One-loop lattice vacuum energy

Volume 147B, number 4,5

PHYSICS LETTERS

8 November 1984

ONE-LOOP LATFICE VACUUM ENERGY R. GUPTA Department of Physics, Northeastern University, Boston, MA 02115, USA and Gregory KILCUP 1 and Stephen SHARPE 2 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 23 July 1984

We propose that lattice Monte Carlo methods be used to measure the vacuum energy due to one-loop gauge boson exchange. Depending on the theory involved, this quantity is related to mZn+ - m~0, to the vacuum alignment problem, or to the masses of composite Higgs bosons.

Lattice Monte Carlo calculations have to date yielded reasonable quantitative results for the gross features o f the hadronic spectrum. The masses o f the lowest lying meson and baryon states in the first few j/~7 channels agree quite well with the experimental values [1,2]. In this letter we point out that certain fine structure in the spectrum, namely the n + - 7r° mass difference, may also be calculated. The analogous computation for groups and representations other than those in QCD is also very interesting - in technicolor theories it feeds in to the vacuum alignment problem [3], while in a new class o f models [4] it gives the mass o f the Higgs boson [5]. In the latter case it would be particularly interesting to perform the calculation with fermions transforming as real or pseudoreal representations o f SU(N). Some of the data for the SU(2) and SU(3) models have already been collected b y lattice spectroscopists. As we have said, the problem is quite general, but to make it easier to discuss we will focus on the pion electromagnetic splitting in QCD with three massless quarks. The generalizations are straightforward. The quantity we propose to measure is the vacuum energy density from one-loop gauge boson exchange (fig. l a ) , 1 NSF Graduate Fellow. 2 Junior Fellow, Harvard Society of Fellows. 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

[a}

[b)

Fig. 1. (a) The one-loop electromagnetic contribution to the vacuum energy, which, as shown in the text, is related to (b) the one-loop electromagnetic contribution to the ~r+--n° difference. The blobs represent QCD interactions to all orders. which gives a contribution e2V =

-ie2f d4x DUV(x)(OIT(jL(x)]R(o))[O)

(1)

where D is the p h o t o n propagator, 10)is the physical vacuum, characterized by its orientation o f the (chiral symmetry breaking) quark condensate, and r E ( j R ) is the left-(right-) handed piece of the electromagnetic current:

•L = ~L,), Q ~ L ~b = (u, d, s ) ,

Q = diag(2/3, - 1 / 3 , - 1 / 3 ) .

Since the vacuum preserves SU(3)V , any SU(3)V rota339

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tion of J will do just as well. In (1) we have already subtracted an infinite contribution (LL + RR) which is independent of the vacuum orientation. In the chiral limit, cancellations between vector and axialvector intermediate states render V itself finite. Current algebra then allows us to express the pion mass splitting (fig. lb) in terms of V, up to corrections of order a:

velope estimate and the brute force method of the lattice. Taking the Weinberg sum rules [7] for the difference of the vector and axial spectral functions and saturating them with the p, A 1 and 7r poles one finds 2_ 2 gp - gA1 '

6m2=_mTr+2 _mro2 = 2 e 2 V / ( f 2 t r a 2 ) ,

c = (3[327r2)(g2[f 4) ln(m 2 / m 2 ) ,

(2)

where f~r = 93 MeV. In the real world, quark mass differences contribute to this splitting in second order of chiral perturbation theory, so we expect additional corrections of order (mq/f~) 2 _ or one percent. There are larger contributions from ~-rr mixing and long distance physics, but we expect (2) to hold to within a few percent. A more convenient way to parametrize our ignorance is the language of the chiral lagrangian. Fluctuations in the SU(3)L X SU(3)R orientation of the condensate are described by the phenomenological lagrangian: .~= ¼f2 T r ( O Z B u Z t ) ,

Y. = exp(2i~aTa/f~),

1 ~ab tr TaT b _- 5v .

(3)

Diagrams (la) and ( l b ) generate pieces of a single chiral symmetry breaking term Z?EM = ce2f~n t r ( a ~ a z + ) .

(4)

Comparing (1) and (4), the quantity we really want to measure is the dimensionless ratio

c = - V / ( f 4 tr 0 2) in QCD and other vector-like theories. According to the "naive dimensional analysis" of ref. [6], c should be of order c = A2/16~'2f 2 ~" 1,

2 2 ~2m2 / 2 mp/mA1 = 1 --I~ p,go , 1

where go is defined by (01 l(ffTUu - dy"d)lp> = eUgp. Then using either (a) the KSSRF relation [8] go = x/~fnmp (good to 20%) or (b)gp = 0.12 (GeV) 2 (from B(p ~ e+e - ) = 4.3 X 10-5), one finds (a)

c=0.89,

mAl=l.09GeV,

(b)

c =0.81,

mA1 = 0 . 9 6 G e V .

Option (a) is the classic result of Das et al. [9]. In either case the good agreement with c(exp) must be considered somewhat fortuitous in view of the less successful predictions for the A 1 parameters [MA1 = 1.23 GeV, gA1 = (0.17 + 0.03) GeV2]. Of course, taking quark masses (which invalidate the sum rules as written) into account and including more than single particle poles in the model of the spectral functions ought to improve things. For theories other than QCD, though, we do not have the luxury of peeking at the rest of the spectrum, and we must rely on the lattice to calculate c. Our central point is that the computation of c is rather straightforward, if one is willing to trust the lattice evaluation of the magnitude of certain correlators, instead of merely measuring ratios of correlators, as one does in extracting particle masses. Rotating to euclidean space, the vacuum energy is V= tr Q2 f d 4 x T~-(o[]L(x)jR(o)[o) 4rt2 .~ X2 /~ v '

where A ~ 41rfn is the loop momentum cutoff. Plugging in the observed mass splitting, we find the serendipitious result

where

c(exp) = 0.79.

TU~ =gUy + ~(gUV _ 2xUx~/x2),

Or, put the other way around, naive dimensional analysis gives the splitting correct to about 20 percent! It is worth mentioning that for QCD we have a third option lying between this naive back-of-the-en-

and where we have rotated and rescaled the current into the convenient form .L 1/~ = ~Lg' dL •

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Conservation of the currents (in the chiral limit) of course guarantees that V is independent of the gauge parameter (~). Likewise, f~r can be measured by taking seriously the magnitudes of the axial vector two-point function: f2=

lim T~o

exp(m~rT)(OIAO(T,p=O)AO(O,p=O)lO)

X 2/mr(aL)3 where (a/) 3 is the lattice volume and Ag is the axial current. Alternatively one can measure the quark condensate and use

f2=2mq(qq)/m 2 . There are a number of comments which should be made: (a) The calculation really makes sense only in a chirally invariant formulation, such as K o g u t Susskind fermions. With Wilson fermions, even at the critical value of the hopping parameter where the pion mass vanishes, quantities such as (~q) and V diverge (like a - 3 and a - 4 respectively) in perturbation theory. For Kogut-Susskind fermions, on the other hand, the discrete chiral symmetry causes the perturbative contribution to vanish. We can then obtain V by the standard method of extrapolating to the chiral limit from a series of measurements at finite quark mass. (b) Away from the chiral limit the numerator in (5) fails to vanish at the origin, and the integral (or rather, sum) blows up at x = 0. To proceed one needs to introduce an ultraviolet cutoff, which may be removed only after extrapolating to the chiral limit. There are two obvious ways of implementing such a regulator: (i) replace the continuum photon propagator with the lattice version, which is finite at the origin, or (ii) simply delete the offending term from the sum. (c) For staggered fermions there is some ambiguity in defining X 2 and x~xv, since the fermions live on several sites. This error is of the same sort as we make in approximating our integral by a Riemann sum, and goes away when we take the lattice spacing to zero. (d) As long as we are in the scaling region, the physical size of the lattice is irrelevant - the lattice spacing drops out of the dimensionless ratio V/f 4. The only problem we foresee in calculating c is the extrapolation to the chiral limit. Away from the chiral limit there are divergent contributions to both V and (~lq) which we must separate from the finite chiral lim-

8 November 1984

it by the extrapolation. The errors introduced have not proved an insurmountable obstacle to the calculation of (Etq) = (250 MeV) 3, but the scale associated to V (about 79 MeV) is much smaller, and may well hide in the noise. As an estimate of these divergent pieces we take the lowest order perturbative contributions. As a check, we note that this estimate o f (~lq) agrees with the Monte Carlo measurement * 1. Expressing everything in terms of the inverse lattice spacing, for mqa = 0.1, we find have recently been made with mqa ,~ 0.1 for SU(3) at fl = 5.7 [11], and for SU(2) [12]. Only a modest increase in statistics would allow V to emerge from the noise. Of course, the only way to actually see whether V can be measured is to attempt a calculation. We did so * 2, but unfortunately had only Wilson propagators at our disposal, so our calculation suffers from the problem mentioned in (a) above, that there are divergent contributions to V even at the critical value of the hopping parameter. Our (physicaLly meaningless) results were (tT:lq)= 3.41 -+0.01 and V = (4.14 -+0.04) X 10 -3. It might be possible, however, to extract a physically meaningful number by adopting the method used to measure the gluon condensate
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scaling term. Though it is poorly founded in theory [15], this idea seems to have worked in practice. TypicaUy the first few perturbative terms are calculated directly, and the rest fit to the data at several values o f /L Since we had data at only one value of/3, we would actually have to calculate the higher order terms, an impossible task. We therefore use our results to check whether the statistical errors allow the calculation to proceed in principle. For (~lq) the error is the same size as the effect, and the lowest order perturbative term is 90 percent o f the total - one may hope that subtracting just a few perturbative terms would expose a quant i t y one could measure with improved statistics. F o r V, the error is an order o f magnitude larger. Furthermore, the zeroth order perturbative term accounts for only 40 percent o f the observed value, indicating that many higher order terms are needed. We conclude that (~lq) is just barely measurable with this technique, while a measurement o f V looks hopeless. In conclusion, we think that while it is not worth pursuing the Wilson fermion approach, it is worth attempting a calculation using K o g u t - S u s s k i n d fermions. We are beginning such a calculation, but think that the quantity is of sufficient interest that it should be measured on the larger lattices already used for spectroscopy. We stress again that this time the p and A 1 correlators are all that is required. We thank David Kaplan for forcing us to think through this problem, and Howard Georgi for constant harrassment.

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References [1] A. Billoire, E. Maxinari and R. Petronzio, CERN-TH3838 (March 1984). [2] H. Lipps et a[, Phys. Lett. 126B (1983) 250; J.P. Gflchrist et at, Phys. Lett. 136B (1984) 87; P. Hasenfratz and I. Montvay, NueL Phys. B237 (1984) 237. [3] J. PreskiU, Nue[ Phys. B177 (1981) 21; M. Peskin, Nucl. Phy~ B175 (1980) 197. [4] D.B. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 187; D.B. Kaplan, H. Georgi and S. Dirnopoulos, Phys. Lett. 136B (1984) 187. [5] H. Georgi, D.B. Kaplan and P. Galison, Phys. Lett. 143B (1984) 152. [6] A. Manohar and H. Georgi, Nucl. Phys~ B234 (1984) 189. [7] S. Weinberg, Phys. Rev. Lett. 18 (1967) 507. [8] K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 (1966) 255; J.J. Sakurai, Phys. Rev. Lett. 17 (1966) 552; Riazuddin and Fayyazuddin, Phys. Rev. 147 (1966) 1071. [9] T. Daset al., Phys. Rev. Lett. 18 (1967) 759. [10] N.A. Campbell, C. Michael and P.E.L. Rakow, Phys. Lett. 139B (1984) 288. [11] I.M. Barbour et al., Phys. Lett. 136B (1984) 80. [12] J. Kogut, J. Shigemitsu and D.K. Sinclair, Phys. Lett. 138B (1984) 283. [13] R. Brower et al., Harvard preprint HUTP-84/A004 (January 1984). [14] A. DiGiacomo and G.C. Rossi, Phys. Lett. 100B (1981) 481; 108B (1982) 327; T. Banks et al., NueL Phys. B190 (1981) 692. [15] F. David, NucL Phys. B234 (1984) 237.