Looking for new gluon physics at the Tevatron

Physics Letters B 323 (1994) 401-407 North-Holland PHYSICS LETTERS B

Looking for new gluon physics at the Tevatron

P e t e r C h o 1

Lauritsen Laboratory. California Institute of Technology, Pasadena, CA 91125, USA

a n d

E l i z a b e t h H. S i m m o n s 2

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 12 August 1993; revised manuscript received 9 December 1993 Editor: H. Georgi

CDF data from the 1988-1989 Tevatron run are used to constrain the characteristic scale of physics beyond the Standard Model which couples to gluons. New gluonic physics would cause the inclusive jet cross section to deviate from that of pure QCD at high transverse jet energies. We find that the CDF data place a lower bound of 2.03 TeV at 95% CL on the characteristic scale of new gluon sector physics.

The inclusive je t cross section da ta f rom the 1988-1989 Fermi lab Tevatron run span seven orders o f magnitude and include the highest t ransverse j e t energy measurements repor ted to date [ 1 ]. These da ta provide a str ingent test o f quan tum chromodynamics and restrict possible new physics beyond the S tandard Model . In this letter, we investigate the l imits that these cross section measurements place upon non-s tandard gluon physics. In part icular , we consider the impac t upon the inclusive je t cross section o f nonrenormal izable gluonic opera tors which may appear in an effective Lagrangian describing strong interact ion physics at Tevatron energies. Similar analyses have been conducted in the past to constrain hypothet ical quark substructure [2 ]. It is impor tan t to emphasize however that the C D F da ta can be used to place l imits on qual i ta t ively different physics which couples to gluons rather than quarks. Our gluon sector findings therefore complement previous quark composi teness studies.

The nonrenormal izab le gluonic operators whose cross section effects we will examine could arise from several physical ly dis t inct sources. Fo r example, suppose there exist new heavy colored bosons or fermions beyond those in the S tandard Model . Such part icles would induce nonlocal interact ions among gluons through loop diagrams. The leading behavior o f these graphs can be reexpressed via an opera tor product expansion in terms o f local but nonrenormal izab le gluon operators . Mternat ive ly , we might speculate that gluons are bound states of some more fundamenta l preon consti tuents. Then as in the case o f composi te quarks, preon exchange could generate new gluon interact ions. In this letter, we will not specify the underlying physics whose effects are reproduced by nonrenormal izable opera tors at low energies. Instead, we s imply seek to place a l imit on its characteris t ic scale A.

To begin, we enumera te the nonrenormal izab le gluon operators whose scattering effects would be easiest to

Work supported in part by the US Department of Energy under DOE Grant no. DE-FG03-92-ER40701 and by a DuBridge Fellowship. Work supported in part by the National Science Foundation under grant no. PHY-9218167 and the Texas National Research Labo- ratory Commission under grant no. RGFY93-278B.

0370-2693/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDIO370-2693(94)OOO71-E

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observe: those of lowest dimension. The leading operators which preserve gauge invariance along with C, P and T are of dimension six [ 3 ]:

g r r:_~, g2'_J' /'2_). l /~ a ).1' 01 = -~-Llab~'-'a,,'-'b).'-'cU, 02 = ~ D Gu~,D).Ga , (1)

where

DU= OU-igGUaTa, Gg" = OUG~ - O"Gg +gfabcGgG'j.

All other dimension-6 gluon operators either vanish or reduce to combinations of the two in eq. ( 1 ). The contributions of O1 and 02 to parton scattering cross sections have been studied in refs. [ 3,4 ]. Operator

O~ mediates gluon-gluon as well as gluon-quark scattering and interferes with the pure QCD amplitudes for these processes. One might expect its impact to become pronounced at high energies due to the sizable gluon content of colliding hadrons at small parton momentum fractions. However, the helicity structure of the O1 amplitude for gg---,gg is orthogonal to that of pure QCD. The two amplitudes therefore do not interfere at O ( 1/ A2). Similarly, the interference terms in the differential cross sections for gg--'q(1 and the partonic processes related to it by crossing vanish in the limit of zero quark mass. Thus O~ affects inclusive jet cross sections starting only at O ( 1/h4). Operator 02, on the other hand, mediates quark-quark scattering because the classical equation of motion

D / , G U " = - g ~, ~/"T,,q (2) f lavors

relates its S-matrix elements to those of a color octet four-quark operator ~1

g 2

02 EOM 2.__~ flav~o~s (:tTuTaq)(~.i, UTaq ) . (3)

We will find that this operator consequently affects inclusive jet production at O (1/A 2). The strongest signals of new gluon physics thus ironically occur in non-gluonic channels.

To complete our operator basis so that it closes under renormalization, we include three more four-quark operators

ga 03 = 2~2 E._~ (¢;,.~'5:r.q) (c~'~'ST,,q),

g2 g2 _ _ _ (:ly,,ySq) (gl?)'ySq)

04= 2!A 2 fla,,o~Z ( ~ , , q ) ( ~ V q ) , O s - 2!A2n~,~o,~

(4)

along with O 1 and 02 into the effective Lagrangian

5

~e~=-~cv+ ~ CR~)O,(/~). i = 1

(5)

These operators are multiplied by dimensionless coefficients Ca whose values depend upon the nature of the underlying short-distance physics. In our work, we assume that the new physics couples to gluons rather than quarks; hence C3, C4 and C5 are zero at high energies (i.e. at scale A). To study the general effects of new gluon sector physics, we adopt the convention used in previous quark substructure studies and define A to be the scale at which the magnitude of the gluon operator coefficients C1 or C2 equals 4n [ 2,6 ].

To determine the operators' coefficients at energies probed by the Tevatron, we evolve their values down from the compositeness scale via the renormalization group. The running of all the dimension-6 operators in the effective Lagrangian is governed by the anomalous dimension matrix

#1 This identification should be understood as a relation among S-matrix elements and not as a true operator identity [ 5 ].

402


01 02 03 04 05 0117+2nf/3 0 0 0 0 \ 021 0 311/36-2nf/3 5/4 0 2/3

7=03 I \ 0 41/36 35/4-2nf/3 2/3 ~ ) - - O4~ 0 4/3 6 11 -2n f /3 O5 0 22/3 0 0 l l - 2 n f / 3

g~ , (6) 87r 2

where nf denotes the number of active quark flavors. The last four rows of 7 describe the mixing among the four- quark operators in our basis and were determined from a straightforward operator mixing computation. The entries in the first row on the other hand were extracted from the highly nontrivial anomalous dimension cal- culations of Narison and Tarrach [ 7 ] and Morozov [ 8 ]. Notice that the triple gluon field strength operator runs only into itself and does not mix with any of the other four-quark operators. Since O1 neither affects parton scattering at O ( 1/A 2) nor mixes into operators which do, we will not be able to place any limit upon its characteristic scale.

It is convenient to decompose the anomalous dimension matrix as

2 t 7 -

7= ~n2SDS-l +O(g4) , (7)

where the diagonal matrix D contains the eigenvalues 2i of 8~r27/g 2 while the corresponding eigenvectors form the columns of matrix S. The general solution to the coefficients' renormalization group equation can then be written as

/ . . x2j /2b . n _ l . T/OLs(P.) ~ c , ( ~ ) = Z ~ ),Jl=:-_-_-_-_-_-_-_-_v~; (s~)j~c~(A), (8)

j ,k k tXs t..'l ) /

where b = - 11/2 + nf/3 is the coefficient in the one-loop QCD beta function fl(g) = bg3/8n 2 [ 9 ] #2. I f we choose the coefficient values at a compositeness scale ofA = 2000 GeV to be C(A ) = 4n ( - 1, - 1, 0, 0, 0), then we find at a representative Tevatron energy scale of 300 GeV that these coefficients evolve into

C(g = 300 GeV) = 4n ( - 0.7324, - 0.8791, 0.0310, - 0.0003, 0.0160). (9)

The effects of QCD running are therefore not negligible and tend to suppress the gluon operator coefficients. We now turn to computing the inclusive jet cross section in p/~ collisions. Neglecting higher order multi-jet

events, we start with the two-jet differential cross section

da d3a (AB~2je t s ) = ~ 2XaXt, Pr[fa/A(X,,)fb/B(Xb)+ (A'-"B, a~b)] --~ (ab~cd), (10)

d r / 1 dr/2 d P T ,~bca

expressed in terms of the jets ' pseudorapidities 0/1, r/2), their common transverse momentum (PT), and the partons' momentum fractions (xa, xb). The partons' distribution functions fa/a(x,,) and fb/B(Xb) are folded together with the differential cross section da/d[ for the elementary scattering process ab-,cd. The product is then summed over all possible initial and final parton configurations. To convert the two-jet expression (10) into an inclusive single-jet cross section, we integrate over the pseudorapidity range of one jet, average the other over the pseudorapidity interval 0.1 ~< I~/I ~ 0.7 visible to the CDF detector, and multiply by two to count the contributions of both jets to the inclusive cross section:

I d~d~- £~ dr/, d~2d~id~/2dpT. -oo

~2 We choose the constant of integration which enters into the strong interaction fine structure constant to be a,(Mz) = 0.118 [ 10 ].

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The result may then be compared with the measurements reported by CDF. The lowest order QCD predictions for the patton cross sections

dtr(ab~cd)=--~-27(ab~cd) dt

(12)

are well documented in the literature [ 1 1,1 2 ]. They conventionally include initial state color averaging factors and are written in terms of the partonic invariants g, [ and , . We list below the formulae for 27(ab---, cd) in which the nonrenormalizable operators in our effective Lagrangian induce nonvanishing O ( 1/A 2) interference terms:

492-1-" 2 8 (C2+C3)g2+(C2-C3). 2 (~---~) 27(qq'-'qq')= 9 ~ + 9 [A 2 t-O . (13a)

4 • + " 2 8(C2+C3)"2+(C2-C3)[2+0(-~44), (13b) 27(qFl---,q'Ft') = 9 g--Y-- + 9 gA 2

4 ([email protected] g2.~_~2~__ 8 ~2 8C2 (g2.~_.2 g2_~/'2x~ 8C3 (g2_.2 ~2 ~2) 27(qq-~qq)=~\-V - + a2 ] ~ + 9 a 2 \ i + . )+Vh~\ i +

{8(C2+C3) 16(C4 + C s ) ) g 3 ( ~ ) ~7"~ 9A 2 I t . + o , (13c)

4(g2+.2 i'2+.2"~ .2 8C2(g2 +[2+"2)-8C3(g2 . z + p g . 2 ) = s 9A2k [

[8(C2+C3) 16(C, + C , ) ~ "3 " ( ~ ) ~ 9A 2 ) gt + O . (13d)

Here q' denotes a quark not identical in flavor to quark q. We have dropped O( 1/A 4) terms in these formulae since we are not keeping track of any dimension-8 gluon operators whose contributions to S(ab~cd) are of the same order.

We have also neglected all parton masses. At transverse jet energies of a few hundred GeV. this is a good approximation for all partons except the top quark whose mass we assume is rot= 1 40 GeV. Initial state top mass effects may safely be ignored as the proton's top content is negligible. However for processes involving t/'production, we use the heavy flavor QCD cross sections given in ref. [ 1 3 ] and incorporate O ( 1/A 2) interference corrections:

27(qgl~ti-)= 4 g2 + 98 (C2 +C3)"2+ (C2-C3)t2+2C2m2S +o(--~ 2 , (14a)

3 (m2--t)(m2--.) 1 m2(g--4m 2) 27(gg~tf)= 4 g2 -- 24 ( m 2 - f ) ( m 2 - , )

1 (m~- t ) (m2- , ) -2m2(m2+t ) 1 (mE-- f ) (mE-, ) -2m2(m2+,) + 6 ( m 2 - [ ) 2 + 6 ( m E - f ) 2

3 ( m 2 - ~ ( m E - , ) + m 2 ( , - t ) 3 (mE- f ) (m2- f )+m2t({ - , ) - 8 g(m2t-~ - 8 g(mEt-.)

9C1m2[ _ 2 t'2 "~" [ " "~ " 2 4 f"~'" 1 - 3mr g-~-) + O(~-~) (14b) 8 A 2 ~3+2rnt ~

Note that the coefficient of the operator O~ enters into eq. (14b). Since it appears in no other cross section

404


formula at O( 1/.42), the triple gluon field strength operator has no perceptible effect upon the inclusive cross section. We consequently ignore it from here on.

Combining eqs. ( 10 ) - (14 ) , we calculate the single-jet inclusive cross section as a function of transverse jet energy FT. We perform the computation using the leading order parton distribution functions of Morfin and Tung (MT set SL) [ 14] and the CTEQ collaboration (CTEQ set L) [ 15] as well as the next-to-leading order functions of Morfin and Tung (MT sets B 1 and S), Harriman, Martin, Roberts and Stifling (HMRS set B) [ 16 ] and the CTEQ collaboration (CTEQ sets MS). These and other parton distribution functions are available in the PAKPDF package [ 17 ].

Representative results obtained from the MT set SL structure function evaluated at the renormalization scale Q2= E2/2 are compared with the experimental data in fig. 1. The solid curve in the figure illustrates the predictions of pure QCD with no nonrenormalizable operator interactions. Following the example of the CDF analysis [ 1 ], we have multiplied the theoretical predictions by a normalization factor n to align them with the data. A fit for this constant performed over the region 80 ~<Ex ~< 160 GeV where effects from any nonrenormalizable operators are negligible yields n = 1.35 + 0.01. The resulting agreement between the shapes of the QCD and experimental cross section values is striking. There is however a slight suggestion of discrepancy at the highest ET values, where compositeness effects would first show up. We therefore plot in the same figure the differential cross sections obtained after setting C2 (,4) = - 4zt for A = 1 TeV and A = 2 TeV in our effective Lagranglan. The resulting dotted and dashed curves in fig. 1 qualitatively appear to fit the last few data points slightly better.

To be more quantitative, we perform a least squares fit for the compositeness scale `4. First we convert the differential cross sections in each CDF bin into numbers of events:

N=4200 n b - ' × ( A E x f d r / ~ ) n D , d2a ~ ' (15)

We then examine the transformed error bars and consider the statistics obeyed by the binned events. For each bin with transverse energy ET< 115 GeV, the statistical uncertainties are significantly greater than x / ~ and the systematic error bars are very large as well. We therefore exclude these points from our analysis. Bins in the intermediate energy range 115 < E.r < 300 GeV contain many events and follow Gaussian statistics. At the highest transverse energies, the data bins have fewer than 25 events and are described by Poisson statistics. We assign Gaussian error bars to these last points following ref. [ 18 ]. In addition to the statistical fluctuations associated with each bin, we need to consider the systematic errors. Normalizing the theoretical number of events by the

%

102

10 0

10-2

10-4

i0 8

' ~ ' 1 ' ' r . . . . I . . . . P . . . .

. . . . t . . . . / . . . . I . . . . I , , , 100 200 300 400 500

E T (GeV)

Fig. 1. Inclusive jet cross section plotted against transverse jet energy ET. The data points are the experimental measurements reported by CDF. The solid curve represents the predictions of pure QCD with no composite interactions, while the dotted and dashed curves illustrate the effect of the gluon operator 02 with C2 (A) = - 47t for A = 1 TeV and A = 2 TeV respectively. The theoretical results are based upon the leading order MT set SL distribution function evaluated at Q2 =E~./2.

405


factor n removes a large, correlated systematic uncertainty. We therefore subtract an averaged percentage uncertainty from all the bins' systematic error bars. The corrected errors are then small for bins with transverse energies above 80 GeV.

For our least squares fit, we adopt the X 2 function

22= ~ ,4,( V - ' ) u d j , (16) l d

th e x p where Ji = N i - N~ is the difference between the theoretically expected and experimentally measured number of events in the ith bin and V denotes the covariance matrix. The statistical and systematic uncertainties for each bin are summed in quadrature to form the diagonal entries a~ = a~ (stat) + a~ (syst) in V, while the off- diagonal elements which take into account residual bin-to-bin correlations are given by a~ = ai (syst)ai(syst) [ 19 ]. We illustrate in fig. 2 the dependence o f z 2 (22 d.o.f. ) upon A -2 for the MT set SL structure function ~3. Various limits may readily be extracted from the clean parabola appearing in this plot. For instance, we find

A - 2 = 0 . 1 1 3 +_0.080 TeV -2 (17)

by locating the 2~in + 1 points on the parabola. This translates into the asymmetrical 1 a interval

+ 2 5 9 A = 2.98_o:69 TeV (18)

for the scale of new gluon physics. Alternatively, we may quote the more conservative lower bound

A > 2.03 TeV at 95% C L . (19)

Analogous results from the other leading and next-to-leading order distribution functions evaluated at the renormalization scales Q2= E ZT/2 and Q 2 = E 2 are displayed in table 1. We see from the 95% lower limit entries in the last column of the table that the bound in (19) represents a conservative estimate for A.

In summary, the 1988-1989 CDF inclusive jet data provide a strong new limit on non-standard physics that couples to gluons. Our gluon substructure bound compares favorably with the lower limit A q ~ k > 1.4 TeV that the same data provide for the quark compositeness scale [ 1 ]. It should soon be possible to substantially improve upon both these results. The current 1992-1993 Tevatron run will collect a data sample five times larger than the one used in this analysis. Cross sections at higher transverse jet energies will be probed, and sensitivity to

~3 Negative values for A -2 do not imply imaginary values for A. Instead the sign of A -2 is simply absorbed into the dimensionless coefficient of the operator 02.

30

25

2O

15

lo , , J . . . . I . . . . I . . . . I - 0 . 2 0 0,2 0.4

A a (TeV z)

Fig. 2. X 2 for 22 degrees of freedom plotted as a function of A -2.

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Table 1

PHYSICS LETTERS B 17 March 1994

Distribution function Q 2 Normalization factor X ~ .4 - 2 (TeV- 2) A95 (TeV)

MT E2/2 1.35 + 0.01 11.76 0.113 + 0.080 2.03 set SL E 2 1.61 + 0.01 12.17 0.081 +0.067 2.29 MT E2/2 1.17+0.01 11.10 0.104+0.085 2.03 set S E 2 1.39+0.01 11.34 0.081 +0.071 2.25 MT E2/2 1.15+0.01 11.15 0.104+0.085 2.03 set B1 E 2 1.37+0.010 11.45 0.081 +0.072 2.24 HMRS E2/2 1.08 + 0.01 11.52 0.081 _+ 0.092 2.08 set B E 2 1.28+0.01 11.48 0.077+0.077 2.22 CTEQ E2/2 1.25_+0.01 15.82 -0.131 _+0.070 2.02 set L E~ 1.48 _+0.01 15.97 -0.117-+0.067 2.10 CTEQ E2/2 1.21 _+0,01 12.18 0.014_+ 0.084 2.45 set MS E 2 1.43 _+ 0.01 12.22 0.018_+ 0.071 2.65

a n y n o n r e n o r m a l i z a b l e o p e r a t o r s in t he e f fec t ive L a g r a n g i a n wil l b e e n h a n c e d . W e t h e r e f o r e l ook f o r w a r d to

u p d a t i n g o u r f i n d i n g s as n e w d a t a c o m e s f o r t h f r o m B a t a v i a .

We t h a n k S t eve B e h r e n d s , J o h n H u t h , F r a n k P o r t e r a n d M a r k Wise for he lp fu l d i scuss ions . We are a lso gra te -

ful to t he A s p e n C e n t e r for Phys ics , w h e r e t h i s w o r k was c o m p l e t e d , for i ts w a r m hosp i ta l i ty .

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Looking for new gluon physics at the Tevatron