- Email: [email protected]
Higgs boson production by W and Z collisions
Volume 167B, number 4
PHYSICS LETTERS
20 February 1986
H I G G S B O S O N P R O D U C T I O N BY W AND Z C O L L I S I O N S J. L I N D F O R S LAPP, F-74019 Annecy-le-Vteux, France Recewed 14 November 1985
A general formulatmn for the WW ~ H and ZZ ~ H mechamsms of heavy H~ggs boson production m hadron colhders ~s presented The known results m the effectwe W approximation are obtained from the quark-parton model hmlt of the double melasuc scattering conflguratton In addition contributions from the elastic scattering configurations and from a -processes of QCD are estimated
The proposed proton colliders with center-of-mass energy x/~ = 1 0 - 4 0 TeV would provide a first direct probe of electroweak interactions beyond the 1 TeV threshold. In consequence, the following observation has attracted renewed interest recently [1-3] : either the Higgs scalar boson is lighter than about 1 TeV or the Higgs sector including the longitudinal polarization states of W and Z will become strongly interacting [4,5]. The second possibility implies interesting new phenomena with the physics of W's and Z's at 1 TeV reminiscent of pion physics at 103 times smaller center-of-mass energies [2,6]. In this letter we concentrate on the gateway to this new physics. In hadron coUiders the fusion of two longitudinally polarized W's or Z's becomes the dominant production mechanism of Higgs bosons when m H ~ 0.5 TeV [1,7]. Calculations on the pp ~ WW, ZZ ~ H process convolute the cross section for the qq -'- WW, ZZ ~ H subprocess with the quark distributions either in the effective W approximation [8] or without approximations [9]. Here we present a formulation of the pp -~ WW -~ H process, which provides a basis for a more general analysis of the weak gauge-boson fusion channel. The formalism of previous calculations is obtained as the quark-parton model limit of our formulae, but we point out additional although numerically small contributions to heavy Higgs production. In general the cross section for the pp ~ WW ~ H process of fig. 1 can be written in the form
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
p(pl)
>
(~ql)
p(p;~)
>
(~W(q~)
X(k) Fig. 1. The two-gauge-boson fusion process in p r o t o n - p r o t o n
collisions.
o = (2m2/s) (41r)-4 (ct/sin20w)2 xr
d4ql
d4q2
•](q~ _ m2w)2 (q2 ~ m--2w)2
X W v(pl ,---ql)aut~';uV'(ql ,q2)l¥,v,(p2,-q2), (1) where a~U';~"is the absorptive part of the WW ~ X WW amplitude, given for the case of a standard model (perturbative) Higgs boson production by
aUU';vV'(ql, q2) = A2gUU'g vv'21r 8((q 1 + q2 )2 - m 2 ) , (2a)
A = emw/sin 0 w.
(2b)
The hadronic tensor Wuv for the charged weak current has the standard Lorentz decomposition into invariant structure functions
471
Volume 167B, number 4
PHYSICS LETTERS
- q) = -W1 ('guy - quqv/q2)
20 February 1986
ql2mWv(1) q2mWUV(2)
+ (W2/m2)(Pu - qu p'q[q2) (Pv - qv p" q/q2)
= (4XlX2)-1 (FL(1)FL(2)[2q~ql + (ql "q2 )21
+ (i/2m 2) W3 eu~op~q~.
(3)
+ FL(1)F2(2)[4q1" k2 ql .k~ - q2q~]
Eq. (1) is a generalization of a similar formula for twophoton collisions discussed previously in connection with the process pp -~ 7 " / ~ ~+£- [10,11]. We define the usual scaling variables (note the different sign convention of momenta qi from that in deep inelastic scattering)
+ F2(1)FL(2)[4q2" kl q2" k'l - q2 q2]
xi = Q2/(_2pfqi),
X [ 8 k l . k 2 k'l.k' 2 - 8 k l . k ' 2 k2.k'l ] ).
Q/2 = _q2
(i = 1,2),
k~=k i - q i
(i=1,2).
(4b)
Using the identity 1 = f dxi(q2 /xi)6((ki - qi)2) we can now rewrite eq. (1) in a form where the quarkparton model interpretation will be more transparent
o=fdx I d x 2 ( Z s ) - l f ~ 3 ( k l
+k2 = k'1 + k ; +k)
q 2 m W , ( 1 ) q2mW"V(2)
× 7r2(ct/sin20w)2A2 ~
-.._-5-2-~,2 - ~ _ 2 - - . . ~ 2 "
(ql-mW)
(q2-mw)
The phase-space integral f dR 3 is defined as
-
f d R 3 = (2ff)-5 f d4ql 6((kl - ql )2) d4q2 5((k 2 _ q2)2) X 6((ql + q2 )2 - m 2) = (2rr) -5 fd4k'16(k~)d4k'2~(ki2)d4kS(k 2 - m 2) (6)
which can be evaluated for example in CMS(k 1 + k2) using the variables n and ~', E~ = (1 - n) x/~]2 and E~ = (1 - ~') X/~[2, which are especially suitable for taking the forward scattering or effective W approximarion [1,8,9]. Introducing next the structure functions F 1 = mWl , F2, 3 = (Q2/2mx)W2, 3 and F L = F 2 - 2xF 1 we write
-
k I = x~ p~
>
/
k~
k2 -- X2P2
~---)" k > ~ k2
Fig. 2. Diagram of momentum flow in eq. (5). 472
(7)
This equation when inserted into eq. (1) or eq. (5) gives our general formulation of the WW ~ H process. The corresponding formula for ZZ ~ H is obtained with obvious changes of couplings and masses. Note that we have not introduced any particle (parton) interpretation of the momenta k i and k~, and they only indicate momentum flow as shown diagrammatically in fig. 2. Compared to previous formulations of the WW H process we stress the following differences in our approach: our formulation is not restricted to the deep inelastic scattering case; in addition to the F 2 contribution there are terms proportional to F L and F 3 ; there is a Q2-dependence in structure functions to be integrated over. To take the quark-parton model limit we replace in eq. (7) the structure functions by the singlet (s) and non singlet (ns) scaling quark distributions: F212x ~qS(x),F3/2 ~ qnS(x) a n d F L ~ 0 from the CallanGross relation. This gives in the valence-quark approx. imation the cross section (g2 = e/2x/~ sin 0w) -
(5)
X 54(k 1 + k 2 - k'1 - k~ - k),
+ F 3 (1)F3 (2)x I x 2
(4a)
and the four-momenta (light-like for mp ~ O)
k i = x i p i,
+ F2(1)F2(2 ) [8k 1 "k 2 k i • k~ + 8k 1 • k'2 k2"k' 1 ]
Volume 167B, number 4
PHYSICS LETTERS
O = f d x 1 dx 2 q(xl)q(x2)(2s')-I
q2 = - ( 1 - ~') (1 - cos 02)g/2 ,
, , 2 _ m2)-2 X f d R 3 A 2 g 4 6 4 k l . k 2 kl.k2(ql
(13 cont'd)
the invariant phase space integral is proportional to
X (q2 _ m 2 ) - 2
(8)
3(ql:- row) 2 - 2 ( q 22 - m 2 ) - 2
f
cc f
in agreement with Cahn and Dawson (ref. [1]). In eq. (1) we effectively integrate Q12 and Q22 over
the range 0
20 February 1986
j
< e - m:.. The minimal Qi or
can be increased by fixing the transverse m o m e n t u m k T of the Higgs boson, but for the total cross section of eq. (1) we must take into account the elastic scattering configurations in addition to the double inelastic configuration discussed above. The inelastic threshold condition
Q2 >>.[xi/(1 _ xi)][( m + mTr)2 _ m 2 ]
(9)
separates the total pp ~ WW ~ H contribution into four terms: Otot = OEL_E L + OEL_INEL + OlNEL_EL + OlNEL_INEL, (10) corresponding to each of the two pW vertices being either elastic or inelastic. The structure functions F/ (i = 1,2, 3) in eq. (7) are replaced for the elastic vertex by the proton form factors Ge(Q2), Gm(Q 2) and Ga(Q2 ) according to F 1 ~ 6(1 - x)[Gm(Q2) 2 + Ga(Q2) 2 ]/2, F L -" 8(1 - x ) [ G e ( Q 2 ) 2 - Gm(Q2) 2 ]/(1 + Q2/4m2),
d cos01
d cos02 (14)
(1 -- cos01 + el )2 (1 - cos02 +e2 )2 '
where e I = 2m2W/S(1 - n),
e 2 = 2m2/s(1 - ~').
Neglecting Q2-dependence of the inelastic structure functions the above integrals give factors 1
f
d cos Oi(1 - cos 0i + el)-2 ~-- lie i cc 1/m2w
in the forward scattering approximation. The l/m2w dependence reflects the constancy of the W-propagator ( 2 1/m 4) over the range 0 < Q2 < m 2 or in the forward cone 0 i < x/~r However, for the elastic scattering case the Q2-dependence of the proton form factor is much stronger than the scaling violations for the inelastic case and cannot be neglected. With the parametrizations of eq. (12) we have in the forward scattering approximation instead of eq. (15) 1 2 2 -4 f d cos0i(1 - cos0i + ei)-2 (1 + Qi/my) -1
(mv/mw)2/e i • F 2 ~ 2 F 1 + F L, F 3 ~ 6(1 -- x)2Ga(Q2)Gm(Q2).
(11)
Typical form factor parametrizations are Gm(Q 2) = Up Go(Q2) =/ap/(l + Q2/m2)2, Ga(Q2 ) = 1.23/(1 + Q2/m2)2,
(12)
with m v = 0.84 GeV]e 2, m a = 0.89 GeV/e 2 and #p = 2.79. To assess the relative importance of the elastic and inelastic configurations we will here discuss results of calculations in the effective W approximation. Defining cos 0 i as the polar scattering angle of k; in CMS(k 1 + k2) ql2 = --(1 -- '7)(1 -- cos 01)s/2,
(13)
(15)
-1
(I 6)
The form factor cuts off the integral already at Q2 < m2v or to a cone 0 i < ~ mv/m w, and the elastic vertex is suppressed by a factor (mv/mw) 2 ~-- 10 -4 relative to the inelastic vertex. We recall that in the case of the process pp ~ 77 -" £+~- m w is effectively replaced by m(proton), and the elastic contribution is as important as the inelastic one. Having disposed of the elastic contribution we elaborate more on the inelastic case. Using the variables n and ~"introduced after eq. (6) the phase space integral gives in the effective W approximation
f~3(q21_ =
2 )- 2 (q22_ m2w)-2 mw
2-71r-3(,~m4) - l f u n
6(n
- m2/O.
(17)
473
Volume 167B, number 4
PHYSICS LETTERS
The coefficients of the structure functions in eq. (7) reduce to simple forms proportional to products of
fleL/F2(r/) = (ot2/2rr)4(1
-- r/)/*/,
flgL/FL(rD = (ot2/2rr)r/,
(18)
where a2 = g2/47r. The coefficient of the F3 F3-term vanishes for the forward-scattering kinematics. Finally the cross section of eq. (5) can be written in the factorized form
20 February 1986
Finally we present a numerical estimate for the inelastic versus elastic o(pp ~ WW ~ H) cross sections using the effective t¢ approximation and a simple model parametrization of F2(x ) "" 2x q S(x). Defining the luminosity functions dL r d-Tr(pp -~ WW) 1 dr' [", d L .
.-1
r LT. ~rTr'{'pp "+ F2 F2 'O --f7
o = f a x I dx 2 f d n d~ &Wr WL~H X { [FL(X 1 )/2x I ] fWL/Ft(rl)
where
+ [F2(Xl)/2x I ] fleL/F2(rl)}
r ~L r (pp ~ F 2 F 2 ) 1
X ([FL(XE)/2X2]flCL/FL(~) + [F 2 (XE)/2x2
]fWljf 2(~)},
= f -~- [F2(x)/2 ] [F2(r/x)/2], (19) dL 7" ~-~'r(F2 F2 -~WW)
where
b(m2 /~rl~) =A2mw4(Ir/4)~(1 - m 2 /~rl~)
(20)
is the Higgs production cross section by the fusion of longitudinally polarized W's. The contribution of transverse W's is suppressed by a factor (mw/mrl) 4. fWI_JF2 and fWT/Fhave the interpretation of WL L~. l-, distribution functions reside F 2 and F L. Alternatively we can write eq. (19) as
o = f d z I dz 2 WL(Z1)WL(g2) OWLWL~H ,
(21)
where we defined the W L distribution function inside a proton 1
1
WL(Z)=f dx f 0
d r / 8 ( z - xr/)
0
X { [F2(x)/2x]fWL/F 2(~1)+ [FL(X)/2x]fICL/F1(r/)}. (22) In perturbative QCD R = FL/F 2 = O(Ots) [12], but the vanishing offlCL/FL(rl) at 77 = 0 further suppresses the contribution of the FL-term in eq. (22). For the set 1 parametrization of ref. [13] with Q2 = m 2 we fred F L to contribute less than 1% of l~L(Z ) for z ~> 0.01. 474
(24a)
T
1
=fox [XfWL/F2(X)] [(7"/X)fWL/F2(7./X)] 7"
= (%/2r02. 16[(1 + r)ln(1/r) - 2(1 - r)],
(24b)
we can write the pp -+ WW -+ H cross section as 1
=
=
-- r T
r'
~r'
(pp -+ WW) ?r(r/r').
(25)
In our numerical calculation we take for the inelastic vertex the model parametrization Xqs(X) = 2(1 - x ) 2 , which gives a reasonable description of the x-dependence and normalization o f f 2. For the elastic vertex we assume F 2 + ½b(mv/mw)2 8(1 - x), where b = Ge(0)2 + Ga(0) 2 = 2.5 and the suppression factor was discussed before. Inserting the above parametrizations into eq. (24a) we can calculate order of magnitude estimates for the different terms of Otot in eq. (10). They are plotted for m H = 1 TeV/c 2 in fig. 3, where for normalization reasons we have not multiplied the elastic vertex by the W-propagator suppression factor. The double inelastic curve agrees in magnitude with previous estimates [9], and as expected the elastic terms are suppressed by factors of 10 -4
PHYSICS LETTERS
Volume 167B, number 4 lO
in eq. (10) with correct integrations over Q2 is needed and will be presented elsewhere.
(mwlmv)2x (Inet-eI+el-inel)
o- [pb]
ol
I would like to thank P. Aurenche and K. Kajantle for discussions. This research was supported by le Minist6re des Relations Ext6rieures (France) and by the Herman Rosenberg Foundation at Helsinki University.
References
/ii//t/ ! //~
mH= 1 TeV
/ 0 01 0
20 February 1986
I
I0
210
30 I
~0
[TeV] Fig. 3. The double inelastae versus the elastic contributions to a(pp ~ WW~ H) for mH = 1 TeV/c2 as a function of the center-of-mass energy (elastic-inelastic) or 10 - 8 (elastic-elastic) in the energy range x/~ = 1 0 - 4 0 TeV. We conclude that neither the elastic contributions nor the higher order QCD terms implicit in F L will change drastically the previous estimates on the gauge-boson fusion cross sections. This statement is verified above in the forward-scattering approximation and neglecting the Q2-dependence of structure functions. However, as an enhancement by a factor of two in the W and Z pair production rate may signal the onset of the new strong interactions [2], it is important to have a reliable absolute normalization of the WW and ZZ fusion cross sections. To this end a complete numerical analysis of the different terms
[1] R.N. Cahn and S. Dawson, Phys. Lett. 136B (1984) 196; 138B (1984) 464(E). [2] M.S. Chanowitz and M.K. GaiUard, Phys. Lett. 142B (1984) 85 ; preprint LBL-19470 (1985), to be published in Nucl. Phys. B. [3] E. Eichten, I. Hmchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579; L. Roberts, Phys. Rev. D32 (1985) 701; D.A. Dlcus and S S.D. Wfllenbrock, preprint UUTG12-85 (1985). [4] M. Veltman, Acta Physica Polomca B8 (1977) 475. [5] B.W. Lee, C. Quigg and M. Thaeker, Phys. Rev. D16 (1977) 1519. [6 ] M. Veltman, TASI Lectures in Elementary particle physics (1984), ed D.N. Williams(University of Michigan, Ann Arbor, MI, 1985) p. 86, M. Einhorn, TASI Lectures in Elementary particle physics (1984), ed. D.N. Williams(Universityof Michigan, Ann Arbor, MI, 1985) p. 113. [7] D.R.T. Jones and S.T. Petcov, Phys. Lett. 84B (1979) 440. [8] S. Dawson, Nucl. Phys. B242 (1985) 42; G.L. Kane, W.W. Repko and W.B. Rolrdck, Phys. Lett. 148B (1984) 367; J. Lindfors, Z. Phys. C28 (1985) 427. [9] R.N. Cahn, Nucl. Phys. B255 (1985) 341. [10] Min-ShihChen, 13. Muzinich, H. Terazawa and T.P. Cheng, Phys. Rev. D7 (1973) 3485; V.M. Budnev, I.F. Ginzburg, G.V. Meledin and V.G. Serbo, Phys. Rep. 15C (1975) 181 ; R. Moore, Z. Phys. C5 (1980) 351. [11 ] B. Schrempp and F. Schrempp, Nucl. Phys. B182 (1981) 393. [12] G. Altarelli, Phys. Rep. 81 (1982) 1. [13] D.W. Duke and J.F. Owens, Phys. Rev. D30 (1984) 49.
475
PHYSICS LETTERS
20 February 1986
H I G G S B O S O N P R O D U C T I O N BY W AND Z C O L L I S I O N S J. L I N D F O R S LAPP, F-74019 Annecy-le-Vteux, France Recewed 14 November 1985
A general formulatmn for the WW ~ H and ZZ ~ H mechamsms of heavy H~ggs boson production m hadron colhders ~s presented The known results m the effectwe W approximation are obtained from the quark-parton model hmlt of the double melasuc scattering conflguratton In addition contributions from the elastic scattering configurations and from a -processes of QCD are estimated
The proposed proton colliders with center-of-mass energy x/~ = 1 0 - 4 0 TeV would provide a first direct probe of electroweak interactions beyond the 1 TeV threshold. In consequence, the following observation has attracted renewed interest recently [1-3] : either the Higgs scalar boson is lighter than about 1 TeV or the Higgs sector including the longitudinal polarization states of W and Z will become strongly interacting [4,5]. The second possibility implies interesting new phenomena with the physics of W's and Z's at 1 TeV reminiscent of pion physics at 103 times smaller center-of-mass energies [2,6]. In this letter we concentrate on the gateway to this new physics. In hadron coUiders the fusion of two longitudinally polarized W's or Z's becomes the dominant production mechanism of Higgs bosons when m H ~ 0.5 TeV [1,7]. Calculations on the pp ~ WW, ZZ ~ H process convolute the cross section for the qq -'- WW, ZZ ~ H subprocess with the quark distributions either in the effective W approximation [8] or without approximations [9]. Here we present a formulation of the pp -~ WW -~ H process, which provides a basis for a more general analysis of the weak gauge-boson fusion channel. The formalism of previous calculations is obtained as the quark-parton model limit of our formulae, but we point out additional although numerically small contributions to heavy Higgs production. In general the cross section for the pp ~ WW ~ H process of fig. 1 can be written in the form
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
p(pl)
>
(~ql)
p(p;~)
>
(~W(q~)
X(k) Fig. 1. The two-gauge-boson fusion process in p r o t o n - p r o t o n
collisions.
o = (2m2/s) (41r)-4 (ct/sin20w)2 xr
d4ql
d4q2
•](q~ _ m2w)2 (q2 ~ m--2w)2
X W v(pl ,---ql)aut~';uV'(ql ,q2)l¥,v,(p2,-q2), (1) where a~U';~"is the absorptive part of the WW ~ X WW amplitude, given for the case of a standard model (perturbative) Higgs boson production by
aUU';vV'(ql, q2) = A2gUU'g vv'21r 8((q 1 + q2 )2 - m 2 ) , (2a)
A = emw/sin 0 w.
(2b)
The hadronic tensor Wuv for the charged weak current has the standard Lorentz decomposition into invariant structure functions
471
Volume 167B, number 4
PHYSICS LETTERS
- q) = -W1 ('guy - quqv/q2)
20 February 1986
ql2mWv(1) q2mWUV(2)
+ (W2/m2)(Pu - qu p'q[q2) (Pv - qv p" q/q2)
= (4XlX2)-1 (FL(1)FL(2)[2q~ql + (ql "q2 )21
+ (i/2m 2) W3 eu~op~q~.
(3)
+ FL(1)F2(2)[4q1" k2 ql .k~ - q2q~]
Eq. (1) is a generalization of a similar formula for twophoton collisions discussed previously in connection with the process pp -~ 7 " / ~ ~+£- [10,11]. We define the usual scaling variables (note the different sign convention of momenta qi from that in deep inelastic scattering)
+ F2(1)FL(2)[4q2" kl q2" k'l - q2 q2]
xi = Q2/(_2pfqi),
X [ 8 k l . k 2 k'l.k' 2 - 8 k l . k ' 2 k2.k'l ] ).
Q/2 = _q2
(i = 1,2),
k~=k i - q i
(i=1,2).
(4b)
Using the identity 1 = f dxi(q2 /xi)6((ki - qi)2) we can now rewrite eq. (1) in a form where the quarkparton model interpretation will be more transparent
o=fdx I d x 2 ( Z s ) - l f ~ 3 ( k l
+k2 = k'1 + k ; +k)
q 2 m W , ( 1 ) q2mW"V(2)
× 7r2(ct/sin20w)2A2 ~
-.._-5-2-~,2 - ~ _ 2 - - . . ~ 2 "
(ql-mW)
(q2-mw)
The phase-space integral f dR 3 is defined as
-
f d R 3 = (2ff)-5 f d4ql 6((kl - ql )2) d4q2 5((k 2 _ q2)2) X 6((ql + q2 )2 - m 2) = (2rr) -5 fd4k'16(k~)d4k'2~(ki2)d4kS(k 2 - m 2) (6)
which can be evaluated for example in CMS(k 1 + k2) using the variables n and ~', E~ = (1 - n) x/~]2 and E~ = (1 - ~') X/~[2, which are especially suitable for taking the forward scattering or effective W approximarion [1,8,9]. Introducing next the structure functions F 1 = mWl , F2, 3 = (Q2/2mx)W2, 3 and F L = F 2 - 2xF 1 we write
-
k I = x~ p~
>
/
k~
k2 -- X2P2
~---)" k > ~ k2
Fig. 2. Diagram of momentum flow in eq. (5). 472
(7)
This equation when inserted into eq. (1) or eq. (5) gives our general formulation of the WW ~ H process. The corresponding formula for ZZ ~ H is obtained with obvious changes of couplings and masses. Note that we have not introduced any particle (parton) interpretation of the momenta k i and k~, and they only indicate momentum flow as shown diagrammatically in fig. 2. Compared to previous formulations of the WW H process we stress the following differences in our approach: our formulation is not restricted to the deep inelastic scattering case; in addition to the F 2 contribution there are terms proportional to F L and F 3 ; there is a Q2-dependence in structure functions to be integrated over. To take the quark-parton model limit we replace in eq. (7) the structure functions by the singlet (s) and non singlet (ns) scaling quark distributions: F212x ~qS(x),F3/2 ~ qnS(x) a n d F L ~ 0 from the CallanGross relation. This gives in the valence-quark approx. imation the cross section (g2 = e/2x/~ sin 0w) -
(5)
X 54(k 1 + k 2 - k'1 - k~ - k),
+ F 3 (1)F3 (2)x I x 2
(4a)
and the four-momenta (light-like for mp ~ O)
k i = x i p i,
+ F2(1)F2(2 ) [8k 1 "k 2 k i • k~ + 8k 1 • k'2 k2"k' 1 ]
Volume 167B, number 4
PHYSICS LETTERS
O = f d x 1 dx 2 q(xl)q(x2)(2s')-I
q2 = - ( 1 - ~') (1 - cos 02)g/2 ,
, , 2 _ m2)-2 X f d R 3 A 2 g 4 6 4 k l . k 2 kl.k2(ql
(13 cont'd)
the invariant phase space integral is proportional to
X (q2 _ m 2 ) - 2
(8)
3(ql:- row) 2 - 2 ( q 22 - m 2 ) - 2
f
cc f
in agreement with Cahn and Dawson (ref. [1]). In eq. (1) we effectively integrate Q12 and Q22 over
the range 0
20 February 1986
j
< e - m:.. The minimal Qi or
can be increased by fixing the transverse m o m e n t u m k T of the Higgs boson, but for the total cross section of eq. (1) we must take into account the elastic scattering configurations in addition to the double inelastic configuration discussed above. The inelastic threshold condition
Q2 >>.[xi/(1 _ xi)][( m + mTr)2 _ m 2 ]
(9)
separates the total pp ~ WW ~ H contribution into four terms: Otot = OEL_E L + OEL_INEL + OlNEL_EL + OlNEL_INEL, (10) corresponding to each of the two pW vertices being either elastic or inelastic. The structure functions F/ (i = 1,2, 3) in eq. (7) are replaced for the elastic vertex by the proton form factors Ge(Q2), Gm(Q 2) and Ga(Q2 ) according to F 1 ~ 6(1 - x)[Gm(Q2) 2 + Ga(Q2) 2 ]/2, F L -" 8(1 - x ) [ G e ( Q 2 ) 2 - Gm(Q2) 2 ]/(1 + Q2/4m2),
d cos01
d cos02 (14)
(1 -- cos01 + el )2 (1 - cos02 +e2 )2 '
where e I = 2m2W/S(1 - n),
e 2 = 2m2/s(1 - ~').
Neglecting Q2-dependence of the inelastic structure functions the above integrals give factors 1
f
d cos Oi(1 - cos 0i + el)-2 ~-- lie i cc 1/m2w
in the forward scattering approximation. The l/m2w dependence reflects the constancy of the W-propagator ( 2 1/m 4) over the range 0 < Q2 < m 2 or in the forward cone 0 i < x/~r However, for the elastic scattering case the Q2-dependence of the proton form factor is much stronger than the scaling violations for the inelastic case and cannot be neglected. With the parametrizations of eq. (12) we have in the forward scattering approximation instead of eq. (15) 1 2 2 -4 f d cos0i(1 - cos0i + ei)-2 (1 + Qi/my) -1
(mv/mw)2/e i • F 2 ~ 2 F 1 + F L, F 3 ~ 6(1 -- x)2Ga(Q2)Gm(Q2).
(11)
Typical form factor parametrizations are Gm(Q 2) = Up Go(Q2) =/ap/(l + Q2/m2)2, Ga(Q2 ) = 1.23/(1 + Q2/m2)2,
(12)
with m v = 0.84 GeV]e 2, m a = 0.89 GeV/e 2 and #p = 2.79. To assess the relative importance of the elastic and inelastic configurations we will here discuss results of calculations in the effective W approximation. Defining cos 0 i as the polar scattering angle of k; in CMS(k 1 + k2) ql2 = --(1 -- '7)(1 -- cos 01)s/2,
(13)
(15)
-1
(I 6)
The form factor cuts off the integral already at Q2 < m2v or to a cone 0 i < ~ mv/m w, and the elastic vertex is suppressed by a factor (mv/mw) 2 ~-- 10 -4 relative to the inelastic vertex. We recall that in the case of the process pp ~ 77 -" £+~- m w is effectively replaced by m(proton), and the elastic contribution is as important as the inelastic one. Having disposed of the elastic contribution we elaborate more on the inelastic case. Using the variables n and ~"introduced after eq. (6) the phase space integral gives in the effective W approximation
f~3(q21_ =
2 )- 2 (q22_ m2w)-2 mw
2-71r-3(,~m4) - l f u n
6(n
- m2/O.
(17)
473
Volume 167B, number 4
PHYSICS LETTERS
The coefficients of the structure functions in eq. (7) reduce to simple forms proportional to products of
fleL/F2(r/) = (ot2/2rr)4(1
-- r/)/*/,
flgL/FL(rD = (ot2/2rr)r/,
(18)
where a2 = g2/47r. The coefficient of the F3 F3-term vanishes for the forward-scattering kinematics. Finally the cross section of eq. (5) can be written in the factorized form
20 February 1986
Finally we present a numerical estimate for the inelastic versus elastic o(pp ~ WW ~ H) cross sections using the effective t¢ approximation and a simple model parametrization of F2(x ) "" 2x q S(x). Defining the luminosity functions dL r d-Tr(pp -~ WW) 1 dr' [", d L .
.-1
r LT. ~rTr'{'pp "+ F2 F2 'O --f7
o = f a x I dx 2 f d n d~ &Wr WL~H X { [FL(X 1 )/2x I ] fWL/Ft(rl)
where
+ [F2(Xl)/2x I ] fleL/F2(rl)}
r ~L r (pp ~ F 2 F 2 ) 1
X ([FL(XE)/2X2]flCL/FL(~) + [F 2 (XE)/2x2
]fWljf 2(~)},
= f -~- [F2(x)/2 ] [F2(r/x)/2], (19) dL 7" ~-~'r(F2 F2 -~WW)
where
b(m2 /~rl~) =A2mw4(Ir/4)~(1 - m 2 /~rl~)
(20)
is the Higgs production cross section by the fusion of longitudinally polarized W's. The contribution of transverse W's is suppressed by a factor (mw/mrl) 4. fWI_JF2 and fWT/Fhave the interpretation of WL L~. l-, distribution functions reside F 2 and F L. Alternatively we can write eq. (19) as
o = f d z I dz 2 WL(Z1)WL(g2) OWLWL~H ,
(21)
where we defined the W L distribution function inside a proton 1
1
WL(Z)=f dx f 0
d r / 8 ( z - xr/)
0
X { [F2(x)/2x]fWL/F 2(~1)+ [FL(X)/2x]fICL/F1(r/)}. (22) In perturbative QCD R = FL/F 2 = O(Ots) [12], but the vanishing offlCL/FL(rl) at 77 = 0 further suppresses the contribution of the FL-term in eq. (22). For the set 1 parametrization of ref. [13] with Q2 = m 2 we fred F L to contribute less than 1% of l~L(Z ) for z ~> 0.01. 474
(24a)
T
1
=fox [XfWL/F2(X)] [(7"/X)fWL/F2(7./X)] 7"
= (%/2r02. 16[(1 + r)ln(1/r) - 2(1 - r)],
(24b)
we can write the pp -+ WW -+ H cross section as 1
=
=
-- r T
r'
~r'
(pp -+ WW) ?r(r/r').
(25)
In our numerical calculation we take for the inelastic vertex the model parametrization Xqs(X) = 2(1 - x ) 2 , which gives a reasonable description of the x-dependence and normalization o f f 2. For the elastic vertex we assume F 2 + ½b(mv/mw)2 8(1 - x), where b = Ge(0)2 + Ga(0) 2 = 2.5 and the suppression factor was discussed before. Inserting the above parametrizations into eq. (24a) we can calculate order of magnitude estimates for the different terms of Otot in eq. (10). They are plotted for m H = 1 TeV/c 2 in fig. 3, where for normalization reasons we have not multiplied the elastic vertex by the W-propagator suppression factor. The double inelastic curve agrees in magnitude with previous estimates [9], and as expected the elastic terms are suppressed by factors of 10 -4
PHYSICS LETTERS
Volume 167B, number 4 lO
in eq. (10) with correct integrations over Q2 is needed and will be presented elsewhere.
(mwlmv)2x (Inet-eI+el-inel)
o- [pb]
ol
I would like to thank P. Aurenche and K. Kajantle for discussions. This research was supported by le Minist6re des Relations Ext6rieures (France) and by the Herman Rosenberg Foundation at Helsinki University.
References
/ii//t/ ! //~
mH= 1 TeV
/ 0 01 0
20 February 1986
I
I0
210
30 I
~0
[TeV] Fig. 3. The double inelastae versus the elastic contributions to a(pp ~ WW~ H) for mH = 1 TeV/c2 as a function of the center-of-mass energy (elastic-inelastic) or 10 - 8 (elastic-elastic) in the energy range x/~ = 1 0 - 4 0 TeV. We conclude that neither the elastic contributions nor the higher order QCD terms implicit in F L will change drastically the previous estimates on the gauge-boson fusion cross sections. This statement is verified above in the forward-scattering approximation and neglecting the Q2-dependence of structure functions. However, as an enhancement by a factor of two in the W and Z pair production rate may signal the onset of the new strong interactions [2], it is important to have a reliable absolute normalization of the WW and ZZ fusion cross sections. To this end a complete numerical analysis of the different terms
[1] R.N. Cahn and S. Dawson, Phys. Lett. 136B (1984) 196; 138B (1984) 464(E). [2] M.S. Chanowitz and M.K. GaiUard, Phys. Lett. 142B (1984) 85 ; preprint LBL-19470 (1985), to be published in Nucl. Phys. B. [3] E. Eichten, I. Hmchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579; L. Roberts, Phys. Rev. D32 (1985) 701; D.A. Dlcus and S S.D. Wfllenbrock, preprint UUTG12-85 (1985). [4] M. Veltman, Acta Physica Polomca B8 (1977) 475. [5] B.W. Lee, C. Quigg and M. Thaeker, Phys. Rev. D16 (1977) 1519. [6 ] M. Veltman, TASI Lectures in Elementary particle physics (1984), ed D.N. Williams(University of Michigan, Ann Arbor, MI, 1985) p. 86, M. Einhorn, TASI Lectures in Elementary particle physics (1984), ed. D.N. Williams(Universityof Michigan, Ann Arbor, MI, 1985) p. 113. [7] D.R.T. Jones and S.T. Petcov, Phys. Lett. 84B (1979) 440. [8] S. Dawson, Nucl. Phys. B242 (1985) 42; G.L. Kane, W.W. Repko and W.B. Rolrdck, Phys. Lett. 148B (1984) 367; J. Lindfors, Z. Phys. C28 (1985) 427. [9] R.N. Cahn, Nucl. Phys. B255 (1985) 341. [10] Min-ShihChen, 13. Muzinich, H. Terazawa and T.P. Cheng, Phys. Rev. D7 (1973) 3485; V.M. Budnev, I.F. Ginzburg, G.V. Meledin and V.G. Serbo, Phys. Rep. 15C (1975) 181 ; R. Moore, Z. Phys. C5 (1980) 351. [11 ] B. Schrempp and F. Schrempp, Nucl. Phys. B182 (1981) 393. [12] G. Altarelli, Phys. Rep. 81 (1982) 1. [13] D.W. Duke and J.F. Owens, Phys. Rev. D30 (1984) 49.
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