Strong-coupling expansions for the off-axis glueball masses

PHYSICS LETTERS

Volume 113B, number 4

STRONG-COUPLING

24June1982

EXPANSIONS FOR THE OFF-AXIS GLUEBALL MASSES

Hikaru KAWAI and Ryuichi NAKAYAMA Department

of Physics, University of Tokyo, Tokyo 113, Japan

Received 25 February 1982

Strong-coupling expansions, up to order gm4, for the off-axis glueball masses are developed in fourdimensional spacetime for lattice gauge theories with gauge groups SU (2) SU (3), Zz, 23. Glueball mass spectra for the states O++,2++, l+are obtained. Restoration of Lorentz invariance is discussed.

In lattice gauge theories [l] Lorentz (rotational) invariance is explicitly broken. Recent Monte Carlo calculations, however, suggest that the continuum limit of four-dimensional lattice QCD is both asymptotically free and confining [2] . This indicates that the Lorentz invariance is restored in the continuum limit. One possible method to investigate the recovery of the Lorentz invariance is to study the directional dependence of the masses of glueballs. (For the calculation of the directional asymmetry of a string tension in hamiltonian lattice gauge theories, see ref. [3] .) Previous calculations of the glueball masses by strong coupling expansions * ’ and Monte Carlo methods [6] are for the on-axis glueball masses, and the calculations of the off-axis glueball masses have not yet been done. In this paper we calculate the off-axis glueball masses by strong-coupling expansion, and compare them with on-axis results. Moreover we obtain masses for various glueball states, O++, 2++, l+-. We use the euclidean action given by Wilson [l]

S =-!- C g2 p

[tr(l

- U(P)) + c.c.] ,

(1)

where U(P) is an ordered product of gauge field matrices on the boundary of a plaquette P, and the sum extends over all unoriented plaquettes P. The * ’ In hamiltonian lattice gauge theory: ref. [4] , in euchdean lattice gauge theory: ref. [5] _ 0 031-9163/82/0000-0000/$02.75

0 1982 North-Holland

usual expansion = 2 tr( 1)/g2. The off-axis The connected two plaquettes

parameter fl is related to g2 by fl

glueball masses are defined as follows. correlation function G(r, n) between P,, P2 at a distance r in the direction decays exponentially as r + 00 nZ(nl,f12,n3,n4) G(r, n) E (tr U(P,) *tr U(P,)> - (tr U(P,))(tr

U(P2)>

+A2(r, n) exp(-m2(n)r) f ... , (2) where the Ai(r, n) are some powers of Y.We define ml(n), m2(n), -.a as the masses of glueballs in the direction n . The calculation of the on-axis mass was carried out by exponentiating the strong-coupling series of G into the form A(r) expf-mr) [4,.5] . This prescription, however, does not work for off-axis correlations because of the following two reasons: (i) For this prescription to work it is necessary to make a projection onto the eigenstates of each mass mi in the space of orientations of plaquettes. These eigenstates are complicated for general n, and we do not know a priori how to construct them. (ii) In the case of the on-axis correlation function, the only diagram which contributes to the leadingorder of strong-coupling expansions is the long straight tube connecting Pi and Pz (fig.‘la). On the other hand, 329

Volume 113B, number 4

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24 June 1982

II 11

la)

P2 (a)

(b)

P2 161

Fig. 2. (a) Leading contribution to G(k,/3): sum over random walk of a tube. (b) Higher-order contribution to G(k,/3): sum over random walk of a tube with diagrams (Di) inserted.

Fig. 1. (a) Leading contribution in the strong-coupling expansion for the on-axis plaquette-plaquette correlation. (b) The same as (a) but for the off-axis correlation. in the off axis case, many zigzag tubes contribute to the leading order, the number of which is difficult to calculate (fig. l b ) . Thus we compute the mass as the pole o f the correlation function G. We calculate G in the m o m e n t u m space; we multiply each diagram by exp(ik -x) (x is the displacement vector between P1 and P2, and k is a four-momentum vector), and sum over x and diagrams. F o r that purpose it is convenient to make G a matrix with respect to six orientations of plaquettes (in four-dimensional s p a c e - t i m e ) . First, the leading term for the correlation function G(k, {3) is given by the sum over the random walk of a tube (fig. 2a)

G(k, ~) = [1 - u(f3)aM(k)] - 1 ,

where u (/3) is the expansion coefficient for the fundamental representation in the character expansion. M(k) is a six b y six matrix and shown in tables 1 and 2. The indices o f matrix elements are six orientations of plaquettes; (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Table 1 is a matrix for the correlation o f [tr U(P) + tr U~ ( P ) ] / V ~ (charge conjugation even). Table 2 is for the correlation of [tr U(P) - tr U t ( p ) ] / x / ~ i (charge conjugation odd). Note the minus signs in • 1 • 1 M(1,2)(2,3) = - 4 smsk I sm~k3, ... etc. F o r chargeconjugation odd-states plaquettes are oriented; (/,]) = • 1 • 1 - ( ] , i). And M(i,] ) (i, l) = 4 sm ~k] sm ~k I (] ¢ l), 1 • 1 hence M(1, 2) (2,3) = --M(2,1) (2,3) = - 4 sin ~k 1 sin gk3, ... etc. If the gauge group is SU(2) or Z2, charge-conjugation odd states do not exist and the matrix o f

(3)

Table 1

M(k) for C-even states. (1, 2) (1, 2) (1, 3)

(1, 4)

"2 cos k3 + 2 cos k 4

(1, 3) 1

4 cos ~k2

1 X cos~k 3

2 cos k 2

(1, 4) 1

4 cos ~k2

1 X cos~k 4 1

4 cos ~k 3

1 X cos~k 4

+2cosk 4

2 cos k 2 + 2 cos k 3

(3, 4)

330

(2, 4)

1 4 cos ~k 1

1 4 cos ~k 1

1

X cos~k 3

0

(3, 4) 0

1

0

1

+ 2 cosk 4 symmetric

1

X cos ~k 4

1

4 cos {kl I X cos~k 2

2cosk 1

(2, 3)

(2, 4)

(2, 3)

4 cos ~k 1 1 X cos~k 4 1

4 cos ~k 1 X cos½k2

4 cos ~k 1 X cos½k3

1 4 cos ~k3 1 X cos ~k 4

1 4 cos~k2 1 × cos~k 4

2 cos k 1

4 cos ~k 2

+2 cosk 3

1

1 × cos ~k3

2 cos k 1 + 2 cos k

Volume 113B, number 4

24 June 1982

PHYSICS LETTERS

Table 2

M(k) for C-odd states•

(1, 2)

1

~2 cos k 3 + 2 cos k 4

(1, 3) (1, 4)

(2, 3)

(1,4)

(1, 3)

(1, 2)

•

(2, 4)

1

1

4 sin ~k 2 1 X sin ~k 3

4 sm ~k 2 • 1 X sm ~k4

- 4 sin ~k I 1 X sin ~k 3

2 cos k 2 + 2 cos k 4

4 sin ~k 3 1 X sin ~k4

1

4 sin ~kl • 1 X sm ~k 2

0

2 cos k 2 + 2 cos k 3

0

4 sin ~kl 1 X sin ~k2

- 4 sin ~k 1 1 X Sill~ k 4

1

1

1

4 sin ~k3 1 X sin ~ k 4

+ 2 cos k 4

(2, 4)

2 cos kl

symmetric

+ 2 cos k 3

(a)

(c)

JJJ

/

// (d)

1

- 4 sin ~k 2 . 1 X sm ~k 4 •

1

4 sm ~k2 • 1 X s m ~ka

a sum over the random walk o f these tubes with diagrams Di's inserted. The final form for G - l ( k , [3) (the inverse o f G ) is G - l(k,/~) = 1 -

- ~ ~ i

/

1

4 sin ~kl 1 X sin ~k3

+ 2 cos k 2

table 2 is not necessary because tr U(P)= tr US(p). Next consider higher-order corrections to G(k, [3). First construct various diagrams (Di) such as those in fig. 3. Remove from each diagram two arbitrary plaquettes which are identified as an entrance and an exit, and connect them b y simple tubes (fig. 2b). G is

/

1

- 4 sm ~kl 1 X sin ~k 4

2 cosk 1

(3, 4)

(b)

0

.

2 cos k l

(2, 3)

(3, 4)

1

Fig. 3. Diagrams Di contributing to G (k, ~) up to order ~2. (a) A partition wall of a tube. (b), (c) Diagrams with twelve plaquettes. (d), (e) Diagrams with sixteen plaquettes.

exp(ik'Ax)Di(fl)+Dcancel,

(4)

where Ax is a displacement vector between the entrance and the exit of a diagram Di, Di([3) is a matrix corresponding to the diagram Di, and Dcancel is a matrix which cancels the overlapping effects o f the tube with itself or with vacuum bubbles. Details of the method will be explained elsewhere [7]. In fig. 3 diagrams which contribute up to order/3 2 are shown. Up to this order there is no diagram contributing to Deancel. Matrix elements of G - 1 (k, r) up tp order/32 are presented in tables 3, 4. Now we compute the mass spectra of glueballs by extracting the poles of G(k, [J).This is done b y the following procedure• The condition for a pole is det

(e)

-+Ax

u(fl)aM(k)

G-l(k,

0) = 0 .

(5)

This is not sufficient for the determination o f the mom e n t u m k. We are seeking for the mass in the space-time direction n = ( n l , n2, n3, n4) ( ~ 4 = 1 n~2= l,nv>~O,v = 1,2, 3, 4). One might be tempted to put k / / n . In 331

V o l u m e 113B, n u m b e r 4

PHYSICS LETTERS

24 June 1982

Table 3 Matrix elements of G -I (k, ~) for charge-conjugation even states u p to order ~2. G-1 is a symmetric matrix. E l e m e n t s which are n o t listed below can be obtained b y p e r m u t a t i o n s of indices. Z+ is det'med in table 5. -1 G(1,2)(1,2) = (1 + Z + ) -1 - 2u4(cos k 3 + cos k4) - 48u t0 cos k 3 cos k 4 - 8 u l 4 ( c o s k l c o s k 2 + cos k 1 c o s k 3 + c o s k 1 cos k 4 + c o s k 2 c o s k 3 + c o s k 2 cos k4) ( c o s k 3 + c o s k 4 ) -1 G(1,2)(1,3) = - [ 4 u 4 + 8 0 u 10 c o s k 4 + 1 6 u l 4 ( c O S k l c o s k 2 + c o s k 1 c o s k a + 3 c o s k 1 c o s k 4 + cos k 2 cos k3 + cos k 2 cos k 4 + cos k 3 cos k 4 + cos2k4 1] cos ~k 2 1 cos ~k 31 G(-112)(3,4)=-

1 6 0 u l O + 6 4 u 14 ~ c o s k # #=1

cos~k 1 cos~k 2 eos~k3 cos~k 4

Table 4 Matrix elements of G-1 (k, ~) for charge-conjugation o d d s t a t e s u p to order ~2. G-1 is a symmetric matrix. Elements which are n o t listed below can be obtained by p e r m u t a t i o n s of indices u p to signs. The signs m u s t be determined in the same way as those in table 2. Z _ is defined in table 5 -1 G(1,2)(1,2) = (1 + Z _ ) - I

_ 2u4(cos k3 + c o s k 4 ) _ 4 8 u 1 0 c o s k 3 c o s k 4

- 8 u l 4 ( c o s k 1 c o s k 2 + cos k 1 cos k 3 + cos k 1 cos k 4 + c o s , k 2 cos k 3 + c o s k 2 c o s k 4 ) (cos k 3 + c o s k 4 ) -1 G(1,2)(1,3) = - [ 4 u 4 + 4 8 u 10 cos k 4 + 16u 14(cos k I cos k 2 + cos k 1 cos k 3 + cos k 1 cos k 4 + cos k 2 cos k 3 + cos k 2 cos k 4 + cos k 3 cos k 4 - cos2k4)] sin ~k 2 1 sin ~k 31 -1 G(1,2)(3,4) = 0

lattice theories, however, this condition cannot be consistently imposed due to the lack of rotational invariance. Instead we impose the following condition * ~,

(0/0k) det G - l(k, ~ ) / / n .

(6)

Suppose that eqs. (5) and (6) are solved for k. Then the mass is given in the following way,

4:2 In the real space G is given by

4

First integrate over k4. In the complex k4-plane the pole k 4 = k 4 ( k l , k2, k 3) is given by the solution of eq. (5). Thus

From tables 3 , 4 and eqs. (5), (6) and (7), we can calculate gluebaU masses. For general n, eqs. (5) and (6) can be solved only numerically. We present in this paper glueball masses only for the direction n = (½,½, 1 1 ~,~). Before presenting the results for n = (~ , ~1, ~J, ~ a) , w e discuss the on-axis masses; n = (1,0, 0, 0). To leading order, the solution to eqs. (5) and (6) is k 1 = i In u 03) - 4 + O031), k 2 = k 3 = k 4 = 0 for both eigenstates of charge conjugation, and the on-axis mass is, by using

d3k

G(r,n)=f

- ~ (2~r)a

[" 1 3 ]] -exp[ J r [ G m k ¢ + n 4 f c 4 ( k l , k 2 , k 3 )

)1

I. \i=1 " °

x R (kl, k2, ka), where R is the residue. A s r -~ ~, t h e saddle point/~i for k i integration is given by t h e solution o f n i + n 4 (O/Oki) × k 4 ( k l , k 2, k3) = 0 (i = 1, 2, 3), which are equivalent to eq. (6) u n d e r the constraint of eq. (51. T h e n the mass in the direction n is [see eq. (2)], (/_~_ m = -i 1

[eq. (7)1. 332

^ ~ ) 4 x tlgk~ ni[ci+n4k4(kl,fc2,fca) = - i /2=1

~ kun u . /a=l

(7)

G(r,n) = ? d4----~kexp(irn . k ) G ( k ) . --~r (2rr)4

m = -ik .n = -i

eq. (7), Mon_axis = - 4 In U(fl) + O(31),

(8)

which agrees with that obtained by the exponentiation method. The mass eigenstates are the eigenvectors of G - 1 with eigenvalue 0, and given by (0, 0, 0, 1,0, 0),

Volume 113B, number 4

PHYSICS LETTERS

24 June 1982

Table 5 1 1 1 1. Strong-coupling series for the gluebaU masses in the direction (1) n = (~, ~, ~, ~ ) and (2) n = (l, 0, 0, 0) up to order #2. u = tanh # for Z2, u = [exp[~ #) - 1]/[exp @3) + 2] for Z3, u = h 03)/]1 (~), v = I3 (~)/I103) for SU(2), u, o 1 and v2 for SU(3) are the same as those used by Mtinster [4]. u is 0(.61) and the v is O (¢/2). (1)

1 1 1 1

n = (~,~,~,~) 71

m I = - 8 l n u - 2 In 6 - 2 ln(1 +Z+) - ~ u rn2 = - 8 l n u - 2 In 2 - 2 ln(1 +Z+) - ~3u

2

2

m 3 = - 8 l n u - 4 In 2 - 2 In(1 +Z_) - ~ u 2 (2)

n = (1, 0, 0, 0) m~ =m~ = - 4 l n u - ln(1 +Z+) m~ = - 4 in u - ln(1 + Z_ )a) Z2 Z+

-u

Z

Z3 2

SU (2)

u-2u

2

3o-4u

SU (3) 2

3 u + 6 o 1+ 8v2 - 18u 2 - 3 u - 601 + 802

-u

a) m3 ' m~ do not exist for SU(2) and Z2.

other is C-odd.) The masses and the eigenstates for the groups SU(2), SU(3), Z 2 a n d Z 3 up to order/32 are summarized in tables 5, 6. j p c o f glueballs m 1, m 2 and rn 3 appear to be 0 ++, 2 ++ a n d 1+ - , correspondence to the on-axis masses is taken i n t o account. I n b o t h on-axis and o f f a x i s cases, the n u m b e r o f mass eigenstates is smaller t h a n twelve. Other solutions o f eqs. (5) and (6) with a n o m a l o u s behavior [ e x p ( - i k u ) = O ( ~ - x ) , ?t > 4] m a y exist [ 7 ] . I f this is true, more diagrams are needed to calculate these masses. I n fig. 4 glueball masses o f S U ( 2 ) and S U ( 3 ) theory f o r n = ( !2 , 51, 5 1, 5 )1 ' are plotted. F o r b o t h groups, the lowest off-axis mass m 1 comes close to the on-axis mass m ~ near the crossover region [the crossover p o i n t is/3 2 for SU(2) and/3 ~ 5 - 6 for S U ( 3 ) ] . F o r the dis-

(0, 0, 0, 0, 1 , 0 ) and (0, 0, 0, 0, 0, 1) (in the same matrix n o t a t i o n as tables 1 - 4 for b o t h eigenstates o f charge conjugation; i.e. three orientations perpendicular to the direction of the correlation. On-axis masses u p to order 132 are listed in part (2) o f table 5. The three Codd eigenstates form an axial vector (1 + - ) triplet [4,5]. On the other h a n d , the three-fold degeneracy o f C-even eigenstates becomes partially lifted in the order/34 [7] ; the singlet state with the lowest mass [the so-called scalar (0 ++) glueballJ is n o w (0, 0, 0, 1, 1, 1) + O(f12), and the remaining two states form part o f a tensor (2 ++) multiplet. _ 1 1 1 1 N o w consider the masses for n - (5, ~, ~, 5). I n this case G - 1 can be easily diagonalized and three different masses appear. (Two o f them are C-even and the

Table 6 .1 1 1 1 Mass eigenstates for masses m l , m2 and m 3 in the direction n = (~, ~, ~, ~). (1, 2) m~(O++) m2(2 +')

m3(1 +-)

( 1

(1, 3)

(1, 4)

(2, 4)

(3, 4)

1

1

1

l

1 )

0

0 0

0

-1 ) 0 )

0

0 )

1

0 1

1 ) 0 )

0

1

(0

1

0 0

(0

0

i

( 1 (0 ( 1

0 1 -1

-1 -1

( 1

(2, 3)

0

-1 I

-1

-1

) 333

Volume 113B, number 4

PHYSICS LETTERS

24 June 1982

strong-coupling series of off-axis masses are calculated up to order ~2. The results are consistent with the restoration of Lorentz invariance.

0m0 , 1 , 1 . 1 . 1 , 1 . 1 , 1 , 1 ,

10 m

%, 7

1.0

2.0

p =~/u2

(a)

3.0

!

2

3

4

5

6

?

8

1"~=6192 (b)

We would like to thank Professor T. Eguchi for useful advice and careful reading of the manuscript. We would also like to thank Professor K. Igi for encouragement. The computer calculation for this work has been financially supported by the Institute for Nuclear Study, University o f Tokyo.

1 1 1 1

Fig. 4. GluebaU masses m i for n = (:, : , : , : ) in four-dimensional space-time. The lattice spacing is set equal to unity. (a) is for SU (2) and (b) is for SU (3). The solid lines represent the strongcoupling series of off-axis masses up to order ~ . The dotted lines show the on-axis masses rn~. The strong-coupling series for m i and m~ axe given in table 3a. cussion o f the recovery o f Lorentz invariance, however, it is necessary to consider higher-order corrections. The calculation of fourth-order corrections requires estimations of very m a n y diagrams and is under study [7]. In the fourth-order calculation, it is more clearly seen that the on- and off-axis masses come close to each other. To summarize, we proposed a m e t h o d of calculating off-axis glueball masses. The problem of the projection onto mass eigenstates is reduced to the diagonalization o f the matrix G - 1, and the counting problem o f zigzag tubes is replaced by eqs. (5) and (6). The

334

References

[1] K.G. Wilson, Phys. Rev. D10 (1974~ 2445; [2] M. Creutz, Phys. Rev. Lett. 43 (1979) 553; Phys. Rev. D21 (1980) 2308. [3] J.B. Kogut, D.K. Sinclair, R.B. Pearson, J.L. Richardson and J. Shigemitsu, Phys. Rev. D23 (1981) 2945; J.B. Kogut and D.K. Sinclair, preprint ILL-(TH)-81-25 (1981). [4] J. Kogut, D.K. Sinclair and L. Susskind, Nucl. Phys. BII4 (1976) 199. [5] G. Mtinster, Nucl. Phys. B190 [FS 3] (1981) 439; N. Kimura and A. Ukawa, preprint INS-rep.-431 (1981). [6] B. Berg, Phys. Lett. 97B (1980) 401; G. Bhanot and C. Rebbi, Nucl. Phys. B189 (1981) 469; J. Engels, F. Karsh, H. Satz and I. Montvay, Phys. Lett. 102B (1981) 332; K. Ishikawa, M. Teper and G. Schierholz, Phys. Lett. ll0B (1982) 399. [7] R. Nakayama, in preparation.