Self-energy and flavor interpretation of staggered fermions

Nuclear Physics B245 (1984) 61-88 © North-Holland Pubhshlng Company

SELF-ENERGY

AND FLAVOR INTERPRETATION FERMIONS

OF STAGGERED

Maarten FL GOLTERMAN and Jan SMIT lnsntute of Theoretwal Phystcs, Umverslt) of Amsterdam, Valkemerstraat 65, 1018 X E Amsterdam. The Netherlands

Recewed 3 Aprd 1984 We investigate the symmetries of the staggered fermlon formulation and the construchon of local hadron fields on the lattice The one-loop fermlon self-energy ~s calculated w~th a general non-degenerate mass matrix and the ~mphcatlons of the results are discussed

1. I n t r o d u c t i o n Two c o m m o n l y e m p l o y e d f e r m l o n f o r m u l a t i o n s in lattice Q C D are W l l s o n ' s m e t h o d [1] a n d the m e t h o d o f s t a g g e r e d f e r m l o n s [2-10] In the first f o r m u l a t i o n chlral s y m m e t r y IS c o m p l e t e l y b r o k e n on the lattice, w h e r e a s in the latter t h e r e r e m a i n s a s y m m e t r y g r o u p with c h a r a c t e r i s t i c s o f chlral s y m m e t r y On the o t h e r h a n d , with W l l s o n ' s m e t h o d the species d o u b l i n g p r o b l e m is c o m p l e t e l y solved for Q C D - h k e theories, w h e r e a s in the s t a g g e r e d case we are left with some d o u b l i n g r e m a i n i n g . It is, however, t e m p t i n g to Interpret the s p e c t r u m as d e s c r i b i n g two or f o u r flavors, as has b e e n suggested using v a r i o u s f o r m u l a t i o n s o f the t h e o r y [2-10] O n e o f the i m p o r t a n t questions is w h e t h e r the lattice s y m m e t r y g r o u p g u a r a n t e e s that the a p p r o p r i a t e c o u n t e r t e r m s for o b t a i n i n g r e n o r m a h z e d c o n t i n u u m G r e e n f u n c t i o n s can be p r o d u c e d by a d j u s t i n g a restricted set o f p a r a m e t e r s in the lattice a c t i o n It is t h e r e f o r e instructive to c a l c u l a t e the f e r m l o n self-energy in the o n e - l o o p a p p r o x i m a t i o n with a general mass m a t r i x Thts is d o n e in this p a p e r for a e u c l i d e a n S U ( N ) lattice g a u g e t h e o r y c o u p l e d to s t a g g e r e d f e r m l o n s in the four-flavor a n d two-flavor formulations. A g e n e r a l s t a g g e r e d f e r m l o n mass term b r e a k s the lattice s y m m e t r i e s Previous c a l c u l a t i o n s o f the o n e - l o o p self-energy [3, 11] i n v o l v e d o n l y a flavor d e g e n e r a t e mass term which d o e s not b r e a k the r o t a t i o n - I n v e r s i o n - t r a n s l a t i o n g r o u p on the lattice T h e results m this relatively s t m p l e s i t u a t i o n i n d i c a t e d that the f e r m l o n self-mass v a n i s h e s w h e n the b a r e m a s s is zero, as in the c o n t i n u u m theory. (This follows a l r e a d y from c o n s i d e r a t i o n s in the naive f e r m l o n f o r m u l a t i o n [12] b e c a u s e o f the e q u i v a l e n c e to f o u r s t a g g e r e d f e r m l o n s [4] ) S i m i l a r b e h a v i o r has been o b s e r v e d at strong c o u p l i n g in the e q u a t i o n for the b a r y o n mass m the n o n - d e g e n e r a t e t w o - f l a v o r case [10] These results h o l d o n l y for the o r l g m a l form o f the staggered f e r m l o n a c t i o n In the Dlrac,--Kahler f o r m u l a t t o n [5, 7], a n a t u r a l g a u g e field c o u p l i n g 61

M F L Golterman, J Srmt/ Staggered [ermtons

62

[7] leads to a f e r m i o n self-mass even when the b a r e mass vanishes [11] a n d the e x p e c t e d G o l d s t o n e b o s o n at strong c o u p l i n g [4, 13, 14] is a b s e n t [15] The lattice s y m m e t r y g r o u p is s m a l l e r in this m o d e l W e exhibit the lattice symmetries, i n c l u d i n g parity a n d charge c o n j u g a t i o n The lattice s y m m e t r y g r o u p is t a k e n as it stands a n d we do not try to Identify flavors on the lattice We discuss its role in restricting the f e r m i o n self-energy to a form differing from the c o n t i n u u m e x p r e s s m n by c o v a r i a n t c o n t a c t terms only a n d argue that the lattice s y m m e t r i e s e n l a r g e to the c o n t i n u u m s y m m e t r i e s in the h m l t o f zero lattice s p a c i n g The c o n s t r u c t i o n o f local h a d r o n o p e r a t o r s t r a n s f o r m i n g i r r e d u c i b l y u n d e r the lattice s y m m e t r y g r o u p is i n d i c a t e d This p a p e r is o r g a n i z e d as follows In sect 2 the action a n d its symmetries are e x h i b i t e d a n d the f o r m a h s m i n t r o d u c e d m ref [10] is r e w e w e d G u i d e d by the s y m m e t r i e s in the classical c o n t i n u u m limit a g e n e r a l mass term ~s c o n s t r u c t e d in sect 3 a n d the t r a n s f o r m a t i o n to the c o n v e n t i o n a l four-flavor a c t m n ~s given We outline the c o n s t r u c t i o n o f m e s o n fields on the lattice which have the correct c o n t i n u u m limit In sect 4 we p r e s e n t the o n e - l o o p c o r r e c t i o n to the f e r m m n self-energy a n d in sect 5 we &scuss the r e n o r m a h z a t i o n o f the mass p a r a m e t e r s Sect 6 c o n t a i n s a & s c u s s i o n on how the lattice s y m m e t r i e s restrict the form o f the c o n t a c t terms In sect 7 the two-flavor case ~s t r e a t e d in so far as ~t differs from the four-flavor case Sect 8 c o n t a i n s o u r conclusions T h e r e are three a p p e n d i c e s , A gives the F e y n m a n rules used in sects 4 a n d 7 a n d B c o n t a i n s the e v a l u a t i o n o f the c o n t a c t terms for the f e r m I o n self-energy A p p e n d i x C deals with a technical detail

2. Staggered fermions We start from the a c t i o n for e u c l i d e a n s t a g g e r e d f e r m l o n s

S = - E ½~q~(~c)[f((x)U~,(x)x(x+a.)-~fx +a.) U~(x)x(x)]+

(21)

where ~/,(x)=l.

~h(x)=(-l)

~,,

- q ~ ( x ) = ( - 1 ) ~,+'2,

~ / 4 ( x ) = ( - 1 ) ~,+'-'+~" ( 2 2 )

The f e r m l o n field X only has a gauge g r o u p i n d e x We take the lattice gauge field m the f u n d a m e n t a l r e p r e s e n t a t i o n o f the c o l o r g r o u p M a s s terms will be c o n s i d e r e d later on W e list the s y m m e t r : e s o f this action (1) shift i n v a r l a n c e

X(x)~.(x)x(x+ao),

2(x)-~o(x)2(x+%).

U~(x)--. O~(x +ap),

(23)

M FL Golterman, J Smtt/ Staggeredferrmons

63

w~th ~ , ( x ) = ( - 1 ) ~ + ~ +~.,

s r 2 ( x ) = ( - 1 ) ~+'~,

~3(x)=(-l)'~,

~4(x) = 1 , (24)

(li) rotational l n v a n a n c e

X(x) ~ S R ( R - ' x ) x ( R - ' x ) ,

~(x) ~ S R ( R - ' x ) 2 ( R - ' x ) ,

(25)

U(x, y) ~ U ( R - ' x , R - ~ y ) , where R

=-

R (°°-' 1s the rotation xp ~ x~, x~ ~ -xp, x. ~ x . r ~ p, ~r,

SR(X) = ½It + n ~ ( x ) n . ( x ) ~ ~ ( x ) ~ ( x ) + n ~ ( x ) n ~ ( x ) ~ ( x ) ~ ( x ) ] ,

p~o-,

x=y+a.,

( U~.(y),

(26) (27)

(this lnvarlance follows from

SR( R - I x)n~(X)SR( R - I x + R - l au) = R.~n~( R - ' x) ) ,

(28)

fin) axis reversal

X(x)-~(-1)*oX(Ix) ,

~(x)~(-1)*.2(Ix),

U(x, y)-~ U( Ix, Iy) , where I ~ I ~p) is the axis reversal x ~ - x o , We have two continuous invarlances (1) U ( I ) mvarlance

X(x)+e'~x(x),

x~x.

(29)

r#p

£(x) + e-'~f(x),

(2 10)

$(x)-.e'm'~12(x) ,

(2 11)

(11) U ( I ) . mvarlance

X(x)~e'~c~IX(x), where

e ( x ) = ( - 1 ) ",+~'-+~,+~

(2 12)

Furthermore there as a dtscrete symmetry interchanging X and )~

x(x)~(,c)

T,

£ ( x ) ~ x ( x ) ~,

U.(x)-~ US(x)

(2 13)

The path integral measure is l n v a n a n t u n d e r these transformations provided that ( a n n ) - p e r l o d l c b o u n d a r y conditions are used and that the l a m c e contains an even n u m b e r o f sites The action (2 1) has spectrum d o u b h n g as becomes clear when we transform to m o m e n t u m space We write the signs -q.(x), ft.(x) and e(x) as exponentials

,?~(x)=exp(tTr~x),

st.(x) = exp (rzr~ x) ,

e(x)=exp(1%x),

(2 14)

64

M F L Golterman, J Smzt / Staggeredfermtons

with obvious choices for 7rn., 7re. and 7r~ (listed in [10]) E x p a n d i n g U u ( x ) = 1 + zgA'~(x)tm + , with t~ the hermltlan generators of the gauge group, we obtain the reverse free f e r m m n propagator in m o m e n t u m space /~(k. -1) = ~ 6(k - I + 7r~.)~ sin k.

(2 15)

tL

Here 6(k) IS (27r) 4 times the periodic delta function We divide m o m e n t u m space Into 16 regions,

k=p+rc~

(mod 2 I r ) ,

-~r < k~. ~< It,

A=I,

,16,

-½7r < p . <~ ~Tr,

(2 16)

where rrA lS one of the sixteen vectors (0, 0, 0, 0), (~r, 0, 0.0), (~r, 7r, 0, 0), etc and introduce 16-component fields (as m [10])

~A(p)=~(p+TrA),

OL(p)=x(p+rcA)

(2 17)

Here we do not Interpret the index A as flavor, Dlrac or whatever, yet We now want to write the Inverse p r o p a g a t o r (2 15) in &-language, for which some further preparations are needed In [10] an abehan group o f sixteen 16 x 16 matrices (~A)B( was introduced ~A IS defined as follows {;, (~A)B~ =

~r.+~rB+1r¢ = 0 ,

(mod27r),

otherwise

(2 18)

As explained in [10] these may be written as tensor products o f the Pauh m a m c e s ~r~, ~-~, p~, w~ and 1 ( 2 x 2 identity) Note that ~%., ~-~,, and ~-~ are elements of this group. Furthermore sign factors S A are defined S A = exp ( ~ A . )

(2 19)

Using (2.16), (2 18) and (2 19) the inverse p r o p a g a t o r (2 15) can be written as

19(p +Try, - q +TrB) =~. 6 ( p - q +~'A +TrB + % . ) 1 sin (p + ~rA). = Y~ gfp

-

q)( ~rn )ABSAt sin p .

g

= g(p - q) ~ (S.~rn.)ABl sin Pu

(2 20)

~x

In the last line of (2 20) diagonal matrices S. are introduced, (~)AB

4 = S,uf~AB

(2 21 )

The c o m m u t a t i o n rules for S~, and ~'a can be derived by c o m p a r i n g the results o f (1) a translation X(X) ~ X(X + a . ) followed by a multiplication with exp ( t~A X ),

M F L Golterman, J Smtt/ Staggered fermlon~

65

and (il) these two o p e r a t m n s m reverse order The results of (l) and (fi) &ffer by the sign factor S A, l e. in m o m e n t u m space, m &-language, ,~ASIx = S . I r^ A S .A

(2.22)

S~-k.. = ~ m G e ~ . ,

(2 23)

S . : q , = ~; S . e ~ . ,

(2 24)

In p a m c u l a r we obtain

e.~= S~.-

_ S,.c, = { 1 , l ,

/z~ v

(225)

Two more sets of 16 x 16 matrices F~ and ~ . were defined G =

--'~ = S~-

&~.~,

(2

26)

which obey (using (2.23) and (2 24))

{G, F.} = 2 G . ,

Ca

= F u~ _ -_ F ~T ,

{-=., -%} = 2 G . , (2 27)

[ G , -%] = 0

The reverse fermion p r o p a g a t o r (2 20) can be written as (2.28)

q) • d'~ sm p .

I~AB(p , - q ) = g ( p -

In an exphclt representation for .k.., .kc., ~ and S. [10], "/'r~ = 1 ,

~r~_. = O'j ,

:re, = o-1091 ,

~

S 1 = 0-3 g 3 p - ~ t o 3 ,

7r,7~ - - 71 ,

= rl09j ,

7"g.4 = P 1 ,

7~¢4 = 1 ,

:re, = P~091 ,

S2 = r3P309 3 ,

$3 = P3W3

7~'e --- t(O 1 ,

,

S 4 = W3,

(2 29)

we have, using (2 26) -El = 0°37"3103093 ,

--'~1 = --0027"3/93092 ,

F 2 = 0017"3193093 ,

~"-2 ~-- - 7-2P3092 ,

/"3 = 7-1P3093 , /~4 = p l 0 9 3

"~3 = --P2092 , ,

(2 30)

24=093

The symmetries (2 3)-(2 13) can be expressed in m o m e n t u m space as follows 0) shift m v a r l a n c e ~bA(p)-'->e%'(.~)ABdp.(p)

;

(2 31)

01) 90 ° rotaUons & ( p ) ~ exp

exp ( ~ T r ~ o ~ ) & ( R - ' p ) .

(2 32)

M F L Golterman, J $mlt / ~taggered/ermums

66 (m) axis reversal

g(p) -~ 1;v~_,,=_,g( lp),

(233)

where*

F~ = -F~F2FJ~ , (w) U(1) i n v a n a n c e ~(p)~e

'~(p),

(2 34)

(v) U(1)~ i n v a r i a n c e g ( P ) --, q~( p ) e ' ~ z J , ,

(235)

where F5-~5 = ~-~ = w~ was used The discrete s y m m e t r y (2 13) b e c o m e s in m o m e n t u m space & ( p ) ~ 4~(-p) v ,

4~(p)-~ ~ ( - p ) v

(236)

We shall d e n o t e the set o f 16 matrices consisting o f the F - m a t r i c e s a n d their p r o d u c t s by -~,, A = 1, , 16

3. Symmetries and mass matrix It is further e x p l a i n e d in [10] that, using the f o r m a l i s m o f sect 2, the tree g r a p h c o n t i n u u m limit o f the action can be written as

S ~ - f dxda(x)F.[~. +,ga~(x)tm]O(x)

(3 1)

F r o m this form we see that in the classical c o n t i n u u m hmlt the s y m m e t r y g r o u p (2 31 )-(2 35) enlarges to a c o n t i n u o u s g r o u p o f U ( 4 ) x U(4) t r a n s f o r m a t m n s

(32) ~(p)~

exp (aw.,F.I,)~(R i . + inversions

'p),

R u ~ = ( e o, )., (3 3 )

This suggests that we c o n s i d e r the - ll~F,,(~ # v) as the g e n e r a t o r s o f the c o n t i n u u m e u c l i d e a n r o t a t i o n g r o u p a n d the ~ as the g e n e r a t o r s o f the U(4) flavor g r o u p F o l l o w i n g the lines o f ref [10] we shall now construct a general mass matrix M, lifting c o m p l e t e l y the d e g e n e r a c y o f the f o u r flavors which are e x p e c t e d to c o m e out m the c o n t i n u u m limit We d o this in two steps First we derive the general form t h a t M m u s t have in the c o n t i n u u m , a n d t h e n show h o w an a p p r o p r i a t e lattice a c t m n for th~s m a t r i x can be c o n s t r u c t e d * This &fiefs by a sign from the definmon m [10]

M F L Golterman, J S m t t / Staggered fermwns

67

In the c o n t i n u u m one d e m a n d s the actmn to be rotatmnally mvarlant, which lmphes that M has to c o m m u t e with F~,F~. This restricts M to be o f the form

M=m+m

~ +tm~.~(_t~ ~ ) -

5 ~ =.-

5=

+ s l m d a r terms multlphed by -~F5

(3 4)

Since a -~.-matrax can be p r o d u c e d by a shift operation (cf (2 3), (2 31)) we define hermltlan m a m c e s Eu(x, y) on the lattice (x and y are lattice points) which p r o d u c e these shifts when acting on X

E.(x, y) = E ~'. (z)½[ U . ( z ) 8 ( x - z)6(y - z - a. ) + U*.(z)6(y - z)8(x - z - a . ) ] z

(35) This matrix depends on the gauge fields U . in order to maintain gauge m v a r m n c e We n o w define a mass term for the actmn

S . . . . = -Y~ rn~(x)x(x) - Y~ rn.2(x)Eu(x, Y)X(Y) +it E m...~(x)E.(x, y)E, (y, z ) x ( z ) +~t • +~

m~su.,,,¢./2(w)E~(w, x)Et3(x,y)E~(y, z ) x ( z )

Y. mSeo,o.,,8~(v)Eo,(v,w)Et3(w,x)E.y(x,y)E~(Y,Z)X(z)

(36)

vv~ z(~,z

S u m m a t i o n s over ~z, ~, a,/3, 7, 8 are u n d e r s t o o d ; rn.~ is ant~symmemc m ~z ~ The Fs~5 terms in ( 3 4 ) can be p r o d u c e d by ( 3 6 ) with an overall e(x) plugged in We shall, however, not consider such terms m the rest o f th~s paper The condition free f e r m m n (mass)2>~ 0 imphes that M has to be hermltian and hence rn, rn. etc have to be real As an example we write out the second term in ( 3 6 )

E mo..~(x)Eo.(x, Y)X(Y)

= ~ Z C.(z)[a?(z) V.(z)x(z +a.) +~(z +a.) U~(z)x(z)]

(37)

:/z

The choice for a symmetric shift o p e r a t o r IS m a d e since we expect that it leads to a h e r m m a n transfer matrix and because it p r o d u c e s cosines (with definite symmetry u n d e r p~, ~ - p . ) rather than exponentials m the free fermlon p r o p a g a t o r G

G-'(k, - l ) = g ( k - l + 7 % . ) , sin k. + g ( k - l ) m + g ( k - l + ~'c. ) m . cos k. + etc (The complete expression is given m a p p e n d i x A (A 15) )

(3 8)

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68

Staggered Jerrmons

Writing k = p + 7rA, l = q + rr~ we o b t a i n m a w a y similar to the d e r w a t l o n o f (2 28) the f o l l o w i n g e x p r e s s m n for the inverse f e r m i o n p r o p a g a t o r

G I(p+rr~,--q+TrR)=6(p--q)SatB(p),

(39)

S i(p) = tI'~p. +m +~ m . ~ . c o s p . pt

+ ~ ~ m . . ( - ~ - = . -=, ) cos p . cos p . ~tt

+ V

rnSt.= = s c o s p . cospt3 cospvIeu,,t~v]

~xaBy

~5 COS Pl COS P2 COS P3 COS P4

(3 10)

The mass term (3 6) b r e a k s lattice r o t a t i o n l n v a r i a n c e H o w e v e r as the various mass p a r a m e t e r s m (3 6) are a c c o m p a m e d by a factor a ( l a m c e & s t a n c e ) we expect this b r e a k i n g only to show up in O ( a ) terms o f lattice G r e e n f u n c t m n s a n d h e n c e r e s t o r a t i o n o f r o t a t m n l n v a r l a n c e in the c o n t i n u u m hm~t as b e f o r e This p o i n t will be & s c u s s e d further in sect 6 We shall now m a k e c o n t a c t with the c o n v e n t i o n a l form o f a four-flavor D i r a c a c n o n In the classical c o n t i n u u m limit o u r action has the form (use (3 I ) a n d (3 4)) S~ont=-f dx{ck(x)F.[8.+igA~(x)tm]6(x)+dS(x)M&(xt}

(3 11)

With a u m t a r y m a t r i x

V = V2 V1, I Vi = exp ( 117./.0.21_302) exp (lJTrr2p3) exp (t47ro21_1p2w3),

' V . = e x p ( ltcrp~) exp ( t~Trp~) _ exp ( - l~17r~o3) , _

(3 12)

the I" a n d w matrices given by (2 30) t r a n s f o r m to 3`1 : 0"~1_3 ,

~1 = PlCO? ,

y~_ = cr 11_~,

~._ = p ~ t o ~ ,

3`4 = o'21"~,

~4 :

o9~,

(3 13)

i~. =- V~-. V

(3 14)

under

3'. =- VF. V- ,

F r o m (3 13) we see that if we define tb" a n d 0 by

6= v'~o,

~=Ov,

(3.15)

M FL Golterman, J Srmt/ Staggered ferrmon~

69

where ~b carries a spin and a flavor index on which the y and ~: matrices act, respectwely, the action (3.11) can be written as S~o~t = - I

dx[~%.(c~ + t g A ~ t m ) ~ +

(3 16)

with M = m + rn~.~u + ~rn~,~(- I~:.~¢~)+ rn~t~:.~s + m5~:5

(3.17)

One could, of course, transform to other representattons for the y- and (-matrices The symmetry transformations (2 31)-(2 35) translate directly into ~0-1anguage, replacing -"~ by ~c and Fu by 3% Of particular interest is panty. This may be defined as the product of three axis reversals (2 9), I~1~I~2~I ~3), followed by a shift (2 3) in the 4-dlrectton The effect on ~b and ~ is gwen by ~(p)--)e'P.l'a~(pp),

~ b ( p ) ~ q b ( P p ) I 4 e-'p4 ,

p = I~1~I~2~I ~3~ ,

(3 18a) (3 18b)

which translates into the usual continuum form ~0(x). y46(Px),

f(x)~t~(Px)y4

(3 19)

This parity transformation ~s a symmetry of the full lattice action, including the mass terms (3 6) The transformation (2.13) interchanging X and ,f ~s related to charge conjugation Its effect on the mass term (3 6) 1s given by (m, m~., m.~, m~, m 5) 5 _ m 5) and subsequent apphcatlon of a U(1)~ transformation ( - m , - m . , rn~, m~, (211) with /3 =~Tr ~ then gives (m, -rn~., - m . , , rn., 5 rn 5) The signs of rn. and m~.~ can be restored by shifts, but only for specml choices of M wall m .5 and rn 5 remain unaffected so that we have an mvarlance of the action So let us look at some examples Particularly simple mass matrices are obtained by choosing a subset of four commuting matrices from the set ~:a, A = 1, ., 16 (this is always a maximally commuting subset) Then ~t ~s easy to dmgonallze the mass matrix one simply chooses a representation m which the four sO-matrices are diagonal M then has the form M = a +bp3 +cw3 + dp3~3

(3 20)

Up to lattice d~rect~on permutations there are only two poss~bihtles for the commuting subset, namely m = rn + m4~_..~4+ m 1 2 ( - t ~ ~2) + m ~ t ~ 3 ~ 5 ,

(3 21)

M = m -f- m12(-i~'1~'2) --1-m34(-/,-~3,..~4) + rnS~5

(3 22)

These choices were considered respectively in refs [9, 16] and [7] The first choice

70

M FL Golterman, J Srmt / $taggeredJerm~on~

leads to an actmn density with a smaller spread over the lattice [ 16] A non-degenerate mass matrix m v o l w n g the mlmmal n u m b e r of hnk c o u p h n g s needs a n o n - c o m m u t i n g subset w~th m, m . and m.,, non-zero, for example M = rn + m 2 ~ 2 + m4~,, + m12( --l--~j--~2) + m ~ 4 ( - t ~ a )

(3 23)

Returning to the discussion of the charge conjugaUon transformation, we see that m the examples (3 21)-(3 23) above the signs of m~. and m~.~ can be flipped by shifts without affecting m~5 and m s For example, a shift m the 2-direction followed by a shift m the 4-direction has the desired effect m all three cases O f course, we can always add or take away shifts leawng M mvarmnt In a representation where M is diagonal, these "httle group shifts" correspond to dmgonal matrices, 1 e phase transformations on each flavor c o m p o n e n t of It will now be shown that the conventional charge conjugation transformation on the qJ-fields follows from the lattice transformation, using (3 21) as an example Let the lattice charge conjugation be defined by (J) the transformation (2 13). (n) a shift m the 2-direction followed by a shift m the 4-direction, and (in) a U(I )~ transformation with 13 =½It The effect on the 4,'s and ~'s is given by (cf (3 12)(3 15)) p)

e ~

~ ( p) - - e

[cb(-p)tl s_~_4=2] 'p'- 'P4[tw-zZ.FsZsg( - p

qJ(p)~[t~(-p)C]*,

,

)]T,

q~(p)~-[C

(3 24)

~(_p)]V,

(3 25)

C ' y ~ C j = -y~,

(3 26)

with C = VIFs~5,.~4~,_.2V T = - t ' y 4 ,

We have put exp ( t p 2 + l p J ~ 1 m the transformation on 4' and qJ, as is appropriate for the c o n t i n u u m hmlt It is straightforward to construct local meson fields on the lattice We take again the mass term (3 21) as an lllustratwe example and use a basis m flavor space such that the U(l )F Goldstone b o s o n of the massless theory is closely related to the plons The transformation (3 14) brings the mass matrix (3 21) into the diagonal form (3 20), with a = m , b = m l 2 , c = m 4 , d = m ~ The U(1). Goldstone b o s o n can be described by the field [4] 2( x ) le( x ) x( x ) ~ $( x ) vys~stp( x ) ,

(3 27)

with ~:5=w~, and we take the triplet o f c o n t i n u u m pion fields to be Try(x)= t~(x)zyswk( 1 + p3)q~(x) This supposes that ( 1 +p~) projects onto the u p - d o w n sector and ( 1 - p3) projects onto the s t r a n g e - c h a r m sector, 1 e we have to choose a, ,d such that m u = - a + b + c + d , md=a+b-c-d, m,=a-b+c-d, m~=a-b-c+d

M FL Golterman, J Srmt/ Staggeredfermlons

71

X+O+ l 02+O 4 X+O2~ X

X+al+Cl4 X+C1I

Fig 1 Graphical representatmn of (3 29)

For example, the ~r + and F

meson fields are given by

7r+(x) = ~(x)z-/5(to I + tto2)(1 +p3)~b(x) = q~(x)Iy5~:5( 1 - ~:4- z~:,~2 - t~:3~:5)0(x),

(3.28a)

F ( x ) = t~(x)tTs(tO , +u02)(1 -p3)~b(x)

= qT(x)lys~s(1 - ~4 + l(, ~2 + t ( , ~ , ) O ( x )

(3 28b)

These operators can be obtained as the c o n t i n u u m hmlt o f fields on the lattice, via the "rules" Y5~5 ~ F5-~5 ~ e ( x ) , ~. ~ --"-. -~ shift in the /x-direction For example, 4JYs~s~:3~:s0 arises as the continuum limit o f

- X ( X ) e ( X ) ~ l ( x ) Ul(x)~2(x ÷ a l ) U2(x +a~) x ff4(x + al + a2) U4(x + al + a . ) x ( x + al + a2 + a4),

(3 29)

which Is dlustrated in fig 1. One can define other operators, eqmvalent m the sense that they also p r o d u c e the m a m x Y5~:3,by taking other paths from x to Xl + at + a2 + a4 or by taking other diagonally opposite sites in fig 1 for X and )~ (eg. x + a j , x + a2 + a4) If we want operators to transform irreducibly u n d e r the lattice symmetry group, then the average over all such equivalent operators has to be taken In particular, the resulting meson fields then transform correctly u n d e r p a n t y and charge c o n j u g a t m n (e g 7r+~ - ~ ' + under parity and 7 r ~ ~r- u n d e r charge conjugatmn) F r o m (3 28) we see that our example fields are constructed as a s u p e r p o s m o n o f zero-, one-, two- and three-hnk ~rreduoble lattice operators, which do not transform into each other u n d e r the lattice symmetry operatmns (as they revolve a different n u m b e r o f links) Hence, the coeffioents m this s u p e r p o s m o n presumably have to be renormalized as a consequence o f the fluctuating link-gauge fields

4. Fermion self-energy to one-loop order (4-flavor case) We consider the o n e - l o o p contribution to the fermlon self-energy illustrated in fig. 2 It wdl be s h o w n that the o n e - l o o p contribution has the form X a B ( p ) g ( p - q)

72

M F L Golterman, J Smtt / StaggeredJerrmons

(

p'

q'

q+rr B (b)

(Q)

Fig 2 One-loop contnbutmn to the fermmn self-energy

The F e y n m a n rules are hsted m a p p e n d i x A F o r each vertex s h o w n m fig 2 there are five types, one c o m i n g from the kinetic term (2 1) o f the action a n d f o u r f r o m the mass term (3 6) So there are c o n t r i b u t i o n s X(") m which the two vertices are b o t h o f the kinetic type, b o t h o f the m a s s t y p e a n d o f the k i n e t i c - m a s s t y p e We shall now p r e s e n t in some detail the c a l c u l a t m n o f the k m e n c c o n t r i b u t i o n to the first d i a g r a m o f fig 2 X, ~'~ U s i n g (A 2), (A 15) a n d (A 16), w o r k i n g in the F e y n m a n gauge, we have

X(gba)(P)6(P-q)=g2 N2-12---~a i ~ fe, o,,k6(ap_p,+k +xa + x . . ) I

ip'

4

x_fie . + S . e ~"P.) G( p , -q')6( q' - aq - k + erB + r r . . ) -

r

x ~ ( S . e 'aq- +e-'qd,)D(k) I

B

(4 l)

H e r e t o G ( p ' , - q ' ) is the free f e r m l o n p r o p a g a t o r a n d D(k) r e p r e s e n t s the g l u o n p r o p a g a t o r The i n t e g r a t i o n over l o o p m o m e n t a is written in a s h o r t h a n d n o t a t i o n

f( P )=-(~)4

d4p' f ( P ') ,

(42)

We s c a l e d the l o o p m o m e n t a in o r d e r not to have a - d e p e n d e n c e in the i n t e g r a t i o n regions In a c c o r d a n c e with the f o r m a h s m o f sect 2 we e x t r a c t e d the vectors 7rA a n d ~rB out o f the external m o m e n t a Defining _~

N ~- l

g_=g2 2~I

'

(43)

we o b t a i n for (4 1), after integrating over the m o m e n t a p ' a n d q'

v(a°ba'(p)g(p

f

,oa.+

t~ dk

x (e'"q- + e ""q*k).)G(ap +k +era +er.,., - a q - k +err + v r . ) D ( k )

(44)

Let err = ZrA + vr.. a n d erO = err + er.. Then, using (A 15), we o b t a i n for the reverse

M F L Golterman, J Smlt / Staggered fermmn~

73

f e r m i o n p r o p a g a t o r ( p u t t i n g a = 1)

a l ( p + k + r r c , --q--k+rrD)=~, g ( p - - q +rrc +~'o+~.o)S c P

X 1 sin ( p + k ) o + e t c

= g(p - q)Sc~D(p + k ) ,

(4 5)

w h e r e S-~(p +k) is given b y (3 10) F r o m this it follows that

G ( p + k + rrc, - q - k + rrD) = g(p - q)ScD( p + k) ,

(4 6)

a n d using (as rrc = 77"A 4" '/'/'-0g, ")TD = '/TB "~ 7r'r/u )

ScD(P +k) = ( ~r. S(p +k)~n,~)AB ,

(4 7)

we get

S A G ( p + k + rrA + 7r,.. --q -- k + 7rs + ~rn.)S~ = ( F , S (p +k)F~)ASt~(p - q)

(4.8)

F i n a l l y the e x p r e s s i o n for X t" g~ b e c o m e s

X,t"k)(p)=g2a-'~fg½[l+cos(k+2ap.)JF.S(k+ap)FgO(k),

(49)

where the a - d e p e n d e n c e has b e e n restored. This e x p r e s s i o n for X,~a'k)(p) has the s a m e f o r m as In the naive f e r m i o n case E x p r e s s i o n (4 9) was e v a l u a t e d b y splitting the I n t e g r a t i o n r e g i o n into an i n n e r a n d an o u t e r region, [k[ < 6 a n d [k[ > 8 respectively, with 6 small b u t 6 ~> a (first let a ~ 0, t h e n 6 ~ 0). F o r 6 ~ 0 there will be a d i v e r g e n c e in b o t h integrals but t h e y will c a n c e l in the s u m b e c a u s e the total e x p r e s s i o n is i n d e p e n d e n t o f 8 In the o u t e r region an e x p a n s i o n is m a d e in a a n d terms are k e p t u p to o r d e r a °, while in the i n n e r r e g i o n the l n t e g r a n d is r e p l a c e d b y its c o v a r l a n t form, w h i c h IS a l l o w e d for 6 ~ 0 T h e total result is a c o n t i n u u m - h k e e x p r e s s i o n c o m i n g from the Inner r e g i o n plus a c o v a r i a n t c o n t a c t term c o m i n g f r o m the o u t e r region We e x p l a i n a bit m o r e explicitly h o w the Inner region integral was c a l c u l a t e d . First we m a k e the t r a n s f o r m a t i o n (3 14) a n d d i a g o n a h z e the mass m a t r i x (3 17) Mo = d l a g ( M j , M2, M3, M4) = WoMW*o,

(4 10)

w h e r e Wo is a m a t r i x In ~:-space (1 e. can be written as a s u p e r p o s i t l o n o f E-matrices) As Wo c o m m u t e s with y . the e x p r e s s i o n for the Inner region Integral is

Xl~ k~( ~

[

21%(k + ap) o +4aMa

tuner,P, = g 2a-l W; -Ikl<~ [(k + ap) 2 + a2M~](k 2 + a2A 2) Wo + O ( a , ~) (4 11) w h e r e the y - m a t r i x m u l t i p l i c a t i o n s have b e e n c a r r i e d out A n I n f r a r e d r e g u l a t i n g

M F L Golterman, d Smlt / Staggered Jermton~

74

mass )t has been a d d e d in the gluon p r o p a g a t o r N o t e that (4 11 ) is just the c o n t i n u u m form with a s p h e r i c a l cutoff The terms with ap. a n d aMo in the n u m e r a t o r are l o g a r i t h m i c a l l y divergent, i e o f o r d e r In a so that a F e y n m a n p a r a m e t e r x can be i n t r o d u c e d a n d the i n t e g r a t i o n variable can be shifted to k +axp T h e r e is no n e e d to shift the integration region b e c a u s e this differs o n l y in o r d e r a In a corrections, which are d r o p p e d in the c o n t i n u u m h m l t T h e term with k o in the n u m e r a t o r is linearly divergent a n d we first s u b t r a c t

I Ikl ~ k~[(k +ap)2 +a2Md] =I

I(/t2+a2)t2)-l--

I¸Ikl

k,(k: + a2A ~) ~

k"[-2akp+a2('~2-M~-p2] x [ ( k +ap)2 +a2M~]-,(k,_+a2A2) 2

(4 12)

The s e c o n d integral on the l e f t - h a n d side equals zero a n d we are a g a i n left with an integral which is only l o g a r i t h m i c a l l y d i v e r g e n t The result o f e v a l u a t i n g all d i a g r a m s leads to the f o l l o w i n g e x p r e s s i o n for the o n e - l o o p self-energy v ( p ) = ~2{ IYu p~.T + too-s + muGo'v + ½mj.. ( - iI~sc~) crv q

+ m-~l~sO'A + m- bC:sOp 1 W~

831.2

A = KM~ +(1

Xo1d x [ t y . p u ( l - x ) + 2 M d ] l n ( a Z A ) W o }

-X)A2+X(1 - x ) p

2,

(4 13a) (4 13b)

where terms o f o r d e r a In a have b e e n d r o p p e d The e x p r e s s i o n s for r a n d %, ) = S, V, T, A, P are h s t e d in a p p e n d i x B with their n u m e r i c a l values T h e y consist m a i n l y o f integrals over p e r i o d i c functions ( o u t e r region integrals in the limit 6 ~ 0 c o m b i n e d with the ~ - - 0 d i v e r g e n c e c o m i n g from the inner region Integral) The % a n d ~- are i n d e p e n d e n t o f p, M a n d a N o t e that no ~:-matrIces turn up m the y . p . c o n t a c t term a n d that for each t y p e o f mass term (S, V, T, A. P) there is only an overall c o n s t a n t ~ The reason for this is e x p l a i n e d in sect 6

5. Renormalization In this section we define wave f u n c t i o n a n d mass r e n o r m a h z a t l o n constants Z2 a n d Zm,, J = S, V, T, A, P The b a r e Inverse p r o p a g a t o r So t, with Mo d e n o t i n g the

M F L Golterman, J Smlt / Staggeredfermlons

75

bare mass matrix*,

Mo = ~ Moj , 3

Mos = mo,

Mov = mo~.~:~ ,

MOA = m ~ . t ~ , ,

i MOT = ~mo.~(--t~.~,,),

M o p = rnoSsC5,

(5 l)

can be written as

So'(p) = t%.p. + Mo +..~(p) .

(5 2)

The o n e - l o o p expression for 2 Is given m (4 13). For each Mo.J = S, introduce a mass r e n o r m a h z a t i o n Zmj and a r e n o r m a h z e d mass Mj Moj -- Zm;Mj,

Ms = m ,

Mv = m.~:~.,

etc

, P, we

(5 3)

We n o w have to choose the Zmj and a wave function renormalization Z2 such that the r e n o r m a h z e d inverse p r o p a g a t o r S ~ defined by

S '(p) = Z2SoI(p)IMo=E,Z.,,,M,

(5.4)

lS finite in the limit a ~ 0 W h e n we absorb the contact terms in Z2 and Z,.j this leads to the following expressions for Z2 and Zm,, Zz = 1 + g

1 - ~ 2 1 n (a~p.2)-~'+z2

+O(g4),

Z,. = l + g Z [ 1 - - ~ z l n ( a 2 ~ 2 ) - t r j + z + z j ]

(5.5)

+O(g4),

(56)

j = S, V, T , A , P , where /~ is the renormalization scale parameter The addition o f the constants z2 and zj constitutes a finite renormallzation If we substitute the expressions (5 5) and (5 6) in (5 4) we obtain for the renormalized inverse p r o p a g a t o r to one-loop order

S-~(p)=ly.p. +2Mj j

l+g2z~-8~.---5 Wo*

l+g2zj---W*o 47r 2

dx(l-x)

ln 4

dxln---TWo tz

Wo

,

(57)

where use has been m a d e o f Moj = Mj + O ( g 2) As S -j is h e r m m a n it is possible to find a unitary matrix W that dlagonahzes S ~, and which equals Wo in lowest order. We show in a p p e n d i x C that a W i n d e p e n d e n t o f p can be f o u n d If the zj m (5 6) * In this section m, m~

in the lattice actmn

etc denote the r e n o r m a h z e d mass parameters and too, mo~ etc the parameters

76

M F L Golterman, J Smlt / Staggered Ierrnton~

are chosen independent of j, z s - - v ~,, Mj simultaneously

= ze, then W d m g o n a h z e s S ~ and M =

So i = W S I W*

[

- Io

_~

3 2 _- 1

= 1%.p~. 1 +g-z2--87r2

dx ( l - x )

ln~]

+ M d [ 1 + g2ZS -- 4g~2 fro1 dx l n ~ 5 ] , Md = W M W T

(5 8a) (5 8b)

From (5 6) it is clear that when we want to relate c o n t i n u u m physical masses to the bare mass parameters o f the lattice theory (whmh e g are relevant m c o m p u t e r slmulatmns) the % and z will turn up m these relatmns This IS &scussed m ref [17] for the case of Wilson fermmns and for both Wilson and staggered fermlons m ref [18]

6. Fermion self-energy and the role of lattice symmetries In this section we show how the result (4 13) can be understood as the most general o u t c o m e consistent with the symmetries of the action (2 I ) + ( 3 6) As is clear from the derivation o f (4 13) the log &vergent part (integral over F e y n m a n parameter) comes from a c o n t m u u m - h k e mtegral with spherical c u t o f 8, and hence we shall discuss here m particular how the form of the contact terms m Z ( p ) is determined by the latttce s y m m e m e s F r o m dimensional arguments we expect the contact terms to be of the form p~.X,~ + a J Y o + m Y + m , . Y +½m~., Y~. + m .5Y ~ .5+

lTl5

Ys+O(a),

(6l)

where X~., Yo, Y etc are m a m c e s i n d e p e n d e n t of a, p and mass parameters They can be written as a s u p e r p o s m o n of 1, /'~, F,f,,, F,,Fs, I~, ~ . , , 3 5 and thmr products* In d~scussing the restrictions on the matrices ~mplled by the symmetry group of the latUce we shall use the nomenclature scalar, vector, tensor, axtal vector and pseudo-scalar for the ~rreducIble representattons o f the lattme 90 ° rotatmn group (plus mverstons) This is appropriate since these Irreducible representatmns of 0 ( 4 ) remain irreducible u n d e r the hypercublcal g r o u p [19] We may consider M as a set of external fields coupled to different bfllnear operators build from h' and )~ From (3 6) we see that m order to have m v a n a n c e of the action under lattice rotatmns (2 5) and reflectmns (2 9) we have to consider * For convemence we discuss here the contact terms tn v before applying the tran~formatton (3 14)

M F L Golterman, J Smlt / Staggeredferrmons

77

the c o m p o n e n t s m, m,., m.~, m s5 and m 5 as transforming as scalar, vector, etc u n d e r the lattice rotation-reflection g r o u p F~rst we discuss the Implications o f the s y m m e m e s for the form o f X . m (6 1) In the absence o f a mass term we have shift symmetry, which implies that X . has to c o m m u t e with - ~ for all v Furthermore X . should transform as a vector, and we c o n c l u d e that X . must be p r o p o m o n a l to F . One o f the symmetries that remains u n b r o k e n with the general mass term (3 6) is the s y m m e t r y

with fiR,,,- = --pp,~, /~. = p . r # p, o', which can be obtained as the p r o d u c t o f two shifts (2 31) and (2 32) squared The mass part o f (6 1) thus has to c o m m u t e with FRF~, hence the Y's m (6 1) will be superposltlOnS o f the -~a, A = 1, , 16 (Fs-matrlces can be ruled out using parity) F r o m m v a n a n c e u n d e r the rotation-reflection g r o u p (considering the mass parameters as external fields) we deduce that YoOC1, y o c 1, Y~. oc_~, y~.~ ~ etc (remember that ='-'. transforms as a vector as ~'.(x) does obey eq (2.8) w a h ¢.(x) substituted for r / . ( x ) ) In the massless case there IS U(1). lnvarlance (2 11) which lmphes that Yo has to anticommute wlth F5-~5 (cf (2 35)) and hence Yo vanishes We thus arrive at the general form for the contact term

cp~,F~. + Csm + Cvm.-~ + CT½m..(--l~,,~. ) + etc ,

(6.3)

where c, Cs, Cv, are constants In deriving (6.3) the existence of shift symmetry is essential because it would be lmposs~ble to " c o n s t r u c t " the symmetry (6 2) The mass contact terms then could obtain all sorts o f scalars built from F and z" matrices, e g ~,. F~,_~ ~' M o r e o v e r hnear divergences p r o p o m o n a l to these scalars could (and will [11]) develop We expect these considerations to go t h r o u g h for h~gher orders in perturbation theory The divergence (m a) structure originates from "'inner region" type integrals, which are just c o n t i n u u m expressions with a spherical cutoff The specific properties o f the lattice regularlzatlon are delegated to the contact terms, and to them the considerations above apply F r o m the role played by the lattice symmetries it follows that these can be interpreted as a s u b g r o u p o f the SU(4) × SU(4) × U(1) × C z P × (euchdean Pomcar6 group) s y m m e t r y g r o u p o f the q u a n t u m c o n t i n u u m theory Note that a slnglet axial U(1) is not needed to a c c o m m o d a t e the l a m c e s y m m e m e s Which lattice symmetries are to be interpreted as axial s y m m e m e s depends on the choice o f the mass matrix In particular it is consistent to consider U(I)~ as a (non-slnglet) axial s y m m e t r y when a scalar mass is present, but it is also possible to choose a mass matrix which antlcommutes with F5-~5 and thus an action for which U(1)~ symmetry behaves as a vector symmetry (This is necessary m the

M F L Golterman, I Smtt/ Staggered ]ermmns

78

two-flavor t h e o r y [10] ) H o w e v e r , it seems sensible to use a n o n - z e r o scalar mass m the four-flavor case as strong c o u p l i n g suggests that such a mass is g e n e r a t e d d y n a m i c a l l y [4, 14] The lattice rotaUon g r o u p hes in a " d m g o n a l " s u b g r o u p o f the c o n t i n u u m r o t a t i o n a n d flavor groups, as follows from the form o f the lattice rotations, which c o n t a i n s F a n d -~ matrices in a s y m m e m c w a y (cf (2 32)) This will also be clear from the following interesung example C o n s i d e r the special mass matrix MT o f the t e n s o r type only, in a r e p r e s e n t a t i o n m w h i c h -1~:~2 = p~ a n d - 1 ~ : 4 = -p~0)~ (e g ~:~ = ~:~w2, so4= 0)1)

Mx = ~rn(- t ~ , - = 2 -

I~3-=4) ~

½mp3t 1 -

o)3)

=

t°

0 m

(

-m

)

(64)

The u n b r o k e n rotations are rotations in the (12) a n d (34) p l a n e s a n d a r o t a t i o n in the (13) a n d (24) p l a n e s s i m u l t a n e o u s l y This last s y m m e t r y is b r o k e n when the coefficients of-~j ~2 a n d ~3-=4 are c h o s e n u n e q u a l The e x p h c i t f o r m o f this s y m m e t r y is given b y

&(p)oexp[~rr(F,I'~+F2F4)]exp

[~4~r(-~" R 1¢- '-~- + -=2--~4)]q~( "

'p)

(65)

The m i n u s sign in (6 4) can be t r a n s f o r m e d a w a y by a chlral t r a n s f o r m a t i o n (note that F ~ s c o m m u t e s w~th MT and h e n c e generates axial t r a n s f o r m a t m n s l 4~ ~ exp [,~rr( 1 + '-=,-~2)F5~514;, 2 --' ~ exp [1~,~'( 1_

+ 1--,~"-=~ ) F ~ -=, ] ,

(6 6)

which is an e l e m e n t o f S U ( 4 ) x S U ( 4 ) a n d thus n o n - a n o m a l o u s This turns the s y m m e t r y (6 5) into a s y m m e t r y which is a p r o d u c t o f a r o t a t i o n a n d an axial flavor transformation

(o(p)--> exp

[¼~-(I',F3 + I ~ F 4 ) ] exp [~7r(-=~-=4 + ~3-=2)Fs]d~(R ' p ) ,

exp [ ~ - ( ~ , -=4 + --~-=2 )Fs] --, !( 2 1-w3)-½(l+0)~)tp,y~

(67)

The r o t a t i o n s in the (12) and (34) p l a n e s are unaffected b y the t r a n s f o r m a t i o n (6 6) As this residual s y m m e t r y g r o u p f o r b i d s a mass for the two quarks in the w~ = +1 sector, we e x p e c t that in the c o n t i n u u m limit an SU(2) m u l t l p l e t o f G o l d s t o n e b o s o n s will show up In the case o f different coefficients in (6 4) the s y m m e t r y (6 6) is b r o k e n , but from the c o n s l d e r a U o n o f this section It follows that no t u n i n g o f the b a r e masses is n e e d e d in o r d e r to k e e p mass ratios fixed (Only one %, n a m e l y ~rv, a p p e a r s m the r e n o r m a h z a t l o n constants.)

M F L Golterman, J Srmt/ Staggeredfermtons

79

7. The two-flavor case In this section we shall repeat the foregoing discussion for the case of reduced staggered fermlons As this resembles very much the four-flavor case and furthermore has already been discussed in [10] we shall be short on It In the reduced case one restricts the fields )f(x) to the even sites of the lattice (e(x) = +1) and the fields X(x) to the odd sites (e(x)= - 1 ) Since there now can be no confusion we omit the bar on )~ The action then becomes

S = - ~ I E %(x)xT(x) ~,.

2

U~,(x) 4

U*(x) X ( x + % )

2

(71)

When the gauge group is S O ( N ) the e(x)-dependence disappears and the reduced case is equivalent to considering only one (real) field per site A few things change for the symmetries we have shift lnvariance under

g(x)--> G(x)g(x +%),

U~(x)~ U*(x +%),

(7 2)

and only U(1)~ lnvarlance (which was shown in ref [10] to be a vector U(I) symmetry). Because of the transformation law for the gauge fields in (7 2) the tree graph continuum hmlt

S - -½ f dx qb(x)XF~,[au + zgG'~(x)(t~'- ~ d 2 ' ) ] 6 ( x ) ,

(73)

IS only lnvariant under an even number of shifts (~, = Fs-~5). t~ ~ and t~ ~ are the antlsymmetrlc and symmetric parts of the generators tin, 1 e

t~'=½(tm--tT),

t(~)=l(tm +t~)

(7 4)

Restricting ourselves to the S U ( N ) case (see [10] for the S O ( N ) case) the general form of the mass matrix is

M = m~t--~,s + m~,---~,~sFs,

(75)

which leads to a mass term of the action" S ....

l- (y)2 +E:¢x'y)

=I~{I'~.¥T(x)E(x)[EI.t(x,y) -

i t3.~XT(x)[ + zrn,~ae.o,

(E,~E~E~,)(x, l+e(V)']

+( E~Et3E~)*(x, Y) - - - - ~ J x t y

1 -- 2e(y) , ,]

)~ ,

(76)

M F L Goherman, J Smlt / Staggered Jermton~

80

where E. denotes the matrix (3 5) and E . E ~ E v is the m a m x obtained by muluplymg E., E~ and Ev The tree-level continuum hmlt in conventional Dlrac form was extensively discussed m [10] Meson operators can be defined just as m the four-flavor case We now come to the calculation of the one-loop fermlon self-energy The expressmn for the first diagram of fig 2, using the rules gwen m (A 18)-(A 19), is X<~,(p)~(p_q)=g

2N-1

I

~ . ~ B ~ ,
x[e '"~ +e 'lk+~q~-]D(k) × ] G ( k + a p + rra + 7r.7., - k - aq +TrR +Tr~. ) N +2 N

G ( k + ap + 7r.t + re.. + 7L, - k - aq + 7"rB+ 7r.~,~+ 7r~)[ _j

(7 7) where we used N-I , ~ t~)t~'=0, .. 4 ,n It is easy to verify that for general k, l, ~,

(a) (a)

t,. t,. =

V t~)t~ ' ,.

G ( k + 7r~, - l + "nF ) = - G ( k ,

N-I 4

N+2 N

-l),

(7 8)

(7 9)

and using this fact (7 7) reduces to an expressmn similar to (4 9) The final result for X(p) is just (4 13) with Ma a 2 ×2 matrix and a mass contact term

Expressmns and numerical values for ~v and ~A are gwen m appendix B 8. Conclusion The results of a one-loop calculaUon of the fermlon self-energy support the flavor interpretation of the spectrum doubhng exhibited by staggered fermlons These flavors can be Identified m the continuum hmlt, when the latUce symmetry group generated by latUce rotauons, reversions, shifts and U( 1) × U(1 )~ transformaUons, enlarges to the SU(4) × SU(4) × U( 1) flavor lnvanance and Pomcar6 mvarmnce The degeneracy m the fermlon masses can be hfted completely by adding appropriate mass terms to the actmn Although the lamce symmetry group ~s partially broken by these mass terms, thxs breaking becomes sufficiently weak m the continuum hmlt for recovenng the expected lnvarlance group of the continuum In particular, no hnear dwergencles develop m the fermlon self-energy and the lamce symmemes restrict its form m accordance with chlral symmetry The continuous U( 1). symmetry becomes a flavor non-smglet axial symmetry, provided that the mass m a m x is chosen to break it This chmce ~s natural, since at strong couphng the U ( I L symmetry is dynamically broken Pamy and charge conjugaUon can be defined as latUce symmetries which take the usual form in the continuum hm~t

M F L Golterman, J Srmt/ Staggered fermlons

81

There ms a possible drawback in the staggered fermxon formulation which may be important for computer simulations the tuning of parameters m the mass terms of the action The quark masses are given as superposltlons with g-dependent coefficients (In general even as much more comphcated functions) of the parameters m the mass terms of the action [18] The g-dependence is caused by the mass renormahzatIon strengths %, which take different values for each j , j = S, V, T, A, P corresponds to zero-, one-, two-, three-, four-link mass terms m the action, respectively This effect can be understood lntumvely as a reduction of the effective strengths of these mass terms due to the fluctuations of the gauge field It grows with the number of hnk varmbles revolved and becomes relatively large for ~rA and trp [18] The signs of the ~r, are m accordance with this picture. The remarks above concerned the four-flavor formulation, but they apply with suitable modifications also to the reduced, two-flavor theory In particular, there the U(I)~ symmetry has to be interpreted as a vector symmetry [10] Of course, to describe four-flavor Q C D also a doublet of reduced fields could be chosen, which already possesses a continuous SU(2) vector flavor mvanance on the lattice For computer simulations beyond the quenched approximation, this formulation has the &sadvantage of a complex fermlon determinant [10] We have only briefly touched on the construction of local hadron fields on the lattice There may also be a tuning problem in the association of physical particles to fields transforming in Irreducible representations of the lattice symmetry group For example, it may be difficult to create an F-meson out of the vacuum without also exciting a pion Further investigation is needed here Compared with the Wilson fermion method, the staggered formulation has the advantage of a less severe breaking of chlral symmetry However, with a multlplet of Wilson fermions the full vector part of the flavor symmetry is kept on the lattice and the adjustment problem for mass parameters, as well as the construction of hadron fields appears to be much simpler Our formahsm makes it possible to calculate other vertex functions in perturbation theory, which would be useful for further verification of the statements made here The &scusslon on the form of the contact terms suggests that the results generalize to higher orders This work is financially supported by the "Stlchtmg voor Fundamenteel Onderzoek der Materle (FOM)"

Appendix A FEYNMAN RULES

In this appen&x we list the vertex functions in momentum space for the four-flavor action (2.1) +(3 6) and for the two-flavor action (7 1) +(7 6) First consider the case

M

82

FL Golterman. d Smtt / Staggeredfermtons k 1 /11 m 1

k n .u n m n

p

q

Fig 3 Vertices of four flavors We parametrlze

U . ( x ) = exp [lgaa"u'(x)tm].

(A 1)

where t~ are the hermltlan generators of the gauge group In the fundamental representation We put the lattice distance a equal to 1 From the kinetic part (2 1) we obtain the vertex functions (fig 3) F~',' ~'.:'(p. - q . kl

k.) ~km'

=-(tg)"t .....

Z6".,

. , , g ( p - q + k + 7 % . ) e " ~ +',

(A2)

+k.,

(A3)

#

w~th

k=kl+k2+

e ." =- e~(p, " k ) = ~~[e 'Ip+~È + ( - 1 ) " ttl

.,, = 8 ~ , 6 ~ .

e 'P-],

(A4)

8.,.,,,

tm I m,, = Ln~ | p tmPItmP2

(A5) (A6)

tmpn

where ~ p is the sum over all permutations of {1,2, , n} The vertex functions coming from the mass terms (3 6) may be obtained as follows The nth derivative o f the shift operator E. with respect to the gauge field A~' leads to funcUons similar to (A 2) . tp,-q)=(g)

.,,, .,

tin,

~,,g(p-q+k+Tr~,,)

x½[e 'q, + ( - 1 ) " e 'P,,].

(A 7)

where we have abbreviated the gauge field labels (kl. m~./xl ), , (k., m.. # . ) of E. by (n) The vector-type mass term vertex functions can be written as ~ , l n),'

v tp.-q)=-VmpL,

~ ' ( n J/

tP.-q)

(A8)

p

The T. A and P mass terms mvolve E~Et~. E~Et3E ~ and E..Et~E.E~ , whose vertex funcUons follow from (A 7). for example

( ~ t~,

~i

tP.-q)

,'°"

=

Y~ nl+n2=n

f

E~n"(P.-r)E~n'-'(r.-q)[ r

"

....

(A9)

83

M F L Golterman, J Smtt/ Staggeredferrmons

where "symm" means a symmetnzatlon m the gauge field radices (kt, ml,/~l) (k., m., ~n) The T, A and P mass terms lead to the following vertex functions in the case ~t~ = ~2 . . . .

/z.

(A 10)

F~x")(p, - q ) = l½ ~ m ~ e t 3 ~ g ( p - q + k +rc;.~+~r~.) ct

n

x ( l g ) " t . . . . . [6 m ~ , e ~ c o s ( p + k ) ¢ + 6 ~ , J

F~A"'(p,--q)

lg

n

~,cos(p~)e~].

(All)

5

a/3-rp

× ~ ( p - q +k +Trc. +Trc~ +1rc~)(tg)"t . . . . . [t3~, ..e2 )
~.,, cos (p~.)e"~ cos (p + k ) ~

~, c o s p ~ c o s ( p ~ ) e vn] ,

+8~,

(A12)

ot l3 Tt5

x ( z g ) " t . . . . . [ t~a~, ~,e. c o s ( p + k ) ~ c o s ( p + k ) v c o s ( p + k ) ~

+8~

.°cos(p~)e~cos(p+k)vcos(p+k)~ n

-v

+6~., . , c o s p ~ n

xcos(pt3)evcos(p+k)~+6~,

.,cosp~cospt~cos(pv)e~] ,

(A13)

where e2 is given m (A,4) and e.,. is defined m (2 25) The sign factors e~.~ a p p e a r naturally when expressing e g r and q in (A 9) m terms o f p and k The expression for F~p"~ was furthermore simplified by means of the identity

e~ae~e~e~t~ea~ea~ea~ = Ie ~ a I

(A. 14)

In this p a p e r the highest n needed is n = 2 (diagram 2b), for which the above expressions are adequate in the F e y n m a n gauge The inverse fermion p r o p a g a t o r corresponds to n = 0 in the expressions above, it is given by

G-~(p, - q ) = ~ [ 6 ( p - q +rr~ )~ s t o p . + m ~ . 8 ( p - q + ~-~.) cos p~] + rng( p - q) - t½ Z m~.~e~,g(p - q + ~r¢. + ~r;~) cos p~. cos p~ 1

-~

~

5

--

rn ~.eu~t~e~.~e~e~t36( p - q + Tr~ + rrt~ + Tr~)

oel3 yp~

× c o s p~ cosp~ c o s p v - m S S ( p - q + r r ; , × c o s p~ cos P2 COS P3 COS P4

+ ~r¢~+ rt; + ~r~.) ( A 15)

M F L Golterman, J Smtt/ Staggered fermlons

84

The d e p e n d e n c e on the l a m c e d i s t a n c e a m a y be r e s t o r e d by m u l t i p l y i n g all m o m e n t a a n d masses by a, b y the r e p l a c e m e n t g--, ag a n d an overall factor a ~ for the reverse p r o p a g a t o r a n d other vertex f u n c u o n s The gauge field p r o p a g a t o r m the. F e y n m a n gauge is gwen by

D.,,(k)=~.,,

A-~+4

sln-(~k.)

~6.,D(k).

(A16)

where an i n f r a r e d r e g u l a t m g mass A has b e e n m s e r t e d C o n s i d e r next the two-flavor case Because o f the e ( x ) d e p e n d e n c e m (7 1) the vertex functions s p h t up into two parts, one w~th a n d one w i t h o u t a ~-~ in the d e l t a - f u n c t m n , where 7r~ = (Tr, 7r, 7r, 7r) ,

~(x) = e '~, "

(A17)

These two types o f terms are a c c o m p a n i e d by the s y m m e t r i c a n d a n t i s y m m e m c parts o f the g e n e r a t o r p r o d u c t s (A 6), so we define

t~, . . . . =½(t . . . . . .

+t,v. . . . . ) ,

t ~r n I

--

m n

~

l( t~, _

m,,

vI tm

m,,

),

(A 18)

where T m e a n s t r a n s p o s e d U n d e r the restriction (A 10), the vertex functions are o b t a i n e d from ( A 2), (A 4), (A 8), (A 1 1 ) - ( A 13) by the following r e p l a c e m e n t s (1) for n o d d , (a) tin, ~ t,., ~ , (b) t.,, --. t ~.... a n d p ~ p + 7r~, a d d (a) a n d (b), (u) for n e v e n , ( a ) t.,, -~t~, ~ , ( b ) tin, ~ t t~ ~, a n d p ~ p + T r , a d d ( a ) a n d (b) Ira) for the rh. vertices, s u b s m u t e again p--. p + 7r~ a n d m . --. - r h . The reverse f e r m l o n p r o p a g a t o r is g w e n b y G - ~(p, - q ) = v [ g ( p - q + ~ - )t sm p~ + rfiug(p - q + rr~,. + 7r~ ) cos p . /x

-- 16 ~

m~.e~..t~.el3,,e~.,e~t~3( p - q +Tr;,, +Trcl~ +Trc~ )

c~/3-/

x c o s p . cos p~ cos Pv]

(A 19)

Appendix B NUMERICAL RFSULTS In this a p p e n d i x we give the full e x p r e s s i o n s a n d n u m e r i c a l values o f the cr a n d coefficlent~ m the o n e - l o o p self-energy results (4 13) a n d (7 10) The c o n t r i b u t i o n c o m i n g from the tuner region for the four-flavor case, using m e t h o d s as e x p l a i n e d m the text, is g1 + I n 62)[ty~.p~ + 4 ( m + m . ~ . ~ ( P ) . . . . . = 167r2{(-2

+½m.,(--t~, ) +m~vs~s+mS~5) ]

Io1d x [ l y . p .

6)

-2Wo

1-~)+2Md]ln(a2A)Wo}+O(a,

(B 1)

M F L Golterman, J Smlt / Staggeredfermmns

85

The contribution from the outer region consists of integrals over comphcated functions of the loop momentum For each term an Integral proportmnal to 1

can be separated off, leaving an integral for which the hmlt 6 ~ 0 can be taken In (B 2) an abbrevlaUon was introduced S. = E s m " (½k.)

(B 3)

/x

The log 62 divergence m (B 1) cancels against the dwergence m integral (B.2) and a constant K [20] is left K =l~m ( l l n 6 =4

Io{ dxx

2 + ( ~ )18 ) e-4~/o(X) 4

}

1 [1 - ( 1 +½x) e ~/2] 4 7.r2X 2

= 0 4855321

(B 4)

This constant K is related to the constant Foooo defined in ref [21] by zr2K = Foooo+ 1 - y(Euler) Combining the contact terms coming from (B.1), the K contribution and the convergent part (in the hmtt 6-* 0) of the outer regmn integrals, we obtain 1

-y

2+ K -

=

+ (s2

q- . - S 2 - ~ S 4 - S 2 S 4 @ S 4 " 2 ° 2

2°2'J4-~-S°@S286-2S8

(B 5)

$2($2 - S4) 2 where now ( ) denotes lntegrauon over the whole momentum region For the mass terms we obtain, with j = S, V, T, A, P 1 [

o,j=~-

~5_~ +4K

\S2/

( 4 ' 4 \

(S2_S4)S2/

+ (./3j + 4( Y s - 1 ) - S2Tj~] 7--S4~2 /J

(B 6)

The expressmns for %, /3j and yj are hsted in table 1 The integrals in (B 5) and (B 6) were calculated with the M o n t e Carlo routine V E G A S [22] The ($2 ~) integral was calculated with V E G A S and independently using Bessel funcUons and numerical mtegratmn to check the ability o f V E G A S to handle the (mtegrable) smgularmes contained m (B 5) and (B 6), we have

($2 ~) = 0 6197336

(B 7)

M FL Golterman, J Smtt/ Staggered/ermtons

86

TABLE 1 E x p r e s s i o n s for %, fl~ and 71 m f o r m u l a (B 6), % - - c o s ku

j

%

/3,

~,

S V T A P

0 -1 3 -5 7

0 0 (I c~)t: 11 - t~)c:(¢~ + 1) (1 - ~)e2[~s(t 4 + 1 ) + 1 ]

1 el tlQ ¢1Q¢~ qcz~

F o r the two-flavor case the c a l c u l a t i o n s are c o m p l e t e l y a n a l o g o u s , the e x p r e s s i o n for r is the s a m e as for the four-flavor case a n d similar e x p r e s s i o n s for #A a n d - # v are o b t a i n e d , with

fl,X=(1--C~)C2(C~+I),

6~,:--5,

C/v= 1 ,

/3v:0,

~/~=ClC~C~,

~v=cl

(BS)

We hst the values o b t a i n e d for all these coefficients r = -0,044566( 13), trs = 0 19745(3) , cry = 0 00385(4) ,

~v=-0

01109(4),

efT=--0 11713(5) , ¢rA : - 0

20911(4) = f f a ,

Crp = --0 29384(6)

(B 9)

Appendix C EXISTENCE

OF W

In this a p p e n d i x we show that a u n i t a r y m a t r i x W i n d e p e n d e n t o f p can be f o u n d , which & a g o n a l i z e s S -~ given m (5 7) W e split the term ~ j Mj( 1 +~2zj) in (5 7) into two parts MA a n d MB, with MA m d e p e n d e n t o f %, using (5 3)

M~,={l-~,-[l~-~21n(a21x2)+rl}~Moj, ~

3

M~ = ~2 ~ M0j%

(Cla)

(C l b )

3

As MA lS p r o p o r t i o n a l to Mo, It is & a g o n a h z e d by Wo W e p a r a m e t r l z e W as w -- [1 + zgeH + O(g~)] Wo,

(C 2)

M F L Golterman, J Smlt / Staggered fermton~

87

which is unitary to this order If H IS hermitlan W Is a matrix in ~-space We now have to find an H such that W ( M A + M B ) W* IS diagonal to order g2 Using (C 2) we obtain the following equation for the off-diagonal piece of this matrix W ( M A + MB) W#loff_dtagonal= O(g 4)

¢ = ~2{~[H, Moo] + WoMB Wo}on-d,,go~a~

(C 3)

This equation for H can be solved for general MB and Mod provided a good chmce for Wo IS made in the case of degenerate diagonal elements in Mod We choose a basis in the space of 4 x 4 matrices diagonal matrices (Dr),j = ~5r,6rj ;

(C.4a)

off-diagonal matrices (E~),j = 6,,,6&,

a #/3 ,

(C.4b)

with commutation rule (C 5)

[E~, D~,] = (6t3~,- 6 ~ ) E ~ . Parametrlze W o M B Wo[off_dlagonal = ~ Co43E¢3 ,

(C 6a)

Mod = 2 Mo~D~ , 3,

(C.6b)

H = 2 H.t~E~

(C 6c)

Using (C 5) an equation for H~e is obtained from (C 3) IH,~t3( Mo, ~ - Mot3) = C ~ ,

(C 7)

which can immediately be solved for H~e in the case Mo~ # Mo¢ for all c~ #/3 Now consider e g the case M~ -- M2, M3 # M4 and M3, M4 not equal to M~ Then (C 7) has a solution If C~2 = 0 We can arrange for this by making use of a freedom in the choice of Wo, which exists in the case of a degenerate eigenvalue If Wo &agonahzes Mo, Mo is also diagonahzed by liZo=WoWx

with

W,~=( X

01) ,

(C8)

where X is any unitary 2 x 2 matrix X can be chosen to dIagonahze the 2 ×2 left upper block of WoMr~W~ and we thus have C~2 = 0

88

M F L Golterman, J Smlt / Staggeredfermtons

Note added in proof T h e s t a g g e r e d f e r m l o n s e l f - e n e r g y h a s a l s o b e e n c a l c u l a t e d b y G o c k e l e r [23]

References [1] K G Wilson, in New phenomena m subnuclear physics, ed A Zlchlchl (Plenum, New York, 1977) (Ence, 1975) [2] T B a n k s e t a l , Phys Re,~ DI5 (1977) 1111 L Susskmd, Phys Rev D16 (1977) 3031 [3] H S Sharatchandra, H J Thun and P Welsz, Nucl Phys B192 (1981) 205 [4] N Kawamoto and J Smlt, Nucl Phys B192 (1981) 100 [5] J M Rabm, Nucl Phys B201 (1982) 315 [6] A Duncan, R Roskses and H Valdya, Phys Lett l14B (1982) 439, F Ghozm, Nucl Phys B204 (1982) 419 [7] P Becherand H Joos, Z Phys C15(1982) 343, Lett NuovoClm 38 (1983) 293 [8] H Kluberg-Stern, A Morel, O Napoly and B Petersson, Nucl Phys B220 [FS8] (1983) 447 [9] P Mltra, Nucl Phys B227 (1983) 349 [10] C P van den Doel and J Smlt, Nucl Phys B228 (1983) 122 [11] P Mltraand P Welsz Phys Lett 126B(1983~ 355 [12] LH Karstenand J Smlt, Nucl Ph'cs B183 (1981) 103 [13] J-M Blalron, R Brout, F EnglertandJ Greenslte, Nucl Phys BI80[FS2](1981)439 [14] H Kluberg-Stern, A Morel, O Napoly and B Petersson, Nucl Phys BI90 [FS3] (1981) 504 [t5] O Napoly, Phys Lett 132B (i983)145 [16] A N Burkltt, A Kenway and R D Kenway, Phys Lett 128B (1983) 83 [17] A Gonzalez-Arroyo, FJ Ynduram a n d G Martmelh, Phys Lett 117BI1982) 437, H BW Hamberand Chl MmWu, Phys Lett 133B(1983) 351 [18] M F L Golterman a n d J Smlt, Phys Lett 140B (1984) 392 [19] M Baake, B Gemundenand R Oedmgen, J Math Phys 24 (1983) 1021 [20] R Groot, J Hoek and J Smlt, Nucl Phys B237 (1984) 111 [21] A Gonzalez-Arroyo and C P Korthals Altes, Nucl Phys B205 [FS5] (1982) 46 [22] G P Lepage, J Comp Phys 27 (1978~ 192 [23] M Gockeler, DESY preprmt 84-025