Nuclear Physics B (Proc. Suppl.) 17 (1990) 599-602North-Holland

t-EXPANSION CALCULATION OF THE TENSOR GLUEBALL IN SU(2)

David HORN and Yael SHADMI

School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, ISRAEL

We use the t-expansion to calculate the mass of the 2++ glueball in the SU(2) lattice gauge theory toorder H7 . The asymptotic behaviour of the t-expansion series is investigated by means of two methods :the exponential fit and Padi approximants . The results obtained from the two methods are in agreementwith each other indicating that the tensor mass is bigger than the scalar one by about 20%.

1 . INTRODUCTIONWe investigate the tensor glueball in SU(2) by

using the t-expansion method for the Hamiltonianformulation of this theory. This method proved itsusefulness in the investigation of other glueballs inthe past, including the scalar glueball in SU(2) 1and SU(3) 2 and various odd charge-conjugationstates in SU(3) 3.

After a short formulation of the problem andthe method we discuss the approximation schemeswhich we employ. All indicate that the tensor massis heavier than the scalar one.

2 . THE METHODWe use the Kogut-Susskind Hamiltonian

H=2+xE(2-trUp)]P

where Ér is the electric color-field defined on thelink f and tr UP is the magnetic color-field of a pla-quette p. g is the coupling and x = 4/g4 . We willalso use the variable y = Vx = 2/g2 .

For calculational purposes it is useful to workwith

H=JÉf-xJ: trUP

and to start the t-expansion procedure from thestrong-coupling vacuum, which is the state obeying

Ér10) =0 .

This means that one calculates the energy-function

E(t,g2)

_(0 IH e-éH 10)(01 e-iy 10)

0920-5632/90/$3.50 © Elsevier Science Publishers B.V .North-Holland

which in the limit t --+ oo turns into the correctvacuum-energy. Expanding this energy-function inpowersof t one performs a cluster-expansion whosecoefficients are connected matrix-elements ofH:

E(t,g2) = 2

(n!)" w+l)c+ Z Nn gP "

NP is the number of plaquettes, which is taken toinfinity.

From the energy-function one can deduce a t-expansion for the scalar mass, since this state lies inthe same sector of Hilbert space as the vacuuml :

Ms(t) - - 1! -02 &

In[-ôôtt) ]

The tensor glueball requires a separate calcualtion,starting from a trial-state which has the requiredquantum numbers . We choose the one- plaque-tte state which belongs to the basis of the E-representation of the cubic symmetry group :

IT) = E(trU�z - trU=s) 10)

We have performed the calculation to order H7(t6) . The difference of the energy-functions in thetensor sector and the vacuum sector leads to thefollowing series for the mass of the tensor glueball :

MT = 2[3 + tX2 - 1 t2X2 - 1 t3(16x2 - X4)+

+ 24t4(143x2 - 2.5X4)-

- 11 -t5(856.75X2 - 32.125x4 - 41x6)+

+ 7~0t 6(4472.75X2 + 379.375X4 - 822.5x6)]

This can be compared with the known results for

the scalar 1,4:

sMS = 2[3 + tXa - 2taXS-

As can be seen, the three first terms in the tensorseries and scalar series are identical. This leadsto the degeneracy of the two masses in the strongcoupling region, manifest in the various fits whichwe present.

3. ANALYSISTo extract physical information from the t-

series it proved useful 1 to apply D-Padé ap-proximants to the scaling ratio of M2lo . Thestring tension Q is calculated separately by startingwith an infinite string state in the strongcouplinglimit. This has led to a very smooth result for

Msl,lb, 3.5 in the region 1 < y < -2 whichis our scaling window. The same analysis for thetensor-state leads to singularities in this region ofy. They are evident in Fig. 1 where we displayMT/MS as obtained from the ratio of the D-Padé

T® S

1.20

1 .15

1 .10

1 .05

1 .00

- 61t3(25X2 + 17X4) + 24t4(215X2 -F 110X4)-

120t5(1250.5Xa -I- 447x4 - 626X6)+

+71 ts(6306

.5X2 +1651X4 -10430x6)]

0 0.5 1 1.5 2 2.5 3y=2/g

2

Fig. 1

D. Horn, Y Shadmi/t-Expansion calculation

approximants 0/5 and 1/4 to the scaling ratios ofthe tensor and the scalar . This figure seems toindicate a mass-ratio in between 1.1 and 1.2 in thecross-over region .

It would clearly be preferable to have a reliableapproximation method which does not lead to asingular behavior in y for the mass of the tensorstate. This turns out to be the case in the expfitmethod which has been developed in the past fewyears 5,6 . To introduce it let us note first that theenergy-function can be represented by the logarith-mic derivative of the norm-function

Z(t) = (101 e-Ht

p)

The main idea of the expfit method is to repre-sent this matrix-element by a sum of decreasingexponentials. Using a simple quantum-mechanicalinterpretation one is led to the fit

PZ(t) a;e-E;t

with the conditions that the ai are real and posi-tive and sum up to 1 and the e; are real . The latterrepresent the eigenvalues of a system with p levelswhich leads to the same energy-function, given thefinite-order to which it was calculated . The p lev-els can account for an expansion of E(t) to orderH2P'i. In a finite-volume problem this is equiva-lent to the method of Bessis and Villani7 who haveproved that the e; form upper-Emits on the energyeigenvalues of the original problem.

The t-expansion is formulated in a fashionwhich is directly applicable to the infinite volumelimit . It is however a linked cluster expansion whichis applied to a mean-field state. This implies thatin a finite calculaticae, in which the largest diagramhas volume St, one obtains the same energy-densityfor any finite lattice (with periodic boundary con-ditions) whose volume V > 11 . One possibility istherefore to choose V = 11 and apply to it the exp-fit procedure. It is then guaranteed that all condi-tions on the a; will be obeyed . However the fit maybe of poor quality because we replace a very largeHilbert space by a small number of energy levels.The alternative is to imagine that the volume Vis constructed out of small non-interacting cells of

size v, leading to the expfit

If v < l it is not guaranteed that the conditions onthe coefficients and energy-levels can be fulfilled . Ifthey are, it means we can project our problem onanother quantum system which can be used as anapproximate representation . It involves now (V/v)Plevels and has therefore a better chance for repre-senting correctly the spectrum, although its lowestenergy may no longer be an upper-bound on thetrue vacuum . It follows therefore that v can beregarded as a parameter which can be varied toobtain the optimal fit . The lowest energy valuesfollow usually from the lowest possible v values6 .

Using these criteria for the vacuum en trgy-density, calculated to order H', we find that thesmallest v is 2.7 plaquettes. The results for theenergy-density (per plaquette) are shown in Fig.2 . The dotted line indicates the expected value atg = 0 . Clearly the approximation breaks down be-fore y = 3 however it does display a transition fromstrong to weak-coupling behavior with the crossoveroccuring in the region 1 < y < 1.5 .

2.5E2.0

0.5

0 .0

PZ(t) ;:t: JE aie-git]/" .

:=1

0 0.5 1 1.5 2 2.5y=2/g"

D. Horn, Y. S6admi /t-Expansion calculation

Fig . 2Applying such an analysis to the tensor mass

we have to make a choice of V since we are nowinterested in an intensive quantity which is obtained

by subtracting the vacuum energy from that ofthe ground-state of the tensor sector. Sine thelargest diagram in this calculation is defined on11 = 24 plaquettes, we have chosen for these fitsV = 25 v = 2.5. This means that the ground-state energy of the tensor sector is derived forET(V) = VE .I. MT through the expfit method,and the approximated MT is obtained by subtract-ing from it the approximated value for the vacuumenergy E�a. = Vt. The result is a curve whichgoes through a minimum of 3.703 at y ;t; 1 . Thiscan be viewed also as an approximation scheme "inthe background of E..," . As such it can be re-peated also for the scalar mass . Both are shownin Fig. 3, where they are also compared to Msas derived from ez - et in the expfit of the vac-uum sector. The latter is the lowest curve. It ispresumably closest to the correct physical curvewhich should display the exponential decrease ofasymptotic- freedom beyond the cross-over region .Clearly the increase observed in the mass curves isan artifact of our fit and our short series. Trying toextract physical conclusions we compare the mini-mum of the MT curve with that of the MS whichwas obtained in the same procedure (it is 3.02, ob-tained at y =1.1). The two are presented as dottedlines on Fig. 3, and correspond to a ratio of 1.23 .

10

8

8

4

2

0

Fig . 3

601

602

This result should be regarded as a qualitative indi-cation which agrees with the one derived from Fig.1.

4. DISCUSSIONThe t-expansion technique involves computa-

tion of diagrams and fitting the resulting series byan appropriate approximation scheme . Our experi-ence shows that on both accounts the calculationof the tensor glueball is much more complicatedthan the scalar one. The D-Padé approximants ofthe tensor-mass series (to order ts) display singu-larities in y which did not appear in the case ofthe scalar . These results indicate that the tensor isheavier than the scalar by 10 to 20 % in the cross-over region . The other fitting procedure which weapply, the expfit method, does not lead to singular-ities in y but, at this stage, cannot give an unam-biguous result either.

The qualitative result, that the tensor-state isheavier than the scalar one, is similar to that ob-tained by Hamer 8 in a Hamiltonian strong couplingexpansion in SU(3) . A reliable quantitative resultrequires presumably a higher-order calculation, orone which is based on other wave-functions, so thatmuch larger clusters will be involved in the struc-ture of the tensor glueball .

REFERENCES1. D . Horn, M. Karliner and M . Weinstein, Phys .

Rev. D31 (1985) 2589 .

2 . C. P. van den Doel and D. Horn, Phys. Rev.D33 (1986) 3011 .

3. C. P. van den Doel and D. Horn, Phys . Rev.D35 (1987) 2824 .

4. G . J. Mathews, N . J. Snyderman and S. D.Bloom, Phys . Rev . D36 (1987) 2553 .

5. A. Krasnitz and E . G . Klepfish Phys . Rev .D37 (1988) 2300 .

6. D . Horn, Int . Jour. Mod. Phys.A4 (1989)2147 .

7. D. Bessis and M. Villani, Int. J. Math .Phys.16 (1975) 462.

8. C . J . Hamer Phys . Lett.B224 (1989) 339.

D. Horn, Y. Shadmi /t-Expansion calculation