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Perturbation theory in terms of superfields
Volume 53B, number 1
PHYSICS LETTERS
P E R T U R B A T I O N T H E O R Y IN T E R M S O F
11 lqovcml~r 1974
SUPERFIELDS
F. KRAUSE and M. SCHEUNERT Physikalisches Institut der Universitiit Bonn, Germany J. HONERKAMP and M. SCHLINDWEIN Fakultiit j~arPhysik der Universitiit, Freiburg, Germany Received 15 September 1974 A perturbation theory in terms of superfieldshas been developed for a class of super-symmetricLagrangians which contain massless spin zero boson and spin-l/2 Majorana spinor fields. The construction of super-symmetric Lagrangian field theories is much simplified by the concept of the superfield [ 1]. Various models in terms of superfields have been constructed in recent publications [2] in order to become familiar with these techniques and to find some connection with realistic models. Now one may suggest [3] that given a super-symmetric Lagrangian one should formulate the perturbation theory in terms of the superfields too, thus calculating various Feynman diagrams at once. The purpose of this note is to demonstate the usefullness of this approach in a model which contains as interaction Lagrangian an arbitrary functional of the chiral superfields ~± which are defined by [3] : qb±(x, 0) = exp (~- ¼0~TS0)~0± (x, 0) = ~0±(x -T-¼ 07T50,0±),
(1)
~o±= A± (x) + -0± ~b±(x) + ½ 0~: O±F+(x) ,
O~ =21(1 + iT5)0 . For the massless component fields we take A* = A , F * = F , field. The Lagrangian now reads [3]: L = Lo + Lint ,
4± = ½ (1-+ iT5)~ where ~b is a Majorana spinor
Lo = -~ (DO)2 q5+~_ ,
(2)
Lint = - ½ g D D (V(~+) + V ( ~ _ ) ) ,
(V"(~) = 0).
The covariant derivative is given by Da = O/O~a - ½ i(~,F0)~au;
/ ~ = (C-1)~aD a .
(3)
Special cases are V ( ¢ ) = ¢3, which is part of a Lagrangian studied by Wess and Zumino [4] and V(~) = ~
[exp 0e~) - 1 - J'¢ - ½( f ~ ) 2 ] ,
(4)
which gives rise to a Lagrangian being nonpolynomial in the component fields A ±. Ofcourse, this Lagrangian is probably non-renormalizable, though a tremendous cancellation of divergences is achieved by super-symmetry. It can be shown [5] with the method outlined in this paper that indeed an infinite number of super-symmetric counterterms is needed to absorb the divergences arising from the two vertex graphs. The interesting point however is that if one is willing to use the superpropagator method to determine the corresponding free constants the problem is reduced to the definition of the pure massless superpropagator for a scalar field with exponential coupling [5, 6].
60
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
To develop the perturbation theory for the Lagrangian (2) we start from the generating functional:
z(J+,s_) = f[d*+]
[dO_ ] exp [i8 (~+,~_) + i ~ ( x , 0) (J+(x,O)c~+(x,O) +J_(x, O)~_(x, 0))] ,
(s)
where the generalized integration measure:
fl(x,O)=fd'x (-½
(6)
has been introduced. For our calculations it will be essential that for any chiral superfield ~± one may translate the x-variable:
fd4x*~(x,O) =fd4x ¢,(x; ¼~-/-/s0,0~) = fd4y,i(y,O,),
(7)
and that one can interchange j'd4x and/~D-operators in the following sense:
fd4x (- ½30)¢~(x,0) = -½ (BD)o fd4x¢i(x,O±)=fd4x F . ( x ) ,
(8)
where
(BD)o=(C_I)~a~a O As usual we define a background field ~)+ as the solution of the classical equations of motion:
-½DD~; +gV'(c~±)+d± = 0 .
(9)
Notice that the functional derivative has to be defined by:
8r~+(x,O
(y, tg)g(c~t(.y,O))=g'(c~+(x,O)) .
Expanding the exponent in (5) around q)± we first obtain a term which is independent of X± = O± - ~± giving rise to the tree functional. The terms linear in X+ vanish. The quadratic term can be written as:
½xaS,ab(~+,~_)X b ;
(a, b =
+,-),
(10)
where we use the following notations and definitions:
8,,, =(
#++ v"(~+) 8_+
8+_ #__v"(*_) ) "
O Oa)
The generalized 8-functions: 8+_(x, 0 ly, O) = 8 ((x - ¼G7$0) - (y + ¼ ~ S O ) + i0+~O_),
8_+ (x, 0 ly, O) = 8+_(y, 0 Ix, 0),
(11)
~ ~(x, 0 ly, o) = ½ (~- i})~(0-oL8 ((x; ¼ ~ s 0 ) - (y ~ ¼ ~ s o ) ) , are bilocal chiral superfields obeying the important relations:
f,+_¢_=-½3o,_,
(12)
Analogous formulae hold for the other admissible index combinations. Of course, the symbolic multiplication as 61
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
used in (1 O) has always to be done with the generalized integration measure ~ , e.g. X+8,++ X+ in (10) reads explicitely:
~(x, o)~ (y, ~)×+(~, o) 5÷+(x,o)y, ~)g v"(~+(y, ~))×÷(y, ~).
(10b)
To obtain the inverse of S,a b = S,°ab + o~b .~,int we need relations like:
8+_ = - ½ / ) O 8 -
;
5_+ = - ½ / 5 0 5 + + ,
(13)
and the inverse of S~b = (~÷6o-) defined by ~°G° = 1 ;
(l)ab =
5.+ 0 ~ 0 ~t__] "
(14)
One obtains: G.~ =
(2-+
;
A+_ -
- -
1
~+_.
(15)
a2-i0
Now the inverse of S~b can be calculated as usual by means of the introduced symbolic multiplication. The oneloop functional reads: - ~r tr log (1 + s i n t G ° ) .
(16)
It does not contain a tadpole contribution and gives rise to a two vertex graph contribution proportional to: + i0+7~_)] g 2)~(x,O):(y,O)V . (~+(x,O+))V (qo_(y,O_))fd4qld4q2 exP [ - i ( q l - q 2 ) ( x - Y . . . . ^ , (17) (ql2 + iO) (q2 + iO) requiring a counterterm proportional to:
:~(x, 0) (-¼59)r"(~+(x, 0))r'"(~ (x, 0)),
(18)
which for the special case V((I)) = (b3 is given by [4]. The higher order derivatives of the action are handled in the usual way and lead to the higher order diagrams. A general higher order graph G of our theory contains n+ vertices of type V(~+) and n_ vertices of type V(~_) and the vertex i of the first type is connected with the vertex j of the second type by Xi/superfield-propagators A+_. The corresponding amplitude is given by:
n+
n_
(19)
x l-I 1-I r("i)(6+(xi, 0i)) r(vp(~_(y/, oj)) (A+_(x~, 0~ly/, oi))~/. i=1j=1
K G is a combinatorical factor. Furthermore lai = Y,/~7,~ 2; el = ZiXi/~ 2. To transform the above expression we use the properties of chiral superfields given in (1), (7): We introduce a Fourier-representation for each propagator calling by q~ the momentum of the k-th line which connects the vertices i and j. Furthermore let Qi/= Y'kq k-- Q/i" This is the total four-momentum flowing from vertex i to vertex/. All together we f'md:
62
Volume 53B, number 1 n÷
PHYSICS LETTERS
11 November 1974
n-
AG--KG
d%f d%
x v(.,'(&÷(x,,o0)
?'i/ xi/
J d4q
oi)) [
exp
[-iQil(xi-y I + i0i÷7t~]_)]
(20)
l'l:~l((qb2 + i0)
We apply (8) to perform all (/)D)o-operations. The integrand depends on 0 i and O/only via Oi÷,0]_. So (8) demands that we pick up just the quadratic term in eachof these variables which is composed of contributions of a vertex factor and the exponentials exp [Qi]Oi 70s ]. The latter connect the quadratic terms for different vertices in a nontrivial way. The evaluation is straight+f'or~vard and the results may be expressed by some simple graphical rules: To each graph G there belongs a set of "momentum diagrams" M which are constructed in the following way: a) M contains the same vertices as G, b) From each vertex of M there originate at most two lines. (We call a vertex of M "isolated" if there originates no line, "saturated" if there originate two lines and an "endpoint vertex", if there originates just one line); c) M must not contain any "tadpole"; d) Two vertices may be connected in M only if they are connected in G too (Xi! > 0). To each momentum diagram M of G we associate an amplitude AG(M ) so that finally: A G = M~AG(M). Example: Let G be the boxgraph. Then we have associated momentum diagrams of fig. 1, AG(M) has the following structure:
n÷ n_ ,.',i-'-I" ]-[ [-[
AG(M)=KG
~i/ ~il
rd4x/M rd4y' )" l["/÷((x"M)l 2 ~O ] "-O'/'
exp[_iO.ll(Xi.Y/)]p
f ~l d4q~i Hic((q//k)2+i0)
. (Q0"M)(22)
The vertex factors are given by the following rules
Vke (x, M) = F~k(x) for an isolated vertex, VICe(X,M) ffiA ±Vic(x) for a saturated vertex, VIce(X, M) ffi -~IC(x) or - OVic(x) for an endpoint vertex. Notice that VO'i)(~o+(x,0+)is a chiral superfield with canonical decomposition: V(#i)(q+(x,8+)) = A V+i(x) + O+ ~lVi (x) + ½0+O +FVi(x) .
(23)
The momentum function P is constructed as follows: e) If vertices and] are connected by just one line in M, write down a factor Q~Tt~ = QiP/in P. O If verticesi and ] ate connected by two lines in M, write down a factor Q~ in P. g) For each closed loop of M with more than two verticesmultiply the factors Q7 of e) in the order given by
0 0 ,
Mj--
C:>
0
S3--o, ----1
,
,
0
,
,
•
•
•
•
F i g . 1. M o m e n t u m d i a g r a m s a s s o c i a t e d t o t h e b o x d i a g r a m .
63
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
the loop and write down the following fator for P: - Tr((TQ1)(7Q2) ... (TQ2k)~ (1 + i75)) •
(24)
Take ½(1 + i75) corresponding to whether the vertex ' T ' is of the -+ type. (It is easy to convince oneself that both directions of the loop and all choices of the vertex ' T ' lead to the same expression for P). h) For each chain iQI2Q2] ... -Qk-lkof M multiply the factors Q3' of e) with the vertex factors of the endpoint vertices i and k to the total expression: ~Vl (1) (TQ1) ... (TQk-1)~b Vk(k)
•
(25)
Again the order of the chain is irrelevant. For each contribution AG(M ) to A G we find for the superficial degree of divergence: d(G, M) = d(G) + I(M). d(G) is the superficial degree of divergence of the graph G where all internal lines of G are taken as spin zero boson lines. I(M) is the "length of M", that is the number of all internal lines of M. Of course, in most cases the graph is less divergent as indicated by this superficial degree as can be seen from the two vertex graph (d = 2) and the box graph (d = 0) which are logarithmically divergent resp. convergent. The ultraviolett behaviour of each graph has to be derived directly from eq. (20). A more detailed paper with applications and examples is, in preparation [5].
References
[1] A. Salam and J. Strathdee, Trieste preprint IC/74/11 (1974); S. Ferrara, J. Wess and B. Zumino, CERN preprint TH. 1863 (1974), [2] A. Salam and J. Strathdee, Trieste preprint IC/74/36 (1974); R. Delbourgo, A. Salam and J. Strathdee, Trieste preprint IC/74/45 (1974). [3] A. Salam and J. Strathdee, Trieste preprint IC/74/42 (1974). [4] J. Wessand B. Zumino, CERN preprint TH. 1794 (1973). [5] F. Krause et al., in preparation. [6] H. Lehmann and K. Pohlmeyer, Comm. Math. Phys. 20 (1971) 101; F. Kranse and M. Scheunert, Nuovo Cimento 12A (1972) 221.
64
PHYSICS LETTERS
P E R T U R B A T I O N T H E O R Y IN T E R M S O F
11 lqovcml~r 1974
SUPERFIELDS
F. KRAUSE and M. SCHEUNERT Physikalisches Institut der Universitiit Bonn, Germany J. HONERKAMP and M. SCHLINDWEIN Fakultiit j~arPhysik der Universitiit, Freiburg, Germany Received 15 September 1974 A perturbation theory in terms of superfieldshas been developed for a class of super-symmetricLagrangians which contain massless spin zero boson and spin-l/2 Majorana spinor fields. The construction of super-symmetric Lagrangian field theories is much simplified by the concept of the superfield [ 1]. Various models in terms of superfields have been constructed in recent publications [2] in order to become familiar with these techniques and to find some connection with realistic models. Now one may suggest [3] that given a super-symmetric Lagrangian one should formulate the perturbation theory in terms of the superfields too, thus calculating various Feynman diagrams at once. The purpose of this note is to demonstate the usefullness of this approach in a model which contains as interaction Lagrangian an arbitrary functional of the chiral superfields ~± which are defined by [3] : qb±(x, 0) = exp (~- ¼0~TS0)~0± (x, 0) = ~0±(x -T-¼ 07T50,0±),
(1)
~o±= A± (x) + -0± ~b±(x) + ½ 0~: O±F+(x) ,
O~ =21(1 + iT5)0 . For the massless component fields we take A* = A , F * = F , field. The Lagrangian now reads [3]: L = Lo + Lint ,
4± = ½ (1-+ iT5)~ where ~b is a Majorana spinor
Lo = -~ (DO)2 q5+~_ ,
(2)
Lint = - ½ g D D (V(~+) + V ( ~ _ ) ) ,
(V"(~) = 0).
The covariant derivative is given by Da = O/O~a - ½ i(~,F0)~au;
/ ~ = (C-1)~aD a .
(3)
Special cases are V ( ¢ ) = ¢3, which is part of a Lagrangian studied by Wess and Zumino [4] and V(~) = ~
[exp 0e~) - 1 - J'¢ - ½( f ~ ) 2 ] ,
(4)
which gives rise to a Lagrangian being nonpolynomial in the component fields A ±. Ofcourse, this Lagrangian is probably non-renormalizable, though a tremendous cancellation of divergences is achieved by super-symmetry. It can be shown [5] with the method outlined in this paper that indeed an infinite number of super-symmetric counterterms is needed to absorb the divergences arising from the two vertex graphs. The interesting point however is that if one is willing to use the superpropagator method to determine the corresponding free constants the problem is reduced to the definition of the pure massless superpropagator for a scalar field with exponential coupling [5, 6].
60
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
To develop the perturbation theory for the Lagrangian (2) we start from the generating functional:
z(J+,s_) = f[d*+]
[dO_ ] exp [i8 (~+,~_) + i ~ ( x , 0) (J+(x,O)c~+(x,O) +J_(x, O)~_(x, 0))] ,
(s)
where the generalized integration measure:
fl(x,O)=fd'x (-½
(6)
has been introduced. For our calculations it will be essential that for any chiral superfield ~± one may translate the x-variable:
fd4x*~(x,O) =fd4x ¢,(x; ¼~-/-/s0,0~) = fd4y,i(y,O,),
(7)
and that one can interchange j'd4x and/~D-operators in the following sense:
fd4x (- ½30)¢~(x,0) = -½ (BD)o fd4x¢i(x,O±)=fd4x F . ( x ) ,
(8)
where
(BD)o=(C_I)~a~a O As usual we define a background field ~)+ as the solution of the classical equations of motion:
-½DD~; +gV'(c~±)+d± = 0 .
(9)
Notice that the functional derivative has to be defined by:
8r~+(x,O
(y, tg)g(c~t(.y,O))=g'(c~+(x,O)) .
Expanding the exponent in (5) around q)± we first obtain a term which is independent of X± = O± - ~± giving rise to the tree functional. The terms linear in X+ vanish. The quadratic term can be written as:
½xaS,ab(~+,~_)X b ;
(a, b =
+,-),
(10)
where we use the following notations and definitions:
8,,, =(
#++ v"(~+) 8_+
8+_ #__v"(*_) ) "
O Oa)
The generalized 8-functions: 8+_(x, 0 ly, O) = 8 ((x - ¼G7$0) - (y + ¼ ~ S O ) + i0+~O_),
8_+ (x, 0 ly, O) = 8+_(y, 0 Ix, 0),
(11)
~ ~(x, 0 ly, o) = ½ (~- i})~(0-oL8 ((x; ¼ ~ s 0 ) - (y ~ ¼ ~ s o ) ) , are bilocal chiral superfields obeying the important relations:
f,+_¢_=-½3o,_,
(12)
Analogous formulae hold for the other admissible index combinations. Of course, the symbolic multiplication as 61
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
used in (1 O) has always to be done with the generalized integration measure ~ , e.g. X+8,++ X+ in (10) reads explicitely:
~(x, o)~ (y, ~)×+(~, o) 5÷+(x,o)y, ~)g v"(~+(y, ~))×÷(y, ~).
(10b)
To obtain the inverse of S,a b = S,°ab + o~b .~,int we need relations like:
8+_ = - ½ / ) O 8 -
;
5_+ = - ½ / 5 0 5 + + ,
(13)
and the inverse of S~b = (~÷6o-) defined by ~°G° = 1 ;
(l)ab =
5.+ 0 ~ 0 ~t__] "
(14)
One obtains: G.~ =
(2-+
;
A+_ -
- -
1
~+_.
(15)
a2-i0
Now the inverse of S~b can be calculated as usual by means of the introduced symbolic multiplication. The oneloop functional reads: - ~r tr log (1 + s i n t G ° ) .
(16)
It does not contain a tadpole contribution and gives rise to a two vertex graph contribution proportional to: + i0+7~_)] g 2)~(x,O):(y,O)V . (~+(x,O+))V (qo_(y,O_))fd4qld4q2 exP [ - i ( q l - q 2 ) ( x - Y . . . . ^ , (17) (ql2 + iO) (q2 + iO) requiring a counterterm proportional to:
:~(x, 0) (-¼59)r"(~+(x, 0))r'"(~ (x, 0)),
(18)
which for the special case V((I)) = (b3 is given by [4]. The higher order derivatives of the action are handled in the usual way and lead to the higher order diagrams. A general higher order graph G of our theory contains n+ vertices of type V(~+) and n_ vertices of type V(~_) and the vertex i of the first type is connected with the vertex j of the second type by Xi/superfield-propagators A+_. The corresponding amplitude is given by:
n+
n_
(19)
x l-I 1-I r("i)(6+(xi, 0i)) r(vp(~_(y/, oj)) (A+_(x~, 0~ly/, oi))~/. i=1j=1
K G is a combinatorical factor. Furthermore lai = Y,/~7,~ 2; el = ZiXi/~ 2. To transform the above expression we use the properties of chiral superfields given in (1), (7): We introduce a Fourier-representation for each propagator calling by q~ the momentum of the k-th line which connects the vertices i and j. Furthermore let Qi/= Y'kq k-- Q/i" This is the total four-momentum flowing from vertex i to vertex/. All together we f'md:
62
Volume 53B, number 1 n÷
PHYSICS LETTERS
11 November 1974
n-
AG--KG
d%f d%
x v(.,'(&÷(x,,o0)
?'i/ xi/
J d4q
oi)) [
exp
[-iQil(xi-y I + i0i÷7t~]_)]
(20)
l'l:~l((qb2 + i0)
We apply (8) to perform all (/)D)o-operations. The integrand depends on 0 i and O/only via Oi÷,0]_. So (8) demands that we pick up just the quadratic term in eachof these variables which is composed of contributions of a vertex factor and the exponentials exp [Qi]Oi 70s ]. The latter connect the quadratic terms for different vertices in a nontrivial way. The evaluation is straight+f'or~vard and the results may be expressed by some simple graphical rules: To each graph G there belongs a set of "momentum diagrams" M which are constructed in the following way: a) M contains the same vertices as G, b) From each vertex of M there originate at most two lines. (We call a vertex of M "isolated" if there originates no line, "saturated" if there originate two lines and an "endpoint vertex", if there originates just one line); c) M must not contain any "tadpole"; d) Two vertices may be connected in M only if they are connected in G too (Xi! > 0). To each momentum diagram M of G we associate an amplitude AG(M ) so that finally: A G = M~AG(M). Example: Let G be the boxgraph. Then we have associated momentum diagrams of fig. 1, AG(M) has the following structure:
n÷ n_ ,.',i-'-I" ]-[ [-[
AG(M)=KG
~i/ ~il
rd4x/M rd4y' )" l["/÷((x"M)l 2 ~O ] "-O'/'
exp[_iO.ll(Xi.Y/)]p
f ~l d4q~i Hic((q//k)2+i0)
. (Q0"M)(22)
The vertex factors are given by the following rules
Vke (x, M) = F~k(x) for an isolated vertex, VICe(X,M) ffiA ±Vic(x) for a saturated vertex, VIce(X, M) ffi -~IC(x) or - OVic(x) for an endpoint vertex. Notice that VO'i)(~o+(x,0+)is a chiral superfield with canonical decomposition: V(#i)(q+(x,8+)) = A V+i(x) + O+ ~lVi (x) + ½0+O +FVi(x) .
(23)
The momentum function P is constructed as follows: e) If vertices and] are connected by just one line in M, write down a factor Q~Tt~ = QiP/in P. O If verticesi and ] ate connected by two lines in M, write down a factor Q~ in P. g) For each closed loop of M with more than two verticesmultiply the factors Q7 of e) in the order given by
0 0 ,
Mj--
C:>
0
S3--o, ----1
,
,
0
,
,
•
•
•
•
F i g . 1. M o m e n t u m d i a g r a m s a s s o c i a t e d t o t h e b o x d i a g r a m .
63
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
the loop and write down the following fator for P: - Tr((TQ1)(7Q2) ... (TQ2k)~ (1 + i75)) •
(24)
Take ½(1 + i75) corresponding to whether the vertex ' T ' is of the -+ type. (It is easy to convince oneself that both directions of the loop and all choices of the vertex ' T ' lead to the same expression for P). h) For each chain iQI2Q2] ... -Qk-lkof M multiply the factors Q3' of e) with the vertex factors of the endpoint vertices i and k to the total expression: ~Vl (1) (TQ1) ... (TQk-1)~b Vk(k)
•
(25)
Again the order of the chain is irrelevant. For each contribution AG(M ) to A G we find for the superficial degree of divergence: d(G, M) = d(G) + I(M). d(G) is the superficial degree of divergence of the graph G where all internal lines of G are taken as spin zero boson lines. I(M) is the "length of M", that is the number of all internal lines of M. Of course, in most cases the graph is less divergent as indicated by this superficial degree as can be seen from the two vertex graph (d = 2) and the box graph (d = 0) which are logarithmically divergent resp. convergent. The ultraviolett behaviour of each graph has to be derived directly from eq. (20). A more detailed paper with applications and examples is, in preparation [5].
References
[1] A. Salam and J. Strathdee, Trieste preprint IC/74/11 (1974); S. Ferrara, J. Wess and B. Zumino, CERN preprint TH. 1863 (1974), [2] A. Salam and J. Strathdee, Trieste preprint IC/74/36 (1974); R. Delbourgo, A. Salam and J. Strathdee, Trieste preprint IC/74/45 (1974). [3] A. Salam and J. Strathdee, Trieste preprint IC/74/42 (1974). [4] J. Wessand B. Zumino, CERN preprint TH. 1794 (1973). [5] F. Krause et al., in preparation. [6] H. Lehmann and K. Pohlmeyer, Comm. Math. Phys. 20 (1971) 101; F. Kranse and M. Scheunert, Nuovo Cimento 12A (1972) 221.
64