P u b l i s h e d f o r S IS S A b y S p r i n g e r R e c e i v e d: September 21, 2015
R e v i s e d: February 10, 2016 A c c e p t e d: February 21, 2016 P u b l i s h e d: March 11, 2016
Soft gluon resummation for associated ttH production at the LH C
Anna Kulesza,a Leszek Motyka,b Tomasz Stebelb and Vincent Theeuwesa aIn stitu te fo r Theoretical P hysics, W W U M unster,
M unster, D-Ą81Ą9 G erm any
bIn stitu te o f P hysics, Jagellonian U niversity, S. Ł ojasiewicza 11, Kraków, 30-348 Poland
E - m a i l : anna.kulesza@uni-muenster.de, leszekm@th.if.uj.edu.pl,
tomasz.stebel@uj.edu.pl, vthee_01@uni-muenster.de
A b s t r a c t : We perform resum m ation of soft gluon corrections to th e to ta l cross section for th e process pp ^ t t H . T he resu m m ation is carried ou t at next-to-leading-logarithm ic (NLL) accuracy using th e M ellin space technique, extending its application to th e class of 2 ^ 3 processes. We present an an alytical result for th e soft anom alous dim ension for a hadronic pro d uctio n of two coloured m assive particles in association w ith a colour singlet.
We discuss th e im pact of resu m m atio n on th e num erical prediction for th e associated Higgs boson p ro d uction w ith to p quarks a t th e LHC.
Ke y w o r d s: R esum m ation, Higgs Physics, QCD
ArXiv ePr in t: 1509.02780
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Contents
1 I n tr o d u c tio n 1
2 R e s u m m a t io n for 2 ^ 3 p r o c e s s e s w it h tw o m a ss iv e c o lo r e d p a r tic le s in
3 T h e o r e tic a l p r e d ic tio n s for t h e p p ^ t t H p r o c e ss at N L O + N L L a c c u r a c y 6
1 Introduction
E stablish in g th e prop erties of th e Higgs boson discovered a t th e LHC in 2012 [1, 2], in p a rticu la r its couplings to th e S tan d ard M odel (SM) particles, is one of th e m ain tasks of th e cu rren t LHC run. Since th e SM Higgs boson couples to ferm ions prop ortion ally to th e ir masses, th e top-H iggs Yukawa coupling is expected to be especially sensitive to th e underlying physics. A d irect way to probe th e stre n g th of th e coupling w ith o u t m aking any assum ptions regarding its n a tu re is provided by th e m easurem ent of Higgs p ro d u ction rates in th e p p ^ t t H process. A lthough th e productio n cross section is low and th e collision en
ergy and th e lum inosity available so far have not been sufficient enough to m easure a Higgs signal in R u n 1 [3- 7], such a m easurem ent in R u n 2 is eagerly aw aited. Correspondingly, precision predictions for th e pp ^ t t H pro du ction process are of great im p ortance and a lot of effort has been invested in th e recent years to im prove th e th eoretical accuracy.
T he next-to-leading-order (NLO) QCD, i.e. O ( a f a ) predictions are already known for some tim e [8- 13] and have been newly recalculated and m atched to p a rto n showers in [14
17]. As of late, th e m ixed QCD-weak corrections [18] and Q C D -E W corrections [19, 20] of
O ( a 2 a 2) are also available. Furth erm ore, th e NLO QCD corrections to th e hadronic t t H
pro d u ction w ith to p and a n tito p quarks decaying into b o tto m quarks and leptons have been recently obtain ed [21]. C oncurrently, new m ethods for a b e tte r m easurem ent of th e process have been proposed e.g. in [22] or in [23]. In general, for th e LHC collision energies of R u n 2, th e NLO QCD corrections are ~ 20%, w hereas th e size of th e (electro)w eak correction is m ore th a n ten tim es sm aller. T he scale u n certain ty of th e NLO QCD corrections is estim ated to be ~ 10% [8- 13, 24]. W hile m atching fixed-order predictions to p a rto n showers pursued recently by m any groups in such fram ew orks as aMC@NLO [14, 15, 25], P O W H E G BO X [16, 17, 26] or SH ER PA [27] allows for a m ore accurate description of final s ta te characteristics, it does not change th e predictions for th e overall p ro duction rates. An im provem ent in th e accuracy w ith which these rates are known can only be achieved by calculating higher order corrections. However, calculations of th e next-to-next-to-leading- order corrections are cu rrently technically o u t of reach. It is nevertheless interestin g to
t h e fin a l s t a t e 2
4 S u m m a r y 10
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ask th e question w hat is th e size and th e effect of c e rtain classes of corrections of higher th a n N L O accuracy. In p articu lar, we focus here on tak in g into account contrib utio ns from soft gluon em ission to all orders in p e rtu rb a tio n theory. T he tra d itio n al (Mellin- space) resu m m atio n form alism which is applied in this ty p e of calculations has been very well developed and copiously employed for description of th e 2 ^ 2 ty p e processes at th e B o rn level. T he universality of resum m ation concepts w arran ts th e ir applications to scatterin g processes w ith m any p arto n s in th e final sta te , as shown in a general analytical tre a tm e n t developed for a rb itra ry num ber of p a rto n s [28- 30]. Recently, th e soft gluon resu m m atio n technique in th e soft collinear effective theory (SC ET ) fram ew ork was applied to pp ^ ttW ± [31]. So far, however, no calculations in th e tra d itio n a l resum m ation fram ew ork for processes involving 2 ^ 3 scatterin g a t th e B orn level have been perform ed.
In th is p a p e r we tak e th e first step in th is direction by developing th e M ellin-space th resh old resum m ation form alism a t th e next-to-leading-logarithm ic (NLL) accuracy for th e case of 2 ^ 3 processes w ith two coloured m assive particles in th e final sta te . We th e n apply th is form alism in order to estim ate th e im pact of soft gluon corrections on th e predictions for th e to ta l t t H p ro d uction rate. In this p a rticu la r case, th e threshold region is reached w hen th e square of th e p artonic center-of-m ass (c.o.m .) energy, \/S, approaches M = 2m t + m H , w here m t is th e to p q uark m ass and m H is th e Higgs boson m ass. In th e threshold region, th e cross section receives enhancem ent in th e form of logarithm ic corrections in fi = \ J 1 — M 2/ S . T he q u a n tity fi m easures th e distance from absolute p ro d uctio n threshold and can be related to th e m axim al velocity of th e t t system . A dditionally, in th e threshold region th e v irtu al QCD corrections are also enhanced due to C oulom b-type interactions betw een th e two final sta te to p quarks which becom e large w hen th e to p q u ark velocity in th e t t c.o.m. fram e fikl ^ 0 w ith fikl = a/1 — 4 m 2 / s k i and Skl = (pt + p*)2. However, th e co n tributions to th e to ta l cross section from th e threshold region are strongly suppressed by th e fi4 factor originating from th e m assive th re e particle phase space. N evertheless, one expects th a t th e threshold corrections can still have a non-negligible im pact on th e predictions.
T he associated p ro d uction of a Higgs boson w ith a ttt p air involves four coloured p arto ns a t th e B orn level and as such is characterized by a non-trivial colour flow. T he colour stru c tu re influences th e con tribu tio n s from wide-angle soft gluon emissions which have to be included a t th e NLL accuracy. T he evolution of th e colour exchange at NLL is governed by th e one-loop soft anom alous dim ension [28, 32- 36]. S ta rtin g from four coloured p a rto n s in th e process, th e soft anom alous dim ension is a m a trix and is known for heavy- q u ark [32, 33, 37], dijet [34- 36] and supersym m etric particle p ro d uctio n [38- 40, 43], as well as for th e general case of 2 ^ n QCD processes [28- 30]. Here we ad o p t th e calculations of th e soft anom alous dim ension for th e case of 2 ^ 3 processes w ith two coloured massive particles in th e final state.
2 Resum m ation for 2 ^ 3 processes w ith two massive colored particles in the final state
T he resum m ation of soft gluon corrections to th e to ta l cross section app^ t is perform ed in M ellin space, w here th e M ellin m om ents are tak en w .r.t. th e variable p = M 2/ S . A t th e
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parto n ic level, th e M ellin m om ents for th e process i j ^ k l B , w here i, j denote massless coloured parton s, k, l two m assive quarks and B a massive colour-singlet particle, is given by
< T i j ^ k i B , N ( m k, m i , m B, K f , ^ r ) = / d pp N - 1 & i j ^ k l B ( P , m k , p | ) (2.1)
J0
w ith p = 1 — P2.
At LO, th e t t H p ro d u ctio n receives co ntribu tion s from th e q q and g g channels. We analyze th e colour stru c tu re of th e underlying processes in th e s-channel color bases, {cq} and { c g}, w ith c? = 5 a a 5 a k a i , c% = T £ i a . a i , c? = ^ aj 5 a k a i , c | s = T ba i a k d ba*aj, C8A = i T ^a k f baiaj • In th is basis th e soft anom alous dim ension m atrix becomes diagonal in th e pro du ction threshold lim it [32, 33] and th e NLL resum m ed cross section in th e N -space has th e form [32, 33, 37]
- (res) = ^(°) C Ai A j A (int) (2 2)
° i j ^ k l B , N = ° i j ^ k l B , I , N C i j ^ k l B ,I A N+1A N+1 A i j ^ k l B , I , N+1, (2-2)
I
w here we suppress explicit dependence on th e scales. T he index I in eq. (2.2) d istin guishes betw een co ntrib u tio ns from different colour channels. T he colour-channel-depend
ent co ntrib u tio n s to th e LO p arto n ic cross sections in M ellin-m om ent space are denoted by kl B I n . T he radiative factors AN describe th e effect of th e soft gluon rad iatio n collinear to th e initial s ta te p arto n s and are universal. Large-angle soft gluon emission is accounted for by th e factors A j ] ^ I N which depend on th e p arto n ic process under consideration and th e colour configuration of th e p a rticip a tin g particles. T he expressions for th e rad iativ e factors in th e MS factorisation scheme read (see e.g. [37])
, Ai f \ z N - 1 — 1 f M 2 ( 1 - z ) 2 d q 2 A f f 2 , ,
lnAN = d z — --- - 2-A i ( a s ( q ) ) ,
J
° 1 — z q 2,, , p 1 z N — 1 1
ln A i^fclB I N = d z ~ j ---D i j ^ k l B , I ( a s ( M 2 ( 1 — z )2 )). (2.3)
J 0 1 — z
T he coefficients A i , D ij^ klB, I are power series in th e coupling c o n stan t a s,
A i =
(
A i (1) +( ^)2
A i (2) + . . . , D i j , k l B I =(
D j \ k l B J + . . . (2.4) T he universal LL and NLL coefficients A(1), A(2) are well known [44, 45] and given by A(1) = Ci, A (2) = 1 C i( ( § — £ ) C a — 5f w ith C g = Ca = 3, and C q = C F = 4/3 .T he NLL coefficients D ij ^ k l B ,I are o b tained by tak in g th e threshold lim it s ^ M 2 =
( m k + m l + m B ) 2 of th e gauge-invariant soft anom alous dim ension m atrices r i j ^ k l B. In th is lim it r i j ^ klB = O fdiag(yj , . . . ) and D ij ^ k l B, I = 2Re(yIj ). T he calculations of r i j ^ klB
apply th e m ethods developed in th e heavy q u ark pair-p ro d u ctio n [32, 33] to th e process at
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hand, tak in g into account 2 ^ 3 kinem atics, and yield
(2.5)
(2.6)
w ith
r « l = - C f(l p m + 1),
r22 = r f l = 1 ( ( C a - 2 Cf) (l p m + 1) + CAA3), where
A3 = ( T i ( m k) + T 2 ( m i) + U i ( m i) + U2 ( m k ))/2 , Q3 = ( T i ( m k) + T 2 ( m i) - U i ( m i) - U2( m k))/2 , and
(2.7)
(2.8)
(2.9) (2.10) U2 = ( p j - P k)2. (2.11) Eqs. (2.5) , (2.6) reproduce th e known results for heavy q u a rk -a n tiq u a rk (sq uark-antisquark) pair- pro du ction soft anom alous dim ension [32, 33, 38- 40] in th e lim it p B ^ 0. Also, our result for Y q q - k W agrees w ith th e result obtain ed in th e SC E T fram ew ork in [41, 42]. It can be also explicitly seen th a t in th e lim it s ^ (2m t + m H) 2 th e non-diagonal elem ents vanish and th e diagonal elem ents give D q q - k i B j = {0, - N c } , D g g - k i B j = {0, - N c , - N c } , which are th e same coefficients as for th e heavy-quark pair prod uctio n D ij ^ k i . T his confirms a sim ple physical in tu itio n th a t th e properties of th e soft em ission in th e absolute threshold lim it are only driven by th e colour s tru c tu re of th e subprocesses and do not depend on th e th e ir kinem atics.
T k2 + ^fci A ( K - P k l \ , . \ L f k = ""2K a T
(M
K T & ) + ” V ' Ti(m ) = 2 (ln ((m2 - t i ) 2 / ( m 2 s ) ) - 1 + in ) ,U ( m ) = 2 (ln ((m2 — U i ) 2/ ( m 2s)) — 1 + in ) ,
k = \ / l - ( m k - m i ) 2 / s ki , Ski = ( p k + P i) 2,
t i = (Pi - Pk)2, t2 = ( P j - P i ) 2 , u i = ( p i - p i ) 2 ,
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T q q ^ k l B = O l - C F ( Li3,ki + 1) CA O3
n _ 2Q3 2[(CA - 2C F) ( L fi,ki + 1) + CA^3 + (8C f - 3C a )0 3] _
r i g 0 o
r g g ^ kiB = 0 r gg Nc O
r n 0 12 2 2 O3 i
2O n 2—4 O r gg
2O3 ~ Y N 7 O3 I 33
For com pletness we display th e explicit NLL expressions for th e resum m ed factors in th e M ellin space, which were used in our num erical im plem entation:
ln AN N= L g(1) (b° a y (^ R )ln N) ln N + g(2) (b° a y (^ R )ln N , M 2, M r , M f ) , (2.12)
w ith
(2.13)
(2.14)
w here b° and b1 are th e first two coefficients of th e QCD N-function,
T he coefficients
(2.15)
(2.16)
(2.17)
contain all non-logarithm ic co ntrib u tion s to th e NLO cross section tak en in th e threshold lim it. M ore specifically, these consist of Coulom b corrections, N -in d ep en d en t hard con
trib u tio n s from v irtu a l corrections and N -in d ep en d en t non-logarithm ic contrib u tio n s from soft emissions. A lthough form ally th e coefficients C ij ^ k l B ,I begin to c o n trib u te at NNLL accuracy, in our num erical studies of th e pp ^ t t H process we consider b o th th e case of
C ij ^ k l B ,I = 1, i.e. w ith th e first-order corrections to th e coefficients neglected, as well as th e case w ith these corrections included. In th e la tte r case we tre a t th e Coulom b corrections and th e hard contrib ution s additively, i.e.
C (1) = c (1>hard) + c (1>Coul)
C i j ^ k l B , I = C i j ^ k l B , I + C i j ^ k l B , I.
For k, l d enoting m assive quarks th e Coulom b corrections are C ^ ^ Ob i = C Fn 2/(2Nkl) and j k O B s = (C f - C a / 2 ) n 2/ ( 2 ^ kl). T he additive tre a tm e n t is consistent w ith NLL
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ln ^ (j-ik l B , I , N _ kl B , I (5o a s ( p 2R ) ln N )
A(1)
g* (A) — 2 n b0A [2A + ( 1 - 2A) l n ( 1 - 2A)I ’ A(1)
g (2)(A, M 2, mR, mF ) — — l n ( 1 - 2A)
a(1)b r i
+ 2 * ^ 5 3 2A + l n ( 1 - 2A) + 2 ln2 ( 1 - 2A) A(2)
- [ 2 A + l n ( 1 - 2A)I
A(1) ( m 2 \
- [ 2 A + l n ( 1 - 2 A ) | l ” ( m 2) + A(1) 2A ln ( M2f \
+ 2 n b 3 2A lH ’ h (2) (A) ln( 1 - 2A) D
h i j ^ k l B , I(A) — 2n b3 V i j ^ k l B J ,
, 11CA - 4 Tru , 17C3 - 10Ca Tr u, - 6 CfTr u,
5 3 — — i2n — ■ 5 1 — ---2^ --- • c ________ _1 1 c (1) +
C ij^ klB>I _ 1 + n C i j ^ k l B , I + • • •
resu m m atio n and m atching to NLO. We note th a t in general Coulom b corrections can also be resum m ed [46- 50]. A com bined resu m m ation of Coulom b and soft corrections is, however, beyond th e scope of th is paper.
3 Theoretical predictions for the p p ^ t t H process at NLO +NLL accuracy
T he resum m ation-im proved N L O + N L L cross sections for th e pp ^ t t H process are ob
tain e d th ro u g h m atching th e NLL resum m ed expressions w ith th e full NLO cross sections
^ (NLO+NLL) / 2 2 \ ~ (NLO) , 2 2 \ , ~ (res-exp) , 2 2 \
a h 1h 2^ k l ( p . F , = (Th 1h 2 ^ k l B( p . F , + r ) + (Th 1h 2 ^ k l B( p . F , .r)
w ith Tr(res-exP) = ^ f r d N p - N f (N+1) (..2 ) f (N+1) (..2 ) w ith a h i h 2^ k l B = ^ 2 n i P f i / h i F) f j / h 2 ( . F )
i , j C
x j u BJ N (.F > v R ) - J ’ (. F > v R ) L J > ( N L O ) (3-1)
w here v ij F k w n is given in eq. (2.2) and v jF k l B n |(NLO) represents its p e rtu rb a tiv e ex pan
sion tru n c a te d a t NLO. T he m om ents of th e p a rto n d istrib u tio n functions (pdf) f i / h ( x , vF) are defined in th e sta n d a rd way ń / h (v F ) = /o1 d x x N 1f i / h ( x , V2F ). T he inverse Mellin tran sfo rm (3.1) is evaluated num erically using a contou r C in th e com plex-N space accord
ing to th e “M inim al P rescrip tio n ” m eth od developed in ref. [51].
As m entioned in th e previous section, th e calculation of first-order con tribu tion s to th e coefficients C ij F t t H , i requires knowledge of th e NLO real corrections in th e threshold lim it as well as v irtu a l corrections. In our calculations we follow th e m ethodology of [52, 53], w here th e case of two m assive coloured particle in th e final s ta te was considered. We have explicitly checked th a t adding a m assive colour singlet particle in th e final sta te does not introduce any e x tra term s dependent on th e m ass of th e added particle. T hus th e N-space results for th e p air-p rod uctio n process of two m assive coloured particles are also applicable in our 2 ^ 3 case. T his way, th e problem of calculating th e C (j^ tfH i coefficients reduces to calculation of v irtu a l corrections to th e process. We e x tra ct th em num erically using th e publicly available P O W H E G im plem entation of th e t t H process [17], based on th e calculations developed in [10- 13]. T he results were th e n cross-checked using th e stan dalon e M adLoop im plem entation in aMC@NLO [14]. Since th e q q channel receives only colour-octet contributions, th e ex tra cte d value c o n trib u tin g to C ^ t t H 8 is exact. In th e g g channel, however, b o th th e singlet and o ctet prod uctio n m odes con trib ute. T he im plem entation of th e v irtu a l corrections to g g ^ t t H in P O W H E G and in aMC@NLO does not allow for th e ir sep arate e x tra ctio n in each colour channel. Instead, we e x tra ct th e value which co ntrib u tes to th e coefficient C g g ^ H 1 averaged over colour channels and use th e sam e value to fu rth e r calculate C (1 ’ g g ^ t t H , 1 and C (1 ’ hat 5 a . In order to m easure th e sizeg g ^ t t H , 8
of th e error introduced by th is procedure, we th e n rescale this value by th e ratios of th e corresponding colour-channel d ep endent and colour averaged coefficients found for g g ^ t t
in [54]. T he scale dependence of th e C j F kl B i can be fully deduced from renorm alization
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Figure 1. Scale dependence of th e LO, NLO and N L O + N L L cross sections a t %/S = 8 an d V S = 14T eV LH C collision energy. T he resu lts are o b ta in e d while sim ultaneously varying m f an d MR, M = M f = MR.
group argum ents, in th e sam e way as for th e full NLO result. We have checked th a t num erical results o b tain ed w ith th e procedure which we use to e x tra ct th e values of th e coefficients a t mo = Mf = MR show th e sam e scale dependence as expected from exact analy tical expressions.
In our phenom enological analysis we use m t = 173 GeV, m H = 125 GeV and choose th e central scale MF,0 = MR,0 = mt + m H/ 2, in accordance w ith [24]. T he NLO cross section is calculated using th e aMC@NLO code [25]. In th e im plem entation of th e resum m ation form ula, eq. (2.2) , we num erically tak e a M ellin tran sfo rm of th e LO cross sections and th e c j +t t H j coefficient term s which are b o th calculated in th e x space. We perform the cu rren t analysis em ploying M M HT2014 [55] pdfs and use th e corresponding values of as .
Beside presenting th e full result including non-zero C tfH j coefficients, we also show the results w ith CiJ^ tfH, j = 1.
We begin our num erical stu d y by analysing th e scale dependence of th e resum m ed to ta l cross section for pp ^ t t H a t V S = 8 and 14 TeV, varying sim ultaneously th e factorization and renorm alization scales, MF and mR. As d em o n strated in figure 1, adding th e soft gluon corrections stabilizes th e dependence on m = MF = MR of th e N L O +N L L predictions w ith respect to NLO. As an exam ple, th e central values and th e scale erro r a t V S = 8 TeV changes from 132+9+% fb at NLO to 141+4*2% fb a t N L O +N L L (w ith c j %tfH j coefficients included) and correspondingly, from 613+94% fb to 650++2% fb at V S = 14 TeV. It is also clear from figure 1 th a t th e coefficients C tfH strongly im pact th e predictions, especially a t higher scales.
In o rder to u n d e rsta n d these effects b e tte r, in figure 2 we analyse th e dependence on th e facto rization and renorm alization scale sep arately for th e case stu d y of V S = 14 TeV. We observe th a t th e weak scale dependence present w hen th e scales are varied sim ultaneously is a result of th e cancellations betw een renorm alization and factorization scale dependencies.
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Figure 2. F acto rizatio n and ren o rm alizatio n scale dependence of th e LO, NLO an d N L O + N L L cross sections a t %/S = 14 TeV LH C collision energy. T he resu lts are o b ta in e d w ith = p 0 for variatio n an d an d = p 0 for variation.
A sim ilar effect of th e opposite behaviour of th e to ta l cross section u n der and p R vari
ations was previously shown for th e to ta l cross section for th e inclusive Higgs p roduction in th e gluon-fusion process [57]. T he typical decrease of th e cross section w ith increasing
Pr originates from ru n nin g of a s. T he behaviour under variation of th e factorization scale, on th e o th er h a n d , is related to th e effect of scaling violation of pdfs a t probed values of x. In th is context, it is interesting to observe th a t th e N L O + N L L predictions in figure 2 show very little dependence aro und th e central scale, in agreem ent w ith ex p ectatio n of th e factorizatio n scale dependence in th e resum m ed exponential and in th e pdfs cancelling each oth er, here up to NLL. T he relatively strong dependence on p F of th e N L O + N L L p re
dictions w ith non-zero t f H/ can be th e n easily understood: th e resum m ed expression will tak e into account higher o rder scale dependent term s which involve b o th C t?H p and logarithm s of N . These term s do not have th e ir equivalent in th e p d f evolution since th e pdfs do not carry any process-specific inform ation. C orrespondingly, th ey are not cancelled and can lead to strong effects if th e coefficients C j1^ tfH p are num erically su b stan tial. As these term s can only provide a p a rt of th e full scale dependence a t higher orders, it is to be expected th a t th e ir im pact will be significantly modified w hen NNLO corrections are known.
Given th e argum ents above, we choose to estim ate th e th eo retical u n certain ty due to scale v ariation using th e 7-point m ethod, w here th e m inim um and m axim um values ob
tain e d w ith (p F / p 0, p R/ p 0) = (0.5, 0.5), (0.5,1), (1, 0.5), (1 ,1 ), (1, 2), (2 ,1), (2, 2) are con
sidered. T he effect of including NLL corrections is sum m arized in tab le 1 for th e LH C collision energy of 8, 13 and 14 TeV. T he N L O + N L L predictions show a significant reduc
tio n of th e scale uncertainty, com pared to NLO results. T he reduction of th e positive and negative scale errors am ounts to around 20-30% of th e NLO erro r for V S = 13,14 TeV and to arou n d 25-35% for V S = 8 TeV. T his general redu ction tre n d is not sustained for th e
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V S [TeV] NLO [fb] N L O + N L L N L O + N L L w ith e p d f error Value [fb] K -factor Value [fb] K -factor
8 132+3.9%
2—9.3% 135+3.o%
135-5.9% 1.03 141+7.7%
141-4.6% 1.07 +3.0%
-2.7%
13 506+5.9%
6—9.4% 516+4.6%
6—6.5% 1.02 537+8.2%
537-5.5% 1.06 +2.3%
-2.3%
14 613+6.2%
6 —9.4% 62 5+4.6%
6 —6.7% 1.02 650+7.9%
650-5.7% 1.06 +2.3%
- 2.2%
T a b le 1. N L O + N L L an d NLO to ta l cross sections for pp ^ t t H for various LH C collision energies.
T he erro r ranges given to g eth er w ith th e NLO an d N L O + N L L resu lts in d icate th e scale u ncertainty.
positive error after including th e C j tfH j coefficients. M ore specifically, th e negative error is fu rth e r slightly reduced, while th e positive erro r is increased. T he origin of this increase can be traced back to th e su b stan tial dependence on p , p of th e resum m ed predictions w ith non-zero tfH j coefficients, m anifesting itself at larger scales. However, even after th e red istrib u tio n of th e error betw een th e positive and negative p a rts, th e overall size of th e scale error, corresponding to th e size of th e error bar, is reduced after resu m m ation by aro u nd 7% at 8T eV and 10 (13)% a t 13 (14) TeV w ith respect to th e NLO u ncertainties.
T he scale error of th e predictions is still a few tim es larger th a n th e p d f error, cf. tab le 1.
For simplicity, th e p d f erro r shown in tab le 1 is calculated for th e NLO predictions, however adding th e soft gluon correction can only m inim ally influence th e value of th e p d f e rro r.1
As expected on th e basis of large phase-space suppression in th e threshold regime, th e predictions for to ta l cross section a t N L O + N L L are only m oderately increased by 2-3%
w .r.t. th e full NLO result. In tro d u cin g th e coefficients C j1^ tfH j leads to an increase in the K -fac to r of up to 6-7% , indicating th e im portance of c o n stan t term s in th e th resh old lim it.
Since th e im pact of soft corrections is bigger for processes tak in g place closer to threshold th e K -fac to r gets slightly higher for sm aller collider energies. We also check th e im pact of our appro xim ated tre a tm e n t of keeping p a rts of egg ( “ H 1 and egg ( “ h 8 coefficients com ing from th e v irtu a l corrections equal to th e colour channel averaged value, by rescaling at p , p =
p,R = th e averaged Cgg ( “ H coefficient w ith ratios e g g( a + / e g g , t a k e n from [54].
T he procedure is m otivated by obvious sim ilarities betw een th e colour stru c tu re s of th e pp ^ t t and p p ^ t i H cross sections considered a t th reshold. We find th a t such rescaling of th e hadronic t t H cross section leads to a 3 per mille effect a t 14 TeV, or a 5% effect on th e correction itself. T herefore we do not expect th a t th e exact knowledge of th e c g g ( tfH 1 and eggLt t H 8 coefficients will have a significant im pact on th e hadronic N L O + N L L predictions.
However, we stress th a t because of th e large phase-space suppression in th e threshold regim e th e resum m ed results, while system atically tak in g into account a well defined class of correction, should not be used to e stim ate th e size of th e NNLO to ta l cross section, by e.g. m ethods of expansion of th e resum m ed exponential.
1It should be however noted that the size of the pdf error presented here does not represent the un
certainty related to using fixed-order cross sections in the pdf fits, as opposed to using resummed cross sections. Recent results by the NNPDF collaboration [56] indicate that this effect would need to be inves
tigated in the full phenomenological study of the ttH production and its theoretical uncertainty, which is a task beyond the scope of this paper.
J H E P 0 3 ( 2 0 1 6 ) 0 6 5
In m ore detail, QCD corrections to th e t t H cross section m ay be divided into loga
rithm ically enhanced (up to th e NLL accuracy) soft gluon corrections and th e form ally subleading pieces i.e. corrections th a t en ter beyond th e NLL accuracy at th e absolute th resh old lim it. A d irect num erical analysis of relative im portance of th e two classes of corrections in th e NLO correction shows th a t th e soft gluon logarithm s do not dom i
n a te th e exact NLO correction. In th e window of renorm alisation and facto risation scales 1 /2 v 0 < V F = V R < 2v 0 th e NLL result expanded to th e NLO accuracy differs from th e NLO cross-section by ab o u t 10%, which should be com pared to th e typical relative m agn itu de of th e exact NLO correction in th is scale window of up to 20%. We took into account some of these form ally non-leading corrections via th e C (1)-coefficient determ ined in th e absolute threshold lim it. T his approach, however, also does not provide a satisfac
to ry appro x im ation to th e exact NLO correction. We conclude th a t a good approxim ation of th e exact NLO correction requires inclusion of subleading pieces in th e NLL expansion beyond th e absolute threshold lim it. T herefore our results should be viewed as an all-order im provem ent of a well defined sub-class of p e rtu rb a tiv e corrections to th e t t H cross-section, which, however, om its o th er possibly im p o rta n t contrib u tio n s in th e full p e rtu rb a tiv e ex
pansion.
4 Summary
We have investigated th e im pact of th e soft gluon em ission effects on th e to ta l cross section for th e process pp ^ t t H a t th e LHC. T he resum m ation of soft gluon em ission has been perform ed using th e M ellin-m om ent resum m ation technique a t th e N L O + N L L accuracy.
To th e best of our knowledge, th is is th e first application of this m ethod to a 2 ^ 3 process. Supplem enting th e NLO predictions w ith NLL corrections results in m oderate m odifications of th e overall size of th e to ta l rates. T h e size of these m odifications, as well as th e size of th e theo retical erro r due to scale variation is strongly influenced by th e inclusion of th e first-order hard m atching coefficients into th e resum m atio n fram ew ork.
T he overall size of th e th eo retical scale erro r becomes sm aller after resum m ation, albeit th e reduction is relatively m odest w hen th e non-zero first-order h ard m atching coefficients are considered.
N o t e a d d e d . A fter th e arX iv publication of this paper, ref. [58] appeared. T he results of [58] seem to su p p o rt our conclusion regarding th e im p o rtance of th e corrections from beyond th e absolute threshold region.
Acknowledgm ents
T his work has been sup p o rted in p a rt by th e D FG gran t K U 3103/1. S u pp ort of th e Polish N ation al Science C entre gran ts no. D E C -2 0 1 4 /1 3 /B /S T 2 /0 2 4 8 6 is gratefully acknowledged.
TS acknowledges su p p o rt in th e form of a scholarship of M arian Smoluchowski R esearch C onsortium M a tte r E nergy F u tu re from K N O W funding.
J H E P 0 3 ( 2 0 1 6 ) 0 6 5
O p e n A c c e s s . This article is d istrib u ted under th e term s of th e C reative Com m ons A ttrib u tio n License (CC -B Y 4.0) , which perm its any use, d istrib u tio n and reprodu ction in any m edium , provided th e original au th o r(s) and source are credited.
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