BFKL Pomeron loop contribution in diffractive photoproduction and inclusive hadroproduction of $J/\psi$ and $\Upsilon$

P u b l i s h e d f o r SISSA b y S p r i n g e r

Received: May 6, 2019 A ccepted: July 12, 2019 Published: July 23, 2019

B F K L Pomeron loop contribution in diffractive photoproduction and inclusive hadroproduction of J / ψ and Y

P io tr K o tko ,“ Leszek M o ty k a

,6

M ariusz Sadzikowski

6

and Anna M . S tastoc aIn stitu te o f N u c lear P h y s ic s P olish A c a d e m y o f Sciences,

E. Radzikowskiego 152, 3 1 -3 4 2 Kraków, P o la n d bIn stitu te o f Ph y sic s, Jagellonian University,

S. L o ja sie w ic za 11, 3 0 -3 4 8 Kraków, P o la n d

cD e p a r t m e n t o f Ph y sic s, The P e n n s y lv a n ia S ta te University, U n iversity P ark, P A 16802, U.S.A.

E - m a i l : piotr.kotko@ifj.edu.pl, leszekm@th.if.uj.edu.pl, ufsadzik@th.if.uj.edu.pl, ams52@psu.edu

A b s t r a c t : W e an a ly z e c o n trib u tio n s to th e h eav y v e c to r m eso n p ro d u c tio n w ith large tra n s v e rs e m o m e n tu m in p ro to n -p ro to n a n d d iffractiv e p h o to n -p ro to n s c a tte rin g d riv e n by a n ex ch an g e of tw o B a litsk y -F a d in -K u ra e v -L ip a to v P o m e ro n s in th e sq u a re d a m p litu d e s . T h e P o m e ro n s cou p le to a single p a r to n a n d fo rm a P o m e ro n lo op closed by th e v ec to r m eson im p a c t fa c to rs . F o r th e p h o to n -p ro to n case th e d iffractiv e c u t of th e P o m e ro n loo p c o n trib u te s , a n d for th e inclusive h a d ro p ro d u c tio n on e finds th e loo p w ith tw o c u t P o m e ro n s. W e c o m p u te b o th of th e s e P o m e ro n loop c o n trib u tio n s a n d s tu d y in d e ta il th e ir p ro p e rtie s. T h e re su lts a re th e n used to c a lc u la te th e cross sectio n s for d iffractiv e J / 0 p h o ­ to p ro d u c tio n w ith larg e tra n s v e rs e m o m e n tu m a t H E R A a n d th e c o rre la te d tw o P o m e ro n c o n trib u tio n for inclusive J / 0 a n d Y p ro d u c tio n cross sectio n s a t th e L H C . W ith in a unified a p p ro a c h a go o d d e s c rip tio n of th e p h o to p ro d u c tio n d a t a is fo un d, b u t c o rre la te d tw o P o m e ro n m ech an ism gives only a sm all c o n trib u tio n to h a d ro p ro d u c tio n of th e v e c to r m esons a t th e L H C .

K e y w o r d s : Q C D P h e n o m en o lo g y

A r X i v e P r i n t : 1905.00130

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

C o n te n ts

1 I n t r o d u c t i o n 1

2 D i f f r a c t i v e h e a v y v e c t o r m e s o n p r o d u c t i o n 5 3 T w o P o m e r o n c o n t r i b u t i o n t o h e a v y v e c t o r m e s o n h a d r o p r o d u c t i o n 10

3.1 D ire c t a p p ro a c h 10

3.2 T h e color fa c to rs 12

3.3 V ecto r m eson h a d ro p ro d u c tio n in co n fo rm al re p re s e n ta tio n of th e B F K L

P o m e ro n 14

4 P r o p e r t i e s o f t h e P o m e r o n lo o p a t t h e p a r t o n le v e l 17

5 N u m e r i c a l r e s u l t s 19

6 D i s c u s s i o n 25

7 S u m m a r y a n d c o n c l u s i o n s 2 7

1 In tr o d u c tio n

T h e h eav y v e c to r m esons w ith n e g a tiv e C -p a r ity — c h a rm o n ia a n d b o tto m o n ia — are clas­

sical p ro b e s of th e Q C D ex ch an g e a t h ig h energies. T h e signals o f J / 0 , ^ a n d Y m esons in th e ir lep to n ic d ecay c h a n n els are cle a r a n d allow for a c c u ra te m e a su re m e n ts of th e co r­

re sp o n d in g d iffe ren tial cross sections. T h e u n d e rly in g p ro d u c tio n d y n a m ic s is d riv en by g luonic degrees o f freedom a n d th e ir Q C D ev o lu tio n . T h e c u rre n tly a c c e p te d p ic tu re s of th e h eav y v e c to r m eson p ro d u c tio n m ech an ism s in d iffractiv e p h o to n -h a d ro n a n d inclusive h a d ro n -h a d ro n collisions are, how ever, q u ite d ifferen t. T h e d iffractiv e p h o to p ro d u c tio n d a t a a t high energ ies a n d large tra n s v e rs e m o m e n tu m o b ta in e d by H1 [1] a n d ZE U S [2]

c o lla b o ra tio n s a t H E R A h ave b ee n successfully d esc rib ed [3- 7] assu m in g an ex ch an g e of th e n o n -fo rw ard B a litsk y -F a d in -K u ra e v -L ip a to v (B F K L ) P o m e ro n [8- 17] in th e d iffractive a m p litu d e . O n th e o th e r h a n d , in th e h ig h en e rg y inclusive h a d ro p ro d u c tio n o f J / 0 an d Y , a good d e sc rip tio n of d a t a from h a d ro n collid ers re q u ires a d o p tin g th e C olo r O c te t M odel (C O M ) [18- 25], see also refs. [2 6 , 27] for a review . In th e p re se n t p a p e r we shall in v e stig a te th e d iffractiv e p h o to p ro d u c tio n a n d a c o n trib u tio n to th e inclusive h ad ro p ro - d u c tio n o f h eav y v e c to r m esons assu m in g th e sam e u n d e rly in g Q C D d y n am ics of tw o B F K L P o m e ro n exchange.

T h e s ta n d a r d p ic tu re o f d iffractiv e p h o to p ro d u c tio n of v e c to r m esons a t H E R A a t large m o m e n tu m tra n s f e r assu m es ex ch an g e of g luo nic h a rd color sin g let across larg e ra p id ity

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d is ta n c e b etw e en th e in co m in g p h o to n — o u tg o in g m eson v e rte x a n d th e d iffractiv e re m ­ n a n t of th e p ro to n . T h e k in e m a tic s of th is p ro cess allow s to a p p ly th e h ig h en e rg y lim it in w h ich th e d o m in a n t c o n trib u tio n to th e color sin g let a m p litu d e is given by th e B F K L P o m e ro n [9- 14]. B y th e B F K L P o m e ro n one u n d e rs ta n d s th e sy ste m o f tw o R eggeized g luons in th e t-c h a n n e l in te ra c tin g by ex ch an g e of u su al glu on s. T h e p ro p a g a tio n of th e R eggeized g luons a n d th e effective in te ra c tio n s b etw e en th e m are deriv ed in Q C D in th e h ig h en e rg y lim it. In m ore d e ta il, th e ex c h ang e a m p litu d e is d e sc rib e d by th e B F K L evo­

lu tio n e q u a tio n t h a t fo rm a lly re su m s lo g a rith m ic a lly e n h a n c e d p e r tu r b a tiv e c o rre c tio n s to all o rd ers. In th e B F K L a p p ro a c h one re su m s lo g a rith m s of a ra tio o f a larg e collision en e rg y y / s a n d o th e r, m uch sm aller m ass scales e.g. th e m eso n m ass o r th e m o m e n tu m tra n s fe r. T h ese lo g a rith m s a re re la te d to th e ra p id ity d is ta n c e Y b etw e en th e p ro je c tile a n d th e ta r g e t in th e h ig h en e rg y s c a tte rin g process. So fa r th e B F K L re s u m m a tio n in Q C D w as p erfo rm ed a t th e lead in g lo g a rith m ic (LL) [9- 14] a n d n e x t-to -le a d in g lo g a rith ­ m ic (N L L) a p p ro x im a tio n [15, 16, 28- 32]. In th e LL a p p ro x im a tio n one re su m s te rm s

^ l l ~ a ^ Y ” w hile ~ a™+ l Y n te rm s are re su m m ed by th e N LL B F K L ev o lu tio n . T h e B F K L fo rm alism assu m es high en e rg y (or fcy) fa c to riz a tio n in w hich h a rd m a trix elem en ts a re fa cto rized in ra p id ity space from th e B F K L ev o lu tio n . In a d d itio n , m a trix elem en ts a re off-shell, w ith in itia l q u a rk s o r glu on s c a rry in g n o n -zero tra n s v e rs e m o m e n ta unlike in th e s ta n d a r d co llin ear a p p ro x im a tio n . H ence also th e B F K L P o m e ro n m ay c a rry n o n -zero tra n s v e rs e m o m e n tu m , a n d th e co rre sp o n d in g a m p litu d e is governed by th e no n -fo rw ard B F K L P o m e ro n . T h is fo rm alism w as ap p lied [5 , 6, 33- 35] som e tim e ago to th e d a t a from H E R A o n J / 0 , p a n d $ d iffractiv e m esons p h o to p ro d u c tio n [1, 2 , 36] w ith larg e tra n s v e rs e m o m e n tu m p T a n d w as show n to d e sc rib e th e d a t a well. In th is p a p e r we re v isit th e d iffrac­

tiv e J / 0 p h o to p ro d u c tio n a t H E R A a n d use th e e sta b lish e d d e s c rip tio n o f th is p rocess to e s tim a te th e B F K L P o m e ro n loop c o n trib u tio n to inclusive v e c to r m eson h a d ro p ro d u c tio n .

T h e C O M o f inclusive heav y m eso n h a d ro p ro d u c tio n assu m es n o n -zero a m p litu d e s for a ch an g e of th e q u a n tu m n u m b e rs (in p a r tic u la r color a n d a n g u la r m o m e n tu m ) b etw e en th e p a rto n ic p h a se a n d th e m eson [18- 20]. M o re specifically, in th e p a rto n ic su b p ro c ess heav y q u a r k -a n tiq u a rk p a ir Q Q is p ro d u c e d w ith a n a r b itr a r y color a n d a n g u la r m o m e n tu m q u a n tu m n u m b ers, a n d th e tra n s itio n to th e final s ta te m eson is govern ed by s e p a ra te m u ltip lic a tiv e coefficients for each set o f th e p a rto n ic q u a n tu m n u m b e rs. T h e v alu es of th e se coefficients are o b ta in e d by fittin g th e p re d ic te d cross sectio n s to e x p e rim e n ta l d a ta . T h e th e o re tic a l basis for th is m ech a n ism com es from tw o c o m p le m e n ta ry sources. F ir s t, w ith in h eav y q u a rk effective th e o ry on e finds no n -zero a m p litu d e s of h ig h er Fock s ta te s in th e h eav y v e c to r m eson wave fu n c tio n [18], for in s ta n c e a s ta te Q Q g w ith a n a d d itio n a l g lu o n g. O bviously, th e q u a n tu m n u m b e rs o f Q Q in th is s ta te do n o t m a tc h th e q u a n tu m n u m b e rs o f th e m eson. F ro m a n o th e r p e rsp e c tiv e , befo re th e p a rto n ic Q Q m akes th e m eson a pro cess of h a d ro n iz a tio n o ccu rs, in w hich th e q u a n tu m n u m b e rs m ay change.

T h is is a p ic tu re of fra g m e n ta tio n of th e p rim o rd ia l Q Q s ta te in to th e h ea v y m eso n [22].

T h e h a d ro n iz a tio n p rocess does n o t have s a tis fa c to ry p e r tu r b a tiv e d e s c rip tio n as it o ccurs a t low (h a d ro n ic ) scales a n d d e p e n d s on n o n -p e rtu rb a tiv e p ro p e rtie s of th e Q C D v acu u m . In b o th scenarios th e tra n s itio n a m p litu d e s o f Q Q to th e m eso n c a n n o t b e d eriv ed from th e o ry — o n ly o rd e r-o f-m a g n itu d e e s tim a te s c a n b e o b ta in e d . N ev erth eless th e C O M has

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a solid th e o re tic a l b asis, a n d it is s tro n g ly s u p p o rte d by th e successful fits of its p re d ic tio n s to th e b u lk of e x p e rim e n ta l d a ta . H ow ever, since th e p a ra m e te rs o f th e m od el are fitte d , th e re is still ro o m for o th e r th a n C O M po ssib le m ech a n ism s of th e inclusive v e c to r m eson h a d ro p ro d u c tio n .

T h e classical a lte r n a tiv e to th e C O M o f th e h ea v y m eso n p ro d u c tio n is th e C olor S ing let M ech an ism (C SM ) [37- 39]. In fa c t, th e C SM w as co n sid ered to b e th e s ta n d a r d Q C D p re d ic tio n before it w as c o n tra d ic te d by th e T e v a tro n d a t a [4 0 , 41]. In th is a p p ro a c h one assu m es th e e x a c t m a tc h in g o f th e q u a n tu m n u m b e rs o f th e p ro d u c e d p a rto n ic Q Q s ta te a n d th e final s ta te m eson. T h e m a in a d v a n ta g e of th is m ech a n ism is its co m p leten e ss w ith in p e r tu r b a tiv e Q C D a n d no need for a d d itio n a l p a ra m e te rs . F or th e C -o d d m esons V a t th e lead in g tw ist th e C SM is d riv en by th e g + g ^ V + g p a rto n ic s c a tte rin g . T h e p re d ic tio n s o f th e C SM , how ever, fail b a d ly in d esc rib in g th e J / 0 h a d ro p ro d u c tio n a t th e T e v a tro n a n d th e L H C , see e.g. [2 6 , 27]. F o r th e to ta l inclusive v e c to r m eson h a d ro p ro d u c tio n cross sectio n s th e C SM b o th a t th e lead in g o rd e r (L O ) a n d a t th e n e x t-to -le a d in g o rd e r (N L O ) are m ore th a n one o rd e r of m a g n itu d e below th e d a t a [2 6 , 27]. M oreover, th e C SM p re d ic tio n s lead to th e d is trib u tio n in th e m eson tra n s v e rs e m o m e n tu m p t w hich is m uch to o soft, w hile th e C O M is ab le to d esc rib e well th e p T d e p e n d e n c e o f th e m eson p ro d u c tio n cross sectio n . T h e C SM a n d C O M a p p ro a c h e s w ere also e x te n d e d from th e co llin ear a p p ro x im a tio n to th e k T-fa c to riz a tio n fram ew o rk [42- 48]. I t w as fo u n d t h a t also in th e k T-fa c to riz a tio n a p p ro a c h th e C SM m od el is m u ch below th e d a t a for th e d ire c t J / 0 p ro d u c tio n a t larg e tra n s v e rs e m o m en ta.

B ey o n d th e lead in g tw ist a p p ro x im a tio n th e C SM m ay be realized also w ith a fusion of th re e in itia l s ta te gluons. A t th e p a rto n ic level th e m eso n fo rm a tio n o ccu rs by g + g + g ^ V d ia g ra m s w ith th e co u p lin g th ro u g h th e h ea v y q u a rk loop. T h e T h re e G lu o n F u sio n (3 G F ) m ech a n ism w as co n sid ered in ref. [49] as a c o n trib u tio n to J / 0 h a d ro p ro d u c tio n , a n d in ref. [50] it w as p ro p o se d as a p o ssib le lead in g c o n trib u tio n to h ea v y v e c to r m eso n h ad ro p ro - d u c tio n . T h e n it w as fu r th e r stu d ie d in refs. [51- 53]. M oreover, re c e n tly c o n trib u tio n s of m u ltip le glu o n co u p lin g s in th e J / 0 p ro d u c tio n w ere co m b in ed w ith th e C O M [54, 55].

In th is a p p ro a c h a good d e sc rip tio n of th e J / 0 h a d ro p ro d u c tio n d a t a w as o b ta in e d , in ­ clu d in g th e m eson p o la riz a tio n . A lth o u g h th e u n c o rre la te d 3 G F m ech a n ism is a n im p licit c o n trib u tio n of th is fram ew ork, a t la rg e r p t th e cross sec tio n is d o m in a te d by th e C O M c o n trib u tio n s .

In th e 3 G F m ech a n ism one of th e g luo ns com es from o ne h a d ro n ic b e a m (th e p ro ­ je c tile ), a n d tw o o th e r ones from th e o th e r b e a m (th e ta r g e t) . T h e se tw o glu on s in th e t-c h a n n e l c a n b e ta k e n e ith e r as co m p letely in d e p e n d e n t (u n c o rre la te d ) or as co m in g from a single p a r to n o f th e ta r g e t (c o rre la te d ). T h e tw o scenario s co rre sp o n d to a n u n c o rre la te d d o u b le glu o n d is trib u tio n in th e ta rg e t, a n d to th e c o rre la te d c o n trib u tio n in th e d o u b le g lu o n d is trib u tio n , respectively. A d e ta ile d s tu d y of th e u n c o rre la te d c o n trib u tio n to th e 3 G F m ech an ism show ed t h a t it m ay c o n trib u te to th e t o ta l J / 0 h a d ro p ro d u c tio n cross sec tio n as a fra c tio n of a b o u t 20-25% [51]. T h e o b ta in e d p T -d e p e n d e n c e of th e m eson p ro d u c tio n d iffe ren tial cross sec tio n d a / d p T w as fo u n d to be m u ch s te e p e r th a n th e e x p e ri­

m e n ta l d a ta . Specifically, a t la rg e r values o f p t th e e x p e rim e n ta l d a t a c a n b e a p p ro x im a te d w ith a pow er law: d a / d p T ~ 1 /P n w ith n ~ 5, w hile w ith th e u n c o rre la te d 3 G F m ech an ism

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n > 8 is o b ta in e d . T h is r a th e r ste e p p T d e p e n d e n c e is well u n d e rs to o d as th e u n c o rre la te d 3 G F c o n trib u tio n e n te rs a t a h ig h er tw ist, a n d it is su p p re sse d a t larg e p T by a n a d d itio n a l fa c to r o f ( A h / p T ) 2 w ith re sp e c t to th e lead in g tw ist, w h ere Ah is a sm all h a d ro n ic scale.

In th is p a p e r we an a ly z e in d e ta il th e c o rre la te d 3 G F m ech an ism . T h e tw o g luons t h a t e n te r th e v e c to r m eson p ro d u c tio n v e rte x from th e ta r g e t side are a ssu m ed to com e from a s p littin g of a single p a rto n : q u a rk o r glu o n in th e ta r g e t. Since th e p a re n t p a r to n is point-like, it do es n o t in tro d u c e an y a d d itio n a l h a d ro n ic scale a n d o ne e x p e c ts a h a rd e r d e p e n d e n c e of th e m eson p T -d is trib u tio n th a n it w as for th e u n c o rre la te d 3 G F . A t th e low est o rd e r th is m ech a n ism o ccu rs th ro u g h g + g ^ V + g (o r g + q ^ V + q) p a rto n ic p ro cess w ith an ex ch an g e of tw o g lu on s b etw e en th e g ^ V tra n s itio n v e rte x a t th e side o f th e p ro je c tile a n d th e g ^ g (o r q ^ q) s c a tte rin g a t th e ta r g e t side. A t th e a m p litu d e level th e tw o glu o n ex ch an g e in th e t-c h a n n e l c a rrie s th e sy m m e tric color o c te t. T h e im p o rta n t fe a tu re of th e p rocess a t th e low est o rd e r is a flat d e p e n d e n c e o n th e ra p id ity d is ta n c e Y b etw e en th e p ro je c tile g lu o n a n d th e ta r g e t gluon. B ey o n d th e low est o rd e r a p p ro x im a tio n th e a m p litu d e of th is su b p ro c ess is m odified by Q C D ra d ia tiv e co rrec tio n s.

T h ese c o rre c tio n s c a n be re su m m ed by a Q C D e v o lu tio n e q u a tio n . In th e high en e rg y lim it co rre sp o n d in g to Y » 1, th e cross sec tio n of c o rre la te d 3 G F cross sec tio n is d riv en by an ex ch an g e of fo u r R eggeized g lu ons in th e t o ta l color sin glet s ta te t h a t in te ra c t w ith B F K L kernels. T h is sy stem is d esc rib ed by B artels-K w ieciń sk i-P rasz ało w icz (B K P ) ev o lu tio n e q u a tio n [56- 58]. I t w as show n in ref. [59] t h a t in th e large N c lim it th e sy stem of fo u r R eggeized g luons in th e color singlet c h a n n e l m ay be a p p ro x im a te d by tw o in d e p e n d e n t B F K L P o m e ro n s. In con seq u en ce one e x p e c ts a s tro n g e n h a n c e m e n t ~ e x p (2 A p ) o f th e cross sec tio n a t large Y by th e d o u b le B F K L e v o lu tio n w ith th e B F K L P o m e ro n in te rc e p t A p ~ 0.3. In a d d itio n , th e an o m alo u s d im en sio n s of th e B F K L G re e n ’s fu n c tio n s could lead to a less ste e p p T d e p e n d en ce . So, d e s p ite th is c o n trib u tio n e n te rs a t th e O ( a ^ ) o rd e r a n d is a well defined p a r t of th e N N L O c o rre c tio n to th e C SM c o n trib u tio n , it m ay b e im p o rta n t d u e to s tro n g effects of th e B K P ev o lu tio n . T h e first e s tim a te of th is c o n trib u tio n [50]

su g g ested t h a t it m ay re p ro d u c e th e inclusive J/ 0 h a d ro p ro d u c tio n cross sec tio n d a t a from th e T e v a tro n . In th is p a p e r we p erfo rm a d e ta ile d c a lc u la tio n of th e c o rre la te d 3 G F cross sec tio n to verify its im p o rta n c e .

In th e an a ly sis we sh all use a c o n n e c tio n b etw e en th e c o rre la te d 3 G F m ech an ism an d th e d iffractiv e p h o to p ro d u c tio n of v e c to r m esons a t larg e p T . T h is c o n n e c tio n o rig in a te s from th e k in e m a tic a l id e n tity o f th e im p a c t fa c to rs co rre sp o n d in g to y + (2g) ^ V an d g + (2g) ^ V tra n s itio n s . A t th e lead in g o rd e r th e difference b etw e en th e s e tw o im p a c t fa c to rs com es o n ly from th e color fa c to rs a n d th e gau g e co u p lin g c o n s ta n ts . H ence, th e cross sectio n s for a v e c to r m eson p ro d u c tio n by glu o n s c a tte rin g off a p a rto n ic ta r g e t an d for th e m eson d iffractiv e p h o to p ro d u c tio n a re p ro p o rtio n a l to each o th e r. T h is re la tio n im poses im p o rta n t c o n s tra in ts on th e c o rre la te d 3 G F m ech an ism . W h e n th e B K P /B F K L e v o lu tio n is ta k e n in to ac c o u n t how ever, th e d iffractiv e p h o to p ro d u c tio n a n d th e 3 G F m ech a n ism are d ifferen t d u e to th e d ifferen t color flow. T h e d iffractiv e p h o to p ro d u c tio n cross sec tio n c o rre sp o n d s to th e d iffractiv e c u t of tw o B F K L P o m e ro n exch ang e, a n d th e c o rre la te d 3 G F cross sec tio n to a n ex ch an g e of tw o c u t B F K L P o m e ro n s. T h u s in d e p e n d e n t e v a lu a tio n is n ecessary for th e s e tw o pro cesses. In b o th cases, how ever, a t th e level o f cross

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

sec tio n one finds th e to p o lo g y of th e B F K L P o m e ro n loo p sp a n n e d b etw e en th e m eson im p a c t fa c to rs a t th e p ro je c tile side a n d th e p a r to n im p a c t fa c to r a t th e ta r g e t side. So, b esid es e v a lu a tin g th e c o rre la te d 3 G F c o n trib u tio n w ith th e B K P /B F K L e v o lu tio n effects in clu d ed , we sh all also re v isit th e d iffractiv e p h o to p ro d u c tio n case a n d an a ly se in d e ta il th e p ro p e rtie s of th e B F K L P o m e ro n loop in th e t-ch a n n el.

T h e p a p e r is o rg a n iz ed as follows. In th e n e x t sec tio n we d esc rib e th e th e o re tic a l fra m e ­ w ork for th e d iffractiv e heav y v e c to r m eson p ro d u c tio n in D IS, a n d th e n o n -fo rw ard B F K L ev o lu tio n . In sec tio n 3 th e tw o -P o m e ro n c o n trib u tio n to th e v e c to r m eson h a d ro p ro d u c tio n is an a ly z ed . T h e d e ta ils o f th e color fa c to rs are d iscu ssed a n d th e B F K L P o m e ro n in th e co n fo rm al re p re s e n ta tio n is d esc rib ed . In sec tio n 4 th e p ro p e rtie s of th e P o m e ro n loop a t th e p a r to n level a re an a ly z ed , in p a r tic u la r th e d e p e n d e n c e o n th e tra n s v e rs e m o m e n tu m . In sec tio n 5 we p re se n t th e co m p a riso n o f o u r n u m eric al c a lc u la tio n s w ith th e v e c to r m eson p ro d u c tio n in d iffractiv e D IS as well as in h a d ro p ro d u c tio n , a n d we d iscuss th e re su lts in sec tio n 6. F inally, in sec tio n 7 we give su m m a ry a n d conclusions.

2 D iffra c tiv e h e a v y v e c to r m eso n p r o d u c tio n

W e b eg in w ith a s h o rt reco llectio n o f th e p e r tu r b a tiv e Q C D a p p ro a c h to th e h a rd color singlet ex ch an g e in th e d iffractiv e heav y v e c to r m eson V p h o to p ro d u c tio n off a p ro to n a t larg e m o m e n tu m tra n s f e r t. T h e p ro cess w as in v e stig a te d in d e ta il a t H E R A in e± p collisions w ith in v a ria n t c.m .s. en e rg y V S = 318 G eV . In th e m e a su re m e n t of th e process e± p ^ e ± V X , a larg e ra p id ity g a p dev oid o f p a rtic le s is re q u ired . I t s e p a ra te s th e p ro d u c e d v e c to r m eson V a n d th e d isso c ia te d p ro to n re m n a n t X . T h e e ± p cross sec tio n m ay b e fa cto rized in to a u n iv ersa l flux fu n c tio n o f q u a si-re a l p h o to n s in th e e le c tro n an d th e cross sec tio n for th e d iffractiv e p h o to p ro d u c tio n su b p ro cess,

YP ^ V X , (2.1)

w ith th e ra p id ity gap. In th is p rocess th e p h o to n -p ro to n in v a ria n t m ass, y f s = V z S , is assu m ed to b e m uch la rg e r th a n all th e o th e r scales p re s e n t in th e process, hence s » |t|

a n d s » My , w h ere M y is th e m eso n m ass. T h e a p p lic a b ility o f th e p e r tu r b a tiv e Q C D is e n su re d by th e c o n d itio n s |t| » AQc d a n d M y » AQc d . T h e d iffractiv ely p ro d u c e d C -o d d s ta te V is a ssu m ed to b e a h ea v y v e c to r q u a rk o n iu m . In th is w ork we focus o n th e J / 0 m eson. Since in th e available d a t a th e p h o to n flux is s tro n g ly d o m in a te d by v ery low v irtu a litie s —q2 = Q2 we ta k e th e lim it Q2 ^ 0 in c a lc u la tio n s of th e Q C D a m p litu d e s .

W ith in th e k in e m a tic regim e specified above, th e color sin g let ex ch an g e in th e t-c h a n n e l o f th e p rocess (2.1) m ay b e d e sc rib e d in Q C D as th e p e r tu r b a tiv e P o m e ro n exch an g e (figure 1) , t h a t is governed by th e n o n -fo rw ard B F K L e q u a tio n [9- 11, 13, 14]. D u e to th e larg e m o m e n tu m tra n s fe r, th e cross sec tio n c a n b e fa cto rized in to a p a rto n ic cross sectio n (d o m in a te d by th e P o m e ro n ex ch ang e) a n d th e co llin ear p a r to n d is trib u tio n fu n c tio n s (P D F s ), t h a t d esc rib e th e s tru c tu r e of th e p ro to n ta rg e t:

d a = ^ J d x f i ( x , ^ ) d a Yi ( s , t , ^ ) , (2.2) i=g,q,q

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

F ig u r e 1. Diffractive photoproduction of a vector meson V . The zigzag line corresponds to a perturbative Pomeron coupling to individual partons inside the proton.

w h ere s = x s is th e p h o to n -p a r to n in v a ria n t m ass sq u ared , a n d f i is th e P D F for th e p a r to n i w hich m ay b e a q u a rk (a n ti-q u a rk ) o r a gluon . A n a tu r a l choice for th e fa c to riz a tio n scale is p ~ vTt"- H ere x is th e lo n g itu d in a l fra c tio n of th e p ro to n lig ht co ne m o m e n tu m c a rrie d by th e q u a rk o r gluon. In w h a t follows, th e tra n s v e rs e tw o -v ecto rs in th e lig ht cone basis a re d e n o te d by th e b o ld c h a ra c te rs; for ex a m p le, we d e n o te th e v e c to r m eson tra n s v e rse m o m e n tu m as p . In th e h ig h en e rg y k in em atics we have t ~ — |p|2 = —p 2. In eq. (2.2) th e co u p lin g of th e h ig h -t P o m e ro n to th e p ro to n is a ssu m ed to o c c u r th ro u g h co u p lin g to th e in d iv id u a l p a rto n s . T h is a p p ro x im a tio n w as stu d ie d in d e ta il a n d m o tiv a te d in [3 5 , 6 0 , 61].

In th e h ig h en e rg y lim it, th e k in e m a tic p a r t of th e B F K L P o m e ro n co u p lin g to q u a rk s a n d g luons is th e sam e. T h e o n ly difference b etw e en th e q u a rk a n d g lu o n p a rto n ic ta rg e t com es from th e color fa cto rs,

d a Yi = C Yi d a l - p , (2.3)

w h ere C Yi, i = q , g , is th e color fa c to r an d

d a i _p = ¹

1 6 n s 2 |A ( = t = —p2 )|2 d2P (2.4)

w ith A b ein g th e a m p litu d e to p ro d u c e th e v e c to r m eson th ro u g h a single P o m e ro n ex ­ ch a n g e (figure 2) . T h e a m p litu d e is d o m in a te d by th e im a g in a ry p a r t, a n d th e co rre c tio n co m in g from th e re al p a r t e n te rs o n ly a t a s u b le a d in g o rd e r in th e lo g a rith m ic e x p a n sio n a n d m ay b e neg lected . T h e im a g in a ry p a r t of th e a m p litu d e re ad s

Irn A ( M = —p2) = s

/

^{d2k i ( k i , p )} ( y , k i , p )

2n ( k2 + so) [(p — k i) 2 + so] (2.5) w h ere k i a n d p — k i = k2 are tra n s v e rs e m o m e n ta o f th e ex c h an g ed glu on s, a n d p is th e tra n s v e rs e m o m e n tu m c a rrie d by th e P o m e ro n . $ y , are th e im p a c t fa c to rs for th e v e c to r m eson a n d for th e q u a rk , respectively. T h e y are s trip p e d off th e color fa cto rs. In a d d itio n , th e q u a rk im p a c t fa c to r is evolved to ra p id ity y usin g th e B F K L ev o lu tio n .

t

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

F ig u re 2 . The diagram s contributing to the cross section for diffractive vector meson V photopro­

duction off a parton: a gluon (left) and a quark (right). The non-forward BFKL G reen’s function G is evolved along the rapidity gap between the vector meson and the proton rem nants.

T h e e x a c t form o f th e im p a c t fa c to rs will b e d e sc rib e d below in th is section . T h e ra p id ity e v o lu tio n le n g th y o f th e q u a rk im p a c t fa c to r is defined by th e re la tio n

(2 .6) y = loM

w h ere we set A = E t = \ JMV + p 2 • T h is choice is d ifferen t th a n th e choice m a d e in ea rlie r stu d ie s [4- 7], w h ere A = M V w as used. T h e la t t e r value w as selected as a re s u lt of fits d riv en m o stly by th e light v e c to r m eson h ig h -t p h o to p ro d u c tio n d a ta . T h e J / 0 d a ta , how ever, w ere well d esc rib ed b o th w ith A = E t a n d A = M V. H ence in th is p a p e r we use A = E t , th e value w ith a s tra ig h tfo rw a rd k in e m a tic m o tiv a tio n . T h is choice w as used e.g.

in ref. [3].

T h e sm all p a r a m e te r s 0 is a p h en o m en o lo g ical in fra re d cu to ff t h a t m im icks th e ef­

fects of th e color confinem ent; th e re su lts are, how ever, fin ite for s 0 ^ 0. W e in tro d u c e th is p a r a m e te r follow ing th e a p p ro a c h p ro p o se d in ref. [62]. T h e a m p litu d e does n o t d e­

p e n d on th e p ro d u c e d m eson d ire c tio n in th e tra n s v e rs e p lan e, h en ce th e p h a se space d 2p in (2 .4 ) m ay be triv ia lly in te g ra te d over th e a z im u th a l angle. W e p re se n t th e re su lts in th is form to keep th e n o ta tio n u n ifo rm w ith th e m ore c o m p lic a te d case of th e v e c to r m eson h a d ro p ro d u c tio n , w h ich we sh all d e sc rib e in sec tio n 3 .

In th e c a lc u la tio n s we ta k e th e n o n -re la tiv istic a p p ro x im a tio n for th e m eson wave fu n c tio n . W ith th is a s s u m p tio n th e low est o rd e r p h o to n to v e c to r m eson im p a c t fa c to r re a d s [33 , 6 3 , 64]

5ab

$av (ki, p)

⁼ (k u p ) - ^ , (2.7) w h ere a, b are color indices of th e ex c h an g ed gluons, a n d th e k in e m a tic p a r t o f th e im p a c t fa c to r re ad s

w here

(2.8)

(

²

.

⁹

)

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

w ith eq b ein g th e ch a rg e of th e q u a rk in th e m eso n in u n its of th e e le m e n ta ry ch a rg e e, My

— th e m ass of th e v e c to r m eson, a n d r y its le p to n ic d ecay w id th . T h e p h o to n to v ec to r m eson im p a c t fa c to r (2.8) is valid for th e tra n sv e rs e p o la riz a tio n s o f th e p h o to n a n d of th e v e c to r m eson. T h e inco m ing p h o to n is q u a si-re a l h ence its p o la riz a tio n is c o n s tra in e d to b e tra n sv e rse . T h e a m p litu d e of th e tra n s v e rs e p h o to n to lo n g itu d in a ly p o lariz ed v e c to r m eson w as e s tim a te d in ref. [33] to b e sm all. T h erefo re th e a m p litu d e d e sc rib e d by (2 .8) p ro v id es th e d o m in a n t c o n trib u tio n to d iffractiv e v e c to r m eson p h o to p ro d u c tio n .

T h e q u a rk im p a c t fa c to r in th e color sin glet ch a n n el <fe\’ab (y, k i , p ) is also fa cto rized in to th e color p a r t a n d th e k in e m a tic p a r t,

( 2 . 10)

w h ere th e n o ta tio n is th e sam e as in th e case o f th e p h o to n -v e c to r m eson im p a c t fa cto r.

T h e d iffractiv e glu o n im p a c t fa c to r differs from th e d iffractiv e q u a rk im p a c t fa c to r o n ly by th e color fa cto r,

(2.11) T h e k in e m a tic p a r t of th e q u a rk im p a c t fa c to r $q (y, k i , p ) in (2.5) is th e so lu tio n of th e n o n -fo rw ard B F K L e q u a tio n w ith th e in itia l c o n d itio n given by th e lead in g o rd e r q u a rk im p a c t fa c to r [62]

$q,o (k i, p ) = as . (2.12)

A t a given m o m e n tu m tra n s f e r p th ro u g h th e P o m e ro n , th e evolved q u a rk im p a c t fa c to r

$q (y, k i , p ) m ay b e re p re se n te d as co n v o lu tio n of th e lead in g o rd e r im p a c t fa c to r $q,0 w ith th e n o n -fo rw ard B F K L G re e n ’s fu n c tio n Gy:

$q (y, k i, p ) = J d 2k'i $q,o ( k, p ) Gy ( k i, k[ ; p ) . ^(2.13) In w h a t follows, we sh all use tra n s v e rs e m o m e n tu m v aria b les k , k ' t h a t reflect th e s y m m e try of th e p ro b lem : k = ( k 2 — k i ) / 2 , k ' = (k? — k ' ) / 2 . T h e tra n s v e rs e m o m e n ta of g luons ta k e th e form

k i = — — k , k 2 = p + k , k ' = — — k ', k? = — + k '.

i 2 ’ 2 2 i 2 ’ ? 2 (2.14)

T h e ex p licit form of th e lead in g lo g a rith m ic B F K L e q u a tio n t h a t defines th e q u a rk im p a c t fa c to r $ q (y, k i , p ) is th e follow ing

w ith a s = a sN c/ n .

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

Sab T la b { y , k h p ) = T q { y , k h p ) 2 N ,

< q a‘ ( v ,k i , p ) = T q ( y , k ! ,p ) N N n ja ‘ .

f y f d ? k ' 1

T q (V, k U p ) = T q,0 ( k l> p ) + a s dv ' / 772 ,---- J0 J ( k ' — k ) 2 + so

\ \ k 2 k% 2 ( k ' — k ) 2 + so 1 i , , , )

( [fci^T + so + H f + io —p (k i'2 + so) (*22 -+ » o )J T q ( y , k l , p )

k 2 k2 1 1

— — 7--- 1---2---1--- 5--- 2--- 2--- T q (y ', k ! , p ) >, (2.15) k '2 + ( k ' — k ) 2 + so k22 + ( k ' — k ) 2 + so_ q VV’ ’^ J ’ v ;

T h e n o n -fo rw ard B F K L e q u a tio n given in ( 2.15) is solved n um erically, using an a p ­ p ro x im a tio n in tro d u c e d in [62]. T h e id ea is to use th e F o u rie r d e c o m p o sitio n o f th e im p a c t fa c to r w .r.t. th e an gle b etw e en k a n d p ,

$q(y, k i , p) = Y ^

$im)(y,k2,P2)cos(m^fc)

m = 0

w h ere th e F o u rie r coefficients

(2.16)

$ i 0)( y , k 2,P 2) = J 2 k $ q ( y , k i , P ) , (2.17)

an d

$qm )(y,fc2, p 2) = [ — $q (y, k i , p )c o s(m ^ fc) J 0 n

(2.18)

for m > 0. W e have checked t h a t th e full so lu tio n $ q(y, k i , p ) is well a p p ro x im a te d by th e lead in g co m p o n e n t T^0)(y, k 2, p 2), in ac co rd a n c e w ith re s u lts [5 , 62]. Since th is a p p ro x im a ­ tio n leads to m uch g re a te r n u m eric al efficiency, w ith a n egligible effect o n accuracy , we use it in th e e stim a te s of th e cross section . T h e lead in g F o u rie r co m p o n e n t w ith m = 0 o beys th e e q u a tio n ,

(2.19)

T h e in d e p e n d e n c e o f th e lead in g o rd e r q u a rk im p a c t fa c to r o n th e angles allow s to set

$q,0( k 2, p 2) = $q,0(k , p ) = Os. A nalo g o u sly we define $ ( ° ( k 2, P 2) = / 02n[ d ^ k/(2 n )] $ y ( k i , p ).

O ne finds t h a t $ ( ° ( k 2 , p 2) = $ y ( k i , p ) | fcl=p/ 2-fc.

F inally, let us n o te t h a t th e cross sec tio n (2.2) c a n b e w ritte n in te rm s o f ( 2 .4) as follows:

d a = j d x j C i q ^ [ f q ( x , p ) + f g ( x , p)] + C i g f g ( x , p ) \ d a — (x s, t) (2.2 0)

U sin g eqs. (2.7) , (2.10) a n d (2.11) it is s tra ig h tfo rw a rd to o b ta in th e color fa c to rs for th e d iffractiv e p ro d u c tio n off th e q u a rk a n d gluo n. T h e re su lts re ad

(2 .2 1 )

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

< > (y , k V ) = ¢ ,,0 ( k k J ? ) - < X , / ^ l ' dy’ j ^ k )2 + s ,

{ [ + k f c - p 2 ( 5 5 ^ + - ¾ ¾ ] ^ ( y ' k ’ 2,p 2)

- _fc;2 + ( k '- k ) 2 + so + k'22 + ( k ' - k ) 2 + S0_ ^ ' (y ,k ,p ) } •

/ N 2 — 1 \ 2

C Y, = ( N nT ) , C ’ “ = 1 •

3 T w o P o m e r o n c o n tr ib u tio n to h e a v y v e c to r m eso n h a d ro p ro d u c tio n

3 .1 D i r e c t a p p r o a c h

L e t us now co n sid er th e heav y v e c to r m eso n h a d ro p ro d u c tio n th ro u g h th e (c u t) d o u b le P o m e ro n exchange. W e focus on a h a d ro n ic an a lo g u e of th e d iffractiv e v e c to r m eson p h o to p ro d u c tio n — w h ere th e p ro je c tile is a gluon, a n d th e tw o P o m e ro n s cou ple to a single p a r to n in th e ta rg e t. T h e g en e ral d ia g ra m s c o n trib u tin g to th e p a rto n ic cross section , d a 2 -p, a re d e p ic te d in figure 3 . N o te t h a t a t th e low est o rd ers, th e top o lo g ies o f th e co n sid ered p a rto n ic p rocesses are th e sam e as for th e d iffractiv e p h o to p ro d u c tio n . A fte r inclu sio n of th e ev o lu tio n , how ever, th e tw o processes co rre sp o n d to d ifferen t c u ts th ro u g h th e tw o P o m e ro n s; one finds th e d iffractiv e c u t in th e p h o to p ro d u c tio n , a n d th e d o u b le c u t P o m e ro n s in th e h a d ro p ro d u c tio n . T h e v e c to r m eson im p a c t fa c to r d esc rib es th e fusion of th re e g luons in to th e m eson. T h e co u p lin g o f b o th P o m e ro n s to th e single p a r to n in th e ta r g e t leads to a c o rre la tio n o f th e g lu o n d is trib u tio n s in th e ta r g e t, t h a t e n te r th e m eson im p a c t fa c to r. T h is sh o u ld b e c o n tra s te d w ith a n o th e r p o ssib le c o n trib u tio n , w h ere th e g luons in th e ta r g e t are u n c o rre la te d , see [51].

In th e c a lc u la tio n s, th e in com ing p a rto n s fro m th e p ro je c tile a n d ta r g e t p ro to n s are tr e a te d as co llin ear a n d c a rry lo n g itu d in a l m o m e n tu m fra c tio n s x1 a n d x 2, w h e reas th e g lu o n ex ch an g e in th e t-c h a n n e l is d e c rib e d in th e fcy -facto rizatio n fram ew ork, a ssu m in g th e h ig h en e rg y lim it. H ence th e v e c to r m eso n im p a c t fa c to r t h a t e n te rs th e c a lc u la tio n d esc rib es th e tra n s itio n from th e co llin ear p ro je c tile g lu o n to th e v e c to r m eson by co u plin g o f tw o t-c h a n n e l g luo ns w ith no n -zero tra n s v e rs e m o m e n ta . T h is im p a c t fa c to r m ay be o b ta in e d from th e im p a c t fa c to r d esc rib in g th e fu sio n of th re e g luo ns in to th e v e c to r m eson t h a t w as d eriv e d in [64] by ta k in g th e co llin ear lim it for th e p ro je c tile g lu on . B ecau se of th e o d d C -p a r ity o f th e m eson, th e im p a c t fa c to r is fully sy m m e tric in color indices o f th e gluons, a n d th e fu n c tio n a l d e p e n d e n c e o n th e e x te rn a l b o so n m o m e n ta is th e sam e as in th e d iffractiv e im p a c t fa c to r for th e exclusive v e c to r m eson p h o to p ro d u c tio n . Specifically, in th e co llin ear lim it for th e p ro je c tile gluon, th e th re e -g lu o n im p a c t fa c to r for inclusive v e c to r m eson h a d ro p ro d u c tio n re ad s

g da i“2C

* a r c ( k i, p ) = -gs- ( k i, p ) — , (3.1)

q 2 —c

w h ere g s is th e s tro n g co u p lin g c o n s ta n t ( a s = g 2s / 4 n ) , d aia2C is th e fully sy m m e tric color te n so r, a n w ith n = 1,2, a re th e co lo r indices of th e t-c h a n n e l glu on s, a n d c is th e color in d ex of th e p ro je c tile gluon.

T h e co rre sp o n d in g lead in g o rd e r im p a c t fa c to rs of th e ta r g e t q u a rk (i = q) a n d g lu o n (i = g ) m ay b e ex p ressed in th e follow ing way:

( k i , p ) = $ q,0 ( k i , p ) T R1 % , (3.2) w h ere m a tric e s tR ) are th e g e n e ra to rs of th e co lo r g ro u p in th e color ch a rg e re p re s e n ta tio n R i of th e ta r g e t p a rtic le . N o te t h a t a fte r th e p ro je c tio n o n th e co lo r sing let ch a n n el p erfo rm ed by T ^ T r2T % ) /d im ( R i ), th e sin glet color im p a c t fa c to rs ¢^0^ 62 a re recovered.

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

F ig u r e 3. Correlated contributions of 1 + 2 gluons fusion to hadroproduction cross section of a heavy vector meson V w ith partonic targets: a gluon (left) and a quark (right). The blobs w ith G denote the BFKL gluon G reen’s functions. The two BFKL G reen’s functions are evolved independently. Gluons 14 and 23 are projected onto color singlet states.

U sin g th e n o ta tio n from th e p re v io u s sec tio n we c a n ex p ress th e cross sec tio n for th e inclusive v e c to r m eson p ro d u c tio n in th e co n sid ered tw o P o m e ro n m ech an ism as

d a = I d x i d x 2 | f g ( x i , ^ ) d a2 - p ( x ix 2S , t , y )

x (yCgq ^ [fq ( x2, ^ ) + f (x2,ju)] + Cgg f g ( x 2 , v ^ + [xi o x2] j , (3.3)

w h ere S is th e h a d ro n ic collision en e rg y sq u ared , th e color fa c to rs C gq a n d C gg acco m m o ­ d a te th e color s tr u c tu r e of th e v e c to r m eson im p a c t fa c to r a n d th e color p ro je c tio n o n to th e tw o -P o m e ro n s ta te . T h e ir c a lc u la tio n is stra ig h tfo rw a rd , b u t re q u ires c e rta in a ssu m p tio n s o n how th e p ro je c tio n is m ad e. W e shall d iscuss th is issue la te r in th is sec tio n a n d give th e values of th e color fa cto rs.

T h e d ia g ra m s in figure 3 , s trip p e d off th e color fa cto rs, give th e follow ing ex p re s­

sion for th e p a rto n ic cross sec tio n for th e v e c to r m eson p ro d u c tio n w ith th e tra n s v e rse m o m e n tu m p :

(3.4) w h ere we set th e ra p id ity e v o lu tio n le n g th to y = lo g (x i x 2S / ( M y + p 2)), a n d th e re m a in in g n o ta tio n is th e sam e as in th e p rev io u s section . In fa c t, th e ab ov e e q u a tio n d esc rib es th e B F K L P o m e ro n loop w ith c u t th ro u g h b o th P o m e ro n s, a n d th e u p p e r a n d lower P o m e ro n co u p lin g s given by th e m eson a n d q u a rk fo u r-g lu o n im p a c t fa cto rs. T h e color p a r t of th e m eson im p a c t fa c to r fo rb id s th e co u p lin g of th e g lu o n p a ir in th e color sin glet s ta te to th e m eson, so th e P o m e ro n s are fo rm ed o n ly b etw e en th e g lu on s a t th e o p p o site sides

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

d a 2 -p ( x i x 2 S ,t ,^ ) a s f d 2k i f d 2k 2 f 2 & n , 1 ^ d p--- = 1 6 ^ 0 ^ / - 2 n d ] - 2 n d ] d q 8 ( k i + k 2 - P )

w ( k i , k i +k2) $ y ( q -k i , - k i - k 2) (y, k i , q ) ( y , k2, - q )

X r n r n ,

( k i — q)2 + s 0 ( k 2 + q)2 + s 0 (¾2 + s o) (k 2 + So)

o f th e u n ita r ity c u t. T h e tra n s v e rse m o m e n tu m of th e P o m e ro n s in th e loop is ± q . In

N o te t h a t we a p p ly in eq. (3.4) th e v e c to r m eson im p a c t fa c to r $ V co rre sp o n d in g to a tra n s itio n o f a tra n s v e rse ly p o la riz e d g lu o n to a tra n sv e rse ly p o lariz ed m eson. T h e glu o n is tr e a te d w ith in th e co llin ear a p p ro x im a tio n so it c a n n o t c a rry th e lo n g itu d in a l p o la riz a tio n . As in th e d iffractiv e p h o to p ro d u c tio n case th e tra n s itio n of a tra n s v e rs e ly p o lariz ed glu on to a lo n g itu d in a ly p o lariz ed m eson is s tro n g ly su p p re sse d [33].

S pecial a tte n tio n sh o u ld be p a id to th e ta r g e t q u a rk a n d g lu o n im p a c t fa cto rs. A t th e lead in g o rd e r th e p a r to n fo u r-g lu o n im p a c t fa c to r is a c o n s ta n t fu n c tio n of th e gluo n m o m e n ta , as it is for th e tw o -g lu o n im p a c t fa c to r. T h is follows d ire c tly fro m th e p o in t-lik e n a tu r e of th e p a rto n s . As a re su lt, th e fo u r-g lu o n p a r to n im p a c t fa c to r is p ro p o rtio n a l to th e p ro d u c t of tw o -g lu o n im p a c t fa cto rs. N e x t, in o u r a p p ro a c h we a p p ro x im a te th e full B a rte ls - K w ieciń sk i-P raszało w icz (B K P ) [56- 58] e v o lu tio n of th e fo u r g lu o n t-c h a n n e l s ta te in th e a m p litu d e sq u a re d by th e in d e p e n d e n t e v o lu tio n o f tw o P o m e ro n s. T h is a p p ro x im a tio n is valid in th e larg e N c lim it, as color re c o n n e c tio n b etw e en tw o P o m e ro n s is su p p re sse d by 1 /N c2 [59]. H ence, w ith th e fa cto rized fo rm of th e p a r to n im p a c t fa c to r, a n d w ith th e in d e p e n d e n t P o m e ro n ev o lu tio n s, also th e evolved fo u r-g lu o n p a r to n im p a c t fa c to r m ay b e fa cto rized (u p to a c o n s ta n t fa c to r) in to a p ro d u c t o f tw o -g lu o n evolved im p a c t fa cto rs. T h u s, in eq. (3.4) b o th q u a rk im p a c t fa c to rs are evolved in d e p e n d e n tly acco rd in g to eq. (2.19) .

L e t us now d iscu ss th e low est o rd e r c o n trib u tio n to th e c o rre la te d cross sec tio n ( 3.3) , o b ta in e d by s e ttin g (y, kj , ± q ) ^ $ q,0(k j, ± q ) in eq. (3.4) . T h e co rre sp o n d in g d ia g ra m s a re d e p ic te d in figure 4 . W e see t h a t th e se d ia g ra m s a re v irtu a lly th e sam e as for th e d iffractiv e p h o to p ro d u c tio n a t th e low est o rd e r, ex c ep t for th e v e c to r m eso n v erte x , w hich h ere c o n ta in s a n in co m in g g lu o n in ste a d o f a p h o to n . G iv en th e s y m m e try o f th e color p a r t o f th e im p a c t fa c to r, th e low est o rd e r gq ^ V q a n d g g ^ V g cross section s m ay b e o b ta in e d from th e Yq ^ V q a n d yg ^ V g cross sectio n s by s u ita b le m o d ificatio n s of th e color fa c to rs a n d th e co u p lin g c o n s ta n ts , w hile th e m o m e n tu m d e p e n d e n t p a r t re m a in s th e sam e. T h e ex p licit re s u lt re ad s

T h is re su lt is a useful b e n c h m a rk for th e cross sec tio n ( 3.4) w ith th e B F K L ev o lu tio n in clu d ed .

3 .2 T h e c o l o r f a c t o r s

In te rm s of th e B K P e q u a tio n e ig e n sta te s, eq. (3.4) re p re se n ts ex ch an g e of th e lead in g tw o P o m e ro n s ta te (w h ere th e larg e N c lim it is im p licitly a ssu m ed ). T h e B K P e q u a tio n for n u m eric al e s tim a te s o f (3.4) we shall use th e a p p ro x im a tio n o f th e q u a rk im p a c t fa c to rs by th e lead in g F o u rie r co m p o n en ts, (y, k j, q ) ^ $^0) (y, ( k j — q / 2 )2, q2) , a n d $ V (k j, q ) ^

$V0) ((k j — q /2)2, q2) , cf. th e discu ssio n p erfo rm ed for th e d iffractiv e p h o to p ro d u c tio n .

(3.5)

w here

(3.6)

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

F ig u r e 4. The lowest order diagram s for the am plitude squared for the production of the heavy vector meson through the coupling of 1 + 2 gluons off a quark (left) and off a gluon (right).

fo u r gluons, how ever, allow s also for o th e r so lu tio n s, like e.g. a single P o m e ro n exchan ge, in w hich th e e le m e n ta ry gluons a re p a ire d in to g lu o n R eggeons. In o rd e r to p ro p e rly p ro je c t th e im p a c t fa c to rs on th e tw o P o m e ro n B K P eig e n sta te s, we a p p ly th e follow ing p ro c ed u re . W e s t a r t from th e B ose s y m m e try p ro p e rtie s o f th e B K P e ig e n sta te s. G iv en th e sy m m e tric k in e m a tic p a r t of th e im p a c t fa cto rs, it im po ses th e color s y m m e try b etw e en th e p a irs of g luons a t th e sam e side o f th e u n ita r ity c u t. Also, th e color p a r t sh o u ld b e in v a ria n t u n d e r th e in te rc h a n g e of th e g luons a t th e left a n d rig h t side o f th e u n ita r ity c u t. E m p lo y in g th e re la tio n s b etw e en in v a ria n t te n s o rs of Q C D , we find t h a t po ssib le co lo r te n so rs for th e t-c h a n n e l g luons are

P o ( K } ) = Óaia26“3“4, P2({fln}) = 6a i“35 a2a4 + 5 a ia 45 a2a3, P d ( { a n } ) = d aia2cd a3a4c, (3.7) w h ere a n , n = 1 , . . . , 4, a n d c a re co lo r indices in th e a d jo in t re p re s e n ta tio n , a n d esc rib e th e t-c h a n n e l g luons in th e n a tu r a l o rd e r. T h e sc a la r p ro d u c t o n th e sp ace of th e color te n so rs m ay b e defined as

( P a \P b ) = E P a ({an } ) Pb ({ a n } ), A , B = 0,2,d . (3.8) {an}

I t is co n v en ien t to use th e n o rm alize d color te n s o rs Pa({an } ) P A ({an }) =

V ( Pa\Pa)

(3.9)

T h e color te n so rs P0({ a n }), P2({ a n }) a n d P d({ a n }) are n o rm alize d to one, a n d o rth o g o n a l u p to 1/N ,2 c o rrec tio n s, ( P a \ P b ) = 6aB + O Q /N ,2). Since th e an a ly sis is p erfo rm ed a t th e lead in g o rd e r in A c, th e te n s o rs i^A({ai }) m ay b e tr e a te d as a n o th o n o rm a l basis. T h erefo re th e p ro je c to rs m ay b e d efined on th e co lo r te n so rs, co rre sp o n d in g to th e B K P eig en sta te s:

Pa ({an }) Pa ({bn})

Pa ({ a n }, {bn}) = (3.10)

(Pa\Pa)

In o u r c a lc u la tio n we p ro je c t th e im p a c t fa c to rs o n th e color s ta te P2 co rre sp o n d in g to th e ex ch an g e of tw o c u t P o m e ro n s. T h e color te n s o rs a sso c ia te d w ith th e u p p e r a n d lower im p a c t fa c to r a re d e n o te d by C a ({ a n }) a n d C g({bn }) co rresp o n d in g ly , an d we define th e color fa c to r by

C a g = E E C a ( { a n } ) P2({an}, {bn})C g({bn}) , (3.11) {an } {bn }

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

w h ere th e color p ro je c to r ta k e s th e follow ing form

P ( { a 4 , {bn}) = ---^ r (ba i“3b“ 2“ 4 + ba i“4b“2“3) ( 4 lb34 2&4 + 4 lb4bb2b3) . (3.12) 2 A C2 ( A C2 - 1)

T h e color te n so rs e n te rin g th e cross sectio n s a re th e following:

1____ _ da1a2c da3a4C

C'a ( a i , a 2, a 3, a 4) = A 2 _ 1 ^ ---4A 2--- , (3 ' 13) for th e color av erag ed g lu o n to m eso n tra n s itio n a m p litu d e sq u a re d , an d

^ 1.62. ^ 4) =

4 ^

¹⁴ ( ¾ ¾ ¾ ¾ ) ■ (3 4 4 )

for th e color av erag ed s c a tte rin g a m p litu d e sq u a re d of th e p a r to n in color re p re s e n ta tio n R j. T h e ex p licit form of color fa c to rs C ^g follows fro m eqs. (3.11) , (3.13) , (3.14) a n d re ad s

C ( A 2 _ 4 ) ( A ? _ 2 ) C 3 N 2 _ 4

= 1 6 N J . Cgg = 8 (A ? _ 1) N | ' ( )

3 .3 V e c t o r m e s o n h a d r o p r o d u c t i o n i n c o n f o r m a l r e p r e s e n t a t i o n o f t h e B F K L P o m e r o n

In w h a t follows we shall recall th e L ip a to v so lu tio n to th e n o n -fo rw ard lead in g lo g a rith m ic B F K L e q u a tio n by m ean s of th e co n fo rm al eig en fu n ctio n s [14], a n d th e a p p lic a tio n to d iffractiv e s c a tte rin g . T h e n we shall em ploy th e fo rm alism to d e sc rib e th e v e c to r m eson h a d ro p ro d u c tio n in th e tw o P o m e ro n ex ch an g e m ech an ism . T h e so lu tio n to th e n o n ­ fo rw ard B F K L e v o lu tio n e q u a tio n c a n be p re se n te d in th e m o m e n tu m o r in th e c o o rd in a te space. To th is en d , for a gen eric 2 -d im en sio n al tra n s v e rs e m o m e n tu m k = (k x ,k y) we in tro d u c e th e com plexified m o m e n ta (k,fc), w here

k = kx + ik y, fc = kx _ ik y . (3.16)

Sim ilarly, for th e tra n s v e rs e c o o rd in a te space, th e 2 -d im en sio n al c o o rd in a te s p = (px , p y) w ill be tra d e d for th e co m p lex v aria b les

p = px + ip y. p = px _ ip y . (3 .17)

In w h a t follows we set s 0 = 0 a n d use th e o rig in al form o f th e n o n -fo rw ard B F K L e q u a tio n . I t w as show n by L ev L ip a to v [14, 17] t h a t th e B F K L e q u a tio n in th e lead in g lo g a rith ­ m ic a p p ro x im a tio n is in v a ria n t u n d e r th e co n fo rm al tra n s fo rm a tio n s of th e com plexified tra n s v e rs e p o sitio n s of th e R eggeized gluons,

a p + 6 ,

P ; , (3.18)

' cp + d v 7

for a r b itr a r y co m p lex p a ra m e te rs a, 6, c, d. In an a ly se s of h ig h en e rg y s c a tte rin g a m p litu d e s A ( s , t ) it is c u s to m a ry to use th e M ellin m o m en ts w c o n ju g a te to s [17],

A ( s , t ) = |s| / -dW A ( w ,f ) s w. (3.19)

J 2 n i

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

T h e so lu tio n for th e glu o n G re e n ’s fu n c tio n is th e n re p re se n te d as

X f d 2P0 (Pl'Bi P'2'o) , (3.20)

J W - W n ( v)

w h ere wn (v) is th e LL B F K L eigenvalue

(3.21)

a n d ^ a re p o ly g a m m a fu n c tio n s, th e fu n c tio n s E n ,v are co n fo rm al eig en fu n ctio n s defined as

E n v ( P li'>2) = ( p p 2) ( P i t ) 1 (3 '22)

w h ere th e pow ers h - h a re th e co n fo rm al w eights

1 n ~ 1 n , ,

h = 2 + 2 + iV| h = 2 - 2 + i v ' ( )

T h e abo v e form o f th e B F K L G re e n ’s fu n c tio n m ay b e F o u rie r tra n sfo rm e d to th e tra n s v e rs e m o m e n tu m re p re s e n ta tio n , a n d in v erse -M ellin -tran sfo rm ed in w to th e ra p id ity space

n n k > ) = 1 + « f + « d v 2 + n 2/ 4

V( 1 - 11 q ) ( 2 n )6 n^ o 7 - « V (v 2 + (n - 1)2/ 4 ) ( v 2 + (n + 1)2/4 )

x ex p (w n (v )y ) E ,v(k i , q) E ,v(k , q ) , (3.24) w here

E n v (k i, q ) = d 2p i d 2p2 e x p ( i k i p i + i ( q — k ) p2) , (3.25) a re th e co n fo rm al eig en fu n ctio n s in th e m o m e n tu m re p re s e n ta tio n . T h e a n a ly tic form of th e eig en fu n ctio n s E nv ( k i , q) w as d eriv e d in [65]. It is r a th e r lengthy, so in s te a d o f listin g it h ere we refer th e re a d e r to th e o rig in al p a p e r. In th e lim it y ^ 0, on e finds

^ k/, q) ^ ^ ) 2 • (3.26)

T h e d o m in a n t im a g in a ry p a r t of th e a m p litu d e for th e B F K L P o m e ro n ex ch an g e b e ­ tw ee n lead in g o rd e r im p a c t fa cto rs $A ,0( k i , q) a n d $ B)0( k / , q) re ad s

/

d 2 k i d 2 k /

— ^ $ A,0 ( k i , q ) Gy ( k i , k i , q ) ¢ B,0 ( k i , q ) • (3.27)

I t is co n v en ien t to define th e p ro je c tio n of th e im p a c t fa cto rs on th e co n fo rm al eigen ­ fu n c tio n s

/

^{d2 k} ¢ A,o( k l , q ) E nv ( k l , q ), (3 .28)

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

, , f + « V 2 + n 2/ 4

G " (Pl 1 P2; P l - P2) = n £ 2 - ~ dV ( v 2 + ( n - 1)2/ 4 ) ( v 2 + ( n + 1)2/ 4 )

, , N c a s I" . . / |n| 1 \ , / Ini 1 \ Wn (v) = 2t (1) - ^ 2 ^ + 2 + w ) - T + 2 - %v) 1

a n d an a lo g o u sly for th e in d ex B . T h is leads to th e follow ing form of th e P o m e ro n exch an g e a m p litu d e

_ 2\ - 1 f v 2 + n 2/ 4

m A ( s , t = - q ) = d v ( v 2 + ( n - 1)2/ 4 ) ( v 2 + ( n + 1)2/4 )

x ex p (wra(v )y ) I A V ( q ) [ I Bv (q )]- . (3.29) T h e im p a c t fa cto rs I q v (q) a n d I y v(q ) w ere c o m p u te d in th e a n a ly tic form in refs. [3 , 5 , 35 , 60]. In th e c a lc u la tio n of v e c to r m eson h a d ro p ro d u c tio n we sh all need th e q u a rk im p a c t fa c to r, w hich is tr e a te d w ith in th e M u eller-T an g schem e [3 5 , 60]. I t ta k e s th e form

4 n a si n ( ł Y V p -in4>v r ( 1 / 2 + n / 2 — i v) (3 30)

n q \ 4 J r ( 1 / 2 + ( / 2 + i v ) , (3 ' 30)

w h ere Gq is th e p o la r an g le of th e v e c to r q in th e tra n s v e rs e p lane.

T h e fo rm alism m ay b e also ap p lied to th e tw o -P o m e ro n ex ch an g e process, assu m in g in d e p e n d e n t B F K L e v o lu tio n o f th e P o m e ro n s. H ence, we re w rite eq. ( 3.4) as

d a 2- p (X1X2S , t , i B ) a s f d 2k i d 2k [ f d 2k 2 d 2k'2 [ 2 ,-2/, , , n

— d p — = 1 6 ^ 2 ^ / - 2 T - y - 2 r - y d q * <k i + k2 - p

x $ y ( k 1, k 1 + k 2) $ y (q — k 1, —k 1 — k 2)

x G y( k y k [ , q ) G y( k2, k 2 , —q ) $q,o ( k , q) $q,o (k 2 , —q ) . (3.31) U sin g th e re p re s e n ta tio n of th e B F K L G re e n ’s fu n c tio n s by th e co n fo rm al eig en fu n ctio n s in th e m o m e n tu m space th is m ay b e re w ritte n as,

d a 2 - p ( x 1x 2S , t , p ) d 2p

A new n o n -triv ia l o b je c t t h a t a p p e a rs in th is e q u a tio n , l y ' ^ n2,V2 (p, q) is a p ro je c tio n of a p a ir o f th e v e c to r m eson im p a c t fa c to rs (co m ing from th e a m p litu d e a n d its co m p lex c o n ju g a te ) on th e co n fo rm al eigenfunction s:

/ ^{d2kq f cfik-q}

( 2 ^ 7 (2 n )2 * 2 ( k 1 + k 2 — P ) E ni,V1 ( k1, q ) E^n2,V2 ( k2, —q )

x $ y ( k 1, p ) $ y (q — k 1, p ) . (3.33) N o te t h a t in c o n tra s t to th e d iffractiv e s c a tte rin g , in th is ex p ressio n th e v e c to r m eson im p a c t fa c to rs $ y a re co u p led to tw o d ifferen t co n fo rm al eig en fu n ctio n s each. In o th e r

J H E P 0 7 ( 2 0 1 9 ) 1 2 9

= a s f d 2q + ^ f _d)1_______________ v i2+ « 1 / 4 ___________

1 6 n 2a e m e 2q J n ~ - (XJ - ™ (2^ ) 3 (v 2 + («1 — 1)2/ 4 ) ( v 2 + (( 1 + 1 )2/4 )

f dv2 v2 + « 2 /4 , , , ,

X n ( 2 ^ + (( 2 — 1 ) 2 / 4 ) 0 2 + (( 2 + 1 )2 /4 ) ^ '+ (Vq))]

x i n i 'U n„ . 2 (p , q) [I?.1,V1 (q ) I?.„V2 ( —q ) ] - . (3.32)