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Technicolor in the presence of extended technicolor interactions
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
TECHNICOLOR IN THE PRESENCE OF EXTENDED TECHNICOLOR INTERACTIONS Stephen F. K I N G and Douglas A. ROSS
PhysicsDepartment, The University,SouthamptonS09 5NH, UK Received 29 September 1989; revised manuscript received 22 November 1989
We study technicolor dynamics in the presence of heavy extended technicolor gauge boson exchange. We derive the effective potential which is then studied numerically in order to obtain the criticality curve, and consider the effect of making the fourfermion approximation. The gap equation is then solved numerically and the solutions X(p) are used to determine (~,~') and F,. Compared to corresponding results in the four-fermion approximation, we find a decreased sensitivity in F~ and (~u~) to extended technicolor (ETC) interactions for the same value of the dimensionless ETC coupling.
Technicolor ( T C ) [ 1 ], a new QCD-like gauge force which confines at ATC~ 1 TeV, is an alternative mechanism for breaking electroweak symmetry which does not rely on Higgs bosons. The role o f Yukawa couplings is played by heavy extended technicolor ( E T C ) [ 2 ] gauge bosons which couple fermions ( i.e. quarks and leptons) to technifermions. The d y n a m ically generated technifermion mass X(p) is fed-down radiatively to fermions which acquire a mass mr ~ ( 1/ M 2) (~'~u), where M is the ETC boson mass and (~7~') is the technifermion condensate. In this paper we consider another feature of ETC theories, which is that the ETC bosons may also couple technifermions to technifermions ~1. As we shall see, if such couplings are reasonably large they can have a considerable influence on technicolor d y n a m ics at A T c ~ 1 TeV despite the fact that M>> ATC. This problem has been recently studied in the idealised situation in which ETC boson exchange is approxim a t e d by a four-fermion interaction [ 3-5 ]. Here we use the full heavy gauge boson p r o p a g a t o r to derive the effective potential and hence the criticality curve, which is c o m p a r e d to the previous curve in the fourfermion and static coupling a p p r o x i m a t i o n s [3,4]. A new gap equation including the ETC gauge boson ex~ Such interactions arise from the broken U( 1) generators in the embedding SU (N)vc c SU ( N+ M) ETC.If orthogonal gauge groups are used such interactions are absent since the embedding SO(N)Tc = SU (N+ M)ETC involves no broken U( 1)'s.
change term is obtained and solved numerically, and the solution X(p) is used to d e t e r m i n e ( q ~ , ) and F=. C o m p a r e d to corresponding results in the four-fermion a p p r o x i m a t i o n [ 5 ], we find a decreased sensitivity in F~ and ( ~ ' ) to the ETC interaction for the same value of the dimensionless ETC coupling. O u r starting point is the S c h w i n g e r - D y s o n equation which, in ladder a p p r o x i m a t i o n , yields the following pair of coupled equations:
S(p)=_g2 f
d4k 7~Guv(p-k)7"S(k) (2zr)4 Z2(k)k2 S2(k ) ,
(1)
Z(p)-i g2( m
d4k Tr[yUGu~(p_k)p.kT~p.p]Z(k) 4 J (27~) 4 p2[Z2(k)k2-SZ(k)]
{2) where S(p) is the technifermion self-energy function, Z ( p ) is the wavefunction and G~,,(p-k) is the lowest order gauge boson propagator. In the present case the right-hand sides o f these equations consist o f two terms, one from massless TC gluon exchange and one from massive ETC gauge boson exchange. In the massless case, in Landau gauge Z = 1 for all p to all orders in X(p). This can be verified by performing the angular integrals in eq. ( 2 ) by contour integration which for a massless gauge boson propagator in Landau gauge cancel. However, for a massive gauge boson the above cancellation is spoiled, and the calculation o f Z(p) in Landau gauge becomes exceed-
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Volume 236, number 3
PHYSICS LETTERS B
ingly messy. For these reasons we find it convenient to work in Landau gauge for the massless TC gluons and Feynman gauge for the massive ETC gauge bosons ~2. Using these gauges we find, after continuing to euclidean space performing the angular integrations by standard methods, and introducing a cut-off,
22 February 1990
curve we need to examine the stability of the symmetric solution Z'(p) = 0 , by studying the second derivative of the effective potential in eq. (A.I), 02V
O£(p)OX(q)
X=O
M 2
1 3C2 f
X(p)=~-
k2d k 2 =c$(p2-- q 2)
J max(k2, p2) 0
S(k)
gZTc 42l. 2
× o~(max (k 2, p2) ) k2-+-"'~2 ( k )
F(p 2,
3C2 o~(max(p 2, q2) ) 47t max(p2, q2)
q2,
m2).
(6)
M 2
1g2TC f ~ -
k2d
k2 F(k2, p 2, M2 ) k 2 +Z'(k) X2(k)
,
0
(3) ~//2
g2TC ~ Z(P)-I=
1~5~2
k2dk2(
k2
k2+X2(k )
1
(k2+M2)21-p 2
0
-
-
(k2+p 2+M2 ) 2 ") ) p2 F(k , p ~ , 312)_ ,
(4)
where Z(p) has been subtracted so that Z(p)-, 1 as p2-~0, and F(k 2, p2 M 2 ) is given by
F(k2, p 2 , M 2 ) = - 1
2pZk 2
- ~2+M2)Z--4k2p
[ (kZ+p2+M 2) 51 .
(5)
Note that for p2 <
328
The first two terms on the right-hand side are just the usual terms considered in a previous study of the effective potential [7 ], and the third term is the new term originating from ETC gauge boson exchange contributions. As in the previous analysis [7] the coupling is held constant for P < A T c at some value c~z, and for p > A T c varies according to the usual oneloop renormalisation group expression. For a given fixed value o f 2 = g 2 v c / 4 g 2 and ~z, we examine the stability of the potential at the origin. This is done in numerical approximation by discretisingp 2 and q2 in eq. (6) and constructing the corresponding matrix and obtaining its eigenvalues. For sufficiently small values of c~z all the eigenvalues are positive corresponding to the symmetric solution X = 0 being a m i n i m u m of the potential and hence stable. For some critical value ~x---C~cone of the eigenvalues flips sign and the turning point at X = 0 becomes a saddle point and hence unstable. In this way the criticality curve for chiral symmetry breaking can be mapped out in the ,a.-c~cplane. In fig. 1 we show the resulting criticality curves for a typical TC theory based on SU (4) with nf= 12 Dirac technifermion flavours. Criticality curve (a) is obtained using the four-fermion approximation in which F-, 1/M 2 in eq. (6), and curve (b) is obtained using the full heavy gauge boson propagator. Also shown is criticality curve (c) corresponding to the four-fermion approximation F ~ 1 / M 2 with a static coupling oe. Criticality curve (c) has previously been obtained analytically [ 3,4 ], and is included here for comparison. Notice that in our numerical treatment the expected plateau at CZc=~z/3C2=0.56 in the region 0 < 2 < ] is slightly rounded in our numerical treatment, and intersects at c~c= 0.57 instead of 0.56. This
0,7 ~
, , , I
I
I
'
i
0,6 ~ 0,59
7"6',~............ 0,5 -
I I
""-........
I I I I
0,3
b}
1{a}
{c} '",
O( C
"X
I I I I
"... ~
0,2
"., I X~
0,1 0 ~
22 February 1990
PHYSICS LETTERS B
Volume 236, number 3
~ ~ ~ I 0,5
I
1,0
I
I
I
\
1,5
N Fig. 1. The criticalily curves for SU(4) corresponding to {a) nr= 12, in the four-fermion approximation, (b) nr= 12 using the full ETC gaugeboson propagator, (c) nf--22 in the four-fermion approximation, which corresponds to the static coupling limit. minor discrepancy reflects the level of accuracy of our numerical approximation. Criticality curve (a) corresponding to the four-fermion approximation exhibits a plateau at a c = 0 . 5 9 in the region 0 < 2 < 0 . 8 9 . As noted in ref. [7] for pure TC theories ( 2 = 0 ) the effect of a running coupling is to increase the value o f a c by a few per cent relative to zc/3C2 in order to compensate for the diminishing coupling at large p. The extent of the plateau region depends on the rate of running of the coupling. The plateau region shrinks as nr is increased until it extends out only as far a s 2 ~ 0 . 2 5 in the static coupling limit (criticality curve (c) ). Over the plateau region the four-fermion interaction apparently contributes nothing to the process of chiral symmetry breaking, while beyond this region the critical gauge coupling ac required to trigger the symmetry breaking is smoothly reduced to zero as 2-~1, which is the N a m b u - J o n a - L a s i n i o limit. Criticality curve (b) corresponding to using the full heavy ETC gauge boson propagator exhibits an extended plateau region 0 < 2 < 1.58. The value of ac smoothly tends to zero as 2-~ 1.77, which is significantly larger than unity. This effect can be understood from the fact that 1 / M 2 is replaced by F ( k 2, 1)2, M 2 ) in eqs. ( 5 ), (6) which is numerically smaller than 1//12 for all values o f k 2 and pZ. Hence a larger value of 2 is required to trigger chiral symmetry
breaking for a given value of ac, in order to compensate for this reduction. Having determined the criticality curves from the effective potential, we now turn to the task of solving the gap equation in (3). This equation is solved numerically using the same method as for the pure TC theory [ 7 ]. Above the chiral symmetry breaking scale p = 2X(0) the TC coupling evolves according to the one-loop result a(p>/x)=
a(#) 1+ b a ( l L ) l n ( p / p ) '
(7)
where 1
b = 2~ [ ~ C 2 ( a d j ) - ~ T ( R ) r t f ]
(8)
,
and C2(adj ) is the Casimir of the adjoint representation, T ( R ) is the index of the fermion representation R and nf is the number of Dirac fermions. Confinement dynamics is crudely approximated by introducing some infrared cut-off or "confinement" scale ATC SO that for p < ATC the coupling takes a constant value ao. In the intermediate region ATC < P < P (if it exists) the fermions do not contribute and the coupling evolves according to ao a(Arc
(9)
where
bo= ~l ~q~(adj) .
(10)
The gap equation therefore determines its own value of X(0) and hence y - 2 X ( 0 ) . Therefore for each chosen value o f a o , a corresponding value of a ( p ) is determined. Once the solution X(p 2) is found, the condensate is determined from M 2
d(R) f p2dp2X(p) (Y~¢ff),~/= 47t2 J Z 2 ( p ) p 2 + Z 2 ( p ) '
(11)
where d ( R ) is the dimension of the fermion representation (rep) R, and Z ( p 2) is given in terms of S ( k 2 ) in eq. (4). Similarly F 2 is calculated from the standard results [4] whose approximate form is ,'vl2
F~
X2(p)-½P~(P)~5,
•
(12)
329
Volume 236, number 3
PHYSICS LETTERS B
T h e r e s u l t s are s h o w n in t a b l e 1 f o r t h e t h e o r y G T c = S U ( 4 ) w i t h n f = 12, c o r r e s p o n d i n g to t h e criticality c u r v e s ( a ) a n d ( b ) in fig. 1. In t h e first colu m n t h e v a l u e o f 2 = g / T C / 4 ~ Z 2 is g i v e n , t h e s e c o n d c o l u m n s h o w s t h e c o r r e s p o n d i n g v a l u e o f c~(/z) = c~. The next two columns show Z'(0)/ATc and S ( M ) / A r c w h e r e M/ATc= 2 5 0 0 a n d A r c is set e q u a l to u n ity to set t h e scale. N o t e t h a t t h e i n t e r m e d i a t e r e g i o n
/t/ATc is e q u a l
to t w i c e X ( 0 ) ~ A r c . T h e f i n a l t w o colu m n s s h o w scaled v a l u e s o f F~ a n d (qT~u) M w h i c h are d e r i v e d f r o m X(p) as d i s c u s s e d a b o v e . In t a b l e 1 ( f i r s t p a r t ) t h e r e s u l t s f o r S U ( 4 ) w i t h n f = 12 a r e s h o w n u s i n g t h e f o u r - f e r m i o n a p p r o x i m a t i o n . O v e r t h e p l a t e a u region o f t h e criticality c u r v e 0 < 2 < 0 . 8 9 , X(O)/ATc a n d FJATc r e m a i n c o n s t a n t w h i l e X(M) ~Arc a n d (q)~ff)/MZATc g r o w b y s e v e r a l o r d e r s o f m a g n i t u d e . F o r 2 > 0.89 b o t h Z ' ( 0 ) / A T c a n d FJATc g r o w q u i c k l y w i t h i n c r e a s i n g 2 c o r r e s p o n d i n g to t h e r a p i d f a l l - o f f o f c~ in c r i t i c a l i t y c u r v e ( a ) o f fig. 1, w h i l e X(M)/ATc a n d (~IJ)M/M2ATc g r o w e v e r m o r e q u i c k l y w i t h i n c r e a s i n g 2 so t h a t b y 2 = 0.94
22 February 1990
t h e f u n c t i o n S(p) is q u i t e flat. T h r o u g h o u t we c h o s e c % = 2 . 3 5 w h i c h l e a d s to c~(/z) =c~c w h e r e ~ is obt a i n e d f r o m t h e c r i t i c a l i t y c u r v e s (fig. 1 ). T h e c o n d i t i o n c~ (/1) = c~ is a p p r o x i m a t e l y m a i n t a i n e d e v e n b e y o n d t h e p l a t e a u region w h e r e c~ is f a l l i n g w i t h o u t a d j u s t i n g c% #4. N o t e t h a t t h e o n s e t o f t h e i n c r e a s e in X(O) ~Arc c o r r e s p o n d s to t h e v a l u e o f 2 for w h i c h o~ b e g i n s to fall. T a b l e 1 ( l o w e r p a r t ) d i s p l a y s t h e r e s u l t s for t h e s a m e S U ( 4 ) t h e o r y w i t h n r = 12 b u t u s i n g t h e full E T C gauge b o s o n p r o p a g a t o r . O v e r t h e p l a t e a u r e g i o n ~4 This can be understood from the definition p ---2,5"(0) coupled with our ansatz for the behaviour of the running coupling in eqs. (7)-(10). As X(0) (and hence jl) increases, with Arc and o~0fixed, so the value o f a ( p ) drops according to eq. (9). The solutions for 2 >> 0.89 arrange themselves so that o~(#) = e~c is approximately maintained. Strictly speaking we cannot identify ~(#) with ~c from the criticality curves, because the behaviour of the running coupling is slightly different in the important low momentum region. But since e¢o is an arbitrary parameter we find it convenient to set it equal to 2.35 in order to obtain ~(p) =~c-
Table 1 The results of a numerical study of chiral symmetry breaking for SU(4) with ?If= 12 is shown in two cases, using the four-fermion approximation, and using the full ETC gauge boson propagator. We choose the same value So= 2.35 and keep it fixed in both cases. This maintains the condition that ~ (/2) = ~x~,where ~c is read-off from the appropriate criticality curve in fig. 2, and/1 = 2S(0 ) is the scale of chiral symmetry breaking. In both cases X(O)/ATc and FJATc are constant over the plateau regions of the criticality curves then grow steeply in response to the rapidly falling ~c. S(M) ~Arcand (q)~') M/M2ATcgrow steadily across the plateau region and grow dramatically beyond the plateau region. Note that we choose M/ATc=2500 with A'rc= 1 (fixed) rather than fixing F~ as in ref. [5 ]. Case
2
o~(#) = a ¢
Z(0)/ATC
Z(M)/AT c
FJATc
(~N)M/M2ATc
Fermi approximation
0.00 0.50 0.80 0.89 0.90 0.91 0.92 0.93 0.94
0.59 0.59 0.59 0.59 0.52 0.34 0.24 0.17 0.11
0.86 0.86 0.86 0.86 0.97 1.4 2.4 5.3 18
2.2X 10 8 5.8X 10 -7 4.0)< 10 6 4.6X 10 -5 1.5XI0 2 1.3)< 10 -~ 6.1 )< 10-L 2.4 12
0.19 0.19 0.19 0.19 0.22 0.43 1.3 3.7 14
4,6× 10 -8 1.1 × 10 -7 4.7X 10 -7 5.0)< 10 -6 1.6X10 -3 1.4)< 10 -2 6.5× 10 -2 2.5)< 10 -~ 1.2
full gauge propagator
0.00 0.50 1.00 1.50 1.58 1.60 1.62 1.64 1.66 1.68
0.59 0.59 0.59 0.59 0.59 0.42 0.27 0.19 0.15 0.11
0.86 0.86 0.86 0.86 0.86 1.25 1.9 3.9 6.5 23
2.2x 10 8
0.19 0.19 0.19 0.19 0.19 0.34 0.78 2.4 4.5 17
4.6×10 -8 7.2×10 8
330
2.0X 10 -7 7 . 3 X 10 -7 9 . 3 × 10 -6 1.3)< 10 -4
0.5)<10 ~ 2.0)< 10- ~ 9.1 )< 10 -~ 1.9 9.5
1.4X 10 -7
1.3X I0 -~ 1.7)< 10 -5 6.7X 10 -3 2.6 × 10 -2 1.2)< 10 -~ 2.5× 10 -~ 1.2
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
0 < 2 < 1.58, X (0)/ATc and F,~/ATcare constant while X(M)/ATc and ((/q.,)/M2ATc grow by several orders o f magnitude, as in the four-fermion approximation. For 2 > 1.58 both X ( 0 ) / A T c and F,~/ATcgrow quickly with increasing 2 corresponding to the rapid f a l l - o f f o f o~c in criticality curve ( b ) in fig. 1, while X(M)/ATc and (~tg/)M/M2A.rc grow ever more quickly with increasing 2, so that by 2 = 1.68 the function X(p) is quite flat. In other words the results for the full theory m i m i c those obtained in the four-fermion a p p r o x i m a t i o n , but occur for correspondingly larger values o f 2. This results from using F(k 2, p2 M 2) r a t h e r t h a n 1 / M 2 i n e q s . ( 3 ) - ( 5 ) . In conclusion ETC interactions can have an important effect on (gTg/) and F~. O u r results, based on using the full ETC gauge boson propagator, confirm the results of other authors based on the four-fermion approximation [ 3-5 ]. The effect of using the full ETC propagator is to replace 1/M 2 by F(k 2, p2 M 2) in eqs. ( 3 ) - ( 6 ) which has the effect o f increasing the value of 2 required for chiral symmetry breaking, since F< 1/M 2 for all k 2, p2. We wish to emphasise that these results have important physical implications for technicolor theories. F o r instance it has been suggested that isospin-violating techniquark condensates resulting from ETC interactions may be responsible for the c h a r m - s t r a n g e quark mass splitting [8 ].
Acknowledgement We thank T i m Morris for useful discussions. One o f us ( S . F . K ) gratefully acknowledges the support o f an SERC A d v a n c e d Fellowship.
(o)
{b)
Fig. 2. Graphs contributing to Vin the ladder approximation. In (a) a massless technigluon is exchanged across the technifermion bubble, while in (b) a massive ETC gauge boson is exchanged.
V[X,Z]= 4~z2CA - ½ In
k2 dk21 Z(k,)kZt +X2(kt)
[ZX(k,)k~+X2(k,)])
f k 2 dk 2 k 2 dk2X(k,)X(k2)
J [l,-~+S2(k, ) 1 [k~ +$2(k2) ] ×
3C2 c~(max(k~2, k22) ) 47~ max(k 2, k22) g~Tc F ( k ~ 4 7 t z , k~ , M 2 ) ) J
,
(A. 1)
where the function F i s given by eq. (5). Note that in the second term Z is set to 1 consistent with this a p p r o x i m a t i o n . The first graph is calculated in the Landau gauge so that Z only has a contribution from massive gauge boson exchange (eq. ( 4 ) ) . The gap equation (eq. ( 3 ) ) may be obtained from the above effective action by equating to zero the functional derivative with respect to X and neglecting terms proportional to 8z/SX~Xa/M 2, which is assumed to be negligible.
Appendix
References
The effective potential, V, in the ladder approximation, is obtained from the two graphs shown in fig. 2. In fig. 2a the gauge boson exchanged is a massless technigluon whereas in fig. 2b it is a massive ETC gauge boson. The fermion lines are understood to be the full propagators. V is in general a functional o f both _F and Z given by
[ 1] S. Weinberg, Phys. Rev. D 19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. [2] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. B 90 (1980) 125. [3] K. Kondo, H. Mino and K. Yamawaki, Phys. Rev. D 39 (1989) 1. [4]T. Appelquist, M. Soldate, T. Takeuchi and L.C.R. Wijewardhana, Effective four-fermion interactions and chiral symmetry breaking, in: Proc. 12th Johns Hopkins Workshop on Current problems in particle theory (Baltimore, MD, June 1988). 331
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[5]T. Appelquist, T. Takeuchi, M. Einhorn and L.C.R. Wijewardhana, Phys. Lett. B 220 (1989) 223. [6] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D 10 (1974) 2428;
332
22 Feoruary 1990
M.E. Peskin, in: Recent advances in field theory and statistical mechanics, Les Houches Lectures (1982), eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984 ). [7] S. King and D. Ross, Phys. Lett. B 228 (1989) 363. [8] B. Holdom, Phys. Lett. B 226 (1989) 137.
PHYSICS LETTERS B
22 February 1990
TECHNICOLOR IN THE PRESENCE OF EXTENDED TECHNICOLOR INTERACTIONS Stephen F. K I N G and Douglas A. ROSS
PhysicsDepartment, The University,SouthamptonS09 5NH, UK Received 29 September 1989; revised manuscript received 22 November 1989
We study technicolor dynamics in the presence of heavy extended technicolor gauge boson exchange. We derive the effective potential which is then studied numerically in order to obtain the criticality curve, and consider the effect of making the fourfermion approximation. The gap equation is then solved numerically and the solutions X(p) are used to determine (~,~') and F,. Compared to corresponding results in the four-fermion approximation, we find a decreased sensitivity in F~ and (~u~) to extended technicolor (ETC) interactions for the same value of the dimensionless ETC coupling.
Technicolor ( T C ) [ 1 ], a new QCD-like gauge force which confines at ATC~ 1 TeV, is an alternative mechanism for breaking electroweak symmetry which does not rely on Higgs bosons. The role o f Yukawa couplings is played by heavy extended technicolor ( E T C ) [ 2 ] gauge bosons which couple fermions ( i.e. quarks and leptons) to technifermions. The d y n a m ically generated technifermion mass X(p) is fed-down radiatively to fermions which acquire a mass mr ~ ( 1/ M 2) (~'~u), where M is the ETC boson mass and (~7~') is the technifermion condensate. In this paper we consider another feature of ETC theories, which is that the ETC bosons may also couple technifermions to technifermions ~1. As we shall see, if such couplings are reasonably large they can have a considerable influence on technicolor d y n a m ics at A T c ~ 1 TeV despite the fact that M>> ATC. This problem has been recently studied in the idealised situation in which ETC boson exchange is approxim a t e d by a four-fermion interaction [ 3-5 ]. Here we use the full heavy gauge boson p r o p a g a t o r to derive the effective potential and hence the criticality curve, which is c o m p a r e d to the previous curve in the fourfermion and static coupling a p p r o x i m a t i o n s [3,4]. A new gap equation including the ETC gauge boson ex~ Such interactions arise from the broken U( 1) generators in the embedding SU (N)vc c SU ( N+ M) ETC.If orthogonal gauge groups are used such interactions are absent since the embedding SO(N)Tc = SU (N+ M)ETC involves no broken U( 1)'s.
change term is obtained and solved numerically, and the solution X(p) is used to d e t e r m i n e ( q ~ , ) and F=. C o m p a r e d to corresponding results in the four-fermion a p p r o x i m a t i o n [ 5 ], we find a decreased sensitivity in F~ and ( ~ ' ) to the ETC interaction for the same value of the dimensionless ETC coupling. O u r starting point is the S c h w i n g e r - D y s o n equation which, in ladder a p p r o x i m a t i o n , yields the following pair of coupled equations:
S(p)=_g2 f
d4k 7~Guv(p-k)7"S(k) (2zr)4 Z2(k)k2 S2(k ) ,
(1)
Z(p)-i g2( m
d4k Tr[yUGu~(p_k)p.kT~p.p]Z(k) 4 J (27~) 4 p2[Z2(k)k2-SZ(k)]
{2) where S(p) is the technifermion self-energy function, Z ( p ) is the wavefunction and G~,,(p-k) is the lowest order gauge boson propagator. In the present case the right-hand sides o f these equations consist o f two terms, one from massless TC gluon exchange and one from massive ETC gauge boson exchange. In the massless case, in Landau gauge Z = 1 for all p to all orders in X(p). This can be verified by performing the angular integrals in eq. ( 2 ) by contour integration which for a massless gauge boson propagator in Landau gauge cancel. However, for a massive gauge boson the above cancellation is spoiled, and the calculation o f Z(p) in Landau gauge becomes exceed-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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Volume 236, number 3
PHYSICS LETTERS B
ingly messy. For these reasons we find it convenient to work in Landau gauge for the massless TC gluons and Feynman gauge for the massive ETC gauge bosons ~2. Using these gauges we find, after continuing to euclidean space performing the angular integrations by standard methods, and introducing a cut-off,
22 February 1990
curve we need to examine the stability of the symmetric solution Z'(p) = 0 , by studying the second derivative of the effective potential in eq. (A.I), 02V
O£(p)OX(q)
X=O
M 2
1 3C2 f
X(p)=~-
k2d k 2 =c$(p2-- q 2)
J max(k2, p2) 0
S(k)
gZTc 42l. 2
× o~(max (k 2, p2) ) k2-+-"'~2 ( k )
F(p 2,
3C2 o~(max(p 2, q2) ) 47t max(p2, q2)
q2,
m2).
(6)
M 2
1g2TC f ~ -
k2d
k2 F(k2, p 2, M2 ) k 2 +Z'(k) X2(k)
,
0
(3) ~//2
g2TC ~ Z(P)-I=
1~5~2
k2dk2(
k2
k2+X2(k )
1
(k2+M2)21-p 2
0
-
-
(k2+p 2+M2 ) 2 ") ) p2 F(k , p ~ , 312)_ ,
(4)
where Z(p) has been subtracted so that Z(p)-, 1 as p2-~0, and F(k 2, p2 M 2 ) is given by
F(k2, p 2 , M 2 ) = - 1
2pZk 2
- ~2+M2)Z--4k2p
[ (kZ+p2+M 2) 51 .
(5)
Note that for p2 <
328
The first two terms on the right-hand side are just the usual terms considered in a previous study of the effective potential [7 ], and the third term is the new term originating from ETC gauge boson exchange contributions. As in the previous analysis [7] the coupling is held constant for P < A T c at some value c~z, and for p > A T c varies according to the usual oneloop renormalisation group expression. For a given fixed value o f 2 = g 2 v c / 4 g 2 and ~z, we examine the stability of the potential at the origin. This is done in numerical approximation by discretisingp 2 and q2 in eq. (6) and constructing the corresponding matrix and obtaining its eigenvalues. For sufficiently small values of c~z all the eigenvalues are positive corresponding to the symmetric solution X = 0 being a m i n i m u m of the potential and hence stable. For some critical value ~x---C~cone of the eigenvalues flips sign and the turning point at X = 0 becomes a saddle point and hence unstable. In this way the criticality curve for chiral symmetry breaking can be mapped out in the ,a.-c~cplane. In fig. 1 we show the resulting criticality curves for a typical TC theory based on SU (4) with nf= 12 Dirac technifermion flavours. Criticality curve (a) is obtained using the four-fermion approximation in which F-, 1/M 2 in eq. (6), and curve (b) is obtained using the full heavy gauge boson propagator. Also shown is criticality curve (c) corresponding to the four-fermion approximation F ~ 1 / M 2 with a static coupling oe. Criticality curve (c) has previously been obtained analytically [ 3,4 ], and is included here for comparison. Notice that in our numerical treatment the expected plateau at CZc=~z/3C2=0.56 in the region 0 < 2 < ] is slightly rounded in our numerical treatment, and intersects at c~c= 0.57 instead of 0.56. This
0,7 ~
, , , I
I
I
'
i
0,6 ~ 0,59
7"6',~............ 0,5 -
I I
""-........
I I I I
0,3
b}
1{a}
{c} '",
O( C
"X
I I I I
"... ~
0,2
"., I X~
0,1 0 ~
22 February 1990
PHYSICS LETTERS B
Volume 236, number 3
~ ~ ~ I 0,5
I
1,0
I
I
I
\
1,5
N Fig. 1. The criticalily curves for SU(4) corresponding to {a) nr= 12, in the four-fermion approximation, (b) nr= 12 using the full ETC gaugeboson propagator, (c) nf--22 in the four-fermion approximation, which corresponds to the static coupling limit. minor discrepancy reflects the level of accuracy of our numerical approximation. Criticality curve (a) corresponding to the four-fermion approximation exhibits a plateau at a c = 0 . 5 9 in the region 0 < 2 < 0 . 8 9 . As noted in ref. [7] for pure TC theories ( 2 = 0 ) the effect of a running coupling is to increase the value o f a c by a few per cent relative to zc/3C2 in order to compensate for the diminishing coupling at large p. The extent of the plateau region depends on the rate of running of the coupling. The plateau region shrinks as nr is increased until it extends out only as far a s 2 ~ 0 . 2 5 in the static coupling limit (criticality curve (c) ). Over the plateau region the four-fermion interaction apparently contributes nothing to the process of chiral symmetry breaking, while beyond this region the critical gauge coupling ac required to trigger the symmetry breaking is smoothly reduced to zero as 2-~1, which is the N a m b u - J o n a - L a s i n i o limit. Criticality curve (b) corresponding to using the full heavy ETC gauge boson propagator exhibits an extended plateau region 0 < 2 < 1.58. The value of ac smoothly tends to zero as 2-~ 1.77, which is significantly larger than unity. This effect can be understood from the fact that 1 / M 2 is replaced by F ( k 2, 1)2, M 2 ) in eqs. ( 5 ), (6) which is numerically smaller than 1//12 for all values o f k 2 and pZ. Hence a larger value of 2 is required to trigger chiral symmetry
breaking for a given value of ac, in order to compensate for this reduction. Having determined the criticality curves from the effective potential, we now turn to the task of solving the gap equation in (3). This equation is solved numerically using the same method as for the pure TC theory [ 7 ]. Above the chiral symmetry breaking scale p = 2X(0) the TC coupling evolves according to the one-loop result a(p>/x)=
a(#) 1+ b a ( l L ) l n ( p / p ) '
(7)
where 1
b = 2~ [ ~ C 2 ( a d j ) - ~ T ( R ) r t f ]
(8)
,
and C2(adj ) is the Casimir of the adjoint representation, T ( R ) is the index of the fermion representation R and nf is the number of Dirac fermions. Confinement dynamics is crudely approximated by introducing some infrared cut-off or "confinement" scale ATC SO that for p < ATC the coupling takes a constant value ao. In the intermediate region ATC < P < P (if it exists) the fermions do not contribute and the coupling evolves according to ao a(Arc
(9)
where
bo= ~l ~q~(adj) .
(10)
The gap equation therefore determines its own value of X(0) and hence y - 2 X ( 0 ) . Therefore for each chosen value o f a o , a corresponding value of a ( p ) is determined. Once the solution X(p 2) is found, the condensate is determined from M 2
d(R) f p2dp2X(p) (Y~¢ff),~/= 47t2 J Z 2 ( p ) p 2 + Z 2 ( p ) '
(11)
where d ( R ) is the dimension of the fermion representation (rep) R, and Z ( p 2) is given in terms of S ( k 2 ) in eq. (4). Similarly F 2 is calculated from the standard results [4] whose approximate form is ,'vl2
F~
X2(p)-½P~(P)~5,
•
(12)
329
Volume 236, number 3
PHYSICS LETTERS B
T h e r e s u l t s are s h o w n in t a b l e 1 f o r t h e t h e o r y G T c = S U ( 4 ) w i t h n f = 12, c o r r e s p o n d i n g to t h e criticality c u r v e s ( a ) a n d ( b ) in fig. 1. In t h e first colu m n t h e v a l u e o f 2 = g / T C / 4 ~ Z 2 is g i v e n , t h e s e c o n d c o l u m n s h o w s t h e c o r r e s p o n d i n g v a l u e o f c~(/z) = c~. The next two columns show Z'(0)/ATc and S ( M ) / A r c w h e r e M/ATc= 2 5 0 0 a n d A r c is set e q u a l to u n ity to set t h e scale. N o t e t h a t t h e i n t e r m e d i a t e r e g i o n
/t/ATc is e q u a l
to t w i c e X ( 0 ) ~ A r c . T h e f i n a l t w o colu m n s s h o w scaled v a l u e s o f F~ a n d (qT~u) M w h i c h are d e r i v e d f r o m X(p) as d i s c u s s e d a b o v e . In t a b l e 1 ( f i r s t p a r t ) t h e r e s u l t s f o r S U ( 4 ) w i t h n f = 12 a r e s h o w n u s i n g t h e f o u r - f e r m i o n a p p r o x i m a t i o n . O v e r t h e p l a t e a u region o f t h e criticality c u r v e 0 < 2 < 0 . 8 9 , X(O)/ATc a n d FJATc r e m a i n c o n s t a n t w h i l e X(M) ~Arc a n d (q)~ff)/MZATc g r o w b y s e v e r a l o r d e r s o f m a g n i t u d e . F o r 2 > 0.89 b o t h Z ' ( 0 ) / A T c a n d FJATc g r o w q u i c k l y w i t h i n c r e a s i n g 2 c o r r e s p o n d i n g to t h e r a p i d f a l l - o f f o f c~ in c r i t i c a l i t y c u r v e ( a ) o f fig. 1, w h i l e X(M)/ATc a n d (~IJ)M/M2ATc g r o w e v e r m o r e q u i c k l y w i t h i n c r e a s i n g 2 so t h a t b y 2 = 0.94
22 February 1990
t h e f u n c t i o n S(p) is q u i t e flat. T h r o u g h o u t we c h o s e c % = 2 . 3 5 w h i c h l e a d s to c~(/z) =c~c w h e r e ~ is obt a i n e d f r o m t h e c r i t i c a l i t y c u r v e s (fig. 1 ). T h e c o n d i t i o n c~ (/1) = c~ is a p p r o x i m a t e l y m a i n t a i n e d e v e n b e y o n d t h e p l a t e a u region w h e r e c~ is f a l l i n g w i t h o u t a d j u s t i n g c% #4. N o t e t h a t t h e o n s e t o f t h e i n c r e a s e in X(O) ~Arc c o r r e s p o n d s to t h e v a l u e o f 2 for w h i c h o~ b e g i n s to fall. T a b l e 1 ( l o w e r p a r t ) d i s p l a y s t h e r e s u l t s for t h e s a m e S U ( 4 ) t h e o r y w i t h n r = 12 b u t u s i n g t h e full E T C gauge b o s o n p r o p a g a t o r . O v e r t h e p l a t e a u r e g i o n ~4 This can be understood from the definition p ---2,5"(0) coupled with our ansatz for the behaviour of the running coupling in eqs. (7)-(10). As X(0) (and hence jl) increases, with Arc and o~0fixed, so the value o f a ( p ) drops according to eq. (9). The solutions for 2 >> 0.89 arrange themselves so that o~(#) = e~c is approximately maintained. Strictly speaking we cannot identify ~(#) with ~c from the criticality curves, because the behaviour of the running coupling is slightly different in the important low momentum region. But since e¢o is an arbitrary parameter we find it convenient to set it equal to 2.35 in order to obtain ~(p) =~c-
Table 1 The results of a numerical study of chiral symmetry breaking for SU(4) with ?If= 12 is shown in two cases, using the four-fermion approximation, and using the full ETC gauge boson propagator. We choose the same value So= 2.35 and keep it fixed in both cases. This maintains the condition that ~ (/2) = ~x~,where ~c is read-off from the appropriate criticality curve in fig. 2, and/1 = 2S(0 ) is the scale of chiral symmetry breaking. In both cases X(O)/ATc and FJATc are constant over the plateau regions of the criticality curves then grow steeply in response to the rapidly falling ~c. S(M) ~Arcand (q)~') M/M2ATcgrow steadily across the plateau region and grow dramatically beyond the plateau region. Note that we choose M/ATc=2500 with A'rc= 1 (fixed) rather than fixing F~ as in ref. [5 ]. Case
2
o~(#) = a ¢
Z(0)/ATC
Z(M)/AT c
FJATc
(~N)M/M2ATc
Fermi approximation
0.00 0.50 0.80 0.89 0.90 0.91 0.92 0.93 0.94
0.59 0.59 0.59 0.59 0.52 0.34 0.24 0.17 0.11
0.86 0.86 0.86 0.86 0.97 1.4 2.4 5.3 18
2.2X 10 8 5.8X 10 -7 4.0)< 10 6 4.6X 10 -5 1.5XI0 2 1.3)< 10 -~ 6.1 )< 10-L 2.4 12
0.19 0.19 0.19 0.19 0.22 0.43 1.3 3.7 14
4,6× 10 -8 1.1 × 10 -7 4.7X 10 -7 5.0)< 10 -6 1.6X10 -3 1.4)< 10 -2 6.5× 10 -2 2.5)< 10 -~ 1.2
full gauge propagator
0.00 0.50 1.00 1.50 1.58 1.60 1.62 1.64 1.66 1.68
0.59 0.59 0.59 0.59 0.59 0.42 0.27 0.19 0.15 0.11
0.86 0.86 0.86 0.86 0.86 1.25 1.9 3.9 6.5 23
2.2x 10 8
0.19 0.19 0.19 0.19 0.19 0.34 0.78 2.4 4.5 17
4.6×10 -8 7.2×10 8
330
2.0X 10 -7 7 . 3 X 10 -7 9 . 3 × 10 -6 1.3)< 10 -4
0.5)<10 ~ 2.0)< 10- ~ 9.1 )< 10 -~ 1.9 9.5
1.4X 10 -7
1.3X I0 -~ 1.7)< 10 -5 6.7X 10 -3 2.6 × 10 -2 1.2)< 10 -~ 2.5× 10 -~ 1.2
Volume 236, number 3
PHYSICS LETTERS B
22 February 1990
0 < 2 < 1.58, X (0)/ATc and F,~/ATcare constant while X(M)/ATc and ((/q.,)/M2ATc grow by several orders o f magnitude, as in the four-fermion approximation. For 2 > 1.58 both X ( 0 ) / A T c and F,~/ATcgrow quickly with increasing 2 corresponding to the rapid f a l l - o f f o f o~c in criticality curve ( b ) in fig. 1, while X(M)/ATc and (~tg/)M/M2A.rc grow ever more quickly with increasing 2, so that by 2 = 1.68 the function X(p) is quite flat. In other words the results for the full theory m i m i c those obtained in the four-fermion a p p r o x i m a t i o n , but occur for correspondingly larger values o f 2. This results from using F(k 2, p2 M 2) r a t h e r t h a n 1 / M 2 i n e q s . ( 3 ) - ( 5 ) . In conclusion ETC interactions can have an important effect on (gTg/) and F~. O u r results, based on using the full ETC gauge boson propagator, confirm the results of other authors based on the four-fermion approximation [ 3-5 ]. The effect of using the full ETC propagator is to replace 1/M 2 by F(k 2, p2 M 2) in eqs. ( 3 ) - ( 6 ) which has the effect o f increasing the value of 2 required for chiral symmetry breaking, since F< 1/M 2 for all k 2, p2. We wish to emphasise that these results have important physical implications for technicolor theories. F o r instance it has been suggested that isospin-violating techniquark condensates resulting from ETC interactions may be responsible for the c h a r m - s t r a n g e quark mass splitting [8 ].
Acknowledgement We thank T i m Morris for useful discussions. One o f us ( S . F . K ) gratefully acknowledges the support o f an SERC A d v a n c e d Fellowship.
(o)
{b)
Fig. 2. Graphs contributing to Vin the ladder approximation. In (a) a massless technigluon is exchanged across the technifermion bubble, while in (b) a massive ETC gauge boson is exchanged.
V[X,Z]= 4~z2CA - ½ In
k2 dk21 Z(k,)kZt +X2(kt)
[ZX(k,)k~+X2(k,)])
f k 2 dk 2 k 2 dk2X(k,)X(k2)
J [l,-~+S2(k, ) 1 [k~ +$2(k2) ] ×
3C2 c~(max(k~2, k22) ) 47~ max(k 2, k22) g~Tc F ( k ~ 4 7 t z , k~ , M 2 ) ) J
,
(A. 1)
where the function F i s given by eq. (5). Note that in the second term Z is set to 1 consistent with this a p p r o x i m a t i o n . The first graph is calculated in the Landau gauge so that Z only has a contribution from massive gauge boson exchange (eq. ( 4 ) ) . The gap equation (eq. ( 3 ) ) may be obtained from the above effective action by equating to zero the functional derivative with respect to X and neglecting terms proportional to 8z/SX~Xa/M 2, which is assumed to be negligible.
Appendix
References
The effective potential, V, in the ladder approximation, is obtained from the two graphs shown in fig. 2. In fig. 2a the gauge boson exchanged is a massless technigluon whereas in fig. 2b it is a massive ETC gauge boson. The fermion lines are understood to be the full propagators. V is in general a functional o f both _F and Z given by
[ 1] S. Weinberg, Phys. Rev. D 19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. [2] S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. B 90 (1980) 125. [3] K. Kondo, H. Mino and K. Yamawaki, Phys. Rev. D 39 (1989) 1. [4]T. Appelquist, M. Soldate, T. Takeuchi and L.C.R. Wijewardhana, Effective four-fermion interactions and chiral symmetry breaking, in: Proc. 12th Johns Hopkins Workshop on Current problems in particle theory (Baltimore, MD, June 1988). 331
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[5]T. Appelquist, T. Takeuchi, M. Einhorn and L.C.R. Wijewardhana, Phys. Lett. B 220 (1989) 223. [6] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D 10 (1974) 2428;
332
22 Feoruary 1990
M.E. Peskin, in: Recent advances in field theory and statistical mechanics, Les Houches Lectures (1982), eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984 ). [7] S. King and D. Ross, Phys. Lett. B 228 (1989) 363. [8] B. Holdom, Phys. Lett. B 226 (1989) 137.