Study of the chiral transition in a U(1) gauge-higgs-fermion theory with yukawa couplings

Volume 199, number 4

PHYSICS LETTERS B

31 December 1987

STUDY OF THE CHIRAL TRANSITION IN A U(1) G A U G E - H I G G S - F E R M I O N T H E O R Y W I T H YUKAWA C O U P L I N G S I-Hsiu LEE

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA and R o b e r t E. S H R O C K

Institutefor TheoreticalPhysics. State Universityof New York, Stony Brook. NY 11794. USA Received 28 September 1987

We study the (zero-temperature) chiral transition in a U(I ) gauge-Higgs-fermion theory with Yukawa couplings involving fermions of two different charges. Using analytic methods in the strong coupling limit, we find that for a range of sufficiently weak Yukawa couplings, this transition continues to exist and be of second-order, but is removed as the strength of the Yukawa coupling increases through a critical value. Some implications for the continuum limit of this model, and for electroweak interactions, are noted.

It has recently been established that an i m p o r t a n t feature o f lattice gauge-Higgs models with fermions is the ( z e r o - t e m p e r a t u r e ) chiral transition [ 1 - 5 ] separating a phase where there is spontaneous chiral symmetry breaking (SzSB) from phase(s) where this s y m m e t r y is realized explicitly. This has i m p o r t a n t i m p l i c a t i o n s for the u n d e r s t a n d i n g o f lattice models o f weak interactions. In this paper we present the first study o f the chiral transition in a lattice g a u g e H i g g s - f e r m i o n theory which includes a Yukawa interaction ~J. It is clearly desirable to study the effect o f a Yukawa coupling since such couplings a p p e a r in the current s t a n d a r d electroweak theory, as well as generalizations thereof to higher unification. We find that it has a very interesting effect on the chiral transition. F o r this study we choose a U ( 1 ) lattice gauge model with a Higgs field 0 having charge q , = 1 and two different fermions fields, ~ and {, with charges qz = 1 and q¢ = 0. One m a y think o f this as a simplified, abelian m o d e l o f the S U ( 2 ) part o f the stan-

d a r d electroweak theory, with )~ corresponding to the I = 1 fermions, { corresponding to the I = 0 fermions, and 0 to the I = ½ Higgs. Although the fermions are coupled in a vectorial manner to the gauge fields, this does not reduce the physical relevance o f the model, since the S U ( 2 ) sector o f the s t a n d a r d electroweak theory can be written in a vectorlike m a n n e r [7] (essentially because the representations o f SU (2) are real). Our m o d e l is defined in usual n o t a t i o n by the discretized path integral

Z = f M dU,,,¢, dO,, dO,*, d z . d~,,d~., d ~ . e - s ,

(la)

pl,l*

where S = SG + SH + SF + Sv, with

So=fig ~

[1-P],

(lb)

S. = - 2/~h Z Re{0~* g,,4,0n +~.,,},

(lc)

plaq.

o,]t

8, Interesting studies of Yukawa couplings in scalar-fermion models (without gauge fields) are reported in ref. [6].

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

541

Volume 199, number 4

PHYSICS LETTERS B

1

s~= ~ Y. q..,,[(2ou,,.z.+~,,-~.+~,, u*..,,zo) n

,p

+(~.~,,+e,,--~.+,.,,~.)] + ~ (rnz2.Z,,+m~.~,,),

(ld)

31 December 1987

tegrating out the fermions first, and thereby obtaining a nonlocal fermion determinant with its attendant complications, we integrate first over Higgs and gauge fields. This yields a path integral with a fermionic action which is bilinear, except for a term ~ ( 1 - r 2) E ~.Z.fG,+,,D~.+e,,,

n

n 4~

(h2.~.c~.+h*~.z.O*.).

Sv=Z

(le)

n

fig= 1/g~, and P = R e { UpJaq}. The sites n lie on a (hyper)cubic d-dimensional lattice; we shall retain d in our results but focus on the physical case d = 4. The units are chosen so that the lattice spacing a = 1. The Higgs fields satisfy 9',¢, = 1. As is well known, this "fixed-length" condition on the lattice Higgs fields does not imply that the continuum Higgs fields 0caCa- ~ (where a is the lattice spacing, which vanishes in the continuum limit) are of fixed length. With no loss of generality, we take fh i> 0. We use Kogut-Susskind fermions [8], which are advantageous for studies ofchiral symmetry. The q~.~= { I for]z= 1; ( - 1 ) " ~ + + ..... for 2 ~<]z~
where

r-I, ( 2f h)/Io( 2flh),

(2)

with Iv(x) the modified Bessel function, r increases from 0 to 1 as flh increases from 0 to ~ and serves as an equivalent (positive) variable, which we shall use. For r = 1, the gauge field is frozen out, and the theory can be exactly re-expressed as a theory of two free fermions f1.2 with masses mf,,~ = 1

[mz+m~+{(mz-m~)2+4h2}l/2].

(3)

Further, (~X) = (~)

= 0 for r = 1 and m z = m ~ = 0 .

(4)

We proceed to the nontrivial case 0 ~
~ ~n~.. n

That this method should be quite reliable has been discussed before and confirmed by comparison with numerical simulations [ 3,5 ]. In particular, we have shown that it goes beyond the quenched approximation. In a spin model, such a mean field method would reduce the calculation essentially to a singlesite integration. Here one still has a coupled, infinite set of fermionic integrations to perform, but since the fermionic action is now bilinear, these can in fact be done exactly. Using this result, we then calculate the various condensates in the usual manner, e.g. (~?~) =Of/Omz, where f=--limg~N~ -~ l n Z is the reduced free energy and the limit mz--,0 is subsequently taken to define the spontaneous condensate. For simplicity, we set rnx=rn~=0 so that the Yukawa interaction is the sole source of explicit chiral symmetry breaking. A physical motivation for this is that in the standard electroweak theory, fermion masses arise entirely from the Yukawa couplings.

Volume 199, number 4

PHYSICS LETTERS B

We find that

(ZX) =A J (dp)sF,

(5a)

where

A=ld(l -r2) (2X),

(5b)

f (dp)=-(2zO -a i ddp,

(5C)

--It d

s = ~ sin2p/,,

(5d)

AL= 1

F= [(h 2 +rs) 2 +A2s]-I

(5e)

(note that (5a) is symmetric under ( ~ Z ) - - ' 2)~ 7; by convention, we choose (~)~)/>0.) We first show the existence of a phase transition analytically. Observe that the integral in (5a) is bounded above, independently of h, by r-eJa, where Je=f(dp)s i is finite for d > 2. Hence,

RHS(5a)<½d(r-2-1)Ja

(~2) =c (~Z).

(6)

Then for any E>0, it follows that c < e for r 2 in the interval 1 >~r2> [1 +2e/(dJa)] -1. Hence, the only solution of (5a) in this interval is ( 2 ~ ) = 0 . Next, expanding the equation for small r, we obtain ( ~ Z ) =½d ( ~ X ) x(f

(dp)sFo-2h2rf

(dp)(sFo)2+O(r2)), (7a)

where Fo = [h4 +{½d ( ~ X ) } 2 S ] - ' For r = 0 and h~ = (½d) 1/2,

h
(7b)

where (8)

(7a) (or (5a)) has a nonzero solution for ( ~ X ) . Furthermore, because of the analyticity in r of the expansion (7), there is a nonzero solution for ( Z Z ) in a neighborhood of r = 0 . Since we have shown that (a) ( 2 X ) is zero over a finite interval in r adjacent to and less than 1; and (b) ( ~ X ) is nonvanishing at r = 0 and, indeed, in a neighborhood of r~>0, is follows that ( Z Z ) must vanish non-analytically at a critical value rc~(0,1) (depending on d and h 2) and remain zero for re
31 December 1987

ergy is, of course, also non-analytic at rc. Thus, the theory must undergo a phase transition at this point. Note that this result applies, in particular, to the h = 0 case studied before [ 3,5]. We next show that as h increases through he, ( Z Z ) vanishes everywhere. First, RHS (5a) is a monotonically decreasing function of r (in the physical range 0~r~< 1), as can be seen by explicit evaluation (see below). Since ( Z Z ) ( r = 0 ) vanishes as h increases through he, it therefore follows that ( Z Z ) vanishes for all r as h increases through hc, and hence the chiral transition is removed. Thus, both in the absence of any explicit zSB and in the presence of sufficiently weak explicit zSB by the Yukawa interaction (le) the theory has a chiral phase transition. This is quite interesting, and contrasts, e.g., with the typical situation in statistical mechanics: consider a spin model with a (global) symmetry group G and a spontaneous symmetry breaking transition characterized by a magnetization M(fl) which vanishes non-analytically as the inverse temperature fl decreases through fie and remains zero forfl
V o l u m e 199, n u m b e r 4

t.0

"', ,, " ' - , \ "',,, \\

0.8

\,\,

"",

o.7 \ '\ 0.5

'', \\

'I

\

X(f

\ ",

\

0.4

0.0

I ii 0.I

02

i ',

I !1

0.3

0.4

I

I 0.5

06

0.7

0.8

0.9

1.0

Io2 Fig. 1.

Mx~(~d) ~/2 (~?~)

for d=4

and h2=0

(solid);

0.25

(dashed); 0.5 (dotted); and 1.0 (dot-dashed). Here, h~ =2. Interestingly, through its Yukawa interaction with the charged fermion, the neutral fermion picks up a condensate:

=-h2a f

(dp)F.

(9)

( ¢¢ ) is plotted in fig. 2 for several values of h 2. This plot shows that I ( ~ ¢ ) 1 (where it is nonzero) de-

0.3- ' \ , \ \, \ \

\

,

I ,,' ....... i ......... O0 0.1

',

i ......... 0.E

\

\

i ,,I . . . . . . . i ,,h . . . . . . . 0.3 0.4 0.5

E2

Fig. 2. - M x ~ - ( ~ ) for d=4 and h2=0.25 (dashed); 0.5 (dotted); and 1.0 (dot-dashed).

544

(~X)

(dp)Fo-2h2rf (dp)sF2+O(r2)).

(10)

As is clear from (9), when r increases through rc, since ( f¢)¢) vanishes continuously but non-analytically and remains zero for r>rc, this causes ( ~ ¢ ) to do the same. For a fixed r
-2 {)~X) [ 1 - ½ d ( ~ z ) 2 ] ,

atr=0.

(11) For general r, since f(dp)sF (h2/d) ()~)~). In addition to purely fermionic condensates, this model exhibits a new type of condensate:

Y=-(2/h)

"'-....

0.1

0.0

1987

(12)

Y is plotted in fig. 3 for several values of h 2. Since for h C 0 there is a source term for the operator ( 2 ¢ ~ + h . c . ) in the action, it has a nonvanishing vacu u m expectation value for all r, in contrast to the pure fermionic condensates, for which there are no source terms and which therefore have to be dynamically generated to be nonzero. From (12) follows the simple relation

0.4-

0.2-

31 D e c e m b e r

Y=(fC{O+h.e.)=-2h I (dp)(h2 +rs)F.

0.5-

I

B

(~¢)------½ dh2

\

',,

o.a,

LETTERS

creases as a function of r. For small r, this can be seen from the expansion

~-~

0.9.

~

PHYSICS

[1-½d(~)¢)2],

atr=0.

(13)

For small h , - Y increases as r increases from zero, has a cusp at r=rc, and then decreases as r moves further upward to 1. Moreover,

-Y=2hj(dp) (h2+rs) -~,

forh>hc,

(14)

which is analytic and decreases from the value 2/h at r = 0 to a nonzero value at r = 1. The latter result

Volume ! 99, number 4

PHYSICS LETTERS B

sure of which is the nearest-neighbor correlation gauge-invariant function { L ) - { Re {0'~U,,,¢,O.+ c,,} ). We find

1.2 II

/

I0

/\.

\.

0.9

-~..

/

,"'- "~'-

,

0.7 ::~

\

/

0.5

~.~_

/

/ '

~

/

,"

/ 0.3

/

/

/ /

0.2-

/

/

O,lO'O

......

00

] .........

0.1

~ .........

0.2

I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . I . . . . . . . . . r . . . . . . . . . I ........

0.3

0.4

0.5 F

0.6

0.7

0.8

0.9

=r+

(rd) -

×[1-(l+r

-~

/ /

/' 04

(L)

/..~

,""

0.6

31 December 1987

1.0

2

Fig. 3. -Y----(Z~0+h.c.) for d=4 and h2=0.25 (dashed); 0.5 (dotted); 1.0 (dot-dashed); and 4.0 (solid). shows that this quantity exists in the absence of any gauge-fermion interaction. The fact that Y is nonvanishing for h > hc contrasts with the behavior of the (t'f) condensates. The analyticity of Y and the vanishing of the (]~f) condensates for h>h~ are in accord with the result that the theory has no phase transition there. In the large-d limit, we can obtain exact solutions for the condensates since (a) we can solve the meanfield equations (5a), (9), and (12) exactly, and (b) the mean field approximation should be exact in this limit. For h hc, the Yukawa coupling is never negligible. The chiral transition has an important effect on the short-range order in the bosonic sector, a mea-

Jr'(2flh)

2) ½d ( X Z ) 2 + ½ h Y ] ,

(15)

where r'(x)=dr/dx. In particular, for h
Volume 199, number 4

PHYSICS LETTERS B

a n a l o g o u s to the S U (2) p a r t o f the Y u k a w a interactions in the s t a n d a r d e l e c t r o w e a k S U ( 2 ) × U ( I ) e l e c t r o w e a k theory, in t h a t b o t h c o u p l i n g s i n v o l v e a n o n s i n g l e t f e r m i o n c o u p l e d w i t h a singlet f e r m i o n . O u r n e w f i n d i n g s s h o u l d t h e r e f o r e be useful in gene r a l i z i n g the lattice studies o f the b o s o n i c sector o f the e l e c t r o w e a k lattice t h e o r y [ 10,2] ( a n d o f simplified g a u g e - H i g g s m o d e l s ~2) to i n c l u d e f e r m i o n s . T h i s r e s e a r c h was s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy a n d the N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r contracts D E - A C 0 2 - 7 6 C H 0 0 0 1 6 ( I H L ) and NSF-PHY-85-07-627 (RES). ~2 For a comprehensive review of early work, see ref. [ 11 ].

References [ 1] I-H. Lee and J. Shigemitsu, Phys. Lett. B 178 (1986) 93.

546

[2] [3] [4] [ 5]

31 December 1987

I-H. Lee and J. Shigemitsu, Nucl. Phys. B 276 (1986) 580. I-H. Lee and R.E. Shrock, Phys. Rev. Lett. 59 (1987) 14. I-H. Lee and R.E. Shrock, Phys. Lett. B 196 (1987) 82. I-H. Lee and R.E. Shrock, Nucl. Phys. B 290 [FS20] (1987), to be published. [6] J. Shigemitsu, Phys. Lett. B 189 (1987) 164; OSU preprint DOE/ER/01545-397. [7] H.M. Georgi, as cited in: M, Claudson, E. Farhi and R.L. Jaffe, Phys. Rev. D 34 (1986) 873. [ 8 ] T. Banks, L. Susskind and J. Kogut, Phys. Rev. D 13 ( 1977) 1043; L. Susskind, Phys. Rev. D 16 (1977) 3031; J. Kogut, Rev. Mod. Phys. 55 (1983) 775. [9] N. Kawamoto and J. Smit, Nucl. Phys. B 192 (1981) 100. [10] R.E. Shrock, Phys. Lett. B 162 (1985) 165; Nucl. Phys. B 267 (1986) 301; Phys. Lett. B 180 (1986) 269; Phys. Rev. Lett. 56 (1986) 2124; Nucl. Phys. B 278 (1986) 380; D.J.E. Callaway and R. Petronzio, Nucl. Phys. B 292 (1987) 497. [ 11 ] J. Jers~ik, in: Proc. Wuppertal Lattice gauge theory Conf. (1985).