Composite leptoquarks

Volume 167B, number 3

PHYSICS LETTERS

13 February 1986

COMPOSITE LEPTOQUARKS ' Jos6 W U D K A Centerfor TheoreticalPhysics, Laboratoryfor Nuclear Science and Department of Pl~vsics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 22 July 1985; revised manuscript received 6 November 1985

We study leptoquarks which may exist if quarks and leptons are composite, and which are not Goldstone bosons. A discussion of their production and detection at HERA is given, emphasizing their effect on polarized electron scattering where striking signatures are to be found.

Among the various proposed theories of elementary particles that describe physics at and beyond the electroweak scale (AFermi ~ 300 GeV), the composite models predict new exciting phenomena that may be uncovered by the next generation of particle accelerators such as HERA [ 1 ]. One outstanding characteristic of these models is the richness of their spectrum which contains many new particles such as excited vector bosons and leptoquarks (particles carrying lepton and baryon number). Leptoquarks appear in various models as Goldstone bosons [2], however we wish to emphasize that this characterization is not true in general (see below). This means that leptoquarks need not couple derivatively to the fermion fields, thus opening a new range of possibilities for their production and decay. For definiteness we shall study leptoquarks in the context of the A b b o t t - F a r h i ( A F ) m o d e l [3] which we briefly describe. The lagrangian for the AF model is identical with the one in the Weinberg-Salam model, with the same quantum numbers assigned to the various fields. However the parameters in the potential for the scalars are such that no spontaneous symmetry breaking occurs, and the SU(2)L gauge symmetry is confining. It follows that all physical particles must be SU(2)L singlets; thus, for example, the left-handed fer:': This work is supported in part through funds provided by the US Department of Energy (DOE) under contract DEAC02-76ER03069. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

mions are bound states of a left-handed preonic fermion and the fundamental scalar. In contrast, the righthanded fermions are point-like. In this model all lowenergy weak phenomena are produced by residual interactions, nonetheless under certain conditions it is indistinguishable from the standard model below the Fermi scale [4]. Let us call the left-handed preons ~b~ (where a is a flavor index which runs from 1 to 12 for 3 generations), which transform according to the (0, 1/2) representation of the Lorentz group, and belongs to a 2 of SU(2)L. Leptoquarks in the AF model are bound states of two ffL, or of f L and ~bL. An SU(2)L singlet made out of two ~bL belongs to the (0,0) or to the (0,1) representations of the Lorentz group; the SU(2)L singlet ffL--~-t state belongs to the (1/2, I/2) representation. (We have not included bound state wavefunctions containing derivatives of ~bL since we expect the corresponding coupling to the physical fermions to be down by a factor of m/AFermi, where m is a typical fermion mass, for each derivative.) We define S ab, K~,b, and Vgb as the interpolating fields for the (0,0), (0,1) and (1/2, 1/2) states respectively. Due to Fermi-Dirac statistics S ab is antisymmetric in its flavor indices while K~ b is symmetric. Furthermore K~ b is a self-dual antisymmetric Lorentz two tensor. The S, V, and K mulfiplets contain dileptons, diquarks as well as leptoquarks; their quantum numbers are summarized in table 1. It is clear that there is no a priori reason for these multiplets to couple derivatively to the fermions. 337

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Table 1 Electric charge and SU (3)Color representations corresponding to the members of the S, V, and K multiplets. The representations of the Lorentz group are indicated in the first column. Dilepton

Leptoquark

Diquark

-1,0 0,0 -1,0

-1/3,3 2/3,3 -1/3,3

1/3,3 0,1~98 1/3,6

S(0,0) V(1/2,1/2) K(0,1)

13 February 1986

/~ - ~ u r + - L a ' .0 = ,~7 1 3 a . 2 .em I u - 2 V ~ [ L 7u~r L - S i n Owl u ] ,

to obtain a total effective four-fermion lagrangian whose coefficients we can compare with experiment [6]. This way we obtain bounds o n M / X andM/v. The most stringent constraints come from neutrino-nucleon scattering and give ,1 M > 275X (GeV),

a and Kuv ab The low-energy interactions of S ab, VUub, with the physical left-handed fermions are described by the lagrangians (we use the conventions of Itzykson and Zuber [5]) 1

"P

£S = -2-X(SabL

aT

2

b

Cr L + h.c.),

£V = ~1-.¢v#btFa,,,~lb ~,a ~ - ~ +h'c'), - 1

/av~" aT

£K-~O(Kab

L

2

b

Cr ouvL + h . c . ) ,

(1)

where L a are the fields for the physical left-handed doublets under the global SU(2) symmetry of the model [4], C is the charge conjugation matrix, and r i are the usual Pauli matrices. Note that S ab and K~ b couple the up-component of one doublet with the down-component of the other; this and the antisymmerry in its flavor indices implies that S ab does not contribute to processes like e+e - -+/x+/a- or like the K 0L - K 0S mass difference. The coupling constants are undetermined, however we shall argue below that X and v are of order one. From (1) we can obtain the current-current effective lagrangians produced by the exchange of the three types of leptoquarks. The final results are, after a Fierz transformation, £~ff = _(~218MZ)[([a.),/a.cLa) 2 _ (/a3,uLa)2 ] , £ ~ f = _(v2/SM 2) [(£a3,uiLa)2 + (~TlaLa)2] ,

£~f= 0,

(2)

where M and M V are the masses of S and V respectively and where we have ignored a mixing of V with the photon. We can now use (2) together with the usual result from W and Z exchange, elf = --fa 2/8Mw)[S 2 .+ "J- + ~(j0)2] , £WZ (3) 338

(3cont'd)

MV > 415v (GeV).

(4)

In the AF model the W and the Z are bound states of two fundamental scalars in a P-wave. The interactions are assumed to be such that g, which measures the strength of the residual low-energy interactions, turns out to be ~ 0 . 7 ; therefore it is natural to expect "~ 1, v ~ 1. The reader should be aware that a common practice in the literature [8,9] is to set X2/327r 1 [which would increase the bounds in (4) by a factor of 10], however we believe this is not appropriate in the present case. Below we shall restrict ourselves to the study of scalar leptoquarks, the effects of V and K will be presented in a future publication. Using (1) we can calculate the production cross section for scalar leptoquarks. The main production reaction of S ab is through L a - L b fusion. For electronquark fusion in an electron-proton reaction, assuming the quark and electron are massless we obtain

o hard = (rr)t2x/4M 2) 6 (x - M2/s)

- aox~ (x

-

M2/s),

(5)

where x/s is the CM energy of the e ~ c t r o n - p r o t o n system and x is the fraction of the momentum of the proton carried by the quark. The measured cross section is obtained from (5) by averaging with the quark distribution functions at Q2 = M 2. For the reaction where an up-quark and an electron create a leptoquark, using the two sets of distribution functions of ref. [9], we obtain the cross section as a function of M presented in fig. 1, where we have taken x/S-= 314 GeV, as is expected for HERA, and X = 0.6. The shaded region in fig. 1 reflects the uncertainty in the quark distribution functions (see the discussion at the end of .1 A complete study of this type of experimental constraints will be given in ref. [7].

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PHYSICS LETTERS

t.2

I

I

I

the paper). Note that the same leptoquark can be created through ue-d fusion. We now study the modifications due to the presence of S in the reactions

1200

'

:0.6 Luminocity= '1 ~ t.0

1000

0.8

800

06

6oo

0.4

400

0.2

200

:"

b

I

190

240

i

290

M (GeV) !

Fig. 1. Production cross section for scalar leptoquarks as a function of their mass for x/s-= 314 GeV, X = 0.6.

-~ e

,,

.~ s u

e

A e

~,

v

d

,,

e

r

"k tt

e-+u~e-+u,

(6a)

e - + u ~ Pe + d ,

(6b)

e- + d~e-

(6c)

+ d,

which will occur in an e - p collider such as HERA. For the first reaction we consider the cross section for polarized electrons due to S, Z and photon exchanges (see fig. 2a), for the second reaction S and W - exchanges will be included (see fig. 2b), finally in reaction (6c) only Z and photon exchanges contribute. The reason we include (6c), even though S does not contribute to it, is that we will be interested in final states of the type lepton + (quark-jet), therefore both (6a) and (6c) must be included when the final lepton is an electron. Aside from (6c), in this model there are no other significant background reactions for this type of final state. For M < 300 GeV leptoquarks will be produced at HERA; therefore as S appears in the s-channel of reactions (6), resonance effects arise. It follows that we must include a decay width in the expression for the S propagator DS. To lowest order in X this is done by including the imaginary part of the diagrams in fig. 2c. This gives DS = i ( p 2 - M 2 - i p 2 F 2 / M ) -1 , F2 =- ~.2M/87r,

(7)

where 1~2 is the two-body total decay width of S when the fermions are assumed massless. Since X is small and the phase space for two body decays is much larger than that for n body decays (n > 2), this expression for the propagator is expected to be a good approximation to the exact one at the energies we are interested in. The hard cross sections for left- and right-polarized electrons may be expressed in the following way

d

/~ e

13 February 1986

~

d a V d / d ~ = (/'/4/r) 2 {a 2 + [(s + t)/s] 2132} , .....

+

..... 0 ....

+

""0""'(::3"-

+

etc.

daLhard/d~2 = (l,/4r02 {~,2 + ~-2 + [(s + t)/s] 262},

(8)

where Fig. 2. (a) Diagrams contributing to the reaction (6a). Co) Diagrams contributing to the reaction (6b). (c) One-loop corrections to the S propagator.

=- -q/t + (gv + gA)(g~¢ + g~A)/(t -- # 2 ) , t3 =- --q/t + (gv + gA)(g~¢ --gtA)/(t --/a2),

(9) 339

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13 February 1986

Table 2 Values of the parameters in (8) for the two reactions (5). e is the charge of the proton and 0 w the Weinberg angle,Mz,M w are respectively the Z and W masses. Reaction

q

e + u -o e + u e + u ~ ve + d e + d --* e + d

gv

2e2/3 0 -e2/3

gA

e Cot 20 w g/2 e Cot 20 w

- e Csc 20 w -g/2 - e Csc 20 w

7 = - q / t + ( g v - gA)(g~ r + g'A)/(t - / 2 2 ) -- K (s - M 2 ) / [ ( s - M2) 2 + ()t2s/gn') 2] ,

-

-q/t + ( g v

- gA)(g~¢

g'A)/(t--/22),

-

-- K Qt2s/Slr)/[(s - M 2 ) 2 + (~k2s/87r) 2] , l ' -= l - [(s - 412)/4ls] t , t = - s [1 + (s/4l 2) c o t 2 ( 0 / 2 ) ] - 1 .

(9 c o n t ' d )

The values o f q , g v , g~r, gA, g'A,/2 and K for each o f the reactions (6) are presented in table 2 where we ass u m e d the standard m o d e l couplings for the W, Z and

~

t0 -7

'

I

"'t-

i

gA

(gA - 2gv)/3 g/2 (gv - 2gA)/3

-gA 0 gA

#

K

MZ MW MZ

X2/2 h2/2 0

p h o t o n . In these formulas 0 is the angle b e t w e e n the i n c o m i n g electron and the outgoing lepton, I and l' are the m o m e n t a o f the initial and final leptons respectively. In (8) and (9), the Mandelstam variables s, t, u refer to the q u a r k - e l e c t r o n system. To obtain the m e a s u r e d cross section, the hard cross section m u s t be averaged using the quark distribution functions. We shall use, for purposes o f comparison, two different distribution functions as presented in ref. [9] with Q2 = - t . The energy and m o m e n t u m o f the initial electron and p r o t o n are set to 30 G e V and 820 GeV respectively, as these will be the operating conditions at H E R A . The plots o f the unpolarized cross section for three values o f k and f o r M = 300 GeV are presented in fig. 3a for reaction e + p ~ e +

M3 =00 I

t

g~v

tO -e

I

GeV

I

I

I

b

I

M= 300

I

i

GeV

X=O b~

bc~

X=O

|0

'

i 30

I 60

~

I 90

I

I t20

0 (Degrees)

~

1 t50

r t80

t0"

30

60

90

120

150

180

0 (Degrees)

Fig. 3. (a) Differential cross section for the reactions (6a) + (6c) for h = 0, 0.6, 1.2. (b) Differential cross section for the reaction (6b) for h = 0, 0.6, 1.2. (l = 30 GeV and x/~ = 314 GeV.)

340

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(quark-jet) [(6a) + (6c)] and in fig. 3b for reaction (6b). The uncertainty in the graphs is a reflection of the uncertainty in the quark distribution functions and will be discussed below. Another quantity of interest is the left-right asymmetry defined by _ daR -- doL doR + doL

2xtO-r

e-+ p - e ' + jet A

E

t0-7

b

I

J

I

I

I

M = 300

=

1

.

2

~

I 300

~. ~..~

I

T T T "v -e q , , h ~ r ."Z . . . . . . . .

I 400

I

I 500

I

I 600

M (GeV)

a

3xtO-e

e'*p-v

.Q

+jet

X:O.6... ~ ~ -- --

E

~

b lO-S

-2.4 14 GeV

- X=t 2_.....~ I

7xt0-9

..

I

I

300

I

I

400

I

500

I

600

M (GeV)

t .00 ~ = 0 "

=~,

. . . . . . . . .

0.50

~ E

0.00

o°

. . . . . .

f=,

i=.

.*o

. . . . .

° . .

. . . . . . .

°°

~ ~ ~

),=0,6 s / S ' "

-0.50

-I.00

GeV

X

X=0.6 . . . . . .

4 x t0 -8

I

.

X= 0 • ...........

~>' ,¢E

t.O0

Ot4 GeV

(10)

where da are the averaged cross sections. The variation of _~ with 0 for reactions (6a) + (6c) is presented in fig. 4 where the parameters are fLxed as above. For (6b) _q~ is identically minus one since both W and S couple only to left-handed fermions. To study the M dependence we have plotted in fig. 5 the variation of the total cross sections and the asymmetry with M for three values of X. The cuts are conservative estimates based on the performance of the detectors at HERA [1 ], except for the lower cut in the rapidity y for (6a) + (6c), which is designed to exclude the effect of the photon. From figs. 3 - 5 it is quite clear that scalar leptoquarks may have dramatic effects in both the cross section and the asymmetry. Nevertheless, one must be

13 February 1986

~'=I'21"'~= 300

i

t

400

M (GeV)

t

0==t70. I

500

600 c

Fig. 5. (a) Total cross section for the reactions (6a)+ (6c) as a 0.50

function of M for h = 0, 0.6, 1.2. (b) Total cross section for the reaction (6b) as a function of M for X = 0, 0.6, 1.2. (l = 30 GeV and ,v/S-= 314 GeV.) (c) Left-right asymmetry for (6a)+ (6c) at 0 = 170 ° as a function of M for X = 0, 0.6, 1.2.

X=0

E

-0.50

- t.00

I

30

r

I

60

~

I

90

,

I

120

~

I

150

,

t80

8 (Degrees)

Fig. 4. L e f t - r i g h t a s y m m e t r y for the reactions (6a)+ (6c) for X = 0, 0.6, 1.2. (l = 30 G e V and x/~-= 314 GeV.)

aware of the following caveats. In the first place we have not studied rigorously the effects o f a variation AQCD on the cross sections, we have only estimated such a variation by comparing the results of using the two distribution functions presented in ref. [8] which take AQCD = 200 and 290 GeV. Another related problem is the value of Q2 taken for such functions, it is known [9,10] that the choice is ambiguous to the order in QCD at which the distribution functions are evaluated, this might give rise to an error of up to 20%. In the second place, for reaction (6b) there is the problem of reconstructing the neutrino's trajectory given the observed jet, this cannot be done with less than 341

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about 15% of error [11 ]. Finally, the measurement of ~q is usually never done with a precision higher than ~25%. All these problems restrict the range of observable M at HERA to less than about 600 GeV. However, i f M is of the order of 300 GeV, the effect of S should be quite striking. I am very grateful for the help I received from M. Claudson, E. Farhi, R. Jaffe, C. Korpa, R. Peccei and Z. Ryzak during the elaboration of this work. I would also like to thank P. Haridas for interesting discussions.

References [1] Proc. Workshop on Experimentation at HERA (Amsterdam, 1983) DESY-HERA 83/20 (1983). [2] B. Schrempp and F. Schrempp, Phys. Lett. 153B (1985) 101;

[3] [4] [5] [6]

[7] [8] [9] [10]

[11]

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13 February 1986

S. Chadha, J. Proudfoot and D.H. Saxon, Rutherford Laboratory preprint RL-83-071-mc (Aug. 1983); R.J. Cashmore, Oxford University preprint OXFORDNP77/83 (1983). L.F. Abbott and E. Farhi, Phys. Lett. 101B (1981) 69; Nucl. Phys. B189 (1981) 547. M. Claudson, E. Farhi and R. Jaffe, to be published. C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). P.Q. Hung and J.J. Sakurai, Annu. Rev. Nucl. Part. Sci. 31 (1981) 375, J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys. 53 (1981) 211. C. Korpa and Z. Ryzak, in preparation. G. Altarelli, B. Mele and R. Riickl, CERN preprint CERN-TH.3932/84. E. Eichten, I. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579. I. Hinchliffe, Particles and fields-1981 : Testing the standard model, eds. C.A. Heusch and W.T. Kirk (American Institute of Physics, New York, 1982) pp. 173. J.P. Berge et al., Nucl. Phys. B184 (1981) 13.