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Higgs physics on the F4 lattice
649
Nuclear Physics B (Proc. Suppl .) 17 (1990) 649-652 North-Holland
HIGGS PHYSICS ON THE F4 LATTICE Urs M. Heller Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 22306-4052 U.S.A.
With the goal of investigating the cut-off dependence of the Higgs mass bound we formulate the O(4) scalar model on the F4 lattice. We describe the structure of the F4 lattice and construct the transfer matrix . In order to compare different cut-off schemes we investigate the cut-off dependence of various physical observables. We compute the sigma width and the differential cross-section of Goldstone boson scattering at tree level, including all cut-off effects. Comparing with the hypercubic lattice allows us to define a relation between the two cut-offs. Finally we discuss finite volume effects to be exploited in the numerical simulation. 1. INTRODUCTION
As you have heard from H. Neuberger's review talk,' because of triviality the observable predictions of the minimal standard model will necessarily cv,tain_ some weak cut-off dependence wfietce "-`$eve" physics sets in. To obtain a bound on the Higgs mass we work, as usual, in the pure O(4) symmetric scalar sector. The bound on the Higgs mass is the highest mass for which the cutoff effects are smaller than some previously demanded value. A lattice model for which the leading cut-off effects are the same as those possible in the real world is described by the action S = -2rc y',% j(x) j(x)+S ~P (x) n .n .
+
[de
x) _
l2 ]
x
[42
3
.
Here the x's are sites on an F4 lattice.2 This lattice has enough symmetry to eliminate the order 4 primitive invariant r,,, 04 which is available on the hypercubic lattice, and contributes to order ~ but is absent in the real world. In practice we have to do the following: work in the vicinity of the O(4) symmetry breaking critical surface. Estimate some physically observable rut-off effects and 0920-5632/90/$3.50 © Elsevier
North-Holland
Science Publishers B.V.
limit them. This defines a precise neighborhood of the critical surface on the broken side in which we need to find the c-particle mass and fir, which sets the scale. In this talk, I shall consider the first part of this program. For the second part, see G. Bhanot's talk.3
In contradistinction to asymptotically free theories we don't believe that a simple universal ratio relation between different cut-off schemes exists . Rather, it is only through the numerical value of cut-off effects on physically relevant observables that different cut-off schemes can be meaningfully compared. We will use perturbation theory for the computation of such cut-off effects. In order to define an "on-shell" concept on the lattice we also have to set up a transfer matrix formalism. Some finite size effects are also discussed for the F4 lattice.
STRUCTURE AND THE TRANSFER MATRIX With a slight abuse of notation we can define in any dimension d >_ 2 the lattice 2. F4
Fd =
x (x =
o se(,,) ,
xpeZ,
x;, = even
(2) for the direction . A basis With e(,,) a unit vector in it Fd lattice is
650
U.M. Heller /Higgs physics on the F4 lattice
b(i) =e(i)
- e(i+1),
i = 1, ...,d-1
TI
b(d) =e(i) -}- e(d) .
The unit cell of the Fd lattice has volume 2. Finite symmetrical lattices Fd,L can be obtained as Fd,L =Fdl ^'s
z, pcPdt
x-y4* z-y=
zub(,up x;, =0modL .
Fourier transforms can easily be defined with the use of the reciprocal lattice and periodic boundary conditions are imposed along the basis vectors .4 For general d, Fd is the lattice associated with the Lie Algebra Cd (SP2d) . For d = 4 the lattice is also the root lattice of F4 and the lattice symmetry becomes bigger? This can be seen, for example, in the free inverse propagator, I'(2) (k) = g (k) -}- m where
2
; [2 - cos (k,, - h.,) - cos (k,, + k,.)] 2(d- 1) ~
9 (k) =
To 3efine a scattering theory on the lattice following Barata and Fredenhagen, 5 one needs a transfer matrix formalism . On the F4 lattice this is a little less straight forward than on a hypercubic one . The F4 lattice can be regarded as a stack of 3-d (spatial) f.c.c. lattices where the corner of a cube lies above the center of a cube in the previous "time slice ." Therefore, the transfer matrix T also generates a shift in "space" . It is not selfadjoint and its eigenvalues come in complex conjugate pairs related by the parity transformation . However, this difference to the hypercubic transfer matrix turns out immateriU.4 All real quantities come out real as they should . For given 3-momentum q there are one-p:zrticle states (q > with energy w (q) defined by
~
4 -w (q-1) + _ E qj j+2
I q> .
(6)
We choose the normalization of the "relativistic" type and the phase such that the wave function renormalization (0101gJ = Z= (q-)
(7)
is real. Z (q-;) and w (q-) can be computed in perturbation theory.4 The expressions for w (q-) and Z (q-) are needed when extracting the invariant matrix elements from a Euclidean correlation function in momentum space . We now can take c,,--r to the F4 lattice the definitions and expressions from hypercubic lattice scattering theory and obtain for example 7çÎ (Pi iP2 *- P3 eP4)1 CM =
)4 16x2
Zw (P)
IPF 121M
a d 4 _ (d1 + 3 E + O (k6) . ~ ) 4 [ 1) k~J
The subleading Euclidean invariance breaking term disppears at d = 4. We expect therefore that at d = 4 we shall have less violation of Euclidean invariance than for the hype ;ubic lattice.
q->= exp
(of ~- PJ
12
-1 ) ( 1A - b I lo
P (A - %
3. CUT-OFF EFFECTS IN PERTURBATION THEORY
Having set up the scattering theory on the F4 lattice we can consider pion-pion scattering. This is physically relevant because of the "equivalence theorem which states that up to O (AM SF) it describes the scattering of longitudinal gauge boson. 6 Here E is a typical energy and MW the W boson mass. For the hypercubic, lattice the relevant cross section has been worked out to leading order in perturbation theory by Löscher and Weisz.7 In complete analogy we obtain4 1M12 = (Z(P)Z(P)) 2 {in --2)A2 (S)
with
(A () + A (t) + A (a)) 21' 1
2 2 (cosh MR - 1) A (z) = 39R m2R
x
2 (cosh MR - 1) - x' (10)
U.M. Heller/Higgs physics on the F4 lattice and s, t and û given in terms of the function g(k) of Eq. (5) . We have computed bc (W, 0,MR) -
( an dir
latt -1
Re
of (Pl - P) (
= 0.
1
(12)
The perturbative computation to leading (1- loop) order by Löscher and Weisz7 can be easily taken overfrom the hypercubic to the F4 lattice. The resulting cut-o$ dependence measured by b (MR) = (rô -1 is shown
~
cont
in Figure 2. To leading order in MR we find4
for various center of mass energies W = 2w (p-) at Wand _ 1 plot it at constant R = 4 in Figure 1. For small SF4 240 mR + O (76R) R (13) MR we find b = O(m4R) while for the hypercubic lattice 7 SHC mR a = O(mR) showing the smaller breaking of Euclidean 1920 + O ( R) ' lattice . invariance on the F4 They are of the same order because the averaging over all directions of the decay products erases the Lorentz violating effects in the propagator. We can compare the cut-off effects by equating SF4 with 6HC and obtain v
C0 u
MF4 MHC R = R
C b
w
b b y
0.01
W b b
(8
r,, F'4 0.001
0 .2
_.1.,MHC R
(14)
Similar values are obtained when including higher orders in MR but limiting the cut-off effects to b < 0.05 or b < 0.1.
J0
â b
1)146
0 .5
MR /A,
1
LATT.
2
FIGURE 1 (da for GoldDeviation a(mR) _ (~llatt / 3a) coat - 1 stone boson scattering at Wand constant R = 4 We can use Figure 1 and a similar figure for the hypercubic lattice? to numerically find a ratio between the smallest allowed cut-offs when tolerating a certain maximum cut-off effect for R's below some maximal Rn ax. For example, for b < 0.05 and R < 4 we obtain ^- 1.4, while increasing the tolerance to ô < 0.1 4
gives ~ - 1.5.
After the mass, the most important property of the Higgs particle is the width. For a heavy Higgs the width can again be worked out in the pure A04 theory. It is given by the location of the complex pole in the analytically continued o, propagator, i.e., by
10-5 10-° 0.1
0.2
0.5
MR
1
2
5
FIGURE 2 Deviation b(MR) = (r,)latt / (ro)cont - 1 of the o7particle width .
From the above consideration we can conclude that the ratio of hypercubic to F4 cut-off lies in the range = 1.1 to 1.5. 4
U.M. Heller /Higgs physics on the F4 lattice
652 4. FINITE SIZE EFFECTS
e study of the Higgs mass bound necessarily requires some nonperturbative information . We use Monte Carlo methods to obtain it . The numerical stud ies are limited to finite (small) volume and thus affected by systematic errors that need to be studied theoretically.
4.1. Symmetric Phase Here the finite size corrections have been studied by Lùscher . 8 They are easily taken over to the F4 lattice . The leading correction to the mass gap, (L), comes around the world by the from virtual particles looping shortest paths. For our spatial Ec.c. lattices these shortest paths have length V2-L. Therefore, we find (L) -
(oo) oc (%î2L) a e-fLA~(co).
(15)
One can also define a finite four volume "mass" mg (L) from the inverse propagator. It's leading finite size corrections are ml (L) - m~ « (.`2-L) ' e- /2-Lri`R .
(16)
They are smaller than on the hypercubic lattice for the same physical cut-off. 4.2. Broken Phase Due to the appearance ofmassless Goldstone bosons at infinite volume, the finite size effects in the broken phase are much stronger, power law like. These complications are compensated by the fact that the soft Goldstone bosons interact weakly and hence the leading finite size effects are due to almost free particles .
Defining the magnetization by M = A,, & $( x) we find in a finite volume, assuming that at L --+ oo we end up in the broken phase, 9
y (M)L =0
y 1oPL 2(N2_ 1) Zlr*F4 + 4 C (M2 )L =(MZ )oo +
(17)
where (42)00 = ZRfx by current algebra and ap4 = 0.105346 . This formula will be our main tool for determining fs in an MC simulation.
Another important finite size effect in the broken phase is related to the instability of the a-particle resonance in the infinite volume theory. One expects strong mixing effects when the energy of two pions is close to mg, but a quantitative analysis is not yet available . ACKNOWLEDGMENT It's a pleasure to thank my collaborators G. Bhanot, K. Bitar and H. Neuberger for numerous discussions and the organizers for an interesting and stimulating confer ence. This work has been supported in part by the U.S. Department of Energy through Contract No. DE-FC0585ER250000. REFERENCES 1. H. Neuberger, these proceedings . 2. H. S. M. Coxeter, Regular Polytopes (Dover, 1973) . H. S. M. Coxeter,Regular Complex Polytopes (Cambridge University Press, 1974) . 3. G. Bhanot, these proceedings . 4. G. Bhanot, K. Bitar, U. M. Heller, and H. Neuberger, work in progress . 5. J. C. A. Barata and K. Fredenhagen, Nucl. Phys. B (Proc . Suppl.) 4 (1988) 639. 6. M. Chanowitz and M. K. Gaillard, Nucl. Phys. B261 (1985) 379 . 7. M. Lûscher and P. Weisz, Nucl. Phys. B318 (1989) 705. 8. M. Lâcher, Commun . Math. Phys. 104 (9186) 177. 9. H. Neuberger, Phys. Rev . Lett. 60 (1988) 889 .
Nuclear Physics B (Proc. Suppl .) 17 (1990) 649-652 North-Holland
HIGGS PHYSICS ON THE F4 LATTICE Urs M. Heller Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 22306-4052 U.S.A.
With the goal of investigating the cut-off dependence of the Higgs mass bound we formulate the O(4) scalar model on the F4 lattice. We describe the structure of the F4 lattice and construct the transfer matrix . In order to compare different cut-off schemes we investigate the cut-off dependence of various physical observables. We compute the sigma width and the differential cross-section of Goldstone boson scattering at tree level, including all cut-off effects. Comparing with the hypercubic lattice allows us to define a relation between the two cut-offs. Finally we discuss finite volume effects to be exploited in the numerical simulation. 1. INTRODUCTION
As you have heard from H. Neuberger's review talk,' because of triviality the observable predictions of the minimal standard model will necessarily cv,tain_ some weak cut-off dependence wfietce "-`$eve" physics sets in. To obtain a bound on the Higgs mass we work, as usual, in the pure O(4) symmetric scalar sector. The bound on the Higgs mass is the highest mass for which the cutoff effects are smaller than some previously demanded value. A lattice model for which the leading cut-off effects are the same as those possible in the real world is described by the action S = -2rc y',% j(x) j(x)+S ~P (x) n .n .
+
[de
x) _
l2 ]
x
[42
3
.
Here the x's are sites on an F4 lattice.2 This lattice has enough symmetry to eliminate the order 4 primitive invariant r,,, 04 which is available on the hypercubic lattice, and contributes to order ~ but is absent in the real world. In practice we have to do the following: work in the vicinity of the O(4) symmetry breaking critical surface. Estimate some physically observable rut-off effects and 0920-5632/90/$3.50 © Elsevier
North-Holland
Science Publishers B.V.
limit them. This defines a precise neighborhood of the critical surface on the broken side in which we need to find the c-particle mass and fir, which sets the scale. In this talk, I shall consider the first part of this program. For the second part, see G. Bhanot's talk.3
In contradistinction to asymptotically free theories we don't believe that a simple universal ratio relation between different cut-off schemes exists . Rather, it is only through the numerical value of cut-off effects on physically relevant observables that different cut-off schemes can be meaningfully compared. We will use perturbation theory for the computation of such cut-off effects. In order to define an "on-shell" concept on the lattice we also have to set up a transfer matrix formalism. Some finite size effects are also discussed for the F4 lattice.
STRUCTURE AND THE TRANSFER MATRIX With a slight abuse of notation we can define in any dimension d >_ 2 the lattice 2. F4
Fd =
x (x =
o se(,,) ,
xpeZ,
x;, = even
(2) for the direction . A basis With e(,,) a unit vector in it Fd lattice is
650
U.M. Heller /Higgs physics on the F4 lattice
b(i) =e(i)
- e(i+1),
i = 1, ...,d-1
TI
b(d) =e(i) -}- e(d) .
The unit cell of the Fd lattice has volume 2. Finite symmetrical lattices Fd,L can be obtained as Fd,L =Fdl ^'s
z, pcPdt
x-y4* z-y=
zub(,up x;, =0modL .
Fourier transforms can easily be defined with the use of the reciprocal lattice and periodic boundary conditions are imposed along the basis vectors .4 For general d, Fd is the lattice associated with the Lie Algebra Cd (SP2d) . For d = 4 the lattice is also the root lattice of F4 and the lattice symmetry becomes bigger? This can be seen, for example, in the free inverse propagator, I'(2) (k) = g (k) -}- m where
2
; [2 - cos (k,, - h.,) - cos (k,, + k,.)] 2(d- 1) ~
9 (k) =
To 3efine a scattering theory on the lattice following Barata and Fredenhagen, 5 one needs a transfer matrix formalism . On the F4 lattice this is a little less straight forward than on a hypercubic one . The F4 lattice can be regarded as a stack of 3-d (spatial) f.c.c. lattices where the corner of a cube lies above the center of a cube in the previous "time slice ." Therefore, the transfer matrix T also generates a shift in "space" . It is not selfadjoint and its eigenvalues come in complex conjugate pairs related by the parity transformation . However, this difference to the hypercubic transfer matrix turns out immateriU.4 All real quantities come out real as they should . For given 3-momentum q there are one-p:zrticle states (q > with energy w (q) defined by
~
4 -w (q-1) + _ E qj j+2
I q> .
(6)
We choose the normalization of the "relativistic" type and the phase such that the wave function renormalization (0101gJ = Z= (q-)
(7)
is real. Z (q-;) and w (q-) can be computed in perturbation theory.4 The expressions for w (q-) and Z (q-) are needed when extracting the invariant matrix elements from a Euclidean correlation function in momentum space . We now can take c,,--r to the F4 lattice the definitions and expressions from hypercubic lattice scattering theory and obtain for example 7çÎ (Pi iP2 *- P3 eP4)1 CM =
)4 16x2
Zw (P)
IPF 121M
a d 4 _ (d1 + 3 E + O (k6) . ~ ) 4 [ 1) k~J
The subleading Euclidean invariance breaking term disppears at d = 4. We expect therefore that at d = 4 we shall have less violation of Euclidean invariance than for the hype ;ubic lattice.
q->= exp
(of ~- PJ
12
-1 ) ( 1A - b I lo
P (A - %
3. CUT-OFF EFFECTS IN PERTURBATION THEORY
Having set up the scattering theory on the F4 lattice we can consider pion-pion scattering. This is physically relevant because of the "equivalence theorem which states that up to O (AM SF) it describes the scattering of longitudinal gauge boson. 6 Here E is a typical energy and MW the W boson mass. For the hypercubic, lattice the relevant cross section has been worked out to leading order in perturbation theory by Löscher and Weisz.7 In complete analogy we obtain4 1M12 = (Z(P)Z(P)) 2 {in --2)A2 (S)
with
(A () + A (t) + A (a)) 21' 1
2 2 (cosh MR - 1) A (z) = 39R m2R
x
2 (cosh MR - 1) - x' (10)
U.M. Heller/Higgs physics on the F4 lattice and s, t and û given in terms of the function g(k) of Eq. (5) . We have computed bc (W, 0,MR) -
( an dir
latt -1
Re
of (Pl - P) (
= 0.
1
(12)
The perturbative computation to leading (1- loop) order by Löscher and Weisz7 can be easily taken overfrom the hypercubic to the F4 lattice. The resulting cut-o$ dependence measured by b (MR) = (rô -1 is shown
~
cont
in Figure 2. To leading order in MR we find4
for various center of mass energies W = 2w (p-) at Wand _ 1 plot it at constant R = 4 in Figure 1. For small SF4 240 mR + O (76R) R (13) MR we find b = O(m4R) while for the hypercubic lattice 7 SHC mR a = O(mR) showing the smaller breaking of Euclidean 1920 + O ( R) ' lattice . invariance on the F4 They are of the same order because the averaging over all directions of the decay products erases the Lorentz violating effects in the propagator. We can compare the cut-off effects by equating SF4 with 6HC and obtain v
C0 u
MF4 MHC R = R
C b
w
b b y
0.01
W b b
(8
r,, F'4 0.001
0 .2
_.1.,MHC R
(14)
Similar values are obtained when including higher orders in MR but limiting the cut-off effects to b < 0.05 or b < 0.1.
J0
â b
1)146
0 .5
MR /A,
1
LATT.
2
FIGURE 1 (da for GoldDeviation a(mR) _ (~llatt / 3a) coat - 1 stone boson scattering at Wand constant R = 4 We can use Figure 1 and a similar figure for the hypercubic lattice? to numerically find a ratio between the smallest allowed cut-offs when tolerating a certain maximum cut-off effect for R's below some maximal Rn ax. For example, for b < 0.05 and R < 4 we obtain ^- 1.4, while increasing the tolerance to ô < 0.1 4
gives ~ - 1.5.
After the mass, the most important property of the Higgs particle is the width. For a heavy Higgs the width can again be worked out in the pure A04 theory. It is given by the location of the complex pole in the analytically continued o, propagator, i.e., by
10-5 10-° 0.1
0.2
0.5
MR
1
2
5
FIGURE 2 Deviation b(MR) = (r,)latt / (ro)cont - 1 of the o7particle width .
From the above consideration we can conclude that the ratio of hypercubic to F4 cut-off lies in the range = 1.1 to 1.5. 4
U.M. Heller /Higgs physics on the F4 lattice
652 4. FINITE SIZE EFFECTS
e study of the Higgs mass bound necessarily requires some nonperturbative information . We use Monte Carlo methods to obtain it . The numerical stud ies are limited to finite (small) volume and thus affected by systematic errors that need to be studied theoretically.
4.1. Symmetric Phase Here the finite size corrections have been studied by Lùscher . 8 They are easily taken over to the F4 lattice . The leading correction to the mass gap, (L), comes around the world by the from virtual particles looping shortest paths. For our spatial Ec.c. lattices these shortest paths have length V2-L. Therefore, we find (L) -
(oo) oc (%î2L) a e-fLA~(co).
(15)
One can also define a finite four volume "mass" mg (L) from the inverse propagator. It's leading finite size corrections are ml (L) - m~ « (.`2-L) ' e- /2-Lri`R .
(16)
They are smaller than on the hypercubic lattice for the same physical cut-off. 4.2. Broken Phase Due to the appearance ofmassless Goldstone bosons at infinite volume, the finite size effects in the broken phase are much stronger, power law like. These complications are compensated by the fact that the soft Goldstone bosons interact weakly and hence the leading finite size effects are due to almost free particles .
Defining the magnetization by M = A,, & $( x) we find in a finite volume, assuming that at L --+ oo we end up in the broken phase, 9
y (M)L =0
y 1oPL 2(N2_ 1) Zlr*F4 + 4 C (M2 )L =(MZ )oo +
(17)
where (42)00 = ZRfx by current algebra and ap4 = 0.105346 . This formula will be our main tool for determining fs in an MC simulation.
Another important finite size effect in the broken phase is related to the instability of the a-particle resonance in the infinite volume theory. One expects strong mixing effects when the energy of two pions is close to mg, but a quantitative analysis is not yet available . ACKNOWLEDGMENT It's a pleasure to thank my collaborators G. Bhanot, K. Bitar and H. Neuberger for numerous discussions and the organizers for an interesting and stimulating confer ence. This work has been supported in part by the U.S. Department of Energy through Contract No. DE-FC0585ER250000. REFERENCES 1. H. Neuberger, these proceedings . 2. H. S. M. Coxeter, Regular Polytopes (Dover, 1973) . H. S. M. Coxeter,Regular Complex Polytopes (Cambridge University Press, 1974) . 3. G. Bhanot, these proceedings . 4. G. Bhanot, K. Bitar, U. M. Heller, and H. Neuberger, work in progress . 5. J. C. A. Barata and K. Fredenhagen, Nucl. Phys. B (Proc . Suppl.) 4 (1988) 639. 6. M. Chanowitz and M. K. Gaillard, Nucl. Phys. B261 (1985) 379 . 7. M. Lûscher and P. Weisz, Nucl. Phys. B318 (1989) 705. 8. M. Lâcher, Commun . Math. Phys. 104 (9186) 177. 9. H. Neuberger, Phys. Rev . Lett. 60 (1988) 889 .