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Quark-Loop Amplitudes for W^+- H^-+ Associated Hadroproduction

a r X i v :h e p -p h /9909502v 1 24 S e p 1999

DESY 99-121ISSN 0418-9833

KEK-TH-638hep-ph/9909502September 1999

Quark-Loop Amplitudes for W ±H ?Associated

Hadroproduction

A.A.Barrientos Bendez′u 1and

B.A.Kniehl 2,?

II.Institut f¨u r Theoretische Physik,Universit¨a t Hamburg,Luruper Chaussee 149,22761Hamburg,Germany

High Energy Accelerator Research Organization (KEK),Theory Division,

1-1Oho,Tsukuba-shi,Ibaraki-ken,305-0801Japan

Abstract

In this addendum to our paper entitled W ±H ?Associated Production at the Large Hadron Collider [Phys.Rev.D 59,015009(1999)],we list analytic results for the helicity amplitudes of the partonic subprocess gg →W ?H +induced by virtual quarks.

PACS numbers:12.60.Fr,12.60.Jv,13.85.-t

In a recent paper[1],we studied the hadroproduction of a charged Higgs boson in association with a W boson at the CERN Large Hadron Collider(LHC)in the context of the two-Higgs-doublet model of type II,which serves as the Higgs sector for the minimal supersymmetric extension of the standard model(SM).This reaction dominantly proceeds via the partonic subprocesses bˉb→W±H?at the tree level(see Fig.1in Ref.[1])and gg→W±H?,which is mediated by triangle-and box-type diagrams involving virtual top and bottom quarks(see Fig.2in Ref.[1]).In Ref.[1],we presented analytic expressions for the cross section of bˉb→W±H?and the transition-matrix element of gg→W±H?arising from the quark triangles.However,we refrained from listing our formulas for the quark box contributions because we found that were somewhat lengthy.In the meantime, a signal-versus-background analysis of W±H?associated production at the LHC was carried out by Moretti and Odagiri[2],who generated the signal cross section by using the formulas published in Ref.[1],thus omitting the quark box contributions.This motivated us to further compactify our expressions for the latter by introducing helicity amplitudes. The purpose of this brief report is to provide these results,which may be useful for other authors as well.

Calling the four-momenta of the two gluons and the W boson p a,p b,and p W,respec-tively,we de?ne the partonic Mandelstam variables as s=(p a+p b)2,t=(p a?p W)2,and u=(p b?p W)2.Furthermore,we introduce the following short-hand notations:w=m2W, h=m2H,d=t?u,t1=t?w,t2=t?h,u1=u?w,u2=u?h,N=tu?wh,λ=s2+w2+h2?2(sw+wh+hs),and q=m2t?m2b.We label the helicity states of the two gluons and the W boson in the partonic center-of-mass frame byλa=?1/2,1/2,λb=?1/2,1/2,andλW=?1,0,1.For reference,we?rst list the helicity amplitudes

for the quark triangle contributions,M△λ

aλbλW .They may be extracted from Eq.(5)of

Ref.[1]and read

M△λaλb0=s √

m W

[(1+λaλb)Σ(s)?(λa+λb)Π(s)],(1) whereΣandΠare the vector and axial-vector form factors given in Eq.(6)of Ref.[1].In this case,the W boson can only be longitudinally polarized because it couples to two Higgs

bosons,so that M△λ

aλbλW =0forλW=±1.As for the quark box contributions,all twelve

helicity amplitudes,M2λ

aλbλW ,contribute.Due to Bose and weak-isospin symmetry,they

are related by

M2λaλb0 t,u,m2b,m2t,tanβ =M2λbλa0 u,t,m2b,m2t,tanβ ,

M2λaλbλW t,u,m2b,m2t,tanβ =?M2λbλaλW u,t,m2b,m2t,tanβ ,

M2λaλb0 t,u,m2b,m2t,tanβ =?M2?λa?λb0 t,u,m2t,m2b,cotβ ,

M2λaλbλW t,u,m2b,m2t,tanβ =M2?λb?λa?λW u,t,m2t,m2b,cotβ .(2) KeepingλW=±1generic,we thus only need to specify four expressions.These read: M2++0=2λ m2b tanβ+m2t cotβ F0+++m2t cotβG0+++(t?u) ,

M 2+?0

m 2b

tan β+

m 2t cot β

F 0+?+m 2t cot β

G 0

t ?u,m 2b ?m 2

t ,tan β?cot β

,M 2++λW

m 2b tan β+m 2

t cot β

√√

√

F 1

+?λ+λW F 2+?

+m 2

t cot β G 1+?λ

+λW G 2+?

t ?u,m 2b ?m 2

t ,tan β?cot β,λW →?λW

,(3)

where F i +±and G i +±,with i =0,1,2,are complex functions of t ,u ,m 2b ,and m 2

t .The

normalization of M △λa λb λW and M 2

λa λb λW is such that the di?erential cross section of gg →W ?H +is given by

dσ

256(4π)3s 2

λa ,λb ,λW

M △

λa λb λW

M 2

λa λb λW

2.(4)

We now express the form factors F i +±and G i

+±in terms of the standard scalar two-,three-,and four-point functions,

B 0

p 21,m 20,m 2

d D q

(q 2?m 20+i?)[(q +p 1)2?m 21+i?]

C 0

p 21,(p 2?p 1)

,p 22,m 20,m 21,m 2

d D q (q 2?m 20+i?)[(q +p 1)2?m 21+i?][(q +p 2)2?m 22+i?]

D 0

p 21,(p 2?p 1)2,(p 3?p 2)

,p 23,p 2

2,(p 3

?p 1)

,m 20,m 21,m 22,m 2

d D q

(q 2?m 20+i?)[(q +p 1)2?m 21+i?][(q +p 2)2?m 22+i?][(q +p 3)2?m 23

+i?],(5)

where D is the space-time dimensionality.The B 0function is ultraviolet (UV)diver-gent in the physical limit D →4,while the C 0and D 0functions are UV ?nite in this limit.We evaluate the B 0,C 0,and D 0functions numerically with the aid of the program package FF [3].To simplify to notation,we introduce the abbreviations C ab ijk (c )=C 0 a,b,c,m 2i ,m 2j ,m 2k and D abcd ijkl (e,f )=D 0 a,b,c,d,e,f,m 2i ,m 2j ,m 2k ,m 2

l .We ?nd

F 0++

=?2s (t 1+u 1)

m 2b C 00

bbb (s )

m 2t C 00

ttt (s )

+f 1(t,u,q )

t 2C h 0

btt (t )

t 1C w 0

tbb (t )

+f 1(u,t,q )

t 2C h 0

tbb (t )

t 1C w 0

btt (t )

?f 1(t,u,q )

N +

s (m 2b

m 2t )

D h 0w 0

bttb (t,u )

?2sm 2b [2wt 2+f 1(t,u,q )]D hw 00btbb (s,t )+2sm 2t [2wu 2?f 1(t,u,q )]D hw 00

tbtt (s,t ),

G 0++

=?2s 2

(t +u )C 00

ttt (s )

?t 2f 1(?t 2,u 2,h )

C h 0

btt (t )

C h 0

tbb (t )

?t 1f 1(?t 2,u 2,h )

C w 0

btt (t )

C w 0

tbb (t )

2sλm 2b

+(N +sq )f 1(?t 2,u 2,h )

D h 0w 0

bttb (t,u )

2sλm 2b D hw 00

btbb (s,t )

+2s

2swh ?

sm 2b (t

+u )+

m 2t f 1(?t 2,u 2,h )

D hw 00

tbtt (s,t ),

F 0+?=2s {2wN +(t +u )[N ?f 2(0,t,λ)]+qf 2(t,u,2λ)+2q 2(t 1+u 1)}C 00

bbb (s )

?2t 2[wh (t 1+u 1)+qf 1(2u,2h,t )]C h 0

btt (t )+2t 2[w (hd ?2tt 2?2N )

?qf 1(2u,2h,t )]C h 0tbb (t )?2t 1[w (hd ?2ut 2)+qf 1(2t,2h,t )]C w 0

btt (t )

+2t 1[w (hd ?2tt 2)?qf 1(2t,2h,t )]C w 0

tbb (t )?2{λ(ud ?2wt 2)?(t 1+u 1)

×[N (t +u )+q (d 2

+2N )]}C hw

btb (s )

?[ud ?2N ?q (t 1+u 1)]

m 2b

m 2t

+sq

D h 0w 0bttb (t,u )? 2N

wN ?

m 2b f 3(t,u,2w )

?Nq

ud +

2u 21

2m 2b (t 1

+u 1)

?q 2

[du 1(t 1+u 1)?t 2f 3(t,u,w )]?sq 3

(t 1+u 1)

D h 0w 0tbbt (t,u )?2{stw (hd ?2tt 2)

2Nm 2b f 2(0,t,λ)

sN (t +u )?stf 2(t,0,2λ)+

2Nm 2b (t 1

+u 1)

?sq 2

f 2(t,0,λ)?sq 3

(t 1+u 1)

D hw 00btbb (s,t )

+2(t 1+u 1)

stwh +2uNm 2t

+q st (t +2u )?N

s ?

2m 2t

+sq 2

(2t +u )+sq

D hw 00

tbtt (s,t ),

G 0+?

=2s

(t +u )(d 2

+2N )?2λq C 00

bbb

(s )+2

t 2

(t +u )?wh (3t ?u )

t 2

C h 0

btt (t )

C h 0

tbb (t )

t 1

C w 0

btt (t )

C w 0

tbb (t )

?2λ(d 2+2N )C hw

btb (s )

?λ

m 2b

m 2t

+sq

D h 0w 0bttb (t,u )

2N 2wN ?λm 2b

?λq (N +sq )

D h 0w 0tbbt (t,u )

?2f 4

m 2b ,m 2

D hw 00

btbb

(s,t )?2f 4

m 2t ,m 2

D hw 00

tbtt (s,t ),F 1

=2s 2

m 2b C 00

bbb (s )?

m 2t C 00

ttt (s )

?f 5

w,m 2b ,m 2

t 2C h 0

btt (t )

t 1C w 0tbb (t )

+f 5

w,m 2t ,m 2

t 2C h 0tbb (t )

t 1C w 0

btt (t )

N +

s (m 2b

m 2t )

f 5

w,m 2b ,m 2

D h 0w 0

bttb (t,u )

+2sf 6

m 2b ,m 2

D hw 00

btbb (s,t )

?2sf 6

m 2t ,m 2

D hw 00

tbtt (s,t ),

F 2

=?(N ?sq )

t 2C h 0

btt (t )

t 1C w 0

btt (t )

+(N +sq )

t 2C h 0

tbb (t )

t 1C w 0

tbb (t )

+ N

N +

2s (m 2b

m 2t )

+s 2q

D h 0w 0

bttb (t,u ),

G 1++

=?sd (t 2+u 2)C 00

ttt (s )

+t 2f 3(t,u,h )

C h 0

btt (t )

C h 0

tbb (t )

?t 1f 3(u,t,h )

C w 0

btt (t )

+C w 0

tbb (t )

+(N +sq )f 3(u,t,h )D h 0w 0

bttb (t,u )+s (t 2+u 2)[2N +d (t +q )]

×D hw 00tbtt (s,t ),

G 2++

?sdC 00

ttt (s )

t 2u 2

C h 0

btt (t )

C h 0

tbb (t )

+(st +N )C w 0btt (t )?t 1t 2C w 0

tbb (t )

+t 2(N +sq )D h 0w 0bttb (t,u )+s [2N +d (t +q )]D hw 00

tbtt (s,t ),

F 1

=?4sdNB 0

s,m 2b ,m 2

?2s {sd (t 2+u 2)?N [4su +d (w ?h )+λ]

?2sq[d(t+u)?2N]+2sdq2}C00bbb(s)+2t2 2sN(t+u)+f7 m2b,m2t C h0btt(t)

?2t2f7 m2t,m2b C h0tbb(t)?2t1 2t2N(t1+u1)+f8 m2b,m2t C w0btt(t)

+2t1f8 m2t,m2b C w0tbb(t)?2sλ[N?d(t+u?2q)]C hw btb(s)+f5 h,m2t,m2b

× N m2b+m2t +sq2 D h0w0bttb(t,u)? N2 f3(t,u,w)+2m2t(3s+w?h)

?Nq sλ?N(t1+u1)?sd t1+2m2t +2sNq2(2s+u2)+s2dq3 D h0w0tbbt(t,u)

?2s(t?q) Nf3(u,t,h)?d st(t?2m2t)?2t1t2m2b+sq2 D hw00btbb(s,t)

+2s stdN?(ud+λ) st2+2Nm2t ?q sd(t(t+2u)?N)+2stλ+2dNm2t

?sq2[d(2t+u)+λ]?sdq3 D hw00tbtt(s,t),

F2+?=?4sNB0 s,m2b,m2b ?2s s(t2+u2)+N(t2?u1)+4Nm2b?2sq(t+u)+2sq2 ×C00bbb(s)?2t2 u1N?f9 t2,m2b,m2t C h0btt(t)?2t2 u1N+f9 ?t2,m2t,m2b

×C h0tbb(t)+2t1f9 t2,m2b,m2t C w0btt(t)?2t1f9 ?t2,m2t,m2b, C w0tbb(t)+2s[d2(t+u) +N(t+3u)?2q(d2+2N)]C hw btb(s)?q N2+2sN m2b+m2t +s2q2 D h0w0bttb(t,u)? u1N2+Nq tu1+ut2+u21+4sm2t ?2sNq2+s2q3 D h0w0tbbt(t,u)

?2f10 t2,m2b,m2t D hw00btbb(s,t)+2f10 s,m2t,m2b D hw00tbtt(s,t),

G1+?=?2f3(u,t,h) 2sC00bbb(s)+t1 C w0btt(t)+C w0tbb(t) +2t2f3(t,u,h) C h0btt(t)+C h0tbb(t) ?[dN(t1+u1)+sλq]D h0w0tbbt(t,u)+2s(t?q)f3(u,t,h)D hw00btbb(s,t)

?2s(t+q)f3(t,u,h)D hw00tbtt(s,t),

G2+?=2t2 2sC00bbb(s)+u2 C h0btt(t)+C h0tbb(t) +t1 C w0btt(t)+C w0tbb(t)

+f5 w,m2b,m2t D h0w0tbbt(t,u)?2st2(t?q)D hw00btbb(s,t)?2su2(t+q)D hw00tbtt(s,t),(6) where we have used the auxiliary functions

f1(t,u,q)=?w(t?u)+q(t1+u1),

f2(t,u,λ)=λ?(3t+u)(t1+u1),

f3(t,u,h)=2N?(t?u)(u?h),

f4 m2b,m2t =?st[N(t+u)?tλ]?s(m2b?m2t)[N(t+u)?2tλ]

+λ 2Nm2b+s m2b?m2t 2 ,

f5 w,m2b,m2t =N(t+u?2w)?sd m2b?m2t ,

f6 m2b,m2t =?sm2b 2N+d t+m2b?m2t ,

f7 m2b,m2t =sd(t2+N)? m2b?m2t [2std+N(3s+w?h)],

f8 m2b,m2t =2N2?d(st2?t2N)+ m2b?m2t [d(2st+N)?2t2N],

f9 t2,m2b,m2t =st2?t2N? m2b?m2t (2st+N),

f10 t2,m2b,m2t =?s t+m2b?m2t st t+2m2b?2m2t +N t2+4m2b +sq2 .(7) Here it is understood that all variables appearing on the right-hand sides are to be taken as independent. E.g.,N should be treated as independent of t,u,w,and h.Notice

in Eq.(2). that the UV divergences of F1+?and F2+?cancel in the expression for M+?λ

Finally,we remark that we recover the SM result for dσ(gg→ZH)/dt due to one quark ?avour[4]from Eq.(4)by substituting m W=m Z,m b=m t,and tanβ=1and adjusting the strengths of the axial-vector and Yukawa couplings.In particular,the contribution proportional to the weak vector coupling then vanishes as required by charge-conjugation invariance.This serves as a useful check for our analytical and numerical analyses.

Acknowledgements

B.A.K.thanks the KEK Theory Division for the hospitality extended to him during a visit when this paper was prepared.The work of A.A.B.B.was supported by the Friedrich-Ebert-Stiftung through Grant No.219747.The II.Institut f¨u r Theoretische Physik is supported by the Bundesministerium f¨u r Bildung und Forschung under Contract No. 05HT9GUA3,and by the European Commission through the Research Training Net-work Quantum Chromodynamics and the Deep Structure of Elementary Particles under Contract No.ERBFMRXCT980194.

References

[1]A.A.Barrientos Bendez′u and B.A.Kniehl,Phys.Rev.D59,015009(1999).

[2]S.Moretti and K.Odagiri,Phys.Rev.D59,055008(1999).

[3]G.J.van Oldenborgh,https://www.sodocs.net/doc/4d9885269.html,mun.66,1(1991).

[4]B.A.Kniehl,Phys.Rev.D42,2253(1990);42,3100(1990).

Quark-Loop Amplitudes for W^+- H^-+ Associated Hadroproduction

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