtain

˜λ



₌

λ

20

(

z21

+

x21

)

2

(

3

+ λ

0

)

2

=

λ

2₀

(

1

−

y2₁

)

2

(

3

+ λ

0

)

2

,

(12)

where z1 isthecomponentofthe unitvector z

ˆ

 alongthe z axis

andtheidentityx2₁

+

y2₁

+

z2₁

=

1 isused.Thus,

˜λ

isinvariantunder arotationalongthe y axis.Forthecoplanarcase, y1

=

0 and

˜λ

is

invariantunderrotationalonganyaxis.

Inananalogousfashion,onecanshowtheinvarianceof

G

and

λ

θ underthe rotation alongthe x and z axis, respectively. Using

Eq. (9),Eq. (5) becomes

G

=

1

+ λ

0

−

2

λ

0sin2

θ

1cos2

φ

1

3

+ λ

0

=

1

+ λ

0

−

2

λ

0x2₁ 3

+ λ

0

,

(13)

where x1

=

sin

θ

1cos

φ

1 is the component of the unit vector

ˆ

z

alongthex axisinthedileptonrestframe.Similarly,fromEq. (9), theparameter

λ

θ canbewrittenas

λ

θ

=

−λ

0

+

3

λ

0cos2

θ

1 2

+ λ

0

− λ

0cos2

θ

1

=

−λ

0

+

3

λ

0z21 2

+ λ

0

− λ

0z2₁

,

(14)

wherez1

=

cos

θ

1 isthecomponentoftheunitvector

ˆ

zalongthe z axis inthe dilepton restframe. From Eq. (13) and Eq. (14) we note that

G

and

λ

θ are invariant under the rotation along the x

andz axis,respectively.

Using the above results one can see that despite the nonlin-ear transformation of

λ

θ

,

λ

θ φ and

λ

φ under rotations, the linear

combination1

− λ

θ

−

4

λ

φremainszeroinallotherrotatedframes

if itis zeroin one particular frame,aswas observedfor specific rotations in [22]. Ifthecombinationisnonzero however,then its value willchange underrotations, even around the y axis. From Eq. (9), itfollowsthatthequantity 1

− λ

θ

−

4

λ

θ φ isnot invariant

underrotationsalongthe y axis.Ontheotherhand,thequantity,

(

1

− λ

θ

−

4

λ

φ

)/(

3

+ λ

θ

)

,is invariantundersuchrotations,namely 1

− λ

θ

−

4

λ

φ 3

+ λ

θ

=

1

−

2

F

=

1

− λ

0

+

4

λ

0y 2 1 3

+ λ

0

.

(15)

Therefore,toexaminetheamountoftheviolationoftheLam–Tung relation,thequantity,

(

1

− λ

θ

−

4

λ

φ

)/(

3

+ λ

θ

)

,ispreferred.

OftenintheliteraturefortheDrell–Yanprocess,anothersetof angularcoefficientsareconsidered: A0

,

A1

,

A2,where

d

σ

d



∝ (

1

+

cos 2

_{θ )}

₊

A0 2

(

1

−

3 cos 2

_{θ )}

₊

_A 1sin 2

θ

cos

φ

+

A2 2 sin 2

_θ

_{cos 2}

_φ.

₍₁₆₎

The Lam–Tung relation isthen expressedas A0

=

A2. The

viola-tionof theLam–Tung relation, A0

−

A2

=

2

(

1

−

2

F)

, is

rotation-allyinvariantaround the y axis.Onthe otherhand,the quantity

LT

=

1

−

A2

/

A0of[33] isnot.

In conclusion, we have presented an intuitive derivation for rotation-invariant quantities for lepton angular distributions in Drell–Yanandvectorquarkonium production.Byexpressingthese quantitiesintermsofthe

λ

0andthex,y andz componentsofthe

unitvectorofthequarkmomentuminthedileptonrestframe,the invariant properties of thesequantities become transparent. This approachoffersausefulinsightregardingtherolesof

λ

0 andthe

acoplanarityofthe partonic subprocesses indetermining the ap-plicabilityandvalues oftheseinvariant quantities. Thisapproach could also be extended to other hard processes, such as hadron pairproduction in e+e− annihilation,which is closelyconnected totheDrell–Yanandvectorquarkoniumproduction.

Thiswork was supported in partby the U.S. NationalScience Foundation and the Ministry of Science and Technology of Tai-wan. It was also supported in part by the U.S. Department of Energy,OfficeofScience,OfficeofNuclear Physicsundercontract DE-AC05-060R23177.

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