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Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach
© 2021 he Au ho s
Published e sion
Li, Meijian
Li, M. (2021). Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach. In
Ha dP obes2020 : 10 h In e na ional Con e ence on Ha d and Elec omagne ic P obes o High-
Ene gy Nuclea Collisions (A icle 105). Sissa Medialab. POS P oceedings o Science, 387.
h ps://doi.o g/10.22323/1.387.0105
2021
PoS(Ha dP obes2020)105
Fo wa d qua k je -nucleus sca e ing in a ligh - on
Hamil onian app oach
Meijian Lia,b,c,∗
aDepa men o Physics, Uni e si y o Jy äskylä,
P.O. Box 35, FI-40014 Finland
bHelsinki Ins i u e o Physics, Uni e si y o Helsinki,
P.O. Box 64, FI-00014, Finland
cDepa men o Physics and As onomy, Iowa S a e Uni e si y,
Ames, IA, 50011, USA
E-mail: [email p o ec ed]
We in es iga e he sca e ing o a qua k je on a high-ene gy hea y nucleus using he ime-
dependen ligh - on Hamil onian app oach. We simula e a eal- ime e olu ion o he qua k in a
s ong classical colo ield o he ela i is ic nucleus, desc ibed as he Colo Glass Condensa e. We
s udy he sub-eikonal e ec by le ing he qua k je ca y ealis ic ini e longi udinal momen a, and
we ind sizeable changes on he ans e se coo dina e dis ibu ion o he qua k. We also obse e
he ene gy loss o he qua k h ough gluon emissions in he |qi+|qgiFock space. This app oach
p o ides us wi h an oppo uni y o s udy sca e ing p ocesses om non-pe u ba i e aspec s.
Ha dP obes2020
1-6 June 2020
Aus in, Texas
∗Speake
©Copy igh owned by he au ho (s) unde he e ms o he C ea i e Commons
A ibu ion-NonComme cial-NoDe i a i es 4.0 In e na ional License (CC BY-NC-ND 4.0). h ps://pos.sissa.i /
PoS(Ha dP obes2020)105
Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach Meijian Li
1. In oduc ion
Sca e ing o an ul a ela i is ic qua k o a hea y nucleus is one o he mos di ec ways o
s udy he s uc u e o he cold nuclea ma e a low alues o Bjo ken’s x. In a ecen wo k [1],
we in es iga ed he sub-eikonal non-pe u ba i e co ec ions o he qua k-nucleus sca e ing using a
ligh - on Hamil onian o malism, he ime-dependen basis ligh - on quan iza ion ( BLFQ) [2].
In his p oceeding, we ex end he in es iga ion by including one dynamical gluon.
2. Time-dependen basis ligh - on quan iza ion ( BLFQ)
We conside sca e ing o a high-ene gy qua k mo ing in he posi i e zdi ec ion, on a high-
ene gy nucleus mo ing in he nega i e zdi ec ion. The qua k has momen um pµand p+>> p−,p⊥
whe eas he nucleus has momen um Pµand P−>> P+,P⊥. We ea he qua k s a e a he ampli ude
le el and he nucleus as an ex e nal backg ound ield. The qua k in e ac s wi h he nuclea ield o
a ini e wid h along x+.
The QCD Lag angian wi h a backg ound gluon ield eads,
L=−1
4Fµν aFa
µν +Ψ(iγµDµ−m)Ψ,(1)
whe e Dµ≡∂µI+igAµ,m=mqI(mqis he qua k mass) and Iis he 3 by 3 uni ma ix
in colo space. Fµν
a≡∂µCν
a−∂νCµ
a−g abcCµ
bCν
cis he ield enso , and Dµ≡∂µI+igCµ.
Cµ=Aµ+Aµis he summa ion o he quan um gauge ield Aµ=AaµTaand he backg ound
gluon ield Aµ=AaµTa. The backg ound ield is desc ibed by he CGC o malism [3]. The local
densi y o cha ge cha ge in he nucleus is ea ed as a s ochas ic a iable sa is ying he co ela ion
ela ion,
ρa(®
x⊥,x+)ρb(®
y⊥,y+)=g2˜µ2δabδ2(®
x⊥−®
y⊥)δ(x+−y+).(2)
The ield is sol ed om (m2
g− ∇2
⊥)A−
a(®
x⊥,x+)=ρa(®
x⊥,x+)in he co a ian gauge o ∂µAµ=0
and i has only one nonze o componen A−.
The ligh - on Hamil onian can be de i ed om he Lag angian h ough he s anda d Legend e
ans o ma ion [4]. We w i e i in o wo pa s as P−(x+)=P−
KE +V(x+).P−
KE is he kine ic ene gy
o he qua k and he dynamical gluon. V(x+)a e he emaining in e ac ion e ms, which in gene al,
could ha e a ime dependence a ising om he ex e nal ield. The qua k s a e obeys he e olu ion
equa ion on he ligh on , which in he in e ac ion pic u e eads,
i∂
∂x+|ψ;x+iI=1
2VI(x+) |ψ;x+iI,(3)
whe e VI(x+)=ei1
2P−
K E x+V(x+)e−i1
2P−
K E x+is he in e ac ion Hamil onian in he in e ac ion pic u e.
We sol e Eq. (3) by decomposing he e olu ion ime x+in o ns eps o size δx+≡x+/n,
|ψ;x+iI=T
+exp "−i
2∫x+
0
dz+VI(z+)#|ψ; 0iI=T
+lim
n→∞
n
Ö
k=11−i
2VI(x+
k)x+
n|ψ; 0iI
=lim
n→∞ 1−i
2VI(x+
n)δx+. . . 1−i
2VI(x+
1)δx+|ψ; 0iI,
(4)
2
PoS(Ha dP obes2020)105
Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach Meijian Li
whe e T
+is he ligh - on ime o de ing. This p oduc expansion is exac in he limi o n→ ∞.
The nume ical calcula ion is ca ied ou on he si es o a 3-dimensional disc e e space. The
2-dimensional ans e se space is a la ice ex ending om −L⊥ o L⊥ o each side, wi h pe iodic
bounda y condi ions. The e a e a numbe o 2N⊥la ice si es on each side, wi h indices nx,ny=
−N⊥,−N⊥+1, . . . , N⊥−1. The la ice spacing is a=L⊥/N⊥. The co esponding momen um space
is also in a pe iodic la ice ex ending om −π/a o π/awi h spacing dp≡π/L⊥. The backg ound
ield ex ends om 0 o Lηalong x+, and i is disc e ized in o a numbe o Nηlaye s. The laye
index is k=1,2, . . . , Nη, and each laye has a leng h o τ=Lη/Nη. In his disc e ized space, he
co ela ion ela ion o he colo cha ge as de ined in Eq. (2) also akes a disc e e o m as,
ρa(nx,ny,k)ρb(n0
x,n0
y,k0)=g2˜µ2δab
δnx,n0
xδny,n0
y
a2
δk,k0
τ.(5)
3. Calcula ion in he Fock space o |qi
To s a wi h, we unca e he Fock space o he qua k o he leading sec o as |qi. In his case,
he QCD Lag angian in Eq. (1) educes o Lq=Ψ(iγµDµ−m)Ψ. The in e ac ion e m in he
Hamil onian is be ween he backg ound ield and he qua k,
P−
KE =∫dx−d2x⊥
1
2Ψγ+m2− ∇2
⊥
2i∂−
Ψ,V(x+)=∫dx−d2x⊥g¯
Ψγ+TaΨAa
+.(6)
Conside ing ha he backg ound ield in e ac s wi h he qua k in he ans e se space and he colo
space, we cons uc he basis s a e as |βi=|kx,ky,ci, whe e kxand kya e he ans e se momen um
o he qua k and cis he colo o he qua k. The qua k s a e is hen expanded as
|ψ;x+iI=Õβcβ(x+) |βi,
whe e cβ(x+) ≡ hβ|ψ;x+iIa e he basis coe icien s. The sca e ing o he qua k on a CGC ield is
simula ed acco ding o he non-pe u ba i e me hod explained in Sec ion 2.
We ind ha he esul ing qua k’s c oss sec ions ag ee wi h he analy ical eikonal expec a ions
no only in he eikonal limi o p+=∞, bu also in cases wi h e y small p+[1]. The non-eikonal
e ec is obse ed in he ans e se coo dina e space. As shown in Fig. 1, in he eikonal limi o
p+=∞, he qua k does no change i s ans e se loca ion. In he case wi h a ini e alue o p+, he
qua k sp eads ou in i s ans e se coo dina e dis ibu ion.
4. Calcula ion in he Fock space o |qi+|qgi
In he |qi+|qgiFock space, he QCD Lag angian in Eq. (1) is es o ed. The in e ac ion pa
con ains, he in e ac ion o he ex e nal ield wi h he qua k, ha be ween he dynamical gluon and
he qua k, and ha o he ex e nal ield wi h he dynamical gluon. The ins an aneous e ms in he
|qgisec o a e excluded by he “gauge cu o ” o mula ion [5].
P−
KE =∫dx−d2x⊥−1
2Aj
a(i∇)2
⊥Aa
j+1
2Ψγ+m2− ∇2
⊥
2i∂−
Ψ,
V(x+)=∫dx−d2x⊥+g¯
Ψγ+TaΨAa
++g¯
ΨγµTaΨAa
µ+g abc ∂+Ai
bAc
iAa
+.
(7)
3
PoS(Ha dP obes2020)105
Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach Meijian Li
��(��)
��(��)
(a) E olu ion o he qua k’s ans e se coo dina e dis ibu ion a p+=∞
��(��)
��(��)
(b) E olu ion o he qua k’s ans e se coo dina e dis ibu ion a p+=10 GeV
Figu e 1: The e olu ion o he qua k’s ans e se coo dina e dis ibu ion a di e en p+, (a) p+=∞,
(b) p+=10 GeV. The ini ial s a e o he qua k is dis ibu ed as Ce−|® ⊥|2/(0.2∗50 GeV−1)2, whe e Cis he
no maliza ion coe icien . F om le o igh , he ans e se coo dina e dis ibu ions o he qua k a e shown a
a sequen ial in e ac ion ime. Pa ame e s in hose panels: Lη=50 GeV−1,Nη=4,mg=0.1GeV, N⊥=18,
L⊥=50 GeV−1,g2˜µ=0.486 GeV−3/2. (Figu e adap ed om Re . [1])
The qua k s a e is expanded on he disc e ized momen um basis as
|ψ;x+iI=Õβq
cβq(x+) |βqi+Õβqg
cβqg (x+) |βqgi.
cβq(x+) ≡ hβq|ψ;x+iIand cβqg (x+) ≡ hβqg |ψ;x+iIa e he basis coe icien s. The basis s a es
a e |βqi=|kx,ky,k+,s,ciand |βqgi=|kx
q,ky
q,k+
q,sq,cq,kx
g,ky
g,k+
g,sg,cgi, o he |qiand he |qgi
sec o espec i ely. Compa ed wi h ba e qua k basis |βi, hese basis s a es also speci y he spin
p ojec ion and he longi udinal momen um o he pa ons, since hese quan um numbe s could
change ia he gluon emission/abso p ion. The x−di ec ion is ea ed as a ci cle o leng h 2L, and
he longi udinal momen um p+in he basis s a es akes he disc e e alues p+=(2π/L)k+, whe e
k+=1,2,3, . . . o bosons (neglec ing he ze o mode) and k+=1/2,3/2,5/2, . . . o e mions. We
de ine Kmax as he o al k+o he sys em, and i emains he same du ing he e olu ion.
In Fig. 2, we p esen he e olu ion o he p+dis ibu ion in he |qi+|qgiFock space. The
dec easing p obabili y o he single qua k s a e and he eme ging o he one-gluon-d essed qua k
indica e ha he qua k loses ene gy h ough he e olu ion, and ha is ia gluon emissions.
5. Conclusions
We applied he BLFQ o malism o he qua k-nucleus sca e ing and s udy sub-eikonal e ec s
om non-pe u ba i e aspec s. We a e able o access he wa e unc ion o he qua k a in e media e
ime du ing he e olu ion. We o esee mo e applica ions o sca e ing p ocesses in he nea u u e.
4

PoS(Ha dP obes2020)105
Fo wa d qua k je -nucleus sca e ing in a ligh - on Hamil onian app oach Meijian Li
��
+(��� ���)
�/�
� �� �� �� �� ��
����
����
����
����
����
�+(���-�)
�����������
|�〉
(a) E olu ion o p+in he |qisec o
�+(��� ���)
��
+=�/�� ��
+=�
��
+=�/�� ��
+=�
��
+=�/�� ��
+=�
��
+=�/�� ��
+=�
� �� �� �� �� ��
����
����
����
����
����
�+(���-�)
�����������
|��〉
(b) E olu ion o p+in he |qgisec o
Figu e 2: The e olu ion o he p+dis ibu ion in (a) he |qisec o and (b) he |qgisec o . Pa ame e s in
hose panels: Lη=50 GeV−1,Nη=8,L=0.01 ∗2πGeV−1,Kmax =4.5,mg=0.1GeV, mq=4.2GeV,
N⊥=4,L⊥=50 GeV−1,g2˜µ=0.018 GeV−3/2. The ini ial s a e is a ba e qua k wi h px=py=0,
p+=4.5×100 GeV, spin up, and single colo c=1.
Acknowledgmen s
This wo k was suppo ed by he US Depa men o Ene gy (DOE) unde G an Nos. DE-FG02-
87ER40371, DE-SC0018223 (SciDAC-4/NUCLEI), DE-SC0015376 (DOE Topical Collabo a ion
in Nuclea Theo y o Double-Be a Decay and Fundamen al Symme ies). This esea ch used e-
sou ces o he Na ional Ene gy Resea ch Scien i ic Compu ing Cen e (NERSC), a U.S. Depa men
o Ene gy O ice o Science Use Facili y ope a ed unde Con ac No. DE-AC02-05CH11231. This
wo k has been suppo ed by he Eu opean Resea ch Council (ERC) unde he Eu opean Union’s
Ho izon 2020 esea ch and inno a ion p og amme (g an ag eemen No ERC-2015-CoG-681707).
The con en o his a icle does no e lec he o icial opinion o he Eu opean Union and esponsi-
bili y o he in o ma ion and iews exp essed he ein lies en i ely wi h he au ho s.
Re e ences
[1] M. Li, X. Zhao, P. Ma is, G. Chen, Y. Li, K. Tuchin e al., Ul a ela i is ic qua k-nucleus
sca e ing in a ligh - on Hamil onian app oach,Phys. Re . D 101 (2020) 076016
[2002.09757].
[2] X. Zhao, A. Ilde on, P. Ma is and J.P. Va y, Sca e ing in Time-Dependen Basis Ligh -F on
Quan iza ion,Phys. Re . D88 (2013) 065014 [1303.3273].
[3] J. Jalilian-Ma ian, Elas ic sca e ing o a qua k om a colo ield: longi udinal momen um
exchange,Phys. Re . D96 (2017) 074020 [1708.07533].
[4] S.J. B odsky, H.-C. Pauli and S.S. Pinsky, Quan um ch omodynamics and o he ield heo ies
on he ligh cone,Phys. Rep . 301 (1998) 299 [hep-ph/9705477].
[5] A.C. Tang, S.J. B odsky and H.C. Pauli, Disc e ized ligh cone quan iza ion: Fo malism o
quan um elec odynamics,Phys. Re . D44 (1991) 1842.
5